LMFDB - Embedded newform 6032.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 159x^9 - 827x^8 - 362x^7 + 3029x^6 - 142x^5 - 4316x^4 + 208x^3 + 2228x^2 + 376x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.993990$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.993990 q^{3} +3.24551 q^{5} -4.74513 q^{7} -2.01198 q^{9} +O(q^{10})$	$q-0.993990 q^{3} +3.24551 q^{5} -4.74513 q^{7} -2.01198 q^{9} -5.30260 q^{11} -1.00000 q^{13} -3.22601 q^{15} -5.32782 q^{17} -5.67520 q^{19} +4.71661 q^{21} -3.62267 q^{23} +5.53335 q^{25} +4.98186 q^{27} +1.00000 q^{29} +0.521785 q^{31} +5.27073 q^{33} -15.4004 q^{35} -6.10527 q^{37} +0.993990 q^{39} -5.95988 q^{41} +11.9372 q^{43} -6.52992 q^{45} +1.32143 q^{47} +15.5163 q^{49} +5.29579 q^{51} -4.90353 q^{53} -17.2097 q^{55} +5.64108 q^{57} -13.3080 q^{59} +12.3207 q^{61} +9.54713 q^{63} -3.24551 q^{65} +1.21689 q^{67} +3.60090 q^{69} +14.4018 q^{71} -0.437718 q^{73} -5.50009 q^{75} +25.1615 q^{77} +11.8568 q^{79} +1.08404 q^{81} -3.97402 q^{83} -17.2915 q^{85} -0.993990 q^{87} +0.468634 q^{89} +4.74513 q^{91} -0.518649 q^{93} -18.4189 q^{95} -8.41190 q^{97} +10.6687 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * 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
$3$	−0.993990	−0.573880	−0.286940	−	0.957949i	$-0.592638\pi$
$3$	−0.993990	−0.573880	−0.286940	+	0.957949i	$0.592638\pi$
$4$	0	0
$5$	3.24551	1.45144	0.725719	−	0.687992i	$-0.241507\pi$
$5$	3.24551	1.45144	0.725719	+	0.687992i	$0.241507\pi$
$6$	0	0
$7$	−4.74513	−1.79349	−0.896745	−	0.442547i	$-0.854075\pi$
$7$	−4.74513	−1.79349	−0.896745	+	0.442547i	$0.854075\pi$
$8$	0	0
$9$	−2.01198	−0.670662
$10$	0	0
$11$	−5.30260	−1.59879	−0.799397	−	0.600803i	$-0.794848\pi$
$11$	−5.30260	−1.59879	−0.799397	+	0.600803i	$0.794848\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.22601	−0.832951
$16$	0	0
$17$	−5.32782	−1.29219	−0.646093	−	0.763259i	$-0.723598\pi$
$17$	−5.32782	−1.29219	−0.646093	+	0.763259i	$0.723598\pi$
$18$	0	0
$19$	−5.67520	−1.30198	−0.650990	−	0.759087i	$-0.725646\pi$
$19$	−5.67520	−1.30198	−0.650990	+	0.759087i	$0.725646\pi$
$20$	0	0
$21$	4.71661	1.02925
$22$	0	0
$23$	−3.62267	−0.755380	−0.377690	−	0.925932i	$-0.623281\pi$
$23$	−3.62267	−0.755380	−0.377690	+	0.925932i	$0.623281\pi$
$24$	0	0
$25$	5.53335	1.10667
$26$	0	0
$27$	4.98186	0.958760
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.521785	0.0937154	0.0468577	−	0.998902i	$-0.485079\pi$
$31$	0.521785	0.0937154	0.0468577	+	0.998902i	$0.485079\pi$
$32$	0	0
$33$	5.27073	0.917516
$34$	0	0
$35$	−15.4004	−2.60314
$36$	0	0
$37$	−6.10527	−1.00370	−0.501850	−	0.864955i	$-0.667347\pi$
$37$	−6.10527	−1.00370	−0.501850	+	0.864955i	$0.667347\pi$
$38$	0	0
$39$	0.993990	0.159166
$40$	0	0
$41$	−5.95988	−0.930777	−0.465389	−	0.885106i	$-0.654085\pi$
$41$	−5.95988	−0.930777	−0.465389	+	0.885106i	$0.654085\pi$
$42$	0	0
$43$	11.9372	1.82040	0.910202	−	0.414165i	$-0.135926\pi$
$43$	11.9372	1.82040	0.910202	+	0.414165i	$0.135926\pi$
$44$	0	0
$45$	−6.52992	−0.973423
$46$	0	0
$47$	1.32143	0.192750	0.0963749	−	0.995345i	$-0.469275\pi$
$47$	1.32143	0.192750	0.0963749	+	0.995345i	$0.469275\pi$
$48$	0	0
$49$	15.5163	2.21661
$50$	0	0
$51$	5.29579	0.741559
$52$	0	0
$53$	−4.90353	−0.673552	−0.336776	−	0.941585i	$-0.609337\pi$
$53$	−4.90353	−0.673552	−0.336776	+	0.941585i	$0.609337\pi$
$54$	0	0
$55$	−17.2097	−2.32055
$56$	0	0
$57$	5.64108	0.747180
$58$	0	0
$59$	−13.3080	−1.73255	−0.866277	−	0.499564i	$-0.833494\pi$
$59$	−13.3080	−1.73255	−0.866277	+	0.499564i	$0.833494\pi$
$60$	0	0
$61$	12.3207	1.57751	0.788753	−	0.614711i	$-0.210727\pi$
$61$	12.3207	1.57751	0.788753	+	0.614711i	$0.210727\pi$
$62$	0	0
$63$	9.54713	1.20283
$64$	0	0
$65$	−3.24551	−0.402556
$66$	0	0
$67$	1.21689	0.148667	0.0743333	−	0.997233i	$-0.476317\pi$
$67$	1.21689	0.148667	0.0743333	+	0.997233i	$0.476317\pi$
$68$	0	0
$69$	3.60090	0.433498
$70$	0	0
$71$	14.4018	1.70918	0.854589	−	0.519305i	$-0.173809\pi$
$71$	14.4018	1.70918	0.854589	+	0.519305i	$0.173809\pi$
$72$	0	0
$73$	−0.437718	−0.0512311	−0.0256155	−	0.999672i	$-0.508155\pi$
$73$	−0.437718	−0.0512311	−0.0256155	+	0.999672i	$0.508155\pi$
$74$	0	0
$75$	−5.50009	−0.635096
$76$	0	0
$77$	25.1615	2.86742
$78$	0	0
$79$	11.8568	1.33400	0.666999	−	0.745059i	$-0.267578\pi$
$79$	11.8568	1.33400	0.666999	+	0.745059i	$0.267578\pi$
$80$	0	0
$81$	1.08404	0.120449
$82$	0	0
$83$	−3.97402	−0.436206	−0.218103	−	0.975926i	$-0.569987\pi$
$83$	−3.97402	−0.436206	−0.218103	+	0.975926i	$0.569987\pi$
$84$	0	0
$85$	−17.2915	−1.87553
$86$	0	0
$87$	−0.993990	−0.106567
$88$	0	0
$89$	0.468634	0.0496751	0.0248376	−	0.999692i	$-0.492093\pi$
$89$	0.468634	0.0496751	0.0248376	+	0.999692i	$0.492093\pi$
$90$	0	0
$91$	4.74513	0.497425
$92$	0	0
$93$	−0.518649	−0.0537814
$94$	0	0
$95$	−18.4189	−1.88974
$96$	0	0
$97$	−8.41190	−0.854099	−0.427050	−	0.904228i	$-0.640447\pi$
$97$	−8.41190	−0.854099	−0.427050	+	0.904228i	$0.640447\pi$
$98$	0	0
$99$	10.6687	1.07225
$100$	0	0
$101$	7.89952	0.786032	0.393016	−	0.919532i	$-0.371432\pi$
$101$	7.89952	0.786032	0.393016	+	0.919532i	$0.371432\pi$
$102$	0	0
$103$	17.3650	1.71102	0.855512	−	0.517783i	$-0.173243\pi$
$103$	17.3650	1.71102	0.855512	+	0.517783i	$0.173243\pi$
$104$	0	0
$105$	15.3078	1.49389
$106$	0	0
$107$	16.5434	1.59931	0.799653	−	0.600462i	$-0.205017\pi$
$107$	16.5434	1.59931	0.799653	+	0.600462i	$0.205017\pi$
$108$	0	0
$109$	−6.05928	−0.580374	−0.290187	−	0.956970i	$-0.593718\pi$
$109$	−6.05928	−0.580374	−0.290187	+	0.956970i	$0.593718\pi$
$110$	0	0
$111$	6.06857	0.576004
$112$	0	0
$113$	−10.3165	−0.970496	−0.485248	−	0.874376i	$-0.661271\pi$
$113$	−10.3165	−0.970496	−0.485248	+	0.874376i	$0.661271\pi$
$114$	0	0
$115$	−11.7574	−1.09639
$116$	0	0
$117$	2.01198	0.186008
$118$	0	0
$119$	25.2812	2.31752
$120$	0	0
$121$	17.1176	1.55614
$122$	0	0
$123$	5.92406	0.534155
$124$	0	0
$125$	1.73100	0.154825
$126$	0	0
$127$	19.4832	1.72885	0.864426	−	0.502760i	$-0.167682\pi$
$127$	19.4832	1.72885	0.864426	+	0.502760i	$0.167682\pi$
$128$	0	0
$129$	−11.8654	−1.04469
$130$	0	0
$131$	−12.5424	−1.09583	−0.547916	−	0.836533i	$-0.684579\pi$
$131$	−12.5424	−1.09583	−0.547916	+	0.836533i	$0.684579\pi$
$132$	0	0
$133$	26.9295	2.33509
$134$	0	0
$135$	16.1687	1.39158
$136$	0	0
$137$	−1.65679	−0.141549	−0.0707744	−	0.997492i	$-0.522547\pi$
$137$	−1.65679	−0.141549	−0.0707744	+	0.997492i	$0.522547\pi$
$138$	0	0
$139$	−17.2151	−1.46017	−0.730084	−	0.683358i	$-0.760519\pi$
$139$	−17.2151	−1.46017	−0.730084	+	0.683358i	$0.760519\pi$
$140$	0	0
$141$	−1.31348	−0.110615
$142$	0	0
$143$	5.30260	0.443426
$144$	0	0
$145$	3.24551	0.269525
$146$	0	0
$147$	−15.4230	−1.27207
$148$	0	0
$149$	1.89236	0.155028	0.0775141	−	0.996991i	$-0.475302\pi$
$149$	1.89236	0.155028	0.0775141	+	0.996991i	$0.475302\pi$
$150$	0	0
$151$	−4.88983	−0.397929	−0.198964	−	0.980007i	$-0.563758\pi$
$151$	−4.88983	−0.397929	−0.198964	+	0.980007i	$0.563758\pi$
$152$	0	0
$153$	10.7195	0.866619
$154$	0	0
$155$	1.69346	0.136022
$156$	0	0
$157$	0.522636	0.0417109	0.0208554	−	0.999783i	$-0.493361\pi$
$157$	0.522636	0.0417109	0.0208554	+	0.999783i	$0.493361\pi$
$158$	0	0
$159$	4.87406	0.386538
$160$	0	0
$161$	17.1901	1.35477
$162$	0	0
$163$	5.02867	0.393876	0.196938	−	0.980416i	$-0.436900\pi$
$163$	5.02867	0.393876	0.196938	+	0.980416i	$0.436900\pi$
$164$	0	0
$165$	17.1062	1.33172
$166$	0	0
$167$	9.36761	0.724888	0.362444	−	0.932006i	$-0.381942\pi$
$167$	9.36761	0.724888	0.362444	+	0.932006i	$0.381942\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	11.4184	0.873187
$172$	0	0
$173$	−12.8009	−0.973232	−0.486616	−	0.873616i	$-0.661769\pi$
$173$	−12.8009	−0.973232	−0.486616	+	0.873616i	$0.661769\pi$
$174$	0	0
$175$	−26.2565	−1.98480
$176$	0	0
$177$	13.2280	0.994278
$178$	0	0
$179$	−20.2166	−1.51106	−0.755530	−	0.655114i	$-0.772620\pi$
$179$	−20.2166	−1.51106	−0.755530	+	0.655114i	$0.772620\pi$
$180$	0	0
$181$	8.19673	0.609258	0.304629	−	0.952471i	$-0.401467\pi$
$181$	8.19673	0.609258	0.304629	+	0.952471i	$0.401467\pi$
$182$	0	0
$183$	−12.2467	−0.905299
$184$	0	0
$185$	−19.8147	−1.45681
$186$	0	0
$187$	28.2513	2.06594
$188$	0	0
$189$	−23.6396	−1.71953
$190$	0	0
$191$	−5.14402	−0.372208	−0.186104	−	0.982530i	$-0.559586\pi$
$191$	−5.14402	−0.372208	−0.186104	+	0.982530i	$0.559586\pi$
$192$	0	0
$193$	0.608196	0.0437789	0.0218895	−	0.999760i	$-0.493032\pi$
$193$	0.608196	0.0437789	0.0218895	+	0.999760i	$0.493032\pi$
$194$	0	0
$195$	3.22601	0.231019
$196$	0	0
$197$	−9.18588	−0.654467	−0.327234	−	0.944944i	$-0.606116\pi$
$197$	−9.18588	−0.654467	−0.327234	+	0.944944i	$0.606116\pi$
$198$	0	0
$199$	−26.1806	−1.85590	−0.927948	−	0.372709i	$-0.878429\pi$
$199$	−26.1806	−1.85590	−0.927948	+	0.372709i	$0.878429\pi$
$200$	0	0
$201$	−1.20957	−0.0853168
$202$	0	0
$203$	−4.74513	−0.333043
$204$	0	0
$205$	−19.3429	−1.35096
$206$	0	0
$207$	7.28877	0.506604
$208$	0	0
$209$	30.0933	2.08160
$210$	0	0
$211$	1.87843	0.129316	0.0646582	−	0.997907i	$-0.479404\pi$
$211$	1.87843	0.129316	0.0646582	+	0.997907i	$0.479404\pi$
$212$	0	0
$213$	−14.3152	−0.980863
$214$	0	0
$215$	38.7423	2.64220
$216$	0	0
$217$	−2.47594	−0.168078
$218$	0	0
$219$	0.435087	0.0294005
$220$	0	0
$221$	5.32782	0.358388
$222$	0	0
$223$	−22.6218	−1.51487	−0.757433	−	0.652912i	$-0.773547\pi$
$223$	−22.6218	−1.51487	−0.757433	+	0.652912i	$0.773547\pi$
$224$	0	0
$225$	−11.1330	−0.742201
$226$	0	0
$227$	−14.4017	−0.955876	−0.477938	−	0.878394i	$-0.658616\pi$
$227$	−14.4017	−0.955876	−0.477938	+	0.878394i	$0.658616\pi$
$228$	0	0
$229$	−2.87041	−0.189682	−0.0948410	−	0.995492i	$-0.530234\pi$
$229$	−2.87041	−0.189682	−0.0948410	+	0.995492i	$0.530234\pi$
$230$	0	0
$231$	−25.0103	−1.64556
$232$	0	0
$233$	2.06555	0.135319	0.0676594	−	0.997708i	$-0.478447\pi$
$233$	2.06555	0.135319	0.0676594	+	0.997708i	$0.478447\pi$
$234$	0	0
$235$	4.28871	0.279764
$236$	0	0
$237$	−11.7856	−0.765555
$238$	0	0
$239$	−10.0631	−0.650927	−0.325463	−	0.945555i	$-0.605520\pi$
$239$	−10.0631	−0.650927	−0.325463	+	0.945555i	$0.605520\pi$
$240$	0	0
$241$	−10.1053	−0.650937	−0.325469	−	0.945553i	$-0.605522\pi$
$241$	−10.1053	−0.650937	−0.325469	+	0.945553i	$0.605522\pi$
$242$	0	0
$243$	−16.0231	−1.02788
$244$	0	0
$245$	50.3582	3.21727
$246$	0	0
$247$	5.67520	0.361104
$248$	0	0
$249$	3.95014	0.250330
$250$	0	0
$251$	−8.41869	−0.531383	−0.265692	−	0.964058i	$-0.585600\pi$
$251$	−8.41869	−0.531383	−0.265692	+	0.964058i	$0.585600\pi$
$252$	0	0
$253$	19.2096	1.20770
$254$	0	0
$255$	17.1876	1.07633
$256$	0	0
$257$	2.92955	0.182740	0.0913701	−	0.995817i	$-0.470875\pi$
$257$	2.92955	0.182740	0.0913701	+	0.995817i	$0.470875\pi$
$258$	0	0
$259$	28.9703	1.80013
$260$	0	0
$261$	−2.01198	−0.124539
$262$	0	0
$263$	−17.8723	−1.10205	−0.551027	−	0.834487i	$-0.685764\pi$
$263$	−17.8723	−1.10205	−0.551027	+	0.834487i	$0.685764\pi$
$264$	0	0
$265$	−15.9145	−0.977619
$266$	0	0
$267$	−0.465818	−0.0285076
$268$	0	0
$269$	12.9391	0.788913	0.394456	−	0.918915i	$-0.370933\pi$
$269$	12.9391	0.788913	0.394456	+	0.918915i	$0.370933\pi$
$270$	0	0
$271$	−3.17135	−0.192646	−0.0963230	−	0.995350i	$-0.530708\pi$
$271$	−3.17135	−0.192646	−0.0963230	+	0.995350i	$0.530708\pi$
$272$	0	0
$273$	−4.71661	−0.285462
$274$	0	0
$275$	−29.3411	−1.76934
$276$	0	0
$277$	9.82837	0.590530	0.295265	−	0.955415i	$-0.404592\pi$
$277$	9.82837	0.590530	0.295265	+	0.955415i	$0.404592\pi$
$278$	0	0
$279$	−1.04982	−0.0628513
$280$	0	0
$281$	2.50570	0.149478	0.0747389	−	0.997203i	$-0.476188\pi$
$281$	2.50570	0.149478	0.0747389	+	0.997203i	$0.476188\pi$
$282$	0	0
$283$	−13.1824	−0.783610	−0.391805	−	0.920048i	$-0.628149\pi$
$283$	−13.1824	−0.783610	−0.391805	+	0.920048i	$0.628149\pi$
$284$	0	0
$285$	18.3082	1.08448
$286$	0	0
$287$	28.2804	1.66934
$288$	0	0
$289$	11.3856	0.669742
$290$	0	0
$291$	8.36135	0.490151
$292$	0	0
$293$	32.6521	1.90756	0.953778	−	0.300513i	$-0.0971580\pi$
$293$	32.6521	1.90756	0.953778	+	0.300513i	$0.0971580\pi$
$294$	0	0
$295$	−43.1913	−2.51469
$296$	0	0
$297$	−26.4168	−1.53286
$298$	0	0
$299$	3.62267	0.209505
$300$	0	0
$301$	−56.6435	−3.26488
$302$	0	0
$303$	−7.85204	−0.451088
$304$	0	0
$305$	39.9870	2.28965
$306$	0	0
$307$	−2.93805	−0.167683	−0.0838416	−	0.996479i	$-0.526719\pi$
$307$	−2.93805	−0.167683	−0.0838416	+	0.996479i	$0.526719\pi$
$308$	0	0
$309$	−17.2606	−0.981922
$310$	0	0
$311$	10.2003	0.578407	0.289204	−	0.957268i	$-0.406610\pi$
$311$	10.2003	0.578407	0.289204	+	0.957268i	$0.406610\pi$
$312$	0	0
$313$	−15.5282	−0.877704	−0.438852	−	0.898559i	$-0.644615\pi$
$313$	−15.5282	−0.877704	−0.438852	+	0.898559i	$0.644615\pi$
$314$	0	0
$315$	30.9853	1.74583
$316$	0	0
$317$	−18.2360	−1.02424	−0.512119	−	0.858915i	$-0.671139\pi$
$317$	−18.2360	−1.02424	−0.512119	+	0.858915i	$0.671139\pi$
$318$	0	0
$319$	−5.30260	−0.296889
$320$	0	0
$321$	−16.4439	−0.917810
$322$	0	0
$323$	30.2364	1.68240
$324$	0	0
$325$	−5.53335	−0.306935
$326$	0	0
$327$	6.02287	0.333065
$328$	0	0
$329$	−6.27034	−0.345695
$330$	0	0
$331$	−11.5366	−0.634110	−0.317055	−	0.948407i	$-0.602694\pi$
$331$	−11.5366	−0.634110	−0.317055	+	0.948407i	$0.602694\pi$
$332$	0	0
$333$	12.2837	0.673143
$334$	0	0
$335$	3.94942	0.215780
$336$	0	0
$337$	16.7495	0.912404	0.456202	−	0.889876i	$-0.349210\pi$
$337$	16.7495	0.912404	0.456202	+	0.889876i	$0.349210\pi$
$338$	0	0
$339$	10.2545	0.556949
$340$	0	0
$341$	−2.76682	−0.149832
$342$	0	0
$343$	−40.4107	−2.18197
$344$	0	0
$345$	11.6868	0.629194
$346$	0	0
$347$	29.6756	1.59307	0.796534	−	0.604594i	$-0.206665\pi$
$347$	29.6756	1.59307	0.796534	+	0.604594i	$0.206665\pi$
$348$	0	0
$349$	−18.3722	−0.983441	−0.491721	−	0.870753i	$-0.663632\pi$
$349$	−18.3722	−0.983441	−0.491721	+	0.870753i	$0.663632\pi$
$350$	0	0
$351$	−4.98186	−0.265912
$352$	0	0
$353$	−14.4800	−0.770690	−0.385345	−	0.922772i	$-0.625918\pi$
$353$	−14.4800	−0.770690	−0.385345	+	0.922772i	$0.625918\pi$
$354$	0	0
$355$	46.7412	2.48076
$356$	0	0
$357$	−25.1292	−1.32998
$358$	0	0
$359$	−15.2061	−0.802545	−0.401272	−	0.915959i	$-0.631432\pi$
$359$	−15.2061	−0.802545	−0.401272	+	0.915959i	$0.631432\pi$
$360$	0	0
$361$	13.2078	0.695150
$362$	0	0
$363$	−17.0147	−0.893039
$364$	0	0
$365$	−1.42062	−0.0743587
$366$	0	0
$367$	17.4349	0.910097	0.455048	−	0.890467i	$-0.349622\pi$
$367$	17.4349	0.910097	0.455048	+	0.890467i	$0.349622\pi$
$368$	0	0
$369$	11.9912	0.624237
$370$	0	0
$371$	23.2679	1.20801
$372$	0	0
$373$	16.4192	0.850154	0.425077	−	0.905157i	$-0.360247\pi$
$373$	16.4192	0.850154	0.425077	+	0.905157i	$0.360247\pi$
$374$	0	0
$375$	−1.72059	−0.0888510
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	26.5740	1.36501	0.682507	−	0.730879i	$-0.260890\pi$
$379$	26.5740	1.36501	0.682507	+	0.730879i	$0.260890\pi$
$380$	0	0
$381$	−19.3661	−0.992154
$382$	0	0
$383$	−16.6042	−0.848436	−0.424218	−	0.905560i	$-0.639451\pi$
$383$	−16.6042	−0.848436	−0.424218	+	0.905560i	$0.639451\pi$
$384$	0	0
$385$	81.6620	4.16188
$386$	0	0
$387$	−24.0174	−1.22087
$388$	0	0
$389$	21.7346	1.10199	0.550995	−	0.834509i	$-0.314248\pi$
$389$	21.7346	1.10199	0.550995	+	0.834509i	$0.314248\pi$
$390$	0	0
$391$	19.3009	0.976091
$392$	0	0
$393$	12.4670	0.628876
$394$	0	0
$395$	38.4815	1.93621
$396$	0	0
$397$	−3.32736	−0.166996	−0.0834978	−	0.996508i	$-0.526609\pi$
$397$	−3.32736	−0.166996	−0.0834978	+	0.996508i	$0.526609\pi$
$398$	0	0
$399$	−26.7677	−1.34006
$400$	0	0
$401$	21.3016	1.06375	0.531875	−	0.846823i	$-0.321487\pi$
$401$	21.3016	1.06375	0.531875	+	0.846823i	$0.321487\pi$
$402$	0	0
$403$	−0.521785	−0.0259920
$404$	0	0
$405$	3.51825	0.174823
$406$	0	0
$407$	32.3738	1.60471
$408$	0	0
$409$	−4.09213	−0.202343	−0.101172	−	0.994869i	$-0.532259\pi$
$409$	−4.09213	−0.202343	−0.101172	+	0.994869i	$0.532259\pi$
$410$	0	0
$411$	1.64683	0.0812320
$412$	0	0
$413$	63.1482	3.10732
$414$	0	0
$415$	−12.8977	−0.633125
$416$	0	0
$417$	17.1116	0.837961
$418$	0	0
$419$	−36.1480	−1.76595	−0.882974	−	0.469423i	$-0.844462\pi$
$419$	−36.1480	−1.76595	−0.882974	+	0.469423i	$0.844462\pi$
$420$	0	0
$421$	−8.20338	−0.399809	−0.199904	−	0.979815i	$-0.564063\pi$
$421$	−8.20338	−0.399809	−0.199904	+	0.979815i	$0.564063\pi$
$422$	0	0
$423$	−2.65869	−0.129270
$424$	0	0
$425$	−29.4807	−1.43002
$426$	0	0
$427$	−58.4634	−2.82924
$428$	0	0
$429$	−5.27073	−0.254473
$430$	0	0
$431$	32.2764	1.55470	0.777351	−	0.629067i	$-0.216563\pi$
$431$	32.2764	1.55470	0.777351	+	0.629067i	$0.216563\pi$
$432$	0	0
$433$	−9.68205	−0.465290	−0.232645	−	0.972562i	$-0.574738\pi$
$433$	−9.68205	−0.465290	−0.232645	+	0.972562i	$0.574738\pi$
$434$	0	0
$435$	−3.22601	−0.154675
$436$	0	0
$437$	20.5594	0.983489
$438$	0	0
$439$	10.7287	0.512054	0.256027	−	0.966670i	$-0.417586\pi$
$439$	10.7287	0.512054	0.256027	+	0.966670i	$0.417586\pi$
$440$	0	0
$441$	−31.2185	−1.48659
$442$	0	0
$443$	18.6593	0.886532	0.443266	−	0.896390i	$-0.353820\pi$
$443$	18.6593	0.886532	0.443266	+	0.896390i	$0.353820\pi$
$444$	0	0
$445$	1.52096	0.0721003
$446$	0	0
$447$	−1.88099	−0.0889676
$448$	0	0
$449$	−9.38017	−0.442678	−0.221339	−	0.975197i	$-0.571043\pi$
$449$	−9.38017	−0.442678	−0.221339	+	0.975197i	$0.571043\pi$
$450$	0	0
$451$	31.6029	1.48812
$452$	0	0
$453$	4.86044	0.228363
$454$	0	0
$455$	15.4004	0.721981
$456$	0	0
$457$	−17.2875	−0.808674	−0.404337	−	0.914610i	$-0.632498\pi$
$457$	−17.2875	−0.808674	−0.404337	+	0.914610i	$0.632498\pi$
$458$	0	0
$459$	−26.5424	−1.23889
$460$	0	0
$461$	−10.7317	−0.499824	−0.249912	−	0.968269i	$-0.580402\pi$
$461$	−10.7317	−0.499824	−0.249912	+	0.968269i	$0.580402\pi$
$462$	0	0
$463$	−17.7590	−0.825329	−0.412665	−	0.910883i	$-0.635402\pi$
$463$	−17.7590	−0.825329	−0.412665	+	0.910883i	$0.635402\pi$
$464$	0	0
$465$	−1.68328	−0.0780604
$466$	0	0
$467$	−4.28628	−0.198345	−0.0991726	−	0.995070i	$-0.531620\pi$
$467$	−4.28628	−0.198345	−0.0991726	+	0.995070i	$0.531620\pi$
$468$	0	0
$469$	−5.77429	−0.266632
$470$	0	0
$471$	−0.519495	−0.0239370
$472$	0	0
$473$	−63.2981	−2.91045
$474$	0	0
$475$	−31.4028	−1.44086
$476$	0	0
$477$	9.86583	0.451726
$478$	0	0
$479$	−13.9860	−0.639036	−0.319518	−	0.947580i	$-0.603521\pi$
$479$	−13.9860	−0.639036	−0.319518	+	0.947580i	$0.603521\pi$
$480$	0	0
$481$	6.10527	0.278376
$482$	0	0
$483$	−17.0867	−0.777474
$484$	0	0
$485$	−27.3009	−1.23967
$486$	0	0
$487$	−22.1363	−1.00309	−0.501546	−	0.865131i	$-0.667235\pi$
$487$	−22.1363	−1.00309	−0.501546	+	0.865131i	$0.667235\pi$
$488$	0	0
$489$	−4.99845	−0.226038
$490$	0	0
$491$	36.8492	1.66298	0.831490	−	0.555540i	$-0.187489\pi$
$491$	36.8492	1.66298	0.831490	+	0.555540i	$0.187489\pi$
$492$	0	0
$493$	−5.32782	−0.239953
$494$	0	0
$495$	34.6256	1.55630
$496$	0	0
$497$	−68.3384	−3.06539
$498$	0	0
$499$	7.59699	0.340088	0.170044	−	0.985436i	$-0.445609\pi$
$499$	7.59699	0.340088	0.170044	+	0.985436i	$0.445609\pi$
$500$	0	0
$501$	−9.31131	−0.415999
$502$	0	0
$503$	23.0924	1.02964	0.514819	−	0.857299i	$-0.327859\pi$
$503$	23.0924	1.02964	0.514819	+	0.857299i	$0.327859\pi$
$504$	0	0
$505$	25.6380	1.14088
$506$	0	0
$507$	−0.993990	−0.0441446
$508$	0	0
$509$	−27.9737	−1.23991	−0.619957	−	0.784636i	$-0.712850\pi$
$509$	−27.9737	−1.23991	−0.619957	+	0.784636i	$0.712850\pi$
$510$	0	0
$511$	2.07703	0.0918824
$512$	0	0
$513$	−28.2730	−1.24828
$514$	0	0
$515$	56.3583	2.48344
$516$	0	0
$517$	−7.00700	−0.308167
$518$	0	0
$519$	12.7239	0.558518
$520$	0	0
$521$	3.28433	0.143889	0.0719446	−	0.997409i	$-0.477080\pi$
$521$	3.28433	0.143889	0.0719446	+	0.997409i	$0.477080\pi$
$522$	0	0
$523$	16.0950	0.703787	0.351893	−	0.936040i	$-0.385538\pi$
$523$	16.0950	0.703787	0.351893	+	0.936040i	$0.385538\pi$
$524$	0	0
$525$	26.0987	1.13904
$526$	0	0
$527$	−2.77998	−0.121098
$528$	0	0
$529$	−9.87623	−0.429401
$530$	0	0
$531$	26.7755	1.16196
$532$	0	0
$533$	5.95988	0.258151
$534$	0	0
$535$	53.6917	2.32129
$536$	0	0
$537$	20.0951	0.867167
$538$	0	0
$539$	−82.2765	−3.54390
$540$	0	0
$541$	22.9524	0.986802	0.493401	−	0.869802i	$-0.335753\pi$
$541$	22.9524	0.986802	0.493401	+	0.869802i	$0.335753\pi$
$542$	0	0
$543$	−8.14747	−0.349641
$544$	0	0
$545$	−19.6655	−0.842377
$546$	0	0
$547$	−9.88969	−0.422853	−0.211426	−	0.977394i	$-0.567811\pi$
$547$	−9.88969	−0.422853	−0.211426	+	0.977394i	$0.567811\pi$
$548$	0	0
$549$	−24.7891	−1.05797
$550$	0	0
$551$	−5.67520	−0.241771
$552$	0	0
$553$	−56.2622	−2.39251
$554$	0	0
$555$	19.6956	0.836033
$556$	0	0
$557$	33.6618	1.42630	0.713148	−	0.701013i	$-0.247269\pi$
$557$	33.6618	1.42630	0.713148	+	0.701013i	$0.247269\pi$
$558$	0	0
$559$	−11.9372	−0.504889
$560$	0	0
$561$	−28.0815	−1.18560
$562$	0	0
$563$	−14.2522	−0.600659	−0.300330	−	0.953835i	$-0.597097\pi$
$563$	−14.2522	−0.600659	−0.300330	+	0.953835i	$0.597097\pi$
$564$	0	0
$565$	−33.4824	−1.40861
$566$	0	0
$567$	−5.14389	−0.216023
$568$	0	0
$569$	−21.8378	−0.915489	−0.457744	−	0.889084i	$-0.651343\pi$
$569$	−21.8378	−0.915489	−0.457744	+	0.889084i	$0.651343\pi$
$570$	0	0
$571$	−1.13118	−0.0473385	−0.0236693	−	0.999720i	$-0.507535\pi$
$571$	−1.13118	−0.0473385	−0.0236693	+	0.999720i	$0.507535\pi$
$572$	0	0
$573$	5.11310	0.213603
$574$	0	0
$575$	−20.0455	−0.835956
$576$	0	0
$577$	20.4377	0.850833	0.425416	−	0.904998i	$-0.360128\pi$
$577$	20.4377	0.850833	0.425416	+	0.904998i	$0.360128\pi$
$578$	0	0
$579$	−0.604541	−0.0251239
$580$	0	0
$581$	18.8573	0.782331
$582$	0	0
$583$	26.0015	1.07687
$584$	0	0
$585$	6.52992	0.269979
$586$	0	0
$587$	18.7479	0.773811	0.386905	−	0.922119i	$-0.373544\pi$
$587$	18.7479	0.773811	0.386905	+	0.922119i	$0.373544\pi$
$588$	0	0
$589$	−2.96123	−0.122016
$590$	0	0
$591$	9.13067	0.375586
$592$	0	0
$593$	34.0931	1.40003	0.700017	−	0.714126i	$-0.253176\pi$
$593$	34.0931	1.40003	0.700017	+	0.714126i	$0.253176\pi$
$594$	0	0
$595$	82.0504	3.36374
$596$	0	0
$597$	26.0233	1.06506
$598$	0	0
$599$	−7.18565	−0.293598	−0.146799	−	0.989166i	$-0.546897\pi$
$599$	−7.18565	−0.293598	−0.146799	+	0.989166i	$0.546897\pi$
$600$	0	0
$601$	−42.3294	−1.72665	−0.863327	−	0.504645i	$-0.831623\pi$
$601$	−42.3294	−1.72665	−0.863327	+	0.504645i	$0.831623\pi$
$602$	0	0
$603$	−2.44836	−0.0997049
$604$	0	0
$605$	55.5553	2.25864
$606$	0	0
$607$	21.3647	0.867165	0.433583	−	0.901114i	$-0.357249\pi$
$607$	21.3647	0.867165	0.433583	+	0.901114i	$0.357249\pi$
$608$	0	0
$609$	4.71661	0.191127
$610$	0	0
$611$	−1.32143	−0.0534592
$612$	0	0
$613$	22.3507	0.902737	0.451368	−	0.892338i	$-0.350936\pi$
$613$	22.3507	0.902737	0.451368	+	0.892338i	$0.350936\pi$
$614$	0	0
$615$	19.2266	0.775292
$616$	0	0
$617$	35.3967	1.42502	0.712508	−	0.701664i	$-0.247559\pi$
$617$	35.3967	1.42502	0.712508	+	0.701664i	$0.247559\pi$
$618$	0	0
$619$	30.1770	1.21292	0.606458	−	0.795116i	$-0.292590\pi$
$619$	30.1770	1.21292	0.606458	+	0.795116i	$0.292590\pi$
$620$	0	0
$621$	−18.0477	−0.724228
$622$	0	0
$623$	−2.22373	−0.0890919
$624$	0	0
$625$	−22.0488	−0.881951
$626$	0	0
$627$	−29.9124	−1.19459
$628$	0	0
$629$	32.5277	1.29697
$630$	0	0
$631$	46.2340	1.84055	0.920274	−	0.391274i	$-0.127966\pi$
$631$	46.2340	1.84055	0.920274	+	0.391274i	$0.127966\pi$
$632$	0	0
$633$	−1.86714	−0.0742122
$634$	0	0
$635$	63.2329	2.50932
$636$	0	0
$637$	−15.5163	−0.614776
$638$	0	0
$639$	−28.9762	−1.14628
$640$	0	0
$641$	38.3175	1.51345	0.756726	−	0.653732i	$-0.226798\pi$
$641$	38.3175	1.51345	0.756726	+	0.653732i	$0.226798\pi$
$642$	0	0
$643$	22.6571	0.893507	0.446754	−	0.894657i	$-0.352580\pi$
$643$	22.6571	0.893507	0.446754	+	0.894657i	$0.352580\pi$
$644$	0	0
$645$	−38.5094	−1.51631
$646$	0	0
$647$	16.9739	0.667312	0.333656	−	0.942695i	$-0.391718\pi$
$647$	16.9739	0.667312	0.333656	+	0.942695i	$0.391718\pi$
$648$	0	0
$649$	70.5670	2.77000
$650$	0	0
$651$	2.46106	0.0964565
$652$	0	0
$653$	−35.6770	−1.39615	−0.698074	−	0.716025i	$-0.745959\pi$
$653$	−35.6770	−1.39615	−0.698074	+	0.716025i	$0.745959\pi$
$654$	0	0
$655$	−40.7064	−1.59053
$656$	0	0
$657$	0.880683	0.0343587
$658$	0	0
$659$	18.7581	0.730711	0.365355	−	0.930868i	$-0.380948\pi$
$659$	18.7581	0.730711	0.365355	+	0.930868i	$0.380948\pi$
$660$	0	0
$661$	0.155240	0.00603812	0.00301906	−	0.999995i	$-0.499039\pi$
$661$	0.155240	0.00603812	0.00301906	+	0.999995i	$0.499039\pi$
$662$	0	0
$663$	−5.29579	−0.205672
$664$	0	0
$665$	87.4001	3.38923
$666$	0	0
$667$	−3.62267	−0.140271
$668$	0	0
$669$	22.4858	0.869352
$670$	0	0
$671$	−65.3318	−2.52211
$672$	0	0
$673$	42.6487	1.64399	0.821993	−	0.569497i	$-0.192862\pi$
$673$	42.6487	1.64399	0.821993	+	0.569497i	$0.192862\pi$
$674$	0	0
$675$	27.5664	1.06103
$676$	0	0
$677$	38.7103	1.48776	0.743880	−	0.668314i	$-0.232984\pi$
$677$	38.7103	1.48776	0.743880	+	0.668314i	$0.232984\pi$
$678$	0	0
$679$	39.9156	1.53182
$680$	0	0
$681$	14.3152	0.548558
$682$	0	0
$683$	−27.7990	−1.06370	−0.531849	−	0.846839i	$-0.678503\pi$
$683$	−27.7990	−1.06370	−0.531849	+	0.846839i	$0.678503\pi$
$684$	0	0
$685$	−5.37712	−0.205449
$686$	0	0
$687$	2.85316	0.108855
$688$	0	0
$689$	4.90353	0.186810
$690$	0	0
$691$	34.3799	1.30787	0.653937	−	0.756549i	$-0.273116\pi$
$691$	34.3799	1.30787	0.653937	+	0.756549i	$0.273116\pi$
$692$	0	0
$693$	−50.6246	−1.92307
$694$	0	0
$695$	−55.8719	−2.11934
$696$	0	0
$697$	31.7532	1.20274
$698$	0	0
$699$	−2.05314	−0.0776567
$700$	0	0
$701$	−16.3043	−0.615806	−0.307903	−	0.951418i	$-0.599627\pi$
$701$	−16.3043	−0.615806	−0.307903	+	0.951418i	$0.599627\pi$
$702$	0	0
$703$	34.6486	1.30680
$704$	0	0
$705$	−4.26293	−0.160551
$706$	0	0
$707$	−37.4843	−1.40974
$708$	0	0
$709$	38.5472	1.44767	0.723835	−	0.689973i	$-0.242378\pi$
$709$	38.5472	1.44767	0.723835	+	0.689973i	$0.242378\pi$
$710$	0	0
$711$	−23.8558	−0.894661
$712$	0	0
$713$	−1.89026	−0.0707908
$714$	0	0
$715$	17.2097	0.643605
$716$	0	0
$717$	10.0026	0.373554
$718$	0	0
$719$	15.1886	0.566438	0.283219	−	0.959055i	$-0.408598\pi$
$719$	15.1886	0.566438	0.283219	+	0.959055i	$0.408598\pi$
$720$	0	0
$721$	−82.3991	−3.06870
$722$	0	0
$723$	10.0445	0.373560
$724$	0	0
$725$	5.53335	0.205503
$726$	0	0
$727$	1.21252	0.0449700	0.0224850	−	0.999747i	$-0.492842\pi$
$727$	1.21252	0.0449700	0.0224850	+	0.999747i	$0.492842\pi$
$728$	0	0
$729$	12.6747	0.469433
$730$	0	0
$731$	−63.5991	−2.35230
$732$	0	0
$733$	−39.5028	−1.45907	−0.729535	−	0.683943i	$-0.760264\pi$
$733$	−39.5028	−1.45907	−0.729535	+	0.683943i	$0.760264\pi$
$734$	0	0
$735$	−50.0555	−1.84633
$736$	0	0
$737$	−6.45267	−0.237687
$738$	0	0
$739$	3.21870	0.118402	0.0592008	−	0.998246i	$-0.481145\pi$
$739$	3.21870	0.118402	0.0592008	+	0.998246i	$0.481145\pi$
$740$	0	0
$741$	−5.64108	−0.207230
$742$	0	0
$743$	−15.4550	−0.566989	−0.283494	−	0.958974i	$-0.591494\pi$
$743$	−15.4550	−0.566989	−0.283494	+	0.958974i	$0.591494\pi$
$744$	0	0
$745$	6.14168	0.225014
$746$	0	0
$747$	7.99567	0.292546
$748$	0	0
$749$	−78.5004	−2.86834
$750$	0	0
$751$	6.88626	0.251283	0.125642	−	0.992076i	$-0.459901\pi$
$751$	6.88626	0.251283	0.125642	+	0.992076i	$0.459901\pi$
$752$	0	0
$753$	8.36809	0.304950
$754$	0	0
$755$	−15.8700	−0.577569
$756$	0	0
$757$	−46.2391	−1.68059	−0.840295	−	0.542130i	$-0.817618\pi$
$757$	−46.2391	−1.68059	−0.840295	+	0.542130i	$0.817618\pi$
$758$	0	0
$759$	−19.0941	−0.693073
$760$	0	0
$761$	50.0455	1.81415	0.907074	−	0.420971i	$-0.138311\pi$
$761$	50.0455	1.81415	0.907074	+	0.420971i	$0.138311\pi$
$762$	0	0
$763$	28.7521	1.04090
$764$	0	0
$765$	34.7902	1.25784
$766$	0	0
$767$	13.3080	0.480524
$768$	0	0
$769$	10.5333	0.379842	0.189921	−	0.981799i	$-0.439177\pi$
$769$	10.5333	0.379842	0.189921	+	0.981799i	$0.439177\pi$
$770$	0	0
$771$	−2.91194	−0.104871
$772$	0	0
$773$	26.8989	0.967485	0.483743	−	0.875210i	$-0.339277\pi$
$773$	26.8989	0.967485	0.483743	+	0.875210i	$0.339277\pi$
$774$	0	0
$775$	2.88722	0.103712
$776$	0	0
$777$	−28.7962	−1.03306
$778$	0	0
$779$	33.8235	1.21185
$780$	0	0
$781$	−76.3669	−2.73262
$782$	0	0
$783$	4.98186	0.178037
$784$	0	0
$785$	1.69622	0.0605407
$786$	0	0
$787$	37.6289	1.34133	0.670663	−	0.741763i	$-0.266010\pi$
$787$	37.6289	1.34133	0.670663	+	0.741763i	$0.266010\pi$
$788$	0	0
$789$	17.7649	0.632447
$790$	0	0
$791$	48.9532	1.74058
$792$	0	0
$793$	−12.3207	−0.437521
$794$	0	0
$795$	15.8188	0.561036
$796$	0	0
$797$	10.9795	0.388913	0.194456	−	0.980911i	$-0.437706\pi$
$797$	10.9795	0.388913	0.194456	+	0.980911i	$0.437706\pi$
$798$	0	0
$799$	−7.04032	−0.249069
$800$	0	0
$801$	−0.942885	−0.0333152
$802$	0	0
$803$	2.32105	0.0819079
$804$	0	0
$805$	55.7906	1.96636
$806$	0	0
$807$	−12.8614	−0.452741
$808$	0	0
$809$	−37.8352	−1.33021	−0.665107	−	0.746748i	$-0.731614\pi$
$809$	−37.8352	−1.33021	−0.665107	+	0.746748i	$0.731614\pi$
$810$	0	0
$811$	48.1617	1.69119	0.845593	−	0.533828i	$-0.179247\pi$
$811$	48.1617	1.69119	0.845593	+	0.533828i	$0.179247\pi$
$812$	0	0
$813$	3.15229	0.110556
$814$	0	0
$815$	16.3206	0.571686
$816$	0	0
$817$	−67.7459	−2.37013
$818$	0	0
$819$	−9.54713	−0.333604
$820$	0	0
$821$	−2.07943	−0.0725726	−0.0362863	−	0.999341i	$-0.511553\pi$
$821$	−2.07943	−0.0725726	−0.0362863	+	0.999341i	$0.511553\pi$
$822$	0	0
$823$	−36.9221	−1.28702	−0.643512	−	0.765436i	$-0.722523\pi$
$823$	−36.9221	−1.28702	−0.643512	+	0.765436i	$0.722523\pi$
$824$	0	0
$825$	29.1648	1.01539
$826$	0	0
$827$	25.7556	0.895609	0.447805	−	0.894131i	$-0.352206\pi$
$827$	25.7556	0.895609	0.447805	+	0.894131i	$0.352206\pi$
$828$	0	0
$829$	25.2232	0.876037	0.438018	−	0.898966i	$-0.355680\pi$
$829$	25.2232	0.876037	0.438018	+	0.898966i	$0.355680\pi$
$830$	0	0
$831$	−9.76930	−0.338893
$832$	0	0
$833$	−82.6677	−2.86427
$834$	0	0
$835$	30.4027	1.05213
$836$	0	0
$837$	2.59946	0.0898506
$838$	0	0
$839$	−12.8416	−0.443340	−0.221670	−	0.975122i	$-0.571151\pi$
$839$	−12.8416	−0.443340	−0.221670	+	0.975122i	$0.571151\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.49064	−0.0857823
$844$	0	0
$845$	3.24551	0.111649
$846$	0	0
$847$	−81.2250	−2.79093
$848$	0	0
$849$	13.1031	0.449698
$850$	0	0
$851$	22.1174	0.758175
$852$	0	0
$853$	52.9787	1.81396	0.906978	−	0.421178i	$-0.138383\pi$
$853$	52.9787	1.81396	0.906978	+	0.421178i	$0.138383\pi$
$854$	0	0
$855$	37.0586	1.26738
$856$	0	0
$857$	25.2165	0.861378	0.430689	−	0.902501i	$-0.358271\pi$
$857$	25.2165	0.861378	0.430689	+	0.902501i	$0.358271\pi$
$858$	0	0
$859$	−53.1342	−1.81291	−0.906457	−	0.422298i	$-0.861224\pi$
$859$	−53.1342	−1.81291	−0.906457	+	0.422298i	$0.861224\pi$
$860$	0	0
$861$	−28.1104	−0.958001
$862$	0	0
$863$	−51.0906	−1.73914	−0.869572	−	0.493806i	$-0.835605\pi$
$863$	−51.0906	−1.73914	−0.869572	+	0.493806i	$0.835605\pi$
$864$	0	0
$865$	−41.5454	−1.41258
$866$	0	0
$867$	−11.3172	−0.384352
$868$	0	0
$869$	−62.8721	−2.13279
$870$	0	0
$871$	−1.21689	−0.0412327
$872$	0	0
$873$	16.9246	0.572812
$874$	0	0
$875$	−8.21380	−0.277677
$876$	0	0
$877$	31.3606	1.05897	0.529487	−	0.848318i	$-0.322385\pi$
$877$	31.3606	1.05897	0.529487	+	0.848318i	$0.322385\pi$
$878$	0	0
$879$	−32.4558	−1.09471
$880$	0	0
$881$	12.8522	0.433003	0.216501	−	0.976282i	$-0.430535\pi$
$881$	12.8522	0.433003	0.216501	+	0.976282i	$0.430535\pi$
$882$	0	0
$883$	24.9594	0.839952	0.419976	−	0.907535i	$-0.362039\pi$
$883$	24.9594	0.839952	0.419976	+	0.907535i	$0.362039\pi$
$884$	0	0
$885$	42.9317	1.44313
$886$	0	0
$887$	−18.1445	−0.609232	−0.304616	−	0.952475i	$-0.598528\pi$
$887$	−18.1445	−0.609232	−0.304616	+	0.952475i	$0.598528\pi$
$888$	0	0
$889$	−92.4502	−3.10068
$890$	0	0
$891$	−5.74821	−0.192572
$892$	0	0
$893$	−7.49935	−0.250956
$894$	0	0
$895$	−65.6132	−2.19321
$896$	0	0
$897$	−3.60090	−0.120231
$898$	0	0
$899$	0.521785	0.0174025
$900$	0	0
$901$	26.1251	0.870354
$902$	0	0
$903$	56.3030	1.87365
$904$	0	0
$905$	26.6026	0.884300
$906$	0	0
$907$	18.6733	0.620037	0.310019	−	0.950730i	$-0.399665\pi$
$907$	18.6733	0.620037	0.310019	+	0.950730i	$0.399665\pi$
$908$	0	0
$909$	−15.8937	−0.527161
$910$	0	0
$911$	−26.8390	−0.889216	−0.444608	−	0.895725i	$-0.646657\pi$
$911$	−26.8390	−0.889216	−0.444608	+	0.895725i	$0.646657\pi$
$912$	0	0
$913$	21.0727	0.697403
$914$	0	0
$915$	−39.7467	−1.31398
$916$	0	0
$917$	59.5152	1.96536
$918$	0	0
$919$	−6.97398	−0.230051	−0.115025	−	0.993363i	$-0.536695\pi$
$919$	−6.97398	−0.230051	−0.115025	+	0.993363i	$0.536695\pi$
$920$	0	0
$921$	2.92039	0.0962300
$922$	0	0
$923$	−14.4018	−0.474041
$924$	0	0
$925$	−33.7826	−1.11076
$926$	0	0
$927$	−34.9381	−1.14752
$928$	0	0
$929$	2.78489	0.0913692	0.0456846	−	0.998956i	$-0.485453\pi$
$929$	2.78489	0.0913692	0.0456846	+	0.998956i	$0.485453\pi$
$930$	0	0
$931$	−88.0578	−2.88598
$932$	0	0
$933$	−10.1390	−0.331936
$934$	0	0
$935$	91.6899	2.99858
$936$	0	0
$937$	−60.1641	−1.96548	−0.982738	−	0.185003i	$-0.940770\pi$
$937$	−60.1641	−1.96548	−0.982738	+	0.185003i	$0.940770\pi$
$938$	0	0
$939$	15.4348	0.503697
$940$	0	0
$941$	−18.6235	−0.607110	−0.303555	−	0.952814i	$-0.598174\pi$
$941$	−18.6235	−0.607110	−0.303555	+	0.952814i	$0.598174\pi$
$942$	0	0
$943$	21.5907	0.703090
$944$	0	0
$945$	−76.7225	−2.49578
$946$	0	0
$947$	52.6959	1.71239	0.856194	−	0.516654i	$-0.172823\pi$
$947$	52.6959	1.71239	0.856194	+	0.516654i	$0.172823\pi$
$948$	0	0
$949$	0.437718	0.0142089
$950$	0	0
$951$	18.1264	0.587790
$952$	0	0
$953$	−4.81931	−0.156113	−0.0780563	−	0.996949i	$-0.524871\pi$
$953$	−4.81931	−0.156113	−0.0780563	+	0.996949i	$0.524871\pi$
$954$	0	0
$955$	−16.6950	−0.540237
$956$	0	0
$957$	5.27073	0.170378
$958$	0	0
$959$	7.86166	0.253866
$960$	0	0
$961$	−30.7277	−0.991217
$962$	0	0
$963$	−33.2850	−1.07259
$964$	0	0
$965$	1.97391	0.0635424
$966$	0	0
$967$	44.2362	1.42254	0.711269	−	0.702920i	$-0.248121\pi$
$967$	44.2362	1.42254	0.711269	+	0.702920i	$0.248121\pi$
$968$	0	0
$969$	−30.0547	−0.965495
$970$	0	0
$971$	47.8099	1.53429	0.767146	−	0.641472i	$-0.221676\pi$
$971$	47.8099	1.53429	0.767146	+	0.641472i	$0.221676\pi$
$972$	0	0
$973$	81.6880	2.61880
$974$	0	0
$975$	5.50009	0.176144
$976$	0	0
$977$	−40.0574	−1.28155	−0.640774	−	0.767729i	$-0.721386\pi$
$977$	−40.0574	−1.28155	−0.640774	+	0.767729i	$0.721386\pi$
$978$	0	0
$979$	−2.48498	−0.0794203
$980$	0	0
$981$	12.1912	0.389235
$982$	0	0
$983$	2.87278	0.0916274	0.0458137	−	0.998950i	$-0.485412\pi$
$983$	2.87278	0.0916274	0.0458137	+	0.998950i	$0.485412\pi$
$984$	0	0
$985$	−29.8129	−0.949918
$986$	0	0
$987$	6.23265	0.198388
$988$	0	0
$989$	−43.2445	−1.37510
$990$	0	0
$991$	−24.0259	−0.763208	−0.381604	−	0.924326i	$-0.624628\pi$
$991$	−24.0259	−0.763208	−0.381604	+	0.924326i	$0.624628\pi$
$992$	0	0
$993$	11.4673	0.363903
$994$	0	0
$995$	−84.9696	−2.69372
$996$	0	0
$997$	−42.9532	−1.36034	−0.680170	−	0.733054i	$-0.738094\pi$
$997$	−42.9532	−1.36034	−0.680170	+	0.733054i	$0.738094\pi$
$998$	0	0
$999$	−30.4156	−0.962307

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.4		13
4.3	odd	2		3016.2.a.k.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.10	✓	13	4.3	odd	2
6032.2.a.be.1.4		13	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-0.993990 q^{3} +3.24551 q^{5} -4.74513 q^{7} -2.01198 q^{9} +O(q^{10})\)	\(q-0.993990 q^{3} +3.24551 q^{5} -4.74513 q^{7} -2.01198 q^{9} -5.30260 q^{11} -1.00000 q^{13} -3.22601 q^{15} -5.32782 q^{17} -5.67520 q^{19} +4.71661 q^{21} -3.62267 q^{23} +5.53335 q^{25} +4.98186 q^{27} +1.00000 q^{29} +0.521785 q^{31} +5.27073 q^{33} -15.4004 q^{35} -6.10527 q^{37} +0.993990 q^{39} -5.95988 q^{41} +11.9372 q^{43} -6.52992 q^{45} +1.32143 q^{47} +15.5163 q^{49} +5.29579 q^{51} -4.90353 q^{53} -17.2097 q^{55} +5.64108 q^{57} -13.3080 q^{59} +12.3207 q^{61} +9.54713 q^{63} -3.24551 q^{65} +1.21689 q^{67} +3.60090 q^{69} +14.4018 q^{71} -0.437718 q^{73} -5.50009 q^{75} +25.1615 q^{77} +11.8568 q^{79} +1.08404 q^{81} -3.97402 q^{83} -17.2915 q^{85} -0.993990 q^{87} +0.468634 q^{89} +4.74513 q^{91} -0.518649 q^{93} -18.4189 q^{95} -8.41190 q^{97} +10.6687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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