LMFDB - Embedded newform 8032.2.a.j.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.369540 q^{3} -3.94367 q^{5} +3.05773 q^{7} -2.86344 q^{9} +O(q^{10})$	$q+0.369540 q^{3} -3.94367 q^{5} +3.05773 q^{7} -2.86344 q^{9} +5.92402 q^{11} +4.39427 q^{13} -1.45734 q^{15} +6.07590 q^{17} +1.60708 q^{19} +1.12995 q^{21} +7.25543 q^{23} +10.5525 q^{25} -2.16677 q^{27} +2.83135 q^{29} -6.95069 q^{31} +2.18916 q^{33} -12.0587 q^{35} +7.42175 q^{37} +1.62386 q^{39} -3.32782 q^{41} +1.53185 q^{43} +11.2925 q^{45} -3.21836 q^{47} +2.34971 q^{49} +2.24529 q^{51} -9.95078 q^{53} -23.3624 q^{55} +0.593879 q^{57} +3.98601 q^{59} +9.96893 q^{61} -8.75563 q^{63} -17.3296 q^{65} -7.23376 q^{67} +2.68117 q^{69} +2.54385 q^{71} +9.26426 q^{73} +3.89958 q^{75} +18.1141 q^{77} -16.8973 q^{79} +7.78961 q^{81} +18.0497 q^{83} -23.9614 q^{85} +1.04629 q^{87} -8.06826 q^{89} +13.4365 q^{91} -2.56856 q^{93} -6.33778 q^{95} +0.446048 q^{97} -16.9631 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * 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
$3$	0.369540	0.213354	0.106677	−	0.994294i	$-0.465979\pi$
$3$	0.369540	0.213354	0.106677	+	0.994294i	$0.465979\pi$
$4$	0	0
$5$	−3.94367	−1.76366	−0.881832	−	0.471565i	$-0.843689\pi$
$5$	−3.94367	−1.76366	−0.881832	+	0.471565i	$0.843689\pi$
$6$	0	0
$7$	3.05773	1.15571	0.577857	−	0.816138i	$-0.303889\pi$
$7$	3.05773	1.15571	0.577857	+	0.816138i	$0.303889\pi$
$8$	0	0
$9$	−2.86344	−0.954480
$10$	0	0
$11$	5.92402	1.78616	0.893080	−	0.449899i	$-0.148540\pi$
$11$	5.92402	1.78616	0.893080	+	0.449899i	$0.148540\pi$
$12$	0	0
$13$	4.39427	1.21875	0.609376	−	0.792881i	$-0.291420\pi$
$13$	4.39427	1.21875	0.609376	+	0.792881i	$0.291420\pi$
$14$	0	0
$15$	−1.45734	−0.376284
$16$	0	0
$17$	6.07590	1.47362	0.736811	−	0.676098i	$-0.236331\pi$
$17$	6.07590	1.47362	0.736811	+	0.676098i	$0.236331\pi$
$18$	0	0
$19$	1.60708	0.368689	0.184344	−	0.982862i	$-0.440984\pi$
$19$	1.60708	0.368689	0.184344	+	0.982862i	$0.440984\pi$
$20$	0	0
$21$	1.12995	0.246576
$22$	0	0
$23$	7.25543	1.51286	0.756431	−	0.654073i	$-0.226941\pi$
$23$	7.25543	1.51286	0.756431	+	0.654073i	$0.226941\pi$
$24$	0	0
$25$	10.5525	2.11051
$26$	0	0
$27$	−2.16677	−0.416996
$28$	0	0
$29$	2.83135	0.525768	0.262884	−	0.964827i	$-0.415326\pi$
$29$	2.83135	0.525768	0.262884	+	0.964827i	$0.415326\pi$
$30$	0	0
$31$	−6.95069	−1.24838	−0.624191	−	0.781272i	$-0.714571\pi$
$31$	−6.95069	−1.24838	−0.624191	+	0.781272i	$0.714571\pi$
$32$	0	0
$33$	2.18916	0.381084
$34$	0	0
$35$	−12.0587	−2.03829
$36$	0	0
$37$	7.42175	1.22013	0.610064	−	0.792352i	$-0.291144\pi$
$37$	7.42175	1.22013	0.610064	+	0.792352i	$0.291144\pi$
$38$	0	0
$39$	1.62386	0.260025
$40$	0	0
$41$	−3.32782	−0.519718	−0.259859	−	0.965647i	$-0.583676\pi$
$41$	−3.32782	−0.519718	−0.259859	+	0.965647i	$0.583676\pi$
$42$	0	0
$43$	1.53185	0.233605	0.116802	−	0.993155i	$-0.462736\pi$
$43$	1.53185	0.233605	0.116802	+	0.993155i	$0.462736\pi$
$44$	0	0
$45$	11.2925	1.68338
$46$	0	0
$47$	−3.21836	−0.469447	−0.234723	−	0.972062i	$-0.575418\pi$
$47$	−3.21836	−0.469447	−0.234723	+	0.972062i	$0.575418\pi$
$48$	0	0
$49$	2.34971	0.335673
$50$	0	0
$51$	2.24529	0.314403
$52$	0	0
$53$	−9.95078	−1.36684	−0.683422	−	0.730023i	$-0.739509\pi$
$53$	−9.95078	−1.36684	−0.683422	+	0.730023i	$0.739509\pi$
$54$	0	0
$55$	−23.3624	−3.15018
$56$	0	0
$57$	0.593879	0.0786611
$58$	0	0
$59$	3.98601	0.518934	0.259467	−	0.965752i	$-0.416453\pi$
$59$	3.98601	0.518934	0.259467	+	0.965752i	$0.416453\pi$
$60$	0	0
$61$	9.96893	1.27639	0.638195	−	0.769874i	$-0.279681\pi$
$61$	9.96893	1.27639	0.638195	+	0.769874i	$0.279681\pi$
$62$	0	0
$63$	−8.75563	−1.10311
$64$	0	0
$65$	−17.3296	−2.14947
$66$	0	0
$67$	−7.23376	−0.883745	−0.441872	−	0.897078i	$-0.645686\pi$
$67$	−7.23376	−0.883745	−0.441872	+	0.897078i	$0.645686\pi$
$68$	0	0
$69$	2.68117	0.322775
$70$	0	0
$71$	2.54385	0.301900	0.150950	−	0.988541i	$-0.451767\pi$
$71$	2.54385	0.301900	0.150950	+	0.988541i	$0.451767\pi$
$72$	0	0
$73$	9.26426	1.08430	0.542150	−	0.840282i	$-0.317611\pi$
$73$	9.26426	1.08430	0.542150	+	0.840282i	$0.317611\pi$
$74$	0	0
$75$	3.89958	0.450285
$76$	0	0
$77$	18.1141	2.06429
$78$	0	0
$79$	−16.8973	−1.90110	−0.950548	−	0.310579i	$-0.899477\pi$
$79$	−16.8973	−1.90110	−0.950548	+	0.310579i	$0.899477\pi$
$80$	0	0
$81$	7.78961	0.865513
$82$	0	0
$83$	18.0497	1.98121	0.990607	−	0.136743i	$-0.0436633\pi$
$83$	18.0497	1.98121	0.990607	+	0.136743i	$0.0436633\pi$
$84$	0	0
$85$	−23.9614	−2.59897
$86$	0	0
$87$	1.04629	0.112175
$88$	0	0
$89$	−8.06826	−0.855234	−0.427617	−	0.903960i	$-0.640647\pi$
$89$	−8.06826	−0.855234	−0.427617	+	0.903960i	$0.640647\pi$
$90$	0	0
$91$	13.4365	1.40853
$92$	0	0
$93$	−2.56856	−0.266347
$94$	0	0
$95$	−6.33778	−0.650243
$96$	0	0
$97$	0.446048	0.0452893	0.0226447	−	0.999744i	$-0.492791\pi$
$97$	0.446048	0.0452893	0.0226447	+	0.999744i	$0.492791\pi$
$98$	0	0
$99$	−16.9631	−1.70485
$100$	0	0
$101$	−1.60941	−0.160142	−0.0800710	−	0.996789i	$-0.525515\pi$
$101$	−1.60941	−0.160142	−0.0800710	+	0.996789i	$0.525515\pi$
$102$	0	0
$103$	−4.83414	−0.476322	−0.238161	−	0.971226i	$-0.576545\pi$
$103$	−4.83414	−0.476322	−0.238161	+	0.971226i	$0.576545\pi$
$104$	0	0
$105$	−4.45616	−0.434877
$106$	0	0
$107$	−11.3410	−1.09638	−0.548190	−	0.836354i	$-0.684683\pi$
$107$	−11.3410	−1.09638	−0.548190	+	0.836354i	$0.684683\pi$
$108$	0	0
$109$	−19.1000	−1.82944	−0.914722	−	0.404083i	$-0.867591\pi$
$109$	−19.1000	−1.82944	−0.914722	+	0.404083i	$0.867591\pi$
$110$	0	0
$111$	2.74263	0.260319
$112$	0	0
$113$	−3.26781	−0.307410	−0.153705	−	0.988117i	$-0.549121\pi$
$113$	−3.26781	−0.307410	−0.153705	+	0.988117i	$0.549121\pi$
$114$	0	0
$115$	−28.6130	−2.66818
$116$	0	0
$117$	−12.5827	−1.16327
$118$	0	0
$119$	18.5785	1.70309
$120$	0	0
$121$	24.0940	2.19036
$122$	0	0
$123$	−1.22976	−0.110884
$124$	0	0
$125$	−21.8974	−1.95856
$126$	0	0
$127$	−4.63876	−0.411624	−0.205812	−	0.978592i	$-0.565983\pi$
$127$	−4.63876	−0.411624	−0.205812	+	0.978592i	$0.565983\pi$
$128$	0	0
$129$	0.566079	0.0498405
$130$	0	0
$131$	14.0180	1.22476	0.612381	−	0.790563i	$-0.290212\pi$
$131$	14.0180	1.22476	0.612381	+	0.790563i	$0.290212\pi$
$132$	0	0
$133$	4.91401	0.426098
$134$	0	0
$135$	8.54504	0.735440
$136$	0	0
$137$	16.2541	1.38868	0.694339	−	0.719648i	$-0.255697\pi$
$137$	16.2541	1.38868	0.694339	+	0.719648i	$0.255697\pi$
$138$	0	0
$139$	9.16009	0.776949	0.388475	−	0.921459i	$-0.373002\pi$
$139$	9.16009	0.776949	0.388475	+	0.921459i	$0.373002\pi$
$140$	0	0
$141$	−1.18931	−0.100158
$142$	0	0
$143$	26.0318	2.17688
$144$	0	0
$145$	−11.1659	−0.927277
$146$	0	0
$147$	0.868311	0.0716171
$148$	0	0
$149$	15.6797	1.28453	0.642267	−	0.766481i	$-0.277994\pi$
$149$	15.6797	1.28453	0.642267	+	0.766481i	$0.277994\pi$
$150$	0	0
$151$	−2.47612	−0.201504	−0.100752	−	0.994912i	$-0.532125\pi$
$151$	−2.47612	−0.201504	−0.100752	+	0.994912i	$0.532125\pi$
$152$	0	0
$153$	−17.3980	−1.40654
$154$	0	0
$155$	27.4112	2.20172
$156$	0	0
$157$	−8.48157	−0.676903	−0.338452	−	0.940984i	$-0.609903\pi$
$157$	−8.48157	−0.676903	−0.338452	+	0.940984i	$0.609903\pi$
$158$	0	0
$159$	−3.67721	−0.291622
$160$	0	0
$161$	22.1852	1.74843
$162$	0	0
$163$	−7.23635	−0.566795	−0.283397	−	0.959003i	$-0.591461\pi$
$163$	−7.23635	−0.566795	−0.283397	+	0.959003i	$0.591461\pi$
$164$	0	0
$165$	−8.63333	−0.672103
$166$	0	0
$167$	−16.3252	−1.26328	−0.631640	−	0.775262i	$-0.717618\pi$
$167$	−16.3252	−1.26328	−0.631640	+	0.775262i	$0.717618\pi$
$168$	0	0
$169$	6.30963	0.485356
$170$	0	0
$171$	−4.60177	−0.351906
$172$	0	0
$173$	25.6835	1.95268	0.976340	−	0.216243i	$-0.0693803\pi$
$173$	25.6835	1.95268	0.976340	+	0.216243i	$0.0693803\pi$
$174$	0	0
$175$	32.2668	2.43914
$176$	0	0
$177$	1.47299	0.110717
$178$	0	0
$179$	20.4746	1.53034	0.765171	−	0.643827i	$-0.222654\pi$
$179$	20.4746	1.53034	0.765171	+	0.643827i	$0.222654\pi$
$180$	0	0
$181$	−8.49134	−0.631156	−0.315578	−	0.948900i	$-0.602198\pi$
$181$	−8.49134	−0.631156	−0.315578	+	0.948900i	$0.602198\pi$
$182$	0	0
$183$	3.68391	0.272323
$184$	0	0
$185$	−29.2689	−2.15190
$186$	0	0
$187$	35.9938	2.63212
$188$	0	0
$189$	−6.62541	−0.481927
$190$	0	0
$191$	−8.17997	−0.591882	−0.295941	−	0.955206i	$-0.595633\pi$
$191$	−8.17997	−0.591882	−0.295941	+	0.955206i	$0.595633\pi$
$192$	0	0
$193$	−23.2476	−1.67340	−0.836701	−	0.547660i	$-0.815519\pi$
$193$	−23.2476	−1.67340	−0.836701	+	0.547660i	$0.815519\pi$
$194$	0	0
$195$	−6.40396	−0.458597
$196$	0	0
$197$	7.00357	0.498983	0.249492	−	0.968377i	$-0.419736\pi$
$197$	7.00357	0.498983	0.249492	+	0.968377i	$0.419736\pi$
$198$	0	0
$199$	−7.47544	−0.529920	−0.264960	−	0.964259i	$-0.585359\pi$
$199$	−7.47544	−0.529920	−0.264960	+	0.964259i	$0.585359\pi$
$200$	0	0
$201$	−2.67316	−0.188550
$202$	0	0
$203$	8.65749	0.607637
$204$	0	0
$205$	13.1238	0.916608
$206$	0	0
$207$	−20.7755	−1.44400
$208$	0	0
$209$	9.52036	0.658537
$210$	0	0
$211$	9.20708	0.633842	0.316921	−	0.948452i	$-0.397351\pi$
$211$	9.20708	0.633842	0.316921	+	0.948452i	$0.397351\pi$
$212$	0	0
$213$	0.940055	0.0644115
$214$	0	0
$215$	−6.04111	−0.412000
$216$	0	0
$217$	−21.2533	−1.44277
$218$	0	0
$219$	3.42351	0.231339
$220$	0	0
$221$	26.6992	1.79598
$222$	0	0
$223$	26.6574	1.78511	0.892556	−	0.450937i	$-0.148910\pi$
$223$	26.6574	1.78511	0.892556	+	0.450937i	$0.148910\pi$
$224$	0	0
$225$	−30.2166	−2.01444
$226$	0	0
$227$	−4.10867	−0.272702	−0.136351	−	0.990661i	$-0.543537\pi$
$227$	−4.10867	−0.272702	−0.136351	+	0.990661i	$0.543537\pi$
$228$	0	0
$229$	−13.3217	−0.880322	−0.440161	−	0.897919i	$-0.645079\pi$
$229$	−13.3217	−0.880322	−0.440161	+	0.897919i	$0.645079\pi$
$230$	0	0
$231$	6.69386	0.440424
$232$	0	0
$233$	−13.6800	−0.896207	−0.448104	−	0.893982i	$-0.647900\pi$
$233$	−13.6800	−0.896207	−0.448104	+	0.893982i	$0.647900\pi$
$234$	0	0
$235$	12.6922	0.827946
$236$	0	0
$237$	−6.24422	−0.405606
$238$	0	0
$239$	8.59306	0.555839	0.277919	−	0.960604i	$-0.410355\pi$
$239$	8.59306	0.555839	0.277919	+	0.960604i	$0.410355\pi$
$240$	0	0
$241$	−14.4861	−0.933129	−0.466565	−	0.884487i	$-0.654508\pi$
$241$	−14.4861	−0.933129	−0.466565	+	0.884487i	$0.654508\pi$
$242$	0	0
$243$	9.37889	0.601656
$244$	0	0
$245$	−9.26648	−0.592014
$246$	0	0
$247$	7.06193	0.449340
$248$	0	0
$249$	6.67008	0.422699
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	42.9813	2.70221
$254$	0	0
$255$	−8.85467	−0.554501
$256$	0	0
$257$	11.1794	0.697351	0.348676	−	0.937243i	$-0.386632\pi$
$257$	11.1794	0.697351	0.348676	+	0.937243i	$0.386632\pi$
$258$	0	0
$259$	22.6937	1.41012
$260$	0	0
$261$	−8.10739	−0.501835
$262$	0	0
$263$	−31.7585	−1.95831	−0.979156	−	0.203110i	$-0.934895\pi$
$263$	−31.7585	−1.95831	−0.979156	+	0.203110i	$0.934895\pi$
$264$	0	0
$265$	39.2426	2.41065
$266$	0	0
$267$	−2.98154	−0.182467
$268$	0	0
$269$	0.305820	0.0186462	0.00932308	−	0.999957i	$-0.497032\pi$
$269$	0.305820	0.0186462	0.00932308	+	0.999957i	$0.497032\pi$
$270$	0	0
$271$	26.9138	1.63490	0.817449	−	0.576001i	$-0.195387\pi$
$271$	26.9138	1.63490	0.817449	+	0.576001i	$0.195387\pi$
$272$	0	0
$273$	4.96532	0.300515
$274$	0	0
$275$	62.5134	3.76970
$276$	0	0
$277$	−7.43808	−0.446911	−0.223455	−	0.974714i	$-0.571734\pi$
$277$	−7.43808	−0.446911	−0.223455	+	0.974714i	$0.571734\pi$
$278$	0	0
$279$	19.9029	1.19156
$280$	0	0
$281$	10.6121	0.633063	0.316531	−	0.948582i	$-0.397482\pi$
$281$	10.6121	0.633063	0.316531	+	0.948582i	$0.397482\pi$
$282$	0	0
$283$	−16.1412	−0.959496	−0.479748	−	0.877406i	$-0.659272\pi$
$283$	−16.1412	−0.959496	−0.479748	+	0.877406i	$0.659272\pi$
$284$	0	0
$285$	−2.34206	−0.138732
$286$	0	0
$287$	−10.1756	−0.600645
$288$	0	0
$289$	19.9166	1.17156
$290$	0	0
$291$	0.164832	0.00966265
$292$	0	0
$293$	−4.63688	−0.270889	−0.135445	−	0.990785i	$-0.543246\pi$
$293$	−4.63688	−0.270889	−0.135445	+	0.990785i	$0.543246\pi$
$294$	0	0
$295$	−15.7195	−0.915225
$296$	0	0
$297$	−12.8360	−0.744821
$298$	0	0
$299$	31.8823	1.84380
$300$	0	0
$301$	4.68398	0.269980
$302$	0	0
$303$	−0.594740	−0.0341669
$304$	0	0
$305$	−39.3142	−2.25112
$306$	0	0
$307$	0.239627	0.0136762	0.00683811	−	0.999977i	$-0.497823\pi$
$307$	0.239627	0.0136762	0.00683811	+	0.999977i	$0.497823\pi$
$308$	0	0
$309$	−1.78641	−0.101625
$310$	0	0
$311$	15.8574	0.899193	0.449597	−	0.893232i	$-0.351568\pi$
$311$	15.8574	0.899193	0.449597	+	0.893232i	$0.351568\pi$
$312$	0	0
$313$	5.84043	0.330121	0.165060	−	0.986283i	$-0.447218\pi$
$313$	5.84043	0.330121	0.165060	+	0.986283i	$0.447218\pi$
$314$	0	0
$315$	34.5293	1.94551
$316$	0	0
$317$	18.9494	1.06430	0.532151	−	0.846649i	$-0.321384\pi$
$317$	18.9494	1.06430	0.532151	+	0.846649i	$0.321384\pi$
$318$	0	0
$319$	16.7730	0.939105
$320$	0	0
$321$	−4.19096	−0.233917
$322$	0	0
$323$	9.76444	0.543308
$324$	0	0
$325$	46.3707	2.57218
$326$	0	0
$327$	−7.05819	−0.390319
$328$	0	0
$329$	−9.84089	−0.542546
$330$	0	0
$331$	29.2478	1.60760	0.803801	−	0.594898i	$-0.202808\pi$
$331$	29.2478	1.60760	0.803801	+	0.594898i	$0.202808\pi$
$332$	0	0
$333$	−21.2517	−1.16459
$334$	0	0
$335$	28.5276	1.55863
$336$	0	0
$337$	19.8088	1.07905	0.539526	−	0.841969i	$-0.318603\pi$
$337$	19.8088	1.07905	0.539526	+	0.841969i	$0.318603\pi$
$338$	0	0
$339$	−1.20759	−0.0655871
$340$	0	0
$341$	−41.1761	−2.22981
$342$	0	0
$343$	−14.2193	−0.767772
$344$	0	0
$345$	−10.5736	−0.569266
$346$	0	0
$347$	16.3124	0.875697	0.437848	−	0.899049i	$-0.355741\pi$
$347$	16.3124	0.875697	0.437848	+	0.899049i	$0.355741\pi$
$348$	0	0
$349$	−6.90115	−0.369410	−0.184705	−	0.982794i	$-0.559133\pi$
$349$	−6.90115	−0.369410	−0.184705	+	0.982794i	$0.559133\pi$
$350$	0	0
$351$	−9.52139	−0.508214
$352$	0	0
$353$	−5.27935	−0.280992	−0.140496	−	0.990081i	$-0.544870\pi$
$353$	−5.27935	−0.280992	−0.140496	+	0.990081i	$0.544870\pi$
$354$	0	0
$355$	−10.0321	−0.532450
$356$	0	0
$357$	6.86548	0.363360
$358$	0	0
$359$	11.5465	0.609403	0.304701	−	0.952448i	$-0.401443\pi$
$359$	11.5465	0.609403	0.304701	+	0.952448i	$0.401443\pi$
$360$	0	0
$361$	−16.4173	−0.864069
$362$	0	0
$363$	8.90369	0.467323
$364$	0	0
$365$	−36.5352	−1.91234
$366$	0	0
$367$	−29.0407	−1.51591	−0.757956	−	0.652306i	$-0.773802\pi$
$367$	−29.0407	−1.51591	−0.757956	+	0.652306i	$0.773802\pi$
$368$	0	0
$369$	9.52902	0.496061
$370$	0	0
$371$	−30.4268	−1.57968
$372$	0	0
$373$	10.4300	0.540045	0.270022	−	0.962854i	$-0.412969\pi$
$373$	10.4300	0.540045	0.270022	+	0.962854i	$0.412969\pi$
$374$	0	0
$375$	−8.09195	−0.417866
$376$	0	0
$377$	12.4417	0.640780
$378$	0	0
$379$	−0.849878	−0.0436553	−0.0218276	−	0.999762i	$-0.506949\pi$
$379$	−0.849878	−0.0436553	−0.0218276	+	0.999762i	$0.506949\pi$
$380$	0	0
$381$	−1.71421	−0.0878214
$382$	0	0
$383$	−36.5197	−1.86607	−0.933035	−	0.359786i	$-0.882850\pi$
$383$	−36.5197	−1.86607	−0.933035	+	0.359786i	$0.882850\pi$
$384$	0	0
$385$	−71.4358	−3.64071
$386$	0	0
$387$	−4.38636	−0.222971
$388$	0	0
$389$	18.0677	0.916071	0.458036	−	0.888934i	$-0.348553\pi$
$389$	18.0677	0.916071	0.458036	+	0.888934i	$0.348553\pi$
$390$	0	0
$391$	44.0833	2.22939
$392$	0	0
$393$	5.18022	0.261308
$394$	0	0
$395$	66.6374	3.35289
$396$	0	0
$397$	−32.8086	−1.64661	−0.823307	−	0.567596i	$-0.807874\pi$
$397$	−32.8086	−1.64661	−0.823307	+	0.567596i	$0.807874\pi$
$398$	0	0
$399$	1.81592	0.0909097
$400$	0	0
$401$	−9.82305	−0.490540	−0.245270	−	0.969455i	$-0.578877\pi$
$401$	−9.82305	−0.490540	−0.245270	+	0.969455i	$0.578877\pi$
$402$	0	0
$403$	−30.5432	−1.52147
$404$	0	0
$405$	−30.7197	−1.52647
$406$	0	0
$407$	43.9666	2.17934
$408$	0	0
$409$	−10.1477	−0.501769	−0.250885	−	0.968017i	$-0.580721\pi$
$409$	−10.1477	−0.501769	−0.250885	+	0.968017i	$0.580721\pi$
$410$	0	0
$411$	6.00652	0.296280
$412$	0	0
$413$	12.1881	0.599739
$414$	0	0
$415$	−71.1821	−3.49419
$416$	0	0
$417$	3.38502	0.165765
$418$	0	0
$419$	11.6913	0.571157	0.285578	−	0.958355i	$-0.407814\pi$
$419$	11.6913	0.571157	0.285578	+	0.958355i	$0.407814\pi$
$420$	0	0
$421$	20.9140	1.01929	0.509644	−	0.860385i	$-0.329777\pi$
$421$	20.9140	1.01929	0.509644	+	0.860385i	$0.329777\pi$
$422$	0	0
$423$	9.21559	0.448077
$424$	0	0
$425$	64.1162	3.11009
$426$	0	0
$427$	30.4823	1.47514
$428$	0	0
$429$	9.61976	0.464447
$430$	0	0
$431$	−33.0950	−1.59413	−0.797065	−	0.603894i	$-0.793615\pi$
$431$	−33.0950	−1.59413	−0.797065	+	0.603894i	$0.793615\pi$
$432$	0	0
$433$	13.4372	0.645751	0.322875	−	0.946442i	$-0.395351\pi$
$433$	13.4372	0.645751	0.322875	+	0.946442i	$0.395351\pi$
$434$	0	0
$435$	−4.12624	−0.197838
$436$	0	0
$437$	11.6600	0.557775
$438$	0	0
$439$	3.13478	0.149615	0.0748075	−	0.997198i	$-0.476166\pi$
$439$	3.13478	0.149615	0.0748075	+	0.997198i	$0.476166\pi$
$440$	0	0
$441$	−6.72826	−0.320393
$442$	0	0
$443$	36.3033	1.72482	0.862412	−	0.506207i	$-0.168953\pi$
$443$	36.3033	1.72482	0.862412	+	0.506207i	$0.168953\pi$
$444$	0	0
$445$	31.8186	1.50834
$446$	0	0
$447$	5.79429	0.274060
$448$	0	0
$449$	29.2575	1.38075	0.690373	−	0.723454i	$-0.257446\pi$
$449$	29.2575	1.38075	0.690373	+	0.723454i	$0.257446\pi$
$450$	0	0
$451$	−19.7141	−0.928300
$452$	0	0
$453$	−0.915026	−0.0429917
$454$	0	0
$455$	−52.9891	−2.48417
$456$	0	0
$457$	7.72948	0.361570	0.180785	−	0.983523i	$-0.442136\pi$
$457$	7.72948	0.361570	0.180785	+	0.983523i	$0.442136\pi$
$458$	0	0
$459$	−13.1651	−0.614494
$460$	0	0
$461$	−2.75635	−0.128376	−0.0641881	−	0.997938i	$-0.520446\pi$
$461$	−2.75635	−0.128376	−0.0641881	+	0.997938i	$0.520446\pi$
$462$	0	0
$463$	34.9847	1.62588	0.812939	−	0.582349i	$-0.197866\pi$
$463$	34.9847	1.62588	0.812939	+	0.582349i	$0.197866\pi$
$464$	0	0
$465$	10.1295	0.469746
$466$	0	0
$467$	4.60272	0.212988	0.106494	−	0.994313i	$-0.466037\pi$
$467$	4.60272	0.212988	0.106494	+	0.994313i	$0.466037\pi$
$468$	0	0
$469$	−22.1189	−1.02136
$470$	0	0
$471$	−3.13428	−0.144420
$472$	0	0
$473$	9.07471	0.417255
$474$	0	0
$475$	16.9587	0.778120
$476$	0	0
$477$	28.4935	1.30463
$478$	0	0
$479$	−34.6775	−1.58445	−0.792227	−	0.610226i	$-0.791079\pi$
$479$	−34.6775	−1.58445	−0.792227	+	0.610226i	$0.791079\pi$
$480$	0	0
$481$	32.6132	1.48703
$482$	0	0
$483$	8.19829	0.373035
$484$	0	0
$485$	−1.75907	−0.0798751
$486$	0	0
$487$	−9.75884	−0.442215	−0.221108	−	0.975249i	$-0.570967\pi$
$487$	−9.75884	−0.442215	−0.221108	+	0.975249i	$0.570967\pi$
$488$	0	0
$489$	−2.67412	−0.120928
$490$	0	0
$491$	−6.80005	−0.306882	−0.153441	−	0.988158i	$-0.549036\pi$
$491$	−6.80005	−0.306882	−0.153441	+	0.988158i	$0.549036\pi$
$492$	0	0
$493$	17.2030	0.774783
$494$	0	0
$495$	66.8968	3.00679
$496$	0	0
$497$	7.77842	0.348910
$498$	0	0
$499$	−28.3178	−1.26768	−0.633839	−	0.773465i	$-0.718522\pi$
$499$	−28.3178	−1.26768	−0.633839	+	0.773465i	$0.718522\pi$
$500$	0	0
$501$	−6.03280	−0.269525
$502$	0	0
$503$	5.62529	0.250819	0.125410	−	0.992105i	$-0.459975\pi$
$503$	5.62529	0.250819	0.125410	+	0.992105i	$0.459975\pi$
$504$	0	0
$505$	6.34697	0.282437
$506$	0	0
$507$	2.33166	0.103553
$508$	0	0
$509$	−9.56364	−0.423901	−0.211951	−	0.977280i	$-0.567982\pi$
$509$	−9.56364	−0.423901	−0.211951	+	0.977280i	$0.567982\pi$
$510$	0	0
$511$	28.3276	1.25314
$512$	0	0
$513$	−3.48217	−0.153742
$514$	0	0
$515$	19.0643	0.840072
$516$	0	0
$517$	−19.0657	−0.838506
$518$	0	0
$519$	9.49106	0.416612
$520$	0	0
$521$	−11.8396	−0.518704	−0.259352	−	0.965783i	$-0.583509\pi$
$521$	−11.8396	−0.518704	−0.259352	+	0.965783i	$0.583509\pi$
$522$	0	0
$523$	−1.35740	−0.0593548	−0.0296774	−	0.999560i	$-0.509448\pi$
$523$	−1.35740	−0.0593548	−0.0296774	+	0.999560i	$0.509448\pi$
$524$	0	0
$525$	11.9239	0.520400
$526$	0	0
$527$	−42.2317	−1.83964
$528$	0	0
$529$	29.6413	1.28875
$530$	0	0
$531$	−11.4137	−0.495313
$532$	0	0
$533$	−14.6233	−0.633408
$534$	0	0
$535$	44.7253	1.93364
$536$	0	0
$537$	7.56617	0.326504
$538$	0	0
$539$	13.9197	0.599565
$540$	0	0
$541$	2.43302	0.104604	0.0523019	−	0.998631i	$-0.483344\pi$
$541$	2.43302	0.104604	0.0523019	+	0.998631i	$0.483344\pi$
$542$	0	0
$543$	−3.13789	−0.134660
$544$	0	0
$545$	75.3240	3.22652
$546$	0	0
$547$	13.1639	0.562847	0.281423	−	0.959584i	$-0.409193\pi$
$547$	13.1639	0.562847	0.281423	+	0.959584i	$0.409193\pi$
$548$	0	0
$549$	−28.5454	−1.21829
$550$	0	0
$551$	4.55019	0.193845
$552$	0	0
$553$	−51.6674	−2.19712
$554$	0	0
$555$	−10.8160	−0.459115
$556$	0	0
$557$	−25.8324	−1.09455	−0.547276	−	0.836952i	$-0.684335\pi$
$557$	−25.8324	−1.09455	−0.547276	+	0.836952i	$0.684335\pi$
$558$	0	0
$559$	6.73136	0.284706
$560$	0	0
$561$	13.3011	0.561574
$562$	0	0
$563$	−24.3036	−1.02428	−0.512138	−	0.858903i	$-0.671146\pi$
$563$	−24.3036	−1.02428	−0.512138	+	0.858903i	$0.671146\pi$
$564$	0	0
$565$	12.8872	0.542168
$566$	0	0
$567$	23.8185	1.00028
$568$	0	0
$569$	34.1845	1.43309	0.716545	−	0.697540i	$-0.245722\pi$
$569$	34.1845	1.43309	0.716545	+	0.697540i	$0.245722\pi$
$570$	0	0
$571$	0.293924	0.0123004	0.00615018	−	0.999981i	$-0.498042\pi$
$571$	0.293924	0.0123004	0.00615018	+	0.999981i	$0.498042\pi$
$572$	0	0
$573$	−3.02282	−0.126280
$574$	0	0
$575$	76.5632	3.19291
$576$	0	0
$577$	11.6086	0.483272	0.241636	−	0.970367i	$-0.422316\pi$
$577$	11.6086	0.483272	0.241636	+	0.970367i	$0.422316\pi$
$578$	0	0
$579$	−8.59092	−0.357027
$580$	0	0
$581$	55.1911	2.28971
$582$	0	0
$583$	−58.9486	−2.44140
$584$	0	0
$585$	49.6222	2.05162
$586$	0	0
$587$	−8.32180	−0.343478	−0.171739	−	0.985143i	$-0.554938\pi$
$587$	−8.32180	−0.343478	−0.171739	+	0.985143i	$0.554938\pi$
$588$	0	0
$589$	−11.1703	−0.460264
$590$	0	0
$591$	2.58810	0.106460
$592$	0	0
$593$	−31.7923	−1.30555	−0.652776	−	0.757551i	$-0.726396\pi$
$593$	−31.7923	−1.30555	−0.652776	+	0.757551i	$0.726396\pi$
$594$	0	0
$595$	−73.2673	−3.00367
$596$	0	0
$597$	−2.76247	−0.113060
$598$	0	0
$599$	−23.6163	−0.964935	−0.482468	−	0.875914i	$-0.660259\pi$
$599$	−23.6163	−0.964935	−0.482468	+	0.875914i	$0.660259\pi$
$600$	0	0
$601$	−34.2352	−1.39648	−0.698241	−	0.715863i	$-0.746034\pi$
$601$	−34.2352	−1.39648	−0.698241	+	0.715863i	$0.746034\pi$
$602$	0	0
$603$	20.7134	0.843517
$604$	0	0
$605$	−95.0188	−3.86307
$606$	0	0
$607$	−40.5223	−1.64475	−0.822375	−	0.568946i	$-0.807351\pi$
$607$	−40.5223	−1.64475	−0.822375	+	0.568946i	$0.807351\pi$
$608$	0	0
$609$	3.19929	0.129642
$610$	0	0
$611$	−14.1424	−0.572139
$612$	0	0
$613$	35.4084	1.43013	0.715067	−	0.699056i	$-0.246396\pi$
$613$	35.4084	1.43013	0.715067	+	0.699056i	$0.246396\pi$
$614$	0	0
$615$	4.84977	0.195562
$616$	0	0
$617$	−26.8652	−1.08155	−0.540776	−	0.841167i	$-0.681869\pi$
$617$	−26.8652	−1.08155	−0.540776	+	0.841167i	$0.681869\pi$
$618$	0	0
$619$	−3.94200	−0.158442	−0.0792211	−	0.996857i	$-0.525243\pi$
$619$	−3.94200	−0.158442	−0.0792211	+	0.996857i	$0.525243\pi$
$620$	0	0
$621$	−15.7209	−0.630857
$622$	0	0
$623$	−24.6706	−0.988405
$624$	0	0
$625$	33.5933	1.34373
$626$	0	0
$627$	3.51815	0.140501
$628$	0	0
$629$	45.0938	1.79801
$630$	0	0
$631$	24.4000	0.971350	0.485675	−	0.874140i	$-0.338574\pi$
$631$	24.4000	0.971350	0.485675	+	0.874140i	$0.338574\pi$
$632$	0	0
$633$	3.40238	0.135233
$634$	0	0
$635$	18.2937	0.725965
$636$	0	0
$637$	10.3253	0.409102
$638$	0	0
$639$	−7.28417	−0.288157
$640$	0	0
$641$	35.6826	1.40938	0.704688	−	0.709517i	$-0.251087\pi$
$641$	35.6826	1.40938	0.704688	+	0.709517i	$0.251087\pi$
$642$	0	0
$643$	4.84993	0.191263	0.0956314	−	0.995417i	$-0.469513\pi$
$643$	4.84993	0.191263	0.0956314	+	0.995417i	$0.469513\pi$
$644$	0	0
$645$	−2.23243	−0.0879018
$646$	0	0
$647$	13.8494	0.544478	0.272239	−	0.962230i	$-0.412236\pi$
$647$	13.8494	0.544478	0.272239	+	0.962230i	$0.412236\pi$
$648$	0	0
$649$	23.6132	0.926899
$650$	0	0
$651$	−7.85395	−0.307821
$652$	0	0
$653$	24.6033	0.962803	0.481402	−	0.876500i	$-0.340128\pi$
$653$	24.6033	0.962803	0.481402	+	0.876500i	$0.340128\pi$
$654$	0	0
$655$	−55.2825	−2.16007
$656$	0	0
$657$	−26.5277	−1.03494
$658$	0	0
$659$	−6.19378	−0.241275	−0.120638	−	0.992697i	$-0.538494\pi$
$659$	−6.19378	−0.241275	−0.120638	+	0.992697i	$0.538494\pi$
$660$	0	0
$661$	−0.647856	−0.0251987	−0.0125993	−	0.999921i	$-0.504011\pi$
$661$	−0.647856	−0.0251987	−0.0125993	+	0.999921i	$0.504011\pi$
$662$	0	0
$663$	9.86640	0.383179
$664$	0	0
$665$	−19.3792	−0.751494
$666$	0	0
$667$	20.5426	0.795414
$668$	0	0
$669$	9.85096	0.380860
$670$	0	0
$671$	59.0561	2.27984
$672$	0	0
$673$	−28.0680	−1.08194	−0.540970	−	0.841042i	$-0.681943\pi$
$673$	−28.0680	−1.08194	−0.540970	+	0.841042i	$0.681943\pi$
$674$	0	0
$675$	−22.8650	−0.880073
$676$	0	0
$677$	−4.37469	−0.168133	−0.0840666	−	0.996460i	$-0.526791\pi$
$677$	−4.37469	−0.168133	−0.0840666	+	0.996460i	$0.526791\pi$
$678$	0	0
$679$	1.36389	0.0523415
$680$	0	0
$681$	−1.51832	−0.0581820
$682$	0	0
$683$	−3.67172	−0.140495	−0.0702473	−	0.997530i	$-0.522379\pi$
$683$	−3.67172	−0.140495	−0.0702473	+	0.997530i	$0.522379\pi$
$684$	0	0
$685$	−64.1007	−2.44916
$686$	0	0
$687$	−4.92289	−0.187820
$688$	0	0
$689$	−43.7264	−1.66584
$690$	0	0
$691$	−7.74468	−0.294621	−0.147311	−	0.989090i	$-0.547062\pi$
$691$	−7.74468	−0.294621	−0.147311	+	0.989090i	$0.547062\pi$
$692$	0	0
$693$	−51.8685	−1.97032
$694$	0	0
$695$	−36.1244	−1.37028
$696$	0	0
$697$	−20.2195	−0.765869
$698$	0	0
$699$	−5.05530	−0.191209
$700$	0	0
$701$	−46.6675	−1.76261	−0.881304	−	0.472550i	$-0.843334\pi$
$701$	−46.6675	−1.76261	−0.881304	+	0.472550i	$0.843334\pi$
$702$	0	0
$703$	11.9273	0.449848
$704$	0	0
$705$	4.69026	0.176645
$706$	0	0
$707$	−4.92113	−0.185078
$708$	0	0
$709$	36.9614	1.38811	0.694057	−	0.719920i	$-0.255822\pi$
$709$	36.9614	1.38811	0.694057	+	0.719920i	$0.255822\pi$
$710$	0	0
$711$	48.3844	1.81456
$712$	0	0
$713$	−50.4303	−1.88863
$714$	0	0
$715$	−102.661	−3.83929
$716$	0	0
$717$	3.17548	0.118590
$718$	0	0
$719$	30.6710	1.14383	0.571917	−	0.820312i	$-0.306200\pi$
$719$	30.6710	1.14383	0.571917	+	0.820312i	$0.306200\pi$
$720$	0	0
$721$	−14.7815	−0.550492
$722$	0	0
$723$	−5.35317	−0.199087
$724$	0	0
$725$	29.8779	1.10964
$726$	0	0
$727$	−0.277935	−0.0103080	−0.00515401	−	0.999987i	$-0.501641\pi$
$727$	−0.277935	−0.0103080	−0.00515401	+	0.999987i	$0.501641\pi$
$728$	0	0
$729$	−19.9030	−0.737147
$730$	0	0
$731$	9.30737	0.344245
$732$	0	0
$733$	33.2250	1.22719	0.613597	−	0.789619i	$-0.289722\pi$
$733$	33.2250	1.22719	0.613597	+	0.789619i	$0.289722\pi$
$734$	0	0
$735$	−3.42433	−0.126308
$736$	0	0
$737$	−42.8529	−1.57851
$738$	0	0
$739$	16.7893	0.617606	0.308803	−	0.951126i	$-0.400072\pi$
$739$	16.7893	0.617606	0.308803	+	0.951126i	$0.400072\pi$
$740$	0	0
$741$	2.60966	0.0958684
$742$	0	0
$743$	44.6535	1.63818	0.819088	−	0.573667i	$-0.194480\pi$
$743$	44.6535	1.63818	0.819088	+	0.573667i	$0.194480\pi$
$744$	0	0
$745$	−61.8357	−2.26549
$746$	0	0
$747$	−51.6843	−1.89103
$748$	0	0
$749$	−34.6778	−1.26710
$750$	0	0
$751$	−18.1547	−0.662475	−0.331238	−	0.943547i	$-0.607466\pi$
$751$	−18.1547	−0.662475	−0.331238	+	0.943547i	$0.607466\pi$
$752$	0	0
$753$	−0.369540	−0.0134668
$754$	0	0
$755$	9.76502	0.355385
$756$	0	0
$757$	−19.6217	−0.713162	−0.356581	−	0.934264i	$-0.616058\pi$
$757$	−19.6217	−0.713162	−0.356581	+	0.934264i	$0.616058\pi$
$758$	0	0
$759$	15.8833	0.576527
$760$	0	0
$761$	41.3648	1.49947	0.749737	−	0.661736i	$-0.230180\pi$
$761$	41.3648	1.49947	0.749737	+	0.661736i	$0.230180\pi$
$762$	0	0
$763$	−58.4025	−2.11431
$764$	0	0
$765$	68.6119	2.48067
$766$	0	0
$767$	17.5156	0.632452
$768$	0	0
$769$	−28.6446	−1.03295	−0.516476	−	0.856302i	$-0.672756\pi$
$769$	−28.6446	−1.03295	−0.516476	+	0.856302i	$0.672756\pi$
$770$	0	0
$771$	4.13123	0.148783
$772$	0	0
$773$	−38.2503	−1.37577	−0.687884	−	0.725820i	$-0.741460\pi$
$773$	−38.2503	−1.37577	−0.687884	+	0.725820i	$0.741460\pi$
$774$	0	0
$775$	−73.3475	−2.63472
$776$	0	0
$777$	8.38623	0.300854
$778$	0	0
$779$	−5.34806	−0.191614
$780$	0	0
$781$	15.0698	0.539241
$782$	0	0
$783$	−6.13489	−0.219243
$784$	0	0
$785$	33.4485	1.19383
$786$	0	0
$787$	−28.7664	−1.02541	−0.512706	−	0.858564i	$-0.671357\pi$
$787$	−28.7664	−1.02541	−0.512706	+	0.858564i	$0.671357\pi$
$788$	0	0
$789$	−11.7360	−0.417813
$790$	0	0
$791$	−9.99209	−0.355278
$792$	0	0
$793$	43.8062	1.55560
$794$	0	0
$795$	14.5017	0.514322
$796$	0	0
$797$	−34.5254	−1.22295	−0.611477	−	0.791262i	$-0.709424\pi$
$797$	−34.5254	−1.22295	−0.611477	+	0.791262i	$0.709424\pi$
$798$	0	0
$799$	−19.5545	−0.691787
$800$	0	0
$801$	23.1030	0.816304
$802$	0	0
$803$	54.8817	1.93673
$804$	0	0
$805$	−87.4909	−3.08365
$806$	0	0
$807$	0.113013	0.00397823
$808$	0	0
$809$	26.8812	0.945094	0.472547	−	0.881306i	$-0.343335\pi$
$809$	26.8812	0.945094	0.472547	+	0.881306i	$0.343335\pi$
$810$	0	0
$811$	47.7704	1.67745	0.838724	−	0.544557i	$-0.183302\pi$
$811$	47.7704	1.67745	0.838724	+	0.544557i	$0.183302\pi$
$812$	0	0
$813$	9.94572	0.348812
$814$	0	0
$815$	28.5378	0.999635
$816$	0	0
$817$	2.46180	0.0861275
$818$	0	0
$819$	−38.4746	−1.34441
$820$	0	0
$821$	−4.81603	−0.168081	−0.0840403	−	0.996462i	$-0.526782\pi$
$821$	−4.81603	−0.168081	−0.0840403	+	0.996462i	$0.526782\pi$
$822$	0	0
$823$	27.0380	0.942486	0.471243	−	0.882003i	$-0.343805\pi$
$823$	27.0380	0.942486	0.471243	+	0.882003i	$0.343805\pi$
$824$	0	0
$825$	23.1012	0.804280
$826$	0	0
$827$	37.4622	1.30269	0.651344	−	0.758783i	$-0.274206\pi$
$827$	37.4622	1.30269	0.651344	+	0.758783i	$0.274206\pi$
$828$	0	0
$829$	−11.3425	−0.393942	−0.196971	−	0.980409i	$-0.563110\pi$
$829$	−11.3425	−0.393942	−0.196971	+	0.980409i	$0.563110\pi$
$830$	0	0
$831$	−2.74867	−0.0953501
$832$	0	0
$833$	14.2766	0.494655
$834$	0	0
$835$	64.3811	2.22800
$836$	0	0
$837$	15.0606	0.520570
$838$	0	0
$839$	44.4942	1.53611	0.768055	−	0.640384i	$-0.221225\pi$
$839$	44.4942	1.53611	0.768055	+	0.640384i	$0.221225\pi$
$840$	0	0
$841$	−20.9835	−0.723568
$842$	0	0
$843$	3.92158	0.135066
$844$	0	0
$845$	−24.8831	−0.856004
$846$	0	0
$847$	73.6730	2.53143
$848$	0	0
$849$	−5.96482	−0.204712
$850$	0	0
$851$	53.8480	1.84589
$852$	0	0
$853$	38.9758	1.33450	0.667252	−	0.744832i	$-0.267470\pi$
$853$	38.9758	1.33450	0.667252	+	0.744832i	$0.267470\pi$
$854$	0	0
$855$	18.1479	0.620644
$856$	0	0
$857$	42.6471	1.45680	0.728398	−	0.685154i	$-0.240265\pi$
$857$	42.6471	1.45680	0.728398	+	0.685154i	$0.240265\pi$
$858$	0	0
$859$	−3.81457	−0.130151	−0.0650757	−	0.997880i	$-0.520729\pi$
$859$	−3.81457	−0.130151	−0.0650757	+	0.997880i	$0.520729\pi$
$860$	0	0
$861$	−3.76028	−0.128150
$862$	0	0
$863$	20.6253	0.702093	0.351046	−	0.936358i	$-0.385826\pi$
$863$	20.6253	0.702093	0.351046	+	0.936358i	$0.385826\pi$
$864$	0	0
$865$	−101.287	−3.44387
$866$	0	0
$867$	7.35997	0.249958
$868$	0	0
$869$	−100.100	−3.39566
$870$	0	0
$871$	−31.7871	−1.07707
$872$	0	0
$873$	−1.27723	−0.0432278
$874$	0	0
$875$	−66.9562	−2.26353
$876$	0	0
$877$	7.84715	0.264979	0.132490	−	0.991184i	$-0.457703\pi$
$877$	7.84715	0.264979	0.132490	+	0.991184i	$0.457703\pi$
$878$	0	0
$879$	−1.71351	−0.0577952
$880$	0	0
$881$	22.5772	0.760645	0.380322	−	0.924854i	$-0.375813\pi$
$881$	22.5772	0.760645	0.380322	+	0.924854i	$0.375813\pi$
$882$	0	0
$883$	−48.7772	−1.64148	−0.820742	−	0.571299i	$-0.806440\pi$
$883$	−48.7772	−1.64148	−0.820742	+	0.571299i	$0.806440\pi$
$884$	0	0
$885$	−5.80898	−0.195267
$886$	0	0
$887$	−29.0804	−0.976425	−0.488212	−	0.872725i	$-0.662351\pi$
$887$	−29.0804	−0.976425	−0.488212	+	0.872725i	$0.662351\pi$
$888$	0	0
$889$	−14.1841	−0.475719
$890$	0	0
$891$	46.1458	1.54594
$892$	0	0
$893$	−5.17216	−0.173080
$894$	0	0
$895$	−80.7450	−2.69901
$896$	0	0
$897$	11.7818	0.393382
$898$	0	0
$899$	−19.6798	−0.656359
$900$	0	0
$901$	−60.4600	−2.01421
$902$	0	0
$903$	1.73092	0.0576013
$904$	0	0
$905$	33.4870	1.11315
$906$	0	0
$907$	1.57390	0.0522603	0.0261302	−	0.999659i	$-0.491682\pi$
$907$	1.57390	0.0522603	0.0261302	+	0.999659i	$0.491682\pi$
$908$	0	0
$909$	4.60844	0.152852
$910$	0	0
$911$	−10.3570	−0.343142	−0.171571	−	0.985172i	$-0.554884\pi$
$911$	−10.3570	−0.343142	−0.171571	+	0.985172i	$0.554884\pi$
$912$	0	0
$913$	106.927	3.53876
$914$	0	0
$915$	−14.5281	−0.480286
$916$	0	0
$917$	42.8634	1.41547
$918$	0	0
$919$	−9.32307	−0.307540	−0.153770	−	0.988107i	$-0.549141\pi$
$919$	−9.32307	−0.307540	−0.153770	+	0.988107i	$0.549141\pi$
$920$	0	0
$921$	0.0885515	0.00291787
$922$	0	0
$923$	11.1784	0.367941
$924$	0	0
$925$	78.3183	2.57509
$926$	0	0
$927$	13.8423	0.454640
$928$	0	0
$929$	16.0062	0.525147	0.262574	−	0.964912i	$-0.415429\pi$
$929$	16.0062	0.525147	0.262574	+	0.964912i	$0.415429\pi$
$930$	0	0
$931$	3.77617	0.123759
$932$	0	0
$933$	5.85995	0.191846
$934$	0	0
$935$	−141.948	−4.64218
$936$	0	0
$937$	−54.5331	−1.78152	−0.890760	−	0.454474i	$-0.849827\pi$
$937$	−54.5331	−1.78152	−0.890760	+	0.454474i	$0.849827\pi$
$938$	0	0
$939$	2.15827	0.0704325
$940$	0	0
$941$	−54.2011	−1.76691	−0.883453	−	0.468520i	$-0.844787\pi$
$941$	−54.2011	−1.76691	−0.883453	+	0.468520i	$0.844787\pi$
$942$	0	0
$943$	−24.1448	−0.786262
$944$	0	0
$945$	26.1284	0.849958
$946$	0	0
$947$	−36.2903	−1.17928	−0.589639	−	0.807667i	$-0.700730\pi$
$947$	−36.2903	−1.17928	−0.589639	+	0.807667i	$0.700730\pi$
$948$	0	0
$949$	40.7097	1.32149
$950$	0	0
$951$	7.00254	0.227073
$952$	0	0
$953$	32.6572	1.05787	0.528935	−	0.848662i	$-0.322591\pi$
$953$	32.6572	1.05787	0.528935	+	0.848662i	$0.322591\pi$
$954$	0	0
$955$	32.2591	1.04388
$956$	0	0
$957$	6.19827	0.200362
$958$	0	0
$959$	49.7006	1.60491
$960$	0	0
$961$	17.3121	0.558456
$962$	0	0
$963$	32.4744	1.04647
$964$	0	0
$965$	91.6810	2.95132
$966$	0	0
$967$	−1.46771	−0.0471984	−0.0235992	−	0.999721i	$-0.507513\pi$
$967$	−1.46771	−0.0471984	−0.0235992	+	0.999721i	$0.507513\pi$
$968$	0	0
$969$	3.60835	0.115917
$970$	0	0
$971$	−4.54950	−0.146000	−0.0730001	−	0.997332i	$-0.523257\pi$
$971$	−4.54950	−0.146000	−0.0730001	+	0.997332i	$0.523257\pi$
$972$	0	0
$973$	28.0091	0.897930
$974$	0	0
$975$	17.1358	0.548785
$976$	0	0
$977$	−39.7858	−1.27286	−0.636430	−	0.771334i	$-0.719590\pi$
$977$	−39.7858	−1.27286	−0.636430	+	0.771334i	$0.719590\pi$
$978$	0	0
$979$	−47.7966	−1.52758
$980$	0	0
$981$	54.6916	1.74617
$982$	0	0
$983$	−52.6839	−1.68036	−0.840178	−	0.542311i	$-0.817549\pi$
$983$	−52.6839	−1.68036	−0.840178	+	0.542311i	$0.817549\pi$
$984$	0	0
$985$	−27.6198	−0.880039
$986$	0	0
$987$	−3.63660	−0.115754
$988$	0	0
$989$	11.1142	0.353412
$990$	0	0
$991$	−6.55691	−0.208287	−0.104144	−	0.994562i	$-0.533210\pi$
$991$	−6.55691	−0.208287	−0.104144	+	0.994562i	$0.533210\pi$
$992$	0	0
$993$	10.8082	0.342988
$994$	0	0
$995$	29.4807	0.934600
$996$	0	0
$997$	−18.1868	−0.575982	−0.287991	−	0.957633i	$-0.592987\pi$
$997$	−18.1868	−0.575982	−0.287991	+	0.957633i	$0.592987\pi$
$998$	0	0
$999$	−16.0813	−0.508788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.15	yes	30
4.3	odd	2		8032.2.a.g.1.16	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.16	✓	30	4.3	odd	2
8032.2.a.j.1.15	yes	30	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q+0.369540 q^{3} -3.94367 q^{5} +3.05773 q^{7} -2.86344 q^{9} +O(q^{10})\)	\(q+0.369540 q^{3} -3.94367 q^{5} +3.05773 q^{7} -2.86344 q^{9} +5.92402 q^{11} +4.39427 q^{13} -1.45734 q^{15} +6.07590 q^{17} +1.60708 q^{19} +1.12995 q^{21} +7.25543 q^{23} +10.5525 q^{25} -2.16677 q^{27} +2.83135 q^{29} -6.95069 q^{31} +2.18916 q^{33} -12.0587 q^{35} +7.42175 q^{37} +1.62386 q^{39} -3.32782 q^{41} +1.53185 q^{43} +11.2925 q^{45} -3.21836 q^{47} +2.34971 q^{49} +2.24529 q^{51} -9.95078 q^{53} -23.3624 q^{55} +0.593879 q^{57} +3.98601 q^{59} +9.96893 q^{61} -8.75563 q^{63} -17.3296 q^{65} -7.23376 q^{67} +2.68117 q^{69} +2.54385 q^{71} +9.26426 q^{73} +3.89958 q^{75} +18.1141 q^{77} -16.8973 q^{79} +7.78961 q^{81} +18.0497 q^{83} -23.9614 q^{85} +1.04629 q^{87} -8.06826 q^{89} +13.4365 q^{91} -2.56856 q^{93} -6.33778 q^{95} +0.446048 q^{97} -16.9631 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

Modular forms

Varieties

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