LMFDB - Embedded newform 8008.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46260 q^{3} +3.86081 q^{5} -1.00000 q^{7} -0.860806 q^{9} +O(q^{10})$	$q+1.46260 q^{3} +3.86081 q^{5} -1.00000 q^{7} -0.860806 q^{9} -1.00000 q^{11} +1.00000 q^{13} +5.64681 q^{15} -7.78600 q^{17} -4.18421 q^{19} -1.46260 q^{21} -9.11982 q^{23} +9.90582 q^{25} -5.64681 q^{27} +4.79641 q^{29} -8.32340 q^{31} -1.46260 q^{33} -3.86081 q^{35} -0.323404 q^{37} +1.46260 q^{39} +11.0152 q^{41} -2.58242 q^{43} -3.32340 q^{45} +8.36842 q^{47} +1.00000 q^{49} -11.3878 q^{51} -4.98959 q^{53} -3.86081 q^{55} -6.11982 q^{57} -4.97021 q^{59} +1.13919 q^{61} +0.860806 q^{63} +3.86081 q^{65} +1.18421 q^{67} -13.3386 q^{69} -10.7860 q^{71} +1.44322 q^{73} +14.4882 q^{75} +1.00000 q^{77} -2.33382 q^{79} -5.67660 q^{81} -6.98062 q^{83} -30.0602 q^{85} +7.01523 q^{87} -2.90582 q^{89} -1.00000 q^{91} -12.1738 q^{93} -16.1544 q^{95} -14.5422 q^{97} +0.860806 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} - 13 q^{17} + q^{19} - 2 q^{21} - 13 q^{23} + 5 q^{25} - q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} + 7 q^{37} + 2 q^{39} - 10 q^{41} + 9 q^{43} - 2 q^{45} - 2 q^{47} + 3 q^{49} - 27 q^{51} - 11 q^{53} - 6 q^{55} - 4 q^{57} + 9 q^{59} + 9 q^{61} - 3 q^{63} + 6 q^{65} - 10 q^{67} + 11 q^{69} - 22 q^{71} - 18 q^{73} + 2 q^{75} + 3 q^{77} - 3 q^{79} - 25 q^{81} - q^{83} - 28 q^{85} - 22 q^{87} + 16 q^{89} - 3 q^{91} - 19 q^{93} - 11 q^{95} + q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 - 13 * q^17 + q^19 - 2 * q^21 - 13 * q^23 + 5 * q^25 - q^27 + 8 * q^29 - 17 * q^31 - 2 * q^33 - 6 * q^35 + 7 * q^37 + 2 * q^39 - 10 * q^41 + 9 * q^43 - 2 * q^45 - 2 * q^47 + 3 * q^49 - 27 * q^51 - 11 * q^53 - 6 * q^55 - 4 * q^57 + 9 * q^59 + 9 * q^61 - 3 * q^63 + 6 * q^65 - 10 * q^67 + 11 * q^69 - 22 * q^71 - 18 * q^73 + 2 * q^75 + 3 * q^77 - 3 * q^79 - 25 * q^81 - q^83 - 28 * q^85 - 22 * q^87 + 16 * q^89 - 3 * q^91 - 19 * q^93 - 11 * q^95 + q^97 - 3 * 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
$3$	1.46260	0.844432	0.422216	−	0.906495i	$-0.361252\pi$
$3$	1.46260	0.844432	0.422216	+	0.906495i	$0.361252\pi$
$4$	0	0
$5$	3.86081	1.72660	0.863302	−	0.504687i	$-0.168392\pi$
$5$	3.86081	1.72660	0.863302	+	0.504687i	$0.168392\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.860806	−0.286935
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	5.64681	1.45800
$16$	0	0
$17$	−7.78600	−1.88838	−0.944192	−	0.329397i	$-0.893155\pi$
$17$	−7.78600	−1.88838	−0.944192	+	0.329397i	$0.893155\pi$
$18$	0	0
$19$	−4.18421	−0.959924	−0.479962	−	0.877289i	$-0.659349\pi$
$19$	−4.18421	−0.959924	−0.479962	+	0.877289i	$0.659349\pi$
$20$	0	0
$21$	−1.46260	−0.319165
$22$	0	0
$23$	−9.11982	−1.90161	−0.950807	−	0.309784i	$-0.899743\pi$
$23$	−9.11982	−1.90161	−0.950807	+	0.309784i	$0.899743\pi$
$24$	0	0
$25$	9.90582	1.98116
$26$	0	0
$27$	−5.64681	−1.08673
$28$	0	0
$29$	4.79641	0.890672	0.445336	−	0.895364i	$-0.353084\pi$
$29$	4.79641	0.890672	0.445336	+	0.895364i	$0.353084\pi$
$30$	0	0
$31$	−8.32340	−1.49493	−0.747464	−	0.664303i	$-0.768729\pi$
$31$	−8.32340	−1.49493	−0.747464	+	0.664303i	$0.768729\pi$
$32$	0	0
$33$	−1.46260	−0.254606
$34$	0	0
$35$	−3.86081	−0.652595
$36$	0	0
$37$	−0.323404	−0.0531673	−0.0265837	−	0.999647i	$-0.508463\pi$
$37$	−0.323404	−0.0531673	−0.0265837	+	0.999647i	$0.508463\pi$
$38$	0	0
$39$	1.46260	0.234203
$40$	0	0
$41$	11.0152	1.72029	0.860145	−	0.510050i	$-0.170373\pi$
$41$	11.0152	1.72029	0.860145	+	0.510050i	$0.170373\pi$
$42$	0	0
$43$	−2.58242	−0.393815	−0.196907	−	0.980422i	$-0.563090\pi$
$43$	−2.58242	−0.393815	−0.196907	+	0.980422i	$0.563090\pi$
$44$	0	0
$45$	−3.32340	−0.495424
$46$	0	0
$47$	8.36842	1.22066	0.610330	−	0.792147i	$-0.291037\pi$
$47$	8.36842	1.22066	0.610330	+	0.792147i	$0.291037\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−11.3878	−1.59461
$52$	0	0
$53$	−4.98959	−0.685373	−0.342686	−	0.939450i	$-0.611337\pi$
$53$	−4.98959	−0.685373	−0.342686	+	0.939450i	$0.611337\pi$
$54$	0	0
$55$	−3.86081	−0.520591
$56$	0	0
$57$	−6.11982	−0.810590
$58$	0	0
$59$	−4.97021	−0.647067	−0.323533	−	0.946217i	$-0.604871\pi$
$59$	−4.97021	−0.647067	−0.323533	+	0.946217i	$0.604871\pi$
$60$	0	0
$61$	1.13919	0.145859	0.0729294	−	0.997337i	$-0.476765\pi$
$61$	1.13919	0.145859	0.0729294	+	0.997337i	$0.476765\pi$
$62$	0	0
$63$	0.860806	0.108451
$64$	0	0
$65$	3.86081	0.478874
$66$	0	0
$67$	1.18421	0.144674	0.0723371	−	0.997380i	$-0.476954\pi$
$67$	1.18421	0.144674	0.0723371	+	0.997380i	$0.476954\pi$
$68$	0	0
$69$	−13.3386	−1.60578
$70$	0	0
$71$	−10.7860	−1.28006	−0.640032	−	0.768349i	$-0.721079\pi$
$71$	−10.7860	−1.28006	−0.640032	+	0.768349i	$0.721079\pi$
$72$	0	0
$73$	1.44322	0.168917	0.0844583	−	0.996427i	$-0.473084\pi$
$73$	1.44322	0.168917	0.0844583	+	0.996427i	$0.473084\pi$
$74$	0	0
$75$	14.4882	1.67296
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.33382	−0.262575	−0.131287	−	0.991344i	$-0.541911\pi$
$79$	−2.33382	−0.262575	−0.131287	+	0.991344i	$0.541911\pi$
$80$	0	0
$81$	−5.67660	−0.630733
$82$	0	0
$83$	−6.98062	−0.766223	−0.383112	−	0.923702i	$-0.625148\pi$
$83$	−6.98062	−0.766223	−0.383112	+	0.923702i	$0.625148\pi$
$84$	0	0
$85$	−30.0602	−3.26049
$86$	0	0
$87$	7.01523	0.752111
$88$	0	0
$89$	−2.90582	−0.308016	−0.154008	−	0.988070i	$-0.549218\pi$
$89$	−2.90582	−0.308016	−0.154008	+	0.988070i	$0.549218\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−12.1738	−1.26236
$94$	0	0
$95$	−16.1544	−1.65741
$96$	0	0
$97$	−14.5422	−1.47654	−0.738269	−	0.674506i	$-0.764357\pi$
$97$	−14.5422	−1.47654	−0.738269	+	0.674506i	$0.764357\pi$
$98$	0	0
$99$	0.860806	0.0865142
$100$	0	0
$101$	−5.53740	−0.550992	−0.275496	−	0.961302i	$-0.588842\pi$
$101$	−5.53740	−0.550992	−0.275496	+	0.961302i	$0.588842\pi$
$102$	0	0
$103$	5.29362	0.521596	0.260798	−	0.965393i	$-0.416014\pi$
$103$	5.29362	0.521596	0.260798	+	0.965393i	$0.416014\pi$
$104$	0	0
$105$	−5.64681	−0.551072
$106$	0	0
$107$	15.1004	1.45982	0.729908	−	0.683546i	$-0.239563\pi$
$107$	15.1004	1.45982	0.729908	+	0.683546i	$0.239563\pi$
$108$	0	0
$109$	0.0941782	0.00902063	0.00451032	−	0.999990i	$-0.498564\pi$
$109$	0.0941782	0.00902063	0.00451032	+	0.999990i	$0.498564\pi$
$110$	0	0
$111$	−0.473011	−0.0448962
$112$	0	0
$113$	−11.2140	−1.05492	−0.527462	−	0.849579i	$-0.676856\pi$
$113$	−11.2140	−1.05492	−0.527462	+	0.849579i	$0.676856\pi$
$114$	0	0
$115$	−35.2099	−3.28334
$116$	0	0
$117$	−0.860806	−0.0795815
$118$	0	0
$119$	7.78600	0.713742
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.1109	1.45267
$124$	0	0
$125$	18.9404	1.69408
$126$	0	0
$127$	9.13023	0.810177	0.405089	−	0.914277i	$-0.367241\pi$
$127$	9.13023	0.810177	0.405089	+	0.914277i	$0.367241\pi$
$128$	0	0
$129$	−3.77704	−0.332550
$130$	0	0
$131$	10.4972	0.917145	0.458572	−	0.888657i	$-0.348361\pi$
$131$	10.4972	0.917145	0.458572	+	0.888657i	$0.348361\pi$
$132$	0	0
$133$	4.18421	0.362817
$134$	0	0
$135$	−21.8012	−1.87635
$136$	0	0
$137$	−19.6170	−1.67600	−0.837998	−	0.545674i	$-0.816274\pi$
$137$	−19.6170	−1.67600	−0.837998	+	0.545674i	$0.816274\pi$
$138$	0	0
$139$	8.64681	0.733413	0.366706	−	0.930337i	$-0.380485\pi$
$139$	8.64681	0.733413	0.366706	+	0.930337i	$0.380485\pi$
$140$	0	0
$141$	12.2396	1.03076
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	18.5180	1.53784
$146$	0	0
$147$	1.46260	0.120633
$148$	0	0
$149$	−2.98062	−0.244182	−0.122091	−	0.992519i	$-0.538960\pi$
$149$	−2.98062	−0.244182	−0.122091	+	0.992519i	$0.538960\pi$
$150$	0	0
$151$	3.57201	0.290686	0.145343	−	0.989381i	$-0.453571\pi$
$151$	3.57201	0.290686	0.145343	+	0.989381i	$0.453571\pi$
$152$	0	0
$153$	6.70224	0.541844
$154$	0	0
$155$	−32.1350	−2.58115
$156$	0	0
$157$	5.37738	0.429162	0.214581	−	0.976706i	$-0.431161\pi$
$157$	5.37738	0.429162	0.214581	+	0.976706i	$0.431161\pi$
$158$	0	0
$159$	−7.29776	−0.578750
$160$	0	0
$161$	9.11982	0.718742
$162$	0	0
$163$	−1.53740	−0.120419	−0.0602093	−	0.998186i	$-0.519177\pi$
$163$	−1.53740	−0.120419	−0.0602093	+	0.998186i	$0.519177\pi$
$164$	0	0
$165$	−5.64681	−0.439603
$166$	0	0
$167$	−18.5824	−1.43795	−0.718975	−	0.695036i	$-0.755389\pi$
$167$	−18.5824	−1.43795	−0.718975	+	0.695036i	$0.755389\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.60179	0.275436
$172$	0	0
$173$	4.70224	0.357504	0.178752	−	0.983894i	$-0.442794\pi$
$173$	4.70224	0.357504	0.178752	+	0.983894i	$0.442794\pi$
$174$	0	0
$175$	−9.90582	−0.748810
$176$	0	0
$177$	−7.26943	−0.546403
$178$	0	0
$179$	−8.73057	−0.652554	−0.326277	−	0.945274i	$-0.605794\pi$
$179$	−8.73057	−0.652554	−0.326277	+	0.945274i	$0.605794\pi$
$180$	0	0
$181$	−9.48824	−0.705255	−0.352628	−	0.935764i	$-0.614712\pi$
$181$	−9.48824	−0.705255	−0.352628	+	0.935764i	$0.614712\pi$
$182$	0	0
$183$	1.66618	0.123168
$184$	0	0
$185$	−1.24860	−0.0917990
$186$	0	0
$187$	7.78600	0.569369
$188$	0	0
$189$	5.64681	0.410745
$190$	0	0
$191$	−13.7666	−0.996118	−0.498059	−	0.867143i	$-0.665954\pi$
$191$	−13.7666	−0.996118	−0.498059	+	0.867143i	$0.665954\pi$
$192$	0	0
$193$	12.8012	0.921453	0.460726	−	0.887542i	$-0.347589\pi$
$193$	12.8012	0.921453	0.460726	+	0.887542i	$0.347589\pi$
$194$	0	0
$195$	5.64681	0.404376
$196$	0	0
$197$	−27.3193	−1.94642	−0.973208	−	0.229925i	$-0.926152\pi$
$197$	−27.3193	−1.94642	−0.973208	+	0.229925i	$0.926152\pi$
$198$	0	0
$199$	−10.1634	−0.720463	−0.360232	−	0.932863i	$-0.617302\pi$
$199$	−10.1634	−0.720463	−0.360232	+	0.932863i	$0.617302\pi$
$200$	0	0
$201$	1.73202	0.122168
$202$	0	0
$203$	−4.79641	−0.336642
$204$	0	0
$205$	42.5277	2.97026
$206$	0	0
$207$	7.85039	0.545640
$208$	0	0
$209$	4.18421	0.289428
$210$	0	0
$211$	−4.30403	−0.296302	−0.148151	−	0.988965i	$-0.547332\pi$
$211$	−4.30403	−0.296302	−0.148151	+	0.988965i	$0.547332\pi$
$212$	0	0
$213$	−15.7756	−1.08093
$214$	0	0
$215$	−9.97021	−0.679963
$216$	0	0
$217$	8.32340	0.565029
$218$	0	0
$219$	2.11086	0.142638
$220$	0	0
$221$	−7.78600	−0.523743
$222$	0	0
$223$	−11.4674	−0.767915	−0.383957	−	0.923351i	$-0.625439\pi$
$223$	−11.4674	−0.767915	−0.383957	+	0.923351i	$0.625439\pi$
$224$	0	0
$225$	−8.52699	−0.568466
$226$	0	0
$227$	13.4868	0.895150	0.447575	−	0.894246i	$-0.352288\pi$
$227$	13.4868	0.895150	0.447575	+	0.894246i	$0.352288\pi$
$228$	0	0
$229$	23.0256	1.52158	0.760789	−	0.649000i	$-0.224812\pi$
$229$	23.0256	1.52158	0.760789	+	0.649000i	$0.224812\pi$
$230$	0	0
$231$	1.46260	0.0962319
$232$	0	0
$233$	11.6620	0.764005	0.382003	−	0.924161i	$-0.375234\pi$
$233$	11.6620	0.764005	0.382003	+	0.924161i	$0.375234\pi$
$234$	0	0
$235$	32.3088	2.10760
$236$	0	0
$237$	−3.41344	−0.221727
$238$	0	0
$239$	14.7756	0.955753	0.477877	−	0.878427i	$-0.341407\pi$
$239$	14.7756	0.955753	0.477877	+	0.878427i	$0.341407\pi$
$240$	0	0
$241$	2.36842	0.152563	0.0762817	−	0.997086i	$-0.475695\pi$
$241$	2.36842	0.152563	0.0762817	+	0.997086i	$0.475695\pi$
$242$	0	0
$243$	8.63785	0.554118
$244$	0	0
$245$	3.86081	0.246658
$246$	0	0
$247$	−4.18421	−0.266235
$248$	0	0
$249$	−10.2099	−0.647023
$250$	0	0
$251$	19.2445	1.21470	0.607350	−	0.794435i	$-0.292233\pi$
$251$	19.2445	1.21470	0.607350	+	0.794435i	$0.292233\pi$
$252$	0	0
$253$	9.11982	0.573358
$254$	0	0
$255$	−43.9661	−2.75326
$256$	0	0
$257$	10.6676	0.665429	0.332714	−	0.943028i	$-0.392036\pi$
$257$	10.6676	0.665429	0.332714	+	0.943028i	$0.392036\pi$
$258$	0	0
$259$	0.323404	0.0200954
$260$	0	0
$261$	−4.12878	−0.255565
$262$	0	0
$263$	−18.4238	−1.13606	−0.568032	−	0.823007i	$-0.692295\pi$
$263$	−18.4238	−1.13606	−0.568032	+	0.823007i	$0.692295\pi$
$264$	0	0
$265$	−19.2638	−1.18337
$266$	0	0
$267$	−4.25005	−0.260099
$268$	0	0
$269$	8.70638	0.530838	0.265419	−	0.964133i	$-0.414490\pi$
$269$	8.70638	0.530838	0.265419	+	0.964133i	$0.414490\pi$
$270$	0	0
$271$	−26.3732	−1.60206	−0.801030	−	0.598624i	$-0.795714\pi$
$271$	−26.3732	−1.60206	−0.801030	+	0.598624i	$0.795714\pi$
$272$	0	0
$273$	−1.46260	−0.0885205
$274$	0	0
$275$	−9.90582	−0.597344
$276$	0	0
$277$	−5.48197	−0.329380	−0.164690	−	0.986345i	$-0.552662\pi$
$277$	−5.48197	−0.329380	−0.164690	+	0.986345i	$0.552662\pi$
$278$	0	0
$279$	7.16484	0.428947
$280$	0	0
$281$	15.8969	0.948327	0.474164	−	0.880437i	$-0.342751\pi$
$281$	15.8969	0.948327	0.474164	+	0.880437i	$0.342751\pi$
$282$	0	0
$283$	−16.3892	−0.974239	−0.487120	−	0.873335i	$-0.661952\pi$
$283$	−16.3892	−0.974239	−0.487120	+	0.873335i	$0.661952\pi$
$284$	0	0
$285$	−23.6274	−1.39957
$286$	0	0
$287$	−11.0152	−0.650208
$288$	0	0
$289$	43.6218	2.56599
$290$	0	0
$291$	−21.2694	−1.24684
$292$	0	0
$293$	14.5180	0.848152	0.424076	−	0.905627i	$-0.360599\pi$
$293$	14.5180	0.848152	0.424076	+	0.905627i	$0.360599\pi$
$294$	0	0
$295$	−19.1890	−1.11723
$296$	0	0
$297$	5.64681	0.327661
$298$	0	0
$299$	−9.11982	−0.527413
$300$	0	0
$301$	2.58242	0.148848
$302$	0	0
$303$	−8.09899	−0.465275
$304$	0	0
$305$	4.39821	0.251841
$306$	0	0
$307$	17.8760	1.02024	0.510120	−	0.860103i	$-0.329601\pi$
$307$	17.8760	1.02024	0.510120	+	0.860103i	$0.329601\pi$
$308$	0	0
$309$	7.74244	0.440452
$310$	0	0
$311$	−13.1690	−0.746744	−0.373372	−	0.927682i	$-0.621799\pi$
$311$	−13.1690	−0.746744	−0.373372	+	0.927682i	$0.621799\pi$
$312$	0	0
$313$	−23.4674	−1.32646	−0.663228	−	0.748417i	$-0.730814\pi$
$313$	−23.4674	−1.32646	−0.663228	+	0.748417i	$0.730814\pi$
$314$	0	0
$315$	3.32340	0.187253
$316$	0	0
$317$	−3.67660	−0.206498	−0.103249	−	0.994656i	$-0.532924\pi$
$317$	−3.67660	−0.206498	−0.103249	+	0.994656i	$0.532924\pi$
$318$	0	0
$319$	−4.79641	−0.268548
$320$	0	0
$321$	22.0859	1.23271
$322$	0	0
$323$	32.5783	1.81270
$324$	0	0
$325$	9.90582	0.549476
$326$	0	0
$327$	0.137745	0.00761731
$328$	0	0
$329$	−8.36842	−0.461366
$330$	0	0
$331$	−7.86081	−0.432069	−0.216034	−	0.976386i	$-0.569312\pi$
$331$	−7.86081	−0.432069	−0.216034	+	0.976386i	$0.569312\pi$
$332$	0	0
$333$	0.278388	0.0152556
$334$	0	0
$335$	4.57201	0.249795
$336$	0	0
$337$	−26.9557	−1.46837	−0.734184	−	0.678951i	$-0.762435\pi$
$337$	−26.9557	−1.46837	−0.734184	+	0.678951i	$0.762435\pi$
$338$	0	0
$339$	−16.4016	−0.890811
$340$	0	0
$341$	8.32340	0.450738
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−51.4979	−2.77255
$346$	0	0
$347$	17.9058	0.961235	0.480617	−	0.876930i	$-0.340413\pi$
$347$	17.9058	0.961235	0.480617	+	0.876930i	$0.340413\pi$
$348$	0	0
$349$	−13.2244	−0.707886	−0.353943	−	0.935267i	$-0.615159\pi$
$349$	−13.2244	−0.707886	−0.353943	+	0.935267i	$0.615159\pi$
$350$	0	0
$351$	−5.64681	−0.301404
$352$	0	0
$353$	−1.63785	−0.0871737	−0.0435869	−	0.999050i	$-0.513879\pi$
$353$	−1.63785	−0.0871737	−0.0435869	+	0.999050i	$0.513879\pi$
$354$	0	0
$355$	−41.6427	−2.21016
$356$	0	0
$357$	11.3878	0.602706
$358$	0	0
$359$	−23.4045	−1.23524	−0.617620	−	0.786476i	$-0.711903\pi$
$359$	−23.4045	−1.23524	−0.617620	+	0.786476i	$0.711903\pi$
$360$	0	0
$361$	−1.49239	−0.0785466
$362$	0	0
$363$	1.46260	0.0767665
$364$	0	0
$365$	5.57201	0.291652
$366$	0	0
$367$	11.6814	0.609765	0.304882	−	0.952390i	$-0.401383\pi$
$367$	11.6814	0.609765	0.304882	+	0.952390i	$0.401383\pi$
$368$	0	0
$369$	−9.48197	−0.493612
$370$	0	0
$371$	4.98959	0.259046
$372$	0	0
$373$	30.8448	1.59708	0.798542	−	0.601940i	$-0.205605\pi$
$373$	30.8448	1.59708	0.798542	+	0.601940i	$0.205605\pi$
$374$	0	0
$375$	27.7022	1.43054
$376$	0	0
$377$	4.79641	0.247028
$378$	0	0
$379$	−10.7354	−0.551440	−0.275720	−	0.961238i	$-0.588916\pi$
$379$	−10.7354	−0.551440	−0.275720	+	0.961238i	$0.588916\pi$
$380$	0	0
$381$	13.3539	0.684139
$382$	0	0
$383$	−24.4343	−1.24853	−0.624266	−	0.781212i	$-0.714602\pi$
$383$	−24.4343	−1.24853	−0.624266	+	0.781212i	$0.714602\pi$
$384$	0	0
$385$	3.86081	0.196765
$386$	0	0
$387$	2.22296	0.112999
$388$	0	0
$389$	11.2445	0.570116	0.285058	−	0.958510i	$-0.407987\pi$
$389$	11.2445	0.570116	0.285058	+	0.958510i	$0.407987\pi$
$390$	0	0
$391$	71.0069	3.59098
$392$	0	0
$393$	15.3532	0.774466
$394$	0	0
$395$	−9.01041	−0.453363
$396$	0	0
$397$	0.621168	0.0311755	0.0155878	−	0.999879i	$-0.495038\pi$
$397$	0.621168	0.0311755	0.0155878	+	0.999879i	$0.495038\pi$
$398$	0	0
$399$	6.11982	0.306374
$400$	0	0
$401$	28.7577	1.43609	0.718045	−	0.695997i	$-0.245037\pi$
$401$	28.7577	1.43609	0.718045	+	0.695997i	$0.245037\pi$
$402$	0	0
$403$	−8.32340	−0.414618
$404$	0	0
$405$	−21.9162	−1.08903
$406$	0	0
$407$	0.323404	0.0160306
$408$	0	0
$409$	10.3088	0.509740	0.254870	−	0.966975i	$-0.417967\pi$
$409$	10.3088	0.509740	0.254870	+	0.966975i	$0.417967\pi$
$410$	0	0
$411$	−28.6918	−1.41526
$412$	0	0
$413$	4.97021	0.244568
$414$	0	0
$415$	−26.9508	−1.32296
$416$	0	0
$417$	12.6468	0.619317
$418$	0	0
$419$	26.1752	1.27874	0.639372	−	0.768897i	$-0.279194\pi$
$419$	26.1752	1.27874	0.639372	+	0.768897i	$0.279194\pi$
$420$	0	0
$421$	35.4737	1.72888	0.864441	−	0.502735i	$-0.167673\pi$
$421$	35.4737	1.72888	0.864441	+	0.502735i	$0.167673\pi$
$422$	0	0
$423$	−7.20359	−0.350250
$424$	0	0
$425$	−77.1268	−3.74120
$426$	0	0
$427$	−1.13919	−0.0551295
$428$	0	0
$429$	−1.46260	−0.0706149
$430$	0	0
$431$	32.5872	1.56967	0.784836	−	0.619704i	$-0.212747\pi$
$431$	32.5872	1.56967	0.784836	+	0.619704i	$0.212747\pi$
$432$	0	0
$433$	−16.5839	−0.796970	−0.398485	−	0.917175i	$-0.630464\pi$
$433$	−16.5839	−0.796970	−0.398485	+	0.917175i	$0.630464\pi$
$434$	0	0
$435$	27.0844	1.29860
$436$	0	0
$437$	38.1592	1.82540
$438$	0	0
$439$	−0.427995	−0.0204271	−0.0102135	−	0.999948i	$-0.503251\pi$
$439$	−0.427995	−0.0204271	−0.0102135	+	0.999948i	$0.503251\pi$
$440$	0	0
$441$	−0.860806	−0.0409908
$442$	0	0
$443$	37.9259	1.80191	0.900956	−	0.433910i	$-0.142866\pi$
$443$	37.9259	1.80191	0.900956	+	0.433910i	$0.142866\pi$
$444$	0	0
$445$	−11.2188	−0.531823
$446$	0	0
$447$	−4.35946	−0.206195
$448$	0	0
$449$	20.2251	0.954481	0.477240	−	0.878773i	$-0.341637\pi$
$449$	20.2251	0.954481	0.477240	+	0.878773i	$0.341637\pi$
$450$	0	0
$451$	−11.0152	−0.518687
$452$	0	0
$453$	5.22441	0.245464
$454$	0	0
$455$	−3.86081	−0.180997
$456$	0	0
$457$	28.4931	1.33285	0.666424	−	0.745573i	$-0.267824\pi$
$457$	28.4931	1.33285	0.666424	+	0.745573i	$0.267824\pi$
$458$	0	0
$459$	43.9661	2.05216
$460$	0	0
$461$	−2.45845	−0.114502	−0.0572508	−	0.998360i	$-0.518233\pi$
$461$	−2.45845	−0.114502	−0.0572508	+	0.998360i	$0.518233\pi$
$462$	0	0
$463$	−42.0450	−1.95400	−0.976999	−	0.213245i	$-0.931597\pi$
$463$	−42.0450	−1.95400	−0.976999	+	0.213245i	$0.931597\pi$
$464$	0	0
$465$	−47.0007	−2.17960
$466$	0	0
$467$	−3.31299	−0.153307	−0.0766535	−	0.997058i	$-0.524424\pi$
$467$	−3.31299	−0.153307	−0.0766535	+	0.997058i	$0.524424\pi$
$468$	0	0
$469$	−1.18421	−0.0546817
$470$	0	0
$471$	7.86495	0.362398
$472$	0	0
$473$	2.58242	0.118740
$474$	0	0
$475$	−41.4480	−1.90177
$476$	0	0
$477$	4.29507	0.196658
$478$	0	0
$479$	23.7562	1.08545	0.542725	−	0.839911i	$-0.317393\pi$
$479$	23.7562	1.08545	0.542725	+	0.839911i	$0.317393\pi$
$480$	0	0
$481$	−0.323404	−0.0147460
$482$	0	0
$483$	13.3386	0.606929
$484$	0	0
$485$	−56.1447	−2.54940
$486$	0	0
$487$	7.33863	0.332545	0.166273	−	0.986080i	$-0.446827\pi$
$487$	7.33863	0.332545	0.166273	+	0.986080i	$0.446827\pi$
$488$	0	0
$489$	−2.24860	−0.101685
$490$	0	0
$491$	28.6468	1.29281	0.646406	−	0.762993i	$-0.276271\pi$
$491$	28.6468	1.29281	0.646406	+	0.762993i	$0.276271\pi$
$492$	0	0
$493$	−37.3449	−1.68193
$494$	0	0
$495$	3.32340	0.149376
$496$	0	0
$497$	10.7860	0.483818
$498$	0	0
$499$	−14.9419	−0.668890	−0.334445	−	0.942415i	$-0.608549\pi$
$499$	−14.9419	−0.668890	−0.334445	+	0.942415i	$0.608549\pi$
$500$	0	0
$501$	−27.1786	−1.21425
$502$	0	0
$503$	18.3892	0.819936	0.409968	−	0.912100i	$-0.365540\pi$
$503$	18.3892	0.819936	0.409968	+	0.912100i	$0.365540\pi$
$504$	0	0
$505$	−21.3788	−0.951346
$506$	0	0
$507$	1.46260	0.0649563
$508$	0	0
$509$	8.39261	0.371996	0.185998	−	0.982550i	$-0.440448\pi$
$509$	8.39261	0.371996	0.185998	+	0.982550i	$0.440448\pi$
$510$	0	0
$511$	−1.44322	−0.0638444
$512$	0	0
$513$	23.6274	1.04318
$514$	0	0
$515$	20.4376	0.900589
$516$	0	0
$517$	−8.36842	−0.368043
$518$	0	0
$519$	6.87748	0.301888
$520$	0	0
$521$	−40.5935	−1.77843	−0.889217	−	0.457486i	$-0.848750\pi$
$521$	−40.5935	−1.77843	−0.889217	+	0.457486i	$0.848750\pi$
$522$	0	0
$523$	−12.8352	−0.561243	−0.280621	−	0.959819i	$-0.590540\pi$
$523$	−12.8352	−0.561243	−0.280621	+	0.959819i	$0.590540\pi$
$524$	0	0
$525$	−14.4882	−0.632319
$526$	0	0
$527$	64.8060	2.82300
$528$	0	0
$529$	60.1711	2.61613
$530$	0	0
$531$	4.27839	0.185666
$532$	0	0
$533$	11.0152	0.477123
$534$	0	0
$535$	58.2999	2.52052
$536$	0	0
$537$	−12.7693	−0.551037
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	29.7008	1.27694	0.638468	−	0.769648i	$-0.279568\pi$
$541$	29.7008	1.27694	0.638468	+	0.769648i	$0.279568\pi$
$542$	0	0
$543$	−13.8775	−0.595540
$544$	0	0
$545$	0.363604	0.0155751
$546$	0	0
$547$	20.4376	0.873850	0.436925	−	0.899498i	$-0.356068\pi$
$547$	20.4376	0.873850	0.436925	+	0.899498i	$0.356068\pi$
$548$	0	0
$549$	−0.980625	−0.0418521
$550$	0	0
$551$	−20.0692	−0.854977
$552$	0	0
$553$	2.33382	0.0992440
$554$	0	0
$555$	−1.82620	−0.0775180
$556$	0	0
$557$	−13.7770	−0.583752	−0.291876	−	0.956456i	$-0.594279\pi$
$557$	−13.7770	−0.583752	−0.291876	+	0.956456i	$0.594279\pi$
$558$	0	0
$559$	−2.58242	−0.109225
$560$	0	0
$561$	11.3878	0.480793
$562$	0	0
$563$	−45.2132	−1.90551	−0.952755	−	0.303740i	$-0.901765\pi$
$563$	−45.2132	−1.90551	−0.952755	+	0.303740i	$0.901765\pi$
$564$	0	0
$565$	−43.2951	−1.82144
$566$	0	0
$567$	5.67660	0.238395
$568$	0	0
$569$	−24.5180	−1.02785	−0.513925	−	0.857835i	$-0.671809\pi$
$569$	−24.5180	−1.02785	−0.513925	+	0.857835i	$0.671809\pi$
$570$	0	0
$571$	11.9446	0.499865	0.249932	−	0.968263i	$-0.419592\pi$
$571$	11.9446	0.499865	0.249932	+	0.968263i	$0.419592\pi$
$572$	0	0
$573$	−20.1350	−0.841154
$574$	0	0
$575$	−90.3393	−3.76741
$576$	0	0
$577$	−33.5735	−1.39768	−0.698841	−	0.715277i	$-0.746300\pi$
$577$	−33.5735	−1.39768	−0.698841	+	0.715277i	$0.746300\pi$
$578$	0	0
$579$	18.7231	0.778104
$580$	0	0
$581$	6.98062	0.289605
$582$	0	0
$583$	4.98959	0.206648
$584$	0	0
$585$	−3.32340	−0.137406
$586$	0	0
$587$	−7.87122	−0.324880	−0.162440	−	0.986718i	$-0.551936\pi$
$587$	−7.87122	−0.324880	−0.162440	+	0.986718i	$0.551936\pi$
$588$	0	0
$589$	34.8269	1.43502
$590$	0	0
$591$	−39.9571	−1.64362
$592$	0	0
$593$	−47.9196	−1.96782	−0.983911	−	0.178659i	$-0.942824\pi$
$593$	−47.9196	−1.96782	−0.983911	+	0.178659i	$0.942824\pi$
$594$	0	0
$595$	30.0602	1.23235
$596$	0	0
$597$	−14.8650	−0.608382
$598$	0	0
$599$	−11.6137	−0.474521	−0.237261	−	0.971446i	$-0.576250\pi$
$599$	−11.6137	−0.474521	−0.237261	+	0.971446i	$0.576250\pi$
$600$	0	0
$601$	7.52844	0.307091	0.153546	−	0.988142i	$-0.450931\pi$
$601$	7.52844	0.307091	0.153546	+	0.988142i	$0.450931\pi$
$602$	0	0
$603$	−1.01938	−0.0415122
$604$	0	0
$605$	3.86081	0.156964
$606$	0	0
$607$	30.9765	1.25730	0.628648	−	0.777690i	$-0.283609\pi$
$607$	30.9765	1.25730	0.628648	+	0.777690i	$0.283609\pi$
$608$	0	0
$609$	−7.01523	−0.284271
$610$	0	0
$611$	8.36842	0.338550
$612$	0	0
$613$	−41.0665	−1.65866	−0.829330	−	0.558759i	$-0.811278\pi$
$613$	−41.0665	−1.65866	−0.829330	+	0.558759i	$0.811278\pi$
$614$	0	0
$615$	62.2009	2.50818
$616$	0	0
$617$	26.9765	1.08603	0.543016	−	0.839722i	$-0.317282\pi$
$617$	26.9765	1.08603	0.543016	+	0.839722i	$0.317282\pi$
$618$	0	0
$619$	−5.43696	−0.218530	−0.109265	−	0.994013i	$-0.534850\pi$
$619$	−5.43696	−0.218530	−0.109265	+	0.994013i	$0.534850\pi$
$620$	0	0
$621$	51.4979	2.06654
$622$	0	0
$623$	2.90582	0.116419
$624$	0	0
$625$	23.5962	0.943848
$626$	0	0
$627$	6.11982	0.244402
$628$	0	0
$629$	2.51803	0.100400
$630$	0	0
$631$	46.5083	1.85147	0.925733	−	0.378178i	$-0.123449\pi$
$631$	46.5083	1.85147	0.925733	+	0.378178i	$0.123449\pi$
$632$	0	0
$633$	−6.29507	−0.250206
$634$	0	0
$635$	35.2501	1.39886
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	9.28465	0.367295
$640$	0	0
$641$	−16.7383	−0.661123	−0.330561	−	0.943785i	$-0.607238\pi$
$641$	−16.7383	−0.661123	−0.330561	+	0.943785i	$0.607238\pi$
$642$	0	0
$643$	−6.43426	−0.253742	−0.126871	−	0.991919i	$-0.540494\pi$
$643$	−6.43426	−0.253742	−0.126871	+	0.991919i	$0.540494\pi$
$644$	0	0
$645$	−14.5824	−0.574182
$646$	0	0
$647$	−24.7514	−0.973078	−0.486539	−	0.873659i	$-0.661741\pi$
$647$	−24.7514	−0.973078	−0.486539	+	0.873659i	$0.661741\pi$
$648$	0	0
$649$	4.97021	0.195098
$650$	0	0
$651$	12.1738	0.477129
$652$	0	0
$653$	44.8046	1.75334	0.876670	−	0.481093i	$-0.159760\pi$
$653$	44.8046	1.75334	0.876670	+	0.481093i	$0.159760\pi$
$654$	0	0
$655$	40.5277	1.58355
$656$	0	0
$657$	−1.24234	−0.0484681
$658$	0	0
$659$	−23.6156	−0.919932	−0.459966	−	0.887937i	$-0.652138\pi$
$659$	−23.6156	−0.919932	−0.459966	+	0.887937i	$0.652138\pi$
$660$	0	0
$661$	43.9300	1.70868	0.854340	−	0.519715i	$-0.173962\pi$
$661$	43.9300	1.70868	0.854340	+	0.519715i	$0.173962\pi$
$662$	0	0
$663$	−11.3878	−0.442265
$664$	0	0
$665$	16.1544	0.626442
$666$	0	0
$667$	−43.7424	−1.69371
$668$	0	0
$669$	−16.7722	−0.648452
$670$	0	0
$671$	−1.13919	−0.0439781
$672$	0	0
$673$	−30.0096	−1.15679	−0.578393	−	0.815758i	$-0.696320\pi$
$673$	−30.0096	−1.15679	−0.578393	+	0.815758i	$0.696320\pi$
$674$	0	0
$675$	−55.9363	−2.15299
$676$	0	0
$677$	9.17794	0.352737	0.176369	−	0.984324i	$-0.443565\pi$
$677$	9.17794	0.352737	0.176369	+	0.984324i	$0.443565\pi$
$678$	0	0
$679$	14.5422	0.558079
$680$	0	0
$681$	19.7258	0.755893
$682$	0	0
$683$	32.3941	1.23952	0.619762	−	0.784790i	$-0.287229\pi$
$683$	32.3941	1.23952	0.619762	+	0.784790i	$0.287229\pi$
$684$	0	0
$685$	−75.7375	−2.89378
$686$	0	0
$687$	33.6773	1.28487
$688$	0	0
$689$	−4.98959	−0.190088
$690$	0	0
$691$	20.9910	0.798537	0.399268	−	0.916834i	$-0.369264\pi$
$691$	20.9910	0.798537	0.399268	+	0.916834i	$0.369264\pi$
$692$	0	0
$693$	−0.860806	−0.0326993
$694$	0	0
$695$	33.3836	1.26631
$696$	0	0
$697$	−85.7646	−3.24857
$698$	0	0
$699$	17.0569	0.645150
$700$	0	0
$701$	−11.5028	−0.434455	−0.217227	−	0.976121i	$-0.569701\pi$
$701$	−11.5028	−0.434455	−0.217227	+	0.976121i	$0.569701\pi$
$702$	0	0
$703$	1.35319	0.0510366
$704$	0	0
$705$	47.2549	1.77972
$706$	0	0
$707$	5.53740	0.208255
$708$	0	0
$709$	−35.3207	−1.32650	−0.663249	−	0.748399i	$-0.730823\pi$
$709$	−35.3207	−1.32650	−0.663249	+	0.748399i	$0.730823\pi$
$710$	0	0
$711$	2.00896	0.0753420
$712$	0	0
$713$	75.9079	2.84277
$714$	0	0
$715$	−3.86081	−0.144386
$716$	0	0
$717$	21.6108	0.807068
$718$	0	0
$719$	−42.7867	−1.59567	−0.797837	−	0.602873i	$-0.794022\pi$
$719$	−42.7867	−1.59567	−0.797837	+	0.602873i	$0.794022\pi$
$720$	0	0
$721$	−5.29362	−0.197145
$722$	0	0
$723$	3.46405	0.128829
$724$	0	0
$725$	47.5124	1.76457
$726$	0	0
$727$	−40.1517	−1.48915	−0.744573	−	0.667542i	$-0.767347\pi$
$727$	−40.1517	−1.48915	−0.744573	+	0.667542i	$0.767347\pi$
$728$	0	0
$729$	29.6635	1.09865
$730$	0	0
$731$	20.1067	0.743674
$732$	0	0
$733$	34.1109	1.25991	0.629957	−	0.776630i	$-0.283073\pi$
$733$	34.1109	1.25991	0.629957	+	0.776630i	$0.283073\pi$
$734$	0	0
$735$	5.64681	0.208286
$736$	0	0
$737$	−1.18421	−0.0436209
$738$	0	0
$739$	−23.4045	−0.860947	−0.430474	−	0.902603i	$-0.641653\pi$
$739$	−23.4045	−0.860947	−0.430474	+	0.902603i	$0.641653\pi$
$740$	0	0
$741$	−6.11982	−0.224817
$742$	0	0
$743$	−16.8864	−0.619504	−0.309752	−	0.950817i	$-0.600246\pi$
$743$	−16.8864	−0.619504	−0.309752	+	0.950817i	$0.600246\pi$
$744$	0	0
$745$	−11.5076	−0.421606
$746$	0	0
$747$	6.00896	0.219856
$748$	0	0
$749$	−15.1004	−0.551758
$750$	0	0
$751$	−49.0394	−1.78947	−0.894737	−	0.446594i	$-0.852637\pi$
$751$	−49.0394	−1.78947	−0.894737	+	0.446594i	$0.852637\pi$
$752$	0	0
$753$	28.1469	1.02573
$754$	0	0
$755$	13.7908	0.501899
$756$	0	0
$757$	24.3434	0.884778	0.442389	−	0.896823i	$-0.354131\pi$
$757$	24.3434	0.884778	0.442389	+	0.896823i	$0.354131\pi$
$758$	0	0
$759$	13.3386	0.484162
$760$	0	0
$761$	8.90437	0.322783	0.161392	−	0.986890i	$-0.448402\pi$
$761$	8.90437	0.322783	0.161392	+	0.986890i	$0.448402\pi$
$762$	0	0
$763$	−0.0941782	−0.00340948
$764$	0	0
$765$	25.8760	0.935550
$766$	0	0
$767$	−4.97021	−0.179464
$768$	0	0
$769$	−40.5901	−1.46372	−0.731859	−	0.681456i	$-0.761347\pi$
$769$	−40.5901	−1.46372	−0.731859	+	0.681456i	$0.761347\pi$
$770$	0	0
$771$	15.6025	0.561909
$772$	0	0
$773$	−33.1517	−1.19238	−0.596192	−	0.802842i	$-0.703320\pi$
$773$	−33.1517	−1.19238	−0.596192	+	0.802842i	$0.703320\pi$
$774$	0	0
$775$	−82.4502	−2.96170
$776$	0	0
$777$	0.473011	0.0169692
$778$	0	0
$779$	−46.0900	−1.65135
$780$	0	0
$781$	10.7860	0.385954
$782$	0	0
$783$	−27.0844	−0.967919
$784$	0	0
$785$	20.7610	0.740993
$786$	0	0
$787$	15.2501	0.543606	0.271803	−	0.962353i	$-0.412380\pi$
$787$	15.2501	0.543606	0.271803	+	0.962353i	$0.412380\pi$
$788$	0	0
$789$	−26.9467	−0.959328
$790$	0	0
$791$	11.2140	0.398724
$792$	0	0
$793$	1.13919	0.0404540
$794$	0	0
$795$	−28.1752	−0.999273
$796$	0	0
$797$	46.5243	1.64797	0.823987	−	0.566608i	$-0.191745\pi$
$797$	46.5243	1.64797	0.823987	+	0.566608i	$0.191745\pi$
$798$	0	0
$799$	−65.1565	−2.30507
$800$	0	0
$801$	2.50135	0.0883808
$802$	0	0
$803$	−1.44322	−0.0509302
$804$	0	0
$805$	35.2099	1.24098
$806$	0	0
$807$	12.7339	0.448256
$808$	0	0
$809$	−39.5845	−1.39172	−0.695859	−	0.718178i	$-0.744976\pi$
$809$	−39.5845	−1.39172	−0.695859	+	0.718178i	$0.744976\pi$
$810$	0	0
$811$	−7.99585	−0.280772	−0.140386	−	0.990097i	$-0.544834\pi$
$811$	−7.99585	−0.280772	−0.140386	+	0.990097i	$0.544834\pi$
$812$	0	0
$813$	−38.5735	−1.35283
$814$	0	0
$815$	−5.93561	−0.207915
$816$	0	0
$817$	10.8054	0.378032
$818$	0	0
$819$	0.860806	0.0300790
$820$	0	0
$821$	27.6669	0.965580	0.482790	−	0.875736i	$-0.339623\pi$
$821$	27.6669	0.965580	0.482790	+	0.875736i	$0.339623\pi$
$822$	0	0
$823$	−8.49383	−0.296076	−0.148038	−	0.988982i	$-0.547296\pi$
$823$	−8.49383	−0.296076	−0.148038	+	0.988982i	$0.547296\pi$
$824$	0	0
$825$	−14.4882	−0.504416
$826$	0	0
$827$	−11.1648	−0.388239	−0.194120	−	0.980978i	$-0.562185\pi$
$827$	−11.1648	−0.388239	−0.194120	+	0.980978i	$0.562185\pi$
$828$	0	0
$829$	0.382979	0.0133014	0.00665070	−	0.999978i	$-0.497883\pi$
$829$	0.382979	0.0133014	0.00665070	+	0.999978i	$0.497883\pi$
$830$	0	0
$831$	−8.01793	−0.278139
$832$	0	0
$833$	−7.78600	−0.269769
$834$	0	0
$835$	−71.7431	−2.48277
$836$	0	0
$837$	47.0007	1.62458
$838$	0	0
$839$	−4.02419	−0.138931	−0.0694653	−	0.997584i	$-0.522129\pi$
$839$	−4.02419	−0.138931	−0.0694653	+	0.997584i	$0.522129\pi$
$840$	0	0
$841$	−5.99440	−0.206704
$842$	0	0
$843$	23.2507	0.800797
$844$	0	0
$845$	3.86081	0.132816
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−23.9709	−0.822679
$850$	0	0
$851$	2.94939	0.101104
$852$	0	0
$853$	−54.1205	−1.85305	−0.926525	−	0.376233i	$-0.877219\pi$
$853$	−54.1205	−1.85305	−0.926525	+	0.376233i	$0.877219\pi$
$854$	0	0
$855$	13.9058	0.475569
$856$	0	0
$857$	−0.225859	−0.00771519	−0.00385759	−	0.999993i	$-0.501228\pi$
$857$	−0.225859	−0.00771519	−0.00385759	+	0.999993i	$0.501228\pi$
$858$	0	0
$859$	−19.1586	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$859$	−19.1586	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$860$	0	0
$861$	−16.1109	−0.549057
$862$	0	0
$863$	−3.70079	−0.125976	−0.0629881	−	0.998014i	$-0.520063\pi$
$863$	−3.70079	−0.125976	−0.0629881	+	0.998014i	$0.520063\pi$
$864$	0	0
$865$	18.1544	0.617269
$866$	0	0
$867$	63.8012	2.16680
$868$	0	0
$869$	2.33382	0.0791693
$870$	0	0
$871$	1.18421	0.0401254
$872$	0	0
$873$	12.5180	0.423671
$874$	0	0
$875$	−18.9404	−0.640303
$876$	0	0
$877$	−42.2445	−1.42649	−0.713247	−	0.700913i	$-0.752776\pi$
$877$	−42.2445	−1.42649	−0.713247	+	0.700913i	$0.752776\pi$
$878$	0	0
$879$	21.2340	0.716206
$880$	0	0
$881$	3.46742	0.116820	0.0584101	−	0.998293i	$-0.481397\pi$
$881$	3.46742	0.116820	0.0584101	+	0.998293i	$0.481397\pi$
$882$	0	0
$883$	−3.68286	−0.123938	−0.0619691	−	0.998078i	$-0.519738\pi$
$883$	−3.68286	−0.123938	−0.0619691	+	0.998078i	$0.519738\pi$
$884$	0	0
$885$	−28.0658	−0.943423
$886$	0	0
$887$	−40.7368	−1.36781	−0.683905	−	0.729571i	$-0.739720\pi$
$887$	−40.7368	−1.36781	−0.683905	+	0.729571i	$0.739720\pi$
$888$	0	0
$889$	−9.13023	−0.306218
$890$	0	0
$891$	5.67660	0.190173
$892$	0	0
$893$	−35.0152	−1.17174
$894$	0	0
$895$	−33.7071	−1.12670
$896$	0	0
$897$	−13.3386	−0.445364
$898$	0	0
$899$	−39.9225	−1.33149
$900$	0	0
$901$	38.8489	1.29425
$902$	0	0
$903$	3.77704	0.125692
$904$	0	0
$905$	−36.6323	−1.21770
$906$	0	0
$907$	−46.9140	−1.55775	−0.778877	−	0.627177i	$-0.784210\pi$
$907$	−46.9140	−1.55775	−0.778877	+	0.627177i	$0.784210\pi$
$908$	0	0
$909$	4.76663	0.158099
$910$	0	0
$911$	−29.4640	−0.976187	−0.488094	−	0.872791i	$-0.662308\pi$
$911$	−29.4640	−0.976187	−0.488094	+	0.872791i	$0.662308\pi$
$912$	0	0
$913$	6.98062	0.231025
$914$	0	0
$915$	6.43281	0.212662
$916$	0	0
$917$	−10.4972	−0.346648
$918$	0	0
$919$	41.5284	1.36990	0.684948	−	0.728592i	$-0.259825\pi$
$919$	41.5284	1.36990	0.684948	+	0.728592i	$0.259825\pi$
$920$	0	0
$921$	26.1455	0.861522
$922$	0	0
$923$	−10.7860	−0.355026
$924$	0	0
$925$	−3.20359	−0.105333
$926$	0	0
$927$	−4.55678	−0.149664
$928$	0	0
$929$	−39.6281	−1.30016	−0.650078	−	0.759868i	$-0.725264\pi$
$929$	−39.6281	−1.30016	−0.650078	+	0.759868i	$0.725264\pi$
$930$	0	0
$931$	−4.18421	−0.137132
$932$	0	0
$933$	−19.2609	−0.630575
$934$	0	0
$935$	30.0602	0.983075
$936$	0	0
$937$	3.92665	0.128278	0.0641390	−	0.997941i	$-0.479570\pi$
$937$	3.92665	0.128278	0.0641390	+	0.997941i	$0.479570\pi$
$938$	0	0
$939$	−34.3234	−1.12010
$940$	0	0
$941$	10.5180	0.342878	0.171439	−	0.985195i	$-0.445158\pi$
$941$	10.5180	0.342878	0.171439	+	0.985195i	$0.445158\pi$
$942$	0	0
$943$	−100.457	−3.27133
$944$	0	0
$945$	21.8012	0.709194
$946$	0	0
$947$	−41.8407	−1.35964	−0.679819	−	0.733380i	$-0.737942\pi$
$947$	−41.8407	−1.35964	−0.679819	+	0.733380i	$0.737942\pi$
$948$	0	0
$949$	1.44322	0.0468490
$950$	0	0
$951$	−5.37738	−0.174374
$952$	0	0
$953$	−17.5415	−0.568226	−0.284113	−	0.958791i	$-0.591699\pi$
$953$	−17.5415	−0.568226	−0.284113	+	0.958791i	$0.591699\pi$
$954$	0	0
$955$	−53.1503	−1.71990
$956$	0	0
$957$	−7.01523	−0.226770
$958$	0	0
$959$	19.6170	0.633467
$960$	0	0
$961$	38.2791	1.23481
$962$	0	0
$963$	−12.9986	−0.418872
$964$	0	0
$965$	49.4231	1.59098
$966$	0	0
$967$	29.1857	0.938548	0.469274	−	0.883053i	$-0.344516\pi$
$967$	29.1857	0.938548	0.469274	+	0.883053i	$0.344516\pi$
$968$	0	0
$969$	47.6489	1.53070
$970$	0	0
$971$	−3.57952	−0.114872	−0.0574361	−	0.998349i	$-0.518293\pi$
$971$	−3.57952	−0.114872	−0.0574361	+	0.998349i	$0.518293\pi$
$972$	0	0
$973$	−8.64681	−0.277204
$974$	0	0
$975$	14.4882	0.463995
$976$	0	0
$977$	0.709750	0.0227069	0.0113535	−	0.999936i	$-0.496386\pi$
$977$	0.709750	0.0227069	0.0113535	+	0.999936i	$0.496386\pi$
$978$	0	0
$979$	2.90582	0.0928705
$980$	0	0
$981$	−0.0810691	−0.00258834
$982$	0	0
$983$	−5.60739	−0.178848	−0.0894240	−	0.995994i	$-0.528503\pi$
$983$	−5.60739	−0.178848	−0.0894240	+	0.995994i	$0.528503\pi$
$984$	0	0
$985$	−105.474	−3.36069
$986$	0	0
$987$	−12.2396	−0.389592
$988$	0	0
$989$	23.5512	0.748884
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−11.4972	−0.364853
$994$	0	0
$995$	−39.2389	−1.24396
$996$	0	0
$997$	57.1190	1.80898	0.904489	−	0.426497i	$-0.140252\pi$
$997$	57.1190	1.80898	0.904489	+	0.426497i	$0.140252\pi$
$998$	0	0
$999$	1.82620	0.0577785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.i.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.i.1.2	✓	3	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46260 q^{3} +3.86081 q^{5} -1.00000 q^{7} -0.860806 q^{9} +O(q^{10})\)	\(q+1.46260 q^{3} +3.86081 q^{5} -1.00000 q^{7} -0.860806 q^{9} -1.00000 q^{11} +1.00000 q^{13} +5.64681 q^{15} -7.78600 q^{17} -4.18421 q^{19} -1.46260 q^{21} -9.11982 q^{23} +9.90582 q^{25} -5.64681 q^{27} +4.79641 q^{29} -8.32340 q^{31} -1.46260 q^{33} -3.86081 q^{35} -0.323404 q^{37} +1.46260 q^{39} +11.0152 q^{41} -2.58242 q^{43} -3.32340 q^{45} +8.36842 q^{47} +1.00000 q^{49} -11.3878 q^{51} -4.98959 q^{53} -3.86081 q^{55} -6.11982 q^{57} -4.97021 q^{59} +1.13919 q^{61} +0.860806 q^{63} +3.86081 q^{65} +1.18421 q^{67} -13.3386 q^{69} -10.7860 q^{71} +1.44322 q^{73} +14.4882 q^{75} +1.00000 q^{77} -2.33382 q^{79} -5.67660 q^{81} -6.98062 q^{83} -30.0602 q^{85} +7.01523 q^{87} -2.90582 q^{89} -1.00000 q^{91} -12.1738 q^{93} -16.1544 q^{95} -14.5422 q^{97} +0.860806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} - 13 q^{17} + q^{19} - 2 q^{21} - 13 q^{23} + 5 q^{25} - q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} + 7 q^{37} + 2 q^{39} - 10 q^{41} + 9 q^{43} - 2 q^{45} - 2 q^{47} + 3 q^{49} - 27 q^{51} - 11 q^{53} - 6 q^{55} - 4 q^{57} + 9 q^{59} + 9 q^{61} - 3 q^{63} + 6 q^{65} - 10 q^{67} + 11 q^{69} - 22 q^{71} - 18 q^{73} + 2 q^{75} + 3 q^{77} - 3 q^{79} - 25 q^{81} - q^{83} - 28 q^{85} - 22 q^{87} + 16 q^{89} - 3 q^{91} - 19 q^{93} - 11 q^{95} + q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 - 13 * q^17 + q^19 - 2 * q^21 - 13 * q^23 + 5 * q^25 - q^27 + 8 * q^29 - 17 * q^31 - 2 * q^33 - 6 * q^35 + 7 * q^37 + 2 * q^39 - 10 * q^41 + 9 * q^43 - 2 * q^45 - 2 * q^47 + 3 * q^49 - 27 * q^51 - 11 * q^53 - 6 * q^55 - 4 * q^57 + 9 * q^59 + 9 * q^61 - 3 * q^63 + 6 * q^65 - 10 * q^67 + 11 * q^69 - 22 * q^71 - 18 * q^73 + 2 * q^75 + 3 * q^77 - 3 * q^79 - 25 * q^81 - q^83 - 28 * q^85 - 22 * q^87 + 16 * q^89 - 3 * q^91 - 19 * q^93 - 11 * q^95 + q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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