LMFDB - Embedded newform 1881.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, 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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.245526$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.18524 q^{2} +2.77529 q^{4} +3.43077 q^{5} +3.93972 q^{7} +1.69419 q^{8} +7.49706 q^{10} -1.00000 q^{11} +3.31182 q^{13} +8.60924 q^{14} -1.84836 q^{16} -2.80637 q^{17} -1.00000 q^{19} +9.52137 q^{20} -2.18524 q^{22} -6.88998 q^{23} +6.77018 q^{25} +7.23713 q^{26} +10.9338 q^{28} -5.67979 q^{29} +2.51864 q^{31} -7.42749 q^{32} -6.13259 q^{34} +13.5163 q^{35} -6.39893 q^{37} -2.18524 q^{38} +5.81238 q^{40} -0.560629 q^{41} -9.40080 q^{43} -2.77529 q^{44} -15.0563 q^{46} +12.1742 q^{47} +8.52137 q^{49} +14.7945 q^{50} +9.19126 q^{52} -5.68316 q^{53} -3.43077 q^{55} +6.67463 q^{56} -12.4117 q^{58} -4.35730 q^{59} -3.56412 q^{61} +5.50384 q^{62} -12.5342 q^{64} +11.3621 q^{65} -9.95563 q^{67} -7.78847 q^{68} +29.5363 q^{70} +11.4671 q^{71} -8.95834 q^{73} -13.9832 q^{74} -2.77529 q^{76} -3.93972 q^{77} +8.49105 q^{79} -6.34128 q^{80} -1.22511 q^{82} +5.21960 q^{83} -9.62799 q^{85} -20.5430 q^{86} -1.69419 q^{88} +7.28423 q^{89} +13.0476 q^{91} -19.1217 q^{92} +26.6036 q^{94} -3.43077 q^{95} +10.6574 q^{97} +18.6213 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

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 $

$2$	2.18524	1.54520	0.772600	−	0.634893i	$-0.218956\pi$
$2$	2.18524	1.54520	0.772600	+	0.634893i	$0.218956\pi$
$3$	0	0
$3$	0	0
$4$	2.77529	1.38764
$5$	3.43077	1.53429	0.767143	−	0.641476i	$-0.221678\pi$
$5$	3.43077	1.53429	0.767143	+	0.641476i	$0.221678\pi$
$6$	0	0
$7$	3.93972	1.48907	0.744537	−	0.667582i	$-0.232671\pi$
$7$	3.93972	1.48907	0.744537	+	0.667582i	$0.232671\pi$
$8$	1.69419	0.598987
$9$	0	0
$10$	7.49706	2.37078
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.31182	0.918534	0.459267	−	0.888298i	$-0.348112\pi$
$13$	3.31182	0.918534	0.459267	+	0.888298i	$0.348112\pi$
$14$	8.60924	2.30092
$15$	0	0
$16$	−1.84836	−0.462089
$17$	−2.80637	−0.680644	−0.340322	−	0.940309i	$-0.610536\pi$
$17$	−2.80637	−0.680644	−0.340322	+	0.940309i	$0.610536\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	9.52137	2.12904
$21$	0	0
$22$	−2.18524	−0.465895
$23$	−6.88998	−1.43666	−0.718330	−	0.695702i	$-0.755093\pi$
$23$	−6.88998	−1.43666	−0.718330	+	0.695702i	$0.755093\pi$
$24$	0	0
$25$	6.77018	1.35404
$26$	7.23713	1.41932
$27$	0	0
$28$	10.9338	2.06630
$29$	−5.67979	−1.05471	−0.527355	−	0.849645i	$-0.676816\pi$
$29$	−5.67979	−1.05471	−0.527355	+	0.849645i	$0.676816\pi$
$30$	0	0
$31$	2.51864	0.452361	0.226180	−	0.974085i	$-0.427376\pi$
$31$	2.51864	0.452361	0.226180	+	0.974085i	$0.427376\pi$
$32$	−7.42749	−1.31301
$33$	0	0
$34$	−6.13259	−1.05173
$35$	13.5163	2.28466
$36$	0	0
$37$	−6.39893	−1.05198	−0.525989	−	0.850491i	$-0.676305\pi$
$37$	−6.39893	−1.05198	−0.525989	+	0.850491i	$0.676305\pi$
$38$	−2.18524	−0.354493
$39$	0	0
$40$	5.81238	0.919018
$41$	−0.560629	−0.0875555	−0.0437778	−	0.999041i	$-0.513939\pi$
$41$	−0.560629	−0.0875555	−0.0437778	+	0.999041i	$0.513939\pi$
$42$	0	0
$43$	−9.40080	−1.43361	−0.716805	−	0.697274i	$-0.754396\pi$
$43$	−9.40080	−1.43361	−0.716805	+	0.697274i	$0.754396\pi$
$44$	−2.77529	−0.418390
$45$	0	0
$46$	−15.0563	−2.21993
$47$	12.1742	1.77579	0.887896	−	0.460044i	$-0.152166\pi$
$47$	12.1742	1.77579	0.887896	+	0.460044i	$0.152166\pi$
$48$	0	0
$49$	8.52137	1.21734
$50$	14.7945	2.09226
$51$	0	0
$52$	9.19126	1.27460
$53$	−5.68316	−0.780643	−0.390321	−	0.920679i	$-0.627636\pi$
$53$	−5.68316	−0.780643	−0.390321	+	0.920679i	$0.627636\pi$
$54$	0	0
$55$	−3.43077	−0.462605
$56$	6.67463	0.891935
$57$	0	0
$58$	−12.4117	−1.62974
$59$	−4.35730	−0.567273	−0.283636	−	0.958932i	$-0.591541\pi$
$59$	−4.35730	−0.567273	−0.283636	+	0.958932i	$0.591541\pi$
$60$	0	0
$61$	−3.56412	−0.456339	−0.228169	−	0.973621i	$-0.573274\pi$
$61$	−3.56412	−0.456339	−0.228169	+	0.973621i	$0.573274\pi$
$62$	5.50384	0.698988
$63$	0	0
$64$	−12.5342	−1.56677
$65$	11.3621	1.40929
$66$	0	0
$67$	−9.95563	−1.21627	−0.608137	−	0.793832i	$-0.708083\pi$
$67$	−9.95563	−1.21627	−0.608137	+	0.793832i	$0.708083\pi$
$68$	−7.78847	−0.944491
$69$	0	0
$70$	29.5363	3.53026
$71$	11.4671	1.36089	0.680447	−	0.732797i	$-0.261786\pi$
$71$	11.4671	1.36089	0.680447	+	0.732797i	$0.261786\pi$
$72$	0	0
$73$	−8.95834	−1.04849	−0.524247	−	0.851566i	$-0.675653\pi$
$73$	−8.95834	−1.04849	−0.524247	+	0.851566i	$0.675653\pi$
$74$	−13.9832	−1.62552
$75$	0	0
$76$	−2.77529	−0.318347
$77$	−3.93972	−0.448972
$78$	0	0
$79$	8.49105	0.955318	0.477659	−	0.878545i	$-0.341485\pi$
$79$	8.49105	0.955318	0.477659	+	0.878545i	$0.341485\pi$
$80$	−6.34128	−0.708977
$81$	0	0
$82$	−1.22511	−0.135291
$83$	5.21960	0.572926	0.286463	−	0.958091i	$-0.407520\pi$
$83$	5.21960	0.572926	0.286463	+	0.958091i	$0.407520\pi$
$84$	0	0
$85$	−9.62799	−1.04430
$86$	−20.5430	−2.21521
$87$	0	0
$88$	−1.69419	−0.180601
$89$	7.28423	0.772127	0.386064	−	0.922472i	$-0.373834\pi$
$89$	7.28423	0.772127	0.386064	+	0.922472i	$0.373834\pi$
$90$	0	0
$91$	13.0476	1.36776
$92$	−19.1217	−1.99357
$93$	0	0
$94$	26.6036	2.74395
$95$	−3.43077	−0.351989
$96$	0	0
$97$	10.6574	1.08209	0.541045	−	0.840993i	$-0.318029\pi$
$97$	10.6574	1.08209	0.541045	+	0.840993i	$0.318029\pi$
$98$	18.6213	1.88103
$99$	0	0
$100$	18.7892	1.87892
$101$	11.4716	1.14147	0.570735	−	0.821134i	$-0.306658\pi$
$101$	11.4716	1.14147	0.570735	+	0.821134i	$0.306658\pi$
$102$	0	0
$103$	18.3034	1.80349	0.901745	−	0.432268i	$-0.142286\pi$
$103$	18.3034	1.80349	0.901745	+	0.432268i	$0.142286\pi$
$104$	5.61086	0.550190
$105$	0	0
$106$	−12.4191	−1.20625
$107$	−1.38838	−0.134220	−0.0671100	−	0.997746i	$-0.521378\pi$
$107$	−1.38838	−0.134220	−0.0671100	+	0.997746i	$0.521378\pi$
$108$	0	0
$109$	−0.412113	−0.0394732	−0.0197366	−	0.999805i	$-0.506283\pi$
$109$	−0.412113	−0.0394732	−0.0197366	+	0.999805i	$0.506283\pi$
$110$	−7.49706	−0.714817
$111$	0	0
$112$	−7.28200	−0.688084
$113$	6.54003	0.615234	0.307617	−	0.951510i	$-0.400468\pi$
$113$	6.54003	0.615234	0.307617	+	0.951510i	$0.400468\pi$
$114$	0	0
$115$	−23.6379	−2.20425
$116$	−15.7630	−1.46356
$117$	0	0
$118$	−9.52177	−0.876550
$119$	−11.0563	−1.01353
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.78847	−0.705135
$123$	0	0
$124$	6.98995	0.627716
$125$	6.07307	0.543192
$126$	0	0
$127$	9.08005	0.805724	0.402862	−	0.915261i	$-0.368015\pi$
$127$	9.08005	0.805724	0.402862	+	0.915261i	$0.368015\pi$
$128$	−12.5352	−1.10797
$129$	0	0
$130$	24.8289	2.17764
$131$	10.3876	0.907571	0.453785	−	0.891111i	$-0.350073\pi$
$131$	10.3876	0.907571	0.453785	+	0.891111i	$0.350073\pi$
$132$	0	0
$133$	−3.93972	−0.341617
$134$	−21.7555	−1.87939
$135$	0	0
$136$	−4.75452	−0.407697
$137$	0.798293	0.0682028	0.0341014	−	0.999418i	$-0.489143\pi$
$137$	0.798293	0.0682028	0.0341014	+	0.999418i	$0.489143\pi$
$138$	0	0
$139$	5.03184	0.426795	0.213398	−	0.976965i	$-0.431547\pi$
$139$	5.03184	0.426795	0.213398	+	0.976965i	$0.431547\pi$
$140$	37.5115	3.17030
$141$	0	0
$142$	25.0584	2.10285
$143$	−3.31182	−0.276948
$144$	0	0
$145$	−19.4860	−1.61823
$146$	−19.5761	−1.62013
$147$	0	0
$148$	−17.7589	−1.45977
$149$	−19.8351	−1.62496	−0.812479	−	0.582991i	$-0.801882\pi$
$149$	−19.8351	−1.62496	−0.812479	+	0.582991i	$0.801882\pi$
$150$	0	0
$151$	22.5447	1.83466	0.917331	−	0.398125i	$-0.130339\pi$
$151$	22.5447	1.83466	0.917331	+	0.398125i	$0.130339\pi$
$152$	−1.69419	−0.137417
$153$	0	0
$154$	−8.60924	−0.693752
$155$	8.64087	0.694051
$156$	0	0
$157$	−11.8013	−0.941843	−0.470921	−	0.882175i	$-0.656078\pi$
$157$	−11.8013	−0.941843	−0.470921	+	0.882175i	$0.656078\pi$
$158$	18.5550	1.47616
$159$	0	0
$160$	−25.4820	−2.01453
$161$	−27.1446	−2.13929
$162$	0	0
$163$	−24.8395	−1.94558	−0.972789	−	0.231691i	$-0.925574\pi$
$163$	−24.8395	−1.94558	−0.972789	+	0.231691i	$0.925574\pi$
$164$	−1.55591	−0.121496
$165$	0	0
$166$	11.4061	0.885285
$167$	2.79938	0.216623	0.108311	−	0.994117i	$-0.465456\pi$
$167$	2.79938	0.216623	0.108311	+	0.994117i	$0.465456\pi$
$168$	0	0
$169$	−2.03184	−0.156295
$170$	−21.0395	−1.61366
$171$	0	0
$172$	−26.0899	−1.98934
$173$	6.43926	0.489568	0.244784	−	0.969578i	$-0.421283\pi$
$173$	6.43926	0.489568	0.244784	+	0.969578i	$0.421283\pi$
$174$	0	0
$175$	26.6726	2.01626
$176$	1.84836	0.139325
$177$	0	0
$178$	15.9178	1.19309
$179$	−12.5241	−0.936095	−0.468048	−	0.883703i	$-0.655042\pi$
$179$	−12.5241	−0.936095	−0.468048	+	0.883703i	$0.655042\pi$
$180$	0	0
$181$	13.7515	1.02214	0.511071	−	0.859538i	$-0.329249\pi$
$181$	13.7515	1.02214	0.511071	+	0.859538i	$0.329249\pi$
$182$	28.5123	2.11347
$183$	0	0
$184$	−11.6729	−0.860541
$185$	−21.9533	−1.61404
$186$	0	0
$187$	2.80637	0.205222
$188$	33.7869	2.46417
$189$	0	0
$190$	−7.49706	−0.543894
$191$	−3.10678	−0.224799	−0.112399	−	0.993663i	$-0.535854\pi$
$191$	−3.10678	−0.224799	−0.112399	+	0.993663i	$0.535854\pi$
$192$	0	0
$193$	−0.747815	−0.0538289	−0.0269144	−	0.999638i	$-0.508568\pi$
$193$	−0.747815	−0.0538289	−0.0269144	+	0.999638i	$0.508568\pi$
$194$	23.2889	1.67205
$195$	0	0
$196$	23.6492	1.68923
$197$	3.41798	0.243521	0.121761	−	0.992559i	$-0.461146\pi$
$197$	3.41798	0.243521	0.121761	+	0.992559i	$0.461146\pi$
$198$	0	0
$199$	5.36785	0.380517	0.190258	−	0.981734i	$-0.439067\pi$
$199$	5.36785	0.380517	0.190258	+	0.981734i	$0.439067\pi$
$200$	11.4700	0.811049
$201$	0	0
$202$	25.0683	1.76380
$203$	−22.3768	−1.57054
$204$	0	0
$205$	−1.92339	−0.134335
$206$	39.9974	2.78675
$207$	0	0
$208$	−6.12142	−0.424444
$209$	1.00000	0.0691714
$210$	0	0
$211$	2.55492	0.175888	0.0879441	−	0.996125i	$-0.471970\pi$
$211$	2.55492	0.175888	0.0879441	+	0.996125i	$0.471970\pi$
$212$	−15.7724	−1.08325
$213$	0	0
$214$	−3.03395	−0.207397
$215$	−32.2520	−2.19957
$216$	0	0
$217$	9.92272	0.673598
$218$	−0.900566	−0.0609940
$219$	0	0
$220$	−9.52137	−0.641931
$221$	−9.29418	−0.625194
$222$	0	0
$223$	−24.9404	−1.67013	−0.835066	−	0.550149i	$-0.814571\pi$
$223$	−24.9404	−1.67013	−0.835066	+	0.550149i	$0.814571\pi$
$224$	−29.2622	−1.95516
$225$	0	0
$226$	14.2915	0.950660
$227$	22.8254	1.51497	0.757487	−	0.652851i	$-0.226427\pi$
$227$	22.8254	1.51497	0.757487	+	0.652851i	$0.226427\pi$
$228$	0	0
$229$	−0.603546	−0.0398834	−0.0199417	−	0.999801i	$-0.506348\pi$
$229$	−0.603546	−0.0398834	−0.0199417	+	0.999801i	$0.506348\pi$
$230$	−51.6546	−3.40601
$231$	0	0
$232$	−9.62264	−0.631757
$233$	−17.3705	−1.13798	−0.568988	−	0.822346i	$-0.692665\pi$
$233$	−17.3705	−1.13798	−0.568988	+	0.822346i	$0.692665\pi$
$234$	0	0
$235$	41.7669	2.72457
$236$	−12.0928	−0.787172
$237$	0	0
$238$	−24.1607	−1.56610
$239$	−7.23486	−0.467984	−0.233992	−	0.972238i	$-0.575179\pi$
$239$	−7.23486	−0.467984	−0.233992	+	0.972238i	$0.575179\pi$
$240$	0	0
$241$	−12.2034	−0.786090	−0.393045	−	0.919519i	$-0.628578\pi$
$241$	−12.2034	−0.786090	−0.393045	+	0.919519i	$0.628578\pi$
$242$	2.18524	0.140473
$243$	0	0
$244$	−9.89146	−0.633236
$245$	29.2349	1.86775
$246$	0	0
$247$	−3.31182	−0.210726
$248$	4.26705	0.270958
$249$	0	0
$250$	13.2711	0.839340
$251$	14.0923	0.889499	0.444750	−	0.895655i	$-0.353293\pi$
$251$	14.0923	0.889499	0.444750	+	0.895655i	$0.353293\pi$
$252$	0	0
$253$	6.88998	0.433169
$254$	19.8421	1.24501
$255$	0	0
$256$	−2.32415	−0.145259
$257$	−0.440920	−0.0275038	−0.0137519	−	0.999905i	$-0.504378\pi$
$257$	−0.440920	−0.0275038	−0.0137519	+	0.999905i	$0.504378\pi$
$258$	0	0
$259$	−25.2100	−1.56647
$260$	31.5331	1.95560
$261$	0	0
$262$	22.6995	1.40238
$263$	15.0661	0.929016	0.464508	−	0.885569i	$-0.346231\pi$
$263$	15.0661	0.929016	0.464508	+	0.885569i	$0.346231\pi$
$264$	0	0
$265$	−19.4976	−1.19773
$266$	−8.60924	−0.527866
$267$	0	0
$268$	−27.6297	−1.68775
$269$	7.25751	0.442498	0.221249	−	0.975217i	$-0.428987\pi$
$269$	7.25751	0.442498	0.221249	+	0.975217i	$0.428987\pi$
$270$	0	0
$271$	−16.8878	−1.02586	−0.512931	−	0.858430i	$-0.671440\pi$
$271$	−16.8878	−1.02586	−0.512931	+	0.858430i	$0.671440\pi$
$272$	5.18716	0.314518
$273$	0	0
$274$	1.74446	0.105387
$275$	−6.77018	−0.408257
$276$	0	0
$277$	−15.3818	−0.924204	−0.462102	−	0.886827i	$-0.652905\pi$
$277$	−15.3818	−0.924204	−0.462102	+	0.886827i	$0.652905\pi$
$278$	10.9958	0.659484
$279$	0	0
$280$	22.8991	1.36848
$281$	25.8974	1.54491	0.772456	−	0.635069i	$-0.219028\pi$
$281$	25.8974	1.54491	0.772456	+	0.635069i	$0.219028\pi$
$282$	0	0
$283$	18.6882	1.11090	0.555450	−	0.831550i	$-0.312546\pi$
$283$	18.6882	1.11090	0.555450	+	0.831550i	$0.312546\pi$
$284$	31.8245	1.88844
$285$	0	0
$286$	−7.23713	−0.427941
$287$	−2.20872	−0.130377
$288$	0	0
$289$	−9.12431	−0.536724
$290$	−42.5817	−2.50049
$291$	0	0
$292$	−24.8620	−1.45494
$293$	−26.7471	−1.56258	−0.781291	−	0.624167i	$-0.785439\pi$
$293$	−26.7471	−1.56258	−0.781291	+	0.624167i	$0.785439\pi$
$294$	0	0
$295$	−14.9489	−0.870359
$296$	−10.8410	−0.630121
$297$	0	0
$298$	−43.3446	−2.51088
$299$	−22.8184	−1.31962
$300$	0	0
$301$	−37.0365	−2.13475
$302$	49.2657	2.83492
$303$	0	0
$304$	1.84836	0.106010
$305$	−12.2277	−0.700155
$306$	0	0
$307$	22.6415	1.29222	0.646109	−	0.763245i	$-0.276395\pi$
$307$	22.6415	1.29222	0.646109	+	0.763245i	$0.276395\pi$
$308$	−10.9338	−0.623014
$309$	0	0
$310$	18.8824	1.07245
$311$	1.38723	0.0786628	0.0393314	−	0.999226i	$-0.487477\pi$
$311$	1.38723	0.0786628	0.0393314	+	0.999226i	$0.487477\pi$
$312$	0	0
$313$	12.9018	0.729255	0.364627	−	0.931153i	$-0.381196\pi$
$313$	12.9018	0.729255	0.364627	+	0.931153i	$0.381196\pi$
$314$	−25.7886	−1.45534
$315$	0	0
$316$	23.5651	1.32564
$317$	19.0712	1.07114	0.535572	−	0.844489i	$-0.320096\pi$
$317$	19.0712	1.07114	0.535572	+	0.844489i	$0.320096\pi$
$318$	0	0
$319$	5.67979	0.318007
$320$	−43.0018	−2.40387
$321$	0	0
$322$	−59.3175	−3.30564
$323$	2.80637	0.156150
$324$	0	0
$325$	22.4216	1.24373
$326$	−54.2803	−3.00631
$327$	0	0
$328$	−0.949812	−0.0524446
$329$	47.9630	2.64428
$330$	0	0
$331$	−12.4616	−0.684952	−0.342476	−	0.939527i	$-0.611265\pi$
$331$	−12.4616	−0.684952	−0.342476	+	0.939527i	$0.611265\pi$
$332$	14.4859	0.795017
$333$	0	0
$334$	6.11733	0.334725
$335$	−34.1555	−1.86611
$336$	0	0
$337$	−0.401035	−0.0218458	−0.0109229	−	0.999940i	$-0.503477\pi$
$337$	−0.401035	−0.0218458	−0.0109229	+	0.999940i	$0.503477\pi$
$338$	−4.44006	−0.241508
$339$	0	0
$340$	−26.7204	−1.44912
$341$	−2.51864	−0.136392
$342$	0	0
$343$	5.99377	0.323633
$344$	−15.9268	−0.858713
$345$	0	0
$346$	14.0713	0.756480
$347$	−1.06608	−0.0572304	−0.0286152	−	0.999591i	$-0.509110\pi$
$347$	−1.06608	−0.0572304	−0.0286152	+	0.999591i	$0.509110\pi$
$348$	0	0
$349$	22.2695	1.19206	0.596029	−	0.802963i	$-0.296744\pi$
$349$	22.2695	1.19206	0.596029	+	0.802963i	$0.296744\pi$
$350$	58.2861	3.11552
$351$	0	0
$352$	7.42749	0.395886
$353$	−12.3631	−0.658023	−0.329012	−	0.944326i	$-0.606716\pi$
$353$	−12.3631	−0.658023	−0.329012	+	0.944326i	$0.606716\pi$
$354$	0	0
$355$	39.3410	2.08800
$356$	20.2158	1.07144
$357$	0	0
$358$	−27.3682	−1.44645
$359$	−32.1914	−1.69899	−0.849497	−	0.527593i	$-0.823095\pi$
$359$	−32.1914	−1.69899	−0.849497	+	0.527593i	$0.823095\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	30.0504	1.57941
$363$	0	0
$364$	36.2109	1.89797
$365$	−30.7340	−1.60869
$366$	0	0
$367$	−18.9777	−0.990628	−0.495314	−	0.868714i	$-0.664947\pi$
$367$	−18.9777	−0.990628	−0.495314	+	0.868714i	$0.664947\pi$
$368$	12.7351	0.663865
$369$	0	0
$370$	−47.9732	−2.49401
$371$	−22.3901	−1.16243
$372$	0	0
$373$	1.15238	0.0596678	0.0298339	−	0.999555i	$-0.490502\pi$
$373$	1.15238	0.0596678	0.0298339	+	0.999555i	$0.490502\pi$
$374$	6.13259	0.317109
$375$	0	0
$376$	20.6254	1.06368
$377$	−18.8104	−0.968787
$378$	0	0
$379$	13.5365	0.695325	0.347663	−	0.937620i	$-0.386975\pi$
$379$	13.5365	0.695325	0.347663	+	0.937620i	$0.386975\pi$
$380$	−9.52137	−0.488436
$381$	0	0
$382$	−6.78907	−0.347359
$383$	−13.1671	−0.672809	−0.336404	−	0.941718i	$-0.609211\pi$
$383$	−13.1671	−0.672809	−0.336404	+	0.941718i	$0.609211\pi$
$384$	0	0
$385$	−13.5163	−0.688852
$386$	−1.63416	−0.0831764
$387$	0	0
$388$	29.5772	1.50156
$389$	−0.223588	−0.0113364	−0.00566819	−	0.999984i	$-0.501804\pi$
$389$	−0.223588	−0.0113364	−0.00566819	+	0.999984i	$0.501804\pi$
$390$	0	0
$391$	19.3358	0.977854
$392$	14.4368	0.729170
$393$	0	0
$394$	7.46913	0.376289
$395$	29.1308	1.46573
$396$	0	0
$397$	−21.6504	−1.08660	−0.543301	−	0.839538i	$-0.682826\pi$
$397$	−21.6504	−1.08660	−0.543301	+	0.839538i	$0.682826\pi$
$398$	11.7301	0.587975
$399$	0	0
$400$	−12.5137	−0.625685
$401$	−1.30180	−0.0650089	−0.0325045	−	0.999472i	$-0.510348\pi$
$401$	−1.30180	−0.0650089	−0.0325045	+	0.999472i	$0.510348\pi$
$402$	0	0
$403$	8.34128	0.415509
$404$	31.8371	1.58395
$405$	0	0
$406$	−48.8986	−2.42680
$407$	6.39893	0.317183
$408$	0	0
$409$	−2.15631	−0.106622	−0.0533112	−	0.998578i	$-0.516978\pi$
$409$	−2.15631	−0.106622	−0.0533112	+	0.998578i	$0.516978\pi$
$410$	−4.20307	−0.207575
$411$	0	0
$412$	50.7973	2.50260
$413$	−17.1665	−0.844710
$414$	0	0
$415$	17.9073	0.879032
$416$	−24.5985	−1.20604
$417$	0	0
$418$	2.18524	0.106884
$419$	29.6335	1.44769	0.723844	−	0.689963i	$-0.242373\pi$
$419$	29.6335	1.44769	0.723844	+	0.689963i	$0.242373\pi$
$420$	0	0
$421$	−5.25385	−0.256057	−0.128028	−	0.991770i	$-0.540865\pi$
$421$	−5.25385	−0.256057	−0.128028	+	0.991770i	$0.540865\pi$
$422$	5.58313	0.271782
$423$	0	0
$424$	−9.62837	−0.467595
$425$	−18.9996	−0.921616
$426$	0	0
$427$	−14.0416	−0.679522
$428$	−3.85316	−0.186249
$429$	0	0
$430$	−70.4784	−3.39877
$431$	−17.8489	−0.859752	−0.429876	−	0.902888i	$-0.641443\pi$
$431$	−17.8489	−0.859752	−0.429876	+	0.902888i	$0.641443\pi$
$432$	0	0
$433$	−0.696383	−0.0334660	−0.0167330	−	0.999860i	$-0.505327\pi$
$433$	−0.696383	−0.0334660	−0.0167330	+	0.999860i	$0.505327\pi$
$434$	21.6836	1.04084
$435$	0	0
$436$	−1.14373	−0.0547748
$437$	6.88998	0.329593
$438$	0	0
$439$	−24.8210	−1.18464	−0.592321	−	0.805702i	$-0.701788\pi$
$439$	−24.8210	−1.18464	−0.592321	+	0.805702i	$0.701788\pi$
$440$	−5.81238	−0.277094
$441$	0	0
$442$	−20.3100	−0.966050
$443$	3.93424	0.186922	0.0934608	−	0.995623i	$-0.470207\pi$
$443$	3.93424	0.186922	0.0934608	+	0.995623i	$0.470207\pi$
$444$	0	0
$445$	24.9905	1.18466
$446$	−54.5008	−2.58069
$447$	0	0
$448$	−49.3810	−2.33303
$449$	−6.13394	−0.289479	−0.144739	−	0.989470i	$-0.546234\pi$
$449$	−6.13394	−0.289479	−0.144739	+	0.989470i	$0.546234\pi$
$450$	0	0
$451$	0.560629	0.0263990
$452$	18.1505	0.853725
$453$	0	0
$454$	49.8790	2.34094
$455$	44.7634	2.09854
$456$	0	0
$457$	8.40189	0.393024	0.196512	−	0.980501i	$-0.437039\pi$
$457$	8.40189	0.393024	0.196512	+	0.980501i	$0.437039\pi$
$458$	−1.31889	−0.0616279
$459$	0	0
$460$	−65.6021	−3.05871
$461$	−25.9903	−1.21049	−0.605246	−	0.796039i	$-0.706925\pi$
$461$	−25.9903	−1.21049	−0.605246	+	0.796039i	$0.706925\pi$
$462$	0	0
$463$	19.3724	0.900311	0.450155	−	0.892950i	$-0.351369\pi$
$463$	19.3724	0.900311	0.450155	+	0.892950i	$0.351369\pi$
$464$	10.4983	0.487370
$465$	0	0
$466$	−37.9587	−1.75840
$467$	34.6720	1.60443	0.802214	−	0.597037i	$-0.203655\pi$
$467$	34.6720	1.60443	0.802214	+	0.597037i	$0.203655\pi$
$468$	0	0
$469$	−39.2224	−1.81112
$470$	91.2709	4.21001
$471$	0	0
$472$	−7.38210	−0.339789
$473$	9.40080	0.432249
$474$	0	0
$475$	−6.77018	−0.310637
$476$	−30.6844	−1.40642
$477$	0	0
$478$	−15.8099	−0.723130
$479$	−23.2352	−1.06164	−0.530821	−	0.847484i	$-0.678117\pi$
$479$	−23.2352	−1.06164	−0.530821	+	0.847484i	$0.678117\pi$
$480$	0	0
$481$	−21.1921	−0.966277
$482$	−26.6674	−1.21467
$483$	0	0
$484$	2.77529	0.126149
$485$	36.5629	1.66024
$486$	0	0
$487$	14.2680	0.646545	0.323272	−	0.946306i	$-0.395217\pi$
$487$	14.2680	0.646545	0.323272	+	0.946306i	$0.395217\pi$
$488$	−6.03830	−0.273341
$489$	0	0
$490$	63.8853	2.88604
$491$	−41.6168	−1.87814	−0.939071	−	0.343724i	$-0.888311\pi$
$491$	−41.6168	−1.87814	−0.939071	+	0.343724i	$0.888311\pi$
$492$	0	0
$493$	15.9396	0.717882
$494$	−7.23713	−0.325614
$495$	0	0
$496$	−4.65534	−0.209031
$497$	45.1771	2.02647
$498$	0	0
$499$	19.9664	0.893820	0.446910	−	0.894579i	$-0.352524\pi$
$499$	19.9664	0.893820	0.446910	+	0.894579i	$0.352524\pi$
$500$	16.8545	0.753757
$501$	0	0
$502$	30.7951	1.37445
$503$	−2.31362	−0.103159	−0.0515797	−	0.998669i	$-0.516426\pi$
$503$	−2.31362	−0.103159	−0.0515797	+	0.998669i	$0.516426\pi$
$504$	0	0
$505$	39.3565	1.75134
$506$	15.0563	0.669334
$507$	0	0
$508$	25.1998	1.11806
$509$	13.2009	0.585120	0.292560	−	0.956247i	$-0.405493\pi$
$509$	13.2009	0.585120	0.292560	+	0.956247i	$0.405493\pi$
$510$	0	0
$511$	−35.2933	−1.56128
$512$	19.9916	0.883511
$513$	0	0
$514$	−0.963517	−0.0424989
$515$	62.7948	2.76707
$516$	0	0
$517$	−12.1742	−0.535421
$518$	−55.0899	−2.42051
$519$	0	0
$520$	19.2496	0.844149
$521$	15.6498	0.685629	0.342814	−	0.939403i	$-0.388620\pi$
$521$	15.6498	0.685629	0.342814	+	0.939403i	$0.388620\pi$
$522$	0	0
$523$	−17.7350	−0.775498	−0.387749	−	0.921765i	$-0.626747\pi$
$523$	−17.7350	−0.775498	−0.387749	+	0.921765i	$0.626747\pi$
$524$	28.8286	1.25938
$525$	0	0
$526$	32.9231	1.43552
$527$	−7.06822	−0.307897
$528$	0	0
$529$	24.4719	1.06399
$530$	−42.6071	−1.85073
$531$	0	0
$532$	−10.9338	−0.474042
$533$	−1.85670	−0.0804227
$534$	0	0
$535$	−4.76322	−0.205932
$536$	−16.8667	−0.728532
$537$	0	0
$538$	15.8594	0.683748
$539$	−8.52137	−0.367041
$540$	0	0
$541$	−34.5196	−1.48412	−0.742058	−	0.670336i	$-0.766150\pi$
$541$	−34.5196	−1.48412	−0.742058	+	0.670336i	$0.766150\pi$
$542$	−36.9040	−1.58516
$543$	0	0
$544$	20.8442	0.893690
$545$	−1.41386	−0.0605632
$546$	0	0
$547$	1.99561	0.0853263	0.0426632	−	0.999090i	$-0.486416\pi$
$547$	1.99561	0.0853263	0.0426632	+	0.999090i	$0.486416\pi$
$548$	2.21549	0.0946411
$549$	0	0
$550$	−14.7945	−0.630839
$551$	5.67979	0.241967
$552$	0	0
$553$	33.4523	1.42254
$554$	−33.6130	−1.42808
$555$	0	0
$556$	13.9648	0.592239
$557$	23.5272	0.996881	0.498440	−	0.866924i	$-0.333906\pi$
$557$	23.5272	0.996881	0.498440	+	0.866924i	$0.333906\pi$
$558$	0	0
$559$	−31.1338	−1.31682
$560$	−24.9829	−1.05572
$561$	0	0
$562$	56.5922	2.38720
$563$	−14.2122	−0.598972	−0.299486	−	0.954101i	$-0.596815\pi$
$563$	−14.2122	−0.598972	−0.299486	+	0.954101i	$0.596815\pi$
$564$	0	0
$565$	22.4373	0.943945
$566$	40.8383	1.71656
$567$	0	0
$568$	19.4275	0.815158
$569$	−22.1906	−0.930277	−0.465138	−	0.885238i	$-0.653995\pi$
$569$	−22.1906	−0.930277	−0.465138	+	0.885238i	$0.653995\pi$
$570$	0	0
$571$	8.67954	0.363227	0.181614	−	0.983370i	$-0.441868\pi$
$571$	8.67954	0.363227	0.181614	+	0.983370i	$0.441868\pi$
$572$	−9.19126	−0.384306
$573$	0	0
$574$	−4.82659	−0.201458
$575$	−46.6464	−1.94529
$576$	0	0
$577$	40.9150	1.70331	0.851657	−	0.524100i	$-0.175598\pi$
$577$	40.9150	1.70331	0.851657	+	0.524100i	$0.175598\pi$
$578$	−19.9388	−0.829346
$579$	0	0
$580$	−54.0794	−2.24552
$581$	20.5638	0.853128
$582$	0	0
$583$	5.68316	0.235373
$584$	−15.1771	−0.628034
$585$	0	0
$586$	−58.4489	−2.41450
$587$	21.2684	0.877840	0.438920	−	0.898526i	$-0.355361\pi$
$587$	21.2684	0.877840	0.438920	+	0.898526i	$0.355361\pi$
$588$	0	0
$589$	−2.51864	−0.103779
$590$	−32.6670	−1.34488
$591$	0	0
$592$	11.8275	0.486107
$593$	23.0212	0.945368	0.472684	−	0.881232i	$-0.343285\pi$
$593$	23.0212	0.945368	0.472684	+	0.881232i	$0.343285\pi$
$594$	0	0
$595$	−37.9316	−1.55504
$596$	−55.0482	−2.25486
$597$	0	0
$598$	−49.8637	−2.03908
$599$	41.6249	1.70075	0.850374	−	0.526179i	$-0.176376\pi$
$599$	41.6249	1.70075	0.850374	+	0.526179i	$0.176376\pi$
$600$	0	0
$601$	−20.8629	−0.851016	−0.425508	−	0.904955i	$-0.639905\pi$
$601$	−20.8629	−0.851016	−0.425508	+	0.904955i	$0.639905\pi$
$602$	−80.9338	−3.29861
$603$	0	0
$604$	62.5680	2.54586
$605$	3.43077	0.139481
$606$	0	0
$607$	−0.146703	−0.00595450	−0.00297725	−	0.999996i	$-0.500948\pi$
$607$	−0.146703	−0.00595450	−0.00297725	+	0.999996i	$0.500948\pi$
$608$	7.42749	0.301224
$609$	0	0
$610$	−26.7204	−1.08188
$611$	40.3188	1.63113
$612$	0	0
$613$	−18.3393	−0.740716	−0.370358	−	0.928889i	$-0.620765\pi$
$613$	−18.3393	−0.740716	−0.370358	+	0.928889i	$0.620765\pi$
$614$	49.4772	1.99674
$615$	0	0
$616$	−6.67463	−0.268929
$617$	−4.12050	−0.165885	−0.0829425	−	0.996554i	$-0.526432\pi$
$617$	−4.12050	−0.165885	−0.0829425	+	0.996554i	$0.526432\pi$
$618$	0	0
$619$	28.5699	1.14832	0.574161	−	0.818743i	$-0.305328\pi$
$619$	28.5699	1.14832	0.574161	+	0.818743i	$0.305328\pi$
$620$	23.9809	0.963096
$621$	0	0
$622$	3.03144	0.121550
$623$	28.6978	1.14975
$624$	0	0
$625$	−13.0156	−0.520624
$626$	28.1937	1.12684
$627$	0	0
$628$	−32.7519	−1.30694
$629$	17.9577	0.716022
$630$	0	0
$631$	33.9323	1.35082	0.675412	−	0.737441i	$-0.263966\pi$
$631$	33.9323	1.35082	0.675412	+	0.737441i	$0.263966\pi$
$632$	14.3855	0.572223
$633$	0	0
$634$	41.6752	1.65513
$635$	31.1516	1.23621
$636$	0	0
$637$	28.2213	1.11817
$638$	12.4117	0.491385
$639$	0	0
$640$	−43.0054	−1.69994
$641$	−19.7694	−0.780844	−0.390422	−	0.920636i	$-0.627671\pi$
$641$	−19.7694	−0.780844	−0.390422	+	0.920636i	$0.627671\pi$
$642$	0	0
$643$	36.7642	1.44984	0.724920	−	0.688833i	$-0.241877\pi$
$643$	36.7642	1.44984	0.724920	+	0.688833i	$0.241877\pi$
$644$	−75.3340	−2.96858
$645$	0	0
$646$	6.13259	0.241284
$647$	−24.9430	−0.980610	−0.490305	−	0.871551i	$-0.663115\pi$
$647$	−24.9430	−0.980610	−0.490305	+	0.871551i	$0.663115\pi$
$648$	0	0
$649$	4.35730	0.171039
$650$	48.9967	1.92181
$651$	0	0
$652$	−68.9367	−2.69977
$653$	−8.60802	−0.336858	−0.168429	−	0.985714i	$-0.553869\pi$
$653$	−8.60802	−0.336858	−0.168429	+	0.985714i	$0.553869\pi$
$654$	0	0
$655$	35.6375	1.39247
$656$	1.03624	0.0404584
$657$	0	0
$658$	104.811	4.08595
$659$	−27.7805	−1.08217	−0.541087	−	0.840967i	$-0.681987\pi$
$659$	−27.7805	−1.08217	−0.541087	+	0.840967i	$0.681987\pi$
$660$	0	0
$661$	30.8037	1.19812	0.599062	−	0.800702i	$-0.295540\pi$
$661$	30.8037	1.19812	0.599062	+	0.800702i	$0.295540\pi$
$662$	−27.2316	−1.05839
$663$	0	0
$664$	8.84300	0.343175
$665$	−13.5163	−0.524138
$666$	0	0
$667$	39.1336	1.51526
$668$	7.76909	0.300595
$669$	0	0
$670$	−74.6380	−2.88352
$671$	3.56412	0.137591
$672$	0	0
$673$	15.2015	0.585974	0.292987	−	0.956116i	$-0.405351\pi$
$673$	15.2015	0.585974	0.292987	+	0.956116i	$0.405351\pi$
$674$	−0.876359	−0.0337561
$675$	0	0
$676$	−5.63894	−0.216882
$677$	9.49027	0.364741	0.182370	−	0.983230i	$-0.441623\pi$
$677$	9.49027	0.364741	0.182370	+	0.983230i	$0.441623\pi$
$678$	0	0
$679$	41.9870	1.61131
$680$	−16.3117	−0.625523
$681$	0	0
$682$	−5.50384	−0.210753
$683$	−15.0960	−0.577631	−0.288816	−	0.957385i	$-0.593261\pi$
$683$	−15.0960	−0.577631	−0.288816	+	0.957385i	$0.593261\pi$
$684$	0	0
$685$	2.73876	0.104643
$686$	13.0978	0.500078
$687$	0	0
$688$	17.3760	0.662455
$689$	−18.8216	−0.717047
$690$	0	0
$691$	3.46673	0.131881	0.0659403	−	0.997824i	$-0.478995\pi$
$691$	3.46673	0.131881	0.0659403	+	0.997824i	$0.478995\pi$
$692$	17.8708	0.679345
$693$	0	0
$694$	−2.32965	−0.0884325
$695$	17.2631	0.654826
$696$	0	0
$697$	1.57333	0.0595941
$698$	48.6643	1.84197
$699$	0	0
$700$	74.0241	2.79785
$701$	26.2612	0.991872	0.495936	−	0.868359i	$-0.334825\pi$
$701$	26.2612	0.991872	0.495936	+	0.868359i	$0.334825\pi$
$702$	0	0
$703$	6.39893	0.241340
$704$	12.5342	0.472399
$705$	0	0
$706$	−27.0165	−1.01678
$707$	45.1950	1.69973
$708$	0	0
$709$	−20.1488	−0.756704	−0.378352	−	0.925662i	$-0.623509\pi$
$709$	−20.1488	−0.756704	−0.378352	+	0.925662i	$0.623509\pi$
$710$	85.9696	3.22638
$711$	0	0
$712$	12.3409	0.462494
$713$	−17.3534	−0.649889
$714$	0	0
$715$	−11.3621	−0.424918
$716$	−34.7580	−1.29897
$717$	0	0
$718$	−70.3459	−2.62529
$719$	43.4738	1.62130	0.810650	−	0.585532i	$-0.199114\pi$
$719$	43.4738	1.62130	0.810650	+	0.585532i	$0.199114\pi$
$720$	0	0
$721$	72.1103	2.68553
$722$	2.18524	0.0813263
$723$	0	0
$724$	38.1644	1.41837
$725$	−38.4532	−1.42811
$726$	0	0
$727$	9.55640	0.354427	0.177214	−	0.984172i	$-0.443292\pi$
$727$	9.55640	0.354427	0.177214	+	0.984172i	$0.443292\pi$
$728$	22.1052	0.819273
$729$	0	0
$730$	−67.1612	−2.48575
$731$	26.3821	0.975777
$732$	0	0
$733$	−20.3072	−0.750065	−0.375032	−	0.927012i	$-0.622368\pi$
$733$	−20.3072	−0.750065	−0.375032	+	0.927012i	$0.622368\pi$
$734$	−41.4709	−1.53072
$735$	0	0
$736$	51.1753	1.88635
$737$	9.95563	0.366720
$738$	0	0
$739$	−6.60855	−0.243099	−0.121550	−	0.992585i	$-0.538786\pi$
$739$	−6.60855	−0.243099	−0.121550	+	0.992585i	$0.538786\pi$
$740$	−60.9266	−2.23971
$741$	0	0
$742$	−48.9277	−1.79619
$743$	39.1431	1.43602	0.718010	−	0.696033i	$-0.245053\pi$
$743$	39.1431	1.43602	0.718010	+	0.696033i	$0.245053\pi$
$744$	0	0
$745$	−68.0498	−2.49315
$746$	2.51822	0.0921988
$747$	0	0
$748$	7.78847	0.284775
$749$	−5.46983	−0.199863
$750$	0	0
$751$	43.2173	1.57702	0.788511	−	0.615020i	$-0.210852\pi$
$751$	43.2173	1.57702	0.788511	+	0.615020i	$0.210852\pi$
$752$	−22.5023	−0.820574
$753$	0	0
$754$	−41.1054	−1.49697
$755$	77.3457	2.81490
$756$	0	0
$757$	9.93714	0.361171	0.180586	−	0.983559i	$-0.442201\pi$
$757$	9.93714	0.361171	0.180586	+	0.983559i	$0.442201\pi$
$758$	29.5806	1.07442
$759$	0	0
$760$	−5.81238	−0.210837
$761$	−14.5829	−0.528630	−0.264315	−	0.964436i	$-0.585146\pi$
$761$	−14.5829	−0.528630	−0.264315	+	0.964436i	$0.585146\pi$
$762$	0	0
$763$	−1.62361	−0.0587785
$764$	−8.62221	−0.311941
$765$	0	0
$766$	−28.7734	−1.03962
$767$	−14.4306	−0.521059
$768$	0	0
$769$	−11.8924	−0.428853	−0.214426	−	0.976740i	$-0.568788\pi$
$769$	−11.8924	−0.428853	−0.214426	+	0.976740i	$0.568788\pi$
$770$	−29.5363	−1.06441
$771$	0	0
$772$	−2.07540	−0.0746953
$773$	−15.0935	−0.542876	−0.271438	−	0.962456i	$-0.587499\pi$
$773$	−15.0935	−0.542876	−0.271438	+	0.962456i	$0.587499\pi$
$774$	0	0
$775$	17.0516	0.612513
$776$	18.0556	0.648158
$777$	0	0
$778$	−0.488595	−0.0175170
$779$	0.560629	0.0200866
$780$	0	0
$781$	−11.4671	−0.410325
$782$	42.2534	1.51098
$783$	0	0
$784$	−15.7505	−0.562519
$785$	−40.4874	−1.44506
$786$	0	0
$787$	−5.98214	−0.213240	−0.106620	−	0.994300i	$-0.534003\pi$
$787$	−5.98214	−0.213240	−0.106620	+	0.994300i	$0.534003\pi$
$788$	9.48589	0.337921
$789$	0	0
$790$	63.6580	2.26485
$791$	25.7659	0.916128
$792$	0	0
$793$	−11.8037	−0.419163
$794$	−47.3114	−1.67902
$795$	0	0
$796$	14.8973	0.528022
$797$	−24.2718	−0.859751	−0.429876	−	0.902888i	$-0.641443\pi$
$797$	−24.2718	−0.859751	−0.429876	+	0.902888i	$0.641443\pi$
$798$	0	0
$799$	−34.1653	−1.20868
$800$	−50.2854	−1.77786
$801$	0	0
$802$	−2.84476	−0.100452
$803$	8.95834	0.316133
$804$	0	0
$805$	−93.1268	−3.28229
$806$	18.2277	0.642044
$807$	0	0
$808$	19.4351	0.683726
$809$	17.9557	0.631289	0.315644	−	0.948878i	$-0.397779\pi$
$809$	17.9557	0.631289	0.315644	+	0.948878i	$0.397779\pi$
$810$	0	0
$811$	5.48300	0.192534	0.0962671	−	0.995356i	$-0.469310\pi$
$811$	5.48300	0.192534	0.0962671	+	0.995356i	$0.469310\pi$
$812$	−62.1019	−2.17935
$813$	0	0
$814$	13.9832	0.490111
$815$	−85.2185	−2.98508
$816$	0	0
$817$	9.40080	0.328892
$818$	−4.71205	−0.164753
$819$	0	0
$820$	−5.33795	−0.186409
$821$	−52.6324	−1.83688	−0.918441	−	0.395558i	$-0.870551\pi$
$821$	−52.6324	−1.83688	−0.918441	+	0.395558i	$0.870551\pi$
$822$	0	0
$823$	−20.4784	−0.713831	−0.356916	−	0.934137i	$-0.616172\pi$
$823$	−20.4784	−0.713831	−0.356916	+	0.934137i	$0.616172\pi$
$824$	31.0095	1.08027
$825$	0	0
$826$	−37.5131	−1.30525
$827$	0.359755	0.0125099	0.00625496	−	0.999980i	$-0.498009\pi$
$827$	0.359755	0.0125099	0.00625496	+	0.999980i	$0.498009\pi$
$828$	0	0
$829$	22.7422	0.789870	0.394935	−	0.918709i	$-0.370767\pi$
$829$	22.7422	0.789870	0.394935	+	0.918709i	$0.370767\pi$
$830$	39.1317	1.35828
$831$	0	0
$832$	−41.5109	−1.43913
$833$	−23.9141	−0.828574
$834$	0	0
$835$	9.60403	0.332361
$836$	2.77529	0.0959853
$837$	0	0
$838$	64.7563	2.23697
$839$	−13.1616	−0.454388	−0.227194	−	0.973849i	$-0.572955\pi$
$839$	−13.1616	−0.454388	−0.227194	+	0.973849i	$0.572955\pi$
$840$	0	0
$841$	3.25998	0.112413
$842$	−11.4809	−0.395659
$843$	0	0
$844$	7.09065	0.244070
$845$	−6.97077	−0.239802
$846$	0	0
$847$	3.93972	0.135370
$848$	10.5045	0.360726
$849$	0	0
$850$	−41.5187	−1.42408
$851$	44.0885	1.51133
$852$	0	0
$853$	−36.9102	−1.26378	−0.631891	−	0.775057i	$-0.717721\pi$
$853$	−36.9102	−1.26378	−0.631891	+	0.775057i	$0.717721\pi$
$854$	−30.6844	−1.05000
$855$	0	0
$856$	−2.35218	−0.0803960
$857$	−24.3615	−0.832174	−0.416087	−	0.909325i	$-0.636599\pi$
$857$	−24.3615	−0.832174	−0.416087	+	0.909325i	$0.636599\pi$
$858$	0	0
$859$	−7.69750	−0.262635	−0.131318	−	0.991340i	$-0.541921\pi$
$859$	−7.69750	−0.262635	−0.131318	+	0.991340i	$0.541921\pi$
$860$	−89.5085	−3.05222
$861$	0	0
$862$	−39.0042	−1.32849
$863$	29.0225	0.987937	0.493969	−	0.869480i	$-0.335546\pi$
$863$	29.0225	0.987937	0.493969	+	0.869480i	$0.335546\pi$
$864$	0	0
$865$	22.0916	0.751137
$866$	−1.52177	−0.0517117
$867$	0	0
$868$	27.5384	0.934714
$869$	−8.49105	−0.288039
$870$	0	0
$871$	−32.9713	−1.11719
$872$	−0.698197	−0.0236439
$873$	0	0
$874$	15.0563	0.509286
$875$	23.9262	0.808852
$876$	0	0
$877$	−36.0010	−1.21567	−0.607834	−	0.794064i	$-0.707961\pi$
$877$	−36.0010	−1.21567	−0.607834	+	0.794064i	$0.707961\pi$
$878$	−54.2399	−1.83051
$879$	0	0
$880$	6.34128	0.213765
$881$	15.3587	0.517448	0.258724	−	0.965951i	$-0.416698\pi$
$881$	15.3587	0.517448	0.258724	+	0.965951i	$0.416698\pi$
$882$	0	0
$883$	40.2870	1.35577	0.677883	−	0.735170i	$-0.262898\pi$
$883$	40.2870	1.35577	0.677883	+	0.735170i	$0.262898\pi$
$884$	−25.7940	−0.867547
$885$	0	0
$886$	8.59728	0.288831
$887$	7.09855	0.238346	0.119173	−	0.992874i	$-0.461976\pi$
$887$	7.09855	0.238346	0.119173	+	0.992874i	$0.461976\pi$
$888$	0	0
$889$	35.7728	1.19978
$890$	54.6104	1.83054
$891$	0	0
$892$	−69.2168	−2.31755
$893$	−12.1742	−0.407395
$894$	0	0
$895$	−42.9673	−1.43624
$896$	−49.3851	−1.64984
$897$	0	0
$898$	−13.4042	−0.447302
$899$	−14.3053	−0.477110
$900$	0	0
$901$	15.9490	0.531339
$902$	1.22511	0.0407917
$903$	0	0
$904$	11.0801	0.368517
$905$	47.1783	1.56826
$906$	0	0
$907$	28.4435	0.944452	0.472226	−	0.881478i	$-0.343451\pi$
$907$	28.4435	0.944452	0.472226	+	0.881478i	$0.343451\pi$
$908$	63.3470	2.10224
$909$	0	0
$910$	97.8190	3.24267
$911$	−49.5740	−1.64246	−0.821230	−	0.570598i	$-0.806712\pi$
$911$	−49.5740	−1.64246	−0.821230	+	0.570598i	$0.806712\pi$
$912$	0	0
$913$	−5.21960	−0.172744
$914$	18.3602	0.607301
$915$	0	0
$916$	−1.67501	−0.0553440
$917$	40.9243	1.35144
$918$	0	0
$919$	17.8421	0.588556	0.294278	−	0.955720i	$-0.404921\pi$
$919$	17.8421	0.588556	0.294278	+	0.955720i	$0.404921\pi$
$920$	−40.0472	−1.32032
$921$	0	0
$922$	−56.7952	−1.87045
$923$	37.9770	1.25003
$924$	0	0
$925$	−43.3219	−1.42441
$926$	42.3334	1.39116
$927$	0	0
$928$	42.1865	1.38484
$929$	−36.0244	−1.18192	−0.590961	−	0.806700i	$-0.701251\pi$
$929$	−36.0244	−1.18192	−0.590961	+	0.806700i	$0.701251\pi$
$930$	0	0
$931$	−8.52137	−0.279277
$932$	−48.2080	−1.57911
$933$	0	0
$934$	75.7667	2.47916
$935$	9.62799	0.314869
$936$	0	0
$937$	−3.50371	−0.114461	−0.0572307	−	0.998361i	$-0.518227\pi$
$937$	−3.50371	−0.114461	−0.0572307	+	0.998361i	$0.518227\pi$
$938$	−85.7104	−2.79854
$939$	0	0
$940$	115.915	3.78074
$941$	−44.2074	−1.44112	−0.720560	−	0.693392i	$-0.756115\pi$
$941$	−44.2074	−1.44112	−0.720560	+	0.693392i	$0.756115\pi$
$942$	0	0
$943$	3.86272	0.125788
$944$	8.05385	0.262130
$945$	0	0
$946$	20.5430	0.667912
$947$	−37.9515	−1.23326	−0.616629	−	0.787254i	$-0.711502\pi$
$947$	−37.9515	−1.23326	−0.616629	+	0.787254i	$0.711502\pi$
$948$	0	0
$949$	−29.6684	−0.963077
$950$	−14.7945	−0.479996
$951$	0	0
$952$	−18.7315	−0.607090
$953$	−13.0303	−0.422094	−0.211047	−	0.977476i	$-0.567687\pi$
$953$	−13.0303	−0.422094	−0.211047	+	0.977476i	$0.567687\pi$
$954$	0	0
$955$	−10.6586	−0.344906
$956$	−20.0788	−0.649396
$957$	0	0
$958$	−50.7745	−1.64045
$959$	3.14505	0.101559
$960$	0	0
$961$	−24.6565	−0.795370
$962$	−46.3099	−1.49309
$963$	0	0
$964$	−33.8679	−1.09081
$965$	−2.56558	−0.0825889
$966$	0	0
$967$	24.0504	0.773409	0.386705	−	0.922204i	$-0.373613\pi$
$967$	24.0504	0.773409	0.386705	+	0.922204i	$0.373613\pi$
$968$	1.69419	0.0544534
$969$	0	0
$970$	79.8989	2.56540
$971$	−54.1335	−1.73723	−0.868614	−	0.495490i	$-0.834988\pi$
$971$	−54.1335	−1.73723	−0.868614	+	0.495490i	$0.834988\pi$
$972$	0	0
$973$	19.8240	0.635529
$974$	31.1791	0.999041
$975$	0	0
$976$	6.58776	0.210869
$977$	50.2507	1.60766	0.803832	−	0.594857i	$-0.202791\pi$
$977$	50.2507	1.60766	0.803832	+	0.594857i	$0.202791\pi$
$978$	0	0
$979$	−7.28423	−0.232805
$980$	81.1351	2.59177
$981$	0	0
$982$	−90.9429	−2.90210
$983$	−6.64633	−0.211985	−0.105993	−	0.994367i	$-0.533802\pi$
$983$	−6.64633	−0.211985	−0.105993	+	0.994367i	$0.533802\pi$
$984$	0	0
$985$	11.7263	0.373631
$986$	34.8318	1.10927
$987$	0	0
$988$	−9.19126	−0.292413
$989$	64.7714	2.05961
$990$	0	0
$991$	51.7993	1.64546	0.822729	−	0.568434i	$-0.192451\pi$
$991$	51.7993	1.64546	0.822729	+	0.568434i	$0.192451\pi$
$992$	−18.7072	−0.593953
$993$	0	0
$994$	98.7230	3.13130
$995$	18.4159	0.583822
$996$	0	0
$997$	−14.2042	−0.449851	−0.224925	−	0.974376i	$-0.572214\pi$
$997$	−14.2042	−0.449851	−0.224925	+	0.974376i	$0.572214\pi$
$998$	43.6315	1.38113
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.k.1.5		5
3.2	odd	2		209.2.a.c.1.1	✓	5
12.11	even	2		3344.2.a.t.1.2		5
15.14	odd	2		5225.2.a.h.1.5		5
33.32	even	2		2299.2.a.n.1.5		5
57.56	even	2		3971.2.a.h.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.1	✓	5	3.2	odd	2
1881.2.a.k.1.5		5	1.1	even	1	trivial
2299.2.a.n.1.5		5	33.32	even	2
3344.2.a.t.1.2		5	12.11	even	2
3971.2.a.h.1.5		5	57.56	even	2
5225.2.a.h.1.5		5	15.14	odd	2

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:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18524 q^{2} +2.77529 q^{4} +3.43077 q^{5} +3.93972 q^{7} +1.69419 q^{8} +7.49706 q^{10} -1.00000 q^{11} +3.31182 q^{13} +8.60924 q^{14} -1.84836 q^{16} -2.80637 q^{17} -1.00000 q^{19} +9.52137 q^{20} -2.18524 q^{22} -6.88998 q^{23} +6.77018 q^{25} +7.23713 q^{26} +10.9338 q^{28} -5.67979 q^{29} +2.51864 q^{31} -7.42749 q^{32} -6.13259 q^{34} +13.5163 q^{35} -6.39893 q^{37} -2.18524 q^{38} +5.81238 q^{40} -0.560629 q^{41} -9.40080 q^{43} -2.77529 q^{44} -15.0563 q^{46} +12.1742 q^{47} +8.52137 q^{49} +14.7945 q^{50} +9.19126 q^{52} -5.68316 q^{53} -3.43077 q^{55} +6.67463 q^{56} -12.4117 q^{58} -4.35730 q^{59} -3.56412 q^{61} +5.50384 q^{62} -12.5342 q^{64} +11.3621 q^{65} -9.95563 q^{67} -7.78847 q^{68} +29.5363 q^{70} +11.4671 q^{71} -8.95834 q^{73} -13.9832 q^{74} -2.77529 q^{76} -3.93972 q^{77} +8.49105 q^{79} -6.34128 q^{80} -1.22511 q^{82} +5.21960 q^{83} -9.62799 q^{85} -20.5430 q^{86} -1.69419 q^{88} +7.28423 q^{89} +13.0476 q^{91} -19.1217 q^{92} +26.6036 q^{94} -3.43077 q^{95} +10.6574 q^{97} +18.6213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more