LMFDB - Embedded newform 4056.2.c.r.337.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4056,2,Mod(337,4056)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [12,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,-18,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 601x^{8} + 4599x^{6} + 17849x^{4} + 31203x^{2} + 16129 $ x^12 + 39x^10 + 601x^8 + 4599x^6 + 17849x^4 + 31203*x^2 + 16129
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$1.88227i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.r.337.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.88227i q^{5} -1.29156i q^{7} +1.00000 q^{9} -4.19367i q^{11} +1.88227i q^{15} +1.68421 q^{17} +1.19025i q^{19} -1.29156i q^{21} -3.01499 q^{23} +1.45707 q^{25} +1.00000 q^{27} +3.15000 q^{29} -1.47463i q^{31} -4.19367i q^{33} +2.43106 q^{35} -8.15343i q^{37} -10.0181i q^{41} -10.7587 q^{43} +1.88227i q^{45} +5.31909i q^{47} +5.33187 q^{49} +1.68421 q^{51} +8.29590 q^{53} +7.89361 q^{55} +1.19025i q^{57} +3.03369i q^{59} +7.84487 q^{61} -1.29156i q^{63} +4.50497i q^{67} -3.01499 q^{69} -3.85816i q^{71} -1.76494i q^{73} +1.45707 q^{75} -5.41637 q^{77} -12.6593 q^{79} +1.00000 q^{81} -15.6280i q^{83} +3.17013i q^{85} +3.15000 q^{87} -10.6679i q^{89} -1.47463i q^{93} -2.24036 q^{95} -2.38541i q^{97} -4.19367i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 12 q^{9} - 18 q^{17} - 24 q^{23} - 18 q^{25} + 12 q^{27} + 14 q^{29} + 12 q^{35} - 30 q^{43} - 26 q^{49} - 18 q^{51} + 44 q^{53} + 6 q^{55} + 50 q^{61} - 24 q^{69} - 18 q^{75} - 90 q^{77}+ \cdots - 94 q^{95}+O(q^{100}) $ 12 * q + 12 * q^3 + 12 * q^9 - 18 * q^17 - 24 * q^23 - 18 * q^25 + 12 * q^27 + 14 * q^29 + 12 * q^35 - 30 * q^43 - 26 * q^49 - 18 * q^51 + 44 * q^53 + 6 * q^55 + 50 * q^61 - 24 * q^69 - 18 * q^75 - 90 * q^77 + 28 * q^79 + 12 * q^81 + 14 * q^87 - 94 * q^95

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
$5$	1.88227i	0.841776i	0.907113	+	0.420888i	$0.138281\pi$
$5$	1.88227i	0.841776i	−0.907113	+	0.420888i	$0.861719\pi$
$6$	0	0
$7$	− 1.29156i	− 0.488164i	−0.969755	−	0.244082i	$-0.921513\pi$
$7$	− 1.29156i	− 0.488164i	0.969755	−	0.244082i	$-0.0784866\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 4.19367i	− 1.26444i	−0.774790	−	0.632219i	$-0.782144\pi$
$11$	− 4.19367i	− 1.26444i	0.774790	−	0.632219i	$-0.217856\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.88227i	0.486000i
$16$	0	0
$17$	1.68421	0.408480	0.204240	−	0.978921i	$-0.434528\pi$
$17$	1.68421	0.408480	0.204240	+	0.978921i	$0.434528\pi$
$18$	0	0
$19$	1.19025i	0.273061i	0.990636	+	0.136531i	$0.0435952\pi$
$19$	1.19025i	0.273061i	−0.990636	+	0.136531i	$0.956405\pi$
$20$	0	0
$21$	− 1.29156i	− 0.281841i
$22$	0	0
$23$	−3.01499	−0.628670	−0.314335	−	0.949312i	$-0.601781\pi$
$23$	−3.01499	−0.628670	−0.314335	+	0.949312i	$0.601781\pi$
$24$	0	0
$25$	1.45707	0.291413
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.15000	0.584941	0.292471	−	0.956275i	$-0.405523\pi$
$29$	3.15000	0.584941	0.292471	+	0.956275i	$0.405523\pi$
$30$	0	0
$31$	− 1.47463i	− 0.264851i	−0.991193	−	0.132425i	$-0.957723\pi$
$31$	− 1.47463i	− 0.264851i	0.991193	−	0.132425i	$-0.0422765\pi$
$32$	0	0
$33$	− 4.19367i	− 0.730024i
$34$	0	0
$35$	2.43106	0.410924
$36$	0	0
$37$	− 8.15343i	− 1.34041i	−0.742174	−	0.670207i	$-0.766205\pi$
$37$	− 8.15343i	− 1.34041i	0.742174	−	0.670207i	$-0.233795\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.0181i	− 1.56457i	−0.622921	−	0.782285i	$-0.714054\pi$
$41$	− 10.0181i	− 1.56457i	0.622921	−	0.782285i	$-0.285946\pi$
$42$	0	0
$43$	−10.7587	−1.64069	−0.820345	−	0.571869i	$-0.806218\pi$
$43$	−10.7587	−1.64069	−0.820345	+	0.571869i	$0.806218\pi$
$44$	0	0
$45$	1.88227i	0.280592i
$46$	0	0
$47$	5.31909i	0.775869i	0.921687	+	0.387934i	$0.126811\pi$
$47$	5.31909i	0.775869i	−0.921687	+	0.387934i	$0.873189\pi$
$48$	0	0
$49$	5.33187	0.761696
$50$	0	0
$51$	1.68421	0.235836
$52$	0	0
$53$	8.29590	1.13953	0.569765	−	0.821808i	$-0.307034\pi$
$53$	8.29590	1.13953	0.569765	+	0.821808i	$0.307034\pi$
$54$	0	0
$55$	7.89361	1.06437
$56$	0	0
$57$	1.19025i	0.157652i
$58$	0	0
$59$	3.03369i	0.394953i	0.980308	+	0.197477i	$0.0632746\pi$
$59$	3.03369i	0.394953i	−0.980308	+	0.197477i	$0.936725\pi$
$60$	0	0
$61$	7.84487	1.00443	0.502216	−	0.864742i	$-0.332518\pi$
$61$	7.84487	1.00443	0.502216	+	0.864742i	$0.332518\pi$
$62$	0	0
$63$	− 1.29156i	− 0.162721i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.50497i	0.550370i	0.961391	+	0.275185i	$0.0887390\pi$
$67$	4.50497i	0.550370i	−0.961391	+	0.275185i	$0.911261\pi$
$68$	0	0
$69$	−3.01499	−0.362963
$70$	0	0
$71$	− 3.85816i	− 0.457880i	−0.973441	−	0.228940i	$-0.926474\pi$
$71$	− 3.85816i	− 0.457880i	0.973441	−	0.228940i	$-0.0735259\pi$
$72$	0	0
$73$	− 1.76494i	− 0.206571i	−0.994652	−	0.103285i	$-0.967064\pi$
$73$	− 1.76494i	− 0.206571i	0.994652	−	0.103285i	$-0.0329355\pi$
$74$	0	0
$75$	1.45707	0.168248
$76$	0	0
$77$	−5.41637	−0.617253
$78$	0	0
$79$	−12.6593	−1.42428	−0.712140	−	0.702037i	$-0.752274\pi$
$79$	−12.6593	−1.42428	−0.712140	+	0.702037i	$0.752274\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 15.6280i	− 1.71540i	−0.514149	−	0.857701i	$-0.671892\pi$
$83$	− 15.6280i	− 1.71540i	0.514149	−	0.857701i	$-0.328108\pi$
$84$	0	0
$85$	3.17013i	0.343849i
$86$	0	0
$87$	3.15000	0.337716
$88$	0	0
$89$	− 10.6679i	− 1.13080i	−0.824818	−	0.565398i	$-0.808722\pi$
$89$	− 10.6679i	− 1.13080i	0.824818	−	0.565398i	$-0.191278\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 1.47463i	− 0.152912i
$94$	0	0
$95$	−2.24036	−0.229856
$96$	0	0
$97$	− 2.38541i	− 0.242201i	−0.992640	−	0.121101i	$-0.961358\pi$
$97$	− 2.38541i	− 0.242201i	0.992640	−	0.121101i	$-0.0386424\pi$
$98$	0	0
$99$	− 4.19367i	− 0.421479i
$100$	0	0
$101$	0.880061	0.0875693	0.0437847	−	0.999041i	$-0.486058\pi$
$101$	0.880061	0.0875693	0.0437847	+	0.999041i	$0.486058\pi$
$102$	0	0
$103$	−5.21107	−0.513462	−0.256731	−	0.966483i	$-0.582645\pi$
$103$	−5.21107	−0.513462	−0.256731	+	0.966483i	$0.582645\pi$
$104$	0	0
$105$	2.43106	0.237247
$106$	0	0
$107$	19.9637	1.92996	0.964981	−	0.262319i	$-0.0844874\pi$
$107$	19.9637	1.92996	0.964981	+	0.262319i	$0.0844874\pi$
$108$	0	0
$109$	− 18.5577i	− 1.77751i	−0.458386	−	0.888753i	$-0.651572\pi$
$109$	− 18.5577i	− 1.77751i	0.458386	−	0.888753i	$-0.348428\pi$
$110$	0	0
$111$	− 8.15343i	− 0.773889i
$112$	0	0
$113$	9.98409	0.939224	0.469612	−	0.882873i	$-0.344394\pi$
$113$	9.98409	0.939224	0.469612	+	0.882873i	$0.344394\pi$
$114$	0	0
$115$	− 5.67503i	− 0.529199i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2.17525i	− 0.199405i
$120$	0	0
$121$	−6.58685	−0.598805
$122$	0	0
$123$	− 10.0181i	− 0.903305i
$124$	0	0
$125$	12.1539i	1.08708i
$126$	0	0
$127$	14.7930	1.31267	0.656335	−	0.754469i	$-0.272106\pi$
$127$	14.7930	1.31267	0.656335	+	0.754469i	$0.272106\pi$
$128$	0	0
$129$	−10.7587	−0.947252
$130$	0	0
$131$	−0.834088	−0.0728746	−0.0364373	−	0.999336i	$-0.511601\pi$
$131$	−0.834088	−0.0728746	−0.0364373	+	0.999336i	$0.511601\pi$
$132$	0	0
$133$	1.53727	0.133299
$134$	0	0
$135$	1.88227i	0.162000i
$136$	0	0
$137$	− 6.20377i	− 0.530024i	−0.964245	−	0.265012i	$-0.914624\pi$
$137$	− 6.20377i	− 0.530024i	0.964245	−	0.265012i	$-0.0853759\pi$
$138$	0	0
$139$	−20.1616	−1.71008	−0.855042	−	0.518559i	$-0.826469\pi$
$139$	−20.1616	−1.71008	−0.855042	+	0.518559i	$0.826469\pi$
$140$	0	0
$141$	5.31909i	0.447948i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.92915i	0.492389i
$146$	0	0
$147$	5.33187	0.439766
$148$	0	0
$149$	16.2561i	1.33175i	0.746063	+	0.665876i	$0.231942\pi$
$149$	16.2561i	1.33175i	−0.746063	+	0.665876i	$0.768058\pi$
$150$	0	0
$151$	18.0930i	1.47239i	0.676769	+	0.736195i	$0.263379\pi$
$151$	18.0930i	1.47239i	−0.676769	+	0.736195i	$0.736621\pi$
$152$	0	0
$153$	1.68421	0.136160
$154$	0	0
$155$	2.77564	0.222945
$156$	0	0
$157$	15.0559	1.20159	0.600795	−	0.799403i	$-0.294851\pi$
$157$	15.0559	1.20159	0.600795	+	0.799403i	$0.294851\pi$
$158$	0	0
$159$	8.29590	0.657907
$160$	0	0
$161$	3.89404i	0.306894i
$162$	0	0
$163$	− 7.31660i	− 0.573080i	−0.958068	−	0.286540i	$-0.907495\pi$
$163$	− 7.31660i	− 0.573080i	0.958068	−	0.286540i	$-0.0925051\pi$
$164$	0	0
$165$	7.89361	0.614517
$166$	0	0
$167$	11.9586i	0.925385i	0.886519	+	0.462693i	$0.153117\pi$
$167$	11.9586i	0.925385i	−0.886519	+	0.462693i	$0.846883\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.19025i	0.0910204i
$172$	0	0
$173$	16.6231	1.26383	0.631915	−	0.775038i	$-0.282269\pi$
$173$	16.6231	1.26383	0.631915	+	0.775038i	$0.282269\pi$
$174$	0	0
$175$	− 1.88189i	− 0.142257i
$176$	0	0
$177$	3.03369i	0.228026i
$178$	0	0
$179$	−1.89790	−0.141856	−0.0709278	−	0.997481i	$-0.522596\pi$
$179$	−1.89790	−0.141856	−0.0709278	+	0.997481i	$0.522596\pi$
$180$	0	0
$181$	−16.3850	−1.21789	−0.608944	−	0.793213i	$-0.708406\pi$
$181$	−16.3850	−1.21789	−0.608944	+	0.793213i	$0.708406\pi$
$182$	0	0
$183$	7.84487	0.579909
$184$	0	0
$185$	15.3469	1.12833
$186$	0	0
$187$	− 7.06300i	− 0.516498i
$188$	0	0
$189$	− 1.29156i	− 0.0939471i
$190$	0	0
$191$	13.5859	0.983042	0.491521	−	0.870866i	$-0.336441\pi$
$191$	13.5859	0.983042	0.491521	+	0.870866i	$0.336441\pi$
$192$	0	0
$193$	− 24.8588i	− 1.78938i	−0.446689	−	0.894689i	$-0.647397\pi$
$193$	− 24.8588i	− 1.78938i	0.446689	−	0.894689i	$-0.352603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.47333i	0.318711i	0.987221	+	0.159356i	$0.0509416\pi$
$197$	4.47333i	0.318711i	−0.987221	+	0.159356i	$0.949058\pi$
$198$	0	0
$199$	4.49522	0.318658	0.159329	−	0.987226i	$-0.449067\pi$
$199$	4.49522	0.318658	0.159329	+	0.987226i	$0.449067\pi$
$200$	0	0
$201$	4.50497i	0.317756i
$202$	0	0
$203$	− 4.06842i	− 0.285547i
$204$	0	0
$205$	18.8568	1.31702
$206$	0	0
$207$	−3.01499	−0.209557
$208$	0	0
$209$	4.99150	0.345269
$210$	0	0
$211$	21.7861	1.49982	0.749908	−	0.661542i	$-0.230098\pi$
$211$	21.7861	1.49982	0.749908	+	0.661542i	$0.230098\pi$
$212$	0	0
$213$	− 3.85816i	− 0.264357i
$214$	0	0
$215$	− 20.2508i	− 1.38109i
$216$	0	0
$217$	−1.90457	−0.129291
$218$	0	0
$219$	− 1.76494i	− 0.119264i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.0555395i	0.00371920i	0.999998	+	0.00185960i	$0.000591929\pi$
$223$	0.0555395i	0.00371920i	−0.999998	+	0.00185960i	$0.999408\pi$
$224$	0	0
$225$	1.45707	0.0971378
$226$	0	0
$227$	− 5.24936i	− 0.348412i	−0.984709	−	0.174206i	$-0.944264\pi$
$227$	− 5.24936i	− 0.348412i	0.984709	−	0.174206i	$-0.0557359\pi$
$228$	0	0
$229$	21.6951i	1.43365i	0.697252	+	0.716826i	$0.254406\pi$
$229$	21.6951i	1.43365i	−0.697252	+	0.716826i	$0.745594\pi$
$230$	0	0
$231$	−5.41637	−0.356371
$232$	0	0
$233$	−9.34933	−0.612495	−0.306248	−	0.951952i	$-0.599074\pi$
$233$	−9.34933	−0.612495	−0.306248	+	0.951952i	$0.599074\pi$
$234$	0	0
$235$	−10.0119	−0.653107
$236$	0	0
$237$	−12.6593	−0.822309
$238$	0	0
$239$	− 14.7781i	− 0.955913i	−0.878383	−	0.477957i	$-0.841378\pi$
$239$	− 14.7781i	− 0.955913i	0.878383	−	0.477957i	$-0.158622\pi$
$240$	0	0
$241$	16.7668i	1.08004i	0.841651	+	0.540022i	$0.181584\pi$
$241$	16.7668i	1.08004i	−0.841651	+	0.540022i	$0.818416\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	10.0360i	0.641178i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 15.6280i	− 0.990387i
$250$	0	0
$251$	−1.44048	−0.0909222	−0.0454611	−	0.998966i	$-0.514476\pi$
$251$	−1.44048	−0.0909222	−0.0454611	+	0.998966i	$0.514476\pi$
$252$	0	0
$253$	12.6439i	0.794914i
$254$	0	0
$255$	3.17013i	0.198521i
$256$	0	0
$257$	21.0265	1.31160	0.655799	−	0.754935i	$-0.272332\pi$
$257$	21.0265	1.31160	0.655799	+	0.754935i	$0.272332\pi$
$258$	0	0
$259$	−10.5306	−0.654342
$260$	0	0
$261$	3.15000	0.194980
$262$	0	0
$263$	10.6833	0.658760	0.329380	−	0.944197i	$-0.393160\pi$
$263$	10.6833	0.658760	0.329380	+	0.944197i	$0.393160\pi$
$264$	0	0
$265$	15.6151i	0.959228i
$266$	0	0
$267$	− 10.6679i	− 0.652866i
$268$	0	0
$269$	18.3247	1.11728	0.558638	−	0.829412i	$-0.311324\pi$
$269$	18.3247	1.11728	0.558638	+	0.829412i	$0.311324\pi$
$270$	0	0
$271$	3.64076i	0.221161i	0.993867	+	0.110580i	$0.0352709\pi$
$271$	3.64076i	0.221161i	−0.993867	+	0.110580i	$0.964729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.11045i	− 0.368474i
$276$	0	0
$277$	−22.3601	−1.34349	−0.671744	−	0.740783i	$-0.734454\pi$
$277$	−22.3601	−1.34349	−0.671744	+	0.740783i	$0.734454\pi$
$278$	0	0
$279$	− 1.47463i	− 0.0882836i
$280$	0	0
$281$	− 2.39777i	− 0.143039i	−0.997439	−	0.0715195i	$-0.977215\pi$
$281$	− 2.39777i	− 0.143039i	0.997439	−	0.0715195i	$-0.0227848\pi$
$282$	0	0
$283$	1.64168	0.0975879	0.0487939	−	0.998809i	$-0.484462\pi$
$283$	1.64168	0.0975879	0.0487939	+	0.998809i	$0.484462\pi$
$284$	0	0
$285$	−2.24036	−0.132708
$286$	0	0
$287$	−12.9390	−0.763766
$288$	0	0
$289$	−14.1635	−0.833144
$290$	0	0
$291$	− 2.38541i	− 0.139835i
$292$	0	0
$293$	− 1.35348i	− 0.0790709i	−0.999218	−	0.0395355i	$-0.987412\pi$
$293$	− 1.35348i	− 0.0790709i	0.999218	−	0.0395355i	$-0.0125878\pi$
$294$	0	0
$295$	−5.71022	−0.332462
$296$	0	0
$297$	− 4.19367i	− 0.243341i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13.8955i	0.800925i
$302$	0	0
$303$	0.880061	0.0505582
$304$	0	0
$305$	14.7661i	0.845507i
$306$	0	0
$307$	23.9864i	1.36898i	0.729023	+	0.684489i	$0.239975\pi$
$307$	23.9864i	1.36898i	−0.729023	+	0.684489i	$0.760025\pi$
$308$	0	0
$309$	−5.21107	−0.296447
$310$	0	0
$311$	−6.23414	−0.353506	−0.176753	−	0.984255i	$-0.556559\pi$
$311$	−6.23414	−0.353506	−0.176753	+	0.984255i	$0.556559\pi$
$312$	0	0
$313$	−28.7120	−1.62290	−0.811450	−	0.584422i	$-0.801321\pi$
$313$	−28.7120	−1.62290	−0.811450	+	0.584422i	$0.801321\pi$
$314$	0	0
$315$	2.43106	0.136975
$316$	0	0
$317$	13.8082i	0.775545i	0.921755	+	0.387772i	$0.126755\pi$
$317$	13.8082i	0.775545i	−0.921755	+	0.387772i	$0.873245\pi$
$318$	0	0
$319$	− 13.2101i	− 0.739622i
$320$	0	0
$321$	19.9637	1.11426
$322$	0	0
$323$	2.00462i	0.111540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.5577i	− 1.02624i
$328$	0	0
$329$	6.86992	0.378751
$330$	0	0
$331$	7.06203i	0.388165i	0.980985	+	0.194082i	$0.0621729\pi$
$331$	7.06203i	0.388165i	−0.980985	+	0.194082i	$0.937827\pi$
$332$	0	0
$333$	− 8.15343i	− 0.446805i
$334$	0	0
$335$	−8.47956	−0.463288
$336$	0	0
$337$	15.9635	0.869585	0.434792	−	0.900531i	$-0.356822\pi$
$337$	15.9635	0.869585	0.434792	+	0.900531i	$0.356822\pi$
$338$	0	0
$339$	9.98409	0.542261
$340$	0	0
$341$	−6.18410	−0.334888
$342$	0	0
$343$	− 15.9274i	− 0.859996i
$344$	0	0
$345$	− 5.67503i	− 0.305533i
$346$	0	0
$347$	17.1822	0.922388	0.461194	−	0.887299i	$-0.347421\pi$
$347$	17.1822	0.922388	0.461194	+	0.887299i	$0.347421\pi$
$348$	0	0
$349$	− 28.2999i	− 1.51486i	−0.652918	−	0.757429i	$-0.726455\pi$
$349$	− 28.2999i	− 1.51486i	0.652918	−	0.757429i	$-0.273545\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.313161i	0.0166679i	0.999965	+	0.00833393i	$0.00265280\pi$
$353$	0.313161i	0.0166679i	−0.999965	+	0.00833393i	$0.997347\pi$
$354$	0	0
$355$	7.26210	0.385432
$356$	0	0
$357$	− 2.17525i	− 0.115127i
$358$	0	0
$359$	− 12.3342i	− 0.650977i	−0.945546	−	0.325488i	$-0.894471\pi$
$359$	− 12.3342i	− 0.650977i	0.945546	−	0.325488i	$-0.105529\pi$
$360$	0	0
$361$	17.5833	0.925438
$362$	0	0
$363$	−6.58685	−0.345720
$364$	0	0
$365$	3.32209	0.173886
$366$	0	0
$367$	−9.08368	−0.474164	−0.237082	−	0.971490i	$-0.576191\pi$
$367$	−9.08368	−0.474164	−0.237082	+	0.971490i	$0.576191\pi$
$368$	0	0
$369$	− 10.0181i	− 0.521523i
$370$	0	0
$371$	− 10.7146i	− 0.556277i
$372$	0	0
$373$	−16.5431	−0.856569	−0.428285	−	0.903644i	$-0.640882\pi$
$373$	−16.5431	−0.856569	−0.428285	+	0.903644i	$0.640882\pi$
$374$	0	0
$375$	12.1539i	0.627626i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.4212i	0.535299i	0.963516	+	0.267650i	$0.0862469\pi$
$379$	10.4212i	0.535299i	−0.963516	+	0.267650i	$0.913753\pi$
$380$	0	0
$381$	14.7930	0.757871
$382$	0	0
$383$	− 15.5767i	− 0.795931i	−0.917400	−	0.397965i	$-0.869716\pi$
$383$	− 15.5767i	− 0.795931i	0.917400	−	0.397965i	$-0.130284\pi$
$384$	0	0
$385$	− 10.1951i	− 0.519589i
$386$	0	0
$387$	−10.7587	−0.546896
$388$	0	0
$389$	−13.0121	−0.659742	−0.329871	−	0.944026i	$-0.607005\pi$
$389$	−13.0121	−0.659742	−0.329871	+	0.944026i	$0.607005\pi$
$390$	0	0
$391$	−5.07787	−0.256799
$392$	0	0
$393$	−0.834088	−0.0420742
$394$	0	0
$395$	− 23.8282i	− 1.19892i
$396$	0	0
$397$	33.9449i	1.70364i	0.523831	+	0.851822i	$0.324502\pi$
$397$	33.9449i	1.70364i	−0.523831	+	0.851822i	$0.675498\pi$
$398$	0	0
$399$	1.53727	0.0769600
$400$	0	0
$401$	− 37.0787i	− 1.85162i	−0.377988	−	0.925811i	$-0.623384\pi$
$401$	− 37.0787i	− 1.85162i	0.377988	−	0.925811i	$-0.376616\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.88227i	0.0935307i
$406$	0	0
$407$	−34.1928	−1.69487
$408$	0	0
$409$	31.0280i	1.53424i	0.641505	+	0.767118i	$0.278310\pi$
$409$	31.0280i	1.53424i	−0.641505	+	0.767118i	$0.721690\pi$
$410$	0	0
$411$	− 6.20377i	− 0.306009i
$412$	0	0
$413$	3.91819	0.192802
$414$	0	0
$415$	29.4162	1.44398
$416$	0	0
$417$	−20.1616	−0.987317
$418$	0	0
$419$	−27.5226	−1.34457	−0.672284	−	0.740293i	$-0.734687\pi$
$419$	−27.5226	−1.34457	−0.672284	+	0.740293i	$0.734687\pi$
$420$	0	0
$421$	− 10.2919i	− 0.501599i	−0.968039	−	0.250799i	$-0.919307\pi$
$421$	− 10.2919i	− 0.501599i	0.968039	−	0.250799i	$-0.0806934\pi$
$422$	0	0
$423$	5.31909i	0.258623i
$424$	0	0
$425$	2.45400	0.119036
$426$	0	0
$427$	− 10.1321i	− 0.490327i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 30.9340i	− 1.49004i	−0.667043	−	0.745020i	$-0.732440\pi$
$431$	− 30.9340i	− 1.49004i	0.667043	−	0.745020i	$-0.267560\pi$
$432$	0	0
$433$	−9.59342	−0.461030	−0.230515	−	0.973069i	$-0.574041\pi$
$433$	−9.59342	−0.461030	−0.230515	+	0.973069i	$0.574041\pi$
$434$	0	0
$435$	5.92915i	0.284281i
$436$	0	0
$437$	− 3.58859i	− 0.171665i
$438$	0	0
$439$	−18.3735	−0.876918	−0.438459	−	0.898751i	$-0.644476\pi$
$439$	−18.3735	−0.876918	−0.438459	+	0.898751i	$0.644476\pi$
$440$	0	0
$441$	5.33187	0.253899
$442$	0	0
$443$	−16.7144	−0.794125	−0.397062	−	0.917792i	$-0.629970\pi$
$443$	−16.7144	−0.794125	−0.397062	+	0.917792i	$0.629970\pi$
$444$	0	0
$445$	20.0799	0.951878
$446$	0	0
$447$	16.2561i	0.768887i
$448$	0	0
$449$	− 32.4482i	− 1.53132i	−0.643243	−	0.765662i	$-0.722412\pi$
$449$	− 32.4482i	− 1.53132i	0.643243	−	0.765662i	$-0.277588\pi$
$450$	0	0
$451$	−42.0127	−1.97830
$452$	0	0
$453$	18.0930i	0.850085i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16.5767i	− 0.775427i	−0.921780	−	0.387713i	$-0.873265\pi$
$457$	− 16.5767i	− 0.775427i	0.921780	−	0.387713i	$-0.126735\pi$
$458$	0	0
$459$	1.68421	0.0786120
$460$	0	0
$461$	− 2.43380i	− 0.113353i	−0.998393	−	0.0566767i	$-0.981950\pi$
$461$	− 2.43380i	− 0.113353i	0.998393	−	0.0566767i	$-0.0180504\pi$
$462$	0	0
$463$	30.9325i	1.43755i	0.695240	+	0.718777i	$0.255298\pi$
$463$	30.9325i	1.43755i	−0.695240	+	0.718777i	$0.744702\pi$
$464$	0	0
$465$	2.77564	0.128717
$466$	0	0
$467$	−19.4582	−0.900417	−0.450208	−	0.892924i	$-0.648650\pi$
$467$	−19.4582	−0.900417	−0.450208	+	0.892924i	$0.648650\pi$
$468$	0	0
$469$	5.81844	0.268670
$470$	0	0
$471$	15.0559	0.693738
$472$	0	0
$473$	45.1185i	2.07455i
$474$	0	0
$475$	1.73427i	0.0795737i
$476$	0	0
$477$	8.29590	0.379843
$478$	0	0
$479$	21.0622i	0.962356i	0.876623	+	0.481178i	$0.159791\pi$
$479$	21.0622i	0.962356i	−0.876623	+	0.481178i	$0.840209\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.89404i	0.177185i
$484$	0	0
$485$	4.48997	0.203879
$486$	0	0
$487$	− 37.6092i	− 1.70424i	−0.523349	−	0.852118i	$-0.675318\pi$
$487$	− 37.6092i	− 1.70424i	0.523349	−	0.852118i	$-0.324682\pi$
$488$	0	0
$489$	− 7.31660i	− 0.330868i
$490$	0	0
$491$	−36.5335	−1.64873	−0.824367	−	0.566056i	$-0.808469\pi$
$491$	−36.5335	−1.64873	−0.824367	+	0.566056i	$0.808469\pi$
$492$	0	0
$493$	5.30526	0.238937
$494$	0	0
$495$	7.89361	0.354791
$496$	0	0
$497$	−4.98305	−0.223520
$498$	0	0
$499$	22.4840i	1.00652i	0.864134	+	0.503262i	$0.167867\pi$
$499$	22.4840i	1.00652i	−0.864134	+	0.503262i	$0.832133\pi$
$500$	0	0
$501$	11.9586i	0.534271i
$502$	0	0
$503$	−2.21863	−0.0989237	−0.0494619	−	0.998776i	$-0.515751\pi$
$503$	−2.21863	−0.0989237	−0.0494619	+	0.998776i	$0.515751\pi$
$504$	0	0
$505$	1.65651i	0.0737137i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.2456i	1.25196i	0.779837	+	0.625982i	$0.215302\pi$
$509$	28.2456i	1.25196i	−0.779837	+	0.625982i	$0.784698\pi$
$510$	0	0
$511$	−2.27953	−0.100840
$512$	0	0
$513$	1.19025i	0.0525507i
$514$	0	0
$515$	− 9.80863i	− 0.432220i
$516$	0	0
$517$	22.3065	0.981038
$518$	0	0
$519$	16.6231	0.729673
$520$	0	0
$521$	28.1688	1.23410	0.617050	−	0.786924i	$-0.288328\pi$
$521$	28.1688	1.23410	0.617050	+	0.786924i	$0.288328\pi$
$522$	0	0
$523$	4.70961	0.205937	0.102968	−	0.994685i	$-0.467166\pi$
$523$	4.70961	0.205937	0.102968	+	0.994685i	$0.467166\pi$
$524$	0	0
$525$	− 1.88189i	− 0.0821323i
$526$	0	0
$527$	− 2.48358i	− 0.108186i
$528$	0	0
$529$	−13.9098	−0.604774
$530$	0	0
$531$	3.03369i	0.131651i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	37.5770i	1.62460i
$536$	0	0
$537$	−1.89790	−0.0819004
$538$	0	0
$539$	− 22.3601i	− 0.963118i
$540$	0	0
$541$	3.17651i	0.136569i	0.997666	+	0.0682844i	$0.0217525\pi$
$541$	3.17651i	0.136569i	−0.997666	+	0.0682844i	$0.978247\pi$
$542$	0	0
$543$	−16.3850	−0.703148
$544$	0	0
$545$	34.9306	1.49626
$546$	0	0
$547$	−25.4331	−1.08744	−0.543721	−	0.839266i	$-0.682985\pi$
$547$	−25.4331	−1.08744	−0.543721	+	0.839266i	$0.682985\pi$
$548$	0	0
$549$	7.84487	0.334811
$550$	0	0
$551$	3.74928i	0.159725i
$552$	0	0
$553$	16.3502i	0.695282i
$554$	0	0
$555$	15.3469	0.651441
$556$	0	0
$557$	20.1670i	0.854505i	0.904132	+	0.427252i	$0.140518\pi$
$557$	20.1670i	0.854505i	−0.904132	+	0.427252i	$0.859482\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 7.06300i	− 0.298200i
$562$	0	0
$563$	18.2498	0.769136	0.384568	−	0.923097i	$-0.374350\pi$
$563$	18.2498	0.769136	0.384568	+	0.923097i	$0.374350\pi$
$564$	0	0
$565$	18.7927i	0.790616i
$566$	0	0
$567$	− 1.29156i	− 0.0542404i
$568$	0	0
$569$	−27.4857	−1.15226	−0.576129	−	0.817359i	$-0.695438\pi$
$569$	−27.4857	−1.15226	−0.576129	+	0.817359i	$0.695438\pi$
$570$	0	0
$571$	0.0250476	0.00104821	0.000524106	−	1.00000i	$-0.499833\pi$
$571$	0.0250476	0.00104821	0.000524106		1.00000i	$0.499833\pi$
$572$	0	0
$573$	13.5859	0.567559
$574$	0	0
$575$	−4.39305	−0.183203
$576$	0	0
$577$	21.0269i	0.875362i	0.899130	+	0.437681i	$0.144200\pi$
$577$	21.0269i	0.875362i	−0.899130	+	0.437681i	$0.855800\pi$
$578$	0	0
$579$	− 24.8588i	− 1.03310i
$580$	0	0
$581$	−20.1846	−0.837397
$582$	0	0
$583$	− 34.7902i	− 1.44086i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 29.4551i	− 1.21574i	−0.794035	−	0.607872i	$-0.792024\pi$
$587$	− 29.4551i	− 1.21574i	0.794035	−	0.607872i	$-0.207976\pi$
$588$	0	0
$589$	1.75517	0.0723206
$590$	0	0
$591$	4.47333i	0.184008i
$592$	0	0
$593$	3.93477i	0.161582i	0.996731	+	0.0807909i	$0.0257446\pi$
$593$	3.93477i	0.161582i	−0.996731	+	0.0807909i	$0.974255\pi$
$594$	0	0
$595$	4.09441	0.167854
$596$	0	0
$597$	4.49522	0.183977
$598$	0	0
$599$	35.1931	1.43795	0.718975	−	0.695036i	$-0.244612\pi$
$599$	35.1931	1.43795	0.718975	+	0.695036i	$0.244612\pi$
$600$	0	0
$601$	−14.9780	−0.610967	−0.305484	−	0.952197i	$-0.598818\pi$
$601$	−14.9780	−0.610967	−0.305484	+	0.952197i	$0.598818\pi$
$602$	0	0
$603$	4.50497i	0.183457i
$604$	0	0
$605$	− 12.3982i	− 0.504059i
$606$	0	0
$607$	−24.6169	−0.999170	−0.499585	−	0.866265i	$-0.666514\pi$
$607$	−24.6169	−0.999170	−0.499585	+	0.866265i	$0.666514\pi$
$608$	0	0
$609$	− 4.06842i	− 0.164861i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	36.1429i	1.45980i	0.683554	+	0.729900i	$0.260433\pi$
$613$	36.1429i	1.45980i	−0.683554	+	0.729900i	$0.739567\pi$
$614$	0	0
$615$	18.8568	0.760380
$616$	0	0
$617$	− 12.0808i	− 0.486356i	−0.969982	−	0.243178i	$-0.921810\pi$
$617$	− 12.0808i	− 0.486356i	0.969982	−	0.243178i	$-0.0781898\pi$
$618$	0	0
$619$	− 23.1257i	− 0.929500i	−0.885442	−	0.464750i	$-0.846144\pi$
$619$	− 23.1257i	− 0.929500i	0.885442	−	0.464750i	$-0.153856\pi$
$620$	0	0
$621$	−3.01499	−0.120988
$622$	0	0
$623$	−13.7782	−0.552014
$624$	0	0
$625$	−15.5916	−0.623665
$626$	0	0
$627$	4.99150	0.199341
$628$	0	0
$629$	− 13.7320i	− 0.547533i
$630$	0	0
$631$	43.7565i	1.74192i	0.491354	+	0.870960i	$0.336502\pi$
$631$	43.7565i	1.74192i	−0.491354	+	0.870960i	$0.663498\pi$
$632$	0	0
$633$	21.7861	0.865920
$634$	0	0
$635$	27.8445i	1.10497i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 3.85816i	− 0.152627i
$640$	0	0
$641$	7.76562	0.306724	0.153362	−	0.988170i	$-0.450990\pi$
$641$	7.76562	0.306724	0.153362	+	0.988170i	$0.450990\pi$
$642$	0	0
$643$	− 9.44419i	− 0.372442i	−0.982508	−	0.186221i	$-0.940376\pi$
$643$	− 9.44419i	− 0.372442i	0.982508	−	0.186221i	$-0.0596241\pi$
$644$	0	0
$645$	− 20.2508i	− 0.797374i
$646$	0	0
$647$	41.8606	1.64571	0.822856	−	0.568250i	$-0.192379\pi$
$647$	41.8606	1.64571	0.822856	+	0.568250i	$0.192379\pi$
$648$	0	0
$649$	12.7223	0.499394
$650$	0	0
$651$	−1.90457	−0.0746460
$652$	0	0
$653$	−43.8823	−1.71725	−0.858624	−	0.512605i	$-0.828680\pi$
$653$	−43.8823	−1.71725	−0.858624	+	0.512605i	$0.828680\pi$
$654$	0	0
$655$	− 1.56998i	− 0.0613441i
$656$	0	0
$657$	− 1.76494i	− 0.0688569i
$658$	0	0
$659$	−27.3533	−1.06553	−0.532767	−	0.846262i	$-0.678848\pi$
$659$	−27.3533	−1.06553	−0.532767	+	0.846262i	$0.678848\pi$
$660$	0	0
$661$	46.8238i	1.82123i	0.413251	+	0.910617i	$0.364393\pi$
$661$	46.8238i	1.82123i	−0.413251	+	0.910617i	$0.635607\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.89356i	0.112208i
$666$	0	0
$667$	−9.49724	−0.367735
$668$	0	0
$669$	0.0555395i	0.00214728i
$670$	0	0
$671$	− 32.8988i	− 1.27004i
$672$	0	0
$673$	−36.2574	−1.39762	−0.698811	−	0.715307i	$-0.746287\pi$
$673$	−36.2574	−1.39762	−0.698811	+	0.715307i	$0.746287\pi$
$674$	0	0
$675$	1.45707	0.0560825
$676$	0	0
$677$	51.1630	1.96636	0.983178	−	0.182652i	$-0.0584682\pi$
$677$	51.1630	1.96636	0.983178	+	0.182652i	$0.0584682\pi$
$678$	0	0
$679$	−3.08089	−0.118234
$680$	0	0
$681$	− 5.24936i	− 0.201156i
$682$	0	0
$683$	30.8240i	1.17945i	0.807605	+	0.589724i	$0.200763\pi$
$683$	30.8240i	1.17945i	−0.807605	+	0.589724i	$0.799237\pi$
$684$	0	0
$685$	11.6772	0.446161
$686$	0	0
$687$	21.6951i	0.827720i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	4.12305i	0.156848i	0.996920	+	0.0784241i	$0.0249888\pi$
$691$	4.12305i	0.156848i	−0.996920	+	0.0784241i	$0.975011\pi$
$692$	0	0
$693$	−5.41637	−0.205751
$694$	0	0
$695$	− 37.9495i	− 1.43951i
$696$	0	0
$697$	− 16.8726i	− 0.639095i
$698$	0	0
$699$	−9.34933	−0.353624
$700$	0	0
$701$	−6.97922	−0.263602	−0.131801	−	0.991276i	$-0.542076\pi$
$701$	−6.97922	−0.263602	−0.131801	+	0.991276i	$0.542076\pi$
$702$	0	0
$703$	9.70459	0.366015
$704$	0	0
$705$	−10.0119	−0.377072
$706$	0	0
$707$	− 1.13665i	− 0.0427482i
$708$	0	0
$709$	23.1455i	0.869247i	0.900612	+	0.434623i	$0.143118\pi$
$709$	23.1455i	0.869247i	−0.900612	+	0.434623i	$0.856882\pi$
$710$	0	0
$711$	−12.6593	−0.474760
$712$	0	0
$713$	4.44599i	0.166504i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 14.7781i	− 0.551897i
$718$	0	0
$719$	−21.8249	−0.813930	−0.406965	−	0.913444i	$-0.633413\pi$
$719$	−21.8249	−0.813930	−0.406965	+	0.913444i	$0.633413\pi$
$720$	0	0
$721$	6.73041i	0.250653i
$722$	0	0
$723$	16.7668i	0.623563i
$724$	0	0
$725$	4.58976	0.170460
$726$	0	0
$727$	43.0916	1.59818	0.799090	−	0.601211i	$-0.205315\pi$
$727$	43.0916	1.59818	0.799090	+	0.601211i	$0.205315\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.1199	−0.670189
$732$	0	0
$733$	− 23.5077i	− 0.868277i	−0.900846	−	0.434138i	$-0.857053\pi$
$733$	− 23.5077i	− 0.868277i	0.900846	−	0.434138i	$-0.142947\pi$
$734$	0	0
$735$	10.0360i	0.370184i
$736$	0	0
$737$	18.8923	0.695909
$738$	0	0
$739$	23.9187i	0.879862i	0.898032	+	0.439931i	$0.144997\pi$
$739$	23.9187i	0.879862i	−0.898032	+	0.439931i	$0.855003\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 0.613067i	− 0.0224912i	−0.999937	−	0.0112456i	$-0.996420\pi$
$743$	− 0.613067i	− 0.0224912i	0.999937	−	0.0112456i	$-0.00357967\pi$
$744$	0	0
$745$	−30.5983	−1.12104
$746$	0	0
$747$	− 15.6280i	− 0.571800i
$748$	0	0
$749$	− 25.7843i	− 0.942137i
$750$	0	0
$751$	−0.945022	−0.0344843	−0.0172422	−	0.999851i	$-0.505489\pi$
$751$	−0.945022	−0.0344843	−0.0172422	+	0.999851i	$0.505489\pi$
$752$	0	0
$753$	−1.44048	−0.0524939
$754$	0	0
$755$	−34.0559	−1.23942
$756$	0	0
$757$	−14.8014	−0.537966	−0.268983	−	0.963145i	$-0.586688\pi$
$757$	−14.8014	−0.537966	−0.268983	+	0.963145i	$0.586688\pi$
$758$	0	0
$759$	12.6439i	0.458944i
$760$	0	0
$761$	11.6688i	0.422994i	0.977379	+	0.211497i	$0.0678339\pi$
$761$	11.6688i	0.422994i	−0.977379	+	0.211497i	$0.932166\pi$
$762$	0	0
$763$	−23.9684	−0.867714
$764$	0	0
$765$	3.17013i	0.114616i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.64412i	0.0953494i	0.998863	+	0.0476747i	$0.0151811\pi$
$769$	2.64412i	0.0953494i	−0.998863	+	0.0476747i	$0.984819\pi$
$770$	0	0
$771$	21.0265	0.757252
$772$	0	0
$773$	− 0.0776248i	− 0.00279197i	−0.999999	−	0.00139599i	$-0.999556\pi$
$773$	− 0.0776248i	− 0.00279197i	0.999999	−	0.00139599i	$-0.000444356\pi$
$774$	0	0
$775$	− 2.14863i	− 0.0771811i
$776$	0	0
$777$	−10.5306	−0.377784
$778$	0	0
$779$	11.9240	0.427223
$780$	0	0
$781$	−16.1799	−0.578961
$782$	0	0
$783$	3.15000	0.112572
$784$	0	0
$785$	28.3392i	1.01147i
$786$	0	0
$787$	24.3434i	0.867749i	0.900973	+	0.433874i	$0.142854\pi$
$787$	24.3434i	0.867749i	−0.900973	+	0.433874i	$0.857146\pi$
$788$	0	0
$789$	10.6833	0.380335
$790$	0	0
$791$	− 12.8950i	− 0.458495i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	15.6151i	0.553811i
$796$	0	0
$797$	46.4833	1.64652	0.823261	−	0.567663i	$-0.192152\pi$
$797$	46.4833	1.64652	0.823261	+	0.567663i	$0.192152\pi$
$798$	0	0
$799$	8.95844i	0.316927i
$800$	0	0
$801$	− 10.6679i	− 0.376932i
$802$	0	0
$803$	−7.40158	−0.261196
$804$	0	0
$805$	−7.32964	−0.258336
$806$	0	0
$807$	18.3247	0.645060
$808$	0	0
$809$	17.4581	0.613796	0.306898	−	0.951742i	$-0.400709\pi$
$809$	17.4581	0.613796	0.306898	+	0.951742i	$0.400709\pi$
$810$	0	0
$811$	− 13.3281i	− 0.468012i	−0.972235	−	0.234006i	$-0.924817\pi$
$811$	− 13.3281i	− 0.468012i	0.972235	−	0.234006i	$-0.0751835\pi$
$812$	0	0
$813$	3.64076i	0.127687i
$814$	0	0
$815$	13.7718	0.482405
$816$	0	0
$817$	− 12.8055i	− 0.448009i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.0294i	0.384928i	0.981304	+	0.192464i	$0.0616479\pi$
$821$	11.0294i	0.384928i	−0.981304	+	0.192464i	$0.938352\pi$
$822$	0	0
$823$	12.1161	0.422340	0.211170	−	0.977449i	$-0.432273\pi$
$823$	12.1161	0.422340	0.211170	+	0.977449i	$0.432273\pi$
$824$	0	0
$825$	− 6.11045i	− 0.212739i
$826$	0	0
$827$	− 23.6413i	− 0.822087i	−0.911616	−	0.411044i	$-0.865164\pi$
$827$	− 23.6413i	− 0.822087i	0.911616	−	0.411044i	$-0.134836\pi$
$828$	0	0
$829$	43.4519	1.50915	0.754574	−	0.656215i	$-0.227843\pi$
$829$	43.4519	1.50915	0.754574	+	0.656215i	$0.227843\pi$
$830$	0	0
$831$	−22.3601	−0.775663
$832$	0	0
$833$	8.97997	0.311138
$834$	0	0
$835$	−22.5093	−0.778967
$836$	0	0
$837$	− 1.47463i	− 0.0509706i
$838$	0	0
$839$	17.2176i	0.594416i	0.954813	+	0.297208i	$0.0960555\pi$
$839$	17.2176i	0.594416i	−0.954813	+	0.297208i	$0.903945\pi$
$840$	0	0
$841$	−19.0775	−0.657844
$842$	0	0
$843$	− 2.39777i	− 0.0825836i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.50731i	0.292315i
$848$	0	0
$849$	1.64168	0.0563424
$850$	0	0
$851$	24.5825i	0.842678i
$852$	0	0
$853$	− 19.5329i	− 0.668793i	−0.942432	−	0.334397i	$-0.891467\pi$
$853$	− 19.5329i	− 0.668793i	0.942432	−	0.334397i	$-0.108533\pi$
$854$	0	0
$855$	−2.24036	−0.0766188
$856$	0	0
$857$	19.9607	0.681843	0.340922	−	0.940092i	$-0.389261\pi$
$857$	19.9607	0.681843	0.340922	+	0.940092i	$0.389261\pi$
$858$	0	0
$859$	36.0681	1.23063	0.615315	−	0.788282i	$-0.289029\pi$
$859$	36.0681	1.23063	0.615315	+	0.788282i	$0.289029\pi$
$860$	0	0
$861$	−12.9390	−0.440960
$862$	0	0
$863$	− 3.06724i	− 0.104410i	−0.998636	−	0.0522051i	$-0.983375\pi$
$863$	− 3.06724i	− 0.104410i	0.998636	−	0.0522051i	$-0.0166249\pi$
$864$	0	0
$865$	31.2891i	1.06386i
$866$	0	0
$867$	−14.1635	−0.481016
$868$	0	0
$869$	53.0888i	1.80091i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.38541i	− 0.0807337i
$874$	0	0
$875$	15.6975	0.530673
$876$	0	0
$877$	− 26.5958i	− 0.898075i	−0.893513	−	0.449038i	$-0.851767\pi$
$877$	− 26.5958i	− 0.898075i	0.893513	−	0.449038i	$-0.148233\pi$
$878$	0	0
$879$	− 1.35348i	− 0.0456516i
$880$	0	0
$881$	−56.4634	−1.90230	−0.951151	−	0.308727i	$-0.900097\pi$
$881$	−56.4634	−1.90230	−0.951151	+	0.308727i	$0.900097\pi$
$882$	0	0
$883$	−29.6728	−0.998568	−0.499284	−	0.866438i	$-0.666404\pi$
$883$	−29.6728	−0.998568	−0.499284	+	0.866438i	$0.666404\pi$
$884$	0	0
$885$	−5.71022	−0.191947
$886$	0	0
$887$	−42.4301	−1.42466	−0.712332	−	0.701843i	$-0.752361\pi$
$887$	−42.4301	−1.42466	−0.712332	+	0.701843i	$0.752361\pi$
$888$	0	0
$889$	− 19.1061i	− 0.640798i
$890$	0	0
$891$	− 4.19367i	− 0.140493i
$892$	0	0
$893$	−6.33103	−0.211860
$894$	0	0
$895$	− 3.57236i	− 0.119411i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.64508i	− 0.154922i
$900$	0	0
$901$	13.9720	0.465475
$902$	0	0
$903$	13.8955i	0.462414i
$904$	0	0
$905$	− 30.8410i	− 1.02519i
$906$	0	0
$907$	11.1465	0.370114	0.185057	−	0.982728i	$-0.440753\pi$
$907$	11.1465	0.370114	0.185057	+	0.982728i	$0.440753\pi$
$908$	0	0
$909$	0.880061	0.0291898
$910$	0	0
$911$	−44.4798	−1.47368	−0.736841	−	0.676066i	$-0.763683\pi$
$911$	−44.4798	−1.47368	−0.736841	+	0.676066i	$0.763683\pi$
$912$	0	0
$913$	−65.5389	−2.16902
$914$	0	0
$915$	14.7661i	0.488154i
$916$	0	0
$917$	1.07727i	0.0355747i
$918$	0	0
$919$	−31.2712	−1.03154	−0.515771	−	0.856726i	$-0.672495\pi$
$919$	−31.2712	−1.03154	−0.515771	+	0.856726i	$0.672495\pi$
$920$	0	0
$921$	23.9864i	0.790380i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 11.8801i	− 0.390615i
$926$	0	0
$927$	−5.21107	−0.171154
$928$	0	0
$929$	19.2303i	0.630927i	0.948938	+	0.315464i	$0.102160\pi$
$929$	19.2303i	0.630927i	−0.948938	+	0.315464i	$0.897840\pi$
$930$	0	0
$931$	6.34625i	0.207990i
$932$	0	0
$933$	−6.23414	−0.204097
$934$	0	0
$935$	13.2945	0.434775
$936$	0	0
$937$	−40.1991	−1.31325	−0.656624	−	0.754218i	$-0.728016\pi$
$937$	−40.1991	−1.31325	−0.656624	+	0.754218i	$0.728016\pi$
$938$	0	0
$939$	−28.7120	−0.936982
$940$	0	0
$941$	57.3886i	1.87082i	0.353570	+	0.935408i	$0.384968\pi$
$941$	57.3886i	1.87082i	−0.353570	+	0.935408i	$0.615032\pi$
$942$	0	0
$943$	30.2046i	0.983597i
$944$	0	0
$945$	2.43106	0.0790824
$946$	0	0
$947$	12.1040i	0.393328i	0.980471	+	0.196664i	$0.0630108\pi$
$947$	12.1040i	0.393328i	−0.980471	+	0.196664i	$0.936989\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.8082i	0.447761i
$952$	0	0
$953$	−23.5095	−0.761547	−0.380773	−	0.924668i	$-0.624342\pi$
$953$	−23.5095	−0.761547	−0.380773	+	0.924668i	$0.624342\pi$
$954$	0	0
$955$	25.5723i	0.827501i
$956$	0	0
$957$	− 13.2101i	− 0.427021i
$958$	0	0
$959$	−8.01254	−0.258738
$960$	0	0
$961$	28.8255	0.929854
$962$	0	0
$963$	19.9637	0.643321
$964$	0	0
$965$	46.7910	1.50626
$966$	0	0
$967$	45.1235i	1.45107i	0.688183	+	0.725537i	$0.258409\pi$
$967$	45.1235i	1.45107i	−0.688183	+	0.725537i	$0.741591\pi$
$968$	0	0
$969$	2.00462i	0.0643977i
$970$	0	0
$971$	39.7833	1.27671	0.638353	−	0.769744i	$-0.279616\pi$
$971$	39.7833	1.27671	0.638353	+	0.769744i	$0.279616\pi$
$972$	0	0
$973$	26.0399i	0.834801i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.4379i	0.333937i	0.985962	+	0.166968i	$0.0533978\pi$
$977$	10.4379i	0.333937i	−0.985962	+	0.166968i	$0.946602\pi$
$978$	0	0
$979$	−44.7377	−1.42982
$980$	0	0
$981$	− 18.5577i	− 0.592502i
$982$	0	0
$983$	1.56154i	0.0498052i	0.999690	+	0.0249026i	$0.00792757\pi$
$983$	1.56154i	0.0498052i	−0.999690	+	0.0249026i	$0.992072\pi$
$984$	0	0
$985$	−8.42000	−0.268283
$986$	0	0
$987$	6.86992	0.218672
$988$	0	0
$989$	32.4375	1.03145
$990$	0	0
$991$	−3.79782	−0.120642	−0.0603209	−	0.998179i	$-0.519212\pi$
$991$	−3.79782	−0.120642	−0.0603209	+	0.998179i	$0.519212\pi$
$992$	0	0
$993$	7.06203i	0.224107i
$994$	0	0
$995$	8.46121i	0.268238i
$996$	0	0
$997$	−19.2540	−0.609780	−0.304890	−	0.952388i	$-0.598620\pi$
$997$	−19.2540	−0.609780	−0.304890	+	0.952388i	$0.598620\pi$
$998$	0	0
$999$	− 8.15343i	− 0.257963i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.r.337.8		12
13.5	odd	4		4056.2.a.bi.1.4	yes	6
13.8	odd	4		4056.2.a.bh.1.3	✓	6
13.12	even	2	inner	4056.2.c.r.337.5		12
52.31	even	4		8112.2.a.cu.1.4		6
52.47	even	4		8112.2.a.ct.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.3	✓	6	13.8	odd	4
4056.2.a.bi.1.4	yes	6	13.5	odd	4
4056.2.c.r.337.5		12	13.12	even	2	inner
4056.2.c.r.337.8		12	1.1	even	1	trivial
8112.2.a.ct.1.3		6	52.47	even	4
8112.2.a.cu.1.4		6	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.88227i q^{5} -1.29156i q^{7} +1.00000 q^{9} -4.19367i q^{11} +1.88227i q^{15} +1.68421 q^{17} +1.19025i q^{19} -1.29156i q^{21} -3.01499 q^{23} +1.45707 q^{25} +1.00000 q^{27} +3.15000 q^{29} -1.47463i q^{31} -4.19367i q^{33} +2.43106 q^{35} -8.15343i q^{37} -10.0181i q^{41} -10.7587 q^{43} +1.88227i q^{45} +5.31909i q^{47} +5.33187 q^{49} +1.68421 q^{51} +8.29590 q^{53} +7.89361 q^{55} +1.19025i q^{57} +3.03369i q^{59} +7.84487 q^{61} -1.29156i q^{63} +4.50497i q^{67} -3.01499 q^{69} -3.85816i q^{71} -1.76494i q^{73} +1.45707 q^{75} -5.41637 q^{77} -12.6593 q^{79} +1.00000 q^{81} -15.6280i q^{83} +3.17013i q^{85} +3.15000 q^{87} -10.6679i q^{89} -1.47463i q^{93} -2.24036 q^{95} -2.38541i q^{97} -4.19367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 12 q^{9} - 18 q^{17} - 24 q^{23} - 18 q^{25} + 12 q^{27} + 14 q^{29} + 12 q^{35} - 30 q^{43} - 26 q^{49} - 18 q^{51} + 44 q^{53} + 6 q^{55} + 50 q^{61} - 24 q^{69} - 18 q^{75} - 90 q^{77}+ \cdots - 94 q^{95}+O(q^{100}) \) 12 * q + 12 * q^3 + 12 * q^9 - 18 * q^17 - 24 * q^23 - 18 * q^25 + 12 * q^27 + 14 * q^29 + 12 * q^35 - 30 * q^43 - 26 * q^49 - 18 * q^51 + 44 * q^53 + 6 * q^55 + 50 * q^61 - 24 * q^69 - 18 * q^75 - 90 * q^77 + 28 * q^79 + 12 * q^81 + 14 * q^87 - 94 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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