LMFDB - Embedded newform 4008.2.a.m.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.79706$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-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	$-0.168748\pi$
$7$	4.56518	1.72547	0.862737	+	0.505652i	$0.168748\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.91073	0.877618	0.438809	−	0.898580i	$-0.355401\pi$
$11$	2.91073	0.877618	0.438809	+	0.898580i	$0.355401\pi$
$12$	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	$-0.415105\pi$
$19$	2.29761	0.527108	0.263554	+	0.964645i	$0.415105\pi$
$20$	0	0
$21$	4.56518	0.996203
$22$	0	0
$23$	2.74788	0.572972	0.286486	−	0.958084i	$-0.407513\pi$
$23$	2.74788	0.572972	0.286486	+	0.958084i	$0.407513\pi$
$24$	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	$-0.718608\pi$
$31$	−7.06047	−1.26810	−0.634049	+	0.773293i	$0.718608\pi$
$32$	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	$-0.162989\pi$
$43$	11.4328	1.74349	0.871745	+	0.489960i	$0.162989\pi$
$44$	0	0
$45$	2.79706	0.416961
$46$	0	0
$47$	12.1758	1.77602	0.888010	−	0.459824i	$-0.152088\pi$
$47$	12.1758	1.77602	0.888010	+	0.459824i	$0.152088\pi$
$48$	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	$-0.576180\pi$
$59$	−3.64159	−0.474094	−0.237047	+	0.971498i	$0.576180\pi$
$60$	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	$-0.829289\pi$
$67$	−14.0723	−1.71921	−0.859603	+	0.510963i	$0.829289\pi$
$68$	0	0
$69$	2.74788	0.330806
$70$	0	0
$71$	−0.961653	−0.114127	−0.0570636	−	0.998371i	$-0.518174\pi$
$71$	−0.961653	−0.114127	−0.0570636	+	0.998371i	$0.518174\pi$
$72$	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	$-0.450224\pi$
$79$	2.76849	0.311479	0.155740	+	0.987798i	$0.450224\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.8645	−1.85112	−0.925562	−	0.378596i	$-0.876407\pi$
$83$	−16.8645	−1.85112	−0.925562	+	0.378596i	$0.876407\pi$
$84$	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	$-0.551033\pi$
$103$	−3.24034	−0.319280	−0.159640	+	0.987175i	$0.551033\pi$
$104$	0	0
$105$	12.7691	1.24613
$106$	0	0
$107$	−12.4272	−1.20138	−0.600690	−	0.799482i	$-0.705107\pi$
$107$	−12.4272	−1.20138	−0.600690	+	0.799482i	$0.705107\pi$
$108$	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	$-0.477796\pi$
$127$	1.57092	0.139397	0.0696984	+	0.997568i	$0.477796\pi$
$128$	0	0
$129$	11.4328	1.00660
$130$	0	0
$131$	1.67110	0.146005	0.0730025	−	0.997332i	$-0.476742\pi$
$131$	1.67110	0.146005	0.0730025	+	0.997332i	$0.476742\pi$
$132$	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	$-0.680403\pi$
$139$	−12.6598	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$140$	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	$-0.318127\pi$
$151$	13.2906	1.08157	0.540786	+	0.841160i	$0.318127\pi$
$152$	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	$-0.415261\pi$
$163$	6.71759	0.526163	0.263081	+	0.964774i	$0.415261\pi$
$164$	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	$-0.507582\pi$
$179$	−0.637299	−0.0476340	−0.0238170	+	0.999716i	$0.507582\pi$
$180$	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	$-0.763394\pi$
$191$	−20.3497	−1.47245	−0.736225	+	0.676737i	$0.763394\pi$
$192$	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	$-0.463702\pi$
$199$	3.21029	0.227572	0.113786	+	0.993505i	$0.463702\pi$
$200$	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	$-0.812273\pi$
$211$	−24.1441	−1.66215	−0.831073	+	0.556164i	$0.812273\pi$
$212$	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	$-0.685058\pi$
$223$	−16.4018	−1.09835	−0.549174	+	0.835708i	$0.685058\pi$
$224$	0	0
$225$	2.82355	0.188237
$226$	0	0
$227$	18.8181	1.24900	0.624500	−	0.781025i	$-0.285303\pi$
$227$	18.8181	1.24900	0.624500	+	0.781025i	$0.285303\pi$
$228$	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	$-0.526119\pi$
$239$	−2.53425	−0.163927	−0.0819637	+	0.996635i	$0.526119\pi$
$240$	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	$-0.428180\pi$
$251$	7.08878	0.447440	0.223720	+	0.974653i	$0.428180\pi$
$252$	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	$-0.372617\pi$
$263$	12.6362	0.779181	0.389591	+	0.920988i	$0.372617\pi$
$264$	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	$-0.437919\pi$
$271$	6.38067	0.387598	0.193799	+	0.981041i	$0.437919\pi$
$272$	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	$-0.324190\pi$
$283$	17.6525	1.04933	0.524666	+	0.851308i	$0.324190\pi$
$284$	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	$-0.226359\pi$
$307$	26.5494	1.51525	0.757627	+	0.652688i	$0.226359\pi$
$308$	0	0
$309$	−3.24034	−0.184336
$310$	0	0
$311$	−6.57114	−0.372615	−0.186308	−	0.982491i	$-0.559652\pi$
$311$	−6.57114	−0.372615	−0.186308	+	0.982491i	$0.559652\pi$
$312$	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	$-0.815187\pi$
$331$	−30.4241	−1.67226	−0.836130	+	0.548532i	$0.815187\pi$
$332$	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	$-0.693291\pi$
$347$	−21.2584	−1.14121	−0.570604	+	0.821225i	$0.693291\pi$
$348$	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	$-0.550591\pi$
$359$	−5.99756	−0.316539	−0.158270	+	0.987396i	$0.550591\pi$
$360$	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	$-0.606340\pi$
$367$	−12.5632	−0.655793	−0.327896	+	0.944714i	$0.606340\pi$
$368$	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	$-0.363179\pi$
$379$	16.2254	0.833444	0.416722	+	0.909034i	$0.363179\pi$
$380$	0	0
$381$	1.57092	0.0804808
$382$	0	0
$383$	12.9406	0.661232	0.330616	−	0.943765i	$-0.392743\pi$
$383$	12.9406	0.661232	0.330616	+	0.943765i	$0.392743\pi$
$384$	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	$-0.416015\pi$
$419$	10.6768	0.521594	0.260797	+	0.965394i	$0.416015\pi$
$420$	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	$-0.687025\pi$
$431$	−23.0164	−1.10866	−0.554330	+	0.832297i	$0.687025\pi$
$432$	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	$-0.447799\pi$
$439$	6.84142	0.326523	0.163262	+	0.986583i	$0.447799\pi$
$440$	0	0
$441$	13.8408	0.659087
$442$	0	0
$443$	−22.9161	−1.08878	−0.544389	−	0.838833i	$-0.683238\pi$
$443$	−22.9161	−1.08878	−0.544389	+	0.838833i	$0.683238\pi$
$444$	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	$-0.880348\pi$
$463$	−40.0301	−1.86036	−0.930178	+	0.367108i	$0.880348\pi$
$464$	0	0
$465$	−19.7486	−0.915817
$466$	0	0
$467$	19.8776	0.919825	0.459913	−	0.887964i	$-0.347881\pi$
$467$	19.8776	0.919825	0.459913	+	0.887964i	$0.347881\pi$
$468$	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	$-0.666767\pi$
$479$	−21.8981	−1.00055	−0.500274	+	0.865867i	$0.666767\pi$
$480$	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	$-0.247758\pi$
$487$	31.4280	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$488$	0	0
$489$	6.71759	0.303780
$490$	0	0
$491$	26.9355	1.21558	0.607792	−	0.794097i	$-0.292056\pi$
$491$	26.9355	1.21558	0.607792	+	0.794097i	$0.292056\pi$
$492$	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	$-0.688453\pi$
$499$	−24.9321	−1.11611	−0.558057	+	0.829802i	$0.688453\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−23.9427	−1.06755	−0.533776	−	0.845626i	$-0.679227\pi$
$503$	−23.9427	−1.06755	−0.533776	+	0.845626i	$0.679227\pi$
$504$	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	$-0.250728\pi$
$523$	32.2678	1.41097	0.705487	+	0.708723i	$0.250728\pi$
$524$	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	$-0.280681\pi$
$547$	29.7390	1.27155	0.635774	+	0.771875i	$0.280681\pi$
$548$	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	$-0.771368\pi$
$563$	−35.7313	−1.50589	−0.752946	+	0.658082i	$0.771368\pi$
$564$	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	$-0.593778\pi$
$571$	−13.8771	−0.580740	−0.290370	+	0.956914i	$0.593778\pi$
$572$	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	$-0.493817\pi$
$587$	0.941202	0.0388476	0.0194238	+	0.999811i	$0.493817\pi$
$588$	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	$-0.685625\pi$
$599$	−26.9543	−1.10132	−0.550662	+	0.834728i	$0.685625\pi$
$600$	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	$-0.467760\pi$
$607$	4.98231	0.202226	0.101113	+	0.994875i	$0.467760\pi$
$608$	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	$-0.371941\pi$
$619$	19.4829	0.783086	0.391543	+	0.920160i	$0.371941\pi$
$620$	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	$-0.395700\pi$
$631$	16.1689	0.643674	0.321837	+	0.946795i	$0.395700\pi$
$632$	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	$-0.645789\pi$
$643$	−22.4242	−0.884325	−0.442163	+	0.896935i	$0.645789\pi$
$644$	0	0
$645$	31.9783	1.25915
$646$	0	0
$647$	−6.63740	−0.260943	−0.130472	−	0.991452i	$-0.541649\pi$
$647$	−6.63740	−0.260943	−0.130472	+	0.991452i	$0.541649\pi$
$648$	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	$-0.337769\pi$
$659$	25.0489	0.975766	0.487883	+	0.872909i	$0.337769\pi$
$660$	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	$-0.687929\pi$
$683$	−29.0973	−1.11338	−0.556689	+	0.830721i	$0.687929\pi$
$684$	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	$-0.351937\pi$
$691$	23.5824	0.897118	0.448559	+	0.893753i	$0.351937\pi$
$692$	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	$-0.654435\pi$
$719$	−25.0101	−0.932721	−0.466360	+	0.884595i	$0.654435\pi$
$720$	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	$-0.725734\pi$
$727$	−35.1165	−1.30240	−0.651199	+	0.758907i	$0.725734\pi$
$728$	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	$-0.394944\pi$
$739$	17.6202	0.648169	0.324085	+	0.946028i	$0.394944\pi$
$740$	0	0
$741$	−1.95397	−0.0717808
$742$	0	0
$743$	−42.3274	−1.55284	−0.776421	−	0.630214i	$-0.782967\pi$
$743$	−42.3274	−1.55284	−0.776421	+	0.630214i	$0.782967\pi$
$744$	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	$-0.553812\pi$
$751$	−9.22171	−0.336505	−0.168253	+	0.985744i	$0.553812\pi$
$752$	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	$-0.466382\pi$
$787$	5.91461	0.210833	0.105416	+	0.994428i	$0.466382\pi$
$788$	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	$-0.455738\pi$
$811$	7.89452	0.277214	0.138607	+	0.990347i	$0.455738\pi$
$812$	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	$-0.492228\pi$
$823$	1.40078	0.0488280	0.0244140	+	0.999702i	$0.492228\pi$
$824$	0	0
$825$	8.21859	0.286135
$826$	0	0
$827$	−16.7472	−0.582358	−0.291179	−	0.956669i	$-0.594048\pi$
$827$	−16.7472	−0.582358	−0.291179	+	0.956669i	$0.594048\pi$
$828$	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	$-0.293824\pi$
$839$	34.9539	1.20674	0.603371	+	0.797461i	$0.293824\pi$
$840$	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	$-0.330145\pi$
$859$	29.8157	1.01730	0.508650	+	0.860974i	$0.330145\pi$
$860$	0	0
$861$	22.6861	0.773140
$862$	0	0
$863$	17.5960	0.598974	0.299487	−	0.954100i	$-0.403184\pi$
$863$	17.5960	0.598974	0.299487	+	0.954100i	$0.403184\pi$
$864$	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	$-0.576387\pi$
$883$	−14.1255	−0.475360	−0.237680	+	0.971344i	$0.576387\pi$
$884$	0	0
$885$	−10.1857	−0.342390
$886$	0	0
$887$	−14.1172	−0.474010	−0.237005	−	0.971508i	$-0.576166\pi$
$887$	−14.1172	−0.474010	−0.237005	+	0.971508i	$0.576166\pi$
$888$	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	$-0.788034\pi$
$907$	−47.3644	−1.57271	−0.786354	+	0.617776i	$0.788034\pi$
$908$	0	0
$909$	1.35436	0.0449214
$910$	0	0
$911$	−32.8073	−1.08696	−0.543478	−	0.839423i	$-0.682893\pi$
$911$	−32.8073	−1.08696	−0.543478	+	0.839423i	$0.682893\pi$
$912$	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	$-0.473039\pi$
$919$	5.12933	0.169201	0.0846005	+	0.996415i	$0.473039\pi$
$920$	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	$-0.512851\pi$
$947$	−2.48420	−0.0807256	−0.0403628	+	0.999185i	$0.512851\pi$
$948$	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	$-0.573909\pi$
$967$	−14.3114	−0.460224	−0.230112	+	0.973164i	$0.573909\pi$
$968$	0	0
$969$	−17.9144	−0.575494
$970$	0	0
$971$	7.01096	0.224992	0.112496	−	0.993652i	$-0.464115\pi$
$971$	7.01096	0.224992	0.112496	+	0.993652i	$0.464115\pi$
$972$	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	$-0.483309\pi$
$983$	3.28649	0.104823	0.0524114	+	0.998626i	$0.483309\pi$
$984$	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	$-0.417842\pi$
$991$	16.0707	0.510502	0.255251	+	0.966875i	$0.417842\pi$
$992$	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	4008.2.a.m.1.10	✓	13
4.3	odd	2		8016.2.a.bg.1.10		13

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

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.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

Introduction

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