LMFDB - Embedded newform 1920.2.m.m.959.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(959,1920)]

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([1, 1, 1, 1]))

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

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

chi := DirichletCharacter("1920.959");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1920 = 2^{7} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1920.m (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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			959.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1920.959
Dual form			1920.2.m.m.959.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.00000i) q^{3} +(1.00000 - 2.00000i) q^{5} -2.82843 q^{7} +(1.00000 + 2.82843i) q^{9} -2.82843i q^{11} +2.82843 q^{13} +(-3.41421 + 1.82843i) q^{15} +2.82843 q^{17} +4.00000 q^{19} +(4.00000 + 2.82843i) q^{21} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +(1.41421 - 5.00000i) q^{27} +6.00000 q^{29} +2.82843i q^{31} +(-2.82843 + 4.00000i) q^{33} +(-2.82843 + 5.65685i) q^{35} -8.48528 q^{37} +(-4.00000 - 2.82843i) q^{39} -5.65685i q^{41} +2.00000i q^{43} +(6.65685 + 0.828427i) q^{45} -10.0000i q^{47} +1.00000 q^{49} +(-4.00000 - 2.82843i) q^{51} -4.00000i q^{53} +(-5.65685 - 2.82843i) q^{55} +(-5.65685 - 4.00000i) q^{57} -2.82843i q^{59} -5.65685i q^{61} +(-2.82843 - 8.00000i) q^{63} +(2.82843 - 5.65685i) q^{65} +10.0000i q^{67} +(-6.00000 + 8.48528i) q^{69} -16.0000 q^{71} +16.0000i q^{73} +(0.242641 + 8.65685i) q^{75} +8.00000i q^{77} +14.1421i q^{79} +(-7.00000 + 5.65685i) q^{81} +8.48528 q^{83} +(2.82843 - 5.65685i) q^{85} +(-8.48528 - 6.00000i) q^{87} -8.00000 q^{91} +(2.82843 - 4.00000i) q^{93} +(4.00000 - 8.00000i) q^{95} -8.00000i q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{5} + 4 q^{9} - 8 q^{15} + 16 q^{19} + 16 q^{21} - 12 q^{25} + 24 q^{29} - 16 q^{39} + 4 q^{45} + 4 q^{49} - 16 q^{51} - 24 q^{69} - 64 q^{71} - 16 q^{75} - 28 q^{81} - 32 q^{91} + 16 q^{95} + 32 q^{99}+O(q^{100})$ 4 * q + 4 * q^5 + 4 * q^9 - 8 * q^15 + 16 * q^19 + 16 * q^21 - 12 * q^25 + 24 * q^29 - 16 * q^39 + 4 * q^45 + 4 * q^49 - 16 * q^51 - 24 * q^69 - 64 * q^71 - 16 * q^75 - 28 * q^81 - 32 * q^91 + 16 * q^95 + 32 * q^99

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
$3$	−1.41421	−	1.00000i	−0.816497	−	0.577350i
$3$	−1.41421	−	1.00000i	−0.816497	−	0.577350i
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$	−2.82843			−1.06904			−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843			−1.06904			−0.534522	+	0.845154i	$0.679509\pi$
$8$		0			0
$9$	1.00000	+	2.82843i	0.333333	+	0.942809i
$10$		0			0
$11$		−	2.82843i		−	0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$		−	2.82843i		−	0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$		0			0
$13$	2.82843			0.784465			0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843			0.784465			0.392232	+	0.919866i	$0.371703\pi$
$14$		0			0
$15$	−3.41421	+	1.82843i	−0.881546	+	0.472098i
$16$		0			0
$17$	2.82843			0.685994			0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843			0.685994			0.342997	+	0.939336i	$0.388558\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	4.00000	+	2.82843i	0.872872	+	0.617213i
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$	1.41421	−	5.00000i	0.272166	−	0.962250i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$			2.82843i			0.508001i	0.967204	+	0.254000i	$0.0817464\pi$
$31$			2.82843i			0.508001i	−0.967204	+	0.254000i	$0.918254\pi$
$32$		0			0
$33$	−2.82843	+	4.00000i	−0.492366	+	0.696311i
$34$		0			0
$35$	−2.82843	+	5.65685i	−0.478091	+	0.956183i
$36$		0			0
$37$	−8.48528			−1.39497			−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528			−1.39497			−0.697486	+	0.716599i	$0.745698\pi$
$38$		0			0
$39$	−4.00000	−	2.82843i	−0.640513	−	0.452911i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	6.65685	+	0.828427i	0.992345	+	0.123495i
$46$		0			0
$47$		−	10.0000i		−	1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$		−	10.0000i		−	1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−4.00000	−	2.82843i	−0.560112	−	0.396059i
$52$		0			0
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$	−5.65685	−	2.82843i	−0.762770	−	0.381385i
$56$		0			0
$57$	−5.65685	−	4.00000i	−0.749269	−	0.529813i
$58$		0			0
$59$		−	2.82843i		−	0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$		−	2.82843i		−	0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$		0			0
$61$		−	5.65685i		−	0.724286i	−0.932123	−	0.362143i	$-0.882045\pi$
$61$		−	5.65685i		−	0.724286i	0.932123	−	0.362143i	$-0.117955\pi$
$62$		0			0
$63$	−2.82843	−	8.00000i	−0.356348	−	1.00791i
$64$		0			0
$65$	2.82843	−	5.65685i	0.350823	−	0.701646i
$66$		0			0
$67$			10.0000i			1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$			10.0000i			1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$		0			0
$69$	−6.00000	+	8.48528i	−0.722315	+	1.02151i
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$			16.0000i			1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$			16.0000i			1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	0.242641	+	8.65685i	0.0280177	+	0.999607i
$76$		0			0
$77$			8.00000i			0.911685i
$78$		0			0
$79$			14.1421i			1.59111i	0.605878	+	0.795557i	$0.292822\pi$
$79$			14.1421i			1.59111i	−0.605878	+	0.795557i	$0.707178\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	8.48528			0.931381			0.465690	−	0.884948i	$-0.345806\pi$
$83$	8.48528			0.931381			0.465690	+	0.884948i	$0.345806\pi$
$84$		0			0
$85$	2.82843	−	5.65685i	0.306786	−	0.613572i
$86$		0			0
$87$	−8.48528	−	6.00000i	−0.909718	−	0.643268i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		0			0
$93$	2.82843	−	4.00000i	0.293294	−	0.414781i
$94$		0			0
$95$	4.00000	−	8.00000i	0.410391	−	0.820783i
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$		0			0
$99$	8.00000	−	2.82843i	0.804030	−	0.284268i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	−14.1421			−1.39347			−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421			−1.39347			−0.696733	+	0.717331i	$0.745364\pi$
$104$		0			0
$105$	9.65685	−	5.17157i	0.942412	−	0.504694i
$106$		0			0
$107$	−2.82843			−0.273434			−0.136717	−	0.990610i	$-0.543655\pi$
$107$	−2.82843			−0.273434			−0.136717	+	0.990610i	$0.543655\pi$
$108$		0			0
$109$		−	16.9706i		−	1.62549i	−0.582623	−	0.812743i	$-0.697974\pi$
$109$		−	16.9706i		−	1.62549i	0.582623	−	0.812743i	$-0.302026\pi$
$110$		0			0
$111$	12.0000	+	8.48528i	1.13899	+	0.805387i
$112$		0			0
$113$	−8.48528			−0.798228			−0.399114	−	0.916901i	$-0.630682\pi$
$113$	−8.48528			−0.798228			−0.399114	+	0.916901i	$0.630682\pi$
$114$		0			0
$115$	−12.0000	−	6.00000i	−1.11901	−	0.559503i
$116$		0			0
$117$	2.82843	+	8.00000i	0.261488	+	0.739600i
$118$		0			0
$119$	−8.00000			−0.733359
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	−5.65685	+	8.00000i	−0.510061	+	0.721336i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$	−8.48528			−0.752947			−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528			−0.752947			−0.376473	+	0.926427i	$0.622863\pi$
$128$		0			0
$129$	2.00000	−	2.82843i	0.176090	−	0.249029i
$130$		0			0
$131$			8.48528i			0.741362i	0.928760	+	0.370681i	$0.120876\pi$
$131$			8.48528i			0.741362i	−0.928760	+	0.370681i	$0.879124\pi$
$132$		0			0
$133$	−11.3137			−0.981023
$134$		0			0
$135$	−8.58579	−	7.82843i	−0.738947	−	0.673764i
$136$		0			0
$137$	14.1421			1.20824			0.604122	−	0.796892i	$-0.293524\pi$
$137$	14.1421			1.20824			0.604122	+	0.796892i	$0.293524\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$	−10.0000	+	14.1421i	−0.842152	+	1.19098i
$142$		0			0
$143$		−	8.00000i		−	0.668994i
$144$		0			0
$145$	6.00000	−	12.0000i	0.498273	−	0.996546i
$146$		0			0
$147$	−1.41421	−	1.00000i	−0.116642	−	0.0824786i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		−	8.48528i		−	0.690522i	−0.938507	−	0.345261i	$-0.887790\pi$
$151$		−	8.48528i		−	0.690522i	0.938507	−	0.345261i	$-0.112210\pi$
$152$		0			0
$153$	2.82843	+	8.00000i	0.228665	+	0.646762i
$154$		0			0
$155$	5.65685	+	2.82843i	0.454369	+	0.227185i
$156$		0			0
$157$	−8.48528			−0.677199			−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528			−0.677199			−0.338600	+	0.940931i	$0.609953\pi$
$158$		0			0
$159$	−4.00000	+	5.65685i	−0.317221	+	0.448618i
$160$		0			0
$161$			16.9706i			1.33747i
$162$		0			0
$163$		−	18.0000i		−	1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$		−	18.0000i		−	1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$		0			0
$165$	5.17157	+	9.65685i	0.402606	+	0.751785i
$166$		0			0
$167$		−	2.00000i		−	0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$		−	2.00000i		−	0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	4.00000	+	11.3137i	0.305888	+	0.865181i
$172$		0			0
$173$			12.0000i			0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$			12.0000i			0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$		0			0
$175$	8.48528	+	11.3137i	0.641427	+	0.855236i
$176$		0			0
$177$	−2.82843	+	4.00000i	−0.212598	+	0.300658i
$178$		0			0
$179$		−	25.4558i		−	1.90266i	−0.308175	−	0.951330i	$-0.599718\pi$
$179$		−	25.4558i		−	1.90266i	0.308175	−	0.951330i	$-0.400282\pi$
$180$		0			0
$181$		−	22.6274i		−	1.68188i	−0.541126	−	0.840941i	$-0.682002\pi$
$181$		−	22.6274i		−	1.68188i	0.541126	−	0.840941i	$-0.317998\pi$
$182$		0			0
$183$	−5.65685	+	8.00000i	−0.418167	+	0.591377i
$184$		0			0
$185$	−8.48528	+	16.9706i	−0.623850	+	1.24770i
$186$		0			0
$187$		−	8.00000i		−	0.585018i
$188$		0			0
$189$	−4.00000	+	14.1421i	−0.290957	+	1.02869i
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	−9.65685	+	5.17157i	−0.691542	+	0.370344i
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$		−	19.7990i		−	1.40351i	−0.712417	−	0.701757i	$-0.752399\pi$
$199$		−	19.7990i		−	1.40351i	0.712417	−	0.701757i	$-0.247601\pi$
$200$		0			0
$201$	10.0000	−	14.1421i	0.705346	−	0.997509i
$202$		0			0
$203$	−16.9706			−1.19110
$204$		0			0
$205$	−11.3137	−	5.65685i	−0.790184	−	0.395092i
$206$		0			0
$207$	16.9706	−	6.00000i	1.17954	−	0.417029i
$208$		0			0
$209$		−	11.3137i		−	0.782586i
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	22.6274	+	16.0000i	1.55041	+	1.09630i
$214$		0			0
$215$	4.00000	+	2.00000i	0.272798	+	0.136399i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		0			0
$219$	16.0000	−	22.6274i	1.08118	−	1.52902i
$220$		0			0
$221$	8.00000			0.538138
$222$		0			0
$223$	8.48528			0.568216			0.284108	−	0.958792i	$-0.408302\pi$
$223$	8.48528			0.568216			0.284108	+	0.958792i	$0.408302\pi$
$224$		0			0
$225$	8.31371	−	12.4853i	0.554247	−	0.832352i
$226$		0			0
$227$	−2.82843			−0.187729			−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843			−0.187729			−0.0938647	+	0.995585i	$0.529922\pi$
$228$		0			0
$229$		−	11.3137i		−	0.747631i	−0.927503	−	0.373815i	$-0.878049\pi$
$229$		−	11.3137i		−	0.747631i	0.927503	−	0.373815i	$-0.121951\pi$
$230$		0			0
$231$	8.00000	−	11.3137i	0.526361	−	0.744387i
$232$		0			0
$233$	−19.7990			−1.29707			−0.648537	−	0.761183i	$-0.724619\pi$
$233$	−19.7990			−1.29707			−0.648537	+	0.761183i	$0.724619\pi$
$234$		0			0
$235$	−20.0000	−	10.0000i	−1.30466	−	0.652328i
$236$		0			0
$237$	14.1421	−	20.0000i	0.918630	−	1.29914i
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$	15.5563	−	1.00000i	0.997940	−	0.0641500i
$244$		0			0
$245$	1.00000	−	2.00000i	0.0638877	−	0.127775i
$246$		0			0
$247$	11.3137			0.719874
$248$		0			0
$249$	−12.0000	−	8.48528i	−0.760469	−	0.537733i
$250$		0			0
$251$			19.7990i			1.24970i	0.780744	+	0.624851i	$0.214840\pi$
$251$			19.7990i			1.24970i	−0.780744	+	0.624851i	$0.785160\pi$
$252$		0			0
$253$	−16.9706			−1.06693
$254$		0			0
$255$	−9.65685	+	5.17157i	−0.604736	+	0.323856i
$256$		0			0
$257$	2.82843			0.176432			0.0882162	−	0.996101i	$-0.471883\pi$
$257$	2.82843			0.176432			0.0882162	+	0.996101i	$0.471883\pi$
$258$		0			0
$259$	24.0000			1.49129
$260$		0			0
$261$	6.00000	+	16.9706i	0.371391	+	1.05045i
$262$		0			0
$263$			14.0000i			0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$			14.0000i			0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$		0			0
$265$	−8.00000	−	4.00000i	−0.491436	−	0.245718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$			25.4558i			1.54633i	0.634203	+	0.773166i	$0.281328\pi$
$271$			25.4558i			1.54633i	−0.634203	+	0.773166i	$0.718672\pi$
$272$		0			0
$273$	11.3137	+	8.00000i	0.684737	+	0.484182i
$274$		0			0
$275$	−11.3137	+	8.48528i	−0.682242	+	0.511682i
$276$		0			0
$277$	25.4558			1.52949			0.764747	−	0.644331i	$-0.222864\pi$
$277$	25.4558			1.52949			0.764747	+	0.644331i	$0.222864\pi$
$278$		0			0
$279$	−8.00000	+	2.82843i	−0.478947	+	0.169334i
$280$		0			0
$281$		−	16.9706i		−	1.01238i	−0.862422	−	0.506189i	$-0.831054\pi$
$281$		−	16.9706i		−	1.01238i	0.862422	−	0.506189i	$-0.168946\pi$
$282$		0			0
$283$			18.0000i			1.06999i	0.844856	+	0.534994i	$0.179686\pi$
$283$			18.0000i			1.06999i	−0.844856	+	0.534994i	$0.820314\pi$
$284$		0			0
$285$	−13.6569	+	7.31371i	−0.808962	+	0.433227i
$286$		0			0
$287$			16.0000i			0.944450i
$288$		0			0
$289$	−9.00000			−0.529412
$290$		0			0
$291$	−8.00000	+	11.3137i	−0.468968	+	0.663221i
$292$		0			0
$293$		−	12.0000i		−	0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$		−	12.0000i		−	0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$		0			0
$295$	−5.65685	−	2.82843i	−0.329355	−	0.164677i
$296$		0			0
$297$	−14.1421	−	4.00000i	−0.820610	−	0.232104i
$298$		0			0
$299$		−	16.9706i		−	0.981433i
$300$		0			0
$301$		−	5.65685i		−	0.326056i
$302$		0			0
$303$	8.48528	+	6.00000i	0.487467	+	0.344691i
$304$		0			0
$305$	−11.3137	−	5.65685i	−0.647821	−	0.323911i
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$	20.0000	+	14.1421i	1.13776	+	0.804518i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$	−18.8284	−	2.34315i	−1.06086	−	0.132021i
$316$		0			0
$317$		−	4.00000i		−	0.224662i	−0.993671	−	0.112331i	$-0.964168\pi$
$317$		−	4.00000i		−	0.224662i	0.993671	−	0.112331i	$-0.0358318\pi$
$318$		0			0
$319$		−	16.9706i		−	0.950169i
$320$		0			0
$321$	4.00000	+	2.82843i	0.223258	+	0.157867i
$322$		0			0
$323$	11.3137			0.629512
$324$		0			0
$325$	−8.48528	−	11.3137i	−0.470679	−	0.627572i
$326$		0			0
$327$	−16.9706	+	24.0000i	−0.938474	+	1.32720i
$328$		0			0
$329$			28.2843i			1.55936i
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$		0			0
$333$	−8.48528	−	24.0000i	−0.464991	−	1.31519i
$334$		0			0
$335$	20.0000	+	10.0000i	1.09272	+	0.546358i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$	12.0000	+	8.48528i	0.651751	+	0.460857i
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$	16.9706			0.916324
$344$		0			0
$345$	10.9706	+	20.4853i	0.590635	+	1.10289i
$346$		0			0
$347$	31.1127			1.67022			0.835109	−	0.550085i	$-0.185405\pi$
$347$	31.1127			1.67022			0.835109	+	0.550085i	$0.185405\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	4.00000	−	14.1421i	0.213504	−	0.754851i
$352$		0			0
$353$	2.82843			0.150542			0.0752710	−	0.997163i	$-0.476018\pi$
$353$	2.82843			0.150542			0.0752710	+	0.997163i	$0.476018\pi$
$354$		0			0
$355$	−16.0000	+	32.0000i	−0.849192	+	1.69838i
$356$		0			0
$357$	11.3137	+	8.00000i	0.598785	+	0.423405i
$358$		0			0
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−4.24264	−	3.00000i	−0.222681	−	0.157459i
$364$		0			0
$365$	32.0000	+	16.0000i	1.67496	+	0.837478i
$366$		0			0
$367$	8.48528			0.442928			0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528			0.442928			0.221464	+	0.975169i	$0.428916\pi$
$368$		0			0
$369$	16.0000	−	5.65685i	0.832927	−	0.294484i
$370$		0			0
$371$			11.3137i			0.587378i
$372$		0			0
$373$	−8.48528			−0.439351			−0.219676	−	0.975573i	$-0.570500\pi$
$373$	−8.48528			−0.439351			−0.219676	+	0.975573i	$0.570500\pi$
$374$		0			0
$375$	17.5563	+	8.17157i	0.906606	+	0.421978i
$376$		0			0
$377$	16.9706			0.874028
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$	12.0000	+	8.48528i	0.614779	+	0.434714i
$382$		0			0
$383$			18.0000i			0.919757i	0.887982	+	0.459879i	$0.152107\pi$
$383$			18.0000i			0.919757i	−0.887982	+	0.459879i	$0.847893\pi$
$384$		0			0
$385$	16.0000	+	8.00000i	0.815436	+	0.407718i
$386$		0			0
$387$	−5.65685	+	2.00000i	−0.287554	+	0.101666i
$388$		0			0
$389$	2.00000			0.101404			0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000			0.101404			0.0507020	+	0.998714i	$0.483854\pi$
$390$		0			0
$391$		−	16.9706i		−	0.858238i
$392$		0			0
$393$	8.48528	−	12.0000i	0.428026	−	0.605320i
$394$		0			0
$395$	28.2843	+	14.1421i	1.42314	+	0.711568i
$396$		0			0
$397$	−8.48528			−0.425864			−0.212932	−	0.977067i	$-0.568301\pi$
$397$	−8.48528			−0.425864			−0.212932	+	0.977067i	$0.568301\pi$
$398$		0			0
$399$	16.0000	+	11.3137i	0.801002	+	0.566394i
$400$		0			0
$401$			22.6274i			1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$			22.6274i			1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$		0			0
$403$			8.00000i			0.398508i
$404$		0			0
$405$	4.31371	+	19.6569i	0.214350	+	0.976757i
$406$		0			0
$407$			24.0000i			1.18964i
$408$		0			0
$409$	38.0000			1.87898			0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000			1.87898			0.939490	+	0.342578i	$0.111300\pi$
$410$		0			0
$411$	−20.0000	−	14.1421i	−0.986527	−	0.697580i
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$	8.48528	−	16.9706i	0.416526	−	0.833052i
$416$		0			0
$417$	−28.2843	−	20.0000i	−1.38509	−	0.979404i
$418$		0			0
$419$		−	25.4558i		−	1.24360i	−0.783176	−	0.621800i	$-0.786402\pi$
$419$		−	25.4558i		−	1.24360i	0.783176	−	0.621800i	$-0.213598\pi$
$420$		0			0
$421$			5.65685i			0.275698i	0.990453	+	0.137849i	$0.0440189\pi$
$421$			5.65685i			0.275698i	−0.990453	+	0.137849i	$0.955981\pi$
$422$		0			0
$423$	28.2843	−	10.0000i	1.37523	−	0.486217i
$424$		0			0
$425$	−8.48528	−	11.3137i	−0.411597	−	0.548795i
$426$		0			0
$427$			16.0000i			0.774294i
$428$		0			0
$429$	−8.00000	+	11.3137i	−0.386244	+	0.546231i
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$		−	8.00000i		−	0.384455i	−0.981350	−	0.192228i	$-0.938429\pi$
$433$		−	8.00000i		−	0.384455i	0.981350	−	0.192228i	$-0.0615712\pi$
$434$		0			0
$435$	−20.4853	+	10.9706i	−0.982194	+	0.525998i
$436$		0			0
$437$		−	24.0000i		−	1.14808i
$438$		0			0
$439$			14.1421i			0.674967i	0.941331	+	0.337484i	$0.109576\pi$
$439$			14.1421i			0.674967i	−0.941331	+	0.337484i	$0.890424\pi$
$440$		0			0
$441$	1.00000	+	2.82843i	0.0476190	+	0.134687i
$442$		0			0
$443$	−31.1127			−1.47821			−0.739104	−	0.673591i	$-0.764751\pi$
$443$	−31.1127			−1.47821			−0.739104	+	0.673591i	$0.764751\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	14.1421	+	10.0000i	0.668900	+	0.472984i
$448$		0			0
$449$			5.65685i			0.266963i	0.991051	+	0.133482i	$0.0426157\pi$
$449$			5.65685i			0.266963i	−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$	−8.48528	+	12.0000i	−0.398673	+	0.563809i
$454$		0			0
$455$	−8.00000	+	16.0000i	−0.375046	+	0.750092i
$456$		0			0
$457$			16.0000i			0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$			16.0000i			0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$		0			0
$459$	4.00000	−	14.1421i	0.186704	−	0.660098i
$460$		0			0
$461$	−10.0000			−0.465746			−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000			−0.465746			−0.232873	+	0.972507i	$0.574813\pi$
$462$		0			0
$463$	25.4558			1.18303			0.591517	−	0.806293i	$-0.298529\pi$
$463$	25.4558			1.18303			0.591517	+	0.806293i	$0.298529\pi$
$464$		0			0
$465$	−5.17157	−	9.65685i	−0.239826	−	0.447826i
$466$		0			0
$467$	−31.1127			−1.43972			−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−31.1127			−1.43972			−0.719862	+	0.694117i	$0.755795\pi$
$468$		0			0
$469$		−	28.2843i		−	1.30605i
$470$		0			0
$471$	12.0000	+	8.48528i	0.552931	+	0.390981i
$472$		0			0
$473$	5.65685			0.260102
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$		0			0
$477$	11.3137	−	4.00000i	0.518019	−	0.183147i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	−24.0000			−1.09431
$482$		0			0
$483$	16.9706	−	24.0000i	0.772187	−	1.09204i
$484$		0			0
$485$	−16.0000	−	8.00000i	−0.726523	−	0.363261i
$486$		0			0
$487$	25.4558			1.15351			0.576757	−	0.816916i	$-0.304318\pi$
$487$	25.4558			1.15351			0.576757	+	0.816916i	$0.304318\pi$
$488$		0			0
$489$	−18.0000	+	25.4558i	−0.813988	+	1.15115i
$490$		0			0
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$	16.9706			0.764316
$494$		0			0
$495$	2.34315	−	18.8284i	0.105317	−	0.846275i
$496$		0			0
$497$	45.2548			2.02996
$498$		0			0
$499$	44.0000			1.96971			0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000			1.96971			0.984855	+	0.173379i	$0.0554684\pi$
$500$		0			0
$501$	−2.00000	+	2.82843i	−0.0893534	+	0.126365i
$502$		0			0
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$		0			0
$505$	−6.00000	+	12.0000i	−0.266996	+	0.533993i
$506$		0			0
$507$	7.07107	+	5.00000i	0.314037	+	0.222058i
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$		−	45.2548i		−	2.00196i
$512$		0			0
$513$	5.65685	−	20.0000i	0.249756	−	0.883022i
$514$		0			0
$515$	−14.1421	+	28.2843i	−0.623177	+	1.24635i
$516$		0			0
$517$	−28.2843			−1.24394
$518$		0			0
$519$	12.0000	−	16.9706i	0.526742	−	0.744925i
$520$		0			0
$521$		−	22.6274i		−	0.991325i	−0.868515	−	0.495663i	$-0.834925\pi$
$521$		−	22.6274i		−	0.991325i	0.868515	−	0.495663i	$-0.165075\pi$
$522$		0			0
$523$			6.00000i			0.262362i	0.991358	+	0.131181i	$0.0418769\pi$
$523$			6.00000i			0.262362i	−0.991358	+	0.131181i	$0.958123\pi$
$524$		0			0
$525$	−0.686292	−	24.4853i	−0.0299522	−	1.06863i
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	8.00000	−	2.82843i	0.347170	−	0.122743i
$532$		0			0
$533$		−	16.0000i		−	0.693037i
$534$		0			0
$535$	−2.82843	+	5.65685i	−0.122284	+	0.244567i
$536$		0			0
$537$	−25.4558	+	36.0000i	−1.09850	+	1.55351i
$538$		0			0
$539$		−	2.82843i		−	0.121829i
$540$		0			0
$541$		−	33.9411i		−	1.45924i	−0.683851	−	0.729621i	$-0.739696\pi$
$541$		−	33.9411i		−	1.45924i	0.683851	−	0.729621i	$-0.260304\pi$
$542$		0			0
$543$	−22.6274	+	32.0000i	−0.971035	+	1.37325i
$544$		0			0
$545$	−33.9411	−	16.9706i	−1.45388	−	0.726939i
$546$		0			0
$547$		−	22.0000i		−	0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$		−	22.0000i		−	0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$		0			0
$549$	16.0000	−	5.65685i	0.682863	−	0.241429i
$550$		0			0
$551$	24.0000			1.02243
$552$		0			0
$553$		−	40.0000i		−	1.70097i
$554$		0			0
$555$	28.9706	−	15.5147i	1.22973	−	0.658563i
$556$		0			0
$557$			44.0000i			1.86434i	0.362021	+	0.932170i	$0.382087\pi$
$557$			44.0000i			1.86434i	−0.362021	+	0.932170i	$0.617913\pi$
$558$		0			0
$559$			5.65685i			0.239259i
$560$		0			0
$561$	−8.00000	+	11.3137i	−0.337760	+	0.477665i
$562$		0			0
$563$	36.7696			1.54965			0.774826	−	0.632175i	$-0.217837\pi$
$563$	36.7696			1.54965			0.774826	+	0.632175i	$0.217837\pi$
$564$		0			0
$565$	−8.48528	+	16.9706i	−0.356978	+	0.713957i
$566$		0			0
$567$	19.7990	−	16.0000i	0.831479	−	0.671937i
$568$		0			0
$569$		−	39.5980i		−	1.66003i	−0.557738	−	0.830017i	$-0.688331\pi$
$569$		−	39.5980i		−	1.66003i	0.557738	−	0.830017i	$-0.311669\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	11.3137	+	8.00000i	0.472637	+	0.334205i
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		0			0
$579$	16.0000	−	22.6274i	0.664937	−	0.940363i
$580$		0			0
$581$	−24.0000			−0.995688
$582$		0			0
$583$	−11.3137			−0.468566
$584$		0			0
$585$	18.8284	+	2.34315i	0.778460	+	0.0968772i
$586$		0			0
$587$	−14.1421			−0.583708			−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421			−0.583708			−0.291854	+	0.956463i	$0.594272\pi$
$588$		0			0
$589$			11.3137i			0.466173i
$590$		0			0
$591$	−12.0000	+	16.9706i	−0.493614	+	0.698076i
$592$		0			0
$593$	−8.48528			−0.348449			−0.174224	−	0.984706i	$-0.555742\pi$
$593$	−8.48528			−0.348449			−0.174224	+	0.984706i	$0.555742\pi$
$594$		0			0
$595$	−8.00000	+	16.0000i	−0.327968	+	0.655936i
$596$		0			0
$597$	−19.7990	+	28.0000i	−0.810319	+	1.14596i
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$	−28.2843	+	10.0000i	−1.15182	+	0.407231i
$604$		0			0
$605$	3.00000	−	6.00000i	0.121967	−	0.243935i
$606$		0			0
$607$	−25.4558			−1.03322			−0.516610	−	0.856221i	$-0.672806\pi$
$607$	−25.4558			−1.03322			−0.516610	+	0.856221i	$0.672806\pi$
$608$		0			0
$609$	24.0000	+	16.9706i	0.972529	+	0.687682i
$610$		0			0
$611$		−	28.2843i		−	1.14426i
$612$		0			0
$613$	14.1421			0.571195			0.285598	−	0.958350i	$-0.407808\pi$
$613$	14.1421			0.571195			0.285598	+	0.958350i	$0.407808\pi$
$614$		0			0
$615$	10.3431	+	19.3137i	0.417076	+	0.778804i
$616$		0			0
$617$	14.1421			0.569341			0.284670	−	0.958625i	$-0.408116\pi$
$617$	14.1421			0.569341			0.284670	+	0.958625i	$0.408116\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	−30.0000	−	8.48528i	−1.20386	−	0.340503i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$	−11.3137	+	16.0000i	−0.451826	+	0.638978i
$628$		0			0
$629$	−24.0000			−0.956943
$630$		0			0
$631$			36.7696i			1.46377i	0.681427	+	0.731886i	$0.261360\pi$
$631$			36.7696i			1.46377i	−0.681427	+	0.731886i	$0.738640\pi$
$632$		0			0
$633$	5.65685	+	4.00000i	0.224840	+	0.158986i
$634$		0			0
$635$	−8.48528	+	16.9706i	−0.336728	+	0.673456i
$636$		0			0
$637$	2.82843			0.112066
$638$		0			0
$639$	−16.0000	−	45.2548i	−0.632950	−	1.79025i
$640$		0			0
$641$		−	28.2843i		−	1.11716i	−0.829450	−	0.558581i	$-0.811346\pi$
$641$		−	28.2843i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$642$		0			0
$643$		−	34.0000i		−	1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$		−	34.0000i		−	1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$		0			0
$645$	−3.65685	−	6.82843i	−0.143988	−	0.268869i
$646$		0			0
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$		0			0
$649$	−8.00000			−0.314027
$650$		0			0
$651$	−8.00000	+	11.3137i	−0.313545	+	0.443419i
$652$		0			0
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$		0			0
$655$	16.9706	+	8.48528i	0.663095	+	0.331547i
$656$		0			0
$657$	−45.2548	+	16.0000i	−1.76556	+	0.624219i
$658$		0			0
$659$		−	2.82843i		−	0.110180i	−0.998481	−	0.0550899i	$-0.982455\pi$
$659$		−	2.82843i		−	0.110180i	0.998481	−	0.0550899i	$-0.0175446\pi$
$660$		0			0
$661$			16.9706i			0.660078i	0.943967	+	0.330039i	$0.107062\pi$
$661$			16.9706i			0.660078i	−0.943967	+	0.330039i	$0.892938\pi$
$662$		0			0
$663$	−11.3137	−	8.00000i	−0.439388	−	0.310694i
$664$		0			0
$665$	−11.3137	+	22.6274i	−0.438727	+	0.877454i
$666$		0			0
$667$		−	36.0000i		−	1.39393i
$668$		0			0
$669$	−12.0000	−	8.48528i	−0.463947	−	0.328060i
$670$		0			0
$671$	−16.0000			−0.617673
$672$		0			0
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$		0			0
$675$	−24.2426	+	9.34315i	−0.933100	+	0.359618i
$676$		0			0
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$		0			0
$679$			22.6274i			0.868361i
$680$		0			0
$681$	4.00000	+	2.82843i	0.153280	+	0.108386i
$682$		0			0
$683$	−14.1421			−0.541134			−0.270567	−	0.962701i	$-0.587211\pi$
$683$	−14.1421			−0.541134			−0.270567	+	0.962701i	$0.587211\pi$
$684$		0			0
$685$	14.1421	−	28.2843i	0.540343	−	1.08069i
$686$		0			0
$687$	−11.3137	+	16.0000i	−0.431645	+	0.610438i
$688$		0			0
$689$		−	11.3137i		−	0.431018i
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$		0			0
$693$	−22.6274	+	8.00000i	−0.859544	+	0.303895i
$694$		0			0
$695$	20.0000	−	40.0000i	0.758643	−	1.51729i
$696$		0			0
$697$		−	16.0000i		−	0.606043i
$698$		0			0
$699$	28.0000	+	19.7990i	1.05906	+	0.748867i
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	−33.9411			−1.28011
$704$		0			0
$705$	18.2843	+	34.1421i	0.688625	+	1.28587i
$706$		0			0
$707$	16.9706			0.638244
$708$		0			0
$709$			22.6274i			0.849790i	0.905243	+	0.424895i	$0.139689\pi$
$709$			22.6274i			0.849790i	−0.905243	+	0.424895i	$0.860311\pi$
$710$		0			0
$711$	−40.0000	+	14.1421i	−1.50012	+	0.530372i
$712$		0			0
$713$	16.9706			0.635553
$714$		0			0
$715$	−16.0000	−	8.00000i	−0.598366	−	0.299183i
$716$		0			0
$717$	−33.9411	−	24.0000i	−1.26755	−	0.896296i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	40.0000			1.48968
$722$		0			0
$723$	2.82843	+	2.00000i	0.105190	+	0.0743808i
$724$		0			0
$725$	−18.0000	−	24.0000i	−0.668503	−	0.891338i
$726$		0			0
$727$	2.82843			0.104901			0.0524503	−	0.998624i	$-0.483297\pi$
$727$	2.82843			0.104901			0.0524503	+	0.998624i	$0.483297\pi$
$728$		0			0
$729$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$730$		0			0
$731$			5.65685i			0.209226i
$732$		0			0
$733$	14.1421			0.522352			0.261176	−	0.965291i	$-0.415890\pi$
$733$	14.1421			0.522352			0.261176	+	0.965291i	$0.415890\pi$
$734$		0			0
$735$	−3.41421	+	1.82843i	−0.125935	+	0.0674426i
$736$		0			0
$737$	28.2843			1.04186
$738$		0			0
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$	−16.0000	−	11.3137i	−0.587775	−	0.415619i
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$	−10.0000	+	20.0000i	−0.366372	+	0.732743i
$746$		0			0
$747$	8.48528	+	24.0000i	0.310460	+	0.878114i
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$		−	8.48528i		−	0.309632i	−0.987943	−	0.154816i	$-0.950521\pi$
$751$		−	8.48528i		−	0.309632i	0.987943	−	0.154816i	$-0.0494785\pi$
$752$		0			0
$753$	19.7990	−	28.0000i	0.721515	−	1.02038i
$754$		0			0
$755$	−16.9706	−	8.48528i	−0.617622	−	0.308811i
$756$		0			0
$757$	−42.4264			−1.54201			−0.771007	−	0.636827i	$-0.780247\pi$
$757$	−42.4264			−1.54201			−0.771007	+	0.636827i	$0.780247\pi$
$758$		0			0
$759$	24.0000	+	16.9706i	0.871145	+	0.615992i
$760$		0			0
$761$			33.9411i			1.23036i	0.788385	+	0.615182i	$0.210918\pi$
$761$			33.9411i			1.23036i	−0.788385	+	0.615182i	$0.789082\pi$
$762$		0			0
$763$			48.0000i			1.73772i
$764$		0			0
$765$	18.8284	+	2.34315i	0.680743	+	0.0847166i
$766$		0			0
$767$		−	8.00000i		−	0.288863i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	−4.00000	−	2.82843i	−0.144056	−	0.101863i
$772$		0			0
$773$		−	12.0000i		−	0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$		−	12.0000i		−	0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$		0			0
$775$	11.3137	−	8.48528i	0.406400	−	0.304800i
$776$		0			0
$777$	−33.9411	−	24.0000i	−1.21763	−	0.860995i
$778$		0			0
$779$		−	22.6274i		−	0.810711i
$780$		0			0
$781$			45.2548i			1.61935i
$782$		0			0
$783$	8.48528	−	30.0000i	0.303239	−	1.07211i
$784$		0			0
$785$	−8.48528	+	16.9706i	−0.302853	+	0.605705i
$786$		0			0
$787$			14.0000i			0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$			14.0000i			0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$		0			0
$789$	14.0000	−	19.7990i	0.498413	−	0.704863i
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$		−	16.0000i		−	0.568177i
$794$		0			0
$795$	7.31371	+	13.6569i	0.259391	+	0.484359i
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$		−	28.2843i		−	1.00063i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	45.2548			1.59701
$804$		0			0
$805$	33.9411	+	16.9706i	1.19627	+	0.598134i
$806$		0			0
$807$	−19.7990	−	14.0000i	−0.696957	−	0.492823i
$808$		0			0
$809$			22.6274i			0.795538i	0.917486	+	0.397769i	$0.130215\pi$
$809$			22.6274i			0.795538i	−0.917486	+	0.397769i	$0.869785\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$		0			0
$813$	25.4558	−	36.0000i	0.892775	−	1.26258i
$814$		0			0
$815$	−36.0000	−	18.0000i	−1.26102	−	0.630512i
$816$		0			0
$817$			8.00000i			0.279885i
$818$		0			0
$819$	−8.00000	−	22.6274i	−0.279543	−	0.790666i
$820$		0			0
$821$	−10.0000			−0.349002			−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000			−0.349002			−0.174501	+	0.984657i	$0.555831\pi$
$822$		0			0
$823$	8.48528			0.295778			0.147889	−	0.989004i	$-0.452752\pi$
$823$	8.48528			0.295778			0.147889	+	0.989004i	$0.452752\pi$
$824$		0			0
$825$	24.4853	−	0.686292i	0.852468	−	0.0238936i
$826$		0			0
$827$	25.4558			0.885186			0.442593	−	0.896723i	$-0.354059\pi$
$827$	25.4558			0.885186			0.442593	+	0.896723i	$0.354059\pi$
$828$		0			0
$829$		−	5.65685i		−	0.196471i	−0.995163	−	0.0982353i	$-0.968680\pi$
$829$		−	5.65685i		−	0.196471i	0.995163	−	0.0982353i	$-0.0313198\pi$
$830$		0			0
$831$	−36.0000	−	25.4558i	−1.24883	−	0.883053i
$832$		0			0
$833$	2.82843			0.0979992
$834$		0			0
$835$	−4.00000	−	2.00000i	−0.138426	−	0.0692129i
$836$		0			0
$837$	14.1421	+	4.00000i	0.488824	+	0.138260i
$838$		0			0
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−16.9706	+	24.0000i	−0.584497	+	0.826604i
$844$		0			0
$845$	−5.00000	+	10.0000i	−0.172005	+	0.344010i
$846$		0			0
$847$	−8.48528			−0.291558
$848$		0			0
$849$	18.0000	−	25.4558i	0.617758	−	0.873642i
$850$		0			0
$851$			50.9117i			1.74523i
$852$		0			0
$853$	2.82843			0.0968435			0.0484218	−	0.998827i	$-0.484581\pi$
$853$	2.82843			0.0968435			0.0484218	+	0.998827i	$0.484581\pi$
$854$		0			0
$855$	26.6274	+	3.31371i	0.910638	+	0.113326i
$856$		0			0
$857$	25.4558			0.869555			0.434778	−	0.900538i	$-0.356827\pi$
$857$	25.4558			0.869555			0.434778	+	0.900538i	$0.356827\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	16.0000	−	22.6274i	0.545279	−	0.771140i
$862$		0			0
$863$		−	26.0000i		−	0.885050i	−0.896756	−	0.442525i	$-0.854083\pi$
$863$		−	26.0000i		−	0.885050i	0.896756	−	0.442525i	$-0.145917\pi$
$864$		0			0
$865$	24.0000	+	12.0000i	0.816024	+	0.408012i
$866$		0			0
$867$	12.7279	+	9.00000i	0.432263	+	0.305656i
$868$		0			0
$869$	40.0000			1.35691
$870$		0			0
$871$			28.2843i			0.958376i
$872$		0			0
$873$	22.6274	−	8.00000i	0.765822	−	0.270759i
$874$		0			0
$875$	31.1127	−	5.65685i	1.05180	−	0.191237i
$876$		0			0
$877$	36.7696			1.24162			0.620810	−	0.783961i	$-0.286804\pi$
$877$	36.7696			1.24162			0.620810	+	0.783961i	$0.286804\pi$
$878$		0			0
$879$	−12.0000	+	16.9706i	−0.404750	+	0.572403i
$880$		0			0
$881$		−	5.65685i		−	0.190584i	−0.995449	−	0.0952921i	$-0.969621\pi$
$881$		−	5.65685i		−	0.190584i	0.995449	−	0.0952921i	$-0.0303785\pi$
$882$		0			0
$883$		−	34.0000i		−	1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$		−	34.0000i		−	1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$		0			0
$885$	5.17157	+	9.65685i	0.173841	+	0.324612i
$886$		0			0
$887$		−	34.0000i		−	1.14161i	−0.821086	−	0.570804i	$-0.806632\pi$
$887$		−	34.0000i		−	1.14161i	0.821086	−	0.570804i	$-0.193368\pi$
$888$		0			0
$889$	24.0000			0.804934
$890$		0			0
$891$	16.0000	+	19.7990i	0.536020	+	0.663291i
$892$		0			0
$893$		−	40.0000i		−	1.33855i
$894$		0			0
$895$	−50.9117	−	25.4558i	−1.70179	−	0.850895i
$896$		0			0
$897$	−16.9706	+	24.0000i	−0.566631	+	0.801337i
$898$		0			0
$899$			16.9706i			0.566000i
$900$		0			0
$901$		−	11.3137i		−	0.376914i
$902$		0			0
$903$	−5.65685	+	8.00000i	−0.188248	+	0.266223i
$904$		0			0
$905$	−45.2548	−	22.6274i	−1.50432	−	0.752161i
$906$		0			0
$907$		−	14.0000i		−	0.464862i	−0.972613	−	0.232431i	$-0.925332\pi$
$907$		−	14.0000i		−	0.464862i	0.972613	−	0.232431i	$-0.0746680\pi$
$908$		0			0
$909$	−6.00000	−	16.9706i	−0.199007	−	0.562878i
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$		−	24.0000i		−	0.794284i
$914$		0			0
$915$	10.3431	+	19.3137i	0.341934	+	0.638492i
$916$		0			0
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$			25.4558i			0.839711i	0.907591	+	0.419855i	$0.137919\pi$
$919$			25.4558i			0.839711i	−0.907591	+	0.419855i	$0.862081\pi$
$920$		0			0
$921$	−2.00000	+	2.82843i	−0.0659022	+	0.0931998i
$922$		0			0
$923$	−45.2548			−1.48958
$924$		0			0
$925$	25.4558	+	33.9411i	0.836983	+	1.11598i
$926$		0			0
$927$	−14.1421	−	40.0000i	−0.464489	−	1.31377i
$928$		0			0
$929$			16.9706i			0.556786i	0.960467	+	0.278393i	$0.0898018\pi$
$929$			16.9706i			0.556786i	−0.960467	+	0.278393i	$0.910198\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.0000	−	8.00000i	−0.523256	−	0.261628i
$936$		0			0
$937$			24.0000i			0.784046i	0.919955	+	0.392023i	$0.128225\pi$
$937$			24.0000i			0.784046i	−0.919955	+	0.392023i	$0.871775\pi$
$938$		0			0
$939$	−8.00000	+	11.3137i	−0.261070	+	0.369209i
$940$		0			0
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$	−33.9411			−1.10528
$944$		0			0
$945$	24.2843	+	22.1421i	0.789967	+	0.720284i
$946$		0			0
$947$	−19.7990			−0.643381			−0.321690	−	0.946845i	$-0.604251\pi$
$947$	−19.7990			−0.643381			−0.321690	+	0.946845i	$0.604251\pi$
$948$		0			0
$949$			45.2548i			1.46903i
$950$		0			0
$951$	−4.00000	+	5.65685i	−0.129709	+	0.183436i
$952$		0			0
$953$	−31.1127			−1.00784			−0.503920	−	0.863751i	$-0.668109\pi$
$953$	−31.1127			−1.00784			−0.503920	+	0.863751i	$0.668109\pi$
$954$		0			0
$955$	−8.00000	+	16.0000i	−0.258874	+	0.517748i
$956$		0			0
$957$	−16.9706	+	24.0000i	−0.548580	+	0.775810i
$958$		0			0
$959$	−40.0000			−1.29167
$960$		0			0
$961$	23.0000			0.741935
$962$		0			0
$963$	−2.82843	−	8.00000i	−0.0911448	−	0.257796i
$964$		0			0
$965$	32.0000	+	16.0000i	1.03012	+	0.515058i
$966$		0			0
$967$	−8.48528			−0.272868			−0.136434	−	0.990649i	$-0.543564\pi$
$967$	−8.48528			−0.272868			−0.136434	+	0.990649i	$0.543564\pi$
$968$		0			0
$969$	−16.0000	−	11.3137i	−0.513994	−	0.363449i
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$	−56.5685			−1.81350
$974$		0			0
$975$	0.686292	+	24.4853i	0.0219789	+	0.784157i
$976$		0			0
$977$	14.1421			0.452447			0.226224	−	0.974075i	$-0.427362\pi$
$977$	14.1421			0.452447			0.226224	+	0.974075i	$0.427362\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	48.0000	−	16.9706i	1.53252	−	0.541828i
$982$		0			0
$983$		−	54.0000i		−	1.72233i	−0.508323	−	0.861166i	$-0.669735\pi$
$983$		−	54.0000i		−	1.72233i	0.508323	−	0.861166i	$-0.330265\pi$
$984$		0			0
$985$	−24.0000	−	12.0000i	−0.764704	−	0.382352i
$986$		0			0
$987$	28.2843	−	40.0000i	0.900298	−	1.27321i
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$		−	8.48528i		−	0.269544i	−0.990877	−	0.134772i	$-0.956970\pi$
$991$		−	8.48528i		−	0.269544i	0.990877	−	0.134772i	$-0.0430302\pi$
$992$		0			0
$993$	16.9706	+	12.0000i	0.538545	+	0.380808i
$994$		0			0
$995$	−39.5980	−	19.7990i	−1.25534	−	0.627670i
$996$		0			0
$997$	48.0833			1.52281			0.761406	−	0.648275i	$-0.224509\pi$
$997$	48.0833			1.52281			0.761406	+	0.648275i	$0.224509\pi$
$998$		0			0
$999$	−12.0000	+	42.4264i	−0.379663	+	1.34231i

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.m.m.959.1	yes	4
3.2	odd	2		1920.2.m.j.959.3	yes	4
4.3	odd	2		1920.2.m.n.959.4	yes	4
5.4	even	2	inner	1920.2.m.m.959.4	yes	4
8.3	odd	2		1920.2.m.j.959.1	yes	4
8.5	even	2		1920.2.m.i.959.4	yes	4
12.11	even	2		1920.2.m.i.959.2	yes	4
15.14	odd	2		1920.2.m.j.959.2	yes	4
20.19	odd	2		1920.2.m.n.959.1	yes	4
24.5	odd	2		1920.2.m.n.959.2	yes	4
24.11	even	2	inner	1920.2.m.m.959.3	yes	4
40.19	odd	2		1920.2.m.j.959.4	yes	4
40.29	even	2		1920.2.m.i.959.1	✓	4
60.59	even	2		1920.2.m.i.959.3	yes	4
120.29	odd	2		1920.2.m.n.959.3	yes	4
120.59	even	2	inner	1920.2.m.m.959.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.m.i.959.1	✓	4	40.29	even	2
1920.2.m.i.959.2	yes	4	12.11	even	2
1920.2.m.i.959.3	yes	4	60.59	even	2
1920.2.m.i.959.4	yes	4	8.5	even	2
1920.2.m.j.959.1	yes	4	8.3	odd	2
1920.2.m.j.959.2	yes	4	15.14	odd	2
1920.2.m.j.959.3	yes	4	3.2	odd	2
1920.2.m.j.959.4	yes	4	40.19	odd	2
1920.2.m.m.959.1	yes	4	1.1	even	1	trivial
1920.2.m.m.959.2	yes	4	120.59	even	2	inner
1920.2.m.m.959.3	yes	4	24.11	even	2	inner
1920.2.m.m.959.4	yes	4	5.4	even	2	inner
1920.2.m.n.959.1	yes	4	20.19	odd	2
1920.2.m.n.959.2	yes	4	24.5	odd	2
1920.2.m.n.959.3	yes	4	120.29	odd	2
1920.2.m.n.959.4	yes	4	4.3	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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.m.m.959.1

Level	$1920$
Weight	$2$
Character	1920.959
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$