LMFDB - Embedded newform 8037.2.a.o.1.15

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			1.15
Root			$-1.61470$ 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)

$f(q)$	$=$	$q+1.61470 q^{2} +0.607250 q^{4} -2.02448 q^{5} -1.29797 q^{7} -2.24887 q^{8} +O(q^{10})$	\(q+1.61470 q^{2} +0.607250 q^{4} -2.02448 q^{5} -1.29797 q^{7} -2.24887 q^{8} -3.26892 q^{10} -5.65826 q^{11} -2.24299 q^{13} -2.09584 q^{14} -4.84575 q^{16} -2.06910 q^{17} -1.00000 q^{19} -1.22936 q^{20} -9.13638 q^{22} +2.15918 q^{23} -0.901494 q^{25} -3.62175 q^{26} -0.788195 q^{28} +7.29374 q^{29} +3.56193 q^{31} -3.32668 q^{32} -3.34097 q^{34} +2.62772 q^{35} +0.880650 q^{37} -1.61470 q^{38} +4.55279 q^{40} -10.1558 q^{41} -8.07785 q^{43} -3.43598 q^{44} +3.48643 q^{46} -1.00000 q^{47} -5.31526 q^{49} -1.45564 q^{50} -1.36206 q^{52} -8.45701 q^{53} +11.4550 q^{55} +2.91898 q^{56} +11.7772 q^{58} +8.85943 q^{59} -0.251067 q^{61} +5.75145 q^{62} +4.31991 q^{64} +4.54088 q^{65} +6.22660 q^{67} -1.25646 q^{68} +4.24297 q^{70} +10.7701 q^{71} -8.67240 q^{73} +1.42198 q^{74} -0.607250 q^{76} +7.34428 q^{77} -5.08930 q^{79} +9.81010 q^{80} -16.3986 q^{82} -7.62221 q^{83} +4.18884 q^{85} -13.0433 q^{86} +12.7247 q^{88} +7.55270 q^{89} +2.91135 q^{91} +1.31116 q^{92} -1.61470 q^{94} +2.02448 q^{95} +17.7937 q^{97} -8.58254 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.61470	1.14176	0.570882	−	0.821032i	$-0.306601\pi$
$2$	1.61470	1.14176	0.570882	+	0.821032i	$0.306601\pi$
$3$	0	0
$3$	0	0
$4$	0.607250	0.303625
$5$	−2.02448	−0.905373	−0.452687	−	0.891670i	$-0.649534\pi$
$5$	−2.02448	−0.905373	−0.452687	+	0.891670i	$0.649534\pi$
$6$	0	0
$7$	−1.29797	−0.490588	−0.245294	−	0.969449i	$-0.578885\pi$
$7$	−1.29797	−0.490588	−0.245294	+	0.969449i	$0.578885\pi$
$8$	−2.24887	−0.795096
$9$	0	0
$10$	−3.26892	−1.03372
$11$	−5.65826	−1.70603	−0.853014	−	0.521887i	$-0.825228\pi$
$11$	−5.65826	−1.70603	−0.853014	+	0.521887i	$0.825228\pi$
$12$	0	0
$13$	−2.24299	−0.622094	−0.311047	−	0.950395i	$-0.600680\pi$
$13$	−2.24299	−0.622094	−0.311047	+	0.950395i	$0.600680\pi$
$14$	−2.09584	−0.560136
$15$	0	0
$16$	−4.84575	−1.21144
$17$	−2.06910	−0.501830	−0.250915	−	0.968009i	$-0.580731\pi$
$17$	−2.06910	−0.501830	−0.250915	+	0.968009i	$0.580731\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.22936	−0.274894
$21$	0	0
$22$	−9.13638	−1.94788
$23$	2.15918	0.450221	0.225110	−	0.974333i	$-0.427726\pi$
$23$	2.15918	0.450221	0.225110	+	0.974333i	$0.427726\pi$
$24$	0	0
$25$	−0.901494	−0.180299
$26$	−3.62175	−0.710285
$27$	0	0
$28$	−0.788195	−0.148955
$29$	7.29374	1.35441	0.677206	−	0.735793i	$-0.263190\pi$
$29$	7.29374	1.35441	0.677206	+	0.735793i	$0.263190\pi$
$30$	0	0
$31$	3.56193	0.639742	0.319871	−	0.947461i	$-0.396360\pi$
$31$	3.56193	0.639742	0.319871	+	0.947461i	$0.396360\pi$
$32$	−3.32668	−0.588079
$33$	0	0
$34$	−3.34097	−0.572972
$35$	2.62772	0.444166
$36$	0	0
$37$	0.880650	0.144778	0.0723890	−	0.997376i	$-0.476938\pi$
$37$	0.880650	0.144778	0.0723890	+	0.997376i	$0.476938\pi$
$38$	−1.61470	−0.261939
$39$	0	0
$40$	4.55279	0.719859
$41$	−10.1558	−1.58607	−0.793035	−	0.609176i	$-0.791500\pi$
$41$	−10.1558	−1.58607	−0.793035	+	0.609176i	$0.791500\pi$
$42$	0	0
$43$	−8.07785	−1.23186	−0.615930	−	0.787801i	$-0.711220\pi$
$43$	−8.07785	−1.23186	−0.615930	+	0.787801i	$0.711220\pi$
$44$	−3.43598	−0.517993
$45$	0	0
$46$	3.48643	0.514046
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.31526	−0.759323
$50$	−1.45564	−0.205859
$51$	0	0
$52$	−1.36206	−0.188883
$53$	−8.45701	−1.16166	−0.580830	−	0.814025i	$-0.697272\pi$
$53$	−8.45701	−1.16166	−0.580830	+	0.814025i	$0.697272\pi$
$54$	0	0
$55$	11.4550	1.54459
$56$	2.91898	0.390065
$57$	0	0
$58$	11.7772	1.54642
$59$	8.85943	1.15340	0.576699	−	0.816956i	$-0.304340\pi$
$59$	8.85943	1.15340	0.576699	+	0.816956i	$0.304340\pi$
$60$	0	0
$61$	−0.251067	−0.0321458	−0.0160729	−	0.999871i	$-0.505116\pi$
$61$	−0.251067	−0.0321458	−0.0160729	+	0.999871i	$0.505116\pi$
$62$	5.75145	0.730434
$63$	0	0
$64$	4.31991	0.539989
$65$	4.54088	0.563227
$66$	0	0
$67$	6.22660	0.760700	0.380350	−	0.924843i	$-0.375803\pi$
$67$	6.22660	0.760700	0.380350	+	0.924843i	$0.375803\pi$
$68$	−1.25646	−0.152368
$69$	0	0
$70$	4.24297	0.507132
$71$	10.7701	1.27817	0.639085	−	0.769136i	$-0.279313\pi$
$71$	10.7701	1.27817	0.639085	+	0.769136i	$0.279313\pi$
$72$	0	0
$73$	−8.67240	−1.01503	−0.507514	−	0.861644i	$-0.669435\pi$
$73$	−8.67240	−1.01503	−0.507514	+	0.861644i	$0.669435\pi$
$74$	1.42198	0.165302
$75$	0	0
$76$	−0.607250	−0.0696564
$77$	7.34428	0.836958
$78$	0	0
$79$	−5.08930	−0.572591	−0.286295	−	0.958141i	$-0.592424\pi$
$79$	−5.08930	−0.572591	−0.286295	+	0.958141i	$0.592424\pi$
$80$	9.81010	1.09680
$81$	0	0
$82$	−16.3986	−1.81092
$83$	−7.62221	−0.836646	−0.418323	−	0.908298i	$-0.637382\pi$
$83$	−7.62221	−0.836646	−0.418323	+	0.908298i	$0.637382\pi$
$84$	0	0
$85$	4.18884	0.454344
$86$	−13.0433	−1.40649
$87$	0	0
$88$	12.7247	1.35646
$89$	7.55270	0.800585	0.400292	−	0.916388i	$-0.368909\pi$
$89$	7.55270	0.800585	0.400292	+	0.916388i	$0.368909\pi$
$90$	0	0
$91$	2.91135	0.305192
$92$	1.31116	0.136698
$93$	0	0
$94$	−1.61470	−0.166543
$95$	2.02448	0.207707
$96$	0	0
$97$	17.7937	1.80668	0.903340	−	0.428926i	$-0.141108\pi$
$97$	17.7937	1.80668	0.903340	+	0.428926i	$0.141108\pi$
$98$	−8.58254	−0.866968
$99$	0	0
$100$	−0.547433	−0.0547433
$101$	8.13642	0.809604	0.404802	−	0.914404i	$-0.367340\pi$
$101$	8.13642	0.809604	0.404802	+	0.914404i	$0.367340\pi$
$102$	0	0
$103$	9.93632	0.979055	0.489528	−	0.871988i	$-0.337169\pi$
$103$	9.93632	0.979055	0.489528	+	0.871988i	$0.337169\pi$
$104$	5.04420	0.494624
$105$	0	0
$106$	−13.6555	−1.32634
$107$	−10.5523	−1.02013	−0.510064	−	0.860137i	$-0.670378\pi$
$107$	−10.5523	−1.02013	−0.510064	+	0.860137i	$0.670378\pi$
$108$	0	0
$109$	11.8944	1.13928	0.569640	−	0.821894i	$-0.307083\pi$
$109$	11.8944	1.13928	0.569640	+	0.821894i	$0.307083\pi$
$110$	18.4964	1.76356
$111$	0	0
$112$	6.28966	0.594317
$113$	−10.3982	−0.978176	−0.489088	−	0.872235i	$-0.662670\pi$
$113$	−10.3982	−0.978176	−0.489088	+	0.872235i	$0.662670\pi$
$114$	0	0
$115$	−4.37121	−0.407618
$116$	4.42912	0.411234
$117$	0	0
$118$	14.3053	1.31691
$119$	2.68564	0.246192
$120$	0	0
$121$	21.0159	1.91053
$122$	−0.405397	−0.0367030
$123$	0	0
$124$	2.16298	0.194242
$125$	11.9474	1.06861
$126$	0	0
$127$	−19.3925	−1.72081	−0.860403	−	0.509615i	$-0.829788\pi$
$127$	−19.3925	−1.72081	−0.860403	+	0.509615i	$0.829788\pi$
$128$	13.6287	1.20462
$129$	0	0
$130$	7.33216	0.643073
$131$	−10.8807	−0.950651	−0.475325	−	0.879810i	$-0.657670\pi$
$131$	−10.8807	−0.950651	−0.475325	+	0.879810i	$0.657670\pi$
$132$	0	0
$133$	1.29797	0.112549
$134$	10.0541	0.868540
$135$	0	0
$136$	4.65314	0.399003
$137$	12.6541	1.08111	0.540554	−	0.841309i	$-0.318214\pi$
$137$	12.6541	1.08111	0.540554	+	0.841309i	$0.318214\pi$
$138$	0	0
$139$	−18.0343	−1.52965	−0.764824	−	0.644239i	$-0.777174\pi$
$139$	−18.0343	−1.52965	−0.764824	+	0.644239i	$0.777174\pi$
$140$	1.59568	0.134860
$141$	0	0
$142$	17.3904	1.45937
$143$	12.6914	1.06131
$144$	0	0
$145$	−14.7660	−1.22625
$146$	−14.0033	−1.15892
$147$	0	0
$148$	0.534775	0.0439582
$149$	6.95914	0.570115	0.285057	−	0.958510i	$-0.407987\pi$
$149$	6.95914	0.570115	0.285057	+	0.958510i	$0.407987\pi$
$150$	0	0
$151$	8.76112	0.712969	0.356485	−	0.934301i	$-0.383975\pi$
$151$	8.76112	0.712969	0.356485	+	0.934301i	$0.383975\pi$
$152$	2.24887	0.182407
$153$	0	0
$154$	11.8588	0.955608
$155$	−7.21105	−0.579205
$156$	0	0
$157$	9.39020	0.749419	0.374710	−	0.927142i	$-0.377742\pi$
$157$	9.39020	0.749419	0.374710	+	0.927142i	$0.377742\pi$
$158$	−8.21768	−0.653764
$159$	0	0
$160$	6.73478	0.532431
$161$	−2.80256	−0.220873
$162$	0	0
$163$	24.3695	1.90877	0.954385	−	0.298580i	$-0.0965130\pi$
$163$	24.3695	1.90877	0.954385	+	0.298580i	$0.0965130\pi$
$164$	−6.16712	−0.481571
$165$	0	0
$166$	−12.3076	−0.955252
$167$	18.3394	1.41914	0.709571	−	0.704634i	$-0.248889\pi$
$167$	18.3394	1.41914	0.709571	+	0.704634i	$0.248889\pi$
$168$	0	0
$169$	−7.96899	−0.612999
$170$	6.76372	0.518753
$171$	0	0
$172$	−4.90527	−0.374024
$173$	9.95870	0.757146	0.378573	−	0.925571i	$-0.376415\pi$
$173$	9.95870	0.757146	0.378573	+	0.925571i	$0.376415\pi$
$174$	0	0
$175$	1.17012	0.0884525
$176$	27.4185	2.06675
$177$	0	0
$178$	12.1953	0.914079
$179$	−0.725424	−0.0542207	−0.0271104	−	0.999632i	$-0.508631\pi$
$179$	−0.725424	−0.0542207	−0.0271104	+	0.999632i	$0.508631\pi$
$180$	0	0
$181$	−0.280028	−0.0208143	−0.0104072	−	0.999946i	$-0.503313\pi$
$181$	−0.280028	−0.0208143	−0.0104072	+	0.999946i	$0.503313\pi$
$182$	4.70095	0.348457
$183$	0	0
$184$	−4.85572	−0.357969
$185$	−1.78286	−0.131078
$186$	0	0
$187$	11.7075	0.856137
$188$	−0.607250	−0.0442883
$189$	0	0
$190$	3.26892	0.237152
$191$	−3.68397	−0.266563	−0.133281	−	0.991078i	$-0.542551\pi$
$191$	−3.68397	−0.266563	−0.133281	+	0.991078i	$0.542551\pi$
$192$	0	0
$193$	−9.80802	−0.705997	−0.352998	−	0.935624i	$-0.614838\pi$
$193$	−9.80802	−0.705997	−0.352998	+	0.935624i	$0.614838\pi$
$194$	28.7315	2.06280
$195$	0	0
$196$	−3.22769	−0.230550
$197$	25.1388	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$197$	25.1388	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$198$	0	0
$199$	23.8129	1.68805	0.844027	−	0.536301i	$-0.180179\pi$
$199$	23.8129	1.68805	0.844027	+	0.536301i	$0.180179\pi$
$200$	2.02734	0.143355
$201$	0	0
$202$	13.1379	0.924377
$203$	−9.46709	−0.664459
$204$	0	0
$205$	20.5602	1.43599
$206$	16.0442	1.11785
$207$	0	0
$208$	10.8690	0.753628
$209$	5.65826	0.391390
$210$	0	0
$211$	−11.1139	−0.765115	−0.382557	−	0.923932i	$-0.624957\pi$
$211$	−11.1139	−0.765115	−0.382557	+	0.923932i	$0.624957\pi$
$212$	−5.13552	−0.352709
$213$	0	0
$214$	−17.0387	−1.16474
$215$	16.3534	1.11529
$216$	0	0
$217$	−4.62330	−0.313850
$218$	19.2059	1.30079
$219$	0	0
$220$	6.95606	0.468977
$221$	4.64097	0.312186
$222$	0	0
$223$	−17.8740	−1.19693	−0.598465	−	0.801149i	$-0.704222\pi$
$223$	−17.8740	−1.19693	−0.598465	+	0.801149i	$0.704222\pi$
$224$	4.31794	0.288505
$225$	0	0
$226$	−16.7899	−1.11685
$227$	−2.95716	−0.196274	−0.0981368	−	0.995173i	$-0.531288\pi$
$227$	−2.95716	−0.196274	−0.0981368	+	0.995173i	$0.531288\pi$
$228$	0	0
$229$	−10.1915	−0.673476	−0.336738	−	0.941598i	$-0.609324\pi$
$229$	−10.1915	−0.673476	−0.336738	+	0.941598i	$0.609324\pi$
$230$	−7.05819	−0.465403
$231$	0	0
$232$	−16.4027	−1.07689
$233$	−1.38084	−0.0904620	−0.0452310	−	0.998977i	$-0.514402\pi$
$233$	−1.38084	−0.0904620	−0.0452310	+	0.998977i	$0.514402\pi$
$234$	0	0
$235$	2.02448	0.132062
$236$	5.37989	0.350201
$237$	0	0
$238$	4.33649	0.281093
$239$	2.09251	0.135353	0.0676766	−	0.997707i	$-0.478441\pi$
$239$	2.09251	0.135353	0.0676766	+	0.997707i	$0.478441\pi$
$240$	0	0
$241$	−25.0795	−1.61552	−0.807758	−	0.589515i	$-0.799319\pi$
$241$	−25.0795	−1.61552	−0.807758	+	0.589515i	$0.799319\pi$
$242$	33.9343	2.18138
$243$	0	0
$244$	−0.152460	−0.00976028
$245$	10.7606	0.687471
$246$	0	0
$247$	2.24299	0.142718
$248$	−8.01032	−0.508656
$249$	0	0
$250$	19.2915	1.22010
$251$	−13.7132	−0.865567	−0.432783	−	0.901498i	$-0.642468\pi$
$251$	−13.7132	−0.865567	−0.432783	+	0.901498i	$0.642468\pi$
$252$	0	0
$253$	−12.2172	−0.768089
$254$	−31.3130	−1.96475
$255$	0	0
$256$	13.3664	0.835402
$257$	−12.8312	−0.800391	−0.400195	−	0.916430i	$-0.631058\pi$
$257$	−12.8312	−0.800391	−0.400195	+	0.916430i	$0.631058\pi$
$258$	0	0
$259$	−1.14306	−0.0710264
$260$	2.75745	0.171010
$261$	0	0
$262$	−17.5690	−1.08542
$263$	−0.648094	−0.0399632	−0.0199816	−	0.999800i	$-0.506361\pi$
$263$	−0.648094	−0.0399632	−0.0199816	+	0.999800i	$0.506361\pi$
$264$	0	0
$265$	17.1210	1.05174
$266$	2.09584	0.128504
$267$	0	0
$268$	3.78110	0.230968
$269$	−5.65164	−0.344586	−0.172293	−	0.985046i	$-0.555118\pi$
$269$	−5.65164	−0.344586	−0.172293	+	0.985046i	$0.555118\pi$
$270$	0	0
$271$	−28.8907	−1.75499	−0.877493	−	0.479589i	$-0.840786\pi$
$271$	−28.8907	−1.75499	−0.877493	+	0.479589i	$0.840786\pi$
$272$	10.0263	0.607936
$273$	0	0
$274$	20.4325	1.23437
$275$	5.10089	0.307595
$276$	0	0
$277$	−5.36596	−0.322409	−0.161205	−	0.986921i	$-0.551538\pi$
$277$	−5.36596	−0.322409	−0.161205	+	0.986921i	$0.551538\pi$
$278$	−29.1199	−1.74650
$279$	0	0
$280$	−5.90940	−0.353154
$281$	−16.6840	−0.995285	−0.497643	−	0.867382i	$-0.665801\pi$
$281$	−16.6840	−0.995285	−0.497643	+	0.867382i	$0.665801\pi$
$282$	0	0
$283$	−18.5234	−1.10110	−0.550551	−	0.834802i	$-0.685582\pi$
$283$	−18.5234	−1.10110	−0.550551	+	0.834802i	$0.685582\pi$
$284$	6.54012	0.388085
$285$	0	0
$286$	20.4928	1.21177
$287$	13.1820	0.778108
$288$	0	0
$289$	−12.7188	−0.748166
$290$	−23.8426	−1.40009
$291$	0	0
$292$	−5.26632	−0.308188
$293$	10.8297	0.632675	0.316338	−	0.948647i	$-0.397547\pi$
$293$	10.8297	0.632675	0.316338	+	0.948647i	$0.397547\pi$
$294$	0	0
$295$	−17.9357	−1.04426
$296$	−1.98047	−0.115112
$297$	0	0
$298$	11.2369	0.650936
$299$	−4.84303	−0.280080
$300$	0	0
$301$	10.4848	0.604336
$302$	14.1466	0.814043
$303$	0	0
$304$	4.84575	0.277923
$305$	0.508279	0.0291040
$306$	0	0
$307$	−5.01505	−0.286224	−0.143112	−	0.989706i	$-0.545711\pi$
$307$	−5.01505	−0.286224	−0.143112	+	0.989706i	$0.545711\pi$
$308$	4.45981	0.254121
$309$	0	0
$310$	−11.6437	−0.661316
$311$	16.5708	0.939646	0.469823	−	0.882761i	$-0.344318\pi$
$311$	16.5708	0.939646	0.469823	+	0.882761i	$0.344318\pi$
$312$	0	0
$313$	−4.54679	−0.257000	−0.128500	−	0.991710i	$-0.541016\pi$
$313$	−4.54679	−0.257000	−0.128500	+	0.991710i	$0.541016\pi$
$314$	15.1623	0.855660
$315$	0	0
$316$	−3.09048	−0.173853
$317$	−2.98855	−0.167854	−0.0839268	−	0.996472i	$-0.526746\pi$
$317$	−2.98855	−0.167854	−0.0839268	+	0.996472i	$0.526746\pi$
$318$	0	0
$319$	−41.2698	−2.31067
$320$	−8.74556	−0.488892
$321$	0	0
$322$	−4.52529	−0.252185
$323$	2.06910	0.115128
$324$	0	0
$325$	2.02204	0.112163
$326$	39.3494	2.17936
$327$	0	0
$328$	22.8391	1.26108
$329$	1.29797	0.0715597
$330$	0	0
$331$	7.28943	0.400663	0.200332	−	0.979728i	$-0.435798\pi$
$331$	7.28943	0.400663	0.200332	+	0.979728i	$0.435798\pi$
$332$	−4.62859	−0.254027
$333$	0	0
$334$	29.6125	1.62033
$335$	−12.6056	−0.688718
$336$	0	0
$337$	25.0555	1.36486	0.682431	−	0.730950i	$-0.260923\pi$
$337$	25.0555	1.36486	0.682431	+	0.730950i	$0.260923\pi$
$338$	−12.8675	−0.699900
$339$	0	0
$340$	2.54368	0.137950
$341$	−20.1543	−1.09142
$342$	0	0
$343$	15.9849	0.863103
$344$	18.1660	0.979447
$345$	0	0
$346$	16.0803	0.864482
$347$	14.0097	0.752080	0.376040	−	0.926603i	$-0.377286\pi$
$347$	14.0097	0.752080	0.376040	+	0.926603i	$0.377286\pi$
$348$	0	0
$349$	−16.1144	−0.862584	−0.431292	−	0.902212i	$-0.641942\pi$
$349$	−16.1144	−0.862584	−0.431292	+	0.902212i	$0.641942\pi$
$350$	1.88939	0.100992
$351$	0	0
$352$	18.8232	1.00328
$353$	−2.24569	−0.119526	−0.0597631	−	0.998213i	$-0.519035\pi$
$353$	−2.24569	−0.119526	−0.0597631	+	0.998213i	$0.519035\pi$
$354$	0	0
$355$	−21.8037	−1.15722
$356$	4.58638	0.243078
$357$	0	0
$358$	−1.17134	−0.0619073
$359$	−1.32232	−0.0697896	−0.0348948	−	0.999391i	$-0.511110\pi$
$359$	−1.32232	−0.0697896	−0.0348948	+	0.999391i	$0.511110\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.452161	−0.0237650
$363$	0	0
$364$	1.76792	0.0926640
$365$	17.5571	0.918979
$366$	0	0
$367$	1.71125	0.0893264	0.0446632	−	0.999002i	$-0.485779\pi$
$367$	1.71125	0.0893264	0.0446632	+	0.999002i	$0.485779\pi$
$368$	−10.4629	−0.545414
$369$	0	0
$370$	−2.87877	−0.149660
$371$	10.9770	0.569897
$372$	0	0
$373$	34.6459	1.79390	0.896950	−	0.442133i	$-0.145778\pi$
$373$	34.6459	1.79390	0.896950	+	0.442133i	$0.145778\pi$
$374$	18.9041	0.977506
$375$	0	0
$376$	2.24887	0.115977
$377$	−16.3598	−0.842572
$378$	0	0
$379$	−13.8054	−0.709136	−0.354568	−	0.935030i	$-0.615372\pi$
$379$	−13.8054	−0.709136	−0.354568	+	0.935030i	$0.615372\pi$
$380$	1.22936	0.0630650
$381$	0	0
$382$	−5.94850	−0.304351
$383$	−9.01040	−0.460410	−0.230205	−	0.973142i	$-0.573940\pi$
$383$	−9.01040	−0.460410	−0.230205	+	0.973142i	$0.573940\pi$
$384$	0	0
$385$	−14.8683	−0.757759
$386$	−15.8370	−0.806082
$387$	0	0
$388$	10.8052	0.548553
$389$	−8.28237	−0.419933	−0.209966	−	0.977709i	$-0.567335\pi$
$389$	−8.28237	−0.419933	−0.209966	+	0.977709i	$0.567335\pi$
$390$	0	0
$391$	−4.46756	−0.225934
$392$	11.9533	0.603735
$393$	0	0
$394$	40.5916	2.04498
$395$	10.3032	0.518409
$396$	0	0
$397$	15.7665	0.791299	0.395650	−	0.918402i	$-0.370519\pi$
$397$	15.7665	0.791299	0.395650	+	0.918402i	$0.370519\pi$
$398$	38.4507	1.92736
$399$	0	0
$400$	4.36841	0.218421
$401$	−19.9926	−0.998383	−0.499192	−	0.866492i	$-0.666370\pi$
$401$	−19.9926	−0.998383	−0.499192	+	0.866492i	$0.666370\pi$
$402$	0	0
$403$	−7.98939	−0.397980
$404$	4.94084	0.245816
$405$	0	0
$406$	−15.2865	−0.758656
$407$	−4.98295	−0.246995
$408$	0	0
$409$	−6.91090	−0.341722	−0.170861	−	0.985295i	$-0.554655\pi$
$409$	−6.91090	−0.341722	−0.170861	+	0.985295i	$0.554655\pi$
$410$	33.1985	1.63956
$411$	0	0
$412$	6.03384	0.297266
$413$	−11.4993	−0.565844
$414$	0	0
$415$	15.4310	0.757477
$416$	7.46171	0.365841
$417$	0	0
$418$	9.13638	0.446875
$419$	20.5914	1.00596	0.502978	−	0.864299i	$-0.332238\pi$
$419$	20.5914	1.00596	0.502978	+	0.864299i	$0.332238\pi$
$420$	0	0
$421$	−8.43478	−0.411086	−0.205543	−	0.978648i	$-0.565896\pi$
$421$	−8.43478	−0.411086	−0.205543	+	0.978648i	$0.565896\pi$
$422$	−17.9456	−0.873580
$423$	0	0
$424$	19.0187	0.923630
$425$	1.86528	0.0904794
$426$	0	0
$427$	0.325879	0.0157704
$428$	−6.40787	−0.309736
$429$	0	0
$430$	26.4058	1.27340
$431$	−2.02392	−0.0974890	−0.0487445	−	0.998811i	$-0.515522\pi$
$431$	−2.02392	−0.0974890	−0.0487445	+	0.998811i	$0.515522\pi$
$432$	0	0
$433$	−11.1375	−0.535234	−0.267617	−	0.963525i	$-0.586236\pi$
$433$	−11.1375	−0.535234	−0.267617	+	0.963525i	$0.586236\pi$
$434$	−7.46523	−0.358343
$435$	0	0
$436$	7.22290	0.345914
$437$	−2.15918	−0.103288
$438$	0	0
$439$	25.3927	1.21193	0.605963	−	0.795493i	$-0.292788\pi$
$439$	25.3927	1.21193	0.605963	+	0.795493i	$0.292788\pi$
$440$	−25.7608	−1.22810
$441$	0	0
$442$	7.49377	0.356442
$443$	19.4492	0.924060	0.462030	−	0.886864i	$-0.347121\pi$
$443$	19.4492	0.924060	0.462030	+	0.886864i	$0.347121\pi$
$444$	0	0
$445$	−15.2903	−0.724828
$446$	−28.8611	−1.36661
$447$	0	0
$448$	−5.60714	−0.264912
$449$	−4.27088	−0.201555	−0.100778	−	0.994909i	$-0.532133\pi$
$449$	−4.27088	−0.201555	−0.100778	+	0.994909i	$0.532133\pi$
$450$	0	0
$451$	57.4642	2.70588
$452$	−6.31428	−0.296999
$453$	0	0
$454$	−4.77492	−0.224098
$455$	−5.89395	−0.276313
$456$	0	0
$457$	−35.1274	−1.64319	−0.821595	−	0.570072i	$-0.806915\pi$
$457$	−35.1274	−1.64319	−0.821595	+	0.570072i	$0.806915\pi$
$458$	−16.4563	−0.768951
$459$	0	0
$460$	−2.65442	−0.123763
$461$	−0.601971	−0.0280366	−0.0140183	−	0.999902i	$-0.504462\pi$
$461$	−0.601971	−0.0280366	−0.0140183	+	0.999902i	$0.504462\pi$
$462$	0	0
$463$	−6.88515	−0.319980	−0.159990	−	0.987119i	$-0.551146\pi$
$463$	−6.88515	−0.319980	−0.159990	+	0.987119i	$0.551146\pi$
$464$	−35.3436	−1.64079
$465$	0	0
$466$	−2.22964	−0.103286
$467$	21.4295	0.991638	0.495819	−	0.868426i	$-0.334868\pi$
$467$	21.4295	0.991638	0.495819	+	0.868426i	$0.334868\pi$
$468$	0	0
$469$	−8.08197	−0.373191
$470$	3.26892	0.150784
$471$	0	0
$472$	−19.9237	−0.917063
$473$	45.7065	2.10159
$474$	0	0
$475$	0.901494	0.0413634
$476$	1.63085	0.0747501
$477$	0	0
$478$	3.37877	0.154541
$479$	−7.84172	−0.358298	−0.179149	−	0.983822i	$-0.557334\pi$
$479$	−7.84172	−0.358298	−0.179149	+	0.983822i	$0.557334\pi$
$480$	0	0
$481$	−1.97529	−0.0900655
$482$	−40.4959	−1.84454
$483$	0	0
$484$	12.7619	0.580086
$485$	−36.0230	−1.63572
$486$	0	0
$487$	20.2916	0.919499	0.459750	−	0.888049i	$-0.347939\pi$
$487$	20.2916	0.919499	0.459750	+	0.888049i	$0.347939\pi$
$488$	0.564617	0.0255590
$489$	0	0
$490$	17.3752	0.784930
$491$	−40.1405	−1.81152	−0.905759	−	0.423794i	$-0.860698\pi$
$491$	−40.1405	−1.81152	−0.905759	+	0.423794i	$0.860698\pi$
$492$	0	0
$493$	−15.0915	−0.679685
$494$	3.62175	0.162950
$495$	0	0
$496$	−17.2602	−0.775007
$497$	−13.9793	−0.627056
$498$	0	0
$499$	23.5434	1.05395	0.526975	−	0.849881i	$-0.323326\pi$
$499$	23.5434	1.05395	0.526975	+	0.849881i	$0.323326\pi$
$500$	7.25508	0.324457
$501$	0	0
$502$	−22.1426	−0.988273
$503$	−34.9979	−1.56048	−0.780239	−	0.625481i	$-0.784903\pi$
$503$	−34.9979	−1.56048	−0.780239	+	0.625481i	$0.784903\pi$
$504$	0	0
$505$	−16.4720	−0.732994
$506$	−19.7271	−0.876977
$507$	0	0
$508$	−11.7761	−0.522480
$509$	26.1894	1.16083	0.580413	−	0.814322i	$-0.302891\pi$
$509$	26.1894	1.16083	0.580413	+	0.814322i	$0.302891\pi$
$510$	0	0
$511$	11.2566	0.497961
$512$	−5.67467	−0.250787
$513$	0	0
$514$	−20.7186	−0.913857
$515$	−20.1159	−0.886411
$516$	0	0
$517$	5.65826	0.248850
$518$	−1.84570	−0.0810954
$519$	0	0
$520$	−10.2119	−0.447820
$521$	33.6777	1.47545	0.737724	−	0.675102i	$-0.235901\pi$
$521$	33.6777	1.47545	0.737724	+	0.675102i	$0.235901\pi$
$522$	0	0
$523$	10.1938	0.445742	0.222871	−	0.974848i	$-0.428457\pi$
$523$	10.1938	0.445742	0.222871	+	0.974848i	$0.428457\pi$
$524$	−6.60730	−0.288641
$525$	0	0
$526$	−1.04648	−0.0456285
$527$	−7.36999	−0.321042
$528$	0	0
$529$	−18.3379	−0.797301
$530$	27.6453	1.20083
$531$	0	0
$532$	0.788195	0.0341726
$533$	22.7794	0.986685
$534$	0	0
$535$	21.3628	0.923596
$536$	−14.0028	−0.604830
$537$	0	0
$538$	−9.12569	−0.393436
$539$	30.0751	1.29543
$540$	0	0
$541$	−25.7666	−1.10779	−0.553897	−	0.832585i	$-0.686860\pi$
$541$	−25.7666	−1.10779	−0.553897	+	0.832585i	$0.686860\pi$
$542$	−46.6498	−2.00378
$543$	0	0
$544$	6.88323	0.295116
$545$	−24.0800	−1.03147
$546$	0	0
$547$	36.3119	1.55258	0.776291	−	0.630374i	$-0.217099\pi$
$547$	36.3119	1.55258	0.776291	+	0.630374i	$0.217099\pi$
$548$	7.68418	0.328252
$549$	0	0
$550$	8.23639	0.351201
$551$	−7.29374	−0.310724
$552$	0	0
$553$	6.60578	0.280906
$554$	−8.66440	−0.368115
$555$	0	0
$556$	−10.9513	−0.464440
$557$	−12.3573	−0.523595	−0.261797	−	0.965123i	$-0.584315\pi$
$557$	−12.3573	−0.523595	−0.261797	+	0.965123i	$0.584315\pi$
$558$	0	0
$559$	18.1185	0.766333
$560$	−12.7333	−0.538079
$561$	0	0
$562$	−26.9397	−1.13638
$563$	−30.7831	−1.29735	−0.648676	−	0.761065i	$-0.724677\pi$
$563$	−30.7831	−1.29735	−0.648676	+	0.761065i	$0.724677\pi$
$564$	0	0
$565$	21.0508	0.885614
$566$	−29.9097	−1.25720
$567$	0	0
$568$	−24.2205	−1.01627
$569$	7.28604	0.305447	0.152723	−	0.988269i	$-0.451196\pi$
$569$	7.28604	0.305447	0.152723	+	0.988269i	$0.451196\pi$
$570$	0	0
$571$	5.96886	0.249789	0.124894	−	0.992170i	$-0.460141\pi$
$571$	5.96886	0.249789	0.124894	+	0.992170i	$0.460141\pi$
$572$	7.70687	0.322240
$573$	0	0
$574$	21.2849	0.888416
$575$	−1.94649	−0.0811743
$576$	0	0
$577$	18.0277	0.750503	0.375252	−	0.926923i	$-0.377556\pi$
$577$	18.0277	0.750503	0.375252	+	0.926923i	$0.377556\pi$
$578$	−20.5371	−0.854230
$579$	0	0
$580$	−8.96666	−0.372320
$581$	9.89343	0.410449
$582$	0	0
$583$	47.8519	1.98182
$584$	19.5031	0.807044
$585$	0	0
$586$	17.4866	0.722366
$587$	27.7028	1.14341	0.571707	−	0.820458i	$-0.306281\pi$
$587$	27.7028	1.14341	0.571707	+	0.820458i	$0.306281\pi$
$588$	0	0
$589$	−3.56193	−0.146767
$590$	−28.9607	−1.19229
$591$	0	0
$592$	−4.26741	−0.175389
$593$	−29.5914	−1.21517	−0.607587	−	0.794253i	$-0.707862\pi$
$593$	−29.5914	−1.21517	−0.607587	+	0.794253i	$0.707862\pi$
$594$	0	0
$595$	−5.43701	−0.222896
$596$	4.22594	0.173101
$597$	0	0
$598$	−7.82003	−0.319785
$599$	−19.7717	−0.807850	−0.403925	−	0.914792i	$-0.632354\pi$
$599$	−19.7717	−0.807850	−0.403925	+	0.914792i	$0.632354\pi$
$600$	0	0
$601$	24.3203	0.992044	0.496022	−	0.868310i	$-0.334794\pi$
$601$	24.3203	0.992044	0.496022	+	0.868310i	$0.334794\pi$
$602$	16.9299	0.690009
$603$	0	0
$604$	5.32019	0.216475
$605$	−42.5462	−1.72975
$606$	0	0
$607$	28.0641	1.13909	0.569544	−	0.821961i	$-0.307120\pi$
$607$	28.0641	1.13909	0.569544	+	0.821961i	$0.307120\pi$
$608$	3.32668	0.134915
$609$	0	0
$610$	0.820717	0.0332299
$611$	2.24299	0.0907417
$612$	0	0
$613$	−25.7334	−1.03936	−0.519682	−	0.854360i	$-0.673949\pi$
$613$	−25.7334	−1.03936	−0.519682	+	0.854360i	$0.673949\pi$
$614$	−8.09780	−0.326801
$615$	0	0
$616$	−16.5163	−0.665462
$617$	21.1564	0.851725	0.425862	−	0.904788i	$-0.359971\pi$
$617$	21.1564	0.851725	0.425862	+	0.904788i	$0.359971\pi$
$618$	0	0
$619$	−42.1291	−1.69331	−0.846656	−	0.532140i	$-0.821388\pi$
$619$	−42.1291	−1.69331	−0.846656	+	0.532140i	$0.821388\pi$
$620$	−4.37891	−0.175861
$621$	0	0
$622$	26.7569	1.07285
$623$	−9.80321	−0.392757
$624$	0	0
$625$	−19.6798	−0.787193
$626$	−7.34169	−0.293433
$627$	0	0
$628$	5.70220	0.227543
$629$	−1.82215	−0.0726540
$630$	0	0
$631$	46.1768	1.83827	0.919134	−	0.393945i	$-0.128890\pi$
$631$	46.1768	1.83827	0.919134	+	0.393945i	$0.128890\pi$
$632$	11.4452	0.455265
$633$	0	0
$634$	−4.82561	−0.191649
$635$	39.2596	1.55797
$636$	0	0
$637$	11.9221	0.472370
$638$	−66.6383	−2.63824
$639$	0	0
$640$	−27.5910	−1.09063
$641$	17.7068	0.699375	0.349687	−	0.936866i	$-0.386288\pi$
$641$	17.7068	0.699375	0.349687	+	0.936866i	$0.386288\pi$
$642$	0	0
$643$	45.1799	1.78172	0.890861	−	0.454277i	$-0.150102\pi$
$643$	45.1799	1.78172	0.890861	+	0.454277i	$0.150102\pi$
$644$	−1.70186	−0.0670626
$645$	0	0
$646$	3.34097	0.131449
$647$	28.2089	1.10901	0.554503	−	0.832182i	$-0.312909\pi$
$647$	28.2089	1.10901	0.554503	+	0.832182i	$0.312909\pi$
$648$	0	0
$649$	−50.1289	−1.96773
$650$	3.26499	0.128063
$651$	0	0
$652$	14.7984	0.579550
$653$	−18.0898	−0.707910	−0.353955	−	0.935262i	$-0.615163\pi$
$653$	−18.0898	−0.707910	−0.353955	+	0.935262i	$0.615163\pi$
$654$	0	0
$655$	22.0277	0.860694
$656$	49.2125	1.92142
$657$	0	0
$658$	2.09584	0.0817042
$659$	8.87496	0.345719	0.172860	−	0.984946i	$-0.444699\pi$
$659$	8.87496	0.345719	0.172860	+	0.984946i	$0.444699\pi$
$660$	0	0
$661$	−8.88881	−0.345735	−0.172867	−	0.984945i	$-0.555303\pi$
$661$	−8.88881	−0.345735	−0.172867	+	0.984945i	$0.555303\pi$
$662$	11.7702	0.457463
$663$	0	0
$664$	17.1414	0.665214
$665$	−2.62772	−0.101899
$666$	0	0
$667$	15.7485	0.609785
$668$	11.1366	0.430887
$669$	0	0
$670$	−20.3543	−0.786353
$671$	1.42060	0.0548417
$672$	0	0
$673$	−14.4374	−0.556519	−0.278260	−	0.960506i	$-0.589758\pi$
$673$	−14.4374	−0.556519	−0.278260	+	0.960506i	$0.589758\pi$
$674$	40.4571	1.55835
$675$	0	0
$676$	−4.83917	−0.186122
$677$	−18.1303	−0.696802	−0.348401	−	0.937346i	$-0.613275\pi$
$677$	−18.1303	−0.696802	−0.348401	+	0.937346i	$0.613275\pi$
$678$	0	0
$679$	−23.0958	−0.886336
$680$	−9.42016	−0.361247
$681$	0	0
$682$	−32.5432	−1.24614
$683$	−20.3519	−0.778743	−0.389372	−	0.921081i	$-0.627308\pi$
$683$	−20.3519	−0.778743	−0.389372	+	0.921081i	$0.627308\pi$
$684$	0	0
$685$	−25.6178	−0.978807
$686$	25.8108	0.985460
$687$	0	0
$688$	39.1432	1.49232
$689$	18.9690	0.722661
$690$	0	0
$691$	−21.2883	−0.809845	−0.404923	−	0.914351i	$-0.632702\pi$
$691$	−21.2883	−0.809845	−0.404923	+	0.914351i	$0.632702\pi$
$692$	6.04743	0.229889
$693$	0	0
$694$	22.6214	0.858698
$695$	36.5100	1.38490
$696$	0	0
$697$	21.0134	0.795938
$698$	−26.0199	−0.984867
$699$	0	0
$700$	0.710554	0.0268564
$701$	30.6149	1.15631	0.578154	−	0.815928i	$-0.303773\pi$
$701$	30.6149	1.15631	0.578154	+	0.815928i	$0.303773\pi$
$702$	0	0
$703$	−0.880650	−0.0332144
$704$	−24.4432	−0.921237
$705$	0	0
$706$	−3.62611	−0.136471
$707$	−10.5609	−0.397182
$708$	0	0
$709$	18.2628	0.685875	0.342938	−	0.939358i	$-0.388578\pi$
$709$	18.2628	0.685875	0.342938	+	0.939358i	$0.388578\pi$
$710$	−35.2064	−1.32127
$711$	0	0
$712$	−16.9850	−0.636541
$713$	7.69086	0.288025
$714$	0	0
$715$	−25.6935	−0.960882
$716$	−0.440514	−0.0164628
$717$	0	0
$718$	−2.13515	−0.0796832
$719$	46.5020	1.73423	0.867117	−	0.498105i	$-0.165971\pi$
$719$	46.5020	1.73423	0.867117	+	0.498105i	$0.165971\pi$
$720$	0	0
$721$	−12.8971	−0.480313
$722$	1.61470	0.0600928
$723$	0	0
$724$	−0.170047	−0.00631975
$725$	−6.57526	−0.244199
$726$	0	0
$727$	52.4483	1.94520	0.972600	−	0.232485i	$-0.0746856\pi$
$727$	52.4483	1.94520	0.972600	+	0.232485i	$0.0746856\pi$
$728$	−6.54724	−0.242657
$729$	0	0
$730$	28.3494	1.04926
$731$	16.7139	0.618185
$732$	0	0
$733$	−15.0502	−0.555890	−0.277945	−	0.960597i	$-0.589653\pi$
$733$	−15.0502	−0.555890	−0.277945	+	0.960597i	$0.589653\pi$
$734$	2.76315	0.101990
$735$	0	0
$736$	−7.18291	−0.264765
$737$	−35.2317	−1.29778
$738$	0	0
$739$	38.0704	1.40044	0.700221	−	0.713926i	$-0.253085\pi$
$739$	38.0704	1.40044	0.700221	+	0.713926i	$0.253085\pi$
$740$	−1.08264	−0.0397986
$741$	0	0
$742$	17.7245	0.650687
$743$	−50.3952	−1.84882	−0.924410	−	0.381400i	$-0.875442\pi$
$743$	−50.3952	−1.84882	−0.924410	+	0.381400i	$0.875442\pi$
$744$	0	0
$745$	−14.0886	−0.516167
$746$	55.9427	2.04821
$747$	0	0
$748$	7.10938	0.259945
$749$	13.6966	0.500463
$750$	0	0
$751$	−50.5340	−1.84401	−0.922005	−	0.387178i	$-0.873450\pi$
$751$	−50.5340	−1.84401	−0.922005	+	0.387178i	$0.873450\pi$
$752$	4.84575	0.176706
$753$	0	0
$754$	−26.4161	−0.962019
$755$	−17.7367	−0.645504
$756$	0	0
$757$	34.5273	1.25492	0.627458	−	0.778650i	$-0.284095\pi$
$757$	34.5273	1.25492	0.627458	+	0.778650i	$0.284095\pi$
$758$	−22.2916	−0.809666
$759$	0	0
$760$	−4.55279	−0.165147
$761$	−3.56856	−0.129360	−0.0646801	−	0.997906i	$-0.520603\pi$
$761$	−3.56856	−0.129360	−0.0646801	+	0.997906i	$0.520603\pi$
$762$	0	0
$763$	−15.4387	−0.558918
$764$	−2.23709	−0.0809351
$765$	0	0
$766$	−14.5491	−0.525679
$767$	−19.8716	−0.717523
$768$	0	0
$769$	44.0828	1.58967	0.794833	−	0.606829i	$-0.207559\pi$
$769$	44.0828	1.58967	0.794833	+	0.606829i	$0.207559\pi$
$770$	−24.0078	−0.865182
$771$	0	0
$772$	−5.95592	−0.214358
$773$	−24.3177	−0.874646	−0.437323	−	0.899304i	$-0.644073\pi$
$773$	−24.3177	−0.874646	−0.437323	+	0.899304i	$0.644073\pi$
$774$	0	0
$775$	−3.21106	−0.115345
$776$	−40.0158	−1.43648
$777$	0	0
$778$	−13.3735	−0.479464
$779$	10.1558	0.363870
$780$	0	0
$781$	−60.9398	−2.18060
$782$	−7.21376	−0.257964
$783$	0	0
$784$	25.7564	0.919872
$785$	−19.0102	−0.678504
$786$	0	0
$787$	37.8348	1.34867	0.674333	−	0.738428i	$-0.264431\pi$
$787$	37.8348	1.34867	0.674333	+	0.738428i	$0.264431\pi$
$788$	15.2656	0.543813
$789$	0	0
$790$	16.6365	0.591900
$791$	13.4965	0.479882
$792$	0	0
$793$	0.563141	0.0199977
$794$	25.4582	0.903477
$795$	0	0
$796$	14.4604	0.512535
$797$	3.39172	0.120141	0.0600705	−	0.998194i	$-0.480867\pi$
$797$	3.39172	0.120141	0.0600705	+	0.998194i	$0.480867\pi$
$798$	0	0
$799$	2.06910	0.0731995
$800$	2.99898	0.106030
$801$	0	0
$802$	−32.2820	−1.13992
$803$	49.0707	1.73167
$804$	0	0
$805$	5.67372	0.199973
$806$	−12.9004	−0.454399
$807$	0	0
$808$	−18.2978	−0.643713
$809$	−55.7004	−1.95832	−0.979162	−	0.203083i	$-0.934904\pi$
$809$	−55.7004	−1.95832	−0.979162	+	0.203083i	$0.934904\pi$
$810$	0	0
$811$	−18.3284	−0.643596	−0.321798	−	0.946808i	$-0.604287\pi$
$811$	−18.3284	−0.643596	−0.321798	+	0.946808i	$0.604287\pi$
$812$	−5.74889	−0.201746
$813$	0	0
$814$	−8.04595	−0.282011
$815$	−49.3356	−1.72815
$816$	0	0
$817$	8.07785	0.282608
$818$	−11.1590	−0.390166
$819$	0	0
$820$	12.4852	0.436002
$821$	1.17574	0.0410335	0.0205168	−	0.999790i	$-0.493469\pi$
$821$	1.17574	0.0410335	0.0205168	+	0.999790i	$0.493469\pi$
$822$	0	0
$823$	44.2302	1.54177	0.770884	−	0.636976i	$-0.219815\pi$
$823$	44.2302	1.54177	0.770884	+	0.636976i	$0.219815\pi$
$824$	−22.3455	−0.778443
$825$	0	0
$826$	−18.5679	−0.646060
$827$	−31.7564	−1.10428	−0.552139	−	0.833752i	$-0.686188\pi$
$827$	−31.7564	−1.10428	−0.552139	+	0.833752i	$0.686188\pi$
$828$	0	0
$829$	−20.4542	−0.710403	−0.355202	−	0.934790i	$-0.615588\pi$
$829$	−20.4542	−0.710403	−0.355202	+	0.934790i	$0.615588\pi$
$830$	24.9164	0.864860
$831$	0	0
$832$	−9.68953	−0.335924
$833$	10.9978	0.381051
$834$	0	0
$835$	−37.1276	−1.28485
$836$	3.43598	0.118836
$837$	0	0
$838$	33.2489	1.14856
$839$	−22.3698	−0.772292	−0.386146	−	0.922438i	$-0.626194\pi$
$839$	−22.3698	−0.772292	−0.386146	+	0.922438i	$0.626194\pi$
$840$	0	0
$841$	24.1986	0.834435
$842$	−13.6196	−0.469363
$843$	0	0
$844$	−6.74894	−0.232308
$845$	16.1330	0.554993
$846$	0	0
$847$	−27.2781	−0.937286
$848$	40.9805	1.40728
$849$	0	0
$850$	3.01187	0.103306
$851$	1.90148	0.0651820
$852$	0	0
$853$	−3.78841	−0.129713	−0.0648564	−	0.997895i	$-0.520659\pi$
$853$	−3.78841	−0.129713	−0.0648564	+	0.997895i	$0.520659\pi$
$854$	0.526196	0.0180060
$855$	0	0
$856$	23.7307	0.811099
$857$	2.30816	0.0788453	0.0394226	−	0.999223i	$-0.487448\pi$
$857$	2.30816	0.0788453	0.0394226	+	0.999223i	$0.487448\pi$
$858$	0	0
$859$	42.7754	1.45948	0.729739	−	0.683726i	$-0.239642\pi$
$859$	42.7754	1.45948	0.729739	+	0.683726i	$0.239642\pi$
$860$	9.93061	0.338631
$861$	0	0
$862$	−3.26803	−0.111309
$863$	−17.6133	−0.599565	−0.299783	−	0.954008i	$-0.596914\pi$
$863$	−17.6133	−0.599565	−0.299783	+	0.954008i	$0.596914\pi$
$864$	0	0
$865$	−20.1612	−0.685500
$866$	−17.9837	−0.611111
$867$	0	0
$868$	−2.80750	−0.0952927
$869$	28.7966	0.976857
$870$	0	0
$871$	−13.9662	−0.473227
$872$	−26.7490	−0.905837
$873$	0	0
$874$	−3.48643	−0.117930
$875$	−15.5075	−0.524248
$876$	0	0
$877$	14.3313	0.483933	0.241966	−	0.970285i	$-0.422208\pi$
$877$	14.3313	0.483933	0.241966	+	0.970285i	$0.422208\pi$
$878$	41.0015	1.38373
$879$	0	0
$880$	−55.5081	−1.87118
$881$	41.1302	1.38571	0.692856	−	0.721076i	$-0.256352\pi$
$881$	41.1302	1.38571	0.692856	+	0.721076i	$0.256352\pi$
$882$	0	0
$883$	3.39173	0.114141	0.0570704	−	0.998370i	$-0.481824\pi$
$883$	3.39173	0.114141	0.0570704	+	0.998370i	$0.481824\pi$
$884$	2.81823	0.0947874
$885$	0	0
$886$	31.4046	1.05506
$887$	−39.6614	−1.33170	−0.665850	−	0.746085i	$-0.731931\pi$
$887$	−39.6614	−1.33170	−0.665850	+	0.746085i	$0.731931\pi$
$888$	0	0
$889$	25.1710	0.844207
$890$	−24.6892	−0.827583
$891$	0	0
$892$	−10.8540	−0.363418
$893$	1.00000	0.0334637
$894$	0	0
$895$	1.46860	0.0490900
$896$	−17.6897	−0.590972
$897$	0	0
$898$	−6.89619	−0.230129
$899$	25.9798	0.866475
$900$	0	0
$901$	17.4984	0.582956
$902$	92.7873	3.08948
$903$	0	0
$904$	23.3841	0.777743
$905$	0.566910	0.0188447
$906$	0	0
$907$	−10.5417	−0.350030	−0.175015	−	0.984566i	$-0.555997\pi$
$907$	−10.5417	−0.350030	−0.175015	+	0.984566i	$0.555997\pi$
$908$	−1.79574	−0.0595936
$909$	0	0
$910$	−9.51696	−0.315484
$911$	−7.61354	−0.252248	−0.126124	−	0.992014i	$-0.540254\pi$
$911$	−7.61354	−0.252248	−0.126124	+	0.992014i	$0.540254\pi$
$912$	0	0
$913$	43.1284	1.42734
$914$	−56.7201	−1.87613
$915$	0	0
$916$	−6.18881	−0.204484
$917$	14.1229	0.466378
$918$	0	0
$919$	−43.8345	−1.44597	−0.722983	−	0.690866i	$-0.757229\pi$
$919$	−43.8345	−1.44597	−0.722983	+	0.690866i	$0.757229\pi$
$920$	9.83029	0.324095
$921$	0	0
$922$	−0.972002	−0.0320112
$923$	−24.1572	−0.795142
$924$	0	0
$925$	−0.793901	−0.0261033
$926$	−11.1174	−0.365341
$927$	0	0
$928$	−24.2639	−0.796502
$929$	−28.7342	−0.942737	−0.471368	−	0.881936i	$-0.656240\pi$
$929$	−28.7342	−0.942737	−0.471368	+	0.881936i	$0.656240\pi$
$930$	0	0
$931$	5.31526	0.174201
$932$	−0.838517	−0.0274665
$933$	0	0
$934$	34.6021	1.13222
$935$	−23.7015	−0.775124
$936$	0	0
$937$	40.8205	1.33355	0.666773	−	0.745261i	$-0.267675\pi$
$937$	40.8205	1.33355	0.666773	+	0.745261i	$0.267675\pi$
$938$	−13.0499	−0.426096
$939$	0	0
$940$	1.22936	0.0400974
$941$	−18.7218	−0.610313	−0.305156	−	0.952302i	$-0.598709\pi$
$941$	−18.7218	−0.610313	−0.305156	+	0.952302i	$0.598709\pi$
$942$	0	0
$943$	−21.9282	−0.714082
$944$	−42.9305	−1.39727
$945$	0	0
$946$	73.8023	2.39952
$947$	−16.1113	−0.523547	−0.261773	−	0.965129i	$-0.584307\pi$
$947$	−16.1113	−0.523547	−0.261773	+	0.965129i	$0.584307\pi$
$948$	0	0
$949$	19.4521	0.631443
$950$	1.45564	0.0472272
$951$	0	0
$952$	−6.03965	−0.195746
$953$	22.3285	0.723291	0.361645	−	0.932316i	$-0.382215\pi$
$953$	22.3285	0.723291	0.361645	+	0.932316i	$0.382215\pi$
$954$	0	0
$955$	7.45811	0.241339
$956$	1.27068	0.0410966
$957$	0	0
$958$	−12.6620	−0.409091
$959$	−16.4246	−0.530379
$960$	0	0
$961$	−18.3126	−0.590730
$962$	−3.18950	−0.102834
$963$	0	0
$964$	−15.2296	−0.490511
$965$	19.8561	0.639191
$966$	0	0
$967$	44.6654	1.43634	0.718171	−	0.695867i	$-0.244980\pi$
$967$	44.6654	1.43634	0.718171	+	0.695867i	$0.244980\pi$
$968$	−47.2620	−1.51906
$969$	0	0
$970$	−58.1662	−1.86761
$971$	−29.1722	−0.936180	−0.468090	−	0.883681i	$-0.655058\pi$
$971$	−29.1722	−0.936180	−0.468090	+	0.883681i	$0.655058\pi$
$972$	0	0
$973$	23.4080	0.750428
$974$	32.7648	1.04985
$975$	0	0
$976$	1.21661	0.0389426
$977$	48.6586	1.55673	0.778363	−	0.627815i	$-0.216051\pi$
$977$	48.6586	1.55673	0.778363	+	0.627815i	$0.216051\pi$
$978$	0	0
$979$	−42.7351	−1.36582
$980$	6.53439	0.208733
$981$	0	0
$982$	−64.8149	−2.06833
$983$	−12.1825	−0.388561	−0.194280	−	0.980946i	$-0.562237\pi$
$983$	−12.1825	−0.388561	−0.194280	+	0.980946i	$0.562237\pi$
$984$	0	0
$985$	−50.8930	−1.62158
$986$	−24.3682	−0.776040
$987$	0	0
$988$	1.36206	0.0433328
$989$	−17.4415	−0.554609
$990$	0	0
$991$	58.0138	1.84287	0.921434	−	0.388534i	$-0.127018\pi$
$991$	58.0138	1.84287	0.921434	+	0.388534i	$0.127018\pi$
$992$	−11.8494	−0.376219
$993$	0	0
$994$	−22.5723	−0.715950
$995$	−48.2087	−1.52832
$996$	0	0
$997$	−57.4558	−1.81964	−0.909822	−	0.414998i	$-0.863782\pi$
$997$	−57.4558	−1.81964	−0.909822	+	0.414998i	$0.863782\pi$
$998$	38.0156	1.20336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.15		18
3.2	odd	2		893.2.a.c.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.4	✓	18	3.2	odd	2
8037.2.a.o.1.15		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)

\(f(q)\)	\(=\)	\(q+1.61470 q^{2} +0.607250 q^{4} -2.02448 q^{5} -1.29797 q^{7} -2.24887 q^{8} +O(q^{10})\)	\(q+1.61470 q^{2} +0.607250 q^{4} -2.02448 q^{5} -1.29797 q^{7} -2.24887 q^{8} -3.26892 q^{10} -5.65826 q^{11} -2.24299 q^{13} -2.09584 q^{14} -4.84575 q^{16} -2.06910 q^{17} -1.00000 q^{19} -1.22936 q^{20} -9.13638 q^{22} +2.15918 q^{23} -0.901494 q^{25} -3.62175 q^{26} -0.788195 q^{28} +7.29374 q^{29} +3.56193 q^{31} -3.32668 q^{32} -3.34097 q^{34} +2.62772 q^{35} +0.880650 q^{37} -1.61470 q^{38} +4.55279 q^{40} -10.1558 q^{41} -8.07785 q^{43} -3.43598 q^{44} +3.48643 q^{46} -1.00000 q^{47} -5.31526 q^{49} -1.45564 q^{50} -1.36206 q^{52} -8.45701 q^{53} +11.4550 q^{55} +2.91898 q^{56} +11.7772 q^{58} +8.85943 q^{59} -0.251067 q^{61} +5.75145 q^{62} +4.31991 q^{64} +4.54088 q^{65} +6.22660 q^{67} -1.25646 q^{68} +4.24297 q^{70} +10.7701 q^{71} -8.67240 q^{73} +1.42198 q^{74} -0.607250 q^{76} +7.34428 q^{77} -5.08930 q^{79} +9.81010 q^{80} -16.3986 q^{82} -7.62221 q^{83} +4.18884 q^{85} -13.0433 q^{86} +12.7247 q^{88} +7.55270 q^{89} +2.91135 q^{91} +1.31116 q^{92} -1.61470 q^{94} +2.02448 q^{95} +17.7937 q^{97} -8.58254 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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