LMFDB - Embedded newform 1881.2.a.k.1.1

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			1.1
Root			$2.71457$ 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)

$f(q)$	$=$	\(q-2.65432 q^{2} +5.04540 q^{4} +1.06025 q^{5} -3.36889 q^{7} -8.08346 q^{8} -2.81425 q^{10} -1.00000 q^{11} -2.15993 q^{13} +8.94209 q^{14} +11.3653 q^{16} -3.67288 q^{17} -1.00000 q^{19} +5.34940 q^{20} +2.65432 q^{22} -3.15468 q^{23} -3.87587 q^{25} +5.73313 q^{26} -16.9974 q^{28} -7.17849 q^{29} +4.65295 q^{31} -14.0001 q^{32} +9.74900 q^{34} -3.57187 q^{35} +2.27446 q^{37} +2.65432 q^{38} -8.57050 q^{40} +11.3852 q^{41} +9.38838 q^{43} -5.04540 q^{44} +8.37352 q^{46} +5.77094 q^{47} +4.34940 q^{49} +10.2878 q^{50} -10.8977 q^{52} +5.65820 q^{53} -1.06025 q^{55} +27.2322 q^{56} +19.0540 q^{58} +13.7944 q^{59} +6.98152 q^{61} -12.3504 q^{62} +14.4301 q^{64} -2.29007 q^{65} -4.81332 q^{67} -18.5312 q^{68} +9.48087 q^{70} -15.2629 q^{71} -8.08806 q^{73} -6.03713 q^{74} -5.04540 q^{76} +3.36889 q^{77} +13.4291 q^{79} +12.0500 q^{80} -30.2199 q^{82} -9.96666 q^{83} -3.89418 q^{85} -24.9197 q^{86} +8.08346 q^{88} +4.61626 q^{89} +7.27655 q^{91} -15.9166 q^{92} -15.3179 q^{94} -1.06025 q^{95} -4.09907 q^{97} -11.5447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.65432	−1.87689	−0.938443	−	0.345435i	$-0.887732\pi$
$2$	−2.65432	−1.87689	−0.938443	+	0.345435i	$0.887732\pi$
$3$	0	0
$3$	0	0
$4$	5.04540	2.52270
$5$	1.06025	0.474159	0.237080	−	0.971490i	$-0.423810\pi$
$5$	1.06025	0.474159	0.237080	+	0.971490i	$0.423810\pi$
$6$	0	0
$7$	−3.36889	−1.27332	−0.636660	−	0.771145i	$-0.719684\pi$
$7$	−3.36889	−1.27332	−0.636660	+	0.771145i	$0.719684\pi$
$8$	−8.08346	−2.85793
$9$	0	0
$10$	−2.81425	−0.889943
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.15993	−0.599056	−0.299528	−	0.954087i	$-0.596829\pi$
$13$	−2.15993	−0.599056	−0.299528	+	0.954087i	$0.596829\pi$
$14$	8.94209	2.38987
$15$	0	0
$16$	11.3653	2.84131
$17$	−3.67288	−0.890805	−0.445402	−	0.895330i	$-0.646939\pi$
$17$	−3.67288	−0.890805	−0.445402	+	0.895330i	$0.646939\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	5.34940	1.19616
$21$	0	0
$22$	2.65432	0.565902
$23$	−3.15468	−0.657797	−0.328898	−	0.944365i	$-0.606677\pi$
$23$	−3.15468	−0.657797	−0.328898	+	0.944365i	$0.606677\pi$
$24$	0	0
$25$	−3.87587	−0.775173
$26$	5.73313	1.12436
$27$	0	0
$28$	−16.9974	−3.21220
$29$	−7.17849	−1.33301	−0.666506	−	0.745499i	$-0.732211\pi$
$29$	−7.17849	−1.33301	−0.666506	+	0.745499i	$0.732211\pi$
$30$	0	0
$31$	4.65295	0.835694	0.417847	−	0.908517i	$-0.362785\pi$
$31$	4.65295	0.835694	0.417847	+	0.908517i	$0.362785\pi$
$32$	−14.0001	−2.47489
$33$	0	0
$34$	9.74900	1.67194
$35$	−3.57187	−0.603756
$36$	0	0
$37$	2.27446	0.373918	0.186959	−	0.982368i	$-0.440137\pi$
$37$	2.27446	0.373918	0.186959	+	0.982368i	$0.440137\pi$
$38$	2.65432	0.430587
$39$	0	0
$40$	−8.57050	−1.35512
$41$	11.3852	1.77807	0.889034	−	0.457841i	$-0.151377\pi$
$41$	11.3852	1.77807	0.889034	+	0.457841i	$0.151377\pi$
$42$	0	0
$43$	9.38838	1.43171	0.715857	−	0.698247i	$-0.246036\pi$
$43$	9.38838	1.43171	0.715857	+	0.698247i	$0.246036\pi$
$44$	−5.04540	−0.760623
$45$	0	0
$46$	8.37352	1.23461
$47$	5.77094	0.841778	0.420889	−	0.907112i	$-0.361718\pi$
$47$	5.77094	0.841778	0.420889	+	0.907112i	$0.361718\pi$
$48$	0	0
$49$	4.34940	0.621342
$50$	10.2878	1.45491
$51$	0	0
$52$	−10.8977	−1.51124
$53$	5.65820	0.777213	0.388607	−	0.921404i	$-0.372957\pi$
$53$	5.65820	0.777213	0.388607	+	0.921404i	$0.372957\pi$
$54$	0	0
$55$	−1.06025	−0.142964
$56$	27.2322	3.63906
$57$	0	0
$58$	19.0540	2.50191
$59$	13.7944	1.79588	0.897939	−	0.440121i	$-0.145064\pi$
$59$	13.7944	1.79588	0.897939	+	0.440121i	$0.145064\pi$
$60$	0	0
$61$	6.98152	0.893892	0.446946	−	0.894561i	$-0.352512\pi$
$61$	6.98152	0.893892	0.446946	+	0.894561i	$0.352512\pi$
$62$	−12.3504	−1.56850
$63$	0	0
$64$	14.4301	1.80377
$65$	−2.29007	−0.284048
$66$	0	0
$67$	−4.81332	−0.588041	−0.294020	−	0.955799i	$-0.594993\pi$
$67$	−4.81332	−0.588041	−0.294020	+	0.955799i	$0.594993\pi$
$68$	−18.5312	−2.24723
$69$	0	0
$70$	9.48087	1.13318
$71$	−15.2629	−1.81137	−0.905685	−	0.423951i	$-0.860643\pi$
$71$	−15.2629	−1.81137	−0.905685	+	0.423951i	$0.860643\pi$
$72$	0	0
$73$	−8.08806	−0.946636	−0.473318	−	0.880892i	$-0.656944\pi$
$73$	−8.08806	−0.946636	−0.473318	+	0.880892i	$0.656944\pi$
$74$	−6.03713	−0.701802
$75$	0	0
$76$	−5.04540	−0.578747
$77$	3.36889	0.383920
$78$	0	0
$79$	13.4291	1.51090	0.755448	−	0.655209i	$-0.227419\pi$
$79$	13.4291	1.51090	0.755448	+	0.655209i	$0.227419\pi$
$80$	12.0500	1.34724
$81$	0	0
$82$	−30.2199	−3.33723
$83$	−9.96666	−1.09398	−0.546992	−	0.837138i	$-0.684227\pi$
$83$	−9.96666	−1.09398	−0.546992	+	0.837138i	$0.684227\pi$
$84$	0	0
$85$	−3.89418	−0.422383
$86$	−24.9197	−2.68716
$87$	0	0
$88$	8.08346	0.861699
$89$	4.61626	0.489323	0.244661	−	0.969609i	$-0.421323\pi$
$89$	4.61626	0.489323	0.244661	+	0.969609i	$0.421323\pi$
$90$	0	0
$91$	7.27655	0.762790
$92$	−15.9166	−1.65942
$93$	0	0
$94$	−15.3179	−1.57992
$95$	−1.06025	−0.108780
$96$	0	0
$97$	−4.09907	−0.416197	−0.208099	−	0.978108i	$-0.566727\pi$
$97$	−4.09907	−0.416197	−0.208099	+	0.978108i	$0.566727\pi$
$98$	−11.5447	−1.16619
$99$	0	0
$100$	−19.5553	−1.95553
$101$	10.8730	1.08190	0.540950	−	0.841055i	$-0.318065\pi$
$101$	10.8730	1.08190	0.540950	+	0.841055i	$0.318065\pi$
$102$	0	0
$103$	8.85864	0.872867	0.436434	−	0.899736i	$-0.356241\pi$
$103$	8.85864	0.872867	0.436434	+	0.899736i	$0.356241\pi$
$104$	17.4597	1.71206
$105$	0	0
$106$	−15.0186	−1.45874
$107$	18.1669	1.75626	0.878131	−	0.478421i	$-0.158791\pi$
$107$	18.1669	1.75626	0.878131	+	0.478421i	$0.158791\pi$
$108$	0	0
$109$	−0.211303	−0.0202392	−0.0101196	−	0.999949i	$-0.503221\pi$
$109$	−0.211303	−0.0202392	−0.0101196	+	0.999949i	$0.503221\pi$
$110$	2.81425	0.268328
$111$	0	0
$112$	−38.2883	−3.61790
$113$	0.198345	0.0186587	0.00932935	−	0.999956i	$-0.497030\pi$
$113$	0.198345	0.0186587	0.00932935	+	0.999956i	$0.497030\pi$
$114$	0	0
$115$	−3.34476	−0.311900
$116$	−36.2184	−3.36279
$117$	0	0
$118$	−36.6147	−3.37066
$119$	12.3735	1.13428
$120$	0	0
$121$	1.00000	0.0909091
$122$	−18.5312	−1.67773
$123$	0	0
$124$	23.4760	2.10821
$125$	−9.41066	−0.841715
$126$	0	0
$127$	−3.60331	−0.319742	−0.159871	−	0.987138i	$-0.551108\pi$
$127$	−3.60331	−0.319742	−0.159871	+	0.987138i	$0.551108\pi$
$128$	−10.3020	−0.910578
$129$	0	0
$130$	6.07857	0.533126
$131$	−16.8741	−1.47430	−0.737150	−	0.675729i	$-0.763829\pi$
$131$	−16.8741	−1.47430	−0.737150	+	0.675729i	$0.763829\pi$
$132$	0	0
$133$	3.36889	0.292119
$134$	12.7761	1.10369
$135$	0	0
$136$	29.6896	2.54586
$137$	6.10839	0.521875	0.260937	−	0.965356i	$-0.415968\pi$
$137$	6.10839	0.521875	0.260937	+	0.965356i	$0.415968\pi$
$138$	0	0
$139$	11.3347	0.961398	0.480699	−	0.876886i	$-0.340383\pi$
$139$	11.3347	0.961398	0.480699	+	0.876886i	$0.340383\pi$
$140$	−18.0215	−1.52310
$141$	0	0
$142$	40.5125	3.39973
$143$	2.15993	0.180622
$144$	0	0
$145$	−7.61101	−0.632060
$146$	21.4683	1.77673
$147$	0	0
$148$	11.4755	0.943284
$149$	−6.75698	−0.553554	−0.276777	−	0.960934i	$-0.589266\pi$
$149$	−6.75698	−0.553554	−0.276777	+	0.960934i	$0.589266\pi$
$150$	0	0
$151$	6.46231	0.525895	0.262948	−	0.964810i	$-0.415305\pi$
$151$	6.46231	0.525895	0.262948	+	0.964810i	$0.415305\pi$
$152$	8.08346	0.655655
$153$	0	0
$154$	−8.94209	−0.720574
$155$	4.93330	0.396252
$156$	0	0
$157$	0.248382	0.0198230	0.00991152	−	0.999951i	$-0.496845\pi$
$157$	0.248382	0.0198230	0.00991152	+	0.999951i	$0.496845\pi$
$158$	−35.6452	−2.83578
$159$	0	0
$160$	−14.8436	−1.17349
$161$	10.6278	0.837585
$162$	0	0
$163$	−16.9710	−1.32927	−0.664637	−	0.747167i	$-0.731414\pi$
$163$	−16.9710	−1.32927	−0.664637	+	0.747167i	$0.731414\pi$
$164$	57.4428	4.48553
$165$	0	0
$166$	26.4547	2.05328
$167$	0.865538	0.0669773	0.0334887	−	0.999439i	$-0.489338\pi$
$167$	0.865538	0.0669773	0.0334887	+	0.999439i	$0.489338\pi$
$168$	0	0
$169$	−8.33471	−0.641132
$170$	10.3364	0.792765
$171$	0	0
$172$	47.3681	3.61178
$173$	4.14483	0.315125	0.157563	−	0.987509i	$-0.449636\pi$
$173$	4.14483	0.315125	0.157563	+	0.987509i	$0.449636\pi$
$174$	0	0
$175$	13.0574	0.987043
$176$	−11.3653	−0.856688
$177$	0	0
$178$	−12.2530	−0.918403
$179$	−2.04584	−0.152913	−0.0764567	−	0.997073i	$-0.524361\pi$
$179$	−2.04584	−0.152913	−0.0764567	+	0.997073i	$0.524361\pi$
$180$	0	0
$181$	5.27519	0.392101	0.196051	−	0.980594i	$-0.437188\pi$
$181$	5.27519	0.392101	0.196051	+	0.980594i	$0.437188\pi$
$182$	−19.3143	−1.43167
$183$	0	0
$184$	25.5007	1.87994
$185$	2.41150	0.177297
$186$	0	0
$187$	3.67288	0.268588
$188$	29.1167	2.12355
$189$	0	0
$190$	2.81425	0.204167
$191$	16.1899	1.17146	0.585729	−	0.810507i	$-0.300808\pi$
$191$	16.1899	1.17146	0.585729	+	0.810507i	$0.300808\pi$
$192$	0	0
$193$	−20.0620	−1.44409	−0.722045	−	0.691846i	$-0.756798\pi$
$193$	−20.0620	−1.44409	−0.722045	+	0.691846i	$0.756798\pi$
$194$	10.8802	0.781155
$195$	0	0
$196$	21.9444	1.56746
$197$	23.8398	1.69851	0.849257	−	0.527979i	$-0.177050\pi$
$197$	23.8398	1.69851	0.849257	+	0.527979i	$0.177050\pi$
$198$	0	0
$199$	−1.90194	−0.134825	−0.0674125	−	0.997725i	$-0.521474\pi$
$199$	−1.90194	−0.134825	−0.0674125	+	0.997725i	$0.521474\pi$
$200$	31.3304	2.21539
$201$	0	0
$202$	−28.8603	−2.03060
$203$	24.1835	1.69735
$204$	0	0
$205$	12.0712	0.843087
$206$	−23.5136	−1.63827
$207$	0	0
$208$	−24.5481	−1.70211
$209$	1.00000	0.0691714
$210$	0	0
$211$	12.3048	0.847100	0.423550	−	0.905873i	$-0.360784\pi$
$211$	12.3048	0.847100	0.423550	+	0.905873i	$0.360784\pi$
$212$	28.5479	1.96068
$213$	0	0
$214$	−48.2207	−3.29630
$215$	9.95405	0.678860
$216$	0	0
$217$	−15.6753	−1.06411
$218$	0.560865	0.0379866
$219$	0	0
$220$	−5.34940	−0.360656
$221$	7.93316	0.533642
$222$	0	0
$223$	22.8473	1.52997	0.764985	−	0.644048i	$-0.222746\pi$
$223$	22.8473	1.52997	0.764985	+	0.644048i	$0.222746\pi$
$224$	47.1647	3.15132
$225$	0	0
$226$	−0.526470	−0.0350202
$227$	−4.00654	−0.265923	−0.132962	−	0.991121i	$-0.542449\pi$
$227$	−4.00654	−0.265923	−0.132962	+	0.991121i	$0.542449\pi$
$228$	0	0
$229$	−24.6024	−1.62577	−0.812887	−	0.582422i	$-0.802105\pi$
$229$	−24.6024	−1.62577	−0.812887	+	0.582422i	$0.802105\pi$
$230$	8.87805	0.585401
$231$	0	0
$232$	58.0270	3.80966
$233$	−16.2994	−1.06781	−0.533903	−	0.845545i	$-0.679275\pi$
$233$	−16.2994	−1.06781	−0.533903	+	0.845545i	$0.679275\pi$
$234$	0	0
$235$	6.11865	0.399137
$236$	69.5982	4.53046
$237$	0	0
$238$	−32.8433	−2.12891
$239$	15.4182	0.997324	0.498662	−	0.866797i	$-0.333825\pi$
$239$	15.4182	0.997324	0.498662	+	0.866797i	$0.333825\pi$
$240$	0	0
$241$	24.1441	1.55526	0.777629	−	0.628723i	$-0.216422\pi$
$241$	24.1441	1.55526	0.777629	+	0.628723i	$0.216422\pi$
$242$	−2.65432	−0.170626
$243$	0	0
$244$	35.2245	2.25502
$245$	4.61146	0.294615
$246$	0	0
$247$	2.15993	0.137433
$248$	−37.6119	−2.38836
$249$	0	0
$250$	24.9789	1.57980
$251$	19.9099	1.25670	0.628350	−	0.777931i	$-0.283731\pi$
$251$	19.9099	1.25670	0.628350	+	0.777931i	$0.283731\pi$
$252$	0	0
$253$	3.15468	0.198333
$254$	9.56433	0.600119
$255$	0	0
$256$	−1.51547	−0.0947166
$257$	22.3126	1.39182	0.695911	−	0.718128i	$-0.255001\pi$
$257$	22.3126	1.39182	0.695911	+	0.718128i	$0.255001\pi$
$258$	0	0
$259$	−7.66239	−0.476118
$260$	−11.5543	−0.716568
$261$	0	0
$262$	44.7893	2.76709
$263$	−13.7963	−0.850716	−0.425358	−	0.905025i	$-0.639852\pi$
$263$	−13.7963	−0.850716	−0.425358	+	0.905025i	$0.639852\pi$
$264$	0	0
$265$	5.99912	0.368523
$266$	−8.94209	−0.548275
$267$	0	0
$268$	−24.2851	−1.48345
$269$	11.2028	0.683048	0.341524	−	0.939873i	$-0.389057\pi$
$269$	11.2028	0.683048	0.341524	+	0.939873i	$0.389057\pi$
$270$	0	0
$271$	17.1234	1.04017	0.520085	−	0.854115i	$-0.325900\pi$
$271$	17.1234	1.04017	0.520085	+	0.854115i	$0.325900\pi$
$272$	−41.7433	−2.53106
$273$	0	0
$274$	−16.2136	−0.979499
$275$	3.87587	0.233723
$276$	0	0
$277$	−13.7127	−0.823919	−0.411960	−	0.911202i	$-0.635156\pi$
$277$	−13.7127	−0.823919	−0.411960	+	0.911202i	$0.635156\pi$
$278$	−30.0859	−1.80443
$279$	0	0
$280$	28.8730	1.72549
$281$	−25.8599	−1.54267	−0.771337	−	0.636427i	$-0.780412\pi$
$281$	−25.8599	−1.54267	−0.771337	+	0.636427i	$0.780412\pi$
$282$	0	0
$283$	28.8711	1.71621	0.858103	−	0.513477i	$-0.171643\pi$
$283$	28.8711	1.71621	0.858103	+	0.513477i	$0.171643\pi$
$284$	−77.0073	−4.56954
$285$	0	0
$286$	−5.73313	−0.339007
$287$	−38.3554	−2.26405
$288$	0	0
$289$	−3.50993	−0.206467
$290$	20.2020	1.18630
$291$	0	0
$292$	−40.8075	−2.38808
$293$	−6.37971	−0.372707	−0.186353	−	0.982483i	$-0.559667\pi$
$293$	−6.37971	−0.372707	−0.186353	+	0.982483i	$0.559667\pi$
$294$	0	0
$295$	14.6255	0.851532
$296$	−18.3855	−1.06863
$297$	0	0
$298$	17.9352	1.03896
$299$	6.81389	0.394057
$300$	0	0
$301$	−31.6284	−1.82303
$302$	−17.1530	−0.987045
$303$	0	0
$304$	−11.3653	−0.651842
$305$	7.40217	0.423847
$306$	0	0
$307$	10.4299	0.595264	0.297632	−	0.954681i	$-0.403803\pi$
$307$	10.4299	0.595264	0.297632	+	0.954681i	$0.403803\pi$
$308$	16.9974	0.968515
$309$	0	0
$310$	−13.0945	−0.743720
$311$	8.26224	0.468508	0.234254	−	0.972175i	$-0.424735\pi$
$311$	8.26224	0.468508	0.234254	+	0.972175i	$0.424735\pi$
$312$	0	0
$313$	−24.1167	−1.36315	−0.681577	−	0.731746i	$-0.738706\pi$
$313$	−24.1167	−1.36315	−0.681577	+	0.731746i	$0.738706\pi$
$314$	−0.659285	−0.0372056
$315$	0	0
$316$	67.7554	3.81154
$317$	11.7330	0.658989	0.329495	−	0.944157i	$-0.393122\pi$
$317$	11.7330	0.658989	0.329495	+	0.944157i	$0.393122\pi$
$318$	0	0
$319$	7.17849	0.401918
$320$	15.2996	0.855273
$321$	0	0
$322$	−28.2095	−1.57205
$323$	3.67288	0.204365
$324$	0	0
$325$	8.37159	0.464372
$326$	45.0465	2.49489
$327$	0	0
$328$	−92.0317	−5.08160
$329$	−19.4416	−1.07185
$330$	0	0
$331$	−6.95223	−0.382129	−0.191065	−	0.981577i	$-0.561194\pi$
$331$	−6.95223	−0.382129	−0.191065	+	0.981577i	$0.561194\pi$
$332$	−50.2858	−2.75979
$333$	0	0
$334$	−2.29741	−0.125709
$335$	−5.10333	−0.278825
$336$	0	0
$337$	29.4416	1.60379	0.801893	−	0.597468i	$-0.203826\pi$
$337$	29.4416	1.60379	0.801893	+	0.597468i	$0.203826\pi$
$338$	22.1230	1.20333
$339$	0	0
$340$	−19.6477	−1.06555
$341$	−4.65295	−0.251971
$342$	0	0
$343$	8.92959	0.482152
$344$	−75.8905	−4.09174
$345$	0	0
$346$	−11.0017	−0.591454
$347$	17.2180	0.924311	0.462155	−	0.886799i	$-0.347076\pi$
$347$	17.2180	0.924311	0.462155	+	0.886799i	$0.347076\pi$
$348$	0	0
$349$	4.32405	0.231461	0.115731	−	0.993281i	$-0.463079\pi$
$349$	4.32405	0.231461	0.115731	+	0.993281i	$0.463079\pi$
$350$	−34.6583	−1.85257
$351$	0	0
$352$	14.0001	0.746207
$353$	−23.4421	−1.24770	−0.623848	−	0.781545i	$-0.714432\pi$
$353$	−23.4421	−1.24770	−0.623848	+	0.781545i	$0.714432\pi$
$354$	0	0
$355$	−16.1825	−0.858878
$356$	23.2909	1.23441
$357$	0	0
$358$	5.43031	0.287001
$359$	−8.20544	−0.433066	−0.216533	−	0.976275i	$-0.569475\pi$
$359$	−8.20544	−0.433066	−0.216533	+	0.976275i	$0.569475\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.0020	−0.735930
$363$	0	0
$364$	36.7131	1.92429
$365$	−8.57539	−0.448856
$366$	0	0
$367$	33.1494	1.73039	0.865193	−	0.501439i	$-0.167196\pi$
$367$	33.1494	1.73039	0.865193	+	0.501439i	$0.167196\pi$
$368$	−35.8538	−1.86901
$369$	0	0
$370$	−6.40088	−0.332766
$371$	−19.0618	−0.989640
$372$	0	0
$373$	−16.5840	−0.858685	−0.429343	−	0.903142i	$-0.641255\pi$
$373$	−16.5840	−0.858685	−0.429343	+	0.903142i	$0.641255\pi$
$374$	−9.74900	−0.504109
$375$	0	0
$376$	−46.6492	−2.40575
$377$	15.5050	0.798550
$378$	0	0
$379$	5.79467	0.297652	0.148826	−	0.988863i	$-0.452451\pi$
$379$	5.79467	0.297652	0.148826	+	0.988863i	$0.452451\pi$
$380$	−5.34940	−0.274418
$381$	0	0
$382$	−42.9730	−2.19869
$383$	−5.89016	−0.300973	−0.150487	−	0.988612i	$-0.548084\pi$
$383$	−5.89016	−0.300973	−0.150487	+	0.988612i	$0.548084\pi$
$384$	0	0
$385$	3.57187	0.182039
$386$	53.2508	2.71039
$387$	0	0
$388$	−20.6814	−1.04994
$389$	−13.8055	−0.699964	−0.349982	−	0.936756i	$-0.613812\pi$
$389$	−13.8055	−0.699964	−0.349982	+	0.936756i	$0.613812\pi$
$390$	0	0
$391$	11.5868	0.585968
$392$	−35.1581	−1.77575
$393$	0	0
$394$	−63.2784	−3.18792
$395$	14.2383	0.716405
$396$	0	0
$397$	6.48471	0.325458	0.162729	−	0.986671i	$-0.447970\pi$
$397$	6.48471	0.325458	0.162729	+	0.986671i	$0.447970\pi$
$398$	5.04835	0.253051
$399$	0	0
$400$	−44.0502	−2.20251
$401$	−3.70806	−0.185172	−0.0925858	−	0.995705i	$-0.529513\pi$
$401$	−3.70806	−0.185172	−0.0925858	+	0.995705i	$0.529513\pi$
$402$	0	0
$403$	−10.0500	−0.500628
$404$	54.8584	2.72931
$405$	0	0
$406$	−64.1908	−3.18573
$407$	−2.27446	−0.112741
$408$	0	0
$409$	31.0569	1.53567	0.767833	−	0.640650i	$-0.221335\pi$
$409$	31.0569	1.53567	0.767833	+	0.640650i	$0.221335\pi$
$410$	−32.0407	−1.58238
$411$	0	0
$412$	44.6954	2.20198
$413$	−46.4717	−2.28673
$414$	0	0
$415$	−10.5672	−0.518722
$416$	30.2392	1.48260
$417$	0	0
$418$	−2.65432	−0.129827
$419$	11.2871	0.551410	0.275705	−	0.961242i	$-0.411089\pi$
$419$	11.2871	0.551410	0.275705	+	0.961242i	$0.411089\pi$
$420$	0	0
$421$	34.5194	1.68237	0.841186	−	0.540746i	$-0.181858\pi$
$421$	34.5194	1.68237	0.841186	+	0.540746i	$0.181858\pi$
$422$	−32.6609	−1.58991
$423$	0	0
$424$	−45.7378	−2.22122
$425$	14.2356	0.690528
$426$	0	0
$427$	−23.5199	−1.13821
$428$	91.6593	4.43052
$429$	0	0
$430$	−26.4212	−1.27414
$431$	−18.1654	−0.874996	−0.437498	−	0.899219i	$-0.644135\pi$
$431$	−18.1654	−0.874996	−0.437498	+	0.899219i	$0.644135\pi$
$432$	0	0
$433$	10.7804	0.518075	0.259037	−	0.965867i	$-0.416595\pi$
$433$	10.7804	0.518075	0.259037	+	0.965867i	$0.416595\pi$
$434$	41.6071	1.99720
$435$	0	0
$436$	−1.06611	−0.0510573
$437$	3.15468	0.150909
$438$	0	0
$439$	−1.38742	−0.0662180	−0.0331090	−	0.999452i	$-0.510541\pi$
$439$	−1.38742	−0.0662180	−0.0331090	+	0.999452i	$0.510541\pi$
$440$	8.57050	0.408583
$441$	0	0
$442$	−21.0571	−1.00159
$443$	7.26792	0.345310	0.172655	−	0.984982i	$-0.444766\pi$
$443$	7.26792	0.345310	0.172655	+	0.984982i	$0.444766\pi$
$444$	0	0
$445$	4.89440	0.232017
$446$	−60.6441	−2.87158
$447$	0	0
$448$	−48.6135	−2.29677
$449$	−21.4298	−1.01134	−0.505668	−	0.862728i	$-0.668754\pi$
$449$	−21.4298	−1.01134	−0.505668	+	0.862728i	$0.668754\pi$
$450$	0	0
$451$	−11.3852	−0.536108
$452$	1.00073	0.0470703
$453$	0	0
$454$	10.6346	0.499108
$455$	7.71498	0.361684
$456$	0	0
$457$	−17.6312	−0.824754	−0.412377	−	0.911013i	$-0.635301\pi$
$457$	−17.6312	−0.824754	−0.412377	+	0.911013i	$0.635301\pi$
$458$	65.3026	3.05139
$459$	0	0
$460$	−16.8756	−0.786831
$461$	−34.2073	−1.59320	−0.796598	−	0.604510i	$-0.793369\pi$
$461$	−34.2073	−1.59320	−0.796598	+	0.604510i	$0.793369\pi$
$462$	0	0
$463$	−11.0009	−0.511253	−0.255626	−	0.966776i	$-0.582282\pi$
$463$	−11.0009	−0.511253	−0.255626	+	0.966776i	$0.582282\pi$
$464$	−81.5854	−3.78751
$465$	0	0
$466$	43.2637	2.00415
$467$	−4.38453	−0.202892	−0.101446	−	0.994841i	$-0.532347\pi$
$467$	−4.38453	−0.202892	−0.101446	+	0.994841i	$0.532347\pi$
$468$	0	0
$469$	16.2155	0.748764
$470$	−16.2408	−0.749134
$471$	0	0
$472$	−111.506	−5.13250
$473$	−9.38838	−0.431678
$474$	0	0
$475$	3.87587	0.177837
$476$	62.4294	2.86145
$477$	0	0
$478$	−40.9249	−1.87186
$479$	14.3683	0.656505	0.328253	−	0.944590i	$-0.393540\pi$
$479$	14.3683	0.656505	0.328253	+	0.944590i	$0.393540\pi$
$480$	0	0
$481$	−4.91266	−0.223998
$482$	−64.0861	−2.91904
$483$	0	0
$484$	5.04540	0.229336
$485$	−4.34604	−0.197344
$486$	0	0
$487$	1.01712	0.0460899	0.0230449	−	0.999734i	$-0.492664\pi$
$487$	1.01712	0.0460899	0.0230449	+	0.999734i	$0.492664\pi$
$488$	−56.4348	−2.55468
$489$	0	0
$490$	−12.2403	−0.552959
$491$	2.41045	0.108782	0.0543910	−	0.998520i	$-0.482678\pi$
$491$	2.41045	0.108782	0.0543910	+	0.998520i	$0.482678\pi$
$492$	0	0
$493$	26.3658	1.18745
$494$	−5.73313	−0.257946
$495$	0	0
$496$	52.8820	2.37447
$497$	51.4189	2.30645
$498$	0	0
$499$	4.07426	0.182389	0.0911945	−	0.995833i	$-0.470931\pi$
$499$	4.07426	0.182389	0.0911945	+	0.995833i	$0.470931\pi$
$500$	−47.4805	−2.12339
$501$	0	0
$502$	−52.8471	−2.35868
$503$	19.9553	0.889762	0.444881	−	0.895590i	$-0.353246\pi$
$503$	19.9553	0.889762	0.444881	+	0.895590i	$0.353246\pi$
$504$	0	0
$505$	11.5281	0.512993
$506$	−8.37352	−0.372249
$507$	0	0
$508$	−18.1801	−0.806613
$509$	8.79239	0.389716	0.194858	−	0.980831i	$-0.437575\pi$
$509$	8.79239	0.389716	0.194858	+	0.980831i	$0.437575\pi$
$510$	0	0
$511$	27.2478	1.20537
$512$	24.6266	1.08835
$513$	0	0
$514$	−59.2247	−2.61229
$515$	9.39239	0.413878
$516$	0	0
$517$	−5.77094	−0.253806
$518$	20.3384	0.893618
$519$	0	0
$520$	18.5117	0.811790
$521$	−27.3483	−1.19815	−0.599075	−	0.800693i	$-0.704465\pi$
$521$	−27.3483	−1.19815	−0.599075	+	0.800693i	$0.704465\pi$
$522$	0	0
$523$	−41.7043	−1.82360	−0.911800	−	0.410634i	$-0.865307\pi$
$523$	−41.7043	−1.82360	−0.911800	+	0.410634i	$0.865307\pi$
$524$	−85.1368	−3.71922
$525$	0	0
$526$	36.6197	1.59670
$527$	−17.0897	−0.744441
$528$	0	0
$529$	−13.0480	−0.567304
$530$	−15.9236	−0.691675
$531$	0	0
$532$	16.9974	0.736930
$533$	−24.5912	−1.06516
$534$	0	0
$535$	19.2615	0.832748
$536$	38.9083	1.68058
$537$	0	0
$538$	−29.7358	−1.28200
$539$	−4.34940	−0.187342
$540$	0	0
$541$	−20.7584	−0.892472	−0.446236	−	0.894915i	$-0.647236\pi$
$541$	−20.7584	−0.892472	−0.446236	+	0.894915i	$0.647236\pi$
$542$	−45.4508	−1.95228
$543$	0	0
$544$	51.4207	2.20464
$545$	−0.224035	−0.00959659
$546$	0	0
$547$	5.39396	0.230629	0.115315	−	0.993329i	$-0.463212\pi$
$547$	5.39396	0.230629	0.115315	+	0.993329i	$0.463212\pi$
$548$	30.8193	1.31653
$549$	0	0
$550$	−10.2878	−0.438672
$551$	7.17849	0.305814
$552$	0	0
$553$	−45.2412	−1.92385
$554$	36.3980	1.54640
$555$	0	0
$556$	57.1881	2.42532
$557$	38.0076	1.61043	0.805217	−	0.592981i	$-0.202049\pi$
$557$	38.0076	1.61043	0.805217	+	0.592981i	$0.202049\pi$
$558$	0	0
$559$	−20.2782	−0.857677
$560$	−40.5952	−1.71546
$561$	0	0
$562$	68.6404	2.89542
$563$	26.4313	1.11395	0.556973	−	0.830530i	$-0.311963\pi$
$563$	26.4313	1.11395	0.556973	+	0.830530i	$0.311963\pi$
$564$	0	0
$565$	0.210295	0.00884719
$566$	−76.6330	−3.22112
$567$	0	0
$568$	123.377	5.17677
$569$	−40.6105	−1.70248	−0.851239	−	0.524778i	$-0.824148\pi$
$569$	−40.6105	−1.70248	−0.851239	+	0.524778i	$0.824148\pi$
$570$	0	0
$571$	8.44299	0.353328	0.176664	−	0.984271i	$-0.443469\pi$
$571$	8.44299	0.353328	0.176664	+	0.984271i	$0.443469\pi$
$572$	10.8977	0.455656
$573$	0	0
$574$	101.807	4.24936
$575$	12.2271	0.509906
$576$	0	0
$577$	20.9771	0.873288	0.436644	−	0.899634i	$-0.356167\pi$
$577$	20.9771	0.873288	0.436644	+	0.899634i	$0.356167\pi$
$578$	9.31648	0.387514
$579$	0	0
$580$	−38.4006	−1.59450
$581$	33.5766	1.39299
$582$	0	0
$583$	−5.65820	−0.234339
$584$	65.3795	2.70542
$585$	0	0
$586$	16.9338	0.699528
$587$	21.1452	0.872756	0.436378	−	0.899764i	$-0.356261\pi$
$587$	21.1452	0.872756	0.436378	+	0.899764i	$0.356261\pi$
$588$	0	0
$589$	−4.65295	−0.191721
$590$	−38.8208	−1.59823
$591$	0	0
$592$	25.8498	1.06242
$593$	−27.7731	−1.14051	−0.570253	−	0.821469i	$-0.693155\pi$
$593$	−27.7731	−1.14051	−0.570253	+	0.821469i	$0.693155\pi$
$594$	0	0
$595$	13.1191	0.537829
$596$	−34.0917	−1.39645
$597$	0	0
$598$	−18.0862	−0.739600
$599$	−23.0129	−0.940281	−0.470140	−	0.882592i	$-0.655797\pi$
$599$	−23.0129	−0.940281	−0.470140	+	0.882592i	$0.655797\pi$
$600$	0	0
$601$	23.3322	0.951738	0.475869	−	0.879516i	$-0.342134\pi$
$601$	23.3322	0.951738	0.475869	+	0.879516i	$0.342134\pi$
$602$	83.9517	3.42162
$603$	0	0
$604$	32.6049	1.32668
$605$	1.06025	0.0431054
$606$	0	0
$607$	−3.22857	−0.131044	−0.0655218	−	0.997851i	$-0.520871\pi$
$607$	−3.22857	−0.131044	−0.0655218	+	0.997851i	$0.520871\pi$
$608$	14.0001	0.567778
$609$	0	0
$610$	−19.6477	−0.795512
$611$	−12.4648	−0.504273
$612$	0	0
$613$	−17.2202	−0.695519	−0.347759	−	0.937584i	$-0.613057\pi$
$613$	−17.2202	−0.695519	−0.347759	+	0.937584i	$0.613057\pi$
$614$	−27.6842	−1.11724
$615$	0	0
$616$	−27.2322	−1.09722
$617$	27.4776	1.10621	0.553103	−	0.833113i	$-0.313443\pi$
$617$	27.4776	1.10621	0.553103	+	0.833113i	$0.313443\pi$
$618$	0	0
$619$	−9.53870	−0.383393	−0.191696	−	0.981454i	$-0.561399\pi$
$619$	−9.53870	−0.383393	−0.191696	+	0.981454i	$0.561399\pi$
$620$	24.8905	0.999625
$621$	0	0
$622$	−21.9306	−0.879337
$623$	−15.5517	−0.623064
$624$	0	0
$625$	9.40166	0.376066
$626$	64.0133	2.55848
$627$	0	0
$628$	1.25319	0.0500076
$629$	−8.35381	−0.333088
$630$	0	0
$631$	1.16647	0.0464363	0.0232182	−	0.999730i	$-0.492609\pi$
$631$	1.16647	0.0464363	0.0232182	+	0.999730i	$0.492609\pi$
$632$	−108.554	−4.31804
$633$	0	0
$634$	−31.1430	−1.23685
$635$	−3.82042	−0.151609
$636$	0	0
$637$	−9.39438	−0.372219
$638$	−19.0540	−0.754355
$639$	0	0
$640$	−10.9227	−0.431759
$641$	−10.0249	−0.395960	−0.197980	−	0.980206i	$-0.563438\pi$
$641$	−10.0249	−0.395960	−0.197980	+	0.980206i	$0.563438\pi$
$642$	0	0
$643$	−40.7197	−1.60583	−0.802915	−	0.596094i	$-0.796719\pi$
$643$	−40.7197	−1.60583	−0.802915	+	0.596094i	$0.796719\pi$
$644$	53.6213	2.11298
$645$	0	0
$646$	−9.74900	−0.383569
$647$	16.6460	0.654423	0.327211	−	0.944951i	$-0.393891\pi$
$647$	16.6460	0.654423	0.327211	+	0.944951i	$0.393891\pi$
$648$	0	0
$649$	−13.7944	−0.541477
$650$	−22.2209	−0.871574
$651$	0	0
$652$	−85.6256	−3.35336
$653$	34.7416	1.35954	0.679771	−	0.733424i	$-0.262079\pi$
$653$	34.7416	1.35954	0.679771	+	0.733424i	$0.262079\pi$
$654$	0	0
$655$	−17.8908	−0.699053
$656$	129.396	5.05205
$657$	0	0
$658$	51.6043	2.01175
$659$	22.9537	0.894148	0.447074	−	0.894497i	$-0.352466\pi$
$659$	22.9537	0.894148	0.447074	+	0.894497i	$0.352466\pi$
$660$	0	0
$661$	−22.7141	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$661$	−22.7141	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$662$	18.4534	0.717213
$663$	0	0
$664$	80.5651	3.12653
$665$	3.57187	0.138511
$666$	0	0
$667$	22.6459	0.876851
$668$	4.36698	0.168964
$669$	0	0
$670$	13.5459	0.523322
$671$	−6.98152	−0.269518
$672$	0	0
$673$	20.9448	0.807364	0.403682	−	0.914899i	$-0.367730\pi$
$673$	20.9448	0.807364	0.403682	+	0.914899i	$0.367730\pi$
$674$	−78.1473	−3.01012
$675$	0	0
$676$	−42.0519	−1.61738
$677$	−11.4153	−0.438726	−0.219363	−	0.975643i	$-0.570398\pi$
$677$	−11.4153	−0.438726	−0.219363	+	0.975643i	$0.570398\pi$
$678$	0	0
$679$	13.8093	0.529952
$680$	31.4784	1.20714
$681$	0	0
$682$	12.3504	0.472921
$683$	34.9550	1.33752	0.668759	−	0.743479i	$-0.266826\pi$
$683$	34.9550	1.33752	0.668759	+	0.743479i	$0.266826\pi$
$684$	0	0
$685$	6.47643	0.247452
$686$	−23.7020	−0.904945
$687$	0	0
$688$	106.701	4.06795
$689$	−12.2213	−0.465594
$690$	0	0
$691$	−15.8717	−0.603789	−0.301894	−	0.953341i	$-0.597619\pi$
$691$	−15.8717	−0.603789	−0.301894	+	0.953341i	$0.597619\pi$
$692$	20.9123	0.794967
$693$	0	0
$694$	−45.7020	−1.73483
$695$	12.0177	0.455855
$696$	0	0
$697$	−41.8165	−1.58391
$698$	−11.4774	−0.434426
$699$	0	0
$700$	65.8795	2.49001
$701$	18.5830	0.701869	0.350935	−	0.936400i	$-0.385864\pi$
$701$	18.5830	0.701869	0.350935	+	0.936400i	$0.385864\pi$
$702$	0	0
$703$	−2.27446	−0.0857828
$704$	−14.4301	−0.543857
$705$	0	0
$706$	62.2228	2.34178
$707$	−36.6298	−1.37760
$708$	0	0
$709$	23.8063	0.894064	0.447032	−	0.894518i	$-0.352481\pi$
$709$	23.8063	0.894064	0.447032	+	0.894518i	$0.352481\pi$
$710$	42.9535	1.61202
$711$	0	0
$712$	−37.3153	−1.39845
$713$	−14.6786	−0.549717
$714$	0	0
$715$	2.29007	0.0856437
$716$	−10.3221	−0.385755
$717$	0	0
$718$	21.7798	0.812816
$719$	−37.0144	−1.38040	−0.690202	−	0.723616i	$-0.742478\pi$
$719$	−37.0144	−1.38040	−0.690202	+	0.723616i	$0.742478\pi$
$720$	0	0
$721$	−29.8437	−1.11144
$722$	−2.65432	−0.0987835
$723$	0	0
$724$	26.6154	0.989154
$725$	27.8229	1.03332
$726$	0	0
$727$	−22.0744	−0.818694	−0.409347	−	0.912379i	$-0.634243\pi$
$727$	−22.0744	−0.818694	−0.409347	+	0.912379i	$0.634243\pi$
$728$	−58.8197	−2.18000
$729$	0	0
$730$	22.7618	0.842452
$731$	−34.4824	−1.27538
$732$	0	0
$733$	−18.7090	−0.691031	−0.345515	−	0.938413i	$-0.612296\pi$
$733$	−18.7090	−0.691031	−0.345515	+	0.938413i	$0.612296\pi$
$734$	−87.9891	−3.24774
$735$	0	0
$736$	44.1658	1.62797
$737$	4.81332	0.177301
$738$	0	0
$739$	−5.91630	−0.217635	−0.108817	−	0.994062i	$-0.534706\pi$
$739$	−5.91630	−0.217635	−0.108817	+	0.994062i	$0.534706\pi$
$740$	12.1670	0.447267
$741$	0	0
$742$	50.5961	1.85744
$743$	−14.0422	−0.515159	−0.257579	−	0.966257i	$-0.582925\pi$
$743$	−14.0422	−0.515159	−0.257579	+	0.966257i	$0.582925\pi$
$744$	0	0
$745$	−7.16411	−0.262473
$746$	44.0191	1.61165
$747$	0	0
$748$	18.5312	0.677566
$749$	−61.2023	−2.23628
$750$	0	0
$751$	4.91166	0.179229	0.0896145	−	0.995977i	$-0.471437\pi$
$751$	4.91166	0.179229	0.0896145	+	0.995977i	$0.471437\pi$
$752$	65.5882	2.39176
$753$	0	0
$754$	−41.1553	−1.49879
$755$	6.85168	0.249358
$756$	0	0
$757$	49.2829	1.79122	0.895609	−	0.444843i	$-0.146741\pi$
$757$	49.2829	1.79122	0.895609	+	0.444843i	$0.146741\pi$
$758$	−15.3809	−0.558659
$759$	0	0
$760$	8.57050	0.310885
$761$	7.25922	0.263146	0.131573	−	0.991306i	$-0.457997\pi$
$761$	7.25922	0.263146	0.131573	+	0.991306i	$0.457997\pi$
$762$	0	0
$763$	0.711856	0.0257709
$764$	81.6843	2.95524
$765$	0	0
$766$	15.6344	0.564892
$767$	−29.7949	−1.07583
$768$	0	0
$769$	−21.5123	−0.775755	−0.387877	−	0.921711i	$-0.626792\pi$
$769$	−21.5123	−0.775755	−0.387877	+	0.921711i	$0.626792\pi$
$770$	−9.48087	−0.341667
$771$	0	0
$772$	−101.221	−3.64301
$773$	−4.54038	−0.163306	−0.0816530	−	0.996661i	$-0.526020\pi$
$773$	−4.54038	−0.163306	−0.0816530	+	0.996661i	$0.526020\pi$
$774$	0	0
$775$	−18.0342	−0.647808
$776$	33.1346	1.18946
$777$	0	0
$778$	36.6440	1.31375
$779$	−11.3852	−0.407917
$780$	0	0
$781$	15.2629	0.546149
$782$	−30.7550	−1.09980
$783$	0	0
$784$	49.4320	1.76543
$785$	0.263348	0.00939928
$786$	0	0
$787$	−7.25028	−0.258444	−0.129222	−	0.991616i	$-0.541248\pi$
$787$	−7.25028	−0.258444	−0.129222	+	0.991616i	$0.541248\pi$
$788$	120.281	4.28484
$789$	0	0
$790$	−37.7929	−1.34461
$791$	−0.668201	−0.0237585
$792$	0	0
$793$	−15.0796	−0.535491
$794$	−17.2125	−0.610848
$795$	0	0
$796$	−9.59605	−0.340123
$797$	−18.8674	−0.668319	−0.334159	−	0.942517i	$-0.608452\pi$
$797$	−18.8674	−0.668319	−0.334159	+	0.942517i	$0.608452\pi$
$798$	0	0
$799$	−21.1960	−0.749860
$800$	54.2624	1.91847
$801$	0	0
$802$	9.84236	0.347546
$803$	8.08806	0.285422
$804$	0	0
$805$	11.2681	0.397149
$806$	26.6760	0.939622
$807$	0	0
$808$	−87.8911	−3.09200
$809$	19.4948	0.685399	0.342700	−	0.939445i	$-0.388659\pi$
$809$	19.4948	0.685399	0.342700	+	0.939445i	$0.388659\pi$
$810$	0	0
$811$	4.28635	0.150514	0.0752570	−	0.997164i	$-0.476022\pi$
$811$	4.28635	0.150514	0.0752570	+	0.997164i	$0.476022\pi$
$812$	122.016	4.28191
$813$	0	0
$814$	6.03713	0.211601
$815$	−17.9936	−0.630287
$816$	0	0
$817$	−9.38838	−0.328458
$818$	−82.4350	−2.88227
$819$	0	0
$820$	60.9039	2.12686
$821$	−5.37419	−0.187561	−0.0937803	−	0.995593i	$-0.529895\pi$
$821$	−5.37419	−0.187561	−0.0937803	+	0.995593i	$0.529895\pi$
$822$	0	0
$823$	26.3196	0.917445	0.458723	−	0.888580i	$-0.348307\pi$
$823$	26.3196	0.917445	0.458723	+	0.888580i	$0.348307\pi$
$824$	−71.6084	−2.49460
$825$	0	0
$826$	123.351	4.29192
$827$	8.11186	0.282077	0.141039	−	0.990004i	$-0.454956\pi$
$827$	8.11186	0.282077	0.141039	+	0.990004i	$0.454956\pi$
$828$	0	0
$829$	−21.5948	−0.750017	−0.375009	−	0.927021i	$-0.622360\pi$
$829$	−21.5948	−0.750017	−0.375009	+	0.927021i	$0.622360\pi$
$830$	28.0486	0.973582
$831$	0	0
$832$	−31.1681	−1.08056
$833$	−15.9748	−0.553495
$834$	0	0
$835$	0.917688	0.0317579
$836$	5.04540	0.174499
$837$	0	0
$838$	−29.9595	−1.03493
$839$	4.35109	0.150216	0.0751082	−	0.997175i	$-0.476070\pi$
$839$	4.35109	0.150216	0.0751082	+	0.997175i	$0.476070\pi$
$840$	0	0
$841$	22.5308	0.776923
$842$	−91.6254	−3.15762
$843$	0	0
$844$	62.0828	2.13698
$845$	−8.83689	−0.303998
$846$	0	0
$847$	−3.36889	−0.115756
$848$	64.3069	2.20831
$849$	0	0
$850$	−37.7858	−1.29604
$851$	−7.17519	−0.245962
$852$	0	0
$853$	19.5022	0.667744	0.333872	−	0.942618i	$-0.391645\pi$
$853$	19.5022	0.667744	0.333872	+	0.942618i	$0.391645\pi$
$854$	62.4294	2.13629
$855$	0	0
$856$	−146.851	−5.01928
$857$	−13.3462	−0.455898	−0.227949	−	0.973673i	$-0.573202\pi$
$857$	−13.3462	−0.455898	−0.227949	+	0.973673i	$0.573202\pi$
$858$	0	0
$859$	21.6016	0.737036	0.368518	−	0.929621i	$-0.379865\pi$
$859$	21.6016	0.737036	0.368518	+	0.929621i	$0.379865\pi$
$860$	50.2221	1.71256
$861$	0	0
$862$	48.2167	1.64227
$863$	2.72425	0.0927345	0.0463673	−	0.998924i	$-0.485236\pi$
$863$	2.72425	0.0927345	0.0463673	+	0.998924i	$0.485236\pi$
$864$	0	0
$865$	4.39456	0.149420
$866$	−28.6147	−0.972367
$867$	0	0
$868$	−79.0879	−2.68442
$869$	−13.4291	−0.455552
$870$	0	0
$871$	10.3964	0.352269
$872$	1.70806	0.0578422
$873$	0	0
$874$	−8.37352	−0.283239
$875$	31.7034	1.07177
$876$	0	0
$877$	1.29130	0.0436041	0.0218021	−	0.999762i	$-0.493060\pi$
$877$	1.29130	0.0436041	0.0218021	+	0.999762i	$0.493060\pi$
$878$	3.68266	0.124284
$879$	0	0
$880$	−12.0500	−0.406207
$881$	−24.9873	−0.841844	−0.420922	−	0.907097i	$-0.638293\pi$
$881$	−24.9873	−0.841844	−0.420922	+	0.907097i	$0.638293\pi$
$882$	0	0
$883$	15.5046	0.521772	0.260886	−	0.965370i	$-0.415985\pi$
$883$	15.5046	0.521772	0.260886	+	0.965370i	$0.415985\pi$
$884$	40.0260	1.34622
$885$	0	0
$886$	−19.2914	−0.648107
$887$	9.98029	0.335105	0.167553	−	0.985863i	$-0.446414\pi$
$887$	9.98029	0.335105	0.167553	+	0.985863i	$0.446414\pi$
$888$	0	0
$889$	12.1391	0.407134
$890$	−12.9913	−0.435469
$891$	0	0
$892$	115.274	3.85966
$893$	−5.77094	−0.193117
$894$	0	0
$895$	−2.16911	−0.0725053
$896$	34.7063	1.15946
$897$	0	0
$898$	56.8815	1.89816
$899$	−33.4012	−1.11399
$900$	0	0
$901$	−20.7819	−0.692345
$902$	30.2199	1.00621
$903$	0	0
$904$	−1.60331	−0.0533253
$905$	5.59303	0.185919
$906$	0	0
$907$	11.8887	0.394758	0.197379	−	0.980327i	$-0.436757\pi$
$907$	11.8887	0.394758	0.197379	+	0.980327i	$0.436757\pi$
$908$	−20.2146	−0.670845
$909$	0	0
$910$	−20.4780	−0.678839
$911$	23.7166	0.785767	0.392883	−	0.919588i	$-0.371478\pi$
$911$	23.7166	0.785767	0.392883	+	0.919588i	$0.371478\pi$
$912$	0	0
$913$	9.96666	0.329848
$914$	46.7989	1.54797
$915$	0	0
$916$	−124.129	−4.10134
$917$	56.8471	1.87726
$918$	0	0
$919$	18.1426	0.598470	0.299235	−	0.954179i	$-0.403269\pi$
$919$	18.1426	0.598470	0.299235	+	0.954179i	$0.403269\pi$
$920$	27.0372	0.891390
$921$	0	0
$922$	90.7972	2.99024
$923$	32.9667	1.08511
$924$	0	0
$925$	−8.81549	−0.289852
$926$	29.1997	0.959563
$927$	0	0
$928$	100.500	3.29906
$929$	50.5019	1.65691	0.828457	−	0.560053i	$-0.189219\pi$
$929$	50.5019	1.65691	0.828457	+	0.560053i	$0.189219\pi$
$930$	0	0
$931$	−4.34940	−0.142546
$932$	−82.2368	−2.69376
$933$	0	0
$934$	11.6379	0.380805
$935$	3.89418	0.127353
$936$	0	0
$937$	−39.6587	−1.29559	−0.647797	−	0.761813i	$-0.724309\pi$
$937$	−39.6587	−1.29559	−0.647797	+	0.761813i	$0.724309\pi$
$938$	−43.0412	−1.40534
$939$	0	0
$940$	30.8711	1.00690
$941$	34.8809	1.13709	0.568543	−	0.822653i	$-0.307507\pi$
$941$	34.8809	1.13709	0.568543	+	0.822653i	$0.307507\pi$
$942$	0	0
$943$	−35.9166	−1.16961
$944$	156.777	5.10265
$945$	0	0
$946$	24.9197	0.810210
$947$	30.3067	0.984835	0.492417	−	0.870359i	$-0.336113\pi$
$947$	30.3067	0.984835	0.492417	+	0.870359i	$0.336113\pi$
$948$	0	0
$949$	17.4696	0.567088
$950$	−10.2878	−0.333780
$951$	0	0
$952$	−100.021	−3.24169
$953$	6.65804	0.215675	0.107838	−	0.994169i	$-0.465607\pi$
$953$	6.65804	0.215675	0.107838	+	0.994169i	$0.465607\pi$
$954$	0	0
$955$	17.1653	0.555457
$956$	77.7912	2.51595
$957$	0	0
$958$	−38.1381	−1.23219
$959$	−20.5785	−0.664513
$960$	0	0
$961$	−9.35006	−0.301615
$962$	13.0398	0.420419
$963$	0	0
$964$	121.817	3.92345
$965$	−21.2707	−0.684729
$966$	0	0
$967$	−20.0020	−0.643221	−0.321611	−	0.946872i	$-0.604224\pi$
$967$	−20.0020	−0.643221	−0.321611	+	0.946872i	$0.604224\pi$
$968$	−8.08346	−0.259812
$969$	0	0
$970$	11.5358	0.370392
$971$	0.491941	0.0157871	0.00789357	−	0.999969i	$-0.497487\pi$
$971$	0.491941	0.0157871	0.00789357	+	0.999969i	$0.497487\pi$
$972$	0	0
$973$	−38.1853	−1.22417
$974$	−2.69975	−0.0865055
$975$	0	0
$976$	79.3467	2.53983
$977$	−23.6515	−0.756679	−0.378339	−	0.925667i	$-0.623505\pi$
$977$	−23.6515	−0.756679	−0.378339	+	0.925667i	$0.623505\pi$
$978$	0	0
$979$	−4.61626	−0.147536
$980$	23.2666	0.743225
$981$	0	0
$982$	−6.39809	−0.204171
$983$	−39.5609	−1.26180	−0.630899	−	0.775865i	$-0.717314\pi$
$983$	−39.5609	−1.26180	−0.630899	+	0.775865i	$0.717314\pi$
$984$	0	0
$985$	25.2762	0.805366
$986$	−69.9831	−2.22872
$987$	0	0
$988$	10.8977	0.346702
$989$	−29.6173	−0.941777
$990$	0	0
$991$	−13.6096	−0.432322	−0.216161	−	0.976358i	$-0.569354\pi$
$991$	−13.6096	−0.432322	−0.216161	+	0.976358i	$0.569354\pi$
$992$	−65.1417	−2.06825
$993$	0	0
$994$	−136.482	−4.32895
$995$	−2.01654	−0.0639285
$996$	0	0
$997$	41.1821	1.30425	0.652125	−	0.758111i	$-0.273878\pi$
$997$	41.1821	1.30425	0.652125	+	0.758111i	$0.273878\pi$
$998$	−10.8144	−0.342323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.k.1.1		5
3.2	odd	2		209.2.a.c.1.5	✓	5
12.11	even	2		3344.2.a.t.1.3		5
15.14	odd	2		5225.2.a.h.1.1		5
33.32	even	2		2299.2.a.n.1.1		5
57.56	even	2		3971.2.a.h.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.5	✓	5	3.2	odd	2
1881.2.a.k.1.1		5	1.1	even	1	trivial
2299.2.a.n.1.1		5	33.32	even	2
3344.2.a.t.1.3		5	12.11	even	2
3971.2.a.h.1.1		5	57.56	even	2
5225.2.a.h.1.1		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)

\(f(q)\)	\(=\)	\(q-2.65432 q^{2} +5.04540 q^{4} +1.06025 q^{5} -3.36889 q^{7} -8.08346 q^{8} -2.81425 q^{10} -1.00000 q^{11} -2.15993 q^{13} +8.94209 q^{14} +11.3653 q^{16} -3.67288 q^{17} -1.00000 q^{19} +5.34940 q^{20} +2.65432 q^{22} -3.15468 q^{23} -3.87587 q^{25} +5.73313 q^{26} -16.9974 q^{28} -7.17849 q^{29} +4.65295 q^{31} -14.0001 q^{32} +9.74900 q^{34} -3.57187 q^{35} +2.27446 q^{37} +2.65432 q^{38} -8.57050 q^{40} +11.3852 q^{41} +9.38838 q^{43} -5.04540 q^{44} +8.37352 q^{46} +5.77094 q^{47} +4.34940 q^{49} +10.2878 q^{50} -10.8977 q^{52} +5.65820 q^{53} -1.06025 q^{55} +27.2322 q^{56} +19.0540 q^{58} +13.7944 q^{59} +6.98152 q^{61} -12.3504 q^{62} +14.4301 q^{64} -2.29007 q^{65} -4.81332 q^{67} -18.5312 q^{68} +9.48087 q^{70} -15.2629 q^{71} -8.08806 q^{73} -6.03713 q^{74} -5.04540 q^{76} +3.36889 q^{77} +13.4291 q^{79} +12.0500 q^{80} -30.2199 q^{82} -9.96666 q^{83} -3.89418 q^{85} -24.9197 q^{86} +8.08346 q^{88} +4.61626 q^{89} +7.27655 q^{91} -15.9166 q^{92} -15.3179 q^{94} -1.06025 q^{95} -4.09907 q^{97} -11.5447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) \) 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