LMFDB - Embedded newform 8037.2.a.i.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.5476681.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 19x^{3} + 32x^{2} - 44x - 8 $ x^6 - 2x^5 - 11x^4 + 19x^3 + 32x^2 - 44*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			$1.40754$ 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)

$f(q)$	$=$	$q-2.24698 q^{2} +3.04892 q^{4} +1.32682 q^{5} +1.40754 q^{7} -2.35690 q^{8} +O(q^{10})$	\(q-2.24698 q^{2} +3.04892 q^{4} +1.32682 q^{5} +1.40754 q^{7} -2.35690 q^{8} -2.98134 q^{10} +3.24698 q^{11} -2.03008 q^{13} -3.16271 q^{14} -0.801938 q^{16} -0.145448 q^{17} -1.00000 q^{19} +4.04536 q^{20} -7.29590 q^{22} +4.01141 q^{23} -3.23955 q^{25} +4.56154 q^{26} +4.29146 q^{28} +0.962495 q^{29} -3.87142 q^{31} +6.51573 q^{32} +0.326819 q^{34} +1.86755 q^{35} -6.74481 q^{37} +2.24698 q^{38} -3.12717 q^{40} -8.07843 q^{41} +8.20332 q^{43} +9.89977 q^{44} -9.01356 q^{46} +1.00000 q^{47} -5.01884 q^{49} +7.27921 q^{50} -6.18954 q^{52} -7.02209 q^{53} +4.30815 q^{55} -3.31742 q^{56} -2.16271 q^{58} +0.958232 q^{59} -5.59747 q^{61} +8.69900 q^{62} -13.0368 q^{64} -2.69354 q^{65} +9.39760 q^{67} -0.443459 q^{68} -4.19634 q^{70} +2.46388 q^{71} -4.63077 q^{73} +15.1555 q^{74} -3.04892 q^{76} +4.57024 q^{77} -8.02107 q^{79} -1.06403 q^{80} +18.1521 q^{82} +8.49351 q^{83} -0.192983 q^{85} -18.4327 q^{86} -7.65279 q^{88} +3.26469 q^{89} -2.85741 q^{91} +12.2305 q^{92} -2.24698 q^{94} -1.32682 q^{95} -13.1343 q^{97} +11.2772 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 4 * q^5 + 2 * q^7 - 6 * q^8	$ 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 10 q^{11} - 8 q^{13} + q^{14} + 4 q^{16} + 3 q^{17} - 6 q^{19} - 7 q^{20} - 16 q^{22} + 10 q^{26} - 7 q^{28} - 2 q^{31} + 14 q^{32} - 2 q^{34} - 7 q^{35} - 9 q^{37} + 4 q^{38} - 4 q^{40} - 18 q^{41} + 9 q^{43} + 14 q^{44} - 7 q^{46} + 6 q^{47} - 16 q^{49} + 7 q^{50} - 21 q^{52} - 20 q^{53} + 2 q^{55} + 5 q^{56} + 7 q^{58} - 19 q^{59} + 13 q^{62} - 22 q^{64} + 22 q^{65} - 11 q^{67} - 14 q^{70} - 4 q^{71} - 7 q^{73} + 27 q^{74} + q^{77} - 16 q^{79} + 5 q^{80} - 2 q^{82} - 7 q^{83} - 25 q^{85} + q^{86} - 10 q^{88} + 33 q^{89} - 4 q^{91} + 21 q^{92} - 4 q^{94} - 4 q^{95} - 34 q^{97} + 20 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 4 * q^5 + 2 * q^7 - 6 * q^8 + 2 * q^10 + 10 * q^11 - 8 * q^13 + q^14 + 4 * q^16 + 3 * q^17 - 6 * q^19 - 7 * q^20 - 16 * q^22 + 10 * q^26 - 7 * q^28 - 2 * q^31 + 14 * q^32 - 2 * q^34 - 7 * q^35 - 9 * q^37 + 4 * q^38 - 4 * q^40 - 18 * q^41 + 9 * q^43 + 14 * q^44 - 7 * q^46 + 6 * q^47 - 16 * q^49 + 7 * q^50 - 21 * q^52 - 20 * q^53 + 2 * q^55 + 5 * q^56 + 7 * q^58 - 19 * q^59 + 13 * q^62 - 22 * q^64 + 22 * q^65 - 11 * q^67 - 14 * q^70 - 4 * q^71 - 7 * q^73 + 27 * q^74 + q^77 - 16 * q^79 + 5 * q^80 - 2 * q^82 - 7 * q^83 - 25 * q^85 + q^86 - 10 * q^88 + 33 * q^89 - 4 * q^91 + 21 * q^92 - 4 * q^94 - 4 * q^95 - 34 * 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 $

$2$	−2.24698	−1.58885	−0.794427	−	0.607359i	$-0.792229\pi$
$2$	−2.24698	−1.58885	−0.794427	+	0.607359i	$0.792229\pi$
$3$	0	0
$3$	0	0
$4$	3.04892	1.52446
$5$	1.32682	0.593372	0.296686	−	0.954975i	$-0.404119\pi$
$5$	1.32682	0.593372	0.296686	+	0.954975i	$0.404119\pi$
$6$	0	0
$7$	1.40754	0.531999	0.265999	−	0.963973i	$-0.414298\pi$
$7$	1.40754	0.531999	0.265999	+	0.963973i	$0.414298\pi$
$8$	−2.35690	−0.833289
$9$	0	0
$10$	−2.98134	−0.942781
$11$	3.24698	0.979001	0.489501	−	0.872003i	$-0.337179\pi$
$11$	3.24698	0.979001	0.489501	+	0.872003i	$0.337179\pi$
$12$	0	0
$13$	−2.03008	−0.563042	−0.281521	−	0.959555i	$-0.590839\pi$
$13$	−2.03008	−0.563042	−0.281521	+	0.959555i	$0.590839\pi$
$14$	−3.16271	−0.845269
$15$	0	0
$16$	−0.801938	−0.200484
$17$	−0.145448	−0.0352764	−0.0176382	−	0.999844i	$-0.505615\pi$
$17$	−0.145448	−0.0352764	−0.0176382	+	0.999844i	$0.505615\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.04536	0.904570
$21$	0	0
$22$	−7.29590	−1.55549
$23$	4.01141	0.836437	0.418219	−	0.908346i	$-0.362655\pi$
$23$	4.01141	0.836437	0.418219	+	0.908346i	$0.362655\pi$
$24$	0	0
$25$	−3.23955	−0.647910
$26$	4.56154	0.894592
$27$	0	0
$28$	4.29146	0.811010
$29$	0.962495	0.178731	0.0893654	−	0.995999i	$-0.471516\pi$
$29$	0.962495	0.178731	0.0893654	+	0.995999i	$0.471516\pi$
$30$	0	0
$31$	−3.87142	−0.695327	−0.347664	−	0.937619i	$-0.613025\pi$
$31$	−3.87142	−0.695327	−0.347664	+	0.937619i	$0.613025\pi$
$32$	6.51573	1.15183
$33$	0	0
$34$	0.326819	0.0560490
$35$	1.86755	0.315673
$36$	0	0
$37$	−6.74481	−1.10884	−0.554420	−	0.832237i	$-0.687060\pi$
$37$	−6.74481	−1.10884	−0.554420	+	0.832237i	$0.687060\pi$
$38$	2.24698	0.364508
$39$	0	0
$40$	−3.12717	−0.494450
$41$	−8.07843	−1.26164	−0.630820	−	0.775930i	$-0.717281\pi$
$41$	−8.07843	−1.26164	−0.630820	+	0.775930i	$0.717281\pi$
$42$	0	0
$43$	8.20332	1.25099	0.625497	−	0.780227i	$-0.284896\pi$
$43$	8.20332	1.25099	0.625497	+	0.780227i	$0.284896\pi$
$44$	9.89977	1.49245
$45$	0	0
$46$	−9.01356	−1.32898
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−5.01884	−0.716977
$50$	7.27921	1.02944
$51$	0	0
$52$	−6.18954	−0.858334
$53$	−7.02209	−0.964558	−0.482279	−	0.876018i	$-0.660191\pi$
$53$	−7.02209	−0.964558	−0.482279	+	0.876018i	$0.660191\pi$
$54$	0	0
$55$	4.30815	0.580911
$56$	−3.31742	−0.443309
$57$	0	0
$58$	−2.16271	−0.283977
$59$	0.958232	0.124751	0.0623756	−	0.998053i	$-0.480132\pi$
$59$	0.958232	0.124751	0.0623756	+	0.998053i	$0.480132\pi$
$60$	0	0
$61$	−5.59747	−0.716682	−0.358341	−	0.933591i	$-0.616658\pi$
$61$	−5.59747	−0.716682	−0.358341	+	0.933591i	$0.616658\pi$
$62$	8.69900	1.10477
$63$	0	0
$64$	−13.0368	−1.62960
$65$	−2.69354	−0.334093
$66$	0	0
$67$	9.39760	1.14810	0.574050	−	0.818820i	$-0.305371\pi$
$67$	9.39760	1.14810	0.574050	+	0.818820i	$0.305371\pi$
$68$	−0.443459	−0.0537774
$69$	0	0
$70$	−4.19634	−0.501558
$71$	2.46388	0.292409	0.146205	−	0.989254i	$-0.453294\pi$
$71$	2.46388	0.292409	0.146205	+	0.989254i	$0.453294\pi$
$72$	0	0
$73$	−4.63077	−0.541991	−0.270995	−	0.962581i	$-0.587353\pi$
$73$	−4.63077	−0.541991	−0.270995	+	0.962581i	$0.587353\pi$
$74$	15.1555	1.76179
$75$	0	0
$76$	−3.04892	−0.349735
$77$	4.57024	0.520828
$78$	0	0
$79$	−8.02107	−0.902441	−0.451220	−	0.892413i	$-0.649011\pi$
$79$	−8.02107	−0.902441	−0.451220	+	0.892413i	$0.649011\pi$
$80$	−1.06403	−0.118962
$81$	0	0
$82$	18.1521	2.00456
$83$	8.49351	0.932284	0.466142	−	0.884710i	$-0.345644\pi$
$83$	8.49351	0.932284	0.466142	+	0.884710i	$0.345644\pi$
$84$	0	0
$85$	−0.192983	−0.0209320
$86$	−18.4327	−1.98765
$87$	0	0
$88$	−7.65279	−0.815790
$89$	3.26469	0.346056	0.173028	−	0.984917i	$-0.444645\pi$
$89$	3.26469	0.346056	0.173028	+	0.984917i	$0.444645\pi$
$90$	0	0
$91$	−2.85741	−0.299538
$92$	12.2305	1.27511
$93$	0	0
$94$	−2.24698	−0.231758
$95$	−1.32682	−0.136129
$96$	0	0
$97$	−13.1343	−1.33359	−0.666794	−	0.745242i	$-0.732334\pi$
$97$	−13.1343	−1.33359	−0.666794	+	0.745242i	$0.732334\pi$
$98$	11.2772	1.13917
$99$	0	0
$100$	−9.87712	−0.987712
$101$	−11.9871	−1.19276	−0.596379	−	0.802703i	$-0.703395\pi$
$101$	−11.9871	−1.19276	−0.596379	+	0.802703i	$0.703395\pi$
$102$	0	0
$103$	−11.2609	−1.10957	−0.554786	−	0.831993i	$-0.687200\pi$
$103$	−11.2609	−1.10957	−0.554786	+	0.831993i	$0.687200\pi$
$104$	4.78468	0.469176
$105$	0	0
$106$	15.7785	1.53254
$107$	4.03350	0.389933	0.194967	−	0.980810i	$-0.437540\pi$
$107$	4.03350	0.389933	0.194967	+	0.980810i	$0.437540\pi$
$108$	0	0
$109$	−18.3270	−1.75540	−0.877702	−	0.479206i	$-0.840925\pi$
$109$	−18.3270	−1.75540	−0.877702	+	0.479206i	$0.840925\pi$
$110$	−9.68033	−0.922984
$111$	0	0
$112$	−1.12876	−0.106657
$113$	0.426516	0.0401232	0.0200616	−	0.999799i	$-0.493614\pi$
$113$	0.426516	0.0401232	0.0200616	+	0.999799i	$0.493614\pi$
$114$	0	0
$115$	5.32242	0.496318
$116$	2.93457	0.272468
$117$	0	0
$118$	−2.15313	−0.198212
$119$	−0.204724	−0.0187670
$120$	0	0
$121$	−0.457123	−0.0415567
$122$	12.5774	1.13870
$123$	0	0
$124$	−11.8036	−1.06000
$125$	−10.9324	−0.977823
$126$	0	0
$127$	−8.81895	−0.782555	−0.391277	−	0.920273i	$-0.627967\pi$
$127$	−8.81895	−0.782555	−0.391277	+	0.920273i	$0.627967\pi$
$128$	16.2620	1.43738
$129$	0	0
$130$	6.05234	0.530825
$131$	−2.39979	−0.209671	−0.104836	−	0.994490i	$-0.533432\pi$
$131$	−2.39979	−0.209671	−0.104836	+	0.994490i	$0.533432\pi$
$132$	0	0
$133$	−1.40754	−0.122049
$134$	−21.1162	−1.82416
$135$	0	0
$136$	0.342806	0.0293954
$137$	−0.788724	−0.0673852	−0.0336926	−	0.999432i	$-0.510727\pi$
$137$	−0.788724	−0.0673852	−0.0336926	+	0.999432i	$0.510727\pi$
$138$	0	0
$139$	5.19429	0.440574	0.220287	−	0.975435i	$-0.429301\pi$
$139$	5.19429	0.440574	0.220287	+	0.975435i	$0.429301\pi$
$140$	5.69399	0.481230
$141$	0	0
$142$	−5.53630	−0.464596
$143$	−6.59162	−0.551219
$144$	0	0
$145$	1.27706	0.106054
$146$	10.4053	0.861145
$147$	0	0
$148$	−20.5644	−1.69038
$149$	−10.7040	−0.876906	−0.438453	−	0.898754i	$-0.644473\pi$
$149$	−10.7040	−0.876906	−0.438453	+	0.898754i	$0.644473\pi$
$150$	0	0
$151$	22.8781	1.86179	0.930896	−	0.365285i	$-0.119028\pi$
$151$	22.8781	1.86179	0.930896	+	0.365285i	$0.119028\pi$
$152$	2.35690	0.191169
$153$	0	0
$154$	−10.2692	−0.827519
$155$	−5.13667	−0.412587
$156$	0	0
$157$	8.89020	0.709515	0.354758	−	0.934958i	$-0.384563\pi$
$157$	8.89020	0.709515	0.354758	+	0.934958i	$0.384563\pi$
$158$	18.0232	1.43385
$159$	0	0
$160$	8.64519	0.683463
$161$	5.64621	0.444984
$162$	0	0
$163$	−12.6207	−0.988533	−0.494266	−	0.869310i	$-0.664563\pi$
$163$	−12.6207	−0.988533	−0.494266	+	0.869310i	$0.664563\pi$
$164$	−24.6305	−1.92332
$165$	0	0
$166$	−19.0847	−1.48126
$167$	−20.2072	−1.56368	−0.781841	−	0.623478i	$-0.785719\pi$
$167$	−20.2072	−1.56368	−0.781841	+	0.623478i	$0.785719\pi$
$168$	0	0
$169$	−8.87879	−0.682984
$170$	0.433630	0.0332579
$171$	0	0
$172$	25.0112	1.90709
$173$	−3.38197	−0.257127	−0.128563	−	0.991701i	$-0.541037\pi$
$173$	−3.38197	−0.257127	−0.128563	+	0.991701i	$0.541037\pi$
$174$	0	0
$175$	−4.55979	−0.344688
$176$	−2.60388	−0.196274
$177$	0	0
$178$	−7.33569	−0.549833
$179$	−21.1281	−1.57919	−0.789594	−	0.613630i	$-0.789709\pi$
$179$	−21.1281	−1.57919	−0.789594	+	0.613630i	$0.789709\pi$
$180$	0	0
$181$	24.6698	1.83369	0.916846	−	0.399240i	$-0.130726\pi$
$181$	24.6698	1.83369	0.916846	+	0.399240i	$0.130726\pi$
$182$	6.42054	0.475922
$183$	0	0
$184$	−9.45448	−0.696994
$185$	−8.94914	−0.657954
$186$	0	0
$187$	−0.472267	−0.0345356
$188$	3.04892	0.222365
$189$	0	0
$190$	2.98134	0.216289
$191$	2.38451	0.172537	0.0862684	−	0.996272i	$-0.472506\pi$
$191$	2.38451	0.172537	0.0862684	+	0.996272i	$0.472506\pi$
$192$	0	0
$193$	−10.8591	−0.781653	−0.390826	−	0.920464i	$-0.627811\pi$
$193$	−10.8591	−0.781653	−0.390826	+	0.920464i	$0.627811\pi$
$194$	29.5126	2.11888
$195$	0	0
$196$	−15.3020	−1.09300
$197$	−15.6594	−1.11568	−0.557842	−	0.829947i	$-0.688371\pi$
$197$	−15.6594	−1.11568	−0.557842	+	0.829947i	$0.688371\pi$
$198$	0	0
$199$	22.9413	1.62626	0.813132	−	0.582080i	$-0.197761\pi$
$199$	22.9413	1.62626	0.813132	+	0.582080i	$0.197761\pi$
$200$	7.63528	0.539896
$201$	0	0
$202$	26.9347	1.89512
$203$	1.35475	0.0950846
$204$	0	0
$205$	−10.7186	−0.748621
$206$	25.3031	1.76295
$207$	0	0
$208$	1.62800	0.112881
$209$	−3.24698	−0.224598
$210$	0	0
$211$	−5.81683	−0.400447	−0.200224	−	0.979750i	$-0.564167\pi$
$211$	−5.81683	−0.400447	−0.200224	+	0.979750i	$0.564167\pi$
$212$	−21.4098	−1.47043
$213$	0	0
$214$	−9.06319	−0.619547
$215$	10.8843	0.742304
$216$	0	0
$217$	−5.44916	−0.369913
$218$	41.1803	2.78908
$219$	0	0
$220$	13.1352	0.885575
$221$	0.295271	0.0198621
$222$	0	0
$223$	−4.28265	−0.286787	−0.143394	−	0.989666i	$-0.545801\pi$
$223$	−4.28265	−0.286787	−0.143394	+	0.989666i	$0.545801\pi$
$224$	9.17113	0.612772
$225$	0	0
$226$	−0.958372	−0.0637500
$227$	−10.2984	−0.683526	−0.341763	−	0.939786i	$-0.611024\pi$
$227$	−10.2984	−0.683526	−0.341763	+	0.939786i	$0.611024\pi$
$228$	0	0
$229$	4.56467	0.301642	0.150821	−	0.988561i	$-0.451808\pi$
$229$	4.56467	0.301642	0.150821	+	0.988561i	$0.451808\pi$
$230$	−11.9594	−0.788577
$231$	0	0
$232$	−2.26850	−0.148934
$233$	−9.63613	−0.631284	−0.315642	−	0.948878i	$-0.602220\pi$
$233$	−9.63613	−0.631284	−0.315642	+	0.948878i	$0.602220\pi$
$234$	0	0
$235$	1.32682	0.0865521
$236$	2.92157	0.190178
$237$	0	0
$238$	0.460010	0.0298180
$239$	24.2892	1.57114	0.785570	−	0.618772i	$-0.212370\pi$
$239$	24.2892	1.57114	0.785570	+	0.618772i	$0.212370\pi$
$240$	0	0
$241$	−8.27948	−0.533328	−0.266664	−	0.963789i	$-0.585921\pi$
$241$	−8.27948	−0.533328	−0.266664	+	0.963789i	$0.585921\pi$
$242$	1.02715	0.0660275
$243$	0	0
$244$	−17.0662	−1.09255
$245$	−6.65909	−0.425434
$246$	0	0
$247$	2.03008	0.129171
$248$	9.12453	0.579408
$249$	0	0
$250$	24.5649	1.55362
$251$	15.4805	0.977123	0.488561	−	0.872529i	$-0.337522\pi$
$251$	15.4805	0.977123	0.488561	+	0.872529i	$0.337522\pi$
$252$	0	0
$253$	13.0250	0.818873
$254$	19.8160	1.24337
$255$	0	0
$256$	−10.4668	−0.654176
$257$	6.19342	0.386335	0.193167	−	0.981166i	$-0.438124\pi$
$257$	6.19342	0.386335	0.193167	+	0.981166i	$0.438124\pi$
$258$	0	0
$259$	−9.49357	−0.589902
$260$	−8.21239	−0.509311
$261$	0	0
$262$	5.39229	0.333137
$263$	−5.72229	−0.352851	−0.176426	−	0.984314i	$-0.556454\pi$
$263$	−5.72229	−0.352851	−0.176426	+	0.984314i	$0.556454\pi$
$264$	0	0
$265$	−9.31704	−0.572341
$266$	3.16271	0.193918
$267$	0	0
$268$	28.6525	1.75023
$269$	17.7238	1.08064	0.540318	−	0.841461i	$-0.318304\pi$
$269$	17.7238	1.08064	0.540318	+	0.841461i	$0.318304\pi$
$270$	0	0
$271$	24.0273	1.45955	0.729777	−	0.683685i	$-0.239624\pi$
$271$	24.0273	1.45955	0.729777	+	0.683685i	$0.239624\pi$
$272$	0.116640	0.00707236
$273$	0	0
$274$	1.77225	0.107065
$275$	−10.5188	−0.634305
$276$	0	0
$277$	−22.4694	−1.35006	−0.675028	−	0.737792i	$-0.735869\pi$
$277$	−22.4694	−1.35006	−0.675028	+	0.737792i	$0.735869\pi$
$278$	−11.6715	−0.700008
$279$	0	0
$280$	−4.40161	−0.263047
$281$	−2.03230	−0.121237	−0.0606186	−	0.998161i	$-0.519307\pi$
$281$	−2.03230	−0.121237	−0.0606186	+	0.998161i	$0.519307\pi$
$282$	0	0
$283$	9.98148	0.593337	0.296669	−	0.954980i	$-0.404124\pi$
$283$	9.98148	0.593337	0.296669	+	0.954980i	$0.404124\pi$
$284$	7.51218	0.445766
$285$	0	0
$286$	14.8112	0.875806
$287$	−11.3707	−0.671191
$288$	0	0
$289$	−16.9788	−0.998756
$290$	−2.86952	−0.168504
$291$	0	0
$292$	−14.1188	−0.826243
$293$	8.76511	0.512063	0.256032	−	0.966668i	$-0.417585\pi$
$293$	8.76511	0.512063	0.256032	+	0.966668i	$0.417585\pi$
$294$	0	0
$295$	1.27140	0.0740238
$296$	15.8968	0.923984
$297$	0	0
$298$	24.0517	1.39328
$299$	−8.14347	−0.470949
$300$	0	0
$301$	11.5465	0.665527
$302$	−51.4066	−2.95812
$303$	0	0
$304$	0.801938	0.0459943
$305$	−7.42683	−0.425259
$306$	0	0
$307$	7.87403	0.449395	0.224697	−	0.974429i	$-0.427861\pi$
$307$	7.87403	0.449395	0.224697	+	0.974429i	$0.427861\pi$
$308$	13.9343	0.793980
$309$	0	0
$310$	11.5420	0.655541
$311$	−14.2710	−0.809234	−0.404617	−	0.914486i	$-0.632595\pi$
$311$	−14.2710	−0.809234	−0.404617	+	0.914486i	$0.632595\pi$
$312$	0	0
$313$	−3.57391	−0.202009	−0.101005	−	0.994886i	$-0.532206\pi$
$313$	−3.57391	−0.202009	−0.101005	+	0.994886i	$0.532206\pi$
$314$	−19.9761	−1.12732
$315$	0	0
$316$	−24.4556	−1.37573
$317$	−9.61863	−0.540236	−0.270118	−	0.962827i	$-0.587063\pi$
$317$	−9.61863	−0.540236	−0.270118	+	0.962827i	$0.587063\pi$
$318$	0	0
$319$	3.12520	0.174978
$320$	−17.2975	−0.966961
$321$	0	0
$322$	−12.6869	−0.707014
$323$	0.145448	0.00809295
$324$	0	0
$325$	6.57654	0.364801
$326$	28.3586	1.57064
$327$	0	0
$328$	19.0400	1.05131
$329$	1.40754	0.0776000
$330$	0	0
$331$	−20.3642	−1.11932	−0.559660	−	0.828723i	$-0.689068\pi$
$331$	−20.3642	−1.11932	−0.559660	+	0.828723i	$0.689068\pi$
$332$	25.8960	1.42123
$333$	0	0
$334$	45.4052	2.48446
$335$	12.4689	0.681250
$336$	0	0
$337$	−22.8612	−1.24533	−0.622665	−	0.782489i	$-0.713950\pi$
$337$	−22.8612	−1.24533	−0.622665	+	0.782489i	$0.713950\pi$
$338$	19.9505	1.08516
$339$	0	0
$340$	−0.588390	−0.0319099
$341$	−12.5704	−0.680726
$342$	0	0
$343$	−16.9170	−0.913430
$344$	−19.3344	−1.04244
$345$	0	0
$346$	7.59922	0.408537
$347$	−5.62766	−0.302108	−0.151054	−	0.988525i	$-0.548267\pi$
$347$	−5.62766	−0.302108	−0.151054	+	0.988525i	$0.548267\pi$
$348$	0	0
$349$	17.3420	0.928296	0.464148	−	0.885758i	$-0.346361\pi$
$349$	17.3420	0.928296	0.464148	+	0.885758i	$0.346361\pi$
$350$	10.2457	0.547658
$351$	0	0
$352$	21.1564	1.12764
$353$	−19.8082	−1.05428	−0.527141	−	0.849778i	$-0.676736\pi$
$353$	−19.8082	−1.05428	−0.527141	+	0.849778i	$0.676736\pi$
$354$	0	0
$355$	3.26913	0.173507
$356$	9.95376	0.527548
$357$	0	0
$358$	47.4744	2.50910
$359$	30.8551	1.62847	0.814235	−	0.580536i	$-0.197157\pi$
$359$	30.8551	1.62847	0.814235	+	0.580536i	$0.197157\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−55.4326	−2.91347
$363$	0	0
$364$	−8.71200	−0.456633
$365$	−6.14420	−0.321602
$366$	0	0
$367$	−4.77864	−0.249443	−0.124721	−	0.992192i	$-0.539804\pi$
$367$	−4.77864	−0.249443	−0.124721	+	0.992192i	$0.539804\pi$
$368$	−3.21690	−0.167693
$369$	0	0
$370$	20.1085	1.04539
$371$	−9.88385	−0.513144
$372$	0	0
$373$	30.6065	1.58475	0.792373	−	0.610037i	$-0.208845\pi$
$373$	30.6065	1.58475	0.792373	+	0.610037i	$0.208845\pi$
$374$	1.06117	0.0548720
$375$	0	0
$376$	−2.35690	−0.121548
$377$	−1.95394	−0.100633
$378$	0	0
$379$	10.8171	0.555635	0.277818	−	0.960634i	$-0.410389\pi$
$379$	10.8171	0.555635	0.277818	+	0.960634i	$0.410389\pi$
$380$	−4.04536	−0.207523
$381$	0	0
$382$	−5.35794	−0.274136
$383$	−12.7455	−0.651262	−0.325631	−	0.945497i	$-0.605577\pi$
$383$	−12.7455	−0.651262	−0.325631	+	0.945497i	$0.605577\pi$
$384$	0	0
$385$	6.06389	0.309044
$386$	24.4001	1.24193
$387$	0	0
$388$	−40.0455	−2.03300
$389$	36.9677	1.87434	0.937169	−	0.348874i	$-0.113436\pi$
$389$	36.9677	1.87434	0.937169	+	0.348874i	$0.113436\pi$
$390$	0	0
$391$	−0.583452	−0.0295065
$392$	11.8289	0.597449
$393$	0	0
$394$	35.1863	1.77266
$395$	−10.6425	−0.535483
$396$	0	0
$397$	5.58714	0.280411	0.140205	−	0.990122i	$-0.455224\pi$
$397$	5.58714	0.280411	0.140205	+	0.990122i	$0.455224\pi$
$398$	−51.5486	−2.58390
$399$	0	0
$400$	2.59792	0.129896
$401$	18.0486	0.901306	0.450653	−	0.892699i	$-0.351191\pi$
$401$	18.0486	0.901306	0.450653	+	0.892699i	$0.351191\pi$
$402$	0	0
$403$	7.85928	0.391499
$404$	−36.5476	−1.81831
$405$	0	0
$406$	−3.04409	−0.151076
$407$	−21.9003	−1.08556
$408$	0	0
$409$	4.07855	0.201671	0.100836	−	0.994903i	$-0.467848\pi$
$409$	4.07855	0.201671	0.100836	+	0.994903i	$0.467848\pi$
$410$	24.0845	1.18945
$411$	0	0
$412$	−34.3337	−1.69150
$413$	1.34875	0.0663675
$414$	0	0
$415$	11.2693	0.553191
$416$	−13.2274	−0.648528
$417$	0	0
$418$	7.29590	0.356854
$419$	−29.9725	−1.46425	−0.732127	−	0.681168i	$-0.761472\pi$
$419$	−29.9725	−1.46425	−0.732127	+	0.681168i	$0.761472\pi$
$420$	0	0
$421$	36.5696	1.78229	0.891146	−	0.453716i	$-0.149902\pi$
$421$	36.5696	1.78229	0.891146	+	0.453716i	$0.149902\pi$
$422$	13.0703	0.636252
$423$	0	0
$424$	16.5503	0.803755
$425$	0.471187	0.0228559
$426$	0	0
$427$	−7.87864	−0.381274
$428$	12.2978	0.594437
$429$	0	0
$430$	−24.4568	−1.17941
$431$	26.9745	1.29932	0.649659	−	0.760226i	$-0.274912\pi$
$431$	26.9745	1.29932	0.649659	+	0.760226i	$0.274912\pi$
$432$	0	0
$433$	−12.5615	−0.603666	−0.301833	−	0.953361i	$-0.597599\pi$
$433$	−12.5615	−0.603666	−0.301833	+	0.953361i	$0.597599\pi$
$434$	12.2442	0.587739
$435$	0	0
$436$	−55.8774	−2.67604
$437$	−4.01141	−0.191892
$438$	0	0
$439$	−17.8134	−0.850186	−0.425093	−	0.905150i	$-0.639759\pi$
$439$	−17.8134	−0.850186	−0.425093	+	0.905150i	$0.639759\pi$
$440$	−10.1539	−0.484067
$441$	0	0
$442$	−0.663468	−0.0315579
$443$	22.4118	1.06482	0.532408	−	0.846488i	$-0.321287\pi$
$443$	22.4118	1.06482	0.532408	+	0.846488i	$0.321287\pi$
$444$	0	0
$445$	4.33165	0.205340
$446$	9.62302	0.455663
$447$	0	0
$448$	−18.3498	−0.866948
$449$	23.1151	1.09087	0.545434	−	0.838154i	$-0.316365\pi$
$449$	23.1151	1.09087	0.545434	+	0.838154i	$0.316365\pi$
$450$	0	0
$451$	−26.2305	−1.23515
$452$	1.30041	0.0611662
$453$	0	0
$454$	23.1402	1.08602
$455$	−3.79126	−0.177737
$456$	0	0
$457$	17.5303	0.820035	0.410018	−	0.912078i	$-0.365523\pi$
$457$	17.5303	0.820035	0.410018	+	0.912078i	$0.365523\pi$
$458$	−10.2567	−0.479266
$459$	0	0
$460$	16.2276	0.756616
$461$	−3.01863	−0.140592	−0.0702959	−	0.997526i	$-0.522394\pi$
$461$	−3.01863	−0.140592	−0.0702959	+	0.997526i	$0.522394\pi$
$462$	0	0
$463$	−26.3155	−1.22299	−0.611493	−	0.791250i	$-0.709431\pi$
$463$	−26.3155	−1.22299	−0.611493	+	0.791250i	$0.709431\pi$
$464$	−0.771861	−0.0358327
$465$	0	0
$466$	21.6522	1.00302
$467$	18.3360	0.848490	0.424245	−	0.905548i	$-0.360540\pi$
$467$	18.3360	0.848490	0.424245	+	0.905548i	$0.360540\pi$
$468$	0	0
$469$	13.2275	0.610788
$470$	−2.98134	−0.137519
$471$	0	0
$472$	−2.25845	−0.103954
$473$	26.6360	1.22472
$474$	0	0
$475$	3.23955	0.148641
$476$	−0.624185	−0.0286095
$477$	0	0
$478$	−54.5774	−2.49631
$479$	−29.6500	−1.35474	−0.677372	−	0.735640i	$-0.736881\pi$
$479$	−29.6500	−1.35474	−0.677372	+	0.735640i	$0.736881\pi$
$480$	0	0
$481$	13.6925	0.624324
$482$	18.6038	0.847381
$483$	0	0
$484$	−1.39373	−0.0633514
$485$	−17.4269	−0.791313
$486$	0	0
$487$	11.6890	0.529681	0.264840	−	0.964292i	$-0.414681\pi$
$487$	11.6890	0.529681	0.264840	+	0.964292i	$0.414681\pi$
$488$	13.1926	0.597203
$489$	0	0
$490$	14.9628	0.675953
$491$	14.7797	0.666996	0.333498	−	0.942751i	$-0.391771\pi$
$491$	14.7797	0.666996	0.333498	+	0.942751i	$0.391771\pi$
$492$	0	0
$493$	−0.139993	−0.00630497
$494$	−4.56154	−0.205233
$495$	0	0
$496$	3.10464	0.139402
$497$	3.46801	0.155561
$498$	0	0
$499$	0.726775	0.0325349	0.0162675	−	0.999868i	$-0.494822\pi$
$499$	0.726775	0.0325349	0.0162675	+	0.999868i	$0.494822\pi$
$500$	−33.3320	−1.49065
$501$	0	0
$502$	−34.7845	−1.55251
$503$	36.0441	1.60713	0.803564	−	0.595219i	$-0.202935\pi$
$503$	36.0441	1.60713	0.803564	+	0.595219i	$0.202935\pi$
$504$	0	0
$505$	−15.9047	−0.707749
$506$	−29.2668	−1.30107
$507$	0	0
$508$	−26.8882	−1.19297
$509$	35.3559	1.56712	0.783561	−	0.621314i	$-0.213401\pi$
$509$	35.3559	1.56712	0.783561	+	0.621314i	$0.213401\pi$
$510$	0	0
$511$	−6.51798	−0.288339
$512$	−9.00538	−0.397985
$513$	0	0
$514$	−13.9165	−0.613830
$515$	−14.9412	−0.658389
$516$	0	0
$517$	3.24698	0.142802
$518$	21.3319	0.937268
$519$	0	0
$520$	6.34840	0.278396
$521$	−3.40392	−0.149128	−0.0745641	−	0.997216i	$-0.523757\pi$
$521$	−3.40392	−0.149128	−0.0745641	+	0.997216i	$0.523757\pi$
$522$	0	0
$523$	13.0824	0.572052	0.286026	−	0.958222i	$-0.407666\pi$
$523$	13.0824	0.572052	0.286026	+	0.958222i	$0.407666\pi$
$524$	−7.31677	−0.319635
$525$	0	0
$526$	12.8579	0.560630
$527$	0.563091	0.0245286
$528$	0	0
$529$	−6.90857	−0.300373
$530$	20.9352	0.909367
$531$	0	0
$532$	−4.29146	−0.186059
$533$	16.3998	0.710356
$534$	0	0
$535$	5.35172	0.231375
$536$	−22.1492	−0.956699
$537$	0	0
$538$	−39.8249	−1.71697
$539$	−16.2961	−0.701922
$540$	0	0
$541$	−11.9123	−0.512150	−0.256075	−	0.966657i	$-0.582429\pi$
$541$	−11.9123	−0.512150	−0.256075	+	0.966657i	$0.582429\pi$
$542$	−53.9888	−2.31902
$543$	0	0
$544$	−0.947701	−0.0406323
$545$	−24.3166	−1.04161
$546$	0	0
$547$	−25.3861	−1.08543	−0.542717	−	0.839916i	$-0.682604\pi$
$547$	−25.3861	−1.08543	−0.542717	+	0.839916i	$0.682604\pi$
$548$	−2.40475	−0.102726
$549$	0	0
$550$	23.6354	1.00782
$551$	−0.962495	−0.0410037
$552$	0	0
$553$	−11.2899	−0.480097
$554$	50.4883	2.14504
$555$	0	0
$556$	15.8370	0.671636
$557$	−4.84063	−0.205104	−0.102552	−	0.994728i	$-0.532701\pi$
$557$	−4.84063	−0.205104	−0.102552	+	0.994728i	$0.532701\pi$
$558$	0	0
$559$	−16.6534	−0.704362
$560$	−1.49766	−0.0632875
$561$	0	0
$562$	4.56655	0.192628
$563$	−19.3530	−0.815630	−0.407815	−	0.913065i	$-0.633709\pi$
$563$	−19.3530	−0.815630	−0.407815	+	0.913065i	$0.633709\pi$
$564$	0	0
$565$	0.565909	0.0238080
$566$	−22.4282	−0.942727
$567$	0	0
$568$	−5.80712	−0.243661
$569$	−22.6909	−0.951252	−0.475626	−	0.879648i	$-0.657778\pi$
$569$	−22.6909	−0.951252	−0.475626	+	0.879648i	$0.657778\pi$
$570$	0	0
$571$	38.1859	1.59803	0.799015	−	0.601310i	$-0.205354\pi$
$571$	38.1859	1.59803	0.799015	+	0.601310i	$0.205354\pi$
$572$	−20.0973	−0.840310
$573$	0	0
$574$	25.5497	1.06642
$575$	−12.9952	−0.541936
$576$	0	0
$577$	15.6296	0.650670	0.325335	−	0.945599i	$-0.394523\pi$
$577$	15.6296	0.650670	0.325335	+	0.945599i	$0.394523\pi$
$578$	38.1511	1.58688
$579$	0	0
$580$	3.89364	0.161675
$581$	11.9549	0.495974
$582$	0	0
$583$	−22.8006	−0.944303
$584$	10.9142	0.451635
$585$	0	0
$586$	−19.6950	−0.813594
$587$	−9.77404	−0.403418	−0.201709	−	0.979446i	$-0.564649\pi$
$587$	−9.77404	−0.403418	−0.201709	+	0.979446i	$0.564649\pi$
$588$	0	0
$589$	3.87142	0.159519
$590$	−2.85681	−0.117613
$591$	0	0
$592$	5.40892	0.222305
$593$	10.0580	0.413031	0.206516	−	0.978443i	$-0.433788\pi$
$593$	10.0580	0.413031	0.206516	+	0.978443i	$0.433788\pi$
$594$	0	0
$595$	−0.271631	−0.0111358
$596$	−32.6356	−1.33681
$597$	0	0
$598$	18.2982	0.748270
$599$	−17.6220	−0.720014	−0.360007	−	0.932950i	$-0.617226\pi$
$599$	−17.6220	−0.720014	−0.360007	+	0.932950i	$0.617226\pi$
$600$	0	0
$601$	−38.9534	−1.58894	−0.794472	−	0.607301i	$-0.792252\pi$
$601$	−38.9534	−1.58894	−0.794472	+	0.607301i	$0.792252\pi$
$602$	−25.9447	−1.05743
$603$	0	0
$604$	69.7534	2.83822
$605$	−0.606520	−0.0246585
$606$	0	0
$607$	31.0856	1.26173	0.630863	−	0.775894i	$-0.282701\pi$
$607$	31.0856	1.26173	0.630863	+	0.775894i	$0.282701\pi$
$608$	−6.51573	−0.264248
$609$	0	0
$610$	16.6879	0.675675
$611$	−2.03008	−0.0821281
$612$	0	0
$613$	−41.5140	−1.67674	−0.838368	−	0.545105i	$-0.816490\pi$
$613$	−41.5140	−1.67674	−0.838368	+	0.545105i	$0.816490\pi$
$614$	−17.6928	−0.714023
$615$	0	0
$616$	−10.7716	−0.434000
$617$	17.7389	0.714141	0.357071	−	0.934077i	$-0.383776\pi$
$617$	17.7389	0.714141	0.357071	+	0.934077i	$0.383776\pi$
$618$	0	0
$619$	20.8517	0.838102	0.419051	−	0.907963i	$-0.362363\pi$
$619$	20.8517	0.838102	0.419051	+	0.907963i	$0.362363\pi$
$620$	−15.6613	−0.628973
$621$	0	0
$622$	32.0666	1.28576
$623$	4.59517	0.184102
$624$	0	0
$625$	1.69245	0.0676979
$626$	8.03050	0.320963
$627$	0	0
$628$	27.1055	1.08163
$629$	0.981020	0.0391158
$630$	0	0
$631$	−12.9630	−0.516048	−0.258024	−	0.966139i	$-0.583071\pi$
$631$	−12.9630	−0.516048	−0.258024	+	0.966139i	$0.583071\pi$
$632$	18.9048	0.751993
$633$	0	0
$634$	21.6129	0.858356
$635$	−11.7011	−0.464346
$636$	0	0
$637$	10.1886	0.403688
$638$	−7.02226	−0.278014
$639$	0	0
$640$	21.5768	0.852898
$641$	−13.4368	−0.530721	−0.265360	−	0.964149i	$-0.585491\pi$
$641$	−13.4368	−0.530721	−0.265360	+	0.964149i	$0.585491\pi$
$642$	0	0
$643$	9.02193	0.355790	0.177895	−	0.984049i	$-0.443071\pi$
$643$	9.02193	0.355790	0.177895	+	0.984049i	$0.443071\pi$
$644$	17.2148	0.678359
$645$	0	0
$646$	−0.326819	−0.0128585
$647$	−27.9487	−1.09878	−0.549388	−	0.835567i	$-0.685139\pi$
$647$	−27.9487	−1.09878	−0.549388	+	0.835567i	$0.685139\pi$
$648$	0	0
$649$	3.11136	0.122132
$650$	−14.7773	−0.579615
$651$	0	0
$652$	−38.4796	−1.50698
$653$	−9.76725	−0.382222	−0.191111	−	0.981568i	$-0.561209\pi$
$653$	−9.76725	−0.382222	−0.191111	+	0.981568i	$0.561209\pi$
$654$	0	0
$655$	−3.18409	−0.124413
$656$	6.47840	0.252939
$657$	0	0
$658$	−3.16271	−0.123295
$659$	9.83481	0.383110	0.191555	−	0.981482i	$-0.438647\pi$
$659$	9.83481	0.383110	0.191555	+	0.981482i	$0.438647\pi$
$660$	0	0
$661$	−10.9779	−0.426992	−0.213496	−	0.976944i	$-0.568485\pi$
$661$	−10.9779	−0.426992	−0.213496	+	0.976944i	$0.568485\pi$
$662$	45.7580	1.77844
$663$	0	0
$664$	−20.0183	−0.776861
$665$	−1.86755	−0.0724203
$666$	0	0
$667$	3.86096	0.149497
$668$	−61.6102	−2.38377
$669$	0	0
$670$	−28.0174	−1.08241
$671$	−18.1749	−0.701633
$672$	0	0
$673$	−15.6852	−0.604618	−0.302309	−	0.953210i	$-0.597758\pi$
$673$	−15.6852	−0.604618	−0.302309	+	0.953210i	$0.597758\pi$
$674$	51.3687	1.97865
$675$	0	0
$676$	−27.0707	−1.04118
$677$	7.11441	0.273429	0.136714	−	0.990611i	$-0.456346\pi$
$677$	7.11441	0.273429	0.136714	+	0.990611i	$0.456346\pi$
$678$	0	0
$679$	−18.4870	−0.709468
$680$	0.454842	0.0174424
$681$	0	0
$682$	28.2455	1.08158
$683$	−12.2451	−0.468547	−0.234274	−	0.972171i	$-0.575271\pi$
$683$	−12.2451	−0.468547	−0.234274	+	0.972171i	$0.575271\pi$
$684$	0	0
$685$	−1.04649	−0.0399845
$686$	38.0121	1.45131
$687$	0	0
$688$	−6.57855	−0.250805
$689$	14.2554	0.543087
$690$	0	0
$691$	−5.23155	−0.199017	−0.0995087	−	0.995037i	$-0.531727\pi$
$691$	−5.23155	−0.199017	−0.0995087	+	0.995037i	$0.531727\pi$
$692$	−10.3114	−0.391979
$693$	0	0
$694$	12.6452	0.480006
$695$	6.89188	0.261424
$696$	0	0
$697$	1.17499	0.0445060
$698$	−38.9671	−1.47493
$699$	0	0
$700$	−13.9024	−0.525462
$701$	13.0028	0.491110	0.245555	−	0.969383i	$-0.421030\pi$
$701$	13.0028	0.491110	0.245555	+	0.969383i	$0.421030\pi$
$702$	0	0
$703$	6.74481	0.254385
$704$	−42.3303	−1.59538
$705$	0	0
$706$	44.5086	1.67510
$707$	−16.8723	−0.634546
$708$	0	0
$709$	−37.6285	−1.41317	−0.706585	−	0.707629i	$-0.749765\pi$
$709$	−37.6285	−1.41317	−0.706585	+	0.707629i	$0.749765\pi$
$710$	−7.34567	−0.275678
$711$	0	0
$712$	−7.69453	−0.288365
$713$	−15.5299	−0.581598
$714$	0	0
$715$	−8.74588	−0.327078
$716$	−64.4178	−2.40741
$717$	0	0
$718$	−69.3308	−2.58740
$719$	44.9949	1.67803	0.839014	−	0.544111i	$-0.183133\pi$
$719$	44.9949	1.67803	0.839014	+	0.544111i	$0.183133\pi$
$720$	0	0
$721$	−15.8502	−0.590292
$722$	−2.24698	−0.0836239
$723$	0	0
$724$	75.2162	2.79539
$725$	−3.11805	−0.115802
$726$	0	0
$727$	−39.2707	−1.45647	−0.728236	−	0.685327i	$-0.759659\pi$
$727$	−39.2707	−1.45647	−0.728236	+	0.685327i	$0.759659\pi$
$728$	6.73461	0.249601
$729$	0	0
$730$	13.8059	0.510979
$731$	−1.19316	−0.0441305
$732$	0	0
$733$	0.190186	0.00702467	0.00351233	−	0.999994i	$-0.498882\pi$
$733$	0.190186	0.00702467	0.00351233	+	0.999994i	$0.498882\pi$
$734$	10.7375	0.396328
$735$	0	0
$736$	26.1373	0.963433
$737$	30.5138	1.12399
$738$	0	0
$739$	−11.8560	−0.436132	−0.218066	−	0.975934i	$-0.569975\pi$
$739$	−11.8560	−0.436132	−0.218066	+	0.975934i	$0.569975\pi$
$740$	−27.2852	−1.00302
$741$	0	0
$742$	22.2088	0.815311
$743$	11.2491	0.412689	0.206345	−	0.978479i	$-0.433843\pi$
$743$	11.2491	0.412689	0.206345	+	0.978479i	$0.433843\pi$
$744$	0	0
$745$	−14.2023	−0.520331
$746$	−68.7722	−2.51793
$747$	0	0
$748$	−1.43990	−0.0526481
$749$	5.67730	0.207444
$750$	0	0
$751$	−18.4799	−0.674340	−0.337170	−	0.941444i	$-0.609470\pi$
$751$	−18.4799	−0.674340	−0.337170	+	0.941444i	$0.609470\pi$
$752$	−0.801938	−0.0292437
$753$	0	0
$754$	4.39046	0.159891
$755$	30.3551	1.10473
$756$	0	0
$757$	−0.739852	−0.0268904	−0.0134452	−	0.999910i	$-0.504280\pi$
$757$	−0.739852	−0.0268904	−0.0134452	+	0.999910i	$0.504280\pi$
$758$	−24.3057	−0.882824
$759$	0	0
$760$	3.12717	0.113435
$761$	−50.0217	−1.81328	−0.906642	−	0.421900i	$-0.861363\pi$
$761$	−50.0217	−1.81328	−0.906642	+	0.421900i	$0.861363\pi$
$762$	0	0
$763$	−25.7959	−0.933873
$764$	7.27016	0.263025
$765$	0	0
$766$	28.6388	1.03476
$767$	−1.94529	−0.0702402
$768$	0	0
$769$	−41.9778	−1.51376	−0.756879	−	0.653555i	$-0.773277\pi$
$769$	−41.9778	−1.51376	−0.756879	+	0.653555i	$0.773277\pi$
$770$	−13.6254	−0.491026
$771$	0	0
$772$	−33.1084	−1.19160
$773$	−24.3993	−0.877583	−0.438791	−	0.898589i	$-0.644593\pi$
$773$	−24.3993	−0.877583	−0.438791	+	0.898589i	$0.644593\pi$
$774$	0	0
$775$	12.5417	0.450510
$776$	30.9562	1.11126
$777$	0	0
$778$	−83.0658	−2.97805
$779$	8.07843	0.289440
$780$	0	0
$781$	8.00018	0.286269
$782$	1.31101	0.0468815
$783$	0	0
$784$	4.02480	0.143743
$785$	11.7957	0.421006
$786$	0	0
$787$	16.9154	0.602967	0.301484	−	0.953471i	$-0.402518\pi$
$787$	16.9154	0.602967	0.301484	+	0.953471i	$0.402518\pi$
$788$	−47.7442	−1.70082
$789$	0	0
$790$	23.9135	0.850804
$791$	0.600337	0.0213455
$792$	0	0
$793$	11.3633	0.403522
$794$	−12.5542	−0.445532
$795$	0	0
$796$	69.9460	2.47917
$797$	−7.64524	−0.270808	−0.135404	−	0.990790i	$-0.543233\pi$
$797$	−7.64524	−0.270808	−0.135404	+	0.990790i	$0.543233\pi$
$798$	0	0
$799$	−0.145448	−0.00514559
$800$	−21.1080	−0.746282
$801$	0	0
$802$	−40.5549	−1.43204
$803$	−15.0360	−0.530610
$804$	0	0
$805$	7.49150	0.264041
$806$	−17.6596	−0.622034
$807$	0	0
$808$	28.2523	0.993912
$809$	25.3595	0.891591	0.445795	−	0.895135i	$-0.352921\pi$
$809$	25.3595	0.891591	0.445795	+	0.895135i	$0.352921\pi$
$810$	0	0
$811$	−16.8198	−0.590622	−0.295311	−	0.955401i	$-0.595423\pi$
$811$	−16.8198	−0.590622	−0.295311	+	0.955401i	$0.595423\pi$
$812$	4.13051	0.144953
$813$	0	0
$814$	49.2094	1.72479
$815$	−16.7454	−0.586567
$816$	0	0
$817$	−8.20332	−0.286998
$818$	−9.16442	−0.320427
$819$	0	0
$820$	−32.6802	−1.14124
$821$	36.6474	1.27900	0.639501	−	0.768790i	$-0.279141\pi$
$821$	36.6474	1.27900	0.639501	+	0.768790i	$0.279141\pi$
$822$	0	0
$823$	−29.3743	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$823$	−29.3743	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$824$	26.5409	0.924594
$825$	0	0
$826$	−3.03061	−0.105448
$827$	19.9977	0.695388	0.347694	−	0.937608i	$-0.386965\pi$
$827$	19.9977	0.695388	0.347694	+	0.937608i	$0.386965\pi$
$828$	0	0
$829$	−20.2638	−0.703790	−0.351895	−	0.936039i	$-0.614463\pi$
$829$	−20.2638	−0.703790	−0.351895	+	0.936039i	$0.614463\pi$
$830$	−25.3220	−0.878939
$831$	0	0
$832$	26.4658	0.917536
$833$	0.729981	0.0252923
$834$	0	0
$835$	−26.8113	−0.927844
$836$	−9.89977	−0.342391
$837$	0	0
$838$	67.3477	2.32649
$839$	−11.0047	−0.379926	−0.189963	−	0.981791i	$-0.560837\pi$
$839$	−11.0047	−0.379926	−0.189963	+	0.981791i	$0.560837\pi$
$840$	0	0
$841$	−28.0736	−0.968055
$842$	−82.1711	−2.83180
$843$	0	0
$844$	−17.7350	−0.610465
$845$	−11.7805	−0.405263
$846$	0	0
$847$	−0.643418	−0.0221081
$848$	5.63128	0.193379
$849$	0	0
$850$	−1.05875	−0.0363147
$851$	−27.0562	−0.927475
$852$	0	0
$853$	−50.0074	−1.71222	−0.856111	−	0.516792i	$-0.827126\pi$
$853$	−50.0074	−1.71222	−0.856111	+	0.516792i	$0.827126\pi$
$854$	17.7031	0.605789
$855$	0	0
$856$	−9.50654	−0.324927
$857$	18.8153	0.642717	0.321359	−	0.946958i	$-0.395861\pi$
$857$	18.8153	0.642717	0.321359	+	0.946958i	$0.395861\pi$
$858$	0	0
$859$	28.2421	0.963608	0.481804	−	0.876279i	$-0.339982\pi$
$859$	28.2421	0.963608	0.481804	+	0.876279i	$0.339982\pi$
$860$	33.1854	1.13161
$861$	0	0
$862$	−60.6112	−2.06443
$863$	−11.8722	−0.404135	−0.202068	−	0.979372i	$-0.564766\pi$
$863$	−11.8722	−0.404135	−0.202068	+	0.979372i	$0.564766\pi$
$864$	0	0
$865$	−4.48727	−0.152572
$866$	28.2254	0.959138
$867$	0	0
$868$	−16.6141	−0.563918
$869$	−26.0442	−0.883490
$870$	0	0
$871$	−19.0779	−0.646429
$872$	43.1948	1.46276
$873$	0	0
$874$	9.01356	0.304888
$875$	−15.3877	−0.520201
$876$	0	0
$877$	−18.6459	−0.629627	−0.314813	−	0.949154i	$-0.601942\pi$
$877$	−18.6459	−0.629627	−0.314813	+	0.949154i	$0.601942\pi$
$878$	40.0263	1.35082
$879$	0	0
$880$	−3.45487	−0.116464
$881$	36.5209	1.23042	0.615210	−	0.788363i	$-0.289071\pi$
$881$	36.5209	1.23042	0.615210	+	0.788363i	$0.289071\pi$
$882$	0	0
$883$	29.4991	0.992723	0.496361	−	0.868116i	$-0.334669\pi$
$883$	29.4991	0.992723	0.496361	+	0.868116i	$0.334669\pi$
$884$	0.900257	0.0302789
$885$	0	0
$886$	−50.3588	−1.69184
$887$	10.5787	0.355200	0.177600	−	0.984103i	$-0.443167\pi$
$887$	10.5787	0.355200	0.177600	+	0.984103i	$0.443167\pi$
$888$	0	0
$889$	−12.4130	−0.416318
$890$	−9.73313	−0.326255
$891$	0	0
$892$	−13.0574	−0.437195
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−28.0332	−0.937045
$896$	22.8894	0.764682
$897$	0	0
$898$	−51.9391	−1.73323
$899$	−3.72622	−0.124276
$900$	0	0
$901$	1.02135	0.0340261
$902$	58.9394	1.96247
$903$	0	0
$904$	−1.00525	−0.0334342
$905$	32.7324	1.08806
$906$	0	0
$907$	−51.2097	−1.70039	−0.850195	−	0.526468i	$-0.823516\pi$
$907$	−51.2097	−1.70039	−0.850195	+	0.526468i	$0.823516\pi$
$908$	−31.3988	−1.04201
$909$	0	0
$910$	8.51889	0.282398
$911$	−26.0979	−0.864663	−0.432332	−	0.901715i	$-0.642309\pi$
$911$	−26.0979	−0.864663	−0.432332	+	0.901715i	$0.642309\pi$
$912$	0	0
$913$	27.5783	0.912707
$914$	−39.3903	−1.30292
$915$	0	0
$916$	13.9173	0.459841
$917$	−3.37780	−0.111545
$918$	0	0
$919$	−40.8587	−1.34780	−0.673902	−	0.738821i	$-0.735383\pi$
$919$	−40.8587	−1.34780	−0.673902	+	0.738821i	$0.735383\pi$
$920$	−12.5444	−0.413576
$921$	0	0
$922$	6.78281	0.223380
$923$	−5.00188	−0.164639
$924$	0	0
$925$	21.8502	0.718429
$926$	59.1305	1.94315
$927$	0	0
$928$	6.27136	0.205867
$929$	−44.8256	−1.47068	−0.735340	−	0.677698i	$-0.762978\pi$
$929$	−44.8256	−1.47068	−0.735340	+	0.677698i	$0.762978\pi$
$930$	0	0
$931$	5.01884	0.164486
$932$	−29.3798	−0.962367
$933$	0	0
$934$	−41.2007	−1.34813
$935$	−0.626613	−0.0204924
$936$	0	0
$937$	−5.05879	−0.165264	−0.0826318	−	0.996580i	$-0.526333\pi$
$937$	−5.05879	−0.165264	−0.0826318	+	0.996580i	$0.526333\pi$
$938$	−29.7219	−0.970453
$939$	0	0
$940$	4.04536	0.131945
$941$	−28.8180	−0.939440	−0.469720	−	0.882815i	$-0.655645\pi$
$941$	−28.8180	−0.939440	−0.469720	+	0.882815i	$0.655645\pi$
$942$	0	0
$943$	−32.4059	−1.05528
$944$	−0.768443	−0.0250107
$945$	0	0
$946$	−59.8506	−1.94591
$947$	−8.96734	−0.291399	−0.145700	−	0.989329i	$-0.546543\pi$
$947$	−8.96734	−0.291399	−0.145700	+	0.989329i	$0.546543\pi$
$948$	0	0
$949$	9.40082	0.305164
$950$	−7.27921	−0.236169
$951$	0	0
$952$	0.482512	0.0156383
$953$	52.1262	1.68853	0.844267	−	0.535923i	$-0.180036\pi$
$953$	52.1262	1.68853	0.844267	+	0.535923i	$0.180036\pi$
$954$	0	0
$955$	3.16381	0.102378
$956$	74.0559	2.39514
$957$	0	0
$958$	66.6230	2.15249
$959$	−1.11016	−0.0358489
$960$	0	0
$961$	−16.0121	−0.516520
$962$	−30.7667	−0.991959
$963$	0	0
$964$	−25.2435	−0.813037
$965$	−14.4080	−0.463811
$966$	0	0
$967$	35.4368	1.13957	0.569785	−	0.821794i	$-0.307027\pi$
$967$	35.4368	1.13957	0.569785	+	0.821794i	$0.307027\pi$
$968$	1.07739	0.0346287
$969$	0	0
$970$	39.1578	1.25728
$971$	−10.0373	−0.322113	−0.161056	−	0.986945i	$-0.551490\pi$
$971$	−10.0373	−0.322113	−0.161056	+	0.986945i	$0.551490\pi$
$972$	0	0
$973$	7.31115	0.234385
$974$	−26.2650	−0.841586
$975$	0	0
$976$	4.48882	0.143684
$977$	−26.1070	−0.835236	−0.417618	−	0.908623i	$-0.637135\pi$
$977$	−26.1070	−0.835236	−0.417618	+	0.908623i	$0.637135\pi$
$978$	0	0
$979$	10.6004	0.338789
$980$	−20.3030	−0.648556
$981$	0	0
$982$	−33.2096	−1.05976
$983$	−29.7339	−0.948363	−0.474182	−	0.880427i	$-0.657256\pi$
$983$	−29.7339	−0.948363	−0.474182	+	0.880427i	$0.657256\pi$
$984$	0	0
$985$	−20.7772	−0.662016
$986$	0.314562	0.0100177
$987$	0	0
$988$	6.18954	0.196915
$989$	32.9069	1.04638
$990$	0	0
$991$	49.1307	1.56069	0.780344	−	0.625351i	$-0.215044\pi$
$991$	49.1307	1.56069	0.780344	+	0.625351i	$0.215044\pi$
$992$	−25.2251	−0.800898
$993$	0	0
$994$	−7.79254	−0.247164
$995$	30.4389	0.964978
$996$	0	0
$997$	−36.7048	−1.16245	−0.581226	−	0.813742i	$-0.697427\pi$
$997$	−36.7048	−1.16245	−0.581226	+	0.813742i	$0.697427\pi$
$998$	−1.63305	−0.0516933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.i.1.2		6
3.2	odd	2		2679.2.a.i.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.i.1.5	✓	6	3.2	odd	2
8037.2.a.i.1.2		6	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)

\(f(q)\)	\(=\)	\(q-2.24698 q^{2} +3.04892 q^{4} +1.32682 q^{5} +1.40754 q^{7} -2.35690 q^{8} +O(q^{10})\)	\(q-2.24698 q^{2} +3.04892 q^{4} +1.32682 q^{5} +1.40754 q^{7} -2.35690 q^{8} -2.98134 q^{10} +3.24698 q^{11} -2.03008 q^{13} -3.16271 q^{14} -0.801938 q^{16} -0.145448 q^{17} -1.00000 q^{19} +4.04536 q^{20} -7.29590 q^{22} +4.01141 q^{23} -3.23955 q^{25} +4.56154 q^{26} +4.29146 q^{28} +0.962495 q^{29} -3.87142 q^{31} +6.51573 q^{32} +0.326819 q^{34} +1.86755 q^{35} -6.74481 q^{37} +2.24698 q^{38} -3.12717 q^{40} -8.07843 q^{41} +8.20332 q^{43} +9.89977 q^{44} -9.01356 q^{46} +1.00000 q^{47} -5.01884 q^{49} +7.27921 q^{50} -6.18954 q^{52} -7.02209 q^{53} +4.30815 q^{55} -3.31742 q^{56} -2.16271 q^{58} +0.958232 q^{59} -5.59747 q^{61} +8.69900 q^{62} -13.0368 q^{64} -2.69354 q^{65} +9.39760 q^{67} -0.443459 q^{68} -4.19634 q^{70} +2.46388 q^{71} -4.63077 q^{73} +15.1555 q^{74} -3.04892 q^{76} +4.57024 q^{77} -8.02107 q^{79} -1.06403 q^{80} +18.1521 q^{82} +8.49351 q^{83} -0.192983 q^{85} -18.4327 q^{86} -7.65279 q^{88} +3.26469 q^{89} -2.85741 q^{91} +12.2305 q^{92} -2.24698 q^{94} -1.32682 q^{95} -13.1343 q^{97} +11.2772 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 4 * q^5 + 2 * q^7 - 6 * q^8	\( 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 10 q^{11} - 8 q^{13} + q^{14} + 4 q^{16} + 3 q^{17} - 6 q^{19} - 7 q^{20} - 16 q^{22} + 10 q^{26} - 7 q^{28} - 2 q^{31} + 14 q^{32} - 2 q^{34} - 7 q^{35} - 9 q^{37} + 4 q^{38} - 4 q^{40} - 18 q^{41} + 9 q^{43} + 14 q^{44} - 7 q^{46} + 6 q^{47} - 16 q^{49} + 7 q^{50} - 21 q^{52} - 20 q^{53} + 2 q^{55} + 5 q^{56} + 7 q^{58} - 19 q^{59} + 13 q^{62} - 22 q^{64} + 22 q^{65} - 11 q^{67} - 14 q^{70} - 4 q^{71} - 7 q^{73} + 27 q^{74} + q^{77} - 16 q^{79} + 5 q^{80} - 2 q^{82} - 7 q^{83} - 25 q^{85} + q^{86} - 10 q^{88} + 33 q^{89} - 4 q^{91} + 21 q^{92} - 4 q^{94} - 4 q^{95} - 34 q^{97} + 20 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 4 * q^5 + 2 * q^7 - 6 * q^8 + 2 * q^10 + 10 * q^11 - 8 * q^13 + q^14 + 4 * q^16 + 3 * q^17 - 6 * q^19 - 7 * q^20 - 16 * q^22 + 10 * q^26 - 7 * q^28 - 2 * q^31 + 14 * q^32 - 2 * q^34 - 7 * q^35 - 9 * q^37 + 4 * q^38 - 4 * q^40 - 18 * q^41 + 9 * q^43 + 14 * q^44 - 7 * q^46 + 6 * q^47 - 16 * q^49 + 7 * q^50 - 21 * q^52 - 20 * q^53 + 2 * q^55 + 5 * q^56 + 7 * q^58 - 19 * q^59 + 13 * q^62 - 22 * q^64 + 22 * q^65 - 11 * q^67 - 14 * q^70 - 4 * q^71 - 7 * q^73 + 27 * q^74 + q^77 - 16 * q^79 + 5 * q^80 - 2 * q^82 - 7 * q^83 - 25 * q^85 + q^86 - 10 * q^88 + 33 * q^89 - 4 * q^91 + 21 * q^92 - 4 * q^94 - 4 * q^95 - 34 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more