LMFDB - Embedded newform 1248.4.m.a.337.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,4,Mod(337,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1248.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1248.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$73.6343836872$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.9
Character	$\chi$	$=$	1248.337
Dual form			1248.4.m.a.337.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -4.56929 q^{5} +23.0002i q^{7} -9.00000 q^{9} -9.02801 q^{11} +(-3.63901 + 46.7307i) q^{13} +13.7079i q^{15} -59.4245 q^{17} -27.0061 q^{19} +69.0006 q^{21} -35.7672 q^{23} -104.122 q^{25} +27.0000i q^{27} +142.184i q^{29} -303.839i q^{31} +27.0840i q^{33} -105.095i q^{35} +241.300 q^{37} +(140.192 + 10.9170i) q^{39} +280.464i q^{41} -198.269i q^{43} +41.1236 q^{45} -319.409i q^{47} -186.009 q^{49} +178.273i q^{51} +531.123i q^{53} +41.2516 q^{55} +81.0182i q^{57} +378.115 q^{59} -878.946i q^{61} -207.002i q^{63} +(16.6277 - 213.526i) q^{65} +247.600 q^{67} +107.302i q^{69} -182.156i q^{71} -815.886i q^{73} +312.365i q^{75} -207.646i q^{77} +817.462 q^{79} +81.0000 q^{81} -246.607 q^{83} +271.528 q^{85} +426.551 q^{87} -355.903i q^{89} +(-1074.81 - 83.6980i) q^{91} -911.517 q^{93} +123.399 q^{95} -506.709i q^{97} +81.2521 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) $ 84 * q - 756 * q^9 - 104 * q^17 + 2188 * q^25 - 3396 * q^49 + 1616 * q^55 + 696 * q^65 - 3160 * q^79 + 6804 * q^81 + 2088 * q^87 - 2480 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−4.56929			−0.408690			−0.204345	−	0.978899i	$-0.565506\pi$
$5$	−4.56929			−0.408690			−0.204345	+	0.978899i	$0.565506\pi$
$6$		0			0
$7$			23.0002i			1.24189i	0.783853	+	0.620947i	$0.213252\pi$
$7$			23.0002i			1.24189i	−0.783853	+	0.620947i	$0.786748\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−9.02801			−0.247459			−0.123729	−	0.992316i	$-0.539485\pi$
$11$	−9.02801			−0.247459			−0.123729	+	0.992316i	$0.539485\pi$
$12$		0			0
$13$	−3.63901	+	46.7307i	−0.0776370	+	0.996982i
$13$	−3.63901	+	46.7307i	−0.0776370	+	0.996982i
$14$		0			0
$15$			13.7079i			0.235957i
$16$		0			0
$17$	−59.4245			−0.847797			−0.423899	−	0.905710i	$-0.639339\pi$
$17$	−59.4245			−0.847797			−0.423899	+	0.905710i	$0.639339\pi$
$18$		0			0
$19$	−27.0061			−0.326085			−0.163043	−	0.986619i	$-0.552131\pi$
$19$	−27.0061			−0.326085			−0.163043	+	0.986619i	$0.552131\pi$
$20$		0			0
$21$	69.0006			0.717008
$22$		0			0
$23$	−35.7672			−0.324260			−0.162130	−	0.986769i	$-0.551836\pi$
$23$	−35.7672			−0.324260			−0.162130	+	0.986769i	$0.551836\pi$
$24$		0			0
$25$	−104.122			−0.832972
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$			142.184i			0.910442i	0.890378	+	0.455221i	$0.150440\pi$
$29$			142.184i			0.910442i	−0.890378	+	0.455221i	$0.849560\pi$
$30$		0			0
$31$		−	303.839i		−	1.76036i	−0.474643	−	0.880179i	$-0.657423\pi$
$31$		−	303.839i		−	1.76036i	0.474643	−	0.880179i	$-0.342577\pi$
$32$		0			0
$33$			27.0840i			0.142870i
$34$		0			0
$35$		−	105.095i		−	0.507550i
$36$		0			0
$37$	241.300			1.07215			0.536074	−	0.844171i	$-0.319907\pi$
$37$	241.300			1.07215			0.536074	+	0.844171i	$0.319907\pi$
$38$		0			0
$39$	140.192	+	10.9170i	0.575608	+	0.0448237i
$40$		0			0
$41$			280.464i			1.06832i	0.845383	+	0.534160i	$0.179372\pi$
$41$			280.464i			1.06832i	−0.845383	+	0.534160i	$0.820628\pi$
$42$		0			0
$43$		−	198.269i		−	0.703156i	−0.936159	−	0.351578i	$-0.885645\pi$
$43$		−	198.269i		−	0.703156i	0.936159	−	0.351578i	$-0.114355\pi$
$44$		0			0
$45$	41.1236			0.136230
$46$		0			0
$47$		−	319.409i		−	0.991288i	−0.868526	−	0.495644i	$-0.834932\pi$
$47$		−	319.409i		−	0.991288i	0.868526	−	0.495644i	$-0.165068\pi$
$48$		0			0
$49$	−186.009			−0.542300
$50$		0			0
$51$			178.273i			0.489476i
$52$		0			0
$53$			531.123i			1.37652i	0.725466	+	0.688258i	$0.241624\pi$
$53$			531.123i			1.37652i	−0.725466	+	0.688258i	$0.758376\pi$
$54$		0			0
$55$	41.2516			0.101134
$56$		0			0
$57$			81.0182i			0.188265i
$58$		0			0
$59$	378.115			0.834345			0.417172	−	0.908827i	$-0.363021\pi$
$59$	378.115			0.834345			0.417172	+	0.908827i	$0.363021\pi$
$60$		0			0
$61$		−	878.946i		−	1.84488i	−0.386145	−	0.922438i	$-0.626193\pi$
$61$		−	878.946i		−	1.84488i	0.386145	−	0.922438i	$-0.373807\pi$
$62$		0			0
$63$		−	207.002i		−	0.413965i
$64$		0			0
$65$	16.6277	−	213.526i	0.0317295	−	0.407457i
$66$		0			0
$67$	247.600			0.451480			0.225740	−	0.974188i	$-0.427520\pi$
$67$	247.600			0.451480			0.225740	+	0.974188i	$0.427520\pi$
$68$		0			0
$69$			107.302i			0.187211i
$70$		0			0
$71$		−	182.156i		−	0.304478i	−0.988344	−	0.152239i	$-0.951352\pi$
$71$		−	182.156i		−	0.304478i	0.988344	−	0.152239i	$-0.0486483\pi$
$72$		0			0
$73$		−	815.886i		−	1.30811i	−0.756446	−	0.654057i	$-0.773066\pi$
$73$		−	815.886i		−	1.30811i	0.756446	−	0.654057i	$-0.226934\pi$
$74$		0			0
$75$			312.365i			0.480917i
$76$		0			0
$77$		−	207.646i		−	0.307317i
$78$		0			0
$79$	817.462			1.16420			0.582099	−	0.813118i	$-0.302231\pi$
$79$	817.462			1.16420			0.582099	+	0.813118i	$0.302231\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−246.607			−0.326128			−0.163064	−	0.986616i	$-0.552138\pi$
$83$	−246.607			−0.326128			−0.163064	+	0.986616i	$0.552138\pi$
$84$		0			0
$85$	271.528			0.346486
$86$		0			0
$87$	426.551			0.525644
$88$		0			0
$89$		−	355.903i		−	0.423883i	−0.977282	−	0.211942i	$-0.932021\pi$
$89$		−	355.903i		−	0.423883i	0.977282	−	0.211942i	$-0.0679787\pi$
$90$		0			0
$91$	−1074.81	−	83.6980i	−1.23815	−	0.0964168i
$92$		0			0
$93$	−911.517			−1.01634
$94$		0			0
$95$	123.399			0.133268
$96$		0			0
$97$		−	506.709i		−	0.530397i	−0.964194	−	0.265199i	$-0.914563\pi$
$97$		−	506.709i		−	0.530397i	0.964194	−	0.265199i	$-0.0854375\pi$
$98$		0			0
$99$	81.2521			0.0824863
$100$		0			0
$101$			877.066i			0.864073i	0.901856	+	0.432036i	$0.142205\pi$
$101$			877.066i			0.864073i	−0.901856	+	0.432036i	$0.857795\pi$
$102$		0			0
$103$	1177.42			1.12636			0.563179	−	0.826335i	$-0.309578\pi$
$103$	1177.42			1.12636			0.563179	+	0.826335i	$0.309578\pi$
$104$		0			0
$105$	−315.284			−0.293034
$106$		0			0
$107$		−	1125.19i		−	1.01660i	−0.861179	−	0.508301i	$-0.830274\pi$
$107$		−	1125.19i		−	1.01660i	0.861179	−	0.508301i	$-0.169726\pi$
$108$		0			0
$109$	788.108			0.692542			0.346271	−	0.938135i	$-0.387448\pi$
$109$	788.108			0.692542			0.346271	+	0.938135i	$0.387448\pi$
$110$		0			0
$111$		−	723.900i		−	0.619005i
$112$		0			0
$113$	−1182.33			−0.984284			−0.492142	−	0.870515i	$-0.663786\pi$
$113$	−1182.33			−0.984284			−0.492142	+	0.870515i	$0.663786\pi$
$114$		0			0
$115$	163.431			0.132522
$116$		0			0
$117$	32.7511	−	420.576i	0.0258790	−	0.332327i
$118$		0			0
$119$		−	1366.77i		−	1.05287i
$120$		0			0
$121$	−1249.50			−0.938764
$122$		0			0
$123$	841.392			0.616795
$124$		0			0
$125$	1046.92			0.749118
$126$		0			0
$127$	−1038.87			−0.725864			−0.362932	−	0.931816i	$-0.618224\pi$
$127$	−1038.87			−0.725864			−0.362932	+	0.931816i	$0.618224\pi$
$128$		0			0
$129$	−594.806			−0.405967
$130$		0			0
$131$		−	142.564i		−	0.0950833i	−0.998869	−	0.0475417i	$-0.984861\pi$
$131$		−	142.564i		−	0.0950833i	0.998869	−	0.0475417i	$-0.0151387\pi$
$132$		0			0
$133$		−	621.145i		−	0.404963i
$134$		0			0
$135$		−	123.371i		−	0.0786524i
$136$		0			0
$137$		−	2413.02i		−	1.50481i	−0.658703	−	0.752403i	$-0.728895\pi$
$137$		−	2413.02i		−	1.50481i	0.658703	−	0.752403i	$-0.271105\pi$
$138$		0			0
$139$		−	827.007i		−	0.504646i	−0.967643	−	0.252323i	$-0.918805\pi$
$139$		−	827.007i		−	0.504646i	0.967643	−	0.252323i	$-0.0811946\pi$
$140$		0			0
$141$	−958.226			−0.572320
$142$		0			0
$143$	32.8530	−	421.885i	0.0192119	−	0.246712i
$144$		0			0
$145$		−	649.679i		−	0.372089i
$146$		0			0
$147$			558.026i			0.313097i
$148$		0			0
$149$	−3151.97			−1.73302			−0.866509	−	0.499162i	$-0.833641\pi$
$149$	−3151.97			−1.73302			−0.866509	+	0.499162i	$0.833641\pi$
$150$		0			0
$151$			934.341i			0.503547i	0.967786	+	0.251774i	$0.0810138\pi$
$151$			934.341i			0.503547i	−0.967786	+	0.251774i	$0.918986\pi$
$152$		0			0
$153$	534.820			0.282599
$154$		0			0
$155$			1388.33i			0.719441i
$156$		0			0
$157$			307.214i			0.156168i	0.996947	+	0.0780840i	$0.0248802\pi$
$157$			307.214i			0.156168i	−0.996947	+	0.0780840i	$0.975120\pi$
$158$		0			0
$159$	1593.37			0.794732
$160$		0			0
$161$		−	822.652i		−	0.402696i
$162$		0			0
$163$	−368.699			−0.177170			−0.0885851	−	0.996069i	$-0.528235\pi$
$163$	−368.699			−0.177170			−0.0885851	+	0.996069i	$0.528235\pi$
$164$		0			0
$165$		−	123.755i		−	0.0583897i
$166$		0			0
$167$			402.412i			0.186464i	0.995644	+	0.0932322i	$0.0297199\pi$
$167$			402.412i			0.186464i	−0.995644	+	0.0932322i	$0.970280\pi$
$168$		0			0
$169$	−2170.52	−	340.107i	−0.987945	−	0.154805i
$170$		0			0
$171$	243.055			0.108695
$172$		0			0
$173$			3541.26i			1.55629i	0.628088	+	0.778143i	$0.283838\pi$
$173$			3541.26i			1.55629i	−0.628088	+	0.778143i	$0.716162\pi$
$174$		0			0
$175$		−	2394.82i		−	1.03446i
$176$		0			0
$177$		−	1134.34i		−	0.481709i
$178$		0			0
$179$		−	3755.60i		−	1.56819i	−0.620640	−	0.784096i	$-0.713127\pi$
$179$		−	3755.60i		−	1.56819i	0.620640	−	0.784096i	$-0.286873\pi$
$180$		0			0
$181$			1146.21i			0.470701i	0.971911	+	0.235350i	$0.0756238\pi$
$181$			1146.21i			0.470701i	−0.971911	+	0.235350i	$0.924376\pi$
$182$		0			0
$183$	−2636.84			−1.06514
$184$		0			0
$185$	−1102.57			−0.438176
$186$		0			0
$187$	536.485			0.209795
$188$		0			0
$189$	−621.005			−0.239003
$190$		0			0
$191$	−2181.45			−0.826410			−0.413205	−	0.910638i	$-0.635591\pi$
$191$	−2181.45			−0.826410			−0.413205	+	0.910638i	$0.635591\pi$
$192$		0			0
$193$		−	4438.58i		−	1.65542i	−0.561158	−	0.827709i	$-0.689644\pi$
$193$		−	4438.58i		−	1.65542i	0.561158	−	0.827709i	$-0.310356\pi$
$194$		0			0
$195$	−640.579	−	49.8831i	−0.235245	−	0.0183190i
$196$		0			0
$197$	−3788.49			−1.37014			−0.685072	−	0.728475i	$-0.740229\pi$
$197$	−3788.49			−1.37014			−0.685072	+	0.728475i	$0.740229\pi$
$198$		0			0
$199$	−1535.85			−0.547102			−0.273551	−	0.961858i	$-0.588198\pi$
$199$	−1535.85			−0.547102			−0.273551	+	0.961858i	$0.588198\pi$
$200$		0			0
$201$		−	742.800i		−	0.260662i
$202$		0			0
$203$	−3270.25			−1.13067
$204$		0			0
$205$		−	1281.52i		−	0.436612i
$206$		0			0
$207$	321.905			0.108087
$208$		0			0
$209$	243.811			0.0806927
$210$		0			0
$211$			5166.95i			1.68582i	0.538055	+	0.842909i	$0.319159\pi$
$211$			5166.95i			1.68582i	−0.538055	+	0.842909i	$0.680841\pi$
$212$		0			0
$213$	−546.467			−0.175790
$214$		0			0
$215$			905.948i			0.287373i
$216$		0			0
$217$	6988.35			2.18618
$218$		0			0
$219$	−2447.66			−0.755240
$220$		0			0
$221$	216.246	−	2776.95i	0.0658204	−	0.845238i
$222$		0			0
$223$		−	4812.46i		−	1.44514i	−0.691297	−	0.722570i	$-0.742961\pi$
$223$		−	4812.46i		−	1.44514i	0.691297	−	0.722570i	$-0.257039\pi$
$224$		0			0
$225$	937.094			0.277657
$226$		0			0
$227$	−584.946			−0.171032			−0.0855159	−	0.996337i	$-0.527254\pi$
$227$	−584.946			−0.171032			−0.0855159	+	0.996337i	$0.527254\pi$
$228$		0			0
$229$	4584.28			1.32287			0.661436	−	0.750001i	$-0.269947\pi$
$229$	4584.28			1.32287			0.661436	+	0.750001i	$0.269947\pi$
$230$		0			0
$231$	−622.938			−0.177430
$232$		0			0
$233$	4302.16			1.20963			0.604815	−	0.796366i	$-0.293247\pi$
$233$	4302.16			1.20963			0.604815	+	0.796366i	$0.293247\pi$
$234$		0			0
$235$			1459.47i			0.405130i
$236$		0			0
$237$		−	2452.39i		−	0.672150i
$238$		0			0
$239$		−	7132.42i		−	1.93037i	−0.261573	−	0.965184i	$-0.584241\pi$
$239$		−	7132.42i		−	1.93037i	0.261573	−	0.965184i	$-0.415759\pi$
$240$		0			0
$241$			813.188i			0.217353i	0.994077	+	0.108676i	$0.0346612\pi$
$241$			813.188i			0.217353i	−0.994077	+	0.108676i	$0.965339\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	849.929			0.221632
$246$		0			0
$247$	98.2755	−	1262.01i	0.0253163	−	0.325101i
$248$		0			0
$249$			739.820i			0.188290i
$250$		0			0
$251$		−	98.3380i		−	0.0247292i	−0.999924	−	0.0123646i	$-0.996064\pi$
$251$		−	98.3380i		−	0.0247292i	0.999924	−	0.0123646i	$-0.00393588\pi$
$252$		0			0
$253$	322.906			0.0802409
$254$		0			0
$255$		−	814.584i		−	0.200044i
$256$		0			0
$257$	2348.27			0.569966			0.284983	−	0.958533i	$-0.408012\pi$
$257$	2348.27			0.569966			0.284983	+	0.958533i	$0.408012\pi$
$258$		0			0
$259$			5549.94i			1.33149i
$260$		0			0
$261$		−	1279.65i		−	0.303481i
$262$		0			0
$263$	3653.13			0.856507			0.428254	−	0.903659i	$-0.359129\pi$
$263$	3653.13			0.856507			0.428254	+	0.903659i	$0.359129\pi$
$264$		0			0
$265$		−	2426.86i		−	0.562568i
$266$		0			0
$267$	−1067.71			−0.244729
$268$		0			0
$269$			496.183i			0.112464i	0.998418	+	0.0562320i	$0.0179086\pi$
$269$			496.183i			0.112464i	−0.998418	+	0.0562320i	$0.982091\pi$
$270$		0			0
$271$		−	1164.70i		−	0.261072i	−0.991444	−	0.130536i	$-0.958330\pi$
$271$		−	1164.70i		−	0.261072i	0.991444	−	0.130536i	$-0.0416698\pi$
$272$		0			0
$273$	−251.094	+	3224.44i	−0.0556663	+	0.714843i
$274$		0			0
$275$	940.010			0.206126
$276$		0			0
$277$			3541.96i			0.768289i	0.923273	+	0.384144i	$0.125504\pi$
$277$			3541.96i			0.768289i	−0.923273	+	0.384144i	$0.874496\pi$
$278$		0			0
$279$			2734.55i			0.586786i
$280$		0			0
$281$		−	1001.92i		−	0.212703i	−0.994329	−	0.106352i	$-0.966083\pi$
$281$		−	1001.92i		−	0.212703i	0.994329	−	0.106352i	$-0.0339169\pi$
$282$		0			0
$283$		−	4471.46i		−	0.939226i	−0.882872	−	0.469613i	$-0.844393\pi$
$283$		−	4471.46i		−	0.939226i	0.882872	−	0.469613i	$-0.155607\pi$
$284$		0			0
$285$		−	370.196i		−	0.0769422i
$286$		0			0
$287$	−6450.73			−1.32674
$288$		0			0
$289$	−1381.73			−0.281240
$290$		0			0
$291$	−1520.13			−0.306225
$292$		0			0
$293$	−3292.99			−0.656582			−0.328291	−	0.944577i	$-0.606473\pi$
$293$	−3292.99			−0.656582			−0.328291	+	0.944577i	$0.606473\pi$
$294$		0			0
$295$	−1727.72			−0.340988
$296$		0			0
$297$		−	243.756i		−	0.0476235i
$298$		0			0
$299$	130.157	−	1671.43i	0.0251745	−	0.323281i
$300$		0			0
$301$	4560.22			0.873245
$302$		0			0
$303$	2631.20			0.498873
$304$		0			0
$305$			4016.16i			0.753983i
$306$		0			0
$307$	3274.18			0.608688			0.304344	−	0.952562i	$-0.401563\pi$
$307$	3274.18			0.608688			0.304344	+	0.952562i	$0.401563\pi$
$308$		0			0
$309$		−	3532.27i		−	0.650303i
$310$		0			0
$311$	−3191.33			−0.581876			−0.290938	−	0.956742i	$-0.593967\pi$
$311$	−3191.33			−0.581876			−0.290938	+	0.956742i	$0.593967\pi$
$312$		0			0
$313$	5529.90			0.998621			0.499311	−	0.866423i	$-0.333587\pi$
$313$	5529.90			0.998621			0.499311	+	0.866423i	$0.333587\pi$
$314$		0			0
$315$			945.852i			0.169183i
$316$		0			0
$317$	6475.59			1.14734			0.573668	−	0.819088i	$-0.305520\pi$
$317$	6475.59			1.14734			0.573668	+	0.819088i	$0.305520\pi$
$318$		0			0
$319$		−	1283.63i		−	0.225297i
$320$		0			0
$321$	−3375.58			−0.586936
$322$		0			0
$323$	1604.82			0.276454
$324$		0			0
$325$	378.900	−	4865.67i	0.0646694	−	0.830458i
$326$		0			0
$327$		−	2364.32i		−	0.399839i
$328$		0			0
$329$	7346.46			1.23107
$330$		0			0
$331$	10016.3			1.66327			0.831637	−	0.555319i	$-0.187404\pi$
$331$	10016.3			1.66327			0.831637	+	0.555319i	$0.187404\pi$
$332$		0			0
$333$	−2171.70			−0.357382
$334$		0			0
$335$	−1131.36			−0.184515
$336$		0			0
$337$	−6602.82			−1.06730			−0.533648	−	0.845707i	$-0.679179\pi$
$337$	−6602.82			−1.06730			−0.533648	+	0.845707i	$0.679179\pi$
$338$		0			0
$339$			3546.99i			0.568277i
$340$		0			0
$341$			2743.06i			0.435616i
$342$		0			0
$343$			3610.83i			0.568415i
$344$		0			0
$345$		−	490.292i		−	0.0765115i
$346$		0			0
$347$		−	8226.15i		−	1.27263i	−0.771429	−	0.636315i	$-0.780458\pi$
$347$		−	8226.15i		−	1.27263i	0.771429	−	0.636315i	$-0.219542\pi$
$348$		0			0
$349$	−5987.66			−0.918373			−0.459187	−	0.888340i	$-0.651859\pi$
$349$	−5987.66			−0.918373			−0.459187	+	0.888340i	$0.651859\pi$
$350$		0			0
$351$	−1261.73	−	98.2533i	−0.191869	−	0.0149412i
$352$		0			0
$353$		−	8372.28i		−	1.26236i	−0.775638	−	0.631178i	$-0.782572\pi$
$353$		−	8372.28i		−	1.26236i	0.775638	−	0.631178i	$-0.217428\pi$
$354$		0			0
$355$			832.323i			0.124437i
$356$		0			0
$357$	−4100.32			−0.607877
$358$		0			0
$359$			5365.50i			0.788803i	0.918938	+	0.394401i	$0.129048\pi$
$359$			5365.50i			0.788803i	−0.918938	+	0.394401i	$0.870952\pi$
$360$		0			0
$361$	−6129.67			−0.893668
$362$		0			0
$363$			3748.49i			0.541996i
$364$		0			0
$365$			3728.02i			0.534613i
$366$		0			0
$367$	−2032.50			−0.289089			−0.144545	−	0.989498i	$-0.546172\pi$
$367$	−2032.50			−0.289089			−0.144545	+	0.989498i	$0.546172\pi$
$368$		0			0
$369$		−	2524.18i		−	0.356107i
$370$		0			0
$371$	−12215.9			−1.70949
$372$		0			0
$373$		−	13867.4i		−	1.92500i	−0.271282	−	0.962500i	$-0.587448\pi$
$373$		−	13867.4i		−	1.92500i	0.271282	−	0.962500i	$-0.412552\pi$
$374$		0			0
$375$		−	3140.77i		−	0.432503i
$376$		0			0
$377$	−6644.34	−	517.408i	−0.907694	−	0.0706840i
$378$		0			0
$379$	−4445.76			−0.602542			−0.301271	−	0.953539i	$-0.597411\pi$
$379$	−4445.76			−0.602542			−0.301271	+	0.953539i	$0.597411\pi$
$380$		0			0
$381$			3116.61i			0.419078i
$382$		0			0
$383$		−	1882.41i		−	0.251141i	−0.992085	−	0.125570i	$-0.959924\pi$
$383$		−	1882.41i		−	0.251141i	0.992085	−	0.125570i	$-0.0400761\pi$
$384$		0			0
$385$			948.795i			0.125598i
$386$		0			0
$387$			1784.42i			0.234385i
$388$		0			0
$389$			4490.10i			0.585237i	0.956229	+	0.292619i	$0.0945266\pi$
$389$			4490.10i			0.585237i	−0.956229	+	0.292619i	$0.905473\pi$
$390$		0			0
$391$	2125.45			0.274907
$392$		0			0
$393$	−427.693			−0.0548964
$394$		0			0
$395$	−3735.23			−0.475796
$396$		0			0
$397$	9505.67			1.20170			0.600852	−	0.799361i	$-0.294828\pi$
$397$	9505.67			1.20170			0.600852	+	0.799361i	$0.294828\pi$
$398$		0			0
$399$	−1863.44			−0.233806
$400$		0			0
$401$			9181.79i			1.14343i	0.820451	+	0.571716i	$0.193722\pi$
$401$			9181.79i			1.14343i	−0.820451	+	0.571716i	$0.806278\pi$
$402$		0			0
$403$	14198.6	+	1105.67i	1.75504	+	0.136669i
$404$		0			0
$405$	−370.113			−0.0454100
$406$		0			0
$407$	−2178.46			−0.265312
$408$		0			0
$409$			3469.65i			0.419470i	0.977758	+	0.209735i	$0.0672601\pi$
$409$			3469.65i			0.419470i	−0.977758	+	0.209735i	$0.932740\pi$
$410$		0			0
$411$	−7239.07			−0.868800
$412$		0			0
$413$			8696.71i			1.03617i
$414$		0			0
$415$	1126.82			0.133285
$416$		0			0
$417$	−2481.02			−0.291358
$418$		0			0
$419$		−	12868.1i		−	1.50035i	−0.661241	−	0.750174i	$-0.729970\pi$
$419$		−	12868.1i		−	1.50035i	0.661241	−	0.750174i	$-0.270030\pi$
$420$		0			0
$421$	4414.39			0.511031			0.255516	−	0.966805i	$-0.417755\pi$
$421$	4414.39			0.511031			0.255516	+	0.966805i	$0.417755\pi$
$422$		0			0
$423$			2874.68i			0.330429i
$424$		0			0
$425$	6187.37			0.706192
$426$		0			0
$427$	20215.9			2.29114
$428$		0			0
$429$	−1265.65	−	98.5591i	−0.142439	−	0.0110920i
$430$		0			0
$431$			11981.4i			1.33903i	0.742797	+	0.669517i	$0.233499\pi$
$431$			11981.4i			1.33903i	−0.742797	+	0.669517i	$0.766501\pi$
$432$		0			0
$433$	4268.40			0.473733			0.236866	−	0.971542i	$-0.423880\pi$
$433$	4268.40			0.473733			0.236866	+	0.971542i	$0.423880\pi$
$434$		0			0
$435$	−1949.04			−0.214826
$436$		0			0
$437$	965.932			0.105736
$438$		0			0
$439$	−7424.25			−0.807152			−0.403576	−	0.914946i	$-0.632233\pi$
$439$	−7424.25			−0.807152			−0.403576	+	0.914946i	$0.632233\pi$
$440$		0			0
$441$	1674.08			0.180767
$442$		0			0
$443$			3110.29i			0.333576i	0.985993	+	0.166788i	$0.0533395\pi$
$443$			3110.29i			0.333576i	−0.985993	+	0.166788i	$0.946660\pi$
$444$		0			0
$445$			1626.22i			0.173237i
$446$		0			0
$447$			9455.91i			1.00056i
$448$		0			0
$449$		−	5412.37i		−	0.568876i	−0.958694	−	0.284438i	$-0.908193\pi$
$449$		−	5412.37i		−	0.568876i	0.958694	−	0.284438i	$-0.0918071\pi$
$450$		0			0
$451$		−	2532.03i		−	0.264365i
$452$		0			0
$453$	2803.02			0.290723
$454$		0			0
$455$	4911.14	+	382.441i	0.506018	+	0.0394046i
$456$		0			0
$457$			3771.11i			0.386007i	0.981198	+	0.193003i	$0.0618228\pi$
$457$			3771.11i			0.386007i	−0.981198	+	0.193003i	$0.938177\pi$
$458$		0			0
$459$		−	1604.46i		−	0.163159i
$460$		0			0
$461$	−1276.30			−0.128944			−0.0644722	−	0.997920i	$-0.520536\pi$
$461$	−1276.30			−0.128944			−0.0644722	+	0.997920i	$0.520536\pi$
$462$		0			0
$463$			10359.5i			1.03984i	0.854216	+	0.519919i	$0.174038\pi$
$463$			10359.5i			1.03984i	−0.854216	+	0.519919i	$0.825962\pi$
$464$		0			0
$465$	4164.99			0.415369
$466$		0			0
$467$			18643.9i			1.84740i	0.383112	+	0.923702i	$0.374852\pi$
$467$			18643.9i			1.84740i	−0.383112	+	0.923702i	$0.625148\pi$
$468$		0			0
$469$			5694.85i			0.560690i
$470$		0			0
$471$	921.643			0.0901636
$472$		0			0
$473$			1789.97i			0.174002i
$474$		0			0
$475$	2811.92			0.271620
$476$		0			0
$477$		−	4780.10i		−	0.458838i
$478$		0			0
$479$			4124.15i			0.393397i	0.980464	+	0.196699i	$0.0630221\pi$
$479$			4124.15i			0.393397i	−0.980464	+	0.196699i	$0.936978\pi$
$480$		0			0
$481$	−878.093	+	11276.1i	−0.0832383	+	1.06891i
$482$		0			0
$483$	−2467.96			−0.232497
$484$		0			0
$485$			2315.30i			0.216768i
$486$		0			0
$487$		−	3948.63i		−	0.367412i	−0.982981	−	0.183706i	$-0.941191\pi$
$487$		−	3948.63i		−	0.367412i	0.982981	−	0.183706i	$-0.0588095\pi$
$488$		0			0
$489$			1106.10i			0.102289i
$490$		0			0
$491$		−	9699.68i		−	0.891528i	−0.895150	−	0.445764i	$-0.852932\pi$
$491$		−	9699.68i		−	0.891528i	0.895150	−	0.445764i	$-0.147068\pi$
$492$		0			0
$493$		−	8449.19i		−	0.771871i
$494$		0			0
$495$	−371.265			−0.0337113
$496$		0			0
$497$	4189.62			0.378129
$498$		0			0
$499$	10351.2			0.928620			0.464310	−	0.885673i	$-0.346302\pi$
$499$	10351.2			0.928620			0.464310	+	0.885673i	$0.346302\pi$
$500$		0			0
$501$	1207.23			0.107655
$502$		0			0
$503$	14291.2			1.26682			0.633412	−	0.773815i	$-0.281654\pi$
$503$	14291.2			1.26682			0.633412	+	0.773815i	$0.281654\pi$
$504$		0			0
$505$		−	4007.57i		−	0.353138i
$506$		0			0
$507$	−1020.32	+	6511.55i	−0.0893769	+	0.570390i
$508$		0			0
$509$	−6030.07			−0.525104			−0.262552	−	0.964918i	$-0.584564\pi$
$509$	−6030.07			−0.525104			−0.262552	+	0.964918i	$0.584564\pi$
$510$		0			0
$511$	18765.5			1.62454
$512$		0			0
$513$		−	729.164i		−	0.0627551i
$514$		0			0
$515$	−5379.99			−0.460331
$516$		0			0
$517$			2883.62i			0.245303i
$518$		0			0
$519$	10623.8			0.898522
$520$		0			0
$521$	−2617.24			−0.220083			−0.110042	−	0.993927i	$-0.535098\pi$
$521$	−2617.24			−0.220083			−0.110042	+	0.993927i	$0.535098\pi$
$522$		0			0
$523$		−	21188.4i		−	1.77152i	−0.464144	−	0.885760i	$-0.653638\pi$
$523$		−	21188.4i		−	1.77152i	0.464144	−	0.885760i	$-0.346362\pi$
$524$		0			0
$525$	−7184.45			−0.597248
$526$		0			0
$527$			18055.5i			1.49243i
$528$		0			0
$529$	−10887.7			−0.894856
$530$		0			0
$531$	−3403.03			−0.278115
$532$		0			0
$533$	−13106.3	−	1020.61i	−1.06510	−	0.0829411i
$534$		0			0
$535$			5141.33i			0.415475i
$536$		0			0
$537$	−11266.8			−0.905396
$538$		0			0
$539$	1679.29			0.134197
$540$		0			0
$541$	5104.29			0.405639			0.202819	−	0.979216i	$-0.434990\pi$
$541$	5104.29			0.405639			0.202819	+	0.979216i	$0.434990\pi$
$542$		0			0
$543$	3438.62			0.271759
$544$		0			0
$545$	−3601.10			−0.283035
$546$		0			0
$547$		−	10142.1i		−	0.792770i	−0.918084	−	0.396385i	$-0.870265\pi$
$547$		−	10142.1i		−	0.792770i	0.918084	−	0.396385i	$-0.129735\pi$
$548$		0			0
$549$			7910.51i			0.614959i
$550$		0			0
$551$		−	3839.82i		−	0.296882i
$552$		0			0
$553$			18801.8i			1.44581i
$554$		0			0
$555$			3307.71i			0.252981i
$556$		0			0
$557$	−1339.31			−0.101882			−0.0509409	−	0.998702i	$-0.516222\pi$
$557$	−1339.31			−0.101882			−0.0509409	+	0.998702i	$0.516222\pi$
$558$		0			0
$559$	9265.24	+	721.503i	0.701034	+	0.0545909i
$560$		0			0
$561$		−	1609.45i		−	0.121125i
$562$		0			0
$563$			7894.26i			0.590947i	0.955351	+	0.295474i	$0.0954775\pi$
$563$			7894.26i			0.590947i	−0.955351	+	0.295474i	$0.904523\pi$
$564$		0			0
$565$	5402.41			0.402267
$566$		0			0
$567$			1863.02i			0.137988i
$568$		0			0
$569$	−5150.43			−0.379468			−0.189734	−	0.981836i	$-0.560763\pi$
$569$	−5150.43			−0.379468			−0.189734	+	0.981836i	$0.560763\pi$
$570$		0			0
$571$			9653.04i			0.707473i	0.935345	+	0.353737i	$0.115089\pi$
$571$			9653.04i			0.707473i	−0.935345	+	0.353737i	$0.884911\pi$
$572$		0			0
$573$			6544.35i			0.477128i
$574$		0			0
$575$	3724.14			0.270099
$576$		0			0
$577$		−	6097.68i		−	0.439948i	−0.975506	−	0.219974i	$-0.929403\pi$
$577$		−	6097.68i		−	0.439948i	0.975506	−	0.219974i	$-0.0705972\pi$
$578$		0			0
$579$	−13315.7			−0.955756
$580$		0			0
$581$		−	5672.00i		−	0.405016i
$582$		0			0
$583$		−	4794.98i		−	0.340631i
$584$		0			0
$585$	−149.649	+	1921.74i	−0.0105765	+	0.135819i
$586$		0			0
$587$	−18721.5			−1.31639			−0.658195	−	0.752847i	$-0.728680\pi$
$587$	−18721.5			−1.31639			−0.658195	+	0.752847i	$0.728680\pi$
$588$		0			0
$589$			8205.50i			0.574027i
$590$		0			0
$591$			11365.5i			0.791053i
$592$		0			0
$593$			27503.7i			1.90462i	0.305127	+	0.952312i	$0.401301\pi$
$593$			27503.7i			1.90462i	−0.305127	+	0.952312i	$0.598699\pi$
$594$		0			0
$595$			6245.19i			0.430299i
$596$		0			0
$597$			4607.54i			0.315869i
$598$		0			0
$599$	−6924.86			−0.472357			−0.236179	−	0.971710i	$-0.575895\pi$
$599$	−6924.86			−0.472357			−0.236179	+	0.971710i	$0.575895\pi$
$600$		0			0
$601$	−438.105			−0.0297349			−0.0148674	−	0.999889i	$-0.504733\pi$
$601$	−438.105			−0.0297349			−0.0148674	+	0.999889i	$0.504733\pi$
$602$		0			0
$603$	−2228.40			−0.150493
$604$		0			0
$605$	5709.31			0.383664
$606$		0			0
$607$	130.867			0.00875076			0.00437538	−	0.999990i	$-0.498607\pi$
$607$	130.867			0.00875076			0.00437538	+	0.999990i	$0.498607\pi$
$608$		0			0
$609$			9810.75i			0.652794i
$610$		0			0
$611$	14926.2	+	1162.33i	0.988296	+	0.0769606i
$612$		0			0
$613$	14591.0			0.961382			0.480691	−	0.876890i	$-0.340386\pi$
$613$	14591.0			0.961382			0.480691	+	0.876890i	$0.340386\pi$
$614$		0			0
$615$	−3844.57			−0.252078
$616$		0			0
$617$			26154.6i			1.70656i	0.521457	+	0.853278i	$0.325389\pi$
$617$			26154.6i			1.70656i	−0.521457	+	0.853278i	$0.674611\pi$
$618$		0			0
$619$	−5541.38			−0.359817			−0.179909	−	0.983683i	$-0.557580\pi$
$619$	−5541.38			−0.359817			−0.179909	+	0.983683i	$0.557580\pi$
$620$		0			0
$621$		−	965.714i		−	0.0624038i
$622$		0			0
$623$	8185.83			0.526418
$624$		0			0
$625$	8231.49			0.526815
$626$		0			0
$627$		−	731.433i		−	0.0465879i
$628$		0			0
$629$	−14339.1			−0.908963
$630$		0			0
$631$		−	3022.53i		−	0.190689i	−0.995444	−	0.0953447i	$-0.969605\pi$
$631$		−	3022.53i		−	0.190689i	0.995444	−	0.0953447i	$-0.0303953\pi$
$632$		0			0
$633$	15500.9			0.973308
$634$		0			0
$635$	4746.90			0.296653
$636$		0			0
$637$	676.888	−	8692.32i	0.0421025	−	0.540663i
$638$		0			0
$639$			1639.40i			0.101493i
$640$		0			0
$641$	−24068.8			−1.48309			−0.741545	−	0.670904i	$-0.765906\pi$
$641$	−24068.8			−1.48309			−0.741545	+	0.670904i	$0.765906\pi$
$642$		0			0
$643$	−10923.5			−0.669952			−0.334976	−	0.942227i	$-0.608728\pi$
$643$	−10923.5			−0.669952			−0.334976	+	0.942227i	$0.608728\pi$
$644$		0			0
$645$	2717.84			0.165915
$646$		0			0
$647$	6403.53			0.389102			0.194551	−	0.980892i	$-0.437675\pi$
$647$	6403.53			0.389102			0.194551	+	0.980892i	$0.437675\pi$
$648$		0			0
$649$	−3413.62			−0.206466
$650$		0			0
$651$		−	20965.1i		−	1.26219i
$652$		0			0
$653$		−	3560.37i		−	0.213366i	−0.994293	−	0.106683i	$-0.965977\pi$
$653$		−	3560.37i		−	0.213366i	0.994293	−	0.106683i	$-0.0340230\pi$
$654$		0			0
$655$			651.419i			0.0388596i
$656$		0			0
$657$			7342.98i			0.436038i
$658$		0			0
$659$			18983.0i			1.12212i	0.827776	+	0.561058i	$0.189606\pi$
$659$			18983.0i			1.12212i	−0.827776	+	0.561058i	$0.810394\pi$
$660$		0			0
$661$	6588.80			0.387707			0.193854	−	0.981030i	$-0.437901\pi$
$661$	6588.80			0.387707			0.193854	+	0.981030i	$0.437901\pi$
$662$		0			0
$663$	−8330.84	−	648.739i	−0.487999	−	0.0380014i
$664$		0			0
$665$			2838.19i			0.165504i
$666$		0			0
$667$		−	5085.51i		−	0.295220i
$668$		0			0
$669$	−14437.4			−0.834352
$670$		0			0
$671$			7935.13i			0.456531i
$672$		0			0
$673$	−11113.6			−0.636547			−0.318273	−	0.947999i	$-0.603103\pi$
$673$	−11113.6			−0.636547			−0.318273	+	0.947999i	$0.603103\pi$
$674$		0			0
$675$		−	2811.28i		−	0.160306i
$676$		0			0
$677$			7932.48i			0.450325i	0.974321	+	0.225162i	$0.0722913\pi$
$677$			7932.48i			0.450325i	−0.974321	+	0.225162i	$0.927709\pi$
$678$		0			0
$679$	11654.4			0.658697
$680$		0			0
$681$			1754.84i			0.0987453i
$682$		0			0
$683$	3650.03			0.204487			0.102244	−	0.994759i	$-0.467398\pi$
$683$	3650.03			0.204487			0.102244	+	0.994759i	$0.467398\pi$
$684$		0			0
$685$			11025.8i			0.614999i
$686$		0			0
$687$		−	13752.8i		−	0.763761i
$688$		0			0
$689$	−24819.7	−	1932.76i	−1.37236	−	0.106868i
$690$		0			0
$691$	7823.65			0.430717			0.215359	−	0.976535i	$-0.430908\pi$
$691$	7823.65			0.430717			0.215359	+	0.976535i	$0.430908\pi$
$692$		0			0
$693$			1868.81i			0.102439i
$694$		0			0
$695$			3778.84i			0.206244i
$696$		0			0
$697$		−	16666.4i		−	0.905719i
$698$		0			0
$699$		−	12906.5i		−	0.698380i
$700$		0			0
$701$		−	30382.7i		−	1.63700i	−0.574507	−	0.818500i	$-0.694806\pi$
$701$		−	30382.7i		−	1.63700i	0.574507	−	0.818500i	$-0.305194\pi$
$702$		0			0
$703$	−6516.56			−0.349611
$704$		0			0
$705$	4378.42			0.233902
$706$		0			0
$707$	−20172.7			−1.07309
$708$		0			0
$709$	19960.9			1.05733			0.528665	−	0.848830i	$-0.322693\pi$
$709$	19960.9			1.05733			0.528665	+	0.848830i	$0.322693\pi$
$710$		0			0
$711$	−7357.16			−0.388066
$712$		0			0
$713$			10867.5i			0.570813i
$714$		0			0
$715$	−150.115	+	1927.72i	−0.00785173	+	0.100829i
$716$		0			0
$717$	−21397.3			−1.11450
$718$		0			0
$719$	−10658.4			−0.552838			−0.276419	−	0.961037i	$-0.589148\pi$
$719$	−10658.4			−0.552838			−0.276419	+	0.961037i	$0.589148\pi$
$720$		0			0
$721$			27080.9i			1.39882i
$722$		0			0
$723$	2439.56			0.125489
$724$		0			0
$725$		−	14804.4i		−	0.758373i
$726$		0			0
$727$	−20458.2			−1.04368			−0.521838	−	0.853045i	$-0.674753\pi$
$727$	−20458.2			−1.04368			−0.521838	+	0.853045i	$0.674753\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			11782.0i			0.596134i
$732$		0			0
$733$	31681.2			1.59642			0.798208	−	0.602382i	$-0.205782\pi$
$733$	31681.2			1.59642			0.798208	+	0.602382i	$0.205782\pi$
$734$		0			0
$735$		−	2549.79i		−	0.127960i
$736$		0			0
$737$	−2235.33			−0.111723
$738$		0			0
$739$	33577.7			1.67142			0.835709	−	0.549173i	$-0.185057\pi$
$739$	33577.7			1.67142			0.835709	+	0.549173i	$0.185057\pi$
$740$		0			0
$741$	−3786.04	−	294.826i	−0.187697	−	0.0146164i
$742$		0			0
$743$		−	19963.9i		−	0.985740i	−0.870103	−	0.492870i	$-0.835948\pi$
$743$		−	19963.9i		−	0.985740i	0.870103	−	0.492870i	$-0.164052\pi$
$744$		0			0
$745$	14402.3			0.708267
$746$		0			0
$747$	2219.46			0.108709
$748$		0			0
$749$	25879.6			1.26251
$750$		0			0
$751$	−34167.4			−1.66017			−0.830083	−	0.557640i	$-0.811707\pi$
$751$	−34167.4			−1.66017			−0.830083	+	0.557640i	$0.811707\pi$
$752$		0			0
$753$	−295.014			−0.0142774
$754$		0			0
$755$		−	4269.28i		−	0.205795i
$756$		0			0
$757$			35511.1i			1.70498i	0.522741	+	0.852492i	$0.324910\pi$
$757$			35511.1i			1.70498i	−0.522741	+	0.852492i	$0.675090\pi$
$758$		0			0
$759$		−	968.719i		−	0.0463271i
$760$		0			0
$761$		−	29278.7i		−	1.39468i	−0.716741	−	0.697340i	$-0.754367\pi$
$761$		−	29278.7i		−	1.39468i	0.716741	−	0.697340i	$-0.245633\pi$
$762$		0			0
$763$			18126.6i			0.860063i
$764$		0			0
$765$	−2443.75			−0.115495
$766$		0			0
$767$	−1375.96	+	17669.6i	−0.0647760	+	0.831826i
$768$		0			0
$769$		−	26102.0i		−	1.22401i	−0.790855	−	0.612003i	$-0.790364\pi$
$769$		−	26102.0i		−	1.22401i	0.790855	−	0.612003i	$-0.209636\pi$
$770$		0			0
$771$		−	7044.82i		−	0.329070i
$772$		0			0
$773$	−29899.4			−1.39121			−0.695605	−	0.718424i	$-0.744864\pi$
$773$	−29899.4			−1.39121			−0.695605	+	0.718424i	$0.744864\pi$
$774$		0			0
$775$			31636.2i			1.46633i
$776$		0			0
$777$	16649.8			0.768738
$778$		0			0
$779$		−	7574.23i		−	0.348363i
$780$		0			0
$781$			1644.50i			0.0753457i
$782$		0			0
$783$	−3838.96			−0.175215
$784$		0			0
$785$		−	1403.75i		−	0.0638243i
$786$		0			0
$787$	−12011.5			−0.544047			−0.272024	−	0.962291i	$-0.587693\pi$
$787$	−12011.5			−0.544047			−0.272024	+	0.962291i	$0.587693\pi$
$788$		0			0
$789$		−	10959.4i		−	0.494505i
$790$		0			0
$791$		−	27193.8i		−	1.22238i
$792$		0			0
$793$	41073.7	+	3198.49i	1.83931	+	0.143231i
$794$		0			0
$795$	−7280.57			−0.324799
$796$		0			0
$797$		−	41415.5i		−	1.84067i	−0.391132	−	0.920335i	$-0.627916\pi$
$797$		−	41415.5i		−	1.84067i	0.391132	−	0.920335i	$-0.372084\pi$
$798$		0			0
$799$			18980.7i			0.840411i
$800$		0			0
$801$			3203.12i			0.141294i
$802$		0			0
$803$			7365.83i			0.323704i
$804$		0			0
$805$			3758.94i			0.164578i
$806$		0			0
$807$	1488.55			0.0649311
$808$		0			0
$809$	31305.7			1.36051			0.680254	−	0.732977i	$-0.261869\pi$
$809$	31305.7			1.36051			0.680254	+	0.732977i	$0.261869\pi$
$810$		0			0
$811$	13622.4			0.589822			0.294911	−	0.955525i	$-0.404710\pi$
$811$	13622.4			0.589822			0.294911	+	0.955525i	$0.404710\pi$
$812$		0			0
$813$	−3494.10			−0.150730
$814$		0			0
$815$	1684.69			0.0724077
$816$		0			0
$817$			5354.46i			0.229289i
$818$		0			0
$819$	9673.33	+	753.282i	0.412715	+	0.0321389i
$820$		0			0
$821$	−21506.4			−0.914226			−0.457113	−	0.889409i	$-0.651117\pi$
$821$	−21506.4			−0.914226			−0.457113	+	0.889409i	$0.651117\pi$
$822$		0			0
$823$	23500.9			0.995369			0.497684	−	0.867358i	$-0.334184\pi$
$823$	23500.9			0.995369			0.497684	+	0.867358i	$0.334184\pi$
$824$		0			0
$825$		−	2820.03i		−	0.119007i
$826$		0			0
$827$	−11952.1			−0.502559			−0.251279	−	0.967915i	$-0.580851\pi$
$827$	−11952.1			−0.502559			−0.251279	+	0.967915i	$0.580851\pi$
$828$		0			0
$829$			11126.3i			0.466144i	0.972459	+	0.233072i	$0.0748778\pi$
$829$			11126.3i			0.466144i	−0.972459	+	0.233072i	$0.925122\pi$
$830$		0			0
$831$	10625.9			0.443572
$832$		0			0
$833$	11053.5			0.459760
$834$		0			0
$835$		−	1838.74i		−	0.0762061i
$836$		0			0
$837$	8203.65			0.338781
$838$		0			0
$839$			13125.0i			0.540077i	0.962850	+	0.270039i	$0.0870364\pi$
$839$			13125.0i			0.540077i	−0.962850	+	0.270039i	$0.912964\pi$
$840$		0			0
$841$	4172.82			0.171094
$842$		0			0
$843$	−3005.76			−0.122804
$844$		0			0
$845$	9917.72	+	1554.05i	0.403763	+	0.0632674i
$846$		0			0
$847$		−	28738.6i		−	1.16585i
$848$		0			0
$849$	−13414.4			−0.542262
$850$		0			0
$851$	−8630.62			−0.347654
$852$		0			0
$853$	40716.0			1.63434			0.817169	−	0.576398i	$-0.195542\pi$
$853$	40716.0			1.63434			0.817169	+	0.576398i	$0.195542\pi$
$854$		0			0
$855$	−1110.59			−0.0444226
$856$		0			0
$857$	34628.7			1.38027			0.690136	−	0.723680i	$-0.257551\pi$
$857$	34628.7			1.38027			0.690136	+	0.723680i	$0.257551\pi$
$858$		0			0
$859$		−	22608.8i		−	0.898024i	−0.893526	−	0.449012i	$-0.851776\pi$
$859$		−	22608.8i		−	0.898024i	0.893526	−	0.449012i	$-0.148224\pi$
$860$		0			0
$861$			19352.2i			0.765994i
$862$		0			0
$863$		−	4002.67i		−	0.157882i	−0.996879	−	0.0789412i	$-0.974846\pi$
$863$		−	4002.67i		−	0.157882i	0.996879	−	0.0789412i	$-0.0251539\pi$
$864$		0			0
$865$		−	16181.1i		−	0.636038i
$866$		0			0
$867$			4145.19i			0.162374i
$868$		0			0
$869$	−7380.05			−0.288091
$870$		0			0
$871$	−901.020	+	11570.5i	−0.0350515	+	0.450117i
$872$		0			0
$873$			4560.38i			0.176799i
$874$		0			0
$875$			24079.4i			0.930324i
$876$		0			0
$877$	−39433.0			−1.51831			−0.759154	−	0.650911i	$-0.774387\pi$
$877$	−39433.0			−1.51831			−0.759154	+	0.650911i	$0.774387\pi$
$878$		0			0
$879$			9878.97i			0.379078i
$880$		0			0
$881$	−31588.5			−1.20799			−0.603997	−	0.796987i	$-0.706426\pi$
$881$	−31588.5			−1.20799			−0.603997	+	0.796987i	$0.706426\pi$
$882$		0			0
$883$		−	21145.2i		−	0.805879i	−0.915227	−	0.402939i	$-0.867989\pi$
$883$		−	21145.2i		−	0.805879i	0.915227	−	0.402939i	$-0.132011\pi$
$884$		0			0
$885$			5183.15i			0.196870i
$886$		0			0
$887$	−47339.2			−1.79199			−0.895996	−	0.444063i	$-0.853537\pi$
$887$	−47339.2			−1.79199			−0.895996	+	0.444063i	$0.853537\pi$
$888$		0			0
$889$		−	23894.2i		−	0.901445i
$890$		0			0
$891$	−731.269			−0.0274954
$892$		0			0
$893$			8625.98i			0.323244i
$894$		0			0
$895$			17160.4i			0.640904i
$896$		0			0
$897$	−5014.28	−	390.472i	−0.186646	−	0.0145345i
$898$		0			0
$899$	43200.9			1.60270
$900$		0			0
$901$		−	31561.7i		−	1.16701i
$902$		0			0
$903$		−	13680.7i		−	0.504168i
$904$		0			0
$905$		−	5237.35i		−	0.192371i
$906$		0			0
$907$			13454.8i			0.492568i	0.969198	+	0.246284i	$0.0792097\pi$
$907$			13454.8i			0.492568i	−0.969198	+	0.246284i	$0.920790\pi$
$908$		0			0
$909$		−	7893.60i		−	0.288024i
$910$		0			0
$911$	2696.73			0.0980753			0.0490376	−	0.998797i	$-0.484385\pi$
$911$	2696.73			0.0980753			0.0490376	+	0.998797i	$0.484385\pi$
$912$		0			0
$913$	2226.37			0.0807031
$914$		0			0
$915$	12048.5			0.435312
$916$		0			0
$917$	3279.01			0.118083
$918$		0			0
$919$	20653.8			0.741355			0.370678	−	0.928762i	$-0.379125\pi$
$919$	20653.8			0.741355			0.370678	+	0.928762i	$0.379125\pi$
$920$		0			0
$921$		−	9822.53i		−	0.351426i
$922$		0			0
$923$	8512.26	+	662.867i	0.303559	+	0.0236387i
$924$		0			0
$925$	−25124.5			−0.893069
$926$		0			0
$927$	−10596.8			−0.375453
$928$		0			0
$929$		−	29376.6i		−	1.03748i	−0.854933	−	0.518738i	$-0.826402\pi$
$929$		−	29376.6i		−	1.03748i	0.854933	−	0.518738i	$-0.173598\pi$
$930$		0			0
$931$	5023.37			0.176836
$932$		0			0
$933$			9573.98i			0.335946i
$934$		0			0
$935$	−2451.36			−0.0857411
$936$		0			0
$937$	26554.0			0.925808			0.462904	−	0.886408i	$-0.346807\pi$
$937$	26554.0			0.925808			0.462904	+	0.886408i	$0.346807\pi$
$938$		0			0
$939$		−	16589.7i		−	0.576554i
$940$		0			0
$941$	−2991.18			−0.103623			−0.0518117	−	0.998657i	$-0.516500\pi$
$941$	−2991.18			−0.103623			−0.0518117	+	0.998657i	$0.516500\pi$
$942$		0			0
$943$		−	10031.4i		−	0.346413i
$944$		0			0
$945$	2837.56			0.0976780
$946$		0			0
$947$	−26370.3			−0.904878			−0.452439	−	0.891795i	$-0.649446\pi$
$947$	−26370.3			−0.904878			−0.452439	+	0.891795i	$0.649446\pi$
$948$		0			0
$949$	38126.9	+	2969.02i	1.30417	+	0.101558i
$950$		0			0
$951$		−	19426.8i		−	0.662415i
$952$		0			0
$953$	37295.1			1.26769			0.633845	−	0.773460i	$-0.281476\pi$
$953$	37295.1			1.26769			0.633845	+	0.773460i	$0.281476\pi$
$954$		0			0
$955$	9967.69			0.337746
$956$		0			0
$957$	−3850.90			−0.130075
$958$		0			0
$959$	55500.0			1.86881
$960$		0			0
$961$	−62527.1			−2.09886
$962$		0			0
$963$			10126.7i			0.338867i
$964$		0			0
$965$			20281.2i			0.676553i
$966$		0			0
$967$		−	55463.6i		−	1.84446i	−0.386647	−	0.922228i	$-0.626367\pi$
$967$		−	55463.6i		−	1.84446i	0.386647	−	0.922228i	$-0.373633\pi$
$968$		0			0
$969$		−	4814.47i		−	0.159611i
$970$		0			0
$971$			41859.8i			1.38347i	0.722153	+	0.691733i	$0.243153\pi$
$971$			41859.8i			1.38347i	−0.722153	+	0.691733i	$0.756847\pi$
$972$		0			0
$973$	19021.3			0.626717
$974$		0			0
$975$	−14597.0	−	1136.70i	−0.479465	−	0.0373369i
$976$		0			0
$977$		−	5960.78i		−	0.195192i	−0.995226	−	0.0975958i	$-0.968885\pi$
$977$		−	5960.78i		−	0.195192i	0.995226	−	0.0975958i	$-0.0311152\pi$
$978$		0			0
$979$			3213.09i			0.104894i
$980$		0			0
$981$	−7092.97			−0.230847
$982$		0			0
$983$			6517.91i			0.211484i	0.994394	+	0.105742i	$0.0337218\pi$
$983$			6517.91i			0.211484i	−0.994394	+	0.105742i	$0.966278\pi$
$984$		0			0
$985$	17310.7			0.559964
$986$		0			0
$987$		−	22039.4i		−	0.710761i
$988$		0			0
$989$			7091.52i			0.228005i
$990$		0			0
$991$	−34170.6			−1.09532			−0.547661	−	0.836701i	$-0.684482\pi$
$991$	−34170.6			−1.09532			−0.547661	+	0.836701i	$0.684482\pi$
$992$		0			0
$993$		−	30048.8i		−	0.960292i
$994$		0			0
$995$	7017.73			0.223595
$996$		0			0
$997$		−	29232.1i		−	0.928577i	−0.885684	−	0.464289i	$-0.846310\pi$
$997$		−	29232.1i		−	0.928577i	0.885684	−	0.464289i	$-0.153690\pi$
$998$		0			0
$999$			6515.10i			0.206335i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.4.m.a.337.9		84
4.3	odd	2		312.4.m.a.181.45	yes	84
8.3	odd	2		312.4.m.a.181.39	✓	84
8.5	even	2	inner	1248.4.m.a.337.12		84
13.12	even	2	inner	1248.4.m.a.337.10		84
52.51	odd	2		312.4.m.a.181.40	yes	84
104.51	odd	2		312.4.m.a.181.46	yes	84
104.77	even	2	inner	1248.4.m.a.337.11		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.m.a.181.39	✓	84	8.3	odd	2
312.4.m.a.181.40	yes	84	52.51	odd	2
312.4.m.a.181.45	yes	84	4.3	odd	2
312.4.m.a.181.46	yes	84	104.51	odd	2
1248.4.m.a.337.9		84	1.1	even	1	trivial
1248.4.m.a.337.10		84	13.12	even	2	inner
1248.4.m.a.337.11		84	104.77	even	2	inner
1248.4.m.a.337.12		84	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1248.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -4.56929 q^{5} +23.0002i q^{7} -9.00000 q^{9} -9.02801 q^{11} +(-3.63901 + 46.7307i) q^{13} +13.7079i q^{15} -59.4245 q^{17} -27.0061 q^{19} +69.0006 q^{21} -35.7672 q^{23} -104.122 q^{25} +27.0000i q^{27} +142.184i q^{29} -303.839i q^{31} +27.0840i q^{33} -105.095i q^{35} +241.300 q^{37} +(140.192 + 10.9170i) q^{39} +280.464i q^{41} -198.269i q^{43} +41.1236 q^{45} -319.409i q^{47} -186.009 q^{49} +178.273i q^{51} +531.123i q^{53} +41.2516 q^{55} +81.0182i q^{57} +378.115 q^{59} -878.946i q^{61} -207.002i q^{63} +(16.6277 - 213.526i) q^{65} +247.600 q^{67} +107.302i q^{69} -182.156i q^{71} -815.886i q^{73} +312.365i q^{75} -207.646i q^{77} +817.462 q^{79} +81.0000 q^{81} -246.607 q^{83} +271.528 q^{85} +426.551 q^{87} -355.903i q^{89} +(-1074.81 - 83.6980i) q^{91} -911.517 q^{93} +123.399 q^{95} -506.709i q^{97} +81.2521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) \) 84 * q - 756 * q^9 - 104 * q^17 + 2188 * q^25 - 3396 * q^49 + 1616 * q^55 + 696 * q^65 - 3160 * q^79 + 6804 * q^81 + 2088 * q^87 - 2480 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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