LMFDB - Embedded newform 2208.2.n.b.367.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2208,2,Mod(367,2208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2208, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2208.367");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.n (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:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			367.20
Character	$\chi$	$=$	2208.367
Dual form			2208.2.n.b.367.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.80468 q^{5} +0.415199 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.80468 q^{5} +0.415199 q^{7} +1.00000 q^{9} -5.54747i q^{11} -3.19024i q^{13} +2.80468 q^{15} +2.01349i q^{17} +2.50622i q^{19} +0.415199 q^{21} +(2.87533 + 3.83829i) q^{23} +2.86624 q^{25} +1.00000 q^{27} -4.69915i q^{29} -8.16330i q^{31} -5.54747i q^{33} +1.16450 q^{35} +3.59504 q^{37} -3.19024i q^{39} +2.52410 q^{41} +1.80649i q^{43} +2.80468 q^{45} -3.58339i q^{47} -6.82761 q^{49} +2.01349i q^{51} -0.270501 q^{53} -15.5589i q^{55} +2.50622i q^{57} +0.657857 q^{59} -11.8024 q^{61} +0.415199 q^{63} -8.94761i q^{65} +5.41363i q^{67} +(2.87533 + 3.83829i) q^{69} -6.09004i q^{71} -2.82330 q^{73} +2.86624 q^{75} -2.30330i q^{77} +15.0378 q^{79} +1.00000 q^{81} +11.1848i q^{83} +5.64719i q^{85} -4.69915i q^{87} +4.02672i q^{89} -1.32459i q^{91} -8.16330i q^{93} +7.02915i q^{95} +14.5135i q^{97} -5.54747i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 24 * q^9	$ 24 q + 24 q^{3} + 24 q^{9} + 24 q^{25} + 24 q^{27} + 56 q^{49} + 32 q^{73} + 24 q^{75} + 24 q^{81}+O(q^{100}) $ 24 * q + 24 * q^3 + 24 * q^9 + 24 * q^25 + 24 * q^27 + 56 * q^49 + 32 * q^73 + 24 * q^75 + 24 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2208\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$737$	$1381$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	2.80468			1.25429			0.627146	−	0.778902i	$-0.284223\pi$
$5$	2.80468			1.25429			0.627146	+	0.778902i	$0.284223\pi$
$6$		0			0
$7$	0.415199			0.156930			0.0784652	−	0.996917i	$-0.474998\pi$
$7$	0.415199			0.156930			0.0784652	+	0.996917i	$0.474998\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.54747i		−	1.67263i	−0.548253	−	0.836313i	$-0.684707\pi$
$11$		−	5.54747i		−	1.67263i	0.548253	−	0.836313i	$-0.315293\pi$
$12$		0			0
$13$		−	3.19024i		−	0.884814i	−0.896814	−	0.442407i	$-0.854125\pi$
$13$		−	3.19024i		−	0.884814i	0.896814	−	0.442407i	$-0.145875\pi$
$14$		0			0
$15$	2.80468			0.724166
$16$		0			0
$17$			2.01349i			0.488342i	0.969732	+	0.244171i	$0.0785158\pi$
$17$			2.01349i			0.488342i	−0.969732	+	0.244171i	$0.921484\pi$
$18$		0			0
$19$			2.50622i			0.574966i	0.957786	+	0.287483i	$0.0928185\pi$
$19$			2.50622i			0.574966i	−0.957786	+	0.287483i	$0.907182\pi$
$20$		0			0
$21$	0.415199			0.0906038
$22$		0			0
$23$	2.87533	+	3.83829i	0.599548	+	0.800339i
$23$	2.87533	+	3.83829i	0.599548	+	0.800339i
$24$		0			0
$25$	2.86624			0.573248
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	4.69915i		−	0.872609i	−0.899799	−	0.436305i	$-0.856287\pi$
$29$		−	4.69915i		−	0.872609i	0.899799	−	0.436305i	$-0.143713\pi$
$30$		0			0
$31$		−	8.16330i		−	1.46617i	−0.680136	−	0.733086i	$-0.738079\pi$
$31$		−	8.16330i		−	1.46617i	0.680136	−	0.733086i	$-0.261921\pi$
$32$		0			0
$33$		−	5.54747i		−	0.965691i
$34$		0			0
$35$	1.16450			0.196837
$36$		0			0
$37$	3.59504			0.591020			0.295510	−	0.955340i	$-0.404510\pi$
$37$	3.59504			0.591020			0.295510	+	0.955340i	$0.404510\pi$
$38$		0			0
$39$		−	3.19024i		−	0.510848i
$40$		0			0
$41$	2.52410			0.394198			0.197099	−	0.980384i	$-0.436848\pi$
$41$	2.52410			0.394198			0.197099	+	0.980384i	$0.436848\pi$
$42$		0			0
$43$			1.80649i			0.275487i	0.990468	+	0.137743i	$0.0439849\pi$
$43$			1.80649i			0.275487i	−0.990468	+	0.137743i	$0.956015\pi$
$44$		0			0
$45$	2.80468			0.418097
$46$		0			0
$47$		−	3.58339i		−	0.522690i	−0.965245	−	0.261345i	$-0.915834\pi$
$47$		−	3.58339i		−	0.522690i	0.965245	−	0.261345i	$-0.0841661\pi$
$48$		0			0
$49$	−6.82761			−0.975373
$50$		0			0
$51$			2.01349i			0.281944i
$52$		0			0
$53$	−0.270501			−0.0371562			−0.0185781	−	0.999827i	$-0.505914\pi$
$53$	−0.270501			−0.0371562			−0.0185781	+	0.999827i	$0.505914\pi$
$54$		0			0
$55$		−	15.5589i		−	2.09796i
$56$		0			0
$57$			2.50622i			0.331957i
$58$		0			0
$59$	0.657857			0.0856456			0.0428228	−	0.999083i	$-0.486365\pi$
$59$	0.657857			0.0856456			0.0428228	+	0.999083i	$0.486365\pi$
$60$		0			0
$61$	−11.8024			−1.51114			−0.755572	−	0.655065i	$-0.772641\pi$
$61$	−11.8024			−1.51114			−0.755572	+	0.655065i	$0.772641\pi$
$62$		0			0
$63$	0.415199			0.0523102
$64$		0			0
$65$		−	8.94761i		−	1.10981i
$66$		0			0
$67$			5.41363i			0.661380i	0.943739	+	0.330690i	$0.107281\pi$
$67$			5.41363i			0.661380i	−0.943739	+	0.330690i	$0.892719\pi$
$68$		0			0
$69$	2.87533	+	3.83829i	0.346149	+	0.462076i
$70$		0			0
$71$		−	6.09004i		−	0.722755i	−0.932420	−	0.361377i	$-0.882307\pi$
$71$		−	6.09004i		−	0.722755i	0.932420	−	0.361377i	$-0.117693\pi$
$72$		0			0
$73$	−2.82330			−0.330442			−0.165221	−	0.986257i	$-0.552834\pi$
$73$	−2.82330			−0.330442			−0.165221	+	0.986257i	$0.552834\pi$
$74$		0			0
$75$	2.86624			0.330965
$76$		0			0
$77$		−	2.30330i		−	0.262486i
$78$		0			0
$79$	15.0378			1.69188			0.845942	−	0.533276i	$-0.179039\pi$
$79$	15.0378			1.69188			0.845942	+	0.533276i	$0.179039\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			11.1848i			1.22769i	0.789428	+	0.613844i	$0.210377\pi$
$83$			11.1848i			1.22769i	−0.789428	+	0.613844i	$0.789623\pi$
$84$		0			0
$85$			5.64719i			0.612523i
$86$		0			0
$87$		−	4.69915i		−	0.503801i
$88$		0			0
$89$			4.02672i			0.426831i	0.976962	+	0.213416i	$0.0684588\pi$
$89$			4.02672i			0.426831i	−0.976962	+	0.213416i	$0.931541\pi$
$90$		0			0
$91$		−	1.32459i		−	0.138854i
$92$		0			0
$93$		−	8.16330i		−	0.846494i
$94$		0			0
$95$			7.02915i			0.721175i
$96$		0			0
$97$			14.5135i			1.47362i	0.676100	+	0.736810i	$0.263669\pi$
$97$			14.5135i			1.47362i	−0.676100	+	0.736810i	$0.736331\pi$
$98$		0			0
$99$		−	5.54747i		−	0.557542i
$100$		0			0
$101$			7.00245i			0.696770i	0.937352	+	0.348385i	$0.113270\pi$
$101$			7.00245i			0.696770i	−0.937352	+	0.348385i	$0.886730\pi$
$102$		0			0
$103$	18.4881			1.82169			0.910845	−	0.412750i	$-0.135432\pi$
$103$	18.4881			1.82169			0.910845	+	0.412750i	$0.135432\pi$
$104$		0			0
$105$	1.16450			0.113644
$106$		0			0
$107$			2.34324i			0.226530i	0.993565	+	0.113265i	$0.0361308\pi$
$107$			2.34324i			0.226530i	−0.993565	+	0.113265i	$0.963869\pi$
$108$		0			0
$109$	9.34569			0.895155			0.447578	−	0.894245i	$-0.352287\pi$
$109$	9.34569			0.895155			0.447578	+	0.894245i	$0.352287\pi$
$110$		0			0
$111$	3.59504			0.341226
$112$		0			0
$113$			18.5203i			1.74225i	0.491064	+	0.871123i	$0.336608\pi$
$113$			18.5203i			1.74225i	−0.491064	+	0.871123i	$0.663392\pi$
$114$		0			0
$115$	8.06438	+	10.7652i	0.752008	+	1.00386i
$116$		0			0
$117$		−	3.19024i		−	0.294938i
$118$		0			0
$119$			0.835997i			0.0766357i
$120$		0			0
$121$	−19.7744			−1.79768
$122$		0			0
$123$	2.52410			0.227590
$124$		0			0
$125$	−5.98452			−0.535272
$126$		0			0
$127$		−	5.31443i		−	0.471580i	−0.971804	−	0.235790i	$-0.924232\pi$
$127$		−	5.31443i		−	0.471580i	0.971804	−	0.235790i	$-0.0757677\pi$
$128$		0			0
$129$			1.80649i			0.159052i
$130$		0			0
$131$	−1.82236			−0.159220			−0.0796101	−	0.996826i	$-0.525368\pi$
$131$	−1.82236			−0.159220			−0.0796101	+	0.996826i	$0.525368\pi$
$132$		0			0
$133$			1.04058i			0.0902297i
$134$		0			0
$135$	2.80468			0.241389
$136$		0			0
$137$		−	18.7457i		−	1.60155i	−0.598962	−	0.800777i	$-0.704420\pi$
$137$		−	18.7457i		−	1.60155i	0.598962	−	0.800777i	$-0.295580\pi$
$138$		0			0
$139$	7.64330			0.648296			0.324148	−	0.946006i	$-0.394922\pi$
$139$	7.64330			0.648296			0.324148	+	0.946006i	$0.394922\pi$
$140$		0			0
$141$		−	3.58339i		−	0.301775i
$142$		0			0
$143$	−17.6978			−1.47996
$144$		0			0
$145$		−	13.1796i		−	1.09451i
$146$		0			0
$147$	−6.82761			−0.563132
$148$		0			0
$149$	10.0586			0.824032			0.412016	−	0.911177i	$-0.364825\pi$
$149$	10.0586			0.824032			0.412016	+	0.911177i	$0.364825\pi$
$150$		0			0
$151$		−	16.6249i		−	1.35292i	−0.736480	−	0.676459i	$-0.763514\pi$
$151$		−	16.6249i		−	1.35292i	0.736480	−	0.676459i	$-0.236486\pi$
$152$		0			0
$153$			2.01349i			0.162781i
$154$		0			0
$155$		−	22.8954i		−	1.83901i
$156$		0			0
$157$	6.29506			0.502401			0.251200	−	0.967935i	$-0.419175\pi$
$157$	6.29506			0.502401			0.251200	+	0.967935i	$0.419175\pi$
$158$		0			0
$159$	−0.270501			−0.0214521
$160$		0			0
$161$	1.19383	+	1.59365i	0.0940873	+	0.125598i
$162$		0			0
$163$	−18.3381			−1.43635			−0.718175	−	0.695862i	$-0.755022\pi$
$163$	−18.3381			−1.43635			−0.718175	+	0.695862i	$0.755022\pi$
$164$		0			0
$165$		−	15.5589i		−	1.21126i
$166$		0			0
$167$			19.7559i			1.52876i	0.644766	+	0.764380i	$0.276955\pi$
$167$			19.7559i			1.52876i	−0.644766	+	0.764380i	$0.723045\pi$
$168$		0			0
$169$	2.82236			0.217104
$170$		0			0
$171$			2.50622i			0.191655i
$172$		0			0
$173$		−	11.3869i		−	0.865731i	−0.901459	−	0.432865i	$-0.857503\pi$
$173$		−	11.3869i		−	0.865731i	0.901459	−	0.432865i	$-0.142497\pi$
$174$		0			0
$175$	1.19006			0.0899601
$176$		0			0
$177$	0.657857			0.0494475
$178$		0			0
$179$	−19.1156			−1.42877			−0.714385	−	0.699753i	$-0.753293\pi$
$179$	−19.1156			−1.42877			−0.714385	+	0.699753i	$0.753293\pi$
$180$		0			0
$181$	−5.60596			−0.416688			−0.208344	−	0.978056i	$-0.566807\pi$
$181$	−5.60596			−0.416688			−0.208344	+	0.978056i	$0.566807\pi$
$182$		0			0
$183$	−11.8024			−0.872460
$184$		0			0
$185$	10.0829			0.741312
$186$		0			0
$187$	11.1698			0.816813
$188$		0			0
$189$	0.415199			0.0302013
$190$		0			0
$191$	−25.0291			−1.81104			−0.905522	−	0.424298i	$-0.860521\pi$
$191$	−25.0291			−1.81104			−0.905522	+	0.424298i	$0.860521\pi$
$192$		0			0
$193$	24.6407			1.77367			0.886837	−	0.462082i	$-0.152898\pi$
$193$	24.6407			1.77367			0.886837	+	0.462082i	$0.152898\pi$
$194$		0			0
$195$		−	8.94761i		−	0.640752i
$196$		0			0
$197$			2.39584i			0.170697i	0.996351	+	0.0853483i	$0.0272003\pi$
$197$			2.39584i			0.170697i	−0.996351	+	0.0853483i	$0.972800\pi$
$198$		0			0
$199$	−3.76352			−0.266789			−0.133395	−	0.991063i	$-0.542588\pi$
$199$	−3.76352			−0.266789			−0.133395	+	0.991063i	$0.542588\pi$
$200$		0			0
$201$			5.41363i			0.381848i
$202$		0			0
$203$		−	1.95108i		−	0.136939i
$204$		0			0
$205$	7.07929			0.494439
$206$		0			0
$207$	2.87533	+	3.83829i	0.199849	+	0.266780i
$208$		0			0
$209$	13.9032			0.961703
$210$		0			0
$211$	17.0238			1.17197			0.585983	−	0.810323i	$-0.300708\pi$
$211$	17.0238			1.17197			0.585983	+	0.810323i	$0.300708\pi$
$212$		0			0
$213$		−	6.09004i		−	0.417283i
$214$		0			0
$215$			5.06662i			0.345541i
$216$		0			0
$217$		−	3.38939i		−	0.230087i
$218$		0			0
$219$	−2.82330			−0.190781
$220$		0			0
$221$	6.42351			0.432092
$222$		0			0
$223$			15.8920i			1.06421i	0.846678	+	0.532105i	$0.178599\pi$
$223$			15.8920i			1.06421i	−0.846678	+	0.532105i	$0.821401\pi$
$224$		0			0
$225$	2.86624			0.191083
$226$		0			0
$227$		−	0.778468i		−	0.0516687i	−0.999666	−	0.0258344i	$-0.991776\pi$
$227$		−	0.778468i		−	0.0516687i	0.999666	−	0.0258344i	$-0.00822425\pi$
$228$		0			0
$229$	−18.3034			−1.20952			−0.604761	−	0.796407i	$-0.706731\pi$
$229$	−18.3034			−1.20952			−0.604761	+	0.796407i	$0.706731\pi$
$230$		0			0
$231$		−	2.30330i		−	0.151546i
$232$		0			0
$233$	10.1288			0.663558			0.331779	−	0.943357i	$-0.392351\pi$
$233$	10.1288			0.663558			0.331779	+	0.943357i	$0.392351\pi$
$234$		0			0
$235$		−	10.0503i		−	0.655606i
$236$		0			0
$237$	15.0378			0.976809
$238$		0			0
$239$			22.8436i			1.47763i	0.673910	+	0.738813i	$0.264613\pi$
$239$			22.8436i			1.47763i	−0.673910	+	0.738813i	$0.735387\pi$
$240$		0			0
$241$		−	26.9055i		−	1.73314i	−0.499059	−	0.866568i	$-0.666321\pi$
$241$		−	26.9055i		−	1.73314i	0.499059	−	0.866568i	$-0.333679\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−19.1493			−1.22340
$246$		0			0
$247$	7.99544			0.508738
$248$		0			0
$249$			11.1848i			0.708806i
$250$		0			0
$251$			25.7950i			1.62817i	0.580747	+	0.814084i	$0.302761\pi$
$251$			25.7950i			1.62817i	−0.580747	+	0.814084i	$0.697239\pi$
$252$		0			0
$253$	21.2928	−	15.9508i	1.33867	−	1.00282i
$254$		0			0
$255$			5.64719i			0.353641i
$256$		0			0
$257$	−24.9514			−1.55642			−0.778212	−	0.628002i	$-0.783873\pi$
$257$	−24.9514			−1.55642			−0.778212	+	0.628002i	$0.783873\pi$
$258$		0			0
$259$	1.49265			0.0927491
$260$		0			0
$261$		−	4.69915i		−	0.290870i
$262$		0			0
$263$	4.92026			0.303396			0.151698	−	0.988427i	$-0.451526\pi$
$263$	4.92026			0.303396			0.151698	+	0.988427i	$0.451526\pi$
$264$		0			0
$265$	−0.758669			−0.0466047
$266$		0			0
$267$			4.02672i			0.246431i
$268$		0			0
$269$			8.77633i			0.535102i	0.963544	+	0.267551i	$0.0862144\pi$
$269$			8.77633i			0.535102i	−0.963544	+	0.267551i	$0.913786\pi$
$270$		0			0
$271$		−	7.39559i		−	0.449250i	−0.974445	−	0.224625i	$-0.927884\pi$
$271$		−	7.39559i		−	0.449250i	0.974445	−	0.224625i	$-0.0721158\pi$
$272$		0			0
$273$		−	1.32459i		−	0.0801675i
$274$		0			0
$275$		−	15.9004i		−	0.958829i
$276$		0			0
$277$			8.71879i			0.523861i	0.965087	+	0.261931i	$0.0843592\pi$
$277$			8.71879i			0.523861i	−0.965087	+	0.261931i	$0.915641\pi$
$278$		0			0
$279$		−	8.16330i		−	0.488724i
$280$		0			0
$281$			5.18858i			0.309525i	0.987952	+	0.154762i	$0.0494612\pi$
$281$			5.18858i			0.309525i	−0.987952	+	0.154762i	$0.950539\pi$
$282$		0			0
$283$		−	26.9752i		−	1.60351i	−0.597653	−	0.801755i	$-0.703900\pi$
$283$		−	26.9752i		−	1.60351i	0.597653	−	0.801755i	$-0.296100\pi$
$284$		0			0
$285$			7.02915i			0.416371i
$286$		0			0
$287$	1.04800			0.0618616
$288$		0			0
$289$	12.9459			0.761522
$290$		0			0
$291$			14.5135i			0.850795i
$292$		0			0
$293$	22.7008			1.32619			0.663096	−	0.748534i	$-0.269242\pi$
$293$	22.7008			1.32619			0.663096	+	0.748534i	$0.269242\pi$
$294$		0			0
$295$	1.84508			0.107425
$296$		0			0
$297$		−	5.54747i		−	0.321897i
$298$		0			0
$299$	12.2451	−	9.17300i	0.708151	−	0.530488i
$300$		0			0
$301$			0.750052i			0.0432323i
$302$		0			0
$303$			7.00245i			0.402280i
$304$		0			0
$305$	−33.1020			−1.89542
$306$		0			0
$307$	−4.84075			−0.276276			−0.138138	−	0.990413i	$-0.544112\pi$
$307$	−4.84075			−0.276276			−0.138138	+	0.990413i	$0.544112\pi$
$308$		0			0
$309$	18.4881			1.05175
$310$		0			0
$311$			9.16116i			0.519482i	0.965678	+	0.259741i	$0.0836372\pi$
$311$			9.16116i			0.519482i	−0.965678	+	0.259741i	$0.916363\pi$
$312$		0			0
$313$			17.8518i			1.00904i	0.863399	+	0.504522i	$0.168331\pi$
$313$			17.8518i			1.00904i	−0.863399	+	0.504522i	$0.831669\pi$
$314$		0			0
$315$	1.16450			0.0656122
$316$		0			0
$317$			31.8091i			1.78658i	0.449485	+	0.893288i	$0.351607\pi$
$317$			31.8091i			1.78658i	−0.449485	+	0.893288i	$0.648393\pi$
$318$		0			0
$319$	−26.0684			−1.45955
$320$		0			0
$321$			2.34324i			0.130787i
$322$		0			0
$323$	−5.04624			−0.280780
$324$		0			0
$325$		−	9.14400i		−	0.507218i
$326$		0			0
$327$	9.34569			0.516818
$328$		0			0
$329$		−	1.48782i		−	0.0820260i
$330$		0			0
$331$	3.34810			0.184028			0.0920140	−	0.995758i	$-0.470670\pi$
$331$	3.34810			0.184028			0.0920140	+	0.995758i	$0.470670\pi$
$332$		0			0
$333$	3.59504			0.197007
$334$		0			0
$335$			15.1835i			0.829564i
$336$		0			0
$337$			14.8251i			0.807574i	0.914853	+	0.403787i	$0.132306\pi$
$337$			14.8251i			0.807574i	−0.914853	+	0.403787i	$0.867694\pi$
$338$		0			0
$339$			18.5203i			1.00589i
$340$		0			0
$341$	−45.2856			−2.45235
$342$		0			0
$343$	−5.74121			−0.309996
$344$		0			0
$345$	8.06438	+	10.7652i	0.434172	+	0.579578i
$346$		0			0
$347$	−32.8500			−1.76348			−0.881740	−	0.471736i	$-0.843628\pi$
$347$	−32.8500			−1.76348			−0.881740	+	0.471736i	$0.843628\pi$
$348$		0			0
$349$		−	16.1566i		−	0.864844i	−0.901671	−	0.432422i	$-0.857659\pi$
$349$		−	16.1566i		−	0.864844i	0.901671	−	0.432422i	$-0.142341\pi$
$350$		0			0
$351$		−	3.19024i		−	0.170283i
$352$		0			0
$353$	1.52645			0.0812448			0.0406224	−	0.999175i	$-0.487066\pi$
$353$	1.52645			0.0812448			0.0406224	+	0.999175i	$0.487066\pi$
$354$		0			0
$355$		−	17.0806i		−	0.906546i
$356$		0			0
$357$			0.835997i			0.0442457i
$358$		0			0
$359$	30.2471			1.59638			0.798190	−	0.602405i	$-0.205791\pi$
$359$	30.2471			1.59638			0.798190	+	0.602405i	$0.205791\pi$
$360$		0			0
$361$	12.7189			0.669414
$362$		0			0
$363$	−19.7744			−1.03789
$364$		0			0
$365$	−7.91846			−0.414471
$366$		0			0
$367$	−1.47265			−0.0768718			−0.0384359	−	0.999261i	$-0.512238\pi$
$367$	−1.47265			−0.0768718			−0.0384359	+	0.999261i	$0.512238\pi$
$368$		0			0
$369$	2.52410			0.131399
$370$		0			0
$371$	−0.112312			−0.00583093
$372$		0			0
$373$	12.4114			0.642639			0.321320	−	0.946971i	$-0.395874\pi$
$373$	12.4114			0.642639			0.321320	+	0.946971i	$0.395874\pi$
$374$		0			0
$375$	−5.98452			−0.309039
$376$		0			0
$377$	−14.9914			−0.772097
$378$		0			0
$379$			27.0869i			1.39136i	0.718352	+	0.695679i	$0.244897\pi$
$379$			27.0869i			1.39136i	−0.718352	+	0.695679i	$0.755103\pi$
$380$		0			0
$381$		−	5.31443i		−	0.272267i
$382$		0			0
$383$	−30.4292			−1.55486			−0.777430	−	0.628970i	$-0.783477\pi$
$383$	−30.4292			−1.55486			−0.777430	+	0.628970i	$0.783477\pi$
$384$		0			0
$385$		−	6.46003i		−	0.329234i
$386$		0			0
$387$			1.80649i			0.0918290i
$388$		0			0
$389$	−16.9108			−0.857413			−0.428706	−	0.903444i	$-0.641030\pi$
$389$	−16.9108			−0.857413			−0.428706	+	0.903444i	$0.641030\pi$
$390$		0			0
$391$	−7.72835	+	5.78944i	−0.390839	+	0.292784i
$392$		0			0
$393$	−1.82236			−0.0919258
$394$		0			0
$395$	42.1762			2.12212
$396$		0			0
$397$		−	24.9043i		−	1.24991i	−0.780661	−	0.624955i	$-0.785117\pi$
$397$		−	24.9043i		−	1.24991i	0.780661	−	0.624955i	$-0.214883\pi$
$398$		0			0
$399$			1.04058i			0.0520941i
$400$		0			0
$401$			9.00074i			0.449476i	0.974419	+	0.224738i	$0.0721526\pi$
$401$			9.00074i			0.449476i	−0.974419	+	0.224738i	$0.927847\pi$
$402$		0			0
$403$	−26.0429			−1.29729
$404$		0			0
$405$	2.80468			0.139366
$406$		0			0
$407$		−	19.9434i		−	0.988555i
$408$		0			0
$409$	−3.97216			−0.196411			−0.0982053	−	0.995166i	$-0.531310\pi$
$409$	−3.97216			−0.196411			−0.0982053	+	0.995166i	$0.531310\pi$
$410$		0			0
$411$		−	18.7457i		−	0.924658i
$412$		0			0
$413$	0.273141			0.0134404
$414$		0			0
$415$			31.3697i			1.53988i
$416$		0			0
$417$	7.64330			0.374294
$418$		0			0
$419$			19.3554i			0.945572i	0.881177	+	0.472786i	$0.156752\pi$
$419$			19.3554i			0.945572i	−0.881177	+	0.472786i	$0.843248\pi$
$420$		0			0
$421$	−14.5334			−0.708316			−0.354158	−	0.935186i	$-0.615232\pi$
$421$	−14.5334			−0.708316			−0.354158	+	0.935186i	$0.615232\pi$
$422$		0			0
$423$		−	3.58339i		−	0.174230i
$424$		0			0
$425$			5.77113i			0.279941i
$426$		0			0
$427$	−4.90035			−0.237145
$428$		0			0
$429$	−17.6978			−0.854456
$430$		0			0
$431$	−11.2407			−0.541443			−0.270722	−	0.962658i	$-0.587262\pi$
$431$	−11.2407			−0.541443			−0.270722	+	0.962658i	$0.587262\pi$
$432$		0			0
$433$			5.81482i			0.279442i	0.990191	+	0.139721i	$0.0446206\pi$
$433$			5.81482i			0.279442i	−0.990191	+	0.139721i	$0.955379\pi$
$434$		0			0
$435$		−	13.1796i		−	0.631914i
$436$		0			0
$437$	−9.61960	+	7.20621i	−0.460168	+	0.344720i
$438$		0			0
$439$		−	41.2559i		−	1.96904i	−0.175278	−	0.984519i	$-0.556082\pi$
$439$		−	41.2559i		−	1.96904i	0.175278	−	0.984519i	$-0.443918\pi$
$440$		0			0
$441$	−6.82761			−0.325124
$442$		0			0
$443$	1.67570			0.0796148			0.0398074	−	0.999207i	$-0.487326\pi$
$443$	1.67570			0.0796148			0.0398074	+	0.999207i	$0.487326\pi$
$444$		0			0
$445$			11.2937i			0.535371i
$446$		0			0
$447$	10.0586			0.475755
$448$		0			0
$449$	−17.2494			−0.814049			−0.407024	−	0.913417i	$-0.633434\pi$
$449$	−17.2494			−0.814049			−0.407024	+	0.913417i	$0.633434\pi$
$450$		0			0
$451$		−	14.0024i		−	0.659345i
$452$		0			0
$453$		−	16.6249i		−	0.781108i
$454$		0			0
$455$		−	3.71504i		−	0.174164i
$456$		0			0
$457$		−	22.5158i		−	1.05325i	−0.850099	−	0.526623i	$-0.823458\pi$
$457$		−	22.5158i		−	1.05325i	0.850099	−	0.526623i	$-0.176542\pi$
$458$		0			0
$459$			2.01349i			0.0939815i
$460$		0			0
$461$			34.5037i			1.60700i	0.595308	+	0.803498i	$0.297030\pi$
$461$			34.5037i			1.60700i	−0.595308	+	0.803498i	$0.702970\pi$
$462$		0			0
$463$			28.5673i			1.32763i	0.747895	+	0.663817i	$0.231064\pi$
$463$			28.5673i			1.32763i	−0.747895	+	0.663817i	$0.768936\pi$
$464$		0			0
$465$		−	22.8954i		−	1.06175i
$466$		0			0
$467$			4.59322i			0.212549i	0.994337	+	0.106274i	$0.0338922\pi$
$467$			4.59322i			0.212549i	−0.994337	+	0.106274i	$0.966108\pi$
$468$		0			0
$469$			2.24773i			0.103791i
$470$		0			0
$471$	6.29506			0.290061
$472$		0			0
$473$	10.0214			0.460786
$474$		0			0
$475$			7.18342i			0.329598i
$476$		0			0
$477$	−0.270501			−0.0123854
$478$		0			0
$479$	−6.78356			−0.309949			−0.154974	−	0.987918i	$-0.549530\pi$
$479$	−6.78356			−0.309949			−0.154974	+	0.987918i	$0.549530\pi$
$480$		0			0
$481$		−	11.4690i		−	0.522943i
$482$		0			0
$483$	1.19383	+	1.59365i	0.0543213	+	0.0725138i
$484$		0			0
$485$			40.7057i			1.84835i
$486$		0			0
$487$			2.73804i			0.124073i	0.998074	+	0.0620363i	$0.0197595\pi$
$487$			2.73804i			0.124073i	−0.998074	+	0.0620363i	$0.980241\pi$
$488$		0			0
$489$	−18.3381			−0.829277
$490$		0			0
$491$	−26.6421			−1.20234			−0.601170	−	0.799121i	$-0.705299\pi$
$491$	−26.6421			−1.20234			−0.601170	+	0.799121i	$0.705299\pi$
$492$		0			0
$493$	9.46166			0.426132
$494$		0			0
$495$		−	15.5589i		−	0.699320i
$496$		0			0
$497$		−	2.52858i		−	0.113422i
$498$		0			0
$499$	−13.3190			−0.596240			−0.298120	−	0.954528i	$-0.596360\pi$
$499$	−13.3190			−0.596240			−0.298120	+	0.954528i	$0.596360\pi$
$500$		0			0
$501$			19.7559i			0.882630i
$502$		0			0
$503$	−26.7050			−1.19072			−0.595359	−	0.803460i	$-0.702990\pi$
$503$	−26.7050			−1.19072			−0.595359	+	0.803460i	$0.702990\pi$
$504$		0			0
$505$			19.6396i			0.873953i
$506$		0			0
$507$	2.82236			0.125345
$508$		0			0
$509$			28.4096i			1.25923i	0.776906	+	0.629617i	$0.216788\pi$
$509$			28.4096i			1.25923i	−0.776906	+	0.629617i	$0.783212\pi$
$510$		0			0
$511$	−1.17223			−0.0518564
$512$		0			0
$513$			2.50622i			0.110652i
$514$		0			0
$515$	51.8533			2.28493
$516$		0			0
$517$	−19.8787			−0.874265
$518$		0			0
$519$		−	11.3869i		−	0.499830i
$520$		0			0
$521$		−	24.7986i		−	1.08645i	−0.839588	−	0.543224i	$-0.817203\pi$
$521$		−	24.7986i		−	1.08645i	0.839588	−	0.543224i	$-0.182797\pi$
$522$		0			0
$523$			26.1728i			1.14446i	0.820094	+	0.572229i	$0.193921\pi$
$523$			26.1728i			1.14446i	−0.820094	+	0.572229i	$0.806079\pi$
$524$		0			0
$525$	1.19006			0.0519385
$526$		0			0
$527$	16.4367			0.715993
$528$		0			0
$529$	−6.46496	+	22.0727i	−0.281085	+	0.959683i
$530$		0			0
$531$	0.657857			0.0285485
$532$		0			0
$533$		−	8.05248i		−	0.348792i
$534$		0			0
$535$			6.57204i			0.284134i
$536$		0			0
$537$	−19.1156			−0.824900
$538$		0			0
$539$			37.8760i			1.63143i
$540$		0			0
$541$			21.3299i			0.917046i	0.888683	+	0.458523i	$0.151621\pi$
$541$			21.3299i			0.917046i	−0.888683	+	0.458523i	$0.848379\pi$
$542$		0			0
$543$	−5.60596			−0.240575
$544$		0			0
$545$	26.2117			1.12279
$546$		0			0
$547$	6.09639			0.260663			0.130331	−	0.991470i	$-0.458396\pi$
$547$	6.09639			0.260663			0.130331	+	0.991470i	$0.458396\pi$
$548$		0			0
$549$	−11.8024			−0.503715
$550$		0			0
$551$	11.7771			0.501721
$552$		0			0
$553$	6.24367			0.265508
$554$		0			0
$555$	10.0829			0.427997
$556$		0			0
$557$	−21.1916			−0.897915			−0.448958	−	0.893553i	$-0.648205\pi$
$557$	−21.1916			−0.897915			−0.448958	+	0.893553i	$0.648205\pi$
$558$		0			0
$559$	5.76313			0.243755
$560$		0			0
$561$	11.1698			0.471587
$562$		0			0
$563$		−	14.2901i		−	0.602254i	−0.953584	−	0.301127i	$-0.902637\pi$
$563$		−	14.2901i		−	0.602254i	0.953584	−	0.301127i	$-0.0973628\pi$
$564$		0			0
$565$			51.9436i			2.18529i
$566$		0			0
$567$	0.415199			0.0174367
$568$		0			0
$569$		−	31.9346i		−	1.33877i	−0.742917	−	0.669384i	$-0.766558\pi$
$569$		−	31.9346i		−	1.33877i	0.742917	−	0.669384i	$-0.233442\pi$
$570$		0			0
$571$			25.5699i			1.07007i	0.844831	+	0.535034i	$0.179701\pi$
$571$			25.5699i			1.07007i	−0.844831	+	0.535034i	$0.820299\pi$
$572$		0			0
$573$	−25.0291			−1.04561
$574$		0			0
$575$	8.24138	+	11.0015i	0.343689	+	0.458793i
$576$		0			0
$577$	25.3417			1.05499			0.527494	−	0.849558i	$-0.323132\pi$
$577$	25.3417			1.05499			0.527494	+	0.849558i	$0.323132\pi$
$578$		0			0
$579$	24.6407			1.02403
$580$		0			0
$581$			4.64390i			0.192662i
$582$		0			0
$583$			1.50060i			0.0621483i
$584$		0			0
$585$		−	8.94761i		−	0.369938i
$586$		0			0
$587$	−11.6750			−0.481878			−0.240939	−	0.970540i	$-0.577455\pi$
$587$	−11.6750			−0.481878			−0.240939	+	0.970540i	$0.577455\pi$
$588$		0			0
$589$	20.4590			0.842999
$590$		0			0
$591$			2.39584i			0.0985518i
$592$		0			0
$593$	−39.6476			−1.62813			−0.814065	−	0.580774i	$-0.802750\pi$
$593$	−39.6476			−1.62813			−0.814065	+	0.580774i	$0.802750\pi$
$594$		0			0
$595$			2.34471i			0.0961236i
$596$		0			0
$597$	−3.76352			−0.154031
$598$		0			0
$599$		−	20.5586i		−	0.840003i	−0.907523	−	0.420002i	$-0.862030\pi$
$599$		−	20.5586i		−	0.840003i	0.907523	−	0.420002i	$-0.137970\pi$
$600$		0			0
$601$	−35.6416			−1.45385			−0.726926	−	0.686716i	$-0.759052\pi$
$601$	−35.6416			−1.45385			−0.726926	+	0.686716i	$0.759052\pi$
$602$		0			0
$603$			5.41363i			0.220460i
$604$		0			0
$605$	−55.4610			−2.25481
$606$		0			0
$607$			23.1724i			0.940539i	0.882523	+	0.470269i	$0.155843\pi$
$607$			23.1724i			0.940539i	−0.882523	+	0.470269i	$0.844157\pi$
$608$		0			0
$609$		−	1.95108i		−	0.0790618i
$610$		0			0
$611$	−11.4319			−0.462484
$612$		0			0
$613$	24.7805			1.00087			0.500437	−	0.865773i	$-0.333173\pi$
$613$	24.7805			1.00087			0.500437	+	0.865773i	$0.333173\pi$
$614$		0			0
$615$	7.07929			0.285464
$616$		0			0
$617$		−	17.9944i		−	0.724428i	−0.932095	−	0.362214i	$-0.882021\pi$
$617$		−	17.9944i		−	0.724428i	0.932095	−	0.362214i	$-0.117979\pi$
$618$		0			0
$619$		−	45.4728i		−	1.82771i	−0.406043	−	0.913854i	$-0.633092\pi$
$619$		−	45.4728i		−	1.82771i	0.406043	−	0.913854i	$-0.366908\pi$
$620$		0			0
$621$	2.87533	+	3.83829i	0.115383	+	0.154025i
$622$		0			0
$623$			1.67189i			0.0669828i
$624$		0			0
$625$	−31.1159			−1.24463
$626$		0			0
$627$	13.9032			0.555239
$628$		0			0
$629$			7.23855i			0.288620i
$630$		0			0
$631$	38.7133			1.54115			0.770576	−	0.637349i	$-0.219969\pi$
$631$	38.7133			1.54115			0.770576	+	0.637349i	$0.219969\pi$
$632$		0			0
$633$	17.0238			0.676635
$634$		0			0
$635$		−	14.9053i		−	0.591499i
$636$		0			0
$637$			21.7817i			0.863023i
$638$		0			0
$639$		−	6.09004i		−	0.240918i
$640$		0			0
$641$			16.6975i			0.659512i	0.944066	+	0.329756i	$0.106967\pi$
$641$			16.6975i			0.659512i	−0.944066	+	0.329756i	$0.893033\pi$
$642$		0			0
$643$		−	4.54910i		−	0.179399i	−0.995969	−	0.0896994i	$-0.971409\pi$
$643$		−	4.54910i		−	0.179399i	0.995969	−	0.0896994i	$-0.0285906\pi$
$644$		0			0
$645$			5.06662i			0.199498i
$646$		0			0
$647$		−	24.8216i		−	0.975838i	−0.872889	−	0.487919i	$-0.837756\pi$
$647$		−	24.8216i		−	0.975838i	0.872889	−	0.487919i	$-0.162244\pi$
$648$		0			0
$649$		−	3.64944i		−	0.143253i
$650$		0			0
$651$		−	3.38939i		−	0.132841i
$652$		0			0
$653$		−	20.1907i		−	0.790122i	−0.918655	−	0.395061i	$-0.870723\pi$
$653$		−	20.1907i		−	0.790122i	0.918655	−	0.395061i	$-0.129277\pi$
$654$		0			0
$655$	−5.11113			−0.199708
$656$		0			0
$657$	−2.82330			−0.110147
$658$		0			0
$659$		−	33.2939i		−	1.29695i	−0.761237	−	0.648474i	$-0.775408\pi$
$659$		−	33.2939i		−	1.29695i	0.761237	−	0.648474i	$-0.224592\pi$
$660$		0			0
$661$	−32.2531			−1.25450			−0.627251	−	0.778818i	$-0.715820\pi$
$661$	−32.2531			−1.25450			−0.627251	+	0.778818i	$0.715820\pi$
$662$		0			0
$663$	6.42351			0.249468
$664$		0			0
$665$			2.91849i			0.113174i
$666$		0			0
$667$	18.0367	−	13.5116i	0.698383	−	0.523171i
$668$		0			0
$669$			15.8920i			0.614422i
$670$		0			0
$671$			65.4735i			2.52758i
$672$		0			0
$673$	−39.3450			−1.51664			−0.758319	−	0.651884i	$-0.773979\pi$
$673$	−39.3450			−1.51664			−0.758319	+	0.651884i	$0.773979\pi$
$674$		0			0
$675$	2.86624			0.110322
$676$		0			0
$677$	15.4597			0.594165			0.297083	−	0.954852i	$-0.403986\pi$
$677$	15.4597			0.594165			0.297083	+	0.954852i	$0.403986\pi$
$678$		0			0
$679$			6.02598i			0.231256i
$680$		0			0
$681$		−	0.778468i		−	0.0298310i
$682$		0			0
$683$	−10.8401			−0.414783			−0.207392	−	0.978258i	$-0.566497\pi$
$683$	−10.8401			−0.414783			−0.207392	+	0.978258i	$0.566497\pi$
$684$		0			0
$685$		−	52.5758i		−	2.00882i
$686$		0			0
$687$	−18.3034			−0.698318
$688$		0			0
$689$			0.862963i			0.0328763i
$690$		0			0
$691$	31.6585			1.20435			0.602173	−	0.798366i	$-0.294302\pi$
$691$	31.6585			1.20435			0.602173	+	0.798366i	$0.294302\pi$
$692$		0			0
$693$		−	2.30330i		−	0.0874953i
$694$		0			0
$695$	21.4370			0.813153
$696$		0			0
$697$			5.08223i			0.192503i
$698$		0			0
$699$	10.1288			0.383105
$700$		0			0
$701$	25.0600			0.946504			0.473252	−	0.880927i	$-0.343080\pi$
$701$	25.0600			0.946504			0.473252	+	0.880927i	$0.343080\pi$
$702$		0			0
$703$			9.00995i			0.339817i
$704$		0			0
$705$		−	10.0503i		−	0.378514i
$706$		0			0
$707$			2.90741i			0.109344i
$708$		0			0
$709$	1.51261			0.0568072			0.0284036	−	0.999597i	$-0.490958\pi$
$709$	1.51261			0.0568072			0.0284036	+	0.999597i	$0.490958\pi$
$710$		0			0
$711$	15.0378			0.563961
$712$		0			0
$713$	31.3331	−	23.4722i	1.17343	−	0.879040i
$714$		0			0
$715$	−49.6366			−1.85630
$716$		0			0
$717$			22.8436i			0.853108i
$718$		0			0
$719$		−	14.5869i		−	0.543999i	−0.962297	−	0.272000i	$-0.912315\pi$
$719$		−	14.5869i		−	0.543999i	0.962297	−	0.272000i	$-0.0876850\pi$
$720$		0			0
$721$	7.67625			0.285878
$722$		0			0
$723$		−	26.9055i		−	1.00063i
$724$		0			0
$725$		−	13.4689i		−	0.500222i
$726$		0			0
$727$	−21.7360			−0.806142			−0.403071	−	0.915169i	$-0.632057\pi$
$727$	−21.7360			−0.806142			−0.403071	+	0.915169i	$0.632057\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−3.63734			−0.134532
$732$		0			0
$733$	−8.74545			−0.323021			−0.161510	−	0.986871i	$-0.551637\pi$
$733$	−8.74545			−0.323021			−0.161510	+	0.986871i	$0.551637\pi$
$734$		0			0
$735$	−19.1493			−0.706332
$736$		0			0
$737$	30.0319			1.10624
$738$		0			0
$739$	7.34669			0.270253			0.135126	−	0.990828i	$-0.456856\pi$
$739$	7.34669			0.270253			0.135126	+	0.990828i	$0.456856\pi$
$740$		0			0
$741$	7.99544			0.293720
$742$		0			0
$743$	−28.2818			−1.03756			−0.518779	−	0.854908i	$-0.673613\pi$
$743$	−28.2818			−1.03756			−0.518779	+	0.854908i	$0.673613\pi$
$744$		0			0
$745$	28.2111			1.03358
$746$		0			0
$747$			11.1848i			0.409229i
$748$		0			0
$749$			0.972911i			0.0355494i
$750$		0			0
$751$	−45.1818			−1.64871			−0.824353	−	0.566076i	$-0.808461\pi$
$751$	−45.1818			−1.64871			−0.824353	+	0.566076i	$0.808461\pi$
$752$		0			0
$753$			25.7950i			0.940024i
$754$		0			0
$755$		−	46.6277i		−	1.69695i
$756$		0			0
$757$	11.8830			0.431894			0.215947	−	0.976405i	$-0.430716\pi$
$757$	11.8830			0.431894			0.215947	+	0.976405i	$0.430716\pi$
$758$		0			0
$759$	21.2928	−	15.9508i	0.772880	−	0.578978i
$760$		0			0
$761$	29.0606			1.05345			0.526723	−	0.850037i	$-0.323421\pi$
$761$	29.0606			1.05345			0.526723	+	0.850037i	$0.323421\pi$
$762$		0			0
$763$	3.88032			0.140477
$764$		0			0
$765$			5.64719i			0.204174i
$766$		0			0
$767$		−	2.09872i		−	0.0757804i
$768$		0			0
$769$			0.831518i			0.0299853i	0.999888	+	0.0149927i	$0.00477249\pi$
$769$			0.831518i			0.0299853i	−0.999888	+	0.0149927i	$0.995228\pi$
$770$		0			0
$771$	−24.9514			−0.898602
$772$		0			0
$773$	35.9982			1.29477			0.647383	−	0.762165i	$-0.275864\pi$
$773$	35.9982			1.29477			0.647383	+	0.762165i	$0.275864\pi$
$774$		0			0
$775$		−	23.3980i		−	0.840480i
$776$		0			0
$777$	1.49265			0.0535487
$778$		0			0
$779$			6.32594i			0.226650i
$780$		0			0
$781$	−33.7843			−1.20890
$782$		0			0
$783$		−	4.69915i		−	0.167934i
$784$		0			0
$785$	17.6556			0.630157
$786$		0			0
$787$			17.0308i			0.607081i	0.952818	+	0.303541i	$0.0981688\pi$
$787$			17.0308i			0.607081i	−0.952818	+	0.303541i	$0.901831\pi$
$788$		0			0
$789$	4.92026			0.175166
$790$		0			0
$791$			7.68962i			0.273412i
$792$		0			0
$793$			37.6526i			1.33708i
$794$		0			0
$795$	−0.758669			−0.0269072
$796$		0			0
$797$	−9.38264			−0.332350			−0.166175	−	0.986096i	$-0.553142\pi$
$797$	−9.38264			−0.332350			−0.166175	+	0.986096i	$0.553142\pi$
$798$		0			0
$799$	7.21510			0.255252
$800$		0			0
$801$			4.02672i			0.142277i
$802$		0			0
$803$			15.6622i			0.552706i
$804$		0			0
$805$	3.34832	+	4.46969i	0.118013	+	0.157536i
$806$		0			0
$807$			8.77633i			0.308941i
$808$		0			0
$809$	−4.57653			−0.160902			−0.0804511	−	0.996759i	$-0.525636\pi$
$809$	−4.57653			−0.160902			−0.0804511	+	0.996759i	$0.525636\pi$
$810$		0			0
$811$	−23.5951			−0.828537			−0.414269	−	0.910155i	$-0.635963\pi$
$811$	−23.5951			−0.828537			−0.414269	+	0.910155i	$0.635963\pi$
$812$		0			0
$813$		−	7.39559i		−	0.259375i
$814$		0			0
$815$	−51.4325			−1.80160
$816$		0			0
$817$	−4.52745			−0.158396
$818$		0			0
$819$		−	1.32459i		−	0.0462847i
$820$		0			0
$821$			19.9688i			0.696917i	0.937324	+	0.348458i	$0.113295\pi$
$821$			19.9688i			0.696917i	−0.937324	+	0.348458i	$0.886705\pi$
$822$		0			0
$823$			8.28082i			0.288651i	0.989530	+	0.144326i	$0.0461012\pi$
$823$			8.28082i			0.288651i	−0.989530	+	0.144326i	$0.953899\pi$
$824$		0			0
$825$		−	15.9004i		−	0.553580i
$826$		0			0
$827$			9.29195i			0.323113i	0.986863	+	0.161556i	$0.0516514\pi$
$827$			9.29195i			0.323113i	−0.986863	+	0.161556i	$0.948349\pi$
$828$		0			0
$829$		−	42.2445i		−	1.46721i	−0.679576	−	0.733605i	$-0.737836\pi$
$829$		−	42.2445i		−	1.46721i	0.679576	−	0.733605i	$-0.262164\pi$
$830$		0			0
$831$			8.71879i			0.302452i
$832$		0			0
$833$		−	13.7473i		−	0.476316i
$834$		0			0
$835$			55.4091i			1.91751i
$836$		0			0
$837$		−	8.16330i		−	0.282165i
$838$		0			0
$839$	2.44511			0.0844145			0.0422072	−	0.999109i	$-0.486561\pi$
$839$	2.44511			0.0844145			0.0422072	+	0.999109i	$0.486561\pi$
$840$		0			0
$841$	6.91803			0.238553
$842$		0			0
$843$			5.18858i			0.178704i
$844$		0			0
$845$	7.91581			0.272312
$846$		0			0
$847$	−8.21032			−0.282110
$848$		0			0
$849$		−	26.9752i		−	0.925787i
$850$		0			0
$851$	10.3369	+	13.7988i	0.354345	+	0.473017i
$852$		0			0
$853$			12.3852i			0.424060i	0.977263	+	0.212030i	$0.0680075\pi$
$853$			12.3852i			0.424060i	−0.977263	+	0.212030i	$0.931992\pi$
$854$		0			0
$855$			7.02915i			0.240392i
$856$		0			0
$857$	−39.9980			−1.36631			−0.683153	−	0.730275i	$-0.739392\pi$
$857$	−39.9980			−1.36631			−0.683153	+	0.730275i	$0.739392\pi$
$858$		0			0
$859$	9.74110			0.332362			0.166181	−	0.986095i	$-0.446856\pi$
$859$	9.74110			0.332362			0.166181	+	0.986095i	$0.446856\pi$
$860$		0			0
$861$	1.04800			0.0357158
$862$		0			0
$863$			11.8577i			0.403641i	0.979423	+	0.201820i	$0.0646857\pi$
$863$			11.8577i			0.403641i	−0.979423	+	0.201820i	$0.935314\pi$
$864$		0			0
$865$		−	31.9367i		−	1.08588i
$866$		0			0
$867$	12.9459			0.439665
$868$		0			0
$869$		−	83.4217i		−	2.82989i
$870$		0			0
$871$	17.2708			0.585198
$872$		0			0
$873$			14.5135i			0.491206i
$874$		0			0
$875$	−2.48477			−0.0840004
$876$		0			0
$877$		−	44.9851i		−	1.51904i	−0.650484	−	0.759520i	$-0.725434\pi$
$877$		−	44.9851i		−	1.51904i	0.650484	−	0.759520i	$-0.274566\pi$
$878$		0			0
$879$	22.7008			0.765678
$880$		0			0
$881$			52.2081i			1.75893i	0.475959	+	0.879467i	$0.342101\pi$
$881$			52.2081i			1.75893i	−0.475959	+	0.879467i	$0.657899\pi$
$882$		0			0
$883$	21.1374			0.711329			0.355664	−	0.934614i	$-0.384255\pi$
$883$	21.1374			0.711329			0.355664	+	0.934614i	$0.384255\pi$
$884$		0			0
$885$	1.84508			0.0620216
$886$		0			0
$887$		−	38.9103i		−	1.30648i	−0.757151	−	0.653240i	$-0.773409\pi$
$887$		−	38.9103i		−	1.30648i	0.757151	−	0.653240i	$-0.226591\pi$
$888$		0			0
$889$		−	2.20655i		−	0.0740052i
$890$		0			0
$891$		−	5.54747i		−	0.185847i
$892$		0			0
$893$	8.98075			0.300529
$894$		0			0
$895$	−53.6133			−1.79209
$896$		0			0
$897$	12.2451	−	9.17300i	0.408851	−	0.306277i
$898$		0			0
$899$	−38.3605			−1.27939
$900$		0			0
$901$		−	0.544650i		−	0.0181449i
$902$		0			0
$903$			0.750052i			0.0249602i
$904$		0			0
$905$	−15.7229			−0.522648
$906$		0			0
$907$			26.4527i			0.878349i	0.898402	+	0.439174i	$0.144729\pi$
$907$			26.4527i			0.878349i	−0.898402	+	0.439174i	$0.855271\pi$
$908$		0			0
$909$			7.00245i			0.232257i
$910$		0			0
$911$	44.2857			1.46725			0.733625	−	0.679554i	$-0.237827\pi$
$911$	44.2857			1.46725			0.733625	+	0.679554i	$0.237827\pi$
$912$		0			0
$913$	62.0471			2.05346
$914$		0			0
$915$	−33.1020			−1.09432
$916$		0			0
$917$	−0.756641			−0.0249865
$918$		0			0
$919$	−7.94858			−0.262199			−0.131100	−	0.991369i	$-0.541851\pi$
$919$	−7.94858			−0.262199			−0.131100	+	0.991369i	$0.541851\pi$
$920$		0			0
$921$	−4.84075			−0.159508
$922$		0			0
$923$	−19.4287			−0.639504
$924$		0			0
$925$	10.3042			0.338801
$926$		0			0
$927$	18.4881			0.607230
$928$		0			0
$929$	−12.3596			−0.405504			−0.202752	−	0.979230i	$-0.564988\pi$
$929$	−12.3596			−0.405504			−0.202752	+	0.979230i	$0.564988\pi$
$930$		0			0
$931$		−	17.1115i		−	0.560806i
$932$		0			0
$933$			9.16116i			0.299923i
$934$		0			0
$935$	31.3276			1.02452
$936$		0			0
$937$			37.8690i			1.23713i	0.785735	+	0.618563i	$0.212285\pi$
$937$			37.8690i			1.23713i	−0.785735	+	0.618563i	$0.787715\pi$
$938$		0			0
$939$			17.8518i			0.582571i
$940$		0			0
$941$	40.8202			1.33070			0.665350	−	0.746532i	$-0.268282\pi$
$941$	40.8202			1.33070			0.665350	+	0.746532i	$0.268282\pi$
$942$		0			0
$943$	7.25761	+	9.68822i	0.236340	+	0.315492i
$944$		0			0
$945$	1.16450			0.0378812
$946$		0			0
$947$	35.2377			1.14507			0.572535	−	0.819880i	$-0.305960\pi$
$947$	35.2377			1.14507			0.572535	+	0.819880i	$0.305960\pi$
$948$		0			0
$949$			9.00701i			0.292380i
$950$		0			0
$951$			31.8091i			1.03148i
$952$		0			0
$953$			37.0278i			1.19945i	0.800207	+	0.599723i	$0.204723\pi$
$953$			37.0278i			1.19945i	−0.800207	+	0.599723i	$0.795277\pi$
$954$		0			0
$955$	−70.1988			−2.27158
$956$		0			0
$957$	−26.0684			−0.842671
$958$		0			0
$959$		−	7.78320i		−	0.251333i
$960$		0			0
$961$	−35.6394			−1.14966
$962$		0			0
$963$			2.34324i			0.0755099i
$964$		0			0
$965$	69.1092			2.22470
$966$		0			0
$967$		−	8.73002i		−	0.280738i	−0.990099	−	0.140369i	$-0.955171\pi$
$967$		−	8.73002i		−	0.280738i	0.990099	−	0.140369i	$-0.0448290\pi$
$968$		0			0
$969$	−5.04624			−0.162108
$970$		0			0
$971$		−	41.7254i		−	1.33903i	−0.742798	−	0.669516i	$-0.766502\pi$
$971$		−	41.7254i		−	1.33903i	0.742798	−	0.669516i	$-0.233498\pi$
$972$		0			0
$973$	3.17349			0.101737
$974$		0			0
$975$		−	9.14400i		−	0.292842i
$976$		0			0
$977$			30.5474i			0.977297i	0.872481	+	0.488648i	$0.162510\pi$
$977$			30.5474i			0.977297i	−0.872481	+	0.488648i	$0.837490\pi$
$978$		0			0
$979$	22.3381			0.713929
$980$		0			0
$981$	9.34569			0.298385
$982$		0			0
$983$	44.5757			1.42174			0.710872	−	0.703322i	$-0.248301\pi$
$983$	44.5757			1.42174			0.710872	+	0.703322i	$0.248301\pi$
$984$		0			0
$985$			6.71957i			0.214103i
$986$		0			0
$987$		−	1.48782i		−	0.0473578i
$988$		0			0
$989$	−6.93383	+	5.19425i	−0.220483	+	0.165168i
$990$		0			0
$991$		−	19.0485i		−	0.605096i	−0.953134	−	0.302548i	$-0.902163\pi$
$991$		−	19.0485i		−	0.605096i	0.953134	−	0.302548i	$-0.0978372\pi$
$992$		0			0
$993$	3.34810			0.106249
$994$		0			0
$995$	−10.5555			−0.334631
$996$		0			0
$997$		−	47.6147i		−	1.50797i	−0.656889	−	0.753987i	$-0.728128\pi$
$997$		−	47.6147i		−	1.50797i	0.656889	−	0.753987i	$-0.271872\pi$
$998$		0			0
$999$	3.59504			0.113742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.n.b.367.20		24
4.3	odd	2		552.2.n.b.91.22	yes	24
8.3	odd	2	inner	2208.2.n.b.367.6		24
8.5	even	2		552.2.n.b.91.23	yes	24
23.22	odd	2	inner	2208.2.n.b.367.5		24
92.91	even	2		552.2.n.b.91.21	✓	24
184.45	odd	2		552.2.n.b.91.24	yes	24
184.91	even	2	inner	2208.2.n.b.367.19		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.n.b.91.21	✓	24	92.91	even	2
552.2.n.b.91.22	yes	24	4.3	odd	2
552.2.n.b.91.23	yes	24	8.5	even	2
552.2.n.b.91.24	yes	24	184.45	odd	2
2208.2.n.b.367.5		24	23.22	odd	2	inner
2208.2.n.b.367.6		24	8.3	odd	2	inner
2208.2.n.b.367.19		24	184.91	even	2	inner
2208.2.n.b.367.20		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2208 = 2^{5} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2208.n (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.80468 q^{5} +0.415199 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.80468 q^{5} +0.415199 q^{7} +1.00000 q^{9} -5.54747i q^{11} -3.19024i q^{13} +2.80468 q^{15} +2.01349i q^{17} +2.50622i q^{19} +0.415199 q^{21} +(2.87533 + 3.83829i) q^{23} +2.86624 q^{25} +1.00000 q^{27} -4.69915i q^{29} -8.16330i q^{31} -5.54747i q^{33} +1.16450 q^{35} +3.59504 q^{37} -3.19024i q^{39} +2.52410 q^{41} +1.80649i q^{43} +2.80468 q^{45} -3.58339i q^{47} -6.82761 q^{49} +2.01349i q^{51} -0.270501 q^{53} -15.5589i q^{55} +2.50622i q^{57} +0.657857 q^{59} -11.8024 q^{61} +0.415199 q^{63} -8.94761i q^{65} +5.41363i q^{67} +(2.87533 + 3.83829i) q^{69} -6.09004i q^{71} -2.82330 q^{73} +2.86624 q^{75} -2.30330i q^{77} +15.0378 q^{79} +1.00000 q^{81} +11.1848i q^{83} +5.64719i q^{85} -4.69915i q^{87} +4.02672i q^{89} -1.32459i q^{91} -8.16330i q^{93} +7.02915i q^{95} +14.5135i q^{97} -5.54747i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 24 * q^9	\( 24 q + 24 q^{3} + 24 q^{9} + 24 q^{25} + 24 q^{27} + 56 q^{49} + 32 q^{73} + 24 q^{75} + 24 q^{81}+O(q^{100}) \) 24 * q + 24 * q^3 + 24 * q^9 + 24 * q^25 + 24 * q^27 + 56 * q^49 + 32 * q^73 + 24 * q^75 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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