LMFDB - Embedded newform 4400.2.b.r.4049.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(4049,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-10,0,-4,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.r.4049.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302776i q^{3} +0.697224i q^{7} +2.90833 q^{9} -1.00000 q^{11} +1.60555i q^{13} -4.30278i q^{17} -1.00000 q^{19} -0.211103 q^{21} -3.90833i q^{23} +1.78890i q^{27} +1.69722 q^{29} -1.60555 q^{31} -0.302776i q^{33} -7.60555i q^{37} -0.486122 q^{39} +8.21110 q^{41} -7.21110i q^{43} +5.60555i q^{47} +6.51388 q^{49} +1.30278 q^{51} -6.51388i q^{53} -0.302776i q^{57} -5.60555 q^{59} +3.30278 q^{61} +2.02776i q^{63} +8.00000i q^{67} +1.18335 q^{69} -2.60555 q^{71} -1.90833i q^{73} -0.697224i q^{77} +6.30278 q^{79} +8.18335 q^{81} -6.90833i q^{83} +0.513878i q^{87} +5.09167 q^{89} -1.11943 q^{91} -0.486122i q^{93} +7.11943i q^{97} -2.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{9} - 4 q^{11} - 4 q^{19} + 28 q^{21} + 14 q^{29} + 8 q^{31} - 38 q^{39} + 4 q^{41} - 10 q^{49} - 2 q^{51} - 8 q^{59} + 6 q^{61} + 48 q^{69} + 4 q^{71} + 18 q^{79} + 76 q^{81} + 42 q^{89} + 46 q^{91}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q - 10 * q^9 - 4 * q^11 - 4 * q^19 + 28 * q^21 + 14 * q^29 + 8 * q^31 - 38 * q^39 + 4 * q^41 - 10 * q^49 - 2 * q^51 - 8 * q^59 + 6 * q^61 + 48 * q^69 + 4 * q^71 + 18 * q^79 + 76 * q^81 + 42 * q^89 + 46 * q^91 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	0.302776i	0.174808i	0.996173	+	0.0874038i	$0.0278570\pi$
$3$	0.302776i	0.174808i	−0.996173	+	0.0874038i	$0.972143\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.697224i	0.263526i	0.991281	+	0.131763i	$0.0420638\pi$
$7$	0.697224i	0.263526i	−0.991281	+	0.131763i	$0.957936\pi$
$8$	0	0
$9$	2.90833	0.969442
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.60555i	0.445300i	0.974898	+	0.222650i	$0.0714707\pi$
$13$	1.60555i	0.445300i	−0.974898	+	0.222650i	$0.928529\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.30278i	− 1.04358i	−0.853075	−	0.521788i	$-0.825265\pi$
$17$	− 4.30278i	− 1.04358i	0.853075	−	0.521788i	$-0.174735\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−0.211103	−0.0460664
$22$	0	0
$23$	− 3.90833i	− 0.814942i	−0.913218	−	0.407471i	$-0.866411\pi$
$23$	− 3.90833i	− 0.814942i	0.913218	−	0.407471i	$-0.133589\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.78890i	0.344273i
$28$	0	0
$29$	1.69722	0.315167	0.157583	−	0.987506i	$-0.449630\pi$
$29$	1.69722	0.315167	0.157583	+	0.987506i	$0.449630\pi$
$30$	0	0
$31$	−1.60555	−0.288366	−0.144183	−	0.989551i	$-0.546055\pi$
$31$	−1.60555	−0.288366	−0.144183	+	0.989551i	$0.546055\pi$
$32$	0	0
$33$	− 0.302776i	− 0.0527065i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.60555i	− 1.25034i	−0.780487	−	0.625172i	$-0.785029\pi$
$37$	− 7.60555i	− 1.25034i	0.780487	−	0.625172i	$-0.214971\pi$
$38$	0	0
$39$	−0.486122	−0.0778418
$40$	0	0
$41$	8.21110	1.28236	0.641179	−	0.767391i	$-0.278445\pi$
$41$	8.21110	1.28236	0.641179	+	0.767391i	$0.278445\pi$
$42$	0	0
$43$	− 7.21110i	− 1.09968i	−0.835269	−	0.549841i	$-0.814688\pi$
$43$	− 7.21110i	− 1.09968i	0.835269	−	0.549841i	$-0.185312\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.60555i	0.817654i	0.912612	+	0.408827i	$0.134062\pi$
$47$	5.60555i	0.817654i	−0.912612	+	0.408827i	$0.865938\pi$
$48$	0	0
$49$	6.51388	0.930554
$50$	0	0
$51$	1.30278	0.182425
$52$	0	0
$53$	− 6.51388i	− 0.894750i	−0.894346	−	0.447375i	$-0.852359\pi$
$53$	− 6.51388i	− 0.894750i	0.894346	−	0.447375i	$-0.147641\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.302776i	− 0.0401036i
$58$	0	0
$59$	−5.60555	−0.729781	−0.364890	−	0.931051i	$-0.618894\pi$
$59$	−5.60555	−0.729781	−0.364890	+	0.931051i	$0.618894\pi$
$60$	0	0
$61$	3.30278	0.422877	0.211439	−	0.977391i	$-0.432185\pi$
$61$	3.30278	0.422877	0.211439	+	0.977391i	$0.432185\pi$
$62$	0	0
$63$	2.02776i	0.255473i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	1.18335	0.142458
$70$	0	0
$71$	−2.60555	−0.309222	−0.154611	−	0.987975i	$-0.549412\pi$
$71$	−2.60555	−0.309222	−0.154611	+	0.987975i	$0.549412\pi$
$72$	0	0
$73$	− 1.90833i	− 0.223353i	−0.993745	−	0.111676i	$-0.964378\pi$
$73$	− 1.90833i	− 0.223353i	0.993745	−	0.111676i	$-0.0356220\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.697224i	− 0.0794561i
$78$	0	0
$79$	6.30278	0.709118	0.354559	−	0.935034i	$-0.384631\pi$
$79$	6.30278	0.709118	0.354559	+	0.935034i	$0.384631\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	− 6.90833i	− 0.758287i	−0.925338	−	0.379144i	$-0.876219\pi$
$83$	− 6.90833i	− 0.758287i	0.925338	−	0.379144i	$-0.123781\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.513878i	0.0550935i
$88$	0	0
$89$	5.09167	0.539716	0.269858	−	0.962900i	$-0.413023\pi$
$89$	5.09167	0.539716	0.269858	+	0.962900i	$0.413023\pi$
$90$	0	0
$91$	−1.11943	−0.117348
$92$	0	0
$93$	− 0.486122i	− 0.0504085i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.11943i	0.722869i	0.932398	+	0.361434i	$0.117713\pi$
$97$	7.11943i	0.722869i	−0.932398	+	0.361434i	$0.882287\pi$
$98$	0	0
$99$	−2.90833	−0.292298
$100$	0	0
$101$	−6.51388	−0.648155	−0.324078	−	0.946031i	$-0.605054\pi$
$101$	−6.51388	−0.648155	−0.324078	+	0.946031i	$0.605054\pi$
$102$	0	0
$103$	− 12.3028i	− 1.21223i	−0.795378	−	0.606114i	$-0.792727\pi$
$103$	− 12.3028i	− 1.21223i	0.795378	−	0.606114i	$-0.207273\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.21110i	0.213755i	0.994272	+	0.106878i	$0.0340853\pi$
$107$	2.21110i	0.213755i	−0.994272	+	0.106878i	$0.965915\pi$
$108$	0	0
$109$	−8.90833	−0.853263	−0.426631	−	0.904426i	$-0.640300\pi$
$109$	−8.90833	−0.853263	−0.426631	+	0.904426i	$0.640300\pi$
$110$	0	0
$111$	2.30278	0.218570
$112$	0	0
$113$	− 5.60555i	− 0.527326i	−0.964615	−	0.263663i	$-0.915069\pi$
$113$	− 5.60555i	− 0.527326i	0.964615	−	0.263663i	$-0.0849307\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.66947i	0.431692i
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.48612i	0.224166i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.9083i	1.32290i	0.749989	+	0.661450i	$0.230059\pi$
$127$	14.9083i	1.32290i	−0.749989	+	0.661450i	$0.769941\pi$
$128$	0	0
$129$	2.18335	0.192233
$130$	0	0
$131$	4.69722	0.410398	0.205199	−	0.978720i	$-0.434216\pi$
$131$	4.69722	0.410398	0.205199	+	0.978720i	$0.434216\pi$
$132$	0	0
$133$	− 0.697224i	− 0.0604570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.908327i	0.0776036i	0.999247	+	0.0388018i	$0.0123541\pi$
$137$	0.908327i	0.0776036i	−0.999247	+	0.0388018i	$0.987646\pi$
$138$	0	0
$139$	16.2111	1.37501	0.687504	−	0.726181i	$-0.258706\pi$
$139$	16.2111	1.37501	0.687504	+	0.726181i	$0.258706\pi$
$140$	0	0
$141$	−1.69722	−0.142932
$142$	0	0
$143$	− 1.60555i	− 0.134263i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.97224i	0.162668i
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	14.8167	1.20576	0.602881	−	0.797831i	$-0.294019\pi$
$151$	14.8167	1.20576	0.602881	+	0.797831i	$0.294019\pi$
$152$	0	0
$153$	− 12.5139i	− 1.01169i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2111i	1.21398i	0.794710	+	0.606989i	$0.207623\pi$
$157$	15.2111i	1.21398i	−0.794710	+	0.606989i	$0.792377\pi$
$158$	0	0
$159$	1.97224	0.156409
$160$	0	0
$161$	2.72498	0.214759
$162$	0	0
$163$	− 3.30278i	− 0.258693i	−0.991599	−	0.129347i	$-0.958712\pi$
$163$	− 3.30278i	− 0.258693i	0.991599	−	0.129347i	$-0.0412880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.6333i	− 1.90618i	−0.302683	−	0.953091i	$-0.597882\pi$
$167$	− 24.6333i	− 1.90618i	0.302683	−	0.953091i	$-0.402118\pi$
$168$	0	0
$169$	10.4222	0.801708
$170$	0	0
$171$	−2.90833	−0.222405
$172$	0	0
$173$	− 9.78890i	− 0.744236i	−0.928185	−	0.372118i	$-0.878632\pi$
$173$	− 9.78890i	− 0.744236i	0.928185	−	0.372118i	$-0.121368\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.69722i	− 0.127571i
$178$	0	0
$179$	8.09167	0.604800	0.302400	−	0.953181i	$-0.402212\pi$
$179$	8.09167	0.604800	0.302400	+	0.953181i	$0.402212\pi$
$180$	0	0
$181$	−7.90833	−0.587821	−0.293911	−	0.955833i	$-0.594957\pi$
$181$	−7.90833	−0.587821	−0.293911	+	0.955833i	$0.594957\pi$
$182$	0	0
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.30278i	0.314650i
$188$	0	0
$189$	−1.24726	−0.0907250
$190$	0	0
$191$	−12.9083	−0.934014	−0.467007	−	0.884254i	$-0.654668\pi$
$191$	−12.9083	−0.934014	−0.467007	+	0.884254i	$0.654668\pi$
$192$	0	0
$193$	− 2.42221i	− 0.174354i	−0.996193	−	0.0871771i	$-0.972215\pi$
$193$	− 2.42221i	− 0.174354i	0.996193	−	0.0871771i	$-0.0277846\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.9083i	1.56090i	0.625216	+	0.780452i	$0.285011\pi$
$197$	21.9083i	1.56090i	−0.625216	+	0.780452i	$0.714989\pi$
$198$	0	0
$199$	2.90833	0.206166	0.103083	−	0.994673i	$-0.467129\pi$
$199$	2.90833	0.206166	0.103083	+	0.994673i	$0.467129\pi$
$200$	0	0
$201$	−2.42221	−0.170849
$202$	0	0
$203$	1.18335i	0.0830546i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 11.3667i	− 0.790040i
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	21.6056	1.48739	0.743694	−	0.668520i	$-0.233072\pi$
$211$	21.6056	1.48739	0.743694	+	0.668520i	$0.233072\pi$
$212$	0	0
$213$	− 0.788897i	− 0.0540544i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.11943i	− 0.0759918i
$218$	0	0
$219$	0.577795	0.0390438
$220$	0	0
$221$	6.90833	0.464704
$222$	0	0
$223$	− 13.6056i	− 0.911095i	−0.890211	−	0.455548i	$-0.849443\pi$
$223$	− 13.6056i	− 0.911095i	0.890211	−	0.455548i	$-0.150557\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 25.9361i	− 1.72144i	−0.509080	−	0.860719i	$-0.670014\pi$
$227$	− 25.9361i	− 1.72144i	0.509080	−	0.860719i	$-0.329986\pi$
$228$	0	0
$229$	−26.1194	−1.72602	−0.863010	−	0.505186i	$-0.831424\pi$
$229$	−26.1194	−1.72602	−0.863010	+	0.505186i	$0.831424\pi$
$230$	0	0
$231$	0.211103	0.0138895
$232$	0	0
$233$	− 14.0917i	− 0.923176i	−0.887094	−	0.461588i	$-0.847280\pi$
$233$	− 14.0917i	− 0.923176i	0.887094	−	0.461588i	$-0.152720\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.90833i	0.123959i
$238$	0	0
$239$	14.3305	0.926965	0.463483	−	0.886106i	$-0.346600\pi$
$239$	14.3305	0.926965	0.463483	+	0.886106i	$0.346600\pi$
$240$	0	0
$241$	−8.30278	−0.534829	−0.267414	−	0.963582i	$-0.586169\pi$
$241$	−8.30278	−0.534829	−0.267414	+	0.963582i	$0.586169\pi$
$242$	0	0
$243$	7.84441i	0.503219i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.60555i	− 0.102159i
$248$	0	0
$249$	2.09167	0.132554
$250$	0	0
$251$	23.3305	1.47261	0.736305	−	0.676650i	$-0.236569\pi$
$251$	23.3305	1.47261	0.736305	+	0.676650i	$0.236569\pi$
$252$	0	0
$253$	3.90833i	0.245714i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 28.4222i	− 1.77293i	−0.462797	−	0.886464i	$-0.653154\pi$
$257$	− 28.4222i	− 1.77293i	0.462797	−	0.886464i	$-0.346846\pi$
$258$	0	0
$259$	5.30278	0.329498
$260$	0	0
$261$	4.93608	0.305536
$262$	0	0
$263$	10.8167i	0.666983i	0.942753	+	0.333492i	$0.108227\pi$
$263$	10.8167i	0.666983i	−0.942753	+	0.333492i	$0.891773\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.54163i	0.0943465i
$268$	0	0
$269$	−3.51388	−0.214245	−0.107122	−	0.994246i	$-0.534164\pi$
$269$	−3.51388	−0.214245	−0.107122	+	0.994246i	$0.534164\pi$
$270$	0	0
$271$	29.4222	1.78727	0.893636	−	0.448793i	$-0.148146\pi$
$271$	29.4222	1.78727	0.893636	+	0.448793i	$0.148146\pi$
$272$	0	0
$273$	− 0.338936i	− 0.0205133i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.2111i	1.81521i	0.419826	+	0.907605i	$0.362091\pi$
$277$	30.2111i	1.81521i	−0.419826	+	0.907605i	$0.637909\pi$
$278$	0	0
$279$	−4.66947	−0.279554
$280$	0	0
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	− 23.5139i	− 1.39775i	−0.715241	−	0.698877i	$-0.753683\pi$
$283$	− 23.5139i	− 1.39775i	0.715241	−	0.698877i	$-0.246317\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.72498i	0.337935i
$288$	0	0
$289$	−1.51388	−0.0890517
$290$	0	0
$291$	−2.15559	−0.126363
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.78890i	− 0.103802i
$298$	0	0
$299$	6.27502	0.362894
$300$	0	0
$301$	5.02776	0.289795
$302$	0	0
$303$	− 1.97224i	− 0.113302i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.3305i	− 0.874960i	−0.899228	−	0.437480i	$-0.855871\pi$
$307$	− 15.3305i	− 0.874960i	0.899228	−	0.437480i	$-0.144129\pi$
$308$	0	0
$309$	3.72498	0.211907
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	22.6056i	1.27774i	0.769314	+	0.638871i	$0.220598\pi$
$313$	22.6056i	1.27774i	−0.769314	+	0.638871i	$0.779402\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.6972i	1.10631i	0.833080	+	0.553153i	$0.186576\pi$
$317$	19.6972i	1.10631i	−0.833080	+	0.553153i	$0.813424\pi$
$318$	0	0
$319$	−1.69722	−0.0950263
$320$	0	0
$321$	−0.669468	−0.0373661
$322$	0	0
$323$	4.30278i	0.239413i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.69722i	− 0.149157i
$328$	0	0
$329$	−3.90833	−0.215473
$330$	0	0
$331$	−14.6333	−0.804319	−0.402160	−	0.915570i	$-0.631740\pi$
$331$	−14.6333	−0.804319	−0.402160	+	0.915570i	$0.631740\pi$
$332$	0	0
$333$	− 22.1194i	− 1.21214i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.78890i	0.260868i	0.991457	+	0.130434i	$0.0416370\pi$
$337$	4.78890i	0.260868i	−0.991457	+	0.130434i	$0.958363\pi$
$338$	0	0
$339$	1.69722	0.0921806
$340$	0	0
$341$	1.60555	0.0869455
$342$	0	0
$343$	9.42221i	0.508751i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.9083i	1.17610i	0.808824	+	0.588050i	$0.200104\pi$
$347$	21.9083i	1.17610i	−0.808824	+	0.588050i	$0.799896\pi$
$348$	0	0
$349$	−3.42221	−0.183186	−0.0915932	−	0.995797i	$-0.529196\pi$
$349$	−3.42221	−0.183186	−0.0915932	+	0.995797i	$0.529196\pi$
$350$	0	0
$351$	−2.87217	−0.153305
$352$	0	0
$353$	− 2.21110i	− 0.117685i	−0.998267	−	0.0588426i	$-0.981259\pi$
$353$	− 2.21110i	− 0.117685i	0.998267	−	0.0588426i	$-0.0187410\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.908327i	0.0480738i
$358$	0	0
$359$	16.4222	0.866731	0.433365	−	0.901218i	$-0.357326\pi$
$359$	16.4222	0.866731	0.433365	+	0.901218i	$0.357326\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0.302776i	0.0158916i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.3028i	1.73839i	0.494469	+	0.869195i	$0.335363\pi$
$367$	33.3028i	1.73839i	−0.494469	+	0.869195i	$0.664637\pi$
$368$	0	0
$369$	23.8806	1.24317
$370$	0	0
$371$	4.54163	0.235790
$372$	0	0
$373$	− 14.0278i	− 0.726330i	−0.931725	−	0.363165i	$-0.881696\pi$
$373$	− 14.0278i	− 0.726330i	0.931725	−	0.363165i	$-0.118304\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.72498i	0.140344i
$378$	0	0
$379$	−0.211103	−0.0108436	−0.00542180	−	0.999985i	$-0.501726\pi$
$379$	−0.211103	−0.0108436	−0.00542180	+	0.999985i	$0.501726\pi$
$380$	0	0
$381$	−4.51388	−0.231253
$382$	0	0
$383$	− 0.788897i	− 0.0403108i	−0.999797	−	0.0201554i	$-0.993584\pi$
$383$	− 0.788897i	− 0.0403108i	0.999797	−	0.0201554i	$-0.00641609\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 20.9722i	− 1.06608i
$388$	0	0
$389$	27.6333	1.40106	0.700532	−	0.713621i	$-0.252946\pi$
$389$	27.6333	1.40106	0.700532	+	0.713621i	$0.252946\pi$
$390$	0	0
$391$	−16.8167	−0.850455
$392$	0	0
$393$	1.42221i	0.0717408i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4.48612i	− 0.225152i	−0.993643	−	0.112576i	$-0.964090\pi$
$397$	− 4.48612i	− 0.225152i	0.993643	−	0.112576i	$-0.0359102\pi$
$398$	0	0
$399$	0.211103	0.0105683
$400$	0	0
$401$	11.2111	0.559856	0.279928	−	0.960021i	$-0.409689\pi$
$401$	11.2111	0.559856	0.279928	+	0.960021i	$0.409689\pi$
$402$	0	0
$403$	− 2.57779i	− 0.128409i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.60555i	0.376993i
$408$	0	0
$409$	−30.0278	−1.48478	−0.742388	−	0.669970i	$-0.766307\pi$
$409$	−30.0278	−1.48478	−0.742388	+	0.669970i	$0.766307\pi$
$410$	0	0
$411$	−0.275019	−0.0135657
$412$	0	0
$413$	− 3.90833i	− 0.192316i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.90833i	0.240362i
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−6.09167	−0.296890	−0.148445	−	0.988921i	$-0.547427\pi$
$421$	−6.09167	−0.296890	−0.148445	+	0.988921i	$0.547427\pi$
$422$	0	0
$423$	16.3028i	0.792668i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.30278i	0.111439i
$428$	0	0
$429$	0.486122	0.0234702
$430$	0	0
$431$	−3.78890	−0.182505	−0.0912524	−	0.995828i	$-0.529087\pi$
$431$	−3.78890	−0.182505	−0.0912524	+	0.995828i	$0.529087\pi$
$432$	0	0
$433$	29.0000i	1.39365i	0.717241	+	0.696826i	$0.245405\pi$
$433$	29.0000i	1.39365i	−0.717241	+	0.696826i	$0.754595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.90833i	0.186961i
$438$	0	0
$439$	7.09167	0.338467	0.169234	−	0.985576i	$-0.445871\pi$
$439$	7.09167	0.338467	0.169234	+	0.985576i	$0.445871\pi$
$440$	0	0
$441$	18.9445	0.902118
$442$	0	0
$443$	− 8.60555i	− 0.408862i	−0.978881	−	0.204431i	$-0.934466\pi$
$443$	− 8.60555i	− 0.408862i	0.978881	−	0.204431i	$-0.0655344\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.63331i	0.171850i
$448$	0	0
$449$	2.09167	0.0987122	0.0493561	−	0.998781i	$-0.484283\pi$
$449$	2.09167	0.0987122	0.0493561	+	0.998781i	$0.484283\pi$
$450$	0	0
$451$	−8.21110	−0.386646
$452$	0	0
$453$	4.48612i	0.210776i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 42.1472i	− 1.97156i	−0.168035	−	0.985781i	$-0.553742\pi$
$457$	− 42.1472i	− 1.97156i	0.168035	−	0.985781i	$-0.446258\pi$
$458$	0	0
$459$	7.69722	0.359276
$460$	0	0
$461$	−22.8167	−1.06268	−0.531339	−	0.847159i	$-0.678311\pi$
$461$	−22.8167	−1.06268	−0.531339	+	0.847159i	$0.678311\pi$
$462$	0	0
$463$	− 39.6611i	− 1.84321i	−0.388134	−	0.921603i	$-0.626880\pi$
$463$	− 39.6611i	− 1.84321i	0.388134	−	0.921603i	$-0.373120\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.02776i	0.186382i	0.995648	+	0.0931912i	$0.0297068\pi$
$467$	4.02776i	0.186382i	−0.995648	+	0.0931912i	$0.970293\pi$
$468$	0	0
$469$	−5.57779	−0.257559
$470$	0	0
$471$	−4.60555	−0.212213
$472$	0	0
$473$	7.21110i	0.331567i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 18.9445i	− 0.867408i
$478$	0	0
$479$	−26.4500	−1.20853	−0.604265	−	0.796784i	$-0.706533\pi$
$479$	−26.4500	−1.20853	−0.604265	+	0.796784i	$0.706533\pi$
$480$	0	0
$481$	12.2111	0.556778
$482$	0	0
$483$	0.825058i	0.0375414i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.60555i	0.344640i	0.985041	+	0.172320i	$0.0551264\pi$
$487$	7.60555i	0.344640i	−0.985041	+	0.172320i	$0.944874\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	−12.6333	−0.570133	−0.285067	−	0.958508i	$-0.592016\pi$
$491$	−12.6333	−0.570133	−0.285067	+	0.958508i	$0.592016\pi$
$492$	0	0
$493$	− 7.30278i	− 0.328900i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.81665i	− 0.0814881i
$498$	0	0
$499$	−31.9083	−1.42841	−0.714206	−	0.699935i	$-0.753212\pi$
$499$	−31.9083	−1.42841	−0.714206	+	0.699935i	$0.753212\pi$
$500$	0	0
$501$	7.45837	0.333215
$502$	0	0
$503$	7.81665i	0.348527i	0.984699	+	0.174264i	$0.0557545\pi$
$503$	7.81665i	0.348527i	−0.984699	+	0.174264i	$0.944245\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.15559i	0.140145i
$508$	0	0
$509$	−13.3028	−0.589635	−0.294818	−	0.955554i	$-0.595259\pi$
$509$	−13.3028	−0.589635	−0.294818	+	0.955554i	$0.595259\pi$
$510$	0	0
$511$	1.33053	0.0588593
$512$	0	0
$513$	− 1.78890i	− 0.0789818i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.60555i	− 0.246532i
$518$	0	0
$519$	2.96384	0.130098
$520$	0	0
$521$	32.6056	1.42848	0.714238	−	0.699903i	$-0.246774\pi$
$521$	32.6056	1.42848	0.714238	+	0.699903i	$0.246774\pi$
$522$	0	0
$523$	− 41.6333i	− 1.82050i	−0.414062	−	0.910249i	$-0.635890\pi$
$523$	− 41.6333i	− 1.82050i	0.414062	−	0.910249i	$-0.364110\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.90833i	0.300931i
$528$	0	0
$529$	7.72498	0.335869
$530$	0	0
$531$	−16.3028	−0.707480
$532$	0	0
$533$	13.1833i	0.571034i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.44996i	0.105724i
$538$	0	0
$539$	−6.51388	−0.280573
$540$	0	0
$541$	−30.3305	−1.30401	−0.652006	−	0.758214i	$-0.726072\pi$
$541$	−30.3305	−1.30401	−0.652006	+	0.758214i	$0.726072\pi$
$542$	0	0
$543$	− 2.39445i	− 0.102756i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0917i	0.944572i	0.881445	+	0.472286i	$0.156571\pi$
$547$	22.0917i	0.944572i	−0.881445	+	0.472286i	$0.843429\pi$
$548$	0	0
$549$	9.60555	0.409955
$550$	0	0
$551$	−1.69722	−0.0723042
$552$	0	0
$553$	4.39445i	0.186871i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.8167i	1.22100i	0.792016	+	0.610500i	$0.209032\pi$
$557$	28.8167i	1.22100i	−0.792016	+	0.610500i	$0.790968\pi$
$558$	0	0
$559$	11.5778	0.489689
$560$	0	0
$561$	−1.30278	−0.0550032
$562$	0	0
$563$	39.9083i	1.68194i	0.541085	+	0.840968i	$0.318014\pi$
$563$	39.9083i	1.68194i	−0.541085	+	0.840968i	$0.681986\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.70563i	0.239614i
$568$	0	0
$569$	45.3583	1.90152	0.950759	−	0.309931i	$-0.100306\pi$
$569$	45.3583	1.90152	0.950759	+	0.309931i	$0.100306\pi$
$570$	0	0
$571$	8.93608	0.373963	0.186982	−	0.982363i	$-0.440129\pi$
$571$	8.93608	0.373963	0.186982	+	0.982363i	$0.440129\pi$
$572$	0	0
$573$	− 3.90833i	− 0.163273i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 28.7250i	− 1.19584i	−0.801557	−	0.597918i	$-0.795995\pi$
$577$	− 28.7250i	− 1.19584i	0.801557	−	0.597918i	$-0.204005\pi$
$578$	0	0
$579$	0.733385	0.0304784
$580$	0	0
$581$	4.81665	0.199828
$582$	0	0
$583$	6.51388i	0.269777i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.7250i	1.35070i	0.737495	+	0.675352i	$0.236008\pi$
$587$	32.7250i	1.35070i	−0.737495	+	0.675352i	$0.763992\pi$
$588$	0	0
$589$	1.60555	0.0661556
$590$	0	0
$591$	−6.63331	−0.272858
$592$	0	0
$593$	26.2111i	1.07636i	0.842830	+	0.538180i	$0.180888\pi$
$593$	26.2111i	1.07636i	−0.842830	+	0.538180i	$0.819112\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.880571i	0.0360394i
$598$	0	0
$599$	−44.7250	−1.82741	−0.913707	−	0.406375i	$-0.866793\pi$
$599$	−44.7250	−1.82741	−0.913707	+	0.406375i	$0.866793\pi$
$600$	0	0
$601$	−9.48612	−0.386947	−0.193473	−	0.981106i	$-0.561975\pi$
$601$	−9.48612	−0.386947	−0.193473	+	0.981106i	$0.561975\pi$
$602$	0	0
$603$	23.2666i	0.947490i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.42221i	− 0.341847i	−0.985284	−	0.170923i	$-0.945325\pi$
$607$	− 8.42221i	− 0.341847i	0.985284	−	0.170923i	$-0.0546751\pi$
$608$	0	0
$609$	−0.358288	−0.0145186
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	− 15.3305i	− 0.619194i	−0.950868	−	0.309597i	$-0.899806\pi$
$613$	− 15.3305i	− 0.619194i	0.950868	−	0.309597i	$-0.100194\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 43.0278i	− 1.73223i	−0.499843	−	0.866116i	$-0.666609\pi$
$617$	− 43.0278i	− 1.73223i	0.499843	−	0.866116i	$-0.333391\pi$
$618$	0	0
$619$	−14.8167	−0.595532	−0.297766	−	0.954639i	$-0.596241\pi$
$619$	−14.8167	−0.595532	−0.297766	+	0.954639i	$0.596241\pi$
$620$	0	0
$621$	6.99160	0.280563
$622$	0	0
$623$	3.55004i	0.142229i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.302776i	0.0120917i
$628$	0	0
$629$	−32.7250	−1.30483
$630$	0	0
$631$	−38.5139	−1.53321	−0.766607	−	0.642117i	$-0.778056\pi$
$631$	−38.5139	−1.53321	−0.766607	+	0.642117i	$0.778056\pi$
$632$	0	0
$633$	6.54163i	0.260007i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.4584i	0.414376i
$638$	0	0
$639$	−7.57779	−0.299773
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	8.42221i	0.332139i	0.986114	+	0.166070i	$0.0531077\pi$
$643$	8.42221i	0.332139i	−0.986114	+	0.166070i	$0.946892\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.0000i	− 1.29736i	−0.761060	−	0.648682i	$-0.775321\pi$
$647$	− 33.0000i	− 1.29736i	0.761060	−	0.648682i	$-0.224679\pi$
$648$	0	0
$649$	5.60555	0.220037
$650$	0	0
$651$	0.338936	0.0132839
$652$	0	0
$653$	18.1194i	0.709068i	0.935043	+	0.354534i	$0.115360\pi$
$653$	18.1194i	0.709068i	−0.935043	+	0.354534i	$0.884640\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.55004i	− 0.216528i
$658$	0	0
$659$	36.3583	1.41632	0.708159	−	0.706053i	$-0.249526\pi$
$659$	36.3583	1.41632	0.708159	+	0.706053i	$0.249526\pi$
$660$	0	0
$661$	−20.8167	−0.809674	−0.404837	−	0.914389i	$-0.632672\pi$
$661$	−20.8167	−0.809674	−0.404837	+	0.914389i	$0.632672\pi$
$662$	0	0
$663$	2.09167i	0.0812339i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.63331i	− 0.256843i
$668$	0	0
$669$	4.11943	0.159266
$670$	0	0
$671$	−3.30278	−0.127502
$672$	0	0
$673$	35.0000i	1.34915i	0.738206	+	0.674575i	$0.235673\pi$
$673$	35.0000i	1.34915i	−0.738206	+	0.674575i	$0.764327\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 24.2389i	− 0.931575i	−0.884897	−	0.465788i	$-0.845771\pi$
$677$	− 24.2389i	− 0.931575i	0.884897	−	0.465788i	$-0.154229\pi$
$678$	0	0
$679$	−4.96384	−0.190495
$680$	0	0
$681$	7.85281	0.300920
$682$	0	0
$683$	5.60555i	0.214490i	0.994233	+	0.107245i	$0.0342030\pi$
$683$	5.60555i	0.214490i	−0.994233	+	0.107245i	$0.965797\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.90833i	− 0.301721i
$688$	0	0
$689$	10.4584	0.398432
$690$	0	0
$691$	−7.48612	−0.284785	−0.142393	−	0.989810i	$-0.545480\pi$
$691$	−7.48612	−0.284785	−0.142393	+	0.989810i	$0.545480\pi$
$692$	0	0
$693$	− 2.02776i	− 0.0770281i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 35.3305i	− 1.33824i
$698$	0	0
$699$	4.26662	0.161378
$700$	0	0
$701$	−5.21110	−0.196821	−0.0984103	−	0.995146i	$-0.531376\pi$
$701$	−5.21110	−0.196821	−0.0984103	+	0.995146i	$0.531376\pi$
$702$	0	0
$703$	7.60555i	0.286849i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.54163i	− 0.170806i
$708$	0	0
$709$	−35.6333	−1.33824	−0.669118	−	0.743156i	$-0.733328\pi$
$709$	−35.6333	−1.33824	−0.669118	+	0.743156i	$0.733328\pi$
$710$	0	0
$711$	18.3305	0.687449
$712$	0	0
$713$	6.27502i	0.235001i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.33894i	0.162041i
$718$	0	0
$719$	−16.5778	−0.618247	−0.309124	−	0.951022i	$-0.600036\pi$
$719$	−16.5778	−0.618247	−0.309124	+	0.951022i	$0.600036\pi$
$720$	0	0
$721$	8.57779	0.319454
$722$	0	0
$723$	− 2.51388i	− 0.0934921i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.9083i	0.997975i	0.866609	+	0.498987i	$0.166295\pi$
$727$	26.9083i	0.997975i	−0.866609	+	0.498987i	$0.833705\pi$
$728$	0	0
$729$	22.1749	0.821294
$730$	0	0
$731$	−31.0278	−1.14760
$732$	0	0
$733$	9.42221i	0.348017i	0.984744	+	0.174009i	$0.0556720\pi$
$733$	9.42221i	0.348017i	−0.984744	+	0.174009i	$0.944328\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−0.880571	−0.0323923	−0.0161962	−	0.999869i	$-0.505156\pi$
$739$	−0.880571	−0.0323923	−0.0161962	+	0.999869i	$0.505156\pi$
$740$	0	0
$741$	0.486122	0.0178581
$742$	0	0
$743$	− 9.90833i	− 0.363501i	−0.983345	−	0.181751i	$-0.941824\pi$
$743$	− 9.90833i	− 0.363501i	0.983345	−	0.181751i	$-0.0581764\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 20.0917i	− 0.735116i
$748$	0	0
$749$	−1.54163	−0.0563301
$750$	0	0
$751$	22.3583	0.815866	0.407933	−	0.913012i	$-0.366250\pi$
$751$	22.3583	0.815866	0.407933	+	0.913012i	$0.366250\pi$
$752$	0	0
$753$	7.06392i	0.257423i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.0000i	0.908640i	0.890838	+	0.454320i	$0.150118\pi$
$757$	25.0000i	0.908640i	−0.890838	+	0.454320i	$0.849882\pi$
$758$	0	0
$759$	−1.18335	−0.0429527
$760$	0	0
$761$	−43.2666	−1.56841	−0.784207	−	0.620500i	$-0.786930\pi$
$761$	−43.2666	−1.56841	−0.784207	+	0.620500i	$0.786930\pi$
$762$	0	0
$763$	− 6.21110i	− 0.224857i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.00000i	− 0.324971i
$768$	0	0
$769$	−1.36669	−0.0492842	−0.0246421	−	0.999696i	$-0.507845\pi$
$769$	−1.36669	−0.0492842	−0.0246421	+	0.999696i	$0.507845\pi$
$770$	0	0
$771$	8.60555	0.309921
$772$	0	0
$773$	8.33053i	0.299628i	0.988714	+	0.149814i	$0.0478676\pi$
$773$	8.33053i	0.299628i	−0.988714	+	0.149814i	$0.952132\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.60555i	0.0575988i
$778$	0	0
$779$	−8.21110	−0.294193
$780$	0	0
$781$	2.60555	0.0932340
$782$	0	0
$783$	3.03616i	0.108504i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.8444i	0.921254i	0.887594	+	0.460627i	$0.152375\pi$
$787$	25.8444i	0.921254i	−0.887594	+	0.460627i	$0.847625\pi$
$788$	0	0
$789$	−3.27502	−0.116594
$790$	0	0
$791$	3.90833	0.138964
$792$	0	0
$793$	5.30278i	0.188307i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5.09167i	− 0.180356i	−0.995926	−	0.0901782i	$-0.971256\pi$
$797$	− 5.09167i	− 0.180356i	0.995926	−	0.0901782i	$-0.0287436\pi$
$798$	0	0
$799$	24.1194	0.853284
$800$	0	0
$801$	14.8082	0.523224
$802$	0	0
$803$	1.90833i	0.0673434i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.06392i	− 0.0374516i
$808$	0	0
$809$	3.63331	0.127740	0.0638701	−	0.997958i	$-0.479656\pi$
$809$	3.63331	0.127740	0.0638701	+	0.997958i	$0.479656\pi$
$810$	0	0
$811$	9.60555	0.337297	0.168648	−	0.985676i	$-0.446060\pi$
$811$	9.60555	0.337297	0.168648	+	0.985676i	$0.446060\pi$
$812$	0	0
$813$	8.90833i	0.312429i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.21110i	0.252285i
$818$	0	0
$819$	−3.25567	−0.113762
$820$	0	0
$821$	−3.63331	−0.126803	−0.0634017	−	0.997988i	$-0.520195\pi$
$821$	−3.63331	−0.126803	−0.0634017	+	0.997988i	$0.520195\pi$
$822$	0	0
$823$	31.6333i	1.10267i	0.834285	+	0.551334i	$0.185881\pi$
$823$	31.6333i	1.10267i	−0.834285	+	0.551334i	$0.814119\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 44.6056i	− 1.55109i	−0.631294	−	0.775543i	$-0.717476\pi$
$827$	− 44.6056i	− 1.55109i	0.631294	−	0.775543i	$-0.282524\pi$
$828$	0	0
$829$	−26.1194	−0.907165	−0.453583	−	0.891214i	$-0.649854\pi$
$829$	−26.1194	−0.907165	−0.453583	+	0.891214i	$0.649854\pi$
$830$	0	0
$831$	−9.14719	−0.317312
$832$	0	0
$833$	− 28.0278i	− 0.971104i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.87217i	− 0.0992766i
$838$	0	0
$839$	26.0917	0.900785	0.450392	−	0.892831i	$-0.351284\pi$
$839$	26.0917	0.900785	0.450392	+	0.892831i	$0.351284\pi$
$840$	0	0
$841$	−26.1194	−0.900670
$842$	0	0
$843$	6.35829i	0.218991i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.697224i	0.0239569i
$848$	0	0
$849$	7.11943	0.244338
$850$	0	0
$851$	−29.7250	−1.01896
$852$	0	0
$853$	0.302776i	0.0103668i	0.999987	+	0.00518342i	$0.00164994\pi$
$853$	0.302776i	0.0103668i	−0.999987	+	0.00518342i	$0.998350\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.02776i	− 0.137586i	−0.997631	−	0.0687928i	$-0.978085\pi$
$857$	− 4.02776i	− 0.137586i	0.997631	−	0.0687928i	$-0.0219147\pi$
$858$	0	0
$859$	10.6056	0.361857	0.180928	−	0.983496i	$-0.442090\pi$
$859$	10.6056	0.361857	0.180928	+	0.983496i	$0.442090\pi$
$860$	0	0
$861$	−1.73338	−0.0590736
$862$	0	0
$863$	52.2666i	1.77918i	0.456764	+	0.889588i	$0.349009\pi$
$863$	52.2666i	1.77918i	−0.456764	+	0.889588i	$0.650991\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 0.458365i	− 0.0155669i
$868$	0	0
$869$	−6.30278	−0.213807
$870$	0	0
$871$	−12.8444	−0.435216
$872$	0	0
$873$	20.7056i	0.700779i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.6333i	1.06818i	0.845427	+	0.534090i	$0.179346\pi$
$877$	31.6333i	1.06818i	−0.845427	+	0.534090i	$0.820654\pi$
$878$	0	0
$879$	5.44996	0.183823
$880$	0	0
$881$	12.7527	0.429651	0.214825	−	0.976652i	$-0.431082\pi$
$881$	12.7527	0.429651	0.214825	+	0.976652i	$0.431082\pi$
$882$	0	0
$883$	18.2111i	0.612852i	0.951894	+	0.306426i	$0.0991333\pi$
$883$	18.2111i	0.612852i	−0.951894	+	0.306426i	$0.900867\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 11.6056i	− 0.389676i	−0.980835	−	0.194838i	$-0.937582\pi$
$887$	− 11.6056i	− 0.389676i	0.980835	−	0.194838i	$-0.0624182\pi$
$888$	0	0
$889$	−10.3944	−0.348619
$890$	0	0
$891$	−8.18335	−0.274152
$892$	0	0
$893$	− 5.60555i	− 0.187583i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.89992i	0.0634366i
$898$	0	0
$899$	−2.72498	−0.0908832
$900$	0	0
$901$	−28.0278	−0.933740
$902$	0	0
$903$	1.52228i	0.0506584i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.57779i	− 0.185208i	−0.995703	−	0.0926038i	$-0.970481\pi$
$907$	− 5.57779i	− 0.185208i	0.995703	−	0.0926038i	$-0.0295190\pi$
$908$	0	0
$909$	−18.9445	−0.628349
$910$	0	0
$911$	33.6333	1.11432	0.557161	−	0.830405i	$-0.311891\pi$
$911$	33.6333	1.11432	0.557161	+	0.830405i	$0.311891\pi$
$912$	0	0
$913$	6.90833i	0.228632i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.27502i	0.108151i
$918$	0	0
$919$	−42.0555	−1.38728	−0.693642	−	0.720320i	$-0.743995\pi$
$919$	−42.0555	−1.38728	−0.693642	+	0.720320i	$0.743995\pi$
$920$	0	0
$921$	4.64171	0.152950
$922$	0	0
$923$	− 4.18335i	− 0.137697i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 35.7805i	− 1.17519i
$928$	0	0
$929$	−31.6611	−1.03877	−0.519383	−	0.854542i	$-0.673838\pi$
$929$	−31.6611	−1.03877	−0.519383	+	0.854542i	$0.673838\pi$
$930$	0	0
$931$	−6.51388	−0.213484
$932$	0	0
$933$	2.72498i	0.0892119i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.0000i	0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$	16.0000i	0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$	0	0
$939$	−6.84441	−0.223359
$940$	0	0
$941$	4.57779	0.149232	0.0746159	−	0.997212i	$-0.476227\pi$
$941$	4.57779	0.149232	0.0746159	+	0.997212i	$0.476227\pi$
$942$	0	0
$943$	− 32.0917i	− 1.04505i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.4500i	− 1.24946i	−0.780843	−	0.624728i	$-0.785210\pi$
$947$	− 38.4500i	− 1.24946i	0.780843	−	0.624728i	$-0.214790\pi$
$948$	0	0
$949$	3.06392	0.0994589
$950$	0	0
$951$	−5.96384	−0.193391
$952$	0	0
$953$	− 33.6333i	− 1.08949i	−0.838602	−	0.544745i	$-0.816627\pi$
$953$	− 33.6333i	− 1.08949i	0.838602	−	0.544745i	$-0.183373\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 0.513878i	− 0.0166113i
$958$	0	0
$959$	−0.633308	−0.0204506
$960$	0	0
$961$	−28.4222	−0.916845
$962$	0	0
$963$	6.43061i	0.207223i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 17.9361i	− 0.576786i	−0.957512	−	0.288393i	$-0.906879\pi$
$967$	− 17.9361i	− 0.576786i	0.957512	−	0.288393i	$-0.0931209\pi$
$968$	0	0
$969$	−1.30278	−0.0418512
$970$	0	0
$971$	51.9083	1.66582	0.832909	−	0.553410i	$-0.186674\pi$
$971$	51.9083	1.66582	0.832909	+	0.553410i	$0.186674\pi$
$972$	0	0
$973$	11.3028i	0.362350i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 16.8167i	− 0.538012i	−0.963138	−	0.269006i	$-0.913305\pi$
$977$	− 16.8167i	− 0.538012i	0.963138	−	0.269006i	$-0.0866952\pi$
$978$	0	0
$979$	−5.09167	−0.162731
$980$	0	0
$981$	−25.9083	−0.827189
$982$	0	0
$983$	39.6333i	1.26411i	0.774925	+	0.632053i	$0.217788\pi$
$983$	39.6333i	1.26411i	−0.774925	+	0.632053i	$0.782212\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.18335i	− 0.0376663i
$988$	0	0
$989$	−28.1833	−0.896178
$990$	0	0
$991$	−35.7527	−1.13572	−0.567862	−	0.823124i	$-0.692229\pi$
$991$	−35.7527	−1.13572	−0.567862	+	0.823124i	$0.692229\pi$
$992$	0	0
$993$	− 4.43061i	− 0.140601i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.7805i	1.60823i	0.594471	+	0.804117i	$0.297362\pi$
$997$	50.7805i	1.60823i	−0.594471	+	0.804117i	$0.702638\pi$
$998$	0	0
$999$	13.6056	0.430461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.r.4049.3		4
4.3	odd	2		1100.2.b.e.749.2		4
5.2	odd	4		4400.2.a.bf.1.2		2
5.3	odd	4		4400.2.a.bw.1.1		2
5.4	even	2	inner	4400.2.b.r.4049.2		4
12.11	even	2		9900.2.c.r.5149.2		4
20.3	even	4		1100.2.a.f.1.2	✓	2
20.7	even	4		1100.2.a.i.1.1	yes	2
20.19	odd	2		1100.2.b.e.749.3		4
60.23	odd	4		9900.2.a.bg.1.2		2
60.47	odd	4		9900.2.a.by.1.1		2
60.59	even	2		9900.2.c.r.5149.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.f.1.2	✓	2	20.3	even	4
1100.2.a.i.1.1	yes	2	20.7	even	4
1100.2.b.e.749.2		4	4.3	odd	2
1100.2.b.e.749.3		4	20.19	odd	2
4400.2.a.bf.1.2		2	5.2	odd	4
4400.2.a.bw.1.1		2	5.3	odd	4
4400.2.b.r.4049.2		4	5.4	even	2	inner
4400.2.b.r.4049.3		4	1.1	even	1	trivial
9900.2.a.bg.1.2		2	60.23	odd	4
9900.2.a.by.1.1		2	60.47	odd	4
9900.2.c.r.5149.2		4	12.11	even	2
9900.2.c.r.5149.3		4	60.59	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.302776i q^{3} +0.697224i q^{7} +2.90833 q^{9} -1.00000 q^{11} +1.60555i q^{13} -4.30278i q^{17} -1.00000 q^{19} -0.211103 q^{21} -3.90833i q^{23} +1.78890i q^{27} +1.69722 q^{29} -1.60555 q^{31} -0.302776i q^{33} -7.60555i q^{37} -0.486122 q^{39} +8.21110 q^{41} -7.21110i q^{43} +5.60555i q^{47} +6.51388 q^{49} +1.30278 q^{51} -6.51388i q^{53} -0.302776i q^{57} -5.60555 q^{59} +3.30278 q^{61} +2.02776i q^{63} +8.00000i q^{67} +1.18335 q^{69} -2.60555 q^{71} -1.90833i q^{73} -0.697224i q^{77} +6.30278 q^{79} +8.18335 q^{81} -6.90833i q^{83} +0.513878i q^{87} +5.09167 q^{89} -1.11943 q^{91} -0.486122i q^{93} +7.11943i q^{97} -2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{9} - 4 q^{11} - 4 q^{19} + 28 q^{21} + 14 q^{29} + 8 q^{31} - 38 q^{39} + 4 q^{41} - 10 q^{49} - 2 q^{51} - 8 q^{59} + 6 q^{61} + 48 q^{69} + 4 q^{71} + 18 q^{79} + 76 q^{81} + 42 q^{89} + 46 q^{91}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q - 10 * q^9 - 4 * q^11 - 4 * q^19 + 28 * q^21 + 14 * q^29 + 8 * q^31 - 38 * q^39 + 4 * q^41 - 10 * q^49 - 2 * q^51 - 8 * q^59 + 6 * q^61 + 48 * q^69 + 4 * q^71 + 18 * q^79 + 76 * q^81 + 42 * q^89 + 46 * q^91 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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