LMFDB - Embedded newform 8001.2.a.x.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [22,0,0,10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	\(q-0.684852 q^{2} -1.53098 q^{4} -0.182709 q^{5} +1.00000 q^{7} +2.41820 q^{8} +0.125129 q^{10} +4.23638 q^{11} +3.25646 q^{13} -0.684852 q^{14} +1.40585 q^{16} -2.73216 q^{17} +5.66272 q^{19} +0.279724 q^{20} -2.90129 q^{22} +3.08998 q^{23} -4.96662 q^{25} -2.23019 q^{26} -1.53098 q^{28} -6.70003 q^{29} -9.55396 q^{31} -5.79919 q^{32} +1.87112 q^{34} -0.182709 q^{35} -7.75698 q^{37} -3.87812 q^{38} -0.441827 q^{40} +0.852638 q^{41} -8.16565 q^{43} -6.48580 q^{44} -2.11618 q^{46} +5.27194 q^{47} +1.00000 q^{49} +3.40140 q^{50} -4.98556 q^{52} -5.94870 q^{53} -0.774025 q^{55} +2.41820 q^{56} +4.58853 q^{58} -7.70514 q^{59} -7.28184 q^{61} +6.54305 q^{62} +1.15989 q^{64} -0.594985 q^{65} +8.52437 q^{67} +4.18288 q^{68} +0.125129 q^{70} -7.08011 q^{71} -1.43264 q^{73} +5.31238 q^{74} -8.66950 q^{76} +4.23638 q^{77} +12.5663 q^{79} -0.256862 q^{80} -0.583930 q^{82} -11.1087 q^{83} +0.499191 q^{85} +5.59226 q^{86} +10.2444 q^{88} -9.91467 q^{89} +3.25646 q^{91} -4.73069 q^{92} -3.61050 q^{94} -1.03463 q^{95} -7.39495 q^{97} -0.684852 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52}+ \cdots - 56 q^{97}+O(q^{100}) $ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

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 $

$2$	−0.684852	−0.484263	−0.242132	−	0.970243i	$-0.577847\pi$
$2$	−0.684852	−0.484263	−0.242132	+	0.970243i	$0.577847\pi$
$3$	0	0
$3$	0	0
$4$	−1.53098	−0.765489
$5$	−0.182709	−0.0817100	−0.0408550	−	0.999165i	$-0.513008\pi$
$5$	−0.182709	−0.0817100	−0.0408550	+	0.999165i	$0.513008\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.41820	0.854962
$9$	0	0
$10$	0.125129	0.0395692
$11$	4.23638	1.27732	0.638658	−	0.769491i	$-0.279490\pi$
$11$	4.23638	1.27732	0.638658	+	0.769491i	$0.279490\pi$
$12$	0	0
$13$	3.25646	0.903179	0.451589	−	0.892226i	$-0.350857\pi$
$13$	3.25646	0.903179	0.451589	+	0.892226i	$0.350857\pi$
$14$	−0.684852	−0.183034
$15$	0	0
$16$	1.40585	0.351462
$17$	−2.73216	−0.662646	−0.331323	−	0.943517i	$-0.607495\pi$
$17$	−2.73216	−0.662646	−0.331323	+	0.943517i	$0.607495\pi$
$18$	0	0
$19$	5.66272	1.29912	0.649558	−	0.760312i	$-0.274954\pi$
$19$	5.66272	1.29912	0.649558	+	0.760312i	$0.274954\pi$
$20$	0.279724	0.0625481
$21$	0	0
$22$	−2.90129	−0.618557
$23$	3.08998	0.644306	0.322153	−	0.946688i	$-0.395593\pi$
$23$	3.08998	0.644306	0.322153	+	0.946688i	$0.395593\pi$
$24$	0	0
$25$	−4.96662	−0.993323
$26$	−2.23019	−0.437376
$27$	0	0
$28$	−1.53098	−0.289328
$29$	−6.70003	−1.24417	−0.622083	−	0.782952i	$-0.713713\pi$
$29$	−6.70003	−1.24417	−0.622083	+	0.782952i	$0.713713\pi$
$30$	0	0
$31$	−9.55396	−1.71594	−0.857971	−	0.513698i	$-0.828275\pi$
$31$	−9.55396	−1.71594	−0.857971	+	0.513698i	$0.828275\pi$
$32$	−5.79919	−1.02516
$33$	0	0
$34$	1.87112	0.320895
$35$	−0.182709	−0.0308835
$36$	0	0
$37$	−7.75698	−1.27524	−0.637620	−	0.770351i	$-0.720081\pi$
$37$	−7.75698	−1.27524	−0.637620	+	0.770351i	$0.720081\pi$
$38$	−3.87812	−0.629115
$39$	0	0
$40$	−0.441827	−0.0698590
$41$	0.852638	0.133160	0.0665798	−	0.997781i	$-0.478791\pi$
$41$	0.852638	0.133160	0.0665798	+	0.997781i	$0.478791\pi$
$42$	0	0
$43$	−8.16565	−1.24525	−0.622625	−	0.782521i	$-0.713934\pi$
$43$	−8.16565	−1.24525	−0.622625	+	0.782521i	$0.713934\pi$
$44$	−6.48580	−0.977771
$45$	0	0
$46$	−2.11618	−0.312014
$47$	5.27194	0.768991	0.384496	−	0.923127i	$-0.374375\pi$
$47$	5.27194	0.768991	0.384496	+	0.923127i	$0.374375\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.40140	0.481030
$51$	0	0
$52$	−4.98556	−0.691373
$53$	−5.94870	−0.817116	−0.408558	−	0.912732i	$-0.633968\pi$
$53$	−5.94870	−0.817116	−0.408558	+	0.912732i	$0.633968\pi$
$54$	0	0
$55$	−0.774025	−0.104370
$56$	2.41820	0.323145
$57$	0	0
$58$	4.58853	0.602504
$59$	−7.70514	−1.00312	−0.501562	−	0.865122i	$-0.667241\pi$
$59$	−7.70514	−1.00312	−0.501562	+	0.865122i	$0.667241\pi$
$60$	0	0
$61$	−7.28184	−0.932345	−0.466172	−	0.884694i	$-0.654367\pi$
$61$	−7.28184	−0.932345	−0.466172	+	0.884694i	$0.654367\pi$
$62$	6.54305	0.830968
$63$	0	0
$64$	1.15989	0.144986
$65$	−0.594985	−0.0737988
$66$	0	0
$67$	8.52437	1.04142	0.520709	−	0.853734i	$-0.325668\pi$
$67$	8.52437	1.04142	0.520709	+	0.853734i	$0.325668\pi$
$68$	4.18288	0.507248
$69$	0	0
$70$	0.125129	0.0149557
$71$	−7.08011	−0.840254	−0.420127	−	0.907465i	$-0.638014\pi$
$71$	−7.08011	−0.840254	−0.420127	+	0.907465i	$0.638014\pi$
$72$	0	0
$73$	−1.43264	−0.167678	−0.0838388	−	0.996479i	$-0.526718\pi$
$73$	−1.43264	−0.167678	−0.0838388	+	0.996479i	$0.526718\pi$
$74$	5.31238	0.617552
$75$	0	0
$76$	−8.66950	−0.994459
$77$	4.23638	0.482780
$78$	0	0
$79$	12.5663	1.41382	0.706910	−	0.707303i	$-0.250089\pi$
$79$	12.5663	1.41382	0.706910	+	0.707303i	$0.250089\pi$
$80$	−0.256862	−0.0287180
$81$	0	0
$82$	−0.583930	−0.0644843
$83$	−11.1087	−1.21934	−0.609670	−	0.792655i	$-0.708698\pi$
$83$	−11.1087	−1.21934	−0.609670	+	0.792655i	$0.708698\pi$
$84$	0	0
$85$	0.499191	0.0541448
$86$	5.59226	0.603029
$87$	0	0
$88$	10.2444	1.09206
$89$	−9.91467	−1.05095	−0.525476	−	0.850808i	$-0.676113\pi$
$89$	−9.91467	−1.05095	−0.525476	+	0.850808i	$0.676113\pi$
$90$	0	0
$91$	3.25646	0.341369
$92$	−4.73069	−0.493209
$93$	0	0
$94$	−3.61050	−0.372394
$95$	−1.03463	−0.106151
$96$	0	0
$97$	−7.39495	−0.750843	−0.375422	−	0.926854i	$-0.622502\pi$
$97$	−7.39495	−0.750843	−0.375422	+	0.926854i	$0.622502\pi$
$98$	−0.684852	−0.0691805
$99$	0	0
$100$	7.60378	0.760378
$101$	12.5594	1.24971	0.624853	−	0.780742i	$-0.285159\pi$
$101$	12.5594	1.24971	0.624853	+	0.780742i	$0.285159\pi$
$102$	0	0
$103$	−1.37085	−0.135074	−0.0675369	−	0.997717i	$-0.521514\pi$
$103$	−1.37085	−0.135074	−0.0675369	+	0.997717i	$0.521514\pi$
$104$	7.87475	0.772183
$105$	0	0
$106$	4.07398	0.395700
$107$	7.25917	0.701771	0.350885	−	0.936418i	$-0.385881\pi$
$107$	7.25917	0.701771	0.350885	+	0.936418i	$0.385881\pi$
$108$	0	0
$109$	5.47375	0.524290	0.262145	−	0.965028i	$-0.415570\pi$
$109$	5.47375	0.524290	0.262145	+	0.965028i	$0.415570\pi$
$110$	0.530093	0.0505423
$111$	0	0
$112$	1.40585	0.132840
$113$	−6.45405	−0.607146	−0.303573	−	0.952808i	$-0.598180\pi$
$113$	−6.45405	−0.607146	−0.303573	+	0.952808i	$0.598180\pi$
$114$	0	0
$115$	−0.564568	−0.0526463
$116$	10.2576	0.952395
$117$	0	0
$118$	5.27688	0.485776
$119$	−2.73216	−0.250457
$120$	0	0
$121$	6.94689	0.631536
$122$	4.98698	0.451500
$123$	0	0
$124$	14.6269	1.31354
$125$	1.82099	0.162875
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	10.8040	0.954951
$129$	0	0
$130$	0.407476	0.0357380
$131$	19.0900	1.66790	0.833951	−	0.551839i	$-0.186074\pi$
$131$	19.0900	1.66790	0.833951	+	0.551839i	$0.186074\pi$
$132$	0	0
$133$	5.66272	0.491020
$134$	−5.83793	−0.504321
$135$	0	0
$136$	−6.60690	−0.566537
$137$	8.01951	0.685153	0.342576	−	0.939490i	$-0.388700\pi$
$137$	8.01951	0.685153	0.342576	+	0.939490i	$0.388700\pi$
$138$	0	0
$139$	−11.3101	−0.959306	−0.479653	−	0.877458i	$-0.659237\pi$
$139$	−11.3101	−0.959306	−0.479653	+	0.877458i	$0.659237\pi$
$140$	0.279724	0.0236410
$141$	0	0
$142$	4.84882	0.406904
$143$	13.7956	1.15364
$144$	0	0
$145$	1.22416	0.101661
$146$	0.981145	0.0812001
$147$	0	0
$148$	11.8758	0.976182
$149$	−1.63779	−0.134173	−0.0670866	−	0.997747i	$-0.521370\pi$
$149$	−1.63779	−0.134173	−0.0670866	+	0.997747i	$0.521370\pi$
$150$	0	0
$151$	−21.2446	−1.72886	−0.864429	−	0.502756i	$-0.832320\pi$
$151$	−21.2446	−1.72886	−0.864429	+	0.502756i	$0.832320\pi$
$152$	13.6936	1.11069
$153$	0	0
$154$	−2.90129	−0.233793
$155$	1.74560	0.140210
$156$	0	0
$157$	−16.3492	−1.30481	−0.652405	−	0.757870i	$-0.726240\pi$
$157$	−16.3492	−1.30481	−0.652405	+	0.757870i	$0.726240\pi$
$158$	−8.60606	−0.684661
$159$	0	0
$160$	1.05957	0.0837660
$161$	3.08998	0.243525
$162$	0	0
$163$	9.20865	0.721277	0.360638	−	0.932706i	$-0.382559\pi$
$163$	9.20865	0.721277	0.360638	+	0.932706i	$0.382559\pi$
$164$	−1.30537	−0.101932
$165$	0	0
$166$	7.60782	0.590482
$167$	11.5029	0.890120	0.445060	−	0.895501i	$-0.353182\pi$
$167$	11.5029	0.890120	0.445060	+	0.895501i	$0.353182\pi$
$168$	0	0
$169$	−2.39549	−0.184268
$170$	−0.341872	−0.0262204
$171$	0	0
$172$	12.5014	0.953225
$173$	−23.7780	−1.80781	−0.903903	−	0.427737i	$-0.859311\pi$
$173$	−23.7780	−1.80781	−0.903903	+	0.427737i	$0.859311\pi$
$174$	0	0
$175$	−4.96662	−0.375441
$176$	5.95571	0.448928
$177$	0	0
$178$	6.79008	0.508938
$179$	19.0210	1.42169	0.710846	−	0.703347i	$-0.248312\pi$
$179$	19.0210	1.42169	0.710846	+	0.703347i	$0.248312\pi$
$180$	0	0
$181$	−11.1013	−0.825151	−0.412575	−	0.910923i	$-0.635371\pi$
$181$	−11.1013	−0.825151	−0.412575	+	0.910923i	$0.635371\pi$
$182$	−2.23019	−0.165313
$183$	0	0
$184$	7.47219	0.550857
$185$	1.41727	0.104200
$186$	0	0
$187$	−11.5745	−0.846408
$188$	−8.07122	−0.588654
$189$	0	0
$190$	0.708569	0.0514050
$191$	−17.3402	−1.25469	−0.627347	−	0.778740i	$-0.715859\pi$
$191$	−17.3402	−1.25469	−0.627347	+	0.778740i	$0.715859\pi$
$192$	0	0
$193$	−16.7667	−1.20689	−0.603445	−	0.797404i	$-0.706206\pi$
$193$	−16.7667	−1.20689	−0.603445	+	0.797404i	$0.706206\pi$
$194$	5.06445	0.363606
$195$	0	0
$196$	−1.53098	−0.109356
$197$	5.78061	0.411852	0.205926	−	0.978568i	$-0.433979\pi$
$197$	5.78061	0.411852	0.205926	+	0.978568i	$0.433979\pi$
$198$	0	0
$199$	−2.52069	−0.178687	−0.0893433	−	0.996001i	$-0.528477\pi$
$199$	−2.52069	−0.178687	−0.0893433	+	0.996001i	$0.528477\pi$
$200$	−12.0103	−0.849254
$201$	0	0
$202$	−8.60133	−0.605187
$203$	−6.70003	−0.470250
$204$	0	0
$205$	−0.155785	−0.0108805
$206$	0.938829	0.0654113
$207$	0	0
$208$	4.57809	0.317433
$209$	23.9894	1.65938
$210$	0	0
$211$	−11.8740	−0.817437	−0.408719	−	0.912660i	$-0.634024\pi$
$211$	−11.8740	−0.817437	−0.408719	+	0.912660i	$0.634024\pi$
$212$	9.10732	0.625494
$213$	0	0
$214$	−4.97146	−0.339842
$215$	1.49194	0.101749
$216$	0	0
$217$	−9.55396	−0.648565
$218$	−3.74871	−0.253894
$219$	0	0
$220$	1.18502	0.0798937
$221$	−8.89716	−0.598488
$222$	0	0
$223$	−26.7449	−1.79097	−0.895485	−	0.445092i	$-0.853171\pi$
$223$	−26.7449	−1.79097	−0.895485	+	0.445092i	$0.853171\pi$
$224$	−5.79919	−0.387475
$225$	0	0
$226$	4.42007	0.294018
$227$	25.5670	1.69694	0.848472	−	0.529241i	$-0.177523\pi$
$227$	25.5670	1.69694	0.848472	+	0.529241i	$0.177523\pi$
$228$	0	0
$229$	−26.4599	−1.74852	−0.874261	−	0.485457i	$-0.838653\pi$
$229$	−26.4599	−1.74852	−0.874261	+	0.485457i	$0.838653\pi$
$230$	0.386646	0.0254947
$231$	0	0
$232$	−16.2020	−1.06371
$233$	−10.1130	−0.662526	−0.331263	−	0.943538i	$-0.607475\pi$
$233$	−10.1130	−0.662526	−0.331263	+	0.943538i	$0.607475\pi$
$234$	0	0
$235$	−0.963231	−0.0628343
$236$	11.7964	0.767880
$237$	0	0
$238$	1.87112	0.121287
$239$	5.64416	0.365090	0.182545	−	0.983197i	$-0.441566\pi$
$239$	5.64416	0.365090	0.182545	+	0.983197i	$0.441566\pi$
$240$	0	0
$241$	−2.78563	−0.179438	−0.0897192	−	0.995967i	$-0.528597\pi$
$241$	−2.78563	−0.179438	−0.0897192	+	0.995967i	$0.528597\pi$
$242$	−4.75759	−0.305830
$243$	0	0
$244$	11.1483	0.713699
$245$	−0.182709	−0.0116729
$246$	0	0
$247$	18.4404	1.17333
$248$	−23.1034	−1.46707
$249$	0	0
$250$	−1.24711	−0.0788742
$251$	15.8489	1.00038	0.500188	−	0.865917i	$-0.333264\pi$
$251$	15.8489	1.00038	0.500188	+	0.865917i	$0.333264\pi$
$252$	0	0
$253$	13.0903	0.822982
$254$	−0.684852	−0.0429714
$255$	0	0
$256$	−9.71894	−0.607434
$257$	−23.0355	−1.43691	−0.718457	−	0.695571i	$-0.755151\pi$
$257$	−23.0355	−1.43691	−0.718457	+	0.695571i	$0.755151\pi$
$258$	0	0
$259$	−7.75698	−0.481995
$260$	0.910908	0.0564921
$261$	0	0
$262$	−13.0738	−0.807704
$263$	21.0237	1.29638	0.648189	−	0.761479i	$-0.275527\pi$
$263$	21.0237	1.29638	0.648189	+	0.761479i	$0.275527\pi$
$264$	0	0
$265$	1.08688	0.0667666
$266$	−3.87812	−0.237783
$267$	0	0
$268$	−13.0506	−0.797194
$269$	25.1761	1.53501	0.767507	−	0.641040i	$-0.221497\pi$
$269$	25.1761	1.53501	0.767507	+	0.641040i	$0.221497\pi$
$270$	0	0
$271$	7.93418	0.481967	0.240983	−	0.970529i	$-0.422530\pi$
$271$	7.93418	0.481967	0.240983	+	0.970529i	$0.422530\pi$
$272$	−3.84100	−0.232895
$273$	0	0
$274$	−5.49217	−0.331794
$275$	−21.0405	−1.26879
$276$	0	0
$277$	−13.7272	−0.824786	−0.412393	−	0.911006i	$-0.635307\pi$
$277$	−13.7272	−0.824786	−0.412393	+	0.911006i	$0.635307\pi$
$278$	7.74571	0.464557
$279$	0	0
$280$	−0.441827	−0.0264042
$281$	−10.3547	−0.617711	−0.308855	−	0.951109i	$-0.599946\pi$
$281$	−10.3547	−0.617711	−0.308855	+	0.951109i	$0.599946\pi$
$282$	0	0
$283$	11.3223	0.673038	0.336519	−	0.941677i	$-0.390750\pi$
$283$	11.3223	0.673038	0.336519	+	0.941677i	$0.390750\pi$
$284$	10.8395	0.643205
$285$	0	0
$286$	−9.44793	−0.558668
$287$	0.852638	0.0503296
$288$	0	0
$289$	−9.53530	−0.560900
$290$	−0.838367	−0.0492306
$291$	0	0
$292$	2.19334	0.128355
$293$	−0.941792	−0.0550200	−0.0275100	−	0.999622i	$-0.508758\pi$
$293$	−0.941792	−0.0550200	−0.0275100	+	0.999622i	$0.508758\pi$
$294$	0	0
$295$	1.40780	0.0819653
$296$	−18.7579	−1.09028
$297$	0	0
$298$	1.12165	0.0649752
$299$	10.0624	0.581923
$300$	0	0
$301$	−8.16565	−0.470660
$302$	14.5494	0.837222
$303$	0	0
$304$	7.96093	0.456590
$305$	1.33046	0.0761819
$306$	0	0
$307$	−10.0831	−0.575473	−0.287737	−	0.957710i	$-0.592903\pi$
$307$	−10.0831	−0.575473	−0.287737	+	0.957710i	$0.592903\pi$
$308$	−6.48580	−0.369563
$309$	0	0
$310$	−1.19548	−0.0678984
$311$	6.79471	0.385292	0.192646	−	0.981268i	$-0.438293\pi$
$311$	6.79471	0.385292	0.192646	+	0.981268i	$0.438293\pi$
$312$	0	0
$313$	3.80105	0.214848	0.107424	−	0.994213i	$-0.465740\pi$
$313$	3.80105	0.214848	0.107424	+	0.994213i	$0.465740\pi$
$314$	11.1968	0.631872
$315$	0	0
$316$	−19.2387	−1.08226
$317$	−0.0825552	−0.00463676	−0.00231838	−	0.999997i	$-0.500738\pi$
$317$	−0.0825552	−0.00463676	−0.00231838	+	0.999997i	$0.500738\pi$
$318$	0	0
$319$	−28.3839	−1.58919
$320$	−0.211922	−0.0118468
$321$	0	0
$322$	−2.11618	−0.117930
$323$	−15.4714	−0.860854
$324$	0	0
$325$	−16.1736	−0.897149
$326$	−6.30656	−0.349288
$327$	0	0
$328$	2.06185	0.113846
$329$	5.27194	0.290651
$330$	0	0
$331$	−31.6223	−1.73812	−0.869059	−	0.494709i	$-0.835275\pi$
$331$	−31.6223	−1.73812	−0.869059	+	0.494709i	$0.835275\pi$
$332$	17.0072	0.933391
$333$	0	0
$334$	−7.87777	−0.431053
$335$	−1.55748	−0.0850943
$336$	0	0
$337$	21.0391	1.14607	0.573037	−	0.819530i	$-0.305765\pi$
$337$	21.0391	1.14607	0.573037	+	0.819530i	$0.305765\pi$
$338$	1.64055	0.0892344
$339$	0	0
$340$	−0.764250	−0.0414473
$341$	−40.4742	−2.19180
$342$	0	0
$343$	1.00000	0.0539949
$344$	−19.7461	−1.06464
$345$	0	0
$346$	16.2844	0.875454
$347$	−0.860099	−0.0461725	−0.0230863	−	0.999733i	$-0.507349\pi$
$347$	−0.860099	−0.0461725	−0.0230863	+	0.999733i	$0.507349\pi$
$348$	0	0
$349$	−6.72368	−0.359910	−0.179955	−	0.983675i	$-0.557595\pi$
$349$	−6.72368	−0.359910	−0.179955	+	0.983675i	$0.557595\pi$
$350$	3.40140	0.181812
$351$	0	0
$352$	−24.5676	−1.30946
$353$	7.98795	0.425156	0.212578	−	0.977144i	$-0.431814\pi$
$353$	7.98795	0.425156	0.212578	+	0.977144i	$0.431814\pi$
$354$	0	0
$355$	1.29360	0.0686572
$356$	15.1791	0.804492
$357$	0	0
$358$	−13.0265	−0.688474
$359$	23.6994	1.25080	0.625402	−	0.780302i	$-0.284935\pi$
$359$	23.6994	1.25080	0.625402	+	0.780302i	$0.284935\pi$
$360$	0	0
$361$	13.0664	0.687704
$362$	7.60273	0.399590
$363$	0	0
$364$	−4.98556	−0.261315
$365$	0.261756	0.0137009
$366$	0	0
$367$	9.85990	0.514683	0.257341	−	0.966321i	$-0.417154\pi$
$367$	9.85990	0.514683	0.257341	+	0.966321i	$0.417154\pi$
$368$	4.34405	0.226449
$369$	0	0
$370$	−0.970621	−0.0504602
$371$	−5.94870	−0.308841
$372$	0	0
$373$	−16.6669	−0.862982	−0.431491	−	0.902117i	$-0.642012\pi$
$373$	−16.6669	−0.862982	−0.431491	+	0.902117i	$0.642012\pi$
$374$	7.92679	0.409885
$375$	0	0
$376$	12.7486	0.657458
$377$	−21.8184	−1.12370
$378$	0	0
$379$	13.6371	0.700491	0.350245	−	0.936658i	$-0.386098\pi$
$379$	13.6371	0.700491	0.350245	+	0.936658i	$0.386098\pi$
$380$	1.58400	0.0812573
$381$	0	0
$382$	11.8755	0.607603
$383$	12.0694	0.616718	0.308359	−	0.951270i	$-0.400220\pi$
$383$	12.0694	0.616718	0.308359	+	0.951270i	$0.400220\pi$
$384$	0	0
$385$	−0.774025	−0.0394480
$386$	11.4827	0.584453
$387$	0	0
$388$	11.3215	0.574762
$389$	−30.2684	−1.53467	−0.767335	−	0.641246i	$-0.778418\pi$
$389$	−30.2684	−1.53467	−0.767335	+	0.641246i	$0.778418\pi$
$390$	0	0
$391$	−8.44233	−0.426947
$392$	2.41820	0.122137
$393$	0	0
$394$	−3.95886	−0.199445
$395$	−2.29598	−0.115523
$396$	0	0
$397$	10.2326	0.513561	0.256780	−	0.966470i	$-0.417338\pi$
$397$	10.2326	0.513561	0.256780	+	0.966470i	$0.417338\pi$
$398$	1.72630	0.0865314
$399$	0	0
$400$	−6.98231	−0.349116
$401$	−22.6896	−1.13307	−0.566533	−	0.824039i	$-0.691716\pi$
$401$	−22.6896	−1.13307	−0.566533	+	0.824039i	$0.691716\pi$
$402$	0	0
$403$	−31.1121	−1.54980
$404$	−19.2282	−0.956637
$405$	0	0
$406$	4.58853	0.227725
$407$	−32.8615	−1.62888
$408$	0	0
$409$	29.9744	1.48214	0.741069	−	0.671429i	$-0.234319\pi$
$409$	29.9744	1.48214	0.741069	+	0.671429i	$0.234319\pi$
$410$	0.106689	0.00526902
$411$	0	0
$412$	2.09874	0.103397
$413$	−7.70514	−0.379145
$414$	0	0
$415$	2.02966	0.0996323
$416$	−18.8848	−0.925904
$417$	0	0
$418$	−16.4292	−0.803578
$419$	2.69907	0.131858	0.0659290	−	0.997824i	$-0.478999\pi$
$419$	2.69907	0.131858	0.0659290	+	0.997824i	$0.478999\pi$
$420$	0	0
$421$	−0.141470	−0.00689482	−0.00344741	−	0.999994i	$-0.501097\pi$
$421$	−0.141470	−0.00689482	−0.00344741	+	0.999994i	$0.501097\pi$
$422$	8.13191	0.395855
$423$	0	0
$424$	−14.3851	−0.698603
$425$	13.5696	0.658222
$426$	0	0
$427$	−7.28184	−0.352393
$428$	−11.1136	−0.537198
$429$	0	0
$430$	−1.02176	−0.0492735
$431$	30.3884	1.46376	0.731880	−	0.681434i	$-0.238643\pi$
$431$	30.3884	1.46376	0.731880	+	0.681434i	$0.238643\pi$
$432$	0	0
$433$	13.7113	0.658923	0.329461	−	0.944169i	$-0.393133\pi$
$433$	13.7113	0.658923	0.329461	+	0.944169i	$0.393133\pi$
$434$	6.54305	0.314076
$435$	0	0
$436$	−8.38019	−0.401338
$437$	17.4977	0.837028
$438$	0	0
$439$	20.3026	0.968992	0.484496	−	0.874794i	$-0.339003\pi$
$439$	20.3026	0.968992	0.484496	+	0.874794i	$0.339003\pi$
$440$	−1.87175	−0.0892320
$441$	0	0
$442$	6.09324	0.289826
$443$	37.7781	1.79489	0.897445	−	0.441126i	$-0.145421\pi$
$443$	37.7781	1.79489	0.897445	+	0.441126i	$0.145421\pi$
$444$	0	0
$445$	1.81150	0.0858734
$446$	18.3163	0.867301
$447$	0	0
$448$	1.15989	0.0547996
$449$	−4.11605	−0.194248	−0.0971242	−	0.995272i	$-0.530964\pi$
$449$	−4.11605	−0.194248	−0.0971242	+	0.995272i	$0.530964\pi$
$450$	0	0
$451$	3.61209	0.170087
$452$	9.88100	0.464763
$453$	0	0
$454$	−17.5096	−0.821767
$455$	−0.594985	−0.0278933
$456$	0	0
$457$	−10.6147	−0.496533	−0.248267	−	0.968692i	$-0.579861\pi$
$457$	−10.6147	−0.496533	−0.248267	+	0.968692i	$0.579861\pi$
$458$	18.1211	0.846745
$459$	0	0
$460$	0.864342	0.0403001
$461$	−36.1999	−1.68600	−0.842998	−	0.537916i	$-0.819212\pi$
$461$	−36.1999	−1.68600	−0.842998	+	0.537916i	$0.819212\pi$
$462$	0	0
$463$	−3.39773	−0.157906	−0.0789530	−	0.996878i	$-0.525158\pi$
$463$	−3.39773	−0.157906	−0.0789530	+	0.996878i	$0.525158\pi$
$464$	−9.41924	−0.437277
$465$	0	0
$466$	6.92592	0.320837
$467$	−25.8244	−1.19501	−0.597506	−	0.801865i	$-0.703841\pi$
$467$	−25.8244	−1.19501	−0.597506	+	0.801865i	$0.703841\pi$
$468$	0	0
$469$	8.52437	0.393619
$470$	0.659671	0.0304283
$471$	0	0
$472$	−18.6326	−0.857633
$473$	−34.5928	−1.59058
$474$	0	0
$475$	−28.1246	−1.29044
$476$	4.18288	0.191722
$477$	0	0
$478$	−3.86541	−0.176800
$479$	−0.309258	−0.0141304	−0.00706518	−	0.999975i	$-0.502249\pi$
$479$	−0.309258	−0.0141304	−0.00706518	+	0.999975i	$0.502249\pi$
$480$	0	0
$481$	−25.2603	−1.15177
$482$	1.90775	0.0868954
$483$	0	0
$484$	−10.6355	−0.483434
$485$	1.35113	0.0613514
$486$	0	0
$487$	−4.17161	−0.189034	−0.0945169	−	0.995523i	$-0.530131\pi$
$487$	−4.17161	−0.189034	−0.0945169	+	0.995523i	$0.530131\pi$
$488$	−17.6089	−0.797119
$489$	0	0
$490$	0.125129	0.00565274
$491$	−40.2311	−1.81560	−0.907801	−	0.419401i	$-0.862240\pi$
$491$	−40.2311	−1.81560	−0.907801	+	0.419401i	$0.862240\pi$
$492$	0	0
$493$	18.3056	0.824441
$494$	−12.6289	−0.568203
$495$	0	0
$496$	−13.4314	−0.603089
$497$	−7.08011	−0.317586
$498$	0	0
$499$	7.20374	0.322484	0.161242	−	0.986915i	$-0.448450\pi$
$499$	7.20374	0.322484	0.161242	+	0.986915i	$0.448450\pi$
$500$	−2.78790	−0.124679
$501$	0	0
$502$	−10.8542	−0.484445
$503$	10.8960	0.485830	0.242915	−	0.970048i	$-0.421896\pi$
$503$	10.8960	0.485830	0.242915	+	0.970048i	$0.421896\pi$
$504$	0	0
$505$	−2.29472	−0.102114
$506$	−8.96494	−0.398540
$507$	0	0
$508$	−1.53098	−0.0679262
$509$	7.55836	0.335019	0.167509	−	0.985871i	$-0.446428\pi$
$509$	7.55836	0.335019	0.167509	+	0.985871i	$0.446428\pi$
$510$	0	0
$511$	−1.43264	−0.0633762
$512$	−14.9520	−0.660793
$513$	0	0
$514$	15.7759	0.695845
$515$	0.250467	0.0110369
$516$	0	0
$517$	22.3339	0.982244
$518$	5.31238	0.233413
$519$	0	0
$520$	−1.43879	−0.0630951
$521$	7.79661	0.341576	0.170788	−	0.985308i	$-0.445369\pi$
$521$	7.79661	0.341576	0.170788	+	0.985308i	$0.445369\pi$
$522$	0	0
$523$	−4.65311	−0.203466	−0.101733	−	0.994812i	$-0.532439\pi$
$523$	−4.65311	−0.203466	−0.101733	+	0.994812i	$0.532439\pi$
$524$	−29.2264	−1.27676
$525$	0	0
$526$	−14.3981	−0.627789
$527$	26.1030	1.13706
$528$	0	0
$529$	−13.4520	−0.584870
$530$	−0.744353	−0.0323326
$531$	0	0
$532$	−8.66950	−0.375870
$533$	2.77658	0.120267
$534$	0	0
$535$	−1.32632	−0.0573417
$536$	20.6136	0.890372
$537$	0	0
$538$	−17.2419	−0.743351
$539$	4.23638	0.182474
$540$	0	0
$541$	19.1327	0.822578	0.411289	−	0.911505i	$-0.365079\pi$
$541$	19.1327	0.822578	0.411289	+	0.911505i	$0.365079\pi$
$542$	−5.43374	−0.233399
$543$	0	0
$544$	15.8443	0.679320
$545$	−1.00010	−0.0428398
$546$	0	0
$547$	28.3544	1.21235	0.606173	−	0.795333i	$-0.292704\pi$
$547$	28.3544	1.21235	0.606173	+	0.795333i	$0.292704\pi$
$548$	−12.2777	−0.524477
$549$	0	0
$550$	14.4096	0.614428
$551$	−37.9404	−1.61632
$552$	0	0
$553$	12.5663	0.534374
$554$	9.40108	0.399414
$555$	0	0
$556$	17.3154	0.734338
$557$	−9.66963	−0.409715	−0.204858	−	0.978792i	$-0.565673\pi$
$557$	−9.66963	−0.409715	−0.204858	+	0.978792i	$0.565673\pi$
$558$	0	0
$559$	−26.5911	−1.12468
$560$	−0.256862	−0.0108544
$561$	0	0
$562$	7.09145	0.299135
$563$	−8.16696	−0.344196	−0.172098	−	0.985080i	$-0.555055\pi$
$563$	−8.16696	−0.344196	−0.172098	+	0.985080i	$0.555055\pi$
$564$	0	0
$565$	1.17921	0.0496099
$566$	−7.75406	−0.325928
$567$	0	0
$568$	−17.1211	−0.718385
$569$	30.6856	1.28641	0.643203	−	0.765696i	$-0.277605\pi$
$569$	30.6856	1.28641	0.643203	+	0.765696i	$0.277605\pi$
$570$	0	0
$571$	−5.06856	−0.212113	−0.106056	−	0.994360i	$-0.533822\pi$
$571$	−5.06856	−0.212113	−0.106056	+	0.994360i	$0.533822\pi$
$572$	−21.1207	−0.883102
$573$	0	0
$574$	−0.583930	−0.0243728
$575$	−15.3468	−0.640004
$576$	0	0
$577$	−4.50282	−0.187455	−0.0937274	−	0.995598i	$-0.529878\pi$
$577$	−4.50282	−0.187455	−0.0937274	+	0.995598i	$0.529878\pi$
$578$	6.53027	0.271623
$579$	0	0
$580$	−1.87416	−0.0778202
$581$	−11.1087	−0.460867
$582$	0	0
$583$	−25.2009	−1.04372
$584$	−3.46440	−0.143358
$585$	0	0
$586$	0.644988	0.0266442
$587$	26.0055	1.07336	0.536681	−	0.843785i	$-0.319678\pi$
$587$	26.0055	1.07336	0.536681	+	0.843785i	$0.319678\pi$
$588$	0	0
$589$	−54.1014	−2.22921
$590$	−0.964135	−0.0396928
$591$	0	0
$592$	−10.9051	−0.448199
$593$	−48.2944	−1.98321	−0.991607	−	0.129291i	$-0.958730\pi$
$593$	−48.2944	−1.98321	−0.991607	+	0.129291i	$0.958730\pi$
$594$	0	0
$595$	0.499191	0.0204648
$596$	2.50743	0.102708
$597$	0	0
$598$	−6.89125	−0.281804
$599$	−14.1896	−0.579772	−0.289886	−	0.957061i	$-0.593617\pi$
$599$	−14.1896	−0.579772	−0.289886	+	0.957061i	$0.593617\pi$
$600$	0	0
$601$	−12.2962	−0.501572	−0.250786	−	0.968043i	$-0.580689\pi$
$601$	−12.2962	−0.501572	−0.250786	+	0.968043i	$0.580689\pi$
$602$	5.59226	0.227923
$603$	0	0
$604$	32.5249	1.32342
$605$	−1.26926	−0.0516028
$606$	0	0
$607$	21.2861	0.863978	0.431989	−	0.901879i	$-0.357812\pi$
$607$	21.2861	0.863978	0.431989	+	0.901879i	$0.357812\pi$
$608$	−32.8392	−1.33180
$609$	0	0
$610$	−0.911168	−0.0368921
$611$	17.1678	0.694536
$612$	0	0
$613$	−22.9001	−0.924928	−0.462464	−	0.886638i	$-0.653034\pi$
$613$	−22.9001	−0.924928	−0.462464	+	0.886638i	$0.653034\pi$
$614$	6.90543	0.278681
$615$	0	0
$616$	10.2444	0.412758
$617$	−26.1239	−1.05171	−0.525854	−	0.850575i	$-0.676254\pi$
$617$	−26.1239	−1.05171	−0.525854	+	0.850575i	$0.676254\pi$
$618$	0	0
$619$	2.74080	0.110162	0.0550811	−	0.998482i	$-0.482458\pi$
$619$	2.74080	0.110162	0.0550811	+	0.998482i	$0.482458\pi$
$620$	−2.67247	−0.107329
$621$	0	0
$622$	−4.65337	−0.186583
$623$	−9.91467	−0.397223
$624$	0	0
$625$	24.5004	0.980015
$626$	−2.60315	−0.104043
$627$	0	0
$628$	25.0303	0.998818
$629$	21.1933	0.845032
$630$	0	0
$631$	−18.8494	−0.750381	−0.375191	−	0.926948i	$-0.622423\pi$
$631$	−18.8494	−0.750381	−0.375191	+	0.926948i	$0.622423\pi$
$632$	30.3878	1.20876
$633$	0	0
$634$	0.0565381	0.00224541
$635$	−0.182709	−0.00725059
$636$	0	0
$637$	3.25646	0.129026
$638$	19.4387	0.769587
$639$	0	0
$640$	−1.97400	−0.0780290
$641$	9.48946	0.374811	0.187406	−	0.982283i	$-0.439992\pi$
$641$	9.48946	0.374811	0.187406	+	0.982283i	$0.439992\pi$
$642$	0	0
$643$	−23.4153	−0.923408	−0.461704	−	0.887034i	$-0.652762\pi$
$643$	−23.4153	−0.923408	−0.461704	+	0.887034i	$0.652762\pi$
$644$	−4.73069	−0.186415
$645$	0	0
$646$	10.5957	0.416880
$647$	16.1609	0.635352	0.317676	−	0.948199i	$-0.397098\pi$
$647$	16.1609	0.635352	0.317676	+	0.948199i	$0.397098\pi$
$648$	0	0
$649$	−32.6419	−1.28131
$650$	11.0765	0.434456
$651$	0	0
$652$	−14.0982	−0.552129
$653$	48.0044	1.87856	0.939279	−	0.343156i	$-0.111496\pi$
$653$	48.0044	1.87856	0.939279	+	0.343156i	$0.111496\pi$
$654$	0	0
$655$	−3.48792	−0.136284
$656$	1.19868	0.0468006
$657$	0	0
$658$	−3.61050	−0.140752
$659$	22.3312	0.869899	0.434949	−	0.900455i	$-0.356766\pi$
$659$	22.3312	0.869899	0.434949	+	0.900455i	$0.356766\pi$
$660$	0	0
$661$	−14.5861	−0.567335	−0.283667	−	0.958923i	$-0.591551\pi$
$661$	−14.5861	−0.567335	−0.283667	+	0.958923i	$0.591551\pi$
$662$	21.6566	0.841707
$663$	0	0
$664$	−26.8631	−1.04249
$665$	−1.03463	−0.0401213
$666$	0	0
$667$	−20.7030	−0.801623
$668$	−17.6107	−0.681377
$669$	0	0
$670$	1.06664	0.0412081
$671$	−30.8486	−1.19090
$672$	0	0
$673$	35.1286	1.35411	0.677054	−	0.735933i	$-0.263256\pi$
$673$	35.1286	1.35411	0.677054	+	0.735933i	$0.263256\pi$
$674$	−14.4087	−0.555002
$675$	0	0
$676$	3.66744	0.141055
$677$	38.9180	1.49574	0.747871	−	0.663844i	$-0.231076\pi$
$677$	38.9180	1.49574	0.747871	+	0.663844i	$0.231076\pi$
$678$	0	0
$679$	−7.39495	−0.283792
$680$	1.20714	0.0462918
$681$	0	0
$682$	27.7188	1.06141
$683$	−18.3659	−0.702751	−0.351376	−	0.936235i	$-0.614286\pi$
$683$	−18.3659	−0.702751	−0.351376	+	0.936235i	$0.614286\pi$
$684$	0	0
$685$	−1.46524	−0.0559839
$686$	−0.684852	−0.0261478
$687$	0	0
$688$	−11.4797	−0.437658
$689$	−19.3717	−0.738002
$690$	0	0
$691$	−17.9973	−0.684650	−0.342325	−	0.939582i	$-0.611214\pi$
$691$	−17.9973	−0.684650	−0.342325	+	0.939582i	$0.611214\pi$
$692$	36.4036	1.38386
$693$	0	0
$694$	0.589040	0.0223597
$695$	2.06645	0.0783849
$696$	0	0
$697$	−2.32954	−0.0882377
$698$	4.60472	0.174291
$699$	0	0
$700$	7.60378	0.287396
$701$	21.7226	0.820451	0.410226	−	0.911984i	$-0.365450\pi$
$701$	21.7226	0.820451	0.410226	+	0.911984i	$0.365450\pi$
$702$	0	0
$703$	−43.9256	−1.65668
$704$	4.91373	0.185193
$705$	0	0
$706$	−5.47056	−0.205887
$707$	12.5594	0.472345
$708$	0	0
$709$	11.5399	0.433392	0.216696	−	0.976239i	$-0.430472\pi$
$709$	11.5399	0.433392	0.216696	+	0.976239i	$0.430472\pi$
$710$	−0.885925	−0.0332482
$711$	0	0
$712$	−23.9756	−0.898524
$713$	−29.5216	−1.10559
$714$	0	0
$715$	−2.52058	−0.0942643
$716$	−29.1207	−1.08829
$717$	0	0
$718$	−16.2306	−0.605719
$719$	−7.43315	−0.277210	−0.138605	−	0.990348i	$-0.544262\pi$
$719$	−7.43315	−0.277210	−0.138605	+	0.990348i	$0.544262\pi$
$720$	0	0
$721$	−1.37085	−0.0510531
$722$	−8.94853	−0.333030
$723$	0	0
$724$	16.9958	0.631644
$725$	33.2765	1.23586
$726$	0	0
$727$	4.16165	0.154347	0.0771736	−	0.997018i	$-0.475410\pi$
$727$	4.16165	0.154347	0.0771736	+	0.997018i	$0.475410\pi$
$728$	7.87475	0.291858
$729$	0	0
$730$	−0.179264	−0.00663486
$731$	22.3098	0.825160
$732$	0	0
$733$	−18.6578	−0.689143	−0.344571	−	0.938760i	$-0.611976\pi$
$733$	−18.6578	−0.689143	−0.344571	+	0.938760i	$0.611976\pi$
$734$	−6.75257	−0.249242
$735$	0	0
$736$	−17.9194	−0.660518
$737$	36.1125	1.33022
$738$	0	0
$739$	39.7826	1.46343	0.731713	−	0.681613i	$-0.238721\pi$
$739$	39.7826	1.46343	0.731713	+	0.681613i	$0.238721\pi$
$740$	−2.16981	−0.0797638
$741$	0	0
$742$	4.07398	0.149560
$743$	30.1346	1.10553	0.552765	−	0.833337i	$-0.313573\pi$
$743$	30.1346	1.10553	0.552765	+	0.833337i	$0.313573\pi$
$744$	0	0
$745$	0.299240	0.0109633
$746$	11.4144	0.417911
$747$	0	0
$748$	17.7202	0.647916
$749$	7.25917	0.265244
$750$	0	0
$751$	2.66769	0.0973454	0.0486727	−	0.998815i	$-0.484501\pi$
$751$	2.66769	0.0973454	0.0486727	+	0.998815i	$0.484501\pi$
$752$	7.41155	0.270271
$753$	0	0
$754$	14.9424	0.544168
$755$	3.88158	0.141265
$756$	0	0
$757$	−19.7069	−0.716259	−0.358130	−	0.933672i	$-0.616585\pi$
$757$	−19.7069	−0.716259	−0.358130	+	0.933672i	$0.616585\pi$
$758$	−9.33939	−0.339222
$759$	0	0
$760$	−2.50194	−0.0907549
$761$	11.8208	0.428502	0.214251	−	0.976779i	$-0.431269\pi$
$761$	11.8208	0.428502	0.214251	+	0.976779i	$0.431269\pi$
$762$	0	0
$763$	5.47375	0.198163
$764$	26.5475	0.960455
$765$	0	0
$766$	−8.26575	−0.298654
$767$	−25.0915	−0.906000
$768$	0	0
$769$	−31.0031	−1.11800	−0.559001	−	0.829167i	$-0.688815\pi$
$769$	−31.0031	−1.11800	−0.559001	+	0.829167i	$0.688815\pi$
$770$	0.530093	0.0191032
$771$	0	0
$772$	25.6694	0.923861
$773$	25.2317	0.907520	0.453760	−	0.891124i	$-0.350082\pi$
$773$	25.2317	0.907520	0.453760	+	0.891124i	$0.350082\pi$
$774$	0	0
$775$	47.4509	1.70449
$776$	−17.8824	−0.641942
$777$	0	0
$778$	20.7294	0.743185
$779$	4.82825	0.172990
$780$	0	0
$781$	−29.9940	−1.07327
$782$	5.78174	0.206755
$783$	0	0
$784$	1.40585	0.0502089
$785$	2.98716	0.106616
$786$	0	0
$787$	43.2540	1.54184	0.770920	−	0.636932i	$-0.219797\pi$
$787$	43.2540	1.54184	0.770920	+	0.636932i	$0.219797\pi$
$788$	−8.84999	−0.315268
$789$	0	0
$790$	1.57241	0.0559437
$791$	−6.45405	−0.229480
$792$	0	0
$793$	−23.7130	−0.842074
$794$	−7.00783	−0.248699
$795$	0	0
$796$	3.85911	0.136783
$797$	−47.4345	−1.68022	−0.840108	−	0.542419i	$-0.817508\pi$
$797$	−47.4345	−1.68022	−0.840108	+	0.542419i	$0.817508\pi$
$798$	0	0
$799$	−14.4038	−0.509569
$800$	28.8024	1.01832
$801$	0	0
$802$	15.5390	0.548702
$803$	−6.06919	−0.214177
$804$	0	0
$805$	−0.564568	−0.0198984
$806$	21.3072	0.750513
$807$	0	0
$808$	30.3711	1.06845
$809$	21.8218	0.767212	0.383606	−	0.923497i	$-0.374682\pi$
$809$	21.8218	0.767212	0.383606	+	0.923497i	$0.374682\pi$
$810$	0	0
$811$	24.8994	0.874336	0.437168	−	0.899380i	$-0.355982\pi$
$811$	24.8994	0.874336	0.437168	+	0.899380i	$0.355982\pi$
$812$	10.2576	0.359971
$813$	0	0
$814$	22.5053	0.788809
$815$	−1.68250	−0.0589356
$816$	0	0
$817$	−46.2397	−1.61772
$818$	−20.5280	−0.717745
$819$	0	0
$820$	0.238503	0.00832888
$821$	−38.3151	−1.33721	−0.668603	−	0.743619i	$-0.733108\pi$
$821$	−38.3151	−1.33721	−0.668603	+	0.743619i	$0.733108\pi$
$822$	0	0
$823$	−7.09096	−0.247175	−0.123588	−	0.992334i	$-0.539440\pi$
$823$	−7.09096	−0.247175	−0.123588	+	0.992334i	$0.539440\pi$
$824$	−3.31498	−0.115483
$825$	0	0
$826$	5.27688	0.183606
$827$	−16.2005	−0.563346	−0.281673	−	0.959510i	$-0.590889\pi$
$827$	−16.2005	−0.563346	−0.281673	+	0.959510i	$0.590889\pi$
$828$	0	0
$829$	13.9552	0.484684	0.242342	−	0.970191i	$-0.422084\pi$
$829$	13.9552	0.484684	0.242342	+	0.970191i	$0.422084\pi$
$830$	−1.39002	−0.0482483
$831$	0	0
$832$	3.77713	0.130948
$833$	−2.73216	−0.0946637
$834$	0	0
$835$	−2.10168	−0.0727317
$836$	−36.7273	−1.27024
$837$	0	0
$838$	−1.84846	−0.0638540
$839$	−28.7918	−0.994004	−0.497002	−	0.867749i	$-0.665566\pi$
$839$	−28.7918	−0.994004	−0.497002	+	0.867749i	$0.665566\pi$
$840$	0	0
$841$	15.8905	0.547947
$842$	0.0968860	0.00333891
$843$	0	0
$844$	18.1788	0.625739
$845$	0.437678	0.0150566
$846$	0	0
$847$	6.94689	0.238698
$848$	−8.36297	−0.287186
$849$	0	0
$850$	−9.29316	−0.318753
$851$	−23.9689	−0.821644
$852$	0	0
$853$	−20.3538	−0.696902	−0.348451	−	0.937327i	$-0.613292\pi$
$853$	−20.3538	−0.696902	−0.348451	+	0.937327i	$0.613292\pi$
$854$	4.98698	0.170651
$855$	0	0
$856$	17.5541	0.599987
$857$	2.41015	0.0823291	0.0411645	−	0.999152i	$-0.486893\pi$
$857$	2.41015	0.0823291	0.0411645	+	0.999152i	$0.486893\pi$
$858$	0	0
$859$	−5.00391	−0.170731	−0.0853656	−	0.996350i	$-0.527206\pi$
$859$	−5.00391	−0.170731	−0.0853656	+	0.996350i	$0.527206\pi$
$860$	−2.28413	−0.0778880
$861$	0	0
$862$	−20.8116	−0.708845
$863$	−53.9732	−1.83727	−0.918634	−	0.395110i	$-0.870707\pi$
$863$	−53.9732	−1.83727	−0.918634	+	0.395110i	$0.870707\pi$
$864$	0	0
$865$	4.34446	0.147716
$866$	−9.39021	−0.319092
$867$	0	0
$868$	14.6269	0.496470
$869$	53.2356	1.80589
$870$	0	0
$871$	27.7593	0.940586
$872$	13.2366	0.448248
$873$	0	0
$874$	−11.9833	−0.405342
$875$	1.82099	0.0615608
$876$	0	0
$877$	−8.77331	−0.296254	−0.148127	−	0.988968i	$-0.547324\pi$
$877$	−8.77331	−0.296254	−0.148127	+	0.988968i	$0.547324\pi$
$878$	−13.9043	−0.469247
$879$	0	0
$880$	−1.08816	−0.0366820
$881$	59.0141	1.98823	0.994117	−	0.108311i	$-0.0345442\pi$
$881$	59.0141	1.98823	0.994117	+	0.108311i	$0.0345442\pi$
$882$	0	0
$883$	−38.4425	−1.29369	−0.646847	−	0.762620i	$-0.723913\pi$
$883$	−38.4425	−1.29369	−0.646847	+	0.762620i	$0.723913\pi$
$884$	13.6214	0.458136
$885$	0	0
$886$	−25.8724	−0.869200
$887$	46.8555	1.57325	0.786627	−	0.617428i	$-0.211825\pi$
$887$	46.8555	1.57325	0.786627	+	0.617428i	$0.211825\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−1.24061	−0.0415853
$891$	0	0
$892$	40.9458	1.37097
$893$	29.8535	0.999009
$894$	0	0
$895$	−3.47530	−0.116167
$896$	10.8040	0.360937
$897$	0	0
$898$	2.81889	0.0940674
$899$	64.0119	2.13492
$900$	0	0
$901$	16.2528	0.541459
$902$	−2.47375	−0.0823669
$903$	0	0
$904$	−15.6072	−0.519086
$905$	2.02830	0.0674231
$906$	0	0
$907$	−34.3917	−1.14196	−0.570979	−	0.820965i	$-0.693436\pi$
$907$	−34.3917	−1.14196	−0.570979	+	0.820965i	$0.693436\pi$
$908$	−39.1426	−1.29899
$909$	0	0
$910$	0.407476	0.0135077
$911$	5.39247	0.178661	0.0893303	−	0.996002i	$-0.471527\pi$
$911$	5.39247	0.178661	0.0893303	+	0.996002i	$0.471527\pi$
$912$	0	0
$913$	−47.0607	−1.55748
$914$	7.26948	0.240453
$915$	0	0
$916$	40.5096	1.33847
$917$	19.0900	0.630407
$918$	0	0
$919$	−4.62144	−0.152447	−0.0762236	−	0.997091i	$-0.524286\pi$
$919$	−4.62144	−0.152447	−0.0762236	+	0.997091i	$0.524286\pi$
$920$	−1.36524	−0.0450105
$921$	0	0
$922$	24.7916	0.816466
$923$	−23.0561	−0.758900
$924$	0	0
$925$	38.5259	1.26673
$926$	2.32694	0.0764681
$927$	0	0
$928$	38.8548	1.27547
$929$	−39.2036	−1.28623	−0.643115	−	0.765770i	$-0.722358\pi$
$929$	−39.2036	−1.28623	−0.643115	+	0.765770i	$0.722358\pi$
$930$	0	0
$931$	5.66272	0.185588
$932$	15.4828	0.507157
$933$	0	0
$934$	17.6859	0.578700
$935$	2.11476	0.0691601
$936$	0	0
$937$	25.9890	0.849024	0.424512	−	0.905422i	$-0.360446\pi$
$937$	25.9890	0.849024	0.424512	+	0.905422i	$0.360446\pi$
$938$	−5.83793	−0.190615
$939$	0	0
$940$	1.47469	0.0480990
$941$	−50.0404	−1.63127	−0.815635	−	0.578566i	$-0.803612\pi$
$941$	−50.0404	−1.63127	−0.815635	+	0.578566i	$0.803612\pi$
$942$	0	0
$943$	2.63463	0.0857955
$944$	−10.8323	−0.352560
$945$	0	0
$946$	23.6909	0.770258
$947$	−34.5608	−1.12307	−0.561537	−	0.827452i	$-0.689790\pi$
$947$	−34.5608	−1.12307	−0.561537	+	0.827452i	$0.689790\pi$
$948$	0	0
$949$	−4.66532	−0.151443
$950$	19.2612	0.624914
$951$	0	0
$952$	−6.60690	−0.214131
$953$	−49.0514	−1.58893	−0.794465	−	0.607310i	$-0.792248\pi$
$953$	−49.0514	−1.58893	−0.794465	+	0.607310i	$0.792248\pi$
$954$	0	0
$955$	3.16822	0.102521
$956$	−8.64108	−0.279472
$957$	0	0
$958$	0.211796	0.00684281
$959$	8.01951	0.258963
$960$	0	0
$961$	60.2782	1.94446
$962$	17.2995	0.557760
$963$	0	0
$964$	4.26474	0.137358
$965$	3.06342	0.0986151
$966$	0	0
$967$	23.4155	0.752992	0.376496	−	0.926418i	$-0.377129\pi$
$967$	23.4155	0.752992	0.376496	+	0.926418i	$0.377129\pi$
$968$	16.7990	0.539939
$969$	0	0
$970$	−0.925321	−0.0297103
$971$	−2.89055	−0.0927623	−0.0463811	−	0.998924i	$-0.514769\pi$
$971$	−2.89055	−0.0927623	−0.0463811	+	0.998924i	$0.514769\pi$
$972$	0	0
$973$	−11.3101	−0.362584
$974$	2.85694	0.0915421
$975$	0	0
$976$	−10.2372	−0.327684
$977$	8.69803	0.278275	0.139137	−	0.990273i	$-0.455567\pi$
$977$	8.69803	0.278275	0.139137	+	0.990273i	$0.455567\pi$
$978$	0	0
$979$	−42.0023	−1.34240
$980$	0.279724	0.00893545
$981$	0	0
$982$	27.5523	0.879230
$983$	37.6232	1.20000	0.599998	−	0.800002i	$-0.295168\pi$
$983$	37.6232	1.20000	0.599998	+	0.800002i	$0.295168\pi$
$984$	0	0
$985$	−1.05617	−0.0336524
$986$	−12.5366	−0.399247
$987$	0	0
$988$	−28.2318	−0.898174
$989$	−25.2317	−0.802321
$990$	0	0
$991$	−35.3616	−1.12330	−0.561650	−	0.827375i	$-0.689833\pi$
$991$	−35.3616	−1.12330	−0.561650	+	0.827375i	$0.689833\pi$
$992$	55.4053	1.75912
$993$	0	0
$994$	4.84882	0.153795
$995$	0.460552	0.0146005
$996$	0	0
$997$	26.4674	0.838231	0.419115	−	0.907933i	$-0.362340\pi$
$997$	26.4674	0.838231	0.419115	+	0.907933i	$0.362340\pi$
$998$	−4.93349	−0.156167
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.9	✓	22
3.2	odd	2	inner	8001.2.a.x.1.14	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.9	✓	22	1.1	even	1	trivial
8001.2.a.x.1.14	yes	22	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.684852 q^{2} -1.53098 q^{4} -0.182709 q^{5} +1.00000 q^{7} +2.41820 q^{8} +0.125129 q^{10} +4.23638 q^{11} +3.25646 q^{13} -0.684852 q^{14} +1.40585 q^{16} -2.73216 q^{17} +5.66272 q^{19} +0.279724 q^{20} -2.90129 q^{22} +3.08998 q^{23} -4.96662 q^{25} -2.23019 q^{26} -1.53098 q^{28} -6.70003 q^{29} -9.55396 q^{31} -5.79919 q^{32} +1.87112 q^{34} -0.182709 q^{35} -7.75698 q^{37} -3.87812 q^{38} -0.441827 q^{40} +0.852638 q^{41} -8.16565 q^{43} -6.48580 q^{44} -2.11618 q^{46} +5.27194 q^{47} +1.00000 q^{49} +3.40140 q^{50} -4.98556 q^{52} -5.94870 q^{53} -0.774025 q^{55} +2.41820 q^{56} +4.58853 q^{58} -7.70514 q^{59} -7.28184 q^{61} +6.54305 q^{62} +1.15989 q^{64} -0.594985 q^{65} +8.52437 q^{67} +4.18288 q^{68} +0.125129 q^{70} -7.08011 q^{71} -1.43264 q^{73} +5.31238 q^{74} -8.66950 q^{76} +4.23638 q^{77} +12.5663 q^{79} -0.256862 q^{80} -0.583930 q^{82} -11.1087 q^{83} +0.499191 q^{85} +5.59226 q^{86} +10.2444 q^{88} -9.91467 q^{89} +3.25646 q^{91} -4.73069 q^{92} -3.61050 q^{94} -1.03463 q^{95} -7.39495 q^{97} -0.684852 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52}+ \cdots - 56 q^{97}+O(q^{100}) \) 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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