LMFDB - Embedded newform 1503.2.c.a.1502.14

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			1502.14
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46644i q^{2} -4.08332 q^{4} +1.88017 q^{5} -1.78260 q^{7} -5.13837i q^{8} +O(q^{10})$	\(q+2.46644i q^{2} -4.08332 q^{4} +1.88017 q^{5} -1.78260 q^{7} -5.13837i q^{8} +4.63731i q^{10} -2.67232i q^{11} +0.750764i q^{13} -4.39667i q^{14} +4.50684 q^{16} -7.52352 q^{17} +2.22688 q^{19} -7.67731 q^{20} +6.59111 q^{22} -2.75669 q^{23} -1.46498 q^{25} -1.85171 q^{26} +7.27891 q^{28} -3.94225i q^{29} -3.47745 q^{31} +0.839104i q^{32} -18.5563i q^{34} -3.35158 q^{35} -9.27001i q^{37} +5.49245i q^{38} -9.66099i q^{40} +0.931686 q^{41} +0.511334i q^{43} +10.9119i q^{44} -6.79921i q^{46} -1.75935i q^{47} -3.82235 q^{49} -3.61327i q^{50} -3.06561i q^{52} -6.00813 q^{53} -5.02441i q^{55} +9.15965i q^{56} +9.72332 q^{58} -1.86367 q^{59} +8.16224 q^{61} -8.57691i q^{62} +6.94409 q^{64} +1.41156i q^{65} +4.56295i q^{67} +30.7209 q^{68} -8.26646i q^{70} -7.96925 q^{71} -2.71188i q^{73} +22.8639 q^{74} -9.09304 q^{76} +4.76367i q^{77} -12.9736i q^{79} +8.47361 q^{80} +2.29795i q^{82} -5.03470 q^{83} -14.1455 q^{85} -1.26117 q^{86} -13.7314 q^{88} +4.33603i q^{89} -1.33831i q^{91} +11.2564 q^{92} +4.33932 q^{94} +4.18690 q^{95} +15.9545 q^{97} -9.42758i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			2.46644i			1.74404i	0.489475	+	0.872018i	$0.337189\pi$
$2$			2.46644i			1.74404i	−0.489475	+	0.872018i	$0.662811\pi$
$3$		0			0
$3$		0			0
$4$	−4.08332			−2.04166
$5$	1.88017			0.840836			0.420418	−	0.907331i	$-0.361883\pi$
$5$	1.88017			0.840836			0.420418	+	0.907331i	$0.361883\pi$
$6$		0			0
$7$	−1.78260			−0.673759			−0.336879	−	0.941548i	$-0.609371\pi$
$7$	−1.78260			−0.673759			−0.336879	+	0.941548i	$0.609371\pi$
$8$		−	5.13837i		−	1.81669i
$9$		0			0
$10$			4.63731i			1.46645i
$11$		−	2.67232i		−	0.805735i	−0.915258	−	0.402868i	$-0.868014\pi$
$11$		−	2.67232i		−	0.805735i	0.915258	−	0.402868i	$-0.131986\pi$
$12$		0			0
$13$			0.750764i			0.208224i	0.994566	+	0.104112i	$0.0332001\pi$
$13$			0.750764i			0.208224i	−0.994566	+	0.104112i	$0.966800\pi$
$14$		−	4.39667i		−	1.17506i
$15$		0			0
$16$	4.50684			1.12671
$17$	−7.52352			−1.82472			−0.912361	−	0.409387i	$-0.865743\pi$
$17$	−7.52352			−1.82472			−0.912361	+	0.409387i	$0.865743\pi$
$18$		0			0
$19$	2.22688			0.510880			0.255440	−	0.966825i	$-0.417780\pi$
$19$	2.22688			0.510880			0.255440	+	0.966825i	$0.417780\pi$
$20$	−7.67731			−1.71670
$21$		0			0
$22$	6.59111			1.40523
$23$	−2.75669			−0.574810			−0.287405	−	0.957809i	$-0.592793\pi$
$23$	−2.75669			−0.574810			−0.287405	+	0.957809i	$0.592793\pi$
$24$		0			0
$25$	−1.46498			−0.292995
$26$	−1.85171			−0.363151
$27$		0			0
$28$	7.27891			1.37558
$29$		−	3.94225i		−	0.732058i	−0.930603	−	0.366029i	$-0.880717\pi$
$29$		−	3.94225i		−	0.732058i	0.930603	−	0.366029i	$-0.119283\pi$
$30$		0			0
$31$	−3.47745			−0.624568			−0.312284	−	0.949989i	$-0.601094\pi$
$31$	−3.47745			−0.624568			−0.312284	+	0.949989i	$0.601094\pi$
$32$			0.839104i			0.148334i
$33$		0			0
$34$		−	18.5563i		−	3.18238i
$35$	−3.35158			−0.566520
$36$		0			0
$37$		−	9.27001i		−	1.52398i	−0.647589	−	0.761990i	$-0.724223\pi$
$37$		−	9.27001i		−	1.52398i	0.647589	−	0.761990i	$-0.275777\pi$
$38$			5.49245i			0.890993i
$39$		0			0
$40$		−	9.66099i		−	1.52754i
$41$	0.931686			0.145505			0.0727524	−	0.997350i	$-0.476822\pi$
$41$	0.931686			0.145505			0.0727524	+	0.997350i	$0.476822\pi$
$42$		0			0
$43$			0.511334i			0.0779777i	0.999240	+	0.0389888i	$0.0124137\pi$
$43$			0.511334i			0.0779777i	−0.999240	+	0.0389888i	$0.987586\pi$
$44$			10.9119i			1.64504i
$45$		0			0
$46$		−	6.79921i		−	1.00249i
$47$		−	1.75935i		−	0.256627i	−0.991734	−	0.128314i	$-0.959044\pi$
$47$		−	1.75935i		−	0.256627i	0.991734	−	0.128314i	$-0.0409564\pi$
$48$		0			0
$49$	−3.82235			−0.546049
$50$		−	3.61327i		−	0.510994i
$51$		0			0
$52$		−	3.06561i		−	0.425123i
$53$	−6.00813			−0.825280			−0.412640	−	0.910894i	$-0.635393\pi$
$53$	−6.00813			−0.825280			−0.412640	+	0.910894i	$0.635393\pi$
$54$		0			0
$55$		−	5.02441i		−	0.677491i
$56$			9.15965i			1.22401i
$57$		0			0
$58$	9.72332			1.27673
$59$	−1.86367			−0.242629			−0.121314	−	0.992614i	$-0.538711\pi$
$59$	−1.86367			−0.242629			−0.121314	+	0.992614i	$0.538711\pi$
$60$		0			0
$61$	8.16224			1.04507			0.522534	−	0.852618i	$-0.324987\pi$
$61$	8.16224			1.04507			0.522534	+	0.852618i	$0.324987\pi$
$62$		−	8.57691i		−	1.08927i
$63$		0			0
$64$	6.94409			0.868011
$65$			1.41156i			0.175082i
$66$		0			0
$67$			4.56295i			0.557453i	0.960370	+	0.278727i	$0.0899124\pi$
$67$			4.56295i			0.557453i	−0.960370	+	0.278727i	$0.910088\pi$
$68$	30.7209			3.72546
$69$		0			0
$70$		−	8.26646i		−	0.988031i
$71$	−7.96925			−0.945776			−0.472888	−	0.881122i	$-0.656789\pi$
$71$	−7.96925			−0.945776			−0.472888	+	0.881122i	$0.656789\pi$
$72$		0			0
$73$		−	2.71188i		−	0.317402i	−0.987327	−	0.158701i	$-0.949269\pi$
$73$		−	2.71188i		−	0.317402i	0.987327	−	0.158701i	$-0.0507305\pi$
$74$	22.8639			2.65787
$75$		0			0
$76$	−9.09304			−1.04304
$77$			4.76367i			0.542871i
$78$		0			0
$79$		−	12.9736i		−	1.45964i	−0.683637	−	0.729822i	$-0.739603\pi$
$79$		−	12.9736i		−	1.45964i	0.683637	−	0.729822i	$-0.260397\pi$
$80$	8.47361			0.947378
$81$		0			0
$82$			2.29795i			0.253766i
$83$	−5.03470			−0.552630			−0.276315	−	0.961067i	$-0.589113\pi$
$83$	−5.03470			−0.552630			−0.276315	+	0.961067i	$0.589113\pi$
$84$		0			0
$85$	−14.1455			−1.53429
$86$	−1.26117			−0.135996
$87$		0			0
$88$	−13.7314			−1.46377
$89$			4.33603i			0.459619i	0.973236	+	0.229809i	$0.0738103\pi$
$89$			4.33603i			0.459619i	−0.973236	+	0.229809i	$0.926190\pi$
$90$		0			0
$91$		−	1.33831i		−	0.140293i
$92$	11.2564			1.17357
$93$		0			0
$94$	4.33932			0.447567
$95$	4.18690			0.429566
$96$		0			0
$97$	15.9545			1.61993			0.809967	−	0.586475i	$-0.199485\pi$
$97$	15.9545			1.61993			0.809967	+	0.586475i	$0.199485\pi$
$98$		−	9.42758i		−	0.952329i
$99$		0			0
$100$	5.98196			0.598196
$101$	−9.53989			−0.949254			−0.474627	−	0.880187i	$-0.657417\pi$
$101$	−9.53989			−0.949254			−0.474627	+	0.880187i	$0.657417\pi$
$102$		0			0
$103$			17.0846i			1.68339i	0.539950	+	0.841697i	$0.318443\pi$
$103$			17.0846i			1.68339i	−0.539950	+	0.841697i	$0.681557\pi$
$104$	3.85770			0.378279
$105$		0			0
$106$		−	14.8187i		−	1.43932i
$107$			3.60674i			0.348677i	0.984686	+	0.174338i	$0.0557787\pi$
$107$			3.60674i			0.348677i	−0.984686	+	0.174338i	$0.944221\pi$
$108$		0			0
$109$		−	6.26692i		−	0.600262i	−0.953898	−	0.300131i	$-0.902970\pi$
$109$		−	6.26692i		−	0.600262i	0.953898	−	0.300131i	$-0.0970303\pi$
$110$	12.3924			1.18157
$111$		0			0
$112$	−8.03389			−0.759131
$113$	8.95133			0.842070			0.421035	−	0.907044i	$-0.361667\pi$
$113$	8.95133			0.842070			0.421035	+	0.907044i	$0.361667\pi$
$114$		0			0
$115$	−5.18304			−0.483321
$116$			16.0975i			1.49461i
$117$		0			0
$118$		−	4.59662i		−	0.423153i
$119$	13.4114			1.22942
$120$		0			0
$121$	3.85870			0.350791
$122$			20.1317i			1.82264i
$123$		0			0
$124$	14.1995			1.27515
$125$	−12.1552			−1.08720
$126$		0			0
$127$	10.2363			0.908325			0.454163	−	0.890919i	$-0.349939\pi$
$127$	10.2363			0.908325			0.454163	+	0.890919i	$0.349939\pi$
$128$			18.8054i			1.66218i
$129$		0			0
$130$	−3.48152			−0.305350
$131$	−16.1037			−1.40698			−0.703492	−	0.710703i	$-0.748377\pi$
$131$	−16.1037			−1.40698			−0.703492	+	0.710703i	$0.748377\pi$
$132$		0			0
$133$	−3.96962			−0.344210
$134$	−11.2542			−0.972218
$135$		0			0
$136$			38.6586i			3.31495i
$137$		−	14.3749i		−	1.22813i	−0.789257	−	0.614063i	$-0.789534\pi$
$137$		−	14.3749i		−	1.22813i	0.789257	−	0.614063i	$-0.210466\pi$
$138$		0			0
$139$			6.19023i			0.525048i	0.964925	+	0.262524i	$0.0845550\pi$
$139$			6.19023i			0.525048i	−0.964925	+	0.262524i	$0.915445\pi$
$140$	13.6856			1.15664
$141$		0			0
$142$		−	19.6557i		−	1.64947i
$143$	2.00628			0.167774
$144$		0			0
$145$		−	7.41209i		−	0.615540i
$146$	6.68869			0.553560
$147$		0			0
$148$			37.8524i			3.11145i
$149$	5.80999			0.475973			0.237986	−	0.971268i	$-0.423513\pi$
$149$	5.80999			0.475973			0.237986	+	0.971268i	$0.423513\pi$
$150$		0			0
$151$			11.5955i			0.943629i	0.881698	+	0.471814i	$0.156401\pi$
$151$			11.5955i			0.943629i	−0.881698	+	0.471814i	$0.843599\pi$
$152$		−	11.4425i		−	0.928111i
$153$		0			0
$154$	−11.7493			−0.946786
$155$	−6.53818			−0.525159
$156$		0			0
$157$	−0.301646			−0.0240740			−0.0120370	−	0.999928i	$-0.503832\pi$
$157$	−0.301646			−0.0240740			−0.0120370	+	0.999928i	$0.503832\pi$
$158$	31.9986			2.54567
$159$		0			0
$160$			1.57765i			0.124725i
$161$	4.91407			0.387283
$162$		0			0
$163$			10.2262i			0.800980i	0.916301	+	0.400490i	$0.131160\pi$
$163$			10.2262i			0.800980i	−0.916301	+	0.400490i	$0.868840\pi$
$164$	−3.80437			−0.297071
$165$		0			0
$166$		−	12.4178i		−	0.963806i
$167$	−11.8771	+	5.09271i	−0.919074	+	0.394085i
$167$	−11.8771	+	5.09271i	−0.919074	+	0.394085i
$168$		0			0
$169$	12.4364			0.956643
$170$		−	34.8889i		−	2.67586i
$171$		0			0
$172$		−	2.08794i		−	0.159204i
$173$		−	0.854838i		−	0.0649921i	−0.999472	−	0.0324961i	$-0.989654\pi$
$173$		−	0.854838i		−	0.0649921i	0.999472	−	0.0324961i	$-0.0103456\pi$
$174$		0			0
$175$	2.61146			0.197408
$176$		−	12.0437i		−	0.907830i
$177$		0			0
$178$	−10.6946			−0.801591
$179$		−	3.15429i		−	0.235763i	−0.993028	−	0.117881i	$-0.962390\pi$
$179$		−	3.15429i		−	0.235763i	0.993028	−	0.117881i	$-0.0376102\pi$
$180$		0			0
$181$	−18.4799			−1.37360			−0.686799	−	0.726847i	$-0.740985\pi$
$181$	−18.4799			−1.37360			−0.686799	+	0.726847i	$0.740985\pi$
$182$	3.30086			0.244676
$183$		0			0
$184$			14.1649i			1.04425i
$185$		−	17.4292i		−	1.28142i
$186$		0			0
$187$			20.1053i			1.47024i
$188$			7.18398i			0.523945i
$189$		0			0
$190$			10.3267i			0.749179i
$191$		−	12.2528i		−	0.886579i	−0.896378	−	0.443290i	$-0.853811\pi$
$191$		−	12.2528i		−	0.886579i	0.896378	−	0.443290i	$-0.146189\pi$
$192$		0			0
$193$			15.6977i			1.12994i	0.825111	+	0.564971i	$0.191113\pi$
$193$			15.6977i			1.12994i	−0.825111	+	0.564971i	$0.808887\pi$
$194$			39.3508i			2.82522i
$195$		0			0
$196$	15.6078			1.11485
$197$	−15.6201			−1.11289			−0.556443	−	0.830886i	$-0.687834\pi$
$197$	−15.6201			−1.11289			−0.556443	+	0.830886i	$0.687834\pi$
$198$		0			0
$199$	−10.1556			−0.719909			−0.359954	−	0.932970i	$-0.617208\pi$
$199$	−10.1556			−0.719909			−0.359954	+	0.932970i	$0.617208\pi$
$200$			7.52760i			0.532281i
$201$		0			0
$202$		−	23.5295i		−	1.65553i
$203$			7.02745i			0.493230i
$204$		0			0
$205$	1.75172			0.122346
$206$	−42.1381			−2.93590
$207$		0			0
$208$			3.38357i			0.234609i
$209$		−	5.95093i		−	0.411634i
$210$		0			0
$211$	−14.3157			−0.985535			−0.492767	−	0.870161i	$-0.664015\pi$
$211$	−14.3157			−0.985535			−0.492767	+	0.870161i	$0.664015\pi$
$212$	24.5331			1.68494
$213$		0			0
$214$	−8.89580			−0.608105
$215$			0.961392i			0.0655664i
$216$		0			0
$217$	6.19889			0.420808
$218$	15.4570			1.04688
$219$		0			0
$220$			20.5162i			1.38320i
$221$		−	5.64838i		−	0.379951i
$222$		0			0
$223$	−7.03160			−0.470871			−0.235435	−	0.971890i	$-0.575652\pi$
$223$	−7.03160			−0.470871			−0.235435	+	0.971890i	$0.575652\pi$
$224$		−	1.49578i		−	0.0999413i
$225$		0			0
$226$			22.0779i			1.46860i
$227$	−27.1306			−1.80072			−0.900360	−	0.435146i	$-0.856697\pi$
$227$	−27.1306			−1.80072			−0.900360	+	0.435146i	$0.856697\pi$
$228$		0			0
$229$	13.9054			0.918894			0.459447	−	0.888205i	$-0.348048\pi$
$229$	13.9054			0.918894			0.459447	+	0.888205i	$0.348048\pi$
$230$		−	12.7836i		−	0.842928i
$231$		0			0
$232$	−20.2568			−1.32992
$233$		−	15.2195i		−	0.997063i	−0.866872	−	0.498531i	$-0.833873\pi$
$233$		−	15.2195i		−	0.997063i	0.866872	−	0.498531i	$-0.166127\pi$
$234$		0			0
$235$		−	3.30787i		−	0.215781i
$236$	7.60994			0.495365
$237$		0			0
$238$			33.0784i			2.14415i
$239$			10.9788i			0.710162i	0.934836	+	0.355081i	$0.115547\pi$
$239$			10.9788i			0.710162i	−0.934836	+	0.355081i	$0.884453\pi$
$240$		0			0
$241$		−	3.45962i		−	0.222854i	−0.993773	−	0.111427i	$-0.964458\pi$
$241$		−	3.45962i		−	0.222854i	0.993773	−	0.111427i	$-0.0355421\pi$
$242$			9.51725i			0.611792i
$243$		0			0
$244$	−33.3290			−2.13367
$245$	−7.18664			−0.459138
$246$		0			0
$247$			1.67186i			0.106378i
$248$			17.8684i			1.13465i
$249$		0			0
$250$		−	29.9801i		−	1.89611i
$251$		−	18.4662i		−	1.16558i	−0.812623	−	0.582790i	$-0.801961\pi$
$251$		−	18.4662i		−	1.16558i	0.812623	−	0.582790i	$-0.198039\pi$
$252$		0			0
$253$			7.36677i			0.463145i
$254$			25.2472i			1.58415i
$255$		0			0
$256$	−32.4941			−2.03088
$257$	7.44498			0.464405			0.232202	−	0.972667i	$-0.425407\pi$
$257$	7.44498			0.464405			0.232202	+	0.972667i	$0.425407\pi$
$258$		0			0
$259$			16.5247i			1.02679i
$260$		−	5.76385i		−	0.357459i
$261$		0			0
$262$		−	39.7187i		−	2.45383i
$263$			1.98507i			0.122404i	0.998125	+	0.0612022i	$0.0194934\pi$
$263$			1.98507i			0.122404i	−0.998125	+	0.0612022i	$0.980507\pi$
$264$		0			0
$265$	−11.2963			−0.693925
$266$		−	9.79083i		−	0.600314i
$267$		0			0
$268$		−	18.6320i		−	1.13813i
$269$	19.2134			1.17146			0.585731	−	0.810505i	$-0.300807\pi$
$269$	19.2134			1.17146			0.585731	+	0.810505i	$0.300807\pi$
$270$		0			0
$271$			3.05334i			0.185477i	0.995690	+	0.0927386i	$0.0295621\pi$
$271$			3.05334i			0.185477i	−0.995690	+	0.0927386i	$0.970438\pi$
$272$	−33.9073			−2.05593
$273$		0			0
$274$	35.4547			2.14190
$275$			3.91489i			0.236077i
$276$		0			0
$277$		−	2.93651i		−	0.176438i	−0.996101	−	0.0882190i	$-0.971882\pi$
$277$		−	2.93651i		−	0.176438i	0.996101	−	0.0882190i	$-0.0281175\pi$
$278$	−15.2678			−0.915703
$279$		0			0
$280$			17.2217i			1.02919i
$281$			0.514648i			0.0307013i	0.999882	+	0.0153507i	$0.00488646\pi$
$281$			0.514648i			0.0307013i	−0.999882	+	0.0153507i	$0.995114\pi$
$282$		0			0
$283$	−16.8609			−1.00228			−0.501140	−	0.865366i	$-0.667086\pi$
$283$	−16.8609			−1.00228			−0.501140	+	0.865366i	$0.667086\pi$
$284$	32.5410			1.93095
$285$		0			0
$286$			4.94837i			0.292603i
$287$	−1.66082			−0.0980352
$288$		0			0
$289$	39.6034			2.32961
$290$	18.2815			1.07352
$291$		0			0
$292$			11.0735i			0.648026i
$293$			16.7705i			0.979744i	0.871794	+	0.489872i	$0.162956\pi$
$293$			16.7705i			0.979744i	−0.871794	+	0.489872i	$0.837044\pi$
$294$		0			0
$295$	−3.50400			−0.204011
$296$	−47.6327			−2.76860
$297$		0			0
$298$			14.3300i			0.830114i
$299$		−	2.06962i		−	0.119689i
$300$		0			0
$301$		−	0.911502i		−	0.0525381i
$302$	−28.5996			−1.64572
$303$		0			0
$304$	10.0362			0.575614
$305$	15.3464			0.878731
$306$		0			0
$307$		−	0.769261i		−	0.0439040i	−0.999759	−	0.0219520i	$-0.993012\pi$
$307$		−	0.769261i		−	0.0439040i	0.999759	−	0.0219520i	$-0.00698810\pi$
$308$		−	19.4516i		−	1.10836i
$309$		0			0
$310$		−	16.1260i		−	0.915896i
$311$		−	11.4539i		−	0.649490i	−0.945802	−	0.324745i	$-0.894722\pi$
$311$		−	11.4539i		−	0.649490i	0.945802	−	0.324745i	$-0.105278\pi$
$312$		0			0
$313$			2.82982i			0.159951i	0.996797	+	0.0799755i	$0.0254842\pi$
$313$			2.82982i			0.159951i	−0.996797	+	0.0799755i	$0.974516\pi$
$314$		−	0.743992i		−	0.0419859i
$315$		0			0
$316$			52.9753i			2.98010i
$317$			20.5176i			1.15238i	0.817315	+	0.576191i	$0.195462\pi$
$317$			20.5176i			1.15238i	−0.817315	+	0.576191i	$0.804538\pi$
$318$		0			0
$319$	−10.5350			−0.589845
$320$	13.0560			0.729854
$321$		0			0
$322$			12.1203i			0.675436i
$323$	−16.7539			−0.932214
$324$		0			0
$325$		−	1.09985i		−	0.0610088i
$326$	−25.2223			−1.39694
$327$		0			0
$328$		−	4.78735i		−	0.264337i
$329$			3.13621i			0.172905i
$330$		0			0
$331$		−	34.1340i		−	1.87617i	−0.346401	−	0.938086i	$-0.612596\pi$
$331$		−	34.1340i		−	1.87617i	0.346401	−	0.938086i	$-0.387404\pi$
$332$	20.5583			1.12828
$333$		0			0
$334$	−12.5608	−	29.2940i	−0.687299	−	1.60290i
$335$			8.57911i			0.468727i
$336$		0			0
$337$	−3.84232			−0.209304			−0.104652	−	0.994509i	$-0.533373\pi$
$337$	−3.84232			−0.209304			−0.104652	+	0.994509i	$0.533373\pi$
$338$			30.6735i			1.66842i
$339$		0			0
$340$	57.7604			3.13250
$341$			9.29286i			0.503236i
$342$		0			0
$343$	19.2919			1.04166
$344$	2.62742			0.141661
$345$		0			0
$346$	2.10841			0.113349
$347$	−20.1590			−1.08219			−0.541097	−	0.840960i	$-0.681991\pi$
$347$	−20.1590			−1.08219			−0.541097	+	0.840960i	$0.681991\pi$
$348$		0			0
$349$		−	22.2471i		−	1.19086i	−0.803408	−	0.595429i	$-0.796982\pi$
$349$		−	22.2471i		−	1.19086i	0.803408	−	0.595429i	$-0.203018\pi$
$350$			6.44101i			0.344287i
$351$		0			0
$352$	2.24236			0.119518
$353$		−	31.7158i		−	1.68806i	−0.536294	−	0.844031i	$-0.680176\pi$
$353$		−	31.7158i		−	1.68806i	0.536294	−	0.844031i	$-0.319824\pi$
$354$		0			0
$355$	−14.9835			−0.795243
$356$		−	17.7054i		−	0.938384i
$357$		0			0
$358$	7.77986			0.411178
$359$		−	15.3622i		−	0.810787i	−0.914142	−	0.405393i	$-0.867135\pi$
$359$		−	15.3622i		−	0.810787i	0.914142	−	0.405393i	$-0.132865\pi$
$360$		0			0
$361$	−14.0410			−0.739001
$362$		−	45.5795i		−	2.39560i
$363$		0			0
$364$			5.46474i			0.286430i
$365$		−	5.09878i		−	0.266883i
$366$		0			0
$367$	−27.1635			−1.41792			−0.708961	−	0.705247i	$-0.750836\pi$
$367$	−27.1635			−1.41792			−0.708961	+	0.705247i	$0.750836\pi$
$368$	−12.4240			−0.647645
$369$		0			0
$370$	42.9879			2.23484
$371$	10.7101			0.556039
$372$		0			0
$373$		−	36.0983i		−	1.86910i	−0.355831	−	0.934550i	$-0.615802\pi$
$373$		−	36.0983i		−	1.86910i	0.355831	−	0.934550i	$-0.384198\pi$
$374$	−49.5884			−2.56415
$375$		0			0
$376$	−9.04019			−0.466212
$377$	2.95970			0.152432
$378$		0			0
$379$			6.70019i			0.344166i	0.985082	+	0.172083i	$0.0550497\pi$
$379$			6.70019i			0.344166i	−0.985082	+	0.172083i	$0.944950\pi$
$380$	−17.0964			−0.877028
$381$		0			0
$382$	30.2207			1.54622
$383$			18.1255i			0.926170i	0.886314	+	0.463085i	$0.153258\pi$
$383$			18.1255i			0.926170i	−0.886314	+	0.463085i	$0.846742\pi$
$384$		0			0
$385$			8.95649i			0.456465i
$386$	−38.7173			−1.97066
$387$		0			0
$388$	−65.1473			−3.30735
$389$	25.8706			1.31169			0.655845	−	0.754895i	$-0.272312\pi$
$389$	25.8706			1.31169			0.655845	+	0.754895i	$0.272312\pi$
$390$		0			0
$391$	20.7400			1.04887
$392$			19.6406i			0.992002i
$393$		0			0
$394$		−	38.5260i		−	1.94091i
$395$		−	24.3925i		−	1.22732i
$396$		0			0
$397$	6.20453			0.311396			0.155698	−	0.987805i	$-0.450237\pi$
$397$	6.20453			0.311396			0.155698	+	0.987805i	$0.450237\pi$
$398$		−	25.0481i		−	1.25555i
$399$		0			0
$400$	−6.60242			−0.330121
$401$	−29.1606			−1.45621			−0.728105	−	0.685466i	$-0.759599\pi$
$401$	−29.1606			−1.45621			−0.728105	+	0.685466i	$0.759599\pi$
$402$		0			0
$403$		−	2.61074i		−	0.130050i
$404$	38.9544			1.93805
$405$		0			0
$406$	−17.3328			−0.860211
$407$	−24.7724			−1.22792
$408$		0			0
$409$	25.5780			1.26475			0.632375	−	0.774662i	$-0.282080\pi$
$409$	25.5780			1.26475			0.632375	+	0.774662i	$0.282080\pi$
$410$			4.32052i			0.213375i
$411$		0			0
$412$		−	69.7618i		−	3.43692i
$413$	3.32217			0.163473
$414$		0			0
$415$	−9.46607			−0.464671
$416$	−0.629969			−0.0308868
$417$		0			0
$418$	14.6776			0.717904
$419$			3.24332i			0.158447i	0.996857	+	0.0792233i	$0.0252440\pi$
$419$			3.24332i			0.158447i	−0.996857	+	0.0792233i	$0.974756\pi$
$420$		0			0
$421$	16.7610			0.816882			0.408441	−	0.912785i	$-0.366073\pi$
$421$	16.7610			0.816882			0.408441	+	0.912785i	$0.366073\pi$
$422$		−	35.3088i		−	1.71881i
$423$		0			0
$424$			30.8720i			1.49928i
$425$	11.0218			0.534635
$426$		0			0
$427$	−14.5500			−0.704124
$428$		−	14.7275i		−	0.711879i
$429$		0			0
$430$	−2.37121			−0.114350
$431$			28.4448i			1.37014i	0.728477	+	0.685070i	$0.240228\pi$
$431$			28.4448i			1.37014i	−0.728477	+	0.685070i	$0.759772\pi$
$432$		0			0
$433$	−4.98354			−0.239494			−0.119747	−	0.992804i	$-0.538208\pi$
$433$	−4.98354			−0.239494			−0.119747	+	0.992804i	$0.538208\pi$
$434$			15.2892i			0.733904i
$435$		0			0
$436$			25.5898i			1.22553i
$437$	−6.13881			−0.293659
$438$		0			0
$439$		−	7.46286i		−	0.356183i	−0.984014	−	0.178091i	$-0.943008\pi$
$439$		−	7.46286i		−	0.356183i	0.984014	−	0.178091i	$-0.0569923\pi$
$440$	−25.8173			−1.23079
$441$		0			0
$442$	13.9314			0.662649
$443$	1.62256			0.0770900			0.0385450	−	0.999257i	$-0.487728\pi$
$443$	1.62256			0.0770900			0.0385450	+	0.999257i	$0.487728\pi$
$444$		0			0
$445$			8.15246i			0.386464i
$446$		−	17.3430i		−	0.821215i
$447$		0			0
$448$	−12.3785			−0.584830
$449$		−	26.7267i		−	1.26131i	−0.776063	−	0.630655i	$-0.782786\pi$
$449$		−	26.7267i		−	1.26131i	0.776063	−	0.630655i	$-0.217214\pi$
$450$		0			0
$451$		−	2.48976i		−	0.117238i
$452$	−36.5511			−1.71922
$453$		0			0
$454$		−	66.9159i		−	3.14052i
$455$		−	2.51624i		−	0.117963i
$456$		0			0
$457$			28.1255i			1.31566i	0.753168	+	0.657828i	$0.228525\pi$
$457$			28.1255i			1.31566i	−0.753168	+	0.657828i	$0.771475\pi$
$458$			34.2968i			1.60258i
$459$		0			0
$460$	21.1640			0.986776
$461$		−	0.378077i		−	0.0176088i	−0.999961	−	0.00880440i	$-0.997197\pi$
$461$		−	0.378077i		−	0.0176088i	0.999961	−	0.00880440i	$-0.00280256\pi$
$462$		0			0
$463$		−	12.9923i		−	0.603805i	−0.953339	−	0.301902i	$-0.902378\pi$
$463$		−	12.9923i		−	0.603805i	0.953339	−	0.301902i	$-0.0976217\pi$
$464$		−	17.7671i		−	0.824817i
$465$		0			0
$466$	37.5380			1.73891
$467$		−	1.83806i		−	0.0850551i	−0.999095	−	0.0425276i	$-0.986459\pi$
$467$		−	1.83806i		−	0.0850551i	0.999095	−	0.0425276i	$-0.0135410\pi$
$468$		0			0
$469$		−	8.13391i		−	0.375589i
$470$	8.15865			0.376330
$471$		0			0
$472$			9.57621i			0.440781i
$473$	1.36645			0.0628293
$474$		0			0
$475$	−3.26232			−0.149686
$476$	−54.7630			−2.51006
$477$		0			0
$478$	−27.0786			−1.23855
$479$	−18.6039			−0.850035			−0.425017	−	0.905185i	$-0.639732\pi$
$479$	−18.6039			−0.850035			−0.425017	+	0.905185i	$0.639732\pi$
$480$		0			0
$481$	6.95958			0.317330
$482$	8.53294			0.388665
$483$		0			0
$484$	−15.7563			−0.716195
$485$	29.9971			1.36210
$486$		0			0
$487$			31.5486i			1.42960i	0.699328	+	0.714801i	$0.253483\pi$
$487$			31.5486i			1.42960i	−0.699328	+	0.714801i	$0.746517\pi$
$488$		−	41.9406i		−	1.89856i
$489$		0			0
$490$		−	17.7254i		−	0.800753i
$491$		−	38.0408i		−	1.71676i	−0.513015	−	0.858379i	$-0.671472\pi$
$491$		−	38.0408i		−	1.71676i	0.513015	−	0.858379i	$-0.328528\pi$
$492$		0			0
$493$			29.6596i			1.33580i
$494$	−4.12353			−0.185526
$495$		0			0
$496$	−15.6723			−0.703707
$497$	14.2060			0.637225
$498$		0			0
$499$			0.100634i			0.00450499i	0.999997	+	0.00225250i	$0.000716992\pi$
$499$			0.100634i			0.00450499i	−0.999997	+	0.00225250i	$0.999283\pi$
$500$	49.6336			2.21968
$501$		0			0
$502$	45.5459			2.03281
$503$			29.7377i			1.32594i	0.748646	+	0.662970i	$0.230704\pi$
$503$			29.7377i			1.32594i	−0.748646	+	0.662970i	$0.769296\pi$
$504$		0			0
$505$	−17.9366			−0.798167
$506$	−18.1697			−0.807741
$507$		0			0
$508$	−41.7981			−1.85449
$509$			27.2285i			1.20688i	0.797408	+	0.603441i	$0.206204\pi$
$509$			27.2285i			1.20688i	−0.797408	+	0.603441i	$0.793796\pi$
$510$		0			0
$511$			4.83419i			0.213852i
$512$		−	42.5340i		−	1.87975i
$513$		0			0
$514$			18.3626i			0.809939i
$515$			32.1219i			1.41546i
$516$		0			0
$517$	−4.70154			−0.206774
$518$	−40.7571			−1.79077
$519$		0			0
$520$	7.25312			0.318070
$521$	6.44589			0.282399			0.141200	−	0.989981i	$-0.454904\pi$
$521$	6.44589			0.282399			0.141200	+	0.989981i	$0.454904\pi$
$522$		0			0
$523$	6.87620			0.300675			0.150338	−	0.988635i	$-0.451964\pi$
$523$	6.87620			0.300675			0.150338	+	0.988635i	$0.451964\pi$
$524$	65.7564			2.87258
$525$		0			0
$526$	−4.89604			−0.213478
$527$	26.1626			1.13966
$528$		0			0
$529$	−15.4006			−0.669593
$530$		−	27.8616i		−	1.21023i
$531$		0			0
$532$	16.2092			0.702759
$533$			0.699476i			0.0302977i
$534$		0			0
$535$			6.78127i			0.293180i
$536$	23.4462			1.01272
$537$		0			0
$538$			47.3887i			2.04307i
$539$			10.2145i			0.439971i
$540$		0			0
$541$			3.39826i			0.146103i	0.997328	+	0.0730514i	$0.0232737\pi$
$541$			3.39826i			0.146103i	−0.997328	+	0.0730514i	$0.976726\pi$
$542$	−7.53087			−0.323479
$543$		0			0
$544$		−	6.31302i		−	0.270668i
$545$		−	11.7828i		−	0.504721i
$546$		0			0
$547$			19.8728i			0.849700i	0.905264	+	0.424850i	$0.139673\pi$
$547$			19.8728i			0.849700i	−0.905264	+	0.424850i	$0.860327\pi$
$548$			58.6971i			2.50741i
$549$		0			0
$550$	−9.65583			−0.411726
$551$		−	8.77890i		−	0.373994i
$552$		0			0
$553$			23.1267i			0.983448i
$554$	7.24273			0.307714
$555$		0			0
$556$		−	25.2767i		−	1.07197i
$557$			15.6094i			0.661391i	0.943738	+	0.330695i	$0.107283\pi$
$557$			15.6094i			0.661391i	−0.943738	+	0.330695i	$0.892717\pi$
$558$		0			0
$559$	−0.383891			−0.0162368
$560$	−15.1050			−0.638304
$561$		0			0
$562$	−1.26935			−0.0535442
$563$			7.24290i			0.305252i	0.988284	+	0.152626i	$0.0487729\pi$
$563$			7.24290i			0.305252i	−0.988284	+	0.152626i	$0.951227\pi$
$564$		0			0
$565$	16.8300			0.708043
$566$		−	41.5865i		−	1.74801i
$567$		0			0
$568$			40.9490i			1.71818i
$569$	18.2603			0.765513			0.382756	−	0.923849i	$-0.374975\pi$
$569$	18.2603			0.765513			0.382756	+	0.923849i	$0.374975\pi$
$570$		0			0
$571$			27.7174i			1.15994i	0.814640	+	0.579968i	$0.196935\pi$
$571$			27.7174i			1.15994i	−0.814640	+	0.579968i	$0.803065\pi$
$572$	−8.19228			−0.342536
$573$		0			0
$574$		−	4.09631i		−	0.170977i
$575$	4.03849			0.168417
$576$		0			0
$577$	−32.4893			−1.35255			−0.676274	−	0.736651i	$-0.736406\pi$
$577$	−32.4893			−1.35255			−0.676274	+	0.736651i	$0.736406\pi$
$578$			97.6792i			4.06292i
$579$		0			0
$580$			30.2659i			1.25672i
$581$	8.97484			0.372339
$582$		0			0
$583$			16.0556i			0.664957i
$584$	−13.9346			−0.576620
$585$		0			0
$586$	−41.3634			−1.70871
$587$	−21.2335			−0.876402			−0.438201	−	0.898877i	$-0.644384\pi$
$587$	−21.2335			−0.876402			−0.438201	+	0.898877i	$0.644384\pi$
$588$		0			0
$589$	−7.74384			−0.319080
$590$		−	8.64240i		−	0.355802i
$591$		0			0
$592$		−	41.7785i		−	1.71708i
$593$	14.8912			0.611510			0.305755	−	0.952110i	$-0.401091\pi$
$593$	14.8912			0.611510			0.305755	+	0.952110i	$0.401091\pi$
$594$		0			0
$595$	25.2157			1.03374
$596$	−23.7240			−0.971774
$597$		0			0
$598$	5.10460			0.208743
$599$		−	25.1057i		−	1.02579i	−0.858452	−	0.512895i	$-0.828573\pi$
$599$		−	25.1057i		−	1.02579i	0.858452	−	0.512895i	$-0.171427\pi$
$600$		0			0
$601$	2.03895			0.0831704			0.0415852	−	0.999135i	$-0.486759\pi$
$601$	2.03895			0.0831704			0.0415852	+	0.999135i	$0.486759\pi$
$602$	2.24816			0.0916283
$603$		0			0
$604$		−	47.3481i		−	1.92657i
$605$	7.25500			0.294958
$606$		0			0
$607$			41.1220i			1.66909i	0.550938	+	0.834546i	$0.314270\pi$
$607$			41.1220i			1.66909i	−0.550938	+	0.834546i	$0.685730\pi$
$608$			1.86858i			0.0757809i
$609$		0			0
$610$			37.8509i			1.53254i
$611$	1.32085			0.0534361
$612$		0			0
$613$	0.798966			0.0322699			0.0161350	−	0.999870i	$-0.494864\pi$
$613$	0.798966			0.0322699			0.0161350	+	0.999870i	$0.494864\pi$
$614$	1.89733			0.0765701
$615$		0			0
$616$	24.4775			0.986227
$617$			42.2812i			1.70218i	0.525022	+	0.851089i	$0.324057\pi$
$617$			42.2812i			1.70218i	−0.525022	+	0.851089i	$0.675943\pi$
$618$		0			0
$619$			24.8495i			0.998787i	0.866375	+	0.499393i	$0.166444\pi$
$619$			24.8495i			0.998787i	−0.866375	+	0.499393i	$0.833556\pi$
$620$	26.6975			1.07220
$621$		0			0
$622$	28.2503			1.13273
$623$		−	7.72940i		−	0.309672i
$624$		0			0
$625$	−15.5290			−0.621158
$626$	−6.97958			−0.278960
$627$		0			0
$628$	1.23172			0.0491509
$629$			69.7431i			2.78084i
$630$		0			0
$631$	48.4101			1.92717			0.963587	−	0.267393i	$-0.0861623\pi$
$631$	48.4101			1.92717			0.963587	+	0.267393i	$0.0861623\pi$
$632$	−66.6632			−2.65172
$633$		0			0
$634$	−50.6053			−2.00979
$635$	19.2459			0.763752
$636$		0			0
$637$		−	2.86968i		−	0.113701i
$638$		−	25.9838i		−	1.02871i
$639$		0			0
$640$			35.3572i			1.39762i
$641$	−31.9701			−1.26274			−0.631372	−	0.775480i	$-0.717508\pi$
$641$	−31.9701			−1.26274			−0.631372	+	0.775480i	$0.717508\pi$
$642$		0			0
$643$			18.6155i			0.734123i	0.930197	+	0.367062i	$0.119636\pi$
$643$			18.6155i			0.734123i	−0.930197	+	0.367062i	$0.880364\pi$
$644$	−20.0657			−0.790700
$645$		0			0
$646$		−	41.3226i		−	1.62581i
$647$	41.5598			1.63388			0.816941	−	0.576721i	$-0.195668\pi$
$647$	41.5598			1.63388			0.816941	+	0.576721i	$0.195668\pi$
$648$		0			0
$649$			4.98031i			0.195494i
$650$	2.71271			0.106401
$651$		0			0
$652$		−	41.7569i		−	1.63533i
$653$		−	23.7388i		−	0.928972i	−0.885580	−	0.464486i	$-0.846239\pi$
$653$		−	23.7388i		−	0.928972i	0.885580	−	0.464486i	$-0.153761\pi$
$654$		0			0
$655$	−30.2776			−1.18304
$656$	4.19896			0.163942
$657$		0			0
$658$	−7.73527			−0.301552
$659$	−14.7333			−0.573927			−0.286963	−	0.957942i	$-0.592646\pi$
$659$	−14.7333			−0.573927			−0.286963	+	0.957942i	$0.592646\pi$
$660$		0			0
$661$			5.30741i			0.206434i	0.994659	+	0.103217i	$0.0329137\pi$
$661$			5.30741i			0.206434i	−0.994659	+	0.103217i	$0.967086\pi$
$662$	84.1893			3.27211
$663$		0			0
$664$			25.8702i			1.00396i
$665$	−7.46355			−0.289424
$666$		0			0
$667$			10.8676i			0.420794i
$668$	48.4978	−	20.7951i	1.87643	−	0.804588i
$669$		0			0
$670$	−21.1598			−0.817476
$671$		−	21.8121i		−	0.842048i
$672$		0			0
$673$			28.8784i			1.11318i	0.830787	+	0.556590i	$0.187891\pi$
$673$			28.8784i			1.11318i	−0.830787	+	0.556590i	$0.812109\pi$
$674$		−	9.47683i		−	0.365034i
$675$		0			0
$676$	−50.7816			−1.95314
$677$			12.4105i			0.476975i	0.971146	+	0.238488i	$0.0766517\pi$
$677$			12.4105i			0.476975i	−0.971146	+	0.238488i	$0.923348\pi$
$678$		0			0
$679$	−28.4405			−1.09144
$680$			72.6847i			2.78733i
$681$		0			0
$682$	−22.9203			−0.877662
$683$	−11.6266			−0.444879			−0.222439	−	0.974946i	$-0.571402\pi$
$683$	−11.6266			−0.444879			−0.222439	+	0.974946i	$0.571402\pi$
$684$		0			0
$685$		−	27.0271i		−	1.03265i
$686$			47.5822i			1.81670i
$687$		0			0
$688$			2.30450i			0.0878583i
$689$		−	4.51068i		−	0.171843i
$690$		0			0
$691$		−	44.1705i		−	1.68033i	−0.542334	−	0.840163i	$-0.682459\pi$
$691$		−	44.1705i		−	1.68033i	0.542334	−	0.840163i	$-0.317541\pi$
$692$			3.49057i			0.132692i
$693$		0			0
$694$		−	49.7210i		−	1.88738i
$695$			11.6387i			0.441479i
$696$		0			0
$697$	−7.00956			−0.265506
$698$	54.8710			2.07690
$699$		0			0
$700$	−10.6634			−0.403040
$701$			28.1525i			1.06331i	0.846962	+	0.531653i	$0.178429\pi$
$701$			28.1525i			1.06331i	−0.846962	+	0.531653i	$0.821571\pi$
$702$		0			0
$703$		−	20.6432i		−	0.778571i
$704$		−	18.5568i		−	0.699387i
$705$		0			0
$706$	78.2251			2.94404
$707$	17.0058			0.639568
$708$		0			0
$709$		−	24.8827i		−	0.934490i	−0.884128	−	0.467245i	$-0.845247\pi$
$709$		−	24.8827i		−	0.934490i	0.884128	−	0.467245i	$-0.154753\pi$
$710$		−	36.9559i		−	1.38693i
$711$		0			0
$712$	22.2802			0.834984
$713$	9.58625			0.359008
$714$		0			0
$715$	3.77214			0.141070
$716$			12.8800i			0.481347i
$717$		0			0
$718$	37.8900			1.41404
$719$	50.4512			1.88151			0.940756	−	0.339085i	$-0.110117\pi$
$719$	50.4512			1.88151			0.940756	+	0.339085i	$0.110117\pi$
$720$		0			0
$721$		−	30.4549i		−	1.13420i
$722$		−	34.6313i		−	1.28884i
$723$		0			0
$724$	75.4592			2.80442
$725$			5.77531i			0.214490i
$726$		0			0
$727$		−	6.45940i		−	0.239566i	−0.992800	−	0.119783i	$-0.961780\pi$
$727$		−	6.45940i		−	0.239566i	0.992800	−	0.119783i	$-0.0382198\pi$
$728$	−6.87673			−0.254869
$729$		0			0
$730$	12.5758			0.465453
$731$		−	3.84703i		−	0.142288i
$732$		0			0
$733$	−38.0563			−1.40564			−0.702821	−	0.711367i	$-0.748076\pi$
$733$	−38.0563			−1.40564			−0.702821	+	0.711367i	$0.748076\pi$
$734$		−	66.9971i		−	2.47291i
$735$		0			0
$736$		−	2.31315i		−	0.0852639i
$737$	12.1937			0.449160
$738$		0			0
$739$		−	26.9686i		−	0.992057i	−0.868306	−	0.496028i	$-0.834791\pi$
$739$		−	26.9686i		−	0.992057i	0.868306	−	0.496028i	$-0.165209\pi$
$740$			71.1687i			2.61621i
$741$		0			0
$742$			26.4157i			0.969752i
$743$		−	12.7487i		−	0.467706i	−0.972272	−	0.233853i	$-0.924867\pi$
$743$		−	12.7487i		−	0.467706i	0.972272	−	0.233853i	$-0.0751335\pi$
$744$		0			0
$745$	10.9237			0.400215
$746$	89.0343			3.25978
$747$		0			0
$748$		−	82.0962i		−	3.00173i
$749$		−	6.42937i		−	0.234924i
$750$		0			0
$751$		−	5.86669i		−	0.214078i	−0.994255	−	0.107039i	$-0.965863\pi$
$751$		−	5.86669i		−	0.214078i	0.994255	−	0.107039i	$-0.0341370\pi$
$752$		−	7.92911i		−	0.289145i
$753$		0			0
$754$			7.29991i			0.265847i
$755$			21.8015i			0.793437i
$756$		0			0
$757$	−45.2094			−1.64316			−0.821581	−	0.570091i	$-0.806908\pi$
$757$	−45.2094			−1.64316			−0.821581	+	0.570091i	$0.806908\pi$
$758$	−16.5256			−0.600237
$759$		0			0
$760$		−	21.5138i		−	0.780388i
$761$			28.3522i			1.02777i	0.857860	+	0.513884i	$0.171794\pi$
$761$			28.3522i			1.02777i	−0.857860	+	0.513884i	$0.828206\pi$
$762$		0			0
$763$			11.1714i			0.404431i
$764$			50.0319i			1.81009i
$765$		0			0
$766$	−44.7054			−1.61527
$767$		−	1.39917i		−	0.0505212i
$768$		0			0
$769$		−	52.0847i		−	1.87822i	−0.343615	−	0.939111i	$-0.611651\pi$
$769$		−	52.0847i		−	1.87822i	0.343615	−	0.939111i	$-0.388349\pi$
$770$	−22.0906			−0.796091
$771$		0			0
$772$		−	64.0985i		−	2.30696i
$773$	3.95516			0.142257			0.0711286	−	0.997467i	$-0.477340\pi$
$773$	3.95516			0.142257			0.0711286	+	0.997467i	$0.477340\pi$
$774$		0			0
$775$	5.09438			0.182996
$776$		−	81.9802i		−	2.94292i
$777$		0			0
$778$			63.8082i			2.28763i
$779$	2.07475			0.0743356
$780$		0			0
$781$			21.2964i			0.762045i
$782$			51.1540i			1.82926i
$783$		0			0
$784$	−17.2267			−0.615240
$785$	−0.567145			−0.0202423
$786$		0			0
$787$			16.2928i			0.580774i	0.956909	+	0.290387i	$0.0937841\pi$
$787$			16.2928i			0.580774i	−0.956909	+	0.290387i	$0.906216\pi$
$788$	63.7818			2.27213
$789$		0			0
$790$	60.1627			2.14049
$791$	−15.9566			−0.567352
$792$		0			0
$793$			6.12791i			0.217609i
$794$			15.3031i			0.543086i
$795$		0			0
$796$	41.4684			1.46981
$797$	24.5863			0.870893			0.435446	−	0.900215i	$-0.356591\pi$
$797$	24.5863			0.870893			0.435446	+	0.900215i	$0.356591\pi$
$798$		0			0
$799$			13.2365i			0.468274i
$800$		−	1.22927i		−	0.0434612i
$801$		0			0
$802$		−	71.9228i		−	2.53968i
$803$	−7.24702			−0.255742
$804$		0			0
$805$	9.23927			0.325642
$806$	6.43923			0.226812
$807$		0			0
$808$			49.0195i			1.72450i
$809$		−	2.06268i		−	0.0725200i	−0.999342	−	0.0362600i	$-0.988456\pi$
$809$		−	2.06268i		−	0.0725200i	0.999342	−	0.0362600i	$-0.0115444\pi$
$810$		0			0
$811$			41.5088i			1.45757i	0.684743	+	0.728785i	$0.259915\pi$
$811$			41.5088i			1.45757i	−0.684743	+	0.728785i	$0.740085\pi$
$812$		−	28.6953i		−	1.00701i
$813$		0			0
$814$		−	61.0997i		−	2.14154i
$815$			19.2270i			0.673492i
$816$		0			0
$817$			1.13868i			0.0398373i
$818$			63.0866i			2.20577i
$819$		0			0
$820$	−7.15284			−0.249788
$821$	−22.2064			−0.775007			−0.387504	−	0.921868i	$-0.626663\pi$
$821$	−22.2064			−0.775007			−0.387504	+	0.921868i	$0.626663\pi$
$822$		0			0
$823$		−	20.0699i		−	0.699592i	−0.936826	−	0.349796i	$-0.886251\pi$
$823$		−	20.0699i		−	0.699592i	0.936826	−	0.349796i	$-0.113749\pi$
$824$	87.7870			3.05820
$825$		0			0
$826$			8.19392i			0.285103i
$827$	44.5952			1.55073			0.775363	−	0.631515i	$-0.217567\pi$
$827$	44.5952			1.55073			0.775363	+	0.631515i	$0.217567\pi$
$828$		0			0
$829$		−	28.5816i		−	0.992679i	−0.868129	−	0.496339i	$-0.834677\pi$
$829$		−	28.5816i		−	0.992679i	0.868129	−	0.496339i	$-0.165323\pi$
$830$		−	23.3475i		−	0.810403i
$831$		0			0
$832$			5.21337i			0.180741i
$833$	28.7575			0.996388
$834$		0			0
$835$	−22.3308	+	9.57513i	−0.772790	+	0.331361i
$836$			24.2995i			0.840416i
$837$		0			0
$838$	−7.99945			−0.276336
$839$		−	57.6140i		−	1.98906i	−0.104472	−	0.994528i	$-0.533315\pi$
$839$		−	57.6140i		−	1.98906i	0.104472	−	0.994528i	$-0.466685\pi$
$840$		0			0
$841$	13.4587			0.464091
$842$			41.3400i			1.42467i
$843$		0			0
$844$	58.4556			2.01213
$845$	23.3824			0.804379
$846$		0			0
$847$	−6.87851			−0.236348
$848$	−27.0777			−0.929851
$849$		0			0
$850$			27.1845i			0.932422i
$851$			25.5546i			0.875999i
$852$		0			0
$853$	7.80626			0.267281			0.133641	−	0.991030i	$-0.457333\pi$
$853$	7.80626			0.267281			0.133641	+	0.991030i	$0.457333\pi$
$854$		−	35.8867i		−	1.22802i
$855$		0			0
$856$	18.5328			0.633437
$857$		−	25.4240i		−	0.868466i	−0.900801	−	0.434233i	$-0.857020\pi$
$857$		−	25.4240i		−	0.868466i	0.900801	−	0.434233i	$-0.142980\pi$
$858$		0			0
$859$	50.5112			1.72342			0.861710	−	0.507402i	$-0.169394\pi$
$859$	50.5112			1.72342			0.861710	+	0.507402i	$0.169394\pi$
$860$		−	3.92567i		−	0.133864i
$861$		0			0
$862$	−70.1574			−2.38957
$863$		−	48.7020i		−	1.65784i	−0.559370	−	0.828918i	$-0.688957\pi$
$863$		−	48.7020i		−	1.65784i	0.559370	−	0.828918i	$-0.311043\pi$
$864$		0			0
$865$		−	1.60724i		−	0.0546477i
$866$		−	12.2916i		−	0.417685i
$867$		0			0
$868$	−25.3120			−0.859146
$869$	−34.6696			−1.17609
$870$		0			0
$871$	−3.42570			−0.116075
$872$	−32.2017			−1.09049
$873$		0			0
$874$		−	15.1410i		−	0.512152i
$875$	21.6679			0.732508
$876$		0			0
$877$	−4.10735			−0.138695			−0.0693477	−	0.997593i	$-0.522092\pi$
$877$	−4.10735			−0.138695			−0.0693477	+	0.997593i	$0.522092\pi$
$878$	18.4067			0.621195
$879$		0			0
$880$		−	22.6442i		−	0.763336i
$881$	−37.8888			−1.27651			−0.638254	−	0.769826i	$-0.720343\pi$
$881$	−37.8888			−1.27651			−0.638254	+	0.769826i	$0.720343\pi$
$882$		0			0
$883$	37.0855			1.24803			0.624013	−	0.781414i	$-0.285501\pi$
$883$	37.0855			1.24803			0.624013	+	0.781414i	$0.285501\pi$
$884$			23.0641i			0.775731i
$885$		0			0
$886$			4.00194i			0.134448i
$887$	12.7950			0.429614			0.214807	−	0.976657i	$-0.431088\pi$
$887$	12.7950			0.429614			0.214807	+	0.976657i	$0.431088\pi$
$888$		0			0
$889$	−18.2472			−0.611992
$890$	−20.1075			−0.674007
$891$		0			0
$892$	28.7123			0.961357
$893$		−	3.91785i		−	0.131106i
$894$		0			0
$895$		−	5.93059i		−	0.198238i
$896$		−	33.5224i		−	1.11990i
$897$		0			0
$898$	65.9197			2.19977
$899$			13.7090i			0.457220i
$900$		0			0
$901$	45.2023			1.50591
$902$	6.14085			0.204468
$903$		0			0
$904$		−	45.9953i		−	1.52978i
$905$	−34.7452			−1.15497
$906$		0			0
$907$	−41.7951			−1.38778			−0.693892	−	0.720079i	$-0.744106\pi$
$907$	−41.7951			−1.38778			−0.693892	+	0.720079i	$0.744106\pi$
$908$	110.783			3.67645
$909$		0			0
$910$	6.20616			0.205732
$911$			22.2008i			0.735546i	0.929916	+	0.367773i	$0.119880\pi$
$911$			22.2008i			0.735546i	−0.929916	+	0.367773i	$0.880120\pi$
$912$		0			0
$913$			13.4543i			0.445273i
$914$	−69.3699			−2.29455
$915$		0			0
$916$	−56.7801			−1.87607
$917$	28.7064			0.947968
$918$		0			0
$919$	22.5435			0.743643			0.371821	−	0.928304i	$-0.378733\pi$
$919$	22.5435			0.743643			0.371821	+	0.928304i	$0.378733\pi$
$920$			26.6324i			0.878044i
$921$		0			0
$922$	0.932503			0.0307104
$923$		−	5.98303i		−	0.196934i
$924$		0			0
$925$			13.5803i			0.446519i
$926$	32.0448			1.05306
$927$		0			0
$928$	3.30796			0.108589
$929$			37.1188i			1.21783i	0.793237	+	0.608913i	$0.208394\pi$
$929$			37.1188i			1.21783i	−0.793237	+	0.608913i	$0.791606\pi$
$930$		0			0
$931$	−8.51189			−0.278966
$932$			62.1461i			2.03566i
$933$		0			0
$934$	4.53345			0.148339
$935$			37.8012i			1.23623i
$936$		0			0
$937$		−	38.6811i		−	1.26366i	−0.775108	−	0.631829i	$-0.782305\pi$
$937$		−	38.6811i		−	1.26366i	0.775108	−	0.631829i	$-0.217695\pi$
$938$	20.0618			0.655040
$939$		0			0
$940$			13.5071i			0.440552i
$941$	53.0388			1.72902			0.864508	−	0.502618i	$-0.167630\pi$
$941$	53.0388			1.72902			0.864508	+	0.502618i	$0.167630\pi$
$942$		0			0
$943$	−2.56837			−0.0836377
$944$	−8.39925			−0.273372
$945$		0			0
$946$			3.37026i			0.109577i
$947$		−	44.5574i		−	1.44792i	−0.689841	−	0.723961i	$-0.742319\pi$
$947$		−	44.5574i		−	1.44792i	0.689841	−	0.723961i	$-0.257681\pi$
$948$		0			0
$949$	2.03598			0.0660907
$950$		−	8.04631i		−	0.261057i
$951$		0			0
$952$		−	68.9128i		−	2.23348i
$953$	−31.1275			−1.00832			−0.504160	−	0.863610i	$-0.668198\pi$
$953$	−31.1275			−1.00832			−0.504160	+	0.863610i	$0.668198\pi$
$954$		0			0
$955$		−	23.0372i		−	0.745467i
$956$		−	44.8300i		−	1.44991i
$957$		0			0
$958$		−	45.8854i		−	1.48249i
$959$			25.6246i			0.827461i
$960$		0			0
$961$	−18.9074			−0.609915
$962$			17.1654i			0.553434i
$963$		0			0
$964$			14.1267i			0.454991i
$965$			29.5142i			0.950096i
$966$		0			0
$967$	14.7044			0.472861			0.236430	−	0.971648i	$-0.424023\pi$
$967$	14.7044			0.472861			0.236430	+	0.971648i	$0.424023\pi$
$968$		−	19.8274i		−	0.637278i
$969$		0			0
$970$			73.9860i			2.37555i
$971$	3.48397			0.111806			0.0559029	−	0.998436i	$-0.482196\pi$
$971$	3.48397			0.111806			0.0559029	+	0.998436i	$0.482196\pi$
$972$		0			0
$973$		−	11.0347i		−	0.353756i
$974$	−77.8126			−2.49328
$975$		0			0
$976$	36.7859			1.17749
$977$	56.0472			1.79311			0.896555	−	0.442932i	$-0.146062\pi$
$977$	56.0472			1.79311			0.896555	+	0.442932i	$0.146062\pi$
$978$		0			0
$979$	11.5873			0.370331
$980$	29.3453			0.937403
$981$		0			0
$982$	93.8254			2.99409
$983$	26.0456			0.830725			0.415363	−	0.909656i	$-0.363655\pi$
$983$	26.0456			0.830725			0.415363	+	0.909656i	$0.363655\pi$
$984$		0			0
$985$	−29.3684			−0.935755
$986$	−73.1536			−2.32969
$987$		0			0
$988$		−	6.82672i		−	0.217187i
$989$		−	1.40959i		−	0.0448223i
$990$		0			0
$991$		−	9.01428i		−	0.286348i	−0.989698	−	0.143174i	$-0.954269\pi$
$991$		−	9.01428i		−	0.286348i	0.989698	−	0.143174i	$-0.0457309\pi$
$992$		−	2.91794i		−	0.0926447i
$993$		0			0
$994$			35.0382i			1.11134i
$995$	−19.0941			−0.605325
$996$		0			0
$997$	31.5434			0.998991			0.499496	−	0.866316i	$-0.333519\pi$
$997$	31.5434			0.998991			0.499496	+	0.866316i	$0.333519\pi$
$998$	−0.248207			−0.00785686
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.14	yes	56
3.2	odd	2	inner	1503.2.c.a.1502.43	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.44	yes	56
501.500	even	2	inner	1503.2.c.a.1502.13	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.13	✓	56	501.500	even	2	inner
1503.2.c.a.1502.14	yes	56	1.1	even	1	trivial
1503.2.c.a.1502.43	yes	56	3.2	odd	2	inner
1503.2.c.a.1502.44	yes	56	167.166	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)

\(f(q)\)	\(=\)	\(q+2.46644i q^{2} -4.08332 q^{4} +1.88017 q^{5} -1.78260 q^{7} -5.13837i q^{8} +O(q^{10})\)	\(q+2.46644i q^{2} -4.08332 q^{4} +1.88017 q^{5} -1.78260 q^{7} -5.13837i q^{8} +4.63731i q^{10} -2.67232i q^{11} +0.750764i q^{13} -4.39667i q^{14} +4.50684 q^{16} -7.52352 q^{17} +2.22688 q^{19} -7.67731 q^{20} +6.59111 q^{22} -2.75669 q^{23} -1.46498 q^{25} -1.85171 q^{26} +7.27891 q^{28} -3.94225i q^{29} -3.47745 q^{31} +0.839104i q^{32} -18.5563i q^{34} -3.35158 q^{35} -9.27001i q^{37} +5.49245i q^{38} -9.66099i q^{40} +0.931686 q^{41} +0.511334i q^{43} +10.9119i q^{44} -6.79921i q^{46} -1.75935i q^{47} -3.82235 q^{49} -3.61327i q^{50} -3.06561i q^{52} -6.00813 q^{53} -5.02441i q^{55} +9.15965i q^{56} +9.72332 q^{58} -1.86367 q^{59} +8.16224 q^{61} -8.57691i q^{62} +6.94409 q^{64} +1.41156i q^{65} +4.56295i q^{67} +30.7209 q^{68} -8.26646i q^{70} -7.96925 q^{71} -2.71188i q^{73} +22.8639 q^{74} -9.09304 q^{76} +4.76367i q^{77} -12.9736i q^{79} +8.47361 q^{80} +2.29795i q^{82} -5.03470 q^{83} -14.1455 q^{85} -1.26117 q^{86} -13.7314 q^{88} +4.33603i q^{89} -1.33831i q^{91} +11.2564 q^{92} +4.33932 q^{94} +4.18690 q^{95} +15.9545 q^{97} -9.42758i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 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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