LMFDB - Embedded newform 90.5.g.e.73.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,5,Mod(37,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, 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("90.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	90.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:	$9.30329667755$
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:	$\Z[a_1, \ldots, a_{7}]$
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			73.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	90.73
Dual form			90.5.g.e.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-24.6742 + 4.02270i) q^{5} +(44.4393 - 44.4393i) q^{7} +(-16.0000 - 16.0000i) q^{8} +O(q^{10})$	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-24.6742 + 4.02270i) q^{5} +(44.4393 - 44.4393i) q^{7} +(-16.0000 - 16.0000i) q^{8} +(-41.3031 + 57.3939i) q^{10} -167.439 q^{11} +(-223.621 - 223.621i) q^{13} -177.757i q^{14} -64.0000 q^{16} +(71.0148 - 71.0148i) q^{17} +272.211i q^{19} +(32.1816 + 197.394i) q^{20} +(-334.879 + 334.879i) q^{22} +(-87.6821 - 87.6821i) q^{23} +(592.636 - 198.514i) q^{25} -894.484 q^{26} +(-355.514 - 355.514i) q^{28} -839.832i q^{29} +403.787 q^{31} +(-128.000 + 128.000i) q^{32} -284.059i q^{34} +(-917.739 + 1275.27i) q^{35} +(-207.589 + 207.589i) q^{37} +(544.422 + 544.422i) q^{38} +(459.151 + 330.424i) q^{40} +1386.39 q^{41} +(568.393 + 568.393i) q^{43} +1339.51i q^{44} -350.729 q^{46} +(2642.77 - 2642.77i) q^{47} -1548.70i q^{49} +(788.243 - 1582.30i) q^{50} +(-1788.97 + 1788.97i) q^{52} +(1378.64 + 1378.64i) q^{53} +(4131.44 - 673.559i) q^{55} -1422.06 q^{56} +(-1679.66 - 1679.66i) q^{58} -327.503i q^{59} -4205.66 q^{61} +(807.573 - 807.573i) q^{62} +512.000i q^{64} +(6417.24 + 4618.11i) q^{65} +(2832.99 - 2832.99i) q^{67} +(-568.118 - 568.118i) q^{68} +(715.064 + 4386.02i) q^{70} -5339.03 q^{71} +(6865.24 + 6865.24i) q^{73} +830.357i q^{74} +2177.69 q^{76} +(-7440.88 + 7440.88i) q^{77} -4666.69i q^{79} +(1579.15 - 257.453i) q^{80} +(2772.78 - 2772.78i) q^{82} +(-4111.98 - 4111.98i) q^{83} +(-1466.56 + 2037.91i) q^{85} +2273.57 q^{86} +(2679.03 + 2679.03i) q^{88} -9911.62i q^{89} -19875.1 q^{91} +(-701.457 + 701.457i) q^{92} -10571.1i q^{94} +(-1095.03 - 6716.60i) q^{95} +(-10688.9 + 10688.9i) q^{97} +(-3097.40 - 3097.40i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} - 84 q^{5} - 28 q^{7} - 64 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 - 84 * q^5 - 28 * q^7 - 64 * q^8	$ 4 q + 8 q^{2} - 84 q^{5} - 28 q^{7} - 64 q^{8} - 224 q^{10} - 464 q^{11} - 336 q^{13} - 256 q^{16} - 392 q^{17} - 224 q^{20} - 928 q^{22} - 968 q^{23} + 1136 q^{25} - 1344 q^{26} + 224 q^{28} - 560 q^{31} - 512 q^{32} + 2296 q^{35} + 2256 q^{37} - 1232 q^{38} + 896 q^{40} - 392 q^{41} + 216 q^{43} - 3872 q^{46} + 9072 q^{47} + 3976 q^{50} - 2688 q^{52} + 4280 q^{53} + 10500 q^{55} + 896 q^{56} - 2192 q^{58} - 4536 q^{61} - 1120 q^{62} + 12912 q^{65} - 2248 q^{67} + 3136 q^{68} + 9856 q^{70} - 18064 q^{71} + 20524 q^{73} - 4928 q^{76} - 7336 q^{77} + 5376 q^{80} - 784 q^{82} + 336 q^{83} + 15944 q^{85} + 864 q^{86} + 7424 q^{88} - 52752 q^{91} - 7744 q^{92} - 23104 q^{95} - 40404 q^{97} - 23912 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 - 84 * q^5 - 28 * q^7 - 64 * q^8 - 224 * q^10 - 464 * q^11 - 336 * q^13 - 256 * q^16 - 392 * q^17 - 224 * q^20 - 928 * q^22 - 968 * q^23 + 1136 * q^25 - 1344 * q^26 + 224 * q^28 - 560 * q^31 - 512 * q^32 + 2296 * q^35 + 2256 * q^37 - 1232 * q^38 + 896 * q^40 - 392 * q^41 + 216 * q^43 - 3872 * q^46 + 9072 * q^47 + 3976 * q^50 - 2688 * q^52 + 4280 * q^53 + 10500 * q^55 + 896 * q^56 - 2192 * q^58 - 4536 * q^61 - 1120 * q^62 + 12912 * q^65 - 2248 * q^67 + 3136 * q^68 + 9856 * q^70 - 18064 * q^71 + 20524 * q^73 - 4928 * q^76 - 7336 * q^77 + 5376 * q^80 - 784 * q^82 + 336 * q^83 + 15944 * q^85 + 864 * q^86 + 7424 * q^88 - 52752 * q^91 - 7744 * q^92 - 23104 * q^95 - 40404 * q^97 - 23912 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	−24.6742	+	4.02270i	−0.986969	+	0.160908i
$5$	−24.6742	+	4.02270i	−0.986969	+	0.160908i
$6$		0			0
$7$	44.4393	−	44.4393i	0.906924	−	0.906924i	−0.0890986	−	0.996023i	$-0.528399\pi$
$7$	44.4393	−	44.4393i	0.906924	−	0.906924i	0.996023	+	0.0890986i	$0.0283986\pi$
$8$	−16.0000	−	16.0000i	−0.250000	−	0.250000i
$9$		0			0
$10$	−41.3031	+	57.3939i	−0.413031	+	0.573939i
$11$	−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	−0.911170	−	0.412031i	$-0.864820\pi$
$13$	−223.621	−	223.621i	−1.32320	−	1.32320i	−0.412031	−	0.911170i	$-0.635180\pi$
$14$		−	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	$-0.694410\pi$
$17$	71.0148	−	71.0148i	0.245726	−	0.245726i	0.819214	+	0.573488i	$0.194410\pi$
$18$		0			0
$19$			272.211i			0.754048i	0.926204	+	0.377024i	$0.123053\pi$
$19$			272.211i			0.754048i	−0.926204	+	0.377024i	$0.876947\pi$
$20$	32.1816	+	197.394i	0.0804541	+	0.493485i
$21$		0			0
$22$	−334.879	+	334.879i	−0.691898	+	0.691898i
$23$	−87.6821	−	87.6821i	−0.165751	−	0.165751i	0.619358	−	0.785109i	$-0.287393\pi$
$23$	−87.6821	−	87.6821i	−0.165751	−	0.165751i	−0.785109	+	0.619358i	$0.787393\pi$
$24$		0			0
$25$	592.636	−	198.514i	0.948217	−	0.317623i
$26$	−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	$-0.833588\pi$
$29$		−	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
$33$		0			0
$34$		−	284.059i		−	0.245726i
$35$	−917.739	+	1275.27i	−0.749175	+	1.04104i
$36$		0			0
$37$	−207.589	+	207.589i	−0.151636	+	0.151636i	−0.778848	−	0.627212i	$-0.784196\pi$
$37$	−207.589	+	207.589i	−0.151636	+	0.151636i	0.627212	+	0.778848i	$0.284196\pi$
$38$	544.422	+	544.422i	0.377024	+	0.377024i
$39$		0			0
$40$	459.151	+	330.424i	0.286969	+	0.206515i
$41$	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	$-0.180253\pi$
$43$	568.393	+	568.393i	0.307406	+	0.307406i	−0.536497	+	0.843902i	$0.680253\pi$
$44$			1339.51i			0.691898i
$45$		0			0
$46$	−350.729			−0.165751
$47$	2642.77	−	2642.77i	1.19637	−	1.19637i	0.221119	−	0.975247i	$-0.429029\pi$
$47$	2642.77	−	2642.77i	1.19637	−	1.19637i	0.975247	−	0.221119i	$-0.0709709\pi$
$48$		0			0
$49$		−	1548.70i		−	0.645023i
$50$	788.243	−	1582.30i	0.315297	−	0.632920i
$51$		0			0
$52$	−1788.97	+	1788.97i	−0.661600	+	0.661600i
$53$	1378.64	+	1378.64i	0.490792	+	0.490792i	0.908556	−	0.417763i	$-0.137186\pi$
$53$	1378.64	+	1378.64i	0.490792	+	0.490792i	−0.417763	+	0.908556i	$0.637186\pi$
$54$		0			0
$55$	4131.44	−	673.559i	1.36576	−	0.222664i
$56$	−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	$-0.985021\pi$
$59$		−	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
$63$		0			0
$64$			512.000i			0.125000i
$65$	6417.24	+	4618.11i	1.51887	+	1.09304i
$66$		0			0
$67$	2832.99	−	2832.99i	0.631097	−	0.631097i	−0.317246	−	0.948343i	$-0.602758\pi$
$67$	2832.99	−	2832.99i	0.631097	−	0.631097i	0.948343	+	0.317246i	$0.102758\pi$
$68$	−568.118	−	568.118i	−0.122863	−	0.122863i
$69$		0			0
$70$	715.064	+	4386.02i	0.145931	+	0.895106i
$71$	−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	0.935832	+	0.352447i	$0.114650\pi$
$73$	6865.24	+	6865.24i	1.28828	+	1.28828i	0.352447	+	0.935832i	$0.385350\pi$
$74$			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	$-0.878029\pi$
$79$		−	4666.69i		−	0.747748i	0.927480	−	0.373874i	$-0.121971\pi$
$80$	1579.15	−	257.453i	0.246742	−	0.0402270i
$81$		0			0
$82$	2772.78	−	2772.78i	0.412371	−	0.412371i
$83$	−4111.98	−	4111.98i	−0.596890	−	0.596890i	0.342594	−	0.939484i	$-0.388695\pi$
$83$	−4111.98	−	4111.98i	−0.596890	−	0.596890i	−0.939484	+	0.342594i	$0.888695\pi$
$84$		0			0
$85$	−1466.56	+	2037.91i	−0.202985	+	0.282063i
$86$	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	$-0.784832\pi$
$89$		−	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
$93$		0			0
$94$		−	10571.1i		−	1.19637i
$95$	−1095.03	−	6716.60i	−0.121332	−	0.744222i
$96$		0			0
$97$	−10688.9	+	10688.9i	−1.13603	+	1.13603i	−0.146871	+	0.989156i	$0.546920\pi$
$97$	−10688.9	+	10688.9i	−1.13603	+	1.13603i	−0.989156	+	0.146871i	$0.953080\pi$
$98$	−3097.40	−	3097.40i	−0.322511	−	0.322511i
$99$		0			0
$100$	−1588.11	−	4741.09i	−0.158811	−	0.474109i
$101$	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	$-0.301444\pi$
$103$	−2414.27	−	2414.27i	−0.227568	−	0.227568i	−0.811676	+	0.584108i	$0.801444\pi$
$104$			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	$-0.835134\pi$
$107$	−4279.03	+	4279.03i	−0.373747	+	0.373747i	0.495093	+	0.868840i	$0.335134\pi$
$108$		0			0
$109$		−	11819.5i		−	0.994827i	−0.867514	−	0.497413i	$-0.834283\pi$
$109$		−	11819.5i		−	0.994827i	0.867514	−	0.497413i	$-0.165717\pi$
$110$	6915.76	−	9609.99i	0.571550	−	0.794214i
$111$		0			0
$112$	−2844.11	+	2844.11i	−0.226731	+	0.226731i
$113$	−1620.60	−	1620.60i	−0.126917	−	0.126917i	0.640795	−	0.767712i	$-0.278605\pi$
$113$	−1620.60	−	1620.60i	−0.126917	−	0.126917i	−0.767712	+	0.640795i	$0.778605\pi$
$114$		0			0
$115$	2516.21	+	1810.77i	0.190262	+	0.136920i
$116$	−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
$125$	−13824.3	+	7282.19i	−0.884753	+	0.466060i
$126$		0			0
$127$	−11572.5	+	11572.5i	−0.717494	+	0.717494i	−0.968091	−	0.250597i	$-0.919373\pi$
$127$	−11572.5	+	11572.5i	−0.717494	+	0.717494i	0.250597	+	0.968091i	$0.419373\pi$
$128$	1024.00	+	1024.00i	0.0625000	+	0.0625000i
$129$		0			0
$130$	22070.7	−	3598.24i	1.30596	−	0.212914i
$131$	−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	$-0.756949\pi$
$137$	−579.439	+	579.439i	−0.0308721	+	0.0308721i	0.691502	+	0.722374i	$0.256949\pi$
$138$		0			0
$139$		−	36158.2i		−	1.87145i	−0.352734	−	0.935724i	$-0.614748\pi$
$139$		−	36158.2i		−	1.87145i	0.352734	−	0.935724i	$-0.385252\pi$
$140$	10202.2	+	7341.91i	0.520519	+	0.374587i
$141$		0			0
$142$	−10678.1	+	10678.1i	−0.529560	+	0.529560i
$143$	37442.9	+	37442.9i	1.83104	+	1.83104i
$144$		0			0
$145$	3378.40	+	20722.2i	0.160685	+	0.985599i
$146$	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	$-0.984625\pi$
$149$		−	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
$153$		0			0
$154$			29763.5i			1.25500i
$155$	−9963.13	+	1624.31i	−0.414698	+	0.0676093i
$156$		0			0
$157$	18032.9	−	18032.9i	0.731586	−	0.731586i	−0.239348	−	0.970934i	$-0.576934\pi$
$157$	18032.9	−	18032.9i	0.731586	−	0.731586i	0.970934	+	0.239348i	$0.0769337\pi$
$158$	−9333.39	−	9333.39i	−0.373874	−	0.373874i
$159$		0			0
$160$	2643.40	−	3673.21i	0.103258	−	0.143485i
$161$	−7793.06			−0.300647
$162$		0			0
$163$	−6859.00	−	6859.00i	−0.258158	−	0.258158i	0.566147	−	0.824305i	$-0.308434\pi$
$163$	−6859.00	−	6859.00i	−0.258158	−	0.258158i	−0.824305	+	0.566147i	$0.808434\pi$
$164$		−	11091.1i		−	0.412371i
$165$		0			0
$166$	−16447.9			−0.596890
$167$	37439.2	−	37439.2i	1.34244	−	1.34244i	0.448811	−	0.893627i	$-0.351848\pi$
$167$	37439.2	−	37439.2i	1.34244	−	1.34244i	0.893627	−	0.448811i	$-0.148152\pi$
$168$		0			0
$169$			71451.6i			2.50172i
$170$	1142.69	+	7008.94i	0.0395393	+	0.242524i
$171$		0			0
$172$	4547.14	−	4547.14i	0.153703	−	0.153703i
$173$	32638.2	+	32638.2i	1.09052	+	1.09052i	0.995472	+	0.0950500i	$0.0303011\pi$
$173$	32638.2	+	32638.2i	1.09052	+	1.09052i	0.0950500	+	0.995472i	$0.469699\pi$
$174$		0			0
$175$	17514.5	−	35158.1i	0.571901	−	1.14802i
$176$	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	$-0.968176\pi$
$179$		−	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
$183$		0			0
$184$			2805.83i			0.0828754i
$185$	4287.04	−	5957.18i	0.125260	−	0.174059i
$186$		0			0
$187$	−11890.7	+	11890.7i	−0.340034	+	0.340034i
$188$	−21142.2	−	21142.2i	−0.598183	−	0.598183i
$189$		0			0
$190$	−15623.3	−	11243.2i	−0.432777	−	0.311445i
$191$	−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	$-0.398874\pi$
$193$	−23749.1	−	23749.1i	−0.637577	−	0.637577i	−0.949957	+	0.312380i	$0.898874\pi$
$194$			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	$-0.734189\pi$
$197$	2725.09	−	2725.09i	0.0702181	−	0.0702181i	0.741344	+	0.671126i	$0.234189\pi$
$198$		0			0
$199$		−	63212.5i		−	1.59624i	−0.602502	−	0.798118i	$-0.705829\pi$
$199$		−	63212.5i		−	1.59624i	0.602502	−	0.798118i	$-0.294171\pi$
$200$	−12658.4	−	6305.94i	−0.316460	−	0.157649i
$201$		0			0
$202$	3526.19	−	3526.19i	0.0864177	−	0.0864177i
$203$	−37321.5	−	37321.5i	−0.905665	−	0.905665i
$204$		0			0
$205$	−34208.1	+	5577.04i	−0.813995	+	0.132708i
$206$	−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
$215$	−16311.1	−	11738.2i	−0.352864	−	0.253936i
$216$		0			0
$217$	17944.0	−	17944.0i	0.381065	−	0.381065i
$218$	−23639.1	−	23639.1i	−0.497413	−	0.497413i
$219$		0			0
$220$	−5388.47	−	33051.5i	−0.111332	−	0.682882i
$221$	−31760.8			−0.650289
$222$		0			0
$223$	28817.8	+	28817.8i	0.579496	+	0.579496i	0.934764	−	0.355268i	$-0.115611\pi$
$223$	28817.8	+	28817.8i	0.579496	+	0.579496i	−0.355268	+	0.934764i	$0.615611\pi$
$224$			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	$-0.753862\pi$
$227$	−884.237	+	884.237i	−0.0171600	+	0.0171600i	0.698475	+	0.715635i	$0.253862\pi$
$228$		0			0
$229$			12076.4i			0.230285i	0.993349	+	0.115143i	$0.0367325\pi$
$229$			12076.4i			0.230285i	−0.993349	+	0.115143i	$0.963268\pi$
$230$	8653.96	−	1410.88i	0.163591	−	0.0266706i
$231$		0			0
$232$	−13437.3	+	13437.3i	−0.249653	+	0.249653i
$233$	24695.8	+	24695.8i	0.454894	+	0.454894i	0.896975	−	0.442081i	$-0.145760\pi$
$233$	24695.8	+	24695.8i	0.454894	+	0.454894i	−0.442081	+	0.896975i	$0.645760\pi$
$234$		0			0
$235$	−54577.3	+	75839.5i	−0.988271	+	1.37328i
$236$	−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	$0.138603\pi$
$239$			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
$243$		0			0
$244$			33645.3i			0.565125i
$245$	6229.96	+	38213.0i	0.103789	+	0.636618i
$246$		0			0
$247$	60872.1	−	60872.1i	0.997756	−	0.997756i
$248$	−6460.59	−	6460.59i	−0.105043	−	0.105043i
$249$		0			0
$250$	−13084.2	+	42212.9i	−0.209347	+	0.675407i
$251$	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	$-0.710318\pi$
$257$	11614.4	−	11614.4i	0.175846	−	0.175846i	0.789542	+	0.613696i	$0.210318\pi$
$258$		0			0
$259$			18450.2i			0.275044i
$260$	36944.9	−	51337.9i	0.546522	−	0.759436i
$261$		0			0
$262$	−131.246	+	131.246i	−0.00191198	+	0.00191198i
$263$	−86811.6	−	86811.6i	−1.25507	−	1.25507i	−0.953420	−	0.301646i	$-0.902464\pi$
$263$	−86811.6	−	86811.6i	−1.25507	−	1.25507i	−0.301646	−	0.953420i	$-0.597536\pi$
$264$		0			0
$265$	−39562.6	−	28470.9i	−0.563370	−	0.405425i
$266$	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	$0.223412\pi$
$269$			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
$273$		0			0
$274$			2317.76i			0.0308721i
$275$	−99230.5	+	33239.1i	−1.31214	+	0.439525i
$276$		0			0
$277$	65214.7	−	65214.7i	0.849935	−	0.849935i	−0.140189	−	0.990125i	$-0.544771\pi$
$277$	65214.7	−	65214.7i	0.849935	−	0.849935i	0.990125	+	0.140189i	$0.0447711\pi$
$278$	−72316.5	−	72316.5i	−0.935724	−	0.935724i
$279$		0			0
$280$	35088.2	−	5720.51i	0.447553	−	0.0729657i
$281$	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	$-0.317237\pi$
$283$	−23747.0	−	23747.0i	−0.296508	−	0.296508i	−0.839645	+	0.543136i	$0.817237\pi$
$284$			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
$290$	48201.2	+	34687.6i	0.573142	+	0.412457i
$291$		0			0
$292$	54921.9	−	54921.9i	0.644139	−	0.644139i
$293$	55077.2	+	55077.2i	0.641559	+	0.641559i	0.950939	−	0.309380i	$-0.100121\pi$
$293$	55077.2	+	55077.2i	0.641559	+	0.641559i	−0.309380	+	0.950939i	$0.600121\pi$
$294$		0			0
$295$	1317.45	+	8080.88i	0.0151387	+	0.0928569i
$296$	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
$305$	103771.	−	16918.1i	1.11552	−	0.181866i
$306$		0			0
$307$	11961.1	−	11961.1i	0.126910	−	0.126910i	−0.640799	−	0.767709i	$-0.721397\pi$
$307$	11961.1	−	11961.1i	0.126910	−	0.126910i	0.767709	+	0.640799i	$0.221397\pi$
$308$	59527.1	+	59527.1i	0.627499	+	0.627499i
$309$		0			0
$310$	−16677.6	+	23174.9i	−0.173545	+	0.241154i
$311$	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	$-0.264751\pi$
$313$	−6418.45	−	6418.45i	−0.0655151	−	0.0655151i	−0.739105	+	0.673590i	$0.764751\pi$
$314$		−	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	$-0.799702\pi$
$317$	−22100.1	+	22100.1i	−0.219926	+	0.219926i	0.588541	+	0.808467i	$0.299702\pi$
$318$		0			0
$319$			140621.i			1.38187i
$320$	−2059.62	−	12633.2i	−0.0201135	−	0.123371i
$321$		0			0
$322$	−15586.1	+	15586.1i	−0.150323	+	0.150323i
$323$	19331.0	+	19331.0i	0.185289	+	0.185289i
$324$		0			0
$325$	−176918.	−	88133.8i	−1.67496	−	0.834403i
$326$	−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
$335$	−58505.6	+	81298.2i	−0.521324	+	0.724422i
$336$		0			0
$337$	−142816.	+	142816.i	−1.25753	+	1.25753i	−0.305260	+	0.952269i	$0.598743\pi$
$337$	−142816.	+	142816.i	−1.25753	+	1.25753i	−0.952269	+	0.305260i	$0.901257\pi$
$338$	142903.	+	142903.i	1.25086	+	1.25086i
$339$		0			0
$340$	16303.3	+	11732.5i	0.141032	+	0.101492i
$341$	−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	$-0.709891\pi$
$347$	21400.3	−	21400.3i	0.177730	−	0.177730i	0.790366	+	0.612635i	$0.209891\pi$
$348$		0			0
$349$			34091.2i			0.279893i	0.990159	+	0.139946i	$0.0446930\pi$
$349$			34091.2i			0.279893i	−0.990159	+	0.139946i	$0.955307\pi$
$350$	−35287.3	−	105345.i	−0.288060	−	0.859961i
$351$		0			0
$352$	21432.2	−	21432.2i	0.172974	−	0.172974i
$353$	−28.7539	−	28.7539i	−0.000230753	−	0.000230753i	0.706991	−	0.707222i	$-0.250052\pi$
$353$	−28.7539	−	28.7539i	−0.000230753	−	0.000230753i	−0.707222	+	0.706991i	$0.750052\pi$
$354$		0			0
$355$	131736.	−	21477.3i	1.04532	−	0.170421i
$356$	−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	$0.235778\pi$
$359$			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
$363$		0			0
$364$			159001.i			1.20004i
$365$	−197011.	−	141778.i	−1.47879	−	1.06420i
$366$		0			0
$367$	−41954.1	+	41954.1i	−0.311489	+	0.311489i	−0.845486	−	0.533997i	$-0.820689\pi$
$367$	−41954.1	+	41954.1i	−0.311489	+	0.311489i	0.533997	+	0.845486i	$0.320689\pi$
$368$	5611.66	+	5611.66i	0.0414377	+	0.0414377i
$369$		0			0
$370$	−3340.28	−	20488.4i	−0.0243994	−	0.149660i
$371$	122531.			0.890223
$372$		0			0
$373$	−65875.2	−	65875.2i	−0.473483	−	0.473483i	0.429557	−	0.903040i	$-0.358670\pi$
$373$	−65875.2	−	65875.2i	−0.473483	−	0.473483i	−0.903040	+	0.429557i	$0.858670\pi$
$374$			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	$0.0120103\pi$
$379$			10837.0i			0.0754448i	−0.999288	+	0.0377224i	$0.987990\pi$
$380$	−53732.8	+	8760.20i	−0.372111	+	0.0606662i
$381$		0			0
$382$	−79898.0	+	79898.0i	−0.547531	+	0.547531i
$383$	−22327.5	−	22327.5i	−0.152210	−	0.152210i	0.626894	−	0.779104i	$-0.284326\pi$
$383$	−22327.5	−	22327.5i	−0.152210	−	0.152210i	−0.779104	+	0.626894i	$0.784326\pi$
$384$		0			0
$385$	153666.	−	213531.i	1.03671	−	1.44058i
$386$	−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	$-0.764713\pi$
$389$		−	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
$393$		0			0
$394$		−	10900.4i		−	0.0702181i
$395$	18772.7	+	115147.i	0.120319	+	0.738004i
$396$		0			0
$397$	117299.	−	117299.i	0.744238	−	0.744238i	−0.229153	−	0.973390i	$-0.573596\pi$
$397$	117299.	−	117299.i	0.744238	−	0.744238i	0.973390	+	0.229153i	$0.0735955\pi$
$398$	−126425.	−	126425.i	−0.798118	−	0.798118i
$399$		0			0
$400$	−37928.7	+	12704.9i	−0.237054	+	0.0794057i
$401$	−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	$-0.812252\pi$
$409$		−	186090.i		−	1.11244i	0.831036	−	0.556219i	$-0.187748\pi$
$410$	−57262.2	+	79570.3i	−0.340644	+	0.473351i
$411$		0			0
$412$	−19314.1	+	19314.1i	−0.113784	+	0.113784i
$413$	−14554.0	−	14554.0i	−0.0853261	−	0.0853261i
$414$		0			0
$415$	118001.	+	84918.6i	0.685157	+	0.493068i
$416$	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	$-0.906465\pi$
$419$		−	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
$423$		0			0
$424$		−	44116.3i		−	0.245396i
$425$	27988.5	−	56183.4i	0.154953	−	0.311050i
$426$		0			0
$427$	−186897.	+	186897.i	−1.02505	+	1.02505i
$428$	34232.2	+	34232.2i	0.186873	+	0.186873i
$429$		0			0
$430$	−56098.6	+	9145.90i	−0.303400	+	0.0494641i
$431$	−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	$-0.0959040\pi$
$433$	123405.	+	123405.i	0.658201	+	0.658201i	−0.296754	+	0.954954i	$0.595904\pi$
$434$		−	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	$-0.758494\pi$
$439$		−	265180.i		−	1.37598i	0.725721	−	0.687989i	$-0.241506\pi$
$440$	−76879.9	−	55326.0i	−0.397107	−	0.285775i
$441$		0			0
$442$	−63521.6	+	63521.6i	−0.325145	+	0.325145i
$443$	99815.9	+	99815.9i	0.508619	+	0.508619i	0.914102	−	0.405484i	$-0.132897\pi$
$443$	99815.9	+	99815.9i	0.508619	+	0.508619i	−0.405484	+	0.914102i	$0.632897\pi$
$444$		0			0
$445$	39871.5	+	244562.i	0.201346	+	1.23500i
$446$	115271.			0.579496
$447$		0			0
$448$	22752.9	+	22752.9i	0.113366	+	0.113366i
$449$			182934.i			0.907408i	0.891152	+	0.453704i	$0.149898\pi$
$449$			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
$453$		0			0
$454$			3536.95i			0.0171600i
$455$	490403.	−	79951.7i	2.36881	−	0.386193i
$456$		0			0
$457$	147497.	−	147497.i	0.706239	−	0.706239i	−0.259503	−	0.965742i	$-0.583559\pi$
$457$	147497.	−	147497.i	0.706239	−	0.706239i	0.965742	+	0.259503i	$0.0835588\pi$
$458$	24152.8	+	24152.8i	0.115143	+	0.115143i
$459$		0			0
$460$	14486.2	−	20129.7i	0.0684601	−	0.0951308i
$461$	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	$-0.0444090\pi$
$463$	182475.	+	182475.i	0.851221	+	0.851221i	−0.139063	+	0.990284i	$0.544409\pi$
$464$			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	$-0.754488\pi$
$467$	−4348.30	+	4348.30i	−0.0199382	+	0.0199382i	0.697067	+	0.717006i	$0.254488\pi$
$468$		0			0
$469$		−	251792.i		−	1.14471i
$470$	42524.4	+	260833.i	0.192505	+	1.18078i
$471$		0			0
$472$	−5240.04	+	5240.04i	−0.0235207	+	0.0235207i
$473$	−95171.3	−	95171.3i	−0.425386	−	0.425386i
$474$		0			0
$475$	54037.8	+	161322.i	0.239503	+	0.715001i
$476$	−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	$-0.946091\pi$
$479$		−	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
$483$		0			0
$484$		−	107159.i		−	0.457445i
$485$	220742.	−	306738.i	0.938428	−	1.30402i
$486$		0			0
$487$	77225.6	−	77225.6i	0.325614	−	0.325614i	−0.525302	−	0.850916i	$-0.676048\pi$
$487$	77225.6	−	77225.6i	0.325614	−	0.325614i	0.850916	+	0.525302i	$0.176048\pi$
$488$	67290.6	+	67290.6i	0.282562	+	0.282562i
$489$		0			0
$490$	88885.9	+	63966.1i	0.370204	+	0.266414i
$491$	−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	$0.231442\pi$
$499$			331024.i			1.32941i	−0.747108	+	0.664703i	$0.768558\pi$
$500$	58257.5	+	110594.i	0.233030	+	0.442377i
$501$		0			0
$502$	139336.	−	139336.i	0.552910	−	0.552910i
$503$	123301.	+	123301.i	0.487338	+	0.487338i	0.907465	−	0.420127i	$-0.138015\pi$
$503$	123301.	+	123301.i	0.487338	+	0.487338i	−0.420127	+	0.907465i	$0.638015\pi$
$504$		0			0
$505$	−43503.0	+	7092.40i	−0.170583	+	0.0278106i
$506$	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	$0.248555\pi$
$509$			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
$513$		0			0
$514$		−	46457.8i		−	0.175846i
$515$	69282.1	+	49858.3i	0.261220	+	0.187985i
$516$		0			0
$517$	−442504.	+	442504.i	−1.65553	+	1.65553i
$518$	36900.5	+	36900.5i	0.137522	+	0.137522i
$519$		0			0
$520$	−28785.9	−	176566.i	−0.106457	−	0.652979i
$521$	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	$-0.189857\pi$
$523$	72655.5	+	72655.5i	0.265623	+	0.265623i	−0.561711	+	0.827334i	$0.689857\pi$
$524$			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
$530$	−136067.	+	22183.4i	−0.484397	+	0.0789725i
$531$		0			0
$532$	96775.0	−	96775.0i	0.341932	−	0.341932i
$533$	−310026.	−	310026.i	−1.09130	−	1.09130i
$534$		0			0
$535$	88368.4	−	122795.i	0.308738	−	0.429015i
$536$	−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
$545$	47546.5	+	291638.i	0.160076	+	0.981863i
$546$		0			0
$547$	−9895.50	+	9895.50i	−0.0330722	+	0.0330722i	−0.723450	−	0.690377i	$-0.757445\pi$
$547$	−9895.50	+	9895.50i	−0.0330722	+	0.0330722i	0.690377	+	0.723450i	$0.257445\pi$
$548$	4635.51	+	4635.51i	0.0154361	+	0.0154361i
$549$		0			0
$550$	−131983.	+	264939.i	−0.436307	+	0.875832i
$551$	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	$-0.503519\pi$
$557$	306800.	−	306800.i	0.988883	−	0.988883i	0.999939	+	0.0110558i	$0.00351924\pi$
$558$		0			0
$559$		−	254209.i		−	0.813518i
$560$	58735.3	−	81617.4i	0.187294	−	0.260259i
$561$		0			0
$562$	17338.7	−	17338.7i	0.0548963	−	0.0548963i
$563$	−331132.	−	331132.i	−1.04468	−	1.04468i	−0.998954	−	0.0457279i	$-0.985439\pi$
$563$	−331132.	−	331132.i	−1.04468	−	1.04468i	−0.0457279	−	0.998954i	$-0.514561\pi$
$564$		0			0
$565$	46506.4	+	33467.9i	0.145685	+	0.104841i
$566$	−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	$-0.869873\pi$
$569$		−	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
$573$		0			0
$574$		−	246441.i		−	0.747978i
$575$	−69369.7	−	34557.4i	−0.209814	−	0.104521i
$576$		0			0
$577$	−23928.8	+	23928.8i	−0.0718736	+	0.0718736i	−0.742130	−	0.670256i	$-0.766184\pi$
$577$	−23928.8	+	23928.8i	−0.0718736	+	0.0718736i	0.670256	+	0.742130i	$0.266184\pi$
$578$	146870.	+	146870.i	0.439619	+	0.439619i
$579$		0			0
$580$	165778.	−	27027.2i	0.492799	−	0.0803424i
$581$	−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	$-0.871231\pi$
$587$	−181136.	+	181136.i	−0.525690	+	0.525690i	0.393595	+	0.919284i	$0.371231\pi$
$588$		0			0
$589$			109915.i			0.316831i
$590$	18796.6	+	13526.9i	0.0539978	+	0.0388591i
$591$		0			0
$592$	13285.7	−	13285.7i	0.0379089	−	0.0379089i
$593$	235890.	+	235890.i	0.670812	+	0.670812i	0.957903	−	0.287091i	$-0.0926883\pi$
$593$	235890.	+	235890.i	0.670812	+	0.670812i	−0.287091	+	0.957903i	$0.592688\pi$
$594$		0			0
$595$	25390.1	+	155736.i	0.0717183	+	0.439902i
$596$	−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	$0.0632381\pi$
$599$			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
$603$		0			0
$604$			71524.7i			0.196057i
$605$	−330509.	+	53883.8i	−0.902969	+	0.147213i
$606$		0			0
$607$	247253.	−	247253.i	0.671064	−	0.671064i	−0.286898	−	0.957961i	$-0.592624\pi$
$607$	247253.	−	247253.i	0.671064	−	0.671064i	0.957961	+	0.286898i	$0.0926240\pi$
$608$	−34843.0	−	34843.0i	−0.0942560	−	0.0942560i
$609$		0			0
$610$	173707.	−	241379.i	0.466828	−	0.648694i
$611$	−1.18196e6			−3.16606
$612$		0			0
$613$	−163758.	−	163758.i	−0.435794	−	0.435794i	0.454800	−	0.890594i	$-0.349711\pi$
$613$	−163758.	−	163758.i	−0.435794	−	0.435794i	−0.890594	+	0.454800i	$0.849711\pi$
$614$		−	47844.5i		−	0.126910i
$615$		0			0
$616$	238108.			0.627499
$617$	−273782.	+	273782.i	−0.719176	+	0.719176i	−0.968437	−	0.249260i	$-0.919812\pi$
$617$	−273782.	+	273782.i	−0.719176	+	0.719176i	0.249260	+	0.968437i	$0.419812\pi$
$618$		0			0
$619$			428345.i			1.11792i	0.829194	+	0.558962i	$0.188800\pi$
$619$			428345.i			1.11792i	−0.829194	+	0.558962i	$0.811200\pi$
$620$	12994.5	+	79705.0i	0.0338047	+	0.207349i
$621$		0			0
$622$	130569.	−	130569.i	0.337489	−	0.337489i
$623$	−440465.	−	440465.i	−1.13484	−	1.13484i
$624$		0			0
$625$	311809.	−	235293.i	0.798231	−	0.602351i
$626$	−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
$635$	238989.	−	332094.i	0.592694	−	0.823595i
$636$		0			0
$637$	−346322.	+	346322.i	−0.853495	+	0.853495i
$638$	281242.	+	281242.i	0.690937	+	0.690937i
$639$		0			0
$640$	−29385.7	−	21147.2i	−0.0717423	−	0.0516288i
$641$	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	$-0.314266\pi$
$643$	−117251.	−	117251.i	−0.283592	−	0.283592i	−0.834540	+	0.550948i	$0.814266\pi$
$644$			62344.5i			0.150323i
$645$		0			0
$646$	77324.1			0.185289
$647$	362031.	−	362031.i	0.864843	−	0.864843i	−0.127053	−	0.991896i	$-0.540552\pi$
$647$	362031.	−	362031.i	0.864843	−	0.864843i	0.991896	+	0.127053i	$0.0405519\pi$
$648$		0			0
$649$			54836.8i			0.130192i
$650$	−530103.	+	177568.i	−1.25468	+	0.420279i
$651$		0			0
$652$	−54872.0	+	54872.0i	−0.129079	+	0.129079i
$653$	411293.	+	411293.i	0.964552	+	0.964552i	0.999393	−	0.0348413i	$-0.0110926\pi$
$653$	411293.	+	411293.i	0.964552	+	0.964552i	−0.0348413	+	0.999393i	$0.511093\pi$
$654$		0			0
$655$	1619.20	−	263.982i	0.00377413	−	0.000615306i
$656$	−88729.0			−0.206185
$657$		0			0
$658$	−469772.	−	469772.i	−1.08501	−	1.08501i
$659$			291952.i			0.672265i	0.941815	+	0.336133i	$0.109119\pi$
$659$			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
$663$		0			0
$664$			131583.i			0.298445i
$665$	−347143.	−	249819.i	−0.784992	−	0.564914i
$666$		0			0
$667$	−73638.3	+	73638.3i	−0.165521	+	0.165521i
$668$	−299514.	−	299514.i	−0.671219	−	0.671219i
$669$		0			0
$670$	45585.2	+	279608.i	0.101549	+	0.622873i
$671$	704193.			1.56404
$672$		0			0
$673$	310335.	+	310335.i	0.685173	+	0.685173i	0.961161	−	0.275988i	$-0.0890050\pi$
$673$	310335.	+	310335.i	0.685173	+	0.685173i	−0.275988	+	0.961161i	$0.589005\pi$
$674$			571265.i			1.25753i
$675$		0			0
$676$	571613.			1.25086
$677$	−9203.97	+	9203.97i	−0.0200816	+	0.0200816i	−0.717076	−	0.696995i	$-0.754520\pi$
$677$	−9203.97	+	9203.97i	−0.0200816	+	0.0200816i	0.696995	+	0.717076i	$0.254520\pi$
$678$		0			0
$679$			950012.i			2.06058i
$680$	56071.5	−	9141.49i	0.121262	−	0.0197697i
$681$		0			0
$682$	−135220.	+	135220.i	−0.290717	+	0.290717i
$683$	421590.	+	421590.i	0.903750	+	0.903750i	0.995758	−	0.0920080i	$-0.0293285\pi$
$683$	421590.	+	421590.i	0.903750	+	0.903750i	−0.0920080	+	0.995758i	$0.529329\pi$
$684$		0			0
$685$	11966.3	−	16628.1i	0.0255023	−	0.0354374i
$686$	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
$695$	145454.	+	892177.i	0.301131	+	1.84706i
$696$		0			0
$697$	98454.3	−	98454.3i	0.202660	−	0.202660i
$698$	68182.4	+	68182.4i	0.139946	+	0.139946i
$699$		0			0
$700$	−281265.	−	140116.i	−0.574010	−	0.285951i
$701$	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	$0.0523870\pi$
$709$			164715.i			0.327673i	−0.986487	+	0.163837i	$0.947613\pi$
$710$	220518.	−	306428.i	0.437449	−	0.607871i
$711$		0			0
$712$	−158586.	+	158586.i	−0.312827	+	0.312827i
$713$	−35404.9	−	35404.9i	−0.0696441	−	0.0696441i
$714$		0			0
$715$	−1.07450e6	−	773254.i	−2.10181	−	1.51255i
$716$	−51169.5			−0.0998126
$717$		0			0
$718$	347885.	+	347885.i	0.674818	+	0.674818i
$719$		−	782798.i		−	1.51423i	−0.653282	−	0.757115i	$-0.726608\pi$
$719$		−	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
$723$		0			0
$724$		−	392208.i		−	0.748236i
$725$	−166719.	−	497715.i	−0.317182	−	0.946900i
$726$		0			0
$727$	659110.	−	659110.i	1.24706	−	1.24706i	0.290054	−	0.957010i	$-0.406327\pi$
$727$	659110.	−	659110.i	1.24706	−	1.24706i	0.957010	−	0.290054i	$-0.0936732\pi$
$728$	318002.	+	318002.i	0.600021	+	0.600021i
$729$		0			0
$730$	−677578.	+	110467.i	−1.27149	+	0.207295i
$731$	80728.6			0.151075
$732$		0			0
$733$	356952.	+	356952.i	0.664358	+	0.664358i	0.956404	−	0.292046i	$-0.0943361\pi$
$733$	356952.	+	356952.i	0.664358	+	0.664358i	−0.292046	+	0.956404i	$0.594336\pi$
$734$			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	$-0.931952\pi$
$739$		−	231725.i		−	0.424311i	0.977236	−	0.212156i	$-0.0680483\pi$
$740$	−47657.4	−	34296.3i	−0.0870296	−	0.0626302i
$741$		0			0
$742$	245062.	−	245062.i	0.445111	−	0.445111i
$743$	35964.9	+	35964.9i	0.0651480	+	0.0651480i	0.738930	−	0.673782i	$-0.235331\pi$
$743$	35964.9	+	35964.9i	0.0651480	+	0.0651480i	−0.673782	+	0.738930i	$0.735331\pi$
$744$		0			0
$745$	8624.43	+	52900.1i	0.0155388	+	0.0953111i
$746$	−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
$755$	220602.	−	35965.3i	0.387004	−	0.0630943i
$756$		0			0
$757$	−111880.	+	111880.i	−0.195236	+	0.195236i	−0.797954	−	0.602718i	$-0.794084\pi$
$757$	−111880.	+	111880.i	−0.195236	+	0.195236i	0.602718	+	0.797954i	$0.294084\pi$
$758$	21673.9	+	21673.9i	0.0377224	+	0.0377224i
$759$		0			0
$760$	−89945.3	+	124986.i	−0.155722	+	0.216389i
$761$	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	$-0.869298\pi$
$769$		−	472110.i		−	0.798344i	0.916876	−	0.399172i	$-0.130702\pi$
$770$	−119730.	−	734392.i	−0.201939	−	1.23864i
$771$		0			0
$772$	−189993.	+	189993.i	−0.318789	+	0.318789i
$773$	20353.7	+	20353.7i	0.0340631	+	0.0340631i	0.723933	−	0.689870i	$-0.242332\pi$
$773$	20353.7	+	20353.7i	0.0340631	+	0.0340631i	−0.689870	+	0.723933i	$0.742332\pi$
$774$		0			0
$775$	239298.	−	80157.4i	0.398416	−	0.133457i
$776$	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
$785$	−372406.	+	517488.i	−0.604335	+	0.839771i
$786$		0			0
$787$	640630.	−	640630.i	1.03433	−	1.03433i	0.0349373	−	0.999390i	$-0.488877\pi$
$787$	640630.	−	640630.i	1.03433	−	1.03433i	0.999390	−	0.0349373i	$-0.0111232\pi$
$788$	−21800.7	−	21800.7i	−0.0351090	−	0.0351090i
$789$		0			0
$790$	267840.	+	192749.i	0.429161	+	0.308843i
$791$	−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	$-0.956515\pi$
$797$	−542782.	+	542782.i	−0.854494	+	0.854494i	0.136189	+	0.990683i	$0.456515\pi$
$798$		0			0
$799$		−	375352.i		−	0.587956i
$800$	−50447.5	+	101267.i	−0.0788243	+	0.158230i
$801$		0			0
$802$	−86186.2	+	86186.2i	−0.133995	+	0.133995i
$803$	−1.14951e6	−	1.14951e6i	−1.78271	−	1.78271i
$804$		0			0
$805$	192288.	−	31349.2i	0.296729	−	0.0483765i
$806$	−361181.			−0.555974
$807$		0			0
$808$	−28209.5	−	28209.5i	−0.0432088	−	0.0432088i
$809$			1.09699e6i			1.67613i	0.545571	+	0.838065i	$0.316313\pi$
$809$			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
$813$		0			0
$814$		−	139034.i		−	0.209833i
$815$	196832.	+	141649.i	0.296334	+	0.213254i
$816$		0			0
$817$	−154723.	+	154723.i	−0.231798	+	0.231798i
$818$	−372179.	−	372179.i	−0.556219	−	0.556219i
$819$		0			0
$820$	44616.3	+	273665.i	0.0663538	+	0.406997i
$821$	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	−0.999998	−	0.00176504i	$-0.999438\pi$
$823$	−678523.	−	678523.i	−1.00176	−	1.00176i	−0.00176504	−	0.999998i	$-0.500562\pi$
$824$			77256.6i			0.113784i
$825$		0			0
$826$	−58215.9			−0.0853261
$827$	822466.	−	822466.i	1.20256	−	1.20256i	0.229174	−	0.973385i	$-0.426397\pi$
$827$	822466.	−	822466.i	1.20256	−	1.20256i	0.973385	−	0.229174i	$-0.0736026\pi$
$828$		0			0
$829$			1.26805e6i			1.84514i	0.385832	+	0.922569i	$0.373914\pi$
$829$			1.26805e6i			1.84514i	−0.385832	+	0.922569i	$0.626086\pi$
$830$	405839.	−	66165.0i	0.589112	−	0.0960445i
$831$		0			0
$832$	114494.	−	114494.i	0.165400	−	0.165400i
$833$	−109981.	−	109981.i	−0.158499	−	0.158499i
$834$		0			0
$835$	−773178.	+	1.07439e6i	−1.10894	+	1.54095i
$836$	−364631.			−0.521724
$837$		0			0
$838$	−203397.	−	203397.i	−0.289639	−	0.289639i
$839$		−	487091.i		−	0.691968i	−0.938240	−	0.345984i	$-0.887545\pi$
$839$		−	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
$843$		0			0
$844$		−	123761.i		−	0.173740i
$845$	−287429.	−	1.76301e6i	−0.402547	−	2.46912i
$846$		0			0
$847$	595260.	−	595260.i	0.829736	−	0.829736i
$848$	−88232.7	−	88232.7i	−0.122698	−	0.122698i
$849$		0			0
$850$	−56389.8	−	168344.i	−0.0780482	−	0.233002i
$851$	36403.7			0.0502675
$852$		0			0
$853$	10645.4	+	10645.4i	0.0146307	+	0.0146307i	0.714384	−	0.699754i	$-0.246707\pi$
$853$	10645.4	+	10645.4i	0.0146307	+	0.0146307i	−0.699754	+	0.714384i	$0.746707\pi$
$854$			747586.i			1.02505i
$855$		0			0
$856$	136929.			0.186873
$857$	692563.	−	692563.i	0.942970	−	0.942970i	−0.0554896	−	0.998459i	$-0.517672\pi$
$857$	692563.	−	692563.i	0.942970	−	0.942970i	0.998459	+	0.0554896i	$0.0176720\pi$
$858$		0			0
$859$		−	745974.i		−	1.01097i	−0.862836	−	0.505484i	$-0.831314\pi$
$859$		−	745974.i		−	1.01097i	0.862836	−	0.505484i	$-0.168686\pi$
$860$	−93905.5	+	130489.i	−0.126968	+	0.176432i
$861$		0			0
$862$	−421196.	+	421196.i	−0.566852	+	0.566852i
$863$	−259362.	−	259362.i	−0.348245	−	0.348245i	0.511211	−	0.859456i	$-0.329197\pi$
$863$	−259362.	−	259362.i	−0.348245	−	0.348245i	−0.859456	+	0.511211i	$0.829197\pi$
$864$		0			0
$865$	−936618.	−	674030.i	−1.25179	−	0.900838i
$866$	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
$875$	−290725.	+	937956.i	−0.379723	+	1.22509i
$876$		0			0
$877$	−191986.	+	191986.i	−0.249615	+	0.249615i	−0.820813	−	0.571197i	$-0.806479\pi$
$877$	−191986.	+	191986.i	−0.249615	+	0.249615i	0.571197	+	0.820813i	$0.306479\pi$
$878$	−530360.	−	530360.i	−0.687989	−	0.687989i
$879$		0			0
$880$	−264412.	+	43107.8i	−0.341441	+	0.0556660i
$881$	−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	$-0.163909\pi$
$883$	294604.	+	294604.i	0.377848	+	0.377848i	−0.492477	+	0.870325i	$0.663909\pi$
$884$			254086.i			0.325145i
$885$		0			0
$886$	399264.			0.508619
$887$	373328.	−	373328.i	0.474507	−	0.474507i	−0.428862	−	0.903370i	$-0.641085\pi$
$887$	373328.	−	373328.i	0.474507	−	0.474507i	0.903370	+	0.428862i	$0.141085\pi$
$888$		0			0
$889$			1.02854e6i			1.30143i
$890$	568866.	+	409380.i	0.718175	+	0.516829i
$891$		0			0
$892$	230542.	−	230542.i	0.289748	−	0.289748i
$893$	719392.	+	719392.i	0.902117	+	0.902117i
$894$		0			0
$895$	25730.0	+	157821.i	0.0321213	+	0.197024i
$896$	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
$905$	−1.20968e6	+	197217.i	−1.47697	+	0.240795i
$906$		0			0
$907$	−342694.	+	342694.i	−0.416574	+	0.416574i	−0.884021	−	0.467447i	$-0.845174\pi$
$907$	−342694.	+	342694.i	−0.416574	+	0.416574i	0.467447	+	0.884021i	$0.345174\pi$
$908$	7073.89	+	7073.89i	0.00857999	+	0.00857999i
$909$		0			0
$910$	820903.	−	1.14071e6i	0.991309	−	1.37750i
$911$	−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	$0.0600465\pi$
$919$			316752.i			0.375050i	−0.982260	+	0.187525i	$0.939953\pi$
$920$	−11287.0	−	69231.7i	−0.0133353	−	0.0817955i
$921$		0			0
$922$	709825.	−	709825.i	0.835006	−	0.835006i
$923$	1.19392e6	+	1.19392e6i	1.40143	+	1.40143i
$924$		0			0
$925$	−81815.4	+	164234.i	−0.0956206	+	0.191947i
$926$	729901.			0.851221
$927$		0			0
$928$	107499.	+	107499.i	0.124826	+	0.124826i
$929$		−	1.39646e6i		−	1.61807i	−0.587759	−	0.809036i	$-0.699990\pi$
$929$		−	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
$933$		0			0
$934$			17393.2i			0.0199382i
$935$	245560.	−	341226.i	0.280889	−	0.390318i
$936$		0			0
$937$	81884.2	−	81884.2i	0.0932655	−	0.0932655i	−0.658935	−	0.752200i	$-0.728993\pi$
$937$	81884.2	−	81884.2i	0.0932655	−	0.0932655i	0.752200	+	0.658935i	$0.228993\pi$
$938$	−503585.	−	503585.i	−0.572357	−	0.572357i
$939$		0			0
$940$	606716.	+	436618.i	0.686641	+	0.494136i
$941$	−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	$-0.932988\pi$
$947$	−689602.	+	689602.i	−0.768950	+	0.768950i	0.208971	+	0.977922i	$0.432988\pi$
$948$		0			0
$949$		−	3.07042e6i		−	3.40930i
$950$	430720.	+	214569.i	0.477252	+	0.237749i
$951$		0			0
$952$	−100987.	+	100987.i	−0.111427	+	0.111427i
$953$	−150337.	−	150337.i	−0.165532	−	0.165532i	0.619480	−	0.785012i	$-0.287343\pi$
$953$	−150337.	−	150337.i	−0.165532	−	0.165532i	−0.785012	+	0.619480i	$0.787343\pi$
$954$		0			0
$955$	985710.	−	160703.i	1.08079	−	0.176204i
$956$	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
$965$	681527.	+	490456.i	0.731861	+	0.526678i
$966$		0			0
$967$	−312723.	+	312723.i	−0.334432	+	0.334432i	−0.854267	−	0.519835i	$-0.825993\pi$
$967$	−312723.	+	312723.i	−0.334432	+	0.334432i	0.519835	+	0.854267i	$0.325993\pi$
$968$	−214319.	−	214319.i	−0.228723	−	0.228723i
$969$		0			0
$970$	−171993.	−	1.05496e6i	−0.182796	−	1.12122i
$971$	−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	$-0.562927\pi$
$977$	748461.	−	748461.i	0.784115	−	0.784115i	0.980522	+	0.196407i	$0.0629274\pi$
$978$		0			0
$979$			1.65959e6i			1.73156i
$980$	305704.	−	49839.7i	0.318309	−	0.0518947i
$981$		0			0
$982$	−680434.	+	680434.i	−0.705607	+	0.705607i
$983$	542812.	+	542812.i	0.561749	+	0.561749i	0.929804	−	0.368055i	$-0.119976\pi$
$983$	542812.	+	542812.i	0.561749	+	0.561749i	−0.368055	+	0.929804i	$0.619976\pi$
$984$		0			0
$985$	−56277.3	+	78201.8i	−0.0580044	+	0.0806017i
$986$	−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
$995$	254285.	+	1.55972e6i	0.256847	+	1.57544i
$996$		0			0
$997$	25593.2	−	25593.2i	0.0257475	−	0.0257475i	−0.694116	−	0.719863i	$-0.744204\pi$
$997$	25593.2	−	25593.2i	0.0257475	−	0.0257475i	0.719863	+	0.694116i	$0.244204\pi$
$998$	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	90.5.g.e.73.1		4
3.2	odd	2		30.5.f.a.13.1	yes	4
5.2	odd	4	inner	90.5.g.e.37.1		4
5.3	odd	4		450.5.g.f.307.1		4
5.4	even	2		450.5.g.f.343.1		4
12.11	even	2		240.5.bg.b.193.2		4
15.2	even	4		30.5.f.a.7.1	✓	4
15.8	even	4		150.5.f.e.7.2		4
15.14	odd	2		150.5.f.e.43.2		4
60.47	odd	4		240.5.bg.b.97.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.5.f.a.7.1	✓	4	15.2	even	4
30.5.f.a.13.1	yes	4	3.2	odd	2
90.5.g.e.37.1		4	5.2	odd	4	inner
90.5.g.e.73.1		4	1.1	even	1	trivial
150.5.f.e.7.2		4	15.8	even	4
150.5.f.e.43.2		4	15.14	odd	2
240.5.bg.b.97.2		4	60.47	odd	4
240.5.bg.b.193.2		4	12.11	even	2
450.5.g.f.307.1		4	5.3	odd	4
450.5.g.f.343.1		4	5.4	even	2

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	90.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-24.6742 + 4.02270i) q^{5} +(44.4393 - 44.4393i) q^{7} +(-16.0000 - 16.0000i) q^{8} +O(q^{10})\)	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-24.6742 + 4.02270i) q^{5} +(44.4393 - 44.4393i) q^{7} +(-16.0000 - 16.0000i) q^{8} +(-41.3031 + 57.3939i) q^{10} -167.439 q^{11} +(-223.621 - 223.621i) q^{13} -177.757i q^{14} -64.0000 q^{16} +(71.0148 - 71.0148i) q^{17} +272.211i q^{19} +(32.1816 + 197.394i) q^{20} +(-334.879 + 334.879i) q^{22} +(-87.6821 - 87.6821i) q^{23} +(592.636 - 198.514i) q^{25} -894.484 q^{26} +(-355.514 - 355.514i) q^{28} -839.832i q^{29} +403.787 q^{31} +(-128.000 + 128.000i) q^{32} -284.059i q^{34} +(-917.739 + 1275.27i) q^{35} +(-207.589 + 207.589i) q^{37} +(544.422 + 544.422i) q^{38} +(459.151 + 330.424i) q^{40} +1386.39 q^{41} +(568.393 + 568.393i) q^{43} +1339.51i q^{44} -350.729 q^{46} +(2642.77 - 2642.77i) q^{47} -1548.70i q^{49} +(788.243 - 1582.30i) q^{50} +(-1788.97 + 1788.97i) q^{52} +(1378.64 + 1378.64i) q^{53} +(4131.44 - 673.559i) q^{55} -1422.06 q^{56} +(-1679.66 - 1679.66i) q^{58} -327.503i q^{59} -4205.66 q^{61} +(807.573 - 807.573i) q^{62} +512.000i q^{64} +(6417.24 + 4618.11i) q^{65} +(2832.99 - 2832.99i) q^{67} +(-568.118 - 568.118i) q^{68} +(715.064 + 4386.02i) q^{70} -5339.03 q^{71} +(6865.24 + 6865.24i) q^{73} +830.357i q^{74} +2177.69 q^{76} +(-7440.88 + 7440.88i) q^{77} -4666.69i q^{79} +(1579.15 - 257.453i) q^{80} +(2772.78 - 2772.78i) q^{82} +(-4111.98 - 4111.98i) q^{83} +(-1466.56 + 2037.91i) q^{85} +2273.57 q^{86} +(2679.03 + 2679.03i) q^{88} -9911.62i q^{89} -19875.1 q^{91} +(-701.457 + 701.457i) q^{92} -10571.1i q^{94} +(-1095.03 - 6716.60i) q^{95} +(-10688.9 + 10688.9i) q^{97} +(-3097.40 - 3097.40i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} - 84 q^{5} - 28 q^{7} - 64 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 - 84 * q^5 - 28 * q^7 - 64 * q^8	\( 4 q + 8 q^{2} - 84 q^{5} - 28 q^{7} - 64 q^{8} - 224 q^{10} - 464 q^{11} - 336 q^{13} - 256 q^{16} - 392 q^{17} - 224 q^{20} - 928 q^{22} - 968 q^{23} + 1136 q^{25} - 1344 q^{26} + 224 q^{28} - 560 q^{31} - 512 q^{32} + 2296 q^{35} + 2256 q^{37} - 1232 q^{38} + 896 q^{40} - 392 q^{41} + 216 q^{43} - 3872 q^{46} + 9072 q^{47} + 3976 q^{50} - 2688 q^{52} + 4280 q^{53} + 10500 q^{55} + 896 q^{56} - 2192 q^{58} - 4536 q^{61} - 1120 q^{62} + 12912 q^{65} - 2248 q^{67} + 3136 q^{68} + 9856 q^{70} - 18064 q^{71} + 20524 q^{73} - 4928 q^{76} - 7336 q^{77} + 5376 q^{80} - 784 q^{82} + 336 q^{83} + 15944 q^{85} + 864 q^{86} + 7424 q^{88} - 52752 q^{91} - 7744 q^{92} - 23104 q^{95} - 40404 q^{97} - 23912 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 - 84 * q^5 - 28 * q^7 - 64 * q^8 - 224 * q^10 - 464 * q^11 - 336 * q^13 - 256 * q^16 - 392 * q^17 - 224 * q^20 - 928 * q^22 - 968 * q^23 + 1136 * q^25 - 1344 * q^26 + 224 * q^28 - 560 * q^31 - 512 * q^32 + 2296 * q^35 + 2256 * q^37 - 1232 * q^38 + 896 * q^40 - 392 * q^41 + 216 * q^43 - 3872 * q^46 + 9072 * q^47 + 3976 * q^50 - 2688 * q^52 + 4280 * q^53 + 10500 * q^55 + 896 * q^56 - 2192 * q^58 - 4536 * q^61 - 1120 * q^62 + 12912 * q^65 - 2248 * q^67 + 3136 * q^68 + 9856 * q^70 - 18064 * q^71 + 20524 * q^73 - 4928 * q^76 - 7336 * q^77 + 5376 * q^80 - 784 * q^82 + 336 * q^83 + 15944 * q^85 + 864 * q^86 + 7424 * q^88 - 52752 * q^91 - 7744 * q^92 - 23104 * q^95 - 40404 * q^97 - 23912 * q^98

Introduction

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