LMFDB - Embedded newform 1503.2.c.a.1502.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1502,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1503.1502");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.c (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:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1502.11
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.118553i q^{2} +1.98595 q^{4} +3.37331 q^{5} +2.05489 q^{7} -0.472544i q^{8} +O(q^{10})$	\(q-0.118553i q^{2} +1.98595 q^{4} +3.37331 q^{5} +2.05489 q^{7} -0.472544i q^{8} -0.399915i q^{10} +4.02548i q^{11} -0.580620i q^{13} -0.243613i q^{14} +3.91587 q^{16} +0.463155 q^{17} -2.47692 q^{19} +6.69922 q^{20} +0.477231 q^{22} -2.29934 q^{23} +6.37925 q^{25} -0.0688340 q^{26} +4.08091 q^{28} +8.19699i q^{29} -3.26496 q^{31} -1.40932i q^{32} -0.0549081i q^{34} +6.93180 q^{35} -2.44279i q^{37} +0.293645i q^{38} -1.59404i q^{40} -2.34714 q^{41} -7.95176i q^{43} +7.99438i q^{44} +0.272592i q^{46} -2.26027i q^{47} -2.77741 q^{49} -0.756277i q^{50} -1.15308i q^{52} -8.16096 q^{53} +13.5792i q^{55} -0.971028i q^{56} +0.971773 q^{58} -6.67660 q^{59} +0.344788 q^{61} +0.387069i q^{62} +7.66466 q^{64} -1.95861i q^{65} -10.1378i q^{67} +0.919800 q^{68} -0.821783i q^{70} -8.32237 q^{71} -7.15709i q^{73} -0.289599 q^{74} -4.91902 q^{76} +8.27193i q^{77} -12.1532i q^{79} +13.2095 q^{80} +0.278260i q^{82} -11.1343 q^{83} +1.56237 q^{85} -0.942701 q^{86} +1.90222 q^{88} -1.54245i q^{89} -1.19311i q^{91} -4.56636 q^{92} -0.267960 q^{94} -8.35542 q^{95} -2.52821 q^{97} +0.329269i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 48 q^{4}+O(q^{10}) $ 56 * q - 48 * q^4	$ 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) $ 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	0.118553i		−	0.0838293i	−0.999121	−	0.0419147i	$-0.986654\pi$
$2$		−	0.118553i		−	0.0838293i	0.999121	−	0.0419147i	$-0.0133458\pi$
$3$		0			0
$3$		0			0
$4$	1.98595			0.992973
$5$	3.37331			1.50859			0.754296	−	0.656534i	$-0.227978\pi$
$5$	3.37331			1.50859			0.754296	+	0.656534i	$0.227978\pi$
$6$		0			0
$7$	2.05489			0.776677			0.388338	−	0.921517i	$-0.373049\pi$
$7$	2.05489			0.776677			0.388338	+	0.921517i	$0.373049\pi$
$8$		−	0.472544i		−	0.167070i
$9$		0			0
$10$		−	0.399915i		−	0.126464i
$11$			4.02548i			1.21373i	0.794806	+	0.606864i	$0.207573\pi$
$11$			4.02548i			1.21373i	−0.794806	+	0.606864i	$0.792427\pi$
$12$		0			0
$13$		−	0.580620i		−	0.161035i	−0.996753	−	0.0805175i	$-0.974343\pi$
$13$		−	0.580620i		−	0.161035i	0.996753	−	0.0805175i	$-0.0256573\pi$
$14$		−	0.243613i		−	0.0651083i
$15$		0			0
$16$	3.91587			0.978967
$17$	0.463155			0.112331			0.0561657	−	0.998421i	$-0.482112\pi$
$17$	0.463155			0.112331			0.0561657	+	0.998421i	$0.482112\pi$
$18$		0			0
$19$	−2.47692			−0.568243			−0.284122	−	0.958788i	$-0.591702\pi$
$19$	−2.47692			−0.568243			−0.284122	+	0.958788i	$0.591702\pi$
$20$	6.69922			1.49799
$21$		0			0
$22$	0.477231			0.101746
$23$	−2.29934			−0.479445			−0.239722	−	0.970841i	$-0.577056\pi$
$23$	−2.29934			−0.479445			−0.239722	+	0.970841i	$0.577056\pi$
$24$		0			0
$25$	6.37925			1.27585
$26$	−0.0688340			−0.0134995
$27$		0			0
$28$	4.08091			0.771219
$29$			8.19699i			1.52214i	0.648669	+	0.761071i	$0.275326\pi$
$29$			8.19699i			1.52214i	−0.648669	+	0.761071i	$0.724674\pi$
$30$		0			0
$31$	−3.26496			−0.586404			−0.293202	−	0.956050i	$-0.594721\pi$
$31$	−3.26496			−0.586404			−0.293202	+	0.956050i	$0.594721\pi$
$32$		−	1.40932i		−	0.249136i
$33$		0			0
$34$		−	0.0549081i		−	0.00941667i
$35$	6.93180			1.17169
$36$		0			0
$37$		−	2.44279i		−	0.401593i	−0.979633	−	0.200796i	$-0.935647\pi$
$37$		−	2.44279i		−	0.401593i	0.979633	−	0.200796i	$-0.0643530\pi$
$38$			0.293645i			0.0476354i
$39$		0			0
$40$		−	1.59404i		−	0.252040i
$41$	−2.34714			−0.366562			−0.183281	−	0.983061i	$-0.558672\pi$
$41$	−2.34714			−0.366562			−0.183281	+	0.983061i	$0.558672\pi$
$42$		0			0
$43$		−	7.95176i		−	1.21263i	−0.795224	−	0.606316i	$-0.792647\pi$
$43$		−	7.95176i		−	1.21263i	0.795224	−	0.606316i	$-0.207353\pi$
$44$			7.99438i			1.20520i
$45$		0			0
$46$			0.272592i			0.0401915i
$47$		−	2.26027i		−	0.329694i	−0.986319	−	0.164847i	$-0.947287\pi$
$47$		−	2.26027i		−	0.329694i	0.986319	−	0.164847i	$-0.0527130\pi$
$48$		0			0
$49$	−2.77741			−0.396773
$50$		−	0.756277i		−	0.106954i
$51$		0			0
$52$		−	1.15308i		−	0.159903i
$53$	−8.16096			−1.12099			−0.560497	−	0.828157i	$-0.689390\pi$
$53$	−8.16096			−1.12099			−0.560497	+	0.828157i	$0.689390\pi$
$54$		0			0
$55$			13.5792i			1.83102i
$56$		−	0.971028i		−	0.129759i
$57$		0			0
$58$	0.971773			0.127600
$59$	−6.67660			−0.869220			−0.434610	−	0.900619i	$-0.643114\pi$
$59$	−6.67660			−0.869220			−0.434610	+	0.900619i	$0.643114\pi$
$60$		0			0
$61$	0.344788			0.0441456			0.0220728	−	0.999756i	$-0.492973\pi$
$61$	0.344788			0.0441456			0.0220728	+	0.999756i	$0.492973\pi$
$62$			0.387069i			0.0491578i
$63$		0			0
$64$	7.66466			0.958082
$65$		−	1.95861i		−	0.242936i
$66$		0			0
$67$		−	10.1378i		−	1.23853i	−0.785183	−	0.619264i	$-0.787431\pi$
$67$		−	10.1378i		−	1.23853i	0.785183	−	0.619264i	$-0.212569\pi$
$68$	0.919800			0.111542
$69$		0			0
$70$		−	0.821783i		−	0.0982218i
$71$	−8.32237			−0.987684			−0.493842	−	0.869552i	$-0.664408\pi$
$71$	−8.32237			−0.987684			−0.493842	+	0.869552i	$0.664408\pi$
$72$		0			0
$73$		−	7.15709i		−	0.837674i	−0.908061	−	0.418837i	$-0.862438\pi$
$73$		−	7.15709i		−	0.837674i	0.908061	−	0.418837i	$-0.137562\pi$
$74$	−0.289599			−0.0336653
$75$		0			0
$76$	−4.91902			−0.564250
$77$			8.27193i			0.942674i
$78$		0			0
$79$		−	12.1532i		−	1.36734i	−0.729792	−	0.683669i	$-0.760383\pi$
$79$		−	12.1532i		−	1.36734i	0.729792	−	0.683669i	$-0.239617\pi$
$80$	13.2095			1.47686
$81$		0			0
$82$			0.278260i			0.0307287i
$83$	−11.1343			−1.22215			−0.611077	−	0.791571i	$-0.709263\pi$
$83$	−11.1343			−1.22215			−0.611077	+	0.791571i	$0.709263\pi$
$84$		0			0
$85$	1.56237			0.169462
$86$	−0.942701			−0.101654
$87$		0			0
$88$	1.90222			0.202777
$89$		−	1.54245i		−	0.163500i	−0.996653	−	0.0817498i	$-0.973949\pi$
$89$		−	1.54245i		−	0.163500i	0.996653	−	0.0817498i	$-0.0260508\pi$
$90$		0			0
$91$		−	1.19311i		−	0.125072i
$92$	−4.56636			−0.476076
$93$		0			0
$94$	−0.267960			−0.0276380
$95$	−8.35542			−0.857248
$96$		0			0
$97$	−2.52821			−0.256701			−0.128351	−	0.991729i	$-0.540968\pi$
$97$	−2.52821			−0.256701			−0.128351	+	0.991729i	$0.540968\pi$
$98$			0.329269i			0.0332612i
$99$		0			0
$100$	12.6688			1.26688
$101$	13.1353			1.30701			0.653507	−	0.756920i	$-0.273297\pi$
$101$	13.1353			1.30701			0.653507	+	0.756920i	$0.273297\pi$
$102$		0			0
$103$			6.58733i			0.649069i	0.945874	+	0.324534i	$0.105208\pi$
$103$			6.58733i			0.649069i	−0.945874	+	0.324534i	$0.894792\pi$
$104$	−0.274369			−0.0269041
$105$		0			0
$106$			0.967502i			0.0939721i
$107$			0.944190i			0.0912783i	0.998958	+	0.0456392i	$0.0145324\pi$
$107$			0.944190i			0.0912783i	−0.998958	+	0.0456392i	$0.985468\pi$
$108$		0			0
$109$			12.9599i			1.24133i	0.784076	+	0.620664i	$0.213137\pi$
$109$			12.9599i			1.24133i	−0.784076	+	0.620664i	$0.786863\pi$
$110$	1.60985			0.153493
$111$		0			0
$112$	8.04670			0.760341
$113$	15.1145			1.42186			0.710928	−	0.703265i	$-0.248275\pi$
$113$	15.1145			1.42186			0.710928	+	0.703265i	$0.248275\pi$
$114$		0			0
$115$	−7.75639			−0.723287
$116$			16.2788i			1.51145i
$117$		0			0
$118$			0.791528i			0.0728661i
$119$	0.951733			0.0872453
$120$		0			0
$121$	−5.20448			−0.473134
$122$		−	0.0408755i		−	0.00370069i
$123$		0			0
$124$	−6.48403			−0.582283
$125$	4.65266			0.416146
$126$		0			0
$127$	13.9986			1.24218			0.621089	−	0.783740i	$-0.286691\pi$
$127$	13.9986			1.24218			0.621089	+	0.783740i	$0.286691\pi$
$128$		−	3.72731i		−	0.329451i
$129$		0			0
$130$	−0.232199			−0.0203652
$131$	10.4992			0.917323			0.458661	−	0.888611i	$-0.348329\pi$
$131$	10.4992			0.917323			0.458661	+	0.888611i	$0.348329\pi$
$132$		0			0
$133$	−5.08980			−0.441341
$134$	−1.20186			−0.103825
$135$		0			0
$136$		−	0.218861i		−	0.0187672i
$137$		−	11.7548i		−	1.00428i	−0.864786	−	0.502140i	$-0.832546\pi$
$137$		−	11.7548i		−	1.00428i	0.864786	−	0.502140i	$-0.167454\pi$
$138$		0			0
$139$		−	14.4454i		−	1.22524i	−0.790377	−	0.612621i	$-0.790115\pi$
$139$		−	14.4454i		−	1.22524i	0.790377	−	0.612621i	$-0.209885\pi$
$140$	13.7662			1.16345
$141$		0			0
$142$			0.986638i			0.0827969i
$143$	2.33727			0.195453
$144$		0			0
$145$			27.6510i			2.29629i
$146$	−0.848491			−0.0702216
$147$		0			0
$148$		−	4.85126i		−	0.398771i
$149$	10.7985			0.884649			0.442324	−	0.896855i	$-0.354154\pi$
$149$	10.7985			0.884649			0.442324	+	0.896855i	$0.354154\pi$
$150$		0			0
$151$		−	12.1834i		−	0.991469i	−0.868474	−	0.495735i	$-0.834899\pi$
$151$		−	12.1834i		−	0.991469i	0.868474	−	0.495735i	$-0.165101\pi$
$152$			1.17045i			0.0949361i
$153$		0			0
$154$	0.980658			0.0790237
$155$	−11.0137			−0.884645
$156$		0			0
$157$	19.8699			1.58579			0.792894	−	0.609360i	$-0.208574\pi$
$157$	19.8699			1.58579			0.792894	+	0.609360i	$0.208574\pi$
$158$	−1.44079			−0.114623
$159$		0			0
$160$		−	4.75409i		−	0.375844i
$161$	−4.72489			−0.372374
$162$		0			0
$163$			12.3631i			0.968350i	0.874971	+	0.484175i	$0.160880\pi$
$163$			12.3631i			0.968350i	−0.874971	+	0.484175i	$0.839120\pi$
$164$	−4.66130			−0.363986
$165$		0			0
$166$			1.32001i			0.102452i
$167$	−5.81641	+	11.5399i	−0.450088	+	0.892984i
$167$	−5.81641	+	11.5399i	−0.450088	+	0.892984i
$168$		0			0
$169$	12.6629			0.974068
$170$		−	0.185222i		−	0.0142059i
$171$		0			0
$172$		−	15.7918i		−	1.20411i
$173$		−	6.01242i		−	0.457115i	−0.973530	−	0.228558i	$-0.926599\pi$
$173$		−	6.01242i		−	0.457115i	0.973530	−	0.228558i	$-0.0734010\pi$
$174$		0			0
$175$	13.1087			0.990924
$176$			15.7632i			1.18820i
$177$		0			0
$178$	−0.182862			−0.0137060
$179$		−	9.53548i		−	0.712715i	−0.934350	−	0.356358i	$-0.884018\pi$
$179$		−	9.53548i		−	0.712715i	0.934350	−	0.356358i	$-0.115982\pi$
$180$		0			0
$181$	−10.8695			−0.807925			−0.403963	−	0.914775i	$-0.632367\pi$
$181$	−10.8695			−0.807925			−0.403963	+	0.914775i	$0.632367\pi$
$182$	−0.141447			−0.0104847
$183$		0			0
$184$			1.08654i			0.0801006i
$185$		−	8.24032i		−	0.605840i
$186$		0			0
$187$			1.86442i			0.136340i
$188$		−	4.48877i		−	0.327377i
$189$		0			0
$190$			0.990556i			0.0718625i
$191$		−	11.6553i		−	0.843351i	−0.906747	−	0.421676i	$-0.861442\pi$
$191$		−	11.6553i		−	0.843351i	0.906747	−	0.421676i	$-0.138558\pi$
$192$		0			0
$193$			6.40358i			0.460940i	0.973079	+	0.230470i	$0.0740263\pi$
$193$			6.40358i			0.460940i	−0.973079	+	0.230470i	$0.925974\pi$
$194$			0.299726i			0.0215191i
$195$		0			0
$196$	−5.51579			−0.393985
$197$	15.9583			1.13698			0.568489	−	0.822691i	$-0.307528\pi$
$197$	15.9583			1.13698			0.568489	+	0.822691i	$0.307528\pi$
$198$		0			0
$199$	6.04918			0.428815			0.214408	−	0.976744i	$-0.431218\pi$
$199$	6.04918			0.428815			0.214408	+	0.976744i	$0.431218\pi$
$200$		−	3.01448i		−	0.213156i
$201$		0			0
$202$		−	1.55723i		−	0.109566i
$203$			16.8439i			1.18221i
$204$		0			0
$205$	−7.91766			−0.552993
$206$	0.780945			0.0544110
$207$		0			0
$208$		−	2.27363i		−	0.157648i
$209$		−	9.97077i		−	0.689693i
$210$		0			0
$211$	−14.0976			−0.970522			−0.485261	−	0.874369i	$-0.661275\pi$
$211$	−14.0976			−0.970522			−0.485261	+	0.874369i	$0.661275\pi$
$212$	−16.2072			−1.11312
$213$		0			0
$214$	0.111936			0.00765180
$215$		−	26.8238i		−	1.82937i
$216$		0			0
$217$	−6.70915			−0.455447
$218$	1.53642			0.104060
$219$		0			0
$220$			26.9676i			1.81815i
$221$		−	0.268917i		−	0.0180893i
$222$		0			0
$223$	−14.5255			−0.972697			−0.486348	−	0.873765i	$-0.661671\pi$
$223$	−14.5255			−0.972697			−0.486348	+	0.873765i	$0.661671\pi$
$224$		−	2.89601i		−	0.193498i
$225$		0			0
$226$		−	1.79187i		−	0.119193i
$227$	2.03496			0.135065			0.0675326	−	0.997717i	$-0.478487\pi$
$227$	2.03496			0.135065			0.0675326	+	0.997717i	$0.478487\pi$
$228$		0			0
$229$	−12.0749			−0.797933			−0.398966	−	0.916966i	$-0.630631\pi$
$229$	−12.0749			−0.797933			−0.398966	+	0.916966i	$0.630631\pi$
$230$			0.919539i			0.0606326i
$231$		0			0
$232$	3.87344			0.254303
$233$			9.84600i			0.645033i	0.946564	+	0.322516i	$0.104529\pi$
$233$			9.84600i			0.645033i	−0.946564	+	0.322516i	$0.895471\pi$
$234$		0			0
$235$		−	7.62459i		−	0.497374i
$236$	−13.2594			−0.863112
$237$		0			0
$238$		−	0.112830i		−	0.00731371i
$239$		−	26.4972i		−	1.71396i	−0.515350	−	0.856980i	$-0.672338\pi$
$239$		−	26.4972i		−	1.71396i	0.515350	−	0.856980i	$-0.327662\pi$
$240$		0			0
$241$			15.3557i			0.989149i	0.869135	+	0.494574i	$0.164676\pi$
$241$			15.3557i			0.989149i	−0.869135	+	0.494574i	$0.835324\pi$
$242$			0.617004i			0.0396625i
$243$		0			0
$244$	0.684730			0.0438353
$245$	−9.36908			−0.598569
$246$		0			0
$247$			1.43815i			0.0915071i
$248$			1.54284i			0.0979702i
$249$		0			0
$250$		−	0.551584i		−	0.0348852i
$251$			4.79529i			0.302676i	0.988482	+	0.151338i	$0.0483582\pi$
$251$			4.79529i			0.302676i	−0.988482	+	0.151338i	$0.951642\pi$
$252$		0			0
$253$		−	9.25593i		−	0.581915i
$254$		−	1.65957i		−	0.104131i
$255$		0			0
$256$	14.8874			0.930465
$257$	5.22489			0.325919			0.162960	−	0.986633i	$-0.447896\pi$
$257$	5.22489			0.325919			0.162960	+	0.986633i	$0.447896\pi$
$258$		0			0
$259$		−	5.01968i		−	0.311908i
$260$		−	3.88970i		−	0.241229i
$261$		0			0
$262$		−	1.24471i		−	0.0768985i
$263$			9.74914i			0.601158i	0.953757	+	0.300579i	$0.0971799\pi$
$263$			9.74914i			0.601158i	−0.953757	+	0.300579i	$0.902820\pi$
$264$		0			0
$265$	−27.5295			−1.69112
$266$			0.603409i			0.0369973i
$267$		0			0
$268$		−	20.1331i		−	1.22982i
$269$	−31.0937			−1.89582			−0.947908	−	0.318544i	$-0.896806\pi$
$269$	−31.0937			−1.89582			−0.947908	+	0.318544i	$0.896806\pi$
$270$		0			0
$271$			15.3090i			0.929953i	0.885323	+	0.464976i	$0.153937\pi$
$271$			15.3090i			0.929953i	−0.885323	+	0.464976i	$0.846063\pi$
$272$	1.81365			0.109969
$273$		0			0
$274$	−1.39356			−0.0841881
$275$			25.6795i			1.54853i
$276$		0			0
$277$		−	10.4686i		−	0.628999i	−0.949258	−	0.314500i	$-0.898163\pi$
$277$		−	10.4686i		−	0.628999i	0.949258	−	0.314500i	$-0.101837\pi$
$278$	−1.71254			−0.102711
$279$		0			0
$280$		−	3.27558i		−	0.195753i
$281$		−	14.0775i		−	0.839792i	−0.907572	−	0.419896i	$-0.862067\pi$
$281$		−	14.0775i		−	0.839792i	0.907572	−	0.419896i	$-0.137933\pi$
$282$		0			0
$283$	11.1584			0.663300			0.331650	−	0.943402i	$-0.392395\pi$
$283$	11.1584			0.663300			0.331650	+	0.943402i	$0.392395\pi$
$284$	−16.5278			−0.980743
$285$		0			0
$286$		−	0.277090i		−	0.0163847i
$287$	−4.82313			−0.284700
$288$		0			0
$289$	−16.7855			−0.987382
$290$	3.27810			0.192497
$291$		0			0
$292$		−	14.2136i		−	0.831788i
$293$			19.8777i			1.16127i	0.814164	+	0.580635i	$0.197196\pi$
$293$			19.8777i			1.16127i	−0.814164	+	0.580635i	$0.802804\pi$
$294$		0			0
$295$	−22.5223			−1.31130
$296$	−1.15433			−0.0670939
$297$		0			0
$298$		−	1.28019i		−	0.0741595i
$299$			1.33504i			0.0772074i
$300$		0			0
$301$		−	16.3400i		−	0.941823i
$302$	−1.44437			−0.0831142
$303$		0			0
$304$	−9.69928			−0.556292
$305$	1.16308			0.0665976
$306$		0			0
$307$			18.3005i			1.04446i	0.852804	+	0.522232i	$0.174900\pi$
$307$			18.3005i			1.04446i	−0.852804	+	0.522232i	$0.825100\pi$
$308$			16.4276i			0.936050i
$309$		0			0
$310$			1.30571i			0.0741591i
$311$			18.9469i			1.07438i	0.843462	+	0.537189i	$0.180514\pi$
$311$			18.9469i			1.07438i	−0.843462	+	0.537189i	$0.819486\pi$
$312$		0			0
$313$			28.4711i			1.60928i	0.593762	+	0.804641i	$0.297642\pi$
$313$			28.4711i			1.60928i	−0.593762	+	0.804641i	$0.702358\pi$
$314$		−	2.35562i		−	0.132935i
$315$		0			0
$316$		−	24.1355i		−	1.35773i
$317$			9.61414i			0.539984i	0.962863	+	0.269992i	$0.0870210\pi$
$317$			9.61414i			0.539984i	−0.962863	+	0.269992i	$0.912979\pi$
$318$		0			0
$319$	−32.9968			−1.84747
$320$	25.8553			1.44536
$321$		0			0
$322$			0.560148i			0.0312158i
$323$	−1.14719			−0.0638316
$324$		0			0
$325$		−	3.70392i		−	0.205457i
$326$	1.46567			0.0811761
$327$		0			0
$328$			1.10913i			0.0612414i
$329$		−	4.64461i		−	0.256066i
$330$		0			0
$331$			33.9506i			1.86609i	0.359757	+	0.933046i	$0.382860\pi$
$331$			33.9506i			1.86609i	−0.359757	+	0.933046i	$0.617140\pi$
$332$	−22.1122			−1.21357
$333$		0			0
$334$	1.36808	+	0.689550i	0.0748583	+	0.0377305i
$335$		−	34.1979i		−	1.86843i
$336$		0			0
$337$	12.7183			0.692810			0.346405	−	0.938085i	$-0.387402\pi$
$337$	12.7183			0.692810			0.346405	+	0.938085i	$0.387402\pi$
$338$		−	1.50122i		−	0.0816554i
$339$		0			0
$340$	3.10277			0.168272
$341$		−	13.1430i		−	0.711735i
$342$		0			0
$343$	−20.0915			−1.08484
$344$	−3.75755			−0.202594
$345$		0			0
$346$	−0.712787			−0.0383197
$347$	−1.61855			−0.0868883			−0.0434441	−	0.999056i	$-0.513833\pi$
$347$	−1.61855			−0.0868883			−0.0434441	+	0.999056i	$0.513833\pi$
$348$		0			0
$349$			30.5772i			1.63676i	0.574678	+	0.818380i	$0.305127\pi$
$349$			30.5772i			1.63676i	−0.574678	+	0.818380i	$0.694873\pi$
$350$		−	1.55407i		−	0.0830684i
$351$		0			0
$352$	5.67320			0.302383
$353$		−	35.8828i		−	1.90985i	−0.296849	−	0.954925i	$-0.595936\pi$
$353$		−	35.8828i		−	1.90985i	0.296849	−	0.954925i	$-0.404064\pi$
$354$		0			0
$355$	−28.0740			−1.49001
$356$		−	3.06322i		−	0.162351i
$357$		0			0
$358$	−1.13046			−0.0597464
$359$			1.86137i			0.0982395i	0.998793	+	0.0491197i	$0.0156416\pi$
$359$			1.86137i			0.0982395i	−0.998793	+	0.0491197i	$0.984358\pi$
$360$		0			0
$361$	−12.8649			−0.677099
$362$			1.28861i			0.0677278i
$363$		0			0
$364$		−	2.36946i		−	0.124193i
$365$		−	24.1431i		−	1.26371i
$366$		0			0
$367$	−13.0548			−0.681457			−0.340729	−	0.940162i	$-0.610674\pi$
$367$	−13.0548			−0.681457			−0.340729	+	0.940162i	$0.610674\pi$
$368$	−9.00390			−0.469361
$369$		0			0
$370$	−0.976910			−0.0507871
$371$	−16.7699			−0.870650
$372$		0			0
$373$		−	19.6682i		−	1.01838i	−0.860654	−	0.509191i	$-0.829945\pi$
$373$		−	19.6682i		−	1.01838i	0.860654	−	0.509191i	$-0.170055\pi$
$374$	0.221032			0.0114293
$375$		0			0
$376$	−1.06808			−0.0550818
$377$	4.75934			0.245118
$378$		0			0
$379$			5.15236i			0.264659i	0.991206	+	0.132329i	$0.0422457\pi$
$379$			5.15236i			0.264659i	−0.991206	+	0.132329i	$0.957754\pi$
$380$	−16.5934			−0.851223
$381$		0			0
$382$	−1.38177			−0.0706976
$383$		−	16.3244i		−	0.834139i	−0.908874	−	0.417070i	$-0.863057\pi$
$383$		−	16.3244i		−	0.834139i	0.908874	−	0.417070i	$-0.136943\pi$
$384$		0			0
$385$			27.9038i			1.42211i
$386$	0.759161			0.0386403
$387$		0			0
$388$	−5.02089			−0.254897
$389$	2.31423			0.117336			0.0586681	−	0.998278i	$-0.481315\pi$
$389$	2.31423			0.117336			0.0586681	+	0.998278i	$0.481315\pi$
$390$		0			0
$391$	−1.06495			−0.0538567
$392$			1.31245i			0.0662887i
$393$		0			0
$394$		−	1.89189i		−	0.0953121i
$395$		−	40.9965i		−	2.06276i
$396$		0			0
$397$	0.625961			0.0314161			0.0157081	−	0.999877i	$-0.495000\pi$
$397$	0.625961			0.0314161			0.0157081	+	0.999877i	$0.495000\pi$
$398$		−	0.717146i		−	0.0359473i
$399$		0			0
$400$	24.9803			1.24902
$401$	14.5423			0.726208			0.363104	−	0.931749i	$-0.381717\pi$
$401$	14.5423			0.726208			0.363104	+	0.931749i	$0.381717\pi$
$402$		0			0
$403$			1.89570i			0.0944316i
$404$	26.0861			1.29783
$405$		0			0
$406$	1.99689			0.0991040
$407$	9.83342			0.487424
$408$		0			0
$409$	11.8264			0.584776			0.292388	−	0.956300i	$-0.405550\pi$
$409$	11.8264			0.584776			0.292388	+	0.956300i	$0.405550\pi$
$410$			0.938658i			0.0463570i
$411$		0			0
$412$			13.0821i			0.644508i
$413$	−13.7197			−0.675103
$414$		0			0
$415$	−37.5597			−1.84373
$416$	−0.818282			−0.0401196
$417$		0			0
$418$	−1.18206			−0.0578164
$419$		−	10.1514i		−	0.495926i	−0.968769	−	0.247963i	$-0.920239\pi$
$419$		−	10.1514i		−	0.495926i	0.968769	−	0.247963i	$-0.0797612\pi$
$420$		0			0
$421$	8.42832			0.410771			0.205386	−	0.978681i	$-0.434155\pi$
$421$	8.42832			0.410771			0.205386	+	0.978681i	$0.434155\pi$
$422$			1.67131i			0.0813582i
$423$		0			0
$424$			3.85641i			0.187284i
$425$	2.95458			0.143318
$426$		0			0
$427$	0.708502			0.0342868
$428$			1.87511i			0.0906369i
$429$		0			0
$430$	−3.18003			−0.153355
$431$			2.67224i			0.128717i	0.997927	+	0.0643585i	$0.0205001\pi$
$431$			2.67224i			0.128717i	−0.997927	+	0.0643585i	$0.979500\pi$
$432$		0			0
$433$	31.1633			1.49761			0.748806	−	0.662789i	$-0.230627\pi$
$433$	31.1633			1.49761			0.748806	+	0.662789i	$0.230627\pi$
$434$			0.795386i			0.0381798i
$435$		0			0
$436$			25.7376i			1.23261i
$437$	5.69526			0.272441
$438$		0			0
$439$			26.1263i			1.24694i	0.781848	+	0.623469i	$0.214277\pi$
$439$			26.1263i			1.24694i	−0.781848	+	0.623469i	$0.785723\pi$
$440$	6.41677			0.305908
$441$		0			0
$442$	−0.0318808			−0.00151641
$443$	−12.5354			−0.595574			−0.297787	−	0.954632i	$-0.596248\pi$
$443$	−12.5354			−0.595574			−0.297787	+	0.954632i	$0.596248\pi$
$444$		0			0
$445$		−	5.20317i		−	0.246654i
$446$			1.72203i			0.0815405i
$447$		0			0
$448$	15.7501			0.744120
$449$		−	19.3041i		−	0.911016i	−0.890232	−	0.455508i	$-0.849458\pi$
$449$		−	19.3041i		−	0.911016i	0.890232	−	0.455508i	$-0.150542\pi$
$450$		0			0
$451$		−	9.44838i		−	0.444907i
$452$	30.0166			1.41186
$453$		0			0
$454$		−	0.241250i		−	0.0113224i
$455$		−	4.02475i		−	0.188683i
$456$		0			0
$457$		−	37.8418i		−	1.77016i	−0.465435	−	0.885082i	$-0.654102\pi$
$457$		−	37.8418i		−	1.77016i	0.465435	−	0.885082i	$-0.345898\pi$
$458$			1.43151i			0.0668901i
$459$		0			0
$460$	−15.4038			−0.718204
$461$		−	8.58712i		−	0.399942i	−0.979802	−	0.199971i	$-0.935915\pi$
$461$		−	8.58712i		−	0.399942i	0.979802	−	0.199971i	$-0.0640848\pi$
$462$		0			0
$463$			13.3946i			0.622499i	0.950328	+	0.311250i	$0.100748\pi$
$463$			13.3946i			0.622499i	−0.950328	+	0.311250i	$0.899252\pi$
$464$			32.0983i			1.49013i
$465$		0			0
$466$	1.16727			0.0540727
$467$			18.7467i			0.867495i	0.901034	+	0.433748i	$0.142809\pi$
$467$			18.7467i			0.867495i	−0.901034	+	0.433748i	$0.857191\pi$
$468$		0			0
$469$		−	20.8321i		−	0.961935i
$470$	−0.903915			−0.0416945
$471$		0			0
$472$			3.15499i			0.145220i
$473$	32.0096			1.47180
$474$		0			0
$475$	−15.8009			−0.724994
$476$	1.89009			0.0866322
$477$		0			0
$478$	−3.14131			−0.143680
$479$	−24.6337			−1.12554			−0.562771	−	0.826613i	$-0.690265\pi$
$479$	−24.6337			−1.12554			−0.562771	+	0.826613i	$0.690265\pi$
$480$		0			0
$481$	−1.41834			−0.0646706
$482$	1.82046			0.0829197
$483$		0			0
$484$	−10.3358			−0.469809
$485$	−8.52846			−0.387257
$486$		0			0
$487$		−	13.0382i		−	0.590819i	−0.955371	−	0.295409i	$-0.904544\pi$
$487$		−	13.0382i		−	0.590819i	0.955371	−	0.295409i	$-0.0954560\pi$
$488$		−	0.162927i		−	0.00737538i
$489$		0			0
$490$			1.11073i			0.0501776i
$491$			16.5488i			0.746837i	0.927663	+	0.373418i	$0.121814\pi$
$491$			16.5488i			0.746837i	−0.927663	+	0.373418i	$0.878186\pi$
$492$		0			0
$493$			3.79647i			0.170984i
$494$	0.170496			0.00767098
$495$		0			0
$496$	−12.7852			−0.574070
$497$	−17.1016			−0.767111
$498$		0			0
$499$			40.2933i			1.80377i	0.431971	+	0.901887i	$0.357818\pi$
$499$			40.2933i			1.80377i	−0.431971	+	0.901887i	$0.642182\pi$
$500$	9.23992			0.413222
$501$		0			0
$502$	0.568494			0.0253731
$503$		−	21.9987i		−	0.980872i	−0.871477	−	0.490436i	$-0.836838\pi$
$503$		−	21.9987i		−	0.980872i	0.871477	−	0.490436i	$-0.163162\pi$
$504$		0			0
$505$	44.3096			1.97175
$506$	−1.09731			−0.0487816
$507$		0			0
$508$	27.8005			1.23345
$509$			17.4225i			0.772239i	0.922449	+	0.386119i	$0.126185\pi$
$509$			17.4225i			0.772239i	−0.922449	+	0.386119i	$0.873815\pi$
$510$		0			0
$511$		−	14.7071i		−	0.650602i
$512$		−	9.21957i		−	0.407451i
$513$		0			0
$514$		−	0.619423i		−	0.0273216i
$515$			22.2211i			0.979180i
$516$		0			0
$517$	9.09866			0.400159
$518$	−0.595096			−0.0261470
$519$		0			0
$520$	−0.925532			−0.0405872
$521$	8.46186			0.370721			0.185360	−	0.982671i	$-0.440655\pi$
$521$	8.46186			0.370721			0.185360	+	0.982671i	$0.440655\pi$
$522$		0			0
$523$	−19.5811			−0.856220			−0.428110	−	0.903727i	$-0.640820\pi$
$523$	−19.5811			−0.856220			−0.428110	+	0.903727i	$0.640820\pi$
$524$	20.8509			0.910876
$525$		0			0
$526$	1.15579			0.0503946
$527$	−1.51218			−0.0658716
$528$		0			0
$529$	−17.7131			−0.770133
$530$			3.26369i			0.141766i
$531$		0			0
$532$	−10.1081			−0.438240
$533$			1.36280i			0.0590294i
$534$		0			0
$535$			3.18505i			0.137702i
$536$	−4.79055			−0.206920
$537$		0			0
$538$			3.68624i			0.158925i
$539$		−	11.1804i		−	0.481574i
$540$		0			0
$541$		−	4.69202i		−	0.201726i	−0.994900	−	0.100863i	$-0.967840\pi$
$541$		−	4.69202i		−	0.201726i	0.994900	−	0.100863i	$-0.0321603\pi$
$542$	1.81491			0.0779573
$543$		0			0
$544$		−	0.652735i		−	0.0279858i
$545$			43.7177i			1.87266i
$546$		0			0
$547$			33.0384i			1.41262i	0.707902	+	0.706311i	$0.249642\pi$
$547$			33.0384i			1.41262i	−0.707902	+	0.706311i	$0.750358\pi$
$548$		−	23.3444i		−	0.997223i
$549$		0			0
$550$	3.04438			0.129813
$551$		−	20.3032i		−	0.864947i
$552$		0			0
$553$		−	24.9735i		−	1.06198i
$554$	−1.24108			−0.0527286
$555$		0			0
$556$		−	28.6878i		−	1.21663i
$557$			9.68186i			0.410234i	0.978738	+	0.205117i	$0.0657574\pi$
$557$			9.68186i			0.410234i	−0.978738	+	0.205117i	$0.934243\pi$
$558$		0			0
$559$	−4.61695			−0.195276
$560$	27.1440			1.14704
$561$		0			0
$562$	−1.66892			−0.0703992
$563$			24.3475i			1.02612i	0.858352	+	0.513062i	$0.171489\pi$
$563$			24.3475i			1.02612i	−0.858352	+	0.513062i	$0.828511\pi$
$564$		0			0
$565$	50.9861			2.14500
$566$		−	1.32286i		−	0.0556040i
$567$		0			0
$568$			3.93269i			0.165012i
$569$	10.1841			0.426942			0.213471	−	0.976949i	$-0.431523\pi$
$569$	10.1841			0.426942			0.213471	+	0.976949i	$0.431523\pi$
$570$		0			0
$571$		−	13.3401i		−	0.558266i	−0.960252	−	0.279133i	$-0.909953\pi$
$571$		−	13.3401i		−	0.558266i	0.960252	−	0.279133i	$-0.0900470\pi$
$572$	4.64170			0.194079
$573$		0			0
$574$			0.571794i			0.0238662i
$575$	−14.6681			−0.611700
$576$		0			0
$577$	−8.52219			−0.354783			−0.177392	−	0.984140i	$-0.556766\pi$
$577$	−8.52219			−0.354783			−0.177392	+	0.984140i	$0.556766\pi$
$578$			1.98996i			0.0827715i
$579$		0			0
$580$			54.9134i			2.28015i
$581$	−22.8799			−0.949218
$582$		0			0
$583$		−	32.8518i		−	1.36058i
$584$	−3.38204			−0.139950
$585$		0			0
$586$	2.35656			0.0973485
$587$	1.70252			0.0702703			0.0351352	−	0.999383i	$-0.488814\pi$
$587$	1.70252			0.0702703			0.0351352	+	0.999383i	$0.488814\pi$
$588$		0			0
$589$	8.08703			0.333220
$590$			2.67007i			0.109925i
$591$		0			0
$592$		−	9.56566i		−	0.393146i
$593$	32.2861			1.32583			0.662916	−	0.748693i	$-0.269318\pi$
$593$	32.2861			1.32583			0.662916	+	0.748693i	$0.269318\pi$
$594$		0			0
$595$	3.21050			0.131618
$596$	21.4453			0.878432
$597$		0			0
$598$	0.158273			0.00647225
$599$			34.4826i			1.40892i	0.709743	+	0.704461i	$0.248811\pi$
$599$			34.4826i			1.40892i	−0.709743	+	0.704461i	$0.751189\pi$
$600$		0			0
$601$	33.1107			1.35062			0.675308	−	0.737536i	$-0.264011\pi$
$601$	33.1107			1.35062			0.675308	+	0.737536i	$0.264011\pi$
$602$	−1.93715			−0.0789524
$603$		0			0
$604$		−	24.1955i		−	0.984502i
$605$	−17.5563			−0.713767
$606$		0			0
$607$		−	5.64008i		−	0.228924i	−0.993428	−	0.114462i	$-0.963486\pi$
$607$		−	5.64008i		−	0.228924i	0.993428	−	0.114462i	$-0.0365144\pi$
$608$			3.49078i			0.141570i
$609$		0			0
$610$		−	0.137886i		−	0.00558283i
$611$	−1.31236			−0.0530923
$612$		0			0
$613$	15.1021			0.609967			0.304983	−	0.952358i	$-0.401349\pi$
$613$	15.1021			0.609967			0.304983	+	0.952358i	$0.401349\pi$
$614$	2.16957			0.0875566
$615$		0			0
$616$	3.90885			0.157492
$617$			24.4177i			0.983019i	0.870872	+	0.491509i	$0.163555\pi$
$617$			24.4177i			0.983019i	−0.870872	+	0.491509i	$0.836445\pi$
$618$		0			0
$619$		−	32.2634i		−	1.29678i	−0.761309	−	0.648389i	$-0.775443\pi$
$619$		−	32.2634i		−	1.29678i	0.761309	−	0.648389i	$-0.224557\pi$
$620$	−21.8727			−0.878428
$621$		0			0
$622$	2.24620			0.0900644
$623$		−	3.16957i		−	0.126986i
$624$		0			0
$625$	−16.2014			−0.648056
$626$	3.37532			0.134905
$627$		0			0
$628$	39.4605			1.57464
$629$		−	1.13139i		−	0.0451115i
$630$		0			0
$631$	−19.7475			−0.786136			−0.393068	−	0.919509i	$-0.628586\pi$
$631$	−19.7475			−0.786136			−0.393068	+	0.919509i	$0.628586\pi$
$632$	−5.74291			−0.228441
$633$		0			0
$634$	1.13978			0.0452664
$635$	47.2218			1.87394
$636$		0			0
$637$			1.61262i			0.0638944i
$638$			3.91185i			0.154872i
$639$		0			0
$640$		−	12.5734i		−	0.497007i
$641$	−45.6058			−1.80132			−0.900661	−	0.434523i	$-0.856917\pi$
$641$	−45.6058			−1.80132			−0.900661	+	0.434523i	$0.856917\pi$
$642$		0			0
$643$		−	30.0179i		−	1.18379i	−0.806016	−	0.591894i	$-0.798380\pi$
$643$		−	30.0179i		−	1.18379i	0.806016	−	0.591894i	$-0.201620\pi$
$644$	−9.38338			−0.369757
$645$		0			0
$646$			0.136003i			0.00535096i
$647$	−38.0213			−1.49477			−0.747385	−	0.664391i	$-0.768691\pi$
$647$	−38.0213			−1.49477			−0.747385	+	0.664391i	$0.768691\pi$
$648$		0			0
$649$		−	26.8765i		−	1.05500i
$650$	−0.439110			−0.0172233
$651$		0			0
$652$			24.5524i			0.961545i
$653$			43.8556i			1.71620i	0.513481	+	0.858101i	$0.328356\pi$
$653$			43.8556i			1.71620i	−0.513481	+	0.858101i	$0.671644\pi$
$654$		0			0
$655$	35.4172			1.38387
$656$	−9.19111			−0.358853
$657$		0			0
$658$	−0.550630			−0.0214658
$659$	13.5649			0.528413			0.264206	−	0.964466i	$-0.414890\pi$
$659$	13.5649			0.528413			0.264206	+	0.964466i	$0.414890\pi$
$660$		0			0
$661$		−	9.54412i		−	0.371223i	−0.982623	−	0.185612i	$-0.940573\pi$
$661$		−	9.54412i		−	0.371223i	0.982623	−	0.185612i	$-0.0594266\pi$
$662$	4.02493			0.156433
$663$		0			0
$664$			5.26147i			0.204185i
$665$	−17.1695			−0.665804
$666$		0			0
$667$		−	18.8476i		−	0.729783i
$668$	−11.5511	+	22.9176i	−0.446925	+	0.886709i
$669$		0			0
$670$	−4.05425			−0.156629
$671$			1.38794i			0.0535807i
$672$		0			0
$673$			12.7781i			0.492558i	0.969199	+	0.246279i	$0.0792080\pi$
$673$			12.7781i			0.492558i	−0.969199	+	0.246279i	$0.920792\pi$
$674$		−	1.50779i		−	0.0580778i
$675$		0			0
$676$	25.1478			0.967223
$677$			35.0629i			1.34758i	0.738924	+	0.673789i	$0.235334\pi$
$677$			35.0629i			1.34758i	−0.738924	+	0.673789i	$0.764666\pi$
$678$		0			0
$679$	−5.19521			−0.199374
$680$		−	0.738287i		−	0.0283120i
$681$		0			0
$682$	−1.55814			−0.0596642
$683$	−41.8232			−1.60032			−0.800161	−	0.599785i	$-0.795253\pi$
$683$	−41.8232			−1.60032			−0.800161	+	0.599785i	$0.795253\pi$
$684$		0			0
$685$		−	39.6527i		−	1.51505i
$686$			2.38190i			0.0909415i
$687$		0			0
$688$		−	31.1380i		−	1.18713i
$689$			4.73842i			0.180519i
$690$		0			0
$691$		−	18.1541i		−	0.690615i	−0.938490	−	0.345308i	$-0.887775\pi$
$691$		−	18.1541i		−	0.690615i	0.938490	−	0.345308i	$-0.112225\pi$
$692$		−	11.9403i		−	0.453903i
$693$		0			0
$694$			0.191883i			0.00728378i
$695$		−	48.7289i		−	1.84839i
$696$		0			0
$697$	−1.08709			−0.0411765
$698$	3.62500			0.137208
$699$		0			0
$700$	26.0331			0.983960
$701$		−	32.4413i		−	1.22529i	−0.790358	−	0.612645i	$-0.790106\pi$
$701$		−	32.4413i		−	1.22529i	0.790358	−	0.612645i	$-0.209894\pi$
$702$		0			0
$703$			6.05060i			0.228203i
$704$			30.8539i			1.16285i
$705$		0			0
$706$	−4.25400			−0.160101
$707$	26.9917			1.01513
$708$		0			0
$709$		−	7.11018i		−	0.267028i	−0.991047	−	0.133514i	$-0.957374\pi$
$709$		−	7.11018i		−	0.267028i	0.991047	−	0.133514i	$-0.0426262\pi$
$710$			3.32824i			0.124907i
$711$		0			0
$712$	−0.728876			−0.0273158
$713$	7.50724			0.281148
$714$		0			0
$715$	7.88436			0.294858
$716$		−	18.9369i		−	0.707707i
$717$		0			0
$718$	0.220670			0.00823535
$719$	−31.9727			−1.19238			−0.596189	−	0.802844i	$-0.703319\pi$
$719$	−31.9727			−1.19238			−0.596189	+	0.802844i	$0.703319\pi$
$720$		0			0
$721$			13.5363i			0.504117i
$722$			1.52517i			0.0567608i
$723$		0			0
$724$	−21.5863			−0.802248
$725$			52.2906i			1.94203i
$726$		0			0
$727$		−	36.3017i		−	1.34636i	−0.739481	−	0.673178i	$-0.764929\pi$
$727$		−	36.3017i		−	1.34636i	0.739481	−	0.673178i	$-0.235071\pi$
$728$	−0.563798			−0.0208958
$729$		0			0
$730$	−2.86223			−0.105936
$731$		−	3.68289i		−	0.136217i
$732$		0			0
$733$	27.3820			1.01138			0.505688	−	0.862716i	$-0.331239\pi$
$733$	27.3820			1.01138			0.505688	+	0.862716i	$0.331239\pi$
$734$			1.54768i			0.0571261i
$735$		0			0
$736$			3.24051i			0.119447i
$737$	40.8094			1.50323
$738$		0			0
$739$			9.34284i			0.343682i	0.985125	+	0.171841i	$0.0549715\pi$
$739$			9.34284i			0.343682i	−0.985125	+	0.171841i	$0.945028\pi$
$740$		−	16.3648i		−	0.601583i
$741$		0			0
$742$			1.98811i			0.0729860i
$743$			6.86237i			0.251756i	0.992046	+	0.125878i	$0.0401748\pi$
$743$			6.86237i			0.251756i	−0.992046	+	0.125878i	$0.959825\pi$
$744$		0			0
$745$	36.4268			1.33457
$746$	−2.33172			−0.0853703
$747$		0			0
$748$			3.70263i			0.135382i
$749$			1.94021i			0.0708938i
$750$		0			0
$751$			45.1764i			1.64851i	0.566219	+	0.824255i	$0.308406\pi$
$751$			45.1764i			1.64851i	−0.566219	+	0.824255i	$0.691594\pi$
$752$		−	8.85091i		−	0.322760i
$753$		0			0
$754$		−	0.564231i		−	0.0205481i
$755$		−	41.0984i		−	1.49572i
$756$		0			0
$757$	32.8680			1.19461			0.597304	−	0.802015i	$-0.296239\pi$
$757$	32.8680			1.19461			0.597304	+	0.802015i	$0.296239\pi$
$758$	0.610825			0.0221862
$759$		0			0
$760$			3.94830i			0.143220i
$761$		−	10.6770i		−	0.387040i	−0.981096	−	0.193520i	$-0.938010\pi$
$761$		−	10.6770i		−	0.387040i	0.981096	−	0.193520i	$-0.0619905\pi$
$762$		0			0
$763$			26.6311i			0.964111i
$764$		−	23.1469i		−	0.837425i
$765$		0			0
$766$	−1.93530			−0.0699253
$767$			3.87657i			0.139975i
$768$		0			0
$769$		−	0.392212i		−	0.0141435i	−0.999975	−	0.00707176i	$-0.997749\pi$
$769$		−	0.392212i		−	0.0141435i	0.999975	−	0.00707176i	$-0.00225103\pi$
$770$	3.30807			0.119215
$771$		0			0
$772$			12.7172i			0.457701i
$773$	−20.9889			−0.754917			−0.377458	−	0.926027i	$-0.623202\pi$
$773$	−20.9889			−0.754917			−0.377458	+	0.926027i	$0.623202\pi$
$774$		0			0
$775$	−20.8280			−0.748164
$776$			1.19469i			0.0428869i
$777$		0			0
$778$		−	0.274358i		−	0.00983621i
$779$	5.81368			0.208297
$780$		0			0
$781$		−	33.5015i		−	1.19878i
$782$			0.126252i			0.00451477i
$783$		0			0
$784$	−10.8760			−0.388428
$785$	67.0273			2.39231
$786$		0			0
$787$		−	1.85506i		−	0.0661256i	−0.999453	−	0.0330628i	$-0.989474\pi$
$787$		−	1.85506i		−	0.0661256i	0.999453	−	0.0330628i	$-0.0105261\pi$
$788$	31.6922			1.12899
$789$		0			0
$790$	−4.86024			−0.172919
$791$	31.0588			1.10432
$792$		0			0
$793$		−	0.200191i		−	0.00710898i
$794$		−	0.0742093i		−	0.00263359i
$795$		0			0
$796$	12.0133			0.425802
$797$	−40.5573			−1.43661			−0.718306	−	0.695727i	$-0.755082\pi$
$797$	−40.5573			−1.43661			−0.718306	+	0.695727i	$0.755082\pi$
$798$		0			0
$799$		−	1.04685i		−	0.0370350i
$800$		−	8.99043i		−	0.317860i
$801$		0			0
$802$		−	1.72403i		−	0.0608775i
$803$	28.8107			1.01671
$804$		0			0
$805$	−15.9386			−0.561760
$806$	0.224740			0.00791614
$807$		0			0
$808$		−	6.20702i		−	0.218362i
$809$			22.6422i			0.796057i	0.917373	+	0.398029i	$0.130306\pi$
$809$			22.6422i			0.796057i	−0.917373	+	0.398029i	$0.869694\pi$
$810$		0			0
$811$			48.4520i			1.70138i	0.525667	+	0.850691i	$0.323816\pi$
$811$			48.4520i			1.70138i	−0.525667	+	0.850691i	$0.676184\pi$
$812$			33.4511i			1.17390i
$813$		0			0
$814$		−	1.16578i		−	0.0408604i
$815$			41.7045i			1.46085i
$816$		0			0
$817$			19.6958i			0.689070i
$818$		−	1.40205i		−	0.0490214i
$819$		0			0
$820$	−15.7240			−0.549107
$821$	−52.7568			−1.84122			−0.920612	−	0.390479i	$-0.872309\pi$
$821$	−52.7568			−1.84122			−0.920612	+	0.390479i	$0.872309\pi$
$822$		0			0
$823$			32.7533i			1.14171i	0.821052	+	0.570854i	$0.193388\pi$
$823$			32.7533i			1.14171i	−0.821052	+	0.570854i	$0.806612\pi$
$824$	3.11280			0.108440
$825$		0			0
$826$			1.62651i			0.0565934i
$827$	−10.8865			−0.378560			−0.189280	−	0.981923i	$-0.560615\pi$
$827$	−10.8865			−0.378560			−0.189280	+	0.981923i	$0.560615\pi$
$828$		0			0
$829$		−	14.4187i		−	0.500783i	−0.968145	−	0.250392i	$-0.919441\pi$
$829$		−	14.4187i		−	0.500783i	0.968145	−	0.250392i	$-0.0805593\pi$
$830$			4.45279i			0.154559i
$831$		0			0
$832$		−	4.45026i		−	0.154285i
$833$	−1.28637			−0.0445701
$834$		0			0
$835$	−19.6206	+	38.9277i	−0.678999	+	1.34715i
$836$		−	19.8014i		−	0.684846i
$837$		0			0
$838$	−1.20347			−0.0415731
$839$		−	10.0744i		−	0.347806i	−0.984763	−	0.173903i	$-0.944362\pi$
$839$		−	10.0744i		−	0.347806i	0.984763	−	0.173903i	$-0.0556379\pi$
$840$		0			0
$841$	−38.1906			−1.31692
$842$		−	0.999199i		−	0.0344347i
$843$		0			0
$844$	−27.9972			−0.963702
$845$	42.7159			1.46947
$846$		0			0
$847$	−10.6946			−0.367472
$848$	−31.9572			−1.09742
$849$		0			0
$850$		−	0.350273i		−	0.0120143i
$851$			5.61681i			0.192542i
$852$		0			0
$853$	−35.3325			−1.20976			−0.604881	−	0.796316i	$-0.706779\pi$
$853$	−35.3325			−1.20976			−0.604881	+	0.796316i	$0.706779\pi$
$854$		−	0.0839947i		−	0.00287424i
$855$		0			0
$856$	0.446171			0.0152498
$857$			37.1234i			1.26811i	0.773288	+	0.634055i	$0.218611\pi$
$857$			37.1234i			1.26811i	−0.773288	+	0.634055i	$0.781389\pi$
$858$		0			0
$859$	31.9133			1.08887			0.544434	−	0.838804i	$-0.316745\pi$
$859$	31.9133			1.08887			0.544434	+	0.838804i	$0.316745\pi$
$860$		−	53.2706i		−	1.81651i
$861$		0			0
$862$	0.316800			0.0107903
$863$		−	13.5193i		−	0.460203i	−0.973167	−	0.230101i	$-0.926094\pi$
$863$		−	13.5193i		−	0.460203i	0.973167	−	0.230101i	$-0.0739057\pi$
$864$		0			0
$865$		−	20.2818i		−	0.689601i
$866$		−	3.69449i		−	0.125544i
$867$		0			0
$868$	−13.3240			−0.452246
$869$	48.9223			1.65958
$870$		0			0
$871$	−5.88620			−0.199446
$872$	6.12410			0.207388
$873$		0			0
$874$		−	0.675188i		−	0.0228386i
$875$	9.56071			0.323211
$876$		0			0
$877$	10.7585			0.363289			0.181645	−	0.983364i	$-0.441858\pi$
$877$	10.7585			0.363289			0.181645	+	0.983364i	$0.441858\pi$
$878$	3.09734			0.104530
$879$		0			0
$880$			53.1744i			1.79251i
$881$	9.85932			0.332169			0.166085	−	0.986112i	$-0.446888\pi$
$881$	9.85932			0.332169			0.166085	+	0.986112i	$0.446888\pi$
$882$		0			0
$883$	42.7640			1.43912			0.719562	−	0.694428i	$-0.244343\pi$
$883$	42.7640			1.43912			0.719562	+	0.694428i	$0.244343\pi$
$884$		−	0.534054i		−	0.0179622i
$885$		0			0
$886$			1.48610i			0.0499265i
$887$	22.7137			0.762651			0.381325	−	0.924441i	$-0.375468\pi$
$887$	22.7137			0.762651			0.381325	+	0.924441i	$0.375468\pi$
$888$		0			0
$889$	28.7657			0.964771
$890$	−0.616849			−0.0206768
$891$		0			0
$892$	−28.8468			−0.965861
$893$			5.59849i			0.187346i
$894$		0			0
$895$		−	32.1662i		−	1.07520i
$896$		−	7.65923i		−	0.255877i
$897$		0			0
$898$	−2.28855			−0.0763698
$899$		−	26.7628i		−	0.892590i
$900$		0			0
$901$	−3.77978			−0.125923
$902$	−1.12013			−0.0372962
$903$		0			0
$904$		−	7.14228i		−	0.237549i
$905$	−36.6663			−1.21883
$906$		0			0
$907$	34.4261			1.14310			0.571550	−	0.820567i	$-0.306342\pi$
$907$	34.4261			1.14310			0.571550	+	0.820567i	$0.306342\pi$
$908$	4.04133			0.134116
$909$		0			0
$910$	−0.477144			−0.0158172
$911$			0.951674i			0.0315304i	0.999876	+	0.0157652i	$0.00501842\pi$
$911$			0.951674i			0.0315304i	−0.999876	+	0.0157652i	$0.994982\pi$
$912$		0			0
$913$		−	44.8211i		−	1.48336i
$914$	−4.48624			−0.148392
$915$		0			0
$916$	−23.9801			−0.792325
$917$	21.5748			0.712463
$918$		0			0
$919$	54.9946			1.81411			0.907053	−	0.421017i	$-0.138327\pi$
$919$	54.9946			1.81411			0.907053	+	0.421017i	$0.138327\pi$
$920$			3.66523i			0.120839i
$921$		0			0
$922$	−1.01803			−0.0335269
$923$			4.83214i			0.159052i
$924$		0			0
$925$		−	15.5832i		−	0.512373i
$926$	1.58796			0.0521837
$927$		0			0
$928$	11.5522			0.379220
$929$			27.3360i			0.896866i	0.893816	+	0.448433i	$0.148018\pi$
$929$			27.3360i			0.896866i	−0.893816	+	0.448433i	$0.851982\pi$
$930$		0			0
$931$	6.87941			0.225464
$932$			19.5536i			0.640500i
$933$		0			0
$934$	2.22247			0.0727215
$935$			6.28927i			0.205681i
$936$		0			0
$937$			48.7114i			1.59133i	0.605735	+	0.795666i	$0.292879\pi$
$937$			48.7114i			1.59133i	−0.605735	+	0.795666i	$0.707121\pi$
$938$	−2.46969			−0.0806384
$939$		0			0
$940$		−	15.1420i		−	0.493878i
$941$	−3.68551			−0.120144			−0.0600721	−	0.998194i	$-0.519133\pi$
$941$	−3.68551			−0.120144			−0.0600721	+	0.998194i	$0.519133\pi$
$942$		0			0
$943$	5.39687			0.175746
$944$	−26.1447			−0.850938
$945$		0			0
$946$		−	3.79482i		−	0.123380i
$947$		−	8.30571i		−	0.269899i	−0.990852	−	0.134950i	$-0.956913\pi$
$947$		−	8.30571i		−	0.269899i	0.990852	−	0.134950i	$-0.0430873\pi$
$948$		0			0
$949$	−4.15555			−0.134895
$950$			1.87323i			0.0607757i
$951$		0			0
$952$		−	0.449736i		−	0.0145760i
$953$	−24.0079			−0.777692			−0.388846	−	0.921303i	$-0.627126\pi$
$953$	−24.0079			−0.777692			−0.388846	+	0.921303i	$0.627126\pi$
$954$		0			0
$955$		−	39.3172i		−	1.27227i
$956$		−	52.6219i		−	1.70192i
$957$		0			0
$958$			2.92039i			0.0943534i
$959$		−	24.1549i		−	0.780002i
$960$		0			0
$961$	−20.3400			−0.656130
$962$			0.168147i			0.00542129i
$963$		0			0
$964$			30.4956i			0.982198i
$965$			21.6013i			0.695370i
$966$		0			0
$967$	46.3110			1.48926			0.744630	−	0.667477i	$-0.232626\pi$
$967$	46.3110			1.48926			0.744630	+	0.667477i	$0.232626\pi$
$968$			2.45934i			0.0790463i
$969$		0			0
$970$			1.01107i			0.0324635i
$971$	−0.433495			−0.0139115			−0.00695576	−	0.999976i	$-0.502214\pi$
$971$	−0.433495			−0.0139115			−0.00695576	+	0.999976i	$0.502214\pi$
$972$		0			0
$973$		−	29.6837i		−	0.951617i
$974$	−1.54572			−0.0495279
$975$		0			0
$976$	1.35014			0.0432171
$977$	19.0337			0.608943			0.304472	−	0.952521i	$-0.401520\pi$
$977$	19.0337			0.608943			0.304472	+	0.952521i	$0.401520\pi$
$978$		0			0
$979$	6.20910			0.198444
$980$	−18.6065			−0.594362
$981$		0			0
$982$	1.96190			0.0626068
$983$	38.8638			1.23956			0.619782	−	0.784774i	$-0.287221\pi$
$983$	38.8638			1.23956			0.619782	+	0.784774i	$0.287221\pi$
$984$		0			0
$985$	53.8322			1.71524
$986$	0.450081			0.0143335
$987$		0			0
$988$			2.85608i			0.0908641i
$989$			18.2838i			0.581390i
$990$		0			0
$991$			24.6235i			0.782191i	0.920350	+	0.391096i	$0.127904\pi$
$991$			24.6235i			0.782191i	−0.920350	+	0.391096i	$0.872096\pi$
$992$			4.60139i			0.146094i
$993$		0			0
$994$			2.02744i			0.0643064i
$995$	20.4058			0.646907
$996$		0			0
$997$	29.3680			0.930095			0.465047	−	0.885286i	$-0.346037\pi$
$997$	29.3680			0.930095			0.465047	+	0.885286i	$0.346037\pi$
$998$	4.77687			0.151209
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.11	✓	56
3.2	odd	2	inner	1503.2.c.a.1502.46	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.45	yes	56
501.500	even	2	inner	1503.2.c.a.1502.12	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.11	✓	56	1.1	even	1	trivial
1503.2.c.a.1502.12	yes	56	501.500	even	2	inner
1503.2.c.a.1502.45	yes	56	167.166	odd	2	inner
1503.2.c.a.1502.46	yes	56	3.2	odd	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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.118553i q^{2} +1.98595 q^{4} +3.37331 q^{5} +2.05489 q^{7} -0.472544i q^{8} +O(q^{10})\)	\(q-0.118553i q^{2} +1.98595 q^{4} +3.37331 q^{5} +2.05489 q^{7} -0.472544i q^{8} -0.399915i q^{10} +4.02548i q^{11} -0.580620i q^{13} -0.243613i q^{14} +3.91587 q^{16} +0.463155 q^{17} -2.47692 q^{19} +6.69922 q^{20} +0.477231 q^{22} -2.29934 q^{23} +6.37925 q^{25} -0.0688340 q^{26} +4.08091 q^{28} +8.19699i q^{29} -3.26496 q^{31} -1.40932i q^{32} -0.0549081i q^{34} +6.93180 q^{35} -2.44279i q^{37} +0.293645i q^{38} -1.59404i q^{40} -2.34714 q^{41} -7.95176i q^{43} +7.99438i q^{44} +0.272592i q^{46} -2.26027i q^{47} -2.77741 q^{49} -0.756277i q^{50} -1.15308i q^{52} -8.16096 q^{53} +13.5792i q^{55} -0.971028i q^{56} +0.971773 q^{58} -6.67660 q^{59} +0.344788 q^{61} +0.387069i q^{62} +7.66466 q^{64} -1.95861i q^{65} -10.1378i q^{67} +0.919800 q^{68} -0.821783i q^{70} -8.32237 q^{71} -7.15709i q^{73} -0.289599 q^{74} -4.91902 q^{76} +8.27193i q^{77} -12.1532i q^{79} +13.2095 q^{80} +0.278260i q^{82} -11.1343 q^{83} +1.56237 q^{85} -0.942701 q^{86} +1.90222 q^{88} -1.54245i q^{89} -1.19311i q^{91} -4.56636 q^{92} -0.267960 q^{94} -8.35542 q^{95} -2.52821 q^{97} +0.329269i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 56 * q - 48 * q^4	\( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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