LMFDB - Embedded newform 7938.2.a.bx.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:	$1$
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			$0.239123$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.76088 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.76088 q^{5} -1.00000 q^{8} -1.76088 q^{10} -6.12476 q^{11} -0.760877 q^{13} +1.00000 q^{16} +6.84213 q^{17} +1.94282 q^{19} +1.76088 q^{20} +6.12476 q^{22} +0.421067 q^{23} -1.89931 q^{25} +0.760877 q^{26} +1.46457 q^{29} -7.70370 q^{31} -1.00000 q^{32} -6.84213 q^{34} -2.88564 q^{37} -1.94282 q^{38} -1.76088 q^{40} +6.94282 q^{41} -8.66019 q^{43} -6.12476 q^{44} -0.421067 q^{46} +1.66019 q^{47} +1.89931 q^{50} -0.760877 q^{52} -0.225450 q^{53} -10.7850 q^{55} -1.46457 q^{58} +1.98633 q^{59} +10.3502 q^{61} +7.70370 q^{62} +1.00000 q^{64} -1.33981 q^{65} +6.78495 q^{67} +6.84213 q^{68} -10.7850 q^{71} +0.306707 q^{73} +2.88564 q^{74} +1.94282 q^{76} -13.4451 q^{79} +1.76088 q^{80} -6.94282 q^{82} +3.12476 q^{83} +12.0482 q^{85} +8.66019 q^{86} +6.12476 q^{88} -2.60301 q^{89} +0.421067 q^{92} -1.66019 q^{94} +3.42107 q^{95} -3.63611 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.76088	0.787488	0.393744	−	0.919220i	$-0.371180\pi$
$5$	1.76088	0.787488	0.393744	+	0.919220i	$0.371180\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.76088	−0.556838
$11$	−6.12476	−1.84669	−0.923343	−	0.383977i	$-0.874554\pi$
$11$	−6.12476	−1.84669	−0.923343	+	0.383977i	$0.874554\pi$
$12$	0	0
$13$	−0.760877	−0.211029	−0.105515	−	0.994418i	$-0.533649\pi$
$13$	−0.760877	−0.211029	−0.105515	+	0.994418i	$0.533649\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.84213	1.65946	0.829731	−	0.558164i	$-0.188494\pi$
$17$	6.84213	1.65946	0.829731	+	0.558164i	$0.188494\pi$
$18$	0	0
$19$	1.94282	0.445713	0.222857	−	0.974851i	$-0.428462\pi$
$19$	1.94282	0.445713	0.222857	+	0.974851i	$0.428462\pi$
$20$	1.76088	0.393744
$21$	0	0
$22$	6.12476	1.30580
$23$	0.421067	0.0877985	0.0438992	−	0.999036i	$-0.486022\pi$
$23$	0.421067	0.0877985	0.0438992	+	0.999036i	$0.486022\pi$
$24$	0	0
$25$	−1.89931	−0.379863
$26$	0.760877	0.149220
$27$	0	0
$28$	0	0
$29$	1.46457	0.271964	0.135982	−	0.990711i	$-0.456581\pi$
$29$	1.46457	0.271964	0.135982	+	0.990711i	$0.456581\pi$
$30$	0	0
$31$	−7.70370	−1.38362	−0.691812	−	0.722077i	$-0.743187\pi$
$31$	−7.70370	−1.38362	−0.691812	+	0.722077i	$0.743187\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.84213	−1.17342
$35$	0	0
$36$	0	0
$37$	−2.88564	−0.474396	−0.237198	−	0.971461i	$-0.576229\pi$
$37$	−2.88564	−0.474396	−0.237198	+	0.971461i	$0.576229\pi$
$38$	−1.94282	−0.315167
$39$	0	0
$40$	−1.76088	−0.278419
$41$	6.94282	1.08429	0.542143	−	0.840286i	$-0.317613\pi$
$41$	6.94282	1.08429	0.542143	+	0.840286i	$0.317613\pi$
$42$	0	0
$43$	−8.66019	−1.32067	−0.660333	−	0.750973i	$-0.729585\pi$
$43$	−8.66019	−1.32067	−0.660333	+	0.750973i	$0.729585\pi$
$44$	−6.12476	−0.923343
$45$	0	0
$46$	−0.421067	−0.0620829
$47$	1.66019	0.242164	0.121082	−	0.992643i	$-0.461364\pi$
$47$	1.66019	0.242164	0.121082	+	0.992643i	$0.461364\pi$
$48$	0	0
$49$	0	0
$50$	1.89931	0.268603
$51$	0	0
$52$	−0.760877	−0.105515
$53$	−0.225450	−0.0309680	−0.0154840	−	0.999880i	$-0.504929\pi$
$53$	−0.225450	−0.0309680	−0.0154840	+	0.999880i	$0.504929\pi$
$54$	0	0
$55$	−10.7850	−1.45424
$56$	0	0
$57$	0	0
$58$	−1.46457	−0.192308
$59$	1.98633	0.258598	0.129299	−	0.991606i	$-0.458727\pi$
$59$	1.98633	0.258598	0.129299	+	0.991606i	$0.458727\pi$
$60$	0	0
$61$	10.3502	1.32521	0.662605	−	0.748969i	$-0.269451\pi$
$61$	10.3502	1.32521	0.662605	+	0.748969i	$0.269451\pi$
$62$	7.70370	0.978370
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.33981	−0.166183
$66$	0	0
$67$	6.78495	0.828914	0.414457	−	0.910069i	$-0.363972\pi$
$67$	6.78495	0.828914	0.414457	+	0.910069i	$0.363972\pi$
$68$	6.84213	0.829731
$69$	0	0
$70$	0	0
$71$	−10.7850	−1.27994	−0.639969	−	0.768401i	$-0.721053\pi$
$71$	−10.7850	−1.27994	−0.639969	+	0.768401i	$0.721053\pi$
$72$	0	0
$73$	0.306707	0.0358973	0.0179487	−	0.999839i	$-0.494286\pi$
$73$	0.306707	0.0358973	0.0179487	+	0.999839i	$0.494286\pi$
$74$	2.88564	0.335449
$75$	0	0
$76$	1.94282	0.222857
$77$	0	0
$78$	0	0
$79$	−13.4451	−1.51270	−0.756348	−	0.654169i	$-0.773018\pi$
$79$	−13.4451	−1.51270	−0.756348	+	0.654169i	$0.773018\pi$
$80$	1.76088	0.196872
$81$	0	0
$82$	−6.94282	−0.766706
$83$	3.12476	0.342987	0.171494	−	0.985185i	$-0.445141\pi$
$83$	3.12476	0.342987	0.171494	+	0.985185i	$0.445141\pi$
$84$	0	0
$85$	12.0482	1.30681
$86$	8.66019	0.933852
$87$	0	0
$88$	6.12476	0.652902
$89$	−2.60301	−0.275919	−0.137959	−	0.990438i	$-0.544054\pi$
$89$	−2.60301	−0.275919	−0.137959	+	0.990438i	$0.544054\pi$
$90$	0	0
$91$	0	0
$92$	0.421067	0.0438992
$93$	0	0
$94$	−1.66019	−0.171236
$95$	3.42107	0.350994
$96$	0	0
$97$	−3.63611	−0.369191	−0.184596	−	0.982815i	$-0.559098\pi$
$97$	−3.63611	−0.369191	−0.184596	+	0.982815i	$0.559098\pi$
$98$	0	0
$99$	0	0
$100$	−1.89931	−0.189931
$101$	−8.01040	−0.797065	−0.398532	−	0.917154i	$-0.630480\pi$
$101$	−8.01040	−0.797065	−0.398532	+	0.917154i	$0.630480\pi$
$102$	0	0
$103$	6.82846	0.672828	0.336414	−	0.941714i	$-0.390786\pi$
$103$	6.82846	0.672828	0.336414	+	0.941714i	$0.390786\pi$
$104$	0.760877	0.0746101
$105$	0	0
$106$	0.225450	0.0218977
$107$	3.54583	0.342788	0.171394	−	0.985203i	$-0.445173\pi$
$107$	3.54583	0.342788	0.171394	+	0.985203i	$0.445173\pi$
$108$	0	0
$109$	−0.703697	−0.0674019	−0.0337010	−	0.999432i	$-0.510729\pi$
$109$	−0.703697	−0.0674019	−0.0337010	+	0.999432i	$0.510729\pi$
$110$	10.7850	1.02830
$111$	0	0
$112$	0	0
$113$	8.50232	0.799831	0.399916	−	0.916552i	$-0.369039\pi$
$113$	8.50232	0.799831	0.399916	+	0.916552i	$0.369039\pi$
$114$	0	0
$115$	0.741446	0.0691402
$116$	1.46457	0.135982
$117$	0	0
$118$	−1.98633	−0.182856
$119$	0	0
$120$	0	0
$121$	26.5127	2.41025
$122$	−10.3502	−0.937064
$123$	0	0
$124$	−7.70370	−0.691812
$125$	−12.1488	−1.08663
$126$	0	0
$127$	−18.9532	−1.68183	−0.840913	−	0.541170i	$-0.817982\pi$
$127$	−18.9532	−1.68183	−0.840913	+	0.541170i	$0.817982\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.33981	0.117509
$131$	−7.29303	−0.637195	−0.318598	−	0.947890i	$-0.603212\pi$
$131$	−7.29303	−0.637195	−0.318598	+	0.947890i	$0.603212\pi$
$132$	0	0
$133$	0	0
$134$	−6.78495	−0.586131
$135$	0	0
$136$	−6.84213	−0.586708
$137$	8.18194	0.699031	0.349515	−	0.936931i	$-0.386346\pi$
$137$	8.18194	0.699031	0.349515	+	0.936931i	$0.386346\pi$
$138$	0	0
$139$	−12.4646	−1.05723	−0.528616	−	0.848861i	$-0.677289\pi$
$139$	−12.4646	−1.05723	−0.528616	+	0.848861i	$0.677289\pi$
$140$	0	0
$141$	0	0
$142$	10.7850	0.905053
$143$	4.66019	0.389705
$144$	0	0
$145$	2.57893	0.214169
$146$	−0.306707	−0.0253832
$147$	0	0
$148$	−2.88564	−0.237198
$149$	−8.82846	−0.723256	−0.361628	−	0.932323i	$-0.617779\pi$
$149$	−8.82846	−0.723256	−0.361628	+	0.932323i	$0.617779\pi$
$150$	0	0
$151$	−14.9863	−1.21957	−0.609785	−	0.792567i	$-0.708744\pi$
$151$	−14.9863	−1.21957	−0.609785	+	0.792567i	$0.708744\pi$
$152$	−1.94282	−0.157584
$153$	0	0
$154$	0	0
$155$	−13.5653	−1.08959
$156$	0	0
$157$	−18.9806	−1.51481	−0.757407	−	0.652943i	$-0.773534\pi$
$157$	−18.9806	−1.51481	−0.757407	+	0.652943i	$0.773534\pi$
$158$	13.4451	1.06964
$159$	0	0
$160$	−1.76088	−0.139210
$161$	0	0
$162$	0	0
$163$	15.0377	1.17785	0.588924	−	0.808189i	$-0.299552\pi$
$163$	15.0377	1.17785	0.588924	+	0.808189i	$0.299552\pi$
$164$	6.94282	0.542143
$165$	0	0
$166$	−3.12476	−0.242529
$167$	−1.14419	−0.0885404	−0.0442702	−	0.999020i	$-0.514096\pi$
$167$	−1.14419	−0.0885404	−0.0442702	+	0.999020i	$0.514096\pi$
$168$	0	0
$169$	−12.4211	−0.955467
$170$	−12.0482	−0.924051
$171$	0	0
$172$	−8.66019	−0.660333
$173$	0.497677	0.0378377	0.0189188	−	0.999821i	$-0.493978\pi$
$173$	0.497677	0.0378377	0.0189188	+	0.999821i	$0.493978\pi$
$174$	0	0
$175$	0	0
$176$	−6.12476	−0.461671
$177$	0	0
$178$	2.60301	0.195104
$179$	8.82846	0.659870	0.329935	−	0.944004i	$-0.392973\pi$
$179$	8.82846	0.659870	0.329935	+	0.944004i	$0.392973\pi$
$180$	0	0
$181$	−1.32941	−0.0988140	−0.0494070	−	0.998779i	$-0.515733\pi$
$181$	−1.32941	−0.0988140	−0.0494070	+	0.998779i	$0.515733\pi$
$182$	0	0
$183$	0	0
$184$	−0.421067	−0.0310414
$185$	−5.08126	−0.373581
$186$	0	0
$187$	−41.9064	−3.06450
$188$	1.66019	0.121082
$189$	0	0
$190$	−3.42107	−0.248190
$191$	16.1683	1.16989	0.584947	−	0.811071i	$-0.301115\pi$
$191$	16.1683	1.16989	0.584947	+	0.811071i	$0.301115\pi$
$192$	0	0
$193$	−14.1683	−1.01985	−0.509927	−	0.860218i	$-0.670328\pi$
$193$	−14.1683	−1.01985	−0.509927	+	0.860218i	$0.670328\pi$
$194$	3.63611	0.261058
$195$	0	0
$196$	0	0
$197$	−15.8421	−1.12871	−0.564353	−	0.825534i	$-0.690874\pi$
$197$	−15.8421	−1.12871	−0.564353	+	0.825534i	$0.690874\pi$
$198$	0	0
$199$	−8.94282	−0.633940	−0.316970	−	0.948436i	$-0.602665\pi$
$199$	−8.94282	−0.633940	−0.316970	+	0.948436i	$0.602665\pi$
$200$	1.89931	0.134302
$201$	0	0
$202$	8.01040	0.563610
$203$	0	0
$204$	0	0
$205$	12.2255	0.853862
$206$	−6.82846	−0.475761
$207$	0	0
$208$	−0.760877	−0.0527573
$209$	−11.8993	−0.823093
$210$	0	0
$211$	−22.7713	−1.56764	−0.783820	−	0.620988i	$-0.786731\pi$
$211$	−22.7713	−1.56764	−0.783820	+	0.620988i	$0.786731\pi$
$212$	−0.225450	−0.0154840
$213$	0	0
$214$	−3.54583	−0.242388
$215$	−15.2495	−1.04001
$216$	0	0
$217$	0	0
$218$	0.703697	0.0476604
$219$	0	0
$220$	−10.7850	−0.727121
$221$	−5.20602	−0.350195
$222$	0	0
$223$	−12.8856	−0.862886	−0.431443	−	0.902140i	$-0.641995\pi$
$223$	−12.8856	−0.862886	−0.431443	+	0.902140i	$0.641995\pi$
$224$	0	0
$225$	0	0
$226$	−8.50232	−0.565566
$227$	21.9967	1.45997	0.729987	−	0.683461i	$-0.239526\pi$
$227$	21.9967	1.45997	0.729987	+	0.683461i	$0.239526\pi$
$228$	0	0
$229$	3.79863	0.251020	0.125510	−	0.992092i	$-0.459943\pi$
$229$	3.79863	0.251020	0.125510	+	0.992092i	$0.459943\pi$
$230$	−0.741446	−0.0488895
$231$	0	0
$232$	−1.46457	−0.0961540
$233$	−6.67059	−0.437005	−0.218503	−	0.975836i	$-0.570117\pi$
$233$	−6.67059	−0.437005	−0.218503	+	0.975836i	$0.570117\pi$
$234$	0	0
$235$	2.92339	0.190701
$236$	1.98633	0.129299
$237$	0	0
$238$	0	0
$239$	−15.6408	−1.01172	−0.505858	−	0.862617i	$-0.668824\pi$
$239$	−15.6408	−1.01172	−0.505858	+	0.862617i	$0.668824\pi$
$240$	0	0
$241$	−21.4120	−1.37927	−0.689635	−	0.724157i	$-0.742229\pi$
$241$	−21.4120	−1.37927	−0.689635	+	0.724157i	$0.742229\pi$
$242$	−26.5127	−1.70430
$243$	0	0
$244$	10.3502	0.662605
$245$	0	0
$246$	0	0
$247$	−1.47825	−0.0940586
$248$	7.70370	0.489185
$249$	0	0
$250$	12.1488	0.768360
$251$	−23.6030	−1.48981	−0.744904	−	0.667171i	$-0.767505\pi$
$251$	−23.6030	−1.48981	−0.744904	+	0.667171i	$0.767505\pi$
$252$	0	0
$253$	−2.57893	−0.162136
$254$	18.9532	1.18923
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.2599	1.26378	0.631890	−	0.775058i	$-0.282279\pi$
$257$	20.2599	1.26378	0.631890	+	0.775058i	$0.282279\pi$
$258$	0	0
$259$	0	0
$260$	−1.33981	−0.0830915
$261$	0	0
$262$	7.29303	0.450565
$263$	22.4887	1.38671	0.693355	−	0.720596i	$-0.256132\pi$
$263$	22.4887	1.38671	0.693355	+	0.720596i	$0.256132\pi$
$264$	0	0
$265$	−0.396990	−0.0243869
$266$	0	0
$267$	0	0
$268$	6.78495	0.414457
$269$	25.3412	1.54508	0.772540	−	0.634966i	$-0.218986\pi$
$269$	25.3412	1.54508	0.772540	+	0.634966i	$0.218986\pi$
$270$	0	0
$271$	−13.7576	−0.835715	−0.417858	−	0.908513i	$-0.637219\pi$
$271$	−13.7576	−0.835715	−0.417858	+	0.908513i	$0.637219\pi$
$272$	6.84213	0.414865
$273$	0	0
$274$	−8.18194	−0.494289
$275$	11.6328	0.701487
$276$	0	0
$277$	−3.28263	−0.197234	−0.0986171	−	0.995125i	$-0.531442\pi$
$277$	−3.28263	−0.197234	−0.0986171	+	0.995125i	$0.531442\pi$
$278$	12.4646	0.747575
$279$	0	0
$280$	0	0
$281$	1.26896	0.0756996	0.0378498	−	0.999283i	$-0.487949\pi$
$281$	1.26896	0.0756996	0.0378498	+	0.999283i	$0.487949\pi$
$282$	0	0
$283$	8.19235	0.486984	0.243492	−	0.969903i	$-0.421707\pi$
$283$	8.19235	0.486984	0.243492	+	0.969903i	$0.421707\pi$
$284$	−10.7850	−0.639969
$285$	0	0
$286$	−4.66019	−0.275563
$287$	0	0
$288$	0	0
$289$	29.8148	1.75381
$290$	−2.57893	−0.151440
$291$	0	0
$292$	0.306707	0.0179487
$293$	−15.4509	−0.902651	−0.451326	−	0.892359i	$-0.649049\pi$
$293$	−15.4509	−0.902651	−0.451326	+	0.892359i	$0.649049\pi$
$294$	0	0
$295$	3.49768	0.203643
$296$	2.88564	0.167724
$297$	0	0
$298$	8.82846	0.511419
$299$	−0.320380	−0.0185280
$300$	0	0
$301$	0	0
$302$	14.9863	0.862366
$303$	0	0
$304$	1.94282	0.111428
$305$	18.2255	1.04359
$306$	0	0
$307$	−4.89931	−0.279619	−0.139809	−	0.990178i	$-0.544649\pi$
$307$	−4.89931	−0.279619	−0.139809	+	0.990178i	$0.544649\pi$
$308$	0	0
$309$	0	0
$310$	13.5653	0.770455
$311$	−7.69002	−0.436061	−0.218031	−	0.975942i	$-0.569963\pi$
$311$	−7.69002	−0.436061	−0.218031	+	0.975942i	$0.569963\pi$
$312$	0	0
$313$	1.72313	0.0973969	0.0486985	−	0.998814i	$-0.484493\pi$
$313$	1.72313	0.0973969	0.0486985	+	0.998814i	$0.484493\pi$
$314$	18.9806	1.07114
$315$	0	0
$316$	−13.4451	−0.756348
$317$	−33.2028	−1.86485	−0.932426	−	0.361361i	$-0.882312\pi$
$317$	−33.2028	−1.86485	−0.932426	+	0.361361i	$0.882312\pi$
$318$	0	0
$319$	−8.97017	−0.502233
$320$	1.76088	0.0984360
$321$	0	0
$322$	0	0
$323$	13.2930	0.739644
$324$	0	0
$325$	1.44514	0.0801621
$326$	−15.0377	−0.832864
$327$	0	0
$328$	−6.94282	−0.383353
$329$	0	0
$330$	0	0
$331$	2.88891	0.158789	0.0793944	−	0.996843i	$-0.474701\pi$
$331$	2.88891	0.158789	0.0793944	+	0.996843i	$0.474701\pi$
$332$	3.12476	0.171494
$333$	0	0
$334$	1.14419	0.0626075
$335$	11.9475	0.652760
$336$	0	0
$337$	8.72313	0.475179	0.237590	−	0.971366i	$-0.423643\pi$
$337$	8.72313	0.475179	0.237590	+	0.971366i	$0.423643\pi$
$338$	12.4211	0.675617
$339$	0	0
$340$	12.0482	0.653403
$341$	47.1833	2.55512
$342$	0	0
$343$	0	0
$344$	8.66019	0.466926
$345$	0	0
$346$	−0.497677	−0.0267553
$347$	−9.69467	−0.520437	−0.260219	−	0.965550i	$-0.583795\pi$
$347$	−9.69467	−0.520437	−0.260219	+	0.965550i	$0.583795\pi$
$348$	0	0
$349$	28.3984	1.52013	0.760065	−	0.649847i	$-0.225167\pi$
$349$	28.3984	1.52013	0.760065	+	0.649847i	$0.225167\pi$
$350$	0	0
$351$	0	0
$352$	6.12476	0.326451
$353$	−4.39372	−0.233854	−0.116927	−	0.993141i	$-0.537304\pi$
$353$	−4.39372	−0.233854	−0.116927	+	0.993141i	$0.537304\pi$
$354$	0	0
$355$	−18.9910	−1.00794
$356$	−2.60301	−0.137959
$357$	0	0
$358$	−8.82846	−0.466599
$359$	32.1592	1.69730	0.848650	−	0.528955i	$-0.177416\pi$
$359$	32.1592	1.69730	0.848650	+	0.528955i	$0.177416\pi$
$360$	0	0
$361$	−15.2255	−0.801339
$362$	1.32941	0.0698721
$363$	0	0
$364$	0	0
$365$	0.540073	0.0282687
$366$	0	0
$367$	−34.6030	−1.80626	−0.903131	−	0.429365i	$-0.858738\pi$
$367$	−34.6030	−1.80626	−0.903131	+	0.429365i	$0.858738\pi$
$368$	0.421067	0.0219496
$369$	0	0
$370$	5.08126	0.264162
$371$	0	0
$372$	0	0
$373$	10.9759	0.568312	0.284156	−	0.958778i	$-0.408287\pi$
$373$	10.9759	0.568312	0.284156	+	0.958778i	$0.408287\pi$
$374$	41.9064	2.16693
$375$	0	0
$376$	−1.66019	−0.0856178
$377$	−1.11436	−0.0573925
$378$	0	0
$379$	33.9877	1.74583	0.872916	−	0.487871i	$-0.162226\pi$
$379$	33.9877	1.74583	0.872916	+	0.487871i	$0.162226\pi$
$380$	3.42107	0.175497
$381$	0	0
$382$	−16.1683	−0.827241
$383$	−21.0241	−1.07428	−0.537140	−	0.843493i	$-0.680495\pi$
$383$	−21.0241	−1.07428	−0.537140	+	0.843493i	$0.680495\pi$
$384$	0	0
$385$	0	0
$386$	14.1683	0.721146
$387$	0	0
$388$	−3.63611	−0.184596
$389$	−13.7382	−0.696553	−0.348277	−	0.937392i	$-0.613233\pi$
$389$	−13.7382	−0.696553	−0.348277	+	0.937392i	$0.613233\pi$
$390$	0	0
$391$	2.88099	0.145698
$392$	0	0
$393$	0	0
$394$	15.8421	0.798115
$395$	−23.6752	−1.19123
$396$	0	0
$397$	−7.15787	−0.359243	−0.179622	−	0.983736i	$-0.557487\pi$
$397$	−7.15787	−0.359243	−0.179622	+	0.983736i	$0.557487\pi$
$398$	8.94282	0.448263
$399$	0	0
$400$	−1.89931	−0.0949657
$401$	9.27936	0.463389	0.231695	−	0.972789i	$-0.425573\pi$
$401$	9.27936	0.463389	0.231695	+	0.972789i	$0.425573\pi$
$402$	0	0
$403$	5.86156	0.291985
$404$	−8.01040	−0.398532
$405$	0	0
$406$	0	0
$407$	17.6739	0.876061
$408$	0	0
$409$	−15.1683	−0.750023	−0.375011	−	0.927020i	$-0.622361\pi$
$409$	−15.1683	−0.750023	−0.375011	+	0.927020i	$0.622361\pi$
$410$	−12.2255	−0.603772
$411$	0	0
$412$	6.82846	0.336414
$413$	0	0
$414$	0	0
$415$	5.50232	0.270098
$416$	0.760877	0.0373051
$417$	0	0
$418$	11.8993	0.582014
$419$	8.33654	0.407267	0.203633	−	0.979047i	$-0.434725\pi$
$419$	8.33654	0.407267	0.203633	+	0.979047i	$0.434725\pi$
$420$	0	0
$421$	7.00465	0.341386	0.170693	−	0.985324i	$-0.445399\pi$
$421$	7.00465	0.341386	0.170693	+	0.985324i	$0.445399\pi$
$422$	22.7713	1.10849
$423$	0	0
$424$	0.225450	0.0109488
$425$	−12.9954	−0.630367
$426$	0	0
$427$	0	0
$428$	3.54583	0.171394
$429$	0	0
$430$	15.2495	0.735397
$431$	−3.45090	−0.166224	−0.0831120	−	0.996540i	$-0.526486\pi$
$431$	−3.45090	−0.166224	−0.0831120	+	0.996540i	$0.526486\pi$
$432$	0	0
$433$	−28.2599	−1.35809	−0.679043	−	0.734099i	$-0.737605\pi$
$433$	−28.2599	−1.35809	−0.679043	+	0.734099i	$0.737605\pi$
$434$	0	0
$435$	0	0
$436$	−0.703697	−0.0337010
$437$	0.818057	0.0391330
$438$	0	0
$439$	28.8960	1.37913	0.689566	−	0.724222i	$-0.257801\pi$
$439$	28.8960	1.37913	0.689566	+	0.724222i	$0.257801\pi$
$440$	10.7850	0.514152
$441$	0	0
$442$	5.20602	0.247625
$443$	13.7609	0.653799	0.326899	−	0.945059i	$-0.393996\pi$
$443$	13.7609	0.653799	0.326899	+	0.945059i	$0.393996\pi$
$444$	0	0
$445$	−4.58358	−0.217283
$446$	12.8856	0.610153
$447$	0	0
$448$	0	0
$449$	20.2003	0.953309	0.476655	−	0.879091i	$-0.341849\pi$
$449$	20.2003	0.953309	0.476655	+	0.879091i	$0.341849\pi$
$450$	0	0
$451$	−42.5231	−2.00234
$452$	8.50232	0.399916
$453$	0	0
$454$	−21.9967	−1.03236
$455$	0	0
$456$	0	0
$457$	20.0298	0.936956	0.468478	−	0.883475i	$-0.344803\pi$
$457$	20.0298	0.936956	0.468478	+	0.883475i	$0.344803\pi$
$458$	−3.79863	−0.177498
$459$	0	0
$460$	0.741446	0.0345701
$461$	−11.9532	−0.556717	−0.278359	−	0.960477i	$-0.589790\pi$
$461$	−11.9532	−0.556717	−0.278359	+	0.960477i	$0.589790\pi$
$462$	0	0
$463$	−13.2905	−0.617664	−0.308832	−	0.951117i	$-0.599938\pi$
$463$	−13.2905	−0.617664	−0.308832	+	0.951117i	$0.599938\pi$
$464$	1.46457	0.0679911
$465$	0	0
$466$	6.67059	0.309009
$467$	11.2301	0.519667	0.259833	−	0.965653i	$-0.416332\pi$
$467$	11.2301	0.519667	0.259833	+	0.965653i	$0.416332\pi$
$468$	0	0
$469$	0	0
$470$	−2.92339	−0.134846
$471$	0	0
$472$	−1.98633	−0.0914281
$473$	53.0416	2.43886
$474$	0	0
$475$	−3.69002	−0.169310
$476$	0	0
$477$	0	0
$478$	15.6408	0.715392
$479$	−32.6271	−1.49077	−0.745385	−	0.666634i	$-0.767734\pi$
$479$	−32.6271	−1.49077	−0.745385	+	0.666634i	$0.767734\pi$
$480$	0	0
$481$	2.19562	0.100111
$482$	21.4120	0.975292
$483$	0	0
$484$	26.5127	1.20512
$485$	−6.40275	−0.290734
$486$	0	0
$487$	−3.69794	−0.167570	−0.0837848	−	0.996484i	$-0.526701\pi$
$487$	−3.69794	−0.167570	−0.0837848	+	0.996484i	$0.526701\pi$
$488$	−10.3502	−0.468532
$489$	0	0
$490$	0	0
$491$	−37.5609	−1.69510	−0.847549	−	0.530717i	$-0.821923\pi$
$491$	−37.5609	−1.69510	−0.847549	+	0.530717i	$0.821923\pi$
$492$	0	0
$493$	10.0208	0.451314
$494$	1.47825	0.0665095
$495$	0	0
$496$	−7.70370	−0.345906
$497$	0	0
$498$	0	0
$499$	−31.7954	−1.42336	−0.711678	−	0.702506i	$-0.752064\pi$
$499$	−31.7954	−1.42336	−0.711678	+	0.702506i	$0.752064\pi$
$500$	−12.1488	−0.543313
$501$	0	0
$502$	23.6030	1.05345
$503$	30.8252	1.37443	0.687214	−	0.726455i	$-0.258834\pi$
$503$	30.8252	1.37443	0.687214	+	0.726455i	$0.258834\pi$
$504$	0	0
$505$	−14.1053	−0.627679
$506$	2.57893	0.114648
$507$	0	0
$508$	−18.9532	−0.840913
$509$	8.01616	0.355310	0.177655	−	0.984093i	$-0.443149\pi$
$509$	8.01616	0.355310	0.177655	+	0.984093i	$0.443149\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.2599	−0.893627
$515$	12.0241	0.529844
$516$	0	0
$517$	−10.1683	−0.447200
$518$	0	0
$519$	0	0
$520$	1.33981	0.0587546
$521$	−29.7292	−1.30246	−0.651229	−	0.758881i	$-0.725746\pi$
$521$	−29.7292	−1.30246	−0.651229	+	0.758881i	$0.725746\pi$
$522$	0	0
$523$	26.9396	1.17798	0.588992	−	0.808139i	$-0.299525\pi$
$523$	26.9396	1.17798	0.588992	+	0.808139i	$0.299525\pi$
$524$	−7.29303	−0.318598
$525$	0	0
$526$	−22.4887	−0.980552
$527$	−52.7097	−2.29607
$528$	0	0
$529$	−22.8227	−0.992291
$530$	0.396990	0.0172441
$531$	0	0
$532$	0	0
$533$	−5.28263	−0.228816
$534$	0	0
$535$	6.24377	0.269942
$536$	−6.78495	−0.293065
$537$	0	0
$538$	−25.3412	−1.09254
$539$	0	0
$540$	0	0
$541$	−14.3114	−0.615293	−0.307647	−	0.951501i	$-0.599541\pi$
$541$	−14.3114	−0.615293	−0.307647	+	0.951501i	$0.599541\pi$
$542$	13.7576	0.590940
$543$	0	0
$544$	−6.84213	−0.293354
$545$	−1.23912	−0.0530782
$546$	0	0
$547$	−2.04926	−0.0876202	−0.0438101	−	0.999040i	$-0.513950\pi$
$547$	−2.04926	−0.0876202	−0.0438101	+	0.999040i	$0.513950\pi$
$548$	8.18194	0.349515
$549$	0	0
$550$	−11.6328	−0.496026
$551$	2.84540	0.121218
$552$	0	0
$553$	0	0
$554$	3.28263	0.139466
$555$	0	0
$556$	−12.4646	−0.528616
$557$	17.6868	0.749412	0.374706	−	0.927144i	$-0.377744\pi$
$557$	17.6868	0.749412	0.374706	+	0.927144i	$0.377744\pi$
$558$	0	0
$559$	6.58934	0.278699
$560$	0	0
$561$	0	0
$562$	−1.26896	−0.0535277
$563$	0.937063	0.0394925	0.0197462	−	0.999805i	$-0.493714\pi$
$563$	0.937063	0.0394925	0.0197462	+	0.999805i	$0.493714\pi$
$564$	0	0
$565$	14.9715	0.629858
$566$	−8.19235	−0.344350
$567$	0	0
$568$	10.7850	0.452527
$569$	−23.5264	−0.986278	−0.493139	−	0.869951i	$-0.664151\pi$
$569$	−23.5264	−0.986278	−0.493139	+	0.869951i	$0.664151\pi$
$570$	0	0
$571$	−0.484004	−0.0202549	−0.0101275	−	0.999949i	$-0.503224\pi$
$571$	−0.484004	−0.0202549	−0.0101275	+	0.999949i	$0.503224\pi$
$572$	4.66019	0.194852
$573$	0	0
$574$	0	0
$575$	−0.799737	−0.0333514
$576$	0	0
$577$	−4.46130	−0.185727	−0.0928633	−	0.995679i	$-0.529602\pi$
$577$	−4.46130	−0.185727	−0.0928633	+	0.995679i	$0.529602\pi$
$578$	−29.8148	−1.24013
$579$	0	0
$580$	2.57893	0.107084
$581$	0	0
$582$	0	0
$583$	1.38083	0.0571881
$584$	−0.306707	−0.0126916
$585$	0	0
$586$	15.4509	0.638271
$587$	−16.6304	−0.686408	−0.343204	−	0.939261i	$-0.611512\pi$
$587$	−16.6304	−0.686408	−0.343204	+	0.939261i	$0.611512\pi$
$588$	0	0
$589$	−14.9669	−0.616700
$590$	−3.49768	−0.143997
$591$	0	0
$592$	−2.88564	−0.118599
$593$	−41.5264	−1.70528	−0.852642	−	0.522495i	$-0.825001\pi$
$593$	−41.5264	−1.70528	−0.852642	+	0.522495i	$0.825001\pi$
$594$	0	0
$595$	0	0
$596$	−8.82846	−0.361628
$597$	0	0
$598$	0.320380	0.0131013
$599$	−15.0766	−0.616014	−0.308007	−	0.951384i	$-0.599662\pi$
$599$	−15.0766	−0.616014	−0.308007	+	0.951384i	$0.599662\pi$
$600$	0	0
$601$	−16.1111	−0.657185	−0.328593	−	0.944472i	$-0.606574\pi$
$601$	−16.1111	−0.657185	−0.328593	+	0.944472i	$0.606574\pi$
$602$	0	0
$603$	0	0
$604$	−14.9863	−0.609785
$605$	46.6856	1.89804
$606$	0	0
$607$	−19.5732	−0.794451	−0.397225	−	0.917721i	$-0.630027\pi$
$607$	−19.5732	−0.794451	−0.397225	+	0.917721i	$0.630027\pi$
$608$	−1.94282	−0.0787918
$609$	0	0
$610$	−18.2255	−0.737927
$611$	−1.26320	−0.0511036
$612$	0	0
$613$	5.55159	0.224226	0.112113	−	0.993695i	$-0.464238\pi$
$613$	5.55159	0.224226	0.112113	+	0.993695i	$0.464238\pi$
$614$	4.89931	0.197720
$615$	0	0
$616$	0	0
$617$	1.26896	0.0510863	0.0255431	−	0.999674i	$-0.491868\pi$
$617$	1.26896	0.0510863	0.0255431	+	0.999674i	$0.491868\pi$
$618$	0	0
$619$	−4.50232	−0.180964	−0.0904818	−	0.995898i	$-0.528841\pi$
$619$	−4.50232	−0.180964	−0.0904818	+	0.995898i	$0.528841\pi$
$620$	−13.5653	−0.544794
$621$	0	0
$622$	7.69002	0.308342
$623$	0	0
$624$	0	0
$625$	−11.8960	−0.475842
$626$	−1.72313	−0.0688700
$627$	0	0
$628$	−18.9806	−0.757407
$629$	−19.7439	−0.787242
$630$	0	0
$631$	−1.69905	−0.0676381	−0.0338191	−	0.999428i	$-0.510767\pi$
$631$	−1.69905	−0.0676381	−0.0338191	+	0.999428i	$0.510767\pi$
$632$	13.4451	0.534819
$633$	0	0
$634$	33.2028	1.31865
$635$	−33.3743	−1.32442
$636$	0	0
$637$	0	0
$638$	8.97017	0.355132
$639$	0	0
$640$	−1.76088	−0.0696048
$641$	0.948577	0.0374666	0.0187333	−	0.999825i	$-0.494037\pi$
$641$	0.948577	0.0374666	0.0187333	+	0.999825i	$0.494037\pi$
$642$	0	0
$643$	−19.6979	−0.776811	−0.388405	−	0.921489i	$-0.626974\pi$
$643$	−19.6979	−0.776811	−0.388405	+	0.921489i	$0.626974\pi$
$644$	0	0
$645$	0	0
$646$	−13.2930	−0.523007
$647$	−23.4542	−0.922079	−0.461039	−	0.887380i	$-0.652523\pi$
$647$	−23.4542	−0.922079	−0.461039	+	0.887380i	$0.652523\pi$
$648$	0	0
$649$	−12.1658	−0.477549
$650$	−1.44514	−0.0566832
$651$	0	0
$652$	15.0377	0.588924
$653$	−22.7907	−0.891869	−0.445935	−	0.895065i	$-0.647129\pi$
$653$	−22.7907	−0.891869	−0.445935	+	0.895065i	$0.647129\pi$
$654$	0	0
$655$	−12.8421	−0.501784
$656$	6.94282	0.271072
$657$	0	0
$658$	0	0
$659$	−26.4796	−1.03150	−0.515750	−	0.856739i	$-0.672487\pi$
$659$	−26.4796	−1.03150	−0.515750	+	0.856739i	$0.672487\pi$
$660$	0	0
$661$	26.7382	1.03999	0.519997	−	0.854168i	$-0.325933\pi$
$661$	26.7382	1.03999	0.519997	+	0.854168i	$0.325933\pi$
$662$	−2.88891	−0.112281
$663$	0	0
$664$	−3.12476	−0.121264
$665$	0	0
$666$	0	0
$667$	0.616683	0.0238781
$668$	−1.14419	−0.0442702
$669$	0	0
$670$	−11.9475	−0.461571
$671$	−63.3926	−2.44724
$672$	0	0
$673$	20.7713	0.800674	0.400337	−	0.916368i	$-0.368893\pi$
$673$	20.7713	0.800674	0.400337	+	0.916368i	$0.368893\pi$
$674$	−8.72313	−0.336002
$675$	0	0
$676$	−12.4211	−0.477733
$677$	−20.6979	−0.795486	−0.397743	−	0.917497i	$-0.630207\pi$
$677$	−20.6979	−0.795486	−0.397743	+	0.917497i	$0.630207\pi$
$678$	0	0
$679$	0	0
$680$	−12.0482	−0.462026
$681$	0	0
$682$	−47.1833	−1.80674
$683$	28.5836	1.09372	0.546860	−	0.837224i	$-0.315823\pi$
$683$	28.5836	1.09372	0.546860	+	0.837224i	$0.315823\pi$
$684$	0	0
$685$	14.4074	0.550478
$686$	0	0
$687$	0	0
$688$	−8.66019	−0.330167
$689$	0.171540	0.00653515
$690$	0	0
$691$	6.69794	0.254802	0.127401	−	0.991851i	$-0.459337\pi$
$691$	6.69794	0.254802	0.127401	+	0.991851i	$0.459337\pi$
$692$	0.497677	0.0189188
$693$	0	0
$694$	9.69467	0.368005
$695$	−21.9486	−0.832557
$696$	0	0
$697$	47.5037	1.79933
$698$	−28.3984	−1.07489
$699$	0	0
$700$	0	0
$701$	25.1442	0.949683	0.474842	−	0.880071i	$-0.342505\pi$
$701$	25.1442	0.949683	0.474842	+	0.880071i	$0.342505\pi$
$702$	0	0
$703$	−5.60628	−0.211445
$704$	−6.12476	−0.230836
$705$	0	0
$706$	4.39372	0.165360
$707$	0	0
$708$	0	0
$709$	8.86621	0.332977	0.166489	−	0.986043i	$-0.446757\pi$
$709$	8.86621	0.332977	0.166489	+	0.986043i	$0.446757\pi$
$710$	18.9910	0.712719
$711$	0	0
$712$	2.60301	0.0975519
$713$	−3.24377	−0.121480
$714$	0	0
$715$	8.20602	0.306888
$716$	8.82846	0.329935
$717$	0	0
$718$	−32.1592	−1.20017
$719$	−23.6030	−0.880244	−0.440122	−	0.897938i	$-0.645065\pi$
$719$	−23.6030	−0.880244	−0.440122	+	0.897938i	$0.645065\pi$
$720$	0	0
$721$	0	0
$722$	15.2255	0.566633
$723$	0	0
$724$	−1.32941	−0.0494070
$725$	−2.78168	−0.103309
$726$	0	0
$727$	6.51384	0.241585	0.120792	−	0.992678i	$-0.461456\pi$
$727$	6.51384	0.241585	0.120792	+	0.992678i	$0.461456\pi$
$728$	0	0
$729$	0	0
$730$	−0.540073	−0.0199890
$731$	−59.2542	−2.19159
$732$	0	0
$733$	23.1981	0.856842	0.428421	−	0.903579i	$-0.359070\pi$
$733$	23.1981	0.856842	0.428421	+	0.903579i	$0.359070\pi$
$734$	34.6030	1.27722
$735$	0	0
$736$	−0.421067	−0.0155207
$737$	−41.5562	−1.53074
$738$	0	0
$739$	15.1568	0.557550	0.278775	−	0.960356i	$-0.410072\pi$
$739$	15.1568	0.557550	0.278775	+	0.960356i	$0.410072\pi$
$740$	−5.08126	−0.186791
$741$	0	0
$742$	0	0
$743$	−10.4347	−0.382813	−0.191407	−	0.981511i	$-0.561305\pi$
$743$	−10.4347	−0.382813	−0.191407	+	0.981511i	$0.561305\pi$
$744$	0	0
$745$	−15.5458	−0.569555
$746$	−10.9759	−0.401857
$747$	0	0
$748$	−41.9064	−1.53225
$749$	0	0
$750$	0	0
$751$	40.2118	1.46735	0.733674	−	0.679501i	$-0.237804\pi$
$751$	40.2118	1.46735	0.733674	+	0.679501i	$0.237804\pi$
$752$	1.66019	0.0605409
$753$	0	0
$754$	1.11436	0.0405826
$755$	−26.3891	−0.960397
$756$	0	0
$757$	−21.5206	−0.782181	−0.391091	−	0.920352i	$-0.627902\pi$
$757$	−21.5206	−0.782181	−0.391091	+	0.920352i	$0.627902\pi$
$758$	−33.9877	−1.23449
$759$	0	0
$760$	−3.42107	−0.124095
$761$	−23.6627	−0.857771	−0.428886	−	0.903359i	$-0.641094\pi$
$761$	−23.6627	−0.857771	−0.428886	+	0.903359i	$0.641094\pi$
$762$	0	0
$763$	0	0
$764$	16.1683	0.584947
$765$	0	0
$766$	21.0241	0.759631
$767$	−1.51135	−0.0545717
$768$	0	0
$769$	−11.2553	−0.405876	−0.202938	−	0.979192i	$-0.565049\pi$
$769$	−11.2553	−0.405876	−0.202938	+	0.979192i	$0.565049\pi$
$770$	0	0
$771$	0	0
$772$	−14.1683	−0.509927
$773$	−0.277984	−0.00999839	−0.00499919	−	0.999988i	$-0.501591\pi$
$773$	−0.277984	−0.00999839	−0.00499919	+	0.999988i	$0.501591\pi$
$774$	0	0
$775$	14.6317	0.525587
$776$	3.63611	0.130529
$777$	0	0
$778$	13.7382	0.492538
$779$	13.4887	0.483281
$780$	0	0
$781$	66.0553	2.36364
$782$	−2.88099	−0.103024
$783$	0	0
$784$	0	0
$785$	−33.4224	−1.19290
$786$	0	0
$787$	29.3880	1.04757	0.523784	−	0.851851i	$-0.324520\pi$
$787$	29.3880	1.04757	0.523784	+	0.851851i	$0.324520\pi$
$788$	−15.8421	−0.564353
$789$	0	0
$790$	23.6752	0.842327
$791$	0	0
$792$	0	0
$793$	−7.87524	−0.279658
$794$	7.15787	0.254023
$795$	0	0
$796$	−8.94282	−0.316970
$797$	−0.866210	−0.0306827	−0.0153414	−	0.999882i	$-0.504883\pi$
$797$	−0.866210	−0.0306827	−0.0153414	+	0.999882i	$0.504883\pi$
$798$	0	0
$799$	11.3592	0.401861
$800$	1.89931	0.0671509
$801$	0	0
$802$	−9.27936	−0.327666
$803$	−1.87851	−0.0662910
$804$	0	0
$805$	0	0
$806$	−5.86156	−0.206465
$807$	0	0
$808$	8.01040	0.281805
$809$	19.3341	0.679749	0.339875	−	0.940471i	$-0.389615\pi$
$809$	19.3341	0.679749	0.339875	+	0.940471i	$0.389615\pi$
$810$	0	0
$811$	47.0391	1.65177	0.825884	−	0.563841i	$-0.190677\pi$
$811$	47.0391	1.65177	0.825884	+	0.563841i	$0.190677\pi$
$812$	0	0
$813$	0	0
$814$	−17.6739	−0.619469
$815$	26.4796	0.927541
$816$	0	0
$817$	−16.8252	−0.588639
$818$	15.1683	0.530346
$819$	0	0
$820$	12.2255	0.426931
$821$	−1.41066	−0.0492325	−0.0246162	−	0.999697i	$-0.507836\pi$
$821$	−1.41066	−0.0492325	−0.0246162	+	0.999697i	$0.507836\pi$
$822$	0	0
$823$	−35.0391	−1.22139	−0.610694	−	0.791867i	$-0.709109\pi$
$823$	−35.0391	−1.22139	−0.610694	+	0.791867i	$0.709109\pi$
$824$	−6.82846	−0.237881
$825$	0	0
$826$	0	0
$827$	18.5997	0.646776	0.323388	−	0.946266i	$-0.395178\pi$
$827$	18.5997	0.646776	0.323388	+	0.946266i	$0.395178\pi$
$828$	0	0
$829$	38.1696	1.32569	0.662843	−	0.748758i	$-0.269350\pi$
$829$	38.1696	1.32569	0.662843	+	0.748758i	$0.269350\pi$
$830$	−5.50232	−0.190988
$831$	0	0
$832$	−0.760877	−0.0263787
$833$	0	0
$834$	0	0
$835$	−2.01478	−0.0697245
$836$	−11.8993	−0.411546
$837$	0	0
$838$	−8.33654	−0.287981
$839$	−34.7382	−1.19930	−0.599648	−	0.800264i	$-0.704693\pi$
$839$	−34.7382	−1.19930	−0.599648	+	0.800264i	$0.704693\pi$
$840$	0	0
$841$	−26.8550	−0.926035
$842$	−7.00465	−0.241396
$843$	0	0
$844$	−22.7713	−0.783820
$845$	−21.8720	−0.752419
$846$	0	0
$847$	0	0
$848$	−0.225450	−0.00774199
$849$	0	0
$850$	12.9954	0.445737
$851$	−1.21505	−0.0416513
$852$	0	0
$853$	−42.3171	−1.44891	−0.724455	−	0.689322i	$-0.757909\pi$
$853$	−42.3171	−1.44891	−0.724455	+	0.689322i	$0.757909\pi$
$854$	0	0
$855$	0	0
$856$	−3.54583	−0.121194
$857$	14.9234	0.509773	0.254887	−	0.966971i	$-0.417962\pi$
$857$	14.9234	0.509773	0.254887	+	0.966971i	$0.417962\pi$
$858$	0	0
$859$	−19.4132	−0.662368	−0.331184	−	0.943566i	$-0.607448\pi$
$859$	−19.4132	−0.662368	−0.331184	+	0.943566i	$0.607448\pi$
$860$	−15.2495	−0.520005
$861$	0	0
$862$	3.45090	0.117538
$863$	−1.08453	−0.0369177	−0.0184588	−	0.999830i	$-0.505876\pi$
$863$	−1.08453	−0.0369177	−0.0184588	+	0.999830i	$0.505876\pi$
$864$	0	0
$865$	0.876348	0.0297967
$866$	28.2599	0.960312
$867$	0	0
$868$	0	0
$869$	82.3483	2.79348
$870$	0	0
$871$	−5.16251	−0.174925
$872$	0.703697	0.0238302
$873$	0	0
$874$	−0.818057	−0.0276712
$875$	0	0
$876$	0	0
$877$	−28.5699	−0.964737	−0.482369	−	0.875968i	$-0.660223\pi$
$877$	−28.5699	−0.964737	−0.482369	+	0.875968i	$0.660223\pi$
$878$	−28.8960	−0.975194
$879$	0	0
$880$	−10.7850	−0.363561
$881$	45.9967	1.54967	0.774835	−	0.632164i	$-0.217833\pi$
$881$	45.9967	1.54967	0.774835	+	0.632164i	$0.217833\pi$
$882$	0	0
$883$	32.9384	1.10847	0.554233	−	0.832361i	$-0.313012\pi$
$883$	32.9384	1.10847	0.554233	+	0.832361i	$0.313012\pi$
$884$	−5.20602	−0.175097
$885$	0	0
$886$	−13.7609	−0.462306
$887$	−28.3398	−0.951558	−0.475779	−	0.879565i	$-0.657834\pi$
$887$	−28.3398	−0.951558	−0.475779	+	0.879565i	$0.657834\pi$
$888$	0	0
$889$	0	0
$890$	4.58358	0.153642
$891$	0	0
$892$	−12.8856	−0.431443
$893$	3.22545	0.107936
$894$	0	0
$895$	15.5458	0.519640
$896$	0	0
$897$	0	0
$898$	−20.2003	−0.674091
$899$	−11.2826	−0.376297
$900$	0	0
$901$	−1.54256	−0.0513901
$902$	42.5231	1.41587
$903$	0	0
$904$	−8.50232	−0.282783
$905$	−2.34092	−0.0778149
$906$	0	0
$907$	7.94747	0.263891	0.131946	−	0.991257i	$-0.457878\pi$
$907$	7.94747	0.263891	0.131946	+	0.991257i	$0.457878\pi$
$908$	21.9967	0.729987
$909$	0	0
$910$	0	0
$911$	−8.01616	−0.265587	−0.132794	−	0.991144i	$-0.542395\pi$
$911$	−8.01616	−0.265587	−0.132794	+	0.991144i	$0.542395\pi$
$912$	0	0
$913$	−19.1384	−0.633390
$914$	−20.0298	−0.662528
$915$	0	0
$916$	3.79863	0.125510
$917$	0	0
$918$	0	0
$919$	24.0449	0.793168	0.396584	−	0.917999i	$-0.370196\pi$
$919$	24.0449	0.793168	0.396584	+	0.917999i	$0.370196\pi$
$920$	−0.741446	−0.0244448
$921$	0	0
$922$	11.9532	0.393658
$923$	8.20602	0.270104
$924$	0	0
$925$	5.48073	0.180205
$926$	13.2905	0.436754
$927$	0	0
$928$	−1.46457	−0.0480770
$929$	27.8662	0.914261	0.457130	−	0.889400i	$-0.348877\pi$
$929$	27.8662	0.914261	0.457130	+	0.889400i	$0.348877\pi$
$930$	0	0
$931$	0	0
$932$	−6.67059	−0.218503
$933$	0	0
$934$	−11.2301	−0.367460
$935$	−73.7921	−2.41326
$936$	0	0
$937$	−53.2211	−1.73866	−0.869328	−	0.494235i	$-0.835448\pi$
$937$	−53.2211	−1.73866	−0.869328	+	0.494235i	$0.835448\pi$
$938$	0	0
$939$	0	0
$940$	2.92339	0.0953505
$941$	−30.0482	−0.979542	−0.489771	−	0.871851i	$-0.662920\pi$
$941$	−30.0482	−0.979542	−0.489771	+	0.871851i	$0.662920\pi$
$942$	0	0
$943$	2.92339	0.0951987
$944$	1.98633	0.0646494
$945$	0	0
$946$	−53.0416	−1.72453
$947$	39.6889	1.28972	0.644858	−	0.764302i	$-0.276916\pi$
$947$	39.6889	1.28972	0.644858	+	0.764302i	$0.276916\pi$
$948$	0	0
$949$	−0.233366	−0.00757538
$950$	3.69002	0.119720
$951$	0	0
$952$	0	0
$953$	−23.0643	−0.747126	−0.373563	−	0.927605i	$-0.621864\pi$
$953$	−23.0643	−0.747126	−0.373563	+	0.927605i	$0.621864\pi$
$954$	0	0
$955$	28.4703	0.921278
$956$	−15.6408	−0.505858
$957$	0	0
$958$	32.6271	1.05413
$959$	0	0
$960$	0	0
$961$	28.3469	0.914418
$962$	−2.19562	−0.0707895
$963$	0	0
$964$	−21.4120	−0.689635
$965$	−24.9486	−0.803123
$966$	0	0
$967$	−30.5803	−0.983396	−0.491698	−	0.870766i	$-0.663624\pi$
$967$	−30.5803	−0.983396	−0.491698	+	0.870766i	$0.663624\pi$
$968$	−26.5127	−0.852151
$969$	0	0
$970$	6.40275	0.205580
$971$	−26.2060	−0.840991	−0.420496	−	0.907295i	$-0.638144\pi$
$971$	−26.2060	−0.840991	−0.420496	+	0.907295i	$0.638144\pi$
$972$	0	0
$973$	0	0
$974$	3.69794	0.118490
$975$	0	0
$976$	10.3502	0.331302
$977$	−21.0539	−0.673574	−0.336787	−	0.941581i	$-0.609340\pi$
$977$	−21.0539	−0.673574	−0.336787	+	0.941581i	$0.609340\pi$
$978$	0	0
$979$	15.9428	0.509535
$980$	0	0
$981$	0	0
$982$	37.5609	1.19862
$983$	−19.5297	−0.622900	−0.311450	−	0.950263i	$-0.600815\pi$
$983$	−19.5297	−0.622900	−0.311450	+	0.950263i	$0.600815\pi$
$984$	0	0
$985$	−27.8960	−0.888842
$986$	−10.0208	−0.319128
$987$	0	0
$988$	−1.47825	−0.0470293
$989$	−3.64652	−0.115952
$990$	0	0
$991$	14.9967	0.476387	0.238193	−	0.971218i	$-0.423445\pi$
$991$	14.9967	0.476387	0.238193	+	0.971218i	$0.423445\pi$
$992$	7.70370	0.244593
$993$	0	0
$994$	0	0
$995$	−15.7472	−0.499220
$996$	0	0
$997$	58.5641	1.85475	0.927373	−	0.374139i	$-0.122062\pi$
$997$	58.5641	1.85475	0.927373	+	0.374139i	$0.122062\pi$
$998$	31.7954	1.00646
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bx.1.2		3
3.2	odd	2		7938.2.a.by.1.2		3
7.3	odd	6		1134.2.g.n.163.2		6
7.5	odd	6		1134.2.g.n.487.2		6
7.6	odd	2		7938.2.a.bu.1.2		3
9.2	odd	6		882.2.f.m.589.3		6
9.4	even	3		2646.2.f.n.883.2		6
9.5	odd	6		882.2.f.m.295.3		6
9.7	even	3		2646.2.f.n.1765.2		6
21.5	even	6		1134.2.g.k.487.2		6
21.17	even	6		1134.2.g.k.163.2		6
21.20	even	2		7938.2.a.cb.1.2		3
63.2	odd	6		882.2.h.o.67.2		6
63.4	even	3		2646.2.h.p.667.2		6
63.5	even	6		126.2.e.d.25.3	✓	6
63.11	odd	6		882.2.e.p.373.1		6
63.13	odd	6		2646.2.f.o.883.2		6
63.16	even	3		2646.2.h.p.361.2		6
63.20	even	6		882.2.f.l.589.1		6
63.23	odd	6		882.2.e.p.655.1		6
63.25	even	3		2646.2.e.o.1549.2		6
63.31	odd	6		378.2.h.d.289.2		6
63.32	odd	6		882.2.h.o.79.2		6
63.34	odd	6		2646.2.f.o.1765.2		6
63.38	even	6		126.2.e.d.121.3	yes	6
63.40	odd	6		378.2.e.c.235.2		6
63.41	even	6		882.2.f.l.295.1		6
63.47	even	6		126.2.h.c.67.2	yes	6
63.52	odd	6		378.2.e.c.37.2		6
63.58	even	3		2646.2.e.o.2125.2		6
63.59	even	6		126.2.h.c.79.2	yes	6
63.61	odd	6		378.2.h.d.361.2		6
252.31	even	6		3024.2.t.g.289.2		6
252.47	odd	6		1008.2.t.g.193.2		6
252.59	odd	6		1008.2.t.g.961.2		6
252.103	even	6		3024.2.q.h.2881.2		6
252.115	even	6		3024.2.q.h.2305.2		6
252.131	odd	6		1008.2.q.h.529.1		6
252.187	even	6		3024.2.t.g.1873.2		6
252.227	odd	6		1008.2.q.h.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.d.25.3	✓	6	63.5	even	6
126.2.e.d.121.3	yes	6	63.38	even	6
126.2.h.c.67.2	yes	6	63.47	even	6
126.2.h.c.79.2	yes	6	63.59	even	6
378.2.e.c.37.2		6	63.52	odd	6
378.2.e.c.235.2		6	63.40	odd	6
378.2.h.d.289.2		6	63.31	odd	6
378.2.h.d.361.2		6	63.61	odd	6
882.2.e.p.373.1		6	63.11	odd	6
882.2.e.p.655.1		6	63.23	odd	6
882.2.f.l.295.1		6	63.41	even	6
882.2.f.l.589.1		6	63.20	even	6
882.2.f.m.295.3		6	9.5	odd	6
882.2.f.m.589.3		6	9.2	odd	6
882.2.h.o.67.2		6	63.2	odd	6
882.2.h.o.79.2		6	63.32	odd	6
1008.2.q.h.529.1		6	252.131	odd	6
1008.2.q.h.625.1		6	252.227	odd	6
1008.2.t.g.193.2		6	252.47	odd	6
1008.2.t.g.961.2		6	252.59	odd	6
1134.2.g.k.163.2		6	21.17	even	6
1134.2.g.k.487.2		6	21.5	even	6
1134.2.g.n.163.2		6	7.3	odd	6
1134.2.g.n.487.2		6	7.5	odd	6
2646.2.e.o.1549.2		6	63.25	even	3
2646.2.e.o.2125.2		6	63.58	even	3
2646.2.f.n.883.2		6	9.4	even	3
2646.2.f.n.1765.2		6	9.7	even	3
2646.2.f.o.883.2		6	63.13	odd	6
2646.2.f.o.1765.2		6	63.34	odd	6
2646.2.h.p.361.2		6	63.16	even	3
2646.2.h.p.667.2		6	63.4	even	3
3024.2.q.h.2305.2		6	252.115	even	6
3024.2.q.h.2881.2		6	252.103	even	6
3024.2.t.g.289.2		6	252.31	even	6
3024.2.t.g.1873.2		6	252.187	even	6
7938.2.a.bu.1.2		3	7.6	odd	2
7938.2.a.bx.1.2		3	1.1	even	1	trivial
7938.2.a.by.1.2		3	3.2	odd	2
7938.2.a.cb.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.76088 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.76088 q^{5} -1.00000 q^{8} -1.76088 q^{10} -6.12476 q^{11} -0.760877 q^{13} +1.00000 q^{16} +6.84213 q^{17} +1.94282 q^{19} +1.76088 q^{20} +6.12476 q^{22} +0.421067 q^{23} -1.89931 q^{25} +0.760877 q^{26} +1.46457 q^{29} -7.70370 q^{31} -1.00000 q^{32} -6.84213 q^{34} -2.88564 q^{37} -1.94282 q^{38} -1.76088 q^{40} +6.94282 q^{41} -8.66019 q^{43} -6.12476 q^{44} -0.421067 q^{46} +1.66019 q^{47} +1.89931 q^{50} -0.760877 q^{52} -0.225450 q^{53} -10.7850 q^{55} -1.46457 q^{58} +1.98633 q^{59} +10.3502 q^{61} +7.70370 q^{62} +1.00000 q^{64} -1.33981 q^{65} +6.78495 q^{67} +6.84213 q^{68} -10.7850 q^{71} +0.306707 q^{73} +2.88564 q^{74} +1.94282 q^{76} -13.4451 q^{79} +1.76088 q^{80} -6.94282 q^{82} +3.12476 q^{83} +12.0482 q^{85} +8.66019 q^{86} +6.12476 q^{88} -2.60301 q^{89} +0.421067 q^{92} -1.66019 q^{94} +3.42107 q^{95} -3.63611 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{8}+O(q^{10}) \) 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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