LMFDB - Embedded newform 8037.2.a.r.1.18

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:	$23$
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q+1.55024 q^{2} +0.403250 q^{4} +0.718559 q^{5} -2.40005 q^{7} -2.47535 q^{8} +O(q^{10})$	\(q+1.55024 q^{2} +0.403250 q^{4} +0.718559 q^{5} -2.40005 q^{7} -2.47535 q^{8} +1.11394 q^{10} +0.568929 q^{11} +2.33712 q^{13} -3.72066 q^{14} -4.64389 q^{16} +3.98562 q^{17} +1.00000 q^{19} +0.289759 q^{20} +0.881978 q^{22} -1.61973 q^{23} -4.48367 q^{25} +3.62310 q^{26} -0.967819 q^{28} -1.27721 q^{29} +7.06204 q^{31} -2.24845 q^{32} +6.17868 q^{34} -1.72458 q^{35} +9.07284 q^{37} +1.55024 q^{38} -1.77868 q^{40} -6.46276 q^{41} -8.21650 q^{43} +0.229420 q^{44} -2.51098 q^{46} +1.00000 q^{47} -1.23976 q^{49} -6.95078 q^{50} +0.942442 q^{52} -5.36571 q^{53} +0.408809 q^{55} +5.94096 q^{56} -1.97998 q^{58} +6.83549 q^{59} +6.65890 q^{61} +10.9479 q^{62} +5.80213 q^{64} +1.67936 q^{65} +2.04326 q^{67} +1.60720 q^{68} -2.67351 q^{70} -7.46730 q^{71} +14.0826 q^{73} +14.0651 q^{74} +0.403250 q^{76} -1.36546 q^{77} +13.8709 q^{79} -3.33691 q^{80} -10.0188 q^{82} -12.5137 q^{83} +2.86391 q^{85} -12.7376 q^{86} -1.40830 q^{88} +7.47983 q^{89} -5.60920 q^{91} -0.653157 q^{92} +1.55024 q^{94} +0.718559 q^{95} -8.73484 q^{97} -1.92192 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) $ 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7	$ 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) $ 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7 + 5 * q^10 - 2 * q^11 + 13 * q^13 - 6 * q^14 + 35 * q^16 - 12 * q^17 + 23 * q^19 - 3 * q^20 - 4 * q^22 - 13 * q^23 + 46 * q^25 + 7 * q^26 + 11 * q^28 + 18 * q^29 + 22 * q^31 + 4 * q^32 - 20 * q^34 - 25 * q^35 + 8 * q^37 - q^38 - 16 * q^40 + 16 * q^41 + 68 * q^43 + 18 * q^44 + 13 * q^46 + 23 * q^47 + 52 * q^49 + 21 * q^50 + 54 * q^52 + 7 * q^53 + 32 * q^55 - 33 * q^56 + 6 * q^58 - 10 * q^59 + 28 * q^61 + 24 * q^62 + 40 * q^64 + 18 * q^65 + 55 * q^67 - 41 * q^68 - 40 * q^70 + 3 * q^71 + 48 * q^73 + 55 * q^74 + 31 * q^76 + 14 * q^77 - 31 * q^79 + 3 * q^80 + 48 * q^82 - 47 * q^83 - 5 * q^85 + 71 * q^86 + 9 * q^88 + 6 * q^89 + 40 * q^91 - 35 * q^92 - q^94 - q^95 - 18 * q^97 + 68 * 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 $

$2$	1.55024	1.09619	0.548093	−	0.836417i	$-0.315354\pi$
$2$	1.55024	1.09619	0.548093	+	0.836417i	$0.315354\pi$
$3$	0	0
$3$	0	0
$4$	0.403250	0.201625
$5$	0.718559	0.321349	0.160675	−	0.987007i	$-0.448633\pi$
$5$	0.718559	0.321349	0.160675	+	0.987007i	$0.448633\pi$
$6$	0	0
$7$	−2.40005	−0.907134	−0.453567	−	0.891222i	$-0.649849\pi$
$7$	−2.40005	−0.907134	−0.453567	+	0.891222i	$0.649849\pi$
$8$	−2.47535	−0.875168
$9$	0	0
$10$	1.11394	0.352259
$11$	0.568929	0.171539	0.0857693	−	0.996315i	$-0.472665\pi$
$11$	0.568929	0.171539	0.0857693	+	0.996315i	$0.472665\pi$
$12$	0	0
$13$	2.33712	0.648200	0.324100	−	0.946023i	$-0.394939\pi$
$13$	2.33712	0.648200	0.324100	+	0.946023i	$0.394939\pi$
$14$	−3.72066	−0.994388
$15$	0	0
$16$	−4.64389	−1.16097
$17$	3.98562	0.966656	0.483328	−	0.875439i	$-0.339428\pi$
$17$	3.98562	0.966656	0.483328	+	0.875439i	$0.339428\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.289759	0.0647920
$21$	0	0
$22$	0.881978	0.188038
$23$	−1.61973	−0.337738	−0.168869	−	0.985639i	$-0.554011\pi$
$23$	−1.61973	−0.337738	−0.168869	+	0.985639i	$0.554011\pi$
$24$	0	0
$25$	−4.48367	−0.896735
$26$	3.62310	0.710548
$27$	0	0
$28$	−0.967819	−0.182901
$29$	−1.27721	−0.237171	−0.118586	−	0.992944i	$-0.537836\pi$
$29$	−1.27721	−0.237171	−0.118586	+	0.992944i	$0.537836\pi$
$30$	0	0
$31$	7.06204	1.26838	0.634190	−	0.773177i	$-0.281334\pi$
$31$	7.06204	1.26838	0.634190	+	0.773177i	$0.281334\pi$
$32$	−2.24845	−0.397474
$33$	0	0
$34$	6.17868	1.05963
$35$	−1.72458	−0.291507
$36$	0	0
$37$	9.07284	1.49157	0.745783	−	0.666189i	$-0.232076\pi$
$37$	9.07284	1.49157	0.745783	+	0.666189i	$0.232076\pi$
$38$	1.55024	0.251482
$39$	0	0
$40$	−1.77868	−0.281235
$41$	−6.46276	−1.00931	−0.504657	−	0.863320i	$-0.668381\pi$
$41$	−6.46276	−1.00931	−0.504657	+	0.863320i	$0.668381\pi$
$42$	0	0
$43$	−8.21650	−1.25300	−0.626502	−	0.779420i	$-0.715514\pi$
$43$	−8.21650	−1.25300	−0.626502	+	0.779420i	$0.715514\pi$
$44$	0.229420	0.0345864
$45$	0	0
$46$	−2.51098	−0.370224
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−1.23976	−0.177108
$50$	−6.95078	−0.982988
$51$	0	0
$52$	0.942442	0.130693
$53$	−5.36571	−0.737037	−0.368519	−	0.929620i	$-0.620135\pi$
$53$	−5.36571	−0.737037	−0.368519	+	0.929620i	$0.620135\pi$
$54$	0	0
$55$	0.408809	0.0551238
$56$	5.94096	0.793895
$57$	0	0
$58$	−1.97998	−0.259984
$59$	6.83549	0.889905	0.444953	−	0.895554i	$-0.353220\pi$
$59$	6.83549	0.889905	0.444953	+	0.895554i	$0.353220\pi$
$60$	0	0
$61$	6.65890	0.852585	0.426293	−	0.904585i	$-0.359819\pi$
$61$	6.65890	0.852585	0.426293	+	0.904585i	$0.359819\pi$
$62$	10.9479	1.39038
$63$	0	0
$64$	5.80213	0.725267
$65$	1.67936	0.208299
$66$	0	0
$67$	2.04326	0.249623	0.124812	−	0.992180i	$-0.460167\pi$
$67$	2.04326	0.249623	0.124812	+	0.992180i	$0.460167\pi$
$68$	1.60720	0.194902
$69$	0	0
$70$	−2.67351	−0.319546
$71$	−7.46730	−0.886205	−0.443102	−	0.896471i	$-0.646122\pi$
$71$	−7.46730	−0.886205	−0.443102	+	0.896471i	$0.646122\pi$
$72$	0	0
$73$	14.0826	1.64824	0.824122	−	0.566413i	$-0.191669\pi$
$73$	14.0826	1.64824	0.824122	+	0.566413i	$0.191669\pi$
$74$	14.0651	1.63503
$75$	0	0
$76$	0.403250	0.0462559
$77$	−1.36546	−0.155608
$78$	0	0
$79$	13.8709	1.56060	0.780299	−	0.625407i	$-0.215067\pi$
$79$	13.8709	1.56060	0.780299	+	0.625407i	$0.215067\pi$
$80$	−3.33691	−0.373078
$81$	0	0
$82$	−10.0188	−1.10640
$83$	−12.5137	−1.37355	−0.686776	−	0.726869i	$-0.740975\pi$
$83$	−12.5137	−1.37355	−0.686776	+	0.726869i	$0.740975\pi$
$84$	0	0
$85$	2.86391	0.310634
$86$	−12.7376	−1.37353
$87$	0	0
$88$	−1.40830	−0.150125
$89$	7.47983	0.792860	0.396430	−	0.918065i	$-0.370249\pi$
$89$	7.47983	0.792860	0.396430	+	0.918065i	$0.370249\pi$
$90$	0	0
$91$	−5.60920	−0.588004
$92$	−0.653157	−0.0680963
$93$	0	0
$94$	1.55024	0.159895
$95$	0.718559	0.0737226
$96$	0	0
$97$	−8.73484	−0.886889	−0.443445	−	0.896302i	$-0.646244\pi$
$97$	−8.73484	−0.886889	−0.443445	+	0.896302i	$0.646244\pi$
$98$	−1.92192	−0.194144
$99$	0	0
$100$	−1.80804	−0.180804
$101$	12.4309	1.23692	0.618460	−	0.785816i	$-0.287757\pi$
$101$	12.4309	1.23692	0.618460	+	0.785816i	$0.287757\pi$
$102$	0	0
$103$	13.2524	1.30580	0.652898	−	0.757446i	$-0.273553\pi$
$103$	13.2524	1.30580	0.652898	+	0.757446i	$0.273553\pi$
$104$	−5.78518	−0.567284
$105$	0	0
$106$	−8.31815	−0.807930
$107$	17.8742	1.72797	0.863984	−	0.503520i	$-0.167962\pi$
$107$	17.8742	1.72797	0.863984	+	0.503520i	$0.167962\pi$
$108$	0	0
$109$	13.3754	1.28113	0.640564	−	0.767905i	$-0.278701\pi$
$109$	13.3754	1.28113	0.640564	+	0.767905i	$0.278701\pi$
$110$	0.633753	0.0604260
$111$	0	0
$112$	11.1456	1.05316
$113$	−15.1415	−1.42440	−0.712198	−	0.701978i	$-0.752300\pi$
$113$	−15.1415	−1.42440	−0.712198	+	0.701978i	$0.752300\pi$
$114$	0	0
$115$	−1.16387	−0.108532
$116$	−0.515033	−0.0478196
$117$	0	0
$118$	10.5967	0.975502
$119$	−9.56570	−0.876886
$120$	0	0
$121$	−10.6763	−0.970575
$122$	10.3229	0.934592
$123$	0	0
$124$	2.84777	0.255737
$125$	−6.81458	−0.609515
$126$	0	0
$127$	21.5698	1.91401	0.957007	−	0.290064i	$-0.0936766\pi$
$127$	21.5698	1.91401	0.957007	+	0.290064i	$0.0936766\pi$
$128$	13.4916	1.19250
$129$	0	0
$130$	2.60341	0.228334
$131$	−10.2511	−0.895644	−0.447822	−	0.894123i	$-0.647800\pi$
$131$	−10.2511	−0.895644	−0.447822	+	0.894123i	$0.647800\pi$
$132$	0	0
$133$	−2.40005	−0.208111
$134$	3.16754	0.273634
$135$	0	0
$136$	−9.86581	−0.845986
$137$	−9.33690	−0.797705	−0.398853	−	0.917015i	$-0.630592\pi$
$137$	−9.33690	−0.797705	−0.398853	+	0.917015i	$0.630592\pi$
$138$	0	0
$139$	15.2093	1.29004	0.645018	−	0.764167i	$-0.276850\pi$
$139$	15.2093	1.29004	0.645018	+	0.764167i	$0.276850\pi$
$140$	−0.695436	−0.0587750
$141$	0	0
$142$	−11.5761	−0.971446
$143$	1.32965	0.111191
$144$	0	0
$145$	−0.917749	−0.0762149
$146$	21.8314	1.80678
$147$	0	0
$148$	3.65862	0.300737
$149$	22.5644	1.84855	0.924273	−	0.381733i	$-0.124673\pi$
$149$	22.5644	1.84855	0.924273	+	0.381733i	$0.124673\pi$
$150$	0	0
$151$	4.68865	0.381557	0.190779	−	0.981633i	$-0.438899\pi$
$151$	4.68865	0.381557	0.190779	+	0.981633i	$0.438899\pi$
$152$	−2.47535	−0.200777
$153$	0	0
$154$	−2.11679	−0.170576
$155$	5.07450	0.407593
$156$	0	0
$157$	−20.2312	−1.61463	−0.807314	−	0.590121i	$-0.799080\pi$
$157$	−20.2312	−1.61463	−0.807314	+	0.590121i	$0.799080\pi$
$158$	21.5032	1.71071
$159$	0	0
$160$	−1.61565	−0.127728
$161$	3.88744	0.306373
$162$	0	0
$163$	21.9116	1.71625	0.858124	−	0.513442i	$-0.171630\pi$
$163$	21.9116	1.71625	0.858124	+	0.513442i	$0.171630\pi$
$164$	−2.60611	−0.203503
$165$	0	0
$166$	−19.3992	−1.50567
$167$	15.1659	1.17358	0.586788	−	0.809741i	$-0.300392\pi$
$167$	15.1659	1.17358	0.586788	+	0.809741i	$0.300392\pi$
$168$	0	0
$169$	−7.53788	−0.579837
$170$	4.43975	0.340513
$171$	0	0
$172$	−3.31330	−0.252637
$173$	17.6086	1.33876	0.669378	−	0.742922i	$-0.266560\pi$
$173$	17.6086	1.33876	0.669378	+	0.742922i	$0.266560\pi$
$174$	0	0
$175$	10.7610	0.813458
$176$	−2.64204	−0.199151
$177$	0	0
$178$	11.5955	0.869123
$179$	8.07398	0.603477	0.301739	−	0.953391i	$-0.402433\pi$
$179$	8.07398	0.603477	0.301739	+	0.953391i	$0.402433\pi$
$180$	0	0
$181$	−0.990436	−0.0736185	−0.0368092	−	0.999322i	$-0.511719\pi$
$181$	−0.990436	−0.0736185	−0.0368092	+	0.999322i	$0.511719\pi$
$182$	−8.69562	−0.644562
$183$	0	0
$184$	4.00941	0.295577
$185$	6.51937	0.479314
$186$	0	0
$187$	2.26754	0.165819
$188$	0.403250	0.0294100
$189$	0	0
$190$	1.11394	0.0808137
$191$	20.3144	1.46990	0.734950	−	0.678121i	$-0.237205\pi$
$191$	20.3144	1.46990	0.734950	+	0.678121i	$0.237205\pi$
$192$	0	0
$193$	17.5737	1.26498	0.632492	−	0.774567i	$-0.282032\pi$
$193$	17.5737	1.26498	0.632492	+	0.774567i	$0.282032\pi$
$194$	−13.5411	−0.972196
$195$	0	0
$196$	−0.499932	−0.0357094
$197$	−5.94023	−0.423224	−0.211612	−	0.977354i	$-0.567871\pi$
$197$	−5.94023	−0.423224	−0.211612	+	0.977354i	$0.567871\pi$
$198$	0	0
$199$	−6.92848	−0.491147	−0.245574	−	0.969378i	$-0.578976\pi$
$199$	−6.92848	−0.491147	−0.245574	+	0.969378i	$0.578976\pi$
$200$	11.0987	0.784793
$201$	0	0
$202$	19.2709	1.35590
$203$	3.06536	0.215146
$204$	0	0
$205$	−4.64388	−0.324343
$206$	20.5444	1.43140
$207$	0	0
$208$	−10.8533	−0.752542
$209$	0.568929	0.0393536
$210$	0	0
$211$	−2.16833	−0.149274	−0.0746369	−	0.997211i	$-0.523780\pi$
$211$	−2.16833	−0.149274	−0.0746369	+	0.997211i	$0.523780\pi$
$212$	−2.16372	−0.148605
$213$	0	0
$214$	27.7094	1.89417
$215$	−5.90404	−0.402652
$216$	0	0
$217$	−16.9493	−1.15059
$218$	20.7350	1.40435
$219$	0	0
$220$	0.164852	0.0111143
$221$	9.31487	0.626586
$222$	0	0
$223$	16.4848	1.10390	0.551952	−	0.833876i	$-0.313883\pi$
$223$	16.4848	1.10390	0.551952	+	0.833876i	$0.313883\pi$
$224$	5.39640	0.360562
$225$	0	0
$226$	−23.4730	−1.56140
$227$	−5.52501	−0.366708	−0.183354	−	0.983047i	$-0.558695\pi$
$227$	−5.52501	−0.366708	−0.183354	+	0.983047i	$0.558695\pi$
$228$	0	0
$229$	−12.4474	−0.822545	−0.411272	−	0.911512i	$-0.634915\pi$
$229$	−12.4474	−0.822545	−0.411272	+	0.911512i	$0.634915\pi$
$230$	−1.80429	−0.118971
$231$	0	0
$232$	3.16153	0.207565
$233$	24.1683	1.58332	0.791658	−	0.610964i	$-0.209218\pi$
$233$	24.1683	1.58332	0.791658	+	0.610964i	$0.209218\pi$
$234$	0	0
$235$	0.718559	0.0468736
$236$	2.75641	0.179427
$237$	0	0
$238$	−14.8291	−0.961231
$239$	−2.21428	−0.143230	−0.0716148	−	0.997432i	$-0.522815\pi$
$239$	−2.21428	−0.143230	−0.0716148	+	0.997432i	$0.522815\pi$
$240$	0	0
$241$	−20.7905	−1.33923	−0.669617	−	0.742707i	$-0.733542\pi$
$241$	−20.7905	−1.33923	−0.669617	+	0.742707i	$0.733542\pi$
$242$	−16.5509	−1.06393
$243$	0	0
$244$	2.68520	0.171902
$245$	−0.890839	−0.0569136
$246$	0	0
$247$	2.33712	0.148707
$248$	−17.4810	−1.11005
$249$	0	0
$250$	−10.5642	−0.668142
$251$	−1.57738	−0.0995630	−0.0497815	−	0.998760i	$-0.515853\pi$
$251$	−1.57738	−0.0995630	−0.0497815	+	0.998760i	$0.515853\pi$
$252$	0	0
$253$	−0.921513	−0.0579350
$254$	33.4385	2.09812
$255$	0	0
$256$	9.31100	0.581937
$257$	−16.1442	−1.00705	−0.503524	−	0.863981i	$-0.667964\pi$
$257$	−16.1442	−1.00705	−0.503524	+	0.863981i	$0.667964\pi$
$258$	0	0
$259$	−21.7753	−1.35305
$260$	0.677200	0.0419982
$261$	0	0
$262$	−15.8917	−0.981793
$263$	29.6113	1.82591	0.912957	−	0.408056i	$-0.133793\pi$
$263$	29.6113	1.82591	0.912957	+	0.408056i	$0.133793\pi$
$264$	0	0
$265$	−3.85558	−0.236846
$266$	−3.72066	−0.228128
$267$	0	0
$268$	0.823942	0.0503303
$269$	19.9939	1.21905	0.609524	−	0.792768i	$-0.291361\pi$
$269$	19.9939	1.21905	0.609524	+	0.792768i	$0.291361\pi$
$270$	0	0
$271$	−2.93935	−0.178553	−0.0892765	−	0.996007i	$-0.528455\pi$
$271$	−2.93935	−0.178553	−0.0892765	+	0.996007i	$0.528455\pi$
$272$	−18.5088	−1.12226
$273$	0	0
$274$	−14.4745	−0.874434
$275$	−2.55089	−0.153825
$276$	0	0
$277$	20.2613	1.21738	0.608691	−	0.793408i	$-0.291695\pi$
$277$	20.2613	1.21738	0.608691	+	0.793408i	$0.291695\pi$
$278$	23.5781	1.41412
$279$	0	0
$280$	4.26893	0.255118
$281$	−9.90719	−0.591013	−0.295507	−	0.955341i	$-0.595488\pi$
$281$	−9.90719	−0.591013	−0.295507	+	0.955341i	$0.595488\pi$
$282$	0	0
$283$	−26.7217	−1.58844	−0.794219	−	0.607632i	$-0.792120\pi$
$283$	−26.7217	−1.58844	−0.794219	+	0.607632i	$0.792120\pi$
$284$	−3.01118	−0.178681
$285$	0	0
$286$	2.06129	0.121886
$287$	15.5110	0.915583
$288$	0	0
$289$	−1.11481	−0.0655769
$290$	−1.42273	−0.0835457
$291$	0	0
$292$	5.67880	0.332327
$293$	−25.1172	−1.46736	−0.733682	−	0.679493i	$-0.762200\pi$
$293$	−25.1172	−1.46736	−0.733682	+	0.679493i	$0.762200\pi$
$294$	0	0
$295$	4.91171	0.285971
$296$	−22.4584	−1.30537
$297$	0	0
$298$	34.9802	2.02635
$299$	−3.78551	−0.218922
$300$	0	0
$301$	19.7200	1.13664
$302$	7.26855	0.418258
$303$	0	0
$304$	−4.64389	−0.266345
$305$	4.78482	0.273978
$306$	0	0
$307$	10.4363	0.595633	0.297817	−	0.954623i	$-0.403742\pi$
$307$	10.4363	0.595633	0.297817	+	0.954623i	$0.403742\pi$
$308$	−0.550621	−0.0313745
$309$	0	0
$310$	7.86670	0.446798
$311$	−11.2964	−0.640562	−0.320281	−	0.947323i	$-0.603777\pi$
$311$	−11.2964	−0.640562	−0.320281	+	0.947323i	$0.603777\pi$
$312$	0	0
$313$	20.0211	1.13166	0.565829	−	0.824523i	$-0.308556\pi$
$313$	20.0211	1.13166	0.565829	+	0.824523i	$0.308556\pi$
$314$	−31.3633	−1.76993
$315$	0	0
$316$	5.59343	0.314655
$317$	−13.2984	−0.746910	−0.373455	−	0.927648i	$-0.621827\pi$
$317$	−13.2984	−0.746910	−0.373455	+	0.927648i	$0.621827\pi$
$318$	0	0
$319$	−0.726640	−0.0406840
$320$	4.16918	0.233064
$321$	0	0
$322$	6.02647	0.335842
$323$	3.98562	0.221766
$324$	0	0
$325$	−10.4789	−0.581263
$326$	33.9683	1.88133
$327$	0	0
$328$	15.9976	0.883320
$329$	−2.40005	−0.132319
$330$	0	0
$331$	−1.30805	−0.0718971	−0.0359485	−	0.999354i	$-0.511445\pi$
$331$	−1.30805	−0.0718971	−0.0359485	+	0.999354i	$0.511445\pi$
$332$	−5.04613	−0.276942
$333$	0	0
$334$	23.5109	1.28646
$335$	1.46820	0.0802164
$336$	0	0
$337$	11.7882	0.642146	0.321073	−	0.947054i	$-0.395956\pi$
$337$	11.7882	0.642146	0.321073	+	0.947054i	$0.395956\pi$
$338$	−11.6855	−0.635610
$339$	0	0
$340$	1.15487	0.0626316
$341$	4.01780	0.217576
$342$	0	0
$343$	19.7758	1.06779
$344$	20.3387	1.09659
$345$	0	0
$346$	27.2976	1.46753
$347$	−25.9230	−1.39162	−0.695809	−	0.718227i	$-0.744954\pi$
$347$	−25.9230	−1.39162	−0.695809	+	0.718227i	$0.744954\pi$
$348$	0	0
$349$	−1.98134	−0.106059	−0.0530294	−	0.998593i	$-0.516888\pi$
$349$	−1.98134	−0.106059	−0.0530294	+	0.998593i	$0.516888\pi$
$350$	16.6822	0.891702
$351$	0	0
$352$	−1.27921	−0.0681821
$353$	−1.46660	−0.0780593	−0.0390296	−	0.999238i	$-0.512427\pi$
$353$	−1.46660	−0.0780593	−0.0390296	+	0.999238i	$0.512427\pi$
$354$	0	0
$355$	−5.36569	−0.284781
$356$	3.01624	0.159860
$357$	0	0
$358$	12.5166	0.661524
$359$	−20.1559	−1.06379	−0.531893	−	0.846812i	$-0.678519\pi$
$359$	−20.1559	−1.06379	−0.531893	+	0.846812i	$0.678519\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.53541	−0.0806996
$363$	0	0
$364$	−2.26191	−0.118556
$365$	10.1192	0.529662
$366$	0	0
$367$	10.7731	0.562349	0.281174	−	0.959657i	$-0.409276\pi$
$367$	10.7731	0.562349	0.281174	+	0.959657i	$0.409276\pi$
$368$	7.52186	0.392104
$369$	0	0
$370$	10.1066	0.525417
$371$	12.8780	0.668591
$372$	0	0
$373$	13.9648	0.723069	0.361534	−	0.932359i	$-0.382253\pi$
$373$	13.9648	0.723069	0.361534	+	0.932359i	$0.382253\pi$
$374$	3.51523	0.181768
$375$	0	0
$376$	−2.47535	−0.127656
$377$	−2.98498	−0.153734
$378$	0	0
$379$	18.7788	0.964604	0.482302	−	0.876005i	$-0.339801\pi$
$379$	18.7788	0.964604	0.482302	+	0.876005i	$0.339801\pi$
$380$	0.289759	0.0148643
$381$	0	0
$382$	31.4923	1.61129
$383$	−16.7408	−0.855416	−0.427708	−	0.903917i	$-0.640679\pi$
$383$	−16.7408	−0.855416	−0.427708	+	0.903917i	$0.640679\pi$
$384$	0	0
$385$	−0.981163	−0.0500047
$386$	27.2435	1.38666
$387$	0	0
$388$	−3.52232	−0.178819
$389$	0.670058	0.0339733	0.0169867	−	0.999856i	$-0.494593\pi$
$389$	0.670058	0.0339733	0.0169867	+	0.999856i	$0.494593\pi$
$390$	0	0
$391$	−6.45565	−0.326476
$392$	3.06883	0.154999
$393$	0	0
$394$	−9.20879	−0.463932
$395$	9.96706	0.501497
$396$	0	0
$397$	−6.27877	−0.315123	−0.157561	−	0.987509i	$-0.550363\pi$
$397$	−6.27877	−0.315123	−0.157561	+	0.987509i	$0.550363\pi$
$398$	−10.7408	−0.538389
$399$	0	0
$400$	20.8217	1.04108
$401$	30.8655	1.54135	0.770675	−	0.637229i	$-0.219919\pi$
$401$	30.8655	1.54135	0.770675	+	0.637229i	$0.219919\pi$
$402$	0	0
$403$	16.5048	0.822164
$404$	5.01275	0.249394
$405$	0	0
$406$	4.75205	0.235840
$407$	5.16180	0.255861
$408$	0	0
$409$	15.3790	0.760445	0.380222	−	0.924895i	$-0.375847\pi$
$409$	15.3790	0.760445	0.380222	+	0.924895i	$0.375847\pi$
$410$	−7.19913	−0.355540
$411$	0	0
$412$	5.34402	0.263281
$413$	−16.4055	−0.807263
$414$	0	0
$415$	−8.99180	−0.441390
$416$	−5.25490	−0.257643
$417$	0	0
$418$	0.881978	0.0431389
$419$	−29.9298	−1.46217	−0.731084	−	0.682287i	$-0.760985\pi$
$419$	−29.9298	−1.46217	−0.731084	+	0.682287i	$0.760985\pi$
$420$	0	0
$421$	−29.7419	−1.44953	−0.724765	−	0.688996i	$-0.758052\pi$
$421$	−29.7419	−1.44953	−0.724765	+	0.688996i	$0.758052\pi$
$422$	−3.36143	−0.163632
$423$	0	0
$424$	13.2820	0.645031
$425$	−17.8702	−0.866833
$426$	0	0
$427$	−15.9817	−0.773409
$428$	7.20778	0.348401
$429$	0	0
$430$	−9.15269	−0.441382
$431$	22.5861	1.08793	0.543967	−	0.839107i	$-0.316922\pi$
$431$	22.5861	1.08793	0.543967	+	0.839107i	$0.316922\pi$
$432$	0	0
$433$	−17.7527	−0.853143	−0.426571	−	0.904454i	$-0.640279\pi$
$433$	−17.7527	−0.853143	−0.426571	+	0.904454i	$0.640279\pi$
$434$	−26.2755	−1.26126
$435$	0	0
$436$	5.39361	0.258307
$437$	−1.61973	−0.0774823
$438$	0	0
$439$	−2.90310	−0.138557	−0.0692786	−	0.997597i	$-0.522070\pi$
$439$	−2.90310	−0.138557	−0.0692786	+	0.997597i	$0.522070\pi$
$440$	−1.01195	−0.0482426
$441$	0	0
$442$	14.4403	0.686855
$443$	−6.62928	−0.314966	−0.157483	−	0.987522i	$-0.550338\pi$
$443$	−6.62928	−0.314966	−0.157483	+	0.987522i	$0.550338\pi$
$444$	0	0
$445$	5.37470	0.254785
$446$	25.5554	1.21008
$447$	0	0
$448$	−13.9254	−0.657914
$449$	−30.6571	−1.44680	−0.723399	−	0.690431i	$-0.757421\pi$
$449$	−30.6571	−1.44680	−0.723399	+	0.690431i	$0.757421\pi$
$450$	0	0
$451$	−3.67685	−0.173136
$452$	−6.10582	−0.287194
$453$	0	0
$454$	−8.56510	−0.401980
$455$	−4.03054	−0.188955
$456$	0	0
$457$	11.3940	0.532987	0.266494	−	0.963837i	$-0.414135\pi$
$457$	11.3940	0.532987	0.266494	+	0.963837i	$0.414135\pi$
$458$	−19.2964	−0.901662
$459$	0	0
$460$	−0.469332	−0.0218827
$461$	−17.3601	−0.808538	−0.404269	−	0.914640i	$-0.632474\pi$
$461$	−17.3601	−0.808538	−0.404269	+	0.914640i	$0.632474\pi$
$462$	0	0
$463$	−8.38572	−0.389717	−0.194859	−	0.980831i	$-0.562425\pi$
$463$	−8.38572	−0.389717	−0.194859	+	0.980831i	$0.562425\pi$
$464$	5.93121	0.275349
$465$	0	0
$466$	37.4667	1.73561
$467$	−35.5413	−1.64466	−0.822328	−	0.569014i	$-0.807325\pi$
$467$	−35.5413	−1.64466	−0.822328	+	0.569014i	$0.807325\pi$
$468$	0	0
$469$	−4.90392	−0.226442
$470$	1.11394	0.0513822
$471$	0	0
$472$	−16.9202	−0.778817
$473$	−4.67460	−0.214939
$474$	0	0
$475$	−4.48367	−0.205725
$476$	−3.85736	−0.176802
$477$	0	0
$478$	−3.43266	−0.157006
$479$	−5.44216	−0.248659	−0.124329	−	0.992241i	$-0.539678\pi$
$479$	−5.44216	−0.248659	−0.124329	+	0.992241i	$0.539678\pi$
$480$	0	0
$481$	21.2043	0.966832
$482$	−32.2303	−1.46805
$483$	0	0
$484$	−4.30522	−0.195692
$485$	−6.27650	−0.285001
$486$	0	0
$487$	22.0105	0.997390	0.498695	−	0.866778i	$-0.333813\pi$
$487$	22.0105	0.997390	0.498695	+	0.866778i	$0.333813\pi$
$488$	−16.4831	−0.746155
$489$	0	0
$490$	−1.38102	−0.0623880
$491$	−31.8384	−1.43685	−0.718423	−	0.695606i	$-0.755136\pi$
$491$	−31.8384	−1.43685	−0.718423	+	0.695606i	$0.755136\pi$
$492$	0	0
$493$	−5.09047	−0.229263
$494$	3.62310	0.163011
$495$	0	0
$496$	−32.7953	−1.47255
$497$	17.9219	0.803907
$498$	0	0
$499$	−0.404739	−0.0181186	−0.00905930	−	0.999959i	$-0.502884\pi$
$499$	−0.404739	−0.0181186	−0.00905930	+	0.999959i	$0.502884\pi$
$500$	−2.74798	−0.122893
$501$	0	0
$502$	−2.44531	−0.109140
$503$	−26.9115	−1.19993	−0.599963	−	0.800028i	$-0.704818\pi$
$503$	−26.9115	−1.19993	−0.599963	+	0.800028i	$0.704818\pi$
$504$	0	0
$505$	8.93234	0.397484
$506$	−1.42857	−0.0635076
$507$	0	0
$508$	8.69803	0.385913
$509$	20.8512	0.924212	0.462106	−	0.886825i	$-0.347094\pi$
$509$	20.8512	0.924212	0.462106	+	0.886825i	$0.347094\pi$
$510$	0	0
$511$	−33.7989	−1.49518
$512$	−12.5489	−0.554590
$513$	0	0
$514$	−25.0274	−1.10391
$515$	9.52263	0.419617
$516$	0	0
$517$	0.568929	0.0250215
$518$	−33.7569	−1.48319
$519$	0	0
$520$	−4.15700	−0.182296
$521$	20.2442	0.886914	0.443457	−	0.896296i	$-0.353752\pi$
$521$	20.2442	0.886914	0.443457	+	0.896296i	$0.353752\pi$
$522$	0	0
$523$	27.9183	1.22078	0.610391	−	0.792100i	$-0.291012\pi$
$523$	27.9183	1.22078	0.610391	+	0.792100i	$0.291012\pi$
$524$	−4.13376	−0.180584
$525$	0	0
$526$	45.9048	2.00154
$527$	28.1466	1.22609
$528$	0	0
$529$	−20.3765	−0.885933
$530$	−5.97708	−0.259628
$531$	0	0
$532$	−0.967819	−0.0419603
$533$	−15.1042	−0.654237
$534$	0	0
$535$	12.8437	0.555281
$536$	−5.05777	−0.218462
$537$	0	0
$538$	30.9953	1.33630
$539$	−0.705334	−0.0303809
$540$	0	0
$541$	−34.3276	−1.47586	−0.737930	−	0.674877i	$-0.764197\pi$
$541$	−34.3276	−1.47586	−0.737930	+	0.674877i	$0.764197\pi$
$542$	−4.55671	−0.195727
$543$	0	0
$544$	−8.96148	−0.384220
$545$	9.61099	0.411690
$546$	0	0
$547$	17.3678	0.742591	0.371296	−	0.928515i	$-0.378914\pi$
$547$	17.3678	0.742591	0.371296	+	0.928515i	$0.378914\pi$
$548$	−3.76510	−0.160837
$549$	0	0
$550$	−3.95450	−0.168620
$551$	−1.27721	−0.0544109
$552$	0	0
$553$	−33.2908	−1.41567
$554$	31.4099	1.33448
$555$	0	0
$556$	6.13315	0.260103
$557$	11.5505	0.489411	0.244705	−	0.969597i	$-0.421309\pi$
$557$	11.5505	0.489411	0.244705	+	0.969597i	$0.421309\pi$
$558$	0	0
$559$	−19.2029	−0.812197
$560$	8.00875	0.338431
$561$	0	0
$562$	−15.3585	−0.647861
$563$	30.2358	1.27429	0.637144	−	0.770745i	$-0.280116\pi$
$563$	30.2358	1.27429	0.637144	+	0.770745i	$0.280116\pi$
$564$	0	0
$565$	−10.8801	−0.457729
$566$	−41.4250	−1.74122
$567$	0	0
$568$	18.4842	0.775578
$569$	−18.9050	−0.792541	−0.396270	−	0.918134i	$-0.629696\pi$
$569$	−18.9050	−0.792541	−0.396270	+	0.918134i	$0.629696\pi$
$570$	0	0
$571$	38.6805	1.61873	0.809364	−	0.587308i	$-0.199812\pi$
$571$	38.6805	1.61873	0.809364	+	0.587308i	$0.199812\pi$
$572$	0.536182	0.0224189
$573$	0	0
$574$	24.0457	1.00365
$575$	7.26235	0.302861
$576$	0	0
$577$	−8.06706	−0.335836	−0.167918	−	0.985801i	$-0.553704\pi$
$577$	−8.06706	−0.335836	−0.167918	+	0.985801i	$0.553704\pi$
$578$	−1.72822	−0.0718845
$579$	0	0
$580$	−0.370082	−0.0153668
$581$	30.0334	1.24600
$582$	0	0
$583$	−3.05271	−0.126430
$584$	−34.8593	−1.44249
$585$	0	0
$586$	−38.9378	−1.60850
$587$	23.3387	0.963292	0.481646	−	0.876366i	$-0.340039\pi$
$587$	23.3387	0.963292	0.481646	+	0.876366i	$0.340039\pi$
$588$	0	0
$589$	7.06204	0.290986
$590$	7.61433	0.313477
$591$	0	0
$592$	−42.1332	−1.73167
$593$	45.2932	1.85997	0.929983	−	0.367601i	$-0.119821\pi$
$593$	45.2932	1.85997	0.929983	+	0.367601i	$0.119821\pi$
$594$	0	0
$595$	−6.87352	−0.281787
$596$	9.09907	0.372713
$597$	0	0
$598$	−5.86845	−0.239979
$599$	−37.3612	−1.52654	−0.763268	−	0.646082i	$-0.776406\pi$
$599$	−37.3612	−1.52654	−0.763268	+	0.646082i	$0.776406\pi$
$600$	0	0
$601$	0.103362	0.00421622	0.00210811	−	0.999998i	$-0.499329\pi$
$601$	0.103362	0.00421622	0.00210811	+	0.999998i	$0.499329\pi$
$602$	30.5708	1.24597
$603$	0	0
$604$	1.89070	0.0769314
$605$	−7.67157	−0.311894
$606$	0	0
$607$	3.74615	0.152052	0.0760258	−	0.997106i	$-0.475777\pi$
$607$	3.74615	0.152052	0.0760258	+	0.997106i	$0.475777\pi$
$608$	−2.24845	−0.0911868
$609$	0	0
$610$	7.41762	0.300331
$611$	2.33712	0.0945497
$612$	0	0
$613$	−21.0081	−0.848509	−0.424255	−	0.905543i	$-0.639464\pi$
$613$	−21.0081	−0.848509	−0.424255	+	0.905543i	$0.639464\pi$
$614$	16.1788	0.652925
$615$	0	0
$616$	3.37999	0.136184
$617$	−26.7047	−1.07509	−0.537545	−	0.843235i	$-0.680648\pi$
$617$	−26.7047	−1.07509	−0.537545	+	0.843235i	$0.680648\pi$
$618$	0	0
$619$	33.6361	1.35195	0.675975	−	0.736925i	$-0.263723\pi$
$619$	33.6361	1.35195	0.675975	+	0.736925i	$0.263723\pi$
$620$	2.04629	0.0821809
$621$	0	0
$622$	−17.5122	−0.702175
$623$	−17.9520	−0.719230
$624$	0	0
$625$	17.5217	0.700867
$626$	31.0375	1.24051
$627$	0	0
$628$	−8.15824	−0.325549
$629$	36.1609	1.44183
$630$	0	0
$631$	−43.9693	−1.75039	−0.875196	−	0.483769i	$-0.839268\pi$
$631$	−43.9693	−1.75039	−0.875196	+	0.483769i	$0.839268\pi$
$632$	−34.3353	−1.36578
$633$	0	0
$634$	−20.6157	−0.818753
$635$	15.4992	0.615067
$636$	0	0
$637$	−2.89746	−0.114802
$638$	−1.12647	−0.0445973
$639$	0	0
$640$	9.69452	0.383210
$641$	−25.3699	−1.00205	−0.501025	−	0.865433i	$-0.667043\pi$
$641$	−25.3699	−1.00205	−0.501025	+	0.865433i	$0.667043\pi$
$642$	0	0
$643$	−4.60891	−0.181758	−0.0908788	−	0.995862i	$-0.528968\pi$
$643$	−4.60891	−0.181758	−0.0908788	+	0.995862i	$0.528968\pi$
$644$	1.56761	0.0617725
$645$	0	0
$646$	6.17868	0.243097
$647$	13.5440	0.532469	0.266234	−	0.963908i	$-0.414220\pi$
$647$	13.5440	0.532469	0.266234	+	0.963908i	$0.414220\pi$
$648$	0	0
$649$	3.88891	0.152653
$650$	−16.2448	−0.637173
$651$	0	0
$652$	8.83584	0.346038
$653$	−40.3301	−1.57824	−0.789119	−	0.614240i	$-0.789462\pi$
$653$	−40.3301	−1.57824	−0.789119	+	0.614240i	$0.789462\pi$
$654$	0	0
$655$	−7.36603	−0.287815
$656$	30.0124	1.17179
$657$	0	0
$658$	−3.72066	−0.145046
$659$	9.73894	0.379375	0.189688	−	0.981845i	$-0.439252\pi$
$659$	9.73894	0.379375	0.189688	+	0.981845i	$0.439252\pi$
$660$	0	0
$661$	21.9357	0.853201	0.426600	−	0.904440i	$-0.359711\pi$
$661$	21.9357	0.853201	0.426600	+	0.904440i	$0.359711\pi$
$662$	−2.02780	−0.0788126
$663$	0	0
$664$	30.9757	1.20209
$665$	−1.72458	−0.0668763
$666$	0	0
$667$	2.06873	0.0801017
$668$	6.11566	0.236622
$669$	0	0
$670$	2.27607	0.0879321
$671$	3.78844	0.146251
$672$	0	0
$673$	−24.0369	−0.926555	−0.463278	−	0.886213i	$-0.653327\pi$
$673$	−24.0369	−0.926555	−0.463278	+	0.886213i	$0.653327\pi$
$674$	18.2746	0.703912
$675$	0	0
$676$	−3.03965	−0.116910
$677$	−20.1463	−0.774283	−0.387142	−	0.922020i	$-0.626538\pi$
$677$	−20.1463	−0.774283	−0.387142	+	0.922020i	$0.626538\pi$
$678$	0	0
$679$	20.9641	0.804527
$680$	−7.08917	−0.271857
$681$	0	0
$682$	6.22856	0.238504
$683$	6.90964	0.264390	0.132195	−	0.991224i	$-0.457797\pi$
$683$	6.90964	0.264390	0.132195	+	0.991224i	$0.457797\pi$
$684$	0	0
$685$	−6.70912	−0.256342
$686$	30.6573	1.17050
$687$	0	0
$688$	38.1565	1.45470
$689$	−12.5403	−0.477747
$690$	0	0
$691$	19.7080	0.749729	0.374864	−	0.927080i	$-0.377689\pi$
$691$	19.7080	0.749729	0.374864	+	0.927080i	$0.377689\pi$
$692$	7.10066	0.269927
$693$	0	0
$694$	−40.1869	−1.52547
$695$	10.9288	0.414553
$696$	0	0
$697$	−25.7581	−0.975659
$698$	−3.07156	−0.116260
$699$	0	0
$700$	4.33939	0.164013
$701$	19.6020	0.740356	0.370178	−	0.928961i	$-0.379297\pi$
$701$	19.6020	0.740356	0.370178	+	0.928961i	$0.379297\pi$
$702$	0	0
$703$	9.07284	0.342189
$704$	3.30100	0.124411
$705$	0	0
$706$	−2.27359	−0.0855675
$707$	−29.8348	−1.12205
$708$	0	0
$709$	−29.1612	−1.09517	−0.547587	−	0.836749i	$-0.684453\pi$
$709$	−29.1612	−1.09517	−0.547587	+	0.836749i	$0.684453\pi$
$710$	−8.31812	−0.312174
$711$	0	0
$712$	−18.5152	−0.693886
$713$	−11.4386	−0.428380
$714$	0	0
$715$	0.955435	0.0357312
$716$	3.25583	0.121676
$717$	0	0
$718$	−31.2465	−1.16611
$719$	0.714742	0.0266554	0.0133277	−	0.999911i	$-0.495758\pi$
$719$	0.714742	0.0266554	0.0133277	+	0.999911i	$0.495758\pi$
$720$	0	0
$721$	−31.8064	−1.18453
$722$	1.55024	0.0576940
$723$	0	0
$724$	−0.399393	−0.0148433
$725$	5.72658	0.212680
$726$	0	0
$727$	14.7449	0.546859	0.273429	−	0.961892i	$-0.411842\pi$
$727$	14.7449	0.546859	0.273429	+	0.961892i	$0.411842\pi$
$728$	13.8847	0.514602
$729$	0	0
$730$	15.6872	0.580608
$731$	−32.7479	−1.21122
$732$	0	0
$733$	15.6372	0.577575	0.288787	−	0.957393i	$-0.406748\pi$
$733$	15.6372	0.577575	0.288787	+	0.957393i	$0.406748\pi$
$734$	16.7008	0.616439
$735$	0	0
$736$	3.64189	0.134242
$737$	1.16247	0.0428200
$738$	0	0
$739$	−33.1065	−1.21784	−0.608922	−	0.793230i	$-0.708398\pi$
$739$	−33.1065	−1.21784	−0.608922	+	0.793230i	$0.708398\pi$
$740$	2.62893	0.0966415
$741$	0	0
$742$	19.9640	0.732901
$743$	−11.6300	−0.426663	−0.213331	−	0.976980i	$-0.568431\pi$
$743$	−11.6300	−0.426663	−0.213331	+	0.976980i	$0.568431\pi$
$744$	0	0
$745$	16.2138	0.594029
$746$	21.6488	0.792618
$747$	0	0
$748$	0.914383	0.0334332
$749$	−42.8991	−1.56750
$750$	0	0
$751$	50.8647	1.85608	0.928039	−	0.372483i	$-0.121494\pi$
$751$	50.8647	1.85608	0.928039	+	0.372483i	$0.121494\pi$
$752$	−4.64389	−0.169345
$753$	0	0
$754$	−4.62745	−0.168522
$755$	3.36907	0.122613
$756$	0	0
$757$	−44.6005	−1.62103	−0.810516	−	0.585716i	$-0.800813\pi$
$757$	−44.6005	−1.62103	−0.810516	+	0.585716i	$0.800813\pi$
$758$	29.1117	1.05739
$759$	0	0
$760$	−1.77868	−0.0645197
$761$	30.1214	1.09190	0.545950	−	0.837818i	$-0.316169\pi$
$761$	30.1214	1.09190	0.545950	+	0.837818i	$0.316169\pi$
$762$	0	0
$763$	−32.1015	−1.16215
$764$	8.19179	0.296368
$765$	0	0
$766$	−25.9523	−0.937695
$767$	15.9754	0.576837
$768$	0	0
$769$	−28.0520	−1.01158	−0.505790	−	0.862657i	$-0.668799\pi$
$769$	−28.0520	−1.01158	−0.505790	+	0.862657i	$0.668799\pi$
$770$	−1.52104	−0.0548145
$771$	0	0
$772$	7.08659	0.255052
$773$	−17.2866	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$773$	−17.2866	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$774$	0	0
$775$	−31.6639	−1.13740
$776$	21.6218	0.776177
$777$	0	0
$778$	1.03875	0.0372411
$779$	−6.46276	−0.231553
$780$	0	0
$781$	−4.24836	−0.152018
$782$	−10.0078	−0.357879
$783$	0	0
$784$	5.75730	0.205618
$785$	−14.5373	−0.518860
$786$	0	0
$787$	−39.8704	−1.42123	−0.710614	−	0.703582i	$-0.751583\pi$
$787$	−39.8704	−1.42123	−0.710614	+	0.703582i	$0.751583\pi$
$788$	−2.39539	−0.0853324
$789$	0	0
$790$	15.4513	0.549734
$791$	36.3405	1.29212
$792$	0	0
$793$	15.5626	0.552646
$794$	−9.73362	−0.345433
$795$	0	0
$796$	−2.79391	−0.0990274
$797$	−22.9226	−0.811960	−0.405980	−	0.913882i	$-0.633070\pi$
$797$	−22.9226	−0.811960	−0.405980	+	0.913882i	$0.633070\pi$
$798$	0	0
$799$	3.98562	0.141001
$800$	10.0813	0.356429
$801$	0	0
$802$	47.8490	1.68961
$803$	8.01200	0.282737
$804$	0	0
$805$	2.79336	0.0984529
$806$	25.5865	0.901245
$807$	0	0
$808$	−30.7708	−1.08251
$809$	41.3740	1.45463	0.727317	−	0.686302i	$-0.240767\pi$
$809$	41.3740	1.45463	0.727317	+	0.686302i	$0.240767\pi$
$810$	0	0
$811$	50.5868	1.77634	0.888171	−	0.459513i	$-0.151976\pi$
$811$	50.5868	1.77634	0.888171	+	0.459513i	$0.151976\pi$
$812$	1.23611	0.0433788
$813$	0	0
$814$	8.00204	0.280471
$815$	15.7448	0.551515
$816$	0	0
$817$	−8.21650	−0.287459
$818$	23.8412	0.833589
$819$	0	0
$820$	−1.87264	−0.0653955
$821$	−8.64283	−0.301637	−0.150818	−	0.988561i	$-0.548191\pi$
$821$	−8.64283	−0.301637	−0.150818	+	0.988561i	$0.548191\pi$
$822$	0	0
$823$	−12.2404	−0.426673	−0.213336	−	0.976979i	$-0.568433\pi$
$823$	−12.2404	−0.426673	−0.213336	+	0.976979i	$0.568433\pi$
$824$	−32.8043	−1.14279
$825$	0	0
$826$	−25.4325	−0.884911
$827$	−5.98628	−0.208163	−0.104082	−	0.994569i	$-0.533190\pi$
$827$	−5.98628	−0.208163	−0.104082	+	0.994569i	$0.533190\pi$
$828$	0	0
$829$	11.8453	0.411405	0.205702	−	0.978615i	$-0.434052\pi$
$829$	11.8453	0.411405	0.205702	+	0.978615i	$0.434052\pi$
$830$	−13.9395	−0.483846
$831$	0	0
$832$	13.5603	0.470118
$833$	−4.94121	−0.171203
$834$	0	0
$835$	10.8976	0.377128
$836$	0.229420	0.00793467
$837$	0	0
$838$	−46.3985	−1.60281
$839$	−19.6373	−0.677954	−0.338977	−	0.940795i	$-0.610081\pi$
$839$	−19.6373	−0.677954	−0.338977	+	0.940795i	$0.610081\pi$
$840$	0	0
$841$	−27.3687	−0.943750
$842$	−46.1071	−1.58896
$843$	0	0
$844$	−0.874378	−0.0300973
$845$	−5.41641	−0.186330
$846$	0	0
$847$	25.6237	0.880441
$848$	24.9178	0.855680
$849$	0	0
$850$	−27.7032	−0.950211
$851$	−14.6956	−0.503758
$852$	0	0
$853$	24.0222	0.822503	0.411251	−	0.911522i	$-0.365092\pi$
$853$	24.0222	0.822503	0.411251	+	0.911522i	$0.365092\pi$
$854$	−24.7755	−0.847800
$855$	0	0
$856$	−44.2450	−1.51226
$857$	−37.6638	−1.28657	−0.643285	−	0.765627i	$-0.722429\pi$
$857$	−37.6638	−1.28657	−0.643285	+	0.765627i	$0.722429\pi$
$858$	0	0
$859$	−21.5386	−0.734887	−0.367443	−	0.930046i	$-0.619767\pi$
$859$	−21.5386	−0.734887	−0.367443	+	0.930046i	$0.619767\pi$
$860$	−2.38080	−0.0811847
$861$	0	0
$862$	35.0139	1.19258
$863$	37.2731	1.26879	0.634395	−	0.773009i	$-0.281249\pi$
$863$	37.2731	1.26879	0.634395	+	0.773009i	$0.281249\pi$
$864$	0	0
$865$	12.6528	0.430209
$866$	−27.5211	−0.935203
$867$	0	0
$868$	−6.83478	−0.231988
$869$	7.89155	0.267703
$870$	0	0
$871$	4.77533	0.161806
$872$	−33.1087	−1.12120
$873$	0	0
$874$	−2.51098	−0.0849351
$875$	16.3553	0.552911
$876$	0	0
$877$	35.0865	1.18479	0.592393	−	0.805649i	$-0.298183\pi$
$877$	35.0865	1.18479	0.592393	+	0.805649i	$0.298183\pi$
$878$	−4.50050	−0.151885
$879$	0	0
$880$	−1.89846	−0.0639972
$881$	−26.7469	−0.901126	−0.450563	−	0.892745i	$-0.648777\pi$
$881$	−26.7469	−0.901126	−0.450563	+	0.892745i	$0.648777\pi$
$882$	0	0
$883$	47.6608	1.60391	0.801957	−	0.597382i	$-0.203793\pi$
$883$	47.6608	1.60391	0.801957	+	0.597382i	$0.203793\pi$
$884$	3.75622	0.126335
$885$	0	0
$886$	−10.2770	−0.345262
$887$	−17.2199	−0.578189	−0.289095	−	0.957301i	$-0.593354\pi$
$887$	−17.2199	−0.578189	−0.289095	+	0.957301i	$0.593354\pi$
$888$	0	0
$889$	−51.7687	−1.73627
$890$	8.33208	0.279292
$891$	0	0
$892$	6.64749	0.222574
$893$	1.00000	0.0334637
$894$	0	0
$895$	5.80163	0.193927
$896$	−32.3806	−1.08176
$897$	0	0
$898$	−47.5259	−1.58596
$899$	−9.01969	−0.300824
$900$	0	0
$901$	−21.3857	−0.712461
$902$	−5.70001	−0.189790
$903$	0	0
$904$	37.4806	1.24659
$905$	−0.711687	−0.0236573
$906$	0	0
$907$	37.1260	1.23275	0.616374	−	0.787454i	$-0.288601\pi$
$907$	37.1260	1.23275	0.616374	+	0.787454i	$0.288601\pi$
$908$	−2.22796	−0.0739374
$909$	0	0
$910$	−6.24831	−0.207130
$911$	31.3258	1.03787	0.518936	−	0.854813i	$-0.326328\pi$
$911$	31.3258	1.03787	0.518936	+	0.854813i	$0.326328\pi$
$912$	0	0
$913$	−7.11938	−0.235617
$914$	17.6634	0.584254
$915$	0	0
$916$	−5.01939	−0.165845
$917$	24.6032	0.812469
$918$	0	0
$919$	28.1413	0.928295	0.464148	−	0.885758i	$-0.346361\pi$
$919$	28.1413	0.928295	0.464148	+	0.885758i	$0.346361\pi$
$920$	2.88099	0.0949836
$921$	0	0
$922$	−26.9123	−0.886309
$923$	−17.4519	−0.574438
$924$	0	0
$925$	−40.6796	−1.33754
$926$	−12.9999	−0.427203
$927$	0	0
$928$	2.87174	0.0942695
$929$	−14.5856	−0.478537	−0.239268	−	0.970953i	$-0.576908\pi$
$929$	−14.5856	−0.478537	−0.239268	+	0.970953i	$0.576908\pi$
$930$	0	0
$931$	−1.23976	−0.0406314
$932$	9.74585	0.319236
$933$	0	0
$934$	−55.0976	−1.80285
$935$	1.62936	0.0532857
$936$	0	0
$937$	−17.6195	−0.575605	−0.287802	−	0.957690i	$-0.592925\pi$
$937$	−17.6195	−0.575605	−0.287802	+	0.957690i	$0.592925\pi$
$938$	−7.60226	−0.248223
$939$	0	0
$940$	0.289759	0.00945089
$941$	14.0237	0.457161	0.228581	−	0.973525i	$-0.426592\pi$
$941$	14.0237	0.457161	0.228581	+	0.973525i	$0.426592\pi$
$942$	0	0
$943$	10.4680	0.340883
$944$	−31.7433	−1.03316
$945$	0	0
$946$	−7.24677	−0.235613
$947$	30.0087	0.975151	0.487576	−	0.873081i	$-0.337881\pi$
$947$	30.0087	0.975151	0.487576	+	0.873081i	$0.337881\pi$
$948$	0	0
$949$	32.9127	1.06839
$950$	−6.95078	−0.225513
$951$	0	0
$952$	23.6784	0.767423
$953$	−6.03281	−0.195422	−0.0977109	−	0.995215i	$-0.531152\pi$
$953$	−6.03281	−0.195422	−0.0977109	+	0.995215i	$0.531152\pi$
$954$	0	0
$955$	14.5971	0.472352
$956$	−0.892906	−0.0288787
$957$	0	0
$958$	−8.43667	−0.272576
$959$	22.4090	0.723626
$960$	0	0
$961$	18.8725	0.608789
$962$	32.8718	1.05983
$963$	0	0
$964$	−8.38376	−0.270023
$965$	12.6278	0.406502
$966$	0	0
$967$	−4.39824	−0.141438	−0.0707189	−	0.997496i	$-0.522529\pi$
$967$	−4.39824	−0.141438	−0.0707189	+	0.997496i	$0.522529\pi$
$968$	26.4276	0.849416
$969$	0	0
$970$	−9.73010	−0.312415
$971$	−18.6418	−0.598245	−0.299122	−	0.954215i	$-0.596694\pi$
$971$	−18.6418	−0.598245	−0.299122	+	0.954215i	$0.596694\pi$
$972$	0	0
$973$	−36.5031	−1.17024
$974$	34.1216	1.09333
$975$	0	0
$976$	−30.9232	−0.989828
$977$	−3.95857	−0.126646	−0.0633229	−	0.997993i	$-0.520170\pi$
$977$	−3.95857	−0.126646	−0.0633229	+	0.997993i	$0.520170\pi$
$978$	0	0
$979$	4.25549	0.136006
$980$	−0.359231	−0.0114752
$981$	0	0
$982$	−49.3572	−1.57505
$983$	−42.8044	−1.36525	−0.682625	−	0.730769i	$-0.739162\pi$
$983$	−42.8044	−1.36525	−0.682625	+	0.730769i	$0.739162\pi$
$984$	0	0
$985$	−4.26840	−0.136003
$986$	−7.89145	−0.251315
$987$	0	0
$988$	0.942442	0.0299831
$989$	13.3085	0.423187
$990$	0	0
$991$	−9.55590	−0.303553	−0.151777	−	0.988415i	$-0.548499\pi$
$991$	−9.55590	−0.303553	−0.151777	+	0.988415i	$0.548499\pi$
$992$	−15.8787	−0.504148
$993$	0	0
$994$	27.7833	0.881231
$995$	−4.97852	−0.157830
$996$	0	0
$997$	−35.6104	−1.12779	−0.563897	−	0.825845i	$-0.690698\pi$
$997$	−35.6104	−1.12779	−0.563897	+	0.825845i	$0.690698\pi$
$998$	−0.627443	−0.0198614
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.r.1.18		23
3.2	odd	2		893.2.a.d.1.6	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.d.1.6	✓	23	3.2	odd	2
8037.2.a.r.1.18		23	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)

\(f(q)\)	\(=\)	\(q+1.55024 q^{2} +0.403250 q^{4} +0.718559 q^{5} -2.40005 q^{7} -2.47535 q^{8} +O(q^{10})\)	\(q+1.55024 q^{2} +0.403250 q^{4} +0.718559 q^{5} -2.40005 q^{7} -2.47535 q^{8} +1.11394 q^{10} +0.568929 q^{11} +2.33712 q^{13} -3.72066 q^{14} -4.64389 q^{16} +3.98562 q^{17} +1.00000 q^{19} +0.289759 q^{20} +0.881978 q^{22} -1.61973 q^{23} -4.48367 q^{25} +3.62310 q^{26} -0.967819 q^{28} -1.27721 q^{29} +7.06204 q^{31} -2.24845 q^{32} +6.17868 q^{34} -1.72458 q^{35} +9.07284 q^{37} +1.55024 q^{38} -1.77868 q^{40} -6.46276 q^{41} -8.21650 q^{43} +0.229420 q^{44} -2.51098 q^{46} +1.00000 q^{47} -1.23976 q^{49} -6.95078 q^{50} +0.942442 q^{52} -5.36571 q^{53} +0.408809 q^{55} +5.94096 q^{56} -1.97998 q^{58} +6.83549 q^{59} +6.65890 q^{61} +10.9479 q^{62} +5.80213 q^{64} +1.67936 q^{65} +2.04326 q^{67} +1.60720 q^{68} -2.67351 q^{70} -7.46730 q^{71} +14.0826 q^{73} +14.0651 q^{74} +0.403250 q^{76} -1.36546 q^{77} +13.8709 q^{79} -3.33691 q^{80} -10.0188 q^{82} -12.5137 q^{83} +2.86391 q^{85} -12.7376 q^{86} -1.40830 q^{88} +7.47983 q^{89} -5.60920 q^{91} -0.653157 q^{92} +1.55024 q^{94} +0.718559 q^{95} -8.73484 q^{97} -1.92192 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) \) 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7	\( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) \) 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7 + 5 * q^10 - 2 * q^11 + 13 * q^13 - 6 * q^14 + 35 * q^16 - 12 * q^17 + 23 * q^19 - 3 * q^20 - 4 * q^22 - 13 * q^23 + 46 * q^25 + 7 * q^26 + 11 * q^28 + 18 * q^29 + 22 * q^31 + 4 * q^32 - 20 * q^34 - 25 * q^35 + 8 * q^37 - q^38 - 16 * q^40 + 16 * q^41 + 68 * q^43 + 18 * q^44 + 13 * q^46 + 23 * q^47 + 52 * q^49 + 21 * q^50 + 54 * q^52 + 7 * q^53 + 32 * q^55 - 33 * q^56 + 6 * q^58 - 10 * q^59 + 28 * q^61 + 24 * q^62 + 40 * q^64 + 18 * q^65 + 55 * q^67 - 41 * q^68 - 40 * q^70 + 3 * q^71 + 48 * q^73 + 55 * q^74 + 31 * q^76 + 14 * q^77 - 31 * q^79 + 3 * q^80 + 48 * q^82 - 47 * q^83 - 5 * q^85 + 71 * q^86 + 9 * q^88 + 6 * q^89 + 40 * q^91 - 35 * q^92 - q^94 - q^95 - 18 * q^97 + 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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