LMFDB - Embedded newform 400.4.c.g.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -39.0000 q^{11} -84.0000i q^{13} -61.0000i q^{17} +151.000 q^{19} +10.0000 q^{21} -58.0000i q^{23} +145.000i q^{27} -192.000 q^{29} +18.0000 q^{31} -195.000i q^{33} -138.000i q^{37} +420.000 q^{39} +229.000 q^{41} -164.000i q^{43} +212.000i q^{47} +339.000 q^{49} +305.000 q^{51} -578.000i q^{53} +755.000i q^{57} -336.000 q^{59} +858.000 q^{61} -4.00000i q^{63} +209.000i q^{67} +290.000 q^{69} +780.000 q^{71} +403.000i q^{73} +78.0000i q^{77} -230.000 q^{79} -671.000 q^{81} -1293.00i q^{83} -960.000i q^{87} +1369.00 q^{89} -168.000 q^{91} +90.0000i q^{93} +382.000i q^{97} -78.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 78 q^{11} + 302 q^{19} + 20 q^{21} - 384 q^{29} + 36 q^{31} + 840 q^{39} + 458 q^{41} + 678 q^{49} + 610 q^{51} - 672 q^{59} + 1716 q^{61} + 580 q^{69} + 1560 q^{71} - 460 q^{79} - 1342 q^{81} + 2738 q^{89} - 336 q^{91} - 156 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 78 * q^11 + 302 * q^19 + 20 * q^21 - 384 * q^29 + 36 * q^31 + 840 * q^39 + 458 * q^41 + 678 * q^49 + 610 * q^51 - 672 * q^59 + 1716 * q^61 + 580 * q^69 + 1560 * q^71 - 460 * q^79 - 1342 * q^81 + 2738 * q^89 - 336 * q^91 - 156 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	5.00000i	0.962250i	0.876652	+	0.481125i	$0.159772\pi$
$3$	5.00000i	0.962250i	−0.876652	+	0.481125i	$0.840228\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.107990i	−0.998541	−	0.0539949i	$-0.982805\pi$
$7$	− 2.00000i	− 0.107990i	0.998541	−	0.0539949i	$-0.0171955\pi$
$8$	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
$13$	− 84.0000i	− 1.79211i	−0.443945	−	0.896054i	$-0.646421\pi$
$13$	− 84.0000i	− 1.79211i	0.443945	−	0.896054i	$-0.353579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 61.0000i	− 0.870275i	−0.900364	−	0.435137i	$-0.856700\pi$
$17$	− 61.0000i	− 0.870275i	0.900364	−	0.435137i	$-0.143300\pi$
$18$	0	0
$19$	151.000	1.82325	0.911626	−	0.411021i	$-0.134828\pi$
$19$	151.000	1.82325	0.911626	+	0.411021i	$0.134828\pi$
$20$	0	0
$21$	10.0000	0.103913
$22$	0	0
$23$	− 58.0000i	− 0.525819i	−0.964821	−	0.262909i	$-0.915318\pi$
$23$	− 58.0000i	− 0.525819i	0.964821	−	0.262909i	$-0.0846821\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	145.000i	1.03353i
$28$	0	0
$29$	−192.000	−1.22943	−0.614716	−	0.788749i	$-0.710729\pi$
$29$	−192.000	−1.22943	−0.614716	+	0.788749i	$0.710729\pi$
$30$	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
$33$	− 195.000i	− 1.02864i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 138.000i	− 0.613164i	−0.951844	−	0.306582i	$-0.900815\pi$
$37$	− 138.000i	− 0.613164i	0.951844	−	0.306582i	$-0.0991853\pi$
$38$	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
$43$	− 164.000i	− 0.581622i	−0.956780	−	0.290811i	$-0.906075\pi$
$43$	− 164.000i	− 0.581622i	0.956780	−	0.290811i	$-0.0939252\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	212.000i	0.657944i	0.944340	+	0.328972i	$0.106702\pi$
$47$	212.000i	0.657944i	−0.944340	+	0.328972i	$0.893298\pi$
$48$	0	0
$49$	339.000	0.988338
$50$	0	0
$51$	305.000	0.837422
$52$	0	0
$53$	− 578.000i	− 1.49801i	−0.662566	−	0.749004i	$-0.730532\pi$
$53$	− 578.000i	− 1.49801i	0.662566	−	0.749004i	$-0.269468\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	755.000i	1.75442i
$58$	0	0
$59$	−336.000	−0.741415	−0.370707	−	0.928750i	$-0.620885\pi$
$59$	−336.000	−0.741415	−0.370707	+	0.928750i	$0.620885\pi$
$60$	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
$63$	− 4.00000i	− 0.00799925i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	209.000i	0.381096i	0.981678	+	0.190548i	$0.0610264\pi$
$67$	209.000i	0.381096i	−0.981678	+	0.190548i	$0.938974\pi$
$68$	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
$73$	403.000i	0.646131i	0.946377	+	0.323066i	$0.104713\pi$
$73$	403.000i	0.646131i	−0.946377	+	0.323066i	$0.895287\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	78.0000i	0.115441i
$78$	0	0
$79$	−230.000	−0.327557	−0.163779	−	0.986497i	$-0.552368\pi$
$79$	−230.000	−0.327557	−0.163779	+	0.986497i	$0.552368\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	− 1293.00i	− 1.70994i	−0.518676	−	0.854971i	$-0.673575\pi$
$83$	− 1293.00i	− 1.70994i	0.518676	−	0.854971i	$-0.326425\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 960.000i	− 1.18302i
$88$	0	0
$89$	1369.00	1.63049	0.815246	−	0.579115i	$-0.196602\pi$
$89$	1369.00	1.63049	0.815246	+	0.579115i	$0.196602\pi$
$90$	0	0
$91$	−168.000	−0.193530
$92$	0	0
$93$	90.0000i	0.100350i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	382.000i	0.399858i	0.979810	+	0.199929i	$0.0640711\pi$
$97$	382.000i	0.399858i	−0.979810	+	0.199929i	$0.935929\pi$
$98$	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
$103$	− 1348.00i	− 1.28954i	−0.764378	−	0.644769i	$-0.776954\pi$
$103$	− 1348.00i	− 1.28954i	0.764378	−	0.644769i	$-0.223046\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	775.000i	0.700206i	0.936711	+	0.350103i	$0.113853\pi$
$107$	775.000i	0.700206i	−0.936711	+	0.350103i	$0.886147\pi$
$108$	0	0
$109$	−446.000	−0.391918	−0.195959	−	0.980612i	$-0.562782\pi$
$109$	−446.000	−0.391918	−0.195959	+	0.980612i	$0.562782\pi$
$110$	0	0
$111$	690.000	0.590017
$112$	0	0
$113$	231.000i	0.192307i	0.995367	+	0.0961533i	$0.0306539\pi$
$113$	231.000i	0.192307i	−0.995367	+	0.0961533i	$0.969346\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 168.000i	− 0.132749i
$118$	0	0
$119$	−122.000	−0.0939809
$120$	0	0
$121$	190.000	0.142750
$122$	0	0
$123$	1145.00i	0.839359i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2386.00i	− 1.66711i	−0.552435	−	0.833556i	$-0.686301\pi$
$127$	− 2386.00i	− 1.66711i	0.552435	−	0.833556i	$-0.313699\pi$
$128$	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
$133$	− 302.000i	− 0.196893i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1125.00i	− 0.701571i	−0.936456	−	0.350786i	$-0.885915\pi$
$137$	− 1125.00i	− 0.701571i	0.936456	−	0.350786i	$-0.114085\pi$
$138$	0	0
$139$	−1377.00	−0.840256	−0.420128	−	0.907465i	$-0.638015\pi$
$139$	−1377.00	−0.840256	−0.420128	+	0.907465i	$0.638015\pi$
$140$	0	0
$141$	−1060.00	−0.633107
$142$	0	0
$143$	3276.00i	1.91575i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1695.00i	0.951029i
$148$	0	0
$149$	−1920.00	−1.05565	−0.527827	−	0.849352i	$-0.676993\pi$
$149$	−1920.00	−1.05565	−0.527827	+	0.849352i	$0.676993\pi$
$150$	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
$153$	− 122.000i	− 0.0644648i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 634.000i	− 0.322285i	−0.986931	−	0.161142i	$-0.948482\pi$
$157$	− 634.000i	− 0.322285i	0.986931	−	0.161142i	$-0.0515178\pi$
$158$	0	0
$159$	2890.00	1.44146
$160$	0	0
$161$	−116.000	−0.0567831
$162$	0	0
$163$	103.000i	0.0494944i	0.999694	+	0.0247472i	$0.00787808\pi$
$163$	103.000i	0.0494944i	−0.999694	+	0.0247472i	$0.992122\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 44.0000i	− 0.0203882i	−0.999948	−	0.0101941i	$-0.996755\pi$
$167$	− 44.0000i	− 0.0203882i	0.999948	−	0.0101941i	$-0.00324493\pi$
$168$	0	0
$169$	−4859.00	−2.21165
$170$	0	0
$171$	302.000	0.135056
$172$	0	0
$173$	1128.00i	0.495724i	0.968795	+	0.247862i	$0.0797280\pi$
$173$	1128.00i	0.495724i	−0.968795	+	0.247862i	$0.920272\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1680.00i	− 0.713427i
$178$	0	0
$179$	−2245.00	−0.937426	−0.468713	−	0.883351i	$-0.655282\pi$
$179$	−2245.00	−0.937426	−0.468713	+	0.883351i	$0.655282\pi$
$180$	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
$183$	4290.00i	1.73293i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2379.00i	0.930319i
$188$	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
$193$	3357.00i	1.25203i	0.779810	+	0.626016i	$0.215316\pi$
$193$	3357.00i	1.25203i	−0.779810	+	0.626016i	$0.784684\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 166.000i	− 0.0600356i	−0.999549	−	0.0300178i	$-0.990444\pi$
$197$	− 166.000i	− 0.0600356i	0.999549	−	0.0300178i	$-0.00955640\pi$
$198$	0	0
$199$	3372.00	1.20118	0.600590	−	0.799557i	$-0.294932\pi$
$199$	3372.00	1.20118	0.600590	+	0.799557i	$0.294932\pi$
$200$	0	0
$201$	−1045.00	−0.366710
$202$	0	0
$203$	384.000i	0.132766i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 116.000i	− 0.0389496i
$208$	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
$213$	3900.00i	1.25457i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 36.0000i	− 0.0112619i
$218$	0	0
$219$	−2015.00	−0.621740
$220$	0	0
$221$	−5124.00	−1.55963
$222$	0	0
$223$	828.000i	0.248641i	0.992242	+	0.124321i	$0.0396751\pi$
$223$	828.000i	0.248641i	−0.992242	+	0.124321i	$0.960325\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2388.00i	0.698225i	0.937081	+	0.349113i	$0.113517\pi$
$227$	2388.00i	0.698225i	−0.937081	+	0.349113i	$0.886483\pi$
$228$	0	0
$229$	2844.00	0.820685	0.410342	−	0.911932i	$-0.365409\pi$
$229$	2844.00	0.820685	0.410342	+	0.911932i	$0.365409\pi$
$230$	0	0
$231$	−390.000	−0.111083
$232$	0	0
$233$	− 5962.00i	− 1.67632i	−0.545421	−	0.838162i	$-0.683630\pi$
$233$	− 5962.00i	− 1.67632i	0.545421	−	0.838162i	$-0.316370\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1150.00i	− 0.315192i
$238$	0	0
$239$	−4320.00	−1.16919	−0.584597	−	0.811324i	$-0.698748\pi$
$239$	−4320.00	−1.16919	−0.584597	+	0.811324i	$0.698748\pi$
$240$	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
$243$	560.000i	0.147835i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 12684.0i	− 3.26746i
$248$	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
$253$	2262.00i	0.562098i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2130.00i	0.516987i	0.966013	+	0.258494i	$0.0832261\pi$
$257$	2130.00i	0.516987i	−0.966013	+	0.258494i	$0.916774\pi$
$258$	0	0
$259$	−276.000	−0.0662155
$260$	0	0
$261$	−384.000	−0.0910690
$262$	0	0
$263$	− 3066.00i	− 0.718850i	−0.933174	−	0.359425i	$-0.882973\pi$
$263$	− 3066.00i	− 0.718850i	0.933174	−	0.359425i	$-0.117027\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6845.00i	1.56894i
$268$	0	0
$269$	3744.00	0.848609	0.424304	−	0.905520i	$-0.360519\pi$
$269$	3744.00	0.848609	0.424304	+	0.905520i	$0.360519\pi$
$270$	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
$273$	− 840.000i	− 0.186224i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7040.00i	1.52705i	0.645779	+	0.763525i	$0.276533\pi$
$277$	7040.00i	1.52705i	−0.645779	+	0.763525i	$0.723467\pi$
$278$	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
$283$	− 6001.00i	− 1.26050i	−0.776391	−	0.630252i	$-0.782952\pi$
$283$	− 6001.00i	− 1.26050i	0.776391	−	0.630252i	$-0.217048\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 458.000i	− 0.0941982i
$288$	0	0
$289$	1192.00	0.242622
$290$	0	0
$291$	−1910.00	−0.384764
$292$	0	0
$293$	− 4802.00i	− 0.957460i	−0.877962	−	0.478730i	$-0.841097\pi$
$293$	− 4802.00i	− 0.957460i	0.877962	−	0.478730i	$-0.158903\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5655.00i	− 1.10484i
$298$	0	0
$299$	−4872.00	−0.942325
$300$	0	0
$301$	−328.000	−0.0628093
$302$	0	0
$303$	− 3970.00i	− 0.752708i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6149.00i	− 1.14313i	−0.820556	−	0.571567i	$-0.806336\pi$
$307$	− 6149.00i	− 1.14313i	0.820556	−	0.571567i	$-0.193664\pi$
$308$	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
$313$	4042.00i	0.729928i	0.931022	+	0.364964i	$0.118919\pi$
$313$	4042.00i	0.729928i	−0.931022	+	0.364964i	$0.881081\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3844.00i	0.681074i	0.940231	+	0.340537i	$0.110609\pi$
$317$	3844.00i	0.681074i	−0.940231	+	0.340537i	$0.889391\pi$
$318$	0	0
$319$	7488.00	1.31426
$320$	0	0
$321$	−3875.00	−0.673774
$322$	0	0
$323$	− 9211.00i	− 1.58673i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2230.00i	− 0.377123i
$328$	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
$333$	− 276.000i	− 0.0454195i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1603.00i	− 0.259113i	−0.991572	−	0.129556i	$-0.958645\pi$
$337$	− 1603.00i	− 0.259113i	0.991572	−	0.129556i	$-0.0413553\pi$
$338$	0	0
$339$	−1155.00	−0.185047
$340$	0	0
$341$	−702.000	−0.111482
$342$	0	0
$343$	− 1364.00i	− 0.214720i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11607.0i	1.79567i	0.440335	+	0.897833i	$0.354860\pi$
$347$	11607.0i	1.79567i	−0.440335	+	0.897833i	$0.645140\pi$
$348$	0	0
$349$	−4030.00	−0.618112	−0.309056	−	0.951044i	$-0.600013\pi$
$349$	−4030.00	−0.618112	−0.309056	+	0.951044i	$0.600013\pi$
$350$	0	0
$351$	12180.0	1.85219
$352$	0	0
$353$	− 2106.00i	− 0.317538i	−0.987316	−	0.158769i	$-0.949247\pi$
$353$	− 2106.00i	− 0.317538i	0.987316	−	0.158769i	$-0.0507526\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 610.000i	− 0.0904331i
$358$	0	0
$359$	7394.00	1.08702	0.543510	−	0.839402i	$-0.317095\pi$
$359$	7394.00	1.08702	0.543510	+	0.839402i	$0.317095\pi$
$360$	0	0
$361$	15942.0	2.32425
$362$	0	0
$363$	950.000i	0.137361i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6940.00i	0.987098i	0.869718	+	0.493549i	$0.164301\pi$
$367$	6940.00i	0.987098i	−0.869718	+	0.493549i	$0.835699\pi$
$368$	0	0
$369$	458.000	0.0646139
$370$	0	0
$371$	−1156.00	−0.161770
$372$	0	0
$373$	− 7486.00i	− 1.03917i	−0.854419	−	0.519585i	$-0.826087\pi$
$373$	− 7486.00i	− 1.03917i	0.854419	−	0.519585i	$-0.173913\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16128.0i	2.20327i
$378$	0	0
$379$	1285.00	0.174158	0.0870792	−	0.996201i	$-0.472247\pi$
$379$	1285.00	0.174158	0.0870792	+	0.996201i	$0.472247\pi$
$380$	0	0
$381$	11930.0	1.60418
$382$	0	0
$383$	9622.00i	1.28371i	0.766826	+	0.641855i	$0.221835\pi$
$383$	9622.00i	1.28371i	−0.766826	+	0.641855i	$0.778165\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 328.000i	− 0.0430831i
$388$	0	0
$389$	−1974.00	−0.257290	−0.128645	−	0.991691i	$-0.541063\pi$
$389$	−1974.00	−0.257290	−0.128645	+	0.991691i	$0.541063\pi$
$390$	0	0
$391$	−3538.00	−0.457607
$392$	0	0
$393$	− 12260.0i	− 1.57363i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8084.00i	− 1.02198i	−0.859588	−	0.510988i	$-0.829280\pi$
$397$	− 8084.00i	− 1.02198i	0.859588	−	0.510988i	$-0.170720\pi$
$398$	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
$403$	− 1512.00i	− 0.186894i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5382.00i	0.655469i
$408$	0	0
$409$	4835.00	0.584536	0.292268	−	0.956336i	$-0.405590\pi$
$409$	4835.00	0.584536	0.292268	+	0.956336i	$0.405590\pi$
$410$	0	0
$411$	5625.00	0.675087
$412$	0	0
$413$	672.000i	0.0800653i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6885.00i	− 0.808537i
$418$	0	0
$419$	4619.00	0.538551	0.269276	−	0.963063i	$-0.413216\pi$
$419$	4619.00	0.538551	0.269276	+	0.963063i	$0.413216\pi$
$420$	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
$423$	424.000i	0.0487366i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1716.00i	− 0.194480i
$428$	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
$433$	2029.00i	0.225191i	0.993641	+	0.112595i	$0.0359164\pi$
$433$	2029.00i	0.225191i	−0.993641	+	0.112595i	$0.964084\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8758.00i	− 0.958700i
$438$	0	0
$439$	3208.00	0.348769	0.174384	−	0.984678i	$-0.444206\pi$
$439$	3208.00	0.348769	0.174384	+	0.984678i	$0.444206\pi$
$440$	0	0
$441$	678.000	0.0732102
$442$	0	0
$443$	13227.0i	1.41859i	0.704914	+	0.709293i	$0.250986\pi$
$443$	13227.0i	1.41859i	−0.704914	+	0.709293i	$0.749014\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 9600.00i	− 1.01580i
$448$	0	0
$449$	−3617.00	−0.380171	−0.190086	−	0.981768i	$-0.560877\pi$
$449$	−3617.00	−0.380171	−0.190086	+	0.981768i	$0.560877\pi$
$450$	0	0
$451$	−8931.00	−0.932471
$452$	0	0
$453$	− 9270.00i	− 0.961463i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6215.00i	0.636161i	0.948064	+	0.318080i	$0.103038\pi$
$457$	6215.00i	0.636161i	−0.948064	+	0.318080i	$0.896962\pi$
$458$	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
$463$	− 3364.00i	− 0.337664i	−0.985645	−	0.168832i	$-0.946000\pi$
$463$	− 3364.00i	− 0.337664i	0.985645	−	0.168832i	$-0.0539995\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18964.0i	1.87912i	0.342384	+	0.939560i	$0.388766\pi$
$467$	18964.0i	1.87912i	−0.342384	+	0.939560i	$0.611234\pi$
$468$	0	0
$469$	418.000	0.0411545
$470$	0	0
$471$	3170.00	0.310119
$472$	0	0
$473$	6396.00i	0.621751i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1156.00i	− 0.110964i
$478$	0	0
$479$	−10926.0	−1.04222	−0.521108	−	0.853491i	$-0.674481\pi$
$479$	−10926.0	−1.04222	−0.521108	+	0.853491i	$0.674481\pi$
$480$	0	0
$481$	−11592.0	−1.09886
$482$	0	0
$483$	− 580.000i	− 0.0546396i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4350.00i	0.404758i	0.979307	+	0.202379i	$0.0648673\pi$
$487$	4350.00i	0.404758i	−0.979307	+	0.202379i	$0.935133\pi$
$488$	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
$493$	11712.0i	1.06994i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1560.00i	− 0.140796i
$498$	0	0
$499$	9068.00	0.813506	0.406753	−	0.913538i	$-0.366661\pi$
$499$	9068.00	0.813506	0.406753	+	0.913538i	$0.366661\pi$
$500$	0	0
$501$	220.000	0.0196185
$502$	0	0
$503$	− 19836.0i	− 1.75834i	−0.476511	−	0.879169i	$-0.658099\pi$
$503$	− 19836.0i	− 1.75834i	0.476511	−	0.879169i	$-0.341901\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 24295.0i	− 2.12816i
$508$	0	0
$509$	−2682.00	−0.233551	−0.116776	−	0.993158i	$-0.537256\pi$
$509$	−2682.00	−0.233551	−0.116776	+	0.993158i	$0.537256\pi$
$510$	0	0
$511$	806.000	0.0697756
$512$	0	0
$513$	21895.0i	1.88438i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8268.00i	− 0.703339i
$518$	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
$523$	7701.00i	0.643865i	0.946763	+	0.321932i	$0.104332\pi$
$523$	7701.00i	0.643865i	−0.946763	+	0.321932i	$0.895668\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1098.00i	− 0.0907583i
$528$	0	0
$529$	8803.00	0.723514
$530$	0	0
$531$	−672.000	−0.0549196
$532$	0	0
$533$	− 19236.0i	− 1.56323i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11225.0i	− 0.902038i
$538$	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
$543$	15250.0i	1.20523i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 19541.0i	− 1.52745i	−0.645544	−	0.763723i	$-0.723369\pi$
$547$	− 19541.0i	− 1.52745i	0.645544	−	0.763723i	$-0.276631\pi$
$548$	0	0
$549$	1716.00	0.133401
$550$	0	0
$551$	−28992.0	−2.24156
$552$	0	0
$553$	460.000i	0.0353729i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13508.0i	1.02756i	0.857921	+	0.513781i	$0.171756\pi$
$557$	13508.0i	1.02756i	−0.857921	+	0.513781i	$0.828244\pi$
$558$	0	0
$559$	−13776.0	−1.04233
$560$	0	0
$561$	−11895.0	−0.895200
$562$	0	0
$563$	− 8712.00i	− 0.652162i	−0.945342	−	0.326081i	$-0.894272\pi$
$563$	− 8712.00i	− 0.652162i	0.945342	−	0.326081i	$-0.105728\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1342.00i	0.0993981i
$568$	0	0
$569$	9623.00	0.708993	0.354497	−	0.935057i	$-0.384652\pi$
$569$	9623.00	0.708993	0.354497	+	0.935057i	$0.384652\pi$
$570$	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
$573$	21110.0i	1.53906i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3629.00i	0.261832i	0.991393	+	0.130916i	$0.0417919\pi$
$577$	3629.00i	0.261832i	−0.991393	+	0.130916i	$0.958208\pi$
$578$	0	0
$579$	−16785.0	−1.20477
$580$	0	0
$581$	−2586.00	−0.184656
$582$	0	0
$583$	22542.0i	1.60136i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9219.00i	− 0.648226i	−0.946018	−	0.324113i	$-0.894934\pi$
$587$	− 9219.00i	− 0.648226i	0.946018	−	0.324113i	$-0.105066\pi$
$588$	0	0
$589$	2718.00	0.190141
$590$	0	0
$591$	830.000	0.0577693
$592$	0	0
$593$	19111.0i	1.32343i	0.749755	+	0.661716i	$0.230171\pi$
$593$	19111.0i	1.32343i	−0.749755	+	0.661716i	$0.769829\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16860.0i	1.15584i
$598$	0	0
$599$	−17086.0	−1.16547	−0.582734	−	0.812663i	$-0.698017\pi$
$599$	−17086.0	−1.16547	−0.582734	+	0.812663i	$0.698017\pi$
$600$	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
$603$	418.000i	0.0282293i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14784.0i	0.988573i	0.869299	+	0.494287i	$0.164571\pi$
$607$	14784.0i	0.988573i	−0.869299	+	0.494287i	$0.835429\pi$
$608$	0	0
$609$	−1920.00	−0.127754
$610$	0	0
$611$	17808.0	1.17911
$612$	0	0
$613$	17846.0i	1.17585i	0.808917	+	0.587923i	$0.200054\pi$
$613$	17846.0i	1.17585i	−0.808917	+	0.587923i	$0.799946\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11618.0i	0.758060i	0.925384	+	0.379030i	$0.123742\pi$
$617$	11618.0i	0.758060i	−0.925384	+	0.379030i	$0.876258\pi$
$618$	0	0
$619$	−9556.00	−0.620498	−0.310249	−	0.950655i	$-0.600412\pi$
$619$	−9556.00	−0.620498	−0.310249	+	0.950655i	$0.600412\pi$
$620$	0	0
$621$	8410.00	0.543449
$622$	0	0
$623$	− 2738.00i	− 0.176076i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 29445.0i	− 1.87547i
$628$	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
$633$	− 28005.0i	− 1.75845i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 28476.0i	− 1.77121i
$638$	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
$643$	27036.0i	1.65816i	0.559131	+	0.829079i	$0.311135\pi$
$643$	27036.0i	1.65816i	−0.559131	+	0.829079i	$0.688865\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17556.0i	1.06677i	0.845874	+	0.533383i	$0.179080\pi$
$647$	17556.0i	1.06677i	−0.845874	+	0.533383i	$0.820920\pi$
$648$	0	0
$649$	13104.0	0.792569
$650$	0	0
$651$	180.000	0.0108368
$652$	0	0
$653$	17262.0i	1.03448i	0.855841	+	0.517239i	$0.173040\pi$
$653$	17262.0i	1.03448i	−0.855841	+	0.517239i	$0.826960\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	806.000i	0.0478616i
$658$	0	0
$659$	10517.0	0.621675	0.310838	−	0.950463i	$-0.399390\pi$
$659$	10517.0	0.621675	0.310838	+	0.950463i	$0.399390\pi$
$660$	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
$663$	− 25620.0i	− 1.50075i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11136.0i	0.646458i
$668$	0	0
$669$	−4140.00	−0.239255
$670$	0	0
$671$	−33462.0	−1.92517
$672$	0	0
$673$	9626.00i	0.551345i	0.961252	+	0.275672i	$0.0889005\pi$
$673$	9626.00i	0.551345i	−0.961252	+	0.275672i	$0.911100\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28464.0i	1.61589i	0.589255	+	0.807947i	$0.299421\pi$
$677$	28464.0i	1.61589i	−0.589255	+	0.807947i	$0.700579\pi$
$678$	0	0
$679$	764.000	0.0431806
$680$	0	0
$681$	−11940.0	−0.671868
$682$	0	0
$683$	− 3963.00i	− 0.222020i	−0.993819	−	0.111010i	$-0.964591\pi$
$683$	− 3963.00i	− 0.222020i	0.993819	−	0.111010i	$-0.0354086\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14220.0i	0.789704i
$688$	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
$693$	156.000i	0.00855115i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 13969.0i	− 0.759130i
$698$	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
$703$	− 20838.0i	− 1.11795i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1588.00i	0.0844737i
$708$	0	0
$709$	35228.0	1.86603	0.933015	−	0.359837i	$-0.117168\pi$
$709$	35228.0	1.86603	0.933015	+	0.359837i	$0.117168\pi$
$710$	0	0
$711$	−460.000	−0.0242635
$712$	0	0
$713$	− 1044.00i	− 0.0548361i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 21600.0i	− 1.12506i
$718$	0	0
$719$	−8658.00	−0.449081	−0.224540	−	0.974465i	$-0.572088\pi$
$719$	−8658.00	−0.449081	−0.224540	+	0.974465i	$0.572088\pi$
$720$	0	0
$721$	−2696.00	−0.139257
$722$	0	0
$723$	19285.0i	0.992001i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 5728.00i	− 0.292214i	−0.989269	−	0.146107i	$-0.953326\pi$
$727$	− 5728.00i	− 0.292214i	0.989269	−	0.146107i	$-0.0466744\pi$
$728$	0	0
$729$	−20917.0	−1.06269
$730$	0	0
$731$	−10004.0	−0.506171
$732$	0	0
$733$	− 21460.0i	− 1.08137i	−0.841226	−	0.540684i	$-0.818165\pi$
$733$	− 21460.0i	− 1.08137i	0.841226	−	0.540684i	$-0.181835\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8151.00i	− 0.407389i
$738$	0	0
$739$	29164.0	1.45171	0.725856	−	0.687847i	$-0.241444\pi$
$739$	29164.0	1.45171	0.725856	+	0.687847i	$0.241444\pi$
$740$	0	0
$741$	63420.0	3.14412
$742$	0	0
$743$	29478.0i	1.45551i	0.685838	+	0.727754i	$0.259436\pi$
$743$	29478.0i	1.45551i	−0.685838	+	0.727754i	$0.740564\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2586.00i	− 0.126662i
$748$	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
$753$	1435.00i	0.0694480i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2880.00i	0.138277i	0.997607	+	0.0691383i	$0.0220250\pi$
$757$	2880.00i	0.138277i	−0.997607	+	0.0691383i	$0.977975\pi$
$758$	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
$763$	892.000i	0.0423232i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28224.0i	1.32870i
$768$	0	0
$769$	−26421.0	−1.23897	−0.619484	−	0.785010i	$-0.712658\pi$
$769$	−26421.0	−1.23897	−0.619484	+	0.785010i	$0.712658\pi$
$770$	0	0
$771$	−10650.0	−0.497471
$772$	0	0
$773$	− 32504.0i	− 1.51240i	−0.654339	−	0.756202i	$-0.727053\pi$
$773$	− 32504.0i	− 1.51240i	0.654339	−	0.756202i	$-0.272947\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1380.00i	− 0.0637159i
$778$	0	0
$779$	34579.0	1.59040
$780$	0	0
$781$	−30420.0	−1.39374
$782$	0	0
$783$	− 27840.0i	− 1.27065i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 996.000i	− 0.0451125i	−0.999746	−	0.0225563i	$-0.992820\pi$
$787$	− 996.000i	− 0.0451125i	0.999746	−	0.0225563i	$-0.00718049\pi$
$788$	0	0
$789$	15330.0	0.691714
$790$	0	0
$791$	462.000	0.0207672
$792$	0	0
$793$	− 72072.0i	− 3.22743i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15134.0i	0.672615i	0.941752	+	0.336307i	$0.109178\pi$
$797$	15134.0i	0.672615i	−0.941752	+	0.336307i	$0.890822\pi$
$798$	0	0
$799$	12932.0	0.572592
$800$	0	0
$801$	2738.00	0.120777
$802$	0	0
$803$	− 15717.0i	− 0.690711i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18720.0i	0.816574i
$808$	0	0
$809$	36942.0	1.60545	0.802727	−	0.596347i	$-0.203382\pi$
$809$	36942.0	1.60545	0.802727	+	0.596347i	$0.203382\pi$
$810$	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
$813$	16730.0i	0.721706i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 24764.0i	− 1.06044i
$818$	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
$823$	− 6788.00i	− 0.287503i	−0.989614	−	0.143751i	$-0.954083\pi$
$823$	− 6788.00i	− 0.287503i	0.989614	−	0.143751i	$-0.0459166\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 33011.0i	− 1.38803i	−0.719958	−	0.694017i	$-0.755839\pi$
$827$	− 33011.0i	− 1.38803i	0.719958	−	0.694017i	$-0.244161\pi$
$828$	0	0
$829$	−17732.0	−0.742892	−0.371446	−	0.928454i	$-0.621138\pi$
$829$	−17732.0	−0.742892	−0.371446	+	0.928454i	$0.621138\pi$
$830$	0	0
$831$	−35200.0	−1.46940
$832$	0	0
$833$	− 20679.0i	− 0.860126i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2610.00i	0.107784i
$838$	0	0
$839$	8480.00	0.348942	0.174471	−	0.984662i	$-0.444179\pi$
$839$	8480.00	0.348942	0.174471	+	0.984662i	$0.444179\pi$
$840$	0	0
$841$	12475.0	0.511501
$842$	0	0
$843$	− 15050.0i	− 0.614887i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 380.000i	− 0.0154155i
$848$	0	0
$849$	30005.0	1.21292
$850$	0	0
$851$	−8004.00	−0.322413
$852$	0	0
$853$	− 30014.0i	− 1.20476i	−0.798210	−	0.602380i	$-0.794219\pi$
$853$	− 30014.0i	− 1.20476i	0.798210	−	0.602380i	$-0.205781\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21643.0i	0.862673i	0.902191	+	0.431337i	$0.141958\pi$
$857$	21643.0i	0.862673i	−0.902191	+	0.431337i	$0.858042\pi$
$858$	0	0
$859$	2799.00	0.111177	0.0555883	−	0.998454i	$-0.482297\pi$
$859$	2799.00	0.111177	0.0555883	+	0.998454i	$0.482297\pi$
$860$	0	0
$861$	2290.00	0.0906423
$862$	0	0
$863$	− 19384.0i	− 0.764588i	−0.924041	−	0.382294i	$-0.875134\pi$
$863$	− 19384.0i	− 0.764588i	0.924041	−	0.382294i	$-0.124866\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5960.00i	0.233463i
$868$	0	0
$869$	8970.00	0.350157
$870$	0	0
$871$	17556.0	0.682965
$872$	0	0
$873$	764.000i	0.0296191i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5132.00i	0.197600i	0.995107	+	0.0988001i	$0.0315004\pi$
$877$	5132.00i	0.197600i	−0.995107	+	0.0988001i	$0.968500\pi$
$878$	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
$883$	24317.0i	0.926764i	0.886159	+	0.463382i	$0.153364\pi$
$883$	24317.0i	0.926764i	−0.886159	+	0.463382i	$0.846636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26100.0i	0.987996i	0.869463	+	0.493998i	$0.164465\pi$
$887$	26100.0i	0.987996i	−0.869463	+	0.493998i	$0.835535\pi$
$888$	0	0
$889$	−4772.00	−0.180031
$890$	0	0
$891$	26169.0	0.983944
$892$	0	0
$893$	32012.0i	1.19960i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24360.0i	− 0.906752i
$898$	0	0
$899$	−3456.00	−0.128214
$900$	0	0
$901$	−35258.0	−1.30368
$902$	0	0
$903$	− 1640.00i	− 0.0604383i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 24356.0i	− 0.891651i	−0.895120	−	0.445826i	$-0.852910\pi$
$907$	− 24356.0i	− 0.891651i	0.895120	−	0.445826i	$-0.147090\pi$
$908$	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
$913$	50427.0i	1.82792i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4904.00i	0.176602i
$918$	0	0
$919$	−34838.0	−1.25049	−0.625245	−	0.780429i	$-0.715001\pi$
$919$	−34838.0	−1.25049	−0.625245	+	0.780429i	$0.715001\pi$
$920$	0	0
$921$	30745.0	1.09998
$922$	0	0
$923$	− 65520.0i	− 2.33653i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2696.00i	− 0.0955213i
$928$	0	0
$929$	−26334.0	−0.930022	−0.465011	−	0.885305i	$-0.653950\pi$
$929$	−26334.0	−0.930022	−0.465011	+	0.885305i	$0.653950\pi$
$930$	0	0
$931$	51189.0	1.80199
$932$	0	0
$933$	4390.00i	0.154043i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30949.0i	1.07904i	0.841973	+	0.539520i	$0.181394\pi$
$937$	30949.0i	1.07904i	−0.841973	+	0.539520i	$0.818606\pi$
$938$	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
$943$	− 13282.0i	− 0.458665i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1216.00i	0.0417262i	0.999782	+	0.0208631i	$0.00664141\pi$
$947$	1216.00i	0.0417262i	−0.999782	+	0.0208631i	$0.993359\pi$
$948$	0	0
$949$	33852.0	1.15794
$950$	0	0
$951$	−19220.0	−0.655364
$952$	0	0
$953$	6033.00i	0.205066i	0.994730	+	0.102533i	$0.0326947\pi$
$953$	6033.00i	0.205066i	−0.994730	+	0.102533i	$0.967305\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	37440.0i	1.26464i
$958$	0	0
$959$	−2250.00	−0.0757626
$960$	0	0
$961$	−29467.0	−0.989124
$962$	0	0
$963$	1550.00i	0.0518671i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41792.0i	1.38980i	0.719105	+	0.694902i	$0.244552\pi$
$967$	41792.0i	1.38980i	−0.719105	+	0.694902i	$0.755448\pi$
$968$	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
$973$	2754.00i	0.0907391i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30119.0i	0.986277i	0.869951	+	0.493138i	$0.164150\pi$
$977$	30119.0i	0.986277i	−0.869951	+	0.493138i	$0.835850\pi$
$978$	0	0
$979$	−53391.0	−1.74299
$980$	0	0
$981$	−892.000	−0.0290310
$982$	0	0
$983$	18438.0i	0.598251i	0.954214	+	0.299126i	$0.0966949\pi$
$983$	18438.0i	0.598251i	−0.954214	+	0.299126i	$0.903305\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2120.00i	0.0683691i
$988$	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
$993$	13585.0i	0.434146i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 6804.00i	− 0.216133i	−0.994144	−	0.108067i	$-0.965534\pi$
$997$	− 6804.00i	− 0.216133i	0.994144	−	0.108067i	$-0.0344660\pi$
$998$	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.c.g.49.2		2
4.3	odd	2		200.4.c.d.49.1		2
5.2	odd	4		400.4.a.q.1.1		1
5.3	odd	4		400.4.a.f.1.1		1
5.4	even	2	inner	400.4.c.g.49.1		2
12.11	even	2		1800.4.f.c.649.2		2
20.3	even	4		200.4.a.h.1.1	yes	1
20.7	even	4		200.4.a.c.1.1	✓	1
20.19	odd	2		200.4.c.d.49.2		2
40.3	even	4		1600.4.a.m.1.1		1
40.13	odd	4		1600.4.a.bo.1.1		1
40.27	even	4		1600.4.a.bn.1.1		1
40.37	odd	4		1600.4.a.n.1.1		1
60.23	odd	4		1800.4.a.t.1.1		1
60.47	odd	4		1800.4.a.p.1.1		1
60.59	even	2		1800.4.f.c.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.4.a.c.1.1	✓	1	20.7	even	4
200.4.a.h.1.1	yes	1	20.3	even	4
200.4.c.d.49.1		2	4.3	odd	2
200.4.c.d.49.2		2	20.19	odd	2
400.4.a.f.1.1		1	5.3	odd	4
400.4.a.q.1.1		1	5.2	odd	4
400.4.c.g.49.1		2	5.4	even	2	inner
400.4.c.g.49.2		2	1.1	even	1	trivial
1600.4.a.m.1.1		1	40.3	even	4
1600.4.a.n.1.1		1	40.37	odd	4
1600.4.a.bn.1.1		1	40.27	even	4
1600.4.a.bo.1.1		1	40.13	odd	4
1800.4.a.p.1.1		1	60.47	odd	4
1800.4.a.t.1.1		1	60.23	odd	4
1800.4.f.c.649.1		2	60.59	even	2
1800.4.f.c.649.2		2	12.11	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -39.0000 q^{11} -84.0000i q^{13} -61.0000i q^{17} +151.000 q^{19} +10.0000 q^{21} -58.0000i q^{23} +145.000i q^{27} -192.000 q^{29} +18.0000 q^{31} -195.000i q^{33} -138.000i q^{37} +420.000 q^{39} +229.000 q^{41} -164.000i q^{43} +212.000i q^{47} +339.000 q^{49} +305.000 q^{51} -578.000i q^{53} +755.000i q^{57} -336.000 q^{59} +858.000 q^{61} -4.00000i q^{63} +209.000i q^{67} +290.000 q^{69} +780.000 q^{71} +403.000i q^{73} +78.0000i q^{77} -230.000 q^{79} -671.000 q^{81} -1293.00i q^{83} -960.000i q^{87} +1369.00 q^{89} -168.000 q^{91} +90.0000i q^{93} +382.000i q^{97} -78.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 78 q^{11} + 302 q^{19} + 20 q^{21} - 384 q^{29} + 36 q^{31} + 840 q^{39} + 458 q^{41} + 678 q^{49} + 610 q^{51} - 672 q^{59} + 1716 q^{61} + 580 q^{69} + 1560 q^{71} - 460 q^{79} - 1342 q^{81} + 2738 q^{89} - 336 q^{91} - 156 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 78 * q^11 + 302 * q^19 + 20 * q^21 - 384 * q^29 + 36 * q^31 + 840 * q^39 + 458 * q^41 + 678 * q^49 + 610 * q^51 - 672 * q^59 + 1716 * q^61 + 580 * q^69 + 1560 * q^71 - 460 * q^79 - 1342 * q^81 + 2738 * q^89 - 336 * q^91 - 156 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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