LMFDB - Embedded newform 8001.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66267 q^{2} +5.08984 q^{4} -1.08542 q^{5} +1.00000 q^{7} -8.22723 q^{8} +O(q^{10})$	\(q-2.66267 q^{2} +5.08984 q^{4} -1.08542 q^{5} +1.00000 q^{7} -8.22723 q^{8} +2.89013 q^{10} -4.06877 q^{11} -1.24352 q^{13} -2.66267 q^{14} +11.7268 q^{16} -1.93951 q^{17} -4.54013 q^{19} -5.52463 q^{20} +10.8338 q^{22} -4.23345 q^{23} -3.82185 q^{25} +3.31110 q^{26} +5.08984 q^{28} -10.6103 q^{29} -7.36913 q^{31} -14.7701 q^{32} +5.16429 q^{34} -1.08542 q^{35} -7.14143 q^{37} +12.0889 q^{38} +8.93004 q^{40} -1.34649 q^{41} +0.657166 q^{43} -20.7094 q^{44} +11.2723 q^{46} +1.90807 q^{47} +1.00000 q^{49} +10.1764 q^{50} -6.32934 q^{52} -7.63321 q^{53} +4.41634 q^{55} -8.22723 q^{56} +28.2518 q^{58} -11.0918 q^{59} -5.42916 q^{61} +19.6216 q^{62} +15.8744 q^{64} +1.34975 q^{65} +3.79898 q^{67} -9.87179 q^{68} +2.89013 q^{70} +5.32761 q^{71} +7.99461 q^{73} +19.0153 q^{74} -23.1085 q^{76} -4.06877 q^{77} -2.53211 q^{79} -12.7285 q^{80} +3.58525 q^{82} -11.8780 q^{83} +2.10519 q^{85} -1.74982 q^{86} +33.4747 q^{88} +11.3787 q^{89} -1.24352 q^{91} -21.5476 q^{92} -5.08056 q^{94} +4.92797 q^{95} -5.06823 q^{97} -2.66267 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.66267	−1.88280	−0.941398	−	0.337299i	$-0.890487\pi$
$2$	−2.66267	−1.88280	−0.941398	+	0.337299i	$0.890487\pi$
$3$	0	0
$3$	0	0
$4$	5.08984	2.54492
$5$	−1.08542	−0.485417	−0.242708	−	0.970099i	$-0.578036\pi$
$5$	−1.08542	−0.485417	−0.242708	+	0.970099i	$0.578036\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−8.22723	−2.90876
$9$	0	0
$10$	2.89013	0.913940
$11$	−4.06877	−1.22678	−0.613390	−	0.789780i	$-0.710195\pi$
$11$	−4.06877	−1.22678	−0.613390	+	0.789780i	$0.710195\pi$
$12$	0	0
$13$	−1.24352	−0.344892	−0.172446	−	0.985019i	$-0.555167\pi$
$13$	−1.24352	−0.344892	−0.172446	+	0.985019i	$0.555167\pi$
$14$	−2.66267	−0.711630
$15$	0	0
$16$	11.7268	2.93169
$17$	−1.93951	−0.470400	−0.235200	−	0.971947i	$-0.575575\pi$
$17$	−1.93951	−0.470400	−0.235200	+	0.971947i	$0.575575\pi$
$18$	0	0
$19$	−4.54013	−1.04158	−0.520789	−	0.853686i	$-0.674362\pi$
$19$	−4.54013	−1.04158	−0.520789	+	0.853686i	$0.674362\pi$
$20$	−5.52463	−1.23535
$21$	0	0
$22$	10.8338	2.30977
$23$	−4.23345	−0.882736	−0.441368	−	0.897326i	$-0.645507\pi$
$23$	−4.23345	−0.882736	−0.441368	+	0.897326i	$0.645507\pi$
$24$	0	0
$25$	−3.82185	−0.764371
$26$	3.31110	0.649360
$27$	0	0
$28$	5.08984	0.961889
$29$	−10.6103	−1.97029	−0.985143	−	0.171734i	$-0.945063\pi$
$29$	−10.6103	−1.97029	−0.985143	+	0.171734i	$0.945063\pi$
$30$	0	0
$31$	−7.36913	−1.32353	−0.661767	−	0.749709i	$-0.730193\pi$
$31$	−7.36913	−1.32353	−0.661767	+	0.749709i	$0.730193\pi$
$32$	−14.7701	−2.61101
$33$	0	0
$34$	5.16429	0.885668
$35$	−1.08542	−0.183470
$36$	0	0
$37$	−7.14143	−1.17404	−0.587022	−	0.809571i	$-0.699700\pi$
$37$	−7.14143	−1.17404	−0.587022	+	0.809571i	$0.699700\pi$
$38$	12.0889	1.96108
$39$	0	0
$40$	8.93004	1.41196
$41$	−1.34649	−0.210286	−0.105143	−	0.994457i	$-0.533530\pi$
$41$	−1.34649	−0.210286	−0.105143	+	0.994457i	$0.533530\pi$
$42$	0	0
$43$	0.657166	0.100217	0.0501084	−	0.998744i	$-0.484043\pi$
$43$	0.657166	0.100217	0.0501084	+	0.998744i	$0.484043\pi$
$44$	−20.7094	−3.12205
$45$	0	0
$46$	11.2723	1.66201
$47$	1.90807	0.278320	0.139160	−	0.990270i	$-0.455560\pi$
$47$	1.90807	0.278320	0.139160	+	0.990270i	$0.455560\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.1764	1.43915
$51$	0	0
$52$	−6.32934	−0.877721
$53$	−7.63321	−1.04850	−0.524251	−	0.851564i	$-0.675655\pi$
$53$	−7.63321	−1.04850	−0.524251	+	0.851564i	$0.675655\pi$
$54$	0	0
$55$	4.41634	0.595499
$56$	−8.22723	−1.09941
$57$	0	0
$58$	28.2518	3.70965
$59$	−11.0918	−1.44402	−0.722012	−	0.691880i	$-0.756783\pi$
$59$	−11.0918	−1.44402	−0.722012	+	0.691880i	$0.756783\pi$
$60$	0	0
$61$	−5.42916	−0.695132	−0.347566	−	0.937655i	$-0.612992\pi$
$61$	−5.42916	−0.695132	−0.347566	+	0.937655i	$0.612992\pi$
$62$	19.6216	2.49195
$63$	0	0
$64$	15.8744	1.98430
$65$	1.34975	0.167416
$66$	0	0
$67$	3.79898	0.464120	0.232060	−	0.972701i	$-0.425453\pi$
$67$	3.79898	0.464120	0.232060	+	0.972701i	$0.425453\pi$
$68$	−9.87179	−1.19713
$69$	0	0
$70$	2.89013	0.345437
$71$	5.32761	0.632271	0.316135	−	0.948714i	$-0.397615\pi$
$71$	5.32761	0.632271	0.316135	+	0.948714i	$0.397615\pi$
$72$	0	0
$73$	7.99461	0.935698	0.467849	−	0.883808i	$-0.345029\pi$
$73$	7.99461	0.935698	0.467849	+	0.883808i	$0.345029\pi$
$74$	19.0153	2.21048
$75$	0	0
$76$	−23.1085	−2.65073
$77$	−4.06877	−0.463679
$78$	0	0
$79$	−2.53211	−0.284884	−0.142442	−	0.989803i	$-0.545495\pi$
$79$	−2.53211	−0.284884	−0.142442	+	0.989803i	$0.545495\pi$
$80$	−12.7285	−1.42309
$81$	0	0
$82$	3.58525	0.395925
$83$	−11.8780	−1.30378	−0.651889	−	0.758315i	$-0.726023\pi$
$83$	−11.8780	−1.30378	−0.651889	+	0.758315i	$0.726023\pi$
$84$	0	0
$85$	2.10519	0.228340
$86$	−1.74982	−0.188688
$87$	0	0
$88$	33.4747	3.56841
$89$	11.3787	1.20614	0.603069	−	0.797689i	$-0.293944\pi$
$89$	11.3787	1.20614	0.603069	+	0.797689i	$0.293944\pi$
$90$	0	0
$91$	−1.24352	−0.130357
$92$	−21.5476	−2.24649
$93$	0	0
$94$	−5.08056	−0.524020
$95$	4.92797	0.505599
$96$	0	0
$97$	−5.06823	−0.514600	−0.257300	−	0.966332i	$-0.582833\pi$
$97$	−5.06823	−0.514600	−0.257300	+	0.966332i	$0.582833\pi$
$98$	−2.66267	−0.268971
$99$	0	0
$100$	−19.4526	−1.94526
$101$	3.92878	0.390928	0.195464	−	0.980711i	$-0.437379\pi$
$101$	3.92878	0.390928	0.195464	+	0.980711i	$0.437379\pi$
$102$	0	0
$103$	−14.4451	−1.42332	−0.711661	−	0.702523i	$-0.752057\pi$
$103$	−14.4451	−1.42332	−0.711661	+	0.702523i	$0.752057\pi$
$104$	10.2308	1.00321
$105$	0	0
$106$	20.3247	1.97411
$107$	9.48196	0.916656	0.458328	−	0.888783i	$-0.348449\pi$
$107$	9.48196	0.916656	0.458328	+	0.888783i	$0.348449\pi$
$108$	0	0
$109$	−2.81462	−0.269592	−0.134796	−	0.990873i	$-0.543038\pi$
$109$	−2.81462	−0.269592	−0.134796	+	0.990873i	$0.543038\pi$
$110$	−11.7593	−1.12120
$111$	0	0
$112$	11.7268	1.10807
$113$	−11.6187	−1.09299	−0.546497	−	0.837461i	$-0.684039\pi$
$113$	−11.6187	−1.09299	−0.546497	+	0.837461i	$0.684039\pi$
$114$	0	0
$115$	4.59509	0.428495
$116$	−54.0048	−5.01422
$117$	0	0
$118$	29.5338	2.71880
$119$	−1.93951	−0.177795
$120$	0	0
$121$	5.55487	0.504988
$122$	14.4561	1.30879
$123$	0	0
$124$	−37.5077	−3.36829
$125$	9.57546	0.856455
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−12.7283	−1.12503
$129$	0	0
$130$	−3.59395	−0.315210
$131$	15.1975	1.32781	0.663905	−	0.747817i	$-0.268898\pi$
$131$	15.1975	1.32781	0.663905	+	0.747817i	$0.268898\pi$
$132$	0	0
$133$	−4.54013	−0.393679
$134$	−10.1155	−0.873843
$135$	0	0
$136$	15.9568	1.36828
$137$	13.8972	1.18732	0.593659	−	0.804717i	$-0.297683\pi$
$137$	13.8972	1.18732	0.593659	+	0.804717i	$0.297683\pi$
$138$	0	0
$139$	−9.34261	−0.792430	−0.396215	−	0.918158i	$-0.629677\pi$
$139$	−9.34261	−0.792430	−0.396215	+	0.918158i	$0.629677\pi$
$140$	−5.52463	−0.466917
$141$	0	0
$142$	−14.1857	−1.19044
$143$	5.05961	0.423106
$144$	0	0
$145$	11.5167	0.956410
$146$	−21.2870	−1.76173
$147$	0	0
$148$	−36.3487	−2.98784
$149$	16.5381	1.35485	0.677427	−	0.735590i	$-0.263095\pi$
$149$	16.5381	1.35485	0.677427	+	0.735590i	$0.263095\pi$
$150$	0	0
$151$	1.55578	0.126607	0.0633036	−	0.997994i	$-0.479836\pi$
$151$	1.55578	0.126607	0.0633036	+	0.997994i	$0.479836\pi$
$152$	37.3527	3.02970
$153$	0	0
$154$	10.8338	0.873013
$155$	7.99864	0.642466
$156$	0	0
$157$	2.31798	0.184995	0.0924974	−	0.995713i	$-0.470515\pi$
$157$	2.31798	0.184995	0.0924974	+	0.995713i	$0.470515\pi$
$158$	6.74217	0.536379
$159$	0	0
$160$	16.0318	1.26743
$161$	−4.23345	−0.333643
$162$	0	0
$163$	23.4129	1.83384	0.916920	−	0.399072i	$-0.130668\pi$
$163$	23.4129	1.83384	0.916920	+	0.399072i	$0.130668\pi$
$164$	−6.85339	−0.535160
$165$	0	0
$166$	31.6272	2.45475
$167$	−17.0759	−1.32137	−0.660686	−	0.750662i	$-0.729734\pi$
$167$	−17.0759	−1.32137	−0.660686	+	0.750662i	$0.729734\pi$
$168$	0	0
$169$	−11.4536	−0.881050
$170$	−5.60544	−0.429918
$171$	0	0
$172$	3.34487	0.255044
$173$	21.1194	1.60568	0.802840	−	0.596195i	$-0.203321\pi$
$173$	21.1194	1.60568	0.802840	+	0.596195i	$0.203321\pi$
$174$	0	0
$175$	−3.82185	−0.288905
$176$	−47.7135	−3.59654
$177$	0	0
$178$	−30.2977	−2.27091
$179$	−10.7336	−0.802270	−0.401135	−	0.916019i	$-0.631384\pi$
$179$	−10.7336	−0.802270	−0.401135	+	0.916019i	$0.631384\pi$
$180$	0	0
$181$	13.1049	0.974083	0.487041	−	0.873379i	$-0.338076\pi$
$181$	13.1049	0.974083	0.487041	+	0.873379i	$0.338076\pi$
$182$	3.31110	0.245435
$183$	0	0
$184$	34.8296	2.56767
$185$	7.75148	0.569900
$186$	0	0
$187$	7.89142	0.577078
$188$	9.71175	0.708303
$189$	0	0
$190$	−13.1216	−0.951940
$191$	14.8406	1.07383	0.536915	−	0.843637i	$-0.319590\pi$
$191$	14.8406	1.07383	0.536915	+	0.843637i	$0.319590\pi$
$192$	0	0
$193$	−5.91805	−0.425991	−0.212995	−	0.977053i	$-0.568322\pi$
$193$	−5.91805	−0.425991	−0.212995	+	0.977053i	$0.568322\pi$
$194$	13.4950	0.968887
$195$	0	0
$196$	5.08984	0.363560
$197$	−25.9987	−1.85233	−0.926164	−	0.377121i	$-0.876914\pi$
$197$	−25.9987	−1.85233	−0.926164	+	0.377121i	$0.876914\pi$
$198$	0	0
$199$	23.3666	1.65641	0.828207	−	0.560422i	$-0.189361\pi$
$199$	23.3666	1.65641	0.828207	+	0.560422i	$0.189361\pi$
$200$	31.4433	2.22337
$201$	0	0
$202$	−10.4611	−0.736038
$203$	−10.6103	−0.744698
$204$	0	0
$205$	1.46151	0.102076
$206$	38.4627	2.67982
$207$	0	0
$208$	−14.5825	−1.01112
$209$	18.4727	1.27779
$210$	0	0
$211$	−9.76256	−0.672082	−0.336041	−	0.941847i	$-0.609088\pi$
$211$	−9.76256	−0.672082	−0.336041	+	0.941847i	$0.609088\pi$
$212$	−38.8518	−2.66835
$213$	0	0
$214$	−25.2474	−1.72588
$215$	−0.713304	−0.0486469
$216$	0	0
$217$	−7.36913	−0.500249
$218$	7.49442	0.507586
$219$	0	0
$220$	22.4785	1.51550
$221$	2.41183	0.162237
$222$	0	0
$223$	12.5108	0.837783	0.418892	−	0.908036i	$-0.362419\pi$
$223$	12.5108	0.837783	0.418892	+	0.908036i	$0.362419\pi$
$224$	−14.7701	−0.986868
$225$	0	0
$226$	30.9368	2.05789
$227$	7.64749	0.507582	0.253791	−	0.967259i	$-0.418322\pi$
$227$	7.64749	0.507582	0.253791	+	0.967259i	$0.418322\pi$
$228$	0	0
$229$	9.09367	0.600927	0.300463	−	0.953793i	$-0.402859\pi$
$229$	9.09367	0.600927	0.300463	+	0.953793i	$0.402859\pi$
$230$	−12.2352	−0.806768
$231$	0	0
$232$	87.2935	5.73110
$233$	2.47517	0.162154	0.0810769	−	0.996708i	$-0.474164\pi$
$233$	2.47517	0.162154	0.0810769	+	0.996708i	$0.474164\pi$
$234$	0	0
$235$	−2.07106	−0.135101
$236$	−56.4552	−3.67492
$237$	0	0
$238$	5.16429	0.334751
$239$	4.95751	0.320674	0.160337	−	0.987062i	$-0.448742\pi$
$239$	4.95751	0.320674	0.160337	+	0.987062i	$0.448742\pi$
$240$	0	0
$241$	7.55514	0.486669	0.243335	−	0.969942i	$-0.421759\pi$
$241$	7.55514	0.486669	0.243335	+	0.969942i	$0.421759\pi$
$242$	−14.7908	−0.950790
$243$	0	0
$244$	−27.6335	−1.76906
$245$	−1.08542	−0.0693452
$246$	0	0
$247$	5.64577	0.359231
$248$	60.6275	3.84985
$249$	0	0
$250$	−25.4963	−1.61253
$251$	−9.08990	−0.573749	−0.286875	−	0.957968i	$-0.592616\pi$
$251$	−9.08990	−0.573749	−0.286875	+	0.957968i	$0.592616\pi$
$252$	0	0
$253$	17.2249	1.08292
$254$	2.66267	0.167071
$255$	0	0
$256$	2.14234	0.133896
$257$	−28.5045	−1.77806	−0.889032	−	0.457845i	$-0.848622\pi$
$257$	−28.5045	−1.77806	−0.889032	+	0.457845i	$0.848622\pi$
$258$	0	0
$259$	−7.14143	−0.443747
$260$	6.87002	0.426060
$261$	0	0
$262$	−40.4659	−2.49999
$263$	−13.7059	−0.845141	−0.422571	−	0.906330i	$-0.638872\pi$
$263$	−13.7059	−0.845141	−0.422571	+	0.906330i	$0.638872\pi$
$264$	0	0
$265$	8.28527	0.508960
$266$	12.0889	0.741218
$267$	0	0
$268$	19.3362	1.18115
$269$	12.5100	0.762749	0.381374	−	0.924421i	$-0.375451\pi$
$269$	12.5100	0.762749	0.381374	+	0.924421i	$0.375451\pi$
$270$	0	0
$271$	14.1490	0.859489	0.429744	−	0.902951i	$-0.358604\pi$
$271$	14.1490	0.859489	0.429744	+	0.902951i	$0.358604\pi$
$272$	−22.7442	−1.37907
$273$	0	0
$274$	−37.0037	−2.23548
$275$	15.5502	0.937714
$276$	0	0
$277$	13.8654	0.833094	0.416547	−	0.909114i	$-0.363240\pi$
$277$	13.8654	0.833094	0.416547	+	0.909114i	$0.363240\pi$
$278$	24.8763	1.49198
$279$	0	0
$280$	8.93004	0.533672
$281$	15.2262	0.908321	0.454161	−	0.890920i	$-0.349939\pi$
$281$	15.2262	0.908321	0.454161	+	0.890920i	$0.349939\pi$
$282$	0	0
$283$	−20.2480	−1.20362	−0.601808	−	0.798641i	$-0.705553\pi$
$283$	−20.2480	−1.20362	−0.601808	+	0.798641i	$0.705553\pi$
$284$	27.1166	1.60908
$285$	0	0
$286$	−13.4721	−0.796622
$287$	−1.34649	−0.0794806
$288$	0	0
$289$	−13.2383	−0.778723
$290$	−30.6652	−1.80072
$291$	0	0
$292$	40.6913	2.38128
$293$	−4.62289	−0.270072	−0.135036	−	0.990841i	$-0.543115\pi$
$293$	−4.62289	−0.270072	−0.135036	+	0.990841i	$0.543115\pi$
$294$	0	0
$295$	12.0393	0.700953
$296$	58.7542	3.41502
$297$	0	0
$298$	−44.0356	−2.55091
$299$	5.26440	0.304448
$300$	0	0
$301$	0.657166	0.0378784
$302$	−4.14252	−0.238375
$303$	0	0
$304$	−53.2410	−3.05358
$305$	5.89294	0.337429
$306$	0	0
$307$	−30.9294	−1.76523	−0.882616	−	0.470095i	$-0.844220\pi$
$307$	−30.9294	−1.76523	−0.882616	+	0.470095i	$0.844220\pi$
$308$	−20.7094	−1.18003
$309$	0	0
$310$	−21.2978	−1.20963
$311$	17.9086	1.01551	0.507753	−	0.861503i	$-0.330476\pi$
$311$	17.9086	1.01551	0.507753	+	0.861503i	$0.330476\pi$
$312$	0	0
$313$	5.30817	0.300036	0.150018	−	0.988683i	$-0.452067\pi$
$313$	5.30817	0.300036	0.150018	+	0.988683i	$0.452067\pi$
$314$	−6.17202	−0.348307
$315$	0	0
$316$	−12.8880	−0.725007
$317$	−25.8361	−1.45110	−0.725550	−	0.688170i	$-0.758414\pi$
$317$	−25.8361	−1.45110	−0.725550	+	0.688170i	$0.758414\pi$
$318$	0	0
$319$	43.1709	2.41711
$320$	−17.2305	−0.963214
$321$	0	0
$322$	11.2723	0.628181
$323$	8.80564	0.489959
$324$	0	0
$325$	4.75257	0.263625
$326$	−62.3409	−3.45274
$327$	0	0
$328$	11.0779	0.611672
$329$	1.90807	0.105195
$330$	0	0
$331$	−13.4453	−0.739019	−0.369510	−	0.929227i	$-0.620474\pi$
$331$	−13.4453	−0.739019	−0.369510	+	0.929227i	$0.620474\pi$
$332$	−60.4570	−3.31801
$333$	0	0
$334$	45.4675	2.48787
$335$	−4.12351	−0.225291
$336$	0	0
$337$	−11.9299	−0.649862	−0.324931	−	0.945738i	$-0.605341\pi$
$337$	−11.9299	−0.649862	−0.324931	+	0.945738i	$0.605341\pi$
$338$	30.4973	1.65884
$339$	0	0
$340$	10.7151	0.581107
$341$	29.9833	1.62369
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.40666	−0.291507
$345$	0	0
$346$	−56.2341	−3.02317
$347$	−11.2275	−0.602725	−0.301363	−	0.953510i	$-0.597442\pi$
$347$	−11.2275	−0.602725	−0.301363	+	0.953510i	$0.597442\pi$
$348$	0	0
$349$	3.16012	0.169157	0.0845787	−	0.996417i	$-0.473046\pi$
$349$	3.16012	0.169157	0.0845787	+	0.996417i	$0.473046\pi$
$350$	10.1764	0.543949
$351$	0	0
$352$	60.0961	3.20313
$353$	−13.7567	−0.732196	−0.366098	−	0.930576i	$-0.619306\pi$
$353$	−13.7567	−0.732196	−0.366098	+	0.930576i	$0.619306\pi$
$354$	0	0
$355$	−5.78272	−0.306915
$356$	57.9157	3.06952
$357$	0	0
$358$	28.5802	1.51051
$359$	1.39139	0.0734347	0.0367173	−	0.999326i	$-0.488310\pi$
$359$	1.39139	0.0734347	0.0367173	+	0.999326i	$0.488310\pi$
$360$	0	0
$361$	1.61280	0.0848842
$362$	−34.8942	−1.83400
$363$	0	0
$364$	−6.32934	−0.331747
$365$	−8.67755	−0.454204
$366$	0	0
$367$	−28.4894	−1.48713	−0.743566	−	0.668662i	$-0.766867\pi$
$367$	−28.4894	−1.48713	−0.743566	+	0.668662i	$0.766867\pi$
$368$	−49.6447	−2.58791
$369$	0	0
$370$	−20.6397	−1.07301
$371$	−7.63321	−0.396296
$372$	0	0
$373$	17.9120	0.927448	0.463724	−	0.885980i	$-0.346513\pi$
$373$	17.9120	0.927448	0.463724	+	0.885980i	$0.346513\pi$
$374$	−21.0123	−1.08652
$375$	0	0
$376$	−15.6981	−0.809568
$377$	13.1942	0.679535
$378$	0	0
$379$	8.45471	0.434289	0.217145	−	0.976139i	$-0.430326\pi$
$379$	8.45471	0.434289	0.217145	+	0.976139i	$0.430326\pi$
$380$	25.0826	1.28671
$381$	0	0
$382$	−39.5157	−2.02180
$383$	24.7148	1.26287	0.631433	−	0.775430i	$-0.282467\pi$
$383$	24.7148	1.26287	0.631433	+	0.775430i	$0.282467\pi$
$384$	0	0
$385$	4.41634	0.225078
$386$	15.7578	0.802053
$387$	0	0
$388$	−25.7964	−1.30962
$389$	−5.12073	−0.259631	−0.129816	−	0.991538i	$-0.541439\pi$
$389$	−5.12073	−0.259631	−0.129816	+	0.991538i	$0.541439\pi$
$390$	0	0
$391$	8.21083	0.415239
$392$	−8.22723	−0.415538
$393$	0	0
$394$	69.2260	3.48755
$395$	2.74841	0.138288
$396$	0	0
$397$	−12.9227	−0.648571	−0.324286	−	0.945959i	$-0.605124\pi$
$397$	−12.9227	−0.648571	−0.324286	+	0.945959i	$0.605124\pi$
$398$	−62.2177	−3.11869
$399$	0	0
$400$	−44.8180	−2.24090
$401$	11.1859	0.558597	0.279298	−	0.960204i	$-0.409898\pi$
$401$	11.1859	0.558597	0.279298	+	0.960204i	$0.409898\pi$
$402$	0	0
$403$	9.16369	0.456476
$404$	19.9968	0.994880
$405$	0	0
$406$	28.2518	1.40211
$407$	29.0568	1.44029
$408$	0	0
$409$	−5.48933	−0.271430	−0.135715	−	0.990748i	$-0.543333\pi$
$409$	−5.48933	−0.271430	−0.135715	+	0.990748i	$0.543333\pi$
$410$	−3.89152	−0.192189
$411$	0	0
$412$	−73.5234	−3.62224
$413$	−11.0918	−0.545790
$414$	0	0
$415$	12.8927	0.632875
$416$	18.3670	0.900515
$417$	0	0
$418$	−49.1869	−2.40581
$419$	−9.85487	−0.481442	−0.240721	−	0.970594i	$-0.577384\pi$
$419$	−9.85487	−0.481442	−0.240721	+	0.970594i	$0.577384\pi$
$420$	0	0
$421$	−16.2444	−0.791705	−0.395853	−	0.918314i	$-0.629551\pi$
$421$	−16.2444	−0.791705	−0.395853	+	0.918314i	$0.629551\pi$
$422$	25.9945	1.26539
$423$	0	0
$424$	62.8001	3.04984
$425$	7.41253	0.359560
$426$	0	0
$427$	−5.42916	−0.262735
$428$	48.2616	2.33281
$429$	0	0
$430$	1.89930	0.0915922
$431$	−3.91365	−0.188514	−0.0942570	−	0.995548i	$-0.530048\pi$
$431$	−3.91365	−0.188514	−0.0942570	+	0.995548i	$0.530048\pi$
$432$	0	0
$433$	−37.5633	−1.80518	−0.902589	−	0.430503i	$-0.858336\pi$
$433$	−37.5633	−1.80518	−0.902589	+	0.430503i	$0.858336\pi$
$434$	19.6216	0.941867
$435$	0	0
$436$	−14.3260	−0.686089
$437$	19.2204	0.919438
$438$	0	0
$439$	−10.4633	−0.499386	−0.249693	−	0.968325i	$-0.580330\pi$
$439$	−10.4633	−0.499386	−0.249693	+	0.968325i	$0.580330\pi$
$440$	−36.3342	−1.73217
$441$	0	0
$442$	−6.42192	−0.305459
$443$	−0.803311	−0.0381664	−0.0190832	−	0.999818i	$-0.506075\pi$
$443$	−0.803311	−0.0381664	−0.0190832	+	0.999818i	$0.506075\pi$
$444$	0	0
$445$	−12.3507	−0.585480
$446$	−33.3121	−1.57737
$447$	0	0
$448$	15.8744	0.749996
$449$	12.8391	0.605914	0.302957	−	0.953004i	$-0.402026\pi$
$449$	12.8391	0.605914	0.302957	+	0.953004i	$0.402026\pi$
$450$	0	0
$451$	5.47854	0.257974
$452$	−59.1372	−2.78158
$453$	0	0
$454$	−20.3628	−0.955673
$455$	1.34975	0.0632774
$456$	0	0
$457$	−18.6387	−0.871884	−0.435942	−	0.899975i	$-0.643585\pi$
$457$	−18.6387	−0.871884	−0.435942	+	0.899975i	$0.643585\pi$
$458$	−24.2135	−1.13142
$459$	0	0
$460$	23.3883	1.09048
$461$	−37.8290	−1.76187	−0.880936	−	0.473235i	$-0.843086\pi$
$461$	−37.8290	−1.76187	−0.880936	+	0.473235i	$0.843086\pi$
$462$	0	0
$463$	−10.6617	−0.495493	−0.247747	−	0.968825i	$-0.579690\pi$
$463$	−10.6617	−0.495493	−0.247747	+	0.968825i	$0.579690\pi$
$464$	−124.425	−5.77627
$465$	0	0
$466$	−6.59058	−0.305303
$467$	−26.9543	−1.24729	−0.623647	−	0.781706i	$-0.714350\pi$
$467$	−26.9543	−1.24729	−0.623647	+	0.781706i	$0.714350\pi$
$468$	0	0
$469$	3.79898	0.175421
$470$	5.51457	0.254368
$471$	0	0
$472$	91.2545	4.20033
$473$	−2.67386	−0.122944
$474$	0	0
$475$	17.3517	0.796151
$476$	−9.87179	−0.452473
$477$	0	0
$478$	−13.2002	−0.603764
$479$	17.9192	0.818749	0.409375	−	0.912366i	$-0.365747\pi$
$479$	17.9192	0.818749	0.409375	+	0.912366i	$0.365747\pi$
$480$	0	0
$481$	8.88054	0.404918
$482$	−20.1169	−0.916299
$483$	0	0
$484$	28.2734	1.28515
$485$	5.50118	0.249796
$486$	0	0
$487$	−26.2459	−1.18931	−0.594657	−	0.803979i	$-0.702712\pi$
$487$	−26.2459	−1.18931	−0.594657	+	0.803979i	$0.702712\pi$
$488$	44.6669	2.02198
$489$	0	0
$490$	2.89013	0.130563
$491$	4.27840	0.193081	0.0965407	−	0.995329i	$-0.469222\pi$
$491$	4.27840	0.193081	0.0965407	+	0.995329i	$0.469222\pi$
$492$	0	0
$493$	20.5788	0.926824
$494$	−15.0328	−0.676359
$495$	0	0
$496$	−86.4160	−3.88019
$497$	5.32761	0.238976
$498$	0	0
$499$	−32.3012	−1.44600	−0.723001	−	0.690847i	$-0.757238\pi$
$499$	−32.3012	−1.44600	−0.723001	+	0.690847i	$0.757238\pi$
$500$	48.7375	2.17961
$501$	0	0
$502$	24.2034	1.08025
$503$	−21.5675	−0.961648	−0.480824	−	0.876817i	$-0.659662\pi$
$503$	−21.5675	−0.961648	−0.480824	+	0.876817i	$0.659662\pi$
$504$	0	0
$505$	−4.26440	−0.189763
$506$	−45.8644	−2.03892
$507$	0	0
$508$	−5.08984	−0.225825
$509$	−34.9642	−1.54976	−0.774880	−	0.632108i	$-0.782190\pi$
$509$	−34.9642	−1.54976	−0.774880	+	0.632108i	$0.782190\pi$
$510$	0	0
$511$	7.99461	0.353661
$512$	19.7521	0.872930
$513$	0	0
$514$	75.8983	3.34773
$515$	15.6791	0.690904
$516$	0	0
$517$	−7.76349	−0.341438
$518$	19.0153	0.835484
$519$	0	0
$520$	−11.1047	−0.486974
$521$	18.7179	0.820047	0.410024	−	0.912075i	$-0.365520\pi$
$521$	18.7179	0.820047	0.410024	+	0.912075i	$0.365520\pi$
$522$	0	0
$523$	16.1244	0.705073	0.352536	−	0.935798i	$-0.385319\pi$
$523$	16.1244	0.705073	0.352536	+	0.935798i	$0.385319\pi$
$524$	77.3527	3.37917
$525$	0	0
$526$	36.4943	1.59123
$527$	14.2925	0.622591
$528$	0	0
$529$	−5.07788	−0.220778
$530$	−22.0610	−0.958268
$531$	0	0
$532$	−23.1085	−1.00188
$533$	1.67439	0.0725258
$534$	0	0
$535$	−10.2920	−0.444960
$536$	−31.2551	−1.35002
$537$	0	0
$538$	−33.3101	−1.43610
$539$	−4.06877	−0.175254
$540$	0	0
$541$	24.6295	1.05890	0.529451	−	0.848340i	$-0.322398\pi$
$541$	24.6295	1.05890	0.529451	+	0.848340i	$0.322398\pi$
$542$	−37.6741	−1.61824
$543$	0	0
$544$	28.6468	1.22822
$545$	3.05506	0.130864
$546$	0	0
$547$	−17.2816	−0.738908	−0.369454	−	0.929249i	$-0.620455\pi$
$547$	−17.2816	−0.738908	−0.369454	+	0.929249i	$0.620455\pi$
$548$	70.7344	3.02163
$549$	0	0
$550$	−41.4052	−1.76552
$551$	48.1722	2.05221
$552$	0	0
$553$	−2.53211	−0.107676
$554$	−36.9192	−1.56855
$555$	0	0
$556$	−47.5524	−2.01667
$557$	2.42166	0.102609	0.0513045	−	0.998683i	$-0.483662\pi$
$557$	2.42166	0.102609	0.0513045	+	0.998683i	$0.483662\pi$
$558$	0	0
$559$	−0.817202	−0.0345640
$560$	−12.7285	−0.537878
$561$	0	0
$562$	−40.5425	−1.71018
$563$	6.25746	0.263721	0.131860	−	0.991268i	$-0.457905\pi$
$563$	6.25746	0.263721	0.131860	+	0.991268i	$0.457905\pi$
$564$	0	0
$565$	12.6112	0.530558
$566$	53.9137	2.26616
$567$	0	0
$568$	−43.8314	−1.83913
$569$	−34.4175	−1.44286	−0.721429	−	0.692489i	$-0.756514\pi$
$569$	−34.4175	−1.44286	−0.721429	+	0.692489i	$0.756514\pi$
$570$	0	0
$571$	12.8008	0.535698	0.267849	−	0.963461i	$-0.413687\pi$
$571$	12.8008	0.535698	0.267849	+	0.963461i	$0.413687\pi$
$572$	25.7526	1.07677
$573$	0	0
$574$	3.58525	0.149646
$575$	16.1796	0.674737
$576$	0	0
$577$	22.7284	0.946195	0.473097	−	0.881010i	$-0.343136\pi$
$577$	22.7284	0.946195	0.473097	+	0.881010i	$0.343136\pi$
$578$	35.2493	1.46618
$579$	0	0
$580$	58.6181	2.43399
$581$	−11.8780	−0.492782
$582$	0	0
$583$	31.0577	1.28628
$584$	−65.7735	−2.72173
$585$	0	0
$586$	12.3092	0.508490
$587$	−31.8929	−1.31636	−0.658179	−	0.752861i	$-0.728673\pi$
$587$	−31.8929	−1.31636	−0.658179	+	0.752861i	$0.728673\pi$
$588$	0	0
$589$	33.4568	1.37856
$590$	−32.0567	−1.31975
$591$	0	0
$592$	−83.7458	−3.44193
$593$	43.8653	1.80133	0.900665	−	0.434514i	$-0.143080\pi$
$593$	43.8653	1.80133	0.900665	+	0.434514i	$0.143080\pi$
$594$	0	0
$595$	2.10519	0.0863045
$596$	84.1763	3.44799
$597$	0	0
$598$	−14.0174	−0.573214
$599$	−10.7591	−0.439606	−0.219803	−	0.975544i	$-0.570541\pi$
$599$	−10.7591	−0.439606	−0.219803	+	0.975544i	$0.570541\pi$
$600$	0	0
$601$	−45.1901	−1.84334	−0.921672	−	0.387970i	$-0.873176\pi$
$601$	−45.1901	−1.84334	−0.921672	+	0.387970i	$0.873176\pi$
$602$	−1.74982	−0.0713173
$603$	0	0
$604$	7.91864	0.322205
$605$	−6.02939	−0.245130
$606$	0	0
$607$	28.4027	1.15283	0.576415	−	0.817157i	$-0.304451\pi$
$607$	28.4027	1.15283	0.576415	+	0.817157i	$0.304451\pi$
$608$	67.0582	2.71957
$609$	0	0
$610$	−15.6910	−0.635310
$611$	−2.37273	−0.0959904
$612$	0	0
$613$	−45.8426	−1.85156	−0.925782	−	0.378058i	$-0.876592\pi$
$613$	−45.8426	−1.85156	−0.925782	+	0.378058i	$0.876592\pi$
$614$	82.3548	3.32357
$615$	0	0
$616$	33.4747	1.34873
$617$	44.0778	1.77451	0.887253	−	0.461283i	$-0.152611\pi$
$617$	44.0778	1.77451	0.887253	+	0.461283i	$0.152611\pi$
$618$	0	0
$619$	4.54339	0.182614	0.0913072	−	0.995823i	$-0.470895\pi$
$619$	4.54339	0.182614	0.0913072	+	0.995823i	$0.470895\pi$
$620$	40.7117	1.63502
$621$	0	0
$622$	−47.6849	−1.91199
$623$	11.3787	0.455877
$624$	0	0
$625$	8.71583	0.348633
$626$	−14.1339	−0.564906
$627$	0	0
$628$	11.7981	0.470797
$629$	13.8509	0.552271
$630$	0	0
$631$	42.8474	1.70573	0.852863	−	0.522134i	$-0.174864\pi$
$631$	42.8474	1.70573	0.852863	+	0.522134i	$0.174864\pi$
$632$	20.8322	0.828661
$633$	0	0
$634$	68.7931	2.73212
$635$	1.08542	0.0430738
$636$	0	0
$637$	−1.24352	−0.0492702
$638$	−114.950	−4.55092
$639$	0	0
$640$	13.8156	0.546108
$641$	−39.9212	−1.57679	−0.788395	−	0.615169i	$-0.789088\pi$
$641$	−39.9212	−1.57679	−0.788395	+	0.615169i	$0.789088\pi$
$642$	0	0
$643$	−1.17611	−0.0463813	−0.0231907	−	0.999731i	$-0.507382\pi$
$643$	−1.17611	−0.0463813	−0.0231907	+	0.999731i	$0.507382\pi$
$644$	−21.5476	−0.849094
$645$	0	0
$646$	−23.4465	−0.922492
$647$	−8.63451	−0.339458	−0.169729	−	0.985491i	$-0.554289\pi$
$647$	−8.63451	−0.339458	−0.169729	+	0.985491i	$0.554289\pi$
$648$	0	0
$649$	45.1298	1.77150
$650$	−12.6545	−0.496352
$651$	0	0
$652$	119.168	4.66697
$653$	−19.3336	−0.756581	−0.378290	−	0.925687i	$-0.623488\pi$
$653$	−19.3336	−0.756581	−0.378290	+	0.925687i	$0.623488\pi$
$654$	0	0
$655$	−16.4957	−0.644541
$656$	−15.7899	−0.616493
$657$	0	0
$658$	−5.08056	−0.198061
$659$	−5.24046	−0.204139	−0.102070	−	0.994777i	$-0.532546\pi$
$659$	−5.24046	−0.204139	−0.102070	+	0.994777i	$0.532546\pi$
$660$	0	0
$661$	−30.6339	−1.19152	−0.595760	−	0.803162i	$-0.703149\pi$
$661$	−30.6339	−1.19152	−0.595760	+	0.803162i	$0.703149\pi$
$662$	35.8004	1.39142
$663$	0	0
$664$	97.7229	3.79238
$665$	4.92797	0.191099
$666$	0	0
$667$	44.9183	1.73924
$668$	−86.9135	−3.36278
$669$	0	0
$670$	10.9796	0.424178
$671$	22.0900	0.852774
$672$	0	0
$673$	−22.1599	−0.854203	−0.427102	−	0.904204i	$-0.640465\pi$
$673$	−22.1599	−0.854203	−0.427102	+	0.904204i	$0.640465\pi$
$674$	31.7654	1.22356
$675$	0	0
$676$	−58.2972	−2.24220
$677$	14.7843	0.568208	0.284104	−	0.958794i	$-0.408304\pi$
$677$	14.7843	0.568208	0.284104	+	0.958794i	$0.408304\pi$
$678$	0	0
$679$	−5.06823	−0.194501
$680$	−17.3199	−0.664188
$681$	0	0
$682$	−79.8357	−3.05707
$683$	−13.8370	−0.529457	−0.264728	−	0.964323i	$-0.585282\pi$
$683$	−13.8370	−0.529457	−0.264728	+	0.964323i	$0.585282\pi$
$684$	0	0
$685$	−15.0844	−0.576344
$686$	−2.66267	−0.101661
$687$	0	0
$688$	7.70643	0.293805
$689$	9.49208	0.361619
$690$	0	0
$691$	3.17287	0.120702	0.0603508	−	0.998177i	$-0.480778\pi$
$691$	3.17287	0.120702	0.0603508	+	0.998177i	$0.480778\pi$
$692$	107.494	4.08632
$693$	0	0
$694$	29.8953	1.13481
$695$	10.1407	0.384659
$696$	0	0
$697$	2.61152	0.0989185
$698$	−8.41437	−0.318489
$699$	0	0
$700$	−19.4526	−0.735239
$701$	−38.9325	−1.47046	−0.735230	−	0.677817i	$-0.762926\pi$
$701$	−38.9325	−1.47046	−0.735230	+	0.677817i	$0.762926\pi$
$702$	0	0
$703$	32.4230	1.22286
$704$	−64.5894	−2.43430
$705$	0	0
$706$	36.6296	1.37857
$707$	3.92878	0.147757
$708$	0	0
$709$	−30.0296	−1.12779	−0.563893	−	0.825848i	$-0.690697\pi$
$709$	−30.0296	−1.12779	−0.563893	+	0.825848i	$0.690697\pi$
$710$	15.3975	0.577857
$711$	0	0
$712$	−93.6151	−3.50837
$713$	31.1969	1.16833
$714$	0	0
$715$	−5.49183	−0.205383
$716$	−54.6325	−2.04171
$717$	0	0
$718$	−3.70481	−0.138262
$719$	16.7389	0.624256	0.312128	−	0.950040i	$-0.398958\pi$
$719$	16.7389	0.624256	0.312128	+	0.950040i	$0.398958\pi$
$720$	0	0
$721$	−14.4451	−0.537965
$722$	−4.29436	−0.159820
$723$	0	0
$724$	66.7020	2.47896
$725$	40.5511	1.50603
$726$	0	0
$727$	−32.8844	−1.21961	−0.609807	−	0.792550i	$-0.708753\pi$
$727$	−32.8844	−1.21961	−0.609807	+	0.792550i	$0.708753\pi$
$728$	10.2308	0.379177
$729$	0	0
$730$	23.1055	0.855172
$731$	−1.27458	−0.0471421
$732$	0	0
$733$	4.38602	0.162001	0.0810006	−	0.996714i	$-0.474188\pi$
$733$	4.38602	0.162001	0.0810006	+	0.996714i	$0.474188\pi$
$734$	75.8579	2.79997
$735$	0	0
$736$	62.5285	2.30483
$737$	−15.4572	−0.569373
$738$	0	0
$739$	43.9166	1.61550	0.807749	−	0.589526i	$-0.200686\pi$
$739$	43.9166	1.61550	0.807749	+	0.589526i	$0.200686\pi$
$740$	39.4538	1.45035
$741$	0	0
$742$	20.3247	0.746145
$743$	−23.4591	−0.860632	−0.430316	−	0.902678i	$-0.641598\pi$
$743$	−23.4591	−0.860632	−0.430316	+	0.902678i	$0.641598\pi$
$744$	0	0
$745$	−17.9509	−0.657669
$746$	−47.6938	−1.74619
$747$	0	0
$748$	40.1660	1.46862
$749$	9.48196	0.346463
$750$	0	0
$751$	−3.93554	−0.143610	−0.0718050	−	0.997419i	$-0.522876\pi$
$751$	−3.93554	−0.143610	−0.0718050	+	0.997419i	$0.522876\pi$
$752$	22.3755	0.815949
$753$	0	0
$754$	−35.1318	−1.27943
$755$	−1.68868	−0.0614573
$756$	0	0
$757$	28.6072	1.03975	0.519874	−	0.854243i	$-0.325979\pi$
$757$	28.6072	1.03975	0.519874	+	0.854243i	$0.325979\pi$
$758$	−22.5121	−0.817678
$759$	0	0
$760$	−40.5435	−1.47067
$761$	23.8266	0.863715	0.431858	−	0.901942i	$-0.357858\pi$
$761$	23.8266	0.863715	0.431858	+	0.901942i	$0.357858\pi$
$762$	0	0
$763$	−2.81462	−0.101896
$764$	75.5363	2.73281
$765$	0	0
$766$	−65.8074	−2.37772
$767$	13.7929	0.498032
$768$	0	0
$769$	24.4356	0.881172	0.440586	−	0.897710i	$-0.354771\pi$
$769$	24.4356	0.881172	0.440586	+	0.897710i	$0.354771\pi$
$770$	−11.7593	−0.423775
$771$	0	0
$772$	−30.1219	−1.08411
$773$	1.90835	0.0686387	0.0343193	−	0.999411i	$-0.489074\pi$
$773$	1.90835	0.0686387	0.0343193	+	0.999411i	$0.489074\pi$
$774$	0	0
$775$	28.1637	1.01167
$776$	41.6975	1.49685
$777$	0	0
$778$	13.6348	0.488832
$779$	6.11323	0.219029
$780$	0	0
$781$	−21.6768	−0.775657
$782$	−21.8628	−0.781811
$783$	0	0
$784$	11.7268	0.418813
$785$	−2.51599	−0.0897996
$786$	0	0
$787$	−39.3716	−1.40345	−0.701724	−	0.712449i	$-0.747586\pi$
$787$	−39.3716	−1.40345	−0.701724	+	0.712449i	$0.747586\pi$
$788$	−132.329	−4.71402
$789$	0	0
$790$	−7.31812	−0.260367
$791$	−11.6187	−0.413113
$792$	0	0
$793$	6.75129	0.239745
$794$	34.4089	1.22113
$795$	0	0
$796$	118.932	4.21544
$797$	19.6650	0.696570	0.348285	−	0.937389i	$-0.386764\pi$
$797$	19.6650	0.696570	0.348285	+	0.937389i	$0.386764\pi$
$798$	0	0
$799$	−3.70072	−0.130922
$800$	56.4491	1.99578
$801$	0	0
$802$	−29.7844	−1.05172
$803$	−32.5282	−1.14790
$804$	0	0
$805$	4.59509	0.161956
$806$	−24.3999	−0.859451
$807$	0	0
$808$	−32.3230	−1.13712
$809$	−0.491017	−0.0172632	−0.00863162	−	0.999963i	$-0.502748\pi$
$809$	−0.491017	−0.0172632	−0.00863162	+	0.999963i	$0.502748\pi$
$810$	0	0
$811$	−2.86011	−0.100432	−0.0502161	−	0.998738i	$-0.515991\pi$
$811$	−2.86011	−0.100432	−0.0502161	+	0.998738i	$0.515991\pi$
$812$	−54.0048	−1.89520
$813$	0	0
$814$	−77.3688	−2.71178
$815$	−25.4129	−0.890176
$816$	0	0
$817$	−2.98362	−0.104384
$818$	14.6163	0.511047
$819$	0	0
$820$	7.43884	0.259776
$821$	−43.7812	−1.52797	−0.763987	−	0.645232i	$-0.776761\pi$
$821$	−43.7812	−1.52797	−0.763987	+	0.645232i	$0.776761\pi$
$822$	0	0
$823$	−39.4581	−1.37542	−0.687712	−	0.725984i	$-0.741385\pi$
$823$	−39.4581	−1.37542	−0.687712	+	0.725984i	$0.741385\pi$
$824$	118.843	4.14011
$825$	0	0
$826$	29.5338	1.02761
$827$	41.9776	1.45970	0.729851	−	0.683606i	$-0.239589\pi$
$827$	41.9776	1.45970	0.729851	+	0.683606i	$0.239589\pi$
$828$	0	0
$829$	−23.2114	−0.806166	−0.403083	−	0.915163i	$-0.632061\pi$
$829$	−23.2114	−0.806166	−0.403083	+	0.915163i	$0.632061\pi$
$830$	−34.3289	−1.19157
$831$	0	0
$832$	−19.7402	−0.684370
$833$	−1.93951	−0.0672001
$834$	0	0
$835$	18.5346	0.641416
$836$	94.0232	3.25186
$837$	0	0
$838$	26.2403	0.906457
$839$	25.6545	0.885691	0.442845	−	0.896598i	$-0.353969\pi$
$839$	25.6545	0.885691	0.442845	+	0.896598i	$0.353969\pi$
$840$	0	0
$841$	83.5789	2.88203
$842$	43.2536	1.49062
$843$	0	0
$844$	−49.6898	−1.71039
$845$	12.4321	0.427676
$846$	0	0
$847$	5.55487	0.190868
$848$	−89.5128	−3.07388
$849$	0	0
$850$	−19.7371	−0.676978
$851$	30.2329	1.03637
$852$	0	0
$853$	−51.7352	−1.77138	−0.885690	−	0.464278i	$-0.846314\pi$
$853$	−51.7352	−1.77138	−0.885690	+	0.464278i	$0.846314\pi$
$854$	14.4561	0.494677
$855$	0	0
$856$	−78.0103	−2.66634
$857$	−26.1881	−0.894567	−0.447284	−	0.894392i	$-0.647609\pi$
$857$	−26.1881	−0.894567	−0.447284	+	0.894392i	$0.647609\pi$
$858$	0	0
$859$	18.1240	0.618381	0.309191	−	0.951000i	$-0.399942\pi$
$859$	18.1240	0.618381	0.309191	+	0.951000i	$0.399942\pi$
$860$	−3.63060	−0.123803
$861$	0	0
$862$	10.4208	0.354933
$863$	26.7519	0.910645	0.455322	−	0.890327i	$-0.349524\pi$
$863$	26.7519	0.910645	0.455322	+	0.890327i	$0.349524\pi$
$864$	0	0
$865$	−22.9235	−0.779424
$866$	100.019	3.39878
$867$	0	0
$868$	−37.5077	−1.27309
$869$	10.3026	0.349490
$870$	0	0
$871$	−4.72413	−0.160071
$872$	23.1565	0.784179
$873$	0	0
$874$	−51.1778	−1.73111
$875$	9.57546	0.323710
$876$	0	0
$877$	−10.8431	−0.366144	−0.183072	−	0.983100i	$-0.558604\pi$
$877$	−10.8431	−0.366144	−0.183072	+	0.983100i	$0.558604\pi$
$878$	27.8603	0.940241
$879$	0	0
$880$	51.7894	1.74582
$881$	−5.37159	−0.180973	−0.0904867	−	0.995898i	$-0.528842\pi$
$881$	−5.37159	−0.180973	−0.0904867	+	0.995898i	$0.528842\pi$
$882$	0	0
$883$	−21.4864	−0.723074	−0.361537	−	0.932358i	$-0.617748\pi$
$883$	−21.4864	−0.723074	−0.361537	+	0.932358i	$0.617748\pi$
$884$	12.2758	0.412880
$885$	0	0
$886$	2.13895	0.0718596
$887$	26.3311	0.884110	0.442055	−	0.896988i	$-0.354250\pi$
$887$	26.3311	0.884110	0.442055	+	0.896988i	$0.354250\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	32.8859	1.10234
$891$	0	0
$892$	63.6778	2.13209
$893$	−8.66288	−0.289892
$894$	0	0
$895$	11.6506	0.389435
$896$	−12.7283	−0.425221
$897$	0	0
$898$	−34.1863	−1.14081
$899$	78.1888	2.60774
$900$	0	0
$901$	14.8047	0.493216
$902$	−14.5876	−0.485713
$903$	0	0
$904$	95.5897	3.17926
$905$	−14.2244	−0.472836
$906$	0	0
$907$	−26.4922	−0.879660	−0.439830	−	0.898081i	$-0.644961\pi$
$907$	−26.4922	−0.879660	−0.439830	+	0.898081i	$0.644961\pi$
$908$	38.9245	1.29175
$909$	0	0
$910$	−3.59395	−0.119138
$911$	11.9558	0.396114	0.198057	−	0.980190i	$-0.436537\pi$
$911$	11.9558	0.396114	0.198057	+	0.980190i	$0.436537\pi$
$912$	0	0
$913$	48.3287	1.59945
$914$	49.6289	1.64158
$915$	0	0
$916$	46.2853	1.52931
$917$	15.1975	0.501865
$918$	0	0
$919$	34.4504	1.13641	0.568207	−	0.822886i	$-0.307637\pi$
$919$	34.4504	1.13641	0.568207	+	0.822886i	$0.307637\pi$
$920$	−37.8049	−1.24639
$921$	0	0
$922$	100.726	3.31725
$923$	−6.62501	−0.218065
$924$	0	0
$925$	27.2935	0.897404
$926$	28.3887	0.932912
$927$	0	0
$928$	156.715	5.14443
$929$	1.70003	0.0557760	0.0278880	−	0.999611i	$-0.491122\pi$
$929$	1.70003	0.0557760	0.0278880	+	0.999611i	$0.491122\pi$
$930$	0	0
$931$	−4.54013	−0.148797
$932$	12.5982	0.412668
$933$	0	0
$934$	71.7704	2.34840
$935$	−8.56554	−0.280123
$936$	0	0
$937$	38.5457	1.25923	0.629617	−	0.776905i	$-0.283212\pi$
$937$	38.5457	1.25923	0.629617	+	0.776905i	$0.283212\pi$
$938$	−10.1155	−0.330281
$939$	0	0
$940$	−10.5414	−0.343822
$941$	20.0504	0.653625	0.326812	−	0.945089i	$-0.394025\pi$
$941$	20.0504	0.653625	0.326812	+	0.945089i	$0.394025\pi$
$942$	0	0
$943$	5.70028	0.185627
$944$	−130.070	−4.23343
$945$	0	0
$946$	7.11961	0.231478
$947$	−19.4513	−0.632081	−0.316041	−	0.948746i	$-0.602354\pi$
$947$	−19.4513	−0.632081	−0.316041	+	0.948746i	$0.602354\pi$
$948$	0	0
$949$	−9.94149	−0.322715
$950$	−46.2020	−1.49899
$951$	0	0
$952$	15.9568	0.517163
$953$	2.38003	0.0770968	0.0385484	−	0.999257i	$-0.487727\pi$
$953$	2.38003	0.0770968	0.0385484	+	0.999257i	$0.487727\pi$
$954$	0	0
$955$	−16.1084	−0.521255
$956$	25.2329	0.816090
$957$	0	0
$958$	−47.7130	−1.54154
$959$	13.8972	0.448764
$960$	0	0
$961$	23.3041	0.751744
$962$	−23.6460	−0.762377
$963$	0	0
$964$	38.4544	1.23853
$965$	6.42360	0.206783
$966$	0	0
$967$	32.0300	1.03001	0.515007	−	0.857186i	$-0.327789\pi$
$967$	32.0300	1.03001	0.515007	+	0.857186i	$0.327789\pi$
$968$	−45.7012	−1.46889
$969$	0	0
$970$	−14.6478	−0.470314
$971$	13.1325	0.421442	0.210721	−	0.977546i	$-0.432419\pi$
$971$	13.1325	0.421442	0.210721	+	0.977546i	$0.432419\pi$
$972$	0	0
$973$	−9.34261	−0.299510
$974$	69.8842	2.23924
$975$	0	0
$976$	−63.6664	−2.03791
$977$	52.3055	1.67340	0.836701	−	0.547660i	$-0.184481\pi$
$977$	52.3055	1.67340	0.836701	+	0.547660i	$0.184481\pi$
$978$	0	0
$979$	−46.2972	−1.47967
$980$	−5.52463	−0.176478
$981$	0	0
$982$	−11.3920	−0.363533
$983$	6.29854	0.200892	0.100446	−	0.994943i	$-0.467973\pi$
$983$	6.29854	0.200892	0.100446	+	0.994943i	$0.467973\pi$
$984$	0	0
$985$	28.2196	0.899151
$986$	−54.7947	−1.74502
$987$	0	0
$988$	28.7360	0.914215
$989$	−2.78208	−0.0884650
$990$	0	0
$991$	−33.2804	−1.05719	−0.528593	−	0.848875i	$-0.677280\pi$
$991$	−33.2804	−1.05719	−0.528593	+	0.848875i	$0.677280\pi$
$992$	108.843	3.45576
$993$	0	0
$994$	−14.1857	−0.449943
$995$	−25.3627	−0.804051
$996$	0	0
$997$	7.55197	0.239173	0.119587	−	0.992824i	$-0.461843\pi$
$997$	7.55197	0.239173	0.119587	+	0.992824i	$0.461843\pi$
$998$	86.0077	2.72253
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.3	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.38	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.3	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.38	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66267 q^{2} +5.08984 q^{4} -1.08542 q^{5} +1.00000 q^{7} -8.22723 q^{8} +O(q^{10})\)	\(q-2.66267 q^{2} +5.08984 q^{4} -1.08542 q^{5} +1.00000 q^{7} -8.22723 q^{8} +2.89013 q^{10} -4.06877 q^{11} -1.24352 q^{13} -2.66267 q^{14} +11.7268 q^{16} -1.93951 q^{17} -4.54013 q^{19} -5.52463 q^{20} +10.8338 q^{22} -4.23345 q^{23} -3.82185 q^{25} +3.31110 q^{26} +5.08984 q^{28} -10.6103 q^{29} -7.36913 q^{31} -14.7701 q^{32} +5.16429 q^{34} -1.08542 q^{35} -7.14143 q^{37} +12.0889 q^{38} +8.93004 q^{40} -1.34649 q^{41} +0.657166 q^{43} -20.7094 q^{44} +11.2723 q^{46} +1.90807 q^{47} +1.00000 q^{49} +10.1764 q^{50} -6.32934 q^{52} -7.63321 q^{53} +4.41634 q^{55} -8.22723 q^{56} +28.2518 q^{58} -11.0918 q^{59} -5.42916 q^{61} +19.6216 q^{62} +15.8744 q^{64} +1.34975 q^{65} +3.79898 q^{67} -9.87179 q^{68} +2.89013 q^{70} +5.32761 q^{71} +7.99461 q^{73} +19.0153 q^{74} -23.1085 q^{76} -4.06877 q^{77} -2.53211 q^{79} -12.7285 q^{80} +3.58525 q^{82} -11.8780 q^{83} +2.10519 q^{85} -1.74982 q^{86} +33.4747 q^{88} +11.3787 q^{89} -1.24352 q^{91} -21.5476 q^{92} -5.08056 q^{94} +4.92797 q^{95} -5.06823 q^{97} -2.66267 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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