LMFDB - Embedded newform 8016.2.a.bg.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
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} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.79706$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.79706 q^{5} -4.56518 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.79706 q^{5} -4.56518 q^{7} +1.00000 q^{9} -2.91073 q^{11} -0.850435 q^{13} -2.79706 q^{15} -7.79699 q^{17} -2.29761 q^{19} +4.56518 q^{21} -2.74788 q^{23} +2.82355 q^{25} -1.00000 q^{27} -6.57467 q^{29} +7.06047 q^{31} +2.91073 q^{33} -12.7691 q^{35} +8.35845 q^{37} +0.850435 q^{39} +4.96938 q^{41} -11.4328 q^{43} +2.79706 q^{45} -12.1758 q^{47} +13.8408 q^{49} +7.79699 q^{51} -6.03568 q^{53} -8.14149 q^{55} +2.29761 q^{57} +3.64159 q^{59} +3.09676 q^{61} -4.56518 q^{63} -2.37872 q^{65} +14.0723 q^{67} +2.74788 q^{69} +0.961653 q^{71} -6.57980 q^{73} -2.82355 q^{75} +13.2880 q^{77} -2.76849 q^{79} +1.00000 q^{81} +16.8645 q^{83} -21.8086 q^{85} +6.57467 q^{87} +8.20258 q^{89} +3.88239 q^{91} -7.06047 q^{93} -6.42656 q^{95} -8.44292 q^{97} -2.91073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.79706	1.25088	0.625442	−	0.780271i	$-0.284919\pi$
$5$	2.79706	1.25088	0.625442	+	0.780271i	$0.284919\pi$
$6$	0	0
$7$	−4.56518	−1.72547	−0.862737	−	0.505652i	$-0.831252\pi$
$7$	−4.56518	−1.72547	−0.862737	+	0.505652i	$0.831252\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.91073	−0.877618	−0.438809	−	0.898580i	$-0.644599\pi$
$11$	−2.91073	−0.877618	−0.438809	+	0.898580i	$0.644599\pi$
$12$	0	0
$13$	−0.850435	−0.235868	−0.117934	−	0.993021i	$-0.537627\pi$
$13$	−0.850435	−0.235868	−0.117934	+	0.993021i	$0.537627\pi$
$14$	0	0
$15$	−2.79706	−0.722198
$16$	0	0
$17$	−7.79699	−1.89105	−0.945523	−	0.325554i	$-0.894449\pi$
$17$	−7.79699	−1.89105	−0.945523	+	0.325554i	$0.894449\pi$
$18$	0	0
$19$	−2.29761	−0.527108	−0.263554	−	0.964645i	$-0.584895\pi$
$19$	−2.29761	−0.527108	−0.263554	+	0.964645i	$0.584895\pi$
$20$	0	0
$21$	4.56518	0.996203
$22$	0	0
$23$	−2.74788	−0.572972	−0.286486	−	0.958084i	$-0.592487\pi$
$23$	−2.74788	−0.572972	−0.286486	+	0.958084i	$0.592487\pi$
$24$	0	0
$25$	2.82355	0.564710
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.57467	−1.22088	−0.610442	−	0.792061i	$-0.709008\pi$
$29$	−6.57467	−1.22088	−0.610442	+	0.792061i	$0.709008\pi$
$30$	0	0
$31$	7.06047	1.26810	0.634049	−	0.773293i	$-0.281392\pi$
$31$	7.06047	1.26810	0.634049	+	0.773293i	$0.281392\pi$
$32$	0	0
$33$	2.91073	0.506693
$34$	0	0
$35$	−12.7691	−2.15837
$36$	0	0
$37$	8.35845	1.37412	0.687060	−	0.726600i	$-0.258901\pi$
$37$	8.35845	1.37412	0.687060	+	0.726600i	$0.258901\pi$
$38$	0	0
$39$	0.850435	0.136179
$40$	0	0
$41$	4.96938	0.776087	0.388043	−	0.921641i	$-0.373151\pi$
$41$	4.96938	0.776087	0.388043	+	0.921641i	$0.373151\pi$
$42$	0	0
$43$	−11.4328	−1.74349	−0.871745	−	0.489960i	$-0.837011\pi$
$43$	−11.4328	−1.74349	−0.871745	+	0.489960i	$0.837011\pi$
$44$	0	0
$45$	2.79706	0.416961
$46$	0	0
$47$	−12.1758	−1.77602	−0.888010	−	0.459824i	$-0.847912\pi$
$47$	−12.1758	−1.77602	−0.888010	+	0.459824i	$0.847912\pi$
$48$	0	0
$49$	13.8408	1.97726
$50$	0	0
$51$	7.79699	1.09180
$52$	0	0
$53$	−6.03568	−0.829065	−0.414532	−	0.910035i	$-0.636055\pi$
$53$	−6.03568	−0.829065	−0.414532	+	0.910035i	$0.636055\pi$
$54$	0	0
$55$	−8.14149	−1.09780
$56$	0	0
$57$	2.29761	0.304326
$58$	0	0
$59$	3.64159	0.474094	0.237047	−	0.971498i	$-0.423820\pi$
$59$	3.64159	0.474094	0.237047	+	0.971498i	$0.423820\pi$
$60$	0	0
$61$	3.09676	0.396500	0.198250	−	0.980151i	$-0.436474\pi$
$61$	3.09676	0.396500	0.198250	+	0.980151i	$0.436474\pi$
$62$	0	0
$63$	−4.56518	−0.575158
$64$	0	0
$65$	−2.37872	−0.295044
$66$	0	0
$67$	14.0723	1.71921	0.859603	−	0.510963i	$-0.170711\pi$
$67$	14.0723	1.71921	0.859603	+	0.510963i	$0.170711\pi$
$68$	0	0
$69$	2.74788	0.330806
$70$	0	0
$71$	0.961653	0.114127	0.0570636	−	0.998371i	$-0.481826\pi$
$71$	0.961653	0.114127	0.0570636	+	0.998371i	$0.481826\pi$
$72$	0	0
$73$	−6.57980	−0.770108	−0.385054	−	0.922894i	$-0.625817\pi$
$73$	−6.57980	−0.770108	−0.385054	+	0.922894i	$0.625817\pi$
$74$	0	0
$75$	−2.82355	−0.326036
$76$	0	0
$77$	13.2880	1.51431
$78$	0	0
$79$	−2.76849	−0.311479	−0.155740	−	0.987798i	$-0.549776\pi$
$79$	−2.76849	−0.311479	−0.155740	+	0.987798i	$0.549776\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.8645	1.85112	0.925562	−	0.378596i	$-0.123593\pi$
$83$	16.8645	1.85112	0.925562	+	0.378596i	$0.123593\pi$
$84$	0	0
$85$	−21.8086	−2.36548
$86$	0	0
$87$	6.57467	0.704878
$88$	0	0
$89$	8.20258	0.869472	0.434736	−	0.900558i	$-0.356842\pi$
$89$	8.20258	0.869472	0.434736	+	0.900558i	$0.356842\pi$
$90$	0	0
$91$	3.88239	0.406985
$92$	0	0
$93$	−7.06047	−0.732136
$94$	0	0
$95$	−6.42656	−0.659351
$96$	0	0
$97$	−8.44292	−0.857249	−0.428624	−	0.903483i	$-0.641002\pi$
$97$	−8.44292	−0.857249	−0.428624	+	0.903483i	$0.641002\pi$
$98$	0	0
$99$	−2.91073	−0.292539
$100$	0	0
$101$	1.35436	0.134764	0.0673821	−	0.997727i	$-0.478535\pi$
$101$	1.35436	0.134764	0.0673821	+	0.997727i	$0.478535\pi$
$102$	0	0
$103$	3.24034	0.319280	0.159640	−	0.987175i	$-0.448967\pi$
$103$	3.24034	0.319280	0.159640	+	0.987175i	$0.448967\pi$
$104$	0	0
$105$	12.7691	1.24613
$106$	0	0
$107$	12.4272	1.20138	0.600690	−	0.799482i	$-0.294893\pi$
$107$	12.4272	1.20138	0.600690	+	0.799482i	$0.294893\pi$
$108$	0	0
$109$	19.9444	1.91033	0.955163	−	0.296082i	$-0.0956800\pi$
$109$	19.9444	1.91033	0.955163	+	0.296082i	$0.0956800\pi$
$110$	0	0
$111$	−8.35845	−0.793349
$112$	0	0
$113$	5.83295	0.548718	0.274359	−	0.961627i	$-0.411534\pi$
$113$	5.83295	0.548718	0.274359	+	0.961627i	$0.411534\pi$
$114$	0	0
$115$	−7.68598	−0.716722
$116$	0	0
$117$	−0.850435	−0.0786228
$118$	0	0
$119$	35.5946	3.26295
$120$	0	0
$121$	−2.52766	−0.229787
$122$	0	0
$123$	−4.96938	−0.448074
$124$	0	0
$125$	−6.08766	−0.544497
$126$	0	0
$127$	−1.57092	−0.139397	−0.0696984	−	0.997568i	$-0.522204\pi$
$127$	−1.57092	−0.139397	−0.0696984	+	0.997568i	$0.522204\pi$
$128$	0	0
$129$	11.4328	1.00660
$130$	0	0
$131$	−1.67110	−0.146005	−0.0730025	−	0.997332i	$-0.523258\pi$
$131$	−1.67110	−0.146005	−0.0730025	+	0.997332i	$0.523258\pi$
$132$	0	0
$133$	10.4890	0.909511
$134$	0	0
$135$	−2.79706	−0.240733
$136$	0	0
$137$	10.0249	0.856481	0.428240	−	0.903665i	$-0.359134\pi$
$137$	10.0249	0.856481	0.428240	+	0.903665i	$0.359134\pi$
$138$	0	0
$139$	12.6598	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$139$	12.6598	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$140$	0	0
$141$	12.1758	1.02539
$142$	0	0
$143$	2.47539	0.207002
$144$	0	0
$145$	−18.3897	−1.52719
$146$	0	0
$147$	−13.8408	−1.14157
$148$	0	0
$149$	−19.1913	−1.57221	−0.786107	−	0.618090i	$-0.787907\pi$
$149$	−19.1913	−1.57221	−0.786107	+	0.618090i	$0.787907\pi$
$150$	0	0
$151$	−13.2906	−1.08157	−0.540786	−	0.841160i	$-0.681873\pi$
$151$	−13.2906	−1.08157	−0.540786	+	0.841160i	$0.681873\pi$
$152$	0	0
$153$	−7.79699	−0.630349
$154$	0	0
$155$	19.7486	1.58624
$156$	0	0
$157$	−0.624256	−0.0498211	−0.0249105	−	0.999690i	$-0.507930\pi$
$157$	−0.624256	−0.0498211	−0.0249105	+	0.999690i	$0.507930\pi$
$158$	0	0
$159$	6.03568	0.478661
$160$	0	0
$161$	12.5445	0.988649
$162$	0	0
$163$	−6.71759	−0.526163	−0.263081	−	0.964774i	$-0.584739\pi$
$163$	−6.71759	−0.526163	−0.263081	+	0.964774i	$0.584739\pi$
$164$	0	0
$165$	8.14149	0.633814
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.2768	−0.944366
$170$	0	0
$171$	−2.29761	−0.175703
$172$	0	0
$173$	21.4371	1.62983	0.814917	−	0.579578i	$-0.196783\pi$
$173$	21.4371	1.62983	0.814917	+	0.579578i	$0.196783\pi$
$174$	0	0
$175$	−12.8900	−0.974394
$176$	0	0
$177$	−3.64159	−0.273718
$178$	0	0
$179$	0.637299	0.0476340	0.0238170	−	0.999716i	$-0.492418\pi$
$179$	0.637299	0.0476340	0.0238170	+	0.999716i	$0.492418\pi$
$180$	0	0
$181$	−14.5472	−1.08128	−0.540642	−	0.841253i	$-0.681819\pi$
$181$	−14.5472	−1.08128	−0.540642	+	0.841253i	$0.681819\pi$
$182$	0	0
$183$	−3.09676	−0.228919
$184$	0	0
$185$	23.3791	1.71887
$186$	0	0
$187$	22.6949	1.65962
$188$	0	0
$189$	4.56518	0.332068
$190$	0	0
$191$	20.3497	1.47245	0.736225	−	0.676737i	$-0.236606\pi$
$191$	20.3497	1.47245	0.736225	+	0.676737i	$0.236606\pi$
$192$	0	0
$193$	10.0666	0.724612	0.362306	−	0.932059i	$-0.381990\pi$
$193$	10.0666	0.724612	0.362306	+	0.932059i	$0.381990\pi$
$194$	0	0
$195$	2.37872	0.170344
$196$	0	0
$197$	−1.11488	−0.0794322	−0.0397161	−	0.999211i	$-0.512645\pi$
$197$	−1.11488	−0.0794322	−0.0397161	+	0.999211i	$0.512645\pi$
$198$	0	0
$199$	−3.21029	−0.227572	−0.113786	−	0.993505i	$-0.536298\pi$
$199$	−3.21029	−0.227572	−0.113786	+	0.993505i	$0.536298\pi$
$200$	0	0
$201$	−14.0723	−0.992584
$202$	0	0
$203$	30.0145	2.10661
$204$	0	0
$205$	13.8997	0.970794
$206$	0	0
$207$	−2.74788	−0.190991
$208$	0	0
$209$	6.68772	0.462599
$210$	0	0
$211$	24.1441	1.66215	0.831073	−	0.556164i	$-0.187727\pi$
$211$	24.1441	1.66215	0.831073	+	0.556164i	$0.187727\pi$
$212$	0	0
$213$	−0.961653	−0.0658913
$214$	0	0
$215$	−31.9783	−2.18090
$216$	0	0
$217$	−32.2323	−2.18807
$218$	0	0
$219$	6.57980	0.444622
$220$	0	0
$221$	6.63083	0.446038
$222$	0	0
$223$	16.4018	1.09835	0.549174	−	0.835708i	$-0.314942\pi$
$223$	16.4018	1.09835	0.549174	+	0.835708i	$0.314942\pi$
$224$	0	0
$225$	2.82355	0.188237
$226$	0	0
$227$	−18.8181	−1.24900	−0.624500	−	0.781025i	$-0.714697\pi$
$227$	−18.8181	−1.24900	−0.624500	+	0.781025i	$0.714697\pi$
$228$	0	0
$229$	12.7703	0.843885	0.421943	−	0.906623i	$-0.361348\pi$
$229$	12.7703	0.843885	0.421943	+	0.906623i	$0.361348\pi$
$230$	0	0
$231$	−13.2880	−0.874286
$232$	0	0
$233$	−18.2136	−1.19321	−0.596606	−	0.802534i	$-0.703485\pi$
$233$	−18.2136	−1.19321	−0.596606	+	0.802534i	$0.703485\pi$
$234$	0	0
$235$	−34.0564	−2.22159
$236$	0	0
$237$	2.76849	0.179833
$238$	0	0
$239$	2.53425	0.163927	0.0819637	−	0.996635i	$-0.473881\pi$
$239$	2.53425	0.163927	0.0819637	+	0.996635i	$0.473881\pi$
$240$	0	0
$241$	10.9201	0.703426	0.351713	−	0.936108i	$-0.385599\pi$
$241$	10.9201	0.703426	0.351713	+	0.936108i	$0.385599\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	38.7137	2.47333
$246$	0	0
$247$	1.95397	0.124328
$248$	0	0
$249$	−16.8645	−1.06875
$250$	0	0
$251$	−7.08878	−0.447440	−0.223720	−	0.974653i	$-0.571820\pi$
$251$	−7.08878	−0.447440	−0.223720	+	0.974653i	$0.571820\pi$
$252$	0	0
$253$	7.99833	0.502851
$254$	0	0
$255$	21.8086	1.36571
$256$	0	0
$257$	−23.0298	−1.43656	−0.718280	−	0.695754i	$-0.755070\pi$
$257$	−23.0298	−1.43656	−0.718280	+	0.695754i	$0.755070\pi$
$258$	0	0
$259$	−38.1578	−2.37101
$260$	0	0
$261$	−6.57467	−0.406962
$262$	0	0
$263$	−12.6362	−0.779181	−0.389591	−	0.920988i	$-0.627383\pi$
$263$	−12.6362	−0.779181	−0.389591	+	0.920988i	$0.627383\pi$
$264$	0	0
$265$	−16.8822	−1.03706
$266$	0	0
$267$	−8.20258	−0.501990
$268$	0	0
$269$	29.8615	1.82069	0.910343	−	0.413854i	$-0.135818\pi$
$269$	29.8615	1.82069	0.910343	+	0.413854i	$0.135818\pi$
$270$	0	0
$271$	−6.38067	−0.387598	−0.193799	−	0.981041i	$-0.562081\pi$
$271$	−6.38067	−0.387598	−0.193799	+	0.981041i	$0.562081\pi$
$272$	0	0
$273$	−3.88239	−0.234973
$274$	0	0
$275$	−8.21859	−0.495600
$276$	0	0
$277$	23.5063	1.41235	0.706177	−	0.708035i	$-0.250418\pi$
$277$	23.5063	1.41235	0.706177	+	0.708035i	$0.250418\pi$
$278$	0	0
$279$	7.06047	0.422699
$280$	0	0
$281$	−23.4338	−1.39794	−0.698972	−	0.715149i	$-0.746359\pi$
$281$	−23.4338	−1.39794	−0.698972	+	0.715149i	$0.746359\pi$
$282$	0	0
$283$	−17.6525	−1.04933	−0.524666	−	0.851308i	$-0.675810\pi$
$283$	−17.6525	−1.04933	−0.524666	+	0.851308i	$0.675810\pi$
$284$	0	0
$285$	6.42656	0.380676
$286$	0	0
$287$	−22.6861	−1.33912
$288$	0	0
$289$	43.7930	2.57606
$290$	0	0
$291$	8.44292	0.494933
$292$	0	0
$293$	−28.9734	−1.69265	−0.846323	−	0.532671i	$-0.821188\pi$
$293$	−28.9734	−1.69265	−0.846323	+	0.532671i	$0.821188\pi$
$294$	0	0
$295$	10.1857	0.593037
$296$	0	0
$297$	2.91073	0.168898
$298$	0	0
$299$	2.33689	0.135146
$300$	0	0
$301$	52.1929	3.00835
$302$	0	0
$303$	−1.35436	−0.0778062
$304$	0	0
$305$	8.66184	0.495975
$306$	0	0
$307$	−26.5494	−1.51525	−0.757627	−	0.652688i	$-0.773641\pi$
$307$	−26.5494	−1.51525	−0.757627	+	0.652688i	$0.773641\pi$
$308$	0	0
$309$	−3.24034	−0.184336
$310$	0	0
$311$	6.57114	0.372615	0.186308	−	0.982491i	$-0.440348\pi$
$311$	6.57114	0.372615	0.186308	+	0.982491i	$0.440348\pi$
$312$	0	0
$313$	1.39897	0.0790747	0.0395374	−	0.999218i	$-0.487412\pi$
$313$	1.39897	0.0790747	0.0395374	+	0.999218i	$0.487412\pi$
$314$	0	0
$315$	−12.7691	−0.719456
$316$	0	0
$317$	−32.0982	−1.80282	−0.901408	−	0.432971i	$-0.857465\pi$
$317$	−32.0982	−1.80282	−0.901408	+	0.432971i	$0.857465\pi$
$318$	0	0
$319$	19.1371	1.07147
$320$	0	0
$321$	−12.4272	−0.693617
$322$	0	0
$323$	17.9144	0.996786
$324$	0	0
$325$	−2.40125	−0.133197
$326$	0	0
$327$	−19.9444	−1.10293
$328$	0	0
$329$	55.5846	3.06448
$330$	0	0
$331$	30.4241	1.67226	0.836130	−	0.548532i	$-0.184813\pi$
$331$	30.4241	1.67226	0.836130	+	0.548532i	$0.184813\pi$
$332$	0	0
$333$	8.35845	0.458040
$334$	0	0
$335$	39.3611	2.15053
$336$	0	0
$337$	9.18567	0.500375	0.250188	−	0.968197i	$-0.419508\pi$
$337$	9.18567	0.500375	0.250188	+	0.968197i	$0.419508\pi$
$338$	0	0
$339$	−5.83295	−0.316802
$340$	0	0
$341$	−20.5511	−1.11290
$342$	0	0
$343$	−31.2296	−1.68624
$344$	0	0
$345$	7.68598	0.413799
$346$	0	0
$347$	21.2584	1.14121	0.570604	−	0.821225i	$-0.306709\pi$
$347$	21.2584	1.14121	0.570604	+	0.821225i	$0.306709\pi$
$348$	0	0
$349$	−5.23246	−0.280087	−0.140044	−	0.990145i	$-0.544724\pi$
$349$	−5.23246	−0.280087	−0.140044	+	0.990145i	$0.544724\pi$
$350$	0	0
$351$	0.850435	0.0453929
$352$	0	0
$353$	−24.0428	−1.27967	−0.639834	−	0.768513i	$-0.720997\pi$
$353$	−24.0428	−1.27967	−0.639834	+	0.768513i	$0.720997\pi$
$354$	0	0
$355$	2.68980	0.142760
$356$	0	0
$357$	−35.5946	−1.88387
$358$	0	0
$359$	5.99756	0.316539	0.158270	−	0.987396i	$-0.449409\pi$
$359$	5.99756	0.316539	0.158270	+	0.987396i	$0.449409\pi$
$360$	0	0
$361$	−13.7210	−0.722157
$362$	0	0
$363$	2.52766	0.132668
$364$	0	0
$365$	−18.4041	−0.963315
$366$	0	0
$367$	12.5632	0.655793	0.327896	−	0.944714i	$-0.393660\pi$
$367$	12.5632	0.655793	0.327896	+	0.944714i	$0.393660\pi$
$368$	0	0
$369$	4.96938	0.258696
$370$	0	0
$371$	27.5540	1.43053
$372$	0	0
$373$	−0.156554	−0.00810604	−0.00405302	−	0.999992i	$-0.501290\pi$
$373$	−0.156554	−0.00810604	−0.00405302	+	0.999992i	$0.501290\pi$
$374$	0	0
$375$	6.08766	0.314365
$376$	0	0
$377$	5.59133	0.287968
$378$	0	0
$379$	−16.2254	−0.833444	−0.416722	−	0.909034i	$-0.636821\pi$
$379$	−16.2254	−0.833444	−0.416722	+	0.909034i	$0.636821\pi$
$380$	0	0
$381$	1.57092	0.0804808
$382$	0	0
$383$	−12.9406	−0.661232	−0.330616	−	0.943765i	$-0.607257\pi$
$383$	−12.9406	−0.661232	−0.330616	+	0.943765i	$0.607257\pi$
$384$	0	0
$385$	37.1673	1.89422
$386$	0	0
$387$	−11.4328	−0.581163
$388$	0	0
$389$	17.2828	0.876271	0.438136	−	0.898909i	$-0.355639\pi$
$389$	17.2828	0.876271	0.438136	+	0.898909i	$0.355639\pi$
$390$	0	0
$391$	21.4252	1.08352
$392$	0	0
$393$	1.67110	0.0842960
$394$	0	0
$395$	−7.74363	−0.389624
$396$	0	0
$397$	−12.2136	−0.612982	−0.306491	−	0.951873i	$-0.599155\pi$
$397$	−12.2136	−0.612982	−0.306491	+	0.951873i	$0.599155\pi$
$398$	0	0
$399$	−10.4890	−0.525107
$400$	0	0
$401$	26.2780	1.31226	0.656129	−	0.754648i	$-0.272193\pi$
$401$	26.2780	1.31226	0.656129	+	0.754648i	$0.272193\pi$
$402$	0	0
$403$	−6.00447	−0.299104
$404$	0	0
$405$	2.79706	0.138987
$406$	0	0
$407$	−24.3292	−1.20595
$408$	0	0
$409$	−3.32947	−0.164632	−0.0823159	−	0.996606i	$-0.526232\pi$
$409$	−3.32947	−0.164632	−0.0823159	+	0.996606i	$0.526232\pi$
$410$	0	0
$411$	−10.0249	−0.494489
$412$	0	0
$413$	−16.6245	−0.818038
$414$	0	0
$415$	47.1712	2.31554
$416$	0	0
$417$	−12.6598	−0.619953
$418$	0	0
$419$	−10.6768	−0.521594	−0.260797	−	0.965394i	$-0.583985\pi$
$419$	−10.6768	−0.521594	−0.260797	+	0.965394i	$0.583985\pi$
$420$	0	0
$421$	−36.4760	−1.77773	−0.888866	−	0.458167i	$-0.848506\pi$
$421$	−36.4760	−1.77773	−0.888866	+	0.458167i	$0.848506\pi$
$422$	0	0
$423$	−12.1758	−0.592007
$424$	0	0
$425$	−22.0152	−1.06789
$426$	0	0
$427$	−14.1373	−0.684151
$428$	0	0
$429$	−2.47539	−0.119513
$430$	0	0
$431$	23.0164	1.10866	0.554330	−	0.832297i	$-0.312975\pi$
$431$	23.0164	1.10866	0.554330	+	0.832297i	$0.312975\pi$
$432$	0	0
$433$	24.8633	1.19486	0.597428	−	0.801923i	$-0.296190\pi$
$433$	24.8633	1.19486	0.597428	+	0.801923i	$0.296190\pi$
$434$	0	0
$435$	18.3897	0.881721
$436$	0	0
$437$	6.31355	0.302018
$438$	0	0
$439$	−6.84142	−0.326523	−0.163262	−	0.986583i	$-0.552201\pi$
$439$	−6.84142	−0.326523	−0.163262	+	0.986583i	$0.552201\pi$
$440$	0	0
$441$	13.8408	0.659087
$442$	0	0
$443$	22.9161	1.08878	0.544389	−	0.838833i	$-0.316762\pi$
$443$	22.9161	1.08878	0.544389	+	0.838833i	$0.316762\pi$
$444$	0	0
$445$	22.9431	1.08761
$446$	0	0
$447$	19.1913	0.907719
$448$	0	0
$449$	−2.55643	−0.120645	−0.0603226	−	0.998179i	$-0.519213\pi$
$449$	−2.55643	−0.120645	−0.0603226	+	0.998179i	$0.519213\pi$
$450$	0	0
$451$	−14.4645	−0.681107
$452$	0	0
$453$	13.2906	0.624446
$454$	0	0
$455$	10.8593	0.509091
$456$	0	0
$457$	18.3051	0.856277	0.428139	−	0.903713i	$-0.359170\pi$
$457$	18.3051	0.856277	0.428139	+	0.903713i	$0.359170\pi$
$458$	0	0
$459$	7.79699	0.363932
$460$	0	0
$461$	15.9794	0.744236	0.372118	−	0.928185i	$-0.378632\pi$
$461$	15.9794	0.744236	0.372118	+	0.928185i	$0.378632\pi$
$462$	0	0
$463$	40.0301	1.86036	0.930178	−	0.367108i	$-0.119652\pi$
$463$	40.0301	1.86036	0.930178	+	0.367108i	$0.119652\pi$
$464$	0	0
$465$	−19.7486	−0.915817
$466$	0	0
$467$	−19.8776	−0.919825	−0.459913	−	0.887964i	$-0.652119\pi$
$467$	−19.8776	−0.919825	−0.459913	+	0.887964i	$0.652119\pi$
$468$	0	0
$469$	−64.2426	−2.96645
$470$	0	0
$471$	0.624256	0.0287642
$472$	0	0
$473$	33.2779	1.53012
$474$	0	0
$475$	−6.48742	−0.297663
$476$	0	0
$477$	−6.03568	−0.276355
$478$	0	0
$479$	21.8981	1.00055	0.500274	−	0.865867i	$-0.333233\pi$
$479$	21.8981	1.00055	0.500274	+	0.865867i	$0.333233\pi$
$480$	0	0
$481$	−7.10832	−0.324112
$482$	0	0
$483$	−12.5445	−0.570797
$484$	0	0
$485$	−23.6154	−1.07232
$486$	0	0
$487$	−31.4280	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$487$	−31.4280	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$488$	0	0
$489$	6.71759	0.303780
$490$	0	0
$491$	−26.9355	−1.21558	−0.607792	−	0.794097i	$-0.707944\pi$
$491$	−26.9355	−1.21558	−0.607792	+	0.794097i	$0.707944\pi$
$492$	0	0
$493$	51.2626	2.30875
$494$	0	0
$495$	−8.14149	−0.365933
$496$	0	0
$497$	−4.39011	−0.196924
$498$	0	0
$499$	24.9321	1.11611	0.558057	−	0.829802i	$-0.311547\pi$
$499$	24.9321	1.11611	0.558057	+	0.829802i	$0.311547\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	23.9427	1.06755	0.533776	−	0.845626i	$-0.320773\pi$
$503$	23.9427	1.06755	0.533776	+	0.845626i	$0.320773\pi$
$504$	0	0
$505$	3.78824	0.168574
$506$	0	0
$507$	12.2768	0.545230
$508$	0	0
$509$	−11.2763	−0.499813	−0.249906	−	0.968270i	$-0.580400\pi$
$509$	−11.2763	−0.499813	−0.249906	+	0.968270i	$0.580400\pi$
$510$	0	0
$511$	30.0380	1.32880
$512$	0	0
$513$	2.29761	0.101442
$514$	0	0
$515$	9.06343	0.399382
$516$	0	0
$517$	35.4404	1.55867
$518$	0	0
$519$	−21.4371	−0.940985
$520$	0	0
$521$	18.3218	0.802693	0.401347	−	0.915926i	$-0.368542\pi$
$521$	18.3218	0.802693	0.401347	+	0.915926i	$0.368542\pi$
$522$	0	0
$523$	−32.2678	−1.41097	−0.705487	−	0.708723i	$-0.749272\pi$
$523$	−32.2678	−1.41097	−0.705487	+	0.708723i	$0.749272\pi$
$524$	0	0
$525$	12.8900	0.562566
$526$	0	0
$527$	−55.0504	−2.39803
$528$	0	0
$529$	−15.4492	−0.671703
$530$	0	0
$531$	3.64159	0.158031
$532$	0	0
$533$	−4.22614	−0.183054
$534$	0	0
$535$	34.7596	1.50279
$536$	0	0
$537$	−0.637299	−0.0275015
$538$	0	0
$539$	−40.2869	−1.73528
$540$	0	0
$541$	36.6203	1.57443	0.787215	−	0.616678i	$-0.211522\pi$
$541$	36.6203	1.57443	0.787215	+	0.616678i	$0.211522\pi$
$542$	0	0
$543$	14.5472	0.624280
$544$	0	0
$545$	55.7857	2.38960
$546$	0	0
$547$	−29.7390	−1.27155	−0.635774	−	0.771875i	$-0.719319\pi$
$547$	−29.7390	−1.27155	−0.635774	+	0.771875i	$0.719319\pi$
$548$	0	0
$549$	3.09676	0.132167
$550$	0	0
$551$	15.1060	0.643538
$552$	0	0
$553$	12.6386	0.537450
$554$	0	0
$555$	−23.3791	−0.992387
$556$	0	0
$557$	−21.1579	−0.896490	−0.448245	−	0.893911i	$-0.647951\pi$
$557$	−21.1579	−0.896490	−0.448245	+	0.893911i	$0.647951\pi$
$558$	0	0
$559$	9.72288	0.411234
$560$	0	0
$561$	−22.6949	−0.958180
$562$	0	0
$563$	35.7313	1.50589	0.752946	−	0.658082i	$-0.228632\pi$
$563$	35.7313	1.50589	0.752946	+	0.658082i	$0.228632\pi$
$564$	0	0
$565$	16.3151	0.686382
$566$	0	0
$567$	−4.56518	−0.191719
$568$	0	0
$569$	−12.2786	−0.514745	−0.257373	−	0.966312i	$-0.582857\pi$
$569$	−12.2786	−0.514745	−0.257373	+	0.966312i	$0.582857\pi$
$570$	0	0
$571$	13.8771	0.580740	0.290370	−	0.956914i	$-0.406222\pi$
$571$	13.8771	0.580740	0.290370	+	0.956914i	$0.406222\pi$
$572$	0	0
$573$	−20.3497	−0.850120
$574$	0	0
$575$	−7.75878	−0.323563
$576$	0	0
$577$	26.7746	1.11464	0.557321	−	0.830297i	$-0.311829\pi$
$577$	26.7746	1.11464	0.557321	+	0.830297i	$0.311829\pi$
$578$	0	0
$579$	−10.0666	−0.418355
$580$	0	0
$581$	−76.9896	−3.19407
$582$	0	0
$583$	17.5682	0.727602
$584$	0	0
$585$	−2.37872	−0.0983480
$586$	0	0
$587$	−0.941202	−0.0388476	−0.0194238	−	0.999811i	$-0.506183\pi$
$587$	−0.941202	−0.0388476	−0.0194238	+	0.999811i	$0.506183\pi$
$588$	0	0
$589$	−16.2222	−0.668424
$590$	0	0
$591$	1.11488	0.0458602
$592$	0	0
$593$	35.9662	1.47695	0.738477	−	0.674279i	$-0.235545\pi$
$593$	35.9662	1.47695	0.738477	+	0.674279i	$0.235545\pi$
$594$	0	0
$595$	99.5603	4.08158
$596$	0	0
$597$	3.21029	0.131388
$598$	0	0
$599$	26.9543	1.10132	0.550662	−	0.834728i	$-0.314375\pi$
$599$	26.9543	1.10132	0.550662	+	0.834728i	$0.314375\pi$
$600$	0	0
$601$	37.3201	1.52232	0.761159	−	0.648566i	$-0.224631\pi$
$601$	37.3201	1.52232	0.761159	+	0.648566i	$0.224631\pi$
$602$	0	0
$603$	14.0723	0.573069
$604$	0	0
$605$	−7.07002	−0.287437
$606$	0	0
$607$	−4.98231	−0.202226	−0.101113	−	0.994875i	$-0.532240\pi$
$607$	−4.98231	−0.202226	−0.101113	+	0.994875i	$0.532240\pi$
$608$	0	0
$609$	−30.0145	−1.21625
$610$	0	0
$611$	10.3547	0.418907
$612$	0	0
$613$	17.9611	0.725441	0.362720	−	0.931898i	$-0.381848\pi$
$613$	17.9611	0.725441	0.362720	+	0.931898i	$0.381848\pi$
$614$	0	0
$615$	−13.8997	−0.560488
$616$	0	0
$617$	14.8834	0.599184	0.299592	−	0.954067i	$-0.403149\pi$
$617$	14.8834	0.599184	0.299592	+	0.954067i	$0.403149\pi$
$618$	0	0
$619$	−19.4829	−0.783086	−0.391543	−	0.920160i	$-0.628059\pi$
$619$	−19.4829	−0.783086	−0.391543	+	0.920160i	$0.628059\pi$
$620$	0	0
$621$	2.74788	0.110269
$622$	0	0
$623$	−37.4462	−1.50025
$624$	0	0
$625$	−31.1453	−1.24581
$626$	0	0
$627$	−6.68772	−0.267082
$628$	0	0
$629$	−65.1707	−2.59853
$630$	0	0
$631$	−16.1689	−0.643674	−0.321837	−	0.946795i	$-0.604300\pi$
$631$	−16.1689	−0.643674	−0.321837	+	0.946795i	$0.604300\pi$
$632$	0	0
$633$	−24.1441	−0.959640
$634$	0	0
$635$	−4.39397	−0.174369
$636$	0	0
$637$	−11.7707	−0.466374
$638$	0	0
$639$	0.961653	0.0380424
$640$	0	0
$641$	−34.4316	−1.35997	−0.679984	−	0.733227i	$-0.738013\pi$
$641$	−34.4316	−1.35997	−0.679984	+	0.733227i	$0.738013\pi$
$642$	0	0
$643$	22.4242	0.884325	0.442163	−	0.896935i	$-0.354211\pi$
$643$	22.4242	0.884325	0.442163	+	0.896935i	$0.354211\pi$
$644$	0	0
$645$	31.9783	1.25915
$646$	0	0
$647$	6.63740	0.260943	0.130472	−	0.991452i	$-0.458351\pi$
$647$	6.63740	0.260943	0.130472	+	0.991452i	$0.458351\pi$
$648$	0	0
$649$	−10.5997	−0.416074
$650$	0	0
$651$	32.2323	1.26328
$652$	0	0
$653$	−29.1259	−1.13978	−0.569892	−	0.821720i	$-0.693015\pi$
$653$	−29.1259	−1.13978	−0.569892	+	0.821720i	$0.693015\pi$
$654$	0	0
$655$	−4.67418	−0.182635
$656$	0	0
$657$	−6.57980	−0.256703
$658$	0	0
$659$	−25.0489	−0.975766	−0.487883	−	0.872909i	$-0.662231\pi$
$659$	−25.0489	−0.975766	−0.487883	+	0.872909i	$0.662231\pi$
$660$	0	0
$661$	19.1690	0.745589	0.372795	−	0.927914i	$-0.378400\pi$
$661$	19.1690	0.745589	0.372795	+	0.927914i	$0.378400\pi$
$662$	0	0
$663$	−6.63083	−0.257520
$664$	0	0
$665$	29.3384	1.13769
$666$	0	0
$667$	18.0664	0.699533
$668$	0	0
$669$	−16.4018	−0.634132
$670$	0	0
$671$	−9.01384	−0.347975
$672$	0	0
$673$	−28.4432	−1.09640	−0.548202	−	0.836346i	$-0.684687\pi$
$673$	−28.4432	−1.09640	−0.548202	+	0.836346i	$0.684687\pi$
$674$	0	0
$675$	−2.82355	−0.108679
$676$	0	0
$677$	−21.9968	−0.845405	−0.422703	−	0.906268i	$-0.638919\pi$
$677$	−21.9968	−0.845405	−0.422703	+	0.906268i	$0.638919\pi$
$678$	0	0
$679$	38.5434	1.47916
$680$	0	0
$681$	18.8181	0.721111
$682$	0	0
$683$	29.0973	1.11338	0.556689	−	0.830721i	$-0.312071\pi$
$683$	29.0973	1.11338	0.556689	+	0.830721i	$0.312071\pi$
$684$	0	0
$685$	28.0401	1.07136
$686$	0	0
$687$	−12.7703	−0.487217
$688$	0	0
$689$	5.13296	0.195550
$690$	0	0
$691$	−23.5824	−0.897118	−0.448559	−	0.893753i	$-0.648063\pi$
$691$	−23.5824	−0.897118	−0.448559	+	0.893753i	$0.648063\pi$
$692$	0	0
$693$	13.2880	0.504769
$694$	0	0
$695$	35.4102	1.34319
$696$	0	0
$697$	−38.7462	−1.46762
$698$	0	0
$699$	18.2136	0.688902
$700$	0	0
$701$	23.2444	0.877928	0.438964	−	0.898505i	$-0.355346\pi$
$701$	23.2444	0.877928	0.438964	+	0.898505i	$0.355346\pi$
$702$	0	0
$703$	−19.2045	−0.724310
$704$	0	0
$705$	34.0564	1.28264
$706$	0	0
$707$	−6.18291	−0.232532
$708$	0	0
$709$	−18.7991	−0.706016	−0.353008	−	0.935620i	$-0.614841\pi$
$709$	−18.7991	−0.706016	−0.353008	+	0.935620i	$0.614841\pi$
$710$	0	0
$711$	−2.76849	−0.103826
$712$	0	0
$713$	−19.4013	−0.726584
$714$	0	0
$715$	6.92381	0.258936
$716$	0	0
$717$	−2.53425	−0.0946435
$718$	0	0
$719$	25.0101	0.932721	0.466360	−	0.884595i	$-0.345565\pi$
$719$	25.0101	0.932721	0.466360	+	0.884595i	$0.345565\pi$
$720$	0	0
$721$	−14.7927	−0.550910
$722$	0	0
$723$	−10.9201	−0.406123
$724$	0	0
$725$	−18.5639	−0.689446
$726$	0	0
$727$	35.1165	1.30240	0.651199	−	0.758907i	$-0.274266\pi$
$727$	35.1165	1.30240	0.651199	+	0.758907i	$0.274266\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	89.1416	3.29702
$732$	0	0
$733$	31.3756	1.15889	0.579443	−	0.815013i	$-0.303270\pi$
$733$	31.3756	1.15889	0.579443	+	0.815013i	$0.303270\pi$
$734$	0	0
$735$	−38.7137	−1.42798
$736$	0	0
$737$	−40.9607	−1.50881
$738$	0	0
$739$	−17.6202	−0.648169	−0.324085	−	0.946028i	$-0.605056\pi$
$739$	−17.6202	−0.648169	−0.324085	+	0.946028i	$0.605056\pi$
$740$	0	0
$741$	−1.95397	−0.0717808
$742$	0	0
$743$	42.3274	1.55284	0.776421	−	0.630214i	$-0.217033\pi$
$743$	42.3274	1.55284	0.776421	+	0.630214i	$0.217033\pi$
$744$	0	0
$745$	−53.6793	−1.96666
$746$	0	0
$747$	16.8645	0.617041
$748$	0	0
$749$	−56.7323	−2.07295
$750$	0	0
$751$	9.22171	0.336505	0.168253	−	0.985744i	$-0.446188\pi$
$751$	9.22171	0.336505	0.168253	+	0.985744i	$0.446188\pi$
$752$	0	0
$753$	7.08878	0.258329
$754$	0	0
$755$	−37.1746	−1.35292
$756$	0	0
$757$	38.8185	1.41088	0.705441	−	0.708769i	$-0.250749\pi$
$757$	38.8185	1.41088	0.705441	+	0.708769i	$0.250749\pi$
$758$	0	0
$759$	−7.99833	−0.290321
$760$	0	0
$761$	−11.9586	−0.433498	−0.216749	−	0.976227i	$-0.569545\pi$
$761$	−11.9586	−0.433498	−0.216749	+	0.976227i	$0.569545\pi$
$762$	0	0
$763$	−91.0496	−3.29622
$764$	0	0
$765$	−21.8086	−0.788493
$766$	0	0
$767$	−3.09693	−0.111824
$768$	0	0
$769$	33.8376	1.22021	0.610107	−	0.792319i	$-0.291127\pi$
$769$	33.8376	1.22021	0.610107	+	0.792319i	$0.291127\pi$
$770$	0	0
$771$	23.0298	0.829399
$772$	0	0
$773$	18.8998	0.679778	0.339889	−	0.940466i	$-0.389610\pi$
$773$	18.8998	0.679778	0.339889	+	0.940466i	$0.389610\pi$
$774$	0	0
$775$	19.9356	0.716108
$776$	0	0
$777$	38.1578	1.36890
$778$	0	0
$779$	−11.4177	−0.409081
$780$	0	0
$781$	−2.79911	−0.100160
$782$	0	0
$783$	6.57467	0.234959
$784$	0	0
$785$	−1.74608	−0.0623204
$786$	0	0
$787$	−5.91461	−0.210833	−0.105416	−	0.994428i	$-0.533618\pi$
$787$	−5.91461	−0.210833	−0.105416	+	0.994428i	$0.533618\pi$
$788$	0	0
$789$	12.6362	0.449860
$790$	0	0
$791$	−26.6285	−0.946799
$792$	0	0
$793$	−2.63360	−0.0935218
$794$	0	0
$795$	16.8822	0.598749
$796$	0	0
$797$	9.51329	0.336978	0.168489	−	0.985704i	$-0.446111\pi$
$797$	9.51329	0.336978	0.168489	+	0.985704i	$0.446111\pi$
$798$	0	0
$799$	94.9344	3.35854
$800$	0	0
$801$	8.20258	0.289824
$802$	0	0
$803$	19.1520	0.675860
$804$	0	0
$805$	35.0879	1.23668
$806$	0	0
$807$	−29.8615	−1.05117
$808$	0	0
$809$	−10.7944	−0.379510	−0.189755	−	0.981831i	$-0.560769\pi$
$809$	−10.7944	−0.379510	−0.189755	+	0.981831i	$0.560769\pi$
$810$	0	0
$811$	−7.89452	−0.277214	−0.138607	−	0.990347i	$-0.544262\pi$
$811$	−7.89452	−0.277214	−0.138607	+	0.990347i	$0.544262\pi$
$812$	0	0
$813$	6.38067	0.223780
$814$	0	0
$815$	−18.7895	−0.658168
$816$	0	0
$817$	26.2682	0.919007
$818$	0	0
$819$	3.88239	0.135662
$820$	0	0
$821$	8.60687	0.300382	0.150191	−	0.988657i	$-0.452011\pi$
$821$	8.60687	0.300382	0.150191	+	0.988657i	$0.452011\pi$
$822$	0	0
$823$	−1.40078	−0.0488280	−0.0244140	−	0.999702i	$-0.507772\pi$
$823$	−1.40078	−0.0488280	−0.0244140	+	0.999702i	$0.507772\pi$
$824$	0	0
$825$	8.21859	0.286135
$826$	0	0
$827$	16.7472	0.582358	0.291179	−	0.956669i	$-0.405952\pi$
$827$	16.7472	0.582358	0.291179	+	0.956669i	$0.405952\pi$
$828$	0	0
$829$	−15.6416	−0.543256	−0.271628	−	0.962402i	$-0.587562\pi$
$829$	−15.6416	−0.543256	−0.271628	+	0.962402i	$0.587562\pi$
$830$	0	0
$831$	−23.5063	−0.815423
$832$	0	0
$833$	−107.917	−3.73910
$834$	0	0
$835$	2.79706	0.0967963
$836$	0	0
$837$	−7.06047	−0.244045
$838$	0	0
$839$	−34.9539	−1.20674	−0.603371	−	0.797461i	$-0.706176\pi$
$839$	−34.9539	−1.20674	−0.603371	+	0.797461i	$0.706176\pi$
$840$	0	0
$841$	14.2262	0.490560
$842$	0	0
$843$	23.4338	0.807103
$844$	0	0
$845$	−34.3389	−1.18129
$846$	0	0
$847$	11.5392	0.396492
$848$	0	0
$849$	17.6525	0.605832
$850$	0	0
$851$	−22.9680	−0.787333
$852$	0	0
$853$	6.43085	0.220188	0.110094	−	0.993921i	$-0.464885\pi$
$853$	6.43085	0.220188	0.110094	+	0.993921i	$0.464885\pi$
$854$	0	0
$855$	−6.42656	−0.219784
$856$	0	0
$857$	−13.2815	−0.453686	−0.226843	−	0.973931i	$-0.572840\pi$
$857$	−13.2815	−0.453686	−0.226843	+	0.973931i	$0.572840\pi$
$858$	0	0
$859$	−29.8157	−1.01730	−0.508650	−	0.860974i	$-0.669855\pi$
$859$	−29.8157	−1.01730	−0.508650	+	0.860974i	$0.669855\pi$
$860$	0	0
$861$	22.6861	0.773140
$862$	0	0
$863$	−17.5960	−0.598974	−0.299487	−	0.954100i	$-0.596816\pi$
$863$	−17.5960	−0.598974	−0.299487	+	0.954100i	$0.596816\pi$
$864$	0	0
$865$	59.9609	2.03873
$866$	0	0
$867$	−43.7930	−1.48729
$868$	0	0
$869$	8.05832	0.273360
$870$	0	0
$871$	−11.9676	−0.405506
$872$	0	0
$873$	−8.44292	−0.285750
$874$	0	0
$875$	27.7912	0.939515
$876$	0	0
$877$	−27.3636	−0.924003	−0.462001	−	0.886879i	$-0.652868\pi$
$877$	−27.3636	−0.924003	−0.462001	+	0.886879i	$0.652868\pi$
$878$	0	0
$879$	28.9734	0.977249
$880$	0	0
$881$	27.2657	0.918604	0.459302	−	0.888280i	$-0.348100\pi$
$881$	27.2657	0.918604	0.459302	+	0.888280i	$0.348100\pi$
$882$	0	0
$883$	14.1255	0.475360	0.237680	−	0.971344i	$-0.423613\pi$
$883$	14.1255	0.475360	0.237680	+	0.971344i	$0.423613\pi$
$884$	0	0
$885$	−10.1857	−0.342390
$886$	0	0
$887$	14.1172	0.474010	0.237005	−	0.971508i	$-0.423834\pi$
$887$	14.1172	0.474010	0.237005	+	0.971508i	$0.423834\pi$
$888$	0	0
$889$	7.17154	0.240526
$890$	0	0
$891$	−2.91073	−0.0975131
$892$	0	0
$893$	27.9752	0.936154
$894$	0	0
$895$	1.78257	0.0595846
$896$	0	0
$897$	−2.33689	−0.0780266
$898$	0	0
$899$	−46.4202	−1.54820
$900$	0	0
$901$	47.0601	1.56780
$902$	0	0
$903$	−52.1929	−1.73687
$904$	0	0
$905$	−40.6894	−1.35256
$906$	0	0
$907$	47.3644	1.57271	0.786354	−	0.617776i	$-0.211966\pi$
$907$	47.3644	1.57271	0.786354	+	0.617776i	$0.211966\pi$
$908$	0	0
$909$	1.35436	0.0449214
$910$	0	0
$911$	32.8073	1.08696	0.543478	−	0.839423i	$-0.317107\pi$
$911$	32.8073	1.08696	0.543478	+	0.839423i	$0.317107\pi$
$912$	0	0
$913$	−49.0881	−1.62458
$914$	0	0
$915$	−8.66184	−0.286352
$916$	0	0
$917$	7.62888	0.251928
$918$	0	0
$919$	−5.12933	−0.169201	−0.0846005	−	0.996415i	$-0.526961\pi$
$919$	−5.12933	−0.169201	−0.0846005	+	0.996415i	$0.526961\pi$
$920$	0	0
$921$	26.5494	0.874832
$922$	0	0
$923$	−0.817823	−0.0269190
$924$	0	0
$925$	23.6005	0.775980
$926$	0	0
$927$	3.24034	0.106427
$928$	0	0
$929$	57.3874	1.88282	0.941410	−	0.337264i	$-0.109501\pi$
$929$	57.3874	1.88282	0.941410	+	0.337264i	$0.109501\pi$
$930$	0	0
$931$	−31.8008	−1.04223
$932$	0	0
$933$	−6.57114	−0.215129
$934$	0	0
$935$	63.4791	2.07599
$936$	0	0
$937$	23.5812	0.770366	0.385183	−	0.922840i	$-0.374138\pi$
$937$	23.5812	0.770366	0.385183	+	0.922840i	$0.374138\pi$
$938$	0	0
$939$	−1.39897	−0.0456538
$940$	0	0
$941$	12.3255	0.401800	0.200900	−	0.979612i	$-0.435613\pi$
$941$	12.3255	0.401800	0.200900	+	0.979612i	$0.435613\pi$
$942$	0	0
$943$	−13.6552	−0.444676
$944$	0	0
$945$	12.7691	0.415378
$946$	0	0
$947$	2.48420	0.0807256	0.0403628	−	0.999185i	$-0.487149\pi$
$947$	2.48420	0.0807256	0.0403628	+	0.999185i	$0.487149\pi$
$948$	0	0
$949$	5.59570	0.181644
$950$	0	0
$951$	32.0982	1.04086
$952$	0	0
$953$	−35.7243	−1.15722	−0.578612	−	0.815603i	$-0.696405\pi$
$953$	−35.7243	−1.15722	−0.578612	+	0.815603i	$0.696405\pi$
$954$	0	0
$955$	56.9193	1.84186
$956$	0	0
$957$	−19.1371	−0.618614
$958$	0	0
$959$	−45.7652	−1.47784
$960$	0	0
$961$	18.8502	0.608071
$962$	0	0
$963$	12.4272	0.400460
$964$	0	0
$965$	28.1570	0.906405
$966$	0	0
$967$	14.3114	0.460224	0.230112	−	0.973164i	$-0.426091\pi$
$967$	14.3114	0.460224	0.230112	+	0.973164i	$0.426091\pi$
$968$	0	0
$969$	−17.9144	−0.575494
$970$	0	0
$971$	−7.01096	−0.224992	−0.112496	−	0.993652i	$-0.535885\pi$
$971$	−7.01096	−0.224992	−0.112496	+	0.993652i	$0.535885\pi$
$972$	0	0
$973$	−57.7942	−1.85280
$974$	0	0
$975$	2.40125	0.0769015
$976$	0	0
$977$	23.2826	0.744876	0.372438	−	0.928057i	$-0.378522\pi$
$977$	23.2826	0.744876	0.372438	+	0.928057i	$0.378522\pi$
$978$	0	0
$979$	−23.8755	−0.763064
$980$	0	0
$981$	19.9444	0.636775
$982$	0	0
$983$	−3.28649	−0.104823	−0.0524114	−	0.998626i	$-0.516691\pi$
$983$	−3.28649	−0.104823	−0.0524114	+	0.998626i	$0.516691\pi$
$984$	0	0
$985$	−3.11840	−0.0993605
$986$	0	0
$987$	−55.5846	−1.76928
$988$	0	0
$989$	31.4160	0.998971
$990$	0	0
$991$	−16.0707	−0.510502	−0.255251	−	0.966875i	$-0.582158\pi$
$991$	−16.0707	−0.510502	−0.255251	+	0.966875i	$0.582158\pi$
$992$	0	0
$993$	−30.4241	−0.965479
$994$	0	0
$995$	−8.97938	−0.284666
$996$	0	0
$997$	−28.3838	−0.898924	−0.449462	−	0.893299i	$-0.648384\pi$
$997$	−28.3838	−0.898924	−0.449462	+	0.893299i	$0.648384\pi$
$998$	0	0
$999$	−8.35845	−0.264450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bg.1.10		13
4.3	odd	2		4008.2.a.m.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.10	✓	13	4.3	odd	2
8016.2.a.bg.1.10		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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.79706 q^{5} -4.56518 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.79706 q^{5} -4.56518 q^{7} +1.00000 q^{9} -2.91073 q^{11} -0.850435 q^{13} -2.79706 q^{15} -7.79699 q^{17} -2.29761 q^{19} +4.56518 q^{21} -2.74788 q^{23} +2.82355 q^{25} -1.00000 q^{27} -6.57467 q^{29} +7.06047 q^{31} +2.91073 q^{33} -12.7691 q^{35} +8.35845 q^{37} +0.850435 q^{39} +4.96938 q^{41} -11.4328 q^{43} +2.79706 q^{45} -12.1758 q^{47} +13.8408 q^{49} +7.79699 q^{51} -6.03568 q^{53} -8.14149 q^{55} +2.29761 q^{57} +3.64159 q^{59} +3.09676 q^{61} -4.56518 q^{63} -2.37872 q^{65} +14.0723 q^{67} +2.74788 q^{69} +0.961653 q^{71} -6.57980 q^{73} -2.82355 q^{75} +13.2880 q^{77} -2.76849 q^{79} +1.00000 q^{81} +16.8645 q^{83} -21.8086 q^{85} +6.57467 q^{87} +8.20258 q^{89} +3.88239 q^{91} -7.06047 q^{93} -6.42656 q^{95} -8.44292 q^{97} -2.91073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * q^99

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