LMFDB - Embedded newform 320.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("320.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.a (trivial)

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:	yes
Analytic conductor:	$51.3228223402$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} -25.0000 q^{5} +192.000 q^{7} -227.000 q^{9} +148.000 q^{11} -286.000 q^{13} -100.000 q^{15} -1678.00 q^{17} -1060.00 q^{19} +768.000 q^{21} +2976.00 q^{23} +625.000 q^{25} -1880.00 q^{27} +3410.00 q^{29} -2448.00 q^{31} +592.000 q^{33} -4800.00 q^{35} -182.000 q^{37} -1144.00 q^{39} -9398.00 q^{41} +1244.00 q^{43} +5675.00 q^{45} -12088.0 q^{47} +20057.0 q^{49} -6712.00 q^{51} -23846.0 q^{53} -3700.00 q^{55} -4240.00 q^{57} +20020.0 q^{59} -32302.0 q^{61} -43584.0 q^{63} +7150.00 q^{65} -60972.0 q^{67} +11904.0 q^{69} -32648.0 q^{71} -38774.0 q^{73} +2500.00 q^{75} +28416.0 q^{77} -33360.0 q^{79} +47641.0 q^{81} -16716.0 q^{83} +41950.0 q^{85} +13640.0 q^{87} +101370. q^{89} -54912.0 q^{91} -9792.00 q^{93} +26500.0 q^{95} -119038. q^{97} -33596.0 q^{99} +O(q^{100})$

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
$3$	4.00000	0.256600	0.128300	−	0.991735i	$-0.459048\pi$
$3$	4.00000	0.256600	0.128300	+	0.991735i	$0.459048\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	192.000	1.48100	0.740502	−	0.672054i	$-0.234588\pi$
$7$	192.000	1.48100	0.740502	+	0.672054i	$0.234588\pi$
$8$	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
$13$	−286.000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−286.000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	0	0
$15$	−100.000	−0.114755
$16$	0	0
$17$	−1678.00	−1.40822	−0.704109	−	0.710092i	$-0.748653\pi$
$17$	−1678.00	−1.40822	−0.704109	+	0.710092i	$0.748653\pi$
$18$	0	0
$19$	−1060.00	−0.673631	−0.336815	−	0.941571i	$-0.609350\pi$
$19$	−1060.00	−0.673631	−0.336815	+	0.941571i	$0.609350\pi$
$20$	0	0
$21$	768.000	0.380026
$22$	0	0
$23$	2976.00	1.17304	0.586521	−	0.809934i	$-0.300497\pi$
$23$	2976.00	1.17304	0.586521	+	0.809934i	$0.300497\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−1880.00	−0.496305
$28$	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	$-0.573467\pi$
$31$	−2448.00	−0.457517	−0.228758	+	0.973483i	$0.573467\pi$
$32$	0	0
$33$	592.000	0.0946317
$34$	0	0
$35$	−4800.00	−0.662325
$36$	0	0
$37$	−182.000	−0.0218558	−0.0109279	−	0.999940i	$-0.503479\pi$
$37$	−182.000	−0.0218558	−0.0109279	+	0.999940i	$0.503479\pi$
$38$	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
$43$	1244.00	0.102600	0.0513002	−	0.998683i	$-0.483663\pi$
$43$	1244.00	0.102600	0.0513002	+	0.998683i	$0.483663\pi$
$44$	0	0
$45$	5675.00	0.417767
$46$	0	0
$47$	−12088.0	−0.798196	−0.399098	−	0.916908i	$-0.630677\pi$
$47$	−12088.0	−0.798196	−0.399098	+	0.916908i	$0.630677\pi$
$48$	0	0
$49$	20057.0	1.19337
$50$	0	0
$51$	−6712.00	−0.361349
$52$	0	0
$53$	−23846.0	−1.16607	−0.583037	−	0.812446i	$-0.698136\pi$
$53$	−23846.0	−1.16607	−0.583037	+	0.812446i	$0.698136\pi$
$54$	0	0
$55$	−3700.00	−0.164928
$56$	0	0
$57$	−4240.00	−0.172854
$58$	0	0
$59$	20020.0	0.748745	0.374373	−	0.927278i	$-0.377858\pi$
$59$	20020.0	0.748745	0.374373	+	0.927278i	$0.377858\pi$
$60$	0	0
$61$	−32302.0	−1.11149	−0.555744	−	0.831353i	$-0.687567\pi$
$61$	−32302.0	−1.11149	−0.555744	+	0.831353i	$0.687567\pi$
$62$	0	0
$63$	−43584.0	−1.38349
$64$	0	0
$65$	7150.00	0.209905
$66$	0	0
$67$	−60972.0	−1.65937	−0.829685	−	0.558231i	$-0.811480\pi$
$67$	−60972.0	−1.65937	−0.829685	+	0.558231i	$0.811480\pi$
$68$	0	0
$69$	11904.0	0.301003
$70$	0	0
$71$	−32648.0	−0.768618	−0.384309	−	0.923204i	$-0.625560\pi$
$71$	−32648.0	−0.768618	−0.384309	+	0.923204i	$0.625560\pi$
$72$	0	0
$73$	−38774.0	−0.851596	−0.425798	−	0.904818i	$-0.640007\pi$
$73$	−38774.0	−0.851596	−0.425798	+	0.904818i	$0.640007\pi$
$74$	0	0
$75$	2500.00	0.0513200
$76$	0	0
$77$	28416.0	0.546180
$78$	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
$83$	−16716.0	−0.266340	−0.133170	−	0.991093i	$-0.542516\pi$
$83$	−16716.0	−0.266340	−0.133170	+	0.991093i	$0.542516\pi$
$84$	0	0
$85$	41950.0	0.629774
$86$	0	0
$87$	13640.0	0.193204
$88$	0	0
$89$	101370.	1.35655	0.678273	−	0.734810i	$-0.262729\pi$
$89$	101370.	1.35655	0.678273	+	0.734810i	$0.262729\pi$
$90$	0	0
$91$	−54912.0	−0.695126
$92$	0	0
$93$	−9792.00	−0.117399
$94$	0	0
$95$	26500.0	0.301257
$96$	0	0
$97$	−119038.	−1.28457	−0.642283	−	0.766468i	$-0.722013\pi$
$97$	−119038.	−1.28457	−0.642283	+	0.766468i	$0.722013\pi$
$98$	0	0
$99$	−33596.0	−0.344508
$100$	0	0
$101$	89898.0	0.876893	0.438446	−	0.898757i	$-0.355529\pi$
$101$	89898.0	0.876893	0.438446	+	0.898757i	$0.355529\pi$
$102$	0	0
$103$	−19504.0	−0.181147	−0.0905734	−	0.995890i	$-0.528870\pi$
$103$	−19504.0	−0.181147	−0.0905734	+	0.995890i	$0.528870\pi$
$104$	0	0
$105$	−19200.0	−0.169953
$106$	0	0
$107$	−158292.	−1.33659	−0.668297	−	0.743895i	$-0.732976\pi$
$107$	−158292.	−1.33659	−0.668297	+	0.743895i	$0.732976\pi$
$108$	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
$113$	11186.0	0.0824098	0.0412049	−	0.999151i	$-0.486880\pi$
$113$	11186.0	0.0824098	0.0412049	+	0.999151i	$0.486880\pi$
$114$	0	0
$115$	−74400.0	−0.524600
$116$	0	0
$117$	64922.0	0.438457
$118$	0	0
$119$	−322176.	−2.08557
$120$	0	0
$121$	−139147.	−0.863993
$122$	0	0
$123$	−37592.0	−0.224044
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	70552.0	0.388150	0.194075	−	0.980987i	$-0.437829\pi$
$127$	70552.0	0.388150	0.194075	+	0.980987i	$0.437829\pi$
$128$	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.	−0.997650
$134$	0	0
$135$	47000.0	0.221954
$136$	0	0
$137$	−144918.	−0.659661	−0.329831	−	0.944040i	$-0.606992\pi$
$137$	−144918.	−0.659661	−0.329831	+	0.944040i	$0.606992\pi$
$138$	0	0
$139$	−112220.	−0.492644	−0.246322	−	0.969188i	$-0.579222\pi$
$139$	−112220.	−0.492644	−0.246322	+	0.969188i	$0.579222\pi$
$140$	0	0
$141$	−48352.0	−0.204817
$142$	0	0
$143$	−42328.0	−0.173096
$144$	0	0
$145$	−85250.0	−0.336724
$146$	0	0
$147$	80228.0	0.306219
$148$	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	$-0.793615\pi$
$151$	−446648.	−1.59413	−0.797064	+	0.603895i	$0.793615\pi$
$152$	0	0
$153$	380906.	1.31550
$154$	0	0
$155$	61200.0	0.204608
$156$	0	0
$157$	262258.	0.849141	0.424570	−	0.905395i	$-0.360425\pi$
$157$	262258.	0.849141	0.424570	+	0.905395i	$0.360425\pi$
$158$	0	0
$159$	−95384.0	−0.299215
$160$	0	0
$161$	571392.	1.73728
$162$	0	0
$163$	154564.	0.455658	0.227829	−	0.973701i	$-0.426837\pi$
$163$	154564.	0.455658	0.227829	+	0.973701i	$0.426837\pi$
$164$	0	0
$165$	−14800.0	−0.0423206
$166$	0	0
$167$	396672.	1.10063	0.550314	−	0.834958i	$-0.314508\pi$
$167$	396672.	1.10063	0.550314	+	0.834958i	$0.314508\pi$
$168$	0	0
$169$	−289497.	−0.779700
$170$	0	0
$171$	240620.	0.629276
$172$	0	0
$173$	573474.	1.45680	0.728398	−	0.685155i	$-0.240265\pi$
$173$	573474.	1.45680	0.728398	+	0.685155i	$0.240265\pi$
$174$	0	0
$175$	120000.	0.296201
$176$	0	0
$177$	80080.0	0.192128
$178$	0	0
$179$	594460.	1.38672	0.693362	−	0.720589i	$-0.256129\pi$
$179$	594460.	1.38672	0.693362	+	0.720589i	$0.256129\pi$
$180$	0	0
$181$	107098.	0.242988	0.121494	−	0.992592i	$-0.461231\pi$
$181$	107098.	0.242988	0.121494	+	0.992592i	$0.461231\pi$
$182$	0	0
$183$	−129208.	−0.285208
$184$	0	0
$185$	4550.00	0.00977422
$186$	0	0
$187$	−248344.	−0.519337
$188$	0	0
$189$	−360960.	−0.735029
$190$	0	0
$191$	469552.	0.931323	0.465661	−	0.884963i	$-0.345816\pi$
$191$	469552.	0.931323	0.465661	+	0.884963i	$0.345816\pi$
$192$	0	0
$193$	52706.0	0.101851	0.0509257	−	0.998702i	$-0.483783\pi$
$193$	52706.0	0.101851	0.0509257	+	0.998702i	$0.483783\pi$
$194$	0	0
$195$	28600.0	0.0538616
$196$	0	0
$197$	−455862.	−0.836889	−0.418444	−	0.908242i	$-0.637425\pi$
$197$	−455862.	−0.836889	−0.418444	+	0.908242i	$0.637425\pi$
$198$	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.	1.11510
$204$	0	0
$205$	234950.	0.390473
$206$	0	0
$207$	−675552.	−1.09580
$208$	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.	−0.197228
$214$	0	0
$215$	−31100.0	−0.0458843
$216$	0	0
$217$	−470016.	−0.677584
$218$	0	0
$219$	−155096.	−0.218520
$220$	0	0
$221$	479908.	0.660963
$222$	0	0
$223$	1.12158e6	1.51031	0.755156	−	0.655545i	$-0.227561\pi$
$223$	1.12158e6	1.51031	0.755156	+	0.655545i	$0.227561\pi$
$224$	0	0
$225$	−141875.	−0.186831
$226$	0	0
$227$	23348.0	0.0300736	0.0150368	−	0.999887i	$-0.495213\pi$
$227$	23348.0	0.0300736	0.0150368	+	0.999887i	$0.495213\pi$
$228$	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.	−0.585667	−0.292834	−	0.956163i	$-0.594598\pi$
$233$	−485334.	−0.585667	−0.292834	+	0.956163i	$0.594598\pi$
$234$	0	0
$235$	302200.	0.356964
$236$	0	0
$237$	−133440.	−0.154317
$238$	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.	0.703331
$244$	0	0
$245$	−501425.	−0.533692
$246$	0	0
$247$	303160.	0.316176
$248$	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.	0.432607
$254$	0	0
$255$	167800.	0.161600
$256$	0	0
$257$	1.30624e6	1.23365	0.616823	−	0.787102i	$-0.288419\pi$
$257$	1.30624e6	1.23365	0.616823	+	0.787102i	$0.288419\pi$
$258$	0	0
$259$	−34944.0	−0.0323685
$260$	0	0
$261$	−774070.	−0.703362
$262$	0	0
$263$	2.12834e6	1.89736	0.948682	−	0.316231i	$-0.102417\pi$
$263$	2.12834e6	1.89736	0.948682	+	0.316231i	$0.102417\pi$
$264$	0	0
$265$	596150.	0.521484
$266$	0	0
$267$	405480.	0.348090
$268$	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	$-0.512278\pi$
$271$	−93248.0	−0.0771288	−0.0385644	+	0.999256i	$0.512278\pi$
$272$	0	0
$273$	−219648.	−0.178370
$274$	0	0
$275$	92500.0	0.0737581
$276$	0	0
$277$	110298.	0.0863711	0.0431855	−	0.999067i	$-0.486249\pi$
$277$	110298.	0.0863711	0.0431855	+	0.999067i	$0.486249\pi$
$278$	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.	0.246332	0.123166	−	0.992386i	$-0.460695\pi$
$283$	331884.	0.246332	0.123166	+	0.992386i	$0.460695\pi$
$284$	0	0
$285$	106000.	0.0773025
$286$	0	0
$287$	−1.80442e6	−1.29310
$288$	0	0
$289$	1.39583e6	0.983076
$290$	0	0
$291$	−476152.	−0.329620
$292$	0	0
$293$	−2.19481e6	−1.49358	−0.746788	−	0.665063i	$-0.768405\pi$
$293$	−2.19481e6	−1.49358	−0.746788	+	0.665063i	$0.768405\pi$
$294$	0	0
$295$	−500500.	−0.334849
$296$	0	0
$297$	−278240.	−0.183033
$298$	0	0
$299$	−851136.	−0.550581
$300$	0	0
$301$	238848.	0.151952
$302$	0	0
$303$	359592.	0.225011
$304$	0	0
$305$	807550.	0.497073
$306$	0	0
$307$	2.37751e6	1.43971	0.719857	−	0.694123i	$-0.244207\pi$
$307$	2.37751e6	1.43971	0.719857	+	0.694123i	$0.244207\pi$
$308$	0	0
$309$	−78016.0	−0.0464823
$310$	0	0
$311$	−2.37305e6	−1.39125	−0.695626	−	0.718405i	$-0.744873\pi$
$311$	−2.37305e6	−1.39125	−0.695626	+	0.718405i	$0.744873\pi$
$312$	0	0
$313$	−1.42941e6	−0.824702	−0.412351	−	0.911025i	$-0.635292\pi$
$313$	−1.42941e6	−0.824702	−0.412351	+	0.911025i	$0.635292\pi$
$314$	0	0
$315$	1.08960e6	0.618715
$316$	0	0
$317$	−2.12462e6	−1.18750	−0.593750	−	0.804650i	$-0.702353\pi$
$317$	−2.12462e6	−1.18750	−0.593750	+	0.804650i	$0.702353\pi$
$318$	0	0
$319$	504680.	0.277677
$320$	0	0
$321$	−633168.	−0.342970
$322$	0	0
$323$	1.77868e6	0.948618
$324$	0	0
$325$	−178750.	−0.0938723
$326$	0	0
$327$	−147320.	−0.0761890
$328$	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
$333$	41314.0	0.0204168
$334$	0	0
$335$	1.52430e6	0.742093
$336$	0	0
$337$	2.40008e6	1.15120	0.575601	−	0.817731i	$-0.304768\pi$
$337$	2.40008e6	1.15120	0.575601	+	0.817731i	$0.304768\pi$
$338$	0	0
$339$	44744.0	0.0211464
$340$	0	0
$341$	−362304.	−0.168728
$342$	0	0
$343$	624000.	0.286384
$344$	0	0
$345$	−297600.	−0.134612
$346$	0	0
$347$	−1.77741e6	−0.792436	−0.396218	−	0.918156i	$-0.629678\pi$
$347$	−1.77741e6	−0.792436	−0.396218	+	0.918156i	$0.629678\pi$
$348$	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.	−0.282700	−0.141350	−	0.989960i	$-0.545144\pi$
$353$	−661854.	−0.282700	−0.141350	+	0.989960i	$0.545144\pi$
$354$	0	0
$355$	816200.	0.343737
$356$	0	0
$357$	−1.28870e6	−0.535159
$358$	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.	−0.221701
$364$	0	0
$365$	969350.	0.380845
$366$	0	0
$367$	−1.49993e6	−0.581307	−0.290653	−	0.956828i	$-0.593873\pi$
$367$	−1.49993e6	−0.581307	−0.290653	+	0.956828i	$0.593873\pi$
$368$	0	0
$369$	2.13335e6	0.815634
$370$	0	0
$371$	−4.57843e6	−1.72696
$372$	0	0
$373$	2.23807e6	0.832918	0.416459	−	0.909154i	$-0.363271\pi$
$373$	2.23807e6	0.832918	0.416459	+	0.909154i	$0.363271\pi$
$374$	0	0
$375$	−62500.0	−0.0229510
$376$	0	0
$377$	−975260.	−0.353400
$378$	0	0
$379$	−3.15934e6	−1.12979	−0.564896	−	0.825162i	$-0.691084\pi$
$379$	−3.15934e6	−1.12979	−0.564896	+	0.825162i	$0.691084\pi$
$380$	0	0
$381$	282208.	0.0995994
$382$	0	0
$383$	342216.	0.119207	0.0596037	−	0.998222i	$-0.481016\pi$
$383$	342216.	0.119207	0.0596037	+	0.998222i	$0.481016\pi$
$384$	0	0
$385$	−710400.	−0.244259
$386$	0	0
$387$	−282388.	−0.0958449
$388$	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.	−0.0998775
$394$	0	0
$395$	834000.	0.268951
$396$	0	0
$397$	5.45674e6	1.73763	0.868814	−	0.495138i	$-0.164883\pi$
$397$	5.45674e6	1.73763	0.868814	+	0.495138i	$0.164883\pi$
$398$	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.	0.214741
$404$	0	0
$405$	−1.19102e6	−0.360814
$406$	0	0
$407$	−26936.0	−0.00806022
$408$	0	0
$409$	−2.71207e6	−0.801664	−0.400832	−	0.916151i	$-0.631279\pi$
$409$	−2.71207e6	−0.801664	−0.400832	+	0.916151i	$0.631279\pi$
$410$	0	0
$411$	−579672.	−0.169269
$412$	0	0
$413$	3.84384e6	1.10889
$414$	0	0
$415$	417900.	0.119111
$416$	0	0
$417$	−448880.	−0.126413
$418$	0	0
$419$	−3.71746e6	−1.03445	−0.517227	−	0.855848i	$-0.673036\pi$
$419$	−3.71746e6	−1.03445	−0.517227	+	0.855848i	$0.673036\pi$
$420$	0	0
$421$	−3.55250e6	−0.976853	−0.488426	−	0.872605i	$-0.662429\pi$
$421$	−3.55250e6	−0.976853	−0.488426	+	0.872605i	$0.662429\pi$
$422$	0	0
$423$	2.74398e6	0.745640
$424$	0	0
$425$	−1.04875e6	−0.281643
$426$	0	0
$427$	−6.20198e6	−1.64612
$428$	0	0
$429$	−169312.	−0.0444165
$430$	0	0
$431$	−4.06205e6	−1.05330	−0.526650	−	0.850082i	$-0.676552\pi$
$431$	−4.06205e6	−1.05330	−0.526650	+	0.850082i	$0.676552\pi$
$432$	0	0
$433$	7.26287e6	1.86161	0.930804	−	0.365518i	$-0.119108\pi$
$433$	7.26287e6	1.86161	0.930804	+	0.365518i	$0.119108\pi$
$434$	0	0
$435$	−341000.	−0.0864035
$436$	0	0
$437$	−3.15456e6	−0.790197
$438$	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
$443$	6.51524e6	1.57733	0.788663	−	0.614826i	$-0.210774\pi$
$443$	6.51524e6	1.57733	0.788663	+	0.614826i	$0.210774\pi$
$444$	0	0
$445$	−2.53425e6	−0.606666
$446$	0	0
$447$	−1.61500e6	−0.382299
$448$	0	0
$449$	−509950.	−0.119375	−0.0596873	−	0.998217i	$-0.519010\pi$
$449$	−509950.	−0.119375	−0.0596873	+	0.998217i	$0.519010\pi$
$450$	0	0
$451$	−1.39090e6	−0.322000
$452$	0	0
$453$	−1.78659e6	−0.409053
$454$	0	0
$455$	1.37280e6	0.310870
$456$	0	0
$457$	1.22084e6	0.273444	0.136722	−	0.990609i	$-0.456343\pi$
$457$	1.22084e6	0.273444	0.136722	+	0.990609i	$0.456343\pi$
$458$	0	0
$459$	3.15464e6	0.698905
$460$	0	0
$461$	4.07210e6	0.892413	0.446207	−	0.894930i	$-0.352775\pi$
$461$	4.07210e6	0.892413	0.446207	+	0.894930i	$0.352775\pi$
$462$	0	0
$463$	2.02294e6	0.438561	0.219280	−	0.975662i	$-0.429629\pi$
$463$	2.02294e6	0.438561	0.219280	+	0.975662i	$0.429629\pi$
$464$	0	0
$465$	244800.	0.0525024
$466$	0	0
$467$	−3.25097e6	−0.689797	−0.344898	−	0.938640i	$-0.612087\pi$
$467$	−3.25097e6	−0.689797	−0.344898	+	0.938640i	$0.612087\pi$
$468$	0	0
$469$	−1.17066e7	−2.45753
$470$	0	0
$471$	1.04903e6	0.217890
$472$	0	0
$473$	184112.	0.0378381
$474$	0	0
$475$	−662500.	−0.134726
$476$	0	0
$477$	5.41304e6	1.08929
$478$	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
$483$	2.28557e6	0.445786
$484$	0	0
$485$	2.97595e6	0.574475
$486$	0	0
$487$	−8.53197e6	−1.63015	−0.815074	−	0.579357i	$-0.803304\pi$
$487$	−8.53197e6	−1.63015	−0.815074	+	0.579357i	$0.803304\pi$
$488$	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
$493$	−5.72198e6	−1.06030
$494$	0	0
$495$	839900.	0.154069
$496$	0	0
$497$	−6.26842e6	−1.13833
$498$	0	0
$499$	6.49190e6	1.16713	0.583567	−	0.812065i	$-0.301657\pi$
$499$	6.49190e6	1.16713	0.583567	+	0.812065i	$0.301657\pi$
$500$	0	0
$501$	1.58669e6	0.282421
$502$	0	0
$503$	8.61770e6	1.51870	0.759349	−	0.650684i	$-0.225518\pi$
$503$	8.61770e6	1.51870	0.759349	+	0.650684i	$0.225518\pi$
$504$	0	0
$505$	−2.24745e6	−0.392158
$506$	0	0
$507$	−1.15799e6	−0.200071
$508$	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
$513$	1.99280e6	0.334326
$514$	0	0
$515$	487600.	0.0810113
$516$	0	0
$517$	−1.78902e6	−0.294367
$518$	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
$523$	6.89452e6	1.10217	0.551087	−	0.834448i	$-0.314213\pi$
$523$	6.89452e6	1.10217	0.551087	+	0.834448i	$0.314213\pi$
$524$	0	0
$525$	480000.	0.0760051
$526$	0	0
$527$	4.10774e6	0.644283
$528$	0	0
$529$	2.42023e6	0.376026
$530$	0	0
$531$	−4.54454e6	−0.699445
$532$	0	0
$533$	2.68783e6	0.409811
$534$	0	0
$535$	3.95730e6	0.597743
$536$	0	0
$537$	2.37784e6	0.355834
$538$	0	0
$539$	2.96844e6	0.440104
$540$	0	0
$541$	−155502.	−0.0228425	−0.0114212	−	0.999935i	$-0.503636\pi$
$541$	−155502.	−0.0228425	−0.0114212	+	0.999935i	$0.503636\pi$
$542$	0	0
$543$	428392.	0.0623508
$544$	0	0
$545$	920750.	0.132785
$546$	0	0
$547$	−1.26544e7	−1.80831	−0.904157	−	0.427201i	$-0.859500\pi$
$547$	−1.26544e7	−1.80831	−0.904157	+	0.427201i	$0.859500\pi$
$548$	0	0
$549$	7.33255e6	1.03830
$550$	0	0
$551$	−3.61460e6	−0.507202
$552$	0	0
$553$	−6.40512e6	−0.890665
$554$	0	0
$555$	18200.0	0.00250807
$556$	0	0
$557$	7.07786e6	0.966638	0.483319	−	0.875444i	$-0.339431\pi$
$557$	7.07786e6	0.966638	0.483319	+	0.875444i	$0.339431\pi$
$558$	0	0
$559$	−355784.	−0.0481567
$560$	0	0
$561$	−993376.	−0.133262
$562$	0	0
$563$	−846636.	−0.112571	−0.0562854	−	0.998415i	$-0.517926\pi$
$563$	−846636.	−0.112571	−0.0562854	+	0.998415i	$0.517926\pi$
$564$	0	0
$565$	−279650.	−0.0368548
$566$	0	0
$567$	9.14707e6	1.19488
$568$	0	0
$569$	4.96041e6	0.642299	0.321149	−	0.947029i	$-0.395931\pi$
$569$	4.96041e6	0.642299	0.321149	+	0.947029i	$0.395931\pi$
$570$	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
$573$	1.87821e6	0.238978
$574$	0	0
$575$	1.86000e6	0.234608
$576$	0	0
$577$	−2.86080e6	−0.357724	−0.178862	−	0.983874i	$-0.557242\pi$
$577$	−2.86080e6	−0.357724	−0.178862	+	0.983874i	$0.557242\pi$
$578$	0	0
$579$	210824.	0.0261351
$580$	0	0
$581$	−3.20947e6	−0.394451
$582$	0	0
$583$	−3.52921e6	−0.430037
$584$	0	0
$585$	−1.62305e6	−0.196084
$586$	0	0
$587$	6.74027e6	0.807387	0.403694	−	0.914894i	$-0.367726\pi$
$587$	6.74027e6	0.807387	0.403694	+	0.914894i	$0.367726\pi$
$588$	0	0
$589$	2.59488e6	0.308197
$590$	0	0
$591$	−1.82345e6	−0.214746
$592$	0	0
$593$	−1.78609e6	−0.208578	−0.104289	−	0.994547i	$-0.533257\pi$
$593$	−1.78609e6	−0.208578	−0.104289	+	0.994547i	$0.533257\pi$
$594$	0	0
$595$	8.05440e6	0.932697
$596$	0	0
$597$	3.46000e6	0.397320
$598$	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
$603$	1.38406e7	1.55011
$604$	0	0
$605$	3.47868e6	0.386390
$606$	0	0
$607$	7.07999e6	0.779940	0.389970	−	0.920828i	$-0.372485\pi$
$607$	7.07999e6	0.779940	0.389970	+	0.920828i	$0.372485\pi$
$608$	0	0
$609$	2.61888e6	0.286136
$610$	0	0
$611$	3.45717e6	0.374643
$612$	0	0
$613$	−5.09609e6	−0.547754	−0.273877	−	0.961765i	$-0.588306\pi$
$613$	−5.09609e6	−0.547754	−0.273877	+	0.961765i	$0.588306\pi$
$614$	0	0
$615$	939800.	0.100195
$616$	0	0
$617$	−1.30003e7	−1.37480	−0.687400	−	0.726279i	$-0.741248\pi$
$617$	−1.30003e7	−1.37480	−0.687400	+	0.726279i	$0.741248\pi$
$618$	0	0
$619$	−4.84406e6	−0.508139	−0.254070	−	0.967186i	$-0.581769\pi$
$619$	−4.84406e6	−0.508139	−0.254070	+	0.967186i	$0.581769\pi$
$620$	0	0
$621$	−5.59488e6	−0.582186
$622$	0	0
$623$	1.94630e7	2.00905
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−627520.	−0.0637468
$628$	0	0
$629$	305396.	0.0307777
$630$	0	0
$631$	6.22775e6	0.622670	0.311335	−	0.950300i	$-0.399224\pi$
$631$	6.22775e6	0.622670	0.311335	+	0.950300i	$0.399224\pi$
$632$	0	0
$633$	−4.42261e6	−0.438702
$634$	0	0
$635$	−1.76380e6	−0.173586
$636$	0	0
$637$	−5.73630e6	−0.560123
$638$	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
$643$	1.74382e7	1.66332	0.831659	−	0.555287i	$-0.187391\pi$
$643$	1.74382e7	1.66332	0.831659	+	0.555287i	$0.187391\pi$
$644$	0	0
$645$	−124400.	−0.0117739
$646$	0	0
$647$	−4.25469e6	−0.399583	−0.199792	−	0.979838i	$-0.564026\pi$
$647$	−4.25469e6	−0.399583	−0.199792	+	0.979838i	$0.564026\pi$
$648$	0	0
$649$	2.96296e6	0.276130
$650$	0	0
$651$	−1.88006e6	−0.173868
$652$	0	0
$653$	−3.01085e6	−0.276316	−0.138158	−	0.990410i	$-0.544118\pi$
$653$	−3.01085e6	−0.276316	−0.138158	+	0.990410i	$0.544118\pi$
$654$	0	0
$655$	1.91130e6	0.174071
$656$	0	0
$657$	8.80170e6	0.795524
$658$	0	0
$659$	8.11462e6	0.727871	0.363936	−	0.931424i	$-0.381433\pi$
$659$	8.11462e6	0.727871	0.363936	+	0.931424i	$0.381433\pi$
$660$	0	0
$661$	−2.47370e6	−0.220213	−0.110107	−	0.993920i	$-0.535119\pi$
$661$	−2.47370e6	−0.220213	−0.110107	+	0.993920i	$0.535119\pi$
$662$	0	0
$663$	1.91963e6	0.169603
$664$	0	0
$665$	5.08800e6	0.446162
$666$	0	0
$667$	1.01482e7	0.883228
$668$	0	0
$669$	4.48630e6	0.387546
$670$	0	0
$671$	−4.78070e6	−0.409907
$672$	0	0
$673$	5.77063e6	0.491117	0.245559	−	0.969382i	$-0.421029\pi$
$673$	5.77063e6	0.491117	0.245559	+	0.969382i	$0.421029\pi$
$674$	0	0
$675$	−1.17500e6	−0.0992610
$676$	0	0
$677$	−1.67197e7	−1.40203	−0.701014	−	0.713147i	$-0.747269\pi$
$677$	−1.67197e7	−1.40203	−0.701014	+	0.713147i	$0.747269\pi$
$678$	0	0
$679$	−2.28553e7	−1.90245
$680$	0	0
$681$	93392.0	0.00771688
$682$	0	0
$683$	−7.14532e6	−0.586097	−0.293049	−	0.956098i	$-0.594670\pi$
$683$	−7.14532e6	−0.586097	−0.293049	+	0.956098i	$0.594670\pi$
$684$	0	0
$685$	3.62295e6	0.295009
$686$	0	0
$687$	2.38404e6	0.192718
$688$	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
$693$	−6.45043e6	−0.510218
$694$	0	0
$695$	2.80550e6	0.220317
$696$	0	0
$697$	1.57698e7	1.22955
$698$	0	0
$699$	−1.94134e6	−0.150282
$700$	0	0
$701$	1.60141e7	1.23086	0.615428	−	0.788193i	$-0.288983\pi$
$701$	1.60141e7	1.23086	0.615428	+	0.788193i	$0.288983\pi$
$702$	0	0
$703$	192920.	0.0147228
$704$	0	0
$705$	1.20880e6	0.0915971
$706$	0	0
$707$	1.72604e7	1.29868
$708$	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
$713$	−7.28525e6	−0.536686
$714$	0	0
$715$	1.05820e6	0.0774110
$716$	0	0
$717$	−195520.	−0.0142034
$718$	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.	−0.0315316
$724$	0	0
$725$	2.13125e6	0.150588
$726$	0	0
$727$	−1.93264e7	−1.35618	−0.678088	−	0.734981i	$-0.737191\pi$
$727$	−1.93264e7	−1.35618	−0.678088	+	0.734981i	$0.737191\pi$
$728$	0	0
$729$	−8.98715e6	−0.626330
$730$	0	0
$731$	−2.08743e6	−0.144484
$732$	0	0
$733$	−5.26197e6	−0.361733	−0.180866	−	0.983508i	$-0.557890\pi$
$733$	−5.26197e6	−0.361733	−0.180866	+	0.983508i	$0.557890\pi$
$734$	0	0
$735$	−2.00570e6	−0.136945
$736$	0	0
$737$	−9.02386e6	−0.611961
$738$	0	0
$739$	−2.82944e7	−1.90585	−0.952927	−	0.303199i	$-0.901945\pi$
$739$	−2.82944e7	−1.90585	−0.952927	+	0.303199i	$0.901945\pi$
$740$	0	0
$741$	1.21264e6	0.0811309
$742$	0	0
$743$	2.09863e7	1.39464	0.697321	−	0.716759i	$-0.254375\pi$
$743$	2.09863e7	1.39464	0.697321	+	0.716759i	$0.254375\pi$
$744$	0	0
$745$	1.00938e7	0.666288
$746$	0	0
$747$	3.79453e6	0.248804
$748$	0	0
$749$	−3.03921e7	−1.97950
$750$	0	0
$751$	−1.89668e7	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$751$	−1.89668e7	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$752$	0	0
$753$	6.57499e6	0.422579
$754$	0	0
$755$	1.11662e7	0.712915
$756$	0	0
$757$	1.08257e7	0.686617	0.343309	−	0.939223i	$-0.388452\pi$
$757$	1.08257e7	0.686617	0.343309	+	0.939223i	$0.388452\pi$
$758$	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
$763$	−7.07136e6	−0.439736
$764$	0	0
$765$	−9.52265e6	−0.588307
$766$	0	0
$767$	−5.72572e6	−0.351432
$768$	0	0
$769$	−1.57826e7	−0.962415	−0.481208	−	0.876607i	$-0.659802\pi$
$769$	−1.57826e7	−0.962415	−0.481208	+	0.876607i	$0.659802\pi$
$770$	0	0
$771$	5.22497e6	0.316554
$772$	0	0
$773$	2.44049e7	1.46902	0.734510	−	0.678598i	$-0.237412\pi$
$773$	2.44049e7	1.46902	0.734510	+	0.678598i	$0.237412\pi$
$774$	0	0
$775$	−1.53000e6	−0.0915034
$776$	0	0
$777$	−139776.	−0.00830577
$778$	0	0
$779$	9.96188e6	0.588163
$780$	0	0
$781$	−4.83190e6	−0.283459
$782$	0	0
$783$	−6.41080e6	−0.373687
$784$	0	0
$785$	−6.55645e6	−0.379747
$786$	0	0
$787$	−3.37607e7	−1.94301	−0.971505	−	0.237019i	$-0.923830\pi$
$787$	−3.37607e7	−1.94301	−0.971505	+	0.237019i	$0.923830\pi$
$788$	0	0
$789$	8.51334e6	0.486864
$790$	0	0
$791$	2.14771e6	0.122049
$792$	0	0
$793$	9.23837e6	0.521690
$794$	0	0
$795$	2.38460e6	0.133813
$796$	0	0
$797$	−2.19885e7	−1.22617	−0.613083	−	0.790019i	$-0.710071\pi$
$797$	−2.19885e7	−1.22617	−0.613083	+	0.790019i	$0.710071\pi$
$798$	0	0
$799$	2.02837e7	1.12403
$800$	0	0
$801$	−2.30110e7	−1.26723
$802$	0	0
$803$	−5.73855e6	−0.314061
$804$	0	0
$805$	−1.42848e7	−0.776935
$806$	0	0
$807$	5.76436e6	0.311578
$808$	0	0
$809$	−2.93597e7	−1.57717	−0.788587	−	0.614923i	$-0.789187\pi$
$809$	−2.93597e7	−1.57717	−0.788587	+	0.614923i	$0.789187\pi$
$810$	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.	−0.0197912
$814$	0	0
$815$	−3.86410e6	−0.203777
$816$	0	0
$817$	−1.31864e6	−0.0691148
$818$	0	0
$819$	1.24650e7	0.649357
$820$	0	0
$821$	2.71430e6	0.140540	0.0702699	−	0.997528i	$-0.477614\pi$
$821$	2.71430e6	0.140540	0.0702699	+	0.997528i	$0.477614\pi$
$822$	0	0
$823$	−1.25866e7	−0.647753	−0.323877	−	0.946099i	$-0.604986\pi$
$823$	−1.25866e7	−0.647753	−0.323877	+	0.946099i	$0.604986\pi$
$824$	0	0
$825$	370000.	0.0189263
$826$	0	0
$827$	8.72355e6	0.443537	0.221768	−	0.975099i	$-0.428817\pi$
$827$	8.72355e6	0.443537	0.221768	+	0.975099i	$0.428817\pi$
$828$	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
$833$	−3.36556e7	−1.68053
$834$	0	0
$835$	−9.91680e6	−0.492216
$836$	0	0
$837$	4.60224e6	0.227068
$838$	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.	−0.0372597
$844$	0	0
$845$	7.23742e6	0.348692
$846$	0	0
$847$	−2.67162e7	−1.27958
$848$	0	0
$849$	1.32754e6	0.0632087
$850$	0	0
$851$	−541632.	−0.0256378
$852$	0	0
$853$	2.20186e7	1.03613	0.518067	−	0.855340i	$-0.326652\pi$
$853$	2.20186e7	1.03613	0.518067	+	0.855340i	$0.326652\pi$
$854$	0	0
$855$	−6.01550e6	−0.281421
$856$	0	0
$857$	3.16676e7	1.47287	0.736434	−	0.676510i	$-0.236508\pi$
$857$	3.16676e7	1.47287	0.736434	+	0.676510i	$0.236508\pi$
$858$	0	0
$859$	−1.58064e7	−0.730886	−0.365443	−	0.930834i	$-0.619082\pi$
$859$	−1.58064e7	−0.730886	−0.365443	+	0.930834i	$0.619082\pi$
$860$	0	0
$861$	−7.21766e6	−0.331809
$862$	0	0
$863$	−1.44287e7	−0.659476	−0.329738	−	0.944072i	$-0.606960\pi$
$863$	−1.44287e7	−0.659476	−0.329738	+	0.944072i	$0.606960\pi$
$864$	0	0
$865$	−1.43368e7	−0.651499
$866$	0	0
$867$	5.58331e6	0.252257
$868$	0	0
$869$	−4.93728e6	−0.221788
$870$	0	0
$871$	1.74380e7	0.778845
$872$	0	0
$873$	2.70216e7	1.19999
$874$	0	0
$875$	−3.00000e6	−0.132465
$876$	0	0
$877$	−247902.	−0.0108838	−0.00544191	−	0.999985i	$-0.501732\pi$
$877$	−247902.	−0.0108838	−0.00544191	+	0.999985i	$0.501732\pi$
$878$	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
$883$	−4.18015e7	−1.80422	−0.902112	−	0.431503i	$-0.857984\pi$
$883$	−4.18015e7	−1.80422	−0.902112	+	0.431503i	$0.857984\pi$
$884$	0	0
$885$	−2.00200e6	−0.0859223
$886$	0	0
$887$	−2.10476e7	−0.898241	−0.449120	−	0.893471i	$-0.648263\pi$
$887$	−2.10476e7	−0.898241	−0.449120	+	0.893471i	$0.648263\pi$
$888$	0	0
$889$	1.35460e7	0.574852
$890$	0	0
$891$	7.05087e6	0.297542
$892$	0	0
$893$	1.28133e7	0.537690
$894$	0	0
$895$	−1.48615e7	−0.620162
$896$	0	0
$897$	−3.40454e6	−0.141279
$898$	0	0
$899$	−8.34768e6	−0.344482
$900$	0	0
$901$	4.00136e7	1.64208
$902$	0	0
$903$	955392.	0.0389908
$904$	0	0
$905$	−2.67745e6	−0.108668
$906$	0	0
$907$	−7.48309e6	−0.302039	−0.151019	−	0.988531i	$-0.548256\pi$
$907$	−7.48309e6	−0.302039	−0.151019	+	0.988531i	$0.548256\pi$
$908$	0	0
$909$	−2.04068e7	−0.819155
$910$	0	0
$911$	−6.63165e6	−0.264744	−0.132372	−	0.991200i	$-0.542259\pi$
$911$	−6.63165e6	−0.264744	−0.132372	+	0.991200i	$0.542259\pi$
$912$	0	0
$913$	−2.47397e6	−0.0982239
$914$	0	0
$915$	3.23020e6	0.127549
$916$	0	0
$917$	−1.46788e7	−0.576457
$918$	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
$923$	9.33733e6	0.360760
$924$	0	0
$925$	−113750.	−0.00437116
$926$	0	0
$927$	4.42741e6	0.169219
$928$	0	0
$929$	−1.28653e7	−0.489081	−0.244541	−	0.969639i	$-0.578637\pi$
$929$	−1.28653e7	−0.489081	−0.244541	+	0.969639i	$0.578637\pi$
$930$	0	0
$931$	−2.12604e7	−0.803892
$932$	0	0
$933$	−9.49219e6	−0.356995
$934$	0	0
$935$	6.20860e6	0.232255
$936$	0	0
$937$	1.06887e7	0.397718	0.198859	−	0.980028i	$-0.436276\pi$
$937$	1.06887e7	0.397718	0.198859	+	0.980028i	$0.436276\pi$
$938$	0	0
$939$	−5.71766e6	−0.211619
$940$	0	0
$941$	−2.82455e7	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$941$	−2.82455e7	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$942$	0	0
$943$	−2.79684e7	−1.02421
$944$	0	0
$945$	9.02400e6	0.328715
$946$	0	0
$947$	1.70892e7	0.619222	0.309611	−	0.950863i	$-0.399801\pi$
$947$	1.70892e7	0.619222	0.309611	+	0.950863i	$0.399801\pi$
$948$	0	0
$949$	1.10894e7	0.399706
$950$	0	0
$951$	−8.49849e6	−0.304713
$952$	0	0
$953$	2.22259e7	0.792735	0.396367	−	0.918092i	$-0.370271\pi$
$953$	2.22259e7	0.792735	0.396367	+	0.918092i	$0.370271\pi$
$954$	0	0
$955$	−1.17388e7	−0.416500
$956$	0	0
$957$	2.01872e6	0.0712519
$958$	0	0
$959$	−2.78243e7	−0.976961
$960$	0	0
$961$	−2.26364e7	−0.790678
$962$	0	0
$963$	3.59323e7	1.24859
$964$	0	0
$965$	−1.31765e6	−0.0455493
$966$	0	0
$967$	2.41551e7	0.830696	0.415348	−	0.909663i	$-0.363660\pi$
$967$	2.41551e7	0.830696	0.415348	+	0.909663i	$0.363660\pi$
$968$	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
$973$	−2.15462e7	−0.729608
$974$	0	0
$975$	−715000.	−0.0240877
$976$	0	0
$977$	−1.56612e7	−0.524915	−0.262457	−	0.964944i	$-0.584533\pi$
$977$	−1.56612e7	−0.524915	−0.262457	+	0.964944i	$0.584533\pi$
$978$	0	0
$979$	1.50028e7	0.500281
$980$	0	0
$981$	8.36041e6	0.277367
$982$	0	0
$983$	−1.63420e7	−0.539412	−0.269706	−	0.962943i	$-0.586927\pi$
$983$	−1.63420e7	−0.539412	−0.269706	+	0.962943i	$0.586927\pi$
$984$	0	0
$985$	1.13966e7	0.374268
$986$	0	0
$987$	−9.28358e6	−0.303335
$988$	0	0
$989$	3.70214e6	0.120355
$990$	0	0
$991$	1.37576e7	0.444997	0.222498	−	0.974933i	$-0.428579\pi$
$991$	1.37576e7	0.444997	0.222498	+	0.974933i	$0.428579\pi$
$992$	0	0
$993$	−1.23994e7	−0.399050
$994$	0	0
$995$	−2.16250e7	−0.692466
$996$	0	0
$997$	1.29097e7	0.411320	0.205660	−	0.978624i	$-0.434066\pi$
$997$	1.29097e7	0.411320	0.205660	+	0.978624i	$0.434066\pi$
$998$	0	0
$999$	342160.	0.0108471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.a.j.1.1		1
4.3	odd	2		320.6.a.g.1.1		1
8.3	odd	2		80.6.a.e.1.1		1
8.5	even	2		5.6.a.a.1.1	✓	1
24.5	odd	2		45.6.a.b.1.1		1
24.11	even	2		720.6.a.a.1.1		1
40.3	even	4		400.6.c.j.49.2		2
40.13	odd	4		25.6.b.a.24.1		2
40.19	odd	2		400.6.a.g.1.1		1
40.27	even	4		400.6.c.j.49.1		2
40.29	even	2		25.6.a.a.1.1		1
40.37	odd	4		25.6.b.a.24.2		2
56.13	odd	2		245.6.a.b.1.1		1
88.21	odd	2		605.6.a.a.1.1		1
104.77	even	2		845.6.a.b.1.1		1
120.29	odd	2		225.6.a.f.1.1		1
120.53	even	4		225.6.b.e.199.2		2
120.77	even	4		225.6.b.e.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.a.a.1.1	✓	1	8.5	even	2
25.6.a.a.1.1		1	40.29	even	2
25.6.b.a.24.1		2	40.13	odd	4
25.6.b.a.24.2		2	40.37	odd	4
45.6.a.b.1.1		1	24.5	odd	2
80.6.a.e.1.1		1	8.3	odd	2
225.6.a.f.1.1		1	120.29	odd	2
225.6.b.e.199.1		2	120.77	even	4
225.6.b.e.199.2		2	120.53	even	4
245.6.a.b.1.1		1	56.13	odd	2
320.6.a.g.1.1		1	4.3	odd	2
320.6.a.j.1.1		1	1.1	even	1	trivial
400.6.a.g.1.1		1	40.19	odd	2
400.6.c.j.49.1		2	40.27	even	4
400.6.c.j.49.2		2	40.3	even	4
605.6.a.a.1.1		1	88.21	odd	2
720.6.a.a.1.1		1	24.11	even	2
845.6.a.b.1.1		1	104.77	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more