LMFDB - Embedded newform 5780.2.c.h.5201.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5780,2,Mod(5201,5780)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5780.5201"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5780, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-4,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$46.1535323683$
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} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121 $ x^12 + 6x^10 - 16x^9 + 9x^8 - 72x^7 + 114x^6 - 144x^5 + 391x^4 - 484x^3 + 504x^2 - 396x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5201.11
Root			$-1.47591 + 1.40653i$ of defining polynomial
Character	$\chi$	$=$	5780.5201
Dual form			5780.2.c.h.5201.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79518i q^{3} -1.00000i q^{5} -5.23303i q^{7} -0.222687 q^{9} +5.28579i q^{11} -2.63690 q^{13} +1.79518 q^{15} -4.31329 q^{19} +9.39426 q^{21} -4.16595i q^{23} -1.00000 q^{25} +4.98579i q^{27} -2.54692i q^{29} +3.00972i q^{31} -9.48896 q^{33} -5.23303 q^{35} +9.47127i q^{37} -4.73372i q^{39} +1.35277i q^{41} +0.613266 q^{43} +0.222687i q^{45} +5.18546 q^{47} -20.3846 q^{49} +1.77451 q^{53} +5.28579 q^{55} -7.74315i q^{57} -7.97014 q^{59} -3.64785i q^{61} +1.16533i q^{63} +2.63690i q^{65} +11.7984 q^{67} +7.47865 q^{69} -7.19428i q^{71} +6.49108i q^{73} -1.79518i q^{75} +27.6607 q^{77} -1.79108i q^{79} -9.61847 q^{81} -12.8929 q^{83} +4.57220 q^{87} +3.34014 q^{89} +13.7990i q^{91} -5.40301 q^{93} +4.31329i q^{95} +4.62864i q^{97} -1.17708i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 16 q^{13} - 24 q^{19} - 16 q^{21} - 12 q^{25} - 48 q^{33} - 8 q^{35} + 8 q^{43} + 8 q^{47} - 44 q^{49} + 16 q^{53} - 16 q^{55} + 32 q^{67} - 32 q^{69} + 48 q^{77} + 4 q^{81} - 16 q^{83}+ \cdots + 24 q^{93}+O(q^{100}) $ 12 * q - 4 * q^9 - 16 * q^13 - 24 * q^19 - 16 * q^21 - 12 * q^25 - 48 * q^33 - 8 * q^35 + 8 * q^43 + 8 * q^47 - 44 * q^49 + 16 * q^53 - 16 * q^55 + 32 * q^67 - 32 * q^69 + 48 * q^77 + 4 * q^81 - 16 * q^83 + 56 * q^89 + 24 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$	$2891$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.79518i	1.03645i	0.855244	+	0.518225i	$0.173407\pi$
$3$	1.79518i	1.03645i	−0.855244	+	0.518225i	$0.826593\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 5.23303i	− 1.97790i	−0.148248	−	0.988950i	$-0.547363\pi$
$7$	− 5.23303i	− 1.97790i	0.148248	−	0.988950i	$-0.452637\pi$
$8$	0	0
$9$	−0.222687	−0.0742291
$10$	0	0
$11$	5.28579i	1.59372i	0.604161	+	0.796862i	$0.293508\pi$
$11$	5.28579i	1.59372i	−0.604161	+	0.796862i	$0.706492\pi$
$12$	0	0
$13$	−2.63690	−0.731345	−0.365672	−	0.930744i	$-0.619161\pi$
$13$	−2.63690	−0.731345	−0.365672	+	0.930744i	$0.619161\pi$
$14$	0	0
$15$	1.79518	0.463515
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−4.31329	−0.989536	−0.494768	−	0.869025i	$-0.664747\pi$
$19$	−4.31329	−0.989536	−0.494768	+	0.869025i	$0.664747\pi$
$20$	0	0
$21$	9.39426	2.05000
$22$	0	0
$23$	− 4.16595i	− 0.868661i	−0.900754	−	0.434331i	$-0.856985\pi$
$23$	− 4.16595i	− 0.868661i	0.900754	−	0.434331i	$-0.143015\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.98579i	0.959515i
$28$	0	0
$29$	− 2.54692i	− 0.472952i	−0.971637	−	0.236476i	$-0.924008\pi$
$29$	− 2.54692i	− 0.472952i	0.971637	−	0.236476i	$-0.0759925\pi$
$30$	0	0
$31$	3.00972i	0.540562i	0.962781	+	0.270281i	$0.0871166\pi$
$31$	3.00972i	0.540562i	−0.962781	+	0.270281i	$0.912883\pi$
$32$	0	0
$33$	−9.48896	−1.65182
$34$	0	0
$35$	−5.23303	−0.884544
$36$	0	0
$37$	9.47127i	1.55707i	0.627603	+	0.778534i	$0.284036\pi$
$37$	9.47127i	1.55707i	−0.627603	+	0.778534i	$0.715964\pi$
$38$	0	0
$39$	− 4.73372i	− 0.758002i
$40$	0	0
$41$	1.35277i	0.211268i	0.994405	+	0.105634i	$0.0336871\pi$
$41$	1.35277i	0.211268i	−0.994405	+	0.105634i	$0.966313\pi$
$42$	0	0
$43$	0.613266	0.0935222	0.0467611	−	0.998906i	$-0.485110\pi$
$43$	0.613266	0.0935222	0.0467611	+	0.998906i	$0.485110\pi$
$44$	0	0
$45$	0.222687i	0.0331963i
$46$	0	0
$47$	5.18546	0.756378	0.378189	−	0.925728i	$-0.376547\pi$
$47$	5.18546	0.756378	0.378189	+	0.925728i	$0.376547\pi$
$48$	0	0
$49$	−20.3846	−2.91209
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.77451	0.243748	0.121874	−	0.992546i	$-0.461110\pi$
$53$	1.77451	0.243748	0.121874	+	0.992546i	$0.461110\pi$
$54$	0	0
$55$	5.28579	0.712735
$56$	0	0
$57$	− 7.74315i	− 1.02560i
$58$	0	0
$59$	−7.97014	−1.03762	−0.518812	−	0.854888i	$-0.673626\pi$
$59$	−7.97014	−1.03762	−0.518812	+	0.854888i	$0.673626\pi$
$60$	0	0
$61$	− 3.64785i	− 0.467059i	−0.972350	−	0.233530i	$-0.924972\pi$
$61$	− 3.64785i	− 0.467059i	0.972350	−	0.233530i	$-0.0750276\pi$
$62$	0	0
$63$	1.16533i	0.146818i
$64$	0	0
$65$	2.63690i	0.327067i
$66$	0	0
$67$	11.7984	1.44141	0.720703	−	0.693244i	$-0.243819\pi$
$67$	11.7984	1.44141	0.720703	+	0.693244i	$0.243819\pi$
$68$	0	0
$69$	7.47865	0.900324
$70$	0	0
$71$	− 7.19428i	− 0.853804i	−0.904298	−	0.426902i	$-0.859605\pi$
$71$	− 7.19428i	− 0.853804i	0.904298	−	0.426902i	$-0.140395\pi$
$72$	0	0
$73$	6.49108i	0.759724i	0.925043	+	0.379862i	$0.124028\pi$
$73$	6.49108i	0.759724i	−0.925043	+	0.379862i	$0.875972\pi$
$74$	0	0
$75$	− 1.79518i	− 0.207290i
$76$	0	0
$77$	27.6607	3.15223
$78$	0	0
$79$	− 1.79108i	− 0.201513i	−0.994911	−	0.100756i	$-0.967874\pi$
$79$	− 1.79108i	− 0.201513i	0.994911	−	0.100756i	$-0.0321262\pi$
$80$	0	0
$81$	−9.61847	−1.06872
$82$	0	0
$83$	−12.8929	−1.41518	−0.707588	−	0.706625i	$-0.750217\pi$
$83$	−12.8929	−1.41518	−0.707588	+	0.706625i	$0.750217\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.57220	0.490191
$88$	0	0
$89$	3.34014	0.354055	0.177027	−	0.984206i	$-0.443352\pi$
$89$	3.34014	0.354055	0.177027	+	0.984206i	$0.443352\pi$
$90$	0	0
$91$	13.7990i	1.44653i
$92$	0	0
$93$	−5.40301	−0.560266
$94$	0	0
$95$	4.31329i	0.442534i
$96$	0	0
$97$	4.62864i	0.469967i	0.971999	+	0.234983i	$0.0755036\pi$
$97$	4.62864i	0.469967i	−0.971999	+	0.234983i	$0.924496\pi$
$98$	0	0
$99$	− 1.17708i	− 0.118301i
$100$	0	0
$101$	−17.3002	−1.72144	−0.860719	−	0.509081i	$-0.829985\pi$
$101$	−17.3002	−1.72144	−0.860719	+	0.509081i	$0.829985\pi$
$102$	0	0
$103$	18.0638	1.77988	0.889940	−	0.456077i	$-0.150746\pi$
$103$	18.0638	1.77988	0.889940	+	0.456077i	$0.150746\pi$
$104$	0	0
$105$	− 9.39426i	− 0.916786i
$106$	0	0
$107$	12.3333i	1.19231i	0.802870	+	0.596154i	$0.203305\pi$
$107$	12.3333i	1.19231i	−0.802870	+	0.596154i	$0.796695\pi$
$108$	0	0
$109$	5.86139i	0.561419i	0.959793	+	0.280710i	$0.0905698\pi$
$109$	5.86139i	0.561419i	−0.959793	+	0.280710i	$0.909430\pi$
$110$	0	0
$111$	−17.0027	−1.61382
$112$	0	0
$113$	4.39938i	0.413859i	0.978356	+	0.206929i	$0.0663470\pi$
$113$	4.39938i	0.413859i	−0.978356	+	0.206929i	$0.933653\pi$
$114$	0	0
$115$	−4.16595	−0.388477
$116$	0	0
$117$	0.587204	0.0542871
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−16.9395	−1.53996
$122$	0	0
$123$	−2.42848	−0.218968
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−4.13619	−0.367027	−0.183514	−	0.983017i	$-0.558747\pi$
$127$	−4.13619	−0.367027	−0.183514	+	0.983017i	$0.558747\pi$
$128$	0	0
$129$	1.10093i	0.0969311i
$130$	0	0
$131$	22.1002i	1.93090i	0.260581	+	0.965452i	$0.416086\pi$
$131$	22.1002i	1.93090i	−0.260581	+	0.965452i	$0.583914\pi$
$132$	0	0
$133$	22.5716i	1.95720i
$134$	0	0
$135$	4.98579	0.429108
$136$	0	0
$137$	−2.63690	−0.225286	−0.112643	−	0.993636i	$-0.535932\pi$
$137$	−2.63690	−0.225286	−0.112643	+	0.993636i	$0.535932\pi$
$138$	0	0
$139$	12.9759i	1.10060i	0.834966	+	0.550301i	$0.185487\pi$
$139$	12.9759i	1.10060i	−0.834966	+	0.550301i	$0.814513\pi$
$140$	0	0
$141$	9.30887i	0.783948i
$142$	0	0
$143$	− 13.9381i	− 1.16556i
$144$	0	0
$145$	−2.54692	−0.211511
$146$	0	0
$147$	− 36.5942i	− 3.01824i
$148$	0	0
$149$	−5.47255	−0.448329	−0.224164	−	0.974551i	$-0.571965\pi$
$149$	−5.47255	−0.448329	−0.224164	+	0.974551i	$0.571965\pi$
$150$	0	0
$151$	22.6375	1.84221	0.921105	−	0.389314i	$-0.127288\pi$
$151$	22.6375	1.84221	0.921105	+	0.389314i	$0.127288\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.00972	0.241747
$156$	0	0
$157$	−11.0324	−0.880480	−0.440240	−	0.897880i	$-0.645107\pi$
$157$	−11.0324	−0.880480	−0.440240	+	0.897880i	$0.645107\pi$
$158$	0	0
$159$	3.18558i	0.252633i
$160$	0	0
$161$	−21.8006	−1.71813
$162$	0	0
$163$	9.88026i	0.773882i	0.922105	+	0.386941i	$0.126468\pi$
$163$	9.88026i	0.773882i	−0.922105	+	0.386941i	$0.873532\pi$
$164$	0	0
$165$	9.48896i	0.738715i
$166$	0	0
$167$	16.1670i	1.25104i	0.780209	+	0.625519i	$0.215113\pi$
$167$	16.1670i	1.25104i	−0.780209	+	0.625519i	$0.784887\pi$
$168$	0	0
$169$	−6.04675	−0.465135
$170$	0	0
$171$	0.960514	0.0734524
$172$	0	0
$173$	6.01359i	0.457205i	0.973520	+	0.228602i	$0.0734155\pi$
$173$	6.01359i	0.457205i	−0.973520	+	0.228602i	$0.926584\pi$
$174$	0	0
$175$	5.23303i	0.395580i
$176$	0	0
$177$	− 14.3079i	− 1.07545i
$178$	0	0
$179$	12.0604	0.901438	0.450719	−	0.892666i	$-0.351168\pi$
$179$	12.0604	0.901438	0.450719	+	0.892666i	$0.351168\pi$
$180$	0	0
$181$	3.47549i	0.258331i	0.991623	+	0.129166i	$0.0412298\pi$
$181$	3.47549i	0.258331i	−0.991623	+	0.129166i	$0.958770\pi$
$182$	0	0
$183$	6.54856	0.484084
$184$	0	0
$185$	9.47127	0.696342
$186$	0	0
$187$	0	0
$188$	0	0
$189$	26.0908	1.89783
$190$	0	0
$191$	4.77866	0.345772	0.172886	−	0.984942i	$-0.444691\pi$
$191$	4.77866	0.345772	0.172886	+	0.984942i	$0.444691\pi$
$192$	0	0
$193$	22.7383i	1.63674i	0.574691	+	0.818370i	$0.305122\pi$
$193$	22.7383i	1.63674i	−0.574691	+	0.818370i	$0.694878\pi$
$194$	0	0
$195$	−4.73372	−0.338989
$196$	0	0
$197$	− 12.6763i	− 0.903149i	−0.892234	−	0.451574i	$-0.850863\pi$
$197$	− 12.6763i	− 0.903149i	0.892234	−	0.451574i	$-0.149137\pi$
$198$	0	0
$199$	− 6.16604i	− 0.437099i	−0.975826	−	0.218549i	$-0.929867\pi$
$199$	− 6.16604i	− 0.437099i	0.975826	−	0.218549i	$-0.0701325\pi$
$200$	0	0
$201$	21.1803i	1.49395i
$202$	0	0
$203$	−13.3281	−0.935452
$204$	0	0
$205$	1.35277	0.0944818
$206$	0	0
$207$	0.927705i	0.0644799i
$208$	0	0
$209$	− 22.7991i	− 1.57705i
$210$	0	0
$211$	19.7285i	1.35816i	0.734062	+	0.679082i	$0.237622\pi$
$211$	19.7285i	1.35816i	−0.734062	+	0.679082i	$0.762378\pi$
$212$	0	0
$213$	12.9151	0.884926
$214$	0	0
$215$	− 0.613266i	− 0.0418244i
$216$	0	0
$217$	15.7500	1.06918
$218$	0	0
$219$	−11.6527	−0.787416
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−23.7682	−1.59164	−0.795818	−	0.605536i	$-0.792959\pi$
$223$	−23.7682	−1.59164	−0.795818	+	0.605536i	$0.792959\pi$
$224$	0	0
$225$	0.222687	0.0148458
$226$	0	0
$227$	12.9091i	0.856807i	0.903587	+	0.428404i	$0.140924\pi$
$227$	12.9091i	0.856807i	−0.903587	+	0.428404i	$0.859076\pi$
$228$	0	0
$229$	23.9466	1.58244	0.791218	−	0.611534i	$-0.209447\pi$
$229$	23.9466	1.58244	0.791218	+	0.611534i	$0.209447\pi$
$230$	0	0
$231$	49.6560i	3.26713i
$232$	0	0
$233$	9.68125i	0.634239i	0.948386	+	0.317120i	$0.102716\pi$
$233$	9.68125i	0.634239i	−0.948386	+	0.317120i	$0.897284\pi$
$234$	0	0
$235$	− 5.18546i	− 0.338262i
$236$	0	0
$237$	3.21532	0.208858
$238$	0	0
$239$	−18.7624	−1.21364	−0.606820	−	0.794839i	$-0.707555\pi$
$239$	−18.7624	−1.21364	−0.606820	+	0.794839i	$0.707555\pi$
$240$	0	0
$241$	− 0.565814i	− 0.0364473i	−0.999834	−	0.0182236i	$-0.994199\pi$
$241$	− 0.565814i	− 0.0364473i	0.999834	−	0.0182236i	$-0.00580108\pi$
$242$	0	0
$243$	− 2.30957i	− 0.148159i
$244$	0	0
$245$	20.3846i	1.30233i
$246$	0	0
$247$	11.3737	0.723692
$248$	0	0
$249$	− 23.1451i	− 1.46676i
$250$	0	0
$251$	−22.5898	−1.42585	−0.712927	−	0.701238i	$-0.752631\pi$
$251$	−22.5898	−1.42585	−0.712927	+	0.701238i	$0.752631\pi$
$252$	0	0
$253$	22.0203	1.38441
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.4988	1.21630	0.608149	−	0.793823i	$-0.291912\pi$
$257$	19.4988	1.21630	0.608149	+	0.793823i	$0.291912\pi$
$258$	0	0
$259$	49.5635	3.07972
$260$	0	0
$261$	0.567168i	0.0351068i
$262$	0	0
$263$	−3.17067	−0.195512	−0.0977560	−	0.995210i	$-0.531166\pi$
$263$	−3.17067	−0.195512	−0.0977560	+	0.995210i	$0.531166\pi$
$264$	0	0
$265$	− 1.77451i	− 0.109007i
$266$	0	0
$267$	5.99617i	0.366960i
$268$	0	0
$269$	− 15.9119i	− 0.970163i	−0.874469	−	0.485082i	$-0.838790\pi$
$269$	− 15.9119i	− 0.970163i	0.874469	−	0.485082i	$-0.161210\pi$
$270$	0	0
$271$	−12.7342	−0.773546	−0.386773	−	0.922175i	$-0.626410\pi$
$271$	−12.7342	−0.773546	−0.386773	+	0.922175i	$0.626410\pi$
$272$	0	0
$273$	−24.7717	−1.49925
$274$	0	0
$275$	− 5.28579i	− 0.318745i
$276$	0	0
$277$	11.9946i	0.720683i	0.932820	+	0.360342i	$0.117340\pi$
$277$	11.9946i	0.720683i	−0.932820	+	0.360342i	$0.882660\pi$
$278$	0	0
$279$	− 0.670227i	− 0.0401254i
$280$	0	0
$281$	3.10194	0.185046	0.0925232	−	0.995711i	$-0.470507\pi$
$281$	3.10194	0.185046	0.0925232	+	0.995711i	$0.470507\pi$
$282$	0	0
$283$	− 25.9628i	− 1.54333i	−0.636029	−	0.771665i	$-0.719424\pi$
$283$	− 25.9628i	− 1.54333i	0.636029	−	0.771665i	$-0.280576\pi$
$284$	0	0
$285$	−7.74315	−0.458664
$286$	0	0
$287$	7.07911	0.417866
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−8.30926	−0.487097
$292$	0	0
$293$	−4.39037	−0.256488	−0.128244	−	0.991743i	$-0.540934\pi$
$293$	−4.39037	−0.256488	−0.128244	+	0.991743i	$0.540934\pi$
$294$	0	0
$295$	7.97014i	0.464040i
$296$	0	0
$297$	−26.3538	−1.52920
$298$	0	0
$299$	10.9852i	0.635291i
$300$	0	0
$301$	− 3.20924i	− 0.184978i
$302$	0	0
$303$	− 31.0571i	− 1.78418i
$304$	0	0
$305$	−3.64785	−0.208875
$306$	0	0
$307$	14.1398	0.806998	0.403499	−	0.914980i	$-0.367794\pi$
$307$	14.1398	0.806998	0.403499	+	0.914980i	$0.367794\pi$
$308$	0	0
$309$	32.4279i	1.84476i
$310$	0	0
$311$	9.90859i	0.561865i	0.959728	+	0.280932i	$0.0906437\pi$
$311$	9.90859i	0.561865i	−0.959728	+	0.280932i	$0.909356\pi$
$312$	0	0
$313$	− 1.13927i	− 0.0643954i	−0.999482	−	0.0321977i	$-0.989749\pi$
$313$	− 1.13927i	− 0.0643954i	0.999482	−	0.0321977i	$-0.0102506\pi$
$314$	0	0
$315$	1.16533	0.0656589
$316$	0	0
$317$	− 1.69556i	− 0.0952319i	−0.998866	−	0.0476159i	$-0.984838\pi$
$317$	− 1.69556i	− 0.0952319i	0.998866	−	0.0476159i	$-0.0151624\pi$
$318$	0	0
$319$	13.4625	0.753755
$320$	0	0
$321$	−22.1406	−1.23577
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.63690	0.146269
$326$	0	0
$327$	−10.5223	−0.581883
$328$	0	0
$329$	− 27.1357i	− 1.49604i
$330$	0	0
$331$	14.1128	0.775711	0.387856	−	0.921720i	$-0.373216\pi$
$331$	14.1128	0.775711	0.387856	+	0.921720i	$0.373216\pi$
$332$	0	0
$333$	− 2.10913i	− 0.115580i
$334$	0	0
$335$	− 11.7984i	− 0.644617i
$336$	0	0
$337$	3.08406i	0.168000i	0.996466	+	0.0839998i	$0.0267695\pi$
$337$	3.08406i	0.168000i	−0.996466	+	0.0839998i	$0.973230\pi$
$338$	0	0
$339$	−7.89769	−0.428944
$340$	0	0
$341$	−15.9087	−0.861507
$342$	0	0
$343$	70.0422i	3.78192i
$344$	0	0
$345$	− 7.47865i	− 0.402637i
$346$	0	0
$347$	− 6.32536i	− 0.339563i	−0.985482	−	0.169782i	$-0.945694\pi$
$347$	− 6.32536i	− 0.339563i	0.985482	−	0.169782i	$-0.0543062\pi$
$348$	0	0
$349$	7.89867	0.422806	0.211403	−	0.977399i	$-0.432197\pi$
$349$	7.89867	0.422806	0.211403	+	0.977399i	$0.432197\pi$
$350$	0	0
$351$	− 13.1470i	− 0.701737i
$352$	0	0
$353$	−29.9338	−1.59321	−0.796607	−	0.604498i	$-0.793374\pi$
$353$	−29.9338	−1.59321	−0.796607	+	0.604498i	$0.793374\pi$
$354$	0	0
$355$	−7.19428	−0.381833
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.08403	−0.268325	−0.134163	−	0.990959i	$-0.542834\pi$
$359$	−5.08403	−0.268325	−0.134163	+	0.990959i	$0.542834\pi$
$360$	0	0
$361$	−0.395551	−0.0208185
$362$	0	0
$363$	− 30.4096i	− 1.59609i
$364$	0	0
$365$	6.49108	0.339759
$366$	0	0
$367$	− 9.15114i	− 0.477686i	−0.971058	−	0.238843i	$-0.923232\pi$
$367$	− 9.15114i	− 0.477686i	0.971058	−	0.238843i	$-0.0767681\pi$
$368$	0	0
$369$	− 0.301245i	− 0.0156822i
$370$	0	0
$371$	− 9.28609i	− 0.482110i
$372$	0	0
$373$	−9.45731	−0.489681	−0.244840	−	0.969563i	$-0.578736\pi$
$373$	−9.45731	−0.489681	−0.244840	+	0.969563i	$0.578736\pi$
$374$	0	0
$375$	−1.79518	−0.0927029
$376$	0	0
$377$	6.71599i	0.345891i
$378$	0	0
$379$	− 0.461891i	− 0.0237257i	−0.999930	−	0.0118629i	$-0.996224\pi$
$379$	− 0.461891i	− 0.0237257i	0.999930	−	0.0118629i	$-0.00377616\pi$
$380$	0	0
$381$	− 7.42522i	− 0.380406i
$382$	0	0
$383$	−24.2666	−1.23996	−0.619982	−	0.784616i	$-0.712860\pi$
$383$	−24.2666	−1.23996	−0.619982	+	0.784616i	$0.712860\pi$
$384$	0	0
$385$	− 27.6607i	− 1.40972i
$386$	0	0
$387$	−0.136567	−0.00694207
$388$	0	0
$389$	−10.9870	−0.557065	−0.278533	−	0.960427i	$-0.589848\pi$
$389$	−10.9870	−0.557065	−0.278533	+	0.960427i	$0.589848\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−39.6739	−2.00129
$394$	0	0
$395$	−1.79108	−0.0901191
$396$	0	0
$397$	17.2610i	0.866306i	0.901320	+	0.433153i	$0.142599\pi$
$397$	17.2610i	0.866306i	−0.901320	+	0.433153i	$0.857401\pi$
$398$	0	0
$399$	−40.5201	−2.02854
$400$	0	0
$401$	− 15.4421i	− 0.771140i	−0.922679	−	0.385570i	$-0.874005\pi$
$401$	− 15.4421i	− 0.771140i	0.922679	−	0.385570i	$-0.125995\pi$
$402$	0	0
$403$	− 7.93634i	− 0.395337i
$404$	0	0
$405$	9.61847i	0.477946i
$406$	0	0
$407$	−50.0631	−2.48154
$408$	0	0
$409$	−30.1548	−1.49106	−0.745530	−	0.666472i	$-0.767803\pi$
$409$	−30.1548	−1.49106	−0.745530	+	0.666472i	$0.767803\pi$
$410$	0	0
$411$	− 4.73372i	− 0.233497i
$412$	0	0
$413$	41.7080i	2.05232i
$414$	0	0
$415$	12.8929i	0.632886i
$416$	0	0
$417$	−23.2941	−1.14072
$418$	0	0
$419$	1.74566i	0.0852811i	0.999090	+	0.0426405i	$0.0135770\pi$
$419$	1.74566i	0.0852811i	−0.999090	+	0.0426405i	$0.986423\pi$
$420$	0	0
$421$	−27.3370	−1.33233	−0.666163	−	0.745806i	$-0.732064\pi$
$421$	−27.3370	−1.33233	−0.666163	+	0.745806i	$0.732064\pi$
$422$	0	0
$423$	−1.15474	−0.0561452
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19.0893	−0.923797
$428$	0	0
$429$	25.0215	1.20805
$430$	0	0
$431$	− 36.9528i	− 1.77996i	−0.456003	−	0.889978i	$-0.650720\pi$
$431$	− 36.9528i	− 1.77996i	0.456003	−	0.889978i	$-0.349280\pi$
$432$	0	0
$433$	−30.8419	−1.48217	−0.741084	−	0.671413i	$-0.765688\pi$
$433$	−30.8419	−1.48217	−0.741084	+	0.671413i	$0.765688\pi$
$434$	0	0
$435$	− 4.57220i	− 0.219220i
$436$	0	0
$437$	17.9690i	0.859572i
$438$	0	0
$439$	5.38634i	0.257076i	0.991705	+	0.128538i	$0.0410284\pi$
$439$	5.38634i	0.257076i	−0.991705	+	0.128538i	$0.958972\pi$
$440$	0	0
$441$	4.53940	0.216162
$442$	0	0
$443$	26.4761	1.25792	0.628958	−	0.777440i	$-0.283482\pi$
$443$	26.4761	1.25792	0.628958	+	0.777440i	$0.283482\pi$
$444$	0	0
$445$	− 3.34014i	− 0.158338i
$446$	0	0
$447$	− 9.82424i	− 0.464670i
$448$	0	0
$449$	11.3573i	0.535983i	0.963421	+	0.267992i	$0.0863600\pi$
$449$	11.3573i	0.535983i	−0.963421	+	0.267992i	$0.913640\pi$
$450$	0	0
$451$	−7.15047	−0.336703
$452$	0	0
$453$	40.6384i	1.90936i
$454$	0	0
$455$	13.7990	0.646907
$456$	0	0
$457$	3.71394	0.173731	0.0868653	−	0.996220i	$-0.472315\pi$
$457$	3.71394	0.173731	0.0868653	+	0.996220i	$0.472315\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.3511	−0.621821	−0.310910	−	0.950439i	$-0.600634\pi$
$461$	−13.3511	−0.621821	−0.310910	+	0.950439i	$0.600634\pi$
$462$	0	0
$463$	30.5506	1.41981	0.709904	−	0.704298i	$-0.248738\pi$
$463$	30.5506	1.41981	0.709904	+	0.704298i	$0.248738\pi$
$464$	0	0
$465$	5.40301i	0.250558i
$466$	0	0
$467$	12.1558	0.562504	0.281252	−	0.959634i	$-0.409250\pi$
$467$	12.1558	0.562504	0.281252	+	0.959634i	$0.409250\pi$
$468$	0	0
$469$	− 61.7415i	− 2.85096i
$470$	0	0
$471$	− 19.8052i	− 0.912573i
$472$	0	0
$473$	3.24159i	0.149049i
$474$	0	0
$475$	4.31329	0.197907
$476$	0	0
$477$	−0.395162	−0.0180932
$478$	0	0
$479$	− 7.22471i	− 0.330105i	−0.986285	−	0.165053i	$-0.947221\pi$
$479$	− 7.22471i	− 0.330105i	0.986285	−	0.165053i	$-0.0527794\pi$
$480$	0	0
$481$	− 24.9748i	− 1.13875i
$482$	0	0
$483$	− 39.1360i	− 1.78075i
$484$	0	0
$485$	4.62864	0.210176
$486$	0	0
$487$	− 1.15911i	− 0.0525243i	−0.999655	−	0.0262622i	$-0.991640\pi$
$487$	− 1.15911i	− 0.0525243i	0.999655	−	0.0262622i	$-0.00836047\pi$
$488$	0	0
$489$	−17.7369	−0.802090
$490$	0	0
$491$	19.9684	0.901163	0.450581	−	0.892735i	$-0.351217\pi$
$491$	19.9684	0.901163	0.450581	+	0.892735i	$0.351217\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−1.17708	−0.0529057
$496$	0	0
$497$	−37.6479	−1.68874
$498$	0	0
$499$	− 0.666349i	− 0.0298299i	−0.999889	−	0.0149149i	$-0.995252\pi$
$499$	− 0.666349i	− 0.0298299i	0.999889	−	0.0149149i	$-0.00474775\pi$
$500$	0	0
$501$	−29.0227	−1.29664
$502$	0	0
$503$	− 5.67408i	− 0.252995i	−0.991967	−	0.126497i	$-0.959626\pi$
$503$	− 5.67408i	− 0.252995i	0.991967	−	0.126497i	$-0.0403735\pi$
$504$	0	0
$505$	17.3002i	0.769850i
$506$	0	0
$507$	− 10.8550i	− 0.482089i
$508$	0	0
$509$	−13.8648	−0.614545	−0.307273	−	0.951622i	$-0.599416\pi$
$509$	−13.8648	−0.614545	−0.307273	+	0.951622i	$0.599416\pi$
$510$	0	0
$511$	33.9680	1.50266
$512$	0	0
$513$	− 21.5051i	− 0.949475i
$514$	0	0
$515$	− 18.0638i	− 0.795987i
$516$	0	0
$517$	27.4093i	1.20546i
$518$	0	0
$519$	−10.7955	−0.473870
$520$	0	0
$521$	− 30.5723i	− 1.33940i	−0.742633	−	0.669699i	$-0.766423\pi$
$521$	− 30.5723i	− 1.33940i	0.742633	−	0.669699i	$-0.233577\pi$
$522$	0	0
$523$	10.0946	0.441405	0.220703	−	0.975341i	$-0.429165\pi$
$523$	10.0946	0.441405	0.220703	+	0.975341i	$0.429165\pi$
$524$	0	0
$525$	−9.39426	−0.409999
$526$	0	0
$527$	0	0
$528$	0	0
$529$	5.64483	0.245428
$530$	0	0
$531$	1.77485	0.0770219
$532$	0	0
$533$	− 3.56713i	− 0.154510i
$534$	0	0
$535$	12.3333	0.533217
$536$	0	0
$537$	21.6507i	0.934296i
$538$	0	0
$539$	− 107.749i	− 4.64107i
$540$	0	0
$541$	− 25.8933i	− 1.11324i	−0.830767	−	0.556620i	$-0.812098\pi$
$541$	− 25.8933i	− 1.11324i	0.830767	−	0.556620i	$-0.187902\pi$
$542$	0	0
$543$	−6.23914	−0.267747
$544$	0	0
$545$	5.86139	0.251074
$546$	0	0
$547$	− 12.2960i	− 0.525739i	−0.964831	−	0.262869i	$-0.915331\pi$
$547$	− 12.2960i	− 0.525739i	0.964831	−	0.262869i	$-0.0846688\pi$
$548$	0	0
$549$	0.812330i	0.0346694i
$550$	0	0
$551$	10.9856i	0.468003i
$552$	0	0
$553$	−9.37279	−0.398572
$554$	0	0
$555$	17.0027i	0.721724i
$556$	0	0
$557$	−28.6723	−1.21489	−0.607443	−	0.794363i	$-0.707805\pi$
$557$	−28.6723	−1.21489	−0.607443	+	0.794363i	$0.707805\pi$
$558$	0	0
$559$	−1.61712	−0.0683970
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.7196	1.25253	0.626266	−	0.779609i	$-0.284582\pi$
$563$	29.7196	1.25253	0.626266	+	0.779609i	$0.284582\pi$
$564$	0	0
$565$	4.39938	0.185083
$566$	0	0
$567$	50.3338i	2.11382i
$568$	0	0
$569$	−28.0263	−1.17492	−0.587462	−	0.809251i	$-0.699873\pi$
$569$	−28.0263	−1.17492	−0.587462	+	0.809251i	$0.699873\pi$
$570$	0	0
$571$	− 42.3794i	− 1.77352i	−0.462226	−	0.886762i	$-0.652949\pi$
$571$	− 42.3794i	− 1.77352i	0.462226	−	0.886762i	$-0.347051\pi$
$572$	0	0
$573$	8.57858i	0.358375i
$574$	0	0
$575$	4.16595i	0.173732i
$576$	0	0
$577$	8.16367	0.339858	0.169929	−	0.985456i	$-0.445646\pi$
$577$	8.16367	0.339858	0.169929	+	0.985456i	$0.445646\pi$
$578$	0	0
$579$	−40.8195	−1.69640
$580$	0	0
$581$	67.4688i	2.79908i
$582$	0	0
$583$	9.37970i	0.388467i
$584$	0	0
$585$	− 0.587204i	− 0.0242779i
$586$	0	0
$587$	13.4440	0.554894	0.277447	−	0.960741i	$-0.410512\pi$
$587$	13.4440	0.554894	0.277447	+	0.960741i	$0.410512\pi$
$588$	0	0
$589$	− 12.9818i	− 0.534906i
$590$	0	0
$591$	22.7563	0.936069
$592$	0	0
$593$	−14.4936	−0.595182	−0.297591	−	0.954693i	$-0.596183\pi$
$593$	−14.4936	−0.595182	−0.297591	+	0.954693i	$0.596183\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.0692	0.453031
$598$	0	0
$599$	24.1901	0.988382	0.494191	−	0.869353i	$-0.335464\pi$
$599$	24.1901	0.988382	0.494191	+	0.869353i	$0.335464\pi$
$600$	0	0
$601$	− 8.27989i	− 0.337744i	−0.985638	−	0.168872i	$-0.945988\pi$
$601$	− 8.27989i	− 0.337744i	0.985638	−	0.168872i	$-0.0540124\pi$
$602$	0	0
$603$	−2.62736	−0.106994
$604$	0	0
$605$	16.9395i	0.688690i
$606$	0	0
$607$	30.0214i	1.21853i	0.792967	+	0.609265i	$0.208535\pi$
$607$	30.0214i	1.21853i	−0.792967	+	0.609265i	$0.791465\pi$
$608$	0	0
$609$	− 23.9265i	− 0.969549i
$610$	0	0
$611$	−13.6736	−0.553173
$612$	0	0
$613$	1.77264	0.0715963	0.0357982	−	0.999359i	$-0.488603\pi$
$613$	1.77264	0.0715963	0.0357982	+	0.999359i	$0.488603\pi$
$614$	0	0
$615$	2.42848i	0.0979257i
$616$	0	0
$617$	− 27.7453i	− 1.11698i	−0.829510	−	0.558492i	$-0.811380\pi$
$617$	− 27.7453i	− 1.11698i	0.829510	−	0.558492i	$-0.188620\pi$
$618$	0	0
$619$	47.4316i	1.90644i	0.302279	+	0.953219i	$0.402252\pi$
$619$	47.4316i	1.90644i	−0.302279	+	0.953219i	$0.597748\pi$
$620$	0	0
$621$	20.7706	0.833494
$622$	0	0
$623$	− 17.4791i	− 0.700285i
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	40.9286	1.63453
$628$	0	0
$629$	0	0
$630$	0	0
$631$	4.14957	0.165192	0.0825960	−	0.996583i	$-0.473679\pi$
$631$	4.14957	0.165192	0.0825960	+	0.996583i	$0.473679\pi$
$632$	0	0
$633$	−35.4163	−1.40767
$634$	0	0
$635$	4.13619i	0.164140i
$636$	0	0
$637$	53.7522	2.12974
$638$	0	0
$639$	1.60208i	0.0633771i
$640$	0	0
$641$	33.6467i	1.32897i	0.747303	+	0.664483i	$0.231348\pi$
$641$	33.6467i	1.32897i	−0.747303	+	0.664483i	$0.768652\pi$
$642$	0	0
$643$	13.8052i	0.544423i	0.962237	+	0.272212i	$0.0877551\pi$
$643$	13.8052i	0.544423i	−0.962237	+	0.272212i	$0.912245\pi$
$644$	0	0
$645$	1.10093	0.0433489
$646$	0	0
$647$	−24.0014	−0.943594	−0.471797	−	0.881707i	$-0.656395\pi$
$647$	−24.0014	−0.943594	−0.471797	+	0.881707i	$0.656395\pi$
$648$	0	0
$649$	− 42.1285i	− 1.65369i
$650$	0	0
$651$	28.2741i	1.10815i
$652$	0	0
$653$	10.9628i	0.429009i	0.976723	+	0.214505i	$0.0688137\pi$
$653$	10.9628i	0.429009i	−0.976723	+	0.214505i	$0.931186\pi$
$654$	0	0
$655$	22.1002	0.863526
$656$	0	0
$657$	− 1.44548i	− 0.0563936i
$658$	0	0
$659$	13.8658	0.540136	0.270068	−	0.962841i	$-0.412954\pi$
$659$	13.8658	0.540136	0.270068	+	0.962841i	$0.412954\pi$
$660$	0	0
$661$	−2.31432	−0.0900166	−0.0450083	−	0.998987i	$-0.514331\pi$
$661$	−2.31432	−0.0900166	−0.0450083	+	0.998987i	$0.514331\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.5716	0.875288
$666$	0	0
$667$	−10.6104	−0.410835
$668$	0	0
$669$	− 42.6683i	− 1.64965i
$670$	0	0
$671$	19.2818	0.744364
$672$	0	0
$673$	− 1.67123i	− 0.0644213i	−0.999481	−	0.0322106i	$-0.989745\pi$
$673$	− 1.67123i	− 0.0644213i	0.999481	−	0.0322106i	$-0.0102547\pi$
$674$	0	0
$675$	− 4.98579i	− 0.191903i
$676$	0	0
$677$	9.36814i	0.360047i	0.983662	+	0.180023i	$0.0576173\pi$
$677$	9.36814i	0.360047i	−0.983662	+	0.180023i	$0.942383\pi$
$678$	0	0
$679$	24.2218	0.929548
$680$	0	0
$681$	−23.1742	−0.888038
$682$	0	0
$683$	− 34.5712i	− 1.32283i	−0.750020	−	0.661415i	$-0.769956\pi$
$683$	− 34.5712i	− 1.32283i	0.750020	−	0.661415i	$-0.230044\pi$
$684$	0	0
$685$	2.63690i	0.100751i
$686$	0	0
$687$	42.9886i	1.64012i
$688$	0	0
$689$	−4.67922	−0.178264
$690$	0	0
$691$	19.1908i	0.730054i	0.930997	+	0.365027i	$0.118940\pi$
$691$	19.1908i	0.730054i	−0.930997	+	0.365027i	$0.881060\pi$
$692$	0	0
$693$	−6.15968	−0.233987
$694$	0	0
$695$	12.9759	0.492204
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−17.3796	−0.657358
$700$	0	0
$701$	−31.8819	−1.20416	−0.602081	−	0.798435i	$-0.705661\pi$
$701$	−31.8819	−1.20416	−0.602081	+	0.798435i	$0.705661\pi$
$702$	0	0
$703$	− 40.8523i	− 1.54077i
$704$	0	0
$705$	9.30887	0.350592
$706$	0	0
$707$	90.5327i	3.40483i
$708$	0	0
$709$	39.0093i	1.46502i	0.680755	+	0.732512i	$0.261652\pi$
$709$	39.0093i	1.46502i	−0.680755	+	0.732512i	$0.738348\pi$
$710$	0	0
$711$	0.398851i	0.0149581i
$712$	0	0
$713$	12.5384	0.469565
$714$	0	0
$715$	−13.9381	−0.521255
$716$	0	0
$717$	− 33.6820i	− 1.25788i
$718$	0	0
$719$	12.9105i	0.481479i	0.970590	+	0.240740i	$0.0773900\pi$
$719$	12.9105i	0.481479i	−0.970590	+	0.240740i	$0.922610\pi$
$720$	0	0
$721$	− 94.5285i	− 3.52043i
$722$	0	0
$723$	1.01574	0.0377758
$724$	0	0
$725$	2.54692i	0.0945904i
$726$	0	0
$727$	10.4543	0.387728	0.193864	−	0.981028i	$-0.437898\pi$
$727$	10.4543	0.387728	0.193864	+	0.981028i	$0.437898\pi$
$728$	0	0
$729$	−24.7093	−0.915160
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−32.9360	−1.21652	−0.608259	−	0.793739i	$-0.708132\pi$
$733$	−32.9360	−1.21652	−0.608259	+	0.793739i	$0.708132\pi$
$734$	0	0
$735$	−36.5942	−1.34980
$736$	0	0
$737$	62.3639i	2.29721i
$738$	0	0
$739$	18.8281	0.692601	0.346301	−	0.938124i	$-0.387438\pi$
$739$	18.8281	0.692601	0.346301	+	0.938124i	$0.387438\pi$
$740$	0	0
$741$	20.4179i	0.750071i
$742$	0	0
$743$	22.0220i	0.807909i	0.914779	+	0.403955i	$0.132365\pi$
$743$	22.0220i	0.807909i	−0.914779	+	0.403955i	$0.867635\pi$
$744$	0	0
$745$	5.47255i	0.200499i
$746$	0	0
$747$	2.87108	0.105047
$748$	0	0
$749$	64.5407	2.35827
$750$	0	0
$751$	− 15.6631i	− 0.571554i	−0.958296	−	0.285777i	$-0.907748\pi$
$751$	− 15.6631i	− 0.571554i	0.958296	−	0.285777i	$-0.0922516\pi$
$752$	0	0
$753$	− 40.5528i	− 1.47783i
$754$	0	0
$755$	− 22.6375i	− 0.823862i
$756$	0	0
$757$	0.298820	0.0108608	0.00543041	−	0.999985i	$-0.498271\pi$
$757$	0.298820	0.0108608	0.00543041	+	0.999985i	$0.498271\pi$
$758$	0	0
$759$	39.5306i	1.43487i
$760$	0	0
$761$	49.5107	1.79476	0.897381	−	0.441257i	$-0.145467\pi$
$761$	49.5107	1.79476	0.897381	+	0.441257i	$0.145467\pi$
$762$	0	0
$763$	30.6728	1.11043
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.0165	0.758861
$768$	0	0
$769$	28.9566	1.04420	0.522100	−	0.852884i	$-0.325149\pi$
$769$	28.9566	1.04420	0.522100	+	0.852884i	$0.325149\pi$
$770$	0	0
$771$	35.0039i	1.26063i
$772$	0	0
$773$	2.56888	0.0923962	0.0461981	−	0.998932i	$-0.485289\pi$
$773$	2.56888	0.0923962	0.0461981	+	0.998932i	$0.485289\pi$
$774$	0	0
$775$	− 3.00972i	− 0.108112i
$776$	0	0
$777$	88.9756i	3.19198i
$778$	0	0
$779$	− 5.83490i	− 0.209057i
$780$	0	0
$781$	38.0275	1.36073
$782$	0	0
$783$	12.6984	0.453805
$784$	0	0
$785$	11.0324i	0.393762i
$786$	0	0
$787$	− 40.8493i	− 1.45612i	−0.685513	−	0.728060i	$-0.740422\pi$
$787$	− 40.8493i	− 1.45612i	0.685513	−	0.728060i	$-0.259578\pi$
$788$	0	0
$789$	− 5.69194i	− 0.202639i
$790$	0	0
$791$	23.0221	0.818571
$792$	0	0
$793$	9.61902i	0.341581i
$794$	0	0
$795$	3.18558	0.112981
$796$	0	0
$797$	21.4521	0.759873	0.379937	−	0.925013i	$-0.375946\pi$
$797$	21.4521	0.759873	0.379937	+	0.925013i	$0.375946\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−0.743808	−0.0262811
$802$	0	0
$803$	−34.3105	−1.21079
$804$	0	0
$805$	21.8006i	0.768369i
$806$	0	0
$807$	28.5647	1.00553
$808$	0	0
$809$	21.7141i	0.763427i	0.924281	+	0.381714i	$0.124666\pi$
$809$	21.7141i	0.763427i	−0.924281	+	0.381714i	$0.875334\pi$
$810$	0	0
$811$	− 43.0923i	− 1.51317i	−0.653893	−	0.756587i	$-0.726865\pi$
$811$	− 43.0923i	− 1.51317i	0.653893	−	0.756587i	$-0.273135\pi$
$812$	0	0
$813$	− 22.8602i	− 0.801742i
$814$	0	0
$815$	9.88026	0.346090
$816$	0	0
$817$	−2.64519	−0.0925436
$818$	0	0
$819$	− 3.07286i	− 0.107374i
$820$	0	0
$821$	29.4400i	1.02746i	0.857951	+	0.513731i	$0.171737\pi$
$821$	29.4400i	1.02746i	−0.857951	+	0.513731i	$0.828263\pi$
$822$	0	0
$823$	− 27.2734i	− 0.950692i	−0.879799	−	0.475346i	$-0.842323\pi$
$823$	− 27.2734i	− 0.950692i	0.879799	−	0.475346i	$-0.157677\pi$
$824$	0	0
$825$	9.48896	0.330363
$826$	0	0
$827$	3.04713i	0.105959i	0.998596	+	0.0529795i	$0.0168718\pi$
$827$	3.04713i	0.105959i	−0.998596	+	0.0529795i	$0.983128\pi$
$828$	0	0
$829$	−36.6800	−1.27395	−0.636975	−	0.770885i	$-0.719814\pi$
$829$	−36.6800	−1.27395	−0.636975	+	0.770885i	$0.719814\pi$
$830$	0	0
$831$	−21.5325	−0.746952
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.1670	0.559481
$836$	0	0
$837$	−15.0058	−0.518678
$838$	0	0
$839$	− 16.7172i	− 0.577142i	−0.957458	−	0.288571i	$-0.906820\pi$
$839$	− 16.7172i	− 0.577142i	0.957458	−	0.288571i	$-0.0931802\pi$
$840$	0	0
$841$	22.5132	0.776316
$842$	0	0
$843$	5.56856i	0.191791i
$844$	0	0
$845$	6.04675i	0.208015i
$846$	0	0
$847$	88.6452i	3.04588i
$848$	0	0
$849$	46.6081	1.59958
$850$	0	0
$851$	39.4569	1.35256
$852$	0	0
$853$	− 1.26567i	− 0.0433356i	−0.999765	−	0.0216678i	$-0.993102\pi$
$853$	− 1.26567i	− 0.0433356i	0.999765	−	0.0216678i	$-0.00689762\pi$
$854$	0	0
$855$	− 0.960514i	− 0.0328489i
$856$	0	0
$857$	12.4928i	0.426746i	0.976971	+	0.213373i	$0.0684450\pi$
$857$	12.4928i	0.426746i	−0.976971	+	0.213373i	$0.931555\pi$
$858$	0	0
$859$	30.5126	1.04108	0.520538	−	0.853839i	$-0.325731\pi$
$859$	30.5126	1.04108	0.520538	+	0.853839i	$0.325731\pi$
$860$	0	0
$861$	12.7083i	0.433098i
$862$	0	0
$863$	24.4910	0.833685	0.416842	−	0.908979i	$-0.363137\pi$
$863$	24.4910	0.833685	0.416842	+	0.908979i	$0.363137\pi$
$864$	0	0
$865$	6.01359	0.204468
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.46728	0.321155
$870$	0	0
$871$	−31.1113	−1.05417
$872$	0	0
$873$	− 1.03074i	− 0.0348852i
$874$	0	0
$875$	5.23303	0.176909
$876$	0	0
$877$	− 24.9590i	− 0.842805i	−0.906874	−	0.421403i	$-0.861538\pi$
$877$	− 24.9590i	− 0.842805i	0.906874	−	0.421403i	$-0.138462\pi$
$878$	0	0
$879$	− 7.88153i	− 0.265837i
$880$	0	0
$881$	34.3804i	1.15830i	0.815219	+	0.579152i	$0.196616\pi$
$881$	34.3804i	1.15830i	−0.815219	+	0.579152i	$0.803384\pi$
$882$	0	0
$883$	−32.9362	−1.10839	−0.554196	−	0.832386i	$-0.686974\pi$
$883$	−32.9362	−1.10839	−0.554196	+	0.832386i	$0.686974\pi$
$884$	0	0
$885$	−14.3079	−0.480954
$886$	0	0
$887$	− 29.5748i	− 0.993025i	−0.868029	−	0.496513i	$-0.834614\pi$
$887$	− 29.5748i	− 0.993025i	0.868029	−	0.496513i	$-0.165386\pi$
$888$	0	0
$889$	21.6448i	0.725944i
$890$	0	0
$891$	− 50.8412i	− 1.70324i
$892$	0	0
$893$	−22.3664	−0.748463
$894$	0	0
$895$	− 12.0604i	− 0.403135i
$896$	0	0
$897$	−19.7205	−0.658447
$898$	0	0
$899$	7.66553	0.255660
$900$	0	0
$901$	0	0
$902$	0	0
$903$	5.76118	0.191720
$904$	0	0
$905$	3.47549	0.115529
$906$	0	0
$907$	− 2.41586i	− 0.0802173i	−0.999195	−	0.0401086i	$-0.987230\pi$
$907$	− 2.41586i	− 0.0802173i	0.999195	−	0.0401086i	$-0.0127704\pi$
$908$	0	0
$909$	3.85254	0.127781
$910$	0	0
$911$	− 54.1207i	− 1.79310i	−0.442942	−	0.896550i	$-0.646065\pi$
$911$	− 54.1207i	− 1.79310i	0.442942	−	0.896550i	$-0.353935\pi$
$912$	0	0
$913$	− 68.1490i	− 2.25540i
$914$	0	0
$915$	− 6.54856i	− 0.216489i
$916$	0	0
$917$	115.651	3.81914
$918$	0	0
$919$	−9.64435	−0.318138	−0.159069	−	0.987267i	$-0.550849\pi$
$919$	−9.64435	−0.318138	−0.159069	+	0.987267i	$0.550849\pi$
$920$	0	0
$921$	25.3835i	0.836414i
$922$	0	0
$923$	18.9706i	0.624425i
$924$	0	0
$925$	− 9.47127i	− 0.311414i
$926$	0	0
$927$	−4.02258	−0.132119
$928$	0	0
$929$	29.5153i	0.968365i	0.874967	+	0.484182i	$0.160883\pi$
$929$	29.5153i	0.968365i	−0.874967	+	0.484182i	$0.839117\pi$
$930$	0	0
$931$	87.9248	2.88162
$932$	0	0
$933$	−17.7878	−0.582345
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.00891	0.261640	0.130820	−	0.991406i	$-0.458239\pi$
$937$	8.00891	0.261640	0.130820	+	0.991406i	$0.458239\pi$
$938$	0	0
$939$	2.04520	0.0667426
$940$	0	0
$941$	47.2899i	1.54161i	0.637073	+	0.770804i	$0.280145\pi$
$941$	47.2899i	1.54161i	−0.637073	+	0.770804i	$0.719855\pi$
$942$	0	0
$943$	5.63559	0.183520
$944$	0	0
$945$	− 26.0908i	− 0.848734i
$946$	0	0
$947$	27.8278i	0.904282i	0.891947	+	0.452141i	$0.149340\pi$
$947$	27.8278i	0.904282i	−0.891947	+	0.452141i	$0.850660\pi$
$948$	0	0
$949$	− 17.1163i	− 0.555620i
$950$	0	0
$951$	3.04384	0.0987031
$952$	0	0
$953$	−32.2796	−1.04564	−0.522819	−	0.852443i	$-0.675120\pi$
$953$	−32.2796	−1.04564	−0.522819	+	0.852443i	$0.675120\pi$
$954$	0	0
$955$	− 4.77866i	− 0.154634i
$956$	0	0
$957$	24.1677i	0.781230i
$958$	0	0
$959$	13.7990i	0.445593i
$960$	0	0
$961$	21.9416	0.707793
$962$	0	0
$963$	− 2.74648i	− 0.0885040i
$964$	0	0
$965$	22.7383	0.731973
$966$	0	0
$967$	60.1211	1.93337	0.966683	−	0.255978i	$-0.0823974\pi$
$967$	60.1211	1.93337	0.966683	+	0.255978i	$0.0823974\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	56.9650	1.82809	0.914047	−	0.405608i	$-0.132940\pi$
$971$	56.9650	1.82809	0.914047	+	0.405608i	$0.132940\pi$
$972$	0	0
$973$	67.9033	2.17688
$974$	0	0
$975$	4.73372i	0.151600i
$976$	0	0
$977$	24.2881	0.777045	0.388522	−	0.921439i	$-0.372986\pi$
$977$	24.2881	0.777045	0.388522	+	0.921439i	$0.372986\pi$
$978$	0	0
$979$	17.6553i	0.564265i
$980$	0	0
$981$	− 1.30526i	− 0.0416736i
$982$	0	0
$983$	6.00189i	0.191430i	0.995409	+	0.0957152i	$0.0305138\pi$
$983$	6.00189i	0.191430i	−0.995409	+	0.0957152i	$0.969486\pi$
$984$	0	0
$985$	−12.6763	−0.403900
$986$	0	0
$987$	48.7136	1.55057
$988$	0	0
$989$	− 2.55484i	− 0.0812391i
$990$	0	0
$991$	− 38.9660i	− 1.23780i	−0.785471	−	0.618898i	$-0.787579\pi$
$991$	− 38.9660i	− 1.23780i	0.785471	−	0.618898i	$-0.212421\pi$
$992$	0	0
$993$	25.3351i	0.803986i
$994$	0	0
$995$	−6.16604	−0.195477
$996$	0	0
$997$	27.0341i	0.856179i	0.903736	+	0.428090i	$0.140813\pi$
$997$	27.0341i	0.856179i	−0.903736	+	0.428090i	$0.859187\pi$
$998$	0	0
$999$	−47.2218	−1.49403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.c.h.5201.11		12
17.2	even	8		340.2.o.a.81.5	yes	12
17.4	even	4		5780.2.a.m.1.6		6
17.8	even	8		340.2.o.a.21.5	✓	12
17.13	even	4		5780.2.a.n.1.1		6
17.16	even	2	inner	5780.2.c.h.5201.2		12
51.2	odd	8		3060.2.be.b.1441.1		12
51.8	odd	8		3060.2.be.b.361.1		12
68.19	odd	8		1360.2.bt.c.81.2		12
68.59	odd	8		1360.2.bt.c.1041.2		12
85.2	odd	8		1700.2.m.f.149.5		12
85.8	odd	8		1700.2.m.f.1449.5		12
85.19	even	8		1700.2.o.d.1101.2		12
85.42	odd	8		1700.2.m.c.1449.2		12
85.53	odd	8		1700.2.m.c.149.2		12
85.59	even	8		1700.2.o.d.701.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.5	✓	12	17.8	even	8
340.2.o.a.81.5	yes	12	17.2	even	8
1360.2.bt.c.81.2		12	68.19	odd	8
1360.2.bt.c.1041.2		12	68.59	odd	8
1700.2.m.c.149.2		12	85.53	odd	8
1700.2.m.c.1449.2		12	85.42	odd	8
1700.2.m.f.149.5		12	85.2	odd	8
1700.2.m.f.1449.5		12	85.8	odd	8
1700.2.o.d.701.2		12	85.59	even	8
1700.2.o.d.1101.2		12	85.19	even	8
3060.2.be.b.361.1		12	51.8	odd	8
3060.2.be.b.1441.1		12	51.2	odd	8
5780.2.a.m.1.6		6	17.4	even	4
5780.2.a.n.1.1		6	17.13	even	4
5780.2.c.h.5201.2		12	17.16	even	2	inner
5780.2.c.h.5201.11		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.79518i q^{3} -1.00000i q^{5} -5.23303i q^{7} -0.222687 q^{9} +5.28579i q^{11} -2.63690 q^{13} +1.79518 q^{15} -4.31329 q^{19} +9.39426 q^{21} -4.16595i q^{23} -1.00000 q^{25} +4.98579i q^{27} -2.54692i q^{29} +3.00972i q^{31} -9.48896 q^{33} -5.23303 q^{35} +9.47127i q^{37} -4.73372i q^{39} +1.35277i q^{41} +0.613266 q^{43} +0.222687i q^{45} +5.18546 q^{47} -20.3846 q^{49} +1.77451 q^{53} +5.28579 q^{55} -7.74315i q^{57} -7.97014 q^{59} -3.64785i q^{61} +1.16533i q^{63} +2.63690i q^{65} +11.7984 q^{67} +7.47865 q^{69} -7.19428i q^{71} +6.49108i q^{73} -1.79518i q^{75} +27.6607 q^{77} -1.79108i q^{79} -9.61847 q^{81} -12.8929 q^{83} +4.57220 q^{87} +3.34014 q^{89} +13.7990i q^{91} -5.40301 q^{93} +4.31329i q^{95} +4.62864i q^{97} -1.17708i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 16 q^{13} - 24 q^{19} - 16 q^{21} - 12 q^{25} - 48 q^{33} - 8 q^{35} + 8 q^{43} + 8 q^{47} - 44 q^{49} + 16 q^{53} - 16 q^{55} + 32 q^{67} - 32 q^{69} + 48 q^{77} + 4 q^{81} - 16 q^{83}+ \cdots + 24 q^{93}+O(q^{100}) \) 12 * q - 4 * q^9 - 16 * q^13 - 24 * q^19 - 16 * q^21 - 12 * q^25 - 48 * q^33 - 8 * q^35 + 8 * q^43 + 8 * q^47 - 44 * q^49 + 16 * q^53 - 16 * q^55 + 32 * q^67 - 32 * q^69 + 48 * q^77 + 4 * q^81 - 16 * q^83 + 56 * q^89 + 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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