LMFDB - Embedded newform 400.6.c.j.49.2

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
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			400.6.c.j.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} +192.000i q^{7} +227.000 q^{9} +148.000 q^{11} +286.000i q^{13} +1678.00i q^{17} +1060.00 q^{19} -768.000 q^{21} -2976.00i q^{23} +1880.00i q^{27} +3410.00 q^{29} +2448.00 q^{31} +592.000i q^{33} -182.000i q^{37} -1144.00 q^{39} -9398.00 q^{41} +1244.00i q^{43} -12088.0i q^{47} -20057.0 q^{49} -6712.00 q^{51} +23846.0i q^{53} +4240.00i q^{57} -20020.0 q^{59} +32302.0 q^{61} +43584.0i q^{63} +60972.0i q^{67} +11904.0 q^{69} +32648.0 q^{71} -38774.0i q^{73} +28416.0i q^{77} -33360.0 q^{79} +47641.0 q^{81} -16716.0i q^{83} +13640.0i q^{87} -101370. q^{89} -54912.0 q^{91} +9792.00i q^{93} +119038. i q^{97} +33596.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 454 q^{9} + 296 q^{11} + 2120 q^{19} - 1536 q^{21} + 6820 q^{29} + 4896 q^{31} - 2288 q^{39} - 18796 q^{41} - 40114 q^{49} - 13424 q^{51} - 40040 q^{59} + 64604 q^{61} + 23808 q^{69} + 65296 q^{71}+ \cdots + 67192 q^{99}+O(q^{100}) $ 2 * q + 454 * q^9 + 296 * q^11 + 2120 * q^19 - 1536 * q^21 + 6820 * q^29 + 4896 * q^31 - 2288 * q^39 - 18796 * q^41 - 40114 * q^49 - 13424 * q^51 - 40040 * q^59 + 64604 * q^61 + 23808 * q^69 + 65296 * q^71 - 66720 * q^79 + 95282 * q^81 - 202740 * q^89 - 109824 * q^91 + 67192 * q^99

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	0.256600i	0.991735	+	0.128300i	$0.0409521\pi$
$3$	4.00000i	0.256600i	−0.991735	+	0.128300i	$0.959048\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	192.000i	1.48100i	0.672054	+	0.740502i	$0.265412\pi$
$7$	192.000i	1.48100i	−0.672054	+	0.740502i	$0.734588\pi$
$8$	0	0
$9$	227.000	0.934156
$10$	0	0
$11$	148.000	0.368791	0.184395	−	0.982852i	$-0.440967\pi$
$11$	148.000	0.368791	0.184395	+	0.982852i	$0.440967\pi$
$12$	0	0
$13$	286.000i	0.469362i	0.972072	+	0.234681i	$0.0754045\pi$
$13$	286.000i	0.469362i	−0.972072	+	0.234681i	$0.924595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1678.00i	1.40822i	0.710092	+	0.704109i	$0.248653\pi$
$17$	1678.00i	1.40822i	−0.710092	+	0.704109i	$0.751347\pi$
$18$	0	0
$19$	1060.00	0.673631	0.336815	−	0.941571i	$-0.390650\pi$
$19$	1060.00	0.673631	0.336815	+	0.941571i	$0.390650\pi$
$20$	0	0
$21$	−768.000	−0.380026
$22$	0	0
$23$	− 2976.00i	− 1.17304i	−0.809934	−	0.586521i	$-0.800497\pi$
$23$	− 2976.00i	− 1.17304i	0.809934	−	0.586521i	$-0.199503\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1880.00i	0.496305i
$28$	0	0
$29$	3410.00	0.752938	0.376469	−	0.926429i	$-0.377138\pi$
$29$	3410.00	0.752938	0.376469	+	0.926429i	$0.377138\pi$
$30$	0	0
$31$	2448.00	0.457517	0.228758	−	0.973483i	$-0.426533\pi$
$31$	2448.00	0.457517	0.228758	+	0.973483i	$0.426533\pi$
$32$	0	0
$33$	592.000i	0.0946317i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 182.000i	− 0.0218558i	−0.999940	−	0.0109279i	$-0.996521\pi$
$37$	− 182.000i	− 0.0218558i	0.999940	−	0.0109279i	$-0.00347853\pi$
$38$	0	0
$39$	−1144.00	−0.120438
$40$	0	0
$41$	−9398.00	−0.873124	−0.436562	−	0.899674i	$-0.643804\pi$
$41$	−9398.00	−0.873124	−0.436562	+	0.899674i	$0.643804\pi$
$42$	0	0
$43$	1244.00i	0.102600i	0.998683	+	0.0513002i	$0.0163365\pi$
$43$	1244.00i	0.102600i	−0.998683	+	0.0513002i	$0.983663\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12088.0i	− 0.798196i	−0.916908	−	0.399098i	$-0.869323\pi$
$47$	− 12088.0i	− 0.798196i	0.916908	−	0.399098i	$-0.130677\pi$
$48$	0	0
$49$	−20057.0	−1.19337
$50$	0	0
$51$	−6712.00	−0.361349
$52$	0	0
$53$	23846.0i	1.16607i	0.812446	+	0.583037i	$0.198136\pi$
$53$	23846.0i	1.16607i	−0.812446	+	0.583037i	$0.801864\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4240.00i	0.172854i
$58$	0	0
$59$	−20020.0	−0.748745	−0.374373	−	0.927278i	$-0.622142\pi$
$59$	−20020.0	−0.748745	−0.374373	+	0.927278i	$0.622142\pi$
$60$	0	0
$61$	32302.0	1.11149	0.555744	−	0.831353i	$-0.312433\pi$
$61$	32302.0	1.11149	0.555744	+	0.831353i	$0.312433\pi$
$62$	0	0
$63$	43584.0i	1.38349i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	60972.0i	1.65937i	0.558231	+	0.829685i	$0.311480\pi$
$67$	60972.0i	1.65937i	−0.558231	+	0.829685i	$0.688520\pi$
$68$	0	0
$69$	11904.0	0.301003
$70$	0	0
$71$	32648.0	0.768618	0.384309	−	0.923204i	$-0.374440\pi$
$71$	32648.0	0.768618	0.384309	+	0.923204i	$0.374440\pi$
$72$	0	0
$73$	− 38774.0i	− 0.851596i	−0.904818	−	0.425798i	$-0.859993\pi$
$73$	− 38774.0i	− 0.851596i	0.904818	−	0.425798i	$-0.140007\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	28416.0i	0.546180i
$78$	0	0
$79$	−33360.0	−0.601393	−0.300696	−	0.953720i	$-0.597219\pi$
$79$	−33360.0	−0.601393	−0.300696	+	0.953720i	$0.597219\pi$
$80$	0	0
$81$	47641.0	0.806805
$82$	0	0
$83$	− 16716.0i	− 0.266340i	−0.991093	−	0.133170i	$-0.957484\pi$
$83$	− 16716.0i	− 0.266340i	0.991093	−	0.133170i	$-0.0425157\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13640.0i	0.193204i
$88$	0	0
$89$	−101370.	−1.35655	−0.678273	−	0.734810i	$-0.737271\pi$
$89$	−101370.	−1.35655	−0.678273	+	0.734810i	$0.737271\pi$
$90$	0	0
$91$	−54912.0	−0.695126
$92$	0	0
$93$	9792.00i	0.117399i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	119038.i	1.28457i	0.766468	+	0.642283i	$0.222013\pi$
$97$	119038.i	1.28457i	−0.766468	+	0.642283i	$0.777987\pi$
$98$	0	0
$99$	33596.0	0.344508
$100$	0	0
$101$	−89898.0	−0.876893	−0.438446	−	0.898757i	$-0.644471\pi$
$101$	−89898.0	−0.876893	−0.438446	+	0.898757i	$0.644471\pi$
$102$	0	0
$103$	19504.0i	0.181147i	0.995890	+	0.0905734i	$0.0288700\pi$
$103$	19504.0i	0.181147i	−0.995890	+	0.0905734i	$0.971130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	158292.i	1.33659i	0.743895	+	0.668297i	$0.232976\pi$
$107$	158292.i	1.33659i	−0.743895	+	0.668297i	$0.767024\pi$
$108$	0	0
$109$	−36830.0	−0.296917	−0.148459	−	0.988919i	$-0.547431\pi$
$109$	−36830.0	−0.296917	−0.148459	+	0.988919i	$0.547431\pi$
$110$	0	0
$111$	728.000	0.00560821
$112$	0	0
$113$	11186.0i	0.0824098i	0.999151	+	0.0412049i	$0.0131196\pi$
$113$	11186.0i	0.0824098i	−0.999151	+	0.0412049i	$0.986880\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	64922.0i	0.438457i
$118$	0	0
$119$	−322176.	−2.08557
$120$	0	0
$121$	−139147.	−0.863993
$122$	0	0
$123$	− 37592.0i	− 0.224044i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	70552.0i	0.388150i	0.980987	+	0.194075i	$0.0621706\pi$
$127$	70552.0i	0.388150i	−0.980987	+	0.194075i	$0.937829\pi$
$128$	0	0
$129$	−4976.00	−0.0263273
$130$	0	0
$131$	−76452.0	−0.389234	−0.194617	−	0.980879i	$-0.562346\pi$
$131$	−76452.0	−0.389234	−0.194617	+	0.980879i	$0.562346\pi$
$132$	0	0
$133$	203520.i	0.997650i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	144918.i	0.659661i	0.944040	+	0.329831i	$0.106992\pi$
$137$	144918.i	0.659661i	−0.944040	+	0.329831i	$0.893008\pi$
$138$	0	0
$139$	112220.	0.492644	0.246322	−	0.969188i	$-0.420778\pi$
$139$	112220.	0.492644	0.246322	+	0.969188i	$0.420778\pi$
$140$	0	0
$141$	48352.0	0.204817
$142$	0	0
$143$	42328.0i	0.173096i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 80228.0i	− 0.306219i
$148$	0	0
$149$	−403750.	−1.48986	−0.744932	−	0.667140i	$-0.767518\pi$
$149$	−403750.	−1.48986	−0.744932	+	0.667140i	$0.767518\pi$
$150$	0	0
$151$	446648.	1.59413	0.797064	−	0.603895i	$-0.206385\pi$
$151$	446648.	1.59413	0.797064	+	0.603895i	$0.206385\pi$
$152$	0	0
$153$	380906.i	1.31550i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	262258.i	0.849141i	0.905395	+	0.424570i	$0.139575\pi$
$157$	262258.i	0.849141i	−0.905395	+	0.424570i	$0.860425\pi$
$158$	0	0
$159$	−95384.0	−0.299215
$160$	0	0
$161$	571392.	1.73728
$162$	0	0
$163$	154564.i	0.455658i	0.973701	+	0.227829i	$0.0731628\pi$
$163$	154564.i	0.455658i	−0.973701	+	0.227829i	$0.926837\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	396672.i	1.10063i	0.834958	+	0.550314i	$0.185492\pi$
$167$	396672.i	1.10063i	−0.834958	+	0.550314i	$0.814508\pi$
$168$	0	0
$169$	289497.	0.779700
$170$	0	0
$171$	240620.	0.629276
$172$	0	0
$173$	− 573474.i	− 1.45680i	−0.685155	−	0.728398i	$-0.740265\pi$
$173$	− 573474.i	− 1.45680i	0.685155	−	0.728398i	$-0.259735\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 80080.0i	− 0.192128i
$178$	0	0
$179$	−594460.	−1.38672	−0.693362	−	0.720589i	$-0.743871\pi$
$179$	−594460.	−1.38672	−0.693362	+	0.720589i	$0.743871\pi$
$180$	0	0
$181$	−107098.	−0.242988	−0.121494	−	0.992592i	$-0.538769\pi$
$181$	−107098.	−0.242988	−0.121494	+	0.992592i	$0.538769\pi$
$182$	0	0
$183$	129208.i	0.285208i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	248344.i	0.519337i
$188$	0	0
$189$	−360960.	−0.735029
$190$	0	0
$191$	−469552.	−0.931323	−0.465661	−	0.884963i	$-0.654184\pi$
$191$	−469552.	−0.931323	−0.465661	+	0.884963i	$0.654184\pi$
$192$	0	0
$193$	52706.0i	0.101851i	0.998702	+	0.0509257i	$0.0162172\pi$
$193$	52706.0i	0.101851i	−0.998702	+	0.0509257i	$0.983783\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 455862.i	− 0.836889i	−0.908242	−	0.418444i	$-0.862575\pi$
$197$	− 455862.i	− 0.836889i	0.908242	−	0.418444i	$-0.137425\pi$
$198$	0	0
$199$	865000.	1.54840	0.774200	−	0.632940i	$-0.218152\pi$
$199$	865000.	1.54840	0.774200	+	0.632940i	$0.218152\pi$
$200$	0	0
$201$	−243888.	−0.425795
$202$	0	0
$203$	654720.i	1.11510i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 675552.i	− 1.09580i
$208$	0	0
$209$	156880.	0.248429
$210$	0	0
$211$	−1.10565e6	−1.70967	−0.854835	−	0.518900i	$-0.826342\pi$
$211$	−1.10565e6	−1.70967	−0.854835	+	0.518900i	$0.826342\pi$
$212$	0	0
$213$	130592.i	0.197228i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	470016.i	0.677584i
$218$	0	0
$219$	155096.	0.218520
$220$	0	0
$221$	−479908.	−0.660963
$222$	0	0
$223$	− 1.12158e6i	− 1.51031i	−0.655545	−	0.755156i	$-0.727561\pi$
$223$	− 1.12158e6i	− 1.51031i	0.655545	−	0.755156i	$-0.272439\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23348.0i	− 0.0300736i	−0.999887	−	0.0150368i	$-0.995213\pi$
$227$	− 23348.0i	− 0.0300736i	0.999887	−	0.0150368i	$-0.00478654\pi$
$228$	0	0
$229$	596010.	0.751043	0.375522	−	0.926814i	$-0.377464\pi$
$229$	596010.	0.751043	0.375522	+	0.926814i	$0.377464\pi$
$230$	0	0
$231$	−113664.	−0.140150
$232$	0	0
$233$	− 485334.i	− 0.585667i	−0.956163	−	0.292834i	$-0.905402\pi$
$233$	− 485334.i	− 0.585667i	0.956163	−	0.292834i	$-0.0945982\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 133440.i	− 0.154317i
$238$	0	0
$239$	−48880.0	−0.0553524	−0.0276762	−	0.999617i	$-0.508811\pi$
$239$	−48880.0	−0.0553524	−0.0276762	+	0.999617i	$0.508811\pi$
$240$	0	0
$241$	−110798.	−0.122882	−0.0614411	−	0.998111i	$-0.519570\pi$
$241$	−110798.	−0.122882	−0.0614411	+	0.998111i	$0.519570\pi$
$242$	0	0
$243$	647404.i	0.703331i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	303160.i	0.316176i
$248$	0	0
$249$	66864.0	0.0683430
$250$	0	0
$251$	1.64375e6	1.64684	0.823419	−	0.567434i	$-0.192064\pi$
$251$	1.64375e6	1.64684	0.823419	+	0.567434i	$0.192064\pi$
$252$	0	0
$253$	− 440448.i	− 0.432607i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.30624e6i	− 1.23365i	−0.787102	−	0.616823i	$-0.788419\pi$
$257$	− 1.30624e6i	− 1.23365i	0.787102	−	0.616823i	$-0.211581\pi$
$258$	0	0
$259$	34944.0	0.0323685
$260$	0	0
$261$	774070.	0.703362
$262$	0	0
$263$	− 2.12834e6i	− 1.89736i	−0.316231	−	0.948682i	$-0.602417\pi$
$263$	− 2.12834e6i	− 1.89736i	0.316231	−	0.948682i	$-0.397583\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 405480.i	− 0.348090i
$268$	0	0
$269$	1.44109e6	1.21426	0.607128	−	0.794604i	$-0.292321\pi$
$269$	1.44109e6	1.21426	0.607128	+	0.794604i	$0.292321\pi$
$270$	0	0
$271$	93248.0	0.0771288	0.0385644	−	0.999256i	$-0.487722\pi$
$271$	93248.0	0.0771288	0.0385644	+	0.999256i	$0.487722\pi$
$272$	0	0
$273$	− 219648.i	− 0.178370i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	110298.i	0.0863711i	0.999067	+	0.0431855i	$0.0137507\pi$
$277$	110298.i	0.0863711i	−0.999067	+	0.0431855i	$0.986249\pi$
$278$	0	0
$279$	555696.	0.427392
$280$	0	0
$281$	−192198.	−0.145205	−0.0726027	−	0.997361i	$-0.523131\pi$
$281$	−192198.	−0.145205	−0.0726027	+	0.997361i	$0.523131\pi$
$282$	0	0
$283$	331884.i	0.246332i	0.992386	+	0.123166i	$0.0393047\pi$
$283$	331884.i	0.246332i	−0.992386	+	0.123166i	$0.960695\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.80442e6i	− 1.29310i
$288$	0	0
$289$	−1.39583e6	−0.983076
$290$	0	0
$291$	−476152.	−0.329620
$292$	0	0
$293$	2.19481e6i	1.49358i	0.665063	+	0.746788i	$0.268405\pi$
$293$	2.19481e6i	1.49358i	−0.665063	+	0.746788i	$0.731595\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	278240.i	0.183033i
$298$	0	0
$299$	851136.	0.550581
$300$	0	0
$301$	−238848.	−0.151952
$302$	0	0
$303$	− 359592.i	− 0.225011i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.37751e6i	− 1.43971i	−0.694123	−	0.719857i	$-0.744207\pi$
$307$	− 2.37751e6i	− 1.43971i	0.694123	−	0.719857i	$-0.255793\pi$
$308$	0	0
$309$	−78016.0	−0.0464823
$310$	0	0
$311$	2.37305e6	1.39125	0.695626	−	0.718405i	$-0.255127\pi$
$311$	2.37305e6	1.39125	0.695626	+	0.718405i	$0.255127\pi$
$312$	0	0
$313$	− 1.42941e6i	− 0.824702i	−0.911025	−	0.412351i	$-0.864708\pi$
$313$	− 1.42941e6i	− 0.824702i	0.911025	−	0.412351i	$-0.135292\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.12462e6i	− 1.18750i	−0.804650	−	0.593750i	$-0.797647\pi$
$317$	− 2.12462e6i	− 1.18750i	0.804650	−	0.593750i	$-0.202353\pi$
$318$	0	0
$319$	504680.	0.277677
$320$	0	0
$321$	−633168.	−0.342970
$322$	0	0
$323$	1.77868e6i	0.948618i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 147320.i	− 0.0761890i
$328$	0	0
$329$	2.32090e6	1.18213
$330$	0	0
$331$	−3.09985e6	−1.55515	−0.777573	−	0.628793i	$-0.783549\pi$
$331$	−3.09985e6	−1.55515	−0.777573	+	0.628793i	$0.783549\pi$
$332$	0	0
$333$	− 41314.0i	− 0.0204168i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.40008e6i	− 1.15120i	−0.817731	−	0.575601i	$-0.804768\pi$
$337$	− 2.40008e6i	− 1.15120i	0.817731	−	0.575601i	$-0.195232\pi$
$338$	0	0
$339$	−44744.0	−0.0211464
$340$	0	0
$341$	362304.	0.168728
$342$	0	0
$343$	− 624000.i	− 0.286384i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.77741e6i	0.792436i	0.918156	+	0.396218i	$0.129678\pi$
$347$	1.77741e6i	0.792436i	−0.918156	+	0.396218i	$0.870322\pi$
$348$	0	0
$349$	2.14805e6	0.944019	0.472010	−	0.881593i	$-0.343529\pi$
$349$	2.14805e6	0.944019	0.472010	+	0.881593i	$0.343529\pi$
$350$	0	0
$351$	−537680.	−0.232946
$352$	0	0
$353$	− 661854.i	− 0.282700i	−0.989960	−	0.141350i	$-0.954856\pi$
$353$	− 661854.i	− 0.282700i	0.989960	−	0.141350i	$-0.0451443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.28870e6i	− 0.535159i
$358$	0	0
$359$	−259320.	−0.106194	−0.0530970	−	0.998589i	$-0.516909\pi$
$359$	−259320.	−0.106194	−0.0530970	+	0.998589i	$0.516909\pi$
$360$	0	0
$361$	−1.35250e6	−0.546222
$362$	0	0
$363$	− 556588.i	− 0.221701i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.49993e6i	− 0.581307i	−0.956828	−	0.290653i	$-0.906127\pi$
$367$	− 1.49993e6i	− 0.581307i	0.956828	−	0.290653i	$-0.0938726\pi$
$368$	0	0
$369$	−2.13335e6	−0.815634
$370$	0	0
$371$	−4.57843e6	−1.72696
$372$	0	0
$373$	− 2.23807e6i	− 0.832918i	−0.909154	−	0.416459i	$-0.863271\pi$
$373$	− 2.23807e6i	− 0.832918i	0.909154	−	0.416459i	$-0.136729\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	975260.i	0.353400i
$378$	0	0
$379$	3.15934e6	1.12979	0.564896	−	0.825162i	$-0.308916\pi$
$379$	3.15934e6	1.12979	0.564896	+	0.825162i	$0.308916\pi$
$380$	0	0
$381$	−282208.	−0.0995994
$382$	0	0
$383$	− 342216.i	− 0.119207i	−0.998222	−	0.0596037i	$-0.981016\pi$
$383$	− 342216.i	− 0.119207i	0.998222	−	0.0596037i	$-0.0189837\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	282388.i	0.0958449i
$388$	0	0
$389$	−88470.0	−0.0296430	−0.0148215	−	0.999890i	$-0.504718\pi$
$389$	−88470.0	−0.0296430	−0.0148215	+	0.999890i	$0.504718\pi$
$390$	0	0
$391$	4.99373e6	1.65190
$392$	0	0
$393$	− 305808.i	− 0.0998775i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.45674e6i	1.73763i	0.495138	+	0.868814i	$0.335117\pi$
$397$	5.45674e6i	1.73763i	−0.495138	+	0.868814i	$0.664883\pi$
$398$	0	0
$399$	−814080.	−0.255997
$400$	0	0
$401$	4.04680e6	1.25676	0.628378	−	0.777908i	$-0.283719\pi$
$401$	4.04680e6	1.25676	0.628378	+	0.777908i	$0.283719\pi$
$402$	0	0
$403$	700128.i	0.214741i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 26936.0i	− 0.00806022i
$408$	0	0
$409$	2.71207e6	0.801664	0.400832	−	0.916151i	$-0.368721\pi$
$409$	2.71207e6	0.801664	0.400832	+	0.916151i	$0.368721\pi$
$410$	0	0
$411$	−579672.	−0.169269
$412$	0	0
$413$	− 3.84384e6i	− 1.10889i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	448880.i	0.126413i
$418$	0	0
$419$	3.71746e6	1.03445	0.517227	−	0.855848i	$-0.326964\pi$
$419$	3.71746e6	1.03445	0.517227	+	0.855848i	$0.326964\pi$
$420$	0	0
$421$	3.55250e6	0.976853	0.488426	−	0.872605i	$-0.337571\pi$
$421$	3.55250e6	0.976853	0.488426	+	0.872605i	$0.337571\pi$
$422$	0	0
$423$	− 2.74398e6i	− 0.745640i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.20198e6i	1.64612i
$428$	0	0
$429$	−169312.	−0.0444165
$430$	0	0
$431$	4.06205e6	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$431$	4.06205e6	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$432$	0	0
$433$	7.26287e6i	1.86161i	0.365518	+	0.930804i	$0.380892\pi$
$433$	7.26287e6i	1.86161i	−0.365518	+	0.930804i	$0.619108\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.15456e6i	− 0.790197i
$438$	0	0
$439$	−5.41028e6	−1.33986	−0.669928	−	0.742426i	$-0.733675\pi$
$439$	−5.41028e6	−1.33986	−0.669928	+	0.742426i	$0.733675\pi$
$440$	0	0
$441$	−4.55294e6	−1.11480
$442$	0	0
$443$	6.51524e6i	1.57733i	0.614826	+	0.788663i	$0.289226\pi$
$443$	6.51524e6i	1.57733i	−0.614826	+	0.788663i	$0.710774\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.61500e6i	− 0.382299i
$448$	0	0
$449$	509950.	0.119375	0.0596873	−	0.998217i	$-0.480990\pi$
$449$	509950.	0.119375	0.0596873	+	0.998217i	$0.480990\pi$
$450$	0	0
$451$	−1.39090e6	−0.322000
$452$	0	0
$453$	1.78659e6i	0.409053i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.22084e6i	− 0.273444i	−0.990609	−	0.136722i	$-0.956343\pi$
$457$	− 1.22084e6i	− 0.273444i	0.990609	−	0.136722i	$-0.0436568\pi$
$458$	0	0
$459$	−3.15464e6	−0.698905
$460$	0	0
$461$	−4.07210e6	−0.892413	−0.446207	−	0.894930i	$-0.647225\pi$
$461$	−4.07210e6	−0.892413	−0.446207	+	0.894930i	$0.647225\pi$
$462$	0	0
$463$	− 2.02294e6i	− 0.438561i	−0.975662	−	0.219280i	$-0.929629\pi$
$463$	− 2.02294e6i	− 0.438561i	0.975662	−	0.219280i	$-0.0703709\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.25097e6i	0.689797i	0.938640	+	0.344898i	$0.112087\pi$
$467$	3.25097e6i	0.689797i	−0.938640	+	0.344898i	$0.887913\pi$
$468$	0	0
$469$	−1.17066e7	−2.45753
$470$	0	0
$471$	−1.04903e6	−0.217890
$472$	0	0
$473$	184112.i	0.0378381i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.41304e6i	1.08929i
$478$	0	0
$479$	−3.27936e6	−0.653056	−0.326528	−	0.945188i	$-0.605879\pi$
$479$	−3.27936e6	−0.653056	−0.326528	+	0.945188i	$0.605879\pi$
$480$	0	0
$481$	52052.0	0.0102583
$482$	0	0
$483$	2.28557e6i	0.445786i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.53197e6i	− 1.63015i	−0.579357	−	0.815074i	$-0.696696\pi$
$487$	− 8.53197e6i	− 1.63015i	0.579357	−	0.815074i	$-0.303304\pi$
$488$	0	0
$489$	−618256.	−0.116922
$490$	0	0
$491$	−1.51265e6	−0.283162	−0.141581	−	0.989927i	$-0.545219\pi$
$491$	−1.51265e6	−0.283162	−0.141581	+	0.989927i	$0.545219\pi$
$492$	0	0
$493$	5.72198e6i	1.06030i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.26842e6i	1.13833i
$498$	0	0
$499$	−6.49190e6	−1.16713	−0.583567	−	0.812065i	$-0.698343\pi$
$499$	−6.49190e6	−1.16713	−0.583567	+	0.812065i	$0.698343\pi$
$500$	0	0
$501$	−1.58669e6	−0.282421
$502$	0	0
$503$	− 8.61770e6i	− 1.51870i	−0.650684	−	0.759349i	$-0.725518\pi$
$503$	− 8.61770e6i	− 1.51870i	0.650684	−	0.759349i	$-0.274482\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.15799e6i	0.200071i
$508$	0	0
$509$	−2.67323e6	−0.457343	−0.228671	−	0.973504i	$-0.573438\pi$
$509$	−2.67323e6	−0.457343	−0.228671	+	0.973504i	$0.573438\pi$
$510$	0	0
$511$	7.44461e6	1.26122
$512$	0	0
$513$	1.99280e6i	0.334326i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.78902e6i	− 0.294367i
$518$	0	0
$519$	2.29390e6	0.373814
$520$	0	0
$521$	6.18500e6	0.998264	0.499132	−	0.866526i	$-0.333652\pi$
$521$	6.18500e6	0.998264	0.499132	+	0.866526i	$0.333652\pi$
$522$	0	0
$523$	6.89452e6i	1.10217i	0.834448	+	0.551087i	$0.185787\pi$
$523$	6.89452e6i	1.10217i	−0.834448	+	0.551087i	$0.814213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.10774e6i	0.644283i
$528$	0	0
$529$	−2.42023e6	−0.376026
$530$	0	0
$531$	−4.54454e6	−0.699445
$532$	0	0
$533$	− 2.68783e6i	− 0.409811i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.37784e6i	− 0.355834i
$538$	0	0
$539$	−2.96844e6	−0.440104
$540$	0	0
$541$	155502.	0.0228425	0.0114212	−	0.999935i	$-0.496364\pi$
$541$	155502.	0.0228425	0.0114212	+	0.999935i	$0.496364\pi$
$542$	0	0
$543$	− 428392.i	− 0.0623508i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.26544e7i	1.80831i	0.427201	+	0.904157i	$0.359500\pi$
$547$	1.26544e7i	1.80831i	−0.427201	+	0.904157i	$0.640500\pi$
$548$	0	0
$549$	7.33255e6	1.03830
$550$	0	0
$551$	3.61460e6	0.507202
$552$	0	0
$553$	− 6.40512e6i	− 0.890665i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.07786e6i	0.966638i	0.875444	+	0.483319i	$0.160569\pi$
$557$	7.07786e6i	0.966638i	−0.875444	+	0.483319i	$0.839431\pi$
$558$	0	0
$559$	−355784.	−0.0481567
$560$	0	0
$561$	−993376.	−0.133262
$562$	0	0
$563$	− 846636.i	− 0.112571i	−0.998415	−	0.0562854i	$-0.982074\pi$
$563$	− 846636.i	− 0.112571i	0.998415	−	0.0562854i	$-0.0179257\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.14707e6i	1.19488i
$568$	0	0
$569$	−4.96041e6	−0.642299	−0.321149	−	0.947029i	$-0.604069\pi$
$569$	−4.96041e6	−0.642299	−0.321149	+	0.947029i	$0.604069\pi$
$570$	0	0
$571$	−8.96505e6	−1.15070	−0.575351	−	0.817907i	$-0.695134\pi$
$571$	−8.96505e6	−1.15070	−0.575351	+	0.817907i	$0.695134\pi$
$572$	0	0
$573$	− 1.87821e6i	− 0.238978i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.86080e6i	0.357724i	0.983874	+	0.178862i	$0.0572415\pi$
$577$	2.86080e6i	0.357724i	−0.983874	+	0.178862i	$0.942758\pi$
$578$	0	0
$579$	−210824.	−0.0261351
$580$	0	0
$581$	3.20947e6	0.394451
$582$	0	0
$583$	3.52921e6i	0.430037i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.74027e6i	− 0.807387i	−0.914894	−	0.403694i	$-0.867726\pi$
$587$	− 6.74027e6i	− 0.807387i	0.914894	−	0.403694i	$-0.132274\pi$
$588$	0	0
$589$	2.59488e6	0.308197
$590$	0	0
$591$	1.82345e6	0.214746
$592$	0	0
$593$	− 1.78609e6i	− 0.208578i	−0.994547	−	0.104289i	$-0.966743\pi$
$593$	− 1.78609e6i	− 0.208578i	0.994547	−	0.104289i	$-0.0332566\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.46000e6i	0.397320i
$598$	0	0
$599$	4.94620e6	0.563254	0.281627	−	0.959524i	$-0.409126\pi$
$599$	4.94620e6	0.563254	0.281627	+	0.959524i	$0.409126\pi$
$600$	0	0
$601$	−4.58100e6	−0.517337	−0.258669	−	0.965966i	$-0.583284\pi$
$601$	−4.58100e6	−0.517337	−0.258669	+	0.965966i	$0.583284\pi$
$602$	0	0
$603$	1.38406e7i	1.55011i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.07999e6i	0.779940i	0.920828	+	0.389970i	$0.127515\pi$
$607$	7.07999e6i	0.779940i	−0.920828	+	0.389970i	$0.872485\pi$
$608$	0	0
$609$	−2.61888e6	−0.286136
$610$	0	0
$611$	3.45717e6	0.374643
$612$	0	0
$613$	5.09609e6i	0.547754i	0.961765	+	0.273877i	$0.0883061\pi$
$613$	5.09609e6i	0.547754i	−0.961765	+	0.273877i	$0.911694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.30003e7i	1.37480i	0.726279	+	0.687400i	$0.241248\pi$
$617$	1.30003e7i	1.37480i	−0.726279	+	0.687400i	$0.758752\pi$
$618$	0	0
$619$	4.84406e6	0.508139	0.254070	−	0.967186i	$-0.418231\pi$
$619$	4.84406e6	0.508139	0.254070	+	0.967186i	$0.418231\pi$
$620$	0	0
$621$	5.59488e6	0.582186
$622$	0	0
$623$	− 1.94630e7i	− 2.00905i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	627520.i	0.0637468i
$628$	0	0
$629$	305396.	0.0307777
$630$	0	0
$631$	−6.22775e6	−0.622670	−0.311335	−	0.950300i	$-0.600776\pi$
$631$	−6.22775e6	−0.622670	−0.311335	+	0.950300i	$0.600776\pi$
$632$	0	0
$633$	− 4.42261e6i	− 0.438702i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.73630e6i	− 0.560123i
$638$	0	0
$639$	7.41110e6	0.718010
$640$	0	0
$641$	1.53280e6	0.147347	0.0736734	−	0.997282i	$-0.476528\pi$
$641$	1.53280e6	0.147347	0.0736734	+	0.997282i	$0.476528\pi$
$642$	0	0
$643$	1.74382e7i	1.66332i	0.555287	+	0.831659i	$0.312609\pi$
$643$	1.74382e7i	1.66332i	−0.555287	+	0.831659i	$0.687391\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 4.25469e6i	− 0.399583i	−0.979838	−	0.199792i	$-0.935974\pi$
$647$	− 4.25469e6i	− 0.399583i	0.979838	−	0.199792i	$-0.0640265\pi$
$648$	0	0
$649$	−2.96296e6	−0.276130
$650$	0	0
$651$	−1.88006e6	−0.173868
$652$	0	0
$653$	3.01085e6i	0.276316i	0.990410	+	0.138158i	$0.0441181\pi$
$653$	3.01085e6i	0.276316i	−0.990410	+	0.138158i	$0.955882\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8.80170e6i	− 0.795524i
$658$	0	0
$659$	−8.11462e6	−0.727871	−0.363936	−	0.931424i	$-0.618567\pi$
$659$	−8.11462e6	−0.727871	−0.363936	+	0.931424i	$0.618567\pi$
$660$	0	0
$661$	2.47370e6	0.220213	0.110107	−	0.993920i	$-0.464881\pi$
$661$	2.47370e6	0.220213	0.110107	+	0.993920i	$0.464881\pi$
$662$	0	0
$663$	− 1.91963e6i	− 0.169603i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.01482e7i	− 0.883228i
$668$	0	0
$669$	4.48630e6	0.387546
$670$	0	0
$671$	4.78070e6	0.409907
$672$	0	0
$673$	5.77063e6i	0.491117i	0.969382	+	0.245559i	$0.0789714\pi$
$673$	5.77063e6i	0.491117i	−0.969382	+	0.245559i	$0.921029\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.67197e7i	− 1.40203i	−0.713147	−	0.701014i	$-0.752731\pi$
$677$	− 1.67197e7i	− 1.40203i	0.713147	−	0.701014i	$-0.247269\pi$
$678$	0	0
$679$	−2.28553e7	−1.90245
$680$	0	0
$681$	93392.0	0.00771688
$682$	0	0
$683$	− 7.14532e6i	− 0.586097i	−0.956098	−	0.293049i	$-0.905330\pi$
$683$	− 7.14532e6i	− 0.586097i	0.956098	−	0.293049i	$-0.0946698\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.38404e6i	0.192718i
$688$	0	0
$689$	−6.81996e6	−0.547310
$690$	0	0
$691$	8.78395e6	0.699833	0.349917	−	0.936781i	$-0.386210\pi$
$691$	8.78395e6	0.699833	0.349917	+	0.936781i	$0.386210\pi$
$692$	0	0
$693$	6.45043e6i	0.510218i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.57698e7i	− 1.22955i
$698$	0	0
$699$	1.94134e6	0.150282
$700$	0	0
$701$	−1.60141e7	−1.23086	−0.615428	−	0.788193i	$-0.711017\pi$
$701$	−1.60141e7	−1.23086	−0.615428	+	0.788193i	$0.711017\pi$
$702$	0	0
$703$	− 192920.i	− 0.0147228i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.72604e7i	− 1.29868i
$708$	0	0
$709$	1.91354e7	1.42962	0.714811	−	0.699318i	$-0.246513\pi$
$709$	1.91354e7	1.42962	0.714811	+	0.699318i	$0.246513\pi$
$710$	0	0
$711$	−7.57272e6	−0.561795
$712$	0	0
$713$	− 7.28525e6i	− 0.536686i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 195520.i	− 0.0142034i
$718$	0	0
$719$	1.02934e7	0.742566	0.371283	−	0.928520i	$-0.378918\pi$
$719$	1.02934e7	0.742566	0.371283	+	0.928520i	$0.378918\pi$
$720$	0	0
$721$	−3.74477e6	−0.268279
$722$	0	0
$723$	− 443192.i	− 0.0315316i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.93264e7i	− 1.35618i	−0.734981	−	0.678088i	$-0.762809\pi$
$727$	− 1.93264e7i	− 1.35618i	0.734981	−	0.678088i	$-0.237191\pi$
$728$	0	0
$729$	8.98715e6	0.626330
$730$	0	0
$731$	−2.08743e6	−0.144484
$732$	0	0
$733$	5.26197e6i	0.361733i	0.983508	+	0.180866i	$0.0578902\pi$
$733$	5.26197e6i	0.361733i	−0.983508	+	0.180866i	$0.942110\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.02386e6i	0.611961i
$738$	0	0
$739$	2.82944e7	1.90585	0.952927	−	0.303199i	$-0.0980548\pi$
$739$	2.82944e7	1.90585	0.952927	+	0.303199i	$0.0980548\pi$
$740$	0	0
$741$	−1.21264e6	−0.0811309
$742$	0	0
$743$	− 2.09863e7i	− 1.39464i	−0.716759	−	0.697321i	$-0.754375\pi$
$743$	− 2.09863e7i	− 1.39464i	0.716759	−	0.697321i	$-0.245625\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.79453e6i	− 0.248804i
$748$	0	0
$749$	−3.03921e7	−1.97950
$750$	0	0
$751$	1.89668e7	1.22714	0.613572	−	0.789639i	$-0.289732\pi$
$751$	1.89668e7	1.22714	0.613572	+	0.789639i	$0.289732\pi$
$752$	0	0
$753$	6.57499e6i	0.422579i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.08257e7i	0.686617i	0.939223	+	0.343309i	$0.111548\pi$
$757$	1.08257e7i	0.686617i	−0.939223	+	0.343309i	$0.888452\pi$
$758$	0	0
$759$	1.76179e6	0.111007
$760$	0	0
$761$	1.90534e7	1.19264	0.596322	−	0.802745i	$-0.296628\pi$
$761$	1.90534e7	1.19264	0.596322	+	0.802745i	$0.296628\pi$
$762$	0	0
$763$	− 7.07136e6i	− 0.439736i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.72572e6i	− 0.351432i
$768$	0	0
$769$	1.57826e7	0.962415	0.481208	−	0.876607i	$-0.340198\pi$
$769$	1.57826e7	0.962415	0.481208	+	0.876607i	$0.340198\pi$
$770$	0	0
$771$	5.22497e6	0.316554
$772$	0	0
$773$	− 2.44049e7i	− 1.46902i	−0.678598	−	0.734510i	$-0.737412\pi$
$773$	− 2.44049e7i	− 1.46902i	0.678598	−	0.734510i	$-0.262588\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	139776.i	0.00830577i
$778$	0	0
$779$	−9.96188e6	−0.588163
$780$	0	0
$781$	4.83190e6	0.283459
$782$	0	0
$783$	6.41080e6i	0.373687i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.37607e7i	1.94301i	0.237019	+	0.971505i	$0.423830\pi$
$787$	3.37607e7i	1.94301i	−0.237019	+	0.971505i	$0.576170\pi$
$788$	0	0
$789$	8.51334e6	0.486864
$790$	0	0
$791$	−2.14771e6	−0.122049
$792$	0	0
$793$	9.23837e6i	0.521690i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.19885e7i	− 1.22617i	−0.790019	−	0.613083i	$-0.789929\pi$
$797$	− 2.19885e7i	− 1.22617i	0.790019	−	0.613083i	$-0.210071\pi$
$798$	0	0
$799$	2.02837e7	1.12403
$800$	0	0
$801$	−2.30110e7	−1.26723
$802$	0	0
$803$	− 5.73855e6i	− 0.314061i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.76436e6i	0.311578i
$808$	0	0
$809$	2.93597e7	1.57717	0.788587	−	0.614923i	$-0.210813\pi$
$809$	2.93597e7	1.57717	0.788587	+	0.614923i	$0.210813\pi$
$810$	0	0
$811$	−3.17703e7	−1.69617	−0.848083	−	0.529863i	$-0.822243\pi$
$811$	−3.17703e7	−1.69617	−0.848083	+	0.529863i	$0.822243\pi$
$812$	0	0
$813$	372992.i	0.0197912i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.31864e6i	0.0691148i
$818$	0	0
$819$	−1.24650e7	−0.649357
$820$	0	0
$821$	−2.71430e6	−0.140540	−0.0702699	−	0.997528i	$-0.522386\pi$
$821$	−2.71430e6	−0.140540	−0.0702699	+	0.997528i	$0.522386\pi$
$822$	0	0
$823$	1.25866e7i	0.647753i	0.946099	+	0.323877i	$0.104986\pi$
$823$	1.25866e7i	0.647753i	−0.946099	+	0.323877i	$0.895014\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.72355e6i	− 0.443537i	−0.975099	−	0.221768i	$-0.928817\pi$
$827$	− 8.72355e6i	− 0.443537i	0.975099	−	0.221768i	$-0.0711828\pi$
$828$	0	0
$829$	1.06178e7	0.536597	0.268299	−	0.963336i	$-0.413539\pi$
$829$	1.06178e7	0.536597	0.268299	+	0.963336i	$0.413539\pi$
$830$	0	0
$831$	−441192.	−0.0221628
$832$	0	0
$833$	− 3.36556e7i	− 1.68053i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.60224e6i	0.227068i
$838$	0	0
$839$	1.67765e7	0.822805	0.411403	−	0.911454i	$-0.365039\pi$
$839$	1.67765e7	0.822805	0.411403	+	0.911454i	$0.365039\pi$
$840$	0	0
$841$	−8.88305e6	−0.433084
$842$	0	0
$843$	− 768792.i	− 0.0372597i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.67162e7i	− 1.27958i
$848$	0	0
$849$	−1.32754e6	−0.0632087
$850$	0	0
$851$	−541632.	−0.0256378
$852$	0	0
$853$	− 2.20186e7i	− 1.03613i	−0.855340	−	0.518067i	$-0.826652\pi$
$853$	− 2.20186e7i	− 1.03613i	0.855340	−	0.518067i	$-0.173348\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.16676e7i	− 1.47287i	−0.676510	−	0.736434i	$-0.736508\pi$
$857$	− 3.16676e7i	− 1.47287i	0.676510	−	0.736434i	$-0.263492\pi$
$858$	0	0
$859$	1.58064e7	0.730886	0.365443	−	0.930834i	$-0.380918\pi$
$859$	1.58064e7	0.730886	0.365443	+	0.930834i	$0.380918\pi$
$860$	0	0
$861$	7.21766e6	0.331809
$862$	0	0
$863$	1.44287e7i	0.659476i	0.944072	+	0.329738i	$0.106960\pi$
$863$	1.44287e7i	0.659476i	−0.944072	+	0.329738i	$0.893040\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5.58331e6i	− 0.252257i
$868$	0	0
$869$	−4.93728e6	−0.221788
$870$	0	0
$871$	−1.74380e7	−0.778845
$872$	0	0
$873$	2.70216e7i	1.19999i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 247902.i	− 0.0108838i	−0.999985	−	0.00544191i	$-0.998268\pi$
$877$	− 247902.i	− 0.0108838i	0.999985	−	0.00544191i	$-0.00173222\pi$
$878$	0	0
$879$	−8.77922e6	−0.383252
$880$	0	0
$881$	4.10268e7	1.78085	0.890426	−	0.455128i	$-0.150406\pi$
$881$	4.10268e7	1.78085	0.890426	+	0.455128i	$0.150406\pi$
$882$	0	0
$883$	− 4.18015e7i	− 1.80422i	−0.431503	−	0.902112i	$-0.642016\pi$
$883$	− 4.18015e7i	− 1.80422i	0.431503	−	0.902112i	$-0.357984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.10476e7i	− 0.898241i	−0.893471	−	0.449120i	$-0.851737\pi$
$887$	− 2.10476e7i	− 0.898241i	0.893471	−	0.449120i	$-0.148263\pi$
$888$	0	0
$889$	−1.35460e7	−0.574852
$890$	0	0
$891$	7.05087e6	0.297542
$892$	0	0
$893$	− 1.28133e7i	− 0.537690i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.40454e6i	0.141279i
$898$	0	0
$899$	8.34768e6	0.344482
$900$	0	0
$901$	−4.00136e7	−1.64208
$902$	0	0
$903$	− 955392.i	− 0.0389908i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.48309e6i	0.302039i	0.988531	+	0.151019i	$0.0482556\pi$
$907$	7.48309e6i	0.302039i	−0.988531	+	0.151019i	$0.951744\pi$
$908$	0	0
$909$	−2.04068e7	−0.819155
$910$	0	0
$911$	6.63165e6	0.264744	0.132372	−	0.991200i	$-0.457741\pi$
$911$	6.63165e6	0.264744	0.132372	+	0.991200i	$0.457741\pi$
$912$	0	0
$913$	− 2.47397e6i	− 0.0982239i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.46788e7i	− 0.576457i
$918$	0	0
$919$	−1.68976e7	−0.659990	−0.329995	−	0.943983i	$-0.607047\pi$
$919$	−1.68976e7	−0.659990	−0.329995	+	0.943983i	$0.607047\pi$
$920$	0	0
$921$	9.51003e6	0.369431
$922$	0	0
$923$	9.33733e6i	0.360760i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.42741e6i	0.169219i
$928$	0	0
$929$	1.28653e7	0.489081	0.244541	−	0.969639i	$-0.421363\pi$
$929$	1.28653e7	0.489081	0.244541	+	0.969639i	$0.421363\pi$
$930$	0	0
$931$	−2.12604e7	−0.803892
$932$	0	0
$933$	9.49219e6i	0.356995i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.06887e7i	− 0.397718i	−0.980028	−	0.198859i	$-0.936276\pi$
$937$	− 1.06887e7i	− 0.397718i	0.980028	−	0.198859i	$-0.0637236\pi$
$938$	0	0
$939$	5.71766e6	0.211619
$940$	0	0
$941$	2.82455e7	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$941$	2.82455e7	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$942$	0	0
$943$	2.79684e7i	1.02421i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.70892e7i	− 0.619222i	−0.950863	−	0.309611i	$-0.899801\pi$
$947$	− 1.70892e7i	− 0.619222i	0.950863	−	0.309611i	$-0.100199\pi$
$948$	0	0
$949$	1.10894e7	0.399706
$950$	0	0
$951$	8.49849e6	0.304713
$952$	0	0
$953$	2.22259e7i	0.792735i	0.918092	+	0.396367i	$0.129729\pi$
$953$	2.22259e7i	0.792735i	−0.918092	+	0.396367i	$0.870271\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.01872e6i	0.0712519i
$958$	0	0
$959$	−2.78243e7	−0.976961
$960$	0	0
$961$	−2.26364e7	−0.790678
$962$	0	0
$963$	3.59323e7i	1.24859i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.41551e7i	0.830696i	0.909663	+	0.415348i	$0.136340\pi$
$967$	2.41551e7i	0.830696i	−0.909663	+	0.415348i	$0.863660\pi$
$968$	0	0
$969$	−7.11472e6	−0.243416
$970$	0	0
$971$	5.48313e7	1.86630	0.933149	−	0.359491i	$-0.117050\pi$
$971$	5.48313e7	1.86630	0.933149	+	0.359491i	$0.117050\pi$
$972$	0	0
$973$	2.15462e7i	0.729608i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.56612e7i	0.524915i	0.964944	+	0.262457i	$0.0845329\pi$
$977$	1.56612e7i	0.524915i	−0.964944	+	0.262457i	$0.915467\pi$
$978$	0	0
$979$	−1.50028e7	−0.500281
$980$	0	0
$981$	−8.36041e6	−0.277367
$982$	0	0
$983$	1.63420e7i	0.539412i	0.962943	+	0.269706i	$0.0869266\pi$
$983$	1.63420e7i	0.539412i	−0.962943	+	0.269706i	$0.913073\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	9.28358e6i	0.303335i
$988$	0	0
$989$	3.70214e6	0.120355
$990$	0	0
$991$	−1.37576e7	−0.444997	−0.222498	−	0.974933i	$-0.571421\pi$
$991$	−1.37576e7	−0.444997	−0.222498	+	0.974933i	$0.571421\pi$
$992$	0	0
$993$	− 1.23994e7i	− 0.399050i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.29097e7i	0.411320i	0.978624	+	0.205660i	$0.0659341\pi$
$997$	1.29097e7i	0.411320i	−0.978624	+	0.205660i	$0.934066\pi$
$998$	0	0
$999$	342160.	0.0108471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.j.49.2		2
4.3	odd	2		25.6.b.a.24.1		2
5.2	odd	4		80.6.a.e.1.1		1
5.3	odd	4		400.6.a.g.1.1		1
5.4	even	2	inner	400.6.c.j.49.1		2
12.11	even	2		225.6.b.e.199.2		2
15.2	even	4		720.6.a.a.1.1		1
20.3	even	4		25.6.a.a.1.1		1
20.7	even	4		5.6.a.a.1.1	✓	1
20.19	odd	2		25.6.b.a.24.2		2
40.27	even	4		320.6.a.j.1.1		1
40.37	odd	4		320.6.a.g.1.1		1
60.23	odd	4		225.6.a.f.1.1		1
60.47	odd	4		45.6.a.b.1.1		1
60.59	even	2		225.6.b.e.199.1		2
140.27	odd	4		245.6.a.b.1.1		1
220.87	odd	4		605.6.a.a.1.1		1
260.207	even	4		845.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.a.a.1.1	✓	1	20.7	even	4
25.6.a.a.1.1		1	20.3	even	4
25.6.b.a.24.1		2	4.3	odd	2
25.6.b.a.24.2		2	20.19	odd	2
45.6.a.b.1.1		1	60.47	odd	4
80.6.a.e.1.1		1	5.2	odd	4
225.6.a.f.1.1		1	60.23	odd	4
225.6.b.e.199.1		2	60.59	even	2
225.6.b.e.199.2		2	12.11	even	2
245.6.a.b.1.1		1	140.27	odd	4
320.6.a.g.1.1		1	40.37	odd	4
320.6.a.j.1.1		1	40.27	even	4
400.6.a.g.1.1		1	5.3	odd	4
400.6.c.j.49.1		2	5.4	even	2	inner
400.6.c.j.49.2		2	1.1	even	1	trivial
605.6.a.a.1.1		1	220.87	odd	4
720.6.a.a.1.1		1	15.2	even	4
845.6.a.b.1.1		1	260.207	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} +192.000i q^{7} +227.000 q^{9} +148.000 q^{11} +286.000i q^{13} +1678.00i q^{17} +1060.00 q^{19} -768.000 q^{21} -2976.00i q^{23} +1880.00i q^{27} +3410.00 q^{29} +2448.00 q^{31} +592.000i q^{33} -182.000i q^{37} -1144.00 q^{39} -9398.00 q^{41} +1244.00i q^{43} -12088.0i q^{47} -20057.0 q^{49} -6712.00 q^{51} +23846.0i q^{53} +4240.00i q^{57} -20020.0 q^{59} +32302.0 q^{61} +43584.0i q^{63} +60972.0i q^{67} +11904.0 q^{69} +32648.0 q^{71} -38774.0i q^{73} +28416.0i q^{77} -33360.0 q^{79} +47641.0 q^{81} -16716.0i q^{83} +13640.0i q^{87} -101370. q^{89} -54912.0 q^{91} +9792.00i q^{93} +119038. i q^{97} +33596.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 454 q^{9} + 296 q^{11} + 2120 q^{19} - 1536 q^{21} + 6820 q^{29} + 4896 q^{31} - 2288 q^{39} - 18796 q^{41} - 40114 q^{49} - 13424 q^{51} - 40040 q^{59} + 64604 q^{61} + 23808 q^{69} + 65296 q^{71}+ \cdots + 67192 q^{99}+O(q^{100}) \) 2 * q + 454 * q^9 + 296 * q^11 + 2120 * q^19 - 1536 * q^21 + 6820 * q^29 + 4896 * q^31 - 2288 * q^39 - 18796 * q^41 - 40114 * q^49 - 13424 * q^51 - 40040 * q^59 + 64604 * q^61 + 23808 * q^69 + 65296 * q^71 - 66720 * q^79 + 95282 * q^81 - 202740 * q^89 - 109824 * q^91 + 67192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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