LMFDB - Embedded newform 400.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(1,400)]

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, 0]))

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{3} -2.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-5.00000 q^{3} -2.00000 q^{7} -2.00000 q^{9} -39.0000 q^{11} +84.0000 q^{13} -61.0000 q^{17} -151.000 q^{19} +10.0000 q^{21} +58.0000 q^{23} +145.000 q^{27} +192.000 q^{29} +18.0000 q^{31} +195.000 q^{33} -138.000 q^{37} -420.000 q^{39} +229.000 q^{41} +164.000 q^{43} +212.000 q^{47} -339.000 q^{49} +305.000 q^{51} +578.000 q^{53} +755.000 q^{57} +336.000 q^{59} +858.000 q^{61} +4.00000 q^{63} +209.000 q^{67} -290.000 q^{69} +780.000 q^{71} -403.000 q^{73} +78.0000 q^{77} +230.000 q^{79} -671.000 q^{81} +1293.00 q^{83} -960.000 q^{87} -1369.00 q^{89} -168.000 q^{91} -90.0000 q^{93} +382.000 q^{97} +78.0000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−5.00000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−5.00000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	0	0
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	−39.0000	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$11$	−39.0000	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$12$	0	0
$13$	84.0000	1.79211	0.896054	−	0.443945i	$-0.146421\pi$
$13$	84.0000	1.79211	0.896054	+	0.443945i	$0.146421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−61.0000	−0.870275	−0.435137	−	0.900364i	$-0.643300\pi$
$17$	−61.0000	−0.870275	−0.435137	+	0.900364i	$0.643300\pi$
$18$	0	0
$19$	−151.000	−1.82325	−0.911626	−	0.411021i	$-0.865172\pi$
$19$	−151.000	−1.82325	−0.911626	+	0.411021i	$0.865172\pi$
$20$	0	0
$21$	10.0000	0.103913
$22$	0	0
$23$	58.0000	0.525819	0.262909	−	0.964821i	$-0.415318\pi$
$23$	58.0000	0.525819	0.262909	+	0.964821i	$0.415318\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	145.000	1.03353
$28$	0	0
$29$	192.000	1.22943	0.614716	−	0.788749i	$-0.289271\pi$
$29$	192.000	1.22943	0.614716	+	0.788749i	$0.289271\pi$
$30$	0	0
$31$	18.0000	0.104287	0.0521435	−	0.998640i	$-0.483395\pi$
$31$	18.0000	0.104287	0.0521435	+	0.998640i	$0.483395\pi$
$32$	0	0
$33$	195.000	1.02864
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−138.000	−0.613164	−0.306582	−	0.951844i	$-0.599185\pi$
$37$	−138.000	−0.613164	−0.306582	+	0.951844i	$0.599185\pi$
$38$	0	0
$39$	−420.000	−1.72446
$40$	0	0
$41$	229.000	0.872288	0.436144	−	0.899877i	$-0.356344\pi$
$41$	229.000	0.872288	0.436144	+	0.899877i	$0.356344\pi$
$42$	0	0
$43$	164.000	0.581622	0.290811	−	0.956780i	$-0.406075\pi$
$43$	164.000	0.581622	0.290811	+	0.956780i	$0.406075\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	212.000	0.657944	0.328972	−	0.944340i	$-0.393298\pi$
$47$	212.000	0.657944	0.328972	+	0.944340i	$0.393298\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	305.000	0.837422
$52$	0	0
$53$	578.000	1.49801	0.749004	−	0.662566i	$-0.230532\pi$
$53$	578.000	1.49801	0.749004	+	0.662566i	$0.230532\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	755.000	1.75442
$58$	0	0
$59$	336.000	0.741415	0.370707	−	0.928750i	$-0.379115\pi$
$59$	336.000	0.741415	0.370707	+	0.928750i	$0.379115\pi$
$60$	0	0
$61$	858.000	1.80091	0.900456	−	0.434947i	$-0.143233\pi$
$61$	858.000	1.80091	0.900456	+	0.434947i	$0.143233\pi$
$62$	0	0
$63$	4.00000	0.00799925
$64$	0	0
$65$	0	0
$66$	0	0
$67$	209.000	0.381096	0.190548	−	0.981678i	$-0.438974\pi$
$67$	209.000	0.381096	0.190548	+	0.981678i	$0.438974\pi$
$68$	0	0
$69$	−290.000	−0.505970
$70$	0	0
$71$	780.000	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$71$	780.000	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$72$	0	0
$73$	−403.000	−0.646131	−0.323066	−	0.946377i	$-0.604713\pi$
$73$	−403.000	−0.646131	−0.323066	+	0.946377i	$0.604713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	78.0000	0.115441
$78$	0	0
$79$	230.000	0.327557	0.163779	−	0.986497i	$-0.447632\pi$
$79$	230.000	0.327557	0.163779	+	0.986497i	$0.447632\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	1293.00	1.70994	0.854971	−	0.518676i	$-0.173575\pi$
$83$	1293.00	1.70994	0.854971	+	0.518676i	$0.173575\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−960.000	−1.18302
$88$	0	0
$89$	−1369.00	−1.63049	−0.815246	−	0.579115i	$-0.803398\pi$
$89$	−1369.00	−1.63049	−0.815246	+	0.579115i	$0.803398\pi$
$90$	0	0
$91$	−168.000	−0.193530
$92$	0	0
$93$	−90.0000	−0.100350
$94$	0	0
$95$	0	0
$96$	0	0
$97$	382.000	0.399858	0.199929	−	0.979810i	$-0.435929\pi$
$97$	382.000	0.399858	0.199929	+	0.979810i	$0.435929\pi$
$98$	0	0
$99$	78.0000	0.0791848
$100$	0	0
$101$	−794.000	−0.782237	−0.391119	−	0.920340i	$-0.627912\pi$
$101$	−794.000	−0.782237	−0.391119	+	0.920340i	$0.627912\pi$
$102$	0	0
$103$	1348.00	1.28954	0.644769	−	0.764378i	$-0.276954\pi$
$103$	1348.00	1.28954	0.644769	+	0.764378i	$0.276954\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	775.000	0.700206	0.350103	−	0.936711i	$-0.386147\pi$
$107$	775.000	0.700206	0.350103	+	0.936711i	$0.386147\pi$
$108$	0	0
$109$	446.000	0.391918	0.195959	−	0.980612i	$-0.437218\pi$
$109$	446.000	0.391918	0.195959	+	0.980612i	$0.437218\pi$
$110$	0	0
$111$	690.000	0.590017
$112$	0	0
$113$	−231.000	−0.192307	−0.0961533	−	0.995367i	$-0.530654\pi$
$113$	−231.000	−0.192307	−0.0961533	+	0.995367i	$0.530654\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−168.000	−0.132749
$118$	0	0
$119$	122.000	0.0939809
$120$	0	0
$121$	190.000	0.142750
$122$	0	0
$123$	−1145.00	−0.839359
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2386.00	−1.66711	−0.833556	−	0.552435i	$-0.813699\pi$
$127$	−2386.00	−1.66711	−0.833556	+	0.552435i	$0.813699\pi$
$128$	0	0
$129$	−820.000	−0.559666
$130$	0	0
$131$	−2452.00	−1.63536	−0.817680	−	0.575673i	$-0.804740\pi$
$131$	−2452.00	−1.63536	−0.817680	+	0.575673i	$0.804740\pi$
$132$	0	0
$133$	302.000	0.196893
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1125.00	−0.701571	−0.350786	−	0.936456i	$-0.614085\pi$
$137$	−1125.00	−0.701571	−0.350786	+	0.936456i	$0.614085\pi$
$138$	0	0
$139$	1377.00	0.840256	0.420128	−	0.907465i	$-0.361985\pi$
$139$	1377.00	0.840256	0.420128	+	0.907465i	$0.361985\pi$
$140$	0	0
$141$	−1060.00	−0.633107
$142$	0	0
$143$	−3276.00	−1.91575
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1695.00	0.951029
$148$	0	0
$149$	1920.00	1.05565	0.527827	−	0.849352i	$-0.323007\pi$
$149$	1920.00	1.05565	0.527827	+	0.849352i	$0.323007\pi$
$150$	0	0
$151$	−1854.00	−0.999181	−0.499591	−	0.866262i	$-0.666516\pi$
$151$	−1854.00	−0.999181	−0.499591	+	0.866262i	$0.666516\pi$
$152$	0	0
$153$	122.000	0.0644648
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−634.000	−0.322285	−0.161142	−	0.986931i	$-0.551518\pi$
$157$	−634.000	−0.322285	−0.161142	+	0.986931i	$0.551518\pi$
$158$	0	0
$159$	−2890.00	−1.44146
$160$	0	0
$161$	−116.000	−0.0567831
$162$	0	0
$163$	−103.000	−0.0494944	−0.0247472	−	0.999694i	$-0.507878\pi$
$163$	−103.000	−0.0494944	−0.0247472	+	0.999694i	$0.507878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−44.0000	−0.0203882	−0.0101941	−	0.999948i	$-0.503245\pi$
$167$	−44.0000	−0.0203882	−0.0101941	+	0.999948i	$0.503245\pi$
$168$	0	0
$169$	4859.00	2.21165
$170$	0	0
$171$	302.000	0.135056
$172$	0	0
$173$	−1128.00	−0.495724	−0.247862	−	0.968795i	$-0.579728\pi$
$173$	−1128.00	−0.495724	−0.247862	+	0.968795i	$0.579728\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1680.00	−0.713427
$178$	0	0
$179$	2245.00	0.937426	0.468713	−	0.883351i	$-0.344718\pi$
$179$	2245.00	0.937426	0.468713	+	0.883351i	$0.344718\pi$
$180$	0	0
$181$	3050.00	1.25251	0.626256	−	0.779617i	$-0.284586\pi$
$181$	3050.00	1.25251	0.626256	+	0.779617i	$0.284586\pi$
$182$	0	0
$183$	−4290.00	−1.73293
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2379.00	0.930319
$188$	0	0
$189$	−290.000	−0.111611
$190$	0	0
$191$	4222.00	1.59944	0.799720	−	0.600373i	$-0.204981\pi$
$191$	4222.00	1.59944	0.799720	+	0.600373i	$0.204981\pi$
$192$	0	0
$193$	−3357.00	−1.25203	−0.626016	−	0.779810i	$-0.715316\pi$
$193$	−3357.00	−1.25203	−0.626016	+	0.779810i	$0.715316\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−166.000	−0.0600356	−0.0300178	−	0.999549i	$-0.509556\pi$
$197$	−166.000	−0.0600356	−0.0300178	+	0.999549i	$0.509556\pi$
$198$	0	0
$199$	−3372.00	−1.20118	−0.600590	−	0.799557i	$-0.705068\pi$
$199$	−3372.00	−1.20118	−0.600590	+	0.799557i	$0.705068\pi$
$200$	0	0
$201$	−1045.00	−0.366710
$202$	0	0
$203$	−384.000	−0.132766
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−116.000	−0.0389496
$208$	0	0
$209$	5889.00	1.94905
$210$	0	0
$211$	−5601.00	−1.82743	−0.913717	−	0.406350i	$-0.866801\pi$
$211$	−5601.00	−1.82743	−0.913717	+	0.406350i	$0.866801\pi$
$212$	0	0
$213$	−3900.00	−1.25457
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−36.0000	−0.0112619
$218$	0	0
$219$	2015.00	0.621740
$220$	0	0
$221$	−5124.00	−1.55963
$222$	0	0
$223$	−828.000	−0.248641	−0.124321	−	0.992242i	$-0.539675\pi$
$223$	−828.000	−0.248641	−0.124321	+	0.992242i	$0.539675\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2388.00	0.698225	0.349113	−	0.937081i	$-0.386483\pi$
$227$	2388.00	0.698225	0.349113	+	0.937081i	$0.386483\pi$
$228$	0	0
$229$	−2844.00	−0.820685	−0.410342	−	0.911932i	$-0.634591\pi$
$229$	−2844.00	−0.820685	−0.410342	+	0.911932i	$0.634591\pi$
$230$	0	0
$231$	−390.000	−0.111083
$232$	0	0
$233$	5962.00	1.67632	0.838162	−	0.545421i	$-0.183630\pi$
$233$	5962.00	1.67632	0.838162	+	0.545421i	$0.183630\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1150.00	−0.315192
$238$	0	0
$239$	4320.00	1.16919	0.584597	−	0.811324i	$-0.301252\pi$
$239$	4320.00	1.16919	0.584597	+	0.811324i	$0.301252\pi$
$240$	0	0
$241$	3857.00	1.03092	0.515459	−	0.856914i	$-0.327621\pi$
$241$	3857.00	1.03092	0.515459	+	0.856914i	$0.327621\pi$
$242$	0	0
$243$	−560.000	−0.147835
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12684.0	−3.26746
$248$	0	0
$249$	−6465.00	−1.64539
$250$	0	0
$251$	287.000	0.0721724	0.0360862	−	0.999349i	$-0.488511\pi$
$251$	287.000	0.0721724	0.0360862	+	0.999349i	$0.488511\pi$
$252$	0	0
$253$	−2262.00	−0.562098
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2130.00	0.516987	0.258494	−	0.966013i	$-0.416774\pi$
$257$	2130.00	0.516987	0.258494	+	0.966013i	$0.416774\pi$
$258$	0	0
$259$	276.000	0.0662155
$260$	0	0
$261$	−384.000	−0.0910690
$262$	0	0
$263$	3066.00	0.718850	0.359425	−	0.933174i	$-0.382973\pi$
$263$	3066.00	0.718850	0.359425	+	0.933174i	$0.382973\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6845.00	1.56894
$268$	0	0
$269$	−3744.00	−0.848609	−0.424304	−	0.905520i	$-0.639481\pi$
$269$	−3744.00	−0.848609	−0.424304	+	0.905520i	$0.639481\pi$
$270$	0	0
$271$	3346.00	0.750019	0.375009	−	0.927021i	$-0.377640\pi$
$271$	3346.00	0.750019	0.375009	+	0.927021i	$0.377640\pi$
$272$	0	0
$273$	840.000	0.186224
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7040.00	1.52705	0.763525	−	0.645779i	$-0.223467\pi$
$277$	7040.00	1.52705	0.763525	+	0.645779i	$0.223467\pi$
$278$	0	0
$279$	−36.0000	−0.00772496
$280$	0	0
$281$	−3010.00	−0.639009	−0.319505	−	0.947585i	$-0.603516\pi$
$281$	−3010.00	−0.639009	−0.319505	+	0.947585i	$0.603516\pi$
$282$	0	0
$283$	6001.00	1.26050	0.630252	−	0.776391i	$-0.282952\pi$
$283$	6001.00	1.26050	0.630252	+	0.776391i	$0.282952\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−458.000	−0.0941982
$288$	0	0
$289$	−1192.00	−0.242622
$290$	0	0
$291$	−1910.00	−0.384764
$292$	0	0
$293$	4802.00	0.957460	0.478730	−	0.877962i	$-0.341097\pi$
$293$	4802.00	0.957460	0.478730	+	0.877962i	$0.341097\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5655.00	−1.10484
$298$	0	0
$299$	4872.00	0.942325
$300$	0	0
$301$	−328.000	−0.0628093
$302$	0	0
$303$	3970.00	0.752708
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6149.00	−1.14313	−0.571567	−	0.820556i	$-0.693664\pi$
$307$	−6149.00	−1.14313	−0.571567	+	0.820556i	$0.693664\pi$
$308$	0	0
$309$	−6740.00	−1.24086
$310$	0	0
$311$	878.000	0.160086	0.0800431	−	0.996791i	$-0.474494\pi$
$311$	878.000	0.160086	0.0800431	+	0.996791i	$0.474494\pi$
$312$	0	0
$313$	−4042.00	−0.729928	−0.364964	−	0.931022i	$-0.618919\pi$
$313$	−4042.00	−0.729928	−0.364964	+	0.931022i	$0.618919\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3844.00	0.681074	0.340537	−	0.940231i	$-0.389391\pi$
$317$	3844.00	0.681074	0.340537	+	0.940231i	$0.389391\pi$
$318$	0	0
$319$	−7488.00	−1.31426
$320$	0	0
$321$	−3875.00	−0.673774
$322$	0	0
$323$	9211.00	1.58673
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2230.00	−0.377123
$328$	0	0
$329$	−424.000	−0.0710513
$330$	0	0
$331$	2717.00	0.451178	0.225589	−	0.974223i	$-0.427569\pi$
$331$	2717.00	0.451178	0.225589	+	0.974223i	$0.427569\pi$
$332$	0	0
$333$	276.000	0.0454195
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1603.00	−0.259113	−0.129556	−	0.991572i	$-0.541355\pi$
$337$	−1603.00	−0.259113	−0.129556	+	0.991572i	$0.541355\pi$
$338$	0	0
$339$	1155.00	0.185047
$340$	0	0
$341$	−702.000	−0.111482
$342$	0	0
$343$	1364.00	0.214720
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11607.0	1.79567	0.897833	−	0.440335i	$-0.145140\pi$
$347$	11607.0	1.79567	0.897833	+	0.440335i	$0.145140\pi$
$348$	0	0
$349$	4030.00	0.618112	0.309056	−	0.951044i	$-0.399987\pi$
$349$	4030.00	0.618112	0.309056	+	0.951044i	$0.399987\pi$
$350$	0	0
$351$	12180.0	1.85219
$352$	0	0
$353$	2106.00	0.317538	0.158769	−	0.987316i	$-0.449247\pi$
$353$	2106.00	0.317538	0.158769	+	0.987316i	$0.449247\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−610.000	−0.0904331
$358$	0	0
$359$	−7394.00	−1.08702	−0.543510	−	0.839402i	$-0.682905\pi$
$359$	−7394.00	−1.08702	−0.543510	+	0.839402i	$0.682905\pi$
$360$	0	0
$361$	15942.0	2.32425
$362$	0	0
$363$	−950.000	−0.137361
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6940.00	0.987098	0.493549	−	0.869718i	$-0.335699\pi$
$367$	6940.00	0.987098	0.493549	+	0.869718i	$0.335699\pi$
$368$	0	0
$369$	−458.000	−0.0646139
$370$	0	0
$371$	−1156.00	−0.161770
$372$	0	0
$373$	7486.00	1.03917	0.519585	−	0.854419i	$-0.326087\pi$
$373$	7486.00	1.03917	0.519585	+	0.854419i	$0.326087\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16128.0	2.20327
$378$	0	0
$379$	−1285.00	−0.174158	−0.0870792	−	0.996201i	$-0.527753\pi$
$379$	−1285.00	−0.174158	−0.0870792	+	0.996201i	$0.527753\pi$
$380$	0	0
$381$	11930.0	1.60418
$382$	0	0
$383$	−9622.00	−1.28371	−0.641855	−	0.766826i	$-0.721835\pi$
$383$	−9622.00	−1.28371	−0.641855	+	0.766826i	$0.721835\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−328.000	−0.0430831
$388$	0	0
$389$	1974.00	0.257290	0.128645	−	0.991691i	$-0.458937\pi$
$389$	1974.00	0.257290	0.128645	+	0.991691i	$0.458937\pi$
$390$	0	0
$391$	−3538.00	−0.457607
$392$	0	0
$393$	12260.0	1.57363
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8084.00	−1.02198	−0.510988	−	0.859588i	$-0.670720\pi$
$397$	−8084.00	−1.02198	−0.510988	+	0.859588i	$0.670720\pi$
$398$	0	0
$399$	−1510.00	−0.189460
$400$	0	0
$401$	−5667.00	−0.705727	−0.352863	−	0.935675i	$-0.614792\pi$
$401$	−5667.00	−0.705727	−0.352863	+	0.935675i	$0.614792\pi$
$402$	0	0
$403$	1512.00	0.186894
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5382.00	0.655469
$408$	0	0
$409$	−4835.00	−0.584536	−0.292268	−	0.956336i	$-0.594410\pi$
$409$	−4835.00	−0.584536	−0.292268	+	0.956336i	$0.594410\pi$
$410$	0	0
$411$	5625.00	0.675087
$412$	0	0
$413$	−672.000	−0.0800653
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6885.00	−0.808537
$418$	0	0
$419$	−4619.00	−0.538551	−0.269276	−	0.963063i	$-0.586784\pi$
$419$	−4619.00	−0.538551	−0.269276	+	0.963063i	$0.586784\pi$
$420$	0	0
$421$	7476.00	0.865458	0.432729	−	0.901524i	$-0.357551\pi$
$421$	7476.00	0.865458	0.432729	+	0.901524i	$0.357551\pi$
$422$	0	0
$423$	−424.000	−0.0487366
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1716.00	−0.194480
$428$	0	0
$429$	16380.0	1.84344
$430$	0	0
$431$	−7810.00	−0.872841	−0.436420	−	0.899743i	$-0.643754\pi$
$431$	−7810.00	−0.872841	−0.436420	+	0.899743i	$0.643754\pi$
$432$	0	0
$433$	−2029.00	−0.225191	−0.112595	−	0.993641i	$-0.535916\pi$
$433$	−2029.00	−0.225191	−0.112595	+	0.993641i	$0.535916\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8758.00	−0.958700
$438$	0	0
$439$	−3208.00	−0.348769	−0.174384	−	0.984678i	$-0.555794\pi$
$439$	−3208.00	−0.348769	−0.174384	+	0.984678i	$0.555794\pi$
$440$	0	0
$441$	678.000	0.0732102
$442$	0	0
$443$	−13227.0	−1.41859	−0.709293	−	0.704914i	$-0.750986\pi$
$443$	−13227.0	−1.41859	−0.709293	+	0.704914i	$0.750986\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9600.00	−1.01580
$448$	0	0
$449$	3617.00	0.380171	0.190086	−	0.981768i	$-0.439123\pi$
$449$	3617.00	0.380171	0.190086	+	0.981768i	$0.439123\pi$
$450$	0	0
$451$	−8931.00	−0.932471
$452$	0	0
$453$	9270.00	0.961463
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6215.00	0.636161	0.318080	−	0.948064i	$-0.396962\pi$
$457$	6215.00	0.636161	0.318080	+	0.948064i	$0.396962\pi$
$458$	0	0
$459$	−8845.00	−0.899454
$460$	0	0
$461$	−7108.00	−0.718118	−0.359059	−	0.933315i	$-0.616902\pi$
$461$	−7108.00	−0.718118	−0.359059	+	0.933315i	$0.616902\pi$
$462$	0	0
$463$	3364.00	0.337664	0.168832	−	0.985645i	$-0.446000\pi$
$463$	3364.00	0.337664	0.168832	+	0.985645i	$0.446000\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18964.0	1.87912	0.939560	−	0.342384i	$-0.111234\pi$
$467$	18964.0	1.87912	0.939560	+	0.342384i	$0.111234\pi$
$468$	0	0
$469$	−418.000	−0.0411545
$470$	0	0
$471$	3170.00	0.310119
$472$	0	0
$473$	−6396.00	−0.621751
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1156.00	−0.110964
$478$	0	0
$479$	10926.0	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$479$	10926.0	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$480$	0	0
$481$	−11592.0	−1.09886
$482$	0	0
$483$	580.000	0.0546396
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4350.00	0.404758	0.202379	−	0.979307i	$-0.435133\pi$
$487$	4350.00	0.404758	0.202379	+	0.979307i	$0.435133\pi$
$488$	0	0
$489$	515.000	0.0476260
$490$	0	0
$491$	1324.00	0.121693	0.0608465	−	0.998147i	$-0.480620\pi$
$491$	1324.00	0.121693	0.0608465	+	0.998147i	$0.480620\pi$
$492$	0	0
$493$	−11712.0	−1.06994
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1560.00	−0.140796
$498$	0	0
$499$	−9068.00	−0.813506	−0.406753	−	0.913538i	$-0.633339\pi$
$499$	−9068.00	−0.813506	−0.406753	+	0.913538i	$0.633339\pi$
$500$	0	0
$501$	220.000	0.0196185
$502$	0	0
$503$	19836.0	1.75834	0.879169	−	0.476511i	$-0.158099\pi$
$503$	19836.0	1.75834	0.879169	+	0.476511i	$0.158099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−24295.0	−2.12816
$508$	0	0
$509$	2682.00	0.233551	0.116776	−	0.993158i	$-0.462744\pi$
$509$	2682.00	0.233551	0.116776	+	0.993158i	$0.462744\pi$
$510$	0	0
$511$	806.000	0.0697756
$512$	0	0
$513$	−21895.0	−1.88438
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8268.00	−0.703339
$518$	0	0
$519$	5640.00	0.477011
$520$	0	0
$521$	−3035.00	−0.255213	−0.127606	−	0.991825i	$-0.540729\pi$
$521$	−3035.00	−0.255213	−0.127606	+	0.991825i	$0.540729\pi$
$522$	0	0
$523$	−7701.00	−0.643865	−0.321932	−	0.946763i	$-0.604332\pi$
$523$	−7701.00	−0.643865	−0.321932	+	0.946763i	$0.604332\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1098.00	−0.0907583
$528$	0	0
$529$	−8803.00	−0.723514
$530$	0	0
$531$	−672.000	−0.0549196
$532$	0	0
$533$	19236.0	1.56323
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11225.0	−0.902038
$538$	0	0
$539$	13221.0	1.05653
$540$	0	0
$541$	−18112.0	−1.43936	−0.719682	−	0.694304i	$-0.755712\pi$
$541$	−18112.0	−1.43936	−0.719682	+	0.694304i	$0.755712\pi$
$542$	0	0
$543$	−15250.0	−1.20523
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−19541.0	−1.52745	−0.763723	−	0.645544i	$-0.776631\pi$
$547$	−19541.0	−1.52745	−0.763723	+	0.645544i	$0.776631\pi$
$548$	0	0
$549$	−1716.00	−0.133401
$550$	0	0
$551$	−28992.0	−2.24156
$552$	0	0
$553$	−460.000	−0.0353729
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13508.0	1.02756	0.513781	−	0.857921i	$-0.328244\pi$
$557$	13508.0	1.02756	0.513781	+	0.857921i	$0.328244\pi$
$558$	0	0
$559$	13776.0	1.04233
$560$	0	0
$561$	−11895.0	−0.895200
$562$	0	0
$563$	8712.00	0.652162	0.326081	−	0.945342i	$-0.394272\pi$
$563$	8712.00	0.652162	0.326081	+	0.945342i	$0.394272\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1342.00	0.0993981
$568$	0	0
$569$	−9623.00	−0.708993	−0.354497	−	0.935057i	$-0.615348\pi$
$569$	−9623.00	−0.708993	−0.354497	+	0.935057i	$0.615348\pi$
$570$	0	0
$571$	−604.000	−0.0442673	−0.0221336	−	0.999755i	$-0.507046\pi$
$571$	−604.000	−0.0442673	−0.0221336	+	0.999755i	$0.507046\pi$
$572$	0	0
$573$	−21110.0	−1.53906
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3629.00	0.261832	0.130916	−	0.991393i	$-0.458208\pi$
$577$	3629.00	0.261832	0.130916	+	0.991393i	$0.458208\pi$
$578$	0	0
$579$	16785.0	1.20477
$580$	0	0
$581$	−2586.00	−0.184656
$582$	0	0
$583$	−22542.0	−1.60136
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9219.00	−0.648226	−0.324113	−	0.946018i	$-0.605066\pi$
$587$	−9219.00	−0.648226	−0.324113	+	0.946018i	$0.605066\pi$
$588$	0	0
$589$	−2718.00	−0.190141
$590$	0	0
$591$	830.000	0.0577693
$592$	0	0
$593$	−19111.0	−1.32343	−0.661716	−	0.749755i	$-0.730171\pi$
$593$	−19111.0	−1.32343	−0.661716	+	0.749755i	$0.730171\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16860.0	1.15584
$598$	0	0
$599$	17086.0	1.16547	0.582734	−	0.812663i	$-0.301983\pi$
$599$	17086.0	1.16547	0.582734	+	0.812663i	$0.301983\pi$
$600$	0	0
$601$	9035.00	0.613220	0.306610	−	0.951835i	$-0.400805\pi$
$601$	9035.00	0.613220	0.306610	+	0.951835i	$0.400805\pi$
$602$	0	0
$603$	−418.000	−0.0282293
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14784.0	0.988573	0.494287	−	0.869299i	$-0.335429\pi$
$607$	14784.0	0.988573	0.494287	+	0.869299i	$0.335429\pi$
$608$	0	0
$609$	1920.00	0.127754
$610$	0	0
$611$	17808.0	1.17911
$612$	0	0
$613$	−17846.0	−1.17585	−0.587923	−	0.808917i	$-0.700054\pi$
$613$	−17846.0	−1.17585	−0.587923	+	0.808917i	$0.700054\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11618.0	0.758060	0.379030	−	0.925384i	$-0.376258\pi$
$617$	11618.0	0.758060	0.379030	+	0.925384i	$0.376258\pi$
$618$	0	0
$619$	9556.00	0.620498	0.310249	−	0.950655i	$-0.399588\pi$
$619$	9556.00	0.620498	0.310249	+	0.950655i	$0.399588\pi$
$620$	0	0
$621$	8410.00	0.543449
$622$	0	0
$623$	2738.00	0.176076
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−29445.0	−1.87547
$628$	0	0
$629$	8418.00	0.533621
$630$	0	0
$631$	19394.0	1.22355	0.611777	−	0.791030i	$-0.290455\pi$
$631$	19394.0	1.22355	0.611777	+	0.791030i	$0.290455\pi$
$632$	0	0
$633$	28005.0	1.75845
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−28476.0	−1.77121
$638$	0	0
$639$	−1560.00	−0.0965769
$640$	0	0
$641$	12138.0	0.747929	0.373964	−	0.927443i	$-0.377998\pi$
$641$	12138.0	0.747929	0.373964	+	0.927443i	$0.377998\pi$
$642$	0	0
$643$	−27036.0	−1.65816	−0.829079	−	0.559131i	$-0.811135\pi$
$643$	−27036.0	−1.65816	−0.829079	+	0.559131i	$0.811135\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17556.0	1.06677	0.533383	−	0.845874i	$-0.320920\pi$
$647$	17556.0	1.06677	0.533383	+	0.845874i	$0.320920\pi$
$648$	0	0
$649$	−13104.0	−0.792569
$650$	0	0
$651$	180.000	0.0108368
$652$	0	0
$653$	−17262.0	−1.03448	−0.517239	−	0.855841i	$-0.673040\pi$
$653$	−17262.0	−1.03448	−0.517239	+	0.855841i	$0.673040\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	806.000	0.0478616
$658$	0	0
$659$	−10517.0	−0.621675	−0.310838	−	0.950463i	$-0.600610\pi$
$659$	−10517.0	−0.621675	−0.310838	+	0.950463i	$0.600610\pi$
$660$	0	0
$661$	1408.00	0.0828515	0.0414258	−	0.999142i	$-0.486810\pi$
$661$	1408.00	0.0828515	0.0414258	+	0.999142i	$0.486810\pi$
$662$	0	0
$663$	25620.0	1.50075
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11136.0	0.646458
$668$	0	0
$669$	4140.00	0.239255
$670$	0	0
$671$	−33462.0	−1.92517
$672$	0	0
$673$	−9626.00	−0.551345	−0.275672	−	0.961252i	$-0.588900\pi$
$673$	−9626.00	−0.551345	−0.275672	+	0.961252i	$0.588900\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28464.0	1.61589	0.807947	−	0.589255i	$-0.200579\pi$
$677$	28464.0	1.61589	0.807947	+	0.589255i	$0.200579\pi$
$678$	0	0
$679$	−764.000	−0.0431806
$680$	0	0
$681$	−11940.0	−0.671868
$682$	0	0
$683$	3963.00	0.222020	0.111010	−	0.993819i	$-0.464591\pi$
$683$	3963.00	0.222020	0.111010	+	0.993819i	$0.464591\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14220.0	0.789704
$688$	0	0
$689$	48552.0	2.68459
$690$	0	0
$691$	31781.0	1.74965	0.874824	−	0.484442i	$-0.160977\pi$
$691$	31781.0	1.74965	0.874824	+	0.484442i	$0.160977\pi$
$692$	0	0
$693$	−156.000	−0.00855115
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13969.0	−0.759130
$698$	0	0
$699$	−29810.0	−1.61304
$700$	0	0
$701$	−28004.0	−1.50884	−0.754420	−	0.656392i	$-0.772082\pi$
$701$	−28004.0	−1.50884	−0.754420	+	0.656392i	$0.772082\pi$
$702$	0	0
$703$	20838.0	1.11795
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1588.00	0.0844737
$708$	0	0
$709$	−35228.0	−1.86603	−0.933015	−	0.359837i	$-0.882832\pi$
$709$	−35228.0	−1.86603	−0.933015	+	0.359837i	$0.882832\pi$
$710$	0	0
$711$	−460.000	−0.0242635
$712$	0	0
$713$	1044.00	0.0548361
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21600.0	−1.12506
$718$	0	0
$719$	8658.00	0.449081	0.224540	−	0.974465i	$-0.427912\pi$
$719$	8658.00	0.449081	0.224540	+	0.974465i	$0.427912\pi$
$720$	0	0
$721$	−2696.00	−0.139257
$722$	0	0
$723$	−19285.0	−0.992001
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5728.00	−0.292214	−0.146107	−	0.989269i	$-0.546674\pi$
$727$	−5728.00	−0.292214	−0.146107	+	0.989269i	$0.546674\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	−10004.0	−0.506171
$732$	0	0
$733$	21460.0	1.08137	0.540684	−	0.841226i	$-0.318165\pi$
$733$	21460.0	1.08137	0.540684	+	0.841226i	$0.318165\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8151.00	−0.407389
$738$	0	0
$739$	−29164.0	−1.45171	−0.725856	−	0.687847i	$-0.758556\pi$
$739$	−29164.0	−1.45171	−0.725856	+	0.687847i	$0.758556\pi$
$740$	0	0
$741$	63420.0	3.14412
$742$	0	0
$743$	−29478.0	−1.45551	−0.727754	−	0.685838i	$-0.759436\pi$
$743$	−29478.0	−1.45551	−0.727754	+	0.685838i	$0.759436\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2586.00	−0.126662
$748$	0	0
$749$	−1550.00	−0.0756152
$750$	0	0
$751$	−576.000	−0.0279874	−0.0139937	−	0.999902i	$-0.504454\pi$
$751$	−576.000	−0.0279874	−0.0139937	+	0.999902i	$0.504454\pi$
$752$	0	0
$753$	−1435.00	−0.0694480
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2880.00	0.138277	0.0691383	−	0.997607i	$-0.477975\pi$
$757$	2880.00	0.138277	0.0691383	+	0.997607i	$0.477975\pi$
$758$	0	0
$759$	11310.0	0.540879
$760$	0	0
$761$	20789.0	0.990277	0.495138	−	0.868814i	$-0.335117\pi$
$761$	20789.0	0.990277	0.495138	+	0.868814i	$0.335117\pi$
$762$	0	0
$763$	−892.000	−0.0423232
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28224.0	1.32870
$768$	0	0
$769$	26421.0	1.23897	0.619484	−	0.785010i	$-0.287342\pi$
$769$	26421.0	1.23897	0.619484	+	0.785010i	$0.287342\pi$
$770$	0	0
$771$	−10650.0	−0.497471
$772$	0	0
$773$	32504.0	1.51240	0.756202	−	0.654339i	$-0.227053\pi$
$773$	32504.0	1.51240	0.756202	+	0.654339i	$0.227053\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1380.00	−0.0637159
$778$	0	0
$779$	−34579.0	−1.59040
$780$	0	0
$781$	−30420.0	−1.39374
$782$	0	0
$783$	27840.0	1.27065
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−996.000	−0.0451125	−0.0225563	−	0.999746i	$-0.507180\pi$
$787$	−996.000	−0.0451125	−0.0225563	+	0.999746i	$0.507180\pi$
$788$	0	0
$789$	−15330.0	−0.691714
$790$	0	0
$791$	462.000	0.0207672
$792$	0	0
$793$	72072.0	3.22743
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15134.0	0.672615	0.336307	−	0.941752i	$-0.390822\pi$
$797$	15134.0	0.672615	0.336307	+	0.941752i	$0.390822\pi$
$798$	0	0
$799$	−12932.0	−0.572592
$800$	0	0
$801$	2738.00	0.120777
$802$	0	0
$803$	15717.0	0.690711
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18720.0	0.816574
$808$	0	0
$809$	−36942.0	−1.60545	−0.802727	−	0.596347i	$-0.796618\pi$
$809$	−36942.0	−1.60545	−0.802727	+	0.596347i	$0.796618\pi$
$810$	0	0
$811$	11748.0	0.508666	0.254333	−	0.967117i	$-0.418144\pi$
$811$	11748.0	0.508666	0.254333	+	0.967117i	$0.418144\pi$
$812$	0	0
$813$	−16730.0	−0.721706
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24764.0	−1.06044
$818$	0	0
$819$	336.000	0.0143355
$820$	0	0
$821$	−1198.00	−0.0509263	−0.0254631	−	0.999676i	$-0.508106\pi$
$821$	−1198.00	−0.0509263	−0.0254631	+	0.999676i	$0.508106\pi$
$822$	0	0
$823$	6788.00	0.287503	0.143751	−	0.989614i	$-0.454083\pi$
$823$	6788.00	0.287503	0.143751	+	0.989614i	$0.454083\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33011.0	−1.38803	−0.694017	−	0.719958i	$-0.744161\pi$
$827$	−33011.0	−1.38803	−0.694017	+	0.719958i	$0.744161\pi$
$828$	0	0
$829$	17732.0	0.742892	0.371446	−	0.928454i	$-0.378862\pi$
$829$	17732.0	0.742892	0.371446	+	0.928454i	$0.378862\pi$
$830$	0	0
$831$	−35200.0	−1.46940
$832$	0	0
$833$	20679.0	0.860126
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2610.00	0.107784
$838$	0	0
$839$	−8480.00	−0.348942	−0.174471	−	0.984662i	$-0.555821\pi$
$839$	−8480.00	−0.348942	−0.174471	+	0.984662i	$0.555821\pi$
$840$	0	0
$841$	12475.0	0.511501
$842$	0	0
$843$	15050.0	0.614887
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−380.000	−0.0154155
$848$	0	0
$849$	−30005.0	−1.21292
$850$	0	0
$851$	−8004.00	−0.322413
$852$	0	0
$853$	30014.0	1.20476	0.602380	−	0.798210i	$-0.294219\pi$
$853$	30014.0	1.20476	0.602380	+	0.798210i	$0.294219\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21643.0	0.862673	0.431337	−	0.902191i	$-0.358042\pi$
$857$	21643.0	0.862673	0.431337	+	0.902191i	$0.358042\pi$
$858$	0	0
$859$	−2799.00	−0.111177	−0.0555883	−	0.998454i	$-0.517703\pi$
$859$	−2799.00	−0.111177	−0.0555883	+	0.998454i	$0.517703\pi$
$860$	0	0
$861$	2290.00	0.0906423
$862$	0	0
$863$	19384.0	0.764588	0.382294	−	0.924041i	$-0.375134\pi$
$863$	19384.0	0.764588	0.382294	+	0.924041i	$0.375134\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5960.00	0.233463
$868$	0	0
$869$	−8970.00	−0.350157
$870$	0	0
$871$	17556.0	0.682965
$872$	0	0
$873$	−764.000	−0.0296191
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5132.00	0.197600	0.0988001	−	0.995107i	$-0.468500\pi$
$877$	5132.00	0.197600	0.0988001	+	0.995107i	$0.468500\pi$
$878$	0	0
$879$	−24010.0	−0.921316
$880$	0	0
$881$	4430.00	0.169410	0.0847052	−	0.996406i	$-0.473005\pi$
$881$	4430.00	0.169410	0.0847052	+	0.996406i	$0.473005\pi$
$882$	0	0
$883$	−24317.0	−0.926764	−0.463382	−	0.886159i	$-0.653364\pi$
$883$	−24317.0	−0.926764	−0.463382	+	0.886159i	$0.653364\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26100.0	0.987996	0.493998	−	0.869463i	$-0.335535\pi$
$887$	26100.0	0.987996	0.493998	+	0.869463i	$0.335535\pi$
$888$	0	0
$889$	4772.00	0.180031
$890$	0	0
$891$	26169.0	0.983944
$892$	0	0
$893$	−32012.0	−1.19960
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−24360.0	−0.906752
$898$	0	0
$899$	3456.00	0.128214
$900$	0	0
$901$	−35258.0	−1.30368
$902$	0	0
$903$	1640.00	0.0604383
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−24356.0	−0.891651	−0.445826	−	0.895120i	$-0.647090\pi$
$907$	−24356.0	−0.891651	−0.445826	+	0.895120i	$0.647090\pi$
$908$	0	0
$909$	1588.00	0.0579435
$910$	0	0
$911$	29900.0	1.08741	0.543705	−	0.839276i	$-0.317021\pi$
$911$	29900.0	1.08741	0.543705	+	0.839276i	$0.317021\pi$
$912$	0	0
$913$	−50427.0	−1.82792
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4904.00	0.176602
$918$	0	0
$919$	34838.0	1.25049	0.625245	−	0.780429i	$-0.284999\pi$
$919$	34838.0	1.25049	0.625245	+	0.780429i	$0.284999\pi$
$920$	0	0
$921$	30745.0	1.09998
$922$	0	0
$923$	65520.0	2.33653
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2696.00	−0.0955213
$928$	0	0
$929$	26334.0	0.930022	0.465011	−	0.885305i	$-0.346050\pi$
$929$	26334.0	0.930022	0.465011	+	0.885305i	$0.346050\pi$
$930$	0	0
$931$	51189.0	1.80199
$932$	0	0
$933$	−4390.00	−0.154043
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30949.0	1.07904	0.539520	−	0.841973i	$-0.318606\pi$
$937$	30949.0	1.07904	0.539520	+	0.841973i	$0.318606\pi$
$938$	0	0
$939$	20210.0	0.702373
$940$	0	0
$941$	25276.0	0.875637	0.437818	−	0.899063i	$-0.355751\pi$
$941$	25276.0	0.875637	0.437818	+	0.899063i	$0.355751\pi$
$942$	0	0
$943$	13282.0	0.458665
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1216.00	0.0417262	0.0208631	−	0.999782i	$-0.493359\pi$
$947$	1216.00	0.0417262	0.0208631	+	0.999782i	$0.493359\pi$
$948$	0	0
$949$	−33852.0	−1.15794
$950$	0	0
$951$	−19220.0	−0.655364
$952$	0	0
$953$	−6033.00	−0.205066	−0.102533	−	0.994730i	$-0.532695\pi$
$953$	−6033.00	−0.205066	−0.102533	+	0.994730i	$0.532695\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	37440.0	1.26464
$958$	0	0
$959$	2250.00	0.0757626
$960$	0	0
$961$	−29467.0	−0.989124
$962$	0	0
$963$	−1550.00	−0.0518671
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41792.0	1.38980	0.694902	−	0.719105i	$-0.255448\pi$
$967$	41792.0	1.38980	0.694902	+	0.719105i	$0.255448\pi$
$968$	0	0
$969$	−46055.0	−1.52683
$970$	0	0
$971$	2105.00	0.0695702	0.0347851	−	0.999395i	$-0.488925\pi$
$971$	2105.00	0.0695702	0.0347851	+	0.999395i	$0.488925\pi$
$972$	0	0
$973$	−2754.00	−0.0907391
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30119.0	0.986277	0.493138	−	0.869951i	$-0.335850\pi$
$977$	30119.0	0.986277	0.493138	+	0.869951i	$0.335850\pi$
$978$	0	0
$979$	53391.0	1.74299
$980$	0	0
$981$	−892.000	−0.0290310
$982$	0	0
$983$	−18438.0	−0.598251	−0.299126	−	0.954214i	$-0.596695\pi$
$983$	−18438.0	−0.598251	−0.299126	+	0.954214i	$0.596695\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2120.00	0.0683691
$988$	0	0
$989$	9512.00	0.305828
$990$	0	0
$991$	2230.00	0.0714816	0.0357408	−	0.999361i	$-0.488621\pi$
$991$	2230.00	0.0714816	0.0357408	+	0.999361i	$0.488621\pi$
$992$	0	0
$993$	−13585.0	−0.434146
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6804.00	−0.216133	−0.108067	−	0.994144i	$-0.534466\pi$
$997$	−6804.00	−0.216133	−0.108067	+	0.994144i	$0.534466\pi$
$998$	0	0
$999$	−20010.0	−0.633722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.a.f.1.1		1
4.3	odd	2		200.4.a.h.1.1	yes	1
5.2	odd	4		400.4.c.g.49.2		2
5.3	odd	4		400.4.c.g.49.1		2
5.4	even	2		400.4.a.q.1.1		1
8.3	odd	2		1600.4.a.m.1.1		1
8.5	even	2		1600.4.a.bo.1.1		1
12.11	even	2		1800.4.a.t.1.1		1
20.3	even	4		200.4.c.d.49.2		2
20.7	even	4		200.4.c.d.49.1		2
20.19	odd	2		200.4.a.c.1.1	✓	1
40.19	odd	2		1600.4.a.bn.1.1		1
40.29	even	2		1600.4.a.n.1.1		1
60.23	odd	4		1800.4.f.c.649.1		2
60.47	odd	4		1800.4.f.c.649.2		2
60.59	even	2		1800.4.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.4.a.c.1.1	✓	1	20.19	odd	2
200.4.a.h.1.1	yes	1	4.3	odd	2
200.4.c.d.49.1		2	20.7	even	4
200.4.c.d.49.2		2	20.3	even	4
400.4.a.f.1.1		1	1.1	even	1	trivial
400.4.a.q.1.1		1	5.4	even	2
400.4.c.g.49.1		2	5.3	odd	4
400.4.c.g.49.2		2	5.2	odd	4
1600.4.a.m.1.1		1	8.3	odd	2
1600.4.a.n.1.1		1	40.29	even	2
1600.4.a.bn.1.1		1	40.19	odd	2
1600.4.a.bo.1.1		1	8.5	even	2
1800.4.a.p.1.1		1	60.59	even	2
1800.4.a.t.1.1		1	12.11	even	2
1800.4.f.c.649.1		2	60.23	odd	4
1800.4.f.c.649.2		2	60.47	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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