LMFDB - Embedded newform 3969.2.a.s.1.1

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			1.1
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	$-0.785109\pi$
$2$	−2.20800	−1.56129	−0.780644	+	0.624975i	$0.785109\pi$
$3$	0	0
$3$	0	0
$4$	2.87525	1.43762
$5$	3.80779	1.70289	0.851447	−	0.524441i	$-0.175726\pi$
$5$	3.80779	1.70289	0.851447	+	0.524441i	$0.175726\pi$
$6$	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	$-0.726035\pi$
$11$	−4.32433	−1.30384	−0.651918	+	0.758290i	$0.726035\pi$
$12$	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	$-0.662353\pi$
$17$	−4.02595	−0.976436	−0.488218	+	0.872722i	$0.662353\pi$
$18$	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	$-0.410302\pi$
$23$	2.66725	0.556160	0.278080	+	0.960558i	$0.410302\pi$
$24$	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	$-0.522198\pi$
$29$	−0.750492	−0.139363	−0.0696815	+	0.997569i	$0.522198\pi$
$30$	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	$-0.199407\pi$
$41$	10.3745	1.62022	0.810111	+	0.586277i	$0.199407\pi$
$42$	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	$-0.303623\pi$
$47$	7.93254	1.15708	0.578540	+	0.815654i	$0.303623\pi$
$48$	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	$-0.219831\pi$
$53$	11.2238	1.54170	0.770852	+	0.637014i	$0.219831\pi$
$54$	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	$-0.485632\pi$
$59$	0.693198	0.0902468	0.0451234	+	0.998981i	$0.485632\pi$
$60$	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	$-0.568990\pi$
$71$	−3.62399	−0.430088	−0.215044	+	0.976604i	$0.568990\pi$
$72$	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	$-0.615001\pi$
$83$	−6.44067	−0.706956	−0.353478	+	0.935443i	$0.615001\pi$
$84$	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	$-0.504334\pi$
$89$	−0.256874	−0.0272286	−0.0136143	+	0.999907i	$0.504334\pi$
$90$	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	$-0.478353\pi$
$101$	1.36585	0.135907	0.0679534	+	0.997688i	$0.478353\pi$
$102$	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	$-0.627744\pi$
$107$	−8.08149	−0.781267	−0.390634	+	0.920546i	$0.627744\pi$
$108$	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	$-0.541531\pi$
$113$	−2.76606	−0.260209	−0.130104	+	0.991500i	$0.541531\pi$
$114$	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	$-0.779476\pi$
$131$	−17.6138	−1.53893	−0.769463	+	0.638691i	$0.779476\pi$
$132$	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	$-0.620727\pi$
$137$	−8.66725	−0.740493	−0.370247	+	0.928934i	$0.620727\pi$
$138$	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	$-0.560487\pi$
$149$	−4.61122	−0.377766	−0.188883	+	0.982000i	$0.560487\pi$
$150$	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	$-0.648727\pi$
$167$	−11.6415	−0.900848	−0.450424	+	0.892815i	$0.648727\pi$
$168$	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	$-0.761434\pi$
$173$	−19.2571	−1.46409	−0.732045	+	0.681256i	$0.761434\pi$
$174$	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	$-0.539858\pi$
$179$	−3.34187	−0.249783	−0.124891	+	0.992170i	$0.539858\pi$
$180$	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	$-0.548263\pi$
$191$	−4.17490	−0.302085	−0.151043	+	0.988527i	$0.548263\pi$
$192$	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	$-0.585917\pi$
$197$	−7.48520	−0.533299	−0.266649	+	0.963794i	$0.585917\pi$
$198$	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	$-0.545466\pi$
$227$	−4.28949	−0.284703	−0.142352	+	0.989816i	$0.545466\pi$
$228$	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	$-0.539445\pi$
$233$	−3.77342	−0.247205	−0.123603	+	0.992332i	$0.539445\pi$
$234$	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	$-0.430574\pi$
$239$	6.69040	0.432766	0.216383	+	0.976309i	$0.430574\pi$
$240$	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	$-0.681197\pi$
$251$	−17.0787	−1.07800	−0.538999	+	0.842307i	$0.681197\pi$
$252$	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	$-0.413848\pi$
$257$	8.57231	0.534726	0.267363	+	0.963596i	$0.413848\pi$
$258$	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	$-0.556200\pi$
$263$	−5.69685	−0.351283	−0.175641	+	0.984454i	$0.556200\pi$
$264$	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	$-0.342223\pi$
$269$	15.6015	0.951243	0.475621	+	0.879650i	$0.342223\pi$
$270$	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	$-0.326735\pi$
$281$	17.3613	1.03569	0.517844	+	0.855475i	$0.326735\pi$
$282$	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	$-0.601165\pi$
$293$	−10.6981	−0.624990	−0.312495	+	0.949919i	$0.601165\pi$
$294$	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	$-0.375081\pi$
$311$	13.4891	0.764895	0.382447	+	0.923977i	$0.375081\pi$
$312$	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	$-0.358060\pi$
$317$	15.3576	0.862571	0.431286	+	0.902215i	$0.358060\pi$
$318$	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	$-0.462777\pi$
$347$	4.34678	0.233347	0.116674	+	0.993170i	$0.462777\pi$
$348$	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	$-0.732799\pi$
$353$	−25.0967	−1.33576	−0.667881	+	0.744268i	$0.732799\pi$
$354$	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	$-0.819943\pi$
$359$	−31.9918	−1.68846	−0.844231	+	0.535979i	$0.819943\pi$
$360$	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	$-0.638315\pi$
$383$	−16.4776	−0.841968	−0.420984	+	0.907068i	$0.638315\pi$
$384$	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	$-0.199113\pi$
$389$	31.9771	1.62130	0.810651	+	0.585529i	$0.199113\pi$
$390$	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	$-0.258117\pi$
$401$	27.5883	1.37769	0.688847	+	0.724907i	$0.258117\pi$
$402$	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	$-0.430907\pi$
$419$	8.81668	0.430723	0.215362	+	0.976534i	$0.430907\pi$
$420$	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	$-0.631082\pi$
$431$	−16.6194	−0.800530	−0.400265	+	0.916399i	$0.631082\pi$
$432$	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	$-0.218553\pi$
$443$	32.5565	1.54681	0.773404	+	0.633914i	$0.218553\pi$
$444$	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	$-0.501291\pi$
$449$	−0.171881	−0.00811158	−0.00405579	+	0.999992i	$0.501291\pi$
$450$	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	$-0.570483\pi$
$461$	−9.43107	−0.439249	−0.219624	+	0.975584i	$0.570483\pi$
$462$	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	$-0.557035\pi$
$467$	−7.70288	−0.356447	−0.178223	+	0.983990i	$0.557035\pi$
$468$	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	$-0.789946\pi$
$479$	−34.5822	−1.58010	−0.790051	+	0.613041i	$0.789946\pi$
$480$	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	$-0.522671\pi$
$491$	−3.15372	−0.142325	−0.0711627	+	0.997465i	$0.522671\pi$
$492$	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	$-0.658908\pi$
$503$	−21.4742	−0.957487	−0.478744	+	0.877955i	$0.658908\pi$
$504$	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	$-0.562654\pi$
$509$	−8.82435	−0.391133	−0.195566	+	0.980690i	$0.562654\pi$
$510$	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	$-0.725206\pi$
$521$	−29.6703	−1.29988	−0.649939	+	0.759986i	$0.725206\pi$
$522$	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	$-0.421874\pi$
$557$	11.4692	0.485964	0.242982	+	0.970031i	$0.421874\pi$
$558$	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	$-0.320367\pi$
$563$	25.3816	1.06971	0.534854	+	0.844944i	$0.320367\pi$
$564$	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	$-0.558334\pi$
$569$	−8.69404	−0.364473	−0.182237	+	0.983255i	$0.558334\pi$
$570$	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	$-0.718578\pi$
$587$	−30.7200	−1.26795	−0.633975	+	0.773354i	$0.718578\pi$
$588$	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	$-0.139945\pi$
$593$	44.0715	1.80980	0.904901	+	0.425623i	$0.139945\pi$
$594$	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	$-0.311155\pi$
$599$	27.3663	1.11816	0.559078	+	0.829115i	$0.311155\pi$
$600$	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	$-0.352240\pi$
$617$	22.2418	0.895421	0.447710	+	0.894179i	$0.352240\pi$
$618$	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	$-0.445312\pi$
$641$	8.65682	0.341924	0.170962	+	0.985278i	$0.445312\pi$
$642$	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	$-0.423545\pi$
$647$	12.1018	0.475772	0.237886	+	0.971293i	$0.423545\pi$
$648$	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	$-0.249335\pi$
$653$	36.2140	1.41717	0.708583	+	0.705628i	$0.249335\pi$
$654$	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	$-0.777885\pi$
$659$	−39.3414	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$660$	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	$-0.145016\pi$
$677$	46.7308	1.79601	0.898006	+	0.439984i	$0.145016\pi$
$678$	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	$-0.367400\pi$
$683$	21.1494	0.809261	0.404630	+	0.914480i	$0.367400\pi$
$684$	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	$-0.207029\pi$
$701$	42.1420	1.59168	0.795841	+	0.605505i	$0.207029\pi$
$702$	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	$-0.288641\pi$
$719$	33.0498	1.23255	0.616275	+	0.787531i	$0.288641\pi$
$720$	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	$-0.489411\pi$
$743$	1.81318	0.0665192	0.0332596	+	0.999447i	$0.489411\pi$
$744$	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	$-0.425102\pi$
$761$	12.8626	0.466270	0.233135	+	0.972444i	$0.425102\pi$
$762$	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	$-0.439601\pi$
$773$	10.4880	0.377227	0.188614	+	0.982051i	$0.439601\pi$
$774$	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	$-0.306296\pi$
$797$	32.2778	1.14334	0.571669	+	0.820484i	$0.306296\pi$
$798$	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	$-0.323415\pi$
$809$	29.9639	1.05347	0.526737	+	0.850028i	$0.323415\pi$
$810$	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	$-0.544785\pi$
$821$	−8.03611	−0.280462	−0.140231	+	0.990119i	$0.544785\pi$
$822$	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	$-0.793417\pi$
$827$	−45.8218	−1.59338	−0.796690	+	0.604389i	$0.793417\pi$
$828$	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	$-0.710633\pi$
$839$	−35.5972	−1.22895	−0.614476	+	0.788935i	$0.710633\pi$
$840$	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	$-0.645343\pi$
$857$	−25.8147	−0.881814	−0.440907	+	0.897553i	$0.645343\pi$
$858$	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	$-0.380769\pi$
$863$	21.4967	0.731755	0.365877	+	0.930663i	$0.380769\pi$
$864$	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	$-0.281622\pi$
$881$	37.6060	1.26698	0.633490	+	0.773751i	$0.281622\pi$
$882$	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	$-0.646309\pi$
$887$	−26.4248	−0.887259	−0.443630	+	0.896210i	$0.646309\pi$
$888$	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	$-0.525043\pi$
$911$	−4.74439	−0.157189	−0.0785944	+	0.996907i	$0.525043\pi$
$912$	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	$-0.764958\pi$
$929$	−45.0817	−1.47908	−0.739541	+	0.673111i	$0.764958\pi$
$930$	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	$-0.555031\pi$
$941$	−10.5539	−0.344048	−0.172024	+	0.985093i	$0.555031\pi$
$942$	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	$-0.782753\pi$
$947$	−47.7600	−1.55199	−0.775996	+	0.630737i	$0.782753\pi$
$948$	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	$-0.836879\pi$
$953$	−53.8101	−1.74308	−0.871540	+	0.490324i	$0.836879\pi$
$954$	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	$-0.375281\pi$
$971$	23.7986	0.763734	0.381867	+	0.924217i	$0.375281\pi$
$972$	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	$-0.279058\pi$
$977$	39.9903	1.27940	0.639701	+	0.768624i	$0.279058\pi$
$978$	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	$-0.236133\pi$
$983$	46.2286	1.47446	0.737231	+	0.675641i	$0.236133\pi$
$984$	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.s.1.1		4
3.2	odd	2		3969.2.a.x.1.4		4
7.2	even	3		567.2.e.d.487.4	yes	8
7.4	even	3		567.2.e.d.163.4	yes	8
7.6	odd	2		3969.2.a.t.1.1		4
21.2	odd	6		567.2.e.c.487.1	yes	8
21.11	odd	6		567.2.e.c.163.1	✓	8
21.20	even	2		3969.2.a.w.1.4		4
63.2	odd	6		567.2.g.j.109.1		8
63.4	even	3		567.2.g.k.541.4		8
63.11	odd	6		567.2.h.k.352.4		8
63.16	even	3		567.2.g.k.109.4		8
63.23	odd	6		567.2.h.k.298.4		8
63.25	even	3		567.2.h.j.352.1		8
63.32	odd	6		567.2.g.j.541.1		8
63.58	even	3		567.2.h.j.298.1		8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more