LMFDB - Embedded newform 8037.2.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 $ x^12 - x^11 - 15x^10 + 14x^9 + 84x^8 - 76x^7 - 213x^6 + 196x^5 + 225x^4 - 229x^3 - 49x^2 + 83x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.531564$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.531564 q^{2} -1.71744 q^{4} -0.555438 q^{5} +1.54136 q^{7} -1.97606 q^{8} +O(q^{10})$	\(q+0.531564 q^{2} -1.71744 q^{4} -0.555438 q^{5} +1.54136 q^{7} -1.97606 q^{8} -0.295251 q^{10} -0.522534 q^{11} -4.41781 q^{13} +0.819331 q^{14} +2.38448 q^{16} +3.73810 q^{17} +1.00000 q^{19} +0.953931 q^{20} -0.277760 q^{22} -0.679026 q^{23} -4.69149 q^{25} -2.34835 q^{26} -2.64719 q^{28} +2.30786 q^{29} +11.0338 q^{31} +5.21962 q^{32} +1.98704 q^{34} -0.856129 q^{35} -9.40906 q^{37} +0.531564 q^{38} +1.09758 q^{40} +0.130665 q^{41} -3.95911 q^{43} +0.897420 q^{44} -0.360946 q^{46} -1.00000 q^{47} -4.62421 q^{49} -2.49383 q^{50} +7.58732 q^{52} +14.3711 q^{53} +0.290235 q^{55} -3.04581 q^{56} +1.22677 q^{58} +4.45700 q^{59} -4.43167 q^{61} +5.86515 q^{62} -1.99440 q^{64} +2.45382 q^{65} -2.13001 q^{67} -6.41997 q^{68} -0.455087 q^{70} +4.27302 q^{71} +3.15375 q^{73} -5.00152 q^{74} -1.71744 q^{76} -0.805412 q^{77} -14.5652 q^{79} -1.32443 q^{80} +0.0694569 q^{82} +8.63444 q^{83} -2.07628 q^{85} -2.10452 q^{86} +1.03256 q^{88} -4.14514 q^{89} -6.80943 q^{91} +1.16619 q^{92} -0.531564 q^{94} -0.555438 q^{95} +12.1724 q^{97} -2.45806 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

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 $

$2$	0.531564	0.375872	0.187936	−	0.982181i	$-0.439820\pi$
$2$	0.531564	0.375872	0.187936	+	0.982181i	$0.439820\pi$
$3$	0	0
$3$	0	0
$4$	−1.71744	−0.858720
$5$	−0.555438	−0.248399	−0.124200	−	0.992257i	$-0.539636\pi$
$5$	−0.555438	−0.248399	−0.124200	+	0.992257i	$0.539636\pi$
$6$	0	0
$7$	1.54136	0.582579	0.291290	−	0.956635i	$-0.405916\pi$
$7$	1.54136	0.582579	0.291290	+	0.956635i	$0.405916\pi$
$8$	−1.97606	−0.698641
$9$	0	0
$10$	−0.295251	−0.0933664
$11$	−0.522534	−0.157550	−0.0787749	−	0.996892i	$-0.525101\pi$
$11$	−0.522534	−0.157550	−0.0787749	+	0.996892i	$0.525101\pi$
$12$	0	0
$13$	−4.41781	−1.22528	−0.612640	−	0.790362i	$-0.709892\pi$
$13$	−4.41781	−1.22528	−0.612640	+	0.790362i	$0.709892\pi$
$14$	0.819331	0.218975
$15$	0	0
$16$	2.38448	0.596120
$17$	3.73810	0.906623	0.453311	−	0.891352i	$-0.350243\pi$
$17$	3.73810	0.906623	0.453311	+	0.891352i	$0.350243\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.953931	0.213306
$21$	0	0
$22$	−0.277760	−0.0592186
$23$	−0.679026	−0.141587	−0.0707933	−	0.997491i	$-0.522553\pi$
$23$	−0.679026	−0.141587	−0.0707933	+	0.997491i	$0.522553\pi$
$24$	0	0
$25$	−4.69149	−0.938298
$26$	−2.34835	−0.460549
$27$	0	0
$28$	−2.64719	−0.500272
$29$	2.30786	0.428558	0.214279	−	0.976772i	$-0.431260\pi$
$29$	2.30786	0.428558	0.214279	+	0.976772i	$0.431260\pi$
$30$	0	0
$31$	11.0338	1.98172	0.990862	−	0.134881i	$-0.0430651\pi$
$31$	11.0338	1.98172	0.990862	+	0.134881i	$0.0430651\pi$
$32$	5.21962	0.922706
$33$	0	0
$34$	1.98704	0.340774
$35$	−0.856129	−0.144712
$36$	0	0
$37$	−9.40906	−1.54684	−0.773420	−	0.633894i	$-0.781456\pi$
$37$	−9.40906	−1.54684	−0.773420	+	0.633894i	$0.781456\pi$
$38$	0.531564	0.0862310
$39$	0	0
$40$	1.09758	0.173542
$41$	0.130665	0.0204065	0.0102032	−	0.999948i	$-0.496752\pi$
$41$	0.130665	0.0204065	0.0102032	+	0.999948i	$0.496752\pi$
$42$	0	0
$43$	−3.95911	−0.603758	−0.301879	−	0.953346i	$-0.597614\pi$
$43$	−3.95911	−0.603758	−0.301879	+	0.953346i	$0.597614\pi$
$44$	0.897420	0.135291
$45$	0	0
$46$	−0.360946	−0.0532185
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−4.62421	−0.660602
$50$	−2.49383	−0.352680
$51$	0	0
$52$	7.58732	1.05217
$53$	14.3711	1.97403	0.987013	−	0.160639i	$-0.0513556\pi$
$53$	14.3711	1.97403	0.987013	+	0.160639i	$0.0513556\pi$
$54$	0	0
$55$	0.290235	0.0391353
$56$	−3.04581	−0.407014
$57$	0	0
$58$	1.22677	0.161083
$59$	4.45700	0.580252	0.290126	−	0.956988i	$-0.406303\pi$
$59$	4.45700	0.580252	0.290126	+	0.956988i	$0.406303\pi$
$60$	0	0
$61$	−4.43167	−0.567417	−0.283708	−	0.958911i	$-0.591565\pi$
$61$	−4.43167	−0.567417	−0.283708	+	0.958911i	$0.591565\pi$
$62$	5.86515	0.744875
$63$	0	0
$64$	−1.99440	−0.249300
$65$	2.45382	0.304359
$66$	0	0
$67$	−2.13001	−0.260223	−0.130111	−	0.991499i	$-0.541533\pi$
$67$	−2.13001	−0.260223	−0.130111	+	0.991499i	$0.541533\pi$
$68$	−6.41997	−0.778535
$69$	0	0
$70$	−0.455087	−0.0543933
$71$	4.27302	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$71$	4.27302	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$72$	0	0
$73$	3.15375	0.369118	0.184559	−	0.982821i	$-0.440914\pi$
$73$	3.15375	0.369118	0.184559	+	0.982821i	$0.440914\pi$
$74$	−5.00152	−0.581415
$75$	0	0
$76$	−1.71744	−0.197004
$77$	−0.805412	−0.0917852
$78$	0	0
$79$	−14.5652	−1.63871	−0.819357	−	0.573284i	$-0.805669\pi$
$79$	−14.5652	−1.63871	−0.819357	+	0.573284i	$0.805669\pi$
$80$	−1.32443	−0.148076
$81$	0	0
$82$	0.0694569	0.00767023
$83$	8.63444	0.947753	0.473876	−	0.880591i	$-0.342854\pi$
$83$	8.63444	0.947753	0.473876	+	0.880591i	$0.342854\pi$
$84$	0	0
$85$	−2.07628	−0.225205
$86$	−2.10452	−0.226936
$87$	0	0
$88$	1.03256	0.110071
$89$	−4.14514	−0.439384	−0.219692	−	0.975569i	$-0.570505\pi$
$89$	−4.14514	−0.439384	−0.219692	+	0.975569i	$0.570505\pi$
$90$	0	0
$91$	−6.80943	−0.713823
$92$	1.16619	0.121583
$93$	0	0
$94$	−0.531564	−0.0548266
$95$	−0.555438	−0.0569867
$96$	0	0
$97$	12.1724	1.23592	0.617962	−	0.786208i	$-0.287958\pi$
$97$	12.1724	1.23592	0.617962	+	0.786208i	$0.287958\pi$
$98$	−2.45806	−0.248302
$99$	0	0
$100$	8.05735	0.805735
$101$	−4.16812	−0.414744	−0.207372	−	0.978262i	$-0.566491\pi$
$101$	−4.16812	−0.414744	−0.207372	+	0.978262i	$0.566491\pi$
$102$	0	0
$103$	4.24411	0.418184	0.209092	−	0.977896i	$-0.432949\pi$
$103$	4.24411	0.418184	0.209092	+	0.977896i	$0.432949\pi$
$104$	8.72984	0.856031
$105$	0	0
$106$	7.63917	0.741982
$107$	−12.1180	−1.17149	−0.585745	−	0.810495i	$-0.699198\pi$
$107$	−12.1180	−1.17149	−0.585745	+	0.810495i	$0.699198\pi$
$108$	0	0
$109$	−5.72054	−0.547928	−0.273964	−	0.961740i	$-0.588335\pi$
$109$	−5.72054	−0.547928	−0.273964	+	0.961740i	$0.588335\pi$
$110$	0.154278	0.0147099
$111$	0	0
$112$	3.67534	0.347287
$113$	−13.6233	−1.28158	−0.640788	−	0.767717i	$-0.721392\pi$
$113$	−13.6233	−1.28158	−0.640788	+	0.767717i	$0.721392\pi$
$114$	0	0
$115$	0.377157	0.0351700
$116$	−3.96360	−0.368011
$117$	0	0
$118$	2.36918	0.218101
$119$	5.76176	0.528180
$120$	0	0
$121$	−10.7270	−0.975178
$122$	−2.35571	−0.213276
$123$	0	0
$124$	−18.9498	−1.70175
$125$	5.38302	0.481472
$126$	0	0
$127$	11.8039	1.04743	0.523714	−	0.851894i	$-0.324546\pi$
$127$	11.8039	1.04743	0.523714	+	0.851894i	$0.324546\pi$
$128$	−11.4994	−1.01641
$129$	0	0
$130$	1.30436	0.114400
$131$	−4.28857	−0.374694	−0.187347	−	0.982294i	$-0.559989\pi$
$131$	−4.28857	−0.374694	−0.187347	+	0.982294i	$0.559989\pi$
$132$	0	0
$133$	1.54136	0.133653
$134$	−1.13224	−0.0978105
$135$	0	0
$136$	−7.38670	−0.633404
$137$	15.9784	1.36513	0.682564	−	0.730826i	$-0.260865\pi$
$137$	15.9784	1.36513	0.682564	+	0.730826i	$0.260865\pi$
$138$	0	0
$139$	8.82847	0.748821	0.374410	−	0.927263i	$-0.377845\pi$
$139$	8.82847	0.748821	0.374410	+	0.927263i	$0.377845\pi$
$140$	1.47035	0.124267
$141$	0	0
$142$	2.27138	0.190610
$143$	2.30846	0.193043
$144$	0	0
$145$	−1.28187	−0.106454
$146$	1.67642	0.138741
$147$	0	0
$148$	16.1595	1.32830
$149$	−14.1367	−1.15813	−0.579063	−	0.815283i	$-0.696581\pi$
$149$	−14.1367	−1.15813	−0.579063	+	0.815283i	$0.696581\pi$
$150$	0	0
$151$	−1.31989	−0.107411	−0.0537055	−	0.998557i	$-0.517103\pi$
$151$	−1.31989	−0.107411	−0.0537055	+	0.998557i	$0.517103\pi$
$152$	−1.97606	−0.160279
$153$	0	0
$154$	−0.428128	−0.0344995
$155$	−6.12857	−0.492259
$156$	0	0
$157$	−14.8474	−1.18495	−0.592477	−	0.805587i	$-0.701850\pi$
$157$	−14.8474	−1.18495	−0.592477	+	0.805587i	$0.701850\pi$
$158$	−7.74233	−0.615947
$159$	0	0
$160$	−2.89917	−0.229200
$161$	−1.04662	−0.0824854
$162$	0	0
$163$	−3.65804	−0.286520	−0.143260	−	0.989685i	$-0.545758\pi$
$163$	−3.65804	−0.286520	−0.143260	+	0.989685i	$0.545758\pi$
$164$	−0.224410	−0.0175235
$165$	0	0
$166$	4.58975	0.356234
$167$	5.84391	0.452215	0.226108	−	0.974102i	$-0.427400\pi$
$167$	5.84391	0.452215	0.226108	+	0.974102i	$0.427400\pi$
$168$	0	0
$169$	6.51705	0.501312
$170$	−1.10368	−0.0846482
$171$	0	0
$172$	6.79953	0.518459
$173$	−9.81929	−0.746547	−0.373274	−	0.927721i	$-0.621765\pi$
$173$	−9.81929	−0.746547	−0.373274	+	0.927721i	$0.621765\pi$
$174$	0	0
$175$	−7.23127	−0.546633
$176$	−1.24597	−0.0939186
$177$	0	0
$178$	−2.20341	−0.165152
$179$	−20.1486	−1.50598	−0.752989	−	0.658034i	$-0.771388\pi$
$179$	−20.1486	−1.50598	−0.752989	+	0.658034i	$0.771388\pi$
$180$	0	0
$181$	−10.8828	−0.808912	−0.404456	−	0.914557i	$-0.632539\pi$
$181$	−10.8828	−0.808912	−0.404456	+	0.914557i	$0.632539\pi$
$182$	−3.61965	−0.268306
$183$	0	0
$184$	1.34179	0.0989183
$185$	5.22615	0.384234
$186$	0	0
$187$	−1.95328	−0.142838
$188$	1.71744	0.125257
$189$	0	0
$190$	−0.295251	−0.0214197
$191$	−6.38395	−0.461926	−0.230963	−	0.972962i	$-0.574188\pi$
$191$	−6.38395	−0.461926	−0.230963	+	0.972962i	$0.574188\pi$
$192$	0	0
$193$	−10.7538	−0.774078	−0.387039	−	0.922063i	$-0.626502\pi$
$193$	−10.7538	−0.774078	−0.387039	+	0.922063i	$0.626502\pi$
$194$	6.47043	0.464550
$195$	0	0
$196$	7.94181	0.567272
$197$	−13.7090	−0.976727	−0.488364	−	0.872640i	$-0.662406\pi$
$197$	−13.7090	−0.976727	−0.488364	+	0.872640i	$0.662406\pi$
$198$	0	0
$199$	22.0015	1.55965	0.779824	−	0.625999i	$-0.215309\pi$
$199$	22.0015	1.55965	0.779824	+	0.625999i	$0.215309\pi$
$200$	9.27065	0.655534
$201$	0	0
$202$	−2.21562	−0.155891
$203$	3.55723	0.249669
$204$	0	0
$205$	−0.0725764	−0.00506896
$206$	2.25601	0.157184
$207$	0	0
$208$	−10.5342	−0.730414
$209$	−0.522534	−0.0361444
$210$	0	0
$211$	−16.7703	−1.15452	−0.577259	−	0.816561i	$-0.695878\pi$
$211$	−16.7703	−1.15452	−0.577259	+	0.816561i	$0.695878\pi$
$212$	−24.6816	−1.69514
$213$	0	0
$214$	−6.44149	−0.440331
$215$	2.19904	0.149973
$216$	0	0
$217$	17.0070	1.15451
$218$	−3.04083	−0.205951
$219$	0	0
$220$	−0.498461	−0.0336063
$221$	−16.5142	−1.11087
$222$	0	0
$223$	−17.7583	−1.18918	−0.594591	−	0.804028i	$-0.702686\pi$
$223$	−17.7583	−1.18918	−0.594591	+	0.804028i	$0.702686\pi$
$224$	8.04530	0.537549
$225$	0	0
$226$	−7.24168	−0.481709
$227$	6.71033	0.445380	0.222690	−	0.974889i	$-0.428516\pi$
$227$	6.71033	0.445380	0.222690	+	0.974889i	$0.428516\pi$
$228$	0	0
$229$	−2.72852	−0.180306	−0.0901530	−	0.995928i	$-0.528736\pi$
$229$	−2.72852	−0.180306	−0.0901530	+	0.995928i	$0.528736\pi$
$230$	0.200483	0.0132194
$231$	0	0
$232$	−4.56045	−0.299408
$233$	−22.4714	−1.47215	−0.736076	−	0.676899i	$-0.763323\pi$
$233$	−22.4714	−1.47215	−0.736076	+	0.676899i	$0.763323\pi$
$234$	0	0
$235$	0.555438	0.0362328
$236$	−7.65464	−0.498274
$237$	0	0
$238$	3.06274	0.198528
$239$	19.2393	1.24449	0.622245	−	0.782823i	$-0.286221\pi$
$239$	19.2393	1.24449	0.622245	+	0.782823i	$0.286221\pi$
$240$	0	0
$241$	17.5452	1.13019	0.565093	−	0.825027i	$-0.308840\pi$
$241$	17.5452	1.13019	0.565093	+	0.825027i	$0.308840\pi$
$242$	−5.70206	−0.366542
$243$	0	0
$244$	7.61112	0.487252
$245$	2.56846	0.164093
$246$	0	0
$247$	−4.41781	−0.281099
$248$	−21.8034	−1.38451
$249$	0	0
$250$	2.86142	0.180972
$251$	−27.5978	−1.74196	−0.870980	−	0.491318i	$-0.836515\pi$
$251$	−27.5978	−1.74196	−0.870980	+	0.491318i	$0.836515\pi$
$252$	0	0
$253$	0.354814	0.0223070
$254$	6.27454	0.393699
$255$	0	0
$256$	−2.12385	−0.132741
$257$	4.84713	0.302356	0.151178	−	0.988507i	$-0.451693\pi$
$257$	4.84713	0.302356	0.151178	+	0.988507i	$0.451693\pi$
$258$	0	0
$259$	−14.5027	−0.901157
$260$	−4.21429	−0.261359
$261$	0	0
$262$	−2.27965	−0.140837
$263$	3.02814	0.186723	0.0933616	−	0.995632i	$-0.470239\pi$
$263$	3.02814	0.186723	0.0933616	+	0.995632i	$0.470239\pi$
$264$	0	0
$265$	−7.98227	−0.490347
$266$	0.819331	0.0502364
$267$	0	0
$268$	3.65817	0.223458
$269$	−5.34694	−0.326009	−0.163004	−	0.986625i	$-0.552118\pi$
$269$	−5.34694	−0.326009	−0.163004	+	0.986625i	$0.552118\pi$
$270$	0	0
$271$	0.960223	0.0583294	0.0291647	−	0.999575i	$-0.490715\pi$
$271$	0.960223	0.0583294	0.0291647	+	0.999575i	$0.490715\pi$
$272$	8.91343	0.540456
$273$	0	0
$274$	8.49354	0.513114
$275$	2.45146	0.147829
$276$	0	0
$277$	8.32241	0.500045	0.250023	−	0.968240i	$-0.419562\pi$
$277$	8.32241	0.500045	0.250023	+	0.968240i	$0.419562\pi$
$278$	4.69289	0.281461
$279$	0	0
$280$	1.69176	0.101102
$281$	−4.25414	−0.253781	−0.126890	−	0.991917i	$-0.540500\pi$
$281$	−4.25414	−0.253781	−0.126890	+	0.991917i	$0.540500\pi$
$282$	0	0
$283$	9.37629	0.557363	0.278681	−	0.960384i	$-0.410103\pi$
$283$	9.37629	0.557363	0.278681	+	0.960384i	$0.410103\pi$
$284$	−7.33865	−0.435469
$285$	0	0
$286$	1.22709	0.0725594
$287$	0.201402	0.0118884
$288$	0	0
$289$	−3.02659	−0.178035
$290$	−0.681396	−0.0400129
$291$	0	0
$292$	−5.41638	−0.316969
$293$	−30.9934	−1.81065	−0.905327	−	0.424715i	$-0.860374\pi$
$293$	−30.9934	−1.81065	−0.905327	+	0.424715i	$0.860374\pi$
$294$	0	0
$295$	−2.47559	−0.144134
$296$	18.5928	1.08069
$297$	0	0
$298$	−7.51457	−0.435307
$299$	2.99981	0.173483
$300$	0	0
$301$	−6.10240	−0.351737
$302$	−0.701605	−0.0403728
$303$	0	0
$304$	2.38448	0.136759
$305$	2.46152	0.140946
$306$	0	0
$307$	−13.0870	−0.746914	−0.373457	−	0.927647i	$-0.621828\pi$
$307$	−13.0870	−0.746914	−0.373457	+	0.927647i	$0.621828\pi$
$308$	1.38325	0.0788178
$309$	0	0
$310$	−3.25773	−0.185027
$311$	6.14164	0.348260	0.174130	−	0.984723i	$-0.444289\pi$
$311$	6.14164	0.348260	0.174130	+	0.984723i	$0.444289\pi$
$312$	0	0
$313$	−9.24950	−0.522813	−0.261406	−	0.965229i	$-0.584186\pi$
$313$	−9.24950	−0.522813	−0.261406	+	0.965229i	$0.584186\pi$
$314$	−7.89236	−0.445392
$315$	0	0
$316$	25.0149	1.40720
$317$	−14.8766	−0.835553	−0.417776	−	0.908550i	$-0.637190\pi$
$317$	−14.8766	−0.835553	−0.417776	+	0.908550i	$0.637190\pi$
$318$	0	0
$319$	−1.20593	−0.0675193
$320$	1.10777	0.0619260
$321$	0	0
$322$	−0.556347	−0.0310040
$323$	3.73810	0.207994
$324$	0	0
$325$	20.7261	1.14968
$326$	−1.94448	−0.107695
$327$	0	0
$328$	−0.258202	−0.0142568
$329$	−1.54136	−0.0849779
$330$	0	0
$331$	25.8369	1.42013	0.710063	−	0.704138i	$-0.248667\pi$
$331$	25.8369	1.42013	0.710063	+	0.704138i	$0.248667\pi$
$332$	−14.8291	−0.813854
$333$	0	0
$334$	3.10641	0.169975
$335$	1.18309	0.0646391
$336$	0	0
$337$	−27.4969	−1.49785	−0.748927	−	0.662653i	$-0.769430\pi$
$337$	−27.4969	−1.49785	−0.748927	+	0.662653i	$0.769430\pi$
$338$	3.46423	0.188429
$339$	0	0
$340$	3.56589	0.193388
$341$	−5.76552	−0.312220
$342$	0	0
$343$	−17.9171	−0.967432
$344$	7.82342	0.421810
$345$	0	0
$346$	−5.21958	−0.280606
$347$	34.3290	1.84288	0.921439	−	0.388523i	$-0.127014\pi$
$347$	34.3290	1.84288	0.921439	+	0.388523i	$0.127014\pi$
$348$	0	0
$349$	−12.1521	−0.650489	−0.325244	−	0.945630i	$-0.605447\pi$
$349$	−12.1521	−0.650489	−0.325244	+	0.945630i	$0.605447\pi$
$350$	−3.84388	−0.205464
$351$	0	0
$352$	−2.72743	−0.145372
$353$	32.0846	1.70769	0.853846	−	0.520526i	$-0.174264\pi$
$353$	32.0846	1.70769	0.853846	+	0.520526i	$0.174264\pi$
$354$	0	0
$355$	−2.37340	−0.125967
$356$	7.11903	0.377308
$357$	0	0
$358$	−10.7103	−0.566055
$359$	−29.9786	−1.58221	−0.791106	−	0.611679i	$-0.790494\pi$
$359$	−29.9786	−1.58221	−0.791106	+	0.611679i	$0.790494\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.78490	−0.304048
$363$	0	0
$364$	11.6948	0.612974
$365$	−1.75171	−0.0916888
$366$	0	0
$367$	8.25225	0.430764	0.215382	−	0.976530i	$-0.430900\pi$
$367$	8.25225	0.430764	0.215382	+	0.976530i	$0.430900\pi$
$368$	−1.61912	−0.0844027
$369$	0	0
$370$	2.77803	0.144423
$371$	22.1511	1.15003
$372$	0	0
$373$	−32.8499	−1.70090	−0.850451	−	0.526055i	$-0.823671\pi$
$373$	−32.8499	−1.70090	−0.850451	+	0.526055i	$0.823671\pi$
$374$	−1.03830	−0.0536890
$375$	0	0
$376$	1.97606	0.101907
$377$	−10.1957	−0.525104
$378$	0	0
$379$	−9.03631	−0.464164	−0.232082	−	0.972696i	$-0.574554\pi$
$379$	−9.03631	−0.464164	−0.232082	+	0.972696i	$0.574554\pi$
$380$	0.953931	0.0489356
$381$	0	0
$382$	−3.39348	−0.173625
$383$	−24.4978	−1.25178	−0.625891	−	0.779911i	$-0.715264\pi$
$383$	−24.4978	−1.25178	−0.625891	+	0.779911i	$0.715264\pi$
$384$	0	0
$385$	0.447356	0.0227994
$386$	−5.71635	−0.290954
$387$	0	0
$388$	−20.9054	−1.06131
$389$	26.1867	1.32772	0.663860	−	0.747857i	$-0.268917\pi$
$389$	26.1867	1.32772	0.663860	+	0.747857i	$0.268917\pi$
$390$	0	0
$391$	−2.53827	−0.128366
$392$	9.13770	0.461524
$393$	0	0
$394$	−7.28722	−0.367125
$395$	8.09006	0.407055
$396$	0	0
$397$	−4.56535	−0.229128	−0.114564	−	0.993416i	$-0.536547\pi$
$397$	−4.56535	−0.229128	−0.114564	+	0.993416i	$0.536547\pi$
$398$	11.6952	0.586228
$399$	0	0
$400$	−11.1868	−0.559338
$401$	−22.6492	−1.13105	−0.565525	−	0.824731i	$-0.691326\pi$
$401$	−22.6492	−1.13105	−0.565525	+	0.824731i	$0.691326\pi$
$402$	0	0
$403$	−48.7451	−2.42817
$404$	7.15850	0.356149
$405$	0	0
$406$	1.89090	0.0938436
$407$	4.91655	0.243704
$408$	0	0
$409$	−17.3172	−0.856282	−0.428141	−	0.903712i	$-0.640831\pi$
$409$	−17.3172	−0.856282	−0.428141	+	0.903712i	$0.640831\pi$
$410$	−0.0385790	−0.00190528
$411$	0	0
$412$	−7.28900	−0.359103
$413$	6.86984	0.338043
$414$	0	0
$415$	−4.79589	−0.235421
$416$	−23.0593	−1.13057
$417$	0	0
$418$	−0.277760	−0.0135857
$419$	23.1232	1.12964	0.564821	−	0.825213i	$-0.308945\pi$
$419$	23.1232	1.12964	0.564821	+	0.825213i	$0.308945\pi$
$420$	0	0
$421$	−0.0801545	−0.00390649	−0.00195325	−	0.999998i	$-0.500622\pi$
$421$	−0.0801545	−0.00390649	−0.00195325	+	0.999998i	$0.500622\pi$
$422$	−8.91450	−0.433951
$423$	0	0
$424$	−28.3982	−1.37914
$425$	−17.5373	−0.850682
$426$	0	0
$427$	−6.83079	−0.330565
$428$	20.8119	1.00598
$429$	0	0
$430$	1.16893	0.0563707
$431$	−20.7920	−1.00152	−0.500759	−	0.865587i	$-0.666946\pi$
$431$	−20.7920	−1.00152	−0.500759	+	0.865587i	$0.666946\pi$
$432$	0	0
$433$	−22.1945	−1.06660	−0.533299	−	0.845927i	$-0.679048\pi$
$433$	−22.1945	−1.06660	−0.533299	+	0.845927i	$0.679048\pi$
$434$	9.04031	0.433949
$435$	0	0
$436$	9.82468	0.470517
$437$	−0.679026	−0.0324822
$438$	0	0
$439$	13.5123	0.644909	0.322454	−	0.946585i	$-0.395492\pi$
$439$	13.5123	0.644909	0.322454	+	0.946585i	$0.395492\pi$
$440$	−0.573521	−0.0273415
$441$	0	0
$442$	−8.77836	−0.417544
$443$	6.25622	0.297242	0.148621	−	0.988894i	$-0.452517\pi$
$443$	6.25622	0.297242	0.148621	+	0.988894i	$0.452517\pi$
$444$	0	0
$445$	2.30237	0.109143
$446$	−9.43966	−0.446981
$447$	0	0
$448$	−3.07409	−0.145237
$449$	−19.1131	−0.902004	−0.451002	−	0.892523i	$-0.648933\pi$
$449$	−19.1131	−0.902004	−0.451002	+	0.892523i	$0.648933\pi$
$450$	0	0
$451$	−0.0682770	−0.00321504
$452$	23.3973	1.10052
$453$	0	0
$454$	3.56697	0.167406
$455$	3.78222	0.177313
$456$	0	0
$457$	16.8454	0.787996	0.393998	−	0.919111i	$-0.371092\pi$
$457$	16.8454	0.787996	0.393998	+	0.919111i	$0.371092\pi$
$458$	−1.45038	−0.0677720
$459$	0	0
$460$	−0.647744	−0.0302012
$461$	36.1771	1.68494	0.842469	−	0.538745i	$-0.181102\pi$
$461$	36.1771	1.68494	0.842469	+	0.538745i	$0.181102\pi$
$462$	0	0
$463$	−28.7098	−1.33426	−0.667128	−	0.744943i	$-0.732477\pi$
$463$	−28.7098	−1.33426	−0.667128	+	0.744943i	$0.732477\pi$
$464$	5.50304	0.255472
$465$	0	0
$466$	−11.9450	−0.553341
$467$	−17.9593	−0.831058	−0.415529	−	0.909580i	$-0.636403\pi$
$467$	−17.9593	−0.831058	−0.415529	+	0.909580i	$0.636403\pi$
$468$	0	0
$469$	−3.28312	−0.151600
$470$	0.295251	0.0136189
$471$	0	0
$472$	−8.80729	−0.405388
$473$	2.06877	0.0951220
$474$	0	0
$475$	−4.69149	−0.215260
$476$	−9.89547	−0.453558
$477$	0	0
$478$	10.2269	0.467769
$479$	22.0862	1.00914	0.504572	−	0.863370i	$-0.331650\pi$
$479$	22.0862	1.00914	0.504572	+	0.863370i	$0.331650\pi$
$480$	0	0
$481$	41.5675	1.89531
$482$	9.32640	0.424806
$483$	0	0
$484$	18.4229	0.837405
$485$	−6.76104	−0.307003
$486$	0	0
$487$	−14.5313	−0.658476	−0.329238	−	0.944247i	$-0.606792\pi$
$487$	−14.5313	−0.658476	−0.329238	+	0.944247i	$0.606792\pi$
$488$	8.75722	0.396421
$489$	0	0
$490$	1.36530	0.0616780
$491$	21.9470	0.990454	0.495227	−	0.868764i	$-0.335085\pi$
$491$	21.9470	0.990454	0.495227	+	0.868764i	$0.335085\pi$
$492$	0	0
$493$	8.62700	0.388541
$494$	−2.34835	−0.105657
$495$	0	0
$496$	26.3098	1.18135
$497$	6.58625	0.295434
$498$	0	0
$499$	−18.3949	−0.823467	−0.411733	−	0.911304i	$-0.635077\pi$
$499$	−18.3949	−0.823467	−0.411733	+	0.911304i	$0.635077\pi$
$500$	−9.24501	−0.413450
$501$	0	0
$502$	−14.6700	−0.654755
$503$	9.85188	0.439274	0.219637	−	0.975582i	$-0.429513\pi$
$503$	9.85188	0.439274	0.219637	+	0.975582i	$0.429513\pi$
$504$	0	0
$505$	2.31513	0.103022
$506$	0.188606	0.00838457
$507$	0	0
$508$	−20.2725	−0.899448
$509$	−19.1849	−0.850356	−0.425178	−	0.905110i	$-0.639789\pi$
$509$	−19.1849	−0.850356	−0.425178	+	0.905110i	$0.639789\pi$
$510$	0	0
$511$	4.86106	0.215041
$512$	21.8698	0.966518
$513$	0	0
$514$	2.57656	0.113647
$515$	−2.35734	−0.103877
$516$	0	0
$517$	0.522534	0.0229810
$518$	−7.70913	−0.338720
$519$	0	0
$520$	−4.84889	−0.212638
$521$	29.8485	1.30769	0.653843	−	0.756630i	$-0.273156\pi$
$521$	29.8485	1.30769	0.653843	+	0.756630i	$0.273156\pi$
$522$	0	0
$523$	36.5946	1.60017	0.800084	−	0.599888i	$-0.204788\pi$
$523$	36.5946	1.60017	0.800084	+	0.599888i	$0.204788\pi$
$524$	7.36537	0.321758
$525$	0	0
$526$	1.60965	0.0701841
$527$	41.2454	1.79668
$528$	0	0
$529$	−22.5389	−0.979953
$530$	−4.24308	−0.184308
$531$	0	0
$532$	−2.64719	−0.114770
$533$	−0.577254	−0.0250037
$534$	0	0
$535$	6.73079	0.290998
$536$	4.20903	0.181802
$537$	0	0
$538$	−2.84224	−0.122538
$539$	2.41631	0.104078
$540$	0	0
$541$	−21.4163	−0.920761	−0.460380	−	0.887722i	$-0.652287\pi$
$541$	−21.4163	−0.920761	−0.460380	+	0.887722i	$0.652287\pi$
$542$	0.510420	0.0219244
$543$	0	0
$544$	19.5115	0.836547
$545$	3.17740	0.136105
$546$	0	0
$547$	16.6988	0.713989	0.356995	−	0.934106i	$-0.383801\pi$
$547$	16.6988	0.713989	0.356995	+	0.934106i	$0.383801\pi$
$548$	−27.4420	−1.17226
$549$	0	0
$550$	1.30311	0.0555647
$551$	2.30786	0.0983180
$552$	0	0
$553$	−22.4502	−0.954680
$554$	4.42389	0.187953
$555$	0	0
$556$	−15.1624	−0.643027
$557$	−20.4754	−0.867572	−0.433786	−	0.901016i	$-0.642823\pi$
$557$	−20.4754	−0.867572	−0.433786	+	0.901016i	$0.642823\pi$
$558$	0	0
$559$	17.4906	0.739773
$560$	−2.04142	−0.0862659
$561$	0	0
$562$	−2.26135	−0.0953893
$563$	−15.0284	−0.633372	−0.316686	−	0.948530i	$-0.602570\pi$
$563$	−15.0284	−0.633372	−0.316686	+	0.948530i	$0.602570\pi$
$564$	0	0
$565$	7.56692	0.318343
$566$	4.98410	0.209497
$567$	0	0
$568$	−8.44372	−0.354291
$569$	39.4717	1.65474	0.827370	−	0.561658i	$-0.189836\pi$
$569$	39.4717	1.65474	0.827370	+	0.561658i	$0.189836\pi$
$570$	0	0
$571$	−10.5733	−0.442479	−0.221239	−	0.975220i	$-0.571010\pi$
$571$	−10.5733	−0.442479	−0.221239	+	0.975220i	$0.571010\pi$
$572$	−3.96463	−0.165770
$573$	0	0
$574$	0.107058	0.00446852
$575$	3.18564	0.132850
$576$	0	0
$577$	−35.1803	−1.46458	−0.732288	−	0.680996i	$-0.761547\pi$
$577$	−35.1803	−1.46458	−0.732288	+	0.680996i	$0.761547\pi$
$578$	−1.60883	−0.0669184
$579$	0	0
$580$	2.20154	0.0914138
$581$	13.3088	0.552141
$582$	0	0
$583$	−7.50940	−0.311008
$584$	−6.23199	−0.257881
$585$	0	0
$586$	−16.4750	−0.680575
$587$	15.7529	0.650190	0.325095	−	0.945681i	$-0.394604\pi$
$587$	15.7529	0.650190	0.325095	+	0.945681i	$0.394604\pi$
$588$	0	0
$589$	11.0338	0.454639
$590$	−1.31593	−0.0541761
$591$	0	0
$592$	−22.4357	−0.922103
$593$	−19.6667	−0.807613	−0.403806	−	0.914845i	$-0.632313\pi$
$593$	−19.6667	−0.807613	−0.403806	+	0.914845i	$0.632313\pi$
$594$	0	0
$595$	−3.20030	−0.131199
$596$	24.2790	0.994505
$597$	0	0
$598$	1.59459	0.0652076
$599$	−19.2095	−0.784879	−0.392440	−	0.919778i	$-0.628369\pi$
$599$	−19.2095	−0.784879	−0.392440	+	0.919778i	$0.628369\pi$
$600$	0	0
$601$	6.30264	0.257090	0.128545	−	0.991704i	$-0.458969\pi$
$601$	6.30264	0.257090	0.128545	+	0.991704i	$0.458969\pi$
$602$	−3.24382	−0.132208
$603$	0	0
$604$	2.26683	0.0922359
$605$	5.95816	0.242234
$606$	0	0
$607$	−3.13401	−0.127205	−0.0636027	−	0.997975i	$-0.520259\pi$
$607$	−3.13401	−0.127205	−0.0636027	+	0.997975i	$0.520259\pi$
$608$	5.21962	0.211683
$609$	0	0
$610$	1.30845	0.0529777
$611$	4.41781	0.178725
$612$	0	0
$613$	−44.8921	−1.81318	−0.906588	−	0.422016i	$-0.861323\pi$
$613$	−44.8921	−1.81318	−0.906588	+	0.422016i	$0.861323\pi$
$614$	−6.95657	−0.280744
$615$	0	0
$616$	1.59154	0.0641250
$617$	0.579444	0.0233275	0.0116638	−	0.999932i	$-0.496287\pi$
$617$	0.579444	0.0233275	0.0116638	+	0.999932i	$0.496287\pi$
$618$	0	0
$619$	−32.5265	−1.30735	−0.653675	−	0.756775i	$-0.726774\pi$
$619$	−32.5265	−1.30735	−0.653675	+	0.756775i	$0.726774\pi$
$620$	10.5255	0.422713
$621$	0	0
$622$	3.26467	0.130901
$623$	−6.38915	−0.255976
$624$	0	0
$625$	20.4675	0.818700
$626$	−4.91670	−0.196511
$627$	0	0
$628$	25.4996	1.01754
$629$	−35.1720	−1.40240
$630$	0	0
$631$	−0.666189	−0.0265206	−0.0132603	−	0.999912i	$-0.504221\pi$
$631$	−0.666189	−0.0265206	−0.0132603	+	0.999912i	$0.504221\pi$
$632$	28.7817	1.14487
$633$	0	0
$634$	−7.90786	−0.314061
$635$	−6.55635	−0.260181
$636$	0	0
$637$	20.4289	0.809422
$638$	−0.641030	−0.0253786
$639$	0	0
$640$	6.38719	0.252476
$641$	−42.6806	−1.68578	−0.842892	−	0.538083i	$-0.819149\pi$
$641$	−42.6806	−1.68578	−0.842892	+	0.538083i	$0.819149\pi$
$642$	0	0
$643$	−34.5821	−1.36378	−0.681892	−	0.731453i	$-0.738843\pi$
$643$	−34.5821	−1.36378	−0.681892	+	0.731453i	$0.738843\pi$
$644$	1.79751	0.0708319
$645$	0	0
$646$	1.98704	0.0781790
$647$	−39.3014	−1.54510	−0.772549	−	0.634955i	$-0.781019\pi$
$647$	−39.3014	−1.54510	−0.772549	+	0.634955i	$0.781019\pi$
$648$	0	0
$649$	−2.32893	−0.0914187
$650$	11.0172	0.432132
$651$	0	0
$652$	6.28246	0.246040
$653$	−9.45496	−0.370001	−0.185001	−	0.982738i	$-0.559229\pi$
$653$	−9.45496	−0.370001	−0.185001	+	0.982738i	$0.559229\pi$
$654$	0	0
$655$	2.38204	0.0930738
$656$	0.311569	0.0121647
$657$	0	0
$658$	−0.819331	−0.0319408
$659$	2.38097	0.0927496	0.0463748	−	0.998924i	$-0.485233\pi$
$659$	2.38097	0.0927496	0.0463748	+	0.998924i	$0.485233\pi$
$660$	0	0
$661$	49.9090	1.94124	0.970618	−	0.240625i	$-0.0773525\pi$
$661$	49.9090	1.94124	0.970618	+	0.240625i	$0.0773525\pi$
$662$	13.7340	0.533786
$663$	0	0
$664$	−17.0621	−0.662139
$665$	−0.856129	−0.0331993
$666$	0	0
$667$	−1.56709	−0.0606781
$668$	−10.0366	−0.388326
$669$	0	0
$670$	0.628888	0.0242961
$671$	2.31570	0.0893964
$672$	0	0
$673$	26.5913	1.02502	0.512509	−	0.858682i	$-0.328716\pi$
$673$	26.5913	1.02502	0.512509	+	0.858682i	$0.328716\pi$
$674$	−14.6164	−0.563002
$675$	0	0
$676$	−11.1926	−0.430486
$677$	−8.67156	−0.333275	−0.166638	−	0.986018i	$-0.553291\pi$
$677$	−8.67156	−0.333275	−0.166638	+	0.986018i	$0.553291\pi$
$678$	0	0
$679$	18.7621	0.720024
$680$	4.10285	0.157337
$681$	0	0
$682$	−3.06474	−0.117355
$683$	−21.3646	−0.817495	−0.408747	−	0.912648i	$-0.634034\pi$
$683$	−21.3646	−0.817495	−0.408747	+	0.912648i	$0.634034\pi$
$684$	0	0
$685$	−8.87501	−0.339097
$686$	−9.52407	−0.363631
$687$	0	0
$688$	−9.44041	−0.359912
$689$	−63.4889	−2.41874
$690$	0	0
$691$	9.05508	0.344471	0.172236	−	0.985056i	$-0.444901\pi$
$691$	9.05508	0.344471	0.172236	+	0.985056i	$0.444901\pi$
$692$	16.8640	0.641075
$693$	0	0
$694$	18.2481	0.692687
$695$	−4.90366	−0.186007
$696$	0	0
$697$	0.488440	0.0185010
$698$	−6.45963	−0.244501
$699$	0	0
$700$	12.4193	0.469404
$701$	−15.2341	−0.575386	−0.287693	−	0.957723i	$-0.592888\pi$
$701$	−15.2341	−0.575386	−0.287693	+	0.957723i	$0.592888\pi$
$702$	0	0
$703$	−9.40906	−0.354870
$704$	1.04214	0.0392772
$705$	0	0
$706$	17.0550	0.641874
$707$	−6.42457	−0.241621
$708$	0	0
$709$	−23.2859	−0.874521	−0.437261	−	0.899335i	$-0.644051\pi$
$709$	−23.2859	−0.874521	−0.437261	+	0.899335i	$0.644051\pi$
$710$	−1.26161	−0.0473474
$711$	0	0
$712$	8.19103	0.306972
$713$	−7.49222	−0.280586
$714$	0	0
$715$	−1.28220	−0.0479517
$716$	34.6040	1.29321
$717$	0	0
$718$	−15.9356	−0.594710
$719$	−32.4745	−1.21110	−0.605548	−	0.795809i	$-0.707046\pi$
$719$	−32.4745	−1.21110	−0.605548	+	0.795809i	$0.707046\pi$
$720$	0	0
$721$	6.54169	0.243625
$722$	0.531564	0.0197828
$723$	0	0
$724$	18.6906	0.694629
$725$	−10.8273	−0.402115
$726$	0	0
$727$	17.5292	0.650123	0.325061	−	0.945693i	$-0.394615\pi$
$727$	17.5292	0.650123	0.325061	+	0.945693i	$0.394615\pi$
$728$	13.4558	0.498706
$729$	0	0
$730$	−0.931147	−0.0344633
$731$	−14.7995	−0.547381
$732$	0	0
$733$	−23.1819	−0.856242	−0.428121	−	0.903721i	$-0.640824\pi$
$733$	−23.1819	−0.856242	−0.428121	+	0.903721i	$0.640824\pi$
$734$	4.38660	0.161912
$735$	0	0
$736$	−3.54425	−0.130643
$737$	1.11300	0.0409980
$738$	0	0
$739$	3.78256	0.139144	0.0695718	−	0.997577i	$-0.477837\pi$
$739$	3.78256	0.139144	0.0695718	+	0.997577i	$0.477837\pi$
$740$	−8.97560	−0.329950
$741$	0	0
$742$	11.7747	0.432263
$743$	−28.2634	−1.03688	−0.518442	−	0.855113i	$-0.673488\pi$
$743$	−28.2634	−1.03688	−0.518442	+	0.855113i	$0.673488\pi$
$744$	0	0
$745$	7.85207	0.287678
$746$	−17.4618	−0.639322
$747$	0	0
$748$	3.35465	0.122658
$749$	−18.6782	−0.682486
$750$	0	0
$751$	40.6385	1.48292	0.741460	−	0.670997i	$-0.234134\pi$
$751$	40.6385	1.48292	0.741460	+	0.670997i	$0.234134\pi$
$752$	−2.38448	−0.0869530
$753$	0	0
$754$	−5.41965	−0.197372
$755$	0.733116	0.0266808
$756$	0	0
$757$	−10.1633	−0.369390	−0.184695	−	0.982796i	$-0.559130\pi$
$757$	−10.1633	−0.369390	−0.184695	+	0.982796i	$0.559130\pi$
$758$	−4.80337	−0.174466
$759$	0	0
$760$	1.09758	0.0398133
$761$	−41.9537	−1.52082	−0.760410	−	0.649443i	$-0.775002\pi$
$761$	−41.9537	−1.52082	−0.760410	+	0.649443i	$0.775002\pi$
$762$	0	0
$763$	−8.81740	−0.319211
$764$	10.9641	0.396665
$765$	0	0
$766$	−13.0222	−0.470510
$767$	−19.6902	−0.710972
$768$	0	0
$769$	18.5332	0.668324	0.334162	−	0.942516i	$-0.391547\pi$
$769$	18.5332	0.668324	0.334162	+	0.942516i	$0.391547\pi$
$770$	0.237798	0.00856966
$771$	0	0
$772$	18.4691	0.664716
$773$	25.6711	0.923325	0.461663	−	0.887056i	$-0.347253\pi$
$773$	25.6711	0.923325	0.461663	+	0.887056i	$0.347253\pi$
$774$	0	0
$775$	−51.7648	−1.85945
$776$	−24.0534	−0.863468
$777$	0	0
$778$	13.9199	0.499053
$779$	0.130665	0.00468157
$780$	0	0
$781$	−2.23280	−0.0798957
$782$	−1.34925	−0.0482491
$783$	0	0
$784$	−11.0263	−0.393798
$785$	8.24683	0.294342
$786$	0	0
$787$	−27.1905	−0.969236	−0.484618	−	0.874726i	$-0.661041\pi$
$787$	−27.1905	−0.969236	−0.484618	+	0.874726i	$0.661041\pi$
$788$	23.5444	0.838735
$789$	0	0
$790$	4.30038	0.153001
$791$	−20.9985	−0.746620
$792$	0	0
$793$	19.5783	0.695245
$794$	−2.42678	−0.0861230
$795$	0	0
$796$	−37.7863	−1.33930
$797$	53.6797	1.90143	0.950717	−	0.310060i	$-0.100349\pi$
$797$	53.6797	1.90143	0.950717	+	0.310060i	$0.100349\pi$
$798$	0	0
$799$	−3.73810	−0.132245
$800$	−24.4878	−0.865773
$801$	0	0
$802$	−12.0395	−0.425130
$803$	−1.64794	−0.0581546
$804$	0	0
$805$	0.581334	0.0204893
$806$	−25.9111	−0.912681
$807$	0	0
$808$	8.23644	0.289757
$809$	40.7000	1.43094	0.715469	−	0.698645i	$-0.246213\pi$
$809$	40.7000	1.43094	0.715469	+	0.698645i	$0.246213\pi$
$810$	0	0
$811$	−23.3555	−0.820121	−0.410061	−	0.912058i	$-0.634492\pi$
$811$	−23.3555	−0.820121	−0.410061	+	0.912058i	$0.634492\pi$
$812$	−6.10934	−0.214396
$813$	0	0
$814$	2.61346	0.0916018
$815$	2.03181	0.0711713
$816$	0	0
$817$	−3.95911	−0.138512
$818$	−9.20521	−0.321853
$819$	0	0
$820$	0.124646	0.00435282
$821$	45.6510	1.59323	0.796616	−	0.604486i	$-0.206621\pi$
$821$	45.6510	1.59323	0.796616	+	0.604486i	$0.206621\pi$
$822$	0	0
$823$	−49.2240	−1.71584	−0.857920	−	0.513784i	$-0.828243\pi$
$823$	−49.2240	−1.71584	−0.857920	+	0.513784i	$0.828243\pi$
$824$	−8.38659	−0.292161
$825$	0	0
$826$	3.65176	0.127061
$827$	44.7495	1.55609	0.778046	−	0.628207i	$-0.216211\pi$
$827$	44.7495	1.55609	0.778046	+	0.628207i	$0.216211\pi$
$828$	0	0
$829$	41.9255	1.45613	0.728067	−	0.685506i	$-0.240419\pi$
$829$	41.9255	1.45613	0.728067	+	0.685506i	$0.240419\pi$
$830$	−2.54932	−0.0884883
$831$	0	0
$832$	8.81089	0.305463
$833$	−17.2858	−0.598917
$834$	0	0
$835$	−3.24593	−0.112330
$836$	0.897420	0.0310379
$837$	0	0
$838$	12.2915	0.424601
$839$	−6.30443	−0.217653	−0.108827	−	0.994061i	$-0.534709\pi$
$839$	−6.30443	−0.217653	−0.108827	+	0.994061i	$0.534709\pi$
$840$	0	0
$841$	−23.6738	−0.816338
$842$	−0.0426072	−0.00146834
$843$	0	0
$844$	28.8020	0.991407
$845$	−3.61982	−0.124525
$846$	0	0
$847$	−16.5341	−0.568118
$848$	34.2677	1.17676
$849$	0	0
$850$	−9.32217	−0.319748
$851$	6.38900	0.219012
$852$	0	0
$853$	4.94203	0.169212	0.0846060	−	0.996414i	$-0.473037\pi$
$853$	4.94203	0.169212	0.0846060	+	0.996414i	$0.473037\pi$
$854$	−3.63100	−0.124250
$855$	0	0
$856$	23.9458	0.818452
$857$	−21.8963	−0.747963	−0.373981	−	0.927436i	$-0.622008\pi$
$857$	−21.8963	−0.747963	−0.373981	+	0.927436i	$0.622008\pi$
$858$	0	0
$859$	−1.98446	−0.0677089	−0.0338544	−	0.999427i	$-0.510778\pi$
$859$	−1.98446	−0.0677089	−0.0338544	+	0.999427i	$0.510778\pi$
$860$	−3.77671	−0.128785
$861$	0	0
$862$	−11.0523	−0.376443
$863$	44.3763	1.51059	0.755293	−	0.655387i	$-0.227495\pi$
$863$	44.3763	1.51059	0.755293	+	0.655387i	$0.227495\pi$
$864$	0	0
$865$	5.45401	0.185442
$866$	−11.7978	−0.400904
$867$	0	0
$868$	−29.2085	−0.991401
$869$	7.61081	0.258179
$870$	0	0
$871$	9.41000	0.318846
$872$	11.3041	0.382805
$873$	0	0
$874$	−0.360946	−0.0122092
$875$	8.29717	0.280495
$876$	0	0
$877$	25.5115	0.861462	0.430731	−	0.902480i	$-0.358256\pi$
$877$	25.5115	0.861462	0.430731	+	0.902480i	$0.358256\pi$
$878$	7.18267	0.242403
$879$	0	0
$880$	0.692060	0.0233293
$881$	3.47164	0.116963	0.0584813	−	0.998289i	$-0.481374\pi$
$881$	3.47164	0.116963	0.0584813	+	0.998289i	$0.481374\pi$
$882$	0	0
$883$	3.49008	0.117450	0.0587252	−	0.998274i	$-0.481296\pi$
$883$	3.49008	0.117450	0.0587252	+	0.998274i	$0.481296\pi$
$884$	28.3622	0.953924
$885$	0	0
$886$	3.32558	0.111725
$887$	42.6133	1.43081	0.715407	−	0.698708i	$-0.246241\pi$
$887$	42.6133	1.43081	0.715407	+	0.698708i	$0.246241\pi$
$888$	0	0
$889$	18.1941	0.610210
$890$	1.22385	0.0410237
$891$	0	0
$892$	30.4988	1.02117
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	11.1913	0.374084
$896$	−17.7247	−0.592140
$897$	0	0
$898$	−10.1598	−0.339038
$899$	25.4643	0.849284
$900$	0	0
$901$	53.7207	1.78970
$902$	−0.0362936	−0.00120844
$903$	0	0
$904$	26.9205	0.895363
$905$	6.04472	0.200933
$906$	0	0
$907$	4.34584	0.144301	0.0721507	−	0.997394i	$-0.477014\pi$
$907$	4.34584	0.144301	0.0721507	+	0.997394i	$0.477014\pi$
$908$	−11.5246	−0.382457
$909$	0	0
$910$	2.01049	0.0666471
$911$	−26.7952	−0.887763	−0.443882	−	0.896085i	$-0.646399\pi$
$911$	−26.7952	−0.887763	−0.443882	+	0.896085i	$0.646399\pi$
$912$	0	0
$913$	−4.51179	−0.149318
$914$	8.95442	0.296186
$915$	0	0
$916$	4.68608	0.154832
$917$	−6.61023	−0.218289
$918$	0	0
$919$	−40.7872	−1.34545	−0.672724	−	0.739894i	$-0.734876\pi$
$919$	−40.7872	−1.34545	−0.672724	+	0.739894i	$0.734876\pi$
$920$	−0.745283	−0.0245712
$921$	0	0
$922$	19.2305	0.633321
$923$	−18.8774	−0.621357
$924$	0	0
$925$	44.1425	1.45140
$926$	−15.2611	−0.501510
$927$	0	0
$928$	12.0461	0.395433
$929$	−2.80175	−0.0919225	−0.0459613	−	0.998943i	$-0.514635\pi$
$929$	−2.80175	−0.0919225	−0.0459613	+	0.998943i	$0.514635\pi$
$930$	0	0
$931$	−4.62421	−0.151552
$932$	38.5933	1.26417
$933$	0	0
$934$	−9.54652	−0.312372
$935$	1.08493	0.0354809
$936$	0	0
$937$	43.3906	1.41751	0.708754	−	0.705455i	$-0.249257\pi$
$937$	43.3906	1.41751	0.708754	+	0.705455i	$0.249257\pi$
$938$	−1.74519	−0.0569823
$939$	0	0
$940$	−0.953931	−0.0311138
$941$	20.9250	0.682135	0.341067	−	0.940039i	$-0.389212\pi$
$941$	20.9250	0.682135	0.341067	+	0.940039i	$0.389212\pi$
$942$	0	0
$943$	−0.0887251	−0.00288929
$944$	10.6276	0.345900
$945$	0	0
$946$	1.09968	0.0357537
$947$	43.7752	1.42250	0.711252	−	0.702937i	$-0.248129\pi$
$947$	43.7752	1.42250	0.711252	+	0.702937i	$0.248129\pi$
$948$	0	0
$949$	−13.9327	−0.452274
$950$	−2.49383	−0.0809104
$951$	0	0
$952$	−11.3856	−0.369008
$953$	5.96753	0.193307	0.0966537	−	0.995318i	$-0.469186\pi$
$953$	5.96753	0.193307	0.0966537	+	0.995318i	$0.469186\pi$
$954$	0	0
$955$	3.54589	0.114742
$956$	−33.0424	−1.06867
$957$	0	0
$958$	11.7402	0.379309
$959$	24.6285	0.795295
$960$	0	0
$961$	90.7441	2.92723
$962$	22.0958	0.712396
$963$	0	0
$964$	−30.1329	−0.970514
$965$	5.97308	0.192280
$966$	0	0
$967$	25.2342	0.811478	0.405739	−	0.913989i	$-0.367014\pi$
$967$	25.2342	0.811478	0.405739	+	0.913989i	$0.367014\pi$
$968$	21.1971	0.681300
$969$	0	0
$970$	−3.59392	−0.115394
$971$	9.43992	0.302941	0.151471	−	0.988462i	$-0.451599\pi$
$971$	9.43992	0.302941	0.151471	+	0.988462i	$0.451599\pi$
$972$	0	0
$973$	13.6078	0.436247
$974$	−7.72432	−0.247503
$975$	0	0
$976$	−10.5672	−0.338248
$977$	−44.3691	−1.41949	−0.709747	−	0.704457i	$-0.751191\pi$
$977$	−44.3691	−1.41949	−0.709747	+	0.704457i	$0.751191\pi$
$978$	0	0
$979$	2.16597	0.0692248
$980$	−4.41118	−0.140910
$981$	0	0
$982$	11.6662	0.372284
$983$	10.9364	0.348816	0.174408	−	0.984673i	$-0.444199\pi$
$983$	10.9364	0.348816	0.174408	+	0.984673i	$0.444199\pi$
$984$	0	0
$985$	7.61451	0.242618
$986$	4.58580	0.146042
$987$	0	0
$988$	7.58732	0.241385
$989$	2.68834	0.0854841
$990$	0	0
$991$	62.8282	1.99580	0.997902	−	0.0647479i	$-0.0206243\pi$
$991$	62.8282	1.99580	0.997902	+	0.0647479i	$0.0206243\pi$
$992$	57.5920	1.82855
$993$	0	0
$994$	3.50101	0.111045
$995$	−12.2205	−0.387416
$996$	0	0
$997$	20.8243	0.659512	0.329756	−	0.944066i	$-0.393034\pi$
$997$	20.8243	0.659512	0.329756	+	0.944066i	$0.393034\pi$
$998$	−9.77804	−0.309518
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.7		12
3.2	odd	2		893.2.a.a.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.6	✓	12	3.2	odd	2
8037.2.a.m.1.7		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.531564 q^{2} -1.71744 q^{4} -0.555438 q^{5} +1.54136 q^{7} -1.97606 q^{8} +O(q^{10})\)	\(q+0.531564 q^{2} -1.71744 q^{4} -0.555438 q^{5} +1.54136 q^{7} -1.97606 q^{8} -0.295251 q^{10} -0.522534 q^{11} -4.41781 q^{13} +0.819331 q^{14} +2.38448 q^{16} +3.73810 q^{17} +1.00000 q^{19} +0.953931 q^{20} -0.277760 q^{22} -0.679026 q^{23} -4.69149 q^{25} -2.34835 q^{26} -2.64719 q^{28} +2.30786 q^{29} +11.0338 q^{31} +5.21962 q^{32} +1.98704 q^{34} -0.856129 q^{35} -9.40906 q^{37} +0.531564 q^{38} +1.09758 q^{40} +0.130665 q^{41} -3.95911 q^{43} +0.897420 q^{44} -0.360946 q^{46} -1.00000 q^{47} -4.62421 q^{49} -2.49383 q^{50} +7.58732 q^{52} +14.3711 q^{53} +0.290235 q^{55} -3.04581 q^{56} +1.22677 q^{58} +4.45700 q^{59} -4.43167 q^{61} +5.86515 q^{62} -1.99440 q^{64} +2.45382 q^{65} -2.13001 q^{67} -6.41997 q^{68} -0.455087 q^{70} +4.27302 q^{71} +3.15375 q^{73} -5.00152 q^{74} -1.71744 q^{76} -0.805412 q^{77} -14.5652 q^{79} -1.32443 q^{80} +0.0694569 q^{82} +8.63444 q^{83} -2.07628 q^{85} -2.10452 q^{86} +1.03256 q^{88} -4.14514 q^{89} -6.80943 q^{91} +1.16619 q^{92} -0.531564 q^{94} -0.555438 q^{95} +12.1724 q^{97} -2.45806 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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