LMFDB - Embedded newform 608.2.b.c.303.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,2,Mod(303,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.303"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	608.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$4.85490444289$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.319794774016000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + 2x^8 + 8x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.9
Root			$-1.17117 - 0.792696i$ of defining polynomial
Character	$\chi$	$=$	608.303
Dual form			608.2.b.c.303.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26152i q^{3} -3.51876i q^{5} -2.23607i q^{7} +1.40857 q^{9} -2.89511 q^{11} -6.30281 q^{13} +4.43898 q^{15} -4.79021 q^{17} +(-0.895107 - 4.26600i) q^{19} +2.82084 q^{21} -0.524069i q^{23} -7.38164 q^{25} +5.56149i q^{27} +0.415427 q^{29} -1.20271 q^{31} -3.65223i q^{33} -7.86818 q^{35} +5.88738 q^{37} -7.95110i q^{39} -4.87978i q^{41} +10.6853 q^{43} -4.95642i q^{45} -4.27737i q^{47} +2.00000 q^{49} -6.04294i q^{51} +6.05711 q^{53} +10.1872i q^{55} +(5.38164 - 1.12919i) q^{57} -8.08453i q^{59} -8.38042i q^{61} -3.14966i q^{63} +22.1780i q^{65} +9.79353i q^{67} +0.661123 q^{69} -10.2002 q^{71} +7.76328 q^{73} -9.31208i q^{75} +6.47365i q^{77} +7.33578 q^{79} -2.79021 q^{81} -2.00000 q^{83} +16.8556i q^{85} +0.524069i q^{87} -12.1842i q^{89} +14.0935i q^{91} -1.51724i q^{93} +(-15.0110 + 3.14966i) q^{95} +2.19044i q^{97} -4.07796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9} + 12 q^{17} + 24 q^{19} - 44 q^{25} - 40 q^{35} + 24 q^{43} + 24 q^{49} + 20 q^{57} + 4 q^{73} + 36 q^{81} - 24 q^{83} - 64 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 + 12 * q^17 + 24 * q^19 - 44 * q^25 - 40 * q^35 + 24 * q^43 + 24 * q^49 + 20 * q^57 + 4 * q^73 + 36 * q^81 - 24 * q^83 - 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			1.26152i			0.728338i	0.931333	+	0.364169i	$0.118647\pi$
$3$			1.26152i			0.728338i	−0.931333	+	0.364169i	$0.881353\pi$
$4$		0			0
$5$		−	3.51876i		−	1.57364i	−0.617185	−	0.786818i	$-0.711727\pi$
$5$		−	3.51876i		−	1.57364i	0.617185	−	0.786818i	$-0.288273\pi$
$6$		0			0
$7$		−	2.23607i		−	0.845154i	−0.906327	−	0.422577i	$-0.861126\pi$
$7$		−	2.23607i		−	0.845154i	0.906327	−	0.422577i	$-0.138874\pi$
$8$		0			0
$9$	1.40857			0.469524
$10$		0			0
$11$	−2.89511			−0.872907			−0.436454	−	0.899727i	$-0.643766\pi$
$11$	−2.89511			−0.872907			−0.436454	+	0.899727i	$0.643766\pi$
$12$		0			0
$13$	−6.30281			−1.74808			−0.874042	−	0.485851i	$-0.838510\pi$
$13$	−6.30281			−1.74808			−0.874042	+	0.485851i	$0.838510\pi$
$14$		0			0
$15$	4.43898			1.14614
$16$		0			0
$17$	−4.79021			−1.16180			−0.580899	−	0.813976i	$-0.697299\pi$
$17$	−4.79021			−1.16180			−0.580899	+	0.813976i	$0.697299\pi$
$18$		0			0
$19$	−0.895107	−	4.26600i	−0.205352	−	0.978688i
$19$	−0.895107	−	4.26600i	−0.205352	−	0.978688i
$20$		0			0
$21$	2.82084			0.615558
$22$		0			0
$23$		−	0.524069i		−	0.109276i	−0.998506	−	0.0546380i	$-0.982600\pi$
$23$		−	0.524069i		−	0.109276i	0.998506	−	0.0546380i	$-0.0174005\pi$
$24$		0			0
$25$	−7.38164			−1.47633
$26$		0			0
$27$			5.56149i			1.07031i
$28$		0			0
$29$	0.415427			0.0771429			0.0385715	−	0.999256i	$-0.487719\pi$
$29$	0.415427			0.0771429			0.0385715	+	0.999256i	$0.487719\pi$
$30$		0			0
$31$	−1.20271			−0.216013			−0.108006	−	0.994150i	$-0.534447\pi$
$31$	−1.20271			−0.216013			−0.108006	+	0.994150i	$0.534447\pi$
$32$		0			0
$33$		−	3.65223i		−	0.635772i
$34$		0			0
$35$	−7.86818			−1.32996
$36$		0			0
$37$	5.88738			0.967879			0.483939	−	0.875101i	$-0.339205\pi$
$37$	5.88738			0.967879			0.483939	+	0.875101i	$0.339205\pi$
$38$		0			0
$39$		−	7.95110i		−	1.27320i
$40$		0			0
$41$		−	4.87978i		−	0.762093i	−0.924556	−	0.381047i	$-0.875564\pi$
$41$		−	4.87978i		−	0.762093i	0.924556	−	0.381047i	$-0.124436\pi$
$42$		0			0
$43$	10.6853			1.62950			0.814748	−	0.579815i	$-0.196875\pi$
$43$	10.6853			1.62950			0.814748	+	0.579815i	$0.196875\pi$
$44$		0			0
$45$		−	4.95642i		−	0.738859i
$46$		0			0
$47$		−	4.27737i		−	0.623919i	−0.950095	−	0.311960i	$-0.899015\pi$
$47$		−	4.27737i		−	0.623919i	0.950095	−	0.311960i	$-0.100985\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		−	6.04294i		−	0.846181i
$52$		0			0
$53$	6.05711			0.832008			0.416004	−	0.909363i	$-0.363430\pi$
$53$	6.05711			0.832008			0.416004	+	0.909363i	$0.363430\pi$
$54$		0			0
$55$			10.1872i			1.37364i
$56$		0			0
$57$	5.38164	−	1.12919i	0.712816	−	0.149565i
$58$		0			0
$59$		−	8.08453i		−	1.05252i	−0.850325	−	0.526258i	$-0.823595\pi$
$59$		−	8.08453i		−	1.05252i	0.850325	−	0.526258i	$-0.176405\pi$
$60$		0			0
$61$		−	8.38042i		−	1.07300i	−0.843900	−	0.536501i	$-0.819746\pi$
$61$		−	8.38042i		−	1.07300i	0.843900	−	0.536501i	$-0.180254\pi$
$62$		0			0
$63$		−	3.14966i		−	0.396820i
$64$		0			0
$65$			22.1780i			2.75085i
$66$		0			0
$67$			9.79353i			1.19647i	0.801321	+	0.598235i	$0.204131\pi$
$67$			9.79353i			1.19647i	−0.801321	+	0.598235i	$0.795869\pi$
$68$		0			0
$69$	0.661123			0.0795899
$70$		0			0
$71$	−10.2002			−1.21054			−0.605270	−	0.796020i	$-0.706935\pi$
$71$	−10.2002			−1.21054			−0.605270	+	0.796020i	$0.706935\pi$
$72$		0			0
$73$	7.76328			0.908624			0.454312	−	0.890843i	$-0.349885\pi$
$73$	7.76328			0.908624			0.454312	+	0.890843i	$0.349885\pi$
$74$		0			0
$75$		−	9.31208i		−	1.07527i
$76$		0			0
$77$			6.47365i			0.737741i
$78$		0			0
$79$	7.33578			0.825340			0.412670	−	0.910881i	$-0.364596\pi$
$79$	7.33578			0.825340			0.412670	+	0.910881i	$0.364596\pi$
$80$		0			0
$81$	−2.79021			−0.310024
$82$		0			0
$83$	−2.00000			−0.219529			−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000			−0.219529			−0.109764	+	0.993958i	$0.535010\pi$
$84$		0			0
$85$			16.8556i			1.82825i
$86$		0			0
$87$			0.524069i			0.0561861i
$88$		0			0
$89$		−	12.1842i		−	1.29153i	−0.763538	−	0.645763i	$-0.776539\pi$
$89$		−	12.1842i		−	1.29153i	0.763538	−	0.645763i	$-0.223461\pi$
$90$		0			0
$91$			14.0935i			1.47740i
$92$		0			0
$93$		−	1.51724i		−	0.157330i
$94$		0			0
$95$	−15.0110	+	3.14966i	−1.54010	+	0.323148i
$96$		0			0
$97$			2.19044i			0.222406i	0.993798	+	0.111203i	$0.0354703\pi$
$97$			2.19044i			0.222406i	−0.993798	+	0.111203i	$0.964530\pi$
$98$		0			0
$99$	−4.07796			−0.409851
$100$		0			0
$101$			1.51724i			0.150971i	0.997147	+	0.0754853i	$0.0240506\pi$
$101$			1.51724i			0.150971i	−0.997147	+	0.0754853i	$0.975949\pi$
$102$		0			0
$103$	0.830855			0.0818666			0.0409333	−	0.999162i	$-0.486967\pi$
$103$	0.830855			0.0818666			0.0409333	+	0.999162i	$0.486967\pi$
$104$		0			0
$105$		−	9.92585i		−	0.968664i
$106$		0			0
$107$		−	0.780070i		−	0.0754122i	−0.999289	−	0.0377061i	$-0.987995\pi$
$107$		−	0.780070i		−	0.0754122i	0.999289	−	0.0377061i	$-0.0120051\pi$
$108$		0			0
$109$	−6.05711			−0.580166			−0.290083	−	0.957002i	$-0.593683\pi$
$109$	−6.05711			−0.580166			−0.290083	+	0.957002i	$0.593683\pi$
$110$		0			0
$111$			7.42704i			0.704943i
$112$		0			0
$113$			3.58429i			0.337181i	0.985686	+	0.168591i	$0.0539216\pi$
$113$			3.58429i			0.337181i	−0.985686	+	0.168591i	$0.946078\pi$
$114$		0			0
$115$	−1.84407			−0.171961
$116$		0			0
$117$	−8.87795			−0.820767
$118$		0			0
$119$			10.7112i			0.981898i
$120$		0			0
$121$	−2.61836			−0.238033
$122$		0			0
$123$	6.15593			0.555061
$124$		0			0
$125$			8.38042i			0.749567i
$126$		0			0
$127$	−17.0446			−1.51246			−0.756231	−	0.654305i	$-0.772961\pi$
$127$	−17.0446			−1.51246			−0.756231	+	0.654305i	$0.772961\pi$
$128$		0			0
$129$			13.4797i			1.18682i
$130$		0			0
$131$	11.7122			1.02330			0.511652	−	0.859193i	$-0.329034\pi$
$131$	11.7122			1.02330			0.511652	+	0.859193i	$0.329034\pi$
$132$		0			0
$133$	−9.53907	+	2.00152i	−0.827143	+	0.173554i
$134$		0			0
$135$	19.5695			1.68428
$136$		0			0
$137$	0.946141			0.0808343			0.0404171	−	0.999183i	$-0.487131\pi$
$137$	0.946141			0.0808343			0.0404171	+	0.999183i	$0.487131\pi$
$138$		0			0
$139$	0.841247			0.0713537			0.0356768	−	0.999363i	$-0.488641\pi$
$139$	0.841247			0.0713537			0.0356768	+	0.999363i	$0.488641\pi$
$140$		0			0
$141$	5.39599			0.454424
$142$		0			0
$143$	18.2473			1.52592
$144$		0			0
$145$		−	1.46179i		−	0.121395i
$146$		0			0
$147$			2.52304i			0.208097i
$148$		0			0
$149$		−	11.3353i		−	0.928625i	−0.885671	−	0.464313i	$-0.846301\pi$
$149$		−	11.3353i		−	0.928625i	0.885671	−	0.464313i	$-0.153699\pi$
$150$		0			0
$151$	9.36934			0.762466			0.381233	−	0.924479i	$-0.375500\pi$
$151$	9.36934			0.762466			0.381233	+	0.924479i	$0.375500\pi$
$152$		0			0
$153$	−6.74736			−0.545491
$154$		0			0
$155$			4.23203i			0.339925i
$156$		0			0
$157$			17.4194i			1.39022i	0.718902	+	0.695112i	$0.244645\pi$
$157$			17.4194i			1.39022i	−0.718902	+	0.695112i	$0.755355\pi$
$158$		0			0
$159$			7.64115i			0.605983i
$160$		0			0
$161$	−1.17185			−0.0923551
$162$		0			0
$163$	20.3437			1.59344			0.796721	−	0.604347i	$-0.206566\pi$
$163$	20.3437			1.59344			0.796721	+	0.604347i	$0.206566\pi$
$164$		0			0
$165$	−12.8513			−1.00047
$166$		0			0
$167$	−13.8083			−1.06852			−0.534260	−	0.845320i	$-0.679410\pi$
$167$	−13.8083			−1.06852			−0.534260	+	0.845320i	$0.679410\pi$
$168$		0			0
$169$	26.7254			2.05580
$170$		0			0
$171$	−1.26082	−	6.00897i	−0.0964174	−	0.459517i
$172$		0			0
$173$	2.89680			0.220240			0.110120	−	0.993918i	$-0.464876\pi$
$173$	2.89680			0.220240			0.110120	+	0.993918i	$0.464876\pi$
$174$		0			0
$175$			16.5059i			1.24773i
$176$		0			0
$177$	10.1988			0.766588
$178$		0			0
$179$			10.6415i			0.795386i	0.917519	+	0.397693i	$0.130189\pi$
$179$			10.6415i			0.795386i	−0.917519	+	0.397693i	$0.869811\pi$
$180$		0			0
$181$	−2.40541			−0.178793			−0.0893965	−	0.995996i	$-0.528494\pi$
$181$	−2.40541			−0.178793			−0.0893965	+	0.995996i	$0.528494\pi$
$182$		0			0
$183$	10.5720			0.781508
$184$		0			0
$185$		−	20.7162i		−	1.52309i
$186$		0			0
$187$	13.8682			1.01414
$188$		0			0
$189$	12.4359			0.904577
$190$		0			0
$191$		−	10.0526i		−	0.727383i	−0.931520	−	0.363691i	$-0.881516\pi$
$191$		−	10.0526i		−	0.727383i	0.931520	−	0.363691i	$-0.118484\pi$
$192$		0			0
$193$		−	15.0399i		−	1.08259i	−0.840832	−	0.541297i	$-0.817934\pi$
$193$		−	15.0399i		−	1.08259i	0.840832	−	0.541297i	$-0.182066\pi$
$194$		0			0
$195$	−27.9780			−2.00355
$196$		0			0
$197$		−	13.8059i		−	0.983632i	−0.870699	−	0.491816i	$-0.836333\pi$
$197$		−	13.8059i		−	0.983632i	0.870699	−	0.491816i	$-0.163667\pi$
$198$		0			0
$199$			7.36682i			0.522220i	0.965309	+	0.261110i	$0.0840885\pi$
$199$			7.36682i			0.522220i	−0.965309	+	0.261110i	$0.915911\pi$
$200$		0			0
$201$	−12.3547			−0.871434
$202$		0			0
$203$		−	0.928924i		−	0.0651977i
$204$		0			0
$205$	−17.1707			−1.19926
$206$		0			0
$207$		−	0.738189i		−	0.0513077i
$208$		0			0
$209$	2.59143	+	12.3505i	0.179253	+	0.854304i
$210$		0			0
$211$		−	15.6536i		−	1.07764i	−0.842421	−	0.538821i	$-0.818870\pi$
$211$		−	15.6536i		−	1.07764i	0.842421	−	0.538821i	$-0.181130\pi$
$212$		0			0
$213$		−	12.8677i		−	0.881683i
$214$		0			0
$215$		−	37.5990i		−	2.56423i
$216$		0			0
$217$			2.68933i			0.182564i
$218$		0			0
$219$			9.79353i			0.661785i
$220$		0			0
$221$	30.1918			2.03092
$222$		0			0
$223$	11.4029			0.763595			0.381797	−	0.924246i	$-0.375305\pi$
$223$	11.4029			0.763595			0.381797	+	0.924246i	$0.375305\pi$
$224$		0			0
$225$	−10.3976			−0.693171
$226$		0			0
$227$		−	14.4261i		−	0.957494i	−0.877953	−	0.478747i	$-0.841091\pi$
$227$		−	14.4261i		−	0.957494i	0.877953	−	0.478747i	$-0.158909\pi$
$228$		0			0
$229$			18.8419i			1.24511i	0.782576	+	0.622555i	$0.213905\pi$
$229$			18.8419i			1.24511i	−0.782576	+	0.622555i	$0.786095\pi$
$230$		0			0
$231$	−8.16663			−0.537325
$232$		0			0
$233$	5.19878			0.340584			0.170292	−	0.985394i	$-0.445529\pi$
$233$	5.19878			0.340584			0.170292	+	0.985394i	$0.445529\pi$
$234$		0			0
$235$	−15.0510			−0.981821
$236$		0			0
$237$			9.25422i			0.601126i
$238$		0			0
$239$		−	12.2285i		−	0.790995i	−0.918467	−	0.395497i	$-0.870572\pi$
$239$		−	12.2285i		−	0.790995i	0.918467	−	0.395497i	$-0.129428\pi$
$240$		0			0
$241$			29.9134i			1.92689i	0.267899	+	0.963447i	$0.413671\pi$
$241$			29.9134i			1.92689i	−0.267899	+	0.963447i	$0.586329\pi$
$242$		0			0
$243$			13.1646i			0.844508i
$244$		0			0
$245$		−	7.03751i		−	0.449610i
$246$		0			0
$247$	5.64168	+	26.8878i	0.358972	+	1.71083i
$248$		0			0
$249$		−	2.52304i		−	0.159891i
$250$		0			0
$251$	−21.3947			−1.35042			−0.675212	−	0.737624i	$-0.735948\pi$
$251$	−21.3947			−1.35042			−0.675212	+	0.737624i	$0.735948\pi$
$252$		0			0
$253$			1.51724i			0.0953879i
$254$		0			0
$255$	−21.2636			−1.33158
$256$		0			0
$257$			13.5781i			0.846977i	0.905901	+	0.423489i	$0.139195\pi$
$257$			13.5781i			0.846977i	−0.905901	+	0.423489i	$0.860805\pi$
$258$		0			0
$259$		−	13.1646i		−	0.818007i
$260$		0			0
$261$	0.585159			0.0362204
$262$		0			0
$263$			12.0939i			0.745744i	0.927883	+	0.372872i	$0.121627\pi$
$263$			12.0939i			0.745744i	−0.927883	+	0.372872i	$0.878373\pi$
$264$		0			0
$265$		−	21.3135i		−	1.30928i
$266$		0			0
$267$	15.3706			0.940668
$268$		0			0
$269$	21.2379			1.29490			0.647448	−	0.762110i	$-0.275836\pi$
$269$	21.2379			1.29490			0.647448	+	0.762110i	$0.275836\pi$
$270$		0			0
$271$		−	16.1163i		−	0.978997i	−0.872004	−	0.489499i	$-0.837180\pi$
$271$		−	16.1163i		−	0.978997i	0.872004	−	0.489499i	$-0.162820\pi$
$272$		0			0
$273$	−17.7792			−1.07605
$274$		0			0
$275$	21.3706			1.28870
$276$		0			0
$277$		−	3.98785i		−	0.239607i	−0.992798	−	0.119803i	$-0.961774\pi$
$277$		−	3.98785i		−	0.239607i	0.992798	−	0.119803i	$-0.0382264\pi$
$278$		0			0
$279$	−1.69410			−0.101423
$280$		0			0
$281$		−	0.630302i		−	0.0376007i	−0.999823	−	0.0188003i	$-0.994015\pi$
$281$		−	0.630302i		−	0.0376007i	0.999823	−	0.0188003i	$-0.00598468\pi$
$282$		0			0
$283$	−31.0290			−1.84448			−0.922242	−	0.386613i	$-0.873645\pi$
$283$	−31.0290			−1.84448			−0.922242	+	0.386613i	$0.873645\pi$
$284$		0			0
$285$	−3.97336	−	18.9367i	−0.235361	−	1.12171i
$286$		0			0
$287$	−10.9115			−0.644086
$288$		0			0
$289$	5.94614			0.349773
$290$		0			0
$291$	−2.76328			−0.161987
$292$		0			0
$293$	−30.6832			−1.79253			−0.896265	−	0.443519i	$-0.853730\pi$
$293$	−30.6832			−1.79253			−0.896265	+	0.443519i	$0.853730\pi$
$294$		0			0
$295$	−28.4475			−1.65628
$296$		0			0
$297$		−	16.1011i		−	0.934282i
$298$		0			0
$299$			3.30311i			0.191024i
$300$		0			0
$301$		−	23.8931i		−	1.37718i
$302$		0			0
$303$	−1.91402			−0.109958
$304$		0			0
$305$	−29.4886			−1.68851
$306$		0			0
$307$		−	15.6876i		−	0.895339i	−0.894199	−	0.447670i	$-0.852254\pi$
$307$		−	15.6876i		−	0.895339i	0.894199	−	0.447670i	$-0.147746\pi$
$308$		0			0
$309$			1.04814i			0.0596265i
$310$		0			0
$311$		−	28.3998i		−	1.61040i	−0.593001	−	0.805202i	$-0.702057\pi$
$311$		−	28.3998i		−	1.61040i	0.593001	−	0.805202i	$-0.297943\pi$
$312$		0			0
$313$	−5.53757			−0.313002			−0.156501	−	0.987678i	$-0.550021\pi$
$313$	−5.53757			−0.313002			−0.156501	+	0.987678i	$0.550021\pi$
$314$		0			0
$315$	−11.0829			−0.624450
$316$		0			0
$317$	−11.9445			−0.670869			−0.335435	−	0.942063i	$-0.608883\pi$
$317$	−11.9445			−0.670869			−0.335435	+	0.942063i	$0.608883\pi$
$318$		0			0
$319$	−1.20271			−0.0673386
$320$		0			0
$321$	0.984073			0.0549256
$322$		0			0
$323$	4.28775	+	20.4351i	0.238577	+	1.13704i
$324$		0			0
$325$	46.5250			2.58075
$326$		0			0
$327$		−	7.64115i		−	0.422557i
$328$		0			0
$329$	−9.56450			−0.527308
$330$		0			0
$331$		−	15.5378i		−	0.854037i	−0.904243	−	0.427019i	$-0.859564\pi$
$331$		−	15.5378i		−	0.854037i	0.904243	−	0.427019i	$-0.140436\pi$
$332$		0			0
$333$	8.29279			0.454442
$334$		0			0
$335$	34.4610			1.88281
$336$		0			0
$337$			29.4825i			1.60601i	0.595970	+	0.803007i	$0.296768\pi$
$337$			29.4825i			1.60601i	−0.595970	+	0.803007i	$0.703232\pi$
$338$		0			0
$339$	−4.52164			−0.245582
$340$		0			0
$341$	3.48196			0.188559
$342$		0			0
$343$		−	20.1246i		−	1.08663i
$344$		0			0
$345$		−	2.32633i		−	0.125245i
$346$		0			0
$347$	19.3465			1.03858			0.519288	−	0.854599i	$-0.326197\pi$
$347$	19.3465			1.03858			0.519288	+	0.854599i	$0.326197\pi$
$348$		0			0
$349$			12.4630i			0.667131i	0.942727	+	0.333565i	$0.108252\pi$
$349$			12.4630i			0.667131i	−0.942727	+	0.333565i	$0.891748\pi$
$350$		0			0
$351$		−	35.0530i		−	1.87099i
$352$		0			0
$353$	9.22571			0.491035			0.245518	−	0.969392i	$-0.421042\pi$
$353$	9.22571			0.491035			0.245518	+	0.969392i	$0.421042\pi$
$354$		0			0
$355$			35.8920i			1.90495i
$356$		0			0
$357$	−13.5124			−0.715154
$358$		0			0
$359$			18.8765i			0.996262i	0.867102	+	0.498131i	$0.165980\pi$
$359$			18.8765i			0.996262i	−0.867102	+	0.498131i	$0.834020\pi$
$360$		0			0
$361$	−17.3976	+	7.63705i	−0.915662	+	0.401950i
$362$		0			0
$363$		−	3.30311i		−	0.173368i
$364$		0			0
$365$		−	27.3171i		−	1.42984i
$366$		0			0
$367$			20.7230i			1.08173i	0.841109	+	0.540866i	$0.181903\pi$
$367$			20.7230i			1.08173i	−0.841109	+	0.540866i	$0.818097\pi$
$368$		0			0
$369$		−	6.87351i		−	0.357821i
$370$		0			0
$371$		−	13.5441i		−	0.703175i
$372$		0			0
$373$	−4.23686			−0.219376			−0.109688	−	0.993966i	$-0.534985\pi$
$373$	−4.23686			−0.219376			−0.109688	+	0.993966i	$0.534985\pi$
$374$		0			0
$375$	−10.5720			−0.545938
$376$		0			0
$377$	−2.61836			−0.134852
$378$		0			0
$379$			8.91154i			0.457755i	0.973455	+	0.228877i	$0.0735055\pi$
$379$			8.91154i			0.457755i	−0.973455	+	0.228877i	$0.926494\pi$
$380$		0			0
$381$		−	21.5021i		−	1.10158i
$382$		0			0
$383$	24.3480			1.24412			0.622062	−	0.782968i	$-0.286295\pi$
$383$	24.3480			1.24412			0.622062	+	0.782968i	$0.286295\pi$
$384$		0			0
$385$	22.7792			1.16094
$386$		0			0
$387$	15.0510			0.765087
$388$		0			0
$389$		−	3.04966i		−	0.154624i	−0.997007	−	0.0773119i	$-0.975366\pi$
$389$		−	3.04966i		−	0.154624i	0.997007	−	0.0773119i	$-0.0246337\pi$
$390$		0			0
$391$			2.51040i			0.126957i
$392$		0			0
$393$			14.7752i			0.745311i
$394$		0			0
$395$		−	25.8128i		−	1.29878i
$396$		0			0
$397$			14.6389i			0.734704i	0.930082	+	0.367352i	$0.119736\pi$
$397$			14.6389i			0.734704i	−0.930082	+	0.367352i	$0.880264\pi$
$398$		0			0
$399$	−2.52495	−	12.0337i	−0.126406	−	0.602439i
$400$		0			0
$401$		−	28.6859i		−	1.43251i	−0.697841	−	0.716253i	$-0.745856\pi$
$401$		−	28.6859i		−	1.43251i	0.697841	−	0.716253i	$-0.254144\pi$
$402$		0			0
$403$	7.58043			0.377608
$404$		0			0
$405$			9.81808i			0.487864i
$406$		0			0
$407$	−17.0446			−0.844869
$408$		0			0
$409$		−	16.0028i		−	0.791286i	−0.918404	−	0.395643i	$-0.870522\pi$
$409$		−	16.0028i		−	0.791286i	0.918404	−	0.395643i	$-0.129478\pi$
$410$		0			0
$411$			1.19357i			0.0588747i
$412$		0			0
$413$	−18.0776			−0.889539
$414$		0			0
$415$			7.03751i			0.345458i
$416$		0			0
$417$			1.06125i			0.0519696i
$418$		0			0
$419$	−1.68814			−0.0824712			−0.0412356	−	0.999149i	$-0.513129\pi$
$419$	−1.68814			−0.0824712			−0.0412356	+	0.999149i	$0.513129\pi$
$420$		0			0
$421$	15.9178			0.775788			0.387894	−	0.921704i	$-0.373203\pi$
$421$	15.9178			0.775788			0.387894	+	0.921704i	$0.373203\pi$
$422$		0			0
$423$		−	6.02499i		−	0.292945i
$424$		0			0
$425$	35.3596			1.71519
$426$		0			0
$427$	−18.7392			−0.906852
$428$		0			0
$429$			23.0193i			1.11138i
$430$		0			0
$431$	−12.2661			−0.590839			−0.295420	−	0.955368i	$-0.595459\pi$
$431$	−12.2661			−0.590839			−0.295420	+	0.955368i	$0.595459\pi$
$432$		0			0
$433$		−	22.2764i		−	1.07053i	−0.844683	−	0.535267i	$-0.820211\pi$
$433$		−	22.2764i		−	1.07053i	0.844683	−	0.535267i	$-0.179789\pi$
$434$		0			0
$435$	1.84407			0.0884165
$436$		0			0
$437$	−2.23568	+	0.469098i	−0.106947	+	0.0224400i
$438$		0			0
$439$	38.1887			1.82265			0.911323	−	0.411692i	$-0.135062\pi$
$439$	38.1887			1.82265			0.911323	+	0.411692i	$0.135062\pi$
$440$		0			0
$441$	2.81714			0.134150
$442$		0			0
$443$	−9.55632			−0.454035			−0.227017	−	0.973891i	$-0.572897\pi$
$443$	−9.55632			−0.454035			−0.227017	+	0.973891i	$0.572897\pi$
$444$		0			0
$445$	−42.8734			−2.03239
$446$		0			0
$447$	14.2997			0.676353
$448$		0			0
$449$		−	4.08318i		−	0.192697i	−0.995348	−	0.0963485i	$-0.969284\pi$
$449$		−	4.08318i		−	0.192697i	0.995348	−	0.0963485i	$-0.0307163\pi$
$450$		0			0
$451$			14.1275i			0.665237i
$452$		0			0
$453$			11.8196i			0.555333i
$454$		0			0
$455$	49.5916			2.32489
$456$		0			0
$457$	31.4996			1.47349			0.736745	−	0.676170i	$-0.236362\pi$
$457$	31.4996			1.47349			0.736745	+	0.676170i	$0.236362\pi$
$458$		0			0
$459$		−	26.6407i		−	1.24348i
$460$		0			0
$461$		−	11.6044i		−	0.540471i	−0.962794	−	0.270236i	$-0.912898\pi$
$461$		−	11.6044i		−	0.540471i	0.962794	−	0.270236i	$-0.0871016\pi$
$462$		0			0
$463$			33.7855i			1.57015i	0.619403	+	0.785073i	$0.287375\pi$
$463$			33.7855i			1.57015i	−0.619403	+	0.785073i	$0.712625\pi$
$464$		0			0
$465$	−5.33879			−0.247580
$466$		0			0
$467$	−30.6853			−1.41995			−0.709974	−	0.704228i	$-0.751293\pi$
$467$	−30.6853			−1.41995			−0.709974	+	0.704228i	$0.751293\pi$
$468$		0			0
$469$	21.8990			1.01120
$470$		0			0
$471$	−21.9750			−1.01255
$472$		0			0
$473$	−30.9351			−1.42240
$474$		0			0
$475$	6.60736	+	31.4901i	0.303166	+	1.44487i
$476$		0			0
$477$	8.53187			0.390648
$478$		0			0
$479$			30.4463i			1.39113i	0.718464	+	0.695564i	$0.244846\pi$
$479$			30.4463i			1.39113i	−0.718464	+	0.695564i	$0.755154\pi$
$480$		0			0
$481$	−37.1070			−1.69193
$482$		0			0
$483$		−	1.47832i		−	0.0672657i
$484$		0			0
$485$	7.70763			0.349986
$486$		0			0
$487$	−5.15029			−0.233382			−0.116691	−	0.993168i	$-0.537229\pi$
$487$	−5.15029			−0.233382			−0.116691	+	0.993168i	$0.537229\pi$
$488$		0			0
$489$			25.6640i			1.16056i
$490$		0			0
$491$	−20.3437			−0.918099			−0.459049	−	0.888411i	$-0.651810\pi$
$491$	−20.3437			−0.918099			−0.459049	+	0.888411i	$0.651810\pi$
$492$		0			0
$493$	−1.98999			−0.0896245
$494$		0			0
$495$			14.3494i			0.644956i
$496$		0			0
$497$			22.8083i			1.02309i
$498$		0			0
$499$	18.5776			0.831648			0.415824	−	0.909445i	$-0.363493\pi$
$499$	18.5776			0.831648			0.415824	+	0.909445i	$0.363493\pi$
$500$		0			0
$501$		−	17.4194i		−	0.778243i
$502$		0			0
$503$			28.6741i			1.27852i	0.768993	+	0.639258i	$0.220758\pi$
$503$			28.6741i			1.27852i	−0.768993	+	0.639258i	$0.779242\pi$
$504$		0			0
$505$	5.33879			0.237573
$506$		0			0
$507$			33.7145i			1.49731i
$508$		0			0
$509$	−26.8729			−1.19112			−0.595561	−	0.803310i	$-0.703070\pi$
$509$	−26.8729			−1.19112			−0.595561	+	0.803310i	$0.703070\pi$
$510$		0			0
$511$		−	17.3592i		−	0.767927i
$512$		0			0
$513$	23.7254	−	4.97813i	1.04750	−	0.219790i
$514$		0			0
$515$		−	2.92358i		−	0.128828i
$516$		0			0
$517$			12.3835i			0.544624i
$518$		0			0
$519$			3.65437i			0.160409i
$520$		0			0
$521$		−	19.3528i		−	0.847862i	−0.905695	−	0.423931i	$-0.860650\pi$
$521$		−	19.3528i		−	0.847862i	0.905695	−	0.423931i	$-0.139350\pi$
$522$		0			0
$523$		−	33.6806i		−	1.47275i	−0.676575	−	0.736374i	$-0.736536\pi$
$523$		−	33.6806i		−	1.47275i	0.676575	−	0.736374i	$-0.263464\pi$
$524$		0			0
$525$	−20.8224			−0.908766
$526$		0			0
$527$	5.76122			0.250963
$528$		0			0
$529$	22.7254			0.988059
$530$		0			0
$531$		−	11.3876i		−	0.494181i
$532$		0			0
$533$			30.7563i			1.33220i
$534$		0			0
$535$	−2.74488			−0.118671
$536$		0			0
$537$	−13.4245			−0.579310
$538$		0			0
$539$	−5.79021			−0.249402
$540$		0			0
$541$			17.2043i			0.739669i	0.929098	+	0.369834i	$0.120586\pi$
$541$			17.2043i			0.739669i	−0.929098	+	0.369834i	$0.879414\pi$
$542$		0			0
$543$		−	3.03447i		−	0.130222i
$544$		0			0
$545$			21.3135i			0.912970i
$546$		0			0
$547$		−	12.4185i		−	0.530976i	−0.964114	−	0.265488i	$-0.914467\pi$
$547$		−	12.4185i		−	0.530976i	0.964114	−	0.265488i	$-0.0855330\pi$
$548$		0			0
$549$		−	11.8044i		−	0.503800i
$550$		0			0
$551$	−0.371852	−	1.77221i	−0.0158414	−	0.0754989i
$552$		0			0
$553$		−	16.4033i		−	0.697539i
$554$		0			0
$555$	26.1339			1.10932
$556$		0			0
$557$			23.8931i			1.01238i	0.862421	+	0.506192i	$0.168947\pi$
$557$			23.8931i			1.01238i	−0.862421	+	0.506192i	$0.831053\pi$
$558$		0			0
$559$	−67.3475			−2.84850
$560$		0			0
$561$			17.4950i			0.738638i
$562$		0			0
$563$			5.26287i			0.221804i	0.993831	+	0.110902i	$0.0353739\pi$
$563$			5.26287i			0.221804i	−0.993831	+	0.110902i	$0.964626\pi$
$564$		0			0
$565$	12.6122			0.530600
$566$		0			0
$567$			6.23911i			0.262018i
$568$		0			0
$569$		−	42.6949i		−	1.78986i	−0.446202	−	0.894932i	$-0.647224\pi$
$569$		−	42.6949i		−	1.78986i	0.446202	−	0.894932i	$-0.352776\pi$
$570$		0			0
$571$	16.0539			0.671833			0.335917	−	0.941892i	$-0.390954\pi$
$571$	16.0539			0.671833			0.335917	+	0.941892i	$0.390954\pi$
$572$		0			0
$573$	12.6816			0.529780
$574$		0			0
$575$			3.86849i			0.161327i
$576$		0			0
$577$	8.58043			0.357208			0.178604	−	0.983921i	$-0.442842\pi$
$577$	8.58043			0.357208			0.178604	+	0.983921i	$0.442842\pi$
$578$		0			0
$579$	18.9731			0.788494
$580$		0			0
$581$			4.47214i			0.185535i
$582$		0			0
$583$	−17.5360			−0.726266
$584$		0			0
$585$			31.2393i			1.29159i
$586$		0			0
$587$	32.9213			1.35881			0.679404	−	0.733764i	$-0.262238\pi$
$587$	32.9213			1.35881			0.679404	+	0.733764i	$0.262238\pi$
$588$		0			0
$589$	1.07655	+	5.13075i	0.0443585	+	0.211409i
$590$		0			0
$591$	17.4164			0.716416
$592$		0			0
$593$	12.9731			0.532740			0.266370	−	0.963871i	$-0.414176\pi$
$593$	12.9731			0.532740			0.266370	+	0.963871i	$0.414176\pi$
$594$		0			0
$595$	37.6902			1.54515
$596$		0			0
$597$	−9.29338			−0.380353
$598$		0			0
$599$	−12.9775			−0.530245			−0.265122	−	0.964215i	$-0.585412\pi$
$599$	−12.9775			−0.530245			−0.265122	+	0.964215i	$0.585412\pi$
$600$		0			0
$601$		−	0.135889i		−	0.00554301i	−0.999996	−	0.00277150i	$-0.999118\pi$
$601$		−	0.135889i		−	0.00554301i	0.999996	−	0.00277150i	$-0.000882198\pi$
$602$		0			0
$603$			13.7949i			0.561771i
$604$		0			0
$605$			9.21336i			0.374576i
$606$		0			0
$607$	31.1052			1.26252			0.631261	−	0.775571i	$-0.282538\pi$
$607$	31.1052			1.26252			0.631261	+	0.775571i	$0.282538\pi$
$608$		0			0
$609$	1.17185			0.0474860
$610$		0			0
$611$			26.9595i			1.09066i
$612$		0			0
$613$		−	21.4073i		−	0.864633i	−0.901722	−	0.432316i	$-0.857696\pi$
$613$		−	21.4073i		−	0.864633i	0.901722	−	0.432316i	$-0.142304\pi$
$614$		0			0
$615$		−	21.6612i		−	0.873464i
$616$		0			0
$617$	−1.30650			−0.0525978			−0.0262989	−	0.999654i	$-0.508372\pi$
$617$	−1.30650			−0.0525978			−0.0262989	+	0.999654i	$0.508372\pi$
$618$		0			0
$619$	33.9780			1.36569			0.682845	−	0.730563i	$-0.260742\pi$
$619$	33.9780			1.36569			0.682845	+	0.730563i	$0.260742\pi$
$620$		0			0
$621$	2.91461			0.116959
$622$		0			0
$623$	−27.2448			−1.09154
$624$		0			0
$625$	−7.41957			−0.296783
$626$		0			0
$627$	−15.5804	+	3.26914i	−0.622222	+	0.130557i
$628$		0			0
$629$	−28.2018			−1.12448
$630$		0			0
$631$		−	7.50136i		−	0.298625i	−0.988790	−	0.149312i	$-0.952294\pi$
$631$		−	7.50136i		−	0.298625i	0.988790	−	0.149312i	$-0.0477060\pi$
$632$		0			0
$633$	19.7474			0.784887
$634$		0			0
$635$			59.9757i			2.38006i
$636$		0			0
$637$	−12.6056			−0.499452
$638$		0			0
$639$	−14.3677			−0.568378
$640$		0			0
$641$			1.22755i			0.0484852i	0.999706	+	0.0242426i	$0.00771741\pi$
$641$			1.22755i			0.0484852i	−0.999706	+	0.0242426i	$0.992283\pi$
$642$		0			0
$643$	−24.4755			−0.965221			−0.482610	−	0.875835i	$-0.660311\pi$
$643$	−24.4755			−0.965221			−0.482610	+	0.875835i	$0.660311\pi$
$644$		0			0
$645$	47.4319			1.86763
$646$		0			0
$647$		−	0.598399i		−	0.0235255i	−0.999931	−	0.0117627i	$-0.996256\pi$
$647$		−	0.598399i		−	0.0235255i	0.999931	−	0.0117627i	$-0.00374428\pi$
$648$		0			0
$649$			23.4056i			0.918749i
$650$		0			0
$651$	−3.39264			−0.132968
$652$		0			0
$653$			34.6646i			1.35653i	0.734818	+	0.678265i	$0.237268\pi$
$653$			34.6646i			1.35653i	−0.734818	+	0.678265i	$0.762732\pi$
$654$		0			0
$655$		−	41.2125i		−	1.61031i
$656$		0			0
$657$	10.9351			0.426620
$658$		0			0
$659$		−	11.5705i		−	0.450721i	−0.974275	−	0.225361i	$-0.927644\pi$
$659$		−	11.5705i		−	0.450721i	0.974275	−	0.225361i	$-0.0723560\pi$
$660$		0			0
$661$	10.3699			0.403343			0.201672	−	0.979453i	$-0.435363\pi$
$661$	10.3699			0.403343			0.201672	+	0.979453i	$0.435363\pi$
$662$		0			0
$663$			38.0875i			1.47920i
$664$		0			0
$665$	7.04286	+	33.5657i	0.273110	+	1.30162i
$666$		0			0
$667$		−	0.217713i		−	0.00842987i
$668$		0			0
$669$			14.3850i			0.556155i
$670$		0			0
$671$			24.2622i			0.936632i
$672$		0			0
$673$		−	9.16231i		−	0.353181i	−0.984284	−	0.176591i	$-0.943493\pi$
$673$		−	9.16231i		−	0.353181i	0.984284	−	0.176591i	$-0.0565068\pi$
$674$		0			0
$675$		−	41.0530i		−	1.58013i
$676$		0			0
$677$	30.1852			1.16011			0.580055	−	0.814577i	$-0.303031\pi$
$677$	30.1852			1.16011			0.580055	+	0.814577i	$0.303031\pi$
$678$		0			0
$679$	4.89798			0.187967
$680$		0			0
$681$	18.1988			0.697379
$682$		0			0
$683$		−	21.2151i		−	0.811775i	−0.913923	−	0.405887i	$-0.866963\pi$
$683$		−	21.2151i		−	0.811775i	0.913923	−	0.405887i	$-0.133037\pi$
$684$		0			0
$685$		−	3.32924i		−	0.127204i
$686$		0			0
$687$	−23.7694			−0.906860
$688$		0			0
$689$	−38.1768			−1.45442
$690$		0			0
$691$	9.52447			0.362328			0.181164	−	0.983453i	$-0.442014\pi$
$691$	9.52447			0.362328			0.181164	+	0.983453i	$0.442014\pi$
$692$		0			0
$693$			9.11860i			0.346387i
$694$		0			0
$695$		−	2.96014i		−	0.112285i
$696$		0			0
$697$			23.3752i			0.885398i
$698$		0			0
$699$			6.55836i			0.248060i
$700$		0			0
$701$		−	28.7291i		−	1.08508i	−0.840029	−	0.542541i	$-0.817462\pi$
$701$		−	28.7291i		−	1.08508i	0.840029	−	0.542541i	$-0.182538\pi$
$702$		0			0
$703$	−5.26983	−	25.1156i	−0.198755	−	0.947252i
$704$		0			0
$705$		−	18.9872i		−	0.715098i
$706$		0			0
$707$	3.39264			0.127594
$708$		0			0
$709$		−	16.3713i		−	0.614837i	−0.951574	−	0.307419i	$-0.900535\pi$
$709$		−	16.3713i		−	0.614837i	0.951574	−	0.307419i	$-0.0994652\pi$
$710$		0			0
$711$	10.3330			0.387517
$712$		0			0
$713$			0.630302i			0.0236050i
$714$		0			0
$715$		−	64.2078i		−	2.40123i
$716$		0			0
$717$	15.4265			0.576111
$718$		0			0
$719$		−	7.75634i		−	0.289263i	−0.989486	−	0.144631i	$-0.953800\pi$
$719$		−	7.75634i		−	0.289263i	0.989486	−	0.144631i	$-0.0461996\pi$
$720$		0			0
$721$		−	1.85785i		−	0.0691899i
$722$		0			0
$723$	−37.7364			−1.40343
$724$		0			0
$725$	−3.06654			−0.113888
$726$		0			0
$727$			34.3892i			1.27542i	0.770275	+	0.637712i	$0.220119\pi$
$727$			34.3892i			1.27542i	−0.770275	+	0.637712i	$0.779881\pi$
$728$		0			0
$729$	−24.9780			−0.925111
$730$		0			0
$731$	−51.1850			−1.89314
$732$		0			0
$733$		−	42.5350i		−	1.57107i	−0.618819	−	0.785533i	$-0.712389\pi$
$733$		−	42.5350i		−	1.57107i	0.618819	−	0.785533i	$-0.287611\pi$
$734$		0			0
$735$	8.87795			0.327468
$736$		0			0
$737$		−	28.3533i		−	1.04441i
$738$		0			0
$739$	−8.23389			−0.302889			−0.151444	−	0.988466i	$-0.548392\pi$
$739$	−8.23389			−0.302889			−0.151444	+	0.988466i	$0.548392\pi$
$740$		0			0
$741$	−33.9194	+	7.11709i	−1.24606	+	0.261453i
$742$		0			0
$743$	−0.459003			−0.0168392			−0.00841960	−	0.999965i	$-0.502680\pi$
$743$	−0.459003			−0.0168392			−0.00841960	+	0.999965i	$0.502680\pi$
$744$		0			0
$745$	−39.8862			−1.46132
$746$		0			0
$747$	−2.81714			−0.103074
$748$		0			0
$749$	−1.74429			−0.0637350
$750$		0			0
$751$	−4.47136			−0.163162			−0.0815812	−	0.996667i	$-0.525997\pi$
$751$	−4.47136			−0.163162			−0.0815812	+	0.996667i	$0.525997\pi$
$752$		0			0
$753$		−	26.9899i		−	0.983565i
$754$		0			0
$755$		−	32.9684i		−	1.19984i
$756$		0			0
$757$			21.8764i			0.795111i	0.917578	+	0.397556i	$0.130141\pi$
$757$			21.8764i			0.795111i	−0.917578	+	0.397556i	$0.869859\pi$
$758$		0			0
$759$	−1.91402			−0.0694746
$760$		0			0
$761$	13.2416			0.480009			0.240005	−	0.970772i	$-0.422851\pi$
$761$	13.2416			0.480009			0.240005	+	0.970772i	$0.422851\pi$
$762$		0			0
$763$			13.5441i			0.490330i
$764$		0			0
$765$			23.7423i			0.858405i
$766$		0			0
$767$			50.9552i			1.83989i
$768$		0			0
$769$	−4.84407			−0.174682			−0.0873409	−	0.996178i	$-0.527837\pi$
$769$	−4.84407			−0.174682			−0.0873409	+	0.996178i	$0.527837\pi$
$770$		0			0
$771$	−17.1290			−0.616886
$772$		0			0
$773$	−40.0525			−1.44059			−0.720294	−	0.693669i	$-0.755993\pi$
$773$	−40.0525			−1.44059			−0.720294	+	0.693669i	$0.755993\pi$
$774$		0			0
$775$	8.87795			0.318905
$776$		0			0
$777$	16.6074			0.595786
$778$		0			0
$779$	−20.8171	+	4.36792i	−0.745852	+	0.156497i
$780$		0			0
$781$	29.5307			1.05669
$782$		0			0
$783$			2.31040i			0.0825669i
$784$		0			0
$785$	61.2948			2.18770
$786$		0			0
$787$			1.18061i			0.0420842i	0.999779	+	0.0210421i	$0.00669840\pi$
$787$			1.18061i			0.0420842i	−0.999779	+	0.0210421i	$0.993302\pi$
$788$		0			0
$789$	−15.2567			−0.543154
$790$		0			0
$791$	8.01471			0.284970
$792$		0			0
$793$			52.8201i			1.87570i
$794$		0			0
$795$	26.8874			0.953596
$796$		0			0
$797$	47.0231			1.66564			0.832821	−	0.553542i	$-0.186724\pi$
$797$	47.0231			1.66564			0.832821	+	0.553542i	$0.186724\pi$
$798$		0			0
$799$			20.4895i			0.724868i
$800$		0			0
$801$		−	17.1624i		−	0.606402i
$802$		0			0
$803$	−22.4755			−0.793144
$804$		0			0
$805$			4.12347i			0.145333i
$806$		0			0
$807$			26.7920i			0.943122i
$808$		0			0
$809$	−22.8923			−0.804850			−0.402425	−	0.915453i	$-0.631833\pi$
$809$	−22.8923			−0.804850			−0.402425	+	0.915453i	$0.631833\pi$
$810$		0			0
$811$			7.45423i			0.261753i	0.991399	+	0.130877i	$0.0417792\pi$
$811$			7.45423i			0.261753i	−0.991399	+	0.130877i	$0.958221\pi$
$812$		0			0
$813$	20.3310			0.713041
$814$		0			0
$815$		−	71.5845i		−	2.50750i
$816$		0			0
$817$	−9.56450	−	45.5836i	−0.334619	−	1.59477i
$818$		0			0
$819$			19.8517i			0.693675i
$820$		0			0
$821$			51.2254i			1.78778i	0.448288	+	0.893889i	$0.352034\pi$
$821$			51.2254i			1.78778i	−0.448288	+	0.893889i	$0.647966\pi$
$822$		0			0
$823$		−	35.1273i		−	1.22446i	−0.790679	−	0.612231i	$-0.790272\pi$
$823$		−	35.1273i		−	1.22446i	0.790679	−	0.612231i	$-0.209728\pi$
$824$		0			0
$825$			26.9595i			0.938608i
$826$		0			0
$827$			20.5839i			0.715773i	0.933765	+	0.357886i	$0.116503\pi$
$827$			20.5839i			0.715773i	−0.933765	+	0.357886i	$0.883497\pi$
$828$		0			0
$829$	25.0415			0.869727			0.434863	−	0.900496i	$-0.356797\pi$
$829$	25.0415			0.869727			0.434863	+	0.900496i	$0.356797\pi$
$830$		0			0
$831$	5.03075			0.174515
$832$		0			0
$833$	−9.58043			−0.331942
$834$		0			0
$835$			48.5881i			1.68146i
$836$		0			0
$837$		−	6.68885i		−	0.231200i
$838$		0			0
$839$	40.2222			1.38863			0.694313	−	0.719673i	$-0.255708\pi$
$839$	40.2222			1.38863			0.694313	+	0.719673i	$0.255708\pi$
$840$		0			0
$841$	−28.8274			−0.994049
$842$		0			0
$843$	0.795137			0.0273860
$844$		0			0
$845$		−	94.0400i		−	3.23507i
$846$		0			0
$847$			5.85483i			0.201174i
$848$		0			0
$849$		−	39.1437i		−	1.34341i
$850$		0			0
$851$		−	3.08539i		−	0.105766i
$852$		0			0
$853$		−	29.2390i		−	1.00113i	−0.865700	−	0.500563i	$-0.833126\pi$
$853$		−	29.2390i		−	1.00113i	0.865700	−	0.500563i	$-0.166874\pi$
$854$		0			0
$855$	−21.1441	+	4.43652i	−0.723113	+	0.151726i
$856$		0			0
$857$		−	34.2639i		−	1.17043i	−0.810877	−	0.585217i	$-0.801010\pi$
$857$		−	34.2639i		−	1.17043i	0.810877	−	0.585217i	$-0.198990\pi$
$858$		0			0
$859$	−1.50246			−0.0512633			−0.0256317	−	0.999671i	$-0.508160\pi$
$859$	−1.50246			−0.0512633			−0.0256317	+	0.999671i	$0.508160\pi$
$860$		0			0
$861$		−	13.7651i		−	0.469112i
$862$		0			0
$863$	53.5392			1.82249			0.911247	−	0.411860i	$-0.135121\pi$
$863$	53.5392			1.82249			0.911247	+	0.411860i	$0.135121\pi$
$864$		0			0
$865$		−	10.1931i		−	0.346577i
$866$		0			0
$867$			7.50117i			0.254753i
$868$		0			0
$869$	−21.2379			−0.720445
$870$		0			0
$871$		−	61.7267i		−	2.09153i
$872$		0			0
$873$			3.08539i			0.104425i
$874$		0			0
$875$	18.7392			0.633500
$876$		0			0
$877$	−7.96451			−0.268943			−0.134471	−	0.990917i	$-0.542934\pi$
$877$	−7.96451			−0.268943			−0.134471	+	0.990917i	$0.542934\pi$
$878$		0			0
$879$		−	38.7074i		−	1.30557i
$880$		0			0
$881$	−0.779210			−0.0262523			−0.0131261	−	0.999914i	$-0.504178\pi$
$881$	−0.779210			−0.0262523			−0.0131261	+	0.999914i	$0.504178\pi$
$882$		0			0
$883$	−4.42167			−0.148801			−0.0744006	−	0.997228i	$-0.523704\pi$
$883$	−4.42167			−0.148801			−0.0744006	+	0.997228i	$0.523704\pi$
$884$		0			0
$885$		−	35.8870i		−	1.20633i
$886$		0			0
$887$	−15.3505			−0.515419			−0.257709	−	0.966222i	$-0.582968\pi$
$887$	−15.3505			−0.515419			−0.257709	+	0.966222i	$0.582968\pi$
$888$		0			0
$889$			38.1129i			1.27826i
$890$		0			0
$891$	8.07796			0.270622
$892$		0			0
$893$	−18.2473	+	3.82871i	−0.610622	+	0.128123i
$894$		0			0
$895$	37.4450			1.25165
$896$		0			0
$897$	−4.16693			−0.139130
$898$		0			0
$899$	−0.499637			−0.0166638
$900$		0			0
$901$	−29.0148			−0.966625
$902$		0			0
$903$	30.1416			1.00305
$904$		0			0
$905$			8.46406i			0.281355i
$906$		0			0
$907$		−	21.6756i		−	0.719726i	−0.933005	−	0.359863i	$-0.882823\pi$
$907$		−	21.6756i		−	0.719726i	0.933005	−	0.359863i	$-0.117177\pi$
$908$		0			0
$909$			2.13714i			0.0708843i
$910$		0			0
$911$	−20.9922			−0.695502			−0.347751	−	0.937587i	$-0.613054\pi$
$911$	−20.9922			−0.695502			−0.347751	+	0.937587i	$0.613054\pi$
$912$		0			0
$913$	5.79021			0.191628
$914$		0			0
$915$		−	37.2005i		−	1.22981i
$916$		0			0
$917$		−	26.1894i		−	0.864850i
$918$		0			0
$919$		−	12.7719i		−	0.421306i	−0.977561	−	0.210653i	$-0.932441\pi$
$919$		−	12.7719i		−	0.421306i	0.977561	−	0.210653i	$-0.0675591\pi$
$920$		0			0
$921$	19.7902			0.652110
$922$		0			0
$923$	64.2899			2.11613
$924$		0			0
$925$	−43.4585			−1.42891
$926$		0			0
$927$	1.17032			0.0384383
$928$		0			0
$929$	−9.11800			−0.299152			−0.149576	−	0.988750i	$-0.547791\pi$
$929$	−9.11800			−0.299152			−0.149576	+	0.988750i	$0.547791\pi$
$930$		0			0
$931$	−1.79021	−	8.53201i	−0.0586719	−	0.279625i
$932$		0			0
$933$	35.8268			1.17292
$934$		0			0
$935$		−	48.7987i		−	1.59589i
$936$		0			0
$937$	6.18286			0.201985			0.100993	−	0.994887i	$-0.467798\pi$
$937$	6.18286			0.201985			0.100993	+	0.994887i	$0.467798\pi$
$938$		0			0
$939$		−	6.98575i		−	0.227971i
$940$		0			0
$941$	18.9084			0.616397			0.308198	−	0.951322i	$-0.400274\pi$
$941$	18.9084			0.616397			0.308198	+	0.951322i	$0.400274\pi$
$942$		0			0
$943$	−2.55734			−0.0832785
$944$		0			0
$945$		−	43.7588i		−	1.42347i
$946$		0			0
$947$	33.2409			1.08018			0.540092	−	0.841606i	$-0.318389\pi$
$947$	33.2409			1.08018			0.540092	+	0.841606i	$0.318389\pi$
$948$		0			0
$949$	−48.9305			−1.58835
$950$		0			0
$951$		−	15.0682i		−	0.488620i
$952$		0			0
$953$			43.2269i			1.40026i	0.714018	+	0.700128i	$0.246874\pi$
$953$			43.2269i			1.40026i	−0.714018	+	0.700128i	$0.753126\pi$
$954$		0			0
$955$	−35.3727			−1.14463
$956$		0			0
$957$		−	1.51724i		−	0.0490453i
$958$		0			0
$959$		−	2.11564i		−	0.0683174i
$960$		0			0
$961$	−29.5535			−0.953339
$962$		0			0
$963$		−	1.09878i		−	0.0354078i
$964$		0			0
$965$	−52.9216			−1.70361
$966$		0			0
$967$		−	24.5774i		−	0.790356i	−0.918605	−	0.395178i	$-0.870683\pi$
$967$		−	24.5774i		−	0.790356i	0.918605	−	0.395178i	$-0.129317\pi$
$968$		0			0
$969$	−25.7792	+	5.40908i	−0.828148	+	0.173765i
$970$		0			0
$971$		−	31.4083i		−	1.00794i	−0.863721	−	0.503970i	$-0.831872\pi$
$971$		−	31.4083i		−	1.00794i	0.863721	−	0.503970i	$-0.168128\pi$
$972$		0			0
$973$		−	1.88109i		−	0.0603049i
$974$		0			0
$975$			58.6922i			1.87965i
$976$		0			0
$977$			36.9532i			1.18224i	0.806584	+	0.591120i	$0.201314\pi$
$977$			36.9532i			1.18224i	−0.806584	+	0.591120i	$0.798686\pi$
$978$		0			0
$979$			35.2747i			1.12738i
$980$		0			0
$981$	−8.53187			−0.272402
$982$		0			0
$983$	−31.1924			−0.994882			−0.497441	−	0.867498i	$-0.665727\pi$
$983$	−31.1924			−0.994882			−0.497441	+	0.867498i	$0.665727\pi$
$984$		0			0
$985$	−48.5797			−1.54788
$986$		0			0
$987$		−	12.0658i		−	0.384058i
$988$		0			0
$989$		−	5.59985i		−	0.178065i
$990$		0			0
$991$	44.0827			1.40033			0.700166	−	0.713980i	$-0.253109\pi$
$991$	44.0827			1.40033			0.700166	+	0.713980i	$0.253109\pi$
$992$		0			0
$993$	19.6013			0.622028
$994$		0			0
$995$	25.9220			0.821784
$996$		0			0
$997$		−	20.9382i		−	0.663119i	−0.943434	−	0.331560i	$-0.892425\pi$
$997$		−	20.9382i		−	0.663119i	0.943434	−	0.331560i	$-0.107575\pi$
$998$		0			0
$999$			32.7426i			1.03593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.2.b.c.303.9		12
3.2	odd	2		5472.2.e.e.5167.11		12
4.3	odd	2		152.2.b.c.75.10	yes	12
8.3	odd	2	inner	608.2.b.c.303.10		12
8.5	even	2		152.2.b.c.75.4	yes	12
12.11	even	2		1368.2.e.e.379.3		12
19.18	odd	2	inner	608.2.b.c.303.3		12
24.5	odd	2		1368.2.e.e.379.9		12
24.11	even	2		5472.2.e.e.5167.2		12
57.56	even	2		5472.2.e.e.5167.12		12
76.75	even	2		152.2.b.c.75.3	✓	12
152.37	odd	2		152.2.b.c.75.9	yes	12
152.75	even	2	inner	608.2.b.c.303.4		12
228.227	odd	2		1368.2.e.e.379.10		12
456.227	odd	2		5472.2.e.e.5167.1		12
456.341	even	2		1368.2.e.e.379.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.b.c.75.3	✓	12	76.75	even	2
152.2.b.c.75.4	yes	12	8.5	even	2
152.2.b.c.75.9	yes	12	152.37	odd	2
152.2.b.c.75.10	yes	12	4.3	odd	2
608.2.b.c.303.3		12	19.18	odd	2	inner
608.2.b.c.303.4		12	152.75	even	2	inner
608.2.b.c.303.9		12	1.1	even	1	trivial
608.2.b.c.303.10		12	8.3	odd	2	inner
1368.2.e.e.379.3		12	12.11	even	2
1368.2.e.e.379.4		12	456.341	even	2
1368.2.e.e.379.9		12	24.5	odd	2
1368.2.e.e.379.10		12	228.227	odd	2
5472.2.e.e.5167.1		12	456.227	odd	2
5472.2.e.e.5167.2		12	24.11	even	2
5472.2.e.e.5167.11		12	3.2	odd	2
5472.2.e.e.5167.12		12	57.56	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.26152i q^{3} -3.51876i q^{5} -2.23607i q^{7} +1.40857 q^{9} -2.89511 q^{11} -6.30281 q^{13} +4.43898 q^{15} -4.79021 q^{17} +(-0.895107 - 4.26600i) q^{19} +2.82084 q^{21} -0.524069i q^{23} -7.38164 q^{25} +5.56149i q^{27} +0.415427 q^{29} -1.20271 q^{31} -3.65223i q^{33} -7.86818 q^{35} +5.88738 q^{37} -7.95110i q^{39} -4.87978i q^{41} +10.6853 q^{43} -4.95642i q^{45} -4.27737i q^{47} +2.00000 q^{49} -6.04294i q^{51} +6.05711 q^{53} +10.1872i q^{55} +(5.38164 - 1.12919i) q^{57} -8.08453i q^{59} -8.38042i q^{61} -3.14966i q^{63} +22.1780i q^{65} +9.79353i q^{67} +0.661123 q^{69} -10.2002 q^{71} +7.76328 q^{73} -9.31208i q^{75} +6.47365i q^{77} +7.33578 q^{79} -2.79021 q^{81} -2.00000 q^{83} +16.8556i q^{85} +0.524069i q^{87} -12.1842i q^{89} +14.0935i q^{91} -1.51724i q^{93} +(-15.0110 + 3.14966i) q^{95} +2.19044i q^{97} -4.07796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9} + 12 q^{17} + 24 q^{19} - 44 q^{25} - 40 q^{35} + 24 q^{43} + 24 q^{49} + 20 q^{57} + 4 q^{73} + 36 q^{81} - 24 q^{83} - 64 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 + 12 * q^17 + 24 * q^19 - 44 * q^25 - 40 * q^35 + 24 * q^43 + 24 * q^49 + 20 * q^57 + 4 * q^73 + 36 * q^81 - 24 * q^83 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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