LMFDB - Embedded newform 4056.2.c.m.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.44836416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 1 $ x^6 + 12x^4 + 36x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-2.52892i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.m.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.133492i q^{5} -3.92434i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.133492i q^{5} -3.92434i q^{7} +1.00000 q^{9} -5.05784i q^{11} +0.133492i q^{15} +4.92434 q^{17} -6.79085i q^{19} +3.92434i q^{21} +5.05784 q^{23} +4.98218 q^{25} -1.00000 q^{27} -3.86651 q^{29} -1.13349i q^{31} +5.05784i q^{33} -0.523868 q^{35} +6.92434i q^{37} -5.19133i q^{41} +11.9243 q^{43} -0.133492i q^{45} -9.05784i q^{47} -8.40048 q^{49} -4.92434 q^{51} -1.59952 q^{53} -0.675180 q^{55} +6.79085i q^{57} +11.8487i q^{59} +5.79085 q^{61} -3.92434i q^{63} -4.07566i q^{67} -5.05784 q^{69} +1.05784i q^{71} -3.26698i q^{73} -4.98218 q^{75} -19.8487 q^{77} -7.24916 q^{79} +1.00000 q^{81} +11.3248i q^{83} -0.657360i q^{85} +3.86651 q^{87} +7.73302i q^{89} +1.13349i q^{93} -0.906524 q^{95} -7.13349i q^{97} -5.05784i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} - 30 q^{25} - 6 q^{27} - 24 q^{29} + 24 q^{35} + 42 q^{43} - 48 q^{49} - 12 q^{53} - 36 q^{55} + 6 q^{61} + 30 q^{75} - 60 q^{77} + 18 q^{79} + 6 q^{81} + 24 q^{87} + 84 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 30 * q^25 - 6 * q^27 - 24 * q^29 + 24 * q^35 + 42 * q^43 - 48 * q^49 - 12 * q^53 - 36 * q^55 + 6 * q^61 + 30 * q^75 - 60 * q^77 + 18 * q^79 + 6 * q^81 + 24 * q^87 + 84 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	− 0.133492i	− 0.0596994i	−0.999554	−	0.0298497i	$-0.990497\pi$
$5$	− 0.133492i	− 0.0596994i	0.999554	−	0.0298497i	$-0.00950287\pi$
$6$	0	0
$7$	− 3.92434i	− 1.48326i	−0.670808	−	0.741631i	$-0.734052\pi$
$7$	− 3.92434i	− 1.48326i	0.670808	−	0.741631i	$-0.265948\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 5.05784i	− 1.52499i	−0.646991	−	0.762497i	$-0.723973\pi$
$11$	− 5.05784i	− 1.52499i	0.646991	−	0.762497i	$-0.276027\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.133492i	0.0344675i
$16$	0	0
$17$	4.92434	1.19433	0.597164	−	0.802119i	$-0.296294\pi$
$17$	4.92434	1.19433	0.597164	+	0.802119i	$0.296294\pi$
$18$	0	0
$19$	− 6.79085i	− 1.55793i	−0.627068	−	0.778964i	$-0.715745\pi$
$19$	− 6.79085i	− 1.55793i	0.627068	−	0.778964i	$-0.284255\pi$
$20$	0	0
$21$	3.92434i	0.856362i
$22$	0	0
$23$	5.05784	1.05463	0.527316	−	0.849669i	$-0.323198\pi$
$23$	5.05784	1.05463	0.527316	+	0.849669i	$0.323198\pi$
$24$	0	0
$25$	4.98218	0.996436
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.86651	−0.717993	−0.358996	−	0.933339i	$-0.616881\pi$
$29$	−3.86651	−0.717993	−0.358996	+	0.933339i	$0.616881\pi$
$30$	0	0
$31$	− 1.13349i	− 0.203581i	−0.994806	−	0.101791i	$-0.967543\pi$
$31$	− 1.13349i	− 0.203581i	0.994806	−	0.101791i	$-0.0324572\pi$
$32$	0	0
$33$	5.05784i	0.880456i
$34$	0	0
$35$	−0.523868	−0.0885499
$36$	0	0
$37$	6.92434i	1.13836i	0.822215	+	0.569178i	$0.192738\pi$
$37$	6.92434i	1.13836i	−0.822215	+	0.569178i	$0.807262\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.19133i	− 0.810749i	−0.914151	−	0.405375i	$-0.867141\pi$
$41$	− 5.19133i	− 0.810749i	0.914151	−	0.405375i	$-0.132859\pi$
$42$	0	0
$43$	11.9243	1.81845	0.909223	−	0.416310i	$-0.136677\pi$
$43$	11.9243	1.81845	0.909223	+	0.416310i	$0.136677\pi$
$44$	0	0
$45$	− 0.133492i	− 0.0198998i
$46$	0	0
$47$	− 9.05784i	− 1.32122i	−0.750729	−	0.660611i	$-0.770297\pi$
$47$	− 9.05784i	− 1.32122i	0.750729	−	0.660611i	$-0.229703\pi$
$48$	0	0
$49$	−8.40048	−1.20007
$50$	0	0
$51$	−4.92434	−0.689546
$52$	0	0
$53$	−1.59952	−0.219712	−0.109856	−	0.993948i	$-0.535039\pi$
$53$	−1.59952	−0.219712	−0.109856	+	0.993948i	$0.535039\pi$
$54$	0	0
$55$	−0.675180	−0.0910413
$56$	0	0
$57$	6.79085i	0.899470i
$58$	0	0
$59$	11.8487i	1.54257i	0.636491	+	0.771284i	$0.280385\pi$
$59$	11.8487i	1.54257i	−0.636491	+	0.771284i	$0.719615\pi$
$60$	0	0
$61$	5.79085	0.741443	0.370721	−	0.928744i	$-0.379111\pi$
$61$	5.79085	0.741443	0.370721	+	0.928744i	$0.379111\pi$
$62$	0	0
$63$	− 3.92434i	− 0.494421i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.07566i	− 0.497921i	−0.968514	−	0.248960i	$-0.919911\pi$
$67$	− 4.07566i	− 0.497921i	0.968514	−	0.248960i	$-0.0800889\pi$
$68$	0	0
$69$	−5.05784	−0.608892
$70$	0	0
$71$	1.05784i	0.125542i	0.998028	+	0.0627710i	$0.0199938\pi$
$71$	1.05784i	0.125542i	−0.998028	+	0.0627710i	$0.980006\pi$
$72$	0	0
$73$	− 3.26698i	− 0.382372i	−0.981554	−	0.191186i	$-0.938767\pi$
$73$	− 3.26698i	− 0.382372i	0.981554	−	0.191186i	$-0.0612333\pi$
$74$	0	0
$75$	−4.98218	−0.575293
$76$	0	0
$77$	−19.8487	−2.26197
$78$	0	0
$79$	−7.24916	−0.815595	−0.407797	−	0.913072i	$-0.633703\pi$
$79$	−7.24916	−0.815595	−0.407797	+	0.913072i	$0.633703\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3248i	1.24306i	0.783390	+	0.621530i	$0.213489\pi$
$83$	11.3248i	1.24306i	−0.783390	+	0.621530i	$0.786511\pi$
$84$	0	0
$85$	− 0.657360i	− 0.0713007i
$86$	0	0
$87$	3.86651	0.414533
$88$	0	0
$89$	7.73302i	0.819698i	0.912153	+	0.409849i	$0.134419\pi$
$89$	7.73302i	0.819698i	−0.912153	+	0.409849i	$0.865581\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.13349i	0.117538i
$94$	0	0
$95$	−0.906524	−0.0930074
$96$	0	0
$97$	− 7.13349i	− 0.724296i	−0.932121	−	0.362148i	$-0.882043\pi$
$97$	− 7.13349i	− 0.724296i	0.932121	−	0.362148i	$-0.117957\pi$
$98$	0	0
$99$	− 5.05784i	− 0.508332i
$100$	0	0
$101$	−5.71520	−0.568683	−0.284342	−	0.958723i	$-0.591775\pi$
$101$	−5.71520	−0.568683	−0.284342	+	0.958723i	$0.591775\pi$
$102$	0	0
$103$	−6.19133	−0.610050	−0.305025	−	0.952344i	$-0.598665\pi$
$103$	−6.19133	−0.610050	−0.305025	+	0.952344i	$0.598665\pi$
$104$	0	0
$105$	0.523868	0.0511243
$106$	0	0
$107$	−9.05784	−0.875654	−0.437827	−	0.899059i	$-0.644252\pi$
$107$	−9.05784	−0.875654	−0.437827	+	0.899059i	$0.644252\pi$
$108$	0	0
$109$	− 12.7152i	− 1.21789i	−0.793211	−	0.608947i	$-0.791592\pi$
$109$	− 12.7152i	− 1.21789i	0.793211	−	0.608947i	$-0.208408\pi$
$110$	0	0
$111$	− 6.92434i	− 0.657230i
$112$	0	0
$113$	18.9243	1.78025	0.890126	−	0.455714i	$-0.150616\pi$
$113$	18.9243	1.78025	0.890126	+	0.455714i	$0.150616\pi$
$114$	0	0
$115$	− 0.675180i	− 0.0629609i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 19.3248i	− 1.77150i
$120$	0	0
$121$	−14.5817	−1.32561
$122$	0	0
$123$	5.19133i	0.468086i
$124$	0	0
$125$	− 1.33254i	− 0.119186i
$126$	0	0
$127$	7.24916	0.643259	0.321630	−	0.946866i	$-0.395769\pi$
$127$	7.24916	0.643259	0.321630	+	0.946866i	$0.395769\pi$
$128$	0	0
$129$	−11.9243	−1.04988
$130$	0	0
$131$	9.96436	0.870590	0.435295	−	0.900288i	$-0.356644\pi$
$131$	9.96436	0.870590	0.435295	+	0.900288i	$0.356644\pi$
$132$	0	0
$133$	−26.6496	−2.31082
$134$	0	0
$135$	0.133492i	0.0114892i
$136$	0	0
$137$	− 9.45831i	− 0.808078i	−0.914742	−	0.404039i	$-0.867606\pi$
$137$	− 9.45831i	− 0.808078i	0.914742	−	0.404039i	$-0.132394\pi$
$138$	0	0
$139$	−3.40048	−0.288425	−0.144212	−	0.989547i	$-0.546065\pi$
$139$	−3.40048	−0.288425	−0.144212	+	0.989547i	$0.546065\pi$
$140$	0	0
$141$	9.05784i	0.762807i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.516148i	0.0428637i
$146$	0	0
$147$	8.40048	0.692860
$148$	0	0
$149$	3.71520i	0.304361i	0.988353	+	0.152180i	$0.0486295\pi$
$149$	3.71520i	0.304361i	−0.988353	+	0.152180i	$0.951371\pi$
$150$	0	0
$151$	3.32482i	0.270570i	0.990807	+	0.135285i	$0.0431950\pi$
$151$	3.32482i	0.270570i	−0.990807	+	0.135285i	$0.956805\pi$
$152$	0	0
$153$	4.92434	0.398110
$154$	0	0
$155$	−0.151312	−0.0121537
$156$	0	0
$157$	2.32482	0.185541	0.0927704	−	0.995688i	$-0.470428\pi$
$157$	2.32482	0.185541	0.0927704	+	0.995688i	$0.470428\pi$
$158$	0	0
$159$	1.59952	0.126851
$160$	0	0
$161$	− 19.8487i	− 1.56430i
$162$	0	0
$163$	6.86651i	0.537826i	0.963164	+	0.268913i	$0.0866645\pi$
$163$	6.86651i	0.537826i	−0.963164	+	0.268913i	$0.913336\pi$
$164$	0	0
$165$	0.675180	0.0525627
$166$	0	0
$167$	21.8130i	1.68794i	0.536387	+	0.843972i	$0.319789\pi$
$167$	21.8130i	1.68794i	−0.536387	+	0.843972i	$0.680211\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 6.79085i	− 0.519309i
$172$	0	0
$173$	3.58170	0.272312	0.136156	−	0.990687i	$-0.456525\pi$
$173$	3.58170	0.272312	0.136156	+	0.990687i	$0.456525\pi$
$174$	0	0
$175$	− 19.5518i	− 1.47798i
$176$	0	0
$177$	− 11.8487i	− 0.890602i
$178$	0	0
$179$	1.20915	0.0903760	0.0451880	−	0.998979i	$-0.485611\pi$
$179$	1.20915	0.0903760	0.0451880	+	0.998979i	$0.485611\pi$
$180$	0	0
$181$	−18.9243	−1.40664	−0.703318	−	0.710876i	$-0.748299\pi$
$181$	−18.9243	−1.40664	−0.703318	+	0.710876i	$0.748299\pi$
$182$	0	0
$183$	−5.79085	−0.428072
$184$	0	0
$185$	0.924344	0.0679591
$186$	0	0
$187$	− 24.9065i	− 1.82135i
$188$	0	0
$189$	3.92434i	0.285454i
$190$	0	0
$191$	7.84869	0.567911	0.283956	−	0.958837i	$-0.408353\pi$
$191$	7.84869	0.567911	0.283956	+	0.958837i	$0.408353\pi$
$192$	0	0
$193$	− 7.11567i	− 0.512197i	−0.966651	−	0.256099i	$-0.917563\pi$
$193$	− 7.11567i	− 0.512197i	0.966651	−	0.256099i	$-0.0824372\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.4304i	1.66935i	0.550746	+	0.834673i	$0.314343\pi$
$197$	23.4304i	1.66935i	−0.550746	+	0.834673i	$0.685657\pi$
$198$	0	0
$199$	13.6574	0.968145	0.484072	−	0.875028i	$-0.339157\pi$
$199$	13.6574	0.968145	0.484072	+	0.875028i	$0.339157\pi$
$200$	0	0
$201$	4.07566i	0.287475i
$202$	0	0
$203$	15.1735i	1.06497i
$204$	0	0
$205$	−0.693000	−0.0484012
$206$	0	0
$207$	5.05784	0.351544
$208$	0	0
$209$	−34.3470	−2.37583
$210$	0	0
$211$	−19.2492	−1.32517	−0.662584	−	0.748988i	$-0.730540\pi$
$211$	−19.2492	−1.32517	−0.662584	+	0.748988i	$0.730540\pi$
$212$	0	0
$213$	− 1.05784i	− 0.0724817i
$214$	0	0
$215$	− 1.59180i	− 0.108560i
$216$	0	0
$217$	−4.44821	−0.301964
$218$	0	0
$219$	3.26698i	0.220762i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	4.98218	0.332145
$226$	0	0
$227$	− 23.0222i	− 1.52804i	−0.645194	−	0.764018i	$-0.723224\pi$
$227$	− 23.0222i	− 1.52804i	0.645194	−	0.764018i	$-0.276776\pi$
$228$	0	0
$229$	− 8.11567i	− 0.536299i	−0.963377	−	0.268149i	$-0.913588\pi$
$229$	− 8.11567i	− 0.536299i	0.963377	−	0.268149i	$-0.0864121\pi$
$230$	0	0
$231$	19.8487	1.30595
$232$	0	0
$233$	−6.53397	−0.428054	−0.214027	−	0.976828i	$-0.568658\pi$
$233$	−6.53397	−0.428054	−0.214027	+	0.976828i	$0.568658\pi$
$234$	0	0
$235$	−1.20915	−0.0788761
$236$	0	0
$237$	7.24916	0.470884
$238$	0	0
$239$	11.3248i	0.732542i	0.930508	+	0.366271i	$0.119366\pi$
$239$	11.3248i	0.732542i	−0.930508	+	0.366271i	$0.880634\pi$
$240$	0	0
$241$	11.5995i	0.747191i	0.927592	+	0.373596i	$0.121875\pi$
$241$	11.5995i	0.747191i	−0.927592	+	0.373596i	$0.878125\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.12140i	0.0716433i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 11.3248i	− 0.717681i
$250$	0	0
$251$	10.1157	0.638496	0.319248	−	0.947671i	$-0.396570\pi$
$251$	10.1157	0.638496	0.319248	+	0.947671i	$0.396570\pi$
$252$	0	0
$253$	− 25.5817i	− 1.60831i
$254$	0	0
$255$	0.657360i	0.0411655i
$256$	0	0
$257$	−1.19133	−0.0743130	−0.0371565	−	0.999309i	$-0.511830\pi$
$257$	−1.19133	−0.0743130	−0.0371565	+	0.999309i	$0.511830\pi$
$258$	0	0
$259$	27.1735	1.68848
$260$	0	0
$261$	−3.86651	−0.239331
$262$	0	0
$263$	20.9065	1.28915	0.644576	−	0.764540i	$-0.277034\pi$
$263$	20.9065	1.28915	0.644576	+	0.764540i	$0.277034\pi$
$264$	0	0
$265$	0.213524i	0.0131166i
$266$	0	0
$267$	− 7.73302i	− 0.473253i
$268$	0	0
$269$	−5.84869	−0.356601	−0.178300	−	0.983976i	$-0.557060\pi$
$269$	−5.84869	−0.356601	−0.178300	+	0.983976i	$0.557060\pi$
$270$	0	0
$271$	− 23.0979i	− 1.40309i	−0.712623	−	0.701547i	$-0.752493\pi$
$271$	− 23.0979i	− 1.40309i	0.712623	−	0.701547i	$-0.247507\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 25.1990i	− 1.51956i
$276$	0	0
$277$	−4.80867	−0.288925	−0.144463	−	0.989510i	$-0.546145\pi$
$277$	−4.80867	−0.288925	−0.144463	+	0.989510i	$0.546145\pi$
$278$	0	0
$279$	− 1.13349i	− 0.0678604i
$280$	0	0
$281$	− 5.60962i	− 0.334642i	−0.985902	−	0.167321i	$-0.946488\pi$
$281$	− 5.60962i	− 0.334642i	0.985902	−	0.167321i	$-0.0535116\pi$
$282$	0	0
$283$	−25.8887	−1.53892	−0.769462	−	0.638693i	$-0.779475\pi$
$283$	−25.8887	−1.53892	−0.769462	+	0.638693i	$0.779475\pi$
$284$	0	0
$285$	0.906524	0.0536978
$286$	0	0
$287$	−20.3726	−1.20255
$288$	0	0
$289$	7.24916	0.426421
$290$	0	0
$291$	7.13349i	0.418173i
$292$	0	0
$293$	− 25.4482i	− 1.48670i	−0.668902	−	0.743350i	$-0.733236\pi$
$293$	− 25.4482i	− 1.48670i	0.668902	−	0.743350i	$-0.266764\pi$
$294$	0	0
$295$	1.58170	0.0920904
$296$	0	0
$297$	5.05784i	0.293485i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 46.7952i	− 2.69723i
$302$	0	0
$303$	5.71520	0.328329
$304$	0	0
$305$	− 0.773032i	− 0.0442637i
$306$	0	0
$307$	− 0.458312i	− 0.0261572i	−0.999914	−	0.0130786i	$-0.995837\pi$
$307$	− 0.458312i	− 0.0261572i	0.999914	−	0.0130786i	$-0.00416317\pi$
$308$	0	0
$309$	6.19133	0.352212
$310$	0	0
$311$	−22.7909	−1.29235	−0.646175	−	0.763189i	$-0.723633\pi$
$311$	−22.7909	−1.29235	−0.646175	+	0.763189i	$0.723633\pi$
$312$	0	0
$313$	28.5639	1.61453	0.807263	−	0.590192i	$-0.200948\pi$
$313$	28.5639	1.61453	0.807263	+	0.590192i	$0.200948\pi$
$314$	0	0
$315$	−0.523868	−0.0295166
$316$	0	0
$317$	23.9465i	1.34497i	0.740110	+	0.672486i	$0.234773\pi$
$317$	23.9465i	1.34497i	−0.740110	+	0.672486i	$0.765227\pi$
$318$	0	0
$319$	19.5562i	1.09493i
$320$	0	0
$321$	9.05784	0.505559
$322$	0	0
$323$	− 33.4405i	− 1.86068i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.7152i	0.703152i
$328$	0	0
$329$	−35.5461	−1.95972
$330$	0	0
$331$	− 25.1335i	− 1.38146i	−0.723112	−	0.690731i	$-0.757289\pi$
$331$	− 25.1335i	− 1.38146i	0.723112	−	0.690731i	$-0.242711\pi$
$332$	0	0
$333$	6.92434i	0.379452i
$334$	0	0
$335$	−0.544067	−0.0297256
$336$	0	0
$337$	−28.9644	−1.57779	−0.788895	−	0.614529i	$-0.789346\pi$
$337$	−28.9644	−1.57779	−0.788895	+	0.614529i	$0.789346\pi$
$338$	0	0
$339$	−18.9243	−1.02783
$340$	0	0
$341$	−5.73302	−0.310460
$342$	0	0
$343$	5.49595i	0.296753i
$344$	0	0
$345$	0.675180i	0.0363505i
$346$	0	0
$347$	4.67518	0.250977	0.125488	−	0.992095i	$-0.459950\pi$
$347$	4.67518	0.250977	0.125488	+	0.992095i	$0.459950\pi$
$348$	0	0
$349$	27.5161i	1.47291i	0.676489	+	0.736453i	$0.263501\pi$
$349$	27.5161i	1.47291i	−0.676489	+	0.736453i	$0.736499\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 12.5060i	− 0.665630i	−0.942992	−	0.332815i	$-0.892002\pi$
$353$	− 12.5060i	− 0.665630i	0.942992	−	0.332815i	$-0.107998\pi$
$354$	0	0
$355$	0.141213	0.00749478
$356$	0	0
$357$	19.3248i	1.02278i
$358$	0	0
$359$	10.7909i	0.569519i	0.958599	+	0.284760i	$0.0919138\pi$
$359$	10.7909i	0.569519i	−0.958599	+	0.284760i	$0.908086\pi$
$360$	0	0
$361$	−27.1157	−1.42714
$362$	0	0
$363$	14.5817	0.765341
$364$	0	0
$365$	−0.436116	−0.0228274
$366$	0	0
$367$	7.92434	0.413647	0.206824	−	0.978378i	$-0.433687\pi$
$367$	7.92434	0.413647	0.206824	+	0.978378i	$0.433687\pi$
$368$	0	0
$369$	− 5.19133i	− 0.270250i
$370$	0	0
$371$	6.27708i	0.325890i
$372$	0	0
$373$	−33.3726	−1.72797	−0.863983	−	0.503521i	$-0.832037\pi$
$373$	−33.3726	−1.72797	−0.863983	+	0.503521i	$0.832037\pi$
$374$	0	0
$375$	1.33254i	0.0688121i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 14.0501i	− 0.721706i	−0.932623	−	0.360853i	$-0.882486\pi$
$379$	− 14.0501i	− 0.721706i	0.932623	−	0.360853i	$-0.117514\pi$
$380$	0	0
$381$	−7.24916	−0.371386
$382$	0	0
$383$	12.2313i	0.624992i	0.949919	+	0.312496i	$0.101165\pi$
$383$	12.2313i	0.624992i	−0.949919	+	0.312496i	$0.898835\pi$
$384$	0	0
$385$	2.64964i	0.135038i
$386$	0	0
$387$	11.9243	0.606148
$388$	0	0
$389$	28.4805	1.44402	0.722010	−	0.691883i	$-0.243219\pi$
$389$	28.4805	1.44402	0.722010	+	0.691883i	$0.243219\pi$
$390$	0	0
$391$	24.9065	1.25958
$392$	0	0
$393$	−9.96436	−0.502635
$394$	0	0
$395$	0.967705i	0.0486905i
$396$	0	0
$397$	27.2135i	1.36581i	0.730508	+	0.682904i	$0.239283\pi$
$397$	27.2135i	1.36581i	−0.730508	+	0.682904i	$0.760717\pi$
$398$	0	0
$399$	26.6496	1.33415
$400$	0	0
$401$	18.2391i	0.910815i	0.890283	+	0.455408i	$0.150506\pi$
$401$	18.2391i	0.910815i	−0.890283	+	0.455408i	$0.849494\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 0.133492i	− 0.00663327i
$406$	0	0
$407$	35.0222	1.73599
$408$	0	0
$409$	28.0323i	1.38611i	0.720886	+	0.693054i	$0.243735\pi$
$409$	28.0323i	1.38611i	−0.720886	+	0.693054i	$0.756265\pi$
$410$	0	0
$411$	9.45831i	0.466544i
$412$	0	0
$413$	46.4983	2.28803
$414$	0	0
$415$	1.51177	0.0742100
$416$	0	0
$417$	3.40048	0.166522
$418$	0	0
$419$	−15.6974	−0.766867	−0.383433	−	0.923568i	$-0.625258\pi$
$419$	−15.6974	−0.766867	−0.383433	+	0.923568i	$0.625258\pi$
$420$	0	0
$421$	− 11.5239i	− 0.561639i	−0.959761	−	0.280819i	$-0.909394\pi$
$421$	− 11.5239i	− 0.561639i	0.959761	−	0.280819i	$-0.0906062\pi$
$422$	0	0
$423$	− 9.05784i	− 0.440407i
$424$	0	0
$425$	24.5340	1.19007
$426$	0	0
$427$	− 22.7253i	− 1.09975i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	23.3248i	1.12352i	0.827301	+	0.561759i	$0.189875\pi$
$431$	23.3248i	1.12352i	−0.827301	+	0.561759i	$0.810125\pi$
$432$	0	0
$433$	5.23134	0.251402	0.125701	−	0.992068i	$-0.459882\pi$
$433$	5.23134	0.251402	0.125701	+	0.992068i	$0.459882\pi$
$434$	0	0
$435$	− 0.516148i	− 0.0247474i
$436$	0	0
$437$	− 34.3470i	− 1.64304i
$438$	0	0
$439$	−23.7730	−1.13462	−0.567312	−	0.823503i	$-0.692017\pi$
$439$	−23.7730	−1.13462	−0.567312	+	0.823503i	$0.692017\pi$
$440$	0	0
$441$	−8.40048	−0.400023
$442$	0	0
$443$	−10.1157	−0.480610	−0.240305	−	0.970697i	$-0.577247\pi$
$443$	−10.1157	−0.480610	−0.240305	+	0.970697i	$0.577247\pi$
$444$	0	0
$445$	1.03230	0.0489355
$446$	0	0
$447$	− 3.71520i	− 0.175723i
$448$	0	0
$449$	6.53397i	0.308357i	0.988043	+	0.154179i	$0.0492731\pi$
$449$	6.53397i	0.308357i	−0.988043	+	0.154179i	$0.950727\pi$
$450$	0	0
$451$	−26.2569	−1.23639
$452$	0	0
$453$	− 3.32482i	− 0.156214i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.0800i	1.54742i	0.633541	+	0.773709i	$0.281601\pi$
$457$	33.0800i	1.54742i	−0.633541	+	0.773709i	$0.718399\pi$
$458$	0	0
$459$	−4.92434	−0.229849
$460$	0	0
$461$	0.551788i	0.0256993i	0.999917	+	0.0128497i	$0.00409029\pi$
$461$	0.551788i	0.0256993i	−0.999917	+	0.0128497i	$0.995910\pi$
$462$	0	0
$463$	2.34264i	0.108872i	0.998517	+	0.0544359i	$0.0173360\pi$
$463$	2.34264i	0.108872i	−0.998517	+	0.0544359i	$0.982664\pi$
$464$	0	0
$465$	0.151312	0.00701693
$466$	0	0
$467$	−15.1735	−0.702146	−0.351073	−	0.936348i	$-0.614183\pi$
$467$	−15.1735	−0.702146	−0.351073	+	0.936348i	$0.614183\pi$
$468$	0	0
$469$	−15.9943	−0.738547
$470$	0	0
$471$	−2.32482	−0.107122
$472$	0	0
$473$	− 60.3114i	− 2.77312i
$474$	0	0
$475$	− 33.8332i	− 1.55238i
$476$	0	0
$477$	−1.59952	−0.0732372
$478$	0	0
$479$	− 1.58170i	− 0.0722699i	−0.999347	−	0.0361350i	$-0.988495\pi$
$479$	− 1.58170i	− 0.0722699i	0.999347	−	0.0361350i	$-0.0115046\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	19.8487i	0.903147i
$484$	0	0
$485$	−0.952264	−0.0432401
$486$	0	0
$487$	− 17.2091i	− 0.779821i	−0.920853	−	0.389910i	$-0.872506\pi$
$487$	− 17.2091i	− 0.779821i	0.920853	−	0.389910i	$-0.127494\pi$
$488$	0	0
$489$	− 6.86651i	− 0.310514i
$490$	0	0
$491$	23.7075	1.06990	0.534952	−	0.844883i	$-0.320330\pi$
$491$	23.7075	1.06990	0.534952	+	0.844883i	$0.320330\pi$
$492$	0	0
$493$	−19.0400	−0.857519
$494$	0	0
$495$	−0.675180	−0.0303471
$496$	0	0
$497$	4.15131	0.186212
$498$	0	0
$499$	6.11567i	0.273775i	0.990587	+	0.136888i	$0.0437099\pi$
$499$	6.11567i	0.273775i	−0.990587	+	0.136888i	$0.956290\pi$
$500$	0	0
$501$	− 21.8130i	− 0.974535i
$502$	0	0
$503$	−35.7075	−1.59212	−0.796059	−	0.605219i	$-0.793085\pi$
$503$	−35.7075	−1.59212	−0.796059	+	0.605219i	$0.793085\pi$
$504$	0	0
$505$	0.762933i	0.0339501i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	34.2135i	1.51649i	0.651971	+	0.758244i	$0.273942\pi$
$509$	34.2135i	1.51649i	−0.651971	+	0.758244i	$0.726058\pi$
$510$	0	0
$511$	−12.8208	−0.567157
$512$	0	0
$513$	6.79085i	0.299823i
$514$	0	0
$515$	0.826492i	0.0364196i
$516$	0	0
$517$	−45.8130	−2.01486
$518$	0	0
$519$	−3.58170	−0.157219
$520$	0	0
$521$	38.4704	1.68542	0.842710	−	0.538368i	$-0.180959\pi$
$521$	38.4704	1.68542	0.842710	+	0.538368i	$0.180959\pi$
$522$	0	0
$523$	5.44049	0.237896	0.118948	−	0.992900i	$-0.462048\pi$
$523$	5.44049	0.237896	0.118948	+	0.992900i	$0.462048\pi$
$524$	0	0
$525$	19.5518i	0.853310i
$526$	0	0
$527$	− 5.58170i	− 0.243143i
$528$	0	0
$529$	2.58170	0.112248
$530$	0	0
$531$	11.8487i	0.514189i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1.20915i	0.0522760i
$536$	0	0
$537$	−1.20915	−0.0521786
$538$	0	0
$539$	42.4882i	1.83010i
$540$	0	0
$541$	37.3369i	1.60524i	0.596491	+	0.802620i	$0.296561\pi$
$541$	37.3369i	1.60524i	−0.596491	+	0.802620i	$0.703439\pi$
$542$	0	0
$543$	18.9243	0.812121
$544$	0	0
$545$	−1.69738	−0.0727076
$546$	0	0
$547$	−19.2391	−0.822603	−0.411301	−	0.911499i	$-0.634926\pi$
$547$	−19.2391	−0.822603	−0.411301	+	0.911499i	$0.634926\pi$
$548$	0	0
$549$	5.79085	0.247148
$550$	0	0
$551$	26.2569i	1.11858i
$552$	0	0
$553$	28.4482i	1.20974i
$554$	0	0
$555$	−0.924344	−0.0392362
$556$	0	0
$557$	41.7952i	1.77092i	0.464715	+	0.885460i	$0.346157\pi$
$557$	41.7952i	1.77092i	−0.464715	+	0.885460i	$0.653843\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	24.9065i	1.05155i
$562$	0	0
$563$	−31.0121	−1.30700	−0.653502	−	0.756925i	$-0.726701\pi$
$563$	−31.0121	−1.30700	−0.653502	+	0.756925i	$0.726701\pi$
$564$	0	0
$565$	− 2.52625i	− 0.106280i
$566$	0	0
$567$	− 3.92434i	− 0.164807i
$568$	0	0
$569$	35.8130	1.50136	0.750681	−	0.660665i	$-0.229726\pi$
$569$	35.8130	1.50136	0.750681	+	0.660665i	$0.229726\pi$
$570$	0	0
$571$	25.2091	1.05497	0.527485	−	0.849564i	$-0.323135\pi$
$571$	25.2091	1.05497	0.527485	+	0.849564i	$0.323135\pi$
$572$	0	0
$573$	−7.84869	−0.327884
$574$	0	0
$575$	25.1990	1.05087
$576$	0	0
$577$	− 36.8988i	− 1.53612i	−0.640380	−	0.768059i	$-0.721223\pi$
$577$	− 36.8988i	− 1.53612i	0.640380	−	0.768059i	$-0.278777\pi$
$578$	0	0
$579$	7.11567i	0.295717i
$580$	0	0
$581$	44.4425	1.84379
$582$	0	0
$583$	8.09013i	0.335059i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.26698i	− 0.0935684i	−0.998905	−	0.0467842i	$-0.985103\pi$
$587$	− 2.26698i	− 0.0935684i	0.998905	−	0.0467842i	$-0.0148973\pi$
$588$	0	0
$589$	−7.69738	−0.317165
$590$	0	0
$591$	− 23.4304i	− 0.963798i
$592$	0	0
$593$	− 3.49395i	− 0.143479i	−0.997423	−	0.0717397i	$-0.977145\pi$
$593$	− 3.49395i	− 0.143479i	0.997423	−	0.0717397i	$-0.0228551\pi$
$594$	0	0
$595$	−2.57971	−0.105758
$596$	0	0
$597$	−13.6574	−0.558959
$598$	0	0
$599$	17.1990	0.702734	0.351367	−	0.936238i	$-0.385717\pi$
$599$	17.1990	0.702734	0.351367	+	0.936238i	$0.385717\pi$
$600$	0	0
$601$	−2.51615	−0.102636	−0.0513179	−	0.998682i	$-0.516342\pi$
$601$	−2.51615	−0.102636	−0.0513179	+	0.998682i	$0.516342\pi$
$602$	0	0
$603$	− 4.07566i	− 0.165974i
$604$	0	0
$605$	1.94654i	0.0791381i
$606$	0	0
$607$	−5.04774	−0.204881	−0.102441	−	0.994739i	$-0.532665\pi$
$607$	−5.04774	−0.204881	−0.102441	+	0.994739i	$0.532665\pi$
$608$	0	0
$609$	− 15.1735i	− 0.614862i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.25688i	0.131544i	0.997835	+	0.0657722i	$0.0209511\pi$
$613$	3.25688i	0.131544i	−0.997835	+	0.0657722i	$0.979049\pi$
$614$	0	0
$615$	0.693000	0.0279445
$616$	0	0
$617$	24.2391i	0.975828i	0.872892	+	0.487914i	$0.162242\pi$
$617$	24.2391i	0.975828i	−0.872892	+	0.487914i	$0.837758\pi$
$618$	0	0
$619$	8.90215i	0.357808i	0.983867	+	0.178904i	$0.0572551\pi$
$619$	8.90215i	0.357808i	−0.983867	+	0.178904i	$0.942745\pi$
$620$	0	0
$621$	−5.05784	−0.202964
$622$	0	0
$623$	30.3470	1.21583
$624$	0	0
$625$	24.7330	0.989321
$626$	0	0
$627$	34.3470	1.37169
$628$	0	0
$629$	34.0979i	1.35957i
$630$	0	0
$631$	− 20.2969i	− 0.808007i	−0.914757	−	0.404003i	$-0.867618\pi$
$631$	− 20.2969i	− 0.808007i	0.914757	−	0.404003i	$-0.132382\pi$
$632$	0	0
$633$	19.2492	0.765086
$634$	0	0
$635$	− 0.967705i	− 0.0384022i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.05784i	0.0418473i
$640$	0	0
$641$	15.7253	0.621112	0.310556	−	0.950555i	$-0.399485\pi$
$641$	15.7253	0.621112	0.310556	+	0.950555i	$0.399485\pi$
$642$	0	0
$643$	1.66746i	0.0657582i	0.999459	+	0.0328791i	$0.0104676\pi$
$643$	1.66746i	0.0657582i	−0.999459	+	0.0328791i	$0.989532\pi$
$644$	0	0
$645$	1.59180i	0.0626772i
$646$	0	0
$647$	−37.4304	−1.47154	−0.735770	−	0.677231i	$-0.763180\pi$
$647$	−37.4304	−1.47154	−0.735770	+	0.677231i	$0.763180\pi$
$648$	0	0
$649$	59.9287	2.35241
$650$	0	0
$651$	4.44821	0.174339
$652$	0	0
$653$	−46.6140	−1.82415	−0.912073	−	0.410027i	$-0.865519\pi$
$653$	−46.6140	−1.82415	−0.912073	+	0.410027i	$0.865519\pi$
$654$	0	0
$655$	− 1.33016i	− 0.0519737i
$656$	0	0
$657$	− 3.26698i	− 0.127457i
$658$	0	0
$659$	3.46603	0.135017	0.0675087	−	0.997719i	$-0.478495\pi$
$659$	3.46603	0.135017	0.0675087	+	0.997719i	$0.478495\pi$
$660$	0	0
$661$	− 18.9543i	− 0.737235i	−0.929581	−	0.368618i	$-0.879831\pi$
$661$	− 18.9543i	− 0.737235i	0.929581	−	0.368618i	$-0.120169\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.55751i	0.137954i
$666$	0	0
$667$	−19.5562	−0.757218
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	− 29.2892i	− 1.13070i
$672$	0	0
$673$	−13.9321	−0.537042	−0.268521	−	0.963274i	$-0.586535\pi$
$673$	−13.9321	−0.537042	−0.268521	+	0.963274i	$0.586535\pi$
$674$	0	0
$675$	−4.98218	−0.191764
$676$	0	0
$677$	−2.68528	−0.103204	−0.0516018	−	0.998668i	$-0.516433\pi$
$677$	−2.68528	−0.103204	−0.0516018	+	0.998668i	$0.516433\pi$
$678$	0	0
$679$	−27.9943	−1.07432
$680$	0	0
$681$	23.0222i	0.882212i
$682$	0	0
$683$	2.26698i	0.0867437i	0.999059	+	0.0433719i	$0.0138100\pi$
$683$	2.26698i	0.0867437i	−0.999059	+	0.0433719i	$0.986190\pi$
$684$	0	0
$685$	−1.26261	−0.0482418
$686$	0	0
$687$	8.11567i	0.309632i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.49395i	0.0948744i	0.998874	+	0.0474372i	$0.0151054\pi$
$691$	2.49395i	0.0948744i	−0.998874	+	0.0474372i	$0.984895\pi$
$692$	0	0
$693$	−19.8487	−0.753989
$694$	0	0
$695$	0.453936i	0.0172188i
$696$	0	0
$697$	− 25.5639i	− 0.968301i
$698$	0	0
$699$	6.53397	0.247137
$700$	0	0
$701$	37.8487	1.42953	0.714763	−	0.699367i	$-0.246535\pi$
$701$	37.8487	1.42953	0.714763	+	0.699367i	$0.246535\pi$
$702$	0	0
$703$	47.0222	1.77348
$704$	0	0
$705$	1.20915	0.0455391
$706$	0	0
$707$	22.4284i	0.843507i
$708$	0	0
$709$	− 16.4049i	− 0.616097i	−0.951371	−	0.308049i	$-0.900324\pi$
$709$	− 16.4049i	− 0.616097i	0.951371	−	0.308049i	$-0.0996759\pi$
$710$	0	0
$711$	−7.24916	−0.271865
$712$	0	0
$713$	− 5.73302i	− 0.214703i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 11.3248i	− 0.422933i
$718$	0	0
$719$	16.0800	0.599684	0.299842	−	0.953989i	$-0.403066\pi$
$719$	16.0800	0.599684	0.299842	+	0.953989i	$0.403066\pi$
$720$	0	0
$721$	24.2969i	0.904864i
$722$	0	0
$723$	− 11.5995i	− 0.431391i
$724$	0	0
$725$	−19.2636	−0.715434
$726$	0	0
$727$	44.6695	1.65670	0.828349	−	0.560212i	$-0.189280\pi$
$727$	44.6695	1.65670	0.828349	+	0.560212i	$0.189280\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	58.7196	2.17182
$732$	0	0
$733$	27.9422i	1.03207i	0.856568	+	0.516034i	$0.172592\pi$
$733$	27.9422i	1.03207i	−0.856568	+	0.516034i	$0.827408\pi$
$734$	0	0
$735$	− 1.12140i	− 0.0413633i
$736$	0	0
$737$	−20.6140	−0.759326
$738$	0	0
$739$	11.4660i	0.421785i	0.977509	+	0.210892i	$0.0676370\pi$
$739$	11.4660i	0.421785i	−0.977509	+	0.210892i	$0.932363\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19.3147i	0.708588i	0.935134	+	0.354294i	$0.115279\pi$
$743$	19.3147i	0.708588i	−0.935134	+	0.354294i	$0.884721\pi$
$744$	0	0
$745$	0.495949	0.0181702
$746$	0	0
$747$	11.3248i	0.414353i
$748$	0	0
$749$	35.5461i	1.29882i
$750$	0	0
$751$	9.97446	0.363973	0.181987	−	0.983301i	$-0.441747\pi$
$751$	9.97446	0.363973	0.181987	+	0.983301i	$0.441747\pi$
$752$	0	0
$753$	−10.1157	−0.368636
$754$	0	0
$755$	0.443837	0.0161529
$756$	0	0
$757$	−22.2313	−0.808012	−0.404006	−	0.914756i	$-0.632383\pi$
$757$	−22.2313	−0.808012	−0.404006	+	0.914756i	$0.632383\pi$
$758$	0	0
$759$	25.5817i	0.928557i
$760$	0	0
$761$	− 8.26698i	− 0.299678i	−0.988710	−	0.149839i	$-0.952124\pi$
$761$	− 8.26698i	− 0.299678i	0.988710	−	0.149839i	$-0.0478755\pi$
$762$	0	0
$763$	−49.8988	−1.80646
$764$	0	0
$765$	− 0.657360i	− 0.0237669i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.5138i	0.523380i	0.965152	+	0.261690i	$0.0842798\pi$
$769$	14.5138i	0.523380i	−0.965152	+	0.261690i	$0.915720\pi$
$770$	0	0
$771$	1.19133	0.0429046
$772$	0	0
$773$	24.1157i	0.867380i	0.901062	+	0.433690i	$0.142789\pi$
$773$	24.1157i	0.867380i	−0.901062	+	0.433690i	$0.857211\pi$
$774$	0	0
$775$	− 5.64726i	− 0.202856i
$776$	0	0
$777$	−27.1735	−0.974844
$778$	0	0
$779$	−35.2535	−1.26309
$780$	0	0
$781$	5.35036	0.191451
$782$	0	0
$783$	3.86651	0.138178
$784$	0	0
$785$	− 0.310345i	− 0.0110767i
$786$	0	0
$787$	35.0979i	1.25110i	0.780183	+	0.625552i	$0.215126\pi$
$787$	35.0979i	1.25110i	−0.780183	+	0.625552i	$0.784874\pi$
$788$	0	0
$789$	−20.9065	−0.744292
$790$	0	0
$791$	− 74.2656i	− 2.64058i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 0.213524i	− 0.00757290i
$796$	0	0
$797$	45.0323	1.59513	0.797563	−	0.603236i	$-0.206122\pi$
$797$	45.0323	1.59513	0.797563	+	0.603236i	$0.206122\pi$
$798$	0	0
$799$	− 44.6039i	− 1.57797i
$800$	0	0
$801$	7.73302i	0.273233i
$802$	0	0
$803$	−16.5239	−0.583115
$804$	0	0
$805$	−2.64964	−0.0933875
$806$	0	0
$807$	5.84869	0.205884
$808$	0	0
$809$	47.8208	1.68129	0.840644	−	0.541588i	$-0.182177\pi$
$809$	47.8208	1.68129	0.840644	+	0.541588i	$0.182177\pi$
$810$	0	0
$811$	6.56388i	0.230489i	0.993337	+	0.115245i	$0.0367652\pi$
$811$	6.56388i	0.230489i	−0.993337	+	0.115245i	$0.963235\pi$
$812$	0	0
$813$	23.0979i	0.810077i
$814$	0	0
$815$	0.916623	0.0321079
$816$	0	0
$817$	− 80.9765i	− 2.83301i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.81305i	− 0.133076i	−0.997784	−	0.0665381i	$-0.978805\pi$
$821$	− 3.81305i	− 0.133076i	0.997784	−	0.0665381i	$-0.0211954\pi$
$822$	0	0
$823$	10.3470	0.360674	0.180337	−	0.983605i	$-0.442281\pi$
$823$	10.3470	0.360674	0.180337	+	0.983605i	$0.442281\pi$
$824$	0	0
$825$	25.1990i	0.877318i
$826$	0	0
$827$	34.6496i	1.20489i	0.798162	+	0.602443i	$0.205806\pi$
$827$	34.6496i	1.20489i	−0.798162	+	0.602443i	$0.794194\pi$
$828$	0	0
$829$	15.1056	0.524638	0.262319	−	0.964981i	$-0.415513\pi$
$829$	15.1056	0.524638	0.262319	+	0.964981i	$0.415513\pi$
$830$	0	0
$831$	4.80867	0.166811
$832$	0	0
$833$	−41.3668	−1.43328
$834$	0	0
$835$	2.91187	0.100769
$836$	0	0
$837$	1.13349i	0.0391792i
$838$	0	0
$839$	44.8964i	1.55000i	0.631963	+	0.774998i	$0.282249\pi$
$839$	44.8964i	1.55000i	−0.631963	+	0.774998i	$0.717751\pi$
$840$	0	0
$841$	−14.0501	−0.484487
$842$	0	0
$843$	5.60962i	0.193206i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	57.2236i	1.96623i
$848$	0	0
$849$	25.8887	0.888498
$850$	0	0
$851$	35.0222i	1.20055i
$852$	0	0
$853$	− 18.9543i	− 0.648982i	−0.945889	−	0.324491i	$-0.894807\pi$
$853$	− 18.9543i	− 0.648982i	0.945889	−	0.324491i	$-0.105193\pi$
$854$	0	0
$855$	−0.906524	−0.0310025
$856$	0	0
$857$	24.2391	0.827991	0.413995	−	0.910279i	$-0.364133\pi$
$857$	24.2391	0.827991	0.413995	+	0.910279i	$0.364133\pi$
$858$	0	0
$859$	14.1913	0.484202	0.242101	−	0.970251i	$-0.422163\pi$
$859$	14.1913	0.484202	0.242101	+	0.970251i	$0.422163\pi$
$860$	0	0
$861$	20.3726	0.694295
$862$	0	0
$863$	31.7075i	1.07934i	0.841878	+	0.539668i	$0.181450\pi$
$863$	31.7075i	1.07934i	−0.841878	+	0.539668i	$0.818550\pi$
$864$	0	0
$865$	− 0.478129i	− 0.0162569i
$866$	0	0
$867$	−7.24916	−0.246195
$868$	0	0
$869$	36.6651i	1.24378i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 7.13349i	− 0.241432i
$874$	0	0
$875$	−5.22935	−0.176784
$876$	0	0
$877$	− 40.7374i	− 1.37560i	−0.725898	−	0.687802i	$-0.758576\pi$
$877$	− 40.7374i	− 1.37560i	0.725898	−	0.687802i	$-0.241424\pi$
$878$	0	0
$879$	25.4482i	0.858347i
$880$	0	0
$881$	20.3904	0.686969	0.343485	−	0.939158i	$-0.388393\pi$
$881$	20.3904	0.686969	0.343485	+	0.939158i	$0.388393\pi$
$882$	0	0
$883$	3.93444	0.132405	0.0662023	−	0.997806i	$-0.478912\pi$
$883$	3.93444	0.132405	0.0662023	+	0.997806i	$0.478912\pi$
$884$	0	0
$885$	−1.58170	−0.0531684
$886$	0	0
$887$	−15.3147	−0.514218	−0.257109	−	0.966382i	$-0.582770\pi$
$887$	−15.3147	−0.514218	−0.257109	+	0.966382i	$0.582770\pi$
$888$	0	0
$889$	− 28.4482i	− 0.954122i
$890$	0	0
$891$	− 5.05784i	− 0.169444i
$892$	0	0
$893$	−61.5104	−2.05837
$894$	0	0
$895$	− 0.161411i	− 0.00539539i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.38266i	0.146170i
$900$	0	0
$901$	−7.87661	−0.262408
$902$	0	0
$903$	46.7952i	1.55725i
$904$	0	0
$905$	2.52625i	0.0839753i
$906$	0	0
$907$	50.5784	1.67943	0.839713	−	0.543030i	$-0.182723\pi$
$907$	50.5784	1.67943	0.839713	+	0.543030i	$0.182723\pi$
$908$	0	0
$909$	−5.71520	−0.189561
$910$	0	0
$911$	10.3470	0.342812	0.171406	−	0.985200i	$-0.445169\pi$
$911$	10.3470	0.342812	0.171406	+	0.985200i	$0.445169\pi$
$912$	0	0
$913$	57.2791	1.89566
$914$	0	0
$915$	0.773032i	0.0255556i
$916$	0	0
$917$	− 39.1036i	− 1.29131i
$918$	0	0
$919$	2.95226	0.0973862	0.0486931	−	0.998814i	$-0.484494\pi$
$919$	2.95226	0.0973862	0.0486931	+	0.998814i	$0.484494\pi$
$920$	0	0
$921$	0.458312i	0.0151019i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	34.4983i	1.13430i
$926$	0	0
$927$	−6.19133	−0.203350
$928$	0	0
$929$	− 8.35474i	− 0.274110i	−0.990563	−	0.137055i	$-0.956236\pi$
$929$	− 8.35474i	− 0.274110i	0.990563	−	0.137055i	$-0.0437637\pi$
$930$	0	0
$931$	57.0464i	1.86962i
$932$	0	0
$933$	22.7909	0.746139
$934$	0	0
$935$	−3.32482	−0.108733
$936$	0	0
$937$	20.3648	0.665290	0.332645	−	0.943052i	$-0.392059\pi$
$937$	20.3648	0.665290	0.332645	+	0.943052i	$0.392059\pi$
$938$	0	0
$939$	−28.5639	−0.932147
$940$	0	0
$941$	10.5138i	0.342739i	0.985207	+	0.171370i	$0.0548192\pi$
$941$	10.5138i	0.342739i	−0.985207	+	0.171370i	$0.945181\pi$
$942$	0	0
$943$	− 26.2569i	− 0.855042i
$944$	0	0
$945$	0.523868	0.0170414
$946$	0	0
$947$	− 33.5817i	− 1.09126i	−0.838027	−	0.545629i	$-0.816291\pi$
$947$	− 33.5817i	− 1.09126i	0.838027	−	0.545629i	$-0.183709\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 23.9465i	− 0.776520i
$952$	0	0
$953$	−58.9765	−1.91043	−0.955217	−	0.295905i	$-0.904379\pi$
$953$	−58.9765	−1.91043	−0.955217	+	0.295905i	$0.904379\pi$
$954$	0	0
$955$	− 1.04774i	− 0.0339040i
$956$	0	0
$957$	− 19.5562i	− 0.632161i
$958$	0	0
$959$	−37.1177	−1.19859
$960$	0	0
$961$	29.7152	0.958555
$962$	0	0
$963$	−9.05784	−0.291885
$964$	0	0
$965$	−0.949885	−0.0305779
$966$	0	0
$967$	− 58.9509i	− 1.89573i	−0.318667	−	0.947867i	$-0.603235\pi$
$967$	− 58.9509i	− 1.89573i	0.318667	−	0.947867i	$-0.396765\pi$
$968$	0	0
$969$	33.4405i	1.07426i
$970$	0	0
$971$	52.1601	1.67390	0.836948	−	0.547282i	$-0.184338\pi$
$971$	52.1601	1.67390	0.836948	+	0.547282i	$0.184338\pi$
$972$	0	0
$973$	13.3446i	0.427809i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.5538i	0.753552i	0.926304	+	0.376776i	$0.122967\pi$
$977$	23.5538i	0.753552i	−0.926304	+	0.376776i	$0.877033\pi$
$978$	0	0
$979$	39.1123	1.25004
$980$	0	0
$981$	− 12.7152i	− 0.405965i
$982$	0	0
$983$	− 0.453936i	− 0.0144783i	−0.999974	−	0.00723916i	$-0.997696\pi$
$983$	− 0.453936i	− 0.0144783i	0.999974	−	0.00723916i	$-0.00230431\pi$
$984$	0	0
$985$	3.12777	0.0996590
$986$	0	0
$987$	35.5461	1.13144
$988$	0	0
$989$	60.3114	1.91779
$990$	0	0
$991$	−1.72292	−0.0547303	−0.0273651	−	0.999626i	$-0.508712\pi$
$991$	−1.72292	−0.0547303	−0.0273651	+	0.999626i	$0.508712\pi$
$992$	0	0
$993$	25.1335i	0.797587i
$994$	0	0
$995$	− 1.82315i	− 0.0577977i
$996$	0	0
$997$	−12.3248	−0.390331	−0.195165	−	0.980770i	$-0.562524\pi$
$997$	−12.3248	−0.390331	−0.195165	+	0.980770i	$0.562524\pi$
$998$	0	0
$999$	− 6.92434i	− 0.219077i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.m.337.3		6
13.5	odd	4		4056.2.a.y.1.2		3
13.7	odd	12		312.2.q.e.289.2	yes	6
13.8	odd	4		4056.2.a.z.1.2		3
13.11	odd	12		312.2.q.e.217.2	✓	6
13.12	even	2	inner	4056.2.c.m.337.4		6
39.11	even	12		936.2.t.h.217.2		6
39.20	even	12		936.2.t.h.289.2		6
52.7	even	12		624.2.q.j.289.2		6
52.11	even	12		624.2.q.j.529.2		6
52.31	even	4		8112.2.a.cl.1.2		3
52.47	even	4		8112.2.a.ck.1.2		3
156.11	odd	12		1872.2.t.u.1153.2		6
156.59	odd	12		1872.2.t.u.289.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.e.217.2	✓	6	13.11	odd	12
312.2.q.e.289.2	yes	6	13.7	odd	12
624.2.q.j.289.2		6	52.7	even	12
624.2.q.j.529.2		6	52.11	even	12
936.2.t.h.217.2		6	39.11	even	12
936.2.t.h.289.2		6	39.20	even	12
1872.2.t.u.289.2		6	156.59	odd	12
1872.2.t.u.1153.2		6	156.11	odd	12
4056.2.a.y.1.2		3	13.5	odd	4
4056.2.a.z.1.2		3	13.8	odd	4
4056.2.c.m.337.3		6	1.1	even	1	trivial
4056.2.c.m.337.4		6	13.12	even	2	inner
8112.2.a.ck.1.2		3	52.47	even	4
8112.2.a.cl.1.2		3	52.31	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.133492i q^{5} -3.92434i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.133492i q^{5} -3.92434i q^{7} +1.00000 q^{9} -5.05784i q^{11} +0.133492i q^{15} +4.92434 q^{17} -6.79085i q^{19} +3.92434i q^{21} +5.05784 q^{23} +4.98218 q^{25} -1.00000 q^{27} -3.86651 q^{29} -1.13349i q^{31} +5.05784i q^{33} -0.523868 q^{35} +6.92434i q^{37} -5.19133i q^{41} +11.9243 q^{43} -0.133492i q^{45} -9.05784i q^{47} -8.40048 q^{49} -4.92434 q^{51} -1.59952 q^{53} -0.675180 q^{55} +6.79085i q^{57} +11.8487i q^{59} +5.79085 q^{61} -3.92434i q^{63} -4.07566i q^{67} -5.05784 q^{69} +1.05784i q^{71} -3.26698i q^{73} -4.98218 q^{75} -19.8487 q^{77} -7.24916 q^{79} +1.00000 q^{81} +11.3248i q^{83} -0.657360i q^{85} +3.86651 q^{87} +7.73302i q^{89} +1.13349i q^{93} -0.906524 q^{95} -7.13349i q^{97} -5.05784i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} - 30 q^{25} - 6 q^{27} - 24 q^{29} + 24 q^{35} + 42 q^{43} - 48 q^{49} - 12 q^{53} - 36 q^{55} + 6 q^{61} + 30 q^{75} - 60 q^{77} + 18 q^{79} + 6 q^{81} + 24 q^{87} + 84 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 30 * q^25 - 6 * q^27 - 24 * q^29 + 24 * q^35 + 42 * q^43 - 48 * q^49 - 12 * q^53 - 36 * q^55 + 6 * q^61 + 30 * q^75 - 60 * q^77 + 18 * q^79 + 6 * q^81 + 24 * q^87 + 84 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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