LMFDB - Embedded newform 450.5.g.f.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,5,Mod(307,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.307");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	450.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$46.5164833877$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			307.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	450.307
Dual form			450.5.g.f.343.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(-44.4393 - 44.4393i) q^{7} +(16.0000 - 16.0000i) q^{8} +O(q^{10})$	\(q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(-44.4393 - 44.4393i) q^{7} +(16.0000 - 16.0000i) q^{8} -167.439 q^{11} +(223.621 - 223.621i) q^{13} +177.757i q^{14} -64.0000 q^{16} +(-71.0148 - 71.0148i) q^{17} -272.211i q^{19} +(334.879 + 334.879i) q^{22} +(87.6821 - 87.6821i) q^{23} -894.484 q^{26} +(355.514 - 355.514i) q^{28} +839.832i q^{29} +403.787 q^{31} +(128.000 + 128.000i) q^{32} +284.059i q^{34} +(207.589 + 207.589i) q^{37} +(-544.422 + 544.422i) q^{38} +1386.39 q^{41} +(-568.393 + 568.393i) q^{43} -1339.51i q^{44} -350.729 q^{46} +(-2642.77 - 2642.77i) q^{47} +1548.70i q^{49} +(1788.97 + 1788.97i) q^{52} +(-1378.64 + 1378.64i) q^{53} -1422.06 q^{56} +(1679.66 - 1679.66i) q^{58} +327.503i q^{59} -4205.66 q^{61} +(-807.573 - 807.573i) q^{62} -512.000i q^{64} +(-2832.99 - 2832.99i) q^{67} +(568.118 - 568.118i) q^{68} -5339.03 q^{71} +(-6865.24 + 6865.24i) q^{73} -830.357i q^{74} +2177.69 q^{76} +(7440.88 + 7440.88i) q^{77} +4666.69i q^{79} +(-2772.78 - 2772.78i) q^{82} +(4111.98 - 4111.98i) q^{83} +2273.57 q^{86} +(-2679.03 + 2679.03i) q^{88} +9911.62i q^{89} -19875.1 q^{91} +(701.457 + 701.457i) q^{92} +10571.1i q^{94} +(10688.9 + 10688.9i) q^{97} +(3097.40 - 3097.40i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 28 q^{7} + 64 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 28 * q^7 + 64 * q^8	$ 4 q - 8 q^{2} + 28 q^{7} + 64 q^{8} - 464 q^{11} + 336 q^{13} - 256 q^{16} + 392 q^{17} + 928 q^{22} + 968 q^{23} - 1344 q^{26} - 224 q^{28} - 560 q^{31} + 512 q^{32} - 2256 q^{37} + 1232 q^{38} - 392 q^{41} - 216 q^{43} - 3872 q^{46} - 9072 q^{47} + 2688 q^{52} - 4280 q^{53} + 896 q^{56} + 2192 q^{58} - 4536 q^{61} + 1120 q^{62} + 2248 q^{67} - 3136 q^{68} - 18064 q^{71} - 20524 q^{73} - 4928 q^{76} + 7336 q^{77} + 784 q^{82} - 336 q^{83} + 864 q^{86} - 7424 q^{88} - 52752 q^{91} + 7744 q^{92} + 40404 q^{97} + 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 + 928 * q^22 + 968 * q^23 - 1344 * q^26 - 224 * q^28 - 560 * q^31 + 512 * q^32 - 2256 * q^37 + 1232 * q^38 - 392 * q^41 - 216 * q^43 - 3872 * q^46 - 9072 * q^47 + 2688 * q^52 - 4280 * q^53 + 896 * q^56 + 2192 * q^58 - 4536 * q^61 + 1120 * q^62 + 2248 * q^67 - 3136 * q^68 - 18064 * q^71 - 20524 * q^73 - 4928 * q^76 + 7336 * q^77 + 784 * q^82 - 336 * q^83 + 864 * q^86 - 7424 * q^88 - 52752 * q^91 + 7744 * q^92 + 40404 * q^97 + 23912 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

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

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
$3$		0			0
$3$		0			0
$4$			8.00000i			0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−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
$9$		0			0
$10$		0			0
$11$	−167.439			−1.38380			−0.691898	−	0.721995i	$-0.743225\pi$
$11$	−167.439			−1.38380			−0.691898	+	0.721995i	$0.743225\pi$
$12$		0			0
$13$	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	0.573488	−	0.819214i	$-0.305590\pi$
$17$	−71.0148	−	71.0148i	−0.245726	−	0.245726i	−0.819214	+	0.573488i	$0.805590\pi$
$18$		0			0
$19$		−	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
$21$		0			0
$22$	334.879	+	334.879i	0.691898	+	0.691898i
$23$	87.6821	−	87.6821i	0.165751	−	0.165751i	−0.619358	−	0.785109i	$-0.712607\pi$
$23$	87.6821	−	87.6821i	0.165751	−	0.165751i	0.785109	+	0.619358i	$0.212607\pi$
$24$		0			0
$25$		0			0
$26$	−894.484			−1.32320
$27$		0			0
$28$	355.514	−	355.514i	0.453462	−	0.453462i
$29$			839.832i			0.998611i	0.866426	+	0.499306i	$0.166412\pi$
$29$			839.832i			0.998611i	−0.866426	+	0.499306i	$0.833588\pi$
$30$		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
$33$		0			0
$34$			284.059i			0.245726i
$35$		0			0
$36$		0			0
$37$	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
$39$		0			0
$40$		0			0
$41$	1386.39			0.824742			0.412371	−	0.911016i	$-0.364701\pi$
$41$	1386.39			0.824742			0.412371	+	0.911016i	$0.364701\pi$
$42$		0			0
$43$	−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	$-0.929029\pi$
$47$	−2642.77	−	2642.77i	−1.19637	−	1.19637i	−0.221119	−	0.975247i	$-0.570971\pi$
$48$		0			0
$49$			1548.70i			0.645023i
$50$		0			0
$51$		0			0
$52$	1788.97	+	1788.97i	0.661600	+	0.661600i
$53$	−1378.64	+	1378.64i	−0.490792	+	0.490792i	−0.908556	−	0.417763i	$-0.862814\pi$
$53$	−1378.64	+	1378.64i	−0.490792	+	0.490792i	0.417763	+	0.908556i	$0.362814\pi$
$54$		0			0
$55$		0			0
$56$	−1422.06			−0.453462
$57$		0			0
$58$	1679.66	−	1679.66i	0.499306	−	0.499306i
$59$			327.503i			0.0940829i	0.998893	+	0.0470415i	$0.0149793\pi$
$59$			327.503i			0.0940829i	−0.998893	+	0.0470415i	$0.985021\pi$
$60$		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
$63$		0			0
$64$		−	512.000i		−	0.125000i
$65$		0			0
$66$		0			0
$67$	−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
$69$		0			0
$70$		0			0
$71$	−5339.03			−1.05912			−0.529560	−	0.848272i	$-0.677643\pi$
$71$	−5339.03			−1.05912			−0.529560	+	0.848272i	$0.677643\pi$
$72$		0			0
$73$	−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
$78$		0			0
$79$			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
$81$		0			0
$82$	−2772.78	−	2772.78i	−0.412371	−	0.412371i
$83$	4111.98	−	4111.98i	0.596890	−	0.596890i	−0.342594	−	0.939484i	$-0.611305\pi$
$83$	4111.98	−	4111.98i	0.596890	−	0.596890i	0.939484	+	0.342594i	$0.111305\pi$
$84$		0			0
$85$		0			0
$86$	2273.57			0.307406
$87$		0			0
$88$	−2679.03	+	2679.03i	−0.345949	+	0.345949i
$89$			9911.62i			1.25131i	0.780100	+	0.625654i	$0.215168\pi$
$89$			9911.62i			1.25131i	−0.780100	+	0.625654i	$0.784832\pi$
$90$		0			0
$91$	−19875.1			−2.40009
$92$	701.457	+	701.457i	0.0828754	+	0.0828754i
$93$		0			0
$94$			10571.1i			1.19637i
$95$		0			0
$96$		0			0
$97$	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
$99$		0			0
$100$		0			0
$101$	1763.09			0.172835			0.0864177	−	0.996259i	$-0.472458\pi$
$101$	1763.09			0.172835			0.0864177	+	0.996259i	$0.472458\pi$
$102$		0			0
$103$	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	0.868840	−	0.495093i	$-0.164866\pi$
$107$	4279.03	+	4279.03i	0.373747	+	0.373747i	−0.495093	+	0.868840i	$0.664866\pi$
$108$		0			0
$109$			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
$111$		0			0
$112$	2844.11	+	2844.11i	0.226731	+	0.226731i
$113$	1620.60	−	1620.60i	0.126917	−	0.126917i	−0.640795	−	0.767712i	$-0.721395\pi$
$113$	1620.60	−	1620.60i	0.126917	−	0.126917i	0.767712	+	0.640795i	$0.221395\pi$
$114$		0			0
$115$		0			0
$116$	−6718.66			−0.499306
$117$		0			0
$118$	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
$123$		0			0
$124$			3230.29i			0.210087i
$125$		0			0
$126$		0			0
$127$	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
$129$		0			0
$130$		0			0
$131$	−65.6230			−0.00382396			−0.00191198	−	0.999998i	$-0.500609\pi$
$131$	−65.6230			−0.00382396			−0.00191198	+	0.999998i	$0.500609\pi$
$132$		0			0
$133$	−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	0.722374	−	0.691502i	$-0.243051\pi$
$137$	579.439	+	579.439i	0.0308721	+	0.0308721i	−0.691502	+	0.722374i	$0.743051\pi$
$138$		0			0
$139$			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
$141$		0			0
$142$	10678.1	+	10678.1i	0.529560	+	0.529560i
$143$	−37442.9	+	37442.9i	−1.83104	+	1.83104i
$144$		0			0
$145$		0			0
$146$	27461.0			1.28828
$147$		0			0
$148$	−1660.71	+	1660.71i	−0.0758178	+	0.0758178i
$149$			2143.94i			0.0965695i	0.998834	+	0.0482847i	$0.0153755\pi$
$149$			2143.94i			0.0965695i	−0.998834	+	0.0482847i	$0.984625\pi$
$150$		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
$153$		0			0
$154$		−	29763.5i		−	1.25500i
$155$		0			0
$156$		0			0
$157$	−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
$159$		0			0
$160$		0			0
$161$	−7793.06			−0.300647
$162$		0			0
$163$	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	$-0.851848\pi$
$167$	−37439.2	−	37439.2i	−1.34244	−	1.34244i	−0.448811	−	0.893627i	$-0.648152\pi$
$168$		0			0
$169$		−	71451.6i		−	2.50172i
$170$		0			0
$171$		0			0
$172$	−4547.14	−	4547.14i	−0.153703	−	0.153703i
$173$	−32638.2	+	32638.2i	−1.09052	+	1.09052i	−0.0950500	+	0.995472i	$0.530301\pi$
$173$	−32638.2	+	32638.2i	−1.09052	+	1.09052i	−0.995472	+	0.0950500i	$0.969699\pi$
$174$		0			0
$175$		0			0
$176$	10716.1			0.345949
$177$		0			0
$178$	19823.2	−	19823.2i	0.625654	−	0.625654i
$179$			6396.19i			0.199625i	0.995006	+	0.0998126i	$0.0318243\pi$
$179$			6396.19i			0.199625i	−0.995006	+	0.0998126i	$0.968176\pi$
$180$		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
$183$		0			0
$184$		−	2805.83i		−	0.0828754i
$185$		0			0
$186$		0			0
$187$	11890.7	+	11890.7i	0.340034	+	0.340034i
$188$	21142.2	−	21142.2i	0.598183	−	0.598183i
$189$		0			0
$190$		0			0
$191$	−39949.0			−1.09506			−0.547531	−	0.836785i	$-0.684432\pi$
$191$	−39949.0			−1.09506			−0.547531	+	0.836785i	$0.684432\pi$
$192$		0			0
$193$	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	0.671126	−	0.741344i	$-0.265811\pi$
$197$	−2725.09	−	2725.09i	−0.0702181	−	0.0702181i	−0.741344	+	0.671126i	$0.765811\pi$
$198$		0			0
$199$			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
$201$		0			0
$202$	−3526.19	−	3526.19i	−0.0864177	−	0.0864177i
$203$	37321.5	−	37321.5i	0.905665	−	0.905665i
$204$		0			0
$205$		0			0
$206$	−9657.07			−0.227568
$207$		0			0
$208$	−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
$213$		0			0
$214$		−	17116.1i		−	0.373747i
$215$		0			0
$216$		0			0
$217$	−17944.0	−	17944.0i	−0.381065	−	0.381065i
$218$	23639.1	−	23639.1i	0.497413	−	0.497413i
$219$		0			0
$220$		0			0
$221$	−31760.8			−0.650289
$222$		0			0
$223$	−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	0.715635	−	0.698475i	$-0.246138\pi$
$227$	884.237	+	884.237i	0.0171600	+	0.0171600i	−0.698475	+	0.715635i	$0.746138\pi$
$228$		0			0
$229$		−	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
$231$		0			0
$232$	13437.3	+	13437.3i	0.249653	+	0.249653i
$233$	−24695.8	+	24695.8i	−0.454894	+	0.454894i	−0.896975	−	0.442081i	$-0.854240\pi$
$233$	−24695.8	+	24695.8i	−0.454894	+	0.454894i	0.442081	+	0.896975i	$0.354240\pi$
$234$		0			0
$235$		0			0
$236$	−2620.02			−0.0470415
$237$		0			0
$238$	12623.4	−	12623.4i	0.222855	−	0.222855i
$239$		−	48187.8i		−	0.843608i	−0.906687	−	0.421804i	$-0.861397\pi$
$239$		−	48187.8i		−	0.843608i	0.906687	−	0.421804i	$-0.138603\pi$
$240$		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
$243$		0			0
$244$		−	33645.3i		−	0.565125i
$245$		0			0
$246$		0			0
$247$	−60872.1	−	60872.1i	−0.997756	−	0.997756i
$248$	6460.59	−	6460.59i	0.105043	−	0.105043i
$249$		0			0
$250$		0			0
$251$	69667.8			1.10582			0.552910	−	0.833241i	$-0.313517\pi$
$251$	69667.8			1.10582			0.552910	+	0.833241i	$0.313517\pi$
$252$		0			0
$253$	−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	0.613696	−	0.789542i	$-0.289682\pi$
$257$	−11614.4	−	11614.4i	−0.175846	−	0.175846i	−0.789542	+	0.613696i	$0.789682\pi$
$258$		0			0
$259$		−	18450.2i		−	0.275044i
$260$		0			0
$261$		0			0
$262$	131.246	+	131.246i	0.00191198	+	0.00191198i
$263$	86811.6	−	86811.6i	1.25507	−	1.25507i	0.301646	−	0.953420i	$-0.402464\pi$
$263$	86811.6	−	86811.6i	1.25507	−	1.25507i	0.953420	−	0.301646i	$-0.0975360\pi$
$264$		0			0
$265$		0			0
$266$	48387.5			0.683864
$267$		0			0
$268$	22663.9	−	22663.9i	0.315548	−	0.315548i
$269$		−	93439.3i		−	1.29129i	−0.763636	−	0.645647i	$-0.776588\pi$
$269$		−	93439.3i		−	1.29129i	0.763636	−	0.645647i	$-0.223412\pi$
$270$		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
$273$		0			0
$274$		−	2317.76i		−	0.0308721i
$275$		0			0
$276$		0			0
$277$	−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
$279$		0			0
$280$		0			0
$281$	8669.34			0.109793			0.0548963	−	0.998492i	$-0.482517\pi$
$281$	8669.34			0.109793			0.0548963	+	0.998492i	$0.482517\pi$
$282$		0			0
$283$	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
$288$		0			0
$289$		−	73434.8i		−	0.879238i
$290$		0			0
$291$		0			0
$292$	−54921.9	−	54921.9i	−0.644139	−	0.644139i
$293$	−55077.2	+	55077.2i	−0.641559	+	0.641559i	−0.950939	−	0.309380i	$-0.899879\pi$
$293$	−55077.2	+	55077.2i	−0.641559	+	0.641559i	0.309380	+	0.950939i	$0.399879\pi$
$294$		0			0
$295$		0			0
$296$	6642.86			0.0758178
$297$		0			0
$298$	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
$303$		0			0
$304$			17421.5i			0.188512i
$305$		0			0
$306$		0			0
$307$	−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
$309$		0			0
$310$		0			0
$311$	65284.5			0.674977			0.337489	−	0.941330i	$-0.390423\pi$
$311$	65284.5			0.674977			0.337489	+	0.941330i	$0.390423\pi$
$312$		0			0
$313$	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	0.808467	−	0.588541i	$-0.200298\pi$
$317$	22100.1	+	22100.1i	0.219926	+	0.219926i	−0.588541	+	0.808467i	$0.700298\pi$
$318$		0			0
$319$		−	140621.i		−	1.38187i
$320$		0			0
$321$		0			0
$322$	15586.1	+	15586.1i	0.150323	+	0.150323i
$323$	−19331.0	+	19331.0i	−0.185289	+	0.185289i
$324$		0			0
$325$		0			0
$326$	−27436.0			−0.258158
$327$		0			0
$328$	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
$333$		0			0
$334$			149757.i			1.34244i
$335$		0			0
$336$		0			0
$337$	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
$339$		0			0
$340$		0			0
$341$	−67609.8			−0.581434
$342$		0			0
$343$	−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	0.612635	−	0.790366i	$-0.290109\pi$
$347$	−21400.3	−	21400.3i	−0.177730	−	0.177730i	−0.790366	+	0.612635i	$0.790109\pi$
$348$		0			0
$349$		−	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
$351$		0			0
$352$	−21432.2	−	21432.2i	−0.172974	−	0.172974i
$353$	28.7539	−	28.7539i	0.000230753	−	0.000230753i	−0.706991	−	0.707222i	$-0.749948\pi$
$353$	28.7539	−	28.7539i	0.000230753	−	0.000230753i	0.707222	+	0.706991i	$0.249948\pi$
$354$		0			0
$355$		0			0
$356$	−79292.9			−0.625654
$357$		0			0
$358$	12792.4	−	12792.4i	0.0998126	−	0.0998126i
$359$		−	173943.i		−	1.34964i	−0.737984	−	0.674818i	$-0.764222\pi$
$359$		−	173943.i		−	1.34964i	0.737984	−	0.674818i	$-0.235778\pi$
$360$		0			0
$361$	56222.1			0.431412
$362$	−98051.9	−	98051.9i	−0.748236	−	0.748236i
$363$		0			0
$364$		−	159001.i		−	1.20004i
$365$		0			0
$366$		0			0
$367$	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
$369$		0			0
$370$		0			0
$371$	122531.			0.890223
$372$		0			0
$373$	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
$378$		0			0
$379$		−	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
$381$		0			0
$382$	79898.0	+	79898.0i	0.547531	+	0.547531i
$383$	22327.5	−	22327.5i	0.152210	−	0.152210i	−0.626894	−	0.779104i	$-0.715674\pi$
$383$	22327.5	−	22327.5i	0.152210	−	0.152210i	0.779104	+	0.626894i	$0.215674\pi$
$384$		0			0
$385$		0			0
$386$	−94996.5			−0.637577
$387$		0			0
$388$	−85511.0	+	85511.0i	−0.568013	+	0.568013i
$389$			203884.i			1.34736i	0.739023	+	0.673680i	$0.235287\pi$
$389$			203884.i			1.34736i	−0.739023	+	0.673680i	$0.764713\pi$
$390$		0			0
$391$	−12453.5			−0.0814585
$392$	24779.2	+	24779.2i	0.161256	+	0.161256i
$393$		0			0
$394$			10900.4i			0.0702181i
$395$		0			0
$396$		0			0
$397$	−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
$399$		0			0
$400$		0			0
$401$	−43093.1			−0.267990			−0.133995	−	0.990982i	$-0.542781\pi$
$401$	−43093.1			−0.267990			−0.133995	+	0.990982i	$0.542781\pi$
$402$		0			0
$403$	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
$408$		0			0
$409$			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
$411$		0			0
$412$	19314.1	+	19314.1i	0.113784	+	0.113784i
$413$	14554.0	−	14554.0i	0.0853261	−	0.0853261i
$414$		0			0
$415$		0			0
$416$	57247.0			0.330800
$417$		0			0
$418$	91157.7	−	91157.7i	0.521724	−	0.521724i
$419$			101699.i			0.579278i	0.957136	+	0.289639i	$0.0935352\pi$
$419$			101699.i			0.579278i	−0.957136	+	0.289639i	$0.906465\pi$
$420$		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
$423$		0			0
$424$			44116.3i			0.245396i
$425$		0			0
$426$		0			0
$427$	186897.	+	186897.i	1.02505	+	1.02505i
$428$	−34232.2	+	34232.2i	−0.186873	+	0.186873i
$429$		0			0
$430$		0			0
$431$	−210598.			−1.13370			−0.566852	−	0.823820i	$-0.691839\pi$
$431$	−210598.			−1.13370			−0.566852	+	0.823820i	$0.691839\pi$
$432$		0			0
$433$	−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
$438$		0			0
$439$			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
$441$		0			0
$442$	63521.6	+	63521.6i	0.325145	+	0.325145i
$443$	−99815.9	+	99815.9i	−0.508619	+	0.508619i	−0.914102	−	0.405484i	$-0.867103\pi$
$443$	−99815.9	+	99815.9i	−0.508619	+	0.508619i	0.405484	+	0.914102i	$0.367103\pi$
$444$		0			0
$445$		0			0
$446$	115271.			0.579496
$447$		0			0
$448$	−22752.9	+	22752.9i	−0.113366	+	0.113366i
$449$		−	182934.i		−	0.907408i	−0.891152	−	0.453704i	$-0.850102\pi$
$449$		−	182934.i		−	0.907408i	0.891152	−	0.453704i	$-0.149898\pi$
$450$		0			0
$451$	−232136.			−1.14127
$452$	12964.8	+	12964.8i	0.0634585	+	0.0634585i
$453$		0			0
$454$		−	3536.95i		−	0.0171600i
$455$		0			0
$456$		0			0
$457$	−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
$459$		0			0
$460$		0			0
$461$	354912.			1.67001			0.835006	−	0.550241i	$-0.185464\pi$
$461$	354912.			1.67001			0.835006	+	0.550241i	$0.185464\pi$
$462$		0			0
$463$	−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	0.717006	−	0.697067i	$-0.245512\pi$
$467$	4348.30	+	4348.30i	0.0199382	+	0.0199382i	−0.697067	+	0.717006i	$0.745512\pi$
$468$		0			0
$469$			251792.i			1.14471i
$470$		0			0
$471$		0			0
$472$	5240.04	+	5240.04i	0.0235207	+	0.0235207i
$473$	95171.3	−	95171.3i	0.425386	−	0.425386i
$474$		0			0
$475$		0			0
$476$	−50493.5			−0.222855
$477$		0			0
$478$	−96375.5	+	96375.5i	−0.421804	+	0.421804i
$479$			77346.0i			0.337106i	0.985693	+	0.168553i	$0.0539095\pi$
$479$			77346.0i			0.337106i	−0.985693	+	0.168553i	$0.946091\pi$
$480$		0			0
$481$	92842.6			0.401289
$482$	110368.	+	110368.i	0.475062	+	0.475062i
$483$		0			0
$484$			107159.i			0.457445i
$485$		0			0
$486$		0			0
$487$	−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
$489$		0			0
$490$		0			0
$491$	−340217.			−1.41121			−0.705607	−	0.708603i	$-0.749326\pi$
$491$	−340217.			−1.41121			−0.705607	+	0.708603i	$0.749326\pi$
$492$		0			0
$493$	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
$498$		0			0
$499$		−	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
$501$		0			0
$502$	−139336.	−	139336.i	−0.552910	−	0.552910i
$503$	−123301.	+	123301.i	−0.487338	+	0.487338i	−0.907465	−	0.420127i	$-0.861985\pi$
$503$	−123301.	+	123301.i	−0.487338	+	0.487338i	0.420127	+	0.907465i	$0.361985\pi$
$504$		0			0
$505$		0			0
$506$	58725.7			0.229365
$507$		0			0
$508$	−92579.7	+	92579.7i	−0.358747	+	0.358747i
$509$		−	364729.i		−	1.40778i	−0.710309	−	0.703890i	$-0.751445\pi$
$509$		−	364729.i		−	1.40778i	0.710309	−	0.703890i	$-0.248555\pi$
$510$		0			0
$511$	610173.			2.33674
$512$	−8192.00	−	8192.00i	−0.0312500	−	0.0312500i
$513$		0			0
$514$			46457.8i			0.175846i
$515$		0			0
$516$		0			0
$517$	442504.	+	442504.i	1.65553	+	1.65553i
$518$	−36900.5	+	36900.5i	−0.137522	+	0.137522i
$519$		0			0
$520$		0			0
$521$	203461.			0.749560			0.374780	−	0.927114i	$-0.377718\pi$
$521$	203461.			0.749560			0.374780	+	0.927114i	$0.377718\pi$
$522$		0			0
$523$	−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
$528$		0			0
$529$			264465.i			0.945053i
$530$		0			0
$531$		0			0
$532$	−96775.0	−	96775.0i	−0.341932	−	0.341932i
$533$	310026.	−	310026.i	1.09130	−	1.09130i
$534$		0			0
$535$		0			0
$536$	−90655.8			−0.315548
$537$		0			0
$538$	−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
$543$		0			0
$544$		−	18179.8i		−	0.0614315i
$545$		0			0
$546$		0			0
$547$	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
$549$		0			0
$550$		0			0
$551$	228612.			0.753001
$552$		0			0
$553$	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	0.0110558	−	0.999939i	$-0.496481\pi$
$557$	−306800.	−	306800.i	−0.988883	−	0.988883i	−0.999939	+	0.0110558i	$0.996481\pi$
$558$		0			0
$559$			254209.i			0.813518i
$560$		0			0
$561$		0			0
$562$	−17338.7	−	17338.7i	−0.0548963	−	0.0548963i
$563$	331132.	−	331132.i	1.04468	−	1.04468i	0.0457279	−	0.998954i	$-0.485439\pi$
$563$	331132.	−	331132.i	1.04468	−	1.04468i	0.998954	−	0.0457279i	$-0.0145607\pi$
$564$		0			0
$565$		0			0
$566$	−94988.2			−0.296508
$567$		0			0
$568$	−85424.5	+	85424.5i	−0.264780	+	0.264780i
$569$			257399.i			0.795028i	0.917596	+	0.397514i	$0.130127\pi$
$569$			257399.i			0.795028i	−0.917596	+	0.397514i	$0.869873\pi$
$570$		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
$573$		0			0
$574$			246441.i			0.747978i
$575$		0			0
$576$		0			0
$577$	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
$579$		0			0
$580$		0			0
$581$	−365467.			−1.08267
$582$		0			0
$583$	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	0.919284	−	0.393595i	$-0.128769\pi$
$587$	181136.	+	181136.i	0.525690	+	0.525690i	−0.393595	+	0.919284i	$0.628769\pi$
$588$		0			0
$589$		−	109915.i		−	0.316831i
$590$		0			0
$591$		0			0
$592$	−13285.7	−	13285.7i	−0.0379089	−	0.0379089i
$593$	−235890.	+	235890.i	−0.670812	+	0.670812i	−0.957903	−	0.287091i	$-0.907312\pi$
$593$	−235890.	+	235890.i	−0.670812	+	0.670812i	0.287091	+	0.957903i	$0.407312\pi$
$594$		0			0
$595$		0			0
$596$	−17151.5			−0.0482847
$597$		0			0
$598$	−78430.2	+	78430.2i	−0.219322	+	0.219322i
$599$		−	141629.i		−	0.394728i	−0.980330	−	0.197364i	$-0.936762\pi$
$599$		−	141629.i		−	0.394728i	0.980330	−	0.197364i	$-0.0632381\pi$
$600$		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
$603$		0			0
$604$		−	71524.7i		−	0.196057i
$605$		0			0
$606$		0			0
$607$	−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
$609$		0			0
$610$		0			0
$611$	−1.18196e6			−3.16606
$612$		0			0
$613$	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	0.968437	−	0.249260i	$-0.0801875\pi$
$617$	273782.	+	273782.i	0.719176	+	0.719176i	−0.249260	+	0.968437i	$0.580188\pi$
$618$		0			0
$619$		−	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
$621$		0			0
$622$	−130569.	−	130569.i	−0.337489	−	0.337489i
$623$	440465.	−	440465.i	1.13484	−	1.13484i
$624$		0			0
$625$		0			0
$626$	−25673.8			−0.0655151
$627$		0			0
$628$	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
$633$		0			0
$634$		−	88400.5i		−	0.219926i
$635$		0			0
$636$		0			0
$637$	346322.	+	346322.i	0.853495	+	0.853495i
$638$	−281242.	+	281242.i	−0.690937	+	0.690937i
$639$		0			0
$640$		0			0
$641$	501643.			1.22090			0.610448	−	0.792056i	$-0.290989\pi$
$641$	501643.			1.22090			0.610448	+	0.792056i	$0.290989\pi$
$642$		0			0
$643$	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	0.127053	−	0.991896i	$-0.459448\pi$
$647$	−362031.	−	362031.i	−0.864843	−	0.864843i	−0.991896	+	0.127053i	$0.959448\pi$
$648$		0			0
$649$		−	54836.8i		−	0.130192i
$650$		0			0
$651$		0			0
$652$	54872.0	+	54872.0i	0.129079	+	0.129079i
$653$	−411293.	+	411293.i	−0.964552	+	0.964552i	−0.999393	−	0.0348413i	$-0.988907\pi$
$653$	−411293.	+	411293.i	−0.964552	+	0.964552i	0.0348413	+	0.999393i	$0.488907\pi$
$654$		0			0
$655$		0			0
$656$	−88729.0			−0.206185
$657$		0			0
$658$	469772.	−	469772.i	1.08501	−	1.08501i
$659$		−	291952.i		−	0.672265i	−0.941815	−	0.336133i	$-0.890881\pi$
$659$		−	291952.i		−	0.672265i	0.941815	−	0.336133i	$-0.109119\pi$
$660$		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
$663$		0			0
$664$		−	131583.i		−	0.298445i
$665$		0			0
$666$		0			0
$667$	73638.3	+	73638.3i	0.165521	+	0.165521i
$668$	299514.	−	299514.i	0.671219	−	0.671219i
$669$		0			0
$670$		0			0
$671$	704193.			1.56404
$672$		0			0
$673$	−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	0.717076	−	0.696995i	$-0.245480\pi$
$677$	9203.97	+	9203.97i	0.0200816	+	0.0200816i	−0.696995	+	0.717076i	$0.745480\pi$
$678$		0			0
$679$		−	950012.i		−	2.06058i
$680$		0			0
$681$		0			0
$682$	135220.	+	135220.i	0.290717	+	0.290717i
$683$	−421590.	+	421590.i	−0.903750	+	0.903750i	−0.995758	−	0.0920080i	$-0.970671\pi$
$683$	−421590.	+	421590.i	−0.903750	+	0.903750i	0.0920080	+	0.995758i	$0.470671\pi$
$684$		0			0
$685$		0			0
$686$	151502.			0.321937
$687$		0			0
$688$	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
$693$		0			0
$694$			85601.3i			0.177730i
$695$		0			0
$696$		0			0
$697$	−98454.3	−	98454.3i	−0.202660	−	0.202660i
$698$	−68182.4	+	68182.4i	−0.139946	+	0.139946i
$699$		0			0
$700$		0			0
$701$	685359.			1.39471			0.697353	−	0.716728i	$-0.254361\pi$
$701$	685359.			1.39471			0.697353	+	0.716728i	$0.254361\pi$
$702$		0			0
$703$	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
$708$		0			0
$709$		−	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
$711$		0			0
$712$	158586.	+	158586.i	0.312827	+	0.312827i
$713$	35404.9	−	35404.9i	0.0696441	−	0.0696441i
$714$		0			0
$715$		0			0
$716$	−51169.5			−0.0998126
$717$		0			0
$718$	−347885.	+	347885.i	−0.674818	+	0.674818i
$719$			782798.i			1.51423i	0.653282	+	0.757115i	$0.273392\pi$
$719$			782798.i			1.51423i	−0.653282	+	0.757115i	$0.726608\pi$
$720$		0			0
$721$	−214577.			−0.412774
$722$	−112444.	−	112444.i	−0.215706	−	0.215706i
$723$		0			0
$724$			392208.i			0.748236i
$725$		0			0
$726$		0			0
$727$	−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
$729$		0			0
$730$		0			0
$731$	80728.6			0.151075
$732$		0			0
$733$	−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
$738$		0			0
$739$			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
$741$		0			0
$742$	−245062.	−	245062.i	−0.445111	−	0.445111i
$743$	−35964.9	+	35964.9i	−0.0651480	+	0.0651480i	−0.738930	−	0.673782i	$-0.764669\pi$
$743$	−35964.9	+	35964.9i	−0.0651480	+	0.0651480i	0.673782	+	0.738930i	$0.264669\pi$
$744$		0			0
$745$		0			0
$746$	−263501.			−0.473483
$747$		0			0
$748$	−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
$753$		0			0
$754$		−	751216.i		−	1.32136i
$755$		0			0
$756$		0			0
$757$	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
$759$		0			0
$760$		0			0
$761$	579015.			0.999817			0.499908	−	0.866078i	$-0.333367\pi$
$761$	579015.			0.999817			0.499908	+	0.866078i	$0.333367\pi$
$762$		0			0
$763$	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
$768$		0			0
$769$			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
$771$		0			0
$772$	189993.	+	189993.i	0.318789	+	0.318789i
$773$	−20353.7	+	20353.7i	−0.0340631	+	0.0340631i	−0.723933	−	0.689870i	$-0.757668\pi$
$773$	−20353.7	+	20353.7i	−0.0340631	+	0.0340631i	0.689870	+	0.723933i	$0.257668\pi$
$774$		0			0
$775$		0			0
$776$	342044.			0.568013
$777$		0			0
$778$	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
$783$		0			0
$784$		−	99116.8i		−	0.161256i
$785$		0			0
$786$		0			0
$787$	−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
$789$		0			0
$790$		0			0
$791$	−144037.			−0.230208
$792$		0			0
$793$	−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	0.990683	−	0.136189i	$-0.0434854\pi$
$797$	542782.	+	542782.i	0.854494	+	0.854494i	−0.136189	+	0.990683i	$0.543485\pi$
$798$		0			0
$799$			375352.i			0.587956i
$800$		0			0
$801$		0			0
$802$	86186.2	+	86186.2i	0.133995	+	0.133995i
$803$	1.14951e6	−	1.14951e6i	1.78271	−	1.78271i
$804$		0			0
$805$		0			0
$806$	−361181.			−0.555974
$807$		0			0
$808$	28209.5	−	28209.5i	0.0432088	−	0.0432088i
$809$		−	1.09699e6i		−	1.67613i	−0.545571	−	0.838065i	$-0.683687\pi$
$809$		−	1.09699e6i		−	1.67613i	0.545571	−	0.838065i	$-0.316313\pi$
$810$		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
$813$		0			0
$814$			139034.i			0.209833i
$815$		0			0
$816$		0			0
$817$	154723.	+	154723.i	0.231798	+	0.231798i
$818$	372179.	−	372179.i	0.556219	−	0.556219i
$819$		0			0
$820$		0			0
$821$	566243.			0.840071			0.420036	−	0.907508i	$-0.362018\pi$
$821$	566243.			0.840071			0.420036	+	0.907508i	$0.362018\pi$
$822$		0			0
$823$	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	$-0.926397\pi$
$827$	−822466.	−	822466.i	−1.20256	−	1.20256i	−0.229174	−	0.973385i	$-0.573603\pi$
$828$		0			0
$829$		−	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
$831$		0			0
$832$	−114494.	−	114494.i	−0.165400	−	0.165400i
$833$	109981.	−	109981.i	0.158499	−	0.158499i
$834$		0			0
$835$		0			0
$836$	−364631.			−0.521724
$837$		0			0
$838$	203397.	−	203397.i	0.289639	−	0.289639i
$839$			487091.i			0.691968i	0.938240	+	0.345984i	$0.112455\pi$
$839$			487091.i			0.691968i	−0.938240	+	0.345984i	$0.887545\pi$
$840$		0			0
$841$	1962.99			0.00277541
$842$	45065.9	+	45065.9i	0.0635659	+	0.0635659i
$843$		0			0
$844$			123761.i			0.173740i
$845$		0			0
$846$		0			0
$847$	−595260.	−	595260.i	−0.829736	−	0.829736i
$848$	88232.7	−	88232.7i	0.122698	−	0.122698i
$849$		0			0
$850$		0			0
$851$	36403.7			0.0502675
$852$		0			0
$853$	−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	0.0554896	−	0.998459i	$-0.482328\pi$
$857$	−692563.	−	692563.i	−0.942970	−	0.942970i	−0.998459	+	0.0554896i	$0.982328\pi$
$858$		0			0
$859$			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
$861$		0			0
$862$	421196.	+	421196.i	0.566852	+	0.566852i
$863$	259362.	−	259362.i	0.348245	−	0.348245i	−0.511211	−	0.859456i	$-0.670803\pi$
$863$	259362.	−	259362.i	0.348245	−	0.348245i	0.859456	+	0.511211i	$0.170803\pi$
$864$		0			0
$865$		0			0
$866$	493621.			0.658201
$867$		0			0
$868$	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
$873$		0			0
$874$			95472.3i			0.124984i
$875$		0			0
$876$		0			0
$877$	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
$879$		0			0
$880$		0			0
$881$	−1.08074e6			−1.39241			−0.696207	−	0.717841i	$-0.745130\pi$
$881$	−1.08074e6			−1.39241			−0.696207	+	0.717841i	$0.745130\pi$
$882$		0			0
$883$	−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	0.428862	−	0.903370i	$-0.358915\pi$
$887$	−373328.	−	373328.i	−0.474507	−	0.474507i	−0.903370	+	0.428862i	$0.858915\pi$
$888$		0			0
$889$		−	1.02854e6i		−	1.30143i
$890$		0			0
$891$		0			0
$892$	−230542.	−	230542.i	−0.289748	−	0.289748i
$893$	−719392.	+	719392.i	−0.902117	+	0.902117i
$894$		0			0
$895$		0			0
$896$	91011.7			0.113366
$897$		0			0
$898$	−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
$903$		0			0
$904$		−	51859.3i		−	0.0634585i
$905$		0			0
$906$		0			0
$907$	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
$909$		0			0
$910$		0			0
$911$	−653388.			−0.787290			−0.393645	−	0.919263i	$-0.628786\pi$
$911$	−653388.			−0.787290			−0.393645	+	0.919263i	$0.628786\pi$
$912$		0			0
$913$	−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
$918$		0			0
$919$		−	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
$921$		0			0
$922$	−709825.	−	709825.i	−0.835006	−	0.835006i
$923$	−1.19392e6	+	1.19392e6i	−1.40143	+	1.40143i
$924$		0			0
$925$		0			0
$926$	729901.			0.851221
$927$		0			0
$928$	−107499.	+	107499.i	−0.124826	+	0.124826i
$929$			1.39646e6i			1.61807i	0.587759	+	0.809036i	$0.300010\pi$
$929$			1.39646e6i			1.61807i	−0.587759	+	0.809036i	$0.699990\pi$
$930$		0			0
$931$	421574.			0.486378
$932$	−197566.	−	197566.i	−0.227447	−	0.227447i
$933$		0			0
$934$		−	17393.2i		−	0.0199382i
$935$		0			0
$936$		0			0
$937$	−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
$939$		0			0
$940$		0			0
$941$	−1.67094e6			−1.88705			−0.943524	−	0.331305i	$-0.892511\pi$
$941$	−1.67094e6			−1.88705			−0.943524	+	0.331305i	$0.892511\pi$
$942$		0			0
$943$	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	0.977922	−	0.208971i	$-0.0670116\pi$
$947$	689602.	+	689602.i	0.768950	+	0.768950i	−0.208971	+	0.977922i	$0.567012\pi$
$948$		0			0
$949$			3.07042e6i			3.40930i
$950$		0			0
$951$		0			0
$952$	100987.	+	100987.i	0.111427	+	0.111427i
$953$	150337.	−	150337.i	0.165532	−	0.165532i	−0.619480	−	0.785012i	$-0.712657\pi$
$953$	150337.	−	150337.i	0.165532	−	0.165532i	0.785012	+	0.619480i	$0.212657\pi$
$954$		0			0
$955$		0			0
$956$	385502.			0.421804
$957$		0			0
$958$	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
$963$		0			0
$964$		−	441473.i		−	0.475062i
$965$		0			0
$966$		0			0
$967$	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
$969$		0			0
$970$		0			0
$971$	−1.16534e6			−1.23598			−0.617992	−	0.786185i	$-0.712054\pi$
$971$	−1.16534e6			−1.23598			−0.617992	+	0.786185i	$0.712054\pi$
$972$		0			0
$973$	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	0.196407	−	0.980522i	$-0.437073\pi$
$977$	−748461.	−	748461.i	−0.784115	−	0.784115i	−0.980522	+	0.196407i	$0.937073\pi$
$978$		0			0
$979$		−	1.65959e6i		−	1.73156i
$980$		0			0
$981$		0			0
$982$	680434.	+	680434.i	0.705607	+	0.705607i
$983$	−542812.	+	542812.i	−0.561749	+	0.561749i	−0.929804	−	0.368055i	$-0.880024\pi$
$983$	−542812.	+	542812.i	−0.561749	+	0.561749i	0.368055	+	0.929804i	$0.380024\pi$
$984$		0			0
$985$		0			0
$986$	−238562.			−0.245385
$987$		0			0
$988$	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
$993$		0			0
$994$		−	949050.i		−	0.960542i
$995$		0			0
$996$		0			0
$997$	−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
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.5.g.f.307.1		4
3.2	odd	2		150.5.f.e.7.2		4
5.2	odd	4		90.5.g.e.73.1		4
5.3	odd	4	inner	450.5.g.f.343.1		4
5.4	even	2		90.5.g.e.37.1		4
15.2	even	4		30.5.f.a.13.1	yes	4
15.8	even	4		150.5.f.e.43.2		4
15.14	odd	2		30.5.f.a.7.1	✓	4
60.47	odd	4		240.5.bg.b.193.2		4
60.59	even	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	15.14	odd	2
30.5.f.a.13.1	yes	4	15.2	even	4
90.5.g.e.37.1		4	5.4	even	2
90.5.g.e.73.1		4	5.2	odd	4
150.5.f.e.7.2		4	3.2	odd	2
150.5.f.e.43.2		4	15.8	even	4
240.5.bg.b.97.2		4	60.59	even	2
240.5.bg.b.193.2		4	60.47	odd	4
450.5.g.f.307.1		4	1.1	even	1	trivial
450.5.g.f.343.1		4	5.3	odd	4	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(-44.4393 - 44.4393i) q^{7} +(16.0000 - 16.0000i) q^{8} +O(q^{10})\)	\(q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(-44.4393 - 44.4393i) q^{7} +(16.0000 - 16.0000i) q^{8} -167.439 q^{11} +(223.621 - 223.621i) q^{13} +177.757i q^{14} -64.0000 q^{16} +(-71.0148 - 71.0148i) q^{17} -272.211i q^{19} +(334.879 + 334.879i) q^{22} +(87.6821 - 87.6821i) q^{23} -894.484 q^{26} +(355.514 - 355.514i) q^{28} +839.832i q^{29} +403.787 q^{31} +(128.000 + 128.000i) q^{32} +284.059i q^{34} +(207.589 + 207.589i) q^{37} +(-544.422 + 544.422i) q^{38} +1386.39 q^{41} +(-568.393 + 568.393i) q^{43} -1339.51i q^{44} -350.729 q^{46} +(-2642.77 - 2642.77i) q^{47} +1548.70i q^{49} +(1788.97 + 1788.97i) q^{52} +(-1378.64 + 1378.64i) q^{53} -1422.06 q^{56} +(1679.66 - 1679.66i) q^{58} +327.503i q^{59} -4205.66 q^{61} +(-807.573 - 807.573i) q^{62} -512.000i q^{64} +(-2832.99 - 2832.99i) q^{67} +(568.118 - 568.118i) q^{68} -5339.03 q^{71} +(-6865.24 + 6865.24i) q^{73} -830.357i q^{74} +2177.69 q^{76} +(7440.88 + 7440.88i) q^{77} +4666.69i q^{79} +(-2772.78 - 2772.78i) q^{82} +(4111.98 - 4111.98i) q^{83} +2273.57 q^{86} +(-2679.03 + 2679.03i) q^{88} +9911.62i q^{89} -19875.1 q^{91} +(701.457 + 701.457i) q^{92} +10571.1i q^{94} +(10688.9 + 10688.9i) q^{97} +(3097.40 - 3097.40i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 28 q^{7} + 64 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 28 * q^7 + 64 * q^8	\( 4 q - 8 q^{2} + 28 q^{7} + 64 q^{8} - 464 q^{11} + 336 q^{13} - 256 q^{16} + 392 q^{17} + 928 q^{22} + 968 q^{23} - 1344 q^{26} - 224 q^{28} - 560 q^{31} + 512 q^{32} - 2256 q^{37} + 1232 q^{38} - 392 q^{41} - 216 q^{43} - 3872 q^{46} - 9072 q^{47} + 2688 q^{52} - 4280 q^{53} + 896 q^{56} + 2192 q^{58} - 4536 q^{61} + 1120 q^{62} + 2248 q^{67} - 3136 q^{68} - 18064 q^{71} - 20524 q^{73} - 4928 q^{76} + 7336 q^{77} + 784 q^{82} - 336 q^{83} + 864 q^{86} - 7424 q^{88} - 52752 q^{91} + 7744 q^{92} + 40404 q^{97} + 23912 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 28 * q^7 + 64 * q^8 - 464 * q^11 + 336 * q^13 - 256 * q^16 + 392 * q^17 + 928 * q^22 + 968 * q^23 - 1344 * q^26 - 224 * q^28 - 560 * q^31 + 512 * q^32 - 2256 * q^37 + 1232 * q^38 - 392 * q^41 - 216 * q^43 - 3872 * q^46 - 9072 * q^47 + 2688 * q^52 - 4280 * q^53 + 896 * q^56 + 2192 * q^58 - 4536 * q^61 + 1120 * q^62 + 2248 * q^67 - 3136 * q^68 - 18064 * q^71 - 20524 * q^73 - 4928 * q^76 + 7336 * q^77 + 784 * q^82 - 336 * q^83 + 864 * q^86 - 7424 * q^88 - 52752 * q^91 + 7744 * q^92 + 40404 * q^97 + 23912 * q^98

Introduction

L-functions

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