LMFDB - Embedded newform 1520.2.g.g.1519.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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([1, 0, 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("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.g (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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 271x^{12} - 2000x^{10} + 10645x^{8} - 29570x^{6} + 58816x^{4} - 56840x^{2} + 38416 $ x^16 - 20x^14 + 271x^12 - 2000x^10 + 10645x^8 - 29570x^6 + 58816x^4 - 56840*x^2 + 38416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.13
Root			$-2.38641 + 1.37780i$ of defining polynomial
Character	$\chi$	$=$	1520.1519
Dual form			1520.2.g.g.1519.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.75559i q^{3} +(2.03407 - 0.928731i) q^{5} -4.29083 q^{7} -4.59328 q^{9} +O(q^{10})$	$q+2.75559i q^{3} +(2.03407 - 0.928731i) q^{5} -4.29083 q^{7} -4.59328 q^{9} -3.89699i q^{11} -1.27282 q^{13} +(2.55920 + 5.60508i) q^{15} -3.77822i q^{17} +(-4.27492 - 0.851518i) q^{19} -11.8238i q^{21} +1.36603 q^{23} +(3.27492 - 3.77822i) q^{25} -4.39042i q^{27} -4.02978i q^{29} +10.0681 q^{31} +10.7385 q^{33} +(-8.72787 + 3.98503i) q^{35} +8.39208 q^{37} -3.50738i q^{39} -6.20899i q^{41} -0.671571 q^{43} +(-9.34307 + 4.26592i) q^{45} -9.54564 q^{47} +11.4112 q^{49} +10.4112 q^{51} -9.95086 q^{53} +(-3.61926 - 7.92677i) q^{55} +(2.34643 - 11.7799i) q^{57} -3.08836 q^{59} -6.74979 q^{61} +19.7090 q^{63} +(-2.58902 + 1.18211i) q^{65} -13.4147i q^{67} +3.76421i q^{69} -10.0681 q^{71} -1.28769i q^{73} +(10.4112 + 9.02433i) q^{75} +16.7213i q^{77} +4.94974 q^{79} -1.68164 q^{81} +11.1273 q^{83} +(-3.50895 - 7.68517i) q^{85} +11.1044 q^{87} +0.805775i q^{89} +5.46147 q^{91} +27.7437i q^{93} +(-9.48633 + 2.23820i) q^{95} +1.15356 q^{97} +17.9000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) $ 16 * q - 32 * q^9 + 32 * q^15 - 8 * q^19 - 8 * q^25 + 96 * q^31 - 24 * q^45 - 8 * q^49 - 24 * q^51 - 72 * q^59 - 24 * q^61 - 96 * q^71 - 24 * q^75 + 32 * q^79 - 8 * q^81 - 56 * q^91

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
$3$			2.75559i			1.59094i	0.605993	+	0.795470i	$0.292776\pi$
$3$			2.75559i			1.59094i	−0.605993	+	0.795470i	$0.707224\pi$
$4$		0			0
$5$	2.03407	−	0.928731i	0.909666	−	0.415341i
$5$	2.03407	−	0.928731i	0.909666	−	0.415341i
$6$		0			0
$7$	−4.29083			−1.62178			−0.810890	−	0.585198i	$-0.801017\pi$
$7$	−4.29083			−1.62178			−0.810890	+	0.585198i	$0.801017\pi$
$8$		0			0
$9$	−4.59328			−1.53109
$10$		0			0
$11$		−	3.89699i		−	1.17499i	−0.809229	−	0.587494i	$-0.800115\pi$
$11$		−	3.89699i		−	1.17499i	0.809229	−	0.587494i	$-0.199885\pi$
$12$		0			0
$13$	−1.27282			−0.353018			−0.176509	−	0.984299i	$-0.556480\pi$
$13$	−1.27282			−0.353018			−0.176509	+	0.984299i	$0.556480\pi$
$14$		0			0
$15$	2.55920	+	5.60508i	0.660783	+	1.44722i
$16$		0			0
$17$		−	3.77822i		−	0.916352i	−0.888861	−	0.458176i	$-0.848503\pi$
$17$		−	3.77822i		−	0.916352i	0.888861	−	0.458176i	$-0.151497\pi$
$18$		0			0
$19$	−4.27492	−	0.851518i	−0.980733	−	0.195352i
$19$	−4.27492	−	0.851518i	−0.980733	−	0.195352i
$20$		0			0
$21$		−	11.8238i		−	2.58016i
$22$		0			0
$23$	1.36603			0.284836			0.142418	−	0.989807i	$-0.454512\pi$
$23$	1.36603			0.284836			0.142418	+	0.989807i	$0.454512\pi$
$24$		0			0
$25$	3.27492	−	3.77822i	0.654983	−	0.755643i
$26$		0			0
$27$		−	4.39042i		−	0.844936i
$28$		0			0
$29$		−	4.02978i		−	0.748311i	−0.927366	−	0.374156i	$-0.877933\pi$
$29$		−	4.02978i		−	0.748311i	0.927366	−	0.374156i	$-0.122067\pi$
$30$		0			0
$31$	10.0681			1.80829			0.904146	−	0.427223i	$-0.140508\pi$
$31$	10.0681			1.80829			0.904146	+	0.427223i	$0.140508\pi$
$32$		0			0
$33$	10.7385			1.86934
$34$		0			0
$35$	−8.72787	+	3.98503i	−1.47528	+	0.673592i
$36$		0			0
$37$	8.39208			1.37965			0.689825	−	0.723976i	$-0.257688\pi$
$37$	8.39208			1.37965			0.689825	+	0.723976i	$0.257688\pi$
$38$		0			0
$39$		−	3.50738i		−	0.561630i
$40$		0			0
$41$		−	6.20899i		−	0.969681i	−0.874603	−	0.484840i	$-0.838878\pi$
$41$		−	6.20899i		−	0.969681i	0.874603	−	0.484840i	$-0.161122\pi$
$42$		0			0
$43$	−0.671571			−0.102414			−0.0512068	−	0.998688i	$-0.516307\pi$
$43$	−0.671571			−0.102414			−0.0512068	+	0.998688i	$0.516307\pi$
$44$		0			0
$45$	−9.34307	+	4.26592i	−1.39278	+	0.635926i
$46$		0			0
$47$	−9.54564			−1.39238			−0.696188	−	0.717860i	$-0.745122\pi$
$47$	−9.54564			−1.39238			−0.696188	+	0.717860i	$0.745122\pi$
$48$		0			0
$49$	11.4112			1.63017
$50$		0			0
$51$	10.4112			1.45786
$52$		0			0
$53$	−9.95086			−1.36686			−0.683428	−	0.730018i	$-0.739512\pi$
$53$	−9.95086			−1.36686			−0.683428	+	0.730018i	$0.739512\pi$
$54$		0			0
$55$	−3.61926	−	7.92677i	−0.488021	−	1.06885i
$56$		0			0
$57$	2.34643	−	11.7799i	0.310793	−	1.56029i
$58$		0			0
$59$	−3.08836			−0.402071			−0.201035	−	0.979584i	$-0.564431\pi$
$59$	−3.08836			−0.402071			−0.201035	+	0.979584i	$0.564431\pi$
$60$		0			0
$61$	−6.74979			−0.864222			−0.432111	−	0.901820i	$-0.642231\pi$
$61$	−6.74979			−0.864222			−0.432111	+	0.901820i	$0.642231\pi$
$62$		0			0
$63$	19.7090			2.48310
$64$		0			0
$65$	−2.58902	+	1.18211i	−0.321128	+	0.146623i
$66$		0			0
$67$		−	13.4147i		−	1.63887i	−0.573171	−	0.819436i	$-0.694287\pi$
$67$		−	13.4147i		−	1.63887i	0.573171	−	0.819436i	$-0.305713\pi$
$68$		0			0
$69$			3.76421i			0.453157i
$70$		0			0
$71$	−10.0681			−1.19487			−0.597435	−	0.801918i	$-0.703813\pi$
$71$	−10.0681			−1.19487			−0.597435	+	0.801918i	$0.703813\pi$
$72$		0			0
$73$		−	1.28769i		−	0.150713i	−0.997157	−	0.0753565i	$-0.975991\pi$
$73$		−	1.28769i		−	0.150713i	0.997157	−	0.0753565i	$-0.0240095\pi$
$74$		0			0
$75$	10.4112	+	9.02433i	1.20218	+	1.04204i
$76$		0			0
$77$			16.7213i			1.90557i
$78$		0			0
$79$	4.94974			0.556890			0.278445	−	0.960452i	$-0.410181\pi$
$79$	4.94974			0.556890			0.278445	+	0.960452i	$0.410181\pi$
$80$		0			0
$81$	−1.68164			−0.186849
$82$		0			0
$83$	11.1273			1.22138			0.610690	−	0.791870i	$-0.290892\pi$
$83$	11.1273			1.22138			0.610690	+	0.791870i	$0.290892\pi$
$84$		0			0
$85$	−3.50895	−	7.68517i	−0.380599	−	0.833574i
$86$		0			0
$87$	11.1044			1.19052
$88$		0			0
$89$			0.805775i			0.0854120i	0.999088	+	0.0427060i	$0.0135979\pi$
$89$			0.805775i			0.0854120i	−0.999088	+	0.0427060i	$0.986402\pi$
$90$		0			0
$91$	5.46147			0.572518
$92$		0			0
$93$			27.7437i			2.87689i
$94$		0			0
$95$	−9.48633	+	2.23820i	−0.973277	+	0.229634i
$96$		0			0
$97$	1.15356			0.117127			0.0585633	−	0.998284i	$-0.481348\pi$
$97$	1.15356			0.117127			0.0585633	+	0.998284i	$0.481348\pi$
$98$		0			0
$99$			17.9000i			1.79901i
$100$		0			0
$101$	−2.19996			−0.218904			−0.109452	−	0.993992i	$-0.534910\pi$
$101$	−2.19996			−0.218904			−0.109452	+	0.993992i	$0.534910\pi$
$102$		0			0
$103$		−	4.06414i		−	0.400452i	−0.979750	−	0.200226i	$-0.935832\pi$
$103$		−	4.06414i		−	0.400452i	0.979750	−	0.200226i	$-0.0641676\pi$
$104$		0			0
$105$	−10.9811	−	24.0504i	−1.07165	−	2.34708i
$106$		0			0
$107$			10.1451i			0.980763i	0.871508	+	0.490382i	$0.163143\pi$
$107$			10.1451i			0.980763i	−0.871508	+	0.490382i	$0.836857\pi$
$108$		0			0
$109$			11.0445i			1.05788i	0.848661	+	0.528938i	$0.177409\pi$
$109$			11.0445i			1.05788i	−0.848661	+	0.528938i	$0.822591\pi$
$110$		0			0
$111$			23.1251i			2.19494i
$112$		0			0
$113$	2.34643			0.220734			0.110367	−	0.993891i	$-0.464797\pi$
$113$	2.34643			0.220734			0.110367	+	0.993891i	$0.464797\pi$
$114$		0			0
$115$	2.77860	−	1.26867i	0.259105	−	0.118304i
$116$		0			0
$117$	5.84643			0.540503
$118$		0			0
$119$			16.2117i			1.48612i
$120$		0			0
$121$	−4.18655			−0.380596
$122$		0			0
$123$	17.1094			1.54270
$124$		0			0
$125$	3.15248	−	10.7267i	0.281966	−	0.959424i
$126$		0			0
$127$			7.88479i			0.699662i	0.936813	+	0.349831i	$0.113761\pi$
$127$			7.88479i			0.699662i	−0.936813	+	0.349831i	$0.886239\pi$
$128$		0			0
$129$		−	1.85057i		−	0.162934i
$130$		0			0
$131$		−	20.1834i		−	1.76343i	−0.471778	−	0.881717i	$-0.656388\pi$
$131$		−	20.1834i		−	1.76343i	0.471778	−	0.881717i	$-0.343612\pi$
$132$		0			0
$133$	18.3429	+	3.65372i	1.59053	+	0.316818i
$134$		0			0
$135$	−4.07752	−	8.93043i	−0.350937	−	0.768609i
$136$		0			0
$137$		−	15.0496i		−	1.28577i	−0.765962	−	0.642886i	$-0.777737\pi$
$137$		−	15.0496i		−	1.28577i	0.765962	−	0.642886i	$-0.222263\pi$
$138$		0			0
$139$		−	15.3892i		−	1.30529i	−0.757663	−	0.652646i	$-0.773659\pi$
$139$		−	15.3892i		−	1.30529i	0.757663	−	0.652646i	$-0.226341\pi$
$140$		0			0
$141$		−	26.3039i		−	2.21519i
$142$		0			0
$143$			4.96019i			0.414792i
$144$		0			0
$145$	−3.74258	−	8.19687i	−0.310805	−	0.680713i
$146$		0			0
$147$			31.4446i			2.59351i
$148$		0			0
$149$	−4.28173			−0.350773			−0.175387	−	0.984500i	$-0.556118\pi$
$149$	−4.28173			−0.350773			−0.175387	+	0.984500i	$0.556118\pi$
$150$		0			0
$151$	−20.8224			−1.69451			−0.847253	−	0.531190i	$-0.821745\pi$
$151$	−20.8224			−1.69451			−0.847253	+	0.531190i	$0.821745\pi$
$152$		0			0
$153$			17.3544i			1.40302i
$154$		0			0
$155$	20.4794	−	9.35060i	1.64494	−	0.751058i
$156$		0			0
$157$			0.248996i			0.0198721i	0.999951	+	0.00993604i	$0.00316279\pi$
$157$			0.248996i			0.0198721i	−0.999951	+	0.00993604i	$0.996837\pi$
$158$		0			0
$159$		−	27.4205i		−	2.17459i
$160$		0			0
$161$	−5.86138			−0.461941
$162$		0			0
$163$	9.83806			0.770576			0.385288	−	0.922796i	$-0.374102\pi$
$163$	9.83806			0.770576			0.385288	+	0.922796i	$0.374102\pi$
$164$		0			0
$165$	21.8429	−	9.97319i	1.70047	−	0.776412i
$166$		0			0
$167$		−	21.3365i		−	1.65107i	−0.564354	−	0.825533i	$-0.690875\pi$
$167$		−	21.3365i		−	1.65107i	0.564354	−	0.825533i	$-0.309125\pi$
$168$		0			0
$169$	−11.3799			−0.875378
$170$		0			0
$171$	19.6359	+	3.91126i	1.50159	+	0.299101i
$172$		0			0
$173$	−12.3772			−0.941025			−0.470512	−	0.882393i	$-0.655931\pi$
$173$	−12.3772			−0.941025			−0.470512	+	0.882393i	$0.655931\pi$
$174$		0			0
$175$	−14.0521	+	16.2117i	−1.06224	+	1.22549i
$176$		0			0
$177$		−	8.51027i		−	0.639671i
$178$		0			0
$179$	6.63672			0.496052			0.248026	−	0.968753i	$-0.420218\pi$
$179$	6.63672			0.496052			0.248026	+	0.968753i	$0.420218\pi$
$180$		0			0
$181$			24.4268i			1.81563i	0.419375	+	0.907813i	$0.362249\pi$
$181$			24.4268i			1.81563i	−0.419375	+	0.907813i	$0.637751\pi$
$182$		0			0
$183$		−	18.5997i		−	1.37493i
$184$		0			0
$185$	17.0701	−	7.79399i	1.25502	−	0.573025i
$186$		0			0
$187$	−14.7237			−1.07670
$188$		0			0
$189$			18.8385i			1.37030i
$190$		0			0
$191$		−	9.07593i		−	0.656711i	−0.944554	−	0.328356i	$-0.893506\pi$
$191$		−	9.07593i		−	0.656711i	0.944554	−	0.328356i	$-0.106494\pi$
$192$		0			0
$193$	11.5096			0.828482			0.414241	−	0.910167i	$-0.364047\pi$
$193$	11.5096			0.828482			0.414241	+	0.910167i	$0.364047\pi$
$194$		0			0
$195$	−3.25741	−	7.13427i	−0.233268	−	0.510896i
$196$		0			0
$197$			7.81391i			0.556718i	0.960477	+	0.278359i	$0.0897905\pi$
$197$			7.81391i			0.556718i	−0.960477	+	0.278359i	$0.910209\pi$
$198$		0			0
$199$			20.7206i			1.46884i	0.678694	+	0.734422i	$0.262546\pi$
$199$			20.7206i			1.46884i	−0.678694	+	0.734422i	$0.737454\pi$
$200$		0			0
$201$	36.9655			2.60735
$202$		0			0
$203$			17.2911i			1.21360i
$204$		0			0
$205$	−5.76648	−	12.6295i	−0.402748	−	0.882085i
$206$		0			0
$207$	−6.27453			−0.436110
$208$		0			0
$209$	−3.31836	+	16.6593i	−0.229536	+	1.15235i
$210$		0			0
$211$	18.2749			1.25810			0.629049	−	0.777366i	$-0.283445\pi$
$211$	18.2749			1.25810			0.629049	+	0.777366i	$0.283445\pi$
$212$		0			0
$213$		−	27.7437i		−	1.90097i
$214$		0			0
$215$	−1.36603	+	0.623709i	−0.0931621	+	0.0425366i
$216$		0			0
$217$	−43.2007			−2.93266
$218$		0			0
$219$	3.54835			0.239776
$220$		0			0
$221$			4.80900i			0.323489i
$222$		0			0
$223$		−	14.5355i		−	0.973370i	−0.873578	−	0.486685i	$-0.838206\pi$
$223$		−	14.5355i		−	0.973370i	0.873578	−	0.486685i	$-0.161794\pi$
$224$		0			0
$225$	−15.0426	+	17.3544i	−1.00284	+	1.15696i
$226$		0			0
$227$		−	3.53192i		−	0.234422i	−0.993107	−	0.117211i	$-0.962605\pi$
$227$		−	3.53192i		−	0.234422i	0.993107	−	0.117211i	$-0.0373954\pi$
$228$		0			0
$229$	13.2315			0.874360			0.437180	−	0.899374i	$-0.355977\pi$
$229$	13.2315			0.874360			0.437180	+	0.899374i	$0.355977\pi$
$230$		0			0
$231$	−46.0771			−3.03165
$232$		0			0
$233$		−	16.2149i		−	1.06227i	−0.847287	−	0.531135i	$-0.821766\pi$
$233$		−	16.2149i		−	1.06227i	0.847287	−	0.531135i	$-0.178234\pi$
$234$		0			0
$235$	−19.4165	+	8.86534i	−1.26660	+	0.578311i
$236$		0			0
$237$			13.6395i			0.885979i
$238$		0			0
$239$		−	18.7805i		−	1.21481i	−0.794393	−	0.607405i	$-0.792211\pi$
$239$		−	18.7805i		−	1.21481i	0.794393	−	0.607405i	$-0.207789\pi$
$240$		0			0
$241$			23.1339i			1.49018i	0.666961	+	0.745092i	$0.267595\pi$
$241$			23.1339i			1.49018i	−0.666961	+	0.745092i	$0.732405\pi$
$242$		0			0
$243$		−	17.8052i		−	1.14220i
$244$		0			0
$245$	23.2113	−	10.5979i	1.48291	−	0.677078i
$246$		0			0
$247$	5.44122	+	1.08383i	0.346216	+	0.0689626i
$248$		0			0
$249$			30.6623i			1.94314i
$250$		0			0
$251$			19.5803i			1.23590i	0.786219	+	0.617948i	$0.212036\pi$
$251$			19.5803i			1.23590i	−0.786219	+	0.617948i	$0.787964\pi$
$252$		0			0
$253$		−	5.32339i		−	0.334679i
$254$		0			0
$255$	21.1772	−	9.66922i	1.32617	−	0.605510i
$256$		0			0
$257$	−8.47882			−0.528894			−0.264447	−	0.964400i	$-0.585189\pi$
$257$	−8.47882			−0.528894			−0.264447	+	0.964400i	$0.585189\pi$
$258$		0			0
$259$	−36.0090			−2.23749
$260$		0			0
$261$			18.5099i			1.14573i
$262$		0			0
$263$	−21.4834			−1.32472			−0.662361	−	0.749185i	$-0.730445\pi$
$263$	−21.4834			−1.32472			−0.662361	+	0.749185i	$0.730445\pi$
$264$		0			0
$265$	−20.2408	+	9.24167i	−1.24338	+	0.567711i
$266$		0			0
$267$	−2.22039			−0.135885
$268$		0			0
$269$			29.8831i			1.82200i	0.412403	+	0.911001i	$0.364690\pi$
$269$			29.8831i			1.82200i	−0.412403	+	0.911001i	$0.635310\pi$
$270$		0			0
$271$		−	23.6043i		−	1.43386i	−0.697147	−	0.716928i	$-0.745548\pi$
$271$		−	23.6043i		−	1.43386i	0.697147	−	0.716928i	$-0.254452\pi$
$272$		0			0
$273$			15.0496i			0.910842i
$274$		0			0
$275$	−14.7237	−	12.7623i	−0.887871	−	0.769597i
$276$		0			0
$277$		−	9.88282i		−	0.593801i	−0.954908	−	0.296901i	$-0.904047\pi$
$277$		−	9.88282i		−	0.593801i	0.954908	−	0.296901i	$-0.0959530\pi$
$278$		0			0
$279$	−46.2458			−2.76866
$280$		0			0
$281$		−	15.8801i		−	0.947327i	−0.880706	−	0.473664i	$-0.842931\pi$
$281$		−	15.8801i		−	0.947327i	0.880706	−	0.473664i	$-0.157069\pi$
$282$		0			0
$283$	−21.8562			−1.29922			−0.649608	−	0.760270i	$-0.725067\pi$
$283$	−21.8562			−1.29922			−0.649608	+	0.760270i	$0.725067\pi$
$284$		0			0
$285$	−6.16756	−	26.1404i	−0.365334	−	1.54843i
$286$		0			0
$287$			26.6417i			1.57261i
$288$		0			0
$289$	2.72508			0.160299
$290$		0			0
$291$			3.17875i			0.186342i
$292$		0			0
$293$	−19.3366			−1.12966			−0.564828	−	0.825209i	$-0.691057\pi$
$293$	−19.3366			−1.12966			−0.564828	+	0.825209i	$0.691057\pi$
$294$		0			0
$295$	−6.28196	+	2.86826i	−0.365750	+	0.166997i
$296$		0			0
$297$	−17.1094			−0.992789
$298$		0			0
$299$	−1.73871			−0.100552
$300$		0			0
$301$	2.88160			0.166092
$302$		0			0
$303$		−	6.06217i		−	0.348263i
$304$		0			0
$305$	−13.7296	+	6.26874i	−0.786153	+	0.358947i
$306$		0			0
$307$			11.5552i			0.659490i	0.944070	+	0.329745i	$0.106963\pi$
$307$			11.5552i			0.659490i	−0.944070	+	0.329745i	$0.893037\pi$
$308$		0			0
$309$	11.1991			0.637095
$310$		0			0
$311$		−	4.49415i		−	0.254840i	−0.991849	−	0.127420i	$-0.959330\pi$
$311$		−	4.49415i		−	0.254840i	0.991849	−	0.127420i	$-0.0406696\pi$
$312$		0			0
$313$		−	10.8367i		−	0.612524i	−0.951947	−	0.306262i	$-0.900922\pi$
$313$		−	10.8367i		−	0.612524i	0.951947	−	0.306262i	$-0.0990784\pi$
$314$		0			0
$315$	40.0895	−	18.3043i	2.25879	−	1.03133i
$316$		0			0
$317$	23.5603			1.32328			0.661639	−	0.749823i	$-0.269861\pi$
$317$	23.5603			1.32328			0.661639	+	0.749823i	$0.269861\pi$
$318$		0			0
$319$	−15.7040			−0.879257
$320$		0			0
$321$	−27.9557			−1.56034
$322$		0			0
$323$	−3.21722	+	16.1516i	−0.179011	+	0.898697i
$324$		0			0
$325$	−4.16839	+	4.80900i	−0.231221	+	0.266756i
$326$		0			0
$327$	−30.4342			−1.68302
$328$		0			0
$329$	40.9587			2.25813
$330$		0			0
$331$	−11.7609			−0.646436			−0.323218	−	0.946325i	$-0.604765\pi$
$331$	−11.7609			−0.646436			−0.323218	+	0.946325i	$0.604765\pi$
$332$		0			0
$333$	−38.5471			−2.11237
$334$		0			0
$335$	−12.4587	−	27.2866i	−0.680691	−	1.49083i
$336$		0			0
$337$	5.46399			0.297643			0.148821	−	0.988864i	$-0.452452\pi$
$337$	5.46399			0.297643			0.148821	+	0.988864i	$0.452452\pi$
$338$		0			0
$339$			6.46581i			0.351175i
$340$		0			0
$341$		−	39.2355i		−	2.12472i
$342$		0			0
$343$	−18.9278			−1.02200
$344$		0			0
$345$	3.49593	+	7.65667i	0.188215	+	0.412221i
$346$		0			0
$347$	6.05528			0.325065			0.162532	−	0.986703i	$-0.448034\pi$
$347$	6.05528			0.325065			0.162532	+	0.986703i	$0.448034\pi$
$348$		0			0
$349$	20.9093			1.11925			0.559625	−	0.828746i	$-0.310945\pi$
$349$	20.9093			1.11925			0.559625	+	0.828746i	$0.310945\pi$
$350$		0			0
$351$			5.58823i			0.298278i
$352$		0			0
$353$			17.3760i			0.924829i	0.886664	+	0.462415i	$0.153017\pi$
$353$			17.3760i			0.924829i	−0.886664	+	0.462415i	$0.846983\pi$
$354$		0			0
$355$	−20.4794	+	9.35060i	−1.08693	+	0.496278i
$356$		0			0
$357$	−44.6727			−2.36433
$358$		0			0
$359$			24.5861i			1.29760i	0.760957	+	0.648802i	$0.224730\pi$
$359$			24.5861i			1.29760i	−0.760957	+	0.648802i	$0.775270\pi$
$360$		0			0
$361$	17.5498	+	7.28034i	0.923675	+	0.383176i
$362$		0			0
$363$		−	11.5364i		−	0.605505i
$364$		0			0
$365$	−1.19592	−	2.61926i	−0.0625973	−	0.137099i
$366$		0			0
$367$	31.0227			1.61937			0.809685	−	0.586864i	$-0.199638\pi$
$367$	31.0227			1.61937			0.809685	+	0.586864i	$0.199638\pi$
$368$		0			0
$369$			28.5196i			1.48467i
$370$		0			0
$371$	42.6974			2.21674
$372$		0			0
$373$	−1.74201			−0.0901976			−0.0450988	−	0.998983i	$-0.514360\pi$
$373$	−1.74201			−0.0901976			−0.0450988	+	0.998983i	$0.514360\pi$
$374$		0			0
$375$	29.5584	+	8.68694i	1.52639	+	0.448592i
$376$		0			0
$377$			5.12920i			0.264167i
$378$		0			0
$379$	−8.37543			−0.430217			−0.215108	−	0.976590i	$-0.569010\pi$
$379$	−8.37543			−0.430217			−0.215108	+	0.976590i	$0.569010\pi$
$380$		0			0
$381$	−21.7273			−1.11312
$382$		0			0
$383$			29.7541i			1.52036i	0.649711	+	0.760182i	$0.274890\pi$
$383$			29.7541i			1.52036i	−0.649711	+	0.760182i	$0.725110\pi$
$384$		0			0
$385$	15.5296	+	34.0124i	0.791463	+	1.73343i
$386$		0			0
$387$	3.08471			0.156805
$388$		0			0
$389$	−25.6278			−1.29938			−0.649691	−	0.760199i	$-0.725102\pi$
$389$	−25.6278			−1.29938			−0.649691	+	0.760199i	$0.725102\pi$
$390$		0			0
$391$		−	5.16114i		−	0.261010i
$392$		0			0
$393$	55.6173			2.80552
$394$		0			0
$395$	10.0681	−	4.59698i	0.506584	−	0.231299i
$396$		0			0
$397$			28.4992i			1.43033i	0.698955	+	0.715166i	$0.253649\pi$
$397$			28.4992i			1.43033i	−0.698955	+	0.715166i	$0.746351\pi$
$398$		0			0
$399$	−10.0681	+	50.5456i	−0.504038	+	2.53045i
$400$		0			0
$401$		−	0.805775i		−	0.0402385i	−0.999798	−	0.0201192i	$-0.993595\pi$
$401$		−	0.805775i		−	0.0402385i	0.999798	−	0.0201192i	$-0.00640459\pi$
$402$		0			0
$403$	−12.8150			−0.638360
$404$		0			0
$405$	−3.42058	+	1.56179i	−0.169970	+	0.0776061i
$406$		0			0
$407$		−	32.7039i		−	1.62107i
$408$		0			0
$409$			18.6270i			0.921044i	0.887648	+	0.460522i	$0.152338\pi$
$409$			18.6270i			0.921044i	−0.887648	+	0.460522i	$0.847662\pi$
$410$		0			0
$411$	41.4705			2.04559
$412$		0			0
$413$	13.2516			0.652071
$414$		0			0
$415$	22.6338	−	10.3343i	1.11105	−	0.507290i
$416$		0			0
$417$	42.4063			2.07664
$418$		0			0
$419$			6.84362i			0.334333i	0.985929	+	0.167166i	$0.0534617\pi$
$419$			6.84362i			0.334333i	−0.985929	+	0.167166i	$0.946538\pi$
$420$		0			0
$421$			3.51613i			0.171366i	0.996322	+	0.0856829i	$0.0273072\pi$
$421$			3.51613i			0.171366i	−0.996322	+	0.0856829i	$0.972693\pi$
$422$		0			0
$423$	43.8458			2.13185
$424$		0			0
$425$	−14.2749	−	12.3733i	−0.692435	−	0.600195i
$426$		0			0
$427$	28.9622			1.40158
$428$		0			0
$429$	−13.6682			−0.659909
$430$		0			0
$431$	4.78109			0.230297			0.115148	−	0.993348i	$-0.463266\pi$
$431$	4.78109			0.230297			0.115148	+	0.993348i	$0.463266\pi$
$432$		0			0
$433$	−38.6339			−1.85663			−0.928313	−	0.371800i	$-0.878741\pi$
$433$	−38.6339			−1.85663			−0.928313	+	0.371800i	$0.878741\pi$
$434$		0			0
$435$	22.5872	−	10.3130i	1.08297	−	0.494472i
$436$		0			0
$437$	−5.83964	−	1.16319i	−0.279348	−	0.0556432i
$438$		0			0
$439$	−0.854791			−0.0407970			−0.0203985	−	0.999792i	$-0.506493\pi$
$439$	−0.854791			−0.0407970			−0.0203985	+	0.999792i	$0.506493\pi$
$440$		0			0
$441$	−52.4149			−2.49595
$442$		0			0
$443$	−32.4655			−1.54248			−0.771241	−	0.636544i	$-0.780363\pi$
$443$	−32.4655			−1.54248			−0.771241	+	0.636544i	$0.780363\pi$
$444$		0			0
$445$	0.748348	+	1.63901i	0.0354751	+	0.0776964i
$446$		0			0
$447$		−	11.7987i		−	0.558059i
$448$		0			0
$449$		−	27.8623i		−	1.31490i	−0.753496	−	0.657452i	$-0.771634\pi$
$449$		−	27.8623i		−	1.31490i	0.753496	−	0.657452i	$-0.228366\pi$
$450$		0			0
$451$	−24.1964			−1.13936
$452$		0			0
$453$		−	57.3781i		−	2.69586i
$454$		0			0
$455$	11.1090	−	5.07224i	0.520800	−	0.237790i
$456$		0			0
$457$		−	39.3358i		−	1.84005i	−0.391857	−	0.920026i	$-0.628167\pi$
$457$		−	39.3358i		−	1.84005i	0.391857	−	0.920026i	$-0.371833\pi$
$458$		0			0
$459$	−16.5879			−0.774259
$460$		0			0
$461$	12.9417			0.602754			0.301377	−	0.953505i	$-0.402554\pi$
$461$	12.9417			0.602754			0.301377	+	0.953505i	$0.402554\pi$
$462$		0			0
$463$	−2.91845			−0.135632			−0.0678160	−	0.997698i	$-0.521603\pi$
$463$	−2.91845			−0.135632			−0.0678160	+	0.997698i	$0.521603\pi$
$464$		0			0
$465$	25.7664	+	56.4327i	1.19489	+	2.61701i
$466$		0			0
$467$	17.5078			0.810167			0.405083	−	0.914280i	$-0.367243\pi$
$467$	17.5078			0.810167			0.405083	+	0.914280i	$0.367243\pi$
$468$		0			0
$469$			57.5604i			2.65789i
$470$		0			0
$471$	−0.686132			−0.0316153
$472$		0			0
$473$			2.61711i			0.120335i
$474$		0			0
$475$	−17.2172	+	13.3629i	−0.789980	+	0.613132i
$476$		0			0
$477$	45.7070			2.09278
$478$		0			0
$479$		−	17.4158i		−	0.795749i	−0.917440	−	0.397875i	$-0.869748\pi$
$479$		−	17.4158i		−	0.795749i	0.917440	−	0.397875i	$-0.130252\pi$
$480$		0			0
$481$	−10.6816			−0.487041
$482$		0			0
$483$		−	16.1516i		−	0.734921i
$484$		0			0
$485$	2.34643	−	1.07135i	0.106546	−	0.0486475i
$486$		0			0
$487$		−	0.982280i		−	0.0445114i	−0.999752	−	0.0222557i	$-0.992915\pi$
$487$		−	0.982280i		−	0.0445114i	0.999752	−	0.0222557i	$-0.00708479\pi$
$488$		0			0
$489$			27.1097i			1.22594i
$490$		0			0
$491$			9.43701i			0.425886i	0.977065	+	0.212943i	$0.0683049\pi$
$491$			9.43701i			0.425886i	−0.977065	+	0.212943i	$0.931695\pi$
$492$		0			0
$493$	−15.2254			−0.685717
$494$		0			0
$495$	16.6243	+	36.4099i	0.747205	+	1.63650i
$496$		0			0
$497$	43.2007			1.93782
$498$		0			0
$499$		−	20.8524i		−	0.933481i	−0.884394	−	0.466741i	$-0.845428\pi$
$499$		−	20.8524i		−	0.933481i	0.884394	−	0.466741i	$-0.154572\pi$
$500$		0			0
$501$	58.7946			2.62675
$502$		0			0
$503$	12.0491			0.537244			0.268622	−	0.963246i	$-0.413432\pi$
$503$	12.0491			0.537244			0.268622	+	0.963246i	$0.413432\pi$
$504$		0			0
$505$	−4.47487	+	2.04317i	−0.199129	+	0.0909197i
$506$		0			0
$507$		−	31.3584i		−	1.39268i
$508$		0			0
$509$			8.30763i			0.368229i	0.982905	+	0.184115i	$0.0589418\pi$
$509$			8.30763i			0.368229i	−0.982905	+	0.184115i	$0.941058\pi$
$510$		0			0
$511$			5.52527i			0.244424i
$512$		0			0
$513$	−3.73852	+	18.7687i	−0.165060	+	0.828657i
$514$		0			0
$515$	−3.77450	−	8.26677i	−0.166324	−	0.364277i
$516$		0			0
$517$			37.1993i			1.63602i
$518$		0			0
$519$		−	34.1066i		−	1.49711i
$520$		0			0
$521$		−	37.6771i		−	1.65066i	−0.564649	−	0.825331i	$-0.690989\pi$
$521$		−	37.6771i		−	1.65066i	0.564649	−	0.825331i	$-0.309011\pi$
$522$		0			0
$523$		−	30.6871i		−	1.34185i	−0.741524	−	0.670926i	$-0.765897\pi$
$523$		−	30.6871i		−	1.34185i	0.741524	−	0.670926i	$-0.234103\pi$
$524$		0			0
$525$	−44.6727	−	38.7219i	−1.94968	−	1.68996i
$526$		0			0
$527$		−	38.0396i		−	1.65703i
$528$		0			0
$529$	−21.1340			−0.918868
$530$		0			0
$531$	14.1857			0.615607
$532$		0			0
$533$			7.90295i			0.342315i
$534$		0			0
$535$	9.42206	+	20.6359i	0.407351	+	0.892167i
$536$		0			0
$537$			18.2881i			0.789189i
$538$		0			0
$539$		−	44.4694i		−	1.91543i
$540$		0			0
$541$	36.8593			1.58470			0.792352	−	0.610064i	$-0.208856\pi$
$541$	36.8593			1.58470			0.792352	+	0.610064i	$0.208856\pi$
$542$		0			0
$543$	−67.3101			−2.88855
$544$		0			0
$545$	10.2574	+	22.4654i	0.439379	+	0.962313i
$546$		0			0
$547$		−	5.71713i		−	0.244447i	−0.992503	−	0.122223i	$-0.960998\pi$
$547$		−	5.71713i		−	0.244447i	0.992503	−	0.122223i	$-0.0390024\pi$
$548$		0			0
$549$	31.0036			1.32320
$550$		0			0
$551$	−3.43143	+	17.2270i	−0.146184	+	0.733894i
$552$		0			0
$553$	−21.2385			−0.903153
$554$		0			0
$555$	21.4770	+	47.0382i	0.911649	+	1.99666i
$556$		0			0
$557$		−	6.04131i		−	0.255979i	−0.991776	−	0.127989i	$-0.959148\pi$
$557$		−	6.04131i		−	0.255979i	0.991776	−	0.127989i	$-0.0408523\pi$
$558$		0			0
$559$	0.854791			0.0361538
$560$		0			0
$561$		−	40.5724i		−	1.71297i
$562$		0			0
$563$			20.4223i			0.860696i	0.902663	+	0.430348i	$0.141609\pi$
$563$			20.4223i			0.860696i	−0.902663	+	0.430348i	$0.858391\pi$
$564$		0			0
$565$	4.77282	−	2.17921i	0.200794	−	0.0916799i
$566$		0			0
$567$	7.21563			0.303028
$568$		0			0
$569$			10.1848i			0.426967i	0.976947	+	0.213484i	$0.0684810\pi$
$569$			10.1848i			0.426967i	−0.976947	+	0.213484i	$0.931519\pi$
$570$		0			0
$571$			29.8938i			1.25102i	0.780218	+	0.625508i	$0.215108\pi$
$571$			29.8938i			1.25102i	−0.780218	+	0.625508i	$0.784892\pi$
$572$		0			0
$573$	25.0095			1.04479
$574$		0			0
$575$	4.47362	−	5.16114i	0.186563	−	0.215234i
$576$		0			0
$577$			11.5836i			0.482233i	0.970496	+	0.241117i	$0.0775137\pi$
$577$			11.5836i			0.482233i	−0.970496	+	0.241117i	$0.922486\pi$
$578$		0			0
$579$			31.7158i			1.31807i
$580$		0			0
$581$	−47.7454			−1.98081
$582$		0			0
$583$			38.7784i			1.60604i
$584$		0			0
$585$	11.8921	−	5.42976i	0.491677	−	0.224493i
$586$		0			0
$587$	25.3119			1.04473			0.522367	−	0.852720i	$-0.325049\pi$
$587$	25.3119			1.04473			0.522367	+	0.852720i	$0.325049\pi$
$588$		0			0
$589$	−43.0405	−	8.57321i	−1.77345	−	0.353253i
$590$		0			0
$591$	−21.5319			−0.885706
$592$		0			0
$593$		−	24.0578i		−	0.987937i	−0.869480	−	0.493968i	$-0.835546\pi$
$593$		−	24.0578i		−	0.987937i	0.869480	−	0.493968i	$-0.164454\pi$
$594$		0			0
$595$	15.0563	+	32.9758i	0.617248	+	1.35187i
$596$		0			0
$597$	−57.0974			−2.33684
$598$		0			0
$599$	−30.8906			−1.26215			−0.631077	−	0.775720i	$-0.717387\pi$
$599$	−30.8906			−1.26215			−0.631077	+	0.775720i	$0.717387\pi$
$600$		0			0
$601$			21.7970i			0.889116i	0.895750	+	0.444558i	$0.146639\pi$
$601$			21.7970i			0.889116i	−0.895750	+	0.444558i	$0.853361\pi$
$602$		0			0
$603$			61.6176i			2.50926i
$604$		0			0
$605$	−8.51576	+	3.88818i	−0.346215	+	0.158077i
$606$		0			0
$607$			7.21987i			0.293045i	0.989207	+	0.146523i	$0.0468081\pi$
$607$			7.21987i			0.293045i	−0.989207	+	0.146523i	$0.953192\pi$
$608$		0			0
$609$	−47.6472			−1.93076
$610$		0			0
$611$	12.1499			0.491533
$612$		0			0
$613$		−	36.4438i		−	1.47195i	−0.677007	−	0.735976i	$-0.736723\pi$
$613$		−	36.4438i		−	1.47195i	0.677007	−	0.735976i	$-0.263277\pi$
$614$		0			0
$615$	34.8018	−	15.8901i	1.40335	−	0.640749i
$616$		0			0
$617$		−	7.80543i		−	0.314235i	−0.987580	−	0.157117i	$-0.949780\pi$
$617$		−	7.80543i		−	0.314235i	0.987580	−	0.157117i	$-0.0502201\pi$
$618$		0			0
$619$			23.0937i			0.928212i	0.885780	+	0.464106i	$0.153624\pi$
$619$			23.0937i			0.928212i	−0.885780	+	0.464106i	$0.846376\pi$
$620$		0			0
$621$		−	5.99742i		−	0.240668i
$622$		0			0
$623$		−	3.45744i		−	0.138520i
$624$		0			0
$625$	−3.54983	−	24.7467i	−0.141993	−	0.989868i
$626$		0			0
$627$	−45.9063	−	9.14404i	−1.83332	−	0.365178i
$628$		0			0
$629$		−	31.7071i		−	1.26424i
$630$		0			0
$631$			1.50917i			0.0600790i	0.999549	+	0.0300395i	$0.00956332\pi$
$631$			1.50917i			0.0600790i	−0.999549	+	0.0300395i	$0.990437\pi$
$632$		0			0
$633$			50.3582i			2.00156i
$634$		0			0
$635$	7.32285	+	16.0383i	0.290598	+	0.636459i
$636$		0			0
$637$	−14.5245			−0.575480
$638$		0			0
$639$	46.2458			1.82946
$640$		0			0
$641$		−	29.8565i		−	1.17926i	−0.807673	−	0.589631i	$-0.799273\pi$
$641$		−	29.8565i		−	1.17926i	0.807673	−	0.589631i	$-0.200727\pi$
$642$		0			0
$643$	−1.00975			−0.0398207			−0.0199104	−	0.999802i	$-0.506338\pi$
$643$	−1.00975			−0.0398207			−0.0199104	+	0.999802i	$0.506338\pi$
$644$		0			0
$645$	−1.71869	−	3.76421i	−0.0676732	−	0.148215i
$646$		0			0
$647$	24.9903			0.982469			0.491235	−	0.871027i	$-0.336546\pi$
$647$	24.9903			0.982469			0.491235	+	0.871027i	$0.336546\pi$
$648$		0			0
$649$			12.0353i			0.472428i
$650$		0			0
$651$		−	119.043i		−	4.66568i
$652$		0			0
$653$			1.09782i			0.0429609i	0.999769	+	0.0214805i	$0.00683797\pi$
$653$			1.09782i			0.0429609i	−0.999769	+	0.0214805i	$0.993162\pi$
$654$		0			0
$655$	−18.7450	−	41.0546i	−0.732427	−	1.60414i
$656$		0			0
$657$			5.91473i			0.230756i
$658$		0			0
$659$	37.1842			1.44849			0.724246	−	0.689542i	$-0.242188\pi$
$659$	37.1842			1.44849			0.724246	+	0.689542i	$0.242188\pi$
$660$		0			0
$661$		−	6.68613i		−	0.260060i	−0.991510	−	0.130030i	$-0.958493\pi$
$661$		−	6.68613i		−	0.260060i	0.991510	−	0.130030i	$-0.0415074\pi$
$662$		0			0
$663$	−13.2516			−0.514651
$664$		0			0
$665$	40.7042	−	9.60372i	1.57844	−	0.372416i
$666$		0			0
$667$		−	5.50478i		−	0.213146i
$668$		0			0
$669$	40.0539			1.54857
$670$		0			0
$671$			26.3039i			1.01545i
$672$		0			0
$673$	−0.524476			−0.0202170			−0.0101085	−	0.999949i	$-0.503218\pi$
$673$	−0.524476			−0.0202170			−0.0101085	+	0.999949i	$0.503218\pi$
$674$		0			0
$675$	−16.5879	−	14.3783i	−0.638470	−	0.553419i
$676$		0			0
$677$	37.6175			1.44576			0.722878	−	0.690976i	$-0.242819\pi$
$677$	37.6175			1.44576			0.722878	+	0.690976i	$0.242819\pi$
$678$		0			0
$679$	−4.94974			−0.189954
$680$		0			0
$681$	9.73253			0.372952
$682$		0			0
$683$			31.4446i			1.20320i	0.798799	+	0.601598i	$0.205469\pi$
$683$			31.4446i			1.20320i	−0.798799	+	0.601598i	$0.794531\pi$
$684$		0			0
$685$	−13.9770	−	30.6120i	−0.534034	−	1.16962i
$686$		0			0
$687$			36.4605i			1.39106i
$688$		0			0
$689$	12.6657			0.482524
$690$		0			0
$691$			10.2220i			0.388865i	0.980916	+	0.194432i	$0.0622865\pi$
$691$			10.2220i			0.388865i	−0.980916	+	0.194432i	$0.937714\pi$
$692$		0			0
$693$		−	76.8057i		−	2.91761i
$694$		0			0
$695$	−14.2924	−	31.3027i	−0.542142	−	1.18738i
$696$		0			0
$697$	−23.4589			−0.888569
$698$		0			0
$699$	44.6815			1.69001
$700$		0			0
$701$	7.74088			0.292369			0.146184	−	0.989257i	$-0.453301\pi$
$701$	7.74088			0.292369			0.146184	+	0.989257i	$0.453301\pi$
$702$		0			0
$703$	−35.8754	−	7.14601i	−1.35307	−	0.269517i
$704$		0			0
$705$	−24.4292	−	53.5041i	−0.920058	−	2.01508i
$706$		0			0
$707$	9.43963			0.355014
$708$		0			0
$709$	3.15077			0.118330			0.0591648	−	0.998248i	$-0.481156\pi$
$709$	3.15077			0.118330			0.0591648	+	0.998248i	$0.481156\pi$
$710$		0			0
$711$	−22.7355			−0.852650
$712$		0			0
$713$	13.7533			0.515067
$714$		0			0
$715$	4.60668	+	10.0894i	0.172280	+	0.377322i
$716$		0			0
$717$	51.7514			1.93269
$718$		0			0
$719$		−	18.7147i		−	0.697940i	−0.937134	−	0.348970i	$-0.886531\pi$
$719$		−	18.7147i		−	0.697940i	0.937134	−	0.348970i	$-0.113469\pi$
$720$		0			0
$721$			17.4385i			0.649445i
$722$		0			0
$723$	−63.7475			−2.37080
$724$		0			0
$725$	−15.2254	−	13.1972i	−0.565456	−	0.490132i
$726$		0			0
$727$	29.2510			1.08486			0.542430	−	0.840101i	$-0.317504\pi$
$727$	29.2510			1.08486			0.542430	+	0.840101i	$0.317504\pi$
$728$		0			0
$729$	44.0188			1.63033
$730$		0			0
$731$			2.53734i			0.0938469i
$732$		0			0
$733$			19.5959i			0.723792i	0.932219	+	0.361896i	$0.117870\pi$
$733$			19.5959i			0.723792i	−0.932219	+	0.361896i	$0.882130\pi$
$734$		0			0
$735$	29.2036	+	63.9607i	1.07719	+	2.35923i
$736$		0			0
$737$	−52.2772			−1.92565
$738$		0			0
$739$			18.5650i			0.682924i	0.939896	+	0.341462i	$0.110922\pi$
$739$			18.5650i			0.682924i	−0.939896	+	0.341462i	$0.889078\pi$
$740$		0			0
$741$	−2.98660	+	14.9938i	−0.109715	+	0.550810i
$742$		0			0
$743$		−	18.7569i		−	0.688124i	−0.938947	−	0.344062i	$-0.888197\pi$
$743$		−	18.7569i		−	0.688124i	0.938947	−	0.344062i	$-0.111803\pi$
$744$		0			0
$745$	−8.70936	+	3.97658i	−0.319086	+	0.145690i
$746$		0			0
$747$	−51.1108			−1.87005
$748$		0			0
$749$		−	43.5309i		−	1.59058i
$750$		0			0
$751$	−8.25850			−0.301357			−0.150679	−	0.988583i	$-0.548146\pi$
$751$	−8.25850			−0.301357			−0.150679	+	0.988583i	$0.548146\pi$
$752$		0			0
$753$	−53.9552			−1.96624
$754$		0			0
$755$	−42.3544	+	19.3384i	−1.54143	+	0.703798i
$756$		0			0
$757$			33.0671i			1.20184i	0.799308	+	0.600922i	$0.205200\pi$
$757$			33.0671i			1.20184i	−0.799308	+	0.600922i	$0.794800\pi$
$758$		0			0
$759$	14.6691			0.532454
$760$		0			0
$761$	8.64353			0.313328			0.156664	−	0.987652i	$-0.449926\pi$
$761$	8.64353			0.313328			0.156664	+	0.987652i	$0.449926\pi$
$762$		0			0
$763$		−	47.3902i		−	1.71564i
$764$		0			0
$765$	16.1176	+	35.3001i	0.582732	+	1.27628i
$766$		0			0
$767$	3.93094			0.141938
$768$		0			0
$769$	23.7790			0.857492			0.428746	−	0.903425i	$-0.358955\pi$
$769$	23.7790			0.857492			0.428746	+	0.903425i	$0.358955\pi$
$770$		0			0
$771$		−	23.3642i		−	0.841440i
$772$		0			0
$773$	10.5799			0.380534			0.190267	−	0.981732i	$-0.439065\pi$
$773$	10.5799			0.380534			0.190267	+	0.981732i	$0.439065\pi$
$774$		0			0
$775$	32.9724	−	38.0396i	1.18440	−	1.36642i
$776$		0			0
$777$		−	99.2260i		−	3.55971i
$778$		0			0
$779$	−5.28706	+	26.5429i	−0.189429	+	0.950998i
$780$		0			0
$781$			39.2355i			1.40396i
$782$		0			0
$783$	−17.6924			−0.632275
$784$		0			0
$785$	0.231251	+	0.506477i	0.00825369	+	0.0180769i
$786$		0			0
$787$		−	25.2873i		−	0.901395i	−0.892677	−	0.450697i	$-0.851175\pi$
$787$		−	25.2873i		−	0.901395i	0.892677	−	0.450697i	$-0.148825\pi$
$788$		0			0
$789$		−	59.1994i		−	2.10755i
$790$		0			0
$791$	−10.0681			−0.357982
$792$		0			0
$793$	8.59129			0.305086
$794$		0			0
$795$	−25.4663	−	55.7753i	−0.903195	−	1.97815i
$796$		0			0
$797$	28.2317			1.00002			0.500009	−	0.866020i	$-0.333330\pi$
$797$	28.2317			1.00002			0.500009	+	0.866020i	$0.333330\pi$
$798$		0			0
$799$			36.0655i			1.27591i
$800$		0			0
$801$		−	3.70115i		−	0.130774i
$802$		0			0
$803$	−5.01813			−0.177086
$804$		0			0
$805$	−11.9225	+	5.44365i	−0.420212	+	0.191863i
$806$		0			0
$807$	−82.3455			−2.89870
$808$		0			0
$809$	−5.14844			−0.181010			−0.0905048	−	0.995896i	$-0.528848\pi$
$809$	−5.14844			−0.181010			−0.0905048	+	0.995896i	$0.528848\pi$
$810$		0			0
$811$	3.42568			0.120292			0.0601460	−	0.998190i	$-0.480843\pi$
$811$	3.42568			0.120292			0.0601460	+	0.998190i	$0.480843\pi$
$812$		0			0
$813$	65.0436			2.28118
$814$		0			0
$815$	20.0113	−	9.13691i	0.700967	−	0.320052i
$816$		0			0
$817$	2.87091	+	0.571855i	0.100440	+	0.0200067i
$818$		0			0
$819$	−25.0860			−0.876577
$820$		0			0
$821$	−28.1721			−0.983213			−0.491606	−	0.870818i	$-0.663590\pi$
$821$	−28.1721			−0.983213			−0.491606	+	0.870818i	$0.663590\pi$
$822$		0			0
$823$	−1.75153			−0.0610546			−0.0305273	−	0.999534i	$-0.509719\pi$
$823$	−1.75153			−0.0610546			−0.0305273	+	0.999534i	$0.509719\pi$
$824$		0			0
$825$	35.1677	−	40.5724i	1.22438	−	1.41255i
$826$		0			0
$827$			18.6366i			0.648058i	0.946047	+	0.324029i	$0.105037\pi$
$827$			18.6366i			0.648058i	−0.946047	+	0.324029i	$0.894963\pi$
$828$		0			0
$829$			55.5396i			1.92897i	0.264135	+	0.964486i	$0.414914\pi$
$829$			55.5396i			1.92897i	−0.264135	+	0.964486i	$0.585086\pi$
$830$		0			0
$831$	27.2330			0.944703
$832$		0			0
$833$		−	43.1140i		−	1.49381i
$834$		0			0
$835$	−19.8158	−	43.4000i	−0.685755	−	1.50192i
$836$		0			0
$837$		−	44.2034i		−	1.52789i
$838$		0			0
$839$	15.5354			0.536340			0.268170	−	0.963372i	$-0.413581\pi$
$839$	15.5354			0.536340			0.268170	+	0.963372i	$0.413581\pi$
$840$		0			0
$841$	12.7609			0.440030
$842$		0			0
$843$	43.7590			1.50714
$844$		0			0
$845$	−23.1476	+	10.5689i	−0.796302	+	0.363581i
$846$		0			0
$847$	17.9638			0.617243
$848$		0			0
$849$		−	60.2267i		−	2.06697i
$850$		0			0
$851$	11.4638			0.392974
$852$		0			0
$853$			22.7542i			0.779088i	0.921008	+	0.389544i	$0.127367\pi$
$853$			22.7542i			0.779088i	−0.921008	+	0.389544i	$0.872633\pi$
$854$		0			0
$855$	43.5733	−	10.2807i	1.49018	−	0.351591i
$856$		0			0
$857$	31.1273			1.06329			0.531644	−	0.846968i	$-0.321574\pi$
$857$	31.1273			1.06329			0.531644	+	0.846968i	$0.321574\pi$
$858$		0			0
$859$		−	17.5202i		−	0.597781i	−0.954287	−	0.298890i	$-0.903384\pi$
$859$		−	17.5202i		−	0.597781i	0.954287	−	0.298890i	$-0.0966165\pi$
$860$		0			0
$861$	−73.4136			−2.50193
$862$		0			0
$863$			26.4721i			0.901120i	0.892746	+	0.450560i	$0.148776\pi$
$863$			26.4721i			0.901120i	−0.892746	+	0.450560i	$0.851224\pi$
$864$		0			0
$865$	−25.1762	+	11.4951i	−0.856018	+	0.390846i
$866$		0			0
$867$			7.50921i			0.255026i
$868$		0			0
$869$		−	19.2891i		−	0.654339i
$870$		0			0
$871$			17.0746i			0.578551i
$872$		0			0
$873$	−5.29864			−0.179332
$874$		0			0
$875$	−13.5267	+	46.0264i	−0.457288	+	1.55598i
$876$		0			0
$877$	−0.0718259			−0.00242539			−0.00121269	−	0.999999i	$-0.500386\pi$
$877$	−0.0718259			−0.00242539			−0.00121269	+	0.999999i	$0.500386\pi$
$878$		0			0
$879$		−	53.2837i		−	1.79722i
$880$		0			0
$881$	30.4674			1.02647			0.513237	−	0.858247i	$-0.328446\pi$
$881$	30.4674			1.02647			0.513237	+	0.858247i	$0.328446\pi$
$882$		0			0
$883$	42.2906			1.42319			0.711596	−	0.702589i	$-0.247973\pi$
$883$	42.2906			1.42319			0.711596	+	0.702589i	$0.247973\pi$
$884$		0			0
$885$	−7.90375	−	17.3105i	−0.265682	−	0.581887i
$886$		0			0
$887$			29.7541i			0.999045i	0.866301	+	0.499522i	$0.166491\pi$
$887$			29.7541i			0.999045i	−0.866301	+	0.499522i	$0.833509\pi$
$888$		0			0
$889$		−	33.8323i		−	1.13470i
$890$		0			0
$891$			6.55334i			0.219545i
$892$		0			0
$893$	40.8068	+	8.12829i	1.36555	+	0.272003i
$894$		0			0
$895$	13.4996	−	6.16373i	0.451241	−	0.206031i
$896$		0			0
$897$		−	4.79117i		−	0.159973i
$898$		0			0
$899$		−	40.5724i		−	1.35317i
$900$		0			0
$901$			37.5965i			1.25252i
$902$		0			0
$903$			7.94050i			0.264243i
$904$		0			0
$905$	22.6859	+	49.6858i	0.754104	+	1.65161i
$906$		0			0
$907$		−	17.8978i		−	0.594287i	−0.954833	−	0.297143i	$-0.903966\pi$
$907$		−	17.8978i		−	0.594287i	0.954833	−	0.297143i	$-0.0960339\pi$
$908$		0			0
$909$	10.1050			0.335162
$910$		0			0
$911$	14.6457			0.485234			0.242617	−	0.970122i	$-0.421994\pi$
$911$	14.6457			0.485234			0.242617	+	0.970122i	$0.421994\pi$
$912$		0			0
$913$		−	43.3630i		−	1.43511i
$914$		0			0
$915$	−17.2741	−	37.8331i	−0.571063	−	1.25072i
$916$		0			0
$917$			86.6037i			2.85990i
$918$		0			0
$919$		−	33.5202i		−	1.10573i	−0.833271	−	0.552865i	$-0.813535\pi$
$919$		−	33.5202i		−	1.10573i	0.833271	−	0.552865i	$-0.186465\pi$
$920$		0			0
$921$	−31.8414			−1.04921
$922$		0			0
$923$	12.8150			0.421810
$924$		0			0
$925$	27.4834	−	31.7071i	0.903648	−	1.04252i
$926$		0			0
$927$			18.6677i			0.613129i
$928$		0			0
$929$	−23.0479			−0.756178			−0.378089	−	0.925769i	$-0.623419\pi$
$929$	−23.0479			−0.756178			−0.378089	+	0.925769i	$0.623419\pi$
$930$		0			0
$931$	−48.7820	−	9.71685i	−1.59877	−	0.318457i
$932$		0			0
$933$	12.3840			0.405435
$934$		0			0
$935$	−29.9491	+	13.6743i	−0.979439	+	0.447199i
$936$		0			0
$937$			24.2230i			0.791329i	0.918395	+	0.395665i	$0.129486\pi$
$937$			24.2230i			0.791329i	−0.918395	+	0.395665i	$0.870514\pi$
$938$		0			0
$939$	29.8614			0.974489
$940$		0			0
$941$		−	38.3757i		−	1.25101i	−0.780219	−	0.625506i	$-0.784892\pi$
$941$		−	38.3757i		−	1.25101i	0.780219	−	0.625506i	$-0.215108\pi$
$942$		0			0
$943$		−	8.48163i		−	0.276200i
$944$		0			0
$945$	17.4959	+	38.3190i	0.569143	+	1.24652i
$946$		0			0
$947$	35.2906			1.14679			0.573396	−	0.819279i	$-0.305626\pi$
$947$	35.2906			1.14679			0.573396	+	0.819279i	$0.305626\pi$
$948$		0			0
$949$			1.63901i			0.0532044i
$950$		0			0
$951$			64.9225i			2.10526i
$952$		0			0
$953$	−42.8440			−1.38785			−0.693927	−	0.720046i	$-0.744121\pi$
$953$	−42.8440			−1.38785			−0.693927	+	0.720046i	$0.744121\pi$
$954$		0			0
$955$	−8.42910	−	18.4611i	−0.272759	−	0.597388i
$956$		0			0
$957$		−	43.2739i		−	1.39885i
$958$		0			0
$959$			64.5751i			2.08524i
$960$		0			0
$961$	70.3676			2.26992
$962$		0			0
$963$		−	46.5992i		−	1.50164i
$964$		0			0
$965$	23.4115	−	10.6894i	0.753642	−	0.344103i
$966$		0			0
$967$	1.98009			0.0636754			0.0318377	−	0.999493i	$-0.489864\pi$
$967$	1.98009			0.0636754			0.0318377	+	0.999493i	$0.489864\pi$
$968$		0			0
$969$	−44.5071	−	8.86534i	−1.42977	−	0.284796i
$970$		0			0
$971$	31.6227			1.01482			0.507410	−	0.861705i	$-0.330603\pi$
$971$	31.6227			1.01482			0.507410	+	0.861705i	$0.330603\pi$
$972$		0			0
$973$			66.0323i			2.11690i
$974$		0			0
$975$	−13.2516	−	11.4864i	−0.424392	−	0.367859i
$976$		0			0
$977$	−25.4801			−0.815180			−0.407590	−	0.913165i	$-0.633631\pi$
$977$	−25.4801			−0.815180			−0.407590	+	0.913165i	$0.633631\pi$
$978$		0			0
$979$	3.14010			0.100358
$980$		0			0
$981$		−	50.7306i		−	1.61970i
$982$		0			0
$983$			1.67238i			0.0533405i	0.999644	+	0.0266703i	$0.00849041\pi$
$983$			1.67238i			0.0533405i	−0.999644	+	0.0266703i	$0.991510\pi$
$984$		0			0
$985$	7.25702	+	15.8941i	0.231228	+	0.506428i
$986$		0			0
$987$			112.865i			3.59255i
$988$		0			0
$989$	−0.917383			−0.0291711
$990$		0			0
$991$	32.3629			1.02804			0.514021	−	0.857778i	$-0.328155\pi$
$991$	32.3629			1.02804			0.514021	+	0.857778i	$0.328155\pi$
$992$		0			0
$993$		−	32.4081i		−	1.02844i
$994$		0			0
$995$	19.2438	+	42.1472i	0.610071	+	1.33616i
$996$		0			0
$997$			37.0267i			1.17265i	0.810077	+	0.586323i	$0.199425\pi$
$997$			37.0267i			1.17265i	−0.810077	+	0.586323i	$0.800575\pi$
$998$		0			0
$999$		−	36.8447i		−	1.16572i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.g.g.1519.13	yes	16
4.3	odd	2		1520.2.g.e.1519.4	yes	16
5.4	even	2	inner	1520.2.g.g.1519.4	yes	16
19.18	odd	2		1520.2.g.e.1519.3	✓	16
20.19	odd	2		1520.2.g.e.1519.13	yes	16
76.75	even	2	inner	1520.2.g.g.1519.14	yes	16
95.94	odd	2		1520.2.g.e.1519.14	yes	16
380.379	even	2	inner	1520.2.g.g.1519.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.g.e.1519.3	✓	16	19.18	odd	2
1520.2.g.e.1519.4	yes	16	4.3	odd	2
1520.2.g.e.1519.13	yes	16	20.19	odd	2
1520.2.g.e.1519.14	yes	16	95.94	odd	2
1520.2.g.g.1519.3	yes	16	380.379	even	2	inner
1520.2.g.g.1519.4	yes	16	5.4	even	2	inner
1520.2.g.g.1519.13	yes	16	1.1	even	1	trivial
1520.2.g.g.1519.14	yes	16	76.75	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.75559i q^{3} +(2.03407 - 0.928731i) q^{5} -4.29083 q^{7} -4.59328 q^{9} +O(q^{10})\)	\(q+2.75559i q^{3} +(2.03407 - 0.928731i) q^{5} -4.29083 q^{7} -4.59328 q^{9} -3.89699i q^{11} -1.27282 q^{13} +(2.55920 + 5.60508i) q^{15} -3.77822i q^{17} +(-4.27492 - 0.851518i) q^{19} -11.8238i q^{21} +1.36603 q^{23} +(3.27492 - 3.77822i) q^{25} -4.39042i q^{27} -4.02978i q^{29} +10.0681 q^{31} +10.7385 q^{33} +(-8.72787 + 3.98503i) q^{35} +8.39208 q^{37} -3.50738i q^{39} -6.20899i q^{41} -0.671571 q^{43} +(-9.34307 + 4.26592i) q^{45} -9.54564 q^{47} +11.4112 q^{49} +10.4112 q^{51} -9.95086 q^{53} +(-3.61926 - 7.92677i) q^{55} +(2.34643 - 11.7799i) q^{57} -3.08836 q^{59} -6.74979 q^{61} +19.7090 q^{63} +(-2.58902 + 1.18211i) q^{65} -13.4147i q^{67} +3.76421i q^{69} -10.0681 q^{71} -1.28769i q^{73} +(10.4112 + 9.02433i) q^{75} +16.7213i q^{77} +4.94974 q^{79} -1.68164 q^{81} +11.1273 q^{83} +(-3.50895 - 7.68517i) q^{85} +11.1044 q^{87} +0.805775i q^{89} +5.46147 q^{91} +27.7437i q^{93} +(-9.48633 + 2.23820i) q^{95} +1.15356 q^{97} +17.9000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) \) 16 * q - 32 * q^9 + 32 * q^15 - 8 * q^19 - 8 * q^25 + 96 * q^31 - 24 * q^45 - 8 * q^49 - 24 * q^51 - 72 * q^59 - 24 * q^61 - 96 * q^71 - 24 * q^75 + 32 * q^79 - 8 * q^81 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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