LMFDB - Embedded newform 8688.2.a.bf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8688,2,Mod(1,8688)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8688.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8688, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8688 = 2^{4} \cdot 3 \cdot 181 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8688.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,8,0,5,0,3,0,8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.3740292761$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 $ x^8 - 3x^7 - 6x^6 + 21x^5 + 5x^4 - 35x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 543)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.48495$ of defining polynomial
Character	$\chi$	$=$	8688.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.370811 q^{5} +1.96328 q^{7} +1.00000 q^{9} -4.04845 q^{11} +4.06041 q^{13} +0.370811 q^{15} +3.62919 q^{17} -5.64676 q^{19} +1.96328 q^{21} +8.92443 q^{23} -4.86250 q^{25} +1.00000 q^{27} -6.74722 q^{29} +8.80518 q^{31} -4.04845 q^{33} +0.728004 q^{35} -0.921480 q^{37} +4.06041 q^{39} -0.316916 q^{41} +2.22243 q^{43} +0.370811 q^{45} +4.93727 q^{47} -3.14555 q^{49} +3.62919 q^{51} +14.1879 q^{53} -1.50121 q^{55} -5.64676 q^{57} +0.537828 q^{59} -2.35027 q^{61} +1.96328 q^{63} +1.50564 q^{65} -10.6251 q^{67} +8.92443 q^{69} -1.33418 q^{71} +8.35531 q^{73} -4.86250 q^{75} -7.94822 q^{77} +15.5212 q^{79} +1.00000 q^{81} -9.55110 q^{83} +1.34574 q^{85} -6.74722 q^{87} +3.36562 q^{89} +7.97171 q^{91} +8.80518 q^{93} -2.09388 q^{95} +12.3388 q^{97} -4.04845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 5 q^{5} + 3 q^{7} + 8 q^{9} - 4 q^{11} + 13 q^{13} + 5 q^{15} + 27 q^{17} + 10 q^{19} + 3 q^{21} - 3 q^{23} + 13 q^{25} + 8 q^{27} + 16 q^{29} + q^{31} - 4 q^{33} - 14 q^{35} + 3 q^{37}+ \cdots - 4 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + 5 * q^5 + 3 * q^7 + 8 * q^9 - 4 * q^11 + 13 * q^13 + 5 * q^15 + 27 * q^17 + 10 * q^19 + 3 * q^21 - 3 * q^23 + 13 * q^25 + 8 * q^27 + 16 * q^29 + q^31 - 4 * q^33 - 14 * q^35 + 3 * q^37 + 13 * q^39 + 20 * q^41 + 4 * q^43 + 5 * q^45 - 12 * q^47 + 5 * q^49 + 27 * q^51 + 23 * q^53 + 13 * q^55 + 10 * q^57 - 7 * q^59 + q^61 + 3 * q^63 + 10 * q^65 + 25 * q^67 - 3 * q^69 - 11 * q^71 + 23 * q^73 + 13 * q^75 + 20 * q^77 + 26 * q^79 + 8 * q^81 - 16 * q^83 - 33 * q^85 + 16 * q^87 + 14 * q^89 + 35 * q^91 + q^93 - 10 * q^95 + 28 * q^97 - 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.370811	0.165832	0.0829158	−	0.996557i	$-0.473577\pi$
$5$	0.370811	0.165832	0.0829158	+	0.996557i	$0.473577\pi$
$6$	0	0
$7$	1.96328	0.742048	0.371024	−	0.928623i	$-0.379007\pi$
$7$	1.96328	0.742048	0.371024	+	0.928623i	$0.379007\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.04845	−1.22065	−0.610326	−	0.792150i	$-0.708962\pi$
$11$	−4.04845	−1.22065	−0.610326	+	0.792150i	$0.708962\pi$
$12$	0	0
$13$	4.06041	1.12616	0.563078	−	0.826404i	$-0.309617\pi$
$13$	4.06041	1.12616	0.563078	+	0.826404i	$0.309617\pi$
$14$	0	0
$15$	0.370811	0.0957429
$16$	0	0
$17$	3.62919	0.880208	0.440104	−	0.897947i	$-0.354942\pi$
$17$	3.62919	0.880208	0.440104	+	0.897947i	$0.354942\pi$
$18$	0	0
$19$	−5.64676	−1.29545	−0.647727	−	0.761872i	$-0.724280\pi$
$19$	−5.64676	−1.29545	−0.647727	+	0.761872i	$0.724280\pi$
$20$	0	0
$21$	1.96328	0.428422
$22$	0	0
$23$	8.92443	1.86087	0.930436	−	0.366454i	$-0.119428\pi$
$23$	8.92443	1.86087	0.930436	+	0.366454i	$0.119428\pi$
$24$	0	0
$25$	−4.86250	−0.972500
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.74722	−1.25293	−0.626464	−	0.779451i	$-0.715498\pi$
$29$	−6.74722	−1.25293	−0.626464	+	0.779451i	$0.715498\pi$
$30$	0	0
$31$	8.80518	1.58146	0.790729	−	0.612166i	$-0.209702\pi$
$31$	8.80518	1.58146	0.790729	+	0.612166i	$0.209702\pi$
$32$	0	0
$33$	−4.04845	−0.704744
$34$	0	0
$35$	0.728004	0.123055
$36$	0	0
$37$	−0.921480	−0.151490	−0.0757452	−	0.997127i	$-0.524134\pi$
$37$	−0.921480	−0.151490	−0.0757452	+	0.997127i	$0.524134\pi$
$38$	0	0
$39$	4.06041	0.650186
$40$	0	0
$41$	−0.316916	−0.0494939	−0.0247470	−	0.999694i	$-0.507878\pi$
$41$	−0.316916	−0.0494939	−0.0247470	+	0.999694i	$0.507878\pi$
$42$	0	0
$43$	2.22243	0.338917	0.169458	−	0.985537i	$-0.445798\pi$
$43$	2.22243	0.338917	0.169458	+	0.985537i	$0.445798\pi$
$44$	0	0
$45$	0.370811	0.0552772
$46$	0	0
$47$	4.93727	0.720174	0.360087	−	0.932919i	$-0.382747\pi$
$47$	4.93727	0.720174	0.360087	+	0.932919i	$0.382747\pi$
$48$	0	0
$49$	−3.14555	−0.449364
$50$	0	0
$51$	3.62919	0.508188
$52$	0	0
$53$	14.1879	1.94885	0.974426	−	0.224711i	$-0.0721437\pi$
$53$	14.1879	1.94885	0.974426	+	0.224711i	$0.0721437\pi$
$54$	0	0
$55$	−1.50121	−0.202423
$56$	0	0
$57$	−5.64676	−0.747931
$58$	0	0
$59$	0.537828	0.0700192	0.0350096	−	0.999387i	$-0.488854\pi$
$59$	0.537828	0.0700192	0.0350096	+	0.999387i	$0.488854\pi$
$60$	0	0
$61$	−2.35027	−0.300921	−0.150461	−	0.988616i	$-0.548076\pi$
$61$	−2.35027	−0.300921	−0.150461	+	0.988616i	$0.548076\pi$
$62$	0	0
$63$	1.96328	0.247349
$64$	0	0
$65$	1.50564	0.186752
$66$	0	0
$67$	−10.6251	−1.29806	−0.649032	−	0.760761i	$-0.724826\pi$
$67$	−10.6251	−1.29806	−0.649032	+	0.760761i	$0.724826\pi$
$68$	0	0
$69$	8.92443	1.07438
$70$	0	0
$71$	−1.33418	−0.158338	−0.0791691	−	0.996861i	$-0.525227\pi$
$71$	−1.33418	−0.158338	−0.0791691	+	0.996861i	$0.525227\pi$
$72$	0	0
$73$	8.35531	0.977915	0.488958	−	0.872308i	$-0.337377\pi$
$73$	8.35531	0.977915	0.488958	+	0.872308i	$0.337377\pi$
$74$	0	0
$75$	−4.86250	−0.561473
$76$	0	0
$77$	−7.94822	−0.905783
$78$	0	0
$79$	15.5212	1.74627	0.873137	−	0.487475i	$-0.162082\pi$
$79$	15.5212	1.74627	0.873137	+	0.487475i	$0.162082\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.55110	−1.04837	−0.524185	−	0.851604i	$-0.675630\pi$
$83$	−9.55110	−1.04837	−0.524185	+	0.851604i	$0.675630\pi$
$84$	0	0
$85$	1.34574	0.145966
$86$	0	0
$87$	−6.74722	−0.723378
$88$	0	0
$89$	3.36562	0.356755	0.178378	−	0.983962i	$-0.442915\pi$
$89$	3.36562	0.356755	0.178378	+	0.983962i	$0.442915\pi$
$90$	0	0
$91$	7.97171	0.835662
$92$	0	0
$93$	8.80518	0.913055
$94$	0	0
$95$	−2.09388	−0.214827
$96$	0	0
$97$	12.3388	1.25281	0.626407	−	0.779496i	$-0.284525\pi$
$97$	12.3388	1.25281	0.626407	+	0.779496i	$0.284525\pi$
$98$	0	0
$99$	−4.04845	−0.406884
$100$	0	0
$101$	1.21813	0.121208	0.0606042	−	0.998162i	$-0.480697\pi$
$101$	1.21813	0.121208	0.0606042	+	0.998162i	$0.480697\pi$
$102$	0	0
$103$	1.05825	0.104272	0.0521361	−	0.998640i	$-0.483397\pi$
$103$	1.05825	0.104272	0.0521361	+	0.998640i	$0.483397\pi$
$104$	0	0
$105$	0.728004	0.0710459
$106$	0	0
$107$	3.95524	0.382368	0.191184	−	0.981554i	$-0.438767\pi$
$107$	3.95524	0.382368	0.191184	+	0.981554i	$0.438767\pi$
$108$	0	0
$109$	−1.34588	−0.128912	−0.0644562	−	0.997921i	$-0.520531\pi$
$109$	−1.34588	−0.128912	−0.0644562	+	0.997921i	$0.520531\pi$
$110$	0	0
$111$	−0.921480	−0.0874630
$112$	0	0
$113$	−2.24544	−0.211233	−0.105617	−	0.994407i	$-0.533682\pi$
$113$	−2.24544	−0.211233	−0.105617	+	0.994407i	$0.533682\pi$
$114$	0	0
$115$	3.30927	0.308591
$116$	0	0
$117$	4.06041	0.375385
$118$	0	0
$119$	7.12510	0.653157
$120$	0	0
$121$	5.38992	0.489993
$122$	0	0
$123$	−0.316916	−0.0285753
$124$	0	0
$125$	−3.65712	−0.327103
$126$	0	0
$127$	17.6937	1.57006	0.785032	−	0.619455i	$-0.212646\pi$
$127$	17.6937	1.57006	0.785032	+	0.619455i	$0.212646\pi$
$128$	0	0
$129$	2.22243	0.195674
$130$	0	0
$131$	−13.4044	−1.17115	−0.585575	−	0.810618i	$-0.699131\pi$
$131$	−13.4044	−1.17115	−0.585575	+	0.810618i	$0.699131\pi$
$132$	0	0
$133$	−11.0861	−0.961290
$134$	0	0
$135$	0.370811	0.0319143
$136$	0	0
$137$	5.68019	0.485292	0.242646	−	0.970115i	$-0.421985\pi$
$137$	5.68019	0.485292	0.242646	+	0.970115i	$0.421985\pi$
$138$	0	0
$139$	17.0855	1.44917	0.724585	−	0.689185i	$-0.242031\pi$
$139$	17.0855	1.44917	0.724585	+	0.689185i	$0.242031\pi$
$140$	0	0
$141$	4.93727	0.415793
$142$	0	0
$143$	−16.4384	−1.37465
$144$	0	0
$145$	−2.50194	−0.207775
$146$	0	0
$147$	−3.14555	−0.259440
$148$	0	0
$149$	−12.9363	−1.05978	−0.529890	−	0.848067i	$-0.677767\pi$
$149$	−12.9363	−1.05978	−0.529890	+	0.848067i	$0.677767\pi$
$150$	0	0
$151$	10.7174	0.872173	0.436087	−	0.899905i	$-0.356364\pi$
$151$	10.7174	0.872173	0.436087	+	0.899905i	$0.356364\pi$
$152$	0	0
$153$	3.62919	0.293403
$154$	0	0
$155$	3.26506	0.262256
$156$	0	0
$157$	−3.06175	−0.244354	−0.122177	−	0.992508i	$-0.538988\pi$
$157$	−3.06175	−0.244354	−0.122177	+	0.992508i	$0.538988\pi$
$158$	0	0
$159$	14.1879	1.12517
$160$	0	0
$161$	17.5211	1.38086
$162$	0	0
$163$	−5.50865	−0.431471	−0.215735	−	0.976452i	$-0.569215\pi$
$163$	−5.50865	−0.431471	−0.215735	+	0.976452i	$0.569215\pi$
$164$	0	0
$165$	−1.50121	−0.116869
$166$	0	0
$167$	−20.2718	−1.56868	−0.784341	−	0.620330i	$-0.786999\pi$
$167$	−20.2718	−1.56868	−0.784341	+	0.620330i	$0.786999\pi$
$168$	0	0
$169$	3.48696	0.268227
$170$	0	0
$171$	−5.64676	−0.431818
$172$	0	0
$173$	19.9597	1.51751	0.758753	−	0.651378i	$-0.225809\pi$
$173$	19.9597	1.51751	0.758753	+	0.651378i	$0.225809\pi$
$174$	0	0
$175$	−9.54643	−0.721642
$176$	0	0
$177$	0.537828	0.0404256
$178$	0	0
$179$	2.45329	0.183367	0.0916837	−	0.995788i	$-0.470775\pi$
$179$	2.45329	0.183367	0.0916837	+	0.995788i	$0.470775\pi$
$180$	0	0
$181$	1.00000	0.0743294
$181$	1.00000	0.0743294
$182$	0	0
$183$	−2.35027	−0.173737
$184$	0	0
$185$	−0.341695	−0.0251219
$186$	0	0
$187$	−14.6926	−1.07443
$188$	0	0
$189$	1.96328	0.142807
$190$	0	0
$191$	−26.6172	−1.92595	−0.962975	−	0.269590i	$-0.913112\pi$
$191$	−26.6172	−1.92595	−0.962975	+	0.269590i	$0.913112\pi$
$192$	0	0
$193$	9.38507	0.675552	0.337776	−	0.941226i	$-0.390325\pi$
$193$	9.38507	0.675552	0.337776	+	0.941226i	$0.390325\pi$
$194$	0	0
$195$	1.50564	0.107821
$196$	0	0
$197$	−2.17420	−0.154905	−0.0774527	−	0.996996i	$-0.524679\pi$
$197$	−2.17420	−0.154905	−0.0774527	+	0.996996i	$0.524679\pi$
$198$	0	0
$199$	19.5191	1.38367	0.691835	−	0.722055i	$-0.256802\pi$
$199$	19.5191	1.38367	0.691835	+	0.722055i	$0.256802\pi$
$200$	0	0
$201$	−10.6251	−0.749438
$202$	0	0
$203$	−13.2467	−0.929733
$204$	0	0
$205$	−0.117516	−0.00820766
$206$	0	0
$207$	8.92443	0.620291
$208$	0	0
$209$	22.8606	1.58130
$210$	0	0
$211$	8.94618	0.615881	0.307940	−	0.951406i	$-0.400360\pi$
$211$	8.94618	0.615881	0.307940	+	0.951406i	$0.400360\pi$
$212$	0	0
$213$	−1.33418	−0.0914166
$214$	0	0
$215$	0.824100	0.0562031
$216$	0	0
$217$	17.2870	1.17352
$218$	0	0
$219$	8.35531	0.564600
$220$	0	0
$221$	14.7360	0.991251
$222$	0	0
$223$	17.4305	1.16723	0.583615	−	0.812030i	$-0.301638\pi$
$223$	17.4305	1.16723	0.583615	+	0.812030i	$0.301638\pi$
$224$	0	0
$225$	−4.86250	−0.324167
$226$	0	0
$227$	−5.26176	−0.349235	−0.174618	−	0.984636i	$-0.555869\pi$
$227$	−5.26176	−0.349235	−0.174618	+	0.984636i	$0.555869\pi$
$228$	0	0
$229$	24.2475	1.60232	0.801159	−	0.598451i	$-0.204217\pi$
$229$	24.2475	1.60232	0.801159	+	0.598451i	$0.204217\pi$
$230$	0	0
$231$	−7.94822	−0.522954
$232$	0	0
$233$	12.1328	0.794845	0.397422	−	0.917636i	$-0.369905\pi$
$233$	12.1328	0.794845	0.397422	+	0.917636i	$0.369905\pi$
$234$	0	0
$235$	1.83079	0.119428
$236$	0	0
$237$	15.5212	1.00821
$238$	0	0
$239$	−15.1685	−0.981172	−0.490586	−	0.871393i	$-0.663217\pi$
$239$	−15.1685	−0.981172	−0.490586	+	0.871393i	$0.663217\pi$
$240$	0	0
$241$	−18.2872	−1.17798	−0.588990	−	0.808141i	$-0.700474\pi$
$241$	−18.2872	−1.17798	−0.588990	+	0.808141i	$0.700474\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.16640	−0.0745188
$246$	0	0
$247$	−22.9282	−1.45888
$248$	0	0
$249$	−9.55110	−0.605277
$250$	0	0
$251$	1.81307	0.114440	0.0572199	−	0.998362i	$-0.481776\pi$
$251$	1.81307	0.114440	0.0572199	+	0.998362i	$0.481776\pi$
$252$	0	0
$253$	−36.1301	−2.27148
$254$	0	0
$255$	1.34574	0.0842736
$256$	0	0
$257$	−6.67435	−0.416335	−0.208167	−	0.978093i	$-0.566750\pi$
$257$	−6.67435	−0.416335	−0.208167	+	0.978093i	$0.566750\pi$
$258$	0	0
$259$	−1.80912	−0.112413
$260$	0	0
$261$	−6.74722	−0.417642
$262$	0	0
$263$	21.0546	1.29828	0.649140	−	0.760669i	$-0.275129\pi$
$263$	21.0546	1.29828	0.649140	+	0.760669i	$0.275129\pi$
$264$	0	0
$265$	5.26101	0.323181
$266$	0	0
$267$	3.36562	0.205973
$268$	0	0
$269$	5.45589	0.332652	0.166326	−	0.986071i	$-0.446810\pi$
$269$	5.45589	0.332652	0.166326	+	0.986071i	$0.446810\pi$
$270$	0	0
$271$	4.00206	0.243108	0.121554	−	0.992585i	$-0.461212\pi$
$271$	4.00206	0.243108	0.121554	+	0.992585i	$0.461212\pi$
$272$	0	0
$273$	7.97171	0.482470
$274$	0	0
$275$	19.6856	1.18708
$276$	0	0
$277$	22.5668	1.35591	0.677954	−	0.735105i	$-0.262867\pi$
$277$	22.5668	1.35591	0.677954	+	0.735105i	$0.262867\pi$
$278$	0	0
$279$	8.80518	0.527153
$280$	0	0
$281$	−9.87322	−0.588987	−0.294494	−	0.955653i	$-0.595151\pi$
$281$	−9.87322	−0.588987	−0.294494	+	0.955653i	$0.595151\pi$
$282$	0	0
$283$	29.5858	1.75869	0.879347	−	0.476181i	$-0.157979\pi$
$283$	29.5858	1.75869	0.879347	+	0.476181i	$0.157979\pi$
$284$	0	0
$285$	−2.09388	−0.124031
$286$	0	0
$287$	−0.622193	−0.0367269
$288$	0	0
$289$	−3.82899	−0.225234
$290$	0	0
$291$	12.3388	0.723312
$292$	0	0
$293$	−16.2215	−0.947668	−0.473834	−	0.880614i	$-0.657130\pi$
$293$	−16.2215	−0.947668	−0.473834	+	0.880614i	$0.657130\pi$
$294$	0	0
$295$	0.199432	0.0116114
$296$	0	0
$297$	−4.04845	−0.234915
$298$	0	0
$299$	36.2369	2.09563
$300$	0	0
$301$	4.36324	0.251493
$302$	0	0
$303$	1.21813	0.0699798
$304$	0	0
$305$	−0.871506	−0.0499023
$306$	0	0
$307$	−13.1246	−0.749063	−0.374531	−	0.927214i	$-0.622196\pi$
$307$	−13.1246	−0.749063	−0.374531	+	0.927214i	$0.622196\pi$
$308$	0	0
$309$	1.05825	0.0602016
$310$	0	0
$311$	−18.3176	−1.03869	−0.519347	−	0.854563i	$-0.673825\pi$
$311$	−18.3176	−1.03869	−0.519347	+	0.854563i	$0.673825\pi$
$312$	0	0
$313$	25.8600	1.46170	0.730848	−	0.682540i	$-0.239125\pi$
$313$	25.8600	1.46170	0.730848	+	0.682540i	$0.239125\pi$
$314$	0	0
$315$	0.728004	0.0410184
$316$	0	0
$317$	18.3141	1.02862	0.514312	−	0.857603i	$-0.328047\pi$
$317$	18.3141	1.02862	0.514312	+	0.857603i	$0.328047\pi$
$318$	0	0
$319$	27.3158	1.52939
$320$	0	0
$321$	3.95524	0.220760
$322$	0	0
$323$	−20.4931	−1.14027
$324$	0	0
$325$	−19.7438	−1.09519
$326$	0	0
$327$	−1.34588	−0.0744276
$328$	0	0
$329$	9.69321	0.534404
$330$	0	0
$331$	−14.6270	−0.803971	−0.401985	−	0.915646i	$-0.631680\pi$
$331$	−14.6270	−0.803971	−0.401985	+	0.915646i	$0.631680\pi$
$332$	0	0
$333$	−0.921480	−0.0504968
$334$	0	0
$335$	−3.93991	−0.215260
$336$	0	0
$337$	−5.19389	−0.282929	−0.141465	−	0.989943i	$-0.545181\pi$
$337$	−5.19389	−0.282929	−0.141465	+	0.989943i	$0.545181\pi$
$338$	0	0
$339$	−2.24544	−0.121956
$340$	0	0
$341$	−35.6473	−1.93041
$342$	0	0
$343$	−19.9185	−1.07550
$344$	0	0
$345$	3.30927	0.178165
$346$	0	0
$347$	−18.0979	−0.971545	−0.485773	−	0.874085i	$-0.661462\pi$
$347$	−18.0979	−0.971545	−0.485773	+	0.874085i	$0.661462\pi$
$348$	0	0
$349$	21.5687	1.15455	0.577274	−	0.816551i	$-0.304117\pi$
$349$	21.5687	1.15455	0.577274	+	0.816551i	$0.304117\pi$
$350$	0	0
$351$	4.06041	0.216729
$352$	0	0
$353$	−12.6910	−0.675472	−0.337736	−	0.941241i	$-0.609661\pi$
$353$	−12.6910	−0.675472	−0.337736	+	0.941241i	$0.609661\pi$
$354$	0	0
$355$	−0.494729	−0.0262575
$356$	0	0
$357$	7.12510	0.377100
$358$	0	0
$359$	2.09266	0.110446	0.0552231	−	0.998474i	$-0.482413\pi$
$359$	2.09266	0.110446	0.0552231	+	0.998474i	$0.482413\pi$
$360$	0	0
$361$	12.8858	0.678203
$362$	0	0
$363$	5.38992	0.282897
$364$	0	0
$365$	3.09824	0.162169
$366$	0	0
$367$	−27.7795	−1.45008	−0.725039	−	0.688708i	$-0.758178\pi$
$367$	−27.7795	−1.45008	−0.725039	+	0.688708i	$0.758178\pi$
$368$	0	0
$369$	−0.316916	−0.0164980
$370$	0	0
$371$	27.8547	1.44614
$372$	0	0
$373$	−19.1864	−0.993433	−0.496716	−	0.867913i	$-0.665461\pi$
$373$	−19.1864	−0.993433	−0.496716	+	0.867913i	$0.665461\pi$
$374$	0	0
$375$	−3.65712	−0.188853
$376$	0	0
$377$	−27.3965	−1.41099
$378$	0	0
$379$	1.30800	0.0671872	0.0335936	−	0.999436i	$-0.489305\pi$
$379$	1.30800	0.0671872	0.0335936	+	0.999436i	$0.489305\pi$
$380$	0	0
$381$	17.6937	0.906477
$382$	0	0
$383$	−32.9356	−1.68293	−0.841466	−	0.540310i	$-0.818307\pi$
$383$	−32.9356	−1.68293	−0.841466	+	0.540310i	$0.818307\pi$
$384$	0	0
$385$	−2.94728	−0.150208
$386$	0	0
$387$	2.22243	0.112972
$388$	0	0
$389$	−14.6600	−0.743291	−0.371645	−	0.928375i	$-0.621206\pi$
$389$	−14.6600	−0.743291	−0.371645	+	0.928375i	$0.621206\pi$
$390$	0	0
$391$	32.3884	1.63795
$392$	0	0
$393$	−13.4044	−0.676164
$394$	0	0
$395$	5.75544	0.289587
$396$	0	0
$397$	7.01622	0.352134	0.176067	−	0.984378i	$-0.443662\pi$
$397$	7.01622	0.352134	0.176067	+	0.984378i	$0.443662\pi$
$398$	0	0
$399$	−11.0861	−0.555001
$400$	0	0
$401$	21.5827	1.07779	0.538895	−	0.842373i	$-0.318842\pi$
$401$	21.5827	1.07779	0.538895	+	0.842373i	$0.318842\pi$
$402$	0	0
$403$	35.7527	1.78097
$404$	0	0
$405$	0.370811	0.0184257
$406$	0	0
$407$	3.73056	0.184917
$408$	0	0
$409$	−5.30477	−0.262304	−0.131152	−	0.991362i	$-0.541868\pi$
$409$	−5.30477	−0.262304	−0.131152	+	0.991362i	$0.541868\pi$
$410$	0	0
$411$	5.68019	0.280183
$412$	0	0
$413$	1.05590	0.0519577
$414$	0	0
$415$	−3.54165	−0.173853
$416$	0	0
$417$	17.0855	0.836679
$418$	0	0
$419$	30.8119	1.50526	0.752630	−	0.658443i	$-0.228785\pi$
$419$	30.8119	1.50526	0.752630	+	0.658443i	$0.228785\pi$
$420$	0	0
$421$	22.1309	1.07860	0.539298	−	0.842115i	$-0.318690\pi$
$421$	22.1309	1.07860	0.539298	+	0.842115i	$0.318690\pi$
$422$	0	0
$423$	4.93727	0.240058
$424$	0	0
$425$	−17.6469	−0.856002
$426$	0	0
$427$	−4.61423	−0.223298
$428$	0	0
$429$	−16.4384	−0.793652
$430$	0	0
$431$	−21.0033	−1.01169	−0.505846	−	0.862624i	$-0.668820\pi$
$431$	−21.0033	−1.01169	−0.505846	+	0.862624i	$0.668820\pi$
$432$	0	0
$433$	−8.63171	−0.414814	−0.207407	−	0.978255i	$-0.566502\pi$
$433$	−8.63171	−0.414814	−0.207407	+	0.978255i	$0.566502\pi$
$434$	0	0
$435$	−2.50194	−0.119959
$436$	0	0
$437$	−50.3941	−2.41068
$438$	0	0
$439$	36.9023	1.76125	0.880624	−	0.473815i	$-0.157123\pi$
$439$	36.9023	1.76125	0.880624	+	0.473815i	$0.157123\pi$
$440$	0	0
$441$	−3.14555	−0.149788
$442$	0	0
$443$	38.0863	1.80953	0.904767	−	0.425906i	$-0.140045\pi$
$443$	38.0863	1.80953	0.904767	+	0.425906i	$0.140045\pi$
$444$	0	0
$445$	1.24801	0.0591613
$446$	0	0
$447$	−12.9363	−0.611864
$448$	0	0
$449$	10.5002	0.495534	0.247767	−	0.968820i	$-0.420303\pi$
$449$	10.5002	0.495534	0.247767	+	0.968820i	$0.420303\pi$
$450$	0	0
$451$	1.28302	0.0604149
$452$	0	0
$453$	10.7174	0.503549
$454$	0	0
$455$	2.95600	0.138579
$456$	0	0
$457$	−14.4330	−0.675147	−0.337574	−	0.941299i	$-0.609606\pi$
$457$	−14.4330	−0.675147	−0.337574	+	0.941299i	$0.609606\pi$
$458$	0	0
$459$	3.62919	0.169396
$460$	0	0
$461$	24.9671	1.16284	0.581418	−	0.813605i	$-0.302498\pi$
$461$	24.9671	1.16284	0.581418	+	0.813605i	$0.302498\pi$
$462$	0	0
$463$	−32.3333	−1.50266	−0.751329	−	0.659928i	$-0.770587\pi$
$463$	−32.3333	−1.50266	−0.751329	+	0.659928i	$0.770587\pi$
$464$	0	0
$465$	3.26506	0.151413
$466$	0	0
$467$	−32.8660	−1.52086	−0.760428	−	0.649422i	$-0.775011\pi$
$467$	−32.8660	−1.52086	−0.760428	+	0.649422i	$0.775011\pi$
$468$	0	0
$469$	−20.8600	−0.963227
$470$	0	0
$471$	−3.06175	−0.141078
$472$	0	0
$473$	−8.99738	−0.413700
$474$	0	0
$475$	27.4573	1.25983
$476$	0	0
$477$	14.1879	0.649617
$478$	0	0
$479$	−10.2308	−0.467455	−0.233728	−	0.972302i	$-0.575092\pi$
$479$	−10.2308	−0.467455	−0.233728	+	0.972302i	$0.575092\pi$
$480$	0	0
$481$	−3.74159	−0.170602
$482$	0	0
$483$	17.5211	0.797239
$484$	0	0
$485$	4.57535	0.207756
$486$	0	0
$487$	−7.93179	−0.359424	−0.179712	−	0.983719i	$-0.557517\pi$
$487$	−7.93179	−0.359424	−0.179712	+	0.983719i	$0.557517\pi$
$488$	0	0
$489$	−5.50865	−0.249110
$490$	0	0
$491$	−39.3721	−1.77684	−0.888420	−	0.459032i	$-0.848196\pi$
$491$	−39.3721	−1.77684	−0.888420	+	0.459032i	$0.848196\pi$
$492$	0	0
$493$	−24.4869	−1.10284
$494$	0	0
$495$	−1.50121	−0.0674743
$496$	0	0
$497$	−2.61937	−0.117495
$498$	0	0
$499$	−1.78195	−0.0797712	−0.0398856	−	0.999204i	$-0.512699\pi$
$499$	−1.78195	−0.0797712	−0.0398856	+	0.999204i	$0.512699\pi$
$500$	0	0
$501$	−20.2718	−0.905679
$502$	0	0
$503$	37.1023	1.65431	0.827156	−	0.561973i	$-0.189958\pi$
$503$	37.1023	1.65431	0.827156	+	0.561973i	$0.189958\pi$
$504$	0	0
$505$	0.451696	0.0201002
$506$	0	0
$507$	3.48696	0.154861
$508$	0	0
$509$	−18.0275	−0.799054	−0.399527	−	0.916722i	$-0.630826\pi$
$509$	−18.0275	−0.799054	−0.399527	+	0.916722i	$0.630826\pi$
$510$	0	0
$511$	16.4038	0.725660
$512$	0	0
$513$	−5.64676	−0.249310
$514$	0	0
$515$	0.392410	0.0172916
$516$	0	0
$517$	−19.9883	−0.879083
$518$	0	0
$519$	19.9597	0.876133
$520$	0	0
$521$	20.9237	0.916683	0.458342	−	0.888776i	$-0.348444\pi$
$521$	20.9237	0.916683	0.458342	+	0.888776i	$0.348444\pi$
$522$	0	0
$523$	−43.3377	−1.89502	−0.947512	−	0.319720i	$-0.896411\pi$
$523$	−43.3377	−1.89502	−0.947512	+	0.319720i	$0.896411\pi$
$524$	0	0
$525$	−9.54643	−0.416640
$526$	0	0
$527$	31.9557	1.39201
$528$	0	0
$529$	56.6455	2.46285
$530$	0	0
$531$	0.537828	0.0233397
$532$	0	0
$533$	−1.28681	−0.0557379
$534$	0	0
$535$	1.46665	0.0634087
$536$	0	0
$537$	2.45329	0.105867
$538$	0	0
$539$	12.7346	0.548517
$540$	0	0
$541$	29.2960	1.25953	0.629767	−	0.776784i	$-0.283150\pi$
$541$	29.2960	1.25953	0.629767	+	0.776784i	$0.283150\pi$
$542$	0	0
$543$	1.00000	0.0429141
$544$	0	0
$545$	−0.499068	−0.0213777
$546$	0	0
$547$	38.8865	1.66267	0.831334	−	0.555773i	$-0.187578\pi$
$547$	38.8865	1.66267	0.831334	+	0.555773i	$0.187578\pi$
$548$	0	0
$549$	−2.35027	−0.100307
$550$	0	0
$551$	38.0999	1.62311
$552$	0	0
$553$	30.4724	1.29582
$554$	0	0
$555$	−0.341695	−0.0145041
$556$	0	0
$557$	41.8529	1.77336	0.886682	−	0.462379i	$-0.153004\pi$
$557$	41.8529	1.77336	0.886682	+	0.462379i	$0.153004\pi$
$558$	0	0
$559$	9.02397	0.381673
$560$	0	0
$561$	−14.6926	−0.620321
$562$	0	0
$563$	2.68310	0.113079	0.0565396	−	0.998400i	$-0.481993\pi$
$563$	2.68310	0.113079	0.0565396	+	0.998400i	$0.481993\pi$
$564$	0	0
$565$	−0.832633	−0.0350291
$566$	0	0
$567$	1.96328	0.0824498
$568$	0	0
$569$	−9.17154	−0.384491	−0.192246	−	0.981347i	$-0.561577\pi$
$569$	−9.17154	−0.384491	−0.192246	+	0.981347i	$0.561577\pi$
$570$	0	0
$571$	26.6381	1.11477	0.557385	−	0.830254i	$-0.311805\pi$
$571$	26.6381	1.11477	0.557385	+	0.830254i	$0.311805\pi$
$572$	0	0
$573$	−26.6172	−1.11195
$574$	0	0
$575$	−43.3950	−1.80970
$576$	0	0
$577$	−10.2555	−0.426941	−0.213470	−	0.976950i	$-0.568477\pi$
$577$	−10.2555	−0.426941	−0.213470	+	0.976950i	$0.568477\pi$
$578$	0	0
$579$	9.38507	0.390030
$580$	0	0
$581$	−18.7515	−0.777941
$582$	0	0
$583$	−57.4388	−2.37887
$584$	0	0
$585$	1.50564	0.0622507
$586$	0	0
$587$	12.5974	0.519949	0.259974	−	0.965616i	$-0.416286\pi$
$587$	12.5974	0.519949	0.259974	+	0.965616i	$0.416286\pi$
$588$	0	0
$589$	−49.7207	−2.04871
$590$	0	0
$591$	−2.17420	−0.0894347
$592$	0	0
$593$	19.2129	0.788979	0.394490	−	0.918900i	$-0.370921\pi$
$593$	19.2129	0.788979	0.394490	+	0.918900i	$0.370921\pi$
$594$	0	0
$595$	2.64206	0.108314
$596$	0	0
$597$	19.5191	0.798863
$598$	0	0
$599$	20.6159	0.842345	0.421172	−	0.906981i	$-0.361619\pi$
$599$	20.6159	0.842345	0.421172	+	0.906981i	$0.361619\pi$
$600$	0	0
$601$	14.6688	0.598354	0.299177	−	0.954198i	$-0.403288\pi$
$601$	14.6688	0.598354	0.299177	+	0.954198i	$0.403288\pi$
$602$	0	0
$603$	−10.6251	−0.432688
$604$	0	0
$605$	1.99864	0.0812563
$606$	0	0
$607$	−18.7059	−0.759250	−0.379625	−	0.925140i	$-0.623947\pi$
$607$	−18.7059	−0.759250	−0.379625	+	0.925140i	$0.623947\pi$
$608$	0	0
$609$	−13.2467	−0.536781
$610$	0	0
$611$	20.0473	0.811029
$612$	0	0
$613$	−2.57586	−0.104038	−0.0520191	−	0.998646i	$-0.516566\pi$
$613$	−2.57586	−0.104038	−0.0520191	+	0.998646i	$0.516566\pi$
$614$	0	0
$615$	−0.117516	−0.00473869
$616$	0	0
$617$	−19.1221	−0.769828	−0.384914	−	0.922952i	$-0.625769\pi$
$617$	−19.1221	−0.769828	−0.384914	+	0.922952i	$0.625769\pi$
$618$	0	0
$619$	11.0897	0.445733	0.222867	−	0.974849i	$-0.428459\pi$
$619$	11.0897	0.445733	0.222867	+	0.974849i	$0.428459\pi$
$620$	0	0
$621$	8.92443	0.358125
$622$	0	0
$623$	6.60765	0.264730
$624$	0	0
$625$	22.9564	0.918256
$626$	0	0
$627$	22.8606	0.912964
$628$	0	0
$629$	−3.34423	−0.133343
$630$	0	0
$631$	−28.4699	−1.13337	−0.566685	−	0.823935i	$-0.691774\pi$
$631$	−28.4699	−1.13337	−0.566685	+	0.823935i	$0.691774\pi$
$632$	0	0
$633$	8.94618	0.355579
$634$	0	0
$635$	6.56102	0.260366
$636$	0	0
$637$	−12.7722	−0.506054
$638$	0	0
$639$	−1.33418	−0.0527794
$640$	0	0
$641$	−15.8650	−0.626632	−0.313316	−	0.949649i	$-0.601440\pi$
$641$	−15.8650	−0.626632	−0.313316	+	0.949649i	$0.601440\pi$
$642$	0	0
$643$	−5.72748	−0.225870	−0.112935	−	0.993602i	$-0.536025\pi$
$643$	−5.72748	−0.225870	−0.112935	+	0.993602i	$0.536025\pi$
$644$	0	0
$645$	0.824100	0.0324489
$646$	0	0
$647$	−5.88463	−0.231349	−0.115674	−	0.993287i	$-0.536903\pi$
$647$	−5.88463	−0.231349	−0.115674	+	0.993287i	$0.536903\pi$
$648$	0	0
$649$	−2.17737	−0.0854691
$650$	0	0
$651$	17.2870	0.677531
$652$	0	0
$653$	−41.1149	−1.60895	−0.804476	−	0.593985i	$-0.797554\pi$
$653$	−41.1149	−1.60895	−0.804476	+	0.593985i	$0.797554\pi$
$654$	0	0
$655$	−4.97050	−0.194214
$656$	0	0
$657$	8.35531	0.325972
$658$	0	0
$659$	12.1465	0.473162	0.236581	−	0.971612i	$-0.423973\pi$
$659$	12.1465	0.473162	0.236581	+	0.971612i	$0.423973\pi$
$660$	0	0
$661$	−15.9501	−0.620389	−0.310194	−	0.950673i	$-0.600394\pi$
$661$	−15.9501	−0.620389	−0.310194	+	0.950673i	$0.600394\pi$
$662$	0	0
$663$	14.7360	0.572299
$664$	0	0
$665$	−4.11086	−0.159412
$666$	0	0
$667$	−60.2151	−2.33154
$668$	0	0
$669$	17.4305	0.673901
$670$	0	0
$671$	9.51495	0.367321
$672$	0	0
$673$	−28.3034	−1.09102	−0.545508	−	0.838106i	$-0.683663\pi$
$673$	−28.3034	−1.09102	−0.545508	+	0.838106i	$0.683663\pi$
$674$	0	0
$675$	−4.86250	−0.187158
$676$	0	0
$677$	5.99704	0.230485	0.115242	−	0.993337i	$-0.463236\pi$
$677$	5.99704	0.230485	0.115242	+	0.993337i	$0.463236\pi$
$678$	0	0
$679$	24.2244	0.929648
$680$	0	0
$681$	−5.26176	−0.201631
$682$	0	0
$683$	−33.7218	−1.29033	−0.645165	−	0.764043i	$-0.723211\pi$
$683$	−33.7218	−1.29033	−0.645165	+	0.764043i	$0.723211\pi$
$684$	0	0
$685$	2.10628	0.0804767
$686$	0	0
$687$	24.2475	0.925099
$688$	0	0
$689$	57.6085	2.19471
$690$	0	0
$691$	−8.64738	−0.328962	−0.164481	−	0.986380i	$-0.552595\pi$
$691$	−8.64738	−0.328962	−0.164481	+	0.986380i	$0.552595\pi$
$692$	0	0
$693$	−7.94822	−0.301928
$694$	0	0
$695$	6.33548	0.240318
$696$	0	0
$697$	−1.15015	−0.0435649
$698$	0	0
$699$	12.1328	0.458904
$700$	0	0
$701$	12.2904	0.464202	0.232101	−	0.972692i	$-0.425440\pi$
$701$	12.2904	0.464202	0.232101	+	0.972692i	$0.425440\pi$
$702$	0	0
$703$	5.20337	0.196249
$704$	0	0
$705$	1.83079	0.0689516
$706$	0	0
$707$	2.39153	0.0899426
$708$	0	0
$709$	2.80892	0.105491	0.0527457	−	0.998608i	$-0.483203\pi$
$709$	2.80892	0.105491	0.0527457	+	0.998608i	$0.483203\pi$
$710$	0	0
$711$	15.5212	0.582091
$712$	0	0
$713$	78.5813	2.94289
$714$	0	0
$715$	−6.09552	−0.227960
$716$	0	0
$717$	−15.1685	−0.566480
$718$	0	0
$719$	17.4798	0.651888	0.325944	−	0.945389i	$-0.394318\pi$
$719$	17.4798	0.651888	0.325944	+	0.945389i	$0.394318\pi$
$720$	0	0
$721$	2.07763	0.0773751
$722$	0	0
$723$	−18.2872	−0.680106
$724$	0	0
$725$	32.8084	1.21847
$726$	0	0
$727$	26.2700	0.974299	0.487149	−	0.873319i	$-0.338037\pi$
$727$	26.2700	0.974299	0.487149	+	0.873319i	$0.338037\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.06561	0.298317
$732$	0	0
$733$	−29.3942	−1.08570	−0.542849	−	0.839830i	$-0.682655\pi$
$733$	−29.3942	−1.08570	−0.542849	+	0.839830i	$0.682655\pi$
$734$	0	0
$735$	−1.16640	−0.0430234
$736$	0	0
$737$	43.0152	1.58449
$738$	0	0
$739$	2.53688	0.0933208	0.0466604	−	0.998911i	$-0.485142\pi$
$739$	2.53688	0.0933208	0.0466604	+	0.998911i	$0.485142\pi$
$740$	0	0
$741$	−22.9282	−0.842287
$742$	0	0
$743$	37.6157	1.37998	0.689992	−	0.723817i	$-0.257614\pi$
$743$	37.6157	1.37998	0.689992	+	0.723817i	$0.257614\pi$
$744$	0	0
$745$	−4.79690	−0.175745
$746$	0	0
$747$	−9.55110	−0.349457
$748$	0	0
$749$	7.76524	0.283736
$750$	0	0
$751$	−19.3798	−0.707180	−0.353590	−	0.935401i	$-0.615039\pi$
$751$	−19.3798	−0.707180	−0.353590	+	0.935401i	$0.615039\pi$
$752$	0	0
$753$	1.81307	0.0660718
$754$	0	0
$755$	3.97414	0.144634
$756$	0	0
$757$	−0.893853	−0.0324877	−0.0162438	−	0.999868i	$-0.505171\pi$
$757$	−0.893853	−0.0324877	−0.0162438	+	0.999868i	$0.505171\pi$
$758$	0	0
$759$	−36.1301	−1.31144
$760$	0	0
$761$	−20.9488	−0.759392	−0.379696	−	0.925111i	$-0.623971\pi$
$761$	−20.9488	−0.759392	−0.379696	+	0.925111i	$0.623971\pi$
$762$	0	0
$763$	−2.64234	−0.0956592
$764$	0	0
$765$	1.34574	0.0486554
$766$	0	0
$767$	2.18380	0.0788526
$768$	0	0
$769$	−52.9737	−1.91028	−0.955141	−	0.296153i	$-0.904296\pi$
$769$	−52.9737	−1.91028	−0.955141	+	0.296153i	$0.904296\pi$
$770$	0	0
$771$	−6.67435	−0.240371
$772$	0	0
$773$	−4.57701	−0.164624	−0.0823118	−	0.996607i	$-0.526230\pi$
$773$	−4.57701	−0.164624	−0.0823118	+	0.996607i	$0.526230\pi$
$774$	0	0
$775$	−42.8152	−1.53797
$776$	0	0
$777$	−1.80912	−0.0649018
$778$	0	0
$779$	1.78955	0.0641171
$780$	0	0
$781$	5.40136	0.193276
$782$	0	0
$783$	−6.74722	−0.241126
$784$	0	0
$785$	−1.13533	−0.0405217
$786$	0	0
$787$	−10.6325	−0.379008	−0.189504	−	0.981880i	$-0.560688\pi$
$787$	−10.6325	−0.379008	−0.189504	+	0.981880i	$0.560688\pi$
$788$	0	0
$789$	21.0546	0.749563
$790$	0	0
$791$	−4.40842	−0.156745
$792$	0	0
$793$	−9.54307	−0.338885
$794$	0	0
$795$	5.26101	0.186589
$796$	0	0
$797$	−56.1843	−1.99015	−0.995076	−	0.0991189i	$-0.968398\pi$
$797$	−56.1843	−1.99015	−0.995076	+	0.0991189i	$0.968398\pi$
$798$	0	0
$799$	17.9183	0.633903
$800$	0	0
$801$	3.36562	0.118918
$802$	0	0
$803$	−33.8260	−1.19369
$804$	0	0
$805$	6.49702	0.228990
$806$	0	0
$807$	5.45589	0.192057
$808$	0	0
$809$	11.6538	0.409726	0.204863	−	0.978791i	$-0.434325\pi$
$809$	11.6538	0.409726	0.204863	+	0.978791i	$0.434325\pi$
$810$	0	0
$811$	16.4662	0.578206	0.289103	−	0.957298i	$-0.406643\pi$
$811$	16.4662	0.578206	0.289103	+	0.957298i	$0.406643\pi$
$812$	0	0
$813$	4.00206	0.140358
$814$	0	0
$815$	−2.04267	−0.0715515
$816$	0	0
$817$	−12.5495	−0.439051
$818$	0	0
$819$	7.97171	0.278554
$820$	0	0
$821$	43.9803	1.53492	0.767462	−	0.641094i	$-0.221519\pi$
$821$	43.9803	1.53492	0.767462	+	0.641094i	$0.221519\pi$
$822$	0	0
$823$	−15.3338	−0.534504	−0.267252	−	0.963627i	$-0.586116\pi$
$823$	−15.3338	−0.534504	−0.267252	+	0.963627i	$0.586116\pi$
$824$	0	0
$825$	19.6856	0.685364
$826$	0	0
$827$	16.9750	0.590280	0.295140	−	0.955454i	$-0.404634\pi$
$827$	16.9750	0.590280	0.295140	+	0.955454i	$0.404634\pi$
$828$	0	0
$829$	27.5204	0.955823	0.477911	−	0.878408i	$-0.341394\pi$
$829$	27.5204	0.955823	0.477911	+	0.878408i	$0.341394\pi$
$830$	0	0
$831$	22.5668	0.782833
$832$	0	0
$833$	−11.4158	−0.395534
$834$	0	0
$835$	−7.51702	−0.260137
$836$	0	0
$837$	8.80518	0.304352
$838$	0	0
$839$	−15.2470	−0.526384	−0.263192	−	0.964744i	$-0.584775\pi$
$839$	−15.2470	−0.526384	−0.263192	+	0.964744i	$0.584775\pi$
$840$	0	0
$841$	16.5250	0.569827
$842$	0	0
$843$	−9.87322	−0.340052
$844$	0	0
$845$	1.29300	0.0444806
$846$	0	0
$847$	10.5819	0.363598
$848$	0	0
$849$	29.5858	1.01538
$850$	0	0
$851$	−8.22369	−0.281904
$852$	0	0
$853$	−23.7441	−0.812982	−0.406491	−	0.913655i	$-0.633248\pi$
$853$	−23.7441	−0.812982	−0.406491	+	0.913655i	$0.633248\pi$
$854$	0	0
$855$	−2.09388	−0.0716091
$856$	0	0
$857$	4.28871	0.146500	0.0732498	−	0.997314i	$-0.476663\pi$
$857$	4.28871	0.146500	0.0732498	+	0.997314i	$0.476663\pi$
$858$	0	0
$859$	−1.15611	−0.0394458	−0.0197229	−	0.999805i	$-0.506278\pi$
$859$	−1.15611	−0.0394458	−0.0197229	+	0.999805i	$0.506278\pi$
$860$	0	0
$861$	−0.622193	−0.0212043
$862$	0	0
$863$	49.4010	1.68163	0.840815	−	0.541323i	$-0.182076\pi$
$863$	49.4010	1.68163	0.840815	+	0.541323i	$0.182076\pi$
$864$	0	0
$865$	7.40126	0.251651
$866$	0	0
$867$	−3.82899	−0.130039
$868$	0	0
$869$	−62.8368	−2.13159
$870$	0	0
$871$	−43.1424	−1.46182
$872$	0	0
$873$	12.3388	0.417604
$874$	0	0
$875$	−7.17994	−0.242726
$876$	0	0
$877$	43.4495	1.46719	0.733593	−	0.679589i	$-0.237842\pi$
$877$	43.4495	1.46719	0.733593	+	0.679589i	$0.237842\pi$
$878$	0	0
$879$	−16.2215	−0.547137
$880$	0	0
$881$	−51.9644	−1.75073	−0.875363	−	0.483466i	$-0.839378\pi$
$881$	−51.9644	−1.75073	−0.875363	+	0.483466i	$0.839378\pi$
$882$	0	0
$883$	35.2566	1.18648	0.593239	−	0.805026i	$-0.297849\pi$
$883$	35.2566	1.18648	0.593239	+	0.805026i	$0.297849\pi$
$884$	0	0
$885$	0.199432	0.00670384
$886$	0	0
$887$	24.7631	0.831464	0.415732	−	0.909487i	$-0.363525\pi$
$887$	24.7631	0.831464	0.415732	+	0.909487i	$0.363525\pi$
$888$	0	0
$889$	34.7377	1.16506
$890$	0	0
$891$	−4.04845	−0.135628
$892$	0	0
$893$	−27.8795	−0.932953
$894$	0	0
$895$	0.909706	0.0304081
$896$	0	0
$897$	36.2369	1.20991
$898$	0	0
$899$	−59.4105	−1.98145
$900$	0	0
$901$	51.4904	1.71539
$902$	0	0
$903$	4.36324	0.145199
$904$	0	0
$905$	0.370811	0.0123262
$906$	0	0
$907$	55.6804	1.84884	0.924419	−	0.381377i	$-0.124550\pi$
$907$	55.6804	1.84884	0.924419	+	0.381377i	$0.124550\pi$
$908$	0	0
$909$	1.21813	0.0404028
$910$	0	0
$911$	−39.9314	−1.32299	−0.661493	−	0.749951i	$-0.730077\pi$
$911$	−39.9314	−1.32299	−0.661493	+	0.749951i	$0.730077\pi$
$912$	0	0
$913$	38.6671	1.27970
$914$	0	0
$915$	−0.871506	−0.0288111
$916$	0	0
$917$	−26.3166	−0.869050
$918$	0	0
$919$	−27.4981	−0.907080	−0.453540	−	0.891236i	$-0.649839\pi$
$919$	−27.4981	−0.907080	−0.453540	+	0.891236i	$0.649839\pi$
$920$	0	0
$921$	−13.1246	−0.432472
$922$	0	0
$923$	−5.41733	−0.178314
$924$	0	0
$925$	4.48070	0.147324
$926$	0	0
$927$	1.05825	0.0347574
$928$	0	0
$929$	−45.6591	−1.49803	−0.749013	−	0.662556i	$-0.769472\pi$
$929$	−45.6591	−1.49803	−0.749013	+	0.662556i	$0.769472\pi$
$930$	0	0
$931$	17.7621	0.582131
$932$	0	0
$933$	−18.3176	−0.599690
$934$	0	0
$935$	−5.44817	−0.178174
$936$	0	0
$937$	−10.2634	−0.335291	−0.167646	−	0.985847i	$-0.553616\pi$
$937$	−10.2634	−0.335291	−0.167646	+	0.985847i	$0.553616\pi$
$938$	0	0
$939$	25.8600	0.843910
$940$	0	0
$941$	29.9236	0.975482	0.487741	−	0.872988i	$-0.337821\pi$
$941$	29.9236	0.975482	0.487741	+	0.872988i	$0.337821\pi$
$942$	0	0
$943$	−2.82829	−0.0921019
$944$	0	0
$945$	0.728004	0.0236820
$946$	0	0
$947$	−16.8152	−0.546421	−0.273211	−	0.961954i	$-0.588086\pi$
$947$	−16.8152	−0.546421	−0.273211	+	0.961954i	$0.588086\pi$
$948$	0	0
$949$	33.9260	1.10129
$950$	0	0
$951$	18.3141	0.593876
$952$	0	0
$953$	−12.5451	−0.406374	−0.203187	−	0.979140i	$-0.565130\pi$
$953$	−12.5451	−0.406374	−0.203187	+	0.979140i	$0.565130\pi$
$954$	0	0
$955$	−9.86993	−0.319383
$956$	0	0
$957$	27.3158	0.882993
$958$	0	0
$959$	11.1518	0.360110
$960$	0	0
$961$	46.5313	1.50101
$962$	0	0
$963$	3.95524	0.127456
$964$	0	0
$965$	3.48009	0.112028
$966$	0	0
$967$	8.59093	0.276266	0.138133	−	0.990414i	$-0.455890\pi$
$967$	8.59093	0.276266	0.138133	+	0.990414i	$0.455890\pi$
$968$	0	0
$969$	−20.4931	−0.658335
$970$	0	0
$971$	−23.9170	−0.767535	−0.383767	−	0.923430i	$-0.625374\pi$
$971$	−23.9170	−0.767535	−0.383767	+	0.923430i	$0.625374\pi$
$972$	0	0
$973$	33.5435	1.07535
$974$	0	0
$975$	−19.7438	−0.632306
$976$	0	0
$977$	−21.5036	−0.687963	−0.343981	−	0.938977i	$-0.611776\pi$
$977$	−21.5036	−0.687963	−0.343981	+	0.938977i	$0.611776\pi$
$978$	0	0
$979$	−13.6255	−0.435474
$980$	0	0
$981$	−1.34588	−0.0429708
$982$	0	0
$983$	−23.9678	−0.764456	−0.382228	−	0.924068i	$-0.624843\pi$
$983$	−23.9678	−0.764456	−0.382228	+	0.924068i	$0.624843\pi$
$984$	0	0
$985$	−0.806217	−0.0256882
$986$	0	0
$987$	9.69321	0.308538
$988$	0	0
$989$	19.8339	0.630681
$990$	0	0
$991$	−9.12524	−0.289873	−0.144936	−	0.989441i	$-0.546298\pi$
$991$	−9.12524	−0.289873	−0.144936	+	0.989441i	$0.546298\pi$
$992$	0	0
$993$	−14.6270	−0.464173
$994$	0	0
$995$	7.23788	0.229456
$996$	0	0
$997$	−23.5653	−0.746321	−0.373161	−	0.927767i	$-0.621726\pi$
$997$	−23.5653	−0.746321	−0.373161	+	0.927767i	$0.621726\pi$
$998$	0	0
$999$	−0.921480	−0.0291543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8688.2.a.bf.1.4		8
4.3	odd	2		543.2.a.d.1.8	✓	8
12.11	even	2		1629.2.a.e.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.2.a.d.1.8	✓	8	4.3	odd	2
1629.2.a.e.1.1		8	12.11	even	2
8688.2.a.bf.1.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8688 = 2^{4} \cdot 3 \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8688.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.370811 q^{5} +1.96328 q^{7} +1.00000 q^{9} -4.04845 q^{11} +4.06041 q^{13} +0.370811 q^{15} +3.62919 q^{17} -5.64676 q^{19} +1.96328 q^{21} +8.92443 q^{23} -4.86250 q^{25} +1.00000 q^{27} -6.74722 q^{29} +8.80518 q^{31} -4.04845 q^{33} +0.728004 q^{35} -0.921480 q^{37} +4.06041 q^{39} -0.316916 q^{41} +2.22243 q^{43} +0.370811 q^{45} +4.93727 q^{47} -3.14555 q^{49} +3.62919 q^{51} +14.1879 q^{53} -1.50121 q^{55} -5.64676 q^{57} +0.537828 q^{59} -2.35027 q^{61} +1.96328 q^{63} +1.50564 q^{65} -10.6251 q^{67} +8.92443 q^{69} -1.33418 q^{71} +8.35531 q^{73} -4.86250 q^{75} -7.94822 q^{77} +15.5212 q^{79} +1.00000 q^{81} -9.55110 q^{83} +1.34574 q^{85} -6.74722 q^{87} +3.36562 q^{89} +7.97171 q^{91} +8.80518 q^{93} -2.09388 q^{95} +12.3388 q^{97} -4.04845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 5 q^{5} + 3 q^{7} + 8 q^{9} - 4 q^{11} + 13 q^{13} + 5 q^{15} + 27 q^{17} + 10 q^{19} + 3 q^{21} - 3 q^{23} + 13 q^{25} + 8 q^{27} + 16 q^{29} + q^{31} - 4 q^{33} - 14 q^{35} + 3 q^{37}+ \cdots - 4 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + 5 * q^5 + 3 * q^7 + 8 * q^9 - 4 * q^11 + 13 * q^13 + 5 * q^15 + 27 * q^17 + 10 * q^19 + 3 * q^21 - 3 * q^23 + 13 * q^25 + 8 * q^27 + 16 * q^29 + q^31 - 4 * q^33 - 14 * q^35 + 3 * q^37 + 13 * q^39 + 20 * q^41 + 4 * q^43 + 5 * q^45 - 12 * q^47 + 5 * q^49 + 27 * q^51 + 23 * q^53 + 13 * q^55 + 10 * q^57 - 7 * q^59 + q^61 + 3 * q^63 + 10 * q^65 + 25 * q^67 - 3 * q^69 - 11 * q^71 + 23 * q^73 + 13 * q^75 + 20 * q^77 + 26 * q^79 + 8 * q^81 - 16 * q^83 - 33 * q^85 + 16 * q^87 + 14 * q^89 + 35 * q^91 + q^93 - 10 * q^95 + 28 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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