LMFDB - Embedded newform 5780.2.c.h.5201.6

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			5201.6
Root			$-0.411629 + 1.88205i$ of defining polynomial
Character	$\chi$	$=$	5780.5201
Dual form			5780.2.c.h.5201.7

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.111711i q^{3} +1.00000i q^{5} +2.79103i q^{7} +2.98752 q^{9} +1.22301i q^{11} +3.40173 q^{13} +0.111711 q^{15} -3.42309 q^{19} +0.311790 q^{21} -9.54005i q^{23} -1.00000 q^{25} -0.668874i q^{27} -10.0141i q^{29} -7.98033i q^{31} +0.136625 q^{33} -2.79103 q^{35} -5.71862i q^{37} -0.380012i q^{39} -11.6497i q^{41} -7.49527 q^{43} +2.98752i q^{45} -8.61396 q^{47} -0.789873 q^{49} +3.58108 q^{53} -1.22301 q^{55} +0.382398i q^{57} +4.23376 q^{59} -10.8514i q^{61} +8.33827i q^{63} +3.40173i q^{65} +11.1125 q^{67} -1.06573 q^{69} +0.376703i q^{71} -9.86263i q^{73} +0.111711i q^{75} -3.41348 q^{77} +14.4999i q^{79} +8.88784 q^{81} -1.52005 q^{83} -1.11869 q^{87} -4.28461 q^{89} +9.49436i q^{91} -0.891494 q^{93} -3.42309i q^{95} +2.55960i q^{97} +3.65378i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 4 q^{9} - 16 q^{13} - 24 q^{19} - 16 q^{21} - 12 q^{25} - 48 q^{33} - 8 q^{35} + 8 q^{43} + 8 q^{47} - 44 q^{49} + 16 q^{53} - 16 q^{55} + 32 q^{67} - 32 q^{69} + 48 q^{77} + 4 q^{81} - 16 q^{83}+ \cdots + 24 q^{93}+O(q^{100})$ 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
$3$	− 0.111711i	− 0.0644966i	−0.999480	−	0.0322483i	$-0.989733\pi$
$3$	− 0.111711i	− 0.0644966i	0.999480	−	0.0322483i	$-0.0102667\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	2.79103i	1.05491i	0.849582	+	0.527456i	$0.176854\pi$
$7$	2.79103i	1.05491i	−0.849582	+	0.527456i	$0.823146\pi$
$8$	0	0
$9$	2.98752	0.995840
$10$	0	0
$11$	1.22301i	0.368753i	0.982856	+	0.184376i	$0.0590265\pi$
$11$	1.22301i	0.368753i	−0.982856	+	0.184376i	$0.940973\pi$
$12$	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	$-0.628444\pi$
$19$	−3.42309	−0.785311	−0.392656	+	0.919686i	$0.628444\pi$
$20$	0	0
$21$	0.311790	0.0680382
$22$	0	0
$23$	− 9.54005i	− 1.98924i	−0.103599	−	0.994619i	$-0.533036\pi$
$23$	− 9.54005i	− 1.98924i	0.103599	−	0.994619i	$-0.466964\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 0.668874i	− 0.128725i
$28$	0	0
$29$	− 10.0141i	− 1.85958i	−0.368095	−	0.929788i	$-0.619990\pi$
$29$	− 10.0141i	− 1.85958i	0.368095	−	0.929788i	$-0.380010\pi$
$30$	0	0
$31$	− 7.98033i	− 1.43331i	−0.697428	−	0.716655i	$-0.745672\pi$
$31$	− 7.98033i	− 1.43331i	0.697428	−	0.716655i	$-0.254328\pi$
$32$	0	0
$33$	0.136625	0.0237833
$34$	0	0
$35$	−2.79103	−0.471771
$36$	0	0
$37$	− 5.71862i	− 0.940135i	−0.882631	−	0.470067i	$-0.844230\pi$
$37$	− 5.71862i	− 0.940135i	0.882631	−	0.470067i	$-0.155770\pi$
$38$	0	0
$39$	− 0.380012i	− 0.0608507i
$40$	0	0
$41$	− 11.6497i	− 1.81937i	−0.415298	−	0.909685i	$-0.636323\pi$
$41$	− 11.6497i	− 1.81937i	0.415298	−	0.909685i	$-0.363677\pi$
$42$	0	0
$43$	−7.49527	−1.14302	−0.571509	−	0.820596i	$-0.693642\pi$
$43$	−7.49527	−1.14302	−0.571509	+	0.820596i	$0.693642\pi$
$44$	0	0
$45$	2.98752i	0.445353i
$46$	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	$-0.420900\pi$
$53$	3.58108	0.491899	0.245949	+	0.969283i	$0.420900\pi$
$54$	0	0
$55$	−1.22301	−0.164911
$56$	0	0
$57$	0.382398i	0.0506499i
$58$	0	0
$59$	4.23376	0.551189	0.275594	−	0.961274i	$-0.411125\pi$
$59$	4.23376	0.551189	0.275594	+	0.961274i	$0.411125\pi$
$60$	0	0
$61$	− 10.8514i	− 1.38938i	−0.719307	−	0.694692i	$-0.755541\pi$
$61$	− 10.8514i	− 1.38938i	0.719307	−	0.694692i	$-0.244459\pi$
$62$	0	0
$63$	8.33827i	1.05052i
$64$	0	0
$65$	3.40173i	0.421933i
$66$	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
$71$	0.376703i	0.0447064i	0.999750	+	0.0223532i	$0.00711584\pi$
$71$	0.376703i	0.0447064i	−0.999750	+	0.0223532i	$0.992884\pi$
$72$	0	0
$73$	− 9.86263i	− 1.15433i	−0.816626	−	0.577167i	$-0.804158\pi$
$73$	− 9.86263i	− 1.15433i	0.816626	−	0.577167i	$-0.195842\pi$
$74$	0	0
$75$	0.111711i	0.0128993i
$76$	0	0
$77$	−3.41348	−0.389002
$78$	0	0
$79$	14.4999i	1.63136i	0.578502	+	0.815681i	$0.303637\pi$
$79$	14.4999i	1.63136i	−0.578502	+	0.815681i	$0.696363\pi$
$80$	0	0
$81$	8.88784	0.987538
$82$	0	0
$83$	−1.52005	−0.166848	−0.0834238	−	0.996514i	$-0.526586\pi$
$83$	−1.52005	−0.166848	−0.0834238	+	0.996514i	$0.526586\pi$
$84$	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
$91$	9.49436i	0.995279i
$92$	0	0
$93$	−0.891494	−0.0924436
$94$	0	0
$95$	− 3.42309i	− 0.351202i
$96$	0	0
$97$	2.55960i	0.259888i	0.991521	+	0.129944i	$0.0414798\pi$
$97$	2.55960i	0.259888i	−0.991521	+	0.129944i	$0.958520\pi$
$98$	0	0
$99$	3.65378i	0.367219i
$100$	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
$105$	0.311790i	0.0304276i
$106$	0	0
$107$	− 0.626391i	− 0.0605555i	−0.999542	−	0.0302777i	$-0.990361\pi$
$107$	− 0.626391i	− 0.0605555i	0.999542	−	0.0302777i	$-0.00963917\pi$
$108$	0	0
$109$	14.1635i	1.35662i	0.734776	+	0.678309i	$0.237287\pi$
$109$	14.1635i	1.35662i	−0.734776	+	0.678309i	$0.762713\pi$
$110$	0	0
$111$	−0.638834	−0.0606355
$112$	0	0
$113$	10.0727i	0.947558i	0.880644	+	0.473779i	$0.157110\pi$
$113$	10.0727i	0.947558i	−0.880644	+	0.473779i	$0.842890\pi$
$114$	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
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	15.7863	1.40081	0.700403	−	0.713748i	$-0.253004\pi$
$127$	15.7863	1.40081	0.700403	+	0.713748i	$0.253004\pi$
$128$	0	0
$129$	0.837307i	0.0737208i
$130$	0	0
$131$	− 11.6233i	− 1.01554i	−0.861494	−	0.507768i	$-0.830471\pi$
$131$	− 11.6233i	− 1.01554i	0.861494	−	0.507768i	$-0.169529\pi$
$132$	0	0
$133$	− 9.55397i	− 0.828434i
$134$	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
$139$	− 7.73558i	− 0.656124i	−0.944656	−	0.328062i	$-0.893605\pi$
$139$	− 7.73558i	− 0.656124i	0.944656	−	0.328062i	$-0.106395\pi$
$140$	0	0
$141$	0.962277i	0.0810384i
$142$	0	0
$143$	4.16037i	0.347908i
$144$	0	0
$145$	10.0141	0.831628
$146$	0	0
$147$	0.0882378i	0.00727773i
$148$	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	$-0.673802\pi$
$151$	−12.7622	−1.03857	−0.519285	+	0.854601i	$0.673802\pi$
$152$	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
$159$	− 0.400047i	− 0.0317258i
$160$	0	0
$161$	26.6266	2.09847
$162$	0	0
$163$	2.85862i	0.223904i	0.993714	+	0.111952i	$0.0357104\pi$
$163$	2.85862i	0.223904i	−0.993714	+	0.111952i	$0.964290\pi$
$164$	0	0
$165$	0.136625i	0.0106362i
$166$	0	0
$167$	0.0654903i	0.00506779i	0.999997	+	0.00253390i	$0.000806565\pi$
$167$	0.0654903i	0.00506779i	−0.999997	+	0.00253390i	$0.999193\pi$
$168$	0	0
$169$	−1.42820	−0.109862
$170$	0	0
$171$	−10.2266	−0.782045
$172$	0	0
$173$	− 14.5329i	− 1.10492i	−0.833541	−	0.552458i	$-0.813690\pi$
$173$	− 14.5329i	− 1.10492i	0.833541	−	0.552458i	$-0.186310\pi$
$174$	0	0
$175$	− 2.79103i	− 0.210982i
$176$	0	0
$177$	− 0.472959i	− 0.0355498i
$178$	0	0
$179$	−21.5967	−1.61421	−0.807105	−	0.590408i	$-0.798967\pi$
$179$	−21.5967	−1.61421	−0.807105	+	0.590408i	$0.798967\pi$
$180$	0	0
$181$	3.65552i	0.271713i	0.990729	+	0.135856i	$0.0433786\pi$
$181$	3.65552i	0.271713i	−0.990729	+	0.135856i	$0.956621\pi$
$182$	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
$193$	7.20843i	0.518874i	0.965760	+	0.259437i	$0.0835370\pi$
$193$	7.20843i	0.518874i	−0.965760	+	0.259437i	$0.916463\pi$
$194$	0	0
$195$	0.380012	0.0272132
$196$	0	0
$197$	16.0793i	1.14560i	0.819693	+	0.572802i	$0.194144\pi$
$197$	16.0793i	1.14560i	−0.819693	+	0.572802i	$0.805856\pi$
$198$	0	0
$199$	10.5825i	0.750171i	0.926990	+	0.375086i	$0.122387\pi$
$199$	10.5825i	0.750171i	−0.926990	+	0.375086i	$0.877613\pi$
$200$	0	0
$201$	− 1.24139i	− 0.0875608i
$202$	0	0
$203$	27.9498	1.96169
$204$	0	0
$205$	11.6497	0.813647
$206$	0	0
$207$	− 28.5011i	− 1.98096i
$208$	0	0
$209$	− 4.18649i	− 0.289586i
$210$	0	0
$211$	− 4.88039i	− 0.335980i	−0.985789	−	0.167990i	$-0.946272\pi$
$211$	− 4.88039i	− 0.335980i	0.985789	−	0.167990i	$-0.0537277\pi$
$212$	0	0
$213$	0.0420820	0.00288341
$214$	0	0
$215$	− 7.49527i	− 0.511173i
$216$	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	$-0.596539\pi$
$223$	−8.91984	−0.597317	−0.298658	+	0.954360i	$0.596539\pi$
$224$	0	0
$225$	−2.98752	−0.199168
$226$	0	0
$227$	12.9332i	0.858405i	0.903208	+	0.429202i	$0.141205\pi$
$227$	12.9332i	0.858405i	−0.903208	+	0.429202i	$0.858795\pi$
$228$	0	0
$229$	13.9331	0.920726	0.460363	−	0.887731i	$-0.347719\pi$
$229$	13.9331	0.920726	0.460363	+	0.887731i	$0.347719\pi$
$230$	0	0
$231$	0.381324i	0.0250893i
$232$	0	0
$233$	− 15.6010i	− 1.02206i	−0.859564	−	0.511028i	$-0.829265\pi$
$233$	− 15.6010i	− 1.02206i	0.859564	−	0.511028i	$-0.170735\pi$
$234$	0	0
$235$	− 8.61396i	− 0.561913i
$236$	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
$241$	0.317246i	0.0204356i	0.999948	+	0.0102178i	$0.00325248\pi$
$241$	0.317246i	0.0204356i	−0.999948	+	0.0102178i	$0.996748\pi$
$242$	0	0
$243$	− 2.99949i	− 0.192418i
$244$	0	0
$245$	− 0.789873i	− 0.0504632i
$246$	0	0
$247$	−11.6445	−0.740919
$248$	0	0
$249$	0.169807i	0.0107611i
$250$	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	$-0.452075\pi$
$257$	4.80913	0.299985	0.149993	+	0.988687i	$0.452075\pi$
$258$	0	0
$259$	15.9609	0.991759
$260$	0	0
$261$	− 29.9174i	− 1.85184i
$262$	0	0
$263$	−6.06324	−0.373876	−0.186938	−	0.982372i	$-0.559856\pi$
$263$	−6.06324	−0.373876	−0.186938	+	0.982372i	$0.559856\pi$
$264$	0	0
$265$	3.58108i	0.219984i
$266$	0	0
$267$	0.478640i	0.0292923i
$268$	0	0
$269$	− 17.1298i	− 1.04442i	−0.852817	−	0.522210i	$-0.825108\pi$
$269$	− 17.1298i	− 1.04442i	0.852817	−	0.522210i	$-0.174892\pi$
$270$	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
$275$	− 1.22301i	− 0.0737505i
$276$	0	0
$277$	6.38905i	0.383881i	0.981407	+	0.191940i	$0.0614780\pi$
$277$	6.38905i	0.383881i	−0.981407	+	0.191940i	$0.938522\pi$
$278$	0	0
$279$	− 23.8414i	− 1.42735i
$280$	0	0
$281$	−1.36275	−0.0812946	−0.0406473	−	0.999174i	$-0.512942\pi$
$281$	−1.36275	−0.0812946	−0.0406473	+	0.999174i	$0.512942\pi$
$282$	0	0
$283$	24.8930i	1.47974i	0.672751	+	0.739869i	$0.265112\pi$
$283$	24.8930i	1.47974i	−0.672751	+	0.739869i	$0.734888\pi$
$284$	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
$295$	4.23376i	0.246499i
$296$	0	0
$297$	0.818043	0.0474676
$298$	0	0
$299$	− 32.4527i	− 1.87679i
$300$	0	0
$301$	− 20.9196i	− 1.20578i
$302$	0	0
$303$	− 0.316434i	− 0.0181787i
$304$	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
$309$	− 0.560919i	− 0.0319096i
$310$	0	0
$311$	0.717381i	0.0406790i	0.999793	+	0.0203395i	$0.00647471\pi$
$311$	0.717381i	0.0406790i	−0.999793	+	0.0203395i	$0.993525\pi$
$312$	0	0
$313$	10.6077i	0.599582i	0.954005	+	0.299791i	$0.0969169\pi$
$313$	10.6077i	0.599582i	−0.954005	+	0.299791i	$0.903083\pi$
$314$	0	0
$315$	−8.33827	−0.469808
$316$	0	0
$317$	21.3476i	1.19900i	0.800373	+	0.599502i	$0.204635\pi$
$317$	21.3476i	1.19900i	−0.800373	+	0.599502i	$0.795365\pi$
$318$	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
$329$	− 24.0419i	− 1.32547i
$330$	0	0
$331$	−12.4420	−0.683872	−0.341936	−	0.939723i	$-0.611083\pi$
$331$	−12.4420	−0.683872	−0.341936	+	0.939723i	$0.611083\pi$
$332$	0	0
$333$	− 17.0845i	− 0.936224i
$334$	0	0
$335$	11.1125i	0.607139i
$336$	0	0
$337$	10.2204i	0.556741i	0.960474	+	0.278371i	$0.0897943\pi$
$337$	10.2204i	0.556741i	−0.960474	+	0.278371i	$0.910206\pi$
$338$	0	0
$339$	1.12523	0.0611142
$340$	0	0
$341$	9.76006	0.528537
$342$	0	0
$343$	17.3327i	0.935877i
$344$	0	0
$345$	− 1.06573i	− 0.0573771i
$346$	0	0
$347$	17.5246i	0.940768i	0.882462	+	0.470384i	$0.155885\pi$
$347$	17.5246i	0.940768i	−0.882462	+	0.470384i	$0.844115\pi$
$348$	0	0
$349$	−20.4572	−1.09505	−0.547525	−	0.836789i	$-0.684430\pi$
$349$	−20.4572	−1.09505	−0.547525	+	0.836789i	$0.684430\pi$
$350$	0	0
$351$	− 2.27533i	− 0.121448i
$352$	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	$-0.328269\pi$
$359$	19.4670	1.02743	0.513715	+	0.857961i	$0.328269\pi$
$360$	0	0
$361$	−7.28243	−0.383286
$362$	0	0
$363$	− 1.06173i	− 0.0557264i
$364$	0	0
$365$	9.86263	0.516234
$366$	0	0
$367$	− 23.3070i	− 1.21662i	−0.793701	−	0.608308i	$-0.791848\pi$
$367$	− 23.3070i	− 1.21662i	0.793701	−	0.608308i	$-0.208152\pi$
$368$	0	0
$369$	− 34.8036i	− 1.81180i
$370$	0	0
$371$	9.99491i	0.518910i
$372$	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
$377$	− 34.0654i	− 1.75446i
$378$	0	0
$379$	7.55925i	0.388293i	0.980973	+	0.194146i	$0.0621937\pi$
$379$	7.55925i	0.388293i	−0.980973	+	0.194146i	$0.937806\pi$
$380$	0	0
$381$	− 1.76351i	− 0.0903472i
$382$	0	0
$383$	10.1244	0.517333	0.258666	−	0.965967i	$-0.416717\pi$
$383$	10.1244	0.517333	0.258666	+	0.965967i	$0.416717\pi$
$384$	0	0
$385$	− 3.41348i	− 0.173967i
$386$	0	0
$387$	−22.3923	−1.13826
$388$	0	0
$389$	−12.6889	−0.643355	−0.321677	−	0.946849i	$-0.604247\pi$
$389$	−12.6889	−0.643355	−0.321677	+	0.946849i	$0.604247\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.29846	−0.0654986
$394$	0	0
$395$	−14.4999	−0.729567
$396$	0	0
$397$	23.3968i	1.17425i	0.809495	+	0.587127i	$0.199741\pi$
$397$	23.3968i	1.17425i	−0.809495	+	0.587127i	$0.800259\pi$
$398$	0	0
$399$	−1.06729	−0.0534312
$400$	0	0
$401$	11.3052i	0.564554i	0.959333	+	0.282277i	$0.0910897\pi$
$401$	11.3052i	0.564554i	−0.959333	+	0.282277i	$0.908910\pi$
$402$	0	0
$403$	− 27.1470i	− 1.35229i
$404$	0	0
$405$	8.88784i	0.441640i
$406$	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
$411$	− 0.380012i	− 0.0187446i
$412$	0	0
$413$	11.8166i	0.581456i
$414$	0	0
$415$	− 1.52005i	− 0.0746165i
$416$	0	0
$417$	−0.864153	−0.0423177
$418$	0	0
$419$	20.1361i	0.983712i	0.870677	+	0.491856i	$0.163681\pi$
$419$	20.1361i	0.983712i	−0.870677	+	0.491856i	$0.836319\pi$
$420$	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
$431$	− 2.89540i	− 0.139466i	−0.997566	−	0.0697332i	$-0.977785\pi$
$431$	− 2.89540i	− 0.139466i	0.997566	−	0.0697332i	$-0.0222148\pi$
$432$	0	0
$433$	19.7898	0.951038	0.475519	−	0.879705i	$-0.342260\pi$
$433$	19.7898	0.951038	0.475519	+	0.879705i	$0.342260\pi$
$434$	0	0
$435$	− 1.11869i	− 0.0536372i
$436$	0	0
$437$	32.6565i	1.56217i
$438$	0	0
$439$	− 35.4830i	− 1.69351i	−0.531981	−	0.846756i	$-0.678552\pi$
$439$	− 35.4830i	− 1.69351i	0.531981	−	0.846756i	$-0.321448\pi$
$440$	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
$445$	− 4.28461i	− 0.203110i
$446$	0	0
$447$	0.509588i	0.0241027i
$448$	0	0
$449$	− 17.2899i	− 0.815963i	−0.912990	−	0.407981i	$-0.866233\pi$
$449$	− 17.2899i	− 0.815963i	0.912990	−	0.407981i	$-0.133767\pi$
$450$	0	0
$451$	14.2477	0.670898
$452$	0	0
$453$	1.42568i	0.0669842i
$454$	0	0
$455$	−9.49436	−0.445102
$456$	0	0
$457$	−4.94745	−0.231432	−0.115716	−	0.993282i	$-0.536916\pi$
$457$	−4.94745	−0.231432	−0.115716	+	0.993282i	$0.536916\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.14971	−0.426145	−0.213072	−	0.977036i	$-0.568347\pi$
$461$	−9.14971	−0.426145	−0.213072	+	0.977036i	$0.568347\pi$
$462$	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
$465$	− 0.891494i	− 0.0413420i
$466$	0	0
$467$	15.2164	0.704132	0.352066	−	0.935975i	$-0.385479\pi$
$467$	15.2164	0.704132	0.352066	+	0.935975i	$0.385479\pi$
$468$	0	0
$469$	31.0153i	1.43215i
$470$	0	0
$471$	2.50778i	0.115552i
$472$	0	0
$473$	− 9.16682i	− 0.421491i
$474$	0	0
$475$	3.42309	0.157062
$476$	0	0
$477$	10.6985	0.489853
$478$	0	0
$479$	15.7997i	0.721907i	0.932584	+	0.360954i	$0.117549\pi$
$479$	15.7997i	0.721907i	−0.932584	+	0.360954i	$0.882451\pi$
$480$	0	0
$481$	− 19.4532i	− 0.886990i
$482$	0	0
$483$	− 2.97450i	− 0.135344i
$484$	0	0
$485$	−2.55960	−0.116225
$486$	0	0
$487$	− 11.7249i	− 0.531306i	−0.964069	−	0.265653i	$-0.914412\pi$
$487$	− 11.7249i	− 0.531306i	0.964069	−	0.265653i	$-0.0855875\pi$
$488$	0	0
$489$	0.319340	0.0144411
$490$	0	0
$491$	−6.73370	−0.303888	−0.151944	−	0.988389i	$-0.548553\pi$
$491$	−6.73370	−0.303888	−0.151944	+	0.988389i	$0.548553\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3.65378	−0.164225
$496$	0	0
$497$	−1.05139	−0.0471613
$498$	0	0
$499$	− 27.7045i	− 1.24023i	−0.784513	−	0.620113i	$-0.787087\pi$
$499$	− 27.7045i	− 1.24023i	0.784513	−	0.620113i	$-0.212913\pi$
$500$	0	0
$501$	0.00731601	0.000326855 0
$502$	0	0
$503$	− 9.90746i	− 0.441752i	−0.975302	−	0.220876i	$-0.929108\pi$
$503$	− 9.90746i	− 0.441752i	0.975302	−	0.220876i	$-0.0708916\pi$
$504$	0	0
$505$	2.83260i	0.126049i
$506$	0	0
$507$	0.159547i	0.00708572i
$508$	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
$513$	2.28962i	0.101089i
$514$	0	0
$515$	5.02114i	0.221258i
$516$	0	0
$517$	− 10.5350i	− 0.463329i
$518$	0	0
$519$	−1.62349	−0.0712633
$520$	0	0
$521$	9.66006i	0.423215i	0.977355	+	0.211607i	$0.0678698\pi$
$521$	9.66006i	0.423215i	−0.977355	+	0.211607i	$0.932130\pi$
$522$	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
$533$	− 39.6290i	− 1.71652i
$534$	0	0
$535$	0.626391	0.0270812
$536$	0	0
$537$	2.41259i	0.104111i
$538$	0	0
$539$	− 0.966026i	− 0.0416097i
$540$	0	0
$541$	− 34.2065i	− 1.47065i	−0.677714	−	0.735325i	$-0.737029\pi$
$541$	− 34.2065i	− 1.47065i	0.677714	−	0.735325i	$-0.262971\pi$
$542$	0	0
$543$	0.408363	0.0175245
$544$	0	0
$545$	−14.1635	−0.606698
$546$	0	0
$547$	22.3827i	0.957013i	0.878084	+	0.478507i	$0.158822\pi$
$547$	22.3827i	0.957013i	−0.878084	+	0.478507i	$0.841178\pi$
$548$	0	0
$549$	− 32.4189i	− 1.38360i
$550$	0	0
$551$	34.2793i	1.46035i
$552$	0	0
$553$	−40.4696	−1.72094
$554$	0	0
$555$	− 0.638834i	− 0.0271170i
$556$	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	$-0.522535\pi$
$563$	−3.35687	−0.141475	−0.0707376	+	0.997495i	$0.522535\pi$
$564$	0	0
$565$	−10.0727	−0.423761
$566$	0	0
$567$	24.8063i	1.04177i
$568$	0	0
$569$	−42.4229	−1.77846	−0.889231	−	0.457458i	$-0.848760\pi$
$569$	−42.4229	−1.77846	−0.889231	+	0.457458i	$0.848760\pi$
$570$	0	0
$571$	− 13.5399i	− 0.566627i	−0.959027	−	0.283313i	$-0.908566\pi$
$571$	− 13.5399i	− 0.566627i	0.959027	−	0.283313i	$-0.0914337\pi$
$572$	0	0
$573$	2.55463i	0.106721i
$574$	0	0
$575$	9.54005i	0.397848i
$576$	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
$581$	− 4.24252i	− 0.176010i
$582$	0	0
$583$	4.37971i	0.181389i
$584$	0	0
$585$	10.1628i	0.420178i
$586$	0	0
$587$	33.1051	1.36639	0.683197	−	0.730235i	$-0.260589\pi$
$587$	33.1051	1.36639	0.683197	+	0.730235i	$0.260589\pi$
$588$	0	0
$589$	27.3174i	1.12560i
$590$	0	0
$591$	1.79624	0.0738876
$592$	0	0
$593$	11.1110	0.456276	0.228138	−	0.973629i	$-0.426736\pi$
$593$	11.1110	0.456276	0.228138	+	0.973629i	$0.426736\pi$
$594$	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
$601$	14.8768i	0.606837i	0.952857	+	0.303419i	$0.0981281\pi$
$601$	14.8768i	0.606837i	−0.952857	+	0.303419i	$0.901872\pi$
$602$	0	0
$603$	33.1987	1.35196
$604$	0	0
$605$	9.50424i	0.386402i
$606$	0	0
$607$	31.5336i	1.27991i	0.768414	+	0.639954i	$0.221046\pi$
$607$	31.5336i	1.27991i	−0.768414	+	0.639954i	$0.778954\pi$
$608$	0	0
$609$	− 3.12231i	− 0.126522i
$610$	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
$615$	− 1.30140i	− 0.0524775i
$616$	0	0
$617$	− 41.2188i	− 1.65941i	−0.558205	−	0.829703i	$-0.688510\pi$
$617$	− 41.2188i	− 1.65941i	0.558205	−	0.829703i	$-0.311490\pi$
$618$	0	0
$619$	8.58475i	0.345050i	0.985005	+	0.172525i	$0.0551926\pi$
$619$	8.58475i	0.345050i	−0.985005	+	0.172525i	$0.944807\pi$
$620$	0	0
$621$	−6.38109	−0.256064
$622$	0	0
$623$	− 11.9585i	− 0.479107i
$624$	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	$-0.388016\pi$
$631$	17.3123	0.689192	0.344596	+	0.938751i	$0.388016\pi$
$632$	0	0
$633$	−0.545195	−0.0216696
$634$	0	0
$635$	15.7863i	0.626459i
$636$	0	0
$637$	−2.68694	−0.106460
$638$	0	0
$639$	1.12541i	0.0445205i
$640$	0	0
$641$	15.5753i	0.615186i	0.951518	+	0.307593i	$0.0995235\pi$
$641$	15.5753i	0.615186i	−0.951518	+	0.307593i	$0.900476\pi$
$642$	0	0
$643$	11.1342i	0.439090i	0.975602	+	0.219545i	$0.0704573\pi$
$643$	11.1342i	0.439090i	−0.975602	+	0.219545i	$0.929543\pi$
$644$	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
$649$	5.17795i	0.203252i
$650$	0	0
$651$	− 2.48819i	− 0.0975199i
$652$	0	0
$653$	− 26.4486i	− 1.03501i	−0.855679	−	0.517507i	$-0.826860\pi$
$653$	− 26.4486i	− 1.03501i	0.855679	−	0.517507i	$-0.173140\pi$
$654$	0	0
$655$	11.6233	0.454161
$656$	0	0
$657$	− 29.4648i	− 1.14953i
$658$	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	$-0.199183\pi$
$661$	41.6769	1.62105	0.810523	+	0.585707i	$0.199183\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.55397	0.370487
$666$	0	0
$667$	−95.5353	−3.69914
$668$	0	0
$669$	0.996447i	0.0385249i
$670$	0	0
$671$	13.2715	0.512339
$672$	0	0
$673$	− 2.61917i	− 0.100962i	−0.998725	−	0.0504809i	$-0.983925\pi$
$673$	− 2.61917i	− 0.100962i	0.998725	−	0.0504809i	$-0.0160754\pi$
$674$	0	0
$675$	0.668874i	0.0257450i
$676$	0	0
$677$	11.9643i	0.459827i	0.973211	+	0.229914i	$0.0738443\pi$
$677$	11.9643i	0.459827i	−0.973211	+	0.229914i	$0.926156\pi$
$678$	0	0
$679$	−7.14393	−0.274159
$680$	0	0
$681$	1.44478	0.0553642
$682$	0	0
$683$	− 20.0911i	− 0.768765i	−0.923174	−	0.384382i	$-0.874414\pi$
$683$	− 20.0911i	− 0.768765i	0.923174	−	0.384382i	$-0.125586\pi$
$684$	0	0
$685$	3.40173i	0.129974i
$686$	0	0
$687$	− 1.55649i	− 0.0593837i
$688$	0	0
$689$	12.1819	0.464092
$690$	0	0
$691$	21.1162i	0.803297i	0.915794	+	0.401649i	$0.131563\pi$
$691$	21.1162i	0.803297i	−0.915794	+	0.401649i	$0.868437\pi$
$692$	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
$703$	19.5754i	0.738298i
$704$	0	0
$705$	−0.962277	−0.0362415
$706$	0	0
$707$	7.90590i	0.297332i
$708$	0	0
$709$	− 38.8314i	− 1.45834i	−0.684331	−	0.729171i	$-0.739906\pi$
$709$	− 38.8314i	− 1.45834i	0.684331	−	0.729171i	$-0.260094\pi$
$710$	0	0
$711$	43.3186i	1.62458i
$712$	0	0
$713$	−76.1328	−2.85120
$714$	0	0
$715$	−4.16037	−0.155589
$716$	0	0
$717$	− 1.65663i	− 0.0618681i
$718$	0	0
$719$	10.3227i	0.384972i	0.981300	+	0.192486i	$0.0616551\pi$
$719$	10.3227i	0.384972i	−0.981300	+	0.192486i	$0.938345\pi$
$720$	0	0
$721$	14.0142i	0.521916i
$722$	0	0
$723$	0.0354399	0.00131802
$724$	0	0
$725$	10.0141i	0.371915i
$726$	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	$-0.289643\pi$
$733$	33.2356	1.22759	0.613793	+	0.789467i	$0.289643\pi$
$734$	0	0
$735$	−0.0882378	−0.00325470
$736$	0	0
$737$	13.5907i	0.500620i
$738$	0	0
$739$	−11.8455	−0.435745	−0.217872	−	0.975977i	$-0.569912\pi$
$739$	−11.8455	−0.435745	−0.217872	+	0.975977i	$0.569912\pi$
$740$	0	0
$741$	1.30082i	0.0477867i
$742$	0	0
$743$	− 17.5645i	− 0.644380i	−0.946675	−	0.322190i	$-0.895581\pi$
$743$	− 17.5645i	− 0.644380i	0.946675	−	0.322190i	$-0.104419\pi$
$744$	0	0
$745$	− 4.56165i	− 0.167126i
$746$	0	0
$747$	−4.54119	−0.166154
$748$	0	0
$749$	1.74828	0.0638807
$750$	0	0
$751$	− 16.7784i	− 0.612251i	−0.951991	−	0.306126i	$-0.900967\pi$
$751$	− 16.7784i	− 0.612251i	0.951991	−	0.306126i	$-0.0990328\pi$
$752$	0	0
$753$	− 0.753107i	− 0.0274447i
$754$	0	0
$755$	− 12.7622i	− 0.464462i
$756$	0	0
$757$	45.4038	1.65023	0.825114	−	0.564966i	$-0.191111\pi$
$757$	45.4038	1.65023	0.825114	+	0.564966i	$0.191111\pi$
$758$	0	0
$759$	− 1.30341i	− 0.0473106i
$760$	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
$771$	− 0.537234i	− 0.0193480i
$772$	0	0
$773$	14.6221	0.525922	0.262961	−	0.964807i	$-0.415301\pi$
$773$	14.6221	0.525922	0.262961	+	0.964807i	$0.415301\pi$
$774$	0	0
$775$	7.98033i	0.286662i
$776$	0	0
$777$	− 1.78301i	− 0.0639651i
$778$	0	0
$779$	39.8779i	1.42877i
$780$	0	0
$781$	−0.460713	−0.0164856
$782$	0	0
$783$	−6.69819	−0.239374
$784$	0	0
$785$	− 22.4488i	− 0.801230i
$786$	0	0
$787$	42.4836i	1.51438i	0.653196	+	0.757189i	$0.273428\pi$
$787$	42.4836i	1.51438i	−0.653196	+	0.757189i	$0.726572\pi$
$788$	0	0
$789$	0.677333i	0.0241137i
$790$	0	0
$791$	−28.1132	−0.999590
$792$	0	0
$793$	− 36.9137i	− 1.31084i
$794$	0	0
$795$	0.400047	0.0141882
$796$	0	0
$797$	39.4840	1.39860	0.699298	−	0.714831i	$-0.253496\pi$
$797$	39.4840	1.39860	0.699298	+	0.714831i	$0.253496\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.8004	−0.452279
$802$	0	0
$803$	12.0621	0.425664
$804$	0	0
$805$	26.6266i	0.938465i
$806$	0	0
$807$	−1.91359	−0.0673616
$808$	0	0
$809$	− 31.8589i	− 1.12010i	−0.828459	−	0.560050i	$-0.810782\pi$
$809$	− 31.8589i	− 1.12010i	0.828459	−	0.560050i	$-0.189218\pi$
$810$	0	0
$811$	43.4274i	1.52494i	0.647023	+	0.762470i	$0.276014\pi$
$811$	43.4274i	1.52494i	−0.647023	+	0.762470i	$0.723986\pi$
$812$	0	0
$813$	− 2.90459i	− 0.101868i
$814$	0	0
$815$	−2.85862	−0.100133
$816$	0	0
$817$	25.6570	0.897625
$818$	0	0
$819$	28.3646i	0.991139i
$820$	0	0
$821$	9.50872i	0.331857i	0.986138	+	0.165928i	$0.0530620\pi$
$821$	9.50872i	0.331857i	−0.986138	+	0.165928i	$0.946938\pi$
$822$	0	0
$823$	− 21.9555i	− 0.765322i	−0.923889	−	0.382661i	$-0.875008\pi$
$823$	− 21.9555i	− 0.765322i	0.923889	−	0.382661i	$-0.124992\pi$
$824$	0	0
$825$	−0.136625	−0.00475666
$826$	0	0
$827$	− 39.3188i	− 1.36725i	−0.729834	−	0.683625i	$-0.760402\pi$
$827$	− 39.3188i	− 1.36725i	0.729834	−	0.683625i	$-0.239598\pi$
$828$	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
$839$	− 11.4653i	− 0.395827i	−0.980219	−	0.197914i	$-0.936583\pi$
$839$	− 11.4653i	− 0.395827i	0.980219	−	0.197914i	$-0.0634166\pi$
$840$	0	0
$841$	−71.2827	−2.45803
$842$	0	0
$843$	0.152234i	0.00524322i
$844$	0	0
$845$	− 1.42820i	− 0.0491317i
$846$	0	0
$847$	26.5266i	0.911466i
$848$	0	0
$849$	2.78083	0.0954380
$850$	0	0
$851$	−54.5559	−1.87015
$852$	0	0
$853$	27.6709i	0.947432i	0.880678	+	0.473716i	$0.157088\pi$
$853$	27.6709i	0.947432i	−0.880678	+	0.473716i	$0.842912\pi$
$854$	0	0
$855$	− 10.2266i	− 0.349741i
$856$	0	0
$857$	5.36413i	0.183235i	0.995794	+	0.0916176i	$0.0292037\pi$
$857$	5.36413i	0.183235i	−0.995794	+	0.0916176i	$0.970796\pi$
$858$	0	0
$859$	8.60568	0.293622	0.146811	−	0.989165i	$-0.453099\pi$
$859$	8.60568	0.293622	0.146811	+	0.989165i	$0.453099\pi$
$860$	0	0
$861$	− 3.63225i	− 0.123787i
$862$	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
$873$	7.64686i	0.258807i
$874$	0	0
$875$	2.79103	0.0943542
$876$	0	0
$877$	52.5352i	1.77399i	0.461781	+	0.886994i	$0.347211\pi$
$877$	52.5352i	1.77399i	−0.461781	+	0.886994i	$0.652789\pi$
$878$	0	0
$879$	− 1.48210i	− 0.0499901i
$880$	0	0
$881$	− 30.9645i	− 1.04322i	−0.853184	−	0.521610i	$-0.825332\pi$
$881$	− 30.9645i	− 1.04322i	0.853184	−	0.521610i	$-0.174668\pi$
$882$	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
$887$	20.4492i	0.686618i	0.939223	+	0.343309i	$0.111548\pi$
$887$	20.4492i	0.686618i	−0.939223	+	0.343309i	$0.888452\pi$
$888$	0	0
$889$	44.0600i	1.47773i
$890$	0	0
$891$	10.8700i	0.364157i
$892$	0	0
$893$	29.4864	0.986725
$894$	0	0
$895$	− 21.5967i	− 0.721897i
$896$	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
$907$	− 2.97178i	− 0.0986765i	−0.998782	−	0.0493382i	$-0.984289\pi$
$907$	− 2.97178i	− 0.0986765i	0.998782	−	0.0493382i	$-0.0157112\pi$
$908$	0	0
$909$	8.46246	0.280682
$910$	0	0
$911$	− 25.9345i	− 0.859247i	−0.903008	−	0.429623i	$-0.858646\pi$
$911$	− 25.9345i	− 0.859247i	0.903008	−	0.429623i	$-0.141354\pi$
$912$	0	0
$913$	− 1.85905i	− 0.0615255i
$914$	0	0
$915$	− 1.21223i	− 0.0400750i
$916$	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
$921$	2.34379i	0.0772304i
$922$	0	0
$923$	1.28144i	0.0421792i
$924$	0	0
$925$	5.71862i	0.188027i
$926$	0	0
$927$	15.0008	0.492690
$928$	0	0
$929$	− 4.75171i	− 0.155899i	−0.996957	−	0.0779493i	$-0.975163\pi$
$929$	− 4.75171i	− 0.155899i	0.996957	−	0.0779493i	$-0.0248372\pi$
$930$	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	$-0.430249\pi$
$937$	13.3082	0.434759	0.217380	+	0.976087i	$0.430249\pi$
$938$	0	0
$939$	1.18500	0.0386710
$940$	0	0
$941$	4.31138i	0.140547i	0.997528	+	0.0702735i	$0.0223872\pi$
$941$	4.31138i	0.140547i	−0.997528	+	0.0702735i	$0.977613\pi$
$942$	0	0
$943$	−111.138	−3.61916
$944$	0	0
$945$	1.86685i	0.0607286i
$946$	0	0
$947$	54.4971i	1.77092i	0.464716	+	0.885460i	$0.346156\pi$
$947$	54.4971i	1.77092i	−0.464716	+	0.885460i	$0.653844\pi$
$948$	0	0
$949$	− 33.5500i	− 1.08908i
$950$	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
$955$	− 22.8681i	− 0.739995i
$956$	0	0
$957$	− 1.36818i	− 0.0442268i
$958$	0	0
$959$	9.49436i	0.306589i
$960$	0	0
$961$	−32.6857	−1.05438
$962$	0	0
$963$	− 1.87135i	− 0.0603036i
$964$	0	0
$965$	−7.20843	−0.232048
$966$	0	0
$967$	28.4642	0.915347	0.457674	−	0.889120i	$-0.348683\pi$
$967$	28.4642	0.915347	0.457674	+	0.889120i	$0.348683\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−44.9224	−1.44163	−0.720815	−	0.693128i	$-0.756232\pi$
$971$	−44.9224	−1.44163	−0.720815	+	0.693128i	$0.756232\pi$
$972$	0	0
$973$	21.5903	0.692153
$974$	0	0
$975$	0.380012i	0.0121701i
$976$	0	0
$977$	37.8272	1.21020	0.605100	−	0.796150i	$-0.293133\pi$
$977$	37.8272	1.21020	0.605100	+	0.796150i	$0.293133\pi$
$978$	0	0
$979$	− 5.24014i	− 0.167476i
$980$	0	0
$981$	42.3138i	1.35098i
$982$	0	0
$983$	− 37.8564i	− 1.20743i	−0.797200	−	0.603716i	$-0.793686\pi$
$983$	− 37.8564i	− 1.20743i	0.797200	−	0.603716i	$-0.206314\pi$
$984$	0	0
$985$	−16.0793	−0.512330
$986$	0	0
$987$	−2.68575	−0.0854883
$988$	0	0
$989$	71.5053i	2.27374i
$990$	0	0
$991$	17.8452i	0.566871i	0.958991	+	0.283436i	$0.0914742\pi$
$991$	17.8452i	0.566871i	−0.958991	+	0.283436i	$0.908526\pi$
$992$	0	0
$993$	1.38991i	0.0441074i
$994$	0	0
$995$	−10.5825	−0.335487
$996$	0	0
$997$	− 51.4425i	− 1.62920i	−0.580022	−	0.814601i	$-0.696956\pi$
$997$	− 51.4425i	− 1.62920i	0.580022	−	0.814601i	$-0.303044\pi$
$998$	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.c.h.5201.6		12
17.2	even	8		340.2.o.a.81.3	yes	12
17.4	even	4		5780.2.a.n.1.4		6
17.8	even	8		340.2.o.a.21.3	✓	12
17.13	even	4		5780.2.a.m.1.3		6
17.16	even	2	inner	5780.2.c.h.5201.7		12
51.2	odd	8		3060.2.be.b.1441.6		12
51.8	odd	8		3060.2.be.b.361.6		12
68.19	odd	8		1360.2.bt.c.81.4		12
68.59	odd	8		1360.2.bt.c.1041.4		12
85.2	odd	8		1700.2.m.f.149.3		12
85.8	odd	8		1700.2.m.f.1449.3		12
85.19	even	8		1700.2.o.d.1101.4		12
85.42	odd	8		1700.2.m.c.1449.4		12
85.53	odd	8		1700.2.m.c.149.4		12
85.59	even	8		1700.2.o.d.701.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.3	✓	12	17.8	even	8
340.2.o.a.81.3	yes	12	17.2	even	8
1360.2.bt.c.81.4		12	68.19	odd	8
1360.2.bt.c.1041.4		12	68.59	odd	8
1700.2.m.c.149.4		12	85.53	odd	8
1700.2.m.c.1449.4		12	85.42	odd	8
1700.2.m.f.149.3		12	85.2	odd	8
1700.2.m.f.1449.3		12	85.8	odd	8
1700.2.o.d.701.4		12	85.59	even	8
1700.2.o.d.1101.4		12	85.19	even	8
3060.2.be.b.361.6		12	51.8	odd	8
3060.2.be.b.1441.6		12	51.2	odd	8
5780.2.a.m.1.3		6	17.13	even	4
5780.2.a.n.1.4		6	17.4	even	4
5780.2.c.h.5201.6		12	1.1	even	1	trivial
5780.2.c.h.5201.7		12	17.16	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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.c.h.5201.6

Level	$5780$
Weight	$2$
Character	5780.5201
Analytic conductor	$46.154$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$