LMFDB - Embedded newform 7935.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7935,2,Mod(1,7935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7935, 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("7935.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7935 = 3 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7935.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:	$63.3612940039$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.77200$ of defining polynomial
Character	$\chi$	$=$	7935.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} -3.77200 q^{11} -2.00000 q^{12} -4.77200 q^{13} +1.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} -5.77200 q^{19} -2.00000 q^{20} +4.77200 q^{21} +1.00000 q^{25} +1.00000 q^{27} -9.54400 q^{28} +6.00000 q^{29} +5.77200 q^{31} -3.77200 q^{33} +4.77200 q^{35} -2.00000 q^{36} -1.22800 q^{37} -4.77200 q^{39} -9.77200 q^{41} -2.77200 q^{43} +7.54400 q^{44} +1.00000 q^{45} +7.54400 q^{47} +4.00000 q^{48} +15.7720 q^{49} +6.00000 q^{51} +9.54400 q^{52} -13.5440 q^{53} -3.77200 q^{55} -5.77200 q^{57} -2.00000 q^{60} +8.54400 q^{61} +4.77200 q^{63} -8.00000 q^{64} -4.77200 q^{65} +3.22800 q^{67} -12.0000 q^{68} +9.77200 q^{71} +2.00000 q^{73} +1.00000 q^{75} +11.5440 q^{76} -18.0000 q^{77} +13.7720 q^{79} +4.00000 q^{80} +1.00000 q^{81} +13.5440 q^{83} -9.54400 q^{84} +6.00000 q^{85} +6.00000 q^{87} -7.54400 q^{89} -22.7720 q^{91} +5.77200 q^{93} -5.77200 q^{95} +17.5440 q^{97} -3.77200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9} + q^{11} - 4 q^{12} - q^{13} + 2 q^{15} + 8 q^{16} + 12 q^{17} - 3 q^{19} - 4 q^{20} + q^{21} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 12 q^{29} + 3 q^{31} + q^{33} + q^{35} - 4 q^{36} - 11 q^{37} - q^{39} - 11 q^{41} + 3 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{47} + 8 q^{48} + 23 q^{49} + 12 q^{51} + 2 q^{52} - 10 q^{53} + q^{55} - 3 q^{57} - 4 q^{60} + q^{63} - 16 q^{64} - q^{65} + 15 q^{67} - 24 q^{68} + 11 q^{71} + 4 q^{73} + 2 q^{75} + 6 q^{76} - 36 q^{77} + 19 q^{79} + 8 q^{80} + 2 q^{81} + 10 q^{83} - 2 q^{84} + 12 q^{85} + 12 q^{87} + 2 q^{89} - 37 q^{91} + 3 q^{93} - 3 q^{95} + 18 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9 + q^11 - 4 * q^12 - q^13 + 2 * q^15 + 8 * q^16 + 12 * q^17 - 3 * q^19 - 4 * q^20 + q^21 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 12 * q^29 + 3 * q^31 + q^33 + q^35 - 4 * q^36 - 11 * q^37 - q^39 - 11 * q^41 + 3 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^47 + 8 * q^48 + 23 * q^49 + 12 * q^51 + 2 * q^52 - 10 * q^53 + q^55 - 3 * q^57 - 4 * q^60 + q^63 - 16 * q^64 - q^65 + 15 * q^67 - 24 * q^68 + 11 * q^71 + 4 * q^73 + 2 * q^75 + 6 * q^76 - 36 * q^77 + 19 * q^79 + 8 * q^80 + 2 * q^81 + 10 * q^83 - 2 * q^84 + 12 * q^85 + 12 * q^87 + 2 * q^89 - 37 * q^91 + 3 * q^93 - 3 * q^95 + 18 * q^97 + q^99

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.77200	1.80365	0.901824	−	0.432104i	$-0.142229\pi$
$7$	4.77200	1.80365	0.901824	+	0.432104i	$0.142229\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.77200	−1.13730	−0.568651	−	0.822579i	$-0.692534\pi$
$11$	−3.77200	−1.13730	−0.568651	+	0.822579i	$0.692534\pi$
$12$	−2.00000	−0.577350
$13$	−4.77200	−1.32352	−0.661758	−	0.749718i	$-0.730189\pi$
$13$	−4.77200	−1.32352	−0.661758	+	0.749718i	$0.730189\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−5.77200	−1.32419	−0.662094	−	0.749421i	$-0.730332\pi$
$19$	−5.77200	−1.32419	−0.662094	+	0.749421i	$0.730332\pi$
$20$	−2.00000	−0.447214
$21$	4.77200	1.04134
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−9.54400	−1.80365
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.77200	1.03668	0.518341	−	0.855174i	$-0.326550\pi$
$31$	5.77200	1.03668	0.518341	+	0.855174i	$0.326550\pi$
$32$	0	0
$33$	−3.77200	−0.656621
$34$	0	0
$35$	4.77200	0.806616
$36$	−2.00000	−0.333333
$37$	−1.22800	−0.201882	−0.100941	−	0.994892i	$-0.532185\pi$
$37$	−1.22800	−0.201882	−0.100941	+	0.994892i	$0.532185\pi$
$38$	0	0
$39$	−4.77200	−0.764132
$40$	0	0
$41$	−9.77200	−1.52613	−0.763065	−	0.646322i	$-0.776306\pi$
$41$	−9.77200	−1.52613	−0.763065	+	0.646322i	$0.776306\pi$
$42$	0	0
$43$	−2.77200	−0.422726	−0.211363	−	0.977408i	$-0.567790\pi$
$43$	−2.77200	−0.422726	−0.211363	+	0.977408i	$0.567790\pi$
$44$	7.54400	1.13730
$45$	1.00000	0.149071
$46$	0	0
$47$	7.54400	1.10041	0.550203	−	0.835031i	$-0.314550\pi$
$47$	7.54400	1.10041	0.550203	+	0.835031i	$0.314550\pi$
$48$	4.00000	0.577350
$49$	15.7720	2.25314
$50$	0	0
$51$	6.00000	0.840168
$52$	9.54400	1.32352
$53$	−13.5440	−1.86041	−0.930206	−	0.367038i	$-0.880372\pi$
$53$	−13.5440	−1.86041	−0.930206	+	0.367038i	$0.880372\pi$
$54$	0	0
$55$	−3.77200	−0.508617
$56$	0	0
$57$	−5.77200	−0.764520
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	8.54400	1.09395	0.546974	−	0.837150i	$-0.315780\pi$
$61$	8.54400	1.09395	0.546974	+	0.837150i	$0.315780\pi$
$62$	0	0
$63$	4.77200	0.601216
$64$	−8.00000	−1.00000
$65$	−4.77200	−0.591894
$66$	0	0
$67$	3.22800	0.394363	0.197181	−	0.980367i	$-0.436821\pi$
$67$	3.22800	0.394363	0.197181	+	0.980367i	$0.436821\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	0	0
$71$	9.77200	1.15972	0.579862	−	0.814715i	$-0.303107\pi$
$71$	9.77200	1.15972	0.579862	+	0.814715i	$0.303107\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	11.5440	1.32419
$77$	−18.0000	−2.05129
$78$	0	0
$79$	13.7720	1.54947	0.774736	−	0.632285i	$-0.217883\pi$
$79$	13.7720	1.54947	0.774736	+	0.632285i	$0.217883\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	13.5440	1.48665	0.743324	−	0.668932i	$-0.233248\pi$
$83$	13.5440	1.48665	0.743324	+	0.668932i	$0.233248\pi$
$84$	−9.54400	−1.04134
$85$	6.00000	0.650791
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−7.54400	−0.799663	−0.399831	−	0.916589i	$-0.630931\pi$
$89$	−7.54400	−0.799663	−0.399831	+	0.916589i	$0.630931\pi$
$90$	0	0
$91$	−22.7720	−2.38715
$92$	0	0
$93$	5.77200	0.598529
$94$	0	0
$95$	−5.77200	−0.592195
$96$	0	0
$97$	17.5440	1.78132	0.890662	−	0.454666i	$-0.150241\pi$
$97$	17.5440	1.78132	0.890662	+	0.454666i	$0.150241\pi$
$98$	0	0
$99$	−3.77200	−0.379100
$100$	−2.00000	−0.200000
$101$	11.3160	1.12598	0.562992	−	0.826462i	$-0.309650\pi$
$101$	11.3160	1.12598	0.562992	+	0.826462i	$0.309650\pi$
$102$	0	0
$103$	6.31601	0.622335	0.311167	−	0.950355i	$-0.399280\pi$
$103$	6.31601	0.622335	0.311167	+	0.950355i	$0.399280\pi$
$104$	0	0
$105$	4.77200	0.465700
$106$	0	0
$107$	13.5440	1.30935	0.654674	−	0.755911i	$-0.272806\pi$
$107$	13.5440	1.30935	0.654674	+	0.755911i	$0.272806\pi$
$108$	−2.00000	−0.192450
$109$	−7.31601	−0.700746	−0.350373	−	0.936610i	$-0.613945\pi$
$109$	−7.31601	−0.700746	−0.350373	+	0.936610i	$0.613945\pi$
$110$	0	0
$111$	−1.22800	−0.116556
$112$	19.0880	1.80365
$113$	−1.54400	−0.145248	−0.0726238	−	0.997359i	$-0.523137\pi$
$113$	−1.54400	−0.145248	−0.0726238	+	0.997359i	$0.523137\pi$
$114$	0	0
$115$	0	0
$116$	−12.0000	−1.11417
$117$	−4.77200	−0.441172
$118$	0	0
$119$	28.6320	2.62469
$120$	0	0
$121$	3.22800	0.293454
$122$	0	0
$123$	−9.77200	−0.881112
$124$	−11.5440	−1.03668
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.22800	0.108967	0.0544836	−	0.998515i	$-0.482649\pi$
$127$	1.22800	0.108967	0.0544836	+	0.998515i	$0.482649\pi$
$128$	0	0
$129$	−2.77200	−0.244061
$130$	0	0
$131$	−2.22800	−0.194661	−0.0973306	−	0.995252i	$-0.531030\pi$
$131$	−2.22800	−0.194661	−0.0973306	+	0.995252i	$0.531030\pi$
$132$	7.54400	0.656621
$133$	−27.5440	−2.38837
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	7.54400	0.644528	0.322264	−	0.946650i	$-0.395556\pi$
$137$	7.54400	0.644528	0.322264	+	0.946650i	$0.395556\pi$
$138$	0	0
$139$	−16.0880	−1.36457	−0.682283	−	0.731088i	$-0.739013\pi$
$139$	−16.0880	−1.36457	−0.682283	+	0.731088i	$0.739013\pi$
$140$	−9.54400	−0.806616
$141$	7.54400	0.635320
$142$	0	0
$143$	18.0000	1.50524
$144$	4.00000	0.333333
$145$	6.00000	0.498273
$146$	0	0
$147$	15.7720	1.30085
$148$	2.45600	0.201882
$149$	−8.22800	−0.674064	−0.337032	−	0.941493i	$-0.609423\pi$
$149$	−8.22800	−0.674064	−0.337032	+	0.941493i	$0.609423\pi$
$150$	0	0
$151$	−8.54400	−0.695301	−0.347651	−	0.937624i	$-0.613020\pi$
$151$	−8.54400	−0.695301	−0.347651	+	0.937624i	$0.613020\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	5.77200	0.463618
$156$	9.54400	0.764132
$157$	2.45600	0.196010	0.0980049	−	0.995186i	$-0.468754\pi$
$157$	2.45600	0.196010	0.0980049	+	0.995186i	$0.468754\pi$
$158$	0	0
$159$	−13.5440	−1.07411
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.22800	−0.252836	−0.126418	−	0.991977i	$-0.540348\pi$
$163$	−3.22800	−0.252836	−0.126418	+	0.991977i	$0.540348\pi$
$164$	19.5440	1.52613
$165$	−3.77200	−0.293650
$166$	0	0
$167$	−1.54400	−0.119479	−0.0597393	−	0.998214i	$-0.519027\pi$
$167$	−1.54400	−0.119479	−0.0597393	+	0.998214i	$0.519027\pi$
$168$	0	0
$169$	9.77200	0.751692
$170$	0	0
$171$	−5.77200	−0.441396
$172$	5.54400	0.422726
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	4.77200	0.360729
$176$	−15.0880	−1.13730
$177$	0	0
$178$	0	0
$179$	11.3160	0.845798	0.422899	−	0.906177i	$-0.361013\pi$
$179$	11.3160	0.845798	0.422899	+	0.906177i	$0.361013\pi$
$180$	−2.00000	−0.149071
$181$	7.77200	0.577688	0.288844	−	0.957376i	$-0.406729\pi$
$181$	7.77200	0.577688	0.288844	+	0.957376i	$0.406729\pi$
$182$	0	0
$183$	8.54400	0.631591
$184$	0	0
$185$	−1.22800	−0.0902842
$186$	0	0
$187$	−22.6320	−1.65502
$188$	−15.0880	−1.10041
$189$	4.77200	0.347112
$190$	0	0
$191$	10.4560	0.756569	0.378285	−	0.925689i	$-0.376514\pi$
$191$	10.4560	0.756569	0.378285	+	0.925689i	$0.376514\pi$
$192$	−8.00000	−0.577350
$193$	8.77200	0.631423	0.315711	−	0.948855i	$-0.397757\pi$
$193$	8.77200	0.631423	0.315711	+	0.948855i	$0.397757\pi$
$194$	0	0
$195$	−4.77200	−0.341730
$196$	−31.5440	−2.25314
$197$	15.0880	1.07498	0.537488	−	0.843271i	$-0.319373\pi$
$197$	15.0880	1.07498	0.537488	+	0.843271i	$0.319373\pi$
$198$	0	0
$199$	−12.5440	−0.889221	−0.444610	−	0.895724i	$-0.646658\pi$
$199$	−12.5440	−0.889221	−0.444610	+	0.895724i	$0.646658\pi$
$200$	0	0
$201$	3.22800	0.227685
$202$	0	0
$203$	28.6320	2.00957
$204$	−12.0000	−0.840168
$205$	−9.77200	−0.682506
$206$	0	0
$207$	0	0
$208$	−19.0880	−1.32352
$209$	21.7720	1.50600
$210$	0	0
$211$	−20.5440	−1.41431	−0.707154	−	0.707060i	$-0.750021\pi$
$211$	−20.5440	−1.41431	−0.707154	+	0.707060i	$0.750021\pi$
$212$	27.0880	1.86041
$213$	9.77200	0.669567
$214$	0	0
$215$	−2.77200	−0.189049
$216$	0	0
$217$	27.5440	1.86981
$218$	0	0
$219$	2.00000	0.135147
$220$	7.54400	0.508617
$221$	−28.6320	−1.92600
$222$	0	0
$223$	8.77200	0.587417	0.293708	−	0.955895i	$-0.405111\pi$
$223$	8.77200	0.587417	0.293708	+	0.955895i	$0.405111\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−4.45600	−0.295755	−0.147877	−	0.989006i	$-0.547244\pi$
$227$	−4.45600	−0.295755	−0.147877	+	0.989006i	$0.547244\pi$
$228$	11.5440	0.764520
$229$	−8.08801	−0.534471	−0.267235	−	0.963631i	$-0.586110\pi$
$229$	−8.08801	−0.534471	−0.267235	+	0.963631i	$0.586110\pi$
$230$	0	0
$231$	−18.0000	−1.18431
$232$	0	0
$233$	7.54400	0.494224	0.247112	−	0.968987i	$-0.420518\pi$
$233$	7.54400	0.494224	0.247112	+	0.968987i	$0.420518\pi$
$234$	0	0
$235$	7.54400	0.492117
$236$	0	0
$237$	13.7720	0.894588
$238$	0	0
$239$	14.2280	0.920333	0.460166	−	0.887833i	$-0.347790\pi$
$239$	14.2280	0.920333	0.460166	+	0.887833i	$0.347790\pi$
$240$	4.00000	0.258199
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−17.0880	−1.09395
$245$	15.7720	1.00764
$246$	0	0
$247$	27.5440	1.75258
$248$	0	0
$249$	13.5440	0.858316
$250$	0	0
$251$	0.683994	0.0431733	0.0215867	−	0.999767i	$-0.493128\pi$
$251$	0.683994	0.0431733	0.0215867	+	0.999767i	$0.493128\pi$
$252$	−9.54400	−0.601216
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	16.0000	1.00000
$257$	−28.6320	−1.78602	−0.893008	−	0.450041i	$-0.851409\pi$
$257$	−28.6320	−1.78602	−0.893008	+	0.450041i	$0.851409\pi$
$258$	0	0
$259$	−5.86001	−0.364123
$260$	9.54400	0.591894
$261$	6.00000	0.371391
$262$	0	0
$263$	21.0880	1.30034	0.650171	−	0.759788i	$-0.274697\pi$
$263$	21.0880	1.30034	0.650171	+	0.759788i	$0.274697\pi$
$264$	0	0
$265$	−13.5440	−0.832002
$266$	0	0
$267$	−7.54400	−0.461686
$268$	−6.45600	−0.394363
$269$	−3.77200	−0.229983	−0.114992	−	0.993366i	$-0.536684\pi$
$269$	−3.77200	−0.229983	−0.114992	+	0.993366i	$0.536684\pi$
$270$	0	0
$271$	−21.2280	−1.28951	−0.644755	−	0.764390i	$-0.723040\pi$
$271$	−21.2280	−1.28951	−0.644755	+	0.764390i	$0.723040\pi$
$272$	24.0000	1.45521
$273$	−22.7720	−1.37822
$274$	0	0
$275$	−3.77200	−0.227460
$276$	0	0
$277$	17.8600	1.07310	0.536552	−	0.843867i	$-0.319727\pi$
$277$	17.8600	1.07310	0.536552	+	0.843867i	$0.319727\pi$
$278$	0	0
$279$	5.77200	0.345561
$280$	0	0
$281$	−5.31601	−0.317126	−0.158563	−	0.987349i	$-0.550686\pi$
$281$	−5.31601	−0.317126	−0.158563	+	0.987349i	$0.550686\pi$
$282$	0	0
$283$	−14.7720	−0.878104	−0.439052	−	0.898462i	$-0.644686\pi$
$283$	−14.7720	−0.878104	−0.439052	+	0.898462i	$0.644686\pi$
$284$	−19.5440	−1.15972
$285$	−5.77200	−0.341904
$286$	0	0
$287$	−46.6320	−2.75260
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	17.5440	1.02845
$292$	−4.00000	−0.234082
$293$	−13.5440	−0.791249	−0.395625	−	0.918412i	$-0.629472\pi$
$293$	−13.5440	−0.791249	−0.395625	+	0.918412i	$0.629472\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.77200	−0.218874
$298$	0	0
$299$	0	0
$300$	−2.00000	−0.115470
$301$	−13.2280	−0.762449
$302$	0	0
$303$	11.3160	0.650088
$304$	−23.0880	−1.32419
$305$	8.54400	0.489228
$306$	0	0
$307$	19.2280	1.09740	0.548700	−	0.836019i	$-0.315123\pi$
$307$	19.2280	1.09740	0.548700	+	0.836019i	$0.315123\pi$
$308$	36.0000	2.05129
$309$	6.31601	0.359305
$310$	0	0
$311$	−10.4560	−0.592905	−0.296453	−	0.955048i	$-0.595804\pi$
$311$	−10.4560	−0.592905	−0.296453	+	0.955048i	$0.595804\pi$
$312$	0	0
$313$	15.2280	0.860737	0.430369	−	0.902653i	$-0.358384\pi$
$313$	15.2280	0.860737	0.430369	+	0.902653i	$0.358384\pi$
$314$	0	0
$315$	4.77200	0.268872
$316$	−27.5440	−1.54947
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−22.6320	−1.26715
$320$	−8.00000	−0.447214
$321$	13.5440	0.755953
$322$	0	0
$323$	−34.6320	−1.92698
$324$	−2.00000	−0.111111
$325$	−4.77200	−0.264703
$326$	0	0
$327$	−7.31601	−0.404576
$328$	0	0
$329$	36.0000	1.98474
$330$	0	0
$331$	−2.54400	−0.139831	−0.0699155	−	0.997553i	$-0.522273\pi$
$331$	−2.54400	−0.139831	−0.0699155	+	0.997553i	$0.522273\pi$
$332$	−27.0880	−1.48665
$333$	−1.22800	−0.0672939
$334$	0	0
$335$	3.22800	0.176364
$336$	19.0880	1.04134
$337$	−26.7720	−1.45836	−0.729182	−	0.684320i	$-0.760099\pi$
$337$	−26.7720	−1.45836	−0.729182	+	0.684320i	$0.760099\pi$
$338$	0	0
$339$	−1.54400	−0.0838588
$340$	−12.0000	−0.650791
$341$	−21.7720	−1.17902
$342$	0	0
$343$	41.8600	2.26023
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.5440	−1.37127	−0.685637	−	0.727944i	$-0.740476\pi$
$347$	−25.5440	−1.37127	−0.685637	+	0.727944i	$0.740476\pi$
$348$	−12.0000	−0.643268
$349$	5.68399	0.304257	0.152129	−	0.988361i	$-0.451387\pi$
$349$	5.68399	0.304257	0.152129	+	0.988361i	$0.451387\pi$
$350$	0	0
$351$	−4.77200	−0.254711
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	9.77200	0.518644
$356$	15.0880	0.799663
$357$	28.6320	1.51537
$358$	0	0
$359$	3.08801	0.162979	0.0814894	−	0.996674i	$-0.474032\pi$
$359$	3.08801	0.162979	0.0814894	+	0.996674i	$0.474032\pi$
$360$	0	0
$361$	14.3160	0.753474
$362$	0	0
$363$	3.22800	0.169426
$364$	45.5440	2.38715
$365$	2.00000	0.104685
$366$	0	0
$367$	11.5440	0.602592	0.301296	−	0.953531i	$-0.402581\pi$
$367$	11.5440	0.602592	0.301296	+	0.953531i	$0.402581\pi$
$368$	0	0
$369$	−9.77200	−0.508710
$370$	0	0
$371$	−64.6320	−3.35553
$372$	−11.5440	−0.598529
$373$	22.7720	1.17909	0.589545	−	0.807736i	$-0.299307\pi$
$373$	22.7720	1.17909	0.589545	+	0.807736i	$0.299307\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−28.6320	−1.47462
$378$	0	0
$379$	−14.7720	−0.758787	−0.379393	−	0.925235i	$-0.623867\pi$
$379$	−14.7720	−0.758787	−0.379393	+	0.925235i	$0.623867\pi$
$380$	11.5440	0.592195
$381$	1.22800	0.0629122
$382$	0	0
$383$	16.4560	0.840862	0.420431	−	0.907324i	$-0.361879\pi$
$383$	16.4560	0.840862	0.420431	+	0.907324i	$0.361879\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	0	0
$387$	−2.77200	−0.140909
$388$	−35.0880	−1.78132
$389$	29.3160	1.48638	0.743190	−	0.669080i	$-0.233312\pi$
$389$	29.3160	1.48638	0.743190	+	0.669080i	$0.233312\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.22800	−0.112388
$394$	0	0
$395$	13.7720	0.692945
$396$	7.54400	0.379100
$397$	28.3160	1.42114	0.710570	−	0.703627i	$-0.248437\pi$
$397$	28.3160	1.42114	0.710570	+	0.703627i	$0.248437\pi$
$398$	0	0
$399$	−27.5440	−1.37892
$400$	4.00000	0.200000
$401$	29.3160	1.46397	0.731986	−	0.681320i	$-0.238594\pi$
$401$	29.3160	1.46397	0.731986	+	0.681320i	$0.238594\pi$
$402$	0	0
$403$	−27.5440	−1.37206
$404$	−22.6320	−1.12598
$405$	1.00000	0.0496904
$406$	0	0
$407$	4.63201	0.229600
$408$	0	0
$409$	−38.5440	−1.90588	−0.952939	−	0.303162i	$-0.901958\pi$
$409$	−38.5440	−1.90588	−0.952939	+	0.303162i	$0.901958\pi$
$410$	0	0
$411$	7.54400	0.372118
$412$	−12.6320	−0.622335
$413$	0	0
$414$	0	0
$415$	13.5440	0.664849
$416$	0	0
$417$	−16.0880	−0.787833
$418$	0	0
$419$	−5.31601	−0.259704	−0.129852	−	0.991533i	$-0.541450\pi$
$419$	−5.31601	−0.259704	−0.129852	+	0.991533i	$0.541450\pi$
$420$	−9.54400	−0.465700
$421$	3.91199	0.190659	0.0953294	−	0.995446i	$-0.469610\pi$
$421$	3.91199	0.190659	0.0953294	+	0.995446i	$0.469610\pi$
$422$	0	0
$423$	7.54400	0.366802
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	40.7720	1.97310
$428$	−27.0880	−1.30935
$429$	18.0000	0.869048
$430$	0	0
$431$	24.8600	1.19746	0.598732	−	0.800949i	$-0.295671\pi$
$431$	24.8600	1.19746	0.598732	+	0.800949i	$0.295671\pi$
$432$	4.00000	0.192450
$433$	−4.31601	−0.207414	−0.103707	−	0.994608i	$-0.533070\pi$
$433$	−4.31601	−0.207414	−0.103707	+	0.994608i	$0.533070\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	14.6320	0.700746
$437$	0	0
$438$	0	0
$439$	−10.7720	−0.514120	−0.257060	−	0.966395i	$-0.582754\pi$
$439$	−10.7720	−0.514120	−0.257060	+	0.966395i	$0.582754\pi$
$440$	0	0
$441$	15.7720	0.751048
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	2.45600	0.116556
$445$	−7.54400	−0.357620
$446$	0	0
$447$	−8.22800	−0.389171
$448$	−38.1760	−1.80365
$449$	2.91199	0.137425	0.0687127	−	0.997636i	$-0.478111\pi$
$449$	2.91199	0.137425	0.0687127	+	0.997636i	$0.478111\pi$
$450$	0	0
$451$	36.8600	1.73567
$452$	3.08801	0.145248
$453$	−8.54400	−0.401432
$454$	0	0
$455$	−22.7720	−1.06757
$456$	0	0
$457$	21.2280	0.993004	0.496502	−	0.868036i	$-0.334618\pi$
$457$	21.2280	0.993004	0.496502	+	0.868036i	$0.334618\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	−14.2280	−0.662664	−0.331332	−	0.943514i	$-0.607498\pi$
$461$	−14.2280	−0.662664	−0.331332	+	0.943514i	$0.607498\pi$
$462$	0	0
$463$	−3.22800	−0.150018	−0.0750089	−	0.997183i	$-0.523899\pi$
$463$	−3.22800	−0.150018	−0.0750089	+	0.997183i	$0.523899\pi$
$464$	24.0000	1.11417
$465$	5.77200	0.267670
$466$	0	0
$467$	36.1760	1.67403	0.837013	−	0.547183i	$-0.184300\pi$
$467$	36.1760	1.67403	0.837013	+	0.547183i	$0.184300\pi$
$468$	9.54400	0.441172
$469$	15.4040	0.711291
$470$	0	0
$471$	2.45600	0.113166
$472$	0	0
$473$	10.4560	0.480767
$474$	0	0
$475$	−5.77200	−0.264838
$476$	−57.2640	−2.62469
$477$	−13.5440	−0.620137
$478$	0	0
$479$	12.6840	0.579546	0.289773	−	0.957095i	$-0.406420\pi$
$479$	12.6840	0.579546	0.289773	+	0.957095i	$0.406420\pi$
$480$	0	0
$481$	5.86001	0.267193
$482$	0	0
$483$	0	0
$484$	−6.45600	−0.293454
$485$	17.5440	0.796632
$486$	0	0
$487$	−1.68399	−0.0763091	−0.0381545	−	0.999272i	$-0.512148\pi$
$487$	−1.68399	−0.0763091	−0.0381545	+	0.999272i	$0.512148\pi$
$488$	0	0
$489$	−3.22800	−0.145975
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	19.5440	0.881112
$493$	36.0000	1.62136
$494$	0	0
$495$	−3.77200	−0.169539
$496$	23.0880	1.03668
$497$	46.6320	2.09173
$498$	0	0
$499$	−11.4560	−0.512841	−0.256420	−	0.966565i	$-0.582543\pi$
$499$	−11.4560	−0.512841	−0.256420	+	0.966565i	$0.582543\pi$
$500$	−2.00000	−0.0894427
$501$	−1.54400	−0.0689810
$502$	0	0
$503$	22.6320	1.00911	0.504556	−	0.863379i	$-0.331656\pi$
$503$	22.6320	1.00911	0.504556	+	0.863379i	$0.331656\pi$
$504$	0	0
$505$	11.3160	0.503556
$506$	0	0
$507$	9.77200	0.433990
$508$	−2.45600	−0.108967
$509$	22.6320	1.00315	0.501573	−	0.865115i	$-0.332755\pi$
$509$	22.6320	1.00315	0.501573	+	0.865115i	$0.332755\pi$
$510$	0	0
$511$	9.54400	0.422202
$512$	0	0
$513$	−5.77200	−0.254840
$514$	0	0
$515$	6.31601	0.278316
$516$	5.54400	0.244061
$517$	−28.4560	−1.25149
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.3160	−0.495763	−0.247882	−	0.968790i	$-0.579734\pi$
$521$	−11.3160	−0.495763	−0.247882	+	0.968790i	$0.579734\pi$
$522$	0	0
$523$	11.5440	0.504784	0.252392	−	0.967625i	$-0.418783\pi$
$523$	11.5440	0.504784	0.252392	+	0.967625i	$0.418783\pi$
$524$	4.45600	0.194661
$525$	4.77200	0.208267
$526$	0	0
$527$	34.6320	1.50859
$528$	−15.0880	−0.656621
$529$	0	0
$530$	0	0
$531$	0	0
$532$	55.0880	2.38837
$533$	46.6320	2.01986
$534$	0	0
$535$	13.5440	0.585558
$536$	0	0
$537$	11.3160	0.488322
$538$	0	0
$539$	−59.4920	−2.56250
$540$	−2.00000	−0.0860663
$541$	−16.6840	−0.717301	−0.358650	−	0.933472i	$-0.616763\pi$
$541$	−16.6840	−0.717301	−0.358650	+	0.933472i	$0.616763\pi$
$542$	0	0
$543$	7.77200	0.333529
$544$	0	0
$545$	−7.31601	−0.313383
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−15.0880	−0.644528
$549$	8.54400	0.364649
$550$	0	0
$551$	−34.6320	−1.47537
$552$	0	0
$553$	65.7200	2.79470
$554$	0	0
$555$	−1.22800	−0.0521256
$556$	32.1760	1.36457
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	13.2280	0.559485
$560$	19.0880	0.806616
$561$	−22.6320	−0.955524
$562$	0	0
$563$	33.0880	1.39449	0.697247	−	0.716831i	$-0.254408\pi$
$563$	33.0880	1.39449	0.697247	+	0.716831i	$0.254408\pi$
$564$	−15.0880	−0.635320
$565$	−1.54400	−0.0649567
$566$	0	0
$567$	4.77200	0.200405
$568$	0	0
$569$	−35.3160	−1.48052	−0.740262	−	0.672319i	$-0.765299\pi$
$569$	−35.3160	−1.48052	−0.740262	+	0.672319i	$0.765299\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	−36.0000	−1.50524
$573$	10.4560	0.436806
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	−29.5440	−1.22993	−0.614966	−	0.788553i	$-0.710830\pi$
$577$	−29.5440	−1.22993	−0.614966	+	0.788553i	$0.710830\pi$
$578$	0	0
$579$	8.77200	0.364552
$580$	−12.0000	−0.498273
$581$	64.6320	2.68139
$582$	0	0
$583$	51.0880	2.11585
$584$	0	0
$585$	−4.77200	−0.197298
$586$	0	0
$587$	−25.5440	−1.05431	−0.527157	−	0.849768i	$-0.676742\pi$
$587$	−25.5440	−1.05431	−0.527157	+	0.849768i	$0.676742\pi$
$588$	−31.5440	−1.30085
$589$	−33.3160	−1.37276
$590$	0	0
$591$	15.0880	0.620638
$592$	−4.91199	−0.201882
$593$	46.6320	1.91495	0.957474	−	0.288521i	$-0.0931635\pi$
$593$	46.6320	1.91495	0.957474	+	0.288521i	$0.0931635\pi$
$594$	0	0
$595$	28.6320	1.17380
$596$	16.4560	0.674064
$597$	−12.5440	−0.513392
$598$	0	0
$599$	−8.22800	−0.336187	−0.168093	−	0.985771i	$-0.553761\pi$
$599$	−8.22800	−0.336187	−0.168093	+	0.985771i	$0.553761\pi$
$600$	0	0
$601$	36.7200	1.49784	0.748920	−	0.662660i	$-0.230573\pi$
$601$	36.7200	1.49784	0.748920	+	0.662660i	$0.230573\pi$
$602$	0	0
$603$	3.22800	0.131454
$604$	17.0880	0.695301
$605$	3.22800	0.131237
$606$	0	0
$607$	34.3160	1.39284	0.696422	−	0.717633i	$-0.254774\pi$
$607$	34.3160	1.39284	0.696422	+	0.717633i	$0.254774\pi$
$608$	0	0
$609$	28.6320	1.16023
$610$	0	0
$611$	−36.0000	−1.45640
$612$	−12.0000	−0.485071
$613$	−4.31601	−0.174322	−0.0871609	−	0.996194i	$-0.527779\pi$
$613$	−4.31601	−0.174322	−0.0871609	+	0.996194i	$0.527779\pi$
$614$	0	0
$615$	−9.77200	−0.394045
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	−11.5440	−0.463618
$621$	0	0
$622$	0	0
$623$	−36.0000	−1.44231
$624$	−19.0880	−0.764132
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.7720	0.869490
$628$	−4.91199	−0.196010
$629$	−7.36799	−0.293781
$630$	0	0
$631$	−33.6320	−1.33887	−0.669435	−	0.742871i	$-0.733464\pi$
$631$	−33.6320	−1.33887	−0.669435	+	0.742871i	$0.733464\pi$
$632$	0	0
$633$	−20.5440	−0.816551
$634$	0	0
$635$	1.22800	0.0487316
$636$	27.0880	1.07411
$637$	−75.2640	−2.98207
$638$	0	0
$639$	9.77200	0.386574
$640$	0	0
$641$	−42.8600	−1.69287	−0.846434	−	0.532493i	$-0.821255\pi$
$641$	−42.8600	−1.69287	−0.846434	+	0.532493i	$0.821255\pi$
$642$	0	0
$643$	−14.7720	−0.582551	−0.291275	−	0.956639i	$-0.594080\pi$
$643$	−14.7720	−0.582551	−0.291275	+	0.956639i	$0.594080\pi$
$644$	0	0
$645$	−2.77200	−0.109147
$646$	0	0
$647$	28.6320	1.12564	0.562820	−	0.826579i	$-0.309716\pi$
$647$	28.6320	1.12564	0.562820	+	0.826579i	$0.309716\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	27.5440	1.07953
$652$	6.45600	0.252836
$653$	8.91199	0.348753	0.174377	−	0.984679i	$-0.444209\pi$
$653$	8.91199	0.348753	0.174377	+	0.984679i	$0.444209\pi$
$654$	0	0
$655$	−2.22800	−0.0870551
$656$	−39.0880	−1.52613
$657$	2.00000	0.0780274
$658$	0	0
$659$	41.3160	1.60944	0.804722	−	0.593652i	$-0.202315\pi$
$659$	41.3160	1.60944	0.804722	+	0.593652i	$0.202315\pi$
$660$	7.54400	0.293650
$661$	0.227998	0.00886810	0.00443405	−	0.999990i	$-0.498589\pi$
$661$	0.227998	0.00886810	0.00443405	+	0.999990i	$0.498589\pi$
$662$	0	0
$663$	−28.6320	−1.11198
$664$	0	0
$665$	−27.5440	−1.06811
$666$	0	0
$667$	0	0
$668$	3.08801	0.119479
$669$	8.77200	0.339145
$670$	0	0
$671$	−32.2280	−1.24415
$672$	0	0
$673$	−6.31601	−0.243464	−0.121732	−	0.992563i	$-0.538845\pi$
$673$	−6.31601	−0.243464	−0.121732	+	0.992563i	$0.538845\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	−19.5440	−0.751692
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	83.7200	3.21288
$680$	0	0
$681$	−4.45600	−0.170754
$682$	0	0
$683$	28.4560	1.08884	0.544419	−	0.838813i	$-0.316750\pi$
$683$	28.4560	1.08884	0.544419	+	0.838813i	$0.316750\pi$
$684$	11.5440	0.441396
$685$	7.54400	0.288242
$686$	0	0
$687$	−8.08801	−0.308577
$688$	−11.0880	−0.422726
$689$	64.6320	2.46228
$690$	0	0
$691$	−41.6320	−1.58376	−0.791878	−	0.610679i	$-0.790897\pi$
$691$	−41.6320	−1.58376	−0.791878	+	0.610679i	$0.790897\pi$
$692$	0	0
$693$	−18.0000	−0.683763
$694$	0	0
$695$	−16.0880	−0.610253
$696$	0	0
$697$	−58.6320	−2.22085
$698$	0	0
$699$	7.54400	0.285340
$700$	−9.54400	−0.360729
$701$	34.6320	1.30803	0.654017	−	0.756480i	$-0.273083\pi$
$701$	34.6320	1.30803	0.654017	+	0.756480i	$0.273083\pi$
$702$	0	0
$703$	7.08801	0.267329
$704$	30.1760	1.13730
$705$	7.54400	0.284124
$706$	0	0
$707$	54.0000	2.03088
$708$	0	0
$709$	19.8600	0.745858	0.372929	−	0.927860i	$-0.378353\pi$
$709$	19.8600	0.745858	0.372929	+	0.927860i	$0.378353\pi$
$710$	0	0
$711$	13.7720	0.516490
$712$	0	0
$713$	0	0
$714$	0	0
$715$	18.0000	0.673162
$716$	−22.6320	−0.845798
$717$	14.2280	0.531354
$718$	0	0
$719$	51.0880	1.90526	0.952631	−	0.304130i	$-0.0983657\pi$
$719$	51.0880	1.90526	0.952631	+	0.304130i	$0.0983657\pi$
$720$	4.00000	0.149071
$721$	30.1400	1.12247
$722$	0	0
$723$	19.0000	0.706618
$724$	−15.5440	−0.577688
$725$	6.00000	0.222834
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.6320	−0.615157
$732$	−17.0880	−0.631591
$733$	27.4040	1.01219	0.506095	−	0.862478i	$-0.331088\pi$
$733$	27.4040	1.01219	0.506095	+	0.862478i	$0.331088\pi$
$734$	0	0
$735$	15.7720	0.581759
$736$	0	0
$737$	−12.1760	−0.448509
$738$	0	0
$739$	7.91199	0.291047	0.145524	−	0.989355i	$-0.453513\pi$
$739$	7.91199	0.291047	0.145524	+	0.989355i	$0.453513\pi$
$740$	2.45600	0.0902842
$741$	27.5440	1.01185
$742$	0	0
$743$	10.6320	0.390051	0.195025	−	0.980798i	$-0.437521\pi$
$743$	10.6320	0.390051	0.195025	+	0.980798i	$0.437521\pi$
$744$	0	0
$745$	−8.22800	−0.301451
$746$	0	0
$747$	13.5440	0.495549
$748$	45.2640	1.65502
$749$	64.6320	2.36160
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	30.1760	1.10041
$753$	0.683994	0.0249261
$754$	0	0
$755$	−8.54400	−0.310948
$756$	−9.54400	−0.347112
$757$	1.08801	0.0395443	0.0197722	−	0.999805i	$-0.493706\pi$
$757$	1.08801	0.0395443	0.0197722	+	0.999805i	$0.493706\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.4040	−1.17464	−0.587322	−	0.809353i	$-0.699818\pi$
$761$	−32.4040	−1.17464	−0.587322	+	0.809353i	$0.699818\pi$
$762$	0	0
$763$	−34.9120	−1.26390
$764$	−20.9120	−0.756569
$765$	6.00000	0.216930
$766$	0	0
$767$	0	0
$768$	16.0000	0.577350
$769$	−6.54400	−0.235983	−0.117991	−	0.993015i	$-0.537646\pi$
$769$	−6.54400	−0.235983	−0.117991	+	0.993015i	$0.537646\pi$
$770$	0	0
$771$	−28.6320	−1.03116
$772$	−17.5440	−0.631423
$773$	2.91199	0.104737	0.0523685	−	0.998628i	$-0.483323\pi$
$773$	2.91199	0.104737	0.0523685	+	0.998628i	$0.483323\pi$
$774$	0	0
$775$	5.77200	0.207336
$776$	0	0
$777$	−5.86001	−0.210227
$778$	0	0
$779$	56.4040	2.02088
$780$	9.54400	0.341730
$781$	−36.8600	−1.31895
$782$	0	0
$783$	6.00000	0.214423
$784$	63.0880	2.25314
$785$	2.45600	0.0876583
$786$	0	0
$787$	−15.5440	−0.554084	−0.277042	−	0.960858i	$-0.589354\pi$
$787$	−15.5440	−0.554084	−0.277042	+	0.960858i	$0.589354\pi$
$788$	−30.1760	−1.07498
$789$	21.0880	0.750753
$790$	0	0
$791$	−7.36799	−0.261976
$792$	0	0
$793$	−40.7720	−1.44786
$794$	0	0
$795$	−13.5440	−0.480356
$796$	25.0880	0.889221
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	45.2640	1.60133
$800$	0	0
$801$	−7.54400	−0.266554
$802$	0	0
$803$	−7.54400	−0.266222
$804$	−6.45600	−0.227685
$805$	0	0
$806$	0	0
$807$	−3.77200	−0.132781
$808$	0	0
$809$	−39.9480	−1.40450	−0.702249	−	0.711932i	$-0.747821\pi$
$809$	−39.9480	−1.40450	−0.702249	+	0.711932i	$0.747821\pi$
$810$	0	0
$811$	11.0000	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$811$	11.0000	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$812$	−57.2640	−2.00957
$813$	−21.2280	−0.744498
$814$	0	0
$815$	−3.22800	−0.113072
$816$	24.0000	0.840168
$817$	16.0000	0.559769
$818$	0	0
$819$	−22.7720	−0.795718
$820$	19.5440	0.682506
$821$	32.4040	1.13091	0.565454	−	0.824780i	$-0.308701\pi$
$821$	32.4040	1.13091	0.565454	+	0.824780i	$0.308701\pi$
$822$	0	0
$823$	18.6320	0.649471	0.324736	−	0.945805i	$-0.394725\pi$
$823$	18.6320	0.649471	0.324736	+	0.945805i	$0.394725\pi$
$824$	0	0
$825$	−3.77200	−0.131324
$826$	0	0
$827$	−7.54400	−0.262331	−0.131165	−	0.991361i	$-0.541872\pi$
$827$	−7.54400	−0.262331	−0.131165	+	0.991361i	$0.541872\pi$
$828$	0	0
$829$	16.2280	0.563622	0.281811	−	0.959470i	$-0.409065\pi$
$829$	16.2280	0.563622	0.281811	+	0.959470i	$0.409065\pi$
$830$	0	0
$831$	17.8600	0.619557
$832$	38.1760	1.32352
$833$	94.6320	3.27880
$834$	0	0
$835$	−1.54400	−0.0534325
$836$	−43.5440	−1.50600
$837$	5.77200	0.199510
$838$	0	0
$839$	−5.31601	−0.183529	−0.0917644	−	0.995781i	$-0.529251\pi$
$839$	−5.31601	−0.183529	−0.0917644	+	0.995781i	$0.529251\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−5.31601	−0.183093
$844$	41.0880	1.41431
$845$	9.77200	0.336167
$846$	0	0
$847$	15.4040	0.529288
$848$	−54.1760	−1.86041
$849$	−14.7720	−0.506974
$850$	0	0
$851$	0	0
$852$	−19.5440	−0.669567
$853$	−28.9480	−0.991161	−0.495581	−	0.868562i	$-0.665045\pi$
$853$	−28.9480	−0.991161	−0.495581	+	0.868562i	$0.665045\pi$
$854$	0	0
$855$	−5.77200	−0.197398
$856$	0	0
$857$	−34.6320	−1.18301	−0.591503	−	0.806302i	$-0.701465\pi$
$857$	−34.6320	−1.18301	−0.591503	+	0.806302i	$0.701465\pi$
$858$	0	0
$859$	−9.22800	−0.314855	−0.157428	−	0.987531i	$-0.550320\pi$
$859$	−9.22800	−0.314855	−0.157428	+	0.987531i	$0.550320\pi$
$860$	5.54400	0.189049
$861$	−46.6320	−1.58921
$862$	0	0
$863$	−28.6320	−0.974645	−0.487322	−	0.873222i	$-0.662026\pi$
$863$	−28.6320	−0.974645	−0.487322	+	0.873222i	$0.662026\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000	0.645274
$868$	−55.0880	−1.86981
$869$	−51.9480	−1.76222
$870$	0	0
$871$	−15.4040	−0.521945
$872$	0	0
$873$	17.5440	0.593775
$874$	0	0
$875$	4.77200	0.161323
$876$	−4.00000	−0.135147
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	−13.5440	−0.456828
$880$	−15.0880	−0.508617
$881$	22.6320	0.762492	0.381246	−	0.924474i	$-0.375495\pi$
$881$	22.6320	0.762492	0.381246	+	0.924474i	$0.375495\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	57.2640	1.92600
$885$	0	0
$886$	0	0
$887$	−19.5440	−0.656223	−0.328112	−	0.944639i	$-0.606412\pi$
$887$	−19.5440	−0.656223	−0.328112	+	0.944639i	$0.606412\pi$
$888$	0	0
$889$	5.86001	0.196538
$890$	0	0
$891$	−3.77200	−0.126367
$892$	−17.5440	−0.587417
$893$	−43.5440	−1.45714
$894$	0	0
$895$	11.3160	0.378252
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.6320	1.15504
$900$	−2.00000	−0.0666667
$901$	−81.2640	−2.70730
$902$	0	0
$903$	−13.2280	−0.440200
$904$	0	0
$905$	7.77200	0.258350
$906$	0	0
$907$	−11.6840	−0.387961	−0.193980	−	0.981005i	$-0.562140\pi$
$907$	−11.6840	−0.387961	−0.193980	+	0.981005i	$0.562140\pi$
$908$	8.91199	0.295755
$909$	11.3160	0.375328
$910$	0	0
$911$	−50.4040	−1.66996	−0.834980	−	0.550281i	$-0.814521\pi$
$911$	−50.4040	−1.66996	−0.834980	+	0.550281i	$0.814521\pi$
$912$	−23.0880	−0.764520
$913$	−51.0880	−1.69077
$914$	0	0
$915$	8.54400	0.282456
$916$	16.1760	0.534471
$917$	−10.6320	−0.351100
$918$	0	0
$919$	10.0880	0.332773	0.166386	−	0.986061i	$-0.446790\pi$
$919$	10.0880	0.332773	0.166386	+	0.986061i	$0.446790\pi$
$920$	0	0
$921$	19.2280	0.633584
$922$	0	0
$923$	−46.6320	−1.53491
$924$	36.0000	1.18431
$925$	−1.22800	−0.0403763
$926$	0	0
$927$	6.31601	0.207445
$928$	0	0
$929$	−41.3160	−1.35553	−0.677767	−	0.735277i	$-0.737052\pi$
$929$	−41.3160	−1.35553	−0.677767	+	0.735277i	$0.737052\pi$
$930$	0	0
$931$	−91.0360	−2.98359
$932$	−15.0880	−0.494224
$933$	−10.4560	−0.342314
$934$	0	0
$935$	−22.6320	−0.740146
$936$	0	0
$937$	−43.4040	−1.41795	−0.708974	−	0.705235i	$-0.750841\pi$
$937$	−43.4040	−1.41795	−0.708974	+	0.705235i	$0.750841\pi$
$938$	0	0
$939$	15.2280	0.496947
$940$	−15.0880	−0.492117
$941$	−0.860009	−0.0280355	−0.0140178	−	0.999902i	$-0.504462\pi$
$941$	−0.860009	−0.0280355	−0.0140178	+	0.999902i	$0.504462\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.77200	0.155233
$946$	0	0
$947$	31.5440	1.02504	0.512521	−	0.858675i	$-0.328712\pi$
$947$	31.5440	1.02504	0.512521	+	0.858675i	$0.328712\pi$
$948$	−27.5440	−0.894588
$949$	−9.54400	−0.309811
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−57.0880	−1.84926	−0.924631	−	0.380864i	$-0.875627\pi$
$953$	−57.0880	−1.84926	−0.924631	+	0.380864i	$0.875627\pi$
$954$	0	0
$955$	10.4560	0.338348
$956$	−28.4560	−0.920333
$957$	−22.6320	−0.731589
$958$	0	0
$959$	36.0000	1.16250
$960$	−8.00000	−0.258199
$961$	2.31601	0.0747099
$962$	0	0
$963$	13.5440	0.436449
$964$	−38.0000	−1.22390
$965$	8.77200	0.282381
$966$	0	0
$967$	−31.8600	−1.02455	−0.512274	−	0.858822i	$-0.671197\pi$
$967$	−31.8600	−1.02455	−0.512274	+	0.858822i	$0.671197\pi$
$968$	0	0
$969$	−34.6320	−1.11254
$970$	0	0
$971$	−40.6320	−1.30394	−0.651972	−	0.758243i	$-0.726058\pi$
$971$	−40.6320	−1.30394	−0.651972	+	0.758243i	$0.726058\pi$
$972$	−2.00000	−0.0641500
$973$	−76.7720	−2.46120
$974$	0	0
$975$	−4.77200	−0.152826
$976$	34.1760	1.09395
$977$	7.54400	0.241354	0.120677	−	0.992692i	$-0.461493\pi$
$977$	7.54400	0.241354	0.120677	+	0.992692i	$0.461493\pi$
$978$	0	0
$979$	28.4560	0.909458
$980$	−31.5440	−1.00764
$981$	−7.31601	−0.233582
$982$	0	0
$983$	−10.6320	−0.339108	−0.169554	−	0.985521i	$-0.554233\pi$
$983$	−10.6320	−0.339108	−0.169554	+	0.985521i	$0.554233\pi$
$984$	0	0
$985$	15.0880	0.480744
$986$	0	0
$987$	36.0000	1.14589
$988$	−55.0880	−1.75258
$989$	0	0
$990$	0	0
$991$	−19.0880	−0.606351	−0.303175	−	0.952935i	$-0.598047\pi$
$991$	−19.0880	−0.606351	−0.303175	+	0.952935i	$0.598047\pi$
$992$	0	0
$993$	−2.54400	−0.0807315
$994$	0	0
$995$	−12.5440	−0.397672
$996$	−27.0880	−0.858316
$997$	−42.3160	−1.34016	−0.670081	−	0.742288i	$-0.733741\pi$
$997$	−42.3160	−1.34016	−0.670081	+	0.742288i	$0.733741\pi$
$998$	0	0
$999$	−1.22800	−0.0388521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7935.2.a.s.1.2	yes	2
23.22	odd	2		7935.2.a.r.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7935.2.a.r.1.1	✓	2	23.22	odd	2
7935.2.a.s.1.2	yes	2	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:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} -3.77200 q^{11} -2.00000 q^{12} -4.77200 q^{13} +1.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} -5.77200 q^{19} -2.00000 q^{20} +4.77200 q^{21} +1.00000 q^{25} +1.00000 q^{27} -9.54400 q^{28} +6.00000 q^{29} +5.77200 q^{31} -3.77200 q^{33} +4.77200 q^{35} -2.00000 q^{36} -1.22800 q^{37} -4.77200 q^{39} -9.77200 q^{41} -2.77200 q^{43} +7.54400 q^{44} +1.00000 q^{45} +7.54400 q^{47} +4.00000 q^{48} +15.7720 q^{49} +6.00000 q^{51} +9.54400 q^{52} -13.5440 q^{53} -3.77200 q^{55} -5.77200 q^{57} -2.00000 q^{60} +8.54400 q^{61} +4.77200 q^{63} -8.00000 q^{64} -4.77200 q^{65} +3.22800 q^{67} -12.0000 q^{68} +9.77200 q^{71} +2.00000 q^{73} +1.00000 q^{75} +11.5440 q^{76} -18.0000 q^{77} +13.7720 q^{79} +4.00000 q^{80} +1.00000 q^{81} +13.5440 q^{83} -9.54400 q^{84} +6.00000 q^{85} +6.00000 q^{87} -7.54400 q^{89} -22.7720 q^{91} +5.77200 q^{93} -5.77200 q^{95} +17.5440 q^{97} -3.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9} + q^{11} - 4 q^{12} - q^{13} + 2 q^{15} + 8 q^{16} + 12 q^{17} - 3 q^{19} - 4 q^{20} + q^{21} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 12 q^{29} + 3 q^{31} + q^{33} + q^{35} - 4 q^{36} - 11 q^{37} - q^{39} - 11 q^{41} + 3 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{47} + 8 q^{48} + 23 q^{49} + 12 q^{51} + 2 q^{52} - 10 q^{53} + q^{55} - 3 q^{57} - 4 q^{60} + q^{63} - 16 q^{64} - q^{65} + 15 q^{67} - 24 q^{68} + 11 q^{71} + 4 q^{73} + 2 q^{75} + 6 q^{76} - 36 q^{77} + 19 q^{79} + 8 q^{80} + 2 q^{81} + 10 q^{83} - 2 q^{84} + 12 q^{85} + 12 q^{87} + 2 q^{89} - 37 q^{91} + 3 q^{93} - 3 q^{95} + 18 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9 + q^11 - 4 * q^12 - q^13 + 2 * q^15 + 8 * q^16 + 12 * q^17 - 3 * q^19 - 4 * q^20 + q^21 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 12 * q^29 + 3 * q^31 + q^33 + q^35 - 4 * q^36 - 11 * q^37 - q^39 - 11 * q^41 + 3 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^47 + 8 * q^48 + 23 * q^49 + 12 * q^51 + 2 * q^52 - 10 * q^53 + q^55 - 3 * q^57 - 4 * q^60 + q^63 - 16 * q^64 - q^65 + 15 * q^67 - 24 * q^68 + 11 * q^71 + 4 * q^73 + 2 * q^75 + 6 * q^76 - 36 * q^77 + 19 * q^79 + 8 * q^80 + 2 * q^81 + 10 * q^83 - 2 * q^84 + 12 * q^85 + 12 * q^87 + 2 * q^89 - 37 * q^91 + 3 * q^93 - 3 * q^95 + 18 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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