LMFDB - Embedded newform 400.6.c.c.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,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, 6, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$64.1535279252$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.c.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-22.0000i q^{3} +218.000i q^{7} -241.000 q^{9} +O(q^{10})$	$q-22.0000i q^{3} +218.000i q^{7} -241.000 q^{9} +480.000 q^{11} -622.000i q^{13} -186.000i q^{17} -1204.00 q^{19} +4796.00 q^{21} +3186.00i q^{23} -44.0000i q^{27} -5526.00 q^{29} -9356.00 q^{31} -10560.0i q^{33} -5618.00i q^{37} -13684.0 q^{39} -14394.0 q^{41} +370.000i q^{43} +16146.0i q^{47} -30717.0 q^{49} -4092.00 q^{51} -4374.00i q^{53} +26488.0i q^{57} -11748.0 q^{59} +13202.0 q^{61} -52538.0i q^{63} -11542.0i q^{67} +70092.0 q^{69} +29532.0 q^{71} +33698.0i q^{73} +104640. i q^{77} +31208.0 q^{79} -59531.0 q^{81} +38466.0i q^{83} +121572. i q^{87} -119514. q^{89} +135596. q^{91} +205832. i q^{93} -94658.0i q^{97} -115680. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 482 q^{9}+O(q^{10}) $ 2 * q - 482 * q^9	$ 2 q - 482 q^{9} + 960 q^{11} - 2408 q^{19} + 9592 q^{21} - 11052 q^{29} - 18712 q^{31} - 27368 q^{39} - 28788 q^{41} - 61434 q^{49} - 8184 q^{51} - 23496 q^{59} + 26404 q^{61} + 140184 q^{69} + 59064 q^{71} + 62416 q^{79} - 119062 q^{81} - 239028 q^{89} + 271192 q^{91} - 231360 q^{99}+O(q^{100}) $ 2 * q - 482 * q^9 + 960 * q^11 - 2408 * q^19 + 9592 * q^21 - 11052 * q^29 - 18712 * q^31 - 27368 * q^39 - 28788 * q^41 - 61434 * q^49 - 8184 * q^51 - 23496 * q^59 + 26404 * q^61 + 140184 * q^69 + 59064 * q^71 + 62416 * q^79 - 119062 * q^81 - 239028 * q^89 + 271192 * q^91 - 231360 * 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
$3$	− 22.0000i	− 1.41130i	−0.708560	−	0.705650i	$-0.750655\pi$
$3$	− 22.0000i	− 1.41130i	0.708560	−	0.705650i	$-0.249345\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	218.000i	1.68156i	0.541380	+	0.840778i	$0.317902\pi$
$7$	218.000i	1.68156i	−0.541380	+	0.840778i	$0.682098\pi$
$8$	0	0
$9$	−241.000	−0.991770
$10$	0	0
$11$	480.000	1.19608	0.598039	−	0.801467i	$-0.295947\pi$
$11$	480.000	1.19608	0.598039	+	0.801467i	$0.295947\pi$
$12$	0	0
$13$	− 622.000i	− 1.02078i	−0.859943	−	0.510390i	$-0.829501\pi$
$13$	− 622.000i	− 1.02078i	0.859943	−	0.510390i	$-0.170499\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 186.000i	− 0.156096i	−0.996950	−	0.0780478i	$-0.975131\pi$
$17$	− 186.000i	− 0.156096i	0.996950	−	0.0780478i	$-0.0248687\pi$
$18$	0	0
$19$	−1204.00	−0.765143	−0.382571	−	0.923926i	$-0.624961\pi$
$19$	−1204.00	−0.765143	−0.382571	+	0.923926i	$0.624961\pi$
$20$	0	0
$21$	4796.00	2.37318
$22$	0	0
$23$	3186.00i	1.25582i	0.778287	+	0.627908i	$0.216089\pi$
$23$	3186.00i	1.25582i	−0.778287	+	0.627908i	$0.783911\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 44.0000i	− 0.0116156i
$28$	0	0
$29$	−5526.00	−1.22016	−0.610079	−	0.792341i	$-0.708862\pi$
$29$	−5526.00	−1.22016	−0.610079	+	0.792341i	$0.708862\pi$
$30$	0	0
$31$	−9356.00	−1.74858	−0.874291	−	0.485402i	$-0.838673\pi$
$31$	−9356.00	−1.74858	−0.874291	+	0.485402i	$0.838673\pi$
$32$	0	0
$33$	− 10560.0i	− 1.68803i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5618.00i	− 0.674648i	−0.941389	−	0.337324i	$-0.890478\pi$
$37$	− 5618.00i	− 0.674648i	0.941389	−	0.337324i	$-0.109522\pi$
$38$	0	0
$39$	−13684.0	−1.44063
$40$	0	0
$41$	−14394.0	−1.33728	−0.668639	−	0.743587i	$-0.733123\pi$
$41$	−14394.0	−1.33728	−0.668639	+	0.743587i	$0.733123\pi$
$42$	0	0
$43$	370.000i	0.0305162i	0.999884	+	0.0152581i	$0.00485699\pi$
$43$	370.000i	0.0305162i	−0.999884	+	0.0152581i	$0.995143\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16146.0i	1.06615i	0.846066	+	0.533077i	$0.178965\pi$
$47$	16146.0i	1.06615i	−0.846066	+	0.533077i	$0.821035\pi$
$48$	0	0
$49$	−30717.0	−1.82763
$50$	0	0
$51$	−4092.00	−0.220298
$52$	0	0
$53$	− 4374.00i	− 0.213889i	−0.994265	−	0.106945i	$-0.965893\pi$
$53$	− 4374.00i	− 0.213889i	0.994265	−	0.106945i	$-0.0341068\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	26488.0i	1.07985i
$58$	0	0
$59$	−11748.0	−0.439374	−0.219687	−	0.975570i	$-0.570504\pi$
$59$	−11748.0	−0.439374	−0.219687	+	0.975570i	$0.570504\pi$
$60$	0	0
$61$	13202.0	0.454271	0.227136	−	0.973863i	$-0.427064\pi$
$61$	13202.0	0.454271	0.227136	+	0.973863i	$0.427064\pi$
$62$	0	0
$63$	− 52538.0i	− 1.66772i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 11542.0i	− 0.314119i	−0.987589	−	0.157059i	$-0.949799\pi$
$67$	− 11542.0i	− 0.314119i	0.987589	−	0.157059i	$-0.0502014\pi$
$68$	0	0
$69$	70092.0	1.77233
$70$	0	0
$71$	29532.0	0.695260	0.347630	−	0.937632i	$-0.386987\pi$
$71$	29532.0	0.695260	0.347630	+	0.937632i	$0.386987\pi$
$72$	0	0
$73$	33698.0i	0.740111i	0.929010	+	0.370056i	$0.120661\pi$
$73$	33698.0i	0.740111i	−0.929010	+	0.370056i	$0.879339\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	104640.i	2.01127i
$78$	0	0
$79$	31208.0	0.562598	0.281299	−	0.959620i	$-0.409235\pi$
$79$	31208.0	0.562598	0.281299	+	0.959620i	$0.409235\pi$
$80$	0	0
$81$	−59531.0	−1.00816
$82$	0	0
$83$	38466.0i	0.612889i	0.951889	+	0.306444i	$0.0991394\pi$
$83$	38466.0i	0.612889i	−0.951889	+	0.306444i	$0.900861\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	121572.i	1.72201i
$88$	0	0
$89$	−119514.	−1.59935	−0.799675	−	0.600432i	$-0.794995\pi$
$89$	−119514.	−1.59935	−0.799675	+	0.600432i	$0.794995\pi$
$90$	0	0
$91$	135596.	1.71650
$92$	0	0
$93$	205832.i	2.46777i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 94658.0i	− 1.02148i	−0.859737	−	0.510738i	$-0.829372\pi$
$97$	− 94658.0i	− 1.02148i	0.859737	−	0.510738i	$-0.170628\pi$
$98$	0	0
$99$	−115680.	−1.18623
$100$	0	0
$101$	101046.	0.985634	0.492817	−	0.870133i	$-0.335967\pi$
$101$	101046.	0.985634	0.492817	+	0.870133i	$0.335967\pi$
$102$	0	0
$103$	143434.i	1.33217i	0.745877	+	0.666084i	$0.232031\pi$
$103$	143434.i	1.33217i	−0.745877	+	0.666084i	$0.767969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 57054.0i	− 0.481755i	−0.970555	−	0.240878i	$-0.922565\pi$
$107$	− 57054.0i	− 0.481755i	0.970555	−	0.240878i	$-0.0774353\pi$
$108$	0	0
$109$	3118.00	0.0251368	0.0125684	−	0.999921i	$-0.495999\pi$
$109$	3118.00	0.0251368	0.0125684	+	0.999921i	$0.495999\pi$
$110$	0	0
$111$	−123596.	−0.952132
$112$	0	0
$113$	− 54534.0i	− 0.401764i	−0.979615	−	0.200882i	$-0.935619\pi$
$113$	− 54534.0i	− 0.401764i	0.979615	−	0.200882i	$-0.0643808\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	149902.i	1.01238i
$118$	0	0
$119$	40548.0	0.262484
$120$	0	0
$121$	69349.0	0.430603
$122$	0	0
$123$	316668.i	1.88730i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	24698.0i	0.135879i	0.997689	+	0.0679395i	$0.0216425\pi$
$127$	24698.0i	0.135879i	−0.997689	+	0.0679395i	$0.978358\pi$
$128$	0	0
$129$	8140.00	0.0430675
$130$	0	0
$131$	−236640.	−1.20479	−0.602393	−	0.798200i	$-0.705786\pi$
$131$	−236640.	−1.20479	−0.602393	+	0.798200i	$0.705786\pi$
$132$	0	0
$133$	− 262472.i	− 1.28663i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22158.0i	0.100862i	0.998728	+	0.0504312i	$0.0160596\pi$
$137$	22158.0i	0.100862i	−0.998728	+	0.0504312i	$0.983940\pi$
$138$	0	0
$139$	−193204.	−0.848163	−0.424081	−	0.905624i	$-0.639403\pi$
$139$	−193204.	−0.848163	−0.424081	+	0.905624i	$0.639403\pi$
$140$	0	0
$141$	355212.	1.50467
$142$	0	0
$143$	− 298560.i	− 1.22093i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	675774.i	2.57934i
$148$	0	0
$149$	−448554.	−1.65519	−0.827597	−	0.561322i	$-0.810293\pi$
$149$	−448554.	−1.65519	−0.827597	+	0.561322i	$0.810293\pi$
$150$	0	0
$151$	140860.	0.502742	0.251371	−	0.967891i	$-0.419119\pi$
$151$	140860.	0.502742	0.251371	+	0.967891i	$0.419119\pi$
$152$	0	0
$153$	44826.0i	0.154811i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	335878.i	1.08751i	0.839245	+	0.543754i	$0.182998\pi$
$157$	335878.i	1.08751i	−0.839245	+	0.543754i	$0.817002\pi$
$158$	0	0
$159$	−96228.0	−0.301862
$160$	0	0
$161$	−694548.	−2.11173
$162$	0	0
$163$	101650.i	0.299667i	0.988711	+	0.149833i	$0.0478737\pi$
$163$	101650.i	0.299667i	−0.988711	+	0.149833i	$0.952126\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	139242.i	0.386348i	0.981164	+	0.193174i	$0.0618782\pi$
$167$	139242.i	0.386348i	−0.981164	+	0.193174i	$0.938122\pi$
$168$	0	0
$169$	−15591.0	−0.0419911
$170$	0	0
$171$	290164.	0.758845
$172$	0	0
$173$	− 265014.i	− 0.673215i	−0.941645	−	0.336607i	$-0.890721\pi$
$173$	− 265014.i	− 0.673215i	0.941645	−	0.336607i	$-0.109279\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	258456.i	0.620088i
$178$	0	0
$179$	−142812.	−0.333144	−0.166572	−	0.986029i	$-0.553270\pi$
$179$	−142812.	−0.333144	−0.166572	+	0.986029i	$0.553270\pi$
$180$	0	0
$181$	109670.	0.248824	0.124412	−	0.992231i	$-0.460296\pi$
$181$	109670.	0.248824	0.124412	+	0.992231i	$0.460296\pi$
$182$	0	0
$183$	− 290444.i	− 0.641113i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 89280.0i	− 0.186703i
$188$	0	0
$189$	9592.00	0.0195324
$190$	0	0
$191$	−294948.	−0.585008	−0.292504	−	0.956264i	$-0.594489\pi$
$191$	−294948.	−0.585008	−0.292504	+	0.956264i	$0.594489\pi$
$192$	0	0
$193$	1.00303e6i	1.93831i	0.246459	+	0.969153i	$0.420733\pi$
$193$	1.00303e6i	1.93831i	−0.246459	+	0.969153i	$0.579267\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	823998.i	1.51273i	0.654151	+	0.756364i	$0.273026\pi$
$197$	823998.i	1.51273i	−0.654151	+	0.756364i	$0.726974\pi$
$198$	0	0
$199$	−906712.	−1.62307	−0.811534	−	0.584305i	$-0.801367\pi$
$199$	−906712.	−1.62307	−0.811534	+	0.584305i	$0.801367\pi$
$200$	0	0
$201$	−253924.	−0.443316
$202$	0	0
$203$	− 1.20467e6i	− 2.05176i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 767826.i	− 1.24548i
$208$	0	0
$209$	−577920.	−0.915170
$210$	0	0
$211$	−506384.	−0.783022	−0.391511	−	0.920173i	$-0.628047\pi$
$211$	−506384.	−0.783022	−0.391511	+	0.920173i	$0.628047\pi$
$212$	0	0
$213$	− 649704.i	− 0.981220i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.03961e6i	− 2.94034i
$218$	0	0
$219$	741356.	1.04452
$220$	0	0
$221$	−115692.	−0.159339
$222$	0	0
$223$	542050.i	0.729923i	0.931023	+	0.364962i	$0.118918\pi$
$223$	542050.i	0.729923i	−0.931023	+	0.364962i	$0.881082\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.44857e6i	− 1.86585i	−0.360075	−	0.932924i	$-0.617249\pi$
$227$	− 1.44857e6i	− 1.86585i	0.360075	−	0.932924i	$-0.382751\pi$
$228$	0	0
$229$	478786.	0.603327	0.301663	−	0.953414i	$-0.402458\pi$
$229$	478786.	0.603327	0.301663	+	0.953414i	$0.402458\pi$
$230$	0	0
$231$	2.30208e6	2.83851
$232$	0	0
$233$	374106.i	0.451445i	0.974192	+	0.225723i	$0.0724743\pi$
$233$	374106.i	0.451445i	−0.974192	+	0.225723i	$0.927526\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 686576.i	− 0.793995i
$238$	0	0
$239$	169416.	0.191849	0.0959245	−	0.995389i	$-0.469419\pi$
$239$	169416.	0.191849	0.0959245	+	0.995389i	$0.469419\pi$
$240$	0	0
$241$	−353746.	−0.392328	−0.196164	−	0.980571i	$-0.562848\pi$
$241$	−353746.	−0.392328	−0.196164	+	0.980571i	$0.562848\pi$
$242$	0	0
$243$	1.29899e6i	1.41121i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	748888.i	0.781042i
$248$	0	0
$249$	846252.	0.864971
$250$	0	0
$251$	−1.25520e6	−1.25756	−0.628780	−	0.777583i	$-0.716445\pi$
$251$	−1.25520e6	−1.25756	−0.628780	+	0.777583i	$0.716445\pi$
$252$	0	0
$253$	1.52928e6i	1.50205i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.12877e6i	1.06604i	0.846102	+	0.533021i	$0.178943\pi$
$257$	1.12877e6i	1.06604i	−0.846102	+	0.533021i	$0.821057\pi$
$258$	0	0
$259$	1.22472e6	1.13446
$260$	0	0
$261$	1.33177e6	1.21012
$262$	0	0
$263$	263082.i	0.234532i	0.993101	+	0.117266i	$0.0374130\pi$
$263$	263082.i	0.234532i	−0.993101	+	0.117266i	$0.962587\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.62931e6i	2.25717i
$268$	0	0
$269$	1.18774e6	1.00079	0.500393	−	0.865798i	$-0.333189\pi$
$269$	1.18774e6	1.00079	0.500393	+	0.865798i	$0.333189\pi$
$270$	0	0
$271$	−431300.	−0.356744	−0.178372	−	0.983963i	$-0.557083\pi$
$271$	−431300.	−0.356744	−0.178372	+	0.983963i	$0.557083\pi$
$272$	0	0
$273$	− 2.98311e6i	− 2.42250i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 743114.i	− 0.581910i	−0.956737	−	0.290955i	$-0.906027\pi$
$277$	− 743114.i	− 0.581910i	0.956737	−	0.290955i	$-0.0939730\pi$
$278$	0	0
$279$	2.25480e6	1.73419
$280$	0	0
$281$	1.92193e6	1.45201	0.726007	−	0.687687i	$-0.241374\pi$
$281$	1.92193e6	1.45201	0.726007	+	0.687687i	$0.241374\pi$
$282$	0	0
$283$	1.63071e6i	1.21035i	0.796092	+	0.605176i	$0.206897\pi$
$283$	1.63071e6i	1.21035i	−0.796092	+	0.605176i	$0.793103\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.13789e6i	− 2.24871i
$288$	0	0
$289$	1.38526e6	0.975634
$290$	0	0
$291$	−2.08248e6	−1.44161
$292$	0	0
$293$	71250.0i	0.0484859i	0.999706	+	0.0242430i	$0.00771753\pi$
$293$	71250.0i	0.0484859i	−0.999706	+	0.0242430i	$0.992282\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 21120.0i	− 0.0138932i
$298$	0	0
$299$	1.98169e6	1.28191
$300$	0	0
$301$	−80660.0	−0.0513147
$302$	0	0
$303$	− 2.22301e6i	− 1.39103i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.61762e6i	− 0.979560i	−0.871846	−	0.489780i	$-0.837077\pi$
$307$	− 1.61762e6i	− 0.979560i	0.871846	−	0.489780i	$-0.162923\pi$
$308$	0	0
$309$	3.15555e6	1.88009
$310$	0	0
$311$	682788.	0.400299	0.200150	−	0.979765i	$-0.435857\pi$
$311$	682788.	0.400299	0.200150	+	0.979765i	$0.435857\pi$
$312$	0	0
$313$	− 2.70444e6i	− 1.56033i	−0.625574	−	0.780165i	$-0.715135\pi$
$313$	− 2.70444e6i	− 1.56033i	0.625574	−	0.780165i	$-0.284865\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.60347e6i	− 1.45514i	−0.686035	−	0.727568i	$-0.740650\pi$
$317$	− 2.60347e6i	− 1.45514i	0.686035	−	0.727568i	$-0.259350\pi$
$318$	0	0
$319$	−2.65248e6	−1.45940
$320$	0	0
$321$	−1.25519e6	−0.679902
$322$	0	0
$323$	223944.i	0.119435i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 68596.0i	− 0.0354756i
$328$	0	0
$329$	−3.51983e6	−1.79280
$330$	0	0
$331$	661432.	0.331830	0.165915	−	0.986140i	$-0.446942\pi$
$331$	661432.	0.331830	0.165915	+	0.986140i	$0.446942\pi$
$332$	0	0
$333$	1.35394e6i	0.669096i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.71706e6i	− 0.823588i	−0.911277	−	0.411794i	$-0.864902\pi$
$337$	− 1.71706e6i	− 0.823588i	0.911277	−	0.411794i	$-0.135098\pi$
$338$	0	0
$339$	−1.19975e6	−0.567010
$340$	0	0
$341$	−4.49088e6	−2.09144
$342$	0	0
$343$	− 3.03238e6i	− 1.39171i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	131370.i	0.0585696i	0.999571	+	0.0292848i	$0.00932298\pi$
$347$	131370.i	0.0585696i	−0.999571	+	0.0292848i	$0.990677\pi$
$348$	0	0
$349$	−3.50951e6	−1.54235	−0.771175	−	0.636623i	$-0.780331\pi$
$349$	−3.50951e6	−1.54235	−0.771175	+	0.636623i	$0.780331\pi$
$350$	0	0
$351$	−27368.0	−0.0118570
$352$	0	0
$353$	2.21992e6i	0.948202i	0.880470	+	0.474101i	$0.157227\pi$
$353$	2.21992e6i	0.948202i	−0.880470	+	0.474101i	$0.842773\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 892056.i	− 0.370443i
$358$	0	0
$359$	4.39730e6	1.80074	0.900369	−	0.435128i	$-0.143297\pi$
$359$	4.39730e6	1.80074	0.900369	+	0.435128i	$0.143297\pi$
$360$	0	0
$361$	−1.02648e6	−0.414557
$362$	0	0
$363$	− 1.52568e6i	− 0.607710i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.29824e6i	− 0.890697i	−0.895357	−	0.445348i	$-0.853080\pi$
$367$	− 2.29824e6i	− 0.890697i	0.895357	−	0.445348i	$-0.146920\pi$
$368$	0	0
$369$	3.46895e6	1.32627
$370$	0	0
$371$	953532.	0.359667
$372$	0	0
$373$	− 1.73561e6i	− 0.645920i	−0.946413	−	0.322960i	$-0.895322\pi$
$373$	− 1.73561e6i	− 0.645920i	0.946413	−	0.322960i	$-0.104678\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.43717e6i	1.24551i
$378$	0	0
$379$	−5.39115e6	−1.92789	−0.963947	−	0.266094i	$-0.914267\pi$
$379$	−5.39115e6	−1.92789	−0.963947	+	0.266094i	$0.914267\pi$
$380$	0	0
$381$	543356.	0.191766
$382$	0	0
$383$	− 3.27281e6i	− 1.14005i	−0.821627	−	0.570026i	$-0.806933\pi$
$383$	− 3.27281e6i	− 1.14005i	0.821627	−	0.570026i	$-0.193067\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 89170.0i	− 0.0302650i
$388$	0	0
$389$	−603114.	−0.202081	−0.101040	−	0.994882i	$-0.532217\pi$
$389$	−603114.	−0.202081	−0.101040	+	0.994882i	$0.532217\pi$
$390$	0	0
$391$	592596.	0.196027
$392$	0	0
$393$	5.20608e6i	1.70032i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	749422.i	0.238644i	0.992856	+	0.119322i	$0.0380721\pi$
$397$	749422.i	0.238644i	−0.992856	+	0.119322i	$0.961928\pi$
$398$	0	0
$399$	−5.77438e6	−1.81582
$400$	0	0
$401$	5.31357e6	1.65016	0.825079	−	0.565018i	$-0.191131\pi$
$401$	5.31357e6	1.65016	0.825079	+	0.565018i	$0.191131\pi$
$402$	0	0
$403$	5.81943e6i	1.78492i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.69664e6i	− 0.806932i
$408$	0	0
$409$	−999326.	−0.295392	−0.147696	−	0.989033i	$-0.547186\pi$
$409$	−999326.	−0.295392	−0.147696	+	0.989033i	$0.547186\pi$
$410$	0	0
$411$	487476.	0.142347
$412$	0	0
$413$	− 2.56106e6i	− 0.738831i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.25049e6i	1.19701i
$418$	0	0
$419$	2.03740e6	0.566944	0.283472	−	0.958980i	$-0.408514\pi$
$419$	2.03740e6	0.566944	0.283472	+	0.958980i	$0.408514\pi$
$420$	0	0
$421$	−5.11461e6	−1.40640	−0.703198	−	0.710994i	$-0.748245\pi$
$421$	−5.11461e6	−1.40640	−0.703198	+	0.710994i	$0.748245\pi$
$422$	0	0
$423$	− 3.89119e6i	− 1.05738i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.87804e6i	0.763882i
$428$	0	0
$429$	−6.56832e6	−1.72310
$430$	0	0
$431$	3.30404e6	0.856747	0.428374	−	0.903602i	$-0.359087\pi$
$431$	3.30404e6	0.856747	0.428374	+	0.903602i	$0.359087\pi$
$432$	0	0
$433$	− 2.01638e6i	− 0.516836i	−0.966033	−	0.258418i	$-0.916799\pi$
$433$	− 2.01638e6i	− 0.516836i	0.966033	−	0.258418i	$-0.0832012\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.83594e6i	− 0.960879i
$438$	0	0
$439$	6.58321e6	1.63033	0.815166	−	0.579227i	$-0.196645\pi$
$439$	6.58321e6	1.63033	0.815166	+	0.579227i	$0.196645\pi$
$440$	0	0
$441$	7.40280e6	1.81259
$442$	0	0
$443$	4.81783e6i	1.16638i	0.812334	+	0.583192i	$0.198197\pi$
$443$	4.81783e6i	1.16638i	−0.812334	+	0.583192i	$0.801803\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.86819e6i	2.33598i
$448$	0	0
$449$	6.20399e6	1.45230	0.726149	−	0.687538i	$-0.241308\pi$
$449$	6.20399e6	1.45230	0.726149	+	0.687538i	$0.241308\pi$
$450$	0	0
$451$	−6.90912e6	−1.59949
$452$	0	0
$453$	− 3.09892e6i	− 0.709520i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.84383e6i	− 0.636962i	−0.947929	−	0.318481i	$-0.896827\pi$
$457$	− 2.84383e6i	− 0.636962i	0.947929	−	0.318481i	$-0.103173\pi$
$458$	0	0
$459$	−8184.00	−0.00181315
$460$	0	0
$461$	−1.75605e6	−0.384844	−0.192422	−	0.981312i	$-0.561634\pi$
$461$	−1.75605e6	−0.384844	−0.192422	+	0.981312i	$0.561634\pi$
$462$	0	0
$463$	− 7.66857e6i	− 1.66250i	−0.555899	−	0.831250i	$-0.687626\pi$
$463$	− 7.66857e6i	− 1.66250i	0.555899	−	0.831250i	$-0.312374\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.35903e6i	− 0.288361i	−0.989551	−	0.144181i	$-0.953945\pi$
$467$	− 1.35903e6i	− 0.288361i	0.989551	−	0.144181i	$-0.0460546\pi$
$468$	0	0
$469$	2.51616e6	0.528209
$470$	0	0
$471$	7.38932e6	1.53480
$472$	0	0
$473$	177600.i	0.0364998i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.05413e6i	0.212129i
$478$	0	0
$479$	−2.02706e6	−0.403672	−0.201836	−	0.979419i	$-0.564691\pi$
$479$	−2.02706e6	−0.403672	−0.201836	+	0.979419i	$0.564691\pi$
$480$	0	0
$481$	−3.49440e6	−0.688667
$482$	0	0
$483$	1.52801e7i	2.98028i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.46427e6i	0.470833i	0.971895	+	0.235416i	$0.0756454\pi$
$487$	2.46427e6i	0.470833i	−0.971895	+	0.235416i	$0.924355\pi$
$488$	0	0
$489$	2.23630e6	0.422920
$490$	0	0
$491$	−1.03848e7	−1.94399	−0.971996	−	0.234998i	$-0.924492\pi$
$491$	−1.03848e7	−1.94399	−0.971996	+	0.234998i	$0.924492\pi$
$492$	0	0
$493$	1.02784e6i	0.190461i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.43798e6i	1.16912i
$498$	0	0
$499$	6.49416e6	1.16754	0.583769	−	0.811919i	$-0.301577\pi$
$499$	6.49416e6	1.16754	0.583769	+	0.811919i	$0.301577\pi$
$500$	0	0
$501$	3.06332e6	0.545254
$502$	0	0
$503$	1.03565e7i	1.82513i	0.408931	+	0.912565i	$0.365902\pi$
$503$	1.03565e7i	1.82513i	−0.408931	+	0.912565i	$0.634098\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	343002.i	0.0592621i
$508$	0	0
$509$	−5.87305e6	−1.00478	−0.502388	−	0.864643i	$-0.667545\pi$
$509$	−5.87305e6	−1.00478	−0.502388	+	0.864643i	$0.667545\pi$
$510$	0	0
$511$	−7.34616e6	−1.24454
$512$	0	0
$513$	52976.0i	0.00888763i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.75008e6i	1.27520i
$518$	0	0
$519$	−5.83031e6	−0.950108
$520$	0	0
$521$	2.17295e6	0.350717	0.175358	−	0.984505i	$-0.443892\pi$
$521$	2.17295e6	0.350717	0.175358	+	0.984505i	$0.443892\pi$
$522$	0	0
$523$	− 1.07361e6i	− 0.171629i	−0.996311	−	0.0858145i	$-0.972651\pi$
$523$	− 1.07361e6i	− 0.171629i	0.996311	−	0.0858145i	$-0.0273492\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.74022e6i	0.272946i
$528$	0	0
$529$	−3.71425e6	−0.577075
$530$	0	0
$531$	2.83127e6	0.435757
$532$	0	0
$533$	8.95307e6i	1.36507i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.14186e6i	0.470166i
$538$	0	0
$539$	−1.47442e7	−2.18599
$540$	0	0
$541$	7.09033e6	1.04153	0.520767	−	0.853699i	$-0.325646\pi$
$541$	7.09033e6	1.04153	0.520767	+	0.853699i	$0.325646\pi$
$542$	0	0
$543$	− 2.41274e6i	− 0.351165i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.69763e6i	0.957091i	0.878063	+	0.478545i	$0.158836\pi$
$547$	6.69763e6i	0.957091i	−0.878063	+	0.478545i	$0.841164\pi$
$548$	0	0
$549$	−3.18168e6	−0.450532
$550$	0	0
$551$	6.65330e6	0.933595
$552$	0	0
$553$	6.80334e6i	0.946040i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.19008e7i	− 1.62532i	−0.582735	−	0.812662i	$-0.698018\pi$
$557$	− 1.19008e7i	− 1.62532i	0.582735	−	0.812662i	$-0.301982\pi$
$558$	0	0
$559$	230140.	0.0311503
$560$	0	0
$561$	−1.96416e6	−0.263493
$562$	0	0
$563$	− 8.75636e6i	− 1.16427i	−0.813093	−	0.582133i	$-0.802218\pi$
$563$	− 8.75636e6i	− 1.16427i	0.813093	−	0.582133i	$-0.197782\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.29778e7i	− 1.69528i
$568$	0	0
$569$	1.15677e6	0.149784	0.0748922	−	0.997192i	$-0.476139\pi$
$569$	1.15677e6	0.149784	0.0748922	+	0.997192i	$0.476139\pi$
$570$	0	0
$571$	7.07807e6	0.908500	0.454250	−	0.890874i	$-0.349907\pi$
$571$	7.07807e6	0.908500	0.454250	+	0.890874i	$0.349907\pi$
$572$	0	0
$573$	6.48886e6i	0.825623i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.13404e6i	0.391890i	0.980615	+	0.195945i	$0.0627775\pi$
$577$	3.13404e6i	0.391890i	−0.980615	+	0.195945i	$0.937223\pi$
$578$	0	0
$579$	2.20667e7	2.73553
$580$	0	0
$581$	−8.38559e6	−1.03061
$582$	0	0
$583$	− 2.09952e6i	− 0.255828i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.13833e7i	1.36355i	0.731561	+	0.681776i	$0.238792\pi$
$587$	1.13833e7i	1.36355i	−0.731561	+	0.681776i	$0.761208\pi$
$588$	0	0
$589$	1.12646e7	1.33791
$590$	0	0
$591$	1.81280e7	2.13491
$592$	0	0
$593$	− 1.58655e7i	− 1.85275i	−0.376599	−	0.926376i	$-0.622906\pi$
$593$	− 1.58655e7i	− 1.85275i	0.376599	−	0.926376i	$-0.377094\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.99477e7i	2.29064i
$598$	0	0
$599$	−1.50998e7	−1.71951	−0.859756	−	0.510705i	$-0.829385\pi$
$599$	−1.50998e7	−1.71951	−0.859756	+	0.510705i	$0.829385\pi$
$600$	0	0
$601$	−8.08705e6	−0.913280	−0.456640	−	0.889652i	$-0.650947\pi$
$601$	−8.08705e6	−0.913280	−0.456640	+	0.889652i	$0.650947\pi$
$602$	0	0
$603$	2.78162e6i	0.311534i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 710398.i	− 0.0782582i	−0.999234	−	0.0391291i	$-0.987542\pi$
$607$	− 710398.i	− 0.0782582i	0.999234	−	0.0391291i	$-0.0124584\pi$
$608$	0	0
$609$	−2.65027e7	−2.89566
$610$	0	0
$611$	1.00428e7	1.08831
$612$	0	0
$613$	5.96434e6i	0.641078i	0.947235	+	0.320539i	$0.103864\pi$
$613$	5.96434e6i	0.641078i	−0.947235	+	0.320539i	$0.896136\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.48432e7i	− 1.56970i	−0.619689	−	0.784848i	$-0.712741\pi$
$617$	− 1.48432e7i	− 1.56970i	0.619689	−	0.784848i	$-0.287259\pi$
$618$	0	0
$619$	−1.82042e7	−1.90961	−0.954807	−	0.297227i	$-0.903938\pi$
$619$	−1.82042e7	−1.90961	−0.954807	+	0.297227i	$0.903938\pi$
$620$	0	0
$621$	140184.	0.0145871
$622$	0	0
$623$	− 2.60541e7i	− 2.68940i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.27142e7i	1.29158i
$628$	0	0
$629$	−1.04495e6	−0.105310
$630$	0	0
$631$	−1.09461e6	−0.109443	−0.0547214	−	0.998502i	$-0.517427\pi$
$631$	−1.09461e6	−0.109443	−0.0547214	+	0.998502i	$0.517427\pi$
$632$	0	0
$633$	1.11404e7i	1.10508i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.91060e7i	1.86561i
$638$	0	0
$639$	−7.11721e6	−0.689537
$640$	0	0
$641$	7.44046e6	0.715245	0.357622	−	0.933866i	$-0.383587\pi$
$641$	7.44046e6	0.715245	0.357622	+	0.933866i	$0.383587\pi$
$642$	0	0
$643$	1.07915e7i	1.02933i	0.857391	+	0.514665i	$0.172084\pi$
$643$	1.07915e7i	1.02933i	−0.857391	+	0.514665i	$0.827916\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.62998e6i	− 0.904409i	−0.891914	−	0.452204i	$-0.850638\pi$
$647$	− 9.62998e6i	− 0.904409i	0.891914	−	0.452204i	$-0.149362\pi$
$648$	0	0
$649$	−5.63904e6	−0.525525
$650$	0	0
$651$	−4.48714e7	−4.14970
$652$	0	0
$653$	− 1.00019e7i	− 0.917905i	−0.888461	−	0.458953i	$-0.848225\pi$
$653$	− 1.00019e7i	− 0.917905i	0.888461	−	0.458953i	$-0.151775\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8.12122e6i	− 0.734020i
$658$	0	0
$659$	−4.01060e6	−0.359746	−0.179873	−	0.983690i	$-0.557569\pi$
$659$	−4.01060e6	−0.359746	−0.179873	+	0.983690i	$0.557569\pi$
$660$	0	0
$661$	1.20338e7	1.07127	0.535636	−	0.844449i	$-0.320072\pi$
$661$	1.20338e7	1.07127	0.535636	+	0.844449i	$0.320072\pi$
$662$	0	0
$663$	2.54522e6i	0.224876i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.76058e7i	− 1.53229i
$668$	0	0
$669$	1.19251e7	1.03014
$670$	0	0
$671$	6.33696e6	0.543344
$672$	0	0
$673$	2.01231e6i	0.171260i	0.996327	+	0.0856301i	$0.0272903\pi$
$673$	2.01231e6i	0.171260i	−0.996327	+	0.0856301i	$0.972710\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.62410e7i	− 1.36188i	−0.732337	−	0.680942i	$-0.761571\pi$
$677$	− 1.62410e7i	− 1.36188i	0.732337	−	0.680942i	$-0.238429\pi$
$678$	0	0
$679$	2.06354e7	1.71767
$680$	0	0
$681$	−3.18686e7	−2.63327
$682$	0	0
$683$	− 4.62910e6i	− 0.379704i	−0.981813	−	0.189852i	$-0.939199\pi$
$683$	− 4.62910e6i	− 0.379704i	0.981813	−	0.189852i	$-0.0608008\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.05333e7i	− 0.851476i
$688$	0	0
$689$	−2.72063e6	−0.218334
$690$	0	0
$691$	−1.16794e7	−0.930517	−0.465258	−	0.885175i	$-0.654039\pi$
$691$	−1.16794e7	−0.930517	−0.465258	+	0.885175i	$0.654039\pi$
$692$	0	0
$693$	− 2.52182e7i	− 1.99472i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.67728e6i	0.208743i
$698$	0	0
$699$	8.23033e6	0.637125
$700$	0	0
$701$	1.99543e7	1.53370	0.766851	−	0.641825i	$-0.221822\pi$
$701$	1.99543e7	1.53370	0.766851	+	0.641825i	$0.221822\pi$
$702$	0	0
$703$	6.76407e6i	0.516202i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.20280e7i	1.65740i
$708$	0	0
$709$	4.88331e6	0.364837	0.182419	−	0.983221i	$-0.441607\pi$
$709$	4.88331e6	0.364837	0.182419	+	0.983221i	$0.441607\pi$
$710$	0	0
$711$	−7.52113e6	−0.557968
$712$	0	0
$713$	− 2.98082e7i	− 2.19590i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.72715e6i	− 0.270757i
$718$	0	0
$719$	−1.35778e7	−0.979505	−0.489753	−	0.871861i	$-0.662913\pi$
$719$	−1.35778e7	−0.979505	−0.489753	+	0.871861i	$0.662913\pi$
$720$	0	0
$721$	−3.12686e7	−2.24012
$722$	0	0
$723$	7.78241e6i	0.553692i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.42411e6i	0.450792i	0.974267	+	0.225396i	$0.0723677\pi$
$727$	6.42411e6i	0.450792i	−0.974267	+	0.225396i	$0.927632\pi$
$728$	0	0
$729$	1.41117e7	0.983472
$730$	0	0
$731$	68820.0	0.00476345
$732$	0	0
$733$	9.08556e6i	0.624585i	0.949986	+	0.312293i	$0.101097\pi$
$733$	9.08556e6i	0.624585i	−0.949986	+	0.312293i	$0.898903\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.54016e6i	− 0.375711i
$738$	0	0
$739$	2.02457e7	1.36371	0.681854	−	0.731488i	$-0.261174\pi$
$739$	2.02457e7	1.36371	0.681854	+	0.731488i	$0.261174\pi$
$740$	0	0
$741$	1.64755e7	1.10229
$742$	0	0
$743$	5.44831e6i	0.362067i	0.983477	+	0.181034i	$0.0579443\pi$
$743$	5.44831e6i	0.362067i	−0.983477	+	0.181034i	$0.942056\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.27031e6i	− 0.607845i
$748$	0	0
$749$	1.24378e7	0.810099
$750$	0	0
$751$	1.14072e6	0.0738041	0.0369021	−	0.999319i	$-0.488251\pi$
$751$	1.14072e6	0.0738041	0.0369021	+	0.999319i	$0.488251\pi$
$752$	0	0
$753$	2.76144e7i	1.77479i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.90153e7i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$757$	− 1.90153e7i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$758$	0	0
$759$	3.36442e7	2.11985
$760$	0	0
$761$	2.23551e7	1.39931	0.699656	−	0.714480i	$-0.253337\pi$
$761$	2.23551e7	1.39931	0.699656	+	0.714480i	$0.253337\pi$
$762$	0	0
$763$	679724.i	0.0422689i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.30726e6i	0.448504i
$768$	0	0
$769$	1.00704e7	0.614088	0.307044	−	0.951695i	$-0.400660\pi$
$769$	1.00704e7	0.614088	0.307044	+	0.951695i	$0.400660\pi$
$770$	0	0
$771$	2.48330e7	1.50451
$772$	0	0
$773$	− 4.05963e6i	− 0.244364i	−0.992508	−	0.122182i	$-0.961011\pi$
$773$	− 4.05963e6i	− 0.244364i	0.992508	−	0.122182i	$-0.0389892\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.69439e7i	− 1.60106i
$778$	0	0
$779$	1.73304e7	1.02321
$780$	0	0
$781$	1.41754e7	0.831585
$782$	0	0
$783$	243144.i	0.0141729i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.72256e7i	− 0.991372i	−0.868502	−	0.495686i	$-0.834917\pi$
$787$	− 1.72256e7i	− 0.991372i	0.868502	−	0.495686i	$-0.165083\pi$
$788$	0	0
$789$	5.78780e6	0.330995
$790$	0	0
$791$	1.18884e7	0.675589
$792$	0	0
$793$	− 8.21164e6i	− 0.463711i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.10793e7i	1.17547i	0.809055	+	0.587733i	$0.199980\pi$
$797$	2.10793e7i	1.17547i	−0.809055	+	0.587733i	$0.800020\pi$
$798$	0	0
$799$	3.00316e6	0.166422
$800$	0	0
$801$	2.88029e7	1.58619
$802$	0	0
$803$	1.61750e7i	0.885231i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.61303e7i	− 1.41241i
$808$	0	0
$809$	1.87877e7	1.00926	0.504629	−	0.863336i	$-0.331629\pi$
$809$	1.87877e7	1.00926	0.504629	+	0.863336i	$0.331629\pi$
$810$	0	0
$811$	1.32456e7	0.707164	0.353582	−	0.935404i	$-0.384964\pi$
$811$	1.32456e7	0.707164	0.353582	+	0.935404i	$0.384964\pi$
$812$	0	0
$813$	9.48860e6i	0.503473i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 445480.i	− 0.0233493i
$818$	0	0
$819$	−3.26786e7	−1.70237
$820$	0	0
$821$	−7.66925e6	−0.397096	−0.198548	−	0.980091i	$-0.563623\pi$
$821$	−7.66925e6	−0.397096	−0.198548	+	0.980091i	$0.563623\pi$
$822$	0	0
$823$	8.82786e6i	0.454314i	0.973858	+	0.227157i	$0.0729430\pi$
$823$	8.82786e6i	0.454314i	−0.973858	+	0.227157i	$0.927057\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.06923e7i	− 1.56051i	−0.625463	−	0.780254i	$-0.715090\pi$
$827$	− 3.06923e7i	− 1.56051i	0.625463	−	0.780254i	$-0.284910\pi$
$828$	0	0
$829$	−3.28414e7	−1.65972	−0.829860	−	0.557972i	$-0.811580\pi$
$829$	−3.28414e7	−1.65972	−0.829860	+	0.557972i	$0.811580\pi$
$830$	0	0
$831$	−1.63485e7	−0.821250
$832$	0	0
$833$	5.71336e6i	0.285285i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	411664.i	0.0203109i
$838$	0	0
$839$	−8.42117e6	−0.413017	−0.206508	−	0.978445i	$-0.566210\pi$
$839$	−8.42117e6	−0.413017	−0.206508	+	0.978445i	$0.566210\pi$
$840$	0	0
$841$	1.00255e7	0.488784
$842$	0	0
$843$	− 4.22824e7i	− 2.04923i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.51181e7i	0.724083i
$848$	0	0
$849$	3.58757e7	1.70817
$850$	0	0
$851$	1.78989e7	0.847234
$852$	0	0
$853$	2.35126e7i	1.10644i	0.833035	+	0.553221i	$0.186601\pi$
$853$	2.35126e7i	1.10644i	−0.833035	+	0.553221i	$0.813399\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.13050e7i	0.525799i	0.964823	+	0.262900i	$0.0846788\pi$
$857$	1.13050e7i	0.525799i	−0.964823	+	0.262900i	$0.915321\pi$
$858$	0	0
$859$	1.00078e7	0.462758	0.231379	−	0.972864i	$-0.425676\pi$
$859$	1.00078e7	0.462758	0.231379	+	0.972864i	$0.425676\pi$
$860$	0	0
$861$	−6.90336e7	−3.17360
$862$	0	0
$863$	− 2.61429e7i	− 1.19489i	−0.801911	−	0.597443i	$-0.796183\pi$
$863$	− 2.61429e7i	− 1.19489i	0.801911	−	0.597443i	$-0.203817\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3.04757e7i	− 1.37691i
$868$	0	0
$869$	1.49798e7	0.672911
$870$	0	0
$871$	−7.17912e6	−0.320646
$872$	0	0
$873$	2.28126e7i	1.01307i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.92041e6i	0.0843129i	0.999111	+	0.0421565i	$0.0134228\pi$
$877$	1.92041e6i	0.0843129i	−0.999111	+	0.0421565i	$0.986577\pi$
$878$	0	0
$879$	1.56750e6	0.0684282
$880$	0	0
$881$	−2.56594e7	−1.11380	−0.556899	−	0.830580i	$-0.688009\pi$
$881$	−2.56594e7	−1.11380	−0.556899	+	0.830580i	$0.688009\pi$
$882$	0	0
$883$	2.05643e7i	0.887590i	0.896128	+	0.443795i	$0.146368\pi$
$883$	2.05643e7i	0.887590i	−0.896128	+	0.443795i	$0.853632\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.16868e7i	1.35229i	0.736770	+	0.676143i	$0.236350\pi$
$887$	3.16868e7i	1.35229i	−0.736770	+	0.676143i	$0.763650\pi$
$888$	0	0
$889$	−5.38416e6	−0.228488
$890$	0	0
$891$	−2.85749e7	−1.20584
$892$	0	0
$893$	− 1.94398e7i	− 0.815761i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.35972e7i	− 1.80916i
$898$	0	0
$899$	5.17013e7	2.13355
$900$	0	0
$901$	−813564.	−0.0333872
$902$	0	0
$903$	1.77452e6i	0.0724205i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.96963e6i	0.160225i	0.996786	+	0.0801127i	$0.0255280\pi$
$907$	3.96963e6i	0.160225i	−0.996786	+	0.0801127i	$0.974472\pi$
$908$	0	0
$909$	−2.43521e7	−0.977522
$910$	0	0
$911$	1.37945e7	0.550692	0.275346	−	0.961345i	$-0.411208\pi$
$911$	1.37945e7	0.550692	0.275346	+	0.961345i	$0.411208\pi$
$912$	0	0
$913$	1.84637e7i	0.733063i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.15875e7i	− 2.02592i
$918$	0	0
$919$	8.08126e6	0.315639	0.157819	−	0.987468i	$-0.449554\pi$
$919$	8.08126e6	0.315639	0.157819	+	0.987468i	$0.449554\pi$
$920$	0	0
$921$	−3.55877e7	−1.38245
$922$	0	0
$923$	− 1.83689e7i	− 0.709707i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.45676e7i	− 1.32120i
$928$	0	0
$929$	−2.99956e7	−1.14030	−0.570150	−	0.821541i	$-0.693115\pi$
$929$	−2.99956e7	−1.14030	−0.570150	+	0.821541i	$0.693115\pi$
$930$	0	0
$931$	3.69833e7	1.39840
$932$	0	0
$933$	− 1.50213e7i	− 0.564943i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.07620e7i	0.772540i	0.922386	+	0.386270i	$0.126237\pi$
$937$	2.07620e7i	0.772540i	−0.922386	+	0.386270i	$0.873763\pi$
$938$	0	0
$939$	−5.94976e7	−2.20209
$940$	0	0
$941$	−3.47642e6	−0.127985	−0.0639923	−	0.997950i	$-0.520383\pi$
$941$	−3.47642e6	−0.127985	−0.0639923	+	0.997950i	$0.520383\pi$
$942$	0	0
$943$	− 4.58593e7i	− 1.67938i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.86700e6i	0.0676503i	0.999428	+	0.0338252i	$0.0107689\pi$
$947$	1.86700e6i	0.0676503i	−0.999428	+	0.0338252i	$0.989231\pi$
$948$	0	0
$949$	2.09602e7	0.755490
$950$	0	0
$951$	−5.72763e7	−2.05364
$952$	0	0
$953$	− 3.85501e7i	− 1.37497i	−0.726199	−	0.687484i	$-0.758715\pi$
$953$	− 3.85501e7i	− 1.37497i	0.726199	−	0.687484i	$-0.241285\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.83546e7i	2.05966i
$958$	0	0
$959$	−4.83044e6	−0.169606
$960$	0	0
$961$	5.89056e7	2.05754
$962$	0	0
$963$	1.37500e7i	0.477790i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.64875e7i	0.567008i	0.958971	+	0.283504i	$0.0914969\pi$
$967$	1.64875e7i	0.567008i	−0.958971	+	0.283504i	$0.908503\pi$
$968$	0	0
$969$	4.92677e6	0.168559
$970$	0	0
$971$	2.36976e7	0.806597	0.403299	−	0.915068i	$-0.367864\pi$
$971$	2.36976e7	0.806597	0.403299	+	0.915068i	$0.367864\pi$
$972$	0	0
$973$	− 4.21185e7i	− 1.42623i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.77590e7i	1.93590i	0.251143	+	0.967950i	$0.419194\pi$
$977$	5.77590e7i	1.93590i	−0.251143	+	0.967950i	$0.580806\pi$
$978$	0	0
$979$	−5.73667e7	−1.91295
$980$	0	0
$981$	−751438.	−0.0249299
$982$	0	0
$983$	1.10103e7i	0.363425i	0.983352	+	0.181712i	$0.0581640\pi$
$983$	1.10103e7i	0.363425i	−0.983352	+	0.181712i	$0.941836\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.74362e7i	2.53018i
$988$	0	0
$989$	−1.17882e6	−0.0383228
$990$	0	0
$991$	−3.70807e7	−1.19940	−0.599700	−	0.800225i	$-0.704713\pi$
$991$	−3.70807e7	−1.19940	−0.599700	+	0.800225i	$0.704713\pi$
$992$	0	0
$993$	− 1.45515e7i	− 0.468311i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.52935e6i	− 0.144311i	−0.997393	−	0.0721553i	$-0.977012\pi$
$997$	− 4.52935e6i	− 0.144311i	0.997393	−	0.0721553i	$-0.0229877\pi$
$998$	0	0
$999$	−247192.	−0.00783647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.c.49.1		2
4.3	odd	2		100.6.c.a.49.2		2
5.2	odd	4		80.6.a.b.1.1		1
5.3	odd	4		400.6.a.m.1.1		1
5.4	even	2	inner	400.6.c.c.49.2		2
12.11	even	2		900.6.d.h.649.1		2
15.2	even	4		720.6.a.l.1.1		1
20.3	even	4		100.6.a.a.1.1		1
20.7	even	4		20.6.a.a.1.1	✓	1
20.19	odd	2		100.6.c.a.49.1		2
40.27	even	4		320.6.a.c.1.1		1
40.37	odd	4		320.6.a.n.1.1		1
60.23	odd	4		900.6.a.b.1.1		1
60.47	odd	4		180.6.a.e.1.1		1
60.59	even	2		900.6.d.h.649.2		2
140.27	odd	4		980.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.a.a.1.1	✓	1	20.7	even	4
80.6.a.b.1.1		1	5.2	odd	4
100.6.a.a.1.1		1	20.3	even	4
100.6.c.a.49.1		2	20.19	odd	2
100.6.c.a.49.2		2	4.3	odd	2
180.6.a.e.1.1		1	60.47	odd	4
320.6.a.c.1.1		1	40.27	even	4
320.6.a.n.1.1		1	40.37	odd	4
400.6.a.m.1.1		1	5.3	odd	4
400.6.c.c.49.1		2	1.1	even	1	trivial
400.6.c.c.49.2		2	5.4	even	2	inner
720.6.a.l.1.1		1	15.2	even	4
900.6.a.b.1.1		1	60.23	odd	4
900.6.d.h.649.1		2	12.11	even	2
900.6.d.h.649.2		2	60.59	even	2
980.6.a.b.1.1		1	140.27	odd	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-22.0000i q^{3} +218.000i q^{7} -241.000 q^{9} +O(q^{10})\)	\(q-22.0000i q^{3} +218.000i q^{7} -241.000 q^{9} +480.000 q^{11} -622.000i q^{13} -186.000i q^{17} -1204.00 q^{19} +4796.00 q^{21} +3186.00i q^{23} -44.0000i q^{27} -5526.00 q^{29} -9356.00 q^{31} -10560.0i q^{33} -5618.00i q^{37} -13684.0 q^{39} -14394.0 q^{41} +370.000i q^{43} +16146.0i q^{47} -30717.0 q^{49} -4092.00 q^{51} -4374.00i q^{53} +26488.0i q^{57} -11748.0 q^{59} +13202.0 q^{61} -52538.0i q^{63} -11542.0i q^{67} +70092.0 q^{69} +29532.0 q^{71} +33698.0i q^{73} +104640. i q^{77} +31208.0 q^{79} -59531.0 q^{81} +38466.0i q^{83} +121572. i q^{87} -119514. q^{89} +135596. q^{91} +205832. i q^{93} -94658.0i q^{97} -115680. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 482 q^{9}+O(q^{10}) \) 2 * q - 482 * q^9	\( 2 q - 482 q^{9} + 960 q^{11} - 2408 q^{19} + 9592 q^{21} - 11052 q^{29} - 18712 q^{31} - 27368 q^{39} - 28788 q^{41} - 61434 q^{49} - 8184 q^{51} - 23496 q^{59} + 26404 q^{61} + 140184 q^{69} + 59064 q^{71} + 62416 q^{79} - 119062 q^{81} - 239028 q^{89} + 271192 q^{91} - 231360 q^{99}+O(q^{100}) \) 2 * q - 482 * q^9 + 960 * q^11 - 2408 * q^19 + 9592 * q^21 - 11052 * q^29 - 18712 * q^31 - 27368 * q^39 - 28788 * q^41 - 61434 * q^49 - 8184 * q^51 - 23496 * q^59 + 26404 * q^61 + 140184 * q^69 + 59064 * q^71 + 62416 * q^79 - 119062 * q^81 - 239028 * q^89 + 271192 * q^91 - 231360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more