LMFDB - Embedded newform 7938.2.a.cb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.460505 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -0.460505 q^{5} +1.00000 q^{8} -0.460505 q^{10} -3.64766 q^{11} -1.46050 q^{13} +1.00000 q^{16} +3.73385 q^{17} +4.05408 q^{19} -0.460505 q^{20} -3.64766 q^{22} +1.13307 q^{23} -4.78794 q^{25} -1.46050 q^{26} +8.97509 q^{29} -0.514589 q^{31} +1.00000 q^{32} +3.73385 q^{34} +9.10817 q^{37} +4.05408 q^{38} -0.460505 q^{40} +0.945916 q^{41} -9.32743 q^{43} -3.64766 q^{44} +1.13307 q^{46} +2.32743 q^{47} -4.78794 q^{50} -1.46050 q^{52} -12.4356 q^{53} +1.67977 q^{55} +8.97509 q^{58} -12.8961 q^{59} +12.0833 q^{61} -0.514589 q^{62} +1.00000 q^{64} +0.672570 q^{65} -2.32023 q^{67} +3.73385 q^{68} +1.67977 q^{71} +13.2412 q^{73} +9.10817 q^{74} +4.05408 q^{76} -5.00720 q^{79} -0.460505 q^{80} +0.945916 q^{82} -6.64766 q^{83} -1.71946 q^{85} -9.32743 q^{86} -3.64766 q^{88} +2.72665 q^{89} +1.13307 q^{92} +2.32743 q^{94} -1.86693 q^{95} +11.1872 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 5 * q^5 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 5 q^{5} + 3 q^{8} + 5 q^{10} + q^{11} + 2 q^{13} + 3 q^{16} + 4 q^{17} + 3 q^{19} + 5 q^{20} + q^{22} + 7 q^{23} + 2 q^{25} + 2 q^{26} + 5 q^{29} + 14 q^{31} + 3 q^{32} + 4 q^{34} + 9 q^{37} + 3 q^{38} + 5 q^{40} + 12 q^{41} - 18 q^{43} + q^{44} + 7 q^{46} - 3 q^{47} + 2 q^{50} + 2 q^{52} - 9 q^{53} + 7 q^{55} + 5 q^{58} - 4 q^{59} - 4 q^{61} + 14 q^{62} + 3 q^{64} + 12 q^{65} - 5 q^{67} + 4 q^{68} + 7 q^{71} + 25 q^{73} + 9 q^{74} + 3 q^{76} - 7 q^{79} + 5 q^{80} + 12 q^{82} - 8 q^{83} - 14 q^{85} - 18 q^{86} + q^{88} + 9 q^{89} + 7 q^{92} - 3 q^{94} - 2 q^{95} + 28 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 5 * q^5 + 3 * q^8 + 5 * q^10 + q^11 + 2 * q^13 + 3 * q^16 + 4 * q^17 + 3 * q^19 + 5 * q^20 + q^22 + 7 * q^23 + 2 * q^25 + 2 * q^26 + 5 * q^29 + 14 * q^31 + 3 * q^32 + 4 * q^34 + 9 * q^37 + 3 * q^38 + 5 * q^40 + 12 * q^41 - 18 * q^43 + q^44 + 7 * q^46 - 3 * q^47 + 2 * q^50 + 2 * q^52 - 9 * q^53 + 7 * q^55 + 5 * q^58 - 4 * q^59 - 4 * q^61 + 14 * q^62 + 3 * q^64 + 12 * q^65 - 5 * q^67 + 4 * q^68 + 7 * q^71 + 25 * q^73 + 9 * q^74 + 3 * q^76 - 7 * q^79 + 5 * q^80 + 12 * q^82 - 8 * q^83 - 14 * q^85 - 18 * q^86 + q^88 + 9 * q^89 + 7 * q^92 - 3 * q^94 - 2 * q^95 + 28 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.460505	−0.205944	−0.102972	−	0.994684i	$-0.532835\pi$
$5$	−0.460505	−0.205944	−0.102972	+	0.994684i	$0.532835\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.460505	−0.145624
$11$	−3.64766	−1.09981	−0.549906	−	0.835227i	$-0.685336\pi$
$11$	−3.64766	−1.09981	−0.549906	+	0.835227i	$0.685336\pi$
$12$	0	0
$13$	−1.46050	−0.405071	−0.202536	−	0.979275i	$-0.564918\pi$
$13$	−1.46050	−0.405071	−0.202536	+	0.979275i	$0.564918\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	3.73385	0.905592	0.452796	−	0.891614i	$-0.350427\pi$
$17$	3.73385	0.905592	0.452796	+	0.891614i	$0.350427\pi$
$18$	0	0
$19$	4.05408	0.930071	0.465035	−	0.885292i	$-0.346042\pi$
$19$	4.05408	0.930071	0.465035	+	0.885292i	$0.346042\pi$
$20$	−0.460505	−0.102972
$21$	0	0
$22$	−3.64766	−0.777684
$23$	1.13307	0.236262	0.118131	−	0.992998i	$-0.462310\pi$
$23$	1.13307	0.236262	0.118131	+	0.992998i	$0.462310\pi$
$24$	0	0
$25$	−4.78794	−0.957587
$26$	−1.46050	−0.286429
$27$	0	0
$28$	0	0
$29$	8.97509	1.66663	0.833317	−	0.552796i	$-0.186439\pi$
$29$	8.97509	1.66663	0.833317	+	0.552796i	$0.186439\pi$
$30$	0	0
$31$	−0.514589	−0.0924229	−0.0462115	−	0.998932i	$-0.514715\pi$
$31$	−0.514589	−0.0924229	−0.0462115	+	0.998932i	$0.514715\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.73385	0.640350
$35$	0	0
$36$	0	0
$37$	9.10817	1.49737	0.748687	−	0.662924i	$-0.230685\pi$
$37$	9.10817	1.49737	0.748687	+	0.662924i	$0.230685\pi$
$38$	4.05408	0.657659
$39$	0	0
$40$	−0.460505	−0.0728122
$41$	0.945916	0.147727	0.0738636	−	0.997268i	$-0.476467\pi$
$41$	0.945916	0.147727	0.0738636	+	0.997268i	$0.476467\pi$
$42$	0	0
$43$	−9.32743	−1.42242	−0.711210	−	0.702980i	$-0.751852\pi$
$43$	−9.32743	−1.42242	−0.711210	+	0.702980i	$0.751852\pi$
$44$	−3.64766	−0.549906
$45$	0	0
$46$	1.13307	0.167063
$47$	2.32743	0.339491	0.169745	−	0.985488i	$-0.445705\pi$
$47$	2.32743	0.339491	0.169745	+	0.985488i	$0.445705\pi$
$48$	0	0
$49$	0	0
$50$	−4.78794	−0.677116
$51$	0	0
$52$	−1.46050	−0.202536
$53$	−12.4356	−1.70816	−0.854080	−	0.520141i	$-0.825879\pi$
$53$	−12.4356	−1.70816	−0.854080	+	0.520141i	$0.825879\pi$
$54$	0	0
$55$	1.67977	0.226500
$56$	0	0
$57$	0	0
$58$	8.97509	1.17849
$59$	−12.8961	−1.67893	−0.839465	−	0.543414i	$-0.817131\pi$
$59$	−12.8961	−1.67893	−0.839465	+	0.543414i	$0.817131\pi$
$60$	0	0
$61$	12.0833	1.54710	0.773552	−	0.633733i	$-0.218478\pi$
$61$	12.0833	1.54710	0.773552	+	0.633733i	$0.218478\pi$
$62$	−0.514589	−0.0653529
$63$	0	0
$64$	1.00000	0.125000
$65$	0.672570	0.0834220
$66$	0	0
$67$	−2.32023	−0.283462	−0.141731	−	0.989905i	$-0.545267\pi$
$67$	−2.32023	−0.283462	−0.141731	+	0.989905i	$0.545267\pi$
$68$	3.73385	0.452796
$69$	0	0
$70$	0	0
$71$	1.67977	0.199352	0.0996758	−	0.995020i	$-0.468219\pi$
$71$	1.67977	0.199352	0.0996758	+	0.995020i	$0.468219\pi$
$72$	0	0
$73$	13.2412	1.54977	0.774885	−	0.632102i	$-0.217808\pi$
$73$	13.2412	1.54977	0.774885	+	0.632102i	$0.217808\pi$
$74$	9.10817	1.05880
$75$	0	0
$76$	4.05408	0.465035
$77$	0	0
$78$	0	0
$79$	−5.00720	−0.563354	−0.281677	−	0.959509i	$-0.590891\pi$
$79$	−5.00720	−0.563354	−0.281677	+	0.959509i	$0.590891\pi$
$80$	−0.460505	−0.0514860
$81$	0	0
$82$	0.945916	0.104459
$83$	−6.64766	−0.729676	−0.364838	−	0.931071i	$-0.618876\pi$
$83$	−6.64766	−0.729676	−0.364838	+	0.931071i	$0.618876\pi$
$84$	0	0
$85$	−1.71946	−0.186501
$86$	−9.32743	−1.00580
$87$	0	0
$88$	−3.64766	−0.388842
$89$	2.72665	0.289025	0.144512	−	0.989503i	$-0.453839\pi$
$89$	2.72665	0.289025	0.144512	+	0.989503i	$0.453839\pi$
$90$	0	0
$91$	0	0
$92$	1.13307	0.118131
$93$	0	0
$94$	2.32743	0.240056
$95$	−1.86693	−0.191543
$96$	0	0
$97$	11.1872	1.13588	0.567942	−	0.823069i	$-0.307740\pi$
$97$	11.1872	1.13588	0.567942	+	0.823069i	$0.307740\pi$
$98$	0	0
$99$	0	0
$100$	−4.78794	−0.478794
$101$	13.7558	1.36876	0.684378	−	0.729127i	$-0.260074\pi$
$101$	13.7558	1.36876	0.684378	+	0.729127i	$0.260074\pi$
$102$	0	0
$103$	11.1623	1.09985	0.549925	−	0.835214i	$-0.314656\pi$
$103$	11.1623	1.09985	0.549925	+	0.835214i	$0.314656\pi$
$104$	−1.46050	−0.143214
$105$	0	0
$106$	−12.4356	−1.20785
$107$	7.78074	0.752192	0.376096	−	0.926581i	$-0.377266\pi$
$107$	7.78074	0.752192	0.376096	+	0.926581i	$0.377266\pi$
$108$	0	0
$109$	7.51459	0.719767	0.359884	−	0.932997i	$-0.382816\pi$
$109$	7.51459	0.719767	0.359884	+	0.932997i	$0.382816\pi$
$110$	1.67977	0.160159
$111$	0	0
$112$	0	0
$113$	−6.06128	−0.570197	−0.285099	−	0.958498i	$-0.592026\pi$
$113$	−6.06128	−0.570197	−0.285099	+	0.958498i	$0.592026\pi$
$114$	0	0
$115$	−0.521786	−0.0486568
$116$	8.97509	0.833317
$117$	0	0
$118$	−12.8961	−1.18718
$119$	0	0
$120$	0	0
$121$	2.30545	0.209586
$122$	12.0833	1.09397
$123$	0	0
$124$	−0.514589	−0.0462115
$125$	4.50739	0.403153
$126$	0	0
$127$	8.80992	0.781754	0.390877	−	0.920443i	$-0.372172\pi$
$127$	8.80992	0.781754	0.390877	+	0.920443i	$0.372172\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.672570	0.0589883
$131$	21.1373	1.84678	0.923389	−	0.383865i	$-0.125407\pi$
$131$	21.1373	1.84678	0.923389	+	0.383865i	$0.125407\pi$
$132$	0	0
$133$	0	0
$134$	−2.32023	−0.200438
$135$	0	0
$136$	3.73385	0.320175
$137$	−4.40642	−0.376466	−0.188233	−	0.982124i	$-0.560276\pi$
$137$	−4.40642	−0.376466	−0.188233	+	0.982124i	$0.560276\pi$
$138$	0	0
$139$	2.02491	0.171750	0.0858751	−	0.996306i	$-0.472631\pi$
$139$	2.02491	0.171750	0.0858751	+	0.996306i	$0.472631\pi$
$140$	0	0
$141$	0	0
$142$	1.67977	0.140963
$143$	5.32743	0.445502
$144$	0	0
$145$	−4.13307	−0.343233
$146$	13.2412	1.09585
$147$	0	0
$148$	9.10817	0.748687
$149$	−9.16225	−0.750601	−0.375300	−	0.926903i	$-0.622460\pi$
$149$	−9.16225	−0.750601	−0.375300	+	0.926903i	$0.622460\pi$
$150$	0	0
$151$	−0.103896	−0.00845496	−0.00422748	−	0.999991i	$-0.501346\pi$
$151$	−0.103896	−0.00845496	−0.00422748	+	0.999991i	$0.501346\pi$
$152$	4.05408	0.328830
$153$	0	0
$154$	0	0
$155$	0.236971	0.0190340
$156$	0	0
$157$	20.9823	1.67457	0.837285	−	0.546767i	$-0.184142\pi$
$157$	20.9823	1.67457	0.837285	+	0.546767i	$0.184142\pi$
$158$	−5.00720	−0.398351
$159$	0	0
$160$	−0.460505	−0.0364061
$161$	0	0
$162$	0	0
$163$	23.0364	1.80435	0.902174	−	0.431372i	$-0.141970\pi$
$163$	23.0364	1.80435	0.902174	+	0.431372i	$0.141970\pi$
$164$	0.945916	0.0738636
$165$	0	0
$166$	−6.64766	−0.515959
$167$	10.6300	0.822571	0.411285	−	0.911507i	$-0.365080\pi$
$167$	10.6300	0.822571	0.411285	+	0.911507i	$0.365080\pi$
$168$	0	0
$169$	−10.8669	−0.835917
$170$	−1.71946	−0.131876
$171$	0	0
$172$	−9.32743	−0.711210
$173$	2.93872	0.223427	0.111713	−	0.993740i	$-0.464366\pi$
$173$	2.93872	0.223427	0.111713	+	0.993740i	$0.464366\pi$
$174$	0	0
$175$	0	0
$176$	−3.64766	−0.274953
$177$	0	0
$178$	2.72665	0.204371
$179$	9.16225	0.684819	0.342409	−	0.939551i	$-0.388757\pi$
$179$	9.16225	0.684819	0.342409	+	0.939551i	$0.388757\pi$
$180$	0	0
$181$	22.4284	1.66709	0.833545	−	0.552452i	$-0.186308\pi$
$181$	22.4284	1.66709	0.833545	+	0.552452i	$0.186308\pi$
$182$	0	0
$183$	0	0
$184$	1.13307	0.0835314
$185$	−4.19436	−0.308375
$186$	0	0
$187$	−13.6198	−0.995981
$188$	2.32743	0.169745
$189$	0	0
$190$	−1.86693	−0.135441
$191$	2.48968	0.180147	0.0900736	−	0.995935i	$-0.471290\pi$
$191$	2.48968	0.180147	0.0900736	+	0.995935i	$0.471290\pi$
$192$	0	0
$193$	4.48968	0.323174	0.161587	−	0.986858i	$-0.448339\pi$
$193$	4.48968	0.323174	0.161587	+	0.986858i	$0.448339\pi$
$194$	11.1872	0.803191
$195$	0	0
$196$	0	0
$197$	12.7339	0.907249	0.453625	−	0.891193i	$-0.350131\pi$
$197$	12.7339	0.907249	0.453625	+	0.891193i	$0.350131\pi$
$198$	0	0
$199$	2.94592	0.208830	0.104415	−	0.994534i	$-0.466703\pi$
$199$	2.94592	0.208830	0.104415	+	0.994534i	$0.466703\pi$
$200$	−4.78794	−0.338558
$201$	0	0
$202$	13.7558	0.967857
$203$	0	0
$204$	0	0
$205$	−0.435599	−0.0304235
$206$	11.1623	0.777711
$207$	0	0
$208$	−1.46050	−0.101268
$209$	−14.7879	−1.02290
$210$	0	0
$211$	1.21634	0.0837361	0.0418680	−	0.999123i	$-0.486669\pi$
$211$	1.21634	0.0837361	0.0418680	+	0.999123i	$0.486669\pi$
$212$	−12.4356	−0.854080
$213$	0	0
$214$	7.78074	0.531880
$215$	4.29533	0.292939
$216$	0	0
$217$	0	0
$218$	7.51459	0.508952
$219$	0	0
$220$	1.67977	0.113250
$221$	−5.45331	−0.366829
$222$	0	0
$223$	0.891832	0.0597215	0.0298607	−	0.999554i	$-0.490494\pi$
$223$	0.891832	0.0597215	0.0298607	+	0.999554i	$0.490494\pi$
$224$	0	0
$225$	0	0
$226$	−6.06128	−0.403190
$227$	−14.6519	−0.972483	−0.486242	−	0.873824i	$-0.661632\pi$
$227$	−14.6519	−0.972483	−0.486242	+	0.873824i	$0.661632\pi$
$228$	0	0
$229$	−9.57587	−0.632791	−0.316396	−	0.948627i	$-0.602473\pi$
$229$	−9.57587	−0.632791	−0.316396	+	0.948627i	$0.602473\pi$
$230$	−0.521786	−0.0344056
$231$	0	0
$232$	8.97509	0.589244
$233$	−14.4284	−0.945236	−0.472618	−	0.881267i	$-0.656691\pi$
$233$	−14.4284	−0.945236	−0.472618	+	0.881267i	$0.656691\pi$
$234$	0	0
$235$	−1.07179	−0.0699161
$236$	−12.8961	−0.839465
$237$	0	0
$238$	0	0
$239$	18.3097	1.18436	0.592179	−	0.805807i	$-0.298268\pi$
$239$	18.3097	1.18436	0.592179	+	0.805807i	$0.298268\pi$
$240$	0	0
$241$	0.0933847	0.00601544	0.00300772	−	0.999995i	$-0.499043\pi$
$241$	0.0933847	0.00601544	0.00300772	+	0.999995i	$0.499043\pi$
$242$	2.30545	0.148200
$243$	0	0
$244$	12.0833	0.773552
$245$	0	0
$246$	0	0
$247$	−5.92101	−0.376745
$248$	−0.514589	−0.0326764
$249$	0	0
$250$	4.50739	0.285072
$251$	−18.2733	−1.15340	−0.576702	−	0.816955i	$-0.695661\pi$
$251$	−18.2733	−1.15340	−0.576702	+	0.816955i	$0.695661\pi$
$252$	0	0
$253$	−4.13307	−0.259844
$254$	8.80992	0.552783
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.0512	−1.31314	−0.656568	−	0.754267i	$-0.727992\pi$
$257$	−21.0512	−1.31314	−0.656568	+	0.754267i	$0.727992\pi$
$258$	0	0
$259$	0	0
$260$	0.672570	0.0417110
$261$	0	0
$262$	21.1373	1.30587
$263$	−5.16518	−0.318499	−0.159249	−	0.987238i	$-0.550907\pi$
$263$	−5.16518	−0.318499	−0.159249	+	0.987238i	$0.550907\pi$
$264$	0	0
$265$	5.72665	0.351786
$266$	0	0
$267$	0	0
$268$	−2.32023	−0.141731
$269$	−16.8568	−1.02778	−0.513889	−	0.857857i	$-0.671796\pi$
$269$	−16.8568	−1.02778	−0.513889	+	0.857857i	$0.671796\pi$
$270$	0	0
$271$	−25.1124	−1.52547	−0.762736	−	0.646710i	$-0.776144\pi$
$271$	−25.1124	−1.52547	−0.762736	+	0.646710i	$0.776144\pi$
$272$	3.73385	0.226398
$273$	0	0
$274$	−4.40642	−0.266202
$275$	17.4648	1.05317
$276$	0	0
$277$	3.38151	0.203176	0.101588	−	0.994827i	$-0.467608\pi$
$277$	3.38151	0.203176	0.101588	+	0.994827i	$0.467608\pi$
$278$	2.02491	0.121446
$279$	0	0
$280$	0	0
$281$	20.2776	1.20966	0.604831	−	0.796354i	$-0.293241\pi$
$281$	20.2776	1.20966	0.604831	+	0.796354i	$0.293241\pi$
$282$	0	0
$283$	17.3494	1.03132	0.515658	−	0.856795i	$-0.327548\pi$
$283$	17.3494	1.03132	0.515658	+	0.856795i	$0.327548\pi$
$284$	1.67977	0.0996758
$285$	0	0
$286$	5.32743	0.315018
$287$	0	0
$288$	0	0
$289$	−3.05836	−0.179903
$290$	−4.13307	−0.242702
$291$	0	0
$292$	13.2412	0.774885
$293$	9.87120	0.576682	0.288341	−	0.957528i	$-0.406896\pi$
$293$	9.87120	0.576682	0.288341	+	0.957528i	$0.406896\pi$
$294$	0	0
$295$	5.93872	0.345766
$296$	9.10817	0.529402
$297$	0	0
$298$	−9.16225	−0.530755
$299$	−1.65486	−0.0957031
$300$	0	0
$301$	0	0
$302$	−0.103896	−0.00597856
$303$	0	0
$304$	4.05408	0.232518
$305$	−5.56440	−0.318617
$306$	0	0
$307$	7.78794	0.444481	0.222240	−	0.974992i	$-0.428663\pi$
$307$	7.78794	0.444481	0.222240	+	0.974992i	$0.428663\pi$
$308$	0	0
$309$	0	0
$310$	0.236971	0.0134590
$311$	15.4107	0.873860	0.436930	−	0.899495i	$-0.356066\pi$
$311$	15.4107	0.873860	0.436930	+	0.899495i	$0.356066\pi$
$312$	0	0
$313$	8.49688	0.480272	0.240136	−	0.970739i	$-0.422808\pi$
$313$	8.49688	0.480272	0.240136	+	0.970739i	$0.422808\pi$
$314$	20.9823	1.18410
$315$	0	0
$316$	−5.00720	−0.281677
$317$	−14.1052	−0.792229	−0.396115	−	0.918201i	$-0.629642\pi$
$317$	−14.1052	−0.792229	−0.396115	+	0.918201i	$0.629642\pi$
$318$	0	0
$319$	−32.7381	−1.83298
$320$	−0.460505	−0.0257430
$321$	0	0
$322$	0	0
$323$	15.1373	0.842264
$324$	0	0
$325$	6.99280	0.387891
$326$	23.0364	1.27587
$327$	0	0
$328$	0.945916	0.0522295
$329$	0	0
$330$	0	0
$331$	27.5438	1.51394	0.756971	−	0.653448i	$-0.226678\pi$
$331$	27.5438	1.51394	0.756971	+	0.653448i	$0.226678\pi$
$332$	−6.64766	−0.364838
$333$	0	0
$334$	10.6300	0.581645
$335$	1.06848	0.0583772
$336$	0	0
$337$	−1.49688	−0.0815403	−0.0407701	−	0.999169i	$-0.512981\pi$
$337$	−1.49688	−0.0815403	−0.0407701	+	0.999169i	$0.512981\pi$
$338$	−10.8669	−0.591083
$339$	0	0
$340$	−1.71946	−0.0932506
$341$	1.87705	0.101648
$342$	0	0
$343$	0	0
$344$	−9.32743	−0.502901
$345$	0	0
$346$	2.93872	0.157986
$347$	−18.2881	−0.981758	−0.490879	−	0.871228i	$-0.663324\pi$
$347$	−18.2881	−0.981758	−0.490879	+	0.871228i	$0.663324\pi$
$348$	0	0
$349$	7.80272	0.417670	0.208835	−	0.977951i	$-0.433033\pi$
$349$	7.80272	0.417670	0.208835	+	0.977951i	$0.433033\pi$
$350$	0	0
$351$	0	0
$352$	−3.64766	−0.194421
$353$	26.9253	1.43309	0.716544	−	0.697542i	$-0.245723\pi$
$353$	26.9253	1.43309	0.716544	+	0.697542i	$0.245723\pi$
$354$	0	0
$355$	−0.773541	−0.0410553
$356$	2.72665	0.144512
$357$	0	0
$358$	9.16225	0.484240
$359$	6.26322	0.330560	0.165280	−	0.986247i	$-0.447147\pi$
$359$	6.26322	0.330560	0.165280	+	0.986247i	$0.447147\pi$
$360$	0	0
$361$	−2.56440	−0.134968
$362$	22.4284	1.17881
$363$	0	0
$364$	0	0
$365$	−6.09766	−0.319166
$366$	0	0
$367$	29.2733	1.52806	0.764028	−	0.645183i	$-0.223219\pi$
$367$	29.2733	1.52806	0.764028	+	0.645183i	$0.223219\pi$
$368$	1.13307	0.0590656
$369$	0	0
$370$	−4.19436	−0.218054
$371$	0	0
$372$	0	0
$373$	17.8597	0.924742	0.462371	−	0.886687i	$-0.346999\pi$
$373$	17.8597	0.924742	0.462371	+	0.886687i	$0.346999\pi$
$374$	−13.6198	−0.704265
$375$	0	0
$376$	2.32743	0.120028
$377$	−13.1082	−0.675105
$378$	0	0
$379$	−22.4255	−1.15192	−0.575960	−	0.817478i	$-0.695371\pi$
$379$	−22.4255	−1.15192	−0.575960	+	0.817478i	$0.695371\pi$
$380$	−1.86693	−0.0957713
$381$	0	0
$382$	2.48968	0.127383
$383$	−14.1403	−0.722534	−0.361267	−	0.932462i	$-0.617656\pi$
$383$	−14.1403	−0.722534	−0.361267	+	0.932462i	$0.617656\pi$
$384$	0	0
$385$	0	0
$386$	4.48968	0.228519
$387$	0	0
$388$	11.1872	0.567942
$389$	−23.1301	−1.17275	−0.586373	−	0.810041i	$-0.699445\pi$
$389$	−23.1301	−1.17275	−0.586373	+	0.810041i	$0.699445\pi$
$390$	0	0
$391$	4.23073	0.213957
$392$	0	0
$393$	0	0
$394$	12.7339	0.641522
$395$	2.30584	0.116019
$396$	0	0
$397$	10.2661	0.515243	0.257622	−	0.966246i	$-0.417061\pi$
$397$	10.2661	0.515243	0.257622	+	0.966246i	$0.417061\pi$
$398$	2.94592	0.147665
$399$	0	0
$400$	−4.78794	−0.239397
$401$	34.0335	1.69955	0.849775	−	0.527146i	$-0.176738\pi$
$401$	34.0335	1.69955	0.849775	+	0.527146i	$0.176738\pi$
$402$	0	0
$403$	0.751560	0.0374379
$404$	13.7558	0.684378
$405$	0	0
$406$	0	0
$407$	−33.2235	−1.64683
$408$	0	0
$409$	−3.48968	−0.172554	−0.0862769	−	0.996271i	$-0.527497\pi$
$409$	−3.48968	−0.172554	−0.0862769	+	0.996271i	$0.527497\pi$
$410$	−0.435599	−0.0215127
$411$	0	0
$412$	11.1623	0.549925
$413$	0	0
$414$	0	0
$415$	3.06128	0.150272
$416$	−1.46050	−0.0716071
$417$	0	0
$418$	−14.7879	−0.723302
$419$	−28.9794	−1.41573	−0.707867	−	0.706345i	$-0.750343\pi$
$419$	−28.9794	−1.41573	−0.707867	+	0.706345i	$0.750343\pi$
$420$	0	0
$421$	2.12256	0.103447	0.0517237	−	0.998661i	$-0.483528\pi$
$421$	2.12256	0.103447	0.0517237	+	0.998661i	$0.483528\pi$
$422$	1.21634	0.0592104
$423$	0	0
$424$	−12.4356	−0.603926
$425$	−17.8774	−0.867183
$426$	0	0
$427$	0	0
$428$	7.78074	0.376096
$429$	0	0
$430$	4.29533	0.207139
$431$	−21.8712	−1.05350	−0.526749	−	0.850021i	$-0.676589\pi$
$431$	−21.8712	−1.05350	−0.526749	+	0.850021i	$0.676589\pi$
$432$	0	0
$433$	−13.0512	−0.627199	−0.313599	−	0.949555i	$-0.601535\pi$
$433$	−13.0512	−0.627199	−0.313599	+	0.949555i	$0.601535\pi$
$434$	0	0
$435$	0	0
$436$	7.51459	0.359884
$437$	4.59358	0.219741
$438$	0	0
$439$	4.86400	0.232146	0.116073	−	0.993241i	$-0.462969\pi$
$439$	4.86400	0.232146	0.116073	+	0.993241i	$0.462969\pi$
$440$	1.67977	0.0800797
$441$	0	0
$442$	−5.45331	−0.259387
$443$	−11.5395	−0.548258	−0.274129	−	0.961693i	$-0.588390\pi$
$443$	−11.5395	−0.548258	−0.274129	+	0.961693i	$0.588390\pi$
$444$	0	0
$445$	−1.25564	−0.0595229
$446$	0.891832	0.0422294
$447$	0	0
$448$	0	0
$449$	−26.4251	−1.24708	−0.623538	−	0.781793i	$-0.714306\pi$
$449$	−26.4251	−1.24708	−0.623538	+	0.781793i	$0.714306\pi$
$450$	0	0
$451$	−3.45038	−0.162472
$452$	−6.06128	−0.285099
$453$	0	0
$454$	−14.6519	−0.687649
$455$	0	0
$456$	0	0
$457$	−3.73812	−0.174862	−0.0874310	−	0.996171i	$-0.527866\pi$
$457$	−3.73812	−0.174862	−0.0874310	+	0.996171i	$0.527866\pi$
$458$	−9.57587	−0.447451
$459$	0	0
$460$	−0.521786	−0.0243284
$461$	15.8099	0.736341	0.368171	−	0.929758i	$-0.379984\pi$
$461$	15.8099	0.736341	0.368171	+	0.929758i	$0.379984\pi$
$462$	0	0
$463$	−38.3930	−1.78427	−0.892137	−	0.451766i	$-0.850794\pi$
$463$	−38.3930	−1.78427	−0.892137	+	0.451766i	$0.850794\pi$
$464$	8.97509	0.416658
$465$	0	0
$466$	−14.4284	−0.668383
$467$	−6.31304	−0.292132	−0.146066	−	0.989275i	$-0.546661\pi$
$467$	−6.31304	−0.292132	−0.146066	+	0.989275i	$0.546661\pi$
$468$	0	0
$469$	0	0
$470$	−1.07179	−0.0494381
$471$	0	0
$472$	−12.8961	−0.593591
$473$	34.0233	1.56439
$474$	0	0
$475$	−19.4107	−0.890624
$476$	0	0
$477$	0	0
$478$	18.3097	0.837467
$479$	−20.4136	−0.932722	−0.466361	−	0.884594i	$-0.654435\pi$
$479$	−20.4136	−0.932722	−0.466361	+	0.884594i	$0.654435\pi$
$480$	0	0
$481$	−13.3025	−0.606543
$482$	0.0933847	0.00425356
$483$	0	0
$484$	2.30545	0.104793
$485$	−5.15174	−0.233929
$486$	0	0
$487$	−12.3638	−0.560258	−0.280129	−	0.959962i	$-0.590377\pi$
$487$	−12.3638	−0.560258	−0.280129	+	0.959962i	$0.590377\pi$
$488$	12.0833	0.546984
$489$	0	0
$490$	0	0
$491$	−0.414007	−0.0186839	−0.00934194	−	0.999956i	$-0.502974\pi$
$491$	−0.414007	−0.0186839	−0.00934194	+	0.999956i	$0.502974\pi$
$492$	0	0
$493$	33.5117	1.50929
$494$	−5.92101	−0.266399
$495$	0	0
$496$	−0.514589	−0.0231057
$497$	0	0
$498$	0	0
$499$	−0.923935	−0.0413610	−0.0206805	−	0.999786i	$-0.506583\pi$
$499$	−0.923935	−0.0413610	−0.0206805	+	0.999786i	$0.506583\pi$
$500$	4.50739	0.201577
$501$	0	0
$502$	−18.2733	−0.815579
$503$	−23.8142	−1.06182	−0.530911	−	0.847428i	$-0.678150\pi$
$503$	−23.8142	−1.06182	−0.530911	+	0.847428i	$0.678150\pi$
$504$	0	0
$505$	−6.33463	−0.281887
$506$	−4.13307	−0.183738
$507$	0	0
$508$	8.80992	0.390877
$509$	−30.6342	−1.35784	−0.678919	−	0.734213i	$-0.737551\pi$
$509$	−30.6342	−1.35784	−0.678919	+	0.734213i	$0.737551\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−21.0512	−0.928527
$515$	−5.14027	−0.226507
$516$	0	0
$517$	−8.48968	−0.373376
$518$	0	0
$519$	0	0
$520$	0.672570	0.0294941
$521$	26.9037	1.17867	0.589336	−	0.807888i	$-0.299389\pi$
$521$	26.9037	1.17867	0.589336	+	0.807888i	$0.299389\pi$
$522$	0	0
$523$	15.7060	0.686776	0.343388	−	0.939194i	$-0.388425\pi$
$523$	15.7060	0.686776	0.343388	+	0.939194i	$0.388425\pi$
$524$	21.1373	0.923389
$525$	0	0
$526$	−5.16518	−0.225212
$527$	−1.92140	−0.0836975
$528$	0	0
$529$	−21.7161	−0.944180
$530$	5.72665	0.248750
$531$	0	0
$532$	0	0
$533$	−1.38151	−0.0598400
$534$	0	0
$535$	−3.58307	−0.154910
$536$	−2.32023	−0.100219
$537$	0	0
$538$	−16.8568	−0.726748
$539$	0	0
$540$	0	0
$541$	4.11868	0.177076	0.0885379	−	0.996073i	$-0.471781\pi$
$541$	4.11868	0.177076	0.0885379	+	0.996073i	$0.471781\pi$
$542$	−25.1124	−1.07867
$543$	0	0
$544$	3.73385	0.160088
$545$	−3.46050	−0.148232
$546$	0	0
$547$	23.7204	1.01421	0.507106	−	0.861884i	$-0.330715\pi$
$547$	23.7204	1.01421	0.507106	+	0.861884i	$0.330715\pi$
$548$	−4.40642	−0.188233
$549$	0	0
$550$	17.4648	0.744701
$551$	36.3858	1.55009
$552$	0	0
$553$	0	0
$554$	3.38151	0.143667
$555$	0	0
$556$	2.02491	0.0858751
$557$	42.0626	1.78225	0.891125	−	0.453757i	$-0.149917\pi$
$557$	42.0626	1.78225	0.891125	+	0.453757i	$0.149917\pi$
$558$	0	0
$559$	13.6228	0.576181
$560$	0	0
$561$	0	0
$562$	20.2776	0.855360
$563$	11.8243	0.498335	0.249168	−	0.968460i	$-0.419843\pi$
$563$	11.8243	0.498335	0.249168	+	0.968460i	$0.419843\pi$
$564$	0	0
$565$	2.79125	0.117429
$566$	17.3494	0.729250
$567$	0	0
$568$	1.67977	0.0704815
$569$	14.2016	0.595360	0.297680	−	0.954666i	$-0.403787\pi$
$569$	14.2016	0.595360	0.297680	+	0.954666i	$0.403787\pi$
$570$	0	0
$571$	11.9574	0.500401	0.250200	−	0.968194i	$-0.419503\pi$
$571$	11.9574	0.500401	0.250200	+	0.968194i	$0.419503\pi$
$572$	5.32743	0.222751
$573$	0	0
$574$	0	0
$575$	−5.42509	−0.226242
$576$	0	0
$577$	−42.6270	−1.77459	−0.887293	−	0.461206i	$-0.847417\pi$
$577$	−42.6270	−1.77459	−0.887293	+	0.461206i	$0.847417\pi$
$578$	−3.05836	−0.127211
$579$	0	0
$580$	−4.13307	−0.171617
$581$	0	0
$582$	0	0
$583$	45.3609	1.87866
$584$	13.2412	0.547927
$585$	0	0
$586$	9.87120	0.407775
$587$	−41.0656	−1.69496	−0.847478	−	0.530830i	$-0.821880\pi$
$587$	−41.0656	−1.69496	−0.847478	+	0.530830i	$0.821880\pi$
$588$	0	0
$589$	−2.08619	−0.0859599
$590$	5.93872	0.244493
$591$	0	0
$592$	9.10817	0.374343
$593$	−32.2016	−1.32236	−0.661180	−	0.750228i	$-0.729944\pi$
$593$	−32.2016	−1.32236	−0.661180	+	0.750228i	$0.729944\pi$
$594$	0	0
$595$	0	0
$596$	−9.16225	−0.375300
$597$	0	0
$598$	−1.65486	−0.0676723
$599$	19.0718	0.779252	0.389626	−	0.920973i	$-0.372604\pi$
$599$	19.0718	0.779252	0.389626	+	0.920973i	$0.372604\pi$
$600$	0	0
$601$	−8.54377	−0.348508	−0.174254	−	0.984701i	$-0.555751\pi$
$601$	−8.54377	−0.348508	−0.174254	+	0.984701i	$0.555751\pi$
$602$	0	0
$603$	0	0
$604$	−0.103896	−0.00422748
$605$	−1.06167	−0.0431631
$606$	0	0
$607$	38.0115	1.54284	0.771419	−	0.636328i	$-0.219547\pi$
$607$	38.0115	1.54284	0.771419	+	0.636328i	$0.219547\pi$
$608$	4.05408	0.164415
$609$	0	0
$610$	−5.56440	−0.225296
$611$	−3.39922	−0.137518
$612$	0	0
$613$	−22.6591	−0.915194	−0.457597	−	0.889160i	$-0.651290\pi$
$613$	−22.6591	−0.915194	−0.457597	+	0.889160i	$0.651290\pi$
$614$	7.78794	0.314295
$615$	0	0
$616$	0	0
$617$	20.2776	0.816346	0.408173	−	0.912905i	$-0.366166\pi$
$617$	20.2776	0.816346	0.408173	+	0.912905i	$0.366166\pi$
$618$	0	0
$619$	2.06128	0.0828499	0.0414249	−	0.999142i	$-0.486810\pi$
$619$	2.06128	0.0828499	0.0414249	+	0.999142i	$0.486810\pi$
$620$	0.236971	0.00951698
$621$	0	0
$622$	15.4107	0.617912
$623$	0	0
$624$	0	0
$625$	21.8640	0.874560
$626$	8.49688	0.339604
$627$	0	0
$628$	20.9823	0.837285
$629$	34.0085	1.35601
$630$	0	0
$631$	1.63715	0.0651740	0.0325870	−	0.999469i	$-0.489625\pi$
$631$	1.63715	0.0651740	0.0325870	+	0.999469i	$0.489625\pi$
$632$	−5.00720	−0.199176
$633$	0	0
$634$	−14.1052	−0.560191
$635$	−4.05701	−0.160998
$636$	0	0
$637$	0	0
$638$	−32.7381	−1.29611
$639$	0	0
$640$	−0.460505	−0.0182031
$641$	21.9325	0.866281	0.433140	−	0.901326i	$-0.357405\pi$
$641$	21.9325	0.866281	0.433140	+	0.901326i	$0.357405\pi$
$642$	0	0
$643$	28.3638	1.11856	0.559280	−	0.828979i	$-0.311078\pi$
$643$	28.3638	1.11856	0.559280	+	0.828979i	$0.311078\pi$
$644$	0	0
$645$	0	0
$646$	15.1373	0.595571
$647$	−34.7807	−1.36737	−0.683686	−	0.729776i	$-0.739624\pi$
$647$	−34.7807	−1.36737	−0.683686	+	0.729776i	$0.739624\pi$
$648$	0	0
$649$	47.0406	1.84651
$650$	6.99280	0.274280
$651$	0	0
$652$	23.0364	0.902174
$653$	−3.19863	−0.125172	−0.0625860	−	0.998040i	$-0.519935\pi$
$653$	−3.19863	−0.125172	−0.0625860	+	0.998040i	$0.519935\pi$
$654$	0	0
$655$	−9.73385	−0.380333
$656$	0.945916	0.0369318
$657$	0	0
$658$	0	0
$659$	−10.6084	−0.413243	−0.206622	−	0.978421i	$-0.566247\pi$
$659$	−10.6084	−0.413243	−0.206622	+	0.978421i	$0.566247\pi$
$660$	0	0
$661$	10.1301	0.394017	0.197009	−	0.980402i	$-0.436877\pi$
$661$	10.1301	0.394017	0.197009	+	0.980402i	$0.436877\pi$
$662$	27.5438	1.07052
$663$	0	0
$664$	−6.64766	−0.257979
$665$	0	0
$666$	0	0
$667$	10.1694	0.393763
$668$	10.6300	0.411285
$669$	0	0
$670$	1.06848	0.0412789
$671$	−44.0757	−1.70152
$672$	0	0
$673$	−3.21634	−0.123981	−0.0619903	−	0.998077i	$-0.519745\pi$
$673$	−3.21634	−0.123981	−0.0619903	+	0.998077i	$0.519745\pi$
$674$	−1.49688	−0.0576577
$675$	0	0
$676$	−10.8669	−0.417959
$677$	−29.3638	−1.12854	−0.564271	−	0.825589i	$-0.690843\pi$
$677$	−29.3638	−1.12854	−0.564271	+	0.825589i	$0.690843\pi$
$678$	0	0
$679$	0	0
$680$	−1.71946	−0.0659382
$681$	0	0
$682$	1.87705	0.0718759
$683$	−25.2556	−0.966380	−0.483190	−	0.875515i	$-0.660522\pi$
$683$	−25.2556	−0.966380	−0.483190	+	0.875515i	$0.660522\pi$
$684$	0	0
$685$	2.02918	0.0775309
$686$	0	0
$687$	0	0
$688$	−9.32743	−0.355605
$689$	18.1623	0.691927
$690$	0	0
$691$	−15.3638	−0.584467	−0.292233	−	0.956347i	$-0.594398\pi$
$691$	−15.3638	−0.584467	−0.292233	+	0.956347i	$0.594398\pi$
$692$	2.93872	0.111713
$693$	0	0
$694$	−18.2881	−0.694208
$695$	−0.932479	−0.0353709
$696$	0	0
$697$	3.53191	0.133781
$698$	7.80272	0.295337
$699$	0	0
$700$	0	0
$701$	−13.3700	−0.504980	−0.252490	−	0.967600i	$-0.581249\pi$
$701$	−13.3700	−0.504980	−0.252490	+	0.967600i	$0.581249\pi$
$702$	0	0
$703$	36.9253	1.39266
$704$	−3.64766	−0.137476
$705$	0	0
$706$	26.9253	1.01335
$707$	0	0
$708$	0	0
$709$	−1.12588	−0.0422832	−0.0211416	−	0.999776i	$-0.506730\pi$
$709$	−1.12588	−0.0422832	−0.0211416	+	0.999776i	$0.506730\pi$
$710$	−0.773541	−0.0290305
$711$	0	0
$712$	2.72665	0.102186
$713$	−0.583068	−0.0218361
$714$	0	0
$715$	−2.45331	−0.0917485
$716$	9.16225	0.342409
$717$	0	0
$718$	6.26322	0.233741
$719$	−18.2733	−0.681481	−0.340740	−	0.940157i	$-0.610678\pi$
$719$	−18.2733	−0.681481	−0.340740	+	0.940157i	$0.610678\pi$
$720$	0	0
$721$	0	0
$722$	−2.56440	−0.0954371
$723$	0	0
$724$	22.4284	0.833545
$725$	−42.9722	−1.59595
$726$	0	0
$727$	29.6955	1.10135	0.550673	−	0.834721i	$-0.314371\pi$
$727$	29.6955	1.10135	0.550673	+	0.834721i	$0.314371\pi$
$728$	0	0
$729$	0	0
$730$	−6.09766	−0.225684
$731$	−34.8272	−1.28813
$732$	0	0
$733$	19.2278	0.710195	0.355098	−	0.934829i	$-0.384448\pi$
$733$	19.2278	0.710195	0.355098	+	0.934829i	$0.384448\pi$
$734$	29.2733	1.08050
$735$	0	0
$736$	1.13307	0.0417657
$737$	8.46343	0.311754
$738$	0	0
$739$	30.2671	1.11339	0.556697	−	0.830716i	$-0.312069\pi$
$739$	30.2671	1.11339	0.556697	+	0.830716i	$0.312069\pi$
$740$	−4.19436	−0.154188
$741$	0	0
$742$	0	0
$743$	23.7630	0.871781	0.435890	−	0.900000i	$-0.356433\pi$
$743$	23.7630	0.871781	0.435890	+	0.900000i	$0.356433\pi$
$744$	0	0
$745$	4.21926	0.154582
$746$	17.8597	0.653891
$747$	0	0
$748$	−13.6198	−0.497990
$749$	0	0
$750$	0	0
$751$	12.6683	0.462273	0.231136	−	0.972921i	$-0.425756\pi$
$751$	12.6683	0.462273	0.231136	+	0.972921i	$0.425756\pi$
$752$	2.32743	0.0848727
$753$	0	0
$754$	−13.1082	−0.477371
$755$	0.0478448	0.00174125
$756$	0	0
$757$	−29.0799	−1.05693	−0.528464	−	0.848955i	$-0.677232\pi$
$757$	−29.0799	−1.05693	−0.528464	+	0.848955i	$0.677232\pi$
$758$	−22.4255	−0.814530
$759$	0	0
$760$	−1.86693	−0.0677205
$761$	29.2029	1.05860	0.529302	−	0.848433i	$-0.322454\pi$
$761$	29.2029	1.05860	0.529302	+	0.848433i	$0.322454\pi$
$762$	0	0
$763$	0	0
$764$	2.48968	0.0900736
$765$	0	0
$766$	−14.1403	−0.510909
$767$	18.8348	0.680086
$768$	0	0
$769$	−25.1737	−0.907788	−0.453894	−	0.891056i	$-0.649965\pi$
$769$	−25.1737	−0.907788	−0.453894	+	0.891056i	$0.649965\pi$
$770$	0	0
$771$	0	0
$772$	4.48968	0.161587
$773$	1.50408	0.0540979	0.0270490	−	0.999634i	$-0.491389\pi$
$773$	1.50408	0.0540979	0.0270490	+	0.999634i	$0.491389\pi$
$774$	0	0
$775$	2.46382	0.0885030
$776$	11.1872	0.401596
$777$	0	0
$778$	−23.1301	−0.829256
$779$	3.83482	0.137397
$780$	0	0
$781$	−6.12722	−0.219249
$782$	4.23073	0.151291
$783$	0	0
$784$	0	0
$785$	−9.66245	−0.344868
$786$	0	0
$787$	−14.9531	−0.533021	−0.266510	−	0.963832i	$-0.585871\pi$
$787$	−14.9531	−0.533021	−0.266510	+	0.963832i	$0.585871\pi$
$788$	12.7339	0.453625
$789$	0	0
$790$	2.30584	0.0820381
$791$	0	0
$792$	0	0
$793$	−17.6477	−0.626687
$794$	10.2661	0.364332
$795$	0	0
$796$	2.94592	0.104415
$797$	9.12588	0.323255	0.161628	−	0.986852i	$-0.448326\pi$
$797$	9.12588	0.323255	0.161628	+	0.986852i	$0.448326\pi$
$798$	0	0
$799$	8.69028	0.307440
$800$	−4.78794	−0.169279
$801$	0	0
$802$	34.0335	1.20176
$803$	−48.2996	−1.70446
$804$	0	0
$805$	0	0
$806$	0.751560	0.0264726
$807$	0	0
$808$	13.7558	0.483928
$809$	−35.5510	−1.24991	−0.624953	−	0.780663i	$-0.714882\pi$
$809$	−35.5510	−1.24991	−0.624953	+	0.780663i	$0.714882\pi$
$810$	0	0
$811$	−13.5070	−0.474295	−0.237148	−	0.971474i	$-0.576212\pi$
$811$	−13.5070	−0.474295	−0.237148	+	0.971474i	$0.576212\pi$
$812$	0	0
$813$	0	0
$814$	−33.2235	−1.16448
$815$	−10.6084	−0.371595
$816$	0	0
$817$	−37.8142	−1.32295
$818$	−3.48968	−0.122014
$819$	0	0
$820$	−0.435599	−0.0152118
$821$	21.6228	0.754639	0.377320	−	0.926083i	$-0.376846\pi$
$821$	21.6228	0.754639	0.377320	+	0.926083i	$0.376846\pi$
$822$	0	0
$823$	−1.50700	−0.0525308	−0.0262654	−	0.999655i	$-0.508361\pi$
$823$	−1.50700	−0.0525308	−0.0262654	+	0.999655i	$0.508361\pi$
$824$	11.1623	0.388855
$825$	0	0
$826$	0	0
$827$	23.3786	0.812953	0.406477	−	0.913661i	$-0.366757\pi$
$827$	23.3786	0.812953	0.406477	+	0.913661i	$0.366757\pi$
$828$	0	0
$829$	22.0191	0.764753	0.382377	−	0.924007i	$-0.375106\pi$
$829$	22.0191	0.764753	0.382377	+	0.924007i	$0.375106\pi$
$830$	3.06128	0.106259
$831$	0	0
$832$	−1.46050	−0.0506339
$833$	0	0
$834$	0	0
$835$	−4.89515	−0.169404
$836$	−14.7879	−0.511451
$837$	0	0
$838$	−28.9794	−1.00108
$839$	2.13015	0.0735409	0.0367705	−	0.999324i	$-0.488293\pi$
$839$	2.13015	0.0735409	0.0367705	+	0.999324i	$0.488293\pi$
$840$	0	0
$841$	51.5523	1.77767
$842$	2.12256	0.0731483
$843$	0	0
$844$	1.21634	0.0418680
$845$	5.00427	0.172152
$846$	0	0
$847$	0	0
$848$	−12.4356	−0.427040
$849$	0	0
$850$	−17.8774	−0.613191
$851$	10.3202	0.353773
$852$	0	0
$853$	7.00293	0.239776	0.119888	−	0.992787i	$-0.461747\pi$
$853$	7.00293	0.239776	0.119888	+	0.992787i	$0.461747\pi$
$854$	0	0
$855$	0	0
$856$	7.78074	0.265940
$857$	10.9282	0.373300	0.186650	−	0.982426i	$-0.440237\pi$
$857$	10.9282	0.373300	0.186650	+	0.982426i	$0.440237\pi$
$858$	0	0
$859$	−13.9076	−0.474520	−0.237260	−	0.971446i	$-0.576249\pi$
$859$	−13.9076	−0.474520	−0.237260	+	0.971446i	$0.576249\pi$
$860$	4.29533	0.146469
$861$	0	0
$862$	−21.8712	−0.744936
$863$	36.8463	1.25426	0.627131	−	0.778914i	$-0.284229\pi$
$863$	36.8463	1.25426	0.627131	+	0.778914i	$0.284229\pi$
$864$	0	0
$865$	−1.35329	−0.0460134
$866$	−13.0512	−0.443496
$867$	0	0
$868$	0	0
$869$	18.2646	0.619583
$870$	0	0
$871$	3.38871	0.114822
$872$	7.51459	0.254476
$873$	0	0
$874$	4.59358	0.155380
$875$	0	0
$876$	0	0
$877$	−10.3595	−0.349817	−0.174908	−	0.984585i	$-0.555963\pi$
$877$	−10.3595	−0.349817	−0.174908	+	0.984585i	$0.555963\pi$
$878$	4.86400	0.164152
$879$	0	0
$880$	1.67977	0.0566249
$881$	9.34806	0.314944	0.157472	−	0.987523i	$-0.449666\pi$
$881$	9.34806	0.314944	0.157472	+	0.987523i	$0.449666\pi$
$882$	0	0
$883$	2.29494	0.0772308	0.0386154	−	0.999254i	$-0.487705\pi$
$883$	2.29494	0.0772308	0.0386154	+	0.999254i	$0.487705\pi$
$884$	−5.45331	−0.183415
$885$	0	0
$886$	−11.5395	−0.387677
$887$	−27.6726	−0.929154	−0.464577	−	0.885533i	$-0.653794\pi$
$887$	−27.6726	−0.929154	−0.464577	+	0.885533i	$0.653794\pi$
$888$	0	0
$889$	0	0
$890$	−1.25564	−0.0420891
$891$	0	0
$892$	0.891832	0.0298607
$893$	9.43560	0.315750
$894$	0	0
$895$	−4.21926	−0.141034
$896$	0	0
$897$	0	0
$898$	−26.4251	−0.881817
$899$	−4.61849	−0.154035
$900$	0	0
$901$	−46.4327	−1.54690
$902$	−3.45038	−0.114885
$903$	0	0
$904$	−6.06128	−0.201595
$905$	−10.3284	−0.343327
$906$	0	0
$907$	−2.93152	−0.0973396	−0.0486698	−	0.998815i	$-0.515498\pi$
$907$	−2.93152	−0.0973396	−0.0486698	+	0.998815i	$0.515498\pi$
$908$	−14.6519	−0.486242
$909$	0	0
$910$	0	0
$911$	−30.6342	−1.01496	−0.507479	−	0.861664i	$-0.669422\pi$
$911$	−30.6342	−1.01496	−0.507479	+	0.861664i	$0.669422\pi$
$912$	0	0
$913$	24.2484	0.802506
$914$	−3.73812	−0.123646
$915$	0	0
$916$	−9.57587	−0.316396
$917$	0	0
$918$	0	0
$919$	−26.3714	−0.869912	−0.434956	−	0.900452i	$-0.643236\pi$
$919$	−26.3714	−0.869912	−0.434956	+	0.900452i	$0.643236\pi$
$920$	−0.521786	−0.0172028
$921$	0	0
$922$	15.8099	0.520672
$923$	−2.45331	−0.0807516
$924$	0	0
$925$	−43.6093	−1.43387
$926$	−38.3930	−1.26167
$927$	0	0
$928$	8.97509	0.294622
$929$	17.8741	0.586431	0.293215	−	0.956046i	$-0.405275\pi$
$929$	17.8741	0.586431	0.293215	+	0.956046i	$0.405275\pi$
$930$	0	0
$931$	0	0
$932$	−14.4284	−0.472618
$933$	0	0
$934$	−6.31304	−0.206569
$935$	6.27200	0.205116
$936$	0	0
$937$	15.9134	0.519869	0.259934	−	0.965626i	$-0.416299\pi$
$937$	15.9134	0.519869	0.259934	+	0.965626i	$0.416299\pi$
$938$	0	0
$939$	0	0
$940$	−1.07179	−0.0349580
$941$	−16.2805	−0.530731	−0.265365	−	0.964148i	$-0.585493\pi$
$941$	−16.2805	−0.530731	−0.265365	+	0.964148i	$0.585493\pi$
$942$	0	0
$943$	1.07179	0.0349024
$944$	−12.8961	−0.419732
$945$	0	0
$946$	34.0233	1.10619
$947$	−28.5903	−0.929059	−0.464529	−	0.885558i	$-0.653776\pi$
$947$	−28.5903	−0.929059	−0.464529	+	0.885558i	$0.653776\pi$
$948$	0	0
$949$	−19.3389	−0.627767
$950$	−19.4107	−0.629766
$951$	0	0
$952$	0	0
$953$	−29.3537	−0.950859	−0.475430	−	0.879754i	$-0.657707\pi$
$953$	−29.3537	−0.950859	−0.475430	+	0.879754i	$0.657707\pi$
$954$	0	0
$955$	−1.14651	−0.0371002
$956$	18.3097	0.592179
$957$	0	0
$958$	−20.4136	−0.659534
$959$	0	0
$960$	0	0
$961$	−30.7352	−0.991458
$962$	−13.3025	−0.428891
$963$	0	0
$964$	0.0933847	0.00300772
$965$	−2.06752	−0.0665559
$966$	0	0
$967$	9.39630	0.302165	0.151082	−	0.988521i	$-0.451724\pi$
$967$	9.39630	0.302165	0.151082	+	0.988521i	$0.451724\pi$
$968$	2.30545	0.0741000
$969$	0	0
$970$	−5.15174	−0.165412
$971$	−15.5467	−0.498917	−0.249459	−	0.968385i	$-0.580253\pi$
$971$	−15.5467	−0.498917	−0.249459	+	0.968385i	$0.580253\pi$
$972$	0	0
$973$	0	0
$974$	−12.3638	−0.396162
$975$	0	0
$976$	12.0833	0.386776
$977$	−9.59785	−0.307062	−0.153531	−	0.988144i	$-0.549065\pi$
$977$	−9.59785	−0.307062	−0.153531	+	0.988144i	$0.549065\pi$
$978$	0	0
$979$	−9.94592	−0.317873
$980$	0	0
$981$	0	0
$982$	−0.414007	−0.0132115
$983$	−46.8535	−1.49439	−0.747197	−	0.664603i	$-0.768601\pi$
$983$	−46.8535	−1.49439	−0.747197	+	0.664603i	$0.768601\pi$
$984$	0	0
$985$	−5.86400	−0.186843
$986$	33.5117	1.06723
$987$	0	0
$988$	−5.92101	−0.188372
$989$	−10.5687	−0.336064
$990$	0	0
$991$	−21.6519	−0.687796	−0.343898	−	0.939007i	$-0.611748\pi$
$991$	−21.6519	−0.687796	−0.343898	+	0.939007i	$0.611748\pi$
$992$	−0.514589	−0.0163382
$993$	0	0
$994$	0	0
$995$	−1.35661	−0.0430074
$996$	0	0
$997$	−57.2379	−1.81274	−0.906372	−	0.422481i	$-0.861159\pi$
$997$	−57.2379	−1.81274	−0.906372	+	0.422481i	$0.861159\pi$
$998$	−0.923935	−0.0292466
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.cb.1.1		3
3.2	odd	2		7938.2.a.bu.1.3		3
7.2	even	3		1134.2.g.k.487.3		6
7.4	even	3		1134.2.g.k.163.3		6
7.6	odd	2		7938.2.a.by.1.3		3
9.2	odd	6		2646.2.f.o.1765.1		6
9.4	even	3		882.2.f.l.295.2		6
9.5	odd	6		2646.2.f.o.883.1		6
9.7	even	3		882.2.f.l.589.2		6
21.2	odd	6		1134.2.g.n.487.1		6
21.11	odd	6		1134.2.g.n.163.1		6
21.20	even	2		7938.2.a.bx.1.1		3
63.2	odd	6		378.2.h.d.361.3		6
63.4	even	3		126.2.h.c.79.3	yes	6
63.5	even	6		2646.2.e.o.2125.3		6
63.11	odd	6		378.2.e.c.37.1		6
63.13	odd	6		882.2.f.m.295.2		6
63.16	even	3		126.2.h.c.67.3	yes	6
63.20	even	6		2646.2.f.n.1765.3		6
63.23	odd	6		378.2.e.c.235.1		6
63.25	even	3		126.2.e.d.121.2	yes	6
63.31	odd	6		882.2.h.o.79.1		6
63.32	odd	6		378.2.h.d.289.3		6
63.34	odd	6		882.2.f.m.589.2		6
63.38	even	6		2646.2.e.o.1549.3		6
63.40	odd	6		882.2.e.p.655.2		6
63.41	even	6		2646.2.f.n.883.3		6
63.47	even	6		2646.2.h.p.361.1		6
63.52	odd	6		882.2.e.p.373.2		6
63.58	even	3		126.2.e.d.25.2	✓	6
63.59	even	6		2646.2.h.p.667.1		6
63.61	odd	6		882.2.h.o.67.1		6
252.11	even	6		3024.2.q.h.2305.1		6
252.23	even	6		3024.2.q.h.2881.1		6
252.67	odd	6		1008.2.t.g.961.1		6
252.79	odd	6		1008.2.t.g.193.1		6
252.95	even	6		3024.2.t.g.289.3		6
252.151	odd	6		1008.2.q.h.625.2		6
252.191	even	6		3024.2.t.g.1873.3		6
252.247	odd	6		1008.2.q.h.529.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.d.25.2	✓	6	63.58	even	3
126.2.e.d.121.2	yes	6	63.25	even	3
126.2.h.c.67.3	yes	6	63.16	even	3
126.2.h.c.79.3	yes	6	63.4	even	3
378.2.e.c.37.1		6	63.11	odd	6
378.2.e.c.235.1		6	63.23	odd	6
378.2.h.d.289.3		6	63.32	odd	6
378.2.h.d.361.3		6	63.2	odd	6
882.2.e.p.373.2		6	63.52	odd	6
882.2.e.p.655.2		6	63.40	odd	6
882.2.f.l.295.2		6	9.4	even	3
882.2.f.l.589.2		6	9.7	even	3
882.2.f.m.295.2		6	63.13	odd	6
882.2.f.m.589.2		6	63.34	odd	6
882.2.h.o.67.1		6	63.61	odd	6
882.2.h.o.79.1		6	63.31	odd	6
1008.2.q.h.529.2		6	252.247	odd	6
1008.2.q.h.625.2		6	252.151	odd	6
1008.2.t.g.193.1		6	252.79	odd	6
1008.2.t.g.961.1		6	252.67	odd	6
1134.2.g.k.163.3		6	7.4	even	3
1134.2.g.k.487.3		6	7.2	even	3
1134.2.g.n.163.1		6	21.11	odd	6
1134.2.g.n.487.1		6	21.2	odd	6
2646.2.e.o.1549.3		6	63.38	even	6
2646.2.e.o.2125.3		6	63.5	even	6
2646.2.f.n.883.3		6	63.41	even	6
2646.2.f.n.1765.3		6	63.20	even	6
2646.2.f.o.883.1		6	9.5	odd	6
2646.2.f.o.1765.1		6	9.2	odd	6
2646.2.h.p.361.1		6	63.47	even	6
2646.2.h.p.667.1		6	63.59	even	6
3024.2.q.h.2305.1		6	252.11	even	6
3024.2.q.h.2881.1		6	252.23	even	6
3024.2.t.g.289.3		6	252.95	even	6
3024.2.t.g.1873.3		6	252.191	even	6
7938.2.a.bu.1.3		3	3.2	odd	2
7938.2.a.bx.1.1		3	21.20	even	2
7938.2.a.by.1.3		3	7.6	odd	2
7938.2.a.cb.1.1		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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.460505 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.460505 q^{5} +1.00000 q^{8} -0.460505 q^{10} -3.64766 q^{11} -1.46050 q^{13} +1.00000 q^{16} +3.73385 q^{17} +4.05408 q^{19} -0.460505 q^{20} -3.64766 q^{22} +1.13307 q^{23} -4.78794 q^{25} -1.46050 q^{26} +8.97509 q^{29} -0.514589 q^{31} +1.00000 q^{32} +3.73385 q^{34} +9.10817 q^{37} +4.05408 q^{38} -0.460505 q^{40} +0.945916 q^{41} -9.32743 q^{43} -3.64766 q^{44} +1.13307 q^{46} +2.32743 q^{47} -4.78794 q^{50} -1.46050 q^{52} -12.4356 q^{53} +1.67977 q^{55} +8.97509 q^{58} -12.8961 q^{59} +12.0833 q^{61} -0.514589 q^{62} +1.00000 q^{64} +0.672570 q^{65} -2.32023 q^{67} +3.73385 q^{68} +1.67977 q^{71} +13.2412 q^{73} +9.10817 q^{74} +4.05408 q^{76} -5.00720 q^{79} -0.460505 q^{80} +0.945916 q^{82} -6.64766 q^{83} -1.71946 q^{85} -9.32743 q^{86} -3.64766 q^{88} +2.72665 q^{89} +1.13307 q^{92} +2.32743 q^{94} -1.86693 q^{95} +11.1872 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 5 * q^5 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 5 q^{5} + 3 q^{8} + 5 q^{10} + q^{11} + 2 q^{13} + 3 q^{16} + 4 q^{17} + 3 q^{19} + 5 q^{20} + q^{22} + 7 q^{23} + 2 q^{25} + 2 q^{26} + 5 q^{29} + 14 q^{31} + 3 q^{32} + 4 q^{34} + 9 q^{37} + 3 q^{38} + 5 q^{40} + 12 q^{41} - 18 q^{43} + q^{44} + 7 q^{46} - 3 q^{47} + 2 q^{50} + 2 q^{52} - 9 q^{53} + 7 q^{55} + 5 q^{58} - 4 q^{59} - 4 q^{61} + 14 q^{62} + 3 q^{64} + 12 q^{65} - 5 q^{67} + 4 q^{68} + 7 q^{71} + 25 q^{73} + 9 q^{74} + 3 q^{76} - 7 q^{79} + 5 q^{80} + 12 q^{82} - 8 q^{83} - 14 q^{85} - 18 q^{86} + q^{88} + 9 q^{89} + 7 q^{92} - 3 q^{94} - 2 q^{95} + 28 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 5 * q^5 + 3 * q^8 + 5 * q^10 + q^11 + 2 * q^13 + 3 * q^16 + 4 * q^17 + 3 * q^19 + 5 * q^20 + q^22 + 7 * q^23 + 2 * q^25 + 2 * q^26 + 5 * q^29 + 14 * q^31 + 3 * q^32 + 4 * q^34 + 9 * q^37 + 3 * q^38 + 5 * q^40 + 12 * q^41 - 18 * q^43 + q^44 + 7 * q^46 - 3 * q^47 + 2 * q^50 + 2 * q^52 - 9 * q^53 + 7 * q^55 + 5 * q^58 - 4 * q^59 - 4 * q^61 + 14 * q^62 + 3 * q^64 + 12 * q^65 - 5 * q^67 + 4 * q^68 + 7 * q^71 + 25 * q^73 + 9 * q^74 + 3 * q^76 - 7 * q^79 + 5 * q^80 + 12 * q^82 - 8 * q^83 - 14 * q^85 - 18 * q^86 + q^88 + 9 * q^89 + 7 * q^92 - 3 * q^94 - 2 * q^95 + 28 * q^97

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