LMFDB - Embedded newform 1520.2.d.a.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, not 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.a.609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +(4.00000 - 2.00000i) q^{15} -8.00000i q^{17} -1.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} -8.00000i q^{33} +(4.00000 - 2.00000i) q^{35} -6.00000 q^{41} -10.0000i q^{43} +(1.00000 + 2.00000i) q^{45} -6.00000i q^{47} +3.00000 q^{49} +16.0000 q^{51} +(4.00000 + 8.00000i) q^{55} -2.00000i q^{57} -4.00000 q^{59} +6.00000 q^{61} -2.00000i q^{63} +2.00000i q^{67} +12.0000 q^{69} -16.0000 q^{71} -16.0000i q^{73} +(-8.00000 - 6.00000i) q^{75} -8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} -10.0000i q^{83} +(-16.0000 + 8.00000i) q^{85} -4.00000i q^{87} +10.0000 q^{89} +16.0000i q^{93} +(1.00000 + 2.00000i) q^{95} -4.00000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^9	$ 2 q - 2 q^{5} - 2 q^{9} - 8 q^{11} + 8 q^{15} - 2 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} + 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 32 q^{51} + 8 q^{55} - 8 q^{59} + 12 q^{61} + 24 q^{69} - 32 q^{71} - 16 q^{75} + 16 q^{79} - 22 q^{81} - 32 q^{85} + 20 q^{89} + 2 q^{95} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 - 8 * q^11 + 8 * q^15 - 2 * q^19 - 8 * q^21 - 6 * q^25 - 4 * q^29 + 16 * q^31 + 8 * q^35 - 12 * q^41 + 2 * q^45 + 6 * q^49 + 32 * q^51 + 8 * q^55 - 8 * q^59 + 12 * q^61 + 24 * q^69 - 32 * q^71 - 16 * q^75 + 16 * q^79 - 22 * q^81 - 32 * q^85 + 20 * q^89 + 2 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	4.00000	−	2.00000i	1.03280	−	0.516398i
$16$		0			0
$17$		−	8.00000i		−	1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$		−	8.00000i		−	1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	−4.00000			−0.872872
$22$		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
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		−	8.00000i		−	1.39262i
$34$		0			0
$35$	4.00000	−	2.00000i	0.676123	−	0.338062i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	10.0000i		−	1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$		−	10.0000i		−	1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	16.0000			2.24045
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	4.00000	+	8.00000i	0.539360	+	1.07872i
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	12.0000			1.44463
$70$		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	$-0.614200\pi$
$73$		−	16.0000i		−	1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$		0			0
$75$	−8.00000	−	6.00000i	−0.923760	−	0.692820i
$76$		0			0
$77$		−	8.00000i		−	0.911685i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	10.0000i		−	1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$		−	10.0000i		−	1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$		0			0
$85$	−16.0000	+	8.00000i	−1.73544	+	0.867722i
$86$		0			0
$87$		−	4.00000i		−	0.428845i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			16.0000i			1.65912i
$94$		0			0
$95$	1.00000	+	2.00000i	0.102598	+	0.205196i
$96$		0			0
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$	4.00000	+	8.00000i	0.390360	+	0.780720i
$106$		0			0
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	−12.0000	+	6.00000i	−1.11901	+	0.559503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	16.0000			1.46672
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		−	12.0000i		−	1.08200i
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		0			0
$129$	20.0000			1.76090
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		−	2.00000i		−	0.173422i
$134$		0			0
$135$	8.00000	−	4.00000i	0.688530	−	0.344265i
$136$		0			0
$137$		−	16.0000i		−	1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$		−	16.0000i		−	1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	12.0000			1.01058
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000	+	4.00000i	0.166091	+	0.332182i
$146$		0			0
$147$			6.00000i			0.494872i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			8.00000i			0.646762i
$154$		0			0
$155$	−8.00000	−	16.0000i	−0.642575	−	1.28515i
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.0000			0.945732
$162$		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
$165$	−16.0000	+	8.00000i	−1.24560	+	0.622799i
$166$		0			0
$167$			6.00000i			0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$			6.00000i			0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	1.00000			0.0764719
$172$		0			0
$173$			24.0000i			1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$			24.0000i			1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$		0			0
$177$		−	8.00000i		−	0.601317i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$			12.0000i			0.887066i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			32.0000i			2.34007i
$188$		0			0
$189$	−8.00000			−0.581914
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$		−	4.00000i		−	0.280745i
$204$		0			0
$205$	6.00000	+	12.0000i	0.419058	+	0.838116i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$		−	32.0000i		−	2.19260i
$214$		0			0
$215$	−20.0000	+	10.0000i	−1.36399	+	0.681994i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	32.0000			2.16236
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	26.0000i		−	1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$		−	26.0000i		−	1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$		0			0
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$		0			0
$227$			10.0000i			0.663723i	0.943328	+	0.331862i	$0.107677\pi$
$227$			10.0000i			0.663723i	−0.943328	+	0.331862i	$0.892323\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$	16.0000			1.05272
$232$		0			0
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$		0			0
$235$	−12.0000	+	6.00000i	−0.782794	+	0.391397i
$236$		0			0
$237$			16.0000i			1.03931i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$	−3.00000	−	6.00000i	−0.191663	−	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	20.0000			1.26745
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		0			0
$253$			24.0000i			1.50887i
$254$		0			0
$255$	−16.0000	−	32.0000i	−1.00196	−	2.00392i
$256$		0			0
$257$			20.0000i			1.24757i	0.781598	+	0.623783i	$0.214405\pi$
$257$			20.0000i			1.24757i	−0.781598	+	0.623783i	$0.785595\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			20.0000i			1.22398i
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000	−	16.0000i	0.723627	−	0.964836i
$276$		0			0
$277$		−	12.0000i		−	0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$		−	12.0000i		−	0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			14.0000i			0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$			14.0000i			0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$		0			0
$285$	−4.00000	+	2.00000i	−0.236940	+	0.118470i
$286$		0			0
$287$		−	12.0000i		−	0.708338i
$288$		0			0
$289$	−47.0000			−2.76471
$290$		0			0
$291$	8.00000			0.468968
$292$		0			0
$293$		−	16.0000i		−	0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$		−	16.0000i		−	0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$		0			0
$295$	4.00000	+	8.00000i	0.232889	+	0.465778i
$296$		0			0
$297$		−	16.0000i		−	0.928414i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	20.0000			1.15278
$302$		0			0
$303$			12.0000i			0.689382i
$304$		0			0
$305$	−6.00000	−	12.0000i	−0.343559	−	0.687118i
$306$		0			0
$307$		−	6.00000i		−	0.342438i	−0.985233	−	0.171219i	$-0.945229\pi$
$307$		−	6.00000i		−	0.342438i	0.985233	−	0.171219i	$-0.0547706\pi$
$308$		0			0
$309$	−28.0000			−1.59286
$310$		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
$315$	−4.00000	+	2.00000i	−0.225374	+	0.112687i
$316$		0			0
$317$		−	8.00000i		−	0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$		−	8.00000i		−	0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$		0			0
$319$	8.00000			0.447914
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$			8.00000i			0.445132i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	4.00000i		−	0.221201i
$328$		0			0
$329$	12.0000			0.661581
$330$		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
$333$		0			0
$334$		0			0
$335$	4.00000	−	2.00000i	0.218543	−	0.109272i
$336$		0			0
$337$		−	20.0000i		−	1.08947i	−0.838608	−	0.544735i	$-0.816630\pi$
$337$		−	20.0000i		−	1.08947i	0.838608	−	0.544735i	$-0.183370\pi$
$338$		0			0
$339$	24.0000			1.30350
$340$		0			0
$341$	−32.0000			−1.73290
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$	−12.0000	−	24.0000i	−0.646058	−	1.29212i
$346$		0			0
$347$		−	2.00000i		−	0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$		−	2.00000i		−	0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$		0			0
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	16.0000	+	32.0000i	0.849192	+	1.69838i
$356$		0			0
$357$			32.0000i			1.69362i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			10.0000i			0.524864i
$364$		0			0
$365$	−32.0000	+	16.0000i	−1.67496	+	0.837478i
$366$		0			0
$367$		−	6.00000i		−	0.313197i	−0.987662	−	0.156599i	$-0.949947\pi$
$367$		−	6.00000i		−	0.313197i	0.987662	−	0.156599i	$-0.0500529\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$		0			0
$372$		0			0
$373$			16.0000i			0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$			16.0000i			0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$		0			0
$375$	−4.00000	+	22.0000i	−0.206559	+	1.13608i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$			14.0000i			0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$			14.0000i			0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$		0			0
$385$	−16.0000	+	8.00000i	−0.815436	+	0.407718i
$386$		0			0
$387$			10.0000i			0.508329i
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−48.0000			−2.42746
$392$		0			0
$393$		−	8.00000i		−	0.403547i
$394$		0			0
$395$	−8.00000	−	16.0000i	−0.402524	−	0.805047i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$		0			0
$399$	4.00000			0.200250
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	11.0000	+	22.0000i	0.546594	+	1.09319i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	32.0000			1.57844
$412$		0			0
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	−20.0000	+	10.0000i	−0.981761	+	0.490881i
$416$		0			0
$417$		−	8.00000i		−	0.391762i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$			6.00000i			0.291730i
$424$		0			0
$425$	32.0000	+	24.0000i	1.55223	+	1.16417i
$426$		0			0
$427$			12.0000i			0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$			20.0000i			0.961139i	0.876957	+	0.480569i	$0.159570\pi$
$433$			20.0000i			0.961139i	−0.876957	+	0.480569i	$0.840430\pi$
$434$		0			0
$435$	−8.00000	+	4.00000i	−0.383571	+	0.191785i
$436$		0			0
$437$			6.00000i			0.287019i
$438$		0			0
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$			30.0000i			1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$			30.0000i			1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$		0			0
$445$	−10.0000	−	20.0000i	−0.474045	−	0.948091i
$446$		0			0
$447$			28.0000i			1.32435i
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$	32.0000			1.49363
$460$		0			0
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$			18.0000i			0.836531i	0.908325	+	0.418265i	$0.137362\pi$
$463$			18.0000i			0.836531i	−0.908325	+	0.418265i	$0.862638\pi$
$464$		0			0
$465$	32.0000	−	16.0000i	1.48396	−	0.741982i
$466$		0			0
$467$		−	26.0000i		−	1.20314i	−0.798821	−	0.601568i	$-0.794543\pi$
$467$		−	26.0000i		−	1.20314i	0.798821	−	0.601568i	$-0.205457\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$	8.00000			0.368621
$472$		0			0
$473$			40.0000i			1.83920i
$474$		0			0
$475$	3.00000	−	4.00000i	0.137649	−	0.183533i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			24.0000i			1.09204i
$484$		0			0
$485$	−8.00000	+	4.00000i	−0.363261	+	0.181631i
$486$		0			0
$487$		−	10.0000i		−	0.453143i	−0.973995	−	0.226572i	$-0.927248\pi$
$487$		−	10.0000i		−	0.453143i	0.973995	−	0.226572i	$-0.0727517\pi$
$488$		0			0
$489$	36.0000			1.62798
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$			16.0000i			0.720604i
$494$		0			0
$495$	−4.00000	−	8.00000i	−0.179787	−	0.359573i
$496$		0			0
$497$		−	32.0000i		−	1.43540i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	−12.0000			−0.536120
$502$		0			0
$503$			26.0000i			1.15928i	0.814872	+	0.579641i	$0.196807\pi$
$503$			26.0000i			1.15928i	−0.814872	+	0.579641i	$0.803193\pi$
$504$		0			0
$505$	−6.00000	−	12.0000i	−0.266996	−	0.533993i
$506$		0			0
$507$			26.0000i			1.15470i
$508$		0			0
$509$	−2.00000			−0.0886484			−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000			−0.0886484			−0.0443242	+	0.999017i	$0.514113\pi$
$510$		0			0
$511$	32.0000			1.41560
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$	28.0000	−	14.0000i	1.23383	−	0.616914i
$516$		0			0
$517$			24.0000i			1.05552i
$518$		0			0
$519$	−48.0000			−2.10697
$520$		0			0
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$			2.00000i			0.0874539i	0.999044	+	0.0437269i	$0.0139232\pi$
$523$			2.00000i			0.0874539i	−0.999044	+	0.0437269i	$0.986077\pi$
$524$		0			0
$525$	12.0000	−	16.0000i	0.523723	−	0.698297i
$526$		0			0
$527$		−	64.0000i		−	2.78788i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.00000	−	2.00000i	0.172935	−	0.0864675i
$536$		0			0
$537$		−	24.0000i		−	1.03568i
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		−	52.0000i		−	2.23153i
$544$		0			0
$545$	2.00000	+	4.00000i	0.0856706	+	0.171341i
$546$		0			0
$547$		−	6.00000i		−	0.256541i	−0.991739	−	0.128271i	$-0.959057\pi$
$547$		−	6.00000i		−	0.256541i	0.991739	−	0.128271i	$-0.0409426\pi$
$548$		0			0
$549$	−6.00000			−0.256074
$550$		0			0
$551$	2.00000			0.0852029
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			28.0000i			1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$557$			28.0000i			1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−64.0000			−2.70208
$562$		0			0
$563$		−	6.00000i		−	0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$		−	6.00000i		−	0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$		0			0
$565$	−24.0000	+	12.0000i	−1.00969	+	0.504844i
$566$		0			0
$567$		−	22.0000i		−	0.923913i
$568$		0			0
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		−	32.0000i		−	1.33682i
$574$		0			0
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$		0			0
$579$	−8.00000			−0.332469
$580$		0			0
$581$	20.0000			0.829740
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$	−24.0000			−0.987228
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−16.0000	−	32.0000i	−0.655936	−	1.31187i
$596$		0			0
$597$			16.0000i			0.654836i
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$	−5.00000	−	10.0000i	−0.203279	−	0.406558i
$606$		0			0
$607$		−	2.00000i		−	0.0811775i	−0.999176	−	0.0405887i	$-0.987077\pi$
$607$		−	2.00000i		−	0.0811775i	0.999176	−	0.0405887i	$-0.0129233\pi$
$608$		0			0
$609$	8.00000			0.324176
$610$		0			0
$611$		0			0
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$		0			0
$615$	−24.0000	+	12.0000i	−0.967773	+	0.483887i
$616$		0			0
$617$		−	16.0000i		−	0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$		−	16.0000i		−	0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$		0			0
$619$	−36.0000			−1.44696			−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000			−1.44696			−0.723481	+	0.690344i	$0.757459\pi$
$620$		0			0
$621$	24.0000			0.963087
$622$		0			0
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			8.00000i			0.319489i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		−	40.0000i		−	1.58986i
$634$		0			0
$635$	12.0000	−	6.00000i	0.476205	−	0.238103i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	16.0000			0.632950
$640$		0			0
$641$	34.0000			1.34292			0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000			1.34292			0.671460	+	0.741041i	$0.265668\pi$
$642$		0			0
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		0			0
$645$	−20.0000	−	40.0000i	−0.787499	−	1.57500i
$646$		0			0
$647$		−	38.0000i		−	1.49393i	−0.664861	−	0.746967i	$-0.731509\pi$
$647$		−	38.0000i		−	1.49393i	0.664861	−	0.746967i	$-0.268491\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	−32.0000			−1.25418
$652$		0			0
$653$			28.0000i			1.09572i	0.836569	+	0.547862i	$0.184558\pi$
$653$			28.0000i			1.09572i	−0.836569	+	0.547862i	$0.815442\pi$
$654$		0			0
$655$	4.00000	+	8.00000i	0.156293	+	0.312586i
$656$		0			0
$657$			16.0000i			0.624219i
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	38.0000			1.47803			0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000			1.47803			0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.00000	+	2.00000i	−0.155113	+	0.0775567i
$666$		0			0
$667$			12.0000i			0.464642i
$668$		0			0
$669$	52.0000			2.01044
$670$		0			0
$671$	−24.0000			−0.926510
$672$		0			0
$673$			28.0000i			1.07932i	0.841883	+	0.539660i	$0.181447\pi$
$673$			28.0000i			1.07932i	−0.841883	+	0.539660i	$0.818553\pi$
$674$		0			0
$675$	−16.0000	−	12.0000i	−0.615840	−	0.461880i
$676$		0			0
$677$		−	24.0000i		−	0.922395i	−0.887298	−	0.461197i	$-0.847420\pi$
$677$		−	24.0000i		−	0.922395i	0.887298	−	0.461197i	$-0.152580\pi$
$678$		0			0
$679$	8.00000			0.307012
$680$		0			0
$681$	−20.0000			−0.766402
$682$		0			0
$683$		−	30.0000i		−	1.14792i	−0.818884	−	0.573959i	$-0.805407\pi$
$683$		−	30.0000i		−	1.14792i	0.818884	−	0.573959i	$-0.194593\pi$
$684$		0			0
$685$	−32.0000	+	16.0000i	−1.22266	+	0.611329i
$686$		0			0
$687$		−	52.0000i		−	1.98392i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$			8.00000i			0.303895i
$694$		0			0
$695$	4.00000	+	8.00000i	0.151729	+	0.303457i
$696$		0			0
$697$			48.0000i			1.81813i
$698$		0			0
$699$	−16.0000			−0.605176
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−12.0000	−	24.0000i	−0.451946	−	0.903892i
$706$		0			0
$707$			12.0000i			0.451306i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	40.0000			1.49175			0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000			1.49175			0.745874	+	0.666087i	$0.232032\pi$
$720$		0			0
$721$	−28.0000			−1.04277
$722$		0			0
$723$			4.00000i			0.148762i
$724$		0			0
$725$	6.00000	−	8.00000i	0.222834	−	0.297113i
$726$		0			0
$727$			2.00000i			0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$			2.00000i			0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	−80.0000			−2.95891
$732$		0			0
$733$			4.00000i			0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$			4.00000i			0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$		0			0
$735$	12.0000	−	6.00000i	0.442627	−	0.221313i
$736$		0			0
$737$		−	8.00000i		−	0.294684i
$738$		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
$741$		0			0
$742$		0			0
$743$			6.00000i			0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$			6.00000i			0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$		0			0
$745$	−14.0000	−	28.0000i	−0.512920	−	1.02584i
$746$		0			0
$747$			10.0000i			0.365881i
$748$		0			0
$749$	−4.00000			−0.146157
$750$		0			0
$751$	48.0000			1.75154			0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000			1.75154			0.875772	+	0.482724i	$0.160353\pi$
$752$		0			0
$753$		−	56.0000i		−	2.04075i
$754$		0			0
$755$	−8.00000	−	16.0000i	−0.291150	−	0.582300i
$756$		0			0
$757$		−	36.0000i		−	1.30844i	−0.756303	−	0.654221i	$-0.772997\pi$
$757$		−	36.0000i		−	1.30844i	0.756303	−	0.654221i	$-0.227003\pi$
$758$		0			0
$759$	−48.0000			−1.74229
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$		−	4.00000i		−	0.144810i
$764$		0			0
$765$	16.0000	−	8.00000i	0.578481	−	0.289241i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	−40.0000			−1.44056
$772$		0			0
$773$		−	40.0000i		−	1.43870i	−0.694648	−	0.719350i	$-0.744440\pi$
$773$		−	40.0000i		−	1.43870i	0.694648	−	0.719350i	$-0.255560\pi$
$774$		0			0
$775$	−24.0000	+	32.0000i	−0.862105	+	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.00000			0.214972
$780$		0			0
$781$	64.0000			2.29010
$782$		0			0
$783$		−	8.00000i		−	0.285897i
$784$		0			0
$785$	−8.00000	+	4.00000i	−0.285532	+	0.142766i
$786$		0			0
$787$		−	22.0000i		−	0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$		−	22.0000i		−	0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$		0			0
$789$	12.0000			0.427211
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			16.0000i			0.566749i	0.959009	+	0.283375i	$0.0914540\pi$
$797$			16.0000i			0.566749i	−0.959009	+	0.283375i	$0.908546\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$			64.0000i			2.25851i
$804$		0			0
$805$	−12.0000	−	24.0000i	−0.422944	−	0.845889i
$806$		0			0
$807$		−	36.0000i		−	1.26726i
$808$		0			0
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		0			0
$815$	−36.0000	+	18.0000i	−1.26102	+	0.630512i
$816$		0			0
$817$			10.0000i			0.349856i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	30.0000			1.04701			0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000			1.04701			0.523504	+	0.852023i	$0.324625\pi$
$822$		0			0
$823$			2.00000i			0.0697156i	0.999392	+	0.0348578i	$0.0110978\pi$
$823$			2.00000i			0.0697156i	−0.999392	+	0.0348578i	$0.988902\pi$
$824$		0			0
$825$	32.0000	+	24.0000i	1.11410	+	0.835573i
$826$		0			0
$827$		−	6.00000i		−	0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$		−	6.00000i		−	0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	24.0000			0.832551
$832$		0			0
$833$		−	24.0000i		−	0.831551i
$834$		0			0
$835$	12.0000	−	6.00000i	0.415277	−	0.207639i
$836$		0			0
$837$			32.0000i			1.10608i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	12.0000i		−	0.413302i
$844$		0			0
$845$	−13.0000	−	26.0000i	−0.447214	−	0.894427i
$846$		0			0
$847$			10.0000i			0.343604i
$848$		0			0
$849$	−28.0000			−0.960958
$850$		0			0
$851$		0			0
$852$		0			0
$853$			44.0000i			1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$			44.0000i			1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$		0			0
$855$	−1.00000	−	2.00000i	−0.0341993	−	0.0683986i
$856$		0			0
$857$			12.0000i			0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$			12.0000i			0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$		0			0
$859$	−12.0000			−0.409435			−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000			−0.409435			−0.204717	+	0.978821i	$0.565628\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$		0			0
$863$		−	50.0000i		−	1.70202i	−0.525150	−	0.851010i	$-0.675991\pi$
$863$		−	50.0000i		−	1.70202i	0.525150	−	0.851010i	$-0.324009\pi$
$864$		0			0
$865$	48.0000	−	24.0000i	1.63205	−	0.816024i
$866$		0			0
$867$		−	94.0000i		−	3.19241i
$868$		0			0
$869$	−32.0000			−1.08553
$870$		0			0
$871$		0			0
$872$		0			0
$873$			4.00000i			0.135379i
$874$		0			0
$875$	−4.00000	+	22.0000i	−0.135225	+	0.743736i
$876$		0			0
$877$		−	32.0000i		−	1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$		−	32.0000i		−	1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$		0			0
$879$	32.0000			1.07933
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		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
$885$	−16.0000	+	8.00000i	−0.537834	+	0.268917i
$886$		0			0
$887$			30.0000i			1.00730i	0.863907	+	0.503651i	$0.168010\pi$
$887$			30.0000i			1.00730i	−0.863907	+	0.503651i	$0.831990\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$		0			0
$891$	44.0000			1.47406
$892$		0			0
$893$			6.00000i			0.200782i
$894$		0			0
$895$	12.0000	+	24.0000i	0.401116	+	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16.0000			−0.533630
$900$		0			0
$901$		0			0
$902$		0			0
$903$			40.0000i			1.33112i
$904$		0			0
$905$	26.0000	+	52.0000i	0.864269	+	1.72854i
$906$		0			0
$907$			2.00000i			0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$			2.00000i			0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$			40.0000i			1.32381i
$914$		0			0
$915$	24.0000	−	12.0000i	0.793416	−	0.396708i
$916$		0			0
$917$		−	8.00000i		−	0.264183i
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$	12.0000			0.395413
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	14.0000i		−	0.459820i
$928$		0			0
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$	−3.00000			−0.0983210
$932$		0			0
$933$		0			0
$934$		0			0
$935$	64.0000	−	32.0000i	2.09302	−	1.04651i
$936$		0			0
$937$			48.0000i			1.56809i	0.620703	+	0.784046i	$0.286847\pi$
$937$			48.0000i			1.56809i	−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$	16.0000			0.522140
$940$		0			0
$941$	−50.0000			−1.62995			−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000			−1.62995			−0.814977	+	0.579494i	$0.803250\pi$
$942$		0			0
$943$			36.0000i			1.17232i
$944$		0			0
$945$	8.00000	+	16.0000i	0.260240	+	0.520480i
$946$		0			0
$947$		−	26.0000i		−	0.844886i	−0.906389	−	0.422443i	$-0.861173\pi$
$947$		−	26.0000i		−	0.844886i	0.906389	−	0.422443i	$-0.138827\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	16.0000			0.518836
$952$		0			0
$953$		−	12.0000i		−	0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$		−	12.0000i		−	0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$		0			0
$955$	16.0000	+	32.0000i	0.517748	+	1.03550i
$956$		0			0
$957$			16.0000i			0.517207i
$958$		0			0
$959$	32.0000			1.03333
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	2.00000i		−	0.0644491i
$964$		0			0
$965$	8.00000	−	4.00000i	0.257529	−	0.128765i
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		0			0
$969$	−16.0000			−0.513994
$970$		0			0
$971$	−4.00000			−0.128366			−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000			−0.128366			−0.0641831	+	0.997938i	$0.520444\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			52.0000i			1.66363i	0.555055	+	0.831814i	$0.312697\pi$
$977$			52.0000i			1.66363i	−0.555055	+	0.831814i	$0.687303\pi$
$978$		0			0
$979$	−40.0000			−1.27841
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$			6.00000i			0.191370i	0.995412	+	0.0956851i	$0.0305042\pi$
$983$			6.00000i			0.191370i	−0.995412	+	0.0956851i	$0.969496\pi$
$984$		0			0
$985$	24.0000	−	12.0000i	0.764704	−	0.382352i
$986$		0			0
$987$			24.0000i			0.763928i
$988$		0			0
$989$	−60.0000			−1.90789
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		−	24.0000i		−	0.761617i
$994$		0			0
$995$	−8.00000	−	16.0000i	−0.253617	−	0.507234i
$996$		0			0
$997$		−	28.0000i		−	0.886769i	−0.896332	−	0.443384i	$-0.853778\pi$
$997$		−	28.0000i		−	0.886769i	0.896332	−	0.443384i	$-0.146222\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.a.609.2		2
4.3	odd	2		760.2.d.a.609.1	✓	2
5.2	odd	4		7600.2.a.q.1.1		1
5.3	odd	4		7600.2.a.e.1.1		1
5.4	even	2	inner	1520.2.d.a.609.1		2
20.3	even	4		3800.2.a.g.1.1		1
20.7	even	4		3800.2.a.c.1.1		1
20.19	odd	2		760.2.d.a.609.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.a.609.1	✓	2	4.3	odd	2
760.2.d.a.609.2	yes	2	20.19	odd	2
1520.2.d.a.609.1		2	5.4	even	2	inner
1520.2.d.a.609.2		2	1.1	even	1	trivial
3800.2.a.c.1.1		1	20.7	even	4
3800.2.a.g.1.1		1	20.3	even	4
7600.2.a.e.1.1		1	5.3	odd	4
7600.2.a.q.1.1		1	5.2	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +(4.00000 - 2.00000i) q^{15} -8.00000i q^{17} -1.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} -8.00000i q^{33} +(4.00000 - 2.00000i) q^{35} -6.00000 q^{41} -10.0000i q^{43} +(1.00000 + 2.00000i) q^{45} -6.00000i q^{47} +3.00000 q^{49} +16.0000 q^{51} +(4.00000 + 8.00000i) q^{55} -2.00000i q^{57} -4.00000 q^{59} +6.00000 q^{61} -2.00000i q^{63} +2.00000i q^{67} +12.0000 q^{69} -16.0000 q^{71} -16.0000i q^{73} +(-8.00000 - 6.00000i) q^{75} -8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} -10.0000i q^{83} +(-16.0000 + 8.00000i) q^{85} -4.00000i q^{87} +10.0000 q^{89} +16.0000i q^{93} +(1.00000 + 2.00000i) q^{95} -4.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{9} - 8 q^{11} + 8 q^{15} - 2 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} + 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 32 q^{51} + 8 q^{55} - 8 q^{59} + 12 q^{61} + 24 q^{69} - 32 q^{71} - 16 q^{75} + 16 q^{79} - 22 q^{81} - 32 q^{85} + 20 q^{89} + 2 q^{95} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 - 8 * q^11 + 8 * q^15 - 2 * q^19 - 8 * q^21 - 6 * q^25 - 4 * q^29 + 16 * q^31 + 8 * q^35 - 12 * q^41 + 2 * q^45 + 6 * q^49 + 32 * q^51 + 8 * q^55 - 8 * q^59 + 12 * q^61 + 24 * q^69 - 32 * q^71 - 16 * q^75 + 16 * q^79 - 22 * q^81 - 32 * q^85 + 20 * q^89 + 2 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more