LMFDB - Embedded newform 400.6.n.g.207.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 $ x^20 + 271x^18 + 109637x^16 + 25993614x^14 + 5522961902x^12 + 881545050522x^10 + 133816049059481x^8 + 14779507781220031x^6 + 824105698447750789x^4 + 12044868290803250652*x^2 + 579398322543528055824
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{67}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.3
Root			$-10.8505 + 10.2794i$ of defining polynomial
Character	$\chi$	$=$	400.207
Dual form			400.6.n.g.143.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.68301 - 9.68301i) q^{3} +(-48.6629 + 48.6629i) q^{7} -55.4787i q^{9} +O(q^{10})$	$q+(-9.68301 - 9.68301i) q^{3} +(-48.6629 + 48.6629i) q^{7} -55.4787i q^{9} -463.177i q^{11} +(-320.800 + 320.800i) q^{13} +(-1045.30 - 1045.30i) q^{17} +701.290 q^{19} +942.407 q^{21} +(2001.88 + 2001.88i) q^{23} +(-2890.17 + 2890.17i) q^{27} -3567.76i q^{29} -9044.72i q^{31} +(-4484.95 + 4484.95i) q^{33} +(-1642.14 - 1642.14i) q^{37} +6212.62 q^{39} -14338.6 q^{41} +(-3941.99 - 3941.99i) q^{43} +(7944.15 - 7944.15i) q^{47} +12070.8i q^{49} +20243.2i q^{51} +(-11621.9 + 11621.9i) q^{53} +(-6790.60 - 6790.60i) q^{57} +1121.30 q^{59} -29320.4 q^{61} +(2699.75 + 2699.75i) q^{63} +(-9199.75 + 9199.75i) q^{67} -38768.5i q^{69} +52643.9i q^{71} +(27965.6 - 27965.6i) q^{73} +(22539.6 + 22539.6i) q^{77} +82263.7 q^{79} +42489.8 q^{81} +(77236.8 + 77236.8i) q^{83} +(-34546.7 + 34546.7i) q^{87} +145955. i q^{89} -31222.2i q^{91} +(-87580.1 + 87580.1i) q^{93} +(-97856.7 - 97856.7i) q^{97} -25696.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) $ 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−9.68301	−	9.68301i	−0.621165	−	0.621165i	0.324664	−	0.945829i	$-0.394749\pi$
$3$	−9.68301	−	9.68301i	−0.621165	−	0.621165i	−0.945829	+	0.324664i	$0.894749\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−48.6629	+	48.6629i	−0.375364	+	0.375364i	−0.869427	−	0.494062i	$-0.835512\pi$
$7$	−48.6629	+	48.6629i	−0.375364	+	0.375364i	0.494062	+	0.869427i	$0.335512\pi$
$8$		0			0
$9$		−	55.4787i		−	0.228307i
$10$		0			0
$11$		−	463.177i		−	1.15416i	−0.816688	−	0.577080i	$-0.804192\pi$
$11$		−	463.177i		−	1.15416i	0.816688	−	0.577080i	$-0.195808\pi$
$12$		0			0
$13$	−320.800	+	320.800i	−0.526473	+	0.526473i	−0.919519	−	0.393046i	$-0.871421\pi$
$13$	−320.800	+	320.800i	−0.526473	+	0.526473i	0.393046	+	0.919519i	$0.371421\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1045.30	−	1045.30i	−0.877238	−	0.877238i	0.116010	−	0.993248i	$-0.462989\pi$
$17$	−1045.30	−	1045.30i	−0.877238	−	0.877238i	−0.993248	+	0.116010i	$0.962989\pi$
$18$		0			0
$19$	701.290			0.445670			0.222835	−	0.974856i	$-0.428469\pi$
$19$	701.290			0.445670			0.222835	+	0.974856i	$0.428469\pi$
$20$		0			0
$21$	942.407			0.466327
$22$		0			0
$23$	2001.88	+	2001.88i	0.789076	+	0.789076i	0.981343	−	0.192266i	$-0.0615838\pi$
$23$	2001.88	+	2001.88i	0.789076	+	0.789076i	−0.192266	+	0.981343i	$0.561584\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2890.17	+	2890.17i	−0.762982	+	0.762982i
$28$		0			0
$29$		−	3567.76i		−	0.787773i	−0.919159	−	0.393886i	$-0.871130\pi$
$29$		−	3567.76i		−	0.787773i	0.919159	−	0.393886i	$-0.128870\pi$
$30$		0			0
$31$		−	9044.72i		−	1.69041i	−0.534446	−	0.845203i	$-0.679480\pi$
$31$		−	9044.72i		−	1.69041i	0.534446	−	0.845203i	$-0.320520\pi$
$32$		0			0
$33$	−4484.95	+	4484.95i	−0.716924	+	0.716924i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1642.14	−	1642.14i	−0.197199	−	0.197199i	0.601599	−	0.798798i	$-0.294530\pi$
$37$	−1642.14	−	1642.14i	−0.197199	−	0.197199i	−0.798798	+	0.601599i	$0.794530\pi$
$38$		0			0
$39$	6212.62			0.654054
$40$		0			0
$41$	−14338.6			−1.33213			−0.666067	−	0.745892i	$-0.732024\pi$
$41$	−14338.6			−1.33213			−0.666067	+	0.745892i	$0.732024\pi$
$42$		0			0
$43$	−3941.99	−	3941.99i	−0.325120	−	0.325120i	0.525607	−	0.850727i	$-0.323838\pi$
$43$	−3941.99	−	3941.99i	−0.325120	−	0.325120i	−0.850727	+	0.525607i	$0.823838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7944.15	−	7944.15i	0.524569	−	0.524569i	−0.394379	−	0.918948i	$-0.629040\pi$
$47$	7944.15	−	7944.15i	0.524569	−	0.524569i	0.918948	+	0.394379i	$0.129040\pi$
$48$		0			0
$49$			12070.8i			0.718203i
$50$		0			0
$51$			20243.2i			1.08982i
$52$		0			0
$53$	−11621.9	+	11621.9i	−0.568313	+	0.568313i	−0.931656	−	0.363342i	$-0.881636\pi$
$53$	−11621.9	+	11621.9i	−0.568313	+	0.568313i	0.363342	+	0.931656i	$0.381636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6790.60	−	6790.60i	−0.276835	−	0.276835i
$58$		0			0
$59$	1121.30			0.0419365			0.0209683	−	0.999780i	$-0.493325\pi$
$59$	1121.30			0.0419365			0.0209683	+	0.999780i	$0.493325\pi$
$60$		0			0
$61$	−29320.4			−1.00889			−0.504447	−	0.863442i	$-0.668304\pi$
$61$	−29320.4			−1.00889			−0.504447	+	0.863442i	$0.668304\pi$
$62$		0			0
$63$	2699.75	+	2699.75i	0.0856984	+	0.0856984i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9199.75	+	9199.75i	−0.250374	+	0.250374i	−0.821124	−	0.570750i	$-0.806653\pi$
$67$	−9199.75	+	9199.75i	−0.250374	+	0.250374i	0.570750	+	0.821124i	$0.306653\pi$
$68$		0			0
$69$		−	38768.5i		−	0.980294i
$70$		0			0
$71$			52643.9i			1.23937i	0.784849	+	0.619687i	$0.212740\pi$
$71$			52643.9i			1.23937i	−0.784849	+	0.619687i	$0.787260\pi$
$72$		0			0
$73$	27965.6	−	27965.6i	0.614209	−	0.614209i	−0.329831	−	0.944040i	$-0.606992\pi$
$73$	27965.6	−	27965.6i	0.614209	−	0.614209i	0.944040	+	0.329831i	$0.106992\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	22539.6	+	22539.6i	0.433230	+	0.433230i
$78$		0			0
$79$	82263.7			1.48300			0.741499	−	0.670954i	$-0.234115\pi$
$79$	82263.7			1.48300			0.741499	+	0.670954i	$0.234115\pi$
$80$		0			0
$81$	42489.8			0.719569
$82$		0			0
$83$	77236.8	+	77236.8i	1.23063	+	1.23063i	0.963720	+	0.266914i	$0.0860039\pi$
$83$	77236.8	+	77236.8i	1.23063	+	1.23063i	0.266914	+	0.963720i	$0.413996\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−34546.7	+	34546.7i	−0.489337	+	0.489337i
$88$		0			0
$89$			145955.i			1.95318i	0.215100	+	0.976592i	$0.430992\pi$
$89$			145955.i			1.95318i	−0.215100	+	0.976592i	$0.569008\pi$
$90$		0			0
$91$		−	31222.2i		−	0.395239i
$92$		0			0
$93$	−87580.1	+	87580.1i	−1.05002	+	1.05002i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−97856.7	−	97856.7i	−1.05599	−	1.05599i	−0.998336	−	0.0576570i	$-0.981637\pi$
$97$	−97856.7	−	97856.7i	−1.05599	−	1.05599i	−0.0576570	−	0.998336i	$-0.518363\pi$
$98$		0			0
$99$	−25696.5			−0.263503
$100$		0			0
$101$	86555.1			0.844286			0.422143	−	0.906529i	$-0.361278\pi$
$101$	86555.1			0.844286			0.422143	+	0.906529i	$0.361278\pi$
$102$		0			0
$103$	125928.	+	125928.i	1.16958	+	1.16958i	0.982308	+	0.187273i	$0.0599650\pi$
$103$	125928.	+	125928.i	1.16958	+	1.16958i	0.187273	+	0.982308i	$0.440035\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−152309.	+	152309.i	−1.28607	+	1.28607i	−0.348917	+	0.937154i	$0.613450\pi$
$107$	−152309.	+	152309.i	−1.28607	+	1.28607i	−0.937154	+	0.348917i	$0.886550\pi$
$108$		0			0
$109$			62440.1i			0.503382i	0.967808	+	0.251691i	$0.0809866\pi$
$109$			62440.1i			0.503382i	−0.967808	+	0.251691i	$0.919013\pi$
$110$		0			0
$111$			31801.6i			0.244986i
$112$		0			0
$113$	−32875.2	+	32875.2i	−0.242199	+	0.242199i	−0.817759	−	0.575560i	$-0.804784\pi$
$113$	−32875.2	+	32875.2i	−0.242199	+	0.242199i	0.575560	+	0.817759i	$0.304784\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	17797.6	+	17797.6i	0.120198	+	0.120198i
$118$		0			0
$119$	101734.			0.658568
$120$		0			0
$121$	−53482.3			−0.332083
$122$		0			0
$123$	138841.	+	138841.i	0.827476	+	0.827476i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−120503.	+	120503.i	−0.662960	+	0.662960i	−0.956077	−	0.293116i	$-0.905308\pi$
$127$	−120503.	+	120503.i	−0.662960	+	0.662960i	0.293116	+	0.956077i	$0.405308\pi$
$128$		0			0
$129$			76340.6i			0.403907i
$130$		0			0
$131$			88630.4i			0.451237i	0.974216	+	0.225618i	$0.0724402\pi$
$131$			88630.4i			0.451237i	−0.974216	+	0.225618i	$0.927560\pi$
$132$		0			0
$133$	−34126.8	+	34126.8i	−0.167289	+	0.167289i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	46725.7	+	46725.7i	0.212694	+	0.212694i	0.805411	−	0.592717i	$-0.201945\pi$
$137$	46725.7	+	46725.7i	0.212694	+	0.212694i	−0.592717	+	0.805411i	$0.701945\pi$
$138$		0			0
$139$	280242.			1.23026			0.615128	−	0.788427i	$-0.289104\pi$
$139$	280242.			1.23026			0.615128	+	0.788427i	$0.289104\pi$
$140$		0			0
$141$	−153846.			−0.651688
$142$		0			0
$143$	148587.	+	148587.i	0.607634	+	0.607634i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	116882.	−	116882.i	0.446123	−	0.446123i
$148$		0			0
$149$		−	114782.i		−	0.423555i	−0.977318	−	0.211777i	$-0.932075\pi$
$149$		−	114782.i		−	0.423555i	0.977318	−	0.211777i	$-0.0679251\pi$
$150$		0			0
$151$			388996.i			1.38836i	0.719801	+	0.694181i	$0.244233\pi$
$151$			388996.i			1.38836i	−0.719801	+	0.694181i	$0.755767\pi$
$152$		0			0
$153$	−57991.7	+	57991.7i	−0.200280	+	0.200280i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−230322.	−	230322.i	−0.745738	−	0.745738i	0.227938	−	0.973676i	$-0.426802\pi$
$157$	−230322.	−	230322.i	−0.745738	−	0.745738i	−0.973676	+	0.227938i	$0.926802\pi$
$158$		0			0
$159$	225070.			0.706033
$160$		0			0
$161$	−194835.			−0.592382
$162$		0			0
$163$	−334147.	−	334147.i	−0.985073	−	0.985073i	0.0148175	−	0.999890i	$-0.495283\pi$
$163$	−334147.	−	334147.i	−0.985073	−	0.985073i	−0.999890	+	0.0148175i	$0.995283\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	287959.	−	287959.i	0.798987	−	0.798987i	−0.183949	−	0.982936i	$-0.558888\pi$
$167$	287959.	−	287959.i	0.798987	−	0.798987i	0.982936	+	0.183949i	$0.0588880\pi$
$168$		0			0
$169$			165467.i			0.445652i
$170$		0			0
$171$		−	38906.6i		−	0.101750i
$172$		0			0
$173$	−176255.	+	176255.i	−0.447741	+	0.447741i	−0.894603	−	0.446862i	$-0.852542\pi$
$173$	−176255.	+	176255.i	−0.447741	+	0.447741i	0.446862	+	0.894603i	$0.352542\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−10857.6	−	10857.6i	−0.0260495	−	0.0260495i
$178$		0			0
$179$	−409192.			−0.954540			−0.477270	−	0.878757i	$-0.658374\pi$
$179$	−409192.			−0.954540			−0.477270	+	0.878757i	$0.658374\pi$
$180$		0			0
$181$	−607951.			−1.37934			−0.689671	−	0.724123i	$-0.742245\pi$
$181$	−607951.			−1.37934			−0.689671	+	0.724123i	$0.742245\pi$
$182$		0			0
$183$	283910.	+	283910.i	0.626691	+	0.626691i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−484158.	+	484158.i	−1.01247	+	1.01247i
$188$		0			0
$189$		−	281288.i		−	0.572793i
$190$		0			0
$191$		−	67781.7i		−	0.134440i	−0.997738	−	0.0672201i	$-0.978587\pi$
$191$		−	67781.7i		−	0.134440i	0.997738	−	0.0672201i	$-0.0214130\pi$
$192$		0			0
$193$	−373274.	+	373274.i	−0.721330	+	0.721330i	−0.968876	−	0.247546i	$-0.920376\pi$
$193$	−373274.	+	373274.i	−0.721330	+	0.721330i	0.247546	+	0.968876i	$0.420376\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−76924.7	−	76924.7i	−0.141221	−	0.141221i	0.632962	−	0.774183i	$-0.281839\pi$
$197$	−76924.7	−	76924.7i	−0.141221	−	0.141221i	−0.774183	+	0.632962i	$0.781839\pi$
$198$		0			0
$199$	908495.			1.62626			0.813130	−	0.582083i	$-0.197762\pi$
$199$	908495.			1.62626			0.813130	+	0.582083i	$0.197762\pi$
$200$		0			0
$201$	178162.			0.311047
$202$		0			0
$203$	173618.	+	173618.i	0.295702	+	0.295702i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	111062.	−	111062.i	0.180152	−	0.180152i
$208$		0			0
$209$		−	324822.i		−	0.514374i
$210$		0			0
$211$		−	732879.i		−	1.13325i	−0.823975	−	0.566626i	$-0.808249\pi$
$211$		−	732879.i		−	1.13325i	0.823975	−	0.566626i	$-0.191751\pi$
$212$		0			0
$213$	509751.	−	509751.i	0.769856	−	0.769856i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	440143.	+	440143.i	0.634518	+	0.634518i
$218$		0			0
$219$	−541581.			−0.763051
$220$		0			0
$221$	670663.			0.923684
$222$		0			0
$223$	−346705.	−	346705.i	−0.466872	−	0.466872i	0.434027	−	0.900900i	$-0.357092\pi$
$223$	−346705.	−	346705.i	−0.466872	−	0.466872i	−0.900900	+	0.434027i	$0.857092\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−262744.	+	262744.i	−0.338429	+	0.338429i	−0.855776	−	0.517347i	$-0.826920\pi$
$227$	−262744.	+	262744.i	−0.338429	+	0.338429i	0.517347	+	0.855776i	$0.326920\pi$
$228$		0			0
$229$			1.36759e6i			1.72332i	0.507487	+	0.861660i	$0.330575\pi$
$229$			1.36759e6i			1.72332i	−0.507487	+	0.861660i	$0.669425\pi$
$230$		0			0
$231$		−	436502.i		−	0.538215i
$232$		0			0
$233$	−171228.	+	171228.i	−0.206626	+	0.206626i	−0.802832	−	0.596206i	$-0.796674\pi$
$233$	−171228.	+	171228.i	−0.206626	+	0.206626i	0.596206	+	0.802832i	$0.296674\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−796560.	−	796560.i	−0.921187	−	0.921187i
$238$		0			0
$239$	−1.31148e6			−1.48514			−0.742569	−	0.669769i	$-0.766393\pi$
$239$	−1.31148e6			−1.48514			−0.742569	+	0.669769i	$0.766393\pi$
$240$		0			0
$241$	1.31737e6			1.46105			0.730526	−	0.682884i	$-0.239275\pi$
$241$	1.31737e6			1.46105			0.730526	+	0.682884i	$0.239275\pi$
$242$		0			0
$243$	290883.	+	290883.i	0.316011	+	0.316011i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−224974.	+	224974.i	−0.234633	+	0.234633i
$248$		0			0
$249$		−	1.49577e6i		−	1.52885i
$250$		0			0
$251$		−	188218.i		−	0.188572i	−0.995545	−	0.0942859i	$-0.969943\pi$
$251$		−	188218.i		−	0.188572i	0.995545	−	0.0942859i	$-0.0300568\pi$
$252$		0			0
$253$	927227.	−	927227.i	0.910720	−	0.910720i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	784122.	+	784122.i	0.740544	+	0.740544i	0.972683	−	0.232139i	$-0.0745723\pi$
$257$	784122.	+	784122.i	0.740544	+	0.740544i	−0.232139	+	0.972683i	$0.574572\pi$
$258$		0			0
$259$	159822.			0.148043
$260$		0			0
$261$	−197935.			−0.179854
$262$		0			0
$263$	−554379.	−	554379.i	−0.494217	−	0.494217i	0.415415	−	0.909632i	$-0.363636\pi$
$263$	−554379.	−	554379.i	−0.494217	−	0.494217i	−0.909632	+	0.415415i	$0.863636\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.41328e6	−	1.41328e6i	1.21325	−	1.21325i
$268$		0			0
$269$			146638.i			0.123557i	0.998090	+	0.0617785i	$0.0196772\pi$
$269$			146638.i			0.123557i	−0.998090	+	0.0617785i	$0.980323\pi$
$270$		0			0
$271$			620115.i			0.512919i	0.966555	+	0.256460i	$0.0825560\pi$
$271$			620115.i			0.512919i	−0.966555	+	0.256460i	$0.917444\pi$
$272$		0			0
$273$	−302324.	+	302324.i	−0.245509	+	0.245509i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	571067.	+	571067.i	0.447185	+	0.447185i	0.894418	−	0.447233i	$-0.147590\pi$
$277$	571067.	+	571067.i	0.447185	+	0.447185i	−0.447233	+	0.894418i	$0.647590\pi$
$278$		0			0
$279$	−501789.			−0.385932
$280$		0			0
$281$	−623155.			−0.470793			−0.235397	−	0.971899i	$-0.575639\pi$
$281$	−623155.			−0.470793			−0.235397	+	0.971899i	$0.575639\pi$
$282$		0			0
$283$	−1.66016e6	−	1.66016e6i	−1.23221	−	1.23221i	−0.963115	−	0.269091i	$-0.913277\pi$
$283$	−1.66016e6	−	1.66016e6i	−1.23221	−	1.23221i	−0.269091	−	0.963115i	$-0.586723\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	697760.	−	697760.i	0.500036	−	0.500036i
$288$		0			0
$289$			765434.i			0.539092i
$290$		0			0
$291$			1.89509e6i			1.31189i
$292$		0			0
$293$	1.72577e6	−	1.72577e6i	1.17439	−	1.17439i	0.193242	−	0.981151i	$-0.438100\pi$
$293$	1.72577e6	−	1.72577e6i	1.17439	−	1.17439i	0.981151	−	0.193242i	$-0.0619003\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1.33866e6	+	1.33866e6i	0.880602	+	0.880602i
$298$		0			0
$299$	−1.28441e6			−0.830855
$300$		0			0
$301$	383657.			0.244077
$302$		0			0
$303$	−838114.	−	838114.i	−0.524441	−	0.524441i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	693642.	−	693642.i	0.420039	−	0.420039i	−0.465178	−	0.885217i	$-0.654010\pi$
$307$	693642.	−	693642.i	0.420039	−	0.420039i	0.885217	+	0.465178i	$0.154010\pi$
$308$		0			0
$309$		−	2.43873e6i		−	1.45301i
$310$		0			0
$311$		−	839243.i		−	0.492024i	−0.969267	−	0.246012i	$-0.920880\pi$
$311$		−	839243.i		−	0.492024i	0.969267	−	0.246012i	$-0.0791203\pi$
$312$		0			0
$313$	1.45976e6	−	1.45976e6i	0.842208	−	0.842208i	−0.146938	−	0.989146i	$-0.546942\pi$
$313$	1.45976e6	−	1.45976e6i	0.842208	−	0.842208i	0.989146	+	0.146938i	$0.0469417\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−768911.	−	768911.i	−0.429762	−	0.429762i	0.458785	−	0.888547i	$-0.348285\pi$
$317$	−768911.	−	768911.i	−0.429762	−	0.429762i	−0.888547	+	0.458785i	$0.848285\pi$
$318$		0			0
$319$	−1.65251e6			−0.909215
$320$		0			0
$321$	2.94961e6			1.59772
$322$		0			0
$323$	−733056.	−	733056.i	−0.390959	−	0.390959i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	604608.	−	604608.i	0.312683	−	0.312683i
$328$		0			0
$329$			773171.i			0.393809i
$330$		0			0
$331$		−	1.53718e6i		−	0.771179i	−0.922671	−	0.385589i	$-0.873998\pi$
$331$		−	1.53718e6i		−	0.771179i	0.922671	−	0.385589i	$-0.126002\pi$
$332$		0			0
$333$	−91103.5	+	91103.5i	−0.0450220	+	0.0450220i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	600219.	+	600219.i	0.287895	+	0.287895i	0.836248	−	0.548352i	$-0.184745\pi$
$337$	600219.	+	600219.i	0.287895	+	0.287895i	−0.548352	+	0.836248i	$0.684745\pi$
$338$		0			0
$339$	636661.			0.300891
$340$		0			0
$341$	−4.18931e6			−1.95100
$342$		0			0
$343$	−1.40528e6	−	1.40528e6i	−0.644952	−	0.644952i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15540.8	+	15540.8i	−0.00692868	+	0.00692868i	−0.710563	−	0.703634i	$-0.751560\pi$
$347$	−15540.8	+	15540.8i	−0.00692868	+	0.00692868i	0.703634	+	0.710563i	$0.251560\pi$
$348$		0			0
$349$		−	1.37621e6i		−	0.604815i	−0.953179	−	0.302407i	$-0.902210\pi$
$349$		−	1.37621e6i		−	0.604815i	0.953179	−	0.302407i	$-0.0977903\pi$
$350$		0			0
$351$		−	1.85434e6i		−	0.803379i
$352$		0			0
$353$	−2.13095e6	+	2.13095e6i	−0.910198	+	0.910198i	−0.996287	−	0.0860896i	$-0.972563\pi$
$353$	−2.13095e6	+	2.13095e6i	−0.910198	+	0.910198i	0.0860896	+	0.996287i	$0.472563\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−985095.	−	985095.i	−0.409079	−	0.409079i
$358$		0			0
$359$	−346230.			−0.141785			−0.0708923	−	0.997484i	$-0.522585\pi$
$359$	−346230.			−0.141785			−0.0708923	+	0.997484i	$0.522585\pi$
$360$		0			0
$361$	−1.98429e6			−0.801378
$362$		0			0
$363$	517870.	+	517870.i	0.206278	+	0.206278i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−857952.	+	857952.i	−0.332505	+	0.332505i	−0.853537	−	0.521032i	$-0.825547\pi$
$367$	−857952.	+	857952.i	−0.332505	+	0.332505i	0.521032	+	0.853537i	$0.325547\pi$
$368$		0			0
$369$			795488.i			0.304136i
$370$		0			0
$371$		−	1.13111e6i		−	0.426649i
$372$		0			0
$373$	−1.92433e6	+	1.92433e6i	−0.716157	+	0.716157i	−0.967816	−	0.251659i	$-0.919024\pi$
$373$	−1.92433e6	+	1.92433e6i	−0.716157	+	0.716157i	0.251659	+	0.967816i	$0.419024\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.14454e6	+	1.14454e6i	0.414741	+	0.414741i
$378$		0			0
$379$	−5919.00			−0.00211666			−0.00105833	−	0.999999i	$-0.500337\pi$
$379$	−5919.00			−0.00211666			−0.00105833	+	0.999999i	$0.500337\pi$
$380$		0			0
$381$	2.33366e6			0.823616
$382$		0			0
$383$	−121366.	−	121366.i	−0.0422767	−	0.0422767i	0.685652	−	0.727929i	$-0.259517\pi$
$383$	−121366.	−	121366.i	−0.0422767	−	0.0422767i	−0.727929	+	0.685652i	$0.759517\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−218696.	+	218696.i	−0.0742273	+	0.0742273i
$388$		0			0
$389$		−	2.60692e6i		−	0.873482i	−0.899587	−	0.436741i	$-0.856133\pi$
$389$		−	2.60692e6i		−	0.873482i	0.899587	−	0.436741i	$-0.143867\pi$
$390$		0			0
$391$		−	4.18512e6i		−	1.38442i
$392$		0			0
$393$	858209.	−	858209.i	0.280293	−	0.280293i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.66197e6	−	2.66197e6i	−0.847669	−	0.847669i	0.142173	−	0.989842i	$-0.454591\pi$
$397$	−2.66197e6	−	2.66197e6i	−0.847669	−	0.847669i	−0.989842	+	0.142173i	$0.954591\pi$
$398$		0			0
$399$	660901.			0.207828
$400$		0			0
$401$	3.31838e6			1.03054			0.515270	−	0.857028i	$-0.327692\pi$
$401$	3.31838e6			1.03054			0.515270	+	0.857028i	$0.327692\pi$
$402$		0			0
$403$	2.90155e6	+	2.90155e6i	0.889953	+	0.889953i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−760600.	+	760600.i	−0.227599	+	0.227599i
$408$		0			0
$409$			1.71967e6i			0.508319i	0.967162	+	0.254159i	$0.0817988\pi$
$409$			1.71967e6i			0.508319i	−0.967162	+	0.254159i	$0.918201\pi$
$410$		0			0
$411$		−	904891.i		−	0.264236i
$412$		0			0
$413$	−54565.8	+	54565.8i	−0.0157415	+	0.0157415i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.71358e6	−	2.71358e6i	−0.764193	−	0.764193i
$418$		0			0
$419$	2.43886e6			0.678660			0.339330	−	0.940667i	$-0.389800\pi$
$419$	2.43886e6			0.678660			0.339330	+	0.940667i	$0.389800\pi$
$420$		0			0
$421$	−2.76737e6			−0.760961			−0.380480	−	0.924789i	$-0.624241\pi$
$421$	−2.76737e6			−0.760961			−0.380480	+	0.924789i	$0.624241\pi$
$422$		0			0
$423$	−440731.	−	440731.i	−0.119763	−	0.119763i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.42682e6	−	1.42682e6i	0.378703	−	0.378703i
$428$		0			0
$429$		−	2.87755e6i		−	0.754882i
$430$		0			0
$431$			3.98814e6i			1.03413i	0.855945	+	0.517067i	$0.172976\pi$
$431$			3.98814e6i			1.03413i	−0.855945	+	0.517067i	$0.827024\pi$
$432$		0			0
$433$	−386926.	+	386926.i	−0.0991765	+	0.0991765i	−0.754954	−	0.655778i	$-0.772341\pi$
$433$	−386926.	+	386926.i	−0.0991765	+	0.0991765i	0.655778	+	0.754954i	$0.272341\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.40390e6	+	1.40390e6i	0.351668	+	0.351668i
$438$		0			0
$439$	−1.85711e6			−0.459913			−0.229956	−	0.973201i	$-0.573858\pi$
$439$	−1.85711e6			−0.459913			−0.229956	+	0.973201i	$0.573858\pi$
$440$		0			0
$441$	669674.			0.163971
$442$		0			0
$443$	−2.58101e6	−	2.58101e6i	−0.624857	−	0.624857i	0.321913	−	0.946769i	$-0.395674\pi$
$443$	−2.58101e6	−	2.58101e6i	−0.624857	−	0.624857i	−0.946769	+	0.321913i	$0.895674\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.11144e6	+	1.11144e6i	−0.263097	+	0.263097i
$448$		0			0
$449$			4.66935e6i			1.09305i	0.837443	+	0.546525i	$0.184050\pi$
$449$			4.66935e6i			1.09305i	−0.837443	+	0.546525i	$0.815950\pi$
$450$		0			0
$451$			6.64133e6i			1.53749i
$452$		0			0
$453$	3.76665e6	−	3.76665e6i	0.862402	−	0.862402i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.03755e6	+	5.03755e6i	1.12831	+	1.12831i	0.990452	+	0.137858i	$0.0440219\pi$
$457$	5.03755e6	+	5.03755e6i	1.12831	+	1.12831i	0.137858	+	0.990452i	$0.455978\pi$
$458$		0			0
$459$	6.04217e6			1.33863
$460$		0			0
$461$	−2.77546e6			−0.608250			−0.304125	−	0.952632i	$-0.598364\pi$
$461$	−2.77546e6			−0.608250			−0.304125	+	0.952632i	$0.598364\pi$
$462$		0			0
$463$	6.00377e6	+	6.00377e6i	1.30158	+	1.30158i	0.927325	+	0.374258i	$0.122103\pi$
$463$	6.00377e6	+	6.00377e6i	1.30158	+	1.30158i	0.374258	+	0.927325i	$0.377897\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.93586e6	+	2.93586e6i	−0.622935	+	0.622935i	−0.946281	−	0.323346i	$-0.895192\pi$
$467$	−2.93586e6	+	2.93586e6i	−0.622935	+	0.622935i	0.323346	+	0.946281i	$0.395192\pi$
$468$		0			0
$469$		−	895373.i		−	0.187963i
$470$		0			0
$471$			4.46042e6i			0.926453i
$472$		0			0
$473$	−1.82584e6	+	1.82584e6i	−0.375240	+	0.375240i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	644768.	+	644768.i	0.129750	+	0.129750i
$478$		0			0
$479$	−850769.			−0.169423			−0.0847116	−	0.996406i	$-0.526997\pi$
$479$	−850769.			−0.169423			−0.0847116	+	0.996406i	$0.526997\pi$
$480$		0			0
$481$	1.05360e6			0.207640
$482$		0			0
$483$	1.88659e6	+	1.88659e6i	0.367967	+	0.367967i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.08676e6	−	2.08676e6i	0.398704	−	0.398704i	−0.479072	−	0.877776i	$-0.659027\pi$
$487$	2.08676e6	−	2.08676e6i	0.398704	−	0.398704i	0.877776	+	0.479072i	$0.159027\pi$
$488$		0			0
$489$			6.47109e6i			1.22379i
$490$		0			0
$491$			3.55745e6i			0.665940i	0.942937	+	0.332970i	$0.108051\pi$
$491$			3.55745e6i			0.665940i	−0.942937	+	0.332970i	$0.891949\pi$
$492$		0			0
$493$	−3.72937e6	+	3.72937e6i	−0.691064	+	0.691064i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.56181e6	−	2.56181e6i	−0.465217	−	0.465217i
$498$		0			0
$499$	−4.06234e6			−0.730339			−0.365170	−	0.930941i	$-0.618989\pi$
$499$	−4.06234e6			−0.730339			−0.365170	+	0.930941i	$0.618989\pi$
$500$		0			0
$501$	−5.57662e6			−0.992606
$502$		0			0
$503$	−2.25876e6	−	2.25876e6i	−0.398061	−	0.398061i	0.479488	−	0.877549i	$-0.340823\pi$
$503$	−2.25876e6	−	2.25876e6i	−0.398061	−	0.398061i	−0.877549	+	0.479488i	$0.840823\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.60222e6	−	1.60222e6i	0.276824	−	0.276824i
$508$		0			0
$509$			5.20653e6i			0.890746i	0.895345	+	0.445373i	$0.146929\pi$
$509$			5.20653e6i			0.890746i	−0.895345	+	0.445373i	$0.853071\pi$
$510$		0			0
$511$			2.72177e6i			0.461105i
$512$		0			0
$513$	−2.02685e6	+	2.02685e6i	−0.340038	+	0.340038i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−3.67955e6	−	3.67955e6i	−0.605436	−	0.605436i
$518$		0			0
$519$	3.41336e6			0.556242
$520$		0			0
$521$	1.82071e6			0.293864			0.146932	−	0.989147i	$-0.453060\pi$
$521$	1.82071e6			0.293864			0.146932	+	0.989147i	$0.453060\pi$
$522$		0			0
$523$	3.25978e6	+	3.25978e6i	0.521116	+	0.521116i	0.917908	−	0.396793i	$-0.129877\pi$
$523$	3.25978e6	+	3.25978e6i	0.521116	+	0.521116i	−0.396793	+	0.917908i	$0.629877\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.45442e6	+	9.45442e6i	−1.48289	+	1.48289i
$528$		0			0
$529$			1.57872e6i			0.245283i
$530$		0			0
$531$		−	62208.3i		−	0.00957442i
$532$		0			0
$533$	4.59983e6	−	4.59983e6i	0.701333	−	0.701333i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.96221e6	+	3.96221e6i	0.592927	+	0.592927i
$538$		0			0
$539$	5.59094e6			0.828920
$540$		0			0
$541$	6.09633e6			0.895520			0.447760	−	0.894154i	$-0.352222\pi$
$541$	6.09633e6			0.895520			0.447760	+	0.894154i	$0.352222\pi$
$542$		0			0
$543$	5.88679e6	+	5.88679e6i	0.856799	+	0.856799i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.00865e6	−	2.00865e6i	0.287036	−	0.287036i	−0.548871	−	0.835907i	$-0.684942\pi$
$547$	2.00865e6	−	2.00865e6i	0.287036	−	0.287036i	0.835907	+	0.548871i	$0.184942\pi$
$548$		0			0
$549$			1.62666e6i			0.230338i
$550$		0			0
$551$		−	2.50204e6i		−	0.351087i
$552$		0			0
$553$	−4.00319e6	+	4.00319e6i	−0.556665	+	0.556665i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.00540e7	−	1.00540e7i	−1.37309	−	1.37309i	−0.855821	−	0.517272i	$-0.826948\pi$
$557$	−1.00540e7	−	1.00540e7i	−1.37309	−	1.37309i	−0.517272	−	0.855821i	$-0.673052\pi$
$558$		0			0
$559$	2.52918e6			0.342334
$560$		0			0
$561$	9.37621e6			1.25782
$562$		0			0
$563$	6.99182e6	+	6.99182e6i	0.929650	+	0.929650i	0.997683	−	0.0680335i	$-0.0216725\pi$
$563$	6.99182e6	+	6.99182e6i	0.929650	+	0.929650i	−0.0680335	+	0.997683i	$0.521672\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.06768e6	+	2.06768e6i	−0.270100	+	0.270100i
$568$		0			0
$569$		−	3.15878e6i		−	0.409015i	−0.978865	−	0.204507i	$-0.934441\pi$
$569$		−	3.15878e6i		−	0.409015i	0.978865	−	0.204507i	$-0.0655592\pi$
$570$		0			0
$571$		−	9.35381e6i		−	1.20060i	−0.799775	−	0.600300i	$-0.795048\pi$
$571$		−	9.35381e6i		−	1.20060i	0.799775	−	0.600300i	$-0.204952\pi$
$572$		0			0
$573$	−656331.	+	656331.i	−0.0835095	+	0.0835095i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.00683e7	+	1.00683e7i	1.25897	+	1.25897i	0.951583	+	0.307391i	$0.0994559\pi$
$577$	1.00683e7	+	1.00683e7i	1.25897	+	1.25897i	0.307391	+	0.951583i	$0.400544\pi$
$578$		0			0
$579$	7.22883e6			0.896131
$580$		0			0
$581$	−7.51714e6			−0.923873
$582$		0			0
$583$	5.38301e6	+	5.38301e6i	0.655924	+	0.655924i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.87805e6	+	9.87805e6i	−1.18325	+	1.18325i	−0.204351	+	0.978898i	$0.565508\pi$
$587$	−9.87805e6	+	9.87805e6i	−1.18325	+	1.18325i	−0.978898	+	0.204351i	$0.934492\pi$
$588$		0			0
$589$		−	6.34297e6i		−	0.753364i
$590$		0			0
$591$			1.48972e6i			0.175444i
$592$		0			0
$593$	−6.47163e6	+	6.47163e6i	−0.755748	+	0.755748i	−0.975546	−	0.219797i	$-0.929460\pi$
$593$	−6.47163e6	+	6.47163e6i	−0.755748	+	0.755748i	0.219797	+	0.975546i	$0.429460\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.79696e6	−	8.79696e6i	−1.01018	−	1.01018i
$598$		0			0
$599$	5.49595e6			0.625858			0.312929	−	0.949777i	$-0.398690\pi$
$599$	5.49595e6			0.625858			0.312929	+	0.949777i	$0.398690\pi$
$600$		0			0
$601$	−2.92293e6			−0.330090			−0.165045	−	0.986286i	$-0.552777\pi$
$601$	−2.92293e6			−0.330090			−0.165045	+	0.986286i	$0.552777\pi$
$602$		0			0
$603$	510390.	+	510390.i	0.0571622	+	0.0571622i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.15887e6	−	3.15887e6i	0.347984	−	0.347984i	−0.511374	−	0.859358i	$-0.670863\pi$
$607$	3.15887e6	−	3.15887e6i	0.347984	−	0.347984i	0.859358	+	0.511374i	$0.170863\pi$
$608$		0			0
$609$		−	3.36229e6i		−	0.367360i
$610$		0			0
$611$			5.09697e6i			0.552343i
$612$		0			0
$613$	−1.01465e6	+	1.01465e6i	−0.109060	+	0.109060i	−0.759531	−	0.650471i	$-0.774571\pi$
$613$	−1.01465e6	+	1.01465e6i	−0.109060	+	0.109060i	0.650471	+	0.759531i	$0.274571\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.48588e6	−	2.48588e6i	−0.262886	−	0.262886i	0.563340	−	0.826226i	$-0.309516\pi$
$617$	−2.48588e6	−	2.48588e6i	−0.262886	−	0.262886i	−0.826226	+	0.563340i	$0.809516\pi$
$618$		0			0
$619$	−4.49651e6			−0.471682			−0.235841	−	0.971792i	$-0.575784\pi$
$619$	−4.49651e6			−0.471682			−0.235841	+	0.971792i	$0.575784\pi$
$620$		0			0
$621$	−1.15716e7			−1.20410
$622$		0			0
$623$	−7.10259e6	−	7.10259e6i	−0.733156	−	0.733156i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3.14525e6	+	3.14525e6i	−0.319512	+	0.319512i
$628$		0			0
$629$			3.43304e6i			0.345981i
$630$		0			0
$631$		−	8.34293e6i		−	0.834152i	−0.908872	−	0.417076i	$-0.863055\pi$
$631$		−	8.34293e6i		−	0.834152i	0.908872	−	0.417076i	$-0.136945\pi$
$632$		0			0
$633$	−7.09648e6	+	7.09648e6i	−0.703936	+	0.703936i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.87233e6	−	3.87233e6i	−0.378115	−	0.378115i
$638$		0			0
$639$	2.92061e6			0.282958
$640$		0			0
$641$	−4.67634e6			−0.449532			−0.224766	−	0.974413i	$-0.572162\pi$
$641$	−4.67634e6			−0.449532			−0.224766	+	0.974413i	$0.572162\pi$
$642$		0			0
$643$	−1.95020e6	−	1.95020e6i	−0.186017	−	0.186017i	0.607955	−	0.793972i	$-0.291990\pi$
$643$	−1.95020e6	−	1.95020e6i	−0.186017	−	0.186017i	−0.793972	+	0.607955i	$0.791990\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.09333e6	−	7.09333e6i	0.666177	−	0.666177i	−0.290652	−	0.956829i	$-0.593872\pi$
$647$	7.09333e6	−	7.09333e6i	0.666177	−	0.666177i	0.956829	+	0.290652i	$0.0938722\pi$
$648$		0			0
$649$		−	519362.i		−	0.0484014i
$650$		0			0
$651$		−	8.52381e6i		−	0.788281i
$652$		0			0
$653$	1.02700e7	−	1.02700e7i	0.942516	−	0.942516i	−0.0559194	−	0.998435i	$-0.517809\pi$
$653$	1.02700e7	−	1.02700e7i	0.942516	−	0.942516i	0.998435	+	0.0559194i	$0.0178090\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.55149e6	−	1.55149e6i	−0.140228	−	0.140228i
$658$		0			0
$659$	−2.21615e7			−1.98786			−0.993930	−	0.110013i	$-0.964911\pi$
$659$	−2.21615e7			−1.98786			−0.993930	+	0.110013i	$0.964911\pi$
$660$		0			0
$661$	3.85454e6			0.343138			0.171569	−	0.985172i	$-0.445116\pi$
$661$	3.85454e6			0.343138			0.171569	+	0.985172i	$0.445116\pi$
$662$		0			0
$663$	−6.49403e6	−	6.49403e6i	−0.573761	−	0.573761i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.14224e6	−	7.14224e6i	0.621613	−	0.621613i
$668$		0			0
$669$			6.71430e6i			0.580010i
$670$		0			0
$671$			1.35806e7i			1.16443i
$672$		0			0
$673$	−8.70626e6	+	8.70626e6i	−0.740958	+	0.740958i	−0.972762	−	0.231804i	$-0.925537\pi$
$673$	−8.70626e6	+	8.70626e6i	−0.740958	+	0.740958i	0.231804	+	0.972762i	$0.425537\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.81144e6	+	2.81144e6i	0.235753	+	0.235753i	0.815089	−	0.579336i	$-0.196688\pi$
$677$	2.81144e6	+	2.81144e6i	0.235753	+	0.235753i	−0.579336	+	0.815089i	$0.696688\pi$
$678$		0			0
$679$	9.52399e6			0.792765
$680$		0			0
$681$	5.08830e6			0.420441
$682$		0			0
$683$	−8.19141e6	−	8.19141e6i	−0.671903	−	0.671903i	0.286251	−	0.958155i	$-0.407591\pi$
$683$	−8.19141e6	−	8.19141e6i	−0.671903	−	0.671903i	−0.958155	+	0.286251i	$0.907591\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.32423e7	−	1.32423e7i	1.07047	−	1.07047i
$688$		0			0
$689$		−	7.45662e6i		−	0.598403i
$690$		0			0
$691$			6.47639e6i			0.515986i	0.966147	+	0.257993i	$0.0830611\pi$
$691$			6.47639e6i			0.515986i	−0.966147	+	0.257993i	$0.916939\pi$
$692$		0			0
$693$	1.25047e6	−	1.25047e6i	0.0989096	−	0.0989096i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.49881e7	+	1.49881e7i	1.16860	+	1.16860i
$698$		0			0
$699$	3.31600e6			0.256697
$700$		0			0
$701$	−1.17245e7			−0.901155			−0.450578	−	0.892737i	$-0.648782\pi$
$701$	−1.17245e7			−0.901155			−0.450578	+	0.892737i	$0.648782\pi$
$702$		0			0
$703$	−1.15161e6	−	1.15161e6i	−0.0878858	−	0.0878858i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.21203e6	+	4.21203e6i	−0.316915	+	0.316915i
$708$		0			0
$709$			5.40476e6i			0.403795i	0.979407	+	0.201897i	$0.0647107\pi$
$709$			5.40476e6i			0.403795i	−0.979407	+	0.201897i	$0.935289\pi$
$710$		0			0
$711$		−	4.56388e6i		−	0.338579i
$712$		0			0
$713$	1.81065e7	−	1.81065e7i	1.33386	−	1.33386i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.26991e7	+	1.26991e7i	0.922516	+	0.922516i
$718$		0			0
$719$	4.95216e6			0.357250			0.178625	−	0.983917i	$-0.442835\pi$
$719$	4.95216e6			0.357250			0.178625	+	0.983917i	$0.442835\pi$
$720$		0			0
$721$	−1.22561e7			−0.878038
$722$		0			0
$723$	−1.27561e7	−	1.27561e7i	−0.907555	−	0.907555i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	824940.	−	824940.i	0.0578877	−	0.0578877i	−0.677570	−	0.735458i	$-0.736967\pi$
$727$	824940.	−	824940.i	0.0578877	−	0.0578877i	0.735458	+	0.677570i	$0.236967\pi$
$728$		0			0
$729$		−	1.59583e7i		−	1.11216i
$730$		0			0
$731$			8.24109e6i			0.570415i
$732$		0			0
$733$	1.29259e7	−	1.29259e7i	0.888590	−	0.888590i	−0.105798	−	0.994388i	$-0.533740\pi$
$733$	1.29259e7	−	1.29259e7i	0.888590	−	0.888590i	0.994388	+	0.105798i	$0.0337397\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.26111e6	+	4.26111e6i	0.288971	+	0.288971i
$738$		0			0
$739$	−1.48270e7			−0.998714			−0.499357	−	0.866396i	$-0.666430\pi$
$739$	−1.48270e7			−0.998714			−0.499357	+	0.866396i	$0.666430\pi$
$740$		0			0
$741$	4.35685e6			0.291492
$742$		0			0
$743$	2.13856e6	+	2.13856e6i	0.142118	+	0.142118i	0.774586	−	0.632468i	$-0.217958\pi$
$743$	2.13856e6	+	2.13856e6i	0.142118	+	0.142118i	−0.632468	+	0.774586i	$0.717958\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.28499e6	−	4.28499e6i	0.280963	−	0.280963i
$748$		0			0
$749$		−	1.48236e7i		−	0.965491i
$750$		0			0
$751$		−	499674.i		−	0.0323286i	−0.999869	−	0.0161643i	$-0.994855\pi$
$751$		−	499674.i		−	0.0323286i	0.999869	−	0.0161643i	$-0.00514548\pi$
$752$		0			0
$753$	−1.82252e6	+	1.82252e6i	−0.117134	+	0.117134i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.51219e7	−	1.51219e7i	−0.959105	−	0.959105i	0.0400909	−	0.999196i	$-0.487235\pi$
$757$	−1.51219e7	−	1.51219e7i	−0.959105	−	0.959105i	−0.999196	+	0.0400909i	$0.987235\pi$
$758$		0			0
$759$	−1.79567e7			−1.13141
$760$		0			0
$761$	3.08023e7			1.92806			0.964032	−	0.265788i	$-0.0856321\pi$
$761$	3.08023e7			1.92806			0.964032	+	0.265788i	$0.0856321\pi$
$762$		0			0
$763$	−3.03852e6	−	3.03852e6i	−0.188952	−	0.188952i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−359714.	+	359714.i	−0.0220785	+	0.0220785i
$768$		0			0
$769$			6.19605e6i			0.377832i	0.981993	+	0.188916i	$0.0604975\pi$
$769$			6.19605e6i			0.377832i	−0.981993	+	0.188916i	$0.939503\pi$
$770$		0			0
$771$		−	1.51853e7i		−	0.920001i
$772$		0			0
$773$	−6.65596e6	+	6.65596e6i	−0.400647	+	0.400647i	−0.878461	−	0.477814i	$-0.841429\pi$
$773$	−6.65596e6	+	6.65596e6i	−0.400647	+	0.400647i	0.477814	+	0.878461i	$0.341429\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.54756e6	−	1.54756e6i	−0.0919592	−	0.0919592i
$778$		0			0
$779$	−1.00555e7			−0.593693
$780$		0			0
$781$	2.43835e7			1.43043
$782$		0			0
$783$	1.03114e7	+	1.03114e7i	0.601056	+	0.601056i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.68934e6	+	2.68934e6i	−0.154778	+	0.154778i	−0.780248	−	0.625470i	$-0.784907\pi$
$787$	−2.68934e6	+	2.68934e6i	−0.154778	+	0.154778i	0.625470	+	0.780248i	$0.284907\pi$
$788$		0			0
$789$			1.07361e7i			0.613981i
$790$		0			0
$791$		−	3.19961e6i		−	0.181826i
$792$		0			0
$793$	9.40600e6	−	9.40600e6i	0.531156	−	0.531156i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.61839e6	−	3.61839e6i	−0.201776	−	0.201776i	0.598984	−	0.800761i	$-0.295571\pi$
$797$	−3.61839e6	−	3.61839e6i	−0.201776	−	0.201776i	−0.800761	+	0.598984i	$0.795571\pi$
$798$		0			0
$799$	−1.66080e7			−0.920343
$800$		0			0
$801$	8.09738e6			0.445926
$802$		0			0
$803$	−1.29530e7	−	1.29530e7i	−0.708895	−	0.708895i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.41990e6	−	1.41990e6i	0.0767493	−	0.0767493i
$808$		0			0
$809$			1.19735e7i			0.643204i	0.946875	+	0.321602i	$0.104221\pi$
$809$			1.19735e7i			0.643204i	−0.946875	+	0.321602i	$0.895779\pi$
$810$		0			0
$811$		−	1.43266e7i		−	0.764876i	−0.923981	−	0.382438i	$-0.875085\pi$
$811$		−	1.43266e7i		−	0.764876i	0.923981	−	0.382438i	$-0.124915\pi$
$812$		0			0
$813$	6.00458e6	−	6.00458e6i	0.318608	−	0.318608i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.76448e6	−	2.76448e6i	−0.144896	−	0.144896i
$818$		0			0
$819$	−1.73216e6			−0.0902359
$820$		0			0
$821$	6.43974e6			0.333434			0.166717	−	0.986005i	$-0.446683\pi$
$821$	6.43974e6			0.333434			0.166717	+	0.986005i	$0.446683\pi$
$822$		0			0
$823$	−1.25915e7	−	1.25915e7i	−0.648002	−	0.648002i	0.304508	−	0.952510i	$-0.401508\pi$
$823$	−1.25915e7	−	1.25915e7i	−0.648002	−	0.648002i	−0.952510	+	0.304508i	$0.901508\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.91817e6	−	1.91817e6i	0.0975268	−	0.0975268i	−0.656660	−	0.754187i	$-0.728031\pi$
$827$	1.91817e6	−	1.91817e6i	0.0975268	−	0.0975268i	0.754187	+	0.656660i	$0.228031\pi$
$828$		0			0
$829$		−	1.88830e7i		−	0.954300i	−0.878822	−	0.477150i	$-0.841670\pi$
$829$		−	1.88830e7i		−	0.954300i	0.878822	−	0.477150i	$-0.158330\pi$
$830$		0			0
$831$		−	1.10593e7i		−	0.555552i
$832$		0			0
$833$	1.26176e7	−	1.26176e7i	0.630035	−	0.630035i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.61408e7	+	2.61408e7i	1.28975	+	1.28975i
$838$		0			0
$839$	468493.			0.0229772			0.0114886	−	0.999934i	$-0.496343\pi$
$839$	468493.			0.0229772			0.0114886	+	0.999934i	$0.496343\pi$
$840$		0			0
$841$	7.78221e6			0.379414
$842$		0			0
$843$	6.03402e6	+	6.03402e6i	0.292441	+	0.292441i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.60261e6	−	2.60261e6i	0.124652	−	0.124652i
$848$		0			0
$849$			3.21507e7i			1.53081i
$850$		0			0
$851$		−	6.57473e6i		−	0.311210i
$852$		0			0
$853$	−1.66492e7	+	1.66492e7i	−0.783466	+	0.783466i	−0.980414	−	0.196948i	$-0.936897\pi$
$853$	−1.66492e7	+	1.66492e7i	−0.783466	+	0.783466i	0.196948	+	0.980414i	$0.436897\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.05630e7	−	2.05630e7i	−0.956391	−	0.956391i	0.0426973	−	0.999088i	$-0.486405\pi$
$857$	−2.05630e7	−	2.05630e7i	−0.956391	−	0.956391i	−0.999088	+	0.0426973i	$0.986405\pi$
$858$		0			0
$859$	3.68768e7			1.70518			0.852589	−	0.522582i	$-0.175031\pi$
$859$	3.68768e7			1.70518			0.852589	+	0.522582i	$0.175031\pi$
$860$		0			0
$861$	−1.35128e7			−0.621210
$862$		0			0
$863$	−8.56410e6	−	8.56410e6i	−0.391431	−	0.391431i	0.483767	−	0.875197i	$-0.339268\pi$
$863$	−8.56410e6	−	8.56410e6i	−0.391431	−	0.391431i	−0.875197	+	0.483767i	$0.839268\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	7.41170e6	−	7.41170e6i	0.334865	−	0.334865i
$868$		0			0
$869$		−	3.81027e7i		−	1.71162i
$870$		0			0
$871$		−	5.90256e6i		−	0.263630i
$872$		0			0
$873$	−5.42896e6	+	5.42896e6i	−0.241091	+	0.241091i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.83761e7	+	1.83761e7i	0.806779	+	0.806779i	0.984145	−	0.177366i	$-0.0567576\pi$
$877$	1.83761e7	+	1.83761e7i	0.806779	+	0.806779i	−0.177366	+	0.984145i	$0.556758\pi$
$878$		0			0
$879$	−3.34213e7			−1.45898
$880$		0			0
$881$	−1.58839e7			−0.689472			−0.344736	−	0.938700i	$-0.612032\pi$
$881$	−1.58839e7			−0.689472			−0.344736	+	0.938700i	$0.612032\pi$
$882$		0			0
$883$	1.53863e7	+	1.53863e7i	0.664098	+	0.664098i	0.956343	−	0.292246i	$-0.0944025\pi$
$883$	1.53863e7	+	1.53863e7i	0.664098	+	0.664098i	−0.292246	+	0.956343i	$0.594402\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.45205e7	+	2.45205e7i	−1.04645	+	1.04645i	−0.0475864	+	0.998867i	$0.515153\pi$
$887$	−2.45205e7	+	2.45205e7i	−1.04645	+	1.04645i	−0.998867	+	0.0475864i	$0.984847\pi$
$888$		0			0
$889$		−	1.17280e7i		−	0.497703i
$890$		0			0
$891$		−	1.96803e7i		−	0.830496i
$892$		0			0
$893$	5.57115e6	−	5.57115e6i	0.233785	−	0.233785i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.24369e7	+	1.24369e7i	0.516098	+	0.516098i
$898$		0			0
$899$	−3.22694e7			−1.33166
$900$		0			0
$901$	2.42967e7			0.997092
$902$		0			0
$903$	−3.71496e6	−	3.71496e6i	−0.151612	−	0.151612i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.02264e7	−	2.02264e7i	0.816394	−	0.816394i	−0.169189	−	0.985584i	$-0.554115\pi$
$907$	2.02264e7	−	2.02264e7i	0.816394	−	0.816394i	0.985584	+	0.169189i	$0.0541150\pi$
$908$		0			0
$909$		−	4.80196e6i		−	0.192757i
$910$		0			0
$911$			3.37080e7i			1.34567i	0.739795	+	0.672833i	$0.234923\pi$
$911$			3.37080e7i			1.34567i	−0.739795	+	0.672833i	$0.765077\pi$
$912$		0			0
$913$	3.57743e7	−	3.57743e7i	1.42035	−	1.42035i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.31301e6	−	4.31301e6i	−0.169378	−	0.169378i
$918$		0			0
$919$	3.78592e7			1.47871			0.739355	−	0.673315i	$-0.235130\pi$
$919$	3.78592e7			1.47871			0.739355	+	0.673315i	$0.235130\pi$
$920$		0			0
$921$	−1.34331e7			−0.521827
$922$		0			0
$923$	−1.68882e7	−	1.68882e7i	−0.652497	−	0.652497i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	6.98634e6	−	6.98634e6i	0.267024	−	0.267024i
$928$		0			0
$929$		−	1.07311e6i		−	0.0407947i	−0.999792	−	0.0203973i	$-0.993507\pi$
$929$		−	1.07311e6i		−	0.0407947i	0.999792	−	0.0203973i	$-0.00649312\pi$
$930$		0			0
$931$			8.46516e6i			0.320082i
$932$		0			0
$933$	−8.12639e6	+	8.12639e6i	−0.305628	+	0.305628i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.76810e7	−	1.76810e7i	−0.657898	−	0.657898i	0.296985	−	0.954882i	$-0.404019\pi$
$937$	−1.76810e7	−	1.76810e7i	−0.657898	−	0.657898i	−0.954882	+	0.296985i	$0.904019\pi$
$938$		0			0
$939$	−2.82697e7			−1.04630
$940$		0			0
$941$	−2.81767e7			−1.03733			−0.518664	−	0.854978i	$-0.673571\pi$
$941$	−2.81767e7			−1.03733			−0.518664	+	0.854978i	$0.673571\pi$
$942$		0			0
$943$	−2.87042e7	−	2.87042e7i	−1.05116	−	1.05116i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	812310.	−	812310.i	0.0294338	−	0.0294338i	−0.692237	−	0.721671i	$-0.743375\pi$
$947$	812310.	−	812310.i	0.0294338	−	0.0294338i	0.721671	+	0.692237i	$0.243375\pi$
$948$		0			0
$949$			1.79427e7i			0.646729i
$950$		0			0
$951$			1.48907e7i			0.533906i
$952$		0			0
$953$	−2.90628e7	+	2.90628e7i	−1.03659	+	1.03659i	−0.0372807	+	0.999305i	$0.511870\pi$
$953$	−2.90628e7	+	2.90628e7i	−1.03659	+	1.03659i	−0.999305	+	0.0372807i	$0.988130\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.60012e7	+	1.60012e7i	0.564773	+	0.564773i
$958$		0			0
$959$	−4.54762e6			−0.159675
$960$		0			0
$961$	−5.31778e7			−1.85747
$962$		0			0
$963$	8.44988e6	+	8.44988e6i	0.293619	+	0.293619i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.81711e7	−	2.81711e7i	0.968809	−	0.968809i	−0.0307195	−	0.999528i	$-0.509780\pi$
$967$	2.81711e7	−	2.81711e7i	0.968809	−	0.968809i	0.999528	+	0.0307195i	$0.00977986\pi$
$968$		0			0
$969$			1.41964e7i			0.485700i
$970$		0			0
$971$			4.42018e7i			1.50450i	0.658879	+	0.752249i	$0.271031\pi$
$971$			4.42018e7i			1.50450i	−0.658879	+	0.752249i	$0.728969\pi$
$972$		0			0
$973$	−1.36374e7	+	1.36374e7i	−0.461795	+	0.461795i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.48781e7	−	1.48781e7i	−0.498667	−	0.498667i	0.412356	−	0.911023i	$-0.364706\pi$
$977$	−1.48781e7	−	1.48781e7i	−0.498667	−	0.498667i	−0.911023	+	0.412356i	$0.864706\pi$
$978$		0			0
$979$	6.76029e7			2.25429
$980$		0			0
$981$	3.46409e6			0.114926
$982$		0			0
$983$	−955948.	−	955948.i	−0.0315537	−	0.0315537i	0.691154	−	0.722708i	$-0.257103\pi$
$983$	−955948.	−	955948.i	−0.0315537	−	0.0315537i	−0.722708	+	0.691154i	$0.757103\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.48662e6	−	7.48662e6i	0.244621	−	0.244621i
$988$		0			0
$989$		−	1.57828e7i		−	0.513089i
$990$		0			0
$991$		−	2.39902e7i		−	0.775978i	−0.921664	−	0.387989i	$-0.873170\pi$
$991$		−	2.39902e7i		−	0.775978i	0.921664	−	0.387989i	$-0.126830\pi$
$992$		0			0
$993$	−1.48845e7	+	1.48845e7i	−0.479029	+	0.479029i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.04726e7	−	3.04726e7i	−0.970894	−	0.970894i	0.0286939	−	0.999588i	$-0.490865\pi$
$997$	−3.04726e7	−	3.04726e7i	−0.970894	−	0.970894i	−0.999588	+	0.0286939i	$0.990865\pi$
$998$		0			0
$999$	9.49211e6			0.300919

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.g.207.3		20
4.3	odd	2	inner	400.6.n.g.207.8		20
5.2	odd	4		80.6.n.d.63.3	yes	20
5.3	odd	4	inner	400.6.n.g.143.8		20
5.4	even	2		80.6.n.d.47.8	yes	20
20.3	even	4	inner	400.6.n.g.143.3		20
20.7	even	4		80.6.n.d.63.8	yes	20
20.19	odd	2		80.6.n.d.47.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.d.47.3	✓	20	20.19	odd	2
80.6.n.d.47.8	yes	20	5.4	even	2
80.6.n.d.63.3	yes	20	5.2	odd	4
80.6.n.d.63.8	yes	20	20.7	even	4
400.6.n.g.143.3		20	20.3	even	4	inner
400.6.n.g.143.8		20	5.3	odd	4	inner
400.6.n.g.207.3		20	1.1	even	1	trivial
400.6.n.g.207.8		20	4.3	odd	2	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-9.68301 - 9.68301i) q^{3} +(-48.6629 + 48.6629i) q^{7} -55.4787i q^{9} +O(q^{10})\)	\(q+(-9.68301 - 9.68301i) q^{3} +(-48.6629 + 48.6629i) q^{7} -55.4787i q^{9} -463.177i q^{11} +(-320.800 + 320.800i) q^{13} +(-1045.30 - 1045.30i) q^{17} +701.290 q^{19} +942.407 q^{21} +(2001.88 + 2001.88i) q^{23} +(-2890.17 + 2890.17i) q^{27} -3567.76i q^{29} -9044.72i q^{31} +(-4484.95 + 4484.95i) q^{33} +(-1642.14 - 1642.14i) q^{37} +6212.62 q^{39} -14338.6 q^{41} +(-3941.99 - 3941.99i) q^{43} +(7944.15 - 7944.15i) q^{47} +12070.8i q^{49} +20243.2i q^{51} +(-11621.9 + 11621.9i) q^{53} +(-6790.60 - 6790.60i) q^{57} +1121.30 q^{59} -29320.4 q^{61} +(2699.75 + 2699.75i) q^{63} +(-9199.75 + 9199.75i) q^{67} -38768.5i q^{69} +52643.9i q^{71} +(27965.6 - 27965.6i) q^{73} +(22539.6 + 22539.6i) q^{77} +82263.7 q^{79} +42489.8 q^{81} +(77236.8 + 77236.8i) q^{83} +(-34546.7 + 34546.7i) q^{87} +145955. i q^{89} -31222.2i q^{91} +(-87580.1 + 87580.1i) q^{93} +(-97856.7 - 97856.7i) q^{97} -25696.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) \) 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Introduction

L-functions

Modular forms

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Database

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