LMFDB - Embedded newform 8004.2.a.g.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.06377$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.06377 q^{5} +3.59303 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.06377 q^{5} +3.59303 q^{7} +1.00000 q^{9} +3.57105 q^{11} +1.11209 q^{13} +3.06377 q^{15} -5.35191 q^{17} -1.67014 q^{19} -3.59303 q^{21} +1.00000 q^{23} +4.38666 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.98746 q^{31} -3.57105 q^{33} -11.0082 q^{35} -4.76622 q^{37} -1.11209 q^{39} +3.76801 q^{41} -11.7805 q^{43} -3.06377 q^{45} -6.41650 q^{47} +5.90985 q^{49} +5.35191 q^{51} -0.261608 q^{53} -10.9408 q^{55} +1.67014 q^{57} -7.33982 q^{59} -3.81452 q^{61} +3.59303 q^{63} -3.40720 q^{65} -0.212601 q^{67} -1.00000 q^{69} -15.4656 q^{71} +15.2479 q^{73} -4.38666 q^{75} +12.8309 q^{77} +14.1479 q^{79} +1.00000 q^{81} +4.48671 q^{83} +16.3970 q^{85} +1.00000 q^{87} +7.32038 q^{89} +3.99579 q^{91} -5.98746 q^{93} +5.11693 q^{95} +5.66075 q^{97} +3.57105 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * 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
$5$	−3.06377	−1.37016	−0.685079	−	0.728469i	$-0.740232\pi$
$5$	−3.06377	−1.37016	−0.685079	+	0.728469i	$0.740232\pi$
$6$	0	0
$7$	3.59303	1.35804	0.679018	−	0.734121i	$-0.262406\pi$
$7$	3.59303	1.35804	0.679018	+	0.734121i	$0.262406\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.57105	1.07671	0.538355	−	0.842718i	$-0.319046\pi$
$11$	3.57105	1.07671	0.538355	+	0.842718i	$0.319046\pi$
$12$	0	0
$13$	1.11209	0.308440	0.154220	−	0.988037i	$-0.450714\pi$
$13$	1.11209	0.308440	0.154220	+	0.988037i	$0.450714\pi$
$14$	0	0
$15$	3.06377	0.791061
$16$	0	0
$17$	−5.35191	−1.29803	−0.649014	−	0.760776i	$-0.724818\pi$
$17$	−5.35191	−1.29803	−0.649014	+	0.760776i	$0.724818\pi$
$18$	0	0
$19$	−1.67014	−0.383157	−0.191579	−	0.981477i	$-0.561361\pi$
$19$	−1.67014	−0.383157	−0.191579	+	0.981477i	$0.561361\pi$
$20$	0	0
$21$	−3.59303	−0.784063
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.38666	0.877332
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.98746	1.07538	0.537690	−	0.843143i	$-0.319297\pi$
$31$	5.98746	1.07538	0.537690	+	0.843143i	$0.319297\pi$
$32$	0	0
$33$	−3.57105	−0.621639
$34$	0	0
$35$	−11.0082	−1.86072
$36$	0	0
$37$	−4.76622	−0.783562	−0.391781	−	0.920059i	$-0.628141\pi$
$37$	−4.76622	−0.783562	−0.391781	+	0.920059i	$0.628141\pi$
$38$	0	0
$39$	−1.11209	−0.178078
$40$	0	0
$41$	3.76801	0.588464	0.294232	−	0.955734i	$-0.404936\pi$
$41$	3.76801	0.588464	0.294232	+	0.955734i	$0.404936\pi$
$42$	0	0
$43$	−11.7805	−1.79651	−0.898256	−	0.439473i	$-0.855165\pi$
$43$	−11.7805	−1.79651	−0.898256	+	0.439473i	$0.855165\pi$
$44$	0	0
$45$	−3.06377	−0.456719
$46$	0	0
$47$	−6.41650	−0.935943	−0.467971	−	0.883744i	$-0.655015\pi$
$47$	−6.41650	−0.935943	−0.467971	+	0.883744i	$0.655015\pi$
$48$	0	0
$49$	5.90985	0.844264
$50$	0	0
$51$	5.35191	0.749417
$52$	0	0
$53$	−0.261608	−0.0359346	−0.0179673	−	0.999839i	$-0.505719\pi$
$53$	−0.261608	−0.0359346	−0.0179673	+	0.999839i	$0.505719\pi$
$54$	0	0
$55$	−10.9408	−1.47526
$56$	0	0
$57$	1.67014	0.221216
$58$	0	0
$59$	−7.33982	−0.955563	−0.477781	−	0.878479i	$-0.658559\pi$
$59$	−7.33982	−0.955563	−0.477781	+	0.878479i	$0.658559\pi$
$60$	0	0
$61$	−3.81452	−0.488399	−0.244199	−	0.969725i	$-0.578525\pi$
$61$	−3.81452	−0.488399	−0.244199	+	0.969725i	$0.578525\pi$
$62$	0	0
$63$	3.59303	0.452679
$64$	0	0
$65$	−3.40720	−0.422611
$66$	0	0
$67$	−0.212601	−0.0259734	−0.0129867	−	0.999916i	$-0.504134\pi$
$67$	−0.212601	−0.0259734	−0.0129867	+	0.999916i	$0.504134\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−15.4656	−1.83542	−0.917712	−	0.397245i	$-0.869966\pi$
$71$	−15.4656	−1.83542	−0.917712	+	0.397245i	$0.869966\pi$
$72$	0	0
$73$	15.2479	1.78464	0.892318	−	0.451408i	$-0.149078\pi$
$73$	15.2479	1.78464	0.892318	+	0.451408i	$0.149078\pi$
$74$	0	0
$75$	−4.38666	−0.506528
$76$	0	0
$77$	12.8309	1.46221
$78$	0	0
$79$	14.1479	1.59176	0.795879	−	0.605455i	$-0.207009\pi$
$79$	14.1479	1.59176	0.795879	+	0.605455i	$0.207009\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.48671	0.492481	0.246240	−	0.969209i	$-0.420805\pi$
$83$	4.48671	0.492481	0.246240	+	0.969209i	$0.420805\pi$
$84$	0	0
$85$	16.3970	1.77850
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	7.32038	0.775958	0.387979	−	0.921668i	$-0.373173\pi$
$89$	7.32038	0.775958	0.387979	+	0.921668i	$0.373173\pi$
$90$	0	0
$91$	3.99579	0.418872
$92$	0	0
$93$	−5.98746	−0.620871
$94$	0	0
$95$	5.11693	0.524986
$96$	0	0
$97$	5.66075	0.574762	0.287381	−	0.957816i	$-0.407215\pi$
$97$	5.66075	0.574762	0.287381	+	0.957816i	$0.407215\pi$
$98$	0	0
$99$	3.57105	0.358904
$100$	0	0
$101$	−0.0231709	−0.00230559	−0.00115279	−	0.999999i	$-0.500367\pi$
$101$	−0.0231709	−0.00230559	−0.00115279	+	0.999999i	$0.500367\pi$
$102$	0	0
$103$	−5.51046	−0.542962	−0.271481	−	0.962444i	$-0.587513\pi$
$103$	−5.51046	−0.542962	−0.271481	+	0.962444i	$0.587513\pi$
$104$	0	0
$105$	11.0082	1.07429
$106$	0	0
$107$	3.17665	0.307099	0.153549	−	0.988141i	$-0.450930\pi$
$107$	3.17665	0.307099	0.153549	+	0.988141i	$0.450930\pi$
$108$	0	0
$109$	−11.5216	−1.10357	−0.551783	−	0.833987i	$-0.686053\pi$
$109$	−11.5216	−1.10357	−0.551783	+	0.833987i	$0.686053\pi$
$110$	0	0
$111$	4.76622	0.452390
$112$	0	0
$113$	−0.0994853	−0.00935879	−0.00467940	−	0.999989i	$-0.501490\pi$
$113$	−0.0994853	−0.00935879	−0.00467940	+	0.999989i	$0.501490\pi$
$114$	0	0
$115$	−3.06377	−0.285698
$116$	0	0
$117$	1.11209	0.102813
$118$	0	0
$119$	−19.2296	−1.76277
$120$	0	0
$121$	1.75237	0.159306
$122$	0	0
$123$	−3.76801	−0.339750
$124$	0	0
$125$	1.87914	0.168075
$126$	0	0
$127$	4.69936	0.417001	0.208500	−	0.978022i	$-0.433142\pi$
$127$	4.69936	0.417001	0.208500	+	0.978022i	$0.433142\pi$
$128$	0	0
$129$	11.7805	1.03722
$130$	0	0
$131$	−15.2208	−1.32985	−0.664924	−	0.746911i	$-0.731536\pi$
$131$	−15.2208	−1.32985	−0.664924	+	0.746911i	$0.731536\pi$
$132$	0	0
$133$	−6.00087	−0.520342
$134$	0	0
$135$	3.06377	0.263687
$136$	0	0
$137$	−15.8868	−1.35730	−0.678651	−	0.734461i	$-0.737435\pi$
$137$	−15.8868	−1.35730	−0.678651	+	0.734461i	$0.737435\pi$
$138$	0	0
$139$	−4.61451	−0.391397	−0.195699	−	0.980664i	$-0.562697\pi$
$139$	−4.61451	−0.391397	−0.195699	+	0.980664i	$0.562697\pi$
$140$	0	0
$141$	6.41650	0.540367
$142$	0	0
$143$	3.97134	0.332100
$144$	0	0
$145$	3.06377	0.254432
$146$	0	0
$147$	−5.90985	−0.487436
$148$	0	0
$149$	6.38885	0.523395	0.261697	−	0.965150i	$-0.415718\pi$
$149$	6.38885	0.523395	0.261697	+	0.965150i	$0.415718\pi$
$150$	0	0
$151$	−7.13056	−0.580276	−0.290138	−	0.956985i	$-0.593701\pi$
$151$	−7.13056	−0.580276	−0.290138	+	0.956985i	$0.593701\pi$
$152$	0	0
$153$	−5.35191	−0.432676
$154$	0	0
$155$	−18.3442	−1.47344
$156$	0	0
$157$	19.2868	1.53926	0.769628	−	0.638492i	$-0.220442\pi$
$157$	19.2868	1.53926	0.769628	+	0.638492i	$0.220442\pi$
$158$	0	0
$159$	0.261608	0.0207468
$160$	0	0
$161$	3.59303	0.283170
$162$	0	0
$163$	14.3483	1.12384	0.561922	−	0.827190i	$-0.310062\pi$
$163$	14.3483	1.12384	0.561922	+	0.827190i	$0.310062\pi$
$164$	0	0
$165$	10.9408	0.851744
$166$	0	0
$167$	−5.56450	−0.430594	−0.215297	−	0.976549i	$-0.569072\pi$
$167$	−5.56450	−0.430594	−0.215297	+	0.976549i	$0.569072\pi$
$168$	0	0
$169$	−11.7632	−0.904865
$170$	0	0
$171$	−1.67014	−0.127719
$172$	0	0
$173$	2.66312	0.202473	0.101236	−	0.994862i	$-0.467720\pi$
$173$	2.66312	0.202473	0.101236	+	0.994862i	$0.467720\pi$
$174$	0	0
$175$	15.7614	1.19145
$176$	0	0
$177$	7.33982	0.551694
$178$	0	0
$179$	−5.69011	−0.425299	−0.212649	−	0.977129i	$-0.568209\pi$
$179$	−5.69011	−0.425299	−0.212649	+	0.977129i	$0.568209\pi$
$180$	0	0
$181$	−7.03851	−0.523168	−0.261584	−	0.965181i	$-0.584245\pi$
$181$	−7.03851	−0.523168	−0.261584	+	0.965181i	$0.584245\pi$
$182$	0	0
$183$	3.81452	0.281977
$184$	0	0
$185$	14.6026	1.07360
$186$	0	0
$187$	−19.1119	−1.39760
$188$	0	0
$189$	−3.59303	−0.261354
$190$	0	0
$191$	−21.7486	−1.57367	−0.786836	−	0.617162i	$-0.788282\pi$
$191$	−21.7486	−1.57367	−0.786836	+	0.617162i	$0.788282\pi$
$192$	0	0
$193$	4.55878	0.328148	0.164074	−	0.986448i	$-0.447536\pi$
$193$	4.55878	0.328148	0.164074	+	0.986448i	$0.447536\pi$
$194$	0	0
$195$	3.40720	0.243994
$196$	0	0
$197$	2.74978	0.195914	0.0979569	−	0.995191i	$-0.468769\pi$
$197$	2.74978	0.195914	0.0979569	+	0.995191i	$0.468769\pi$
$198$	0	0
$199$	24.3927	1.72915	0.864575	−	0.502503i	$-0.167587\pi$
$199$	24.3927	1.72915	0.864575	+	0.502503i	$0.167587\pi$
$200$	0	0
$201$	0.212601	0.0149957
$202$	0	0
$203$	−3.59303	−0.252181
$204$	0	0
$205$	−11.5443	−0.806289
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−5.96416	−0.412550
$210$	0	0
$211$	−9.28159	−0.638971	−0.319485	−	0.947591i	$-0.603510\pi$
$211$	−9.28159	−0.638971	−0.319485	+	0.947591i	$0.603510\pi$
$212$	0	0
$213$	15.4656	1.05968
$214$	0	0
$215$	36.0927	2.46150
$216$	0	0
$217$	21.5131	1.46041
$218$	0	0
$219$	−15.2479	−1.03036
$220$	0	0
$221$	−5.95183	−0.400363
$222$	0	0
$223$	19.8162	1.32699	0.663497	−	0.748179i	$-0.269072\pi$
$223$	19.8162	1.32699	0.663497	+	0.748179i	$0.269072\pi$
$224$	0	0
$225$	4.38666	0.292444
$226$	0	0
$227$	7.50724	0.498273	0.249136	−	0.968468i	$-0.419853\pi$
$227$	7.50724	0.498273	0.249136	+	0.968468i	$0.419853\pi$
$228$	0	0
$229$	−8.32203	−0.549935	−0.274968	−	0.961453i	$-0.588667\pi$
$229$	−8.32203	−0.549935	−0.274968	+	0.961453i	$0.588667\pi$
$230$	0	0
$231$	−12.8309	−0.844209
$232$	0	0
$233$	13.5592	0.888293	0.444146	−	0.895954i	$-0.353507\pi$
$233$	13.5592	0.888293	0.444146	+	0.895954i	$0.353507\pi$
$234$	0	0
$235$	19.6586	1.28239
$236$	0	0
$237$	−14.1479	−0.919002
$238$	0	0
$239$	−5.02517	−0.325051	−0.162526	−	0.986704i	$-0.551964\pi$
$239$	−5.02517	−0.325051	−0.162526	+	0.986704i	$0.551964\pi$
$240$	0	0
$241$	−2.24679	−0.144729	−0.0723644	−	0.997378i	$-0.523054\pi$
$241$	−2.24679	−0.144729	−0.0723644	+	0.997378i	$0.523054\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−18.1064	−1.15677
$246$	0	0
$247$	−1.85736	−0.118181
$248$	0	0
$249$	−4.48671	−0.284334
$250$	0	0
$251$	3.86119	0.243716	0.121858	−	0.992548i	$-0.461115\pi$
$251$	3.86119	0.243716	0.121858	+	0.992548i	$0.461115\pi$
$252$	0	0
$253$	3.57105	0.224510
$254$	0	0
$255$	−16.3970	−1.02682
$256$	0	0
$257$	−20.1469	−1.25673	−0.628365	−	0.777919i	$-0.716275\pi$
$257$	−20.1469	−1.25673	−0.628365	+	0.777919i	$0.716275\pi$
$258$	0	0
$259$	−17.1252	−1.06411
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−32.1602	−1.98308	−0.991542	−	0.129788i	$-0.958570\pi$
$263$	−32.1602	−1.98308	−0.991542	+	0.129788i	$0.958570\pi$
$264$	0	0
$265$	0.801505	0.0492360
$266$	0	0
$267$	−7.32038	−0.448000
$268$	0	0
$269$	−10.6391	−0.648678	−0.324339	−	0.945941i	$-0.605142\pi$
$269$	−10.6391	−0.648678	−0.324339	+	0.945941i	$0.605142\pi$
$270$	0	0
$271$	−5.40339	−0.328233	−0.164116	−	0.986441i	$-0.552477\pi$
$271$	−5.40339	−0.328233	−0.164116	+	0.986441i	$0.552477\pi$
$272$	0	0
$273$	−3.99579	−0.241836
$274$	0	0
$275$	15.6650	0.944633
$276$	0	0
$277$	−17.0246	−1.02291	−0.511454	−	0.859311i	$-0.670893\pi$
$277$	−17.0246	−1.02291	−0.511454	+	0.859311i	$0.670893\pi$
$278$	0	0
$279$	5.98746	0.358460
$280$	0	0
$281$	2.47389	0.147580	0.0737898	−	0.997274i	$-0.476491\pi$
$281$	2.47389	0.147580	0.0737898	+	0.997274i	$0.476491\pi$
$282$	0	0
$283$	18.6982	1.11150	0.555748	−	0.831351i	$-0.312432\pi$
$283$	18.6982	1.11150	0.555748	+	0.831351i	$0.312432\pi$
$284$	0	0
$285$	−5.11693	−0.303101
$286$	0	0
$287$	13.5386	0.799156
$288$	0	0
$289$	11.6429	0.684878
$290$	0	0
$291$	−5.66075	−0.331839
$292$	0	0
$293$	−19.6458	−1.14772	−0.573861	−	0.818953i	$-0.694555\pi$
$293$	−19.6458	−1.14772	−0.573861	+	0.818953i	$0.694555\pi$
$294$	0	0
$295$	22.4875	1.30927
$296$	0	0
$297$	−3.57105	−0.207213
$298$	0	0
$299$	1.11209	0.0643141
$300$	0	0
$301$	−42.3277	−2.43973
$302$	0	0
$303$	0.0231709	0.00133113
$304$	0	0
$305$	11.6868	0.669183
$306$	0	0
$307$	−12.5654	−0.717144	−0.358572	−	0.933502i	$-0.616736\pi$
$307$	−12.5654	−0.717144	−0.358572	+	0.933502i	$0.616736\pi$
$308$	0	0
$309$	5.51046	0.313479
$310$	0	0
$311$	−11.0036	−0.623954	−0.311977	−	0.950090i	$-0.600991\pi$
$311$	−11.0036	−0.623954	−0.311977	+	0.950090i	$0.600991\pi$
$312$	0	0
$313$	15.4938	0.875762	0.437881	−	0.899033i	$-0.355729\pi$
$313$	15.4938	0.875762	0.437881	+	0.899033i	$0.355729\pi$
$314$	0	0
$315$	−11.0082	−0.620241
$316$	0	0
$317$	4.06492	0.228309	0.114154	−	0.993463i	$-0.463584\pi$
$317$	4.06492	0.228309	0.114154	+	0.993463i	$0.463584\pi$
$318$	0	0
$319$	−3.57105	−0.199940
$320$	0	0
$321$	−3.17665	−0.177304
$322$	0	0
$323$	8.93846	0.497349
$324$	0	0
$325$	4.87838	0.270604
$326$	0	0
$327$	11.5216	0.637145
$328$	0	0
$329$	−23.0547	−1.27104
$330$	0	0
$331$	−14.8554	−0.816524	−0.408262	−	0.912865i	$-0.633865\pi$
$331$	−14.8554	−0.816524	−0.408262	+	0.912865i	$0.633865\pi$
$332$	0	0
$333$	−4.76622	−0.261187
$334$	0	0
$335$	0.651360	0.0355876
$336$	0	0
$337$	−15.2412	−0.830242	−0.415121	−	0.909766i	$-0.636261\pi$
$337$	−15.2412	−0.830242	−0.415121	+	0.909766i	$0.636261\pi$
$338$	0	0
$339$	0.0994853	0.00540330
$340$	0	0
$341$	21.3815	1.15787
$342$	0	0
$343$	−3.91694	−0.211495
$344$	0	0
$345$	3.06377	0.164948
$346$	0	0
$347$	−3.98566	−0.213962	−0.106981	−	0.994261i	$-0.534118\pi$
$347$	−3.98566	−0.213962	−0.106981	+	0.994261i	$0.534118\pi$
$348$	0	0
$349$	−21.3660	−1.14370	−0.571848	−	0.820360i	$-0.693773\pi$
$349$	−21.3660	−1.14370	−0.571848	+	0.820360i	$0.693773\pi$
$350$	0	0
$351$	−1.11209	−0.0593592
$352$	0	0
$353$	1.96364	0.104514	0.0522569	−	0.998634i	$-0.483359\pi$
$353$	1.96364	0.104514	0.0522569	+	0.998634i	$0.483359\pi$
$354$	0	0
$355$	47.3829	2.51482
$356$	0	0
$357$	19.2296	1.01774
$358$	0	0
$359$	−16.1743	−0.853646	−0.426823	−	0.904335i	$-0.640367\pi$
$359$	−16.1743	−0.853646	−0.426823	+	0.904335i	$0.640367\pi$
$360$	0	0
$361$	−16.2106	−0.853191
$362$	0	0
$363$	−1.75237	−0.0919756
$364$	0	0
$365$	−46.7161	−2.44523
$366$	0	0
$367$	8.17933	0.426957	0.213479	−	0.976948i	$-0.431521\pi$
$367$	8.17933	0.426957	0.213479	+	0.976948i	$0.431521\pi$
$368$	0	0
$369$	3.76801	0.196155
$370$	0	0
$371$	−0.939964	−0.0488005
$372$	0	0
$373$	−18.4633	−0.955994	−0.477997	−	0.878361i	$-0.658637\pi$
$373$	−18.4633	−0.955994	−0.477997	+	0.878361i	$0.658637\pi$
$374$	0	0
$375$	−1.87914	−0.0970381
$376$	0	0
$377$	−1.11209	−0.0572758
$378$	0	0
$379$	−9.89915	−0.508485	−0.254243	−	0.967140i	$-0.581826\pi$
$379$	−9.89915	−0.508485	−0.254243	+	0.967140i	$0.581826\pi$
$380$	0	0
$381$	−4.69936	−0.240755
$382$	0	0
$383$	−8.58947	−0.438901	−0.219451	−	0.975624i	$-0.570426\pi$
$383$	−8.58947	−0.438901	−0.219451	+	0.975624i	$0.570426\pi$
$384$	0	0
$385$	−39.3108	−2.00346
$386$	0	0
$387$	−11.7805	−0.598837
$388$	0	0
$389$	−10.3169	−0.523088	−0.261544	−	0.965191i	$-0.584232\pi$
$389$	−10.3169	−0.523088	−0.261544	+	0.965191i	$0.584232\pi$
$390$	0	0
$391$	−5.35191	−0.270658
$392$	0	0
$393$	15.2208	0.767788
$394$	0	0
$395$	−43.3457	−2.18096
$396$	0	0
$397$	−5.68400	−0.285272	−0.142636	−	0.989775i	$-0.545558\pi$
$397$	−5.68400	−0.285272	−0.142636	+	0.989775i	$0.545558\pi$
$398$	0	0
$399$	6.00087	0.300419
$400$	0	0
$401$	6.02992	0.301120	0.150560	−	0.988601i	$-0.451892\pi$
$401$	6.02992	0.301120	0.150560	+	0.988601i	$0.451892\pi$
$402$	0	0
$403$	6.65863	0.331690
$404$	0	0
$405$	−3.06377	−0.152240
$406$	0	0
$407$	−17.0204	−0.843669
$408$	0	0
$409$	8.16164	0.403567	0.201784	−	0.979430i	$-0.435326\pi$
$409$	8.16164	0.403567	0.201784	+	0.979430i	$0.435326\pi$
$410$	0	0
$411$	15.8868	0.783638
$412$	0	0
$413$	−26.3722	−1.29769
$414$	0	0
$415$	−13.7462	−0.674776
$416$	0	0
$417$	4.61451	0.225973
$418$	0	0
$419$	−11.6677	−0.570007	−0.285003	−	0.958527i	$-0.591995\pi$
$419$	−11.6677	−0.570007	−0.285003	+	0.958527i	$0.591995\pi$
$420$	0	0
$421$	−1.91906	−0.0935293	−0.0467647	−	0.998906i	$-0.514891\pi$
$421$	−1.91906	−0.0935293	−0.0467647	+	0.998906i	$0.514891\pi$
$422$	0	0
$423$	−6.41650	−0.311981
$424$	0	0
$425$	−23.4770	−1.13880
$426$	0	0
$427$	−13.7057	−0.663264
$428$	0	0
$429$	−3.97134	−0.191738
$430$	0	0
$431$	16.5396	0.796684	0.398342	−	0.917237i	$-0.369586\pi$
$431$	16.5396	0.796684	0.398342	+	0.917237i	$0.369586\pi$
$432$	0	0
$433$	25.1753	1.20985	0.604923	−	0.796284i	$-0.293204\pi$
$433$	25.1753	1.20985	0.604923	+	0.796284i	$0.293204\pi$
$434$	0	0
$435$	−3.06377	−0.146896
$436$	0	0
$437$	−1.67014	−0.0798938
$438$	0	0
$439$	2.53610	0.121042	0.0605208	−	0.998167i	$-0.480724\pi$
$439$	2.53610	0.121042	0.0605208	+	0.998167i	$0.480724\pi$
$440$	0	0
$441$	5.90985	0.281421
$442$	0	0
$443$	−20.1892	−0.959219	−0.479610	−	0.877482i	$-0.659222\pi$
$443$	−20.1892	−0.959219	−0.479610	+	0.877482i	$0.659222\pi$
$444$	0	0
$445$	−22.4279	−1.06319
$446$	0	0
$447$	−6.38885	−0.302182
$448$	0	0
$449$	−11.7302	−0.553584	−0.276792	−	0.960930i	$-0.589271\pi$
$449$	−11.7302	−0.553584	−0.276792	+	0.960930i	$0.589271\pi$
$450$	0	0
$451$	13.4557	0.633606
$452$	0	0
$453$	7.13056	0.335023
$454$	0	0
$455$	−12.2422	−0.573921
$456$	0	0
$457$	−9.00150	−0.421072	−0.210536	−	0.977586i	$-0.567521\pi$
$457$	−9.00150	−0.421072	−0.210536	+	0.977586i	$0.567521\pi$
$458$	0	0
$459$	5.35191	0.249806
$460$	0	0
$461$	−19.8985	−0.926764	−0.463382	−	0.886159i	$-0.653364\pi$
$461$	−19.8985	−0.926764	−0.463382	+	0.886159i	$0.653364\pi$
$462$	0	0
$463$	−3.02956	−0.140796	−0.0703978	−	0.997519i	$-0.522427\pi$
$463$	−3.02956	−0.140796	−0.0703978	+	0.997519i	$0.522427\pi$
$464$	0	0
$465$	18.3442	0.850691
$466$	0	0
$467$	−5.74598	−0.265892	−0.132946	−	0.991123i	$-0.542444\pi$
$467$	−5.74598	−0.265892	−0.132946	+	0.991123i	$0.542444\pi$
$468$	0	0
$469$	−0.763882	−0.0352728
$470$	0	0
$471$	−19.2868	−0.888690
$472$	0	0
$473$	−42.0688	−1.93432
$474$	0	0
$475$	−7.32635	−0.336156
$476$	0	0
$477$	−0.261608	−0.0119782
$478$	0	0
$479$	14.9178	0.681614	0.340807	−	0.940133i	$-0.389300\pi$
$479$	14.9178	0.681614	0.340807	+	0.940133i	$0.389300\pi$
$480$	0	0
$481$	−5.30049	−0.241681
$482$	0	0
$483$	−3.59303	−0.163488
$484$	0	0
$485$	−17.3432	−0.787514
$486$	0	0
$487$	−29.1042	−1.31884	−0.659418	−	0.751776i	$-0.729197\pi$
$487$	−29.1042	−1.31884	−0.659418	+	0.751776i	$0.729197\pi$
$488$	0	0
$489$	−14.3483	−0.648851
$490$	0	0
$491$	−38.5168	−1.73824	−0.869120	−	0.494602i	$-0.835314\pi$
$491$	−38.5168	−1.73824	−0.869120	+	0.494602i	$0.835314\pi$
$492$	0	0
$493$	5.35191	0.241038
$494$	0	0
$495$	−10.9408	−0.491755
$496$	0	0
$497$	−55.5682	−2.49257
$498$	0	0
$499$	−7.15238	−0.320185	−0.160092	−	0.987102i	$-0.551179\pi$
$499$	−7.15238	−0.320185	−0.160092	+	0.987102i	$0.551179\pi$
$500$	0	0
$501$	5.56450	0.248604
$502$	0	0
$503$	−37.5510	−1.67432	−0.837158	−	0.546961i	$-0.815785\pi$
$503$	−37.5510	−1.67432	−0.837158	+	0.546961i	$0.815785\pi$
$504$	0	0
$505$	0.0709901	0.00315902
$506$	0	0
$507$	11.7632	0.522424
$508$	0	0
$509$	−2.74755	−0.121783	−0.0608914	−	0.998144i	$-0.519394\pi$
$509$	−2.74755	−0.121783	−0.0608914	+	0.998144i	$0.519394\pi$
$510$	0	0
$511$	54.7862	2.42360
$512$	0	0
$513$	1.67014	0.0737386
$514$	0	0
$515$	16.8828	0.743943
$516$	0	0
$517$	−22.9136	−1.00774
$518$	0	0
$519$	−2.66312	−0.116898
$520$	0	0
$521$	−25.5716	−1.12031	−0.560156	−	0.828387i	$-0.689259\pi$
$521$	−25.5716	−1.12031	−0.560156	+	0.828387i	$0.689259\pi$
$522$	0	0
$523$	17.5183	0.766024	0.383012	−	0.923743i	$-0.374887\pi$
$523$	17.5183	0.766024	0.383012	+	0.923743i	$0.374887\pi$
$524$	0	0
$525$	−15.7614	−0.687883
$526$	0	0
$527$	−32.0444	−1.39587
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.33982	−0.318521
$532$	0	0
$533$	4.19039	0.181506
$534$	0	0
$535$	−9.73252	−0.420774
$536$	0	0
$537$	5.69011	0.245546
$538$	0	0
$539$	21.1043	0.909029
$540$	0	0
$541$	−30.9368	−1.33008	−0.665039	−	0.746809i	$-0.731585\pi$
$541$	−30.9368	−1.33008	−0.665039	+	0.746809i	$0.731585\pi$
$542$	0	0
$543$	7.03851	0.302051
$544$	0	0
$545$	35.2994	1.51206
$546$	0	0
$547$	27.4338	1.17299	0.586493	−	0.809955i	$-0.300508\pi$
$547$	27.4338	1.17299	0.586493	+	0.809955i	$0.300508\pi$
$548$	0	0
$549$	−3.81452	−0.162800
$550$	0	0
$551$	1.67014	0.0711505
$552$	0	0
$553$	50.8337	2.16167
$554$	0	0
$555$	−14.6026	−0.619845
$556$	0	0
$557$	18.3290	0.776627	0.388313	−	0.921527i	$-0.373058\pi$
$557$	18.3290	0.776627	0.388313	+	0.921527i	$0.373058\pi$
$558$	0	0
$559$	−13.1010	−0.554115
$560$	0	0
$561$	19.1119	0.806906
$562$	0	0
$563$	6.98199	0.294256	0.147128	−	0.989117i	$-0.452997\pi$
$563$	6.98199	0.294256	0.147128	+	0.989117i	$0.452997\pi$
$564$	0	0
$565$	0.304800	0.0128230
$566$	0	0
$567$	3.59303	0.150893
$568$	0	0
$569$	39.0041	1.63514	0.817569	−	0.575831i	$-0.195321\pi$
$569$	39.0041	1.63514	0.817569	+	0.575831i	$0.195321\pi$
$570$	0	0
$571$	14.1398	0.591730	0.295865	−	0.955230i	$-0.404392\pi$
$571$	14.1398	0.591730	0.295865	+	0.955230i	$0.404392\pi$
$572$	0	0
$573$	21.7486	0.908560
$574$	0	0
$575$	4.38666	0.182936
$576$	0	0
$577$	−41.1894	−1.71474	−0.857369	−	0.514702i	$-0.827903\pi$
$577$	−41.1894	−1.71474	−0.857369	+	0.514702i	$0.827903\pi$
$578$	0	0
$579$	−4.55878	−0.189456
$580$	0	0
$581$	16.1209	0.668807
$582$	0	0
$583$	−0.934213	−0.0386912
$584$	0	0
$585$	−3.40720	−0.140870
$586$	0	0
$587$	27.7763	1.14645	0.573225	−	0.819398i	$-0.305692\pi$
$587$	27.7763	1.14645	0.573225	+	0.819398i	$0.305692\pi$
$588$	0	0
$589$	−9.99992	−0.412040
$590$	0	0
$591$	−2.74978	−0.113111
$592$	0	0
$593$	5.43304	0.223108	0.111554	−	0.993758i	$-0.464417\pi$
$593$	5.43304	0.223108	0.111554	+	0.993758i	$0.464417\pi$
$594$	0	0
$595$	58.9149	2.41527
$596$	0	0
$597$	−24.3927	−0.998326
$598$	0	0
$599$	−1.30603	−0.0533631	−0.0266816	−	0.999644i	$-0.508494\pi$
$599$	−1.30603	−0.0533631	−0.0266816	+	0.999644i	$0.508494\pi$
$600$	0	0
$601$	−34.8724	−1.42248	−0.711238	−	0.702951i	$-0.751865\pi$
$601$	−34.8724	−1.42248	−0.711238	+	0.702951i	$0.751865\pi$
$602$	0	0
$603$	−0.212601	−0.00865779
$604$	0	0
$605$	−5.36885	−0.218275
$606$	0	0
$607$	42.4139	1.72153	0.860764	−	0.509004i	$-0.169986\pi$
$607$	42.4139	1.72153	0.860764	+	0.509004i	$0.169986\pi$
$608$	0	0
$609$	3.59303	0.145597
$610$	0	0
$611$	−7.13576	−0.288682
$612$	0	0
$613$	−8.94979	−0.361479	−0.180739	−	0.983531i	$-0.557849\pi$
$613$	−8.94979	−0.361479	−0.180739	+	0.983531i	$0.557849\pi$
$614$	0	0
$615$	11.5443	0.465511
$616$	0	0
$617$	25.1866	1.01397	0.506986	−	0.861954i	$-0.330759\pi$
$617$	25.1866	1.01397	0.506986	+	0.861954i	$0.330759\pi$
$618$	0	0
$619$	−19.2846	−0.775112	−0.387556	−	0.921846i	$-0.626681\pi$
$619$	−19.2846	−0.775112	−0.387556	+	0.921846i	$0.626681\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	26.3023	1.05378
$624$	0	0
$625$	−27.6905	−1.10762
$626$	0	0
$627$	5.96416	0.238186
$628$	0	0
$629$	25.5084	1.01709
$630$	0	0
$631$	42.6072	1.69616	0.848082	−	0.529865i	$-0.177757\pi$
$631$	42.6072	1.69616	0.848082	+	0.529865i	$0.177757\pi$
$632$	0	0
$633$	9.28159	0.368910
$634$	0	0
$635$	−14.3977	−0.571356
$636$	0	0
$637$	6.57231	0.260405
$638$	0	0
$639$	−15.4656	−0.611808
$640$	0	0
$641$	−32.7436	−1.29329	−0.646647	−	0.762789i	$-0.723829\pi$
$641$	−32.7436	−1.29329	−0.646647	+	0.762789i	$0.723829\pi$
$642$	0	0
$643$	−20.2307	−0.797821	−0.398911	−	0.916990i	$-0.630612\pi$
$643$	−20.2307	−0.797821	−0.398911	+	0.916990i	$0.630612\pi$
$644$	0	0
$645$	−36.0927	−1.42115
$646$	0	0
$647$	19.7182	0.775201	0.387601	−	0.921827i	$-0.373304\pi$
$647$	19.7182	0.775201	0.387601	+	0.921827i	$0.373304\pi$
$648$	0	0
$649$	−26.2108	−1.02886
$650$	0	0
$651$	−21.5131	−0.843166
$652$	0	0
$653$	37.4049	1.46377	0.731884	−	0.681430i	$-0.238641\pi$
$653$	37.4049	1.46377	0.731884	+	0.681430i	$0.238641\pi$
$654$	0	0
$655$	46.6330	1.82210
$656$	0	0
$657$	15.2479	0.594878
$658$	0	0
$659$	−34.1613	−1.33074	−0.665368	−	0.746516i	$-0.731725\pi$
$659$	−34.1613	−1.33074	−0.665368	+	0.746516i	$0.731725\pi$
$660$	0	0
$661$	30.1100	1.17114	0.585572	−	0.810620i	$-0.300870\pi$
$661$	30.1100	1.17114	0.585572	+	0.810620i	$0.300870\pi$
$662$	0	0
$663$	5.95183	0.231150
$664$	0	0
$665$	18.3853	0.712950
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−19.8162	−0.766140
$670$	0	0
$671$	−13.6218	−0.525864
$672$	0	0
$673$	−14.3930	−0.554811	−0.277405	−	0.960753i	$-0.589475\pi$
$673$	−14.3930	−0.554811	−0.277405	+	0.960753i	$0.589475\pi$
$674$	0	0
$675$	−4.38666	−0.168843
$676$	0	0
$677$	9.90351	0.380623	0.190311	−	0.981724i	$-0.439050\pi$
$677$	9.90351	0.380623	0.190311	+	0.981724i	$0.439050\pi$
$678$	0	0
$679$	20.3392	0.780548
$680$	0	0
$681$	−7.50724	−0.287678
$682$	0	0
$683$	24.6562	0.943442	0.471721	−	0.881748i	$-0.343633\pi$
$683$	24.6562	0.943442	0.471721	+	0.881748i	$0.343633\pi$
$684$	0	0
$685$	48.6735	1.85972
$686$	0	0
$687$	8.32203	0.317505
$688$	0	0
$689$	−0.290933	−0.0110836
$690$	0	0
$691$	−13.2353	−0.503496	−0.251748	−	0.967793i	$-0.581005\pi$
$691$	−13.2353	−0.503496	−0.251748	+	0.967793i	$0.581005\pi$
$692$	0	0
$693$	12.8309	0.487404
$694$	0	0
$695$	14.1378	0.536276
$696$	0	0
$697$	−20.1661	−0.763844
$698$	0	0
$699$	−13.5592	−0.512856
$700$	0	0
$701$	0.304707	0.0115086	0.00575432	−	0.999983i	$-0.498168\pi$
$701$	0.304707	0.0115086	0.00575432	+	0.999983i	$0.498168\pi$
$702$	0	0
$703$	7.96027	0.300227
$704$	0	0
$705$	−19.6586	−0.740388
$706$	0	0
$707$	−0.0832536	−0.00313107
$708$	0	0
$709$	37.7708	1.41851	0.709256	−	0.704951i	$-0.249031\pi$
$709$	37.7708	1.41851	0.709256	+	0.704951i	$0.249031\pi$
$710$	0	0
$711$	14.1479	0.530586
$712$	0	0
$713$	5.98746	0.224232
$714$	0	0
$715$	−12.1673	−0.455030
$716$	0	0
$717$	5.02517	0.187668
$718$	0	0
$719$	10.6927	0.398769	0.199385	−	0.979921i	$-0.436106\pi$
$719$	10.6927	0.398769	0.199385	+	0.979921i	$0.436106\pi$
$720$	0	0
$721$	−19.7992	−0.737362
$722$	0	0
$723$	2.24679	0.0835592
$724$	0	0
$725$	−4.38666	−0.162916
$726$	0	0
$727$	−22.2928	−0.826795	−0.413397	−	0.910551i	$-0.635658\pi$
$727$	−22.2928	−0.826795	−0.413397	+	0.910551i	$0.635658\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	63.0482	2.33192
$732$	0	0
$733$	−38.1613	−1.40952	−0.704759	−	0.709447i	$-0.748945\pi$
$733$	−38.1613	−1.40952	−0.704759	+	0.709447i	$0.748945\pi$
$734$	0	0
$735$	18.1064	0.667864
$736$	0	0
$737$	−0.759209	−0.0279658
$738$	0	0
$739$	−45.0827	−1.65839	−0.829197	−	0.558957i	$-0.811202\pi$
$739$	−45.0827	−1.65839	−0.829197	+	0.558957i	$0.811202\pi$
$740$	0	0
$741$	1.85736	0.0682318
$742$	0	0
$743$	32.4523	1.19056	0.595280	−	0.803518i	$-0.297041\pi$
$743$	32.4523	1.19056	0.595280	+	0.803518i	$0.297041\pi$
$744$	0	0
$745$	−19.5739	−0.717133
$746$	0	0
$747$	4.48671	0.164160
$748$	0	0
$749$	11.4138	0.417051
$750$	0	0
$751$	−51.5238	−1.88013	−0.940065	−	0.340994i	$-0.889236\pi$
$751$	−51.5238	−1.88013	−0.940065	+	0.340994i	$0.889236\pi$
$752$	0	0
$753$	−3.86119	−0.140709
$754$	0	0
$755$	21.8464	0.795070
$756$	0	0
$757$	24.0031	0.872409	0.436205	−	0.899848i	$-0.356322\pi$
$757$	24.0031	0.872409	0.436205	+	0.899848i	$0.356322\pi$
$758$	0	0
$759$	−3.57105	−0.129621
$760$	0	0
$761$	−11.1063	−0.402605	−0.201302	−	0.979529i	$-0.564517\pi$
$761$	−11.1063	−0.402605	−0.201302	+	0.979529i	$0.564517\pi$
$762$	0	0
$763$	−41.3973	−1.49868
$764$	0	0
$765$	16.3970	0.592835
$766$	0	0
$767$	−8.16257	−0.294733
$768$	0	0
$769$	30.5474	1.10157	0.550784	−	0.834648i	$-0.314329\pi$
$769$	30.5474	1.10157	0.550784	+	0.834648i	$0.314329\pi$
$770$	0	0
$771$	20.1469	0.725573
$772$	0	0
$773$	−35.2240	−1.26692	−0.633460	−	0.773775i	$-0.718366\pi$
$773$	−35.2240	−1.26692	−0.633460	+	0.773775i	$0.718366\pi$
$774$	0	0
$775$	26.2650	0.943465
$776$	0	0
$777$	17.1252	0.614362
$778$	0	0
$779$	−6.29312	−0.225474
$780$	0	0
$781$	−55.2282	−1.97622
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−59.0903	−2.10902
$786$	0	0
$787$	−8.36305	−0.298111	−0.149055	−	0.988829i	$-0.547623\pi$
$787$	−8.36305	−0.298111	−0.149055	+	0.988829i	$0.547623\pi$
$788$	0	0
$789$	32.1602	1.14493
$790$	0	0
$791$	−0.357454	−0.0127096
$792$	0	0
$793$	−4.24210	−0.150642
$794$	0	0
$795$	−0.801505	−0.0284264
$796$	0	0
$797$	−39.6446	−1.40429	−0.702143	−	0.712036i	$-0.747773\pi$
$797$	−39.6446	−1.40429	−0.702143	+	0.712036i	$0.747773\pi$
$798$	0	0
$799$	34.3405	1.21488
$800$	0	0
$801$	7.32038	0.258653
$802$	0	0
$803$	54.4511	1.92154
$804$	0	0
$805$	−11.0082	−0.387988
$806$	0	0
$807$	10.6391	0.374514
$808$	0	0
$809$	0.856343	0.0301074	0.0150537	−	0.999887i	$-0.495208\pi$
$809$	0.856343	0.0301074	0.0150537	+	0.999887i	$0.495208\pi$
$810$	0	0
$811$	−34.9984	−1.22896	−0.614480	−	0.788932i	$-0.710634\pi$
$811$	−34.9984	−1.22896	−0.614480	+	0.788932i	$0.710634\pi$
$812$	0	0
$813$	5.40339	0.189505
$814$	0	0
$815$	−43.9597	−1.53984
$816$	0	0
$817$	19.6752	0.688346
$818$	0	0
$819$	3.99579	0.139624
$820$	0	0
$821$	−6.38652	−0.222891	−0.111445	−	0.993771i	$-0.535548\pi$
$821$	−6.38652	−0.222891	−0.111445	+	0.993771i	$0.535548\pi$
$822$	0	0
$823$	−5.71924	−0.199360	−0.0996802	−	0.995020i	$-0.531782\pi$
$823$	−5.71924	−0.199360	−0.0996802	+	0.995020i	$0.531782\pi$
$824$	0	0
$825$	−15.6650	−0.545384
$826$	0	0
$827$	8.17831	0.284388	0.142194	−	0.989839i	$-0.454584\pi$
$827$	8.17831	0.284388	0.142194	+	0.989839i	$0.454584\pi$
$828$	0	0
$829$	33.1778	1.15231	0.576156	−	0.817340i	$-0.304552\pi$
$829$	33.1778	1.15231	0.576156	+	0.817340i	$0.304552\pi$
$830$	0	0
$831$	17.0246	0.590576
$832$	0	0
$833$	−31.6290	−1.09588
$834$	0	0
$835$	17.0483	0.589982
$836$	0	0
$837$	−5.98746	−0.206957
$838$	0	0
$839$	22.5616	0.778914	0.389457	−	0.921045i	$-0.372663\pi$
$839$	22.5616	0.778914	0.389457	+	0.921045i	$0.372663\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.47389	−0.0852052
$844$	0	0
$845$	36.0398	1.23981
$846$	0	0
$847$	6.29632	0.216344
$848$	0	0
$849$	−18.6982	−0.641722
$850$	0	0
$851$	−4.76622	−0.163384
$852$	0	0
$853$	15.1924	0.520177	0.260088	−	0.965585i	$-0.416248\pi$
$853$	15.1924	0.520177	0.260088	+	0.965585i	$0.416248\pi$
$854$	0	0
$855$	5.11693	0.174995
$856$	0	0
$857$	6.28683	0.214754	0.107377	−	0.994218i	$-0.465755\pi$
$857$	6.28683	0.214754	0.107377	+	0.994218i	$0.465755\pi$
$858$	0	0
$859$	9.15460	0.312351	0.156175	−	0.987729i	$-0.450083\pi$
$859$	9.15460	0.312351	0.156175	+	0.987729i	$0.450083\pi$
$860$	0	0
$861$	−13.5386	−0.461393
$862$	0	0
$863$	−35.4016	−1.20508	−0.602542	−	0.798087i	$-0.705845\pi$
$863$	−35.4016	−1.20508	−0.602542	+	0.798087i	$0.705845\pi$
$864$	0	0
$865$	−8.15916	−0.277420
$866$	0	0
$867$	−11.6429	−0.395415
$868$	0	0
$869$	50.5227	1.71386
$870$	0	0
$871$	−0.236433	−0.00801122
$872$	0	0
$873$	5.66075	0.191587
$874$	0	0
$875$	6.75179	0.228252
$876$	0	0
$877$	−45.5745	−1.53894	−0.769471	−	0.638682i	$-0.779480\pi$
$877$	−45.5745	−1.53894	−0.769471	+	0.638682i	$0.779480\pi$
$878$	0	0
$879$	19.6458	0.662638
$880$	0	0
$881$	−3.80769	−0.128284	−0.0641422	−	0.997941i	$-0.520431\pi$
$881$	−3.80769	−0.128284	−0.0641422	+	0.997941i	$0.520431\pi$
$882$	0	0
$883$	−50.1228	−1.68677	−0.843384	−	0.537312i	$-0.819440\pi$
$883$	−50.1228	−1.68677	−0.843384	+	0.537312i	$0.819440\pi$
$884$	0	0
$885$	−22.4875	−0.755908
$886$	0	0
$887$	29.2728	0.982884	0.491442	−	0.870910i	$-0.336470\pi$
$887$	29.2728	0.982884	0.491442	+	0.870910i	$0.336470\pi$
$888$	0	0
$889$	16.8849	0.566302
$890$	0	0
$891$	3.57105	0.119635
$892$	0	0
$893$	10.7165	0.358613
$894$	0	0
$895$	17.4332	0.582727
$896$	0	0
$897$	−1.11209	−0.0371318
$898$	0	0
$899$	−5.98746	−0.199693
$900$	0	0
$901$	1.40010	0.0466441
$902$	0	0
$903$	42.3277	1.40858
$904$	0	0
$905$	21.5643	0.716823
$906$	0	0
$907$	0.0482385	0.00160173	0.000800866	−	1.00000i	$-0.499745\pi$
$907$	0.0482385	0.00160173	0.000800866		1.00000i	$0.499745\pi$
$908$	0	0
$909$	−0.0231709	−0.000768529 0
$910$	0	0
$911$	−59.2456	−1.96289	−0.981447	−	0.191734i	$-0.938589\pi$
$911$	−59.2456	−1.96289	−0.981447	+	0.191734i	$0.938589\pi$
$912$	0	0
$913$	16.0223	0.530259
$914$	0	0
$915$	−11.6868	−0.386353
$916$	0	0
$917$	−54.6888	−1.80598
$918$	0	0
$919$	32.5343	1.07321	0.536604	−	0.843834i	$-0.319707\pi$
$919$	32.5343	1.07321	0.536604	+	0.843834i	$0.319707\pi$
$920$	0	0
$921$	12.5654	0.414043
$922$	0	0
$923$	−17.1992	−0.566118
$924$	0	0
$925$	−20.9078	−0.687443
$926$	0	0
$927$	−5.51046	−0.180987
$928$	0	0
$929$	37.1472	1.21876	0.609380	−	0.792878i	$-0.291418\pi$
$929$	37.1472	1.21876	0.609380	+	0.792878i	$0.291418\pi$
$930$	0	0
$931$	−9.87030	−0.323486
$932$	0	0
$933$	11.0036	0.360240
$934$	0	0
$935$	58.5544	1.91493
$936$	0	0
$937$	−13.3829	−0.437199	−0.218599	−	0.975815i	$-0.570149\pi$
$937$	−13.3829	−0.437199	−0.218599	+	0.975815i	$0.570149\pi$
$938$	0	0
$939$	−15.4938	−0.505622
$940$	0	0
$941$	3.67507	0.119804	0.0599019	−	0.998204i	$-0.480921\pi$
$941$	3.67507	0.119804	0.0599019	+	0.998204i	$0.480921\pi$
$942$	0	0
$943$	3.76801	0.122703
$944$	0	0
$945$	11.0082	0.358097
$946$	0	0
$947$	15.2302	0.494916	0.247458	−	0.968899i	$-0.420405\pi$
$947$	15.2302	0.494916	0.247458	+	0.968899i	$0.420405\pi$
$948$	0	0
$949$	16.9571	0.550452
$950$	0	0
$951$	−4.06492	−0.131814
$952$	0	0
$953$	23.3029	0.754855	0.377428	−	0.926039i	$-0.376809\pi$
$953$	23.3029	0.754855	0.377428	+	0.926039i	$0.376809\pi$
$954$	0	0
$955$	66.6326	2.15618
$956$	0	0
$957$	3.57105	0.115436
$958$	0	0
$959$	−57.0817	−1.84327
$960$	0	0
$961$	4.84972	0.156442
$962$	0	0
$963$	3.17665	0.102366
$964$	0	0
$965$	−13.9670	−0.449614
$966$	0	0
$967$	−38.5543	−1.23982	−0.619911	−	0.784672i	$-0.712831\pi$
$967$	−38.5543	−1.23982	−0.619911	+	0.784672i	$0.712831\pi$
$968$	0	0
$969$	−8.93846	−0.287145
$970$	0	0
$971$	46.9412	1.50642	0.753208	−	0.657782i	$-0.228505\pi$
$971$	46.9412	1.50642	0.753208	+	0.657782i	$0.228505\pi$
$972$	0	0
$973$	−16.5800	−0.531532
$974$	0	0
$975$	−4.87838	−0.156233
$976$	0	0
$977$	−16.4403	−0.525973	−0.262986	−	0.964800i	$-0.584707\pi$
$977$	−16.4403	−0.525973	−0.262986	+	0.964800i	$0.584707\pi$
$978$	0	0
$979$	26.1414	0.835483
$980$	0	0
$981$	−11.5216	−0.367856
$982$	0	0
$983$	14.4253	0.460095	0.230048	−	0.973179i	$-0.426112\pi$
$983$	14.4253	0.460095	0.230048	+	0.973179i	$0.426112\pi$
$984$	0	0
$985$	−8.42468	−0.268433
$986$	0	0
$987$	23.0547	0.733838
$988$	0	0
$989$	−11.7805	−0.374599
$990$	0	0
$991$	30.6353	0.973162	0.486581	−	0.873635i	$-0.338244\pi$
$991$	30.6353	0.973162	0.486581	+	0.873635i	$0.338244\pi$
$992$	0	0
$993$	14.8554	0.471420
$994$	0	0
$995$	−74.7334	−2.36921
$996$	0	0
$997$	40.4323	1.28051	0.640253	−	0.768164i	$-0.278830\pi$
$997$	40.4323	1.28051	0.640253	+	0.768164i	$0.278830\pi$
$998$	0	0
$999$	4.76622	0.150797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.3	✓	12	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.06377 q^{5} +3.59303 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.06377 q^{5} +3.59303 q^{7} +1.00000 q^{9} +3.57105 q^{11} +1.11209 q^{13} +3.06377 q^{15} -5.35191 q^{17} -1.67014 q^{19} -3.59303 q^{21} +1.00000 q^{23} +4.38666 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.98746 q^{31} -3.57105 q^{33} -11.0082 q^{35} -4.76622 q^{37} -1.11209 q^{39} +3.76801 q^{41} -11.7805 q^{43} -3.06377 q^{45} -6.41650 q^{47} +5.90985 q^{49} +5.35191 q^{51} -0.261608 q^{53} -10.9408 q^{55} +1.67014 q^{57} -7.33982 q^{59} -3.81452 q^{61} +3.59303 q^{63} -3.40720 q^{65} -0.212601 q^{67} -1.00000 q^{69} -15.4656 q^{71} +15.2479 q^{73} -4.38666 q^{75} +12.8309 q^{77} +14.1479 q^{79} +1.00000 q^{81} +4.48671 q^{83} +16.3970 q^{85} +1.00000 q^{87} +7.32038 q^{89} +3.99579 q^{91} -5.98746 q^{93} +5.11693 q^{95} +5.66075 q^{97} +3.57105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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