LMFDB - Embedded newform 8024.2.a.bc.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08632 q^{3} -3.87621 q^{5} +3.88634 q^{7} +1.35275 q^{9} +O(q^{10})$	$q-2.08632 q^{3} -3.87621 q^{5} +3.88634 q^{7} +1.35275 q^{9} -5.53713 q^{11} -0.0369407 q^{13} +8.08702 q^{15} -1.00000 q^{17} -2.67352 q^{19} -8.10816 q^{21} -4.08160 q^{23} +10.0250 q^{25} +3.43670 q^{27} +0.828376 q^{29} -5.75629 q^{31} +11.5523 q^{33} -15.0643 q^{35} -5.90293 q^{37} +0.0770702 q^{39} -11.2441 q^{41} +0.186789 q^{43} -5.24352 q^{45} +5.96312 q^{47} +8.10365 q^{49} +2.08632 q^{51} -5.41884 q^{53} +21.4631 q^{55} +5.57782 q^{57} -1.00000 q^{59} -2.57193 q^{61} +5.25723 q^{63} +0.143190 q^{65} -1.85818 q^{67} +8.51553 q^{69} -8.95239 q^{71} +4.56909 q^{73} -20.9154 q^{75} -21.5192 q^{77} +14.7598 q^{79} -11.2283 q^{81} -14.9278 q^{83} +3.87621 q^{85} -1.72826 q^{87} -9.69363 q^{89} -0.143564 q^{91} +12.0095 q^{93} +10.3631 q^{95} -8.84898 q^{97} -7.49034 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−2.08632	−1.20454	−0.602270	−	0.798293i	$-0.705737\pi$
$3$	−2.08632	−1.20454	−0.602270	+	0.798293i	$0.705737\pi$
$4$	0	0
$5$	−3.87621	−1.73349	−0.866746	−	0.498749i	$-0.833793\pi$
$5$	−3.87621	−1.73349	−0.866746	+	0.498749i	$0.833793\pi$
$6$	0	0
$7$	3.88634	1.46890	0.734449	−	0.678663i	$-0.237440\pi$
$7$	3.88634	1.46890	0.734449	+	0.678663i	$0.237440\pi$
$8$	0	0
$9$	1.35275	0.450915
$10$	0	0
$11$	−5.53713	−1.66951	−0.834755	−	0.550622i	$-0.814390\pi$
$11$	−5.53713	−1.66951	−0.834755	+	0.550622i	$0.814390\pi$
$12$	0	0
$13$	−0.0369407	−0.0102455	−0.00512275	−	0.999987i	$-0.501631\pi$
$13$	−0.0369407	−0.0102455	−0.00512275	+	0.999987i	$0.501631\pi$
$14$	0	0
$15$	8.08702	2.08806
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.67352	−0.613347	−0.306673	−	0.951815i	$-0.599216\pi$
$19$	−2.67352	−0.613347	−0.306673	+	0.951815i	$0.599216\pi$
$20$	0	0
$21$	−8.10816	−1.76935
$22$	0	0
$23$	−4.08160	−0.851072	−0.425536	−	0.904942i	$-0.639914\pi$
$23$	−4.08160	−0.851072	−0.425536	+	0.904942i	$0.639914\pi$
$24$	0	0
$25$	10.0250	2.00500
$26$	0	0
$27$	3.43670	0.661394
$28$	0	0
$29$	0.828376	0.153826	0.0769128	−	0.997038i	$-0.475494\pi$
$29$	0.828376	0.153826	0.0769128	+	0.997038i	$0.475494\pi$
$30$	0	0
$31$	−5.75629	−1.03386	−0.516930	−	0.856027i	$-0.672925\pi$
$31$	−5.75629	−1.03386	−0.516930	+	0.856027i	$0.672925\pi$
$32$	0	0
$33$	11.5523	2.01099
$34$	0	0
$35$	−15.0643	−2.54633
$36$	0	0
$37$	−5.90293	−0.970436	−0.485218	−	0.874393i	$-0.661260\pi$
$37$	−5.90293	−0.970436	−0.485218	+	0.874393i	$0.661260\pi$
$38$	0	0
$39$	0.0770702	0.0123411
$40$	0	0
$41$	−11.2441	−1.75603	−0.878017	−	0.478629i	$-0.841134\pi$
$41$	−11.2441	−1.75603	−0.878017	+	0.478629i	$0.841134\pi$
$42$	0	0
$43$	0.186789	0.0284851	0.0142425	−	0.999899i	$-0.495466\pi$
$43$	0.186789	0.0284851	0.0142425	+	0.999899i	$0.495466\pi$
$44$	0	0
$45$	−5.24352	−0.781658
$46$	0	0
$47$	5.96312	0.869810	0.434905	−	0.900476i	$-0.356782\pi$
$47$	5.96312	0.869810	0.434905	+	0.900476i	$0.356782\pi$
$48$	0	0
$49$	8.10365	1.15766
$50$	0	0
$51$	2.08632	0.292144
$52$	0	0
$53$	−5.41884	−0.744335	−0.372167	−	0.928166i	$-0.621385\pi$
$53$	−5.41884	−0.744335	−0.372167	+	0.928166i	$0.621385\pi$
$54$	0	0
$55$	21.4631	2.89408
$56$	0	0
$57$	5.57782	0.738800
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−2.57193	−0.329302	−0.164651	−	0.986352i	$-0.552650\pi$
$61$	−2.57193	−0.329302	−0.164651	+	0.986352i	$0.552650\pi$
$62$	0	0
$63$	5.25723	0.662349
$64$	0	0
$65$	0.143190	0.0177605
$66$	0	0
$67$	−1.85818	−0.227013	−0.113506	−	0.993537i	$-0.536208\pi$
$67$	−1.85818	−0.227013	−0.113506	+	0.993537i	$0.536208\pi$
$68$	0	0
$69$	8.51553	1.02515
$70$	0	0
$71$	−8.95239	−1.06245	−0.531227	−	0.847230i	$-0.678269\pi$
$71$	−8.95239	−1.06245	−0.531227	+	0.847230i	$0.678269\pi$
$72$	0	0
$73$	4.56909	0.534772	0.267386	−	0.963590i	$-0.413840\pi$
$73$	4.56909	0.534772	0.267386	+	0.963590i	$0.413840\pi$
$74$	0	0
$75$	−20.9154	−2.41510
$76$	0	0
$77$	−21.5192	−2.45234
$78$	0	0
$79$	14.7598	1.66060	0.830302	−	0.557314i	$-0.188168\pi$
$79$	14.7598	1.66060	0.830302	+	0.557314i	$0.188168\pi$
$80$	0	0
$81$	−11.2283	−1.24759
$82$	0	0
$83$	−14.9278	−1.63854	−0.819269	−	0.573409i	$-0.805621\pi$
$83$	−14.9278	−1.63854	−0.819269	+	0.573409i	$0.805621\pi$
$84$	0	0
$85$	3.87621	0.420434
$86$	0	0
$87$	−1.72826	−0.185289
$88$	0	0
$89$	−9.69363	−1.02752	−0.513761	−	0.857933i	$-0.671748\pi$
$89$	−9.69363	−1.02752	−0.513761	+	0.857933i	$0.671748\pi$
$90$	0	0
$91$	−0.143564	−0.0150496
$92$	0	0
$93$	12.0095	1.24533
$94$	0	0
$95$	10.3631	1.06323
$96$	0	0
$97$	−8.84898	−0.898478	−0.449239	−	0.893412i	$-0.648305\pi$
$97$	−8.84898	−0.898478	−0.449239	+	0.893412i	$0.648305\pi$
$98$	0	0
$99$	−7.49034	−0.752807
$100$	0	0
$101$	9.18508	0.913950	0.456975	−	0.889480i	$-0.348933\pi$
$101$	9.18508	0.913950	0.456975	+	0.889480i	$0.348933\pi$
$102$	0	0
$103$	11.3249	1.11587	0.557937	−	0.829883i	$-0.311593\pi$
$103$	11.3249	1.11587	0.557937	+	0.829883i	$0.311593\pi$
$104$	0	0
$105$	31.4289	3.06715
$106$	0	0
$107$	−8.36621	−0.808792	−0.404396	−	0.914584i	$-0.632518\pi$
$107$	−8.36621	−0.808792	−0.404396	+	0.914584i	$0.632518\pi$
$108$	0	0
$109$	−13.8969	−1.33108	−0.665542	−	0.746360i	$-0.731800\pi$
$109$	−13.8969	−1.33108	−0.665542	+	0.746360i	$0.731800\pi$
$110$	0	0
$111$	12.3154	1.16893
$112$	0	0
$113$	−19.8787	−1.87003	−0.935017	−	0.354603i	$-0.884616\pi$
$113$	−19.8787	−1.87003	−0.935017	+	0.354603i	$0.884616\pi$
$114$	0	0
$115$	15.8211	1.47533
$116$	0	0
$117$	−0.0499713	−0.00461985
$118$	0	0
$119$	−3.88634	−0.356260
$120$	0	0
$121$	19.6599	1.78726
$122$	0	0
$123$	23.4588	2.11521
$124$	0	0
$125$	−19.4779	−1.74215
$126$	0	0
$127$	−13.3518	−1.18478	−0.592392	−	0.805650i	$-0.701816\pi$
$127$	−13.3518	−1.18478	−0.592392	+	0.805650i	$0.701816\pi$
$128$	0	0
$129$	−0.389703	−0.0343114
$130$	0	0
$131$	−6.88274	−0.601348	−0.300674	−	0.953727i	$-0.597212\pi$
$131$	−6.88274	−0.601348	−0.300674	+	0.953727i	$0.597212\pi$
$132$	0	0
$133$	−10.3902	−0.900944
$134$	0	0
$135$	−13.3214	−1.14652
$136$	0	0
$137$	20.7960	1.77672	0.888362	−	0.459144i	$-0.151844\pi$
$137$	20.7960	1.77672	0.888362	+	0.459144i	$0.151844\pi$
$138$	0	0
$139$	−14.0710	−1.19349	−0.596743	−	0.802433i	$-0.703539\pi$
$139$	−14.0710	−1.19349	−0.596743	+	0.802433i	$0.703539\pi$
$140$	0	0
$141$	−12.4410	−1.04772
$142$	0	0
$143$	0.204546	0.0171050
$144$	0	0
$145$	−3.21096	−0.266656
$146$	0	0
$147$	−16.9068	−1.39445
$148$	0	0
$149$	−22.9462	−1.87982	−0.939912	−	0.341418i	$-0.889093\pi$
$149$	−22.9462	−1.87982	−0.939912	+	0.341418i	$0.889093\pi$
$150$	0	0
$151$	−23.2859	−1.89498	−0.947488	−	0.319791i	$-0.896387\pi$
$151$	−23.2859	−1.89498	−0.947488	+	0.319791i	$0.896387\pi$
$152$	0	0
$153$	−1.35275	−0.109363
$154$	0	0
$155$	22.3126	1.79219
$156$	0	0
$157$	0.396183	0.0316189	0.0158094	−	0.999875i	$-0.494967\pi$
$157$	0.396183	0.0316189	0.0158094	+	0.999875i	$0.494967\pi$
$158$	0	0
$159$	11.3055	0.896581
$160$	0	0
$161$	−15.8625	−1.25014
$162$	0	0
$163$	−3.71607	−0.291065	−0.145532	−	0.989353i	$-0.546490\pi$
$163$	−3.71607	−0.291065	−0.145532	+	0.989353i	$0.546490\pi$
$164$	0	0
$165$	−44.7789	−3.48604
$166$	0	0
$167$	2.33513	0.180697	0.0903487	−	0.995910i	$-0.471202\pi$
$167$	2.33513	0.180697	0.0903487	+	0.995910i	$0.471202\pi$
$168$	0	0
$169$	−12.9986	−0.999895
$170$	0	0
$171$	−3.61659	−0.276567
$172$	0	0
$173$	3.83262	0.291389	0.145694	−	0.989330i	$-0.453458\pi$
$173$	3.83262	0.291389	0.145694	+	0.989330i	$0.453458\pi$
$174$	0	0
$175$	38.9605	2.94514
$176$	0	0
$177$	2.08632	0.156818
$178$	0	0
$179$	10.9415	0.817805	0.408903	−	0.912578i	$-0.365912\pi$
$179$	10.9415	0.817805	0.408903	+	0.912578i	$0.365912\pi$
$180$	0	0
$181$	−0.726326	−0.0539874	−0.0269937	−	0.999636i	$-0.508593\pi$
$181$	−0.726326	−0.0539874	−0.0269937	+	0.999636i	$0.508593\pi$
$182$	0	0
$183$	5.36588	0.396657
$184$	0	0
$185$	22.8810	1.68224
$186$	0	0
$187$	5.53713	0.404915
$188$	0	0
$189$	13.3562	0.971521
$190$	0	0
$191$	−16.1152	−1.16605	−0.583027	−	0.812453i	$-0.698132\pi$
$191$	−16.1152	−1.16605	−0.583027	+	0.812453i	$0.698132\pi$
$192$	0	0
$193$	20.7269	1.49195	0.745976	−	0.665973i	$-0.231983\pi$
$193$	20.7269	1.49195	0.745976	+	0.665973i	$0.231983\pi$
$194$	0	0
$195$	−0.298740	−0.0213932
$196$	0	0
$197$	26.8778	1.91496	0.957480	−	0.288499i	$-0.0931562\pi$
$197$	26.8778	1.91496	0.957480	+	0.288499i	$0.0931562\pi$
$198$	0	0
$199$	−14.9687	−1.06110	−0.530550	−	0.847654i	$-0.678015\pi$
$199$	−14.9687	−1.06110	−0.530550	+	0.847654i	$0.678015\pi$
$200$	0	0
$201$	3.87676	0.273446
$202$	0	0
$203$	3.21935	0.225954
$204$	0	0
$205$	43.5845	3.04407
$206$	0	0
$207$	−5.52136	−0.383761
$208$	0	0
$209$	14.8036	1.02399
$210$	0	0
$211$	0.257587	0.0177330	0.00886651	−	0.999961i	$-0.497178\pi$
$211$	0.257587	0.0177330	0.00886651	+	0.999961i	$0.497178\pi$
$212$	0	0
$213$	18.6776	1.27977
$214$	0	0
$215$	−0.724034	−0.0493787
$216$	0	0
$217$	−22.3709	−1.51864
$218$	0	0
$219$	−9.53261	−0.644154
$220$	0	0
$221$	0.0369407	0.00248490
$222$	0	0
$223$	6.39059	0.427945	0.213973	−	0.976840i	$-0.431360\pi$
$223$	6.39059	0.427945	0.213973	+	0.976840i	$0.431360\pi$
$224$	0	0
$225$	13.5613	0.904084
$226$	0	0
$227$	−4.51478	−0.299657	−0.149828	−	0.988712i	$-0.547872\pi$
$227$	−4.51478	−0.299657	−0.149828	+	0.988712i	$0.547872\pi$
$228$	0	0
$229$	21.2632	1.40511	0.702556	−	0.711628i	$-0.252042\pi$
$229$	21.2632	1.40511	0.702556	+	0.711628i	$0.252042\pi$
$230$	0	0
$231$	44.8960	2.95394
$232$	0	0
$233$	18.3599	1.20280	0.601398	−	0.798950i	$-0.294611\pi$
$233$	18.3599	1.20280	0.601398	+	0.798950i	$0.294611\pi$
$234$	0	0
$235$	−23.1143	−1.50781
$236$	0	0
$237$	−30.7937	−2.00026
$238$	0	0
$239$	10.5472	0.682239	0.341119	−	0.940020i	$-0.389194\pi$
$239$	10.5472	0.682239	0.341119	+	0.940020i	$0.389194\pi$
$240$	0	0
$241$	0.934526	0.0601981	0.0300991	−	0.999547i	$-0.490418\pi$
$241$	0.934526	0.0601981	0.0300991	+	0.999547i	$0.490418\pi$
$242$	0	0
$243$	13.1158	0.841378
$244$	0	0
$245$	−31.4114	−2.00680
$246$	0	0
$247$	0.0987615	0.00628404
$248$	0	0
$249$	31.1442	1.97368
$250$	0	0
$251$	−6.61221	−0.417359	−0.208679	−	0.977984i	$-0.566917\pi$
$251$	−6.61221	−0.417359	−0.208679	+	0.977984i	$0.566917\pi$
$252$	0	0
$253$	22.6004	1.42087
$254$	0	0
$255$	−8.08702	−0.506429
$256$	0	0
$257$	16.5965	1.03526	0.517631	−	0.855604i	$-0.326814\pi$
$257$	16.5965	1.03526	0.517631	+	0.855604i	$0.326814\pi$
$258$	0	0
$259$	−22.9408	−1.42547
$260$	0	0
$261$	1.12058	0.0693623
$262$	0	0
$263$	7.87502	0.485594	0.242797	−	0.970077i	$-0.421935\pi$
$263$	7.87502	0.485594	0.242797	+	0.970077i	$0.421935\pi$
$264$	0	0
$265$	21.0045	1.29030
$266$	0	0
$267$	20.2240	1.23769
$268$	0	0
$269$	−15.3080	−0.933344	−0.466672	−	0.884430i	$-0.654547\pi$
$269$	−15.3080	−0.933344	−0.466672	+	0.884430i	$0.654547\pi$
$270$	0	0
$271$	−22.2366	−1.35078	−0.675389	−	0.737462i	$-0.736024\pi$
$271$	−22.2366	−1.35078	−0.675389	+	0.737462i	$0.736024\pi$
$272$	0	0
$273$	0.299521	0.0181278
$274$	0	0
$275$	−55.5097	−3.34736
$276$	0	0
$277$	−13.5999	−0.817138	−0.408569	−	0.912727i	$-0.633972\pi$
$277$	−13.5999	−0.817138	−0.408569	+	0.912727i	$0.633972\pi$
$278$	0	0
$279$	−7.78680	−0.466184
$280$	0	0
$281$	−13.3387	−0.795717	−0.397859	−	0.917447i	$-0.630247\pi$
$281$	−13.3387	−0.795717	−0.397859	+	0.917447i	$0.630247\pi$
$282$	0	0
$283$	−2.54705	−0.151406	−0.0757031	−	0.997130i	$-0.524120\pi$
$283$	−2.54705	−0.151406	−0.0757031	+	0.997130i	$0.524120\pi$
$284$	0	0
$285$	−21.6208	−1.28070
$286$	0	0
$287$	−43.6984	−2.57944
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	18.4618	1.08225
$292$	0	0
$293$	25.6675	1.49951	0.749756	−	0.661714i	$-0.230171\pi$
$293$	25.6675	1.49951	0.749756	+	0.661714i	$0.230171\pi$
$294$	0	0
$295$	3.87621	0.225682
$296$	0	0
$297$	−19.0295	−1.10420
$298$	0	0
$299$	0.150777	0.00871966
$300$	0	0
$301$	0.725927	0.0418417
$302$	0	0
$303$	−19.1630	−1.10089
$304$	0	0
$305$	9.96933	0.570842
$306$	0	0
$307$	−14.1868	−0.809681	−0.404841	−	0.914387i	$-0.632673\pi$
$307$	−14.1868	−0.809681	−0.404841	+	0.914387i	$0.632673\pi$
$308$	0	0
$309$	−23.6274	−1.34411
$310$	0	0
$311$	−5.98504	−0.339381	−0.169690	−	0.985497i	$-0.554277\pi$
$311$	−5.98504	−0.339381	−0.169690	+	0.985497i	$0.554277\pi$
$312$	0	0
$313$	20.3177	1.14842	0.574211	−	0.818707i	$-0.305309\pi$
$313$	20.3177	1.14842	0.574211	+	0.818707i	$0.305309\pi$
$314$	0	0
$315$	−20.3781	−1.14818
$316$	0	0
$317$	29.9450	1.68188	0.840940	−	0.541128i	$-0.182003\pi$
$317$	29.9450	1.68188	0.840940	+	0.541128i	$0.182003\pi$
$318$	0	0
$319$	−4.58683	−0.256813
$320$	0	0
$321$	17.4546	0.974222
$322$	0	0
$323$	2.67352	0.148758
$324$	0	0
$325$	−0.370330	−0.0205422
$326$	0	0
$327$	28.9935	1.60334
$328$	0	0
$329$	23.1747	1.27766
$330$	0	0
$331$	10.5923	0.582203	0.291101	−	0.956692i	$-0.405978\pi$
$331$	10.5923	0.582203	0.291101	+	0.956692i	$0.405978\pi$
$332$	0	0
$333$	−7.98516	−0.437584
$334$	0	0
$335$	7.20268	0.393525
$336$	0	0
$337$	−18.5679	−1.01146	−0.505730	−	0.862692i	$-0.668777\pi$
$337$	−18.5679	−1.01146	−0.505730	+	0.862692i	$0.668777\pi$
$338$	0	0
$339$	41.4735	2.25253
$340$	0	0
$341$	31.8734	1.72604
$342$	0	0
$343$	4.28915	0.231592
$344$	0	0
$345$	−33.0080	−1.77709
$346$	0	0
$347$	−25.3742	−1.36216	−0.681078	−	0.732211i	$-0.738489\pi$
$347$	−25.3742	−1.36216	−0.681078	+	0.732211i	$0.738489\pi$
$348$	0	0
$349$	29.7099	1.59033	0.795166	−	0.606391i	$-0.207383\pi$
$349$	29.7099	1.59033	0.795166	+	0.606391i	$0.207383\pi$
$350$	0	0
$351$	−0.126954	−0.00677631
$352$	0	0
$353$	23.9828	1.27647	0.638237	−	0.769840i	$-0.279664\pi$
$353$	23.9828	1.27647	0.638237	+	0.769840i	$0.279664\pi$
$354$	0	0
$355$	34.7013	1.84175
$356$	0	0
$357$	8.10816	0.429130
$358$	0	0
$359$	−0.611071	−0.0322511	−0.0161256	−	0.999870i	$-0.505133\pi$
$359$	−0.611071	−0.0322511	−0.0161256	+	0.999870i	$0.505133\pi$
$360$	0	0
$361$	−11.8523	−0.623806
$362$	0	0
$363$	−41.0168	−2.15283
$364$	0	0
$365$	−17.7107	−0.927023
$366$	0	0
$367$	1.59986	0.0835120	0.0417560	−	0.999128i	$-0.486705\pi$
$367$	1.59986	0.0835120	0.0417560	+	0.999128i	$0.486705\pi$
$368$	0	0
$369$	−15.2104	−0.791823
$370$	0	0
$371$	−21.0595	−1.09335
$372$	0	0
$373$	29.2901	1.51658	0.758292	−	0.651915i	$-0.226034\pi$
$373$	29.2901	1.51658	0.758292	+	0.651915i	$0.226034\pi$
$374$	0	0
$375$	40.6371	2.09849
$376$	0	0
$377$	−0.0306008	−0.00157602
$378$	0	0
$379$	−9.65831	−0.496114	−0.248057	−	0.968745i	$-0.579792\pi$
$379$	−9.65831	−0.496114	−0.248057	+	0.968745i	$0.579792\pi$
$380$	0	0
$381$	27.8563	1.42712
$382$	0	0
$383$	10.2784	0.525203	0.262601	−	0.964904i	$-0.415420\pi$
$383$	10.2784	0.525203	0.262601	+	0.964904i	$0.415420\pi$
$384$	0	0
$385$	83.4129	4.25111
$386$	0	0
$387$	0.252678	0.0128444
$388$	0	0
$389$	−15.0423	−0.762673	−0.381337	−	0.924436i	$-0.624536\pi$
$389$	−15.0423	−0.762673	−0.381337	+	0.924436i	$0.624536\pi$
$390$	0	0
$391$	4.08160	0.206415
$392$	0	0
$393$	14.3596	0.724347
$394$	0	0
$395$	−57.2119	−2.87865
$396$	0	0
$397$	7.29599	0.366175	0.183088	−	0.983097i	$-0.441391\pi$
$397$	7.29599	0.366175	0.183088	+	0.983097i	$0.441391\pi$
$398$	0	0
$399$	21.6773	1.08522
$400$	0	0
$401$	−14.2985	−0.714034	−0.357017	−	0.934098i	$-0.616206\pi$
$401$	−14.2985	−0.714034	−0.357017	+	0.934098i	$0.616206\pi$
$402$	0	0
$403$	0.212641	0.0105924
$404$	0	0
$405$	43.5233	2.16269
$406$	0	0
$407$	32.6853	1.62015
$408$	0	0
$409$	−4.29024	−0.212139	−0.106069	−	0.994359i	$-0.533827\pi$
$409$	−4.29024	−0.212139	−0.106069	+	0.994359i	$0.533827\pi$
$410$	0	0
$411$	−43.3872	−2.14013
$412$	0	0
$413$	−3.88634	−0.191234
$414$	0	0
$415$	57.8632	2.84039
$416$	0	0
$417$	29.3566	1.43760
$418$	0	0
$419$	9.34134	0.456354	0.228177	−	0.973620i	$-0.426723\pi$
$419$	9.34134	0.456354	0.228177	+	0.973620i	$0.426723\pi$
$420$	0	0
$421$	−17.4768	−0.851769	−0.425884	−	0.904778i	$-0.640037\pi$
$421$	−17.4768	−0.851769	−0.425884	+	0.904778i	$0.640037\pi$
$422$	0	0
$423$	8.06658	0.392211
$424$	0	0
$425$	−10.0250	−0.486283
$426$	0	0
$427$	−9.99539	−0.483711
$428$	0	0
$429$	−0.426748	−0.0206036
$430$	0	0
$431$	40.7440	1.96257	0.981284	−	0.192567i	$-0.0616812\pi$
$431$	40.7440	1.96257	0.981284	+	0.192567i	$0.0616812\pi$
$432$	0	0
$433$	33.4343	1.60675	0.803375	−	0.595474i	$-0.203036\pi$
$433$	33.4343	1.60675	0.803375	+	0.595474i	$0.203036\pi$
$434$	0	0
$435$	6.69910	0.321197
$436$	0	0
$437$	10.9122	0.522002
$438$	0	0
$439$	−3.72249	−0.177665	−0.0888323	−	0.996047i	$-0.528314\pi$
$439$	−3.72249	−0.177665	−0.0888323	+	0.996047i	$0.528314\pi$
$440$	0	0
$441$	10.9622	0.522008
$442$	0	0
$443$	30.0925	1.42974	0.714869	−	0.699258i	$-0.246486\pi$
$443$	30.0925	1.42974	0.714869	+	0.699258i	$0.246486\pi$
$444$	0	0
$445$	37.5745	1.78120
$446$	0	0
$447$	47.8731	2.26432
$448$	0	0
$449$	−21.9166	−1.03431	−0.517154	−	0.855892i	$-0.673009\pi$
$449$	−21.9166	−1.03431	−0.517154	+	0.855892i	$0.673009\pi$
$450$	0	0
$451$	62.2601	2.93171
$452$	0	0
$453$	48.5818	2.28257
$454$	0	0
$455$	0.556484	0.0260884
$456$	0	0
$457$	−33.9777	−1.58941	−0.794705	−	0.606996i	$-0.792374\pi$
$457$	−33.9777	−1.58941	−0.794705	+	0.606996i	$0.792374\pi$
$458$	0	0
$459$	−3.43670	−0.160412
$460$	0	0
$461$	−36.6204	−1.70558	−0.852792	−	0.522251i	$-0.825092\pi$
$461$	−36.6204	−1.70558	−0.852792	+	0.522251i	$0.825092\pi$
$462$	0	0
$463$	−8.44773	−0.392600	−0.196300	−	0.980544i	$-0.562893\pi$
$463$	−8.44773	−0.392600	−0.196300	+	0.980544i	$0.562893\pi$
$464$	0	0
$465$	−46.5513	−2.15876
$466$	0	0
$467$	−3.30244	−0.152819	−0.0764093	−	0.997077i	$-0.524346\pi$
$467$	−3.30244	−0.152819	−0.0764093	+	0.997077i	$0.524346\pi$
$468$	0	0
$469$	−7.22151	−0.333459
$470$	0	0
$471$	−0.826567	−0.0380862
$472$	0	0
$473$	−1.03428	−0.0475561
$474$	0	0
$475$	−26.8019	−1.22976
$476$	0	0
$477$	−7.33031	−0.335632
$478$	0	0
$479$	14.6823	0.670850	0.335425	−	0.942067i	$-0.391120\pi$
$479$	14.6823	0.670850	0.335425	+	0.942067i	$0.391120\pi$
$480$	0	0
$481$	0.218058	0.00994260
$482$	0	0
$483$	33.0943	1.50584
$484$	0	0
$485$	34.3005	1.55750
$486$	0	0
$487$	40.5995	1.83974	0.919870	−	0.392223i	$-0.128294\pi$
$487$	40.5995	1.83974	0.919870	+	0.392223i	$0.128294\pi$
$488$	0	0
$489$	7.75292	0.350599
$490$	0	0
$491$	−12.9692	−0.585293	−0.292646	−	0.956221i	$-0.594536\pi$
$491$	−12.9692	−0.585293	−0.292646	+	0.956221i	$0.594536\pi$
$492$	0	0
$493$	−0.828376	−0.0373082
$494$	0	0
$495$	29.0341	1.30499
$496$	0	0
$497$	−34.7920	−1.56064
$498$	0	0
$499$	2.71851	0.121697	0.0608486	−	0.998147i	$-0.480619\pi$
$499$	2.71851	0.121697	0.0608486	+	0.998147i	$0.480619\pi$
$500$	0	0
$501$	−4.87183	−0.217657
$502$	0	0
$503$	−15.7310	−0.701412	−0.350706	−	0.936486i	$-0.614058\pi$
$503$	−15.7310	−0.701412	−0.350706	+	0.936486i	$0.614058\pi$
$504$	0	0
$505$	−35.6033	−1.58432
$506$	0	0
$507$	27.1194	1.20441
$508$	0	0
$509$	25.0765	1.11150	0.555749	−	0.831350i	$-0.312432\pi$
$509$	25.0765	1.11150	0.555749	+	0.831350i	$0.312432\pi$
$510$	0	0
$511$	17.7571	0.785526
$512$	0	0
$513$	−9.18808	−0.405664
$514$	0	0
$515$	−43.8976	−1.93436
$516$	0	0
$517$	−33.0186	−1.45216
$518$	0	0
$519$	−7.99608	−0.350989
$520$	0	0
$521$	22.5840	0.989423	0.494712	−	0.869057i	$-0.335274\pi$
$521$	22.5840	0.989423	0.494712	+	0.869057i	$0.335274\pi$
$522$	0	0
$523$	−33.7375	−1.47524	−0.737619	−	0.675217i	$-0.764050\pi$
$523$	−33.7375	−1.47524	−0.737619	+	0.675217i	$0.764050\pi$
$524$	0	0
$525$	−81.2842	−3.54753
$526$	0	0
$527$	5.75629	0.250748
$528$	0	0
$529$	−6.34057	−0.275677
$530$	0	0
$531$	−1.35275	−0.0587042
$532$	0	0
$533$	0.415365	0.0179914
$534$	0	0
$535$	32.4292	1.40203
$536$	0	0
$537$	−22.8275	−0.985079
$538$	0	0
$539$	−44.8710	−1.93273
$540$	0	0
$541$	26.3327	1.13213	0.566066	−	0.824360i	$-0.308465\pi$
$541$	26.3327	1.13213	0.566066	+	0.824360i	$0.308465\pi$
$542$	0	0
$543$	1.51535	0.0650299
$544$	0	0
$545$	53.8674	2.30742
$546$	0	0
$547$	35.4411	1.51535	0.757676	−	0.652631i	$-0.226335\pi$
$547$	35.4411	1.51535	0.757676	+	0.652631i	$0.226335\pi$
$548$	0	0
$549$	−3.47917	−0.148487
$550$	0	0
$551$	−2.21468	−0.0943484
$552$	0	0
$553$	57.3615	2.43926
$554$	0	0
$555$	−47.7371	−2.02633
$556$	0	0
$557$	3.97246	0.168318	0.0841592	−	0.996452i	$-0.473180\pi$
$557$	3.97246	0.168318	0.0841592	+	0.996452i	$0.473180\pi$
$558$	0	0
$559$	−0.00690012	−0.000291844 0
$560$	0	0
$561$	−11.5523	−0.487737
$562$	0	0
$563$	24.4483	1.03037	0.515187	−	0.857078i	$-0.327722\pi$
$563$	24.4483	1.03037	0.515187	+	0.857078i	$0.327722\pi$
$564$	0	0
$565$	77.0541	3.24169
$566$	0	0
$567$	−43.6371	−1.83258
$568$	0	0
$569$	43.9669	1.84319	0.921594	−	0.388156i	$-0.126888\pi$
$569$	43.9669	1.84319	0.921594	+	0.388156i	$0.126888\pi$
$570$	0	0
$571$	−18.0304	−0.754549	−0.377274	−	0.926102i	$-0.623139\pi$
$571$	−18.0304	−0.754549	−0.377274	+	0.926102i	$0.623139\pi$
$572$	0	0
$573$	33.6215	1.40456
$574$	0	0
$575$	−40.9179	−1.70640
$576$	0	0
$577$	−12.2089	−0.508263	−0.254131	−	0.967170i	$-0.581790\pi$
$577$	−12.2089	−0.508263	−0.254131	+	0.967170i	$0.581790\pi$
$578$	0	0
$579$	−43.2429	−1.79712
$580$	0	0
$581$	−58.0145	−2.40685
$582$	0	0
$583$	30.0048	1.24267
$584$	0	0
$585$	0.193699	0.00800848
$586$	0	0
$587$	19.7646	0.815773	0.407886	−	0.913033i	$-0.366266\pi$
$587$	19.7646	0.815773	0.407886	+	0.913033i	$0.366266\pi$
$588$	0	0
$589$	15.3895	0.634115
$590$	0	0
$591$	−56.0757	−2.30665
$592$	0	0
$593$	31.3434	1.28712	0.643559	−	0.765396i	$-0.277457\pi$
$593$	31.3434	1.28712	0.643559	+	0.765396i	$0.277457\pi$
$594$	0	0
$595$	15.0643	0.617575
$596$	0	0
$597$	31.2295	1.27814
$598$	0	0
$599$	10.6312	0.434381	0.217190	−	0.976129i	$-0.430311\pi$
$599$	10.6312	0.434381	0.217190	+	0.976129i	$0.430311\pi$
$600$	0	0
$601$	−2.60298	−0.106178	−0.0530888	−	0.998590i	$-0.516907\pi$
$601$	−2.60298	−0.106178	−0.0530888	+	0.998590i	$0.516907\pi$
$602$	0	0
$603$	−2.51364	−0.102363
$604$	0	0
$605$	−76.2057	−3.09820
$606$	0	0
$607$	35.6985	1.44896	0.724479	−	0.689297i	$-0.242080\pi$
$607$	35.6985	1.44896	0.724479	+	0.689297i	$0.242080\pi$
$608$	0	0
$609$	−6.71661	−0.272171
$610$	0	0
$611$	−0.220282	−0.00891164
$612$	0	0
$613$	49.3466	1.99309	0.996545	−	0.0830491i	$-0.0264658\pi$
$613$	49.3466	1.99309	0.996545	+	0.0830491i	$0.0264658\pi$
$614$	0	0
$615$	−90.9313	−3.66670
$616$	0	0
$617$	19.4006	0.781040	0.390520	−	0.920594i	$-0.372295\pi$
$617$	19.4006	0.781040	0.390520	+	0.920594i	$0.372295\pi$
$618$	0	0
$619$	−2.15293	−0.0865336	−0.0432668	−	0.999064i	$-0.513777\pi$
$619$	−2.15293	−0.0865336	−0.0432668	+	0.999064i	$0.513777\pi$
$620$	0	0
$621$	−14.0272	−0.562894
$622$	0	0
$623$	−37.6727	−1.50933
$624$	0	0
$625$	25.3753	1.01501
$626$	0	0
$627$	−30.8851	−1.23343
$628$	0	0
$629$	5.90293	0.235365
$630$	0	0
$631$	−10.8025	−0.430042	−0.215021	−	0.976609i	$-0.568982\pi$
$631$	−10.8025	−0.430042	−0.215021	+	0.976609i	$0.568982\pi$
$632$	0	0
$633$	−0.537410	−0.0213601
$634$	0	0
$635$	51.7545	2.05381
$636$	0	0
$637$	−0.299354	−0.0118608
$638$	0	0
$639$	−12.1103	−0.479076
$640$	0	0
$641$	6.65398	0.262816	0.131408	−	0.991328i	$-0.458050\pi$
$641$	6.65398	0.262816	0.131408	+	0.991328i	$0.458050\pi$
$642$	0	0
$643$	−15.8482	−0.624993	−0.312497	−	0.949919i	$-0.601165\pi$
$643$	−15.8482	−0.624993	−0.312497	+	0.949919i	$0.601165\pi$
$644$	0	0
$645$	1.51057	0.0594786
$646$	0	0
$647$	0.453623	0.0178338	0.00891688	−	0.999960i	$-0.497162\pi$
$647$	0.453623	0.0178338	0.00891688	+	0.999960i	$0.497162\pi$
$648$	0	0
$649$	5.53713	0.217352
$650$	0	0
$651$	46.6730	1.82926
$652$	0	0
$653$	16.2312	0.635174	0.317587	−	0.948229i	$-0.397127\pi$
$653$	16.2312	0.635174	0.317587	+	0.948229i	$0.397127\pi$
$654$	0	0
$655$	26.6789	1.04243
$656$	0	0
$657$	6.18082	0.241137
$658$	0	0
$659$	−30.7083	−1.19623	−0.598113	−	0.801412i	$-0.704083\pi$
$659$	−30.7083	−1.19623	−0.598113	+	0.801412i	$0.704083\pi$
$660$	0	0
$661$	46.3898	1.80436	0.902178	−	0.431365i	$-0.141968\pi$
$661$	46.3898	1.80436	0.902178	+	0.431365i	$0.141968\pi$
$662$	0	0
$663$	−0.0770702	−0.00299316
$664$	0	0
$665$	40.2745	1.56178
$666$	0	0
$667$	−3.38110	−0.130917
$668$	0	0
$669$	−13.3328	−0.515477
$670$	0	0
$671$	14.2411	0.549772
$672$	0	0
$673$	−13.3192	−0.513417	−0.256708	−	0.966489i	$-0.582638\pi$
$673$	−13.3192	−0.513417	−0.256708	+	0.966489i	$0.582638\pi$
$674$	0	0
$675$	34.4529	1.32609
$676$	0	0
$677$	−21.0567	−0.809276	−0.404638	−	0.914477i	$-0.632602\pi$
$677$	−21.0567	−0.809276	−0.404638	+	0.914477i	$0.632602\pi$
$678$	0	0
$679$	−34.3902	−1.31977
$680$	0	0
$681$	9.41929	0.360948
$682$	0	0
$683$	−20.6139	−0.788771	−0.394385	−	0.918945i	$-0.629042\pi$
$683$	−20.6139	−0.788771	−0.394385	+	0.918945i	$0.629042\pi$
$684$	0	0
$685$	−80.6097	−3.07994
$686$	0	0
$687$	−44.3619	−1.69251
$688$	0	0
$689$	0.200176	0.00762608
$690$	0	0
$691$	0.944254	0.0359211	0.0179606	−	0.999839i	$-0.494283\pi$
$691$	0.944254	0.0359211	0.0179606	+	0.999839i	$0.494283\pi$
$692$	0	0
$693$	−29.1100	−1.10580
$694$	0	0
$695$	54.5421	2.06890
$696$	0	0
$697$	11.2441	0.425901
$698$	0	0
$699$	−38.3047	−1.44881
$700$	0	0
$701$	23.7878	0.898452	0.449226	−	0.893418i	$-0.351700\pi$
$701$	23.7878	0.898452	0.449226	+	0.893418i	$0.351700\pi$
$702$	0	0
$703$	15.7816	0.595213
$704$	0	0
$705$	48.2239	1.81622
$706$	0	0
$707$	35.6964	1.34250
$708$	0	0
$709$	−31.8720	−1.19698	−0.598489	−	0.801131i	$-0.704232\pi$
$709$	−31.8720	−1.19698	−0.598489	+	0.801131i	$0.704232\pi$
$710$	0	0
$711$	19.9662	0.748792
$712$	0	0
$713$	23.4949	0.879890
$714$	0	0
$715$	−0.792861	−0.0296513
$716$	0	0
$717$	−22.0048	−0.821783
$718$	0	0
$719$	−24.0389	−0.896500	−0.448250	−	0.893908i	$-0.647953\pi$
$719$	−24.0389	−0.896500	−0.448250	+	0.893908i	$0.647953\pi$
$720$	0	0
$721$	44.0123	1.63911
$722$	0	0
$723$	−1.94972	−0.0725110
$724$	0	0
$725$	8.30446	0.308420
$726$	0	0
$727$	−25.1006	−0.930930	−0.465465	−	0.885066i	$-0.654113\pi$
$727$	−25.1006	−0.930930	−0.465465	+	0.885066i	$0.654113\pi$
$728$	0	0
$729$	6.32118	0.234118
$730$	0	0
$731$	−0.186789	−0.00690865
$732$	0	0
$733$	1.18280	0.0436876	0.0218438	−	0.999761i	$-0.493046\pi$
$733$	1.18280	0.0436876	0.0218438	+	0.999761i	$0.493046\pi$
$734$	0	0
$735$	65.5344	2.41727
$736$	0	0
$737$	10.2890	0.379000
$738$	0	0
$739$	−26.1663	−0.962544	−0.481272	−	0.876571i	$-0.659825\pi$
$739$	−26.1663	−0.962544	−0.481272	+	0.876571i	$0.659825\pi$
$740$	0	0
$741$	−0.206048	−0.00756938
$742$	0	0
$743$	−13.4560	−0.493652	−0.246826	−	0.969060i	$-0.579388\pi$
$743$	−13.4560	−0.493652	−0.246826	+	0.969060i	$0.579388\pi$
$744$	0	0
$745$	88.9441	3.25866
$746$	0	0
$747$	−20.1935	−0.738842
$748$	0	0
$749$	−32.5139	−1.18803
$750$	0	0
$751$	35.7632	1.30502	0.652509	−	0.757781i	$-0.273717\pi$
$751$	35.7632	1.30502	0.652509	+	0.757781i	$0.273717\pi$
$752$	0	0
$753$	13.7952	0.502725
$754$	0	0
$755$	90.2608	3.28493
$756$	0	0
$757$	−21.7771	−0.791500	−0.395750	−	0.918358i	$-0.629515\pi$
$757$	−21.7771	−0.791500	−0.395750	+	0.918358i	$0.629515\pi$
$758$	0	0
$759$	−47.1516	−1.71150
$760$	0	0
$761$	3.12047	0.113117	0.0565584	−	0.998399i	$-0.481987\pi$
$761$	3.12047	0.113117	0.0565584	+	0.998399i	$0.481987\pi$
$762$	0	0
$763$	−54.0082	−1.95523
$764$	0	0
$765$	5.24352	0.189580
$766$	0	0
$767$	0.0369407	0.00133385
$768$	0	0
$769$	−18.0400	−0.650539	−0.325270	−	0.945621i	$-0.605455\pi$
$769$	−18.0400	−0.650539	−0.325270	+	0.945621i	$0.605455\pi$
$770$	0	0
$771$	−34.6257	−1.24702
$772$	0	0
$773$	4.94504	0.177861	0.0889303	−	0.996038i	$-0.471655\pi$
$773$	4.94504	0.177861	0.0889303	+	0.996038i	$0.471655\pi$
$774$	0	0
$775$	−57.7067	−2.07289
$776$	0	0
$777$	47.8619	1.71704
$778$	0	0
$779$	30.0613	1.07706
$780$	0	0
$781$	49.5706	1.77378
$782$	0	0
$783$	2.84688	0.101739
$784$	0	0
$785$	−1.53569	−0.0548111
$786$	0	0
$787$	32.3053	1.15156	0.575779	−	0.817605i	$-0.304699\pi$
$787$	32.3053	1.15156	0.575779	+	0.817605i	$0.304699\pi$
$788$	0	0
$789$	−16.4298	−0.584918
$790$	0	0
$791$	−77.2555	−2.74689
$792$	0	0
$793$	0.0950088	0.00337386
$794$	0	0
$795$	−43.8223	−1.55422
$796$	0	0
$797$	22.8207	0.808352	0.404176	−	0.914681i	$-0.367558\pi$
$797$	22.8207	0.808352	0.404176	+	0.914681i	$0.367558\pi$
$798$	0	0
$799$	−5.96312	−0.210960
$800$	0	0
$801$	−13.1130	−0.463326
$802$	0	0
$803$	−25.2997	−0.892806
$804$	0	0
$805$	61.4862	2.16711
$806$	0	0
$807$	31.9374	1.12425
$808$	0	0
$809$	18.2339	0.641070	0.320535	−	0.947237i	$-0.396137\pi$
$809$	18.2339	0.641070	0.320535	+	0.947237i	$0.396137\pi$
$810$	0	0
$811$	−21.0171	−0.738011	−0.369006	−	0.929427i	$-0.620302\pi$
$811$	−21.0171	−0.738011	−0.369006	+	0.929427i	$0.620302\pi$
$812$	0	0
$813$	46.3928	1.62707
$814$	0	0
$815$	14.4042	0.504559
$816$	0	0
$817$	−0.499384	−0.0174712
$818$	0	0
$819$	−0.194206	−0.00678610
$820$	0	0
$821$	−26.3854	−0.920855	−0.460428	−	0.887697i	$-0.652304\pi$
$821$	−26.3854	−0.920855	−0.460428	+	0.887697i	$0.652304\pi$
$822$	0	0
$823$	−45.9439	−1.60150	−0.800752	−	0.598996i	$-0.795567\pi$
$823$	−45.9439	−1.60150	−0.800752	+	0.598996i	$0.795567\pi$
$824$	0	0
$825$	115.811	4.03203
$826$	0	0
$827$	−48.6904	−1.69313	−0.846566	−	0.532284i	$-0.821334\pi$
$827$	−48.6904	−1.69313	−0.846566	+	0.532284i	$0.821334\pi$
$828$	0	0
$829$	42.5168	1.47667	0.738335	−	0.674434i	$-0.235612\pi$
$829$	42.5168	1.47667	0.738335	+	0.674434i	$0.235612\pi$
$830$	0	0
$831$	28.3738	0.984276
$832$	0	0
$833$	−8.10365	−0.280775
$834$	0	0
$835$	−9.05143	−0.313238
$836$	0	0
$837$	−19.7827	−0.683790
$838$	0	0
$839$	14.3934	0.496915	0.248457	−	0.968643i	$-0.420076\pi$
$839$	14.3934	0.496915	0.248457	+	0.968643i	$0.420076\pi$
$840$	0	0
$841$	−28.3138	−0.976338
$842$	0	0
$843$	27.8287	0.958473
$844$	0	0
$845$	50.3854	1.73331
$846$	0	0
$847$	76.4049	2.62530
$848$	0	0
$849$	5.31396	0.182375
$850$	0	0
$851$	24.0934	0.825910
$852$	0	0
$853$	−25.1517	−0.861177	−0.430588	−	0.902548i	$-0.641694\pi$
$853$	−25.1517	−0.861177	−0.430588	+	0.902548i	$0.641694\pi$
$854$	0	0
$855$	14.0186	0.479427
$856$	0	0
$857$	−18.3432	−0.626591	−0.313295	−	0.949656i	$-0.601433\pi$
$857$	−18.3432	−0.626591	−0.313295	+	0.949656i	$0.601433\pi$
$858$	0	0
$859$	20.7888	0.709306	0.354653	−	0.934998i	$-0.384599\pi$
$859$	20.7888	0.709306	0.354653	+	0.934998i	$0.384599\pi$
$860$	0	0
$861$	91.1691	3.10703
$862$	0	0
$863$	−47.9496	−1.63222	−0.816112	−	0.577894i	$-0.803875\pi$
$863$	−47.9496	−1.63222	−0.816112	+	0.577894i	$0.803875\pi$
$864$	0	0
$865$	−14.8560	−0.505120
$866$	0	0
$867$	−2.08632	−0.0708553
$868$	0	0
$869$	−81.7269	−2.77239
$870$	0	0
$871$	0.0686424	0.00232586
$872$	0	0
$873$	−11.9704	−0.405137
$874$	0	0
$875$	−75.6976	−2.55905
$876$	0	0
$877$	47.2960	1.59707	0.798536	−	0.601947i	$-0.205608\pi$
$877$	47.2960	1.59707	0.798536	+	0.601947i	$0.205608\pi$
$878$	0	0
$879$	−53.5507	−1.80622
$880$	0	0
$881$	7.23007	0.243587	0.121794	−	0.992555i	$-0.461135\pi$
$881$	7.23007	0.243587	0.121794	+	0.992555i	$0.461135\pi$
$882$	0	0
$883$	−14.6874	−0.494272	−0.247136	−	0.968981i	$-0.579489\pi$
$883$	−14.6874	−0.494272	−0.247136	+	0.968981i	$0.579489\pi$
$884$	0	0
$885$	−8.08702	−0.271842
$886$	0	0
$887$	−44.0362	−1.47859	−0.739296	−	0.673381i	$-0.764841\pi$
$887$	−44.0362	−1.47859	−0.739296	+	0.673381i	$0.764841\pi$
$888$	0	0
$889$	−51.8898	−1.74033
$890$	0	0
$891$	62.1727	2.08286
$892$	0	0
$893$	−15.9425	−0.533495
$894$	0	0
$895$	−42.4115	−1.41766
$896$	0	0
$897$	−0.314569	−0.0105032
$898$	0	0
$899$	−4.76838	−0.159034
$900$	0	0
$901$	5.41884	0.180528
$902$	0	0
$903$	−1.51452	−0.0504000
$904$	0	0
$905$	2.81539	0.0935867
$906$	0	0
$907$	−17.1648	−0.569948	−0.284974	−	0.958535i	$-0.591985\pi$
$907$	−17.1648	−0.569948	−0.284974	+	0.958535i	$0.591985\pi$
$908$	0	0
$909$	12.4251	0.412114
$910$	0	0
$911$	5.14241	0.170376	0.0851879	−	0.996365i	$-0.472851\pi$
$911$	5.14241	0.170376	0.0851879	+	0.996365i	$0.472851\pi$
$912$	0	0
$913$	82.6572	2.73555
$914$	0	0
$915$	−20.7992	−0.687602
$916$	0	0
$917$	−26.7487	−0.883319
$918$	0	0
$919$	50.3726	1.66164	0.830819	−	0.556542i	$-0.187872\pi$
$919$	50.3726	1.66164	0.830819	+	0.556542i	$0.187872\pi$
$920$	0	0
$921$	29.5982	0.975293
$922$	0	0
$923$	0.330707	0.0108854
$924$	0	0
$925$	−59.1768	−1.94572
$926$	0	0
$927$	15.3197	0.503164
$928$	0	0
$929$	21.7284	0.712885	0.356443	−	0.934317i	$-0.383990\pi$
$929$	21.7284	0.712885	0.356443	+	0.934317i	$0.383990\pi$
$930$	0	0
$931$	−21.6652	−0.710049
$932$	0	0
$933$	12.4867	0.408797
$934$	0	0
$935$	−21.4631	−0.701918
$936$	0	0
$937$	−1.02043	−0.0333361	−0.0166681	−	0.999861i	$-0.505306\pi$
$937$	−1.02043	−0.0333361	−0.0166681	+	0.999861i	$0.505306\pi$
$938$	0	0
$939$	−42.3892	−1.38332
$940$	0	0
$941$	4.20375	0.137038	0.0685191	−	0.997650i	$-0.478173\pi$
$941$	4.20375	0.137038	0.0685191	+	0.997650i	$0.478173\pi$
$942$	0	0
$943$	45.8939	1.49451
$944$	0	0
$945$	−51.7714	−1.68412
$946$	0	0
$947$	17.3787	0.564731	0.282365	−	0.959307i	$-0.408881\pi$
$947$	17.3787	0.564731	0.282365	+	0.959307i	$0.408881\pi$
$948$	0	0
$949$	−0.168785	−0.00547901
$950$	0	0
$951$	−62.4750	−2.02589
$952$	0	0
$953$	−15.0335	−0.486984	−0.243492	−	0.969903i	$-0.578293\pi$
$953$	−15.0335	−0.486984	−0.243492	+	0.969903i	$0.578293\pi$
$954$	0	0
$955$	62.4658	2.02135
$956$	0	0
$957$	9.56961	0.309342
$958$	0	0
$959$	80.8204	2.60983
$960$	0	0
$961$	2.13492	0.0688685
$962$	0	0
$963$	−11.3174	−0.364697
$964$	0	0
$965$	−80.3416	−2.58629
$966$	0	0
$967$	5.31753	0.171000	0.0855001	−	0.996338i	$-0.472751\pi$
$967$	5.31753	0.171000	0.0855001	+	0.996338i	$0.472751\pi$
$968$	0	0
$969$	−5.57782	−0.179185
$970$	0	0
$971$	23.0486	0.739664	0.369832	−	0.929099i	$-0.379415\pi$
$971$	23.0486	0.739664	0.369832	+	0.929099i	$0.379415\pi$
$972$	0	0
$973$	−54.6847	−1.75311
$974$	0	0
$975$	0.772627	0.0247439
$976$	0	0
$977$	−58.8117	−1.88155	−0.940776	−	0.339028i	$-0.889902\pi$
$977$	−58.8117	−1.88155	−0.940776	+	0.339028i	$0.889902\pi$
$978$	0	0
$979$	53.6749	1.71546
$980$	0	0
$981$	−18.7990	−0.600206
$982$	0	0
$983$	−57.7568	−1.84216	−0.921079	−	0.389377i	$-0.872690\pi$
$983$	−57.7568	−1.84216	−0.921079	+	0.389377i	$0.872690\pi$
$984$	0	0
$985$	−104.184	−3.31957
$986$	0	0
$987$	−48.3499	−1.53900
$988$	0	0
$989$	−0.762398	−0.0242429
$990$	0	0
$991$	20.5744	0.653568	0.326784	−	0.945099i	$-0.394035\pi$
$991$	20.5744	0.653568	0.326784	+	0.945099i	$0.394035\pi$
$992$	0	0
$993$	−22.0989	−0.701286
$994$	0	0
$995$	58.0216	1.83941
$996$	0	0
$997$	46.6907	1.47871	0.739355	−	0.673316i	$-0.235131\pi$
$997$	46.6907	1.47871	0.739355	+	0.673316i	$0.235131\pi$
$998$	0	0
$999$	−20.2866	−0.641840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.9	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.9	✓	33	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08632 q^{3} -3.87621 q^{5} +3.88634 q^{7} +1.35275 q^{9} +O(q^{10})\)	\(q-2.08632 q^{3} -3.87621 q^{5} +3.88634 q^{7} +1.35275 q^{9} -5.53713 q^{11} -0.0369407 q^{13} +8.08702 q^{15} -1.00000 q^{17} -2.67352 q^{19} -8.10816 q^{21} -4.08160 q^{23} +10.0250 q^{25} +3.43670 q^{27} +0.828376 q^{29} -5.75629 q^{31} +11.5523 q^{33} -15.0643 q^{35} -5.90293 q^{37} +0.0770702 q^{39} -11.2441 q^{41} +0.186789 q^{43} -5.24352 q^{45} +5.96312 q^{47} +8.10365 q^{49} +2.08632 q^{51} -5.41884 q^{53} +21.4631 q^{55} +5.57782 q^{57} -1.00000 q^{59} -2.57193 q^{61} +5.25723 q^{63} +0.143190 q^{65} -1.85818 q^{67} +8.51553 q^{69} -8.95239 q^{71} +4.56909 q^{73} -20.9154 q^{75} -21.5192 q^{77} +14.7598 q^{79} -11.2283 q^{81} -14.9278 q^{83} +3.87621 q^{85} -1.72826 q^{87} -9.69363 q^{89} -0.143564 q^{91} +12.0095 q^{93} +10.3631 q^{95} -8.84898 q^{97} -7.49034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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