LMFDB - Embedded newform 7938.2.a.bw.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.593579 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -0.593579 q^{5} -1.00000 q^{8} +0.593579 q^{10} -0.593579 q^{11} -2.51459 q^{13} +1.00000 q^{16} +2.92101 q^{17} +5.38151 q^{19} -0.593579 q^{20} +0.593579 q^{22} +4.46050 q^{23} -4.64766 q^{25} +2.51459 q^{26} +6.19436 q^{29} +7.86693 q^{31} -1.00000 q^{32} -2.92101 q^{34} -1.00000 q^{37} -5.38151 q^{38} +0.593579 q^{40} -0.273346 q^{41} +11.1623 q^{43} -0.593579 q^{44} -4.46050 q^{46} -12.1623 q^{47} +4.64766 q^{50} -2.51459 q^{52} -8.05408 q^{53} +0.352336 q^{55} -6.19436 q^{58} -8.64766 q^{59} +6.64766 q^{61} -7.86693 q^{62} +1.00000 q^{64} +1.49261 q^{65} -1.91381 q^{67} +2.92101 q^{68} -14.4107 q^{71} +7.91381 q^{73} +1.00000 q^{74} +5.38151 q^{76} -9.24844 q^{79} -0.593579 q^{80} +0.273346 q^{82} +7.70175 q^{83} -1.73385 q^{85} -11.1623 q^{86} +0.593579 q^{88} -12.4356 q^{89} +4.46050 q^{92} +12.1623 q^{94} -3.19436 q^{95} +11.7339 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^5 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{8} - q^{10} + q^{11} + 8 q^{13} + 3 q^{16} - 4 q^{17} - 3 q^{19} + q^{20} - q^{22} + 7 q^{23} - 2 q^{25} - 8 q^{26} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34} - 3 q^{37} + 3 q^{38} - q^{40} + 6 q^{43} + q^{44} - 7 q^{46} - 9 q^{47} + 2 q^{50} + 8 q^{52} - 15 q^{53} + 13 q^{55} - 5 q^{58} - 14 q^{59} + 8 q^{61} - 20 q^{62} + 3 q^{64} + 12 q^{65} - q^{67} - 4 q^{68} + 7 q^{71} + 19 q^{73} + 3 q^{74} - 3 q^{76} - 5 q^{79} + q^{80} + 2 q^{83} + 2 q^{85} - 6 q^{86} - q^{88} - 9 q^{89} + 7 q^{92} + 9 q^{94} + 4 q^{95} + 28 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^5 - 3 * q^8 - q^10 + q^11 + 8 * q^13 + 3 * q^16 - 4 * q^17 - 3 * q^19 + q^20 - q^22 + 7 * q^23 - 2 * q^25 - 8 * q^26 + 5 * q^29 + 20 * q^31 - 3 * q^32 + 4 * q^34 - 3 * q^37 + 3 * q^38 - q^40 + 6 * q^43 + q^44 - 7 * q^46 - 9 * q^47 + 2 * q^50 + 8 * q^52 - 15 * q^53 + 13 * q^55 - 5 * q^58 - 14 * q^59 + 8 * q^61 - 20 * q^62 + 3 * q^64 + 12 * q^65 - q^67 - 4 * q^68 + 7 * q^71 + 19 * q^73 + 3 * q^74 - 3 * q^76 - 5 * q^79 + q^80 + 2 * q^83 + 2 * q^85 - 6 * q^86 - q^88 - 9 * q^89 + 7 * q^92 + 9 * q^94 + 4 * 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
$5$	−0.593579	−0.265457	−0.132728	−	0.991152i	$-0.542374\pi$
$5$	−0.593579	−0.265457	−0.132728	+	0.991152i	$0.542374\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.593579	0.187706
$11$	−0.593579	−0.178971	−0.0894855	−	0.995988i	$-0.528522\pi$
$11$	−0.593579	−0.178971	−0.0894855	+	0.995988i	$0.528522\pi$
$12$	0	0
$13$	−2.51459	−0.697422	−0.348711	−	0.937230i	$-0.613380\pi$
$13$	−2.51459	−0.697422	−0.348711	+	0.937230i	$0.613380\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.92101	0.708449	0.354224	−	0.935160i	$-0.384745\pi$
$17$	2.92101	0.708449	0.354224	+	0.935160i	$0.384745\pi$
$18$	0	0
$19$	5.38151	1.23460	0.617302	−	0.786726i	$-0.288226\pi$
$19$	5.38151	1.23460	0.617302	+	0.786726i	$0.288226\pi$
$20$	−0.593579	−0.132728
$21$	0	0
$22$	0.593579	0.126552
$23$	4.46050	0.930080	0.465040	−	0.885290i	$-0.346040\pi$
$23$	4.46050	0.930080	0.465040	+	0.885290i	$0.346040\pi$
$24$	0	0
$25$	−4.64766	−0.929533
$26$	2.51459	0.493151
$27$	0	0
$28$	0	0
$29$	6.19436	1.15026	0.575132	−	0.818061i	$-0.304951\pi$
$29$	6.19436	1.15026	0.575132	+	0.818061i	$0.304951\pi$
$30$	0	0
$31$	7.86693	1.41294	0.706471	−	0.707742i	$-0.250286\pi$
$31$	7.86693	1.41294	0.706471	+	0.707742i	$0.250286\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.92101	−0.500949
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−5.38151	−0.872997
$39$	0	0
$40$	0.593579	0.0938531
$41$	−0.273346	−0.0426895	−0.0213448	−	0.999772i	$-0.506795\pi$
$41$	−0.273346	−0.0426895	−0.0213448	+	0.999772i	$0.506795\pi$
$42$	0	0
$43$	11.1623	1.70223	0.851114	−	0.524981i	$-0.175928\pi$
$43$	11.1623	1.70223	0.851114	+	0.524981i	$0.175928\pi$
$44$	−0.593579	−0.0894855
$45$	0	0
$46$	−4.46050	−0.657666
$47$	−12.1623	−1.77405	−0.887023	−	0.461724i	$-0.847231\pi$
$47$	−12.1623	−1.77405	−0.887023	+	0.461724i	$0.847231\pi$
$48$	0	0
$49$	0	0
$50$	4.64766	0.657279
$51$	0	0
$52$	−2.51459	−0.348711
$53$	−8.05408	−1.10631	−0.553157	−	0.833077i	$-0.686577\pi$
$53$	−8.05408	−1.10631	−0.553157	+	0.833077i	$0.686577\pi$
$54$	0	0
$55$	0.352336	0.0475090
$56$	0	0
$57$	0	0
$58$	−6.19436	−0.813359
$59$	−8.64766	−1.12583	−0.562915	−	0.826515i	$-0.690320\pi$
$59$	−8.64766	−1.12583	−0.562915	+	0.826515i	$0.690320\pi$
$60$	0	0
$61$	6.64766	0.851146	0.425573	−	0.904924i	$-0.360073\pi$
$61$	6.64766	0.851146	0.425573	+	0.904924i	$0.360073\pi$
$62$	−7.86693	−0.999101
$63$	0	0
$64$	1.00000	0.125000
$65$	1.49261	0.185135
$66$	0	0
$67$	−1.91381	−0.233809	−0.116905	−	0.993143i	$-0.537297\pi$
$67$	−1.91381	−0.233809	−0.116905	+	0.993143i	$0.537297\pi$
$68$	2.92101	0.354224
$69$	0	0
$70$	0	0
$71$	−14.4107	−1.71023	−0.855117	−	0.518435i	$-0.826515\pi$
$71$	−14.4107	−1.71023	−0.855117	+	0.518435i	$0.826515\pi$
$72$	0	0
$73$	7.91381	0.926242	0.463121	−	0.886295i	$-0.346730\pi$
$73$	7.91381	0.926242	0.463121	+	0.886295i	$0.346730\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	5.38151	0.617302
$77$	0	0
$78$	0	0
$79$	−9.24844	−1.04053	−0.520265	−	0.854005i	$-0.674167\pi$
$79$	−9.24844	−1.04053	−0.520265	+	0.854005i	$0.674167\pi$
$80$	−0.593579	−0.0663642
$81$	0	0
$82$	0.273346	0.0301860
$83$	7.70175	0.845377	0.422688	−	0.906275i	$-0.361087\pi$
$83$	7.70175	0.845377	0.422688	+	0.906275i	$0.361087\pi$
$84$	0	0
$85$	−1.73385	−0.188063
$86$	−11.1623	−1.20366
$87$	0	0
$88$	0.593579	0.0632758
$89$	−12.4356	−1.31817	−0.659085	−	0.752068i	$-0.729056\pi$
$89$	−12.4356	−1.31817	−0.659085	+	0.752068i	$0.729056\pi$
$90$	0	0
$91$	0	0
$92$	4.46050	0.465040
$93$	0	0
$94$	12.1623	1.25444
$95$	−3.19436	−0.327734
$96$	0	0
$97$	11.7339	1.19139	0.595696	−	0.803210i	$-0.296876\pi$
$97$	11.7339	1.19139	0.595696	+	0.803210i	$0.296876\pi$
$98$	0	0
$99$	0	0
$100$	−4.64766	−0.464766
$101$	1.62276	0.161470	0.0807352	−	0.996736i	$-0.474273\pi$
$101$	1.62276	0.161470	0.0807352	+	0.996736i	$0.474273\pi$
$102$	0	0
$103$	−6.38151	−0.628789	−0.314395	−	0.949292i	$-0.601802\pi$
$103$	−6.38151	−0.628789	−0.314395	+	0.949292i	$0.601802\pi$
$104$	2.51459	0.246576
$105$	0	0
$106$	8.05408	0.782282
$107$	−18.7089	−1.80866	−0.904331	−	0.426832i	$-0.859630\pi$
$107$	−18.7089	−1.80866	−0.904331	+	0.426832i	$0.859630\pi$
$108$	0	0
$109$	2.86693	0.274602	0.137301	−	0.990529i	$-0.456157\pi$
$109$	2.86693	0.274602	0.137301	+	0.990529i	$0.456157\pi$
$110$	−0.352336	−0.0335940
$111$	0	0
$112$	0	0
$113$	12.3202	1.15899	0.579495	−	0.814976i	$-0.303250\pi$
$113$	12.3202	1.15899	0.579495	+	0.814976i	$0.303250\pi$
$114$	0	0
$115$	−2.64766	−0.246896
$116$	6.19436	0.575132
$117$	0	0
$118$	8.64766	0.796082
$119$	0	0
$120$	0	0
$121$	−10.6477	−0.967969
$122$	−6.64766	−0.601851
$123$	0	0
$124$	7.86693	0.706471
$125$	5.72665	0.512207
$126$	0	0
$127$	12.3346	1.09452	0.547261	−	0.836962i	$-0.315671\pi$
$127$	12.3346	1.09452	0.547261	+	0.836962i	$0.315671\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.49261	−0.130910
$131$	1.18716	0.103723	0.0518613	−	0.998654i	$-0.483485\pi$
$131$	1.18716	0.103723	0.0518613	+	0.998654i	$0.483485\pi$
$132$	0	0
$133$	0	0
$134$	1.91381	0.165328
$135$	0	0
$136$	−2.92101	−0.250475
$137$	2.52179	0.215451	0.107725	−	0.994181i	$-0.465643\pi$
$137$	2.52179	0.215451	0.107725	+	0.994181i	$0.465643\pi$
$138$	0	0
$139$	4.91381	0.416784	0.208392	−	0.978045i	$-0.433177\pi$
$139$	4.91381	0.416784	0.208392	+	0.978045i	$0.433177\pi$
$140$	0	0
$141$	0	0
$142$	14.4107	1.20932
$143$	1.49261	0.124818
$144$	0	0
$145$	−3.67684	−0.305345
$146$	−7.91381	−0.654952
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	18.0512	1.47881	0.739404	−	0.673262i	$-0.235107\pi$
$149$	18.0512	1.47881	0.739404	+	0.673262i	$0.235107\pi$
$150$	0	0
$151$	1.64766	0.134085	0.0670425	−	0.997750i	$-0.478644\pi$
$151$	1.64766	0.134085	0.0670425	+	0.997750i	$0.478644\pi$
$152$	−5.38151	−0.436498
$153$	0	0
$154$	0	0
$155$	−4.66964	−0.375075
$156$	0	0
$157$	6.60078	0.526799	0.263400	−	0.964687i	$-0.415156\pi$
$157$	6.60078	0.526799	0.263400	+	0.964687i	$0.415156\pi$
$158$	9.24844	0.735766
$159$	0	0
$160$	0.593579	0.0469266
$161$	0	0
$162$	0	0
$163$	5.98229	0.468569	0.234285	−	0.972168i	$-0.424725\pi$
$163$	5.98229	0.468569	0.234285	+	0.972168i	$0.424725\pi$
$164$	−0.273346	−0.0213448
$165$	0	0
$166$	−7.70175	−0.597772
$167$	7.46050	0.577311	0.288656	−	0.957433i	$-0.406792\pi$
$167$	7.46050	0.577311	0.288656	+	0.957433i	$0.406792\pi$
$168$	0	0
$169$	−6.67684	−0.513603
$170$	1.73385	0.132980
$171$	0	0
$172$	11.1623	0.851114
$173$	25.6591	1.95083	0.975414	−	0.220381i	$-0.0707301\pi$
$173$	25.6591	1.95083	0.975414	+	0.220381i	$0.0707301\pi$
$174$	0	0
$175$	0	0
$176$	−0.593579	−0.0447427
$177$	0	0
$178$	12.4356	0.932088
$179$	−15.0364	−1.12387	−0.561936	−	0.827181i	$-0.689943\pi$
$179$	−15.0364	−1.12387	−0.561936	+	0.827181i	$0.689943\pi$
$180$	0	0
$181$	0.0861875	0.00640627	0.00320313	−	0.999995i	$-0.498980\pi$
$181$	0.0861875	0.00640627	0.00320313	+	0.999995i	$0.498980\pi$
$182$	0	0
$183$	0	0
$184$	−4.46050	−0.328833
$185$	0.593579	0.0436408
$186$	0	0
$187$	−1.73385	−0.126792
$188$	−12.1623	−0.887023
$189$	0	0
$190$	3.19436	0.231743
$191$	3.98229	0.288148	0.144074	−	0.989567i	$-0.453980\pi$
$191$	3.98229	0.288148	0.144074	+	0.989567i	$0.453980\pi$
$192$	0	0
$193$	6.78074	0.488088	0.244044	−	0.969764i	$-0.421526\pi$
$193$	6.78074	0.488088	0.244044	+	0.969764i	$0.421526\pi$
$194$	−11.7339	−0.842441
$195$	0	0
$196$	0	0
$197$	11.0584	0.787875	0.393938	−	0.919137i	$-0.371113\pi$
$197$	11.0584	0.787875	0.393938	+	0.919137i	$0.371113\pi$
$198$	0	0
$199$	5.61849	0.398284	0.199142	−	0.979971i	$-0.436185\pi$
$199$	5.61849	0.398284	0.199142	+	0.979971i	$0.436185\pi$
$200$	4.64766	0.328639
$201$	0	0
$202$	−1.62276	−0.114177
$203$	0	0
$204$	0	0
$205$	0.162253	0.0113322
$206$	6.38151	0.444621
$207$	0	0
$208$	−2.51459	−0.174355
$209$	−3.19436	−0.220958
$210$	0	0
$211$	−19.3245	−1.33035	−0.665177	−	0.746686i	$-0.731644\pi$
$211$	−19.3245	−1.33035	−0.665177	+	0.746686i	$0.731644\pi$
$212$	−8.05408	−0.553157
$213$	0	0
$214$	18.7089	1.27892
$215$	−6.62568	−0.451868
$216$	0	0
$217$	0	0
$218$	−2.86693	−0.194173
$219$	0	0
$220$	0.352336	0.0237545
$221$	−7.34514	−0.494088
$222$	0	0
$223$	25.3245	1.69585	0.847927	−	0.530113i	$-0.177850\pi$
$223$	25.3245	1.69585	0.847927	+	0.530113i	$0.177850\pi$
$224$	0	0
$225$	0	0
$226$	−12.3202	−0.819530
$227$	−4.81711	−0.319723	−0.159862	−	0.987139i	$-0.551105\pi$
$227$	−4.81711	−0.319723	−0.159862	+	0.987139i	$0.551105\pi$
$228$	0	0
$229$	9.29533	0.614253	0.307126	−	0.951669i	$-0.400633\pi$
$229$	9.29533	0.614253	0.307126	+	0.951669i	$0.400633\pi$
$230$	2.64766	0.174582
$231$	0	0
$232$	−6.19436	−0.406679
$233$	−0.194356	−0.0127327	−0.00636634	−	0.999980i	$-0.502026\pi$
$233$	−0.194356	−0.0127327	−0.00636634	+	0.999980i	$0.502026\pi$
$234$	0	0
$235$	7.21926	0.470933
$236$	−8.64766	−0.562915
$237$	0	0
$238$	0	0
$239$	13.6549	0.883260	0.441630	−	0.897197i	$-0.354400\pi$
$239$	13.6549	0.883260	0.441630	+	0.897197i	$0.354400\pi$
$240$	0	0
$241$	13.0000	0.837404	0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000	0.837404	0.418702	+	0.908124i	$0.362485\pi$
$242$	10.6477	0.684458
$243$	0	0
$244$	6.64766	0.425573
$245$	0	0
$246$	0	0
$247$	−13.5323	−0.861039
$248$	−7.86693	−0.499550
$249$	0	0
$250$	−5.72665	−0.362185
$251$	19.5438	1.23359	0.616796	−	0.787123i	$-0.288430\pi$
$251$	19.5438	1.23359	0.616796	+	0.787123i	$0.288430\pi$
$252$	0	0
$253$	−2.64766	−0.166457
$254$	−12.3346	−0.773943
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.32743	−0.519451	−0.259725	−	0.965683i	$-0.583632\pi$
$257$	−8.32743	−0.519451	−0.259725	+	0.965683i	$0.583632\pi$
$258$	0	0
$259$	0	0
$260$	1.49261	0.0925676
$261$	0	0
$262$	−1.18716	−0.0733429
$263$	−17.0905	−1.05384	−0.526921	−	0.849914i	$-0.676654\pi$
$263$	−17.0905	−1.05384	−0.526921	+	0.849914i	$0.676654\pi$
$264$	0	0
$265$	4.78074	0.293678
$266$	0	0
$267$	0	0
$268$	−1.91381	−0.116905
$269$	−10.0144	−0.610588	−0.305294	−	0.952258i	$-0.598755\pi$
$269$	−10.0144	−0.610588	−0.305294	+	0.952258i	$0.598755\pi$
$270$	0	0
$271$	10.2091	0.620161	0.310081	−	0.950710i	$-0.399644\pi$
$271$	10.2091	0.620161	0.310081	+	0.950710i	$0.399644\pi$
$272$	2.92101	0.177112
$273$	0	0
$274$	−2.52179	−0.152347
$275$	2.75876	0.166359
$276$	0	0
$277$	19.3422	1.16216	0.581081	−	0.813846i	$-0.302630\pi$
$277$	19.3422	1.16216	0.581081	+	0.813846i	$0.302630\pi$
$278$	−4.91381	−0.294711
$279$	0	0
$280$	0	0
$281$	12.8027	0.763746	0.381873	−	0.924215i	$-0.375279\pi$
$281$	12.8027	0.763746	0.381873	+	0.924215i	$0.375279\pi$
$282$	0	0
$283$	16.3523	0.972046	0.486023	−	0.873946i	$-0.338447\pi$
$283$	16.3523	0.972046	0.486023	+	0.873946i	$0.338447\pi$
$284$	−14.4107	−0.855117
$285$	0	0
$286$	−1.49261	−0.0882598
$287$	0	0
$288$	0	0
$289$	−8.46770	−0.498100
$290$	3.67684	0.215912
$291$	0	0
$292$	7.91381	0.463121
$293$	−20.7778	−1.21385	−0.606926	−	0.794758i	$-0.707598\pi$
$293$	−20.7778	−1.21385	−0.606926	+	0.794758i	$0.707598\pi$
$294$	0	0
$295$	5.13307	0.298859
$296$	1.00000	0.0581238
$297$	0	0
$298$	−18.0512	−1.04568
$299$	−11.2163	−0.648658
$300$	0	0
$301$	0	0
$302$	−1.64766	−0.0948124
$303$	0	0
$304$	5.38151	0.308651
$305$	−3.94592	−0.225942
$306$	0	0
$307$	22.6768	1.29424	0.647118	−	0.762390i	$-0.275974\pi$
$307$	22.6768	1.29424	0.647118	+	0.762390i	$0.275974\pi$
$308$	0	0
$309$	0	0
$310$	4.66964	0.265218
$311$	6.51459	0.369408	0.184704	−	0.982794i	$-0.440867\pi$
$311$	6.51459	0.369408	0.184704	+	0.982794i	$0.440867\pi$
$312$	0	0
$313$	−0.266149	−0.0150436	−0.00752181	−	0.999972i	$-0.502394\pi$
$313$	−0.266149	−0.0150436	−0.00752181	+	0.999972i	$0.502394\pi$
$314$	−6.60078	−0.372503
$315$	0	0
$316$	−9.24844	−0.520265
$317$	15.7237	0.883133	0.441566	−	0.897229i	$-0.354423\pi$
$317$	15.7237	0.883133	0.441566	+	0.897229i	$0.354423\pi$
$318$	0	0
$319$	−3.67684	−0.205864
$320$	−0.593579	−0.0331821
$321$	0	0
$322$	0	0
$323$	15.7195	0.874654
$324$	0	0
$325$	11.6870	0.648276
$326$	−5.98229	−0.331328
$327$	0	0
$328$	0.273346	0.0150930
$329$	0	0
$330$	0	0
$331$	−25.1623	−1.38304	−0.691521	−	0.722356i	$-0.743059\pi$
$331$	−25.1623	−1.38304	−0.691521	+	0.722356i	$0.743059\pi$
$332$	7.70175	0.422688
$333$	0	0
$334$	−7.46050	−0.408221
$335$	1.13600	0.0620663
$336$	0	0
$337$	18.7339	1.02050	0.510249	−	0.860027i	$-0.329553\pi$
$337$	18.7339	1.02050	0.510249	+	0.860027i	$0.329553\pi$
$338$	6.67684	0.363172
$339$	0	0
$340$	−1.73385	−0.0940313
$341$	−4.66964	−0.252875
$342$	0	0
$343$	0	0
$344$	−11.1623	−0.601828
$345$	0	0
$346$	−25.6591	−1.37944
$347$	22.5438	1.21021	0.605106	−	0.796145i	$-0.293131\pi$
$347$	22.5438	1.21021	0.605106	+	0.796145i	$0.293131\pi$
$348$	0	0
$349$	3.79086	0.202920	0.101460	−	0.994840i	$-0.467649\pi$
$349$	3.79086	0.202920	0.101460	+	0.994840i	$0.467649\pi$
$350$	0	0
$351$	0	0
$352$	0.593579	0.0316379
$353$	−6.83482	−0.363781	−0.181890	−	0.983319i	$-0.558222\pi$
$353$	−6.83482	−0.363781	−0.181890	+	0.983319i	$0.558222\pi$
$354$	0	0
$355$	8.55389	0.453993
$356$	−12.4356	−0.659085
$357$	0	0
$358$	15.0364	0.794697
$359$	12.6447	0.667364	0.333682	−	0.942686i	$-0.391709\pi$
$359$	12.6447	0.667364	0.333682	+	0.942686i	$0.391709\pi$
$360$	0	0
$361$	9.96070	0.524247
$362$	−0.0861875	−0.00452991
$363$	0	0
$364$	0	0
$365$	−4.69748	−0.245877
$366$	0	0
$367$	−6.54377	−0.341582	−0.170791	−	0.985307i	$-0.554632\pi$
$367$	−6.54377	−0.341582	−0.170791	+	0.985307i	$0.554632\pi$
$368$	4.46050	0.232520
$369$	0	0
$370$	−0.593579	−0.0308587
$371$	0	0
$372$	0	0
$373$	9.42840	0.488184	0.244092	−	0.969752i	$-0.421510\pi$
$373$	9.42840	0.488184	0.244092	+	0.969752i	$0.421510\pi$
$374$	1.73385	0.0896553
$375$	0	0
$376$	12.1623	0.627220
$377$	−15.5763	−0.802218
$378$	0	0
$379$	−7.27762	−0.373826	−0.186913	−	0.982376i	$-0.559848\pi$
$379$	−7.27762	−0.373826	−0.186913	+	0.982376i	$0.559848\pi$
$380$	−3.19436	−0.163867
$381$	0	0
$382$	−3.98229	−0.203752
$383$	24.0833	1.23060	0.615299	−	0.788294i	$-0.289035\pi$
$383$	24.0833	1.23060	0.615299	+	0.788294i	$0.289035\pi$
$384$	0	0
$385$	0	0
$386$	−6.78074	−0.345130
$387$	0	0
$388$	11.7339	0.595696
$389$	−16.2983	−0.826354	−0.413177	−	0.910651i	$-0.635581\pi$
$389$	−16.2983	−0.826354	−0.413177	+	0.910651i	$0.635581\pi$
$390$	0	0
$391$	13.0292	0.658914
$392$	0	0
$393$	0	0
$394$	−11.0584	−0.557112
$395$	5.48968	0.276216
$396$	0	0
$397$	−12.1724	−0.610914	−0.305457	−	0.952206i	$-0.598809\pi$
$397$	−12.1724	−0.610914	−0.305457	+	0.952206i	$0.598809\pi$
$398$	−5.61849	−0.281629
$399$	0	0
$400$	−4.64766	−0.232383
$401$	−33.3609	−1.66596	−0.832981	−	0.553301i	$-0.813368\pi$
$401$	−33.3609	−1.66596	−0.832981	+	0.553301i	$0.813368\pi$
$402$	0	0
$403$	−19.7821	−0.985416
$404$	1.62276	0.0807352
$405$	0	0
$406$	0	0
$407$	0.593579	0.0294226
$408$	0	0
$409$	5.78074	0.285839	0.142920	−	0.989734i	$-0.454351\pi$
$409$	5.78074	0.285839	0.142920	+	0.989734i	$0.454351\pi$
$410$	−0.162253	−0.00801309
$411$	0	0
$412$	−6.38151	−0.314395
$413$	0	0
$414$	0	0
$415$	−4.57160	−0.224411
$416$	2.51459	0.123288
$417$	0	0
$418$	3.19436	0.156241
$419$	30.8712	1.50816	0.754078	−	0.656784i	$-0.228084\pi$
$419$	30.8712	1.50816	0.754078	+	0.656784i	$0.228084\pi$
$420$	0	0
$421$	3.73385	0.181977	0.0909884	−	0.995852i	$-0.470997\pi$
$421$	3.73385	0.181977	0.0909884	+	0.995852i	$0.470997\pi$
$422$	19.3245	0.940702
$423$	0	0
$424$	8.05408	0.391141
$425$	−13.5759	−0.658526
$426$	0	0
$427$	0	0
$428$	−18.7089	−0.904331
$429$	0	0
$430$	6.62568	0.319519
$431$	28.1957	1.35814	0.679070	−	0.734074i	$-0.262383\pi$
$431$	28.1957	1.35814	0.679070	+	0.734074i	$0.262383\pi$
$432$	0	0
$433$	−12.5438	−0.602815	−0.301407	−	0.953495i	$-0.597456\pi$
$433$	−12.5438	−0.602815	−0.301407	+	0.953495i	$0.597456\pi$
$434$	0	0
$435$	0	0
$436$	2.86693	0.137301
$437$	24.0043	1.14828
$438$	0	0
$439$	−26.0406	−1.24285	−0.621426	−	0.783473i	$-0.713446\pi$
$439$	−26.0406	−1.24285	−0.621426	+	0.783473i	$0.713446\pi$
$440$	−0.352336	−0.0167970
$441$	0	0
$442$	7.34514	0.349373
$443$	−23.5729	−1.11998	−0.559992	−	0.828498i	$-0.689196\pi$
$443$	−23.5729	−1.11998	−0.559992	+	0.828498i	$0.689196\pi$
$444$	0	0
$445$	7.38151	0.349917
$446$	−25.3245	−1.19915
$447$	0	0
$448$	0	0
$449$	13.6870	0.645928	0.322964	−	0.946411i	$-0.395321\pi$
$449$	13.6870	0.645928	0.322964	+	0.946411i	$0.395321\pi$
$450$	0	0
$451$	0.162253	0.00764018
$452$	12.3202	0.579495
$453$	0	0
$454$	4.81711	0.226078
$455$	0	0
$456$	0	0
$457$	−22.3523	−1.04560	−0.522799	−	0.852456i	$-0.675112\pi$
$457$	−22.3523	−1.04560	−0.522799	+	0.852456i	$0.675112\pi$
$458$	−9.29533	−0.434342
$459$	0	0
$460$	−2.64766	−0.123448
$461$	−7.97509	−0.371437	−0.185719	−	0.982603i	$-0.559461\pi$
$461$	−7.97509	−0.371437	−0.185719	+	0.982603i	$0.559461\pi$
$462$	0	0
$463$	28.7352	1.33544	0.667719	−	0.744413i	$-0.267271\pi$
$463$	28.7352	1.33544	0.667719	+	0.744413i	$0.267271\pi$
$464$	6.19436	0.287566
$465$	0	0
$466$	0.194356	0.00900336
$467$	33.5657	1.55324	0.776619	−	0.629971i	$-0.216933\pi$
$467$	33.5657	1.55324	0.776619	+	0.629971i	$0.216933\pi$
$468$	0	0
$469$	0	0
$470$	−7.21926	−0.333000
$471$	0	0
$472$	8.64766	0.398041
$473$	−6.62568	−0.304649
$474$	0	0
$475$	−25.0115	−1.14760
$476$	0	0
$477$	0	0
$478$	−13.6549	−0.624559
$479$	−0.367120	−0.0167741	−0.00838707	−	0.999965i	$-0.502670\pi$
$479$	−0.367120	−0.0167741	−0.00838707	+	0.999965i	$0.502670\pi$
$480$	0	0
$481$	2.51459	0.114655
$482$	−13.0000	−0.592134
$483$	0	0
$484$	−10.6477	−0.483985
$485$	−6.96497	−0.316263
$486$	0	0
$487$	29.9076	1.35524	0.677621	−	0.735412i	$-0.263011\pi$
$487$	29.9076	1.35524	0.677621	+	0.735412i	$0.263011\pi$
$488$	−6.64766	−0.300926
$489$	0	0
$490$	0	0
$491$	0.510317	0.0230303	0.0115151	−	0.999934i	$-0.496335\pi$
$491$	0.510317	0.0230303	0.0115151	+	0.999934i	$0.496335\pi$
$492$	0	0
$493$	18.0938	0.814903
$494$	13.5323	0.608847
$495$	0	0
$496$	7.86693	0.353235
$497$	0	0
$498$	0	0
$499$	−19.0191	−0.851410	−0.425705	−	0.904862i	$-0.639974\pi$
$499$	−19.0191	−0.851410	−0.425705	+	0.904862i	$0.639974\pi$
$500$	5.72665	0.256104
$501$	0	0
$502$	−19.5438	−0.872281
$503$	37.7807	1.68456	0.842280	−	0.539040i	$-0.181213\pi$
$503$	37.7807	1.68456	0.842280	+	0.539040i	$0.181213\pi$
$504$	0	0
$505$	−0.963235	−0.0428634
$506$	2.64766	0.117703
$507$	0	0
$508$	12.3346	0.547261
$509$	11.2163	0.497155	0.248578	−	0.968612i	$-0.420037\pi$
$509$	11.2163	0.497155	0.248578	+	0.968612i	$0.420037\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.32743	0.367307
$515$	3.78794	0.166916
$516$	0	0
$517$	7.21926	0.317503
$518$	0	0
$519$	0	0
$520$	−1.49261	−0.0654552
$521$	−27.4720	−1.20357	−0.601785	−	0.798658i	$-0.705543\pi$
$521$	−27.4720	−1.20357	−0.601785	+	0.798658i	$0.705543\pi$
$522$	0	0
$523$	22.1838	0.970032	0.485016	−	0.874505i	$-0.338814\pi$
$523$	22.1838	0.970032	0.485016	+	0.874505i	$0.338814\pi$
$524$	1.18716	0.0518613
$525$	0	0
$526$	17.0905	0.745179
$527$	22.9794	1.00100
$528$	0	0
$529$	−3.10390	−0.134952
$530$	−4.78074	−0.207662
$531$	0	0
$532$	0	0
$533$	0.687353	0.0297726
$534$	0	0
$535$	11.1052	0.480122
$536$	1.91381	0.0826641
$537$	0	0
$538$	10.0144	0.431751
$539$	0	0
$540$	0	0
$541$	−29.8492	−1.28332	−0.641659	−	0.766990i	$-0.721754\pi$
$541$	−29.8492	−1.28332	−0.641659	+	0.766990i	$0.721754\pi$
$542$	−10.2091	−0.438520
$543$	0	0
$544$	−2.92101	−0.125237
$545$	−1.70175	−0.0728949
$546$	0	0
$547$	−17.6870	−0.756240	−0.378120	−	0.925757i	$-0.623429\pi$
$547$	−17.6870	−0.756240	−0.378120	+	0.925757i	$0.623429\pi$
$548$	2.52179	0.107725
$549$	0	0
$550$	−2.75876	−0.117634
$551$	33.3350	1.42012
$552$	0	0
$553$	0	0
$554$	−19.3422	−0.821772
$555$	0	0
$556$	4.91381	0.208392
$557$	−30.1301	−1.27666	−0.638328	−	0.769765i	$-0.720374\pi$
$557$	−30.1301	−1.27666	−0.638328	+	0.769765i	$0.720374\pi$
$558$	0	0
$559$	−28.0685	−1.18717
$560$	0	0
$561$	0	0
$562$	−12.8027	−0.540050
$563$	−4.09766	−0.172696	−0.0863478	−	0.996265i	$-0.527520\pi$
$563$	−4.09766	−0.172696	−0.0863478	+	0.996265i	$0.527520\pi$
$564$	0	0
$565$	−7.31304	−0.307662
$566$	−16.3523	−0.687340
$567$	0	0
$568$	14.4107	0.604659
$569$	6.23697	0.261467	0.130734	−	0.991418i	$-0.458267\pi$
$569$	6.23697	0.261467	0.130734	+	0.991418i	$0.458267\pi$
$570$	0	0
$571$	35.6021	1.48990	0.744951	−	0.667119i	$-0.232473\pi$
$571$	35.6021	1.48990	0.744951	+	0.667119i	$0.232473\pi$
$572$	1.49261	0.0624091
$573$	0	0
$574$	0	0
$575$	−20.7309	−0.864539
$576$	0	0
$577$	46.2776	1.92656	0.963281	−	0.268494i	$-0.0865261\pi$
$577$	46.2776	1.92656	0.963281	+	0.268494i	$0.0865261\pi$
$578$	8.46770	0.352210
$579$	0	0
$580$	−3.67684	−0.152673
$581$	0	0
$582$	0	0
$583$	4.78074	0.197998
$584$	−7.91381	−0.327476
$585$	0	0
$586$	20.7778	0.858324
$587$	2.26322	0.0934132	0.0467066	−	0.998909i	$-0.485127\pi$
$587$	2.26322	0.0934132	0.0467066	+	0.998909i	$0.485127\pi$
$588$	0	0
$589$	42.3360	1.74442
$590$	−5.13307	−0.211325
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	46.1957	1.89703	0.948515	−	0.316732i	$-0.102586\pi$
$593$	46.1957	1.89703	0.948515	+	0.316732i	$0.102586\pi$
$594$	0	0
$595$	0	0
$596$	18.0512	0.739404
$597$	0	0
$598$	11.2163	0.458670
$599$	−16.7807	−0.685642	−0.342821	−	0.939401i	$-0.611382\pi$
$599$	−16.7807	−0.685642	−0.342821	+	0.939401i	$0.611382\pi$
$600$	0	0
$601$	−11.3992	−0.464984	−0.232492	−	0.972598i	$-0.574688\pi$
$601$	−11.3992	−0.464984	−0.232492	+	0.972598i	$0.574688\pi$
$602$	0	0
$603$	0	0
$604$	1.64766	0.0670425
$605$	6.32023	0.256954
$606$	0	0
$607$	14.4284	0.585631	0.292815	−	0.956169i	$-0.405408\pi$
$607$	14.4284	0.585631	0.292815	+	0.956169i	$0.405408\pi$
$608$	−5.38151	−0.218249
$609$	0	0
$610$	3.94592	0.159765
$611$	30.5831	1.23726
$612$	0	0
$613$	−24.4107	−0.985939	−0.492969	−	0.870047i	$-0.664089\pi$
$613$	−24.4107	−0.985939	−0.492969	+	0.870047i	$0.664089\pi$
$614$	−22.6768	−0.915163
$615$	0	0
$616$	0	0
$617$	−48.9397	−1.97024	−0.985119	−	0.171876i	$-0.945017\pi$
$617$	−48.9397	−1.97024	−0.985119	+	0.171876i	$0.945017\pi$
$618$	0	0
$619$	44.6591	1.79500	0.897501	−	0.441012i	$-0.145380\pi$
$619$	44.6591	1.79500	0.897501	+	0.441012i	$0.145380\pi$
$620$	−4.66964	−0.187537
$621$	0	0
$622$	−6.51459	−0.261211
$623$	0	0
$624$	0	0
$625$	19.8391	0.793564
$626$	0.266149	0.0106375
$627$	0	0
$628$	6.60078	0.263400
$629$	−2.92101	−0.116468
$630$	0	0
$631$	33.2852	1.32506	0.662532	−	0.749034i	$-0.269482\pi$
$631$	33.2852	1.32506	0.662532	+	0.749034i	$0.269482\pi$
$632$	9.24844	0.367883
$633$	0	0
$634$	−15.7237	−0.624469
$635$	−7.32158	−0.290548
$636$	0	0
$637$	0	0
$638$	3.67684	0.145568
$639$	0	0
$640$	0.593579	0.0234633
$641$	30.7879	1.21605	0.608025	−	0.793918i	$-0.291962\pi$
$641$	30.7879	1.21605	0.608025	+	0.793918i	$0.291962\pi$
$642$	0	0
$643$	−27.4690	−1.08327	−0.541637	−	0.840613i	$-0.682195\pi$
$643$	−27.4690	−1.08327	−0.541637	+	0.840613i	$0.682195\pi$
$644$	0	0
$645$	0	0
$646$	−15.7195	−0.618474
$647$	−13.2704	−0.521714	−0.260857	−	0.965377i	$-0.584005\pi$
$647$	−13.2704	−0.521714	−0.260857	+	0.965377i	$0.584005\pi$
$648$	0	0
$649$	5.13307	0.201491
$650$	−11.6870	−0.458400
$651$	0	0
$652$	5.98229	0.234285
$653$	−17.1416	−0.670803	−0.335402	−	0.942075i	$-0.608872\pi$
$653$	−17.1416	−0.670803	−0.335402	+	0.942075i	$0.608872\pi$
$654$	0	0
$655$	−0.704673	−0.0275338
$656$	−0.273346	−0.0106724
$657$	0	0
$658$	0	0
$659$	−8.52179	−0.331962	−0.165981	−	0.986129i	$-0.553079\pi$
$659$	−8.52179	−0.331962	−0.165981	+	0.986129i	$0.553079\pi$
$660$	0	0
$661$	−34.3360	−1.33551	−0.667757	−	0.744379i	$-0.732746\pi$
$661$	−34.3360	−1.33551	−0.667757	+	0.744379i	$0.732746\pi$
$662$	25.1623	0.977959
$663$	0	0
$664$	−7.70175	−0.298886
$665$	0	0
$666$	0	0
$667$	27.6300	1.06984
$668$	7.46050	0.288656
$669$	0	0
$670$	−1.13600	−0.0438875
$671$	−3.94592	−0.152330
$672$	0	0
$673$	15.4031	0.593746	0.296873	−	0.954917i	$-0.404056\pi$
$673$	15.4031	0.593746	0.296873	+	0.954917i	$0.404056\pi$
$674$	−18.7339	−0.721601
$675$	0	0
$676$	−6.67684	−0.256802
$677$	7.38151	0.283695	0.141847	−	0.989889i	$-0.454696\pi$
$677$	7.38151	0.283695	0.141847	+	0.989889i	$0.454696\pi$
$678$	0	0
$679$	0	0
$680$	1.73385	0.0664902
$681$	0	0
$682$	4.66964	0.178810
$683$	−9.59785	−0.367252	−0.183626	−	0.982996i	$-0.558783\pi$
$683$	−9.59785	−0.367252	−0.183626	+	0.982996i	$0.558783\pi$
$684$	0	0
$685$	−1.49688	−0.0571929
$686$	0	0
$687$	0	0
$688$	11.1623	0.425557
$689$	20.2527	0.771567
$690$	0	0
$691$	14.1445	0.538084	0.269042	−	0.963128i	$-0.413293\pi$
$691$	14.1445	0.538084	0.269042	+	0.963128i	$0.413293\pi$
$692$	25.6591	0.975414
$693$	0	0
$694$	−22.5438	−0.855750
$695$	−2.91674	−0.110638
$696$	0	0
$697$	−0.798447	−0.0302433
$698$	−3.79086	−0.143486
$699$	0	0
$700$	0	0
$701$	37.3753	1.41164	0.705822	−	0.708389i	$-0.250578\pi$
$701$	37.3753	1.41164	0.705822	+	0.708389i	$0.250578\pi$
$702$	0	0
$703$	−5.38151	−0.202968
$704$	−0.593579	−0.0223714
$705$	0	0
$706$	6.83482	0.257232
$707$	0	0
$708$	0	0
$709$	−10.4868	−0.393838	−0.196919	−	0.980420i	$-0.563094\pi$
$709$	−10.4868	−0.393838	−0.196919	+	0.980420i	$0.563094\pi$
$710$	−8.55389	−0.321022
$711$	0	0
$712$	12.4356	0.466044
$713$	35.0905	1.31415
$714$	0	0
$715$	−0.885981	−0.0331338
$716$	−15.0364	−0.561936
$717$	0	0
$718$	−12.6447	−0.471897
$719$	2.23990	0.0835340	0.0417670	−	0.999127i	$-0.486701\pi$
$719$	2.23990	0.0835340	0.0417670	+	0.999127i	$0.486701\pi$
$720$	0	0
$721$	0	0
$722$	−9.96070	−0.370699
$723$	0	0
$724$	0.0861875	0.00320313
$725$	−28.7893	−1.06921
$726$	0	0
$727$	0.370045	0.0137242	0.00686211	−	0.999976i	$-0.497816\pi$
$727$	0.370045	0.0137242	0.00686211	+	0.999976i	$0.497816\pi$
$728$	0	0
$729$	0	0
$730$	4.69748	0.173861
$731$	32.6050	1.20594
$732$	0	0
$733$	−14.0191	−0.517806	−0.258903	−	0.965903i	$-0.583361\pi$
$733$	−14.0191	−0.517806	−0.258903	+	0.965903i	$0.583361\pi$
$734$	6.54377	0.241535
$735$	0	0
$736$	−4.46050	−0.164416
$737$	1.13600	0.0418451
$738$	0	0
$739$	−26.7745	−0.984916	−0.492458	−	0.870336i	$-0.663901\pi$
$739$	−26.7745	−0.984916	−0.492458	+	0.870336i	$0.663901\pi$
$740$	0.593579	0.0218204
$741$	0	0
$742$	0	0
$743$	10.0934	0.370290	0.185145	−	0.982711i	$-0.440724\pi$
$743$	10.0934	0.370290	0.185145	+	0.982711i	$0.440724\pi$
$744$	0	0
$745$	−10.7148	−0.392560
$746$	−9.42840	−0.345198
$747$	0	0
$748$	−1.73385	−0.0633959
$749$	0	0
$750$	0	0
$751$	11.5146	0.420173	0.210087	−	0.977683i	$-0.432625\pi$
$751$	11.5146	0.420173	0.210087	+	0.977683i	$0.432625\pi$
$752$	−12.1623	−0.443512
$753$	0	0
$754$	15.5763	0.567254
$755$	−0.978019	−0.0355938
$756$	0	0
$757$	−15.2484	−0.554214	−0.277107	−	0.960839i	$-0.589376\pi$
$757$	−15.2484	−0.554214	−0.277107	+	0.960839i	$0.589376\pi$
$758$	7.27762	0.264335
$759$	0	0
$760$	3.19436	0.115871
$761$	1.70175	0.0616883	0.0308442	−	0.999524i	$-0.490180\pi$
$761$	1.70175	0.0616883	0.0308442	+	0.999524i	$0.490180\pi$
$762$	0	0
$763$	0	0
$764$	3.98229	0.144074
$765$	0	0
$766$	−24.0833	−0.870164
$767$	21.7453	0.785178
$768$	0	0
$769$	48.2422	1.73966	0.869829	−	0.493353i	$-0.164229\pi$
$769$	48.2422	1.73966	0.869829	+	0.493353i	$0.164229\pi$
$770$	0	0
$771$	0	0
$772$	6.78074	0.244044
$773$	6.20487	0.223174	0.111587	−	0.993755i	$-0.464407\pi$
$773$	6.20487	0.223174	0.111587	+	0.993755i	$0.464407\pi$
$774$	0	0
$775$	−36.5628	−1.31338
$776$	−11.7339	−0.421221
$777$	0	0
$778$	16.2983	0.584321
$779$	−1.47102	−0.0527046
$780$	0	0
$781$	8.55389	0.306082
$782$	−13.0292	−0.465922
$783$	0	0
$784$	0	0
$785$	−3.91808	−0.139842
$786$	0	0
$787$	−6.09766	−0.217358	−0.108679	−	0.994077i	$-0.534662\pi$
$787$	−6.09766	−0.217358	−0.108679	+	0.994077i	$0.534662\pi$
$788$	11.0584	0.393938
$789$	0	0
$790$	−5.48968	−0.195314
$791$	0	0
$792$	0	0
$793$	−16.7161	−0.593608
$794$	12.1724	0.431981
$795$	0	0
$796$	5.61849	0.199142
$797$	−12.4572	−0.441256	−0.220628	−	0.975358i	$-0.570811\pi$
$797$	−12.4572	−0.441256	−0.220628	+	0.975358i	$0.570811\pi$
$798$	0	0
$799$	−35.5261	−1.25682
$800$	4.64766	0.164320
$801$	0	0
$802$	33.3609	1.17801
$803$	−4.69748	−0.165770
$804$	0	0
$805$	0	0
$806$	19.7821	0.696794
$807$	0	0
$808$	−1.62276	−0.0570884
$809$	5.63288	0.198042	0.0990208	−	0.995085i	$-0.468429\pi$
$809$	5.63288	0.198042	0.0990208	+	0.995085i	$0.468429\pi$
$810$	0	0
$811$	45.6414	1.60269	0.801344	−	0.598204i	$-0.204119\pi$
$811$	45.6414	1.60269	0.801344	+	0.598204i	$0.204119\pi$
$812$	0	0
$813$	0	0
$814$	−0.593579	−0.0208049
$815$	−3.55096	−0.124385
$816$	0	0
$817$	60.0698	2.10158
$818$	−5.78074	−0.202119
$819$	0	0
$820$	0.162253	0.00566611
$821$	32.6946	1.14105	0.570524	−	0.821281i	$-0.306740\pi$
$821$	32.6946	1.14105	0.570524	+	0.821281i	$0.306740\pi$
$822$	0	0
$823$	−10.4399	−0.363911	−0.181956	−	0.983307i	$-0.558243\pi$
$823$	−10.4399	−0.363911	−0.181956	+	0.983307i	$0.558243\pi$
$824$	6.38151	0.222311
$825$	0	0
$826$	0	0
$827$	16.7060	0.580925	0.290463	−	0.956886i	$-0.406191\pi$
$827$	16.7060	0.580925	0.290463	+	0.956886i	$0.406191\pi$
$828$	0	0
$829$	−26.2091	−0.910281	−0.455141	−	0.890420i	$-0.650411\pi$
$829$	−26.2091	−0.910281	−0.455141	+	0.890420i	$0.650411\pi$
$830$	4.57160	0.158682
$831$	0	0
$832$	−2.51459	−0.0871777
$833$	0	0
$834$	0	0
$835$	−4.42840	−0.153251
$836$	−3.19436	−0.110479
$837$	0	0
$838$	−30.8712	−1.06643
$839$	22.3772	0.772548	0.386274	−	0.922384i	$-0.373762\pi$
$839$	22.3772	0.772548	0.386274	+	0.922384i	$0.373762\pi$
$840$	0	0
$841$	9.37005	0.323105
$842$	−3.73385	−0.128677
$843$	0	0
$844$	−19.3245	−0.665177
$845$	3.96324	0.136339
$846$	0	0
$847$	0	0
$848$	−8.05408	−0.276578
$849$	0	0
$850$	13.5759	0.465649
$851$	−4.46050	−0.152904
$852$	0	0
$853$	9.92528	0.339835	0.169918	−	0.985458i	$-0.445650\pi$
$853$	9.92528	0.339835	0.169918	+	0.985458i	$0.445650\pi$
$854$	0	0
$855$	0	0
$856$	18.7089	0.639459
$857$	−7.79552	−0.266290	−0.133145	−	0.991097i	$-0.542508\pi$
$857$	−7.79552	−0.266290	−0.133145	+	0.991097i	$0.542508\pi$
$858$	0	0
$859$	−16.3422	−0.557589	−0.278795	−	0.960351i	$-0.589935\pi$
$859$	−16.3422	−0.557589	−0.278795	+	0.960351i	$0.589935\pi$
$860$	−6.62568	−0.225934
$861$	0	0
$862$	−28.1957	−0.960349
$863$	−1.46050	−0.0497162	−0.0248581	−	0.999691i	$-0.507913\pi$
$863$	−1.46050	−0.0497162	−0.0248581	+	0.999691i	$0.507913\pi$
$864$	0	0
$865$	−15.2307	−0.517860
$866$	12.5438	0.426255
$867$	0	0
$868$	0	0
$869$	5.48968	0.186225
$870$	0	0
$871$	4.81245	0.163064
$872$	−2.86693	−0.0970863
$873$	0	0
$874$	−24.0043	−0.811957
$875$	0	0
$876$	0	0
$877$	−2.40935	−0.0813578	−0.0406789	−	0.999172i	$-0.512952\pi$
$877$	−2.40935	−0.0813578	−0.0406789	+	0.999172i	$0.512952\pi$
$878$	26.0406	0.878829
$879$	0	0
$880$	0.352336	0.0118773
$881$	18.9607	0.638802	0.319401	−	0.947620i	$-0.396518\pi$
$881$	18.9607	0.638802	0.319401	+	0.947620i	$0.396518\pi$
$882$	0	0
$883$	3.64008	0.122498	0.0612492	−	0.998123i	$-0.480492\pi$
$883$	3.64008	0.122498	0.0612492	+	0.998123i	$0.480492\pi$
$884$	−7.34514	−0.247044
$885$	0	0
$886$	23.5729	0.791949
$887$	24.4572	0.821192	0.410596	−	0.911817i	$-0.365321\pi$
$887$	24.4572	0.821192	0.410596	+	0.911817i	$0.365321\pi$
$888$	0	0
$889$	0	0
$890$	−7.38151	−0.247429
$891$	0	0
$892$	25.3245	0.847927
$893$	−65.4513	−2.19025
$894$	0	0
$895$	8.92528	0.298339
$896$	0	0
$897$	0	0
$898$	−13.6870	−0.456740
$899$	48.7305	1.62525
$900$	0	0
$901$	−23.5261	−0.783767
$902$	−0.162253	−0.00540242
$903$	0	0
$904$	−12.3202	−0.409765
$905$	−0.0511591	−0.00170059
$906$	0	0
$907$	10.0368	0.333265	0.166633	−	0.986019i	$-0.446711\pi$
$907$	10.0368	0.333265	0.166633	+	0.986019i	$0.446711\pi$
$908$	−4.81711	−0.159862
$909$	0	0
$910$	0	0
$911$	−22.8918	−0.758440	−0.379220	−	0.925306i	$-0.623808\pi$
$911$	−22.8918	−0.758440	−0.379220	+	0.925306i	$0.623808\pi$
$912$	0	0
$913$	−4.57160	−0.151298
$914$	22.3523	0.739350
$915$	0	0
$916$	9.29533	0.307126
$917$	0	0
$918$	0	0
$919$	−21.7821	−0.718525	−0.359262	−	0.933237i	$-0.616972\pi$
$919$	−21.7821	−0.718525	−0.359262	+	0.933237i	$0.616972\pi$
$920$	2.64766	0.0872909
$921$	0	0
$922$	7.97509	0.262646
$923$	36.2370	1.19275
$924$	0	0
$925$	4.64766	0.152814
$926$	−28.7352	−0.944297
$927$	0	0
$928$	−6.19436	−0.203340
$929$	32.8377	1.07737	0.538686	−	0.842507i	$-0.318921\pi$
$929$	32.8377	1.07737	0.538686	+	0.842507i	$0.318921\pi$
$930$	0	0
$931$	0	0
$932$	−0.194356	−0.00636634
$933$	0	0
$934$	−33.5657	−1.09830
$935$	1.02918	0.0336577
$936$	0	0
$937$	8.78074	0.286854	0.143427	−	0.989661i	$-0.454188\pi$
$937$	8.78074	0.286854	0.143427	+	0.989661i	$0.454188\pi$
$938$	0	0
$939$	0	0
$940$	7.21926	0.235466
$941$	−4.26615	−0.139072	−0.0695362	−	0.997579i	$-0.522152\pi$
$941$	−4.26615	−0.139072	−0.0695362	+	0.997579i	$0.522152\pi$
$942$	0	0
$943$	−1.21926	−0.0397046
$944$	−8.64766	−0.281457
$945$	0	0
$946$	6.62568	0.215420
$947$	−23.0584	−0.749296	−0.374648	−	0.927167i	$-0.622236\pi$
$947$	−23.0584	−0.749296	−0.374648	+	0.927167i	$0.622236\pi$
$948$	0	0
$949$	−19.9000	−0.645981
$950$	25.0115	0.811479
$951$	0	0
$952$	0	0
$953$	−36.5552	−1.18414	−0.592070	−	0.805886i	$-0.701689\pi$
$953$	−36.5552	−1.18414	−0.592070	+	0.805886i	$0.701689\pi$
$954$	0	0
$955$	−2.36381	−0.0764910
$956$	13.6549	0.441630
$957$	0	0
$958$	0.367120	0.0118611
$959$	0	0
$960$	0	0
$961$	30.8885	0.996404
$962$	−2.51459	−0.0810736
$963$	0	0
$964$	13.0000	0.418702
$965$	−4.02491	−0.129566
$966$	0	0
$967$	−53.5438	−1.72185	−0.860926	−	0.508731i	$-0.830115\pi$
$967$	−53.5438	−1.72185	−0.860926	+	0.508731i	$0.830115\pi$
$968$	10.6477	0.342229
$969$	0	0
$970$	6.96497	0.223632
$971$	−31.9794	−1.02627	−0.513133	−	0.858309i	$-0.671515\pi$
$971$	−31.9794	−1.02627	−0.513133	+	0.858309i	$0.671515\pi$
$972$	0	0
$973$	0	0
$974$	−29.9076	−0.958300
$975$	0	0
$976$	6.64766	0.212787
$977$	−27.4208	−0.877270	−0.438635	−	0.898665i	$-0.644538\pi$
$977$	−27.4208	−0.877270	−0.438635	+	0.898665i	$0.644538\pi$
$978$	0	0
$979$	7.38151	0.235914
$980$	0	0
$981$	0	0
$982$	−0.510317	−0.0162849
$983$	59.1564	1.88680	0.943398	−	0.331662i	$-0.107610\pi$
$983$	59.1564	1.88680	0.943398	+	0.331662i	$0.107610\pi$
$984$	0	0
$985$	−6.56401	−0.209147
$986$	−18.0938	−0.576223
$987$	0	0
$988$	−13.5323	−0.430520
$989$	49.7893	1.58321
$990$	0	0
$991$	−12.8377	−0.407804	−0.203902	−	0.978991i	$-0.565362\pi$
$991$	−12.8377	−0.407804	−0.203902	+	0.978991i	$0.565362\pi$
$992$	−7.86693	−0.249775
$993$	0	0
$994$	0	0
$995$	−3.33502	−0.105727
$996$	0	0
$997$	5.78074	0.183078	0.0915389	−	0.995802i	$-0.470821\pi$
$997$	5.78074	0.183078	0.0915389	+	0.995802i	$0.470821\pi$
$998$	19.0191	0.602038
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bw.1.2		3
3.2	odd	2		7938.2.a.bz.1.2		3
7.3	odd	6		1134.2.g.m.163.2		6
7.5	odd	6		1134.2.g.m.487.2		6
7.6	odd	2		7938.2.a.bv.1.2		3
9.2	odd	6		2646.2.f.m.1765.2		6
9.4	even	3		882.2.f.o.295.3		6
9.5	odd	6		2646.2.f.m.883.2		6
9.7	even	3		882.2.f.o.589.3		6
21.5	even	6		1134.2.g.l.487.2		6
21.17	even	6		1134.2.g.l.163.2		6
21.20	even	2		7938.2.a.ca.1.2		3
63.2	odd	6		2646.2.h.o.361.2		6
63.4	even	3		882.2.h.p.79.2		6
63.5	even	6		378.2.e.d.235.2		6
63.11	odd	6		2646.2.e.p.1549.2		6
63.13	odd	6		882.2.f.n.295.1		6
63.16	even	3		882.2.h.p.67.2		6
63.20	even	6		2646.2.f.l.1765.2		6
63.23	odd	6		2646.2.e.p.2125.2		6
63.25	even	3		882.2.e.o.373.2		6
63.31	odd	6		126.2.h.d.79.2	yes	6
63.32	odd	6		2646.2.h.o.667.2		6
63.34	odd	6		882.2.f.n.589.1		6
63.38	even	6		378.2.e.d.37.2		6
63.40	odd	6		126.2.e.c.25.2	✓	6
63.41	even	6		2646.2.f.l.883.2		6
63.47	even	6		378.2.h.c.361.2		6
63.52	odd	6		126.2.e.c.121.2	yes	6
63.58	even	3		882.2.e.o.655.2		6
63.59	even	6		378.2.h.c.289.2		6
63.61	odd	6		126.2.h.d.67.2	yes	6
252.31	even	6		1008.2.t.h.961.2		6
252.47	odd	6		3024.2.t.h.1873.2		6
252.59	odd	6		3024.2.t.h.289.2		6
252.103	even	6		1008.2.q.g.529.2		6
252.115	even	6		1008.2.q.g.625.2		6
252.131	odd	6		3024.2.q.g.2881.2		6
252.187	even	6		1008.2.t.h.193.2		6
252.227	odd	6		3024.2.q.g.2305.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.c.25.2	✓	6	63.40	odd	6
126.2.e.c.121.2	yes	6	63.52	odd	6
126.2.h.d.67.2	yes	6	63.61	odd	6
126.2.h.d.79.2	yes	6	63.31	odd	6
378.2.e.d.37.2		6	63.38	even	6
378.2.e.d.235.2		6	63.5	even	6
378.2.h.c.289.2		6	63.59	even	6
378.2.h.c.361.2		6	63.47	even	6
882.2.e.o.373.2		6	63.25	even	3
882.2.e.o.655.2		6	63.58	even	3
882.2.f.n.295.1		6	63.13	odd	6
882.2.f.n.589.1		6	63.34	odd	6
882.2.f.o.295.3		6	9.4	even	3
882.2.f.o.589.3		6	9.7	even	3
882.2.h.p.67.2		6	63.16	even	3
882.2.h.p.79.2		6	63.4	even	3
1008.2.q.g.529.2		6	252.103	even	6
1008.2.q.g.625.2		6	252.115	even	6
1008.2.t.h.193.2		6	252.187	even	6
1008.2.t.h.961.2		6	252.31	even	6
1134.2.g.l.163.2		6	21.17	even	6
1134.2.g.l.487.2		6	21.5	even	6
1134.2.g.m.163.2		6	7.3	odd	6
1134.2.g.m.487.2		6	7.5	odd	6
2646.2.e.p.1549.2		6	63.11	odd	6
2646.2.e.p.2125.2		6	63.23	odd	6
2646.2.f.l.883.2		6	63.41	even	6
2646.2.f.l.1765.2		6	63.20	even	6
2646.2.f.m.883.2		6	9.5	odd	6
2646.2.f.m.1765.2		6	9.2	odd	6
2646.2.h.o.361.2		6	63.2	odd	6
2646.2.h.o.667.2		6	63.32	odd	6
3024.2.q.g.2305.2		6	252.227	odd	6
3024.2.q.g.2881.2		6	252.131	odd	6
3024.2.t.h.289.2		6	252.59	odd	6
3024.2.t.h.1873.2		6	252.47	odd	6
7938.2.a.bv.1.2		3	7.6	odd	2
7938.2.a.bw.1.2		3	1.1	even	1	trivial
7938.2.a.bz.1.2		3	3.2	odd	2
7938.2.a.ca.1.2		3	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.593579 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.593579 q^{5} -1.00000 q^{8} +0.593579 q^{10} -0.593579 q^{11} -2.51459 q^{13} +1.00000 q^{16} +2.92101 q^{17} +5.38151 q^{19} -0.593579 q^{20} +0.593579 q^{22} +4.46050 q^{23} -4.64766 q^{25} +2.51459 q^{26} +6.19436 q^{29} +7.86693 q^{31} -1.00000 q^{32} -2.92101 q^{34} -1.00000 q^{37} -5.38151 q^{38} +0.593579 q^{40} -0.273346 q^{41} +11.1623 q^{43} -0.593579 q^{44} -4.46050 q^{46} -12.1623 q^{47} +4.64766 q^{50} -2.51459 q^{52} -8.05408 q^{53} +0.352336 q^{55} -6.19436 q^{58} -8.64766 q^{59} +6.64766 q^{61} -7.86693 q^{62} +1.00000 q^{64} +1.49261 q^{65} -1.91381 q^{67} +2.92101 q^{68} -14.4107 q^{71} +7.91381 q^{73} +1.00000 q^{74} +5.38151 q^{76} -9.24844 q^{79} -0.593579 q^{80} +0.273346 q^{82} +7.70175 q^{83} -1.73385 q^{85} -11.1623 q^{86} +0.593579 q^{88} -12.4356 q^{89} +4.46050 q^{92} +12.1623 q^{94} -3.19436 q^{95} +11.7339 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^5 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{8} - q^{10} + q^{11} + 8 q^{13} + 3 q^{16} - 4 q^{17} - 3 q^{19} + q^{20} - q^{22} + 7 q^{23} - 2 q^{25} - 8 q^{26} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34} - 3 q^{37} + 3 q^{38} - q^{40} + 6 q^{43} + q^{44} - 7 q^{46} - 9 q^{47} + 2 q^{50} + 8 q^{52} - 15 q^{53} + 13 q^{55} - 5 q^{58} - 14 q^{59} + 8 q^{61} - 20 q^{62} + 3 q^{64} + 12 q^{65} - q^{67} - 4 q^{68} + 7 q^{71} + 19 q^{73} + 3 q^{74} - 3 q^{76} - 5 q^{79} + q^{80} + 2 q^{83} + 2 q^{85} - 6 q^{86} - q^{88} - 9 q^{89} + 7 q^{92} + 9 q^{94} + 4 q^{95} + 28 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^5 - 3 * q^8 - q^10 + q^11 + 8 * q^13 + 3 * q^16 - 4 * q^17 - 3 * q^19 + q^20 - q^22 + 7 * q^23 - 2 * q^25 - 8 * q^26 + 5 * q^29 + 20 * q^31 - 3 * q^32 + 4 * q^34 - 3 * q^37 + 3 * q^38 - q^40 + 6 * q^43 + q^44 - 7 * q^46 - 9 * q^47 + 2 * q^50 + 8 * q^52 - 15 * q^53 + 13 * q^55 - 5 * q^58 - 14 * q^59 + 8 * q^61 - 20 * q^62 + 3 * q^64 + 12 * q^65 - q^67 - 4 * q^68 + 7 * q^71 + 19 * q^73 + 3 * q^74 - 3 * q^76 - 5 * q^79 + q^80 + 2 * q^83 + 2 * q^85 - 6 * q^86 - q^88 - 9 * q^89 + 7 * q^92 + 9 * q^94 + 4 * q^95 + 28 * q^97

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