LMFDB - Embedded newform 400.4.c.i.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not 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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.i.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} +24.0000i q^{7} +11.0000 q^{9} +O(q^{10})$	$q+4.00000i q^{3} +24.0000i q^{7} +11.0000 q^{9} +44.0000 q^{11} +22.0000i q^{13} -50.0000i q^{17} +44.0000 q^{19} -96.0000 q^{21} +56.0000i q^{23} +152.000i q^{27} -198.000 q^{29} +160.000 q^{31} +176.000i q^{33} +162.000i q^{37} -88.0000 q^{39} -198.000 q^{41} -52.0000i q^{43} +528.000i q^{47} -233.000 q^{49} +200.000 q^{51} -242.000i q^{53} +176.000i q^{57} -668.000 q^{59} +550.000 q^{61} +264.000i q^{63} +188.000i q^{67} -224.000 q^{69} -728.000 q^{71} +154.000i q^{73} +1056.00i q^{77} -656.000 q^{79} -311.000 q^{81} -236.000i q^{83} -792.000i q^{87} -714.000 q^{89} -528.000 q^{91} +640.000i q^{93} +478.000i q^{97} +484.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{9}+O(q^{10}) $ 2 * q + 22 * q^9	$ 2 q + 22 q^{9} + 88 q^{11} + 88 q^{19} - 192 q^{21} - 396 q^{29} + 320 q^{31} - 176 q^{39} - 396 q^{41} - 466 q^{49} + 400 q^{51} - 1336 q^{59} + 1100 q^{61} - 448 q^{69} - 1456 q^{71} - 1312 q^{79} - 622 q^{81} - 1428 q^{89} - 1056 q^{91} + 968 q^{99}+O(q^{100}) $ 2 * q + 22 * q^9 + 88 * q^11 + 88 * q^19 - 192 * q^21 - 396 * q^29 + 320 * q^31 - 176 * q^39 - 396 * q^41 - 466 * q^49 + 400 * q^51 - 1336 * q^59 + 1100 * q^61 - 448 * q^69 - 1456 * q^71 - 1312 * q^79 - 622 * q^81 - 1428 * q^89 - 1056 * q^91 + 968 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	4.00000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	4.00000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	24.0000i	1.29588i	0.761692	+	0.647939i	$0.224369\pi$
$7$	24.0000i	1.29588i	−0.761692	+	0.647939i	$0.775631\pi$
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	44.0000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	44.0000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	22.0000i	0.469362i	0.972072	+	0.234681i	$0.0754045\pi$
$13$	22.0000i	0.469362i	−0.972072	+	0.234681i	$0.924595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 50.0000i	− 0.713340i	−0.934230	−	0.356670i	$-0.883912\pi$
$17$	− 50.0000i	− 0.713340i	0.934230	−	0.356670i	$-0.116088\pi$
$18$	0	0
$19$	44.0000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	0	0
$23$	56.0000i	0.507687i	0.967245	+	0.253844i	$0.0816949\pi$
$23$	56.0000i	0.507687i	−0.967245	+	0.253844i	$0.918305\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	152.000i	1.08342i
$28$	0	0
$29$	−198.000	−1.26785	−0.633925	−	0.773394i	$-0.718557\pi$
$29$	−198.000	−1.26785	−0.633925	+	0.773394i	$0.718557\pi$
$30$	0	0
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	0	0
$33$	176.000i	0.928414i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	162.000i	0.719801i	0.932991	+	0.359900i	$0.117189\pi$
$37$	162.000i	0.719801i	−0.932991	+	0.359900i	$0.882811\pi$
$38$	0	0
$39$	−88.0000	−0.361315
$40$	0	0
$41$	−198.000	−0.754205	−0.377102	−	0.926172i	$-0.623080\pi$
$41$	−198.000	−0.754205	−0.377102	+	0.926172i	$0.623080\pi$
$42$	0	0
$43$	− 52.0000i	− 0.184417i	−0.995740	−	0.0922084i	$-0.970607\pi$
$43$	− 52.0000i	− 0.184417i	0.995740	−	0.0922084i	$-0.0293926\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	528.000i	1.63865i	0.573327	+	0.819327i	$0.305653\pi$
$47$	528.000i	1.63865i	−0.573327	+	0.819327i	$0.694347\pi$
$48$	0	0
$49$	−233.000	−0.679300
$50$	0	0
$51$	200.000	0.549129
$52$	0	0
$53$	− 242.000i	− 0.627194i	−0.949556	−	0.313597i	$-0.898466\pi$
$53$	− 242.000i	− 0.627194i	0.949556	−	0.313597i	$-0.101534\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	176.000i	0.408978i
$58$	0	0
$59$	−668.000	−1.47400	−0.737002	−	0.675891i	$-0.763759\pi$
$59$	−668.000	−1.47400	−0.737002	+	0.675891i	$0.763759\pi$
$60$	0	0
$61$	550.000	1.15443	0.577215	−	0.816592i	$-0.304139\pi$
$61$	550.000	1.15443	0.577215	+	0.816592i	$0.304139\pi$
$62$	0	0
$63$	264.000i	0.527950i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	188.000i	0.342804i	0.985201	+	0.171402i	$0.0548297\pi$
$67$	188.000i	0.342804i	−0.985201	+	0.171402i	$0.945170\pi$
$68$	0	0
$69$	−224.000	−0.390818
$70$	0	0
$71$	−728.000	−1.21687	−0.608435	−	0.793604i	$-0.708202\pi$
$71$	−728.000	−1.21687	−0.608435	+	0.793604i	$0.708202\pi$
$72$	0	0
$73$	154.000i	0.246909i	0.992350	+	0.123454i	$0.0393973\pi$
$73$	154.000i	0.246909i	−0.992350	+	0.123454i	$0.960603\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1056.00i	1.56289i
$78$	0	0
$79$	−656.000	−0.934250	−0.467125	−	0.884191i	$-0.654710\pi$
$79$	−656.000	−0.934250	−0.467125	+	0.884191i	$0.654710\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	− 236.000i	− 0.312101i	−0.987749	−	0.156050i	$-0.950124\pi$
$83$	− 236.000i	− 0.312101i	0.987749	−	0.156050i	$-0.0498762\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 792.000i	− 0.975992i
$88$	0	0
$89$	−714.000	−0.850380	−0.425190	−	0.905104i	$-0.639793\pi$
$89$	−714.000	−0.850380	−0.425190	+	0.905104i	$0.639793\pi$
$90$	0	0
$91$	−528.000	−0.608236
$92$	0	0
$93$	640.000i	0.713601i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	478.000i	0.500346i	0.968201	+	0.250173i	$0.0804875\pi$
$97$	478.000i	0.500346i	−0.968201	+	0.250173i	$0.919513\pi$
$98$	0	0
$99$	484.000	0.491352
$100$	0	0
$101$	1566.00	1.54280	0.771400	−	0.636350i	$-0.219557\pi$
$101$	1566.00	1.54280	0.771400	+	0.636350i	$0.219557\pi$
$102$	0	0
$103$	968.000i	0.926018i	0.886354	+	0.463009i	$0.153230\pi$
$103$	968.000i	0.926018i	−0.886354	+	0.463009i	$0.846770\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 780.000i	− 0.704724i	−0.935864	−	0.352362i	$-0.885379\pi$
$107$	− 780.000i	− 0.704724i	0.935864	−	0.352362i	$-0.114621\pi$
$108$	0	0
$109$	1994.00	1.75221	0.876103	−	0.482123i	$-0.160134\pi$
$109$	1994.00	1.75221	0.876103	+	0.482123i	$0.160134\pi$
$110$	0	0
$111$	−648.000	−0.554103
$112$	0	0
$113$	− 942.000i	− 0.784212i	−0.919920	−	0.392106i	$-0.871747\pi$
$113$	− 942.000i	− 0.784212i	0.919920	−	0.392106i	$-0.128253\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	242.000i	0.191221i
$118$	0	0
$119$	1200.00	0.924402
$120$	0	0
$121$	605.000	0.454545
$122$	0	0
$123$	− 792.000i	− 0.580587i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1408.00i	0.983778i	0.870658	+	0.491889i	$0.163693\pi$
$127$	1408.00i	0.983778i	−0.870658	+	0.491889i	$0.836307\pi$
$128$	0	0
$129$	208.000	0.141964
$130$	0	0
$131$	2692.00	1.79543	0.897714	−	0.440578i	$-0.145227\pi$
$131$	2692.00	1.79543	0.897714	+	0.440578i	$0.145227\pi$
$132$	0	0
$133$	1056.00i	0.688472i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1626.00i	− 1.01400i	−0.861945	−	0.507002i	$-0.830754\pi$
$137$	− 1626.00i	− 1.01400i	0.861945	−	0.507002i	$-0.169246\pi$
$138$	0	0
$139$	−684.000	−0.417382	−0.208691	−	0.977982i	$-0.566920\pi$
$139$	−684.000	−0.417382	−0.208691	+	0.977982i	$0.566920\pi$
$140$	0	0
$141$	−2112.00	−1.26144
$142$	0	0
$143$	968.000i	0.566072i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 932.000i	− 0.522926i
$148$	0	0
$149$	−302.000	−0.166046	−0.0830228	−	0.996548i	$-0.526457\pi$
$149$	−302.000	−0.166046	−0.0830228	+	0.996548i	$0.526457\pi$
$150$	0	0
$151$	−1352.00	−0.728637	−0.364319	−	0.931274i	$-0.618698\pi$
$151$	−1352.00	−0.728637	−0.364319	+	0.931274i	$0.618698\pi$
$152$	0	0
$153$	− 550.000i	− 0.290620i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3142.00i	− 1.59719i	−0.601868	−	0.798595i	$-0.705577\pi$
$157$	− 3142.00i	− 1.59719i	0.601868	−	0.798595i	$-0.294423\pi$
$158$	0	0
$159$	968.000	0.482814
$160$	0	0
$161$	−1344.00	−0.657901
$162$	0	0
$163$	− 3036.00i	− 1.45888i	−0.684043	−	0.729441i	$-0.739780\pi$
$163$	− 3036.00i	− 1.45888i	0.684043	−	0.729441i	$-0.260220\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 264.000i	− 0.122329i	−0.998128	−	0.0611645i	$-0.980519\pi$
$167$	− 264.000i	− 0.122329i	0.998128	−	0.0611645i	$-0.0194814\pi$
$168$	0	0
$169$	1713.00	0.779700
$170$	0	0
$171$	484.000	0.216447
$172$	0	0
$173$	− 2826.00i	− 1.24195i	−0.783832	−	0.620973i	$-0.786737\pi$
$173$	− 2826.00i	− 1.24195i	0.783832	−	0.620973i	$-0.213263\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2672.00i	− 1.13469i
$178$	0	0
$179$	3084.00	1.28776	0.643880	−	0.765127i	$-0.277324\pi$
$179$	3084.00	1.28776	0.643880	+	0.765127i	$0.277324\pi$
$180$	0	0
$181$	−2418.00	−0.992975	−0.496488	−	0.868044i	$-0.665377\pi$
$181$	−2418.00	−0.992975	−0.496488	+	0.868044i	$0.665377\pi$
$182$	0	0
$183$	2200.00i	0.888681i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2200.00i	− 0.860320i
$188$	0	0
$189$	−3648.00	−1.40398
$190$	0	0
$191$	960.000	0.363681	0.181841	−	0.983328i	$-0.441794\pi$
$191$	960.000	0.363681	0.181841	+	0.983328i	$0.441794\pi$
$192$	0	0
$193$	2882.00i	1.07488i	0.843304	+	0.537438i	$0.180608\pi$
$193$	2882.00i	1.07488i	−0.843304	+	0.537438i	$0.819392\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1086.00i	− 0.392763i	−0.980528	−	0.196381i	$-0.937081\pi$
$197$	− 1086.00i	− 0.392763i	0.980528	−	0.196381i	$-0.0629191\pi$
$198$	0	0
$199$	88.0000	0.0313475	0.0156738	−	0.999877i	$-0.495011\pi$
$199$	88.0000	0.0313475	0.0156738	+	0.999877i	$0.495011\pi$
$200$	0	0
$201$	−752.000	−0.263890
$202$	0	0
$203$	− 4752.00i	− 1.64298i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	616.000i	0.206836i
$208$	0	0
$209$	1936.00	0.640746
$210$	0	0
$211$	3476.00	1.13411	0.567056	−	0.823679i	$-0.308082\pi$
$211$	3476.00	1.13411	0.567056	+	0.823679i	$0.308082\pi$
$212$	0	0
$213$	− 2912.00i	− 0.936746i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3840.00i	1.20127i
$218$	0	0
$219$	−616.000	−0.190070
$220$	0	0
$221$	1100.00	0.334815
$222$	0	0
$223$	− 928.000i	− 0.278670i	−0.990245	−	0.139335i	$-0.955503\pi$
$223$	− 928.000i	− 0.278670i	0.990245	−	0.139335i	$-0.0444965\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	156.000i	0.0456127i	0.999740	+	0.0228064i	$0.00726012\pi$
$227$	156.000i	0.0456127i	−0.999740	+	0.0228064i	$0.992740\pi$
$228$	0	0
$229$	1634.00	0.471519	0.235759	−	0.971811i	$-0.424242\pi$
$229$	1634.00	0.471519	0.235759	+	0.971811i	$0.424242\pi$
$230$	0	0
$231$	−4224.00	−1.20311
$232$	0	0
$233$	− 902.000i	− 0.253614i	−0.991927	−	0.126807i	$-0.959527\pi$
$233$	− 902.000i	− 0.253614i	0.991927	−	0.126807i	$-0.0404728\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2624.00i	− 0.719186i
$238$	0	0
$239$	1616.00	0.437365	0.218683	−	0.975796i	$-0.429824\pi$
$239$	1616.00	0.437365	0.218683	+	0.975796i	$0.429824\pi$
$240$	0	0
$241$	4818.00	1.28778	0.643889	−	0.765119i	$-0.277320\pi$
$241$	4818.00	1.28778	0.643889	+	0.765119i	$0.277320\pi$
$242$	0	0
$243$	2860.00i	0.755017i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	968.000i	0.249362i
$248$	0	0
$249$	944.000	0.240255
$250$	0	0
$251$	2140.00	0.538150	0.269075	−	0.963119i	$-0.413282\pi$
$251$	2140.00	0.538150	0.269075	+	0.963119i	$0.413282\pi$
$252$	0	0
$253$	2464.00i	0.612294i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 770.000i	− 0.186892i	−0.995624	−	0.0934461i	$-0.970212\pi$
$257$	− 770.000i	− 0.186892i	0.995624	−	0.0934461i	$-0.0297883\pi$
$258$	0	0
$259$	−3888.00	−0.932774
$260$	0	0
$261$	−2178.00	−0.516532
$262$	0	0
$263$	7400.00i	1.73499i	0.497442	+	0.867497i	$0.334273\pi$
$263$	7400.00i	1.73499i	−0.497442	+	0.867497i	$0.665727\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2856.00i	− 0.654623i
$268$	0	0
$269$	2794.00	0.633283	0.316642	−	0.948545i	$-0.397445\pi$
$269$	2794.00	0.633283	0.316642	+	0.948545i	$0.397445\pi$
$270$	0	0
$271$	−8624.00	−1.93310	−0.966551	−	0.256474i	$-0.917439\pi$
$271$	−8624.00	−1.93310	−0.966551	+	0.256474i	$0.917439\pi$
$272$	0	0
$273$	− 2112.00i	− 0.468220i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1874.00i	0.406490i	0.979128	+	0.203245i	$0.0651488\pi$
$277$	1874.00i	0.406490i	−0.979128	+	0.203245i	$0.934851\pi$
$278$	0	0
$279$	1760.00	0.377665
$280$	0	0
$281$	3338.00	0.708642	0.354321	−	0.935124i	$-0.384712\pi$
$281$	3338.00	0.708642	0.354321	+	0.935124i	$0.384712\pi$
$282$	0	0
$283$	− 7172.00i	− 1.50647i	−0.657751	−	0.753235i	$-0.728492\pi$
$283$	− 7172.00i	− 1.50647i	0.657751	−	0.753235i	$-0.271508\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4752.00i	− 0.977358i
$288$	0	0
$289$	2413.00	0.491146
$290$	0	0
$291$	−1912.00	−0.385166
$292$	0	0
$293$	5214.00i	1.03961i	0.854286	+	0.519804i	$0.173995\pi$
$293$	5214.00i	1.03961i	−0.854286	+	0.519804i	$0.826005\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6688.00i	1.30666i
$298$	0	0
$299$	−1232.00	−0.238289
$300$	0	0
$301$	1248.00	0.238982
$302$	0	0
$303$	6264.00i	1.18765i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	396.000i	0.0736186i	0.999322	+	0.0368093i	$0.0117194\pi$
$307$	396.000i	0.0736186i	−0.999322	+	0.0368093i	$0.988281\pi$
$308$	0	0
$309$	−3872.00	−0.712849
$310$	0	0
$311$	4056.00	0.739533	0.369766	−	0.929125i	$-0.379438\pi$
$311$	4056.00	0.739533	0.369766	+	0.929125i	$0.379438\pi$
$312$	0	0
$313$	2154.00i	0.388982i	0.980904	+	0.194491i	$0.0623055\pi$
$313$	2154.00i	0.388982i	−0.980904	+	0.194491i	$0.937695\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7386.00i	1.30864i	0.756217	+	0.654320i	$0.227045\pi$
$317$	7386.00i	1.30864i	−0.756217	+	0.654320i	$0.772955\pi$
$318$	0	0
$319$	−8712.00	−1.52909
$320$	0	0
$321$	3120.00	0.542497
$322$	0	0
$323$	− 2200.00i	− 0.378982i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7976.00i	1.34885i
$328$	0	0
$329$	−12672.0	−2.12350
$330$	0	0
$331$	1132.00	0.187977	0.0939884	−	0.995573i	$-0.470038\pi$
$331$	1132.00	0.187977	0.0939884	+	0.995573i	$0.470038\pi$
$332$	0	0
$333$	1782.00i	0.293252i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3342.00i	0.540209i	0.962831	+	0.270104i	$0.0870582\pi$
$337$	3342.00i	0.540209i	−0.962831	+	0.270104i	$0.912942\pi$
$338$	0	0
$339$	3768.00	0.603686
$340$	0	0
$341$	7040.00	1.11800
$342$	0	0
$343$	2640.00i	0.415588i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2244.00i	0.347159i	0.984820	+	0.173580i	$0.0555334\pi$
$347$	2244.00i	0.347159i	−0.984820	+	0.173580i	$0.944467\pi$
$348$	0	0
$349$	6522.00	1.00033	0.500164	−	0.865931i	$-0.333273\pi$
$349$	6522.00	1.00033	0.500164	+	0.865931i	$0.333273\pi$
$350$	0	0
$351$	−3344.00	−0.508517
$352$	0	0
$353$	− 11230.0i	− 1.69324i	−0.532200	−	0.846618i	$-0.678635\pi$
$353$	− 11230.0i	− 1.69324i	0.532200	−	0.846618i	$-0.321365\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4800.00i	0.711605i
$358$	0	0
$359$	1848.00	0.271682	0.135841	−	0.990731i	$-0.456626\pi$
$359$	1848.00	0.271682	0.135841	+	0.990731i	$0.456626\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	2420.00i	0.349909i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7120.00i	1.01270i	0.862328	+	0.506350i	$0.169006\pi$
$367$	7120.00i	1.01270i	−0.862328	+	0.506350i	$0.830994\pi$
$368$	0	0
$369$	−2178.00	−0.307269
$370$	0	0
$371$	5808.00	0.812766
$372$	0	0
$373$	6350.00i	0.881476i	0.897636	+	0.440738i	$0.145283\pi$
$373$	6350.00i	0.881476i	−0.897636	+	0.440738i	$0.854717\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4356.00i	− 0.595081i
$378$	0	0
$379$	−7900.00	−1.07070	−0.535351	−	0.844630i	$-0.679821\pi$
$379$	−7900.00	−1.07070	−0.535351	+	0.844630i	$0.679821\pi$
$380$	0	0
$381$	−5632.00	−0.757313
$382$	0	0
$383$	− 10368.0i	− 1.38324i	−0.722263	−	0.691619i	$-0.756898\pi$
$383$	− 10368.0i	− 1.38324i	0.722263	−	0.691619i	$-0.243102\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 572.000i	− 0.0751328i
$388$	0	0
$389$	−8830.00	−1.15090	−0.575448	−	0.817838i	$-0.695172\pi$
$389$	−8830.00	−1.15090	−0.575448	+	0.817838i	$0.695172\pi$
$390$	0	0
$391$	2800.00	0.362154
$392$	0	0
$393$	10768.0i	1.38212i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 9878.00i	− 1.24877i	−0.781116	−	0.624386i	$-0.785349\pi$
$397$	− 9878.00i	− 1.24877i	0.781116	−	0.624386i	$-0.214651\pi$
$398$	0	0
$399$	−4224.00	−0.529986
$400$	0	0
$401$	−13134.0	−1.63561	−0.817806	−	0.575494i	$-0.804810\pi$
$401$	−13134.0	−1.63561	−0.817806	+	0.575494i	$0.804810\pi$
$402$	0	0
$403$	3520.00i	0.435096i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7128.00i	0.868113i
$408$	0	0
$409$	−906.000	−0.109533	−0.0547663	−	0.998499i	$-0.517441\pi$
$409$	−906.000	−0.109533	−0.0547663	+	0.998499i	$0.517441\pi$
$410$	0	0
$411$	6504.00	0.780581
$412$	0	0
$413$	− 16032.0i	− 1.91013i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2736.00i	− 0.321301i
$418$	0	0
$419$	−5412.00	−0.631011	−0.315505	−	0.948924i	$-0.602174\pi$
$419$	−5412.00	−0.631011	−0.315505	+	0.948924i	$0.602174\pi$
$420$	0	0
$421$	−4642.00	−0.537381	−0.268690	−	0.963227i	$-0.586591\pi$
$421$	−4642.00	−0.537381	−0.268690	+	0.963227i	$0.586591\pi$
$422$	0	0
$423$	5808.00i	0.667600i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13200.0i	1.49600i
$428$	0	0
$429$	−3872.00	−0.435762
$430$	0	0
$431$	−656.000	−0.0733142	−0.0366571	−	0.999328i	$-0.511671\pi$
$431$	−656.000	−0.0733142	−0.0366571	+	0.999328i	$0.511671\pi$
$432$	0	0
$433$	9490.00i	1.05326i	0.850096	+	0.526629i	$0.176544\pi$
$433$	9490.00i	1.05326i	−0.850096	+	0.526629i	$0.823456\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2464.00i	0.269723i
$438$	0	0
$439$	5544.00	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$439$	5544.00	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$440$	0	0
$441$	−2563.00	−0.276752
$442$	0	0
$443$	− 7652.00i	− 0.820672i	−0.911935	−	0.410336i	$-0.865412\pi$
$443$	− 7652.00i	− 0.820672i	0.911935	−	0.410336i	$-0.134588\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1208.00i	− 0.127822i
$448$	0	0
$449$	446.000	0.0468776	0.0234388	−	0.999725i	$-0.492539\pi$
$449$	446.000	0.0468776	0.0234388	+	0.999725i	$0.492539\pi$
$450$	0	0
$451$	−8712.00	−0.909605
$452$	0	0
$453$	− 5408.00i	− 0.560905i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1562.00i	− 0.159885i	−0.996799	−	0.0799423i	$-0.974526\pi$
$457$	− 1562.00i	− 0.159885i	0.996799	−	0.0799423i	$-0.0254736\pi$
$458$	0	0
$459$	7600.00	0.772849
$460$	0	0
$461$	10582.0	1.06910	0.534548	−	0.845138i	$-0.320482\pi$
$461$	10582.0	1.06910	0.534548	+	0.845138i	$0.320482\pi$
$462$	0	0
$463$	10768.0i	1.08085i	0.841394	+	0.540423i	$0.181736\pi$
$463$	10768.0i	1.08085i	−0.841394	+	0.540423i	$0.818264\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9876.00i	− 0.978601i	−0.872115	−	0.489301i	$-0.837252\pi$
$467$	− 9876.00i	− 0.978601i	0.872115	−	0.489301i	$-0.162748\pi$
$468$	0	0
$469$	−4512.00	−0.444232
$470$	0	0
$471$	12568.0	1.22952
$472$	0	0
$473$	− 2288.00i	− 0.222415i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2662.00i	− 0.255523i
$478$	0	0
$479$	−352.000	−0.0335768	−0.0167884	−	0.999859i	$-0.505344\pi$
$479$	−352.000	−0.0335768	−0.0167884	+	0.999859i	$0.505344\pi$
$480$	0	0
$481$	−3564.00	−0.337847
$482$	0	0
$483$	− 5376.00i	− 0.506452i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 15176.0i	− 1.41209i	−0.708165	−	0.706047i	$-0.750477\pi$
$487$	− 15176.0i	− 1.41209i	0.708165	−	0.706047i	$-0.249523\pi$
$488$	0	0
$489$	12144.0	1.12305
$490$	0	0
$491$	8844.00	0.812880	0.406440	−	0.913677i	$-0.366770\pi$
$491$	8844.00	0.812880	0.406440	+	0.913677i	$0.366770\pi$
$492$	0	0
$493$	9900.00i	0.904409i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 17472.0i	− 1.57691i
$498$	0	0
$499$	19404.0	1.74077	0.870383	−	0.492375i	$-0.163871\pi$
$499$	19404.0	1.74077	0.870383	+	0.492375i	$0.163871\pi$
$500$	0	0
$501$	1056.00	0.0941689
$502$	0	0
$503$	− 16488.0i	− 1.46156i	−0.682614	−	0.730779i	$-0.739157\pi$
$503$	− 16488.0i	− 1.46156i	0.682614	−	0.730779i	$-0.260843\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6852.00i	0.600213i
$508$	0	0
$509$	12954.0	1.12805	0.564024	−	0.825759i	$-0.309253\pi$
$509$	12954.0	1.12805	0.564024	+	0.825759i	$0.309253\pi$
$510$	0	0
$511$	−3696.00	−0.319964
$512$	0	0
$513$	6688.00i	0.575599i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23232.0i	1.97629i
$518$	0	0
$519$	11304.0	0.956051
$520$	0	0
$521$	10970.0	0.922465	0.461233	−	0.887279i	$-0.347407\pi$
$521$	10970.0	0.922465	0.461233	+	0.887279i	$0.347407\pi$
$522$	0	0
$523$	16940.0i	1.41632i	0.706053	+	0.708159i	$0.250474\pi$
$523$	16940.0i	1.41632i	−0.706053	+	0.708159i	$0.749526\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8000.00i	− 0.661263i
$528$	0	0
$529$	9031.00	0.742254
$530$	0	0
$531$	−7348.00	−0.600520
$532$	0	0
$533$	− 4356.00i	− 0.353995i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12336.0i	0.991318i
$538$	0	0
$539$	−10252.0	−0.819267
$540$	0	0
$541$	198.000	0.0157351	0.00786755	−	0.999969i	$-0.497496\pi$
$541$	198.000	0.0157351	0.00786755	+	0.999969i	$0.497496\pi$
$542$	0	0
$543$	− 9672.00i	− 0.764393i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15268.0i	− 1.19344i	−0.802449	−	0.596721i	$-0.796470\pi$
$547$	− 15268.0i	− 1.19344i	0.802449	−	0.596721i	$-0.203530\pi$
$548$	0	0
$549$	6050.00	0.470324
$550$	0	0
$551$	−8712.00	−0.673582
$552$	0	0
$553$	− 15744.0i	− 1.21067i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20854.0i	− 1.58638i	−0.608976	−	0.793189i	$-0.708419\pi$
$557$	− 20854.0i	− 1.58638i	0.608976	−	0.793189i	$-0.291581\pi$
$558$	0	0
$559$	1144.00	0.0865582
$560$	0	0
$561$	8800.00	0.662275
$562$	0	0
$563$	19316.0i	1.44595i	0.690872	+	0.722977i	$0.257227\pi$
$563$	19316.0i	1.44595i	−0.690872	+	0.722977i	$0.742773\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7464.00i	− 0.552837i
$568$	0	0
$569$	−7018.00	−0.517065	−0.258532	−	0.966003i	$-0.583239\pi$
$569$	−7018.00	−0.517065	−0.258532	+	0.966003i	$0.583239\pi$
$570$	0	0
$571$	−24420.0	−1.78975	−0.894873	−	0.446320i	$-0.852734\pi$
$571$	−24420.0	−1.78975	−0.894873	+	0.446320i	$0.852734\pi$
$572$	0	0
$573$	3840.00i	0.279962i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 23234.0i	− 1.67633i	−0.545415	−	0.838166i	$-0.683628\pi$
$577$	− 23234.0i	− 1.67633i	0.545415	−	0.838166i	$-0.316372\pi$
$578$	0	0
$579$	−11528.0	−0.827439
$580$	0	0
$581$	5664.00	0.404445
$582$	0	0
$583$	− 10648.0i	− 0.756424i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10604.0i	− 0.745611i	−0.927909	−	0.372806i	$-0.878396\pi$
$587$	− 10604.0i	− 0.745611i	0.927909	−	0.372806i	$-0.121604\pi$
$588$	0	0
$589$	7040.00	0.492493
$590$	0	0
$591$	4344.00	0.302349
$592$	0	0
$593$	− 13838.0i	− 0.958277i	−0.877739	−	0.479139i	$-0.840949\pi$
$593$	− 13838.0i	− 0.958277i	0.877739	−	0.479139i	$-0.159051\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	352.000i	0.0241313i
$598$	0	0
$599$	−3960.00	−0.270119	−0.135059	−	0.990837i	$-0.543123\pi$
$599$	−3960.00	−0.270119	−0.135059	+	0.990837i	$0.543123\pi$
$600$	0	0
$601$	−5942.00	−0.403293	−0.201647	−	0.979458i	$-0.564629\pi$
$601$	−5942.00	−0.403293	−0.201647	+	0.979458i	$0.564629\pi$
$602$	0	0
$603$	2068.00i	0.139661i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3040.00i	− 0.203278i	−0.994821	−	0.101639i	$-0.967591\pi$
$607$	− 3040.00i	− 0.203278i	0.994821	−	0.101639i	$-0.0324087\pi$
$608$	0	0
$609$	19008.0	1.26477
$610$	0	0
$611$	−11616.0	−0.769121
$612$	0	0
$613$	− 2530.00i	− 0.166698i	−0.996520	−	0.0833489i	$-0.973438\pi$
$613$	− 2530.00i	− 0.166698i	0.996520	−	0.0833489i	$-0.0265616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19206.0i	1.25317i	0.779354	+	0.626584i	$0.215547\pi$
$617$	19206.0i	1.25317i	−0.779354	+	0.626584i	$0.784453\pi$
$618$	0	0
$619$	10996.0	0.714001	0.357000	−	0.934104i	$-0.383799\pi$
$619$	10996.0	0.714001	0.357000	+	0.934104i	$0.383799\pi$
$620$	0	0
$621$	−8512.00	−0.550040
$622$	0	0
$623$	− 17136.0i	− 1.10199i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7744.00i	0.493247i
$628$	0	0
$629$	8100.00	0.513463
$630$	0	0
$631$	6680.00	0.421437	0.210718	−	0.977547i	$-0.432420\pi$
$631$	6680.00	0.421437	0.210718	+	0.977547i	$0.432420\pi$
$632$	0	0
$633$	13904.0i	0.873040i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5126.00i	− 0.318838i
$638$	0	0
$639$	−8008.00	−0.495761
$640$	0	0
$641$	6274.00	0.386596	0.193298	−	0.981140i	$-0.438082\pi$
$641$	6274.00	0.386596	0.193298	+	0.981140i	$0.438082\pi$
$642$	0	0
$643$	− 9084.00i	− 0.557135i	−0.960417	−	0.278568i	$-0.910140\pi$
$643$	− 9084.00i	− 0.557135i	0.960417	−	0.278568i	$-0.0898596\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 23656.0i	− 1.43742i	−0.695308	−	0.718712i	$-0.744732\pi$
$647$	− 23656.0i	− 1.43742i	0.695308	−	0.718712i	$-0.255268\pi$
$648$	0	0
$649$	−29392.0	−1.77771
$650$	0	0
$651$	−15360.0	−0.924740
$652$	0	0
$653$	− 6762.00i	− 0.405234i	−0.979258	−	0.202617i	$-0.935055\pi$
$653$	− 6762.00i	− 0.405234i	0.979258	−	0.202617i	$-0.0649446\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1694.00i	0.100592i
$658$	0	0
$659$	15276.0	0.902987	0.451494	−	0.892274i	$-0.350891\pi$
$659$	15276.0	0.902987	0.451494	+	0.892274i	$0.350891\pi$
$660$	0	0
$661$	11054.0	0.650455	0.325228	−	0.945636i	$-0.394559\pi$
$661$	11054.0	0.650455	0.325228	+	0.945636i	$0.394559\pi$
$662$	0	0
$663$	4400.00i	0.257740i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11088.0i	− 0.643672i
$668$	0	0
$669$	3712.00	0.214520
$670$	0	0
$671$	24200.0	1.39230
$672$	0	0
$673$	− 21278.0i	− 1.21873i	−0.792889	−	0.609366i	$-0.791424\pi$
$673$	− 21278.0i	− 1.21873i	0.792889	−	0.609366i	$-0.208576\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8926.00i	− 0.506727i	−0.967371	−	0.253363i	$-0.918463\pi$
$677$	− 8926.00i	− 0.506727i	0.967371	−	0.253363i	$-0.0815368\pi$
$678$	0	0
$679$	−11472.0	−0.648387
$680$	0	0
$681$	−624.000	−0.0351127
$682$	0	0
$683$	− 8116.00i	− 0.454685i	−0.973815	−	0.227343i	$-0.926996\pi$
$683$	− 8116.00i	− 0.454685i	0.973815	−	0.227343i	$-0.0730037\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6536.00i	0.362975i
$688$	0	0
$689$	5324.00	0.294381
$690$	0	0
$691$	11764.0	0.647646	0.323823	−	0.946118i	$-0.395032\pi$
$691$	11764.0	0.647646	0.323823	+	0.946118i	$0.395032\pi$
$692$	0	0
$693$	11616.0i	0.636732i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9900.00i	0.538005i
$698$	0	0
$699$	3608.00	0.195232
$700$	0	0
$701$	−4698.00	−0.253126	−0.126563	−	0.991959i	$-0.540395\pi$
$701$	−4698.00	−0.253126	−0.126563	+	0.991959i	$0.540395\pi$
$702$	0	0
$703$	7128.00i	0.382415i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37584.0i	1.99928i
$708$	0	0
$709$	−24638.0	−1.30508	−0.652538	−	0.757756i	$-0.726296\pi$
$709$	−24638.0	−1.30508	−0.652538	+	0.757756i	$0.726296\pi$
$710$	0	0
$711$	−7216.00	−0.380620
$712$	0	0
$713$	8960.00i	0.470624i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6464.00i	0.336684i
$718$	0	0
$719$	16624.0	0.862268	0.431134	−	0.902288i	$-0.358114\pi$
$719$	16624.0	0.862268	0.431134	+	0.902288i	$0.358114\pi$
$720$	0	0
$721$	−23232.0	−1.20001
$722$	0	0
$723$	19272.0i	0.991332i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30216.0i	1.54147i	0.637155	+	0.770735i	$0.280111\pi$
$727$	30216.0i	1.54147i	−0.637155	+	0.770735i	$0.719889\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	−2600.00	−0.131552
$732$	0	0
$733$	− 3322.00i	− 0.167395i	−0.996491	−	0.0836977i	$-0.973327\pi$
$733$	− 3322.00i	− 0.167395i	0.996491	−	0.0836977i	$-0.0266730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8272.00i	0.413437i
$738$	0	0
$739$	−14692.0	−0.731331	−0.365666	−	0.930746i	$-0.619159\pi$
$739$	−14692.0	−0.731331	−0.365666	+	0.930746i	$0.619159\pi$
$740$	0	0
$741$	−3872.00	−0.191959
$742$	0	0
$743$	− 28600.0i	− 1.41216i	−0.708134	−	0.706078i	$-0.750463\pi$
$743$	− 28600.0i	− 1.41216i	0.708134	−	0.706078i	$-0.249537\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2596.00i	− 0.127152i
$748$	0	0
$749$	18720.0	0.913236
$750$	0	0
$751$	29616.0	1.43902	0.719509	−	0.694483i	$-0.244367\pi$
$751$	29616.0	1.43902	0.719509	+	0.694483i	$0.244367\pi$
$752$	0	0
$753$	8560.00i	0.414268i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2894.00i	− 0.138949i	−0.997584	−	0.0694744i	$-0.977868\pi$
$757$	− 2894.00i	− 0.138949i	0.997584	−	0.0694744i	$-0.0221322\pi$
$758$	0	0
$759$	−9856.00	−0.471344
$760$	0	0
$761$	14762.0	0.703183	0.351591	−	0.936154i	$-0.385641\pi$
$761$	14762.0	0.703183	0.351591	+	0.936154i	$0.385641\pi$
$762$	0	0
$763$	47856.0i	2.27065i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 14696.0i	− 0.691841i
$768$	0	0
$769$	7678.00	0.360047	0.180023	−	0.983662i	$-0.442383\pi$
$769$	7678.00	0.360047	0.180023	+	0.983662i	$0.442383\pi$
$770$	0	0
$771$	3080.00	0.143870
$772$	0	0
$773$	27390.0i	1.27445i	0.770678	+	0.637225i	$0.219918\pi$
$773$	27390.0i	1.27445i	−0.770678	+	0.637225i	$0.780082\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 15552.0i	− 0.718050i
$778$	0	0
$779$	−8712.00	−0.400693
$780$	0	0
$781$	−32032.0	−1.46760
$782$	0	0
$783$	− 30096.0i	− 1.37362i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19756.0i	0.894823i	0.894328	+	0.447411i	$0.147654\pi$
$787$	19756.0i	0.894823i	−0.894328	+	0.447411i	$0.852346\pi$
$788$	0	0
$789$	−29600.0	−1.33560
$790$	0	0
$791$	22608.0	1.01624
$792$	0	0
$793$	12100.0i	0.541846i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 38854.0i	− 1.72682i	−0.504499	−	0.863412i	$-0.668323\pi$
$797$	− 38854.0i	− 1.72682i	0.504499	−	0.863412i	$-0.331677\pi$
$798$	0	0
$799$	26400.0	1.16892
$800$	0	0
$801$	−7854.00	−0.346451
$802$	0	0
$803$	6776.00i	0.297783i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11176.0i	0.487502i
$808$	0	0
$809$	14278.0	0.620504	0.310252	−	0.950654i	$-0.399587\pi$
$809$	14278.0	0.620504	0.310252	+	0.950654i	$0.399587\pi$
$810$	0	0
$811$	716.000	0.0310014	0.0155007	−	0.999880i	$-0.495066\pi$
$811$	716.000	0.0310014	0.0155007	+	0.999880i	$0.495066\pi$
$812$	0	0
$813$	− 34496.0i	− 1.48810i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2288.00i	− 0.0979767i
$818$	0	0
$819$	−5808.00	−0.247800
$820$	0	0
$821$	−23538.0	−1.00059	−0.500293	−	0.865856i	$-0.666775\pi$
$821$	−23538.0	−1.00059	−0.500293	+	0.865856i	$0.666775\pi$
$822$	0	0
$823$	6616.00i	0.280218i	0.990136	+	0.140109i	$0.0447453\pi$
$823$	6616.00i	0.280218i	−0.990136	+	0.140109i	$0.955255\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27236.0i	1.14521i	0.819831	+	0.572605i	$0.194067\pi$
$827$	27236.0i	1.14521i	−0.819831	+	0.572605i	$0.805933\pi$
$828$	0	0
$829$	−12070.0	−0.505680	−0.252840	−	0.967508i	$-0.581365\pi$
$829$	−12070.0	−0.505680	−0.252840	+	0.967508i	$0.581365\pi$
$830$	0	0
$831$	−7496.00	−0.312916
$832$	0	0
$833$	11650.0i	0.484572i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	24320.0i	1.00433i
$838$	0	0
$839$	−42024.0	−1.72924	−0.864618	−	0.502429i	$-0.832440\pi$
$839$	−42024.0	−1.72924	−0.864618	+	0.502429i	$0.832440\pi$
$840$	0	0
$841$	14815.0	0.607446
$842$	0	0
$843$	13352.0i	0.545513i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14520.0i	0.589036i
$848$	0	0
$849$	28688.0	1.15968
$850$	0	0
$851$	−9072.00	−0.365434
$852$	0	0
$853$	2414.00i	0.0968978i	0.998826	+	0.0484489i	$0.0154278\pi$
$853$	2414.00i	0.0968978i	−0.998826	+	0.0484489i	$0.984572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37686.0i	1.50213i	0.660226	+	0.751067i	$0.270461\pi$
$857$	37686.0i	1.50213i	−0.660226	+	0.751067i	$0.729539\pi$
$858$	0	0
$859$	40644.0	1.61438	0.807192	−	0.590289i	$-0.200986\pi$
$859$	40644.0	1.61438	0.807192	+	0.590289i	$0.200986\pi$
$860$	0	0
$861$	19008.0	0.752370
$862$	0	0
$863$	18656.0i	0.735872i	0.929851	+	0.367936i	$0.119935\pi$
$863$	18656.0i	0.735872i	−0.929851	+	0.367936i	$0.880065\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9652.00i	0.378084i
$868$	0	0
$869$	−28864.0	−1.12675
$870$	0	0
$871$	−4136.00	−0.160899
$872$	0	0
$873$	5258.00i	0.203845i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13002.0i	0.500623i	0.968165	+	0.250311i	$0.0805330\pi$
$877$	13002.0i	0.500623i	−0.968165	+	0.250311i	$0.919467\pi$
$878$	0	0
$879$	−20856.0	−0.800291
$880$	0	0
$881$	49490.0	1.89258	0.946289	−	0.323323i	$-0.104800\pi$
$881$	49490.0	1.89258	0.946289	+	0.323323i	$0.104800\pi$
$882$	0	0
$883$	− 1100.00i	− 0.0419229i	−0.999780	−	0.0209615i	$-0.993327\pi$
$883$	− 1100.00i	− 0.0419229i	0.999780	−	0.0209615i	$-0.00667273\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 14104.0i	− 0.533896i	−0.963711	−	0.266948i	$-0.913985\pi$
$887$	− 14104.0i	− 0.533896i	0.963711	−	0.266948i	$-0.0860152\pi$
$888$	0	0
$889$	−33792.0	−1.27486
$890$	0	0
$891$	−13684.0	−0.514513
$892$	0	0
$893$	23232.0i	0.870581i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4928.00i	− 0.183435i
$898$	0	0
$899$	−31680.0	−1.17529
$900$	0	0
$901$	−12100.0	−0.447402
$902$	0	0
$903$	4992.00i	0.183968i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12716.0i	− 0.465521i	−0.972534	−	0.232761i	$-0.925224\pi$
$907$	− 12716.0i	− 0.465521i	0.972534	−	0.232761i	$-0.0747759\pi$
$908$	0	0
$909$	17226.0	0.628548
$910$	0	0
$911$	39632.0	1.44135	0.720673	−	0.693275i	$-0.243833\pi$
$911$	39632.0	1.44135	0.720673	+	0.693275i	$0.243833\pi$
$912$	0	0
$913$	− 10384.0i	− 0.376408i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	64608.0i	2.32666i
$918$	0	0
$919$	5704.00	0.204742	0.102371	−	0.994746i	$-0.467357\pi$
$919$	5704.00	0.204742	0.102371	+	0.994746i	$0.467357\pi$
$920$	0	0
$921$	−1584.00	−0.0566716
$922$	0	0
$923$	− 16016.0i	− 0.571152i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10648.0i	0.377267i
$928$	0	0
$929$	−8162.00	−0.288252	−0.144126	−	0.989559i	$-0.546037\pi$
$929$	−8162.00	−0.288252	−0.144126	+	0.989559i	$0.546037\pi$
$930$	0	0
$931$	−10252.0	−0.360898
$932$	0	0
$933$	16224.0i	0.569293i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55110.0i	1.92141i	0.277564	+	0.960707i	$0.410473\pi$
$937$	55110.0i	1.92141i	−0.277564	+	0.960707i	$0.589527\pi$
$938$	0	0
$939$	−8616.00	−0.299438
$940$	0	0
$941$	16374.0	0.567245	0.283622	−	0.958936i	$-0.408464\pi$
$941$	16374.0	0.567245	0.283622	+	0.958936i	$0.408464\pi$
$942$	0	0
$943$	− 11088.0i	− 0.382900i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8460.00i	0.290299i	0.989410	+	0.145149i	$0.0463663\pi$
$947$	8460.00i	0.290299i	−0.989410	+	0.145149i	$0.953634\pi$
$948$	0	0
$949$	−3388.00	−0.115889
$950$	0	0
$951$	−29544.0	−1.00739
$952$	0	0
$953$	− 20502.0i	− 0.696878i	−0.937331	−	0.348439i	$-0.886712\pi$
$953$	− 20502.0i	− 0.696878i	0.937331	−	0.348439i	$-0.113288\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 34848.0i	− 1.17709i
$958$	0	0
$959$	39024.0	1.31403
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	− 8580.00i	− 0.287110i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 36520.0i	− 1.21448i	−0.794518	−	0.607241i	$-0.792276\pi$
$967$	− 36520.0i	− 1.21448i	0.794518	−	0.607241i	$-0.207724\pi$
$968$	0	0
$969$	8800.00	0.291741
$970$	0	0
$971$	−20244.0	−0.669064	−0.334532	−	0.942384i	$-0.608578\pi$
$971$	−20244.0	−0.669064	−0.334532	+	0.942384i	$0.608578\pi$
$972$	0	0
$973$	− 16416.0i	− 0.540876i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 50034.0i	− 1.63841i	−0.573499	−	0.819206i	$-0.694414\pi$
$977$	− 50034.0i	− 1.63841i	0.573499	−	0.819206i	$-0.305586\pi$
$978$	0	0
$979$	−31416.0	−1.02560
$980$	0	0
$981$	21934.0	0.713862
$982$	0	0
$983$	− 37128.0i	− 1.20468i	−0.798240	−	0.602339i	$-0.794235\pi$
$983$	− 37128.0i	− 1.20468i	0.798240	−	0.602339i	$-0.205765\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 50688.0i	− 1.63467i
$988$	0	0
$989$	2912.00	0.0936261
$990$	0	0
$991$	−27808.0	−0.891373	−0.445686	−	0.895189i	$-0.647040\pi$
$991$	−27808.0	−0.891373	−0.445686	+	0.895189i	$0.647040\pi$
$992$	0	0
$993$	4528.00i	0.144705i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28514.0i	0.905765i	0.891570	+	0.452882i	$0.149604\pi$
$997$	28514.0i	0.905765i	−0.891570	+	0.452882i	$0.850396\pi$
$998$	0	0
$999$	−24624.0	−0.779849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.i.49.2		2
4.3	odd	2		200.4.c.e.49.1		2
5.2	odd	4		16.4.a.a.1.1		1
5.3	odd	4		400.4.a.g.1.1		1
5.4	even	2	inner	400.4.c.i.49.1		2
12.11	even	2		1800.4.f.u.649.1		2
15.2	even	4		144.4.a.e.1.1		1
20.3	even	4		200.4.a.g.1.1		1
20.7	even	4		8.4.a.a.1.1	✓	1
20.19	odd	2		200.4.c.e.49.2		2
35.27	even	4		784.4.a.e.1.1		1
40.3	even	4		1600.4.a.o.1.1		1
40.13	odd	4		1600.4.a.bm.1.1		1
40.27	even	4		64.4.a.d.1.1		1
40.37	odd	4		64.4.a.b.1.1		1
55.32	even	4		1936.4.a.l.1.1		1
60.23	odd	4		1800.4.a.d.1.1		1
60.47	odd	4		72.4.a.c.1.1		1
60.59	even	2		1800.4.f.u.649.2		2
80.27	even	4		256.4.b.a.129.2		2
80.37	odd	4		256.4.b.g.129.1		2
80.67	even	4		256.4.b.a.129.1		2
80.77	odd	4		256.4.b.g.129.2		2
120.77	even	4		576.4.a.j.1.1		1
120.107	odd	4		576.4.a.k.1.1		1
140.27	odd	4		392.4.a.e.1.1		1
140.47	odd	12		392.4.i.b.361.1		2
140.67	even	12		392.4.i.g.177.1		2
140.87	odd	12		392.4.i.b.177.1		2
140.107	even	12		392.4.i.g.361.1		2
180.7	even	12		648.4.i.h.433.1		2
180.47	odd	12		648.4.i.e.433.1		2
180.67	even	12		648.4.i.h.217.1		2
180.167	odd	12		648.4.i.e.217.1		2
220.87	odd	4		968.4.a.a.1.1		1
260.207	even	4		1352.4.a.a.1.1		1
340.67	even	4		2312.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.a.a.1.1	✓	1	20.7	even	4
16.4.a.a.1.1		1	5.2	odd	4
64.4.a.b.1.1		1	40.37	odd	4
64.4.a.d.1.1		1	40.27	even	4
72.4.a.c.1.1		1	60.47	odd	4
144.4.a.e.1.1		1	15.2	even	4
200.4.a.g.1.1		1	20.3	even	4
200.4.c.e.49.1		2	4.3	odd	2
200.4.c.e.49.2		2	20.19	odd	2
256.4.b.a.129.1		2	80.67	even	4
256.4.b.a.129.2		2	80.27	even	4
256.4.b.g.129.1		2	80.37	odd	4
256.4.b.g.129.2		2	80.77	odd	4
392.4.a.e.1.1		1	140.27	odd	4
392.4.i.b.177.1		2	140.87	odd	12
392.4.i.b.361.1		2	140.47	odd	12
392.4.i.g.177.1		2	140.67	even	12
392.4.i.g.361.1		2	140.107	even	12
400.4.a.g.1.1		1	5.3	odd	4
400.4.c.i.49.1		2	5.4	even	2	inner
400.4.c.i.49.2		2	1.1	even	1	trivial
576.4.a.j.1.1		1	120.77	even	4
576.4.a.k.1.1		1	120.107	odd	4
648.4.i.e.217.1		2	180.167	odd	12
648.4.i.e.433.1		2	180.47	odd	12
648.4.i.h.217.1		2	180.67	even	12
648.4.i.h.433.1		2	180.7	even	12
784.4.a.e.1.1		1	35.27	even	4
968.4.a.a.1.1		1	220.87	odd	4
1352.4.a.a.1.1		1	260.207	even	4
1600.4.a.o.1.1		1	40.3	even	4
1600.4.a.bm.1.1		1	40.13	odd	4
1800.4.a.d.1.1		1	60.23	odd	4
1800.4.f.u.649.1		2	12.11	even	2
1800.4.f.u.649.2		2	60.59	even	2
1936.4.a.l.1.1		1	55.32	even	4
2312.4.a.a.1.1		1	340.67	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} +24.0000i q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{3} +24.0000i q^{7} +11.0000 q^{9} +44.0000 q^{11} +22.0000i q^{13} -50.0000i q^{17} +44.0000 q^{19} -96.0000 q^{21} +56.0000i q^{23} +152.000i q^{27} -198.000 q^{29} +160.000 q^{31} +176.000i q^{33} +162.000i q^{37} -88.0000 q^{39} -198.000 q^{41} -52.0000i q^{43} +528.000i q^{47} -233.000 q^{49} +200.000 q^{51} -242.000i q^{53} +176.000i q^{57} -668.000 q^{59} +550.000 q^{61} +264.000i q^{63} +188.000i q^{67} -224.000 q^{69} -728.000 q^{71} +154.000i q^{73} +1056.00i q^{77} -656.000 q^{79} -311.000 q^{81} -236.000i q^{83} -792.000i q^{87} -714.000 q^{89} -528.000 q^{91} +640.000i q^{93} +478.000i q^{97} +484.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{9}+O(q^{10}) \) 2 * q + 22 * q^9	\( 2 q + 22 q^{9} + 88 q^{11} + 88 q^{19} - 192 q^{21} - 396 q^{29} + 320 q^{31} - 176 q^{39} - 396 q^{41} - 466 q^{49} + 400 q^{51} - 1336 q^{59} + 1100 q^{61} - 448 q^{69} - 1456 q^{71} - 1312 q^{79} - 622 q^{81} - 1428 q^{89} - 1056 q^{91} + 968 q^{99}+O(q^{100}) \) 2 * q + 22 * q^9 + 88 * q^11 + 88 * q^19 - 192 * q^21 - 396 * q^29 + 320 * q^31 - 176 * q^39 - 396 * q^41 - 466 * q^49 + 400 * q^51 - 1336 * q^59 + 1100 * q^61 - 448 * q^69 - 1456 * q^71 - 1312 * q^79 - 622 * q^81 - 1428 * q^89 - 1056 * q^91 + 968 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more