LMFDB - Embedded newform 8037.2.a.t.1.19

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

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+1.66864 q^{2} +0.784364 q^{4} -2.66351 q^{5} +3.88200 q^{7} -2.02846 q^{8} +O(q^{10})$	\(q+1.66864 q^{2} +0.784364 q^{4} -2.66351 q^{5} +3.88200 q^{7} -2.02846 q^{8} -4.44445 q^{10} +6.21064 q^{11} -4.77724 q^{13} +6.47767 q^{14} -4.95350 q^{16} +7.42923 q^{17} +1.00000 q^{19} -2.08916 q^{20} +10.3633 q^{22} -2.48164 q^{23} +2.09430 q^{25} -7.97150 q^{26} +3.04490 q^{28} -0.647838 q^{29} +4.18007 q^{31} -4.20870 q^{32} +12.3967 q^{34} -10.3398 q^{35} -1.76657 q^{37} +1.66864 q^{38} +5.40283 q^{40} -4.39572 q^{41} +9.69380 q^{43} +4.87141 q^{44} -4.14097 q^{46} +1.00000 q^{47} +8.06995 q^{49} +3.49463 q^{50} -3.74709 q^{52} -6.27665 q^{53} -16.5421 q^{55} -7.87449 q^{56} -1.08101 q^{58} +4.56857 q^{59} -12.2181 q^{61} +6.97503 q^{62} +2.88420 q^{64} +12.7242 q^{65} +7.64533 q^{67} +5.82722 q^{68} -17.2534 q^{70} -14.2690 q^{71} +16.9109 q^{73} -2.94777 q^{74} +0.784364 q^{76} +24.1097 q^{77} +14.1330 q^{79} +13.1937 q^{80} -7.33488 q^{82} +11.1062 q^{83} -19.7878 q^{85} +16.1755 q^{86} -12.5980 q^{88} -12.9855 q^{89} -18.5453 q^{91} -1.94651 q^{92} +1.66864 q^{94} -2.66351 q^{95} +11.3485 q^{97} +13.4659 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) $ 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8	$ 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) $ 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8 + 7 * q^10 + 3 * q^11 + 11 * q^13 - 9 * q^14 + 40 * q^16 + 6 * q^17 + 24 * q^19 + 17 * q^20 + 15 * q^22 + 19 * q^23 + 54 * q^25 + q^26 + 26 * q^28 - 32 * q^29 + 12 * q^31 - 30 * q^32 + 38 * q^34 + 35 * q^35 + 14 * q^37 - 2 * q^38 + 47 * q^40 - 30 * q^41 + 10 * q^43 - q^44 - 3 * q^46 + 24 * q^47 + 56 * q^49 - 44 * q^50 - 10 * q^53 + 24 * q^55 + 6 * q^56 + 41 * q^58 - 11 * q^59 + 10 * q^61 - 2 * q^62 + 61 * q^64 - 35 * q^65 + 12 * q^67 + 22 * q^68 - 20 * q^70 - 44 * q^71 + 33 * q^73 - 36 * q^74 + 32 * q^76 + 23 * q^77 + 50 * q^79 + 13 * q^80 + 5 * q^82 + 72 * q^83 + 33 * q^85 - 44 * q^86 + 38 * q^88 - 48 * q^89 + 47 * q^91 + 29 * q^92 - 2 * q^94 + 2 * q^95 + 50 * q^97 - 22 * 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 $

$2$	1.66864	1.17991	0.589954	−	0.807437i	$-0.299146\pi$
$2$	1.66864	1.17991	0.589954	+	0.807437i	$0.299146\pi$
$3$	0	0
$3$	0	0
$4$	0.784364	0.392182
$5$	−2.66351	−1.19116	−0.595579	−	0.803296i	$-0.703077\pi$
$5$	−2.66351	−1.19116	−0.595579	+	0.803296i	$0.703077\pi$
$6$	0	0
$7$	3.88200	1.46726	0.733630	−	0.679549i	$-0.237825\pi$
$7$	3.88200	1.46726	0.733630	+	0.679549i	$0.237825\pi$
$8$	−2.02846	−0.717169
$9$	0	0
$10$	−4.44445	−1.40546
$11$	6.21064	1.87258	0.936290	−	0.351229i	$-0.114236\pi$
$11$	6.21064	1.87258	0.936290	+	0.351229i	$0.114236\pi$
$12$	0	0
$13$	−4.77724	−1.32497	−0.662484	−	0.749076i	$-0.730498\pi$
$13$	−4.77724	−1.32497	−0.662484	+	0.749076i	$0.730498\pi$
$14$	6.47767	1.73123
$15$	0	0
$16$	−4.95350	−1.23838
$17$	7.42923	1.80185	0.900927	−	0.433971i	$-0.142888\pi$
$17$	7.42923	1.80185	0.900927	+	0.433971i	$0.142888\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.08916	−0.467151
$21$	0	0
$22$	10.3633	2.20947
$23$	−2.48164	−0.517458	−0.258729	−	0.965950i	$-0.583304\pi$
$23$	−2.48164	−0.517458	−0.258729	+	0.965950i	$0.583304\pi$
$24$	0	0
$25$	2.09430	0.418859
$26$	−7.97150	−1.56334
$27$	0	0
$28$	3.04490	0.575433
$29$	−0.647838	−0.120300	−0.0601502	−	0.998189i	$-0.519158\pi$
$29$	−0.647838	−0.120300	−0.0601502	+	0.998189i	$0.519158\pi$
$30$	0	0
$31$	4.18007	0.750762	0.375381	−	0.926871i	$-0.377512\pi$
$31$	4.18007	0.750762	0.375381	+	0.926871i	$0.377512\pi$
$32$	−4.20870	−0.743999
$33$	0	0
$34$	12.3967	2.12602
$35$	−10.3398	−1.74774
$36$	0	0
$37$	−1.76657	−0.290422	−0.145211	−	0.989401i	$-0.546386\pi$
$37$	−1.76657	−0.290422	−0.145211	+	0.989401i	$0.546386\pi$
$38$	1.66864	0.270689
$39$	0	0
$40$	5.40283	0.854262
$41$	−4.39572	−0.686496	−0.343248	−	0.939245i	$-0.611527\pi$
$41$	−4.39572	−0.686496	−0.343248	+	0.939245i	$0.611527\pi$
$42$	0	0
$43$	9.69380	1.47829	0.739145	−	0.673546i	$-0.235230\pi$
$43$	9.69380	1.47829	0.739145	+	0.673546i	$0.235230\pi$
$44$	4.87141	0.734392
$45$	0	0
$46$	−4.14097	−0.610553
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	8.06995	1.15285
$50$	3.49463	0.494215
$51$	0	0
$52$	−3.74709	−0.519628
$53$	−6.27665	−0.862164	−0.431082	−	0.902313i	$-0.641868\pi$
$53$	−6.27665	−0.862164	−0.431082	+	0.902313i	$0.641868\pi$
$54$	0	0
$55$	−16.5421	−2.23054
$56$	−7.87449	−1.05227
$57$	0	0
$58$	−1.08101	−0.141943
$59$	4.56857	0.594777	0.297388	−	0.954757i	$-0.403884\pi$
$59$	4.56857	0.594777	0.297388	+	0.954757i	$0.403884\pi$
$60$	0	0
$61$	−12.2181	−1.56437	−0.782186	−	0.623045i	$-0.785895\pi$
$61$	−12.2181	−1.56437	−0.782186	+	0.623045i	$0.785895\pi$
$62$	6.97503	0.885830
$63$	0	0
$64$	2.88420	0.360525
$65$	12.7242	1.57825
$66$	0	0
$67$	7.64533	0.934026	0.467013	−	0.884251i	$-0.345330\pi$
$67$	7.64533	0.934026	0.467013	+	0.884251i	$0.345330\pi$
$68$	5.82722	0.706655
$69$	0	0
$70$	−17.2534	−2.06217
$71$	−14.2690	−1.69341	−0.846707	−	0.532060i	$-0.821418\pi$
$71$	−14.2690	−1.69341	−0.846707	+	0.532060i	$0.821418\pi$
$72$	0	0
$73$	16.9109	1.97928	0.989638	−	0.143587i	$-0.0458637\pi$
$73$	16.9109	1.97928	0.989638	+	0.143587i	$0.0458637\pi$
$74$	−2.94777	−0.342671
$75$	0	0
$76$	0.784364	0.0899727
$77$	24.1097	2.74756
$78$	0	0
$79$	14.1330	1.59008	0.795042	−	0.606555i	$-0.207449\pi$
$79$	14.1330	1.59008	0.795042	+	0.606555i	$0.207449\pi$
$80$	13.1937	1.47510
$81$	0	0
$82$	−7.33488	−0.810002
$83$	11.1062	1.21906	0.609531	−	0.792762i	$-0.291358\pi$
$83$	11.1062	1.21906	0.609531	+	0.792762i	$0.291358\pi$
$84$	0	0
$85$	−19.7878	−2.14629
$86$	16.1755	1.74425
$87$	0	0
$88$	−12.5980	−1.34296
$89$	−12.9855	−1.37646	−0.688231	−	0.725491i	$-0.741613\pi$
$89$	−12.9855	−1.37646	−0.688231	+	0.725491i	$0.741613\pi$
$90$	0	0
$91$	−18.5453	−1.94407
$92$	−1.94651	−0.202938
$93$	0	0
$94$	1.66864	0.172107
$95$	−2.66351	−0.273271
$96$	0	0
$97$	11.3485	1.15226	0.576132	−	0.817357i	$-0.304561\pi$
$97$	11.3485	1.15226	0.576132	+	0.817357i	$0.304561\pi$
$98$	13.4659	1.36026
$99$	0	0
$100$	1.64269	0.164269
$101$	1.84886	0.183969	0.0919844	−	0.995760i	$-0.470679\pi$
$101$	1.84886	0.183969	0.0919844	+	0.995760i	$0.470679\pi$
$102$	0	0
$103$	−3.56004	−0.350781	−0.175390	−	0.984499i	$-0.556119\pi$
$103$	−3.56004	−0.350781	−0.175390	+	0.984499i	$0.556119\pi$
$104$	9.69044	0.950226
$105$	0	0
$106$	−10.4735	−1.01727
$107$	−8.42408	−0.814387	−0.407193	−	0.913342i	$-0.633492\pi$
$107$	−8.42408	−0.814387	−0.407193	+	0.913342i	$0.633492\pi$
$108$	0	0
$109$	7.12731	0.682673	0.341336	−	0.939941i	$-0.389121\pi$
$109$	7.12731	0.682673	0.341336	+	0.939941i	$0.389121\pi$
$110$	−27.6029	−2.63183
$111$	0	0
$112$	−19.2295	−1.81702
$113$	−15.8697	−1.49290	−0.746448	−	0.665443i	$-0.768242\pi$
$113$	−15.8697	−1.49290	−0.746448	+	0.665443i	$0.768242\pi$
$114$	0	0
$115$	6.60989	0.616375
$116$	−0.508141	−0.0471797
$117$	0	0
$118$	7.62330	0.701781
$119$	28.8403	2.64379
$120$	0	0
$121$	27.5721	2.50655
$122$	−20.3877	−1.84581
$123$	0	0
$124$	3.27869	0.294435
$125$	7.73938	0.692231
$126$	0	0
$127$	8.19876	0.727522	0.363761	−	0.931492i	$-0.381492\pi$
$127$	8.19876	0.727522	0.363761	+	0.931492i	$0.381492\pi$
$128$	13.2301	1.16939
$129$	0	0
$130$	21.2322	1.86218
$131$	4.90476	0.428531	0.214265	−	0.976775i	$-0.431264\pi$
$131$	4.90476	0.428531	0.214265	+	0.976775i	$0.431264\pi$
$132$	0	0
$133$	3.88200	0.336612
$134$	12.7573	1.10206
$135$	0	0
$136$	−15.0699	−1.29223
$137$	−10.5330	−0.899896	−0.449948	−	0.893055i	$-0.648558\pi$
$137$	−10.5330	−0.899896	−0.449948	+	0.893055i	$0.648558\pi$
$138$	0	0
$139$	−7.06659	−0.599380	−0.299690	−	0.954037i	$-0.596883\pi$
$139$	−7.06659	−0.599380	−0.299690	+	0.954037i	$0.596883\pi$
$140$	−8.11014	−0.685432
$141$	0	0
$142$	−23.8098	−1.99807
$143$	−29.6697	−2.48111
$144$	0	0
$145$	1.72552	0.143297
$146$	28.2183	2.33536
$147$	0	0
$148$	−1.38563	−0.113898
$149$	1.03938	0.0851491	0.0425746	−	0.999093i	$-0.486444\pi$
$149$	1.03938	0.0851491	0.0425746	+	0.999093i	$0.486444\pi$
$150$	0	0
$151$	6.25008	0.508624	0.254312	−	0.967122i	$-0.418151\pi$
$151$	6.25008	0.508624	0.254312	+	0.967122i	$0.418151\pi$
$152$	−2.02846	−0.164530
$153$	0	0
$154$	40.2305	3.24187
$155$	−11.1337	−0.894277
$156$	0	0
$157$	21.9470	1.75156	0.875781	−	0.482709i	$-0.160347\pi$
$157$	21.9470	1.75156	0.875781	+	0.482709i	$0.160347\pi$
$158$	23.5829	1.87615
$159$	0	0
$160$	11.2099	0.886221
$161$	−9.63375	−0.759246
$162$	0	0
$163$	20.2647	1.58725	0.793625	−	0.608407i	$-0.208191\pi$
$163$	20.2647	1.58725	0.793625	+	0.608407i	$0.208191\pi$
$164$	−3.44785	−0.269231
$165$	0	0
$166$	18.5322	1.43838
$167$	22.3816	1.73194	0.865970	−	0.500096i	$-0.166702\pi$
$167$	22.3816	1.73194	0.865970	+	0.500096i	$0.166702\pi$
$168$	0	0
$169$	9.82200	0.755538
$170$	−33.0188	−2.53243
$171$	0	0
$172$	7.60347	0.579759
$173$	−16.3531	−1.24330	−0.621651	−	0.783294i	$-0.713538\pi$
$173$	−16.3531	−1.24330	−0.621651	+	0.783294i	$0.713538\pi$
$174$	0	0
$175$	8.13006	0.614575
$176$	−30.7644	−2.31896
$177$	0	0
$178$	−21.6682	−1.62410
$179$	20.9253	1.56403	0.782016	−	0.623259i	$-0.214192\pi$
$179$	20.9253	1.56403	0.782016	+	0.623259i	$0.214192\pi$
$180$	0	0
$181$	5.25776	0.390806	0.195403	−	0.980723i	$-0.437398\pi$
$181$	5.25776	0.390806	0.195403	+	0.980723i	$0.437398\pi$
$182$	−30.9454	−2.29382
$183$	0	0
$184$	5.03391	0.371105
$185$	4.70528	0.345939
$186$	0	0
$187$	46.1403	3.37411
$188$	0.784364	0.0572056
$189$	0	0
$190$	−4.44445	−0.322434
$191$	4.17632	0.302188	0.151094	−	0.988519i	$-0.451720\pi$
$191$	4.17632	0.302188	0.151094	+	0.988519i	$0.451720\pi$
$192$	0	0
$193$	−8.45867	−0.608868	−0.304434	−	0.952533i	$-0.598467\pi$
$193$	−8.45867	−0.608868	−0.304434	+	0.952533i	$0.598467\pi$
$194$	18.9366	1.35957
$195$	0	0
$196$	6.32978	0.452127
$197$	2.55172	0.181803	0.0909013	−	0.995860i	$-0.471025\pi$
$197$	2.55172	0.181803	0.0909013	+	0.995860i	$0.471025\pi$
$198$	0	0
$199$	−14.2207	−1.00808	−0.504039	−	0.863681i	$-0.668153\pi$
$199$	−14.2207	−1.00808	−0.504039	+	0.863681i	$0.668153\pi$
$200$	−4.24820	−0.300393
$201$	0	0
$202$	3.08509	0.217066
$203$	−2.51491	−0.176512
$204$	0	0
$205$	11.7081	0.817726
$206$	−5.94042	−0.413889
$207$	0	0
$208$	23.6641	1.64081
$209$	6.21064	0.429599
$210$	0	0
$211$	11.7570	0.809387	0.404693	−	0.914452i	$-0.367378\pi$
$211$	11.7570	0.809387	0.404693	+	0.914452i	$0.367378\pi$
$212$	−4.92318	−0.338125
$213$	0	0
$214$	−14.0568	−0.960901
$215$	−25.8195	−1.76088
$216$	0	0
$217$	16.2270	1.10156
$218$	11.8929	0.805491
$219$	0	0
$220$	−12.9750	−0.874777
$221$	−35.4912	−2.38740
$222$	0	0
$223$	−25.6505	−1.71768	−0.858842	−	0.512240i	$-0.828816\pi$
$223$	−25.6505	−1.71768	−0.858842	+	0.512240i	$0.828816\pi$
$224$	−16.3382	−1.09164
$225$	0	0
$226$	−26.4809	−1.76148
$227$	−0.906672	−0.0601779	−0.0300890	−	0.999547i	$-0.509579\pi$
$227$	−0.906672	−0.0601779	−0.0300890	+	0.999547i	$0.509579\pi$
$228$	0	0
$229$	16.7752	1.10854	0.554269	−	0.832338i	$-0.312998\pi$
$229$	16.7752	1.10854	0.554269	+	0.832338i	$0.312998\pi$
$230$	11.0295	0.727266
$231$	0	0
$232$	1.31411	0.0862758
$233$	−12.7155	−0.833019	−0.416509	−	0.909131i	$-0.636747\pi$
$233$	−12.7155	−0.833019	−0.416509	+	0.909131i	$0.636747\pi$
$234$	0	0
$235$	−2.66351	−0.173748
$236$	3.58342	0.233261
$237$	0	0
$238$	48.1241	3.11942
$239$	17.2276	1.11436	0.557180	−	0.830392i	$-0.311883\pi$
$239$	17.2276	1.11436	0.557180	+	0.830392i	$0.311883\pi$
$240$	0	0
$241$	11.8506	0.763365	0.381683	−	0.924293i	$-0.375345\pi$
$241$	11.8506	0.763365	0.381683	+	0.924293i	$0.375345\pi$
$242$	46.0079	2.95750
$243$	0	0
$244$	−9.58347	−0.613519
$245$	−21.4944	−1.37323
$246$	0	0
$247$	−4.77724	−0.303968
$248$	−8.47910	−0.538423
$249$	0	0
$250$	12.9142	0.816769
$251$	26.8021	1.69173	0.845867	−	0.533394i	$-0.179084\pi$
$251$	26.8021	1.69173	0.845867	+	0.533394i	$0.179084\pi$
$252$	0	0
$253$	−15.4126	−0.968982
$254$	13.6808	0.858409
$255$	0	0
$256$	16.3079	1.01924
$257$	16.8696	1.05230	0.526148	−	0.850393i	$-0.323636\pi$
$257$	16.8696	1.05230	0.526148	+	0.850393i	$0.323636\pi$
$258$	0	0
$259$	−6.85782	−0.426124
$260$	9.98043	0.618960
$261$	0	0
$262$	8.18428	0.505627
$263$	−3.56019	−0.219531	−0.109765	−	0.993958i	$-0.535010\pi$
$263$	−3.56019	−0.219531	−0.109765	+	0.993958i	$0.535010\pi$
$264$	0	0
$265$	16.7179	1.02697
$266$	6.47767	0.397172
$267$	0	0
$268$	5.99672	0.366308
$269$	23.2472	1.41741	0.708704	−	0.705506i	$-0.249280\pi$
$269$	23.2472	1.41741	0.708704	+	0.705506i	$0.249280\pi$
$270$	0	0
$271$	−12.1545	−0.738334	−0.369167	−	0.929363i	$-0.620357\pi$
$271$	−12.1545	−0.738334	−0.369167	+	0.929363i	$0.620357\pi$
$272$	−36.8007	−2.23137
$273$	0	0
$274$	−17.5758	−1.06179
$275$	13.0069	0.784347
$276$	0	0
$277$	−14.4741	−0.869663	−0.434831	−	0.900512i	$-0.643192\pi$
$277$	−14.4741	−0.869663	−0.434831	+	0.900512i	$0.643192\pi$
$278$	−11.7916	−0.707213
$279$	0	0
$280$	20.9738	1.25342
$281$	3.91617	0.233619	0.116810	−	0.993154i	$-0.462733\pi$
$281$	3.91617	0.233619	0.116810	+	0.993154i	$0.462733\pi$
$282$	0	0
$283$	−15.7530	−0.936416	−0.468208	−	0.883618i	$-0.655100\pi$
$283$	−15.7530	−0.936416	−0.468208	+	0.883618i	$0.655100\pi$
$284$	−11.1921	−0.664126
$285$	0	0
$286$	−49.5081	−2.92748
$287$	−17.0642	−1.00727
$288$	0	0
$289$	38.1935	2.24668
$290$	2.87928	0.169077
$291$	0	0
$292$	13.2643	0.776236
$293$	18.1667	1.06131	0.530654	−	0.847589i	$-0.321947\pi$
$293$	18.1667	1.06131	0.530654	+	0.847589i	$0.321947\pi$
$294$	0	0
$295$	−12.1684	−0.708473
$296$	3.58341	0.208282
$297$	0	0
$298$	1.73435	0.100468
$299$	11.8554	0.685615
$300$	0	0
$301$	37.6314	2.16904
$302$	10.4291	0.600130
$303$	0	0
$304$	−4.95350	−0.284103
$305$	32.5432	1.86342
$306$	0	0
$307$	0.923446	0.0527039	0.0263519	−	0.999653i	$-0.491611\pi$
$307$	0.923446	0.0527039	0.0263519	+	0.999653i	$0.491611\pi$
$308$	18.9108	1.07754
$309$	0	0
$310$	−18.5781	−1.05516
$311$	−8.23178	−0.466781	−0.233391	−	0.972383i	$-0.574982\pi$
$311$	−8.23178	−0.466781	−0.233391	+	0.972383i	$0.574982\pi$
$312$	0	0
$313$	4.99970	0.282599	0.141300	−	0.989967i	$-0.454872\pi$
$313$	4.99970	0.282599	0.141300	+	0.989967i	$0.454872\pi$
$314$	36.6217	2.06668
$315$	0	0
$316$	11.0854	0.623602
$317$	24.0233	1.34928	0.674641	−	0.738146i	$-0.264299\pi$
$317$	24.0233	1.34928	0.674641	+	0.738146i	$0.264299\pi$
$318$	0	0
$319$	−4.02349	−0.225272
$320$	−7.68209	−0.429442
$321$	0	0
$322$	−16.0753	−0.895840
$323$	7.42923	0.413374
$324$	0	0
$325$	−10.0049	−0.554975
$326$	33.8144	1.87281
$327$	0	0
$328$	8.91654	0.492334
$329$	3.88200	0.214022
$330$	0	0
$331$	23.4063	1.28653	0.643264	−	0.765644i	$-0.277580\pi$
$331$	23.4063	1.28653	0.643264	+	0.765644i	$0.277580\pi$
$332$	8.71129	0.478094
$333$	0	0
$334$	37.3469	2.04353
$335$	−20.3634	−1.11257
$336$	0	0
$337$	−28.3767	−1.54578	−0.772889	−	0.634541i	$-0.781189\pi$
$337$	−28.3767	−1.54578	−0.772889	+	0.634541i	$0.781189\pi$
$338$	16.3894	0.891466
$339$	0	0
$340$	−15.5209	−0.841738
$341$	25.9609	1.40586
$342$	0	0
$343$	4.15355	0.224271
$344$	−19.6635	−1.06018
$345$	0	0
$346$	−27.2874	−1.46698
$347$	−7.95720	−0.427165	−0.213582	−	0.976925i	$-0.568513\pi$
$347$	−7.95720	−0.427165	−0.213582	+	0.976925i	$0.568513\pi$
$348$	0	0
$349$	22.2051	1.18861	0.594307	−	0.804238i	$-0.297427\pi$
$349$	22.2051	1.18861	0.594307	+	0.804238i	$0.297427\pi$
$350$	13.5662	0.725142
$351$	0	0
$352$	−26.1387	−1.39320
$353$	16.3431	0.869853	0.434927	−	0.900466i	$-0.356774\pi$
$353$	16.3431	0.869853	0.434927	+	0.900466i	$0.356774\pi$
$354$	0	0
$355$	38.0055	2.01712
$356$	−10.1854	−0.539824
$357$	0	0
$358$	34.9168	1.84541
$359$	−25.7949	−1.36140	−0.680701	−	0.732561i	$-0.738325\pi$
$359$	−25.7949	−1.36140	−0.680701	+	0.732561i	$0.738325\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.77332	0.461115
$363$	0	0
$364$	−14.5462	−0.762430
$365$	−45.0425	−2.35763
$366$	0	0
$367$	−14.2664	−0.744698	−0.372349	−	0.928093i	$-0.621448\pi$
$367$	−14.2664	−0.744698	−0.372349	+	0.928093i	$0.621448\pi$
$368$	12.2928	0.640808
$369$	0	0
$370$	7.85142	0.408176
$371$	−24.3660	−1.26502
$372$	0	0
$373$	17.8523	0.924359	0.462180	−	0.886786i	$-0.347068\pi$
$373$	17.8523	0.924359	0.462180	+	0.886786i	$0.347068\pi$
$374$	76.9916	3.98114
$375$	0	0
$376$	−2.02846	−0.104610
$377$	3.09488	0.159394
$378$	0	0
$379$	−6.14384	−0.315588	−0.157794	−	0.987472i	$-0.550438\pi$
$379$	−6.14384	−0.315588	−0.157794	+	0.987472i	$0.550438\pi$
$380$	−2.08916	−0.107172
$381$	0	0
$382$	6.96879	0.356554
$383$	−24.6919	−1.26170	−0.630848	−	0.775907i	$-0.717293\pi$
$383$	−24.6919	−1.26170	−0.630848	+	0.775907i	$0.717293\pi$
$384$	0	0
$385$	−64.2166	−3.27278
$386$	−14.1145	−0.718408
$387$	0	0
$388$	8.90134	0.451897
$389$	−35.4263	−1.79619	−0.898094	−	0.439805i	$-0.855048\pi$
$389$	−35.4263	−1.79619	−0.898094	+	0.439805i	$0.855048\pi$
$390$	0	0
$391$	−18.4367	−0.932384
$392$	−16.3696	−0.826788
$393$	0	0
$394$	4.25791	0.214510
$395$	−37.6433	−1.89404
$396$	0	0
$397$	1.28718	0.0646015	0.0323008	−	0.999478i	$-0.489717\pi$
$397$	1.28718	0.0646015	0.0323008	+	0.999478i	$0.489717\pi$
$398$	−23.7292	−1.18944
$399$	0	0
$400$	−10.3741	−0.518705
$401$	−31.3824	−1.56716	−0.783580	−	0.621291i	$-0.786608\pi$
$401$	−31.3824	−1.56716	−0.783580	+	0.621291i	$0.786608\pi$
$402$	0	0
$403$	−19.9692	−0.994735
$404$	1.45018	0.0721493
$405$	0	0
$406$	−4.19648	−0.208268
$407$	−10.9715	−0.543838
$408$	0	0
$409$	23.7232	1.17304	0.586518	−	0.809936i	$-0.300498\pi$
$409$	23.7232	1.17304	0.586518	+	0.809936i	$0.300498\pi$
$410$	19.5365	0.964841
$411$	0	0
$412$	−2.79236	−0.137570
$413$	17.7352	0.872692
$414$	0	0
$415$	−29.5814	−1.45210
$416$	20.1059	0.985775
$417$	0	0
$418$	10.3633	0.506887
$419$	11.5287	0.563213	0.281606	−	0.959530i	$-0.409133\pi$
$419$	11.5287	0.563213	0.281606	+	0.959530i	$0.409133\pi$
$420$	0	0
$421$	28.1119	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$421$	28.1119	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$422$	19.6183	0.955002
$423$	0	0
$424$	12.7319	0.618318
$425$	15.5590	0.754723
$426$	0	0
$427$	−47.4309	−2.29534
$428$	−6.60755	−0.319388
$429$	0	0
$430$	−43.0836	−2.07767
$431$	−3.77594	−0.181881	−0.0909403	−	0.995856i	$-0.528987\pi$
$431$	−3.77594	−0.181881	−0.0909403	+	0.995856i	$0.528987\pi$
$432$	0	0
$433$	−7.72158	−0.371076	−0.185538	−	0.982637i	$-0.559403\pi$
$433$	−7.72158	−0.371076	−0.185538	+	0.982637i	$0.559403\pi$
$434$	27.0771	1.29974
$435$	0	0
$436$	5.59041	0.267732
$437$	−2.48164	−0.118713
$438$	0	0
$439$	0.100681	0.00480524	0.00240262	−	0.999997i	$-0.499235\pi$
$439$	0.100681	0.00480524	0.00240262	+	0.999997i	$0.499235\pi$
$440$	33.5550	1.59967
$441$	0	0
$442$	−59.2221	−2.81691
$443$	21.1355	1.00418	0.502088	−	0.864817i	$-0.332565\pi$
$443$	21.1355	1.00418	0.502088	+	0.864817i	$0.332565\pi$
$444$	0	0
$445$	34.5871	1.63959
$446$	−42.8015	−2.02671
$447$	0	0
$448$	11.1965	0.528983
$449$	−20.6820	−0.976042	−0.488021	−	0.872832i	$-0.662281\pi$
$449$	−20.6820	−0.976042	−0.488021	+	0.872832i	$0.662281\pi$
$450$	0	0
$451$	−27.3003	−1.28552
$452$	−12.4476	−0.585487
$453$	0	0
$454$	−1.51291	−0.0710044
$455$	49.3955	2.31570
$456$	0	0
$457$	1.49859	0.0701013	0.0350507	−	0.999386i	$-0.488841\pi$
$457$	1.49859	0.0701013	0.0350507	+	0.999386i	$0.488841\pi$
$458$	27.9918	1.30797
$459$	0	0
$460$	5.18456	0.241731
$461$	5.30507	0.247082	0.123541	−	0.992339i	$-0.460575\pi$
$461$	5.30507	0.247082	0.123541	+	0.992339i	$0.460575\pi$
$462$	0	0
$463$	−28.3898	−1.31938	−0.659692	−	0.751536i	$-0.729313\pi$
$463$	−28.3898	−1.31938	−0.659692	+	0.751536i	$0.729313\pi$
$464$	3.20907	0.148977
$465$	0	0
$466$	−21.2176	−0.982885
$467$	−35.8335	−1.65818	−0.829089	−	0.559116i	$-0.811140\pi$
$467$	−35.8335	−1.65818	−0.829089	+	0.559116i	$0.811140\pi$
$468$	0	0
$469$	29.6792	1.37046
$470$	−4.44445	−0.205007
$471$	0	0
$472$	−9.26715	−0.426555
$473$	60.2047	2.76822
$474$	0	0
$475$	2.09430	0.0960929
$476$	22.6213	1.03685
$477$	0	0
$478$	28.7467	1.31484
$479$	−16.1601	−0.738372	−0.369186	−	0.929356i	$-0.620363\pi$
$479$	−16.1601	−0.738372	−0.369186	+	0.929356i	$0.620363\pi$
$480$	0	0
$481$	8.43932	0.384800
$482$	19.7744	0.900700
$483$	0	0
$484$	21.6266	0.983025
$485$	−30.2268	−1.37253
$486$	0	0
$487$	−27.1893	−1.23207	−0.616033	−	0.787720i	$-0.711261\pi$
$487$	−27.1893	−1.23207	−0.616033	+	0.787720i	$0.711261\pi$
$488$	24.7840	1.12192
$489$	0	0
$490$	−35.8665	−1.62028
$491$	11.1904	0.505015	0.252508	−	0.967595i	$-0.418745\pi$
$491$	11.1904	0.505015	0.252508	+	0.967595i	$0.418745\pi$
$492$	0	0
$493$	−4.81294	−0.216764
$494$	−7.97150	−0.358655
$495$	0	0
$496$	−20.7060	−0.929725
$497$	−55.3921	−2.48468
$498$	0	0
$499$	20.3445	0.910745	0.455372	−	0.890301i	$-0.349506\pi$
$499$	20.3445	0.910745	0.455372	+	0.890301i	$0.349506\pi$
$500$	6.07049	0.271481
$501$	0	0
$502$	44.7231	1.99609
$503$	12.5178	0.558139	0.279069	−	0.960271i	$-0.409974\pi$
$503$	12.5178	0.558139	0.279069	+	0.960271i	$0.409974\pi$
$504$	0	0
$505$	−4.92447	−0.219136
$506$	−25.7181	−1.14331
$507$	0	0
$508$	6.43081	0.285321
$509$	−10.9917	−0.487197	−0.243598	−	0.969876i	$-0.578328\pi$
$509$	−10.9917	−0.487197	−0.243598	+	0.969876i	$0.578328\pi$
$510$	0	0
$511$	65.6483	2.90411
$512$	0.751820	0.0332261
$513$	0	0
$514$	28.1493	1.24161
$515$	9.48220	0.417836
$516$	0	0
$517$	6.21064	0.273144
$518$	−11.4432	−0.502788
$519$	0	0
$520$	−25.8106	−1.13187
$521$	−33.3635	−1.46168	−0.730841	−	0.682548i	$-0.760872\pi$
$521$	−33.3635	−1.46168	−0.730841	+	0.682548i	$0.760872\pi$
$522$	0	0
$523$	6.65718	0.291098	0.145549	−	0.989351i	$-0.453505\pi$
$523$	6.65718	0.291098	0.145549	+	0.989351i	$0.453505\pi$
$524$	3.84712	0.168062
$525$	0	0
$526$	−5.94068	−0.259026
$527$	31.0547	1.35276
$528$	0	0
$529$	−16.8414	−0.732237
$530$	27.8962	1.21173
$531$	0	0
$532$	3.04490	0.132013
$533$	20.9994	0.909585
$534$	0	0
$535$	22.4376	0.970064
$536$	−15.5082	−0.669854
$537$	0	0
$538$	38.7913	1.67241
$539$	50.1196	2.15880
$540$	0	0
$541$	2.89433	0.124437	0.0622185	−	0.998063i	$-0.480182\pi$
$541$	2.89433	0.124437	0.0622185	+	0.998063i	$0.480182\pi$
$542$	−20.2815	−0.871166
$543$	0	0
$544$	−31.2674	−1.34058
$545$	−18.9837	−0.813171
$546$	0	0
$547$	−41.1568	−1.75974	−0.879869	−	0.475216i	$-0.842370\pi$
$547$	−41.1568	−1.75974	−0.879869	+	0.475216i	$0.842370\pi$
$548$	−8.26172	−0.352923
$549$	0	0
$550$	21.7039	0.925457
$551$	−0.647838	−0.0275988
$552$	0	0
$553$	54.8642	2.33306
$554$	−24.1520	−1.02612
$555$	0	0
$556$	−5.54278	−0.235066
$557$	2.02104	0.0856341	0.0428171	−	0.999083i	$-0.486367\pi$
$557$	2.02104	0.0856341	0.0428171	+	0.999083i	$0.486367\pi$
$558$	0	0
$559$	−46.3096	−1.95869
$560$	51.2180	2.16436
$561$	0	0
$562$	6.53469	0.275649
$563$	−0.748930	−0.0315636	−0.0157818	−	0.999875i	$-0.505024\pi$
$563$	−0.748930	−0.0315636	−0.0157818	+	0.999875i	$0.505024\pi$
$564$	0	0
$565$	42.2692	1.77828
$566$	−26.2860	−1.10488
$567$	0	0
$568$	28.9440	1.21446
$569$	−5.00227	−0.209706	−0.104853	−	0.994488i	$-0.533437\pi$
$569$	−5.00227	−0.209706	−0.104853	+	0.994488i	$0.533437\pi$
$570$	0	0
$571$	36.7968	1.53990	0.769948	−	0.638106i	$-0.220282\pi$
$571$	36.7968	1.53990	0.769948	+	0.638106i	$0.220282\pi$
$572$	−23.2719	−0.973045
$573$	0	0
$574$	−28.4740	−1.18848
$575$	−5.19729	−0.216742
$576$	0	0
$577$	−9.38468	−0.390689	−0.195345	−	0.980735i	$-0.562583\pi$
$577$	−9.38468	−0.390689	−0.195345	+	0.980735i	$0.562583\pi$
$578$	63.7312	2.65087
$579$	0	0
$580$	1.35344	0.0561985
$581$	43.1142	1.78868
$582$	0	0
$583$	−38.9820	−1.61447
$584$	−34.3032	−1.41948
$585$	0	0
$586$	30.3136	1.25224
$587$	10.7828	0.445054	0.222527	−	0.974926i	$-0.428569\pi$
$587$	10.7828	0.445054	0.222527	+	0.974926i	$0.428569\pi$
$588$	0	0
$589$	4.18007	0.172237
$590$	−20.3047	−0.835933
$591$	0	0
$592$	8.75070	0.359652
$593$	36.9942	1.51917	0.759585	−	0.650408i	$-0.225402\pi$
$593$	36.9942	1.51917	0.759585	+	0.650408i	$0.225402\pi$
$594$	0	0
$595$	−76.8165	−3.14917
$596$	0.815251	0.0333940
$597$	0	0
$598$	19.7824	0.808963
$599$	−3.12809	−0.127810	−0.0639051	−	0.997956i	$-0.520355\pi$
$599$	−3.12809	−0.127810	−0.0639051	+	0.997956i	$0.520355\pi$
$600$	0	0
$601$	−8.38411	−0.341995	−0.170997	−	0.985271i	$-0.554699\pi$
$601$	−8.38411	−0.341995	−0.170997	+	0.985271i	$0.554699\pi$
$602$	62.7932	2.55926
$603$	0	0
$604$	4.90234	0.199473
$605$	−73.4386	−2.98570
$606$	0	0
$607$	19.8936	0.807455	0.403728	−	0.914879i	$-0.367714\pi$
$607$	19.8936	0.807455	0.403728	+	0.914879i	$0.367714\pi$
$608$	−4.20870	−0.170685
$609$	0	0
$610$	54.3029	2.19866
$611$	−4.77724	−0.193266
$612$	0	0
$613$	−8.72638	−0.352455	−0.176228	−	0.984349i	$-0.556389\pi$
$613$	−8.72638	−0.352455	−0.176228	+	0.984349i	$0.556389\pi$
$614$	1.54090	0.0621857
$615$	0	0
$616$	−48.9056	−1.97046
$617$	8.82898	0.355441	0.177721	−	0.984081i	$-0.443128\pi$
$617$	8.82898	0.355441	0.177721	+	0.984081i	$0.443128\pi$
$618$	0	0
$619$	−9.53800	−0.383365	−0.191682	−	0.981457i	$-0.561394\pi$
$619$	−9.53800	−0.383365	−0.191682	+	0.981457i	$0.561394\pi$
$620$	−8.73284	−0.350719
$621$	0	0
$622$	−13.7359	−0.550759
$623$	−50.4098	−2.01963
$624$	0	0
$625$	−31.0854	−1.24342
$626$	8.34270	0.333441
$627$	0	0
$628$	17.2144	0.686931
$629$	−13.1242	−0.523298
$630$	0	0
$631$	6.56853	0.261489	0.130744	−	0.991416i	$-0.458263\pi$
$631$	6.56853	0.261489	0.130744	+	0.991416i	$0.458263\pi$
$632$	−28.6682	−1.14036
$633$	0	0
$634$	40.0862	1.59203
$635$	−21.8375	−0.866594
$636$	0	0
$637$	−38.5521	−1.52749
$638$	−6.71376	−0.265800
$639$	0	0
$640$	−35.2385	−1.39292
$641$	−16.4665	−0.650387	−0.325193	−	0.945648i	$-0.605429\pi$
$641$	−16.4665	−0.650387	−0.325193	+	0.945648i	$0.605429\pi$
$642$	0	0
$643$	11.5437	0.455238	0.227619	−	0.973750i	$-0.426906\pi$
$643$	11.5437	0.455238	0.227619	+	0.973750i	$0.426906\pi$
$644$	−7.55636	−0.297762
$645$	0	0
$646$	12.3967	0.487743
$647$	−12.2411	−0.481249	−0.240625	−	0.970618i	$-0.577352\pi$
$647$	−12.2411	−0.481249	−0.240625	+	0.970618i	$0.577352\pi$
$648$	0	0
$649$	28.3737	1.11377
$650$	−16.6947	−0.654819
$651$	0	0
$652$	15.8949	0.622491
$653$	−31.0582	−1.21540	−0.607700	−	0.794166i	$-0.707908\pi$
$653$	−31.0582	−1.21540	−0.607700	+	0.794166i	$0.707908\pi$
$654$	0	0
$655$	−13.0639	−0.510448
$656$	21.7742	0.850140
$657$	0	0
$658$	6.47767	0.252526
$659$	5.70225	0.222128	0.111064	−	0.993813i	$-0.464574\pi$
$659$	5.70225	0.222128	0.111064	+	0.993813i	$0.464574\pi$
$660$	0	0
$661$	7.71003	0.299885	0.149943	−	0.988695i	$-0.452091\pi$
$661$	7.71003	0.299885	0.149943	+	0.988695i	$0.452091\pi$
$662$	39.0568	1.51798
$663$	0	0
$664$	−22.5285	−0.874273
$665$	−10.3398	−0.400959
$666$	0	0
$667$	1.60770	0.0622505
$668$	17.5553	0.679236
$669$	0	0
$670$	−33.9793	−1.31273
$671$	−75.8825	−2.92941
$672$	0	0
$673$	−42.9472	−1.65549	−0.827747	−	0.561101i	$-0.810378\pi$
$673$	−42.9472	−1.65549	−0.827747	+	0.561101i	$0.810378\pi$
$674$	−47.3505	−1.82387
$675$	0	0
$676$	7.70402	0.296309
$677$	22.5031	0.864863	0.432432	−	0.901667i	$-0.357656\pi$
$677$	22.5031	0.864863	0.432432	+	0.901667i	$0.357656\pi$
$678$	0	0
$679$	44.0549	1.69067
$680$	40.1389	1.53926
$681$	0	0
$682$	43.3194	1.65879
$683$	−1.29016	−0.0493664	−0.0246832	−	0.999695i	$-0.507858\pi$
$683$	−1.29016	−0.0493664	−0.0246832	+	0.999695i	$0.507858\pi$
$684$	0	0
$685$	28.0548	1.07192
$686$	6.93078	0.264618
$687$	0	0
$688$	−48.0182	−1.83068
$689$	29.9851	1.14234
$690$	0	0
$691$	−43.5775	−1.65776	−0.828882	−	0.559423i	$-0.811023\pi$
$691$	−43.5775	−1.65776	−0.828882	+	0.559423i	$0.811023\pi$
$692$	−12.8268	−0.487601
$693$	0	0
$694$	−13.2777	−0.504015
$695$	18.8219	0.713957
$696$	0	0
$697$	−32.6568	−1.23697
$698$	37.0524	1.40245
$699$	0	0
$700$	6.37693	0.241025
$701$	−11.0816	−0.418545	−0.209273	−	0.977857i	$-0.567110\pi$
$701$	−11.0816	−0.418545	−0.209273	+	0.977857i	$0.567110\pi$
$702$	0	0
$703$	−1.76657	−0.0666274
$704$	17.9127	0.675111
$705$	0	0
$706$	27.2707	1.02635
$707$	7.17729	0.269930
$708$	0	0
$709$	1.48156	0.0556411	0.0278205	−	0.999613i	$-0.491143\pi$
$709$	1.48156	0.0556411	0.0278205	+	0.999613i	$0.491143\pi$
$710$	63.4176	2.38002
$711$	0	0
$712$	26.3406	0.987156
$713$	−10.3734	−0.388488
$714$	0	0
$715$	79.0256	2.95539
$716$	16.4131	0.613385
$717$	0	0
$718$	−43.0424	−1.60633
$719$	−2.07631	−0.0774335	−0.0387167	−	0.999250i	$-0.512327\pi$
$719$	−2.07631	−0.0774335	−0.0387167	+	0.999250i	$0.512327\pi$
$720$	0	0
$721$	−13.8201	−0.514686
$722$	1.66864	0.0621004
$723$	0	0
$724$	4.12400	0.153267
$725$	−1.35676	−0.0503889
$726$	0	0
$727$	−37.5037	−1.39093	−0.695467	−	0.718558i	$-0.744803\pi$
$727$	−37.5037	−1.39093	−0.695467	+	0.718558i	$0.744803\pi$
$728$	37.6183	1.39423
$729$	0	0
$730$	−75.1597	−2.78179
$731$	72.0175	2.66366
$732$	0	0
$733$	−18.5448	−0.684969	−0.342484	−	0.939524i	$-0.611268\pi$
$733$	−18.5448	−0.684969	−0.342484	+	0.939524i	$0.611268\pi$
$734$	−23.8054	−0.878675
$735$	0	0
$736$	10.4445	0.384989
$737$	47.4824	1.74904
$738$	0	0
$739$	−24.9452	−0.917624	−0.458812	−	0.888533i	$-0.651725\pi$
$739$	−24.9452	−0.917624	−0.458812	+	0.888533i	$0.651725\pi$
$740$	3.69065	0.135671
$741$	0	0
$742$	−40.6581	−1.49261
$743$	14.9713	0.549245	0.274622	−	0.961552i	$-0.411447\pi$
$743$	14.9713	0.549245	0.274622	+	0.961552i	$0.411447\pi$
$744$	0	0
$745$	−2.76839	−0.101426
$746$	29.7892	1.09066
$747$	0	0
$748$	36.1908	1.32327
$749$	−32.7023	−1.19492
$750$	0	0
$751$	−21.6848	−0.791288	−0.395644	−	0.918404i	$-0.629479\pi$
$751$	−21.6848	−0.791288	−0.395644	+	0.918404i	$0.629479\pi$
$752$	−4.95350	−0.180636
$753$	0	0
$754$	5.16424	0.188070
$755$	−16.6472	−0.605852
$756$	0	0
$757$	47.6214	1.73083	0.865414	−	0.501058i	$-0.167056\pi$
$757$	47.6214	1.73083	0.865414	+	0.501058i	$0.167056\pi$
$758$	−10.2519	−0.372365
$759$	0	0
$760$	5.40283	0.195981
$761$	16.5443	0.599731	0.299865	−	0.953982i	$-0.403058\pi$
$761$	16.5443	0.599731	0.299865	+	0.953982i	$0.403058\pi$
$762$	0	0
$763$	27.6682	1.00166
$764$	3.27576	0.118513
$765$	0	0
$766$	−41.2019	−1.48868
$767$	−21.8251	−0.788060
$768$	0	0
$769$	26.7449	0.964445	0.482222	−	0.876049i	$-0.339830\pi$
$769$	26.7449	0.964445	0.482222	+	0.876049i	$0.339830\pi$
$770$	−107.154	−3.86158
$771$	0	0
$772$	−6.63468	−0.238787
$773$	−10.4948	−0.377472	−0.188736	−	0.982028i	$-0.560439\pi$
$773$	−10.4948	−0.377472	−0.188736	+	0.982028i	$0.560439\pi$
$774$	0	0
$775$	8.75429	0.314464
$776$	−23.0200	−0.826368
$777$	0	0
$778$	−59.1139	−2.11933
$779$	−4.39572	−0.157493
$780$	0	0
$781$	−88.6194	−3.17105
$782$	−30.7642	−1.10013
$783$	0	0
$784$	−39.9745	−1.42766
$785$	−58.4561	−2.08639
$786$	0	0
$787$	37.3424	1.33111	0.665556	−	0.746348i	$-0.268194\pi$
$787$	37.3424	1.33111	0.665556	+	0.746348i	$0.268194\pi$
$788$	2.00148	0.0712998
$789$	0	0
$790$	−62.8132	−2.23479
$791$	−61.6063	−2.19047
$792$	0	0
$793$	58.3689	2.07274
$794$	2.14784	0.0762238
$795$	0	0
$796$	−11.1542	−0.395350
$797$	5.14554	0.182264	0.0911322	−	0.995839i	$-0.470951\pi$
$797$	5.14554	0.182264	0.0911322	+	0.995839i	$0.470951\pi$
$798$	0	0
$799$	7.42923	0.262827
$800$	−8.81425	−0.311631
$801$	0	0
$802$	−52.3659	−1.84910
$803$	105.028	3.70635
$804$	0	0
$805$	25.6596	0.904382
$806$	−33.3214	−1.17370
$807$	0	0
$808$	−3.75035	−0.131937
$809$	−20.9147	−0.735322	−0.367661	−	0.929960i	$-0.619841\pi$
$809$	−20.9147	−0.735322	−0.367661	+	0.929960i	$0.619841\pi$
$810$	0	0
$811$	−50.9651	−1.78963	−0.894814	−	0.446439i	$-0.852692\pi$
$811$	−50.9651	−1.78963	−0.894814	+	0.446439i	$0.852692\pi$
$812$	−1.97260	−0.0692248
$813$	0	0
$814$	−18.3075	−0.641679
$815$	−53.9751	−1.89067
$816$	0	0
$817$	9.69380	0.339143
$818$	39.5855	1.38407
$819$	0	0
$820$	9.18338	0.320697
$821$	34.4511	1.20235	0.601175	−	0.799117i	$-0.294699\pi$
$821$	34.4511	1.20235	0.601175	+	0.799117i	$0.294699\pi$
$822$	0	0
$823$	8.37207	0.291832	0.145916	−	0.989297i	$-0.453387\pi$
$823$	8.37207	0.291832	0.145916	+	0.989297i	$0.453387\pi$
$824$	7.22139	0.251569
$825$	0	0
$826$	29.5937	1.02970
$827$	−33.1206	−1.15172	−0.575858	−	0.817550i	$-0.695332\pi$
$827$	−33.1206	−1.15172	−0.575858	+	0.817550i	$0.695332\pi$
$828$	0	0
$829$	−29.5848	−1.02752	−0.513761	−	0.857933i	$-0.671748\pi$
$829$	−29.5848	−1.02752	−0.513761	+	0.857933i	$0.671748\pi$
$830$	−49.3608	−1.71334
$831$	0	0
$832$	−13.7785	−0.477683
$833$	59.9535	2.07727
$834$	0	0
$835$	−59.6136	−2.06301
$836$	4.87141	0.168481
$837$	0	0
$838$	19.2372	0.664539
$839$	8.23605	0.284340	0.142170	−	0.989842i	$-0.454592\pi$
$839$	8.23605	0.284340	0.142170	+	0.989842i	$0.454592\pi$
$840$	0	0
$841$	−28.5803	−0.985528
$842$	46.9086	1.61658
$843$	0	0
$844$	9.22179	0.317427
$845$	−26.1610	−0.899966
$846$	0	0
$847$	107.035	3.67776
$848$	31.0914	1.06768
$849$	0	0
$850$	25.9624	0.890503
$851$	4.38399	0.150281
$852$	0	0
$853$	−28.1834	−0.964981	−0.482490	−	0.875901i	$-0.660268\pi$
$853$	−28.1834	−0.964981	−0.482490	+	0.875901i	$0.660268\pi$
$854$	−79.1451	−2.70829
$855$	0	0
$856$	17.0879	0.584053
$857$	37.2230	1.27151	0.635757	−	0.771889i	$-0.280688\pi$
$857$	37.2230	1.27151	0.635757	+	0.771889i	$0.280688\pi$
$858$	0	0
$859$	43.4099	1.48113	0.740563	−	0.671987i	$-0.234559\pi$
$859$	43.4099	1.48113	0.740563	+	0.671987i	$0.234559\pi$
$860$	−20.2519	−0.690585
$861$	0	0
$862$	−6.30069	−0.214602
$863$	1.45104	0.0493939	0.0246970	−	0.999695i	$-0.492138\pi$
$863$	1.45104	0.0493939	0.0246970	+	0.999695i	$0.492138\pi$
$864$	0	0
$865$	43.5566	1.48097
$866$	−12.8846	−0.437835
$867$	0	0
$868$	12.7279	0.432013
$869$	87.7748	2.97756
$870$	0	0
$871$	−36.5236	−1.23755
$872$	−14.4575	−0.489592
$873$	0	0
$874$	−4.14097	−0.140070
$875$	30.0443	1.01568
$876$	0	0
$877$	47.7849	1.61358	0.806791	−	0.590837i	$-0.201202\pi$
$877$	47.7849	1.61358	0.806791	+	0.590837i	$0.201202\pi$
$878$	0.168001	0.00566974
$879$	0	0
$880$	81.9414	2.76224
$881$	24.3577	0.820631	0.410315	−	0.911944i	$-0.365419\pi$
$881$	24.3577	0.820631	0.410315	+	0.911944i	$0.365419\pi$
$882$	0	0
$883$	−2.25144	−0.0757670	−0.0378835	−	0.999282i	$-0.512062\pi$
$883$	−2.25144	−0.0757670	−0.0378835	+	0.999282i	$0.512062\pi$
$884$	−27.8380	−0.936294
$885$	0	0
$886$	35.2675	1.18483
$887$	−14.3538	−0.481952	−0.240976	−	0.970531i	$-0.577468\pi$
$887$	−14.3538	−0.481952	−0.240976	+	0.970531i	$0.577468\pi$
$888$	0	0
$889$	31.8276	1.06746
$890$	57.7134	1.93456
$891$	0	0
$892$	−20.1193	−0.673645
$893$	1.00000	0.0334637
$894$	0	0
$895$	−55.7348	−1.86301
$896$	51.3592	1.71579
$897$	0	0
$898$	−34.5108	−1.15164
$899$	−2.70801	−0.0903170
$900$	0	0
$901$	−46.6307	−1.55349
$902$	−45.5543	−1.51679
$903$	0	0
$904$	32.1911	1.07066
$905$	−14.0041	−0.465512
$906$	0	0
$907$	−30.4440	−1.01088	−0.505438	−	0.862863i	$-0.668669\pi$
$907$	−30.4440	−1.01088	−0.505438	+	0.862863i	$0.668669\pi$
$908$	−0.711161	−0.0236007
$909$	0	0
$910$	82.4234	2.73231
$911$	−12.7089	−0.421065	−0.210532	−	0.977587i	$-0.567520\pi$
$911$	−12.7089	−0.421065	−0.210532	+	0.977587i	$0.567520\pi$
$912$	0	0
$913$	68.9765	2.28279
$914$	2.50062	0.0827131
$915$	0	0
$916$	13.1579	0.434749
$917$	19.0403	0.628766
$918$	0	0
$919$	−19.0036	−0.626870	−0.313435	−	0.949610i	$-0.601480\pi$
$919$	−19.0036	−0.626870	−0.313435	+	0.949610i	$0.601480\pi$
$920$	−13.4079	−0.442045
$921$	0	0
$922$	8.85226	0.291534
$923$	68.1662	2.24372
$924$	0	0
$925$	−3.69972	−0.121646
$926$	−47.3723	−1.55675
$927$	0	0
$928$	2.72655	0.0895035
$929$	39.6219	1.29995	0.649976	−	0.759955i	$-0.274779\pi$
$929$	39.6219	1.29995	0.649976	+	0.759955i	$0.274779\pi$
$930$	0	0
$931$	8.06995	0.264482
$932$	−9.97356	−0.326695
$933$	0	0
$934$	−59.7933	−1.95650
$935$	−122.895	−4.01910
$936$	0	0
$937$	7.76865	0.253791	0.126895	−	0.991916i	$-0.459499\pi$
$937$	7.76865	0.253791	0.126895	+	0.991916i	$0.459499\pi$
$938$	49.5239	1.61701
$939$	0	0
$940$	−2.08916	−0.0681410
$941$	−16.6584	−0.543050	−0.271525	−	0.962431i	$-0.587528\pi$
$941$	−16.6584	−0.543050	−0.271525	+	0.962431i	$0.587528\pi$
$942$	0	0
$943$	10.9086	0.355233
$944$	−22.6304	−0.736557
$945$	0	0
$946$	100.460	3.26624
$947$	−54.9122	−1.78441	−0.892204	−	0.451633i	$-0.850842\pi$
$947$	−54.9122	−1.78441	−0.892204	+	0.451633i	$0.850842\pi$
$948$	0	0
$949$	−80.7876	−2.62248
$950$	3.49463	0.113381
$951$	0	0
$952$	−58.5014	−1.89604
$953$	25.5123	0.826423	0.413212	−	0.910635i	$-0.364407\pi$
$953$	25.5123	0.826423	0.413212	+	0.910635i	$0.364407\pi$
$954$	0	0
$955$	−11.1237	−0.359954
$956$	13.5127	0.437032
$957$	0	0
$958$	−26.9654	−0.871211
$959$	−40.8892	−1.32038
$960$	0	0
$961$	−13.5270	−0.436356
$962$	14.0822	0.454028
$963$	0	0
$964$	9.29520	0.299378
$965$	22.5298	0.725259
$966$	0	0
$967$	18.9369	0.608971	0.304485	−	0.952517i	$-0.401516\pi$
$967$	18.9369	0.608971	0.304485	+	0.952517i	$0.401516\pi$
$968$	−55.9289	−1.79762
$969$	0	0
$970$	−50.4377	−1.61946
$971$	−22.7031	−0.728576	−0.364288	−	0.931286i	$-0.618688\pi$
$971$	−22.7031	−0.728576	−0.364288	+	0.931286i	$0.618688\pi$
$972$	0	0
$973$	−27.4325	−0.879446
$974$	−45.3693	−1.45373
$975$	0	0
$976$	60.5226	1.93728
$977$	−39.0738	−1.25008	−0.625041	−	0.780592i	$-0.714918\pi$
$977$	−39.0738	−1.25008	−0.625041	+	0.780592i	$0.714918\pi$
$978$	0	0
$979$	−80.6484	−2.57754
$980$	−16.8594	−0.538555
$981$	0	0
$982$	18.6727	0.595871
$983$	−22.5498	−0.719227	−0.359614	−	0.933101i	$-0.617091\pi$
$983$	−22.5498	−0.719227	−0.359614	+	0.933101i	$0.617091\pi$
$984$	0	0
$985$	−6.79654	−0.216556
$986$	−8.03107	−0.255761
$987$	0	0
$988$	−3.74709	−0.119211
$989$	−24.0565	−0.764954
$990$	0	0
$991$	28.8595	0.916753	0.458376	−	0.888758i	$-0.348431\pi$
$991$	28.8595	0.916753	0.458376	+	0.888758i	$0.348431\pi$
$992$	−17.5926	−0.558567
$993$	0	0
$994$	−92.4296	−2.93169
$995$	37.8770	1.20078
$996$	0	0
$997$	−39.9582	−1.26549	−0.632745	−	0.774360i	$-0.718072\pi$
$997$	−39.9582	−1.26549	−0.632745	+	0.774360i	$0.718072\pi$
$998$	33.9477	1.07459
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.t.1.19		24
3.2	odd	2		2679.2.a.o.1.6	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.o.1.6	✓	24	3.2	odd	2
8037.2.a.t.1.19		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)

\(f(q)\)	\(=\)	\(q+1.66864 q^{2} +0.784364 q^{4} -2.66351 q^{5} +3.88200 q^{7} -2.02846 q^{8} +O(q^{10})\)	\(q+1.66864 q^{2} +0.784364 q^{4} -2.66351 q^{5} +3.88200 q^{7} -2.02846 q^{8} -4.44445 q^{10} +6.21064 q^{11} -4.77724 q^{13} +6.47767 q^{14} -4.95350 q^{16} +7.42923 q^{17} +1.00000 q^{19} -2.08916 q^{20} +10.3633 q^{22} -2.48164 q^{23} +2.09430 q^{25} -7.97150 q^{26} +3.04490 q^{28} -0.647838 q^{29} +4.18007 q^{31} -4.20870 q^{32} +12.3967 q^{34} -10.3398 q^{35} -1.76657 q^{37} +1.66864 q^{38} +5.40283 q^{40} -4.39572 q^{41} +9.69380 q^{43} +4.87141 q^{44} -4.14097 q^{46} +1.00000 q^{47} +8.06995 q^{49} +3.49463 q^{50} -3.74709 q^{52} -6.27665 q^{53} -16.5421 q^{55} -7.87449 q^{56} -1.08101 q^{58} +4.56857 q^{59} -12.2181 q^{61} +6.97503 q^{62} +2.88420 q^{64} +12.7242 q^{65} +7.64533 q^{67} +5.82722 q^{68} -17.2534 q^{70} -14.2690 q^{71} +16.9109 q^{73} -2.94777 q^{74} +0.784364 q^{76} +24.1097 q^{77} +14.1330 q^{79} +13.1937 q^{80} -7.33488 q^{82} +11.1062 q^{83} -19.7878 q^{85} +16.1755 q^{86} -12.5980 q^{88} -12.9855 q^{89} -18.5453 q^{91} -1.94651 q^{92} +1.66864 q^{94} -2.66351 q^{95} +11.3485 q^{97} +13.4659 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) \) 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8	\( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) \) 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8 + 7 * q^10 + 3 * q^11 + 11 * q^13 - 9 * q^14 + 40 * q^16 + 6 * q^17 + 24 * q^19 + 17 * q^20 + 15 * q^22 + 19 * q^23 + 54 * q^25 + q^26 + 26 * q^28 - 32 * q^29 + 12 * q^31 - 30 * q^32 + 38 * q^34 + 35 * q^35 + 14 * q^37 - 2 * q^38 + 47 * q^40 - 30 * q^41 + 10 * q^43 - q^44 - 3 * q^46 + 24 * q^47 + 56 * q^49 - 44 * q^50 - 10 * q^53 + 24 * q^55 + 6 * q^56 + 41 * q^58 - 11 * q^59 + 10 * q^61 - 2 * q^62 + 61 * q^64 - 35 * q^65 + 12 * q^67 + 22 * q^68 - 20 * q^70 - 44 * q^71 + 33 * q^73 - 36 * q^74 + 32 * q^76 + 23 * q^77 + 50 * q^79 + 13 * q^80 + 5 * q^82 + 72 * q^83 + 33 * q^85 - 44 * q^86 + 38 * q^88 - 48 * q^89 + 47 * q^91 + 29 * q^92 - 2 * q^94 + 2 * q^95 + 50 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more