LMFDB - Embedded newform 608.2.c.b.305.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,2,Mod(305,608)]

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([0, 1, 0]))

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

//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: 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:	$16$
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			305.3
Root			$-0.889165 + 1.09972i$ of defining polynomial
Character	$\chi$	$=$	608.305
Dual form			608.2.c.b.305.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32921i q^{3} +3.13887i q^{5} +0.535658 q^{7} -2.42523 q^{9} +O(q^{10})$	$q-2.32921i q^{3} +3.13887i q^{5} +0.535658 q^{7} -2.42523 q^{9} +0.425227i q^{11} -6.65786i q^{13} +7.31110 q^{15} +7.33239 q^{17} +1.00000i q^{19} -1.24766i q^{21} +5.90033 q^{23} -4.85252 q^{25} -1.33877i q^{27} -0.837037i q^{29} +3.16999 q^{31} +0.990444 q^{33} +1.68136i q^{35} +3.49490i q^{37} -15.5076 q^{39} -0.123059 q^{41} -5.39744i q^{43} -7.61248i q^{45} +2.02928 q^{47} -6.71307 q^{49} -17.0787i q^{51} -5.82785i q^{53} -1.33473 q^{55} +2.32921 q^{57} -5.56633i q^{59} +6.99606i q^{61} -1.29909 q^{63} +20.8982 q^{65} +12.3777i q^{67} -13.7431i q^{69} -12.1786 q^{71} -6.99382 q^{73} +11.3025i q^{75} +0.227776i q^{77} -2.07996 q^{79} -10.3940 q^{81} +11.8227i q^{83} +23.0154i q^{85} -1.94964 q^{87} +13.7684 q^{89} -3.56633i q^{91} -7.38358i q^{93} -3.13887 q^{95} +0.801240 q^{97} -1.03127i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} - 24 q^{9}+O(q^{10}) $ 16 * q + 8 * q^7 - 24 * q^9	$ 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}+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

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
$3$	− 2.32921i	− 1.34477i	−0.740201	−	0.672385i	$-0.765270\pi$
$3$	− 2.32921i	− 1.34477i	0.740201	−	0.672385i	$-0.234730\pi$
$4$	0	0
$5$	3.13887i	1.40375i	0.712302	+	0.701873i	$0.247653\pi$
$5$	3.13887i	1.40375i	−0.712302	+	0.701873i	$0.752347\pi$
$6$	0	0
$7$	0.535658	0.202460	0.101230	−	0.994863i	$-0.467722\pi$
$7$	0.535658	0.202460	0.101230	+	0.994863i	$0.467722\pi$
$8$	0	0
$9$	−2.42523	−0.808409
$10$	0	0
$11$	0.425227i	0.128211i	0.997943	+	0.0641054i	$0.0204194\pi$
$11$	0.425227i	0.128211i	−0.997943	+	0.0641054i	$0.979581\pi$
$12$	0	0
$13$	− 6.65786i	− 1.84656i	−0.384129	−	0.923279i	$-0.625498\pi$
$13$	− 6.65786i	− 1.84656i	0.384129	−	0.923279i	$-0.374502\pi$
$14$	0	0
$15$	7.31110	1.88772
$16$	0	0
$17$	7.33239	1.77837	0.889183	−	0.457552i	$-0.151273\pi$
$17$	7.33239	1.77837	0.889183	+	0.457552i	$0.151273\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 1.24766i	− 0.272262i
$22$	0	0
$23$	5.90033	1.23030	0.615152	−	0.788408i	$-0.289095\pi$
$23$	5.90033	1.23030	0.615152	+	0.788408i	$0.289095\pi$
$24$	0	0
$25$	−4.85252	−0.970503
$26$	0	0
$27$	− 1.33877i	− 0.257646i
$28$	0	0
$29$	− 0.837037i	− 0.155434i	−0.996975	−	0.0777170i	$-0.975237\pi$
$29$	− 0.837037i	− 0.155434i	0.996975	−	0.0777170i	$-0.0247631\pi$
$30$	0	0
$31$	3.16999	0.569347	0.284674	−	0.958625i	$-0.408115\pi$
$31$	3.16999	0.569347	0.284674	+	0.958625i	$0.408115\pi$
$32$	0	0
$33$	0.990444	0.172414
$34$	0	0
$35$	1.68136i	0.284202i
$36$	0	0
$37$	3.49490i	0.574558i	0.957847	+	0.287279i	$0.0927507\pi$
$37$	3.49490i	0.574558i	−0.957847	+	0.287279i	$0.907249\pi$
$38$	0	0
$39$	−15.5076	−2.48320
$40$	0	0
$41$	−0.123059	−0.0192186	−0.00960932	−	0.999954i	$-0.503059\pi$
$41$	−0.123059	−0.0192186	−0.00960932	+	0.999954i	$0.503059\pi$
$42$	0	0
$43$	− 5.39744i	− 0.823101i	−0.911387	−	0.411551i	$-0.864987\pi$
$43$	− 5.39744i	− 0.823101i	0.911387	−	0.411551i	$-0.135013\pi$
$44$	0	0
$45$	− 7.61248i	− 1.13480i
$46$	0	0
$47$	2.02928	0.296001	0.148000	−	0.988987i	$-0.452716\pi$
$47$	2.02928	0.296001	0.148000	+	0.988987i	$0.452716\pi$
$48$	0	0
$49$	−6.71307	−0.959010
$50$	0	0
$51$	− 17.0787i	− 2.39150i
$52$	0	0
$53$	− 5.82785i	− 0.800517i	−0.916402	−	0.400259i	$-0.868920\pi$
$53$	− 5.82785i	− 0.800517i	0.916402	−	0.400259i	$-0.131080\pi$
$54$	0	0
$55$	−1.33473	−0.179975
$56$	0	0
$57$	2.32921	0.308512
$58$	0	0
$59$	− 5.56633i	− 0.724675i	−0.932047	−	0.362337i	$-0.881979\pi$
$59$	− 5.56633i	− 0.724675i	0.932047	−	0.362337i	$-0.118021\pi$
$60$	0	0
$61$	6.99606i	0.895753i	0.894095	+	0.447877i	$0.147820\pi$
$61$	6.99606i	0.895753i	−0.894095	+	0.447877i	$0.852180\pi$
$62$	0	0
$63$	−1.29909	−0.163670
$64$	0	0
$65$	20.8982	2.59210
$66$	0	0
$67$	12.3777i	1.51217i	0.654471	+	0.756087i	$0.272891\pi$
$67$	12.3777i	1.51217i	−0.654471	+	0.756087i	$0.727109\pi$
$68$	0	0
$69$	− 13.7431i	− 1.65448i
$70$	0	0
$71$	−12.1786	−1.44534	−0.722669	−	0.691194i	$-0.757085\pi$
$71$	−12.1786	−1.44534	−0.722669	+	0.691194i	$0.757085\pi$
$72$	0	0
$73$	−6.99382	−0.818565	−0.409282	−	0.912408i	$-0.634221\pi$
$73$	−6.99382	−0.818565	−0.409282	+	0.912408i	$0.634221\pi$
$74$	0	0
$75$	11.3025i	1.30510i
$76$	0	0
$77$	0.227776i	0.0259575i
$78$	0	0
$79$	−2.07996	−0.234014	−0.117007	−	0.993131i	$-0.537330\pi$
$79$	−2.07996	−0.234014	−0.117007	+	0.993131i	$0.537330\pi$
$80$	0	0
$81$	−10.3940	−1.15488
$82$	0	0
$83$	11.8227i	1.29771i	0.760914	+	0.648853i	$0.224751\pi$
$83$	11.8227i	1.29771i	−0.760914	+	0.648853i	$0.775249\pi$
$84$	0	0
$85$	23.0154i	2.49637i
$86$	0	0
$87$	−1.94964	−0.209023
$88$	0	0
$89$	13.7684	1.45945	0.729723	−	0.683743i	$-0.239649\pi$
$89$	13.7684	1.45945	0.729723	+	0.683743i	$0.239649\pi$
$90$	0	0
$91$	− 3.56633i	− 0.373853i
$92$	0	0
$93$	− 7.38358i	− 0.765641i
$94$	0	0
$95$	−3.13887	−0.322041
$96$	0	0
$97$	0.801240	0.0813536	0.0406768	−	0.999172i	$-0.487049\pi$
$97$	0.801240	0.0813536	0.0406768	+	0.999172i	$0.487049\pi$
$98$	0	0
$99$	− 1.03127i	− 0.103647i
$100$	0	0
$101$	12.4171i	1.23555i	0.786356	+	0.617773i	$0.211965\pi$
$101$	12.4171i	1.23555i	−0.786356	+	0.617773i	$0.788035\pi$
$102$	0	0
$103$	2.84869	0.280690	0.140345	−	0.990103i	$-0.455179\pi$
$103$	2.84869	0.280690	0.140345	+	0.990103i	$0.455179\pi$
$104$	0	0
$105$	3.91624	0.382186
$106$	0	0
$107$	0.288142i	0.0278557i	0.999903	+	0.0139278i	$0.00443352\pi$
$107$	0.288142i	0.0278557i	−0.999903	+	0.0139278i	$0.995566\pi$
$108$	0	0
$109$	1.91873i	0.183781i	0.995769	+	0.0918905i	$0.0292910\pi$
$109$	1.91873i	0.183781i	−0.995769	+	0.0918905i	$0.970709\pi$
$110$	0	0
$111$	8.14036	0.772649
$112$	0	0
$113$	−13.4370	−1.26405	−0.632024	−	0.774949i	$-0.717776\pi$
$113$	−13.4370	−1.26405	−0.632024	+	0.774949i	$0.717776\pi$
$114$	0	0
$115$	18.5204i	1.72704i
$116$	0	0
$117$	16.1468i	1.49277i
$118$	0	0
$119$	3.92765	0.360047
$120$	0	0
$121$	10.8192	0.983562
$122$	0	0
$123$	0.286631i	0.0258447i
$124$	0	0
$125$	0.462931i	0.0414058i
$126$	0	0
$127$	18.9992	1.68591	0.842953	−	0.537987i	$-0.180815\pi$
$127$	18.9992	1.68591	0.842953	+	0.537987i	$0.180815\pi$
$128$	0	0
$129$	−12.5718	−1.10688
$130$	0	0
$131$	− 10.1330i	− 0.885322i	−0.896689	−	0.442661i	$-0.854035\pi$
$131$	− 10.1330i	− 0.885322i	0.896689	−	0.442661i	$-0.145965\pi$
$132$	0	0
$133$	0.535658i	0.0464474i
$134$	0	0
$135$	4.20222	0.361670
$136$	0	0
$137$	−3.28221	−0.280418	−0.140209	−	0.990122i	$-0.544778\pi$
$137$	−3.28221	−0.280418	−0.140209	+	0.990122i	$0.544778\pi$
$138$	0	0
$139$	10.5345i	0.893527i	0.894652	+	0.446763i	$0.147423\pi$
$139$	10.5345i	0.893527i	−0.894652	+	0.446763i	$0.852577\pi$
$140$	0	0
$141$	− 4.72662i	− 0.398053i
$142$	0	0
$143$	2.83110	0.236749
$144$	0	0
$145$	2.62735	0.218190
$146$	0	0
$147$	15.6362i	1.28965i
$148$	0	0
$149$	5.93244i	0.486005i	0.970026	+	0.243002i	$0.0781323\pi$
$149$	5.93244i	0.486005i	−0.970026	+	0.243002i	$0.921868\pi$
$150$	0	0
$151$	−4.57907	−0.372639	−0.186320	−	0.982489i	$-0.559656\pi$
$151$	−4.57907	−0.372639	−0.186320	+	0.982489i	$0.559656\pi$
$152$	0	0
$153$	−17.7827	−1.43765
$154$	0	0
$155$	9.95019i	0.799219i
$156$	0	0
$157$	− 3.80861i	− 0.303960i	−0.988384	−	0.151980i	$-0.951435\pi$
$157$	− 3.80861i	− 0.303960i	0.988384	−	0.151980i	$-0.0485649\pi$
$158$	0	0
$159$	−13.5743	−1.07651
$160$	0	0
$161$	3.16056	0.249087
$162$	0	0
$163$	− 17.9293i	− 1.40433i	−0.712016	−	0.702164i	$-0.752217\pi$
$163$	− 17.9293i	− 1.40433i	0.712016	−	0.702164i	$-0.247783\pi$
$164$	0	0
$165$	3.10888i	0.242026i
$166$	0	0
$167$	−18.8189	−1.45625	−0.728123	−	0.685446i	$-0.759607\pi$
$167$	−18.8189	−1.45625	−0.728123	+	0.685446i	$0.759607\pi$
$168$	0	0
$169$	−31.3271	−2.40978
$170$	0	0
$171$	− 2.42523i	− 0.185462i
$172$	0	0
$173$	− 5.87750i	− 0.446858i	−0.974720	−	0.223429i	$-0.928275\pi$
$173$	− 5.87750i	− 0.446858i	0.974720	−	0.223429i	$-0.0717251\pi$
$174$	0	0
$175$	−2.59929	−0.196488
$176$	0	0
$177$	−12.9652	−0.974522
$178$	0	0
$179$	− 18.9245i	− 1.41448i	−0.706973	−	0.707241i	$-0.749940\pi$
$179$	− 18.9245i	− 1.41448i	0.706973	−	0.707241i	$-0.250060\pi$
$180$	0	0
$181$	− 10.5330i	− 0.782914i	−0.920196	−	0.391457i	$-0.871971\pi$
$181$	− 10.5330i	− 0.782914i	0.920196	−	0.391457i	$-0.128029\pi$
$182$	0	0
$183$	16.2953	1.20458
$184$	0	0
$185$	−10.9700	−0.806534
$186$	0	0
$187$	3.11793i	0.228006i
$188$	0	0
$189$	− 0.717121i	− 0.0521629i
$190$	0	0
$191$	−1.60481	−0.116120	−0.0580600	−	0.998313i	$-0.518491\pi$
$191$	−1.60481	−0.116120	−0.0580600	+	0.998313i	$0.518491\pi$
$192$	0	0
$193$	−13.2889	−0.956554	−0.478277	−	0.878209i	$-0.658739\pi$
$193$	−13.2889	−0.956554	−0.478277	+	0.878209i	$0.658739\pi$
$194$	0	0
$195$	− 48.6763i	− 3.48578i
$196$	0	0
$197$	4.06689i	0.289754i	0.989450	+	0.144877i	$0.0462787\pi$
$197$	4.06689i	0.289754i	−0.989450	+	0.144877i	$0.953721\pi$
$198$	0	0
$199$	−12.8273	−0.909307	−0.454653	−	0.890668i	$-0.650237\pi$
$199$	−12.8273	−0.909307	−0.454653	+	0.890668i	$0.650237\pi$
$200$	0	0
$201$	28.8302	2.03353
$202$	0	0
$203$	− 0.448365i	− 0.0314691i
$204$	0	0
$205$	− 0.386268i	− 0.0269781i
$206$	0	0
$207$	−14.3096	−0.994589
$208$	0	0
$209$	−0.425227	−0.0294136
$210$	0	0
$211$	15.9906i	1.10084i	0.834888	+	0.550420i	$0.185532\pi$
$211$	15.9906i	1.10084i	−0.834888	+	0.550420i	$0.814468\pi$
$212$	0	0
$213$	28.3666i	1.94365i
$214$	0	0
$215$	16.9419	1.15543
$216$	0	0
$217$	1.69803	0.115270
$218$	0	0
$219$	16.2901i	1.10078i
$220$	0	0
$221$	− 48.8181i	− 3.28386i
$222$	0	0
$223$	−21.1424	−1.41580	−0.707899	−	0.706314i	$-0.750357\pi$
$223$	−21.1424	−1.41580	−0.707899	+	0.706314i	$0.750357\pi$
$224$	0	0
$225$	11.7685	0.784564
$226$	0	0
$227$	22.2056i	1.47384i	0.675982	+	0.736918i	$0.263720\pi$
$227$	22.2056i	1.47384i	−0.675982	+	0.736918i	$0.736280\pi$
$228$	0	0
$229$	12.6331i	0.834816i	0.908719	+	0.417408i	$0.137061\pi$
$229$	12.6331i	0.834816i	−0.908719	+	0.417408i	$0.862939\pi$
$230$	0	0
$231$	0.530539	0.0349069
$232$	0	0
$233$	3.01852	0.197750	0.0988749	−	0.995100i	$-0.468476\pi$
$233$	3.01852	0.197750	0.0988749	+	0.995100i	$0.468476\pi$
$234$	0	0
$235$	6.36965i	0.415510i
$236$	0	0
$237$	4.84468i	0.314695i
$238$	0	0
$239$	−12.7156	−0.822504	−0.411252	−	0.911522i	$-0.634908\pi$
$239$	−12.7156	−0.822504	−0.411252	+	0.911522i	$0.634908\pi$
$240$	0	0
$241$	−1.05734	−0.0681091	−0.0340545	−	0.999420i	$-0.510842\pi$
$241$	−1.05734	−0.0681091	−0.0340545	+	0.999420i	$0.510842\pi$
$242$	0	0
$243$	20.1934i	1.29541i
$244$	0	0
$245$	− 21.0715i	− 1.34621i
$246$	0	0
$247$	6.65786	0.423630
$248$	0	0
$249$	27.5375	1.74512
$250$	0	0
$251$	− 13.4689i	− 0.850151i	−0.905158	−	0.425076i	$-0.860248\pi$
$251$	− 13.4689i	− 0.850151i	0.905158	−	0.425076i	$-0.139752\pi$
$252$	0	0
$253$	2.50898i	0.157738i
$254$	0	0
$255$	53.6078	3.35705
$256$	0	0
$257$	19.1631	1.19536	0.597680	−	0.801734i	$-0.296089\pi$
$257$	19.1631	1.19536	0.597680	+	0.801734i	$0.296089\pi$
$258$	0	0
$259$	1.87207i	0.116325i
$260$	0	0
$261$	2.03001i	0.125654i
$262$	0	0
$263$	2.55940	0.157819	0.0789097	−	0.996882i	$-0.474856\pi$
$263$	2.55940	0.157819	0.0789097	+	0.996882i	$0.474856\pi$
$264$	0	0
$265$	18.2929	1.12372
$266$	0	0
$267$	− 32.0695i	− 1.96262i
$268$	0	0
$269$	4.88596i	0.297902i	0.988845	+	0.148951i	$0.0475897\pi$
$269$	4.88596i	0.297902i	−0.988845	+	0.148951i	$0.952410\pi$
$270$	0	0
$271$	−32.1299	−1.95175	−0.975875	−	0.218329i	$-0.929940\pi$
$271$	−32.1299	−1.95175	−0.975875	+	0.218329i	$0.929940\pi$
$272$	0	0
$273$	−8.30675	−0.502747
$274$	0	0
$275$	− 2.06342i	− 0.124429i
$276$	0	0
$277$	− 7.46639i	− 0.448612i	−0.974519	−	0.224306i	$-0.927988\pi$
$277$	− 7.46639i	− 0.448612i	0.974519	−	0.224306i	$-0.0720115\pi$
$278$	0	0
$279$	−7.68795	−0.460265
$280$	0	0
$281$	18.9770	1.13208	0.566038	−	0.824379i	$-0.308476\pi$
$281$	18.9770	1.13208	0.566038	+	0.824379i	$0.308476\pi$
$282$	0	0
$283$	− 1.24321i	− 0.0739012i	−0.999317	−	0.0369506i	$-0.988236\pi$
$283$	− 1.24321i	− 0.0739012i	0.999317	−	0.0369506i	$-0.0117644\pi$
$284$	0	0
$285$	7.31110i	0.433072i
$286$	0	0
$287$	−0.0659177	−0.00389100
$288$	0	0
$289$	36.7640	2.16259
$290$	0	0
$291$	− 1.86626i	− 0.109402i
$292$	0	0
$293$	5.35018i	0.312561i	0.987713	+	0.156280i	$0.0499504\pi$
$293$	5.35018i	0.312561i	−0.987713	+	0.156280i	$0.950050\pi$
$294$	0	0
$295$	17.4720	1.01726
$296$	0	0
$297$	0.569280	0.0330330
$298$	0	0
$299$	− 39.2836i	− 2.27183i
$300$	0	0
$301$	− 2.89118i	− 0.166645i
$302$	0	0
$303$	28.9220	1.66153
$304$	0	0
$305$	−21.9597	−1.25741
$306$	0	0
$307$	10.4800i	0.598123i	0.954234	+	0.299062i	$0.0966737\pi$
$307$	10.4800i	0.598123i	−0.954234	+	0.299062i	$0.903326\pi$
$308$	0	0
$309$	− 6.63521i	− 0.377464i
$310$	0	0
$311$	27.3720	1.55212	0.776061	−	0.630658i	$-0.217215\pi$
$311$	27.3720	1.55212	0.776061	+	0.630658i	$0.217215\pi$
$312$	0	0
$313$	−16.9214	−0.956451	−0.478226	−	0.878237i	$-0.658720\pi$
$313$	−16.9214	−0.956451	−0.478226	+	0.878237i	$0.658720\pi$
$314$	0	0
$315$	− 4.07768i	− 0.229751i
$316$	0	0
$317$	− 3.82118i	− 0.214619i	−0.994226	−	0.107309i	$-0.965776\pi$
$317$	− 3.82118i	− 0.214619i	0.994226	−	0.107309i	$-0.0342235\pi$
$318$	0	0
$319$	0.355931	0.0199283
$320$	0	0
$321$	0.671143	0.0374595
$322$	0	0
$323$	7.33239i	0.407985i
$324$	0	0
$325$	32.3074i	1.79209i
$326$	0	0
$327$	4.46913	0.247143
$328$	0	0
$329$	1.08700	0.0599282
$330$	0	0
$331$	26.6082i	1.46252i	0.682101	+	0.731258i	$0.261067\pi$
$331$	26.6082i	1.46252i	−0.682101	+	0.731258i	$0.738933\pi$
$332$	0	0
$333$	− 8.47592i	− 0.464478i
$334$	0	0
$335$	−38.8520	−2.12271
$336$	0	0
$337$	−11.6884	−0.636707	−0.318354	−	0.947972i	$-0.603130\pi$
$337$	−11.6884	−0.636707	−0.318354	+	0.947972i	$0.603130\pi$
$338$	0	0
$339$	31.2976i	1.69985i
$340$	0	0
$341$	1.34797i	0.0729964i
$342$	0	0
$343$	−7.34551	−0.396620
$344$	0	0
$345$	43.1379	2.32247
$346$	0	0
$347$	16.2573i	0.872739i	0.899767	+	0.436370i	$0.143736\pi$
$347$	16.2573i	0.872739i	−0.899767	+	0.436370i	$0.856264\pi$
$348$	0	0
$349$	− 22.5885i	− 1.20913i	−0.796554	−	0.604567i	$-0.793346\pi$
$349$	− 22.5885i	− 1.20913i	0.796554	−	0.604567i	$-0.206654\pi$
$350$	0	0
$351$	−8.91333	−0.475758
$352$	0	0
$353$	−10.2804	−0.547169	−0.273584	−	0.961848i	$-0.588209\pi$
$353$	−10.2804	−0.547169	−0.273584	+	0.961848i	$0.588209\pi$
$354$	0	0
$355$	− 38.2272i	− 2.02889i
$356$	0	0
$357$	− 9.14833i	− 0.484181i
$358$	0	0
$359$	16.1694	0.853386	0.426693	−	0.904397i	$-0.359679\pi$
$359$	16.1694	0.853386	0.426693	+	0.904397i	$0.359679\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 25.2002i	− 1.32267i
$364$	0	0
$365$	− 21.9527i	− 1.14906i
$366$	0	0
$367$	−19.5371	−1.01983	−0.509914	−	0.860225i	$-0.670323\pi$
$367$	−19.5371	−1.01983	−0.509914	+	0.860225i	$0.670323\pi$
$368$	0	0
$369$	0.298447	0.0155365
$370$	0	0
$371$	− 3.12173i	− 0.162072i
$372$	0	0
$373$	12.9672i	0.671418i	0.941966	+	0.335709i	$0.108976\pi$
$373$	12.9672i	0.671418i	−0.941966	+	0.335709i	$0.891024\pi$
$374$	0	0
$375$	1.07826	0.0556813
$376$	0	0
$377$	−5.57288	−0.287018
$378$	0	0
$379$	0.229262i	0.0117764i	0.999983	+	0.00588821i	$0.00187428\pi$
$379$	0.229262i	0.0117764i	−0.999983	+	0.00588821i	$0.998126\pi$
$380$	0	0
$381$	− 44.2532i	− 2.26716i
$382$	0	0
$383$	18.9168	0.966601	0.483301	−	0.875455i	$-0.339438\pi$
$383$	18.9168	0.966601	0.483301	+	0.875455i	$0.339438\pi$
$384$	0	0
$385$	−0.714960	−0.0364377
$386$	0	0
$387$	13.0900i	0.665403i
$388$	0	0
$389$	9.38404i	0.475790i	0.971291	+	0.237895i	$0.0764573\pi$
$389$	9.38404i	0.475790i	−0.971291	+	0.237895i	$0.923543\pi$
$390$	0	0
$391$	43.2636	2.18793
$392$	0	0
$393$	−23.6018	−1.19055
$394$	0	0
$395$	− 6.52874i	− 0.328496i
$396$	0	0
$397$	18.3307i	0.919990i	0.887921	+	0.459995i	$0.152149\pi$
$397$	18.3307i	0.919990i	−0.887921	+	0.459995i	$0.847851\pi$
$398$	0	0
$399$	1.24766	0.0624611
$400$	0	0
$401$	−22.7518	−1.13617	−0.568085	−	0.822970i	$-0.692316\pi$
$401$	−22.7518	−1.13617	−0.568085	+	0.822970i	$0.692316\pi$
$402$	0	0
$403$	− 21.1054i	− 1.05133i
$404$	0	0
$405$	− 32.6253i	− 1.62116i
$406$	0	0
$407$	−1.48613	−0.0736645
$408$	0	0
$409$	−19.3401	−0.956305	−0.478152	−	0.878277i	$-0.658693\pi$
$409$	−19.3401	−0.956305	−0.478152	+	0.878277i	$0.658693\pi$
$410$	0	0
$411$	7.64497i	0.377099i
$412$	0	0
$413$	− 2.98165i	− 0.146717i
$414$	0	0
$415$	−37.1098	−1.82165
$416$	0	0
$417$	24.5371	1.20159
$418$	0	0
$419$	− 28.0225i	− 1.36899i	−0.729017	−	0.684495i	$-0.760023\pi$
$419$	− 28.0225i	− 1.36899i	0.729017	−	0.684495i	$-0.239977\pi$
$420$	0	0
$421$	− 15.5487i	− 0.757798i	−0.925438	−	0.378899i	$-0.876303\pi$
$421$	− 15.5487i	− 0.757798i	0.925438	−	0.378899i	$-0.123697\pi$
$422$	0	0
$423$	−4.92146	−0.239290
$424$	0	0
$425$	−35.5806	−1.72591
$426$	0	0
$427$	3.74749i	0.181354i
$428$	0	0
$429$	− 6.59424i	− 0.318373i
$430$	0	0
$431$	−15.5573	−0.749368	−0.374684	−	0.927153i	$-0.622249\pi$
$431$	−15.5573	−0.749368	−0.374684	+	0.927153i	$0.622249\pi$
$432$	0	0
$433$	4.16219	0.200022	0.100011	−	0.994986i	$-0.468112\pi$
$433$	4.16219	0.200022	0.100011	+	0.994986i	$0.468112\pi$
$434$	0	0
$435$	− 6.11966i	− 0.293415i
$436$	0	0
$437$	5.90033i	0.282251i
$438$	0	0
$439$	−24.6600	−1.17696	−0.588478	−	0.808513i	$-0.700273\pi$
$439$	−24.6600	−1.17696	−0.588478	+	0.808513i	$0.700273\pi$
$440$	0	0
$441$	16.2807	0.775272
$442$	0	0
$443$	19.7595i	0.938801i	0.882985	+	0.469401i	$0.155530\pi$
$443$	19.7595i	0.938801i	−0.882985	+	0.469401i	$0.844470\pi$
$444$	0	0
$445$	43.2172i	2.04869i
$446$	0	0
$447$	13.8179	0.653565
$448$	0	0
$449$	−16.4826	−0.777862	−0.388931	−	0.921267i	$-0.627156\pi$
$449$	−16.4826	−0.777862	−0.388931	+	0.921267i	$0.627156\pi$
$450$	0	0
$451$	− 0.0523282i	− 0.00246404i
$452$	0	0
$453$	10.6656i	0.501115i
$454$	0	0
$455$	11.1943	0.524795
$456$	0	0
$457$	7.44504	0.348264	0.174132	−	0.984722i	$-0.444288\pi$
$457$	7.44504	0.348264	0.174132	+	0.984722i	$0.444288\pi$
$458$	0	0
$459$	− 9.81637i	− 0.458189i
$460$	0	0
$461$	14.9337i	0.695532i	0.937581	+	0.347766i	$0.113060\pi$
$461$	14.9337i	0.695532i	−0.937581	+	0.347766i	$0.886940\pi$
$462$	0	0
$463$	23.6042	1.09698	0.548491	−	0.836156i	$-0.315202\pi$
$463$	23.6042	1.09698	0.548491	+	0.836156i	$0.315202\pi$
$464$	0	0
$465$	23.1761	1.07477
$466$	0	0
$467$	− 0.808996i	− 0.0374359i	−0.999825	−	0.0187179i	$-0.994042\pi$
$467$	− 0.808996i	− 0.0374359i	0.999825	−	0.0187179i	$-0.00595845\pi$
$468$	0	0
$469$	6.63020i	0.306154i
$470$	0	0
$471$	−8.87106	−0.408757
$472$	0	0
$473$	2.29514	0.105530
$474$	0	0
$475$	− 4.85252i	− 0.222649i
$476$	0	0
$477$	14.1339i	0.647145i
$478$	0	0
$479$	12.2547	0.559933	0.279966	−	0.960010i	$-0.409677\pi$
$479$	12.2547	0.559933	0.279966	+	0.960010i	$0.409677\pi$
$480$	0	0
$481$	23.2686	1.06095
$482$	0	0
$483$	− 7.36161i	− 0.334965i
$484$	0	0
$485$	2.51499i	0.114200i
$486$	0	0
$487$	−23.1692	−1.04990	−0.524948	−	0.851134i	$-0.675915\pi$
$487$	−23.1692	−1.04990	−0.524948	+	0.851134i	$0.675915\pi$
$488$	0	0
$489$	−41.7610	−1.88850
$490$	0	0
$491$	− 20.6669i	− 0.932684i	−0.884605	−	0.466342i	$-0.845572\pi$
$491$	− 20.6669i	− 0.932684i	0.884605	−	0.466342i	$-0.154428\pi$
$492$	0	0
$493$	− 6.13749i	− 0.276418i
$494$	0	0
$495$	3.23703	0.145494
$496$	0	0
$497$	−6.52358	−0.292623
$498$	0	0
$499$	5.36134i	0.240007i	0.992773	+	0.120003i	$0.0382905\pi$
$499$	5.36134i	0.240007i	−0.992773	+	0.120003i	$0.961709\pi$
$500$	0	0
$501$	43.8331i	1.95832i
$502$	0	0
$503$	25.6878	1.14536	0.572681	−	0.819778i	$-0.305903\pi$
$503$	25.6878	1.14536	0.572681	+	0.819778i	$0.305903\pi$
$504$	0	0
$505$	−38.9757	−1.73439
$506$	0	0
$507$	72.9675i	3.24060i
$508$	0	0
$509$	16.2273i	0.719261i	0.933095	+	0.359631i	$0.117097\pi$
$509$	16.2273i	0.719261i	−0.933095	+	0.359631i	$0.882903\pi$
$510$	0	0
$511$	−3.74629	−0.165726
$512$	0	0
$513$	1.33877	0.0591080
$514$	0	0
$515$	8.94168i	0.394017i
$516$	0	0
$517$	0.862904i	0.0379505i
$518$	0	0
$519$	−13.6899	−0.600922
$520$	0	0
$521$	32.4863	1.42325	0.711626	−	0.702559i	$-0.247959\pi$
$521$	32.4863	1.42325	0.711626	+	0.702559i	$0.247959\pi$
$522$	0	0
$523$	11.1406i	0.487143i	0.969883	+	0.243571i	$0.0783190\pi$
$523$	11.1406i	0.487143i	−0.969883	+	0.243571i	$0.921681\pi$
$524$	0	0
$525$	6.05429i	0.264231i
$526$	0	0
$527$	23.2436	1.01251
$528$	0	0
$529$	11.8139	0.513649
$530$	0	0
$531$	13.4996i	0.585834i
$532$	0	0
$533$	0.819312i	0.0354884i
$534$	0	0
$535$	−0.904439	−0.0391023
$536$	0	0
$537$	−44.0791	−1.90215
$538$	0	0
$539$	− 2.85458i	− 0.122955i
$540$	0	0
$541$	1.33136i	0.0572394i	0.999590	+	0.0286197i	$0.00911118\pi$
$541$	1.33136i	0.0572394i	−0.999590	+	0.0286197i	$0.990889\pi$
$542$	0	0
$543$	−24.5336	−1.05284
$544$	0	0
$545$	−6.02264	−0.257982
$546$	0	0
$547$	− 21.1345i	− 0.903644i	−0.892108	−	0.451822i	$-0.850774\pi$
$547$	− 21.1345i	− 0.903644i	0.892108	−	0.451822i	$-0.149226\pi$
$548$	0	0
$549$	− 16.9670i	− 0.724135i
$550$	0	0
$551$	0.837037	0.0356590
$552$	0	0
$553$	−1.11415	−0.0473784
$554$	0	0
$555$	25.5515i	1.08460i
$556$	0	0
$557$	− 6.60875i	− 0.280022i	−0.990150	−	0.140011i	$-0.955286\pi$
$557$	− 6.60875i	− 0.280022i	0.990150	−	0.140011i	$-0.0447138\pi$
$558$	0	0
$559$	−35.9354	−1.51991
$560$	0	0
$561$	7.26232	0.306615
$562$	0	0
$563$	4.42073i	0.186312i	0.995652	+	0.0931559i	$0.0296955\pi$
$563$	4.42073i	0.186312i	−0.995652	+	0.0931559i	$0.970305\pi$
$564$	0	0
$565$	− 42.1771i	− 1.77440i
$566$	0	0
$567$	−5.56760	−0.233817
$568$	0	0
$569$	8.59442	0.360297	0.180149	−	0.983639i	$-0.442342\pi$
$569$	8.59442	0.360297	0.180149	+	0.983639i	$0.442342\pi$
$570$	0	0
$571$	− 34.6802i	− 1.45132i	−0.688052	−	0.725661i	$-0.741534\pi$
$571$	− 34.6802i	− 1.45132i	0.688052	−	0.725661i	$-0.258466\pi$
$572$	0	0
$573$	3.73794i	0.156155i
$574$	0	0
$575$	−28.6315	−1.19401
$576$	0	0
$577$	26.2877	1.09437	0.547185	−	0.837012i	$-0.315699\pi$
$577$	26.2877	1.09437	0.547185	+	0.837012i	$0.315699\pi$
$578$	0	0
$579$	30.9526i	1.28635i
$580$	0	0
$581$	6.33290i	0.262733i
$582$	0	0
$583$	2.47816	0.102635
$584$	0	0
$585$	−50.6828	−2.09548
$586$	0	0
$587$	7.01067i	0.289361i	0.989478	+	0.144681i	$0.0462155\pi$
$587$	7.01067i	0.289361i	−0.989478	+	0.144681i	$0.953785\pi$
$588$	0	0
$589$	3.16999i	0.130617i
$590$	0	0
$591$	9.47265	0.389653
$592$	0	0
$593$	34.7115	1.42543	0.712716	−	0.701453i	$-0.247465\pi$
$593$	34.7115	1.42543	0.712716	+	0.701453i	$0.247465\pi$
$594$	0	0
$595$	12.3284i	0.505415i
$596$	0	0
$597$	29.8776i	1.22281i
$598$	0	0
$599$	32.4456	1.32569	0.662845	−	0.748757i	$-0.269349\pi$
$599$	32.4456	1.32569	0.662845	+	0.748757i	$0.269349\pi$
$600$	0	0
$601$	−36.2901	−1.48030	−0.740152	−	0.672440i	$-0.765246\pi$
$601$	−36.2901	−1.48030	−0.740152	+	0.672440i	$0.765246\pi$
$602$	0	0
$603$	− 30.0187i	− 1.22246i
$604$	0	0
$605$	33.9600i	1.38067i
$606$	0	0
$607$	−5.54708	−0.225149	−0.112575	−	0.993643i	$-0.535910\pi$
$607$	−5.54708	−0.225149	−0.112575	+	0.993643i	$0.535910\pi$
$608$	0	0
$609$	−1.04434	−0.0423187
$610$	0	0
$611$	− 13.5107i	− 0.546583i
$612$	0	0
$613$	− 41.7239i	− 1.68521i	−0.538530	−	0.842606i	$-0.681020\pi$
$613$	− 41.7239i	− 1.68521i	0.538530	−	0.842606i	$-0.318980\pi$
$614$	0	0
$615$	−0.899699	−0.0362794
$616$	0	0
$617$	16.0859	0.647593	0.323797	−	0.946127i	$-0.395041\pi$
$617$	16.0859	0.647593	0.323797	+	0.946127i	$0.395041\pi$
$618$	0	0
$619$	− 2.93257i	− 0.117870i	−0.998262	−	0.0589349i	$-0.981230\pi$
$619$	− 2.93257i	− 0.117870i	0.998262	−	0.0589349i	$-0.0187704\pi$
$620$	0	0
$621$	− 7.89918i	− 0.316983i
$622$	0	0
$623$	7.37514	0.295479
$624$	0	0
$625$	−25.7157	−1.02863
$626$	0	0
$627$	0.990444i	0.0395545i
$628$	0	0
$629$	25.6260i	1.02177i
$630$	0	0
$631$	23.0095	0.915994	0.457997	−	0.888954i	$-0.348567\pi$
$631$	23.0095	0.915994	0.457997	+	0.888954i	$0.348567\pi$
$632$	0	0
$633$	37.2455	1.48038
$634$	0	0
$635$	59.6361i	2.36658i
$636$	0	0
$637$	44.6947i	1.77087i
$638$	0	0
$639$	29.5360	1.16842
$640$	0	0
$641$	−0.00476234	−0.000188101 0	−9.40506e−5	−	1.00000i	$-0.500030\pi$
$641$	−0.00476234	−0.000188101 0	−9.40506e−5		1.00000i	$0.500030\pi$
$642$	0	0
$643$	37.6587i	1.48511i	0.669783	+	0.742557i	$0.266387\pi$
$643$	37.6587i	1.48511i	−0.669783	+	0.742557i	$0.733613\pi$
$644$	0	0
$645$	− 39.4612i	− 1.55378i
$646$	0	0
$647$	−25.3744	−0.997570	−0.498785	−	0.866726i	$-0.666220\pi$
$647$	−25.3744	−0.997570	−0.498785	+	0.866726i	$0.666220\pi$
$648$	0	0
$649$	2.36696	0.0929111
$650$	0	0
$651$	− 3.95507i	− 0.155011i
$652$	0	0
$653$	− 22.5548i	− 0.882637i	−0.897350	−	0.441319i	$-0.854511\pi$
$653$	− 22.5548i	− 0.882637i	0.897350	−	0.441319i	$-0.145489\pi$
$654$	0	0
$655$	31.8061	1.24277
$656$	0	0
$657$	16.9616	0.661735
$658$	0	0
$659$	18.3990i	0.716724i	0.933583	+	0.358362i	$0.116665\pi$
$659$	18.3990i	0.716724i	−0.933583	+	0.358362i	$0.883335\pi$
$660$	0	0
$661$	− 39.9383i	− 1.55342i	−0.629859	−	0.776710i	$-0.716887\pi$
$661$	− 39.9383i	− 1.55342i	0.629859	−	0.776710i	$-0.283113\pi$
$662$	0	0
$663$	−113.708	−4.41604
$664$	0	0
$665$	−1.68136	−0.0652004
$666$	0	0
$667$	− 4.93880i	− 0.191231i
$668$	0	0
$669$	49.2451i	1.90392i
$670$	0	0
$671$	−2.97491	−0.114845
$672$	0	0
$673$	−4.04663	−0.155986	−0.0779931	−	0.996954i	$-0.524851\pi$
$673$	−4.04663	−0.155986	−0.0779931	+	0.996954i	$0.524851\pi$
$674$	0	0
$675$	6.49639i	0.250046i
$676$	0	0
$677$	42.8041i	1.64509i	0.568697	+	0.822547i	$0.307448\pi$
$677$	42.8041i	1.64509i	−0.568697	+	0.822547i	$0.692552\pi$
$678$	0	0
$679$	0.429190	0.0164708
$680$	0	0
$681$	51.7215	1.98197
$682$	0	0
$683$	− 6.33556i	− 0.242423i	−0.992627	−	0.121212i	$-0.961322\pi$
$683$	− 6.33556i	− 0.242423i	0.992627	−	0.121212i	$-0.0386780\pi$
$684$	0	0
$685$	− 10.3024i	− 0.393636i
$686$	0	0
$687$	29.4251	1.12264
$688$	0	0
$689$	−38.8010	−1.47820
$690$	0	0
$691$	15.3700i	0.584702i	0.956311	+	0.292351i	$0.0944375\pi$
$691$	15.3700i	0.584702i	−0.956311	+	0.292351i	$0.905562\pi$
$692$	0	0
$693$	− 0.552409i	− 0.0209843i
$694$	0	0
$695$	−33.0665	−1.25428
$696$	0	0
$697$	−0.902320	−0.0341778
$698$	0	0
$699$	− 7.03077i	− 0.265928i
$700$	0	0
$701$	− 36.0602i	− 1.36197i	−0.732296	−	0.680987i	$-0.761551\pi$
$701$	− 36.0602i	− 1.36197i	0.732296	−	0.680987i	$-0.238449\pi$
$702$	0	0
$703$	−3.49490	−0.131813
$704$	0	0
$705$	14.8363	0.558766
$706$	0	0
$707$	6.65131i	0.250148i
$708$	0	0
$709$	− 18.1426i	− 0.681360i	−0.940179	−	0.340680i	$-0.889343\pi$
$709$	− 18.1426i	− 0.681360i	0.940179	−	0.340680i	$-0.110657\pi$
$710$	0	0
$711$	5.04438	0.189179
$712$	0	0
$713$	18.7040	0.700470
$714$	0	0
$715$	8.88647i	0.332335i
$716$	0	0
$717$	29.6173i	1.10608i
$718$	0	0
$719$	38.2801	1.42761	0.713804	−	0.700345i	$-0.246971\pi$
$719$	38.2801	1.42761	0.713804	+	0.700345i	$0.246971\pi$
$720$	0	0
$721$	1.52592	0.0568284
$722$	0	0
$723$	2.46276i	0.0915911i
$724$	0	0
$725$	4.06174i	0.150849i
$726$	0	0
$727$	−23.4550	−0.869897	−0.434949	−	0.900455i	$-0.643233\pi$
$727$	−23.4550	−0.869897	−0.434949	+	0.900455i	$0.643233\pi$
$728$	0	0
$729$	15.8529	0.587144
$730$	0	0
$731$	− 39.5761i	− 1.46378i
$732$	0	0
$733$	19.3814i	0.715870i	0.933747	+	0.357935i	$0.116519\pi$
$733$	19.3814i	0.715870i	−0.933747	+	0.357935i	$0.883481\pi$
$734$	0	0
$735$	−49.0799	−1.81034
$736$	0	0
$737$	−5.26332	−0.193877
$738$	0	0
$739$	− 12.1161i	− 0.445697i	−0.974853	−	0.222848i	$-0.928464\pi$
$739$	− 12.1161i	− 0.445697i	0.974853	−	0.222848i	$-0.0715355\pi$
$740$	0	0
$741$	− 15.5076i	− 0.569685i
$742$	0	0
$743$	29.1148	1.06812	0.534060	−	0.845447i	$-0.320666\pi$
$743$	29.1148	1.06812	0.534060	+	0.845447i	$0.320666\pi$
$744$	0	0
$745$	−18.6212	−0.682227
$746$	0	0
$747$	− 28.6726i	− 1.04908i
$748$	0	0
$749$	0.154345i	0.00563965i
$750$	0	0
$751$	2.09618	0.0764906	0.0382453	−	0.999268i	$-0.487823\pi$
$751$	2.09618	0.0764906	0.0382453	+	0.999268i	$0.487823\pi$
$752$	0	0
$753$	−31.3720	−1.14326
$754$	0	0
$755$	− 14.3731i	− 0.523091i
$756$	0	0
$757$	− 36.7044i	− 1.33404i	−0.745038	−	0.667022i	$-0.767569\pi$
$757$	− 36.7044i	− 1.33404i	0.745038	−	0.667022i	$-0.232431\pi$
$758$	0	0
$759$	5.84395	0.212122
$760$	0	0
$761$	26.8362	0.972812	0.486406	−	0.873733i	$-0.338308\pi$
$761$	26.8362	0.972812	0.486406	+	0.873733i	$0.338308\pi$
$762$	0	0
$763$	1.02778i	0.0372082i
$764$	0	0
$765$	− 55.8177i	− 2.01809i
$766$	0	0
$767$	−37.0599	−1.33815
$768$	0	0
$769$	15.3434	0.553299	0.276649	−	0.960971i	$-0.410776\pi$
$769$	15.3434	0.553299	0.276649	+	0.960971i	$0.410776\pi$
$770$	0	0
$771$	− 44.6349i	− 1.60749i
$772$	0	0
$773$	54.9160i	1.97519i	0.157017	+	0.987596i	$0.449812\pi$
$773$	54.9160i	1.97519i	−0.157017	+	0.987596i	$0.550188\pi$
$774$	0	0
$775$	−15.3824	−0.552553
$776$	0	0
$777$	4.36045	0.156430
$778$	0	0
$779$	− 0.123059i	− 0.00440906i
$780$	0	0
$781$	− 5.17869i	− 0.185308i
$782$	0	0
$783$	−1.12060	−0.0400469
$784$	0	0
$785$	11.9547	0.426683
$786$	0	0
$787$	16.2826i	0.580413i	0.956964	+	0.290206i	$0.0937240\pi$
$787$	16.2826i	0.580413i	−0.956964	+	0.290206i	$0.906276\pi$
$788$	0	0
$789$	− 5.96139i	− 0.212231i
$790$	0	0
$791$	−7.19764	−0.255918
$792$	0	0
$793$	46.5788	1.65406
$794$	0	0
$795$	− 42.6080i	− 1.51115i
$796$	0	0
$797$	20.2945i	0.718867i	0.933171	+	0.359434i	$0.117030\pi$
$797$	20.2945i	0.718867i	−0.933171	+	0.359434i	$0.882970\pi$
$798$	0	0
$799$	14.8795	0.526398
$800$	0	0
$801$	−33.3915	−1.17983
$802$	0	0
$803$	− 2.97396i	− 0.104949i
$804$	0	0
$805$	9.92059i	0.349655i
$806$	0	0
$807$	11.3804	0.400610
$808$	0	0
$809$	−5.57944	−0.196163	−0.0980813	−	0.995178i	$-0.531271\pi$
$809$	−5.57944	−0.196163	−0.0980813	+	0.995178i	$0.531271\pi$
$810$	0	0
$811$	28.4695i	0.999699i	0.866112	+	0.499849i	$0.166611\pi$
$811$	28.4695i	0.999699i	−0.866112	+	0.499849i	$0.833389\pi$
$812$	0	0
$813$	74.8373i	2.62466i
$814$	0	0
$815$	56.2776	1.97132
$816$	0	0
$817$	5.39744	0.188832
$818$	0	0
$819$	8.64917i	0.302226i
$820$	0	0
$821$	− 3.41291i	− 0.119111i	−0.998225	−	0.0595556i	$-0.981032\pi$
$821$	− 3.41291i	− 0.119111i	0.998225	−	0.0595556i	$-0.0189684\pi$
$822$	0	0
$823$	12.8444	0.447729	0.223865	−	0.974620i	$-0.428133\pi$
$823$	12.8444	0.447729	0.223865	+	0.974620i	$0.428133\pi$
$824$	0	0
$825$	−4.80615	−0.167328
$826$	0	0
$827$	− 11.4599i	− 0.398501i	−0.979949	−	0.199251i	$-0.936149\pi$
$827$	− 11.4599i	− 0.398501i	0.979949	−	0.199251i	$-0.0638508\pi$
$828$	0	0
$829$	15.6953i	0.545119i	0.962139	+	0.272559i	$0.0878701\pi$
$829$	15.6953i	0.545119i	−0.962139	+	0.272559i	$0.912130\pi$
$830$	0	0
$831$	−17.3908	−0.603280
$832$	0	0
$833$	−49.2229	−1.70547
$834$	0	0
$835$	− 59.0700i	− 2.04420i
$836$	0	0
$837$	− 4.24388i	− 0.146690i
$838$	0	0
$839$	−40.4629	−1.39693	−0.698467	−	0.715642i	$-0.746134\pi$
$839$	−40.4629	−1.39693	−0.698467	+	0.715642i	$0.746134\pi$
$840$	0	0
$841$	28.2994	0.975840
$842$	0	0
$843$	− 44.2016i	− 1.52238i
$844$	0	0
$845$	− 98.3318i	− 3.38272i
$846$	0	0
$847$	5.79538	0.199131
$848$	0	0
$849$	−2.89570	−0.0993802
$850$	0	0
$851$	20.6211i	0.706881i
$852$	0	0
$853$	1.22071i	0.0417962i	0.999782	+	0.0208981i	$0.00665255\pi$
$853$	1.22071i	0.0417962i	−0.999782	+	0.0208981i	$0.993347\pi$
$854$	0	0
$855$	7.61248	0.260341
$856$	0	0
$857$	−49.3646	−1.68626	−0.843131	−	0.537709i	$-0.819290\pi$
$857$	−49.3646	−1.68626	−0.843131	+	0.537709i	$0.819290\pi$
$858$	0	0
$859$	41.6828i	1.42220i	0.703092	+	0.711099i	$0.251802\pi$
$859$	41.6828i	1.42220i	−0.703092	+	0.711099i	$0.748198\pi$
$860$	0	0
$861$	0.153536i	0.00523250i
$862$	0	0
$863$	22.0429	0.750348	0.375174	−	0.926954i	$-0.377583\pi$
$863$	22.0429	0.750348	0.375174	+	0.926954i	$0.377583\pi$
$864$	0	0
$865$	18.4487	0.627275
$866$	0	0
$867$	− 85.6311i	− 2.90818i
$868$	0	0
$869$	− 0.884457i	− 0.0300031i
$870$	0	0
$871$	82.4089	2.79232
$872$	0	0
$873$	−1.94319	−0.0657670
$874$	0	0
$875$	0.247973i	0.00838300i
$876$	0	0
$877$	− 33.7237i	− 1.13877i	−0.822071	−	0.569385i	$-0.807181\pi$
$877$	− 33.7237i	− 1.13877i	0.822071	−	0.569385i	$-0.192819\pi$
$878$	0	0
$879$	12.4617	0.420323
$880$	0	0
$881$	7.24781	0.244185	0.122092	−	0.992519i	$-0.461040\pi$
$881$	7.24781	0.244185	0.122092	+	0.992519i	$0.461040\pi$
$882$	0	0
$883$	24.1525i	0.812797i	0.913696	+	0.406398i	$0.133215\pi$
$883$	24.1525i	0.812797i	−0.913696	+	0.406398i	$0.866785\pi$
$884$	0	0
$885$	− 40.6960i	− 1.36798i
$886$	0	0
$887$	−20.3957	−0.684821	−0.342411	−	0.939550i	$-0.611243\pi$
$887$	−20.3957	−0.684821	−0.342411	+	0.939550i	$0.611243\pi$
$888$	0	0
$889$	10.1771	0.341328
$890$	0	0
$891$	− 4.41979i	− 0.148069i
$892$	0	0
$893$	2.02928i	0.0679072i
$894$	0	0
$895$	59.4015	1.98557
$896$	0	0
$897$	−91.4998	−3.05509
$898$	0	0
$899$	− 2.65340i	− 0.0884959i
$900$	0	0
$901$	− 42.7321i	− 1.42361i
$902$	0	0
$903$	−6.73416	−0.224099
$904$	0	0
$905$	33.0618	1.09901
$906$	0	0
$907$	− 6.56562i	− 0.218008i	−0.994041	−	0.109004i	$-0.965234\pi$
$907$	− 6.56562i	− 0.218008i	0.994041	−	0.109004i	$-0.0347661\pi$
$908$	0	0
$909$	− 30.1143i	− 0.998827i
$910$	0	0
$911$	−6.98339	−0.231370	−0.115685	−	0.993286i	$-0.536906\pi$
$911$	−6.98339	−0.231370	−0.115685	+	0.993286i	$0.536906\pi$
$912$	0	0
$913$	−5.02732	−0.166380
$914$	0	0
$915$	51.1488i	1.69093i
$916$	0	0
$917$	− 5.42780i	− 0.179242i
$918$	0	0
$919$	−12.4237	−0.409819	−0.204910	−	0.978781i	$-0.565690\pi$
$919$	−12.4237	−0.409819	−0.204910	+	0.978781i	$0.565690\pi$
$920$	0	0
$921$	24.4101	0.804339
$922$	0	0
$923$	81.0837i	2.66890i
$924$	0	0
$925$	− 16.9591i	− 0.557610i
$926$	0	0
$927$	−6.90873	−0.226912
$928$	0	0
$929$	−23.0649	−0.756735	−0.378367	−	0.925655i	$-0.623514\pi$
$929$	−23.0649	−0.756735	−0.378367	+	0.925655i	$0.623514\pi$
$930$	0	0
$931$	− 6.71307i	− 0.220012i
$932$	0	0
$933$	− 63.7551i	− 2.08725i
$934$	0	0
$935$	−9.78679	−0.320062
$936$	0	0
$937$	−20.2747	−0.662347	−0.331173	−	0.943570i	$-0.607445\pi$
$937$	−20.2747	−0.662347	−0.331173	+	0.943570i	$0.607445\pi$
$938$	0	0
$939$	39.4134i	1.28621i
$940$	0	0
$941$	33.8390i	1.10312i	0.834136	+	0.551559i	$0.185967\pi$
$941$	33.8390i	1.10312i	−0.834136	+	0.551559i	$0.814033\pi$
$942$	0	0
$943$	−0.726091	−0.0236448
$944$	0	0
$945$	2.25095	0.0732234
$946$	0	0
$947$	− 55.4208i	− 1.80093i	−0.434926	−	0.900466i	$-0.643225\pi$
$947$	− 55.4208i	− 1.80093i	0.434926	−	0.900466i	$-0.356775\pi$
$948$	0	0
$949$	46.5639i	1.51153i
$950$	0	0
$951$	−8.90033	−0.288613
$952$	0	0
$953$	−19.8530	−0.643102	−0.321551	−	0.946892i	$-0.604204\pi$
$953$	−19.8530	−0.643102	−0.321551	+	0.946892i	$0.604204\pi$
$954$	0	0
$955$	− 5.03729i	− 0.163003i
$956$	0	0
$957$	− 0.829039i	− 0.0267990i
$958$	0	0
$959$	−1.75814	−0.0567734
$960$	0	0
$961$	−20.9512	−0.675844
$962$	0	0
$963$	− 0.698809i	− 0.0225188i
$964$	0	0
$965$	− 41.7121i	− 1.34276i
$966$	0	0
$967$	5.95336	0.191447	0.0957236	−	0.995408i	$-0.469484\pi$
$967$	5.95336	0.191447	0.0957236	+	0.995408i	$0.469484\pi$
$968$	0	0
$969$	17.0787	0.548647
$970$	0	0
$971$	5.33267i	0.171134i	0.996332	+	0.0855668i	$0.0272701\pi$
$971$	5.33267i	0.171134i	−0.996332	+	0.0855668i	$0.972730\pi$
$972$	0	0
$973$	5.64290i	0.180903i
$974$	0	0
$975$	75.2507	2.40995
$976$	0	0
$977$	−12.6063	−0.403312	−0.201656	−	0.979456i	$-0.564632\pi$
$977$	−12.6063	−0.403312	−0.201656	+	0.979456i	$0.564632\pi$
$978$	0	0
$979$	5.85469i	0.187117i
$980$	0	0
$981$	− 4.65335i	− 0.148570i
$982$	0	0
$983$	19.8397	0.632788	0.316394	−	0.948628i	$-0.397528\pi$
$983$	19.8397	0.632788	0.316394	+	0.948628i	$0.397528\pi$
$984$	0	0
$985$	−12.7655	−0.406741
$986$	0	0
$987$	− 2.53185i	− 0.0805897i
$988$	0	0
$989$	− 31.8467i	− 1.01267i
$990$	0	0
$991$	53.9425	1.71354	0.856770	−	0.515699i	$-0.172468\pi$
$991$	53.9425	1.71354	0.856770	+	0.515699i	$0.172468\pi$
$992$	0	0
$993$	61.9760	1.96675
$994$	0	0
$995$	− 40.2634i	− 1.27644i
$996$	0	0
$997$	20.6213i	0.653084i	0.945183	+	0.326542i	$0.105884\pi$
$997$	20.6213i	0.653084i	−0.945183	+	0.326542i	$0.894116\pi$
$998$	0	0
$999$	4.67886	0.148033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.2.c.b.305.3		16
3.2	odd	2		5472.2.g.b.2737.3		16
4.3	odd	2		152.2.c.b.77.13	✓	16
8.3	odd	2		152.2.c.b.77.14	yes	16
8.5	even	2	inner	608.2.c.b.305.14		16
12.11	even	2		1368.2.g.b.685.4		16
16.3	odd	4		4864.2.a.bq.1.2		8
16.5	even	4		4864.2.a.bn.1.2		8
16.11	odd	4		4864.2.a.bo.1.7		8
16.13	even	4		4864.2.a.bp.1.7		8
24.5	odd	2		5472.2.g.b.2737.14		16
24.11	even	2		1368.2.g.b.685.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.13	✓	16	4.3	odd	2
152.2.c.b.77.14	yes	16	8.3	odd	2
608.2.c.b.305.3		16	1.1	even	1	trivial
608.2.c.b.305.14		16	8.5	even	2	inner
1368.2.g.b.685.3		16	24.11	even	2
1368.2.g.b.685.4		16	12.11	even	2
4864.2.a.bn.1.2		8	16.5	even	4
4864.2.a.bo.1.7		8	16.11	odd	4
4864.2.a.bp.1.7		8	16.13	even	4
4864.2.a.bq.1.2		8	16.3	odd	4
5472.2.g.b.2737.3		16	3.2	odd	2
5472.2.g.b.2737.14		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.32921i q^{3} +3.13887i q^{5} +0.535658 q^{7} -2.42523 q^{9} +O(q^{10})\)	\(q-2.32921i q^{3} +3.13887i q^{5} +0.535658 q^{7} -2.42523 q^{9} +0.425227i q^{11} -6.65786i q^{13} +7.31110 q^{15} +7.33239 q^{17} +1.00000i q^{19} -1.24766i q^{21} +5.90033 q^{23} -4.85252 q^{25} -1.33877i q^{27} -0.837037i q^{29} +3.16999 q^{31} +0.990444 q^{33} +1.68136i q^{35} +3.49490i q^{37} -15.5076 q^{39} -0.123059 q^{41} -5.39744i q^{43} -7.61248i q^{45} +2.02928 q^{47} -6.71307 q^{49} -17.0787i q^{51} -5.82785i q^{53} -1.33473 q^{55} +2.32921 q^{57} -5.56633i q^{59} +6.99606i q^{61} -1.29909 q^{63} +20.8982 q^{65} +12.3777i q^{67} -13.7431i q^{69} -12.1786 q^{71} -6.99382 q^{73} +11.3025i q^{75} +0.227776i q^{77} -2.07996 q^{79} -10.3940 q^{81} +11.8227i q^{83} +23.0154i q^{85} -1.94964 q^{87} +13.7684 q^{89} -3.56633i q^{91} -7.38358i q^{93} -3.13887 q^{95} +0.801240 q^{97} -1.03127i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 24 q^{9}+O(q^{10}) \) 16 * q + 8 * q^7 - 24 * q^9	\( 16 q + 8 q^{7} - 24 q^{9} - 8 q^{17} - 24 q^{25} - 16 q^{31} - 8 q^{39} + 16 q^{41} - 24 q^{47} + 24 q^{49} - 16 q^{55} + 32 q^{63} + 16 q^{65} - 48 q^{71} + 48 q^{79} - 16 q^{81} + 48 q^{87} - 16 q^{89} - 16 q^{95} + 32 q^{97}+O(q^{100}) \) 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

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