LMFDB - Embedded newform 7623.2.a.cc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34292 q^{2} +3.48929 q^{4} +0.146365 q^{5} -1.00000 q^{7} +3.48929 q^{8} +O(q^{10})$	$q+2.34292 q^{2} +3.48929 q^{4} +0.146365 q^{5} -1.00000 q^{7} +3.48929 q^{8} +0.342923 q^{10} -4.34292 q^{13} -2.34292 q^{14} +1.19656 q^{16} -0.146365 q^{17} +1.83221 q^{19} +0.510711 q^{20} -8.81079 q^{23} -4.97858 q^{25} -10.1751 q^{26} -3.48929 q^{28} +4.34292 q^{29} +0.292731 q^{31} -4.17513 q^{32} -0.342923 q^{34} -0.146365 q^{35} -3.48929 q^{37} +4.29273 q^{38} +0.510711 q^{40} +2.80344 q^{41} +7.86098 q^{43} -20.6430 q^{46} +0.949808 q^{47} +1.00000 q^{49} -11.6644 q^{50} -15.1537 q^{52} +4.51071 q^{53} -3.48929 q^{56} +10.1751 q^{58} -8.02877 q^{59} -5.43910 q^{61} +0.685846 q^{62} -12.1751 q^{64} -0.635654 q^{65} -7.76060 q^{67} -0.510711 q^{68} -0.342923 q^{70} +3.53948 q^{71} -16.1825 q^{73} -8.17513 q^{74} +6.39312 q^{76} -13.0790 q^{79} +0.175135 q^{80} +6.56825 q^{82} -12.9070 q^{83} -0.0214229 q^{85} +18.4177 q^{86} +9.81079 q^{89} +4.34292 q^{91} -30.7434 q^{92} +2.22533 q^{94} +0.268173 q^{95} -17.4219 q^{97} +2.34292 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - q^5 - 3 * q^7 + 3 * q^8	$ 3 q + q^{2} + 3 q^{4} - q^{5} - 3 q^{7} + 3 q^{8} - 5 q^{10} - 7 q^{13} - q^{14} - q^{16} + q^{17} - 8 q^{19} + 9 q^{20} + 2 q^{23} - 11 q^{26} - 3 q^{28} + 7 q^{29} - 2 q^{31} + 7 q^{32} + 5 q^{34} + q^{35} - 3 q^{37} + 10 q^{38} + 9 q^{40} + 13 q^{41} - 8 q^{43} - 20 q^{46} + 6 q^{47} + 3 q^{49} - 8 q^{50} - 11 q^{52} + 21 q^{53} - 3 q^{56} + 11 q^{58} - 6 q^{59} - 12 q^{61} - 10 q^{62} - 17 q^{64} + 7 q^{65} + 2 q^{67} - 9 q^{68} + 5 q^{70} + 4 q^{73} - 5 q^{74} + 10 q^{76} - 18 q^{79} - 19 q^{80} - 9 q^{82} - 12 q^{83} - 15 q^{85} + 36 q^{86} + q^{89} + 7 q^{91} - 44 q^{92} - 16 q^{94} + 8 q^{95} - 25 q^{97} + q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - q^5 - 3 * q^7 + 3 * q^8 - 5 * q^10 - 7 * q^13 - q^14 - q^16 + q^17 - 8 * q^19 + 9 * q^20 + 2 * q^23 - 11 * q^26 - 3 * q^28 + 7 * q^29 - 2 * q^31 + 7 * q^32 + 5 * q^34 + q^35 - 3 * q^37 + 10 * q^38 + 9 * q^40 + 13 * q^41 - 8 * q^43 - 20 * q^46 + 6 * q^47 + 3 * q^49 - 8 * q^50 - 11 * q^52 + 21 * q^53 - 3 * q^56 + 11 * q^58 - 6 * q^59 - 12 * q^61 - 10 * q^62 - 17 * q^64 + 7 * q^65 + 2 * q^67 - 9 * q^68 + 5 * q^70 + 4 * q^73 - 5 * q^74 + 10 * q^76 - 18 * q^79 - 19 * q^80 - 9 * q^82 - 12 * q^83 - 15 * q^85 + 36 * q^86 + q^89 + 7 * q^91 - 44 * q^92 - 16 * q^94 + 8 * q^95 - 25 * q^97 + q^98

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 $

$2$	2.34292	1.65670	0.828348	−	0.560213i	$-0.189281\pi$
$2$	2.34292	1.65670	0.828348	+	0.560213i	$0.189281\pi$
$3$	0	0
$3$	0	0
$4$	3.48929	1.74464
$5$	0.146365	0.0654566	0.0327283	−	0.999464i	$-0.489580\pi$
$5$	0.146365	0.0654566	0.0327283	+	0.999464i	$0.489580\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.48929	1.23365
$9$	0	0
$10$	0.342923	0.108442
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.34292	−1.20451	−0.602255	−	0.798304i	$-0.705731\pi$
$13$	−4.34292	−1.20451	−0.602255	+	0.798304i	$0.705731\pi$
$14$	−2.34292	−0.626173
$15$	0	0
$16$	1.19656	0.299139
$17$	−0.146365	−0.0354988	−0.0177494	−	0.999842i	$-0.505650\pi$
$17$	−0.146365	−0.0354988	−0.0177494	+	0.999842i	$0.505650\pi$
$18$	0	0
$19$	1.83221	0.420338	0.210169	−	0.977665i	$-0.432599\pi$
$19$	1.83221	0.420338	0.210169	+	0.977665i	$0.432599\pi$
$20$	0.510711	0.114199
$21$	0	0
$22$	0	0
$23$	−8.81079	−1.83718	−0.918588	−	0.395216i	$-0.870670\pi$
$23$	−8.81079	−1.83718	−0.918588	+	0.395216i	$0.870670\pi$
$24$	0	0
$25$	−4.97858	−0.995715
$26$	−10.1751	−1.99551
$27$	0	0
$28$	−3.48929	−0.659414
$29$	4.34292	0.806461	0.403230	−	0.915099i	$-0.367887\pi$
$29$	4.34292	0.806461	0.403230	+	0.915099i	$0.367887\pi$
$30$	0	0
$31$	0.292731	0.0525760	0.0262880	−	0.999654i	$-0.491631\pi$
$31$	0.292731	0.0525760	0.0262880	+	0.999654i	$0.491631\pi$
$32$	−4.17513	−0.738067
$33$	0	0
$34$	−0.342923	−0.0588108
$35$	−0.146365	−0.0247403
$36$	0	0
$37$	−3.48929	−0.573636	−0.286818	−	0.957985i	$-0.592597\pi$
$37$	−3.48929	−0.573636	−0.286818	+	0.957985i	$0.592597\pi$
$38$	4.29273	0.696373
$39$	0	0
$40$	0.510711	0.0807506
$41$	2.80344	0.437824	0.218912	−	0.975745i	$-0.429749\pi$
$41$	2.80344	0.437824	0.218912	+	0.975745i	$0.429749\pi$
$42$	0	0
$43$	7.86098	1.19879	0.599394	−	0.800454i	$-0.295408\pi$
$43$	7.86098	1.19879	0.599394	+	0.800454i	$0.295408\pi$
$44$	0	0
$45$	0	0
$46$	−20.6430	−3.04364
$47$	0.949808	0.138544	0.0692719	−	0.997598i	$-0.477932\pi$
$47$	0.949808	0.138544	0.0692719	+	0.997598i	$0.477932\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.6644	−1.64960
$51$	0	0
$52$	−15.1537	−2.10144
$53$	4.51071	0.619594	0.309797	−	0.950803i	$-0.399739\pi$
$53$	4.51071	0.619594	0.309797	+	0.950803i	$0.399739\pi$
$54$	0	0
$55$	0	0
$56$	−3.48929	−0.466276
$57$	0	0
$58$	10.1751	1.33606
$59$	−8.02877	−1.04526	−0.522628	−	0.852561i	$-0.675048\pi$
$59$	−8.02877	−1.04526	−0.522628	+	0.852561i	$0.675048\pi$
$60$	0	0
$61$	−5.43910	−0.696405	−0.348202	−	0.937419i	$-0.613208\pi$
$61$	−5.43910	−0.696405	−0.348202	+	0.937419i	$0.613208\pi$
$62$	0.685846	0.0871026
$63$	0	0
$64$	−12.1751	−1.52189
$65$	−0.635654	−0.0788432
$66$	0	0
$67$	−7.76060	−0.948108	−0.474054	−	0.880496i	$-0.657210\pi$
$67$	−7.76060	−0.948108	−0.474054	+	0.880496i	$0.657210\pi$
$68$	−0.510711	−0.0619329
$69$	0	0
$70$	−0.342923	−0.0409871
$71$	3.53948	0.420059	0.210030	−	0.977695i	$-0.432644\pi$
$71$	3.53948	0.420059	0.210030	+	0.977695i	$0.432644\pi$
$72$	0	0
$73$	−16.1825	−1.89402	−0.947008	−	0.321210i	$-0.895911\pi$
$73$	−16.1825	−1.89402	−0.947008	+	0.321210i	$0.895911\pi$
$74$	−8.17513	−0.950340
$75$	0	0
$76$	6.39312	0.733341
$77$	0	0
$78$	0	0
$79$	−13.0790	−1.47150	−0.735749	−	0.677254i	$-0.763170\pi$
$79$	−13.0790	−1.47150	−0.735749	+	0.677254i	$0.763170\pi$
$80$	0.175135	0.0195807
$81$	0	0
$82$	6.56825	0.725342
$83$	−12.9070	−1.41672	−0.708362	−	0.705850i	$-0.750565\pi$
$83$	−12.9070	−1.41672	−0.708362	+	0.705850i	$0.750565\pi$
$84$	0	0
$85$	−0.0214229	−0.00232364
$86$	18.4177	1.98603
$87$	0	0
$88$	0	0
$89$	9.81079	1.03994	0.519971	−	0.854184i	$-0.325943\pi$
$89$	9.81079	1.03994	0.519971	+	0.854184i	$0.325943\pi$
$90$	0	0
$91$	4.34292	0.455262
$92$	−30.7434	−3.20522
$93$	0	0
$94$	2.22533	0.229525
$95$	0.268173	0.0275139
$96$	0	0
$97$	−17.4219	−1.76892	−0.884462	−	0.466612i	$-0.845474\pi$
$97$	−17.4219	−1.76892	−0.884462	+	0.466612i	$0.845474\pi$
$98$	2.34292	0.236671
$99$	0	0
$100$	−17.3717	−1.73717
$101$	−15.6357	−1.55581	−0.777903	−	0.628385i	$-0.783716\pi$
$101$	−15.6357	−1.55581	−0.777903	+	0.628385i	$0.783716\pi$
$102$	0	0
$103$	4.16779	0.410664	0.205332	−	0.978692i	$-0.434173\pi$
$103$	4.16779	0.410664	0.205332	+	0.978692i	$0.434173\pi$
$104$	−15.1537	−1.48594
$105$	0	0
$106$	10.5682	1.02648
$107$	−5.49663	−0.531380	−0.265690	−	0.964059i	$-0.585600\pi$
$107$	−5.49663	−0.531380	−0.265690	+	0.964059i	$0.585600\pi$
$108$	0	0
$109$	−5.87819	−0.563029	−0.281514	−	0.959557i	$-0.590837\pi$
$109$	−5.87819	−0.563029	−0.281514	+	0.959557i	$0.590837\pi$
$110$	0	0
$111$	0	0
$112$	−1.19656	−0.113064
$113$	3.88240	0.365226	0.182613	−	0.983185i	$-0.441544\pi$
$113$	3.88240	0.365226	0.182613	+	0.983185i	$0.441544\pi$
$114$	0	0
$115$	−1.28960	−0.120255
$116$	15.1537	1.40699
$117$	0	0
$118$	−18.8108	−1.73167
$119$	0.146365	0.0134173
$120$	0	0
$121$	0	0
$122$	−12.7434	−1.15373
$123$	0	0
$124$	1.02142	0.0917265
$125$	−1.46052	−0.130633
$126$	0	0
$127$	17.6686	1.56784	0.783919	−	0.620863i	$-0.213218\pi$
$127$	17.6686	1.56784	0.783919	+	0.620863i	$0.213218\pi$
$128$	−20.1751	−1.78325
$129$	0	0
$130$	−1.48929	−0.130619
$131$	18.0288	1.57518	0.787590	−	0.616199i	$-0.211328\pi$
$131$	18.0288	1.57518	0.787590	+	0.616199i	$0.211328\pi$
$132$	0	0
$133$	−1.83221	−0.158873
$134$	−18.1825	−1.57073
$135$	0	0
$136$	−0.510711	−0.0437931
$137$	21.5542	1.84150	0.920749	−	0.390156i	$-0.127579\pi$
$137$	21.5542	1.84150	0.920749	+	0.390156i	$0.127579\pi$
$138$	0	0
$139$	−13.5395	−1.14840	−0.574202	−	0.818714i	$-0.694688\pi$
$139$	−13.5395	−1.14840	−0.574202	+	0.818714i	$0.694688\pi$
$140$	−0.510711	−0.0431630
$141$	0	0
$142$	8.29273	0.695911
$143$	0	0
$144$	0	0
$145$	0.635654	0.0527882
$146$	−37.9143	−3.13781
$147$	0	0
$148$	−12.1751	−1.00079
$149$	10.3001	0.843815	0.421908	−	0.906639i	$-0.361361\pi$
$149$	10.3001	0.843815	0.421908	+	0.906639i	$0.361361\pi$
$150$	0	0
$151$	−1.31836	−0.107287	−0.0536435	−	0.998560i	$-0.517083\pi$
$151$	−1.31836	−0.107287	−0.0536435	+	0.998560i	$0.517083\pi$
$152$	6.39312	0.518550
$153$	0	0
$154$	0	0
$155$	0.0428457	0.00344145
$156$	0	0
$157$	9.85677	0.786656	0.393328	−	0.919398i	$-0.371324\pi$
$157$	9.85677	0.786656	0.393328	+	0.919398i	$0.371324\pi$
$158$	−30.6430	−2.43783
$159$	0	0
$160$	−0.611096	−0.0483114
$161$	8.81079	0.694387
$162$	0	0
$163$	−0.292731	−0.0229285	−0.0114642	−	0.999934i	$-0.503649\pi$
$163$	−0.292731	−0.0229285	−0.0114642	+	0.999934i	$0.503649\pi$
$164$	9.78202	0.763847
$165$	0	0
$166$	−30.2400	−2.34708
$167$	−15.7360	−1.21769	−0.608846	−	0.793289i	$-0.708367\pi$
$167$	−15.7360	−1.21769	−0.608846	+	0.793289i	$0.708367\pi$
$168$	0	0
$169$	5.86098	0.450845
$170$	−0.0501921	−0.00384956
$171$	0	0
$172$	27.4292	2.09146
$173$	−17.7360	−1.34845	−0.674223	−	0.738528i	$-0.735521\pi$
$173$	−17.7360	−1.34845	−0.674223	+	0.738528i	$0.735521\pi$
$174$	0	0
$175$	4.97858	0.376345
$176$	0	0
$177$	0	0
$178$	22.9859	1.72287
$179$	23.2285	1.73618	0.868088	−	0.496410i	$-0.165349\pi$
$179$	23.2285	1.73618	0.868088	+	0.496410i	$0.165349\pi$
$180$	0	0
$181$	25.1109	1.86648	0.933238	−	0.359259i	$-0.116970\pi$
$181$	25.1109	1.86648	0.933238	+	0.359259i	$0.116970\pi$
$182$	10.1751	0.754231
$183$	0	0
$184$	−30.7434	−2.26643
$185$	−0.510711	−0.0375483
$186$	0	0
$187$	0	0
$188$	3.31415	0.241710
$189$	0	0
$190$	0.628308	0.0455822
$191$	17.8898	1.29446	0.647228	−	0.762296i	$-0.275928\pi$
$191$	17.8898	1.29446	0.647228	+	0.762296i	$0.275928\pi$
$192$	0	0
$193$	10.3288	0.743487	0.371743	−	0.928336i	$-0.378760\pi$
$193$	10.3288	0.743487	0.371743	+	0.928336i	$0.378760\pi$
$194$	−40.8181	−2.93057
$195$	0	0
$196$	3.48929	0.249235
$197$	14.8610	1.05880	0.529401	−	0.848372i	$-0.322417\pi$
$197$	14.8610	1.05880	0.529401	+	0.848372i	$0.322417\pi$
$198$	0	0
$199$	0.0674041	0.00477815	0.00238908	−	0.999997i	$-0.499240\pi$
$199$	0.0674041	0.00477815	0.00238908	+	0.999997i	$0.499240\pi$
$200$	−17.3717	−1.22836
$201$	0	0
$202$	−36.6331	−2.57750
$203$	−4.34292	−0.304813
$204$	0	0
$205$	0.410327	0.0286585
$206$	9.76481	0.680346
$207$	0	0
$208$	−5.19656	−0.360316
$209$	0	0
$210$	0	0
$211$	13.1323	0.904064	0.452032	−	0.892002i	$-0.350699\pi$
$211$	13.1323	0.904064	0.452032	+	0.892002i	$0.350699\pi$
$212$	15.7392	1.08097
$213$	0	0
$214$	−12.8782	−0.880335
$215$	1.15058	0.0784687
$216$	0	0
$217$	−0.292731	−0.0198719
$218$	−13.7722	−0.932768
$219$	0	0
$220$	0	0
$221$	0.635654	0.0427587
$222$	0	0
$223$	5.37169	0.359715	0.179858	−	0.983693i	$-0.442436\pi$
$223$	5.37169	0.359715	0.179858	+	0.983693i	$0.442436\pi$
$224$	4.17513	0.278963
$225$	0	0
$226$	9.09617	0.605068
$227$	−6.85785	−0.455171	−0.227586	−	0.973758i	$-0.573083\pi$
$227$	−6.85785	−0.455171	−0.227586	+	0.973758i	$0.573083\pi$
$228$	0	0
$229$	−8.76060	−0.578917	−0.289458	−	0.957191i	$-0.593475\pi$
$229$	−8.76060	−0.578917	−0.289458	+	0.957191i	$0.593475\pi$
$230$	−3.02142	−0.199227
$231$	0	0
$232$	15.1537	0.994890
$233$	1.36435	0.0893813	0.0446906	−	0.999001i	$-0.485770\pi$
$233$	1.36435	0.0893813	0.0446906	+	0.999001i	$0.485770\pi$
$234$	0	0
$235$	0.139019	0.00906861
$236$	−28.0147	−1.82360
$237$	0	0
$238$	0.342923	0.0222284
$239$	4.97858	0.322037	0.161019	−	0.986951i	$-0.448522\pi$
$239$	4.97858	0.322037	0.161019	+	0.986951i	$0.448522\pi$
$240$	0	0
$241$	−14.1678	−0.912627	−0.456314	−	0.889819i	$-0.650831\pi$
$241$	−14.1678	−0.912627	−0.456314	+	0.889819i	$0.650831\pi$
$242$	0	0
$243$	0	0
$244$	−18.9786	−1.21498
$245$	0.146365	0.00935095
$246$	0	0
$247$	−7.95715	−0.506302
$248$	1.02142	0.0648604
$249$	0	0
$250$	−3.42188	−0.216419
$251$	2.82908	0.178570	0.0892848	−	0.996006i	$-0.471542\pi$
$251$	2.82908	0.178570	0.0892848	+	0.996006i	$0.471542\pi$
$252$	0	0
$253$	0	0
$254$	41.3963	2.59743
$255$	0	0
$256$	−22.9185	−1.43241
$257$	22.9901	1.43409	0.717043	−	0.697029i	$-0.245495\pi$
$257$	22.9901	1.43409	0.717043	+	0.697029i	$0.245495\pi$
$258$	0	0
$259$	3.48929	0.216814
$260$	−2.21798	−0.137553
$261$	0	0
$262$	42.2400	2.60960
$263$	3.63986	0.224444	0.112222	−	0.993683i	$-0.464203\pi$
$263$	3.63986	0.224444	0.112222	+	0.993683i	$0.464203\pi$
$264$	0	0
$265$	0.660212	0.0405565
$266$	−4.29273	−0.263204
$267$	0	0
$268$	−27.0790	−1.65411
$269$	7.54683	0.460138	0.230069	−	0.973174i	$-0.426105\pi$
$269$	7.54683	0.460138	0.230069	+	0.973174i	$0.426105\pi$
$270$	0	0
$271$	−23.6644	−1.43751	−0.718756	−	0.695263i	$-0.755288\pi$
$271$	−23.6644	−1.43751	−0.718756	+	0.695263i	$0.755288\pi$
$272$	−0.175135	−0.0106191
$273$	0	0
$274$	50.4998	3.05080
$275$	0	0
$276$	0	0
$277$	−31.8929	−1.91626	−0.958129	−	0.286337i	$-0.907562\pi$
$277$	−31.8929	−1.91626	−0.958129	+	0.286337i	$0.907562\pi$
$278$	−31.7220	−1.90256
$279$	0	0
$280$	−0.510711	−0.0305208
$281$	−30.5082	−1.81997	−0.909983	−	0.414645i	$-0.863906\pi$
$281$	−30.5082	−1.81997	−0.909983	+	0.414645i	$0.863906\pi$
$282$	0	0
$283$	−22.6430	−1.34599	−0.672993	−	0.739649i	$-0.734992\pi$
$283$	−22.6430	−1.34599	−0.672993	+	0.739649i	$0.734992\pi$
$284$	12.3503	0.732854
$285$	0	0
$286$	0	0
$287$	−2.80344	−0.165482
$288$	0	0
$289$	−16.9786	−0.998740
$290$	1.48929	0.0874540
$291$	0	0
$292$	−56.4653	−3.30438
$293$	−1.81079	−0.105787	−0.0528937	−	0.998600i	$-0.516844\pi$
$293$	−1.81079	−0.105787	−0.0528937	+	0.998600i	$0.516844\pi$
$294$	0	0
$295$	−1.17513	−0.0684190
$296$	−12.1751	−0.707665
$297$	0	0
$298$	24.1323	1.39795
$299$	38.2646	2.21290
$300$	0	0
$301$	−7.86098	−0.453099
$302$	−3.08883	−0.177742
$303$	0	0
$304$	2.19235	0.125740
$305$	−0.796096	−0.0455843
$306$	0	0
$307$	−29.8469	−1.70345	−0.851726	−	0.523987i	$-0.824444\pi$
$307$	−29.8469	−1.70345	−0.851726	+	0.523987i	$0.824444\pi$
$308$	0	0
$309$	0	0
$310$	0.100384	0.00570144
$311$	5.09304	0.288800	0.144400	−	0.989519i	$-0.453875\pi$
$311$	5.09304	0.288800	0.144400	+	0.989519i	$0.453875\pi$
$312$	0	0
$313$	−22.0319	−1.24532	−0.622658	−	0.782494i	$-0.713947\pi$
$313$	−22.0319	−1.24532	−0.622658	+	0.782494i	$0.713947\pi$
$314$	23.0937	1.30325
$315$	0	0
$316$	−45.6363	−2.56724
$317$	7.08883	0.398148	0.199074	−	0.979984i	$-0.436207\pi$
$317$	7.08883	0.398148	0.199074	+	0.979984i	$0.436207\pi$
$318$	0	0
$319$	0	0
$320$	−1.78202	−0.0996179
$321$	0	0
$322$	20.6430	1.15039
$323$	−0.268173	−0.0149215
$324$	0	0
$325$	21.6216	1.19935
$326$	−0.685846	−0.0379855
$327$	0	0
$328$	9.78202	0.540122
$329$	−0.949808	−0.0523646
$330$	0	0
$331$	−34.6044	−1.90203	−0.951014	−	0.309148i	$-0.899956\pi$
$331$	−34.6044	−1.90203	−0.951014	+	0.309148i	$0.899956\pi$
$332$	−45.0361	−2.47168
$333$	0	0
$334$	−36.8683	−2.01735
$335$	−1.13588	−0.0620599
$336$	0	0
$337$	12.5254	0.682302	0.341151	−	0.940008i	$-0.389183\pi$
$337$	12.5254	0.682302	0.341151	+	0.940008i	$0.389183\pi$
$338$	13.7318	0.746913
$339$	0	0
$340$	−0.0747505	−0.00405392
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	27.4292	1.47889
$345$	0	0
$346$	−41.5542	−2.23397
$347$	27.6461	1.48412	0.742061	−	0.670332i	$-0.233848\pi$
$347$	27.6461	1.48412	0.742061	+	0.670332i	$0.233848\pi$
$348$	0	0
$349$	11.2969	0.604711	0.302356	−	0.953195i	$-0.402227\pi$
$349$	11.2969	0.604711	0.302356	+	0.953195i	$0.402227\pi$
$350$	11.6644	0.623490
$351$	0	0
$352$	0	0
$353$	−16.8034	−0.894357	−0.447178	−	0.894445i	$-0.647571\pi$
$353$	−16.8034	−0.894357	−0.447178	+	0.894445i	$0.647571\pi$
$354$	0	0
$355$	0.518058	0.0274957
$356$	34.2327	1.81433
$357$	0	0
$358$	54.4225	2.87632
$359$	29.0607	1.53376	0.766882	−	0.641788i	$-0.221807\pi$
$359$	29.0607	1.53376	0.766882	+	0.641788i	$0.221807\pi$
$360$	0	0
$361$	−15.6430	−0.823316
$362$	58.8328	3.09218
$363$	0	0
$364$	15.1537	0.794270
$365$	−2.36856	−0.123976
$366$	0	0
$367$	17.6974	0.923797	0.461898	−	0.886933i	$-0.347168\pi$
$367$	17.6974	0.923797	0.461898	+	0.886933i	$0.347168\pi$
$368$	−10.5426	−0.549572
$369$	0	0
$370$	−1.19656	−0.0622061
$371$	−4.51071	−0.234184
$372$	0	0
$373$	12.7820	0.661828	0.330914	−	0.943661i	$-0.392643\pi$
$373$	12.7820	0.661828	0.330914	+	0.943661i	$0.392643\pi$
$374$	0	0
$375$	0	0
$376$	3.31415	0.170914
$377$	−18.8610	−0.971390
$378$	0	0
$379$	−19.6258	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$379$	−19.6258	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$380$	0.935731	0.0480020
$381$	0	0
$382$	41.9143	2.14452
$383$	−29.5500	−1.50993	−0.754966	−	0.655764i	$-0.772347\pi$
$383$	−29.5500	−1.50993	−0.754966	+	0.655764i	$0.772347\pi$
$384$	0	0
$385$	0	0
$386$	24.1997	1.23173
$387$	0	0
$388$	−60.7900	−3.08614
$389$	13.0962	0.664002	0.332001	−	0.943279i	$-0.392276\pi$
$389$	13.0962	0.664002	0.332001	+	0.943279i	$0.392276\pi$
$390$	0	0
$391$	1.28960	0.0652176
$392$	3.48929	0.176236
$393$	0	0
$394$	34.8181	1.75411
$395$	−1.91431	−0.0963193
$396$	0	0
$397$	24.4679	1.22801	0.614003	−	0.789303i	$-0.289558\pi$
$397$	24.4679	1.22801	0.614003	+	0.789303i	$0.289558\pi$
$398$	0.157923	0.00791595
$399$	0	0
$400$	−5.95715	−0.297858
$401$	−24.1751	−1.20725	−0.603624	−	0.797269i	$-0.706277\pi$
$401$	−24.1751	−1.20725	−0.603624	+	0.797269i	$0.706277\pi$
$402$	0	0
$403$	−1.27131	−0.0633284
$404$	−54.5573	−2.71433
$405$	0	0
$406$	−10.1751	−0.504983
$407$	0	0
$408$	0	0
$409$	−5.22112	−0.258168	−0.129084	−	0.991634i	$-0.541204\pi$
$409$	−5.22112	−0.258168	−0.129084	+	0.991634i	$0.541204\pi$
$410$	0.961365	0.0474784
$411$	0	0
$412$	14.5426	0.716463
$413$	8.02877	0.395070
$414$	0	0
$415$	−1.88913	−0.0927339
$416$	18.1323	0.889009
$417$	0	0
$418$	0	0
$419$	22.6718	1.10759	0.553794	−	0.832654i	$-0.313179\pi$
$419$	22.6718	1.10759	0.553794	+	0.832654i	$0.313179\pi$
$420$	0	0
$421$	−2.41454	−0.117677	−0.0588387	−	0.998268i	$-0.518740\pi$
$421$	−2.41454	−0.117677	−0.0588387	+	0.998268i	$0.518740\pi$
$422$	30.7679	1.49776
$423$	0	0
$424$	15.7392	0.764362
$425$	0.728692	0.0353467
$426$	0	0
$427$	5.43910	0.263216
$428$	−19.1793	−0.927069
$429$	0	0
$430$	2.69571	0.129999
$431$	−1.35700	−0.0653644	−0.0326822	−	0.999466i	$-0.510405\pi$
$431$	−1.35700	−0.0653644	−0.0326822	+	0.999466i	$0.510405\pi$
$432$	0	0
$433$	20.6503	0.992392	0.496196	−	0.868210i	$-0.334730\pi$
$433$	20.6503	0.992392	0.496196	+	0.868210i	$0.334730\pi$
$434$	−0.685846	−0.0329217
$435$	0	0
$436$	−20.5107	−0.982285
$437$	−16.1432	−0.772235
$438$	0	0
$439$	20.3074	0.969220	0.484610	−	0.874730i	$-0.338961\pi$
$439$	20.3074	0.969220	0.484610	+	0.874730i	$0.338961\pi$
$440$	0	0
$441$	0	0
$442$	1.48929	0.0708382
$443$	9.81752	0.466444	0.233222	−	0.972423i	$-0.425073\pi$
$443$	9.81752	0.466444	0.233222	+	0.972423i	$0.425073\pi$
$444$	0	0
$445$	1.43596	0.0680711
$446$	12.5855	0.595939
$447$	0	0
$448$	12.1751	0.575221
$449$	−10.0748	−0.475457	−0.237728	−	0.971332i	$-0.576403\pi$
$449$	−10.0748	−0.475457	−0.237728	+	0.971332i	$0.576403\pi$
$450$	0	0
$451$	0	0
$452$	13.5468	0.637189
$453$	0	0
$454$	−16.0674	−0.754081
$455$	0.635654	0.0297999
$456$	0	0
$457$	−14.6791	−0.686660	−0.343330	−	0.939215i	$-0.611555\pi$
$457$	−14.6791	−0.686660	−0.343330	+	0.939215i	$0.611555\pi$
$458$	−20.5254	−0.959089
$459$	0	0
$460$	−4.49977	−0.209803
$461$	27.1396	1.26402	0.632009	−	0.774961i	$-0.282230\pi$
$461$	27.1396	1.26402	0.632009	+	0.774961i	$0.282230\pi$
$462$	0	0
$463$	23.2369	1.07991	0.539955	−	0.841694i	$-0.318441\pi$
$463$	23.2369	1.07991	0.539955	+	0.841694i	$0.318441\pi$
$464$	5.19656	0.241244
$465$	0	0
$466$	3.19656	0.148078
$467$	−23.8568	−1.10396	−0.551980	−	0.833857i	$-0.686127\pi$
$467$	−23.8568	−1.10396	−0.551980	+	0.833857i	$0.686127\pi$
$468$	0	0
$469$	7.76060	0.358351
$470$	0.325711	0.0150239
$471$	0	0
$472$	−28.0147	−1.28948
$473$	0	0
$474$	0	0
$475$	−9.12181	−0.418537
$476$	0.510711	0.0234084
$477$	0	0
$478$	11.6644	0.533518
$479$	−17.2081	−0.786259	−0.393129	−	0.919483i	$-0.628608\pi$
$479$	−17.2081	−0.786259	−0.393129	+	0.919483i	$0.628608\pi$
$480$	0	0
$481$	15.1537	0.690950
$482$	−33.1940	−1.51195
$483$	0	0
$484$	0	0
$485$	−2.54996	−0.115788
$486$	0	0
$487$	−3.16044	−0.143213	−0.0716066	−	0.997433i	$-0.522813\pi$
$487$	−3.16044	−0.143213	−0.0716066	+	0.997433i	$0.522813\pi$
$488$	−18.9786	−0.859120
$489$	0	0
$490$	0.342923	0.0154917
$491$	4.22533	0.190686	0.0953432	−	0.995444i	$-0.469605\pi$
$491$	4.22533	0.190686	0.0953432	+	0.995444i	$0.469605\pi$
$492$	0	0
$493$	−0.635654	−0.0286284
$494$	−18.6430	−0.838788
$495$	0	0
$496$	0.350269	0.0157276
$497$	−3.53948	−0.158767
$498$	0	0
$499$	30.5615	1.36812	0.684061	−	0.729425i	$-0.260212\pi$
$499$	30.5615	1.36812	0.684061	+	0.729425i	$0.260212\pi$
$500$	−5.09617	−0.227908
$501$	0	0
$502$	6.62831	0.295836
$503$	−0.513847	−0.0229113	−0.0114557	−	0.999934i	$-0.503647\pi$
$503$	−0.513847	−0.0229113	−0.0114557	+	0.999934i	$0.503647\pi$
$504$	0	0
$505$	−2.28852	−0.101838
$506$	0	0
$507$	0	0
$508$	61.6510	2.73532
$509$	−24.9217	−1.10463	−0.552316	−	0.833635i	$-0.686256\pi$
$509$	−24.9217	−1.10463	−0.552316	+	0.833635i	$0.686256\pi$
$510$	0	0
$511$	16.1825	0.715871
$512$	−13.3461	−0.589818
$513$	0	0
$514$	53.8641	2.37584
$515$	0.610020	0.0268807
$516$	0	0
$517$	0	0
$518$	8.17513	0.359195
$519$	0	0
$520$	−2.21798	−0.0972649
$521$	31.8855	1.39693	0.698465	−	0.715644i	$-0.253867\pi$
$521$	31.8855	1.39693	0.698465	+	0.715644i	$0.253867\pi$
$522$	0	0
$523$	−5.34713	−0.233814	−0.116907	−	0.993143i	$-0.537298\pi$
$523$	−5.34713	−0.233814	−0.116907	+	0.993143i	$0.537298\pi$
$524$	62.9076	2.74813
$525$	0	0
$526$	8.52792	0.371835
$527$	−0.0428457	−0.00186639
$528$	0	0
$529$	54.6300	2.37522
$530$	1.54683	0.0671899
$531$	0	0
$532$	−6.39312	−0.277177
$533$	−12.1751	−0.527364
$534$	0	0
$535$	−0.804518	−0.0347823
$536$	−27.0790	−1.16963
$537$	0	0
$538$	17.6816	0.762309
$539$	0	0
$540$	0	0
$541$	−27.8610	−1.19784	−0.598919	−	0.800810i	$-0.704403\pi$
$541$	−27.8610	−1.19784	−0.598919	+	0.800810i	$0.704403\pi$
$542$	−55.4439	−2.38152
$543$	0	0
$544$	0.611096	0.0262005
$545$	−0.860365	−0.0368540
$546$	0	0
$547$	30.7005	1.31266	0.656330	−	0.754474i	$-0.272108\pi$
$547$	30.7005	1.31266	0.656330	+	0.754474i	$0.272108\pi$
$548$	75.2087	3.21276
$549$	0	0
$550$	0	0
$551$	7.95715	0.338986
$552$	0	0
$553$	13.0790	0.556174
$554$	−74.7226	−3.17466
$555$	0	0
$556$	−47.2432	−2.00356
$557$	16.7434	0.709440	0.354720	−	0.934973i	$-0.384576\pi$
$557$	16.7434	0.709440	0.354720	+	0.934973i	$0.384576\pi$
$558$	0	0
$559$	−34.1396	−1.44395
$560$	−0.175135	−0.00740079
$561$	0	0
$562$	−71.4783	−3.01513
$563$	−27.6932	−1.16713	−0.583564	−	0.812067i	$-0.698342\pi$
$563$	−27.6932	−1.16713	−0.583564	+	0.812067i	$0.698342\pi$
$564$	0	0
$565$	0.568250	0.0239065
$566$	−53.0508	−2.22989
$567$	0	0
$568$	12.3503	0.518206
$569$	−11.6827	−0.489765	−0.244882	−	0.969553i	$-0.578749\pi$
$569$	−11.6827	−0.489765	−0.244882	+	0.969553i	$0.578749\pi$
$570$	0	0
$571$	24.1151	1.00918	0.504592	−	0.863358i	$-0.331643\pi$
$571$	24.1151	1.00918	0.504592	+	0.863358i	$0.331643\pi$
$572$	0	0
$573$	0	0
$574$	−6.56825	−0.274153
$575$	43.8652	1.82930
$576$	0	0
$577$	44.2572	1.84245	0.921226	−	0.389027i	$-0.127189\pi$
$577$	44.2572	1.84245	0.921226	+	0.389027i	$0.127189\pi$
$578$	−39.7795	−1.65461
$579$	0	0
$580$	2.21798	0.0920966
$581$	12.9070	0.535471
$582$	0	0
$583$	0	0
$584$	−56.4653	−2.33655
$585$	0	0
$586$	−4.24254	−0.175258
$587$	−8.42188	−0.347608	−0.173804	−	0.984780i	$-0.555606\pi$
$587$	−8.42188	−0.347608	−0.173804	+	0.984780i	$0.555606\pi$
$588$	0	0
$589$	0.536345	0.0220997
$590$	−2.75325	−0.113350
$591$	0	0
$592$	−4.17513	−0.171597
$593$	1.94560	0.0798961	0.0399480	−	0.999202i	$-0.487281\pi$
$593$	1.94560	0.0798961	0.0399480	+	0.999202i	$0.487281\pi$
$594$	0	0
$595$	0.0214229	0.000878251 0
$596$	35.9399	1.47216
$597$	0	0
$598$	89.6510	3.66610
$599$	−31.0361	−1.26810	−0.634051	−	0.773292i	$-0.718609\pi$
$599$	−31.0361	−1.26810	−0.634051	+	0.773292i	$0.718609\pi$
$600$	0	0
$601$	5.80031	0.236599	0.118300	−	0.992978i	$-0.462256\pi$
$601$	5.80031	0.236599	0.118300	+	0.992978i	$0.462256\pi$
$602$	−18.4177	−0.750648
$603$	0	0
$604$	−4.60015	−0.187178
$605$	0	0
$606$	0	0
$607$	−2.57560	−0.104540	−0.0522701	−	0.998633i	$-0.516646\pi$
$607$	−2.57560	−0.104540	−0.0522701	+	0.998633i	$0.516646\pi$
$608$	−7.64973	−0.310238
$609$	0	0
$610$	−1.86519	−0.0755194
$611$	−4.12494	−0.166877
$612$	0	0
$613$	−15.8353	−0.639584	−0.319792	−	0.947488i	$-0.603613\pi$
$613$	−15.8353	−0.639584	−0.319792	+	0.947488i	$0.603613\pi$
$614$	−69.9290	−2.82210
$615$	0	0
$616$	0	0
$617$	1.72448	0.0694250	0.0347125	−	0.999397i	$-0.488948\pi$
$617$	1.72448	0.0694250	0.0347125	+	0.999397i	$0.488948\pi$
$618$	0	0
$619$	−17.3288	−0.696505	−0.348253	−	0.937401i	$-0.613225\pi$
$619$	−17.3288	−0.696505	−0.348253	+	0.937401i	$0.613225\pi$
$620$	0.149501	0.00600411
$621$	0	0
$622$	11.9326	0.478454
$623$	−9.81079	−0.393061
$624$	0	0
$625$	24.6791	0.987165
$626$	−51.6191	−2.06311
$627$	0	0
$628$	34.3931	1.37243
$629$	0.510711	0.0203634
$630$	0	0
$631$	6.54683	0.260625	0.130313	−	0.991473i	$-0.458402\pi$
$631$	6.54683	0.260625	0.130313	+	0.991473i	$0.458402\pi$
$632$	−45.6363	−1.81531
$633$	0	0
$634$	16.6086	0.659611
$635$	2.58608	0.102625
$636$	0	0
$637$	−4.34292	−0.172073
$638$	0	0
$639$	0	0
$640$	−2.95294	−0.116725
$641$	−16.0930	−0.635637	−0.317818	−	0.948152i	$-0.602950\pi$
$641$	−16.0930	−0.635637	−0.317818	+	0.948152i	$0.602950\pi$
$642$	0	0
$643$	−3.21377	−0.126739	−0.0633694	−	0.997990i	$-0.520185\pi$
$643$	−3.21377	−0.126739	−0.0633694	+	0.997990i	$0.520185\pi$
$644$	30.7434	1.21146
$645$	0	0
$646$	−0.628308	−0.0247204
$647$	37.8083	1.48640	0.743198	−	0.669071i	$-0.233308\pi$
$647$	37.8083	1.48640	0.743198	+	0.669071i	$0.233308\pi$
$648$	0	0
$649$	0	0
$650$	50.6577	1.98696
$651$	0	0
$652$	−1.02142	−0.0400020
$653$	13.9754	0.546901	0.273451	−	0.961886i	$-0.411835\pi$
$653$	13.9754	0.546901	0.273451	+	0.961886i	$0.411835\pi$
$654$	0	0
$655$	2.63879	0.103106
$656$	3.35448	0.130970
$657$	0	0
$658$	−2.22533	−0.0867523
$659$	8.31729	0.323996	0.161998	−	0.986791i	$-0.448206\pi$
$659$	8.31729	0.323996	0.161998	+	0.986791i	$0.448206\pi$
$660$	0	0
$661$	−38.2474	−1.48765	−0.743825	−	0.668374i	$-0.766990\pi$
$661$	−38.2474	−1.48765	−0.743825	+	0.668374i	$0.766990\pi$
$662$	−81.0754	−3.15108
$663$	0	0
$664$	−45.0361	−1.74774
$665$	−0.268173	−0.0103993
$666$	0	0
$667$	−38.2646	−1.48161
$668$	−54.9076	−2.12444
$669$	0	0
$670$	−2.66129	−0.102815
$671$	0	0
$672$	0	0
$673$	7.86098	0.303019	0.151509	−	0.988456i	$-0.451587\pi$
$673$	7.86098	0.303019	0.151509	+	0.988456i	$0.451587\pi$
$674$	29.3461	1.13037
$675$	0	0
$676$	20.4507	0.786564
$677$	−31.9603	−1.22833	−0.614167	−	0.789176i	$-0.710508\pi$
$677$	−31.9603	−1.22833	−0.614167	+	0.789176i	$0.710508\pi$
$678$	0	0
$679$	17.4219	0.668591
$680$	−0.0747505	−0.00286655
$681$	0	0
$682$	0	0
$683$	−35.1856	−1.34634	−0.673170	−	0.739488i	$-0.735068\pi$
$683$	−35.1856	−1.34634	−0.673170	+	0.739488i	$0.735068\pi$
$684$	0	0
$685$	3.15479	0.120538
$686$	−2.34292	−0.0894532
$687$	0	0
$688$	9.40612	0.358605
$689$	−19.5897	−0.746307
$690$	0	0
$691$	2.05754	0.0782725	0.0391362	−	0.999234i	$-0.487539\pi$
$691$	2.05754	0.0782725	0.0391362	+	0.999234i	$0.487539\pi$
$692$	−61.8862	−2.35256
$693$	0	0
$694$	64.7728	2.45874
$695$	−1.98171	−0.0751706
$696$	0	0
$697$	−0.410327	−0.0155423
$698$	26.4679	1.00182
$699$	0	0
$700$	17.3717	0.656588
$701$	−11.1207	−0.420024	−0.210012	−	0.977699i	$-0.567350\pi$
$701$	−11.1207	−0.420024	−0.210012	+	0.977699i	$0.567350\pi$
$702$	0	0
$703$	−6.39312	−0.241121
$704$	0	0
$705$	0	0
$706$	−39.3692	−1.48168
$707$	15.6357	0.588039
$708$	0	0
$709$	26.6901	1.00237	0.501183	−	0.865341i	$-0.332898\pi$
$709$	26.6901	1.00237	0.501183	+	0.865341i	$0.332898\pi$
$710$	1.21377	0.0455520
$711$	0	0
$712$	34.2327	1.28292
$713$	−2.57919	−0.0965915
$714$	0	0
$715$	0	0
$716$	81.0508	3.02901
$717$	0	0
$718$	68.0869	2.54098
$719$	−19.8996	−0.742130	−0.371065	−	0.928607i	$-0.621007\pi$
$719$	−19.8996	−0.742130	−0.371065	+	0.928607i	$0.621007\pi$
$720$	0	0
$721$	−4.16779	−0.155217
$722$	−36.6503	−1.36398
$723$	0	0
$724$	87.6191	3.25634
$725$	−21.6216	−0.803005
$726$	0	0
$727$	17.3618	0.643915	0.321957	−	0.946754i	$-0.395659\pi$
$727$	17.3618	0.643915	0.321957	+	0.946754i	$0.395659\pi$
$728$	15.1537	0.561634
$729$	0	0
$730$	−5.54935	−0.205391
$731$	−1.15058	−0.0425556
$732$	0	0
$733$	−33.3790	−1.23288	−0.616441	−	0.787401i	$-0.711426\pi$
$733$	−33.3790	−1.23288	−0.616441	+	0.787401i	$0.711426\pi$
$734$	41.4637	1.53045
$735$	0	0
$736$	36.7862	1.35596
$737$	0	0
$738$	0	0
$739$	−15.2369	−0.560498	−0.280249	−	0.959927i	$-0.590417\pi$
$739$	−15.2369	−0.560498	−0.280249	+	0.959927i	$0.590417\pi$
$740$	−1.78202	−0.0655083
$741$	0	0
$742$	−10.5682	−0.387973
$743$	−23.3387	−0.856214	−0.428107	−	0.903728i	$-0.640819\pi$
$743$	−23.3387	−0.856214	−0.428107	+	0.903728i	$0.640819\pi$
$744$	0	0
$745$	1.50758	0.0552333
$746$	29.9473	1.09645
$747$	0	0
$748$	0	0
$749$	5.49663	0.200843
$750$	0	0
$751$	29.6174	1.08075	0.540377	−	0.841423i	$-0.318282\pi$
$751$	29.6174	1.08075	0.540377	+	0.841423i	$0.318282\pi$
$752$	1.13650	0.0414439
$753$	0	0
$754$	−44.1898	−1.60930
$755$	−0.192963	−0.00702265
$756$	0	0
$757$	−25.4360	−0.924486	−0.462243	−	0.886753i	$-0.652955\pi$
$757$	−25.4360	−0.924486	−0.462243	+	0.886753i	$0.652955\pi$
$758$	−45.9817	−1.67013
$759$	0	0
$760$	0.935731	0.0339425
$761$	44.9473	1.62934	0.814669	−	0.579926i	$-0.196919\pi$
$761$	44.9473	1.62934	0.814669	+	0.579926i	$0.196919\pi$
$762$	0	0
$763$	5.87819	0.212805
$764$	62.4225	2.25837
$765$	0	0
$766$	−69.2333	−2.50150
$767$	34.8683	1.25902
$768$	0	0
$769$	32.6749	1.17829	0.589144	−	0.808028i	$-0.299465\pi$
$769$	32.6749	1.17829	0.589144	+	0.808028i	$0.299465\pi$
$770$	0	0
$771$	0	0
$772$	36.0403	1.29712
$773$	49.6222	1.78479	0.892393	−	0.451259i	$-0.149025\pi$
$773$	49.6222	1.78479	0.892393	+	0.451259i	$0.149025\pi$
$774$	0	0
$775$	−1.45738	−0.0523508
$776$	−60.7900	−2.18223
$777$	0	0
$778$	30.6833	1.10005
$779$	5.13650	0.184034
$780$	0	0
$781$	0	0
$782$	3.02142	0.108046
$783$	0	0
$784$	1.19656	0.0427342
$785$	1.44269	0.0514918
$786$	0	0
$787$	−40.2583	−1.43505	−0.717527	−	0.696531i	$-0.754726\pi$
$787$	−40.2583	−1.43505	−0.717527	+	0.696531i	$0.754726\pi$
$788$	51.8543	1.84723
$789$	0	0
$790$	−4.48508	−0.159572
$791$	−3.88240	−0.138042
$792$	0	0
$793$	23.6216	0.838827
$794$	57.3263	2.03444
$795$	0	0
$796$	0.235192	0.00833618
$797$	−17.4862	−0.619391	−0.309696	−	0.950836i	$-0.600227\pi$
$797$	−17.4862	−0.619391	−0.309696	+	0.950836i	$0.600227\pi$
$798$	0	0
$799$	−0.139019	−0.00491814
$800$	20.7862	0.734904
$801$	0	0
$802$	−56.6405	−2.00004
$803$	0	0
$804$	0	0
$805$	1.28960	0.0454523
$806$	−2.97858	−0.104916
$807$	0	0
$808$	−54.5573	−1.91932
$809$	25.9473	0.912258	0.456129	−	0.889914i	$-0.349236\pi$
$809$	25.9473	0.912258	0.456129	+	0.889914i	$0.349236\pi$
$810$	0	0
$811$	18.2829	0.641998	0.320999	−	0.947079i	$-0.395981\pi$
$811$	18.2829	0.641998	0.320999	+	0.947079i	$0.395981\pi$
$812$	−15.1537	−0.531791
$813$	0	0
$814$	0	0
$815$	−0.0428457	−0.00150082
$816$	0	0
$817$	14.4030	0.503897
$818$	−12.2327	−0.427705
$819$	0	0
$820$	1.43175	0.0499989
$821$	−25.3288	−0.883983	−0.441991	−	0.897019i	$-0.645728\pi$
$821$	−25.3288	−0.883983	−0.441991	+	0.897019i	$0.645728\pi$
$822$	0	0
$823$	39.2713	1.36891	0.684456	−	0.729054i	$-0.260040\pi$
$823$	39.2713	1.36891	0.684456	+	0.729054i	$0.260040\pi$
$824$	14.5426	0.506616
$825$	0	0
$826$	18.8108	0.654511
$827$	−27.4047	−0.952954	−0.476477	−	0.879187i	$-0.658086\pi$
$827$	−27.4047	−0.952954	−0.476477	+	0.879187i	$0.658086\pi$
$828$	0	0
$829$	38.5155	1.33770	0.668850	−	0.743397i	$-0.266787\pi$
$829$	38.5155	1.33770	0.668850	+	0.743397i	$0.266787\pi$
$830$	−4.42610	−0.153632
$831$	0	0
$832$	52.8757	1.83313
$833$	−0.146365	−0.00507126
$834$	0	0
$835$	−2.30321	−0.0797060
$836$	0	0
$837$	0	0
$838$	53.1182	1.83494
$839$	17.7276	0.612025	0.306013	−	0.952027i	$-0.401005\pi$
$839$	17.7276	0.612025	0.306013	+	0.952027i	$0.401005\pi$
$840$	0	0
$841$	−10.1390	−0.349621
$842$	−5.65708	−0.194956
$843$	0	0
$844$	45.8223	1.57727
$845$	0.857845	0.0295108
$846$	0	0
$847$	0	0
$848$	5.39733	0.185345
$849$	0	0
$850$	1.70727	0.0585588
$851$	30.7434	1.05387
$852$	0	0
$853$	−15.3032	−0.523972	−0.261986	−	0.965072i	$-0.584377\pi$
$853$	−15.3032	−0.523972	−0.261986	+	0.965072i	$0.584377\pi$
$854$	12.7434	0.436070
$855$	0	0
$856$	−19.1793	−0.655537
$857$	10.7722	0.367970	0.183985	−	0.982929i	$-0.441100\pi$
$857$	10.7722	0.367970	0.183985	+	0.982929i	$0.441100\pi$
$858$	0	0
$859$	47.2516	1.61220	0.806101	−	0.591777i	$-0.201574\pi$
$859$	47.2516	1.61220	0.806101	+	0.591777i	$0.201574\pi$
$860$	4.01469	0.136900
$861$	0	0
$862$	−3.17935	−0.108289
$863$	34.0294	1.15837	0.579187	−	0.815195i	$-0.303370\pi$
$863$	34.0294	1.15837	0.579187	+	0.815195i	$0.303370\pi$
$864$	0	0
$865$	−2.59594	−0.0882647
$866$	48.3822	1.64409
$867$	0	0
$868$	−1.02142	−0.0346694
$869$	0	0
$870$	0	0
$871$	33.7037	1.14201
$872$	−20.5107	−0.694580
$873$	0	0
$874$	−37.8223	−1.27936
$875$	1.46052	0.0493746
$876$	0	0
$877$	3.25410	0.109883	0.0549415	−	0.998490i	$-0.482503\pi$
$877$	3.25410	0.109883	0.0549415	+	0.998490i	$0.482503\pi$
$878$	47.5787	1.60570
$879$	0	0
$880$	0	0
$881$	13.3545	0.449924	0.224962	−	0.974368i	$-0.427774\pi$
$881$	13.3545	0.449924	0.224962	+	0.974368i	$0.427774\pi$
$882$	0	0
$883$	−25.4741	−0.857273	−0.428636	−	0.903477i	$-0.641006\pi$
$883$	−25.4741	−0.857273	−0.428636	+	0.903477i	$0.641006\pi$
$884$	2.21798	0.0745988
$885$	0	0
$886$	23.0017	0.772757
$887$	21.6932	0.728386	0.364193	−	0.931323i	$-0.381345\pi$
$887$	21.6932	0.728386	0.364193	+	0.931323i	$0.381345\pi$
$888$	0	0
$889$	−17.6686	−0.592587
$890$	3.36435	0.112773
$891$	0	0
$892$	18.7434	0.627575
$893$	1.74025	0.0582352
$894$	0	0
$895$	3.39985	0.113644
$896$	20.1751	0.674004
$897$	0	0
$898$	−23.6044	−0.787688
$899$	1.27131	0.0424005
$900$	0	0
$901$	−0.660212	−0.0219949
$902$	0	0
$903$	0	0
$904$	13.5468	0.450561
$905$	3.67536	0.122173
$906$	0	0
$907$	1.21377	0.0403026	0.0201513	−	0.999797i	$-0.493585\pi$
$907$	1.21377	0.0403026	0.0201513	+	0.999797i	$0.493585\pi$
$908$	−23.9290	−0.794112
$909$	0	0
$910$	1.48929	0.0493694
$911$	−57.0080	−1.88876	−0.944379	−	0.328859i	$-0.893336\pi$
$911$	−57.0080	−1.88876	−0.944379	+	0.328859i	$0.893336\pi$
$912$	0	0
$913$	0	0
$914$	−34.3920	−1.13759
$915$	0	0
$916$	−30.5682	−1.01000
$917$	−18.0288	−0.595362
$918$	0	0
$919$	−6.45065	−0.212787	−0.106394	−	0.994324i	$-0.533930\pi$
$919$	−6.45065	−0.212787	−0.106394	+	0.994324i	$0.533930\pi$
$920$	−4.49977	−0.148353
$921$	0	0
$922$	63.5861	2.09410
$923$	−15.3717	−0.505965
$924$	0	0
$925$	17.3717	0.571178
$926$	54.4422	1.78908
$927$	0	0
$928$	−18.1323	−0.595222
$929$	−2.74652	−0.0901104	−0.0450552	−	0.998984i	$-0.514346\pi$
$929$	−2.74652	−0.0901104	−0.0450552	+	0.998984i	$0.514346\pi$
$930$	0	0
$931$	1.83221	0.0600483
$932$	4.76060	0.155939
$933$	0	0
$934$	−55.8946	−1.82893
$935$	0	0
$936$	0	0
$937$	−25.8923	−0.845864	−0.422932	−	0.906162i	$-0.638999\pi$
$937$	−25.8923	−0.845864	−0.422932	+	0.906162i	$0.638999\pi$
$938$	18.1825	0.593679
$939$	0	0
$940$	0.485078	0.0158215
$941$	−9.01890	−0.294008	−0.147004	−	0.989136i	$-0.546963\pi$
$941$	−9.01890	−0.294008	−0.147004	+	0.989136i	$0.546963\pi$
$942$	0	0
$943$	−24.7005	−0.804360
$944$	−9.60688	−0.312677
$945$	0	0
$946$	0	0
$947$	−39.2003	−1.27384	−0.636919	−	0.770930i	$-0.719792\pi$
$947$	−39.2003	−1.27384	−0.636919	+	0.770930i	$0.719792\pi$
$948$	0	0
$949$	70.2793	2.28136
$950$	−21.3717	−0.693389
$951$	0	0
$952$	0.510711	0.0165523
$953$	−55.4464	−1.79609	−0.898043	−	0.439907i	$-0.855011\pi$
$953$	−55.4464	−1.79609	−0.898043	+	0.439907i	$0.855011\pi$
$954$	0	0
$955$	2.61844	0.0847308
$956$	17.3717	0.561841
$957$	0	0
$958$	−40.3173	−1.30259
$959$	−21.5542	−0.696021
$960$	0	0
$961$	−30.9143	−0.997236
$962$	35.5040	1.14469
$963$	0	0
$964$	−49.4355	−1.59221
$965$	1.51179	0.0486661
$966$	0	0
$967$	−23.9467	−0.770073	−0.385037	−	0.922901i	$-0.625811\pi$
$967$	−23.9467	−0.770073	−0.385037	+	0.922901i	$0.625811\pi$
$968$	0	0
$969$	0	0
$970$	−5.97437	−0.191825
$971$	6.42188	0.206088	0.103044	−	0.994677i	$-0.467142\pi$
$971$	6.42188	0.206088	0.103044	+	0.994677i	$0.467142\pi$
$972$	0	0
$973$	13.5395	0.434056
$974$	−7.40467	−0.237261
$975$	0	0
$976$	−6.50819	−0.208322
$977$	−38.5584	−1.23359	−0.616796	−	0.787123i	$-0.711570\pi$
$977$	−38.5584	−1.23359	−0.616796	+	0.787123i	$0.711570\pi$
$978$	0	0
$979$	0	0
$980$	0.510711	0.0163141
$981$	0	0
$982$	9.89962	0.315909
$983$	−21.6069	−0.689153	−0.344576	−	0.938758i	$-0.611977\pi$
$983$	−21.6069	−0.689153	−0.344576	+	0.938758i	$0.611977\pi$
$984$	0	0
$985$	2.17513	0.0693056
$986$	−1.48929	−0.0474286
$987$	0	0
$988$	−27.7648	−0.883316
$989$	−69.2614	−2.20239
$990$	0	0
$991$	34.4507	1.09436	0.547181	−	0.837015i	$-0.315701\pi$
$991$	34.4507	1.09436	0.547181	+	0.837015i	$0.315701\pi$
$992$	−1.22219	−0.0388046
$993$	0	0
$994$	−8.29273	−0.263029
$995$	0.00986564	0.000312762 0
$996$	0	0
$997$	7.76733	0.245994	0.122997	−	0.992407i	$-0.460749\pi$
$997$	7.76733	0.245994	0.122997	+	0.992407i	$0.460749\pi$
$998$	71.6033	2.26656
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cc.1.3		3
3.2	odd	2		2541.2.a.bh.1.1	✓	3
11.10	odd	2		7623.2.a.ca.1.1		3
33.32	even	2		2541.2.a.bj.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bh.1.1	✓	3	3.2	odd	2
2541.2.a.bj.1.3	yes	3	33.32	even	2
7623.2.a.ca.1.1		3	11.10	odd	2
7623.2.a.cc.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34292 q^{2} +3.48929 q^{4} +0.146365 q^{5} -1.00000 q^{7} +3.48929 q^{8} +O(q^{10})\)	\(q+2.34292 q^{2} +3.48929 q^{4} +0.146365 q^{5} -1.00000 q^{7} +3.48929 q^{8} +0.342923 q^{10} -4.34292 q^{13} -2.34292 q^{14} +1.19656 q^{16} -0.146365 q^{17} +1.83221 q^{19} +0.510711 q^{20} -8.81079 q^{23} -4.97858 q^{25} -10.1751 q^{26} -3.48929 q^{28} +4.34292 q^{29} +0.292731 q^{31} -4.17513 q^{32} -0.342923 q^{34} -0.146365 q^{35} -3.48929 q^{37} +4.29273 q^{38} +0.510711 q^{40} +2.80344 q^{41} +7.86098 q^{43} -20.6430 q^{46} +0.949808 q^{47} +1.00000 q^{49} -11.6644 q^{50} -15.1537 q^{52} +4.51071 q^{53} -3.48929 q^{56} +10.1751 q^{58} -8.02877 q^{59} -5.43910 q^{61} +0.685846 q^{62} -12.1751 q^{64} -0.635654 q^{65} -7.76060 q^{67} -0.510711 q^{68} -0.342923 q^{70} +3.53948 q^{71} -16.1825 q^{73} -8.17513 q^{74} +6.39312 q^{76} -13.0790 q^{79} +0.175135 q^{80} +6.56825 q^{82} -12.9070 q^{83} -0.0214229 q^{85} +18.4177 q^{86} +9.81079 q^{89} +4.34292 q^{91} -30.7434 q^{92} +2.22533 q^{94} +0.268173 q^{95} -17.4219 q^{97} +2.34292 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - q^5 - 3 * q^7 + 3 * q^8	\( 3 q + q^{2} + 3 q^{4} - q^{5} - 3 q^{7} + 3 q^{8} - 5 q^{10} - 7 q^{13} - q^{14} - q^{16} + q^{17} - 8 q^{19} + 9 q^{20} + 2 q^{23} - 11 q^{26} - 3 q^{28} + 7 q^{29} - 2 q^{31} + 7 q^{32} + 5 q^{34} + q^{35} - 3 q^{37} + 10 q^{38} + 9 q^{40} + 13 q^{41} - 8 q^{43} - 20 q^{46} + 6 q^{47} + 3 q^{49} - 8 q^{50} - 11 q^{52} + 21 q^{53} - 3 q^{56} + 11 q^{58} - 6 q^{59} - 12 q^{61} - 10 q^{62} - 17 q^{64} + 7 q^{65} + 2 q^{67} - 9 q^{68} + 5 q^{70} + 4 q^{73} - 5 q^{74} + 10 q^{76} - 18 q^{79} - 19 q^{80} - 9 q^{82} - 12 q^{83} - 15 q^{85} + 36 q^{86} + q^{89} + 7 q^{91} - 44 q^{92} - 16 q^{94} + 8 q^{95} - 25 q^{97} + q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - q^5 - 3 * q^7 + 3 * q^8 - 5 * q^10 - 7 * q^13 - q^14 - q^16 + q^17 - 8 * q^19 + 9 * q^20 + 2 * q^23 - 11 * q^26 - 3 * q^28 + 7 * q^29 - 2 * q^31 + 7 * q^32 + 5 * q^34 + q^35 - 3 * q^37 + 10 * q^38 + 9 * q^40 + 13 * q^41 - 8 * q^43 - 20 * q^46 + 6 * q^47 + 3 * q^49 - 8 * q^50 - 11 * q^52 + 21 * q^53 - 3 * q^56 + 11 * q^58 - 6 * q^59 - 12 * q^61 - 10 * q^62 - 17 * q^64 + 7 * q^65 + 2 * q^67 - 9 * q^68 + 5 * q^70 + 4 * q^73 - 5 * q^74 + 10 * q^76 - 18 * q^79 - 19 * q^80 - 9 * q^82 - 12 * q^83 - 15 * q^85 + 36 * q^86 + q^89 + 7 * q^91 - 44 * q^92 - 16 * q^94 + 8 * q^95 - 25 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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