LMFDB - Embedded newform 150.5.f.e.7.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,5,Mod(7,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.7"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	150.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,8,0,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.5054944626$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.2
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	150.7
Dual form			150.5.f.e.43.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.00000 + 2.00000i) q^{2} +(3.67423 - 3.67423i) q^{3} +8.00000i q^{4} +14.6969 q^{6} +(-44.4393 - 44.4393i) q^{7} +(-16.0000 + 16.0000i) q^{8} -27.0000i q^{9} +167.439 q^{11} +(29.3939 + 29.3939i) q^{12} +(223.621 - 223.621i) q^{13} -177.757i q^{14} -64.0000 q^{16} +(71.0148 + 71.0148i) q^{17} +(54.0000 - 54.0000i) q^{18} -272.211i q^{19} -326.561 q^{21} +(334.879 + 334.879i) q^{22} +(-87.6821 + 87.6821i) q^{23} +117.576i q^{24} +894.484 q^{26} +(-99.2043 - 99.2043i) q^{27} +(355.514 - 355.514i) q^{28} -839.832i q^{29} +403.787 q^{31} +(-128.000 - 128.000i) q^{32} +(615.211 - 615.211i) q^{33} +284.059i q^{34} +216.000 q^{36} +(207.589 + 207.589i) q^{37} +(544.422 - 544.422i) q^{38} -1643.27i q^{39} -1386.39 q^{41} +(-653.121 - 653.121i) q^{42} +(-568.393 + 568.393i) q^{43} +1339.51i q^{44} -350.729 q^{46} +(2642.77 + 2642.77i) q^{47} +(-235.151 + 235.151i) q^{48} +1548.70i q^{49} +521.850 q^{51} +(1788.97 + 1788.97i) q^{52} +(1378.64 - 1378.64i) q^{53} -396.817i q^{54} +1422.06 q^{56} +(-1000.17 - 1000.17i) q^{57} +(1679.66 - 1679.66i) q^{58} -327.503i q^{59} -4205.66 q^{61} +(807.573 + 807.573i) q^{62} +(-1199.86 + 1199.86i) q^{63} -512.000i q^{64} +2460.84 q^{66} +(-2832.99 - 2832.99i) q^{67} +(-568.118 + 568.118i) q^{68} +644.330i q^{69} +5339.03 q^{71} +(432.000 + 432.000i) q^{72} +(-6865.24 + 6865.24i) q^{73} +830.357i q^{74} +2177.69 q^{76} +(-7440.88 - 7440.88i) q^{77} +(3286.54 - 3286.54i) q^{78} +4666.69i q^{79} -729.000 q^{81} +(-2772.78 - 2772.78i) q^{82} +(-4111.98 + 4111.98i) q^{83} -2612.49i q^{84} -2273.57 q^{86} +(-3085.74 - 3085.74i) q^{87} +(-2679.03 + 2679.03i) q^{88} -9911.62i q^{89} -19875.1 q^{91} +(-701.457 - 701.457i) q^{92} +(1483.61 - 1483.61i) q^{93} +10571.1i q^{94} -940.604 q^{96} +(10688.9 + 10688.9i) q^{97} +(-3097.40 + 3097.40i) q^{98} -4520.86i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 28 q^{7} - 64 q^{8} + 464 q^{11} + 336 q^{13} - 256 q^{16} - 392 q^{17} + 216 q^{18} - 1512 q^{21} + 928 q^{22} - 968 q^{23} + 1344 q^{26} - 224 q^{28} - 560 q^{31} - 512 q^{32} + 756 q^{33}+ \cdots - 23912 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 28 * q^7 - 64 * q^8 + 464 * q^11 + 336 * q^13 - 256 * q^16 - 392 * q^17 + 216 * q^18 - 1512 * q^21 + 928 * q^22 - 968 * q^23 + 1344 * q^26 - 224 * q^28 - 560 * q^31 - 512 * q^32 + 756 * q^33 + 864 * q^36 - 2256 * q^37 - 1232 * q^38 + 392 * q^41 - 3024 * q^42 - 216 * q^43 - 3872 * q^46 + 9072 * q^47 + 4968 * q^51 + 2688 * q^52 + 4280 * q^53 - 896 * q^56 - 6264 * q^57 + 2192 * q^58 - 4536 * q^61 - 1120 * q^62 + 756 * q^63 + 3024 * q^66 + 2248 * q^67 + 3136 * q^68 + 18064 * q^71 + 1728 * q^72 - 20524 * q^73 - 4928 * q^76 - 7336 * q^77 + 8208 * q^78 - 2916 * q^81 + 784 * q^82 + 336 * q^83 - 864 * q^86 - 8316 * q^87 - 7424 * q^88 - 52752 * q^91 - 7744 * q^92 + 7992 * q^93 + 40404 * q^97 - 23912 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.00000	+	2.00000i	0.500000	+	0.500000i
$2$	2.00000	+	2.00000i	0.500000	+	0.500000i
$3$	3.67423	−	3.67423i	0.408248	−	0.408248i
$3$	3.67423	−	3.67423i	0.408248	−	0.408248i
$4$			8.00000i			0.500000i
$5$		0			0
$5$		0			0
$6$	14.6969			0.408248
$7$	−44.4393	−	44.4393i	−0.906924	−	0.906924i	0.0890986	−	0.996023i	$-0.471601\pi$
$7$	−44.4393	−	44.4393i	−0.906924	−	0.906924i	−0.996023	+	0.0890986i	$0.971601\pi$
$8$	−16.0000	+	16.0000i	−0.250000	+	0.250000i
$9$		−	27.0000i		−	0.333333i
$10$		0			0
$11$	167.439			1.38380			0.691898	−	0.721995i	$-0.256775\pi$
$11$	167.439			1.38380			0.691898	+	0.721995i	$0.256775\pi$
$12$	29.3939	+	29.3939i	0.204124	+	0.204124i
$13$	223.621	−	223.621i	1.32320	−	1.32320i	0.412031	−	0.911170i	$-0.364820\pi$
$13$	223.621	−	223.621i	1.32320	−	1.32320i	0.911170	−	0.412031i	$-0.135180\pi$
$14$		−	177.757i		−	0.906924i
$15$		0			0
$16$	−64.0000			−0.250000
$17$	71.0148	+	71.0148i	0.245726	+	0.245726i	0.819214	−	0.573488i	$-0.194410\pi$
$17$	71.0148	+	71.0148i	0.245726	+	0.245726i	−0.573488	+	0.819214i	$0.694410\pi$
$18$	54.0000	−	54.0000i	0.166667	−	0.166667i
$19$		−	272.211i		−	0.754048i	−0.926204	−	0.377024i	$-0.876947\pi$
$19$		−	272.211i		−	0.754048i	0.926204	−	0.377024i	$-0.123053\pi$
$20$		0			0
$21$	−326.561			−0.740500
$22$	334.879	+	334.879i	0.691898	+	0.691898i
$23$	−87.6821	+	87.6821i	−0.165751	+	0.165751i	−0.785109	−	0.619358i	$-0.787393\pi$
$23$	−87.6821	+	87.6821i	−0.165751	+	0.165751i	0.619358	+	0.785109i	$0.287393\pi$
$24$			117.576i			0.204124i
$25$		0			0
$26$	894.484			1.32320
$27$	−99.2043	−	99.2043i	−0.136083	−	0.136083i
$28$	355.514	−	355.514i	0.453462	−	0.453462i
$29$		−	839.832i		−	0.998611i	−0.866426	−	0.499306i	$-0.833588\pi$
$29$		−	839.832i		−	0.998611i	0.866426	−	0.499306i	$-0.166412\pi$
$30$		0			0
$31$	403.787			0.420173			0.210087	−	0.977683i	$-0.432625\pi$
$31$	403.787			0.420173			0.210087	+	0.977683i	$0.432625\pi$
$32$	−128.000	−	128.000i	−0.125000	−	0.125000i
$33$	615.211	−	615.211i	0.564932	−	0.564932i
$34$			284.059i			0.245726i
$35$		0			0
$36$	216.000			0.166667
$37$	207.589	+	207.589i	0.151636	+	0.151636i	0.778848	−	0.627212i	$-0.215804\pi$
$37$	207.589	+	207.589i	0.151636	+	0.151636i	−0.627212	+	0.778848i	$0.715804\pi$
$38$	544.422	−	544.422i	0.377024	−	0.377024i
$39$		−	1643.27i		−	1.08039i
$40$		0			0
$41$	−1386.39			−0.824742			−0.412371	−	0.911016i	$-0.635299\pi$
$41$	−1386.39			−0.824742			−0.412371	+	0.911016i	$0.635299\pi$
$42$	−653.121	−	653.121i	−0.370250	−	0.370250i
$43$	−568.393	+	568.393i	−0.307406	+	0.307406i	−0.843902	−	0.536497i	$-0.819747\pi$
$43$	−568.393	+	568.393i	−0.307406	+	0.307406i	0.536497	+	0.843902i	$0.319747\pi$
$44$			1339.51i			0.691898i
$45$		0			0
$46$	−350.729			−0.165751
$47$	2642.77	+	2642.77i	1.19637	+	1.19637i	0.975247	+	0.221119i	$0.0709709\pi$
$47$	2642.77	+	2642.77i	1.19637	+	1.19637i	0.221119	+	0.975247i	$0.429029\pi$
$48$	−235.151	+	235.151i	−0.102062	+	0.102062i
$49$			1548.70i			0.645023i
$50$		0			0
$51$	521.850			0.200634
$52$	1788.97	+	1788.97i	0.661600	+	0.661600i
$53$	1378.64	−	1378.64i	0.490792	−	0.490792i	−0.417763	−	0.908556i	$-0.637186\pi$
$53$	1378.64	−	1378.64i	0.490792	−	0.490792i	0.908556	+	0.417763i	$0.137186\pi$
$54$		−	396.817i		−	0.136083i
$55$		0			0
$56$	1422.06			0.453462
$57$	−1000.17	−	1000.17i	−0.307839	−	0.307839i
$58$	1679.66	−	1679.66i	0.499306	−	0.499306i
$59$		−	327.503i		−	0.0940829i	−0.998893	−	0.0470415i	$-0.985021\pi$
$59$		−	327.503i		−	0.0940829i	0.998893	−	0.0470415i	$-0.0149793\pi$
$60$		0			0
$61$	−4205.66			−1.13025			−0.565125	−	0.825005i	$-0.691172\pi$
$61$	−4205.66			−1.13025			−0.565125	+	0.825005i	$0.691172\pi$
$62$	807.573	+	807.573i	0.210087	+	0.210087i
$63$	−1199.86	+	1199.86i	−0.302308	+	0.302308i
$64$		−	512.000i		−	0.125000i
$65$		0			0
$66$	2460.84			0.564932
$67$	−2832.99	−	2832.99i	−0.631097	−	0.631097i	0.317246	−	0.948343i	$-0.397242\pi$
$67$	−2832.99	−	2832.99i	−0.631097	−	0.631097i	−0.948343	+	0.317246i	$0.897242\pi$
$68$	−568.118	+	568.118i	−0.122863	+	0.122863i
$69$			644.330i			0.135335i
$70$		0			0
$71$	5339.03			1.05912			0.529560	−	0.848272i	$-0.322357\pi$
$71$	5339.03			1.05912			0.529560	+	0.848272i	$0.322357\pi$
$72$	432.000	+	432.000i	0.0833333	+	0.0833333i
$73$	−6865.24	+	6865.24i	−1.28828	+	1.28828i	−0.352447	+	0.935832i	$0.614650\pi$
$73$	−6865.24	+	6865.24i	−1.28828	+	1.28828i	−0.935832	+	0.352447i	$0.885350\pi$
$74$			830.357i			0.151636i
$75$		0			0
$76$	2177.69			0.377024
$77$	−7440.88	−	7440.88i	−1.25500	−	1.25500i
$78$	3286.54	−	3286.54i	0.540194	−	0.540194i
$79$			4666.69i			0.747748i	0.927480	+	0.373874i	$0.121971\pi$
$79$			4666.69i			0.747748i	−0.927480	+	0.373874i	$0.878029\pi$
$80$		0			0
$81$	−729.000			−0.111111
$82$	−2772.78	−	2772.78i	−0.412371	−	0.412371i
$83$	−4111.98	+	4111.98i	−0.596890	+	0.596890i	−0.939484	−	0.342594i	$-0.888695\pi$
$83$	−4111.98	+	4111.98i	−0.596890	+	0.596890i	0.342594	+	0.939484i	$0.388695\pi$
$84$		−	2612.49i		−	0.370250i
$85$		0			0
$86$	−2273.57			−0.307406
$87$	−3085.74	−	3085.74i	−0.407681	−	0.407681i
$88$	−2679.03	+	2679.03i	−0.345949	+	0.345949i
$89$		−	9911.62i		−	1.25131i	−0.780100	−	0.625654i	$-0.784832\pi$
$89$		−	9911.62i		−	1.25131i	0.780100	−	0.625654i	$-0.215168\pi$
$90$		0			0
$91$	−19875.1			−2.40009
$92$	−701.457	−	701.457i	−0.0828754	−	0.0828754i
$93$	1483.61	−	1483.61i	0.171535	−	0.171535i
$94$			10571.1i			1.19637i
$95$		0			0
$96$	−940.604			−0.102062
$97$	10688.9	+	10688.9i	1.13603	+	1.13603i	0.989156	+	0.146871i	$0.0469203\pi$
$97$	10688.9	+	10688.9i	1.13603	+	1.13603i	0.146871	+	0.989156i	$0.453080\pi$
$98$	−3097.40	+	3097.40i	−0.322511	+	0.322511i
$99$		−	4520.86i		−	0.461265i
$100$		0			0
$101$	−1763.09			−0.172835			−0.0864177	−	0.996259i	$-0.527542\pi$
$101$	−1763.09			−0.172835			−0.0864177	+	0.996259i	$0.527542\pi$
$102$	1043.70	+	1043.70i	0.100317	+	0.100317i
$103$	2414.27	−	2414.27i	0.227568	−	0.227568i	−0.584108	−	0.811676i	$-0.698556\pi$
$103$	2414.27	−	2414.27i	0.227568	−	0.227568i	0.811676	+	0.584108i	$0.198556\pi$
$104$			7155.87i			0.661600i
$105$		0			0
$106$	5514.54			0.490792
$107$	−4279.03	−	4279.03i	−0.373747	−	0.373747i	0.495093	−	0.868840i	$-0.335134\pi$
$107$	−4279.03	−	4279.03i	−0.373747	−	0.373747i	−0.868840	+	0.495093i	$0.835134\pi$
$108$	793.635	−	793.635i	0.0680414	−	0.0680414i
$109$			11819.5i			0.994827i	0.867514	+	0.497413i	$0.165717\pi$
$109$			11819.5i			0.994827i	−0.867514	+	0.497413i	$0.834283\pi$
$110$		0			0
$111$	1525.46			0.123810
$112$	2844.11	+	2844.11i	0.226731	+	0.226731i
$113$	−1620.60	+	1620.60i	−0.126917	+	0.126917i	−0.767712	−	0.640795i	$-0.778605\pi$
$113$	−1620.60	+	1620.60i	−0.126917	+	0.126917i	0.640795	+	0.767712i	$0.278605\pi$
$114$		−	4000.67i		−	0.307839i
$115$		0			0
$116$	6718.66			0.499306
$117$	−6037.76	−	6037.76i	−0.441067	−	0.441067i
$118$	655.005	−	655.005i	0.0470415	−	0.0470415i
$119$		−	6311.69i		−	0.445710i
$120$		0			0
$121$	13394.9			0.914891
$122$	−8411.32	−	8411.32i	−0.565125	−	0.565125i
$123$	−5093.93	+	5093.93i	−0.336699	+	0.336699i
$124$			3230.29i			0.210087i
$125$		0			0
$126$	−4799.44			−0.302308
$127$	11572.5	+	11572.5i	0.717494	+	0.717494i	0.968091	−	0.250597i	$-0.0806270\pi$
$127$	11572.5	+	11572.5i	0.717494	+	0.717494i	−0.250597	+	0.968091i	$0.580627\pi$
$128$	1024.00	−	1024.00i	0.0625000	−	0.0625000i
$129$			4176.82i			0.250996i
$130$		0			0
$131$	65.6230			0.00382396			0.00191198	−	0.999998i	$-0.499391\pi$
$131$	65.6230			0.00382396			0.00191198	+	0.999998i	$0.499391\pi$
$132$	4921.69	+	4921.69i	0.282466	+	0.282466i
$133$	−12096.9	+	12096.9i	−0.683864	+	0.683864i
$134$		−	11332.0i		−	0.631097i
$135$		0			0
$136$	−2272.47			−0.122863
$137$	−579.439	−	579.439i	−0.0308721	−	0.0308721i	0.691502	−	0.722374i	$-0.256949\pi$
$137$	−579.439	−	579.439i	−0.0308721	−	0.0308721i	−0.722374	+	0.691502i	$0.756949\pi$
$138$	−1288.66	+	1288.66i	−0.0676675	+	0.0676675i
$139$			36158.2i			1.87145i	0.352734	+	0.935724i	$0.385252\pi$
$139$			36158.2i			1.87145i	−0.352734	+	0.935724i	$0.614748\pi$
$140$		0			0
$141$	19420.3			0.976829
$142$	10678.1	+	10678.1i	0.529560	+	0.529560i
$143$	37442.9	−	37442.9i	1.83104	−	1.83104i
$144$			1728.00i			0.0833333i
$145$		0			0
$146$	−27461.0			−1.28828
$147$	5690.29	+	5690.29i	0.263330	+	0.263330i
$148$	−1660.71	+	1660.71i	−0.0758178	+	0.0758178i
$149$		−	2143.94i		−	0.0965695i	−0.998834	−	0.0482847i	$-0.984625\pi$
$149$		−	2143.94i		−	0.0965695i	0.998834	−	0.0482847i	$-0.0153755\pi$
$150$		0			0
$151$	−8940.59			−0.392114			−0.196057	−	0.980593i	$-0.562814\pi$
$151$	−8940.59			−0.392114			−0.196057	+	0.980593i	$0.562814\pi$
$152$	4355.38	+	4355.38i	0.188512	+	0.188512i
$153$	1917.40	−	1917.40i	0.0819086	−	0.0819086i
$154$		−	29763.5i		−	1.25500i
$155$		0			0
$156$	13146.2			0.540194
$157$	−18032.9	−	18032.9i	−0.731586	−	0.731586i	0.239348	−	0.970934i	$-0.423066\pi$
$157$	−18032.9	−	18032.9i	−0.731586	−	0.731586i	−0.970934	+	0.239348i	$0.923066\pi$
$158$	−9333.39	+	9333.39i	−0.373874	+	0.373874i
$159$		−	10130.9i		−	0.400730i
$160$		0			0
$161$	7793.06			0.300647
$162$	−1458.00	−	1458.00i	−0.0555556	−	0.0555556i
$163$	6859.00	−	6859.00i	0.258158	−	0.258158i	−0.566147	−	0.824305i	$-0.691566\pi$
$163$	6859.00	−	6859.00i	0.258158	−	0.258158i	0.824305	+	0.566147i	$0.191566\pi$
$164$		−	11091.1i		−	0.412371i
$165$		0			0
$166$	−16447.9			−0.596890
$167$	37439.2	+	37439.2i	1.34244	+	1.34244i	0.893627	+	0.448811i	$0.148152\pi$
$167$	37439.2	+	37439.2i	1.34244	+	1.34244i	0.448811	+	0.893627i	$0.351848\pi$
$168$	5224.97	−	5224.97i	0.185125	−	0.185125i
$169$		−	71451.6i		−	2.50172i
$170$		0			0
$171$	−7349.70			−0.251349
$172$	−4547.14	−	4547.14i	−0.153703	−	0.153703i
$173$	32638.2	−	32638.2i	1.09052	−	1.09052i	0.0950500	−	0.995472i	$-0.469699\pi$
$173$	32638.2	−	32638.2i	1.09052	−	1.09052i	0.995472	−	0.0950500i	$-0.0303011\pi$
$174$		−	12343.0i		−	0.407681i
$175$		0			0
$176$	−10716.1			−0.345949
$177$	−1203.32	−	1203.32i	−0.0384092	−	0.0384092i
$178$	19823.2	−	19823.2i	0.625654	−	0.625654i
$179$		−	6396.19i		−	0.199625i	−0.995006	−	0.0998126i	$-0.968176\pi$
$179$		−	6396.19i		−	0.199625i	0.995006	−	0.0998126i	$-0.0318243\pi$
$180$		0			0
$181$	49025.9			1.49647			0.748236	−	0.663432i	$-0.230901\pi$
$181$	49025.9			1.49647			0.748236	+	0.663432i	$0.230901\pi$
$182$	−39750.2	−	39750.2i	−1.20004	−	1.20004i
$183$	−15452.6	+	15452.6i	−0.461423	+	0.461423i
$184$		−	2805.83i		−	0.0828754i
$185$		0			0
$186$	5934.43			0.171535
$187$	11890.7	+	11890.7i	0.340034	+	0.340034i
$188$	−21142.2	+	21142.2i	−0.598183	+	0.598183i
$189$			8817.14i			0.246833i
$190$		0			0
$191$	39949.0			1.09506			0.547531	−	0.836785i	$-0.315568\pi$
$191$	39949.0			1.09506			0.547531	+	0.836785i	$0.315568\pi$
$192$	−1881.21	−	1881.21i	−0.0510310	−	0.0510310i
$193$	23749.1	−	23749.1i	0.637577	−	0.637577i	−0.312380	−	0.949957i	$-0.601126\pi$
$193$	23749.1	−	23749.1i	0.637577	−	0.637577i	0.949957	+	0.312380i	$0.101126\pi$
$194$			42755.5i			1.13603i
$195$		0			0
$196$	−12389.6			−0.322511
$197$	2725.09	+	2725.09i	0.0702181	+	0.0702181i	0.741344	−	0.671126i	$-0.234189\pi$
$197$	2725.09	+	2725.09i	0.0702181	+	0.0702181i	−0.671126	+	0.741344i	$0.734189\pi$
$198$	9041.72	−	9041.72i	0.230633	−	0.230633i
$199$			63212.5i			1.59624i	0.602502	+	0.798118i	$0.294171\pi$
$199$			63212.5i			1.59624i	−0.602502	+	0.798118i	$0.705829\pi$
$200$		0			0
$201$	−20818.2			−0.515288
$202$	−3526.19	−	3526.19i	−0.0864177	−	0.0864177i
$203$	−37321.5	+	37321.5i	−0.905665	+	0.905665i
$204$			4174.80i			0.100317i
$205$		0			0
$206$	9657.07			0.227568
$207$	2367.42	+	2367.42i	0.0552502	+	0.0552502i
$208$	−14311.7	+	14311.7i	−0.330800	+	0.330800i
$209$		−	45578.9i		−	1.04345i
$210$		0			0
$211$	15470.2			0.347480			0.173740	−	0.984792i	$-0.444415\pi$
$211$	15470.2			0.347480			0.173740	+	0.984792i	$0.444415\pi$
$212$	11029.1	+	11029.1i	0.245396	+	0.245396i
$213$	19616.8	−	19616.8i	0.432384	−	0.432384i
$214$		−	17116.1i		−	0.373747i
$215$		0			0
$216$	3174.54			0.0680414
$217$	−17944.0	−	17944.0i	−0.381065	−	0.381065i
$218$	−23639.1	+	23639.1i	−0.497413	+	0.497413i
$219$			50449.0i			1.05188i
$220$		0			0
$221$	31760.8			0.650289
$222$	3050.93	+	3050.93i	0.0619050	+	0.0619050i
$223$	−28817.8	+	28817.8i	−0.579496	+	0.579496i	−0.934764	−	0.355268i	$-0.884389\pi$
$223$	−28817.8	+	28817.8i	−0.579496	+	0.579496i	0.355268	+	0.934764i	$0.384389\pi$
$224$			11376.5i			0.226731i
$225$		0			0
$226$	−6482.41			−0.126917
$227$	−884.237	−	884.237i	−0.0171600	−	0.0171600i	0.698475	−	0.715635i	$-0.253862\pi$
$227$	−884.237	−	884.237i	−0.0171600	−	0.0171600i	−0.715635	+	0.698475i	$0.753862\pi$
$228$	8001.34	−	8001.34i	0.153919	−	0.153919i
$229$		−	12076.4i		−	0.230285i	−0.993349	−	0.115143i	$-0.963268\pi$
$229$		−	12076.4i		−	0.230285i	0.993349	−	0.115143i	$-0.0367325\pi$
$230$		0			0
$231$	−54679.1			−1.02470
$232$	13437.3	+	13437.3i	0.249653	+	0.249653i
$233$	24695.8	−	24695.8i	0.454894	−	0.454894i	−0.442081	−	0.896975i	$-0.645760\pi$
$233$	24695.8	−	24695.8i	0.454894	−	0.454894i	0.896975	+	0.442081i	$0.145760\pi$
$234$		−	24151.1i		−	0.441067i
$235$		0			0
$236$	2620.02			0.0470415
$237$	17146.5	+	17146.5i	0.305267	+	0.305267i
$238$	12623.4	−	12623.4i	0.222855	−	0.222855i
$239$			48187.8i			0.843608i	0.906687	+	0.421804i	$0.138603\pi$
$239$			48187.8i			0.843608i	−0.906687	+	0.421804i	$0.861397\pi$
$240$		0			0
$241$	−55184.2			−0.950125			−0.475062	−	0.879952i	$-0.657574\pi$
$241$	−55184.2			−0.950125			−0.475062	+	0.879952i	$0.657574\pi$
$242$	26789.8	+	26789.8i	0.457445	+	0.457445i
$243$	−2678.52	+	2678.52i	−0.0453609	+	0.0453609i
$244$		−	33645.3i		−	0.565125i
$245$		0			0
$246$	−20375.7			−0.336699
$247$	−60872.1	−	60872.1i	−0.997756	−	0.997756i
$248$	−6460.59	+	6460.59i	−0.105043	+	0.105043i
$249$			30216.7i			0.487359i
$250$		0			0
$251$	−69667.8			−1.10582			−0.552910	−	0.833241i	$-0.686483\pi$
$251$	−69667.8			−1.10582			−0.552910	+	0.833241i	$0.686483\pi$
$252$	−9598.89	−	9598.89i	−0.151154	−	0.151154i
$253$	−14681.4	+	14681.4i	−0.229365	+	0.229365i
$254$			46289.8i			0.717494i
$255$		0			0
$256$	4096.00			0.0625000
$257$	11614.4	+	11614.4i	0.175846	+	0.175846i	0.789542	−	0.613696i	$-0.210318\pi$
$257$	11614.4	+	11614.4i	0.175846	+	0.175846i	−0.613696	+	0.789542i	$0.710318\pi$
$258$	−8353.63	+	8353.63i	−0.125498	+	0.125498i
$259$		−	18450.2i		−	0.275044i
$260$		0			0
$261$	−22675.5			−0.332870
$262$	131.246	+	131.246i	0.00191198	+	0.00191198i
$263$	−86811.6	+	86811.6i	−1.25507	+	1.25507i	−0.301646	+	0.953420i	$0.597536\pi$
$263$	−86811.6	+	86811.6i	−1.25507	+	1.25507i	−0.953420	+	0.301646i	$0.902464\pi$
$264$			19686.8i			0.282466i
$265$		0			0
$266$	−48387.5			−0.683864
$267$	−36417.6	−	36417.6i	−0.510845	−	0.510845i
$268$	22663.9	−	22663.9i	0.315548	−	0.315548i
$269$			93439.3i			1.29129i	0.763636	+	0.645647i	$0.223412\pi$
$269$			93439.3i			1.29129i	−0.763636	+	0.645647i	$0.776588\pi$
$270$		0			0
$271$	−10856.5			−0.147826			−0.0739131	−	0.997265i	$-0.523549\pi$
$271$	−10856.5			−0.147826			−0.0739131	+	0.997265i	$0.523549\pi$
$272$	−4544.95	−	4544.95i	−0.0614315	−	0.0614315i
$273$	−73025.8	+	73025.8i	−0.979831	+	0.979831i
$274$		−	2317.76i		−	0.0308721i
$275$		0			0
$276$	−5154.64			−0.0676675
$277$	−65214.7	−	65214.7i	−0.849935	−	0.849935i	0.140189	−	0.990125i	$-0.455229\pi$
$277$	−65214.7	−	65214.7i	−0.849935	−	0.849935i	−0.990125	+	0.140189i	$0.955229\pi$
$278$	−72316.5	+	72316.5i	−0.935724	+	0.935724i
$279$		−	10902.2i		−	0.140058i
$280$		0			0
$281$	−8669.34			−0.109793			−0.0548963	−	0.998492i	$-0.517483\pi$
$281$	−8669.34			−0.109793			−0.0548963	+	0.998492i	$0.517483\pi$
$282$	38840.7	+	38840.7i	0.488414	+	0.488414i
$283$	23747.0	−	23747.0i	0.296508	−	0.296508i	−0.543136	−	0.839645i	$-0.682763\pi$
$283$	23747.0	−	23747.0i	0.296508	−	0.296508i	0.839645	+	0.543136i	$0.182763\pi$
$284$			42712.2i			0.529560i
$285$		0			0
$286$	149772.			1.83104
$287$	61610.2	+	61610.2i	0.747978	+	0.747978i
$288$	−3456.00	+	3456.00i	−0.0416667	+	0.0416667i
$289$		−	73434.8i		−	0.879238i
$290$		0			0
$291$	78546.9			0.927562
$292$	−54921.9	−	54921.9i	−0.644139	−	0.644139i
$293$	55077.2	−	55077.2i	0.641559	−	0.641559i	−0.309380	−	0.950939i	$-0.600121\pi$
$293$	55077.2	−	55077.2i	0.641559	−	0.641559i	0.950939	+	0.309380i	$0.100121\pi$
$294$			22761.1i			0.263330i
$295$		0			0
$296$	−6642.86			−0.0758178
$297$	−16610.7	−	16610.7i	−0.188311	−	0.188311i
$298$	4287.88	−	4287.88i	0.0482847	−	0.0482847i
$299$			39215.1i			0.438643i
$300$		0			0
$301$	50517.9			0.557587
$302$	−17881.2	−	17881.2i	−0.196057	−	0.196057i
$303$	−6478.02	+	6478.02i	−0.0705597	+	0.0705597i
$304$			17421.5i			0.188512i
$305$		0			0
$306$	7669.60			0.0819086
$307$	−11961.1	−	11961.1i	−0.126910	−	0.126910i	0.640799	−	0.767709i	$-0.278603\pi$
$307$	−11961.1	−	11961.1i	−0.126910	−	0.126910i	−0.767709	+	0.640799i	$0.778603\pi$
$308$	59527.1	−	59527.1i	0.627499	−	0.627499i
$309$		−	17741.2i		−	0.185808i
$310$		0			0
$311$	−65284.5			−0.674977			−0.337489	−	0.941330i	$-0.609577\pi$
$311$	−65284.5			−0.674977			−0.337489	+	0.941330i	$0.609577\pi$
$312$	26292.3	+	26292.3i	0.270097	+	0.270097i
$313$	6418.45	−	6418.45i	0.0655151	−	0.0655151i	−0.673590	−	0.739105i	$-0.735249\pi$
$313$	6418.45	−	6418.45i	0.0655151	−	0.0655151i	0.739105	+	0.673590i	$0.235249\pi$
$314$		−	72131.4i		−	0.731586i
$315$		0			0
$316$	−37333.5			−0.373874
$317$	−22100.1	−	22100.1i	−0.219926	−	0.219926i	0.588541	−	0.808467i	$-0.299702\pi$
$317$	−22100.1	−	22100.1i	−0.219926	−	0.219926i	−0.808467	+	0.588541i	$0.799702\pi$
$318$	20261.7	−	20261.7i	0.200365	−	0.200365i
$319$		−	140621.i		−	1.38187i
$320$		0			0
$321$	−31444.3			−0.305163
$322$	15586.1	+	15586.1i	0.150323	+	0.150323i
$323$	19331.0	−	19331.0i	0.185289	−	0.185289i
$324$		−	5832.00i		−	0.0555556i
$325$		0			0
$326$	27436.0			0.258158
$327$	43427.7	+	43427.7i	0.406136	+	0.406136i
$328$	22182.3	−	22182.3i	0.206185	−	0.206185i
$329$		−	234886.i		−	2.17003i
$330$		0			0
$331$	−49108.7			−0.448231			−0.224116	−	0.974563i	$-0.571949\pi$
$331$	−49108.7			−0.448231			−0.224116	+	0.974563i	$0.571949\pi$
$332$	−32895.8	−	32895.8i	−0.298445	−	0.298445i
$333$	5604.91	−	5604.91i	0.0505452	−	0.0505452i
$334$			149757.i			1.34244i
$335$		0			0
$336$	20899.9			0.185125
$337$	142816.	+	142816.i	1.25753	+	1.25753i	0.952269	+	0.305260i	$0.0987434\pi$
$337$	142816.	+	142816.i	1.25753	+	1.25753i	0.305260	+	0.952269i	$0.401257\pi$
$338$	142903.	−	142903.i	1.25086	−	1.25086i
$339$			11909.0i			0.103627i
$340$		0			0
$341$	67609.8			0.581434
$342$	−14699.4	−	14699.4i	−0.125675	−	0.125675i
$343$	−37875.6	+	37875.6i	−0.321937	+	0.321937i
$344$		−	18188.6i		−	0.153703i
$345$		0			0
$346$	130553.			1.09052
$347$	21400.3	+	21400.3i	0.177730	+	0.177730i	0.790366	−	0.612635i	$-0.209891\pi$
$347$	21400.3	+	21400.3i	0.177730	+	0.177730i	−0.612635	+	0.790366i	$0.709891\pi$
$348$	24685.9	−	24685.9i	0.203841	−	0.203841i
$349$		−	34091.2i		−	0.279893i	−0.990159	−	0.139946i	$-0.955307\pi$
$349$		−	34091.2i		−	0.279893i	0.990159	−	0.139946i	$-0.0446930\pi$
$350$		0			0
$351$	−44368.3			−0.360130
$352$	−21432.2	−	21432.2i	−0.172974	−	0.172974i
$353$	−28.7539	+	28.7539i	−0.000230753	+	0.000230753i	−0.707222	−	0.706991i	$-0.750052\pi$
$353$	−28.7539	+	28.7539i	−0.000230753	+	0.000230753i	0.706991	+	0.707222i	$0.250052\pi$
$354$		−	4813.29i		−	0.0384092i
$355$		0			0
$356$	79292.9			0.625654
$357$	−23190.6	−	23190.6i	−0.181960	−	0.181960i
$358$	12792.4	−	12792.4i	0.0998126	−	0.0998126i
$359$			173943.i			1.34964i	0.737984	+	0.674818i	$0.235778\pi$
$359$			173943.i			1.34964i	−0.737984	+	0.674818i	$0.764222\pi$
$360$		0			0
$361$	56222.1			0.431412
$362$	98051.9	+	98051.9i	0.748236	+	0.748236i
$363$	49216.1	−	49216.1i	0.373503	−	0.373503i
$364$		−	159001.i		−	1.20004i
$365$		0			0
$366$	−61810.3			−0.461423
$367$	41954.1	+	41954.1i	0.311489	+	0.311489i	0.845486	−	0.533997i	$-0.179311\pi$
$367$	41954.1	+	41954.1i	0.311489	+	0.311489i	−0.533997	+	0.845486i	$0.679311\pi$
$368$	5611.66	−	5611.66i	0.0414377	−	0.0414377i
$369$			37432.6i			0.274914i
$370$		0			0
$371$	−122531.			−0.890223
$372$	11868.9	+	11868.9i	0.0857676	+	0.0857676i
$373$	65875.2	−	65875.2i	0.473483	−	0.473483i	−0.429557	−	0.903040i	$-0.641330\pi$
$373$	65875.2	−	65875.2i	0.473483	−	0.473483i	0.903040	+	0.429557i	$0.141330\pi$
$374$			47562.7i			0.340034i
$375$		0			0
$376$	−84568.7			−0.598183
$377$	−187804.	−	187804.i	−1.32136	−	1.32136i
$378$	−17634.3	+	17634.3i	−0.123417	+	0.123417i
$379$		−	10837.0i		−	0.0754448i	−0.999288	−	0.0377224i	$-0.987990\pi$
$379$		−	10837.0i		−	0.0754448i	0.999288	−	0.0377224i	$-0.0120103\pi$
$380$		0			0
$381$	85039.9			0.585831
$382$	79898.0	+	79898.0i	0.547531	+	0.547531i
$383$	−22327.5	+	22327.5i	−0.152210	+	0.152210i	−0.779104	−	0.626894i	$-0.784326\pi$
$383$	−22327.5	+	22327.5i	−0.152210	+	0.152210i	0.626894	+	0.779104i	$0.284326\pi$
$384$		−	7524.83i		−	0.0510310i
$385$		0			0
$386$	94996.5			0.637577
$387$	15346.6	+	15346.6i	0.102469	+	0.102469i
$388$	−85511.0	+	85511.0i	−0.568013	+	0.568013i
$389$		−	203884.i		−	1.34736i	−0.739023	−	0.673680i	$-0.764713\pi$
$389$		−	203884.i		−	1.34736i	0.739023	−	0.673680i	$-0.235287\pi$
$390$		0			0
$391$	−12453.5			−0.0814585
$392$	−24779.2	−	24779.2i	−0.161256	−	0.161256i
$393$	241.114	−	241.114i	0.00156113	−	0.00156113i
$394$			10900.4i			0.0702181i
$395$		0			0
$396$	36166.9			0.230633
$397$	−117299.	−	117299.i	−0.744238	−	0.744238i	0.229153	−	0.973390i	$-0.426404\pi$
$397$	−117299.	−	117299.i	−0.744238	−	0.744238i	−0.973390	+	0.229153i	$0.926404\pi$
$398$	−126425.	+	126425.i	−0.798118	+	0.798118i
$399$			88893.5i			0.558373i
$400$		0			0
$401$	43093.1			0.267990			0.133995	−	0.990982i	$-0.457219\pi$
$401$	43093.1			0.267990			0.133995	+	0.990982i	$0.457219\pi$
$402$	−41636.3	−	41636.3i	−0.257644	−	0.257644i
$403$	90295.2	−	90295.2i	0.555974	−	0.555974i
$404$		−	14104.7i		−	0.0864177i
$405$		0			0
$406$	−149286.			−0.905665
$407$	34758.6	+	34758.6i	0.209833	+	0.209833i
$408$	−8349.60	+	8349.60i	−0.0501586	+	0.0501586i
$409$			186090.i			1.11244i	0.831036	+	0.556219i	$0.187748\pi$
$409$			186090.i			1.11244i	−0.831036	+	0.556219i	$0.812252\pi$
$410$		0			0
$411$	−4257.99			−0.0252070
$412$	19314.1	+	19314.1i	0.113784	+	0.113784i
$413$	−14554.0	+	14554.0i	−0.0853261	+	0.0853261i
$414$			9469.67i			0.0552502i
$415$		0			0
$416$	−57247.0			−0.330800
$417$	132854.	+	132854.i	0.764015	+	0.764015i
$418$	91157.7	−	91157.7i	0.521724	−	0.521724i
$419$		−	101699.i		−	0.579278i	−0.957136	−	0.289639i	$-0.906465\pi$
$419$		−	101699.i		−	0.579278i	0.957136	−	0.289639i	$-0.0935352\pi$
$420$		0			0
$421$	−22533.0			−0.127132			−0.0635659	−	0.997978i	$-0.520247\pi$
$421$	−22533.0			−0.127132			−0.0635659	+	0.997978i	$0.520247\pi$
$422$	30940.3	+	30940.3i	0.173740	+	0.173740i
$423$	71354.8	−	71354.8i	0.398789	−	0.398789i
$424$			44116.3i			0.245396i
$425$		0			0
$426$	78467.4			0.432384
$427$	186897.	+	186897.i	1.02505	+	1.02505i
$428$	34232.2	−	34232.2i	0.186873	−	0.186873i
$429$		−	275148.i		−	1.49504i
$430$		0			0
$431$	210598.			1.13370			0.566852	−	0.823820i	$-0.308161\pi$
$431$	210598.			1.13370			0.566852	+	0.823820i	$0.308161\pi$
$432$	6349.08	+	6349.08i	0.0340207	+	0.0340207i
$433$	−123405.	+	123405.i	−0.658201	+	0.658201i	−0.954954	−	0.296754i	$-0.904096\pi$
$433$	−123405.	+	123405.i	−0.658201	+	0.658201i	0.296754	+	0.954954i	$0.404096\pi$
$434$		−	71776.0i		−	0.381065i
$435$		0			0
$436$	−94556.3			−0.497413
$437$	23868.1	+	23868.1i	0.124984	+	0.124984i
$438$	−100898.	+	100898.i	−0.525938	+	0.525938i
$439$			265180.i			1.37598i	0.725721	+	0.687989i	$0.241506\pi$
$439$			265180.i			1.37598i	−0.725721	+	0.687989i	$0.758494\pi$
$440$		0			0
$441$	41814.9			0.215008
$442$	63521.6	+	63521.6i	0.325145	+	0.325145i
$443$	99815.9	−	99815.9i	0.508619	−	0.508619i	−0.405484	−	0.914102i	$-0.632897\pi$
$443$	99815.9	−	99815.9i	0.508619	−	0.508619i	0.914102	+	0.405484i	$0.132897\pi$
$444$			12203.7i			0.0619050i
$445$		0			0
$446$	−115271.			−0.579496
$447$	−7877.34	−	7877.34i	−0.0394243	−	0.0394243i
$448$	−22752.9	+	22752.9i	−0.113366	+	0.113366i
$449$			182934.i			0.907408i	0.891152	+	0.453704i	$0.149898\pi$
$449$			182934.i			0.907408i	−0.891152	+	0.453704i	$0.850102\pi$
$450$		0			0
$451$	−232136.			−1.14127
$452$	−12964.8	−	12964.8i	−0.0634585	−	0.0634585i
$453$	−32849.8	+	32849.8i	−0.160080	+	0.160080i
$454$		−	3536.95i		−	0.0171600i
$455$		0			0
$456$	32005.4			0.153919
$457$	−147497.	−	147497.i	−0.706239	−	0.706239i	0.259503	−	0.965742i	$-0.416441\pi$
$457$	−147497.	−	147497.i	−0.706239	−	0.706239i	−0.965742	+	0.259503i	$0.916441\pi$
$458$	24152.8	−	24152.8i	0.115143	−	0.115143i
$459$		−	14090.0i		−	0.0668781i
$460$		0			0
$461$	−354912.			−1.67001			−0.835006	−	0.550241i	$-0.814536\pi$
$461$	−354912.			−1.67001			−0.835006	+	0.550241i	$0.814536\pi$
$462$	−109358.	−	109358.i	−0.512351	−	0.512351i
$463$	−182475.	+	182475.i	−0.851221	+	0.851221i	−0.990284	−	0.139063i	$-0.955591\pi$
$463$	−182475.	+	182475.i	−0.851221	+	0.851221i	0.139063	+	0.990284i	$0.455591\pi$
$464$			53749.3i			0.249653i
$465$		0			0
$466$	98783.0			0.454894
$467$	−4348.30	−	4348.30i	−0.0199382	−	0.0199382i	0.697067	−	0.717006i	$-0.254488\pi$
$467$	−4348.30	−	4348.30i	−0.0199382	−	0.0199382i	−0.717006	+	0.697067i	$0.754488\pi$
$468$	48302.1	−	48302.1i	0.220533	−	0.220533i
$469$			251792.i			1.14471i
$470$		0			0
$471$	−132514.			−0.597337
$472$	5240.04	+	5240.04i	0.0235207	+	0.0235207i
$473$	−95171.3	+	95171.3i	−0.425386	+	0.425386i
$474$			68586.1i			0.305267i
$475$		0			0
$476$	50493.5			0.222855
$477$	−37223.2	−	37223.2i	−0.163597	−	0.163597i
$478$	−96375.5	+	96375.5i	−0.421804	+	0.421804i
$479$		−	77346.0i		−	0.337106i	−0.985693	−	0.168553i	$-0.946091\pi$
$479$		−	77346.0i		−	0.337106i	0.985693	−	0.168553i	$-0.0539095\pi$
$480$		0			0
$481$	92842.6			0.401289
$482$	−110368.	−	110368.i	−0.475062	−	0.475062i
$483$	28633.5	−	28633.5i	0.122739	−	0.122739i
$484$			107159.i			0.457445i
$485$		0			0
$486$	−10714.1			−0.0453609
$487$	−77225.6	−	77225.6i	−0.325614	−	0.325614i	0.525302	−	0.850916i	$-0.323952\pi$
$487$	−77225.6	−	77225.6i	−0.325614	−	0.325614i	−0.850916	+	0.525302i	$0.823952\pi$
$488$	67290.6	−	67290.6i	0.282562	−	0.282562i
$489$		−	50403.1i		−	0.210785i
$490$		0			0
$491$	340217.			1.41121			0.705607	−	0.708603i	$-0.250674\pi$
$491$	340217.			1.41121			0.705607	+	0.708603i	$0.250674\pi$
$492$	−40751.4	−	40751.4i	−0.168350	−	0.168350i
$493$	59640.5	−	59640.5i	0.245385	−	0.245385i
$494$		−	243488.i		−	0.997756i
$495$		0			0
$496$	−25842.4			−0.105043
$497$	−237263.	−	237263.i	−0.960542	−	0.960542i
$498$	−60433.5	+	60433.5i	−0.243679	+	0.243679i
$499$		−	331024.i		−	1.32941i	−0.747108	−	0.664703i	$-0.768558\pi$
$499$		−	331024.i		−	1.32941i	0.747108	−	0.664703i	$-0.231442\pi$
$500$		0			0
$501$	275121.			1.09610
$502$	−139336.	−	139336.i	−0.552910	−	0.552910i
$503$	123301.	−	123301.i	0.487338	−	0.487338i	−0.420127	−	0.907465i	$-0.638015\pi$
$503$	123301.	−	123301.i	0.487338	−	0.487338i	0.907465	+	0.420127i	$0.138015\pi$
$504$		−	38395.5i		−	0.151154i
$505$		0			0
$506$	−58725.7			−0.229365
$507$	−262530.	−	262530.i	−1.02132	−	1.02132i
$508$	−92579.7	+	92579.7i	−0.358747	+	0.358747i
$509$			364729.i			1.40778i	0.710309	+	0.703890i	$0.248555\pi$
$509$			364729.i			1.40778i	−0.710309	+	0.703890i	$0.751445\pi$
$510$		0			0
$511$	610173.			2.33674
$512$	8192.00	+	8192.00i	0.0312500	+	0.0312500i
$513$	−27004.5	+	27004.5i	−0.102613	+	0.102613i
$514$			46457.8i			0.175846i
$515$		0			0
$516$	−33414.5			−0.125498
$517$	442504.	+	442504.i	1.65553	+	1.65553i
$518$	36900.5	−	36900.5i	0.137522	−	0.137522i
$519$		−	239841.i		−	0.890408i
$520$		0			0
$521$	−203461.			−0.749560			−0.374780	−	0.927114i	$-0.622282\pi$
$521$	−203461.			−0.749560			−0.374780	+	0.927114i	$0.622282\pi$
$522$	−45350.9	−	45350.9i	−0.166435	−	0.166435i
$523$	−72655.5	+	72655.5i	−0.265623	+	0.265623i	−0.827334	−	0.561711i	$-0.810143\pi$
$523$	−72655.5	+	72655.5i	−0.265623	+	0.265623i	0.561711	+	0.827334i	$0.310143\pi$
$524$			524.984i			0.00191198i
$525$		0			0
$526$	−347247.			−1.25507
$527$	28674.8	+	28674.8i	0.103248	+	0.103248i
$528$	−39373.5	+	39373.5i	−0.141233	+	0.141233i
$529$			264465.i			0.945053i
$530$		0			0
$531$	−8842.57			−0.0313610
$532$	−96775.0	−	96775.0i	−0.341932	−	0.341932i
$533$	−310026.	+	310026.i	−1.09130	+	1.09130i
$534$		−	145670.i		−	0.510845i
$535$		0			0
$536$	90655.8			0.315548
$537$	−23501.1	−	23501.1i	−0.0814966	−	0.0814966i
$538$	−186879.	+	186879.i	−0.645647	+	0.645647i
$539$			259313.i			0.892580i
$540$		0			0
$541$	−91367.2			−0.312173			−0.156087	−	0.987743i	$-0.549888\pi$
$541$	−91367.2			−0.312173			−0.156087	+	0.987743i	$0.549888\pi$
$542$	−21713.0	−	21713.0i	−0.0739131	−	0.0739131i
$543$	180133.	−	180133.i	0.610932	−	0.610932i
$544$		−	18179.8i		−	0.0614315i
$545$		0			0
$546$	−292103.			−0.979831
$547$	9895.50	+	9895.50i	0.0330722	+	0.0330722i	0.723450	−	0.690377i	$-0.242555\pi$
$547$	9895.50	+	9895.50i	0.0330722	+	0.0330722i	−0.690377	+	0.723450i	$0.742555\pi$
$548$	4635.51	−	4635.51i	0.0154361	−	0.0154361i
$549$			113553.i			0.376750i
$550$		0			0
$551$	−228612.			−0.753001
$552$	−10309.3	−	10309.3i	−0.0338337	−	0.0338337i
$553$	207384.	−	207384.i	0.678150	−	0.678150i
$554$		−	260859.i		−	0.849935i
$555$		0			0
$556$	−289266.			−0.935724
$557$	306800.	+	306800.i	0.988883	+	0.988883i	0.999939	−	0.0110558i	$-0.00351924\pi$
$557$	306800.	+	306800.i	0.988883	+	0.988883i	−0.0110558	+	0.999939i	$0.503519\pi$
$558$	21804.5	−	21804.5i	0.0700289	−	0.0700289i
$559$			254209.i			0.813518i
$560$		0			0
$561$	87378.2			0.277637
$562$	−17338.7	−	17338.7i	−0.0548963	−	0.0548963i
$563$	−331132.	+	331132.i	−1.04468	+	1.04468i	−0.0457279	+	0.998954i	$0.514561\pi$
$563$	−331132.	+	331132.i	−1.04468	+	1.04468i	−0.998954	+	0.0457279i	$0.985439\pi$
$564$			155363.i			0.488414i
$565$		0			0
$566$	94988.2			0.296508
$567$	32396.2	+	32396.2i	0.100769	+	0.100769i
$568$	−85424.5	+	85424.5i	−0.264780	+	0.264780i
$569$		−	257399.i		−	0.795028i	−0.917596	−	0.397514i	$-0.869873\pi$
$569$		−	257399.i		−	0.795028i	0.917596	−	0.397514i	$-0.130127\pi$
$570$		0			0
$571$	−622495.			−1.90925			−0.954626	−	0.297807i	$-0.903745\pi$
$571$	−622495.			−1.90925			−0.954626	+	0.297807i	$0.903745\pi$
$572$	299543.	+	299543.i	0.915520	+	0.915520i
$573$	146782.	−	146782.i	0.447057	−	0.447057i
$574$			246441.i			0.747978i
$575$		0			0
$576$	−13824.0			−0.0416667
$577$	23928.8	+	23928.8i	0.0718736	+	0.0718736i	0.742130	−	0.670256i	$-0.233816\pi$
$577$	23928.8	+	23928.8i	0.0718736	+	0.0718736i	−0.670256	+	0.742130i	$0.733816\pi$
$578$	146870.	−	146870.i	0.439619	−	0.439619i
$579$		−	174520.i		−	0.520580i
$580$		0			0
$581$	365467.			1.08267
$582$	157094.	+	157094.i	0.463781	+	0.463781i
$583$	230838.	−	230838.i	0.679156	−	0.679156i
$584$		−	219688.i		−	0.644139i
$585$		0			0
$586$	220309.			0.641559
$587$	−181136.	−	181136.i	−0.525690	−	0.525690i	0.393595	−	0.919284i	$-0.371231\pi$
$587$	−181136.	−	181136.i	−0.525690	−	0.525690i	−0.919284	+	0.393595i	$0.871231\pi$
$588$	−45522.3	+	45522.3i	−0.131665	+	0.131665i
$589$		−	109915.i		−	0.316831i
$590$		0			0
$591$	20025.3			0.0573328
$592$	−13285.7	−	13285.7i	−0.0379089	−	0.0379089i
$593$	235890.	−	235890.i	0.670812	−	0.670812i	−0.287091	−	0.957903i	$-0.592688\pi$
$593$	235890.	−	235890.i	0.670812	−	0.670812i	0.957903	+	0.287091i	$0.0926883\pi$
$594$		−	66442.8i		−	0.188311i
$595$		0			0
$596$	17151.5			0.0482847
$597$	232258.	+	232258.i	0.651660	+	0.651660i
$598$	−78430.2	+	78430.2i	−0.219322	+	0.219322i
$599$			141629.i			0.394728i	0.980330	+	0.197364i	$0.0632381\pi$
$599$			141629.i			0.394728i	−0.980330	+	0.197364i	$0.936762\pi$
$600$		0			0
$601$	−564417.			−1.56261			−0.781306	−	0.624148i	$-0.785446\pi$
$601$	−564417.			−1.56261			−0.781306	+	0.624148i	$0.785446\pi$
$602$	101036.	+	101036.i	0.278794	+	0.278794i
$603$	−76490.8	+	76490.8i	−0.210366	+	0.210366i
$604$		−	71524.7i		−	0.196057i
$605$		0			0
$606$	−25912.1			−0.0705597
$607$	−247253.	−	247253.i	−0.671064	−	0.671064i	0.286898	−	0.957961i	$-0.407376\pi$
$607$	−247253.	−	247253.i	−0.671064	−	0.671064i	−0.957961	+	0.286898i	$0.907376\pi$
$608$	−34843.0	+	34843.0i	−0.0942560	+	0.0942560i
$609$			274256.i			0.739472i
$610$		0			0
$611$	1.18196e6			3.16606
$612$	15339.2	+	15339.2i	0.0409543	+	0.0409543i
$613$	163758.	−	163758.i	0.435794	−	0.435794i	−0.454800	−	0.890594i	$-0.650289\pi$
$613$	163758.	−	163758.i	0.435794	−	0.435794i	0.890594	+	0.454800i	$0.150289\pi$
$614$		−	47844.5i		−	0.126910i
$615$		0			0
$616$	238108.			0.627499
$617$	−273782.	−	273782.i	−0.719176	−	0.719176i	0.249260	−	0.968437i	$-0.419812\pi$
$617$	−273782.	−	273782.i	−0.719176	−	0.719176i	−0.968437	+	0.249260i	$0.919812\pi$
$618$	35482.3	−	35482.3i	0.0929042	−	0.0929042i
$619$		−	428345.i		−	1.11792i	−0.829194	−	0.558962i	$-0.811200\pi$
$619$		−	428345.i		−	1.11792i	0.829194	−	0.558962i	$-0.188800\pi$
$620$		0			0
$621$	17396.9			0.0451116
$622$	−130569.	−	130569.i	−0.337489	−	0.337489i
$623$	−440465.	+	440465.i	−1.13484	+	1.13484i
$624$			105169.i			0.270097i
$625$		0			0
$626$	25673.8			0.0655151
$627$	−167467.	−	167467.i	−0.425986	−	0.425986i
$628$	144263.	−	144263.i	0.365793	−	0.365793i
$629$			29483.8i			0.0745216i
$630$		0			0
$631$	108870.			0.273432			0.136716	−	0.990610i	$-0.456345\pi$
$631$	108870.			0.273432			0.136716	+	0.990610i	$0.456345\pi$
$632$	−74667.1	−	74667.1i	−0.186937	−	0.186937i
$633$	56841.0	−	56841.0i	0.141858	−	0.141858i
$634$		−	88400.5i		−	0.219926i
$635$		0			0
$636$	81046.9			0.200365
$637$	346322.	+	346322.i	0.853495	+	0.853495i
$638$	281242.	−	281242.i	0.690937	−	0.690937i
$639$		−	144154.i		−	0.353040i
$640$		0			0
$641$	−501643.			−1.22090			−0.610448	−	0.792056i	$-0.709011\pi$
$641$	−501643.			−1.22090			−0.610448	+	0.792056i	$0.709011\pi$
$642$	−62888.6	−	62888.6i	−0.152581	−	0.152581i
$643$	117251.	−	117251.i	0.283592	−	0.283592i	−0.550948	−	0.834540i	$-0.685734\pi$
$643$	117251.	−	117251.i	0.283592	−	0.283592i	0.834540	+	0.550948i	$0.185734\pi$
$644$			62344.5i			0.150323i
$645$		0			0
$646$	77324.1			0.185289
$647$	362031.	+	362031.i	0.864843	+	0.864843i	0.991896	−	0.127053i	$-0.0405519\pi$
$647$	362031.	+	362031.i	0.864843	+	0.864843i	−0.127053	+	0.991896i	$0.540552\pi$
$648$	11664.0	−	11664.0i	0.0277778	−	0.0277778i
$649$		−	54836.8i		−	0.130192i
$650$		0			0
$651$	−131861.			−0.311139
$652$	54872.0	+	54872.0i	0.129079	+	0.129079i
$653$	411293.	−	411293.i	0.964552	−	0.964552i	−0.0348413	−	0.999393i	$-0.511093\pi$
$653$	411293.	−	411293.i	0.964552	−	0.964552i	0.999393	+	0.0348413i	$0.0110926\pi$
$654$			173711.i			0.406136i
$655$		0			0
$656$	88729.0			0.206185
$657$	185361.	+	185361.i	0.429426	+	0.429426i
$658$	469772.	−	469772.i	1.08501	−	1.08501i
$659$			291952.i			0.672265i	0.941815	+	0.336133i	$0.109119\pi$
$659$			291952.i			0.672265i	−0.941815	+	0.336133i	$0.890881\pi$
$660$		0			0
$661$	757270.			1.73320			0.866599	−	0.499006i	$-0.166301\pi$
$661$	757270.			1.73320			0.866599	+	0.499006i	$0.166301\pi$
$662$	−98217.4	−	98217.4i	−0.224116	−	0.224116i
$663$	116697.	−	116697.i	0.265480	−	0.265480i
$664$		−	131583.i		−	0.298445i
$665$		0			0
$666$	22419.6			0.0505452
$667$	73638.3	+	73638.3i	0.165521	+	0.165521i
$668$	−299514.	+	299514.i	−0.671219	+	0.671219i
$669$			211766.i			0.473156i
$670$		0			0
$671$	−704193.			−1.56404
$672$	41799.8	+	41799.8i	0.0925626	+	0.0925626i
$673$	−310335.	+	310335.i	−0.685173	+	0.685173i	−0.961161	−	0.275988i	$-0.910995\pi$
$673$	−310335.	+	310335.i	−0.685173	+	0.685173i	0.275988	+	0.961161i	$0.410995\pi$
$674$			571265.i			1.25753i
$675$		0			0
$676$	571613.			1.25086
$677$	−9203.97	−	9203.97i	−0.0200816	−	0.0200816i	0.696995	−	0.717076i	$-0.254520\pi$
$677$	−9203.97	−	9203.97i	−0.0200816	−	0.0200816i	−0.717076	+	0.696995i	$0.754520\pi$
$678$	−23817.9	+	23817.9i	−0.0518137	+	0.0518137i
$679$		−	950012.i		−	2.06058i
$680$		0			0
$681$	−6497.79			−0.0140111
$682$	135220.	+	135220.i	0.290717	+	0.290717i
$683$	421590.	−	421590.i	0.903750	−	0.903750i	−0.0920080	−	0.995758i	$-0.529329\pi$
$683$	421590.	−	421590.i	0.903750	−	0.903750i	0.995758	+	0.0920080i	$0.0293285\pi$
$684$		−	58797.6i		−	0.125675i
$685$		0			0
$686$	−151502.			−0.321937
$687$	−44371.4	−	44371.4i	−0.0940135	−	0.0940135i
$688$	36377.1	−	36377.1i	0.0768514	−	0.0768514i
$689$		−	616584.i		−	1.29883i
$690$		0			0
$691$	102275.			0.214197			0.107098	−	0.994248i	$-0.465844\pi$
$691$	102275.			0.214197			0.107098	+	0.994248i	$0.465844\pi$
$692$	261106.	+	261106.i	0.545261	+	0.545261i
$693$	−200904.	+	200904.i	−0.418333	+	0.418333i
$694$			85601.3i			0.177730i
$695$		0			0
$696$	98743.7			0.203841
$697$	−98454.3	−	98454.3i	−0.202660	−	0.202660i
$698$	68182.4	−	68182.4i	0.139946	−	0.139946i
$699$		−	181476.i		−	0.371420i
$700$		0			0
$701$	−685359.			−1.39471			−0.697353	−	0.716728i	$-0.745639\pi$
$701$	−685359.			−1.39471			−0.697353	+	0.716728i	$0.745639\pi$
$702$	−88736.7	−	88736.7i	−0.180065	−	0.180065i
$703$	56508.1	−	56508.1i	0.114341	−	0.114341i
$704$		−	85728.9i		−	0.172974i
$705$		0			0
$706$	−115.016			−0.000230753
$707$	78350.6	+	78350.6i	0.156749	+	0.156749i
$708$	9626.57	−	9626.57i	0.0192046	−	0.0192046i
$709$		−	164715.i		−	0.327673i	−0.986487	−	0.163837i	$-0.947613\pi$
$709$		−	164715.i		−	0.327673i	0.986487	−	0.163837i	$-0.0523870\pi$
$710$		0			0
$711$	126001.			0.249249
$712$	158586.	+	158586.i	0.312827	+	0.312827i
$713$	−35404.9	+	35404.9i	−0.0696441	+	0.0696441i
$714$		−	92762.6i		−	0.181960i
$715$		0			0
$716$	51169.5			0.0998126
$717$	177053.	+	177053.i	0.344402	+	0.344402i
$718$	−347885.	+	347885.i	−0.674818	+	0.674818i
$719$		−	782798.i		−	1.51423i	−0.653282	−	0.757115i	$-0.726608\pi$
$719$		−	782798.i		−	1.51423i	0.653282	−	0.757115i	$-0.273392\pi$
$720$		0			0
$721$	−214577.			−0.412774
$722$	112444.	+	112444.i	0.215706	+	0.215706i
$723$	−202760.	+	202760.i	−0.387887	+	0.387887i
$724$			392208.i			0.748236i
$725$		0			0
$726$	196864.			0.373503
$727$	−659110.	−	659110.i	−1.24706	−	1.24706i	−0.957010	−	0.290054i	$-0.906327\pi$
$727$	−659110.	−	659110.i	−1.24706	−	1.24706i	−0.290054	−	0.957010i	$-0.593673\pi$
$728$	318002.	−	318002.i	0.600021	−	0.600021i
$729$			19683.0i			0.0370370i
$730$		0			0
$731$	−80728.6			−0.151075
$732$	−123621.	−	123621.i	−0.230711	−	0.230711i
$733$	−356952.	+	356952.i	−0.664358	+	0.664358i	−0.956404	−	0.292046i	$-0.905664\pi$
$733$	−356952.	+	356952.i	−0.664358	+	0.664358i	0.292046	+	0.956404i	$0.405664\pi$
$734$			167816.i			0.311489i
$735$		0			0
$736$	22446.6			0.0414377
$737$	−474354.	−	474354.i	−0.873309	−	0.873309i
$738$	−74865.1	+	74865.1i	−0.137457	+	0.137457i
$739$			231725.i			0.424311i	0.977236	+	0.212156i	$0.0680483\pi$
$739$			231725.i			0.424311i	−0.977236	+	0.212156i	$0.931952\pi$
$740$		0			0
$741$	−447317.			−0.814665
$742$	−245062.	−	245062.i	−0.445111	−	0.445111i
$743$	35964.9	−	35964.9i	0.0651480	−	0.0651480i	−0.673782	−	0.738930i	$-0.735331\pi$
$743$	35964.9	−	35964.9i	0.0651480	−	0.0651480i	0.738930	+	0.673782i	$0.235331\pi$
$744$			47475.4i			0.0857676i
$745$		0			0
$746$	263501.			0.473483
$747$	111023.	+	111023.i	0.198963	+	0.198963i
$748$	−95125.3	+	95125.3i	−0.170017	+	0.170017i
$749$			380314.i			0.677920i
$750$		0			0
$751$	702045.			1.24476			0.622379	−	0.782716i	$-0.286166\pi$
$751$	702045.			1.24476			0.622379	+	0.782716i	$0.286166\pi$
$752$	−169137.	−	169137.i	−0.299091	−	0.299091i
$753$	−255976.	+	255976.i	−0.451449	+	0.451449i
$754$		−	751216.i		−	1.32136i
$755$		0			0
$756$	−70537.1			−0.123417
$757$	111880.	+	111880.i	0.195236	+	0.195236i	0.797954	−	0.602718i	$-0.205916\pi$
$757$	111880.	+	111880.i	0.195236	+	0.195236i	−0.602718	+	0.797954i	$0.705916\pi$
$758$	21673.9	−	21673.9i	0.0377224	−	0.0377224i
$759$			107886.i			0.187276i
$760$		0			0
$761$	−579015.			−0.999817			−0.499908	−	0.866078i	$-0.666633\pi$
$761$	−579015.			−0.999817			−0.499908	+	0.866078i	$0.666633\pi$
$762$	170080.	+	170080.i	0.292916	+	0.292916i
$763$	525252.	−	525252.i	0.902232	−	0.902232i
$764$			319592.i			0.547531i
$765$		0			0
$766$	−89310.0			−0.152210
$767$	−73236.4	−	73236.4i	−0.124491	−	0.124491i
$768$	15049.7	−	15049.7i	0.0255155	−	0.0255155i
$769$			472110.i			0.798344i	0.916876	+	0.399172i	$0.130702\pi$
$769$			472110.i			0.798344i	−0.916876	+	0.399172i	$0.869298\pi$
$770$		0			0
$771$	85348.4			0.143578
$772$	189993.	+	189993.i	0.318789	+	0.318789i
$773$	20353.7	−	20353.7i	0.0340631	−	0.0340631i	−0.689870	−	0.723933i	$-0.742332\pi$
$773$	20353.7	−	20353.7i	0.0340631	−	0.0340631i	0.723933	+	0.689870i	$0.242332\pi$
$774$			61386.4i			0.102469i
$775$		0			0
$776$	−342044.			−0.568013
$777$	−67790.5	−	67790.5i	−0.112286	−	0.112286i
$778$	407768.	−	407768.i	0.673680	−	0.673680i
$779$			377391.i			0.621895i
$780$		0			0
$781$	893963.			1.46561
$782$	−24906.9	−	24906.9i	−0.0407293	−	0.0407293i
$783$	−83315.0	+	83315.0i	−0.135894	+	0.135894i
$784$		−	99116.8i		−	0.161256i
$785$		0			0
$786$	964.457			0.00156113
$787$	−640630.	−	640630.i	−1.03433	−	1.03433i	−0.999390	−	0.0349373i	$-0.988877\pi$
$787$	−640630.	−	640630.i	−1.03433	−	1.03433i	−0.0349373	−	0.999390i	$-0.511123\pi$
$788$	−21800.7	+	21800.7i	−0.0351090	+	0.0351090i
$789$			637933.i			1.02476i
$790$		0			0
$791$	144037.			0.230208
$792$	72333.8	+	72333.8i	0.115316	+	0.115316i
$793$	−940474.	+	940474.i	−1.49555	+	1.49555i
$794$		−	469194.i		−	0.744238i
$795$		0			0
$796$	−505700.			−0.798118
$797$	−542782.	−	542782.i	−0.854494	−	0.854494i	0.136189	−	0.990683i	$-0.456515\pi$
$797$	−542782.	−	542782.i	−0.854494	−	0.854494i	−0.990683	+	0.136189i	$0.956515\pi$
$798$	−177787.	+	177787.i	−0.279186	+	0.279186i
$799$			375352.i			0.587956i
$800$		0			0
$801$	−267614.			−0.417103
$802$	86186.2	+	86186.2i	0.133995	+	0.133995i
$803$	−1.14951e6	+	1.14951e6i	−1.78271	+	1.78271i
$804$		−	166545.i		−	0.257644i
$805$		0			0
$806$	361181.			0.555974
$807$	343318.	+	343318.i	0.527168	+	0.527168i
$808$	28209.5	−	28209.5i	0.0432088	−	0.0432088i
$809$			1.09699e6i			1.67613i	0.545571	+	0.838065i	$0.316313\pi$
$809$			1.09699e6i			1.67613i	−0.545571	+	0.838065i	$0.683687\pi$
$810$		0			0
$811$	−711398.			−1.08161			−0.540805	−	0.841148i	$-0.681880\pi$
$811$	−711398.			−1.08161			−0.540805	+	0.841148i	$0.681880\pi$
$812$	−298572.	−	298572.i	−0.452832	−	0.452832i
$813$	−39889.3	+	39889.3i	−0.0603498	+	0.0603498i
$814$			139034.i			0.209833i
$815$		0			0
$816$	−33398.4			−0.0501586
$817$	154723.	+	154723.i	0.231798	+	0.231798i
$818$	−372179.	+	372179.i	−0.556219	+	0.556219i
$819$			536628.i			0.800028i
$820$		0			0
$821$	−566243.			−0.840071			−0.420036	−	0.907508i	$-0.637982\pi$
$821$	−566243.			−0.840071			−0.420036	+	0.907508i	$0.637982\pi$
$822$	−8515.98	−	8515.98i	−0.0126035	−	0.0126035i
$823$	678523.	−	678523.i	1.00176	−	1.00176i	0.00176504	−	0.999998i	$-0.499438\pi$
$823$	678523.	−	678523.i	1.00176	−	1.00176i	0.999998	−	0.00176504i	$-0.000561830\pi$
$824$			77256.6i			0.113784i
$825$		0			0
$826$	−58215.9			−0.0853261
$827$	822466.	+	822466.i	1.20256	+	1.20256i	0.973385	+	0.229174i	$0.0736026\pi$
$827$	822466.	+	822466.i	1.20256	+	1.20256i	0.229174	+	0.973385i	$0.426397\pi$
$828$	−18939.3	+	18939.3i	−0.0276251	+	0.0276251i
$829$		−	1.26805e6i		−	1.84514i	−0.385832	−	0.922569i	$-0.626086\pi$
$829$		−	1.26805e6i		−	1.84514i	0.385832	−	0.922569i	$-0.373914\pi$
$830$		0			0
$831$	−479228.			−0.693969
$832$	−114494.	−	114494.i	−0.165400	−	0.165400i
$833$	−109981.	+	109981.i	−0.158499	+	0.158499i
$834$			531415.i			0.764015i
$835$		0			0
$836$	364631.			0.521724
$837$	−40057.4	−	40057.4i	−0.0571784	−	0.0571784i
$838$	203397.	−	203397.i	0.289639	−	0.289639i
$839$		−	487091.i		−	0.691968i	−0.938240	−	0.345984i	$-0.887545\pi$
$839$		−	487091.i		−	0.691968i	0.938240	−	0.345984i	$-0.112455\pi$
$840$		0			0
$841$	1962.99			0.00277541
$842$	−45065.9	−	45065.9i	−0.0635659	−	0.0635659i
$843$	−31853.2	+	31853.2i	−0.0448227	+	0.0448227i
$844$			123761.i			0.173740i
$845$		0			0
$846$	285419.			0.398789
$847$	−595260.	−	595260.i	−0.829736	−	0.829736i
$848$	−88232.7	+	88232.7i	−0.122698	+	0.122698i
$849$		−	174504.i		−	0.242098i
$850$		0			0
$851$	−36403.7			−0.0502675
$852$	156935.	+	156935.i	0.216192	+	0.216192i
$853$	−10645.4	+	10645.4i	−0.0146307	+	0.0146307i	−0.714384	−	0.699754i	$-0.753293\pi$
$853$	−10645.4	+	10645.4i	−0.0146307	+	0.0146307i	0.699754	+	0.714384i	$0.253293\pi$
$854$			747586.i			1.02505i
$855$		0			0
$856$	136929.			0.186873
$857$	692563.	+	692563.i	0.942970	+	0.942970i	0.998459	−	0.0554896i	$-0.0176720\pi$
$857$	692563.	+	692563.i	0.942970	+	0.942970i	−0.0554896	+	0.998459i	$0.517672\pi$
$858$	550296.	−	550296.i	0.747519	−	0.747519i
$859$			745974.i			1.01097i	0.862836	+	0.505484i	$0.168686\pi$
$859$			745974.i			1.01097i	−0.862836	+	0.505484i	$0.831314\pi$
$860$		0			0
$861$	452741.			0.610722
$862$	421196.	+	421196.i	0.566852	+	0.566852i
$863$	−259362.	+	259362.i	−0.348245	+	0.348245i	−0.859456	−	0.511211i	$-0.829197\pi$
$863$	−259362.	+	259362.i	−0.348245	+	0.348245i	0.511211	+	0.859456i	$0.329197\pi$
$864$			25396.3i			0.0340207i
$865$		0			0
$866$	−493621.			−0.658201
$867$	−269817.	−	269817.i	−0.358947	−	0.358947i
$868$	143552.	−	143552.i	0.190533	−	0.190533i
$869$			781388.i			1.03473i
$870$		0			0
$871$	−1.26703e6			−1.67013
$872$	−189113.	−	189113.i	−0.248707	−	0.248707i
$873$	288600.	−	288600.i	0.378676	−	0.378676i
$874$			95472.3i			0.124984i
$875$		0			0
$876$	−403592.			−0.525938
$877$	191986.	+	191986.i	0.249615	+	0.249615i	0.820813	−	0.571197i	$-0.193521\pi$
$877$	191986.	+	191986.i	0.249615	+	0.249615i	−0.571197	+	0.820813i	$0.693521\pi$
$878$	−530360.	+	530360.i	−0.687989	+	0.687989i
$879$		−	404733.i		−	0.523830i
$880$		0			0
$881$	1.08074e6			1.39241			0.696207	−	0.717841i	$-0.254870\pi$
$881$	1.08074e6			1.39241			0.696207	+	0.717841i	$0.254870\pi$
$882$	83629.8	+	83629.8i	0.107504	+	0.107504i
$883$	−294604.	+	294604.i	−0.377848	+	0.377848i	−0.870325	−	0.492477i	$-0.836091\pi$
$883$	−294604.	+	294604.i	−0.377848	+	0.377848i	0.492477	+	0.870325i	$0.336091\pi$
$884$			254086.i			0.325145i
$885$		0			0
$886$	399264.			0.508619
$887$	373328.	+	373328.i	0.474507	+	0.474507i	0.903370	−	0.428862i	$-0.141085\pi$
$887$	373328.	+	373328.i	0.474507	+	0.474507i	−0.428862	+	0.903370i	$0.641085\pi$
$888$	−24407.4	+	24407.4i	−0.0309525	+	0.0309525i
$889$		−	1.02854e6i		−	1.30143i
$890$		0			0
$891$	−122063.			−0.153755
$892$	−230542.	−	230542.i	−0.289748	−	0.289748i
$893$	719392.	−	719392.i	0.902117	−	0.902117i
$894$		−	31509.3i		−	0.0394243i
$895$		0			0
$896$	−91011.7			−0.113366
$897$	144086.	+	144086.i	0.179075	+	0.179075i
$898$	−365869.	+	365869.i	−0.453704	+	0.453704i
$899$		−	339113.i		−	0.419590i
$900$		0			0
$901$	195807.			0.241201
$902$	−464273.	−	464273.i	−0.570637	−	0.570637i
$903$	185615.	−	185615.i	0.227634	−	0.227634i
$904$		−	51859.3i		−	0.0634585i
$905$		0			0
$906$	−131399.			−0.160080
$907$	342694.	+	342694.i	0.416574	+	0.416574i	0.884021	−	0.467447i	$-0.154826\pi$
$907$	342694.	+	342694.i	0.416574	+	0.416574i	−0.467447	+	0.884021i	$0.654826\pi$
$908$	7073.89	−	7073.89i	0.00857999	−	0.00857999i
$909$			47603.5i			0.0576118i
$910$		0			0
$911$	653388.			0.787290			0.393645	−	0.919263i	$-0.371214\pi$
$911$	653388.			0.787290			0.393645	+	0.919263i	$0.371214\pi$
$912$	64010.7	+	64010.7i	0.0769597	+	0.0769597i
$913$	−688506.	+	688506.i	−0.825974	+	0.825974i
$914$		−	589989.i		−	0.706239i
$915$		0			0
$916$	96611.0			0.115143
$917$	−2916.24	−	2916.24i	−0.00346804	−	0.00346804i
$918$	28179.9	−	28179.9i	0.0334391	−	0.0334391i
$919$		−	316752.i		−	0.375050i	−0.982260	−	0.187525i	$-0.939953\pi$
$919$		−	316752.i		−	0.375050i	0.982260	−	0.187525i	$-0.0600465\pi$
$920$		0			0
$921$	−87896.0			−0.103621
$922$	−709825.	−	709825.i	−0.835006	−	0.835006i
$923$	1.19392e6	−	1.19392e6i	1.40143	−	1.40143i
$924$		−	437433.i		−	0.512351i
$925$		0			0
$926$	−729901.			−0.851221
$927$	−65185.2	−	65185.2i	−0.0758560	−	0.0758560i
$928$	−107499.	+	107499.i	−0.124826	+	0.124826i
$929$		−	1.39646e6i		−	1.61807i	−0.587759	−	0.809036i	$-0.699990\pi$
$929$		−	1.39646e6i		−	1.61807i	0.587759	−	0.809036i	$-0.300010\pi$
$930$		0			0
$931$	421574.			0.486378
$932$	197566.	+	197566.i	0.227447	+	0.227447i
$933$	−239870.	+	239870.i	−0.275558	+	0.275558i
$934$		−	17393.2i		−	0.0199382i
$935$		0			0
$936$	193208.			0.220533
$937$	−81884.2	−	81884.2i	−0.0932655	−	0.0932655i	0.658935	−	0.752200i	$-0.271007\pi$
$937$	−81884.2	−	81884.2i	−0.0932655	−	0.0932655i	−0.752200	+	0.658935i	$0.771007\pi$
$938$	−503585.	+	503585.i	−0.572357	+	0.572357i
$939$		−	47165.7i		−	0.0534928i
$940$		0			0
$941$	1.67094e6			1.88705			0.943524	−	0.331305i	$-0.107489\pi$
$941$	1.67094e6			1.88705			0.943524	+	0.331305i	$0.107489\pi$
$942$	−265028.	−	265028.i	−0.298669	−	0.298669i
$943$	121562.	−	121562.i	0.136702	−	0.136702i
$944$			20960.2i			0.0235207i
$945$		0			0
$946$	−380685.			−0.425386
$947$	−689602.	−	689602.i	−0.768950	−	0.768950i	0.208971	−	0.977922i	$-0.432988\pi$
$947$	−689602.	−	689602.i	−0.768950	−	0.768950i	−0.977922	+	0.208971i	$0.932988\pi$
$948$	−137172.	+	137172.i	−0.152633	+	0.152633i
$949$			3.07042e6i			3.40930i
$950$		0			0
$951$	−162402.			−0.179569
$952$	100987.	+	100987.i	0.111427	+	0.111427i
$953$	−150337.	+	150337.i	−0.165532	+	0.165532i	−0.785012	−	0.619480i	$-0.787343\pi$
$953$	−150337.	+	150337.i	−0.165532	+	0.165532i	0.619480	+	0.785012i	$0.287343\pi$
$954$		−	148893.i		−	0.163597i
$955$		0			0
$956$	−385502.			−0.421804
$957$	−516674.	−	516674.i	−0.564148	−	0.564148i
$958$	154692.	−	154692.i	0.168553	−	0.168553i
$959$			51499.7i			0.0559974i
$960$		0			0
$961$	−760477.			−0.823454
$962$	185685.	+	185685.i	0.200644	+	0.200644i
$963$	−115534.	+	115534.i	−0.124582	+	0.124582i
$964$		−	441473.i		−	0.475062i
$965$		0			0
$966$	114534.			0.122739
$967$	312723.	+	312723.i	0.334432	+	0.334432i	0.854267	−	0.519835i	$-0.174007\pi$
$967$	312723.	+	312723.i	0.334432	+	0.334432i	−0.519835	+	0.854267i	$0.674007\pi$
$968$	−214319.	+	214319.i	−0.228723	+	0.228723i
$969$		−	142053.i		−	0.151288i
$970$		0			0
$971$	1.16534e6			1.23598			0.617992	−	0.786185i	$-0.287946\pi$
$971$	1.16534e6			1.23598			0.617992	+	0.786185i	$0.287946\pi$
$972$	−21428.1	−	21428.1i	−0.0226805	−	0.0226805i
$973$	1.60685e6	−	1.60685e6i	1.69726	−	1.69726i
$974$		−	308903.i		−	0.325614i
$975$		0			0
$976$	269162.			0.282562
$977$	748461.	+	748461.i	0.784115	+	0.784115i	0.980522	−	0.196407i	$-0.0629274\pi$
$977$	748461.	+	748461.i	0.784115	+	0.784115i	−0.196407	+	0.980522i	$0.562927\pi$
$978$	100806.	−	100806.i	0.105393	−	0.105393i
$979$		−	1.65959e6i		−	1.73156i
$980$		0			0
$981$	319127.			0.331609
$982$	680434.	+	680434.i	0.705607	+	0.705607i
$983$	542812.	−	542812.i	0.561749	−	0.561749i	−0.368055	−	0.929804i	$-0.619976\pi$
$983$	542812.	−	542812.i	0.561749	−	0.561749i	0.929804	+	0.368055i	$0.119976\pi$
$984$		−	163006.i		−	0.168350i
$985$		0			0
$986$	238562.			0.245385
$987$	−863025.	−	863025.i	−0.885909	−	0.885909i
$988$	486977.	−	486977.i	0.498878	−	0.498878i
$989$		−	99675.8i		−	0.101905i
$990$		0			0
$991$	1.08269e6			1.10245			0.551224	−	0.834357i	$-0.314161\pi$
$991$	1.08269e6			1.10245			0.551224	+	0.834357i	$0.314161\pi$
$992$	−51684.7	−	51684.7i	−0.0525217	−	0.0525217i
$993$	−180437.	+	180437.i	−0.182990	+	0.182990i
$994$		−	949050.i		−	0.960542i
$995$		0			0
$996$	−241734.			−0.243679
$997$	−25593.2	−	25593.2i	−0.0257475	−	0.0257475i	0.694116	−	0.719863i	$-0.255796\pi$
$997$	−25593.2	−	25593.2i	−0.0257475	−	0.0257475i	−0.719863	+	0.694116i	$0.755796\pi$
$998$	662047.	−	662047.i	0.664703	−	0.664703i
$999$		−	41187.5i		−	0.0412700i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.5.f.e.7.2		4
3.2	odd	2		450.5.g.f.307.1		4
5.2	odd	4		30.5.f.a.13.1	yes	4
5.3	odd	4	inner	150.5.f.e.43.2		4
5.4	even	2		30.5.f.a.7.1	✓	4
15.2	even	4		90.5.g.e.73.1		4
15.8	even	4		450.5.g.f.343.1		4
15.14	odd	2		90.5.g.e.37.1		4
20.7	even	4		240.5.bg.b.193.2		4
20.19	odd	2		240.5.bg.b.97.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.5.f.a.7.1	✓	4	5.4	even	2
30.5.f.a.13.1	yes	4	5.2	odd	4
90.5.g.e.37.1		4	15.14	odd	2
90.5.g.e.73.1		4	15.2	even	4
150.5.f.e.7.2		4	1.1	even	1	trivial
150.5.f.e.43.2		4	5.3	odd	4	inner
240.5.bg.b.97.2		4	20.19	odd	2
240.5.bg.b.193.2		4	20.7	even	4
450.5.g.f.307.1		4	3.2	odd	2
450.5.g.f.343.1		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	150.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 2.00000i) q^{2} +(3.67423 - 3.67423i) q^{3} +8.00000i q^{4} +14.6969 q^{6} +(-44.4393 - 44.4393i) q^{7} +(-16.0000 + 16.0000i) q^{8} -27.0000i q^{9} +167.439 q^{11} +(29.3939 + 29.3939i) q^{12} +(223.621 - 223.621i) q^{13} -177.757i q^{14} -64.0000 q^{16} +(71.0148 + 71.0148i) q^{17} +(54.0000 - 54.0000i) q^{18} -272.211i q^{19} -326.561 q^{21} +(334.879 + 334.879i) q^{22} +(-87.6821 + 87.6821i) q^{23} +117.576i q^{24} +894.484 q^{26} +(-99.2043 - 99.2043i) q^{27} +(355.514 - 355.514i) q^{28} -839.832i q^{29} +403.787 q^{31} +(-128.000 - 128.000i) q^{32} +(615.211 - 615.211i) q^{33} +284.059i q^{34} +216.000 q^{36} +(207.589 + 207.589i) q^{37} +(544.422 - 544.422i) q^{38} -1643.27i q^{39} -1386.39 q^{41} +(-653.121 - 653.121i) q^{42} +(-568.393 + 568.393i) q^{43} +1339.51i q^{44} -350.729 q^{46} +(2642.77 + 2642.77i) q^{47} +(-235.151 + 235.151i) q^{48} +1548.70i q^{49} +521.850 q^{51} +(1788.97 + 1788.97i) q^{52} +(1378.64 - 1378.64i) q^{53} -396.817i q^{54} +1422.06 q^{56} +(-1000.17 - 1000.17i) q^{57} +(1679.66 - 1679.66i) q^{58} -327.503i q^{59} -4205.66 q^{61} +(807.573 + 807.573i) q^{62} +(-1199.86 + 1199.86i) q^{63} -512.000i q^{64} +2460.84 q^{66} +(-2832.99 - 2832.99i) q^{67} +(-568.118 + 568.118i) q^{68} +644.330i q^{69} +5339.03 q^{71} +(432.000 + 432.000i) q^{72} +(-6865.24 + 6865.24i) q^{73} +830.357i q^{74} +2177.69 q^{76} +(-7440.88 - 7440.88i) q^{77} +(3286.54 - 3286.54i) q^{78} +4666.69i q^{79} -729.000 q^{81} +(-2772.78 - 2772.78i) q^{82} +(-4111.98 + 4111.98i) q^{83} -2612.49i q^{84} -2273.57 q^{86} +(-3085.74 - 3085.74i) q^{87} +(-2679.03 + 2679.03i) q^{88} -9911.62i q^{89} -19875.1 q^{91} +(-701.457 - 701.457i) q^{92} +(1483.61 - 1483.61i) q^{93} +10571.1i q^{94} -940.604 q^{96} +(10688.9 + 10688.9i) q^{97} +(-3097.40 + 3097.40i) q^{98} -4520.86i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 28 q^{7} - 64 q^{8} + 464 q^{11} + 336 q^{13} - 256 q^{16} - 392 q^{17} + 216 q^{18} - 1512 q^{21} + 928 q^{22} - 968 q^{23} + 1344 q^{26} - 224 q^{28} - 560 q^{31} - 512 q^{32} + 756 q^{33}+ \cdots - 23912 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 28 * q^7 - 64 * q^8 + 464 * q^11 + 336 * q^13 - 256 * q^16 - 392 * q^17 + 216 * q^18 - 1512 * q^21 + 928 * q^22 - 968 * q^23 + 1344 * q^26 - 224 * q^28 - 560 * q^31 - 512 * q^32 + 756 * q^33 + 864 * q^36 - 2256 * q^37 - 1232 * q^38 + 392 * q^41 - 3024 * q^42 - 216 * q^43 - 3872 * q^46 + 9072 * q^47 + 4968 * q^51 + 2688 * q^52 + 4280 * q^53 - 896 * q^56 - 6264 * q^57 + 2192 * q^58 - 4536 * q^61 - 1120 * q^62 + 756 * q^63 + 3024 * q^66 + 2248 * q^67 + 3136 * q^68 + 18064 * q^71 + 1728 * q^72 - 20524 * q^73 - 4928 * q^76 - 7336 * q^77 + 8208 * q^78 - 2916 * q^81 + 784 * q^82 + 336 * q^83 - 864 * q^86 - 8316 * q^87 - 7424 * q^88 - 52752 * q^91 - 7744 * q^92 + 7992 * q^93 + 40404 * q^97 - 23912 * q^98

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