LMFDB - Embedded newform 608.4.b.b.303.41

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,4,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, 4, 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, 4); N := Newforms(S);

Level:	$N$	$=$	$608 = 2^{5} \cdot 19$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	608.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [56] 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:	$35.8731612835$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.41
Character	$\chi$	$=$	608.303
Dual form			608.4.b.b.303.16

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.03936i q^{3} -5.72550i q^{5} -3.30417i q^{7} +1.60489 q^{9} -18.7882 q^{11} -57.2347 q^{13} +28.8528 q^{15} +27.5535 q^{17} +(77.7299 + 28.5842i) q^{19} +16.6509 q^{21} -188.477i q^{23} +92.2187 q^{25} +144.150i q^{27} +116.587 q^{29} -20.6754 q^{31} -94.6802i q^{33} -18.9180 q^{35} +96.9185 q^{37} -288.426i q^{39} -186.285i q^{41} -240.906 q^{43} -9.18878i q^{45} -205.244i q^{47} +332.082 q^{49} +138.852i q^{51} +386.068 q^{53} +107.572i q^{55} +(-144.046 + 391.709i) q^{57} +23.1750i q^{59} +25.9773i q^{61} -5.30282i q^{63} +327.697i q^{65} -604.479i q^{67} +949.804 q^{69} +828.153 q^{71} +824.991 q^{73} +464.723i q^{75} +62.0793i q^{77} +678.227 q^{79} -683.092 q^{81} -156.548 q^{83} -157.757i q^{85} +587.522i q^{87} +372.489i q^{89} +189.113i q^{91} -104.191i q^{93} +(163.659 - 445.043i) q^{95} +1456.91i q^{97} -30.1529 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$56 q - 528 q^{9} + 40 q^{11} - 184 q^{17} + 84 q^{19} - 1504 q^{25} - 40 q^{35} - 576 q^{43} - 3664 q^{49} - 648 q^{57} - 432 q^{73} + 2296 q^{81} - 5376 q^{83} + 6152 q^{99}+O(q^{100})$ 56 * q - 528 * q^9 + 40 * q^11 - 184 * q^17 + 84 * q^19 - 1504 * q^25 - 40 * q^35 - 576 * q^43 - 3664 * q^49 - 648 * q^57 - 432 * q^73 + 2296 * q^81 - 5376 * q^83 + 6152 * 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
$3$			5.03936i			0.969825i	0.874563	+	0.484912i	$0.161148\pi$
$3$			5.03936i			0.969825i	−0.874563	+	0.484912i	$0.838852\pi$
$4$		0			0
$5$		−	5.72550i		−	0.512104i	−0.966663	−	0.256052i	$-0.917578\pi$
$5$		−	5.72550i		−	0.512104i	0.966663	−	0.256052i	$-0.0824219\pi$
$6$		0			0
$7$		−	3.30417i		−	0.178408i	−0.996013	−	0.0892042i	$-0.971568\pi$
$7$		−	3.30417i		−	0.178408i	0.996013	−	0.0892042i	$-0.0284324\pi$
$8$		0			0
$9$	1.60489			0.0594402
$10$		0			0
$11$	−18.7882			−0.514986			−0.257493	−	0.966280i	$-0.582896\pi$
$11$	−18.7882			−0.514986			−0.257493	+	0.966280i	$0.582896\pi$
$12$		0			0
$13$	−57.2347			−1.22108			−0.610540	−	0.791985i	$-0.709048\pi$
$13$	−57.2347			−1.22108			−0.610540	+	0.791985i	$0.709048\pi$
$14$		0			0
$15$	28.8528			0.496651
$16$		0			0
$17$	27.5535			0.393100			0.196550	−	0.980494i	$-0.437026\pi$
$17$	27.5535			0.393100			0.196550	+	0.980494i	$0.437026\pi$
$18$		0			0
$19$	77.7299	+	28.5842i	0.938551	+	0.345141i
$19$	77.7299	+	28.5842i	0.938551	+	0.345141i
$20$		0			0
$21$	16.6509			0.173025
$22$		0			0
$23$		−	188.477i		−	1.70871i	−0.519694	−	0.854353i	$-0.673954\pi$
$23$		−	188.477i		−	1.70871i	0.519694	−	0.854353i	$-0.326046\pi$
$24$		0			0
$25$	92.2187			0.737749
$26$		0			0
$27$			144.150i			1.02747i
$28$		0			0
$29$	116.587			0.746539			0.373269	−	0.927723i	$-0.378237\pi$
$29$	116.587			0.746539			0.373269	+	0.927723i	$0.378237\pi$
$30$		0			0
$31$	−20.6754			−0.119787			−0.0598937	−	0.998205i	$-0.519076\pi$
$31$	−20.6754			−0.119787			−0.0598937	+	0.998205i	$0.519076\pi$
$32$		0			0
$33$		−	94.6802i		−	0.499446i
$34$		0			0
$35$	−18.9180			−0.0913637
$36$		0			0
$37$	96.9185			0.430630			0.215315	−	0.976545i	$-0.430922\pi$
$37$	96.9185			0.430630			0.215315	+	0.976545i	$0.430922\pi$
$38$		0			0
$39$		−	288.426i		−	1.18423i
$40$		0			0
$41$		−	186.285i		−	0.709582i	−0.934946	−	0.354791i	$-0.884552\pi$
$41$		−	186.285i		−	0.709582i	0.934946	−	0.354791i	$-0.115448\pi$
$42$		0			0
$43$	−240.906			−0.854368			−0.427184	−	0.904165i	$-0.640494\pi$
$43$	−240.906			−0.854368			−0.427184	+	0.904165i	$0.640494\pi$
$44$		0			0
$45$		−	9.18878i		−	0.0304396i
$46$		0			0
$47$		−	205.244i		−	0.636978i	−0.947927	−	0.318489i	$-0.896825\pi$
$47$		−	205.244i		−	0.636978i	0.947927	−	0.318489i	$-0.103175\pi$
$48$		0			0
$49$	332.082			0.968170
$50$		0			0
$51$			138.852i			0.381238i
$52$		0			0
$53$	386.068			1.00057			0.500287	−	0.865859i	$-0.333228\pi$
$53$	386.068			1.00057			0.500287	+	0.865859i	$0.333228\pi$
$54$		0			0
$55$			107.572i			0.263726i
$56$		0			0
$57$	−144.046	+	391.709i	−0.334726	+	0.910230i
$58$		0			0
$59$			23.1750i			0.0511378i	0.999673	+	0.0255689i	$0.00813972\pi$
$59$			23.1750i			0.0511378i	−0.999673	+	0.0255689i	$0.991860\pi$
$60$		0			0
$61$			25.9773i			0.0545255i	0.999628	+	0.0272627i	$0.00867907\pi$
$61$			25.9773i			0.0545255i	−0.999628	+	0.0272627i	$0.991321\pi$
$62$		0			0
$63$		−	5.30282i		−	0.0106046i
$64$		0			0
$65$			327.697i			0.625320i
$66$		0			0
$67$		−	604.479i		−	1.10222i	−0.834432	−	0.551111i	$-0.814204\pi$
$67$		−	604.479i		−	1.10222i	0.834432	−	0.551111i	$-0.185796\pi$
$68$		0			0
$69$	949.804			1.65714
$70$		0			0
$71$	828.153			1.38428			0.692139	−	0.721765i	$-0.256669\pi$
$71$	828.153			1.38428			0.692139	+	0.721765i	$0.256669\pi$
$72$		0			0
$73$	824.991			1.32271			0.661356	−	0.750073i	$-0.269981\pi$
$73$	824.991			1.32271			0.661356	+	0.750073i	$0.269981\pi$
$74$		0			0
$75$			464.723i			0.715487i
$76$		0			0
$77$			62.0793i			0.0918778i
$78$		0			0
$79$	678.227			0.965905			0.482953	−	0.875647i	$-0.339564\pi$
$79$	678.227			0.965905			0.482953	+	0.875647i	$0.339564\pi$
$80$		0			0
$81$	−683.092			−0.937027
$82$		0			0
$83$	−156.548			−0.207029			−0.103514	−	0.994628i	$-0.533009\pi$
$83$	−156.548			−0.207029			−0.103514	+	0.994628i	$0.533009\pi$
$84$		0			0
$85$		−	157.757i		−	0.201308i
$86$		0			0
$87$			587.522i			0.724012i
$88$		0			0
$89$			372.489i			0.443637i	0.975088	+	0.221819i	$0.0711993\pi$
$89$			372.489i			0.443637i	−0.975088	+	0.221819i	$0.928801\pi$
$90$		0			0
$91$			189.113i			0.217851i
$92$		0			0
$93$		−	104.191i		−	0.116173i
$94$		0			0
$95$	163.659	−	445.043i	0.176748	−	0.480636i
$96$		0			0
$97$			1456.91i			1.52502i	0.646974	+	0.762512i	$0.276034\pi$
$97$			1456.91i			1.52502i	−0.646974	+	0.762512i	$0.723966\pi$
$98$		0			0
$99$	−30.1529			−0.0306109
$100$		0			0
$101$			743.632i			0.732615i	0.930494	+	0.366308i	$0.119378\pi$
$101$			743.632i			0.732615i	−0.930494	+	0.366308i	$0.880622\pi$
$102$		0			0
$103$	681.912			0.652338			0.326169	−	0.945311i	$-0.394242\pi$
$103$	681.912			0.652338			0.326169	+	0.945311i	$0.394242\pi$
$104$		0			0
$105$		−	95.3346i		−	0.0886067i
$106$		0			0
$107$			1011.28i			0.913684i	0.889548	+	0.456842i	$0.151020\pi$
$107$			1011.28i			0.913684i	−0.889548	+	0.456842i	$0.848980\pi$
$108$		0			0
$109$	818.210			0.718994			0.359497	−	0.933146i	$-0.382948\pi$
$109$	818.210			0.718994			0.359497	+	0.933146i	$0.382948\pi$
$110$		0			0
$111$			488.407i			0.417636i
$112$		0			0
$113$		−	223.680i		−	0.186213i	−0.995656	−	0.0931064i	$-0.970320\pi$
$113$		−	223.680i		−	0.186213i	0.995656	−	0.0931064i	$-0.0296797\pi$
$114$		0			0
$115$	−1079.13			−0.875035
$116$		0			0
$117$	−91.8552			−0.0725813
$118$		0			0
$119$		−	91.0414i		−	0.0701323i
$120$		0			0
$121$	−978.005			−0.734790
$122$		0			0
$123$	938.758			0.688170
$124$		0			0
$125$		−	1243.69i		−	0.889909i
$126$		0			0
$127$	706.530			0.493657			0.246828	−	0.969059i	$-0.420612\pi$
$127$	706.530			0.493657			0.246828	+	0.969059i	$0.420612\pi$
$128$		0			0
$129$		−	1214.01i		−	0.828587i
$130$		0			0
$131$	2375.61			1.58441			0.792207	−	0.610253i	$-0.208932\pi$
$131$	2375.61			1.58441			0.792207	+	0.610253i	$0.208932\pi$
$132$		0			0
$133$	94.4472	−	256.833i	0.0615760	−	0.167445i
$134$		0			0
$135$	825.332			0.526172
$136$		0			0
$137$	−1067.76			−0.665878			−0.332939	−	0.942948i	$-0.608040\pi$
$137$	−1067.76			−0.665878			−0.332939	+	0.942948i	$0.608040\pi$
$138$		0			0
$139$	2150.99			1.31255			0.656276	−	0.754521i	$-0.272131\pi$
$139$	2150.99			1.31255			0.656276	+	0.754521i	$0.272131\pi$
$140$		0			0
$141$	1034.30			0.617757
$142$		0			0
$143$	1075.33			0.628839
$144$		0			0
$145$		−	667.518i		−	0.382306i
$146$		0			0
$147$			1673.48i			0.938956i
$148$		0			0
$149$		−	484.443i		−	0.266356i	−0.991092	−	0.133178i	$-0.957482\pi$
$149$		−	484.443i		−	0.266356i	0.991092	−	0.133178i	$-0.0425183\pi$
$150$		0			0
$151$	−2021.09			−1.08923			−0.544616	−	0.838686i	$-0.683324\pi$
$151$	−2021.09			−1.08923			−0.544616	+	0.838686i	$0.683324\pi$
$152$		0			0
$153$	44.2202			0.0233660
$154$		0			0
$155$			118.377i			0.0613436i
$156$		0			0
$157$		−	3373.68i		−	1.71496i	−0.514516	−	0.857481i	$-0.672028\pi$
$157$		−	3373.68i		−	1.71496i	0.514516	−	0.857481i	$-0.327972\pi$
$158$		0			0
$159$			1945.53i			0.970382i
$160$		0			0
$161$	−622.761			−0.304847
$162$		0			0
$163$	−1488.88			−0.715450			−0.357725	−	0.933827i	$-0.616447\pi$
$163$	−1488.88			−0.715450			−0.357725	+	0.933827i	$0.616447\pi$
$164$		0			0
$165$	−542.092			−0.255768
$166$		0			0
$167$	12.3994			0.00574548			0.00287274	−	0.999996i	$-0.499086\pi$
$167$	12.3994			0.00574548			0.00287274	+	0.999996i	$0.499086\pi$
$168$		0			0
$169$	1078.81			0.491038
$170$		0			0
$171$	124.748	+	45.8745i	0.0557877	+	0.0205153i
$172$		0			0
$173$	−273.183			−0.120056			−0.0600280	−	0.998197i	$-0.519119\pi$
$173$	−273.183			−0.120056			−0.0600280	+	0.998197i	$0.519119\pi$
$174$		0			0
$175$		−	304.706i		−	0.131621i
$176$		0			0
$177$	−116.787			−0.0495947
$178$		0			0
$179$		−	3358.87i		−	1.40253i	−0.712899	−	0.701267i	$-0.752618\pi$
$179$		−	3358.87i		−	1.40253i	0.712899	−	0.701267i	$-0.247382\pi$
$180$		0			0
$181$	−2753.17			−1.13062			−0.565308	−	0.824880i	$-0.691243\pi$
$181$	−2753.17			−1.13062			−0.565308	+	0.824880i	$0.691243\pi$
$182$		0			0
$183$	−130.909			−0.0528801
$184$		0			0
$185$		−	554.907i		−	0.220527i
$186$		0			0
$187$	−517.679			−0.202441
$188$		0			0
$189$	476.297			0.183309
$190$		0			0
$191$		−	1742.05i		−	0.659950i	−0.943990	−	0.329975i	$-0.892960\pi$
$191$		−	1742.05i		−	0.659950i	0.943990	−	0.329975i	$-0.107040\pi$
$192$		0			0
$193$			1861.80i			0.694379i	0.937795	+	0.347189i	$0.112864\pi$
$193$			1861.80i			0.694379i	−0.937795	+	0.347189i	$0.887136\pi$
$194$		0			0
$195$	−1651.38			−0.606451
$196$		0			0
$197$		−	3038.48i		−	1.09890i	−0.835528	−	0.549448i	$-0.814838\pi$
$197$		−	3038.48i		−	1.09890i	0.835528	−	0.549448i	$-0.185162\pi$
$198$		0			0
$199$		−	3019.80i		−	1.07572i	−0.843035	−	0.537859i	$-0.819233\pi$
$199$		−	3019.80i		−	1.07572i	0.843035	−	0.537859i	$-0.180767\pi$
$200$		0			0
$201$	3046.19			1.06896
$202$		0			0
$203$		−	385.223i		−	0.133189i
$204$		0			0
$205$	−1066.58			−0.363380
$206$		0			0
$207$		−	302.485i		−	0.101566i
$208$		0			0
$209$	−1460.40	−	537.045i	−0.483340	−	0.177743i
$210$		0			0
$211$		−	5180.94i		−	1.69038i	−0.534463	−	0.845192i	$-0.679486\pi$
$211$		−	5180.94i		−	1.69038i	0.534463	−	0.845192i	$-0.320514\pi$
$212$		0			0
$213$			4173.36i			1.34251i
$214$		0			0
$215$			1379.31i			0.437525i
$216$		0			0
$217$			68.3150i			0.0213711i
$218$		0			0
$219$			4157.42i			1.28280i
$220$		0			0
$221$	−1577.02			−0.480007
$222$		0			0
$223$	−6049.57			−1.81663			−0.908317	−	0.418283i	$-0.862632\pi$
$223$	−6049.57			−1.81663			−0.908317	+	0.418283i	$0.862632\pi$
$224$		0			0
$225$	148.001			0.0438520
$226$		0			0
$227$		−	3412.80i		−	0.997867i	−0.866640	−	0.498933i	$-0.833725\pi$
$227$		−	3412.80i		−	0.997867i	0.866640	−	0.498933i	$-0.166275\pi$
$228$		0			0
$229$			5741.64i			1.65685i	0.560101	+	0.828424i	$0.310762\pi$
$229$			5741.64i			1.65685i	−0.560101	+	0.828424i	$0.689238\pi$
$230$		0			0
$231$	−312.840			−0.0891053
$232$		0			0
$233$	2255.67			0.634223			0.317112	−	0.948388i	$-0.397287\pi$
$233$	2255.67			0.634223			0.317112	+	0.948388i	$0.397287\pi$
$234$		0			0
$235$	−1175.13			−0.326199
$236$		0			0
$237$			3417.83i			0.936759i
$238$		0			0
$239$			1011.60i			0.273785i	0.990586	+	0.136893i	$0.0437115\pi$
$239$			1011.60i			0.273785i	−0.990586	+	0.136893i	$0.956288\pi$
$240$		0			0
$241$			4313.46i			1.15292i	0.817125	+	0.576461i	$0.195567\pi$
$241$			4313.46i			1.15292i	−0.817125	+	0.576461i	$0.804433\pi$
$242$		0			0
$243$			449.710i			0.118720i
$244$		0			0
$245$		−	1901.34i		−	0.495804i
$246$		0			0
$247$	−4448.85	−	1636.01i	−1.14605	−	0.421445i
$248$		0			0
$249$		−	788.903i		−	0.200782i
$250$		0			0
$251$	1821.03			0.457938			0.228969	−	0.973434i	$-0.426465\pi$
$251$	1821.03			0.457938			0.228969	+	0.973434i	$0.426465\pi$
$252$		0			0
$253$			3541.14i			0.879959i
$254$		0			0
$255$	794.996			0.195234
$256$		0			0
$257$		−	3365.51i		−	0.816867i	−0.912788	−	0.408434i	$-0.866075\pi$
$257$		−	3365.51i		−	0.816867i	0.912788	−	0.408434i	$-0.133925\pi$
$258$		0			0
$259$		−	320.235i		−	0.0768280i
$260$		0			0
$261$	187.109			0.0443745
$262$		0			0
$263$			1894.24i			0.444122i	0.975033	+	0.222061i	$0.0712784\pi$
$263$			1894.24i			0.444122i	−0.975033	+	0.222061i	$0.928722\pi$
$264$		0			0
$265$		−	2210.43i		−	0.512398i
$266$		0			0
$267$	−1877.10			−0.430250
$268$		0			0
$269$	−5932.53			−1.34466			−0.672329	−	0.740253i	$-0.734706\pi$
$269$	−5932.53			−1.34466			−0.672329	+	0.740253i	$0.734706\pi$
$270$		0			0
$271$			1212.08i			0.271693i	0.990730	+	0.135847i	$0.0433754\pi$
$271$			1212.08i			0.271693i	−0.990730	+	0.135847i	$0.956625\pi$
$272$		0			0
$273$	−953.009			−0.211277
$274$		0			0
$275$	−1732.62			−0.379930
$276$		0			0
$277$		−	1324.62i		−	0.287325i	−0.989627	−	0.143662i	$-0.954112\pi$
$277$		−	1324.62i		−	0.287325i	0.989627	−	0.143662i	$-0.0458879\pi$
$278$		0			0
$279$	−33.1817			−0.00712019
$280$		0			0
$281$		−	2675.75i		−	0.568050i	−0.958817	−	0.284025i	$-0.908330\pi$
$281$		−	2675.75i		−	0.568050i	0.958817	−	0.284025i	$-0.0916699\pi$
$282$		0			0
$283$	−2733.04			−0.574073			−0.287036	−	0.957920i	$-0.592670\pi$
$283$	−2733.04			−0.574073			−0.287036	+	0.957920i	$0.592670\pi$
$284$		0			0
$285$	2242.73	+	824.736i	0.466132	+	0.171415i
$286$		0			0
$287$	−615.518			−0.126595
$288$		0			0
$289$	−4153.81			−0.845472
$290$		0			0
$291$	−7341.91			−1.47901
$292$		0			0
$293$	2897.56			0.577737			0.288869	−	0.957369i	$-0.406721\pi$
$293$	2897.56			0.577737			0.288869	+	0.957369i	$0.406721\pi$
$294$		0			0
$295$	132.689			0.0261879
$296$		0			0
$297$		−	2708.32i		−	0.529133i
$298$		0			0
$299$			10787.4i			2.08647i
$300$		0			0
$301$			795.994i			0.152426i
$302$		0			0
$303$	−3747.43			−0.710508
$304$		0			0
$305$	148.733			0.0279227
$306$		0			0
$307$			578.868i			0.107615i	0.998551	+	0.0538074i	$0.0171357\pi$
$307$			578.868i			0.107615i	−0.998551	+	0.0538074i	$0.982864\pi$
$308$		0			0
$309$			3436.40i			0.632653i
$310$		0			0
$311$			8292.12i			1.51191i	0.654625	+	0.755954i	$0.272826\pi$
$311$			8292.12i			1.51191i	−0.654625	+	0.755954i	$0.727174\pi$
$312$		0			0
$313$	8713.54			1.57354			0.786771	−	0.617245i	$-0.211751\pi$
$313$	8713.54			1.57354			0.786771	+	0.617245i	$0.211751\pi$
$314$		0			0
$315$	−30.3613			−0.00543068
$316$		0			0
$317$	−4416.74			−0.782551			−0.391275	−	0.920274i	$-0.627966\pi$
$317$	−4416.74			−0.782551			−0.391275	+	0.920274i	$0.627966\pi$
$318$		0			0
$319$	−2190.45			−0.384457
$320$		0			0
$321$	−5096.21			−0.886113
$322$		0			0
$323$	2141.73	+	787.595i	0.368944	+	0.135675i
$324$		0			0
$325$	−5278.11			−0.900851
$326$		0			0
$327$			4123.25i			0.697298i
$328$		0			0
$329$	−678.162			−0.113642
$330$		0			0
$331$		−	4318.95i		−	0.717194i	−0.933493	−	0.358597i	$-0.883255\pi$
$331$		−	4318.95i		−	0.717194i	0.933493	−	0.358597i	$-0.116745\pi$
$332$		0			0
$333$	155.543			0.0255968
$334$		0			0
$335$	−3460.95			−0.564453
$336$		0			0
$337$		−	9523.11i		−	1.53934i	−0.638443	−	0.769669i	$-0.720421\pi$
$337$		−	9523.11i		−	1.53934i	0.638443	−	0.769669i	$-0.279579\pi$
$338$		0			0
$339$	1127.20			0.180594
$340$		0			0
$341$	388.453			0.0616888
$342$		0			0
$343$		−	2230.59i		−	0.351138i
$344$		0			0
$345$		−	5438.10i		−	0.848631i
$346$		0			0
$347$	8490.08			1.31346			0.656731	−	0.754125i	$-0.271939\pi$
$347$	8490.08			1.31346			0.656731	+	0.754125i	$0.271939\pi$
$348$		0			0
$349$			11160.9i			1.71184i	0.517110	+	0.855919i	$0.327008\pi$
$349$			11160.9i			1.71184i	−0.517110	+	0.855919i	$0.672992\pi$
$350$		0			0
$351$		−	8250.39i		−	1.25463i
$352$		0			0
$353$	−6494.13			−0.979172			−0.489586	−	0.871955i	$-0.662852\pi$
$353$	−6494.13			−0.979172			−0.489586	+	0.871955i	$0.662852\pi$
$354$		0			0
$355$		−	4741.59i		−	0.708894i
$356$		0			0
$357$	458.790			0.0680161
$358$		0			0
$359$		−	2262.40i		−	0.332604i	−0.986075	−	0.166302i	$-0.946817\pi$
$359$		−	2262.40i		−	0.332604i	0.986075	−	0.166302i	$-0.0531827\pi$
$360$		0			0
$361$	5224.88	+	4443.70i	0.761756	+	0.647864i
$362$		0			0
$363$		−	4928.52i		−	0.712617i
$364$		0			0
$365$		−	4723.49i		−	0.677366i
$366$		0			0
$367$			2224.41i			0.316385i	0.987408	+	0.158192i	$0.0505666\pi$
$367$			2224.41i			0.316385i	−0.987408	+	0.158192i	$0.949433\pi$
$368$		0			0
$369$		−	298.967i		−	0.0421777i
$370$		0			0
$371$		−	1275.63i		−	0.178511i
$372$		0			0
$373$	−63.6398			−0.00883417			−0.00441708	−	0.999990i	$-0.501406\pi$
$373$	−63.6398			−0.00883417			−0.00441708	+	0.999990i	$0.501406\pi$
$374$		0			0
$375$	6267.37			0.863055
$376$		0			0
$377$	−6672.81			−0.911584
$378$		0			0
$379$			3252.80i			0.440858i	0.975403	+	0.220429i	$0.0707458\pi$
$379$			3252.80i			0.440858i	−0.975403	+	0.220429i	$0.929254\pi$
$380$		0			0
$381$			3560.46i			0.478760i
$382$		0			0
$383$	4404.95			0.587682			0.293841	−	0.955854i	$-0.405066\pi$
$383$	4404.95			0.587682			0.293841	+	0.955854i	$0.405066\pi$
$384$		0			0
$385$	355.435			0.0470510
$386$		0			0
$387$	−386.627			−0.0507838
$388$		0			0
$389$		−	10114.5i		−	1.31831i	−0.752006	−	0.659156i	$-0.770914\pi$
$389$		−	10114.5i		−	1.31831i	0.752006	−	0.659156i	$-0.229086\pi$
$390$		0			0
$391$		−	5193.20i		−	0.671692i
$392$		0			0
$393$			11971.6i			1.53660i
$394$		0			0
$395$		−	3883.19i		−	0.494644i
$396$		0			0
$397$		−	15279.7i		−	1.93165i	−0.259186	−	0.965827i	$-0.583454\pi$
$397$		−	15279.7i		−	1.93165i	0.259186	−	0.965827i	$-0.416546\pi$
$398$		0			0
$399$	1294.27	+	475.953i	0.162393	+	0.0597179i
$400$		0			0
$401$			14208.1i			1.76937i	0.466190	+	0.884684i	$0.345626\pi$
$401$			14208.1i			1.76937i	−0.466190	+	0.884684i	$0.654374\pi$
$402$		0			0
$403$	1183.35			0.146270
$404$		0			0
$405$			3911.04i			0.479855i
$406$		0			0
$407$	−1820.92			−0.221768
$408$		0			0
$409$		−	11209.6i		−	1.35520i	−0.735430	−	0.677601i	$-0.763020\pi$
$409$		−	11209.6i		−	1.35520i	0.735430	−	0.677601i	$-0.236980\pi$
$410$		0			0
$411$		−	5380.85i		−	0.645785i
$412$		0			0
$413$	76.5742			0.00912341
$414$		0			0
$415$			896.317i			0.106020i
$416$		0			0
$417$			10839.6i			1.27295i
$418$		0			0
$419$	4914.82			0.573042			0.286521	−	0.958074i	$-0.407501\pi$
$419$	4914.82			0.573042			0.286521	+	0.958074i	$0.407501\pi$
$420$		0			0
$421$	−6104.34			−0.706668			−0.353334	−	0.935497i	$-0.614952\pi$
$421$	−6104.34			−0.706668			−0.353334	+	0.935497i	$0.614952\pi$
$422$		0			0
$423$		−	329.394i		−	0.0378621i
$424$		0			0
$425$	2540.95			0.290009
$426$		0			0
$427$	85.8334			0.00972780
$428$		0			0
$429$			5418.99i			0.609864i
$430$		0			0
$431$	11052.2			1.23519			0.617595	−	0.786497i	$-0.288107\pi$
$431$	11052.2			1.23519			0.617595	+	0.786497i	$0.288107\pi$
$432$		0			0
$433$		−	4407.06i		−	0.489122i	−0.969634	−	0.244561i	$-0.921356\pi$
$433$		−	4407.06i		−	0.489122i	0.969634	−	0.244561i	$-0.0786438\pi$
$434$		0			0
$435$	3363.86			0.370769
$436$		0			0
$437$	5387.48	−	14650.3i	0.589744	−	1.60371i
$438$		0			0
$439$	−14788.9			−1.60782			−0.803911	−	0.594750i	$-0.797251\pi$
$439$	−14788.9			−1.60782			−0.803911	+	0.594750i	$0.797251\pi$
$440$		0			0
$441$	532.955			0.0575483
$442$		0			0
$443$	−13668.3			−1.46592			−0.732958	−	0.680274i	$-0.761861\pi$
$443$	−13668.3			−1.46592			−0.732958	+	0.680274i	$0.761861\pi$
$444$		0			0
$445$	2132.68			0.227189
$446$		0			0
$447$	2441.28			0.258319
$448$		0			0
$449$			4433.66i			0.466007i	0.972476	+	0.233003i	$0.0748554\pi$
$449$			4433.66i			0.466007i	−0.972476	+	0.233003i	$0.925145\pi$
$450$		0			0
$451$			3499.96i			0.365425i
$452$		0			0
$453$		−	10185.0i		−	1.05636i
$454$		0			0
$455$	1082.77			0.111562
$456$		0			0
$457$	8045.08			0.823486			0.411743	−	0.911300i	$-0.364920\pi$
$457$	8045.08			0.823486			0.411743	+	0.911300i	$0.364920\pi$
$458$		0			0
$459$			3971.84i			0.403899i
$460$		0			0
$461$		−	11671.8i		−	1.17919i	−0.807698	−	0.589596i	$-0.799287\pi$
$461$		−	11671.8i		−	1.17919i	0.807698	−	0.589596i	$-0.200713\pi$
$462$		0			0
$463$			6795.37i			0.682090i	0.940047	+	0.341045i	$0.110781\pi$
$463$			6795.37i			0.682090i	−0.940047	+	0.341045i	$0.889219\pi$
$464$		0			0
$465$	−596.544			−0.0594926
$466$		0			0
$467$	7486.56			0.741834			0.370917	−	0.928666i	$-0.379043\pi$
$467$	7486.56			0.741834			0.370917	+	0.928666i	$0.379043\pi$
$468$		0			0
$469$	−1997.30			−0.196646
$470$		0			0
$471$	17001.2			1.66321
$472$		0			0
$473$	4526.18			0.439987
$474$		0			0
$475$	7168.15	+	2636.00i	0.692415	+	0.254627i
$476$		0			0
$477$	619.595			0.0594744
$478$		0			0
$479$			15766.9i			1.50398i	0.659175	+	0.751990i	$0.270906\pi$
$479$			15766.9i			1.50398i	−0.659175	+	0.751990i	$0.729094\pi$
$480$		0			0
$481$	−5547.10			−0.525834
$482$		0			0
$483$		−	3138.31i		−	0.295648i
$484$		0			0
$485$	8341.56			0.780971
$486$		0			0
$487$	−5904.44			−0.549396			−0.274698	−	0.961531i	$-0.588578\pi$
$487$	−5904.44			−0.549396			−0.274698	+	0.961531i	$0.588578\pi$
$488$		0			0
$489$		−	7503.01i		−	0.693861i
$490$		0			0
$491$	15452.6			1.42029			0.710147	−	0.704053i	$-0.248628\pi$
$491$	15452.6			1.42029			0.710147	+	0.704053i	$0.248628\pi$
$492$		0			0
$493$	3212.37			0.293464
$494$		0			0
$495$			172.640i			0.0156760i
$496$		0			0
$497$		−	2736.36i		−	0.246967i
$498$		0			0
$499$	−600.472			−0.0538694			−0.0269347	−	0.999637i	$-0.508575\pi$
$499$	−600.472			−0.0538694			−0.0269347	+	0.999637i	$0.508575\pi$
$500$		0			0
$501$			62.4850i			0.00557210i
$502$		0			0
$503$		−	11018.5i		−	0.976725i	−0.872641	−	0.488362i	$-0.837594\pi$
$503$		−	11018.5i		−	0.976725i	0.872641	−	0.488362i	$-0.162406\pi$
$504$		0			0
$505$	4257.66			0.375175
$506$		0			0
$507$			5436.51i			0.476221i
$508$		0			0
$509$	19114.7			1.66452			0.832262	−	0.554382i	$-0.187045\pi$
$509$	19114.7			1.66452			0.832262	+	0.554382i	$0.187045\pi$
$510$		0			0
$511$		−	2725.91i		−	0.235983i
$512$		0			0
$513$	−4120.42	+	11204.8i	−0.354622	+	0.964334i
$514$		0			0
$515$		−	3904.29i		−	0.334065i
$516$		0			0
$517$			3856.16i			0.328034i
$518$		0			0
$519$		−	1376.66i		−	0.116433i
$520$		0			0
$521$			19202.9i			1.61477i	0.590028	+	0.807383i	$0.299117\pi$
$521$			19202.9i			1.61477i	−0.590028	+	0.807383i	$0.700883\pi$
$522$		0			0
$523$			6024.44i			0.503691i	0.967767	+	0.251846i	$0.0810375\pi$
$523$			6024.44i			0.503691i	−0.967767	+	0.251846i	$0.918962\pi$
$524$		0			0
$525$	1535.52			0.127649
$526$		0			0
$527$	−569.679			−0.0470884
$528$		0			0
$529$	−23356.7			−1.91967
$530$		0			0
$531$			37.1933i			0.00303964i
$532$		0			0
$533$			10662.0i			0.866457i
$534$		0			0
$535$	5790.09			0.467902
$536$		0			0
$537$	16926.5			1.36021
$538$		0			0
$539$	−6239.22			−0.498594
$540$		0			0
$541$			1031.76i			0.0819938i	0.999159	+	0.0409969i	$0.0130534\pi$
$541$			1031.76i			0.0819938i	−0.999159	+	0.0409969i	$0.986947\pi$
$542$		0			0
$543$		−	13874.2i		−	1.09650i
$544$		0			0
$545$		−	4684.66i		−	0.368200i
$546$		0			0
$547$			15015.5i			1.17371i	0.809693	+	0.586853i	$0.199633\pi$
$547$			15015.5i			1.17371i	−0.809693	+	0.586853i	$0.800367\pi$
$548$		0			0
$549$			41.6906i			0.00324101i
$550$		0			0
$551$	9062.28	+	3332.55i	0.700665	+	0.257661i
$552$		0			0
$553$		−	2240.98i		−	0.172326i
$554$		0			0
$555$	2796.37			0.213873
$556$		0			0
$557$			12817.8i			0.975062i	0.873106	+	0.487531i	$0.162102\pi$
$557$			12817.8i			0.975062i	−0.873106	+	0.487531i	$0.837898\pi$
$558$		0			0
$559$	13788.2			1.04325
$560$		0			0
$561$		−	2608.77i		−	0.196332i
$562$		0			0
$563$			8471.22i			0.634137i	0.948403	+	0.317069i	$0.102699\pi$
$563$			8471.22i			0.634137i	−0.948403	+	0.317069i	$0.897301\pi$
$564$		0			0
$565$	−1280.68			−0.0953604
$566$		0			0
$567$			2257.05i			0.167173i
$568$		0			0
$569$		−	6730.08i		−	0.495852i	−0.968779	−	0.247926i	$-0.920251\pi$
$569$		−	6730.08i		−	0.495852i	0.968779	−	0.247926i	$-0.0797489\pi$
$570$		0			0
$571$	−20384.8			−1.49400			−0.747001	−	0.664822i	$-0.768507\pi$
$571$	−20384.8			−1.49400			−0.747001	+	0.664822i	$0.768507\pi$
$572$		0			0
$573$	8778.82			0.640036
$574$		0			0
$575$		−	17381.1i		−	1.26060i
$576$		0			0
$577$	−8621.00			−0.622005			−0.311002	−	0.950409i	$-0.600665\pi$
$577$	−8621.00			−0.622005			−0.311002	+	0.950409i	$0.600665\pi$
$578$		0			0
$579$	−9382.26			−0.673425
$580$		0			0
$581$			517.262i			0.0369357i
$582$		0			0
$583$	−7253.50			−0.515282
$584$		0			0
$585$			525.917i			0.0371692i
$586$		0			0
$587$	9852.46			0.692768			0.346384	−	0.938093i	$-0.387409\pi$
$587$	9852.46			0.692768			0.346384	+	0.938093i	$0.387409\pi$
$588$		0			0
$589$	−1607.10	−	590.990i	−0.112427	−	0.0413435i
$590$		0			0
$591$	15312.0			1.06574
$592$		0			0
$593$	−15120.2			−1.04707			−0.523536	−	0.852004i	$-0.675387\pi$
$593$	−15120.2			−1.04707			−0.523536	+	0.852004i	$0.675387\pi$
$594$		0			0
$595$	−521.257			−0.0359151
$596$		0			0
$597$	15217.8			1.04326
$598$		0			0
$599$	22742.6			1.55131			0.775657	−	0.631155i	$-0.217419\pi$
$599$	22742.6			1.55131			0.775657	+	0.631155i	$0.217419\pi$
$600$		0			0
$601$		−	24084.6i		−	1.63466i	−0.576169	−	0.817331i	$-0.695453\pi$
$601$		−	24084.6i		−	1.63466i	0.576169	−	0.817331i	$-0.304547\pi$
$602$		0			0
$603$		−	970.121i		−	0.0655164i
$604$		0			0
$605$			5599.57i			0.376289i
$606$		0			0
$607$	−12023.0			−0.803954			−0.401977	−	0.915650i	$-0.631677\pi$
$607$	−12023.0			−0.803954			−0.401977	+	0.915650i	$0.631677\pi$
$608$		0			0
$609$	1941.27			0.129170
$610$		0			0
$611$			11747.1i			0.777801i
$612$		0			0
$613$			10246.4i			0.675117i	0.941304	+	0.337559i	$0.109601\pi$
$613$			10246.4i			0.675117i	−0.941304	+	0.337559i	$0.890399\pi$
$614$		0			0
$615$		−	5374.86i		−	0.352415i
$616$		0			0
$617$	21866.3			1.42675			0.713374	−	0.700783i	$-0.247166\pi$
$617$	21866.3			1.42675			0.713374	+	0.700783i	$0.247166\pi$
$618$		0			0
$619$	−12728.3			−0.826482			−0.413241	−	0.910622i	$-0.635603\pi$
$619$	−12728.3			−0.826482			−0.413241	+	0.910622i	$0.635603\pi$
$620$		0			0
$621$	27169.0			1.75565
$622$		0			0
$623$	1230.77			0.0791486
$624$		0			0
$625$	4406.62			0.282023
$626$		0			0
$627$	2706.36	−	7359.49i	0.172379	−	0.468755i
$628$		0			0
$629$	2670.44			0.169281
$630$		0			0
$631$			30417.2i			1.91900i	0.281704	+	0.959501i	$0.409100\pi$
$631$			30417.2i			1.91900i	−0.281704	+	0.959501i	$0.590900\pi$
$632$		0			0
$633$	26108.6			1.63938
$634$		0			0
$635$		−	4045.24i		−	0.252804i
$636$		0			0
$637$	−19006.6			−1.18221
$638$		0			0
$639$	1329.09			0.0822818
$640$		0			0
$641$			21789.6i			1.34265i	0.741165	+	0.671323i	$0.234273\pi$
$641$			21789.6i			1.34265i	−0.741165	+	0.671323i	$0.765727\pi$
$642$		0			0
$643$	−5470.19			−0.335495			−0.167747	−	0.985830i	$-0.553649\pi$
$643$	−5470.19			−0.335495			−0.167747	+	0.985830i	$0.553649\pi$
$644$		0			0
$645$	−6950.82			−0.424323
$646$		0			0
$647$			3431.69i			0.208522i	0.994550	+	0.104261i	$0.0332477\pi$
$647$			3431.69i			0.208522i	−0.994550	+	0.104261i	$0.966752\pi$
$648$		0			0
$649$		−	435.416i		−	0.0263352i
$650$		0			0
$651$	−344.264			−0.0207262
$652$		0			0
$653$			24252.9i			1.45343i	0.686939	+	0.726715i	$0.258954\pi$
$653$			24252.9i			1.45343i	−0.686939	+	0.726715i	$0.741046\pi$
$654$		0			0
$655$		−	13601.6i		−	0.811385i
$656$		0			0
$657$	1324.02			0.0786223
$658$		0			0
$659$			21052.9i			1.24447i	0.782832	+	0.622233i	$0.213775\pi$
$659$			21052.9i			1.24447i	−0.782832	+	0.622233i	$0.786225\pi$
$660$		0			0
$661$	−9819.28			−0.577800			−0.288900	−	0.957359i	$-0.593290\pi$
$661$	−9819.28			−0.577800			−0.288900	+	0.957359i	$0.593290\pi$
$662$		0			0
$663$		−	7947.14i		−	0.465522i
$664$		0			0
$665$	−1470.50	−	540.757i	−0.0857495	−	0.0315333i
$666$		0			0
$667$		−	21974.0i		−	1.27561i
$668$		0			0
$669$		−	30485.9i		−	1.76182i
$670$		0			0
$671$		−	488.066i		−	0.0280798i
$672$		0			0
$673$			1913.83i			0.109618i	0.998497	+	0.0548088i	$0.0174549\pi$
$673$			1913.83i			0.109618i	−0.998497	+	0.0548088i	$0.982545\pi$
$674$		0			0
$675$			13293.3i			0.758016i
$676$		0			0
$677$	−11272.2			−0.639918			−0.319959	−	0.947431i	$-0.603669\pi$
$677$	−11272.2			−0.639918			−0.319959	+	0.947431i	$0.603669\pi$
$678$		0			0
$679$	4813.89			0.272077
$680$		0			0
$681$	17198.3			0.967756
$682$		0			0
$683$		−	30566.3i		−	1.71243i	−0.516622	−	0.856214i	$-0.672811\pi$
$683$		−	30566.3i		−	1.71243i	0.516622	−	0.856214i	$-0.327189\pi$
$684$		0			0
$685$			6113.49i			0.340999i
$686$		0			0
$687$	−28934.2			−1.60685
$688$		0			0
$689$	−22096.5			−1.22178
$690$		0			0
$691$	3125.68			0.172079			0.0860393	−	0.996292i	$-0.472579\pi$
$691$	3125.68			0.172079			0.0860393	+	0.996292i	$0.472579\pi$
$692$		0			0
$693$			99.6302i			0.00546124i
$694$		0			0
$695$		−	12315.5i		−	0.672163i
$696$		0			0
$697$		−	5132.81i		−	0.278937i
$698$		0			0
$699$			11367.1i			0.615085i
$700$		0			0
$701$		−	23361.6i		−	1.25871i	−0.777117	−	0.629356i	$-0.783319\pi$
$701$		−	23361.6i		−	1.25871i	0.777117	−	0.629356i	$-0.216681\pi$
$702$		0			0
$703$	7533.47	+	2770.34i	0.404168	+	0.148628i
$704$		0			0
$705$		−	5921.88i		−	0.316356i
$706$		0			0
$707$	2457.09			0.130705
$708$		0			0
$709$			34656.4i			1.83575i	0.396866	+	0.917877i	$0.370098\pi$
$709$			34656.4i			1.83575i	−0.396866	+	0.917877i	$0.629902\pi$
$710$		0			0
$711$	1088.48			0.0574136
$712$		0			0
$713$			3896.84i			0.204681i
$714$		0			0
$715$		−	6156.83i		−	0.322031i
$716$		0			0
$717$	−5097.79			−0.265524
$718$		0			0
$719$			21824.0i			1.13198i	0.824411	+	0.565992i	$0.191507\pi$
$719$			21824.0i			1.13198i	−0.824411	+	0.565992i	$0.808493\pi$
$720$		0			0
$721$		−	2253.15i		−	0.116383i
$722$		0			0
$723$	−21737.1			−1.11813
$724$		0			0
$725$	10751.5			0.550759
$726$		0			0
$727$		−	32538.0i		−	1.65993i	−0.557817	−	0.829964i	$-0.688361\pi$
$727$		−	32538.0i		−	1.65993i	0.557817	−	0.829964i	$-0.311639\pi$
$728$		0			0
$729$	−20709.7			−1.05216
$730$		0			0
$731$	−6637.80			−0.335852
$732$		0			0
$733$			14353.4i			0.723267i	0.932320	+	0.361633i	$0.117781\pi$
$733$			14353.4i			0.723267i	−0.932320	+	0.361633i	$0.882219\pi$
$734$		0			0
$735$	9581.52			0.480843
$736$		0			0
$737$			11357.1i			0.567629i
$738$		0			0
$739$	20256.9			1.00834			0.504169	−	0.863605i	$-0.331799\pi$
$739$	20256.9			1.00834			0.504169	+	0.863605i	$0.331799\pi$
$740$		0			0
$741$	8244.44	−	22419.3i	0.408727	−	1.11146i
$742$		0			0
$743$	−34092.1			−1.68334			−0.841668	−	0.539995i	$-0.818426\pi$
$743$	−34092.1			−1.68334			−0.841668	+	0.539995i	$0.818426\pi$
$744$		0			0
$745$	−2773.68			−0.136402
$746$		0			0
$747$	−251.242			−0.0123059
$748$		0			0
$749$	3341.44			0.163009
$750$		0			0
$751$	−37961.1			−1.84450			−0.922250	−	0.386595i	$-0.873651\pi$
$751$	−37961.1			−1.84450			−0.922250	+	0.386595i	$0.873651\pi$
$752$		0			0
$753$			9176.82i			0.444119i
$754$		0			0
$755$			11571.7i			0.557800i
$756$		0			0
$757$		−	27811.9i		−	1.33532i	−0.744464	−	0.667662i	$-0.767295\pi$
$757$		−	27811.9i		−	1.33532i	0.744464	−	0.667662i	$-0.232705\pi$
$758$		0			0
$759$	−17845.1			−0.853406
$760$		0			0
$761$	14158.7			0.674447			0.337223	−	0.941425i	$-0.390512\pi$
$761$	14158.7			0.674447			0.337223	+	0.941425i	$0.390512\pi$
$762$		0			0
$763$		−	2703.51i		−	0.128275i
$764$		0			0
$765$		−	253.183i		−	0.0119658i
$766$		0			0
$767$		−	1326.42i		−	0.0624434i
$768$		0			0
$769$	13553.8			0.635584			0.317792	−	0.948160i	$-0.397059\pi$
$769$	13553.8			0.635584			0.317792	+	0.948160i	$0.397059\pi$
$770$		0			0
$771$	16960.0			0.792218
$772$		0			0
$773$	23092.4			1.07448			0.537241	−	0.843429i	$-0.319467\pi$
$773$	23092.4			1.07448			0.537241	+	0.843429i	$0.319467\pi$
$774$		0			0
$775$	−1906.66			−0.0883731
$776$		0			0
$777$	1613.78			0.0745097
$778$		0			0
$779$	5324.82	−	14479.9i	0.244906	−	0.665979i
$780$		0			0
$781$	−15559.5			−0.712883
$782$		0			0
$783$			16806.0i			0.767047i
$784$		0			0
$785$	−19316.0			−0.878239
$786$		0			0
$787$			5587.77i			0.253091i	0.991961	+	0.126545i	$0.0403889\pi$
$787$			5587.77i			0.253091i	−0.991961	+	0.126545i	$0.959611\pi$
$788$		0			0
$789$	−9545.77			−0.430721
$790$		0			0
$791$	−739.077			−0.0332219
$792$		0			0
$793$		−	1486.80i		−	0.0665800i
$794$		0			0
$795$	11139.1			0.496937
$796$		0			0
$797$	−37059.9			−1.64709			−0.823543	−	0.567254i	$-0.808006\pi$
$797$	−37059.9			−1.64709			−0.823543	+	0.567254i	$0.808006\pi$
$798$		0			0
$799$		−	5655.19i		−	0.250396i
$800$		0			0
$801$			597.802i			0.0263699i
$802$		0			0
$803$	−15500.1			−0.681177
$804$		0			0
$805$			3565.62i			0.156114i
$806$		0			0
$807$		−	29896.1i		−	1.30408i
$808$		0			0
$809$	44631.5			1.93963			0.969816	−	0.243840i	$-0.0784071\pi$
$809$	44631.5			1.93963			0.969816	+	0.243840i	$0.0784071\pi$
$810$		0			0
$811$		−	15500.9i		−	0.671158i	−0.942012	−	0.335579i	$-0.891068\pi$
$811$		−	15500.9i		−	0.671158i	0.942012	−	0.335579i	$-0.108932\pi$
$812$		0			0
$813$	−6108.12			−0.263495
$814$		0			0
$815$			8524.60i			0.366385i
$816$		0			0
$817$	−18725.6	−	6886.11i	−0.801868	−	0.294877i
$818$		0			0
$819$			303.505i			0.0129491i
$820$		0			0
$821$			1010.46i			0.0429541i	0.999769	+	0.0214770i	$0.00683688\pi$
$821$			1010.46i			0.0429541i	−0.999769	+	0.0214770i	$0.993163\pi$
$822$		0			0
$823$			23113.5i			0.978962i	0.872014	+	0.489481i	$0.162814\pi$
$823$			23113.5i			0.978962i	−0.872014	+	0.489481i	$0.837186\pi$
$824$		0			0
$825$		−	8731.28i		−	0.368466i
$826$		0			0
$827$		−	8617.64i		−	0.362352i	−0.983451	−	0.181176i	$-0.942010\pi$
$827$		−	8617.64i		−	0.362352i	0.983451	−	0.181176i	$-0.0579903\pi$
$828$		0			0
$829$	−35920.9			−1.50493			−0.752463	−	0.658634i	$-0.771135\pi$
$829$	−35920.9			−1.50493			−0.752463	+	0.658634i	$0.771135\pi$
$830$		0			0
$831$	6675.25			0.278654
$832$		0			0
$833$	9150.03			0.380588
$834$		0			0
$835$		−	70.9927i		−	0.00294228i
$836$		0			0
$837$		−	2980.36i		−	0.123078i
$838$		0			0
$839$	−5510.57			−0.226753			−0.113377	−	0.993552i	$-0.536167\pi$
$839$	−5510.57			−0.226753			−0.113377	+	0.993552i	$0.536167\pi$
$840$		0			0
$841$	−10796.5			−0.442680
$842$		0			0
$843$	13484.1			0.550909
$844$		0			0
$845$		−	6176.73i		−	0.251463i
$846$		0			0
$847$			3231.49i			0.131093i
$848$		0			0
$849$		−	13772.8i		−	0.556750i
$850$		0			0
$851$		−	18266.9i		−	0.735820i
$852$		0			0
$853$			13875.8i			0.556975i	0.960440	+	0.278488i	$0.0898331\pi$
$853$			13875.8i			0.556975i	−0.960440	+	0.278488i	$0.910167\pi$
$854$		0			0
$855$	262.654	−	714.243i	0.0105059	−	0.0285691i
$856$		0			0
$857$		−	29552.2i		−	1.17793i	−0.808160	−	0.588964i	$-0.799536\pi$
$857$		−	29552.2i		−	1.17793i	0.808160	−	0.588964i	$-0.200464\pi$
$858$		0			0
$859$	−5040.17			−0.200196			−0.100098	−	0.994978i	$-0.531916\pi$
$859$	−5040.17			−0.200196			−0.100098	+	0.994978i	$0.531916\pi$
$860$		0			0
$861$		−	3101.82i		−	0.122775i
$862$		0			0
$863$	22745.6			0.897185			0.448593	−	0.893736i	$-0.351925\pi$
$863$	22745.6			0.897185			0.448593	+	0.893736i	$0.351925\pi$
$864$		0			0
$865$			1564.11i			0.0614812i
$866$		0			0
$867$		−	20932.5i		−	0.819960i
$868$		0			0
$869$	−12742.6			−0.497427
$870$		0			0
$871$			34597.2i			1.34590i
$872$		0			0
$873$			2338.18i			0.0906478i
$874$		0			0
$875$	−4109.35			−0.158767
$876$		0			0
$877$	−44944.1			−1.73051			−0.865253	−	0.501336i	$-0.832842\pi$
$877$	−44944.1			−1.73051			−0.865253	+	0.501336i	$0.832842\pi$
$878$		0			0
$879$			14601.8i			0.560304i
$880$		0			0
$881$	−37214.5			−1.42314			−0.711570	−	0.702615i	$-0.752016\pi$
$881$	−37214.5			−1.42314			−0.711570	+	0.702615i	$0.752016\pi$
$882$		0			0
$883$	15692.1			0.598053			0.299027	−	0.954245i	$-0.403338\pi$
$883$	15692.1			0.598053			0.299027	+	0.954245i	$0.403338\pi$
$884$		0			0
$885$			668.665i			0.0253977i
$886$		0			0
$887$	−7657.20			−0.289858			−0.144929	−	0.989442i	$-0.546295\pi$
$887$	−7657.20			−0.289858			−0.144929	+	0.989442i	$0.546295\pi$
$888$		0			0
$889$		−	2334.49i		−	0.0880725i
$890$		0			0
$891$	12834.0			0.482555
$892$		0			0
$893$	5866.75	−	15953.6i	0.219847	−	0.597836i
$894$		0			0
$895$	−19231.2			−0.718243
$896$		0			0
$897$	−54361.7			−2.02351
$898$		0			0
$899$	−2410.48			−0.0894260
$900$		0			0
$901$	10637.5			0.393326
$902$		0			0
$903$	−4011.30			−0.147827
$904$		0			0
$905$			15763.3i			0.578993i
$906$		0			0
$907$		−	34016.8i		−	1.24533i	−0.782490	−	0.622663i	$-0.786051\pi$
$907$		−	34016.8i		−	1.24533i	0.782490	−	0.622663i	$-0.213949\pi$
$908$		0			0
$909$			1193.44i			0.0435468i
$910$		0			0
$911$	20171.8			0.733612			0.366806	−	0.930297i	$-0.380451\pi$
$911$	20171.8			0.733612			0.366806	+	0.930297i	$0.380451\pi$
$912$		0			0
$913$	2941.25			0.106617
$914$		0			0
$915$			749.519i			0.0270801i
$916$		0			0
$917$		−	7849.43i		−	0.282673i
$918$		0			0
$919$			52191.5i			1.87338i	0.350158	+	0.936691i	$0.386128\pi$
$919$			52191.5i			1.87338i	−0.350158	+	0.936691i	$0.613872\pi$
$920$		0			0
$921$	−2917.12			−0.104367
$922$		0			0
$923$	−47399.1			−1.69031
$924$		0			0
$925$	8937.70			0.317697
$926$		0			0
$927$	1094.39			0.0387751
$928$		0			0
$929$	−15362.2			−0.542538			−0.271269	−	0.962504i	$-0.587443\pi$
$929$	−15362.2			−0.542538			−0.271269	+	0.962504i	$0.587443\pi$
$930$		0			0
$931$	25812.7	+	9492.33i	0.908677	+	0.334155i
$932$		0			0
$933$	−41787.0			−1.46628
$934$		0			0
$935$			2963.97i			0.103671i
$936$		0			0
$937$	17961.4			0.626227			0.313113	−	0.949716i	$-0.398628\pi$
$937$	17961.4			0.626227			0.313113	+	0.949716i	$0.398628\pi$
$938$		0			0
$939$			43910.6i			1.52606i
$940$		0			0
$941$	34823.8			1.20640			0.603201	−	0.797589i	$-0.293892\pi$
$941$	34823.8			1.20640			0.603201	+	0.797589i	$0.293892\pi$
$942$		0			0
$943$	−35110.5			−1.21247
$944$		0			0
$945$		−	2727.04i		−	0.0938736i
$946$		0			0
$947$	17274.2			0.592751			0.296375	−	0.955072i	$-0.404222\pi$
$947$	17274.2			0.592751			0.296375	+	0.955072i	$0.404222\pi$
$948$		0			0
$949$	−47218.1			−1.61514
$950$		0			0
$951$		−	22257.5i		−	0.758937i
$952$		0			0
$953$		−	50134.8i		−	1.70412i	−0.523444	−	0.852060i	$-0.675353\pi$
$953$		−	50134.8i		−	1.70412i	0.523444	−	0.852060i	$-0.324647\pi$
$954$		0			0
$955$	−9974.11			−0.337963
$956$		0			0
$957$		−	11038.5i		−	0.372856i
$958$		0			0
$959$			3528.08i			0.118798i
$960$		0			0
$961$	−29363.5			−0.985651
$962$		0			0
$963$			1622.99i			0.0543096i
$964$		0			0
$965$	10659.7			0.355594
$966$		0			0
$967$			12466.9i			0.414590i	0.978278	+	0.207295i	$0.0664660\pi$
$967$			12466.9i			0.414590i	−0.978278	+	0.207295i	$0.933534\pi$
$968$		0			0
$969$	−3968.97	+	10792.9i	−0.131581	+	0.357811i
$970$		0			0
$971$			35033.1i			1.15784i	0.815383	+	0.578922i	$0.196526\pi$
$971$			35033.1i			1.15784i	−0.815383	+	0.578922i	$0.803474\pi$
$972$		0			0
$973$		−	7107.24i		−	0.234170i
$974$		0			0
$975$		−	26598.3i		−	0.873668i
$976$		0			0
$977$			21109.5i			0.691250i	0.938373	+	0.345625i	$0.112333\pi$
$977$			21109.5i			0.691250i	−0.938373	+	0.345625i	$0.887667\pi$
$978$		0			0
$979$		−	6998.38i		−	0.228467i
$980$		0			0
$981$	1313.13			0.0427372
$982$		0			0
$983$	55306.0			1.79449			0.897247	−	0.441530i	$-0.145564\pi$
$983$	55306.0			1.79449			0.897247	+	0.441530i	$0.145564\pi$
$984$		0			0
$985$	−17396.8			−0.562749
$986$		0			0
$987$		−	3417.50i		−	0.110213i
$988$		0			0
$989$			45405.3i			1.45986i
$990$		0			0
$991$	−29330.3			−0.940170			−0.470085	−	0.882621i	$-0.655777\pi$
$991$	−29330.3			−0.940170			−0.470085	+	0.882621i	$0.655777\pi$
$992$		0			0
$993$	21764.7			0.695552
$994$		0			0
$995$	−17289.8			−0.550879
$996$		0			0
$997$			42823.7i			1.36032i	0.733063	+	0.680161i	$0.238090\pi$
$997$			42823.7i			1.36032i	−0.733063	+	0.680161i	$0.761910\pi$
$998$		0			0
$999$			13970.8i			0.442460i

Display

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a_n

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a_n

instead) (See

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instead) Display

a_n

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.b.b.303.41		56
4.3	odd	2		152.4.b.b.75.47	yes	56
8.3	odd	2	inner	608.4.b.b.303.42		56
8.5	even	2		152.4.b.b.75.9	✓	56
19.18	odd	2	inner	608.4.b.b.303.15		56
76.75	even	2		152.4.b.b.75.10	yes	56
152.37	odd	2		152.4.b.b.75.48	yes	56
152.75	even	2	inner	608.4.b.b.303.16		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.b.b.75.9	✓	56	8.5	even	2
152.4.b.b.75.10	yes	56	76.75	even	2
152.4.b.b.75.47	yes	56	4.3	odd	2
152.4.b.b.75.48	yes	56	152.37	odd	2
608.4.b.b.303.15		56	19.18	odd	2	inner
608.4.b.b.303.16		56	152.75	even	2	inner
608.4.b.b.303.41		56	1.1	even	1	trivial
608.4.b.b.303.42		56	8.3	odd	2	inner

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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	608.4.b.b.303.41

Level	$608$
Weight	$4$
Character	608.303
Analytic conductor	$35.873$
Analytic rank	$0$
Dimension	$56$
Inner twists	$4$