LMFDB - Embedded newform 8037.2.a.s.1.8

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:	$24$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.72798 q^{2} +0.985905 q^{4} +4.09118 q^{5} +3.75957 q^{7} +1.75233 q^{8} +O(q^{10})$	\(q-1.72798 q^{2} +0.985905 q^{4} +4.09118 q^{5} +3.75957 q^{7} +1.75233 q^{8} -7.06946 q^{10} -2.93491 q^{11} -2.06402 q^{13} -6.49645 q^{14} -4.99980 q^{16} -4.34212 q^{17} -1.00000 q^{19} +4.03351 q^{20} +5.07147 q^{22} -6.55778 q^{23} +11.7377 q^{25} +3.56658 q^{26} +3.70658 q^{28} -4.94699 q^{29} -7.63052 q^{31} +5.13488 q^{32} +7.50308 q^{34} +15.3811 q^{35} -3.52692 q^{37} +1.72798 q^{38} +7.16911 q^{40} -6.99275 q^{41} +7.07313 q^{43} -2.89355 q^{44} +11.3317 q^{46} +1.00000 q^{47} +7.13434 q^{49} -20.2825 q^{50} -2.03493 q^{52} -7.02450 q^{53} -12.0073 q^{55} +6.58801 q^{56} +8.54829 q^{58} +13.8429 q^{59} +13.1387 q^{61} +13.1854 q^{62} +1.12665 q^{64} -8.44427 q^{65} +7.81340 q^{67} -4.28092 q^{68} -26.5781 q^{70} -13.6388 q^{71} -2.78369 q^{73} +6.09444 q^{74} -0.985905 q^{76} -11.0340 q^{77} +1.75992 q^{79} -20.4551 q^{80} +12.0833 q^{82} -11.3819 q^{83} -17.7644 q^{85} -12.2222 q^{86} -5.14295 q^{88} -2.99155 q^{89} -7.75982 q^{91} -6.46535 q^{92} -1.72798 q^{94} -4.09118 q^{95} +13.2603 q^{97} -12.3280 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8}+O(q^{10}) $ 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8	$ 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8} + 5 q^{10} - 15 q^{11} - 3 q^{13} - 27 q^{14} + 48 q^{16} - 22 q^{17} - 24 q^{19} - 29 q^{20} - 9 q^{22} - 15 q^{23} + 58 q^{25} - q^{26} + 18 q^{28} - 36 q^{29} - 16 q^{31} - 26 q^{32} + 10 q^{34} - 37 q^{35} + 18 q^{37} + 6 q^{38} + 7 q^{40} - 24 q^{41} + 2 q^{43} - 57 q^{44} - 23 q^{46} + 24 q^{47} + 64 q^{49} - 4 q^{50} - 40 q^{52} - 42 q^{53} + 8 q^{55} - 38 q^{56} - 5 q^{58} - 39 q^{59} + 6 q^{61} + 14 q^{62} + 53 q^{64} - 47 q^{65} - 26 q^{67} - 80 q^{68} - 46 q^{70} - 6 q^{71} + 3 q^{73} - 62 q^{74} - 32 q^{76} - 13 q^{77} + 10 q^{79} - 17 q^{80} + 15 q^{82} - 44 q^{83} + 21 q^{85} + 22 q^{86} - 88 q^{88} - 32 q^{89} - 17 q^{91} + 37 q^{92} - 6 q^{94} + 10 q^{95} + 28 q^{97} - 44 q^{98}+O(q^{100}) $ 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8 + 5 * q^10 - 15 * q^11 - 3 * q^13 - 27 * q^14 + 48 * q^16 - 22 * q^17 - 24 * q^19 - 29 * q^20 - 9 * q^22 - 15 * q^23 + 58 * q^25 - q^26 + 18 * q^28 - 36 * q^29 - 16 * q^31 - 26 * q^32 + 10 * q^34 - 37 * q^35 + 18 * q^37 + 6 * q^38 + 7 * q^40 - 24 * q^41 + 2 * q^43 - 57 * q^44 - 23 * q^46 + 24 * q^47 + 64 * q^49 - 4 * q^50 - 40 * q^52 - 42 * q^53 + 8 * q^55 - 38 * q^56 - 5 * q^58 - 39 * q^59 + 6 * q^61 + 14 * q^62 + 53 * q^64 - 47 * q^65 - 26 * q^67 - 80 * q^68 - 46 * q^70 - 6 * q^71 + 3 * q^73 - 62 * q^74 - 32 * q^76 - 13 * q^77 + 10 * q^79 - 17 * q^80 + 15 * q^82 - 44 * q^83 + 21 * q^85 + 22 * q^86 - 88 * q^88 - 32 * q^89 - 17 * q^91 + 37 * q^92 - 6 * q^94 + 10 * q^95 + 28 * q^97 - 44 * 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 $

$2$	−1.72798	−1.22186	−0.610932	−	0.791683i	$-0.709205\pi$
$2$	−1.72798	−1.22186	−0.610932	+	0.791683i	$0.709205\pi$
$3$	0	0
$3$	0	0
$4$	0.985905	0.492952
$5$	4.09118	1.82963	0.914815	−	0.403873i	$-0.132336\pi$
$5$	4.09118	1.82963	0.914815	+	0.403873i	$0.132336\pi$
$6$	0	0
$7$	3.75957	1.42098	0.710491	−	0.703706i	$-0.248473\pi$
$7$	3.75957	1.42098	0.710491	+	0.703706i	$0.248473\pi$
$8$	1.75233	0.619543
$9$	0	0
$10$	−7.06946	−2.23556
$11$	−2.93491	−0.884910	−0.442455	−	0.896791i	$-0.645892\pi$
$11$	−2.93491	−0.884910	−0.442455	+	0.896791i	$0.645892\pi$
$12$	0	0
$13$	−2.06402	−0.572456	−0.286228	−	0.958162i	$-0.592401\pi$
$13$	−2.06402	−0.572456	−0.286228	+	0.958162i	$0.592401\pi$
$14$	−6.49645	−1.73625
$15$	0	0
$16$	−4.99980	−1.24995
$17$	−4.34212	−1.05312	−0.526559	−	0.850138i	$-0.676518\pi$
$17$	−4.34212	−1.05312	−0.526559	+	0.850138i	$0.676518\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.03351	0.901921
$21$	0	0
$22$	5.07147	1.08124
$23$	−6.55778	−1.36739	−0.683696	−	0.729767i	$-0.739629\pi$
$23$	−6.55778	−1.36739	−0.683696	+	0.729767i	$0.739629\pi$
$24$	0	0
$25$	11.7377	2.34755
$26$	3.56658	0.699464
$27$	0	0
$28$	3.70658	0.700477
$29$	−4.94699	−0.918633	−0.459317	−	0.888273i	$-0.651906\pi$
$29$	−4.94699	−0.918633	−0.459317	+	0.888273i	$0.651906\pi$
$30$	0	0
$31$	−7.63052	−1.37048	−0.685241	−	0.728316i	$-0.740303\pi$
$31$	−7.63052	−1.37048	−0.685241	+	0.728316i	$0.740303\pi$
$32$	5.13488	0.907726
$33$	0	0
$34$	7.50308	1.28677
$35$	15.3811	2.59987
$36$	0	0
$37$	−3.52692	−0.579822	−0.289911	−	0.957054i	$-0.593626\pi$
$37$	−3.52692	−0.579822	−0.289911	+	0.957054i	$0.593626\pi$
$38$	1.72798	0.280315
$39$	0	0
$40$	7.16911	1.13354
$41$	−6.99275	−1.09208	−0.546042	−	0.837758i	$-0.683866\pi$
$41$	−6.99275	−1.09208	−0.546042	+	0.837758i	$0.683866\pi$
$42$	0	0
$43$	7.07313	1.07864	0.539321	−	0.842100i	$-0.318681\pi$
$43$	7.07313	1.07864	0.539321	+	0.842100i	$0.318681\pi$
$44$	−2.89355	−0.436219
$45$	0	0
$46$	11.3317	1.67077
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	7.13434	1.01919
$50$	−20.2825	−2.86838
$51$	0	0
$52$	−2.03493	−0.282194
$53$	−7.02450	−0.964890	−0.482445	−	0.875926i	$-0.660251\pi$
$53$	−7.02450	−0.964890	−0.482445	+	0.875926i	$0.660251\pi$
$54$	0	0
$55$	−12.0073	−1.61906
$56$	6.58801	0.880360
$57$	0	0
$58$	8.54829	1.12245
$59$	13.8429	1.80220	0.901099	−	0.433613i	$-0.142762\pi$
$59$	13.8429	1.80220	0.901099	+	0.433613i	$0.142762\pi$
$60$	0	0
$61$	13.1387	1.68224	0.841120	−	0.540848i	$-0.181897\pi$
$61$	13.1387	1.68224	0.841120	+	0.540848i	$0.181897\pi$
$62$	13.1854	1.67454
$63$	0	0
$64$	1.12665	0.140832
$65$	−8.44427	−1.04738
$66$	0	0
$67$	7.81340	0.954558	0.477279	−	0.878752i	$-0.341623\pi$
$67$	7.81340	0.954558	0.477279	+	0.878752i	$0.341623\pi$
$68$	−4.28092	−0.519137
$69$	0	0
$70$	−26.5781	−3.17669
$71$	−13.6388	−1.61863	−0.809316	−	0.587374i	$-0.800162\pi$
$71$	−13.6388	−1.61863	−0.809316	+	0.587374i	$0.800162\pi$
$72$	0	0
$73$	−2.78369	−0.325806	−0.162903	−	0.986642i	$-0.552086\pi$
$73$	−2.78369	−0.325806	−0.162903	+	0.986642i	$0.552086\pi$
$74$	6.09444	0.708464
$75$	0	0
$76$	−0.985905	−0.113091
$77$	−11.0340	−1.25744
$78$	0	0
$79$	1.75992	0.198007	0.0990033	−	0.995087i	$-0.468435\pi$
$79$	1.75992	0.198007	0.0990033	+	0.995087i	$0.468435\pi$
$80$	−20.4551	−2.28695
$81$	0	0
$82$	12.0833	1.33438
$83$	−11.3819	−1.24932	−0.624661	−	0.780896i	$-0.714763\pi$
$83$	−11.3819	−1.24932	−0.624661	+	0.780896i	$0.714763\pi$
$84$	0	0
$85$	−17.7644	−1.92682
$86$	−12.2222	−1.31795
$87$	0	0
$88$	−5.14295	−0.548240
$89$	−2.99155	−0.317104	−0.158552	−	0.987351i	$-0.550683\pi$
$89$	−2.99155	−0.317104	−0.158552	+	0.987351i	$0.550683\pi$
$90$	0	0
$91$	−7.75982	−0.813450
$92$	−6.46535	−0.674059
$93$	0	0
$94$	−1.72798	−0.178227
$95$	−4.09118	−0.419746
$96$	0	0
$97$	13.2603	1.34638	0.673189	−	0.739471i	$-0.264924\pi$
$97$	13.2603	1.34638	0.673189	+	0.739471i	$0.264924\pi$
$98$	−12.3280	−1.24531
$99$	0	0
$100$	11.5723	1.15723
$101$	4.95650	0.493190	0.246595	−	0.969119i	$-0.420688\pi$
$101$	4.95650	0.493190	0.246595	+	0.969119i	$0.420688\pi$
$102$	0	0
$103$	4.04398	0.398466	0.199233	−	0.979952i	$-0.436155\pi$
$103$	4.04398	0.398466	0.199233	+	0.979952i	$0.436155\pi$
$104$	−3.61685	−0.354661
$105$	0	0
$106$	12.1382	1.17896
$107$	11.6973	1.13082	0.565409	−	0.824810i	$-0.308718\pi$
$107$	11.6973	1.13082	0.565409	+	0.824810i	$0.308718\pi$
$108$	0	0
$109$	−10.0647	−0.964022	−0.482011	−	0.876165i	$-0.660093\pi$
$109$	−10.0647	−0.964022	−0.482011	+	0.876165i	$0.660093\pi$
$110$	20.7483	1.97827
$111$	0	0
$112$	−18.7971	−1.77616
$113$	−7.74177	−0.728284	−0.364142	−	0.931343i	$-0.618638\pi$
$113$	−7.74177	−0.728284	−0.364142	+	0.931343i	$0.618638\pi$
$114$	0	0
$115$	−26.8291	−2.50182
$116$	−4.87726	−0.452843
$117$	0	0
$118$	−23.9203	−2.20204
$119$	−16.3245	−1.49646
$120$	0	0
$121$	−2.38627	−0.216934
$122$	−22.7034	−2.05547
$123$	0	0
$124$	−7.52297	−0.675582
$125$	27.5653	2.46551
$126$	0	0
$127$	−18.8354	−1.67137	−0.835687	−	0.549206i	$-0.814930\pi$
$127$	−18.8354	−1.67137	−0.835687	+	0.549206i	$0.814930\pi$
$128$	−12.2166	−1.07980
$129$	0	0
$130$	14.5915	1.27976
$131$	14.6441	1.27946	0.639729	−	0.768601i	$-0.279047\pi$
$131$	14.6441	1.27946	0.639729	+	0.768601i	$0.279047\pi$
$132$	0	0
$133$	−3.75957	−0.325996
$134$	−13.5014	−1.16634
$135$	0	0
$136$	−7.60884	−0.652453
$137$	4.79168	0.409380	0.204690	−	0.978827i	$-0.434381\pi$
$137$	4.79168	0.409380	0.204690	+	0.978827i	$0.434381\pi$
$138$	0	0
$139$	−20.3242	−1.72388	−0.861939	−	0.507012i	$-0.830750\pi$
$139$	−20.3242	−1.72388	−0.861939	+	0.507012i	$0.830750\pi$
$140$	15.1643	1.28161
$141$	0	0
$142$	23.5676	1.97775
$143$	6.05772	0.506572
$144$	0	0
$145$	−20.2390	−1.68076
$146$	4.81015	0.398091
$147$	0	0
$148$	−3.47721	−0.285825
$149$	−11.7799	−0.965044	−0.482522	−	0.875884i	$-0.660279\pi$
$149$	−11.7799	−0.965044	−0.482522	+	0.875884i	$0.660279\pi$
$150$	0	0
$151$	−11.9578	−0.973115	−0.486557	−	0.873649i	$-0.661748\pi$
$151$	−11.9578	−0.973115	−0.486557	+	0.873649i	$0.661748\pi$
$152$	−1.75233	−0.142133
$153$	0	0
$154$	19.0665	1.53642
$155$	−31.2178	−2.50748
$156$	0	0
$157$	−9.45073	−0.754250	−0.377125	−	0.926162i	$-0.623087\pi$
$157$	−9.45073	−0.754250	−0.377125	+	0.926162i	$0.623087\pi$
$158$	−3.04110	−0.241937
$159$	0	0
$160$	21.0077	1.66080
$161$	−24.6544	−1.94304
$162$	0	0
$163$	0.666149	0.0521768	0.0260884	−	0.999660i	$-0.491695\pi$
$163$	0.666149	0.0521768	0.0260884	+	0.999660i	$0.491695\pi$
$164$	−6.89419	−0.538345
$165$	0	0
$166$	19.6676	1.52650
$167$	0.891624	0.0689959	0.0344980	−	0.999405i	$-0.489017\pi$
$167$	0.891624	0.0689959	0.0344980	+	0.999405i	$0.489017\pi$
$168$	0	0
$169$	−8.73982	−0.672294
$170$	30.6964	2.35431
$171$	0	0
$172$	6.97343	0.531719
$173$	−4.09197	−0.311106	−0.155553	−	0.987828i	$-0.549716\pi$
$173$	−4.09197	−0.311106	−0.155553	+	0.987828i	$0.549716\pi$
$174$	0	0
$175$	44.1288	3.33582
$176$	14.6740	1.10609
$177$	0	0
$178$	5.16933	0.387458
$179$	−13.0642	−0.976467	−0.488234	−	0.872713i	$-0.662359\pi$
$179$	−13.0642	−0.976467	−0.488234	+	0.872713i	$0.662359\pi$
$180$	0	0
$181$	−13.7178	−1.01964	−0.509819	−	0.860282i	$-0.670288\pi$
$181$	−13.7178	−1.01964	−0.509819	+	0.860282i	$0.670288\pi$
$182$	13.4088	0.993926
$183$	0	0
$184$	−11.4914	−0.847159
$185$	−14.4293	−1.06086
$186$	0	0
$187$	12.7438	0.931915
$188$	0.985905	0.0719045
$189$	0	0
$190$	7.06946	0.512873
$191$	−12.8655	−0.930917	−0.465458	−	0.885070i	$-0.654111\pi$
$191$	−12.8655	−0.930917	−0.465458	+	0.885070i	$0.654111\pi$
$192$	0	0
$193$	7.82208	0.563046	0.281523	−	0.959555i	$-0.409160\pi$
$193$	7.82208	0.563046	0.281523	+	0.959555i	$0.409160\pi$
$194$	−22.9135	−1.64509
$195$	0	0
$196$	7.03378	0.502413
$197$	17.8935	1.27486	0.637430	−	0.770508i	$-0.279998\pi$
$197$	17.8935	1.27486	0.637430	+	0.770508i	$0.279998\pi$
$198$	0	0
$199$	−14.2623	−1.01103	−0.505515	−	0.862818i	$-0.668697\pi$
$199$	−14.2623	−1.01103	−0.505515	+	0.862818i	$0.668697\pi$
$200$	20.5684	1.45441
$201$	0	0
$202$	−8.56471	−0.602611
$203$	−18.5985	−1.30536
$204$	0	0
$205$	−28.6086	−1.99811
$206$	−6.98791	−0.486871
$207$	0	0
$208$	10.3197	0.715542
$209$	2.93491	0.203012
$210$	0	0
$211$	8.10994	0.558311	0.279156	−	0.960246i	$-0.409946\pi$
$211$	8.10994	0.558311	0.279156	+	0.960246i	$0.409946\pi$
$212$	−6.92549	−0.475645
$213$	0	0
$214$	−20.2126	−1.38171
$215$	28.9374	1.97352
$216$	0	0
$217$	−28.6875	−1.94743
$218$	17.3915	1.17790
$219$	0	0
$220$	−11.8380	−0.798119
$221$	8.96222	0.602864
$222$	0	0
$223$	−16.4490	−1.10151	−0.550754	−	0.834668i	$-0.685660\pi$
$223$	−16.4490	−1.10151	−0.550754	+	0.834668i	$0.685660\pi$
$224$	19.3049	1.28986
$225$	0	0
$226$	13.3776	0.889865
$227$	−6.62280	−0.439571	−0.219785	−	0.975548i	$-0.570536\pi$
$227$	−6.62280	−0.439571	−0.219785	+	0.975548i	$0.570536\pi$
$228$	0	0
$229$	3.87076	0.255787	0.127893	−	0.991788i	$-0.459178\pi$
$229$	3.87076	0.255787	0.127893	+	0.991788i	$0.459178\pi$
$230$	46.3600	3.05689
$231$	0	0
$232$	−8.66878	−0.569133
$233$	−3.44991	−0.226011	−0.113006	−	0.993594i	$-0.536048\pi$
$233$	−3.44991	−0.226011	−0.113006	+	0.993594i	$0.536048\pi$
$234$	0	0
$235$	4.09118	0.266879
$236$	13.6478	0.888398
$237$	0	0
$238$	28.2083	1.82848
$239$	9.23915	0.597631	0.298816	−	0.954311i	$-0.403408\pi$
$239$	9.23915	0.597631	0.298816	+	0.954311i	$0.403408\pi$
$240$	0	0
$241$	−5.27664	−0.339898	−0.169949	−	0.985453i	$-0.554360\pi$
$241$	−5.27664	−0.339898	−0.169949	+	0.985453i	$0.554360\pi$
$242$	4.12343	0.265064
$243$	0	0
$244$	12.9535	0.829264
$245$	29.1879	1.86474
$246$	0	0
$247$	2.06402	0.131330
$248$	−13.3712	−0.849073
$249$	0	0
$250$	−47.6321	−3.01252
$251$	5.22814	0.329997	0.164999	−	0.986294i	$-0.447238\pi$
$251$	5.22814	0.329997	0.164999	+	0.986294i	$0.447238\pi$
$252$	0	0
$253$	19.2465	1.21002
$254$	32.5472	2.04219
$255$	0	0
$256$	18.8567	1.17854
$257$	10.6175	0.662304	0.331152	−	0.943577i	$-0.392563\pi$
$257$	10.6175	0.662304	0.331152	+	0.943577i	$0.392563\pi$
$258$	0	0
$259$	−13.2597	−0.823917
$260$	−8.32525	−0.516310
$261$	0	0
$262$	−25.3046	−1.56332
$263$	−22.2116	−1.36963	−0.684814	−	0.728718i	$-0.740117\pi$
$263$	−22.2116	−1.36963	−0.684814	+	0.728718i	$0.740117\pi$
$264$	0	0
$265$	−28.7385	−1.76539
$266$	6.49645	0.398323
$267$	0	0
$268$	7.70326	0.470552
$269$	27.8202	1.69622	0.848112	−	0.529817i	$-0.177739\pi$
$269$	27.8202	1.69622	0.848112	+	0.529817i	$0.177739\pi$
$270$	0	0
$271$	18.6770	1.13454	0.567272	−	0.823530i	$-0.307999\pi$
$271$	18.6770	1.13454	0.567272	+	0.823530i	$0.307999\pi$
$272$	21.7097	1.31635
$273$	0	0
$274$	−8.27991	−0.500207
$275$	−34.4493	−2.07737
$276$	0	0
$277$	−3.92276	−0.235696	−0.117848	−	0.993032i	$-0.537600\pi$
$277$	−3.92276	−0.235696	−0.117848	+	0.993032i	$0.537600\pi$
$278$	35.1198	2.10635
$279$	0	0
$280$	26.9527	1.61073
$281$	−4.36642	−0.260479	−0.130239	−	0.991483i	$-0.541575\pi$
$281$	−4.36642	−0.260479	−0.130239	+	0.991483i	$0.541575\pi$
$282$	0	0
$283$	27.4414	1.63122	0.815610	−	0.578602i	$-0.196401\pi$
$283$	27.4414	1.63122	0.815610	+	0.578602i	$0.196401\pi$
$284$	−13.4466	−0.797908
$285$	0	0
$286$	−10.4676	−0.618963
$287$	−26.2897	−1.55183
$288$	0	0
$289$	1.85400	0.109059
$290$	34.9726	2.05366
$291$	0	0
$292$	−2.74445	−0.160607
$293$	4.95404	0.289418	0.144709	−	0.989474i	$-0.453775\pi$
$293$	4.95404	0.289418	0.144709	+	0.989474i	$0.453775\pi$
$294$	0	0
$295$	56.6340	3.29736
$296$	−6.18034	−0.359225
$297$	0	0
$298$	20.3553	1.17915
$299$	13.5354	0.782772
$300$	0	0
$301$	26.5919	1.53273
$302$	20.6629	1.18901
$303$	0	0
$304$	4.99980	0.286758
$305$	53.7528	3.07788
$306$	0	0
$307$	−0.540221	−0.0308321	−0.0154160	−	0.999881i	$-0.504907\pi$
$307$	−0.540221	−0.0308321	−0.0154160	+	0.999881i	$0.504907\pi$
$308$	−10.8785	−0.619859
$309$	0	0
$310$	53.9437	3.06379
$311$	1.08728	0.0616542	0.0308271	−	0.999525i	$-0.490186\pi$
$311$	1.08728	0.0616542	0.0308271	+	0.999525i	$0.490186\pi$
$312$	0	0
$313$	11.6140	0.656459	0.328230	−	0.944598i	$-0.393548\pi$
$313$	11.6140	0.656459	0.328230	+	0.944598i	$0.393548\pi$
$314$	16.3306	0.921591
$315$	0	0
$316$	1.73512	0.0976079
$317$	21.1453	1.18764	0.593820	−	0.804598i	$-0.297619\pi$
$317$	21.1453	1.18764	0.593820	+	0.804598i	$0.297619\pi$
$318$	0	0
$319$	14.5190	0.812908
$320$	4.60934	0.257670
$321$	0	0
$322$	42.6023	2.37413
$323$	4.34212	0.241602
$324$	0	0
$325$	−24.2269	−1.34387
$326$	−1.15109	−0.0637530
$327$	0	0
$328$	−12.2536	−0.676593
$329$	3.75957	0.207272
$330$	0	0
$331$	−25.2626	−1.38856	−0.694279	−	0.719706i	$-0.744277\pi$
$331$	−25.2626	−1.38856	−0.694279	+	0.719706i	$0.744277\pi$
$332$	−11.2214	−0.615856
$333$	0	0
$334$	−1.54071	−0.0843037
$335$	31.9660	1.74649
$336$	0	0
$337$	−16.8457	−0.917642	−0.458821	−	0.888529i	$-0.651728\pi$
$337$	−16.8457	−0.917642	−0.458821	+	0.888529i	$0.651728\pi$
$338$	15.1022	0.821452
$339$	0	0
$340$	−17.5140	−0.949830
$341$	22.3949	1.21275
$342$	0	0
$343$	0.505065	0.0272710
$344$	12.3945	0.668266
$345$	0	0
$346$	7.07083	0.380130
$347$	−30.2250	−1.62256	−0.811281	−	0.584657i	$-0.801229\pi$
$347$	−30.2250	−1.62256	−0.811281	+	0.584657i	$0.801229\pi$
$348$	0	0
$349$	−7.57452	−0.405455	−0.202727	−	0.979235i	$-0.564980\pi$
$349$	−7.57452	−0.405455	−0.202727	+	0.979235i	$0.564980\pi$
$350$	−76.2535	−4.07592
$351$	0	0
$352$	−15.0704	−0.803256
$353$	−29.6912	−1.58030	−0.790150	−	0.612913i	$-0.789998\pi$
$353$	−29.6912	−1.58030	−0.790150	+	0.612913i	$0.789998\pi$
$354$	0	0
$355$	−55.7989	−2.96150
$356$	−2.94939	−0.156317
$357$	0	0
$358$	22.5747	1.19311
$359$	−0.0660750	−0.00348730	−0.00174365	−	0.999998i	$-0.500555\pi$
$359$	−0.0660750	−0.00348730	−0.00174365	+	0.999998i	$0.500555\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	23.7041	1.24586
$363$	0	0
$364$	−7.65044	−0.400992
$365$	−11.3886	−0.596105
$366$	0	0
$367$	3.20068	0.167074	0.0835370	−	0.996505i	$-0.473378\pi$
$367$	3.20068	0.167074	0.0835370	+	0.996505i	$0.473378\pi$
$368$	32.7876	1.70917
$369$	0	0
$370$	24.9334	1.29623
$371$	−26.4091	−1.37109
$372$	0	0
$373$	−17.6882	−0.915862	−0.457931	−	0.888988i	$-0.651409\pi$
$373$	−17.6882	−0.915862	−0.457931	+	0.888988i	$0.651409\pi$
$374$	−22.0209	−1.13867
$375$	0	0
$376$	1.75233	0.0903697
$377$	10.2107	0.525877
$378$	0	0
$379$	−15.3169	−0.786778	−0.393389	−	0.919372i	$-0.628697\pi$
$379$	−15.3169	−0.786778	−0.393389	+	0.919372i	$0.628697\pi$
$380$	−4.03351	−0.206915
$381$	0	0
$382$	22.2313	1.13745
$383$	7.87203	0.402242	0.201121	−	0.979566i	$-0.435542\pi$
$383$	7.87203	0.402242	0.201121	+	0.979566i	$0.435542\pi$
$384$	0	0
$385$	−45.1421	−2.30065
$386$	−13.5164	−0.687965
$387$	0	0
$388$	13.0734	0.663700
$389$	−14.7096	−0.745805	−0.372903	−	0.927870i	$-0.621637\pi$
$389$	−14.7096	−0.745805	−0.372903	+	0.927870i	$0.621637\pi$
$390$	0	0
$391$	28.4747	1.44003
$392$	12.5017	0.631433
$393$	0	0
$394$	−30.9196	−1.55771
$395$	7.20015	0.362279
$396$	0	0
$397$	29.6491	1.48804	0.744022	−	0.668155i	$-0.232916\pi$
$397$	29.6491	1.48804	0.744022	+	0.668155i	$0.232916\pi$
$398$	24.6450	1.23534
$399$	0	0
$400$	−58.6863	−2.93432
$401$	−15.6020	−0.779128	−0.389564	−	0.920999i	$-0.627374\pi$
$401$	−15.6020	−0.779128	−0.389564	+	0.920999i	$0.627374\pi$
$402$	0	0
$403$	15.7495	0.784541
$404$	4.88664	0.243119
$405$	0	0
$406$	32.1379	1.59498
$407$	10.3512	0.513090
$408$	0	0
$409$	−25.0516	−1.23872	−0.619360	−	0.785107i	$-0.712608\pi$
$409$	−25.0516	−1.23872	−0.619360	+	0.785107i	$0.712608\pi$
$410$	49.4350	2.44142
$411$	0	0
$412$	3.98698	0.196425
$413$	52.0435	2.56089
$414$	0	0
$415$	−46.5652	−2.28580
$416$	−10.5985	−0.519634
$417$	0	0
$418$	−5.07147	−0.248054
$419$	34.5089	1.68587	0.842936	−	0.538014i	$-0.180825\pi$
$419$	34.5089	1.68587	0.842936	+	0.538014i	$0.180825\pi$
$420$	0	0
$421$	−19.9641	−0.972992	−0.486496	−	0.873683i	$-0.661725\pi$
$421$	−19.9641	−0.972992	−0.486496	+	0.873683i	$0.661725\pi$
$422$	−14.0138	−0.682180
$423$	0	0
$424$	−12.3093	−0.597791
$425$	−50.9666	−2.47225
$426$	0	0
$427$	49.3959	2.39043
$428$	11.5324	0.557440
$429$	0	0
$430$	−50.0032	−2.41137
$431$	−21.6070	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$431$	−21.6070	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$432$	0	0
$433$	37.0942	1.78264	0.891318	−	0.453379i	$-0.149782\pi$
$433$	37.0942	1.78264	0.891318	+	0.453379i	$0.149782\pi$
$434$	49.5713	2.37950
$435$	0	0
$436$	−9.92282	−0.475217
$437$	6.55778	0.313701
$438$	0	0
$439$	4.39008	0.209527	0.104764	−	0.994497i	$-0.466591\pi$
$439$	4.39008	0.209527	0.104764	+	0.994497i	$0.466591\pi$
$440$	−21.0407	−1.00308
$441$	0	0
$442$	−15.4865	−0.736618
$443$	−35.9911	−1.70999	−0.854995	−	0.518637i	$-0.826440\pi$
$443$	−35.9911	−1.70999	−0.854995	+	0.518637i	$0.826440\pi$
$444$	0	0
$445$	−12.2390	−0.580183
$446$	28.4235	1.34589
$447$	0	0
$448$	4.23573	0.200120
$449$	−23.5402	−1.11093	−0.555464	−	0.831540i	$-0.687459\pi$
$449$	−23.5402	−1.11093	−0.555464	+	0.831540i	$0.687459\pi$
$450$	0	0
$451$	20.5231	0.966396
$452$	−7.63265	−0.359010
$453$	0	0
$454$	11.4440	0.537096
$455$	−31.7468	−1.48831
$456$	0	0
$457$	−21.2595	−0.994477	−0.497238	−	0.867614i	$-0.665653\pi$
$457$	−21.2595	−0.994477	−0.497238	+	0.867614i	$0.665653\pi$
$458$	−6.68858	−0.312537
$459$	0	0
$460$	−26.4509	−1.23328
$461$	12.8390	0.597974	0.298987	−	0.954257i	$-0.403351\pi$
$461$	12.8390	0.597974	0.298987	+	0.954257i	$0.403351\pi$
$462$	0	0
$463$	−33.5783	−1.56052	−0.780259	−	0.625457i	$-0.784913\pi$
$463$	−33.5783	−1.56052	−0.780259	+	0.625457i	$0.784913\pi$
$464$	24.7340	1.14825
$465$	0	0
$466$	5.96137	0.276155
$467$	−24.1325	−1.11672	−0.558360	−	0.829599i	$-0.688569\pi$
$467$	−24.1325	−1.11672	−0.558360	+	0.829599i	$0.688569\pi$
$468$	0	0
$469$	29.3750	1.35641
$470$	−7.06946	−0.326090
$471$	0	0
$472$	24.2575	1.11654
$473$	−20.7590	−0.954501
$474$	0	0
$475$	−11.7377	−0.538564
$476$	−16.0944	−0.737685
$477$	0	0
$478$	−15.9650	−0.730224
$479$	−21.0632	−0.962401	−0.481201	−	0.876610i	$-0.659799\pi$
$479$	−21.0632	−0.962401	−0.481201	+	0.876610i	$0.659799\pi$
$480$	0	0
$481$	7.27963	0.331923
$482$	9.11791	0.415310
$483$	0	0
$484$	−2.35264	−0.106938
$485$	54.2502	2.46337
$486$	0	0
$487$	−16.0537	−0.727464	−0.363732	−	0.931504i	$-0.618498\pi$
$487$	−16.0537	−0.727464	−0.363732	+	0.931504i	$0.618498\pi$
$488$	23.0234	1.04222
$489$	0	0
$490$	−50.4359	−2.27846
$491$	0.229697	0.0103661	0.00518305	−	0.999987i	$-0.498350\pi$
$491$	0.229697	0.0103661	0.00518305	+	0.999987i	$0.498350\pi$
$492$	0	0
$493$	21.4804	0.967430
$494$	−3.56658	−0.160468
$495$	0	0
$496$	38.1511	1.71303
$497$	−51.2761	−2.30005
$498$	0	0
$499$	23.1790	1.03763	0.518817	−	0.854886i	$-0.326373\pi$
$499$	23.1790	1.03763	0.518817	+	0.854886i	$0.326373\pi$
$500$	27.1767	1.21538
$501$	0	0
$502$	−9.03411	−0.403212
$503$	11.0500	0.492696	0.246348	−	0.969181i	$-0.420769\pi$
$503$	11.0500	0.492696	0.246348	+	0.969181i	$0.420769\pi$
$504$	0	0
$505$	20.2779	0.902355
$506$	−33.2576	−1.47848
$507$	0	0
$508$	−18.5699	−0.823908
$509$	22.9267	1.01621	0.508104	−	0.861295i	$-0.330346\pi$
$509$	22.9267	1.01621	0.508104	+	0.861295i	$0.330346\pi$
$510$	0	0
$511$	−10.4655	−0.462965
$512$	−8.15072	−0.360215
$513$	0	0
$514$	−18.3469	−0.809245
$515$	16.5447	0.729045
$516$	0	0
$517$	−2.93491	−0.129077
$518$	22.9124	1.00671
$519$	0	0
$520$	−14.7972	−0.648899
$521$	−33.8032	−1.48095	−0.740473	−	0.672086i	$-0.765398\pi$
$521$	−33.8032	−1.48095	−0.740473	+	0.672086i	$0.765398\pi$
$522$	0	0
$523$	20.0223	0.875512	0.437756	−	0.899094i	$-0.355773\pi$
$523$	20.0223	0.875512	0.437756	+	0.899094i	$0.355773\pi$
$524$	14.4377	0.630712
$525$	0	0
$526$	38.3812	1.67350
$527$	33.1326	1.44328
$528$	0	0
$529$	20.0045	0.869762
$530$	49.6595	2.15707
$531$	0	0
$532$	−3.70658	−0.160700
$533$	14.4332	0.625170
$534$	0	0
$535$	47.8557	2.06898
$536$	13.6917	0.591390
$537$	0	0
$538$	−48.0726	−2.07256
$539$	−20.9387	−0.901893
$540$	0	0
$541$	2.35956	0.101446	0.0507228	−	0.998713i	$-0.483848\pi$
$541$	2.35956	0.101446	0.0507228	+	0.998713i	$0.483848\pi$
$542$	−32.2734	−1.38626
$543$	0	0
$544$	−22.2962	−0.955944
$545$	−41.1764	−1.76380
$546$	0	0
$547$	−23.3871	−0.999959	−0.499979	−	0.866037i	$-0.666659\pi$
$547$	−23.3871	−0.999959	−0.499979	+	0.866037i	$0.666659\pi$
$548$	4.72414	0.201805
$549$	0	0
$550$	59.5275	2.53826
$551$	4.94699	0.210749
$552$	0	0
$553$	6.61654	0.281364
$554$	6.77844	0.287988
$555$	0	0
$556$	−20.0378	−0.849790
$557$	42.2498	1.79018	0.895090	−	0.445886i	$-0.147111\pi$
$557$	42.2498	1.79018	0.895090	+	0.445886i	$0.147111\pi$
$558$	0	0
$559$	−14.5991	−0.617475
$560$	−76.9022	−3.24971
$561$	0	0
$562$	7.54508	0.318270
$563$	−21.0343	−0.886488	−0.443244	−	0.896401i	$-0.646173\pi$
$563$	−21.0343	−0.886488	−0.443244	+	0.896401i	$0.646173\pi$
$564$	0	0
$565$	−31.6729	−1.33249
$566$	−47.4181	−1.99313
$567$	0	0
$568$	−23.8998	−1.00281
$569$	−17.3367	−0.726790	−0.363395	−	0.931635i	$-0.618382\pi$
$569$	−17.3367	−0.726790	−0.363395	+	0.931635i	$0.618382\pi$
$570$	0	0
$571$	−33.7687	−1.41317	−0.706587	−	0.707626i	$-0.749766\pi$
$571$	−33.7687	−1.41317	−0.706587	+	0.707626i	$0.749766\pi$
$572$	5.97234	0.249716
$573$	0	0
$574$	45.4280	1.89613
$575$	−76.9735	−3.21002
$576$	0	0
$577$	39.7499	1.65481	0.827404	−	0.561607i	$-0.189817\pi$
$577$	39.7499	1.65481	0.827404	+	0.561607i	$0.189817\pi$
$578$	−3.20367	−0.133255
$579$	0	0
$580$	−19.9538	−0.828535
$581$	−42.7909	−1.77527
$582$	0	0
$583$	20.6163	0.853841
$584$	−4.87795	−0.201851
$585$	0	0
$586$	−8.56046	−0.353629
$587$	−19.8743	−0.820300	−0.410150	−	0.912018i	$-0.634524\pi$
$587$	−19.8743	−0.820300	−0.410150	+	0.912018i	$0.634524\pi$
$588$	0	0
$589$	7.63052	0.314410
$590$	−97.8622	−4.02892
$591$	0	0
$592$	17.6339	0.724749
$593$	14.9043	0.612045	0.306023	−	0.952024i	$-0.401002\pi$
$593$	14.9043	0.612045	0.306023	+	0.952024i	$0.401002\pi$
$594$	0	0
$595$	−66.7864	−2.73797
$596$	−11.6138	−0.475721
$597$	0	0
$598$	−23.3889	−0.956441
$599$	−17.2725	−0.705736	−0.352868	−	0.935673i	$-0.614794\pi$
$599$	−17.2725	−0.705736	−0.352868	+	0.935673i	$0.614794\pi$
$600$	0	0
$601$	17.3275	0.706803	0.353401	−	0.935472i	$-0.385025\pi$
$601$	17.3275	0.706803	0.353401	+	0.935472i	$0.385025\pi$
$602$	−45.9502	−1.87279
$603$	0	0
$604$	−11.7893	−0.479699
$605$	−9.76267	−0.396909
$606$	0	0
$607$	−34.2674	−1.39087	−0.695435	−	0.718589i	$-0.744788\pi$
$607$	−34.2674	−1.39087	−0.695435	+	0.718589i	$0.744788\pi$
$608$	−5.13488	−0.208247
$609$	0	0
$610$	−92.8836	−3.76075
$611$	−2.06402	−0.0835013
$612$	0	0
$613$	15.8977	0.642103	0.321051	−	0.947062i	$-0.395964\pi$
$613$	15.8977	0.642103	0.321051	+	0.947062i	$0.395964\pi$
$614$	0.933490	0.0376726
$615$	0	0
$616$	−19.3353	−0.779040
$617$	47.1761	1.89924	0.949620	−	0.313404i	$-0.101469\pi$
$617$	47.1761	1.89924	0.949620	+	0.313404i	$0.101469\pi$
$618$	0	0
$619$	−27.3031	−1.09740	−0.548702	−	0.836018i	$-0.684878\pi$
$619$	−27.3031	−1.09740	−0.548702	+	0.836018i	$0.684878\pi$
$620$	−30.7778	−1.23607
$621$	0	0
$622$	−1.87880	−0.0753331
$623$	−11.2469	−0.450599
$624$	0	0
$625$	54.0857	2.16343
$626$	−20.0686	−0.802104
$627$	0	0
$628$	−9.31752	−0.371809
$629$	15.3143	0.610621
$630$	0	0
$631$	−3.21711	−0.128071	−0.0640355	−	0.997948i	$-0.520397\pi$
$631$	−3.21711	−0.128071	−0.0640355	+	0.997948i	$0.520397\pi$
$632$	3.08397	0.122674
$633$	0	0
$634$	−36.5387	−1.45114
$635$	−77.0591	−3.05800
$636$	0	0
$637$	−14.7254	−0.583442
$638$	−25.0885	−0.993263
$639$	0	0
$640$	−49.9802	−1.97564
$641$	−22.6060	−0.892885	−0.446442	−	0.894812i	$-0.647309\pi$
$641$	−22.6060	−0.892885	−0.446442	+	0.894812i	$0.647309\pi$
$642$	0	0
$643$	24.3834	0.961588	0.480794	−	0.876834i	$-0.340349\pi$
$643$	24.3834	0.961588	0.480794	+	0.876834i	$0.340349\pi$
$644$	−24.3069	−0.957827
$645$	0	0
$646$	−7.50308	−0.295205
$647$	17.6988	0.695812	0.347906	−	0.937529i	$-0.386893\pi$
$647$	17.6988	0.695812	0.347906	+	0.937529i	$0.386893\pi$
$648$	0	0
$649$	−40.6279	−1.59478
$650$	41.8636	1.64202
$651$	0	0
$652$	0.656759	0.0257207
$653$	−16.3572	−0.640106	−0.320053	−	0.947400i	$-0.603701\pi$
$653$	−16.3572	−0.640106	−0.320053	+	0.947400i	$0.603701\pi$
$654$	0	0
$655$	59.9115	2.34094
$656$	34.9624	1.36505
$657$	0	0
$658$	−6.49645	−0.253258
$659$	−7.42647	−0.289294	−0.144647	−	0.989483i	$-0.546205\pi$
$659$	−7.42647	−0.289294	−0.144647	+	0.989483i	$0.546205\pi$
$660$	0	0
$661$	33.8825	1.31788	0.658938	−	0.752197i	$-0.271006\pi$
$661$	33.8825	1.31788	0.658938	+	0.752197i	$0.271006\pi$
$662$	43.6532	1.69663
$663$	0	0
$664$	−19.9448	−0.774009
$665$	−15.3811	−0.596452
$666$	0	0
$667$	32.4413	1.25613
$668$	0.879056	0.0340117
$669$	0	0
$670$	−55.2365	−2.13397
$671$	−38.5610	−1.48863
$672$	0	0
$673$	−42.3593	−1.63283	−0.816415	−	0.577466i	$-0.804042\pi$
$673$	−42.3593	−1.63283	−0.816415	+	0.577466i	$0.804042\pi$
$674$	29.1089	1.12123
$675$	0	0
$676$	−8.61663	−0.331409
$677$	13.7801	0.529612	0.264806	−	0.964302i	$-0.414692\pi$
$677$	13.7801	0.529612	0.264806	+	0.964302i	$0.414692\pi$
$678$	0	0
$679$	49.8529	1.91318
$680$	−31.1291	−1.19375
$681$	0	0
$682$	−38.6979	−1.48182
$683$	21.6267	0.827524	0.413762	−	0.910385i	$-0.364214\pi$
$683$	21.6267	0.827524	0.413762	+	0.910385i	$0.364214\pi$
$684$	0	0
$685$	19.6036	0.749015
$686$	−0.872741	−0.0333214
$687$	0	0
$688$	−35.3642	−1.34825
$689$	14.4987	0.552357
$690$	0	0
$691$	−48.7493	−1.85451	−0.927256	−	0.374429i	$-0.877839\pi$
$691$	−48.7493	−1.85451	−0.927256	+	0.374429i	$0.877839\pi$
$692$	−4.03429	−0.153361
$693$	0	0
$694$	52.2281	1.98255
$695$	−83.1500	−3.15406
$696$	0	0
$697$	30.3634	1.15009
$698$	13.0886	0.495410
$699$	0	0
$700$	43.5068	1.64440
$701$	−24.4934	−0.925104	−0.462552	−	0.886592i	$-0.653066\pi$
$701$	−24.4934	−0.925104	−0.462552	+	0.886592i	$0.653066\pi$
$702$	0	0
$703$	3.52692	0.133020
$704$	−3.30664	−0.124624
$705$	0	0
$706$	51.3056	1.93091
$707$	18.6343	0.700814
$708$	0	0
$709$	−25.3295	−0.951269	−0.475635	−	0.879643i	$-0.657782\pi$
$709$	−25.3295	−0.951269	−0.475635	+	0.879643i	$0.657782\pi$
$710$	96.4192	3.61855
$711$	0	0
$712$	−5.24220	−0.196460
$713$	50.0393	1.87399
$714$	0	0
$715$	24.7832	0.926840
$716$	−12.8801	−0.481352
$717$	0	0
$718$	0.114176	0.00426101
$719$	12.2310	0.456140	0.228070	−	0.973645i	$-0.426758\pi$
$719$	12.2310	0.456140	0.228070	+	0.973645i	$0.426758\pi$
$720$	0	0
$721$	15.2036	0.566213
$722$	−1.72798	−0.0643086
$723$	0	0
$724$	−13.5245	−0.502633
$725$	−58.0665	−2.15654
$726$	0	0
$727$	6.40863	0.237683	0.118842	−	0.992913i	$-0.462082\pi$
$727$	6.40863	0.237683	0.118842	+	0.992913i	$0.462082\pi$
$728$	−13.5978	−0.503968
$729$	0	0
$730$	19.6792	0.728359
$731$	−30.7124	−1.13594
$732$	0	0
$733$	−16.8061	−0.620749	−0.310375	−	0.950614i	$-0.600454\pi$
$733$	−16.8061	−0.620749	−0.310375	+	0.950614i	$0.600454\pi$
$734$	−5.53070	−0.204142
$735$	0	0
$736$	−33.6734	−1.24122
$737$	−22.9317	−0.844698
$738$	0	0
$739$	41.7550	1.53598	0.767992	−	0.640460i	$-0.221256\pi$
$739$	41.7550	1.53598	0.767992	+	0.640460i	$0.221256\pi$
$740$	−14.2259	−0.522954
$741$	0	0
$742$	45.6343	1.67529
$743$	5.36794	0.196931	0.0984653	−	0.995140i	$-0.468607\pi$
$743$	5.36794	0.196931	0.0984653	+	0.995140i	$0.468607\pi$
$744$	0	0
$745$	−48.1935	−1.76567
$746$	30.5649	1.11906
$747$	0	0
$748$	12.5641	0.459390
$749$	43.9767	1.60687
$750$	0	0
$751$	6.51937	0.237895	0.118948	−	0.992901i	$-0.462048\pi$
$751$	6.51937	0.237895	0.118948	+	0.992901i	$0.462048\pi$
$752$	−4.99980	−0.182324
$753$	0	0
$754$	−17.6438	−0.642551
$755$	−48.9216	−1.78044
$756$	0	0
$757$	49.3554	1.79385	0.896926	−	0.442181i	$-0.145795\pi$
$757$	49.3554	1.79385	0.896926	+	0.442181i	$0.145795\pi$
$758$	26.4673	0.961336
$759$	0	0
$760$	−7.16911	−0.260051
$761$	26.2674	0.952193	0.476097	−	0.879393i	$-0.342051\pi$
$761$	26.2674	0.952193	0.476097	+	0.879393i	$0.342051\pi$
$762$	0	0
$763$	−37.8388	−1.36986
$764$	−12.6842	−0.458898
$765$	0	0
$766$	−13.6027	−0.491485
$767$	−28.5721	−1.03168
$768$	0	0
$769$	27.6184	0.995943	0.497972	−	0.867193i	$-0.334078\pi$
$769$	27.6184	0.995943	0.497972	+	0.867193i	$0.334078\pi$
$770$	78.0045	2.81109
$771$	0	0
$772$	7.71183	0.277555
$773$	45.3005	1.62935	0.814673	−	0.579921i	$-0.196917\pi$
$773$	45.3005	1.62935	0.814673	+	0.579921i	$0.196917\pi$
$774$	0	0
$775$	−89.5650	−3.21727
$776$	23.2364	0.834139
$777$	0	0
$778$	25.4178	0.911273
$779$	6.99275	0.250541
$780$	0	0
$781$	40.0288	1.43234
$782$	−49.2036	−1.75952
$783$	0	0
$784$	−35.6703	−1.27394
$785$	−38.6646	−1.38000
$786$	0	0
$787$	43.7481	1.55945	0.779725	−	0.626122i	$-0.215359\pi$
$787$	43.7481	1.55945	0.779725	+	0.626122i	$0.215359\pi$
$788$	17.6413	0.628445
$789$	0	0
$790$	−12.4417	−0.442656
$791$	−29.1057	−1.03488
$792$	0	0
$793$	−27.1186	−0.963009
$794$	−51.2329	−1.81819
$795$	0	0
$796$	−14.0613	−0.498389
$797$	20.9586	0.742393	0.371196	−	0.928554i	$-0.378948\pi$
$797$	20.9586	0.742393	0.371196	+	0.928554i	$0.378948\pi$
$798$	0	0
$799$	−4.34212	−0.153613
$800$	60.2718	2.13093
$801$	0	0
$802$	26.9600	0.951989
$803$	8.16989	0.288309
$804$	0	0
$805$	−100.866	−3.55505
$806$	−27.2149	−0.958602
$807$	0	0
$808$	8.68544	0.305553
$809$	36.0114	1.26609	0.633047	−	0.774113i	$-0.281804\pi$
$809$	36.0114	1.26609	0.633047	+	0.774113i	$0.281804\pi$
$810$	0	0
$811$	−47.2710	−1.65991	−0.829954	−	0.557831i	$-0.811634\pi$
$811$	−47.2710	−1.65991	−0.829954	+	0.557831i	$0.811634\pi$
$812$	−18.3364	−0.643481
$813$	0	0
$814$	−17.8867	−0.626927
$815$	2.72533	0.0954643
$816$	0	0
$817$	−7.07313	−0.247457
$818$	43.2885	1.51355
$819$	0	0
$820$	−28.2053	−0.984973
$821$	3.38081	0.117991	0.0589956	−	0.998258i	$-0.481210\pi$
$821$	3.38081	0.117991	0.0589956	+	0.998258i	$0.481210\pi$
$822$	0	0
$823$	15.8723	0.553274	0.276637	−	0.960975i	$-0.410780\pi$
$823$	15.8723	0.553274	0.276637	+	0.960975i	$0.410780\pi$
$824$	7.08641	0.246867
$825$	0	0
$826$	−89.9300	−3.12906
$827$	30.3815	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$827$	30.3815	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$828$	0	0
$829$	−12.5111	−0.434530	−0.217265	−	0.976113i	$-0.569714\pi$
$829$	−12.5111	−0.434530	−0.217265	+	0.976113i	$0.569714\pi$
$830$	80.4637	2.79293
$831$	0	0
$832$	−2.32544	−0.0806200
$833$	−30.9782	−1.07333
$834$	0	0
$835$	3.64779	0.126237
$836$	2.89355	0.100075
$837$	0	0
$838$	−59.6306	−2.05991
$839$	4.47176	0.154382	0.0771911	−	0.997016i	$-0.475405\pi$
$839$	4.47176	0.154382	0.0771911	+	0.997016i	$0.475405\pi$
$840$	0	0
$841$	−4.52726	−0.156113
$842$	34.4976	1.18886
$843$	0	0
$844$	7.99563	0.275221
$845$	−35.7562	−1.23005
$846$	0	0
$847$	−8.97136	−0.308259
$848$	35.1211	1.20606
$849$	0	0
$850$	88.0692	3.02075
$851$	23.1288	0.792844
$852$	0	0
$853$	−16.8601	−0.577278	−0.288639	−	0.957438i	$-0.593203\pi$
$853$	−16.8601	−0.577278	−0.288639	+	0.957438i	$0.593203\pi$
$854$	−85.3549	−2.92079
$855$	0	0
$856$	20.4975	0.700591
$857$	33.4154	1.14145	0.570724	−	0.821142i	$-0.306663\pi$
$857$	33.4154	1.14145	0.570724	+	0.821142i	$0.306663\pi$
$858$	0	0
$859$	−49.5460	−1.69049	−0.845243	−	0.534382i	$-0.820544\pi$
$859$	−49.5460	−1.69049	−0.845243	+	0.534382i	$0.820544\pi$
$860$	28.5296	0.972850
$861$	0	0
$862$	37.3364	1.27168
$863$	−27.7204	−0.943614	−0.471807	−	0.881702i	$-0.656398\pi$
$863$	−27.7204	−0.943614	−0.471807	+	0.881702i	$0.656398\pi$
$864$	0	0
$865$	−16.7410	−0.569210
$866$	−64.0980	−2.17814
$867$	0	0
$868$	−28.2831	−0.959991
$869$	−5.16522	−0.175218
$870$	0	0
$871$	−16.1270	−0.546443
$872$	−17.6367	−0.597253
$873$	0	0
$874$	−11.3317	−0.383300
$875$	103.633	3.50345
$876$	0	0
$877$	34.0574	1.15004	0.575018	−	0.818141i	$-0.304995\pi$
$877$	34.0574	1.15004	0.575018	+	0.818141i	$0.304995\pi$
$878$	−7.58596	−0.256014
$879$	0	0
$880$	60.0339	2.02374
$881$	−51.6222	−1.73920	−0.869598	−	0.493760i	$-0.835622\pi$
$881$	−51.6222	−1.73920	−0.869598	+	0.493760i	$0.835622\pi$
$882$	0	0
$883$	15.2907	0.514573	0.257287	−	0.966335i	$-0.417172\pi$
$883$	15.2907	0.514573	0.257287	+	0.966335i	$0.417172\pi$
$884$	8.83590	0.297183
$885$	0	0
$886$	62.1918	2.08938
$887$	−17.5233	−0.588374	−0.294187	−	0.955748i	$-0.595049\pi$
$887$	−17.5233	−0.588374	−0.294187	+	0.955748i	$0.595049\pi$
$888$	0	0
$889$	−70.8131	−2.37499
$890$	21.1487	0.708905
$891$	0	0
$892$	−16.2172	−0.542991
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−53.4481	−1.78657
$896$	−45.9291	−1.53438
$897$	0	0
$898$	40.6769	1.35740
$899$	37.7481	1.25897
$900$	0	0
$901$	30.5012	1.01614
$902$	−35.4635	−1.18081
$903$	0	0
$904$	−13.5662	−0.451204
$905$	−56.1221	−1.86556
$906$	0	0
$907$	−49.9765	−1.65944	−0.829722	−	0.558177i	$-0.811501\pi$
$907$	−49.9765	−1.65944	−0.829722	+	0.558177i	$0.811501\pi$
$908$	−6.52945	−0.216687
$909$	0	0
$910$	54.8577	1.81852
$911$	51.2268	1.69722	0.848609	−	0.529020i	$-0.177440\pi$
$911$	51.2268	1.69722	0.848609	+	0.529020i	$0.177440\pi$
$912$	0	0
$913$	33.4048	1.10554
$914$	36.7359	1.21512
$915$	0	0
$916$	3.81620	0.126091
$917$	55.0553	1.81809
$918$	0	0
$919$	40.6804	1.34192	0.670961	−	0.741492i	$-0.265882\pi$
$919$	40.6804	1.34192	0.670961	+	0.741492i	$0.265882\pi$
$920$	−47.0134	−1.54999
$921$	0	0
$922$	−22.1856	−0.730643
$923$	28.1508	0.926595
$924$	0	0
$925$	−41.3981	−1.36116
$926$	58.0226	1.90674
$927$	0	0
$928$	−25.4022	−0.833868
$929$	5.19328	0.170386	0.0851929	−	0.996364i	$-0.472849\pi$
$929$	5.19328	0.170386	0.0851929	+	0.996364i	$0.472849\pi$
$930$	0	0
$931$	−7.13434	−0.233819
$932$	−3.40129	−0.111413
$933$	0	0
$934$	41.7005	1.36448
$935$	52.1369	1.70506
$936$	0	0
$937$	13.3549	0.436284	0.218142	−	0.975917i	$-0.430000\pi$
$937$	13.3549	0.436284	0.218142	+	0.975917i	$0.430000\pi$
$938$	−50.7593	−1.65735
$939$	0	0
$940$	4.03351	0.131559
$941$	10.8875	0.354921	0.177461	−	0.984128i	$-0.443212\pi$
$941$	10.8875	0.354921	0.177461	+	0.984128i	$0.443212\pi$
$942$	0	0
$943$	45.8569	1.49331
$944$	−69.2120	−2.25266
$945$	0	0
$946$	35.8711	1.16627
$947$	7.03569	0.228629	0.114315	−	0.993445i	$-0.463533\pi$
$947$	7.03569	0.228629	0.114315	+	0.993445i	$0.463533\pi$
$948$	0	0
$949$	5.74559	0.186510
$950$	20.2825	0.658052
$951$	0	0
$952$	−28.6059	−0.927124
$953$	−47.6857	−1.54469	−0.772346	−	0.635203i	$-0.780917\pi$
$953$	−47.6857	−1.54469	−0.772346	+	0.635203i	$0.780917\pi$
$954$	0	0
$955$	−52.6352	−1.70323
$956$	9.10893	0.294604
$957$	0	0
$958$	36.3967	1.17592
$959$	18.0146	0.581723
$960$	0	0
$961$	27.2249	0.878221
$962$	−12.5790	−0.405564
$963$	0	0
$964$	−5.20226	−0.167554
$965$	32.0015	1.03017
$966$	0	0
$967$	−0.0940278	−0.00302373	−0.00151187	−	0.999999i	$-0.500481\pi$
$967$	−0.0940278	−0.00302373	−0.00151187	+	0.999999i	$0.500481\pi$
$968$	−4.18155	−0.134400
$969$	0	0
$970$	−93.7430	−3.00991
$971$	21.6197	0.693809	0.346905	−	0.937900i	$-0.387233\pi$
$971$	21.6197	0.693809	0.346905	+	0.937900i	$0.387233\pi$
$972$	0	0
$973$	−76.4103	−2.44960
$974$	27.7405	0.888862
$975$	0	0
$976$	−65.6910	−2.10272
$977$	19.9360	0.637808	0.318904	−	0.947787i	$-0.396685\pi$
$977$	19.9360	0.637808	0.318904	+	0.947787i	$0.396685\pi$
$978$	0	0
$979$	8.77995	0.280608
$980$	28.7765	0.919230
$981$	0	0
$982$	−0.396912	−0.0126660
$983$	10.0686	0.321138	0.160569	−	0.987025i	$-0.448667\pi$
$983$	10.0686	0.321138	0.160569	+	0.987025i	$0.448667\pi$
$984$	0	0
$985$	73.2055	2.33252
$986$	−37.1177	−1.18207
$987$	0	0
$988$	2.03493	0.0647397
$989$	−46.3841	−1.47493
$990$	0	0
$991$	30.1200	0.956793	0.478396	−	0.878144i	$-0.341218\pi$
$991$	30.1200	0.956793	0.478396	+	0.878144i	$0.341218\pi$
$992$	−39.1818	−1.24402
$993$	0	0
$994$	88.6039	2.81034
$995$	−58.3497	−1.84981
$996$	0	0
$997$	6.18606	0.195914	0.0979572	−	0.995191i	$-0.468769\pi$
$997$	6.18606	0.195914	0.0979572	+	0.995191i	$0.468769\pi$
$998$	−40.0527	−1.26785
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.s.1.8		24
3.2	odd	2		2679.2.a.p.1.17	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.p.1.17	✓	24	3.2	odd	2
8037.2.a.s.1.8		24	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)

\(f(q)\)	\(=\)	\(q-1.72798 q^{2} +0.985905 q^{4} +4.09118 q^{5} +3.75957 q^{7} +1.75233 q^{8} +O(q^{10})\)	\(q-1.72798 q^{2} +0.985905 q^{4} +4.09118 q^{5} +3.75957 q^{7} +1.75233 q^{8} -7.06946 q^{10} -2.93491 q^{11} -2.06402 q^{13} -6.49645 q^{14} -4.99980 q^{16} -4.34212 q^{17} -1.00000 q^{19} +4.03351 q^{20} +5.07147 q^{22} -6.55778 q^{23} +11.7377 q^{25} +3.56658 q^{26} +3.70658 q^{28} -4.94699 q^{29} -7.63052 q^{31} +5.13488 q^{32} +7.50308 q^{34} +15.3811 q^{35} -3.52692 q^{37} +1.72798 q^{38} +7.16911 q^{40} -6.99275 q^{41} +7.07313 q^{43} -2.89355 q^{44} +11.3317 q^{46} +1.00000 q^{47} +7.13434 q^{49} -20.2825 q^{50} -2.03493 q^{52} -7.02450 q^{53} -12.0073 q^{55} +6.58801 q^{56} +8.54829 q^{58} +13.8429 q^{59} +13.1387 q^{61} +13.1854 q^{62} +1.12665 q^{64} -8.44427 q^{65} +7.81340 q^{67} -4.28092 q^{68} -26.5781 q^{70} -13.6388 q^{71} -2.78369 q^{73} +6.09444 q^{74} -0.985905 q^{76} -11.0340 q^{77} +1.75992 q^{79} -20.4551 q^{80} +12.0833 q^{82} -11.3819 q^{83} -17.7644 q^{85} -12.2222 q^{86} -5.14295 q^{88} -2.99155 q^{89} -7.75982 q^{91} -6.46535 q^{92} -1.72798 q^{94} -4.09118 q^{95} +13.2603 q^{97} -12.3280 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8}+O(q^{10}) \) 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8	\( 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8} + 5 q^{10} - 15 q^{11} - 3 q^{13} - 27 q^{14} + 48 q^{16} - 22 q^{17} - 24 q^{19} - 29 q^{20} - 9 q^{22} - 15 q^{23} + 58 q^{25} - q^{26} + 18 q^{28} - 36 q^{29} - 16 q^{31} - 26 q^{32} + 10 q^{34} - 37 q^{35} + 18 q^{37} + 6 q^{38} + 7 q^{40} - 24 q^{41} + 2 q^{43} - 57 q^{44} - 23 q^{46} + 24 q^{47} + 64 q^{49} - 4 q^{50} - 40 q^{52} - 42 q^{53} + 8 q^{55} - 38 q^{56} - 5 q^{58} - 39 q^{59} + 6 q^{61} + 14 q^{62} + 53 q^{64} - 47 q^{65} - 26 q^{67} - 80 q^{68} - 46 q^{70} - 6 q^{71} + 3 q^{73} - 62 q^{74} - 32 q^{76} - 13 q^{77} + 10 q^{79} - 17 q^{80} + 15 q^{82} - 44 q^{83} + 21 q^{85} + 22 q^{86} - 88 q^{88} - 32 q^{89} - 17 q^{91} + 37 q^{92} - 6 q^{94} + 10 q^{95} + 28 q^{97} - 44 q^{98}+O(q^{100}) \) 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8 + 5 * q^10 - 15 * q^11 - 3 * q^13 - 27 * q^14 + 48 * q^16 - 22 * q^17 - 24 * q^19 - 29 * q^20 - 9 * q^22 - 15 * q^23 + 58 * q^25 - q^26 + 18 * q^28 - 36 * q^29 - 16 * q^31 - 26 * q^32 + 10 * q^34 - 37 * q^35 + 18 * q^37 + 6 * q^38 + 7 * q^40 - 24 * q^41 + 2 * q^43 - 57 * q^44 - 23 * q^46 + 24 * q^47 + 64 * q^49 - 4 * q^50 - 40 * q^52 - 42 * q^53 + 8 * q^55 - 38 * q^56 - 5 * q^58 - 39 * q^59 + 6 * q^61 + 14 * q^62 + 53 * q^64 - 47 * q^65 - 26 * q^67 - 80 * q^68 - 46 * q^70 - 6 * q^71 + 3 * q^73 - 62 * q^74 - 32 * q^76 - 13 * q^77 + 10 * q^79 - 17 * q^80 + 15 * q^82 - 44 * q^83 + 21 * q^85 + 22 * q^86 - 88 * q^88 - 32 * q^89 - 17 * q^91 + 37 * q^92 - 6 * q^94 + 10 * q^95 + 28 * q^97 - 44 * q^98

Introduction

L-functions

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Varieties

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