LMFDB - Embedded newform 1920.2.k.k.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(961,1920)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1920.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1920 = 2^{7} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1920.k (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1920.961
Dual form			1920.2.k.k.961.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} -4.82843i q^{11} -4.82843i q^{13} -1.00000 q^{15} -2.82843 q^{17} +2.00000i q^{21} -5.65685 q^{23} -1.00000 q^{25} -1.00000i q^{27} -7.65685i q^{29} -6.82843 q^{31} +4.82843 q^{33} +2.00000i q^{35} -10.4853i q^{37} +4.82843 q^{39} -7.65685 q^{41} +9.65685i q^{43} -1.00000i q^{45} -9.65685 q^{47} -3.00000 q^{49} -2.82843i q^{51} +0.343146i q^{53} +4.82843 q^{55} +0.828427i q^{59} -1.65685i q^{61} -2.00000 q^{63} +4.82843 q^{65} +1.65685i q^{67} -5.65685i q^{69} +2.34315 q^{71} +13.3137 q^{73} -1.00000i q^{75} -9.65685i q^{77} +4.48528 q^{79} +1.00000 q^{81} -2.82843i q^{85} +7.65685 q^{87} +15.6569 q^{89} -9.65685i q^{91} -6.82843i q^{93} -3.65685 q^{97} +4.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{7} - 4 q^{9} - 4 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{39} - 8 q^{41} - 16 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{63} + 8 q^{65} + 32 q^{71} + 8 q^{73} - 16 q^{79} + 4 q^{81} + 8 q^{87}+ \cdots + 8 q^{97}+O(q^{100})$ 4 * q + 8 * q^7 - 4 * q^9 - 4 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^39 - 8 * q^41 - 16 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^63 + 8 * q^65 + 32 * q^71 + 8 * q^73 - 16 * q^79 + 4 * q^81 + 8 * q^87 + 40 * q^89 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$ .

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.82843i	− 1.45583i	−0.685670	−	0.727913i	$-0.740491\pi$
$11$	− 4.82843i	− 1.45583i	0.685670	−	0.727913i	$-0.259509\pi$
$12$	0	0
$13$	− 4.82843i	− 1.33916i	−0.742738	−	0.669582i	$-0.766473\pi$
$13$	− 4.82843i	− 1.33916i	0.742738	−	0.669582i	$-0.233527\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 7.65685i	− 1.42184i	−0.703272	−	0.710921i	$-0.748278\pi$
$29$	− 7.65685i	− 1.42184i	0.703272	−	0.710921i	$-0.251722\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	4.82843	0.840521
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	− 10.4853i	− 1.72377i	−0.507104	−	0.861885i	$-0.669284\pi$
$37$	− 10.4853i	− 1.72377i	0.507104	−	0.861885i	$-0.330716\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	9.65685i	1.47266i	0.676625	+	0.736328i	$0.263442\pi$
$43$	9.65685i	1.47266i	−0.676625	+	0.736328i	$0.736558\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 2.82843i	− 0.396059i
$52$	0	0
$53$	0.343146i	0.0471347i	0.999722	+	0.0235673i	$0.00750241\pi$
$53$	0.343146i	0.0471347i	−0.999722	+	0.0235673i	$0.992498\pi$
$54$	0	0
$55$	4.82843	0.651065
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.828427i	0.107852i	0.998545	+	0.0539260i	$0.0171735\pi$
$59$	0.828427i	0.107852i	−0.998545	+	0.0539260i	$0.982826\pi$
$60$	0	0
$61$	− 1.65685i	− 0.212138i	−0.994359	−	0.106069i	$-0.966173\pi$
$61$	− 1.65685i	− 0.212138i	0.994359	−	0.106069i	$-0.0338265\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	4.82843	0.598893
$66$	0	0
$67$	1.65685i	0.202417i	0.994865	+	0.101208i	$0.0322709\pi$
$67$	1.65685i	0.202417i	−0.994865	+	0.101208i	$0.967729\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	0	0
$73$	13.3137	1.55825	0.779126	−	0.626868i	$-0.215663\pi$
$73$	13.3137	1.55825	0.779126	+	0.626868i	$0.215663\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	− 9.65685i	− 1.10050i
$78$	0	0
$79$	4.48528	0.504634	0.252317	−	0.967645i	$-0.418807\pi$
$79$	4.48528	0.504634	0.252317	+	0.967645i	$0.418807\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 2.82843i	− 0.306786i
$86$	0	0
$87$	7.65685	0.820901
$88$	0	0
$89$	15.6569	1.65962	0.829812	−	0.558044i	$-0.188448\pi$
$89$	15.6569	1.65962	0.829812	+	0.558044i	$0.188448\pi$
$90$	0	0
$91$	− 9.65685i	− 1.01231i
$92$	0	0
$93$	− 6.82843i	− 0.708075i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	4.82843i	0.485275i
$100$	0	0
$101$	− 17.3137i	− 1.72278i	−0.507946	−	0.861389i	$-0.669595\pi$
$101$	− 17.3137i	− 1.72278i	0.507946	−	0.861389i	$-0.330405\pi$
$102$	0	0
$103$	15.6569	1.54272	0.771358	−	0.636402i	$-0.219578\pi$
$103$	15.6569	1.54272	0.771358	+	0.636402i	$0.219578\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	13.6569i	1.32026i	0.751152	+	0.660129i	$0.229498\pi$
$107$	13.6569i	1.32026i	−0.751152	+	0.660129i	$0.770502\pi$
$108$	0	0
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	0	0
$111$	10.4853	0.995219
$112$	0	0
$113$	−3.51472	−0.330637	−0.165318	−	0.986240i	$-0.552865\pi$
$113$	−3.51472	−0.330637	−0.165318	+	0.986240i	$0.552865\pi$
$114$	0	0
$115$	− 5.65685i	− 0.527504i
$116$	0	0
$117$	4.82843i	0.446388i
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−12.3137	−1.11943
$122$	0	0
$123$	− 7.65685i	− 0.690395i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	0	0
$129$	−9.65685	−0.850239
$130$	0	0
$131$	− 2.48528i	− 0.217140i	−0.994089	−	0.108570i	$-0.965373\pi$
$131$	− 2.48528i	− 0.217140i	0.994089	−	0.108570i	$-0.0346272\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.82843	−0.583392	−0.291696	−	0.956511i	$-0.594220\pi$
$137$	−6.82843	−0.583392	−0.291696	+	0.956511i	$0.594220\pi$
$138$	0	0
$139$	− 1.65685i	− 0.140533i	−0.997528	−	0.0702663i	$-0.977615\pi$
$139$	− 1.65685i	− 0.140533i	0.997528	−	0.0702663i	$-0.0223849\pi$
$140$	0	0
$141$	− 9.65685i	− 0.813254i
$142$	0	0
$143$	−23.3137	−1.94959
$144$	0	0
$145$	7.65685	0.635867
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	0.343146i	0.0281116i	0.999901	+	0.0140558i	$0.00447425\pi$
$149$	0.343146i	0.0281116i	−0.999901	+	0.0140558i	$0.995526\pi$
$150$	0	0
$151$	8.48528	0.690522	0.345261	−	0.938507i	$-0.387790\pi$
$151$	8.48528	0.690522	0.345261	+	0.938507i	$0.387790\pi$
$152$	0	0
$153$	2.82843	0.228665
$154$	0	0
$155$	− 6.82843i	− 0.548472i
$156$	0	0
$157$	5.51472i	0.440122i	0.975486	+	0.220061i	$0.0706257\pi$
$157$	5.51472i	0.440122i	−0.975486	+	0.220061i	$0.929374\pi$
$158$	0	0
$159$	−0.343146	−0.0272132
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	4.82843i	0.375893i
$166$	0	0
$167$	10.3431	0.800377	0.400188	−	0.916433i	$-0.368945\pi$
$167$	10.3431	0.800377	0.400188	+	0.916433i	$0.368945\pi$
$168$	0	0
$169$	−10.3137	−0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15.6569i	− 1.19037i	−0.803589	−	0.595184i	$-0.797079\pi$
$173$	− 15.6569i	− 1.19037i	0.803589	−	0.595184i	$-0.202921\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−0.828427	−0.0622684
$178$	0	0
$179$	2.48528i	0.185759i	0.995677	+	0.0928793i	$0.0296071\pi$
$179$	2.48528i	0.185759i	−0.995677	+	0.0928793i	$0.970393\pi$
$180$	0	0
$181$	− 12.0000i	− 0.891953i	−0.895045	−	0.445976i	$-0.852856\pi$
$181$	− 12.0000i	− 0.891953i	0.895045	−	0.445976i	$-0.147144\pi$
$182$	0	0
$183$	1.65685	0.122478
$184$	0	0
$185$	10.4853	0.770893
$186$	0	0
$187$	13.6569i	0.998688i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	−27.3137	−1.97635	−0.988175	−	0.153328i	$-0.951001\pi$
$191$	−27.3137	−1.97635	−0.988175	+	0.153328i	$0.951001\pi$
$192$	0	0
$193$	22.9706	1.65346	0.826729	−	0.562601i	$-0.190199\pi$
$193$	22.9706	1.65346	0.826729	+	0.562601i	$0.190199\pi$
$194$	0	0
$195$	4.82843i	0.345771i
$196$	0	0
$197$	21.3137i	1.51854i	0.650776	+	0.759269i	$0.274444\pi$
$197$	21.3137i	1.51854i	−0.650776	+	0.759269i	$0.725556\pi$
$198$	0	0
$199$	18.8284	1.33471	0.667356	−	0.744739i	$-0.267426\pi$
$199$	18.8284	1.33471	0.667356	+	0.744739i	$0.267426\pi$
$200$	0	0
$201$	−1.65685	−0.116865
$202$	0	0
$203$	− 15.3137i	− 1.07481i
$204$	0	0
$205$	− 7.65685i	− 0.534778i
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	0	0
$210$	0	0
$211$	25.6569i	1.76629i	0.469099	+	0.883145i	$0.344579\pi$
$211$	25.6569i	1.76629i	−0.469099	+	0.883145i	$0.655421\pi$
$212$	0	0
$213$	2.34315i	0.160550i
$214$	0	0
$215$	−9.65685	−0.658592
$216$	0	0
$217$	−13.6569	−0.927088
$218$	0	0
$219$	13.3137i	0.899657i
$220$	0	0
$221$	13.6569i	0.918659i
$222$	0	0
$223$	14.9706	1.00250	0.501252	−	0.865302i	$-0.332873\pi$
$223$	14.9706	1.00250	0.501252	+	0.865302i	$0.332873\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 6.34315i	− 0.421009i	−0.977593	−	0.210505i	$-0.932489\pi$
$227$	− 6.34315i	− 0.421009i	0.977593	−	0.210505i	$-0.0675107\pi$
$228$	0	0
$229$	− 10.3431i	− 0.683494i	−0.939792	−	0.341747i	$-0.888981\pi$
$229$	− 10.3431i	− 0.683494i	0.939792	−	0.341747i	$-0.111019\pi$
$230$	0	0
$231$	9.65685	0.635374
$232$	0	0
$233$	27.7990	1.82117	0.910586	−	0.413319i	$-0.135631\pi$
$233$	27.7990	1.82117	0.910586	+	0.413319i	$0.135631\pi$
$234$	0	0
$235$	− 9.65685i	− 0.629944i
$236$	0	0
$237$	4.48528i	0.291350i
$238$	0	0
$239$	−22.6274	−1.46365	−0.731823	−	0.681495i	$-0.761330\pi$
$239$	−22.6274	−1.46365	−0.731823	+	0.681495i	$0.761330\pi$
$240$	0	0
$241$	−24.6274	−1.58639	−0.793196	−	0.608967i	$-0.791584\pi$
$241$	−24.6274	−1.58639	−0.793196	+	0.608967i	$0.791584\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.17157i	− 0.200188i	−0.994978	−	0.100094i	$-0.968086\pi$
$251$	− 3.17157i	− 0.200188i	0.994978	−	0.100094i	$-0.0319143\pi$
$252$	0	0
$253$	27.3137i	1.71720i
$254$	0	0
$255$	2.82843	0.177123
$256$	0	0
$257$	9.17157	0.572107	0.286053	−	0.958214i	$-0.407656\pi$
$257$	9.17157	0.572107	0.286053	+	0.958214i	$0.407656\pi$
$258$	0	0
$259$	− 20.9706i	− 1.30305i
$260$	0	0
$261$	7.65685i	0.473947i
$262$	0	0
$263$	18.3431	1.13109	0.565543	−	0.824719i	$-0.308666\pi$
$263$	18.3431	1.13109	0.565543	+	0.824719i	$0.308666\pi$
$264$	0	0
$265$	−0.343146	−0.0210793
$266$	0	0
$267$	15.6569i	0.958184i
$268$	0	0
$269$	3.65685i	0.222962i	0.993767	+	0.111481i	$0.0355595\pi$
$269$	3.65685i	0.222962i	−0.993767	+	0.111481i	$0.964441\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	9.65685	0.584459
$274$	0	0
$275$	4.82843i	0.291165i
$276$	0	0
$277$	− 28.8284i	− 1.73213i	−0.499929	−	0.866066i	$-0.666641\pi$
$277$	− 28.8284i	− 1.73213i	0.499929	−	0.866066i	$-0.333359\pi$
$278$	0	0
$279$	6.82843	0.408807
$280$	0	0
$281$	−22.9706	−1.37031	−0.685154	−	0.728398i	$-0.740265\pi$
$281$	−22.9706	−1.37031	−0.685154	+	0.728398i	$0.740265\pi$
$282$	0	0
$283$	20.9706i	1.24657i	0.781995	+	0.623285i	$0.214202\pi$
$283$	20.9706i	1.24657i	−0.781995	+	0.623285i	$0.785798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.3137	−0.903940
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	− 3.65685i	− 0.214369i
$292$	0	0
$293$	5.31371i	0.310430i	0.987881	+	0.155215i	$0.0496071\pi$
$293$	5.31371i	0.310430i	−0.987881	+	0.155215i	$0.950393\pi$
$294$	0	0
$295$	−0.828427	−0.0482329
$296$	0	0
$297$	−4.82843	−0.280174
$298$	0	0
$299$	27.3137i	1.57959i
$300$	0	0
$301$	19.3137i	1.11322i
$302$	0	0
$303$	17.3137	0.994647
$304$	0	0
$305$	1.65685	0.0948712
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	15.6569i	0.890687i
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	10.9706	0.620093	0.310046	−	0.950721i	$-0.399655\pi$
$313$	10.9706	0.620093	0.310046	+	0.950721i	$0.399655\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	0	0
$317$	9.31371i	0.523110i	0.965189	+	0.261555i	$0.0842353\pi$
$317$	9.31371i	0.523110i	−0.965189	+	0.261555i	$0.915765\pi$
$318$	0	0
$319$	−36.9706	−2.06995
$320$	0	0
$321$	−13.6569	−0.762251
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.82843i	0.267833i
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−19.3137	−1.06480
$330$	0	0
$331$	− 32.0000i	− 1.75888i	−0.476011	−	0.879440i	$-0.657918\pi$
$331$	− 32.0000i	− 1.75888i	0.476011	−	0.879440i	$-0.342082\pi$
$332$	0	0
$333$	10.4853i	0.574590i
$334$	0	0
$335$	−1.65685	−0.0905236
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	− 3.51472i	− 0.190893i
$340$	0	0
$341$	32.9706i	1.78546i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	5.65685	0.304555
$346$	0	0
$347$	− 1.65685i	− 0.0889446i	−0.999011	−	0.0444723i	$-0.985839\pi$
$347$	− 1.65685i	− 0.0889446i	0.999011	−	0.0444723i	$-0.0141606\pi$
$348$	0	0
$349$	36.9706i	1.97899i	0.144571	+	0.989494i	$0.453820\pi$
$349$	36.9706i	1.97899i	−0.144571	+	0.989494i	$0.546180\pi$
$350$	0	0
$351$	−4.82843	−0.257722
$352$	0	0
$353$	25.4558	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$353$	25.4558	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$354$	0	0
$355$	2.34315i	0.124361i
$356$	0	0
$357$	− 5.65685i	− 0.299392i
$358$	0	0
$359$	−16.9706	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	− 12.3137i	− 0.646302i
$364$	0	0
$365$	13.3137i	0.696871i
$366$	0	0
$367$	25.3137	1.32136	0.660682	−	0.750666i	$-0.270267\pi$
$367$	25.3137	1.32136	0.660682	+	0.750666i	$0.270267\pi$
$368$	0	0
$369$	7.65685	0.398600
$370$	0	0
$371$	0.686292i	0.0356305i
$372$	0	0
$373$	− 13.5147i	− 0.699766i	−0.936793	−	0.349883i	$-0.886221\pi$
$373$	− 13.5147i	− 0.699766i	0.936793	−	0.349883i	$-0.113779\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−36.9706	−1.90408
$378$	0	0
$379$	− 32.2843i	− 1.65833i	−0.559003	−	0.829166i	$-0.688816\pi$
$379$	− 32.2843i	− 1.65833i	0.559003	−	0.829166i	$-0.311184\pi$
$380$	0	0
$381$	− 4.34315i	− 0.222506i
$382$	0	0
$383$	−17.6569	−0.902223	−0.451112	−	0.892468i	$-0.648972\pi$
$383$	−17.6569	−0.902223	−0.451112	+	0.892468i	$0.648972\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	0	0
$387$	− 9.65685i	− 0.490885i
$388$	0	0
$389$	9.31371i	0.472224i	0.971726	+	0.236112i	$0.0758732\pi$
$389$	9.31371i	0.472224i	−0.971726	+	0.236112i	$0.924127\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	2.48528	0.125366
$394$	0	0
$395$	4.48528i	0.225679i
$396$	0	0
$397$	10.4853i	0.526241i	0.964763	+	0.263121i	$0.0847517\pi$
$397$	10.4853i	0.526241i	−0.964763	+	0.263121i	$0.915248\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.62742	−0.430833	−0.215416	−	0.976522i	$-0.569111\pi$
$401$	−8.62742	−0.430833	−0.215416	+	0.976522i	$0.569111\pi$
$402$	0	0
$403$	32.9706i	1.64238i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	−50.6274	−2.50951
$408$	0	0
$409$	25.3137	1.25168	0.625841	−	0.779951i	$-0.284756\pi$
$409$	25.3137	1.25168	0.625841	+	0.779951i	$0.284756\pi$
$410$	0	0
$411$	− 6.82843i	− 0.336821i
$412$	0	0
$413$	1.65685i	0.0815285i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.65685	0.0811365
$418$	0	0
$419$	14.4853i	0.707652i	0.935311	+	0.353826i	$0.115120\pi$
$419$	14.4853i	0.707652i	−0.935311	+	0.353826i	$0.884880\pi$
$420$	0	0
$421$	16.9706i	0.827095i	0.910483	+	0.413547i	$0.135710\pi$
$421$	16.9706i	0.827095i	−0.910483	+	0.413547i	$0.864290\pi$
$422$	0	0
$423$	9.65685	0.469532
$424$	0	0
$425$	2.82843	0.137199
$426$	0	0
$427$	− 3.31371i	− 0.160362i
$428$	0	0
$429$	− 23.3137i	− 1.12560i
$430$	0	0
$431$	10.3431	0.498212	0.249106	−	0.968476i	$-0.419863\pi$
$431$	10.3431	0.498212	0.249106	+	0.968476i	$0.419863\pi$
$432$	0	0
$433$	12.6274	0.606835	0.303417	−	0.952858i	$-0.401872\pi$
$433$	12.6274	0.606835	0.303417	+	0.952858i	$0.401872\pi$
$434$	0	0
$435$	7.65685i	0.367118i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.14214	0.293148	0.146574	−	0.989200i	$-0.453175\pi$
$439$	6.14214	0.293148	0.146574	+	0.989200i	$0.453175\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	6.34315i	0.301372i	0.988582	+	0.150686i	$0.0481482\pi$
$443$	6.34315i	0.301372i	−0.988582	+	0.150686i	$0.951852\pi$
$444$	0	0
$445$	15.6569i	0.742206i
$446$	0	0
$447$	−0.343146	−0.0162302
$448$	0	0
$449$	−21.3137	−1.00586	−0.502928	−	0.864328i	$-0.667744\pi$
$449$	−21.3137	−1.00586	−0.502928	+	0.864328i	$0.667744\pi$
$450$	0	0
$451$	36.9706i	1.74088i
$452$	0	0
$453$	8.48528i	0.398673i
$454$	0	0
$455$	9.65685	0.452720
$456$	0	0
$457$	−18.9706	−0.887405	−0.443703	−	0.896174i	$-0.646335\pi$
$457$	−18.9706	−0.887405	−0.443703	+	0.896174i	$0.646335\pi$
$458$	0	0
$459$	2.82843i	0.132020i
$460$	0	0
$461$	− 8.34315i	− 0.388579i	−0.980944	−	0.194290i	$-0.937760\pi$
$461$	− 8.34315i	− 0.388579i	0.980944	−	0.194290i	$-0.0622401\pi$
$462$	0	0
$463$	−20.6274	−0.958637	−0.479319	−	0.877641i	$-0.659116\pi$
$463$	−20.6274	−0.958637	−0.479319	+	0.877641i	$0.659116\pi$
$464$	0	0
$465$	6.82843	0.316661
$466$	0	0
$467$	− 35.3137i	− 1.63412i	−0.576550	−	0.817062i	$-0.695601\pi$
$467$	− 35.3137i	− 1.63412i	0.576550	−	0.817062i	$-0.304399\pi$
$468$	0	0
$469$	3.31371i	0.153013i
$470$	0	0
$471$	−5.51472	−0.254105
$472$	0	0
$473$	46.6274	2.14393
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.343146i	− 0.0157116i
$478$	0	0
$479$	21.6569	0.989527	0.494763	−	0.869028i	$-0.335255\pi$
$479$	21.6569	0.989527	0.494763	+	0.869028i	$0.335255\pi$
$480$	0	0
$481$	−50.6274	−2.30841
$482$	0	0
$483$	− 11.3137i	− 0.514792i
$484$	0	0
$485$	− 3.65685i	− 0.166049i
$486$	0	0
$487$	30.9706	1.40341	0.701705	−	0.712468i	$-0.252422\pi$
$487$	30.9706	1.40341	0.701705	+	0.712468i	$0.252422\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	− 32.8284i	− 1.48153i	−0.671766	−	0.740763i	$-0.734464\pi$
$491$	− 32.8284i	− 1.48153i	0.671766	−	0.740763i	$-0.265536\pi$
$492$	0	0
$493$	21.6569i	0.975376i
$494$	0	0
$495$	−4.82843	−0.217022
$496$	0	0
$497$	4.68629	0.210209
$498$	0	0
$499$	− 38.6274i	− 1.72920i	−0.502460	−	0.864600i	$-0.667572\pi$
$499$	− 38.6274i	− 1.72920i	0.502460	−	0.864600i	$-0.332428\pi$
$500$	0	0
$501$	10.3431i	0.462098i
$502$	0	0
$503$	1.65685	0.0738755	0.0369377	−	0.999318i	$-0.488240\pi$
$503$	1.65685	0.0738755	0.0369377	+	0.999318i	$0.488240\pi$
$504$	0	0
$505$	17.3137	0.770450
$506$	0	0
$507$	− 10.3137i	− 0.458048i
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	26.6274	1.17793
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.6569i	0.689923i
$516$	0	0
$517$	46.6274i	2.05067i
$518$	0	0
$519$	15.6569	0.687260
$520$	0	0
$521$	−17.3137	−0.758527	−0.379264	−	0.925289i	$-0.623823\pi$
$521$	−17.3137	−0.758527	−0.379264	+	0.925289i	$0.623823\pi$
$522$	0	0
$523$	29.9411i	1.30923i	0.755961	+	0.654617i	$0.227170\pi$
$523$	29.9411i	1.30923i	−0.755961	+	0.654617i	$0.772830\pi$
$524$	0	0
$525$	− 2.00000i	− 0.0872872i
$526$	0	0
$527$	19.3137	0.841318
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	− 0.828427i	− 0.0359507i
$532$	0	0
$533$	36.9706i	1.60137i
$534$	0	0
$535$	−13.6569	−0.590437
$536$	0	0
$537$	−2.48528	−0.107248
$538$	0	0
$539$	14.4853i	0.623925i
$540$	0	0
$541$	12.0000i	0.515920i	0.966156	+	0.257960i	$0.0830503\pi$
$541$	12.0000i	0.515920i	−0.966156	+	0.257960i	$0.916950\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	− 14.3431i	− 0.613269i	−0.951827	−	0.306634i	$-0.900797\pi$
$547$	− 14.3431i	− 0.613269i	0.951827	−	0.306634i	$-0.0992029\pi$
$548$	0	0
$549$	1.65685i	0.0707128i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.97056	0.381467
$554$	0	0
$555$	10.4853i	0.445075i
$556$	0	0
$557$	− 12.3431i	− 0.522996i	−0.965204	−	0.261498i	$-0.915784\pi$
$557$	− 12.3431i	− 0.522996i	0.965204	−	0.261498i	$-0.0842165\pi$
$558$	0	0
$559$	46.6274	1.97213
$560$	0	0
$561$	−13.6569	−0.576593
$562$	0	0
$563$	− 4.97056i	− 0.209484i	−0.994499	−	0.104742i	$-0.966598\pi$
$563$	− 4.97056i	− 0.209484i	0.994499	−	0.104742i	$-0.0334017\pi$
$564$	0	0
$565$	− 3.51472i	− 0.147865i
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−17.3137	−0.725828	−0.362914	−	0.931823i	$-0.618218\pi$
$569$	−17.3137	−0.725828	−0.362914	+	0.931823i	$0.618218\pi$
$570$	0	0
$571$	6.62742i	0.277349i	0.990338	+	0.138674i	$0.0442841\pi$
$571$	6.62742i	0.277349i	−0.990338	+	0.138674i	$0.955716\pi$
$572$	0	0
$573$	− 27.3137i	− 1.14105i
$574$	0	0
$575$	5.65685	0.235907
$576$	0	0
$577$	−38.9706	−1.62237	−0.811183	−	0.584793i	$-0.801176\pi$
$577$	−38.9706	−1.62237	−0.811183	+	0.584793i	$0.801176\pi$
$578$	0	0
$579$	22.9706i	0.954624i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.65685	0.0686199
$584$	0	0
$585$	−4.82843	−0.199631
$586$	0	0
$587$	18.3431i	0.757103i	0.925580	+	0.378551i	$0.123578\pi$
$587$	18.3431i	0.757103i	−0.925580	+	0.378551i	$0.876422\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21.3137	−0.876729
$592$	0	0
$593$	−0.201010	−0.00825450	−0.00412725	−	0.999991i	$-0.501314\pi$
$593$	−0.201010	−0.00825450	−0.00412725	+	0.999991i	$0.501314\pi$
$594$	0	0
$595$	− 5.65685i	− 0.231908i
$596$	0	0
$597$	18.8284i	0.770596i
$598$	0	0
$599$	−34.3431	−1.40322	−0.701611	−	0.712560i	$-0.747536\pi$
$599$	−34.3431	−1.40322	−0.701611	+	0.712560i	$0.747536\pi$
$600$	0	0
$601$	−39.9411	−1.62923	−0.814616	−	0.580000i	$-0.803052\pi$
$601$	−39.9411	−1.62923	−0.814616	+	0.580000i	$0.803052\pi$
$602$	0	0
$603$	− 1.65685i	− 0.0674723i
$604$	0	0
$605$	− 12.3137i	− 0.500623i
$606$	0	0
$607$	−20.6274	−0.837241	−0.418621	−	0.908161i	$-0.637486\pi$
$607$	−20.6274	−0.837241	−0.418621	+	0.908161i	$0.637486\pi$
$608$	0	0
$609$	15.3137	0.620543
$610$	0	0
$611$	46.6274i	1.88634i
$612$	0	0
$613$	− 26.4853i	− 1.06973i	−0.844937	−	0.534865i	$-0.820362\pi$
$613$	− 26.4853i	− 1.06973i	0.844937	−	0.534865i	$-0.179638\pi$
$614$	0	0
$615$	7.65685	0.308754
$616$	0	0
$617$	−2.82843	−0.113868	−0.0569341	−	0.998378i	$-0.518132\pi$
$617$	−2.82843	−0.113868	−0.0569341	+	0.998378i	$0.518132\pi$
$618$	0	0
$619$	11.0294i	0.443311i	0.975125	+	0.221655i	$0.0711460\pi$
$619$	11.0294i	0.443311i	−0.975125	+	0.221655i	$0.928854\pi$
$620$	0	0
$621$	5.65685i	0.227002i
$622$	0	0
$623$	31.3137	1.25456
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.6569i	1.18250i
$630$	0	0
$631$	−36.4853	−1.45246	−0.726228	−	0.687454i	$-0.758728\pi$
$631$	−36.4853	−1.45246	−0.726228	+	0.687454i	$0.758728\pi$
$632$	0	0
$633$	−25.6569	−1.01977
$634$	0	0
$635$	− 4.34315i	− 0.172352i
$636$	0	0
$637$	14.4853i	0.573928i
$638$	0	0
$639$	−2.34315	−0.0926934
$640$	0	0
$641$	12.3431	0.487525	0.243762	−	0.969835i	$-0.421618\pi$
$641$	12.3431	0.487525	0.243762	+	0.969835i	$0.421618\pi$
$642$	0	0
$643$	2.62742i	0.103615i	0.998657	+	0.0518076i	$0.0164983\pi$
$643$	2.62742i	0.103615i	−0.998657	+	0.0518076i	$0.983502\pi$
$644$	0	0
$645$	− 9.65685i	− 0.380238i
$646$	0	0
$647$	17.6569	0.694163	0.347081	−	0.937835i	$-0.387173\pi$
$647$	17.6569	0.694163	0.347081	+	0.937835i	$0.387173\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	− 13.6569i	− 0.535254i
$652$	0	0
$653$	29.3137i	1.14713i	0.819159	+	0.573567i	$0.194441\pi$
$653$	29.3137i	1.14713i	−0.819159	+	0.573567i	$0.805559\pi$
$654$	0	0
$655$	2.48528	0.0971080
$656$	0	0
$657$	−13.3137	−0.519417
$658$	0	0
$659$	− 39.1716i	− 1.52591i	−0.646453	−	0.762954i	$-0.723748\pi$
$659$	− 39.1716i	− 1.52591i	0.646453	−	0.762954i	$-0.276252\pi$
$660$	0	0
$661$	− 36.9706i	− 1.43799i	−0.695016	−	0.718994i	$-0.744603\pi$
$661$	− 36.9706i	− 1.43799i	0.695016	−	0.718994i	$-0.255397\pi$
$662$	0	0
$663$	−13.6569	−0.530388
$664$	0	0
$665$	0	0
$666$	0	0
$667$	43.3137i	1.67711i
$668$	0	0
$669$	14.9706i	0.578795i
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	33.3137	1.28415	0.642075	−	0.766642i	$-0.278074\pi$
$673$	33.3137	1.28415	0.642075	+	0.766642i	$0.278074\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	− 5.31371i	− 0.204222i	−0.994773	−	0.102111i	$-0.967440\pi$
$677$	− 5.31371i	− 0.204222i	0.994773	−	0.102111i	$-0.0325598\pi$
$678$	0	0
$679$	−7.31371	−0.280674
$680$	0	0
$681$	6.34315	0.243070
$682$	0	0
$683$	− 19.3137i	− 0.739019i	−0.929227	−	0.369509i	$-0.879526\pi$
$683$	− 19.3137i	− 0.739019i	0.929227	−	0.369509i	$-0.120474\pi$
$684$	0	0
$685$	− 6.82843i	− 0.260901i
$686$	0	0
$687$	10.3431	0.394616
$688$	0	0
$689$	1.65685	0.0631211
$690$	0	0
$691$	38.6274i	1.46946i	0.678362	+	0.734728i	$0.262690\pi$
$691$	38.6274i	1.46946i	−0.678362	+	0.734728i	$0.737310\pi$
$692$	0	0
$693$	9.65685i	0.366834i
$694$	0	0
$695$	1.65685	0.0628481
$696$	0	0
$697$	21.6569	0.820312
$698$	0	0
$699$	27.7990i	1.05145i
$700$	0	0
$701$	− 42.0000i	− 1.58632i	−0.609015	−	0.793159i	$-0.708435\pi$
$701$	− 42.0000i	− 1.58632i	0.609015	−	0.793159i	$-0.291565\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	9.65685	0.363698
$706$	0	0
$707$	− 34.6274i	− 1.30230i
$708$	0	0
$709$	− 20.2843i	− 0.761792i	−0.924618	−	0.380896i	$-0.875616\pi$
$709$	− 20.2843i	− 0.761792i	0.924618	−	0.380896i	$-0.124384\pi$
$710$	0	0
$711$	−4.48528	−0.168211
$712$	0	0
$713$	38.6274	1.44661
$714$	0	0
$715$	− 23.3137i	− 0.871883i
$716$	0	0
$717$	− 22.6274i	− 0.845036i
$718$	0	0
$719$	44.2843	1.65152	0.825762	−	0.564018i	$-0.190745\pi$
$719$	44.2843	1.65152	0.825762	+	0.564018i	$0.190745\pi$
$720$	0	0
$721$	31.3137	1.16618
$722$	0	0
$723$	− 24.6274i	− 0.915903i
$724$	0	0
$725$	7.65685i	0.284368i
$726$	0	0
$727$	−17.0294	−0.631587	−0.315793	−	0.948828i	$-0.602271\pi$
$727$	−17.0294	−0.631587	−0.315793	+	0.948828i	$0.602271\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 27.3137i	− 1.01023i
$732$	0	0
$733$	− 17.7990i	− 0.657421i	−0.944431	−	0.328710i	$-0.893386\pi$
$733$	− 17.7990i	− 0.657421i	0.944431	−	0.328710i	$-0.106614\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	− 25.9411i	− 0.954260i	−0.878833	−	0.477130i	$-0.841677\pi$
$739$	− 25.9411i	− 0.954260i	0.878833	−	0.477130i	$-0.158323\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.9706	−1.06283	−0.531413	−	0.847113i	$-0.678339\pi$
$743$	−28.9706	−1.06283	−0.531413	+	0.847113i	$0.678339\pi$
$744$	0	0
$745$	−0.343146	−0.0125719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27.3137i	0.998021i
$750$	0	0
$751$	−5.45584	−0.199087	−0.0995433	−	0.995033i	$-0.531738\pi$
$751$	−5.45584	−0.199087	−0.0995433	+	0.995033i	$0.531738\pi$
$752$	0	0
$753$	3.17157	0.115579
$754$	0	0
$755$	8.48528i	0.308811i
$756$	0	0
$757$	− 17.5147i	− 0.636583i	−0.947993	−	0.318292i	$-0.896891\pi$
$757$	− 17.5147i	− 0.636583i	0.947993	−	0.318292i	$-0.103109\pi$
$758$	0	0
$759$	−27.3137	−0.991425
$760$	0	0
$761$	−10.6863	−0.387378	−0.193689	−	0.981063i	$-0.562045\pi$
$761$	−10.6863	−0.387378	−0.193689	+	0.981063i	$0.562045\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	2.82843i	0.102262i
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	−9.31371	−0.335861	−0.167930	−	0.985799i	$-0.553708\pi$
$769$	−9.31371	−0.335861	−0.167930	+	0.985799i	$0.553708\pi$
$770$	0	0
$771$	9.17157i	0.330306i
$772$	0	0
$773$	− 51.2548i	− 1.84351i	−0.387775	−	0.921754i	$-0.626756\pi$
$773$	− 51.2548i	− 1.84351i	0.387775	−	0.921754i	$-0.373244\pi$
$774$	0	0
$775$	6.82843	0.245284
$776$	0	0
$777$	20.9706	0.752315
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 11.3137i	− 0.404836i
$782$	0	0
$783$	−7.65685	−0.273634
$784$	0	0
$785$	−5.51472	−0.196829
$786$	0	0
$787$	− 39.3137i	− 1.40138i	−0.713465	−	0.700691i	$-0.752875\pi$
$787$	− 39.3137i	− 1.40138i	0.713465	−	0.700691i	$-0.247125\pi$
$788$	0	0
$789$	18.3431i	0.653033i
$790$	0	0
$791$	−7.02944	−0.249938
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	− 0.343146i	− 0.0121701i
$796$	0	0
$797$	− 10.9706i	− 0.388597i	−0.980942	−	0.194299i	$-0.937757\pi$
$797$	− 10.9706i	− 0.388597i	0.980942	−	0.194299i	$-0.0622431\pi$
$798$	0	0
$799$	27.3137	0.966290
$800$	0	0
$801$	−15.6569	−0.553208
$802$	0	0
$803$	− 64.2843i	− 2.26854i
$804$	0	0
$805$	− 11.3137i	− 0.398756i
$806$	0	0
$807$	−3.65685	−0.128727
$808$	0	0
$809$	−12.3431	−0.433962	−0.216981	−	0.976176i	$-0.569621\pi$
$809$	−12.3431	−0.433962	−0.216981	+	0.976176i	$0.569621\pi$
$810$	0	0
$811$	43.5980i	1.53093i	0.643476	+	0.765466i	$0.277492\pi$
$811$	43.5980i	1.53093i	−0.643476	+	0.765466i	$0.722508\pi$
$812$	0	0
$813$	8.48528i	0.297592i
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	9.65685i	0.337438i
$820$	0	0
$821$	1.02944i	0.0359276i	0.999839	+	0.0179638i	$0.00571836\pi$
$821$	1.02944i	0.0359276i	−0.999839	+	0.0179638i	$0.994282\pi$
$822$	0	0
$823$	−38.2843	−1.33451	−0.667253	−	0.744831i	$-0.732530\pi$
$823$	−38.2843	−1.33451	−0.667253	+	0.744831i	$0.732530\pi$
$824$	0	0
$825$	−4.82843	−0.168104
$826$	0	0
$827$	16.6863i	0.580239i	0.956990	+	0.290120i	$0.0936951\pi$
$827$	16.6863i	0.580239i	−0.956990	+	0.290120i	$0.906305\pi$
$828$	0	0
$829$	− 29.9411i	− 1.03990i	−0.854197	−	0.519949i	$-0.825951\pi$
$829$	− 29.9411i	− 1.03990i	0.854197	−	0.519949i	$-0.174049\pi$
$830$	0	0
$831$	28.8284	1.00005
$832$	0	0
$833$	8.48528	0.293998
$834$	0	0
$835$	10.3431i	0.357939i
$836$	0	0
$837$	6.82843i	0.236025i
$838$	0	0
$839$	44.2843	1.52886	0.764431	−	0.644705i	$-0.223020\pi$
$839$	44.2843	1.52886	0.764431	+	0.644705i	$0.223020\pi$
$840$	0	0
$841$	−29.6274	−1.02164
$842$	0	0
$843$	− 22.9706i	− 0.791148i
$844$	0	0
$845$	− 10.3137i	− 0.354802i
$846$	0	0
$847$	−24.6274	−0.846208
$848$	0	0
$849$	−20.9706	−0.719708
$850$	0	0
$851$	59.3137i	2.03325i
$852$	0	0
$853$	− 18.4853i	− 0.632924i	−0.948605	−	0.316462i	$-0.897505\pi$
$853$	− 18.4853i	− 0.632924i	0.948605	−	0.316462i	$-0.102495\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.14214	0.209811	0.104906	−	0.994482i	$-0.466546\pi$
$857$	6.14214	0.209811	0.104906	+	0.994482i	$0.466546\pi$
$858$	0	0
$859$	− 54.9117i	− 1.87356i	−0.349915	−	0.936781i	$-0.613790\pi$
$859$	− 54.9117i	− 1.87356i	0.349915	−	0.936781i	$-0.386210\pi$
$860$	0	0
$861$	− 15.3137i	− 0.521890i
$862$	0	0
$863$	10.3431	0.352085	0.176042	−	0.984383i	$-0.443670\pi$
$863$	10.3431	0.352085	0.176042	+	0.984383i	$0.443670\pi$
$864$	0	0
$865$	15.6569	0.532349
$866$	0	0
$867$	− 9.00000i	− 0.305656i
$868$	0	0
$869$	− 21.6569i	− 0.734658i
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	3.65685	0.123766
$874$	0	0
$875$	− 2.00000i	− 0.0676123i
$876$	0	0
$877$	− 2.48528i	− 0.0839220i	−0.999119	−	0.0419610i	$-0.986639\pi$
$877$	− 2.48528i	− 0.0839220i	0.999119	−	0.0419610i	$-0.0133605\pi$
$878$	0	0
$879$	−5.31371	−0.179227
$880$	0	0
$881$	35.6569	1.20131	0.600655	−	0.799508i	$-0.294907\pi$
$881$	35.6569	1.20131	0.600655	+	0.799508i	$0.294907\pi$
$882$	0	0
$883$	37.9411i	1.27682i	0.769696	+	0.638410i	$0.220408\pi$
$883$	37.9411i	1.27682i	−0.769696	+	0.638410i	$0.779592\pi$
$884$	0	0
$885$	− 0.828427i	− 0.0278473i
$886$	0	0
$887$	39.5980	1.32957	0.664785	−	0.747035i	$-0.268523\pi$
$887$	39.5980	1.32957	0.664785	+	0.747035i	$0.268523\pi$
$888$	0	0
$889$	−8.68629	−0.291329
$890$	0	0
$891$	− 4.82843i	− 0.161758i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.48528	−0.0830738
$896$	0	0
$897$	−27.3137	−0.911978
$898$	0	0
$899$	52.2843i	1.74378i
$900$	0	0
$901$	− 0.970563i	− 0.0323341i
$902$	0	0
$903$	−19.3137	−0.642720
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	− 12.9706i	− 0.430680i	−0.976539	−	0.215340i	$-0.930914\pi$
$907$	− 12.9706i	− 0.430680i	0.976539	−	0.215340i	$-0.0690860\pi$
$908$	0	0
$909$	17.3137i	0.574259i
$910$	0	0
$911$	35.3137	1.17000	0.584998	−	0.811035i	$-0.301095\pi$
$911$	35.3137	1.17000	0.584998	+	0.811035i	$0.301095\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.65685i	0.0547739i
$916$	0	0
$917$	− 4.97056i	− 0.164142i
$918$	0	0
$919$	−44.4853	−1.46743	−0.733717	−	0.679455i	$-0.762216\pi$
$919$	−44.4853	−1.46743	−0.733717	+	0.679455i	$0.762216\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	− 11.3137i	− 0.372395i
$924$	0	0
$925$	10.4853i	0.344754i
$926$	0	0
$927$	−15.6569	−0.514239
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 11.3137i	− 0.370394i
$934$	0	0
$935$	−13.6569	−0.446627
$936$	0	0
$937$	−13.3137	−0.434940	−0.217470	−	0.976067i	$-0.569780\pi$
$937$	−13.3137	−0.434940	−0.217470	+	0.976067i	$0.569780\pi$
$938$	0	0
$939$	10.9706i	0.358011i
$940$	0	0
$941$	− 35.2548i	− 1.14927i	−0.818408	−	0.574637i	$-0.805143\pi$
$941$	− 35.2548i	− 1.14927i	0.818408	−	0.574637i	$-0.194857\pi$
$942$	0	0
$943$	43.3137	1.41049
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	45.9411i	1.49289i	0.665449	+	0.746443i	$0.268240\pi$
$947$	45.9411i	1.49289i	−0.665449	+	0.746443i	$0.731760\pi$
$948$	0	0
$949$	− 64.2843i	− 2.08676i
$950$	0	0
$951$	−9.31371	−0.302018
$952$	0	0
$953$	33.1716	1.07453	0.537266	−	0.843413i	$-0.319457\pi$
$953$	33.1716	1.07453	0.537266	+	0.843413i	$0.319457\pi$
$954$	0	0
$955$	− 27.3137i	− 0.883851i
$956$	0	0
$957$	− 36.9706i	− 1.19509i
$958$	0	0
$959$	−13.6569	−0.441003
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	− 13.6569i	− 0.440086i
$964$	0	0
$965$	22.9706i	0.739449i
$966$	0	0
$967$	−56.9117	−1.83016	−0.915078	−	0.403276i	$-0.867871\pi$
$967$	−56.9117	−1.83016	−0.915078	+	0.403276i	$0.867871\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 15.8579i	− 0.508903i	−0.967086	−	0.254452i	$-0.918105\pi$
$971$	− 15.8579i	− 0.508903i	0.967086	−	0.254452i	$-0.0818949\pi$
$972$	0	0
$973$	− 3.31371i	− 0.106233i
$974$	0	0
$975$	−4.82843	−0.154633
$976$	0	0
$977$	−26.8284	−0.858317	−0.429159	−	0.903229i	$-0.641190\pi$
$977$	−26.8284	−0.858317	−0.429159	+	0.903229i	$0.641190\pi$
$978$	0	0
$979$	− 75.5980i	− 2.41612i
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	0	0
$983$	−19.0294	−0.606945	−0.303472	−	0.952840i	$-0.598146\pi$
$983$	−19.0294	−0.606945	−0.303472	+	0.952840i	$0.598146\pi$
$984$	0	0
$985$	−21.3137	−0.679111
$986$	0	0
$987$	− 19.3137i	− 0.614762i
$988$	0	0
$989$	− 54.6274i	− 1.73705i
$990$	0	0
$991$	15.5147	0.492841	0.246421	−	0.969163i	$-0.420746\pi$
$991$	15.5147	0.492841	0.246421	+	0.969163i	$0.420746\pi$
$992$	0	0
$993$	32.0000	1.01549
$994$	0	0
$995$	18.8284i	0.596901i
$996$	0	0
$997$	− 12.1421i	− 0.384545i	−0.981342	−	0.192273i	$-0.938414\pi$
$997$	− 12.1421i	− 0.384545i	0.981342	−	0.192273i	$-0.0615858\pi$
$998$	0	0
$999$	−10.4853	−0.331740

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	1920.2.k.k.961.3	yes	4
3.2	odd	2		5760.2.k.x.2881.2		4
4.3	odd	2		1920.2.k.j.961.2	✓	4
8.3	odd	2		1920.2.k.j.961.3	yes	4
8.5	even	2	inner	1920.2.k.k.961.2	yes	4
12.11	even	2		5760.2.k.m.2881.1		4
16.3	odd	4		3840.2.a.bm.1.2		2
16.5	even	4		3840.2.a.bj.1.2		2
16.11	odd	4		3840.2.a.bd.1.1		2
16.13	even	4		3840.2.a.bg.1.1		2
24.5	odd	2		5760.2.k.x.2881.3		4
24.11	even	2		5760.2.k.m.2881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.j.961.2	✓	4	4.3	odd	2
1920.2.k.j.961.3	yes	4	8.3	odd	2
1920.2.k.k.961.2	yes	4	8.5	even	2	inner
1920.2.k.k.961.3	yes	4	1.1	even	1	trivial
3840.2.a.bd.1.1		2	16.11	odd	4
3840.2.a.bg.1.1		2	16.13	even	4
3840.2.a.bj.1.2		2	16.5	even	4
3840.2.a.bm.1.2		2	16.3	odd	4
5760.2.k.m.2881.1		4	12.11	even	2
5760.2.k.m.2881.4		4	24.11	even	2
5760.2.k.x.2881.2		4	3.2	odd	2
5760.2.k.x.2881.3		4	24.5	odd	2

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	1920.2.k.k.961.3

Level	$1920$
Weight	$2$
Character	1920.961
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$