LMFDB - Embedded newform 7938.2.a.cd.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7938,2,Mod(1,7938)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,-4,0,4,0,0,0,-4,0,0,-16,0,0,0,0,4,0,0,0,0,0,16,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.3852491245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{15})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 49 $ x^4 - 16*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 882)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.44572$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.44572 q^{5} -1.00000 q^{8} -3.44572 q^{10} -4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -1.41421 q^{17} -6.27415 q^{19} +3.44572 q^{20} +4.00000 q^{22} +8.74597 q^{23} +6.87298 q^{25} +4.24264 q^{26} -1.12702 q^{29} +5.47723 q^{31} -1.00000 q^{32} +1.41421 q^{34} +5.74597 q^{37} +6.27415 q^{38} -3.44572 q^{40} -5.65685 q^{41} +1.12702 q^{43} -4.00000 q^{44} -8.74597 q^{46} +4.06301 q^{47} -6.87298 q^{50} -4.24264 q^{52} -12.6190 q^{53} -13.7829 q^{55} +1.12702 q^{58} +8.30565 q^{59} +6.27415 q^{61} -5.47723 q^{62} +1.00000 q^{64} -14.6190 q^{65} -6.87298 q^{67} -1.41421 q^{68} +9.87298 q^{71} +4.42227 q^{73} -5.74597 q^{74} -6.27415 q^{76} -1.87298 q^{79} +3.44572 q^{80} +5.65685 q^{82} +2.64880 q^{83} -4.87298 q^{85} -1.12702 q^{86} +4.00000 q^{88} -7.07107 q^{89} +8.74597 q^{92} -4.06301 q^{94} -21.6190 q^{95} +15.0175 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 16 q^{11} + 4 q^{16} + 16 q^{22} + 4 q^{23} + 12 q^{25} - 20 q^{29} - 4 q^{32} - 8 q^{37} + 20 q^{43} - 16 q^{44} - 4 q^{46} - 12 q^{50} - 4 q^{53} + 20 q^{58} + 4 q^{64}+ \cdots - 40 q^{95}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 16 * q^11 + 4 * q^16 + 16 * q^22 + 4 * q^23 + 12 * q^25 - 20 * q^29 - 4 * q^32 - 8 * q^37 + 20 * q^43 - 16 * q^44 - 4 * q^46 - 12 * q^50 - 4 * q^53 + 20 * q^58 + 4 * q^64 - 12 * q^65 - 12 * q^67 + 24 * q^71 + 8 * q^74 + 8 * q^79 - 4 * q^85 - 20 * q^86 + 16 * q^88 + 4 * q^92 - 40 * q^95

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
$5$	3.44572	1.54097	0.770486	−	0.637457i	$-0.220013\pi$
$5$	3.44572	1.54097	0.770486	+	0.637457i	$0.220013\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.44572	−1.08963
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−6.27415	−1.43939	−0.719694	−	0.694291i	$-0.755718\pi$
$19$	−6.27415	−1.43939	−0.719694	+	0.694291i	$0.755718\pi$
$20$	3.44572	0.770486
$21$	0	0
$22$	4.00000	0.852803
$23$	8.74597	1.82366	0.911830	−	0.410568i	$-0.134669\pi$
$23$	8.74597	1.82366	0.911830	+	0.410568i	$0.134669\pi$
$24$	0	0
$25$	6.87298	1.37460
$26$	4.24264	0.832050
$27$	0	0
$28$	0	0
$29$	−1.12702	−0.209282	−0.104641	−	0.994510i	$-0.533369\pi$
$29$	−1.12702	−0.209282	−0.104641	+	0.994510i	$0.533369\pi$
$30$	0	0
$31$	5.47723	0.983739	0.491869	−	0.870669i	$-0.336314\pi$
$31$	5.47723	0.983739	0.491869	+	0.870669i	$0.336314\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.41421	0.242536
$35$	0	0
$36$	0	0
$37$	5.74597	0.944631	0.472316	−	0.881430i	$-0.343418\pi$
$37$	5.74597	0.944631	0.472316	+	0.881430i	$0.343418\pi$
$38$	6.27415	1.01780
$39$	0	0
$40$	−3.44572	−0.544816
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	1.12702	0.171868	0.0859342	−	0.996301i	$-0.472613\pi$
$43$	1.12702	0.171868	0.0859342	+	0.996301i	$0.472613\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−8.74597	−1.28952
$47$	4.06301	0.592651	0.296326	−	0.955087i	$-0.404239\pi$
$47$	4.06301	0.592651	0.296326	+	0.955087i	$0.404239\pi$
$48$	0	0
$49$	0	0
$50$	−6.87298	−0.971987
$51$	0	0
$52$	−4.24264	−0.588348
$53$	−12.6190	−1.73335	−0.866673	−	0.498877i	$-0.833746\pi$
$53$	−12.6190	−1.73335	−0.866673	+	0.498877i	$0.833746\pi$
$54$	0	0
$55$	−13.7829	−1.85848
$56$	0	0
$57$	0	0
$58$	1.12702	0.147985
$59$	8.30565	1.08130	0.540652	−	0.841246i	$-0.318178\pi$
$59$	8.30565	1.08130	0.540652	+	0.841246i	$0.318178\pi$
$60$	0	0
$61$	6.27415	0.803322	0.401661	−	0.915788i	$-0.368433\pi$
$61$	6.27415	0.803322	0.401661	+	0.915788i	$0.368433\pi$
$62$	−5.47723	−0.695608
$63$	0	0
$64$	1.00000	0.125000
$65$	−14.6190	−1.81326
$66$	0	0
$67$	−6.87298	−0.839669	−0.419834	−	0.907601i	$-0.637912\pi$
$67$	−6.87298	−0.839669	−0.419834	+	0.907601i	$0.637912\pi$
$68$	−1.41421	−0.171499
$69$	0	0
$70$	0	0
$71$	9.87298	1.17171	0.585854	−	0.810417i	$-0.300759\pi$
$71$	9.87298	1.17171	0.585854	+	0.810417i	$0.300759\pi$
$72$	0	0
$73$	4.42227	0.517587	0.258794	−	0.965933i	$-0.416675\pi$
$73$	4.42227	0.517587	0.258794	+	0.965933i	$0.416675\pi$
$74$	−5.74597	−0.667955
$75$	0	0
$76$	−6.27415	−0.719694
$77$	0	0
$78$	0	0
$79$	−1.87298	−0.210727	−0.105364	−	0.994434i	$-0.533601\pi$
$79$	−1.87298	−0.210727	−0.105364	+	0.994434i	$0.533601\pi$
$80$	3.44572	0.385243
$81$	0	0
$82$	5.65685	0.624695
$83$	2.64880	0.290743	0.145372	−	0.989377i	$-0.453562\pi$
$83$	2.64880	0.290743	0.145372	+	0.989377i	$0.453562\pi$
$84$	0	0
$85$	−4.87298	−0.528549
$86$	−1.12702	−0.121529
$87$	0	0
$88$	4.00000	0.426401
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	8.74597	0.911830
$93$	0	0
$94$	−4.06301	−0.419068
$95$	−21.6190	−2.21806
$96$	0	0
$97$	15.0175	1.52479	0.762396	−	0.647111i	$-0.224023\pi$
$97$	15.0175	1.52479	0.762396	+	0.647111i	$0.224023\pi$
$98$	0	0
$99$	0	0
$100$	6.87298	0.687298
$101$	13.3452	1.32790	0.663949	−	0.747778i	$-0.268879\pi$
$101$	13.3452	1.32790	0.663949	+	0.747778i	$0.268879\pi$
$102$	0	0
$103$	3.88338	0.382641	0.191321	−	0.981528i	$-0.438723\pi$
$103$	3.88338	0.382641	0.191321	+	0.981528i	$0.438723\pi$
$104$	4.24264	0.416025
$105$	0	0
$106$	12.6190	1.22566
$107$	−0.254033	−0.0245583	−0.0122792	−	0.999925i	$-0.503909\pi$
$107$	−0.254033	−0.0245583	−0.0122792	+	0.999925i	$0.503909\pi$
$108$	0	0
$109$	16.8730	1.61614	0.808069	−	0.589087i	$-0.200513\pi$
$109$	16.8730	1.61614	0.808069	+	0.589087i	$0.200513\pi$
$110$	13.7829	1.31415
$111$	0	0
$112$	0	0
$113$	−3.87298	−0.364340	−0.182170	−	0.983267i	$-0.558312\pi$
$113$	−3.87298	−0.364340	−0.182170	+	0.983267i	$0.558312\pi$
$114$	0	0
$115$	30.1361	2.81021
$116$	−1.12702	−0.104641
$117$	0	0
$118$	−8.30565	−0.764597
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−6.27415	−0.568035
$123$	0	0
$124$	5.47723	0.491869
$125$	6.45378	0.577243
$126$	0	0
$127$	−0.745967	−0.0661938	−0.0330969	−	0.999452i	$-0.510537\pi$
$127$	−0.745967	−0.0661938	−0.0330969	+	0.999452i	$0.510537\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	14.6190	1.28217
$131$	−13.3452	−1.16598	−0.582988	−	0.812480i	$-0.698117\pi$
$131$	−13.3452	−1.16598	−0.582988	+	0.812480i	$0.698117\pi$
$132$	0	0
$133$	0	0
$134$	6.87298	0.593735
$135$	0	0
$136$	1.41421	0.121268
$137$	15.4919	1.32357	0.661783	−	0.749696i	$-0.269800\pi$
$137$	15.4919	1.32357	0.661783	+	0.749696i	$0.269800\pi$
$138$	0	0
$139$	19.8774	1.68598	0.842989	−	0.537930i	$-0.180794\pi$
$139$	19.8774	1.68598	0.842989	+	0.537930i	$0.180794\pi$
$140$	0	0
$141$	0	0
$142$	−9.87298	−0.828522
$143$	16.9706	1.41915
$144$	0	0
$145$	−3.88338	−0.322497
$146$	−4.42227	−0.365990
$147$	0	0
$148$	5.74597	0.472316
$149$	15.7460	1.28996	0.644980	−	0.764200i	$-0.276866\pi$
$149$	15.7460	1.28996	0.644980	+	0.764200i	$0.276866\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	6.27415	0.508900
$153$	0	0
$154$	0	0
$155$	18.8730	1.51591
$156$	0	0
$157$	11.9310	0.952198	0.476099	−	0.879392i	$-0.342050\pi$
$157$	11.9310	0.952198	0.476099	+	0.879392i	$0.342050\pi$
$158$	1.87298	0.149007
$159$	0	0
$160$	−3.44572	−0.272408
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−5.65685	−0.441726
$165$	0	0
$166$	−2.64880	−0.205587
$167$	−9.54024	−0.738246	−0.369123	−	0.929381i	$-0.620342\pi$
$167$	−9.54024	−0.738246	−0.369123	+	0.929381i	$0.620342\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	4.87298	0.373741
$171$	0	0
$172$	1.12702	0.0859342
$173$	−12.7279	−0.967686	−0.483843	−	0.875155i	$-0.660759\pi$
$173$	−12.7279	−0.967686	−0.483843	+	0.875155i	$0.660759\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	7.07107	0.529999
$179$	6.87298	0.513711	0.256855	−	0.966450i	$-0.417314\pi$
$179$	6.87298	0.513711	0.256855	+	0.966450i	$0.417314\pi$
$180$	0	0
$181$	−13.3452	−0.991942	−0.495971	−	0.868339i	$-0.665188\pi$
$181$	−13.3452	−0.991942	−0.495971	+	0.868339i	$0.665188\pi$
$182$	0	0
$183$	0	0
$184$	−8.74597	−0.644761
$185$	19.7990	1.45565
$186$	0	0
$187$	5.65685	0.413670
$188$	4.06301	0.296326
$189$	0	0
$190$	21.6190	1.56840
$191$	6.12702	0.443335	0.221668	−	0.975122i	$-0.428850\pi$
$191$	6.12702	0.443335	0.221668	+	0.975122i	$0.428850\pi$
$192$	0	0
$193$	23.8730	1.71841	0.859207	−	0.511627i	$-0.170957\pi$
$193$	23.8730	1.71841	0.859207	+	0.511627i	$0.170957\pi$
$194$	−15.0175	−1.07819
$195$	0	0
$196$	0	0
$197$	−16.6190	−1.18405	−0.592026	−	0.805919i	$-0.701672\pi$
$197$	−16.6190	−1.18405	−0.592026	+	0.805919i	$0.701672\pi$
$198$	0	0
$199$	12.1890	0.864058	0.432029	−	0.901860i	$-0.357798\pi$
$199$	12.1890	0.864058	0.432029	+	0.901860i	$0.357798\pi$
$200$	−6.87298	−0.485993
$201$	0	0
$202$	−13.3452	−0.938966
$203$	0	0
$204$	0	0
$205$	−19.4919	−1.36138
$206$	−3.88338	−0.270568
$207$	0	0
$208$	−4.24264	−0.294174
$209$	25.0966	1.73597
$210$	0	0
$211$	20.6190	1.41947	0.709734	−	0.704470i	$-0.248815\pi$
$211$	20.6190	1.41947	0.709734	+	0.704470i	$0.248815\pi$
$212$	−12.6190	−0.866673
$213$	0	0
$214$	0.254033	0.0173654
$215$	3.88338	0.264845
$216$	0	0
$217$	0	0
$218$	−16.8730	−1.14278
$219$	0	0
$220$	−13.7829	−0.929241
$221$	6.00000	0.403604
$222$	0	0
$223$	−22.4478	−1.50322	−0.751608	−	0.659611i	$-0.770721\pi$
$223$	−22.4478	−1.50322	−0.751608	+	0.659611i	$0.770721\pi$
$224$	0	0
$225$	0	0
$226$	3.87298	0.257627
$227$	−10.3372	−0.686101	−0.343051	−	0.939317i	$-0.611460\pi$
$227$	−10.3372	−0.686101	−0.343051	+	0.939317i	$0.611460\pi$
$228$	0	0
$229$	13.3452	0.881877	0.440938	−	0.897537i	$-0.354646\pi$
$229$	13.3452	0.881877	0.440938	+	0.897537i	$0.354646\pi$
$230$	−30.1361	−1.98712
$231$	0	0
$232$	1.12702	0.0739923
$233$	−7.25403	−0.475228	−0.237614	−	0.971360i	$-0.576365\pi$
$233$	−7.25403	−0.475228	−0.237614	+	0.971360i	$0.576365\pi$
$234$	0	0
$235$	14.0000	0.913259
$236$	8.30565	0.540652
$237$	0	0
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−23.6824	−1.52552	−0.762758	−	0.646684i	$-0.776155\pi$
$241$	−23.6824	−1.52552	−0.762758	+	0.646684i	$0.776155\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	6.27415	0.401661
$245$	0	0
$246$	0	0
$247$	26.6190	1.69372
$248$	−5.47723	−0.347804
$249$	0	0
$250$	−6.45378	−0.408173
$251$	−9.46183	−0.597225	−0.298613	−	0.954374i	$-0.596524\pi$
$251$	−9.46183	−0.597225	−0.298613	+	0.954374i	$0.596524\pi$
$252$	0	0
$253$	−34.9839	−2.19942
$254$	0.745967	0.0468061
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00806	−0.187637	−0.0938187	−	0.995589i	$-0.529907\pi$
$257$	−3.00806	−0.187637	−0.0938187	+	0.995589i	$0.529907\pi$
$258$	0	0
$259$	0	0
$260$	−14.6190	−0.906629
$261$	0	0
$262$	13.3452	0.824470
$263$	17.6190	1.08643	0.543216	−	0.839593i	$-0.317207\pi$
$263$	17.6190	1.08643	0.543216	+	0.839593i	$0.317207\pi$
$264$	0	0
$265$	−43.4814	−2.67104
$266$	0	0
$267$	0	0
$268$	−6.87298	−0.419834
$269$	2.03151	0.123863	0.0619316	−	0.998080i	$-0.480274\pi$
$269$	2.03151	0.123863	0.0619316	+	0.998080i	$0.480274\pi$
$270$	0	0
$271$	7.07107	0.429537	0.214768	−	0.976665i	$-0.431100\pi$
$271$	7.07107	0.429537	0.214768	+	0.976665i	$0.431100\pi$
$272$	−1.41421	−0.0857493
$273$	0	0
$274$	−15.4919	−0.935902
$275$	−27.4919	−1.65783
$276$	0	0
$277$	14.3649	0.863104	0.431552	−	0.902088i	$-0.357966\pi$
$277$	14.3649	0.863104	0.431552	+	0.902088i	$0.357966\pi$
$278$	−19.8774	−1.19217
$279$	0	0
$280$	0	0
$281$	−8.74597	−0.521741	−0.260870	−	0.965374i	$-0.584009\pi$
$281$	−8.74597	−0.521741	−0.260870	+	0.965374i	$0.584009\pi$
$282$	0	0
$283$	5.03956	0.299571	0.149785	−	0.988719i	$-0.452142\pi$
$283$	5.03956	0.299571	0.149785	+	0.988719i	$0.452142\pi$
$284$	9.87298	0.585854
$285$	0	0
$286$	−16.9706	−1.00349
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	3.88338	0.228040
$291$	0	0
$292$	4.42227	0.258794
$293$	0.796921	0.0465566	0.0232783	−	0.999729i	$-0.492590\pi$
$293$	0.796921	0.0465566	0.0232783	+	0.999729i	$0.492590\pi$
$294$	0	0
$295$	28.6190	1.66626
$296$	−5.74597	−0.333978
$297$	0	0
$298$	−15.7460	−0.912139
$299$	−37.1060	−2.14590
$300$	0	0
$301$	0	0
$302$	−11.0000	−0.632979
$303$	0	0
$304$	−6.27415	−0.359847
$305$	21.6190	1.23790
$306$	0	0
$307$	14.2205	0.811609	0.405805	−	0.913960i	$-0.366991\pi$
$307$	14.2205	0.811609	0.405805	+	0.913960i	$0.366991\pi$
$308$	0	0
$309$	0	0
$310$	−18.8730	−1.07191
$311$	1.41421	0.0801927	0.0400963	−	0.999196i	$-0.487234\pi$
$311$	1.41421	0.0801927	0.0400963	+	0.999196i	$0.487234\pi$
$312$	0	0
$313$	15.7360	0.889450	0.444725	−	0.895667i	$-0.353301\pi$
$313$	15.7360	0.889450	0.444725	+	0.895667i	$0.353301\pi$
$314$	−11.9310	−0.673305
$315$	0	0
$316$	−1.87298	−0.105364
$317$	−24.6190	−1.38274	−0.691369	−	0.722502i	$-0.742992\pi$
$317$	−24.6190	−1.38274	−0.691369	+	0.722502i	$0.742992\pi$
$318$	0	0
$319$	4.50807	0.252403
$320$	3.44572	0.192622
$321$	0	0
$322$	0	0
$323$	8.87298	0.493706
$324$	0	0
$325$	−29.1596	−1.61748
$326$	−10.0000	−0.553849
$327$	0	0
$328$	5.65685	0.312348
$329$	0	0
$330$	0	0
$331$	20.3649	1.11936	0.559679	−	0.828710i	$-0.310925\pi$
$331$	20.3649	1.11936	0.559679	+	0.828710i	$0.310925\pi$
$332$	2.64880	0.145372
$333$	0	0
$334$	9.54024	0.522019
$335$	−23.6824	−1.29391
$336$	0	0
$337$	15.7460	0.857737	0.428869	−	0.903367i	$-0.358912\pi$
$337$	15.7460	0.857737	0.428869	+	0.903367i	$0.358912\pi$
$338$	−5.00000	−0.271964
$339$	0	0
$340$	−4.87298	−0.264275
$341$	−21.9089	−1.18643
$342$	0	0
$343$	0	0
$344$	−1.12702	−0.0607647
$345$	0	0
$346$	12.7279	0.684257
$347$	−7.74597	−0.415825	−0.207913	−	0.978147i	$-0.566667\pi$
$347$	−7.74597	−0.415825	−0.207913	+	0.978147i	$0.566667\pi$
$348$	0	0
$349$	−16.4317	−0.879567	−0.439784	−	0.898104i	$-0.644945\pi$
$349$	−16.4317	−0.879567	−0.439784	+	0.898104i	$0.644945\pi$
$350$	0	0
$351$	0	0
$352$	4.00000	0.213201
$353$	−3.88338	−0.206692	−0.103346	−	0.994645i	$-0.532955\pi$
$353$	−3.88338	−0.206692	−0.103346	+	0.994645i	$0.532955\pi$
$354$	0	0
$355$	34.0195	1.80557
$356$	−7.07107	−0.374766
$357$	0	0
$358$	−6.87298	−0.363248
$359$	−18.2379	−0.962560	−0.481280	−	0.876567i	$-0.659828\pi$
$359$	−18.2379	−0.962560	−0.481280	+	0.876567i	$0.659828\pi$
$360$	0	0
$361$	20.3649	1.07184
$362$	13.3452	0.701409
$363$	0	0
$364$	0	0
$365$	15.2379	0.797588
$366$	0	0
$367$	6.89144	0.359730	0.179865	−	0.983691i	$-0.442434\pi$
$367$	6.89144	0.359730	0.179865	+	0.983691i	$0.442434\pi$
$368$	8.74597	0.455915
$369$	0	0
$370$	−19.7990	−1.02930
$371$	0	0
$372$	0	0
$373$	−1.12702	−0.0583547	−0.0291774	−	0.999574i	$-0.509289\pi$
$373$	−1.12702	−0.0583547	−0.0291774	+	0.999574i	$0.509289\pi$
$374$	−5.65685	−0.292509
$375$	0	0
$376$	−4.06301	−0.209534
$377$	4.78153	0.246261
$378$	0	0
$379$	26.3649	1.35427	0.677137	−	0.735857i	$-0.263220\pi$
$379$	26.3649	1.35427	0.677137	+	0.735857i	$0.263220\pi$
$380$	−21.6190	−1.10903
$381$	0	0
$382$	−6.12702	−0.313485
$383$	12.9076	0.659545	0.329773	−	0.944060i	$-0.393028\pi$
$383$	12.9076	0.659545	0.329773	+	0.944060i	$0.393028\pi$
$384$	0	0
$385$	0	0
$386$	−23.8730	−1.21510
$387$	0	0
$388$	15.0175	0.762396
$389$	−13.1270	−0.665566	−0.332783	−	0.943003i	$-0.607988\pi$
$389$	−13.1270	−0.665566	−0.332783	+	0.943003i	$0.607988\pi$
$390$	0	0
$391$	−12.3687	−0.625510
$392$	0	0
$393$	0	0
$394$	16.6190	0.837251
$395$	−6.45378	−0.324725
$396$	0	0
$397$	7.07107	0.354887	0.177443	−	0.984131i	$-0.443217\pi$
$397$	7.07107	0.354887	0.177443	+	0.984131i	$0.443217\pi$
$398$	−12.1890	−0.610981
$399$	0	0
$400$	6.87298	0.343649
$401$	3.87298	0.193408	0.0967038	−	0.995313i	$-0.469170\pi$
$401$	3.87298	0.193408	0.0967038	+	0.995313i	$0.469170\pi$
$402$	0	0
$403$	−23.2379	−1.15756
$404$	13.3452	0.663949
$405$	0	0
$406$	0	0
$407$	−22.9839	−1.13927
$408$	0	0
$409$	23.1435	1.14437	0.572186	−	0.820124i	$-0.306096\pi$
$409$	23.1435	1.14437	0.572186	+	0.820124i	$0.306096\pi$
$410$	19.4919	0.962638
$411$	0	0
$412$	3.88338	0.191321
$413$	0	0
$414$	0	0
$415$	9.12702	0.448028
$416$	4.24264	0.208013
$417$	0	0
$418$	−25.0966	−1.22751
$419$	31.3707	1.53256	0.766280	−	0.642506i	$-0.222105\pi$
$419$	31.3707	1.53256	0.766280	+	0.642506i	$0.222105\pi$
$420$	0	0
$421$	−12.8730	−0.627391	−0.313695	−	0.949524i	$-0.601567\pi$
$421$	−12.8730	−0.627391	−0.313695	+	0.949524i	$0.601567\pi$
$422$	−20.6190	−1.00371
$423$	0	0
$424$	12.6190	0.612830
$425$	−9.71987	−0.471483
$426$	0	0
$427$	0	0
$428$	−0.254033	−0.0122792
$429$	0	0
$430$	−3.88338	−0.187273
$431$	21.4919	1.03523	0.517615	−	0.855614i	$-0.326820\pi$
$431$	21.4919	1.03523	0.517615	+	0.855614i	$0.326820\pi$
$432$	0	0
$433$	−29.6985	−1.42722	−0.713609	−	0.700544i	$-0.752941\pi$
$433$	−29.6985	−1.42722	−0.713609	+	0.700544i	$0.752941\pi$
$434$	0	0
$435$	0	0
$436$	16.8730	0.808069
$437$	−54.8735	−2.62495
$438$	0	0
$439$	−10.9545	−0.522827	−0.261414	−	0.965227i	$-0.584189\pi$
$439$	−10.9545	−0.522827	−0.261414	+	0.965227i	$0.584189\pi$
$440$	13.7829	0.657073
$441$	0	0
$442$	−6.00000	−0.285391
$443$	14.3649	0.682498	0.341249	−	0.939973i	$-0.389150\pi$
$443$	14.3649	0.682498	0.341249	+	0.939973i	$0.389150\pi$
$444$	0	0
$445$	−24.3649	−1.15501
$446$	22.4478	1.06293
$447$	0	0
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	22.6274	1.06548
$452$	−3.87298	−0.182170
$453$	0	0
$454$	10.3372	0.485147
$455$	0	0
$456$	0	0
$457$	26.1270	1.22217	0.611085	−	0.791565i	$-0.290733\pi$
$457$	26.1270	1.22217	0.611085	+	0.791565i	$0.290733\pi$
$458$	−13.3452	−0.623581
$459$	0	0
$460$	30.1361	1.40511
$461$	−28.3627	−1.32098	−0.660491	−	0.750834i	$-0.729652\pi$
$461$	−28.3627	−1.32098	−0.660491	+	0.750834i	$0.729652\pi$
$462$	0	0
$463$	−1.61895	−0.0752390	−0.0376195	−	0.999292i	$-0.511977\pi$
$463$	−1.61895	−0.0752390	−0.0376195	+	0.999292i	$0.511977\pi$
$464$	−1.12702	−0.0523204
$465$	0	0
$466$	7.25403	0.336037
$467$	−6.19574	−0.286705	−0.143352	−	0.989672i	$-0.545788\pi$
$467$	−6.19574	−0.286705	−0.143352	+	0.989672i	$0.545788\pi$
$468$	0	0
$469$	0	0
$470$	−14.0000	−0.645772
$471$	0	0
$472$	−8.30565	−0.382299
$473$	−4.50807	−0.207281
$474$	0	0
$475$	−43.1221	−1.97858
$476$	0	0
$477$	0	0
$478$	−15.0000	−0.686084
$479$	−3.88338	−0.177436	−0.0887182	−	0.996057i	$-0.528277\pi$
$479$	−3.88338	−0.177436	−0.0887182	+	0.996057i	$0.528277\pi$
$480$	0	0
$481$	−24.3781	−1.11154
$482$	23.6824	1.07870
$483$	0	0
$484$	5.00000	0.227273
$485$	51.7460	2.34966
$486$	0	0
$487$	−0.491933	−0.0222916	−0.0111458	−	0.999938i	$-0.503548\pi$
$487$	−0.491933	−0.0222916	−0.0111458	+	0.999938i	$0.503548\pi$
$488$	−6.27415	−0.284017
$489$	0	0
$490$	0	0
$491$	−1.74597	−0.0787944	−0.0393972	−	0.999224i	$-0.512544\pi$
$491$	−1.74597	−0.0787944	−0.0393972	+	0.999224i	$0.512544\pi$
$492$	0	0
$493$	1.59384	0.0717830
$494$	−26.6190	−1.19764
$495$	0	0
$496$	5.47723	0.245935
$497$	0	0
$498$	0	0
$499$	−29.7460	−1.33161	−0.665806	−	0.746125i	$-0.731912\pi$
$499$	−29.7460	−1.33161	−0.665806	+	0.746125i	$0.731912\pi$
$500$	6.45378	0.288622
$501$	0	0
$502$	9.46183	0.422302
$503$	3.88338	0.173152	0.0865758	−	0.996245i	$-0.472408\pi$
$503$	3.88338	0.173152	0.0865758	+	0.996245i	$0.472408\pi$
$504$	0	0
$505$	45.9839	2.04626
$506$	34.9839	1.55522
$507$	0	0
$508$	−0.745967	−0.0330969
$509$	−22.8070	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$509$	−22.8070	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.00806	0.132680
$515$	13.3810	0.589640
$516$	0	0
$517$	−16.2520	−0.714764
$518$	0	0
$519$	0	0
$520$	14.6190	0.641083
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	0	0
$523$	18.2836	0.799484	0.399742	−	0.916628i	$-0.369100\pi$
$523$	18.2836	0.799484	0.399742	+	0.916628i	$0.369100\pi$
$524$	−13.3452	−0.582988
$525$	0	0
$526$	−17.6190	−0.768223
$527$	−7.74597	−0.337420
$528$	0	0
$529$	53.4919	2.32574
$530$	43.4814	1.88871
$531$	0	0
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	−0.875328	−0.0378437
$536$	6.87298	0.296868
$537$	0	0
$538$	−2.03151	−0.0875844
$539$	0	0
$540$	0	0
$541$	−38.1109	−1.63851	−0.819257	−	0.573426i	$-0.805614\pi$
$541$	−38.1109	−1.63851	−0.819257	+	0.573426i	$0.805614\pi$
$542$	−7.07107	−0.303728
$543$	0	0
$544$	1.41421	0.0606339
$545$	58.1396	2.49043
$546$	0	0
$547$	−32.9839	−1.41029	−0.705144	−	0.709064i	$-0.749118\pi$
$547$	−32.9839	−1.41029	−0.705144	+	0.709064i	$0.749118\pi$
$548$	15.4919	0.661783
$549$	0	0
$550$	27.4919	1.17226
$551$	7.07107	0.301238
$552$	0	0
$553$	0	0
$554$	−14.3649	−0.610307
$555$	0	0
$556$	19.8774	0.842989
$557$	−26.6190	−1.12788	−0.563941	−	0.825815i	$-0.690715\pi$
$557$	−26.6190	−1.12788	−0.563941	+	0.825815i	$0.690715\pi$
$558$	0	0
$559$	−4.78153	−0.202237
$560$	0	0
$561$	0	0
$562$	8.74597	0.368926
$563$	21.6509	0.912475	0.456238	−	0.889858i	$-0.349197\pi$
$563$	21.6509	0.912475	0.456238	+	0.889858i	$0.349197\pi$
$564$	0	0
$565$	−13.3452	−0.561437
$566$	−5.03956	−0.211829
$567$	0	0
$568$	−9.87298	−0.414261
$569$	23.4919	0.984833	0.492417	−	0.870360i	$-0.336114\pi$
$569$	23.4919	0.984833	0.492417	+	0.870360i	$0.336114\pi$
$570$	0	0
$571$	31.4919	1.31790	0.658948	−	0.752188i	$-0.271002\pi$
$571$	31.4919	1.31790	0.658948	+	0.752188i	$0.271002\pi$
$572$	16.9706	0.709575
$573$	0	0
$574$	0	0
$575$	60.1109	2.50680
$576$	0	0
$577$	24.2213	1.00834	0.504172	−	0.863603i	$-0.331798\pi$
$577$	24.2213	1.00834	0.504172	+	0.863603i	$0.331798\pi$
$578$	15.0000	0.623918
$579$	0	0
$580$	−3.88338	−0.161249
$581$	0	0
$582$	0	0
$583$	50.4758	2.09049
$584$	−4.42227	−0.182995
$585$	0	0
$586$	−0.796921	−0.0329205
$587$	−8.56369	−0.353461	−0.176731	−	0.984259i	$-0.556552\pi$
$587$	−8.56369	−0.353461	−0.176731	+	0.984259i	$0.556552\pi$
$588$	0	0
$589$	−34.3649	−1.41598
$590$	−28.6190	−1.17822
$591$	0	0
$592$	5.74597	0.236158
$593$	−33.7615	−1.38642	−0.693209	−	0.720736i	$-0.743804\pi$
$593$	−33.7615	−1.38642	−0.693209	+	0.720736i	$0.743804\pi$
$594$	0	0
$595$	0	0
$596$	15.7460	0.644980
$597$	0	0
$598$	37.1060	1.51738
$599$	45.2379	1.84837	0.924185	−	0.381945i	$-0.124745\pi$
$599$	45.2379	1.84837	0.924185	+	0.381945i	$0.124745\pi$
$600$	0	0
$601$	−4.78153	−0.195043	−0.0975213	−	0.995233i	$-0.531091\pi$
$601$	−4.78153	−0.195043	−0.0975213	+	0.995233i	$0.531091\pi$
$602$	0	0
$603$	0	0
$604$	11.0000	0.447584
$605$	17.2286	0.700442
$606$	0	0
$607$	26.6904	1.08333	0.541666	−	0.840594i	$-0.317794\pi$
$607$	26.6904	1.08333	0.541666	+	0.840594i	$0.317794\pi$
$608$	6.27415	0.254450
$609$	0	0
$610$	−21.6190	−0.875326
$611$	−17.2379	−0.697371
$612$	0	0
$613$	26.6190	1.07513	0.537565	−	0.843223i	$-0.319344\pi$
$613$	26.6190	1.07513	0.537565	+	0.843223i	$0.319344\pi$
$614$	−14.2205	−0.573894
$615$	0	0
$616$	0	0
$617$	12.2540	0.493329	0.246664	−	0.969101i	$-0.420665\pi$
$617$	12.2540	0.493329	0.246664	+	0.969101i	$0.420665\pi$
$618$	0	0
$619$	−37.9029	−1.52345	−0.761723	−	0.647902i	$-0.775646\pi$
$619$	−37.9029	−1.52345	−0.761723	+	0.647902i	$0.775646\pi$
$620$	18.8730	0.757957
$621$	0	0
$622$	−1.41421	−0.0567048
$623$	0	0
$624$	0	0
$625$	−12.1270	−0.485081
$626$	−15.7360	−0.628936
$627$	0	0
$628$	11.9310	0.476099
$629$	−8.12602	−0.324006
$630$	0	0
$631$	−4.38105	−0.174407	−0.0872034	−	0.996191i	$-0.527793\pi$
$631$	−4.38105	−0.174407	−0.0872034	+	0.996191i	$0.527793\pi$
$632$	1.87298	0.0745033
$633$	0	0
$634$	24.6190	0.977743
$635$	−2.57039	−0.102003
$636$	0	0
$637$	0	0
$638$	−4.50807	−0.178476
$639$	0	0
$640$	−3.44572	−0.136204
$641$	−17.1109	−0.675839	−0.337920	−	0.941175i	$-0.609723\pi$
$641$	−17.1109	−0.675839	−0.337920	+	0.941175i	$0.609723\pi$
$642$	0	0
$643$	3.52413	0.138978	0.0694890	−	0.997583i	$-0.477863\pi$
$643$	3.52413	0.138978	0.0694890	+	0.997583i	$0.477863\pi$
$644$	0	0
$645$	0	0
$646$	−8.87298	−0.349103
$647$	5.47723	0.215332	0.107666	−	0.994187i	$-0.465662\pi$
$647$	5.47723	0.215332	0.107666	+	0.994187i	$0.465662\pi$
$648$	0	0
$649$	−33.2226	−1.30410
$650$	29.1596	1.14373
$651$	0	0
$652$	10.0000	0.391630
$653$	14.5081	0.567745	0.283872	−	0.958862i	$-0.408381\pi$
$653$	14.5081	0.567745	0.283872	+	0.958862i	$0.408381\pi$
$654$	0	0
$655$	−45.9839	−1.79674
$656$	−5.65685	−0.220863
$657$	0	0
$658$	0	0
$659$	−29.1270	−1.13463	−0.567314	−	0.823502i	$-0.692017\pi$
$659$	−29.1270	−1.13463	−0.567314	+	0.823502i	$0.692017\pi$
$660$	0	0
$661$	−46.2086	−1.79730	−0.898652	−	0.438661i	$-0.855453\pi$
$661$	−46.2086	−1.79730	−0.898652	+	0.438661i	$0.855453\pi$
$662$	−20.3649	−0.791505
$663$	0	0
$664$	−2.64880	−0.102793
$665$	0	0
$666$	0	0
$667$	−9.85685	−0.381659
$668$	−9.54024	−0.369123
$669$	0	0
$670$	23.6824	0.914930
$671$	−25.0966	−0.968843
$672$	0	0
$673$	14.2379	0.548831	0.274415	−	0.961611i	$-0.411516\pi$
$673$	14.2379	0.548831	0.274415	+	0.961611i	$0.411516\pi$
$674$	−15.7460	−0.606512
$675$	0	0
$676$	5.00000	0.192308
$677$	−12.7279	−0.489174	−0.244587	−	0.969627i	$-0.578652\pi$
$677$	−12.7279	−0.489174	−0.244587	+	0.969627i	$0.578652\pi$
$678$	0	0
$679$	0	0
$680$	4.87298	0.186870
$681$	0	0
$682$	21.9089	0.838935
$683$	−7.74597	−0.296391	−0.148196	−	0.988958i	$-0.547347\pi$
$683$	−7.74597	−0.296391	−0.148196	+	0.988958i	$0.547347\pi$
$684$	0	0
$685$	53.3809	2.03958
$686$	0	0
$687$	0	0
$688$	1.12702	0.0429671
$689$	53.5377	2.03962
$690$	0	0
$691$	17.0490	0.648573	0.324287	−	0.945959i	$-0.394876\pi$
$691$	17.0490	0.648573	0.324287	+	0.945959i	$0.394876\pi$
$692$	−12.7279	−0.483843
$693$	0	0
$694$	7.74597	0.294033
$695$	68.4919	2.59805
$696$	0	0
$697$	8.00000	0.303022
$698$	16.4317	0.621948
$699$	0	0
$700$	0	0
$701$	16.2540	0.613906	0.306953	−	0.951725i	$-0.400690\pi$
$701$	16.2540	0.613906	0.306953	+	0.951725i	$0.400690\pi$
$702$	0	0
$703$	−36.0510	−1.35969
$704$	−4.00000	−0.150756
$705$	0	0
$706$	3.88338	0.146153
$707$	0	0
$708$	0	0
$709$	−52.7298	−1.98031	−0.990155	−	0.139974i	$-0.955298\pi$
$709$	−52.7298	−1.98031	−0.990155	+	0.139974i	$0.955298\pi$
$710$	−34.0195	−1.27673
$711$	0	0
$712$	7.07107	0.264999
$713$	47.9036	1.79401
$714$	0	0
$715$	58.4758	2.18687
$716$	6.87298	0.256855
$717$	0	0
$718$	18.2379	0.680632
$719$	23.5027	0.876504	0.438252	−	0.898852i	$-0.355598\pi$
$719$	23.5027	0.876504	0.438252	+	0.898852i	$0.355598\pi$
$720$	0	0
$721$	0	0
$722$	−20.3649	−0.757904
$723$	0	0
$724$	−13.3452	−0.495971
$725$	−7.74597	−0.287678
$726$	0	0
$727$	3.36731	0.124887	0.0624434	−	0.998049i	$-0.480111\pi$
$727$	3.36731	0.124887	0.0624434	+	0.998049i	$0.480111\pi$
$728$	0	0
$729$	0	0
$730$	−15.2379	−0.563980
$731$	−1.59384	−0.0589504
$732$	0	0
$733$	−37.0276	−1.36765	−0.683823	−	0.729648i	$-0.739684\pi$
$733$	−37.0276	−1.36765	−0.683823	+	0.729648i	$0.739684\pi$
$734$	−6.89144	−0.254368
$735$	0	0
$736$	−8.74597	−0.322381
$737$	27.4919	1.01268
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	19.7990	0.727825
$741$	0	0
$742$	0	0
$743$	12.2540	0.449557	0.224778	−	0.974410i	$-0.427834\pi$
$743$	12.2540	0.449557	0.224778	+	0.974410i	$0.427834\pi$
$744$	0	0
$745$	54.2562	1.98779
$746$	1.12702	0.0412630
$747$	0	0
$748$	5.65685	0.206835
$749$	0	0
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	4.06301	0.148163
$753$	0	0
$754$	−4.78153	−0.174133
$755$	37.9029	1.37943
$756$	0	0
$757$	31.2379	1.13536	0.567680	−	0.823249i	$-0.307841\pi$
$757$	31.2379	1.13536	0.567680	+	0.823249i	$0.307841\pi$
$758$	−26.3649	−0.957617
$759$	0	0
$760$	21.6190	0.784202
$761$	30.5738	1.10830	0.554150	−	0.832417i	$-0.313043\pi$
$761$	30.5738	1.10830	0.554150	+	0.832417i	$0.313043\pi$
$762$	0	0
$763$	0	0
$764$	6.12702	0.221668
$765$	0	0
$766$	−12.9076	−0.466369
$767$	−35.2379	−1.27237
$768$	0	0
$769$	−1.23458	−0.0445203	−0.0222601	−	0.999752i	$-0.507086\pi$
$769$	−1.23458	−0.0445203	−0.0222601	+	0.999752i	$0.507086\pi$
$770$	0	0
$771$	0	0
$772$	23.8730	0.859207
$773$	2.03151	0.0730682	0.0365341	−	0.999332i	$-0.488368\pi$
$773$	2.03151	0.0730682	0.0365341	+	0.999332i	$0.488368\pi$
$774$	0	0
$775$	37.6449	1.35224
$776$	−15.0175	−0.539096
$777$	0	0
$778$	13.1270	0.470626
$779$	35.4919	1.27163
$780$	0	0
$781$	−39.4919	−1.41313
$782$	12.3687	0.442303
$783$	0	0
$784$	0	0
$785$	41.1109	1.46731
$786$	0	0
$787$	12.3687	0.440895	0.220448	−	0.975399i	$-0.429248\pi$
$787$	12.3687	0.440895	0.220448	+	0.975399i	$0.429248\pi$
$788$	−16.6190	−0.592026
$789$	0	0
$790$	6.45378	0.229615
$791$	0	0
$792$	0	0
$793$	−26.6190	−0.945267
$794$	−7.07107	−0.250943
$795$	0	0
$796$	12.1890	0.432029
$797$	20.5959	0.729545	0.364772	−	0.931097i	$-0.381147\pi$
$797$	20.5959	0.729545	0.364772	+	0.931097i	$0.381147\pi$
$798$	0	0
$799$	−5.74597	−0.203278
$800$	−6.87298	−0.242997
$801$	0	0
$802$	−3.87298	−0.136760
$803$	−17.6891	−0.624234
$804$	0	0
$805$	0	0
$806$	23.2379	0.818520
$807$	0	0
$808$	−13.3452	−0.469483
$809$	−46.7298	−1.64293	−0.821467	−	0.570256i	$-0.806844\pi$
$809$	−46.7298	−1.64293	−0.821467	+	0.570256i	$0.806844\pi$
$810$	0	0
$811$	−3.00806	−0.105627	−0.0528136	−	0.998604i	$-0.516819\pi$
$811$	−3.00806	−0.105627	−0.0528136	+	0.998604i	$0.516819\pi$
$812$	0	0
$813$	0	0
$814$	22.9839	0.805584
$815$	34.4572	1.20698
$816$	0	0
$817$	−7.07107	−0.247385
$818$	−23.1435	−0.809193
$819$	0	0
$820$	−19.4919	−0.680688
$821$	9.49193	0.331271	0.165635	−	0.986187i	$-0.447033\pi$
$821$	9.49193	0.331271	0.165635	+	0.986187i	$0.447033\pi$
$822$	0	0
$823$	−6.25403	−0.218002	−0.109001	−	0.994042i	$-0.534765\pi$
$823$	−6.25403	−0.218002	−0.109001	+	0.994042i	$0.534765\pi$
$824$	−3.88338	−0.135284
$825$	0	0
$826$	0	0
$827$	−48.8730	−1.69948	−0.849740	−	0.527202i	$-0.823241\pi$
$827$	−48.8730	−1.69948	−0.849740	+	0.527202i	$0.823241\pi$
$828$	0	0
$829$	−10.6180	−0.368779	−0.184389	−	0.982853i	$-0.559031\pi$
$829$	−10.6180	−0.368779	−0.184389	+	0.982853i	$0.559031\pi$
$830$	−9.12702	−0.316803
$831$	0	0
$832$	−4.24264	−0.147087
$833$	0	0
$834$	0	0
$835$	−32.8730	−1.13762
$836$	25.0966	0.867984
$837$	0	0
$838$	−31.3707	−1.08368
$839$	−25.7923	−0.890449	−0.445224	−	0.895419i	$-0.646876\pi$
$839$	−25.7923	−0.890449	−0.445224	+	0.895419i	$0.646876\pi$
$840$	0	0
$841$	−27.7298	−0.956201
$842$	12.8730	0.443632
$843$	0	0
$844$	20.6190	0.709734
$845$	17.2286	0.592682
$846$	0	0
$847$	0	0
$848$	−12.6190	−0.433337
$849$	0	0
$850$	9.71987	0.333389
$851$	50.2540	1.72269
$852$	0	0
$853$	−18.1267	−0.620648	−0.310324	−	0.950631i	$-0.600438\pi$
$853$	−18.1267	−0.620648	−0.310324	+	0.950631i	$0.600438\pi$
$854$	0	0
$855$	0	0
$856$	0.254033	0.00868268
$857$	−25.8151	−0.881827	−0.440914	−	0.897550i	$-0.645345\pi$
$857$	−25.8151	−0.881827	−0.440914	+	0.897550i	$0.645345\pi$
$858$	0	0
$859$	30.5738	1.04317	0.521583	−	0.853201i	$-0.325342\pi$
$859$	30.5738	1.04317	0.521583	+	0.853201i	$0.325342\pi$
$860$	3.88338	0.132422
$861$	0	0
$862$	−21.4919	−0.732018
$863$	43.8730	1.49345	0.746727	−	0.665131i	$-0.231624\pi$
$863$	43.8730	1.49345	0.746727	+	0.665131i	$0.231624\pi$
$864$	0	0
$865$	−43.8569	−1.49118
$866$	29.6985	1.00920
$867$	0	0
$868$	0	0
$869$	7.49193	0.254146
$870$	0	0
$871$	29.1596	0.988035
$872$	−16.8730	−0.571391
$873$	0	0
$874$	54.8735	1.85612
$875$	0	0
$876$	0	0
$877$	31.1270	1.05108	0.525542	−	0.850767i	$-0.323862\pi$
$877$	31.1270	1.05108	0.525542	+	0.850767i	$0.323862\pi$
$878$	10.9545	0.369695
$879$	0	0
$880$	−13.7829	−0.464621
$881$	−26.6904	−0.899223	−0.449612	−	0.893224i	$-0.648438\pi$
$881$	−26.6904	−0.899223	−0.449612	+	0.893224i	$0.648438\pi$
$882$	0	0
$883$	35.4919	1.19440	0.597199	−	0.802093i	$-0.296280\pi$
$883$	35.4919	1.19440	0.597199	+	0.802093i	$0.296280\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−14.3649	−0.482599
$887$	−7.76677	−0.260783	−0.130391	−	0.991463i	$-0.541623\pi$
$887$	−7.76677	−0.260783	−0.130391	+	0.991463i	$0.541623\pi$
$888$	0	0
$889$	0	0
$890$	24.3649	0.816714
$891$	0	0
$892$	−22.4478	−0.751608
$893$	−25.4919	−0.853055
$894$	0	0
$895$	23.6824	0.791614
$896$	0	0
$897$	0	0
$898$	9.00000	0.300334
$899$	−6.17292	−0.205879
$900$	0	0
$901$	17.8459	0.594533
$902$	−22.6274	−0.753411
$903$	0	0
$904$	3.87298	0.128814
$905$	−45.9839	−1.52856
$906$	0	0
$907$	18.6190	0.618232	0.309116	−	0.951024i	$-0.399967\pi$
$907$	18.6190	0.618232	0.309116	+	0.951024i	$0.399967\pi$
$908$	−10.3372	−0.343051
$909$	0	0
$910$	0	0
$911$	38.7460	1.28371	0.641856	−	0.766826i	$-0.278165\pi$
$911$	38.7460	1.28371	0.641856	+	0.766826i	$0.278165\pi$
$912$	0	0
$913$	−10.5952	−0.350650
$914$	−26.1270	−0.864205
$915$	0	0
$916$	13.3452	0.440938
$917$	0	0
$918$	0	0
$919$	−0.635083	−0.0209495	−0.0104747	−	0.999945i	$-0.503334\pi$
$919$	−0.635083	−0.0209495	−0.0104747	+	0.999945i	$0.503334\pi$
$920$	−30.1361	−0.993559
$921$	0	0
$922$	28.3627	0.934075
$923$	−41.8875	−1.37874
$924$	0	0
$925$	39.4919	1.29849
$926$	1.61895	0.0532020
$927$	0	0
$928$	1.12702	0.0369961
$929$	54.4358	1.78598	0.892991	−	0.450075i	$-0.148603\pi$
$929$	54.4358	1.78598	0.892991	+	0.450075i	$0.148603\pi$
$930$	0	0
$931$	0	0
$932$	−7.25403	−0.237614
$933$	0	0
$934$	6.19574	0.202731
$935$	19.4919	0.637454
$936$	0	0
$937$	−45.2320	−1.47767	−0.738833	−	0.673889i	$-0.764623\pi$
$937$	−45.2320	−1.47767	−0.738833	+	0.673889i	$0.764623\pi$
$938$	0	0
$939$	0	0
$940$	14.0000	0.456630
$941$	−25.7139	−0.838249	−0.419124	−	0.907929i	$-0.637663\pi$
$941$	−25.7139	−0.838249	−0.419124	+	0.907929i	$0.637663\pi$
$942$	0	0
$943$	−49.4747	−1.61112
$944$	8.30565	0.270326
$945$	0	0
$946$	4.50807	0.146570
$947$	57.6028	1.87184	0.935920	−	0.352213i	$-0.114571\pi$
$947$	57.6028	1.87184	0.935920	+	0.352213i	$0.114571\pi$
$948$	0	0
$949$	−18.7621	−0.609044
$950$	43.1221	1.39907
$951$	0	0
$952$	0	0
$953$	59.4919	1.92713	0.963566	−	0.267469i	$-0.0861874\pi$
$953$	59.4919	1.92713	0.963566	+	0.267469i	$0.0861874\pi$
$954$	0	0
$955$	21.1120	0.683168
$956$	15.0000	0.485135
$957$	0	0
$958$	3.88338	0.125466
$959$	0	0
$960$	0	0
$961$	−1.00000	−0.0322581
$962$	24.3781	0.785981
$963$	0	0
$964$	−23.6824	−0.762758
$965$	82.2596	2.64803
$966$	0	0
$967$	60.4919	1.94529	0.972645	−	0.232298i	$-0.0746244\pi$
$967$	60.4919	1.94529	0.972645	+	0.232298i	$0.0746244\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−51.7460	−1.66146
$971$	−1.49262	−0.0479005	−0.0239502	−	0.999713i	$-0.507624\pi$
$971$	−1.49262	−0.0479005	−0.0239502	+	0.999713i	$0.507624\pi$
$972$	0	0
$973$	0	0
$974$	0.491933	0.0157626
$975$	0	0
$976$	6.27415	0.200831
$977$	−45.4919	−1.45542	−0.727708	−	0.685887i	$-0.759414\pi$
$977$	−45.4919	−1.45542	−0.727708	+	0.685887i	$0.759414\pi$
$978$	0	0
$979$	28.2843	0.903969
$980$	0	0
$981$	0	0
$982$	1.74597	0.0557160
$983$	9.89949	0.315745	0.157872	−	0.987460i	$-0.449537\pi$
$983$	9.89949	0.315745	0.157872	+	0.987460i	$0.449537\pi$
$984$	0	0
$985$	−57.2642	−1.82459
$986$	−1.59384	−0.0507583
$987$	0	0
$988$	26.6190	0.846862
$989$	9.85685	0.313430
$990$	0	0
$991$	−11.7460	−0.373123	−0.186561	−	0.982443i	$-0.559734\pi$
$991$	−11.7460	−0.373123	−0.186561	+	0.982443i	$0.559734\pi$
$992$	−5.47723	−0.173902
$993$	0	0
$994$	0	0
$995$	42.0000	1.33149
$996$	0	0
$997$	−27.1281	−0.859155	−0.429578	−	0.903030i	$-0.641338\pi$
$997$	−27.1281	−0.859155	−0.429578	+	0.903030i	$0.641338\pi$
$998$	29.7460	0.941592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.cd.1.4		4
3.2	odd	2		7938.2.a.cu.1.1		4
7.6	odd	2	inner	7938.2.a.cd.1.1		4
9.2	odd	6		882.2.f.p.589.4	yes	8
9.4	even	3		2646.2.f.s.883.1		8
9.5	odd	6		882.2.f.p.295.3	yes	8
9.7	even	3		2646.2.f.s.1765.1		8
21.20	even	2		7938.2.a.cu.1.4		4
63.2	odd	6		882.2.h.r.67.1		8
63.4	even	3		2646.2.h.s.667.4		8
63.5	even	6		882.2.e.t.655.2		8
63.11	odd	6		882.2.e.t.373.3		8
63.13	odd	6		2646.2.f.s.883.4		8
63.16	even	3		2646.2.h.s.361.4		8
63.20	even	6		882.2.f.p.589.1	yes	8
63.23	odd	6		882.2.e.t.655.3		8
63.25	even	3		2646.2.e.r.1549.1		8
63.31	odd	6		2646.2.h.s.667.1		8
63.32	odd	6		882.2.h.r.79.1		8
63.34	odd	6		2646.2.f.s.1765.4		8
63.38	even	6		882.2.e.t.373.2		8
63.40	odd	6		2646.2.e.r.2125.4		8
63.41	even	6		882.2.f.p.295.2	✓	8
63.47	even	6		882.2.h.r.67.4		8
63.52	odd	6		2646.2.e.r.1549.4		8
63.58	even	3		2646.2.e.r.2125.1		8
63.59	even	6		882.2.h.r.79.4		8
63.61	odd	6		2646.2.h.s.361.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.e.t.373.2		8	63.38	even	6
882.2.e.t.373.3		8	63.11	odd	6
882.2.e.t.655.2		8	63.5	even	6
882.2.e.t.655.3		8	63.23	odd	6
882.2.f.p.295.2	✓	8	63.41	even	6
882.2.f.p.295.3	yes	8	9.5	odd	6
882.2.f.p.589.1	yes	8	63.20	even	6
882.2.f.p.589.4	yes	8	9.2	odd	6
882.2.h.r.67.1		8	63.2	odd	6
882.2.h.r.67.4		8	63.47	even	6
882.2.h.r.79.1		8	63.32	odd	6
882.2.h.r.79.4		8	63.59	even	6
2646.2.e.r.1549.1		8	63.25	even	3
2646.2.e.r.1549.4		8	63.52	odd	6
2646.2.e.r.2125.1		8	63.58	even	3
2646.2.e.r.2125.4		8	63.40	odd	6
2646.2.f.s.883.1		8	9.4	even	3
2646.2.f.s.883.4		8	63.13	odd	6
2646.2.f.s.1765.1		8	9.7	even	3
2646.2.f.s.1765.4		8	63.34	odd	6
2646.2.h.s.361.1		8	63.61	odd	6
2646.2.h.s.361.4		8	63.16	even	3
2646.2.h.s.667.1		8	63.31	odd	6
2646.2.h.s.667.4		8	63.4	even	3
7938.2.a.cd.1.1		4	7.6	odd	2	inner
7938.2.a.cd.1.4		4	1.1	even	1	trivial
7938.2.a.cu.1.1		4	3.2	odd	2
7938.2.a.cu.1.4		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.44572 q^{5} -1.00000 q^{8} -3.44572 q^{10} -4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -1.41421 q^{17} -6.27415 q^{19} +3.44572 q^{20} +4.00000 q^{22} +8.74597 q^{23} +6.87298 q^{25} +4.24264 q^{26} -1.12702 q^{29} +5.47723 q^{31} -1.00000 q^{32} +1.41421 q^{34} +5.74597 q^{37} +6.27415 q^{38} -3.44572 q^{40} -5.65685 q^{41} +1.12702 q^{43} -4.00000 q^{44} -8.74597 q^{46} +4.06301 q^{47} -6.87298 q^{50} -4.24264 q^{52} -12.6190 q^{53} -13.7829 q^{55} +1.12702 q^{58} +8.30565 q^{59} +6.27415 q^{61} -5.47723 q^{62} +1.00000 q^{64} -14.6190 q^{65} -6.87298 q^{67} -1.41421 q^{68} +9.87298 q^{71} +4.42227 q^{73} -5.74597 q^{74} -6.27415 q^{76} -1.87298 q^{79} +3.44572 q^{80} +5.65685 q^{82} +2.64880 q^{83} -4.87298 q^{85} -1.12702 q^{86} +4.00000 q^{88} -7.07107 q^{89} +8.74597 q^{92} -4.06301 q^{94} -21.6190 q^{95} +15.0175 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 16 q^{11} + 4 q^{16} + 16 q^{22} + 4 q^{23} + 12 q^{25} - 20 q^{29} - 4 q^{32} - 8 q^{37} + 20 q^{43} - 16 q^{44} - 4 q^{46} - 12 q^{50} - 4 q^{53} + 20 q^{58} + 4 q^{64}+ \cdots - 40 q^{95}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 16 * q^11 + 4 * q^16 + 16 * q^22 + 4 * q^23 + 12 * q^25 - 20 * q^29 - 4 * q^32 - 8 * q^37 + 20 * q^43 - 16 * q^44 - 4 * q^46 - 12 * q^50 - 4 * q^53 + 20 * q^58 + 4 * q^64 - 12 * q^65 - 12 * q^67 + 24 * q^71 + 8 * q^74 + 8 * q^79 - 4 * q^85 - 20 * q^86 + 16 * q^88 + 4 * q^92 - 40 * q^95

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