LMFDB - Embedded newform 8037.2.a.e.1.1

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
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			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87939 q^{2} +1.53209 q^{4} -0.532089 q^{5} -1.65270 q^{7} +0.879385 q^{8} +O(q^{10})$	\(q-1.87939 q^{2} +1.53209 q^{4} -0.532089 q^{5} -1.65270 q^{7} +0.879385 q^{8} +1.00000 q^{10} +5.94356 q^{11} -1.34730 q^{13} +3.10607 q^{14} -4.71688 q^{16} +5.63816 q^{17} -1.00000 q^{19} -0.815207 q^{20} -11.1702 q^{22} -4.41147 q^{23} -4.71688 q^{25} +2.53209 q^{26} -2.53209 q^{28} +3.94356 q^{29} -0.0641778 q^{31} +7.10607 q^{32} -10.5963 q^{34} +0.879385 q^{35} +4.36959 q^{37} +1.87939 q^{38} -0.467911 q^{40} -0.943563 q^{41} +5.57398 q^{43} +9.10607 q^{44} +8.29086 q^{46} -1.00000 q^{47} -4.26857 q^{49} +8.86484 q^{50} -2.06418 q^{52} -0.268571 q^{53} -3.16250 q^{55} -1.45336 q^{56} -7.41147 q^{58} -3.30541 q^{59} +5.81521 q^{61} +0.120615 q^{62} -3.92127 q^{64} +0.716881 q^{65} +6.27126 q^{67} +8.63816 q^{68} -1.65270 q^{70} +8.69459 q^{71} +7.24897 q^{73} -8.21213 q^{74} -1.53209 q^{76} -9.82295 q^{77} +2.36959 q^{79} +2.50980 q^{80} +1.77332 q^{82} +11.3054 q^{83} -3.00000 q^{85} -10.4757 q^{86} +5.22668 q^{88} -3.80066 q^{89} +2.22668 q^{91} -6.75877 q^{92} +1.87939 q^{94} +0.532089 q^{95} +6.18984 q^{97} +8.02229 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^5 - 6 * q^7 - 3 * q^8	$ 3 q + 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{10} + 3 q^{11} - 3 q^{13} - 3 q^{14} - 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} - 3 q^{23} - 6 q^{25} + 3 q^{26} - 3 q^{28} - 3 q^{29} + 9 q^{31} + 9 q^{32} - 18 q^{34} - 3 q^{35} + 6 q^{37} - 6 q^{40} + 12 q^{41} + 9 q^{43} + 15 q^{44} + 9 q^{46} - 3 q^{47} - 3 q^{49} + 3 q^{50} + 3 q^{52} + 9 q^{53} - 12 q^{55} + 9 q^{56} - 12 q^{58} - 12 q^{59} + 21 q^{61} + 6 q^{62} - 3 q^{64} - 6 q^{65} + 9 q^{68} - 6 q^{70} + 24 q^{71} + 9 q^{73} - 9 q^{77} + 9 q^{80} + 12 q^{82} + 36 q^{83} - 9 q^{85} - 12 q^{86} + 9 q^{88} + 3 q^{89} - 9 q^{92} - 3 q^{95} + 18 q^{98}+O(q^{100}) $ 3 * q + 3 * q^5 - 6 * q^7 - 3 * q^8 + 3 * q^10 + 3 * q^11 - 3 * q^13 - 3 * q^14 - 6 * q^16 - 3 * q^19 - 6 * q^20 - 12 * q^22 - 3 * q^23 - 6 * q^25 + 3 * q^26 - 3 * q^28 - 3 * q^29 + 9 * q^31 + 9 * q^32 - 18 * q^34 - 3 * q^35 + 6 * q^37 - 6 * q^40 + 12 * q^41 + 9 * q^43 + 15 * q^44 + 9 * q^46 - 3 * q^47 - 3 * q^49 + 3 * q^50 + 3 * q^52 + 9 * q^53 - 12 * q^55 + 9 * q^56 - 12 * q^58 - 12 * q^59 + 21 * q^61 + 6 * q^62 - 3 * q^64 - 6 * q^65 + 9 * q^68 - 6 * q^70 + 24 * q^71 + 9 * q^73 - 9 * q^77 + 9 * q^80 + 12 * q^82 + 36 * q^83 - 9 * q^85 - 12 * q^86 + 9 * q^88 + 3 * q^89 - 9 * q^92 - 3 * q^95 + 18 * 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 $

$2$	−1.87939	−1.32893	−0.664463	−	0.747321i	$-0.731340\pi$
$2$	−1.87939	−1.32893	−0.664463	+	0.747321i	$0.731340\pi$
$3$	0	0
$3$	0	0
$4$	1.53209	0.766044
$5$	−0.532089	−0.237957	−0.118979	−	0.992897i	$-0.537962\pi$
$5$	−0.532089	−0.237957	−0.118979	+	0.992897i	$0.537962\pi$
$6$	0	0
$7$	−1.65270	−0.624663	−0.312332	−	0.949973i	$-0.601110\pi$
$7$	−1.65270	−0.624663	−0.312332	+	0.949973i	$0.601110\pi$
$8$	0.879385	0.310910
$9$	0	0
$10$	1.00000	0.316228
$11$	5.94356	1.79205	0.896026	−	0.444002i	$-0.146442\pi$
$11$	5.94356	1.79205	0.896026	+	0.444002i	$0.146442\pi$
$12$	0	0
$13$	−1.34730	−0.373673	−0.186836	−	0.982391i	$-0.559823\pi$
$13$	−1.34730	−0.373673	−0.186836	+	0.982391i	$0.559823\pi$
$14$	3.10607	0.830131
$15$	0	0
$16$	−4.71688	−1.17922
$17$	5.63816	1.36745	0.683727	−	0.729738i	$-0.260358\pi$
$17$	5.63816	1.36745	0.683727	+	0.729738i	$0.260358\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.815207	−0.182286
$21$	0	0
$22$	−11.1702	−2.38150
$23$	−4.41147	−0.919856	−0.459928	−	0.887956i	$-0.652125\pi$
$23$	−4.41147	−0.919856	−0.459928	+	0.887956i	$0.652125\pi$
$24$	0	0
$25$	−4.71688	−0.943376
$26$	2.53209	0.496583
$27$	0	0
$28$	−2.53209	−0.478520
$29$	3.94356	0.732301	0.366151	−	0.930556i	$-0.380675\pi$
$29$	3.94356	0.732301	0.366151	+	0.930556i	$0.380675\pi$
$30$	0	0
$31$	−0.0641778	−0.0115267	−0.00576333	−	0.999983i	$-0.501835\pi$
$31$	−0.0641778	−0.0115267	−0.00576333	+	0.999983i	$0.501835\pi$
$32$	7.10607	1.25619
$33$	0	0
$34$	−10.5963	−1.81724
$35$	0.879385	0.148643
$36$	0	0
$37$	4.36959	0.718355	0.359178	−	0.933269i	$-0.383057\pi$
$37$	4.36959	0.718355	0.359178	+	0.933269i	$0.383057\pi$
$38$	1.87939	0.304877
$39$	0	0
$40$	−0.467911	−0.0739832
$41$	−0.943563	−0.147360	−0.0736799	−	0.997282i	$-0.523474\pi$
$41$	−0.943563	−0.147360	−0.0736799	+	0.997282i	$0.523474\pi$
$42$	0	0
$43$	5.57398	0.850024	0.425012	−	0.905188i	$-0.360270\pi$
$43$	5.57398	0.850024	0.425012	+	0.905188i	$0.360270\pi$
$44$	9.10607	1.37279
$45$	0	0
$46$	8.29086	1.22242
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−4.26857	−0.609796
$50$	8.86484	1.25368
$51$	0	0
$52$	−2.06418	−0.286250
$53$	−0.268571	−0.0368910	−0.0184455	−	0.999830i	$-0.505872\pi$
$53$	−0.268571	−0.0368910	−0.0184455	+	0.999830i	$0.505872\pi$
$54$	0	0
$55$	−3.16250	−0.426432
$56$	−1.45336	−0.194214
$57$	0	0
$58$	−7.41147	−0.973174
$59$	−3.30541	−0.430327	−0.215164	−	0.976578i	$-0.569029\pi$
$59$	−3.30541	−0.430327	−0.215164	+	0.976578i	$0.569029\pi$
$60$	0	0
$61$	5.81521	0.744561	0.372281	−	0.928120i	$-0.378576\pi$
$61$	5.81521	0.744561	0.372281	+	0.928120i	$0.378576\pi$
$62$	0.120615	0.0153181
$63$	0	0
$64$	−3.92127	−0.490159
$65$	0.716881	0.0889182
$66$	0	0
$67$	6.27126	0.766156	0.383078	−	0.923716i	$-0.374864\pi$
$67$	6.27126	0.766156	0.383078	+	0.923716i	$0.374864\pi$
$68$	8.63816	1.04753
$69$	0	0
$70$	−1.65270	−0.197536
$71$	8.69459	1.03186	0.515929	−	0.856631i	$-0.327447\pi$
$71$	8.69459	1.03186	0.515929	+	0.856631i	$0.327447\pi$
$72$	0	0
$73$	7.24897	0.848428	0.424214	−	0.905562i	$-0.360550\pi$
$73$	7.24897	0.848428	0.424214	+	0.905562i	$0.360550\pi$
$74$	−8.21213	−0.954641
$75$	0	0
$76$	−1.53209	−0.175743
$77$	−9.82295	−1.11943
$78$	0	0
$79$	2.36959	0.266599	0.133300	−	0.991076i	$-0.457443\pi$
$79$	2.36959	0.266599	0.133300	+	0.991076i	$0.457443\pi$
$80$	2.50980	0.280604
$81$	0	0
$82$	1.77332	0.195830
$83$	11.3054	1.24093	0.620465	−	0.784234i	$-0.286944\pi$
$83$	11.3054	1.24093	0.620465	+	0.784234i	$0.286944\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−10.4757	−1.12962
$87$	0	0
$88$	5.22668	0.557166
$89$	−3.80066	−0.402869	−0.201435	−	0.979502i	$-0.564560\pi$
$89$	−3.80066	−0.402869	−0.201435	+	0.979502i	$0.564560\pi$
$90$	0	0
$91$	2.22668	0.233420
$92$	−6.75877	−0.704651
$93$	0	0
$94$	1.87939	0.193844
$95$	0.532089	0.0545912
$96$	0	0
$97$	6.18984	0.628484	0.314242	−	0.949343i	$-0.398250\pi$
$97$	6.18984	0.628484	0.314242	+	0.949343i	$0.398250\pi$
$98$	8.02229	0.810374
$99$	0	0
$100$	−7.22668	−0.722668
$101$	9.28312	0.923705	0.461852	−	0.886957i	$-0.347185\pi$
$101$	9.28312	0.923705	0.461852	+	0.886957i	$0.347185\pi$
$102$	0	0
$103$	−6.31315	−0.622053	−0.311026	−	0.950401i	$-0.600673\pi$
$103$	−6.31315	−0.622053	−0.311026	+	0.950401i	$0.600673\pi$
$104$	−1.18479	−0.116178
$105$	0	0
$106$	0.504748	0.0490254
$107$	12.2909	1.18820	0.594101	−	0.804390i	$-0.297508\pi$
$107$	12.2909	1.18820	0.594101	+	0.804390i	$0.297508\pi$
$108$	0	0
$109$	−2.93851	−0.281458	−0.140729	−	0.990048i	$-0.544945\pi$
$109$	−2.93851	−0.281458	−0.140729	+	0.990048i	$0.544945\pi$
$110$	5.94356	0.566696
$111$	0	0
$112$	7.79561	0.736616
$113$	−13.4834	−1.26841	−0.634205	−	0.773165i	$-0.718673\pi$
$113$	−13.4834	−1.26841	−0.634205	+	0.773165i	$0.718673\pi$
$114$	0	0
$115$	2.34730	0.218887
$116$	6.04189	0.560975
$117$	0	0
$118$	6.21213	0.571873
$119$	−9.31820	−0.854198
$120$	0	0
$121$	24.3259	2.21145
$122$	−10.9290	−0.989467
$123$	0	0
$124$	−0.0983261	−0.00882994
$125$	5.17024	0.462441
$126$	0	0
$127$	−17.7665	−1.57652	−0.788261	−	0.615340i	$-0.789019\pi$
$127$	−17.7665	−1.57652	−0.788261	+	0.615340i	$0.789019\pi$
$128$	−6.84255	−0.604802
$129$	0	0
$130$	−1.34730	−0.118166
$131$	−17.8530	−1.55982	−0.779911	−	0.625890i	$-0.784736\pi$
$131$	−17.8530	−1.55982	−0.779911	+	0.625890i	$0.784736\pi$
$132$	0	0
$133$	1.65270	0.143308
$134$	−11.7861	−1.01816
$135$	0	0
$136$	4.95811	0.425155
$137$	9.32501	0.796689	0.398345	−	0.917236i	$-0.369585\pi$
$137$	9.32501	0.796689	0.398345	+	0.917236i	$0.369585\pi$
$138$	0	0
$139$	−5.21894	−0.442665	−0.221332	−	0.975198i	$-0.571041\pi$
$139$	−5.21894	−0.442665	−0.221332	+	0.975198i	$0.571041\pi$
$140$	1.34730	0.113867
$141$	0	0
$142$	−16.3405	−1.37126
$143$	−8.00774	−0.669641
$144$	0	0
$145$	−2.09833	−0.174256
$146$	−13.6236	−1.12750
$147$	0	0
$148$	6.69459	0.550292
$149$	−3.69728	−0.302893	−0.151447	−	0.988465i	$-0.548393\pi$
$149$	−3.69728	−0.302893	−0.151447	+	0.988465i	$0.548393\pi$
$150$	0	0
$151$	6.10101	0.496494	0.248247	−	0.968697i	$-0.420146\pi$
$151$	6.10101	0.496494	0.248247	+	0.968697i	$0.420146\pi$
$152$	−0.879385	−0.0713276
$153$	0	0
$154$	18.4611	1.48764
$155$	0.0341483	0.00274286
$156$	0	0
$157$	−17.2686	−1.37818	−0.689091	−	0.724675i	$-0.741990\pi$
$157$	−17.2686	−1.37818	−0.689091	+	0.724675i	$0.741990\pi$
$158$	−4.45336	−0.354291
$159$	0	0
$160$	−3.78106	−0.298919
$161$	7.29086	0.574600
$162$	0	0
$163$	−13.5476	−1.06113	−0.530564	−	0.847645i	$-0.678020\pi$
$163$	−13.5476	−1.06113	−0.530564	+	0.847645i	$0.678020\pi$
$164$	−1.44562	−0.112884
$165$	0	0
$166$	−21.2472	−1.64910
$167$	8.96316	0.693590	0.346795	−	0.937941i	$-0.387270\pi$
$167$	8.96316	0.693590	0.346795	+	0.937941i	$0.387270\pi$
$168$	0	0
$169$	−11.1848	−0.860369
$170$	5.63816	0.432427
$171$	0	0
$172$	8.53983	0.651156
$173$	−6.31046	−0.479775	−0.239888	−	0.970801i	$-0.577111\pi$
$173$	−6.31046	−0.479775	−0.239888	+	0.970801i	$0.577111\pi$
$174$	0	0
$175$	7.79561	0.589293
$176$	−28.0351	−2.11322
$177$	0	0
$178$	7.14290	0.535383
$179$	−6.44656	−0.481838	−0.240919	−	0.970545i	$-0.577449\pi$
$179$	−6.44656	−0.481838	−0.240919	+	0.970545i	$0.577449\pi$
$180$	0	0
$181$	−24.9932	−1.85773	−0.928865	−	0.370419i	$-0.879214\pi$
$181$	−24.9932	−1.85773	−0.928865	+	0.370419i	$0.879214\pi$
$182$	−4.18479	−0.310197
$183$	0	0
$184$	−3.87939	−0.285992
$185$	−2.32501	−0.170938
$186$	0	0
$187$	33.5107	2.45055
$188$	−1.53209	−0.111739
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	5.92396	0.428643	0.214321	−	0.976763i	$-0.431246\pi$
$191$	5.92396	0.428643	0.214321	+	0.976763i	$0.431246\pi$
$192$	0	0
$193$	3.76382	0.270926	0.135463	−	0.990782i	$-0.456748\pi$
$193$	3.76382	0.270926	0.135463	+	0.990782i	$0.456748\pi$
$194$	−11.6331	−0.835208
$195$	0	0
$196$	−6.53983	−0.467131
$197$	−2.26083	−0.161077	−0.0805387	−	0.996751i	$-0.525664\pi$
$197$	−2.26083	−0.161077	−0.0805387	+	0.996751i	$0.525664\pi$
$198$	0	0
$199$	−13.8699	−0.983210	−0.491605	−	0.870818i	$-0.663590\pi$
$199$	−13.8699	−0.983210	−0.491605	+	0.870818i	$0.663590\pi$
$200$	−4.14796	−0.293305
$201$	0	0
$202$	−17.4466	−1.22754
$203$	−6.51754	−0.457442
$204$	0	0
$205$	0.502059	0.0350653
$206$	11.8648	0.826662
$207$	0	0
$208$	6.35504	0.440643
$209$	−5.94356	−0.411125
$210$	0	0
$211$	23.9590	1.64941	0.824704	−	0.565564i	$-0.191342\pi$
$211$	23.9590	1.64941	0.824704	+	0.565564i	$0.191342\pi$
$212$	−0.411474	−0.0282602
$213$	0	0
$214$	−23.0993	−1.57903
$215$	−2.96585	−0.202269
$216$	0	0
$217$	0.106067	0.00720029
$218$	5.52259	0.374037
$219$	0	0
$220$	−4.84524	−0.326666
$221$	−7.59627	−0.510980
$222$	0	0
$223$	−9.58946	−0.642158	−0.321079	−	0.947052i	$-0.604045\pi$
$223$	−9.58946	−0.642158	−0.321079	+	0.947052i	$0.604045\pi$
$224$	−11.7442	−0.784694
$225$	0	0
$226$	25.3405	1.68562
$227$	20.6186	1.36850	0.684251	−	0.729247i	$-0.260129\pi$
$227$	20.6186	1.36850	0.684251	+	0.729247i	$0.260129\pi$
$228$	0	0
$229$	−2.94356	−0.194516	−0.0972581	−	0.995259i	$-0.531007\pi$
$229$	−2.94356	−0.194516	−0.0972581	+	0.995259i	$0.531007\pi$
$230$	−4.41147	−0.290884
$231$	0	0
$232$	3.46791	0.227680
$233$	12.9017	0.845217	0.422608	−	0.906312i	$-0.361115\pi$
$233$	12.9017	0.845217	0.422608	+	0.906312i	$0.361115\pi$
$234$	0	0
$235$	0.532089	0.0347097
$236$	−5.06418	−0.329650
$237$	0	0
$238$	17.5125	1.13517
$239$	26.9668	1.74434	0.872168	−	0.489206i	$-0.162713\pi$
$239$	26.9668	1.74434	0.872168	+	0.489206i	$0.162713\pi$
$240$	0	0
$241$	9.31820	0.600238	0.300119	−	0.953902i	$-0.402974\pi$
$241$	9.31820	0.600238	0.300119	+	0.953902i	$0.402974\pi$
$242$	−45.7178	−2.93885
$243$	0	0
$244$	8.90941	0.570367
$245$	2.27126	0.145105
$246$	0	0
$247$	1.34730	0.0857264
$248$	−0.0564370	−0.00358375
$249$	0	0
$250$	−9.71688	−0.614550
$251$	−13.3405	−0.842044	−0.421022	−	0.907050i	$-0.638329\pi$
$251$	−13.3405	−0.842044	−0.421022	+	0.907050i	$0.638329\pi$
$252$	0	0
$253$	−26.2199	−1.64843
$254$	33.3901	2.09508
$255$	0	0
$256$	20.7023	1.29390
$257$	22.5844	1.40878	0.704388	−	0.709815i	$-0.251221\pi$
$257$	22.5844	1.40878	0.704388	+	0.709815i	$0.251221\pi$
$258$	0	0
$259$	−7.22163	−0.448730
$260$	1.09833	0.0681153
$261$	0	0
$262$	33.5526	2.07289
$263$	−21.6905	−1.33749	−0.668746	−	0.743491i	$-0.733169\pi$
$263$	−21.6905	−1.33749	−0.668746	+	0.743491i	$0.733169\pi$
$264$	0	0
$265$	0.142903	0.00877849
$266$	−3.10607	−0.190445
$267$	0	0
$268$	9.60813	0.586910
$269$	17.2831	1.05377	0.526885	−	0.849936i	$-0.323360\pi$
$269$	17.2831	1.05377	0.526885	+	0.849936i	$0.323360\pi$
$270$	0	0
$271$	17.8007	1.08131	0.540657	−	0.841243i	$-0.318176\pi$
$271$	17.8007	1.08131	0.540657	+	0.841243i	$0.318176\pi$
$272$	−26.5945	−1.61253
$273$	0	0
$274$	−17.5253	−1.05874
$275$	−28.0351	−1.69058
$276$	0	0
$277$	−1.78106	−0.107013	−0.0535067	−	0.998567i	$-0.517040\pi$
$277$	−1.78106	−0.107013	−0.0535067	+	0.998567i	$0.517040\pi$
$278$	9.80840	0.588269
$279$	0	0
$280$	0.773318	0.0462146
$281$	−3.27126	−0.195147	−0.0975735	−	0.995228i	$-0.531108\pi$
$281$	−3.27126	−0.195147	−0.0975735	+	0.995228i	$0.531108\pi$
$282$	0	0
$283$	8.07873	0.480230	0.240115	−	0.970744i	$-0.422815\pi$
$283$	8.07873	0.480230	0.240115	+	0.970744i	$0.422815\pi$
$284$	13.3209	0.790449
$285$	0	0
$286$	15.0496	0.889903
$287$	1.55943	0.0920502
$288$	0	0
$289$	14.7888	0.869929
$290$	3.94356	0.231574
$291$	0	0
$292$	11.1061	0.649933
$293$	11.4611	0.669565	0.334782	−	0.942295i	$-0.391337\pi$
$293$	11.4611	0.669565	0.334782	+	0.942295i	$0.391337\pi$
$294$	0	0
$295$	1.75877	0.102400
$296$	3.84255	0.223344
$297$	0	0
$298$	6.94862	0.402522
$299$	5.94356	0.343725
$300$	0	0
$301$	−9.21213	−0.530979
$302$	−11.4662	−0.659803
$303$	0	0
$304$	4.71688	0.270532
$305$	−3.09421	−0.177174
$306$	0	0
$307$	2.41921	0.138072	0.0690359	−	0.997614i	$-0.478008\pi$
$307$	2.41921	0.138072	0.0690359	+	0.997614i	$0.478008\pi$
$308$	−15.0496	−0.857532
$309$	0	0
$310$	−0.0641778	−0.00364505
$311$	−8.74153	−0.495687	−0.247843	−	0.968800i	$-0.579722\pi$
$311$	−8.74153	−0.495687	−0.247843	+	0.968800i	$0.579722\pi$
$312$	0	0
$313$	−11.0155	−0.622632	−0.311316	−	0.950306i	$-0.600770\pi$
$313$	−11.0155	−0.622632	−0.311316	+	0.950306i	$0.600770\pi$
$314$	32.4543	1.83150
$315$	0	0
$316$	3.63041	0.204227
$317$	22.9709	1.29017	0.645087	−	0.764109i	$-0.276821\pi$
$317$	22.9709	1.29017	0.645087	+	0.764109i	$0.276821\pi$
$318$	0	0
$319$	23.4388	1.31232
$320$	2.08647	0.116637
$321$	0	0
$322$	−13.7023	−0.763601
$323$	−5.63816	−0.313715
$324$	0	0
$325$	6.35504	0.352514
$326$	25.4611	1.41016
$327$	0	0
$328$	−0.829755	−0.0458156
$329$	1.65270	0.0911165
$330$	0	0
$331$	−28.9905	−1.59346	−0.796731	−	0.604334i	$-0.793439\pi$
$331$	−28.9905	−1.59346	−0.796731	+	0.604334i	$0.793439\pi$
$332$	17.3209	0.950607
$333$	0	0
$334$	−16.8452	−0.921730
$335$	−3.33687	−0.182313
$336$	0	0
$337$	16.0341	0.873436	0.436718	−	0.899599i	$-0.356141\pi$
$337$	16.0341	0.873436	0.436718	+	0.899599i	$0.356141\pi$
$338$	21.0205	1.14337
$339$	0	0
$340$	−4.59627	−0.249268
$341$	−0.381445	−0.0206564
$342$	0	0
$343$	18.6236	1.00558
$344$	4.90167	0.264281
$345$	0	0
$346$	11.8598	0.637586
$347$	−4.99495	−0.268143	−0.134071	−	0.990972i	$-0.542805\pi$
$347$	−4.99495	−0.268143	−0.134071	+	0.990972i	$0.542805\pi$
$348$	0	0
$349$	19.1753	1.02643	0.513215	−	0.858260i	$-0.328454\pi$
$349$	19.1753	1.02643	0.513215	+	0.858260i	$0.328454\pi$
$350$	−14.6509	−0.783126
$351$	0	0
$352$	42.2354	2.25115
$353$	14.1088	0.750933	0.375467	−	0.926836i	$-0.377482\pi$
$353$	14.1088	0.750933	0.375467	+	0.926836i	$0.377482\pi$
$354$	0	0
$355$	−4.62630	−0.245538
$356$	−5.82295	−0.308616
$357$	0	0
$358$	12.1156	0.640327
$359$	−18.9299	−0.999084	−0.499542	−	0.866290i	$-0.666498\pi$
$359$	−18.9299	−0.999084	−0.499542	+	0.866290i	$0.666498\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	46.9718	2.46878
$363$	0	0
$364$	3.41147	0.178810
$365$	−3.85710	−0.201890
$366$	0	0
$367$	22.1266	1.15500	0.577500	−	0.816391i	$-0.304029\pi$
$367$	22.1266	1.15500	0.577500	+	0.816391i	$0.304029\pi$
$368$	20.8084	1.08471
$369$	0	0
$370$	4.36959	0.227164
$371$	0.443868	0.0230445
$372$	0	0
$373$	−20.2909	−1.05062	−0.525310	−	0.850911i	$-0.676051\pi$
$373$	−20.2909	−1.05062	−0.525310	+	0.850911i	$0.676051\pi$
$374$	−62.9796	−3.25660
$375$	0	0
$376$	−0.879385	−0.0453508
$377$	−5.31315	−0.273641
$378$	0	0
$379$	−5.39961	−0.277360	−0.138680	−	0.990337i	$-0.544286\pi$
$379$	−5.39961	−0.277360	−0.138680	+	0.990337i	$0.544286\pi$
$380$	0.815207	0.0418193
$381$	0	0
$382$	−11.1334	−0.569635
$383$	23.1607	1.18346	0.591729	−	0.806137i	$-0.298445\pi$
$383$	23.1607	1.18346	0.591729	+	0.806137i	$0.298445\pi$
$384$	0	0
$385$	5.22668	0.266376
$386$	−7.07367	−0.360040
$387$	0	0
$388$	9.48339	0.481446
$389$	−16.7452	−0.849013	−0.424507	−	0.905425i	$-0.639552\pi$
$389$	−16.7452	−0.849013	−0.424507	+	0.905425i	$0.639552\pi$
$390$	0	0
$391$	−24.8726	−1.25786
$392$	−3.75372	−0.189591
$393$	0	0
$394$	4.24897	0.214060
$395$	−1.26083	−0.0634392
$396$	0	0
$397$	−15.5553	−0.780699	−0.390349	−	0.920667i	$-0.627646\pi$
$397$	−15.5553	−0.780699	−0.390349	+	0.920667i	$0.627646\pi$
$398$	26.0669	1.30661
$399$	0	0
$400$	22.2490	1.11245
$401$	18.8503	0.941339	0.470669	−	0.882310i	$-0.344013\pi$
$401$	18.8503	0.941339	0.470669	+	0.882310i	$0.344013\pi$
$402$	0	0
$403$	0.0864665	0.00430720
$404$	14.2226	0.707599
$405$	0	0
$406$	12.2490	0.607906
$407$	25.9709	1.28733
$408$	0	0
$409$	27.5577	1.36264	0.681320	−	0.731986i	$-0.261406\pi$
$409$	27.5577	1.36264	0.681320	+	0.731986i	$0.261406\pi$
$410$	−0.943563	−0.0465993
$411$	0	0
$412$	−9.67230	−0.476520
$413$	5.46286	0.268810
$414$	0	0
$415$	−6.01548	−0.295288
$416$	−9.57398	−0.469403
$417$	0	0
$418$	11.1702	0.546355
$419$	−6.39693	−0.312510	−0.156255	−	0.987717i	$-0.549942\pi$
$419$	−6.39693	−0.312510	−0.156255	+	0.987717i	$0.549942\pi$
$420$	0	0
$421$	−35.0087	−1.70622	−0.853109	−	0.521732i	$-0.825286\pi$
$421$	−35.0087	−1.70622	−0.853109	+	0.521732i	$0.825286\pi$
$422$	−45.0283	−2.19194
$423$	0	0
$424$	−0.236177	−0.0114698
$425$	−26.5945	−1.29002
$426$	0	0
$427$	−9.61081	−0.465100
$428$	18.8307	0.910216
$429$	0	0
$430$	5.57398	0.268801
$431$	−34.3141	−1.65285	−0.826426	−	0.563046i	$-0.809629\pi$
$431$	−34.3141	−1.65285	−0.826426	+	0.563046i	$0.809629\pi$
$432$	0	0
$433$	−9.11112	−0.437852	−0.218926	−	0.975741i	$-0.570255\pi$
$433$	−9.11112	−0.437852	−0.218926	+	0.975741i	$0.570255\pi$
$434$	−0.199340	−0.00956865
$435$	0	0
$436$	−4.50206	−0.215610
$437$	4.41147	0.211029
$438$	0	0
$439$	22.9718	1.09639	0.548193	−	0.836352i	$-0.315316\pi$
$439$	22.9718	1.09639	0.548193	+	0.836352i	$0.315316\pi$
$440$	−2.78106	−0.132582
$441$	0	0
$442$	14.2763	0.679055
$443$	−0.212134	−0.0100788	−0.00503939	−	0.999987i	$-0.501604\pi$
$443$	−0.212134	−0.0100788	−0.00503939	+	0.999987i	$0.501604\pi$
$444$	0	0
$445$	2.02229	0.0958657
$446$	18.0223	0.853380
$447$	0	0
$448$	6.48070	0.306185
$449$	4.63816	0.218888	0.109444	−	0.993993i	$-0.465093\pi$
$449$	4.63816	0.218888	0.109444	+	0.993993i	$0.465093\pi$
$450$	0	0
$451$	−5.60813	−0.264076
$452$	−20.6578	−0.971659
$453$	0	0
$454$	−38.7502	−1.81864
$455$	−1.18479	−0.0555439
$456$	0	0
$457$	10.5202	0.492116	0.246058	−	0.969255i	$-0.420865\pi$
$457$	10.5202	0.492116	0.246058	+	0.969255i	$0.420865\pi$
$458$	5.53209	0.258498
$459$	0	0
$460$	3.59627	0.167677
$461$	−35.9932	−1.67637	−0.838185	−	0.545386i	$-0.816383\pi$
$461$	−35.9932	−1.67637	−0.838185	+	0.545386i	$0.816383\pi$
$462$	0	0
$463$	25.3696	1.17902	0.589512	−	0.807759i	$-0.299320\pi$
$463$	25.3696	1.17902	0.589512	+	0.807759i	$0.299320\pi$
$464$	−18.6013	−0.863545
$465$	0	0
$466$	−24.2472	−1.12323
$467$	29.6064	1.37002	0.685010	−	0.728534i	$-0.259798\pi$
$467$	29.6064	1.37002	0.685010	+	0.728534i	$0.259798\pi$
$468$	0	0
$469$	−10.3645	−0.478590
$470$	−1.00000	−0.0461266
$471$	0	0
$472$	−2.90673	−0.133793
$473$	33.1293	1.52329
$474$	0	0
$475$	4.71688	0.216425
$476$	−14.2763	−0.654354
$477$	0	0
$478$	−50.6810	−2.31809
$479$	7.73648	0.353489	0.176744	−	0.984257i	$-0.443443\pi$
$479$	7.73648	0.353489	0.176744	+	0.984257i	$0.443443\pi$
$480$	0	0
$481$	−5.88713	−0.268430
$482$	−17.5125	−0.797672
$483$	0	0
$484$	37.2695	1.69407
$485$	−3.29355	−0.149552
$486$	0	0
$487$	28.2199	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$487$	28.2199	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$488$	5.11381	0.231491
$489$	0	0
$490$	−4.26857	−0.192834
$491$	−20.1803	−0.910726	−0.455363	−	0.890306i	$-0.650491\pi$
$491$	−20.1803	−0.910726	−0.455363	+	0.890306i	$0.650491\pi$
$492$	0	0
$493$	22.2344	1.00139
$494$	−2.53209	−0.113924
$495$	0	0
$496$	0.302719	0.0135925
$497$	−14.3696	−0.644564
$498$	0	0
$499$	−12.2172	−0.546916	−0.273458	−	0.961884i	$-0.588168\pi$
$499$	−12.2172	−0.546916	−0.273458	+	0.961884i	$0.588168\pi$
$500$	7.92127	0.354250
$501$	0	0
$502$	25.0719	1.11901
$503$	23.7442	1.05870	0.529351	−	0.848403i	$-0.322435\pi$
$503$	23.7442	1.05870	0.529351	+	0.848403i	$0.322435\pi$
$504$	0	0
$505$	−4.93944	−0.219802
$506$	49.2772	2.19064
$507$	0	0
$508$	−27.2199	−1.20769
$509$	0.568926	0.0252172	0.0126086	−	0.999921i	$-0.495986\pi$
$509$	0.568926	0.0252172	0.0126086	+	0.999921i	$0.495986\pi$
$510$	0	0
$511$	−11.9804	−0.529982
$512$	−25.2226	−1.11469
$513$	0	0
$514$	−42.4448	−1.87216
$515$	3.35916	0.148022
$516$	0	0
$517$	−5.94356	−0.261398
$518$	13.5722	0.596329
$519$	0	0
$520$	0.630415	0.0276455
$521$	20.7956	0.911072	0.455536	−	0.890217i	$-0.349448\pi$
$521$	20.7956	0.911072	0.455536	+	0.890217i	$0.349448\pi$
$522$	0	0
$523$	28.8203	1.26022	0.630111	−	0.776505i	$-0.283009\pi$
$523$	28.8203	1.26022	0.630111	+	0.776505i	$0.283009\pi$
$524$	−27.3523	−1.19489
$525$	0	0
$526$	40.7648	1.77743
$527$	−0.361844	−0.0157622
$528$	0	0
$529$	−3.53890	−0.153865
$530$	−0.268571	−0.0116660
$531$	0	0
$532$	2.53209	0.109780
$533$	1.27126	0.0550643
$534$	0	0
$535$	−6.53983	−0.282741
$536$	5.51485	0.238205
$537$	0	0
$538$	−32.4816	−1.40038
$539$	−25.3705	−1.09279
$540$	0	0
$541$	−20.0479	−0.861925	−0.430963	−	0.902370i	$-0.641826\pi$
$541$	−20.0479	−0.861925	−0.430963	+	0.902370i	$0.641826\pi$
$542$	−33.4543	−1.43699
$543$	0	0
$544$	40.0651	1.71778
$545$	1.56355	0.0669751
$546$	0	0
$547$	41.8316	1.78859	0.894296	−	0.447477i	$-0.147677\pi$
$547$	41.8316	1.78859	0.894296	+	0.447477i	$0.147677\pi$
$548$	14.2867	0.610299
$549$	0	0
$550$	52.6887	2.24665
$551$	−3.94356	−0.168001
$552$	0	0
$553$	−3.91622	−0.166535
$554$	3.34730	0.142213
$555$	0	0
$556$	−7.99588	−0.339101
$557$	−2.20945	−0.0936172	−0.0468086	−	0.998904i	$-0.514905\pi$
$557$	−2.20945	−0.0936172	−0.0468086	+	0.998904i	$0.514905\pi$
$558$	0	0
$559$	−7.50980	−0.317631
$560$	−4.14796	−0.175283
$561$	0	0
$562$	6.14796	0.259336
$563$	8.86720	0.373708	0.186854	−	0.982388i	$-0.440171\pi$
$563$	8.86720	0.373708	0.186854	+	0.982388i	$0.440171\pi$
$564$	0	0
$565$	7.17436	0.301828
$566$	−15.1830	−0.638191
$567$	0	0
$568$	7.64590	0.320815
$569$	39.5536	1.65817	0.829086	−	0.559122i	$-0.188862\pi$
$569$	39.5536	1.65817	0.829086	+	0.559122i	$0.188862\pi$
$570$	0	0
$571$	27.8280	1.16457	0.582283	−	0.812986i	$-0.302160\pi$
$571$	27.8280	1.16457	0.582283	+	0.812986i	$0.302160\pi$
$572$	−12.2686	−0.512975
$573$	0	0
$574$	−2.93077	−0.122328
$575$	20.8084	0.867770
$576$	0	0
$577$	−2.48482	−0.103445	−0.0517223	−	0.998662i	$-0.516471\pi$
$577$	−2.48482	−0.103445	−0.0517223	+	0.998662i	$0.516471\pi$
$578$	−27.7939	−1.15607
$579$	0	0
$580$	−3.21482	−0.133488
$581$	−18.6845	−0.775163
$582$	0	0
$583$	−1.59627	−0.0661106
$584$	6.37464	0.263784
$585$	0	0
$586$	−21.5398	−0.889802
$587$	13.8990	0.573673	0.286836	−	0.957980i	$-0.407396\pi$
$587$	13.8990	0.573673	0.286836	+	0.957980i	$0.407396\pi$
$588$	0	0
$589$	0.0641778	0.00264440
$590$	−3.30541	−0.136081
$591$	0	0
$592$	−20.6108	−0.847099
$593$	−14.7706	−0.606557	−0.303279	−	0.952902i	$-0.598081\pi$
$593$	−14.7706	−0.606557	−0.303279	+	0.952902i	$0.598081\pi$
$594$	0	0
$595$	4.95811	0.203263
$596$	−5.66456	−0.232030
$597$	0	0
$598$	−11.1702	−0.456785
$599$	−45.0856	−1.84215	−0.921075	−	0.389386i	$-0.872687\pi$
$599$	−45.0856	−1.84215	−0.921075	+	0.389386i	$0.872687\pi$
$600$	0	0
$601$	−21.9905	−0.897011	−0.448506	−	0.893780i	$-0.648044\pi$
$601$	−21.9905	−0.897011	−0.448506	+	0.893780i	$0.648044\pi$
$602$	17.3131	0.705631
$603$	0	0
$604$	9.34730	0.380336
$605$	−12.9436	−0.526231
$606$	0	0
$607$	14.5990	0.592553	0.296277	−	0.955102i	$-0.404255\pi$
$607$	14.5990	0.592553	0.296277	+	0.955102i	$0.404255\pi$
$608$	−7.10607	−0.288189
$609$	0	0
$610$	5.81521	0.235451
$611$	1.34730	0.0545058
$612$	0	0
$613$	25.8753	1.04509	0.522546	−	0.852611i	$-0.324982\pi$
$613$	25.8753	1.04509	0.522546	+	0.852611i	$0.324982\pi$
$614$	−4.54664	−0.183487
$615$	0	0
$616$	−8.63816	−0.348041
$617$	−22.1702	−0.892540	−0.446270	−	0.894898i	$-0.647248\pi$
$617$	−22.1702	−0.892540	−0.446270	+	0.894898i	$0.647248\pi$
$618$	0	0
$619$	39.8408	1.60134	0.800668	−	0.599108i	$-0.204478\pi$
$619$	39.8408	1.60134	0.800668	+	0.599108i	$0.204478\pi$
$620$	0.0523182	0.00210115
$621$	0	0
$622$	16.4287	0.658731
$623$	6.28136	0.251658
$624$	0	0
$625$	20.8334	0.833335
$626$	20.7023	0.827432
$627$	0	0
$628$	−26.4570	−1.05575
$629$	24.6364	0.982318
$630$	0	0
$631$	−18.8280	−0.749531	−0.374765	−	0.927120i	$-0.622277\pi$
$631$	−18.8280	−0.749531	−0.374765	+	0.927120i	$0.622277\pi$
$632$	2.08378	0.0828882
$633$	0	0
$634$	−43.1712	−1.71455
$635$	9.45336	0.375145
$636$	0	0
$637$	5.75103	0.227864
$638$	−44.0506	−1.74398
$639$	0	0
$640$	3.64084	0.143917
$641$	24.3756	0.962777	0.481389	−	0.876507i	$-0.340133\pi$
$641$	24.3756	0.962777	0.481389	+	0.876507i	$0.340133\pi$
$642$	0	0
$643$	−27.8411	−1.09795	−0.548973	−	0.835840i	$-0.684981\pi$
$643$	−27.8411	−1.09795	−0.548973	+	0.835840i	$0.684981\pi$
$644$	11.1702	0.440169
$645$	0	0
$646$	10.5963	0.416905
$647$	−1.25166	−0.0492078	−0.0246039	−	0.999697i	$-0.507832\pi$
$647$	−1.25166	−0.0492078	−0.0246039	+	0.999697i	$0.507832\pi$
$648$	0	0
$649$	−19.6459	−0.771169
$650$	−11.9436	−0.468465
$651$	0	0
$652$	−20.7561	−0.812871
$653$	−8.52435	−0.333584	−0.166792	−	0.985992i	$-0.553341\pi$
$653$	−8.52435	−0.333584	−0.166792	+	0.985992i	$0.553341\pi$
$654$	0	0
$655$	9.49937	0.371171
$656$	4.45067	0.173770
$657$	0	0
$658$	−3.10607	−0.121087
$659$	9.94087	0.387241	0.193621	−	0.981076i	$-0.437977\pi$
$659$	9.94087	0.387241	0.193621	+	0.981076i	$0.437977\pi$
$660$	0	0
$661$	5.62773	0.218893	0.109447	−	0.993993i	$-0.465092\pi$
$661$	5.62773	0.218893	0.109447	+	0.993993i	$0.465092\pi$
$662$	54.4843	2.11759
$663$	0	0
$664$	9.94181	0.385817
$665$	−0.879385	−0.0341011
$666$	0	0
$667$	−17.3969	−0.673612
$668$	13.7324	0.531321
$669$	0	0
$670$	6.27126	0.242280
$671$	34.5631	1.33429
$672$	0	0
$673$	−35.8776	−1.38298	−0.691491	−	0.722386i	$-0.743046\pi$
$673$	−35.8776	−1.38298	−0.691491	+	0.722386i	$0.743046\pi$
$674$	−30.1343	−1.16073
$675$	0	0
$676$	−17.1361	−0.659081
$677$	10.3678	0.398468	0.199234	−	0.979952i	$-0.436155\pi$
$677$	10.3678	0.398468	0.199234	+	0.979952i	$0.436155\pi$
$678$	0	0
$679$	−10.2300	−0.392591
$680$	−2.63816	−0.101169
$681$	0	0
$682$	0.716881	0.0274508
$683$	−4.58347	−0.175382	−0.0876909	−	0.996148i	$-0.527949\pi$
$683$	−4.58347	−0.175382	−0.0876909	+	0.996148i	$0.527949\pi$
$684$	0	0
$685$	−4.96173	−0.189578
$686$	−35.0009	−1.33634
$687$	0	0
$688$	−26.2918	−1.00237
$689$	0.361844	0.0137852
$690$	0	0
$691$	−9.10338	−0.346309	−0.173154	−	0.984895i	$-0.555396\pi$
$691$	−9.10338	−0.346309	−0.173154	+	0.984895i	$0.555396\pi$
$692$	−9.66819	−0.367529
$693$	0	0
$694$	9.38743	0.356342
$695$	2.77694	0.105335
$696$	0	0
$697$	−5.31996	−0.201508
$698$	−36.0378	−1.36405
$699$	0	0
$700$	11.9436	0.451424
$701$	−1.33450	−0.0504035	−0.0252017	−	0.999682i	$-0.508023\pi$
$701$	−1.33450	−0.0504035	−0.0252017	+	0.999682i	$0.508023\pi$
$702$	0	0
$703$	−4.36959	−0.164802
$704$	−23.3063	−0.878391
$705$	0	0
$706$	−26.5158	−0.997935
$707$	−15.3422	−0.577004
$708$	0	0
$709$	42.7579	1.60581	0.802904	−	0.596108i	$-0.203287\pi$
$709$	42.7579	1.60581	0.802904	+	0.596108i	$0.203287\pi$
$710$	8.69459	0.326302
$711$	0	0
$712$	−3.34224	−0.125256
$713$	0.283119	0.0106029
$714$	0	0
$715$	4.26083	0.159346
$716$	−9.87670	−0.369109
$717$	0	0
$718$	35.5767	1.32771
$719$	39.2036	1.46205	0.731023	−	0.682353i	$-0.239043\pi$
$719$	39.2036	1.46205	0.731023	+	0.682353i	$0.239043\pi$
$720$	0	0
$721$	10.4338	0.388574
$722$	−1.87939	−0.0699435
$723$	0	0
$724$	−38.2918	−1.42310
$725$	−18.6013	−0.690836
$726$	0	0
$727$	33.9249	1.25820	0.629102	−	0.777322i	$-0.283423\pi$
$727$	33.9249	1.25820	0.629102	+	0.777322i	$0.283423\pi$
$728$	1.95811	0.0725724
$729$	0	0
$730$	7.24897	0.268296
$731$	31.4270	1.16237
$732$	0	0
$733$	18.2891	0.675523	0.337762	−	0.941232i	$-0.390330\pi$
$733$	18.2891	0.675523	0.337762	+	0.941232i	$0.390330\pi$
$734$	−41.5844	−1.53491
$735$	0	0
$736$	−31.3482	−1.15551
$737$	37.2736	1.37299
$738$	0	0
$739$	37.7543	1.38882	0.694408	−	0.719582i	$-0.255666\pi$
$739$	37.7543	1.38882	0.694408	+	0.719582i	$0.255666\pi$
$740$	−3.56212	−0.130946
$741$	0	0
$742$	−0.834198	−0.0306244
$743$	46.3533	1.70054	0.850269	−	0.526349i	$-0.176440\pi$
$743$	46.3533	1.70054	0.850269	+	0.526349i	$0.176440\pi$
$744$	0	0
$745$	1.96728	0.0720756
$746$	38.1343	1.39620
$747$	0	0
$748$	51.3414	1.87723
$749$	−20.3131	−0.742226
$750$	0	0
$751$	−14.7689	−0.538924	−0.269462	−	0.963011i	$-0.586846\pi$
$751$	−14.7689	−0.538924	−0.269462	+	0.963011i	$0.586846\pi$
$752$	4.71688	0.172007
$753$	0	0
$754$	9.98545	0.363649
$755$	−3.24628	−0.118144
$756$	0	0
$757$	−35.7579	−1.29964	−0.649822	−	0.760086i	$-0.725157\pi$
$757$	−35.7579	−1.29964	−0.649822	+	0.760086i	$0.725157\pi$
$758$	10.1480	0.368590
$759$	0	0
$760$	0.467911	0.0169729
$761$	14.2576	0.516839	0.258420	−	0.966033i	$-0.416798\pi$
$761$	14.2576	0.516839	0.258420	+	0.966033i	$0.416798\pi$
$762$	0	0
$763$	4.85649	0.175817
$764$	9.07604	0.328360
$765$	0	0
$766$	−43.5280	−1.57273
$767$	4.45336	0.160802
$768$	0	0
$769$	−20.3087	−0.732351	−0.366175	−	0.930546i	$-0.619333\pi$
$769$	−20.3087	−0.732351	−0.366175	+	0.930546i	$0.619333\pi$
$770$	−9.82295	−0.353994
$771$	0	0
$772$	5.76651	0.207541
$773$	−6.14290	−0.220945	−0.110472	−	0.993879i	$-0.535236\pi$
$773$	−6.14290	−0.220945	−0.110472	+	0.993879i	$0.535236\pi$
$774$	0	0
$775$	0.302719	0.0108740
$776$	5.44326	0.195402
$777$	0	0
$778$	31.4706	1.12828
$779$	0.943563	0.0338067
$780$	0	0
$781$	51.6769	1.84914
$782$	46.7452	1.67160
$783$	0	0
$784$	20.1343	0.719084
$785$	9.18841	0.327949
$786$	0	0
$787$	12.5790	0.448394	0.224197	−	0.974544i	$-0.428024\pi$
$787$	12.5790	0.448394	0.224197	+	0.974544i	$0.428024\pi$
$788$	−3.46379	−0.123392
$789$	0	0
$790$	2.36959	0.0843061
$791$	22.2841	0.792330
$792$	0	0
$793$	−7.83481	−0.278222
$794$	29.2344	1.03749
$795$	0	0
$796$	−21.2499	−0.753183
$797$	13.2909	0.470786	0.235393	−	0.971900i	$-0.424362\pi$
$797$	13.2909	0.470786	0.235393	+	0.971900i	$0.424362\pi$
$798$	0	0
$799$	−5.63816	−0.199464
$800$	−33.5185	−1.18506
$801$	0	0
$802$	−35.4270	−1.25097
$803$	43.0847	1.52043
$804$	0	0
$805$	−3.87939	−0.136730
$806$	−0.162504	−0.00572395
$807$	0	0
$808$	8.16344	0.287189
$809$	37.1625	1.30656	0.653282	−	0.757115i	$-0.273392\pi$
$809$	37.1625	1.30656	0.653282	+	0.757115i	$0.273392\pi$
$810$	0	0
$811$	−13.9463	−0.489719	−0.244860	−	0.969559i	$-0.578742\pi$
$811$	−13.9463	−0.489719	−0.244860	+	0.969559i	$0.578742\pi$
$812$	−9.98545	−0.350421
$813$	0	0
$814$	−48.8093	−1.71077
$815$	7.20851	0.252503
$816$	0	0
$817$	−5.57398	−0.195009
$818$	−51.7915	−1.81085
$819$	0	0
$820$	0.769200	0.0268616
$821$	38.5276	1.34462	0.672312	−	0.740268i	$-0.265301\pi$
$821$	38.5276	1.34462	0.672312	+	0.740268i	$0.265301\pi$
$822$	0	0
$823$	−2.19698	−0.0765818	−0.0382909	−	0.999267i	$-0.512191\pi$
$823$	−2.19698	−0.0765818	−0.0382909	+	0.999267i	$0.512191\pi$
$824$	−5.55169	−0.193402
$825$	0	0
$826$	−10.2668	−0.357228
$827$	−52.9231	−1.84032	−0.920159	−	0.391545i	$-0.871941\pi$
$827$	−52.9231	−1.84032	−0.920159	+	0.391545i	$0.871941\pi$
$828$	0	0
$829$	35.6732	1.23898	0.619491	−	0.785003i	$-0.287339\pi$
$829$	35.6732	1.23898	0.619491	+	0.785003i	$0.287339\pi$
$830$	11.3054	0.392416
$831$	0	0
$832$	5.28312	0.183159
$833$	−24.0669	−0.833867
$834$	0	0
$835$	−4.76920	−0.165045
$836$	−9.10607	−0.314940
$837$	0	0
$838$	12.0223	0.415303
$839$	−11.5817	−0.399845	−0.199923	−	0.979812i	$-0.564069\pi$
$839$	−11.5817	−0.399845	−0.199923	+	0.979812i	$0.564069\pi$
$840$	0	0
$841$	−13.4483	−0.463735
$842$	65.7948	2.26744
$843$	0	0
$844$	36.7074	1.26352
$845$	5.95130	0.204731
$846$	0	0
$847$	−40.2036	−1.38141
$848$	1.26682	0.0435026
$849$	0	0
$850$	49.9813	1.71435
$851$	−19.2763	−0.660783
$852$	0	0
$853$	52.2354	1.78850	0.894252	−	0.447563i	$-0.147708\pi$
$853$	52.2354	1.78850	0.894252	+	0.447563i	$0.147708\pi$
$854$	18.0624	0.618083
$855$	0	0
$856$	10.8084	0.369424
$857$	44.5417	1.52152	0.760758	−	0.649036i	$-0.224828\pi$
$857$	44.5417	1.52152	0.760758	+	0.649036i	$0.224828\pi$
$858$	0	0
$859$	−12.3108	−0.420039	−0.210019	−	0.977697i	$-0.567353\pi$
$859$	−12.3108	−0.420039	−0.210019	+	0.977697i	$0.567353\pi$
$860$	−4.54395	−0.154947
$861$	0	0
$862$	64.4894	2.19652
$863$	−20.9358	−0.712664	−0.356332	−	0.934359i	$-0.615973\pi$
$863$	−20.9358	−0.712664	−0.356332	+	0.934359i	$0.615973\pi$
$864$	0	0
$865$	3.35773	0.114166
$866$	17.1233	0.581874
$867$	0	0
$868$	0.162504	0.00551574
$869$	14.0838	0.477759
$870$	0	0
$871$	−8.44924	−0.286292
$872$	−2.58408	−0.0875081
$873$	0	0
$874$	−8.29086	−0.280443
$875$	−8.54488	−0.288870
$876$	0	0
$877$	27.8753	0.941281	0.470640	−	0.882325i	$-0.344023\pi$
$877$	27.8753	0.941281	0.470640	+	0.882325i	$0.344023\pi$
$878$	−43.1729	−1.45702
$879$	0	0
$880$	14.9172	0.502857
$881$	44.4175	1.49646	0.748231	−	0.663438i	$-0.230903\pi$
$881$	44.4175	1.49646	0.748231	+	0.663438i	$0.230903\pi$
$882$	0	0
$883$	−41.3628	−1.39197	−0.695984	−	0.718057i	$-0.745032\pi$
$883$	−41.3628	−1.39197	−0.695984	+	0.718057i	$0.745032\pi$
$884$	−11.6382	−0.391434
$885$	0	0
$886$	0.398681	0.0133939
$887$	−38.0948	−1.27910	−0.639549	−	0.768750i	$-0.720879\pi$
$887$	−38.0948	−1.27910	−0.639549	+	0.768750i	$0.720879\pi$
$888$	0	0
$889$	29.3628	0.984796
$890$	−3.80066	−0.127398
$891$	0	0
$892$	−14.6919	−0.491921
$893$	1.00000	0.0334637
$894$	0	0
$895$	3.43014	0.114657
$896$	11.3087	0.377797
$897$	0	0
$898$	−8.71688	−0.290886
$899$	−0.253089	−0.00844099
$900$	0	0
$901$	−1.51424	−0.0504468
$902$	10.5398	0.350938
$903$	0	0
$904$	−11.8571	−0.394361
$905$	13.2986	0.442060
$906$	0	0
$907$	11.6423	0.386575	0.193288	−	0.981142i	$-0.438085\pi$
$907$	11.6423	0.386575	0.193288	+	0.981142i	$0.438085\pi$
$908$	31.5895	1.04833
$909$	0	0
$910$	2.22668	0.0738138
$911$	41.9377	1.38946	0.694729	−	0.719272i	$-0.255524\pi$
$911$	41.9377	1.38946	0.694729	+	0.719272i	$0.255524\pi$
$912$	0	0
$913$	67.1944	2.22381
$914$	−19.7716	−0.653985
$915$	0	0
$916$	−4.50980	−0.149008
$917$	29.5057	0.974363
$918$	0	0
$919$	28.1702	0.929251	0.464625	−	0.885507i	$-0.346189\pi$
$919$	28.1702	0.929251	0.464625	+	0.885507i	$0.346189\pi$
$920$	2.06418	0.0680539
$921$	0	0
$922$	67.6451	2.22777
$923$	−11.7142	−0.385577
$924$	0	0
$925$	−20.6108	−0.677679
$926$	−47.6792	−1.56684
$927$	0	0
$928$	28.0232	0.919907
$929$	4.42366	0.145136	0.0725678	−	0.997363i	$-0.476881\pi$
$929$	4.42366	0.145136	0.0725678	+	0.997363i	$0.476881\pi$
$930$	0	0
$931$	4.26857	0.139897
$932$	19.7665	0.647474
$933$	0	0
$934$	−55.6418	−1.82065
$935$	−17.8307	−0.583126
$936$	0	0
$937$	11.5536	0.377438	0.188719	−	0.982031i	$-0.439566\pi$
$937$	11.5536	0.377438	0.188719	+	0.982031i	$0.439566\pi$
$938$	19.4789	0.636010
$939$	0	0
$940$	0.815207	0.0265891
$941$	27.3131	0.890383	0.445192	−	0.895435i	$-0.353136\pi$
$941$	27.3131	0.890383	0.445192	+	0.895435i	$0.353136\pi$
$942$	0	0
$943$	4.16250	0.135550
$944$	15.5912	0.507451
$945$	0	0
$946$	−62.2627	−2.02433
$947$	12.1676	0.395392	0.197696	−	0.980263i	$-0.436654\pi$
$947$	12.1676	0.395392	0.197696	+	0.980263i	$0.436654\pi$
$948$	0	0
$949$	−9.76651	−0.317034
$950$	−8.86484	−0.287613
$951$	0	0
$952$	−8.19429	−0.265578
$953$	25.8494	0.837343	0.418671	−	0.908138i	$-0.362496\pi$
$953$	25.8494	0.837343	0.418671	+	0.908138i	$0.362496\pi$
$954$	0	0
$955$	−3.15207	−0.101999
$956$	41.3155	1.33624
$957$	0	0
$958$	−14.5398	−0.469761
$959$	−15.4115	−0.497662
$960$	0	0
$961$	−30.9959	−0.999867
$962$	11.0642	0.356723
$963$	0	0
$964$	14.2763	0.459809
$965$	−2.00269	−0.0644688
$966$	0	0
$967$	−3.88207	−0.124839	−0.0624195	−	0.998050i	$-0.519882\pi$
$967$	−3.88207	−0.124839	−0.0624195	+	0.998050i	$0.519882\pi$
$968$	21.3919	0.687561
$969$	0	0
$970$	6.18984	0.198744
$971$	−23.1967	−0.744416	−0.372208	−	0.928149i	$-0.621399\pi$
$971$	−23.1967	−0.744416	−0.372208	+	0.928149i	$0.621399\pi$
$972$	0	0
$973$	8.62536	0.276516
$974$	−53.0360	−1.69938
$975$	0	0
$976$	−27.4296	−0.878002
$977$	4.84161	0.154897	0.0774485	−	0.996996i	$-0.475323\pi$
$977$	4.84161	0.154897	0.0774485	+	0.996996i	$0.475323\pi$
$978$	0	0
$979$	−22.5895	−0.721962
$980$	3.47977	0.111157
$981$	0	0
$982$	37.9267	1.21029
$983$	−6.97596	−0.222498	−0.111249	−	0.993793i	$-0.535485\pi$
$983$	−6.97596	−0.222498	−0.111249	+	0.993793i	$0.535485\pi$
$984$	0	0
$985$	1.20296	0.0383296
$986$	−41.7870	−1.33077
$987$	0	0
$988$	2.06418	0.0656702
$989$	−24.5895	−0.781899
$990$	0	0
$991$	41.4570	1.31692	0.658462	−	0.752614i	$-0.271207\pi$
$991$	41.4570	1.31692	0.658462	+	0.752614i	$0.271207\pi$
$992$	−0.456052	−0.0144797
$993$	0	0
$994$	27.0060	0.856578
$995$	7.38001	0.233962
$996$	0	0
$997$	26.9786	0.854422	0.427211	−	0.904152i	$-0.359496\pi$
$997$	26.9786	0.854422	0.427211	+	0.904152i	$0.359496\pi$
$998$	22.9608	0.726811
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.e.1.1		3
3.2	odd	2		2679.2.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.f.1.3	✓	3	3.2	odd	2
8037.2.a.e.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.87939 q^{2} +1.53209 q^{4} -0.532089 q^{5} -1.65270 q^{7} +0.879385 q^{8} +O(q^{10})\)	\(q-1.87939 q^{2} +1.53209 q^{4} -0.532089 q^{5} -1.65270 q^{7} +0.879385 q^{8} +1.00000 q^{10} +5.94356 q^{11} -1.34730 q^{13} +3.10607 q^{14} -4.71688 q^{16} +5.63816 q^{17} -1.00000 q^{19} -0.815207 q^{20} -11.1702 q^{22} -4.41147 q^{23} -4.71688 q^{25} +2.53209 q^{26} -2.53209 q^{28} +3.94356 q^{29} -0.0641778 q^{31} +7.10607 q^{32} -10.5963 q^{34} +0.879385 q^{35} +4.36959 q^{37} +1.87939 q^{38} -0.467911 q^{40} -0.943563 q^{41} +5.57398 q^{43} +9.10607 q^{44} +8.29086 q^{46} -1.00000 q^{47} -4.26857 q^{49} +8.86484 q^{50} -2.06418 q^{52} -0.268571 q^{53} -3.16250 q^{55} -1.45336 q^{56} -7.41147 q^{58} -3.30541 q^{59} +5.81521 q^{61} +0.120615 q^{62} -3.92127 q^{64} +0.716881 q^{65} +6.27126 q^{67} +8.63816 q^{68} -1.65270 q^{70} +8.69459 q^{71} +7.24897 q^{73} -8.21213 q^{74} -1.53209 q^{76} -9.82295 q^{77} +2.36959 q^{79} +2.50980 q^{80} +1.77332 q^{82} +11.3054 q^{83} -3.00000 q^{85} -10.4757 q^{86} +5.22668 q^{88} -3.80066 q^{89} +2.22668 q^{91} -6.75877 q^{92} +1.87939 q^{94} +0.532089 q^{95} +6.18984 q^{97} +8.02229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^5 - 6 * q^7 - 3 * q^8	\( 3 q + 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{10} + 3 q^{11} - 3 q^{13} - 3 q^{14} - 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} - 3 q^{23} - 6 q^{25} + 3 q^{26} - 3 q^{28} - 3 q^{29} + 9 q^{31} + 9 q^{32} - 18 q^{34} - 3 q^{35} + 6 q^{37} - 6 q^{40} + 12 q^{41} + 9 q^{43} + 15 q^{44} + 9 q^{46} - 3 q^{47} - 3 q^{49} + 3 q^{50} + 3 q^{52} + 9 q^{53} - 12 q^{55} + 9 q^{56} - 12 q^{58} - 12 q^{59} + 21 q^{61} + 6 q^{62} - 3 q^{64} - 6 q^{65} + 9 q^{68} - 6 q^{70} + 24 q^{71} + 9 q^{73} - 9 q^{77} + 9 q^{80} + 12 q^{82} + 36 q^{83} - 9 q^{85} - 12 q^{86} + 9 q^{88} + 3 q^{89} - 9 q^{92} - 3 q^{95} + 18 q^{98}+O(q^{100}) \) 3 * q + 3 * q^5 - 6 * q^7 - 3 * q^8 + 3 * q^10 + 3 * q^11 - 3 * q^13 - 3 * q^14 - 6 * q^16 - 3 * q^19 - 6 * q^20 - 12 * q^22 - 3 * q^23 - 6 * q^25 + 3 * q^26 - 3 * q^28 - 3 * q^29 + 9 * q^31 + 9 * q^32 - 18 * q^34 - 3 * q^35 + 6 * q^37 - 6 * q^40 + 12 * q^41 + 9 * q^43 + 15 * q^44 + 9 * q^46 - 3 * q^47 - 3 * q^49 + 3 * q^50 + 3 * q^52 + 9 * q^53 - 12 * q^55 + 9 * q^56 - 12 * q^58 - 12 * q^59 + 21 * q^61 + 6 * q^62 - 3 * q^64 - 6 * q^65 + 9 * q^68 - 6 * q^70 + 24 * q^71 + 9 * q^73 - 9 * q^77 + 9 * q^80 + 12 * q^82 + 36 * q^83 - 9 * q^85 - 12 * q^86 + 9 * q^88 + 3 * q^89 - 9 * q^92 - 3 * q^95 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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