LMFDB - Embedded newform 7168.2.a.bb.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.9433055232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 $ x^8 - 4x^7 - 6x^6 + 32x^5 + 9x^4 - 76x^3 - 4x^2 + 48*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.02908$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.42180 q^{3} -3.59402 q^{5} +1.00000 q^{7} +8.70871 q^{9} +O(q^{10})$	$q+3.42180 q^{3} -3.59402 q^{5} +1.00000 q^{7} +8.70871 q^{9} -1.08150 q^{11} -1.79014 q^{13} -12.2980 q^{15} -5.65390 q^{17} +0.0630276 q^{19} +3.42180 q^{21} +1.46467 q^{23} +7.91700 q^{25} +19.5341 q^{27} -5.04356 q^{29} -4.75455 q^{31} -3.70069 q^{33} -3.59402 q^{35} -7.19951 q^{37} -6.12551 q^{39} -7.50243 q^{41} -4.56166 q^{43} -31.2993 q^{45} -1.52393 q^{47} +1.00000 q^{49} -19.3465 q^{51} -6.59185 q^{53} +3.88695 q^{55} +0.215668 q^{57} -7.62071 q^{59} +9.62680 q^{61} +8.70871 q^{63} +6.43381 q^{65} -6.97006 q^{67} +5.01180 q^{69} -6.19187 q^{71} +8.59924 q^{73} +27.0904 q^{75} -1.08150 q^{77} +7.84435 q^{79} +40.7155 q^{81} -10.5197 q^{83} +20.3202 q^{85} -17.2580 q^{87} +9.32780 q^{89} -1.79014 q^{91} -16.2691 q^{93} -0.226523 q^{95} +0.485578 q^{97} -9.41852 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9	$ 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9} - 12 q^{11} - 20 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} + 12 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 52 q^{45} + 20 q^{47} + 8 q^{49} - 32 q^{51} - 40 q^{53} + 24 q^{55} - 4 q^{57} - 4 q^{59} + 8 q^{61} + 12 q^{63} + 36 q^{65} - 28 q^{67} - 4 q^{69} - 16 q^{71} + 16 q^{73} + 28 q^{75} - 12 q^{77} + 20 q^{81} + 8 q^{83} - 16 q^{85} + 20 q^{87} + 16 q^{89} - 20 q^{91} - 16 q^{93} - 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9 - 12 * q^11 - 20 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 12 * q^25 + 12 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 - 52 * q^45 + 20 * q^47 + 8 * q^49 - 32 * q^51 - 40 * q^53 + 24 * q^55 - 4 * q^57 - 4 * q^59 + 8 * q^61 + 12 * q^63 + 36 * q^65 - 28 * q^67 - 4 * q^69 - 16 * q^71 + 16 * q^73 + 28 * q^75 - 12 * q^77 + 20 * q^81 + 8 * q^83 - 16 * q^85 + 20 * q^87 + 16 * q^89 - 20 * q^91 - 16 * q^93 - 40 * q^95 - 36 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.42180	1.97558	0.987788	−	0.155801i	$-0.0497958\pi$
$3$	3.42180	1.97558	0.987788	+	0.155801i	$0.0497958\pi$
$4$	0	0
$5$	−3.59402	−1.60730	−0.803648	−	0.595105i	$-0.797110\pi$
$5$	−3.59402	−1.60730	−0.803648	+	0.595105i	$0.797110\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.70871	2.90290
$10$	0	0
$11$	−1.08150	−0.326086	−0.163043	−	0.986619i	$-0.552131\pi$
$11$	−1.08150	−0.326086	−0.163043	+	0.986619i	$0.552131\pi$
$12$	0	0
$13$	−1.79014	−0.496496	−0.248248	−	0.968696i	$-0.579855\pi$
$13$	−1.79014	−0.496496	−0.248248	+	0.968696i	$0.579855\pi$
$14$	0	0
$15$	−12.2980	−3.17534
$16$	0	0
$17$	−5.65390	−1.37127	−0.685636	−	0.727945i	$-0.740476\pi$
$17$	−5.65390	−1.37127	−0.685636	+	0.727945i	$0.740476\pi$
$18$	0	0
$19$	0.0630276	0.0144595	0.00722977	−	0.999974i	$-0.497699\pi$
$19$	0.0630276	0.0144595	0.00722977	+	0.999974i	$0.497699\pi$
$20$	0	0
$21$	3.42180	0.746698
$22$	0	0
$23$	1.46467	0.305404	0.152702	−	0.988272i	$-0.451203\pi$
$23$	1.46467	0.305404	0.152702	+	0.988272i	$0.451203\pi$
$24$	0	0
$25$	7.91700	1.58340
$26$	0	0
$27$	19.5341	3.75933
$28$	0	0
$29$	−5.04356	−0.936565	−0.468282	−	0.883579i	$-0.655127\pi$
$29$	−5.04356	−0.936565	−0.468282	+	0.883579i	$0.655127\pi$
$30$	0	0
$31$	−4.75455	−0.853943	−0.426971	−	0.904265i	$-0.640419\pi$
$31$	−4.75455	−0.853943	−0.426971	+	0.904265i	$0.640419\pi$
$32$	0	0
$33$	−3.70069	−0.644208
$34$	0	0
$35$	−3.59402	−0.607501
$36$	0	0
$37$	−7.19951	−1.18359	−0.591796	−	0.806088i	$-0.701581\pi$
$37$	−7.19951	−1.18359	−0.591796	+	0.806088i	$0.701581\pi$
$38$	0	0
$39$	−6.12551	−0.980867
$40$	0	0
$41$	−7.50243	−1.17168	−0.585841	−	0.810426i	$-0.699236\pi$
$41$	−7.50243	−1.17168	−0.585841	+	0.810426i	$0.699236\pi$
$42$	0	0
$43$	−4.56166	−0.695647	−0.347824	−	0.937560i	$-0.613079\pi$
$43$	−4.56166	−0.695647	−0.347824	+	0.937560i	$0.613079\pi$
$44$	0	0
$45$	−31.2993	−4.66583
$46$	0	0
$47$	−1.52393	−0.222287	−0.111144	−	0.993804i	$-0.535451\pi$
$47$	−1.52393	−0.222287	−0.111144	+	0.993804i	$0.535451\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−19.3465	−2.70905
$52$	0	0
$53$	−6.59185	−0.905460	−0.452730	−	0.891648i	$-0.649550\pi$
$53$	−6.59185	−0.905460	−0.452730	+	0.891648i	$0.649550\pi$
$54$	0	0
$55$	3.88695	0.524117
$56$	0	0
$57$	0.215668	0.0285659
$58$	0	0
$59$	−7.62071	−0.992132	−0.496066	−	0.868285i	$-0.665223\pi$
$59$	−7.62071	−0.992132	−0.496066	+	0.868285i	$0.665223\pi$
$60$	0	0
$61$	9.62680	1.23258	0.616292	−	0.787517i	$-0.288634\pi$
$61$	9.62680	1.23258	0.616292	+	0.787517i	$0.288634\pi$
$62$	0	0
$63$	8.70871	1.09719
$64$	0	0
$65$	6.43381	0.798016
$66$	0	0
$67$	−6.97006	−0.851529	−0.425764	−	0.904834i	$-0.639995\pi$
$67$	−6.97006	−0.851529	−0.425764	+	0.904834i	$0.639995\pi$
$68$	0	0
$69$	5.01180	0.603349
$70$	0	0
$71$	−6.19187	−0.734839	−0.367420	−	0.930055i	$-0.619759\pi$
$71$	−6.19187	−0.734839	−0.367420	+	0.930055i	$0.619759\pi$
$72$	0	0
$73$	8.59924	1.00647	0.503233	−	0.864151i	$-0.332144\pi$
$73$	8.59924	1.00647	0.503233	+	0.864151i	$0.332144\pi$
$74$	0	0
$75$	27.0904	3.12813
$76$	0	0
$77$	−1.08150	−0.123249
$78$	0	0
$79$	7.84435	0.882558	0.441279	−	0.897370i	$-0.354525\pi$
$79$	7.84435	0.882558	0.441279	+	0.897370i	$0.354525\pi$
$80$	0	0
$81$	40.7155	4.52395
$82$	0	0
$83$	−10.5197	−1.15469	−0.577345	−	0.816500i	$-0.695911\pi$
$83$	−10.5197	−1.15469	−0.577345	+	0.816500i	$0.695911\pi$
$84$	0	0
$85$	20.3202	2.20404
$86$	0	0
$87$	−17.2580	−1.85026
$88$	0	0
$89$	9.32780	0.988745	0.494373	−	0.869250i	$-0.335398\pi$
$89$	9.32780	0.988745	0.494373	+	0.869250i	$0.335398\pi$
$90$	0	0
$91$	−1.79014	−0.187658
$92$	0	0
$93$	−16.2691	−1.68703
$94$	0	0
$95$	−0.226523	−0.0232407
$96$	0	0
$97$	0.485578	0.0493030	0.0246515	−	0.999696i	$-0.492152\pi$
$97$	0.485578	0.0493030	0.0246515	+	0.999696i	$0.492152\pi$
$98$	0	0
$99$	−9.41852	−0.946597
$100$	0	0
$101$	−6.44571	−0.641372	−0.320686	−	0.947186i	$-0.603913\pi$
$101$	−6.44571	−0.641372	−0.320686	+	0.947186i	$0.603913\pi$
$102$	0	0
$103$	4.26862	0.420599	0.210300	−	0.977637i	$-0.432556\pi$
$103$	4.26862	0.420599	0.210300	+	0.977637i	$0.432556\pi$
$104$	0	0
$105$	−12.2980	−1.20016
$106$	0	0
$107$	−2.55775	−0.247267	−0.123633	−	0.992328i	$-0.539455\pi$
$107$	−2.55775	−0.247267	−0.123633	+	0.992328i	$0.539455\pi$
$108$	0	0
$109$	15.5780	1.49210	0.746050	−	0.665890i	$-0.231948\pi$
$109$	15.5780	1.49210	0.746050	+	0.665890i	$0.231948\pi$
$110$	0	0
$111$	−24.6353	−2.33828
$112$	0	0
$113$	11.6345	1.09449	0.547243	−	0.836974i	$-0.315677\pi$
$113$	11.6345	1.09449	0.547243	+	0.836974i	$0.315677\pi$
$114$	0	0
$115$	−5.26404	−0.490875
$116$	0	0
$117$	−15.5898	−1.44128
$118$	0	0
$119$	−5.65390	−0.518292
$120$	0	0
$121$	−9.83035	−0.893668
$122$	0	0
$123$	−25.6718	−2.31475
$124$	0	0
$125$	−10.4838	−0.937695
$126$	0	0
$127$	−11.7630	−1.04380	−0.521899	−	0.853007i	$-0.674776\pi$
$127$	−11.7630	−1.04380	−0.521899	+	0.853007i	$0.674776\pi$
$128$	0	0
$129$	−15.6091	−1.37430
$130$	0	0
$131$	15.6871	1.37059	0.685294	−	0.728267i	$-0.259674\pi$
$131$	15.6871	1.37059	0.685294	+	0.728267i	$0.259674\pi$
$132$	0	0
$133$	0.0630276	0.00546519
$134$	0	0
$135$	−70.2059	−6.04236
$136$	0	0
$137$	−7.47159	−0.638341	−0.319170	−	0.947697i	$-0.603404\pi$
$137$	−7.47159	−0.638341	−0.319170	+	0.947697i	$0.603404\pi$
$138$	0	0
$139$	−0.352269	−0.0298790	−0.0149395	−	0.999888i	$-0.504756\pi$
$139$	−0.352269	−0.0298790	−0.0149395	+	0.999888i	$0.504756\pi$
$140$	0	0
$141$	−5.21457	−0.439146
$142$	0	0
$143$	1.93605	0.161901
$144$	0	0
$145$	18.1267	1.50534
$146$	0	0
$147$	3.42180	0.282225
$148$	0	0
$149$	−6.26539	−0.513281	−0.256640	−	0.966507i	$-0.582616\pi$
$149$	−6.26539	−0.513281	−0.256640	+	0.966507i	$0.582616\pi$
$150$	0	0
$151$	−12.4475	−1.01297	−0.506483	−	0.862250i	$-0.669055\pi$
$151$	−12.4475	−1.01297	−0.506483	+	0.862250i	$0.669055\pi$
$152$	0	0
$153$	−49.2382	−3.98067
$154$	0	0
$155$	17.0880	1.37254
$156$	0	0
$157$	−5.24808	−0.418842	−0.209421	−	0.977826i	$-0.567158\pi$
$157$	−5.24808	−0.418842	−0.209421	+	0.977826i	$0.567158\pi$
$158$	0	0
$159$	−22.5560	−1.78881
$160$	0	0
$161$	1.46467	0.115432
$162$	0	0
$163$	−7.72263	−0.604883	−0.302442	−	0.953168i	$-0.597802\pi$
$163$	−7.72263	−0.604883	−0.302442	+	0.953168i	$0.597802\pi$
$164$	0	0
$165$	13.3004	1.03543
$166$	0	0
$167$	8.39368	0.649523	0.324761	−	0.945796i	$-0.394716\pi$
$167$	8.39368	0.649523	0.324761	+	0.945796i	$0.394716\pi$
$168$	0	0
$169$	−9.79539	−0.753491
$170$	0	0
$171$	0.548890	0.0419746
$172$	0	0
$173$	−11.5546	−0.878482	−0.439241	−	0.898369i	$-0.644753\pi$
$173$	−11.5546	−0.878482	−0.439241	+	0.898369i	$0.644753\pi$
$174$	0	0
$175$	7.91700	0.598469
$176$	0	0
$177$	−26.0765	−1.96003
$178$	0	0
$179$	−19.4675	−1.45507	−0.727536	−	0.686070i	$-0.759334\pi$
$179$	−19.4675	−1.45507	−0.727536	+	0.686070i	$0.759334\pi$
$180$	0	0
$181$	11.4791	0.853233	0.426617	−	0.904433i	$-0.359705\pi$
$181$	11.4791	0.853233	0.426617	+	0.904433i	$0.359705\pi$
$182$	0	0
$183$	32.9410	2.43507
$184$	0	0
$185$	25.8752	1.90238
$186$	0	0
$187$	6.11472	0.447153
$188$	0	0
$189$	19.5341	1.42089
$190$	0	0
$191$	−16.5900	−1.20041	−0.600206	−	0.799846i	$-0.704915\pi$
$191$	−16.5900	−1.20041	−0.600206	+	0.799846i	$0.704915\pi$
$192$	0	0
$193$	−1.32261	−0.0952033	−0.0476016	−	0.998866i	$-0.515158\pi$
$193$	−1.32261	−0.0952033	−0.0476016	+	0.998866i	$0.515158\pi$
$194$	0	0
$195$	22.0152	1.57654
$196$	0	0
$197$	16.3452	1.16454	0.582272	−	0.812994i	$-0.302164\pi$
$197$	16.3452	1.16454	0.582272	+	0.812994i	$0.302164\pi$
$198$	0	0
$199$	26.1422	1.85317	0.926587	−	0.376081i	$-0.122728\pi$
$199$	26.1422	1.85317	0.926587	+	0.376081i	$0.122728\pi$
$200$	0	0
$201$	−23.8502	−1.68226
$202$	0	0
$203$	−5.04356	−0.353988
$204$	0	0
$205$	26.9639	1.88324
$206$	0	0
$207$	12.7554	0.886559
$208$	0	0
$209$	−0.0681647	−0.00471505
$210$	0	0
$211$	−24.7253	−1.70216	−0.851081	−	0.525034i	$-0.824053\pi$
$211$	−24.7253	−1.70216	−0.851081	+	0.525034i	$0.824053\pi$
$212$	0	0
$213$	−21.1873	−1.45173
$214$	0	0
$215$	16.3947	1.11811
$216$	0	0
$217$	−4.75455	−0.322760
$218$	0	0
$219$	29.4249	1.98835
$220$	0	0
$221$	10.1213	0.680831
$222$	0	0
$223$	−9.70637	−0.649987	−0.324993	−	0.945716i	$-0.605362\pi$
$223$	−9.70637	−0.649987	−0.324993	+	0.945716i	$0.605362\pi$
$224$	0	0
$225$	68.9469	4.59646
$226$	0	0
$227$	−23.7536	−1.57658	−0.788292	−	0.615301i	$-0.789034\pi$
$227$	−23.7536	−1.57658	−0.788292	+	0.615301i	$0.789034\pi$
$228$	0	0
$229$	−18.6093	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$229$	−18.6093	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$230$	0	0
$231$	−3.70069	−0.243488
$232$	0	0
$233$	−10.7832	−0.706431	−0.353216	−	0.935542i	$-0.614912\pi$
$233$	−10.7832	−0.706431	−0.353216	+	0.935542i	$0.614912\pi$
$234$	0	0
$235$	5.47702	0.357282
$236$	0	0
$237$	26.8418	1.74356
$238$	0	0
$239$	−2.08310	−0.134745	−0.0673724	−	0.997728i	$-0.521462\pi$
$239$	−2.08310	−0.134745	−0.0673724	+	0.997728i	$0.521462\pi$
$240$	0	0
$241$	15.8817	1.02303	0.511516	−	0.859274i	$-0.329084\pi$
$241$	15.8817	1.02303	0.511516	+	0.859274i	$0.329084\pi$
$242$	0	0
$243$	80.7182	5.17808
$244$	0	0
$245$	−3.59402	−0.229614
$246$	0	0
$247$	−0.112828	−0.00717911
$248$	0	0
$249$	−35.9964	−2.28118
$250$	0	0
$251$	24.2151	1.52844	0.764220	−	0.644955i	$-0.223124\pi$
$251$	24.2151	1.52844	0.764220	+	0.644955i	$0.223124\pi$
$252$	0	0
$253$	−1.58404	−0.0995880
$254$	0	0
$255$	69.5318	4.35425
$256$	0	0
$257$	−16.3998	−1.02299	−0.511497	−	0.859285i	$-0.670909\pi$
$257$	−16.3998	−1.02299	−0.511497	+	0.859285i	$0.670909\pi$
$258$	0	0
$259$	−7.19951	−0.447356
$260$	0	0
$261$	−43.9229	−2.71876
$262$	0	0
$263$	24.4978	1.51060	0.755300	−	0.655379i	$-0.227491\pi$
$263$	24.4978	1.51060	0.755300	+	0.655379i	$0.227491\pi$
$264$	0	0
$265$	23.6912	1.45534
$266$	0	0
$267$	31.9179	1.95334
$268$	0	0
$269$	−1.48885	−0.0907768	−0.0453884	−	0.998969i	$-0.514453\pi$
$269$	−1.48885	−0.0907768	−0.0453884	+	0.998969i	$0.514453\pi$
$270$	0	0
$271$	−28.0458	−1.70366	−0.851829	−	0.523820i	$-0.824507\pi$
$271$	−28.0458	−1.70366	−0.851829	+	0.523820i	$0.824507\pi$
$272$	0	0
$273$	−6.12551	−0.370733
$274$	0	0
$275$	−8.56227	−0.516324
$276$	0	0
$277$	−5.08702	−0.305650	−0.152825	−	0.988253i	$-0.548837\pi$
$277$	−5.08702	−0.305650	−0.152825	+	0.988253i	$0.548837\pi$
$278$	0	0
$279$	−41.4060	−2.47891
$280$	0	0
$281$	2.33236	0.139137	0.0695683	−	0.997577i	$-0.477838\pi$
$281$	2.33236	0.139137	0.0695683	+	0.997577i	$0.477838\pi$
$282$	0	0
$283$	6.35544	0.377791	0.188896	−	0.981997i	$-0.439509\pi$
$283$	6.35544	0.377791	0.188896	+	0.981997i	$0.439509\pi$
$284$	0	0
$285$	−0.775116	−0.0459139
$286$	0	0
$287$	−7.50243	−0.442854
$288$	0	0
$289$	14.9666	0.880386
$290$	0	0
$291$	1.66155	0.0974019
$292$	0	0
$293$	1.45568	0.0850415	0.0425207	−	0.999096i	$-0.486461\pi$
$293$	1.45568	0.0850415	0.0425207	+	0.999096i	$0.486461\pi$
$294$	0	0
$295$	27.3890	1.59465
$296$	0	0
$297$	−21.1262	−1.22587
$298$	0	0
$299$	−2.62196	−0.151632
$300$	0	0
$301$	−4.56166	−0.262930
$302$	0	0
$303$	−22.0559	−1.26708
$304$	0	0
$305$	−34.5989	−1.98113
$306$	0	0
$307$	−7.44089	−0.424674	−0.212337	−	0.977196i	$-0.568107\pi$
$307$	−7.44089	−0.424674	−0.212337	+	0.977196i	$0.568107\pi$
$308$	0	0
$309$	14.6064	0.830926
$310$	0	0
$311$	−8.74660	−0.495974	−0.247987	−	0.968763i	$-0.579769\pi$
$311$	−8.74660	−0.495974	−0.247987	+	0.968763i	$0.579769\pi$
$312$	0	0
$313$	−8.77036	−0.495730	−0.247865	−	0.968795i	$-0.579729\pi$
$313$	−8.77036	−0.495730	−0.247865	+	0.968795i	$0.579729\pi$
$314$	0	0
$315$	−31.2993	−1.76352
$316$	0	0
$317$	−21.5025	−1.20770	−0.603851	−	0.797097i	$-0.706368\pi$
$317$	−21.5025	−1.20770	−0.603851	+	0.797097i	$0.706368\pi$
$318$	0	0
$319$	5.45463	0.305401
$320$	0	0
$321$	−8.75210	−0.488495
$322$	0	0
$323$	−0.356352	−0.0198279
$324$	0	0
$325$	−14.1726	−0.786152
$326$	0	0
$327$	53.3047	2.94776
$328$	0	0
$329$	−1.52393	−0.0840167
$330$	0	0
$331$	−6.96891	−0.383046	−0.191523	−	0.981488i	$-0.561343\pi$
$331$	−6.96891	−0.383046	−0.191523	+	0.981488i	$0.561343\pi$
$332$	0	0
$333$	−62.6985	−3.43586
$334$	0	0
$335$	25.0506	1.36866
$336$	0	0
$337$	12.5793	0.685237	0.342619	−	0.939475i	$-0.388686\pi$
$337$	12.5793	0.685237	0.342619	+	0.939475i	$0.388686\pi$
$338$	0	0
$339$	39.8111	2.16224
$340$	0	0
$341$	5.14207	0.278459
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−18.0125	−0.969761
$346$	0	0
$347$	−4.78717	−0.256989	−0.128494	−	0.991710i	$-0.541014\pi$
$347$	−4.78717	−0.256989	−0.128494	+	0.991710i	$0.541014\pi$
$348$	0	0
$349$	35.1475	1.88140	0.940702	−	0.339233i	$-0.110168\pi$
$349$	35.1475	1.88140	0.940702	+	0.339233i	$0.110168\pi$
$350$	0	0
$351$	−34.9688	−1.86650
$352$	0	0
$353$	8.70479	0.463309	0.231655	−	0.972798i	$-0.425586\pi$
$353$	8.70479	0.463309	0.231655	+	0.972798i	$0.425586\pi$
$354$	0	0
$355$	22.2537	1.18110
$356$	0	0
$357$	−19.3465	−1.02393
$358$	0	0
$359$	−4.04735	−0.213611	−0.106805	−	0.994280i	$-0.534062\pi$
$359$	−4.04735	−0.213611	−0.106805	+	0.994280i	$0.534062\pi$
$360$	0	0
$361$	−18.9960	−0.999791
$362$	0	0
$363$	−33.6375	−1.76551
$364$	0	0
$365$	−30.9059	−1.61769
$366$	0	0
$367$	31.6904	1.65423	0.827113	−	0.562036i	$-0.189982\pi$
$367$	31.6904	1.65423	0.827113	+	0.562036i	$0.189982\pi$
$368$	0	0
$369$	−65.3365	−3.40128
$370$	0	0
$371$	−6.59185	−0.342232
$372$	0	0
$373$	−21.4087	−1.10850	−0.554251	−	0.832349i	$-0.686995\pi$
$373$	−21.4087	−1.10850	−0.554251	+	0.832349i	$0.686995\pi$
$374$	0	0
$375$	−35.8733	−1.85249
$376$	0	0
$377$	9.02869	0.465001
$378$	0	0
$379$	−19.5360	−1.00350	−0.501749	−	0.865014i	$-0.667310\pi$
$379$	−19.5360	−1.00350	−0.501749	+	0.865014i	$0.667310\pi$
$380$	0	0
$381$	−40.2507	−2.06210
$382$	0	0
$383$	17.3513	0.886610	0.443305	−	0.896371i	$-0.353806\pi$
$383$	17.3513	0.886610	0.443305	+	0.896371i	$0.353806\pi$
$384$	0	0
$385$	3.88695	0.198097
$386$	0	0
$387$	−39.7262	−2.01940
$388$	0	0
$389$	25.2975	1.28263	0.641316	−	0.767277i	$-0.278389\pi$
$389$	25.2975	1.28263	0.641316	+	0.767277i	$0.278389\pi$
$390$	0	0
$391$	−8.28108	−0.418792
$392$	0	0
$393$	53.6781	2.70770
$394$	0	0
$395$	−28.1928	−1.41853
$396$	0	0
$397$	6.15919	0.309121	0.154560	−	0.987983i	$-0.450604\pi$
$397$	6.15919	0.309121	0.154560	+	0.987983i	$0.450604\pi$
$398$	0	0
$399$	0.215668	0.0107969
$400$	0	0
$401$	−17.6318	−0.880492	−0.440246	−	0.897877i	$-0.645109\pi$
$401$	−17.6318	−0.880492	−0.440246	+	0.897877i	$0.645109\pi$
$402$	0	0
$403$	8.51133	0.423979
$404$	0	0
$405$	−146.333	−7.27132
$406$	0	0
$407$	7.78631	0.385953
$408$	0	0
$409$	24.7264	1.22264	0.611320	−	0.791383i	$-0.290639\pi$
$409$	24.7264	1.22264	0.611320	+	0.791383i	$0.290639\pi$
$410$	0	0
$411$	−25.5663	−1.26109
$412$	0	0
$413$	−7.62071	−0.374991
$414$	0	0
$415$	37.8081	1.85593
$416$	0	0
$417$	−1.20539	−0.0590283
$418$	0	0
$419$	−2.36688	−0.115630	−0.0578149	−	0.998327i	$-0.518413\pi$
$419$	−2.36688	−0.115630	−0.0578149	+	0.998327i	$0.518413\pi$
$420$	0	0
$421$	−31.4952	−1.53498	−0.767492	−	0.641059i	$-0.778495\pi$
$421$	−31.4952	−1.53498	−0.767492	+	0.641059i	$0.778495\pi$
$422$	0	0
$423$	−13.2714	−0.645279
$424$	0	0
$425$	−44.7619	−2.17127
$426$	0	0
$427$	9.62680	0.465873
$428$	0	0
$429$	6.62477	0.319847
$430$	0	0
$431$	23.0028	1.10800	0.554002	−	0.832515i	$-0.313100\pi$
$431$	23.0028	1.10800	0.554002	+	0.832515i	$0.313100\pi$
$432$	0	0
$433$	1.31078	0.0629921	0.0314960	−	0.999504i	$-0.489973\pi$
$433$	1.31078	0.0629921	0.0314960	+	0.999504i	$0.489973\pi$
$434$	0	0
$435$	62.0258	2.97391
$436$	0	0
$437$	0.0923145	0.00441600
$438$	0	0
$439$	36.2799	1.73155	0.865773	−	0.500437i	$-0.166827\pi$
$439$	36.2799	1.73155	0.865773	+	0.500437i	$0.166827\pi$
$440$	0	0
$441$	8.70871	0.414701
$442$	0	0
$443$	−13.5410	−0.643352	−0.321676	−	0.946850i	$-0.604246\pi$
$443$	−13.5410	−0.643352	−0.321676	+	0.946850i	$0.604246\pi$
$444$	0	0
$445$	−33.5243	−1.58921
$446$	0	0
$447$	−21.4389	−1.01403
$448$	0	0
$449$	−3.60363	−0.170066	−0.0850329	−	0.996378i	$-0.527100\pi$
$449$	−3.60363	−0.170066	−0.0850329	+	0.996378i	$0.527100\pi$
$450$	0	0
$451$	8.11391	0.382069
$452$	0	0
$453$	−42.5929	−2.00119
$454$	0	0
$455$	6.43381	0.301622
$456$	0	0
$457$	−6.35277	−0.297170	−0.148585	−	0.988900i	$-0.547472\pi$
$457$	−6.35277	−0.297170	−0.148585	+	0.988900i	$0.547472\pi$
$458$	0	0
$459$	−110.444	−5.15507
$460$	0	0
$461$	5.88792	0.274228	0.137114	−	0.990555i	$-0.456217\pi$
$461$	5.88792	0.274228	0.137114	+	0.990555i	$0.456217\pi$
$462$	0	0
$463$	12.7205	0.591171	0.295585	−	0.955316i	$-0.404485\pi$
$463$	12.7205	0.591171	0.295585	+	0.955316i	$0.404485\pi$
$464$	0	0
$465$	58.4716	2.71156
$466$	0	0
$467$	−21.7910	−1.00836	−0.504182	−	0.863597i	$-0.668206\pi$
$467$	−21.7910	−1.00836	−0.504182	+	0.863597i	$0.668206\pi$
$468$	0	0
$469$	−6.97006	−0.321848
$470$	0	0
$471$	−17.9579	−0.827455
$472$	0	0
$473$	4.93346	0.226841
$474$	0	0
$475$	0.498990	0.0228952
$476$	0	0
$477$	−57.4065	−2.62846
$478$	0	0
$479$	−26.3235	−1.20275	−0.601375	−	0.798967i	$-0.705380\pi$
$479$	−26.3235	−1.20275	−0.601375	+	0.798967i	$0.705380\pi$
$480$	0	0
$481$	12.8882	0.587649
$482$	0	0
$483$	5.01180	0.228045
$484$	0	0
$485$	−1.74518	−0.0792445
$486$	0	0
$487$	4.62575	0.209613	0.104806	−	0.994493i	$-0.466578\pi$
$487$	4.62575	0.209613	0.104806	+	0.994493i	$0.466578\pi$
$488$	0	0
$489$	−26.4253	−1.19499
$490$	0	0
$491$	3.87808	0.175015	0.0875076	−	0.996164i	$-0.472110\pi$
$491$	3.87808	0.175015	0.0875076	+	0.996164i	$0.472110\pi$
$492$	0	0
$493$	28.5157	1.28428
$494$	0	0
$495$	33.8504	1.52146
$496$	0	0
$497$	−6.19187	−0.277743
$498$	0	0
$499$	−0.994539	−0.0445217	−0.0222608	−	0.999752i	$-0.507086\pi$
$499$	−0.994539	−0.0445217	−0.0222608	+	0.999752i	$0.507086\pi$
$500$	0	0
$501$	28.7215	1.28318
$502$	0	0
$503$	23.3204	1.03980	0.519902	−	0.854226i	$-0.325968\pi$
$503$	23.3204	1.03980	0.519902	+	0.854226i	$0.325968\pi$
$504$	0	0
$505$	23.1660	1.03087
$506$	0	0
$507$	−33.5179	−1.48858
$508$	0	0
$509$	6.79254	0.301074	0.150537	−	0.988604i	$-0.451900\pi$
$509$	6.79254	0.301074	0.150537	+	0.988604i	$0.451900\pi$
$510$	0	0
$511$	8.59924	0.380408
$512$	0	0
$513$	1.23119	0.0543582
$514$	0	0
$515$	−15.3415	−0.676027
$516$	0	0
$517$	1.64813	0.0724848
$518$	0	0
$519$	−39.5376	−1.73551
$520$	0	0
$521$	33.2330	1.45596	0.727982	−	0.685596i	$-0.240458\pi$
$521$	33.2330	1.45596	0.727982	+	0.685596i	$0.240458\pi$
$522$	0	0
$523$	28.8901	1.26328	0.631638	−	0.775264i	$-0.282383\pi$
$523$	28.8901	1.26328	0.631638	+	0.775264i	$0.282383\pi$
$524$	0	0
$525$	27.0904	1.18232
$526$	0	0
$527$	26.8818	1.17099
$528$	0	0
$529$	−20.8548	−0.906728
$530$	0	0
$531$	−66.3666	−2.88006
$532$	0	0
$533$	13.4304	0.581736
$534$	0	0
$535$	9.19260	0.397431
$536$	0	0
$537$	−66.6140	−2.87461
$538$	0	0
$539$	−1.08150	−0.0465837
$540$	0	0
$541$	4.95435	0.213004	0.106502	−	0.994312i	$-0.466035\pi$
$541$	4.95435	0.213004	0.106502	+	0.994312i	$0.466035\pi$
$542$	0	0
$543$	39.2791	1.68563
$544$	0	0
$545$	−55.9876	−2.39824
$546$	0	0
$547$	31.0796	1.32887	0.664434	−	0.747347i	$-0.268672\pi$
$547$	31.0796	1.32887	0.664434	+	0.747347i	$0.268672\pi$
$548$	0	0
$549$	83.8370	3.57808
$550$	0	0
$551$	−0.317883	−0.0135423
$552$	0	0
$553$	7.84435	0.333576
$554$	0	0
$555$	88.5398	3.75830
$556$	0	0
$557$	−6.01930	−0.255046	−0.127523	−	0.991836i	$-0.540703\pi$
$557$	−6.01930	−0.255046	−0.127523	+	0.991836i	$0.540703\pi$
$558$	0	0
$559$	8.16603	0.345386
$560$	0	0
$561$	20.9233	0.883384
$562$	0	0
$563$	12.6945	0.535008	0.267504	−	0.963557i	$-0.413801\pi$
$563$	12.6945	0.535008	0.267504	+	0.963557i	$0.413801\pi$
$564$	0	0
$565$	−41.8148	−1.75916
$566$	0	0
$567$	40.7155	1.70989
$568$	0	0
$569$	12.3968	0.519701	0.259851	−	0.965649i	$-0.416327\pi$
$569$	12.3968	0.519701	0.259851	+	0.965649i	$0.416327\pi$
$570$	0	0
$571$	−7.41186	−0.310177	−0.155088	−	0.987901i	$-0.549566\pi$
$571$	−7.41186	−0.310177	−0.155088	+	0.987901i	$0.549566\pi$
$572$	0	0
$573$	−56.7677	−2.37151
$574$	0	0
$575$	11.5958	0.483577
$576$	0	0
$577$	43.5232	1.81189	0.905947	−	0.423390i	$-0.139160\pi$
$577$	43.5232	1.81189	0.905947	+	0.423390i	$0.139160\pi$
$578$	0	0
$579$	−4.52569	−0.188081
$580$	0	0
$581$	−10.5197	−0.436432
$582$	0	0
$583$	7.12912	0.295258
$584$	0	0
$585$	56.0302	2.31657
$586$	0	0
$587$	−3.74658	−0.154638	−0.0773190	−	0.997006i	$-0.524636\pi$
$587$	−3.74658	−0.154638	−0.0773190	+	0.997006i	$0.524636\pi$
$588$	0	0
$589$	−0.299668	−0.0123476
$590$	0	0
$591$	55.9299	2.30065
$592$	0	0
$593$	−5.12318	−0.210384	−0.105192	−	0.994452i	$-0.533546\pi$
$593$	−5.12318	−0.210384	−0.105192	+	0.994452i	$0.533546\pi$
$594$	0	0
$595$	20.3202	0.833048
$596$	0	0
$597$	89.4535	3.66109
$598$	0	0
$599$	39.6945	1.62187	0.810936	−	0.585135i	$-0.198959\pi$
$599$	39.6945	1.62187	0.810936	+	0.585135i	$0.198959\pi$
$600$	0	0
$601$	−19.6274	−0.800620	−0.400310	−	0.916380i	$-0.631097\pi$
$601$	−19.6274	−0.800620	−0.400310	+	0.916380i	$0.631097\pi$
$602$	0	0
$603$	−60.7003	−2.47191
$604$	0	0
$605$	35.3305	1.43639
$606$	0	0
$607$	25.7837	1.04653	0.523264	−	0.852170i	$-0.324714\pi$
$607$	25.7837	1.04653	0.523264	+	0.852170i	$0.324714\pi$
$608$	0	0
$609$	−17.2580	−0.699331
$610$	0	0
$611$	2.72804	0.110365
$612$	0	0
$613$	5.78639	0.233710	0.116855	−	0.993149i	$-0.462719\pi$
$613$	5.78639	0.233710	0.116855	+	0.993149i	$0.462719\pi$
$614$	0	0
$615$	92.2651	3.72049
$616$	0	0
$617$	−10.4658	−0.421337	−0.210668	−	0.977558i	$-0.567564\pi$
$617$	−10.4658	−0.421337	−0.210668	+	0.977558i	$0.567564\pi$
$618$	0	0
$619$	30.4411	1.22353	0.611765	−	0.791040i	$-0.290460\pi$
$619$	30.4411	1.22353	0.611765	+	0.791040i	$0.290460\pi$
$620$	0	0
$621$	28.6109	1.14812
$622$	0	0
$623$	9.32780	0.373711
$624$	0	0
$625$	−1.90615	−0.0762459
$626$	0	0
$627$	−0.233246	−0.00931495
$628$	0	0
$629$	40.7053	1.62303
$630$	0	0
$631$	−0.948167	−0.0377459	−0.0188730	−	0.999822i	$-0.506008\pi$
$631$	−0.948167	−0.0377459	−0.0188730	+	0.999822i	$0.506008\pi$
$632$	0	0
$633$	−84.6052	−3.36275
$634$	0	0
$635$	42.2765	1.67769
$636$	0	0
$637$	−1.79014	−0.0709280
$638$	0	0
$639$	−53.9232	−2.13317
$640$	0	0
$641$	12.1984	0.481808	0.240904	−	0.970549i	$-0.422556\pi$
$641$	12.1984	0.481808	0.240904	+	0.970549i	$0.422556\pi$
$642$	0	0
$643$	−50.4676	−1.99025	−0.995124	−	0.0986311i	$-0.968554\pi$
$643$	−50.4676	−1.99025	−0.995124	+	0.0986311i	$0.968554\pi$
$644$	0	0
$645$	56.0995	2.20891
$646$	0	0
$647$	−29.4906	−1.15939	−0.579697	−	0.814832i	$-0.696829\pi$
$647$	−29.4906	−1.15939	−0.579697	+	0.814832i	$0.696829\pi$
$648$	0	0
$649$	8.24183	0.323520
$650$	0	0
$651$	−16.2691	−0.637637
$652$	0	0
$653$	−43.6924	−1.70982	−0.854908	−	0.518779i	$-0.826387\pi$
$653$	−43.6924	−1.70982	−0.854908	+	0.518779i	$0.826387\pi$
$654$	0	0
$655$	−56.3798	−2.20294
$656$	0	0
$657$	74.8883	2.92167
$658$	0	0
$659$	17.8415	0.695006	0.347503	−	0.937679i	$-0.387030\pi$
$659$	17.8415	0.695006	0.347503	+	0.937679i	$0.387030\pi$
$660$	0	0
$661$	29.0216	1.12881	0.564405	−	0.825498i	$-0.309106\pi$
$661$	29.0216	1.12881	0.564405	+	0.825498i	$0.309106\pi$
$662$	0	0
$663$	34.6330	1.34503
$664$	0	0
$665$	−0.226523	−0.00878418
$666$	0	0
$667$	−7.38713	−0.286031
$668$	0	0
$669$	−33.2133	−1.28410
$670$	0	0
$671$	−10.4114	−0.401929
$672$	0	0
$673$	19.4620	0.750205	0.375103	−	0.926983i	$-0.377607\pi$
$673$	19.4620	0.750205	0.375103	+	0.926983i	$0.377607\pi$
$674$	0	0
$675$	154.651	5.95253
$676$	0	0
$677$	−5.21145	−0.200292	−0.100146	−	0.994973i	$-0.531931\pi$
$677$	−5.21145	−0.200292	−0.100146	+	0.994973i	$0.531931\pi$
$678$	0	0
$679$	0.485578	0.0186348
$680$	0	0
$681$	−81.2802	−3.11466
$682$	0	0
$683$	1.15851	0.0443292	0.0221646	−	0.999754i	$-0.492944\pi$
$683$	1.15851	0.0443292	0.0221646	+	0.999754i	$0.492944\pi$
$684$	0	0
$685$	26.8531	1.02600
$686$	0	0
$687$	−63.6772	−2.42944
$688$	0	0
$689$	11.8003	0.449558
$690$	0	0
$691$	12.8345	0.488247	0.244123	−	0.969744i	$-0.421500\pi$
$691$	12.8345	0.488247	0.244123	+	0.969744i	$0.421500\pi$
$692$	0	0
$693$	−9.41852	−0.357780
$694$	0	0
$695$	1.26606	0.0480244
$696$	0	0
$697$	42.4180	1.60670
$698$	0	0
$699$	−36.8980	−1.39561
$700$	0	0
$701$	−31.9498	−1.20673	−0.603364	−	0.797466i	$-0.706173\pi$
$701$	−31.9498	−1.20673	−0.603364	+	0.797466i	$0.706173\pi$
$702$	0	0
$703$	−0.453768	−0.0171142
$704$	0	0
$705$	18.7413	0.705837
$706$	0	0
$707$	−6.44571	−0.242416
$708$	0	0
$709$	19.2624	0.723415	0.361708	−	0.932292i	$-0.382194\pi$
$709$	19.2624	0.723415	0.361708	+	0.932292i	$0.382194\pi$
$710$	0	0
$711$	68.3142	2.56198
$712$	0	0
$713$	−6.96383	−0.260798
$714$	0	0
$715$	−6.95820	−0.260222
$716$	0	0
$717$	−7.12796	−0.266199
$718$	0	0
$719$	25.9785	0.968836	0.484418	−	0.874837i	$-0.339031\pi$
$719$	25.9785	0.968836	0.484418	+	0.874837i	$0.339031\pi$
$720$	0	0
$721$	4.26862	0.158972
$722$	0	0
$723$	54.3441	2.02108
$724$	0	0
$725$	−39.9298	−1.48296
$726$	0	0
$727$	−27.5763	−1.02275	−0.511375	−	0.859358i	$-0.670864\pi$
$727$	−27.5763	−1.02275	−0.511375	+	0.859358i	$0.670864\pi$
$728$	0	0
$729$	154.055	5.70574
$730$	0	0
$731$	25.7912	0.953922
$732$	0	0
$733$	42.7081	1.57746	0.788730	−	0.614740i	$-0.210739\pi$
$733$	42.7081	1.57746	0.788730	+	0.614740i	$0.210739\pi$
$734$	0	0
$735$	−12.2980	−0.453619
$736$	0	0
$737$	7.53816	0.277672
$738$	0	0
$739$	51.6817	1.90114	0.950571	−	0.310508i	$-0.100499\pi$
$739$	51.6817	1.90114	0.950571	+	0.310508i	$0.100499\pi$
$740$	0	0
$741$	−0.386077	−0.0141829
$742$	0	0
$743$	−43.1375	−1.58256	−0.791281	−	0.611452i	$-0.790586\pi$
$743$	−43.1375	−1.58256	−0.791281	+	0.611452i	$0.790586\pi$
$744$	0	0
$745$	22.5180	0.824994
$746$	0	0
$747$	−91.6133	−3.35196
$748$	0	0
$749$	−2.55775	−0.0934581
$750$	0	0
$751$	−9.04305	−0.329986	−0.164993	−	0.986295i	$-0.552760\pi$
$751$	−9.04305	−0.329986	−0.164993	+	0.986295i	$0.552760\pi$
$752$	0	0
$753$	82.8591	3.01955
$754$	0	0
$755$	44.7367	1.62813
$756$	0	0
$757$	1.51946	0.0552258	0.0276129	−	0.999619i	$-0.491209\pi$
$757$	1.51946	0.0552258	0.0276129	+	0.999619i	$0.491209\pi$
$758$	0	0
$759$	−5.42028	−0.196744
$760$	0	0
$761$	34.1138	1.23662	0.618312	−	0.785932i	$-0.287817\pi$
$761$	34.1138	1.23662	0.618312	+	0.785932i	$0.287817\pi$
$762$	0	0
$763$	15.5780	0.563961
$764$	0	0
$765$	176.963	6.39811
$766$	0	0
$767$	13.6422	0.492590
$768$	0	0
$769$	−54.2612	−1.95671	−0.978354	−	0.206939i	$-0.933650\pi$
$769$	−54.2612	−1.95671	−0.978354	+	0.206939i	$0.933650\pi$
$770$	0	0
$771$	−56.1169	−2.02100
$772$	0	0
$773$	36.3141	1.30613	0.653063	−	0.757303i	$-0.273484\pi$
$773$	36.3141	1.30613	0.653063	+	0.757303i	$0.273484\pi$
$774$	0	0
$775$	−37.6418	−1.35213
$776$	0	0
$777$	−24.6353	−0.883786
$778$	0	0
$779$	−0.472860	−0.0169420
$780$	0	0
$781$	6.69653	0.239621
$782$	0	0
$783$	−98.5212	−3.52086
$784$	0	0
$785$	18.8617	0.673203
$786$	0	0
$787$	−23.1195	−0.824120	−0.412060	−	0.911157i	$-0.635191\pi$
$787$	−23.1195	−0.824120	−0.412060	+	0.911157i	$0.635191\pi$
$788$	0	0
$789$	83.8266	2.98431
$790$	0	0
$791$	11.6345	0.413677
$792$	0	0
$793$	−17.2333	−0.611974
$794$	0	0
$795$	81.0667	2.87514
$796$	0	0
$797$	42.1734	1.49386	0.746929	−	0.664904i	$-0.231528\pi$
$797$	42.1734	1.49386	0.746929	+	0.664904i	$0.231528\pi$
$798$	0	0
$799$	8.61612	0.304816
$800$	0	0
$801$	81.2332	2.87023
$802$	0	0
$803$	−9.30012	−0.328194
$804$	0	0
$805$	−5.26404	−0.185533
$806$	0	0
$807$	−5.09455	−0.179337
$808$	0	0
$809$	33.6378	1.18264	0.591320	−	0.806437i	$-0.298607\pi$
$809$	33.6378	1.18264	0.591320	+	0.806437i	$0.298607\pi$
$810$	0	0
$811$	28.5082	1.00106	0.500528	−	0.865720i	$-0.333139\pi$
$811$	28.5082	1.00106	0.500528	+	0.865720i	$0.333139\pi$
$812$	0	0
$813$	−95.9670	−3.36571
$814$	0	0
$815$	27.7553	0.972226
$816$	0	0
$817$	−0.287511	−0.0100587
$818$	0	0
$819$	−15.5898	−0.544753
$820$	0	0
$821$	−55.8856	−1.95042	−0.975210	−	0.221282i	$-0.928976\pi$
$821$	−55.8856	−1.95042	−0.975210	+	0.221282i	$0.928976\pi$
$822$	0	0
$823$	−9.35542	−0.326109	−0.163055	−	0.986617i	$-0.552135\pi$
$823$	−9.35542	−0.326109	−0.163055	+	0.986617i	$0.552135\pi$
$824$	0	0
$825$	−29.2984	−1.02004
$826$	0	0
$827$	45.6961	1.58901	0.794504	−	0.607259i	$-0.207731\pi$
$827$	45.6961	1.58901	0.794504	+	0.607259i	$0.207731\pi$
$828$	0	0
$829$	7.18848	0.249666	0.124833	−	0.992178i	$-0.460160\pi$
$829$	7.18848	0.249666	0.124833	+	0.992178i	$0.460160\pi$
$830$	0	0
$831$	−17.4068	−0.603834
$832$	0	0
$833$	−5.65390	−0.195896
$834$	0	0
$835$	−30.1671	−1.04397
$836$	0	0
$837$	−92.8758	−3.21026
$838$	0	0
$839$	−1.28514	−0.0443681	−0.0221841	−	0.999754i	$-0.507062\pi$
$839$	−1.28514	−0.0443681	−0.0221841	+	0.999754i	$0.507062\pi$
$840$	0	0
$841$	−3.56255	−0.122846
$842$	0	0
$843$	7.98085	0.274875
$844$	0	0
$845$	35.2048	1.21108
$846$	0	0
$847$	−9.83035	−0.337775
$848$	0	0
$849$	21.7470	0.746356
$850$	0	0
$851$	−10.5449	−0.361474
$852$	0	0
$853$	−13.5876	−0.465232	−0.232616	−	0.972569i	$-0.574729\pi$
$853$	−13.5876	−0.465232	−0.232616	+	0.972569i	$0.574729\pi$
$854$	0	0
$855$	−1.97272	−0.0674657
$856$	0	0
$857$	−35.7439	−1.22099	−0.610494	−	0.792021i	$-0.709029\pi$
$857$	−35.7439	−1.22099	−0.610494	+	0.792021i	$0.709029\pi$
$858$	0	0
$859$	−35.4359	−1.20906	−0.604529	−	0.796583i	$-0.706639\pi$
$859$	−35.4359	−1.20906	−0.604529	+	0.796583i	$0.706639\pi$
$860$	0	0
$861$	−25.6718	−0.874893
$862$	0	0
$863$	−18.5689	−0.632092	−0.316046	−	0.948744i	$-0.602355\pi$
$863$	−18.5689	−0.632092	−0.316046	+	0.948744i	$0.602355\pi$
$864$	0	0
$865$	41.5276	1.41198
$866$	0	0
$867$	51.2126	1.73927
$868$	0	0
$869$	−8.48370	−0.287790
$870$	0	0
$871$	12.4774	0.422781
$872$	0	0
$873$	4.22876	0.143122
$874$	0	0
$875$	−10.4838	−0.354415
$876$	0	0
$877$	−56.4896	−1.90752	−0.953759	−	0.300572i	$-0.902823\pi$
$877$	−56.4896	−1.90752	−0.953759	+	0.300572i	$0.902823\pi$
$878$	0	0
$879$	4.98103	0.168006
$880$	0	0
$881$	−41.1798	−1.38738	−0.693691	−	0.720273i	$-0.744017\pi$
$881$	−41.1798	−1.38738	−0.693691	+	0.720273i	$0.744017\pi$
$882$	0	0
$883$	−38.1836	−1.28498	−0.642490	−	0.766294i	$-0.722099\pi$
$883$	−38.1836	−1.28498	−0.642490	+	0.766294i	$0.722099\pi$
$884$	0	0
$885$	93.7197	3.15035
$886$	0	0
$887$	−47.5720	−1.59731	−0.798656	−	0.601788i	$-0.794455\pi$
$887$	−47.5720	−1.59731	−0.798656	+	0.601788i	$0.794455\pi$
$888$	0	0
$889$	−11.7630	−0.394519
$890$	0	0
$891$	−44.0341	−1.47520
$892$	0	0
$893$	−0.0960494	−0.00321417
$894$	0	0
$895$	69.9667	2.33873
$896$	0	0
$897$	−8.97183	−0.299561
$898$	0	0
$899$	23.9798	0.799773
$900$	0	0
$901$	37.2696	1.24163
$902$	0	0
$903$	−15.6091	−0.519438
$904$	0	0
$905$	−41.2561	−1.37140
$906$	0	0
$907$	33.0681	1.09801	0.549004	−	0.835820i	$-0.315007\pi$
$907$	33.0681	1.09801	0.549004	+	0.835820i	$0.315007\pi$
$908$	0	0
$909$	−56.1338	−1.86184
$910$	0	0
$911$	36.5816	1.21200	0.606002	−	0.795463i	$-0.292772\pi$
$911$	36.5816	1.21200	0.606002	+	0.795463i	$0.292772\pi$
$912$	0	0
$913$	11.3771	0.376528
$914$	0	0
$915$	−118.391	−3.91387
$916$	0	0
$917$	15.6871	0.518033
$918$	0	0
$919$	−16.2676	−0.536618	−0.268309	−	0.963333i	$-0.586465\pi$
$919$	−16.2676	−0.536618	−0.268309	+	0.963333i	$0.586465\pi$
$920$	0	0
$921$	−25.4612	−0.838976
$922$	0	0
$923$	11.0843	0.364845
$924$	0	0
$925$	−56.9985	−1.87410
$926$	0	0
$927$	37.1742	1.22096
$928$	0	0
$929$	11.1820	0.366871	0.183435	−	0.983032i	$-0.441278\pi$
$929$	11.1820	0.366871	0.183435	+	0.983032i	$0.441278\pi$
$930$	0	0
$931$	0.0630276	0.00206565
$932$	0	0
$933$	−29.9291	−0.979835
$934$	0	0
$935$	−21.9764	−0.718706
$936$	0	0
$937$	37.5933	1.22812	0.614059	−	0.789260i	$-0.289536\pi$
$937$	37.5933	1.22812	0.614059	+	0.789260i	$0.289536\pi$
$938$	0	0
$939$	−30.0104	−0.979352
$940$	0	0
$941$	−19.9354	−0.649876	−0.324938	−	0.945735i	$-0.605344\pi$
$941$	−19.9354	−0.649876	−0.324938	+	0.945735i	$0.605344\pi$
$942$	0	0
$943$	−10.9886	−0.357837
$944$	0	0
$945$	−70.2059	−2.28380
$946$	0	0
$947$	−11.2482	−0.365518	−0.182759	−	0.983158i	$-0.558503\pi$
$947$	−11.2482	−0.365518	−0.182759	+	0.983158i	$0.558503\pi$
$948$	0	0
$949$	−15.3939	−0.499706
$950$	0	0
$951$	−73.5774	−2.38591
$952$	0	0
$953$	−29.7695	−0.964329	−0.482164	−	0.876081i	$-0.660149\pi$
$953$	−29.7695	−0.964329	−0.482164	+	0.876081i	$0.660149\pi$
$954$	0	0
$955$	59.6249	1.92942
$956$	0	0
$957$	18.6647	0.603343
$958$	0	0
$959$	−7.47159	−0.241270
$960$	0	0
$961$	−8.39424	−0.270782
$962$	0	0
$963$	−22.2747	−0.717792
$964$	0	0
$965$	4.75348	0.153020
$966$	0	0
$967$	55.3211	1.77901	0.889504	−	0.456927i	$-0.151050\pi$
$967$	55.3211	1.77901	0.889504	+	0.456927i	$0.151050\pi$
$968$	0	0
$969$	−1.21936	−0.0391716
$970$	0	0
$971$	−11.9746	−0.384284	−0.192142	−	0.981367i	$-0.561543\pi$
$971$	−11.9746	−0.384284	−0.192142	+	0.981367i	$0.561543\pi$
$972$	0	0
$973$	−0.352269	−0.0112932
$974$	0	0
$975$	−48.4956	−1.55310
$976$	0	0
$977$	−39.7381	−1.27133	−0.635667	−	0.771963i	$-0.719275\pi$
$977$	−39.7381	−1.27133	−0.635667	+	0.771963i	$0.719275\pi$
$978$	0	0
$979$	−10.0881	−0.322416
$980$	0	0
$981$	135.664	4.33142
$982$	0	0
$983$	5.03380	0.160553	0.0802766	−	0.996773i	$-0.474420\pi$
$983$	5.03380	0.160553	0.0802766	+	0.996773i	$0.474420\pi$
$984$	0	0
$985$	−58.7449	−1.87177
$986$	0	0
$987$	−5.21457	−0.165982
$988$	0	0
$989$	−6.68132	−0.212454
$990$	0	0
$991$	28.5560	0.907110	0.453555	−	0.891228i	$-0.350156\pi$
$991$	28.5560	0.907110	0.453555	+	0.891228i	$0.350156\pi$
$992$	0	0
$993$	−23.8462	−0.756737
$994$	0	0
$995$	−93.9558	−2.97860
$996$	0	0
$997$	−26.4015	−0.836144	−0.418072	−	0.908414i	$-0.637294\pi$
$997$	−26.4015	−0.836144	−0.418072	+	0.908414i	$0.637294\pi$
$998$	0	0
$999$	−140.636	−4.44952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bb.1.8		8
4.3	odd	2		7168.2.a.ba.1.1		8
8.3	odd	2		7168.2.a.be.1.8		8
8.5	even	2		7168.2.a.bf.1.1		8
32.3	odd	8		1792.2.m.f.1345.1	yes	16
32.5	even	8		1792.2.m.e.449.1	✓	16
32.11	odd	8		1792.2.m.f.449.1	yes	16
32.13	even	8		1792.2.m.e.1345.1	yes	16
32.19	odd	8		1792.2.m.g.1345.8	yes	16
32.21	even	8		1792.2.m.h.449.8	yes	16
32.27	odd	8		1792.2.m.g.449.8	yes	16
32.29	even	8		1792.2.m.h.1345.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.1	✓	16	32.5	even	8
1792.2.m.e.1345.1	yes	16	32.13	even	8
1792.2.m.f.449.1	yes	16	32.11	odd	8
1792.2.m.f.1345.1	yes	16	32.3	odd	8
1792.2.m.g.449.8	yes	16	32.27	odd	8
1792.2.m.g.1345.8	yes	16	32.19	odd	8
1792.2.m.h.449.8	yes	16	32.21	even	8
1792.2.m.h.1345.8	yes	16	32.29	even	8
7168.2.a.ba.1.1		8	4.3	odd	2
7168.2.a.bb.1.8		8	1.1	even	1	trivial
7168.2.a.be.1.8		8	8.3	odd	2
7168.2.a.bf.1.1		8	8.5	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q+3.42180 q^{3} -3.59402 q^{5} +1.00000 q^{7} +8.70871 q^{9} +O(q^{10})\)	\(q+3.42180 q^{3} -3.59402 q^{5} +1.00000 q^{7} +8.70871 q^{9} -1.08150 q^{11} -1.79014 q^{13} -12.2980 q^{15} -5.65390 q^{17} +0.0630276 q^{19} +3.42180 q^{21} +1.46467 q^{23} +7.91700 q^{25} +19.5341 q^{27} -5.04356 q^{29} -4.75455 q^{31} -3.70069 q^{33} -3.59402 q^{35} -7.19951 q^{37} -6.12551 q^{39} -7.50243 q^{41} -4.56166 q^{43} -31.2993 q^{45} -1.52393 q^{47} +1.00000 q^{49} -19.3465 q^{51} -6.59185 q^{53} +3.88695 q^{55} +0.215668 q^{57} -7.62071 q^{59} +9.62680 q^{61} +8.70871 q^{63} +6.43381 q^{65} -6.97006 q^{67} +5.01180 q^{69} -6.19187 q^{71} +8.59924 q^{73} +27.0904 q^{75} -1.08150 q^{77} +7.84435 q^{79} +40.7155 q^{81} -10.5197 q^{83} +20.3202 q^{85} -17.2580 q^{87} +9.32780 q^{89} -1.79014 q^{91} -16.2691 q^{93} -0.226523 q^{95} +0.485578 q^{97} -9.41852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9	\( 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9} - 12 q^{11} - 20 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} + 12 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 52 q^{45} + 20 q^{47} + 8 q^{49} - 32 q^{51} - 40 q^{53} + 24 q^{55} - 4 q^{57} - 4 q^{59} + 8 q^{61} + 12 q^{63} + 36 q^{65} - 28 q^{67} - 4 q^{69} - 16 q^{71} + 16 q^{73} + 28 q^{75} - 12 q^{77} + 20 q^{81} + 8 q^{83} - 16 q^{85} + 20 q^{87} + 16 q^{89} - 20 q^{91} - 16 q^{93} - 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9 - 12 * q^11 - 20 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 12 * q^25 + 12 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 - 52 * q^45 + 20 * q^47 + 8 * q^49 - 32 * q^51 - 40 * q^53 + 24 * q^55 - 4 * q^57 - 4 * q^59 + 8 * q^61 + 12 * q^63 + 36 * q^65 - 28 * q^67 - 4 * q^69 - 16 * q^71 + 16 * q^73 + 28 * q^75 - 12 * q^77 + 20 * q^81 + 8 * q^83 - 16 * q^85 + 20 * q^87 + 16 * q^89 - 20 * q^91 - 16 * q^93 - 40 * q^95 - 36 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more