LMFDB - Embedded newform 8037.2.a.o.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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			$1.35283$ 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)

$f(q)$	$=$	$q-1.35283 q^{2} -0.169850 q^{4} -0.632563 q^{5} -0.940374 q^{7} +2.93544 q^{8} +O(q^{10})$	\(q-1.35283 q^{2} -0.169850 q^{4} -0.632563 q^{5} -0.940374 q^{7} +2.93544 q^{8} +0.855750 q^{10} -3.63106 q^{11} +1.19009 q^{13} +1.27217 q^{14} -3.63145 q^{16} -2.84373 q^{17} -1.00000 q^{19} +0.107441 q^{20} +4.91221 q^{22} -8.46472 q^{23} -4.59986 q^{25} -1.61000 q^{26} +0.159722 q^{28} -0.656670 q^{29} +5.51870 q^{31} -0.958140 q^{32} +3.84708 q^{34} +0.594846 q^{35} +3.43845 q^{37} +1.35283 q^{38} -1.85685 q^{40} -6.65661 q^{41} +4.01444 q^{43} +0.616735 q^{44} +11.4513 q^{46} -1.00000 q^{47} -6.11570 q^{49} +6.22284 q^{50} -0.202137 q^{52} -6.14089 q^{53} +2.29687 q^{55} -2.76041 q^{56} +0.888363 q^{58} -6.43661 q^{59} +3.14593 q^{61} -7.46586 q^{62} +8.55910 q^{64} -0.752809 q^{65} -12.6742 q^{67} +0.483007 q^{68} -0.804726 q^{70} -3.87739 q^{71} -9.89083 q^{73} -4.65164 q^{74} +0.169850 q^{76} +3.41456 q^{77} +7.68379 q^{79} +2.29712 q^{80} +9.00527 q^{82} +14.6499 q^{83} +1.79884 q^{85} -5.43085 q^{86} -10.6588 q^{88} +13.7760 q^{89} -1.11913 q^{91} +1.43773 q^{92} +1.35283 q^{94} +0.632563 q^{95} +6.57860 q^{97} +8.27350 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * 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 $

$2$	−1.35283	−0.956596	−0.478298	−	0.878198i	$-0.658746\pi$
$2$	−1.35283	−0.956596	−0.478298	+	0.878198i	$0.658746\pi$
$3$	0	0
$3$	0	0
$4$	−0.169850	−0.0849249
$5$	−0.632563	−0.282891	−0.141445	−	0.989946i	$-0.545175\pi$
$5$	−0.632563	−0.282891	−0.141445	+	0.989946i	$0.545175\pi$
$6$	0	0
$7$	−0.940374	−0.355428	−0.177714	−	0.984082i	$-0.556870\pi$
$7$	−0.940374	−0.355428	−0.177714	+	0.984082i	$0.556870\pi$
$8$	2.93544	1.03783
$9$	0	0
$10$	0.855750	0.270612
$11$	−3.63106	−1.09481	−0.547403	−	0.836869i	$-0.684384\pi$
$11$	−3.63106	−1.09481	−0.547403	+	0.836869i	$0.684384\pi$
$12$	0	0
$13$	1.19009	0.330073	0.165036	−	0.986287i	$-0.447226\pi$
$13$	1.19009	0.330073	0.165036	+	0.986287i	$0.447226\pi$
$14$	1.27217	0.340001
$15$	0	0
$16$	−3.63145	−0.907863
$17$	−2.84373	−0.689706	−0.344853	−	0.938657i	$-0.612071\pi$
$17$	−2.84373	−0.689706	−0.344853	+	0.938657i	$0.612071\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.107441	0.0240245
$21$	0	0
$22$	4.91221	1.04729
$23$	−8.46472	−1.76502	−0.882508	−	0.470297i	$-0.844147\pi$
$23$	−8.46472	−1.76502	−0.882508	+	0.470297i	$0.844147\pi$
$24$	0	0
$25$	−4.59986	−0.919973
$26$	−1.61000	−0.315746
$27$	0	0
$28$	0.159722	0.0301847
$29$	−0.656670	−0.121941	−0.0609703	−	0.998140i	$-0.519419\pi$
$29$	−0.656670	−0.121941	−0.0609703	+	0.998140i	$0.519419\pi$
$30$	0	0
$31$	5.51870	0.991187	0.495594	−	0.868554i	$-0.334951\pi$
$31$	5.51870	0.991187	0.495594	+	0.868554i	$0.334951\pi$
$32$	−0.958140	−0.169377
$33$	0	0
$34$	3.84708	0.659769
$35$	0.594846	0.100547
$36$	0	0
$37$	3.43845	0.565277	0.282639	−	0.959226i	$-0.408790\pi$
$37$	3.43845	0.565277	0.282639	+	0.959226i	$0.408790\pi$
$38$	1.35283	0.219458
$39$	0	0
$40$	−1.85685	−0.293594
$41$	−6.65661	−1.03959	−0.519794	−	0.854292i	$-0.673991\pi$
$41$	−6.65661	−1.03959	−0.519794	+	0.854292i	$0.673991\pi$
$42$	0	0
$43$	4.01444	0.612196	0.306098	−	0.952000i	$-0.400976\pi$
$43$	4.01444	0.612196	0.306098	+	0.952000i	$0.400976\pi$
$44$	0.616735	0.0929763
$45$	0	0
$46$	11.4513	1.68841
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.11570	−0.873671
$50$	6.22284	0.880042
$51$	0	0
$52$	−0.202137	−0.0280314
$53$	−6.14089	−0.843515	−0.421758	−	0.906709i	$-0.638587\pi$
$53$	−6.14089	−0.843515	−0.421758	+	0.906709i	$0.638587\pi$
$54$	0	0
$55$	2.29687	0.309710
$56$	−2.76041	−0.368876
$57$	0	0
$58$	0.888363	0.116648
$59$	−6.43661	−0.837975	−0.418987	−	0.907992i	$-0.637615\pi$
$59$	−6.43661	−0.837975	−0.418987	+	0.907992i	$0.637615\pi$
$60$	0	0
$61$	3.14593	0.402796	0.201398	−	0.979510i	$-0.435452\pi$
$61$	3.14593	0.402796	0.201398	+	0.979510i	$0.435452\pi$
$62$	−7.46586	−0.948165
$63$	0	0
$64$	8.55910	1.06989
$65$	−0.752809	−0.0933745
$66$	0	0
$67$	−12.6742	−1.54841	−0.774203	−	0.632938i	$-0.781849\pi$
$67$	−12.6742	−1.54841	−0.774203	+	0.632938i	$0.781849\pi$
$68$	0.483007	0.0585732
$69$	0	0
$70$	−0.804726	−0.0961831
$71$	−3.87739	−0.460161	−0.230081	−	0.973172i	$-0.573899\pi$
$71$	−3.87739	−0.460161	−0.230081	+	0.973172i	$0.573899\pi$
$72$	0	0
$73$	−9.89083	−1.15763	−0.578817	−	0.815457i	$-0.696486\pi$
$73$	−9.89083	−1.15763	−0.578817	+	0.815457i	$0.696486\pi$
$74$	−4.65164	−0.540742
$75$	0	0
$76$	0.169850	0.0194831
$77$	3.41456	0.389125
$78$	0	0
$79$	7.68379	0.864494	0.432247	−	0.901755i	$-0.357721\pi$
$79$	7.68379	0.864494	0.432247	+	0.901755i	$0.357721\pi$
$80$	2.29712	0.256826
$81$	0	0
$82$	9.00527	0.994465
$83$	14.6499	1.60803	0.804016	−	0.594607i	$-0.202692\pi$
$83$	14.6499	1.60803	0.804016	+	0.594607i	$0.202692\pi$
$84$	0	0
$85$	1.79884	0.195111
$86$	−5.43085	−0.585624
$87$	0	0
$88$	−10.6588	−1.13623
$89$	13.7760	1.46025	0.730127	−	0.683312i	$-0.239461\pi$
$89$	13.7760	1.46025	0.730127	+	0.683312i	$0.239461\pi$
$90$	0	0
$91$	−1.11913	−0.117317
$92$	1.43773	0.149894
$93$	0	0
$94$	1.35283	0.139534
$95$	0.632563	0.0648996
$96$	0	0
$97$	6.57860	0.667956	0.333978	−	0.942581i	$-0.391609\pi$
$97$	6.57860	0.667956	0.333978	+	0.942581i	$0.391609\pi$
$98$	8.27350	0.835750
$99$	0	0
$100$	0.781286	0.0781286
$101$	−6.78570	−0.675202	−0.337601	−	0.941289i	$-0.609616\pi$
$101$	−6.78570	−0.675202	−0.337601	+	0.941289i	$0.609616\pi$
$102$	0	0
$103$	15.7585	1.55273	0.776364	−	0.630284i	$-0.217062\pi$
$103$	15.7585	1.55273	0.776364	+	0.630284i	$0.217062\pi$
$104$	3.49345	0.342561
$105$	0	0
$106$	8.30758	0.806903
$107$	−5.48243	−0.530007	−0.265003	−	0.964247i	$-0.585373\pi$
$107$	−5.48243	−0.530007	−0.265003	+	0.964247i	$0.585373\pi$
$108$	0	0
$109$	−8.66299	−0.829764	−0.414882	−	0.909875i	$-0.636177\pi$
$109$	−8.66299	−0.829764	−0.414882	+	0.909875i	$0.636177\pi$
$110$	−3.10728	−0.296268
$111$	0	0
$112$	3.41492	0.322680
$113$	−15.1993	−1.42983	−0.714915	−	0.699212i	$-0.753534\pi$
$113$	−15.1993	−1.42983	−0.714915	+	0.699212i	$0.753534\pi$
$114$	0	0
$115$	5.35447	0.499307
$116$	0.111535	0.0103558
$117$	0	0
$118$	8.70764	0.801603
$119$	2.67417	0.245141
$120$	0	0
$121$	2.18460	0.198600
$122$	−4.25592	−0.385313
$123$	0	0
$124$	−0.937350	−0.0841765
$125$	6.07252	0.543142
$126$	0	0
$127$	−1.42665	−0.126595	−0.0632975	−	0.997995i	$-0.520162\pi$
$127$	−1.42665	−0.126595	−0.0632975	+	0.997995i	$0.520162\pi$
$128$	−9.66274	−0.854073
$129$	0	0
$130$	1.01842	0.0893216
$131$	−18.7757	−1.64044	−0.820222	−	0.572046i	$-0.806150\pi$
$131$	−18.7757	−1.64044	−0.820222	+	0.572046i	$0.806150\pi$
$132$	0	0
$133$	0.940374	0.0815408
$134$	17.1461	1.48120
$135$	0	0
$136$	−8.34759	−0.715800
$137$	18.5656	1.58616	0.793082	−	0.609115i	$-0.208475\pi$
$137$	18.5656	1.58616	0.793082	+	0.609115i	$0.208475\pi$
$138$	0	0
$139$	−1.98303	−0.168198	−0.0840991	−	0.996457i	$-0.526801\pi$
$139$	−1.98303	−0.168198	−0.0840991	+	0.996457i	$0.526801\pi$
$140$	−0.101034	−0.00853897
$141$	0	0
$142$	5.24545	0.440188
$143$	−4.32130	−0.361366
$144$	0	0
$145$	0.415385	0.0344959
$146$	13.3806	1.10739
$147$	0	0
$148$	−0.584020	−0.0480061
$149$	12.2814	1.00613	0.503067	−	0.864247i	$-0.332205\pi$
$149$	12.2814	1.00613	0.503067	+	0.864247i	$0.332205\pi$
$150$	0	0
$151$	2.21200	0.180010	0.0900050	−	0.995941i	$-0.471312\pi$
$151$	2.21200	0.180010	0.0900050	+	0.995941i	$0.471312\pi$
$152$	−2.93544	−0.238096
$153$	0	0
$154$	−4.61932	−0.372235
$155$	−3.49092	−0.280398
$156$	0	0
$157$	6.99927	0.558603	0.279301	−	0.960204i	$-0.409897\pi$
$157$	6.99927	0.558603	0.279301	+	0.960204i	$0.409897\pi$
$158$	−10.3949	−0.826971
$159$	0	0
$160$	0.606084	0.0479151
$161$	7.96001	0.627336
$162$	0	0
$163$	−2.05852	−0.161236	−0.0806180	−	0.996745i	$-0.525689\pi$
$163$	−2.05852	−0.161236	−0.0806180	+	0.996745i	$0.525689\pi$
$164$	1.13062	0.0882869
$165$	0	0
$166$	−19.8188	−1.53824
$167$	−22.0503	−1.70631	−0.853153	−	0.521660i	$-0.825313\pi$
$167$	−22.0503	−1.70631	−0.853153	+	0.521660i	$0.825313\pi$
$168$	0	0
$169$	−11.5837	−0.891052
$170$	−2.43352	−0.186643
$171$	0	0
$172$	−0.681852	−0.0519907
$173$	13.4916	1.02574	0.512872	−	0.858465i	$-0.328582\pi$
$173$	13.4916	1.02574	0.512872	+	0.858465i	$0.328582\pi$
$174$	0	0
$175$	4.32560	0.326984
$176$	13.1860	0.993934
$177$	0	0
$178$	−18.6366	−1.39687
$179$	−6.55140	−0.489674	−0.244837	−	0.969564i	$-0.578735\pi$
$179$	−6.55140	−0.489674	−0.244837	+	0.969564i	$0.578735\pi$
$180$	0	0
$181$	−4.52556	−0.336382	−0.168191	−	0.985754i	$-0.553793\pi$
$181$	−4.52556	−0.336382	−0.168191	+	0.985754i	$0.553793\pi$
$182$	1.51400	0.112225
$183$	0	0
$184$	−24.8477	−1.83179
$185$	−2.17503	−0.159912
$186$	0	0
$187$	10.3258	0.755094
$188$	0.169850	0.0123876
$189$	0	0
$190$	−0.855750	−0.0620826
$191$	−7.52510	−0.544497	−0.272249	−	0.962227i	$-0.587767\pi$
$191$	−7.52510	−0.544497	−0.272249	+	0.962227i	$0.587767\pi$
$192$	0	0
$193$	3.16589	0.227886	0.113943	−	0.993487i	$-0.463652\pi$
$193$	3.16589	0.227886	0.113943	+	0.993487i	$0.463652\pi$
$194$	−8.89973	−0.638963
$195$	0	0
$196$	1.03875	0.0741964
$197$	3.04497	0.216945	0.108473	−	0.994099i	$-0.465404\pi$
$197$	3.04497	0.216945	0.108473	+	0.994099i	$0.465404\pi$
$198$	0	0
$199$	−2.13810	−0.151566	−0.0757829	−	0.997124i	$-0.524146\pi$
$199$	−2.13810	−0.151566	−0.0757829	+	0.997124i	$0.524146\pi$
$200$	−13.5026	−0.954779
$201$	0	0
$202$	9.17990	0.645896
$203$	0.617516	0.0433411
$204$	0	0
$205$	4.21072	0.294090
$206$	−21.3185	−1.48533
$207$	0	0
$208$	−4.32177	−0.299661
$209$	3.63106	0.251166
$210$	0	0
$211$	−19.6882	−1.35539	−0.677696	−	0.735342i	$-0.737022\pi$
$211$	−19.6882	−1.35539	−0.677696	+	0.735342i	$0.737022\pi$
$212$	1.04303	0.0716355
$213$	0	0
$214$	7.41680	0.507002
$215$	−2.53938	−0.173185
$216$	0	0
$217$	−5.18964	−0.352296
$218$	11.7196	0.793748
$219$	0	0
$220$	−0.390124	−0.0263021
$221$	−3.38430	−0.227653
$222$	0	0
$223$	−9.01049	−0.603387	−0.301694	−	0.953405i	$-0.597552\pi$
$223$	−9.01049	−0.603387	−0.301694	+	0.953405i	$0.597552\pi$
$224$	0.901010	0.0602013
$225$	0	0
$226$	20.5621	1.36777
$227$	−4.95354	−0.328778	−0.164389	−	0.986396i	$-0.552565\pi$
$227$	−4.95354	−0.328778	−0.164389	+	0.986396i	$0.552565\pi$
$228$	0	0
$229$	4.43289	0.292934	0.146467	−	0.989216i	$-0.453210\pi$
$229$	4.43289	0.292934	0.146467	+	0.989216i	$0.453210\pi$
$230$	−7.24369	−0.477634
$231$	0	0
$232$	−1.92762	−0.126554
$233$	−16.6626	−1.09161	−0.545803	−	0.837914i	$-0.683775\pi$
$233$	−16.6626	−1.09161	−0.545803	+	0.837914i	$0.683775\pi$
$234$	0	0
$235$	0.632563	0.0412638
$236$	1.09326	0.0711649
$237$	0	0
$238$	−3.61770	−0.234501
$239$	28.4078	1.83755	0.918774	−	0.394785i	$-0.129181\pi$
$239$	28.4078	1.83755	0.918774	+	0.394785i	$0.129181\pi$
$240$	0	0
$241$	−8.44678	−0.544105	−0.272052	−	0.962282i	$-0.587702\pi$
$241$	−8.44678	−0.544105	−0.272052	+	0.962282i	$0.587702\pi$
$242$	−2.95540	−0.189980
$243$	0	0
$244$	−0.534336	−0.0342074
$245$	3.86856	0.247153
$246$	0	0
$247$	−1.19009	−0.0757239
$248$	16.1998	1.02869
$249$	0	0
$250$	−8.21508	−0.519568
$251$	−5.60589	−0.353841	−0.176920	−	0.984225i	$-0.556614\pi$
$251$	−5.60589	−0.353841	−0.176920	+	0.984225i	$0.556614\pi$
$252$	0	0
$253$	30.7359	1.93235
$254$	1.93002	0.121100
$255$	0	0
$256$	−4.04616	−0.252885
$257$	−20.1295	−1.25565	−0.627823	−	0.778356i	$-0.716054\pi$
$257$	−20.1295	−1.25565	−0.627823	+	0.778356i	$0.716054\pi$
$258$	0	0
$259$	−3.23343	−0.200915
$260$	0.127865	0.00792982
$261$	0	0
$262$	25.4004	1.56924
$263$	11.3941	0.702593	0.351297	−	0.936264i	$-0.385741\pi$
$263$	11.3941	0.702593	0.351297	+	0.936264i	$0.385741\pi$
$264$	0	0
$265$	3.88450	0.238623
$266$	−1.27217	−0.0780016
$267$	0	0
$268$	2.15272	0.131498
$269$	−26.3647	−1.60748	−0.803742	−	0.594978i	$-0.797161\pi$
$269$	−26.3647	−1.60748	−0.803742	+	0.594978i	$0.797161\pi$
$270$	0	0
$271$	23.5865	1.43278	0.716388	−	0.697702i	$-0.245794\pi$
$271$	23.5865	1.43278	0.716388	+	0.697702i	$0.245794\pi$
$272$	10.3269	0.626158
$273$	0	0
$274$	−25.1161	−1.51732
$275$	16.7024	1.00719
$276$	0	0
$277$	28.9733	1.74084	0.870418	−	0.492314i	$-0.163849\pi$
$277$	28.9733	1.74084	0.870418	+	0.492314i	$0.163849\pi$
$278$	2.68270	0.160898
$279$	0	0
$280$	1.74613	0.104351
$281$	31.6189	1.88623	0.943113	−	0.332472i	$-0.107883\pi$
$281$	31.6189	1.88623	0.943113	+	0.332472i	$0.107883\pi$
$282$	0	0
$283$	12.5312	0.744903	0.372451	−	0.928052i	$-0.378517\pi$
$283$	12.5312	0.744903	0.372451	+	0.928052i	$0.378517\pi$
$284$	0.658574	0.0390792
$285$	0	0
$286$	5.84599	0.345681
$287$	6.25971	0.369499
$288$	0	0
$289$	−8.91321	−0.524306
$290$	−0.561946	−0.0329986
$291$	0	0
$292$	1.67996	0.0983120
$293$	−25.1492	−1.46923	−0.734615	−	0.678484i	$-0.762637\pi$
$293$	−25.1492	−1.46923	−0.734615	+	0.678484i	$0.762637\pi$
$294$	0	0
$295$	4.07156	0.237055
$296$	10.0934	0.586664
$297$	0	0
$298$	−16.6147	−0.962464
$299$	−10.0738	−0.582584
$300$	0	0
$301$	−3.77507	−0.217592
$302$	−2.99246	−0.172197
$303$	0	0
$304$	3.63145	0.208278
$305$	−1.99000	−0.113947
$306$	0	0
$307$	21.5317	1.22888	0.614441	−	0.788963i	$-0.289382\pi$
$307$	21.5317	1.22888	0.614441	+	0.788963i	$0.289382\pi$
$308$	−0.579962	−0.0330464
$309$	0	0
$310$	4.72263	0.268227
$311$	2.20809	0.125209	0.0626045	−	0.998038i	$-0.480059\pi$
$311$	2.20809	0.125209	0.0626045	+	0.998038i	$0.480059\pi$
$312$	0	0
$313$	16.9266	0.956746	0.478373	−	0.878157i	$-0.341227\pi$
$313$	16.9266	0.956746	0.478373	+	0.878157i	$0.341227\pi$
$314$	−9.46883	−0.534357
$315$	0	0
$316$	−1.30509	−0.0734171
$317$	12.0942	0.679277	0.339638	−	0.940556i	$-0.389695\pi$
$317$	12.0942	0.679277	0.339638	+	0.940556i	$0.389695\pi$
$318$	0	0
$319$	2.38441	0.133501
$320$	−5.41417	−0.302661
$321$	0	0
$322$	−10.7685	−0.600107
$323$	2.84373	0.158229
$324$	0	0
$325$	−5.47427	−0.303658
$326$	2.78483	0.154238
$327$	0	0
$328$	−19.5401	−1.07892
$329$	0.940374	0.0518445
$330$	0	0
$331$	13.7183	0.754027	0.377013	−	0.926208i	$-0.376951\pi$
$331$	13.7183	0.754027	0.377013	+	0.926208i	$0.376951\pi$
$332$	−2.48828	−0.136562
$333$	0	0
$334$	29.8304	1.63225
$335$	8.01725	0.438029
$336$	0	0
$337$	6.32781	0.344698	0.172349	−	0.985036i	$-0.444864\pi$
$337$	6.32781	0.344698	0.172349	+	0.985036i	$0.444864\pi$
$338$	15.6707	0.852376
$339$	0	0
$340$	−0.305532	−0.0165698
$341$	−20.0387	−1.08516
$342$	0	0
$343$	12.3337	0.665955
$344$	11.7841	0.635358
$345$	0	0
$346$	−18.2518	−0.981222
$347$	−34.9920	−1.87847	−0.939235	−	0.343275i	$-0.888464\pi$
$347$	−34.9920	−1.87847	−0.939235	+	0.343275i	$0.888464\pi$
$348$	0	0
$349$	−31.5929	−1.69113	−0.845564	−	0.533874i	$-0.820736\pi$
$349$	−31.5929	−1.69113	−0.845564	+	0.533874i	$0.820736\pi$
$350$	−5.85180	−0.312792
$351$	0	0
$352$	3.47906	0.185435
$353$	−31.4192	−1.67227	−0.836137	−	0.548520i	$-0.815191\pi$
$353$	−31.4192	−1.67227	−0.836137	+	0.548520i	$0.815191\pi$
$354$	0	0
$355$	2.45269	0.130175
$356$	−2.33985	−0.124012
$357$	0	0
$358$	8.86293	0.468420
$359$	9.52339	0.502625	0.251313	−	0.967906i	$-0.419138\pi$
$359$	9.52339	0.502625	0.251313	+	0.967906i	$0.419138\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.12231	0.321782
$363$	0	0
$364$	0.190085	0.00996315
$365$	6.25657	0.327484
$366$	0	0
$367$	−5.13115	−0.267844	−0.133922	−	0.990992i	$-0.542757\pi$
$367$	−5.13115	−0.267844	−0.133922	+	0.990992i	$0.542757\pi$
$368$	30.7392	1.60239
$369$	0	0
$370$	2.94245	0.152971
$371$	5.77473	0.299809
$372$	0	0
$373$	−13.6778	−0.708208	−0.354104	−	0.935206i	$-0.615214\pi$
$373$	−13.6778	−0.708208	−0.354104	+	0.935206i	$0.615214\pi$
$374$	−13.9690	−0.722319
$375$	0	0
$376$	−2.93544	−0.151384
$377$	−0.781499	−0.0402493
$378$	0	0
$379$	29.1035	1.49494	0.747472	−	0.664293i	$-0.231267\pi$
$379$	29.1035	1.49494	0.747472	+	0.664293i	$0.231267\pi$
$380$	−0.107441	−0.00551159
$381$	0	0
$382$	10.1802	0.520864
$383$	23.3849	1.19491	0.597455	−	0.801902i	$-0.296179\pi$
$383$	23.3849	1.19491	0.597455	+	0.801902i	$0.296179\pi$
$384$	0	0
$385$	−2.15992	−0.110080
$386$	−4.28291	−0.217994
$387$	0	0
$388$	−1.11737	−0.0567261
$389$	9.16251	0.464558	0.232279	−	0.972649i	$-0.425382\pi$
$389$	9.16251	0.464558	0.232279	+	0.972649i	$0.425382\pi$
$390$	0	0
$391$	24.0714	1.21734
$392$	−17.9523	−0.906726
$393$	0	0
$394$	−4.11933	−0.207529
$395$	−4.86048	−0.244557
$396$	0	0
$397$	−6.86561	−0.344575	−0.172288	−	0.985047i	$-0.555116\pi$
$397$	−6.86561	−0.344575	−0.172288	+	0.985047i	$0.555116\pi$
$398$	2.89249	0.144987
$399$	0	0
$400$	16.7042	0.835209
$401$	14.7994	0.739048	0.369524	−	0.929221i	$-0.379521\pi$
$401$	14.7994	0.739048	0.369524	+	0.929221i	$0.379521\pi$
$402$	0	0
$403$	6.56777	0.327164
$404$	1.15255	0.0573415
$405$	0	0
$406$	−0.835394	−0.0414599
$407$	−12.4852	−0.618869
$408$	0	0
$409$	−31.8339	−1.57408	−0.787042	−	0.616899i	$-0.788389\pi$
$409$	−31.8339	−1.57408	−0.787042	+	0.616899i	$0.788389\pi$
$410$	−5.69640	−0.281325
$411$	0	0
$412$	−2.67657	−0.131865
$413$	6.05282	0.297840
$414$	0	0
$415$	−9.26696	−0.454897
$416$	−1.14028	−0.0559067
$417$	0	0
$418$	−4.91221	−0.240264
$419$	25.1533	1.22882	0.614410	−	0.788987i	$-0.289394\pi$
$419$	25.1533	1.22882	0.614410	+	0.788987i	$0.289394\pi$
$420$	0	0
$421$	32.0185	1.56048	0.780242	−	0.625477i	$-0.215096\pi$
$421$	32.0185	1.56048	0.780242	+	0.625477i	$0.215096\pi$
$422$	26.6348	1.29656
$423$	0	0
$424$	−18.0262	−0.875429
$425$	13.0808	0.634510
$426$	0	0
$427$	−2.95836	−0.143165
$428$	0.931190	0.0450108
$429$	0	0
$430$	3.43536	0.165668
$431$	26.8288	1.29230	0.646149	−	0.763211i	$-0.276378\pi$
$431$	26.8288	1.29230	0.646149	+	0.763211i	$0.276378\pi$
$432$	0	0
$433$	−20.5142	−0.985850	−0.492925	−	0.870072i	$-0.664072\pi$
$433$	−20.5142	−0.985850	−0.492925	+	0.870072i	$0.664072\pi$
$434$	7.02071	0.337005
$435$	0	0
$436$	1.47141	0.0704676
$437$	8.46472	0.404922
$438$	0	0
$439$	33.1950	1.58431	0.792155	−	0.610319i	$-0.208959\pi$
$439$	33.1950	1.58431	0.792155	+	0.610319i	$0.208959\pi$
$440$	6.74233	0.321428
$441$	0	0
$442$	4.57839	0.217772
$443$	5.00794	0.237934	0.118967	−	0.992898i	$-0.462042\pi$
$443$	5.00794	0.237934	0.118967	+	0.992898i	$0.462042\pi$
$444$	0	0
$445$	−8.71419	−0.413092
$446$	12.1897	0.577197
$447$	0	0
$448$	−8.04876	−0.380268
$449$	5.50700	0.259892	0.129946	−	0.991521i	$-0.458520\pi$
$449$	5.50700	0.259892	0.129946	+	0.991521i	$0.458520\pi$
$450$	0	0
$451$	24.1706	1.13815
$452$	2.58160	0.121428
$453$	0	0
$454$	6.70130	0.314508
$455$	0.707923	0.0331879
$456$	0	0
$457$	−1.85504	−0.0867753	−0.0433877	−	0.999058i	$-0.513815\pi$
$457$	−1.85504	−0.0867753	−0.0433877	+	0.999058i	$0.513815\pi$
$458$	−5.99695	−0.280219
$459$	0	0
$460$	−0.909455	−0.0424036
$461$	−19.8090	−0.922597	−0.461298	−	0.887245i	$-0.652616\pi$
$461$	−19.8090	−0.922597	−0.461298	+	0.887245i	$0.652616\pi$
$462$	0	0
$463$	−25.4957	−1.18488	−0.592442	−	0.805613i	$-0.701836\pi$
$463$	−25.4957	−1.18488	−0.592442	+	0.805613i	$0.701836\pi$
$464$	2.38467	0.110705
$465$	0	0
$466$	22.5417	1.04422
$467$	−12.5578	−0.581104	−0.290552	−	0.956859i	$-0.593839\pi$
$467$	−12.5578	−0.581104	−0.290552	+	0.956859i	$0.593839\pi$
$468$	0	0
$469$	11.9185	0.550347
$470$	−0.855750	−0.0394728
$471$	0	0
$472$	−18.8943	−0.869679
$473$	−14.5767	−0.670236
$474$	0	0
$475$	4.59986	0.211056
$476$	−0.454207	−0.0208186
$477$	0	0
$478$	−38.4309	−1.75779
$479$	27.9631	1.27767	0.638834	−	0.769345i	$-0.279417\pi$
$479$	27.9631	1.27767	0.638834	+	0.769345i	$0.279417\pi$
$480$	0	0
$481$	4.09208	0.186583
$482$	11.4271	0.520488
$483$	0	0
$484$	−0.371055	−0.0168661
$485$	−4.16138	−0.188958
$486$	0	0
$487$	−36.7637	−1.66592	−0.832961	−	0.553332i	$-0.813356\pi$
$487$	−36.7637	−1.66592	−0.832961	+	0.553332i	$0.813356\pi$
$488$	9.23470	0.418035
$489$	0	0
$490$	−5.23351	−0.236426
$491$	38.3532	1.73086	0.865429	−	0.501032i	$-0.167046\pi$
$491$	38.3532	1.73086	0.865429	+	0.501032i	$0.167046\pi$
$492$	0	0
$493$	1.86739	0.0841031
$494$	1.61000	0.0724371
$495$	0	0
$496$	−20.0409	−0.899862
$497$	3.64620	0.163554
$498$	0	0
$499$	−0.716312	−0.0320665	−0.0160333	−	0.999871i	$-0.505104\pi$
$499$	−0.716312	−0.0320665	−0.0160333	+	0.999871i	$0.505104\pi$
$500$	−1.03142	−0.0461263
$501$	0	0
$502$	7.58382	0.338483
$503$	4.47931	0.199723	0.0998613	−	0.995001i	$-0.468160\pi$
$503$	4.47931	0.199723	0.0998613	+	0.995001i	$0.468160\pi$
$504$	0	0
$505$	4.29238	0.191008
$506$	−41.5805	−1.84848
$507$	0	0
$508$	0.242317	0.0107511
$509$	9.12724	0.404558	0.202279	−	0.979328i	$-0.435165\pi$
$509$	9.12724	0.404558	0.202279	+	0.979328i	$0.435165\pi$
$510$	0	0
$511$	9.30109	0.411456
$512$	24.7992	1.09598
$513$	0	0
$514$	27.2319	1.20115
$515$	−9.96822	−0.439252
$516$	0	0
$517$	3.63106	0.159694
$518$	4.37428	0.192195
$519$	0	0
$520$	−2.20983	−0.0969072
$521$	35.7315	1.56542	0.782712	−	0.622384i	$-0.213836\pi$
$521$	35.7315	1.56542	0.782712	+	0.622384i	$0.213836\pi$
$522$	0	0
$523$	−3.09860	−0.135492	−0.0677462	−	0.997703i	$-0.521581\pi$
$523$	−3.09860	−0.135492	−0.0677462	+	0.997703i	$0.521581\pi$
$524$	3.18905	0.139315
$525$	0	0
$526$	−15.4143	−0.672097
$527$	−15.6937	−0.683627
$528$	0	0
$529$	48.6515	2.11528
$530$	−5.25506	−0.228265
$531$	0	0
$532$	−0.159722	−0.00692485
$533$	−7.92199	−0.343140
$534$	0	0
$535$	3.46798	0.149934
$536$	−37.2045	−1.60699
$537$	0	0
$538$	35.6669	1.53771
$539$	22.2065	0.956500
$540$	0	0
$541$	4.66236	0.200451	0.100225	−	0.994965i	$-0.468044\pi$
$541$	4.66236	0.200451	0.100225	+	0.994965i	$0.468044\pi$
$542$	−31.9085	−1.37059
$543$	0	0
$544$	2.72469	0.116820
$545$	5.47988	0.234732
$546$	0	0
$547$	−9.54530	−0.408127	−0.204064	−	0.978958i	$-0.565415\pi$
$547$	−9.54530	−0.408127	−0.204064	+	0.978958i	$0.565415\pi$
$548$	−3.15336	−0.134705
$549$	0	0
$550$	−22.5955	−0.963475
$551$	0.656670	0.0279751
$552$	0	0
$553$	−7.22564	−0.307266
$554$	−39.1959	−1.66528
$555$	0	0
$556$	0.336817	0.0142842
$557$	26.2921	1.11403	0.557015	−	0.830502i	$-0.311947\pi$
$557$	26.2921	1.11403	0.557015	+	0.830502i	$0.311947\pi$
$558$	0	0
$559$	4.77756	0.202069
$560$	−2.16015	−0.0912832
$561$	0	0
$562$	−42.7750	−1.80436
$563$	−5.67309	−0.239092	−0.119546	−	0.992829i	$-0.538144\pi$
$563$	−5.67309	−0.239092	−0.119546	+	0.992829i	$0.538144\pi$
$564$	0	0
$565$	9.61451	0.404485
$566$	−16.9526	−0.712571
$567$	0	0
$568$	−11.3818	−0.477571
$569$	1.92885	0.0808618	0.0404309	−	0.999182i	$-0.487127\pi$
$569$	1.92885	0.0808618	0.0404309	+	0.999182i	$0.487127\pi$
$570$	0	0
$571$	24.5853	1.02886	0.514430	−	0.857532i	$-0.328003\pi$
$571$	24.5853	1.02886	0.514430	+	0.857532i	$0.328003\pi$
$572$	0.733973	0.0306889
$573$	0	0
$574$	−8.46832	−0.353461
$575$	38.9366	1.62377
$576$	0	0
$577$	32.3399	1.34633	0.673165	−	0.739493i	$-0.264934\pi$
$577$	32.3399	1.34633	0.673165	+	0.739493i	$0.264934\pi$
$578$	12.0581	0.501549
$579$	0	0
$580$	−0.0705531	−0.00292956
$581$	−13.7764	−0.571540
$582$	0	0
$583$	22.2979	0.923486
$584$	−29.0339	−1.20143
$585$	0	0
$586$	34.0225	1.40546
$587$	24.3396	1.00460	0.502301	−	0.864693i	$-0.332487\pi$
$587$	24.3396	1.00460	0.502301	+	0.864693i	$0.332487\pi$
$588$	0	0
$589$	−5.51870	−0.227394
$590$	−5.50813	−0.226766
$591$	0	0
$592$	−12.4866	−0.513194
$593$	22.9378	0.941944	0.470972	−	0.882148i	$-0.343903\pi$
$593$	22.9378	0.941944	0.470972	+	0.882148i	$0.343903\pi$
$594$	0	0
$595$	−1.69158	−0.0693480
$596$	−2.08600	−0.0854459
$597$	0	0
$598$	13.6282	0.557297
$599$	22.5595	0.921755	0.460878	−	0.887464i	$-0.347535\pi$
$599$	22.5595	0.921755	0.460878	+	0.887464i	$0.347535\pi$
$600$	0	0
$601$	−33.3962	−1.36226	−0.681129	−	0.732163i	$-0.738511\pi$
$601$	−33.3962	−1.36226	−0.681129	+	0.732163i	$0.738511\pi$
$602$	5.10704	0.208147
$603$	0	0
$604$	−0.375708	−0.0152873
$605$	−1.38190	−0.0561822
$606$	0	0
$607$	34.5752	1.40337	0.701683	−	0.712489i	$-0.252432\pi$
$607$	34.5752	1.40337	0.701683	+	0.712489i	$0.252432\pi$
$608$	0.958140	0.0388577
$609$	0	0
$610$	2.69213	0.109001
$611$	−1.19009	−0.0481461
$612$	0	0
$613$	9.31254	0.376130	0.188065	−	0.982157i	$-0.439778\pi$
$613$	9.31254	0.376130	0.188065	+	0.982157i	$0.439778\pi$
$614$	−29.1288	−1.17554
$615$	0	0
$616$	10.0232	0.403847
$617$	15.1267	0.608980	0.304490	−	0.952516i	$-0.401514\pi$
$617$	15.1267	0.608980	0.304490	+	0.952516i	$0.401514\pi$
$618$	0	0
$619$	−8.22382	−0.330543	−0.165272	−	0.986248i	$-0.552850\pi$
$619$	−8.22382	−0.330543	−0.165272	+	0.986248i	$0.552850\pi$
$620$	0.592933	0.0238127
$621$	0	0
$622$	−2.98716	−0.119774
$623$	−12.9546	−0.519015
$624$	0	0
$625$	19.1581	0.766323
$626$	−22.8988	−0.915219
$627$	0	0
$628$	−1.18883	−0.0474393
$629$	−9.77801	−0.389875
$630$	0	0
$631$	−27.2747	−1.08579	−0.542895	−	0.839801i	$-0.682672\pi$
$631$	−27.2747	−1.08579	−0.542895	+	0.839801i	$0.682672\pi$
$632$	22.5553	0.897202
$633$	0	0
$634$	−16.3614	−0.649793
$635$	0.902447	0.0358125
$636$	0	0
$637$	−7.27825	−0.288375
$638$	−3.22570	−0.127707
$639$	0	0
$640$	6.11229	0.241609
$641$	−38.4575	−1.51898	−0.759490	−	0.650518i	$-0.774552\pi$
$641$	−38.4575	−1.51898	−0.759490	+	0.650518i	$0.774552\pi$
$642$	0	0
$643$	9.07909	0.358044	0.179022	−	0.983845i	$-0.442707\pi$
$643$	9.07909	0.358044	0.179022	+	0.983845i	$0.442707\pi$
$644$	−1.35201	−0.0532765
$645$	0	0
$646$	−3.84708	−0.151361
$647$	15.1284	0.594757	0.297379	−	0.954760i	$-0.403888\pi$
$647$	15.1284	0.594757	0.297379	+	0.954760i	$0.403888\pi$
$648$	0	0
$649$	23.3717	0.917420
$650$	7.40576	0.290478
$651$	0	0
$652$	0.349640	0.0136929
$653$	5.04223	0.197318	0.0986589	−	0.995121i	$-0.468545\pi$
$653$	5.04223	0.197318	0.0986589	+	0.995121i	$0.468545\pi$
$654$	0	0
$655$	11.8768	0.464066
$656$	24.1732	0.943803
$657$	0	0
$658$	−1.27217	−0.0495942
$659$	−35.8451	−1.39633	−0.698164	−	0.715938i	$-0.745999\pi$
$659$	−35.8451	−1.39633	−0.698164	+	0.715938i	$0.745999\pi$
$660$	0	0
$661$	−20.7428	−0.806802	−0.403401	−	0.915023i	$-0.632172\pi$
$661$	−20.7428	−0.806802	−0.403401	+	0.915023i	$0.632172\pi$
$662$	−18.5586	−0.721299
$663$	0	0
$664$	43.0038	1.66887
$665$	−0.594846	−0.0230671
$666$	0	0
$667$	5.55853	0.215227
$668$	3.74525	0.144908
$669$	0	0
$670$	−10.8460	−0.419017
$671$	−11.4231	−0.440983
$672$	0	0
$673$	−4.11508	−0.158625	−0.0793123	−	0.996850i	$-0.525272\pi$
$673$	−4.11508	−0.158625	−0.0793123	+	0.996850i	$0.525272\pi$
$674$	−8.56046	−0.329737
$675$	0	0
$676$	1.96749	0.0756725
$677$	27.3544	1.05132	0.525658	−	0.850696i	$-0.323819\pi$
$677$	27.3544	1.05132	0.525658	+	0.850696i	$0.323819\pi$
$678$	0	0
$679$	−6.18635	−0.237410
$680$	5.28038	0.202493
$681$	0	0
$682$	27.1090	1.03806
$683$	16.8488	0.644701	0.322350	−	0.946620i	$-0.395527\pi$
$683$	16.8488	0.644701	0.322350	+	0.946620i	$0.395527\pi$
$684$	0	0
$685$	−11.7439	−0.448711
$686$	−16.6854	−0.637050
$687$	0	0
$688$	−14.5782	−0.555790
$689$	−7.30823	−0.278421
$690$	0	0
$691$	−20.0418	−0.762425	−0.381213	−	0.924487i	$-0.624493\pi$
$691$	−20.0418	−0.762425	−0.381213	+	0.924487i	$0.624493\pi$
$692$	−2.29154	−0.0871112
$693$	0	0
$694$	47.3383	1.79694
$695$	1.25439	0.0475817
$696$	0	0
$697$	18.9296	0.717010
$698$	42.7398	1.61773
$699$	0	0
$700$	−0.734702	−0.0277691
$701$	38.1544	1.44107	0.720535	−	0.693418i	$-0.243896\pi$
$701$	38.1544	1.44107	0.720535	+	0.693418i	$0.243896\pi$
$702$	0	0
$703$	−3.43845	−0.129683
$704$	−31.0786	−1.17132
$705$	0	0
$706$	42.5048	1.59969
$707$	6.38110	0.239986
$708$	0	0
$709$	−9.28029	−0.348529	−0.174264	−	0.984699i	$-0.555755\pi$
$709$	−9.28029	−0.348529	−0.174264	+	0.984699i	$0.555755\pi$
$710$	−3.31808	−0.124525
$711$	0	0
$712$	40.4386	1.51550
$713$	−46.7142	−1.74946
$714$	0	0
$715$	2.73350	0.102227
$716$	1.11275	0.0415856
$717$	0	0
$718$	−12.8835	−0.480809
$719$	−0.552059	−0.0205883	−0.0102942	−	0.999947i	$-0.503277\pi$
$719$	−0.552059	−0.0205883	−0.0102942	+	0.999947i	$0.503277\pi$
$720$	0	0
$721$	−14.8189	−0.551883
$722$	−1.35283	−0.0503471
$723$	0	0
$724$	0.768665	0.0285672
$725$	3.02059	0.112182
$726$	0	0
$727$	26.4710	0.981754	0.490877	−	0.871229i	$-0.336676\pi$
$727$	26.4710	0.981754	0.490877	+	0.871229i	$0.336676\pi$
$728$	−3.28515	−0.121756
$729$	0	0
$730$	−8.46408	−0.313270
$731$	−11.4160	−0.422235
$732$	0	0
$733$	−18.7998	−0.694385	−0.347192	−	0.937794i	$-0.612865\pi$
$733$	−18.7998	−0.694385	−0.347192	+	0.937794i	$0.612865\pi$
$734$	6.94158	0.256218
$735$	0	0
$736$	8.11039	0.298953
$737$	46.0210	1.69520
$738$	0	0
$739$	−16.7548	−0.616335	−0.308167	−	0.951332i	$-0.599716\pi$
$739$	−16.7548	−0.616335	−0.308167	+	0.951332i	$0.599716\pi$
$740$	0.369429	0.0135805
$741$	0	0
$742$	−7.81223	−0.286796
$743$	37.2062	1.36496	0.682482	−	0.730902i	$-0.260900\pi$
$743$	37.2062	1.36496	0.682482	+	0.730902i	$0.260900\pi$
$744$	0	0
$745$	−7.76878	−0.284626
$746$	18.5037	0.677468
$747$	0	0
$748$	−1.75383	−0.0641263
$749$	5.15554	0.188379
$750$	0	0
$751$	−8.28140	−0.302193	−0.151096	−	0.988519i	$-0.548280\pi$
$751$	−8.28140	−0.302193	−0.151096	+	0.988519i	$0.548280\pi$
$752$	3.63145	0.132425
$753$	0	0
$754$	1.05724	0.0385023
$755$	−1.39923	−0.0509232
$756$	0	0
$757$	−2.84929	−0.103559	−0.0517797	−	0.998659i	$-0.516489\pi$
$757$	−2.84929	−0.103559	−0.0517797	+	0.998659i	$0.516489\pi$
$758$	−39.3720	−1.43006
$759$	0	0
$760$	1.85685	0.0673550
$761$	39.2518	1.42288	0.711438	−	0.702749i	$-0.248044\pi$
$761$	39.2518	1.42288	0.711438	+	0.702749i	$0.248044\pi$
$762$	0	0
$763$	8.14645	0.294921
$764$	1.27814	0.0462414
$765$	0	0
$766$	−31.6357	−1.14305
$767$	−7.66017	−0.276593
$768$	0	0
$769$	−31.0485	−1.11964	−0.559819	−	0.828615i	$-0.689129\pi$
$769$	−31.0485	−1.11964	−0.559819	+	0.828615i	$0.689129\pi$
$770$	2.92201	0.105302
$771$	0	0
$772$	−0.537726	−0.0193532
$773$	20.8466	0.749799	0.374900	−	0.927065i	$-0.377677\pi$
$773$	20.8466	0.749799	0.374900	+	0.927065i	$0.377677\pi$
$774$	0	0
$775$	−25.3853	−0.911866
$776$	19.3111	0.693227
$777$	0	0
$778$	−12.3953	−0.444394
$779$	6.65661	0.238498
$780$	0	0
$781$	14.0790	0.503788
$782$	−32.5645	−1.16450
$783$	0	0
$784$	22.2089	0.793173
$785$	−4.42748	−0.158023
$786$	0	0
$787$	10.1337	0.361227	0.180613	−	0.983554i	$-0.442192\pi$
$787$	10.1337	0.361227	0.180613	+	0.983554i	$0.442192\pi$
$788$	−0.517188	−0.0184241
$789$	0	0
$790$	6.57541	0.233943
$791$	14.2930	0.508201
$792$	0	0
$793$	3.74396	0.132952
$794$	9.28801	0.329619
$795$	0	0
$796$	0.363156	0.0128717
$797$	−26.4989	−0.938639	−0.469319	−	0.883028i	$-0.655501\pi$
$797$	−26.4989	−0.938639	−0.469319	+	0.883028i	$0.655501\pi$
$798$	0	0
$799$	2.84373	0.100604
$800$	4.40731	0.155822
$801$	0	0
$802$	−20.0211	−0.706970
$803$	35.9142	1.26739
$804$	0	0
$805$	−5.03520	−0.177468
$806$	−8.88508	−0.312964
$807$	0	0
$808$	−19.9190	−0.700748
$809$	32.5665	1.14498	0.572489	−	0.819912i	$-0.305978\pi$
$809$	32.5665	1.14498	0.572489	+	0.819912i	$0.305978\pi$
$810$	0	0
$811$	12.5155	0.439479	0.219739	−	0.975559i	$-0.429479\pi$
$811$	12.5155	0.439479	0.219739	+	0.975559i	$0.429479\pi$
$812$	−0.104885	−0.00368074
$813$	0	0
$814$	16.8904	0.592007
$815$	1.30214	0.0456121
$816$	0	0
$817$	−4.01444	−0.140447
$818$	43.0659	1.50576
$819$	0	0
$820$	−0.715191	−0.0249755
$821$	−5.23036	−0.182541	−0.0912704	−	0.995826i	$-0.529093\pi$
$821$	−5.23036	−0.182541	−0.0912704	+	0.995826i	$0.529093\pi$
$822$	0	0
$823$	−7.50707	−0.261680	−0.130840	−	0.991404i	$-0.541767\pi$
$823$	−7.50707	−0.261680	−0.130840	+	0.991404i	$0.541767\pi$
$824$	46.2580	1.61148
$825$	0	0
$826$	−8.18844	−0.284912
$827$	8.92433	0.310329	0.155165	−	0.987889i	$-0.450409\pi$
$827$	8.92433	0.310329	0.155165	+	0.987889i	$0.450409\pi$
$828$	0	0
$829$	54.8946	1.90657	0.953284	−	0.302077i	$-0.0976800\pi$
$829$	54.8946	1.90657	0.953284	+	0.302077i	$0.0976800\pi$
$830$	12.5366	0.435153
$831$	0	0
$832$	10.1861	0.353141
$833$	17.3914	0.602576
$834$	0	0
$835$	13.9482	0.482698
$836$	−0.616735	−0.0213302
$837$	0	0
$838$	−34.0281	−1.17548
$839$	4.99396	0.172410	0.0862052	−	0.996277i	$-0.472526\pi$
$839$	4.99396	0.172410	0.0862052	+	0.996277i	$0.472526\pi$
$840$	0	0
$841$	−28.5688	−0.985130
$842$	−43.3156	−1.49275
$843$	0	0
$844$	3.34404	0.115107
$845$	7.32740	0.252070
$846$	0	0
$847$	−2.05435	−0.0705882
$848$	22.3003	0.765796
$849$	0	0
$850$	−17.6961	−0.606970
$851$	−29.1055	−0.997723
$852$	0	0
$853$	−26.8651	−0.919843	−0.459922	−	0.887960i	$-0.652122\pi$
$853$	−26.8651	−0.919843	−0.459922	+	0.887960i	$0.652122\pi$
$854$	4.00215	0.136951
$855$	0	0
$856$	−16.0933	−0.550059
$857$	−48.3969	−1.65321	−0.826604	−	0.562784i	$-0.809730\pi$
$857$	−48.3969	−1.65321	−0.826604	+	0.562784i	$0.809730\pi$
$858$	0	0
$859$	6.67113	0.227616	0.113808	−	0.993503i	$-0.463695\pi$
$859$	6.67113	0.227616	0.113808	+	0.993503i	$0.463695\pi$
$860$	0.431314	0.0147077
$861$	0	0
$862$	−36.2948	−1.23621
$863$	−32.0191	−1.08994	−0.544972	−	0.838454i	$-0.683460\pi$
$863$	−32.0191	−1.08994	−0.544972	+	0.838454i	$0.683460\pi$
$864$	0	0
$865$	−8.53425	−0.290173
$866$	27.7522	0.943060
$867$	0	0
$868$	0.881460	0.0299187
$869$	−27.9003	−0.946454
$870$	0	0
$871$	−15.0835	−0.511086
$872$	−25.4297	−0.861157
$873$	0	0
$874$	−11.4513	−0.387347
$875$	−5.71044	−0.193048
$876$	0	0
$877$	44.5319	1.50373	0.751867	−	0.659315i	$-0.229153\pi$
$877$	44.5319	1.50373	0.751867	+	0.659315i	$0.229153\pi$
$878$	−44.9072	−1.51554
$879$	0	0
$880$	−8.34099	−0.281175
$881$	−28.5616	−0.962266	−0.481133	−	0.876648i	$-0.659775\pi$
$881$	−28.5616	−0.962266	−0.481133	+	0.876648i	$0.659775\pi$
$882$	0	0
$883$	−27.7792	−0.934843	−0.467422	−	0.884035i	$-0.654817\pi$
$883$	−27.7792	−0.934843	−0.467422	+	0.884035i	$0.654817\pi$
$884$	0.574824	0.0193334
$885$	0	0
$886$	−6.77489	−0.227607
$887$	18.3790	0.617107	0.308554	−	0.951207i	$-0.400155\pi$
$887$	18.3790	0.617107	0.308554	+	0.951207i	$0.400155\pi$
$888$	0	0
$889$	1.34159	0.0449954
$890$	11.7888	0.395162
$891$	0	0
$892$	1.53043	0.0512426
$893$	1.00000	0.0334637
$894$	0	0
$895$	4.14417	0.138524
$896$	9.08659	0.303562
$897$	0	0
$898$	−7.45004	−0.248611
$899$	−3.62396	−0.120866
$900$	0	0
$901$	17.4630	0.581777
$902$	−32.6987	−1.08875
$903$	0	0
$904$	−44.6166	−1.48393
$905$	2.86270	0.0951593
$906$	0	0
$907$	−15.6002	−0.517997	−0.258999	−	0.965878i	$-0.583392\pi$
$907$	−15.6002	−0.517997	−0.258999	+	0.965878i	$0.583392\pi$
$908$	0.841358	0.0279215
$909$	0	0
$910$	−0.957699	−0.0317474
$911$	33.6478	1.11480	0.557401	−	0.830243i	$-0.311799\pi$
$911$	33.6478	1.11480	0.557401	+	0.830243i	$0.311799\pi$
$912$	0	0
$913$	−53.1946	−1.76048
$914$	2.50956	0.0830089
$915$	0	0
$916$	−0.752925	−0.0248774
$917$	17.6562	0.583060
$918$	0	0
$919$	20.5920	0.679266	0.339633	−	0.940558i	$-0.389697\pi$
$919$	20.5920	0.679266	0.339633	+	0.940558i	$0.389697\pi$
$920$	15.7177	0.518198
$921$	0	0
$922$	26.7982	0.882552
$923$	−4.61446	−0.151887
$924$	0	0
$925$	−15.8164	−0.520040
$926$	34.4913	1.13345
$927$	0	0
$928$	0.629182	0.0206539
$929$	−19.7997	−0.649607	−0.324804	−	0.945782i	$-0.605298\pi$
$929$	−19.7997	−0.649607	−0.324804	+	0.945782i	$0.605298\pi$
$930$	0	0
$931$	6.11570	0.200434
$932$	2.83015	0.0927045
$933$	0	0
$934$	16.9885	0.555882
$935$	−6.53169	−0.213609
$936$	0	0
$937$	11.4220	0.373141	0.186571	−	0.982442i	$-0.440263\pi$
$937$	11.4220	0.373141	0.186571	+	0.982442i	$0.440263\pi$
$938$	−16.1238	−0.526459
$939$	0	0
$940$	−0.107441	−0.00350433
$941$	−7.92382	−0.258309	−0.129154	−	0.991624i	$-0.541226\pi$
$941$	−7.92382	−0.258309	−0.129154	+	0.991624i	$0.541226\pi$
$942$	0	0
$943$	56.3463	1.83489
$944$	23.3742	0.760766
$945$	0	0
$946$	19.7198	0.641145
$947$	30.4170	0.988419	0.494210	−	0.869343i	$-0.335458\pi$
$947$	30.4170	0.988419	0.494210	+	0.869343i	$0.335458\pi$
$948$	0	0
$949$	−11.7710	−0.382104
$950$	−6.22284	−0.201895
$951$	0	0
$952$	7.84986	0.254416
$953$	24.0510	0.779087	0.389544	−	0.921008i	$-0.372633\pi$
$953$	24.0510	0.779087	0.389544	+	0.921008i	$0.372633\pi$
$954$	0	0
$955$	4.76010	0.154033
$956$	−4.82506	−0.156054
$957$	0	0
$958$	−37.8294	−1.22221
$959$	−17.4586	−0.563767
$960$	0	0
$961$	−0.543974	−0.0175475
$962$	−5.53588	−0.178484
$963$	0	0
$964$	1.43468	0.0462081
$965$	−2.00262	−0.0644667
$966$	0	0
$967$	−16.3490	−0.525749	−0.262875	−	0.964830i	$-0.584671\pi$
$967$	−16.3490	−0.525749	−0.262875	+	0.964830i	$0.584671\pi$
$968$	6.41277	0.206114
$969$	0	0
$970$	5.62964	0.180757
$971$	15.2378	0.489004	0.244502	−	0.969649i	$-0.421376\pi$
$971$	15.2378	0.489004	0.244502	+	0.969649i	$0.421376\pi$
$972$	0	0
$973$	1.86479	0.0597824
$974$	49.7350	1.59361
$975$	0	0
$976$	−11.4243	−0.365683
$977$	19.1225	0.611782	0.305891	−	0.952067i	$-0.401046\pi$
$977$	19.1225	0.611782	0.305891	+	0.952067i	$0.401046\pi$
$978$	0	0
$979$	−50.0215	−1.59869
$980$	−0.657075	−0.0209895
$981$	0	0
$982$	−51.8854	−1.65573
$983$	−26.0567	−0.831080	−0.415540	−	0.909575i	$-0.636407\pi$
$983$	−26.0567	−0.831080	−0.415540	+	0.909575i	$0.636407\pi$
$984$	0	0
$985$	−1.92613	−0.0613718
$986$	−2.52626	−0.0804527
$987$	0	0
$988$	0.202137	0.00643084
$989$	−33.9811	−1.08054
$990$	0	0
$991$	30.4506	0.967296	0.483648	−	0.875263i	$-0.339311\pi$
$991$	30.4506	0.967296	0.483648	+	0.875263i	$0.339311\pi$
$992$	−5.28768	−0.167884
$993$	0	0
$994$	−4.93269	−0.156455
$995$	1.35248	0.0428766
$996$	0	0
$997$	36.8960	1.16851	0.584255	−	0.811570i	$-0.301387\pi$
$997$	36.8960	1.16851	0.584255	+	0.811570i	$0.301387\pi$
$998$	0.969049	0.0306747
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.7		18
3.2	odd	2		893.2.a.c.1.12	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.12	✓	18	3.2	odd	2
8037.2.a.o.1.7		18	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)

\(f(q)\)	\(=\)	\(q-1.35283 q^{2} -0.169850 q^{4} -0.632563 q^{5} -0.940374 q^{7} +2.93544 q^{8} +O(q^{10})\)	\(q-1.35283 q^{2} -0.169850 q^{4} -0.632563 q^{5} -0.940374 q^{7} +2.93544 q^{8} +0.855750 q^{10} -3.63106 q^{11} +1.19009 q^{13} +1.27217 q^{14} -3.63145 q^{16} -2.84373 q^{17} -1.00000 q^{19} +0.107441 q^{20} +4.91221 q^{22} -8.46472 q^{23} -4.59986 q^{25} -1.61000 q^{26} +0.159722 q^{28} -0.656670 q^{29} +5.51870 q^{31} -0.958140 q^{32} +3.84708 q^{34} +0.594846 q^{35} +3.43845 q^{37} +1.35283 q^{38} -1.85685 q^{40} -6.65661 q^{41} +4.01444 q^{43} +0.616735 q^{44} +11.4513 q^{46} -1.00000 q^{47} -6.11570 q^{49} +6.22284 q^{50} -0.202137 q^{52} -6.14089 q^{53} +2.29687 q^{55} -2.76041 q^{56} +0.888363 q^{58} -6.43661 q^{59} +3.14593 q^{61} -7.46586 q^{62} +8.55910 q^{64} -0.752809 q^{65} -12.6742 q^{67} +0.483007 q^{68} -0.804726 q^{70} -3.87739 q^{71} -9.89083 q^{73} -4.65164 q^{74} +0.169850 q^{76} +3.41456 q^{77} +7.68379 q^{79} +2.29712 q^{80} +9.00527 q^{82} +14.6499 q^{83} +1.79884 q^{85} -5.43085 q^{86} -10.6588 q^{88} +13.7760 q^{89} -1.11913 q^{91} +1.43773 q^{92} +1.35283 q^{94} +0.632563 q^{95} +6.57860 q^{97} +8.27350 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

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Database

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