LMFDB - Embedded newform 8037.2.a.g.1.2

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:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.679643$ 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)

$f(q)$	$=$	$q-0.858442 q^{2} -1.26308 q^{4} -2.26308 q^{5} -1.17880 q^{7} +2.80116 q^{8} +O(q^{10})$	\(q-0.858442 q^{2} -1.26308 q^{4} -2.26308 q^{5} -1.17880 q^{7} +2.80116 q^{8} +1.94272 q^{10} +2.14156 q^{11} -2.70495 q^{13} +1.01193 q^{14} +0.121519 q^{16} +4.21773 q^{17} +1.00000 q^{19} +2.85844 q^{20} -1.83840 q^{22} +3.95465 q^{23} +0.121519 q^{25} +2.32205 q^{26} +1.48891 q^{28} -4.38460 q^{29} +0.583434 q^{31} -5.70664 q^{32} -3.62067 q^{34} +2.66771 q^{35} +0.942721 q^{37} -0.858442 q^{38} -6.33925 q^{40} +9.93461 q^{41} -8.96972 q^{43} -2.70495 q^{44} -3.39484 q^{46} +1.00000 q^{47} -5.61043 q^{49} -0.104317 q^{50} +3.41657 q^{52} +10.1366 q^{53} -4.84651 q^{55} -3.30201 q^{56} +3.76392 q^{58} +2.35760 q^{59} +1.35118 q^{61} -0.500844 q^{62} +4.65578 q^{64} +6.12152 q^{65} +6.86540 q^{67} -5.32732 q^{68} -2.29008 q^{70} +3.52784 q^{71} -3.72499 q^{73} -0.809271 q^{74} -1.26308 q^{76} -2.52447 q^{77} +3.77585 q^{79} -0.275008 q^{80} -8.52829 q^{82} -0.243039 q^{83} -9.54505 q^{85} +7.69998 q^{86} +5.99885 q^{88} -10.4252 q^{89} +3.18860 q^{91} -4.99503 q^{92} -0.858442 q^{94} -2.26308 q^{95} +0.743342 q^{97} +4.81623 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 - 2 * q^5 - 5 * q^7 - 3 * q^8	$ 4 q + 2 q^{4} - 2 q^{5} - 5 q^{7} - 3 q^{8} - 3 q^{10} + 12 q^{11} + 3 q^{13} + 11 q^{14} - 10 q^{16} + 6 q^{17} + 4 q^{19} + 8 q^{20} + 10 q^{22} + 12 q^{23} - 10 q^{25} + 5 q^{26} - 5 q^{28} - q^{31} - q^{32} - 8 q^{34} - 7 q^{37} - 4 q^{40} + 22 q^{41} - 7 q^{43} + 3 q^{44} - 11 q^{46} + 4 q^{47} - q^{49} - 7 q^{50} + 17 q^{52} + 5 q^{53} - 9 q^{55} + 5 q^{56} + 4 q^{58} + 10 q^{59} - 9 q^{61} + 2 q^{62} + q^{64} + 14 q^{65} - 8 q^{67} + 7 q^{68} + 16 q^{70} - 8 q^{71} - 15 q^{73} + 4 q^{74} + 2 q^{76} - 4 q^{77} + 11 q^{79} - q^{80} - 28 q^{82} + 20 q^{83} + q^{85} - 22 q^{86} - 15 q^{88} - 5 q^{89} + 12 q^{91} + 19 q^{92} - 2 q^{95} + 25 q^{97} - 20 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 - 5 * q^7 - 3 * q^8 - 3 * q^10 + 12 * q^11 + 3 * q^13 + 11 * q^14 - 10 * q^16 + 6 * q^17 + 4 * q^19 + 8 * q^20 + 10 * q^22 + 12 * q^23 - 10 * q^25 + 5 * q^26 - 5 * q^28 - q^31 - q^32 - 8 * q^34 - 7 * q^37 - 4 * q^40 + 22 * q^41 - 7 * q^43 + 3 * q^44 - 11 * q^46 + 4 * q^47 - q^49 - 7 * q^50 + 17 * q^52 + 5 * q^53 - 9 * q^55 + 5 * q^56 + 4 * q^58 + 10 * q^59 - 9 * q^61 + 2 * q^62 + q^64 + 14 * q^65 - 8 * q^67 + 7 * q^68 + 16 * q^70 - 8 * q^71 - 15 * q^73 + 4 * q^74 + 2 * q^76 - 4 * q^77 + 11 * q^79 - q^80 - 28 * q^82 + 20 * q^83 + q^85 - 22 * q^86 - 15 * q^88 - 5 * q^89 + 12 * q^91 + 19 * q^92 - 2 * q^95 + 25 * q^97 - 20 * 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 $

$2$	−0.858442	−0.607010	−0.303505	−	0.952830i	$-0.598157\pi$
$2$	−0.858442	−0.607010	−0.303505	+	0.952830i	$0.598157\pi$
$3$	0	0
$3$	0	0
$4$	−1.26308	−0.631539
$5$	−2.26308	−1.01208	−0.506039	−	0.862510i	$-0.668891\pi$
$5$	−2.26308	−1.01208	−0.506039	+	0.862510i	$0.668891\pi$
$6$	0	0
$7$	−1.17880	−0.445544	−0.222772	−	0.974871i	$-0.571511\pi$
$7$	−1.17880	−0.445544	−0.222772	+	0.974871i	$0.571511\pi$
$8$	2.80116	0.990361
$9$	0	0
$10$	1.94272	0.614342
$11$	2.14156	0.645704	0.322852	−	0.946449i	$-0.395358\pi$
$11$	2.14156	0.645704	0.322852	+	0.946449i	$0.395358\pi$
$12$	0	0
$13$	−2.70495	−0.750219	−0.375110	−	0.926980i	$-0.622395\pi$
$13$	−2.70495	−0.750219	−0.375110	+	0.926980i	$0.622395\pi$
$14$	1.01193	0.270450
$15$	0	0
$16$	0.121519	0.0303798
$17$	4.21773	1.02295	0.511475	−	0.859298i	$-0.329099\pi$
$17$	4.21773	1.02295	0.511475	+	0.859298i	$0.329099\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.85844	0.639167
$21$	0	0
$22$	−1.83840	−0.391949
$23$	3.95465	0.824602	0.412301	−	0.911048i	$-0.364725\pi$
$23$	3.95465	0.824602	0.412301	+	0.911048i	$0.364725\pi$
$24$	0	0
$25$	0.121519	0.0243039
$26$	2.32205	0.455391
$27$	0	0
$28$	1.48891	0.281378
$29$	−4.38460	−0.814199	−0.407100	−	0.913384i	$-0.633460\pi$
$29$	−4.38460	−0.814199	−0.407100	+	0.913384i	$0.633460\pi$
$30$	0	0
$31$	0.583434	0.104788	0.0523939	−	0.998626i	$-0.483315\pi$
$31$	0.583434	0.104788	0.0523939	+	0.998626i	$0.483315\pi$
$32$	−5.70664	−1.00880
$33$	0	0
$34$	−3.62067	−0.620941
$35$	2.66771	0.450926
$36$	0	0
$37$	0.942721	0.154982	0.0774912	−	0.996993i	$-0.475309\pi$
$37$	0.942721	0.154982	0.0774912	+	0.996993i	$0.475309\pi$
$38$	−0.858442	−0.139258
$39$	0	0
$40$	−6.33925	−1.00232
$41$	9.93461	1.55153	0.775763	−	0.631025i	$-0.217365\pi$
$41$	9.93461	1.55153	0.775763	+	0.631025i	$0.217365\pi$
$42$	0	0
$43$	−8.96972	−1.36787	−0.683935	−	0.729543i	$-0.739733\pi$
$43$	−8.96972	−1.36787	−0.683935	+	0.729543i	$0.739733\pi$
$44$	−2.70495	−0.407787
$45$	0	0
$46$	−3.39484	−0.500542
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−5.61043	−0.801490
$50$	−0.104317	−0.0147527
$51$	0	0
$52$	3.41657	0.473792
$53$	10.1366	1.39237	0.696184	−	0.717864i	$-0.254880\pi$
$53$	10.1366	1.39237	0.696184	+	0.717864i	$0.254880\pi$
$54$	0	0
$55$	−4.84651	−0.653503
$56$	−3.30201	−0.441249
$57$	0	0
$58$	3.76392	0.494227
$59$	2.35760	0.306933	0.153467	−	0.988154i	$-0.450956\pi$
$59$	2.35760	0.306933	0.153467	+	0.988154i	$0.450956\pi$
$60$	0	0
$61$	1.35118	0.173001	0.0865003	−	0.996252i	$-0.472432\pi$
$61$	1.35118	0.173001	0.0865003	+	0.996252i	$0.472432\pi$
$62$	−0.500844	−0.0636073
$63$	0	0
$64$	4.65578	0.581973
$65$	6.12152	0.759281
$66$	0	0
$67$	6.86540	0.838742	0.419371	−	0.907815i	$-0.362251\pi$
$67$	6.86540	0.838742	0.419371	+	0.907815i	$0.362251\pi$
$68$	−5.32732	−0.646032
$69$	0	0
$70$	−2.29008	−0.273717
$71$	3.52784	0.418678	0.209339	−	0.977843i	$-0.432869\pi$
$71$	3.52784	0.418678	0.209339	+	0.977843i	$0.432869\pi$
$72$	0	0
$73$	−3.72499	−0.435977	−0.217989	−	0.975951i	$-0.569950\pi$
$73$	−3.72499	−0.435977	−0.217989	+	0.975951i	$0.569950\pi$
$74$	−0.809271	−0.0940758
$75$	0	0
$76$	−1.26308	−0.144885
$77$	−2.52447	−0.287690
$78$	0	0
$79$	3.77585	0.424817	0.212408	−	0.977181i	$-0.431869\pi$
$79$	3.77585	0.424817	0.212408	+	0.977181i	$0.431869\pi$
$80$	−0.275008	−0.0307468
$81$	0	0
$82$	−8.52829	−0.941792
$83$	−0.243039	−0.0266770	−0.0133385	−	0.999911i	$-0.504246\pi$
$83$	−0.243039	−0.0266770	−0.0133385	+	0.999911i	$0.504246\pi$
$84$	0	0
$85$	−9.54505	−1.03531
$86$	7.69998	0.830311
$87$	0	0
$88$	5.99885	0.639480
$89$	−10.4252	−1.10507	−0.552535	−	0.833489i	$-0.686340\pi$
$89$	−10.4252	−1.10507	−0.552535	+	0.833489i	$0.686340\pi$
$90$	0	0
$91$	3.18860	0.334256
$92$	−4.99503	−0.520768
$93$	0	0
$94$	−0.858442	−0.0885415
$95$	−2.26308	−0.232187
$96$	0	0
$97$	0.743342	0.0754750	0.0377375	−	0.999288i	$-0.487985\pi$
$97$	0.743342	0.0754750	0.0377375	+	0.999288i	$0.487985\pi$
$98$	4.81623	0.486513
$99$	0	0
$100$	−0.153488	−0.0153488
$101$	−9.55529	−0.950787	−0.475393	−	0.879773i	$-0.657694\pi$
$101$	−9.55529	−0.950787	−0.475393	+	0.879773i	$0.657694\pi$
$102$	0	0
$103$	−11.9869	−1.18111	−0.590553	−	0.806999i	$-0.701090\pi$
$103$	−11.9869	−1.18111	−0.590553	+	0.806999i	$0.701090\pi$
$104$	−7.57701	−0.742987
$105$	0	0
$106$	−8.70167	−0.845181
$107$	−11.1502	−1.07793	−0.538966	−	0.842328i	$-0.681185\pi$
$107$	−11.1502	−1.07793	−0.538966	+	0.842328i	$0.681185\pi$
$108$	0	0
$109$	−4.08810	−0.391569	−0.195785	−	0.980647i	$-0.562725\pi$
$109$	−4.08810	−0.391569	−0.195785	+	0.980647i	$0.562725\pi$
$110$	4.16045	0.396683
$111$	0	0
$112$	−0.143247	−0.0135356
$113$	−17.2436	−1.62214	−0.811070	−	0.584949i	$-0.801114\pi$
$113$	−17.2436	−1.62214	−0.811070	+	0.584949i	$0.801114\pi$
$114$	0	0
$115$	−8.94968	−0.834562
$116$	5.53809	0.514198
$117$	0	0
$118$	−2.02386	−0.186311
$119$	−4.97185	−0.455769
$120$	0	0
$121$	−6.41373	−0.583066
$122$	−1.15991	−0.105013
$123$	0	0
$124$	−0.736923	−0.0661776
$125$	11.0404	0.987482
$126$	0	0
$127$	−2.85844	−0.253646	−0.126823	−	0.991925i	$-0.540478\pi$
$127$	−2.85844	−0.253646	−0.126823	+	0.991925i	$0.540478\pi$
$128$	7.41657	0.655538
$129$	0	0
$130$	−5.25497	−0.460891
$131$	−6.12963	−0.535548	−0.267774	−	0.963482i	$-0.586288\pi$
$131$	−6.12963	−0.535548	−0.267774	+	0.963482i	$0.586288\pi$
$132$	0	0
$133$	−1.17880	−0.102215
$134$	−5.89355	−0.509125
$135$	0	0
$136$	11.8145	1.01309
$137$	9.31822	0.796110	0.398055	−	0.917362i	$-0.369685\pi$
$137$	9.31822	0.796110	0.398055	+	0.917362i	$0.369685\pi$
$138$	0	0
$139$	−2.53977	−0.215421	−0.107710	−	0.994182i	$-0.534352\pi$
$139$	−2.53977	−0.215421	−0.107710	+	0.994182i	$0.534352\pi$
$140$	−3.36953	−0.284777
$141$	0	0
$142$	−3.02845	−0.254142
$143$	−5.79282	−0.484420
$144$	0	0
$145$	9.92268	0.824034
$146$	3.19769	0.264643
$147$	0	0
$148$	−1.19073	−0.0978773
$149$	19.0993	1.56468	0.782340	−	0.622852i	$-0.214026\pi$
$149$	19.0993	1.56468	0.782340	+	0.622852i	$0.214026\pi$
$150$	0	0
$151$	−8.08259	−0.657752	−0.328876	−	0.944373i	$-0.606670\pi$
$151$	−8.08259	−0.657752	−0.328876	+	0.944373i	$0.606670\pi$
$152$	2.80116	0.227204
$153$	0	0
$154$	2.16711	0.174631
$155$	−1.32036	−0.106054
$156$	0	0
$157$	−6.65815	−0.531379	−0.265689	−	0.964059i	$-0.585600\pi$
$157$	−6.65815	−0.531379	−0.265689	+	0.964059i	$0.585600\pi$
$158$	−3.24135	−0.257868
$159$	0	0
$160$	12.9146	1.02099
$161$	−4.66174	−0.367396
$162$	0	0
$163$	15.2469	1.19423	0.597113	−	0.802157i	$-0.296314\pi$
$163$	15.2469	1.19423	0.597113	+	0.802157i	$0.296314\pi$
$164$	−12.5482	−0.979849
$165$	0	0
$166$	0.208635	0.0161932
$167$	−5.76774	−0.446321	−0.223161	−	0.974782i	$-0.571637\pi$
$167$	−5.76774	−0.446321	−0.223161	+	0.974782i	$0.571637\pi$
$168$	0	0
$169$	−5.68323	−0.437171
$170$	8.19387	0.628441
$171$	0	0
$172$	11.3295	0.863863
$173$	−14.5962	−1.10973	−0.554865	−	0.831941i	$-0.687230\pi$
$173$	−14.5962	−1.10973	−0.554865	+	0.831941i	$0.687230\pi$
$174$	0	0
$175$	−0.143247	−0.0108284
$176$	0.260241	0.0196164
$177$	0	0
$178$	8.94944	0.670789
$179$	9.49632	0.709788	0.354894	−	0.934907i	$-0.384517\pi$
$179$	9.49632	0.709788	0.354894	+	0.934907i	$0.384517\pi$
$180$	0	0
$181$	0.465293	0.0345849	0.0172925	−	0.999850i	$-0.494495\pi$
$181$	0.465293	0.0345849	0.0172925	+	0.999850i	$0.494495\pi$
$182$	−2.73722	−0.202897
$183$	0	0
$184$	11.0776	0.816653
$185$	−2.13345	−0.156854
$186$	0	0
$187$	9.03251	0.660523
$188$	−1.26308	−0.0921194
$189$	0	0
$190$	1.94272	0.140940
$191$	−7.40846	−0.536057	−0.268029	−	0.963411i	$-0.586372\pi$
$191$	−7.40846	−0.536057	−0.268029	+	0.963411i	$0.586372\pi$
$192$	0	0
$193$	23.5978	1.69861	0.849303	−	0.527905i	$-0.177022\pi$
$193$	23.5978	1.69861	0.849303	+	0.527905i	$0.177022\pi$
$194$	−0.638116	−0.0458141
$195$	0	0
$196$	7.08641	0.506172
$197$	21.8119	1.55404	0.777018	−	0.629478i	$-0.216731\pi$
$197$	21.8119	1.55404	0.777018	+	0.629478i	$0.216731\pi$
$198$	0	0
$199$	−2.03487	−0.144248	−0.0721240	−	0.997396i	$-0.522978\pi$
$199$	−2.03487	−0.144248	−0.0721240	+	0.997396i	$0.522978\pi$
$200$	0.340395	0.0240696
$201$	0	0
$202$	8.20266	0.577137
$203$	5.16856	0.362762
$204$	0	0
$205$	−22.4828	−1.57027
$206$	10.2901	0.716944
$207$	0	0
$208$	−0.328704	−0.0227915
$209$	2.14156	0.148135
$210$	0	0
$211$	−19.0820	−1.31366	−0.656829	−	0.754040i	$-0.728103\pi$
$211$	−19.0820	−1.31366	−0.656829	+	0.754040i	$0.728103\pi$
$212$	−12.8033	−0.879334
$213$	0	0
$214$	9.57181	0.654315
$215$	20.2992	1.38439
$216$	0	0
$217$	−0.687752	−0.0466876
$218$	3.50940	0.237686
$219$	0	0
$220$	6.12152	0.412713
$221$	−11.4088	−0.767436
$222$	0	0
$223$	3.22270	0.215808	0.107904	−	0.994161i	$-0.465586\pi$
$223$	3.22270	0.215808	0.107904	+	0.994161i	$0.465586\pi$
$224$	6.72698	0.449465
$225$	0	0
$226$	14.8026	0.984655
$227$	−3.91572	−0.259896	−0.129948	−	0.991521i	$-0.541481\pi$
$227$	−3.91572	−0.259896	−0.129948	+	0.991521i	$0.541481\pi$
$228$	0	0
$229$	−10.6293	−0.702406	−0.351203	−	0.936299i	$-0.614227\pi$
$229$	−10.6293	−0.702406	−0.351203	+	0.936299i	$0.614227\pi$
$230$	7.68278	0.506588
$231$	0	0
$232$	−12.2820	−0.806351
$233$	−16.9956	−1.11342	−0.556708	−	0.830708i	$-0.687936\pi$
$233$	−16.9956	−1.11342	−0.556708	+	0.830708i	$0.687936\pi$
$234$	0	0
$235$	−2.26308	−0.147627
$236$	−2.97783	−0.193840
$237$	0	0
$238$	4.26805	0.276656
$239$	−30.7797	−1.99098	−0.995488	−	0.0948892i	$-0.969750\pi$
$239$	−30.7797	−1.99098	−0.995488	+	0.0948892i	$0.969750\pi$
$240$	0	0
$241$	18.3390	1.18132	0.590660	−	0.806921i	$-0.298868\pi$
$241$	18.3390	1.18132	0.590660	+	0.806921i	$0.298868\pi$
$242$	5.50581	0.353927
$243$	0	0
$244$	−1.70664	−0.109257
$245$	12.6968	0.811172
$246$	0	0
$247$	−2.70495	−0.172112
$248$	1.63429	0.103778
$249$	0	0
$250$	−9.47752	−0.599411
$251$	30.0550	1.89705	0.948527	−	0.316697i	$-0.102574\pi$
$251$	30.0550	1.89705	0.948527	+	0.316697i	$0.102574\pi$
$252$	0	0
$253$	8.46911	0.532449
$254$	2.45381	0.153966
$255$	0	0
$256$	−15.6783	−0.979891
$257$	13.8873	0.866268	0.433134	−	0.901329i	$-0.357408\pi$
$257$	13.8873	0.866268	0.433134	+	0.901329i	$0.357408\pi$
$258$	0	0
$259$	−1.11128	−0.0690514
$260$	−7.73195	−0.479515
$261$	0	0
$262$	5.26193	0.325083
$263$	6.47553	0.399299	0.199649	−	0.979867i	$-0.436020\pi$
$263$	6.47553	0.399299	0.199649	+	0.979867i	$0.436020\pi$
$264$	0	0
$265$	−22.9399	−1.40919
$266$	1.01193	0.0620454
$267$	0	0
$268$	−8.67153	−0.529698
$269$	11.7673	0.717465	0.358732	−	0.933440i	$-0.383209\pi$
$269$	11.7673	0.717465	0.358732	+	0.933440i	$0.383209\pi$
$270$	0	0
$271$	23.2605	1.41298	0.706489	−	0.707724i	$-0.250278\pi$
$271$	23.2605	1.41298	0.706489	+	0.707724i	$0.250278\pi$
$272$	0.512536	0.0310770
$273$	0	0
$274$	−7.99915	−0.483247
$275$	0.260241	0.0156931
$276$	0	0
$277$	−14.7384	−0.885543	−0.442771	−	0.896634i	$-0.646005\pi$
$277$	−14.7384	−0.885543	−0.442771	+	0.896634i	$0.646005\pi$
$278$	2.18025	0.130763
$279$	0	0
$280$	7.47270	0.446579
$281$	25.4200	1.51643	0.758215	−	0.652004i	$-0.226072\pi$
$281$	25.4200	1.51643	0.758215	+	0.652004i	$0.226072\pi$
$282$	0	0
$283$	32.6380	1.94013	0.970063	−	0.242853i	$-0.0780833\pi$
$283$	32.6380	1.94013	0.970063	+	0.242853i	$0.0780833\pi$
$284$	−4.45594	−0.264411
$285$	0	0
$286$	4.97280	0.294048
$287$	−11.7109	−0.691273
$288$	0	0
$289$	0.789232	0.0464254
$290$	−8.51805	−0.500197
$291$	0	0
$292$	4.70495	0.275337
$293$	−6.19982	−0.362198	−0.181099	−	0.983465i	$-0.557965\pi$
$293$	−6.19982	−0.362198	−0.181099	+	0.983465i	$0.557965\pi$
$294$	0	0
$295$	−5.33543	−0.310640
$296$	2.64071	0.153488
$297$	0	0
$298$	−16.3957	−0.949776
$299$	−10.6971	−0.618632
$300$	0	0
$301$	10.5735	0.609446
$302$	6.93843	0.399262
$303$	0	0
$304$	0.121519	0.00696961
$305$	−3.05782	−0.175090
$306$	0	0
$307$	−0.738611	−0.0421548	−0.0210774	−	0.999778i	$-0.506710\pi$
$307$	−0.738611	−0.0421548	−0.0210774	+	0.999778i	$0.506710\pi$
$308$	3.18860	0.181687
$309$	0	0
$310$	1.13345	0.0643756
$311$	26.7733	1.51818	0.759088	−	0.650988i	$-0.225645\pi$
$311$	26.7733	1.51818	0.759088	+	0.650988i	$0.225645\pi$
$312$	0	0
$313$	18.8480	1.06535	0.532677	−	0.846319i	$-0.321186\pi$
$313$	18.8480	1.06535	0.532677	+	0.846319i	$0.321186\pi$
$314$	5.71564	0.322552
$315$	0	0
$316$	−4.76919	−0.268288
$317$	9.17620	0.515387	0.257693	−	0.966227i	$-0.417038\pi$
$317$	9.17620	0.515387	0.257693	+	0.966227i	$0.417038\pi$
$318$	0	0
$319$	−9.38987	−0.525732
$320$	−10.5364	−0.589002
$321$	0	0
$322$	4.00183	0.223013
$323$	4.21773	0.234681
$324$	0	0
$325$	−0.328704	−0.0182332
$326$	−13.0885	−0.724907
$327$	0	0
$328$	27.8285	1.53657
$329$	−1.17880	−0.0649893
$330$	0	0
$331$	4.50656	0.247703	0.123851	−	0.992301i	$-0.460475\pi$
$331$	4.50656	0.247703	0.123851	+	0.992301i	$0.460475\pi$
$332$	0.306977	0.0168475
$333$	0	0
$334$	4.95127	0.270922
$335$	−15.5369	−0.848874
$336$	0	0
$337$	23.3694	1.27301	0.636506	−	0.771272i	$-0.280379\pi$
$337$	23.3694	1.27301	0.636506	+	0.771272i	$0.280379\pi$
$338$	4.87872	0.265367
$339$	0	0
$340$	12.0561	0.653836
$341$	1.24946	0.0676620
$342$	0	0
$343$	14.8652	0.802643
$344$	−25.1256	−1.35468
$345$	0	0
$346$	12.5300	0.673617
$347$	−25.6852	−1.37885	−0.689427	−	0.724355i	$-0.742138\pi$
$347$	−25.6852	−1.37885	−0.689427	+	0.724355i	$0.742138\pi$
$348$	0	0
$349$	−7.17160	−0.383887	−0.191943	−	0.981406i	$-0.561479\pi$
$349$	−7.17160	−0.383887	−0.191943	+	0.981406i	$0.561479\pi$
$350$	0.122969	0.00657298
$351$	0	0
$352$	−12.2211	−0.651387
$353$	−11.6468	−0.619895	−0.309947	−	0.950754i	$-0.600311\pi$
$353$	−11.6468	−0.619895	−0.309947	+	0.950754i	$0.600311\pi$
$354$	0	0
$355$	−7.98378	−0.423735
$356$	13.1679	0.697895
$357$	0	0
$358$	−8.15204	−0.430849
$359$	10.3239	0.544877	0.272438	−	0.962173i	$-0.412170\pi$
$359$	10.3239	0.544877	0.272438	+	0.962173i	$0.412170\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.399427	−0.0209934
$363$	0	0
$364$	−4.02744	−0.211095
$365$	8.42995	0.441244
$366$	0	0
$367$	9.58657	0.500415	0.250208	−	0.968192i	$-0.419501\pi$
$367$	9.58657	0.500415	0.250208	+	0.968192i	$0.419501\pi$
$368$	0.480567	0.0250513
$369$	0	0
$370$	1.83144	0.0952122
$371$	−11.9490	−0.620361
$372$	0	0
$373$	20.6663	1.07006	0.535031	−	0.844832i	$-0.320300\pi$
$373$	20.6663	1.07006	0.535031	+	0.844832i	$0.320300\pi$
$374$	−7.75389	−0.400944
$375$	0	0
$376$	2.80116	0.144459
$377$	11.8601	0.610828
$378$	0	0
$379$	15.6124	0.801956	0.400978	−	0.916088i	$-0.368670\pi$
$379$	15.6124	0.801956	0.400978	+	0.916088i	$0.368670\pi$
$380$	2.85844	0.146635
$381$	0	0
$382$	6.35973	0.325392
$383$	−14.6626	−0.749223	−0.374611	−	0.927182i	$-0.622224\pi$
$383$	−14.6626	−0.749223	−0.374611	+	0.927182i	$0.622224\pi$
$384$	0	0
$385$	5.71306	0.291165
$386$	−20.2573	−1.03107
$387$	0	0
$388$	−0.938899	−0.0476654
$389$	−5.35263	−0.271389	−0.135695	−	0.990751i	$-0.543327\pi$
$389$	−5.35263	−0.271389	−0.135695	+	0.990751i	$0.543327\pi$
$390$	0	0
$391$	16.6796	0.843526
$392$	−15.7157	−0.793765
$393$	0	0
$394$	−18.7243	−0.943316
$395$	−8.54505	−0.429948
$396$	0	0
$397$	28.4052	1.42562	0.712809	−	0.701358i	$-0.247423\pi$
$397$	28.4052	1.42562	0.712809	+	0.701358i	$0.247423\pi$
$398$	1.74682	0.0875600
$399$	0	0
$400$	0.0147670	0.000738348 0
$401$	21.2438	1.06086	0.530432	−	0.847727i	$-0.322030\pi$
$401$	21.2438	1.06086	0.530432	+	0.847727i	$0.322030\pi$
$402$	0	0
$403$	−1.57816	−0.0786139
$404$	12.0691	0.600459
$405$	0	0
$406$	−4.43691	−0.220200
$407$	2.01889	0.100073
$408$	0	0
$409$	−9.59681	−0.474532	−0.237266	−	0.971445i	$-0.576251\pi$
$409$	−9.59681	−0.474532	−0.237266	+	0.971445i	$0.576251\pi$
$410$	19.3002	0.953168
$411$	0	0
$412$	15.1404	0.745914
$413$	−2.77913	−0.136752
$414$	0	0
$415$	0.550015	0.0269992
$416$	15.4362	0.756822
$417$	0	0
$418$	−1.83840	−0.0899192
$419$	23.9363	1.16936	0.584682	−	0.811262i	$-0.301219\pi$
$419$	23.9363	1.16936	0.584682	+	0.811262i	$0.301219\pi$
$420$	0	0
$421$	−2.54315	−0.123946	−0.0619728	−	0.998078i	$-0.519739\pi$
$421$	−2.54315	−0.123946	−0.0619728	+	0.998078i	$0.519739\pi$
$422$	16.3808	0.797404
$423$	0	0
$424$	28.3942	1.37895
$425$	0.512536	0.0248616
$426$	0	0
$427$	−1.59277	−0.0770794
$428$	14.0836	0.680755
$429$	0	0
$430$	−17.4257	−0.840340
$431$	29.6060	1.42607	0.713036	−	0.701128i	$-0.247320\pi$
$431$	29.6060	1.42607	0.713036	+	0.701128i	$0.247320\pi$
$432$	0	0
$433$	−2.16373	−0.103982	−0.0519911	−	0.998648i	$-0.516557\pi$
$433$	−2.16373	−0.103982	−0.0519911	+	0.998648i	$0.516557\pi$
$434$	0.590395	0.0283399
$435$	0	0
$436$	5.16359	0.247291
$437$	3.95465	0.189177
$438$	0	0
$439$	5.59850	0.267202	0.133601	−	0.991035i	$-0.457346\pi$
$439$	5.59850	0.267202	0.133601	+	0.991035i	$0.457346\pi$
$440$	−13.5759	−0.647204
$441$	0	0
$442$	9.79376	0.465842
$443$	−4.27547	−0.203134	−0.101567	−	0.994829i	$-0.532386\pi$
$443$	−4.27547	−0.203134	−0.101567	+	0.994829i	$0.532386\pi$
$444$	0	0
$445$	23.5931	1.11842
$446$	−2.76650	−0.130998
$447$	0	0
$448$	−5.48823	−0.259295
$449$	−5.11371	−0.241331	−0.120666	−	0.992693i	$-0.538503\pi$
$449$	−5.11371	−0.241331	−0.120666	+	0.992693i	$0.538503\pi$
$450$	0	0
$451$	21.2755	1.00183
$452$	21.7800	1.02444
$453$	0	0
$454$	3.36142	0.157759
$455$	−7.21604	−0.338293
$456$	0	0
$457$	−16.3940	−0.766879	−0.383439	−	0.923566i	$-0.625260\pi$
$457$	−16.3940	−0.766879	−0.383439	+	0.923566i	$0.625260\pi$
$458$	9.12466	0.426367
$459$	0	0
$460$	11.3041	0.527058
$461$	9.50734	0.442801	0.221400	−	0.975183i	$-0.428937\pi$
$461$	9.50734	0.442801	0.221400	+	0.975183i	$0.428937\pi$
$462$	0	0
$463$	−20.2375	−0.940515	−0.470257	−	0.882529i	$-0.655839\pi$
$463$	−20.2375	−0.940515	−0.470257	+	0.882529i	$0.655839\pi$
$464$	−0.532813	−0.0247352
$465$	0	0
$466$	14.5897	0.675855
$467$	9.42995	0.436366	0.218183	−	0.975908i	$-0.429987\pi$
$467$	9.42995	0.436366	0.218183	+	0.975908i	$0.429987\pi$
$468$	0	0
$469$	−8.09293	−0.373697
$470$	1.94272	0.0896110
$471$	0	0
$472$	6.60401	0.303974
$473$	−19.2092	−0.883239
$474$	0	0
$475$	0.121519	0.00557569
$476$	6.27984	0.287836
$477$	0	0
$478$	26.4226	1.20854
$479$	−13.1471	−0.600705	−0.300352	−	0.953828i	$-0.597104\pi$
$479$	−13.1471	−0.600705	−0.300352	+	0.953828i	$0.597104\pi$
$480$	0	0
$481$	−2.55002	−0.116271
$482$	−15.7430	−0.717073
$483$	0	0
$484$	8.10104	0.368229
$485$	−1.68224	−0.0763866
$486$	0	0
$487$	16.5926	0.751884	0.375942	−	0.926643i	$-0.377319\pi$
$487$	16.5926	0.751884	0.375942	+	0.926643i	$0.377319\pi$
$488$	3.78487	0.171333
$489$	0	0
$490$	−10.8995	−0.492389
$491$	−26.9925	−1.21816	−0.609078	−	0.793110i	$-0.708460\pi$
$491$	−26.9925	−1.21816	−0.609078	+	0.793110i	$0.708460\pi$
$492$	0	0
$493$	−18.4930	−0.832885
$494$	2.32205	0.104474
$495$	0	0
$496$	0.0708986	0.00318344
$497$	−4.15862	−0.186540
$498$	0	0
$499$	31.0784	1.39126	0.695630	−	0.718400i	$-0.255125\pi$
$499$	31.0784	1.39126	0.695630	+	0.718400i	$0.255125\pi$
$500$	−13.9449	−0.623633
$501$	0	0
$502$	−25.8005	−1.15153
$503$	−31.0389	−1.38396	−0.691979	−	0.721918i	$-0.743261\pi$
$503$	−31.0389	−1.38396	−0.691979	+	0.721918i	$0.743261\pi$
$504$	0	0
$505$	21.6244	0.962271
$506$	−7.27024	−0.323202
$507$	0	0
$508$	3.61043	0.160187
$509$	33.3096	1.47642	0.738210	−	0.674571i	$-0.235671\pi$
$509$	33.3096	1.47642	0.738210	+	0.674571i	$0.235671\pi$
$510$	0	0
$511$	4.39102	0.194247
$512$	−1.37426	−0.0607342
$513$	0	0
$514$	−11.9215	−0.525834
$515$	27.1273	1.19537
$516$	0	0
$517$	2.14156	0.0941856
$518$	0.953968	0.0419149
$519$	0	0
$520$	17.1474	0.751962
$521$	17.3111	0.758413	0.379207	−	0.925312i	$-0.376197\pi$
$521$	17.3111	0.758413	0.379207	+	0.925312i	$0.376197\pi$
$522$	0	0
$523$	−14.0163	−0.612890	−0.306445	−	0.951888i	$-0.599140\pi$
$523$	−14.0163	−0.612890	−0.306445	+	0.951888i	$0.599140\pi$
$524$	7.74219	0.338219
$525$	0	0
$526$	−5.55887	−0.242378
$527$	2.46077	0.107193
$528$	0	0
$529$	−7.36074	−0.320032
$530$	19.6926	0.855390
$531$	0	0
$532$	1.48891	0.0645526
$533$	−26.8727	−1.16398
$534$	0	0
$535$	25.2338	1.09095
$536$	19.2311	0.830657
$537$	0	0
$538$	−10.1015	−0.435508
$539$	−12.0151	−0.517526
$540$	0	0
$541$	−18.9996	−0.816857	−0.408428	−	0.912790i	$-0.633923\pi$
$541$	−18.9996	−0.816857	−0.408428	+	0.912790i	$0.633923\pi$
$542$	−19.9678	−0.857692
$543$	0	0
$544$	−24.0691	−1.03195
$545$	9.25169	0.396299
$546$	0	0
$547$	22.5881	0.965797	0.482899	−	0.875676i	$-0.339584\pi$
$547$	22.5881	0.965797	0.482899	+	0.875676i	$0.339584\pi$
$548$	−11.7696	−0.502774
$549$	0	0
$550$	−0.223402	−0.00952588
$551$	−4.38460	−0.186790
$552$	0	0
$553$	−4.45097	−0.189274
$554$	12.6520	0.537534
$555$	0	0
$556$	3.20793	0.136047
$557$	11.2576	0.477002	0.238501	−	0.971142i	$-0.423344\pi$
$557$	11.2576	0.477002	0.238501	+	0.971142i	$0.423344\pi$
$558$	0	0
$559$	24.2627	1.02620
$560$	0.324179	0.0136991
$561$	0	0
$562$	−21.8216	−0.920489
$563$	1.57671	0.0664505	0.0332253	−	0.999448i	$-0.489422\pi$
$563$	1.57671	0.0664505	0.0332253	+	0.999448i	$0.489422\pi$
$564$	0	0
$565$	39.0236	1.64173
$566$	−28.0178	−1.17768
$567$	0	0
$568$	9.88206	0.414642
$569$	−16.4584	−0.689972	−0.344986	−	0.938608i	$-0.612116\pi$
$569$	−16.4584	−0.689972	−0.344986	+	0.938608i	$0.612116\pi$
$570$	0	0
$571$	25.7759	1.07869	0.539344	−	0.842085i	$-0.318672\pi$
$571$	25.7759	1.07869	0.539344	+	0.842085i	$0.318672\pi$
$572$	7.31677	0.305930
$573$	0	0
$574$	10.0531	0.419610
$575$	0.480567	0.0200410
$576$	0	0
$577$	−1.14494	−0.0476643	−0.0238321	−	0.999716i	$-0.507587\pi$
$577$	−1.14494	−0.0476643	−0.0238321	+	0.999716i	$0.507587\pi$
$578$	−0.677510	−0.0281807
$579$	0	0
$580$	−12.5331	−0.520409
$581$	0.286494	0.0118858
$582$	0	0
$583$	21.7081	0.899057
$584$	−10.4343	−0.431775
$585$	0	0
$586$	5.32219	0.219858
$587$	27.2210	1.12353	0.561766	−	0.827296i	$-0.310122\pi$
$587$	27.2210	1.12353	0.561766	+	0.827296i	$0.310122\pi$
$588$	0	0
$589$	0.583434	0.0240400
$590$	4.58015	0.188562
$591$	0	0
$592$	0.114559	0.00470834
$593$	33.1928	1.36306	0.681532	−	0.731788i	$-0.261314\pi$
$593$	33.1928	1.36306	0.681532	+	0.731788i	$0.261314\pi$
$594$	0	0
$595$	11.2517	0.461274
$596$	−24.1240	−0.988156
$597$	0	0
$598$	9.18288	0.375516
$599$	−33.5430	−1.37053	−0.685264	−	0.728294i	$-0.740313\pi$
$599$	−33.5430	−1.37053	−0.685264	+	0.728294i	$0.740313\pi$
$600$	0	0
$601$	−12.5884	−0.513492	−0.256746	−	0.966479i	$-0.582650\pi$
$601$	−12.5884	−0.513492	−0.256746	+	0.966479i	$0.582650\pi$
$602$	−9.07673	−0.369940
$603$	0	0
$604$	10.2089	0.415396
$605$	14.5148	0.590109
$606$	0	0
$607$	−6.92200	−0.280955	−0.140478	−	0.990084i	$-0.544864\pi$
$607$	−6.92200	−0.280955	−0.140478	+	0.990084i	$0.544864\pi$
$608$	−5.70664	−0.231435
$609$	0	0
$610$	2.62496	0.106282
$611$	−2.70495	−0.109431
$612$	0	0
$613$	40.4091	1.63211	0.816053	−	0.577977i	$-0.196158\pi$
$613$	40.4091	1.63211	0.816053	+	0.577977i	$0.196158\pi$
$614$	0.634055	0.0255884
$615$	0	0
$616$	−7.07144	−0.284916
$617$	36.6358	1.47490	0.737451	−	0.675400i	$-0.236029\pi$
$617$	36.6358	1.47490	0.737451	+	0.675400i	$0.236029\pi$
$618$	0	0
$619$	7.49747	0.301349	0.150674	−	0.988583i	$-0.451856\pi$
$619$	7.49747	0.301349	0.150674	+	0.988583i	$0.451856\pi$
$620$	1.66771	0.0669770
$621$	0	0
$622$	−22.9833	−0.921548
$623$	12.2892	0.492358
$624$	0	0
$625$	−25.5928	−1.02371
$626$	−16.1799	−0.646681
$627$	0	0
$628$	8.40976	0.335586
$629$	3.97614	0.158539
$630$	0	0
$631$	13.7047	0.545576	0.272788	−	0.962074i	$-0.412054\pi$
$631$	13.7047	0.545576	0.272788	+	0.962074i	$0.412054\pi$
$632$	10.5768	0.420722
$633$	0	0
$634$	−7.87724	−0.312845
$635$	6.46888	0.256709
$636$	0	0
$637$	15.1760	0.601294
$638$	8.06066	0.319124
$639$	0	0
$640$	−16.7843	−0.663456
$641$	20.6129	0.814159	0.407080	−	0.913393i	$-0.366547\pi$
$641$	20.6129	0.814159	0.407080	+	0.913393i	$0.366547\pi$
$642$	0	0
$643$	7.54000	0.297349	0.148674	−	0.988886i	$-0.452499\pi$
$643$	7.54000	0.297349	0.148674	+	0.988886i	$0.452499\pi$
$644$	5.88814	0.232025
$645$	0	0
$646$	−3.62067	−0.142454
$647$	−41.0078	−1.61218	−0.806092	−	0.591791i	$-0.798421\pi$
$647$	−41.0078	−1.61218	−0.806092	+	0.591791i	$0.798421\pi$
$648$	0	0
$649$	5.04893	0.198188
$650$	0.282174	0.0110678
$651$	0	0
$652$	−19.2580	−0.754200
$653$	26.6959	1.04469	0.522346	−	0.852734i	$-0.325057\pi$
$653$	26.6959	1.04469	0.522346	+	0.852734i	$0.325057\pi$
$654$	0	0
$655$	13.8718	0.542017
$656$	1.20725	0.0471351
$657$	0	0
$658$	1.01193	0.0394492
$659$	17.9382	0.698773	0.349386	−	0.936979i	$-0.386390\pi$
$659$	17.9382	0.698773	0.349386	+	0.936979i	$0.386390\pi$
$660$	0	0
$661$	−35.9775	−1.39936	−0.699682	−	0.714455i	$-0.746675\pi$
$661$	−35.9775	−1.39936	−0.699682	+	0.714455i	$0.746675\pi$
$662$	−3.86862	−0.150358
$663$	0	0
$664$	−0.680791	−0.0264198
$665$	2.66771	0.103449
$666$	0	0
$667$	−17.3395	−0.671390
$668$	7.28511	0.281869
$669$	0	0
$670$	13.3376	0.515275
$671$	2.89363	0.111707
$672$	0	0
$673$	18.7392	0.722341	0.361171	−	0.932500i	$-0.382377\pi$
$673$	18.7392	0.722341	0.361171	+	0.932500i	$0.382377\pi$
$674$	−20.0613	−0.772731
$675$	0	0
$676$	7.17835	0.276091
$677$	6.53197	0.251044	0.125522	−	0.992091i	$-0.459939\pi$
$677$	6.53197	0.251044	0.125522	+	0.992091i	$0.459939\pi$
$678$	0	0
$679$	−0.876251	−0.0336274
$680$	−26.7372	−1.02533
$681$	0	0
$682$	−1.07259	−0.0410715
$683$	−3.28527	−0.125707	−0.0628537	−	0.998023i	$-0.520020\pi$
$683$	−3.28527	−0.125707	−0.0628537	+	0.998023i	$0.520020\pi$
$684$	0	0
$685$	−21.0879	−0.805726
$686$	−12.7609	−0.487213
$687$	0	0
$688$	−1.08999	−0.0415557
$689$	−27.4190	−1.04458
$690$	0	0
$691$	−17.8110	−0.677561	−0.338780	−	0.940865i	$-0.610014\pi$
$691$	−17.8110	−0.677561	−0.338780	+	0.940865i	$0.610014\pi$
$692$	18.4361	0.700837
$693$	0	0
$694$	22.0493	0.836979
$695$	5.74771	0.218023
$696$	0	0
$697$	41.9015	1.58713
$698$	6.15640	0.233023
$699$	0	0
$700$	0.180932	0.00683858
$701$	23.2341	0.877539	0.438770	−	0.898600i	$-0.355414\pi$
$701$	23.2341	0.877539	0.438770	+	0.898600i	$0.355414\pi$
$702$	0	0
$703$	0.942721	0.0355554
$704$	9.97063	0.375782
$705$	0	0
$706$	9.99807	0.376282
$707$	11.2638	0.423617
$708$	0	0
$709$	20.0153	0.751691	0.375845	−	0.926682i	$-0.377352\pi$
$709$	20.0153	0.751691	0.375845	+	0.926682i	$0.377352\pi$
$710$	6.85361	0.257212
$711$	0	0
$712$	−29.2027	−1.09442
$713$	2.30728	0.0864083
$714$	0	0
$715$	13.1096	0.490271
$716$	−11.9946	−0.448259
$717$	0	0
$718$	−8.86250	−0.330746
$719$	−1.15142	−0.0429407	−0.0214703	−	0.999769i	$-0.506835\pi$
$719$	−1.15142	−0.0429407	−0.0214703	+	0.999769i	$0.506835\pi$
$720$	0	0
$721$	14.1302	0.526235
$722$	−0.858442	−0.0319479
$723$	0	0
$724$	−0.587700	−0.0218417
$725$	−0.532813	−0.0197882
$726$	0	0
$727$	−8.55271	−0.317202	−0.158601	−	0.987343i	$-0.550698\pi$
$727$	−8.55271	−0.317202	−0.158601	+	0.987343i	$0.550698\pi$
$728$	8.93178	0.331034
$729$	0	0
$730$	−7.23662	−0.267839
$731$	−37.8318	−1.39926
$732$	0	0
$733$	−16.5392	−0.610888	−0.305444	−	0.952210i	$-0.598805\pi$
$733$	−16.5392	−0.610888	−0.305444	+	0.952210i	$0.598805\pi$
$734$	−8.22952	−0.303757
$735$	0	0
$736$	−22.5678	−0.831859
$737$	14.7027	0.541579
$738$	0	0
$739$	−44.9355	−1.65298	−0.826490	−	0.562952i	$-0.809666\pi$
$739$	−44.9355	−1.65298	−0.826490	+	0.562952i	$0.809666\pi$
$740$	2.69471	0.0990596
$741$	0	0
$742$	10.2575	0.376565
$743$	−41.9497	−1.53898	−0.769492	−	0.638656i	$-0.779491\pi$
$743$	−41.9497	−1.53898	−0.769492	+	0.638656i	$0.779491\pi$
$744$	0	0
$745$	−43.2233	−1.58358
$746$	−17.7408	−0.649538
$747$	0	0
$748$	−11.4088	−0.417146
$749$	13.1439	0.480266
$750$	0	0
$751$	−36.2202	−1.32169	−0.660846	−	0.750521i	$-0.729802\pi$
$751$	−36.2202	−1.32169	−0.660846	+	0.750521i	$0.729802\pi$
$752$	0.121519	0.00443136
$753$	0	0
$754$	−10.1812	−0.370779
$755$	18.2915	0.665697
$756$	0	0
$757$	23.5730	0.856774	0.428387	−	0.903595i	$-0.359082\pi$
$757$	23.5730	0.856774	0.428387	+	0.903595i	$0.359082\pi$
$758$	−13.4024	−0.486796
$759$	0	0
$760$	−6.33925	−0.229949
$761$	24.3637	0.883184	0.441592	−	0.897216i	$-0.354414\pi$
$761$	24.3637	0.883184	0.441592	+	0.897216i	$0.354414\pi$
$762$	0	0
$763$	4.81905	0.174461
$764$	9.35746	0.338541
$765$	0	0
$766$	12.5870	0.454786
$767$	−6.37719	−0.230267
$768$	0	0
$769$	1.21489	0.0438101	0.0219051	−	0.999760i	$-0.493027\pi$
$769$	1.21489	0.0438101	0.0219051	+	0.999760i	$0.493027\pi$
$770$	−4.90433	−0.176740
$771$	0	0
$772$	−29.8058	−1.07274
$773$	9.01437	0.324224	0.162112	−	0.986772i	$-0.448169\pi$
$773$	9.01437	0.324224	0.162112	+	0.986772i	$0.448169\pi$
$774$	0	0
$775$	0.0708986	0.00254675
$776$	2.08222	0.0747474
$777$	0	0
$778$	4.59492	0.164736
$779$	9.93461	0.355944
$780$	0	0
$781$	7.55508	0.270342
$782$	−14.3185	−0.512029
$783$	0	0
$784$	−0.681776	−0.0243492
$785$	15.0679	0.537797
$786$	0	0
$787$	−0.158316	−0.00564335	−0.00282167	−	0.999996i	$-0.500898\pi$
$787$	−0.158316	−0.00564335	−0.00282167	+	0.999996i	$0.500898\pi$
$788$	−27.5502	−0.981434
$789$	0	0
$790$	7.33543	0.260983
$791$	20.3267	0.722735
$792$	0	0
$793$	−3.65487	−0.129788
$794$	−24.3842	−0.865365
$795$	0	0
$796$	2.57020	0.0910982
$797$	−13.6200	−0.482444	−0.241222	−	0.970470i	$-0.577548\pi$
$797$	−13.6200	−0.482444	−0.241222	+	0.970470i	$0.577548\pi$
$798$	0	0
$799$	4.21773	0.149213
$800$	−0.693468	−0.0245178
$801$	0	0
$802$	−18.2366	−0.643955
$803$	−7.97729	−0.281512
$804$	0	0
$805$	10.5499	0.371834
$806$	1.35476	0.0477194
$807$	0	0
$808$	−26.7659	−0.941622
$809$	−16.1180	−0.566679	−0.283339	−	0.959020i	$-0.591442\pi$
$809$	−16.1180	−0.566679	−0.283339	+	0.959020i	$0.591442\pi$
$810$	0	0
$811$	−41.8564	−1.46978	−0.734889	−	0.678187i	$-0.762766\pi$
$811$	−41.8564	−1.46978	−0.734889	+	0.678187i	$0.762766\pi$
$812$	−6.52829	−0.229098
$813$	0	0
$814$	−1.73310	−0.0607451
$815$	−34.5048	−1.20865
$816$	0	0
$817$	−8.96972	−0.313811
$818$	8.23831	0.288046
$819$	0	0
$820$	28.3975	0.991684
$821$	−1.27050	−0.0443408	−0.0221704	−	0.999754i	$-0.507058\pi$
$821$	−1.27050	−0.0443408	−0.0221704	+	0.999754i	$0.507058\pi$
$822$	0	0
$823$	−17.2019	−0.599621	−0.299810	−	0.953999i	$-0.596923\pi$
$823$	−17.2019	−0.599621	−0.299810	+	0.953999i	$0.596923\pi$
$824$	−33.5773	−1.16972
$825$	0	0
$826$	2.38572	0.0830100
$827$	55.9842	1.94676	0.973381	−	0.229192i	$-0.0736085\pi$
$827$	55.9842	1.94676	0.973381	+	0.229192i	$0.0736085\pi$
$828$	0	0
$829$	−8.18769	−0.284370	−0.142185	−	0.989840i	$-0.545413\pi$
$829$	−8.18769	−0.284370	−0.142185	+	0.989840i	$0.545413\pi$
$830$	−0.472156	−0.0163888
$831$	0	0
$832$	−12.5937	−0.436607
$833$	−23.6633	−0.819884
$834$	0	0
$835$	13.0529	0.451713
$836$	−2.70495	−0.0935528
$837$	0	0
$838$	−20.5479	−0.709816
$839$	5.12848	0.177055	0.0885274	−	0.996074i	$-0.471784\pi$
$839$	5.12848	0.177055	0.0885274	+	0.996074i	$0.471784\pi$
$840$	0	0
$841$	−9.77531	−0.337080
$842$	2.18315	0.0752363
$843$	0	0
$844$	24.1020	0.829626
$845$	12.8616	0.442452
$846$	0	0
$847$	7.56050	0.259782
$848$	1.23179	0.0422999
$849$	0	0
$850$	−0.439982	−0.0150913
$851$	3.72813	0.127799
$852$	0	0
$853$	−46.0747	−1.57757	−0.788784	−	0.614670i	$-0.789289\pi$
$853$	−46.0747	−1.57757	−0.788784	+	0.614670i	$0.789289\pi$
$854$	1.36730	0.0467880
$855$	0	0
$856$	−31.2335	−1.06754
$857$	−13.6215	−0.465302	−0.232651	−	0.972560i	$-0.574740\pi$
$857$	−13.6215	−0.465302	−0.232651	+	0.972560i	$0.574740\pi$
$858$	0	0
$859$	53.0888	1.81137	0.905683	−	0.423956i	$-0.139359\pi$
$859$	53.0888	1.81137	0.905683	+	0.423956i	$0.139359\pi$
$860$	−25.6394	−0.874297
$861$	0	0
$862$	−25.4150	−0.865640
$863$	10.0333	0.341538	0.170769	−	0.985311i	$-0.445375\pi$
$863$	10.0333	0.341538	0.170769	+	0.985311i	$0.445375\pi$
$864$	0	0
$865$	33.0323	1.12313
$866$	1.85744	0.0631183
$867$	0	0
$868$	0.868683	0.0294850
$869$	8.08621	0.274306
$870$	0	0
$871$	−18.5706	−0.629241
$872$	−11.4514	−0.387794
$873$	0	0
$874$	−3.39484	−0.114832
$875$	−13.0144	−0.439967
$876$	0	0
$877$	29.6149	1.00002	0.500012	−	0.866019i	$-0.333329\pi$
$877$	29.6149	1.00002	0.500012	+	0.866019i	$0.333329\pi$
$878$	−4.80599	−0.162194
$879$	0	0
$880$	−0.588945	−0.0198533
$881$	42.9908	1.44840	0.724199	−	0.689591i	$-0.242210\pi$
$881$	42.9908	1.44840	0.724199	+	0.689591i	$0.242210\pi$
$882$	0	0
$883$	−14.2634	−0.480002	−0.240001	−	0.970773i	$-0.577148\pi$
$883$	−14.2634	−0.480002	−0.240001	+	0.970773i	$0.577148\pi$
$884$	14.4101	0.484666
$885$	0	0
$886$	3.67024	0.123304
$887$	−29.4404	−0.988511	−0.494255	−	0.869317i	$-0.664559\pi$
$887$	−29.4404	−0.988511	−0.494255	+	0.869317i	$0.664559\pi$
$888$	0	0
$889$	3.36953	0.113010
$890$	−20.2533	−0.678891
$891$	0	0
$892$	−4.07052	−0.136291
$893$	1.00000	0.0334637
$894$	0	0
$895$	−21.4909	−0.718362
$896$	−8.74264	−0.292071
$897$	0	0
$898$	4.38983	0.146490
$899$	−2.55812	−0.0853182
$900$	0	0
$901$	42.7534	1.42432
$902$	−18.2638	−0.608119
$903$	0	0
$904$	−48.3021	−1.60650
$905$	−1.05299	−0.0350027
$906$	0	0
$907$	5.09122	0.169051	0.0845256	−	0.996421i	$-0.473063\pi$
$907$	5.09122	0.169051	0.0845256	+	0.996421i	$0.473063\pi$
$908$	4.94586	0.164134
$909$	0	0
$910$	6.19455	0.205347
$911$	−22.1362	−0.733406	−0.366703	−	0.930338i	$-0.619513\pi$
$911$	−22.1362	−0.733406	−0.366703	+	0.930338i	$0.619513\pi$
$912$	0	0
$913$	−0.520482	−0.0172254
$914$	14.0733	0.465503
$915$	0	0
$916$	13.4257	0.443596
$917$	7.22560	0.238610
$918$	0	0
$919$	−23.0583	−0.760623	−0.380311	−	0.924859i	$-0.624183\pi$
$919$	−23.0583	−0.760623	−0.380311	+	0.924859i	$0.624183\pi$
$920$	−25.0695	−0.826517
$921$	0	0
$922$	−8.16150	−0.268785
$923$	−9.54265	−0.314100
$924$	0	0
$925$	0.114559	0.00376667
$926$	17.3727	0.570902
$927$	0	0
$928$	25.0213	0.821365
$929$	−25.2452	−0.828269	−0.414135	−	0.910216i	$-0.635916\pi$
$929$	−25.2452	−0.828269	−0.414135	+	0.910216i	$0.635916\pi$
$930$	0	0
$931$	−5.61043	−0.183875
$932$	21.4667	0.703166
$933$	0	0
$934$	−8.09506	−0.264878
$935$	−20.4413	−0.668501
$936$	0	0
$937$	−5.31806	−0.173733	−0.0868667	−	0.996220i	$-0.527685\pi$
$937$	−5.31806	−0.173733	−0.0868667	+	0.996220i	$0.527685\pi$
$938$	6.94731	0.226838
$939$	0	0
$940$	2.85844	0.0932321
$941$	5.78565	0.188607	0.0943034	−	0.995544i	$-0.469938\pi$
$941$	5.78565	0.188607	0.0943034	+	0.995544i	$0.469938\pi$
$942$	0	0
$943$	39.2879	1.27939
$944$	0.286494	0.00932458
$945$	0	0
$946$	16.4900	0.536135
$947$	21.4304	0.696394	0.348197	−	0.937421i	$-0.386794\pi$
$947$	21.4304	0.696394	0.348197	+	0.937421i	$0.386794\pi$
$948$	0	0
$949$	10.0759	0.327079
$950$	−0.104317	−0.00338450
$951$	0	0
$952$	−13.9270	−0.451376
$953$	7.91359	0.256346	0.128173	−	0.991752i	$-0.459089\pi$
$953$	7.91359	0.256346	0.128173	+	0.991752i	$0.459089\pi$
$954$	0	0
$955$	16.7659	0.542532
$956$	38.8772	1.25738
$957$	0	0
$958$	11.2860	0.364634
$959$	−10.9843	−0.354702
$960$	0	0
$961$	−30.6596	−0.989020
$962$	2.18904	0.0705775
$963$	0	0
$964$	−23.1636	−0.746049
$965$	−53.4036	−1.71912
$966$	0	0
$967$	−4.02294	−0.129369	−0.0646845	−	0.997906i	$-0.520604\pi$
$967$	−4.02294	−0.129369	−0.0646845	+	0.997906i	$0.520604\pi$
$968$	−17.9659	−0.577446
$969$	0	0
$970$	1.44411	0.0463675
$971$	46.8894	1.50475	0.752377	−	0.658733i	$-0.228907\pi$
$971$	46.8894	1.50475	0.752377	+	0.658733i	$0.228907\pi$
$972$	0	0
$973$	2.99388	0.0959795
$974$	−14.2438	−0.456401
$975$	0	0
$976$	0.164194	0.00525573
$977$	36.9309	1.18153	0.590763	−	0.806845i	$-0.298827\pi$
$977$	36.9309	1.18153	0.590763	+	0.806845i	$0.298827\pi$
$978$	0	0
$979$	−22.3262	−0.713549
$980$	−16.0371	−0.512286
$981$	0	0
$982$	23.1715	0.739433
$983$	43.6670	1.39276	0.696381	−	0.717672i	$-0.254792\pi$
$983$	43.6670	1.39276	0.696381	+	0.717672i	$0.254792\pi$
$984$	0	0
$985$	−49.3621	−1.57281
$986$	15.8752	0.505569
$987$	0	0
$988$	3.41657	0.108695
$989$	−35.4721	−1.12795
$990$	0	0
$991$	13.3028	0.422579	0.211289	−	0.977424i	$-0.432234\pi$
$991$	13.3028	0.422579	0.211289	+	0.977424i	$0.432234\pi$
$992$	−3.32945	−0.105710
$993$	0	0
$994$	3.56993	0.113231
$995$	4.60507	0.145990
$996$	0	0
$997$	8.66701	0.274487	0.137243	−	0.990537i	$-0.456176\pi$
$997$	8.66701	0.274487	0.137243	+	0.990537i	$0.456176\pi$
$998$	−26.6790	−0.844509
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.g.1.2		4
3.2	odd	2		2679.2.a.h.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.h.1.3	✓	4	3.2	odd	2
8037.2.a.g.1.2		4	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)

\(f(q)\)	\(=\)	\(q-0.858442 q^{2} -1.26308 q^{4} -2.26308 q^{5} -1.17880 q^{7} +2.80116 q^{8} +O(q^{10})\)	\(q-0.858442 q^{2} -1.26308 q^{4} -2.26308 q^{5} -1.17880 q^{7} +2.80116 q^{8} +1.94272 q^{10} +2.14156 q^{11} -2.70495 q^{13} +1.01193 q^{14} +0.121519 q^{16} +4.21773 q^{17} +1.00000 q^{19} +2.85844 q^{20} -1.83840 q^{22} +3.95465 q^{23} +0.121519 q^{25} +2.32205 q^{26} +1.48891 q^{28} -4.38460 q^{29} +0.583434 q^{31} -5.70664 q^{32} -3.62067 q^{34} +2.66771 q^{35} +0.942721 q^{37} -0.858442 q^{38} -6.33925 q^{40} +9.93461 q^{41} -8.96972 q^{43} -2.70495 q^{44} -3.39484 q^{46} +1.00000 q^{47} -5.61043 q^{49} -0.104317 q^{50} +3.41657 q^{52} +10.1366 q^{53} -4.84651 q^{55} -3.30201 q^{56} +3.76392 q^{58} +2.35760 q^{59} +1.35118 q^{61} -0.500844 q^{62} +4.65578 q^{64} +6.12152 q^{65} +6.86540 q^{67} -5.32732 q^{68} -2.29008 q^{70} +3.52784 q^{71} -3.72499 q^{73} -0.809271 q^{74} -1.26308 q^{76} -2.52447 q^{77} +3.77585 q^{79} -0.275008 q^{80} -8.52829 q^{82} -0.243039 q^{83} -9.54505 q^{85} +7.69998 q^{86} +5.99885 q^{88} -10.4252 q^{89} +3.18860 q^{91} -4.99503 q^{92} -0.858442 q^{94} -2.26308 q^{95} +0.743342 q^{97} +4.81623 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 - 2 * q^5 - 5 * q^7 - 3 * q^8	\( 4 q + 2 q^{4} - 2 q^{5} - 5 q^{7} - 3 q^{8} - 3 q^{10} + 12 q^{11} + 3 q^{13} + 11 q^{14} - 10 q^{16} + 6 q^{17} + 4 q^{19} + 8 q^{20} + 10 q^{22} + 12 q^{23} - 10 q^{25} + 5 q^{26} - 5 q^{28} - q^{31} - q^{32} - 8 q^{34} - 7 q^{37} - 4 q^{40} + 22 q^{41} - 7 q^{43} + 3 q^{44} - 11 q^{46} + 4 q^{47} - q^{49} - 7 q^{50} + 17 q^{52} + 5 q^{53} - 9 q^{55} + 5 q^{56} + 4 q^{58} + 10 q^{59} - 9 q^{61} + 2 q^{62} + q^{64} + 14 q^{65} - 8 q^{67} + 7 q^{68} + 16 q^{70} - 8 q^{71} - 15 q^{73} + 4 q^{74} + 2 q^{76} - 4 q^{77} + 11 q^{79} - q^{80} - 28 q^{82} + 20 q^{83} + q^{85} - 22 q^{86} - 15 q^{88} - 5 q^{89} + 12 q^{91} + 19 q^{92} - 2 q^{95} + 25 q^{97} - 20 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 - 5 * q^7 - 3 * q^8 - 3 * q^10 + 12 * q^11 + 3 * q^13 + 11 * q^14 - 10 * q^16 + 6 * q^17 + 4 * q^19 + 8 * q^20 + 10 * q^22 + 12 * q^23 - 10 * q^25 + 5 * q^26 - 5 * q^28 - q^31 - q^32 - 8 * q^34 - 7 * q^37 - 4 * q^40 + 22 * q^41 - 7 * q^43 + 3 * q^44 - 11 * q^46 + 4 * q^47 - q^49 - 7 * q^50 + 17 * q^52 + 5 * q^53 - 9 * q^55 + 5 * q^56 + 4 * q^58 + 10 * q^59 - 9 * q^61 + 2 * q^62 + q^64 + 14 * q^65 - 8 * q^67 + 7 * q^68 + 16 * q^70 - 8 * q^71 - 15 * q^73 + 4 * q^74 + 2 * q^76 - 4 * q^77 + 11 * q^79 - q^80 - 28 * q^82 + 20 * q^83 + q^85 - 22 * q^86 - 15 * q^88 - 5 * q^89 + 12 * q^91 + 19 * q^92 - 2 * q^95 + 25 * q^97 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more