LMFDB - Embedded newform 608.2.b.c.303.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,2,Mod(303,608)]

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

lf = mfeigenbasis(mf)

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")

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

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			303.12
Root			$-1.37364 - 0.336338i$ of defining polynomial
Character	$\chi$	$=$	608.303
Dual form			608.2.b.c.303.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.97320i q^{3} +3.04222i q^{5} +2.23607i q^{7} -5.83991 q^{9} +O(q^{10})$	$q+2.97320i q^{3} +3.04222i q^{5} +2.23607i q^{7} -5.83991 q^{9} +2.29240 q^{11} -3.09769 q^{13} -9.04512 q^{15} +5.58480 q^{17} +(4.29240 - 0.758478i) q^{19} -6.64827 q^{21} -5.51401i q^{23} -4.25511 q^{25} -8.44361i q^{27} +1.85457 q^{29} +4.25142 q^{31} +6.81576i q^{33} -6.80261 q^{35} +1.24312 q^{37} -9.21006i q^{39} -8.33272i q^{41} -4.87720 q^{43} -17.7663i q^{45} +9.36238i q^{47} +2.00000 q^{49} +16.6047i q^{51} -11.4420 q^{53} +6.97399i q^{55} +(2.25511 + 12.7622i) q^{57} +2.49721i q^{59} -2.26613i q^{61} -13.0584i q^{63} -9.42387i q^{65} +4.49015i q^{67} +16.3942 q^{69} -14.6982 q^{71} +1.51021 q^{73} -12.6513i q^{75} +5.12597i q^{77} +11.5314 q^{79} +7.58480 q^{81} -2.00000 q^{83} +16.9902i q^{85} +5.51401i q^{87} +5.29881i q^{89} -6.92665i q^{91} +12.6403i q^{93} +(2.30746 + 13.0584i) q^{95} -1.17375i q^{97} -13.3874 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9}+O(q^{10}) $ 12 * q - 8 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$			2.97320i			1.71658i	0.513168	+	0.858288i	$0.328472\pi$
$3$			2.97320i			1.71658i	−0.513168	+	0.858288i	$0.671528\pi$
$4$		0			0
$5$			3.04222i			1.36052i	0.732970	+	0.680261i	$0.238134\pi$
$5$			3.04222i			1.36052i	−0.732970	+	0.680261i	$0.761866\pi$
$6$		0			0
$7$			2.23607i			0.845154i	0.906327	+	0.422577i	$0.138874\pi$
$7$			2.23607i			0.845154i	−0.906327	+	0.422577i	$0.861126\pi$
$8$		0			0
$9$	−5.83991			−1.94664
$10$		0			0
$11$	2.29240			0.691185			0.345593	−	0.938385i	$-0.387678\pi$
$11$	2.29240			0.691185			0.345593	+	0.938385i	$0.387678\pi$
$12$		0			0
$13$	−3.09769			−0.859146			−0.429573	−	0.903032i	$-0.641336\pi$
$13$	−3.09769			−0.859146			−0.429573	+	0.903032i	$0.641336\pi$
$14$		0			0
$15$	−9.04512			−2.33544
$16$		0			0
$17$	5.58480			1.35451			0.677257	−	0.735747i	$-0.263169\pi$
$17$	5.58480			1.35451			0.677257	+	0.735747i	$0.263169\pi$
$18$		0			0
$19$	4.29240	−	0.758478i	0.984744	−	0.174007i
$19$	4.29240	−	0.758478i	0.984744	−	0.174007i
$20$		0			0
$21$	−6.64827			−1.45077
$22$		0			0
$23$		−	5.51401i		−	1.14975i	−0.818241	−	0.574875i	$-0.805051\pi$
$23$		−	5.51401i		−	1.14975i	0.818241	−	0.574875i	$-0.194949\pi$
$24$		0			0
$25$	−4.25511			−0.851021
$26$		0			0
$27$		−	8.44361i		−	1.62497i
$28$		0			0
$29$	1.85457			0.344385			0.172193	−	0.985063i	$-0.444915\pi$
$29$	1.85457			0.344385			0.172193	+	0.985063i	$0.444915\pi$
$30$		0			0
$31$	4.25142			0.763578			0.381789	−	0.924250i	$-0.375308\pi$
$31$	4.25142			0.763578			0.381789	+	0.924250i	$0.375308\pi$
$32$		0			0
$33$			6.81576i			1.18647i
$34$		0			0
$35$	−6.80261			−1.14985
$36$		0			0
$37$	1.24312			0.204368			0.102184	−	0.994766i	$-0.467417\pi$
$37$	1.24312			0.204368			0.102184	+	0.994766i	$0.467417\pi$
$38$		0			0
$39$		−	9.21006i		−	1.47479i
$40$		0			0
$41$		−	8.33272i		−	1.30135i	−0.759355	−	0.650676i	$-0.774486\pi$
$41$		−	8.33272i		−	1.30135i	0.759355	−	0.650676i	$-0.225514\pi$
$42$		0			0
$43$	−4.87720			−0.743767			−0.371883	−	0.928279i	$-0.621288\pi$
$43$	−4.87720			−0.743767			−0.371883	+	0.928279i	$0.621288\pi$
$44$		0			0
$45$		−	17.7663i		−	2.64844i
$46$		0			0
$47$			9.36238i			1.36564i	0.730585	+	0.682822i	$0.239247\pi$
$47$			9.36238i			1.36564i	−0.730585	+	0.682822i	$0.760753\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$			16.6047i			2.32513i
$52$		0			0
$53$	−11.4420			−1.57168			−0.785838	−	0.618432i	$-0.787768\pi$
$53$	−11.4420			−1.57168			−0.785838	+	0.618432i	$0.787768\pi$
$54$		0			0
$55$			6.97399i			0.940373i
$56$		0			0
$57$	2.25511	+	12.7622i	0.298696	+	1.69039i
$58$		0			0
$59$			2.49721i			0.325110i	0.986700	+	0.162555i	$0.0519734\pi$
$59$			2.49721i			0.325110i	−0.986700	+	0.162555i	$0.948027\pi$
$60$		0			0
$61$		−	2.26613i		−	0.290149i	−0.989421	−	0.145074i	$-0.953658\pi$
$61$		−	2.26613i		−	0.290149i	0.989421	−	0.145074i	$-0.0463422\pi$
$62$		0			0
$63$		−	13.0584i		−	1.64521i
$64$		0			0
$65$		−	9.42387i		−	1.16889i
$66$		0			0
$67$			4.49015i			0.548560i	0.961650	+	0.274280i	$0.0884394\pi$
$67$			4.49015i			0.548560i	−0.961650	+	0.274280i	$0.911561\pi$
$68$		0			0
$69$	16.3942			1.97363
$70$		0			0
$71$	−14.6982			−1.74436			−0.872180	−	0.489186i	$-0.837294\pi$
$71$	−14.6982			−1.74436			−0.872180	+	0.489186i	$0.837294\pi$
$72$		0			0
$73$	1.51021			0.176757			0.0883784	−	0.996087i	$-0.471832\pi$
$73$	1.51021			0.176757			0.0883784	+	0.996087i	$0.471832\pi$
$74$		0			0
$75$		−	12.6513i		−	1.46084i
$76$		0			0
$77$			5.12597i			0.584158i
$78$		0			0
$79$	11.5314			1.29738			0.648690	−	0.761053i	$-0.275317\pi$
$79$	11.5314			1.29738			0.648690	+	0.761053i	$0.275317\pi$
$80$		0			0
$81$	7.58480			0.842756
$82$		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
$85$			16.9902i			1.84285i
$86$		0			0
$87$			5.51401i			0.591164i
$88$		0			0
$89$			5.29881i			0.561673i	0.959756	+	0.280836i	$0.0906118\pi$
$89$			5.29881i			0.561673i	−0.959756	+	0.280836i	$0.909388\pi$
$90$		0			0
$91$		−	6.92665i		−	0.726111i
$92$		0			0
$93$			12.6403i			1.31074i
$94$		0			0
$95$	2.30746	+	13.0584i	0.236740	+	1.33977i
$96$		0			0
$97$		−	1.17375i		−	0.119176i	−0.998223	−	0.0595881i	$-0.981021\pi$
$97$		−	1.17375i		−	0.119176i	0.998223	−	0.0595881i	$-0.0189787\pi$
$98$		0			0
$99$	−13.3874			−1.34549
$100$		0			0
$101$		−	12.6403i		−	1.25776i	−0.777503	−	0.628880i	$-0.783514\pi$
$101$		−	12.6403i		−	1.25776i	0.777503	−	0.628880i	$-0.216486\pi$
$102$		0			0
$103$	3.70914			0.365473			0.182736	−	0.983162i	$-0.441505\pi$
$103$	3.70914			0.365473			0.182736	+	0.983162i	$0.441505\pi$
$104$		0			0
$105$		−	20.2255i		−	1.97381i
$106$		0			0
$107$		−	11.1343i		−	1.07639i	−0.842819	−	0.538197i	$-0.819105\pi$
$107$		−	11.1343i		−	1.07639i	0.842819	−	0.538197i	$-0.180895\pi$
$108$		0			0
$109$	11.4420			1.09594			0.547971	−	0.836497i	$-0.315400\pi$
$109$	11.4420			1.09594			0.547971	+	0.836497i	$0.315400\pi$
$110$		0			0
$111$			3.69605i			0.350814i
$112$		0			0
$113$			17.5348i			1.64954i	0.565471	+	0.824768i	$0.308694\pi$
$113$			17.5348i			1.64954i	−0.565471	+	0.824768i	$0.691306\pi$
$114$		0			0
$115$	16.7748			1.56426
$116$		0			0
$117$	18.0902			1.67244
$118$		0			0
$119$			12.4880i			1.14477i
$120$		0			0
$121$	−5.74489			−0.522263
$122$		0			0
$123$	24.7748			2.23387
$124$		0			0
$125$			2.26613i			0.202689i
$126$		0			0
$127$	2.84974			0.252873			0.126437	−	0.991975i	$-0.459646\pi$
$127$	2.84974			0.252873			0.126437	+	0.991975i	$0.459646\pi$
$128$		0			0
$129$		−	14.5009i		−	1.27673i
$130$		0			0
$131$	−7.97222			−0.696536			−0.348268	−	0.937395i	$-0.613230\pi$
$131$	−7.97222			−0.696536			−0.348268	+	0.937395i	$0.613230\pi$
$132$		0			0
$133$	1.69601	+	9.59810i	0.147063	+	0.832261i
$134$		0			0
$135$	25.6873			2.21081
$136$		0			0
$137$	9.19003			0.785157			0.392578	−	0.919719i	$-0.371583\pi$
$137$	9.19003			0.785157			0.392578	+	0.919719i	$0.371583\pi$
$138$		0			0
$139$	3.89762			0.330592			0.165296	−	0.986244i	$-0.447142\pi$
$139$	3.89762			0.330592			0.165296	+	0.986244i	$0.447142\pi$
$140$		0			0
$141$	−27.8362			−2.34423
$142$		0			0
$143$	−7.10116			−0.593829
$144$		0			0
$145$			5.64202i			0.468544i
$146$		0			0
$147$			5.94640i			0.490451i
$148$		0			0
$149$		−	10.4343i		−	0.854813i	−0.904060	−	0.427406i	$-0.859427\pi$
$149$		−	10.4343i		−	0.854813i	0.904060	−	0.427406i	$-0.140573\pi$
$150$		0			0
$151$	10.9891			0.894279			0.447140	−	0.894464i	$-0.352443\pi$
$151$	10.9891			0.894279			0.447140	+	0.894464i	$0.352443\pi$
$152$		0			0
$153$	−32.6147			−2.63675
$154$		0			0
$155$			12.9338i			1.03886i
$156$		0			0
$157$			5.77980i			0.461278i	0.973039	+	0.230639i	$0.0740816\pi$
$157$			5.77980i			0.461278i	−0.973039	+	0.230639i	$0.925918\pi$
$158$		0			0
$159$		−	34.0193i		−	2.69790i
$160$		0			0
$161$	12.3297			0.971716
$162$		0			0
$163$	−6.65940			−0.521604			−0.260802	−	0.965392i	$-0.583987\pi$
$163$	−6.65940			−0.521604			−0.260802	+	0.965392i	$0.583987\pi$
$164$		0			0
$165$	−20.7351			−1.61422
$166$		0			0
$167$	−1.94397			−0.150429			−0.0752143	−	0.997167i	$-0.523964\pi$
$167$	−1.94397			−0.150429			−0.0752143	+	0.997167i	$0.523964\pi$
$168$		0			0
$169$	−3.40429			−0.261869
$170$		0			0
$171$	−25.0672	+	4.42944i	−1.91694	+	0.338728i
$172$		0			0
$173$	20.5765			1.56440			0.782201	−	0.623026i	$-0.214097\pi$
$173$	20.5765			1.56440			0.782201	+	0.623026i	$0.214097\pi$
$174$		0			0
$175$		−	9.51470i		−	0.719244i
$176$		0			0
$177$	−7.42471			−0.558075
$178$		0			0
$179$		−	8.72610i		−	0.652220i	−0.945332	−	0.326110i	$-0.894262\pi$
$179$		−	8.72610i		−	0.652220i	0.945332	−	0.326110i	$-0.105738\pi$
$180$		0			0
$181$	8.50284			0.632011			0.316006	−	0.948757i	$-0.397658\pi$
$181$	8.50284			0.632011			0.316006	+	0.948757i	$0.397658\pi$
$182$		0			0
$183$	6.73767			0.498063
$184$		0			0
$185$			3.78185i			0.278047i
$186$		0			0
$187$	12.8026			0.936220
$188$		0			0
$189$	18.8805			1.37335
$190$		0			0
$191$		−	11.2405i		−	0.813332i	−0.913577	−	0.406666i	$-0.866691\pi$
$191$		−	11.2405i		−	0.813332i	0.913577	−	0.406666i	$-0.133309\pi$
$192$		0			0
$193$		−	7.76773i		−	0.559134i	−0.960126	−	0.279567i	$-0.909809\pi$
$193$		−	7.76773i		−	0.559134i	0.960126	−	0.279567i	$-0.0901909\pi$
$194$		0			0
$195$	28.0190			2.00648
$196$		0			0
$197$			3.63592i			0.259048i	0.991576	+	0.129524i	$0.0413450\pi$
$197$			3.63592i			0.259048i	−0.991576	+	0.129524i	$0.958655\pi$
$198$		0			0
$199$		−	5.46068i		−	0.387097i	−0.981091	−	0.193549i	$-0.938000\pi$
$199$		−	5.46068i		−	0.387097i	0.981091	−	0.193549i	$-0.0619997\pi$
$200$		0			0
$201$	−13.3501			−0.941645
$202$		0			0
$203$			4.14695i			0.291059i
$204$		0			0
$205$	25.3500			1.77052
$206$		0			0
$207$			32.2013i			2.23815i
$208$		0			0
$209$	9.83991	−	1.73874i	0.680641	−	0.120271i
$210$		0			0
$211$		−	15.3420i		−	1.05619i	−0.849187	−	0.528093i	$-0.822907\pi$
$211$		−	15.3420i		−	1.05619i	0.849187	−	0.528093i	$-0.177093\pi$
$212$		0			0
$213$		−	43.7008i		−	2.99433i
$214$		0			0
$215$		−	14.8375i		−	1.01191i
$216$		0			0
$217$			9.50647i			0.645341i
$218$		0			0
$219$			4.49015i			0.303416i
$220$		0			0
$221$	−17.3000			−1.16372
$222$		0			0
$223$	10.4468			0.699570			0.349785	−	0.936830i	$-0.386255\pi$
$223$	10.4468			0.699570			0.349785	+	0.936830i	$0.386255\pi$
$224$		0			0
$225$	24.8494			1.65663
$226$		0			0
$227$		−	0.193492i		−	0.0128425i	−0.999979	−	0.00642124i	$-0.997956\pi$
$227$		−	0.193492i		−	0.0128425i	0.999979	−	0.00642124i	$-0.00204396\pi$
$228$		0			0
$229$		−	19.3185i		−	1.27660i	−0.769788	−	0.638300i	$-0.779638\pi$
$229$		−	19.3185i		−	1.27660i	0.769788	−	0.638300i	$-0.220362\pi$
$230$		0			0
$231$	−15.2405			−1.00275
$232$		0			0
$233$	−12.4247			−0.813970			−0.406985	−	0.913435i	$-0.633420\pi$
$233$	−12.4247			−0.813970			−0.406985	+	0.913435i	$0.633420\pi$
$234$		0			0
$235$	−28.4824			−1.85799
$236$		0			0
$237$			34.2850i			2.22705i
$238$		0			0
$239$			0.152323i			0.00985293i	0.999988	+	0.00492647i	$0.00156815\pi$
$239$			0.152323i			0.00985293i	−0.999988	+	0.00492647i	$0.998432\pi$
$240$		0			0
$241$			11.9754i			0.771403i	0.922624	+	0.385701i	$0.126041\pi$
$241$			11.9754i			0.771403i	−0.922624	+	0.385701i	$0.873959\pi$
$242$		0			0
$243$		−	2.77971i		−	0.178318i
$244$		0			0
$245$			6.08444i			0.388721i
$246$		0			0
$247$	−13.2965	+	2.34953i	−0.846039	+	0.149497i
$248$		0			0
$249$		−	5.94640i		−	0.376838i
$250$		0			0
$251$	−7.82303			−0.493785			−0.246893	−	0.969043i	$-0.579410\pi$
$251$	−7.82303			−0.493785			−0.246893	+	0.969043i	$0.579410\pi$
$252$		0			0
$253$		−	12.6403i		−	0.794690i
$254$		0			0
$255$	−50.5152			−3.16339
$256$		0			0
$257$			13.4097i			0.836477i	0.908337	+	0.418239i	$0.137352\pi$
$257$			13.4097i			0.836477i	−0.908337	+	0.418239i	$0.862648\pi$
$258$		0			0
$259$			2.77971i			0.172723i
$260$		0			0
$261$	−10.8305			−0.670393
$262$		0			0
$263$			4.11416i			0.253690i	0.991923	+	0.126845i	$0.0404851\pi$
$263$			4.11416i			0.253690i	−0.991923	+	0.126845i	$0.959515\pi$
$264$		0			0
$265$		−	34.8090i		−	2.13830i
$266$		0			0
$267$	−15.7544			−0.964154
$268$		0			0
$269$	−26.4345			−1.61174			−0.805871	−	0.592091i	$-0.798302\pi$
$269$	−26.4345			−1.61174			−0.805871	+	0.592091i	$0.798302\pi$
$270$		0			0
$271$			19.2952i			1.17210i	0.810275	+	0.586050i	$0.199318\pi$
$271$			19.2952i			1.17210i	−0.810275	+	0.586050i	$0.800682\pi$
$272$		0			0
$273$	20.5943			1.24642
$274$		0			0
$275$	−9.75441			−0.588213
$276$		0			0
$277$			26.7106i			1.60488i	0.596731	+	0.802441i	$0.296466\pi$
$277$			26.7106i			1.60488i	−0.596731	+	0.802441i	$0.703534\pi$
$278$		0			0
$279$	−24.8279			−1.48641
$280$		0			0
$281$			23.4424i			1.39846i	0.714899	+	0.699228i	$0.246473\pi$
$281$			23.4424i			1.39846i	−0.714899	+	0.699228i	$0.753527\pi$
$282$		0			0
$283$	11.5366			0.685780			0.342890	−	0.939376i	$-0.388594\pi$
$283$	11.5366			0.685780			0.342890	+	0.939376i	$0.388594\pi$
$284$		0			0
$285$	−38.8253	+	6.86053i	−2.29981	+	0.406383i
$286$		0			0
$287$	18.6325			1.09984
$288$		0			0
$289$	14.1900			0.834707
$290$		0			0
$291$	3.48979			0.204575
$292$		0			0
$293$	−11.7793			−0.688156			−0.344078	−	0.938941i	$-0.611808\pi$
$293$	−11.7793			−0.688156			−0.344078	+	0.938941i	$0.611808\pi$
$294$		0			0
$295$	−7.59707			−0.442319
$296$		0			0
$297$		−	19.3561i		−	1.12316i
$298$		0			0
$299$			17.0807i			0.987803i
$300$		0			0
$301$		−	10.9058i		−	0.628598i
$302$		0			0
$303$	37.5822			2.15904
$304$		0			0
$305$	6.89408			0.394754
$306$		0			0
$307$		−	3.16669i		−	0.180733i	−0.995909	−	0.0903663i	$-0.971196\pi$
$307$		−	3.16669i		−	0.180733i	0.995909	−	0.0903663i	$-0.0288038\pi$
$308$		0			0
$309$			11.0280i			0.627362i
$310$		0			0
$311$		−	9.73483i		−	0.552012i	−0.961156	−	0.276006i	$-0.910989\pi$
$311$		−	9.73483i		−	0.552012i	0.961156	−	0.276006i	$-0.0890109\pi$
$312$		0			0
$313$	−21.0299			−1.18868			−0.594341	−	0.804213i	$-0.702587\pi$
$313$	−21.0299			−1.18868			−0.594341	+	0.804213i	$0.702587\pi$
$314$		0			0
$315$	39.7266			2.23834
$316$		0			0
$317$	10.1989			0.572825			0.286412	−	0.958106i	$-0.407537\pi$
$317$	10.1989			0.572825			0.286412	+	0.958106i	$0.407537\pi$
$318$		0			0
$319$	4.25142			0.238034
$320$		0			0
$321$	33.1045			1.84771
$322$		0			0
$323$	23.9722	−	4.23595i	1.33385	−	0.235695i
$324$		0			0
$325$	13.1810			0.731151
$326$		0			0
$327$			34.0193i			1.88127i
$328$		0			0
$329$	−20.9349			−1.15418
$330$		0			0
$331$			31.4100i			1.72645i	0.504819	+	0.863225i	$0.331559\pi$
$331$			31.4100i			1.72645i	−0.504819	+	0.863225i	$0.668441\pi$
$332$		0			0
$333$	−7.25972			−0.397830
$334$		0			0
$335$	−13.6600			−0.746328
$336$		0			0
$337$		−	23.0554i		−	1.25591i	−0.778251	−	0.627954i	$-0.783893\pi$
$337$		−	23.0554i		−	1.25591i	0.778251	−	0.627954i	$-0.216107\pi$
$338$		0			0
$339$	−52.1345			−2.83156
$340$		0			0
$341$	9.74597			0.527774
$342$		0			0
$343$			20.1246i			1.08663i
$344$		0			0
$345$			49.8749i			2.68517i
$346$		0			0
$347$	−29.3319			−1.57462			−0.787308	−	0.616560i	$-0.788526\pi$
$347$	−29.3319			−1.57462			−0.787308	+	0.616560i	$0.788526\pi$
$348$		0			0
$349$		−	11.9865i		−	0.641622i	−0.947143	−	0.320811i	$-0.896045\pi$
$349$		−	11.9865i		−	0.641622i	0.947143	−	0.320811i	$-0.103955\pi$
$350$		0			0
$351$			26.1557i			1.39609i
$352$		0			0
$353$	−12.5197			−0.666358			−0.333179	−	0.942864i	$-0.608121\pi$
$353$	−12.5197			−0.666358			−0.333179	+	0.942864i	$0.608121\pi$
$354$		0			0
$355$		−	44.7153i		−	2.37324i
$356$		0			0
$357$	−37.1293			−1.96509
$358$		0			0
$359$		−	16.0173i		−	0.845358i	−0.906279	−	0.422679i	$-0.861090\pi$
$359$		−	16.0173i		−	0.845358i	0.906279	−	0.422679i	$-0.138910\pi$
$360$		0			0
$361$	17.8494	−	6.51138i	0.939443	−	0.342704i
$362$		0			0
$363$		−	17.0807i		−	0.896505i
$364$		0			0
$365$			4.59439i			0.240481i
$366$		0			0
$367$		−	28.0338i		−	1.46335i	−0.681652	−	0.731677i	$-0.738738\pi$
$367$		−	28.0338i		−	1.46335i	0.681652	−	0.731677i	$-0.261262\pi$
$368$		0			0
$369$			48.6623i			2.53326i
$370$		0			0
$371$		−	25.5850i		−	1.32831i
$372$		0			0
$373$	13.7697			0.712966			0.356483	−	0.934302i	$-0.383976\pi$
$373$	13.7697			0.712966			0.356483	+	0.934302i	$0.383976\pi$
$374$		0			0
$375$	−6.73767			−0.347932
$376$		0			0
$377$	−5.74489			−0.295877
$378$		0			0
$379$			29.8817i			1.53492i	0.641097	+	0.767460i	$0.278480\pi$
$379$			29.8817i			1.53492i	−0.641097	+	0.767460i	$0.721520\pi$
$380$		0			0
$381$			8.47283i			0.434076i
$382$		0			0
$383$	−8.72800			−0.445980			−0.222990	−	0.974821i	$-0.571582\pi$
$383$	−8.72800			−0.445980			−0.222990	+	0.974821i	$0.571582\pi$
$384$		0			0
$385$	−15.5943			−0.794760
$386$		0			0
$387$	28.4824			1.44784
$388$		0			0
$389$		−	20.6261i		−	1.04579i	−0.852398	−	0.522893i	$-0.824853\pi$
$389$		−	20.6261i		−	1.04579i	0.852398	−	0.522893i	$-0.175147\pi$
$390$		0			0
$391$		−	30.7947i		−	1.55735i
$392$		0			0
$393$		−	23.7030i		−	1.19566i
$394$		0			0
$395$			35.0810i			1.76512i
$396$		0			0
$397$		−	23.3793i		−	1.17337i	−0.809814	−	0.586686i	$-0.800432\pi$
$397$		−	23.3793i		−	1.17337i	0.809814	−	0.586686i	$-0.199568\pi$
$398$		0			0
$399$	−28.5371	+	5.04257i	−1.42864	+	0.252444i
$400$		0			0
$401$			3.17309i			0.158457i	0.996856	+	0.0792283i	$0.0252456\pi$
$401$			3.17309i			0.158457i	−0.996856	+	0.0792283i	$0.974754\pi$
$402$		0			0
$403$	−13.1696			−0.656025
$404$		0			0
$405$			23.0746i			1.14659i
$406$		0			0
$407$	2.84974			0.141256
$408$		0			0
$409$			8.55450i			0.422993i	0.977379	+	0.211496i	$0.0678337\pi$
$409$			8.55450i			0.422993i	−0.977379	+	0.211496i	$0.932166\pi$
$410$		0			0
$411$			27.3238i			1.34778i
$412$		0			0
$413$	−5.58394			−0.274768
$414$		0			0
$415$		−	6.08444i		−	0.298673i
$416$		0			0
$417$			11.5884i			0.567487i
$418$		0			0
$419$	35.5497			1.73671			0.868357	−	0.495939i	$-0.165176\pi$
$419$	35.5497			1.73671			0.868357	+	0.495939i	$0.165176\pi$
$420$		0			0
$421$	28.6265			1.39517			0.697584	−	0.716503i	$-0.254258\pi$
$421$	28.6265			1.39517			0.697584	+	0.716503i	$0.254258\pi$
$422$		0			0
$423$		−	54.6754i		−	2.65841i
$424$		0			0
$425$	−23.7639			−1.15272
$426$		0			0
$427$	5.06723			0.245221
$428$		0			0
$429$		−	21.1132i		−	1.01935i
$430$		0			0
$431$	−31.5656			−1.52046			−0.760230	−	0.649654i	$-0.774914\pi$
$431$	−31.5656			−1.52046			−0.760230	+	0.649654i	$0.774914\pi$
$432$		0			0
$433$		−	18.4868i		−	0.888418i	−0.895923	−	0.444209i	$-0.853485\pi$
$433$		−	18.4868i		−	0.888418i	0.895923	−	0.444209i	$-0.146515\pi$
$434$		0			0
$435$	−16.7748			−0.804292
$436$		0			0
$437$	−4.18225	−	23.6683i	−0.200064	−	1.13221i
$438$		0			0
$439$	10.6256			0.507132			0.253566	−	0.967318i	$-0.418396\pi$
$439$	10.6256			0.507132			0.253566	+	0.967318i	$0.418396\pi$
$440$		0			0
$441$	−11.6798			−0.556182
$442$		0			0
$443$	28.7470			1.36581			0.682907	−	0.730506i	$-0.260716\pi$
$443$	28.7470			1.36581			0.682907	+	0.730506i	$0.260716\pi$
$444$		0			0
$445$	−16.1201			−0.764168
$446$		0			0
$447$	31.0233			1.46735
$448$		0			0
$449$		−	28.2150i		−	1.33155i	−0.746153	−	0.665775i	$-0.768101\pi$
$449$		−	28.2150i		−	1.33155i	0.746153	−	0.665775i	$-0.231899\pi$
$450$		0			0
$451$		−	19.1019i		−	0.899475i
$452$		0			0
$453$			32.6727i			1.53510i
$454$		0			0
$455$	21.0724			0.987890
$456$		0			0
$457$	23.1154			1.08129			0.540647	−	0.841249i	$-0.318179\pi$
$457$	23.1154			1.08129			0.540647	+	0.841249i	$0.318179\pi$
$458$		0			0
$459$		−	47.1559i		−	2.20105i
$460$		0			0
$461$		−	1.90136i		−	0.0885550i	−0.999019	−	0.0442775i	$-0.985901\pi$
$461$		−	1.90136i		−	0.0885550i	0.999019	−	0.0442775i	$-0.0140986\pi$
$462$		0			0
$463$			20.5572i			0.955374i	0.878530	+	0.477687i	$0.158525\pi$
$463$			20.5572i			0.955374i	−0.878530	+	0.477687i	$0.841475\pi$
$464$		0			0
$465$	−38.4546			−1.78329
$466$		0			0
$467$	−15.1228			−0.699800			−0.349900	−	0.936787i	$-0.613784\pi$
$467$	−15.1228			−0.699800			−0.349900	+	0.936787i	$0.613784\pi$
$468$		0			0
$469$	−10.0403			−0.463617
$470$		0			0
$471$	−17.1845			−0.791819
$472$		0			0
$473$	−11.1805			−0.514080
$474$		0			0
$475$	−18.2646	+	3.22740i	−0.838038	+	0.148083i
$476$		0			0
$477$	66.8201			3.05948
$478$		0			0
$479$		−	17.4171i		−	0.795808i	−0.917427	−	0.397904i	$-0.869738\pi$
$479$		−	17.4171i		−	0.795808i	0.917427	−	0.397904i	$-0.130262\pi$
$480$		0			0
$481$	−3.85081			−0.175582
$482$		0			0
$483$			36.6586i			1.66803i
$484$		0			0
$485$	3.57080			0.162142
$486$		0			0
$487$	42.3759			1.92023			0.960117	−	0.279597i	$-0.0902010\pi$
$487$	42.3759			1.92023			0.960117	+	0.279597i	$0.0902010\pi$
$488$		0			0
$489$		−	19.7997i		−	0.895374i
$490$		0			0
$491$	6.65940			0.300534			0.150267	−	0.988645i	$-0.451987\pi$
$491$	6.65940			0.300534			0.150267	+	0.988645i	$0.451987\pi$
$492$		0			0
$493$	10.3574			0.466475
$494$		0			0
$495$		−	40.7275i		−	1.83056i
$496$		0			0
$497$		−	32.8662i		−	1.47425i
$498$		0			0
$499$	19.5028			0.873067			0.436534	−	0.899688i	$-0.356206\pi$
$499$	19.5028			0.873067			0.436534	+	0.899688i	$0.356206\pi$
$500$		0			0
$501$		−	5.77980i		−	0.258222i
$502$		0			0
$503$		−	18.8238i		−	0.839310i	−0.907684	−	0.419655i	$-0.862151\pi$
$503$		−	18.8238i		−	0.839310i	0.907684	−	0.419655i	$-0.137849\pi$
$504$		0			0
$505$	38.4546			1.71121
$506$		0			0
$507$		−	10.1216i		−	0.449518i
$508$		0			0
$509$	−19.8091			−0.878021			−0.439011	−	0.898482i	$-0.644671\pi$
$509$	−19.8091			−0.878021			−0.439011	+	0.898482i	$0.644671\pi$
$510$		0			0
$511$			3.37693i			0.149387i
$512$		0			0
$513$	−6.40429	−	36.2434i	−0.282756	−	1.60018i
$514$		0			0
$515$			11.2840i			0.497234i
$516$		0			0
$517$			21.4623i			0.943913i
$518$		0			0
$519$			61.1780i			2.68542i
$520$		0			0
$521$		−	29.7708i		−	1.30428i	−0.758097	−	0.652141i	$-0.773871\pi$
$521$		−	29.7708i		−	1.30428i	0.758097	−	0.652141i	$-0.226129\pi$
$522$		0			0
$523$		−	2.05365i		−	0.0898000i	−0.998991	−	0.0449000i	$-0.985703\pi$
$523$		−	2.05365i		−	0.0898000i	0.998991	−	0.0449000i	$-0.0142969\pi$
$524$		0			0
$525$	28.2891			1.23464
$526$		0			0
$527$	23.7434			1.03428
$528$		0			0
$529$	−7.40429			−0.321926
$530$		0			0
$531$		−	14.5835i		−	0.632870i
$532$		0			0
$533$			25.8122i			1.11805i
$534$		0			0
$535$	33.8730			1.46446
$536$		0			0
$537$	25.9444			1.11959
$538$		0			0
$539$	4.58480			0.197481
$540$		0			0
$541$		−	24.9916i		−	1.07447i	−0.843432	−	0.537236i	$-0.819468\pi$
$541$		−	24.9916i		−	1.07447i	0.843432	−	0.537236i	$-0.180532\pi$
$542$		0			0
$543$			25.2806i			1.08490i
$544$		0			0
$545$			34.8090i			1.49105i
$546$		0			0
$547$			26.0893i			1.11550i	0.830010	+	0.557749i	$0.188335\pi$
$547$			26.0893i			1.11550i	−0.830010	+	0.557749i	$0.811665\pi$
$548$		0			0
$549$			13.2340i			0.564814i
$550$		0			0
$551$	7.96057	−	1.40665i	0.339131	−	0.0599254i
$552$		0			0
$553$			25.7849i			1.09649i
$554$		0			0
$555$	−11.2442			−0.477290
$556$		0			0
$557$			10.9058i			0.462092i	0.972943	+	0.231046i	$0.0742148\pi$
$557$			10.9058i			0.462092i	−0.972943	+	0.231046i	$0.925785\pi$
$558$		0			0
$559$	15.1081			0.639004
$560$		0			0
$561$			38.0647i			1.60709i
$562$		0			0
$563$		−	27.7390i		−	1.16906i	−0.811372	−	0.584531i	$-0.801279\pi$
$563$		−	27.7390i		−	1.16906i	0.811372	−	0.584531i	$-0.198721\pi$
$564$		0			0
$565$	−53.3448			−2.24423
$566$		0			0
$567$			16.9601i			0.712259i
$568$		0			0
$569$		−	45.2675i		−	1.89771i	−0.315712	−	0.948855i	$-0.602243\pi$
$569$		−	45.2675i		−	1.89771i	0.315712	−	0.948855i	$-0.397757\pi$
$570$		0			0
$571$	7.80997			0.326837			0.163419	−	0.986557i	$-0.447748\pi$
$571$	7.80997			0.326837			0.163419	+	0.986557i	$0.447748\pi$
$572$		0			0
$573$	33.4202			1.39615
$574$		0			0
$575$			23.4627i			0.978462i
$576$		0			0
$577$	−12.1696			−0.506627			−0.253314	−	0.967384i	$-0.581520\pi$
$577$	−12.1696			−0.506627			−0.253314	+	0.967384i	$0.581520\pi$
$578$		0			0
$579$	23.0950			0.959796
$580$		0			0
$581$		−	4.47214i		−	0.185535i
$582$		0			0
$583$	−26.2296			−1.08632
$584$		0			0
$585$			55.0345i			2.27540i
$586$		0			0
$587$	6.84345			0.282459			0.141230	−	0.989977i	$-0.454894\pi$
$587$	6.84345			0.282459			0.141230	+	0.989977i	$0.454894\pi$
$588$		0			0
$589$	18.2488	−	3.22461i	0.751929	−	0.132868i
$590$		0			0
$591$	−10.8103			−0.444676
$592$		0			0
$593$	17.0950			0.702008			0.351004	−	0.936374i	$-0.385840\pi$
$593$	17.0950			0.702008			0.351004	+	0.936374i	$0.385840\pi$
$594$		0			0
$595$	−37.9912			−1.55749
$596$		0			0
$597$	16.2357			0.664482
$598$		0			0
$599$	1.76518			0.0721232			0.0360616	−	0.999350i	$-0.488519\pi$
$599$	1.76518			0.0721232			0.0360616	+	0.999350i	$0.488519\pi$
$600$		0			0
$601$			48.7011i			1.98656i	0.115731	+	0.993281i	$0.463079\pi$
$601$			48.7011i			1.98656i	−0.115731	+	0.993281i	$0.536921\pi$
$602$		0			0
$603$		−	26.2221i		−	1.06785i
$604$		0			0
$605$		−	17.4772i		−	0.710551i
$606$		0			0
$607$	−45.9062			−1.86328			−0.931638	−	0.363387i	$-0.881620\pi$
$607$	−45.9062			−1.86328			−0.931638	+	0.363387i	$0.881620\pi$
$608$		0			0
$609$	−12.3297			−0.499625
$610$		0			0
$611$		−	29.0018i		−	1.17329i
$612$		0			0
$613$			20.9308i			0.845386i	0.906273	+	0.422693i	$0.138915\pi$
$613$			20.9308i			0.845386i	−0.906273	+	0.422693i	$0.861085\pi$
$614$		0			0
$615$			75.3705i			3.03923i
$616$		0			0
$617$	32.8048			1.32067			0.660335	−	0.750971i	$-0.270414\pi$
$617$	32.8048			1.32067			0.660335	+	0.750971i	$0.270414\pi$
$618$		0			0
$619$	−22.0190			−0.885020			−0.442510	−	0.896764i	$-0.645912\pi$
$619$	−22.0190			−0.885020			−0.442510	+	0.896764i	$0.645912\pi$
$620$		0			0
$621$	−46.5581			−1.86831
$622$		0			0
$623$	−11.8485			−0.474700
$624$		0			0
$625$	−28.1696			−1.12678
$626$		0			0
$627$	5.16961	+	29.2560i	0.206454	+	1.16837i
$628$		0			0
$629$	6.94260			0.276819
$630$		0			0
$631$			9.72716i			0.387232i	0.981077	+	0.193616i	$0.0620216\pi$
$631$			9.72716i			0.387232i	−0.981077	+	0.193616i	$0.937978\pi$
$632$		0			0
$633$	45.6147			1.81302
$634$		0			0
$635$			8.66953i			0.344040i
$636$		0			0
$637$	−6.19539			−0.245470
$638$		0			0
$639$	85.8363			3.39563
$640$		0			0
$641$			15.1485i			0.598329i	0.954202	+	0.299165i	$0.0967080\pi$
$641$			15.1485i			0.598329i	−0.954202	+	0.299165i	$0.903292\pi$
$642$		0			0
$643$	1.46201			0.0576560			0.0288280	−	0.999584i	$-0.490822\pi$
$643$	1.46201			0.0576560			0.0288280	+	0.999584i	$0.490822\pi$
$644$		0			0
$645$	44.1149			1.73702
$646$		0			0
$647$			7.90920i			0.310943i	0.987840	+	0.155471i	$0.0496897\pi$
$647$			7.90920i			0.310943i	−0.987840	+	0.155471i	$0.950310\pi$
$648$		0			0
$649$			5.72462i			0.224711i
$650$		0			0
$651$	−28.2646			−1.10778
$652$		0			0
$653$			32.5505i			1.27380i	0.770947	+	0.636900i	$0.219783\pi$
$653$			32.5505i			1.27380i	−0.770947	+	0.636900i	$0.780217\pi$
$654$		0			0
$655$		−	24.2532i		−	0.947653i
$656$		0			0
$657$	−8.81949			−0.344081
$658$		0			0
$659$			12.8731i			0.501463i	0.968057	+	0.250731i	$0.0806711\pi$
$659$			12.8731i			0.501463i	−0.968057	+	0.250731i	$0.919329\pi$
$660$		0			0
$661$	2.01313			0.0783019			0.0391509	−	0.999233i	$-0.487535\pi$
$661$	2.01313			0.0783019			0.0391509	+	0.999233i	$0.487535\pi$
$662$		0			0
$663$		−	51.4364i		−	1.99762i
$664$		0			0
$665$	−29.1995	+	5.15963i	−1.13231	+	0.200082i
$666$		0			0
$667$		−	10.2261i		−	0.395957i
$668$		0			0
$669$			31.0604i			1.20087i
$670$		0			0
$671$		−	5.19489i		−	0.200547i
$672$		0			0
$673$			21.9254i			0.845163i	0.906325	+	0.422582i	$0.138876\pi$
$673$			21.9254i			0.845163i	−0.906325	+	0.422582i	$0.861124\pi$
$674$		0			0
$675$			35.9284i			1.38289i
$676$		0			0
$677$	42.2401			1.62342			0.811710	−	0.584061i	$-0.198537\pi$
$677$	42.2401			1.62342			0.811710	+	0.584061i	$0.198537\pi$
$678$		0			0
$679$	2.62458			0.100722
$680$		0			0
$681$	0.575289			0.0220451
$682$		0			0
$683$		−	6.89837i		−	0.263959i	−0.991252	−	0.131979i	$-0.957867\pi$
$683$		−	6.89837i		−	0.263959i	0.991252	−	0.131979i	$-0.0421332\pi$
$684$		0			0
$685$			27.9581i			1.06822i
$686$		0			0
$687$	57.4376			2.19138
$688$		0			0
$689$	35.4437			1.35030
$690$		0			0
$691$	35.4620			1.34904			0.674519	−	0.738257i	$-0.264351\pi$
$691$	35.4620			1.34904			0.674519	+	0.738257i	$0.264351\pi$
$692$		0			0
$693$		−	29.9352i		−	1.13714i
$694$		0			0
$695$			11.8574i			0.449778i
$696$		0			0
$697$		−	46.5366i		−	1.76270i
$698$		0			0
$699$		−	36.9411i		−	1.39724i
$700$		0			0
$701$		−	10.3586i		−	0.391239i	−0.980680	−	0.195619i	$-0.937328\pi$
$701$		−	10.3586i		−	0.391239i	0.980680	−	0.195619i	$-0.0626717\pi$
$702$		0			0
$703$	5.33598	−	0.942881i	0.201250	−	0.0355614i
$704$		0			0
$705$		−	84.6839i		−	3.18938i
$706$		0			0
$707$	28.2646			1.06300
$708$		0			0
$709$			5.24822i			0.197101i	0.995132	+	0.0985506i	$0.0314206\pi$
$709$			5.24822i			0.197101i	−0.995132	+	0.0985506i	$0.968579\pi$
$710$		0			0
$711$	−67.3421			−2.52553
$712$		0			0
$713$		−	23.4424i		−	0.877924i
$714$		0			0
$715$		−	21.6033i		−	0.807917i
$716$		0			0
$717$	−0.452885			−0.0169133
$718$		0			0
$719$		−	4.31981i		−	0.161102i	−0.996750	−	0.0805509i	$-0.974332\pi$
$719$		−	4.31981i		−	0.161102i	0.996750	−	0.0805509i	$-0.0256680\pi$
$720$		0			0
$721$			8.29390i			0.308881i
$722$		0			0
$723$	−35.6052			−1.32417
$724$		0			0
$725$	−7.89140			−0.293079
$726$		0			0
$727$		−	7.37763i		−	0.273621i	−0.990597	−	0.136811i	$-0.956315\pi$
$727$		−	7.37763i		−	0.273621i	0.990597	−	0.136811i	$-0.0436852\pi$
$728$		0			0
$729$	31.0190			1.14885
$730$		0			0
$731$	−27.2382			−1.00744
$732$		0			0
$733$		−	6.72268i		−	0.248308i	−0.992263	−	0.124154i	$-0.960378\pi$
$733$		−	6.72268i		−	0.248308i	0.992263	−	0.124154i	$-0.0396216\pi$
$734$		0			0
$735$	−18.0902			−0.667269
$736$		0			0
$737$			10.2932i			0.379156i
$738$		0			0
$739$	−36.1622			−1.33025			−0.665125	−	0.746732i	$-0.731622\pi$
$739$	−36.1622			−1.33025			−0.665125	+	0.746732i	$0.731622\pi$
$740$		0			0
$741$	−6.98563	−	39.5333i	−0.256623	−	1.45229i
$742$		0			0
$743$	−11.6697			−0.428120			−0.214060	−	0.976821i	$-0.568669\pi$
$743$	−11.6697			−0.428120			−0.214060	+	0.976821i	$0.568669\pi$
$744$		0			0
$745$	31.7435			1.16299
$746$		0			0
$747$	11.6798			0.427342
$748$		0			0
$749$	24.8971			0.909720
$750$		0			0
$751$	−8.36451			−0.305225			−0.152613	−	0.988286i	$-0.548769\pi$
$751$	−8.36451			−0.305225			−0.152613	+	0.988286i	$0.548769\pi$
$752$		0			0
$753$		−	23.2594i		−	0.847621i
$754$		0			0
$755$			33.4312i			1.21669i
$756$		0			0
$757$		−	44.5991i		−	1.62098i	−0.585751	−	0.810491i	$-0.699200\pi$
$757$		−	44.5991i		−	1.62098i	0.585751	−	0.810491i	$-0.300800\pi$
$758$		0			0
$759$	37.5822			1.36415
$760$		0			0
$761$	−40.6243			−1.47263			−0.736314	−	0.676640i	$-0.763435\pi$
$761$	−40.6243			−1.47263			−0.736314	+	0.676640i	$0.763435\pi$
$762$		0			0
$763$			25.5850i			0.926240i
$764$		0			0
$765$		−	99.2212i		−	3.58735i
$766$		0			0
$767$		−	7.73560i		−	0.279316i
$768$		0			0
$769$	13.7748			0.496733			0.248367	−	0.968666i	$-0.420106\pi$
$769$	13.7748			0.496733			0.248367	+	0.968666i	$0.420106\pi$
$770$		0			0
$771$	−39.8698			−1.43588
$772$		0			0
$773$	−22.7684			−0.818923			−0.409462	−	0.912327i	$-0.634283\pi$
$773$	−22.7684			−0.818923			−0.409462	+	0.912327i	$0.634283\pi$
$774$		0			0
$775$	−18.0902			−0.649821
$776$		0			0
$777$	−8.26462			−0.296492
$778$		0			0
$779$	−6.32018	−	35.7674i	−0.226444	−	1.28150i
$780$		0			0
$781$	−33.6943			−1.20568
$782$		0			0
$783$		−	15.6593i		−	0.559617i
$784$		0			0
$785$	−17.5834			−0.627579
$786$		0			0
$787$		−	6.09611i		−	0.217303i	−0.994080	−	0.108651i	$-0.965347\pi$
$787$		−	6.09611i		−	0.217303i	0.994080	−	0.108651i	$-0.0346532\pi$
$788$		0			0
$789$	−12.2322			−0.435478
$790$		0			0
$791$	−39.2090			−1.39411
$792$		0			0
$793$			7.01979i			0.249280i
$794$		0			0
$795$	103.494			3.67056
$796$		0			0
$797$	−17.2798			−0.612081			−0.306041	−	0.952018i	$-0.599004\pi$
$797$	−17.2798			−0.612081			−0.306041	+	0.952018i	$0.599004\pi$
$798$		0			0
$799$			52.2871i			1.84978i
$800$		0			0
$801$		−	30.9446i		−	1.09337i
$802$		0			0
$803$	3.46201			0.122172
$804$		0			0
$805$			37.5097i			1.32204i
$806$		0			0
$807$		−	78.5951i		−	2.76668i
$808$		0			0
$809$	−39.3801			−1.38453			−0.692264	−	0.721644i	$-0.743387\pi$
$809$	−39.3801			−1.38453			−0.692264	+	0.721644i	$0.743387\pi$
$810$		0			0
$811$			20.9452i			0.735484i	0.929928	+	0.367742i	$0.119869\pi$
$811$			20.9452i			0.735484i	−0.929928	+	0.367742i	$0.880131\pi$
$812$		0			0
$813$	−57.3684			−2.01200
$814$		0			0
$815$		−	20.2594i		−	0.709654i
$816$		0			0
$817$	−20.9349	+	3.69925i	−0.732420	+	0.129420i
$818$		0			0
$819$			40.4510i			1.41347i
$820$		0			0
$821$			52.2181i			1.82243i	0.411936	+	0.911213i	$0.364853\pi$
$821$			52.2181i			1.82243i	−0.411936	+	0.911213i	$0.635147\pi$
$822$		0			0
$823$			39.5789i			1.37963i	0.723983	+	0.689817i	$0.242309\pi$
$823$			39.5789i			1.37963i	−0.723983	+	0.689817i	$0.757691\pi$
$824$		0			0
$825$		−	29.0018i		−	1.00971i
$826$		0			0
$827$		−	19.5172i		−	0.678680i	−0.940664	−	0.339340i	$-0.889796\pi$
$827$		−	19.5172i		−	0.678680i	0.940664	−	0.339340i	$-0.110204\pi$
$828$		0			0
$829$	25.0759			0.870921			0.435461	−	0.900208i	$-0.356586\pi$
$829$	25.0759			0.870921			0.435461	+	0.900208i	$0.356586\pi$
$830$		0			0
$831$	−79.4158			−2.75490
$832$		0			0
$833$	11.1696			0.387004
$834$		0			0
$835$		−	5.91397i		−	0.204661i
$836$		0			0
$837$		−	35.8974i		−	1.24079i
$838$		0			0
$839$	10.0833			0.348115			0.174057	−	0.984736i	$-0.444312\pi$
$839$	10.0833			0.348115			0.174057	+	0.984736i	$0.444312\pi$
$840$		0			0
$841$	−25.5606			−0.881399
$842$		0			0
$843$	−69.6988			−2.40056
$844$		0			0
$845$		−	10.3566i		−	0.356278i
$846$		0			0
$847$		−	12.8460i		−	0.441393i
$848$		0			0
$849$			34.3006i			1.17719i
$850$		0			0
$851$		−	6.85459i		−	0.234972i
$852$		0			0
$853$		−	38.4525i		−	1.31659i	−0.752760	−	0.658295i	$-0.771278\pi$
$853$		−	38.4525i		−	1.31659i	0.752760	−	0.658295i	$-0.228722\pi$
$854$		0			0
$855$	−13.4753	−	76.2600i	−0.460847	−	2.60804i
$856$		0			0
$857$			42.6333i			1.45633i	0.685404	+	0.728163i	$0.259626\pi$
$857$			42.6333i			1.45633i	−0.685404	+	0.728163i	$0.740374\pi$
$858$		0			0
$859$	28.5570			0.974353			0.487176	−	0.873304i	$-0.338027\pi$
$859$	28.5570			0.974353			0.487176	+	0.873304i	$0.338027\pi$
$860$		0			0
$861$			55.3982i			1.88797i
$862$		0			0
$863$	−17.0521			−0.580459			−0.290229	−	0.956957i	$-0.593732\pi$
$863$	−17.0521			−0.580459			−0.290229	+	0.956957i	$0.593732\pi$
$864$		0			0
$865$			62.5982i			2.12840i
$866$		0			0
$867$			42.1898i			1.43284i
$868$		0			0
$869$	26.4345			0.896730
$870$		0			0
$871$		−	13.9091i		−	0.471293i
$872$		0			0
$873$			6.85459i			0.231993i
$874$		0			0
$875$	−5.06723			−0.171304
$876$		0			0
$877$	−10.5160			−0.355099			−0.177550	−	0.984112i	$-0.556817\pi$
$877$	−10.5160			−0.355099			−0.177550	+	0.984112i	$0.556817\pi$
$878$		0			0
$879$		−	35.0223i		−	1.18127i
$880$		0			0
$881$	37.5943			1.26658			0.633292	−	0.773913i	$-0.281703\pi$
$881$	37.5943			1.26658			0.633292	+	0.773913i	$0.281703\pi$
$882$		0			0
$883$	13.2720			0.446638			0.223319	−	0.974745i	$-0.428311\pi$
$883$	13.2720			0.446638			0.223319	+	0.974745i	$0.428311\pi$
$884$		0			0
$885$		−	22.5876i		−	0.759274i
$886$		0			0
$887$	27.6777			0.929325			0.464662	−	0.885488i	$-0.346176\pi$
$887$	27.6777			0.929325			0.464662	+	0.885488i	$0.346176\pi$
$888$		0			0
$889$			6.37220i			0.213717i
$890$		0			0
$891$	17.3874			0.582500
$892$		0			0
$893$	7.10116	+	40.1871i	0.237631	+	1.34481i
$894$		0			0
$895$	26.5467			0.887359
$896$		0			0
$897$	−50.7843			−1.69564
$898$		0			0
$899$	7.88457			0.262965
$900$		0			0
$901$	−63.9012			−2.12886
$902$		0			0
$903$	32.4250			1.07904
$904$		0			0
$905$			25.8675i			0.859866i
$906$		0			0
$907$		−	44.3324i		−	1.47203i	−0.676963	−	0.736017i	$-0.736704\pi$
$907$		−	44.3324i		−	1.47203i	0.676963	−	0.736017i	$-0.263296\pi$
$908$		0			0
$909$			73.8183i			2.44840i
$910$		0			0
$911$	40.9742			1.35754			0.678768	−	0.734353i	$-0.262514\pi$
$911$	40.9742			1.35754			0.678768	+	0.734353i	$0.262514\pi$
$912$		0			0
$913$	−4.58480			−0.151735
$914$		0			0
$915$			20.4975i			0.677625i
$916$		0			0
$917$		−	17.8264i		−	0.588680i
$918$		0			0
$919$			37.2439i			1.22856i	0.789087	+	0.614281i	$0.210554\pi$
$919$			37.2439i			1.22856i	−0.789087	+	0.614281i	$0.789446\pi$
$920$		0			0
$921$	9.41520			0.310241
$922$		0			0
$923$	45.5306			1.49866
$924$		0			0
$925$	−5.28962			−0.173922
$926$		0			0
$927$	−21.6611			−0.711442
$928$		0			0
$929$	−3.86033			−0.126653			−0.0633266	−	0.997993i	$-0.520171\pi$
$929$	−3.86033			−0.126653			−0.0633266	+	0.997993i	$0.520171\pi$
$930$		0			0
$931$	8.58480	−	1.51696i	0.281356	−	0.0497162i
$932$		0			0
$933$	28.9436			0.947570
$934$		0			0
$935$			38.9484i			1.27375i
$936$		0			0
$937$	20.6798			0.675580			0.337790	−	0.941221i	$-0.390321\pi$
$937$	20.6798			0.675580			0.337790	+	0.941221i	$0.390321\pi$
$938$		0			0
$939$		−	62.5262i		−	2.04046i
$940$		0			0
$941$	9.29308			0.302946			0.151473	−	0.988461i	$-0.451598\pi$
$941$	9.29308			0.302946			0.151473	+	0.988461i	$0.451598\pi$
$942$		0			0
$943$	−45.9467			−1.49623
$944$		0			0
$945$			57.4386i			1.86848i
$946$		0			0
$947$	−37.3934			−1.21512			−0.607561	−	0.794273i	$-0.707852\pi$
$947$	−37.3934			−1.21512			−0.607561	+	0.794273i	$0.707852\pi$
$948$		0			0
$949$	−4.67817			−0.151860
$950$		0			0
$951$			30.3232i			0.983298i
$952$		0			0
$953$		−	6.08558i		−	0.197131i	−0.995131	−	0.0985656i	$-0.968575\pi$
$953$		−	6.08558i		−	0.197131i	0.995131	−	0.0985656i	$-0.0314254\pi$
$954$		0			0
$955$	34.1960			1.10656
$956$		0			0
$957$			12.6403i			0.408604i
$958$		0			0
$959$			20.5495i			0.663579i
$960$		0			0
$961$	−12.9254			−0.416949
$962$		0			0
$963$			65.0234i			2.09535i
$964$		0			0
$965$	23.6312			0.760714
$966$		0			0
$967$		−	18.0088i		−	0.579124i	−0.957159	−	0.289562i	$-0.906490\pi$
$967$		−	18.0088i		−	0.579124i	0.957159	−	0.289562i	$-0.0935097\pi$
$968$		0			0
$969$	12.5943	+	71.2742i	0.404588	+	2.28966i
$970$		0			0
$971$			55.6999i			1.78749i	0.448572	+	0.893747i	$0.351933\pi$
$971$			55.6999i			1.78749i	−0.448572	+	0.893747i	$0.648067\pi$
$972$		0			0
$973$			8.71535i			0.279401i
$974$		0			0
$975$			39.1898i			1.25508i
$976$		0			0
$977$		−	33.1269i		−	1.05982i	−0.848053	−	0.529911i	$-0.822225\pi$
$977$		−	33.1269i		−	1.05982i	0.848053	−	0.529911i	$-0.177775\pi$
$978$		0			0
$979$			12.1470i			0.388220i
$980$		0			0
$981$	−66.8201			−2.13340
$982$		0			0
$983$	26.2760			0.838073			0.419037	−	0.907969i	$-0.362368\pi$
$983$	26.2760			0.838073			0.419037	+	0.907969i	$0.362368\pi$
$984$		0			0
$985$	−11.0613			−0.352441
$986$		0			0
$987$		−	62.2437i		−	1.98124i
$988$		0			0
$989$			26.8929i			0.855146i
$990$		0			0
$991$	−47.6714			−1.51433			−0.757166	−	0.653222i	$-0.773417\pi$
$991$	−47.6714			−1.51433			−0.757166	+	0.653222i	$0.773417\pi$
$992$		0			0
$993$	−93.3882			−2.96358
$994$		0			0
$995$	16.6126			0.526654
$996$		0			0
$997$		−	2.73758i		−	0.0866999i	−0.999060	−	0.0433499i	$-0.986197\pi$
$997$		−	2.73758i		−	0.0866999i	0.999060	−	0.0433499i	$-0.0138030\pi$
$998$		0			0
$999$		−	10.4964i		−	0.332093i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.2.b.c.303.12		12
3.2	odd	2		5472.2.e.e.5167.3		12
4.3	odd	2		152.2.b.c.75.12	yes	12
8.3	odd	2	inner	608.2.b.c.303.11		12
8.5	even	2		152.2.b.c.75.2	yes	12
12.11	even	2		1368.2.e.e.379.1		12
19.18	odd	2	inner	608.2.b.c.303.2		12
24.5	odd	2		1368.2.e.e.379.11		12
24.11	even	2		5472.2.e.e.5167.10		12
57.56	even	2		5472.2.e.e.5167.4		12
76.75	even	2		152.2.b.c.75.1	✓	12
152.37	odd	2		152.2.b.c.75.11	yes	12
152.75	even	2	inner	608.2.b.c.303.1		12
228.227	odd	2		1368.2.e.e.379.12		12
456.227	odd	2		5472.2.e.e.5167.9		12
456.341	even	2		1368.2.e.e.379.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.b.c.75.1	✓	12	76.75	even	2
152.2.b.c.75.2	yes	12	8.5	even	2
152.2.b.c.75.11	yes	12	152.37	odd	2
152.2.b.c.75.12	yes	12	4.3	odd	2
608.2.b.c.303.1		12	152.75	even	2	inner
608.2.b.c.303.2		12	19.18	odd	2	inner
608.2.b.c.303.11		12	8.3	odd	2	inner
608.2.b.c.303.12		12	1.1	even	1	trivial
1368.2.e.e.379.1		12	12.11	even	2
1368.2.e.e.379.2		12	456.341	even	2
1368.2.e.e.379.11		12	24.5	odd	2
1368.2.e.e.379.12		12	228.227	odd	2
5472.2.e.e.5167.3		12	3.2	odd	2
5472.2.e.e.5167.4		12	57.56	even	2
5472.2.e.e.5167.9		12	456.227	odd	2
5472.2.e.e.5167.10		12	24.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.97320i q^{3} +3.04222i q^{5} +2.23607i q^{7} -5.83991 q^{9} +O(q^{10})\)	\(q+2.97320i q^{3} +3.04222i q^{5} +2.23607i q^{7} -5.83991 q^{9} +2.29240 q^{11} -3.09769 q^{13} -9.04512 q^{15} +5.58480 q^{17} +(4.29240 - 0.758478i) q^{19} -6.64827 q^{21} -5.51401i q^{23} -4.25511 q^{25} -8.44361i q^{27} +1.85457 q^{29} +4.25142 q^{31} +6.81576i q^{33} -6.80261 q^{35} +1.24312 q^{37} -9.21006i q^{39} -8.33272i q^{41} -4.87720 q^{43} -17.7663i q^{45} +9.36238i q^{47} +2.00000 q^{49} +16.6047i q^{51} -11.4420 q^{53} +6.97399i q^{55} +(2.25511 + 12.7622i) q^{57} +2.49721i q^{59} -2.26613i q^{61} -13.0584i q^{63} -9.42387i q^{65} +4.49015i q^{67} +16.3942 q^{69} -14.6982 q^{71} +1.51021 q^{73} -12.6513i q^{75} +5.12597i q^{77} +11.5314 q^{79} +7.58480 q^{81} -2.00000 q^{83} +16.9902i q^{85} +5.51401i q^{87} +5.29881i q^{89} -6.92665i q^{91} +12.6403i q^{93} +(2.30746 + 13.0584i) q^{95} -1.17375i q^{97} -13.3874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9}+O(q^{10}) \) 12 * q - 8 * q^9	\( 12 q - 8 q^{9} + 12 q^{17} + 24 q^{19} - 44 q^{25} - 40 q^{35} + 24 q^{43} + 24 q^{49} + 20 q^{57} + 4 q^{73} + 36 q^{81} - 24 q^{83} - 64 q^{99}+O(q^{100}) \) 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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