LMFDB - Embedded newform 3969.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.14013.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 3 $ x^4 - x^3 - 6x^2 + 6x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.20800$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20800 q^{2} +2.87525 q^{4} -3.80779 q^{5} +1.93254 q^{8} +O(q^{10})$	$q+2.20800 q^{2} +2.87525 q^{4} -3.80779 q^{5} +1.93254 q^{8} -8.40758 q^{10} +4.32433 q^{11} -2.87525 q^{13} -1.48345 q^{16} +4.02595 q^{17} -1.60821 q^{19} -10.9483 q^{20} +9.54811 q^{22} -2.66725 q^{23} +9.49923 q^{25} -6.34853 q^{26} +0.750492 q^{29} +0.140536 q^{31} -7.14054 q^{32} +8.88928 q^{34} -8.28282 q^{37} -3.55091 q^{38} -7.35870 q^{40} -10.3745 q^{41} +0.267040 q^{43} +12.4335 q^{44} -5.88928 q^{46} -7.93254 q^{47} +20.9743 q^{50} -8.26704 q^{52} -11.2238 q^{53} -16.4661 q^{55} +1.65708 q^{58} -0.693198 q^{59} +2.10744 q^{61} +0.310302 q^{62} -12.7994 q^{64} +10.9483 q^{65} -10.7663 q^{67} +11.5756 q^{68} +3.62399 q^{71} -3.57511 q^{73} -18.2884 q^{74} -4.62399 q^{76} -15.4234 q^{79} +5.64867 q^{80} -22.9068 q^{82} +6.44067 q^{83} -15.3299 q^{85} +0.589624 q^{86} +8.35695 q^{88} +0.256874 q^{89} -7.66900 q^{92} -17.5150 q^{94} +6.12370 q^{95} +1.05856 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 5 q^{4} + 2 q^{5} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + 5 * q^4 + 2 * q^5 - 3 * q^8	$ 4 q + q^{2} + 5 q^{4} + 2 q^{5} - 3 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} - q^{16} + 6 q^{17} - 8 q^{19} - 8 q^{20} - 7 q^{22} - 12 q^{23} + 8 q^{25} + q^{26} - 10 q^{29} - 18 q^{31} - 10 q^{32} - 20 q^{38} - 18 q^{40} - 5 q^{41} - 7 q^{43} + 13 q^{44} + 12 q^{46} - 21 q^{47} + 38 q^{50} - 25 q^{52} - 12 q^{53} - 26 q^{55} - 7 q^{58} + 6 q^{59} - 20 q^{61} + 18 q^{62} - 23 q^{64} + 8 q^{65} - 5 q^{67} + 51 q^{68} - 9 q^{71} - 6 q^{73} + 5 q^{76} - 10 q^{79} - 2 q^{80} - 35 q^{82} + 9 q^{83} - 9 q^{85} - 22 q^{86} + 18 q^{88} - 22 q^{89} - 36 q^{92} - 15 q^{94} - 16 q^{95} - 9 q^{97}+O(q^{100}) $ 4 * q + q^2 + 5 * q^4 + 2 * q^5 - 3 * q^8 - 7 * q^10 + 5 * q^11 - 5 * q^13 - q^16 + 6 * q^17 - 8 * q^19 - 8 * q^20 - 7 * q^22 - 12 * q^23 + 8 * q^25 + q^26 - 10 * q^29 - 18 * q^31 - 10 * q^32 - 20 * q^38 - 18 * q^40 - 5 * q^41 - 7 * q^43 + 13 * q^44 + 12 * q^46 - 21 * q^47 + 38 * q^50 - 25 * q^52 - 12 * q^53 - 26 * q^55 - 7 * q^58 + 6 * q^59 - 20 * q^61 + 18 * q^62 - 23 * q^64 + 8 * q^65 - 5 * q^67 + 51 * q^68 - 9 * q^71 - 6 * q^73 + 5 * q^76 - 10 * q^79 - 2 * q^80 - 35 * q^82 + 9 * q^83 - 9 * q^85 - 22 * q^86 + 18 * q^88 - 22 * q^89 - 36 * q^92 - 15 * q^94 - 16 * q^95 - 9 * 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 $

$2$	2.20800	1.56129	0.780644	−	0.624975i	$-0.214891\pi$
$2$	2.20800	1.56129	0.780644	+	0.624975i	$0.214891\pi$
$3$	0	0
$3$	0	0
$4$	2.87525	1.43762
$5$	−3.80779	−1.70289	−0.851447	−	0.524441i	$-0.824274\pi$
$5$	−3.80779	−1.70289	−0.851447	+	0.524441i	$0.824274\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.93254	0.683256
$9$	0	0
$10$	−8.40758	−2.65871
$11$	4.32433	1.30384	0.651918	−	0.758290i	$-0.273965\pi$
$11$	4.32433	1.30384	0.651918	+	0.758290i	$0.273965\pi$
$12$	0	0
$13$	−2.87525	−0.797450	−0.398725	−	0.917071i	$-0.630547\pi$
$13$	−2.87525	−0.797450	−0.398725	+	0.917071i	$0.630547\pi$
$14$	0	0
$15$	0	0
$16$	−1.48345	−0.370863
$17$	4.02595	0.976436	0.488218	−	0.872722i	$-0.337647\pi$
$17$	4.02595	0.976436	0.488218	+	0.872722i	$0.337647\pi$
$18$	0	0
$19$	−1.60821	−0.368948	−0.184474	−	0.982837i	$-0.559058\pi$
$19$	−1.60821	−0.368948	−0.184474	+	0.982837i	$0.559058\pi$
$20$	−10.9483	−2.44812
$21$	0	0
$22$	9.54811	2.03566
$23$	−2.66725	−0.556160	−0.278080	−	0.960558i	$-0.589698\pi$
$23$	−2.66725	−0.556160	−0.278080	+	0.960558i	$0.589698\pi$
$24$	0	0
$25$	9.49923	1.89985
$26$	−6.34853	−1.24505
$27$	0	0
$28$	0	0
$29$	0.750492	0.139363	0.0696815	−	0.997569i	$-0.477802\pi$
$29$	0.750492	0.139363	0.0696815	+	0.997569i	$0.477802\pi$
$30$	0	0
$31$	0.140536	0.0252410	0.0126205	−	0.999920i	$-0.495983\pi$
$31$	0.140536	0.0252410	0.0126205	+	0.999920i	$0.495983\pi$
$32$	−7.14054	−1.26228
$33$	0	0
$34$	8.88928	1.52450
$35$	0	0
$36$	0	0
$37$	−8.28282	−1.36169	−0.680844	−	0.732429i	$-0.738387\pi$
$37$	−8.28282	−1.36169	−0.680844	+	0.732429i	$0.738387\pi$
$38$	−3.55091	−0.576034
$39$	0	0
$40$	−7.35870	−1.16351
$41$	−10.3745	−1.62022	−0.810111	−	0.586277i	$-0.800593\pi$
$41$	−10.3745	−1.62022	−0.810111	+	0.586277i	$0.800593\pi$
$42$	0	0
$43$	0.267040	0.0407232	0.0203616	−	0.999793i	$-0.493518\pi$
$43$	0.267040	0.0407232	0.0203616	+	0.999793i	$0.493518\pi$
$44$	12.4335	1.87442
$45$	0	0
$46$	−5.88928	−0.868327
$47$	−7.93254	−1.15708	−0.578540	−	0.815654i	$-0.696377\pi$
$47$	−7.93254	−1.15708	−0.578540	+	0.815654i	$0.696377\pi$
$48$	0	0
$49$	0	0
$50$	20.9743	2.96621
$51$	0	0
$52$	−8.26704	−1.14643
$53$	−11.2238	−1.54170	−0.770852	−	0.637014i	$-0.780169\pi$
$53$	−11.2238	−1.54170	−0.770852	+	0.637014i	$0.780169\pi$
$54$	0	0
$55$	−16.4661	−2.22029
$56$	0	0
$57$	0	0
$58$	1.65708	0.217586
$59$	−0.693198	−0.0902468	−0.0451234	−	0.998981i	$-0.514368\pi$
$59$	−0.693198	−0.0902468	−0.0451234	+	0.998981i	$0.514368\pi$
$60$	0	0
$61$	2.10744	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$61$	2.10744	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$62$	0.310302	0.0394085
$63$	0	0
$64$	−12.7994	−1.59992
$65$	10.9483	1.35797
$66$	0	0
$67$	−10.7663	−1.31531	−0.657655	−	0.753319i	$-0.728451\pi$
$67$	−10.7663	−1.31531	−0.657655	+	0.753319i	$0.728451\pi$
$68$	11.5756	1.40375
$69$	0	0
$70$	0	0
$71$	3.62399	0.430088	0.215044	−	0.976604i	$-0.431010\pi$
$71$	3.62399	0.430088	0.215044	+	0.976604i	$0.431010\pi$
$72$	0	0
$73$	−3.57511	−0.418435	−0.209217	−	0.977869i	$-0.567092\pi$
$73$	−3.57511	−0.418435	−0.209217	+	0.977869i	$0.567092\pi$
$74$	−18.2884	−2.12599
$75$	0	0
$76$	−4.62399	−0.530408
$77$	0	0
$78$	0	0
$79$	−15.4234	−1.73526	−0.867632	−	0.497207i	$-0.834359\pi$
$79$	−15.4234	−1.73526	−0.867632	+	0.497207i	$0.834359\pi$
$80$	5.64867	0.631540
$81$	0	0
$82$	−22.9068	−2.52963
$83$	6.44067	0.706956	0.353478	−	0.935443i	$-0.384999\pi$
$83$	6.44067	0.706956	0.353478	+	0.935443i	$0.384999\pi$
$84$	0	0
$85$	−15.3299	−1.66277
$86$	0.589624	0.0635808
$87$	0	0
$88$	8.35695	0.890854
$89$	0.256874	0.0272286	0.0136143	−	0.999907i	$-0.495666\pi$
$89$	0.256874	0.0272286	0.0136143	+	0.999907i	$0.495666\pi$
$90$	0	0
$91$	0	0
$92$	−7.66900	−0.799549
$93$	0	0
$94$	−17.5150	−1.80654
$95$	6.12370	0.628279
$96$	0	0
$97$	1.05856	0.107481	0.0537403	−	0.998555i	$-0.482886\pi$
$97$	1.05856	0.107481	0.0537403	+	0.998555i	$0.482886\pi$
$98$	0	0
$99$	0	0
$100$	27.3126	2.73126
$101$	−1.36585	−0.135907	−0.0679534	−	0.997688i	$-0.521647\pi$
$101$	−1.36585	−0.135907	−0.0679534	+	0.997688i	$0.521647\pi$
$102$	0	0
$103$	−14.9713	−1.47516	−0.737581	−	0.675259i	$-0.764032\pi$
$103$	−14.9713	−1.47516	−0.737581	+	0.675259i	$0.764032\pi$
$104$	−5.55653	−0.544862
$105$	0	0
$106$	−24.7821	−2.40705
$107$	8.08149	0.781267	0.390634	−	0.920546i	$-0.372256\pi$
$107$	8.08149	0.781267	0.390634	+	0.920546i	$0.372256\pi$
$108$	0	0
$109$	−9.10919	−0.872502	−0.436251	−	0.899825i	$-0.643694\pi$
$109$	−9.10919	−0.872502	−0.436251	+	0.899825i	$0.643694\pi$
$110$	−36.3572	−3.46652
$111$	0	0
$112$	0	0
$113$	2.76606	0.260209	0.130104	−	0.991500i	$-0.458469\pi$
$113$	2.76606	0.260209	0.130104	+	0.991500i	$0.458469\pi$
$114$	0	0
$115$	10.1563	0.947082
$116$	2.15785	0.200351
$117$	0	0
$118$	−1.53058	−0.140901
$119$	0	0
$120$	0	0
$121$	7.69986	0.699988
$122$	4.65322	0.421283
$123$	0	0
$124$	0.404075	0.0362870
$125$	−17.1321	−1.53234
$126$	0	0
$127$	13.6573	1.21189	0.605945	−	0.795507i	$-0.292795\pi$
$127$	13.6573	1.21189	0.605945	+	0.795507i	$0.292795\pi$
$128$	−13.9799	−1.23566
$129$	0	0
$130$	24.1739	2.12019
$131$	17.6138	1.53893	0.769463	−	0.638691i	$-0.220524\pi$
$131$	17.6138	1.53893	0.769463	+	0.638691i	$0.220524\pi$
$132$	0	0
$133$	0	0
$134$	−23.7719	−2.05358
$135$	0	0
$136$	7.78031	0.667156
$137$	8.66725	0.740493	0.370247	−	0.928934i	$-0.379273\pi$
$137$	8.66725	0.740493	0.370247	+	0.928934i	$0.379273\pi$
$138$	0	0
$139$	−3.76606	−0.319433	−0.159716	−	0.987163i	$-0.551058\pi$
$139$	−3.76606	−0.319433	−0.159716	+	0.987163i	$0.551058\pi$
$140$	0	0
$141$	0	0
$142$	8.00175	0.671492
$143$	−12.4335	−1.03974
$144$	0	0
$145$	−2.85771	−0.237320
$146$	−7.89383	−0.653298
$147$	0	0
$148$	−23.8152	−1.95759
$149$	4.61122	0.377766	0.188883	−	0.982000i	$-0.439513\pi$
$149$	4.61122	0.377766	0.188883	+	0.982000i	$0.439513\pi$
$150$	0	0
$151$	7.79762	0.634561	0.317281	−	0.948332i	$-0.397230\pi$
$151$	7.79762	0.634561	0.317281	+	0.948332i	$0.397230\pi$
$152$	−3.10792	−0.252086
$153$	0	0
$154$	0	0
$155$	−0.535130	−0.0429827
$156$	0	0
$157$	23.2655	1.85679	0.928395	−	0.371595i	$-0.121189\pi$
$157$	23.2655	1.85679	0.928395	+	0.371595i	$0.121189\pi$
$158$	−34.0547	−2.70925
$159$	0	0
$160$	27.1896	2.14953
$161$	0	0
$162$	0	0
$163$	6.11094	0.478646	0.239323	−	0.970940i	$-0.423075\pi$
$163$	6.11094	0.478646	0.239323	+	0.970940i	$0.423075\pi$
$164$	−29.8292	−2.32927
$165$	0	0
$166$	14.2210	1.10376
$167$	11.6415	0.900848	0.450424	−	0.892815i	$-0.351273\pi$
$167$	11.6415	0.900848	0.450424	+	0.892815i	$0.351273\pi$
$168$	0	0
$169$	−4.73296	−0.364074
$170$	−33.8485	−2.59606
$171$	0	0
$172$	0.767806	0.0585447
$173$	19.2571	1.46409	0.732045	−	0.681256i	$-0.238566\pi$
$173$	19.2571	1.46409	0.732045	+	0.681256i	$0.238566\pi$
$174$	0	0
$175$	0	0
$176$	−6.41494	−0.483544
$177$	0	0
$178$	0.567176	0.0425117
$179$	3.34187	0.249783	0.124891	−	0.992170i	$-0.460142\pi$
$179$	3.34187	0.249783	0.124891	+	0.992170i	$0.460142\pi$
$180$	0	0
$181$	−19.7358	−1.46695	−0.733474	−	0.679717i	$-0.762103\pi$
$181$	−19.7358	−1.46695	−0.733474	+	0.679717i	$0.762103\pi$
$182$	0	0
$183$	0	0
$184$	−5.15457	−0.380000
$185$	31.5392	2.31881
$186$	0	0
$187$	17.4095	1.27311
$188$	−22.8080	−1.66344
$189$	0	0
$190$	13.5211	0.980925
$191$	4.17490	0.302085	0.151043	−	0.988527i	$-0.451737\pi$
$191$	4.17490	0.302085	0.151043	+	0.988527i	$0.451737\pi$
$192$	0	0
$193$	−13.8737	−0.998652	−0.499326	−	0.866414i	$-0.666419\pi$
$193$	−13.8737	−0.998652	−0.499326	+	0.866414i	$0.666419\pi$
$194$	2.33730	0.167808
$195$	0	0
$196$	0	0
$197$	7.48520	0.533299	0.266649	−	0.963794i	$-0.414083\pi$
$197$	7.48520	0.533299	0.266649	+	0.963794i	$0.414083\pi$
$198$	0	0
$199$	−12.6553	−0.897113	−0.448556	−	0.893755i	$-0.648062\pi$
$199$	−12.6553	−0.897113	−0.448556	+	0.893755i	$0.648062\pi$
$200$	18.3576	1.29808
$201$	0	0
$202$	−3.01578	−0.212190
$203$	0	0
$204$	0	0
$205$	39.5038	2.75907
$206$	−33.0565	−2.30315
$207$	0	0
$208$	4.26529	0.295745
$209$	−6.95442	−0.481047
$210$	0	0
$211$	−5.15197	−0.354676	−0.177338	−	0.984150i	$-0.556749\pi$
$211$	−5.15197	−0.354676	−0.177338	+	0.984150i	$0.556749\pi$
$212$	−32.2711	−2.21639
$213$	0	0
$214$	17.8439	1.21978
$215$	−1.01683	−0.0693474
$216$	0	0
$217$	0	0
$218$	−20.1131	−1.36223
$219$	0	0
$220$	−47.3442	−3.19195
$221$	−11.5756	−0.778659
$222$	0	0
$223$	−15.2166	−1.01898	−0.509490	−	0.860476i	$-0.670166\pi$
$223$	−15.2166	−1.01898	−0.509490	+	0.860476i	$0.670166\pi$
$224$	0	0
$225$	0	0
$226$	6.10744	0.406261
$227$	4.28949	0.284703	0.142352	−	0.989816i	$-0.454534\pi$
$227$	4.28949	0.284703	0.142352	+	0.989816i	$0.454534\pi$
$228$	0	0
$229$	0.971251	0.0641821	0.0320910	−	0.999485i	$-0.489783\pi$
$229$	0.971251	0.0641821	0.0320910	+	0.999485i	$0.489783\pi$
$230$	22.4251	1.47867
$231$	0	0
$232$	1.45036	0.0952205
$233$	3.77342	0.247205	0.123603	−	0.992332i	$-0.460555\pi$
$233$	3.77342	0.247205	0.123603	+	0.992332i	$0.460555\pi$
$234$	0	0
$235$	30.2054	1.97038
$236$	−1.99312	−0.129741
$237$	0	0
$238$	0	0
$239$	−6.69040	−0.432766	−0.216383	−	0.976309i	$-0.569426\pi$
$239$	−6.69040	−0.432766	−0.216383	+	0.976309i	$0.569426\pi$
$240$	0	0
$241$	−14.9713	−0.964383	−0.482192	−	0.876066i	$-0.660159\pi$
$241$	−14.9713	−0.964383	−0.482192	+	0.876066i	$0.660159\pi$
$242$	17.0013	1.09288
$243$	0	0
$244$	6.05941	0.387914
$245$	0	0
$246$	0	0
$247$	4.62399	0.294217
$248$	0.271591	0.0172460
$249$	0	0
$250$	−37.8277	−2.39243
$251$	17.0787	1.07800	0.538999	−	0.842307i	$-0.318803\pi$
$251$	17.0787	1.07800	0.538999	+	0.842307i	$0.318803\pi$
$252$	0	0
$253$	−11.5341	−0.725141
$254$	30.1553	1.89211
$255$	0	0
$256$	−5.26879	−0.329299
$257$	−8.57231	−0.534726	−0.267363	−	0.963596i	$-0.586152\pi$
$257$	−8.57231	−0.534726	−0.267363	+	0.963596i	$0.586152\pi$
$258$	0	0
$259$	0	0
$260$	31.4791	1.95225
$261$	0	0
$262$	38.8912	2.40271
$263$	5.69685	0.351283	0.175641	−	0.984454i	$-0.443800\pi$
$263$	5.69685	0.351283	0.175641	+	0.984454i	$0.443800\pi$
$264$	0	0
$265$	42.7377	2.62536
$266$	0	0
$267$	0	0
$268$	−30.9557	−1.89092
$269$	−15.6015	−0.951243	−0.475621	−	0.879650i	$-0.657777\pi$
$269$	−15.6015	−0.951243	−0.475621	+	0.879650i	$0.657777\pi$
$270$	0	0
$271$	−30.7375	−1.86717	−0.933586	−	0.358354i	$-0.883338\pi$
$271$	−30.7375	−1.86717	−0.933586	+	0.358354i	$0.883338\pi$
$272$	−5.97230	−0.362124
$273$	0	0
$274$	19.1373	1.15612
$275$	41.0779	2.47709
$276$	0	0
$277$	−26.5926	−1.59780	−0.798899	−	0.601466i	$-0.794584\pi$
$277$	−26.5926	−1.59780	−0.798899	+	0.601466i	$0.794584\pi$
$278$	−8.31544	−0.498727
$279$	0	0
$280$	0	0
$281$	−17.3613	−1.03569	−0.517844	−	0.855475i	$-0.673265\pi$
$281$	−17.3613	−1.03569	−0.517844	+	0.855475i	$0.673265\pi$
$282$	0	0
$283$	18.3554	1.09112	0.545558	−	0.838073i	$-0.316318\pi$
$283$	18.3554	1.09112	0.545558	+	0.838073i	$0.316318\pi$
$284$	10.4199	0.618305
$285$	0	0
$286$	−27.4532	−1.62334
$287$	0	0
$288$	0	0
$289$	−0.791740	−0.0465730
$290$	−6.30982	−0.370525
$291$	0	0
$292$	−10.2793	−0.601552
$293$	10.6981	0.624990	0.312495	−	0.949919i	$-0.398835\pi$
$293$	10.6981	0.624990	0.312495	+	0.949919i	$0.398835\pi$
$294$	0	0
$295$	2.63955	0.153681
$296$	−16.0069	−0.930381
$297$	0	0
$298$	10.1816	0.589802
$299$	7.66900	0.443510
$300$	0	0
$301$	0	0
$302$	17.2171	0.990733
$303$	0	0
$304$	2.38570	0.136829
$305$	−8.02468	−0.459492
$306$	0	0
$307$	31.3948	1.79180	0.895899	−	0.444258i	$-0.146533\pi$
$307$	31.3948	1.79180	0.895899	+	0.444258i	$0.146533\pi$
$308$	0	0
$309$	0	0
$310$	−1.18157	−0.0671084
$311$	−13.4891	−0.764895	−0.382447	−	0.923977i	$-0.624919\pi$
$311$	−13.4891	−0.764895	−0.382447	+	0.923977i	$0.624919\pi$
$312$	0	0
$313$	−19.3414	−1.09324	−0.546620	−	0.837381i	$-0.684086\pi$
$313$	−19.3414	−1.09324	−0.546620	+	0.837381i	$0.684086\pi$
$314$	51.3701	2.89899
$315$	0	0
$316$	−44.3459	−2.49465
$317$	−15.3576	−0.862571	−0.431286	−	0.902215i	$-0.641940\pi$
$317$	−15.3576	−0.862571	−0.431286	+	0.902215i	$0.641940\pi$
$318$	0	0
$319$	3.24538	0.181706
$320$	48.7373	2.72450
$321$	0	0
$322$	0	0
$323$	−6.47455	−0.360254
$324$	0	0
$325$	−27.3126	−1.51503
$326$	13.4929	0.747304
$327$	0	0
$328$	−20.0491	−1.10703
$329$	0	0
$330$	0	0
$331$	−1.23829	−0.0680627	−0.0340313	−	0.999421i	$-0.510835\pi$
$331$	−1.23829	−0.0680627	−0.0340313	+	0.999421i	$0.510835\pi$
$332$	18.5185	1.01634
$333$	0	0
$334$	25.7044	1.40648
$335$	40.9957	2.23983
$336$	0	0
$337$	11.9086	0.648701	0.324350	−	0.945937i	$-0.394854\pi$
$337$	11.9086	0.648701	0.324350	+	0.945937i	$0.394854\pi$
$338$	−10.4504	−0.568424
$339$	0	0
$340$	−44.0774	−2.39043
$341$	0.607724	0.0329101
$342$	0	0
$343$	0	0
$344$	0.516066	0.0278244
$345$	0	0
$346$	42.5196	2.28587
$347$	−4.34678	−0.233347	−0.116674	−	0.993170i	$-0.537223\pi$
$347$	−4.34678	−0.233347	−0.116674	+	0.993170i	$0.537223\pi$
$348$	0	0
$349$	7.17560	0.384101	0.192051	−	0.981385i	$-0.438486\pi$
$349$	7.17560	0.384101	0.192051	+	0.981385i	$0.438486\pi$
$350$	0	0
$351$	0	0
$352$	−30.8781	−1.64581
$353$	25.0967	1.33576	0.667881	−	0.744268i	$-0.267201\pi$
$353$	25.0967	1.33576	0.667881	+	0.744268i	$0.267201\pi$
$354$	0	0
$355$	−13.7994	−0.732395
$356$	0.738575	0.0391444
$357$	0	0
$358$	7.37883	0.389983
$359$	31.9918	1.68846	0.844231	−	0.535979i	$-0.180057\pi$
$359$	31.9918	1.68846	0.844231	+	0.535979i	$0.180057\pi$
$360$	0	0
$361$	−16.4137	−0.863878
$362$	−43.5765	−2.29033
$363$	0	0
$364$	0	0
$365$	13.6133	0.712550
$366$	0	0
$367$	−11.1248	−0.580707	−0.290354	−	0.956919i	$-0.593773\pi$
$367$	−11.1248	−0.580707	−0.290354	+	0.956919i	$0.593773\pi$
$368$	3.95674	0.206259
$369$	0	0
$370$	69.6385	3.62033
$371$	0	0
$372$	0	0
$373$	16.1546	0.836452	0.418226	−	0.908343i	$-0.362652\pi$
$373$	16.1546	0.836452	0.418226	+	0.908343i	$0.362652\pi$
$374$	38.4402	1.98770
$375$	0	0
$376$	−15.3299	−0.790582
$377$	−2.15785	−0.111135
$378$	0	0
$379$	3.18485	0.163595	0.0817973	−	0.996649i	$-0.473934\pi$
$379$	3.18485	0.163595	0.0817973	+	0.996649i	$0.473934\pi$
$380$	17.6072	0.903228
$381$	0	0
$382$	9.21816	0.471642
$383$	16.4776	0.841968	0.420984	−	0.907068i	$-0.361685\pi$
$383$	16.4776	0.841968	0.420984	+	0.907068i	$0.361685\pi$
$384$	0	0
$385$	0	0
$386$	−30.6331	−1.55918
$387$	0	0
$388$	3.04363	0.154517
$389$	−31.9771	−1.62130	−0.810651	−	0.585529i	$-0.800887\pi$
$389$	−31.9771	−1.62130	−0.810651	+	0.585529i	$0.800887\pi$
$390$	0	0
$391$	−10.7382	−0.543055
$392$	0	0
$393$	0	0
$394$	16.5273	0.832633
$395$	58.7288	2.95497
$396$	0	0
$397$	27.8119	1.39584	0.697919	−	0.716177i	$-0.254110\pi$
$397$	27.8119	1.39584	0.697919	+	0.716177i	$0.254110\pi$
$398$	−27.9429	−1.40065
$399$	0	0
$400$	−14.0917	−0.704583
$401$	−27.5883	−1.37769	−0.688847	−	0.724907i	$-0.741883\pi$
$401$	−27.5883	−1.37769	−0.688847	+	0.724907i	$0.741883\pi$
$402$	0	0
$403$	−0.404075	−0.0201284
$404$	−3.92714	−0.195383
$405$	0	0
$406$	0	0
$407$	−35.8177	−1.77542
$408$	0	0
$409$	−11.8246	−0.584690	−0.292345	−	0.956313i	$-0.594436\pi$
$409$	−11.8246	−0.584690	−0.292345	+	0.956313i	$0.594436\pi$
$410$	87.2242	4.30770
$411$	0	0
$412$	−43.0460	−2.12073
$413$	0	0
$414$	0	0
$415$	−24.5247	−1.20387
$416$	20.5308	1.00661
$417$	0	0
$418$	−15.3553	−0.751054
$419$	−8.81668	−0.430723	−0.215362	−	0.976534i	$-0.569093\pi$
$419$	−8.81668	−0.430723	−0.215362	+	0.976534i	$0.569093\pi$
$420$	0	0
$421$	18.7001	0.911386	0.455693	−	0.890137i	$-0.349391\pi$
$421$	18.7001	0.911386	0.455693	+	0.890137i	$0.349391\pi$
$422$	−11.3755	−0.553752
$423$	0	0
$424$	−21.6904	−1.05338
$425$	38.2434	1.85508
$426$	0	0
$427$	0	0
$428$	23.2363	1.12317
$429$	0	0
$430$	−2.24516	−0.108271
$431$	16.6194	0.800530	0.400265	−	0.916399i	$-0.368918\pi$
$431$	16.6194	0.800530	0.400265	+	0.916399i	$0.368918\pi$
$432$	0	0
$433$	−25.3004	−1.21586	−0.607929	−	0.793992i	$-0.707999\pi$
$433$	−25.3004	−1.21586	−0.607929	+	0.793992i	$0.707999\pi$
$434$	0	0
$435$	0	0
$436$	−26.1912	−1.25433
$437$	4.28949	0.205194
$438$	0	0
$439$	−29.4523	−1.40568	−0.702841	−	0.711347i	$-0.748086\pi$
$439$	−29.4523	−1.40568	−0.702841	+	0.711347i	$0.748086\pi$
$440$	−31.8215	−1.51703
$441$	0	0
$442$	−25.5589	−1.21571
$443$	−32.5565	−1.54681	−0.773404	−	0.633914i	$-0.781447\pi$
$443$	−32.5565	−1.54681	−0.773404	+	0.633914i	$0.781447\pi$
$444$	0	0
$445$	−0.978120	−0.0463674
$446$	−33.5983	−1.59092
$447$	0	0
$448$	0	0
$449$	0.171881	0.00811158	0.00405579	−	0.999992i	$-0.498709\pi$
$449$	0.171881	0.00811158	0.00405579	+	0.999992i	$0.498709\pi$
$450$	0	0
$451$	−44.8627	−2.11250
$452$	7.95309	0.374082
$453$	0	0
$454$	9.47117	0.444504
$455$	0	0
$456$	0	0
$457$	24.5020	1.14616	0.573079	−	0.819500i	$-0.305749\pi$
$457$	24.5020	1.14616	0.573079	+	0.819500i	$0.305749\pi$
$458$	2.14452	0.100207
$459$	0	0
$460$	29.2019	1.36155
$461$	9.43107	0.439249	0.219624	−	0.975584i	$-0.429517\pi$
$461$	9.43107	0.439249	0.219624	+	0.975584i	$0.429517\pi$
$462$	0	0
$463$	2.86121	0.132972	0.0664860	−	0.997787i	$-0.478821\pi$
$463$	2.86121	0.132972	0.0664860	+	0.997787i	$0.478821\pi$
$464$	−1.11332	−0.0516845
$465$	0	0
$466$	8.33170	0.385959
$467$	7.70288	0.356447	0.178223	−	0.983990i	$-0.442965\pi$
$467$	7.70288	0.356447	0.178223	+	0.983990i	$0.442965\pi$
$468$	0	0
$469$	0	0
$470$	66.6934	3.07634
$471$	0	0
$472$	−1.33963	−0.0616616
$473$	1.15477	0.0530964
$474$	0	0
$475$	−15.2767	−0.700944
$476$	0	0
$477$	0	0
$478$	−14.7724	−0.675673
$479$	34.5822	1.58010	0.790051	−	0.613041i	$-0.210054\pi$
$479$	34.5822	1.58010	0.790051	+	0.613041i	$0.210054\pi$
$480$	0	0
$481$	23.8152	1.08588
$482$	−33.0565	−1.50568
$483$	0	0
$484$	22.1390	1.00632
$485$	−4.03078	−0.183028
$486$	0	0
$487$	18.4356	0.835399	0.417699	−	0.908585i	$-0.362837\pi$
$487$	18.4356	0.835399	0.417699	+	0.908585i	$0.362837\pi$
$488$	4.07271	0.184363
$489$	0	0
$490$	0	0
$491$	3.15372	0.142325	0.0711627	−	0.997465i	$-0.477329\pi$
$491$	3.15372	0.142325	0.0711627	+	0.997465i	$0.477329\pi$
$492$	0	0
$493$	3.02144	0.136079
$494$	10.2097	0.459358
$495$	0	0
$496$	−0.208478	−0.00936094
$497$	0	0
$498$	0	0
$499$	28.8798	1.29284	0.646419	−	0.762983i	$-0.276266\pi$
$499$	28.8798	1.29284	0.646419	+	0.762983i	$0.276266\pi$
$500$	−49.2591	−2.20293
$501$	0	0
$502$	37.7097	1.68307
$503$	21.4742	0.957487	0.478744	−	0.877955i	$-0.341092\pi$
$503$	21.4742	0.957487	0.478744	+	0.877955i	$0.341092\pi$
$504$	0	0
$505$	5.20085	0.231435
$506$	−25.4672	−1.13216
$507$	0	0
$508$	39.2681	1.74224
$509$	8.82435	0.391133	0.195566	−	0.980690i	$-0.437346\pi$
$509$	8.82435	0.391133	0.195566	+	0.980690i	$0.437346\pi$
$510$	0	0
$511$	0	0
$512$	16.3263	0.721527
$513$	0	0
$514$	−18.9276	−0.834862
$515$	57.0073	2.51204
$516$	0	0
$517$	−34.3030	−1.50864
$518$	0	0
$519$	0	0
$520$	21.1581	0.927843
$521$	29.6703	1.29988	0.649939	−	0.759986i	$-0.274794\pi$
$521$	29.6703	1.29988	0.649939	+	0.759986i	$0.274794\pi$
$522$	0	0
$523$	−6.93991	−0.303461	−0.151730	−	0.988422i	$-0.548485\pi$
$523$	−6.93991	−0.303461	−0.151730	+	0.988422i	$0.548485\pi$
$524$	50.6441	2.21240
$525$	0	0
$526$	12.5786	0.548454
$527$	0.565790	0.0246462
$528$	0	0
$529$	−15.8858	−0.690686
$530$	94.3648	4.09894
$531$	0	0
$532$	0	0
$533$	29.8292	1.29205
$534$	0	0
$535$	−30.7726	−1.33042
$536$	−20.8063	−0.898693
$537$	0	0
$538$	−34.4481	−1.48516
$539$	0	0
$540$	0	0
$541$	−16.6904	−0.717576	−0.358788	−	0.933419i	$-0.616810\pi$
$541$	−16.6904	−0.717576	−0.358788	+	0.933419i	$0.616810\pi$
$542$	−67.8683	−2.91519
$543$	0	0
$544$	−28.7474	−1.23254
$545$	34.6858	1.48578
$546$	0	0
$547$	3.77924	0.161589	0.0807943	−	0.996731i	$-0.474254\pi$
$547$	3.77924	0.161589	0.0807943	+	0.996731i	$0.474254\pi$
$548$	24.9205	1.06455
$549$	0	0
$550$	90.6997	3.86745
$551$	−1.20695	−0.0514176
$552$	0	0
$553$	0	0
$554$	−58.7164	−2.49462
$555$	0	0
$556$	−10.8283	−0.459224
$557$	−11.4692	−0.485964	−0.242982	−	0.970031i	$-0.578126\pi$
$557$	−11.4692	−0.485964	−0.242982	+	0.970031i	$0.578126\pi$
$558$	0	0
$559$	−0.767806	−0.0324747
$560$	0	0
$561$	0	0
$562$	−38.3337	−1.61701
$563$	−25.3816	−1.06971	−0.534854	−	0.844944i	$-0.679633\pi$
$563$	−25.3816	−1.06971	−0.534854	+	0.844944i	$0.679633\pi$
$564$	0	0
$565$	−10.5325	−0.443108
$566$	40.5287	1.70355
$567$	0	0
$568$	7.00350	0.293860
$569$	8.69404	0.364473	0.182237	−	0.983255i	$-0.441666\pi$
$569$	8.69404	0.364473	0.182237	+	0.983255i	$0.441666\pi$
$570$	0	0
$571$	8.12757	0.340128	0.170064	−	0.985433i	$-0.445603\pi$
$571$	8.12757	0.340128	0.170064	+	0.985433i	$0.445603\pi$
$572$	−35.7494	−1.49476
$573$	0	0
$574$	0	0
$575$	−25.3368	−1.05662
$576$	0	0
$577$	22.0839	0.919367	0.459683	−	0.888083i	$-0.347963\pi$
$577$	22.0839	0.919367	0.459683	+	0.888083i	$0.347963\pi$
$578$	−1.74816	−0.0727138
$579$	0	0
$580$	−8.21663	−0.341177
$581$	0	0
$582$	0	0
$583$	−48.5354	−2.01013
$584$	−6.90904	−0.285898
$585$	0	0
$586$	23.6214	0.975791
$587$	30.7200	1.26795	0.633975	−	0.773354i	$-0.281422\pi$
$587$	30.7200	1.26795	0.633975	+	0.773354i	$0.281422\pi$
$588$	0	0
$589$	−0.226011	−0.00931260
$590$	5.82812	0.239940
$591$	0	0
$592$	12.2872	0.505000
$593$	−44.0715	−1.80980	−0.904901	−	0.425623i	$-0.860055\pi$
$593$	−44.0715	−1.80980	−0.904901	+	0.425623i	$0.860055\pi$
$594$	0	0
$595$	0	0
$596$	13.2584	0.543085
$597$	0	0
$598$	16.9331	0.692447
$599$	−27.3663	−1.11816	−0.559078	−	0.829115i	$-0.688845\pi$
$599$	−27.3663	−1.11816	−0.559078	+	0.829115i	$0.688845\pi$
$600$	0	0
$601$	−7.21816	−0.294435	−0.147217	−	0.989104i	$-0.547032\pi$
$601$	−7.21816	−0.294435	−0.147217	+	0.989104i	$0.547032\pi$
$602$	0	0
$603$	0	0
$604$	22.4201	0.912260
$605$	−29.3194	−1.19200
$606$	0	0
$607$	37.0048	1.50198	0.750989	−	0.660315i	$-0.229577\pi$
$607$	37.0048	1.50198	0.750989	+	0.660315i	$0.229577\pi$
$608$	11.4835	0.465715
$609$	0	0
$610$	−17.7185	−0.717400
$611$	22.8080	0.922713
$612$	0	0
$613$	1.66058	0.0670704	0.0335352	−	0.999438i	$-0.489323\pi$
$613$	1.66058	0.0670704	0.0335352	+	0.999438i	$0.489323\pi$
$614$	69.3197	2.79751
$615$	0	0
$616$	0	0
$617$	−22.2418	−0.895421	−0.447710	−	0.894179i	$-0.647760\pi$
$617$	−22.2418	−0.895421	−0.447710	+	0.894179i	$0.647760\pi$
$618$	0	0
$619$	19.7805	0.795046	0.397523	−	0.917592i	$-0.369870\pi$
$619$	19.7805	0.795046	0.397523	+	0.917592i	$0.369870\pi$
$620$	−1.53863	−0.0617929
$621$	0	0
$622$	−29.7838	−1.19422
$623$	0	0
$624$	0	0
$625$	17.7393	0.709571
$626$	−42.7057	−1.70686
$627$	0	0
$628$	66.8941	2.66936
$629$	−33.3462	−1.32960
$630$	0	0
$631$	49.2569	1.96089	0.980443	−	0.196804i	$-0.0630562\pi$
$631$	49.2569	1.96089	0.980443	+	0.196804i	$0.0630562\pi$
$632$	−29.8063	−1.18563
$633$	0	0
$634$	−33.9096	−1.34672
$635$	−52.0041	−2.06372
$636$	0	0
$637$	0	0
$638$	7.16578	0.283696
$639$	0	0
$640$	53.2324	2.10420
$641$	−8.65682	−0.341924	−0.170962	−	0.985278i	$-0.554688\pi$
$641$	−8.65682	−0.341924	−0.170962	+	0.985278i	$0.554688\pi$
$642$	0	0
$643$	17.7156	0.698637	0.349318	−	0.937004i	$-0.386413\pi$
$643$	17.7156	0.698637	0.349318	+	0.937004i	$0.386413\pi$
$644$	0	0
$645$	0	0
$646$	−14.2958	−0.562460
$647$	−12.1018	−0.475772	−0.237886	−	0.971293i	$-0.576455\pi$
$647$	−12.1018	−0.475772	−0.237886	+	0.971293i	$0.576455\pi$
$648$	0	0
$649$	−2.99762	−0.117667
$650$	−60.3062	−2.36540
$651$	0	0
$652$	17.5705	0.688112
$653$	−36.2140	−1.41717	−0.708583	−	0.705628i	$-0.750665\pi$
$653$	−36.2140	−1.41717	−0.708583	+	0.705628i	$0.750665\pi$
$654$	0	0
$655$	−67.0697	−2.62063
$656$	15.3900	0.600880
$657$	0	0
$658$	0	0
$659$	39.3414	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$659$	39.3414	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$660$	0	0
$661$	−1.66080	−0.0645978	−0.0322989	−	0.999478i	$-0.510283\pi$
$661$	−1.66080	−0.0645978	−0.0322989	+	0.999478i	$0.510283\pi$
$662$	−2.73414	−0.106265
$663$	0	0
$664$	12.4469	0.483032
$665$	0	0
$666$	0	0
$667$	−2.00175	−0.0775081
$668$	33.4722	1.29508
$669$	0	0
$670$	90.5183	3.49703
$671$	9.11327	0.351814
$672$	0	0
$673$	20.8307	0.802965	0.401483	−	0.915867i	$-0.368495\pi$
$673$	20.8307	0.802965	0.401483	+	0.915867i	$0.368495\pi$
$674$	26.2941	1.01281
$675$	0	0
$676$	−13.6084	−0.523401
$677$	−46.7308	−1.79601	−0.898006	−	0.439984i	$-0.854984\pi$
$677$	−46.7308	−1.79601	−0.898006	+	0.439984i	$0.854984\pi$
$678$	0	0
$679$	0	0
$680$	−29.6257	−1.13610
$681$	0	0
$682$	1.34185	0.0513822
$683$	−21.1494	−0.809261	−0.404630	−	0.914480i	$-0.632600\pi$
$683$	−21.1494	−0.809261	−0.404630	+	0.914480i	$0.632600\pi$
$684$	0	0
$685$	−33.0030	−1.26098
$686$	0	0
$687$	0	0
$688$	−0.396141	−0.0151027
$689$	32.2711	1.22943
$690$	0	0
$691$	−23.5970	−0.897672	−0.448836	−	0.893614i	$-0.648161\pi$
$691$	−23.5970	−0.897672	−0.448836	+	0.893614i	$0.648161\pi$
$692$	55.3689	2.10481
$693$	0	0
$694$	−9.59768	−0.364323
$695$	14.3403	0.543960
$696$	0	0
$697$	−41.7671	−1.58204
$698$	15.8437	0.599693
$699$	0	0
$700$	0	0
$701$	−42.1420	−1.59168	−0.795841	−	0.605505i	$-0.792971\pi$
$701$	−42.1420	−1.59168	−0.795841	+	0.605505i	$0.792971\pi$
$702$	0	0
$703$	13.3205	0.502392
$704$	−55.3488	−2.08603
$705$	0	0
$706$	55.4134	2.08551
$707$	0	0
$708$	0	0
$709$	45.9838	1.72696	0.863478	−	0.504386i	$-0.168281\pi$
$709$	45.9838	1.72696	0.863478	+	0.504386i	$0.168281\pi$
$710$	−30.4690	−1.14348
$711$	0	0
$712$	0.496419	0.0186041
$713$	−0.374844	−0.0140380
$714$	0	0
$715$	47.3442	1.77057
$716$	9.60869	0.359094
$717$	0	0
$718$	70.6378	2.63618
$719$	−33.0498	−1.23255	−0.616275	−	0.787531i	$-0.711359\pi$
$719$	−33.0498	−1.23255	−0.616275	+	0.787531i	$0.711359\pi$
$720$	0	0
$721$	0	0
$722$	−36.2413	−1.34876
$723$	0	0
$724$	−56.7452	−2.10892
$725$	7.12910	0.264768
$726$	0	0
$727$	−25.0550	−0.929238	−0.464619	−	0.885511i	$-0.653809\pi$
$727$	−25.0550	−0.929238	−0.464619	+	0.885511i	$0.653809\pi$
$728$	0	0
$729$	0	0
$730$	30.0580	1.11250
$731$	1.07509	0.0397636
$732$	0	0
$733$	14.0125	0.517563	0.258782	−	0.965936i	$-0.416679\pi$
$733$	14.0125	0.517563	0.258782	+	0.965936i	$0.416679\pi$
$734$	−24.5634	−0.906652
$735$	0	0
$736$	19.0456	0.702030
$737$	−46.5570	−1.71495
$738$	0	0
$739$	−38.6013	−1.41997	−0.709987	−	0.704215i	$-0.751299\pi$
$739$	−38.6013	−1.41997	−0.709987	+	0.704215i	$0.751299\pi$
$740$	90.6830	3.33357
$741$	0	0
$742$	0	0
$743$	−1.81318	−0.0665192	−0.0332596	−	0.999447i	$-0.510589\pi$
$743$	−1.81318	−0.0665192	−0.0332596	+	0.999447i	$0.510589\pi$
$744$	0	0
$745$	−17.5586	−0.643296
$746$	35.6692	1.30594
$747$	0	0
$748$	50.0567	1.83026
$749$	0	0
$750$	0	0
$751$	−29.4262	−1.07378	−0.536888	−	0.843653i	$-0.680400\pi$
$751$	−29.4262	−1.07378	−0.536888	+	0.843653i	$0.680400\pi$
$752$	11.7675	0.429118
$753$	0	0
$754$	−4.76452	−0.173514
$755$	−29.6917	−1.08059
$756$	0	0
$757$	22.5455	0.819431	0.409715	−	0.912213i	$-0.365628\pi$
$757$	22.5455	0.819431	0.409715	+	0.912213i	$0.365628\pi$
$758$	7.03213	0.255419
$759$	0	0
$760$	11.8343	0.429275
$761$	−12.8626	−0.466270	−0.233135	−	0.972444i	$-0.574898\pi$
$761$	−12.8626	−0.466270	−0.233135	+	0.972444i	$0.574898\pi$
$762$	0	0
$763$	0	0
$764$	12.0039	0.434285
$765$	0	0
$766$	36.3825	1.31455
$767$	1.99312	0.0719673
$768$	0	0
$769$	−44.7222	−1.61272	−0.806362	−	0.591423i	$-0.798566\pi$
$769$	−44.7222	−1.61272	−0.806362	+	0.591423i	$0.798566\pi$
$770$	0	0
$771$	0	0
$772$	−39.8903	−1.43568
$773$	−10.4880	−0.377227	−0.188614	−	0.982051i	$-0.560399\pi$
$773$	−10.4880	−0.377227	−0.188614	+	0.982051i	$0.560399\pi$
$774$	0	0
$775$	1.33498	0.0479540
$776$	2.04571	0.0734368
$777$	0	0
$778$	−70.6053	−2.53132
$779$	16.6843	0.597777
$780$	0	0
$781$	15.6713	0.560764
$782$	−23.7099	−0.847865
$783$	0	0
$784$	0	0
$785$	−88.5901	−3.16192
$786$	0	0
$787$	32.2074	1.14807	0.574035	−	0.818831i	$-0.305377\pi$
$787$	32.2074	1.14807	0.574035	+	0.818831i	$0.305377\pi$
$788$	21.5218	0.766682
$789$	0	0
$790$	129.673	4.61356
$791$	0	0
$792$	0	0
$793$	−6.05941	−0.215176
$794$	61.4085	2.17931
$795$	0	0
$796$	−36.3872	−1.28971
$797$	−32.2778	−1.14334	−0.571669	−	0.820484i	$-0.693704\pi$
$797$	−32.2778	−1.14334	−0.571669	+	0.820484i	$0.693704\pi$
$798$	0	0
$799$	−31.9360	−1.12981
$800$	−67.8296	−2.39814
$801$	0	0
$802$	−60.9149	−2.15098
$803$	−15.4600	−0.545570
$804$	0	0
$805$	0	0
$806$	−0.892196	−0.0314263
$807$	0	0
$808$	−2.63955	−0.0928591
$809$	−29.9639	−1.05347	−0.526737	−	0.850028i	$-0.676585\pi$
$809$	−29.9639	−1.05347	−0.526737	+	0.850028i	$0.676585\pi$
$810$	0	0
$811$	−13.1971	−0.463414	−0.231707	−	0.972786i	$-0.574431\pi$
$811$	−13.1971	−0.463414	−0.231707	+	0.972786i	$0.574431\pi$
$812$	0	0
$813$	0	0
$814$	−79.0853	−2.77194
$815$	−23.2692	−0.815083
$816$	0	0
$817$	−0.429456	−0.0150247
$818$	−26.1087	−0.912870
$819$	0	0
$820$	113.583	3.96650
$821$	8.03611	0.280462	0.140231	−	0.990119i	$-0.455215\pi$
$821$	8.03611	0.280462	0.140231	+	0.990119i	$0.455215\pi$
$822$	0	0
$823$	19.5656	0.682015	0.341008	−	0.940060i	$-0.389232\pi$
$823$	19.5656	0.682015	0.341008	+	0.940060i	$0.389232\pi$
$824$	−28.9325	−1.00791
$825$	0	0
$826$	0	0
$827$	45.8218	1.59338	0.796690	−	0.604389i	$-0.206583\pi$
$827$	45.8218	1.59338	0.796690	+	0.604389i	$0.206583\pi$
$828$	0	0
$829$	34.1546	1.18624	0.593119	−	0.805115i	$-0.297897\pi$
$829$	34.1546	1.18624	0.593119	+	0.805115i	$0.297897\pi$
$830$	−54.1504	−1.87959
$831$	0	0
$832$	36.8013	1.27586
$833$	0	0
$834$	0	0
$835$	−44.3284	−1.53405
$836$	−19.9957	−0.691565
$837$	0	0
$838$	−19.4672	−0.672483
$839$	35.5972	1.22895	0.614476	−	0.788935i	$-0.289367\pi$
$839$	35.5972	1.22895	0.614476	+	0.788935i	$0.289367\pi$
$840$	0	0
$841$	−28.4368	−0.980578
$842$	41.2897	1.42294
$843$	0	0
$844$	−14.8132	−0.509891
$845$	18.0221	0.619979
$846$	0	0
$847$	0	0
$848$	16.6499	0.571761
$849$	0	0
$850$	84.4413	2.89631
$851$	22.0924	0.757316
$852$	0	0
$853$	33.3946	1.14341	0.571705	−	0.820459i	$-0.306282\pi$
$853$	33.3946	1.14341	0.571705	+	0.820459i	$0.306282\pi$
$854$	0	0
$855$	0	0
$856$	15.6178	0.533806
$857$	25.8147	0.881814	0.440907	−	0.897553i	$-0.354657\pi$
$857$	25.8147	0.881814	0.440907	+	0.897553i	$0.354657\pi$
$858$	0	0
$859$	−43.0007	−1.46716	−0.733582	−	0.679601i	$-0.762153\pi$
$859$	−43.0007	−1.46716	−0.733582	+	0.679601i	$0.762153\pi$
$860$	−2.92364	−0.0996954
$861$	0	0
$862$	36.6956	1.24986
$863$	−21.4967	−0.731755	−0.365877	−	0.930663i	$-0.619231\pi$
$863$	−21.4967	−0.731755	−0.365877	+	0.930663i	$0.619231\pi$
$864$	0	0
$865$	−73.3269	−2.49319
$866$	−55.8631	−1.89830
$867$	0	0
$868$	0	0
$869$	−66.6957	−2.26250
$870$	0	0
$871$	30.9557	1.04889
$872$	−17.6039	−0.596142
$873$	0	0
$874$	9.47117	0.320367
$875$	0	0
$876$	0	0
$877$	14.6841	0.495846	0.247923	−	0.968780i	$-0.420252\pi$
$877$	14.6841	0.495846	0.247923	+	0.968780i	$0.420252\pi$
$878$	−65.0306	−2.19468
$879$	0	0
$880$	24.4267	0.823425
$881$	−37.6060	−1.26698	−0.633490	−	0.773751i	$-0.718378\pi$
$881$	−37.6060	−1.26698	−0.633490	+	0.773751i	$0.718378\pi$
$882$	0	0
$883$	−46.9989	−1.58164	−0.790820	−	0.612049i	$-0.790345\pi$
$883$	−46.9989	−1.58164	−0.790820	+	0.612049i	$0.790345\pi$
$884$	−33.2827	−1.11942
$885$	0	0
$886$	−71.8847	−2.41501
$887$	26.4248	0.887259	0.443630	−	0.896210i	$-0.353691\pi$
$887$	26.4248	0.887259	0.443630	+	0.896210i	$0.353691\pi$
$888$	0	0
$889$	0	0
$890$	−2.15969	−0.0723928
$891$	0	0
$892$	−43.7516	−1.46491
$893$	12.7572	0.426902
$894$	0	0
$895$	−12.7251	−0.425354
$896$	0	0
$897$	0	0
$898$	0.379513	0.0126645
$899$	0.105471	0.00351766
$900$	0	0
$901$	−45.1863	−1.50538
$902$	−99.0567	−3.29823
$903$	0	0
$904$	5.34551	0.177789
$905$	75.1496	2.49806
$906$	0	0
$907$	−0.450052	−0.0149437	−0.00747187	−	0.999972i	$-0.502378\pi$
$907$	−0.450052	−0.0149437	−0.00747187	+	0.999972i	$0.502378\pi$
$908$	12.3333	0.409296
$909$	0	0
$910$	0	0
$911$	4.74439	0.157189	0.0785944	−	0.996907i	$-0.474957\pi$
$911$	4.74439	0.157189	0.0785944	+	0.996907i	$0.474957\pi$
$912$	0	0
$913$	27.8516	0.921754
$914$	54.1004	1.78948
$915$	0	0
$916$	2.79259	0.0922697
$917$	0	0
$918$	0	0
$919$	42.4205	1.39932	0.699662	−	0.714474i	$-0.253334\pi$
$919$	42.4205	1.39932	0.699662	+	0.714474i	$0.253334\pi$
$920$	19.6275	0.647099
$921$	0	0
$922$	20.8238	0.685794
$923$	−10.4199	−0.342974
$924$	0	0
$925$	−78.6805	−2.58700
$926$	6.31755	0.207608
$927$	0	0
$928$	−5.35892	−0.175915
$929$	45.0817	1.47908	0.739541	−	0.673111i	$-0.235042\pi$
$929$	45.0817	1.47908	0.739541	+	0.673111i	$0.235042\pi$
$930$	0	0
$931$	0	0
$932$	10.8495	0.355388
$933$	0	0
$934$	17.0079	0.556517
$935$	−66.2918	−2.16797
$936$	0	0
$937$	19.3045	0.630650	0.315325	−	0.948984i	$-0.397886\pi$
$937$	19.3045	0.630650	0.315325	+	0.948984i	$0.397886\pi$
$938$	0	0
$939$	0	0
$940$	86.8480	2.83267
$941$	10.5539	0.344048	0.172024	−	0.985093i	$-0.444969\pi$
$941$	10.5539	0.344048	0.172024	+	0.985093i	$0.444969\pi$
$942$	0	0
$943$	27.6713	0.901103
$944$	1.02833	0.0334692
$945$	0	0
$946$	2.54973	0.0828989
$947$	47.7600	1.55199	0.775996	−	0.630737i	$-0.217247\pi$
$947$	47.7600	1.55199	0.775996	+	0.630737i	$0.217247\pi$
$948$	0	0
$949$	10.2793	0.333681
$950$	−33.7309	−1.09438
$951$	0	0
$952$	0	0
$953$	53.8101	1.74308	0.871540	−	0.490324i	$-0.163121\pi$
$953$	53.8101	1.74308	0.871540	+	0.490324i	$0.163121\pi$
$954$	0	0
$955$	−15.8971	−0.514419
$956$	−19.2365	−0.622154
$957$	0	0
$958$	76.3574	2.46700
$959$	0	0
$960$	0	0
$961$	−30.9802	−0.999363
$962$	52.5838	1.69537
$963$	0	0
$964$	−43.0460	−1.38642
$965$	52.8281	1.70060
$966$	0	0
$967$	−9.81775	−0.315718	−0.157859	−	0.987462i	$-0.550459\pi$
$967$	−9.81775	−0.315718	−0.157859	+	0.987462i	$0.550459\pi$
$968$	14.8803	0.478271
$969$	0	0
$970$	−8.89994	−0.285760
$971$	−23.7986	−0.763734	−0.381867	−	0.924217i	$-0.624719\pi$
$971$	−23.7986	−0.763734	−0.381867	+	0.924217i	$0.624719\pi$
$972$	0	0
$973$	0	0
$974$	40.7058	1.30430
$975$	0	0
$976$	−3.12629	−0.100070
$977$	−39.9903	−1.27940	−0.639701	−	0.768624i	$-0.720942\pi$
$977$	−39.9903	−1.27940	−0.639701	+	0.768624i	$0.720942\pi$
$978$	0	0
$979$	1.11081	0.0355016
$980$	0	0
$981$	0	0
$982$	6.96340	0.222211
$983$	−46.2286	−1.47446	−0.737231	−	0.675641i	$-0.763867\pi$
$983$	−46.2286	−1.47446	−0.737231	+	0.675641i	$0.763867\pi$
$984$	0	0
$985$	−28.5020	−0.908151
$986$	6.67133	0.212459
$987$	0	0
$988$	13.2951	0.422974
$989$	−0.712263	−0.0226486
$990$	0	0
$991$	14.3964	0.457315	0.228658	−	0.973507i	$-0.426566\pi$
$991$	14.3964	0.457315	0.228658	+	0.973507i	$0.426566\pi$
$992$	−1.00350	−0.0318612
$993$	0	0
$994$	0	0
$995$	48.1888	1.52769
$996$	0	0
$997$	43.8546	1.38889	0.694444	−	0.719547i	$-0.255650\pi$
$997$	43.8546	1.38889	0.694444	+	0.719547i	$0.255650\pi$
$998$	63.7665	2.01849
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.x.1.4		4
3.2	odd	2		3969.2.a.s.1.1		4
7.2	even	3		567.2.e.c.487.1	yes	8
7.4	even	3		567.2.e.c.163.1	✓	8
7.6	odd	2		3969.2.a.w.1.4		4
21.2	odd	6		567.2.e.d.487.4	yes	8
21.11	odd	6		567.2.e.d.163.4	yes	8
21.20	even	2		3969.2.a.t.1.1		4
63.2	odd	6		567.2.g.k.109.4		8
63.4	even	3		567.2.g.j.541.1		8
63.11	odd	6		567.2.h.j.352.1		8
63.16	even	3		567.2.g.j.109.1		8
63.23	odd	6		567.2.h.j.298.1		8
63.25	even	3		567.2.h.k.352.4		8
63.32	odd	6		567.2.g.k.541.4		8
63.58	even	3		567.2.h.k.298.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.c.163.1	✓	8	7.4	even	3
567.2.e.c.487.1	yes	8	7.2	even	3
567.2.e.d.163.4	yes	8	21.11	odd	6
567.2.e.d.487.4	yes	8	21.2	odd	6
567.2.g.j.109.1		8	63.16	even	3
567.2.g.j.541.1		8	63.4	even	3
567.2.g.k.109.4		8	63.2	odd	6
567.2.g.k.541.4		8	63.32	odd	6
567.2.h.j.298.1		8	63.23	odd	6
567.2.h.j.352.1		8	63.11	odd	6
567.2.h.k.298.4		8	63.58	even	3
567.2.h.k.352.4		8	63.25	even	3
3969.2.a.s.1.1		4	3.2	odd	2
3969.2.a.t.1.1		4	21.20	even	2
3969.2.a.w.1.4		4	7.6	odd	2
3969.2.a.x.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20800 q^{2} +2.87525 q^{4} -3.80779 q^{5} +1.93254 q^{8} +O(q^{10})\)	\(q+2.20800 q^{2} +2.87525 q^{4} -3.80779 q^{5} +1.93254 q^{8} -8.40758 q^{10} +4.32433 q^{11} -2.87525 q^{13} -1.48345 q^{16} +4.02595 q^{17} -1.60821 q^{19} -10.9483 q^{20} +9.54811 q^{22} -2.66725 q^{23} +9.49923 q^{25} -6.34853 q^{26} +0.750492 q^{29} +0.140536 q^{31} -7.14054 q^{32} +8.88928 q^{34} -8.28282 q^{37} -3.55091 q^{38} -7.35870 q^{40} -10.3745 q^{41} +0.267040 q^{43} +12.4335 q^{44} -5.88928 q^{46} -7.93254 q^{47} +20.9743 q^{50} -8.26704 q^{52} -11.2238 q^{53} -16.4661 q^{55} +1.65708 q^{58} -0.693198 q^{59} +2.10744 q^{61} +0.310302 q^{62} -12.7994 q^{64} +10.9483 q^{65} -10.7663 q^{67} +11.5756 q^{68} +3.62399 q^{71} -3.57511 q^{73} -18.2884 q^{74} -4.62399 q^{76} -15.4234 q^{79} +5.64867 q^{80} -22.9068 q^{82} +6.44067 q^{83} -15.3299 q^{85} +0.589624 q^{86} +8.35695 q^{88} +0.256874 q^{89} -7.66900 q^{92} -17.5150 q^{94} +6.12370 q^{95} +1.05856 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 5 q^{4} + 2 q^{5} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + 5 * q^4 + 2 * q^5 - 3 * q^8	\( 4 q + q^{2} + 5 q^{4} + 2 q^{5} - 3 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} - q^{16} + 6 q^{17} - 8 q^{19} - 8 q^{20} - 7 q^{22} - 12 q^{23} + 8 q^{25} + q^{26} - 10 q^{29} - 18 q^{31} - 10 q^{32} - 20 q^{38} - 18 q^{40} - 5 q^{41} - 7 q^{43} + 13 q^{44} + 12 q^{46} - 21 q^{47} + 38 q^{50} - 25 q^{52} - 12 q^{53} - 26 q^{55} - 7 q^{58} + 6 q^{59} - 20 q^{61} + 18 q^{62} - 23 q^{64} + 8 q^{65} - 5 q^{67} + 51 q^{68} - 9 q^{71} - 6 q^{73} + 5 q^{76} - 10 q^{79} - 2 q^{80} - 35 q^{82} + 9 q^{83} - 9 q^{85} - 22 q^{86} + 18 q^{88} - 22 q^{89} - 36 q^{92} - 15 q^{94} - 16 q^{95} - 9 q^{97}+O(q^{100}) \) 4 * q + q^2 + 5 * q^4 + 2 * q^5 - 3 * q^8 - 7 * q^10 + 5 * q^11 - 5 * q^13 - q^16 + 6 * q^17 - 8 * q^19 - 8 * q^20 - 7 * q^22 - 12 * q^23 + 8 * q^25 + q^26 - 10 * q^29 - 18 * q^31 - 10 * q^32 - 20 * q^38 - 18 * q^40 - 5 * q^41 - 7 * q^43 + 13 * q^44 + 12 * q^46 - 21 * q^47 + 38 * q^50 - 25 * q^52 - 12 * q^53 - 26 * q^55 - 7 * q^58 + 6 * q^59 - 20 * q^61 + 18 * q^62 - 23 * q^64 + 8 * q^65 - 5 * q^67 + 51 * q^68 - 9 * q^71 - 6 * q^73 + 5 * q^76 - 10 * q^79 - 2 * q^80 - 35 * q^82 + 9 * q^83 - 9 * q^85 - 22 * q^86 + 18 * q^88 - 22 * q^89 - 36 * q^92 - 15 * q^94 - 16 * q^95 - 9 * q^97

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