LMFDB - Embedded newform 8004.2.a.e.1.1

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} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.86352$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.70748 q^{5} -1.59469 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.70748 q^{5} -1.59469 q^{7} +1.00000 q^{9} -2.15216 q^{11} -3.62850 q^{13} +3.70748 q^{15} +6.51792 q^{17} -5.86961 q^{19} +1.59469 q^{21} -1.00000 q^{23} +8.74544 q^{25} -1.00000 q^{27} +1.00000 q^{29} +10.0687 q^{31} +2.15216 q^{33} +5.91228 q^{35} +1.99282 q^{37} +3.62850 q^{39} -2.83754 q^{41} -1.53121 q^{43} -3.70748 q^{45} -1.23502 q^{47} -4.45697 q^{49} -6.51792 q^{51} -7.71284 q^{53} +7.97910 q^{55} +5.86961 q^{57} +7.21271 q^{59} +1.77725 q^{61} -1.59469 q^{63} +13.4526 q^{65} +6.75121 q^{67} +1.00000 q^{69} +6.62781 q^{71} +9.42253 q^{73} -8.74544 q^{75} +3.43203 q^{77} +14.8112 q^{79} +1.00000 q^{81} +11.5780 q^{83} -24.1651 q^{85} -1.00000 q^{87} -9.06238 q^{89} +5.78633 q^{91} -10.0687 q^{93} +21.7615 q^{95} +0.456077 q^{97} -2.15216 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$5$	−3.70748	−1.65804	−0.829019	−	0.559221i	$-0.811100\pi$
$5$	−3.70748	−1.65804	−0.829019	+	0.559221i	$0.811100\pi$
$6$	0	0
$7$	−1.59469	−0.602736	−0.301368	−	0.953508i	$-0.597443\pi$
$7$	−1.59469	−0.602736	−0.301368	+	0.953508i	$0.597443\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.15216	−0.648901	−0.324450	−	0.945903i	$-0.605179\pi$
$11$	−2.15216	−0.648901	−0.324450	+	0.945903i	$0.605179\pi$
$12$	0	0
$13$	−3.62850	−1.00637	−0.503183	−	0.864180i	$-0.667838\pi$
$13$	−3.62850	−1.00637	−0.503183	+	0.864180i	$0.667838\pi$
$14$	0	0
$15$	3.70748	0.957268
$16$	0	0
$17$	6.51792	1.58083	0.790414	−	0.612573i	$-0.209865\pi$
$17$	6.51792	1.58083	0.790414	+	0.612573i	$0.209865\pi$
$18$	0	0
$19$	−5.86961	−1.34658	−0.673291	−	0.739378i	$-0.735120\pi$
$19$	−5.86961	−1.34658	−0.673291	+	0.739378i	$0.735120\pi$
$20$	0	0
$21$	1.59469	0.347990
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.74544	1.74909
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.0687	1.80839	0.904197	−	0.427115i	$-0.140470\pi$
$31$	10.0687	1.80839	0.904197	+	0.427115i	$0.140470\pi$
$32$	0	0
$33$	2.15216	0.374643
$34$	0	0
$35$	5.91228	0.999358
$36$	0	0
$37$	1.99282	0.327618	0.163809	−	0.986492i	$-0.447622\pi$
$37$	1.99282	0.327618	0.163809	+	0.986492i	$0.447622\pi$
$38$	0	0
$39$	3.62850	0.581026
$40$	0	0
$41$	−2.83754	−0.443149	−0.221574	−	0.975143i	$-0.571120\pi$
$41$	−2.83754	−0.443149	−0.221574	+	0.975143i	$0.571120\pi$
$42$	0	0
$43$	−1.53121	−0.233508	−0.116754	−	0.993161i	$-0.537249\pi$
$43$	−1.53121	−0.233508	−0.116754	+	0.993161i	$0.537249\pi$
$44$	0	0
$45$	−3.70748	−0.552679
$46$	0	0
$47$	−1.23502	−0.180147	−0.0900734	−	0.995935i	$-0.528710\pi$
$47$	−1.23502	−0.180147	−0.0900734	+	0.995935i	$0.528710\pi$
$48$	0	0
$49$	−4.45697	−0.636710
$50$	0	0
$51$	−6.51792	−0.912691
$52$	0	0
$53$	−7.71284	−1.05944	−0.529720	−	0.848173i	$-0.677703\pi$
$53$	−7.71284	−1.05944	−0.529720	+	0.848173i	$0.677703\pi$
$54$	0	0
$55$	7.97910	1.07590
$56$	0	0
$57$	5.86961	0.777449
$58$	0	0
$59$	7.21271	0.939015	0.469507	−	0.882929i	$-0.344432\pi$
$59$	7.21271	0.939015	0.469507	+	0.882929i	$0.344432\pi$
$60$	0	0
$61$	1.77725	0.227554	0.113777	−	0.993506i	$-0.463705\pi$
$61$	1.77725	0.227554	0.113777	+	0.993506i	$0.463705\pi$
$62$	0	0
$63$	−1.59469	−0.200912
$64$	0	0
$65$	13.4526	1.66859
$66$	0	0
$67$	6.75121	0.824792	0.412396	−	0.911005i	$-0.364692\pi$
$67$	6.75121	0.824792	0.412396	+	0.911005i	$0.364692\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	6.62781	0.786576	0.393288	−	0.919415i	$-0.371338\pi$
$71$	6.62781	0.786576	0.393288	+	0.919415i	$0.371338\pi$
$72$	0	0
$73$	9.42253	1.10282	0.551412	−	0.834233i	$-0.314089\pi$
$73$	9.42253	1.10282	0.551412	+	0.834233i	$0.314089\pi$
$74$	0	0
$75$	−8.74544	−1.00984
$76$	0	0
$77$	3.43203	0.391116
$78$	0	0
$79$	14.8112	1.66639	0.833194	−	0.552981i	$-0.186510\pi$
$79$	14.8112	1.66639	0.833194	+	0.552981i	$0.186510\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.5780	1.27085	0.635426	−	0.772161i	$-0.280824\pi$
$83$	11.5780	1.27085	0.635426	+	0.772161i	$0.280824\pi$
$84$	0	0
$85$	−24.1651	−2.62107
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−9.06238	−0.960610	−0.480305	−	0.877101i	$-0.659474\pi$
$89$	−9.06238	−0.960610	−0.480305	+	0.877101i	$0.659474\pi$
$90$	0	0
$91$	5.78633	0.606572
$92$	0	0
$93$	−10.0687	−1.04408
$94$	0	0
$95$	21.7615	2.23268
$96$	0	0
$97$	0.456077	0.0463076	0.0231538	−	0.999732i	$-0.492629\pi$
$97$	0.456077	0.0463076	0.0231538	+	0.999732i	$0.492629\pi$
$98$	0	0
$99$	−2.15216	−0.216300
$100$	0	0
$101$	2.55110	0.253844	0.126922	−	0.991913i	$-0.459490\pi$
$101$	2.55110	0.253844	0.126922	+	0.991913i	$0.459490\pi$
$102$	0	0
$103$	6.99010	0.688755	0.344377	−	0.938831i	$-0.388090\pi$
$103$	6.99010	0.688755	0.344377	+	0.938831i	$0.388090\pi$
$104$	0	0
$105$	−5.91228	−0.576980
$106$	0	0
$107$	−7.77034	−0.751188	−0.375594	−	0.926784i	$-0.622561\pi$
$107$	−7.77034	−0.751188	−0.375594	+	0.926784i	$0.622561\pi$
$108$	0	0
$109$	9.83289	0.941820	0.470910	−	0.882181i	$-0.343926\pi$
$109$	9.83289	0.941820	0.470910	+	0.882181i	$0.343926\pi$
$110$	0	0
$111$	−1.99282	−0.189150
$112$	0	0
$113$	−5.13103	−0.482687	−0.241344	−	0.970440i	$-0.577588\pi$
$113$	−5.13103	−0.482687	−0.241344	+	0.970440i	$0.577588\pi$
$114$	0	0
$115$	3.70748	0.345725
$116$	0	0
$117$	−3.62850	−0.335455
$118$	0	0
$119$	−10.3941	−0.952821
$120$	0	0
$121$	−6.36820	−0.578928
$122$	0	0
$123$	2.83754	0.255852
$124$	0	0
$125$	−13.8861	−1.24201
$126$	0	0
$127$	10.5582	0.936889	0.468444	−	0.883493i	$-0.344815\pi$
$127$	10.5582	0.936889	0.468444	+	0.883493i	$0.344815\pi$
$128$	0	0
$129$	1.53121	0.134816
$130$	0	0
$131$	1.14566	0.100097	0.0500483	−	0.998747i	$-0.484062\pi$
$131$	1.14566	0.100097	0.0500483	+	0.998747i	$0.484062\pi$
$132$	0	0
$133$	9.36020	0.811632
$134$	0	0
$135$	3.70748	0.319089
$136$	0	0
$137$	−0.658713	−0.0562777	−0.0281388	−	0.999604i	$-0.508958\pi$
$137$	−0.658713	−0.0562777	−0.0281388	+	0.999604i	$0.508958\pi$
$138$	0	0
$139$	−13.1747	−1.11746	−0.558730	−	0.829350i	$-0.688711\pi$
$139$	−13.1747	−1.11746	−0.558730	+	0.829350i	$0.688711\pi$
$140$	0	0
$141$	1.23502	0.104008
$142$	0	0
$143$	7.80912	0.653032
$144$	0	0
$145$	−3.70748	−0.307890
$146$	0	0
$147$	4.45697	0.367605
$148$	0	0
$149$	1.22464	0.100327	0.0501633	−	0.998741i	$-0.484026\pi$
$149$	1.22464	0.100327	0.0501633	+	0.998741i	$0.484026\pi$
$150$	0	0
$151$	−13.8166	−1.12438	−0.562189	−	0.827009i	$-0.690041\pi$
$151$	−13.8166	−1.12438	−0.562189	+	0.827009i	$0.690041\pi$
$152$	0	0
$153$	6.51792	0.526943
$154$	0	0
$155$	−37.3296	−2.99839
$156$	0	0
$157$	−4.33993	−0.346364	−0.173182	−	0.984890i	$-0.555405\pi$
$157$	−4.33993	−0.346364	−0.173182	+	0.984890i	$0.555405\pi$
$158$	0	0
$159$	7.71284	0.611668
$160$	0	0
$161$	1.59469	0.125679
$162$	0	0
$163$	−19.1345	−1.49873	−0.749363	−	0.662159i	$-0.769640\pi$
$163$	−19.1345	−1.49873	−0.749363	+	0.662159i	$0.769640\pi$
$164$	0	0
$165$	−7.97910	−0.621172
$166$	0	0
$167$	−20.1088	−1.55607	−0.778034	−	0.628223i	$-0.783783\pi$
$167$	−20.1088	−1.55607	−0.778034	+	0.628223i	$0.783783\pi$
$168$	0	0
$169$	0.166035	0.0127719
$170$	0	0
$171$	−5.86961	−0.448860
$172$	0	0
$173$	9.92281	0.754417	0.377209	−	0.926128i	$-0.376884\pi$
$173$	9.92281	0.754417	0.377209	+	0.926128i	$0.376884\pi$
$174$	0	0
$175$	−13.9462	−1.05424
$176$	0	0
$177$	−7.21271	−0.542141
$178$	0	0
$179$	−0.900750	−0.0673252	−0.0336626	−	0.999433i	$-0.510717\pi$
$179$	−0.900750	−0.0673252	−0.0336626	+	0.999433i	$0.510717\pi$
$180$	0	0
$181$	−21.3878	−1.58974	−0.794871	−	0.606778i	$-0.792462\pi$
$181$	−21.3878	−1.58974	−0.794871	+	0.606778i	$0.792462\pi$
$182$	0	0
$183$	−1.77725	−0.131378
$184$	0	0
$185$	−7.38836	−0.543203
$186$	0	0
$187$	−14.0276	−1.02580
$188$	0	0
$189$	1.59469	0.115997
$190$	0	0
$191$	4.06677	0.294261	0.147131	−	0.989117i	$-0.452996\pi$
$191$	4.06677	0.294261	0.147131	+	0.989117i	$0.452996\pi$
$192$	0	0
$193$	5.55739	0.400030	0.200015	−	0.979793i	$-0.435901\pi$
$193$	5.55739	0.400030	0.200015	+	0.979793i	$0.435901\pi$
$194$	0	0
$195$	−13.4526	−0.963362
$196$	0	0
$197$	12.7694	0.909784	0.454892	−	0.890547i	$-0.349678\pi$
$197$	12.7694	0.909784	0.454892	+	0.890547i	$0.349678\pi$
$198$	0	0
$199$	21.2840	1.50878	0.754391	−	0.656425i	$-0.227932\pi$
$199$	21.2840	1.50878	0.754391	+	0.656425i	$0.227932\pi$
$200$	0	0
$201$	−6.75121	−0.476194
$202$	0	0
$203$	−1.59469	−0.111925
$204$	0	0
$205$	10.5201	0.734757
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	12.6323	0.873798
$210$	0	0
$211$	−9.34034	−0.643015	−0.321508	−	0.946907i	$-0.604190\pi$
$211$	−9.34034	−0.643015	−0.321508	+	0.946907i	$0.604190\pi$
$212$	0	0
$213$	−6.62781	−0.454130
$214$	0	0
$215$	5.67695	0.387165
$216$	0	0
$217$	−16.0565	−1.08998
$218$	0	0
$219$	−9.42253	−0.636716
$220$	0	0
$221$	−23.6503	−1.59089
$222$	0	0
$223$	21.2923	1.42584	0.712918	−	0.701247i	$-0.247373\pi$
$223$	21.2923	1.42584	0.712918	+	0.701247i	$0.247373\pi$
$224$	0	0
$225$	8.74544	0.583029
$226$	0	0
$227$	−6.71940	−0.445982	−0.222991	−	0.974820i	$-0.571582\pi$
$227$	−6.71940	−0.445982	−0.222991	+	0.974820i	$0.571582\pi$
$228$	0	0
$229$	15.5515	1.02767	0.513835	−	0.857889i	$-0.328224\pi$
$229$	15.5515	1.02767	0.513835	+	0.857889i	$0.328224\pi$
$230$	0	0
$231$	−3.43203	−0.225811
$232$	0	0
$233$	3.60530	0.236191	0.118096	−	0.993002i	$-0.462321\pi$
$233$	3.60530	0.236191	0.118096	+	0.993002i	$0.462321\pi$
$234$	0	0
$235$	4.57883	0.298690
$236$	0	0
$237$	−14.8112	−0.962089
$238$	0	0
$239$	9.39572	0.607759	0.303879	−	0.952710i	$-0.401718\pi$
$239$	9.39572	0.607759	0.303879	+	0.952710i	$0.401718\pi$
$240$	0	0
$241$	12.5234	0.806704	0.403352	−	0.915045i	$-0.367845\pi$
$241$	12.5234	0.806704	0.403352	+	0.915045i	$0.367845\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	16.5241	1.05569
$246$	0	0
$247$	21.2979	1.35515
$248$	0	0
$249$	−11.5780	−0.733727
$250$	0	0
$251$	−13.9199	−0.878614	−0.439307	−	0.898337i	$-0.644776\pi$
$251$	−13.9199	−0.878614	−0.439307	+	0.898337i	$0.644776\pi$
$252$	0	0
$253$	2.15216	0.135305
$254$	0	0
$255$	24.1651	1.51328
$256$	0	0
$257$	−16.4930	−1.02881	−0.514403	−	0.857548i	$-0.671987\pi$
$257$	−16.4930	−1.02881	−0.514403	+	0.857548i	$0.671987\pi$
$258$	0	0
$259$	−3.17793	−0.197467
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−30.7133	−1.89387	−0.946933	−	0.321431i	$-0.895836\pi$
$263$	−30.7133	−1.89387	−0.946933	+	0.321431i	$0.895836\pi$
$264$	0	0
$265$	28.5952	1.75659
$266$	0	0
$267$	9.06238	0.554609
$268$	0	0
$269$	7.55230	0.460472	0.230236	−	0.973135i	$-0.426050\pi$
$269$	7.55230	0.460472	0.230236	+	0.973135i	$0.426050\pi$
$270$	0	0
$271$	−2.21164	−0.134348	−0.0671739	−	0.997741i	$-0.521398\pi$
$271$	−2.21164	−0.134348	−0.0671739	+	0.997741i	$0.521398\pi$
$272$	0	0
$273$	−5.78633	−0.350205
$274$	0	0
$275$	−18.8216	−1.13498
$276$	0	0
$277$	7.31297	0.439394	0.219697	−	0.975568i	$-0.429493\pi$
$277$	7.31297	0.439394	0.219697	+	0.975568i	$0.429493\pi$
$278$	0	0
$279$	10.0687	0.602798
$280$	0	0
$281$	−32.8318	−1.95858	−0.979290	−	0.202461i	$-0.935106\pi$
$281$	−32.8318	−1.95858	−0.979290	+	0.202461i	$0.935106\pi$
$282$	0	0
$283$	12.8415	0.763347	0.381673	−	0.924297i	$-0.375348\pi$
$283$	12.8415	0.763347	0.381673	+	0.924297i	$0.375348\pi$
$284$	0	0
$285$	−21.7615	−1.28904
$286$	0	0
$287$	4.52498	0.267101
$288$	0	0
$289$	25.4833	1.49902
$290$	0	0
$291$	−0.456077	−0.0267357
$292$	0	0
$293$	−14.4325	−0.843157	−0.421578	−	0.906792i	$-0.638524\pi$
$293$	−14.4325	−0.843157	−0.421578	+	0.906792i	$0.638524\pi$
$294$	0	0
$295$	−26.7410	−1.55692
$296$	0	0
$297$	2.15216	0.124881
$298$	0	0
$299$	3.62850	0.209842
$300$	0	0
$301$	2.44181	0.140744
$302$	0	0
$303$	−2.55110	−0.146557
$304$	0	0
$305$	−6.58914	−0.377293
$306$	0	0
$307$	23.1124	1.31910	0.659548	−	0.751663i	$-0.270748\pi$
$307$	23.1124	1.31910	0.659548	+	0.751663i	$0.270748\pi$
$308$	0	0
$309$	−6.99010	−0.397653
$310$	0	0
$311$	−28.4686	−1.61430	−0.807152	−	0.590344i	$-0.798992\pi$
$311$	−28.4686	−1.61430	−0.807152	+	0.590344i	$0.798992\pi$
$312$	0	0
$313$	−13.2820	−0.750742	−0.375371	−	0.926875i	$-0.622485\pi$
$313$	−13.2820	−0.750742	−0.375371	+	0.926875i	$0.622485\pi$
$314$	0	0
$315$	5.91228	0.333119
$316$	0	0
$317$	−19.8968	−1.11752	−0.558758	−	0.829331i	$-0.688722\pi$
$317$	−19.8968	−1.11752	−0.558758	+	0.829331i	$0.688722\pi$
$318$	0	0
$319$	−2.15216	−0.120498
$320$	0	0
$321$	7.77034	0.433698
$322$	0	0
$323$	−38.2577	−2.12871
$324$	0	0
$325$	−31.7328	−1.76022
$326$	0	0
$327$	−9.83289	−0.543760
$328$	0	0
$329$	1.96948	0.108581
$330$	0	0
$331$	−10.0353	−0.551592	−0.275796	−	0.961216i	$-0.588941\pi$
$331$	−10.0353	−0.551592	−0.275796	+	0.961216i	$0.588941\pi$
$332$	0	0
$333$	1.99282	0.109206
$334$	0	0
$335$	−25.0300	−1.36754
$336$	0	0
$337$	−14.7464	−0.803290	−0.401645	−	0.915796i	$-0.631561\pi$
$337$	−14.7464	−0.803290	−0.401645	+	0.915796i	$0.631561\pi$
$338$	0	0
$339$	5.13103	0.278680
$340$	0	0
$341$	−21.6695	−1.17347
$342$	0	0
$343$	18.2703	0.986503
$344$	0	0
$345$	−3.70748	−0.199604
$346$	0	0
$347$	22.1533	1.18925	0.594625	−	0.804003i	$-0.297300\pi$
$347$	22.1533	1.18925	0.594625	+	0.804003i	$0.297300\pi$
$348$	0	0
$349$	−11.5717	−0.619417	−0.309708	−	0.950832i	$-0.600231\pi$
$349$	−11.5717	−0.619417	−0.309708	+	0.950832i	$0.600231\pi$
$350$	0	0
$351$	3.62850	0.193675
$352$	0	0
$353$	32.8443	1.74813	0.874063	−	0.485813i	$-0.161476\pi$
$353$	32.8443	1.74813	0.874063	+	0.485813i	$0.161476\pi$
$354$	0	0
$355$	−24.5725	−1.30417
$356$	0	0
$357$	10.3941	0.550112
$358$	0	0
$359$	−8.11658	−0.428377	−0.214188	−	0.976792i	$-0.568711\pi$
$359$	−8.11658	−0.428377	−0.214188	+	0.976792i	$0.568711\pi$
$360$	0	0
$361$	15.4523	0.813281
$362$	0	0
$363$	6.36820	0.334244
$364$	0	0
$365$	−34.9339	−1.82852
$366$	0	0
$367$	21.6880	1.13210	0.566051	−	0.824370i	$-0.308470\pi$
$367$	21.6880	1.13210	0.566051	+	0.824370i	$0.308470\pi$
$368$	0	0
$369$	−2.83754	−0.147716
$370$	0	0
$371$	12.2996	0.638562
$372$	0	0
$373$	−24.7291	−1.28042	−0.640211	−	0.768199i	$-0.721153\pi$
$373$	−24.7291	−1.28042	−0.640211	+	0.768199i	$0.721153\pi$
$374$	0	0
$375$	13.8861	0.717078
$376$	0	0
$377$	−3.62850	−0.186877
$378$	0	0
$379$	−9.48141	−0.487027	−0.243514	−	0.969897i	$-0.578300\pi$
$379$	−9.48141	−0.487027	−0.243514	+	0.969897i	$0.578300\pi$
$380$	0	0
$381$	−10.5582	−0.540913
$382$	0	0
$383$	−18.4914	−0.944868	−0.472434	−	0.881366i	$-0.656625\pi$
$383$	−18.4914	−0.944868	−0.472434	+	0.881366i	$0.656625\pi$
$384$	0	0
$385$	−12.7242	−0.648484
$386$	0	0
$387$	−1.53121	−0.0778360
$388$	0	0
$389$	−12.7820	−0.648073	−0.324036	−	0.946045i	$-0.605040\pi$
$389$	−12.7820	−0.648073	−0.324036	+	0.946045i	$0.605040\pi$
$390$	0	0
$391$	−6.51792	−0.329625
$392$	0	0
$393$	−1.14566	−0.0577908
$394$	0	0
$395$	−54.9122	−2.76293
$396$	0	0
$397$	8.93842	0.448607	0.224303	−	0.974519i	$-0.427989\pi$
$397$	8.93842	0.448607	0.224303	+	0.974519i	$0.427989\pi$
$398$	0	0
$399$	−9.36020	−0.468596
$400$	0	0
$401$	−24.4730	−1.22212	−0.611062	−	0.791583i	$-0.709257\pi$
$401$	−24.4730	−1.22212	−0.611062	+	0.791583i	$0.709257\pi$
$402$	0	0
$403$	−36.5344	−1.81991
$404$	0	0
$405$	−3.70748	−0.184226
$406$	0	0
$407$	−4.28888	−0.212592
$408$	0	0
$409$	8.34547	0.412657	0.206329	−	0.978483i	$-0.433848\pi$
$409$	8.34547	0.412657	0.206329	+	0.978483i	$0.433848\pi$
$410$	0	0
$411$	0.658713	0.0324919
$412$	0	0
$413$	−11.5020	−0.565978
$414$	0	0
$415$	−42.9253	−2.10712
$416$	0	0
$417$	13.1747	0.645166
$418$	0	0
$419$	−14.2208	−0.694733	−0.347367	−	0.937729i	$-0.612924\pi$
$419$	−14.2208	−0.694733	−0.347367	+	0.937729i	$0.612924\pi$
$420$	0	0
$421$	28.2369	1.37618	0.688092	−	0.725624i	$-0.258449\pi$
$421$	28.2369	1.37618	0.688092	+	0.725624i	$0.258449\pi$
$422$	0	0
$423$	−1.23502	−0.0600489
$424$	0	0
$425$	57.0021	2.76501
$426$	0	0
$427$	−2.83416	−0.137155
$428$	0	0
$429$	−7.80912	−0.377028
$430$	0	0
$431$	−37.0946	−1.78678	−0.893391	−	0.449280i	$-0.851681\pi$
$431$	−37.0946	−1.78678	−0.893391	+	0.449280i	$0.851681\pi$
$432$	0	0
$433$	−6.57756	−0.316098	−0.158049	−	0.987431i	$-0.550520\pi$
$433$	−6.57756	−0.316098	−0.158049	+	0.987431i	$0.550520\pi$
$434$	0	0
$435$	3.70748	0.177760
$436$	0	0
$437$	5.86961	0.280782
$438$	0	0
$439$	22.0232	1.05111	0.525554	−	0.850760i	$-0.323858\pi$
$439$	22.0232	1.05111	0.525554	+	0.850760i	$0.323858\pi$
$440$	0	0
$441$	−4.45697	−0.212237
$442$	0	0
$443$	39.3018	1.86729	0.933643	−	0.358206i	$-0.116611\pi$
$443$	39.3018	1.86729	0.933643	+	0.358206i	$0.116611\pi$
$444$	0	0
$445$	33.5986	1.59273
$446$	0	0
$447$	−1.22464	−0.0579236
$448$	0	0
$449$	−25.5687	−1.20666	−0.603330	−	0.797492i	$-0.706160\pi$
$449$	−25.5687	−1.20666	−0.603330	+	0.797492i	$0.706160\pi$
$450$	0	0
$451$	6.10683	0.287560
$452$	0	0
$453$	13.8166	0.649160
$454$	0	0
$455$	−21.4527	−1.00572
$456$	0	0
$457$	−22.2440	−1.04053	−0.520265	−	0.854005i	$-0.674167\pi$
$457$	−22.2440	−1.04053	−0.520265	+	0.854005i	$0.674167\pi$
$458$	0	0
$459$	−6.51792	−0.304230
$460$	0	0
$461$	25.4443	1.18506	0.592530	−	0.805548i	$-0.298129\pi$
$461$	25.4443	1.18506	0.592530	+	0.805548i	$0.298129\pi$
$462$	0	0
$463$	38.2248	1.77646	0.888228	−	0.459403i	$-0.151937\pi$
$463$	38.2248	1.77646	0.888228	+	0.459403i	$0.151937\pi$
$464$	0	0
$465$	37.3296	1.73112
$466$	0	0
$467$	−8.60328	−0.398112	−0.199056	−	0.979988i	$-0.563788\pi$
$467$	−8.60328	−0.398112	−0.199056	+	0.979988i	$0.563788\pi$
$468$	0	0
$469$	−10.7661	−0.497131
$470$	0	0
$471$	4.33993	0.199973
$472$	0	0
$473$	3.29542	0.151524
$474$	0	0
$475$	−51.3323	−2.35529
$476$	0	0
$477$	−7.71284	−0.353147
$478$	0	0
$479$	27.0877	1.23767	0.618834	−	0.785522i	$-0.287605\pi$
$479$	27.0877	1.23767	0.618834	+	0.785522i	$0.287605\pi$
$480$	0	0
$481$	−7.23097	−0.329704
$482$	0	0
$483$	−1.59469	−0.0725608
$484$	0	0
$485$	−1.69090	−0.0767798
$486$	0	0
$487$	33.0312	1.49679	0.748394	−	0.663254i	$-0.230825\pi$
$487$	33.0312	1.49679	0.748394	+	0.663254i	$0.230825\pi$
$488$	0	0
$489$	19.1345	0.865290
$490$	0	0
$491$	−10.1438	−0.457783	−0.228892	−	0.973452i	$-0.573510\pi$
$491$	−10.1438	−0.457783	−0.228892	+	0.973452i	$0.573510\pi$
$492$	0	0
$493$	6.51792	0.293552
$494$	0	0
$495$	7.97910	0.358634
$496$	0	0
$497$	−10.5693	−0.474097
$498$	0	0
$499$	30.0106	1.34346	0.671730	−	0.740796i	$-0.265552\pi$
$499$	30.0106	1.34346	0.671730	+	0.740796i	$0.265552\pi$
$500$	0	0
$501$	20.1088	0.898396
$502$	0	0
$503$	−39.1935	−1.74755	−0.873775	−	0.486331i	$-0.838335\pi$
$503$	−39.1935	−1.74755	−0.873775	+	0.486331i	$0.838335\pi$
$504$	0	0
$505$	−9.45816	−0.420882
$506$	0	0
$507$	−0.166035	−0.00737388
$508$	0	0
$509$	−25.1744	−1.11584	−0.557918	−	0.829896i	$-0.688400\pi$
$509$	−25.1744	−1.11584	−0.557918	+	0.829896i	$0.688400\pi$
$510$	0	0
$511$	−15.0260	−0.664711
$512$	0	0
$513$	5.86961	0.259150
$514$	0	0
$515$	−25.9157	−1.14198
$516$	0	0
$517$	2.65797	0.116897
$518$	0	0
$519$	−9.92281	−0.435563
$520$	0	0
$521$	32.4677	1.42244	0.711218	−	0.702971i	$-0.248144\pi$
$521$	32.4677	1.42244	0.711218	+	0.702971i	$0.248144\pi$
$522$	0	0
$523$	15.7931	0.690583	0.345292	−	0.938495i	$-0.387780\pi$
$523$	15.7931	0.690583	0.345292	+	0.938495i	$0.387780\pi$
$524$	0	0
$525$	13.9462	0.608664
$526$	0	0
$527$	65.6271	2.85876
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.21271	0.313005
$532$	0	0
$533$	10.2960	0.445970
$534$	0	0
$535$	28.8084	1.24550
$536$	0	0
$537$	0.900750	0.0388702
$538$	0	0
$539$	9.59212	0.413162
$540$	0	0
$541$	17.7211	0.761891	0.380946	−	0.924597i	$-0.375599\pi$
$541$	17.7211	0.761891	0.380946	+	0.924597i	$0.375599\pi$
$542$	0	0
$543$	21.3878	0.917838
$544$	0	0
$545$	−36.4553	−1.56157
$546$	0	0
$547$	−36.0363	−1.54080	−0.770400	−	0.637560i	$-0.779944\pi$
$547$	−36.0363	−1.54080	−0.770400	+	0.637560i	$0.779944\pi$
$548$	0	0
$549$	1.77725	0.0758513
$550$	0	0
$551$	−5.86961	−0.250054
$552$	0	0
$553$	−23.6192	−1.00439
$554$	0	0
$555$	7.38836	0.313619
$556$	0	0
$557$	−17.0363	−0.721850	−0.360925	−	0.932595i	$-0.617539\pi$
$557$	−17.0363	−0.721850	−0.360925	+	0.932595i	$0.617539\pi$
$558$	0	0
$559$	5.55601	0.234994
$560$	0	0
$561$	14.0276	0.592246
$562$	0	0
$563$	−29.6423	−1.24927	−0.624637	−	0.780915i	$-0.714753\pi$
$563$	−29.6423	−1.24927	−0.624637	+	0.780915i	$0.714753\pi$
$564$	0	0
$565$	19.0232	0.800313
$566$	0	0
$567$	−1.59469	−0.0669706
$568$	0	0
$569$	−2.88610	−0.120991	−0.0604957	−	0.998168i	$-0.519268\pi$
$569$	−2.88610	−0.120991	−0.0604957	+	0.998168i	$0.519268\pi$
$570$	0	0
$571$	−32.5461	−1.36201	−0.681007	−	0.732277i	$-0.738458\pi$
$571$	−32.5461	−1.36201	−0.681007	+	0.732277i	$0.738458\pi$
$572$	0	0
$573$	−4.06677	−0.169892
$574$	0	0
$575$	−8.74544	−0.364710
$576$	0	0
$577$	−8.33309	−0.346911	−0.173456	−	0.984842i	$-0.555493\pi$
$577$	−8.33309	−0.346911	−0.173456	+	0.984842i	$0.555493\pi$
$578$	0	0
$579$	−5.55739	−0.230957
$580$	0	0
$581$	−18.4633	−0.765988
$582$	0	0
$583$	16.5993	0.687471
$584$	0	0
$585$	13.4526	0.556197
$586$	0	0
$587$	9.11774	0.376329	0.188165	−	0.982137i	$-0.439746\pi$
$587$	9.11774	0.376329	0.188165	+	0.982137i	$0.439746\pi$
$588$	0	0
$589$	−59.0994	−2.43515
$590$	0	0
$591$	−12.7694	−0.525264
$592$	0	0
$593$	−43.9819	−1.80612	−0.903061	−	0.429512i	$-0.858685\pi$
$593$	−43.9819	−1.80612	−0.903061	+	0.429512i	$0.858685\pi$
$594$	0	0
$595$	38.5358	1.57981
$596$	0	0
$597$	−21.2840	−0.871096
$598$	0	0
$599$	4.50050	0.183885	0.0919427	−	0.995764i	$-0.470692\pi$
$599$	4.50050	0.183885	0.0919427	+	0.995764i	$0.470692\pi$
$600$	0	0
$601$	−39.4696	−1.61000	−0.804999	−	0.593277i	$-0.797834\pi$
$601$	−39.4696	−1.61000	−0.804999	+	0.593277i	$0.797834\pi$
$602$	0	0
$603$	6.75121	0.274931
$604$	0	0
$605$	23.6100	0.959884
$606$	0	0
$607$	15.2929	0.620718	0.310359	−	0.950619i	$-0.399551\pi$
$607$	15.2929	0.620718	0.310359	+	0.950619i	$0.399551\pi$
$608$	0	0
$609$	1.59469	0.0646200
$610$	0	0
$611$	4.48129	0.181293
$612$	0	0
$613$	−34.1919	−1.38100	−0.690499	−	0.723334i	$-0.742609\pi$
$613$	−34.1919	−1.38100	−0.690499	+	0.723334i	$0.742609\pi$
$614$	0	0
$615$	−10.5201	−0.424212
$616$	0	0
$617$	−44.8672	−1.80629	−0.903143	−	0.429340i	$-0.858746\pi$
$617$	−44.8672	−1.80629	−0.903143	+	0.429340i	$0.858746\pi$
$618$	0	0
$619$	−32.4722	−1.30517	−0.652585	−	0.757716i	$-0.726315\pi$
$619$	−32.4722	−1.30517	−0.652585	+	0.757716i	$0.726315\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	14.4517	0.578994
$624$	0	0
$625$	7.75549	0.310220
$626$	0	0
$627$	−12.6323	−0.504487
$628$	0	0
$629$	12.9891	0.517908
$630$	0	0
$631$	8.91524	0.354910	0.177455	−	0.984129i	$-0.443214\pi$
$631$	8.91524	0.354910	0.177455	+	0.984129i	$0.443214\pi$
$632$	0	0
$633$	9.34034	0.371245
$634$	0	0
$635$	−39.1444	−1.55340
$636$	0	0
$637$	16.1721	0.640763
$638$	0	0
$639$	6.62781	0.262192
$640$	0	0
$641$	−2.82470	−0.111569	−0.0557845	−	0.998443i	$-0.517766\pi$
$641$	−2.82470	−0.111569	−0.0557845	+	0.998443i	$0.517766\pi$
$642$	0	0
$643$	50.0196	1.97258	0.986290	−	0.165023i	$-0.0527697\pi$
$643$	50.0196	1.97258	0.986290	+	0.165023i	$0.0527697\pi$
$644$	0	0
$645$	−5.67695	−0.223530
$646$	0	0
$647$	−17.4253	−0.685058	−0.342529	−	0.939507i	$-0.611283\pi$
$647$	−17.4253	−0.685058	−0.342529	+	0.939507i	$0.611283\pi$
$648$	0	0
$649$	−15.5229	−0.609328
$650$	0	0
$651$	16.0565	0.629302
$652$	0	0
$653$	2.53400	0.0991633	0.0495816	−	0.998770i	$-0.484211\pi$
$653$	2.53400	0.0991633	0.0495816	+	0.998770i	$0.484211\pi$
$654$	0	0
$655$	−4.24751	−0.165964
$656$	0	0
$657$	9.42253	0.367608
$658$	0	0
$659$	46.7865	1.82254	0.911272	−	0.411805i	$-0.135101\pi$
$659$	46.7865	1.82254	0.911272	+	0.411805i	$0.135101\pi$
$660$	0	0
$661$	38.2474	1.48765	0.743827	−	0.668373i	$-0.233009\pi$
$661$	38.2474	1.48765	0.743827	+	0.668373i	$0.233009\pi$
$662$	0	0
$663$	23.6503	0.918501
$664$	0	0
$665$	−34.7028	−1.34572
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−21.2923	−0.823207
$670$	0	0
$671$	−3.82493	−0.147660
$672$	0	0
$673$	9.98712	0.384975	0.192488	−	0.981299i	$-0.438345\pi$
$673$	9.98712	0.384975	0.192488	+	0.981299i	$0.438345\pi$
$674$	0	0
$675$	−8.74544	−0.336612
$676$	0	0
$677$	16.4733	0.633122	0.316561	−	0.948572i	$-0.397472\pi$
$677$	16.4733	0.633122	0.316561	+	0.948572i	$0.397472\pi$
$678$	0	0
$679$	−0.727301	−0.0279112
$680$	0	0
$681$	6.71940	0.257488
$682$	0	0
$683$	13.4867	0.516053	0.258026	−	0.966138i	$-0.416928\pi$
$683$	13.4867	0.516053	0.258026	+	0.966138i	$0.416928\pi$
$684$	0	0
$685$	2.44217	0.0933104
$686$	0	0
$687$	−15.5515	−0.593325
$688$	0	0
$689$	27.9860	1.06618
$690$	0	0
$691$	−18.9990	−0.722758	−0.361379	−	0.932419i	$-0.617694\pi$
$691$	−18.9990	−0.722758	−0.361379	+	0.932419i	$0.617694\pi$
$692$	0	0
$693$	3.43203	0.130372
$694$	0	0
$695$	48.8448	1.85279
$696$	0	0
$697$	−18.4948	−0.700542
$698$	0	0
$699$	−3.60530	−0.136365
$700$	0	0
$701$	−15.3046	−0.578045	−0.289023	−	0.957322i	$-0.593330\pi$
$701$	−15.3046	−0.578045	−0.289023	+	0.957322i	$0.593330\pi$
$702$	0	0
$703$	−11.6971	−0.441165
$704$	0	0
$705$	−4.57883	−0.172449
$706$	0	0
$707$	−4.06821	−0.153001
$708$	0	0
$709$	−23.1417	−0.869104	−0.434552	−	0.900647i	$-0.643093\pi$
$709$	−23.1417	−0.869104	−0.434552	+	0.900647i	$0.643093\pi$
$710$	0	0
$711$	14.8112	0.555462
$712$	0	0
$713$	−10.0687	−0.377076
$714$	0	0
$715$	−28.9522	−1.08275
$716$	0	0
$717$	−9.39572	−0.350890
$718$	0	0
$719$	4.55539	0.169887	0.0849436	−	0.996386i	$-0.472929\pi$
$719$	4.55539	0.169887	0.0849436	+	0.996386i	$0.472929\pi$
$720$	0	0
$721$	−11.1470	−0.415137
$722$	0	0
$723$	−12.5234	−0.465751
$724$	0	0
$725$	8.74544	0.324797
$726$	0	0
$727$	22.0600	0.818159	0.409079	−	0.912499i	$-0.365850\pi$
$727$	22.0600	0.818159	0.409079	+	0.912499i	$0.365850\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.98033	−0.369136
$732$	0	0
$733$	−38.6690	−1.42827	−0.714137	−	0.700006i	$-0.753181\pi$
$733$	−38.6690	−1.42827	−0.714137	+	0.700006i	$0.753181\pi$
$734$	0	0
$735$	−16.5241	−0.609502
$736$	0	0
$737$	−14.5297	−0.535208
$738$	0	0
$739$	24.2883	0.893458	0.446729	−	0.894669i	$-0.352589\pi$
$739$	24.2883	0.893458	0.446729	+	0.894669i	$0.352589\pi$
$740$	0	0
$741$	−21.2979	−0.782398
$742$	0	0
$743$	24.0568	0.882559	0.441279	−	0.897370i	$-0.354525\pi$
$743$	24.0568	0.882559	0.441279	+	0.897370i	$0.354525\pi$
$744$	0	0
$745$	−4.54034	−0.166345
$746$	0	0
$747$	11.5780	0.423618
$748$	0	0
$749$	12.3913	0.452767
$750$	0	0
$751$	−0.261425	−0.00953954	−0.00476977	−	0.999989i	$-0.501518\pi$
$751$	−0.261425	−0.00953954	−0.00476977	+	0.999989i	$0.501518\pi$
$752$	0	0
$753$	13.9199	0.507268
$754$	0	0
$755$	51.2248	1.86426
$756$	0	0
$757$	−40.2089	−1.46142	−0.730709	−	0.682689i	$-0.760810\pi$
$757$	−40.2089	−1.46142	−0.730709	+	0.682689i	$0.760810\pi$
$758$	0	0
$759$	−2.15216	−0.0781185
$760$	0	0
$761$	14.8187	0.537176	0.268588	−	0.963255i	$-0.413443\pi$
$761$	14.8187	0.537176	0.268588	+	0.963255i	$0.413443\pi$
$762$	0	0
$763$	−15.6804	−0.567668
$764$	0	0
$765$	−24.1651	−0.873690
$766$	0	0
$767$	−26.1713	−0.944992
$768$	0	0
$769$	−15.1156	−0.545083	−0.272542	−	0.962144i	$-0.587864\pi$
$769$	−15.1156	−0.545083	−0.272542	+	0.962144i	$0.587864\pi$
$770$	0	0
$771$	16.4930	0.593982
$772$	0	0
$773$	23.2648	0.836777	0.418388	−	0.908268i	$-0.362595\pi$
$773$	23.2648	0.836777	0.418388	+	0.908268i	$0.362595\pi$
$774$	0	0
$775$	88.0553	3.16304
$776$	0	0
$777$	3.17793	0.114008
$778$	0	0
$779$	16.6552	0.596736
$780$	0	0
$781$	−14.2641	−0.510410
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	16.0902	0.574284
$786$	0	0
$787$	−19.9419	−0.710853	−0.355427	−	0.934704i	$-0.615664\pi$
$787$	−19.9419	−0.710853	−0.355427	+	0.934704i	$0.615664\pi$
$788$	0	0
$789$	30.7133	1.09342
$790$	0	0
$791$	8.18240	0.290933
$792$	0	0
$793$	−6.44877	−0.229003
$794$	0	0
$795$	−28.5952	−1.01417
$796$	0	0
$797$	−17.5348	−0.621113	−0.310557	−	0.950555i	$-0.600515\pi$
$797$	−17.5348	−0.621113	−0.310557	+	0.950555i	$0.600515\pi$
$798$	0	0
$799$	−8.04979	−0.284781
$800$	0	0
$801$	−9.06238	−0.320203
$802$	0	0
$803$	−20.2788	−0.715623
$804$	0	0
$805$	−5.91228	−0.208381
$806$	0	0
$807$	−7.55230	−0.265854
$808$	0	0
$809$	13.8664	0.487517	0.243759	−	0.969836i	$-0.421619\pi$
$809$	13.8664	0.487517	0.243759	+	0.969836i	$0.421619\pi$
$810$	0	0
$811$	6.06197	0.212865	0.106432	−	0.994320i	$-0.466057\pi$
$811$	6.06197	0.212865	0.106432	+	0.994320i	$0.466057\pi$
$812$	0	0
$813$	2.21164	0.0775657
$814$	0	0
$815$	70.9407	2.48494
$816$	0	0
$817$	8.98763	0.314437
$818$	0	0
$819$	5.78633	0.202191
$820$	0	0
$821$	−22.6540	−0.790631	−0.395316	−	0.918545i	$-0.629365\pi$
$821$	−22.6540	−0.790631	−0.395316	+	0.918545i	$0.629365\pi$
$822$	0	0
$823$	−20.5866	−0.717605	−0.358803	−	0.933413i	$-0.616815\pi$
$823$	−20.5866	−0.717605	−0.358803	+	0.933413i	$0.616815\pi$
$824$	0	0
$825$	18.8216	0.655284
$826$	0	0
$827$	−2.94930	−0.102557	−0.0512786	−	0.998684i	$-0.516330\pi$
$827$	−2.94930	−0.102557	−0.0512786	+	0.998684i	$0.516330\pi$
$828$	0	0
$829$	−51.3779	−1.78443	−0.892214	−	0.451612i	$-0.850849\pi$
$829$	−51.3779	−1.78443	−0.892214	+	0.451612i	$0.850849\pi$
$830$	0	0
$831$	−7.31297	−0.253684
$832$	0	0
$833$	−29.0502	−1.00653
$834$	0	0
$835$	74.5531	2.58002
$836$	0	0
$837$	−10.0687	−0.348026
$838$	0	0
$839$	46.5085	1.60565	0.802826	−	0.596214i	$-0.203329\pi$
$839$	46.5085	1.60565	0.802826	+	0.596214i	$0.203329\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	32.8318	1.13079
$844$	0	0
$845$	−0.615572	−0.0211763
$846$	0	0
$847$	10.1553	0.348940
$848$	0	0
$849$	−12.8415	−0.440718
$850$	0	0
$851$	−1.99282	−0.0683131
$852$	0	0
$853$	33.7426	1.15532	0.577662	−	0.816276i	$-0.303965\pi$
$853$	33.7426	1.15532	0.577662	+	0.816276i	$0.303965\pi$
$854$	0	0
$855$	21.7615	0.744227
$856$	0	0
$857$	−10.6128	−0.362527	−0.181263	−	0.983435i	$-0.558019\pi$
$857$	−10.6128	−0.362527	−0.181263	+	0.983435i	$0.558019\pi$
$858$	0	0
$859$	−38.9227	−1.32802	−0.664012	−	0.747722i	$-0.731148\pi$
$859$	−38.9227	−1.32802	−0.664012	+	0.747722i	$0.731148\pi$
$860$	0	0
$861$	−4.52498	−0.154211
$862$	0	0
$863$	−13.2174	−0.449924	−0.224962	−	0.974368i	$-0.572226\pi$
$863$	−13.2174	−0.449924	−0.224962	+	0.974368i	$0.572226\pi$
$864$	0	0
$865$	−36.7887	−1.25085
$866$	0	0
$867$	−25.4833	−0.865458
$868$	0	0
$869$	−31.8760	−1.08132
$870$	0	0
$871$	−24.4968	−0.830042
$872$	0	0
$873$	0.456077	0.0154359
$874$	0	0
$875$	22.1441	0.748606
$876$	0	0
$877$	48.3073	1.63122	0.815610	−	0.578602i	$-0.196401\pi$
$877$	48.3073	1.63122	0.815610	+	0.578602i	$0.196401\pi$
$878$	0	0
$879$	14.4325	0.486797
$880$	0	0
$881$	−28.7688	−0.969244	−0.484622	−	0.874724i	$-0.661043\pi$
$881$	−28.7688	−0.969244	−0.484622	+	0.874724i	$0.661043\pi$
$882$	0	0
$883$	7.21826	0.242914	0.121457	−	0.992597i	$-0.461243\pi$
$883$	7.21826	0.242914	0.121457	+	0.992597i	$0.461243\pi$
$884$	0	0
$885$	26.7410	0.898889
$886$	0	0
$887$	31.1158	1.04477	0.522383	−	0.852711i	$-0.325043\pi$
$887$	31.1158	1.04477	0.522383	+	0.852711i	$0.325043\pi$
$888$	0	0
$889$	−16.8370	−0.564696
$890$	0	0
$891$	−2.15216	−0.0721001
$892$	0	0
$893$	7.24911	0.242582
$894$	0	0
$895$	3.33951	0.111628
$896$	0	0
$897$	−3.62850	−0.121152
$898$	0	0
$899$	10.0687	0.335810
$900$	0	0
$901$	−50.2717	−1.67479
$902$	0	0
$903$	−2.44181	−0.0812583
$904$	0	0
$905$	79.2949	2.63585
$906$	0	0
$907$	−11.6633	−0.387272	−0.193636	−	0.981073i	$-0.562028\pi$
$907$	−11.6633	−0.387272	−0.193636	+	0.981073i	$0.562028\pi$
$908$	0	0
$909$	2.55110	0.0846146
$910$	0	0
$911$	−48.7355	−1.61468	−0.807340	−	0.590087i	$-0.799094\pi$
$911$	−48.7355	−1.61468	−0.807340	+	0.590087i	$0.799094\pi$
$912$	0	0
$913$	−24.9178	−0.824658
$914$	0	0
$915$	6.58914	0.217830
$916$	0	0
$917$	−1.82697	−0.0603318
$918$	0	0
$919$	37.0018	1.22058	0.610289	−	0.792179i	$-0.291053\pi$
$919$	37.0018	1.22058	0.610289	+	0.792179i	$0.291053\pi$
$920$	0	0
$921$	−23.1124	−0.761580
$922$	0	0
$923$	−24.0490	−0.791583
$924$	0	0
$925$	17.4281	0.573033
$926$	0	0
$927$	6.99010	0.229585
$928$	0	0
$929$	46.1915	1.51549	0.757747	−	0.652549i	$-0.226300\pi$
$929$	46.1915	1.51549	0.757747	+	0.652549i	$0.226300\pi$
$930$	0	0
$931$	26.1607	0.857382
$932$	0	0
$933$	28.4686	0.932019
$934$	0	0
$935$	52.0071	1.70082
$936$	0	0
$937$	24.0838	0.786785	0.393392	−	0.919371i	$-0.371301\pi$
$937$	24.0838	0.786785	0.393392	+	0.919371i	$0.371301\pi$
$938$	0	0
$939$	13.2820	0.433441
$940$	0	0
$941$	30.9552	1.00911	0.504555	−	0.863380i	$-0.331657\pi$
$941$	30.9552	1.00911	0.504555	+	0.863380i	$0.331657\pi$
$942$	0	0
$943$	2.83754	0.0924029
$944$	0	0
$945$	−5.91228	−0.192327
$946$	0	0
$947$	10.8455	0.352433	0.176216	−	0.984351i	$-0.443614\pi$
$947$	10.8455	0.352433	0.176216	+	0.984351i	$0.443614\pi$
$948$	0	0
$949$	−34.1897	−1.10984
$950$	0	0
$951$	19.8968	0.645198
$952$	0	0
$953$	16.2568	0.526609	0.263304	−	0.964713i	$-0.415188\pi$
$953$	16.2568	0.526609	0.263304	+	0.964713i	$0.415188\pi$
$954$	0	0
$955$	−15.0775	−0.487896
$956$	0	0
$957$	2.15216	0.0695695
$958$	0	0
$959$	1.05044	0.0339205
$960$	0	0
$961$	70.3790	2.27029
$962$	0	0
$963$	−7.77034	−0.250396
$964$	0	0
$965$	−20.6039	−0.663264
$966$	0	0
$967$	−10.7315	−0.345102	−0.172551	−	0.985001i	$-0.555201\pi$
$967$	−10.7315	−0.345102	−0.172551	+	0.985001i	$0.555201\pi$
$968$	0	0
$969$	38.2577	1.22901
$970$	0	0
$971$	−0.672735	−0.0215891	−0.0107946	−	0.999942i	$-0.503436\pi$
$971$	−0.672735	−0.0215891	−0.0107946	+	0.999942i	$0.503436\pi$
$972$	0	0
$973$	21.0095	0.673533
$974$	0	0
$975$	31.7328	1.01626
$976$	0	0
$977$	−1.61692	−0.0517299	−0.0258650	−	0.999665i	$-0.508234\pi$
$977$	−1.61692	−0.0517299	−0.0258650	+	0.999665i	$0.508234\pi$
$978$	0	0
$979$	19.5037	0.623341
$980$	0	0
$981$	9.83289	0.313940
$982$	0	0
$983$	−52.1299	−1.66268	−0.831342	−	0.555761i	$-0.812427\pi$
$983$	−52.1299	−1.66268	−0.831342	+	0.555761i	$0.812427\pi$
$984$	0	0
$985$	−47.3424	−1.50846
$986$	0	0
$987$	−1.96948	−0.0626892
$988$	0	0
$989$	1.53121	0.0486898
$990$	0	0
$991$	57.5804	1.82910	0.914550	−	0.404473i	$-0.132545\pi$
$991$	57.5804	1.82910	0.914550	+	0.404473i	$0.132545\pi$
$992$	0	0
$993$	10.0353	0.318462
$994$	0	0
$995$	−78.9101	−2.50162
$996$	0	0
$997$	−5.70739	−0.180755	−0.0903775	−	0.995908i	$-0.528807\pi$
$997$	−5.70739	−0.180755	−0.0903775	+	0.995908i	$0.528807\pi$
$998$	0	0
$999$	−1.99282	−0.0630502

Twists

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

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

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.70748 q^{5} -1.59469 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.70748 q^{5} -1.59469 q^{7} +1.00000 q^{9} -2.15216 q^{11} -3.62850 q^{13} +3.70748 q^{15} +6.51792 q^{17} -5.86961 q^{19} +1.59469 q^{21} -1.00000 q^{23} +8.74544 q^{25} -1.00000 q^{27} +1.00000 q^{29} +10.0687 q^{31} +2.15216 q^{33} +5.91228 q^{35} +1.99282 q^{37} +3.62850 q^{39} -2.83754 q^{41} -1.53121 q^{43} -3.70748 q^{45} -1.23502 q^{47} -4.45697 q^{49} -6.51792 q^{51} -7.71284 q^{53} +7.97910 q^{55} +5.86961 q^{57} +7.21271 q^{59} +1.77725 q^{61} -1.59469 q^{63} +13.4526 q^{65} +6.75121 q^{67} +1.00000 q^{69} +6.62781 q^{71} +9.42253 q^{73} -8.74544 q^{75} +3.43203 q^{77} +14.8112 q^{79} +1.00000 q^{81} +11.5780 q^{83} -24.1651 q^{85} -1.00000 q^{87} -9.06238 q^{89} +5.78633 q^{91} -10.0687 q^{93} +21.7615 q^{95} +0.456077 q^{97} -2.15216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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