LMFDB - Embedded newform 7605.2.a.ch.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7605,2,Mod(1,7605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7605.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,-4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13824.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 6 $ x^4 - 6*x^2 + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.12603$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.12603 q^{2} -0.732051 q^{4} -1.00000 q^{5} +2.85808 q^{7} -3.07638 q^{8} -1.12603 q^{10} -3.07638 q^{11} +3.21829 q^{14} -2.00000 q^{16} +1.95035 q^{17} -7.42071 q^{19} +0.732051 q^{20} -3.46410 q^{22} +8.79254 q^{23} +1.00000 q^{25} -2.09226 q^{28} +6.68240 q^{29} +3.15276 q^{31} +3.90069 q^{32} +2.19615 q^{34} -2.85808 q^{35} -8.25207 q^{37} -8.35596 q^{38} +3.07638 q^{40} -3.32218 q^{41} +9.54674 q^{43} +2.25207 q^{44} +9.90069 q^{46} +1.79759 q^{47} +1.16864 q^{49} +1.12603 q^{50} -10.6569 q^{53} +3.07638 q^{55} -8.79254 q^{56} +7.52460 q^{58} -4.81805 q^{59} -6.26795 q^{61} +3.55011 q^{62} +8.39230 q^{64} -5.87855 q^{67} -1.42775 q^{68} -3.21829 q^{70} +10.2993 q^{71} -11.1101 q^{73} -9.29209 q^{74} +5.43233 q^{76} -8.79254 q^{77} -11.0968 q^{79} +2.00000 q^{80} -3.74089 q^{82} +6.97707 q^{83} -1.95035 q^{85} +10.7499 q^{86} +9.46410 q^{88} +0.0930509 q^{89} -6.43659 q^{92} +2.02414 q^{94} +7.42071 q^{95} -4.79759 q^{97} +1.31593 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{5} + 12 q^{14} - 8 q^{16} - 12 q^{19} - 4 q^{20} + 4 q^{25} - 12 q^{28} + 12 q^{29} - 12 q^{31} - 12 q^{34} - 24 q^{37} + 12 q^{41} + 16 q^{43} + 24 q^{46} + 24 q^{47} - 4 q^{49} - 12 q^{58}+ \cdots - 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^5 + 12 * q^14 - 8 * q^16 - 12 * q^19 - 4 * q^20 + 4 * q^25 - 12 * q^28 + 12 * q^29 - 12 * q^31 - 12 * q^34 - 24 * q^37 + 12 * q^41 + 16 * q^43 + 24 * q^46 + 24 * q^47 - 4 * q^49 - 12 * q^58 - 12 * q^59 - 32 * q^61 - 8 * q^64 + 12 * q^67 - 12 * q^70 - 12 * q^71 - 24 * q^73 - 24 * q^74 - 24 * q^76 - 8 * q^79 + 8 * q^80 - 12 * q^82 + 12 * q^86 + 24 * q^88 + 12 * q^89 - 24 * q^92 - 12 * q^94 + 12 * q^95 - 36 * q^97 - 24 * q^98

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 $

$2$	1.12603	0.796225	0.398113	−	0.917337i	$-0.369665\pi$
$2$	1.12603	0.796225	0.398113	+	0.917337i	$0.369665\pi$
$3$	0	0
$3$	0	0
$4$	−0.732051	−0.366025
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.85808	1.08025	0.540127	−	0.841584i	$-0.318376\pi$
$7$	2.85808	1.08025	0.540127	+	0.841584i	$0.318376\pi$
$8$	−3.07638	−1.08766
$9$	0	0
$10$	−1.12603	−0.356083
$11$	−3.07638	−0.927563	−0.463781	−	0.885950i	$-0.653508\pi$
$11$	−3.07638	−0.927563	−0.463781	+	0.885950i	$0.653508\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.21829	0.860125
$15$	0	0
$16$	−2.00000	−0.500000
$17$	1.95035	0.473028	0.236514	−	0.971628i	$-0.423995\pi$
$17$	1.95035	0.473028	0.236514	+	0.971628i	$0.423995\pi$
$18$	0	0
$19$	−7.42071	−1.70243	−0.851213	−	0.524820i	$-0.824133\pi$
$19$	−7.42071	−1.70243	−0.851213	+	0.524820i	$0.824133\pi$
$20$	0.732051	0.163692
$21$	0	0
$22$	−3.46410	−0.738549
$23$	8.79254	1.83337	0.916686	−	0.399608i	$-0.130854\pi$
$23$	8.79254	1.83337	0.916686	+	0.399608i	$0.130854\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.09226	−0.395400
$29$	6.68240	1.24089	0.620445	−	0.784250i	$-0.286952\pi$
$29$	6.68240	1.24089	0.620445	+	0.784250i	$0.286952\pi$
$30$	0	0
$31$	3.15276	0.566252	0.283126	−	0.959083i	$-0.408629\pi$
$31$	3.15276	0.566252	0.283126	+	0.959083i	$0.408629\pi$
$32$	3.90069	0.689551
$33$	0	0
$34$	2.19615	0.376637
$35$	−2.85808	−0.483104
$36$	0	0
$37$	−8.25207	−1.35663	−0.678316	−	0.734771i	$-0.737290\pi$
$37$	−8.25207	−1.35663	−0.678316	+	0.734771i	$0.737290\pi$
$38$	−8.35596	−1.35551
$39$	0	0
$40$	3.07638	0.486418
$41$	−3.32218	−0.518838	−0.259419	−	0.965765i	$-0.583531\pi$
$41$	−3.32218	−0.518838	−0.259419	+	0.965765i	$0.583531\pi$
$42$	0	0
$43$	9.54674	1.45586	0.727932	−	0.685649i	$-0.240482\pi$
$43$	9.54674	1.45586	0.727932	+	0.685649i	$0.240482\pi$
$44$	2.25207	0.339512
$45$	0	0
$46$	9.90069	1.45978
$47$	1.79759	0.262205	0.131103	−	0.991369i	$-0.458148\pi$
$47$	1.79759	0.262205	0.131103	+	0.991369i	$0.458148\pi$
$48$	0	0
$49$	1.16864	0.166949
$50$	1.12603	0.159245
$51$	0	0
$52$	0	0
$53$	−10.6569	−1.46384	−0.731918	−	0.681393i	$-0.761375\pi$
$53$	−10.6569	−1.46384	−0.731918	+	0.681393i	$0.761375\pi$
$54$	0	0
$55$	3.07638	0.414819
$56$	−8.79254	−1.17495
$57$	0	0
$58$	7.52460	0.988028
$59$	−4.81805	−0.627257	−0.313629	−	0.949546i	$-0.601545\pi$
$59$	−4.81805	−0.627257	−0.313629	+	0.949546i	$0.601545\pi$
$60$	0	0
$61$	−6.26795	−0.802529	−0.401264	−	0.915962i	$-0.631429\pi$
$61$	−6.26795	−0.802529	−0.401264	+	0.915962i	$0.631429\pi$
$62$	3.55011	0.450864
$63$	0	0
$64$	8.39230	1.04904
$65$	0	0
$66$	0	0
$67$	−5.87855	−0.718179	−0.359090	−	0.933303i	$-0.616913\pi$
$67$	−5.87855	−0.718179	−0.359090	+	0.933303i	$0.616913\pi$
$68$	−1.42775	−0.173140
$69$	0	0
$70$	−3.21829	−0.384660
$71$	10.2993	1.22230	0.611148	−	0.791516i	$-0.290708\pi$
$71$	10.2993	1.22230	0.611148	+	0.791516i	$0.290708\pi$
$72$	0	0
$73$	−11.1101	−1.30034	−0.650172	−	0.759787i	$-0.725303\pi$
$73$	−11.1101	−1.30034	−0.650172	+	0.759787i	$0.725303\pi$
$74$	−9.29209	−1.08018
$75$	0	0
$76$	5.43233	0.623131
$77$	−8.79254	−1.00200
$78$	0	0
$79$	−11.0968	−1.24849	−0.624246	−	0.781228i	$-0.714594\pi$
$79$	−11.0968	−1.24849	−0.624246	+	0.781228i	$0.714594\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−3.74089	−0.413112
$83$	6.97707	0.765833	0.382916	−	0.923783i	$-0.374920\pi$
$83$	6.97707	0.765833	0.382916	+	0.923783i	$0.374920\pi$
$84$	0	0
$85$	−1.95035	−0.211545
$86$	10.7499	1.15920
$87$	0	0
$88$	9.46410	1.00888
$89$	0.0930509	0.00986338	0.00493169	−	0.999988i	$-0.498430\pi$
$89$	0.0930509	0.00986338	0.00493169	+	0.999988i	$0.498430\pi$
$90$	0	0
$91$	0	0
$92$	−6.43659	−0.671061
$93$	0	0
$94$	2.02414	0.208775
$95$	7.42071	0.761348
$96$	0	0
$97$	−4.79759	−0.487121	−0.243561	−	0.969886i	$-0.578316\pi$
$97$	−4.79759	−0.487121	−0.243561	+	0.969886i	$0.578316\pi$
$98$	1.31593	0.132929
$99$	0	0
$100$	−0.732051	−0.0732051
$101$	−2.79254	−0.277869	−0.138934	−	0.990302i	$-0.544368\pi$
$101$	−2.79254	−0.277869	−0.138934	+	0.990302i	$0.544368\pi$
$102$	0	0
$103$	8.35395	0.823139	0.411570	−	0.911378i	$-0.364981\pi$
$103$	8.35395	0.823139	0.411570	+	0.911378i	$0.364981\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−6.35059	−0.613934	−0.306967	−	0.951720i	$-0.599314\pi$
$107$	−6.35059	−0.613934	−0.306967	+	0.951720i	$0.599314\pi$
$108$	0	0
$109$	−11.7996	−1.13020	−0.565098	−	0.825024i	$-0.691162\pi$
$109$	−11.7996	−1.13020	−0.565098	+	0.825024i	$0.691162\pi$
$110$	3.46410	0.330289
$111$	0	0
$112$	−5.71617	−0.540127
$113$	−3.46410	−0.325875	−0.162938	−	0.986636i	$-0.552097\pi$
$113$	−3.46410	−0.325875	−0.162938	+	0.986636i	$0.552097\pi$
$114$	0	0
$115$	−8.79254	−0.819909
$116$	−4.89185	−0.454197
$117$	0	0
$118$	−5.42529	−0.499438
$119$	5.57425	0.510991
$120$	0	0
$121$	−1.53590	−0.139627
$122$	−7.05791	−0.638994
$123$	0	0
$124$	−2.30798	−0.207263
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.9720	−1.32855	−0.664276	−	0.747487i	$-0.731260\pi$
$127$	−14.9720	−1.32855	−0.664276	+	0.747487i	$0.731260\pi$
$128$	1.64863	0.145719
$129$	0	0
$130$	0	0
$131$	10.5831	0.924649	0.462324	−	0.886711i	$-0.347016\pi$
$131$	10.5831	0.924649	0.462324	+	0.886711i	$0.347016\pi$
$132$	0	0
$133$	−21.2090	−1.83905
$134$	−6.61944	−0.571832
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−8.70654	−0.743850	−0.371925	−	0.928263i	$-0.621302\pi$
$137$	−8.70654	−0.743850	−0.371925	+	0.928263i	$0.621302\pi$
$138$	0	0
$139$	−11.6569	−0.988724	−0.494362	−	0.869256i	$-0.664598\pi$
$139$	−11.6569	−0.988724	−0.494362	+	0.869256i	$0.664598\pi$
$140$	2.09226	0.176828
$141$	0	0
$142$	11.5973	0.973223
$143$	0	0
$144$	0	0
$145$	−6.68240	−0.554943
$146$	−12.5104	−1.03537
$147$	0	0
$148$	6.04093	0.496561
$149$	6.82431	0.559070	0.279535	−	0.960136i	$-0.409820\pi$
$149$	6.82431	0.559070	0.279535	+	0.960136i	$0.409820\pi$
$150$	0	0
$151$	12.6810	1.03197	0.515984	−	0.856598i	$-0.327426\pi$
$151$	12.6810	1.03197	0.515984	+	0.856598i	$0.327426\pi$
$152$	22.8289	1.85167
$153$	0	0
$154$	−9.90069	−0.797820
$155$	−3.15276	−0.253235
$156$	0	0
$157$	7.98333	0.637139	0.318569	−	0.947900i	$-0.396798\pi$
$157$	7.98333	0.637139	0.318569	+	0.947900i	$0.396798\pi$
$158$	−12.4954	−0.994081
$159$	0	0
$160$	−3.90069	−0.308377
$161$	25.1298	1.98051
$162$	0	0
$163$	−12.3222	−0.965148	−0.482574	−	0.875855i	$-0.660298\pi$
$163$	−12.3222	−0.965148	−0.482574	+	0.875855i	$0.660298\pi$
$164$	2.43201	0.189908
$165$	0	0
$166$	7.85641	0.609775
$167$	−21.9416	−1.69789	−0.848947	−	0.528478i	$-0.822763\pi$
$167$	−21.9416	−1.69789	−0.848947	+	0.528478i	$0.822763\pi$
$168$	0	0
$169$	0	0
$170$	−2.19615	−0.168437
$171$	0	0
$172$	−6.98870	−0.532883
$173$	−15.0506	−1.14427	−0.572136	−	0.820159i	$-0.693885\pi$
$173$	−15.0506	−1.14427	−0.572136	+	0.820159i	$0.693885\pi$
$174$	0	0
$175$	2.85808	0.216051
$176$	6.15276	0.463781
$177$	0	0
$178$	0.104778	0.00785347
$179$	0.480775	0.0359348	0.0179674	−	0.999839i	$-0.494280\pi$
$179$	0.480775	0.0359348	0.0179674	+	0.999839i	$0.494280\pi$
$180$	0	0
$181$	1.85887	0.138169	0.0690844	−	0.997611i	$-0.477992\pi$
$181$	1.85887	0.138169	0.0690844	+	0.997611i	$0.477992\pi$
$182$	0	0
$183$	0	0
$184$	−27.0492	−1.99409
$185$	8.25207	0.606704
$186$	0	0
$187$	−6.00000	−0.438763
$188$	−1.31593	−0.0959738
$189$	0	0
$190$	8.35596	0.606205
$191$	−8.71875	−0.630866	−0.315433	−	0.948948i	$-0.602150\pi$
$191$	−8.71875	−0.630866	−0.315433	+	0.948948i	$0.602150\pi$
$192$	0	0
$193$	−6.45326	−0.464516	−0.232258	−	0.972654i	$-0.574611\pi$
$193$	−6.45326	−0.464516	−0.232258	+	0.972654i	$0.574611\pi$
$194$	−5.40224	−0.387858
$195$	0	0
$196$	−0.855504	−0.0611074
$197$	−11.5930	−0.825969	−0.412984	−	0.910738i	$-0.635514\pi$
$197$	−11.5930	−0.825969	−0.412984	+	0.910738i	$0.635514\pi$
$198$	0	0
$199$	0.801382	0.0568085	0.0284042	−	0.999597i	$-0.490957\pi$
$199$	0.801382	0.0568085	0.0284042	+	0.999597i	$0.490957\pi$
$200$	−3.07638	−0.217533
$201$	0	0
$202$	−3.14450	−0.221246
$203$	19.0988	1.34048
$204$	0	0
$205$	3.32218	0.232031
$206$	9.40682	0.655404
$207$	0	0
$208$	0	0
$209$	22.8289	1.57911
$210$	0	0
$211$	−8.29209	−0.570851	−0.285426	−	0.958401i	$-0.592135\pi$
$211$	−8.29209	−0.570851	−0.285426	+	0.958401i	$0.592135\pi$
$212$	7.80138	0.535801
$213$	0	0
$214$	−7.15097	−0.488830
$215$	−9.54674	−0.651082
$216$	0	0
$217$	9.01084	0.611696
$218$	−13.2867	−0.899890
$219$	0	0
$220$	−2.25207	−0.151834
$221$	0	0
$222$	0	0
$223$	12.7614	0.854563	0.427282	−	0.904119i	$-0.359471\pi$
$223$	12.7614	0.854563	0.427282	+	0.904119i	$0.359471\pi$
$224$	11.1485	0.744890
$225$	0	0
$226$	−3.90069	−0.259470
$227$	−17.1114	−1.13572	−0.567860	−	0.823125i	$-0.692229\pi$
$227$	−17.1114	−1.13572	−0.567860	+	0.823125i	$0.692229\pi$
$228$	0	0
$229$	6.39993	0.422919	0.211460	−	0.977387i	$-0.432178\pi$
$229$	6.39993	0.422919	0.211460	+	0.977387i	$0.432178\pi$
$230$	−9.90069	−0.652832
$231$	0	0
$232$	−20.5576	−1.34967
$233$	−0.671557	−0.0439952	−0.0219976	−	0.999758i	$-0.507003\pi$
$233$	−0.671557	−0.0439952	−0.0219976	+	0.999758i	$0.507003\pi$
$234$	0	0
$235$	−1.79759	−0.117262
$236$	3.52706	0.229592
$237$	0	0
$238$	6.27679	0.406864
$239$	7.12983	0.461190	0.230595	−	0.973050i	$-0.425933\pi$
$239$	7.12983	0.461190	0.230595	+	0.973050i	$0.425933\pi$
$240$	0	0
$241$	−26.9383	−1.73525	−0.867623	−	0.497223i	$-0.834353\pi$
$241$	−26.9383	−1.73525	−0.867623	+	0.497223i	$0.834353\pi$
$242$	−1.72947	−0.111175
$243$	0	0
$244$	4.58846	0.293746
$245$	−1.16864	−0.0746617
$246$	0	0
$247$	0	0
$248$	−9.69907	−0.615892
$249$	0	0
$250$	−1.12603	−0.0712165
$251$	11.6480	0.735218	0.367609	−	0.929980i	$-0.380176\pi$
$251$	11.6480	0.735218	0.367609	+	0.929980i	$0.380176\pi$
$252$	0	0
$253$	−27.0492	−1.70057
$254$	−16.8590	−1.05783
$255$	0	0
$256$	−14.9282	−0.933013
$257$	7.27879	0.454038	0.227019	−	0.973890i	$-0.427102\pi$
$257$	7.27879	0.454038	0.227019	+	0.973890i	$0.427102\pi$
$258$	0	0
$259$	−23.5851	−1.46551
$260$	0	0
$261$	0	0
$262$	11.9169	0.736229
$263$	−23.9277	−1.47545	−0.737724	−	0.675103i	$-0.764099\pi$
$263$	−23.9277	−1.47545	−0.737724	+	0.675103i	$0.764099\pi$
$264$	0	0
$265$	10.6569	0.654647
$266$	−23.8820	−1.46430
$267$	0	0
$268$	4.30340	0.262872
$269$	−4.38147	−0.267143	−0.133571	−	0.991039i	$-0.542645\pi$
$269$	−4.38147	−0.267143	−0.133571	+	0.991039i	$0.542645\pi$
$270$	0	0
$271$	2.48825	0.151150	0.0755751	−	0.997140i	$-0.475921\pi$
$271$	2.48825	0.151150	0.0755751	+	0.997140i	$0.475921\pi$
$272$	−3.90069	−0.236514
$273$	0	0
$274$	−9.80385	−0.592272
$275$	−3.07638	−0.185513
$276$	0	0
$277$	−21.7061	−1.30419	−0.652096	−	0.758137i	$-0.726110\pi$
$277$	−21.7061	−1.30419	−0.652096	+	0.758137i	$0.726110\pi$
$278$	−13.1260	−0.787247
$279$	0	0
$280$	8.79254	0.525455
$281$	29.7270	1.77336	0.886682	−	0.462379i	$-0.153004\pi$
$281$	29.7270	1.77336	0.886682	+	0.462379i	$0.153004\pi$
$282$	0	0
$283$	27.9995	1.66440	0.832200	−	0.554476i	$-0.187081\pi$
$283$	27.9995	1.66440	0.832200	+	0.554476i	$0.187081\pi$
$284$	−7.53958	−0.447392
$285$	0	0
$286$	0	0
$287$	−9.49508	−0.560477
$288$	0	0
$289$	−13.1962	−0.776244
$290$	−7.52460	−0.441859
$291$	0	0
$292$	8.13319	0.475959
$293$	16.9586	0.990732	0.495366	−	0.868684i	$-0.335034\pi$
$293$	16.9586	0.990732	0.495366	+	0.868684i	$0.335034\pi$
$294$	0	0
$295$	4.81805	0.280518
$296$	25.3865	1.47556
$297$	0	0
$298$	7.68440	0.445145
$299$	0	0
$300$	0	0
$301$	27.2854	1.57270
$302$	14.2793	0.821679
$303$	0	0
$304$	14.8414	0.851213
$305$	6.26795	0.358902
$306$	0	0
$307$	−15.6072	−0.890752	−0.445376	−	0.895344i	$-0.646930\pi$
$307$	−15.6072	−0.890752	−0.445376	+	0.895344i	$0.646930\pi$
$308$	6.43659	0.366759
$309$	0	0
$310$	−3.55011	−0.201632
$311$	24.9941	1.41728	0.708642	−	0.705568i	$-0.249308\pi$
$311$	24.9941	1.41728	0.708642	+	0.705568i	$0.249308\pi$
$312$	0	0
$313$	1.16117	0.0656331	0.0328166	−	0.999461i	$-0.489552\pi$
$313$	1.16117	0.0656331	0.0328166	+	0.999461i	$0.489552\pi$
$314$	8.98949	0.507306
$315$	0	0
$316$	8.12345	0.456980
$317$	8.62570	0.484467	0.242234	−	0.970218i	$-0.422120\pi$
$317$	8.62570	0.484467	0.242234	+	0.970218i	$0.422120\pi$
$318$	0	0
$319$	−20.5576	−1.15100
$320$	−8.39230	−0.469144
$321$	0	0
$322$	28.2970	1.57693
$323$	−14.4729	−0.805296
$324$	0	0
$325$	0	0
$326$	−13.8752	−0.768475
$327$	0	0
$328$	10.2203	0.564321
$329$	5.13766	0.283248
$330$	0	0
$331$	25.6227	1.40835	0.704174	−	0.710027i	$-0.251317\pi$
$331$	25.6227	1.40835	0.704174	+	0.710027i	$0.251317\pi$
$332$	−5.10757	−0.280314
$333$	0	0
$334$	−24.7070	−1.35191
$335$	5.87855	0.321179
$336$	0	0
$337$	20.1770	1.09911	0.549556	−	0.835457i	$-0.314797\pi$
$337$	20.1770	1.09911	0.549556	+	0.835457i	$0.314797\pi$
$338$	0	0
$339$	0	0
$340$	1.42775	0.0774307
$341$	−9.69907	−0.525234
$342$	0	0
$343$	−16.6665	−0.899907
$344$	−29.3694	−1.58349
$345$	0	0
$346$	−16.9474	−0.911099
$347$	−24.1937	−1.29879	−0.649393	−	0.760453i	$-0.724977\pi$
$347$	−24.1937	−1.29879	−0.649393	+	0.760453i	$0.724977\pi$
$348$	0	0
$349$	−25.6761	−1.37441	−0.687206	−	0.726463i	$-0.741163\pi$
$349$	−25.6761	−1.37441	−0.687206	+	0.726463i	$0.741163\pi$
$350$	3.21829	0.172025
$351$	0	0
$352$	−12.0000	−0.639602
$353$	7.84509	0.417552	0.208776	−	0.977963i	$-0.433052\pi$
$353$	7.84509	0.417552	0.208776	+	0.977963i	$0.433052\pi$
$354$	0	0
$355$	−10.2993	−0.546628
$356$	−0.0681180	−0.00361025
$357$	0	0
$358$	0.541368	0.0286122
$359$	−19.7145	−1.04049	−0.520245	−	0.854017i	$-0.674160\pi$
$359$	−19.7145	−1.04049	−0.520245	+	0.854017i	$0.674160\pi$
$360$	0	0
$361$	36.0669	1.89826
$362$	2.09315	0.110014
$363$	0	0
$364$	0	0
$365$	11.1101	0.581532
$366$	0	0
$367$	−11.1761	−0.583388	−0.291694	−	0.956512i	$-0.594219\pi$
$367$	−11.1761	−0.583388	−0.291694	+	0.956512i	$0.594219\pi$
$368$	−17.5851	−0.916686
$369$	0	0
$370$	9.29209	0.483073
$371$	−30.4583	−1.58131
$372$	0	0
$373$	−8.75173	−0.453147	−0.226574	−	0.973994i	$-0.572752\pi$
$373$	−8.75173	−0.453147	−0.226574	+	0.973994i	$0.572752\pi$
$374$	−6.75620	−0.349355
$375$	0	0
$376$	−5.53006	−0.285191
$377$	0	0
$378$	0	0
$379$	−18.7696	−0.964130	−0.482065	−	0.876135i	$-0.660113\pi$
$379$	−18.7696	−0.964130	−0.482065	+	0.876135i	$0.660113\pi$
$380$	−5.43233	−0.278673
$381$	0	0
$382$	−9.81759	−0.502312
$383$	6.07469	0.310402	0.155201	−	0.987883i	$-0.450397\pi$
$383$	6.07469	0.310402	0.155201	+	0.987883i	$0.450397\pi$
$384$	0	0
$385$	8.79254	0.448110
$386$	−7.26658	−0.369859
$387$	0	0
$388$	3.51208	0.178299
$389$	9.66572	0.490072	0.245036	−	0.969514i	$-0.421200\pi$
$389$	9.66572	0.490072	0.245036	+	0.969514i	$0.421200\pi$
$390$	0	0
$391$	17.1485	0.867237
$392$	−3.59518	−0.181584
$393$	0	0
$394$	−13.0541	−0.657657
$395$	11.0968	0.558343
$396$	0	0
$397$	24.3317	1.22117	0.610585	−	0.791950i	$-0.290934\pi$
$397$	24.3317	1.22117	0.610585	+	0.791950i	$0.290934\pi$
$398$	0.902382	0.0452323
$399$	0	0
$400$	−2.00000	−0.100000
$401$	−8.89643	−0.444267	−0.222133	−	0.975016i	$-0.571302\pi$
$401$	−8.89643	−0.444267	−0.222133	+	0.975016i	$0.571302\pi$
$402$	0	0
$403$	0	0
$404$	2.04428	0.101707
$405$	0	0
$406$	21.5059	1.06732
$407$	25.3865	1.25836
$408$	0	0
$409$	−28.2887	−1.39879	−0.699394	−	0.714736i	$-0.746547\pi$
$409$	−28.2887	−1.39879	−0.699394	+	0.714736i	$0.746547\pi$
$410$	3.74089	0.184749
$411$	0	0
$412$	−6.11552	−0.301290
$413$	−13.7704	−0.677597
$414$	0	0
$415$	−6.97707	−0.342491
$416$	0	0
$417$	0	0
$418$	25.7061	1.25733
$419$	−28.8948	−1.41160	−0.705801	−	0.708411i	$-0.749413\pi$
$419$	−28.8948	−1.41160	−0.705801	+	0.708411i	$0.749413\pi$
$420$	0	0
$421$	−12.0134	−0.585498	−0.292749	−	0.956189i	$-0.594570\pi$
$421$	−12.0134	−0.585498	−0.292749	+	0.956189i	$0.594570\pi$
$422$	−9.33717	−0.454526
$423$	0	0
$424$	32.7846	1.59216
$425$	1.95035	0.0946057
$426$	0	0
$427$	−17.9143	−0.866935
$428$	4.64895	0.224716
$429$	0	0
$430$	−10.7499	−0.518408
$431$	−23.0229	−1.10898	−0.554488	−	0.832192i	$-0.687086\pi$
$431$	−23.0229	−1.10898	−0.554488	+	0.832192i	$0.687086\pi$
$432$	0	0
$433$	−26.8795	−1.29174	−0.645872	−	0.763446i	$-0.723506\pi$
$433$	−26.8795	−1.29174	−0.645872	+	0.763446i	$0.723506\pi$
$434$	10.1465	0.487047
$435$	0	0
$436$	8.63790	0.413680
$437$	−65.2469	−3.12118
$438$	0	0
$439$	−31.6544	−1.51078	−0.755392	−	0.655274i	$-0.772553\pi$
$439$	−31.6544	−1.51078	−0.755392	+	0.655274i	$0.772553\pi$
$440$	−9.46410	−0.451183
$441$	0	0
$442$	0	0
$443$	9.89932	0.470331	0.235166	−	0.971955i	$-0.424437\pi$
$443$	9.89932	0.470331	0.235166	+	0.971955i	$0.424437\pi$
$444$	0	0
$445$	−0.0930509	−0.00441104
$446$	14.3697	0.680425
$447$	0	0
$448$	23.9859	1.13323
$449$	15.0083	0.708284	0.354142	−	0.935192i	$-0.384773\pi$
$449$	15.0083	0.708284	0.354142	+	0.935192i	$0.384773\pi$
$450$	0	0
$451$	10.2203	0.481255
$452$	2.53590	0.119279
$453$	0	0
$454$	−19.2679	−0.904290
$455$	0	0
$456$	0	0
$457$	−13.0304	−0.609537	−0.304768	−	0.952427i	$-0.598579\pi$
$457$	−13.0304	−0.609537	−0.304768	+	0.952427i	$0.598579\pi$
$458$	7.20653	0.336739
$459$	0	0
$460$	6.43659	0.300108
$461$	28.7166	1.33746	0.668732	−	0.743503i	$-0.266837\pi$
$461$	28.7166	1.33746	0.668732	+	0.743503i	$0.266837\pi$
$462$	0	0
$463$	−0.460309	−0.0213924	−0.0106962	−	0.999943i	$-0.503405\pi$
$463$	−0.460309	−0.0213924	−0.0106962	+	0.999943i	$0.503405\pi$
$464$	−13.3648	−0.620445
$465$	0	0
$466$	−0.756195	−0.0350301
$467$	34.0634	1.57627	0.788133	−	0.615505i	$-0.211048\pi$
$467$	34.0634	1.57627	0.788133	+	0.615505i	$0.211048\pi$
$468$	0	0
$469$	−16.8014	−0.775816
$470$	−2.02414	−0.0933668
$471$	0	0
$472$	14.8222	0.682245
$473$	−29.3694	−1.35041
$474$	0	0
$475$	−7.42071	−0.340485
$476$	−4.08063	−0.187036
$477$	0	0
$478$	8.02841	0.367211
$479$	30.9311	1.41328	0.706639	−	0.707574i	$-0.250210\pi$
$479$	30.9311	1.41328	0.706639	+	0.707574i	$0.250210\pi$
$480$	0	0
$481$	0	0
$482$	−30.3333	−1.38165
$483$	0	0
$484$	1.12436	0.0511071
$485$	4.79759	0.217847
$486$	0	0
$487$	−16.5380	−0.749409	−0.374704	−	0.927144i	$-0.622256\pi$
$487$	−16.5380	−0.749409	−0.374704	+	0.927144i	$0.622256\pi$
$488$	19.2826	0.872881
$489$	0	0
$490$	−1.31593	−0.0594475
$491$	−18.6824	−0.843125	−0.421562	−	0.906799i	$-0.638518\pi$
$491$	−18.6824	−0.843125	−0.421562	+	0.906799i	$0.638518\pi$
$492$	0	0
$493$	13.0330	0.586976
$494$	0	0
$495$	0	0
$496$	−6.30551	−0.283126
$497$	29.4361	1.32039
$498$	0	0
$499$	−10.9966	−0.492277	−0.246138	−	0.969235i	$-0.579162\pi$
$499$	−10.9966	−0.492277	−0.246138	+	0.969235i	$0.579162\pi$
$500$	0.732051	0.0327383
$501$	0	0
$502$	13.1161	0.585399
$503$	15.9933	0.713104	0.356552	−	0.934275i	$-0.383952\pi$
$503$	15.9933	0.713104	0.356552	+	0.934275i	$0.383952\pi$
$504$	0	0
$505$	2.79254	0.124267
$506$	−30.4583	−1.35404
$507$	0	0
$508$	10.9603	0.486284
$509$	−1.42775	−0.0632840	−0.0316420	−	0.999499i	$-0.510074\pi$
$509$	−1.42775	−0.0632840	−0.0316420	+	0.999499i	$0.510074\pi$
$510$	0	0
$511$	−31.7537	−1.40470
$512$	−20.1069	−0.888608
$513$	0	0
$514$	8.19615	0.361517
$515$	−8.35395	−0.368119
$516$	0	0
$517$	−5.53006	−0.243212
$518$	−26.5576	−1.16687
$519$	0	0
$520$	0	0
$521$	−11.8172	−0.517719	−0.258859	−	0.965915i	$-0.583347\pi$
$521$	−11.8172	−0.517719	−0.258859	+	0.965915i	$0.583347\pi$
$522$	0	0
$523$	28.6126	1.25114	0.625571	−	0.780167i	$-0.284866\pi$
$523$	28.6126	1.25114	0.625571	+	0.780167i	$0.284866\pi$
$524$	−7.74736	−0.338445
$525$	0	0
$526$	−26.9434	−1.17479
$527$	6.14896	0.267853
$528$	0	0
$529$	54.3088	2.36125
$530$	12.0000	0.521247
$531$	0	0
$532$	15.5261	0.673140
$533$	0	0
$534$	0	0
$535$	6.35059	0.274560
$536$	18.0846	0.781137
$537$	0	0
$538$	−4.93367	−0.212706
$539$	−3.59518	−0.154855
$540$	0	0
$541$	33.9315	1.45883	0.729415	−	0.684072i	$-0.239792\pi$
$541$	33.9315	1.45883	0.729415	+	0.684072i	$0.239792\pi$
$542$	2.80185	0.120350
$543$	0	0
$544$	7.60770	0.326177
$545$	11.7996	0.505439
$546$	0	0
$547$	−0.0276116	−0.00118059	−0.000590294	−	1.00000i	$-0.500188\pi$
$547$	−0.0276116	−0.00118059	−0.000590294		1.00000i	$0.500188\pi$
$548$	6.37363	0.272268
$549$	0	0
$550$	−3.46410	−0.147710
$551$	−49.5881	−2.11252
$552$	0	0
$553$	−31.7157	−1.34869
$554$	−24.4417	−1.03843
$555$	0	0
$556$	8.53343	0.361898
$557$	−40.3651	−1.71032	−0.855162	−	0.518360i	$-0.826543\pi$
$557$	−40.3651	−1.71032	−0.855162	+	0.518360i	$0.826543\pi$
$558$	0	0
$559$	0	0
$560$	5.71617	0.241552
$561$	0	0
$562$	33.4736	1.41200
$563$	−14.1210	−0.595129	−0.297564	−	0.954702i	$-0.596174\pi$
$563$	−14.1210	−0.595129	−0.297564	+	0.954702i	$0.596174\pi$
$564$	0	0
$565$	3.46410	0.145736
$566$	31.5284	1.32524
$567$	0	0
$568$	−31.6844	−1.32945
$569$	31.0316	1.30091	0.650456	−	0.759544i	$-0.274578\pi$
$569$	31.0316	1.30091	0.650456	+	0.759544i	$0.274578\pi$
$570$	0	0
$571$	−29.6336	−1.24013	−0.620065	−	0.784551i	$-0.712894\pi$
$571$	−29.6336	−1.24013	−0.620065	+	0.784551i	$0.712894\pi$
$572$	0	0
$573$	0	0
$574$	−10.6918	−0.446266
$575$	8.79254	0.366674
$576$	0	0
$577$	17.7788	0.740140	0.370070	−	0.929004i	$-0.379334\pi$
$577$	17.7788	0.740140	0.370070	+	0.929004i	$0.379334\pi$
$578$	−14.8593	−0.618065
$579$	0	0
$580$	4.89185	0.203123
$581$	19.9410	0.827294
$582$	0	0
$583$	32.7846	1.35780
$584$	34.1790	1.41434
$585$	0	0
$586$	19.0959	0.788846
$587$	−5.56204	−0.229570	−0.114785	−	0.993390i	$-0.536618\pi$
$587$	−5.56204	−0.229570	−0.114785	+	0.993390i	$0.536618\pi$
$588$	0	0
$589$	−23.3957	−0.964002
$590$	5.42529	0.223356
$591$	0	0
$592$	16.5041	0.678316
$593$	33.4290	1.37276	0.686382	−	0.727241i	$-0.259198\pi$
$593$	33.4290	1.37276	0.686382	+	0.727241i	$0.259198\pi$
$594$	0	0
$595$	−5.57425	−0.228522
$596$	−4.99574	−0.204634
$597$	0	0
$598$	0	0
$599$	8.14349	0.332734	0.166367	−	0.986064i	$-0.446796\pi$
$599$	8.14349	0.332734	0.166367	+	0.986064i	$0.446796\pi$
$600$	0	0
$601$	23.1176	0.942987	0.471494	−	0.881869i	$-0.343715\pi$
$601$	23.1176	0.942987	0.471494	+	0.881869i	$0.343715\pi$
$602$	30.7242	1.25223
$603$	0	0
$604$	−9.28316	−0.377726
$605$	1.53590	0.0624431
$606$	0	0
$607$	6.96702	0.282783	0.141391	−	0.989954i	$-0.454842\pi$
$607$	6.96702	0.282783	0.141391	+	0.989954i	$0.454842\pi$
$608$	−28.9459	−1.17391
$609$	0	0
$610$	7.05791	0.285767
$611$	0	0
$612$	0	0
$613$	−34.3851	−1.38880	−0.694401	−	0.719589i	$-0.744330\pi$
$613$	−34.3851	−1.38880	−0.694401	+	0.719589i	$0.744330\pi$
$614$	−17.5742	−0.709239
$615$	0	0
$616$	27.0492	1.08984
$617$	15.6089	0.628392	0.314196	−	0.949358i	$-0.398265\pi$
$617$	15.6089	0.628392	0.314196	+	0.949358i	$0.398265\pi$
$618$	0	0
$619$	−16.0626	−0.645610	−0.322805	−	0.946465i	$-0.604626\pi$
$619$	−16.0626	−0.645610	−0.322805	+	0.946465i	$0.604626\pi$
$620$	2.30798	0.0926906
$621$	0	0
$622$	28.1441	1.12848
$623$	0.265947	0.0106550
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.30751	0.0522588
$627$	0	0
$628$	−5.84420	−0.233209
$629$	−16.0944	−0.641725
$630$	0	0
$631$	−36.6719	−1.45988	−0.729942	−	0.683509i	$-0.760453\pi$
$631$	−36.6719	−1.45988	−0.729942	+	0.683509i	$0.760453\pi$
$632$	34.1381	1.35794
$633$	0	0
$634$	9.71281	0.385745
$635$	14.9720	0.594147
$636$	0	0
$637$	0	0
$638$	−23.1485	−0.916458
$639$	0	0
$640$	−1.64863	−0.0651677
$641$	−26.6422	−1.05230	−0.526152	−	0.850390i	$-0.676366\pi$
$641$	−26.6422	−1.05230	−0.526152	+	0.850390i	$0.676366\pi$
$642$	0	0
$643$	2.06096	0.0812762	0.0406381	−	0.999174i	$-0.487061\pi$
$643$	2.06096	0.0812762	0.0406381	+	0.999174i	$0.487061\pi$
$644$	−18.3963	−0.724916
$645$	0	0
$646$	−16.2970	−0.641197
$647$	15.5271	0.610432	0.305216	−	0.952283i	$-0.401271\pi$
$647$	15.5271	0.610432	0.305216	+	0.952283i	$0.401271\pi$
$648$	0	0
$649$	14.8222	0.581821
$650$	0	0
$651$	0	0
$652$	9.02047	0.353269
$653$	−40.8736	−1.59951	−0.799754	−	0.600328i	$-0.795037\pi$
$653$	−40.8736	−1.59951	−0.799754	+	0.600328i	$0.795037\pi$
$654$	0	0
$655$	−10.5831	−0.413515
$656$	6.64437	0.259419
$657$	0	0
$658$	5.78517	0.225530
$659$	−1.83473	−0.0714708	−0.0357354	−	0.999361i	$-0.511377\pi$
$659$	−1.83473	−0.0714708	−0.0357354	+	0.999361i	$0.511377\pi$
$660$	0	0
$661$	−17.3682	−0.675543	−0.337772	−	0.941228i	$-0.609673\pi$
$661$	−17.3682	−0.675543	−0.337772	+	0.941228i	$0.609673\pi$
$662$	28.8519	1.12136
$663$	0	0
$664$	−21.4641	−0.832969
$665$	21.2090	0.822449
$666$	0	0
$667$	58.7553	2.27501
$668$	16.0624	0.621472
$669$	0	0
$670$	6.61944	0.255731
$671$	19.2826	0.744396
$672$	0	0
$673$	−9.50792	−0.366503	−0.183252	−	0.983066i	$-0.558662\pi$
$673$	−9.50792	−0.366503	−0.183252	+	0.983066i	$0.558662\pi$
$674$	22.7200	0.875141
$675$	0	0
$676$	0	0
$677$	−20.7375	−0.797008	−0.398504	−	0.917167i	$-0.630470\pi$
$677$	−20.7375	−0.797008	−0.398504	+	0.917167i	$0.630470\pi$
$678$	0	0
$679$	−13.7119	−0.526215
$680$	6.00000	0.230089
$681$	0	0
$682$	−10.9215	−0.418205
$683$	−16.6879	−0.638543	−0.319272	−	0.947663i	$-0.603438\pi$
$683$	−16.6879	−0.638543	−0.319272	+	0.947663i	$0.603438\pi$
$684$	0	0
$685$	8.70654	0.332660
$686$	−18.7670	−0.716529
$687$	0	0
$688$	−19.0935	−0.727932
$689$	0	0
$690$	0	0
$691$	22.6459	0.861490	0.430745	−	0.902474i	$-0.358251\pi$
$691$	22.6459	0.861490	0.430745	+	0.902474i	$0.358251\pi$
$692$	11.0178	0.418833
$693$	0	0
$694$	−27.2429	−1.03413
$695$	11.6569	0.442171
$696$	0	0
$697$	−6.47941	−0.245425
$698$	−28.9122	−1.09434
$699$	0	0
$700$	−2.09226	−0.0790801
$701$	−9.33818	−0.352698	−0.176349	−	0.984328i	$-0.556429\pi$
$701$	−9.33818	−0.352698	−0.176349	+	0.984328i	$0.556429\pi$
$702$	0	0
$703$	61.2361	2.30956
$704$	−25.8179	−0.973049
$705$	0	0
$706$	8.83383	0.332465
$707$	−7.98133	−0.300169
$708$	0	0
$709$	28.4424	1.06818	0.534088	−	0.845429i	$-0.320655\pi$
$709$	28.4424	1.06818	0.534088	+	0.845429i	$0.320655\pi$
$710$	−11.5973	−0.435239
$711$	0	0
$712$	−0.286260	−0.0107280
$713$	27.7207	1.03815
$714$	0	0
$715$	0	0
$716$	−0.351951	−0.0131530
$717$	0	0
$718$	−22.1992	−0.828465
$719$	−31.8969	−1.18955	−0.594776	−	0.803891i	$-0.702759\pi$
$719$	−31.8969	−1.18955	−0.594776	+	0.803891i	$0.702759\pi$
$720$	0	0
$721$	23.8763	0.889200
$722$	40.6125	1.51144
$723$	0	0
$724$	−1.36079	−0.0505733
$725$	6.68240	0.248178
$726$	0	0
$727$	−4.68029	−0.173583	−0.0867913	−	0.996227i	$-0.527661\pi$
$727$	−4.68029	−0.173583	−0.0867913	+	0.996227i	$0.527661\pi$
$728$	0	0
$729$	0	0
$730$	12.5104	0.463030
$731$	18.6194	0.688665
$732$	0	0
$733$	−14.1306	−0.521926	−0.260963	−	0.965349i	$-0.584040\pi$
$733$	−14.1306	−0.521926	−0.260963	+	0.965349i	$0.584040\pi$
$734$	−12.5847	−0.464508
$735$	0	0
$736$	34.2970	1.26420
$737$	18.0846	0.666156
$738$	0	0
$739$	11.8882	0.437314	0.218657	−	0.975802i	$-0.429832\pi$
$739$	11.8882	0.437314	0.218657	+	0.975802i	$0.429832\pi$
$740$	−6.04093	−0.222069
$741$	0	0
$742$	−34.2970	−1.25908
$743$	−24.0100	−0.880840	−0.440420	−	0.897792i	$-0.645170\pi$
$743$	−24.0100	−0.880840	−0.440420	+	0.897792i	$0.645170\pi$
$744$	0	0
$745$	−6.82431	−0.250023
$746$	−9.85473	−0.360807
$747$	0	0
$748$	4.39230	0.160599
$749$	−18.1505	−0.663205
$750$	0	0
$751$	15.8698	0.579098	0.289549	−	0.957163i	$-0.406495\pi$
$751$	15.8698	0.579098	0.289549	+	0.957163i	$0.406495\pi$
$752$	−3.59518	−0.131103
$753$	0	0
$754$	0	0
$755$	−12.6810	−0.461510
$756$	0	0
$757$	36.1638	1.31440	0.657198	−	0.753718i	$-0.271742\pi$
$757$	36.1638	1.31440	0.657198	+	0.753718i	$0.271742\pi$
$758$	−21.1352	−0.767665
$759$	0	0
$760$	−22.8289	−0.828091
$761$	14.6181	0.529906	0.264953	−	0.964261i	$-0.414644\pi$
$761$	14.6181	0.529906	0.264953	+	0.964261i	$0.414644\pi$
$762$	0	0
$763$	−33.7242	−1.22090
$764$	6.38256	0.230913
$765$	0	0
$766$	6.84029	0.247150
$767$	0	0
$768$	0	0
$769$	−18.3590	−0.662042	−0.331021	−	0.943623i	$-0.607393\pi$
$769$	−18.3590	−0.662042	−0.331021	+	0.943623i	$0.607393\pi$
$770$	9.90069	0.356796
$771$	0	0
$772$	4.72412	0.170025
$773$	−8.53039	−0.306817	−0.153408	−	0.988163i	$-0.549025\pi$
$773$	−8.53039	−0.306817	−0.153408	+	0.988163i	$0.549025\pi$
$774$	0	0
$775$	3.15276	0.113250
$776$	14.7592	0.529824
$777$	0	0
$778$	10.8839	0.390207
$779$	24.6530	0.883284
$780$	0	0
$781$	−31.6844	−1.13376
$782$	19.3098	0.690516
$783$	0	0
$784$	−2.33728	−0.0834743
$785$	−7.98333	−0.284937
$786$	0	0
$787$	−3.42575	−0.122115	−0.0610574	−	0.998134i	$-0.519447\pi$
$787$	−3.42575	−0.122115	−0.0610574	+	0.998134i	$0.519447\pi$
$788$	8.48668	0.302326
$789$	0	0
$790$	12.4954	0.444567
$791$	−9.90069	−0.352028
$792$	0	0
$793$	0	0
$794$	27.3982	0.972327
$795$	0	0
$796$	−0.586652	−0.0207933
$797$	−28.1009	−0.995387	−0.497693	−	0.867353i	$-0.665819\pi$
$797$	−28.1009	−0.995387	−0.497693	+	0.867353i	$0.665819\pi$
$798$	0	0
$799$	3.50592	0.124031
$800$	3.90069	0.137910
$801$	0	0
$802$	−10.0177	−0.353736
$803$	34.1790	1.20615
$804$	0	0
$805$	−25.1298	−0.885710
$806$	0	0
$807$	0	0
$808$	8.59092	0.302228
$809$	−36.1693	−1.27164	−0.635822	−	0.771836i	$-0.719339\pi$
$809$	−36.1693	−1.27164	−0.635822	+	0.771836i	$0.719339\pi$
$810$	0	0
$811$	13.9825	0.490993	0.245497	−	0.969397i	$-0.421049\pi$
$811$	13.9825	0.490993	0.245497	+	0.969397i	$0.421049\pi$
$812$	−13.9813	−0.490648
$813$	0	0
$814$	28.5860	1.00194
$815$	12.3222	0.431627
$816$	0	0
$817$	−70.8435	−2.47850
$818$	−31.8540	−1.11375
$819$	0	0
$820$	−2.43201	−0.0849294
$821$	−4.23955	−0.147961	−0.0739806	−	0.997260i	$-0.523570\pi$
$821$	−4.23955	−0.147961	−0.0739806	+	0.997260i	$0.523570\pi$
$822$	0	0
$823$	33.6401	1.17262	0.586310	−	0.810087i	$-0.300580\pi$
$823$	33.6401	1.17262	0.586310	+	0.810087i	$0.300580\pi$
$824$	−25.6999	−0.895299
$825$	0	0
$826$	−15.5059	−0.539520
$827$	6.94609	0.241539	0.120770	−	0.992681i	$-0.461464\pi$
$827$	6.94609	0.241539	0.120770	+	0.992681i	$0.461464\pi$
$828$	0	0
$829$	13.3654	0.464201	0.232100	−	0.972692i	$-0.425440\pi$
$829$	13.3654	0.464201	0.232100	+	0.972692i	$0.425440\pi$
$830$	−7.85641	−0.272700
$831$	0	0
$832$	0	0
$833$	2.27925	0.0789714
$834$	0	0
$835$	21.9416	0.759321
$836$	−16.7119	−0.577994
$837$	0	0
$838$	−32.5364	−1.12395
$839$	−20.0721	−0.692967	−0.346483	−	0.938056i	$-0.612624\pi$
$839$	−20.0721	−0.692967	−0.346483	+	0.938056i	$0.612624\pi$
$840$	0	0
$841$	15.6544	0.539808
$842$	−13.5275	−0.466188
$843$	0	0
$844$	6.07023	0.208946
$845$	0	0
$846$	0	0
$847$	−4.38973	−0.150833
$848$	21.3138	0.731918
$849$	0	0
$850$	2.19615	0.0753274
$851$	−72.5566	−2.48721
$852$	0	0
$853$	−54.6353	−1.87068	−0.935338	−	0.353755i	$-0.884905\pi$
$853$	−54.6353	−1.87068	−0.935338	+	0.353755i	$0.884905\pi$
$854$	−20.1721	−0.690275
$855$	0	0
$856$	19.5368	0.667754
$857$	3.66436	0.125172	0.0625860	−	0.998040i	$-0.480065\pi$
$857$	3.66436	0.125172	0.0625860	+	0.998040i	$0.480065\pi$
$858$	0	0
$859$	28.1460	0.960330	0.480165	−	0.877178i	$-0.340577\pi$
$859$	28.1460	0.960330	0.480165	+	0.877178i	$0.340577\pi$
$860$	6.98870	0.238313
$861$	0	0
$862$	−25.9246	−0.882994
$863$	−50.4623	−1.71776	−0.858878	−	0.512180i	$-0.828838\pi$
$863$	−50.4623	−1.71776	−0.858878	+	0.512180i	$0.828838\pi$
$864$	0	0
$865$	15.0506	0.511734
$866$	−30.2671	−1.02852
$867$	0	0
$868$	−6.59639	−0.223896
$869$	34.1381	1.15806
$870$	0	0
$871$	0	0
$872$	36.3000	1.22927
$873$	0	0
$874$	−73.4701	−2.48516
$875$	−2.85808	−0.0966209
$876$	0	0
$877$	23.2087	0.783702	0.391851	−	0.920029i	$-0.371835\pi$
$877$	23.2087	0.783702	0.391851	+	0.920029i	$0.371835\pi$
$878$	−35.6439	−1.20292
$879$	0	0
$880$	−6.15276	−0.207409
$881$	17.1152	0.576624	0.288312	−	0.957536i	$-0.406906\pi$
$881$	17.1152	0.576624	0.288312	+	0.957536i	$0.406906\pi$
$882$	0	0
$883$	−21.3589	−0.718783	−0.359391	−	0.933187i	$-0.617016\pi$
$883$	−21.3589	−0.718783	−0.359391	+	0.933187i	$0.617016\pi$
$884$	0	0
$885$	0	0
$886$	11.1470	0.374489
$887$	−34.3200	−1.15235	−0.576177	−	0.817325i	$-0.695456\pi$
$887$	−34.3200	−1.15235	−0.576177	+	0.817325i	$0.695456\pi$
$888$	0	0
$889$	−42.7913	−1.43517
$890$	−0.104778	−0.00351218
$891$	0	0
$892$	−9.34196	−0.312792
$893$	−13.3394	−0.446385
$894$	0	0
$895$	−0.480775	−0.0160705
$896$	4.71191	0.157414
$897$	0	0
$898$	16.8998	0.563953
$899$	21.0680	0.702656
$900$	0	0
$901$	−20.7846	−0.692436
$902$	11.5084	0.383187
$903$	0	0
$904$	10.6569	0.354443
$905$	−1.85887	−0.0617910
$906$	0	0
$907$	58.9083	1.95602	0.978010	−	0.208560i	$-0.0668777\pi$
$907$	58.9083	1.95602	0.978010	+	0.208560i	$0.0668777\pi$
$908$	12.5264	0.415703
$909$	0	0
$910$	0	0
$911$	55.5007	1.83882	0.919410	−	0.393299i	$-0.128666\pi$
$911$	55.5007	1.83882	0.919410	+	0.393299i	$0.128666\pi$
$912$	0	0
$913$	−21.4641	−0.710358
$914$	−14.6727	−0.485328
$915$	0	0
$916$	−4.68507	−0.154799
$917$	30.2473	0.998855
$918$	0	0
$919$	54.7311	1.80541	0.902707	−	0.430257i	$-0.141577\pi$
$919$	54.7311	1.80541	0.902707	+	0.430257i	$0.141577\pi$
$920$	27.0492	0.891785
$921$	0	0
$922$	32.3358	1.06492
$923$	0	0
$924$	0	0
$925$	−8.25207	−0.271326
$926$	−0.518323	−0.0170332
$927$	0	0
$928$	26.0660	0.855657
$929$	17.1818	0.563718	0.281859	−	0.959456i	$-0.409049\pi$
$929$	17.1818	0.563718	0.281859	+	0.959456i	$0.409049\pi$
$930$	0	0
$931$	−8.67213	−0.284218
$932$	0.491614	0.0161033
$933$	0	0
$934$	38.3565	1.25506
$935$	6.00000	0.196221
$936$	0	0
$937$	39.6401	1.29499	0.647493	−	0.762071i	$-0.275817\pi$
$937$	39.6401	1.29499	0.647493	+	0.762071i	$0.275817\pi$
$938$	−18.9189	−0.617724
$939$	0	0
$940$	1.31593	0.0429208
$941$	33.3796	1.08815	0.544073	−	0.839038i	$-0.316882\pi$
$941$	33.3796	1.08815	0.544073	+	0.839038i	$0.316882\pi$
$942$	0	0
$943$	−29.2105	−0.951223
$944$	9.63611	0.313629
$945$	0	0
$946$	−33.0709	−1.07523
$947$	−20.5599	−0.668108	−0.334054	−	0.942554i	$-0.608417\pi$
$947$	−20.5599	−0.668108	−0.334054	+	0.942554i	$0.608417\pi$
$948$	0	0
$949$	0	0
$950$	−8.35596	−0.271103
$951$	0	0
$952$	−17.1485	−0.555786
$953$	54.8753	1.77758	0.888792	−	0.458311i	$-0.151545\pi$
$953$	54.8753	1.77758	0.888792	+	0.458311i	$0.151545\pi$
$954$	0	0
$955$	8.71875	0.282132
$956$	−5.21939	−0.168807
$957$	0	0
$958$	34.8294	1.12529
$959$	−24.8840	−0.803547
$960$	0	0
$961$	−21.0601	−0.679359
$962$	0	0
$963$	0	0
$964$	19.7202	0.635144
$965$	6.45326	0.207738
$966$	0	0
$967$	−10.9215	−0.351211	−0.175605	−	0.984461i	$-0.556188\pi$
$967$	−10.9215	−0.351211	−0.175605	+	0.984461i	$0.556188\pi$
$968$	4.72500	0.151867
$969$	0	0
$970$	5.40224	0.173456
$971$	−31.4711	−1.00996	−0.504978	−	0.863132i	$-0.668499\pi$
$971$	−31.4711	−1.00996	−0.504978	+	0.863132i	$0.668499\pi$
$972$	0	0
$973$	−33.3164	−1.06807
$974$	−18.6223	−0.596698
$975$	0	0
$976$	12.5359	0.401264
$977$	31.5533	1.00948	0.504740	−	0.863271i	$-0.331588\pi$
$977$	31.5533	1.00948	0.504740	+	0.863271i	$0.331588\pi$
$978$	0	0
$979$	−0.286260	−0.00914890
$980$	0.855504	0.0273281
$981$	0	0
$982$	−21.0370	−0.671317
$983$	43.6214	1.39131	0.695654	−	0.718377i	$-0.255115\pi$
$983$	43.6214	1.39131	0.695654	+	0.718377i	$0.255115\pi$
$984$	0	0
$985$	11.5930	0.369384
$986$	14.6756	0.467365
$987$	0	0
$988$	0	0
$989$	83.9401	2.66914
$990$	0	0
$991$	17.9948	0.571624	0.285812	−	0.958286i	$-0.407737\pi$
$991$	17.9948	0.571624	0.285812	+	0.958286i	$0.407737\pi$
$992$	12.2979	0.390460
$993$	0	0
$994$	33.1460	1.05133
$995$	−0.801382	−0.0254055
$996$	0	0
$997$	−24.2091	−0.766710	−0.383355	−	0.923601i	$-0.625231\pi$
$997$	−24.2091	−0.766710	−0.383355	+	0.923601i	$0.625231\pi$
$998$	−12.3826	−0.391963
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.ch.1.3		4
3.2	odd	2		2535.2.a.bk.1.2		4
13.2	odd	12		585.2.bu.d.316.3		8
13.7	odd	12		585.2.bu.d.361.3		8
13.12	even	2		7605.2.a.ci.1.2		4
39.2	even	12		195.2.bb.b.121.2	✓	8
39.20	even	12		195.2.bb.b.166.2	yes	8
39.38	odd	2		2535.2.a.bj.1.3		4
195.2	odd	12		975.2.w.i.199.3		8
195.59	even	12		975.2.bc.j.751.3		8
195.98	odd	12		975.2.w.i.49.3		8
195.119	even	12		975.2.bc.j.901.3		8
195.137	odd	12		975.2.w.h.49.2		8
195.158	odd	12		975.2.w.h.199.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.b.121.2	✓	8	39.2	even	12
195.2.bb.b.166.2	yes	8	39.20	even	12
585.2.bu.d.316.3		8	13.2	odd	12
585.2.bu.d.361.3		8	13.7	odd	12
975.2.w.h.49.2		8	195.137	odd	12
975.2.w.h.199.2		8	195.158	odd	12
975.2.w.i.49.3		8	195.98	odd	12
975.2.w.i.199.3		8	195.2	odd	12
975.2.bc.j.751.3		8	195.59	even	12
975.2.bc.j.901.3		8	195.119	even	12
2535.2.a.bj.1.3		4	39.38	odd	2
2535.2.a.bk.1.2		4	3.2	odd	2
7605.2.a.ch.1.3		4	1.1	even	1	trivial
7605.2.a.ci.1.2		4	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.12603 q^{2} -0.732051 q^{4} -1.00000 q^{5} +2.85808 q^{7} -3.07638 q^{8} -1.12603 q^{10} -3.07638 q^{11} +3.21829 q^{14} -2.00000 q^{16} +1.95035 q^{17} -7.42071 q^{19} +0.732051 q^{20} -3.46410 q^{22} +8.79254 q^{23} +1.00000 q^{25} -2.09226 q^{28} +6.68240 q^{29} +3.15276 q^{31} +3.90069 q^{32} +2.19615 q^{34} -2.85808 q^{35} -8.25207 q^{37} -8.35596 q^{38} +3.07638 q^{40} -3.32218 q^{41} +9.54674 q^{43} +2.25207 q^{44} +9.90069 q^{46} +1.79759 q^{47} +1.16864 q^{49} +1.12603 q^{50} -10.6569 q^{53} +3.07638 q^{55} -8.79254 q^{56} +7.52460 q^{58} -4.81805 q^{59} -6.26795 q^{61} +3.55011 q^{62} +8.39230 q^{64} -5.87855 q^{67} -1.42775 q^{68} -3.21829 q^{70} +10.2993 q^{71} -11.1101 q^{73} -9.29209 q^{74} +5.43233 q^{76} -8.79254 q^{77} -11.0968 q^{79} +2.00000 q^{80} -3.74089 q^{82} +6.97707 q^{83} -1.95035 q^{85} +10.7499 q^{86} +9.46410 q^{88} +0.0930509 q^{89} -6.43659 q^{92} +2.02414 q^{94} +7.42071 q^{95} -4.79759 q^{97} +1.31593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{5} + 12 q^{14} - 8 q^{16} - 12 q^{19} - 4 q^{20} + 4 q^{25} - 12 q^{28} + 12 q^{29} - 12 q^{31} - 12 q^{34} - 24 q^{37} + 12 q^{41} + 16 q^{43} + 24 q^{46} + 24 q^{47} - 4 q^{49} - 12 q^{58}+ \cdots - 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^5 + 12 * q^14 - 8 * q^16 - 12 * q^19 - 4 * q^20 + 4 * q^25 - 12 * q^28 + 12 * q^29 - 12 * q^31 - 12 * q^34 - 24 * q^37 + 12 * q^41 + 16 * q^43 + 24 * q^46 + 24 * q^47 - 4 * q^49 - 12 * q^58 - 12 * q^59 - 32 * q^61 - 8 * q^64 + 12 * q^67 - 12 * q^70 - 12 * q^71 - 24 * q^73 - 24 * q^74 - 24 * q^76 - 8 * q^79 + 8 * q^80 - 12 * q^82 + 12 * q^86 + 24 * q^88 + 12 * q^89 - 24 * q^92 - 12 * q^94 + 12 * q^95 - 36 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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