LMFDB - Embedded newform 2704.2.f.l.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(337,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2704.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,0,0,0,0,6,0,0,0,0,0,0,0,20,0,0,0,0,0,10] 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:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.l.337.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -0.561553i q^{5} -1.56155i q^{7} -0.561553 q^{9} +1.56155i q^{11} -0.876894i q^{15} +5.00000 q^{17} +1.56155i q^{19} -2.43845i q^{21} +8.68466 q^{23} +4.68466 q^{25} -5.56155 q^{27} -5.00000 q^{29} -8.00000i q^{31} +2.43845i q^{33} -0.876894 q^{35} -1.00000i q^{37} +7.24621i q^{41} -2.43845 q^{43} +0.315342i q^{45} -4.00000i q^{47} +4.56155 q^{49} +7.80776 q^{51} +8.56155 q^{53} +0.876894 q^{55} +2.43845i q^{57} -1.56155i q^{59} +9.24621 q^{61} +0.876894i q^{63} -13.5616i q^{67} +13.5616 q^{69} -4.68466i q^{71} -13.6847i q^{73} +7.31534 q^{75} +2.43845 q^{77} +4.00000 q^{79} -7.00000 q^{81} -14.2462i q^{83} -2.80776i q^{85} -7.80776 q^{87} +2.68466i q^{89} -12.4924i q^{93} +0.876894 q^{95} +17.8078i q^{97} -0.876894i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 6 q^{9} + 20 q^{17} + 10 q^{23} - 6 q^{25} - 14 q^{27} - 20 q^{29} - 20 q^{35} - 18 q^{43} + 10 q^{49} - 10 q^{51} + 26 q^{53} + 20 q^{55} + 4 q^{61} + 46 q^{69} + 54 q^{75} + 18 q^{77}+ \cdots + 20 q^{95}+O(q^{100}) $ 4 * q - 2 * q^3 + 6 * q^9 + 20 * q^17 + 10 * q^23 - 6 * q^25 - 14 * q^27 - 20 * q^29 - 20 * q^35 - 18 * q^43 + 10 * q^49 - 10 * q^51 + 26 * q^53 + 20 * q^55 + 4 * q^61 + 46 * q^69 + 54 * q^75 + 18 * q^77 + 16 * q^79 - 28 * q^81 + 10 * q^87 + 20 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	− 0.561553i	− 0.251134i	−0.992085	−	0.125567i	$-0.959925\pi$
$5$	− 0.561553i	− 0.251134i	0.992085	−	0.125567i	$-0.0400750\pi$
$6$	0	0
$7$	− 1.56155i	− 0.590211i	−0.955465	−	0.295106i	$-0.904645\pi$
$7$	− 1.56155i	− 0.590211i	0.955465	−	0.295106i	$-0.0953549\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	1.56155i	0.470826i	0.971895	+	0.235413i	$0.0756443\pi$
$11$	1.56155i	0.470826i	−0.971895	+	0.235413i	$0.924356\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 0.876894i	− 0.226413i
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.56155i	0.358245i	0.983827	+	0.179122i	$0.0573258\pi$
$19$	1.56155i	0.358245i	−0.983827	+	0.179122i	$0.942674\pi$
$20$	0	0
$21$	− 2.43845i	− 0.532113i
$22$	0	0
$23$	8.68466	1.81088	0.905438	−	0.424478i	$-0.139542\pi$
$23$	8.68466	1.81088	0.905438	+	0.424478i	$0.139542\pi$
$24$	0	0
$25$	4.68466	0.936932
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	− 8.00000i	− 1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$	− 8.00000i	− 1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$	0	0
$33$	2.43845i	0.424479i
$34$	0	0
$35$	−0.876894	−0.148222
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.24621i	1.13167i	0.824519	+	0.565834i	$0.191446\pi$
$41$	7.24621i	1.13167i	−0.824519	+	0.565834i	$0.808554\pi$
$42$	0	0
$43$	−2.43845	−0.371860	−0.185930	−	0.982563i	$-0.559530\pi$
$43$	−2.43845	−0.371860	−0.185930	+	0.982563i	$0.559530\pi$
$44$	0	0
$45$	0.315342i	0.0470083i
$46$	0	0
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	4.56155	0.651650
$50$	0	0
$51$	7.80776	1.09331
$52$	0	0
$53$	8.56155	1.17602	0.588010	−	0.808854i	$-0.299912\pi$
$53$	8.56155	1.17602	0.588010	+	0.808854i	$0.299912\pi$
$54$	0	0
$55$	0.876894	0.118240
$56$	0	0
$57$	2.43845i	0.322980i
$58$	0	0
$59$	− 1.56155i	− 0.203297i	−0.994820	−	0.101648i	$-0.967588\pi$
$59$	− 1.56155i	− 0.203297i	0.994820	−	0.101648i	$-0.0324117\pi$
$60$	0	0
$61$	9.24621	1.18386	0.591928	−	0.805991i	$-0.298367\pi$
$61$	9.24621	1.18386	0.591928	+	0.805991i	$0.298367\pi$
$62$	0	0
$63$	0.876894i	0.110478i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.5616i	− 1.65681i	−0.560132	−	0.828404i	$-0.689249\pi$
$67$	− 13.5616i	− 1.65681i	0.560132	−	0.828404i	$-0.310751\pi$
$68$	0	0
$69$	13.5616	1.63262
$70$	0	0
$71$	− 4.68466i	− 0.555967i	−0.960586	−	0.277983i	$-0.910334\pi$
$71$	− 4.68466i	− 0.555967i	0.960586	−	0.277983i	$-0.0896660\pi$
$72$	0	0
$73$	− 13.6847i	− 1.60167i	−0.598886	−	0.800834i	$-0.704390\pi$
$73$	− 13.6847i	− 1.60167i	0.598886	−	0.800834i	$-0.295610\pi$
$74$	0	0
$75$	7.31534	0.844703
$76$	0	0
$77$	2.43845	0.277887
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 14.2462i	− 1.56372i	−0.623451	−	0.781862i	$-0.714270\pi$
$83$	− 14.2462i	− 1.56372i	0.623451	−	0.781862i	$-0.285730\pi$
$84$	0	0
$85$	− 2.80776i	− 0.304545i
$86$	0	0
$87$	−7.80776	−0.837080
$88$	0	0
$89$	2.68466i	0.284573i	0.989825	+	0.142287i	$0.0454455\pi$
$89$	2.68466i	0.284573i	−0.989825	+	0.142287i	$0.954555\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 12.4924i	− 1.29540i
$94$	0	0
$95$	0.876894	0.0899675
$96$	0	0
$97$	17.8078i	1.80810i	0.427422	+	0.904052i	$0.359422\pi$
$97$	17.8078i	1.80810i	−0.427422	+	0.904052i	$0.640578\pi$
$98$	0	0
$99$	− 0.876894i	− 0.0881312i
$100$	0	0
$101$	11.2462	1.11904	0.559520	−	0.828817i	$-0.310986\pi$
$101$	11.2462	1.11904	0.559520	+	0.828817i	$0.310986\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	−1.36932	−0.133632
$106$	0	0
$107$	−8.68466	−0.839578	−0.419789	−	0.907622i	$-0.637896\pi$
$107$	−8.68466	−0.839578	−0.419789	+	0.907622i	$0.637896\pi$
$108$	0	0
$109$	8.24621i	0.789844i	0.918715	+	0.394922i	$0.129228\pi$
$109$	8.24621i	0.789844i	−0.918715	+	0.394922i	$0.870772\pi$
$110$	0	0
$111$	− 1.56155i	− 0.148216i
$112$	0	0
$113$	−7.24621	−0.681666	−0.340833	−	0.940124i	$-0.610709\pi$
$113$	−7.24621	−0.681666	−0.340833	+	0.940124i	$0.610709\pi$
$114$	0	0
$115$	− 4.87689i	− 0.454773i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 7.80776i	− 0.715737i
$120$	0	0
$121$	8.56155	0.778323
$122$	0	0
$123$	11.3153i	1.02027i
$124$	0	0
$125$	− 5.43845i	− 0.486430i
$126$	0	0
$127$	−7.80776	−0.692827	−0.346414	−	0.938082i	$-0.612601\pi$
$127$	−7.80776	−0.692827	−0.346414	+	0.938082i	$0.612601\pi$
$128$	0	0
$129$	−3.80776	−0.335255
$130$	0	0
$131$	−1.75379	−0.153229	−0.0766146	−	0.997061i	$-0.524411\pi$
$131$	−1.75379	−0.153229	−0.0766146	+	0.997061i	$0.524411\pi$
$132$	0	0
$133$	2.43845	0.211440
$134$	0	0
$135$	3.12311i	0.268794i
$136$	0	0
$137$	11.4924i	0.981864i	0.871198	+	0.490932i	$0.163344\pi$
$137$	11.4924i	0.981864i	−0.871198	+	0.490932i	$0.836656\pi$
$138$	0	0
$139$	−7.31534	−0.620479	−0.310240	−	0.950658i	$-0.600409\pi$
$139$	−7.31534	−0.620479	−0.310240	+	0.950658i	$0.600409\pi$
$140$	0	0
$141$	− 6.24621i	− 0.526026i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.80776i	0.233172i
$146$	0	0
$147$	7.12311	0.587504
$148$	0	0
$149$	3.24621i	0.265940i	0.991120	+	0.132970i	$0.0424514\pi$
$149$	3.24621i	0.265940i	−0.991120	+	0.132970i	$0.957549\pi$
$150$	0	0
$151$	− 20.4924i	− 1.66765i	−0.552029	−	0.833825i	$-0.686146\pi$
$151$	− 20.4924i	− 1.66765i	0.552029	−	0.833825i	$-0.313854\pi$
$152$	0	0
$153$	−2.80776	−0.226994
$154$	0	0
$155$	−4.49242	−0.360840
$156$	0	0
$157$	2.80776	0.224084	0.112042	−	0.993703i	$-0.464261\pi$
$157$	2.80776	0.224084	0.112042	+	0.993703i	$0.464261\pi$
$158$	0	0
$159$	13.3693	1.06026
$160$	0	0
$161$	− 13.5616i	− 1.06880i
$162$	0	0
$163$	22.9309i	1.79608i	0.439909	+	0.898042i	$0.355011\pi$
$163$	22.9309i	1.79608i	−0.439909	+	0.898042i	$0.644989\pi$
$164$	0	0
$165$	1.36932	0.106601
$166$	0	0
$167$	9.56155i	0.739895i	0.929053	+	0.369948i	$0.120624\pi$
$167$	9.56155i	0.739895i	−0.929053	+	0.369948i	$0.879376\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 0.876894i	− 0.0670578i
$172$	0	0
$173$	−12.9309	−0.983116	−0.491558	−	0.870845i	$-0.663572\pi$
$173$	−12.9309	−0.983116	−0.491558	+	0.870845i	$0.663572\pi$
$174$	0	0
$175$	− 7.31534i	− 0.552988i
$176$	0	0
$177$	− 2.43845i	− 0.183285i
$178$	0	0
$179$	16.6847	1.24707	0.623535	−	0.781795i	$-0.285696\pi$
$179$	16.6847	1.24707	0.623535	+	0.781795i	$0.285696\pi$
$180$	0	0
$181$	−10.8078	−0.803335	−0.401667	−	0.915786i	$-0.631569\pi$
$181$	−10.8078	−0.803335	−0.401667	+	0.915786i	$0.631569\pi$
$182$	0	0
$183$	14.4384	1.06732
$184$	0	0
$185$	−0.561553	−0.0412862
$186$	0	0
$187$	7.80776i	0.570960i
$188$	0	0
$189$	8.68466i	0.631716i
$190$	0	0
$191$	16.6847	1.20726	0.603630	−	0.797265i	$-0.293721\pi$
$191$	16.6847	1.20726	0.603630	+	0.797265i	$0.293721\pi$
$192$	0	0
$193$	3.00000i	0.215945i	0.994154	+	0.107972i	$0.0344358\pi$
$193$	3.00000i	0.215945i	−0.994154	+	0.107972i	$0.965564\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.6847i	− 1.04624i	−0.852259	−	0.523119i	$-0.824768\pi$
$197$	− 14.6847i	− 1.04624i	0.852259	−	0.523119i	$-0.175232\pi$
$198$	0	0
$199$	13.5616	0.961353	0.480676	−	0.876898i	$-0.340391\pi$
$199$	13.5616	0.961353	0.480676	+	0.876898i	$0.340391\pi$
$200$	0	0
$201$	− 21.1771i	− 1.49372i
$202$	0	0
$203$	7.80776i	0.547998i
$204$	0	0
$205$	4.06913	0.284200
$206$	0	0
$207$	−4.87689	−0.338968
$208$	0	0
$209$	−2.43845	−0.168671
$210$	0	0
$211$	−24.6847	−1.69936	−0.849681	−	0.527297i	$-0.823205\pi$
$211$	−24.6847	−1.69936	−0.849681	+	0.527297i	$0.823205\pi$
$212$	0	0
$213$	− 7.31534i	− 0.501239i
$214$	0	0
$215$	1.36932i	0.0933866i
$216$	0	0
$217$	−12.4924	−0.848041
$218$	0	0
$219$	− 21.3693i	− 1.44400i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.68466i	0.581568i	0.956789	+	0.290784i	$0.0939161\pi$
$223$	8.68466i	0.581568i	−0.956789	+	0.290784i	$0.906084\pi$
$224$	0	0
$225$	−2.63068	−0.175379
$226$	0	0
$227$	30.0540i	1.99475i	0.0723928	+	0.997376i	$0.476936\pi$
$227$	30.0540i	1.99475i	−0.0723928	+	0.997376i	$0.523064\pi$
$228$	0	0
$229$	12.2462i	0.809252i	0.914482	+	0.404626i	$0.132598\pi$
$229$	12.2462i	0.809252i	−0.914482	+	0.404626i	$0.867402\pi$
$230$	0	0
$231$	3.80776	0.250532
$232$	0	0
$233$	12.2462	0.802276	0.401138	−	0.916018i	$-0.368615\pi$
$233$	12.2462	0.802276	0.401138	+	0.916018i	$0.368615\pi$
$234$	0	0
$235$	−2.24621	−0.146527
$236$	0	0
$237$	6.24621	0.405735
$238$	0	0
$239$	18.2462i	1.18025i	0.807312	+	0.590125i	$0.200921\pi$
$239$	18.2462i	1.18025i	−0.807312	+	0.590125i	$0.799079\pi$
$240$	0	0
$241$	7.49242i	0.482629i	0.970447	+	0.241315i	$0.0775786\pi$
$241$	7.49242i	0.482629i	−0.970447	+	0.241315i	$0.922421\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	− 2.56155i	− 0.163652i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 22.2462i	− 1.40980i
$250$	0	0
$251$	1.06913	0.0674829	0.0337415	−	0.999431i	$-0.489258\pi$
$251$	1.06913	0.0674829	0.0337415	+	0.999431i	$0.489258\pi$
$252$	0	0
$253$	13.5616i	0.852608i
$254$	0	0
$255$	− 4.38447i	− 0.274566i
$256$	0	0
$257$	−5.24621	−0.327250	−0.163625	−	0.986523i	$-0.552319\pi$
$257$	−5.24621	−0.327250	−0.163625	+	0.986523i	$0.552319\pi$
$258$	0	0
$259$	−1.56155	−0.0970302
$260$	0	0
$261$	2.80776	0.173796
$262$	0	0
$263$	11.8078	0.728098	0.364049	−	0.931380i	$-0.381394\pi$
$263$	11.8078	0.728098	0.364049	+	0.931380i	$0.381394\pi$
$264$	0	0
$265$	− 4.80776i	− 0.295339i
$266$	0	0
$267$	4.19224i	0.256561i
$268$	0	0
$269$	−5.80776	−0.354106	−0.177053	−	0.984201i	$-0.556656\pi$
$269$	−5.80776	−0.354106	−0.177053	+	0.984201i	$0.556656\pi$
$270$	0	0
$271$	9.56155i	0.580823i	0.956902	+	0.290411i	$0.0937922\pi$
$271$	9.56155i	0.580823i	−0.956902	+	0.290411i	$0.906208\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.31534i	0.441132i
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	0	0
$279$	4.49242i	0.268954i
$280$	0	0
$281$	2.31534i	0.138122i	0.997612	+	0.0690608i	$0.0220003\pi$
$281$	2.31534i	0.138122i	−0.997612	+	0.0690608i	$0.978000\pi$
$282$	0	0
$283$	−25.5616	−1.51948	−0.759738	−	0.650229i	$-0.774673\pi$
$283$	−25.5616	−1.51948	−0.759738	+	0.650229i	$0.774673\pi$
$284$	0	0
$285$	1.36932	0.0811113
$286$	0	0
$287$	11.3153	0.667923
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	27.8078i	1.63012i
$292$	0	0
$293$	− 21.4924i	− 1.25560i	−0.778374	−	0.627800i	$-0.783955\pi$
$293$	− 21.4924i	− 1.25560i	0.778374	−	0.627800i	$-0.216045\pi$
$294$	0	0
$295$	−0.876894	−0.0510548
$296$	0	0
$297$	− 8.68466i	− 0.503935i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.80776i	0.219476i
$302$	0	0
$303$	17.5616	1.00888
$304$	0	0
$305$	− 5.19224i	− 0.297307i
$306$	0	0
$307$	9.75379i	0.556678i	0.960483	+	0.278339i	$0.0897839\pi$
$307$	9.75379i	0.556678i	−0.960483	+	0.278339i	$0.910216\pi$
$308$	0	0
$309$	−3.50758	−0.199539
$310$	0	0
$311$	2.24621	0.127371	0.0636855	−	0.997970i	$-0.479715\pi$
$311$	2.24621	0.127371	0.0636855	+	0.997970i	$0.479715\pi$
$312$	0	0
$313$	−15.7538	−0.890457	−0.445228	−	0.895417i	$-0.646878\pi$
$313$	−15.7538	−0.890457	−0.445228	+	0.895417i	$0.646878\pi$
$314$	0	0
$315$	0.492423	0.0277449
$316$	0	0
$317$	− 20.5616i	− 1.15485i	−0.816443	−	0.577426i	$-0.804057\pi$
$317$	− 20.5616i	− 1.15485i	0.816443	−	0.577426i	$-0.195943\pi$
$318$	0	0
$319$	− 7.80776i	− 0.437151i
$320$	0	0
$321$	−13.5616	−0.756932
$322$	0	0
$323$	7.80776i	0.434436i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.8769i	0.712094i
$328$	0	0
$329$	−6.24621	−0.344365
$330$	0	0
$331$	− 6.93087i	− 0.380955i	−0.981692	−	0.190478i	$-0.938996\pi$
$331$	− 6.93087i	− 0.380955i	0.981692	−	0.190478i	$-0.0610036\pi$
$332$	0	0
$333$	0.561553i	0.0307729i
$334$	0	0
$335$	−7.61553	−0.416081
$336$	0	0
$337$	−20.5616	−1.12006	−0.560030	−	0.828473i	$-0.689210\pi$
$337$	−20.5616	−1.12006	−0.560030	+	0.828473i	$0.689210\pi$
$338$	0	0
$339$	−11.3153	−0.614565
$340$	0	0
$341$	12.4924	0.676503
$342$	0	0
$343$	− 18.0540i	− 0.974823i
$344$	0	0
$345$	− 7.61553i	− 0.410006i
$346$	0	0
$347$	−2.05398	−0.110263	−0.0551316	−	0.998479i	$-0.517558\pi$
$347$	−2.05398	−0.110263	−0.0551316	+	0.998479i	$0.517558\pi$
$348$	0	0
$349$	16.4384i	0.879930i	0.898015	+	0.439965i	$0.145009\pi$
$349$	16.4384i	0.879930i	−0.898015	+	0.439965i	$0.854991\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.00000i	0.479022i	0.970894	+	0.239511i	$0.0769871\pi$
$353$	9.00000i	0.479022i	−0.970894	+	0.239511i	$0.923013\pi$
$354$	0	0
$355$	−2.63068	−0.139622
$356$	0	0
$357$	− 12.1922i	− 0.645282i
$358$	0	0
$359$	− 28.4924i	− 1.50377i	−0.659293	−	0.751886i	$-0.729144\pi$
$359$	− 28.4924i	− 1.50377i	0.659293	−	0.751886i	$-0.270856\pi$
$360$	0	0
$361$	16.5616	0.871661
$362$	0	0
$363$	13.3693	0.701707
$364$	0	0
$365$	−7.68466	−0.402233
$366$	0	0
$367$	−13.1771	−0.687838	−0.343919	−	0.938999i	$-0.611755\pi$
$367$	−13.1771	−0.687838	−0.343919	+	0.938999i	$0.611755\pi$
$368$	0	0
$369$	− 4.06913i	− 0.211830i
$370$	0	0
$371$	− 13.3693i	− 0.694100i
$372$	0	0
$373$	5.24621	0.271639	0.135819	−	0.990734i	$-0.456633\pi$
$373$	5.24621	0.271639	0.135819	+	0.990734i	$0.456633\pi$
$374$	0	0
$375$	− 8.49242i	− 0.438547i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.93087i	0.150549i	0.997163	+	0.0752743i	$0.0239832\pi$
$379$	2.93087i	0.150549i	−0.997163	+	0.0752743i	$0.976017\pi$
$380$	0	0
$381$	−12.1922	−0.624627
$382$	0	0
$383$	− 29.1771i	− 1.49088i	−0.666573	−	0.745440i	$-0.732240\pi$
$383$	− 29.1771i	− 1.49088i	0.666573	−	0.745440i	$-0.267760\pi$
$384$	0	0
$385$	− 1.36932i	− 0.0697869i
$386$	0	0
$387$	1.36932	0.0696063
$388$	0	0
$389$	29.6847	1.50507	0.752536	−	0.658551i	$-0.228830\pi$
$389$	29.6847	1.50507	0.752536	+	0.658551i	$0.228830\pi$
$390$	0	0
$391$	43.4233	2.19601
$392$	0	0
$393$	−2.73863	−0.138146
$394$	0	0
$395$	− 2.24621i	− 0.113019i
$396$	0	0
$397$	− 9.31534i	− 0.467524i	−0.972294	−	0.233762i	$-0.924896\pi$
$397$	− 9.31534i	− 0.467524i	0.972294	−	0.233762i	$-0.0751036\pi$
$398$	0	0
$399$	3.80776	0.190627
$400$	0	0
$401$	23.4924i	1.17316i	0.809893	+	0.586578i	$0.199525\pi$
$401$	23.4924i	1.17316i	−0.809893	+	0.586578i	$0.800475\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.93087i	0.195326i
$406$	0	0
$407$	1.56155	0.0774033
$408$	0	0
$409$	15.2462i	0.753877i	0.926238	+	0.376938i	$0.123023\pi$
$409$	15.2462i	0.753877i	−0.926238	+	0.376938i	$0.876977\pi$
$410$	0	0
$411$	17.9460i	0.885212i
$412$	0	0
$413$	−2.43845	−0.119988
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−11.4233	−0.559401
$418$	0	0
$419$	−14.9309	−0.729421	−0.364710	−	0.931121i	$-0.618832\pi$
$419$	−14.9309	−0.729421	−0.364710	+	0.931121i	$0.618832\pi$
$420$	0	0
$421$	33.6847i	1.64169i	0.571151	+	0.820845i	$0.306497\pi$
$421$	33.6847i	1.64169i	−0.571151	+	0.820845i	$0.693503\pi$
$422$	0	0
$423$	2.24621i	0.109215i
$424$	0	0
$425$	23.4233	1.13620
$426$	0	0
$427$	− 14.4384i	− 0.698725i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.9309i	1.48989i	0.667127	+	0.744944i	$0.267524\pi$
$431$	30.9309i	1.48989i	−0.667127	+	0.744944i	$0.732476\pi$
$432$	0	0
$433$	−31.4924	−1.51343	−0.756715	−	0.653745i	$-0.773197\pi$
$433$	−31.4924	−1.51343	−0.756715	+	0.653745i	$0.773197\pi$
$434$	0	0
$435$	4.38447i	0.210219i
$436$	0	0
$437$	13.5616i	0.648737i
$438$	0	0
$439$	−9.06913	−0.432846	−0.216423	−	0.976300i	$-0.569439\pi$
$439$	−9.06913	−0.432846	−0.216423	+	0.976300i	$0.569439\pi$
$440$	0	0
$441$	−2.56155	−0.121979
$442$	0	0
$443$	−38.2462	−1.81713	−0.908566	−	0.417741i	$-0.862822\pi$
$443$	−38.2462	−1.81713	−0.908566	+	0.417741i	$0.862822\pi$
$444$	0	0
$445$	1.50758	0.0714660
$446$	0	0
$447$	5.06913i	0.239762i
$448$	0	0
$449$	− 24.0540i	− 1.13518i	−0.823312	−	0.567589i	$-0.807876\pi$
$449$	− 24.0540i	− 1.13518i	0.823312	−	0.567589i	$-0.192124\pi$
$450$	0	0
$451$	−11.3153	−0.532819
$452$	0	0
$453$	− 32.0000i	− 1.50349i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.00000i	0.421002i	0.977594	+	0.210501i	$0.0675096\pi$
$457$	9.00000i	0.421002i	−0.977594	+	0.210501i	$0.932490\pi$
$458$	0	0
$459$	−27.8078	−1.29796
$460$	0	0
$461$	− 23.4924i	− 1.09415i	−0.837083	−	0.547076i	$-0.815741\pi$
$461$	− 23.4924i	− 1.09415i	0.837083	−	0.547076i	$-0.184259\pi$
$462$	0	0
$463$	32.9848i	1.53294i	0.642283	+	0.766468i	$0.277988\pi$
$463$	32.9848i	1.53294i	−0.642283	+	0.766468i	$0.722012\pi$
$464$	0	0
$465$	−7.01515	−0.325320
$466$	0	0
$467$	26.2462	1.21453	0.607265	−	0.794499i	$-0.292267\pi$
$467$	26.2462	1.21453	0.607265	+	0.794499i	$0.292267\pi$
$468$	0	0
$469$	−21.1771	−0.977867
$470$	0	0
$471$	4.38447	0.202026
$472$	0	0
$473$	− 3.80776i	− 0.175081i
$474$	0	0
$475$	7.31534i	0.335651i
$476$	0	0
$477$	−4.80776	−0.220132
$478$	0	0
$479$	20.3002i	0.927539i	0.885956	+	0.463770i	$0.153503\pi$
$479$	20.3002i	0.927539i	−0.885956	+	0.463770i	$0.846497\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 21.1771i	− 0.963591i
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	− 34.4384i	− 1.56056i	−0.625433	−	0.780278i	$-0.715078\pi$
$487$	− 34.4384i	− 1.56056i	0.625433	−	0.780278i	$-0.284922\pi$
$488$	0	0
$489$	35.8078i	1.61928i
$490$	0	0
$491$	−2.05398	−0.0926946	−0.0463473	−	0.998925i	$-0.514758\pi$
$491$	−2.05398	−0.0926946	−0.0463473	+	0.998925i	$0.514758\pi$
$492$	0	0
$493$	−25.0000	−1.12594
$494$	0	0
$495$	−0.492423	−0.0221327
$496$	0	0
$497$	−7.31534	−0.328138
$498$	0	0
$499$	− 16.4924i	− 0.738302i	−0.929369	−	0.369151i	$-0.879648\pi$
$499$	− 16.4924i	− 0.738302i	0.929369	−	0.369151i	$-0.120352\pi$
$500$	0	0
$501$	14.9309i	0.667062i
$502$	0	0
$503$	32.6847	1.45734	0.728668	−	0.684867i	$-0.240140\pi$
$503$	32.6847	1.45734	0.728668	+	0.684867i	$0.240140\pi$
$504$	0	0
$505$	− 6.31534i	− 0.281029i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.0000i	1.46270i	0.682003	+	0.731350i	$0.261109\pi$
$509$	33.0000i	1.46270i	−0.682003	+	0.731350i	$0.738891\pi$
$510$	0	0
$511$	−21.3693	−0.945323
$512$	0	0
$513$	− 8.68466i	− 0.383437i
$514$	0	0
$515$	1.26137i	0.0555824i
$516$	0	0
$517$	6.24621	0.274708
$518$	0	0
$519$	−20.1922	−0.886341
$520$	0	0
$521$	9.05398	0.396662	0.198331	−	0.980135i	$-0.436448\pi$
$521$	9.05398	0.396662	0.198331	+	0.980135i	$0.436448\pi$
$522$	0	0
$523$	12.3002	0.537850	0.268925	−	0.963161i	$-0.413332\pi$
$523$	12.3002	0.537850	0.268925	+	0.963161i	$0.413332\pi$
$524$	0	0
$525$	− 11.4233i	− 0.498553i
$526$	0	0
$527$	− 40.0000i	− 1.74243i
$528$	0	0
$529$	52.4233	2.27927
$530$	0	0
$531$	0.876894i	0.0380540i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.87689i	0.210847i
$536$	0	0
$537$	26.0540	1.12431
$538$	0	0
$539$	7.12311i	0.306814i
$540$	0	0
$541$	− 5.68466i	− 0.244403i	−0.992505	−	0.122201i	$-0.961005\pi$
$541$	− 5.68466i	− 0.244403i	0.992505	−	0.122201i	$-0.0389953\pi$
$542$	0	0
$543$	−16.8769	−0.724257
$544$	0	0
$545$	4.63068	0.198357
$546$	0	0
$547$	−16.4924	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$547$	−16.4924	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$548$	0	0
$549$	−5.19224	−0.221599
$550$	0	0
$551$	− 7.80776i	− 0.332622i
$552$	0	0
$553$	− 6.24621i	− 0.265616i
$554$	0	0
$555$	−0.876894	−0.0372221
$556$	0	0
$557$	31.4924i	1.33438i	0.744889	+	0.667188i	$0.232502\pi$
$557$	31.4924i	1.33438i	−0.744889	+	0.667188i	$0.767498\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.1922i	0.514757i
$562$	0	0
$563$	6.05398	0.255145	0.127572	−	0.991829i	$-0.459282\pi$
$563$	6.05398	0.255145	0.127572	+	0.991829i	$0.459282\pi$
$564$	0	0
$565$	4.06913i	0.171190i
$566$	0	0
$567$	10.9309i	0.459053i
$568$	0	0
$569$	−18.6847	−0.783302	−0.391651	−	0.920114i	$-0.628096\pi$
$569$	−18.6847	−0.783302	−0.391651	+	0.920114i	$0.628096\pi$
$570$	0	0
$571$	−8.49242	−0.355397	−0.177698	−	0.984085i	$-0.556865\pi$
$571$	−8.49242	−0.355397	−0.177698	+	0.984085i	$0.556865\pi$
$572$	0	0
$573$	26.0540	1.08842
$574$	0	0
$575$	40.6847	1.69667
$576$	0	0
$577$	27.4384i	1.14228i	0.820854	+	0.571139i	$0.193498\pi$
$577$	27.4384i	1.14228i	−0.820854	+	0.571139i	$0.806502\pi$
$578$	0	0
$579$	4.68466i	0.194688i
$580$	0	0
$581$	−22.2462	−0.922928
$582$	0	0
$583$	13.3693i	0.553701i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.9309i	0.616263i	0.951344	+	0.308131i	$0.0997037\pi$
$587$	14.9309i	0.616263i	−0.951344	+	0.308131i	$0.900296\pi$
$588$	0	0
$589$	12.4924	0.514741
$590$	0	0
$591$	− 22.9309i	− 0.943250i
$592$	0	0
$593$	− 19.4384i	− 0.798241i	−0.916898	−	0.399121i	$-0.869315\pi$
$593$	− 19.4384i	− 0.798241i	0.916898	−	0.399121i	$-0.130685\pi$
$594$	0	0
$595$	−4.38447	−0.179746
$596$	0	0
$597$	21.1771	0.866720
$598$	0	0
$599$	−26.7386	−1.09251	−0.546255	−	0.837619i	$-0.683947\pi$
$599$	−26.7386	−1.09251	−0.546255	+	0.837619i	$0.683947\pi$
$600$	0	0
$601$	27.9848	1.14153	0.570763	−	0.821115i	$-0.306648\pi$
$601$	27.9848	1.14153	0.570763	+	0.821115i	$0.306648\pi$
$602$	0	0
$603$	7.61553i	0.310128i
$604$	0	0
$605$	− 4.80776i	− 0.195463i
$606$	0	0
$607$	−19.3153	−0.783986	−0.391993	−	0.919968i	$-0.628214\pi$
$607$	−19.3153	−0.783986	−0.391993	+	0.919968i	$0.628214\pi$
$608$	0	0
$609$	12.1922i	0.494054i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	9.00000i	0.363507i	0.983344	+	0.181753i	$0.0581772\pi$
$613$	9.00000i	0.363507i	−0.983344	+	0.181753i	$0.941823\pi$
$614$	0	0
$615$	6.35416	0.256225
$616$	0	0
$617$	− 25.2462i	− 1.01637i	−0.861247	−	0.508187i	$-0.830316\pi$
$617$	− 25.2462i	− 1.01637i	0.861247	−	0.508187i	$-0.169684\pi$
$618$	0	0
$619$	28.9848i	1.16500i	0.812831	+	0.582500i	$0.197925\pi$
$619$	28.9848i	1.16500i	−0.812831	+	0.582500i	$0.802075\pi$
$620$	0	0
$621$	−48.3002	−1.93822
$622$	0	0
$623$	4.19224	0.167958
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	−3.80776	−0.152067
$628$	0	0
$629$	− 5.00000i	− 0.199363i
$630$	0	0
$631$	− 10.0540i	− 0.400242i	−0.979771	−	0.200121i	$-0.935866\pi$
$631$	− 10.0540i	− 0.400242i	0.979771	−	0.200121i	$-0.0641336\pi$
$632$	0	0
$633$	−38.5464	−1.53208
$634$	0	0
$635$	4.38447i	0.173992i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.63068i	0.104068i
$640$	0	0
$641$	1.49242	0.0589471	0.0294736	−	0.999566i	$-0.490617\pi$
$641$	1.49242	0.0589471	0.0294736	+	0.999566i	$0.490617\pi$
$642$	0	0
$643$	20.3002i	0.800561i	0.916393	+	0.400281i	$0.131087\pi$
$643$	20.3002i	0.800561i	−0.916393	+	0.400281i	$0.868913\pi$
$644$	0	0
$645$	2.13826i	0.0841939i
$646$	0	0
$647$	22.9309	0.901506	0.450753	−	0.892649i	$-0.351156\pi$
$647$	22.9309	0.901506	0.450753	+	0.892649i	$0.351156\pi$
$648$	0	0
$649$	2.43845	0.0957174
$650$	0	0
$651$	−19.5076	−0.764562
$652$	0	0
$653$	−24.5464	−0.960575	−0.480287	−	0.877111i	$-0.659468\pi$
$653$	−24.5464	−0.960575	−0.480287	+	0.877111i	$0.659468\pi$
$654$	0	0
$655$	0.984845i	0.0384811i
$656$	0	0
$657$	7.68466i	0.299807i
$658$	0	0
$659$	−21.5616	−0.839919	−0.419959	−	0.907543i	$-0.637956\pi$
$659$	−21.5616	−0.839919	−0.419959	+	0.907543i	$0.637956\pi$
$660$	0	0
$661$	5.73863i	0.223207i	0.993753	+	0.111603i	$0.0355987\pi$
$661$	5.73863i	0.223207i	−0.993753	+	0.111603i	$0.964401\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.36932i	− 0.0530998i
$666$	0	0
$667$	−43.4233	−1.68136
$668$	0	0
$669$	13.5616i	0.524320i
$670$	0	0
$671$	14.4384i	0.557390i
$672$	0	0
$673$	−17.2462	−0.664793	−0.332396	−	0.943140i	$-0.607857\pi$
$673$	−17.2462	−0.664793	−0.332396	+	0.943140i	$0.607857\pi$
$674$	0	0
$675$	−26.0540	−1.00282
$676$	0	0
$677$	−28.7386	−1.10452	−0.552258	−	0.833673i	$-0.686234\pi$
$677$	−28.7386	−1.10452	−0.552258	+	0.833673i	$0.686234\pi$
$678$	0	0
$679$	27.8078	1.06716
$680$	0	0
$681$	46.9309i	1.79839i
$682$	0	0
$683$	− 7.31534i	− 0.279914i	−0.990158	−	0.139957i	$-0.955304\pi$
$683$	− 7.31534i	− 0.279914i	0.990158	−	0.139957i	$-0.0446964\pi$
$684$	0	0
$685$	6.45360	0.246579
$686$	0	0
$687$	19.1231i	0.729592i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 25.5616i	− 0.972407i	−0.873846	−	0.486204i	$-0.838381\pi$
$691$	− 25.5616i	− 0.972407i	0.873846	−	0.486204i	$-0.161619\pi$
$692$	0	0
$693$	−1.36932	−0.0520160
$694$	0	0
$695$	4.10795i	0.155823i
$696$	0	0
$697$	36.2311i	1.37235i
$698$	0	0
$699$	19.1231	0.723302
$700$	0	0
$701$	−32.7386	−1.23652	−0.618261	−	0.785973i	$-0.712162\pi$
$701$	−32.7386	−1.23652	−0.618261	+	0.785973i	$0.712162\pi$
$702$	0	0
$703$	1.56155	0.0588951
$704$	0	0
$705$	−3.50758	−0.132103
$706$	0	0
$707$	− 17.5616i	− 0.660470i
$708$	0	0
$709$	33.7386i	1.26708i	0.773710	+	0.633540i	$0.218399\pi$
$709$	33.7386i	1.26708i	−0.773710	+	0.633540i	$0.781601\pi$
$710$	0	0
$711$	−2.24621	−0.0842395
$712$	0	0
$713$	− 69.4773i	− 2.60194i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	28.4924i	1.06407i
$718$	0	0
$719$	−42.9309	−1.60105	−0.800526	−	0.599299i	$-0.795446\pi$
$719$	−42.9309	−1.60105	−0.800526	+	0.599299i	$0.795446\pi$
$720$	0	0
$721$	3.50758i	0.130629i
$722$	0	0
$723$	11.6998i	0.435121i
$724$	0	0
$725$	−23.4233	−0.869919
$726$	0	0
$727$	6.73863	0.249922	0.124961	−	0.992162i	$-0.460119\pi$
$727$	6.73863	0.249922	0.124961	+	0.992162i	$0.460119\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−12.1922	−0.450946
$732$	0	0
$733$	1.19224i	0.0440362i	0.999758	+	0.0220181i	$0.00700915\pi$
$733$	1.19224i	0.0440362i	−0.999758	+	0.0220181i	$0.992991\pi$
$734$	0	0
$735$	− 4.00000i	− 0.147542i
$736$	0	0
$737$	21.1771	0.780068
$738$	0	0
$739$	− 46.9309i	− 1.72638i	−0.504879	−	0.863190i	$-0.668463\pi$
$739$	− 46.9309i	− 1.72638i	0.504879	−	0.863190i	$-0.331537\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.06913i	− 0.332714i	−0.986066	−	0.166357i	$-0.946800\pi$
$743$	− 9.06913i	− 0.332714i	0.986066	−	0.166357i	$-0.0532004\pi$
$744$	0	0
$745$	1.82292	0.0667866
$746$	0	0
$747$	8.00000i	0.292705i
$748$	0	0
$749$	13.5616i	0.495528i
$750$	0	0
$751$	−48.3002	−1.76250	−0.881249	−	0.472652i	$-0.843297\pi$
$751$	−48.3002	−1.76250	−0.881249	+	0.472652i	$0.843297\pi$
$752$	0	0
$753$	1.66950	0.0608401
$754$	0	0
$755$	−11.5076	−0.418804
$756$	0	0
$757$	−16.5464	−0.601389	−0.300695	−	0.953721i	$-0.597218\pi$
$757$	−16.5464	−0.601389	−0.300695	+	0.953721i	$0.597218\pi$
$758$	0	0
$759$	21.1771i	0.768679i
$760$	0	0
$761$	− 23.0691i	− 0.836255i	−0.908388	−	0.418128i	$-0.862686\pi$
$761$	− 23.0691i	− 0.836255i	0.908388	−	0.418128i	$-0.137314\pi$
$762$	0	0
$763$	12.8769	0.466175
$764$	0	0
$765$	1.57671i	0.0570060i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 0.930870i	− 0.0335680i	−0.999859	−	0.0167840i	$-0.994657\pi$
$769$	− 0.930870i	− 0.0335680i	0.999859	−	0.0167840i	$-0.00534277\pi$
$770$	0	0
$771$	−8.19224	−0.295036
$772$	0	0
$773$	− 34.1922i	− 1.22981i	−0.788601	−	0.614905i	$-0.789194\pi$
$773$	− 34.1922i	− 1.22981i	0.788601	−	0.614905i	$-0.210806\pi$
$774$	0	0
$775$	− 37.4773i	− 1.34622i
$776$	0	0
$777$	−2.43845	−0.0874788
$778$	0	0
$779$	−11.3153	−0.405414
$780$	0	0
$781$	7.31534	0.261764
$782$	0	0
$783$	27.8078	0.993768
$784$	0	0
$785$	− 1.57671i	− 0.0562751i
$786$	0	0
$787$	27.4233i	0.977535i	0.872414	+	0.488767i	$0.162553\pi$
$787$	27.4233i	0.977535i	−0.872414	+	0.488767i	$0.837447\pi$
$788$	0	0
$789$	18.4384	0.656426
$790$	0	0
$791$	11.3153i	0.402327i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 7.50758i	− 0.266266i
$796$	0	0
$797$	−12.9309	−0.458035	−0.229017	−	0.973422i	$-0.573551\pi$
$797$	−12.9309	−0.458035	−0.229017	+	0.973422i	$0.573551\pi$
$798$	0	0
$799$	− 20.0000i	− 0.707549i
$800$	0	0
$801$	− 1.50758i	− 0.0532676i
$802$	0	0
$803$	21.3693	0.754107
$804$	0	0
$805$	−7.61553	−0.268412
$806$	0	0
$807$	−9.06913	−0.319249
$808$	0	0
$809$	−23.2462	−0.817293	−0.408647	−	0.912693i	$-0.633999\pi$
$809$	−23.2462	−0.817293	−0.408647	+	0.912693i	$0.633999\pi$
$810$	0	0
$811$	8.00000i	0.280918i	0.990086	+	0.140459i	$0.0448578\pi$
$811$	8.00000i	0.280918i	−0.990086	+	0.140459i	$0.955142\pi$
$812$	0	0
$813$	14.9309i	0.523648i
$814$	0	0
$815$	12.8769	0.451058
$816$	0	0
$817$	− 3.80776i	− 0.133217i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.8078i	0.481894i	0.970538	+	0.240947i	$0.0774580\pi$
$821$	13.8078i	0.481894i	−0.970538	+	0.240947i	$0.922542\pi$
$822$	0	0
$823$	−1.17708	−0.0410305	−0.0205152	−	0.999790i	$-0.506531\pi$
$823$	−1.17708	−0.0410305	−0.0205152	+	0.999790i	$0.506531\pi$
$824$	0	0
$825$	11.4233i	0.397708i
$826$	0	0
$827$	36.9848i	1.28609i	0.765829	+	0.643045i	$0.222329\pi$
$827$	36.9848i	1.28609i	−0.765829	+	0.643045i	$0.777671\pi$
$828$	0	0
$829$	−6.50758	−0.226018	−0.113009	−	0.993594i	$-0.536049\pi$
$829$	−6.50758	−0.226018	−0.113009	+	0.993594i	$0.536049\pi$
$830$	0	0
$831$	−17.1771	−0.595866
$832$	0	0
$833$	22.8078	0.790242
$834$	0	0
$835$	5.36932	0.185813
$836$	0	0
$837$	44.4924i	1.53788i
$838$	0	0
$839$	− 27.3153i	− 0.943030i	−0.881858	−	0.471515i	$-0.843707\pi$
$839$	− 27.3153i	− 0.943030i	0.881858	−	0.471515i	$-0.156293\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	3.61553i	0.124525i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 13.3693i	− 0.459375i
$848$	0	0
$849$	−39.9157	−1.36990
$850$	0	0
$851$	− 8.68466i	− 0.297706i
$852$	0	0
$853$	− 48.4233i	− 1.65798i	−0.559262	−	0.828991i	$-0.688915\pi$
$853$	− 48.4233i	− 1.65798i	0.559262	−	0.828991i	$-0.311085\pi$
$854$	0	0
$855$	−0.492423	−0.0168405
$856$	0	0
$857$	14.9460	0.510546	0.255273	−	0.966869i	$-0.417835\pi$
$857$	14.9460	0.510546	0.255273	+	0.966869i	$0.417835\pi$
$858$	0	0
$859$	22.2462	0.759031	0.379515	−	0.925185i	$-0.376091\pi$
$859$	22.2462	0.759031	0.379515	+	0.925185i	$0.376091\pi$
$860$	0	0
$861$	17.6695	0.602175
$862$	0	0
$863$	− 21.7538i	− 0.740508i	−0.928931	−	0.370254i	$-0.879271\pi$
$863$	− 21.7538i	− 0.740508i	0.928931	−	0.370254i	$-0.120729\pi$
$864$	0	0
$865$	7.26137i	0.246894i
$866$	0	0
$867$	12.4924	0.424265
$868$	0	0
$869$	6.24621i	0.211888i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	−8.49242	−0.287096
$876$	0	0
$877$	39.7386i	1.34188i	0.741512	+	0.670939i	$0.234109\pi$
$877$	39.7386i	1.34188i	−0.741512	+	0.670939i	$0.765891\pi$
$878$	0	0
$879$	− 33.5616i	− 1.13200i
$880$	0	0
$881$	15.2462	0.513658	0.256829	−	0.966457i	$-0.417322\pi$
$881$	15.2462	0.513658	0.256829	+	0.966457i	$0.417322\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−1.36932	−0.0460291
$886$	0	0
$887$	−20.6847	−0.694523	−0.347261	−	0.937768i	$-0.612888\pi$
$887$	−20.6847	−0.694523	−0.347261	+	0.937768i	$0.612888\pi$
$888$	0	0
$889$	12.1922i	0.408914i
$890$	0	0
$891$	− 10.9309i	− 0.366198i
$892$	0	0
$893$	6.24621	0.209021
$894$	0	0
$895$	− 9.36932i	− 0.313182i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.0000i	1.33407i
$900$	0	0
$901$	42.8078	1.42613
$902$	0	0
$903$	5.94602i	0.197871i
$904$	0	0
$905$	6.06913i	0.201745i
$906$	0	0
$907$	−40.3002	−1.33815	−0.669073	−	0.743197i	$-0.733309\pi$
$907$	−40.3002	−1.33815	−0.669073	+	0.743197i	$0.733309\pi$
$908$	0	0
$909$	−6.31534	−0.209467
$910$	0	0
$911$	30.7386	1.01842	0.509208	−	0.860643i	$-0.329938\pi$
$911$	30.7386	1.01842	0.509208	+	0.860643i	$0.329938\pi$
$912$	0	0
$913$	22.2462	0.736242
$914$	0	0
$915$	− 8.10795i	− 0.268041i
$916$	0	0
$917$	2.73863i	0.0904377i
$918$	0	0
$919$	−58.5464	−1.93127	−0.965634	−	0.259907i	$-0.916308\pi$
$919$	−58.5464	−1.93127	−0.965634	+	0.259907i	$0.916308\pi$
$920$	0	0
$921$	15.2311i	0.501880i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.68466i	− 0.154031i
$926$	0	0
$927$	1.26137	0.0414287
$928$	0	0
$929$	37.9848i	1.24624i	0.782125	+	0.623121i	$0.214136\pi$
$929$	37.9848i	1.24624i	−0.782125	+	0.623121i	$0.785864\pi$
$930$	0	0
$931$	7.12311i	0.233450i
$932$	0	0
$933$	3.50758	0.114833
$934$	0	0
$935$	4.38447	0.143388
$936$	0	0
$937$	−13.6847	−0.447058	−0.223529	−	0.974697i	$-0.571758\pi$
$937$	−13.6847	−0.447058	−0.223529	+	0.974697i	$0.571758\pi$
$938$	0	0
$939$	−24.6004	−0.802803
$940$	0	0
$941$	− 14.0000i	− 0.456387i	−0.973616	−	0.228193i	$-0.926718\pi$
$941$	− 14.0000i	− 0.456387i	0.973616	−	0.228193i	$-0.0732819\pi$
$942$	0	0
$943$	62.9309i	2.04931i
$944$	0	0
$945$	4.87689	0.158645
$946$	0	0
$947$	− 15.4233i	− 0.501190i	−0.968092	−	0.250595i	$-0.919374\pi$
$947$	− 15.4233i	− 0.501190i	0.968092	−	0.250595i	$-0.0806262\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 32.1080i	− 1.04117i
$952$	0	0
$953$	25.8078	0.835995	0.417998	−	0.908448i	$-0.362732\pi$
$953$	25.8078	0.835995	0.417998	+	0.908448i	$0.362732\pi$
$954$	0	0
$955$	− 9.36932i	− 0.303184i
$956$	0	0
$957$	− 12.1922i	− 0.394119i
$958$	0	0
$959$	17.9460	0.579507
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	4.87689	0.157156
$964$	0	0
$965$	1.68466	0.0542311
$966$	0	0
$967$	5.75379i	0.185029i	0.995711	+	0.0925147i	$0.0294905\pi$
$967$	5.75379i	0.185029i	−0.995711	+	0.0925147i	$0.970509\pi$
$968$	0	0
$969$	12.1922i	0.391671i
$970$	0	0
$971$	−19.3153	−0.619859	−0.309929	−	0.950760i	$-0.600305\pi$
$971$	−19.3153	−0.619859	−0.309929	+	0.950760i	$0.600305\pi$
$972$	0	0
$973$	11.4233i	0.366214i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.24621i	− 0.0398698i	−0.999801	−	0.0199349i	$-0.993654\pi$
$977$	− 1.24621i	− 0.0398698i	0.999801	−	0.0199349i	$-0.00634590\pi$
$978$	0	0
$979$	−4.19224	−0.133984
$980$	0	0
$981$	− 4.63068i	− 0.147846i
$982$	0	0
$983$	24.9848i	0.796893i	0.917192	+	0.398446i	$0.130451\pi$
$983$	24.9848i	0.796893i	−0.917192	+	0.398446i	$0.869549\pi$
$984$	0	0
$985$	−8.24621	−0.262746
$986$	0	0
$987$	−9.75379	−0.310467
$988$	0	0
$989$	−21.1771	−0.673392
$990$	0	0
$991$	−3.69981	−0.117528	−0.0587642	−	0.998272i	$-0.518716\pi$
$991$	−3.69981	−0.117528	−0.0587642	+	0.998272i	$0.518716\pi$
$992$	0	0
$993$	− 10.8229i	− 0.343455i
$994$	0	0
$995$	− 7.61553i	− 0.241428i
$996$	0	0
$997$	−43.2462	−1.36962	−0.684811	−	0.728721i	$-0.740115\pi$
$997$	−43.2462	−1.36962	−0.684811	+	0.728721i	$0.740115\pi$
$998$	0	0
$999$	5.56155i	0.175960i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.l.337.3		4
4.3	odd	2		1352.2.f.d.337.1		4
13.5	odd	4		2704.2.a.r.1.2		2
13.7	odd	12		208.2.i.e.81.1		4
13.8	odd	4		2704.2.a.q.1.2		2
13.11	odd	12		208.2.i.e.113.1		4
13.12	even	2	inner	2704.2.f.l.337.4		4
39.11	even	12		1872.2.t.s.1153.1		4
39.20	even	12		1872.2.t.s.289.1		4
52.3	odd	6		1352.2.o.c.1161.3		8
52.7	even	12		104.2.i.b.81.2	yes	4
52.11	even	12		104.2.i.b.9.2	✓	4
52.15	even	12		1352.2.i.e.529.2		4
52.19	even	12		1352.2.i.e.1329.2		4
52.23	odd	6		1352.2.o.c.1161.4		8
52.31	even	4		1352.2.a.h.1.1		2
52.35	odd	6		1352.2.o.c.361.3		8
52.43	odd	6		1352.2.o.c.361.4		8
52.47	even	4		1352.2.a.f.1.1		2
52.51	odd	2		1352.2.f.d.337.2		4
104.11	even	12		832.2.i.o.321.1		4
104.37	odd	12		832.2.i.l.321.2		4
104.59	even	12		832.2.i.o.705.1		4
104.85	odd	12		832.2.i.l.705.2		4
156.11	odd	12		936.2.t.f.217.1		4
156.59	odd	12		936.2.t.f.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.i.b.9.2	✓	4	52.11	even	12
104.2.i.b.81.2	yes	4	52.7	even	12
208.2.i.e.81.1		4	13.7	odd	12
208.2.i.e.113.1		4	13.11	odd	12
832.2.i.l.321.2		4	104.37	odd	12
832.2.i.l.705.2		4	104.85	odd	12
832.2.i.o.321.1		4	104.11	even	12
832.2.i.o.705.1		4	104.59	even	12
936.2.t.f.217.1		4	156.11	odd	12
936.2.t.f.289.1		4	156.59	odd	12
1352.2.a.f.1.1		2	52.47	even	4
1352.2.a.h.1.1		2	52.31	even	4
1352.2.f.d.337.1		4	4.3	odd	2
1352.2.f.d.337.2		4	52.51	odd	2
1352.2.i.e.529.2		4	52.15	even	12
1352.2.i.e.1329.2		4	52.19	even	12
1352.2.o.c.361.3		8	52.35	odd	6
1352.2.o.c.361.4		8	52.43	odd	6
1352.2.o.c.1161.3		8	52.3	odd	6
1352.2.o.c.1161.4		8	52.23	odd	6
1872.2.t.s.289.1		4	39.20	even	12
1872.2.t.s.1153.1		4	39.11	even	12
2704.2.a.q.1.2		2	13.8	odd	4
2704.2.a.r.1.2		2	13.5	odd	4
2704.2.f.l.337.3		4	1.1	even	1	trivial
2704.2.f.l.337.4		4	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -0.561553i q^{5} -1.56155i q^{7} -0.561553 q^{9} +1.56155i q^{11} -0.876894i q^{15} +5.00000 q^{17} +1.56155i q^{19} -2.43845i q^{21} +8.68466 q^{23} +4.68466 q^{25} -5.56155 q^{27} -5.00000 q^{29} -8.00000i q^{31} +2.43845i q^{33} -0.876894 q^{35} -1.00000i q^{37} +7.24621i q^{41} -2.43845 q^{43} +0.315342i q^{45} -4.00000i q^{47} +4.56155 q^{49} +7.80776 q^{51} +8.56155 q^{53} +0.876894 q^{55} +2.43845i q^{57} -1.56155i q^{59} +9.24621 q^{61} +0.876894i q^{63} -13.5616i q^{67} +13.5616 q^{69} -4.68466i q^{71} -13.6847i q^{73} +7.31534 q^{75} +2.43845 q^{77} +4.00000 q^{79} -7.00000 q^{81} -14.2462i q^{83} -2.80776i q^{85} -7.80776 q^{87} +2.68466i q^{89} -12.4924i q^{93} +0.876894 q^{95} +17.8078i q^{97} -0.876894i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 6 q^{9} + 20 q^{17} + 10 q^{23} - 6 q^{25} - 14 q^{27} - 20 q^{29} - 20 q^{35} - 18 q^{43} + 10 q^{49} - 10 q^{51} + 26 q^{53} + 20 q^{55} + 4 q^{61} + 46 q^{69} + 54 q^{75} + 18 q^{77}+ \cdots + 20 q^{95}+O(q^{100}) \) 4 * q - 2 * q^3 + 6 * q^9 + 20 * q^17 + 10 * q^23 - 6 * q^25 - 14 * q^27 - 20 * q^29 - 20 * q^35 - 18 * q^43 + 10 * q^49 - 10 * q^51 + 26 * q^53 + 20 * q^55 + 4 * q^61 + 46 * q^69 + 54 * q^75 + 18 * q^77 + 16 * q^79 - 28 * q^81 + 10 * q^87 + 20 * q^95

Introduction

L-functions

Modular forms

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