LMFDB - Embedded newform 5780.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5780,2,Mod(1,5780)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5780.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$5780 = 2^{2} \cdot 5 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5780.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.1535323683$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.9521152.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 10x^{4} - 8x^{3} + 18x^{2} + 20x + 2$ x^6 - 10x^4 - 8x^3 + 18x^2 + 20x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.111711$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.111711 q^{3} -1.00000 q^{5} +2.79103 q^{7} -2.98752 q^{9} +1.22301 q^{11} +3.40173 q^{13} -0.111711 q^{15} +3.42309 q^{19} +0.311790 q^{21} -9.54005 q^{23} +1.00000 q^{25} -0.668874 q^{27} +10.0141 q^{29} +7.98033 q^{31} +0.136625 q^{33} -2.79103 q^{35} +5.71862 q^{37} +0.380012 q^{39} -11.6497 q^{41} +7.49527 q^{43} +2.98752 q^{45} -8.61396 q^{47} +0.789873 q^{49} -3.58108 q^{53} -1.22301 q^{55} +0.382398 q^{57} -4.23376 q^{59} -10.8514 q^{61} -8.33827 q^{63} -3.40173 q^{65} +11.1125 q^{67} -1.06573 q^{69} -0.376703 q^{71} +9.86263 q^{73} +0.111711 q^{75} +3.41348 q^{77} +14.4999 q^{79} +8.88784 q^{81} +1.52005 q^{83} +1.11869 q^{87} -4.28461 q^{89} +9.49436 q^{91} +0.891494 q^{93} -3.42309 q^{95} -2.55960 q^{97} -3.65378 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{5} + 4 q^{7} + 2 q^{9} + 8 q^{11} - 8 q^{13} + 12 q^{19} - 8 q^{21} + 4 q^{23} + 6 q^{25} - 24 q^{27} + 8 q^{29} + 20 q^{31} - 24 q^{33} - 4 q^{35} + 24 q^{37} + 20 q^{39} + 4 q^{41} - 4 q^{43}+ \cdots + 20 q^{99}+O(q^{100})$ 6 * q - 6 * q^5 + 4 * q^7 + 2 * q^9 + 8 * q^11 - 8 * q^13 + 12 * q^19 - 8 * q^21 + 4 * q^23 + 6 * q^25 - 24 * q^27 + 8 * q^29 + 20 * q^31 - 24 * q^33 - 4 * q^35 + 24 * q^37 + 20 * q^39 + 4 * q^41 - 4 * q^43 - 2 * q^45 + 4 * q^47 + 22 * q^49 - 8 * q^53 - 8 * q^55 - 16 * q^57 + 4 * q^61 + 16 * q^63 + 8 * q^65 + 16 * q^67 - 16 * q^69 + 16 * q^71 + 16 * q^73 - 24 * q^77 + 36 * q^79 + 2 * q^81 + 8 * q^83 + 28 * q^89 - 20 * q^91 - 12 * q^93 - 12 * q^95 + 32 * q^97 + 20 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	0.111711	0.0644966	0.0322483	−	0.999480i	$-0.489733\pi$
$3$	0.111711	0.0644966	0.0322483	+	0.999480i	$0.489733\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.79103	1.05491	0.527456	−	0.849582i	$-0.323146\pi$
$7$	2.79103	1.05491	0.527456	+	0.849582i	$0.323146\pi$
$8$	0	0
$9$	−2.98752	−0.995840
$10$	0	0
$11$	1.22301	0.368753	0.184376	−	0.982856i	$-0.440973\pi$
$11$	1.22301	0.368753	0.184376	+	0.982856i	$0.440973\pi$
$12$	0	0
$13$	3.40173	0.943471	0.471736	−	0.881740i	$-0.343628\pi$
$13$	3.40173	0.943471	0.471736	+	0.881740i	$0.343628\pi$
$14$	0	0
$15$	−0.111711	−0.0288437
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.42309	0.785311	0.392656	−	0.919686i	$-0.371556\pi$
$19$	3.42309	0.785311	0.392656	+	0.919686i	$0.371556\pi$
$20$	0	0
$21$	0.311790	0.0680382
$22$	0	0
$23$	−9.54005	−1.98924	−0.994619	−	0.103599i	$-0.966964\pi$
$23$	−9.54005	−1.98924	−0.994619	+	0.103599i	$0.966964\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.668874	−0.128725
$28$	0	0
$29$	10.0141	1.85958	0.929788	−	0.368095i	$-0.119990\pi$
$29$	10.0141	1.85958	0.929788	+	0.368095i	$0.119990\pi$
$30$	0	0
$31$	7.98033	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$31$	7.98033	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$32$	0	0
$33$	0.136625	0.0237833
$34$	0	0
$35$	−2.79103	−0.471771
$36$	0	0
$37$	5.71862	0.940135	0.470067	−	0.882631i	$-0.344230\pi$
$37$	5.71862	0.940135	0.470067	+	0.882631i	$0.344230\pi$
$38$	0	0
$39$	0.380012	0.0608507
$40$	0	0
$41$	−11.6497	−1.81937	−0.909685	−	0.415298i	$-0.863677\pi$
$41$	−11.6497	−1.81937	−0.909685	+	0.415298i	$0.863677\pi$
$42$	0	0
$43$	7.49527	1.14302	0.571509	−	0.820596i	$-0.306358\pi$
$43$	7.49527	1.14302	0.571509	+	0.820596i	$0.306358\pi$
$44$	0	0
$45$	2.98752	0.445353
$46$	0	0
$47$	−8.61396	−1.25648	−0.628238	−	0.778021i	$-0.716223\pi$
$47$	−8.61396	−1.25648	−0.628238	+	0.778021i	$0.716223\pi$
$48$	0	0
$49$	0.789873	0.112839
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.58108	−0.491899	−0.245949	−	0.969283i	$-0.579100\pi$
$53$	−3.58108	−0.491899	−0.245949	+	0.969283i	$0.579100\pi$
$54$	0	0
$55$	−1.22301	−0.164911
$56$	0	0
$57$	0.382398	0.0506499
$58$	0	0
$59$	−4.23376	−0.551189	−0.275594	−	0.961274i	$-0.588875\pi$
$59$	−4.23376	−0.551189	−0.275594	+	0.961274i	$0.588875\pi$
$60$	0	0
$61$	−10.8514	−1.38938	−0.694692	−	0.719307i	$-0.744459\pi$
$61$	−10.8514	−1.38938	−0.694692	+	0.719307i	$0.744459\pi$
$62$	0	0
$63$	−8.33827	−1.05052
$64$	0	0
$65$	−3.40173	−0.421933
$66$	0	0
$67$	11.1125	1.35760	0.678802	−	0.734322i	$-0.262500\pi$
$67$	11.1125	1.35760	0.678802	+	0.734322i	$0.262500\pi$
$68$	0	0
$69$	−1.06573	−0.128299
$70$	0	0
$71$	−0.376703	−0.0447064	−0.0223532	−	0.999750i	$-0.507116\pi$
$71$	−0.376703	−0.0447064	−0.0223532	+	0.999750i	$0.507116\pi$
$72$	0	0
$73$	9.86263	1.15433	0.577167	−	0.816626i	$-0.304158\pi$
$73$	9.86263	1.15433	0.577167	+	0.816626i	$0.304158\pi$
$74$	0	0
$75$	0.111711	0.0128993
$76$	0	0
$77$	3.41348	0.389002
$78$	0	0
$79$	14.4999	1.63136	0.815681	−	0.578502i	$-0.196363\pi$
$79$	14.4999	1.63136	0.815681	+	0.578502i	$0.196363\pi$
$80$	0	0
$81$	8.88784	0.987538
$82$	0	0
$83$	1.52005	0.166848	0.0834238	−	0.996514i	$-0.473414\pi$
$83$	1.52005	0.166848	0.0834238	+	0.996514i	$0.473414\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.11869	0.119936
$88$	0	0
$89$	−4.28461	−0.454168	−0.227084	−	0.973875i	$-0.572919\pi$
$89$	−4.28461	−0.454168	−0.227084	+	0.973875i	$0.572919\pi$
$90$	0	0
$91$	9.49436	0.995279
$92$	0	0
$93$	0.891494	0.0924436
$94$	0	0
$95$	−3.42309	−0.351202
$96$	0	0
$97$	−2.55960	−0.259888	−0.129944	−	0.991521i	$-0.541480\pi$
$97$	−2.55960	−0.259888	−0.129944	+	0.991521i	$0.541480\pi$
$98$	0	0
$99$	−3.65378	−0.367219
$100$	0	0
$101$	2.83260	0.281855	0.140927	−	0.990020i	$-0.454992\pi$
$101$	2.83260	0.281855	0.140927	+	0.990020i	$0.454992\pi$
$102$	0	0
$103$	5.02114	0.494748	0.247374	−	0.968920i	$-0.420432\pi$
$103$	5.02114	0.494748	0.247374	+	0.968920i	$0.420432\pi$
$104$	0	0
$105$	−0.311790	−0.0304276
$106$	0	0
$107$	0.626391	0.0605555	0.0302777	−	0.999542i	$-0.490361\pi$
$107$	0.626391	0.0605555	0.0302777	+	0.999542i	$0.490361\pi$
$108$	0	0
$109$	14.1635	1.35662	0.678309	−	0.734776i	$-0.262713\pi$
$109$	14.1635	1.35662	0.678309	+	0.734776i	$0.262713\pi$
$110$	0	0
$111$	0.638834	0.0606355
$112$	0	0
$113$	10.0727	0.947558	0.473779	−	0.880644i	$-0.342890\pi$
$113$	10.0727	0.947558	0.473779	+	0.880644i	$0.342890\pi$
$114$	0	0
$115$	9.54005	0.889614
$116$	0	0
$117$	−10.1628	−0.939547
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.50424	−0.864021
$122$	0	0
$123$	−1.30140	−0.117343
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.7863	−1.40081	−0.700403	−	0.713748i	$-0.746996\pi$
$127$	−15.7863	−1.40081	−0.700403	+	0.713748i	$0.746996\pi$
$128$	0	0
$129$	0.837307	0.0737208
$130$	0	0
$131$	11.6233	1.01554	0.507768	−	0.861494i	$-0.330471\pi$
$131$	11.6233	1.01554	0.507768	+	0.861494i	$0.330471\pi$
$132$	0	0
$133$	9.55397	0.828434
$134$	0	0
$135$	0.668874	0.0575675
$136$	0	0
$137$	3.40173	0.290630	0.145315	−	0.989385i	$-0.453580\pi$
$137$	3.40173	0.290630	0.145315	+	0.989385i	$0.453580\pi$
$138$	0	0
$139$	7.73558	0.656124	0.328062	−	0.944656i	$-0.393605\pi$
$139$	7.73558	0.656124	0.328062	+	0.944656i	$0.393605\pi$
$140$	0	0
$141$	−0.962277	−0.0810384
$142$	0	0
$143$	4.16037	0.347908
$144$	0	0
$145$	−10.0141	−0.831628
$146$	0	0
$147$	0.0882378	0.00727773
$148$	0	0
$149$	−4.56165	−0.373705	−0.186853	−	0.982388i	$-0.559829\pi$
$149$	−4.56165	−0.373705	−0.186853	+	0.982388i	$0.559829\pi$
$150$	0	0
$151$	12.7622	1.03857	0.519285	−	0.854601i	$-0.326198\pi$
$151$	12.7622	1.03857	0.519285	+	0.854601i	$0.326198\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.98033	−0.640996
$156$	0	0
$157$	−22.4488	−1.79161	−0.895803	−	0.444452i	$-0.853399\pi$
$157$	−22.4488	−1.79161	−0.895803	+	0.444452i	$0.853399\pi$
$158$	0	0
$159$	−0.400047	−0.0317258
$160$	0	0
$161$	−26.6266	−2.09847
$162$	0	0
$163$	2.85862	0.223904	0.111952	−	0.993714i	$-0.464290\pi$
$163$	2.85862	0.223904	0.111952	+	0.993714i	$0.464290\pi$
$164$	0	0
$165$	−0.136625	−0.0106362
$166$	0	0
$167$	−0.0654903	−0.00506779	−0.00253390	−	0.999997i	$-0.500807\pi$
$167$	−0.0654903	−0.00506779	−0.00253390	+	0.999997i	$0.500807\pi$
$168$	0	0
$169$	−1.42820	−0.109862
$170$	0	0
$171$	−10.2266	−0.782045
$172$	0	0
$173$	14.5329	1.10492	0.552458	−	0.833541i	$-0.313690\pi$
$173$	14.5329	1.10492	0.552458	+	0.833541i	$0.313690\pi$
$174$	0	0
$175$	2.79103	0.210982
$176$	0	0
$177$	−0.472959	−0.0355498
$178$	0	0
$179$	21.5967	1.61421	0.807105	−	0.590408i	$-0.201033\pi$
$179$	21.5967	1.61421	0.807105	+	0.590408i	$0.201033\pi$
$180$	0	0
$181$	3.65552	0.271713	0.135856	−	0.990729i	$-0.456621\pi$
$181$	3.65552	0.271713	0.135856	+	0.990729i	$0.456621\pi$
$182$	0	0
$183$	−1.21223	−0.0896105
$184$	0	0
$185$	−5.71862	−0.420441
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.86685	−0.135793
$190$	0	0
$191$	−22.8681	−1.65468	−0.827340	−	0.561702i	$-0.810147\pi$
$191$	−22.8681	−1.65468	−0.827340	+	0.561702i	$0.810147\pi$
$192$	0	0
$193$	7.20843	0.518874	0.259437	−	0.965760i	$-0.416463\pi$
$193$	7.20843	0.518874	0.259437	+	0.965760i	$0.416463\pi$
$194$	0	0
$195$	−0.380012	−0.0272132
$196$	0	0
$197$	16.0793	1.14560	0.572802	−	0.819693i	$-0.305856\pi$
$197$	16.0793	1.14560	0.572802	+	0.819693i	$0.305856\pi$
$198$	0	0
$199$	−10.5825	−0.750171	−0.375086	−	0.926990i	$-0.622387\pi$
$199$	−10.5825	−0.750171	−0.375086	+	0.926990i	$0.622387\pi$
$200$	0	0
$201$	1.24139	0.0875608
$202$	0	0
$203$	27.9498	1.96169
$204$	0	0
$205$	11.6497	0.813647
$206$	0	0
$207$	28.5011	1.98096
$208$	0	0
$209$	4.18649	0.289586
$210$	0	0
$211$	−4.88039	−0.335980	−0.167990	−	0.985789i	$-0.553728\pi$
$211$	−4.88039	−0.335980	−0.167990	+	0.985789i	$0.553728\pi$
$212$	0	0
$213$	−0.0420820	−0.00288341
$214$	0	0
$215$	−7.49527	−0.511173
$216$	0	0
$217$	22.2734	1.51202
$218$	0	0
$219$	1.10177	0.0744505
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.91984	0.597317	0.298658	−	0.954360i	$-0.403461\pi$
$223$	8.91984	0.597317	0.298658	+	0.954360i	$0.403461\pi$
$224$	0	0
$225$	−2.98752	−0.199168
$226$	0	0
$227$	12.9332	0.858405	0.429202	−	0.903208i	$-0.358795\pi$
$227$	12.9332	0.858405	0.429202	+	0.903208i	$0.358795\pi$
$228$	0	0
$229$	−13.9331	−0.920726	−0.460363	−	0.887731i	$-0.652281\pi$
$229$	−13.9331	−0.920726	−0.460363	+	0.887731i	$0.652281\pi$
$230$	0	0
$231$	0.381324	0.0250893
$232$	0	0
$233$	15.6010	1.02206	0.511028	−	0.859564i	$-0.329265\pi$
$233$	15.6010	1.02206	0.511028	+	0.859564i	$0.329265\pi$
$234$	0	0
$235$	8.61396	0.561913
$236$	0	0
$237$	1.61980	0.105217
$238$	0	0
$239$	14.8296	0.959246	0.479623	−	0.877475i	$-0.340773\pi$
$239$	14.8296	0.959246	0.479623	+	0.877475i	$0.340773\pi$
$240$	0	0
$241$	−0.317246	−0.0204356	−0.0102178	−	0.999948i	$-0.503252\pi$
$241$	−0.317246	−0.0204356	−0.0102178	+	0.999948i	$0.503252\pi$
$242$	0	0
$243$	2.99949	0.192418
$244$	0	0
$245$	−0.789873	−0.0504632
$246$	0	0
$247$	11.6445	0.740919
$248$	0	0
$249$	0.169807	0.0107611
$250$	0	0
$251$	6.74154	0.425522	0.212761	−	0.977104i	$-0.431754\pi$
$251$	6.74154	0.425522	0.212761	+	0.977104i	$0.431754\pi$
$252$	0	0
$253$	−11.6676	−0.733537
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.80913	−0.299985	−0.149993	−	0.988687i	$-0.547925\pi$
$257$	−4.80913	−0.299985	−0.149993	+	0.988687i	$0.547925\pi$
$258$	0	0
$259$	15.9609	0.991759
$260$	0	0
$261$	−29.9174	−1.85184
$262$	0	0
$263$	6.06324	0.373876	0.186938	−	0.982372i	$-0.440144\pi$
$263$	6.06324	0.373876	0.186938	+	0.982372i	$0.440144\pi$
$264$	0	0
$265$	3.58108	0.219984
$266$	0	0
$267$	−0.478640	−0.0292923
$268$	0	0
$269$	17.1298	1.04442	0.522210	−	0.852817i	$-0.325108\pi$
$269$	17.1298	1.04442	0.522210	+	0.852817i	$0.325108\pi$
$270$	0	0
$271$	26.0008	1.57944	0.789719	−	0.613468i	$-0.210226\pi$
$271$	26.0008	1.57944	0.789719	+	0.613468i	$0.210226\pi$
$272$	0	0
$273$	1.06063	0.0641921
$274$	0	0
$275$	1.22301	0.0737505
$276$	0	0
$277$	−6.38905	−0.383881	−0.191940	−	0.981407i	$-0.561478\pi$
$277$	−6.38905	−0.383881	−0.191940	+	0.981407i	$0.561478\pi$
$278$	0	0
$279$	−23.8414	−1.42735
$280$	0	0
$281$	1.36275	0.0812946	0.0406473	−	0.999174i	$-0.487058\pi$
$281$	1.36275	0.0812946	0.0406473	+	0.999174i	$0.487058\pi$
$282$	0	0
$283$	24.8930	1.47974	0.739869	−	0.672751i	$-0.234888\pi$
$283$	24.8930	1.47974	0.739869	+	0.672751i	$0.234888\pi$
$284$	0	0
$285$	−0.382398	−0.0226513
$286$	0	0
$287$	−32.5146	−1.91928
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−0.285936	−0.0167619
$292$	0	0
$293$	13.2673	0.775081	0.387541	−	0.921853i	$-0.373325\pi$
$293$	13.2673	0.775081	0.387541	+	0.921853i	$0.373325\pi$
$294$	0	0
$295$	4.23376	0.246499
$296$	0	0
$297$	−0.818043	−0.0474676
$298$	0	0
$299$	−32.4527	−1.87679
$300$	0	0
$301$	20.9196	1.20578
$302$	0	0
$303$	0.316434	0.0181787
$304$	0	0
$305$	10.8514	0.621351
$306$	0	0
$307$	−20.9807	−1.19743	−0.598717	−	0.800960i	$-0.704323\pi$
$307$	−20.9807	−1.19743	−0.598717	+	0.800960i	$0.704323\pi$
$308$	0	0
$309$	0.560919	0.0319096
$310$	0	0
$311$	−0.717381	−0.0406790	−0.0203395	−	0.999793i	$-0.506475\pi$
$311$	−0.717381	−0.0406790	−0.0203395	+	0.999793i	$0.506475\pi$
$312$	0	0
$313$	10.6077	0.599582	0.299791	−	0.954005i	$-0.403083\pi$
$313$	10.6077	0.599582	0.299791	+	0.954005i	$0.403083\pi$
$314$	0	0
$315$	8.33827	0.469808
$316$	0	0
$317$	21.3476	1.19900	0.599502	−	0.800373i	$-0.295365\pi$
$317$	21.3476	1.19900	0.599502	+	0.800373i	$0.295365\pi$
$318$	0	0
$319$	12.2474	0.685724
$320$	0	0
$321$	0.0699749	0.00390562
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.40173	0.188694
$326$	0	0
$327$	1.58223	0.0874973
$328$	0	0
$329$	−24.0419	−1.32547
$330$	0	0
$331$	12.4420	0.683872	0.341936	−	0.939723i	$-0.388917\pi$
$331$	12.4420	0.683872	0.341936	+	0.939723i	$0.388917\pi$
$332$	0	0
$333$	−17.0845	−0.936224
$334$	0	0
$335$	−11.1125	−0.607139
$336$	0	0
$337$	−10.2204	−0.556741	−0.278371	−	0.960474i	$-0.589794\pi$
$337$	−10.2204	−0.556741	−0.278371	+	0.960474i	$0.589794\pi$
$338$	0	0
$339$	1.12523	0.0611142
$340$	0	0
$341$	9.76006	0.528537
$342$	0	0
$343$	−17.3327	−0.935877
$344$	0	0
$345$	1.06573	0.0573771
$346$	0	0
$347$	17.5246	0.940768	0.470384	−	0.882462i	$-0.344115\pi$
$347$	17.5246	0.940768	0.470384	+	0.882462i	$0.344115\pi$
$348$	0	0
$349$	20.4572	1.09505	0.547525	−	0.836789i	$-0.315570\pi$
$349$	20.4572	1.09505	0.547525	+	0.836789i	$0.315570\pi$
$350$	0	0
$351$	−2.27533	−0.121448
$352$	0	0
$353$	20.0292	1.06605	0.533023	−	0.846101i	$-0.321056\pi$
$353$	20.0292	1.06605	0.533023	+	0.846101i	$0.321056\pi$
$354$	0	0
$355$	0.376703	0.0199933
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.4670	−1.02743	−0.513715	−	0.857961i	$-0.671731\pi$
$359$	−19.4670	−1.02743	−0.513715	+	0.857961i	$0.671731\pi$
$360$	0	0
$361$	−7.28243	−0.383286
$362$	0	0
$363$	−1.06173	−0.0557264
$364$	0	0
$365$	−9.86263	−0.516234
$366$	0	0
$367$	−23.3070	−1.21662	−0.608308	−	0.793701i	$-0.708152\pi$
$367$	−23.3070	−1.21662	−0.608308	+	0.793701i	$0.708152\pi$
$368$	0	0
$369$	34.8036	1.81180
$370$	0	0
$371$	−9.99491	−0.518910
$372$	0	0
$373$	−15.4630	−0.800642	−0.400321	−	0.916375i	$-0.631101\pi$
$373$	−15.4630	−0.800642	−0.400321	+	0.916375i	$0.631101\pi$
$374$	0	0
$375$	−0.111711	−0.00576875
$376$	0	0
$377$	34.0654	1.75446
$378$	0	0
$379$	−7.55925	−0.388293	−0.194146	−	0.980973i	$-0.562194\pi$
$379$	−7.55925	−0.388293	−0.194146	+	0.980973i	$0.562194\pi$
$380$	0	0
$381$	−1.76351	−0.0903472
$382$	0	0
$383$	−10.1244	−0.517333	−0.258666	−	0.965967i	$-0.583283\pi$
$383$	−10.1244	−0.517333	−0.258666	+	0.965967i	$0.583283\pi$
$384$	0	0
$385$	−3.41348	−0.173967
$386$	0	0
$387$	−22.3923	−1.13826
$388$	0	0
$389$	12.6889	0.643355	0.321677	−	0.946849i	$-0.395753\pi$
$389$	12.6889	0.643355	0.321677	+	0.946849i	$0.395753\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.29846	0.0654986
$394$	0	0
$395$	−14.4999	−0.729567
$396$	0	0
$397$	23.3968	1.17425	0.587127	−	0.809495i	$-0.300259\pi$
$397$	23.3968	1.17425	0.587127	+	0.809495i	$0.300259\pi$
$398$	0	0
$399$	1.06729	0.0534312
$400$	0	0
$401$	11.3052	0.564554	0.282277	−	0.959333i	$-0.408910\pi$
$401$	11.3052	0.564554	0.282277	+	0.959333i	$0.408910\pi$
$402$	0	0
$403$	27.1470	1.35229
$404$	0	0
$405$	−8.88784	−0.441640
$406$	0	0
$407$	6.99395	0.346677
$408$	0	0
$409$	−21.1081	−1.04373	−0.521864	−	0.853029i	$-0.674763\pi$
$409$	−21.1081	−1.04373	−0.521864	+	0.853029i	$0.674763\pi$
$410$	0	0
$411$	0.380012	0.0187446
$412$	0	0
$413$	−11.8166	−0.581456
$414$	0	0
$415$	−1.52005	−0.0746165
$416$	0	0
$417$	0.864153	0.0423177
$418$	0	0
$419$	20.1361	0.983712	0.491856	−	0.870677i	$-0.336319\pi$
$419$	20.1361	0.983712	0.491856	+	0.870677i	$0.336319\pi$
$420$	0	0
$421$	−11.6969	−0.570072	−0.285036	−	0.958517i	$-0.592005\pi$
$421$	−11.6969	−0.570072	−0.285036	+	0.958517i	$0.592005\pi$
$422$	0	0
$423$	25.7344	1.25125
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−30.2867	−1.46568
$428$	0	0
$429$	0.464761	0.0224388
$430$	0	0
$431$	−2.89540	−0.139466	−0.0697332	−	0.997566i	$-0.522215\pi$
$431$	−2.89540	−0.139466	−0.0697332	+	0.997566i	$0.522215\pi$
$432$	0	0
$433$	−19.7898	−0.951038	−0.475519	−	0.879705i	$-0.657740\pi$
$433$	−19.7898	−0.951038	−0.475519	+	0.879705i	$0.657740\pi$
$434$	0	0
$435$	−1.11869	−0.0536372
$436$	0	0
$437$	−32.6565	−1.56217
$438$	0	0
$439$	35.4830	1.69351	0.846756	−	0.531981i	$-0.178552\pi$
$439$	35.4830	1.69351	0.846756	+	0.531981i	$0.178552\pi$
$440$	0	0
$441$	−2.35976	−0.112370
$442$	0	0
$443$	−14.7915	−0.702763	−0.351382	−	0.936232i	$-0.614288\pi$
$443$	−14.7915	−0.702763	−0.351382	+	0.936232i	$0.614288\pi$
$444$	0	0
$445$	4.28461	0.203110
$446$	0	0
$447$	−0.509588	−0.0241027
$448$	0	0
$449$	−17.2899	−0.815963	−0.407981	−	0.912990i	$-0.633767\pi$
$449$	−17.2899	−0.815963	−0.407981	+	0.912990i	$0.633767\pi$
$450$	0	0
$451$	−14.2477	−0.670898
$452$	0	0
$453$	1.42568	0.0669842
$454$	0	0
$455$	−9.49436	−0.445102
$456$	0	0
$457$	4.94745	0.231432	0.115716	−	0.993282i	$-0.463084\pi$
$457$	4.94745	0.231432	0.115716	+	0.993282i	$0.463084\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.14971	0.426145	0.213072	−	0.977036i	$-0.431653\pi$
$461$	9.14971	0.426145	0.213072	+	0.977036i	$0.431653\pi$
$462$	0	0
$463$	−9.60773	−0.446509	−0.223255	−	0.974760i	$-0.571668\pi$
$463$	−9.60773	−0.446509	−0.223255	+	0.974760i	$0.571668\pi$
$464$	0	0
$465$	−0.891494	−0.0413420
$466$	0	0
$467$	−15.2164	−0.704132	−0.352066	−	0.935975i	$-0.614521\pi$
$467$	−15.2164	−0.704132	−0.352066	+	0.935975i	$0.614521\pi$
$468$	0	0
$469$	31.0153	1.43215
$470$	0	0
$471$	−2.50778	−0.115552
$472$	0	0
$473$	9.16682	0.421491
$474$	0	0
$475$	3.42309	0.157062
$476$	0	0
$477$	10.6985	0.489853
$478$	0	0
$479$	−15.7997	−0.721907	−0.360954	−	0.932584i	$-0.617549\pi$
$479$	−15.7997	−0.721907	−0.360954	+	0.932584i	$0.617549\pi$
$480$	0	0
$481$	19.4532	0.886990
$482$	0	0
$483$	−2.97450	−0.135344
$484$	0	0
$485$	2.55960	0.116225
$486$	0	0
$487$	−11.7249	−0.531306	−0.265653	−	0.964069i	$-0.585588\pi$
$487$	−11.7249	−0.531306	−0.265653	+	0.964069i	$0.585588\pi$
$488$	0	0
$489$	0.319340	0.0144411
$490$	0	0
$491$	6.73370	0.303888	0.151944	−	0.988389i	$-0.451447\pi$
$491$	6.73370	0.303888	0.151944	+	0.988389i	$0.451447\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.65378	0.164225
$496$	0	0
$497$	−1.05139	−0.0471613
$498$	0	0
$499$	−27.7045	−1.24023	−0.620113	−	0.784513i	$-0.712913\pi$
$499$	−27.7045	−1.24023	−0.620113	+	0.784513i	$0.712913\pi$
$500$	0	0
$501$	−0.00731601	−0.000326855 0
$502$	0	0
$503$	−9.90746	−0.441752	−0.220876	−	0.975302i	$-0.570892\pi$
$503$	−9.90746	−0.441752	−0.220876	+	0.975302i	$0.570892\pi$
$504$	0	0
$505$	−2.83260	−0.126049
$506$	0	0
$507$	−0.159547	−0.00708572
$508$	0	0
$509$	10.2748	0.455423	0.227712	−	0.973729i	$-0.426876\pi$
$509$	10.2748	0.455423	0.227712	+	0.973729i	$0.426876\pi$
$510$	0	0
$511$	27.5269	1.21772
$512$	0	0
$513$	−2.28962	−0.101089
$514$	0	0
$515$	−5.02114	−0.221258
$516$	0	0
$517$	−10.5350	−0.463329
$518$	0	0
$519$	1.62349	0.0712633
$520$	0	0
$521$	9.66006	0.423215	0.211607	−	0.977355i	$-0.432130\pi$
$521$	9.66006	0.423215	0.211607	+	0.977355i	$0.432130\pi$
$522$	0	0
$523$	−8.21602	−0.359262	−0.179631	−	0.983734i	$-0.557490\pi$
$523$	−8.21602	−0.359262	−0.179631	+	0.983734i	$0.557490\pi$
$524$	0	0
$525$	0.311790	0.0136076
$526$	0	0
$527$	0	0
$528$	0	0
$529$	68.0126	2.95707
$530$	0	0
$531$	12.6484	0.548896
$532$	0	0
$533$	−39.6290	−1.71652
$534$	0	0
$535$	−0.626391	−0.0270812
$536$	0	0
$537$	2.41259	0.104111
$538$	0	0
$539$	0.966026	0.0416097
$540$	0	0
$541$	34.2065	1.47065	0.735325	−	0.677714i	$-0.237029\pi$
$541$	34.2065	1.47065	0.735325	+	0.677714i	$0.237029\pi$
$542$	0	0
$543$	0.408363	0.0175245
$544$	0	0
$545$	−14.1635	−0.606698
$546$	0	0
$547$	−22.3827	−0.957013	−0.478507	−	0.878084i	$-0.658822\pi$
$547$	−22.3827	−0.957013	−0.478507	+	0.878084i	$0.658822\pi$
$548$	0	0
$549$	32.4189	1.38360
$550$	0	0
$551$	34.2793	1.46035
$552$	0	0
$553$	40.4696	1.72094
$554$	0	0
$555$	−0.638834	−0.0271170
$556$	0	0
$557$	26.3875	1.11807	0.559036	−	0.829143i	$-0.311171\pi$
$557$	26.3875	1.11807	0.559036	+	0.829143i	$0.311171\pi$
$558$	0	0
$559$	25.4969	1.07840
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.35687	0.141475	0.0707376	−	0.997495i	$-0.477465\pi$
$563$	3.35687	0.141475	0.0707376	+	0.997495i	$0.477465\pi$
$564$	0	0
$565$	−10.0727	−0.423761
$566$	0	0
$567$	24.8063	1.04177
$568$	0	0
$569$	42.4229	1.77846	0.889231	−	0.457458i	$-0.151240\pi$
$569$	42.4229	1.77846	0.889231	+	0.457458i	$0.151240\pi$
$570$	0	0
$571$	−13.5399	−0.566627	−0.283313	−	0.959027i	$-0.591434\pi$
$571$	−13.5399	−0.566627	−0.283313	+	0.959027i	$0.591434\pi$
$572$	0	0
$573$	−2.55463	−0.106721
$574$	0	0
$575$	−9.54005	−0.397848
$576$	0	0
$577$	20.7010	0.861792	0.430896	−	0.902402i	$-0.358198\pi$
$577$	20.7010	0.861792	0.430896	+	0.902402i	$0.358198\pi$
$578$	0	0
$579$	0.805263	0.0334656
$580$	0	0
$581$	4.24252	0.176010
$582$	0	0
$583$	−4.37971	−0.181389
$584$	0	0
$585$	10.1628	0.420178
$586$	0	0
$587$	−33.1051	−1.36639	−0.683197	−	0.730235i	$-0.739411\pi$
$587$	−33.1051	−1.36639	−0.683197	+	0.730235i	$0.739411\pi$
$588$	0	0
$589$	27.3174	1.12560
$590$	0	0
$591$	1.79624	0.0738876
$592$	0	0
$593$	−11.1110	−0.456276	−0.228138	−	0.973629i	$-0.573264\pi$
$593$	−11.1110	−0.456276	−0.228138	+	0.973629i	$0.573264\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.18218	−0.0483835
$598$	0	0
$599$	38.6224	1.57807	0.789034	−	0.614349i	$-0.210581\pi$
$599$	38.6224	1.57807	0.789034	+	0.614349i	$0.210581\pi$
$600$	0	0
$601$	14.8768	0.606837	0.303419	−	0.952857i	$-0.401872\pi$
$601$	14.8768	0.606837	0.303419	+	0.952857i	$0.401872\pi$
$602$	0	0
$603$	−33.1987	−1.35196
$604$	0	0
$605$	9.50424	0.386402
$606$	0	0
$607$	−31.5336	−1.27991	−0.639954	−	0.768414i	$-0.721046\pi$
$607$	−31.5336	−1.27991	−0.639954	+	0.768414i	$0.721046\pi$
$608$	0	0
$609$	3.12231	0.126522
$610$	0	0
$611$	−29.3024	−1.18545
$612$	0	0
$613$	4.65730	0.188107	0.0940533	−	0.995567i	$-0.470018\pi$
$613$	4.65730	0.188107	0.0940533	+	0.995567i	$0.470018\pi$
$614$	0	0
$615$	1.30140	0.0524775
$616$	0	0
$617$	41.2188	1.65941	0.829703	−	0.558205i	$-0.188510\pi$
$617$	41.2188	1.65941	0.829703	+	0.558205i	$0.188510\pi$
$618$	0	0
$619$	8.58475	0.345050	0.172525	−	0.985005i	$-0.444807\pi$
$619$	8.58475	0.345050	0.172525	+	0.985005i	$0.444807\pi$
$620$	0	0
$621$	6.38109	0.256064
$622$	0	0
$623$	−11.9585	−0.479107
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.467679	0.0186773
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−17.3123	−0.689192	−0.344596	−	0.938751i	$-0.611984\pi$
$631$	−17.3123	−0.689192	−0.344596	+	0.938751i	$0.611984\pi$
$632$	0	0
$633$	−0.545195	−0.0216696
$634$	0	0
$635$	15.7863	0.626459
$636$	0	0
$637$	2.68694	0.106460
$638$	0	0
$639$	1.12541	0.0445205
$640$	0	0
$641$	−15.5753	−0.615186	−0.307593	−	0.951518i	$-0.599524\pi$
$641$	−15.5753	−0.615186	−0.307593	+	0.951518i	$0.599524\pi$
$642$	0	0
$643$	−11.1342	−0.439090	−0.219545	−	0.975602i	$-0.570457\pi$
$643$	−11.1342	−0.439090	−0.219545	+	0.975602i	$0.570457\pi$
$644$	0	0
$645$	−0.837307	−0.0329689
$646$	0	0
$647$	15.7228	0.618129	0.309064	−	0.951041i	$-0.399984\pi$
$647$	15.7228	0.618129	0.309064	+	0.951041i	$0.399984\pi$
$648$	0	0
$649$	−5.17795	−0.203252
$650$	0	0
$651$	2.48819	0.0975199
$652$	0	0
$653$	−26.4486	−1.03501	−0.517507	−	0.855679i	$-0.673140\pi$
$653$	−26.4486	−1.03501	−0.517507	+	0.855679i	$0.673140\pi$
$654$	0	0
$655$	−11.6233	−0.454161
$656$	0	0
$657$	−29.4648	−1.14953
$658$	0	0
$659$	−31.7204	−1.23565	−0.617826	−	0.786315i	$-0.711986\pi$
$659$	−31.7204	−1.23565	−0.617826	+	0.786315i	$0.711986\pi$
$660$	0	0
$661$	−41.6769	−1.62105	−0.810523	−	0.585707i	$-0.800817\pi$
$661$	−41.6769	−1.62105	−0.810523	+	0.585707i	$0.800817\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.55397	−0.370487
$666$	0	0
$667$	−95.5353	−3.69914
$668$	0	0
$669$	0.996447	0.0385249
$670$	0	0
$671$	−13.2715	−0.512339
$672$	0	0
$673$	−2.61917	−0.100962	−0.0504809	−	0.998725i	$-0.516075\pi$
$673$	−2.61917	−0.100962	−0.0504809	+	0.998725i	$0.516075\pi$
$674$	0	0
$675$	−0.668874	−0.0257450
$676$	0	0
$677$	−11.9643	−0.459827	−0.229914	−	0.973211i	$-0.573844\pi$
$677$	−11.9643	−0.459827	−0.229914	+	0.973211i	$0.573844\pi$
$678$	0	0
$679$	−7.14393	−0.274159
$680$	0	0
$681$	1.44478	0.0553642
$682$	0	0
$683$	20.0911	0.768765	0.384382	−	0.923174i	$-0.374414\pi$
$683$	20.0911	0.768765	0.384382	+	0.923174i	$0.374414\pi$
$684$	0	0
$685$	−3.40173	−0.129974
$686$	0	0
$687$	−1.55649	−0.0593837
$688$	0	0
$689$	−12.1819	−0.464092
$690$	0	0
$691$	21.1162	0.803297	0.401649	−	0.915794i	$-0.368437\pi$
$691$	21.1162	0.803297	0.401649	+	0.915794i	$0.368437\pi$
$692$	0	0
$693$	−10.1978	−0.387383
$694$	0	0
$695$	−7.73558	−0.293427
$696$	0	0
$697$	0	0
$698$	0	0
$699$	1.74281	0.0659191
$700$	0	0
$701$	35.6241	1.34551	0.672753	−	0.739868i	$-0.265112\pi$
$701$	35.6241	1.34551	0.672753	+	0.739868i	$0.265112\pi$
$702$	0	0
$703$	19.5754	0.738298
$704$	0	0
$705$	0.962277	0.0362415
$706$	0	0
$707$	7.90590	0.297332
$708$	0	0
$709$	38.8314	1.45834	0.729171	−	0.684331i	$-0.239906\pi$
$709$	38.8314	1.45834	0.729171	+	0.684331i	$0.239906\pi$
$710$	0	0
$711$	−43.3186	−1.62458
$712$	0	0
$713$	−76.1328	−2.85120
$714$	0	0
$715$	−4.16037	−0.155589
$716$	0	0
$717$	1.65663	0.0618681
$718$	0	0
$719$	−10.3227	−0.384972	−0.192486	−	0.981300i	$-0.561655\pi$
$719$	−10.3227	−0.384972	−0.192486	+	0.981300i	$0.561655\pi$
$720$	0	0
$721$	14.0142	0.521916
$722$	0	0
$723$	−0.0354399	−0.00131802
$724$	0	0
$725$	10.0141	0.371915
$726$	0	0
$727$	36.4252	1.35094	0.675468	−	0.737389i	$-0.263941\pi$
$727$	36.4252	1.35094	0.675468	+	0.737389i	$0.263941\pi$
$728$	0	0
$729$	−26.3284	−0.975128
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−33.2356	−1.22759	−0.613793	−	0.789467i	$-0.710357\pi$
$733$	−33.2356	−1.22759	−0.613793	+	0.789467i	$0.710357\pi$
$734$	0	0
$735$	−0.0882378	−0.00325470
$736$	0	0
$737$	13.5907	0.500620
$738$	0	0
$739$	11.8455	0.435745	0.217872	−	0.975977i	$-0.430088\pi$
$739$	11.8455	0.435745	0.217872	+	0.975977i	$0.430088\pi$
$740$	0	0
$741$	1.30082	0.0477867
$742$	0	0
$743$	17.5645	0.644380	0.322190	−	0.946675i	$-0.395581\pi$
$743$	17.5645	0.644380	0.322190	+	0.946675i	$0.395581\pi$
$744$	0	0
$745$	4.56165	0.167126
$746$	0	0
$747$	−4.54119	−0.166154
$748$	0	0
$749$	1.74828	0.0638807
$750$	0	0
$751$	16.7784	0.612251	0.306126	−	0.951991i	$-0.400967\pi$
$751$	16.7784	0.612251	0.306126	+	0.951991i	$0.400967\pi$
$752$	0	0
$753$	0.753107	0.0274447
$754$	0	0
$755$	−12.7622	−0.464462
$756$	0	0
$757$	−45.4038	−1.65023	−0.825114	−	0.564966i	$-0.808889\pi$
$757$	−45.4038	−1.65023	−0.825114	+	0.564966i	$0.808889\pi$
$758$	0	0
$759$	−1.30341	−0.0473106
$760$	0	0
$761$	8.72830	0.316401	0.158200	−	0.987407i	$-0.449431\pi$
$761$	8.72830	0.316401	0.158200	+	0.987407i	$0.449431\pi$
$762$	0	0
$763$	39.5309	1.43111
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.4021	−0.520031
$768$	0	0
$769$	0.280789	0.0101255	0.00506276	−	0.999987i	$-0.498388\pi$
$769$	0.280789	0.0101255	0.00506276	+	0.999987i	$0.498388\pi$
$770$	0	0
$771$	−0.537234	−0.0193480
$772$	0	0
$773$	−14.6221	−0.525922	−0.262961	−	0.964807i	$-0.584699\pi$
$773$	−14.6221	−0.525922	−0.262961	+	0.964807i	$0.584699\pi$
$774$	0	0
$775$	7.98033	0.286662
$776$	0	0
$777$	1.78301	0.0639651
$778$	0	0
$779$	−39.8779	−1.42877
$780$	0	0
$781$	−0.460713	−0.0164856
$782$	0	0
$783$	−6.69819	−0.239374
$784$	0	0
$785$	22.4488	0.801230
$786$	0	0
$787$	−42.4836	−1.51438	−0.757189	−	0.653196i	$-0.773428\pi$
$787$	−42.4836	−1.51438	−0.757189	+	0.653196i	$0.773428\pi$
$788$	0	0
$789$	0.677333	0.0241137
$790$	0	0
$791$	28.1132	0.999590
$792$	0	0
$793$	−36.9137	−1.31084
$794$	0	0
$795$	0.400047	0.0141882
$796$	0	0
$797$	−39.4840	−1.39860	−0.699298	−	0.714831i	$-0.746504\pi$
$797$	−39.4840	−1.39860	−0.699298	+	0.714831i	$0.746504\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.8004	0.452279
$802$	0	0
$803$	12.0621	0.425664
$804$	0	0
$805$	26.6266	0.938465
$806$	0	0
$807$	1.91359	0.0673616
$808$	0	0
$809$	−31.8589	−1.12010	−0.560050	−	0.828459i	$-0.689218\pi$
$809$	−31.8589	−1.12010	−0.560050	+	0.828459i	$0.689218\pi$
$810$	0	0
$811$	−43.4274	−1.52494	−0.762470	−	0.647023i	$-0.776014\pi$
$811$	−43.4274	−1.52494	−0.762470	+	0.647023i	$0.776014\pi$
$812$	0	0
$813$	2.90459	0.101868
$814$	0	0
$815$	−2.85862	−0.100133
$816$	0	0
$817$	25.6570	0.897625
$818$	0	0
$819$	−28.3646	−0.991139
$820$	0	0
$821$	−9.50872	−0.331857	−0.165928	−	0.986138i	$-0.553062\pi$
$821$	−9.50872	−0.331857	−0.165928	+	0.986138i	$0.553062\pi$
$822$	0	0
$823$	−21.9555	−0.765322	−0.382661	−	0.923889i	$-0.624992\pi$
$823$	−21.9555	−0.765322	−0.382661	+	0.923889i	$0.624992\pi$
$824$	0	0
$825$	0.136625	0.00475666
$826$	0	0
$827$	−39.3188	−1.36725	−0.683625	−	0.729834i	$-0.739598\pi$
$827$	−39.3188	−1.36725	−0.683625	+	0.729834i	$0.739598\pi$
$828$	0	0
$829$	7.75039	0.269182	0.134591	−	0.990901i	$-0.457028\pi$
$829$	7.75039	0.269182	0.134591	+	0.990901i	$0.457028\pi$
$830$	0	0
$831$	−0.713729	−0.0247590
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.0654903	0.00226639
$836$	0	0
$837$	−5.33784	−0.184503
$838$	0	0
$839$	−11.4653	−0.395827	−0.197914	−	0.980219i	$-0.563417\pi$
$839$	−11.4653	−0.395827	−0.197914	+	0.980219i	$0.563417\pi$
$840$	0	0
$841$	71.2827	2.45803
$842$	0	0
$843$	0.152234	0.00524322
$844$	0	0
$845$	1.42820	0.0491317
$846$	0	0
$847$	−26.5266	−0.911466
$848$	0	0
$849$	2.78083	0.0954380
$850$	0	0
$851$	−54.5559	−1.87015
$852$	0	0
$853$	−27.6709	−0.947432	−0.473716	−	0.880678i	$-0.657088\pi$
$853$	−27.6709	−0.947432	−0.473716	+	0.880678i	$0.657088\pi$
$854$	0	0
$855$	10.2266	0.349741
$856$	0	0
$857$	5.36413	0.183235	0.0916176	−	0.995794i	$-0.470796\pi$
$857$	5.36413	0.183235	0.0916176	+	0.995794i	$0.470796\pi$
$858$	0	0
$859$	−8.60568	−0.293622	−0.146811	−	0.989165i	$-0.546901\pi$
$859$	−8.60568	−0.293622	−0.146811	+	0.989165i	$0.546901\pi$
$860$	0	0
$861$	−3.63225	−0.123787
$862$	0	0
$863$	−54.1029	−1.84168	−0.920841	−	0.389937i	$-0.872497\pi$
$863$	−54.1029	−1.84168	−0.920841	+	0.389937i	$0.872497\pi$
$864$	0	0
$865$	−14.5329	−0.494134
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.7335	0.601569
$870$	0	0
$871$	37.8016	1.28086
$872$	0	0
$873$	7.64686	0.258807
$874$	0	0
$875$	−2.79103	−0.0943542
$876$	0	0
$877$	52.5352	1.77399	0.886994	−	0.461781i	$-0.152789\pi$
$877$	52.5352	1.77399	0.886994	+	0.461781i	$0.152789\pi$
$878$	0	0
$879$	1.48210	0.0499901
$880$	0	0
$881$	30.9645	1.04322	0.521610	−	0.853184i	$-0.325332\pi$
$881$	30.9645	1.04322	0.521610	+	0.853184i	$0.325332\pi$
$882$	0	0
$883$	38.5522	1.29739	0.648693	−	0.761051i	$-0.275316\pi$
$883$	38.5522	1.29739	0.648693	+	0.761051i	$0.275316\pi$
$884$	0	0
$885$	0.472959	0.0158983
$886$	0	0
$887$	−20.4492	−0.686618	−0.343309	−	0.939223i	$-0.611548\pi$
$887$	−20.4492	−0.686618	−0.343309	+	0.939223i	$0.611548\pi$
$888$	0	0
$889$	−44.0600	−1.47773
$890$	0	0
$891$	10.8700	0.364157
$892$	0	0
$893$	−29.4864	−0.986725
$894$	0	0
$895$	−21.5967	−0.721897
$896$	0	0
$897$	−3.62534	−0.121046
$898$	0	0
$899$	79.9161	2.66535
$900$	0	0
$901$	0	0
$902$	0	0
$903$	2.33695	0.0777689
$904$	0	0
$905$	−3.65552	−0.121514
$906$	0	0
$907$	−2.97178	−0.0986765	−0.0493382	−	0.998782i	$-0.515711\pi$
$907$	−2.97178	−0.0986765	−0.0493382	+	0.998782i	$0.515711\pi$
$908$	0	0
$909$	−8.46246	−0.280682
$910$	0	0
$911$	−25.9345	−0.859247	−0.429623	−	0.903008i	$-0.641354\pi$
$911$	−25.9345	−0.859247	−0.429623	+	0.903008i	$0.641354\pi$
$912$	0	0
$913$	1.85905	0.0615255
$914$	0	0
$915$	1.21223	0.0400750
$916$	0	0
$917$	32.4411	1.07130
$918$	0	0
$919$	−13.2875	−0.438313	−0.219157	−	0.975690i	$-0.570331\pi$
$919$	−13.2875	−0.438313	−0.219157	+	0.975690i	$0.570331\pi$
$920$	0	0
$921$	−2.34379	−0.0772304
$922$	0	0
$923$	−1.28144	−0.0421792
$924$	0	0
$925$	5.71862	0.188027
$926$	0	0
$927$	−15.0008	−0.492690
$928$	0	0
$929$	−4.75171	−0.155899	−0.0779493	−	0.996957i	$-0.524837\pi$
$929$	−4.75171	−0.155899	−0.0779493	+	0.996957i	$0.524837\pi$
$930$	0	0
$931$	2.70381	0.0886138
$932$	0	0
$933$	−0.0801396	−0.00262365
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.3082	−0.434759	−0.217380	−	0.976087i	$-0.569751\pi$
$937$	−13.3082	−0.434759	−0.217380	+	0.976087i	$0.569751\pi$
$938$	0	0
$939$	1.18500	0.0386710
$940$	0	0
$941$	4.31138	0.140547	0.0702735	−	0.997528i	$-0.477613\pi$
$941$	4.31138	0.140547	0.0702735	+	0.997528i	$0.477613\pi$
$942$	0	0
$943$	111.138	3.61916
$944$	0	0
$945$	1.86685	0.0607286
$946$	0	0
$947$	−54.4971	−1.77092	−0.885460	−	0.464716i	$-0.846156\pi$
$947$	−54.4971	−1.77092	−0.885460	+	0.464716i	$0.846156\pi$
$948$	0	0
$949$	33.5500	1.08908
$950$	0	0
$951$	2.38477	0.0773316
$952$	0	0
$953$	31.4405	1.01846	0.509229	−	0.860631i	$-0.329931\pi$
$953$	31.4405	1.01846	0.509229	+	0.860631i	$0.329931\pi$
$954$	0	0
$955$	22.8681	0.739995
$956$	0	0
$957$	1.36818	0.0442268
$958$	0	0
$959$	9.49436	0.306589
$960$	0	0
$961$	32.6857	1.05438
$962$	0	0
$963$	−1.87135	−0.0603036
$964$	0	0
$965$	−7.20843	−0.232048
$966$	0	0
$967$	−28.4642	−0.915347	−0.457674	−	0.889120i	$-0.651317\pi$
$967$	−28.4642	−0.915347	−0.457674	+	0.889120i	$0.651317\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.9224	1.44163	0.720815	−	0.693128i	$-0.243768\pi$
$971$	44.9224	1.44163	0.720815	+	0.693128i	$0.243768\pi$
$972$	0	0
$973$	21.5903	0.692153
$974$	0	0
$975$	0.380012	0.0121701
$976$	0	0
$977$	−37.8272	−1.21020	−0.605100	−	0.796150i	$-0.706867\pi$
$977$	−37.8272	−1.21020	−0.605100	+	0.796150i	$0.706867\pi$
$978$	0	0
$979$	−5.24014	−0.167476
$980$	0	0
$981$	−42.3138	−1.35098
$982$	0	0
$983$	37.8564	1.20743	0.603716	−	0.797200i	$-0.293686\pi$
$983$	37.8564	1.20743	0.603716	+	0.797200i	$0.293686\pi$
$984$	0	0
$985$	−16.0793	−0.512330
$986$	0	0
$987$	−2.68575	−0.0854883
$988$	0	0
$989$	−71.5053	−2.27374
$990$	0	0
$991$	−17.8452	−0.566871	−0.283436	−	0.958991i	$-0.591474\pi$
$991$	−17.8452	−0.566871	−0.283436	+	0.958991i	$0.591474\pi$
$992$	0	0
$993$	1.38991	0.0441074
$994$	0	0
$995$	10.5825	0.335487
$996$	0	0
$997$	−51.4425	−1.62920	−0.814601	−	0.580022i	$-0.803044\pi$
$997$	−51.4425	−1.62920	−0.814601	+	0.580022i	$0.803044\pi$
$998$	0	0
$999$	−3.82503	−0.121019

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.m.1.3		6
17.4	even	4		5780.2.c.h.5201.6		12
17.8	even	8		340.2.o.a.81.3	yes	12
17.13	even	4		5780.2.c.h.5201.7		12
17.15	even	8		340.2.o.a.21.3	✓	12
17.16	even	2		5780.2.a.n.1.4		6
51.8	odd	8		3060.2.be.b.1441.6		12
51.32	odd	8		3060.2.be.b.361.6		12
68.15	odd	8		1360.2.bt.c.1041.4		12
68.59	odd	8		1360.2.bt.c.81.4		12
85.8	odd	8		1700.2.m.c.149.4		12
85.32	odd	8		1700.2.m.c.1449.4		12
85.42	odd	8		1700.2.m.f.149.3		12
85.49	even	8		1700.2.o.d.701.4		12
85.59	even	8		1700.2.o.d.1101.4		12
85.83	odd	8		1700.2.m.f.1449.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.3	✓	12	17.15	even	8
340.2.o.a.81.3	yes	12	17.8	even	8
1360.2.bt.c.81.4		12	68.59	odd	8
1360.2.bt.c.1041.4		12	68.15	odd	8
1700.2.m.c.149.4		12	85.8	odd	8
1700.2.m.c.1449.4		12	85.32	odd	8
1700.2.m.f.149.3		12	85.42	odd	8
1700.2.m.f.1449.3		12	85.83	odd	8
1700.2.o.d.701.4		12	85.49	even	8
1700.2.o.d.1101.4		12	85.59	even	8
3060.2.be.b.361.6		12	51.32	odd	8
3060.2.be.b.1441.6		12	51.8	odd	8
5780.2.a.m.1.3		6	1.1	even	1	trivial
5780.2.a.n.1.4		6	17.16	even	2
5780.2.c.h.5201.6		12	17.4	even	4
5780.2.c.h.5201.7		12	17.13	even	4

Overview	Random
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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5780.2.a.m.1.3

Level	$5780$
Weight	$2$
Character	5780.1
Self dual	yes
Analytic conductor	$46.154$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$