LMFDB - Embedded newform 603.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [603,2,Mod(1,603)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(603, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("603.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 603 = 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	603.a (trivial)

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:	yes
Analytic conductor:	$4.81497924188$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.2482793472.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 37x^{2} - 12 $ x^6 - 12x^4 + 37x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.18370$ of defining polynomial
Character	$\chi$	$=$	603.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18370 q^{2} +2.76854 q^{4} -2.68914 q^{5} +0.768540 q^{7} -1.67826 q^{8} +O(q^{10})$	\(q-2.18370 q^{2} +2.76854 q^{4} -2.68914 q^{5} +0.768540 q^{7} -1.67826 q^{8} +5.87227 q^{10} -0.505439 q^{11} -2.64081 q^{13} -1.67826 q^{14} -1.87227 q^{16} +2.12520 q^{17} +0.231460 q^{19} -7.44498 q^{20} +1.10373 q^{22} +0.0584944 q^{23} +2.23146 q^{25} +5.76672 q^{26} +2.12773 q^{28} +2.63064 q^{29} +1.87227 q^{31} +7.44498 q^{32} -4.64081 q^{34} -2.06671 q^{35} +6.10373 q^{37} -0.505439 q^{38} +4.51307 q^{40} -6.43411 q^{41} +5.40935 q^{43} -1.39933 q^{44} -0.127734 q^{46} +6.49260 q^{47} -6.40935 q^{49} -4.87284 q^{50} -7.31118 q^{52} +6.55110 q^{53} +1.35919 q^{55} -1.28981 q^{56} -5.74453 q^{58} +14.4430 q^{59} +10.1779 q^{61} -4.08846 q^{62} -12.5131 q^{64} +7.10149 q^{65} +1.00000 q^{67} +5.88371 q^{68} +4.51307 q^{70} +4.87284 q^{71} +11.0000 q^{73} -13.3287 q^{74} +0.640806 q^{76} -0.388450 q^{77} -1.53708 q^{79} +5.03478 q^{80} +14.0502 q^{82} +8.57285 q^{83} -5.71497 q^{85} -11.8124 q^{86} +0.848258 q^{88} +2.24219 q^{89} -2.02957 q^{91} +0.161944 q^{92} -14.1779 q^{94} -0.622427 q^{95} +4.17789 q^{97} +13.9961 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4}+O(q^{10}) $ 6 * q + 12 * q^4	$ 6 q + 12 q^{4} + 4 q^{10} + 20 q^{13} + 20 q^{16} + 6 q^{19} - 20 q^{22} + 18 q^{25} + 44 q^{28} - 20 q^{31} + 8 q^{34} + 10 q^{37} - 40 q^{40} - 8 q^{43} - 32 q^{46} + 2 q^{49} + 36 q^{52} + 44 q^{55} + 28 q^{58} + 16 q^{61} - 8 q^{64} + 6 q^{67} - 40 q^{70} + 66 q^{73} - 32 q^{76} + 8 q^{82} + 20 q^{85} - 84 q^{88} - 4 q^{91} - 40 q^{94} - 20 q^{97}+O(q^{100}) $ 6 * q + 12 * q^4 + 4 * q^10 + 20 * q^13 + 20 * q^16 + 6 * q^19 - 20 * q^22 + 18 * q^25 + 44 * q^28 - 20 * q^31 + 8 * q^34 + 10 * q^37 - 40 * q^40 - 8 * q^43 - 32 * q^46 + 2 * q^49 + 36 * q^52 + 44 * q^55 + 28 * q^58 + 16 * q^61 - 8 * q^64 + 6 * q^67 - 40 * q^70 + 66 * q^73 - 32 * q^76 + 8 * q^82 + 20 * q^85 - 84 * q^88 - 4 * q^91 - 40 * q^94 - 20 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.18370	−1.54411	−0.772054	−	0.635557i	$-0.780771\pi$
$2$	−2.18370	−1.54411	−0.772054	+	0.635557i	$0.780771\pi$
$3$	0	0
$3$	0	0
$4$	2.76854	1.38427
$5$	−2.68914	−1.20262	−0.601309	−	0.799016i	$-0.705354\pi$
$5$	−2.68914	−1.20262	−0.601309	+	0.799016i	$0.705354\pi$
$6$	0	0
$7$	0.768540	0.290481	0.145240	−	0.989396i	$-0.453604\pi$
$7$	0.768540	0.290481	0.145240	+	0.989396i	$0.453604\pi$
$8$	−1.67826	−0.593355
$9$	0	0
$10$	5.87227	1.85697
$11$	−0.505439	−0.152395	−0.0761977	−	0.997093i	$-0.524278\pi$
$11$	−0.505439	−0.152395	−0.0761977	+	0.997093i	$0.524278\pi$
$12$	0	0
$13$	−2.64081	−0.732428	−0.366214	−	0.930531i	$-0.619346\pi$
$13$	−2.64081	−0.732428	−0.366214	+	0.930531i	$0.619346\pi$
$14$	−1.67826	−0.448534
$15$	0	0
$16$	−1.87227	−0.468066
$17$	2.12520	0.515438	0.257719	−	0.966220i	$-0.417029\pi$
$17$	2.12520	0.515438	0.257719	+	0.966220i	$0.417029\pi$
$18$	0	0
$19$	0.231460	0.0531005	0.0265503	−	0.999647i	$-0.491548\pi$
$19$	0.231460	0.0531005	0.0265503	+	0.999647i	$0.491548\pi$
$20$	−7.44498	−1.66475
$21$	0	0
$22$	1.10373	0.235315
$23$	0.0584944	0.0121969	0.00609846	−	0.999981i	$-0.498059\pi$
$23$	0.0584944	0.0121969	0.00609846	+	0.999981i	$0.498059\pi$
$24$	0	0
$25$	2.23146	0.446292
$26$	5.76672	1.13095
$27$	0	0
$28$	2.12773	0.402104
$29$	2.63064	0.488498	0.244249	−	0.969713i	$-0.421459\pi$
$29$	2.63064	0.488498	0.244249	+	0.969713i	$0.421459\pi$
$30$	0	0
$31$	1.87227	0.336269	0.168134	−	0.985764i	$-0.446226\pi$
$31$	1.87227	0.336269	0.168134	+	0.985764i	$0.446226\pi$
$32$	7.44498	1.31610
$33$	0	0
$34$	−4.64081	−0.795892
$35$	−2.06671	−0.349338
$36$	0	0
$37$	6.10373	1.00345	0.501723	−	0.865028i	$-0.332700\pi$
$37$	6.10373	1.00345	0.501723	+	0.865028i	$0.332700\pi$
$38$	−0.505439	−0.0819930
$39$	0	0
$40$	4.51307	0.713579
$41$	−6.43411	−1.00484	−0.502419	−	0.864624i	$-0.667557\pi$
$41$	−6.43411	−1.00484	−0.502419	+	0.864624i	$0.667557\pi$
$42$	0	0
$43$	5.40935	0.824918	0.412459	−	0.910976i	$-0.364670\pi$
$43$	5.40935	0.824918	0.412459	+	0.910976i	$0.364670\pi$
$44$	−1.39933	−0.210957
$45$	0	0
$46$	−0.127734	−0.0188334
$47$	6.49260	0.947043	0.473522	−	0.880782i	$-0.342983\pi$
$47$	6.49260	0.947043	0.473522	+	0.880782i	$0.342983\pi$
$48$	0	0
$49$	−6.40935	−0.915621
$50$	−4.87284	−0.689123
$51$	0	0
$52$	−7.31118	−1.01388
$53$	6.55110	0.899862	0.449931	−	0.893063i	$-0.351449\pi$
$53$	6.55110	0.899862	0.449931	+	0.893063i	$0.351449\pi$
$54$	0	0
$55$	1.35919	0.183274
$56$	−1.28981	−0.172358
$57$	0	0
$58$	−5.74453	−0.754294
$59$	14.4430	1.88032	0.940161	−	0.340731i	$-0.110675\pi$
$59$	14.4430	1.88032	0.940161	+	0.340731i	$0.110675\pi$
$60$	0	0
$61$	10.1779	1.30314	0.651572	−	0.758586i	$-0.274110\pi$
$61$	10.1779	1.30314	0.651572	+	0.758586i	$0.274110\pi$
$62$	−4.08846	−0.519236
$63$	0	0
$64$	−12.5131	−1.56413
$65$	7.10149	0.880831
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	5.88371	0.713505
$69$	0	0
$70$	4.51307	0.539415
$71$	4.87284	0.578299	0.289150	−	0.957284i	$-0.406627\pi$
$71$	4.87284	0.578299	0.289150	+	0.957284i	$0.406627\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−13.3287	−1.54943
$75$	0	0
$76$	0.640806	0.0735055
$77$	−0.388450	−0.0442680
$78$	0	0
$79$	−1.53708	−0.172935	−0.0864675	−	0.996255i	$-0.527558\pi$
$79$	−1.53708	−0.172935	−0.0864675	+	0.996255i	$0.527558\pi$
$80$	5.03478	0.562906
$81$	0	0
$82$	14.0502	1.55158
$83$	8.57285	0.940993	0.470496	−	0.882402i	$-0.344075\pi$
$83$	8.57285	0.940993	0.470496	+	0.882402i	$0.344075\pi$
$84$	0	0
$85$	−5.71497	−0.619875
$86$	−11.8124	−1.27376
$87$	0	0
$88$	0.848258	0.0904246
$89$	2.24219	0.237672	0.118836	−	0.992914i	$-0.462084\pi$
$89$	2.24219	0.237672	0.118836	+	0.992914i	$0.462084\pi$
$90$	0	0
$91$	−2.02957	−0.212756
$92$	0.161944	0.0168838
$93$	0	0
$94$	−14.1779	−1.46234
$95$	−0.622427	−0.0638597
$96$	0	0
$97$	4.17789	0.424200	0.212100	−	0.977248i	$-0.431970\pi$
$97$	4.17789	0.424200	0.212100	+	0.977248i	$0.431970\pi$
$98$	13.9961	1.41382
$99$	0	0
$100$	6.17789	0.617789
$101$	−3.74497	−0.372638	−0.186319	−	0.982489i	$-0.559656\pi$
$101$	−3.74497	−0.372638	−0.186319	+	0.982489i	$0.559656\pi$
$102$	0	0
$103$	9.10373	0.897017	0.448508	−	0.893779i	$-0.351955\pi$
$103$	9.10373	0.897017	0.448508	+	0.893779i	$0.351955\pi$
$104$	4.43196	0.434589
$105$	0	0
$106$	−14.3056	−1.38948
$107$	10.8600	1.04988	0.524938	−	0.851141i	$-0.324089\pi$
$107$	10.8600	1.04988	0.524938	+	0.851141i	$0.324089\pi$
$108$	0	0
$109$	6.64081	0.636074	0.318037	−	0.948078i	$-0.396976\pi$
$109$	6.64081	0.636074	0.318037	+	0.948078i	$0.396976\pi$
$110$	−2.96807	−0.282994
$111$	0	0
$112$	−1.43891	−0.135964
$113$	−13.6076	−1.28010	−0.640049	−	0.768334i	$-0.721086\pi$
$113$	−13.6076	−1.28010	−0.640049	+	0.768334i	$0.721086\pi$
$114$	0	0
$115$	−0.157299	−0.0146682
$116$	7.28304	0.676213
$117$	0	0
$118$	−31.5392	−2.90342
$119$	1.63330	0.149725
$120$	0	0
$121$	−10.7445	−0.976776
$122$	−22.2254	−2.01220
$123$	0	0
$124$	5.18344	0.465487
$125$	7.44498	0.665900
$126$	0	0
$127$	−18.5872	−1.64935	−0.824675	−	0.565607i	$-0.808642\pi$
$127$	−18.5872	−1.64935	−0.824675	+	0.565607i	$0.808642\pi$
$128$	12.4348	1.09909
$129$	0	0
$130$	−15.5075	−1.36010
$131$	−5.92867	−0.517990	−0.258995	−	0.965879i	$-0.583391\pi$
$131$	−5.92867	−0.517990	−0.258995	+	0.965879i	$0.583391\pi$
$132$	0	0
$133$	0.177886	0.0154247
$134$	−2.18370	−0.188643
$135$	0	0
$136$	−3.56665	−0.305837
$137$	−16.0628	−1.37234	−0.686168	−	0.727443i	$-0.740709\pi$
$137$	−16.0628	−1.37234	−0.686168	+	0.727443i	$0.740709\pi$
$138$	0	0
$139$	−16.9760	−1.43988	−0.719942	−	0.694034i	$-0.755832\pi$
$139$	−16.9760	−1.43988	−0.719942	+	0.694034i	$0.755832\pi$
$140$	−5.72177	−0.483578
$141$	0	0
$142$	−10.6408	−0.892957
$143$	1.33477	0.111619
$144$	0	0
$145$	−7.07416	−0.587477
$146$	−24.0207	−1.98797
$147$	0	0
$148$	16.8984	1.38904
$149$	−6.10415	−0.500072	−0.250036	−	0.968237i	$-0.580442\pi$
$149$	−6.10415	−0.500072	−0.250036	+	0.968237i	$0.580442\pi$
$150$	0	0
$151$	−7.97599	−0.649077	−0.324538	−	0.945872i	$-0.605209\pi$
$151$	−7.97599	−0.649077	−0.324538	+	0.945872i	$0.605209\pi$
$152$	−0.388450	−0.0315074
$153$	0	0
$154$	0.848258	0.0683545
$155$	−5.03478	−0.404403
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	3.35652	0.267030
$159$	0	0
$160$	−20.0206	−1.58277
$161$	0.0449553	0.00354297
$162$	0	0
$163$	14.1539	1.10862	0.554309	−	0.832311i	$-0.312983\pi$
$163$	14.1539	1.10862	0.554309	+	0.832311i	$0.312983\pi$
$164$	−17.8131	−1.39097
$165$	0	0
$166$	−18.7205	−1.45299
$167$	−11.5409	−0.893063	−0.446532	−	0.894768i	$-0.647341\pi$
$167$	−11.5409	−0.893063	−0.446532	+	0.894768i	$0.647341\pi$
$168$	0	0
$169$	−6.02614	−0.463550
$170$	12.4798	0.957154
$171$	0	0
$172$	14.9760	1.14191
$173$	9.84912	0.748815	0.374407	−	0.927264i	$-0.377846\pi$
$173$	9.84912	0.748815	0.374407	+	0.927264i	$0.377846\pi$
$174$	0	0
$175$	1.71497	0.129639
$176$	0.946316	0.0713312
$177$	0	0
$178$	−4.89627	−0.366991
$179$	−14.6185	−1.09264	−0.546319	−	0.837577i	$-0.683971\pi$
$179$	−14.6185	−1.09264	−0.546319	+	0.837577i	$0.683971\pi$
$180$	0	0
$181$	18.9224	1.40649	0.703246	−	0.710946i	$-0.251733\pi$
$181$	18.9224	1.40649	0.703246	+	0.710946i	$0.251733\pi$
$182$	4.43196	0.328519
$183$	0	0
$184$	−0.0981688	−0.00723710
$185$	−16.4138	−1.20676
$186$	0	0
$187$	−1.07416	−0.0785504
$188$	17.9750	1.31096
$189$	0	0
$190$	1.35919	0.0986063
$191$	2.02175	0.146289	0.0731445	−	0.997321i	$-0.476697\pi$
$191$	2.02175	0.146289	0.0731445	+	0.997321i	$0.476697\pi$
$192$	0	0
$193$	23.8187	1.71451	0.857253	−	0.514895i	$-0.172169\pi$
$193$	23.8187	1.71451	0.857253	+	0.514895i	$0.172169\pi$
$194$	−9.12324	−0.655011
$195$	0	0
$196$	−17.7445	−1.26747
$197$	−0.161944	−0.0115380	−0.00576902	−	0.999983i	$-0.501836\pi$
$197$	−0.161944	−0.0115380	−0.00576902	+	0.999983i	$0.501836\pi$
$198$	0	0
$199$	−3.59065	−0.254535	−0.127267	−	0.991868i	$-0.540621\pi$
$199$	−3.59065	−0.254535	−0.127267	+	0.991868i	$0.540621\pi$
$200$	−3.74497	−0.264809
$201$	0	0
$202$	8.17789	0.575394
$203$	2.02175	0.141899
$204$	0	0
$205$	17.3022	1.20844
$206$	−19.8798	−1.38509
$207$	0	0
$208$	4.94429	0.342825
$209$	−0.116989	−0.00809228
$210$	0	0
$211$	0.925840	0.0637374	0.0318687	−	0.999492i	$-0.489854\pi$
$211$	0.925840	0.0637374	0.0318687	+	0.999492i	$0.489854\pi$
$212$	18.1370	1.24565
$213$	0	0
$214$	−23.7150	−1.62112
$215$	−14.5465	−0.992061
$216$	0	0
$217$	1.43891	0.0976797
$218$	−14.5015	−0.982167
$219$	0	0
$220$	3.76298	0.253700
$221$	−5.61225	−0.377521
$222$	0	0
$223$	−11.4835	−0.768992	−0.384496	−	0.923127i	$-0.625625\pi$
$223$	−11.4835	−0.768992	−0.384496	+	0.923127i	$0.625625\pi$
$224$	5.72177	0.382302
$225$	0	0
$226$	29.7150	1.97661
$227$	−8.94776	−0.593884	−0.296942	−	0.954896i	$-0.595967\pi$
$227$	−8.94776	−0.593884	−0.296942	+	0.954896i	$0.595967\pi$
$228$	0	0
$229$	25.7445	1.70125	0.850623	−	0.525776i	$-0.176225\pi$
$229$	25.7445	1.70125	0.850623	+	0.525776i	$0.176225\pi$
$230$	0.343495	0.0226494
$231$	0	0
$232$	−4.41490	−0.289853
$233$	1.91224	0.125275	0.0626374	−	0.998036i	$-0.480049\pi$
$233$	1.91224	0.125275	0.0626374	+	0.998036i	$0.480049\pi$
$234$	0	0
$235$	−17.4595	−1.13893
$236$	39.9861	2.60287
$237$	0	0
$238$	−3.56665	−0.231191
$239$	3.35652	0.217115	0.108558	−	0.994090i	$-0.465377\pi$
$239$	3.35652	0.217115	0.108558	+	0.994090i	$0.465377\pi$
$240$	0	0
$241$	−1.02957	−0.0663201	−0.0331601	−	0.999450i	$-0.510557\pi$
$241$	−1.02957	−0.0663201	−0.0331601	+	0.999450i	$0.510557\pi$
$242$	23.4628	1.50825
$243$	0	0
$244$	28.1779	1.80390
$245$	17.2356	1.10114
$246$	0	0
$247$	−0.611241	−0.0388923
$248$	−3.14215	−0.199527
$249$	0	0
$250$	−16.2576	−1.02822
$251$	−13.2192	−0.834387	−0.417194	−	0.908818i	$-0.636986\pi$
$251$	−13.2192	−0.834387	−0.417194	+	0.908818i	$0.636986\pi$
$252$	0	0
$253$	−0.0295653	−0.00185876
$254$	40.5889	2.54678
$255$	0	0
$256$	−2.12773	−0.132983
$257$	−25.3615	−1.58201	−0.791004	−	0.611811i	$-0.790441\pi$
$257$	−25.3615	−1.58201	−0.791004	+	0.611811i	$0.790441\pi$
$258$	0	0
$259$	4.69096	0.291482
$260$	19.6608	1.21931
$261$	0	0
$262$	12.9464	0.799833
$263$	23.7867	1.46675	0.733376	−	0.679824i	$-0.237944\pi$
$263$	23.7867	1.46675	0.733376	+	0.679824i	$0.237944\pi$
$264$	0	0
$265$	−17.6168	−1.08219
$266$	−0.388450	−0.0238174
$267$	0	0
$268$	2.76854	0.169116
$269$	11.3790	0.693788	0.346894	−	0.937904i	$-0.387236\pi$
$269$	11.3790	0.693788	0.346894	+	0.937904i	$0.387236\pi$
$270$	0	0
$271$	23.2816	1.41426	0.707129	−	0.707085i	$-0.249990\pi$
$271$	23.2816	1.41426	0.707129	+	0.707085i	$0.249990\pi$
$272$	−3.97895	−0.241259
$273$	0	0
$274$	35.0763	2.11904
$275$	−1.12787	−0.0680129
$276$	0	0
$277$	1.05357	0.0633031	0.0316516	−	0.999499i	$-0.489923\pi$
$277$	1.05357	0.0633031	0.0316516	+	0.999499i	$0.489923\pi$
$278$	37.0705	2.22334
$279$	0	0
$280$	3.46848	0.207281
$281$	−2.52719	−0.150760	−0.0753799	−	0.997155i	$-0.524017\pi$
$281$	−2.52719	−0.150760	−0.0753799	+	0.997155i	$0.524017\pi$
$282$	0	0
$283$	23.6614	1.40652	0.703262	−	0.710931i	$-0.251726\pi$
$283$	23.6614	1.40652	0.703262	+	0.710931i	$0.251726\pi$
$284$	13.4906	0.800522
$285$	0	0
$286$	−2.91473	−0.172351
$287$	−4.94487	−0.291886
$288$	0	0
$289$	−12.4835	−0.734324
$290$	15.4478	0.907128
$291$	0	0
$292$	30.4539	1.78218
$293$	23.8587	1.39384	0.696921	−	0.717148i	$-0.254553\pi$
$293$	23.8587	1.39384	0.696921	+	0.717148i	$0.254553\pi$
$294$	0	0
$295$	−38.8393	−2.26131
$296$	−10.2436	−0.595399
$297$	0	0
$298$	13.3296	0.772165
$299$	−0.154472	−0.00893337
$300$	0	0
$301$	4.15730	0.239623
$302$	17.4172	1.00225
$303$	0	0
$304$	−0.433354	−0.0248546
$305$	−27.3697	−1.56719
$306$	0	0
$307$	−1.94643	−0.111088	−0.0555442	−	0.998456i	$-0.517689\pi$
$307$	−1.94643	−0.111088	−0.0555442	+	0.998456i	$0.517689\pi$
$308$	−1.07544	−0.0612788
$309$	0	0
$310$	10.9944	0.624442
$311$	−26.4758	−1.50131	−0.750654	−	0.660696i	$-0.770261\pi$
$311$	−26.4758	−1.50131	−0.750654	+	0.660696i	$0.770261\pi$
$312$	0	0
$313$	0.896274	0.0506604	0.0253302	−	0.999679i	$-0.491936\pi$
$313$	0.896274	0.0506604	0.0253302	+	0.999679i	$0.491936\pi$
$314$	−37.1229	−2.09497
$315$	0	0
$316$	−4.25547	−0.239389
$317$	8.02246	0.450586	0.225293	−	0.974291i	$-0.427666\pi$
$317$	8.02246	0.450586	0.225293	+	0.974291i	$0.427666\pi$
$318$	0	0
$319$	−1.32963	−0.0744449
$320$	33.6494	1.88106
$321$	0	0
$322$	−0.0981688	−0.00547073
$323$	0.491900	0.0273700
$324$	0	0
$325$	−5.89285	−0.326877
$326$	−30.9078	−1.71182
$327$	0	0
$328$	10.7981	0.596226
$329$	4.98982	0.275098
$330$	0	0
$331$	−30.8984	−1.69833	−0.849165	−	0.528127i	$-0.822894\pi$
$331$	−30.8984	−1.69833	−0.849165	+	0.528127i	$0.822894\pi$
$332$	23.7343	1.30259
$333$	0	0
$334$	25.2019	1.37899
$335$	−2.68914	−0.146923
$336$	0	0
$337$	17.3112	0.943000	0.471500	−	0.881866i	$-0.343713\pi$
$337$	17.3112	0.943000	0.471500	+	0.881866i	$0.343713\pi$
$338$	13.1593	0.715771
$339$	0	0
$340$	−15.8221	−0.858075
$341$	−0.946316	−0.0512459
$342$	0	0
$343$	−10.3056	−0.556451
$344$	−9.07829	−0.489469
$345$	0	0
$346$	−21.5075	−1.15625
$347$	−14.2301	−0.763910	−0.381955	−	0.924181i	$-0.624749\pi$
$347$	−14.2301	−0.763910	−0.381955	+	0.924181i	$0.624749\pi$
$348$	0	0
$349$	−18.8688	−1.01003	−0.505013	−	0.863112i	$-0.668512\pi$
$349$	−18.8688	−1.01003	−0.505013	+	0.863112i	$0.668512\pi$
$350$	−3.74497	−0.200177
$351$	0	0
$352$	−3.76298	−0.200568
$353$	21.9465	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$353$	21.9465	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$354$	0	0
$355$	−13.1037	−0.695474
$356$	6.20760	0.329002
$357$	0	0
$358$	31.9224	1.68715
$359$	18.3050	0.966100	0.483050	−	0.875593i	$-0.339529\pi$
$359$	18.3050	0.966100	0.483050	+	0.875593i	$0.339529\pi$
$360$	0	0
$361$	−18.9464	−0.997180
$362$	−41.3209	−2.17178
$363$	0	0
$364$	−5.61893	−0.294512
$365$	−29.5805	−1.54831
$366$	0	0
$367$	17.0261	0.888757	0.444379	−	0.895839i	$-0.353424\pi$
$367$	17.0261	0.888757	0.444379	+	0.895839i	$0.353424\pi$
$368$	−0.109517	−0.00570897
$369$	0	0
$370$	35.8427	1.86337
$371$	5.03478	0.261393
$372$	0	0
$373$	−9.77410	−0.506084	−0.253042	−	0.967455i	$-0.581431\pi$
$373$	−9.77410	−0.506084	−0.253042	+	0.967455i	$0.581431\pi$
$374$	2.34564	0.121290
$375$	0	0
$376$	−10.8963	−0.561932
$377$	−6.94702	−0.357790
$378$	0	0
$379$	−26.5131	−1.36189	−0.680943	−	0.732337i	$-0.738430\pi$
$379$	−26.5131	−1.36189	−0.680943	+	0.732337i	$0.738430\pi$
$380$	−1.72322	−0.0883991
$381$	0	0
$382$	−4.41490	−0.225886
$383$	−12.8682	−0.657535	−0.328768	−	0.944411i	$-0.606633\pi$
$383$	−12.8682	−0.657535	−0.328768	+	0.944411i	$0.606633\pi$
$384$	0	0
$385$	1.04460	0.0532375
$386$	−52.0128	−2.64738
$387$	0	0
$388$	11.5666	0.587207
$389$	4.98982	0.252994	0.126497	−	0.991967i	$-0.459627\pi$
$389$	4.98982	0.252994	0.126497	+	0.991967i	$0.459627\pi$
$390$	0	0
$391$	0.124313	0.00628675
$392$	10.7565	0.543288
$393$	0	0
$394$	0.353637	0.0178160
$395$	4.13342	0.207975
$396$	0	0
$397$	7.59065	0.380964	0.190482	−	0.981691i	$-0.438995\pi$
$397$	7.59065	0.380964	0.190482	+	0.981691i	$0.438995\pi$
$398$	7.84091	0.393029
$399$	0	0
$400$	−4.17789	−0.208894
$401$	21.4411	1.07072	0.535358	−	0.844625i	$-0.320177\pi$
$401$	21.4411	1.07072	0.535358	+	0.844625i	$0.320177\pi$
$402$	0	0
$403$	−4.94429	−0.246293
$404$	−10.3681	−0.515832
$405$	0	0
$406$	−4.41490	−0.219108
$407$	−3.08506	−0.152921
$408$	0	0
$409$	33.6374	1.66326	0.831631	−	0.555329i	$-0.187408\pi$
$409$	33.6374	1.66326	0.831631	+	0.555329i	$0.187408\pi$
$410$	−37.7828	−1.86596
$411$	0	0
$412$	25.2040	1.24171
$413$	11.1000	0.546197
$414$	0	0
$415$	−23.0536	−1.13166
$416$	−19.6608	−0.963948
$417$	0	0
$418$	0.255468	0.0124954
$419$	−37.7963	−1.84647	−0.923236	−	0.384234i	$-0.874466\pi$
$419$	−37.7963	−1.84647	−0.923236	+	0.384234i	$0.874466\pi$
$420$	0	0
$421$	7.46292	0.363720	0.181860	−	0.983324i	$-0.441788\pi$
$421$	7.46292	0.363720	0.181860	+	0.983324i	$0.441788\pi$
$422$	−2.02175	−0.0984175
$423$	0	0
$424$	−10.9944	−0.533937
$425$	4.74231	0.230036
$426$	0	0
$427$	7.82211	0.378539
$428$	30.0663	1.45331
$429$	0	0
$430$	31.7651	1.53185
$431$	−31.9126	−1.53718	−0.768588	−	0.639744i	$-0.779040\pi$
$431$	−31.9126	−1.53718	−0.768588	+	0.639744i	$0.779040\pi$
$432$	0	0
$433$	26.5632	1.27655	0.638274	−	0.769810i	$-0.279649\pi$
$433$	26.5632	1.27655	0.638274	+	0.769810i	$0.279649\pi$
$434$	−3.14215	−0.150828
$435$	0	0
$436$	18.3853	0.880498
$437$	0.0135391	0.000647663 0
$438$	0	0
$439$	5.69096	0.271615	0.135807	−	0.990735i	$-0.456637\pi$
$439$	5.69096	0.271615	0.135807	+	0.990735i	$0.456637\pi$
$440$	−2.28108	−0.108746
$441$	0	0
$442$	12.2555	0.582933
$443$	37.3494	1.77452	0.887261	−	0.461267i	$-0.152605\pi$
$443$	37.3494	1.77452	0.887261	+	0.461267i	$0.152605\pi$
$444$	0	0
$445$	−6.02957	−0.285829
$446$	25.0765	1.18741
$447$	0	0
$448$	−9.61680	−0.454351
$449$	22.8343	1.07762	0.538809	−	0.842428i	$-0.318874\pi$
$449$	22.8343	1.07762	0.538809	+	0.842428i	$0.318874\pi$
$450$	0	0
$451$	3.25205	0.153133
$452$	−37.6733	−1.77200
$453$	0	0
$454$	19.5392	0.917021
$455$	5.45778	0.255865
$456$	0	0
$457$	15.4835	0.724288	0.362144	−	0.932122i	$-0.382045\pi$
$457$	15.4835	0.724288	0.362144	+	0.932122i	$0.382045\pi$
$458$	−56.2183	−2.62691
$459$	0	0
$460$	−0.435490	−0.0203048
$461$	−5.75319	−0.267953	−0.133976	−	0.990985i	$-0.542775\pi$
$461$	−5.75319	−0.267953	−0.133976	+	0.990985i	$0.542775\pi$
$462$	0	0
$463$	19.3318	0.898423	0.449212	−	0.893425i	$-0.351705\pi$
$463$	19.3318	0.898423	0.449212	+	0.893425i	$0.351705\pi$
$464$	−4.92526	−0.228650
$465$	0	0
$466$	−4.17575	−0.193438
$467$	−6.66061	−0.308216	−0.154108	−	0.988054i	$-0.549250\pi$
$467$	−6.66061	−0.308216	−0.154108	+	0.988054i	$0.549250\pi$
$468$	0	0
$469$	0.768540	0.0354879
$470$	38.1263	1.75863
$471$	0	0
$472$	−24.2392	−1.11570
$473$	−2.73409	−0.125714
$474$	0	0
$475$	0.516493	0.0236983
$476$	4.52187	0.207260
$477$	0	0
$478$	−7.32963	−0.335249
$479$	2.67560	0.122251	0.0611256	−	0.998130i	$-0.480531\pi$
$479$	2.67560	0.122251	0.0611256	+	0.998130i	$0.480531\pi$
$480$	0	0
$481$	−16.1188	−0.734952
$482$	2.24826	0.102405
$483$	0	0
$484$	−29.7467	−1.35212
$485$	−11.2349	−0.510151
$486$	0	0
$487$	−37.7171	−1.70913	−0.854563	−	0.519349i	$-0.826175\pi$
$487$	−37.7171	−1.70913	−0.854563	+	0.519349i	$0.826175\pi$
$488$	−17.0811	−0.773227
$489$	0	0
$490$	−37.6374	−1.70028
$491$	36.1630	1.63201	0.816007	−	0.578042i	$-0.196183\pi$
$491$	36.1630	1.63201	0.816007	+	0.578042i	$0.196183\pi$
$492$	0	0
$493$	5.59065	0.251790
$494$	1.33477	0.0600539
$495$	0	0
$496$	−3.50538	−0.157396
$497$	3.74497	0.167985
$498$	0	0
$499$	4.14832	0.185704	0.0928522	−	0.995680i	$-0.470402\pi$
$499$	4.14832	0.185704	0.0928522	+	0.995680i	$0.470402\pi$
$500$	20.6117	0.921785
$501$	0	0
$502$	28.8667	1.28838
$503$	−8.02246	−0.357704	−0.178852	−	0.983876i	$-0.557238\pi$
$503$	−8.02246	−0.357704	−0.178852	+	0.983876i	$0.557238\pi$
$504$	0	0
$505$	10.0707	0.448142
$506$	0.0645618	0.00287012
$507$	0	0
$508$	−51.4595	−2.28315
$509$	31.4792	1.39529	0.697646	−	0.716443i	$-0.254231\pi$
$509$	31.4792	1.39529	0.697646	+	0.716443i	$0.254231\pi$
$510$	0	0
$511$	8.45394	0.373980
$512$	−20.2233	−0.893752
$513$	0	0
$514$	55.3819	2.44279
$515$	−24.4812	−1.07877
$516$	0	0
$517$	−3.28161	−0.144325
$518$	−10.2436	−0.450080
$519$	0	0
$520$	−11.9181	−0.522645
$521$	26.4758	1.15993	0.579964	−	0.814642i	$-0.303067\pi$
$521$	26.4758	1.15993	0.579964	+	0.814642i	$0.303067\pi$
$522$	0	0
$523$	−22.1243	−0.967429	−0.483714	−	0.875226i	$-0.660713\pi$
$523$	−22.1243	−0.967429	−0.483714	+	0.875226i	$0.660713\pi$
$524$	−16.4138	−0.717038
$525$	0	0
$526$	−51.9430	−2.26482
$527$	3.97895	0.173326
$528$	0	0
$529$	−22.9966	−0.999851
$530$	38.4698	1.67102
$531$	0	0
$532$	0.492485	0.0213519
$533$	16.9912	0.735972
$534$	0	0
$535$	−29.2040	−1.26260
$536$	−1.67826	−0.0724898
$537$	0	0
$538$	−24.8483	−1.07128
$539$	3.23953	0.139536
$540$	0	0
$541$	−11.1333	−0.478658	−0.239329	−	0.970939i	$-0.576927\pi$
$541$	−11.1333	−0.478658	−0.239329	+	0.970939i	$0.576927\pi$
$542$	−50.8400	−2.18377
$543$	0	0
$544$	15.8221	0.678368
$545$	−17.8580	−0.764954
$546$	0	0
$547$	−24.3056	−1.03923	−0.519617	−	0.854400i	$-0.673925\pi$
$547$	−24.3056	−1.03923	−0.519617	+	0.854400i	$0.673925\pi$
$548$	−44.4705	−1.89968
$549$	0	0
$550$	2.46292	0.105019
$551$	0.608888	0.0259395
$552$	0	0
$553$	−1.18131	−0.0502343
$554$	−2.30069	−0.0977468
$555$	0	0
$556$	−46.9987	−1.99319
$557$	−16.8472	−0.713837	−0.356919	−	0.934135i	$-0.616173\pi$
$557$	−16.8472	−0.713837	−0.356919	+	0.934135i	$0.616173\pi$
$558$	0	0
$559$	−14.2850	−0.604193
$560$	3.86943	0.163513
$561$	0	0
$562$	5.51863	0.232789
$563$	13.1668	0.554913	0.277456	−	0.960738i	$-0.410509\pi$
$563$	13.1668	0.554913	0.277456	+	0.960738i	$0.410509\pi$
$564$	0	0
$565$	36.5928	1.53947
$566$	−51.6694	−2.17182
$567$	0	0
$568$	−8.17789	−0.343136
$569$	−12.8818	−0.540031	−0.270016	−	0.962856i	$-0.587029\pi$
$569$	−12.8818	−0.540031	−0.270016	+	0.962856i	$0.587029\pi$
$570$	0	0
$571$	37.3579	1.56338	0.781690	−	0.623667i	$-0.214358\pi$
$571$	37.3579	1.56338	0.781690	+	0.623667i	$0.214358\pi$
$572$	3.69535	0.154510
$573$	0	0
$574$	10.7981	0.450704
$575$	0.130528	0.00544339
$576$	0	0
$577$	−34.5632	−1.43889	−0.719443	−	0.694552i	$-0.755603\pi$
$577$	−34.5632	−1.43889	−0.719443	+	0.694552i	$0.755603\pi$
$578$	27.2602	1.13388
$579$	0	0
$580$	−19.5851	−0.813227
$581$	6.58858	0.273340
$582$	0	0
$583$	−3.31118	−0.137135
$584$	−18.4609	−0.763916
$585$	0	0
$586$	−52.1003	−2.15224
$587$	−25.9704	−1.07191	−0.535957	−	0.844245i	$-0.680049\pi$
$587$	−25.9704	−1.07191	−0.535957	+	0.844245i	$0.680049\pi$
$588$	0	0
$589$	0.433354	0.0178561
$590$	84.8133	3.49171
$591$	0	0
$592$	−11.4278	−0.469680
$593$	13.2641	0.544693	0.272346	−	0.962199i	$-0.412200\pi$
$593$	13.2641	0.544693	0.272346	+	0.962199i	$0.412200\pi$
$594$	0	0
$595$	−4.39218	−0.180062
$596$	−16.8996	−0.692234
$597$	0	0
$598$	0.337321	0.0137941
$599$	32.8650	1.34283	0.671414	−	0.741083i	$-0.265687\pi$
$599$	32.8650	1.34283	0.671414	+	0.741083i	$0.265687\pi$
$600$	0	0
$601$	6.00556	0.244972	0.122486	−	0.992470i	$-0.460913\pi$
$601$	6.00556	0.244972	0.122486	+	0.992470i	$0.460913\pi$
$602$	−9.07829	−0.370003
$603$	0	0
$604$	−22.0819	−0.898498
$605$	28.8935	1.17469
$606$	0	0
$607$	4.43335	0.179944	0.0899722	−	0.995944i	$-0.471322\pi$
$607$	4.43335	0.179944	0.0899722	+	0.995944i	$0.471322\pi$
$608$	1.72322	0.0698856
$609$	0	0
$610$	59.7673	2.41991
$611$	−17.1457	−0.693641
$612$	0	0
$613$	−38.1264	−1.53991	−0.769956	−	0.638097i	$-0.779722\pi$
$613$	−38.1264	−1.53991	−0.769956	+	0.638097i	$0.779722\pi$
$614$	4.25041	0.171533
$615$	0	0
$616$	0.651920	0.0262666
$617$	35.2617	1.41958	0.709791	−	0.704413i	$-0.248790\pi$
$617$	35.2617	1.41958	0.709791	+	0.704413i	$0.248790\pi$
$618$	0	0
$619$	−0.433354	−0.0174180	−0.00870899	−	0.999962i	$-0.502772\pi$
$619$	−0.433354	−0.0174180	−0.00870899	+	0.999962i	$0.502772\pi$
$620$	−13.9390	−0.559803
$621$	0	0
$622$	57.8153	2.31818
$623$	1.72322	0.0690392
$624$	0	0
$625$	−31.1779	−1.24712
$626$	−1.95719	−0.0782252
$627$	0	0
$628$	47.0652	1.87811
$629$	12.9717	0.517214
$630$	0	0
$631$	−16.8187	−0.669542	−0.334771	−	0.942300i	$-0.608659\pi$
$631$	−16.8187	−0.669542	−0.334771	+	0.942300i	$0.608659\pi$
$632$	2.57962	0.102612
$633$	0	0
$634$	−17.5186	−0.695754
$635$	49.9836	1.98354
$636$	0	0
$637$	16.9258	0.670626
$638$	2.90351	0.114951
$639$	0	0
$640$	−33.4389	−1.32179
$641$	−21.9465	−0.866835	−0.433417	−	0.901193i	$-0.642692\pi$
$641$	−21.9465	−0.866835	−0.433417	+	0.901193i	$0.642692\pi$
$642$	0	0
$643$	−40.2782	−1.58842	−0.794208	−	0.607645i	$-0.792114\pi$
$643$	−40.2782	−1.58842	−0.794208	+	0.607645i	$0.792114\pi$
$644$	0.124461	0.00490443
$645$	0	0
$646$	−1.07416	−0.0422623
$647$	14.2301	0.559441	0.279721	−	0.960081i	$-0.409758\pi$
$647$	14.2301	0.559441	0.279721	+	0.960081i	$0.409758\pi$
$648$	0	0
$649$	−7.30006	−0.286553
$650$	12.8682	0.504733
$651$	0	0
$652$	39.1856	1.53463
$653$	−32.5861	−1.27519	−0.637596	−	0.770371i	$-0.720071\pi$
$653$	−32.5861	−1.27519	−0.637596	+	0.770371i	$0.720071\pi$
$654$	0	0
$655$	15.9430	0.622945
$656$	12.0464	0.470331
$657$	0	0
$658$	−10.8963	−0.424781
$659$	29.3854	1.14469	0.572347	−	0.820012i	$-0.306033\pi$
$659$	29.3854	1.14469	0.572347	+	0.820012i	$0.306033\pi$
$660$	0	0
$661$	22.2075	0.863770	0.431885	−	0.901929i	$-0.357849\pi$
$661$	22.2075	0.863770	0.431885	+	0.901929i	$0.357849\pi$
$662$	67.4728	2.62241
$663$	0	0
$664$	−14.3875	−0.558342
$665$	−0.478360	−0.0185500
$666$	0	0
$667$	0.153878	0.00595817
$668$	−31.9515	−1.23624
$669$	0	0
$670$	5.87227	0.226865
$671$	−5.14430	−0.198593
$672$	0	0
$673$	−9.71497	−0.374484	−0.187242	−	0.982314i	$-0.559955\pi$
$673$	−9.71497	−0.374484	−0.187242	+	0.982314i	$0.559955\pi$
$674$	−37.8024	−1.45609
$675$	0	0
$676$	−16.6836	−0.641678
$677$	−26.4309	−1.01582	−0.507911	−	0.861410i	$-0.669582\pi$
$677$	−26.4309	−1.01582	−0.507911	+	0.861410i	$0.669582\pi$
$678$	0	0
$679$	3.21087	0.123222
$680$	9.59120	0.367806
$681$	0	0
$682$	2.06647	0.0791292
$683$	−12.0389	−0.460655	−0.230328	−	0.973113i	$-0.573980\pi$
$683$	−12.0389	−0.460655	−0.230328	+	0.973113i	$0.573980\pi$
$684$	0	0
$685$	43.1951	1.65040
$686$	22.5044	0.859221
$687$	0	0
$688$	−10.1277	−0.386116
$689$	−17.3002	−0.659084
$690$	0	0
$691$	−39.0797	−1.48666	−0.743331	−	0.668923i	$-0.766755\pi$
$691$	−39.0797	−1.48666	−0.743331	+	0.668923i	$0.766755\pi$
$692$	27.2677	1.03656
$693$	0	0
$694$	31.0742	1.17956
$695$	45.6508	1.73163
$696$	0	0
$697$	−13.6738	−0.517932
$698$	41.2039	1.55959
$699$	0	0
$700$	4.74795	0.179456
$701$	20.0417	0.756966	0.378483	−	0.925608i	$-0.376446\pi$
$701$	20.0417	0.756966	0.378483	+	0.925608i	$0.376446\pi$
$702$	0	0
$703$	1.41277	0.0532835
$704$	6.32459	0.238367
$705$	0	0
$706$	−47.9246	−1.80366
$707$	−2.87816	−0.108244
$708$	0	0
$709$	17.0947	0.642007	0.321003	−	0.947078i	$-0.395980\pi$
$709$	17.0947	0.642007	0.321003	+	0.947078i	$0.395980\pi$
$710$	28.6146	1.07389
$711$	0	0
$712$	−3.76298	−0.141024
$713$	0.109517	0.00410145
$714$	0	0
$715$	−3.58937	−0.134235
$716$	−40.4719	−1.51251
$717$	0	0
$718$	−39.9726	−1.49176
$719$	−8.28785	−0.309085	−0.154542	−	0.987986i	$-0.549390\pi$
$719$	−8.28785	−0.309085	−0.154542	+	0.987986i	$0.549390\pi$
$720$	0	0
$721$	6.99658	0.260566
$722$	41.3733	1.53975
$723$	0	0
$724$	52.3875	1.94697
$725$	5.87017	0.218013
$726$	0	0
$727$	−8.79811	−0.326304	−0.163152	−	0.986601i	$-0.552166\pi$
$727$	−8.79811	−0.326304	−0.163152	+	0.986601i	$0.552166\pi$
$728$	3.40614	0.126240
$729$	0	0
$730$	64.5949	2.39077
$731$	11.4960	0.425194
$732$	0	0
$733$	11.8517	0.437752	0.218876	−	0.975753i	$-0.429761\pi$
$733$	11.8517	0.437752	0.218876	+	0.975753i	$0.429761\pi$
$734$	−37.1800	−1.37234
$735$	0	0
$736$	0.435490	0.0160524
$737$	−0.505439	−0.0186181
$738$	0	0
$739$	30.2280	1.11196	0.555978	−	0.831197i	$-0.312344\pi$
$739$	30.2280	1.11196	0.555978	+	0.831197i	$0.312344\pi$
$740$	−45.4421	−1.67049
$741$	0	0
$742$	−10.9944	−0.403619
$743$	−6.25256	−0.229384	−0.114692	−	0.993401i	$-0.536588\pi$
$743$	−6.25256	−0.229384	−0.114692	+	0.993401i	$0.536588\pi$
$744$	0	0
$745$	16.4149	0.601396
$746$	21.3437	0.781448
$747$	0	0
$748$	−2.97386	−0.108735
$749$	8.34635	0.304969
$750$	0	0
$751$	−51.6429	−1.88448	−0.942239	−	0.334942i	$-0.891283\pi$
$751$	−51.6429	−1.88448	−0.942239	+	0.334942i	$0.891283\pi$
$752$	−12.1559	−0.443279
$753$	0	0
$754$	15.1702	0.552466
$755$	21.4485	0.780592
$756$	0	0
$757$	11.3407	0.412186	0.206093	−	0.978532i	$-0.433925\pi$
$757$	11.3407	0.412186	0.206093	+	0.978532i	$0.433925\pi$
$758$	57.8966	2.10290
$759$	0	0
$760$	1.04460	0.0378914
$761$	33.2024	1.20359	0.601793	−	0.798652i	$-0.294453\pi$
$761$	33.2024	1.20359	0.601793	+	0.798652i	$0.294453\pi$
$762$	0	0
$763$	5.10373	0.184767
$764$	5.59731	0.202504
$765$	0	0
$766$	28.1003	1.01531
$767$	−38.1412	−1.37720
$768$	0	0
$769$	26.5928	0.958961	0.479480	−	0.877553i	$-0.340825\pi$
$769$	26.5928	0.958961	0.479480	+	0.877553i	$0.340825\pi$
$770$	−2.28108	−0.0822045
$771$	0	0
$772$	65.9430	2.37334
$773$	45.8637	1.64960	0.824802	−	0.565422i	$-0.191287\pi$
$773$	45.8637	1.64960	0.824802	+	0.565422i	$0.191287\pi$
$774$	0	0
$775$	4.17789	0.150074
$776$	−7.01158	−0.251701
$777$	0	0
$778$	−10.8963	−0.390650
$779$	−1.48924	−0.0533575
$780$	0	0
$781$	−2.46292	−0.0881302
$782$	−0.271461	−0.00970743
$783$	0	0
$784$	12.0000	0.428571
$785$	−45.7153	−1.63165
$786$	0	0
$787$	−2.79811	−0.0997417	−0.0498708	−	0.998756i	$-0.515881\pi$
$787$	−2.79811	−0.0997417	−0.0498708	+	0.998756i	$0.515881\pi$
$788$	−0.448349	−0.0159718
$789$	0	0
$790$	−9.02614	−0.321136
$791$	−10.4580	−0.371844
$792$	0	0
$793$	−26.8778	−0.954459
$794$	−16.5757	−0.588250
$795$	0	0
$796$	−9.94087	−0.352345
$797$	−40.1480	−1.42212	−0.711058	−	0.703133i	$-0.751784\pi$
$797$	−40.1480	−1.42212	−0.711058	+	0.703133i	$0.751784\pi$
$798$	0	0
$799$	13.7981	0.488142
$800$	16.6132	0.587365
$801$	0	0
$802$	−46.8208	−1.65330
$803$	−5.55983	−0.196202
$804$	0	0
$805$	−0.120891	−0.00426085
$806$	10.7968	0.380303
$807$	0	0
$808$	6.28503	0.221107
$809$	−46.7516	−1.64370	−0.821849	−	0.569706i	$-0.807057\pi$
$809$	−46.7516	−1.64370	−0.821849	+	0.569706i	$0.807057\pi$
$810$	0	0
$811$	3.51649	0.123481	0.0617404	−	0.998092i	$-0.480335\pi$
$811$	3.51649	0.123481	0.0617404	+	0.998092i	$0.480335\pi$
$812$	5.59731	0.196427
$813$	0	0
$814$	6.73684	0.236126
$815$	−38.0617	−1.33324
$816$	0	0
$817$	1.25205	0.0438036
$818$	−73.4539	−2.56826
$819$	0	0
$820$	47.9018	1.67280
$821$	−7.11503	−0.248316	−0.124158	−	0.992262i	$-0.539623\pi$
$821$	−7.11503	−0.248316	−0.124158	+	0.992262i	$0.539623\pi$
$822$	0	0
$823$	2.69438	0.0939202	0.0469601	−	0.998897i	$-0.485047\pi$
$823$	2.69438	0.0939202	0.0469601	+	0.998897i	$0.485047\pi$
$824$	−15.2784	−0.532249
$825$	0	0
$826$	−24.2392	−0.843388
$827$	−48.6877	−1.69304	−0.846519	−	0.532358i	$-0.821306\pi$
$827$	−48.6877	−1.69304	−0.846519	+	0.532358i	$0.821306\pi$
$828$	0	0
$829$	−6.87227	−0.238684	−0.119342	−	0.992853i	$-0.538078\pi$
$829$	−6.87227	−0.238684	−0.119342	+	0.992853i	$0.538078\pi$
$830$	50.3421	1.74740
$831$	0	0
$832$	33.0446	1.14562
$833$	−13.6212	−0.471946
$834$	0	0
$835$	31.0351	1.07401
$836$	−0.323888	−0.0112019
$837$	0	0
$838$	82.5358	2.85115
$839$	31.2452	1.07871	0.539353	−	0.842080i	$-0.318669\pi$
$839$	31.2452	1.07871	0.539353	+	0.842080i	$0.318669\pi$
$840$	0	0
$841$	−22.0797	−0.761370
$842$	−16.2968	−0.561624
$843$	0	0
$844$	2.56322	0.0882298
$845$	16.2051	0.557473
$846$	0	0
$847$	−8.25760	−0.283735
$848$	−12.2654	−0.421195
$849$	0	0
$850$	−10.3558	−0.355200
$851$	0.357034	0.0122390
$852$	0	0
$853$	30.2610	1.03612	0.518059	−	0.855345i	$-0.326655\pi$
$853$	30.2610	1.03612	0.518059	+	0.855345i	$0.326655\pi$
$854$	−17.0811	−0.584505
$855$	0	0
$856$	−18.2259	−0.622949
$857$	46.6721	1.59429	0.797144	−	0.603790i	$-0.206343\pi$
$857$	46.6721	1.59429	0.797144	+	0.603790i	$0.206343\pi$
$858$	0	0
$859$	17.8153	0.607849	0.303925	−	0.952696i	$-0.401703\pi$
$859$	17.8153	0.607849	0.303925	+	0.952696i	$0.401703\pi$
$860$	−40.2725	−1.37328
$861$	0	0
$862$	69.6875	2.37357
$863$	28.4391	0.968078	0.484039	−	0.875046i	$-0.339169\pi$
$863$	28.4391	0.968078	0.484039	+	0.875046i	$0.339169\pi$
$864$	0	0
$865$	−26.4856	−0.900539
$866$	−58.0061	−1.97113
$867$	0	0
$868$	3.98368	0.135215
$869$	0.776900	0.0263545
$870$	0	0
$871$	−2.64081	−0.0894803
$872$	−11.1450	−0.377417
$873$	0	0
$874$	−0.0295653	−0.00100006
$875$	5.72177	0.193431
$876$	0	0
$877$	38.2130	1.29036	0.645181	−	0.764030i	$-0.276782\pi$
$877$	38.2130	1.29036	0.645181	+	0.764030i	$0.276782\pi$
$878$	−12.4273	−0.419402
$879$	0	0
$880$	−2.54477	−0.0857843
$881$	−32.9684	−1.11074	−0.555368	−	0.831605i	$-0.687422\pi$
$881$	−32.9684	−1.11074	−0.555368	+	0.831605i	$0.687422\pi$
$882$	0	0
$883$	−33.3022	−1.12071	−0.560354	−	0.828253i	$-0.689335\pi$
$883$	−33.3022	−1.12071	−0.560354	+	0.828253i	$0.689335\pi$
$884$	−15.5377	−0.522591
$885$	0	0
$886$	−81.5598	−2.74006
$887$	45.5577	1.52968	0.764839	−	0.644221i	$-0.222818\pi$
$887$	45.5577	1.52968	0.764839	+	0.644221i	$0.222818\pi$
$888$	0	0
$889$	−14.2850	−0.479105
$890$	13.1668	0.441351
$891$	0	0
$892$	−31.7925	−1.06449
$893$	1.50278	0.0502885
$894$	0	0
$895$	39.3112	1.31403
$896$	9.55665	0.319265
$897$	0	0
$898$	−49.8633	−1.66396
$899$	4.92526	0.164267
$900$	0	0
$901$	13.9224	0.463823
$902$	−7.10149	−0.236454
$903$	0	0
$904$	22.8371	0.759552
$905$	−50.8850	−1.69147
$906$	0	0
$907$	−22.1539	−0.735607	−0.367804	−	0.929903i	$-0.619890\pi$
$907$	−22.1539	−0.735607	−0.367804	+	0.929903i	$0.619890\pi$
$908$	−24.7722	−0.822096
$909$	0	0
$910$	−11.9181	−0.395083
$911$	2.24219	0.0742872	0.0371436	−	0.999310i	$-0.488174\pi$
$911$	2.24219	0.0742872	0.0371436	+	0.999310i	$0.488174\pi$
$912$	0	0
$913$	−4.33305	−0.143403
$914$	−33.8113	−1.11838
$915$	0	0
$916$	71.2748	2.35498
$917$	−4.55642	−0.150466
$918$	0	0
$919$	−4.69780	−0.154966	−0.0774831	−	0.996994i	$-0.524688\pi$
$919$	−4.69780	−0.154966	−0.0774831	+	0.996994i	$0.524688\pi$
$920$	0.263989	0.00870347
$921$	0	0
$922$	12.5632	0.413748
$923$	−12.8682	−0.423562
$924$	0	0
$925$	13.6202	0.447830
$926$	−42.2147	−1.38726
$927$	0	0
$928$	19.5851	0.642912
$929$	47.7820	1.56768	0.783839	−	0.620964i	$-0.213259\pi$
$929$	47.7820	1.56768	0.783839	+	0.620964i	$0.213259\pi$
$930$	0	0
$931$	−1.48351	−0.0486200
$932$	5.29411	0.173414
$933$	0	0
$934$	14.5448	0.475919
$935$	2.88857	0.0944662
$936$	0	0
$937$	16.2370	0.530440	0.265220	−	0.964188i	$-0.414555\pi$
$937$	16.2370	0.530440	0.265220	+	0.964188i	$0.414555\pi$
$938$	−1.67826	−0.0547971
$939$	0	0
$940$	−48.3373	−1.57659
$941$	−19.9968	−0.651876	−0.325938	−	0.945391i	$-0.605680\pi$
$941$	−19.9968	−0.651876	−0.325938	+	0.945391i	$0.605680\pi$
$942$	0	0
$943$	−0.376359	−0.0122559
$944$	−27.0412	−0.880116
$945$	0	0
$946$	5.97043	0.194116
$947$	−27.2738	−0.886278	−0.443139	−	0.896453i	$-0.646135\pi$
$947$	−27.2738	−0.886278	−0.443139	+	0.896453i	$0.646135\pi$
$948$	0	0
$949$	−29.0489	−0.942966
$950$	−1.12787	−0.0365928
$951$	0	0
$952$	−2.74111	−0.0888399
$953$	3.75851	0.121750	0.0608750	−	0.998145i	$-0.480611\pi$
$953$	3.75851	0.121750	0.0608750	+	0.998145i	$0.480611\pi$
$954$	0	0
$955$	−5.43678	−0.175930
$956$	9.29266	0.300546
$957$	0	0
$958$	−5.84270	−0.188769
$959$	−12.3449	−0.398638
$960$	0	0
$961$	−27.4946	−0.886923
$962$	35.1985	1.13485
$963$	0	0
$964$	−2.85039	−0.0918050
$965$	−64.0517	−2.06190
$966$	0	0
$967$	32.8483	1.05633	0.528164	−	0.849142i	$-0.322880\pi$
$967$	32.8483	1.05633	0.528164	+	0.849142i	$0.322880\pi$
$968$	18.0321	0.579574
$969$	0	0
$970$	24.5337	0.787728
$971$	−11.9879	−0.384709	−0.192354	−	0.981326i	$-0.561612\pi$
$971$	−11.9879	−0.384709	−0.192354	+	0.981326i	$0.561612\pi$
$972$	0	0
$973$	−13.0467	−0.418259
$974$	82.3628	2.63907
$975$	0	0
$976$	−19.0557	−0.609958
$977$	3.64152	0.116503	0.0582513	−	0.998302i	$-0.481448\pi$
$977$	3.64152	0.116503	0.0582513	+	0.998302i	$0.481448\pi$
$978$	0	0
$979$	−1.13329	−0.0362201
$980$	47.7175	1.52428
$981$	0	0
$982$	−78.9691	−2.52001
$983$	−39.2916	−1.25321	−0.626604	−	0.779338i	$-0.715556\pi$
$983$	−39.2916	−1.25321	−0.626604	+	0.779338i	$0.715556\pi$
$984$	0	0
$985$	0.435490	0.0138759
$986$	−12.2083	−0.388792
$987$	0	0
$988$	−1.69224	−0.0538375
$989$	0.316416	0.0100615
$990$	0	0
$991$	43.8950	1.39437	0.697185	−	0.716891i	$-0.254436\pi$
$991$	43.8950	1.39437	0.697185	+	0.716891i	$0.254436\pi$
$992$	13.9390	0.442563
$993$	0	0
$994$	−8.17789	−0.259387
$995$	9.65576	0.306108
$996$	0	0
$997$	4.30562	0.136360	0.0681802	−	0.997673i	$-0.478281\pi$
$997$	4.30562	0.136360	0.0681802	+	0.997673i	$0.478281\pi$
$998$	−9.05868	−0.286748
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.a.l.1.2	✓	6
3.2	odd	2	inner	603.2.a.l.1.5	yes	6
4.3	odd	2		9648.2.a.cg.1.2		6
12.11	even	2		9648.2.a.cg.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.a.l.1.2	✓	6	1.1	even	1	trivial
603.2.a.l.1.5	yes	6	3.2	odd	2	inner
9648.2.a.cg.1.2		6	4.3	odd	2
9648.2.a.cg.1.5		6	12.11	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:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18370 q^{2} +2.76854 q^{4} -2.68914 q^{5} +0.768540 q^{7} -1.67826 q^{8} +O(q^{10})\)	\(q-2.18370 q^{2} +2.76854 q^{4} -2.68914 q^{5} +0.768540 q^{7} -1.67826 q^{8} +5.87227 q^{10} -0.505439 q^{11} -2.64081 q^{13} -1.67826 q^{14} -1.87227 q^{16} +2.12520 q^{17} +0.231460 q^{19} -7.44498 q^{20} +1.10373 q^{22} +0.0584944 q^{23} +2.23146 q^{25} +5.76672 q^{26} +2.12773 q^{28} +2.63064 q^{29} +1.87227 q^{31} +7.44498 q^{32} -4.64081 q^{34} -2.06671 q^{35} +6.10373 q^{37} -0.505439 q^{38} +4.51307 q^{40} -6.43411 q^{41} +5.40935 q^{43} -1.39933 q^{44} -0.127734 q^{46} +6.49260 q^{47} -6.40935 q^{49} -4.87284 q^{50} -7.31118 q^{52} +6.55110 q^{53} +1.35919 q^{55} -1.28981 q^{56} -5.74453 q^{58} +14.4430 q^{59} +10.1779 q^{61} -4.08846 q^{62} -12.5131 q^{64} +7.10149 q^{65} +1.00000 q^{67} +5.88371 q^{68} +4.51307 q^{70} +4.87284 q^{71} +11.0000 q^{73} -13.3287 q^{74} +0.640806 q^{76} -0.388450 q^{77} -1.53708 q^{79} +5.03478 q^{80} +14.0502 q^{82} +8.57285 q^{83} -5.71497 q^{85} -11.8124 q^{86} +0.848258 q^{88} +2.24219 q^{89} -2.02957 q^{91} +0.161944 q^{92} -14.1779 q^{94} -0.622427 q^{95} +4.17789 q^{97} +13.9961 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4}+O(q^{10}) \) 6 * q + 12 * q^4	\( 6 q + 12 q^{4} + 4 q^{10} + 20 q^{13} + 20 q^{16} + 6 q^{19} - 20 q^{22} + 18 q^{25} + 44 q^{28} - 20 q^{31} + 8 q^{34} + 10 q^{37} - 40 q^{40} - 8 q^{43} - 32 q^{46} + 2 q^{49} + 36 q^{52} + 44 q^{55} + 28 q^{58} + 16 q^{61} - 8 q^{64} + 6 q^{67} - 40 q^{70} + 66 q^{73} - 32 q^{76} + 8 q^{82} + 20 q^{85} - 84 q^{88} - 4 q^{91} - 40 q^{94} - 20 q^{97}+O(q^{100}) \) 6 * q + 12 * q^4 + 4 * q^10 + 20 * q^13 + 20 * q^16 + 6 * q^19 - 20 * q^22 + 18 * q^25 + 44 * q^28 - 20 * q^31 + 8 * q^34 + 10 * q^37 - 40 * q^40 - 8 * q^43 - 32 * q^46 + 2 * q^49 + 36 * q^52 + 44 * q^55 + 28 * q^58 + 16 * q^61 - 8 * q^64 + 6 * q^67 - 40 * q^70 + 66 * q^73 - 32 * q^76 + 8 * q^82 + 20 * q^85 - 84 * q^88 - 4 * q^91 - 40 * q^94 - 20 * q^97

Introduction

L-functions

Modular forms

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