LMFDB - Embedded newform 8004.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 $ x^9 - x^8 - 17x^7 + 4x^6 + 75x^5 + x^4 - 118x^3 - 26x^2 + 60x + 24
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			1.3
Root			$1.51482$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.51482 q^{5} -4.32415 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.51482 q^{5} -4.32415 q^{7} +1.00000 q^{9} +1.39997 q^{11} +0.610599 q^{13} -1.51482 q^{15} +3.34323 q^{17} -3.06893 q^{19} -4.32415 q^{21} -1.00000 q^{23} -2.70531 q^{25} +1.00000 q^{27} -1.00000 q^{29} +6.12118 q^{31} +1.39997 q^{33} +6.55032 q^{35} +9.13429 q^{37} +0.610599 q^{39} -4.45627 q^{41} +9.15076 q^{43} -1.51482 q^{45} +0.760449 q^{47} +11.6983 q^{49} +3.34323 q^{51} +2.50418 q^{53} -2.12071 q^{55} -3.06893 q^{57} +4.37835 q^{59} +5.56134 q^{61} -4.32415 q^{63} -0.924949 q^{65} -9.91868 q^{67} -1.00000 q^{69} -0.00974853 q^{71} -10.2324 q^{73} -2.70531 q^{75} -6.05367 q^{77} -7.12146 q^{79} +1.00000 q^{81} +1.88855 q^{83} -5.06440 q^{85} -1.00000 q^{87} -14.4499 q^{89} -2.64032 q^{91} +6.12118 q^{93} +4.64889 q^{95} -10.5784 q^{97} +1.39997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * 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
$5$	−1.51482	−0.677450	−0.338725	−	0.940885i	$-0.609996\pi$
$5$	−1.51482	−0.677450	−0.338725	+	0.940885i	$0.609996\pi$
$6$	0	0
$7$	−4.32415	−1.63437	−0.817187	−	0.576372i	$-0.804468\pi$
$7$	−4.32415	−1.63437	−0.817187	+	0.576372i	$0.804468\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.39997	0.422106	0.211053	−	0.977475i	$-0.432311\pi$
$11$	1.39997	0.422106	0.211053	+	0.977475i	$0.432311\pi$
$12$	0	0
$13$	0.610599	0.169350	0.0846748	−	0.996409i	$-0.473015\pi$
$13$	0.610599	0.169350	0.0846748	+	0.996409i	$0.473015\pi$
$14$	0	0
$15$	−1.51482	−0.391126
$16$	0	0
$17$	3.34323	0.810852	0.405426	−	0.914128i	$-0.367123\pi$
$17$	3.34323	0.810852	0.405426	+	0.914128i	$0.367123\pi$
$18$	0	0
$19$	−3.06893	−0.704061	−0.352031	−	0.935989i	$-0.614509\pi$
$19$	−3.06893	−0.704061	−0.352031	+	0.935989i	$0.614509\pi$
$20$	0	0
$21$	−4.32415	−0.943606
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.70531	−0.541062
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.12118	1.09940	0.549698	−	0.835363i	$-0.314743\pi$
$31$	6.12118	1.09940	0.549698	+	0.835363i	$0.314743\pi$
$32$	0	0
$33$	1.39997	0.243703
$34$	0	0
$35$	6.55032	1.10721
$36$	0	0
$37$	9.13429	1.50167	0.750834	−	0.660491i	$-0.229652\pi$
$37$	9.13429	1.50167	0.750834	+	0.660491i	$0.229652\pi$
$38$	0	0
$39$	0.610599	0.0977740
$40$	0	0
$41$	−4.45627	−0.695953	−0.347976	−	0.937503i	$-0.613131\pi$
$41$	−4.45627	−0.695953	−0.347976	+	0.937503i	$0.613131\pi$
$42$	0	0
$43$	9.15076	1.39548	0.697739	−	0.716352i	$-0.254189\pi$
$43$	9.15076	1.39548	0.697739	+	0.716352i	$0.254189\pi$
$44$	0	0
$45$	−1.51482	−0.225817
$46$	0	0
$47$	0.760449	0.110923	0.0554614	−	0.998461i	$-0.482337\pi$
$47$	0.760449	0.110923	0.0554614	+	0.998461i	$0.482337\pi$
$48$	0	0
$49$	11.6983	1.67118
$50$	0	0
$51$	3.34323	0.468146
$52$	0	0
$53$	2.50418	0.343976	0.171988	−	0.985099i	$-0.444981\pi$
$53$	2.50418	0.343976	0.171988	+	0.985099i	$0.444981\pi$
$54$	0	0
$55$	−2.12071	−0.285956
$56$	0	0
$57$	−3.06893	−0.406490
$58$	0	0
$59$	4.37835	0.570013	0.285006	−	0.958526i	$-0.408004\pi$
$59$	4.37835	0.570013	0.285006	+	0.958526i	$0.408004\pi$
$60$	0	0
$61$	5.56134	0.712056	0.356028	−	0.934475i	$-0.384131\pi$
$61$	5.56134	0.712056	0.356028	+	0.934475i	$0.384131\pi$
$62$	0	0
$63$	−4.32415	−0.544791
$64$	0	0
$65$	−0.924949	−0.114726
$66$	0	0
$67$	−9.91868	−1.21176	−0.605880	−	0.795556i	$-0.707179\pi$
$67$	−9.91868	−1.21176	−0.605880	+	0.795556i	$0.707179\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−0.00974853	−0.00115694	−0.000578469	−	1.00000i	$-0.500184\pi$
$71$	−0.00974853	−0.00115694	−0.000578469		1.00000i	$0.500184\pi$
$72$	0	0
$73$	−10.2324	−1.19762	−0.598808	−	0.800893i	$-0.704359\pi$
$73$	−10.2324	−1.19762	−0.598808	+	0.800893i	$0.704359\pi$
$74$	0	0
$75$	−2.70531	−0.312382
$76$	0	0
$77$	−6.05367	−0.689880
$78$	0	0
$79$	−7.12146	−0.801227	−0.400613	−	0.916247i	$-0.631203\pi$
$79$	−7.12146	−0.801227	−0.400613	+	0.916247i	$0.631203\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.88855	0.207295	0.103648	−	0.994614i	$-0.466949\pi$
$83$	1.88855	0.207295	0.103648	+	0.994614i	$0.466949\pi$
$84$	0	0
$85$	−5.06440	−0.549312
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−14.4499	−1.53168	−0.765841	−	0.643031i	$-0.777677\pi$
$89$	−14.4499	−1.53168	−0.765841	+	0.643031i	$0.777677\pi$
$90$	0	0
$91$	−2.64032	−0.276781
$92$	0	0
$93$	6.12118	0.634737
$94$	0	0
$95$	4.64889	0.476966
$96$	0	0
$97$	−10.5784	−1.07407	−0.537037	−	0.843559i	$-0.680456\pi$
$97$	−10.5784	−1.07407	−0.537037	+	0.843559i	$0.680456\pi$
$98$	0	0
$99$	1.39997	0.140702
$100$	0	0
$101$	−11.3385	−1.12822	−0.564110	−	0.825699i	$-0.690781\pi$
$101$	−11.3385	−1.12822	−0.564110	+	0.825699i	$0.690781\pi$
$102$	0	0
$103$	−8.30593	−0.818407	−0.409204	−	0.912443i	$-0.634193\pi$
$103$	−8.30593	−0.818407	−0.409204	+	0.912443i	$0.634193\pi$
$104$	0	0
$105$	6.55032	0.639246
$106$	0	0
$107$	−13.7397	−1.32827	−0.664135	−	0.747612i	$-0.731200\pi$
$107$	−13.7397	−1.32827	−0.664135	+	0.747612i	$0.731200\pi$
$108$	0	0
$109$	7.25015	0.694439	0.347219	−	0.937784i	$-0.387126\pi$
$109$	7.25015	0.694439	0.347219	+	0.937784i	$0.387126\pi$
$110$	0	0
$111$	9.13429	0.866988
$112$	0	0
$113$	−13.6977	−1.28857	−0.644287	−	0.764784i	$-0.722846\pi$
$113$	−13.6977	−1.28857	−0.644287	+	0.764784i	$0.722846\pi$
$114$	0	0
$115$	1.51482	0.141258
$116$	0	0
$117$	0.610599	0.0564499
$118$	0	0
$119$	−14.4566	−1.32524
$120$	0	0
$121$	−9.04009	−0.821826
$122$	0	0
$123$	−4.45627	−0.401809
$124$	0	0
$125$	11.6722	1.04399
$126$	0	0
$127$	12.0409	1.06846	0.534228	−	0.845341i	$-0.320603\pi$
$127$	12.0409	1.06846	0.534228	+	0.845341i	$0.320603\pi$
$128$	0	0
$129$	9.15076	0.805680
$130$	0	0
$131$	4.38953	0.383515	0.191758	−	0.981442i	$-0.438581\pi$
$131$	4.38953	0.383515	0.191758	+	0.981442i	$0.438581\pi$
$132$	0	0
$133$	13.2705	1.15070
$134$	0	0
$135$	−1.51482	−0.130375
$136$	0	0
$137$	−21.9510	−1.87540	−0.937701	−	0.347444i	$-0.887050\pi$
$137$	−21.9510	−1.87540	−0.937701	+	0.347444i	$0.887050\pi$
$138$	0	0
$139$	−10.4678	−0.887867	−0.443933	−	0.896060i	$-0.646417\pi$
$139$	−10.4678	−0.887867	−0.443933	+	0.896060i	$0.646417\pi$
$140$	0	0
$141$	0.760449	0.0640413
$142$	0	0
$143$	0.854819	0.0714836
$144$	0	0
$145$	1.51482	0.125799
$146$	0	0
$147$	11.6983	0.964856
$148$	0	0
$149$	−8.19281	−0.671181	−0.335591	−	0.942008i	$-0.608936\pi$
$149$	−8.19281	−0.671181	−0.335591	+	0.942008i	$0.608936\pi$
$150$	0	0
$151$	2.27282	0.184960	0.0924799	−	0.995715i	$-0.470521\pi$
$151$	2.27282	0.184960	0.0924799	+	0.995715i	$0.470521\pi$
$152$	0	0
$153$	3.34323	0.270284
$154$	0	0
$155$	−9.27251	−0.744786
$156$	0	0
$157$	−3.42936	−0.273693	−0.136846	−	0.990592i	$-0.543697\pi$
$157$	−3.42936	−0.273693	−0.136846	+	0.990592i	$0.543697\pi$
$158$	0	0
$159$	2.50418	0.198595
$160$	0	0
$161$	4.32415	0.340791
$162$	0	0
$163$	12.2940	0.962938	0.481469	−	0.876463i	$-0.340103\pi$
$163$	12.2940	0.962938	0.481469	+	0.876463i	$0.340103\pi$
$164$	0	0
$165$	−2.12071	−0.165097
$166$	0	0
$167$	13.3556	1.03349	0.516743	−	0.856141i	$-0.327144\pi$
$167$	13.3556	1.03349	0.516743	+	0.856141i	$0.327144\pi$
$168$	0	0
$169$	−12.6272	−0.971321
$170$	0	0
$171$	−3.06893	−0.234687
$172$	0	0
$173$	3.64447	0.277084	0.138542	−	0.990357i	$-0.455758\pi$
$173$	3.64447	0.277084	0.138542	+	0.990357i	$0.455758\pi$
$174$	0	0
$175$	11.6982	0.884297
$176$	0	0
$177$	4.37835	0.329097
$178$	0	0
$179$	−10.8611	−0.811795	−0.405898	−	0.913919i	$-0.633041\pi$
$179$	−10.8611	−0.811795	−0.405898	+	0.913919i	$0.633041\pi$
$180$	0	0
$181$	−13.3888	−0.995183	−0.497592	−	0.867411i	$-0.665782\pi$
$181$	−13.3888	−0.995183	−0.497592	+	0.867411i	$0.665782\pi$
$182$	0	0
$183$	5.56134	0.411106
$184$	0	0
$185$	−13.8368	−1.01730
$186$	0	0
$187$	4.68042	0.342266
$188$	0	0
$189$	−4.32415	−0.314535
$190$	0	0
$191$	14.7362	1.06627	0.533135	−	0.846030i	$-0.321014\pi$
$191$	14.7362	1.06627	0.533135	+	0.846030i	$0.321014\pi$
$192$	0	0
$193$	14.0677	1.01262	0.506309	−	0.862352i	$-0.331010\pi$
$193$	14.0677	1.01262	0.506309	+	0.862352i	$0.331010\pi$
$194$	0	0
$195$	−0.924949	−0.0662370
$196$	0	0
$197$	−0.713026	−0.0508010	−0.0254005	−	0.999677i	$-0.508086\pi$
$197$	−0.713026	−0.0508010	−0.0254005	+	0.999677i	$0.508086\pi$
$198$	0	0
$199$	0.964843	0.0683959	0.0341979	−	0.999415i	$-0.489112\pi$
$199$	0.964843	0.0683959	0.0341979	+	0.999415i	$0.489112\pi$
$200$	0	0
$201$	−9.91868	−0.699610
$202$	0	0
$203$	4.32415	0.303496
$204$	0	0
$205$	6.75047	0.471473
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.29641	−0.297189
$210$	0	0
$211$	−7.80937	−0.537619	−0.268810	−	0.963193i	$-0.586630\pi$
$211$	−7.80937	−0.537619	−0.268810	+	0.963193i	$0.586630\pi$
$212$	0	0
$213$	−0.00974853	−0.000667958 0
$214$	0	0
$215$	−13.8618	−0.945367
$216$	0	0
$217$	−26.4689	−1.79683
$218$	0	0
$219$	−10.2324	−0.691444
$220$	0	0
$221$	2.04137	0.137317
$222$	0	0
$223$	−12.8260	−0.858892	−0.429446	−	0.903093i	$-0.641291\pi$
$223$	−12.8260	−0.858892	−0.429446	+	0.903093i	$0.641291\pi$
$224$	0	0
$225$	−2.70531	−0.180354
$226$	0	0
$227$	11.9177	0.791007	0.395503	−	0.918465i	$-0.370570\pi$
$227$	11.9177	0.791007	0.395503	+	0.918465i	$0.370570\pi$
$228$	0	0
$229$	−7.88008	−0.520730	−0.260365	−	0.965510i	$-0.583843\pi$
$229$	−7.88008	−0.520730	−0.260365	+	0.965510i	$0.583843\pi$
$230$	0	0
$231$	−6.05367	−0.398302
$232$	0	0
$233$	−10.2646	−0.672455	−0.336227	−	0.941781i	$-0.609151\pi$
$233$	−10.2646	−0.672455	−0.336227	+	0.941781i	$0.609151\pi$
$234$	0	0
$235$	−1.15195	−0.0751447
$236$	0	0
$237$	−7.12146	−0.462589
$238$	0	0
$239$	23.9467	1.54898	0.774492	−	0.632584i	$-0.218006\pi$
$239$	23.9467	1.54898	0.774492	+	0.632584i	$0.218006\pi$
$240$	0	0
$241$	12.0899	0.778776	0.389388	−	0.921074i	$-0.372687\pi$
$241$	12.0899	0.778776	0.389388	+	0.921074i	$0.372687\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−17.7208	−1.13214
$246$	0	0
$247$	−1.87389	−0.119232
$248$	0	0
$249$	1.88855	0.119682
$250$	0	0
$251$	1.67712	0.105859	0.0529294	−	0.998598i	$-0.483144\pi$
$251$	1.67712	0.105859	0.0529294	+	0.998598i	$0.483144\pi$
$252$	0	0
$253$	−1.39997	−0.0880153
$254$	0	0
$255$	−5.06440	−0.317145
$256$	0	0
$257$	5.22319	0.325813	0.162907	−	0.986641i	$-0.447913\pi$
$257$	5.22319	0.325813	0.162907	+	0.986641i	$0.447913\pi$
$258$	0	0
$259$	−39.4980	−2.45429
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−7.02213	−0.433003	−0.216502	−	0.976282i	$-0.569465\pi$
$263$	−7.02213	−0.433003	−0.216502	+	0.976282i	$0.569465\pi$
$264$	0	0
$265$	−3.79340	−0.233027
$266$	0	0
$267$	−14.4499	−0.884316
$268$	0	0
$269$	31.5149	1.92150	0.960749	−	0.277419i	$-0.0894790\pi$
$269$	31.5149	1.92150	0.960749	+	0.277419i	$0.0894790\pi$
$270$	0	0
$271$	−16.2066	−0.984481	−0.492240	−	0.870459i	$-0.663822\pi$
$271$	−16.2066	−0.984481	−0.492240	+	0.870459i	$0.663822\pi$
$272$	0	0
$273$	−2.64032	−0.159799
$274$	0	0
$275$	−3.78735	−0.228386
$276$	0	0
$277$	−1.33764	−0.0803710	−0.0401855	−	0.999192i	$-0.512795\pi$
$277$	−1.33764	−0.0803710	−0.0401855	+	0.999192i	$0.512795\pi$
$278$	0	0
$279$	6.12118	0.366465
$280$	0	0
$281$	−22.6686	−1.35229	−0.676147	−	0.736767i	$-0.736351\pi$
$281$	−22.6686	−1.35229	−0.676147	+	0.736767i	$0.736351\pi$
$282$	0	0
$283$	−28.0802	−1.66920	−0.834598	−	0.550860i	$-0.814300\pi$
$283$	−28.0802	−1.66920	−0.834598	+	0.550860i	$0.814300\pi$
$284$	0	0
$285$	4.64889	0.275376
$286$	0	0
$287$	19.2696	1.13745
$288$	0	0
$289$	−5.82282	−0.342519
$290$	0	0
$291$	−10.5784	−0.620117
$292$	0	0
$293$	6.87386	0.401575	0.200788	−	0.979635i	$-0.435650\pi$
$293$	6.87386	0.401575	0.200788	+	0.979635i	$0.435650\pi$
$294$	0	0
$295$	−6.63243	−0.386155
$296$	0	0
$297$	1.39997	0.0812344
$298$	0	0
$299$	−0.610599	−0.0353118
$300$	0	0
$301$	−39.5693	−2.28073
$302$	0	0
$303$	−11.3385	−0.651379
$304$	0	0
$305$	−8.42445	−0.482383
$306$	0	0
$307$	15.1291	0.863465	0.431733	−	0.902002i	$-0.357902\pi$
$307$	15.1291	0.863465	0.431733	+	0.902002i	$0.357902\pi$
$308$	0	0
$309$	−8.30593	−0.472508
$310$	0	0
$311$	1.59290	0.0903249	0.0451625	−	0.998980i	$-0.485619\pi$
$311$	1.59290	0.0903249	0.0451625	+	0.998980i	$0.485619\pi$
$312$	0	0
$313$	−12.1429	−0.686355	−0.343178	−	0.939271i	$-0.611503\pi$
$313$	−12.1429	−0.686355	−0.343178	+	0.939271i	$0.611503\pi$
$314$	0	0
$315$	6.55032	0.369069
$316$	0	0
$317$	10.4253	0.585543	0.292772	−	0.956182i	$-0.405422\pi$
$317$	10.4253	0.585543	0.292772	+	0.956182i	$0.405422\pi$
$318$	0	0
$319$	−1.39997	−0.0783832
$320$	0	0
$321$	−13.7397	−0.766877
$322$	0	0
$323$	−10.2601	−0.570889
$324$	0	0
$325$	−1.65186	−0.0916286
$326$	0	0
$327$	7.25015	0.400934
$328$	0	0
$329$	−3.28829	−0.181289
$330$	0	0
$331$	15.3210	0.842118	0.421059	−	0.907033i	$-0.361659\pi$
$331$	15.3210	0.842118	0.421059	+	0.907033i	$0.361659\pi$
$332$	0	0
$333$	9.13429	0.500556
$334$	0	0
$335$	15.0251	0.820906
$336$	0	0
$337$	−24.6144	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$337$	−24.6144	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$338$	0	0
$339$	−13.6977	−0.743958
$340$	0	0
$341$	8.56946	0.464062
$342$	0	0
$343$	−20.3160	−1.09696
$344$	0	0
$345$	1.51482	0.0815554
$346$	0	0
$347$	−21.8266	−1.17171	−0.585855	−	0.810416i	$-0.699241\pi$
$347$	−21.8266	−1.17171	−0.585855	+	0.810416i	$0.699241\pi$
$348$	0	0
$349$	−22.9205	−1.22691	−0.613453	−	0.789731i	$-0.710220\pi$
$349$	−22.9205	−1.22691	−0.613453	+	0.789731i	$0.710220\pi$
$350$	0	0
$351$	0.610599	0.0325913
$352$	0	0
$353$	−6.70309	−0.356770	−0.178385	−	0.983961i	$-0.557087\pi$
$353$	−6.70309	−0.356770	−0.178385	+	0.983961i	$0.557087\pi$
$354$	0	0
$355$	0.0147673	0.000783767 0
$356$	0	0
$357$	−14.4566	−0.765125
$358$	0	0
$359$	6.73513	0.355466	0.177733	−	0.984079i	$-0.443124\pi$
$359$	6.73513	0.355466	0.177733	+	0.984079i	$0.443124\pi$
$360$	0	0
$361$	−9.58166	−0.504298
$362$	0	0
$363$	−9.04009	−0.474482
$364$	0	0
$365$	15.5003	0.811325
$366$	0	0
$367$	−7.40494	−0.386535	−0.193267	−	0.981146i	$-0.561908\pi$
$367$	−7.40494	−0.386535	−0.193267	+	0.981146i	$0.561908\pi$
$368$	0	0
$369$	−4.45627	−0.231984
$370$	0	0
$371$	−10.8285	−0.562186
$372$	0	0
$373$	−33.5982	−1.73965	−0.869825	−	0.493361i	$-0.835768\pi$
$373$	−33.5982	−1.73965	−0.869825	+	0.493361i	$0.835768\pi$
$374$	0	0
$375$	11.6722	0.602749
$376$	0	0
$377$	−0.610599	−0.0314474
$378$	0	0
$379$	10.0795	0.517749	0.258874	−	0.965911i	$-0.416648\pi$
$379$	10.0795	0.517749	0.258874	+	0.965911i	$0.416648\pi$
$380$	0	0
$381$	12.0409	0.616873
$382$	0	0
$383$	9.02522	0.461167	0.230584	−	0.973053i	$-0.425936\pi$
$383$	9.02522	0.461167	0.230584	+	0.973053i	$0.425936\pi$
$384$	0	0
$385$	9.17025	0.467359
$386$	0	0
$387$	9.15076	0.465160
$388$	0	0
$389$	−9.75977	−0.494840	−0.247420	−	0.968908i	$-0.579583\pi$
$389$	−9.75977	−0.494840	−0.247420	+	0.968908i	$0.579583\pi$
$390$	0	0
$391$	−3.34323	−0.169074
$392$	0	0
$393$	4.38953	0.221423
$394$	0	0
$395$	10.7878	0.542791
$396$	0	0
$397$	−1.78778	−0.0897259	−0.0448630	−	0.998993i	$-0.514285\pi$
$397$	−1.78778	−0.0897259	−0.0448630	+	0.998993i	$0.514285\pi$
$398$	0	0
$399$	13.2705	0.664357
$400$	0	0
$401$	25.9298	1.29487	0.647436	−	0.762120i	$-0.275841\pi$
$401$	25.9298	1.29487	0.647436	+	0.762120i	$0.275841\pi$
$402$	0	0
$403$	3.73758	0.186182
$404$	0	0
$405$	−1.51482	−0.0752722
$406$	0	0
$407$	12.7877	0.633864
$408$	0	0
$409$	5.29378	0.261761	0.130880	−	0.991398i	$-0.458220\pi$
$409$	5.29378	0.261761	0.130880	+	0.991398i	$0.458220\pi$
$410$	0	0
$411$	−21.9510	−1.08276
$412$	0	0
$413$	−18.9326	−0.931614
$414$	0	0
$415$	−2.86082	−0.140432
$416$	0	0
$417$	−10.4678	−0.512610
$418$	0	0
$419$	−11.7896	−0.575958	−0.287979	−	0.957637i	$-0.592983\pi$
$419$	−11.7896	−0.575958	−0.287979	+	0.957637i	$0.592983\pi$
$420$	0	0
$421$	−2.84387	−0.138602	−0.0693009	−	0.997596i	$-0.522077\pi$
$421$	−2.84387	−0.138602	−0.0693009	+	0.997596i	$0.522077\pi$
$422$	0	0
$423$	0.760449	0.0369743
$424$	0	0
$425$	−9.04446	−0.438721
$426$	0	0
$427$	−24.0481	−1.16377
$428$	0	0
$429$	0.854819	0.0412711
$430$	0	0
$431$	−19.2247	−0.926022	−0.463011	−	0.886352i	$-0.653231\pi$
$431$	−19.2247	−0.926022	−0.463011	+	0.886352i	$0.653231\pi$
$432$	0	0
$433$	5.90838	0.283939	0.141969	−	0.989871i	$-0.454657\pi$
$433$	5.90838	0.283939	0.141969	+	0.989871i	$0.454657\pi$
$434$	0	0
$435$	1.51482	0.0726302
$436$	0	0
$437$	3.06893	0.146807
$438$	0	0
$439$	0.211291	0.0100844	0.00504219	−	0.999987i	$-0.498395\pi$
$439$	0.211291	0.0100844	0.00504219	+	0.999987i	$0.498395\pi$
$440$	0	0
$441$	11.6983	0.557060
$442$	0	0
$443$	−14.4229	−0.685253	−0.342626	−	0.939472i	$-0.611316\pi$
$443$	−14.4229	−0.685253	−0.342626	+	0.939472i	$0.611316\pi$
$444$	0	0
$445$	21.8890	1.03764
$446$	0	0
$447$	−8.19281	−0.387507
$448$	0	0
$449$	−14.7855	−0.697772	−0.348886	−	0.937165i	$-0.613440\pi$
$449$	−14.7855	−0.697772	−0.348886	+	0.937165i	$0.613440\pi$
$450$	0	0
$451$	−6.23864	−0.293766
$452$	0	0
$453$	2.27282	0.106787
$454$	0	0
$455$	3.99962	0.187505
$456$	0	0
$457$	3.17442	0.148493	0.0742467	−	0.997240i	$-0.476345\pi$
$457$	3.17442	0.148493	0.0742467	+	0.997240i	$0.476345\pi$
$458$	0	0
$459$	3.34323	0.156049
$460$	0	0
$461$	−9.38795	−0.437241	−0.218620	−	0.975810i	$-0.570156\pi$
$461$	−9.38795	−0.437241	−0.218620	+	0.975810i	$0.570156\pi$
$462$	0	0
$463$	−31.7277	−1.47451	−0.737256	−	0.675614i	$-0.763879\pi$
$463$	−31.7277	−1.47451	−0.737256	+	0.675614i	$0.763879\pi$
$464$	0	0
$465$	−9.27251	−0.430002
$466$	0	0
$467$	−13.1351	−0.607822	−0.303911	−	0.952700i	$-0.598293\pi$
$467$	−13.1351	−0.607822	−0.303911	+	0.952700i	$0.598293\pi$
$468$	0	0
$469$	42.8898	1.98047
$470$	0	0
$471$	−3.42936	−0.158017
$472$	0	0
$473$	12.8108	0.589041
$474$	0	0
$475$	8.30241	0.380941
$476$	0	0
$477$	2.50418	0.114659
$478$	0	0
$479$	−15.4197	−0.704543	−0.352272	−	0.935898i	$-0.614591\pi$
$479$	−15.4197	−0.704543	−0.352272	+	0.935898i	$0.614591\pi$
$480$	0	0
$481$	5.57738	0.254307
$482$	0	0
$483$	4.32415	0.196756
$484$	0	0
$485$	16.0244	0.727631
$486$	0	0
$487$	1.82235	0.0825785	0.0412893	−	0.999147i	$-0.486853\pi$
$487$	1.82235	0.0825785	0.0412893	+	0.999147i	$0.486853\pi$
$488$	0	0
$489$	12.2940	0.555953
$490$	0	0
$491$	28.0151	1.26430	0.632152	−	0.774844i	$-0.282172\pi$
$491$	28.0151	1.26430	0.632152	+	0.774844i	$0.282172\pi$
$492$	0	0
$493$	−3.34323	−0.150571
$494$	0	0
$495$	−2.12071	−0.0953187
$496$	0	0
$497$	0.0421541	0.00189087
$498$	0	0
$499$	−19.4469	−0.870564	−0.435282	−	0.900294i	$-0.643351\pi$
$499$	−19.4469	−0.870564	−0.435282	+	0.900294i	$0.643351\pi$
$500$	0	0
$501$	13.3556	0.596683
$502$	0	0
$503$	3.00700	0.134075	0.0670377	−	0.997750i	$-0.478645\pi$
$503$	3.00700	0.134075	0.0670377	+	0.997750i	$0.478645\pi$
$504$	0	0
$505$	17.1758	0.764313
$506$	0	0
$507$	−12.6272	−0.560792
$508$	0	0
$509$	9.89327	0.438512	0.219256	−	0.975667i	$-0.429637\pi$
$509$	9.89327	0.438512	0.219256	+	0.975667i	$0.429637\pi$
$510$	0	0
$511$	44.2466	1.95735
$512$	0	0
$513$	−3.06893	−0.135497
$514$	0	0
$515$	12.5820	0.554430
$516$	0	0
$517$	1.06460	0.0468213
$518$	0	0
$519$	3.64447	0.159974
$520$	0	0
$521$	17.0778	0.748193	0.374096	−	0.927390i	$-0.377953\pi$
$521$	17.0778	0.748193	0.374096	+	0.927390i	$0.377953\pi$
$522$	0	0
$523$	−19.7868	−0.865218	−0.432609	−	0.901582i	$-0.642407\pi$
$523$	−19.7868	−0.865218	−0.432609	+	0.901582i	$0.642407\pi$
$524$	0	0
$525$	11.6982	0.510549
$526$	0	0
$527$	20.4645	0.891448
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.37835	0.190004
$532$	0	0
$533$	−2.72099	−0.117859
$534$	0	0
$535$	20.8133	0.899837
$536$	0	0
$537$	−10.8611	−0.468690
$538$	0	0
$539$	16.3772	0.705416
$540$	0	0
$541$	26.6277	1.14481	0.572407	−	0.819969i	$-0.306010\pi$
$541$	26.6277	1.14481	0.572407	+	0.819969i	$0.306010\pi$
$542$	0	0
$543$	−13.3888	−0.574569
$544$	0	0
$545$	−10.9827	−0.470447
$546$	0	0
$547$	−28.1211	−1.20237	−0.601186	−	0.799109i	$-0.705305\pi$
$547$	−28.1211	−1.20237	−0.601186	+	0.799109i	$0.705305\pi$
$548$	0	0
$549$	5.56134	0.237352
$550$	0	0
$551$	3.06893	0.130741
$552$	0	0
$553$	30.7943	1.30950
$554$	0	0
$555$	−13.8368	−0.587341
$556$	0	0
$557$	−1.20662	−0.0511261	−0.0255631	−	0.999673i	$-0.508138\pi$
$557$	−1.20662	−0.0511261	−0.0255631	+	0.999673i	$0.508138\pi$
$558$	0	0
$559$	5.58744	0.236324
$560$	0	0
$561$	4.68042	0.197607
$562$	0	0
$563$	43.0144	1.81284	0.906421	−	0.422376i	$-0.138804\pi$
$563$	43.0144	1.81284	0.906421	+	0.422376i	$0.138804\pi$
$564$	0	0
$565$	20.7496	0.872944
$566$	0	0
$567$	−4.32415	−0.181597
$568$	0	0
$569$	−32.6766	−1.36988	−0.684938	−	0.728602i	$-0.740171\pi$
$569$	−32.6766	−1.36988	−0.684938	+	0.728602i	$0.740171\pi$
$570$	0	0
$571$	−14.8774	−0.622601	−0.311301	−	0.950311i	$-0.600765\pi$
$571$	−14.8774	−0.622601	−0.311301	+	0.950311i	$0.600765\pi$
$572$	0	0
$573$	14.7362	0.615612
$574$	0	0
$575$	2.70531	0.112819
$576$	0	0
$577$	−33.4421	−1.39221	−0.696107	−	0.717938i	$-0.745086\pi$
$577$	−33.4421	−1.39221	−0.696107	+	0.717938i	$0.745086\pi$
$578$	0	0
$579$	14.0677	0.584635
$580$	0	0
$581$	−8.16636	−0.338798
$582$	0	0
$583$	3.50578	0.145195
$584$	0	0
$585$	−0.924949	−0.0382420
$586$	0	0
$587$	12.0315	0.496594	0.248297	−	0.968684i	$-0.420129\pi$
$587$	12.0315	0.496594	0.248297	+	0.968684i	$0.420129\pi$
$588$	0	0
$589$	−18.7855	−0.774042
$590$	0	0
$591$	−0.713026	−0.0293300
$592$	0	0
$593$	42.7186	1.75424	0.877120	−	0.480270i	$-0.159461\pi$
$593$	42.7186	1.75424	0.877120	+	0.480270i	$0.159461\pi$
$594$	0	0
$595$	21.8992	0.897781
$596$	0	0
$597$	0.964843	0.0394884
$598$	0	0
$599$	−32.6667	−1.33473	−0.667363	−	0.744733i	$-0.732577\pi$
$599$	−32.6667	−1.33473	−0.667363	+	0.744733i	$0.732577\pi$
$600$	0	0
$601$	44.0472	1.79672	0.898361	−	0.439257i	$-0.144758\pi$
$601$	44.0472	1.79672	0.898361	+	0.439257i	$0.144758\pi$
$602$	0	0
$603$	−9.91868	−0.403920
$604$	0	0
$605$	13.6941	0.556746
$606$	0	0
$607$	−42.6350	−1.73050	−0.865250	−	0.501340i	$-0.832840\pi$
$607$	−42.6350	−1.73050	−0.865250	+	0.501340i	$0.832840\pi$
$608$	0	0
$609$	4.32415	0.175223
$610$	0	0
$611$	0.464329	0.0187847
$612$	0	0
$613$	16.2818	0.657615	0.328807	−	0.944397i	$-0.393353\pi$
$613$	16.2818	0.657615	0.328807	+	0.944397i	$0.393353\pi$
$614$	0	0
$615$	6.75047	0.272205
$616$	0	0
$617$	14.1096	0.568030	0.284015	−	0.958820i	$-0.408333\pi$
$617$	14.1096	0.568030	0.284015	+	0.958820i	$0.408333\pi$
$618$	0	0
$619$	4.45042	0.178877	0.0894387	−	0.995992i	$-0.471493\pi$
$619$	4.45042	0.178877	0.0894387	+	0.995992i	$0.471493\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	62.4833	2.50334
$624$	0	0
$625$	−4.15476	−0.166190
$626$	0	0
$627$	−4.29641	−0.171582
$628$	0	0
$629$	30.5380	1.21763
$630$	0	0
$631$	−7.22141	−0.287480	−0.143740	−	0.989615i	$-0.545913\pi$
$631$	−7.22141	−0.287480	−0.143740	+	0.989615i	$0.545913\pi$
$632$	0	0
$633$	−7.80937	−0.310394
$634$	0	0
$635$	−18.2398	−0.723825
$636$	0	0
$637$	7.14294	0.283014
$638$	0	0
$639$	−0.00974853	−0.000385646 0
$640$	0	0
$641$	−7.06071	−0.278881	−0.139441	−	0.990230i	$-0.544530\pi$
$641$	−7.06071	−0.278881	−0.139441	+	0.990230i	$0.544530\pi$
$642$	0	0
$643$	−2.71381	−0.107022	−0.0535110	−	0.998567i	$-0.517041\pi$
$643$	−2.71381	−0.107022	−0.0535110	+	0.998567i	$0.517041\pi$
$644$	0	0
$645$	−13.8618	−0.545808
$646$	0	0
$647$	19.9662	0.784952	0.392476	−	0.919762i	$-0.371619\pi$
$647$	19.9662	0.784952	0.392476	+	0.919762i	$0.371619\pi$
$648$	0	0
$649$	6.12956	0.240606
$650$	0	0
$651$	−26.4689	−1.03740
$652$	0	0
$653$	−7.09185	−0.277526	−0.138763	−	0.990326i	$-0.544313\pi$
$653$	−7.09185	−0.277526	−0.138763	+	0.990326i	$0.544313\pi$
$654$	0	0
$655$	−6.64937	−0.259812
$656$	0	0
$657$	−10.2324	−0.399205
$658$	0	0
$659$	−2.93125	−0.114185	−0.0570926	−	0.998369i	$-0.518183\pi$
$659$	−2.93125	−0.114185	−0.0570926	+	0.998369i	$0.518183\pi$
$660$	0	0
$661$	−9.01531	−0.350655	−0.175327	−	0.984510i	$-0.556098\pi$
$661$	−9.01531	−0.350655	−0.175327	+	0.984510i	$0.556098\pi$
$662$	0	0
$663$	2.04137	0.0792803
$664$	0	0
$665$	−20.1025	−0.779541
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−12.8260	−0.495882
$670$	0	0
$671$	7.78570	0.300564
$672$	0	0
$673$	−1.14770	−0.0442406	−0.0221203	−	0.999755i	$-0.507042\pi$
$673$	−1.14770	−0.0442406	−0.0221203	+	0.999755i	$0.507042\pi$
$674$	0	0
$675$	−2.70531	−0.104127
$676$	0	0
$677$	−35.8060	−1.37614	−0.688069	−	0.725646i	$-0.741541\pi$
$677$	−35.8060	−1.37614	−0.688069	+	0.725646i	$0.741541\pi$
$678$	0	0
$679$	45.7426	1.75544
$680$	0	0
$681$	11.9177	0.456688
$682$	0	0
$683$	20.5122	0.784879	0.392439	−	0.919778i	$-0.371631\pi$
$683$	20.5122	0.784879	0.392439	+	0.919778i	$0.371631\pi$
$684$	0	0
$685$	33.2519	1.27049
$686$	0	0
$687$	−7.88008	−0.300644
$688$	0	0
$689$	1.52905	0.0582522
$690$	0	0
$691$	−43.9604	−1.67233	−0.836166	−	0.548476i	$-0.815208\pi$
$691$	−43.9604	−1.67233	−0.836166	+	0.548476i	$0.815208\pi$
$692$	0	0
$693$	−6.05367	−0.229960
$694$	0	0
$695$	15.8569	0.601485
$696$	0	0
$697$	−14.8983	−0.564315
$698$	0	0
$699$	−10.2646	−0.388242
$700$	0	0
$701$	−47.3420	−1.78808	−0.894042	−	0.447983i	$-0.852143\pi$
$701$	−47.3420	−1.78808	−0.894042	+	0.447983i	$0.852143\pi$
$702$	0	0
$703$	−28.0325	−1.05727
$704$	0	0
$705$	−1.15195	−0.0433848
$706$	0	0
$707$	49.0293	1.84394
$708$	0	0
$709$	−16.2945	−0.611954	−0.305977	−	0.952039i	$-0.598983\pi$
$709$	−16.2945	−0.611954	−0.305977	+	0.952039i	$0.598983\pi$
$710$	0	0
$711$	−7.12146	−0.267076
$712$	0	0
$713$	−6.12118	−0.229240
$714$	0	0
$715$	−1.29490	−0.0484265
$716$	0	0
$717$	23.9467	0.894306
$718$	0	0
$719$	−17.0148	−0.634544	−0.317272	−	0.948335i	$-0.602767\pi$
$719$	−17.0148	−0.634544	−0.317272	+	0.948335i	$0.602767\pi$
$720$	0	0
$721$	35.9161	1.33758
$722$	0	0
$723$	12.0899	0.449627
$724$	0	0
$725$	2.70531	0.100473
$726$	0	0
$727$	−13.4533	−0.498954	−0.249477	−	0.968381i	$-0.580259\pi$
$727$	−13.4533	−0.498954	−0.249477	+	0.968381i	$0.580259\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.5931	1.13153
$732$	0	0
$733$	−15.3392	−0.566565	−0.283282	−	0.959037i	$-0.591423\pi$
$733$	−15.3392	−0.566565	−0.283282	+	0.959037i	$0.591423\pi$
$734$	0	0
$735$	−17.7208	−0.653642
$736$	0	0
$737$	−13.8858	−0.511492
$738$	0	0
$739$	36.6826	1.34939	0.674696	−	0.738096i	$-0.264275\pi$
$739$	36.6826	1.34939	0.674696	+	0.738096i	$0.264275\pi$
$740$	0	0
$741$	−1.87389	−0.0688389
$742$	0	0
$743$	6.05643	0.222189	0.111094	−	0.993810i	$-0.464564\pi$
$743$	6.05643	0.222189	0.111094	+	0.993810i	$0.464564\pi$
$744$	0	0
$745$	12.4107	0.454692
$746$	0	0
$747$	1.88855	0.0690984
$748$	0	0
$749$	59.4127	2.17089
$750$	0	0
$751$	−22.8782	−0.834838	−0.417419	−	0.908714i	$-0.637065\pi$
$751$	−22.8782	−0.834838	−0.417419	+	0.908714i	$0.637065\pi$
$752$	0	0
$753$	1.67712	0.0611176
$754$	0	0
$755$	−3.44293	−0.125301
$756$	0	0
$757$	−11.2411	−0.408564	−0.204282	−	0.978912i	$-0.565486\pi$
$757$	−11.2411	−0.408564	−0.204282	+	0.978912i	$0.565486\pi$
$758$	0	0
$759$	−1.39997	−0.0508156
$760$	0	0
$761$	−19.5532	−0.708805	−0.354402	−	0.935093i	$-0.615316\pi$
$761$	−19.5532	−0.708805	−0.354402	+	0.935093i	$0.615316\pi$
$762$	0	0
$763$	−31.3507	−1.13497
$764$	0	0
$765$	−5.06440	−0.183104
$766$	0	0
$767$	2.67342	0.0965314
$768$	0	0
$769$	−46.1221	−1.66321	−0.831604	−	0.555370i	$-0.812577\pi$
$769$	−46.1221	−1.66321	−0.831604	+	0.555370i	$0.812577\pi$
$770$	0	0
$771$	5.22319	0.188108
$772$	0	0
$773$	31.7190	1.14085	0.570426	−	0.821349i	$-0.306778\pi$
$773$	31.7190	1.14085	0.570426	+	0.821349i	$0.306778\pi$
$774$	0	0
$775$	−16.5597	−0.594841
$776$	0	0
$777$	−39.4980	−1.41698
$778$	0	0
$779$	13.6760	0.489993
$780$	0	0
$781$	−0.0136476	−0.000488351 0
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	5.19488	0.185413
$786$	0	0
$787$	52.5322	1.87257	0.936285	−	0.351241i	$-0.114240\pi$
$787$	52.5322	1.87257	0.936285	+	0.351241i	$0.114240\pi$
$788$	0	0
$789$	−7.02213	−0.249994
$790$	0	0
$791$	59.2310	2.10601
$792$	0	0
$793$	3.39575	0.120586
$794$	0	0
$795$	−3.79340	−0.134538
$796$	0	0
$797$	−14.7893	−0.523864	−0.261932	−	0.965086i	$-0.584360\pi$
$797$	−14.7893	−0.523864	−0.261932	+	0.965086i	$0.584360\pi$
$798$	0	0
$799$	2.54235	0.0899420
$800$	0	0
$801$	−14.4499	−0.510560
$802$	0	0
$803$	−14.3251	−0.505522
$804$	0	0
$805$	−6.55032	−0.230869
$806$	0	0
$807$	31.5149	1.10938
$808$	0	0
$809$	−6.11807	−0.215100	−0.107550	−	0.994200i	$-0.534301\pi$
$809$	−6.11807	−0.215100	−0.107550	+	0.994200i	$0.534301\pi$
$810$	0	0
$811$	−16.7462	−0.588038	−0.294019	−	0.955800i	$-0.594993\pi$
$811$	−16.7462	−0.588038	−0.294019	+	0.955800i	$0.594993\pi$
$812$	0	0
$813$	−16.2066	−0.568390
$814$	0	0
$815$	−18.6232	−0.652342
$816$	0	0
$817$	−28.0831	−0.982502
$818$	0	0
$819$	−2.64032	−0.0922602
$820$	0	0
$821$	29.9964	1.04688	0.523440	−	0.852062i	$-0.324648\pi$
$821$	29.9964	1.04688	0.523440	+	0.852062i	$0.324648\pi$
$822$	0	0
$823$	26.0084	0.906595	0.453297	−	0.891359i	$-0.350248\pi$
$823$	26.0084	0.906595	0.453297	+	0.891359i	$0.350248\pi$
$824$	0	0
$825$	−3.78735	−0.131859
$826$	0	0
$827$	20.2426	0.703905	0.351953	−	0.936018i	$-0.385518\pi$
$827$	20.2426	0.703905	0.351953	+	0.936018i	$0.385518\pi$
$828$	0	0
$829$	27.5333	0.956271	0.478136	−	0.878286i	$-0.341313\pi$
$829$	27.5333	0.956271	0.478136	+	0.878286i	$0.341313\pi$
$830$	0	0
$831$	−1.33764	−0.0464022
$832$	0	0
$833$	39.1099	1.35508
$834$	0	0
$835$	−20.2314	−0.700135
$836$	0	0
$837$	6.12118	0.211579
$838$	0	0
$839$	−18.9750	−0.655091	−0.327546	−	0.944835i	$-0.606222\pi$
$839$	−18.9750	−0.655091	−0.327546	+	0.944835i	$0.606222\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−22.6686	−0.780747
$844$	0	0
$845$	19.1279	0.658021
$846$	0	0
$847$	39.0907	1.34317
$848$	0	0
$849$	−28.0802	−0.963711
$850$	0	0
$851$	−9.13429	−0.313119
$852$	0	0
$853$	17.4516	0.597532	0.298766	−	0.954326i	$-0.403425\pi$
$853$	17.4516	0.597532	0.298766	+	0.954326i	$0.403425\pi$
$854$	0	0
$855$	4.64889	0.158989
$856$	0	0
$857$	−19.5316	−0.667185	−0.333593	−	0.942717i	$-0.608261\pi$
$857$	−19.5316	−0.667185	−0.333593	+	0.942717i	$0.608261\pi$
$858$	0	0
$859$	22.9877	0.784332	0.392166	−	0.919895i	$-0.371726\pi$
$859$	22.9877	0.784332	0.392166	+	0.919895i	$0.371726\pi$
$860$	0	0
$861$	19.2696	0.656706
$862$	0	0
$863$	9.78489	0.333082	0.166541	−	0.986035i	$-0.446740\pi$
$863$	9.78489	0.333082	0.166541	+	0.986035i	$0.446740\pi$
$864$	0	0
$865$	−5.52073	−0.187710
$866$	0	0
$867$	−5.82282	−0.197753
$868$	0	0
$869$	−9.96982	−0.338203
$870$	0	0
$871$	−6.05633	−0.205211
$872$	0	0
$873$	−10.5784	−0.358025
$874$	0	0
$875$	−50.4723	−1.70627
$876$	0	0
$877$	−15.4920	−0.523129	−0.261564	−	0.965186i	$-0.584238\pi$
$877$	−15.4920	−0.523129	−0.261564	+	0.965186i	$0.584238\pi$
$878$	0	0
$879$	6.87386	0.231849
$880$	0	0
$881$	26.6221	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$881$	26.6221	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$882$	0	0
$883$	38.6519	1.30074	0.650370	−	0.759617i	$-0.274614\pi$
$883$	38.6519	1.30074	0.650370	+	0.759617i	$0.274614\pi$
$884$	0	0
$885$	−6.63243	−0.222947
$886$	0	0
$887$	36.1945	1.21529	0.607647	−	0.794207i	$-0.292114\pi$
$887$	36.1945	1.21529	0.607647	+	0.794207i	$0.292114\pi$
$888$	0	0
$889$	−52.0665	−1.74626
$890$	0	0
$891$	1.39997	0.0469007
$892$	0	0
$893$	−2.33377	−0.0780965
$894$	0	0
$895$	16.4526	0.549951
$896$	0	0
$897$	−0.610599	−0.0203873
$898$	0	0
$899$	−6.12118	−0.204153
$900$	0	0
$901$	8.37206	0.278914
$902$	0	0
$903$	−39.5693	−1.31678
$904$	0	0
$905$	20.2817	0.674187
$906$	0	0
$907$	−13.1069	−0.435207	−0.217604	−	0.976037i	$-0.569824\pi$
$907$	−13.1069	−0.435207	−0.217604	+	0.976037i	$0.569824\pi$
$908$	0	0
$909$	−11.3385	−0.376074
$910$	0	0
$911$	15.2631	0.505690	0.252845	−	0.967507i	$-0.418634\pi$
$911$	15.2631	0.505690	0.252845	+	0.967507i	$0.418634\pi$
$912$	0	0
$913$	2.64391	0.0875006
$914$	0	0
$915$	−8.42445	−0.278504
$916$	0	0
$917$	−18.9810	−0.626808
$918$	0	0
$919$	12.5748	0.414805	0.207402	−	0.978256i	$-0.433499\pi$
$919$	12.5748	0.414805	0.207402	+	0.978256i	$0.433499\pi$
$920$	0	0
$921$	15.1291	0.498522
$922$	0	0
$923$	−0.00595244	−0.000195927 0
$924$	0	0
$925$	−24.7111	−0.812495
$926$	0	0
$927$	−8.30593	−0.272802
$928$	0	0
$929$	−14.1629	−0.464670	−0.232335	−	0.972636i	$-0.574637\pi$
$929$	−14.1629	−0.464670	−0.232335	+	0.972636i	$0.574637\pi$
$930$	0	0
$931$	−35.9011	−1.17661
$932$	0	0
$933$	1.59290	0.0521491
$934$	0	0
$935$	−7.09001	−0.231868
$936$	0	0
$937$	−26.4583	−0.864355	−0.432178	−	0.901788i	$-0.642255\pi$
$937$	−26.4583	−0.864355	−0.432178	+	0.901788i	$0.642255\pi$
$938$	0	0
$939$	−12.1429	−0.396267
$940$	0	0
$941$	11.4624	0.373665	0.186832	−	0.982392i	$-0.440178\pi$
$941$	11.4624	0.373665	0.186832	+	0.982392i	$0.440178\pi$
$942$	0	0
$943$	4.45627	0.145116
$944$	0	0
$945$	6.55032	0.213082
$946$	0	0
$947$	55.0073	1.78750	0.893748	−	0.448570i	$-0.148066\pi$
$947$	55.0073	1.78750	0.893748	+	0.448570i	$0.148066\pi$
$948$	0	0
$949$	−6.24791	−0.202816
$950$	0	0
$951$	10.4253	0.338064
$952$	0	0
$953$	−37.7834	−1.22392	−0.611962	−	0.790888i	$-0.709619\pi$
$953$	−37.7834	−1.22392	−0.611962	+	0.790888i	$0.709619\pi$
$954$	0	0
$955$	−22.3227	−0.722345
$956$	0	0
$957$	−1.39997	−0.0452546
$958$	0	0
$959$	94.9194	3.06511
$960$	0	0
$961$	6.46884	0.208672
$962$	0	0
$963$	−13.7397	−0.442757
$964$	0	0
$965$	−21.3101	−0.685998
$966$	0	0
$967$	−1.69603	−0.0545406	−0.0272703	−	0.999628i	$-0.508681\pi$
$967$	−1.69603	−0.0545406	−0.0272703	+	0.999628i	$0.508681\pi$
$968$	0	0
$969$	−10.2601	−0.329603
$970$	0	0
$971$	26.1674	0.839751	0.419876	−	0.907582i	$-0.362074\pi$
$971$	26.1674	0.839751	0.419876	+	0.907582i	$0.362074\pi$
$972$	0	0
$973$	45.2643	1.45111
$974$	0	0
$975$	−1.65186	−0.0529018
$976$	0	0
$977$	3.26058	0.104315	0.0521576	−	0.998639i	$-0.483390\pi$
$977$	3.26058	0.104315	0.0521576	+	0.998639i	$0.483390\pi$
$978$	0	0
$979$	−20.2293	−0.646533
$980$	0	0
$981$	7.25015	0.231480
$982$	0	0
$983$	50.4270	1.60837	0.804186	−	0.594378i	$-0.202602\pi$
$983$	50.4270	1.60837	0.804186	+	0.594378i	$0.202602\pi$
$984$	0	0
$985$	1.08011	0.0344151
$986$	0	0
$987$	−3.28829	−0.104668
$988$	0	0
$989$	−9.15076	−0.290977
$990$	0	0
$991$	−21.7399	−0.690589	−0.345295	−	0.938494i	$-0.612221\pi$
$991$	−21.7399	−0.690589	−0.345295	+	0.938494i	$0.612221\pi$
$992$	0	0
$993$	15.3210	0.486197
$994$	0	0
$995$	−1.46157	−0.0463348
$996$	0	0
$997$	43.6208	1.38149	0.690743	−	0.723100i	$-0.257284\pi$
$997$	43.6208	1.38149	0.690743	+	0.723100i	$0.257284\pi$
$998$	0	0
$999$	9.13429	0.288996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.3	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.51482 q^{5} -4.32415 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.51482 q^{5} -4.32415 q^{7} +1.00000 q^{9} +1.39997 q^{11} +0.610599 q^{13} -1.51482 q^{15} +3.34323 q^{17} -3.06893 q^{19} -4.32415 q^{21} -1.00000 q^{23} -2.70531 q^{25} +1.00000 q^{27} -1.00000 q^{29} +6.12118 q^{31} +1.39997 q^{33} +6.55032 q^{35} +9.13429 q^{37} +0.610599 q^{39} -4.45627 q^{41} +9.15076 q^{43} -1.51482 q^{45} +0.760449 q^{47} +11.6983 q^{49} +3.34323 q^{51} +2.50418 q^{53} -2.12071 q^{55} -3.06893 q^{57} +4.37835 q^{59} +5.56134 q^{61} -4.32415 q^{63} -0.924949 q^{65} -9.91868 q^{67} -1.00000 q^{69} -0.00974853 q^{71} -10.2324 q^{73} -2.70531 q^{75} -6.05367 q^{77} -7.12146 q^{79} +1.00000 q^{81} +1.88855 q^{83} -5.06440 q^{85} -1.00000 q^{87} -14.4499 q^{89} -2.64032 q^{91} +6.12118 q^{93} +4.64889 q^{95} -10.5784 q^{97} +1.39997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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