LMFDB - Embedded newform 8004.2.a.j.1.4

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
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			1.4
Root			$-2.34509$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.34509 q^{5} -1.25807 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.34509 q^{5} -1.25807 q^{7} +1.00000 q^{9} -6.44999 q^{11} +1.34717 q^{13} -2.34509 q^{15} -7.39461 q^{17} -4.27627 q^{19} -1.25807 q^{21} +1.00000 q^{23} +0.499463 q^{25} +1.00000 q^{27} -1.00000 q^{29} +10.1300 q^{31} -6.44999 q^{33} +2.95030 q^{35} -5.20798 q^{37} +1.34717 q^{39} -10.6226 q^{41} +0.118668 q^{43} -2.34509 q^{45} +9.08470 q^{47} -5.41725 q^{49} -7.39461 q^{51} +10.4167 q^{53} +15.1258 q^{55} -4.27627 q^{57} -3.39629 q^{59} -2.23961 q^{61} -1.25807 q^{63} -3.15925 q^{65} +7.85081 q^{67} +1.00000 q^{69} -10.7923 q^{71} +7.17561 q^{73} +0.499463 q^{75} +8.11456 q^{77} -4.24131 q^{79} +1.00000 q^{81} -7.91341 q^{83} +17.3410 q^{85} -1.00000 q^{87} -4.52488 q^{89} -1.69484 q^{91} +10.1300 q^{93} +10.0282 q^{95} -0.279080 q^{97} -6.44999 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * 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
$5$	−2.34509	−1.04876	−0.524379	−	0.851485i	$-0.675702\pi$
$5$	−2.34509	−1.04876	−0.524379	+	0.851485i	$0.675702\pi$
$6$	0	0
$7$	−1.25807	−0.475507	−0.237754	−	0.971326i	$-0.576411\pi$
$7$	−1.25807	−0.475507	−0.237754	+	0.971326i	$0.576411\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.44999	−1.94475	−0.972373	−	0.233434i	$-0.925004\pi$
$11$	−6.44999	−1.94475	−0.972373	+	0.233434i	$0.925004\pi$
$12$	0	0
$13$	1.34717	0.373639	0.186819	−	0.982394i	$-0.440182\pi$
$13$	1.34717	0.373639	0.186819	+	0.982394i	$0.440182\pi$
$14$	0	0
$15$	−2.34509	−0.605501
$16$	0	0
$17$	−7.39461	−1.79346	−0.896728	−	0.442582i	$-0.854063\pi$
$17$	−7.39461	−1.79346	−0.896728	+	0.442582i	$0.854063\pi$
$18$	0	0
$19$	−4.27627	−0.981043	−0.490521	−	0.871429i	$-0.663194\pi$
$19$	−4.27627	−0.981043	−0.490521	+	0.871429i	$0.663194\pi$
$20$	0	0
$21$	−1.25807	−0.274534
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.499463	0.0998926
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	10.1300	1.81941	0.909704	−	0.415256i	$-0.136308\pi$
$31$	10.1300	1.81941	0.909704	+	0.415256i	$0.136308\pi$
$32$	0	0
$33$	−6.44999	−1.12280
$34$	0	0
$35$	2.95030	0.498692
$36$	0	0
$37$	−5.20798	−0.856187	−0.428093	−	0.903735i	$-0.640815\pi$
$37$	−5.20798	−0.856187	−0.428093	+	0.903735i	$0.640815\pi$
$38$	0	0
$39$	1.34717	0.215720
$40$	0	0
$41$	−10.6226	−1.65898	−0.829489	−	0.558523i	$-0.811368\pi$
$41$	−10.6226	−1.65898	−0.829489	+	0.558523i	$0.811368\pi$
$42$	0	0
$43$	0.118668	0.0180967	0.00904836	−	0.999959i	$-0.497120\pi$
$43$	0.118668	0.0180967	0.00904836	+	0.999959i	$0.497120\pi$
$44$	0	0
$45$	−2.34509	−0.349586
$46$	0	0
$47$	9.08470	1.32514	0.662570	−	0.749000i	$-0.269466\pi$
$47$	9.08470	1.32514	0.662570	+	0.749000i	$0.269466\pi$
$48$	0	0
$49$	−5.41725	−0.773893
$50$	0	0
$51$	−7.39461	−1.03545
$52$	0	0
$53$	10.4167	1.43085	0.715423	−	0.698691i	$-0.246234\pi$
$53$	10.4167	1.43085	0.715423	+	0.698691i	$0.246234\pi$
$54$	0	0
$55$	15.1258	2.03957
$56$	0	0
$57$	−4.27627	−0.566405
$58$	0	0
$59$	−3.39629	−0.442159	−0.221080	−	0.975256i	$-0.570958\pi$
$59$	−3.39629	−0.442159	−0.221080	+	0.975256i	$0.570958\pi$
$60$	0	0
$61$	−2.23961	−0.286753	−0.143377	−	0.989668i	$-0.545796\pi$
$61$	−2.23961	−0.286753	−0.143377	+	0.989668i	$0.545796\pi$
$62$	0	0
$63$	−1.25807	−0.158502
$64$	0	0
$65$	−3.15925	−0.391857
$66$	0	0
$67$	7.85081	0.959130	0.479565	−	0.877506i	$-0.340795\pi$
$67$	7.85081	0.959130	0.479565	+	0.877506i	$0.340795\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−10.7923	−1.28081	−0.640404	−	0.768038i	$-0.721233\pi$
$71$	−10.7923	−1.28081	−0.640404	+	0.768038i	$0.721233\pi$
$72$	0	0
$73$	7.17561	0.839842	0.419921	−	0.907561i	$-0.362058\pi$
$73$	7.17561	0.839842	0.419921	+	0.907561i	$0.362058\pi$
$74$	0	0
$75$	0.499463	0.0576730
$76$	0	0
$77$	8.11456	0.924740
$78$	0	0
$79$	−4.24131	−0.477185	−0.238592	−	0.971120i	$-0.576686\pi$
$79$	−4.24131	−0.477185	−0.238592	+	0.971120i	$0.576686\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.91341	−0.868609	−0.434305	−	0.900766i	$-0.643006\pi$
$83$	−7.91341	−0.868609	−0.434305	+	0.900766i	$0.643006\pi$
$84$	0	0
$85$	17.3410	1.88090
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−4.52488	−0.479636	−0.239818	−	0.970818i	$-0.577088\pi$
$89$	−4.52488	−0.479636	−0.239818	+	0.970818i	$0.577088\pi$
$90$	0	0
$91$	−1.69484	−0.177668
$92$	0	0
$93$	10.1300	1.05044
$94$	0	0
$95$	10.0282	1.02888
$96$	0	0
$97$	−0.279080	−0.0283363	−0.0141681	−	0.999900i	$-0.504510\pi$
$97$	−0.279080	−0.0283363	−0.0141681	+	0.999900i	$0.504510\pi$
$98$	0	0
$99$	−6.44999	−0.648248
$100$	0	0
$101$	1.46268	0.145542	0.0727709	−	0.997349i	$-0.476816\pi$
$101$	1.46268	0.145542	0.0727709	+	0.997349i	$0.476816\pi$
$102$	0	0
$103$	16.7654	1.65194	0.825970	−	0.563714i	$-0.190628\pi$
$103$	16.7654	1.65194	0.825970	+	0.563714i	$0.190628\pi$
$104$	0	0
$105$	2.95030	0.287920
$106$	0	0
$107$	4.86261	0.470087	0.235043	−	0.971985i	$-0.424477\pi$
$107$	4.86261	0.470087	0.235043	+	0.971985i	$0.424477\pi$
$108$	0	0
$109$	−3.10927	−0.297814	−0.148907	−	0.988851i	$-0.547575\pi$
$109$	−3.10927	−0.297814	−0.148907	+	0.988851i	$0.547575\pi$
$110$	0	0
$111$	−5.20798	−0.494320
$112$	0	0
$113$	−11.8302	−1.11290	−0.556448	−	0.830883i	$-0.687836\pi$
$113$	−11.8302	−1.11290	−0.556448	+	0.830883i	$0.687836\pi$
$114$	0	0
$115$	−2.34509	−0.218681
$116$	0	0
$117$	1.34717	0.124546
$118$	0	0
$119$	9.30296	0.852801
$120$	0	0
$121$	30.6024	2.78203
$122$	0	0
$123$	−10.6226	−0.957812
$124$	0	0
$125$	10.5542	0.943995
$126$	0	0
$127$	11.0935	0.984392	0.492196	−	0.870484i	$-0.336194\pi$
$127$	11.0935	0.984392	0.492196	+	0.870484i	$0.336194\pi$
$128$	0	0
$129$	0.118668	0.0104482
$130$	0	0
$131$	16.1623	1.41211	0.706055	−	0.708157i	$-0.250473\pi$
$131$	16.1623	1.41211	0.706055	+	0.708157i	$0.250473\pi$
$132$	0	0
$133$	5.37986	0.466493
$134$	0	0
$135$	−2.34509	−0.201834
$136$	0	0
$137$	4.08719	0.349192	0.174596	−	0.984640i	$-0.444138\pi$
$137$	4.08719	0.349192	0.174596	+	0.984640i	$0.444138\pi$
$138$	0	0
$139$	8.11732	0.688503	0.344251	−	0.938878i	$-0.388133\pi$
$139$	8.11732	0.688503	0.344251	+	0.938878i	$0.388133\pi$
$140$	0	0
$141$	9.08470	0.765069
$142$	0	0
$143$	−8.68926	−0.726632
$144$	0	0
$145$	2.34509	0.194749
$146$	0	0
$147$	−5.41725	−0.446807
$148$	0	0
$149$	7.82849	0.641335	0.320667	−	0.947192i	$-0.396093\pi$
$149$	7.82849	0.641335	0.320667	+	0.947192i	$0.396093\pi$
$150$	0	0
$151$	−19.0112	−1.54711	−0.773554	−	0.633730i	$-0.781523\pi$
$151$	−19.0112	−1.54711	−0.773554	+	0.633730i	$0.781523\pi$
$152$	0	0
$153$	−7.39461	−0.597819
$154$	0	0
$155$	−23.7559	−1.90812
$156$	0	0
$157$	13.2471	1.05723	0.528615	−	0.848862i	$-0.322712\pi$
$157$	13.2471	1.05723	0.528615	+	0.848862i	$0.322712\pi$
$158$	0	0
$159$	10.4167	0.826100
$160$	0	0
$161$	−1.25807	−0.0991501
$162$	0	0
$163$	−1.77321	−0.138889	−0.0694443	−	0.997586i	$-0.522123\pi$
$163$	−1.77321	−0.138889	−0.0694443	+	0.997586i	$0.522123\pi$
$164$	0	0
$165$	15.1258	1.17754
$166$	0	0
$167$	1.42037	0.109911	0.0549557	−	0.998489i	$-0.482498\pi$
$167$	1.42037	0.109911	0.0549557	+	0.998489i	$0.482498\pi$
$168$	0	0
$169$	−11.1851	−0.860394
$170$	0	0
$171$	−4.27627	−0.327014
$172$	0	0
$173$	13.6543	1.03812	0.519058	−	0.854739i	$-0.326283\pi$
$173$	13.6543	1.03812	0.519058	+	0.854739i	$0.326283\pi$
$174$	0	0
$175$	−0.628361	−0.0474996
$176$	0	0
$177$	−3.39629	−0.255281
$178$	0	0
$179$	−9.98920	−0.746628	−0.373314	−	0.927705i	$-0.621779\pi$
$179$	−9.98920	−0.746628	−0.373314	+	0.927705i	$0.621779\pi$
$180$	0	0
$181$	5.29509	0.393581	0.196790	−	0.980446i	$-0.436948\pi$
$181$	5.29509	0.393581	0.196790	+	0.980446i	$0.436948\pi$
$182$	0	0
$183$	−2.23961	−0.165557
$184$	0	0
$185$	12.2132	0.897932
$186$	0	0
$187$	47.6952	3.48782
$188$	0	0
$189$	−1.25807	−0.0915114
$190$	0	0
$191$	−11.3030	−0.817856	−0.408928	−	0.912567i	$-0.634097\pi$
$191$	−11.3030	−0.817856	−0.408928	+	0.912567i	$0.634097\pi$
$192$	0	0
$193$	24.2983	1.74903	0.874514	−	0.485001i	$-0.161181\pi$
$193$	24.2983	1.74903	0.874514	+	0.485001i	$0.161181\pi$
$194$	0	0
$195$	−3.15925	−0.226238
$196$	0	0
$197$	−0.204036	−0.0145370	−0.00726848	−	0.999974i	$-0.502314\pi$
$197$	−0.204036	−0.0145370	−0.00726848	+	0.999974i	$0.502314\pi$
$198$	0	0
$199$	−3.69588	−0.261994	−0.130997	−	0.991383i	$-0.541818\pi$
$199$	−3.69588	−0.261994	−0.130997	+	0.991383i	$0.541818\pi$
$200$	0	0
$201$	7.85081	0.553754
$202$	0	0
$203$	1.25807	0.0882994
$204$	0	0
$205$	24.9111	1.73987
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	27.5819	1.90788
$210$	0	0
$211$	−10.3858	−0.714989	−0.357495	−	0.933915i	$-0.616369\pi$
$211$	−10.3858	−0.714989	−0.357495	+	0.933915i	$0.616369\pi$
$212$	0	0
$213$	−10.7923	−0.739475
$214$	0	0
$215$	−0.278288	−0.0189791
$216$	0	0
$217$	−12.7443	−0.865142
$218$	0	0
$219$	7.17561	0.484883
$220$	0	0
$221$	−9.96183	−0.670105
$222$	0	0
$223$	2.39453	0.160350	0.0801749	−	0.996781i	$-0.474452\pi$
$223$	2.39453	0.160350	0.0801749	+	0.996781i	$0.474452\pi$
$224$	0	0
$225$	0.499463	0.0332975
$226$	0	0
$227$	−8.00245	−0.531141	−0.265571	−	0.964091i	$-0.585560\pi$
$227$	−8.00245	−0.531141	−0.265571	+	0.964091i	$0.585560\pi$
$228$	0	0
$229$	−10.7641	−0.711312	−0.355656	−	0.934617i	$-0.615743\pi$
$229$	−10.7641	−0.711312	−0.355656	+	0.934617i	$0.615743\pi$
$230$	0	0
$231$	8.11456	0.533899
$232$	0	0
$233$	−0.554990	−0.0363586	−0.0181793	−	0.999835i	$-0.505787\pi$
$233$	−0.554990	−0.0363586	−0.0181793	+	0.999835i	$0.505787\pi$
$234$	0	0
$235$	−21.3045	−1.38975
$236$	0	0
$237$	−4.24131	−0.275503
$238$	0	0
$239$	−17.4662	−1.12979	−0.564897	−	0.825162i	$-0.691084\pi$
$239$	−17.4662	−1.12979	−0.564897	+	0.825162i	$0.691084\pi$
$240$	0	0
$241$	16.5598	1.06671	0.533354	−	0.845892i	$-0.320931\pi$
$241$	16.5598	1.06671	0.533354	+	0.845892i	$0.320931\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	12.7040	0.811626
$246$	0	0
$247$	−5.76087	−0.366556
$248$	0	0
$249$	−7.91341	−0.501492
$250$	0	0
$251$	4.55045	0.287222	0.143611	−	0.989634i	$-0.454129\pi$
$251$	4.55045	0.287222	0.143611	+	0.989634i	$0.454129\pi$
$252$	0	0
$253$	−6.44999	−0.405507
$254$	0	0
$255$	17.3410	1.08594
$256$	0	0
$257$	−18.8900	−1.17833	−0.589163	−	0.808014i	$-0.700543\pi$
$257$	−18.8900	−1.17833	−0.589163	+	0.808014i	$0.700543\pi$
$258$	0	0
$259$	6.55202	0.407123
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−0.490569	−0.0302498	−0.0151249	−	0.999886i	$-0.504815\pi$
$263$	−0.490569	−0.0302498	−0.0151249	+	0.999886i	$0.504815\pi$
$264$	0	0
$265$	−24.4282	−1.50061
$266$	0	0
$267$	−4.52488	−0.276918
$268$	0	0
$269$	19.1733	1.16902	0.584508	−	0.811388i	$-0.301288\pi$
$269$	19.1733	1.16902	0.584508	+	0.811388i	$0.301288\pi$
$270$	0	0
$271$	−28.7636	−1.74726	−0.873631	−	0.486588i	$-0.838241\pi$
$271$	−28.7636	−1.74726	−0.873631	+	0.486588i	$0.838241\pi$
$272$	0	0
$273$	−1.69484	−0.102577
$274$	0	0
$275$	−3.22153	−0.194266
$276$	0	0
$277$	−24.2754	−1.45857	−0.729283	−	0.684212i	$-0.760146\pi$
$277$	−24.2754	−1.45857	−0.729283	+	0.684212i	$0.760146\pi$
$278$	0	0
$279$	10.1300	0.606470
$280$	0	0
$281$	−2.36539	−0.141107	−0.0705537	−	0.997508i	$-0.522477\pi$
$281$	−2.36539	−0.141107	−0.0705537	+	0.997508i	$0.522477\pi$
$282$	0	0
$283$	−2.24568	−0.133492	−0.0667459	−	0.997770i	$-0.521262\pi$
$283$	−2.24568	−0.133492	−0.0667459	+	0.997770i	$0.521262\pi$
$284$	0	0
$285$	10.0282	0.594022
$286$	0	0
$287$	13.3641	0.788856
$288$	0	0
$289$	37.6803	2.21649
$290$	0	0
$291$	−0.279080	−0.0163599
$292$	0	0
$293$	27.7523	1.62131	0.810655	−	0.585525i	$-0.199111\pi$
$293$	27.7523	1.62131	0.810655	+	0.585525i	$0.199111\pi$
$294$	0	0
$295$	7.96461	0.463718
$296$	0	0
$297$	−6.44999	−0.374266
$298$	0	0
$299$	1.34717	0.0779091
$300$	0	0
$301$	−0.149293	−0.00860512
$302$	0	0
$303$	1.46268	0.0840286
$304$	0	0
$305$	5.25210	0.300735
$306$	0	0
$307$	−4.01961	−0.229411	−0.114706	−	0.993400i	$-0.536592\pi$
$307$	−4.01961	−0.229411	−0.114706	+	0.993400i	$0.536592\pi$
$308$	0	0
$309$	16.7654	0.953748
$310$	0	0
$311$	−24.4335	−1.38550	−0.692748	−	0.721180i	$-0.743600\pi$
$311$	−24.4335	−1.38550	−0.692748	+	0.721180i	$0.743600\pi$
$312$	0	0
$313$	−16.4118	−0.927648	−0.463824	−	0.885927i	$-0.653523\pi$
$313$	−16.4118	−0.927648	−0.463824	+	0.885927i	$0.653523\pi$
$314$	0	0
$315$	2.95030	0.166231
$316$	0	0
$317$	21.1766	1.18939	0.594697	−	0.803950i	$-0.297272\pi$
$317$	21.1766	1.18939	0.594697	+	0.803950i	$0.297272\pi$
$318$	0	0
$319$	6.44999	0.361130
$320$	0	0
$321$	4.86261	0.271405
$322$	0	0
$323$	31.6213	1.75946
$324$	0	0
$325$	0.672863	0.0373237
$326$	0	0
$327$	−3.10927	−0.171943
$328$	0	0
$329$	−11.4292	−0.630113
$330$	0	0
$331$	−10.9757	−0.603278	−0.301639	−	0.953422i	$-0.597534\pi$
$331$	−10.9757	−0.603278	−0.301639	+	0.953422i	$0.597534\pi$
$332$	0	0
$333$	−5.20798	−0.285396
$334$	0	0
$335$	−18.4109	−1.00589
$336$	0	0
$337$	20.0132	1.09019	0.545094	−	0.838375i	$-0.316494\pi$
$337$	20.0132	1.09019	0.545094	+	0.838375i	$0.316494\pi$
$338$	0	0
$339$	−11.8302	−0.642531
$340$	0	0
$341$	−65.3387	−3.53829
$342$	0	0
$343$	15.6218	0.843499
$344$	0	0
$345$	−2.34509	−0.126256
$346$	0	0
$347$	16.3958	0.880171	0.440085	−	0.897956i	$-0.354948\pi$
$347$	16.3958	0.880171	0.440085	+	0.897956i	$0.354948\pi$
$348$	0	0
$349$	−5.00885	−0.268117	−0.134059	−	0.990973i	$-0.542801\pi$
$349$	−5.00885	−0.268117	−0.134059	+	0.990973i	$0.542801\pi$
$350$	0	0
$351$	1.34717	0.0719068
$352$	0	0
$353$	−4.40928	−0.234682	−0.117341	−	0.993092i	$-0.537437\pi$
$353$	−4.40928	−0.234682	−0.117341	+	0.993092i	$0.537437\pi$
$354$	0	0
$355$	25.3089	1.34326
$356$	0	0
$357$	9.30296	0.492365
$358$	0	0
$359$	1.94966	0.102899	0.0514495	−	0.998676i	$-0.483616\pi$
$359$	1.94966	0.102899	0.0514495	+	0.998676i	$0.483616\pi$
$360$	0	0
$361$	−0.713549	−0.0375552
$362$	0	0
$363$	30.6024	1.60621
$364$	0	0
$365$	−16.8275	−0.880791
$366$	0	0
$367$	12.5834	0.656847	0.328424	−	0.944531i	$-0.393483\pi$
$367$	12.5834	0.656847	0.328424	+	0.944531i	$0.393483\pi$
$368$	0	0
$369$	−10.6226	−0.552993
$370$	0	0
$371$	−13.1050	−0.680378
$372$	0	0
$373$	−18.3078	−0.947941	−0.473970	−	0.880541i	$-0.657180\pi$
$373$	−18.3078	−0.947941	−0.473970	+	0.880541i	$0.657180\pi$
$374$	0	0
$375$	10.5542	0.545016
$376$	0	0
$377$	−1.34717	−0.0693830
$378$	0	0
$379$	26.2890	1.35038	0.675188	−	0.737646i	$-0.264063\pi$
$379$	26.2890	1.35038	0.675188	+	0.737646i	$0.264063\pi$
$380$	0	0
$381$	11.0935	0.568339
$382$	0	0
$383$	−8.06675	−0.412192	−0.206096	−	0.978532i	$-0.566076\pi$
$383$	−8.06675	−0.412192	−0.206096	+	0.978532i	$0.566076\pi$
$384$	0	0
$385$	−19.0294	−0.969828
$386$	0	0
$387$	0.118668	0.00603224
$388$	0	0
$389$	34.4242	1.74538	0.872688	−	0.488279i	$-0.162375\pi$
$389$	34.4242	1.74538	0.872688	+	0.488279i	$0.162375\pi$
$390$	0	0
$391$	−7.39461	−0.373961
$392$	0	0
$393$	16.1623	0.815283
$394$	0	0
$395$	9.94627	0.500451
$396$	0	0
$397$	−24.7751	−1.24343	−0.621714	−	0.783244i	$-0.713563\pi$
$397$	−24.7751	−1.24343	−0.621714	+	0.783244i	$0.713563\pi$
$398$	0	0
$399$	5.37986	0.269330
$400$	0	0
$401$	4.97629	0.248504	0.124252	−	0.992251i	$-0.460347\pi$
$401$	4.97629	0.248504	0.124252	+	0.992251i	$0.460347\pi$
$402$	0	0
$403$	13.6469	0.679802
$404$	0	0
$405$	−2.34509	−0.116529
$406$	0	0
$407$	33.5914	1.66507
$408$	0	0
$409$	−21.3193	−1.05417	−0.527087	−	0.849811i	$-0.676716\pi$
$409$	−21.3193	−1.05417	−0.527087	+	0.849811i	$0.676716\pi$
$410$	0	0
$411$	4.08719	0.201606
$412$	0	0
$413$	4.27278	0.210250
$414$	0	0
$415$	18.5577	0.910961
$416$	0	0
$417$	8.11732	0.397507
$418$	0	0
$419$	34.4117	1.68112	0.840561	−	0.541717i	$-0.182226\pi$
$419$	34.4117	1.68112	0.840561	+	0.541717i	$0.182226\pi$
$420$	0	0
$421$	−13.3508	−0.650680	−0.325340	−	0.945597i	$-0.605479\pi$
$421$	−13.3508	−0.650680	−0.325340	+	0.945597i	$0.605479\pi$
$422$	0	0
$423$	9.08470	0.441713
$424$	0	0
$425$	−3.69333	−0.179153
$426$	0	0
$427$	2.81760	0.136353
$428$	0	0
$429$	−8.68926	−0.419521
$430$	0	0
$431$	−38.0934	−1.83490	−0.917448	−	0.397856i	$-0.869754\pi$
$431$	−38.0934	−1.83490	−0.917448	+	0.397856i	$0.869754\pi$
$432$	0	0
$433$	22.6448	1.08824	0.544120	−	0.839008i	$-0.316864\pi$
$433$	22.6448	1.08824	0.544120	+	0.839008i	$0.316864\pi$
$434$	0	0
$435$	2.34509	0.112439
$436$	0	0
$437$	−4.27627	−0.204562
$438$	0	0
$439$	29.9826	1.43099	0.715495	−	0.698618i	$-0.246201\pi$
$439$	29.9826	1.43099	0.715495	+	0.698618i	$0.246201\pi$
$440$	0	0
$441$	−5.41725	−0.257964
$442$	0	0
$443$	−14.1631	−0.672910	−0.336455	−	0.941699i	$-0.609228\pi$
$443$	−14.1631	−0.672910	−0.336455	+	0.941699i	$0.609228\pi$
$444$	0	0
$445$	10.6113	0.503022
$446$	0	0
$447$	7.82849	0.370275
$448$	0	0
$449$	33.1865	1.56617	0.783084	−	0.621915i	$-0.213645\pi$
$449$	33.1865	1.56617	0.783084	+	0.621915i	$0.213645\pi$
$450$	0	0
$451$	68.5160	3.22629
$452$	0	0
$453$	−19.0112	−0.893223
$454$	0	0
$455$	3.97457	0.186331
$456$	0	0
$457$	−4.41745	−0.206640	−0.103320	−	0.994648i	$-0.532947\pi$
$457$	−4.41745	−0.206640	−0.103320	+	0.994648i	$0.532947\pi$
$458$	0	0
$459$	−7.39461	−0.345151
$460$	0	0
$461$	39.2457	1.82785	0.913927	−	0.405878i	$-0.133034\pi$
$461$	39.2457	1.82785	0.913927	+	0.405878i	$0.133034\pi$
$462$	0	0
$463$	26.2262	1.21883	0.609416	−	0.792850i	$-0.291404\pi$
$463$	26.2262	1.21883	0.609416	+	0.792850i	$0.291404\pi$
$464$	0	0
$465$	−23.7559	−1.10165
$466$	0	0
$467$	12.1356	0.561571	0.280785	−	0.959771i	$-0.409405\pi$
$467$	12.1356	0.561571	0.280785	+	0.959771i	$0.409405\pi$
$468$	0	0
$469$	−9.87690	−0.456073
$470$	0	0
$471$	13.2471	0.610392
$472$	0	0
$473$	−0.765409	−0.0351935
$474$	0	0
$475$	−2.13584	−0.0979989
$476$	0	0
$477$	10.4167	0.476949
$478$	0	0
$479$	11.9110	0.544229	0.272115	−	0.962265i	$-0.412277\pi$
$479$	11.9110	0.544229	0.272115	+	0.962265i	$0.412277\pi$
$480$	0	0
$481$	−7.01606	−0.319905
$482$	0	0
$483$	−1.25807	−0.0572443
$484$	0	0
$485$	0.654468	0.0297179
$486$	0	0
$487$	19.9032	0.901899	0.450950	−	0.892549i	$-0.351085\pi$
$487$	19.9032	0.901899	0.450950	+	0.892549i	$0.351085\pi$
$488$	0	0
$489$	−1.77321	−0.0801874
$490$	0	0
$491$	18.2515	0.823680	0.411840	−	0.911256i	$-0.364886\pi$
$491$	18.2515	0.823680	0.411840	+	0.911256i	$0.364886\pi$
$492$	0	0
$493$	7.39461	0.333036
$494$	0	0
$495$	15.1258	0.679855
$496$	0	0
$497$	13.5775	0.609033
$498$	0	0
$499$	−5.11958	−0.229184	−0.114592	−	0.993413i	$-0.536556\pi$
$499$	−5.11958	−0.229184	−0.114592	+	0.993413i	$0.536556\pi$
$500$	0	0
$501$	1.42037	0.0634573
$502$	0	0
$503$	−20.4327	−0.911048	−0.455524	−	0.890224i	$-0.650548\pi$
$503$	−20.4327	−0.911048	−0.455524	+	0.890224i	$0.650548\pi$
$504$	0	0
$505$	−3.43011	−0.152638
$506$	0	0
$507$	−11.1851	−0.496749
$508$	0	0
$509$	9.82286	0.435390	0.217695	−	0.976017i	$-0.430146\pi$
$509$	9.82286	0.435390	0.217695	+	0.976017i	$0.430146\pi$
$510$	0	0
$511$	−9.02745	−0.399351
$512$	0	0
$513$	−4.27627	−0.188802
$514$	0	0
$515$	−39.3163	−1.73248
$516$	0	0
$517$	−58.5962	−2.57706
$518$	0	0
$519$	13.6543	0.599356
$520$	0	0
$521$	−26.3946	−1.15637	−0.578184	−	0.815906i	$-0.696238\pi$
$521$	−26.3946	−1.15637	−0.578184	+	0.815906i	$0.696238\pi$
$522$	0	0
$523$	−32.4210	−1.41767	−0.708835	−	0.705375i	$-0.750779\pi$
$523$	−32.4210	−1.41767	−0.708835	+	0.705375i	$0.750779\pi$
$524$	0	0
$525$	−0.628361	−0.0274239
$526$	0	0
$527$	−74.9077	−3.26303
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.39629	−0.147386
$532$	0	0
$533$	−14.3106	−0.619859
$534$	0	0
$535$	−11.4033	−0.493007
$536$	0	0
$537$	−9.98920	−0.431066
$538$	0	0
$539$	34.9412	1.50502
$540$	0	0
$541$	36.1436	1.55393	0.776967	−	0.629541i	$-0.216757\pi$
$541$	36.1436	1.55393	0.776967	+	0.629541i	$0.216757\pi$
$542$	0	0
$543$	5.29509	0.227234
$544$	0	0
$545$	7.29152	0.312334
$546$	0	0
$547$	11.5761	0.494958	0.247479	−	0.968893i	$-0.420398\pi$
$547$	11.5761	0.494958	0.247479	+	0.968893i	$0.420398\pi$
$548$	0	0
$549$	−2.23961	−0.0955844
$550$	0	0
$551$	4.27627	0.182175
$552$	0	0
$553$	5.33588	0.226905
$554$	0	0
$555$	12.2132	0.518422
$556$	0	0
$557$	−1.45992	−0.0618589	−0.0309295	−	0.999522i	$-0.509847\pi$
$557$	−1.45992	−0.0618589	−0.0309295	+	0.999522i	$0.509847\pi$
$558$	0	0
$559$	0.159867	0.00676164
$560$	0	0
$561$	47.6952	2.01369
$562$	0	0
$563$	−41.1215	−1.73307	−0.866533	−	0.499120i	$-0.833657\pi$
$563$	−41.1215	−1.73307	−0.866533	+	0.499120i	$0.833657\pi$
$564$	0	0
$565$	27.7430	1.16716
$566$	0	0
$567$	−1.25807	−0.0528341
$568$	0	0
$569$	−42.9105	−1.79890	−0.899451	−	0.437023i	$-0.856033\pi$
$569$	−42.9105	−1.79890	−0.899451	+	0.437023i	$0.856033\pi$
$570$	0	0
$571$	22.2812	0.932439	0.466220	−	0.884669i	$-0.345616\pi$
$571$	22.2812	0.932439	0.466220	+	0.884669i	$0.345616\pi$
$572$	0	0
$573$	−11.3030	−0.472189
$574$	0	0
$575$	0.499463	0.0208290
$576$	0	0
$577$	−1.03746	−0.0431899	−0.0215950	−	0.999767i	$-0.506874\pi$
$577$	−1.03746	−0.0431899	−0.0215950	+	0.999767i	$0.506874\pi$
$578$	0	0
$579$	24.2983	1.00980
$580$	0	0
$581$	9.95565	0.413030
$582$	0	0
$583$	−67.1878	−2.78263
$584$	0	0
$585$	−3.15925	−0.130619
$586$	0	0
$587$	4.14391	0.171038	0.0855188	−	0.996337i	$-0.472745\pi$
$587$	4.14391	0.171038	0.0855188	+	0.996337i	$0.472745\pi$
$588$	0	0
$589$	−43.3187	−1.78492
$590$	0	0
$591$	−0.204036	−0.00839292
$592$	0	0
$593$	−38.4566	−1.57922	−0.789612	−	0.613607i	$-0.789718\pi$
$593$	−38.4566	−1.57922	−0.789612	+	0.613607i	$0.789718\pi$
$594$	0	0
$595$	−21.8163	−0.894382
$596$	0	0
$597$	−3.69588	−0.151262
$598$	0	0
$599$	1.86570	0.0762303	0.0381151	−	0.999273i	$-0.487865\pi$
$599$	1.86570	0.0762303	0.0381151	+	0.999273i	$0.487865\pi$
$600$	0	0
$601$	29.4674	1.20200	0.600999	−	0.799250i	$-0.294769\pi$
$601$	29.4674	1.20200	0.600999	+	0.799250i	$0.294769\pi$
$602$	0	0
$603$	7.85081	0.319710
$604$	0	0
$605$	−71.7654	−2.91768
$606$	0	0
$607$	18.3829	0.746141	0.373070	−	0.927803i	$-0.378305\pi$
$607$	18.3829	0.746141	0.373070	+	0.927803i	$0.378305\pi$
$608$	0	0
$609$	1.25807	0.0509797
$610$	0	0
$611$	12.2387	0.495123
$612$	0	0
$613$	14.0636	0.568022	0.284011	−	0.958821i	$-0.408335\pi$
$613$	14.0636	0.568022	0.284011	+	0.958821i	$0.408335\pi$
$614$	0	0
$615$	24.9111	1.00451
$616$	0	0
$617$	41.3676	1.66540	0.832699	−	0.553726i	$-0.186795\pi$
$617$	41.3676	1.66540	0.832699	+	0.553726i	$0.186795\pi$
$618$	0	0
$619$	−13.9854	−0.562121	−0.281060	−	0.959690i	$-0.590686\pi$
$619$	−13.9854	−0.562121	−0.281060	+	0.959690i	$0.590686\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	5.69263	0.228070
$624$	0	0
$625$	−27.2479	−1.08991
$626$	0	0
$627$	27.5819	1.10151
$628$	0	0
$629$	38.5110	1.53553
$630$	0	0
$631$	−5.28654	−0.210454	−0.105227	−	0.994448i	$-0.533557\pi$
$631$	−5.28654	−0.210454	−0.105227	+	0.994448i	$0.533557\pi$
$632$	0	0
$633$	−10.3858	−0.412799
$634$	0	0
$635$	−26.0154	−1.03239
$636$	0	0
$637$	−7.29798	−0.289156
$638$	0	0
$639$	−10.7923	−0.426936
$640$	0	0
$641$	6.20988	0.245276	0.122638	−	0.992451i	$-0.460865\pi$
$641$	6.20988	0.245276	0.122638	+	0.992451i	$0.460865\pi$
$642$	0	0
$643$	−16.7158	−0.659205	−0.329603	−	0.944120i	$-0.606915\pi$
$643$	−16.7158	−0.659205	−0.329603	+	0.944120i	$0.606915\pi$
$644$	0	0
$645$	−0.278288	−0.0109576
$646$	0	0
$647$	−36.2631	−1.42565	−0.712825	−	0.701342i	$-0.752584\pi$
$647$	−36.2631	−1.42565	−0.712825	+	0.701342i	$0.752584\pi$
$648$	0	0
$649$	21.9060	0.859887
$650$	0	0
$651$	−12.7443	−0.499490
$652$	0	0
$653$	17.4650	0.683458	0.341729	−	0.939799i	$-0.388988\pi$
$653$	17.4650	0.683458	0.341729	+	0.939799i	$0.388988\pi$
$654$	0	0
$655$	−37.9022	−1.48096
$656$	0	0
$657$	7.17561	0.279947
$658$	0	0
$659$	39.1244	1.52407	0.762034	−	0.647536i	$-0.224201\pi$
$659$	39.1244	1.52407	0.762034	+	0.647536i	$0.224201\pi$
$660$	0	0
$661$	−38.5339	−1.49879	−0.749397	−	0.662121i	$-0.769656\pi$
$661$	−38.5339	−1.49879	−0.749397	+	0.662121i	$0.769656\pi$
$662$	0	0
$663$	−9.96183	−0.386885
$664$	0	0
$665$	−12.6163	−0.489238
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	2.39453	0.0925780
$670$	0	0
$671$	14.4455	0.557662
$672$	0	0
$673$	1.56580	0.0603570	0.0301785	−	0.999545i	$-0.490392\pi$
$673$	1.56580	0.0603570	0.0301785	+	0.999545i	$0.490392\pi$
$674$	0	0
$675$	0.499463	0.0192243
$676$	0	0
$677$	15.1966	0.584051	0.292026	−	0.956410i	$-0.405671\pi$
$677$	15.1966	0.584051	0.292026	+	0.956410i	$0.405671\pi$
$678$	0	0
$679$	0.351103	0.0134741
$680$	0	0
$681$	−8.00245	−0.306655
$682$	0	0
$683$	−26.3797	−1.00939	−0.504695	−	0.863298i	$-0.668395\pi$
$683$	−26.3797	−1.00939	−0.504695	+	0.863298i	$0.668395\pi$
$684$	0	0
$685$	−9.58485	−0.366218
$686$	0	0
$687$	−10.7641	−0.410676
$688$	0	0
$689$	14.0331	0.534620
$690$	0	0
$691$	−15.0803	−0.573681	−0.286840	−	0.957978i	$-0.592605\pi$
$691$	−15.0803	−0.573681	−0.286840	+	0.957978i	$0.592605\pi$
$692$	0	0
$693$	8.11456	0.308247
$694$	0	0
$695$	−19.0359	−0.722072
$696$	0	0
$697$	78.5503	2.97531
$698$	0	0
$699$	−0.554990	−0.0209917
$700$	0	0
$701$	−35.4316	−1.33823	−0.669116	−	0.743158i	$-0.733327\pi$
$701$	−35.4316	−1.33823	−0.669116	+	0.743158i	$0.733327\pi$
$702$	0	0
$703$	22.2707	0.839956
$704$	0	0
$705$	−21.3045	−0.802372
$706$	0	0
$707$	−1.84015	−0.0692061
$708$	0	0
$709$	−19.0617	−0.715877	−0.357939	−	0.933745i	$-0.616520\pi$
$709$	−19.0617	−0.715877	−0.357939	+	0.933745i	$0.616520\pi$
$710$	0	0
$711$	−4.24131	−0.159062
$712$	0	0
$713$	10.1300	0.379373
$714$	0	0
$715$	20.3771	0.762061
$716$	0	0
$717$	−17.4662	−0.652286
$718$	0	0
$719$	−6.27886	−0.234162	−0.117081	−	0.993122i	$-0.537354\pi$
$719$	−6.27886	−0.234162	−0.117081	+	0.993122i	$0.537354\pi$
$720$	0	0
$721$	−21.0920	−0.785509
$722$	0	0
$723$	16.5598	0.615864
$724$	0	0
$725$	−0.499463	−0.0185496
$726$	0	0
$727$	−14.8619	−0.551199	−0.275599	−	0.961273i	$-0.588876\pi$
$727$	−14.8619	−0.551199	−0.275599	+	0.961273i	$0.588876\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.877505	−0.0324557
$732$	0	0
$733$	52.6052	1.94302	0.971509	−	0.237004i	$-0.0761653\pi$
$733$	52.6052	1.94302	0.971509	+	0.237004i	$0.0761653\pi$
$734$	0	0
$735$	12.7040	0.468593
$736$	0	0
$737$	−50.6377	−1.86526
$738$	0	0
$739$	−0.794958	−0.0292430	−0.0146215	−	0.999893i	$-0.504654\pi$
$739$	−0.794958	−0.0292430	−0.0146215	+	0.999893i	$0.504654\pi$
$740$	0	0
$741$	−5.76087	−0.211631
$742$	0	0
$743$	5.73978	0.210572	0.105286	−	0.994442i	$-0.466424\pi$
$743$	5.73978	0.210572	0.105286	+	0.994442i	$0.466424\pi$
$744$	0	0
$745$	−18.3585	−0.672605
$746$	0	0
$747$	−7.91341	−0.289536
$748$	0	0
$749$	−6.11752	−0.223529
$750$	0	0
$751$	−14.7146	−0.536944	−0.268472	−	0.963287i	$-0.586519\pi$
$751$	−14.7146	−0.536944	−0.268472	+	0.963287i	$0.586519\pi$
$752$	0	0
$753$	4.55045	0.165828
$754$	0	0
$755$	44.5830	1.62254
$756$	0	0
$757$	25.8286	0.938756	0.469378	−	0.882997i	$-0.344478\pi$
$757$	25.8286	0.938756	0.469378	+	0.882997i	$0.344478\pi$
$758$	0	0
$759$	−6.44999	−0.234120
$760$	0	0
$761$	20.2387	0.733651	0.366826	−	0.930290i	$-0.380445\pi$
$761$	20.2387	0.733651	0.366826	+	0.930290i	$0.380445\pi$
$762$	0	0
$763$	3.91169	0.141613
$764$	0	0
$765$	17.3410	0.626967
$766$	0	0
$767$	−4.57539	−0.165208
$768$	0	0
$769$	−48.6534	−1.75449	−0.877244	−	0.480045i	$-0.840620\pi$
$769$	−48.6534	−1.75449	−0.877244	+	0.480045i	$0.840620\pi$
$770$	0	0
$771$	−18.8900	−0.680307
$772$	0	0
$773$	23.0404	0.828707	0.414353	−	0.910116i	$-0.364008\pi$
$773$	23.0404	0.828707	0.414353	+	0.910116i	$0.364008\pi$
$774$	0	0
$775$	5.05958	0.181745
$776$	0	0
$777$	6.55202	0.235052
$778$	0	0
$779$	45.4253	1.62753
$780$	0	0
$781$	69.6101	2.49085
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−31.0656	−1.10878
$786$	0	0
$787$	−2.48766	−0.0886756	−0.0443378	−	0.999017i	$-0.514118\pi$
$787$	−2.48766	−0.0886756	−0.0443378	+	0.999017i	$0.514118\pi$
$788$	0	0
$789$	−0.490569	−0.0174647
$790$	0	0
$791$	14.8833	0.529190
$792$	0	0
$793$	−3.01715	−0.107142
$794$	0	0
$795$	−24.4282	−0.866378
$796$	0	0
$797$	−47.1618	−1.67056	−0.835278	−	0.549828i	$-0.814693\pi$
$797$	−47.1618	−1.67056	−0.835278	+	0.549828i	$0.814693\pi$
$798$	0	0
$799$	−67.1778	−2.37658
$800$	0	0
$801$	−4.52488	−0.159879
$802$	0	0
$803$	−46.2826	−1.63328
$804$	0	0
$805$	2.95030	0.103984
$806$	0	0
$807$	19.1733	0.674931
$808$	0	0
$809$	35.7581	1.25719	0.628593	−	0.777734i	$-0.283631\pi$
$809$	35.7581	1.25719	0.628593	+	0.777734i	$0.283631\pi$
$810$	0	0
$811$	−21.0722	−0.739945	−0.369973	−	0.929043i	$-0.620633\pi$
$811$	−21.0722	−0.739945	−0.369973	+	0.929043i	$0.620633\pi$
$812$	0	0
$813$	−28.7636	−1.00878
$814$	0	0
$815$	4.15835	0.145661
$816$	0	0
$817$	−0.507457	−0.0177537
$818$	0	0
$819$	−1.69484	−0.0592226
$820$	0	0
$821$	−29.8141	−1.04052	−0.520260	−	0.854008i	$-0.674165\pi$
$821$	−29.8141	−1.04052	−0.520260	+	0.854008i	$0.674165\pi$
$822$	0	0
$823$	42.3600	1.47658	0.738289	−	0.674485i	$-0.235634\pi$
$823$	42.3600	1.47658	0.738289	+	0.674485i	$0.235634\pi$
$824$	0	0
$825$	−3.22153	−0.112159
$826$	0	0
$827$	−30.7101	−1.06789	−0.533947	−	0.845518i	$-0.679292\pi$
$827$	−30.7101	−1.06789	−0.533947	+	0.845518i	$0.679292\pi$
$828$	0	0
$829$	−1.49418	−0.0518952	−0.0259476	−	0.999663i	$-0.508260\pi$
$829$	−1.49418	−0.0518952	−0.0259476	+	0.999663i	$0.508260\pi$
$830$	0	0
$831$	−24.2754	−0.842104
$832$	0	0
$833$	40.0585	1.38794
$834$	0	0
$835$	−3.33089	−0.115270
$836$	0	0
$837$	10.1300	0.350145
$838$	0	0
$839$	23.4228	0.808643	0.404322	−	0.914617i	$-0.367508\pi$
$839$	23.4228	0.808643	0.404322	+	0.914617i	$0.367508\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.36539	−0.0814685
$844$	0	0
$845$	26.2302	0.902345
$846$	0	0
$847$	−38.5000	−1.32288
$848$	0	0
$849$	−2.24568	−0.0770715
$850$	0	0
$851$	−5.20798	−0.178527
$852$	0	0
$853$	23.4457	0.802764	0.401382	−	0.915911i	$-0.368530\pi$
$853$	23.4457	0.802764	0.401382	+	0.915911i	$0.368530\pi$
$854$	0	0
$855$	10.0282	0.342959
$856$	0	0
$857$	34.2988	1.17163	0.585813	−	0.810446i	$-0.300775\pi$
$857$	34.2988	1.17163	0.585813	+	0.810446i	$0.300775\pi$
$858$	0	0
$859$	39.9851	1.36428	0.682138	−	0.731224i	$-0.261051\pi$
$859$	39.9851	1.36428	0.682138	+	0.731224i	$0.261051\pi$
$860$	0	0
$861$	13.3641	0.455446
$862$	0	0
$863$	16.3589	0.556864	0.278432	−	0.960456i	$-0.410185\pi$
$863$	16.3589	0.556864	0.278432	+	0.960456i	$0.410185\pi$
$864$	0	0
$865$	−32.0206	−1.08873
$866$	0	0
$867$	37.6803	1.27969
$868$	0	0
$869$	27.3564	0.928003
$870$	0	0
$871$	10.5764	0.358368
$872$	0	0
$873$	−0.279080	−0.00944542
$874$	0	0
$875$	−13.2779	−0.448876
$876$	0	0
$877$	21.8906	0.739192	0.369596	−	0.929192i	$-0.379496\pi$
$877$	21.8906	0.739192	0.369596	+	0.929192i	$0.379496\pi$
$878$	0	0
$879$	27.7523	0.936063
$880$	0	0
$881$	10.5595	0.355759	0.177880	−	0.984052i	$-0.443076\pi$
$881$	10.5595	0.355759	0.177880	+	0.984052i	$0.443076\pi$
$882$	0	0
$883$	18.8359	0.633879	0.316939	−	0.948446i	$-0.397345\pi$
$883$	18.8359	0.633879	0.316939	+	0.948446i	$0.397345\pi$
$884$	0	0
$885$	7.96461	0.267728
$886$	0	0
$887$	35.7738	1.20117	0.600584	−	0.799562i	$-0.294935\pi$
$887$	35.7738	1.20117	0.600584	+	0.799562i	$0.294935\pi$
$888$	0	0
$889$	−13.9565	−0.468086
$890$	0	0
$891$	−6.44999	−0.216083
$892$	0	0
$893$	−38.8486	−1.30002
$894$	0	0
$895$	23.4256	0.783032
$896$	0	0
$897$	1.34717	0.0449808
$898$	0	0
$899$	−10.1300	−0.337856
$900$	0	0
$901$	−77.0276	−2.56616
$902$	0	0
$903$	−0.149293	−0.00496817
$904$	0	0
$905$	−12.4175	−0.412771
$906$	0	0
$907$	33.6997	1.11898	0.559491	−	0.828837i	$-0.310997\pi$
$907$	33.6997	1.11898	0.559491	+	0.828837i	$0.310997\pi$
$908$	0	0
$909$	1.46268	0.0485139
$910$	0	0
$911$	−24.7400	−0.819672	−0.409836	−	0.912159i	$-0.634414\pi$
$911$	−24.7400	−0.819672	−0.409836	+	0.912159i	$0.634414\pi$
$912$	0	0
$913$	51.0414	1.68922
$914$	0	0
$915$	5.25210	0.173629
$916$	0	0
$917$	−20.3334	−0.671469
$918$	0	0
$919$	−49.1870	−1.62253	−0.811264	−	0.584680i	$-0.801220\pi$
$919$	−49.1870	−1.62253	−0.811264	+	0.584680i	$0.801220\pi$
$920$	0	0
$921$	−4.01961	−0.132451
$922$	0	0
$923$	−14.5391	−0.478560
$924$	0	0
$925$	−2.60119	−0.0855267
$926$	0	0
$927$	16.7654	0.550646
$928$	0	0
$929$	−32.2541	−1.05822	−0.529112	−	0.848552i	$-0.677475\pi$
$929$	−32.2541	−1.05822	−0.529112	+	0.848552i	$0.677475\pi$
$930$	0	0
$931$	23.1656	0.759222
$932$	0	0
$933$	−24.4335	−0.799916
$934$	0	0
$935$	−111.850	−3.65787
$936$	0	0
$937$	3.64033	0.118924	0.0594622	−	0.998231i	$-0.481061\pi$
$937$	3.64033	0.118924	0.0594622	+	0.998231i	$0.481061\pi$
$938$	0	0
$939$	−16.4118	−0.535578
$940$	0	0
$941$	−43.7029	−1.42467	−0.712336	−	0.701838i	$-0.752363\pi$
$941$	−43.7029	−1.42467	−0.712336	+	0.701838i	$0.752363\pi$
$942$	0	0
$943$	−10.6226	−0.345921
$944$	0	0
$945$	2.95030	0.0959732
$946$	0	0
$947$	−17.2115	−0.559297	−0.279649	−	0.960102i	$-0.590218\pi$
$947$	−17.2115	−0.559297	−0.279649	+	0.960102i	$0.590218\pi$
$948$	0	0
$949$	9.66680	0.313798
$950$	0	0
$951$	21.1766	0.686697
$952$	0	0
$953$	−44.0372	−1.42651	−0.713253	−	0.700907i	$-0.752779\pi$
$953$	−44.0372	−1.42651	−0.713253	+	0.700907i	$0.752779\pi$
$954$	0	0
$955$	26.5066	0.857733
$956$	0	0
$957$	6.44999	0.208499
$958$	0	0
$959$	−5.14199	−0.166043
$960$	0	0
$961$	71.6177	2.31025
$962$	0	0
$963$	4.86261	0.156696
$964$	0	0
$965$	−56.9817	−1.83431
$966$	0	0
$967$	−8.47069	−0.272399	−0.136200	−	0.990681i	$-0.543489\pi$
$967$	−8.47069	−0.272399	−0.136200	+	0.990681i	$0.543489\pi$
$968$	0	0
$969$	31.6213	1.01582
$970$	0	0
$971$	51.8562	1.66414	0.832072	−	0.554667i	$-0.187154\pi$
$971$	51.8562	1.66414	0.832072	+	0.554667i	$0.187154\pi$
$972$	0	0
$973$	−10.2122	−0.327388
$974$	0	0
$975$	0.672863	0.0215489
$976$	0	0
$977$	−35.2841	−1.12884	−0.564420	−	0.825488i	$-0.690900\pi$
$977$	−35.2841	−1.12884	−0.564420	+	0.825488i	$0.690900\pi$
$978$	0	0
$979$	29.1854	0.932770
$980$	0	0
$981$	−3.10927	−0.0992712
$982$	0	0
$983$	−24.4820	−0.780854	−0.390427	−	0.920634i	$-0.627673\pi$
$983$	−24.4820	−0.780854	−0.390427	+	0.920634i	$0.627673\pi$
$984$	0	0
$985$	0.478484	0.0152458
$986$	0	0
$987$	−11.4292	−0.363796
$988$	0	0
$989$	0.118668	0.00377343
$990$	0	0
$991$	−30.8045	−0.978538	−0.489269	−	0.872133i	$-0.662736\pi$
$991$	−30.8045	−0.978538	−0.489269	+	0.872133i	$0.662736\pi$
$992$	0	0
$993$	−10.9757	−0.348303
$994$	0	0
$995$	8.66719	0.274768
$996$	0	0
$997$	−29.1338	−0.922676	−0.461338	−	0.887224i	$-0.652631\pi$
$997$	−29.1338	−0.922676	−0.461338	+	0.887224i	$0.652631\pi$
$998$	0	0
$999$	−5.20798	−0.164773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.4	✓	16	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.34509 q^{5} -1.25807 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.34509 q^{5} -1.25807 q^{7} +1.00000 q^{9} -6.44999 q^{11} +1.34717 q^{13} -2.34509 q^{15} -7.39461 q^{17} -4.27627 q^{19} -1.25807 q^{21} +1.00000 q^{23} +0.499463 q^{25} +1.00000 q^{27} -1.00000 q^{29} +10.1300 q^{31} -6.44999 q^{33} +2.95030 q^{35} -5.20798 q^{37} +1.34717 q^{39} -10.6226 q^{41} +0.118668 q^{43} -2.34509 q^{45} +9.08470 q^{47} -5.41725 q^{49} -7.39461 q^{51} +10.4167 q^{53} +15.1258 q^{55} -4.27627 q^{57} -3.39629 q^{59} -2.23961 q^{61} -1.25807 q^{63} -3.15925 q^{65} +7.85081 q^{67} +1.00000 q^{69} -10.7923 q^{71} +7.17561 q^{73} +0.499463 q^{75} +8.11456 q^{77} -4.24131 q^{79} +1.00000 q^{81} -7.91341 q^{83} +17.3410 q^{85} -1.00000 q^{87} -4.52488 q^{89} -1.69484 q^{91} +10.1300 q^{93} +10.0282 q^{95} -0.279080 q^{97} -6.44999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more