LMFDB - Embedded newform 608.2.c.b.305.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,2,Mod(305,608)] 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([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.305"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [16] 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			305.6
Root			$-1.40771 + 0.135487i$ of defining polynomial
Character	$\chi$	$=$	608.305
Dual form			608.2.c.b.305.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70663i q^{3} +1.66222i q^{5} +1.99556 q^{7} +0.0874066 q^{9} +2.08741i q^{11} +4.77322i q^{13} +2.83680 q^{15} +2.10657 q^{17} -1.00000i q^{19} -3.40569i q^{21} +4.84437 q^{23} +2.23703 q^{25} -5.26907i q^{27} -0.695946i q^{29} -9.77587 q^{31} +3.56244 q^{33} +3.31706i q^{35} -0.0772780i q^{37} +8.14614 q^{39} +10.7743 q^{41} -1.43840i q^{43} +0.145289i q^{45} +2.88133 q^{47} -3.01774 q^{49} -3.59513i q^{51} -9.00264i q^{53} -3.46973 q^{55} -1.70663 q^{57} +11.5253i q^{59} -8.82910i q^{61} +0.174425 q^{63} -7.93414 q^{65} -1.27859i q^{67} -8.26756i q^{69} +4.66655 q^{71} +4.44578 q^{73} -3.81779i q^{75} +4.16555i q^{77} -1.10044 q^{79} -8.73014 q^{81} -2.47419i q^{83} +3.50157i q^{85} -1.18772 q^{87} -15.8259 q^{89} +9.52526i q^{91} +16.6838i q^{93} +1.66222 q^{95} -13.9046 q^{97} +0.182453i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} - 24 q^{9} - 8 q^{17} - 24 q^{25} - 16 q^{31} - 8 q^{39} + 16 q^{41} - 24 q^{47} + 24 q^{49} - 16 q^{55} + 32 q^{63} + 16 q^{65} - 48 q^{71} + 48 q^{79} - 16 q^{81} + 48 q^{87} - 16 q^{89}+ \cdots + 32 q^{97}+O(q^{100}) $ 16 * q + 8 * q^7 - 24 * q^9 - 8 * q^17 - 24 * q^25 - 16 * q^31 - 8 * q^39 + 16 * q^41 - 24 * q^47 + 24 * q^49 - 16 * q^55 + 32 * q^63 + 16 * q^65 - 48 * q^71 + 48 * q^79 - 16 * q^81 + 48 * q^87 - 16 * q^89 - 16 * q^95 + 32 * q^97

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
$3$	− 1.70663i	− 0.985325i	−0.870221	−	0.492662i	$-0.836024\pi$
$3$	− 1.70663i	− 0.985325i	0.870221	−	0.492662i	$-0.163976\pi$
$4$	0	0
$5$	1.66222i	0.743367i	0.928360	+	0.371683i	$0.121219\pi$
$5$	1.66222i	0.743367i	−0.928360	+	0.371683i	$0.878781\pi$
$6$	0	0
$7$	1.99556	0.754251	0.377125	−	0.926162i	$-0.376913\pi$
$7$	1.99556	0.754251	0.377125	+	0.926162i	$0.376913\pi$
$8$	0	0
$9$	0.0874066	0.0291355
$10$	0	0
$11$	2.08741i	0.629377i	0.949195	+	0.314688i	$0.101900\pi$
$11$	2.08741i	0.629377i	−0.949195	+	0.314688i	$0.898100\pi$
$12$	0	0
$13$	4.77322i	1.32385i	0.749568	+	0.661927i	$0.230261\pi$
$13$	4.77322i	1.32385i	−0.749568	+	0.661927i	$0.769739\pi$
$14$	0	0
$15$	2.83680	0.732457
$16$	0	0
$17$	2.10657	0.510917	0.255459	−	0.966820i	$-0.417774\pi$
$17$	2.10657	0.510917	0.255459	+	0.966820i	$0.417774\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 3.40569i	− 0.743182i
$22$	0	0
$23$	4.84437	1.01012	0.505061	−	0.863084i	$-0.331470\pi$
$23$	4.84437	1.01012	0.505061	+	0.863084i	$0.331470\pi$
$24$	0	0
$25$	2.23703	0.447406
$26$	0	0
$27$	− 5.26907i	− 1.01403i
$28$	0	0
$29$	− 0.695946i	− 0.129234i	−0.997910	−	0.0646170i	$-0.979417\pi$
$29$	− 0.695946i	− 0.129234i	0.997910	−	0.0646170i	$-0.0205826\pi$
$30$	0	0
$31$	−9.77587	−1.75580	−0.877899	−	0.478846i	$-0.841055\pi$
$31$	−9.77587	−1.75580	−0.877899	+	0.478846i	$0.841055\pi$
$32$	0	0
$33$	3.56244	0.620140
$34$	0	0
$35$	3.31706i	0.560685i
$36$	0	0
$37$	− 0.0772780i	− 0.0127044i	−0.999980	−	0.00635221i	$-0.997978\pi$
$37$	− 0.0772780i	− 0.0127044i	0.999980	−	0.00635221i	$-0.00202198\pi$
$38$	0	0
$39$	8.14614	1.30443
$40$	0	0
$41$	10.7743	1.68267	0.841334	−	0.540516i	$-0.181771\pi$
$41$	10.7743	1.68267	0.841334	+	0.540516i	$0.181771\pi$
$42$	0	0
$43$	− 1.43840i	− 0.219354i	−0.993967	−	0.109677i	$-0.965018\pi$
$43$	− 1.43840i	− 0.219354i	0.993967	−	0.109677i	$-0.0349817\pi$
$44$	0	0
$45$	0.145289i	0.0216584i
$46$	0	0
$47$	2.88133	0.420285	0.210143	−	0.977671i	$-0.432607\pi$
$47$	2.88133	0.420285	0.210143	+	0.977671i	$0.432607\pi$
$48$	0	0
$49$	−3.01774	−0.431106
$50$	0	0
$51$	− 3.59513i	− 0.503419i
$52$	0	0
$53$	− 9.00264i	− 1.23661i	−0.785939	−	0.618304i	$-0.787820\pi$
$53$	− 9.00264i	− 1.23661i	0.785939	−	0.618304i	$-0.212180\pi$
$54$	0	0
$55$	−3.46973	−0.467858
$56$	0	0
$57$	−1.70663	−0.226049
$58$	0	0
$59$	11.5253i	1.50046i	0.661177	+	0.750230i	$0.270057\pi$
$59$	11.5253i	1.50046i	−0.661177	+	0.750230i	$0.729943\pi$
$60$	0	0
$61$	− 8.82910i	− 1.13045i	−0.824937	−	0.565225i	$-0.808789\pi$
$61$	− 8.82910i	− 1.13045i	0.824937	−	0.565225i	$-0.191211\pi$
$62$	0	0
$63$	0.174425	0.0219755
$64$	0	0
$65$	−7.93414	−0.984109
$66$	0	0
$67$	− 1.27859i	− 0.156205i	−0.996945	−	0.0781023i	$-0.975114\pi$
$67$	− 1.27859i	− 0.156205i	0.996945	−	0.0781023i	$-0.0248861\pi$
$68$	0	0
$69$	− 8.26756i	− 0.995297i
$70$	0	0
$71$	4.66655	0.553818	0.276909	−	0.960896i	$-0.410690\pi$
$71$	4.66655	0.553818	0.276909	+	0.960896i	$0.410690\pi$
$72$	0	0
$73$	4.44578	0.520339	0.260170	−	0.965563i	$-0.416221\pi$
$73$	4.44578	0.520339	0.260170	+	0.965563i	$0.416221\pi$
$74$	0	0
$75$	− 3.81779i	− 0.440840i
$76$	0	0
$77$	4.16555i	0.474708i
$78$	0	0
$79$	−1.10044	−0.123809	−0.0619044	−	0.998082i	$-0.519717\pi$
$79$	−1.10044	−0.123809	−0.0619044	+	0.998082i	$0.519717\pi$
$80$	0	0
$81$	−8.73014	−0.970016
$82$	0	0
$83$	− 2.47419i	− 0.271578i	−0.990738	−	0.135789i	$-0.956643\pi$
$83$	− 2.47419i	− 0.271578i	0.990738	−	0.135789i	$-0.0433569\pi$
$84$	0	0
$85$	3.50157i	0.379799i
$86$	0	0
$87$	−1.18772	−0.127337
$88$	0	0
$89$	−15.8259	−1.67755	−0.838773	−	0.544481i	$-0.816727\pi$
$89$	−15.8259	−1.67755	−0.838773	+	0.544481i	$0.816727\pi$
$90$	0	0
$91$	9.52526i	0.998518i
$92$	0	0
$93$	16.6838i	1.73003i
$94$	0	0
$95$	1.66222	0.170540
$96$	0	0
$97$	−13.9046	−1.41180	−0.705898	−	0.708313i	$-0.749456\pi$
$97$	−13.9046	−1.41180	−0.705898	+	0.708313i	$0.749456\pi$
$98$	0	0
$99$	0.182453i	0.0183372i
$100$	0	0
$101$	− 1.00493i	− 0.0999945i	−0.998749	−	0.0499973i	$-0.984079\pi$
$101$	− 1.00493i	− 0.0999945i	0.998749	−	0.0499973i	$-0.0159213\pi$
$102$	0	0
$103$	−12.4581	−1.22753	−0.613767	−	0.789487i	$-0.710347\pi$
$103$	−12.4581	−1.22753	−0.613767	+	0.789487i	$0.710347\pi$
$104$	0	0
$105$	5.66100	0.552457
$106$	0	0
$107$	15.3004i	1.47915i	0.673076	+	0.739573i	$0.264973\pi$
$107$	15.3004i	1.47915i	−0.673076	+	0.739573i	$0.735027\pi$
$108$	0	0
$109$	7.65850i	0.733551i	0.930309	+	0.366775i	$0.119538\pi$
$109$	7.65850i	0.733551i	−0.930309	+	0.366775i	$0.880462\pi$
$110$	0	0
$111$	−0.131885	−0.0125180
$112$	0	0
$113$	17.3938	1.63627	0.818134	−	0.575028i	$-0.195009\pi$
$113$	17.3938	1.63627	0.818134	+	0.575028i	$0.195009\pi$
$114$	0	0
$115$	8.05240i	0.750891i
$116$	0	0
$117$	0.417211i	0.0385712i
$118$	0	0
$119$	4.20378	0.385360
$120$	0	0
$121$	6.64273	0.603885
$122$	0	0
$123$	− 18.3878i	− 1.65797i
$124$	0	0
$125$	12.0295i	1.07595i
$126$	0	0
$127$	−18.1488	−1.61045	−0.805225	−	0.592970i	$-0.797955\pi$
$127$	−18.1488	−1.61045	−0.805225	+	0.592970i	$0.797955\pi$
$128$	0	0
$129$	−2.45482	−0.216135
$130$	0	0
$131$	9.70500i	0.847930i	0.905679	+	0.423965i	$0.139362\pi$
$131$	9.70500i	0.847930i	−0.905679	+	0.423965i	$0.860638\pi$
$132$	0	0
$133$	− 1.99556i	− 0.173037i
$134$	0	0
$135$	8.75834	0.753798
$136$	0	0
$137$	−20.2253	−1.72797	−0.863983	−	0.503521i	$-0.832038\pi$
$137$	−20.2253	−1.72797	−0.863983	+	0.503521i	$0.832038\pi$
$138$	0	0
$139$	− 16.7746i	− 1.42280i	−0.702786	−	0.711401i	$-0.748061\pi$
$139$	− 16.7746i	− 1.42280i	0.702786	−	0.711401i	$-0.251939\pi$
$140$	0	0
$141$	− 4.91737i	− 0.414117i
$142$	0	0
$143$	−9.96366	−0.833203
$144$	0	0
$145$	1.15681	0.0960682
$146$	0	0
$147$	5.15017i	0.424779i
$148$	0	0
$149$	− 13.6533i	− 1.11853i	−0.828991	−	0.559263i	$-0.811084\pi$
$149$	− 13.6533i	− 1.11853i	0.828991	−	0.559263i	$-0.188916\pi$
$150$	0	0
$151$	−6.77602	−0.551425	−0.275712	−	0.961240i	$-0.588914\pi$
$151$	−6.77602	−0.551425	−0.275712	+	0.961240i	$0.588914\pi$
$152$	0	0
$153$	0.184128	0.0148858
$154$	0	0
$155$	− 16.2496i	− 1.30520i
$156$	0	0
$157$	23.4178i	1.86895i	0.356035	+	0.934473i	$0.384129\pi$
$157$	23.4178i	1.86895i	−0.356035	+	0.934473i	$0.615871\pi$
$158$	0	0
$159$	−15.3642	−1.21846
$160$	0	0
$161$	9.66724	0.761885
$162$	0	0
$163$	1.06973i	0.0837875i	0.999122	+	0.0418938i	$0.0133391\pi$
$163$	1.06973i	0.0837875i	−0.999122	+	0.0418938i	$0.986661\pi$
$164$	0	0
$165$	5.92155i	0.460992i
$166$	0	0
$167$	−13.6217	−1.05408	−0.527039	−	0.849841i	$-0.676698\pi$
$167$	−13.6217	−1.05408	−0.527039	+	0.849841i	$0.676698\pi$
$168$	0	0
$169$	−9.78367	−0.752590
$170$	0	0
$171$	− 0.0874066i	− 0.00668415i
$172$	0	0
$173$	− 11.5684i	− 0.879527i	−0.898114	−	0.439763i	$-0.855062\pi$
$173$	− 11.5684i	− 0.879527i	0.898114	−	0.439763i	$-0.144938\pi$
$174$	0	0
$175$	4.46413	0.337456
$176$	0	0
$177$	19.6694	1.47844
$178$	0	0
$179$	− 20.4983i	− 1.53211i	−0.642772	−	0.766057i	$-0.722216\pi$
$179$	− 20.4983i	− 1.53211i	0.642772	−	0.766057i	$-0.277784\pi$
$180$	0	0
$181$	− 18.7745i	− 1.39550i	−0.716341	−	0.697750i	$-0.754185\pi$
$181$	− 18.7745i	− 1.39550i	0.716341	−	0.697750i	$-0.245815\pi$
$182$	0	0
$183$	−15.0680	−1.11386
$184$	0	0
$185$	0.128453	0.00944404
$186$	0	0
$187$	4.39726i	0.321560i
$188$	0	0
$189$	− 10.5147i	− 0.764835i
$190$	0	0
$191$	14.0392	1.01584	0.507919	−	0.861405i	$-0.330415\pi$
$191$	14.0392	1.01584	0.507919	+	0.861405i	$0.330415\pi$
$192$	0	0
$193$	−23.8700	−1.71820	−0.859099	−	0.511809i	$-0.828976\pi$
$193$	−23.8700	−1.71820	−0.859099	+	0.511809i	$0.828976\pi$
$194$	0	0
$195$	13.5407i	0.969667i
$196$	0	0
$197$	− 1.84890i	− 0.131729i	−0.997829	−	0.0658645i	$-0.979020\pi$
$197$	− 1.84890i	− 0.131729i	0.997829	−	0.0658645i	$-0.0209805\pi$
$198$	0	0
$199$	18.3608	1.30156	0.650781	−	0.759266i	$-0.274442\pi$
$199$	18.3608	1.30156	0.650781	+	0.759266i	$0.274442\pi$
$200$	0	0
$201$	−2.18208	−0.153912
$202$	0	0
$203$	− 1.38880i	− 0.0974748i
$204$	0	0
$205$	17.9093i	1.25084i
$206$	0	0
$207$	0.423430	0.0294304
$208$	0	0
$209$	2.08741	0.144389
$210$	0	0
$211$	− 15.5147i	− 1.06808i	−0.845460	−	0.534039i	$-0.820674\pi$
$211$	− 15.5147i	− 1.06808i	0.845460	−	0.534039i	$-0.179326\pi$
$212$	0	0
$213$	− 7.96409i	− 0.545690i
$214$	0	0
$215$	2.39094	0.163061
$216$	0	0
$217$	−19.5083	−1.32431
$218$	0	0
$219$	− 7.58731i	− 0.512703i
$220$	0	0
$221$	10.0551i	0.676380i
$222$	0	0
$223$	13.9260	0.932553	0.466276	−	0.884639i	$-0.345595\pi$
$223$	13.9260	0.932553	0.466276	+	0.884639i	$0.345595\pi$
$224$	0	0
$225$	0.195531	0.0130354
$226$	0	0
$227$	1.49624i	0.0993088i	0.998766	+	0.0496544i	$0.0158120\pi$
$227$	1.49624i	0.0993088i	−0.998766	+	0.0496544i	$0.984188\pi$
$228$	0	0
$229$	− 1.03706i	− 0.0685310i	−0.999413	−	0.0342655i	$-0.989091\pi$
$229$	− 1.03706i	− 0.0685310i	0.999413	−	0.0342655i	$-0.0109092\pi$
$230$	0	0
$231$	7.10905	0.467741
$232$	0	0
$233$	4.93372	0.323219	0.161609	−	0.986855i	$-0.448332\pi$
$233$	4.93372	0.323219	0.161609	+	0.986855i	$0.448332\pi$
$234$	0	0
$235$	4.78940i	0.312426i
$236$	0	0
$237$	1.87804i	0.121992i
$238$	0	0
$239$	3.84533	0.248733	0.124367	−	0.992236i	$-0.460310\pi$
$239$	3.84533	0.248733	0.124367	+	0.992236i	$0.460310\pi$
$240$	0	0
$241$	−1.41134	−0.0909123	−0.0454562	−	0.998966i	$-0.514474\pi$
$241$	−1.41134	−0.0909123	−0.0454562	+	0.998966i	$0.514474\pi$
$242$	0	0
$243$	− 0.908064i	− 0.0582523i
$244$	0	0
$245$	− 5.01614i	− 0.320469i
$246$	0	0
$247$	4.77322	0.303713
$248$	0	0
$249$	−4.22253	−0.267592
$250$	0	0
$251$	− 8.24640i	− 0.520508i	−0.965540	−	0.260254i	$-0.916194\pi$
$251$	− 8.24640i	− 0.520508i	0.965540	−	0.260254i	$-0.0838063\pi$
$252$	0	0
$253$	10.1122i	0.635747i
$254$	0	0
$255$	5.97590	0.374225
$256$	0	0
$257$	15.0726	0.940200	0.470100	−	0.882613i	$-0.344218\pi$
$257$	15.0726	0.940200	0.470100	+	0.882613i	$0.344218\pi$
$258$	0	0
$259$	− 0.154213i	− 0.00958232i
$260$	0	0
$261$	− 0.0608303i	− 0.00376530i
$262$	0	0
$263$	9.49141	0.585265	0.292633	−	0.956225i	$-0.405469\pi$
$263$	9.49141	0.585265	0.292633	+	0.956225i	$0.405469\pi$
$264$	0	0
$265$	14.9644	0.919253
$266$	0	0
$267$	27.0091i	1.65293i
$268$	0	0
$269$	3.98549i	0.243000i	0.992591	+	0.121500i	$0.0387704\pi$
$269$	3.98549i	0.243000i	−0.992591	+	0.121500i	$0.961230\pi$
$270$	0	0
$271$	6.38427	0.387817	0.193908	−	0.981020i	$-0.437884\pi$
$271$	6.38427	0.387817	0.193908	+	0.981020i	$0.437884\pi$
$272$	0	0
$273$	16.2561	0.983865
$274$	0	0
$275$	4.66959i	0.281587i
$276$	0	0
$277$	− 15.1758i	− 0.911823i	−0.890025	−	0.455912i	$-0.849313\pi$
$277$	− 15.1758i	− 0.911823i	0.890025	−	0.455912i	$-0.150687\pi$
$278$	0	0
$279$	−0.854475	−0.0511561
$280$	0	0
$281$	14.9058	0.889207	0.444603	−	0.895728i	$-0.353345\pi$
$281$	14.9058	0.889207	0.444603	+	0.895728i	$0.353345\pi$
$282$	0	0
$283$	14.9931i	0.891250i	0.895220	+	0.445625i	$0.147019\pi$
$283$	14.9931i	0.891250i	−0.895220	+	0.445625i	$0.852981\pi$
$284$	0	0
$285$	− 2.83680i	− 0.168037i
$286$	0	0
$287$	21.5008	1.26915
$288$	0	0
$289$	−12.5624	−0.738963
$290$	0	0
$291$	23.7300i	1.39108i
$292$	0	0
$293$	− 21.6983i	− 1.26763i	−0.773485	−	0.633814i	$-0.781488\pi$
$293$	− 21.6983i	− 1.26763i	0.773485	−	0.633814i	$-0.218512\pi$
$294$	0	0
$295$	−19.1575	−1.11539
$296$	0	0
$297$	10.9987	0.638208
$298$	0	0
$299$	23.1233i	1.33725i
$300$	0	0
$301$	− 2.87042i	− 0.165448i
$302$	0	0
$303$	−1.71505	−0.0985270
$304$	0	0
$305$	14.6759	0.840339
$306$	0	0
$307$	7.81449i	0.445996i	0.974819	+	0.222998i	$0.0715844\pi$
$307$	7.81449i	0.445996i	−0.974819	+	0.222998i	$0.928416\pi$
$308$	0	0
$309$	21.2614i	1.20952i
$310$	0	0
$311$	−12.8288	−0.727455	−0.363727	−	0.931505i	$-0.618496\pi$
$311$	−12.8288	−0.727455	−0.363727	+	0.931505i	$0.618496\pi$
$312$	0	0
$313$	−4.07578	−0.230377	−0.115188	−	0.993344i	$-0.536747\pi$
$313$	−4.07578	−0.230377	−0.115188	+	0.993344i	$0.536747\pi$
$314$	0	0
$315$	0.289933i	0.0163358i
$316$	0	0
$317$	− 2.26448i	− 0.127186i	−0.997976	−	0.0635931i	$-0.979744\pi$
$317$	− 2.26448i	− 0.127186i	0.997976	−	0.0635931i	$-0.0202560\pi$
$318$	0	0
$319$	1.45272	0.0813368
$320$	0	0
$321$	26.1122	1.45744
$322$	0	0
$323$	− 2.10657i	− 0.117212i
$324$	0	0
$325$	10.6778i	0.592300i
$326$	0	0
$327$	13.0702	0.722786
$328$	0	0
$329$	5.74987	0.317000
$330$	0	0
$331$	7.31171i	0.401888i	0.979603	+	0.200944i	$0.0644009\pi$
$331$	7.31171i	0.401888i	−0.979603	+	0.200944i	$0.935599\pi$
$332$	0	0
$333$	− 0.00675460i	0 0.000370150i
$334$	0	0
$335$	2.12529	0.116117
$336$	0	0
$337$	−6.00356	−0.327035	−0.163517	−	0.986540i	$-0.552284\pi$
$337$	−6.00356	−0.327035	−0.163517	+	0.986540i	$0.552284\pi$
$338$	0	0
$339$	− 29.6848i	− 1.61225i
$340$	0	0
$341$	− 20.4062i	− 1.10506i
$342$	0	0
$343$	−19.9910	−1.07941
$344$	0	0
$345$	13.7425	0.739871
$346$	0	0
$347$	6.66747i	0.357929i	0.983856	+	0.178964i	$0.0572747\pi$
$347$	6.66747i	0.357929i	−0.983856	+	0.178964i	$0.942725\pi$
$348$	0	0
$349$	− 35.5072i	− 1.90066i	−0.311248	−	0.950329i	$-0.600747\pi$
$349$	− 35.5072i	− 1.90066i	0.311248	−	0.950329i	$-0.399253\pi$
$350$	0	0
$351$	25.1504	1.34243
$352$	0	0
$353$	−27.8315	−1.48132	−0.740659	−	0.671881i	$-0.765487\pi$
$353$	−27.8315	−1.48132	−0.740659	+	0.671881i	$0.765487\pi$
$354$	0	0
$355$	7.75683i	0.411690i
$356$	0	0
$357$	− 7.17431i	− 0.379705i
$358$	0	0
$359$	−10.6778	−0.563551	−0.281775	−	0.959480i	$-0.590923\pi$
$359$	−10.6778	−0.563551	−0.281775	+	0.959480i	$0.590923\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 11.3367i	− 0.595023i
$364$	0	0
$365$	7.38986i	0.386803i
$366$	0	0
$367$	−16.8950	−0.881911	−0.440956	−	0.897529i	$-0.645360\pi$
$367$	−16.8950	−0.881911	−0.440956	+	0.897529i	$0.645360\pi$
$368$	0	0
$369$	0.941747	0.0490254
$370$	0	0
$371$	− 17.9653i	− 0.932713i
$372$	0	0
$373$	33.4640i	1.73270i	0.499438	+	0.866350i	$0.333540\pi$
$373$	33.4640i	1.73270i	−0.499438	+	0.866350i	$0.666460\pi$
$374$	0	0
$375$	20.5300	1.06016
$376$	0	0
$377$	3.32191	0.171087
$378$	0	0
$379$	− 37.0251i	− 1.90185i	−0.309423	−	0.950925i	$-0.600136\pi$
$379$	− 37.0251i	− 1.90185i	0.309423	−	0.950925i	$-0.399864\pi$
$380$	0	0
$381$	30.9734i	1.58682i
$382$	0	0
$383$	16.2264	0.829131	0.414566	−	0.910019i	$-0.363934\pi$
$383$	16.2264	0.829131	0.414566	+	0.910019i	$0.363934\pi$
$384$	0	0
$385$	−6.92405	−0.352882
$386$	0	0
$387$	− 0.125726i	− 0.00639100i
$388$	0	0
$389$	− 11.8131i	− 0.598947i	−0.954105	−	0.299473i	$-0.903189\pi$
$389$	− 11.8131i	− 0.598947i	0.954105	−	0.299473i	$-0.0968110\pi$
$390$	0	0
$391$	10.2050	0.516089
$392$	0	0
$393$	16.5629	0.835486
$394$	0	0
$395$	− 1.82917i	− 0.0920353i
$396$	0	0
$397$	10.9654i	0.550340i	0.961396	+	0.275170i	$0.0887341\pi$
$397$	10.9654i	0.550340i	−0.961396	+	0.275170i	$0.911266\pi$
$398$	0	0
$399$	−3.40569	−0.170498
$400$	0	0
$401$	11.9479	0.596650	0.298325	−	0.954464i	$-0.403572\pi$
$401$	11.9479	0.596650	0.298325	+	0.954464i	$0.403572\pi$
$402$	0	0
$403$	− 46.6624i	− 2.32442i
$404$	0	0
$405$	− 14.5114i	− 0.721077i
$406$	0	0
$407$	0.161311	0.00799587
$408$	0	0
$409$	−1.28997	−0.0637848	−0.0318924	−	0.999491i	$-0.510153\pi$
$409$	−1.28997	−0.0637848	−0.0318924	+	0.999491i	$0.510153\pi$
$410$	0	0
$411$	34.5172i	1.70261i
$412$	0	0
$413$	22.9993i	1.13172i
$414$	0	0
$415$	4.11265	0.201882
$416$	0	0
$417$	−28.6281	−1.40192
$418$	0	0
$419$	15.6802i	0.766030i	0.923742	+	0.383015i	$0.125114\pi$
$419$	15.6802i	0.766030i	−0.923742	+	0.383015i	$0.874886\pi$
$420$	0	0
$421$	− 26.2544i	− 1.27956i	−0.768557	−	0.639782i	$-0.779025\pi$
$421$	− 26.2544i	− 1.27956i	0.768557	−	0.639782i	$-0.220975\pi$
$422$	0	0
$423$	0.251847	0.0122452
$424$	0	0
$425$	4.71245	0.228587
$426$	0	0
$427$	− 17.6190i	− 0.852643i
$428$	0	0
$429$	17.0043i	0.820975i
$430$	0	0
$431$	33.4648	1.61194	0.805972	−	0.591954i	$-0.201643\pi$
$431$	33.4648	1.61194	0.805972	+	0.591954i	$0.201643\pi$
$432$	0	0
$433$	5.89119	0.283112	0.141556	−	0.989930i	$-0.454789\pi$
$433$	5.89119	0.283112	0.141556	+	0.989930i	$0.454789\pi$
$434$	0	0
$435$	− 1.97426i	− 0.0946584i
$436$	0	0
$437$	− 4.84437i	− 0.231738i
$438$	0	0
$439$	−25.9770	−1.23982	−0.619908	−	0.784674i	$-0.712830\pi$
$439$	−25.9770	−1.23982	−0.619908	+	0.784674i	$0.712830\pi$
$440$	0	0
$441$	−0.263770	−0.0125605
$442$	0	0
$443$	− 11.0652i	− 0.525722i	−0.964834	−	0.262861i	$-0.915334\pi$
$443$	− 11.0652i	− 0.525722i	0.964834	−	0.262861i	$-0.0846661\pi$
$444$	0	0
$445$	− 26.3062i	− 1.24703i
$446$	0	0
$447$	−23.3012	−1.10211
$448$	0	0
$449$	3.56950	0.168455	0.0842274	−	0.996447i	$-0.473158\pi$
$449$	3.56950	0.168455	0.0842274	+	0.996447i	$0.473158\pi$
$450$	0	0
$451$	22.4904i	1.05903i
$452$	0	0
$453$	11.5642i	0.543332i
$454$	0	0
$455$	−15.8331	−0.742265
$456$	0	0
$457$	−23.3258	−1.09113	−0.545567	−	0.838067i	$-0.683686\pi$
$457$	−23.3258	−1.09113	−0.545567	+	0.838067i	$0.683686\pi$
$458$	0	0
$459$	− 11.0996i	− 0.518087i
$460$	0	0
$461$	8.16887i	0.380462i	0.981739	+	0.190231i	$0.0609237\pi$
$461$	8.16887i	0.380462i	−0.981739	+	0.190231i	$0.939076\pi$
$462$	0	0
$463$	12.8015	0.594936	0.297468	−	0.954732i	$-0.403858\pi$
$463$	12.8015	0.594936	0.297468	+	0.954732i	$0.403858\pi$
$464$	0	0
$465$	−27.7321	−1.28605
$466$	0	0
$467$	− 0.897425i	− 0.0415279i	−0.999784	−	0.0207639i	$-0.993390\pi$
$467$	− 0.897425i	− 0.0415279i	0.999784	−	0.0207639i	$-0.00660984\pi$
$468$	0	0
$469$	− 2.55150i	− 0.117817i
$470$	0	0
$471$	39.9656	1.84152
$472$	0	0
$473$	3.00253	0.138056
$474$	0	0
$475$	− 2.23703i	− 0.102642i
$476$	0	0
$477$	− 0.786890i	− 0.0360292i
$478$	0	0
$479$	−12.9960	−0.593803	−0.296901	−	0.954908i	$-0.595953\pi$
$479$	−12.9960	−0.593803	−0.296901	+	0.954908i	$0.595953\pi$
$480$	0	0
$481$	0.368865	0.0168188
$482$	0	0
$483$	− 16.4984i	− 0.750704i
$484$	0	0
$485$	− 23.1125i	− 1.04948i
$486$	0	0
$487$	−12.0621	−0.546588	−0.273294	−	0.961931i	$-0.588113\pi$
$487$	−12.0621	−0.546588	−0.273294	+	0.961931i	$0.588113\pi$
$488$	0	0
$489$	1.82563	0.0825579
$490$	0	0
$491$	− 30.5337i	− 1.37797i	−0.724778	−	0.688983i	$-0.758058\pi$
$491$	− 30.5337i	− 1.37797i	0.724778	−	0.688983i	$-0.241942\pi$
$492$	0	0
$493$	− 1.46606i	− 0.0660279i
$494$	0	0
$495$	−0.303277	−0.0136313
$496$	0	0
$497$	9.31239	0.417718
$498$	0	0
$499$	40.2124i	1.80015i	0.435731	+	0.900077i	$0.356490\pi$
$499$	40.2124i	1.80015i	−0.435731	+	0.900077i	$0.643510\pi$
$500$	0	0
$501$	23.2472i	1.03861i
$502$	0	0
$503$	−35.7181	−1.59259	−0.796296	−	0.604907i	$-0.793210\pi$
$503$	−35.7181	−1.59259	−0.796296	+	0.604907i	$0.793210\pi$
$504$	0	0
$505$	1.67042	0.0743326
$506$	0	0
$507$	16.6971i	0.741545i
$508$	0	0
$509$	35.3428i	1.56654i	0.621681	+	0.783270i	$0.286450\pi$
$509$	35.3428i	1.56654i	−0.621681	+	0.783270i	$0.713550\pi$
$510$	0	0
$511$	8.87183	0.392466
$512$	0	0
$513$	−5.26907	−0.232635
$514$	0	0
$515$	− 20.7081i	− 0.912508i
$516$	0	0
$517$	6.01450i	0.264518i
$518$	0	0
$519$	−19.7429	−0.866619
$520$	0	0
$521$	2.50662	0.109817	0.0549086	−	0.998491i	$-0.482513\pi$
$521$	2.50662	0.109817	0.0549086	+	0.998491i	$0.482513\pi$
$522$	0	0
$523$	9.95330i	0.435227i	0.976035	+	0.217614i	$0.0698273\pi$
$523$	9.95330i	0.435227i	−0.976035	+	0.217614i	$0.930173\pi$
$524$	0	0
$525$	− 7.61863i	− 0.332504i
$526$	0	0
$527$	−20.5935	−0.897068
$528$	0	0
$529$	0.467940	0.0203452
$530$	0	0
$531$	1.00738i	0.0437167i
$532$	0	0
$533$	51.4283i	2.22761i
$534$	0	0
$535$	−25.4326	−1.09955
$536$	0	0
$537$	−34.9831	−1.50963
$538$	0	0
$539$	− 6.29925i	− 0.271328i
$540$	0	0
$541$	− 34.7061i	− 1.49213i	−0.665873	−	0.746065i	$-0.731941\pi$
$541$	− 34.7061i	− 1.49213i	0.665873	−	0.746065i	$-0.268059\pi$
$542$	0	0
$543$	−32.0412	−1.37502
$544$	0	0
$545$	−12.7301	−0.545297
$546$	0	0
$547$	7.21429i	0.308461i	0.988035	+	0.154230i	$0.0492898\pi$
$547$	7.21429i	0.308461i	−0.988035	+	0.154230i	$0.950710\pi$
$548$	0	0
$549$	− 0.771721i	− 0.0329363i
$550$	0	0
$551$	−0.695946	−0.0296483
$552$	0	0
$553$	−2.19599	−0.0933829
$554$	0	0
$555$	− 0.219222i	− 0.00930545i
$556$	0	0
$557$	− 4.27230i	− 0.181023i	−0.995895	−	0.0905116i	$-0.971150\pi$
$557$	− 4.27230i	− 0.181023i	0.995895	−	0.0905116i	$-0.0288502\pi$
$558$	0	0
$559$	6.86582	0.290393
$560$	0	0
$561$	7.50451	0.316840
$562$	0	0
$563$	− 23.9662i	− 1.01005i	−0.863103	−	0.505027i	$-0.831482\pi$
$563$	− 23.9662i	− 1.01005i	0.863103	−	0.505027i	$-0.168518\pi$
$564$	0	0
$565$	28.9122i	1.21635i
$566$	0	0
$567$	−17.4215	−0.731635
$568$	0	0
$569$	6.53074	0.273783	0.136891	−	0.990586i	$-0.456289\pi$
$569$	6.53074	0.273783	0.136891	+	0.990586i	$0.456289\pi$
$570$	0	0
$571$	− 13.8125i	− 0.578035i	−0.957324	−	0.289017i	$-0.906671\pi$
$571$	− 13.8125i	− 0.578035i	0.957324	−	0.289017i	$-0.0933285\pi$
$572$	0	0
$573$	− 23.9597i	− 1.00093i
$574$	0	0
$575$	10.8370	0.451934
$576$	0	0
$577$	−1.75601	−0.0731035	−0.0365517	−	0.999332i	$-0.511637\pi$
$577$	−1.75601	−0.0731035	−0.0365517	+	0.999332i	$0.511637\pi$
$578$	0	0
$579$	40.7373i	1.69298i
$580$	0	0
$581$	− 4.93740i	− 0.204838i
$582$	0	0
$583$	18.7922	0.778293
$584$	0	0
$585$	−0.693496	−0.0286725
$586$	0	0
$587$	22.1818i	0.915540i	0.889071	+	0.457770i	$0.151352\pi$
$587$	22.1818i	0.915540i	−0.889071	+	0.457770i	$0.848648\pi$
$588$	0	0
$589$	9.77587i	0.402808i
$590$	0	0
$591$	−3.15540	−0.129796
$592$	0	0
$593$	−14.8347	−0.609186	−0.304593	−	0.952483i	$-0.598520\pi$
$593$	−14.8347	−0.609186	−0.304593	+	0.952483i	$0.598520\pi$
$594$	0	0
$595$	6.98760i	0.286464i
$596$	0	0
$597$	− 31.3351i	− 1.28246i
$598$	0	0
$599$	19.9337	0.814468	0.407234	−	0.913324i	$-0.366493\pi$
$599$	19.9337	0.814468	0.407234	+	0.913324i	$0.366493\pi$
$600$	0	0
$601$	31.4765	1.28395	0.641976	−	0.766724i	$-0.278115\pi$
$601$	31.4765	1.28395	0.641976	+	0.766724i	$0.278115\pi$
$602$	0	0
$603$	− 0.111757i	− 0.00455110i
$604$	0	0
$605$	11.0417i	0.448908i
$606$	0	0
$607$	−25.8245	−1.04818	−0.524092	−	0.851662i	$-0.675595\pi$
$607$	−25.8245	−1.04818	−0.524092	+	0.851662i	$0.675595\pi$
$608$	0	0
$609$	−2.37018	−0.0960443
$610$	0	0
$611$	13.7532i	0.556396i
$612$	0	0
$613$	− 0.844965i	− 0.0341278i	−0.999854	−	0.0170639i	$-0.994568\pi$
$613$	− 0.844965i	− 0.0341278i	0.999854	−	0.0170639i	$-0.00543187\pi$
$614$	0	0
$615$	30.5646	1.23248
$616$	0	0
$617$	−10.0344	−0.403970	−0.201985	−	0.979389i	$-0.564739\pi$
$617$	−10.0344	−0.403970	−0.201985	+	0.979389i	$0.564739\pi$
$618$	0	0
$619$	29.2896i	1.17725i	0.808406	+	0.588625i	$0.200331\pi$
$619$	29.2896i	1.17725i	−0.808406	+	0.588625i	$0.799669\pi$
$620$	0	0
$621$	− 25.5253i	− 1.02430i
$622$	0	0
$623$	−31.5816	−1.26529
$624$	0	0
$625$	−8.81055	−0.352422
$626$	0	0
$627$	− 3.56244i	− 0.142270i
$628$	0	0
$629$	− 0.162791i	− 0.00649091i
$630$	0	0
$631$	1.02437	0.0407796	0.0203898	−	0.999792i	$-0.493509\pi$
$631$	1.02437	0.0407796	0.0203898	+	0.999792i	$0.493509\pi$
$632$	0	0
$633$	−26.4779	−1.05240
$634$	0	0
$635$	− 30.1673i	− 1.19715i
$636$	0	0
$637$	− 14.4043i	− 0.570721i
$638$	0	0
$639$	0.407887	0.0161358
$640$	0	0
$641$	−31.8367	−1.25747	−0.628737	−	0.777618i	$-0.716428\pi$
$641$	−31.8367	−1.25747	−0.628737	+	0.777618i	$0.716428\pi$
$642$	0	0
$643$	34.2996i	1.35264i	0.736606	+	0.676322i	$0.236427\pi$
$643$	34.2996i	1.35264i	−0.736606	+	0.676322i	$0.763573\pi$
$644$	0	0
$645$	− 4.08045i	− 0.160668i
$646$	0	0
$647$	−9.45559	−0.371738	−0.185869	−	0.982575i	$-0.559510\pi$
$647$	−9.45559	−0.371738	−0.185869	+	0.982575i	$0.559510\pi$
$648$	0	0
$649$	−24.0579	−0.944355
$650$	0	0
$651$	33.2936i	1.30488i
$652$	0	0
$653$	− 6.33451i	− 0.247888i	−0.992289	−	0.123944i	$-0.960446\pi$
$653$	− 6.33451i	− 0.247888i	0.992289	−	0.123944i	$-0.0395544\pi$
$654$	0	0
$655$	−16.1318	−0.630323
$656$	0	0
$657$	0.388590	0.0151604
$658$	0	0
$659$	− 7.22355i	− 0.281390i	−0.990053	−	0.140695i	$-0.955066\pi$
$659$	− 7.22355i	− 0.281390i	0.990053	−	0.140695i	$-0.0449337\pi$
$660$	0	0
$661$	0.529621i	0.0205999i	0.999947	+	0.0102999i	$0.00327863\pi$
$661$	0.529621i	0.0205999i	−0.999947	+	0.0102999i	$0.996721\pi$
$662$	0	0
$663$	17.1604	0.666454
$664$	0	0
$665$	3.31706	0.128630
$666$	0	0
$667$	− 3.37142i	− 0.130542i
$668$	0	0
$669$	− 23.7665i	− 0.918867i
$670$	0	0
$671$	18.4299	0.711479
$672$	0	0
$673$	41.7434	1.60909	0.804544	−	0.593893i	$-0.202410\pi$
$673$	41.7434	1.60909	0.804544	+	0.593893i	$0.202410\pi$
$674$	0	0
$675$	− 11.7871i	− 0.453684i
$676$	0	0
$677$	33.4165i	1.28430i	0.766578	+	0.642151i	$0.221958\pi$
$677$	33.4165i	1.28430i	−0.766578	+	0.642151i	$0.778042\pi$
$678$	0	0
$679$	−27.7474	−1.06485
$680$	0	0
$681$	2.55353	0.0978514
$682$	0	0
$683$	30.1765i	1.15467i	0.816507	+	0.577336i	$0.195908\pi$
$683$	30.1765i	1.15467i	−0.816507	+	0.577336i	$0.804092\pi$
$684$	0	0
$685$	− 33.6189i	− 1.28451i
$686$	0	0
$687$	−1.76988	−0.0675252
$688$	0	0
$689$	42.9716	1.63709
$690$	0	0
$691$	− 17.6002i	− 0.669544i	−0.942299	−	0.334772i	$-0.891341\pi$
$691$	− 17.6002i	− 0.669544i	0.942299	−	0.334772i	$-0.108659\pi$
$692$	0	0
$693$	0.364096i	0.0138309i
$694$	0	0
$695$	27.8830	1.05766
$696$	0	0
$697$	22.6968	0.859704
$698$	0	0
$699$	− 8.42005i	− 0.318475i
$700$	0	0
$701$	30.6993i	1.15950i	0.814796	+	0.579748i	$0.196849\pi$
$701$	30.6993i	1.15950i	−0.814796	+	0.579748i	$0.803151\pi$
$702$	0	0
$703$	−0.0772780	−0.00291459
$704$	0	0
$705$	8.17374	0.307841
$706$	0	0
$707$	− 2.00540i	− 0.0754210i
$708$	0	0
$709$	8.79437i	0.330279i	0.986270	+	0.165140i	$0.0528075\pi$
$709$	8.79437i	0.330279i	−0.986270	+	0.165140i	$0.947193\pi$
$710$	0	0
$711$	−0.0961853	−0.00360723
$712$	0	0
$713$	−47.3579	−1.77357
$714$	0	0
$715$	− 16.5618i	− 0.619375i
$716$	0	0
$717$	− 6.56256i	− 0.245083i
$718$	0	0
$719$	−12.5991	−0.469867	−0.234933	−	0.972011i	$-0.575487\pi$
$719$	−12.5991	−0.469867	−0.234933	+	0.972011i	$0.575487\pi$
$720$	0	0
$721$	−24.8609	−0.925869
$722$	0	0
$723$	2.40864i	0.0895781i
$724$	0	0
$725$	− 1.55685i	− 0.0578200i
$726$	0	0
$727$	39.7756	1.47519	0.737597	−	0.675241i	$-0.235960\pi$
$727$	39.7756	1.47519	0.737597	+	0.675241i	$0.235960\pi$
$728$	0	0
$729$	−27.7402	−1.02741
$730$	0	0
$731$	− 3.03009i	− 0.112072i
$732$	0	0
$733$	5.50154i	0.203204i	0.994825	+	0.101602i	$0.0323968\pi$
$733$	5.50154i	0.203204i	−0.994825	+	0.101602i	$0.967603\pi$
$734$	0	0
$735$	−8.56071	−0.315766
$736$	0	0
$737$	2.66894	0.0983115
$738$	0	0
$739$	52.6937i	1.93837i	0.246335	+	0.969185i	$0.420774\pi$
$739$	52.6937i	1.93837i	−0.246335	+	0.969185i	$0.579226\pi$
$740$	0	0
$741$	− 8.14614i	− 0.299256i
$742$	0	0
$743$	6.61647	0.242735	0.121367	−	0.992608i	$-0.461272\pi$
$743$	6.61647	0.242735	0.121367	+	0.992608i	$0.461272\pi$
$744$	0	0
$745$	22.6948	0.831474
$746$	0	0
$747$	− 0.216261i	− 0.00791256i
$748$	0	0
$749$	30.5329i	1.11565i
$750$	0	0
$751$	39.9062	1.45620	0.728099	−	0.685472i	$-0.240404\pi$
$751$	39.9062	1.45620	0.728099	+	0.685472i	$0.240404\pi$
$752$	0	0
$753$	−14.0736	−0.512869
$754$	0	0
$755$	− 11.2632i	− 0.409911i
$756$	0	0
$757$	− 34.2645i	− 1.24537i	−0.782475	−	0.622683i	$-0.786043\pi$
$757$	− 34.2645i	− 1.24537i	0.782475	−	0.622683i	$-0.213957\pi$
$758$	0	0
$759$	17.2578	0.626417
$760$	0	0
$761$	−12.4757	−0.452243	−0.226122	−	0.974099i	$-0.572605\pi$
$761$	−12.4757	−0.452243	−0.226122	+	0.974099i	$0.572605\pi$
$762$	0	0
$763$	15.2830i	0.553281i
$764$	0	0
$765$	0.306060i	0.0110656i
$766$	0	0
$767$	−55.0126	−1.98639
$768$	0	0
$769$	3.32384	0.119861	0.0599304	−	0.998203i	$-0.480912\pi$
$769$	3.32384	0.119861	0.0599304	+	0.998203i	$0.480912\pi$
$770$	0	0
$771$	− 25.7233i	− 0.926402i
$772$	0	0
$773$	14.4248i	0.518824i	0.965767	+	0.259412i	$0.0835287\pi$
$773$	14.4248i	0.518824i	−0.965767	+	0.259412i	$0.916471\pi$
$774$	0	0
$775$	−21.8689	−0.785555
$776$	0	0
$777$	−0.263185	−0.00944170
$778$	0	0
$779$	− 10.7743i	− 0.386030i
$780$	0	0
$781$	9.74099i	0.348560i
$782$	0	0
$783$	−3.66699	−0.131047
$784$	0	0
$785$	−38.9255	−1.38931
$786$	0	0
$787$	− 38.6403i	− 1.37738i	−0.725057	−	0.688689i	$-0.758187\pi$
$787$	− 38.6403i	− 1.37738i	0.725057	−	0.688689i	$-0.241813\pi$
$788$	0	0
$789$	− 16.1983i	− 0.576676i
$790$	0	0
$791$	34.7103	1.23416
$792$	0	0
$793$	42.1433	1.49655
$794$	0	0
$795$	− 25.5387i	− 0.905763i
$796$	0	0
$797$	− 30.9316i	− 1.09565i	−0.836592	−	0.547826i	$-0.815456\pi$
$797$	− 30.9316i	− 1.09565i	0.836592	−	0.547826i	$-0.184544\pi$
$798$	0	0
$799$	6.06971	0.214731
$800$	0	0
$801$	−1.38329	−0.0488762
$802$	0	0
$803$	9.28015i	0.327490i
$804$	0	0
$805$	16.0691i	0.566360i
$806$	0	0
$807$	6.80177	0.239434
$808$	0	0
$809$	56.7269	1.99441	0.997206	−	0.0747056i	$-0.0238017\pi$
$809$	56.7269	1.99441	0.997206	+	0.0747056i	$0.0238017\pi$
$810$	0	0
$811$	4.91800i	0.172694i	0.996265	+	0.0863472i	$0.0275195\pi$
$811$	4.91800i	0.172694i	−0.996265	+	0.0863472i	$0.972481\pi$
$812$	0	0
$813$	− 10.8956i	− 0.382125i
$814$	0	0
$815$	−1.77812	−0.0622849
$816$	0	0
$817$	−1.43840	−0.0503233
$818$	0	0
$819$	0.832570i	0.0290924i
$820$	0	0
$821$	− 19.0797i	− 0.665886i	−0.942947	−	0.332943i	$-0.891958\pi$
$821$	− 19.0797i	− 0.665886i	0.942947	−	0.332943i	$-0.108042\pi$
$822$	0	0
$823$	19.6263	0.684131	0.342066	−	0.939676i	$-0.388873\pi$
$823$	19.6263	0.684131	0.342066	+	0.939676i	$0.388873\pi$
$824$	0	0
$825$	7.96928	0.277455
$826$	0	0
$827$	− 28.0107i	− 0.974029i	−0.873394	−	0.487014i	$-0.838086\pi$
$827$	− 28.0107i	− 0.974029i	0.873394	−	0.487014i	$-0.161914\pi$
$828$	0	0
$829$	− 23.1932i	− 0.805533i	−0.915303	−	0.402766i	$-0.868049\pi$
$829$	− 23.1932i	− 0.805533i	0.915303	−	0.402766i	$-0.131951\pi$
$830$	0	0
$831$	−25.8995	−0.898442
$832$	0	0
$833$	−6.35707	−0.220259
$834$	0	0
$835$	− 22.6422i	− 0.783567i
$836$	0	0
$837$	51.5097i	1.78044i
$838$	0	0
$839$	−27.6363	−0.954110	−0.477055	−	0.878873i	$-0.658296\pi$
$839$	−27.6363	−0.954110	−0.477055	+	0.878873i	$0.658296\pi$
$840$	0	0
$841$	28.5157	0.983299
$842$	0	0
$843$	− 25.4388i	− 0.876157i
$844$	0	0
$845$	− 16.2626i	− 0.559450i
$846$	0	0
$847$	13.2560	0.455481
$848$	0	0
$849$	25.5878	0.878171
$850$	0	0
$851$	− 0.374363i	− 0.0128330i
$852$	0	0
$853$	53.0502i	1.81640i	0.418532	+	0.908202i	$0.362545\pi$
$853$	53.0502i	1.81640i	−0.418532	+	0.908202i	$0.637455\pi$
$854$	0	0
$855$	0.145289	0.00496877
$856$	0	0
$857$	−8.06832	−0.275609	−0.137804	−	0.990459i	$-0.544004\pi$
$857$	−8.06832	−0.275609	−0.137804	+	0.990459i	$0.544004\pi$
$858$	0	0
$859$	− 10.1806i	− 0.347357i	−0.984802	−	0.173679i	$-0.944435\pi$
$859$	− 10.1806i	− 0.347357i	0.984802	−	0.173679i	$-0.0555654\pi$
$860$	0	0
$861$	− 36.6940i	− 1.25053i
$862$	0	0
$863$	45.7074	1.55590	0.777949	−	0.628327i	$-0.216260\pi$
$863$	45.7074	1.55590	0.777949	+	0.628327i	$0.216260\pi$
$864$	0	0
$865$	19.2292	0.653811
$866$	0	0
$867$	21.4394i	0.728119i
$868$	0	0
$869$	− 2.29706i	− 0.0779223i
$870$	0	0
$871$	6.10299	0.206792
$872$	0	0
$873$	−1.21535	−0.0411334
$874$	0	0
$875$	24.0056i	0.811539i
$876$	0	0
$877$	− 8.41970i	− 0.284313i	−0.989844	−	0.142157i	$-0.954596\pi$
$877$	− 8.41970i	− 0.284313i	0.989844	−	0.142157i	$-0.0454037\pi$
$878$	0	0
$879$	−37.0310	−1.24903
$880$	0	0
$881$	8.33625	0.280855	0.140428	−	0.990091i	$-0.455152\pi$
$881$	8.33625	0.280855	0.140428	+	0.990091i	$0.455152\pi$
$882$	0	0
$883$	− 26.9797i	− 0.907941i	−0.891017	−	0.453970i	$-0.850007\pi$
$883$	− 26.9797i	− 0.907941i	0.891017	−	0.453970i	$-0.149993\pi$
$884$	0	0
$885$	32.6948i	1.09902i
$886$	0	0
$887$	25.8570	0.868193	0.434097	−	0.900866i	$-0.357068\pi$
$887$	25.8570	0.868193	0.434097	+	0.900866i	$0.357068\pi$
$888$	0	0
$889$	−36.2171	−1.21468
$890$	0	0
$891$	− 18.2234i	− 0.610505i
$892$	0	0
$893$	− 2.88133i	− 0.0964200i
$894$	0	0
$895$	34.0726	1.13892
$896$	0	0
$897$	39.4629	1.31763
$898$	0	0
$899$	6.80348i	0.226909i
$900$	0	0
$901$	− 18.9647i	− 0.631805i
$902$	0	0
$903$	−4.89875	−0.163020
$904$	0	0
$905$	31.2074	1.03737
$906$	0	0
$907$	− 21.7649i	− 0.722690i	−0.932432	−	0.361345i	$-0.882318\pi$
$907$	− 21.7649i	− 0.722690i	0.932432	−	0.361345i	$-0.117682\pi$
$908$	0	0
$909$	− 0.0878377i	− 0.00291339i
$910$	0	0
$911$	13.3098	0.440974	0.220487	−	0.975390i	$-0.429235\pi$
$911$	13.3098	0.440974	0.220487	+	0.975390i	$0.429235\pi$
$912$	0	0
$913$	5.16464	0.170925
$914$	0	0
$915$	− 25.0464i	− 0.828007i
$916$	0	0
$917$	19.3669i	0.639552i
$918$	0	0
$919$	29.7421	0.981103	0.490551	−	0.871412i	$-0.336795\pi$
$919$	29.7421	0.981103	0.490551	+	0.871412i	$0.336795\pi$
$920$	0	0
$921$	13.3365	0.439451
$922$	0	0
$923$	22.2745i	0.733174i
$924$	0	0
$925$	− 0.172873i	− 0.00568403i
$926$	0	0
$927$	−1.08892	−0.0357648
$928$	0	0
$929$	−30.3181	−0.994705	−0.497352	−	0.867549i	$-0.665694\pi$
$929$	−30.3181	−0.994705	−0.497352	+	0.867549i	$0.665694\pi$
$930$	0	0
$931$	3.01774i	0.0989024i
$932$	0	0
$933$	21.8941i	0.716779i
$934$	0	0
$935$	−7.30921	−0.239037
$936$	0	0
$937$	−8.81465	−0.287962	−0.143981	−	0.989580i	$-0.545990\pi$
$937$	−8.81465	−0.287962	−0.143981	+	0.989580i	$0.545990\pi$
$938$	0	0
$939$	6.95586i	0.226996i
$940$	0	0
$941$	42.2936i	1.37873i	0.724413	+	0.689366i	$0.242111\pi$
$941$	42.2936i	1.37873i	−0.724413	+	0.689366i	$0.757889\pi$
$942$	0	0
$943$	52.1949	1.69970
$944$	0	0
$945$	17.4778	0.568553
$946$	0	0
$947$	30.2387i	0.982626i	0.870983	+	0.491313i	$0.163483\pi$
$947$	30.2387i	0.982626i	−0.870983	+	0.491313i	$0.836517\pi$
$948$	0	0
$949$	21.2207i	0.688853i
$950$	0	0
$951$	−3.86464	−0.125320
$952$	0	0
$953$	−1.52937	−0.0495411	−0.0247706	−	0.999693i	$-0.507886\pi$
$953$	−1.52937	−0.0495411	−0.0247706	+	0.999693i	$0.507886\pi$
$954$	0	0
$955$	23.3362i	0.755141i
$956$	0	0
$957$	− 2.47926i	− 0.0801432i
$958$	0	0
$959$	−40.3609	−1.30332
$960$	0	0
$961$	64.5676	2.08283
$962$	0	0
$963$	1.33736i	0.0430957i
$964$	0	0
$965$	− 39.6771i	− 1.27725i
$966$	0	0
$967$	3.67723	0.118252	0.0591259	−	0.998251i	$-0.481169\pi$
$967$	3.67723	0.118252	0.0591259	+	0.998251i	$0.481169\pi$
$968$	0	0
$969$	−3.59513	−0.115492
$970$	0	0
$971$	− 5.89852i	− 0.189293i	−0.995511	−	0.0946463i	$-0.969828\pi$
$971$	− 5.89852i	− 0.189293i	0.995511	−	0.0946463i	$-0.0301720\pi$
$972$	0	0
$973$	− 33.4747i	− 1.07315i
$974$	0	0
$975$	18.2232	0.583608
$976$	0	0
$977$	−19.5003	−0.623871	−0.311935	−	0.950103i	$-0.600977\pi$
$977$	−19.5003	−0.623871	−0.311935	+	0.950103i	$0.600977\pi$
$978$	0	0
$979$	− 33.0352i	− 1.05581i
$980$	0	0
$981$	0.669403i	0.0213724i
$982$	0	0
$983$	2.43384	0.0776274	0.0388137	−	0.999246i	$-0.487642\pi$
$983$	2.43384	0.0776274	0.0388137	+	0.999246i	$0.487642\pi$
$984$	0	0
$985$	3.07328	0.0979229
$986$	0	0
$987$	− 9.81291i	− 0.312348i
$988$	0	0
$989$	− 6.96815i	− 0.221574i
$990$	0	0
$991$	15.1966	0.482736	0.241368	−	0.970434i	$-0.422404\pi$
$991$	15.1966	0.482736	0.241368	+	0.970434i	$0.422404\pi$
$992$	0	0
$993$	12.4784	0.395990
$994$	0	0
$995$	30.5196i	0.967537i
$996$	0	0
$997$	42.5056i	1.34617i	0.739567	+	0.673083i	$0.235030\pi$
$997$	42.5056i	1.34617i	−0.739567	+	0.673083i	$0.764970\pi$
$998$	0	0
$999$	−0.407183	−0.0128827

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.2.c.b.305.6		16
3.2	odd	2		5472.2.g.b.2737.6		16
4.3	odd	2		152.2.c.b.77.8	yes	16
8.3	odd	2		152.2.c.b.77.7	✓	16
8.5	even	2	inner	608.2.c.b.305.11		16
12.11	even	2		1368.2.g.b.685.9		16
16.3	odd	4		4864.2.a.bo.1.3		8
16.5	even	4		4864.2.a.bp.1.3		8
16.11	odd	4		4864.2.a.bq.1.6		8
16.13	even	4		4864.2.a.bn.1.6		8
24.5	odd	2		5472.2.g.b.2737.11		16
24.11	even	2		1368.2.g.b.685.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.7	✓	16	8.3	odd	2
152.2.c.b.77.8	yes	16	4.3	odd	2
608.2.c.b.305.6		16	1.1	even	1	trivial
608.2.c.b.305.11		16	8.5	even	2	inner
1368.2.g.b.685.9		16	12.11	even	2
1368.2.g.b.685.10		16	24.11	even	2
4864.2.a.bn.1.6		8	16.13	even	4
4864.2.a.bo.1.3		8	16.3	odd	4
4864.2.a.bp.1.3		8	16.5	even	4
4864.2.a.bq.1.6		8	16.11	odd	4
5472.2.g.b.2737.6		16	3.2	odd	2
5472.2.g.b.2737.11		16	24.5	odd	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.70663i q^{3} +1.66222i q^{5} +1.99556 q^{7} +0.0874066 q^{9} +2.08741i q^{11} +4.77322i q^{13} +2.83680 q^{15} +2.10657 q^{17} -1.00000i q^{19} -3.40569i q^{21} +4.84437 q^{23} +2.23703 q^{25} -5.26907i q^{27} -0.695946i q^{29} -9.77587 q^{31} +3.56244 q^{33} +3.31706i q^{35} -0.0772780i q^{37} +8.14614 q^{39} +10.7743 q^{41} -1.43840i q^{43} +0.145289i q^{45} +2.88133 q^{47} -3.01774 q^{49} -3.59513i q^{51} -9.00264i q^{53} -3.46973 q^{55} -1.70663 q^{57} +11.5253i q^{59} -8.82910i q^{61} +0.174425 q^{63} -7.93414 q^{65} -1.27859i q^{67} -8.26756i q^{69} +4.66655 q^{71} +4.44578 q^{73} -3.81779i q^{75} +4.16555i q^{77} -1.10044 q^{79} -8.73014 q^{81} -2.47419i q^{83} +3.50157i q^{85} -1.18772 q^{87} -15.8259 q^{89} +9.52526i q^{91} +16.6838i q^{93} +1.66222 q^{95} -13.9046 q^{97} +0.182453i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 24 q^{9} - 8 q^{17} - 24 q^{25} - 16 q^{31} - 8 q^{39} + 16 q^{41} - 24 q^{47} + 24 q^{49} - 16 q^{55} + 32 q^{63} + 16 q^{65} - 48 q^{71} + 48 q^{79} - 16 q^{81} + 48 q^{87} - 16 q^{89}+ \cdots + 32 q^{97}+O(q^{100}) \) 16 * q + 8 * q^7 - 24 * q^9 - 8 * q^17 - 24 * q^25 - 16 * q^31 - 8 * q^39 + 16 * q^41 - 24 * q^47 + 24 * q^49 - 16 * q^55 + 32 * q^63 + 16 * q^65 - 48 * q^71 + 48 * q^79 - 16 * q^81 + 48 * q^87 - 16 * q^89 - 16 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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