LMFDB - Embedded newform 8037.2.a.x.1.25

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:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	$q+1.53643 q^{2} +0.360613 q^{4} +3.08058 q^{5} +0.219575 q^{7} -2.51880 q^{8} +O(q^{10})$	\(q+1.53643 q^{2} +0.360613 q^{4} +3.08058 q^{5} +0.219575 q^{7} -2.51880 q^{8} +4.73310 q^{10} -1.41237 q^{11} -0.0195438 q^{13} +0.337361 q^{14} -4.59118 q^{16} -4.21283 q^{17} +1.00000 q^{19} +1.11090 q^{20} -2.17001 q^{22} +0.718115 q^{23} +4.49000 q^{25} -0.0300276 q^{26} +0.0791816 q^{28} +10.0276 q^{29} -2.01881 q^{31} -2.01643 q^{32} -6.47272 q^{34} +0.676418 q^{35} +9.33436 q^{37} +1.53643 q^{38} -7.75938 q^{40} +5.47188 q^{41} +5.21210 q^{43} -0.509320 q^{44} +1.10333 q^{46} -1.00000 q^{47} -6.95179 q^{49} +6.89857 q^{50} -0.00704775 q^{52} +1.73962 q^{53} -4.35093 q^{55} -0.553065 q^{56} +15.4067 q^{58} +9.73587 q^{59} -1.34047 q^{61} -3.10176 q^{62} +6.08427 q^{64} -0.0602062 q^{65} +3.87426 q^{67} -1.51920 q^{68} +1.03927 q^{70} -1.43964 q^{71} -3.37228 q^{73} +14.3416 q^{74} +0.360613 q^{76} -0.310121 q^{77} +12.1055 q^{79} -14.1435 q^{80} +8.40716 q^{82} -4.31825 q^{83} -12.9780 q^{85} +8.00802 q^{86} +3.55748 q^{88} +9.45332 q^{89} -0.00429132 q^{91} +0.258962 q^{92} -1.53643 q^{94} +3.08058 q^{95} +11.2686 q^{97} -10.6809 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * 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 $

$2$	1.53643	1.08642	0.543210	−	0.839597i	$-0.317209\pi$
$2$	1.53643	1.08642	0.543210	+	0.839597i	$0.317209\pi$
$3$	0	0
$3$	0	0
$4$	0.360613	0.180307
$5$	3.08058	1.37768	0.688840	−	0.724914i	$-0.258120\pi$
$5$	3.08058	1.37768	0.688840	+	0.724914i	$0.258120\pi$
$6$	0	0
$7$	0.219575	0.0829914	0.0414957	−	0.999139i	$-0.486788\pi$
$7$	0.219575	0.0829914	0.0414957	+	0.999139i	$0.486788\pi$
$8$	−2.51880	−0.890531
$9$	0	0
$10$	4.73310	1.49674
$11$	−1.41237	−0.425846	−0.212923	−	0.977069i	$-0.568298\pi$
$11$	−1.41237	−0.425846	−0.212923	+	0.977069i	$0.568298\pi$
$12$	0	0
$13$	−0.0195438	−0.00542047	−0.00271023	−	0.999996i	$-0.500863\pi$
$13$	−0.0195438	−0.00542047	−0.00271023	+	0.999996i	$0.500863\pi$
$14$	0.337361	0.0901635
$15$	0	0
$16$	−4.59118	−1.14780
$17$	−4.21283	−1.02176	−0.510881	−	0.859651i	$-0.670681\pi$
$17$	−4.21283	−1.02176	−0.510881	+	0.859651i	$0.670681\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.11090	0.248405
$21$	0	0
$22$	−2.17001	−0.462647
$23$	0.718115	0.149737	0.0748687	−	0.997193i	$-0.476146\pi$
$23$	0.718115	0.149737	0.0748687	+	0.997193i	$0.476146\pi$
$24$	0	0
$25$	4.49000	0.898001
$26$	−0.0300276	−0.00588890
$27$	0	0
$28$	0.0791816	0.0149639
$29$	10.0276	1.86208	0.931039	−	0.364921i	$-0.118904\pi$
$29$	10.0276	1.86208	0.931039	+	0.364921i	$0.118904\pi$
$30$	0	0
$31$	−2.01881	−0.362589	−0.181295	−	0.983429i	$-0.558029\pi$
$31$	−2.01881	−0.362589	−0.181295	+	0.983429i	$0.558029\pi$
$32$	−2.01643	−0.356457
$33$	0	0
$34$	−6.47272	−1.11006
$35$	0.676418	0.114336
$36$	0	0
$37$	9.33436	1.53456	0.767279	−	0.641313i	$-0.221610\pi$
$37$	9.33436	1.53456	0.767279	+	0.641313i	$0.221610\pi$
$38$	1.53643	0.249242
$39$	0	0
$40$	−7.75938	−1.22687
$41$	5.47188	0.854565	0.427282	−	0.904118i	$-0.359471\pi$
$41$	5.47188	0.854565	0.427282	+	0.904118i	$0.359471\pi$
$42$	0	0
$43$	5.21210	0.794838	0.397419	−	0.917637i	$-0.369906\pi$
$43$	5.21210	0.794838	0.397419	+	0.917637i	$0.369906\pi$
$44$	−0.509320	−0.0767829
$45$	0	0
$46$	1.10333	0.162678
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.95179	−0.993112
$50$	6.89857	0.975605
$51$	0	0
$52$	−0.00704775	−0.000977346 0
$53$	1.73962	0.238956	0.119478	−	0.992837i	$-0.461878\pi$
$53$	1.73962	0.238956	0.119478	+	0.992837i	$0.461878\pi$
$54$	0	0
$55$	−4.35093	−0.586679
$56$	−0.553065	−0.0739064
$57$	0	0
$58$	15.4067	2.02300
$59$	9.73587	1.26750	0.633751	−	0.773537i	$-0.281514\pi$
$59$	9.73587	1.26750	0.633751	+	0.773537i	$0.281514\pi$
$60$	0	0
$61$	−1.34047	−0.171629	−0.0858146	−	0.996311i	$-0.527349\pi$
$61$	−1.34047	−0.171629	−0.0858146	+	0.996311i	$0.527349\pi$
$62$	−3.10176	−0.393924
$63$	0	0
$64$	6.08427	0.760534
$65$	−0.0602062	−0.00746767
$66$	0	0
$67$	3.87426	0.473317	0.236658	−	0.971593i	$-0.423948\pi$
$67$	3.87426	0.473317	0.236658	+	0.971593i	$0.423948\pi$
$68$	−1.51920	−0.184231
$69$	0	0
$70$	1.03927	0.124216
$71$	−1.43964	−0.170854	−0.0854269	−	0.996344i	$-0.527225\pi$
$71$	−1.43964	−0.170854	−0.0854269	+	0.996344i	$0.527225\pi$
$72$	0	0
$73$	−3.37228	−0.394695	−0.197348	−	0.980334i	$-0.563233\pi$
$73$	−3.37228	−0.394695	−0.197348	+	0.980334i	$0.563233\pi$
$74$	14.3416	1.66717
$75$	0	0
$76$	0.360613	0.0413652
$77$	−0.310121	−0.0353416
$78$	0	0
$79$	12.1055	1.36197	0.680987	−	0.732295i	$-0.261551\pi$
$79$	12.1055	1.36197	0.680987	+	0.732295i	$0.261551\pi$
$80$	−14.1435	−1.58130
$81$	0	0
$82$	8.40716	0.928415
$83$	−4.31825	−0.473990	−0.236995	−	0.971511i	$-0.576163\pi$
$83$	−4.31825	−0.473990	−0.236995	+	0.971511i	$0.576163\pi$
$84$	0	0
$85$	−12.9780	−1.40766
$86$	8.00802	0.863527
$87$	0	0
$88$	3.55748	0.379229
$89$	9.45332	1.00205	0.501025	−	0.865433i	$-0.332956\pi$
$89$	9.45332	1.00205	0.501025	+	0.865433i	$0.332956\pi$
$90$	0	0
$91$	−0.00429132	−0.000449852 0
$92$	0.258962	0.0269987
$93$	0	0
$94$	−1.53643	−0.158471
$95$	3.08058	0.316061
$96$	0	0
$97$	11.2686	1.14415	0.572077	−	0.820200i	$-0.306138\pi$
$97$	11.2686	1.14415	0.572077	+	0.820200i	$0.306138\pi$
$98$	−10.6809	−1.07894
$99$	0	0
$100$	1.61916	0.161916
$101$	18.4091	1.83178	0.915888	−	0.401434i	$-0.131488\pi$
$101$	18.4091	1.83178	0.915888	+	0.401434i	$0.131488\pi$
$102$	0	0
$103$	12.0085	1.18323	0.591615	−	0.806221i	$-0.298491\pi$
$103$	12.0085	1.18323	0.591615	+	0.806221i	$0.298491\pi$
$104$	0.0492269	0.00482709
$105$	0	0
$106$	2.67281	0.259606
$107$	11.9581	1.15604	0.578019	−	0.816023i	$-0.303826\pi$
$107$	11.9581	1.15604	0.578019	+	0.816023i	$0.303826\pi$
$108$	0	0
$109$	10.1061	0.967990	0.483995	−	0.875071i	$-0.339185\pi$
$109$	10.1061	0.967990	0.483995	+	0.875071i	$0.339185\pi$
$110$	−6.68489	−0.637379
$111$	0	0
$112$	−1.00811	−0.0952572
$113$	−18.7567	−1.76448	−0.882242	−	0.470797i	$-0.843967\pi$
$113$	−18.7567	−1.76448	−0.882242	+	0.470797i	$0.843967\pi$
$114$	0	0
$115$	2.21221	0.206290
$116$	3.61608	0.335745
$117$	0	0
$118$	14.9585	1.37704
$119$	−0.925032	−0.0847975
$120$	0	0
$121$	−9.00521	−0.818655
$122$	−2.05953	−0.186461
$123$	0	0
$124$	−0.728010	−0.0653773
$125$	−1.57109	−0.140523
$126$	0	0
$127$	17.2987	1.53501	0.767506	−	0.641041i	$-0.221497\pi$
$127$	17.2987	1.53501	0.767506	+	0.641041i	$0.221497\pi$
$128$	13.3809	1.18272
$129$	0	0
$130$	−0.0925026	−0.00811302
$131$	11.4492	1.00032	0.500162	−	0.865932i	$-0.333274\pi$
$131$	11.4492	1.00032	0.500162	+	0.865932i	$0.333274\pi$
$132$	0	0
$133$	0.219575	0.0190395
$134$	5.95253	0.514220
$135$	0	0
$136$	10.6113	0.909911
$137$	−2.80520	−0.239665	−0.119832	−	0.992794i	$-0.538236\pi$
$137$	−2.80520	−0.239665	−0.119832	+	0.992794i	$0.538236\pi$
$138$	0	0
$139$	−2.97814	−0.252603	−0.126301	−	0.991992i	$-0.540311\pi$
$139$	−2.97814	−0.252603	−0.126301	+	0.991992i	$0.540311\pi$
$140$	0.243926	0.0206155
$141$	0	0
$142$	−2.21190	−0.185619
$143$	0.0276031	0.00230828
$144$	0	0
$145$	30.8908	2.56535
$146$	−5.18126	−0.428804
$147$	0	0
$148$	3.36609	0.276691
$149$	−21.3630	−1.75012	−0.875062	−	0.484010i	$-0.839180\pi$
$149$	−21.3630	−1.75012	−0.875062	+	0.484010i	$0.839180\pi$
$150$	0	0
$151$	−6.25084	−0.508686	−0.254343	−	0.967114i	$-0.581859\pi$
$151$	−6.25084	−0.508686	−0.254343	+	0.967114i	$0.581859\pi$
$152$	−2.51880	−0.204302
$153$	0	0
$154$	−0.476479	−0.0383957
$155$	−6.21912	−0.499532
$156$	0	0
$157$	4.36318	0.348219	0.174110	−	0.984726i	$-0.444295\pi$
$157$	4.36318	0.348219	0.174110	+	0.984726i	$0.444295\pi$
$158$	18.5992	1.47968
$159$	0	0
$160$	−6.21177	−0.491084
$161$	0.157680	0.0124269
$162$	0	0
$163$	−5.16556	−0.404598	−0.202299	−	0.979324i	$-0.564841\pi$
$163$	−5.16556	−0.404598	−0.202299	+	0.979324i	$0.564841\pi$
$164$	1.97323	0.154084
$165$	0	0
$166$	−6.63469	−0.514952
$167$	16.7286	1.29450	0.647248	−	0.762280i	$-0.275920\pi$
$167$	16.7286	1.29450	0.647248	+	0.762280i	$0.275920\pi$
$168$	0	0
$169$	−12.9996	−0.999971
$170$	−19.9398	−1.52931
$171$	0	0
$172$	1.87955	0.143315
$173$	−18.5378	−1.40940	−0.704700	−	0.709505i	$-0.748918\pi$
$173$	−18.5378	−1.40940	−0.704700	+	0.709505i	$0.748918\pi$
$174$	0	0
$175$	0.985891	0.0745264
$176$	6.48446	0.488784
$177$	0	0
$178$	14.5244	1.08865
$179$	3.28270	0.245361	0.122680	−	0.992446i	$-0.460851\pi$
$179$	3.28270	0.245361	0.122680	+	0.992446i	$0.460851\pi$
$180$	0	0
$181$	−17.9602	−1.33497	−0.667487	−	0.744621i	$-0.732630\pi$
$181$	−17.9602	−1.33497	−0.667487	+	0.744621i	$0.732630\pi$
$182$	−0.00659330	−0.000488728 0
$183$	0	0
$184$	−1.80879	−0.133346
$185$	28.7553	2.11413
$186$	0	0
$187$	5.95008	0.435113
$188$	−0.360613	−0.0263004
$189$	0	0
$190$	4.73310	0.343375
$191$	−5.95355	−0.430784	−0.215392	−	0.976528i	$-0.569103\pi$
$191$	−5.95355	−0.430784	−0.215392	+	0.976528i	$0.569103\pi$
$192$	0	0
$193$	−2.77593	−0.199816	−0.0999079	−	0.994997i	$-0.531855\pi$
$193$	−2.77593	−0.199816	−0.0999079	+	0.994997i	$0.531855\pi$
$194$	17.3134	1.24303
$195$	0	0
$196$	−2.50691	−0.179065
$197$	0.952377	0.0678540	0.0339270	−	0.999424i	$-0.489199\pi$
$197$	0.952377	0.0678540	0.0339270	+	0.999424i	$0.489199\pi$
$198$	0	0
$199$	16.3777	1.16098	0.580492	−	0.814266i	$-0.302860\pi$
$199$	16.3777	1.16098	0.580492	+	0.814266i	$0.302860\pi$
$200$	−11.3094	−0.799697
$201$	0	0
$202$	28.2843	1.99008
$203$	2.20181	0.154536
$204$	0	0
$205$	16.8566	1.17732
$206$	18.4502	1.28548
$207$	0	0
$208$	0.0897291	0.00622159
$209$	−1.41237	−0.0976957
$210$	0	0
$211$	24.6779	1.69889	0.849447	−	0.527674i	$-0.176936\pi$
$211$	24.6779	1.69889	0.849447	+	0.527674i	$0.176936\pi$
$212$	0.627332	0.0430853
$213$	0	0
$214$	18.3728	1.25594
$215$	16.0563	1.09503
$216$	0	0
$217$	−0.443280	−0.0300918
$218$	15.5273	1.05164
$219$	0	0
$220$	−1.56900	−0.105782
$221$	0.0823347	0.00553843
$222$	0	0
$223$	−8.63387	−0.578167	−0.289083	−	0.957304i	$-0.593351\pi$
$223$	−8.63387	−0.578167	−0.289083	+	0.957304i	$0.593351\pi$
$224$	−0.442756	−0.0295829
$225$	0	0
$226$	−28.8184	−1.91697
$227$	−20.8999	−1.38717	−0.693587	−	0.720373i	$-0.743971\pi$
$227$	−20.8999	−1.38717	−0.693587	+	0.720373i	$0.743971\pi$
$228$	0	0
$229$	−20.2906	−1.34084	−0.670422	−	0.741980i	$-0.733887\pi$
$229$	−20.2906	−1.34084	−0.670422	+	0.741980i	$0.733887\pi$
$230$	3.39891	0.224118
$231$	0	0
$232$	−25.2575	−1.65824
$233$	−16.5819	−1.08632	−0.543158	−	0.839630i	$-0.682772\pi$
$233$	−16.5819	−1.08632	−0.543158	+	0.839630i	$0.682772\pi$
$234$	0	0
$235$	−3.08058	−0.200955
$236$	3.51089	0.228539
$237$	0	0
$238$	−1.42125	−0.0921257
$239$	−16.4587	−1.06462	−0.532311	−	0.846549i	$-0.678676\pi$
$239$	−16.4587	−1.06462	−0.532311	+	0.846549i	$0.678676\pi$
$240$	0	0
$241$	−24.7957	−1.59723	−0.798615	−	0.601842i	$-0.794434\pi$
$241$	−24.7957	−1.59723	−0.798615	+	0.601842i	$0.794434\pi$
$242$	−13.8359	−0.889403
$243$	0	0
$244$	−0.483390	−0.0309459
$245$	−21.4156	−1.36819
$246$	0	0
$247$	−0.0195438	−0.00124354
$248$	5.08498	0.322897
$249$	0	0
$250$	−2.41387	−0.152666
$251$	−16.4784	−1.04011	−0.520054	−	0.854133i	$-0.674088\pi$
$251$	−16.4784	−1.04011	−0.520054	+	0.854133i	$0.674088\pi$
$252$	0	0
$253$	−1.01425	−0.0637650
$254$	26.5782	1.66767
$255$	0	0
$256$	8.39026	0.524391
$257$	−0.278530	−0.0173742	−0.00868712	−	0.999962i	$-0.502765\pi$
$257$	−0.278530	−0.0173742	−0.00868712	+	0.999962i	$0.502765\pi$
$258$	0	0
$259$	2.04959	0.127355
$260$	−0.0217112	−0.00134647
$261$	0	0
$262$	17.5909	1.08677
$263$	−4.63096	−0.285557	−0.142779	−	0.989755i	$-0.545604\pi$
$263$	−4.63096	−0.285557	−0.142779	+	0.989755i	$0.545604\pi$
$264$	0	0
$265$	5.35906	0.329205
$266$	0.337361	0.0206849
$267$	0	0
$268$	1.39711	0.0853421
$269$	−20.2983	−1.23761	−0.618806	−	0.785544i	$-0.712383\pi$
$269$	−20.2983	−1.23761	−0.618806	+	0.785544i	$0.712383\pi$
$270$	0	0
$271$	7.05883	0.428794	0.214397	−	0.976747i	$-0.431221\pi$
$271$	7.05883	0.428794	0.214397	+	0.976747i	$0.431221\pi$
$272$	19.3419	1.17278
$273$	0	0
$274$	−4.30999	−0.260376
$275$	−6.34155	−0.382410
$276$	0	0
$277$	−4.64353	−0.279003	−0.139501	−	0.990222i	$-0.544550\pi$
$277$	−4.64353	−0.279003	−0.139501	+	0.990222i	$0.544550\pi$
$278$	−4.57571	−0.274433
$279$	0	0
$280$	−1.70376	−0.101819
$281$	9.31468	0.555667	0.277834	−	0.960629i	$-0.410384\pi$
$281$	9.31468	0.555667	0.277834	+	0.960629i	$0.410384\pi$
$282$	0	0
$283$	5.10172	0.303266	0.151633	−	0.988437i	$-0.451547\pi$
$283$	5.10172	0.303266	0.151633	+	0.988437i	$0.451547\pi$
$284$	−0.519153	−0.0308061
$285$	0	0
$286$	0.0424101	0.00250776
$287$	1.20149	0.0709215
$288$	0	0
$289$	0.747975	0.0439985
$290$	47.4616	2.78704
$291$	0	0
$292$	−1.21609	−0.0711662
$293$	27.3432	1.59740	0.798702	−	0.601726i	$-0.205520\pi$
$293$	27.3432	1.59740	0.798702	+	0.601726i	$0.205520\pi$
$294$	0	0
$295$	29.9922	1.74621
$296$	−23.5114	−1.36657
$297$	0	0
$298$	−32.8227	−1.90137
$299$	−0.0140347	−0.000811646 0
$300$	0	0
$301$	1.14445	0.0659647
$302$	−9.60397	−0.552646
$303$	0	0
$304$	−4.59118	−0.263323
$305$	−4.12942	−0.236450
$306$	0	0
$307$	6.12756	0.349718	0.174859	−	0.984593i	$-0.444053\pi$
$307$	6.12756	0.349718	0.174859	+	0.984593i	$0.444053\pi$
$308$	−0.111834	−0.00637232
$309$	0	0
$310$	−9.55523	−0.542701
$311$	19.6526	1.11440	0.557198	−	0.830379i	$-0.311876\pi$
$311$	19.6526	1.11440	0.557198	+	0.830379i	$0.311876\pi$
$312$	0	0
$313$	−14.1754	−0.801239	−0.400619	−	0.916245i	$-0.631205\pi$
$313$	−14.1754	−0.801239	−0.400619	+	0.916245i	$0.631205\pi$
$314$	6.70371	0.378312
$315$	0	0
$316$	4.36540	0.245573
$317$	17.4418	0.979627	0.489814	−	0.871827i	$-0.337065\pi$
$317$	17.4418	0.979627	0.489814	+	0.871827i	$0.337065\pi$
$318$	0	0
$319$	−14.1627	−0.792958
$320$	18.7431	1.04777
$321$	0	0
$322$	0.242264	0.0135008
$323$	−4.21283	−0.234408
$324$	0	0
$325$	−0.0877516	−0.00486758
$326$	−7.93651	−0.439563
$327$	0	0
$328$	−13.7826	−0.761016
$329$	−0.219575	−0.0121055
$330$	0	0
$331$	−25.6475	−1.40971	−0.704857	−	0.709349i	$-0.748989\pi$
$331$	−25.6475	−1.40971	−0.704857	+	0.709349i	$0.748989\pi$
$332$	−1.55722	−0.0854636
$333$	0	0
$334$	25.7022	1.40636
$335$	11.9350	0.652078
$336$	0	0
$337$	−16.8442	−0.917564	−0.458782	−	0.888549i	$-0.651714\pi$
$337$	−16.8442	−0.917564	−0.458782	+	0.888549i	$0.651714\pi$
$338$	−19.9730	−1.08639
$339$	0	0
$340$	−4.68004	−0.253811
$341$	2.85131	0.154407
$342$	0	0
$343$	−3.06346	−0.165411
$344$	−13.1282	−0.707828
$345$	0	0
$346$	−28.4820	−1.53120
$347$	16.3652	0.878532	0.439266	−	0.898357i	$-0.355239\pi$
$347$	16.3652	0.878532	0.439266	+	0.898357i	$0.355239\pi$
$348$	0	0
$349$	18.6172	0.996556	0.498278	−	0.867017i	$-0.333966\pi$
$349$	18.6172	0.996556	0.498278	+	0.867017i	$0.333966\pi$
$350$	1.51475	0.0809669
$351$	0	0
$352$	2.84794	0.151796
$353$	−9.35631	−0.497986	−0.248993	−	0.968505i	$-0.580100\pi$
$353$	−9.35631	−0.497986	−0.248993	+	0.968505i	$0.580100\pi$
$354$	0	0
$355$	−4.43493	−0.235382
$356$	3.40899	0.180676
$357$	0	0
$358$	5.04364	0.266564
$359$	−8.31981	−0.439103	−0.219551	−	0.975601i	$-0.570459\pi$
$359$	−8.31981	−0.439103	−0.219551	+	0.975601i	$0.570459\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−27.5946	−1.45034
$363$	0	0
$364$	−0.00154751	−8.11114e−5 0
$365$	−10.3886	−0.543763
$366$	0	0
$367$	18.2491	0.952597	0.476299	−	0.879284i	$-0.341978\pi$
$367$	18.2491	0.952597	0.476299	+	0.879284i	$0.341978\pi$
$368$	−3.29700	−0.171868
$369$	0	0
$370$	44.1804	2.29683
$371$	0.381978	0.0198313
$372$	0	0
$373$	−30.6018	−1.58450	−0.792251	−	0.610195i	$-0.791091\pi$
$373$	−30.6018	−1.58450	−0.792251	+	0.610195i	$0.791091\pi$
$374$	9.14188	0.472715
$375$	0	0
$376$	2.51880	0.129897
$377$	−0.195977	−0.0100933
$378$	0	0
$379$	−31.2860	−1.60705	−0.803526	−	0.595270i	$-0.797045\pi$
$379$	−31.2860	−1.60705	−0.803526	+	0.595270i	$0.797045\pi$
$380$	1.11090	0.0569880
$381$	0	0
$382$	−9.14721	−0.468012
$383$	32.7032	1.67106	0.835528	−	0.549448i	$-0.185162\pi$
$383$	32.7032	1.67106	0.835528	+	0.549448i	$0.185162\pi$
$384$	0	0
$385$	−0.955354	−0.0486893
$386$	−4.26502	−0.217084
$387$	0	0
$388$	4.06361	0.206299
$389$	34.8023	1.76455	0.882273	−	0.470739i	$-0.156013\pi$
$389$	34.8023	1.76455	0.882273	+	0.470739i	$0.156013\pi$
$390$	0	0
$391$	−3.02530	−0.152996
$392$	17.5102	0.884397
$393$	0	0
$394$	1.46326	0.0737179
$395$	37.2920	1.87636
$396$	0	0
$397$	−12.0299	−0.603765	−0.301883	−	0.953345i	$-0.597615\pi$
$397$	−12.0299	−0.603765	−0.301883	+	0.953345i	$0.597615\pi$
$398$	25.1632	1.26131
$399$	0	0
$400$	−20.6144	−1.03072
$401$	1.00698	0.0502860	0.0251430	−	0.999684i	$-0.491996\pi$
$401$	1.00698	0.0502860	0.0251430	+	0.999684i	$0.491996\pi$
$402$	0	0
$403$	0.0394552	0.00196540
$404$	6.63858	0.330281
$405$	0	0
$406$	3.38292	0.167891
$407$	−13.1836	−0.653485
$408$	0	0
$409$	30.4890	1.50758	0.753791	−	0.657114i	$-0.228223\pi$
$409$	30.4890	1.50758	0.753791	+	0.657114i	$0.228223\pi$
$410$	25.8990	1.27906
$411$	0	0
$412$	4.33041	0.213344
$413$	2.13775	0.105192
$414$	0	0
$415$	−13.3027	−0.653006
$416$	0.0394086	0.00193216
$417$	0	0
$418$	−2.17001	−0.106139
$419$	32.9180	1.60815	0.804074	−	0.594529i	$-0.202661\pi$
$419$	32.9180	1.60815	0.804074	+	0.594529i	$0.202661\pi$
$420$	0	0
$421$	6.55205	0.319327	0.159664	−	0.987171i	$-0.448959\pi$
$421$	6.55205	0.319327	0.159664	+	0.987171i	$0.448959\pi$
$422$	37.9158	1.84571
$423$	0	0
$424$	−4.38177	−0.212797
$425$	−18.9156	−0.917543
$426$	0	0
$427$	−0.294333	−0.0142438
$428$	4.31227	0.208441
$429$	0	0
$430$	24.6694	1.18966
$431$	18.8943	0.910105	0.455053	−	0.890465i	$-0.349620\pi$
$431$	18.8943	0.910105	0.455053	+	0.890465i	$0.349620\pi$
$432$	0	0
$433$	11.2917	0.542642	0.271321	−	0.962489i	$-0.412539\pi$
$433$	11.2917	0.542642	0.271321	+	0.962489i	$0.412539\pi$
$434$	−0.681068	−0.0326923
$435$	0	0
$436$	3.64440	0.174535
$437$	0.718115	0.0343521
$438$	0	0
$439$	−8.18134	−0.390474	−0.195237	−	0.980756i	$-0.562548\pi$
$439$	−8.18134	−0.390474	−0.195237	+	0.980756i	$0.562548\pi$
$440$	10.9591	0.522456
$441$	0	0
$442$	0.126501	0.00601706
$443$	7.55471	0.358935	0.179468	−	0.983764i	$-0.442562\pi$
$443$	7.55471	0.358935	0.179468	+	0.983764i	$0.442562\pi$
$444$	0	0
$445$	29.1218	1.38050
$446$	−13.2653	−0.628132
$447$	0	0
$448$	1.33595	0.0631178
$449$	−1.98210	−0.0935412	−0.0467706	−	0.998906i	$-0.514893\pi$
$449$	−1.98210	−0.0935412	−0.0467706	+	0.998906i	$0.514893\pi$
$450$	0	0
$451$	−7.72833	−0.363913
$452$	−6.76392	−0.318148
$453$	0	0
$454$	−32.1112	−1.50705
$455$	−0.0132198	−0.000619752 0
$456$	0	0
$457$	32.9503	1.54135	0.770675	−	0.637228i	$-0.219919\pi$
$457$	32.9503	1.54135	0.770675	+	0.637228i	$0.219919\pi$
$458$	−31.1751	−1.45672
$459$	0	0
$460$	0.797754	0.0371955
$461$	−19.5448	−0.910294	−0.455147	−	0.890416i	$-0.650413\pi$
$461$	−19.5448	−0.910294	−0.455147	+	0.890416i	$0.650413\pi$
$462$	0	0
$463$	19.9693	0.928054	0.464027	−	0.885821i	$-0.346404\pi$
$463$	19.9693	0.928054	0.464027	+	0.885821i	$0.346404\pi$
$464$	−46.0385	−2.13729
$465$	0	0
$466$	−25.4769	−1.18020
$467$	22.3239	1.03303	0.516513	−	0.856279i	$-0.327230\pi$
$467$	22.3239	1.03303	0.516513	+	0.856279i	$0.327230\pi$
$468$	0	0
$469$	0.850690	0.0392812
$470$	−4.73310	−0.218322
$471$	0	0
$472$	−24.5227	−1.12875
$473$	−7.36142	−0.338478
$474$	0	0
$475$	4.49000	0.206015
$476$	−0.333579	−0.0152896
$477$	0	0
$478$	−25.2876	−1.15663
$479$	5.19821	0.237512	0.118756	−	0.992923i	$-0.462109\pi$
$479$	5.19821	0.237512	0.118756	+	0.992923i	$0.462109\pi$
$480$	0	0
$481$	−0.182428	−0.00831802
$482$	−38.0968	−1.73526
$483$	0	0
$484$	−3.24740	−0.147609
$485$	34.7139	1.57628
$486$	0	0
$487$	0.674626	0.0305702	0.0152851	−	0.999883i	$-0.495134\pi$
$487$	0.674626	0.0305702	0.0152851	+	0.999883i	$0.495134\pi$
$488$	3.37637	0.152841
$489$	0	0
$490$	−32.9035	−1.48643
$491$	11.3129	0.510545	0.255272	−	0.966869i	$-0.417835\pi$
$491$	11.3129	0.510545	0.255272	+	0.966869i	$0.417835\pi$
$492$	0	0
$493$	−42.2446	−1.90260
$494$	−0.0300276	−0.00135101
$495$	0	0
$496$	9.26874	0.416179
$497$	−0.316108	−0.0141794
$498$	0	0
$499$	0.266961	0.0119508	0.00597540	−	0.999982i	$-0.498098\pi$
$499$	0.266961	0.0119508	0.00597540	+	0.999982i	$0.498098\pi$
$500$	−0.566556	−0.0253372
$501$	0	0
$502$	−25.3179	−1.12999
$503$	17.4373	0.777492	0.388746	−	0.921345i	$-0.372908\pi$
$503$	17.4373	0.777492	0.388746	+	0.921345i	$0.372908\pi$
$504$	0	0
$505$	56.7109	2.52360
$506$	−1.55832	−0.0692756
$507$	0	0
$508$	6.23815	0.276773
$509$	−17.6333	−0.781582	−0.390791	−	0.920480i	$-0.627798\pi$
$509$	−17.6333	−0.781582	−0.390791	+	0.920480i	$0.627798\pi$
$510$	0	0
$511$	−0.740467	−0.0327563
$512$	−13.8708	−0.613007
$513$	0	0
$514$	−0.427942	−0.0188757
$515$	36.9931	1.63011
$516$	0	0
$517$	1.41237	0.0621160
$518$	3.14905	0.138361
$519$	0	0
$520$	0.151648	0.00665018
$521$	28.5829	1.25224	0.626120	−	0.779727i	$-0.284642\pi$
$521$	28.5829	1.25224	0.626120	+	0.779727i	$0.284642\pi$
$522$	0	0
$523$	1.70881	0.0747211	0.0373605	−	0.999302i	$-0.488105\pi$
$523$	1.70881	0.0747211	0.0373605	+	0.999302i	$0.488105\pi$
$524$	4.12875	0.180365
$525$	0	0
$526$	−7.11514	−0.310235
$527$	8.50492	0.370480
$528$	0	0
$529$	−22.4843	−0.977579
$530$	8.23382	0.357654
$531$	0	0
$532$	0.0791816	0.00343296
$533$	−0.106941	−0.00463214
$534$	0	0
$535$	36.8381	1.59265
$536$	−9.75850	−0.421503
$537$	0	0
$538$	−31.1869	−1.34456
$539$	9.81850	0.422913
$540$	0	0
$541$	19.3602	0.832362	0.416181	−	0.909282i	$-0.363368\pi$
$541$	19.3602	0.832362	0.416181	+	0.909282i	$0.363368\pi$
$542$	10.8454	0.465850
$543$	0	0
$544$	8.49487	0.364215
$545$	31.1327	1.33358
$546$	0	0
$547$	−45.9608	−1.96514	−0.982571	−	0.185885i	$-0.940485\pi$
$547$	−45.9608	−1.96514	−0.982571	+	0.185885i	$0.940485\pi$
$548$	−1.01159	−0.0432131
$549$	0	0
$550$	−9.74334	−0.415457
$551$	10.0276	0.427190
$552$	0	0
$553$	2.65806	0.113032
$554$	−7.13446	−0.303114
$555$	0	0
$556$	−1.07396	−0.0455460
$557$	7.10706	0.301136	0.150568	−	0.988600i	$-0.451890\pi$
$557$	7.10706	0.301136	0.150568	+	0.988600i	$0.451890\pi$
$558$	0	0
$559$	−0.101864	−0.00430839
$560$	−3.10556	−0.131234
$561$	0	0
$562$	14.3113	0.603688
$563$	−23.6627	−0.997265	−0.498632	−	0.866814i	$-0.666164\pi$
$563$	−23.6627	−0.997265	−0.498632	+	0.866814i	$0.666164\pi$
$564$	0	0
$565$	−57.7817	−2.43089
$566$	7.83843	0.329474
$567$	0	0
$568$	3.62617	0.152151
$569$	31.8058	1.33337	0.666684	−	0.745341i	$-0.267713\pi$
$569$	31.8058	1.33337	0.666684	+	0.745341i	$0.267713\pi$
$570$	0	0
$571$	10.0784	0.421770	0.210885	−	0.977511i	$-0.432365\pi$
$571$	10.0784	0.421770	0.210885	+	0.977511i	$0.432365\pi$
$572$	0.00995403	0.000416199 0
$573$	0	0
$574$	1.84600	0.0770505
$575$	3.22434	0.134464
$576$	0	0
$577$	−23.6292	−0.983698	−0.491849	−	0.870680i	$-0.663679\pi$
$577$	−23.6292	−0.983698	−0.491849	+	0.870680i	$0.663679\pi$
$578$	1.14921	0.0478008
$579$	0	0
$580$	11.1397	0.462549
$581$	−0.948179	−0.0393371
$582$	0	0
$583$	−2.45700	−0.101758
$584$	8.49409	0.351488
$585$	0	0
$586$	42.0108	1.73545
$587$	−37.8661	−1.56290	−0.781450	−	0.623968i	$-0.785520\pi$
$587$	−37.8661	−1.56290	−0.781450	+	0.623968i	$0.785520\pi$
$588$	0	0
$589$	−2.01881	−0.0831837
$590$	46.0808	1.89712
$591$	0	0
$592$	−42.8557	−1.76136
$593$	26.6225	1.09325	0.546626	−	0.837377i	$-0.315912\pi$
$593$	26.6225	1.09325	0.546626	+	0.837377i	$0.315912\pi$
$594$	0	0
$595$	−2.84964	−0.116824
$596$	−7.70378	−0.315559
$597$	0	0
$598$	−0.0215633	−0.000881788 0
$599$	−15.1044	−0.617148	−0.308574	−	0.951200i	$-0.599852\pi$
$599$	−15.1044	−0.617148	−0.308574	+	0.951200i	$0.599852\pi$
$600$	0	0
$601$	8.44588	0.344515	0.172257	−	0.985052i	$-0.444894\pi$
$601$	8.44588	0.344515	0.172257	+	0.985052i	$0.444894\pi$
$602$	1.75836	0.0716654
$603$	0	0
$604$	−2.25414	−0.0917195
$605$	−27.7413	−1.12784
$606$	0	0
$607$	15.8308	0.642553	0.321277	−	0.946985i	$-0.395888\pi$
$607$	15.8308	0.642553	0.321277	+	0.946985i	$0.395888\pi$
$608$	−2.01643	−0.0817769
$609$	0	0
$610$	−6.34456	−0.256884
$611$	0.0195438	0.000790656 0
$612$	0	0
$613$	1.84337	0.0744531	0.0372265	−	0.999307i	$-0.488148\pi$
$613$	1.84337	0.0744531	0.0372265	+	0.999307i	$0.488148\pi$
$614$	9.41455	0.379940
$615$	0	0
$616$	0.781133	0.0314727
$617$	9.14679	0.368236	0.184118	−	0.982904i	$-0.441057\pi$
$617$	9.14679	0.368236	0.184118	+	0.982904i	$0.441057\pi$
$618$	0	0
$619$	−42.2604	−1.69859	−0.849295	−	0.527918i	$-0.822973\pi$
$619$	−42.2604	−1.69859	−0.849295	+	0.527918i	$0.822973\pi$
$620$	−2.24270	−0.0900689
$621$	0	0
$622$	30.1948	1.21070
$623$	2.07571	0.0831616
$624$	0	0
$625$	−27.2899	−1.09160
$626$	−21.7794	−0.870481
$627$	0	0
$628$	1.57342	0.0627863
$629$	−39.3241	−1.56795
$630$	0	0
$631$	−24.3430	−0.969082	−0.484541	−	0.874769i	$-0.661013\pi$
$631$	−24.3430	−0.969082	−0.484541	+	0.874769i	$0.661013\pi$
$632$	−30.4913	−1.21288
$633$	0	0
$634$	26.7980	1.06429
$635$	53.2902	2.11476
$636$	0	0
$637$	0.135864	0.00538313
$638$	−21.7599	−0.861485
$639$	0	0
$640$	41.2210	1.62940
$641$	−4.96133	−0.195961	−0.0979803	−	0.995188i	$-0.531238\pi$
$641$	−4.96133	−0.195961	−0.0979803	+	0.995188i	$0.531238\pi$
$642$	0	0
$643$	2.01583	0.0794965	0.0397482	−	0.999210i	$-0.487344\pi$
$643$	2.01583	0.0794965	0.0397482	+	0.999210i	$0.487344\pi$
$644$	0.0568615	0.00224066
$645$	0	0
$646$	−6.47272	−0.254666
$647$	18.7284	0.736289	0.368145	−	0.929769i	$-0.379993\pi$
$647$	18.7284	0.736289	0.368145	+	0.929769i	$0.379993\pi$
$648$	0	0
$649$	−13.7507	−0.539761
$650$	−0.134824	−0.00528823
$651$	0	0
$652$	−1.86277	−0.0729517
$653$	−20.9351	−0.819254	−0.409627	−	0.912253i	$-0.634341\pi$
$653$	−20.9351	−0.819254	−0.409627	+	0.912253i	$0.634341\pi$
$654$	0	0
$655$	35.2703	1.37813
$656$	−25.1224	−0.980866
$657$	0	0
$658$	−0.337361	−0.0131517
$659$	−19.8052	−0.771502	−0.385751	−	0.922603i	$-0.626058\pi$
$659$	−19.8052	−0.771502	−0.385751	+	0.922603i	$0.626058\pi$
$660$	0	0
$661$	−33.0361	−1.28496	−0.642478	−	0.766305i	$-0.722093\pi$
$661$	−33.0361	−1.28496	−0.642478	+	0.766305i	$0.722093\pi$
$662$	−39.4056	−1.53154
$663$	0	0
$664$	10.8768	0.422103
$665$	0.676418	0.0262304
$666$	0	0
$667$	7.20097	0.278823
$668$	6.03254	0.233406
$669$	0	0
$670$	18.3373	0.708431
$671$	1.89324	0.0730876
$672$	0	0
$673$	19.7766	0.762330	0.381165	−	0.924507i	$-0.375523\pi$
$673$	19.7766	0.762330	0.381165	+	0.924507i	$0.375523\pi$
$674$	−25.8800	−0.996859
$675$	0	0
$676$	−4.68784	−0.180301
$677$	37.5472	1.44306	0.721528	−	0.692386i	$-0.243440\pi$
$677$	37.5472	1.44306	0.721528	+	0.692386i	$0.243440\pi$
$678$	0	0
$679$	2.47430	0.0949550
$680$	32.6890	1.25357
$681$	0	0
$682$	4.38084	0.167751
$683$	−26.6660	−1.02035	−0.510174	−	0.860071i	$-0.670419\pi$
$683$	−26.6660	−1.02035	−0.510174	+	0.860071i	$0.670419\pi$
$684$	0	0
$685$	−8.64166	−0.330181
$686$	−4.70679	−0.179706
$687$	0	0
$688$	−23.9297	−0.912312
$689$	−0.0339988	−0.00129525
$690$	0	0
$691$	−26.7172	−1.01637	−0.508185	−	0.861248i	$-0.669684\pi$
$691$	−26.7172	−1.01637	−0.508185	+	0.861248i	$0.669684\pi$
$692$	−6.68497	−0.254124
$693$	0	0
$694$	25.1440	0.954454
$695$	−9.17443	−0.348006
$696$	0	0
$697$	−23.0521	−0.873162
$698$	28.6040	1.08268
$699$	0	0
$700$	0.355525	0.0134376
$701$	31.3461	1.18393	0.591964	−	0.805965i	$-0.298353\pi$
$701$	31.3461	1.18393	0.591964	+	0.805965i	$0.298353\pi$
$702$	0	0
$703$	9.33436	0.352052
$704$	−8.59325	−0.323870
$705$	0	0
$706$	−14.3753	−0.541022
$707$	4.04218	0.152022
$708$	0	0
$709$	41.6520	1.56427	0.782136	−	0.623108i	$-0.214130\pi$
$709$	41.6520	1.56427	0.782136	+	0.623108i	$0.214130\pi$
$710$	−6.81396	−0.255723
$711$	0	0
$712$	−23.8110	−0.892356
$713$	−1.44974	−0.0542932
$714$	0	0
$715$	0.0850335	0.00318007
$716$	1.18379	0.0442402
$717$	0	0
$718$	−12.7828	−0.477050
$719$	−22.3200	−0.832394	−0.416197	−	0.909274i	$-0.636637\pi$
$719$	−22.3200	−0.832394	−0.416197	+	0.909274i	$0.636637\pi$
$720$	0	0
$721$	2.63676	0.0981979
$722$	1.53643	0.0571800
$723$	0	0
$724$	−6.47670	−0.240705
$725$	45.0239	1.67215
$726$	0	0
$727$	14.4045	0.534232	0.267116	−	0.963664i	$-0.413929\pi$
$727$	14.4045	0.534232	0.267116	+	0.963664i	$0.413929\pi$
$728$	0.0108090	0.000400607 0
$729$	0	0
$730$	−15.9613	−0.590755
$731$	−21.9577	−0.812136
$732$	0	0
$733$	−19.0819	−0.704807	−0.352403	−	0.935848i	$-0.614635\pi$
$733$	−19.0819	−0.704807	−0.352403	+	0.935848i	$0.614635\pi$
$734$	28.0385	1.03492
$735$	0	0
$736$	−1.44803	−0.0533750
$737$	−5.47190	−0.201560
$738$	0	0
$739$	−18.4896	−0.680151	−0.340076	−	0.940398i	$-0.610453\pi$
$739$	−18.4896	−0.680151	−0.340076	+	0.940398i	$0.610453\pi$
$740$	10.3695	0.381192
$741$	0	0
$742$	0.586881	0.0215451
$743$	−42.4413	−1.55702	−0.778510	−	0.627632i	$-0.784024\pi$
$743$	−42.4413	−1.55702	−0.778510	+	0.627632i	$0.784024\pi$
$744$	0	0
$745$	−65.8105	−2.41111
$746$	−47.0175	−1.72143
$747$	0	0
$748$	2.14568	0.0784538
$749$	2.62571	0.0959412
$750$	0	0
$751$	21.2323	0.774777	0.387389	−	0.921916i	$-0.373377\pi$
$751$	21.2323	0.774777	0.387389	+	0.921916i	$0.373377\pi$
$752$	4.59118	0.167423
$753$	0	0
$754$	−0.301105	−0.0109656
$755$	−19.2562	−0.700806
$756$	0	0
$757$	−7.60481	−0.276402	−0.138201	−	0.990404i	$-0.544132\pi$
$757$	−7.60481	−0.276402	−0.138201	+	0.990404i	$0.544132\pi$
$758$	−48.0687	−1.74593
$759$	0	0
$760$	−7.75938	−0.281462
$761$	−20.4508	−0.741340	−0.370670	−	0.928765i	$-0.620872\pi$
$761$	−20.4508	−0.741340	−0.370670	+	0.928765i	$0.620872\pi$
$762$	0	0
$763$	2.21905	0.0803349
$764$	−2.14693	−0.0776732
$765$	0	0
$766$	50.2461	1.81547
$767$	−0.190276	−0.00687045
$768$	0	0
$769$	−43.1270	−1.55520	−0.777599	−	0.628760i	$-0.783563\pi$
$769$	−43.1270	−1.55520	−0.777599	+	0.628760i	$0.783563\pi$
$770$	−1.46783	−0.0528970
$771$	0	0
$772$	−1.00104	−0.0360281
$773$	−15.1674	−0.545533	−0.272767	−	0.962080i	$-0.587939\pi$
$773$	−15.1674	−0.545533	−0.272767	+	0.962080i	$0.587939\pi$
$774$	0	0
$775$	−9.06447	−0.325605
$776$	−28.3834	−1.01890
$777$	0	0
$778$	53.4712	1.91704
$779$	5.47188	0.196051
$780$	0	0
$781$	2.03330	0.0727574
$782$	−4.64816	−0.166218
$783$	0	0
$784$	31.9169	1.13989
$785$	13.4411	0.479735
$786$	0	0
$787$	−47.0194	−1.67606	−0.838031	−	0.545623i	$-0.816293\pi$
$787$	−47.0194	−1.67606	−0.838031	+	0.545623i	$0.816293\pi$
$788$	0.343440	0.0122345
$789$	0	0
$790$	57.2965	2.03852
$791$	−4.11850	−0.146437
$792$	0	0
$793$	0.0261978	0.000930310 0
$794$	−18.4831	−0.655942
$795$	0	0
$796$	5.90601	0.209333
$797$	−26.6726	−0.944790	−0.472395	−	0.881387i	$-0.656610\pi$
$797$	−26.6726	−0.944790	−0.472395	+	0.881387i	$0.656610\pi$
$798$	0	0
$799$	4.21283	0.149039
$800$	−9.05376	−0.320099
$801$	0	0
$802$	1.54715	0.0546317
$803$	4.76291	0.168079
$804$	0	0
$805$	0.485746	0.0171203
$806$	0.0606201	0.00213525
$807$	0	0
$808$	−46.3689	−1.63125
$809$	17.5702	0.617736	0.308868	−	0.951105i	$-0.400050\pi$
$809$	17.5702	0.617736	0.308868	+	0.951105i	$0.400050\pi$
$810$	0	0
$811$	−33.8024	−1.18696	−0.593481	−	0.804848i	$-0.702247\pi$
$811$	−33.8024	−1.18696	−0.593481	+	0.804848i	$0.702247\pi$
$812$	0.794001	0.0278640
$813$	0	0
$814$	−20.2556	−0.709959
$815$	−15.9129	−0.557406
$816$	0	0
$817$	5.21210	0.182348
$818$	46.8441	1.63787
$819$	0	0
$820$	6.07872	0.212278
$821$	25.5460	0.891562	0.445781	−	0.895142i	$-0.352926\pi$
$821$	25.5460	0.891562	0.445781	+	0.895142i	$0.352926\pi$
$822$	0	0
$823$	53.5184	1.86553	0.932767	−	0.360480i	$-0.117387\pi$
$823$	53.5184	1.86553	0.932767	+	0.360480i	$0.117387\pi$
$824$	−30.2469	−1.05370
$825$	0	0
$826$	3.28450	0.114282
$827$	−38.8701	−1.35165	−0.675823	−	0.737064i	$-0.736212\pi$
$827$	−38.8701	−1.35165	−0.675823	+	0.737064i	$0.736212\pi$
$828$	0	0
$829$	12.0498	0.418506	0.209253	−	0.977862i	$-0.432897\pi$
$829$	12.0498	0.418506	0.209253	+	0.977862i	$0.432897\pi$
$830$	−20.4387	−0.709439
$831$	0	0
$832$	−0.118910	−0.00412245
$833$	29.2867	1.01472
$834$	0	0
$835$	51.5338	1.78340
$836$	−0.509320	−0.0176152
$837$	0	0
$838$	50.5761	1.74712
$839$	43.8544	1.51402	0.757010	−	0.653403i	$-0.226659\pi$
$839$	43.8544	1.51402	0.757010	+	0.653403i	$0.226659\pi$
$840$	0	0
$841$	71.5526	2.46733
$842$	10.0668	0.346923
$843$	0	0
$844$	8.89917	0.306322
$845$	−40.0464	−1.37764
$846$	0	0
$847$	−1.97732	−0.0679414
$848$	−7.98694	−0.274273
$849$	0	0
$850$	−29.0625	−0.996837
$851$	6.70314	0.229781
$852$	0	0
$853$	20.5086	0.702203	0.351101	−	0.936337i	$-0.385807\pi$
$853$	20.5086	0.702203	0.351101	+	0.936337i	$0.385807\pi$
$854$	−0.452221	−0.0154747
$855$	0	0
$856$	−30.1202	−1.02949
$857$	47.6606	1.62805	0.814027	−	0.580827i	$-0.197271\pi$
$857$	47.6606	1.62805	0.814027	+	0.580827i	$0.197271\pi$
$858$	0	0
$859$	−9.96103	−0.339866	−0.169933	−	0.985456i	$-0.554355\pi$
$859$	−9.96103	−0.339866	−0.169933	+	0.985456i	$0.554355\pi$
$860$	5.79012	0.197442
$861$	0	0
$862$	29.0297	0.988756
$863$	13.7264	0.467253	0.233626	−	0.972326i	$-0.424941\pi$
$863$	13.7264	0.467253	0.233626	+	0.972326i	$0.424941\pi$
$864$	0	0
$865$	−57.1072	−1.94170
$866$	17.3488	0.589537
$867$	0	0
$868$	−0.159853	−0.00542575
$869$	−17.0975	−0.579991
$870$	0	0
$871$	−0.0757177	−0.00256560
$872$	−25.4553	−0.862025
$873$	0	0
$874$	1.10333	0.0373208
$875$	−0.344972	−0.0116622
$876$	0	0
$877$	−27.1320	−0.916183	−0.458092	−	0.888905i	$-0.651467\pi$
$877$	−27.1320	−0.916183	−0.458092	+	0.888905i	$0.651467\pi$
$878$	−12.5700	−0.424218
$879$	0	0
$880$	19.9759	0.673388
$881$	−23.7635	−0.800612	−0.400306	−	0.916382i	$-0.631096\pi$
$881$	−23.7635	−0.800612	−0.400306	+	0.916382i	$0.631096\pi$
$882$	0	0
$883$	42.9315	1.44476	0.722380	−	0.691496i	$-0.243048\pi$
$883$	42.9315	1.44476	0.722380	+	0.691496i	$0.243048\pi$
$884$	0.0296910	0.000998616 0
$885$	0	0
$886$	11.6073	0.389954
$887$	34.7083	1.16539	0.582695	−	0.812691i	$-0.301998\pi$
$887$	34.7083	1.16539	0.582695	+	0.812691i	$0.301998\pi$
$888$	0	0
$889$	3.79836	0.127393
$890$	44.7435	1.49981
$891$	0	0
$892$	−3.11349	−0.104247
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	10.1126	0.338028
$896$	2.93811	0.0981553
$897$	0	0
$898$	−3.04536	−0.101625
$899$	−20.2438	−0.675169
$900$	0	0
$901$	−7.32875	−0.244156
$902$	−11.8740	−0.395362
$903$	0	0
$904$	47.2444	1.57133
$905$	−55.3280	−1.83917
$906$	0	0
$907$	−39.5802	−1.31424	−0.657120	−	0.753786i	$-0.728225\pi$
$907$	−39.5802	−1.31424	−0.657120	+	0.753786i	$0.728225\pi$
$908$	−7.53677	−0.250117
$909$	0	0
$910$	−0.0203112	−0.000673311 0
$911$	−55.3853	−1.83500	−0.917499	−	0.397739i	$-0.869795\pi$
$911$	−55.3853	−1.83500	−0.917499	+	0.397739i	$0.869795\pi$
$912$	0	0
$913$	6.09898	0.201847
$914$	50.6258	1.67455
$915$	0	0
$916$	−7.31707	−0.241763
$917$	2.51396	0.0830184
$918$	0	0
$919$	−3.30764	−0.109109	−0.0545546	−	0.998511i	$-0.517374\pi$
$919$	−3.30764	−0.109109	−0.0545546	+	0.998511i	$0.517374\pi$
$920$	−5.57213	−0.183708
$921$	0	0
$922$	−30.0293	−0.988961
$923$	0.0281360	0.000926107 0
$924$	0	0
$925$	41.9113	1.37803
$926$	30.6815	1.00826
$927$	0	0
$928$	−20.2199	−0.663751
$929$	20.6460	0.677372	0.338686	−	0.940900i	$-0.390018\pi$
$929$	20.6460	0.677372	0.338686	+	0.940900i	$0.390018\pi$
$930$	0	0
$931$	−6.95179	−0.227836
$932$	−5.97966	−0.195870
$933$	0	0
$934$	34.2990	1.12230
$935$	18.3297	0.599447
$936$	0	0
$937$	−41.0457	−1.34090	−0.670452	−	0.741953i	$-0.733900\pi$
$937$	−41.0457	−1.34090	−0.670452	+	0.741953i	$0.733900\pi$
$938$	1.30702	0.0426759
$939$	0	0
$940$	−1.11090	−0.0362336
$941$	26.8186	0.874263	0.437131	−	0.899398i	$-0.355995\pi$
$941$	26.8186	0.874263	0.437131	+	0.899398i	$0.355995\pi$
$942$	0	0
$943$	3.92944	0.127960
$944$	−44.6992	−1.45483
$945$	0	0
$946$	−11.3103	−0.367729
$947$	−36.2933	−1.17937	−0.589687	−	0.807632i	$-0.700749\pi$
$947$	−36.2933	−1.17937	−0.589687	+	0.807632i	$0.700749\pi$
$948$	0	0
$949$	0.0659070	0.00213943
$950$	6.89857	0.223819
$951$	0	0
$952$	2.32997	0.0755148
$953$	−11.6940	−0.378807	−0.189403	−	0.981899i	$-0.560655\pi$
$953$	−11.6940	−0.378807	−0.189403	+	0.981899i	$0.560655\pi$
$954$	0	0
$955$	−18.3404	−0.593482
$956$	−5.93521	−0.191958
$957$	0	0
$958$	7.98667	0.258038
$959$	−0.615951	−0.0198901
$960$	0	0
$961$	−26.9244	−0.868529
$962$	−0.280288	−0.00903686
$963$	0	0
$964$	−8.94165	−0.287991
$965$	−8.55149	−0.275282
$966$	0	0
$967$	−16.8885	−0.543099	−0.271549	−	0.962425i	$-0.587536\pi$
$967$	−16.8885	−0.543099	−0.271549	+	0.962425i	$0.587536\pi$
$968$	22.6823	0.729038
$969$	0	0
$970$	53.3354	1.71250
$971$	−9.03781	−0.290037	−0.145019	−	0.989429i	$-0.546324\pi$
$971$	−9.03781	−0.290037	−0.145019	+	0.989429i	$0.546324\pi$
$972$	0	0
$973$	−0.653925	−0.0209639
$974$	1.03651	0.0332121
$975$	0	0
$976$	6.15433	0.196995
$977$	59.8120	1.91355	0.956777	−	0.290821i	$-0.0939284\pi$
$977$	59.8120	1.91355	0.956777	+	0.290821i	$0.0939284\pi$
$978$	0	0
$979$	−13.3516	−0.426719
$980$	−7.72274	−0.246694
$981$	0	0
$982$	17.3815	0.554665
$983$	−12.9624	−0.413436	−0.206718	−	0.978401i	$-0.566278\pi$
$983$	−12.9624	−0.413436	−0.206718	+	0.978401i	$0.566278\pi$
$984$	0	0
$985$	2.93388	0.0934811
$986$	−64.9058	−2.06702
$987$	0	0
$988$	−0.00704775	−0.000224219 0
$989$	3.74289	0.119017
$990$	0	0
$991$	−0.893965	−0.0283977	−0.0141989	−	0.999899i	$-0.504520\pi$
$991$	−0.893965	−0.0283977	−0.0141989	+	0.999899i	$0.504520\pi$
$992$	4.07079	0.129248
$993$	0	0
$994$	−0.485678	−0.0154048
$995$	50.4529	1.59946
$996$	0	0
$997$	−47.5580	−1.50618	−0.753088	−	0.657919i	$-0.771437\pi$
$997$	−47.5580	−1.50618	−0.753088	+	0.657919i	$0.771437\pi$
$998$	0.410166	0.0129836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.x.1.25	yes	34
3.2	odd	2		8037.2.a.u.1.10	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.10	✓	34	3.2	odd	2
8037.2.a.x.1.25	yes	34	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)

\(f(q)\)	\(=\)	\(q+1.53643 q^{2} +0.360613 q^{4} +3.08058 q^{5} +0.219575 q^{7} -2.51880 q^{8} +O(q^{10})\)	\(q+1.53643 q^{2} +0.360613 q^{4} +3.08058 q^{5} +0.219575 q^{7} -2.51880 q^{8} +4.73310 q^{10} -1.41237 q^{11} -0.0195438 q^{13} +0.337361 q^{14} -4.59118 q^{16} -4.21283 q^{17} +1.00000 q^{19} +1.11090 q^{20} -2.17001 q^{22} +0.718115 q^{23} +4.49000 q^{25} -0.0300276 q^{26} +0.0791816 q^{28} +10.0276 q^{29} -2.01881 q^{31} -2.01643 q^{32} -6.47272 q^{34} +0.676418 q^{35} +9.33436 q^{37} +1.53643 q^{38} -7.75938 q^{40} +5.47188 q^{41} +5.21210 q^{43} -0.509320 q^{44} +1.10333 q^{46} -1.00000 q^{47} -6.95179 q^{49} +6.89857 q^{50} -0.00704775 q^{52} +1.73962 q^{53} -4.35093 q^{55} -0.553065 q^{56} +15.4067 q^{58} +9.73587 q^{59} -1.34047 q^{61} -3.10176 q^{62} +6.08427 q^{64} -0.0602062 q^{65} +3.87426 q^{67} -1.51920 q^{68} +1.03927 q^{70} -1.43964 q^{71} -3.37228 q^{73} +14.3416 q^{74} +0.360613 q^{76} -0.310121 q^{77} +12.1055 q^{79} -14.1435 q^{80} +8.40716 q^{82} -4.31825 q^{83} -12.9780 q^{85} +8.00802 q^{86} +3.55748 q^{88} +9.45332 q^{89} -0.00429132 q^{91} +0.258962 q^{92} -1.53643 q^{94} +3.08058 q^{95} +11.2686 q^{97} -10.6809 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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