LMFDB - Embedded newform 8024.2.a.w.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 $ x^20 - 2x^19 - 29x^18 + 51x^17 + 341x^16 - 514x^15 - 2114x^14 + 2629x^13 + 7532x^12 - 7342x^11 - 15668x^10 + 11260x^9 + 18269x^8 - 9059x^7 - 10680x^6 + 3386x^5 + 2473x^4 - 427x^3 - 193x^2 + 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.06424$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.06424 q^{3} -1.91818 q^{5} -4.17854 q^{7} -1.86738 q^{9} +O(q^{10})$	$q-1.06424 q^{3} -1.91818 q^{5} -4.17854 q^{7} -1.86738 q^{9} +0.160390 q^{11} +3.20750 q^{13} +2.04141 q^{15} +1.00000 q^{17} -0.133106 q^{19} +4.44698 q^{21} -2.36479 q^{23} -1.32058 q^{25} +5.18009 q^{27} -3.65819 q^{29} +4.61088 q^{31} -0.170694 q^{33} +8.01518 q^{35} +0.109732 q^{37} -3.41356 q^{39} -0.894148 q^{41} +6.91279 q^{43} +3.58198 q^{45} +5.35466 q^{47} +10.4602 q^{49} -1.06424 q^{51} -4.17274 q^{53} -0.307657 q^{55} +0.141657 q^{57} -1.00000 q^{59} -2.02739 q^{61} +7.80293 q^{63} -6.15256 q^{65} -14.2501 q^{67} +2.51672 q^{69} +7.18296 q^{71} +4.37690 q^{73} +1.40542 q^{75} -0.670195 q^{77} +12.6900 q^{79} +0.0892795 q^{81} +11.7671 q^{83} -1.91818 q^{85} +3.89321 q^{87} +11.5802 q^{89} -13.4026 q^{91} -4.90711 q^{93} +0.255321 q^{95} +1.44796 q^{97} -0.299510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * 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
$3$	−1.06424	−0.614442	−0.307221	−	0.951638i	$-0.599399\pi$
$3$	−1.06424	−0.614442	−0.307221	+	0.951638i	$0.599399\pi$
$4$	0	0
$5$	−1.91818	−0.857836	−0.428918	−	0.903343i	$-0.641105\pi$
$5$	−1.91818	−0.857836	−0.428918	+	0.903343i	$0.641105\pi$
$6$	0	0
$7$	−4.17854	−1.57934	−0.789669	−	0.613533i	$-0.789748\pi$
$7$	−4.17854	−1.57934	−0.789669	+	0.613533i	$0.789748\pi$
$8$	0	0
$9$	−1.86738	−0.622462
$10$	0	0
$11$	0.160390	0.0483594	0.0241797	−	0.999708i	$-0.492303\pi$
$11$	0.160390	0.0483594	0.0241797	+	0.999708i	$0.492303\pi$
$12$	0	0
$13$	3.20750	0.889600	0.444800	−	0.895630i	$-0.353275\pi$
$13$	3.20750	0.889600	0.444800	+	0.895630i	$0.353275\pi$
$14$	0	0
$15$	2.04141	0.527090
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.133106	−0.0305366	−0.0152683	−	0.999883i	$-0.504860\pi$
$19$	−0.133106	−0.0305366	−0.0152683	+	0.999883i	$0.504860\pi$
$20$	0	0
$21$	4.44698	0.970411
$22$	0	0
$23$	−2.36479	−0.493093	−0.246547	−	0.969131i	$-0.579296\pi$
$23$	−2.36479	−0.493093	−0.246547	+	0.969131i	$0.579296\pi$
$24$	0	0
$25$	−1.32058	−0.264117
$26$	0	0
$27$	5.18009	0.996908
$28$	0	0
$29$	−3.65819	−0.679309	−0.339654	−	0.940550i	$-0.610310\pi$
$29$	−3.65819	−0.679309	−0.339654	+	0.940550i	$0.610310\pi$
$30$	0	0
$31$	4.61088	0.828139	0.414070	−	0.910245i	$-0.364107\pi$
$31$	4.61088	0.828139	0.414070	+	0.910245i	$0.364107\pi$
$32$	0	0
$33$	−0.170694	−0.0297140
$34$	0	0
$35$	8.01518	1.35481
$36$	0	0
$37$	0.109732	0.0180398	0.00901991	−	0.999959i	$-0.497129\pi$
$37$	0.109732	0.0180398	0.00901991	+	0.999959i	$0.497129\pi$
$38$	0	0
$39$	−3.41356	−0.546607
$40$	0	0
$41$	−0.894148	−0.139643	−0.0698213	−	0.997560i	$-0.522243\pi$
$41$	−0.894148	−0.139643	−0.0698213	+	0.997560i	$0.522243\pi$
$42$	0	0
$43$	6.91279	1.05419	0.527096	−	0.849806i	$-0.323281\pi$
$43$	6.91279	1.05419	0.527096	+	0.849806i	$0.323281\pi$
$44$	0	0
$45$	3.58198	0.533970
$46$	0	0
$47$	5.35466	0.781057	0.390529	−	0.920591i	$-0.372292\pi$
$47$	5.35466	0.781057	0.390529	+	0.920591i	$0.372292\pi$
$48$	0	0
$49$	10.4602	1.49431
$50$	0	0
$51$	−1.06424	−0.149024
$52$	0	0
$53$	−4.17274	−0.573170	−0.286585	−	0.958055i	$-0.592520\pi$
$53$	−4.17274	−0.573170	−0.286585	+	0.958055i	$0.592520\pi$
$54$	0	0
$55$	−0.307657	−0.0414844
$56$	0	0
$57$	0.141657	0.0187629
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−2.02739	−0.259580	−0.129790	−	0.991541i	$-0.541430\pi$
$61$	−2.02739	−0.259580	−0.129790	+	0.991541i	$0.541430\pi$
$62$	0	0
$63$	7.80293	0.983077
$64$	0	0
$65$	−6.15256	−0.763131
$66$	0	0
$67$	−14.2501	−1.74092	−0.870462	−	0.492235i	$-0.836180\pi$
$67$	−14.2501	−1.74092	−0.870462	+	0.492235i	$0.836180\pi$
$68$	0	0
$69$	2.51672	0.302977
$70$	0	0
$71$	7.18296	0.852460	0.426230	−	0.904615i	$-0.359841\pi$
$71$	7.18296	0.852460	0.426230	+	0.904615i	$0.359841\pi$
$72$	0	0
$73$	4.37690	0.512278	0.256139	−	0.966640i	$-0.417550\pi$
$73$	4.37690	0.512278	0.256139	+	0.966640i	$0.417550\pi$
$74$	0	0
$75$	1.40542	0.162284
$76$	0	0
$77$	−0.670195	−0.0763758
$78$	0	0
$79$	12.6900	1.42774	0.713868	−	0.700280i	$-0.246942\pi$
$79$	12.6900	1.42774	0.713868	+	0.700280i	$0.246942\pi$
$80$	0	0
$81$	0.0892795	0.00991994
$82$	0	0
$83$	11.7671	1.29160	0.645802	−	0.763505i	$-0.276523\pi$
$83$	11.7671	1.29160	0.645802	+	0.763505i	$0.276523\pi$
$84$	0	0
$85$	−1.91818	−0.208056
$86$	0	0
$87$	3.89321	0.417395
$88$	0	0
$89$	11.5802	1.22750	0.613749	−	0.789501i	$-0.289661\pi$
$89$	11.5802	1.22750	0.613749	+	0.789501i	$0.289661\pi$
$90$	0	0
$91$	−13.4026	−1.40498
$92$	0	0
$93$	−4.90711	−0.508843
$94$	0	0
$95$	0.255321	0.0261954
$96$	0	0
$97$	1.44796	0.147018	0.0735091	−	0.997295i	$-0.476580\pi$
$97$	1.44796	0.147018	0.0735091	+	0.997295i	$0.476580\pi$
$98$	0	0
$99$	−0.299510	−0.0301019
$100$	0	0
$101$	8.08701	0.804687	0.402344	−	0.915489i	$-0.368196\pi$
$101$	8.08701	0.804687	0.402344	+	0.915489i	$0.368196\pi$
$102$	0	0
$103$	7.94206	0.782554	0.391277	−	0.920273i	$-0.372033\pi$
$103$	7.94206	0.782554	0.391277	+	0.920273i	$0.372033\pi$
$104$	0	0
$105$	−8.53011	−0.832454
$106$	0	0
$107$	19.5002	1.88516	0.942579	−	0.333984i	$-0.108393\pi$
$107$	19.5002	1.88516	0.942579	+	0.333984i	$0.108393\pi$
$108$	0	0
$109$	−6.89453	−0.660377	−0.330188	−	0.943915i	$-0.607112\pi$
$109$	−6.89453	−0.660377	−0.330188	+	0.943915i	$0.607112\pi$
$110$	0	0
$111$	−0.116782	−0.0110844
$112$	0	0
$113$	−9.44436	−0.888450	−0.444225	−	0.895915i	$-0.646521\pi$
$113$	−9.44436	−0.888450	−0.444225	+	0.895915i	$0.646521\pi$
$114$	0	0
$115$	4.53610	0.422993
$116$	0	0
$117$	−5.98963	−0.553742
$118$	0	0
$119$	−4.17854	−0.383046
$120$	0	0
$121$	−10.9743	−0.997661
$122$	0	0
$123$	0.951592	0.0858022
$124$	0	0
$125$	12.1240	1.08441
$126$	0	0
$127$	−2.11312	−0.187509	−0.0937545	−	0.995595i	$-0.529887\pi$
$127$	−2.11312	−0.187509	−0.0937545	+	0.995595i	$0.529887\pi$
$128$	0	0
$129$	−7.35690	−0.647739
$130$	0	0
$131$	−0.883783	−0.0772165	−0.0386082	−	0.999254i	$-0.512292\pi$
$131$	−0.883783	−0.0772165	−0.0386082	+	0.999254i	$0.512292\pi$
$132$	0	0
$133$	0.556187	0.0482275
$134$	0	0
$135$	−9.93634	−0.855184
$136$	0	0
$137$	−21.4698	−1.83429	−0.917143	−	0.398558i	$-0.869511\pi$
$137$	−21.4698	−1.83429	−0.917143	+	0.398558i	$0.869511\pi$
$138$	0	0
$139$	−20.0264	−1.69861	−0.849307	−	0.527899i	$-0.822980\pi$
$139$	−20.0264	−1.69861	−0.849307	+	0.527899i	$0.822980\pi$
$140$	0	0
$141$	−5.69867	−0.479914
$142$	0	0
$143$	0.514450	0.0430205
$144$	0	0
$145$	7.01707	0.582736
$146$	0	0
$147$	−11.1322	−0.918165
$148$	0	0
$149$	3.53553	0.289642	0.144821	−	0.989458i	$-0.453739\pi$
$149$	3.53553	0.289642	0.144821	+	0.989458i	$0.453739\pi$
$150$	0	0
$151$	24.2641	1.97458	0.987290	−	0.158926i	$-0.0508033\pi$
$151$	24.2641	1.97458	0.987290	+	0.158926i	$0.0508033\pi$
$152$	0	0
$153$	−1.86738	−0.150969
$154$	0	0
$155$	−8.84451	−0.710408
$156$	0	0
$157$	5.16318	0.412067	0.206033	−	0.978545i	$-0.433944\pi$
$157$	5.16318	0.412067	0.206033	+	0.978545i	$0.433944\pi$
$158$	0	0
$159$	4.44082	0.352180
$160$	0	0
$161$	9.88137	0.778761
$162$	0	0
$163$	−20.6150	−1.61469	−0.807346	−	0.590078i	$-0.799097\pi$
$163$	−20.6150	−1.61469	−0.807346	+	0.590078i	$0.799097\pi$
$164$	0	0
$165$	0.327422	0.0254898
$166$	0	0
$167$	23.9955	1.85683	0.928415	−	0.371545i	$-0.121172\pi$
$167$	23.9955	1.85683	0.928415	+	0.371545i	$0.121172\pi$
$168$	0	0
$169$	−2.71196	−0.208613
$170$	0	0
$171$	0.248560	0.0190078
$172$	0	0
$173$	−10.6639	−0.810759	−0.405380	−	0.914148i	$-0.632861\pi$
$173$	−10.6639	−0.810759	−0.405380	+	0.914148i	$0.632861\pi$
$174$	0	0
$175$	5.51811	0.417130
$176$	0	0
$177$	1.06424	0.0799935
$178$	0	0
$179$	22.2794	1.66524	0.832618	−	0.553847i	$-0.186841\pi$
$179$	22.2794	1.66524	0.832618	+	0.553847i	$0.186841\pi$
$180$	0	0
$181$	−10.2428	−0.761343	−0.380671	−	0.924710i	$-0.624307\pi$
$181$	−10.2428	−0.761343	−0.380671	+	0.924710i	$0.624307\pi$
$182$	0	0
$183$	2.15763	0.159497
$184$	0	0
$185$	−0.210486	−0.0154752
$186$	0	0
$187$	0.160390	0.0117289
$188$	0	0
$189$	−21.6452	−1.57445
$190$	0	0
$191$	−18.5120	−1.33948	−0.669739	−	0.742596i	$-0.733594\pi$
$191$	−18.5120	−1.33948	−0.669739	+	0.742596i	$0.733594\pi$
$192$	0	0
$193$	10.5029	0.756013	0.378006	−	0.925803i	$-0.376610\pi$
$193$	10.5029	0.756013	0.378006	+	0.925803i	$0.376610\pi$
$194$	0	0
$195$	6.54782	0.468899
$196$	0	0
$197$	−0.744946	−0.0530752	−0.0265376	−	0.999648i	$-0.508448\pi$
$197$	−0.744946	−0.0530752	−0.0265376	+	0.999648i	$0.508448\pi$
$198$	0	0
$199$	−12.5826	−0.891955	−0.445978	−	0.895044i	$-0.647144\pi$
$199$	−12.5826	−0.891955	−0.445978	+	0.895044i	$0.647144\pi$
$200$	0	0
$201$	15.1656	1.06970
$202$	0	0
$203$	15.2859	1.07286
$204$	0	0
$205$	1.71514	0.119790
$206$	0	0
$207$	4.41598	0.306932
$208$	0	0
$209$	−0.0213488	−0.00147673
$210$	0	0
$211$	−17.8039	−1.22567	−0.612837	−	0.790210i	$-0.709972\pi$
$211$	−17.8039	−1.22567	−0.612837	+	0.790210i	$0.709972\pi$
$212$	0	0
$213$	−7.64442	−0.523787
$214$	0	0
$215$	−13.2600	−0.904323
$216$	0	0
$217$	−19.2667	−1.30791
$218$	0	0
$219$	−4.65809	−0.314765
$220$	0	0
$221$	3.20750	0.215760
$222$	0	0
$223$	−4.10375	−0.274807	−0.137404	−	0.990515i	$-0.543876\pi$
$223$	−4.10375	−0.274807	−0.137404	+	0.990515i	$0.543876\pi$
$224$	0	0
$225$	2.46604	0.164403
$226$	0	0
$227$	−16.6845	−1.10739	−0.553694	−	0.832720i	$-0.686782\pi$
$227$	−16.6845	−1.10739	−0.553694	+	0.832720i	$0.686782\pi$
$228$	0	0
$229$	11.0375	0.729381	0.364690	−	0.931129i	$-0.381175\pi$
$229$	11.0375	0.729381	0.364690	+	0.931129i	$0.381175\pi$
$230$	0	0
$231$	0.713251	0.0469285
$232$	0	0
$233$	−12.4340	−0.814577	−0.407288	−	0.913300i	$-0.633526\pi$
$233$	−12.4340	−0.814577	−0.407288	+	0.913300i	$0.633526\pi$
$234$	0	0
$235$	−10.2712	−0.670019
$236$	0	0
$237$	−13.5053	−0.877260
$238$	0	0
$239$	−24.1773	−1.56390	−0.781950	−	0.623342i	$-0.785775\pi$
$239$	−24.1773	−1.56390	−0.781950	+	0.623342i	$0.785775\pi$
$240$	0	0
$241$	0.904401	0.0582576	0.0291288	−	0.999576i	$-0.490727\pi$
$241$	0.904401	0.0582576	0.0291288	+	0.999576i	$0.490727\pi$
$242$	0	0
$243$	−15.6353	−1.00300
$244$	0	0
$245$	−20.0645	−1.28187
$246$	0	0
$247$	−0.426936	−0.0271653
$248$	0	0
$249$	−12.5230	−0.793615
$250$	0	0
$251$	−1.87554	−0.118383	−0.0591914	−	0.998247i	$-0.518852\pi$
$251$	−1.87554	−0.118383	−0.0591914	+	0.998247i	$0.518852\pi$
$252$	0	0
$253$	−0.379289	−0.0238457
$254$	0	0
$255$	2.04141	0.127838
$256$	0	0
$257$	0.600680	0.0374694	0.0187347	−	0.999824i	$-0.494036\pi$
$257$	0.600680	0.0374694	0.0187347	+	0.999824i	$0.494036\pi$
$258$	0	0
$259$	−0.458519	−0.0284910
$260$	0	0
$261$	6.83125	0.422844
$262$	0	0
$263$	−6.54078	−0.403322	−0.201661	−	0.979455i	$-0.564634\pi$
$263$	−6.54078	−0.403322	−0.201661	+	0.979455i	$0.564634\pi$
$264$	0	0
$265$	8.00408	0.491686
$266$	0	0
$267$	−12.3241	−0.754226
$268$	0	0
$269$	−18.6272	−1.13572	−0.567861	−	0.823124i	$-0.692229\pi$
$269$	−18.6272	−1.13572	−0.567861	+	0.823124i	$0.692229\pi$
$270$	0	0
$271$	−16.0864	−0.977181	−0.488590	−	0.872513i	$-0.662489\pi$
$271$	−16.0864	−0.977181	−0.488590	+	0.872513i	$0.662489\pi$
$272$	0	0
$273$	14.2637	0.863277
$274$	0	0
$275$	−0.211809	−0.0127725
$276$	0	0
$277$	−12.8508	−0.772132	−0.386066	−	0.922471i	$-0.626166\pi$
$277$	−12.8508	−0.772132	−0.386066	+	0.922471i	$0.626166\pi$
$278$	0	0
$279$	−8.61029	−0.515485
$280$	0	0
$281$	−2.78420	−0.166091	−0.0830457	−	0.996546i	$-0.526465\pi$
$281$	−2.78420	−0.166091	−0.0830457	+	0.996546i	$0.526465\pi$
$282$	0	0
$283$	9.47104	0.562995	0.281498	−	0.959562i	$-0.409169\pi$
$283$	9.47104	0.562995	0.281498	+	0.959562i	$0.409169\pi$
$284$	0	0
$285$	−0.271724	−0.0160955
$286$	0	0
$287$	3.73623	0.220543
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.54098	−0.0903341
$292$	0	0
$293$	14.0503	0.820826	0.410413	−	0.911900i	$-0.365385\pi$
$293$	14.0503	0.820826	0.410413	+	0.911900i	$0.365385\pi$
$294$	0	0
$295$	1.91818	0.111681
$296$	0	0
$297$	0.830834	0.0482099
$298$	0	0
$299$	−7.58506	−0.438656
$300$	0	0
$301$	−28.8854	−1.66492
$302$	0	0
$303$	−8.60655	−0.494433
$304$	0	0
$305$	3.88889	0.222677
$306$	0	0
$307$	−11.1274	−0.635071	−0.317536	−	0.948246i	$-0.602855\pi$
$307$	−11.1274	−0.635071	−0.317536	+	0.948246i	$0.602855\pi$
$308$	0	0
$309$	−8.45229	−0.480834
$310$	0	0
$311$	−14.0420	−0.796248	−0.398124	−	0.917332i	$-0.630339\pi$
$311$	−14.0420	−0.796248	−0.398124	+	0.917332i	$0.630339\pi$
$312$	0	0
$313$	−22.2299	−1.25651	−0.628254	−	0.778008i	$-0.716230\pi$
$313$	−22.2299	−1.25651	−0.628254	+	0.778008i	$0.716230\pi$
$314$	0	0
$315$	−14.9674	−0.843319
$316$	0	0
$317$	−33.6035	−1.88736	−0.943681	−	0.330857i	$-0.892662\pi$
$317$	−33.6035	−1.88736	−0.943681	+	0.330857i	$0.892662\pi$
$318$	0	0
$319$	−0.586737	−0.0328510
$320$	0	0
$321$	−20.7530	−1.15832
$322$	0	0
$323$	−0.133106	−0.00740620
$324$	0	0
$325$	−4.23577	−0.234958
$326$	0	0
$327$	7.33747	0.405763
$328$	0	0
$329$	−22.3746	−1.23355
$330$	0	0
$331$	26.8142	1.47384	0.736921	−	0.675979i	$-0.236279\pi$
$331$	26.8142	1.47384	0.736921	+	0.675979i	$0.236279\pi$
$332$	0	0
$333$	−0.204912	−0.0112291
$334$	0	0
$335$	27.3342	1.49343
$336$	0	0
$337$	16.7091	0.910201	0.455100	−	0.890440i	$-0.349603\pi$
$337$	16.7091	0.910201	0.455100	+	0.890440i	$0.349603\pi$
$338$	0	0
$339$	10.0511	0.545901
$340$	0	0
$341$	0.739539	0.0400483
$342$	0	0
$343$	−14.4584	−0.780679
$344$	0	0
$345$	−4.82751	−0.259905
$346$	0	0
$347$	−14.0969	−0.756761	−0.378381	−	0.925650i	$-0.623519\pi$
$347$	−14.0969	−0.756761	−0.378381	+	0.925650i	$0.623519\pi$
$348$	0	0
$349$	20.0202	1.07166	0.535828	−	0.844327i	$-0.320000\pi$
$349$	20.0202	1.07166	0.535828	+	0.844327i	$0.320000\pi$
$350$	0	0
$351$	16.6151	0.886849
$352$	0	0
$353$	2.09100	0.111292	0.0556462	−	0.998451i	$-0.482278\pi$
$353$	2.09100	0.111292	0.0556462	+	0.998451i	$0.482278\pi$
$354$	0	0
$355$	−13.7782	−0.731271
$356$	0	0
$357$	4.44698	0.235359
$358$	0	0
$359$	−3.17428	−0.167532	−0.0837661	−	0.996485i	$-0.526695\pi$
$359$	−3.17428	−0.167532	−0.0837661	+	0.996485i	$0.526695\pi$
$360$	0	0
$361$	−18.9823	−0.999068
$362$	0	0
$363$	11.6793	0.613005
$364$	0	0
$365$	−8.39569	−0.439451
$366$	0	0
$367$	24.2003	1.26325	0.631623	−	0.775276i	$-0.282389\pi$
$367$	24.2003	1.26325	0.631623	+	0.775276i	$0.282389\pi$
$368$	0	0
$369$	1.66972	0.0869221
$370$	0	0
$371$	17.4360	0.905230
$372$	0	0
$373$	6.80794	0.352502	0.176251	−	0.984345i	$-0.443603\pi$
$373$	6.80794	0.352502	0.176251	+	0.984345i	$0.443603\pi$
$374$	0	0
$375$	−12.9029	−0.666304
$376$	0	0
$377$	−11.7336	−0.604313
$378$	0	0
$379$	−20.6206	−1.05921	−0.529604	−	0.848245i	$-0.677659\pi$
$379$	−20.6206	−1.05921	−0.529604	+	0.848245i	$0.677659\pi$
$380$	0	0
$381$	2.24887	0.115213
$382$	0	0
$383$	−14.5057	−0.741204	−0.370602	−	0.928792i	$-0.620849\pi$
$383$	−14.5057	−0.741204	−0.370602	+	0.928792i	$0.620849\pi$
$384$	0	0
$385$	1.28556	0.0655180
$386$	0	0
$387$	−12.9088	−0.656193
$388$	0	0
$389$	26.5307	1.34516	0.672579	−	0.740025i	$-0.265187\pi$
$389$	26.5307	1.34516	0.672579	+	0.740025i	$0.265187\pi$
$390$	0	0
$391$	−2.36479	−0.119593
$392$	0	0
$393$	0.940561	0.0474450
$394$	0	0
$395$	−24.3417	−1.22476
$396$	0	0
$397$	−17.6572	−0.886192	−0.443096	−	0.896474i	$-0.646120\pi$
$397$	−17.6572	−0.886192	−0.443096	+	0.896474i	$0.646120\pi$
$398$	0	0
$399$	−0.591919	−0.0296330
$400$	0	0
$401$	−37.4327	−1.86930	−0.934650	−	0.355570i	$-0.884287\pi$
$401$	−37.4327	−1.86930	−0.934650	+	0.355570i	$0.884287\pi$
$402$	0	0
$403$	14.7894	0.736712
$404$	0	0
$405$	−0.171254	−0.00850968
$406$	0	0
$407$	0.0175999	0.000872395 0
$408$	0	0
$409$	−4.72211	−0.233493	−0.116747	−	0.993162i	$-0.537247\pi$
$409$	−4.72211	−0.233493	−0.116747	+	0.993162i	$0.537247\pi$
$410$	0	0
$411$	22.8491	1.12706
$412$	0	0
$413$	4.17854	0.205612
$414$	0	0
$415$	−22.5714	−1.10798
$416$	0	0
$417$	21.3129	1.04370
$418$	0	0
$419$	34.3620	1.67869	0.839347	−	0.543597i	$-0.182938\pi$
$419$	34.3620	1.67869	0.839347	+	0.543597i	$0.182938\pi$
$420$	0	0
$421$	6.12297	0.298415	0.149208	−	0.988806i	$-0.452328\pi$
$421$	6.12297	0.298415	0.149208	+	0.988806i	$0.452328\pi$
$422$	0	0
$423$	−9.99921	−0.486178
$424$	0	0
$425$	−1.32058	−0.0640578
$426$	0	0
$427$	8.47150	0.409965
$428$	0	0
$429$	−0.547501	−0.0264336
$430$	0	0
$431$	−16.3859	−0.789279	−0.394640	−	0.918836i	$-0.629131\pi$
$431$	−16.3859	−0.789279	−0.394640	+	0.918836i	$0.629131\pi$
$432$	0	0
$433$	33.7339	1.62115	0.810574	−	0.585637i	$-0.199155\pi$
$433$	33.7339	1.62115	0.810574	+	0.585637i	$0.199155\pi$
$434$	0	0
$435$	−7.46787	−0.358057
$436$	0	0
$437$	0.314767	0.0150574
$438$	0	0
$439$	24.2894	1.15927	0.579634	−	0.814877i	$-0.303196\pi$
$439$	24.2894	1.15927	0.579634	+	0.814877i	$0.303196\pi$
$440$	0	0
$441$	−19.5331	−0.930149
$442$	0	0
$443$	39.0511	1.85537	0.927686	−	0.373360i	$-0.121795\pi$
$443$	39.0511	1.85537	0.927686	+	0.373360i	$0.121795\pi$
$444$	0	0
$445$	−22.2129	−1.05299
$446$	0	0
$447$	−3.76267	−0.177968
$448$	0	0
$449$	−38.3012	−1.80754	−0.903772	−	0.428015i	$-0.859213\pi$
$449$	−38.3012	−1.80754	−0.903772	+	0.428015i	$0.859213\pi$
$450$	0	0
$451$	−0.143412	−0.00675303
$452$	0	0
$453$	−25.8229	−1.21326
$454$	0	0
$455$	25.7087	1.20524
$456$	0	0
$457$	−35.3265	−1.65251	−0.826253	−	0.563300i	$-0.809532\pi$
$457$	−35.3265	−1.65251	−0.826253	+	0.563300i	$0.809532\pi$
$458$	0	0
$459$	5.18009	0.241786
$460$	0	0
$461$	24.1249	1.12361	0.561805	−	0.827270i	$-0.310107\pi$
$461$	24.1249	1.12361	0.561805	+	0.827270i	$0.310107\pi$
$462$	0	0
$463$	40.6068	1.88716	0.943578	−	0.331150i	$-0.107437\pi$
$463$	40.6068	1.88716	0.943578	+	0.331150i	$0.107437\pi$
$464$	0	0
$465$	9.41271	0.436504
$466$	0	0
$467$	−12.3396	−0.571010	−0.285505	−	0.958377i	$-0.592161\pi$
$467$	−12.3396	−0.571010	−0.285505	+	0.958377i	$0.592161\pi$
$468$	0	0
$469$	59.5445	2.74951
$470$	0	0
$471$	−5.49488	−0.253191
$472$	0	0
$473$	1.10874	0.0509800
$474$	0	0
$475$	0.175777	0.00806522
$476$	0	0
$477$	7.79212	0.356777
$478$	0	0
$479$	1.90725	0.0871446	0.0435723	−	0.999050i	$-0.486126\pi$
$479$	1.90725	0.0871446	0.0435723	+	0.999050i	$0.486126\pi$
$480$	0	0
$481$	0.351965	0.0160482
$482$	0	0
$483$	−10.5162	−0.478503
$484$	0	0
$485$	−2.77745	−0.126118
$486$	0	0
$487$	−38.0405	−1.72378	−0.861891	−	0.507094i	$-0.830720\pi$
$487$	−38.0405	−1.72378	−0.861891	+	0.507094i	$0.830720\pi$
$488$	0	0
$489$	21.9394	0.992134
$490$	0	0
$491$	−6.05138	−0.273095	−0.136547	−	0.990634i	$-0.543601\pi$
$491$	−6.05138	−0.273095	−0.136547	+	0.990634i	$0.543601\pi$
$492$	0	0
$493$	−3.65819	−0.164757
$494$	0	0
$495$	0.574514	0.0258225
$496$	0	0
$497$	−30.0142	−1.34632
$498$	0	0
$499$	−10.3737	−0.464390	−0.232195	−	0.972669i	$-0.574591\pi$
$499$	−10.3737	−0.464390	−0.232195	+	0.972669i	$0.574591\pi$
$500$	0	0
$501$	−25.5371	−1.14091
$502$	0	0
$503$	14.1928	0.632827	0.316413	−	0.948621i	$-0.397521\pi$
$503$	14.1928	0.632827	0.316413	+	0.948621i	$0.397521\pi$
$504$	0	0
$505$	−15.5123	−0.690290
$506$	0	0
$507$	2.88619	0.128180
$508$	0	0
$509$	−19.4423	−0.861766	−0.430883	−	0.902408i	$-0.641798\pi$
$509$	−19.4423	−0.861766	−0.430883	+	0.902408i	$0.641798\pi$
$510$	0	0
$511$	−18.2891	−0.809060
$512$	0	0
$513$	−0.689499	−0.0304421
$514$	0	0
$515$	−15.2343	−0.671304
$516$	0	0
$517$	0.858834	0.0377715
$518$	0	0
$519$	11.3490	0.498164
$520$	0	0
$521$	−11.9208	−0.522258	−0.261129	−	0.965304i	$-0.584095\pi$
$521$	−11.9208	−0.522258	−0.261129	+	0.965304i	$0.584095\pi$
$522$	0	0
$523$	17.2111	0.752587	0.376293	−	0.926501i	$-0.377198\pi$
$523$	17.2111	0.752587	0.376293	+	0.926501i	$0.377198\pi$
$524$	0	0
$525$	−5.87261	−0.256302
$526$	0	0
$527$	4.61088	0.200853
$528$	0	0
$529$	−17.4078	−0.756859
$530$	0	0
$531$	1.86738	0.0810376
$532$	0	0
$533$	−2.86798	−0.124226
$534$	0	0
$535$	−37.4049	−1.61716
$536$	0	0
$537$	−23.7107	−1.02319
$538$	0	0
$539$	1.67770	0.0722638
$540$	0	0
$541$	8.79953	0.378321	0.189161	−	0.981946i	$-0.439423\pi$
$541$	8.79953	0.378321	0.189161	+	0.981946i	$0.439423\pi$
$542$	0	0
$543$	10.9009	0.467801
$544$	0	0
$545$	13.2250	0.566495
$546$	0	0
$547$	43.6704	1.86721	0.933605	−	0.358304i	$-0.116645\pi$
$547$	43.6704	1.86721	0.933605	+	0.358304i	$0.116645\pi$
$548$	0	0
$549$	3.78591	0.161579
$550$	0	0
$551$	0.486926	0.0207437
$552$	0	0
$553$	−53.0256	−2.25488
$554$	0	0
$555$	0.224008	0.00950861
$556$	0	0
$557$	4.97577	0.210830	0.105415	−	0.994428i	$-0.466383\pi$
$557$	4.97577	0.210830	0.105415	+	0.994428i	$0.466383\pi$
$558$	0	0
$559$	22.1728	0.937808
$560$	0	0
$561$	−0.170694	−0.00720671
$562$	0	0
$563$	34.6786	1.46153	0.730764	−	0.682631i	$-0.239164\pi$
$563$	34.6786	1.46153	0.730764	+	0.682631i	$0.239164\pi$
$564$	0	0
$565$	18.1160	0.762145
$566$	0	0
$567$	−0.373057	−0.0156669
$568$	0	0
$569$	−11.3371	−0.475274	−0.237637	−	0.971354i	$-0.576373\pi$
$569$	−11.3371	−0.475274	−0.237637	+	0.971354i	$0.576373\pi$
$570$	0	0
$571$	17.9001	0.749097	0.374549	−	0.927207i	$-0.377798\pi$
$571$	17.9001	0.749097	0.374549	+	0.927207i	$0.377798\pi$
$572$	0	0
$573$	19.7012	0.823031
$574$	0	0
$575$	3.12291	0.130234
$576$	0	0
$577$	−7.16496	−0.298281	−0.149141	−	0.988816i	$-0.547651\pi$
$577$	−7.16496	−0.298281	−0.149141	+	0.988816i	$0.547651\pi$
$578$	0	0
$579$	−11.1776	−0.464526
$580$	0	0
$581$	−49.1691	−2.03988
$582$	0	0
$583$	−0.669266	−0.0277182
$584$	0	0
$585$	11.4892	0.475020
$586$	0	0
$587$	39.0347	1.61114	0.805568	−	0.592503i	$-0.201860\pi$
$587$	39.0347	1.61114	0.805568	+	0.592503i	$0.201860\pi$
$588$	0	0
$589$	−0.613735	−0.0252885
$590$	0	0
$591$	0.792804	0.0326116
$592$	0	0
$593$	38.2259	1.56975	0.784875	−	0.619654i	$-0.212727\pi$
$593$	38.2259	1.56975	0.784875	+	0.619654i	$0.212727\pi$
$594$	0	0
$595$	8.01518	0.328590
$596$	0	0
$597$	13.3909	0.548054
$598$	0	0
$599$	15.0850	0.616358	0.308179	−	0.951328i	$-0.400280\pi$
$599$	15.0850	0.616358	0.308179	+	0.951328i	$0.400280\pi$
$600$	0	0
$601$	12.9564	0.528503	0.264252	−	0.964454i	$-0.414875\pi$
$601$	12.9564	0.528503	0.264252	+	0.964454i	$0.414875\pi$
$602$	0	0
$603$	26.6104	1.08366
$604$	0	0
$605$	21.0506	0.855830
$606$	0	0
$607$	7.07147	0.287022	0.143511	−	0.989649i	$-0.454161\pi$
$607$	7.07147	0.287022	0.143511	+	0.989649i	$0.454161\pi$
$608$	0	0
$609$	−16.2679	−0.659208
$610$	0	0
$611$	17.1751	0.694828
$612$	0	0
$613$	26.3286	1.06340	0.531700	−	0.846933i	$-0.321553\pi$
$613$	26.3286	1.06340	0.531700	+	0.846933i	$0.321553\pi$
$614$	0	0
$615$	−1.82532	−0.0736042
$616$	0	0
$617$	−28.9897	−1.16708	−0.583540	−	0.812084i	$-0.698333\pi$
$617$	−28.9897	−1.16708	−0.583540	+	0.812084i	$0.698333\pi$
$618$	0	0
$619$	−9.99213	−0.401618	−0.200809	−	0.979630i	$-0.564357\pi$
$619$	−9.99213	−0.401618	−0.200809	+	0.979630i	$0.564357\pi$
$620$	0	0
$621$	−12.2498	−0.491569
$622$	0	0
$623$	−48.3882	−1.93863
$624$	0	0
$625$	−16.6531	−0.666125
$626$	0	0
$627$	0.0227204	0.000907364 0
$628$	0	0
$629$	0.109732	0.00437530
$630$	0	0
$631$	28.0348	1.11605	0.558024	−	0.829825i	$-0.311560\pi$
$631$	28.0348	1.11605	0.558024	+	0.829825i	$0.311560\pi$
$632$	0	0
$633$	18.9477	0.753105
$634$	0	0
$635$	4.05334	0.160852
$636$	0	0
$637$	33.5509	1.32934
$638$	0	0
$639$	−13.4133	−0.530624
$640$	0	0
$641$	45.3405	1.79084	0.895421	−	0.445220i	$-0.146875\pi$
$641$	45.3405	1.79084	0.895421	+	0.445220i	$0.146875\pi$
$642$	0	0
$643$	−34.6203	−1.36529	−0.682645	−	0.730750i	$-0.739170\pi$
$643$	−34.6203	−1.36529	−0.682645	+	0.730750i	$0.739170\pi$
$644$	0	0
$645$	14.1119	0.555654
$646$	0	0
$647$	−48.8724	−1.92137	−0.960686	−	0.277637i	$-0.910449\pi$
$647$	−48.8724	−1.92137	−0.960686	+	0.277637i	$0.910449\pi$
$648$	0	0
$649$	−0.160390	−0.00629586
$650$	0	0
$651$	20.5045	0.803635
$652$	0	0
$653$	38.4942	1.50639	0.753197	−	0.657795i	$-0.228511\pi$
$653$	38.4942	1.50639	0.753197	+	0.657795i	$0.228511\pi$
$654$	0	0
$655$	1.69526	0.0662391
$656$	0	0
$657$	−8.17337	−0.318873
$658$	0	0
$659$	0.836603	0.0325894	0.0162947	−	0.999867i	$-0.494813\pi$
$659$	0.836603	0.0325894	0.0162947	+	0.999867i	$0.494813\pi$
$660$	0	0
$661$	−47.1025	−1.83207	−0.916037	−	0.401095i	$-0.868630\pi$
$661$	−47.1025	−1.83207	−0.916037	+	0.401095i	$0.868630\pi$
$662$	0	0
$663$	−3.41356	−0.132572
$664$	0	0
$665$	−1.06687	−0.0413713
$666$	0	0
$667$	8.65086	0.334963
$668$	0	0
$669$	4.36739	0.168853
$670$	0	0
$671$	−0.325172	−0.0125531
$672$	0	0
$673$	−1.11443	−0.0429581	−0.0214791	−	0.999769i	$-0.506838\pi$
$673$	−1.11443	−0.0429581	−0.0214791	+	0.999769i	$0.506838\pi$
$674$	0	0
$675$	−6.84074	−0.263300
$676$	0	0
$677$	26.4876	1.01800	0.509000	−	0.860767i	$-0.330015\pi$
$677$	26.4876	1.01800	0.509000	+	0.860767i	$0.330015\pi$
$678$	0	0
$679$	−6.05036	−0.232191
$680$	0	0
$681$	17.7564	0.680425
$682$	0	0
$683$	−19.0638	−0.729458	−0.364729	−	0.931114i	$-0.618838\pi$
$683$	−19.0638	−0.729458	−0.364729	+	0.931114i	$0.618838\pi$
$684$	0	0
$685$	41.1829	1.57352
$686$	0	0
$687$	−11.7466	−0.448162
$688$	0	0
$689$	−13.3841	−0.509892
$690$	0	0
$691$	−34.1379	−1.29867	−0.649334	−	0.760504i	$-0.724952\pi$
$691$	−34.1379	−1.29867	−0.649334	+	0.760504i	$0.724952\pi$
$692$	0	0
$693$	1.25151	0.0475410
$694$	0	0
$695$	38.4142	1.45713
$696$	0	0
$697$	−0.894148	−0.0338683
$698$	0	0
$699$	13.2328	0.500510
$700$	0	0
$701$	−15.4807	−0.584700	−0.292350	−	0.956311i	$-0.594437\pi$
$701$	−15.4807	−0.584700	−0.292350	+	0.956311i	$0.594437\pi$
$702$	0	0
$703$	−0.0146060	−0.000550874 0
$704$	0	0
$705$	10.9311	0.411688
$706$	0	0
$707$	−33.7918	−1.27087
$708$	0	0
$709$	16.2739	0.611177	0.305589	−	0.952164i	$-0.401147\pi$
$709$	16.2739	0.611177	0.305589	+	0.952164i	$0.401147\pi$
$710$	0	0
$711$	−23.6971	−0.888711
$712$	0	0
$713$	−10.9038	−0.408350
$714$	0	0
$715$	−0.986808	−0.0369045
$716$	0	0
$717$	25.7305	0.960925
$718$	0	0
$719$	43.6565	1.62811	0.814056	−	0.580786i	$-0.197255\pi$
$719$	43.6565	1.62811	0.814056	+	0.580786i	$0.197255\pi$
$720$	0	0
$721$	−33.1862	−1.23592
$722$	0	0
$723$	−0.962503	−0.0357959
$724$	0	0
$725$	4.83095	0.179417
$726$	0	0
$727$	−14.6382	−0.542902	−0.271451	−	0.962452i	$-0.587503\pi$
$727$	−14.6382	−0.542902	−0.271451	+	0.962452i	$0.587503\pi$
$728$	0	0
$729$	16.3719	0.606367
$730$	0	0
$731$	6.91279	0.255679
$732$	0	0
$733$	−44.8190	−1.65543	−0.827713	−	0.561151i	$-0.810359\pi$
$733$	−44.8190	−1.65543	−0.827713	+	0.561151i	$0.810359\pi$
$734$	0	0
$735$	21.3535	0.787635
$736$	0	0
$737$	−2.28557	−0.0841901
$738$	0	0
$739$	7.09630	0.261042	0.130521	−	0.991446i	$-0.458335\pi$
$739$	7.09630	0.261042	0.130521	+	0.991446i	$0.458335\pi$
$740$	0	0
$741$	0.454364	0.0166915
$742$	0	0
$743$	4.56646	0.167527	0.0837636	−	0.996486i	$-0.473306\pi$
$743$	4.56646	0.167527	0.0837636	+	0.996486i	$0.473306\pi$
$744$	0	0
$745$	−6.78179	−0.248465
$746$	0	0
$747$	−21.9736	−0.803974
$748$	0	0
$749$	−81.4824	−2.97730
$750$	0	0
$751$	−29.9832	−1.09410	−0.547051	−	0.837099i	$-0.684250\pi$
$751$	−29.9832	−1.09410	−0.547051	+	0.837099i	$0.684250\pi$
$752$	0	0
$753$	1.99603	0.0727393
$754$	0	0
$755$	−46.5428	−1.69387
$756$	0	0
$757$	−25.4267	−0.924150	−0.462075	−	0.886841i	$-0.652895\pi$
$757$	−25.4267	−0.924150	−0.462075	+	0.886841i	$0.652895\pi$
$758$	0	0
$759$	0.403656	0.0146518
$760$	0	0
$761$	25.9863	0.942002	0.471001	−	0.882133i	$-0.343893\pi$
$761$	25.9863	0.942002	0.471001	+	0.882133i	$0.343893\pi$
$762$	0	0
$763$	28.8091	1.04296
$764$	0	0
$765$	3.58198	0.129507
$766$	0	0
$767$	−3.20750	−0.115816
$768$	0	0
$769$	9.53023	0.343669	0.171834	−	0.985126i	$-0.445031\pi$
$769$	9.53023	0.343669	0.171834	+	0.985126i	$0.445031\pi$
$770$	0	0
$771$	−0.639270	−0.0230227
$772$	0	0
$773$	30.1718	1.08520	0.542602	−	0.839990i	$-0.317439\pi$
$773$	30.1718	1.08520	0.542602	+	0.839990i	$0.317439\pi$
$774$	0	0
$775$	−6.08906	−0.218726
$776$	0	0
$777$	0.487976	0.0175060
$778$	0	0
$779$	0.119016	0.00426420
$780$	0	0
$781$	1.15207	0.0412245
$782$	0	0
$783$	−18.9497	−0.677208
$784$	0	0
$785$	−9.90391	−0.353486
$786$	0	0
$787$	−27.9871	−0.997634	−0.498817	−	0.866707i	$-0.666232\pi$
$787$	−27.9871	−0.997634	−0.498817	+	0.866707i	$0.666232\pi$
$788$	0	0
$789$	6.96098	0.247818
$790$	0	0
$791$	39.4636	1.40316
$792$	0	0
$793$	−6.50283	−0.230922
$794$	0	0
$795$	−8.51829	−0.302113
$796$	0	0
$797$	−2.55194	−0.0903942	−0.0451971	−	0.998978i	$-0.514392\pi$
$797$	−2.55194	−0.0903942	−0.0451971	+	0.998978i	$0.514392\pi$
$798$	0	0
$799$	5.35466	0.189434
$800$	0	0
$801$	−21.6247	−0.764070
$802$	0	0
$803$	0.702012	0.0247735
$804$	0	0
$805$	−18.9542	−0.668049
$806$	0	0
$807$	19.8239	0.697835
$808$	0	0
$809$	−42.7084	−1.50155	−0.750774	−	0.660559i	$-0.770319\pi$
$809$	−42.7084	−1.50155	−0.750774	+	0.660559i	$0.770319\pi$
$810$	0	0
$811$	−37.7016	−1.32388	−0.661942	−	0.749555i	$-0.730267\pi$
$811$	−37.7016	−1.32388	−0.661942	+	0.749555i	$0.730267\pi$
$812$	0	0
$813$	17.1199	0.600420
$814$	0	0
$815$	39.5433	1.38514
$816$	0	0
$817$	−0.920133	−0.0321914
$818$	0	0
$819$	25.0279	0.874545
$820$	0	0
$821$	10.1329	0.353640	0.176820	−	0.984243i	$-0.443419\pi$
$821$	10.1329	0.353640	0.176820	+	0.984243i	$0.443419\pi$
$822$	0	0
$823$	13.0800	0.455941	0.227970	−	0.973668i	$-0.426791\pi$
$823$	13.0800	0.455941	0.227970	+	0.973668i	$0.426791\pi$
$824$	0	0
$825$	0.225416	0.00784798
$826$	0	0
$827$	5.91799	0.205789	0.102894	−	0.994692i	$-0.467190\pi$
$827$	5.91799	0.205789	0.102894	+	0.994692i	$0.467190\pi$
$828$	0	0
$829$	−0.495271	−0.0172015	−0.00860074	−	0.999963i	$-0.502738\pi$
$829$	−0.495271	−0.0172015	−0.00860074	+	0.999963i	$0.502738\pi$
$830$	0	0
$831$	13.6764	0.474430
$832$	0	0
$833$	10.4602	0.362423
$834$	0	0
$835$	−46.0278	−1.59286
$836$	0	0
$837$	23.8848	0.825578
$838$	0	0
$839$	23.2876	0.803976	0.401988	−	0.915645i	$-0.368319\pi$
$839$	23.2876	0.803976	0.401988	+	0.915645i	$0.368319\pi$
$840$	0	0
$841$	−15.6177	−0.538540
$842$	0	0
$843$	2.96307	0.102053
$844$	0	0
$845$	5.20204	0.178955
$846$	0	0
$847$	45.8564	1.57564
$848$	0	0
$849$	−10.0795	−0.345928
$850$	0	0
$851$	−0.259493	−0.00889531
$852$	0	0
$853$	−44.0576	−1.50850	−0.754251	−	0.656586i	$-0.772000\pi$
$853$	−44.0576	−1.50850	−0.754251	+	0.656586i	$0.772000\pi$
$854$	0	0
$855$	−0.476782	−0.0163056
$856$	0	0
$857$	1.18705	0.0405490	0.0202745	−	0.999794i	$-0.493546\pi$
$857$	1.18705	0.0405490	0.0202745	+	0.999794i	$0.493546\pi$
$858$	0	0
$859$	−0.901079	−0.0307444	−0.0153722	−	0.999882i	$-0.504893\pi$
$859$	−0.901079	−0.0307444	−0.0153722	+	0.999882i	$0.504893\pi$
$860$	0	0
$861$	−3.97626	−0.135511
$862$	0	0
$863$	31.1368	1.05991	0.529955	−	0.848026i	$-0.322209\pi$
$863$	31.1368	1.05991	0.529955	+	0.848026i	$0.322209\pi$
$864$	0	0
$865$	20.4552	0.695499
$866$	0	0
$867$	−1.06424	−0.0361436
$868$	0	0
$869$	2.03535	0.0690444
$870$	0	0
$871$	−45.7071	−1.54873
$872$	0	0
$873$	−2.70390	−0.0915132
$874$	0	0
$875$	−50.6606	−1.71264
$876$	0	0
$877$	−4.39845	−0.148525	−0.0742625	−	0.997239i	$-0.523660\pi$
$877$	−4.39845	−0.148525	−0.0742625	+	0.997239i	$0.523660\pi$
$878$	0	0
$879$	−14.9529	−0.504349
$880$	0	0
$881$	−28.2072	−0.950325	−0.475162	−	0.879898i	$-0.657611\pi$
$881$	−28.2072	−0.950325	−0.475162	+	0.879898i	$0.657611\pi$
$882$	0	0
$883$	21.5063	0.723745	0.361873	−	0.932228i	$-0.382137\pi$
$883$	21.5063	0.723745	0.361873	+	0.932228i	$0.382137\pi$
$884$	0	0
$885$	−2.04141	−0.0686213
$886$	0	0
$887$	0.494290	0.0165966	0.00829831	−	0.999966i	$-0.497359\pi$
$887$	0.494290	0.0165966	0.00829831	+	0.999966i	$0.497359\pi$
$888$	0	0
$889$	8.82974	0.296140
$890$	0	0
$891$	0.0143195	0.000479722 0
$892$	0	0
$893$	−0.712736	−0.0238508
$894$	0	0
$895$	−42.7358	−1.42850
$896$	0	0
$897$	8.07236	0.269528
$898$	0	0
$899$	−16.8675	−0.562562
$900$	0	0
$901$	−4.17274	−0.139014
$902$	0	0
$903$	30.7411	1.02300
$904$	0	0
$905$	19.6476	0.653108
$906$	0	0
$907$	−13.5811	−0.450953	−0.225477	−	0.974249i	$-0.572394\pi$
$907$	−13.5811	−0.450953	−0.225477	+	0.974249i	$0.572394\pi$
$908$	0	0
$909$	−15.1016	−0.500887
$910$	0	0
$911$	40.9612	1.35711	0.678553	−	0.734551i	$-0.262607\pi$
$911$	40.9612	1.35711	0.678553	+	0.734551i	$0.262607\pi$
$912$	0	0
$913$	1.88732	0.0624612
$914$	0	0
$915$	−4.13873	−0.136822
$916$	0	0
$917$	3.69292	0.121951
$918$	0	0
$919$	−7.72818	−0.254929	−0.127465	−	0.991843i	$-0.540684\pi$
$919$	−7.72818	−0.254929	−0.127465	+	0.991843i	$0.540684\pi$
$920$	0	0
$921$	11.8422	0.390214
$922$	0	0
$923$	23.0393	0.758348
$924$	0	0
$925$	−0.144910	−0.00476462
$926$	0	0
$927$	−14.8309	−0.487110
$928$	0	0
$929$	−51.9285	−1.70372	−0.851860	−	0.523769i	$-0.824525\pi$
$929$	−51.9285	−1.70372	−0.851860	+	0.523769i	$0.824525\pi$
$930$	0	0
$931$	−1.39231	−0.0456310
$932$	0	0
$933$	14.9441	0.489248
$934$	0	0
$935$	−0.307657	−0.0100615
$936$	0	0
$937$	−34.9388	−1.14140	−0.570701	−	0.821158i	$-0.693328\pi$
$937$	−34.9388	−1.14140	−0.570701	+	0.821158i	$0.693328\pi$
$938$	0	0
$939$	23.6580	0.772051
$940$	0	0
$941$	12.9325	0.421588	0.210794	−	0.977530i	$-0.432395\pi$
$941$	12.9325	0.421588	0.210794	+	0.977530i	$0.432395\pi$
$942$	0	0
$943$	2.11447	0.0688568
$944$	0	0
$945$	41.5193	1.35062
$946$	0	0
$947$	−8.42827	−0.273882	−0.136941	−	0.990579i	$-0.543727\pi$
$947$	−8.42827	−0.273882	−0.136941	+	0.990579i	$0.543727\pi$
$948$	0	0
$949$	14.0389	0.455722
$950$	0	0
$951$	35.7623	1.15967
$952$	0	0
$953$	−38.0235	−1.23170	−0.615850	−	0.787863i	$-0.711187\pi$
$953$	−38.0235	−1.23170	−0.615850	+	0.787863i	$0.711187\pi$
$954$	0	0
$955$	35.5093	1.14905
$956$	0	0
$957$	0.624431	0.0201850
$958$	0	0
$959$	89.7122	2.89696
$960$	0	0
$961$	−9.73975	−0.314186
$962$	0	0
$963$	−36.4144	−1.17344
$964$	0	0
$965$	−20.1464	−0.648535
$966$	0	0
$967$	6.08257	0.195602	0.0978011	−	0.995206i	$-0.468819\pi$
$967$	6.08257	0.195602	0.0978011	+	0.995206i	$0.468819\pi$
$968$	0	0
$969$	0.141657	0.00455068
$970$	0	0
$971$	−27.9445	−0.896783	−0.448392	−	0.893837i	$-0.648003\pi$
$971$	−27.9445	−0.896783	−0.448392	+	0.893837i	$0.648003\pi$
$972$	0	0
$973$	83.6809	2.68269
$974$	0	0
$975$	4.50789	0.144368
$976$	0	0
$977$	−7.44219	−0.238097	−0.119048	−	0.992888i	$-0.537984\pi$
$977$	−7.44219	−0.238097	−0.119048	+	0.992888i	$0.537984\pi$
$978$	0	0
$979$	1.85735	0.0593610
$980$	0	0
$981$	12.8747	0.411059
$982$	0	0
$983$	−21.6456	−0.690388	−0.345194	−	0.938531i	$-0.612187\pi$
$983$	−21.6456	−0.690388	−0.345194	+	0.938531i	$0.612187\pi$
$984$	0	0
$985$	1.42894	0.0455298
$986$	0	0
$987$	23.8121	0.757947
$988$	0	0
$989$	−16.3473	−0.519815
$990$	0	0
$991$	−54.3405	−1.72618	−0.863091	−	0.505049i	$-0.831474\pi$
$991$	−54.3405	−1.72618	−0.863091	+	0.505049i	$0.831474\pi$
$992$	0	0
$993$	−28.5369	−0.905590
$994$	0	0
$995$	24.1356	0.765151
$996$	0	0
$997$	20.6937	0.655376	0.327688	−	0.944786i	$-0.393730\pi$
$997$	20.6937	0.655376	0.327688	+	0.944786i	$0.393730\pi$
$998$	0	0
$999$	0.568421	0.0179840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.w.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.w.1.7	✓	20	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.06424 q^{3} -1.91818 q^{5} -4.17854 q^{7} -1.86738 q^{9} +O(q^{10})\)	\(q-1.06424 q^{3} -1.91818 q^{5} -4.17854 q^{7} -1.86738 q^{9} +0.160390 q^{11} +3.20750 q^{13} +2.04141 q^{15} +1.00000 q^{17} -0.133106 q^{19} +4.44698 q^{21} -2.36479 q^{23} -1.32058 q^{25} +5.18009 q^{27} -3.65819 q^{29} +4.61088 q^{31} -0.170694 q^{33} +8.01518 q^{35} +0.109732 q^{37} -3.41356 q^{39} -0.894148 q^{41} +6.91279 q^{43} +3.58198 q^{45} +5.35466 q^{47} +10.4602 q^{49} -1.06424 q^{51} -4.17274 q^{53} -0.307657 q^{55} +0.141657 q^{57} -1.00000 q^{59} -2.02739 q^{61} +7.80293 q^{63} -6.15256 q^{65} -14.2501 q^{67} +2.51672 q^{69} +7.18296 q^{71} +4.37690 q^{73} +1.40542 q^{75} -0.670195 q^{77} +12.6900 q^{79} +0.0892795 q^{81} +11.7671 q^{83} -1.91818 q^{85} +3.89321 q^{87} +11.5802 q^{89} -13.4026 q^{91} -4.90711 q^{93} +0.255321 q^{95} +1.44796 q^{97} -0.299510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more