LMFDB - Embedded newform 1288.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1288,2,Mod(1,1288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1288 = 2^{3} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1288.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:	$10.2847317803$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.89122$ of defining polynomial
Character	$\chi$	$=$	1288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89122 q^{3} -0.776183 q^{5} +1.00000 q^{7} +5.35915 q^{9} +O(q^{10})$	$q-2.89122 q^{3} -0.776183 q^{5} +1.00000 q^{7} +5.35915 q^{9} -4.69175 q^{11} +0.513978 q^{13} +2.24412 q^{15} +5.71205 q^{17} +3.33481 q^{19} -2.89122 q^{21} -1.00000 q^{23} -4.39754 q^{25} -6.82083 q^{27} +3.51258 q^{29} +0.262205 q^{31} +13.5649 q^{33} -0.776183 q^{35} +2.62901 q^{37} -1.48602 q^{39} -0.912918 q^{41} -7.73375 q^{43} -4.15968 q^{45} +0.891220 q^{47} +1.00000 q^{49} -16.5148 q^{51} -13.5649 q^{53} +3.64166 q^{55} -9.64166 q^{57} -4.37724 q^{59} -1.40520 q^{61} +5.35915 q^{63} -0.398941 q^{65} +5.17998 q^{67} +2.89122 q^{69} -13.4498 q^{71} -1.89748 q^{73} +12.7143 q^{75} -4.69175 q^{77} +4.69175 q^{79} +3.64306 q^{81} -18.0406 q^{83} -4.43359 q^{85} -10.1556 q^{87} +8.78870 q^{89} +0.513978 q^{91} -0.758093 q^{93} -2.58842 q^{95} +8.11099 q^{97} -25.1438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 4 * q^7 + q^9	$ 4 q - 3 q^{3} + 4 q^{7} + q^{9} - 10 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 10 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 9 q^{27} - 13 q^{29} + 3 q^{31} + 20 q^{33} - 11 q^{39} + q^{41} - 8 q^{43} + 4 q^{45} - 5 q^{47} + 4 q^{49} - 4 q^{51} - 20 q^{53} - 22 q^{55} - 2 q^{57} - 14 q^{59} + 8 q^{61} + q^{63} - 2 q^{65} - 18 q^{67} + 3 q^{69} - 25 q^{71} + 15 q^{73} - 11 q^{75} - 10 q^{77} + 10 q^{79} - 36 q^{83} - 28 q^{85} + q^{87} + 4 q^{89} - 3 q^{91} + 17 q^{93} - 36 q^{95} + 6 q^{97} - 28 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 4 * q^7 + q^9 - 10 * q^11 - 3 * q^13 - 6 * q^15 - 4 * q^17 - 10 * q^19 - 3 * q^21 - 4 * q^23 + 4 * q^25 - 9 * q^27 - 13 * q^29 + 3 * q^31 + 20 * q^33 - 11 * q^39 + q^41 - 8 * q^43 + 4 * q^45 - 5 * q^47 + 4 * q^49 - 4 * q^51 - 20 * q^53 - 22 * q^55 - 2 * q^57 - 14 * q^59 + 8 * q^61 + q^63 - 2 * q^65 - 18 * q^67 + 3 * q^69 - 25 * q^71 + 15 * q^73 - 11 * q^75 - 10 * q^77 + 10 * q^79 - 36 * q^83 - 28 * q^85 + q^87 + 4 * q^89 - 3 * q^91 + 17 * q^93 - 36 * q^95 + 6 * q^97 - 28 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.89122	−1.66925	−0.834623	−	0.550821i	$-0.814315\pi$
$3$	−2.89122	−1.66925	−0.834623	+	0.550821i	$0.814315\pi$
$4$	0	0
$5$	−0.776183	−0.347120	−0.173560	−	0.984823i	$-0.555527\pi$
$5$	−0.776183	−0.347120	−0.173560	+	0.984823i	$0.555527\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.35915	1.78638
$10$	0	0
$11$	−4.69175	−1.41462	−0.707308	−	0.706906i	$-0.750091\pi$
$11$	−4.69175	−1.41462	−0.707308	+	0.706906i	$0.750091\pi$
$12$	0	0
$13$	0.513978	0.142552	0.0712759	−	0.997457i	$-0.477293\pi$
$13$	0.513978	0.142552	0.0712759	+	0.997457i	$0.477293\pi$
$14$	0	0
$15$	2.24412	0.579428
$16$	0	0
$17$	5.71205	1.38538	0.692688	−	0.721238i	$-0.256427\pi$
$17$	5.71205	1.38538	0.692688	+	0.721238i	$0.256427\pi$
$18$	0	0
$19$	3.33481	0.765057	0.382528	−	0.923944i	$-0.375053\pi$
$19$	3.33481	0.765057	0.382528	+	0.923944i	$0.375053\pi$
$20$	0	0
$21$	−2.89122	−0.630916
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.39754	−0.879508
$26$	0	0
$27$	−6.82083	−1.31267
$28$	0	0
$29$	3.51258	0.652269	0.326135	−	0.945323i	$-0.394254\pi$
$29$	3.51258	0.652269	0.326135	+	0.945323i	$0.394254\pi$
$30$	0	0
$31$	0.262205	0.0470934	0.0235467	−	0.999723i	$-0.492504\pi$
$31$	0.262205	0.0470934	0.0235467	+	0.999723i	$0.492504\pi$
$32$	0	0
$33$	13.5649	2.36134
$34$	0	0
$35$	−0.776183	−0.131199
$36$	0	0
$37$	2.62901	0.432207	0.216104	−	0.976370i	$-0.430665\pi$
$37$	2.62901	0.432207	0.216104	+	0.976370i	$0.430665\pi$
$38$	0	0
$39$	−1.48602	−0.237954
$40$	0	0
$41$	−0.912918	−0.142574	−0.0712869	−	0.997456i	$-0.522711\pi$
$41$	−0.912918	−0.142574	−0.0712869	+	0.997456i	$0.522711\pi$
$42$	0	0
$43$	−7.73375	−1.17939	−0.589693	−	0.807628i	$-0.700751\pi$
$43$	−7.73375	−1.17939	−0.589693	+	0.807628i	$0.700751\pi$
$44$	0	0
$45$	−4.15968	−0.620089
$46$	0	0
$47$	0.891220	0.129998	0.0649989	−	0.997885i	$-0.479296\pi$
$47$	0.891220	0.129998	0.0649989	+	0.997885i	$0.479296\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−16.5148	−2.31253
$52$	0	0
$53$	−13.5649	−1.86328	−0.931640	−	0.363383i	$-0.881622\pi$
$53$	−13.5649	−1.86328	−0.931640	+	0.363383i	$0.881622\pi$
$54$	0	0
$55$	3.64166	0.491041
$56$	0	0
$57$	−9.64166	−1.27707
$58$	0	0
$59$	−4.37724	−0.569868	−0.284934	−	0.958547i	$-0.591972\pi$
$59$	−4.37724	−0.569868	−0.284934	+	0.958547i	$0.591972\pi$
$60$	0	0
$61$	−1.40520	−0.179917	−0.0899586	−	0.995946i	$-0.528673\pi$
$61$	−1.40520	−0.179917	−0.0899586	+	0.995946i	$0.528673\pi$
$62$	0	0
$63$	5.35915	0.675190
$64$	0	0
$65$	−0.398941	−0.0494825
$66$	0	0
$67$	5.17998	0.632835	0.316418	−	0.948620i	$-0.397520\pi$
$67$	5.17998	0.632835	0.316418	+	0.948620i	$0.397520\pi$
$68$	0	0
$69$	2.89122	0.348062
$70$	0	0
$71$	−13.4498	−1.59620	−0.798101	−	0.602523i	$-0.794162\pi$
$71$	−13.4498	−1.59620	−0.798101	+	0.602523i	$0.794162\pi$
$72$	0	0
$73$	−1.89748	−0.222083	−0.111041	−	0.993816i	$-0.535419\pi$
$73$	−1.89748	−0.222083	−0.111041	+	0.993816i	$0.535419\pi$
$74$	0	0
$75$	12.7143	1.46812
$76$	0	0
$77$	−4.69175	−0.534674
$78$	0	0
$79$	4.69175	0.527863	0.263932	−	0.964541i	$-0.414981\pi$
$79$	4.69175	0.527863	0.263932	+	0.964541i	$0.414981\pi$
$80$	0	0
$81$	3.64306	0.404784
$82$	0	0
$83$	−18.0406	−1.98021	−0.990106	−	0.140319i	$-0.955187\pi$
$83$	−18.0406	−1.98021	−0.990106	+	0.140319i	$0.955187\pi$
$84$	0	0
$85$	−4.43359	−0.480891
$86$	0	0
$87$	−10.1556	−1.08880
$88$	0	0
$89$	8.78870	0.931600	0.465800	−	0.884890i	$-0.345767\pi$
$89$	8.78870	0.931600	0.465800	+	0.884890i	$0.345767\pi$
$90$	0	0
$91$	0.513978	0.0538795
$92$	0	0
$93$	−0.758093	−0.0786106
$94$	0	0
$95$	−2.58842	−0.265566
$96$	0	0
$97$	8.11099	0.823546	0.411773	−	0.911286i	$-0.364910\pi$
$97$	8.11099	0.823546	0.411773	+	0.911286i	$0.364910\pi$
$98$	0	0
$99$	−25.1438	−2.52705
$100$	0	0
$101$	−12.6262	−1.25636	−0.628178	−	0.778070i	$-0.716199\pi$
$101$	−12.6262	−1.25636	−0.628178	+	0.778070i	$0.716199\pi$
$102$	0	0
$103$	−9.87173	−0.972690	−0.486345	−	0.873767i	$-0.661670\pi$
$103$	−9.87173	−0.972690	−0.486345	+	0.873767i	$0.661670\pi$
$104$	0	0
$105$	2.24412	0.219003
$106$	0	0
$107$	−14.2707	−1.37960	−0.689799	−	0.724001i	$-0.742301\pi$
$107$	−14.2707	−1.37960	−0.689799	+	0.724001i	$0.742301\pi$
$108$	0	0
$109$	−16.1452	−1.54643	−0.773215	−	0.634144i	$-0.781353\pi$
$109$	−16.1452	−1.54643	−0.773215	+	0.634144i	$0.781353\pi$
$110$	0	0
$111$	−7.60106	−0.721461
$112$	0	0
$113$	−10.8438	−1.02010	−0.510048	−	0.860146i	$-0.670372\pi$
$113$	−10.8438	−1.02010	−0.510048	+	0.860146i	$0.670372\pi$
$114$	0	0
$115$	0.776183	0.0723794
$116$	0	0
$117$	2.75448	0.254652
$118$	0	0
$119$	5.71205	0.523623
$120$	0	0
$121$	11.0125	1.00114
$122$	0	0
$123$	2.63945	0.237991
$124$	0	0
$125$	7.29421	0.652414
$126$	0	0
$127$	−9.05090	−0.803138	−0.401569	−	0.915829i	$-0.631535\pi$
$127$	−9.05090	−0.803138	−0.401569	+	0.915829i	$0.631535\pi$
$128$	0	0
$129$	22.3600	1.96868
$130$	0	0
$131$	−8.49228	−0.741974	−0.370987	−	0.928638i	$-0.620981\pi$
$131$	−8.49228	−0.741974	−0.370987	+	0.928638i	$0.620981\pi$
$132$	0	0
$133$	3.33481	0.289164
$134$	0	0
$135$	5.29421	0.455653
$136$	0	0
$137$	−5.60106	−0.478531	−0.239265	−	0.970954i	$-0.576907\pi$
$137$	−5.60106	−0.478531	−0.239265	+	0.970954i	$0.576907\pi$
$138$	0	0
$139$	11.8396	1.00422	0.502111	−	0.864803i	$-0.332557\pi$
$139$	11.8396	1.00422	0.502111	+	0.864803i	$0.332557\pi$
$140$	0	0
$141$	−2.57671	−0.216998
$142$	0	0
$143$	−2.41145	−0.201656
$144$	0	0
$145$	−2.72640	−0.226415
$146$	0	0
$147$	−2.89122	−0.238464
$148$	0	0
$149$	18.1786	1.48925	0.744624	−	0.667485i	$-0.232629\pi$
$149$	18.1786	1.48925	0.744624	+	0.667485i	$0.232629\pi$
$150$	0	0
$151$	−0.820828	−0.0667980	−0.0333990	−	0.999442i	$-0.510633\pi$
$151$	−0.820828	−0.0667980	−0.0333990	+	0.999442i	$0.510633\pi$
$152$	0	0
$153$	30.6117	2.47481
$154$	0	0
$155$	−0.203519	−0.0163471
$156$	0	0
$157$	1.51815	0.121162	0.0605809	−	0.998163i	$-0.480705\pi$
$157$	1.51815	0.121162	0.0605809	+	0.998163i	$0.480705\pi$
$158$	0	0
$159$	39.2190	3.11027
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	16.2449	1.27240	0.636200	−	0.771524i	$-0.280505\pi$
$163$	16.2449	1.27240	0.636200	+	0.771524i	$0.280505\pi$
$164$	0	0
$165$	−10.5288	−0.819668
$166$	0	0
$167$	14.9701	1.15842	0.579210	−	0.815179i	$-0.303361\pi$
$167$	14.9701	1.15842	0.579210	+	0.815179i	$0.303361\pi$
$168$	0	0
$169$	−12.7358	−0.979679
$170$	0	0
$171$	17.8717	1.36669
$172$	0	0
$173$	2.61370	0.198716	0.0993580	−	0.995052i	$-0.468321\pi$
$173$	2.61370	0.198716	0.0993580	+	0.995052i	$0.468321\pi$
$174$	0	0
$175$	−4.39754	−0.332423
$176$	0	0
$177$	12.6556	0.951251
$178$	0	0
$179$	12.8459	0.960144	0.480072	−	0.877229i	$-0.340610\pi$
$179$	12.8459	0.960144	0.480072	+	0.877229i	$0.340610\pi$
$180$	0	0
$181$	14.7400	1.09562	0.547808	−	0.836604i	$-0.315462\pi$
$181$	14.7400	1.09562	0.547808	+	0.836604i	$0.315462\pi$
$182$	0	0
$183$	4.06273	0.300326
$184$	0	0
$185$	−2.04060	−0.150028
$186$	0	0
$187$	−26.7995	−1.95977
$188$	0	0
$189$	−6.82083	−0.496142
$190$	0	0
$191$	−2.80318	−0.202831	−0.101415	−	0.994844i	$-0.532337\pi$
$191$	−2.80318	−0.202831	−0.101415	+	0.994844i	$0.532337\pi$
$192$	0	0
$193$	−1.00221	−0.0721405	−0.0360703	−	0.999349i	$-0.511484\pi$
$193$	−1.00221	−0.0721405	−0.0360703	+	0.999349i	$0.511484\pi$
$194$	0	0
$195$	1.15343	0.0825985
$196$	0	0
$197$	−23.4498	−1.67073	−0.835366	−	0.549694i	$-0.814744\pi$
$197$	−23.4498	−1.67073	−0.835366	+	0.549694i	$0.814744\pi$
$198$	0	0
$199$	10.7211	0.759999	0.379999	−	0.924987i	$-0.375924\pi$
$199$	10.7211	0.759999	0.379999	+	0.924987i	$0.375924\pi$
$200$	0	0
$201$	−14.9765	−1.05636
$202$	0	0
$203$	3.51258	0.246535
$204$	0	0
$205$	0.708591	0.0494902
$206$	0	0
$207$	−5.35915	−0.372487
$208$	0	0
$209$	−15.6461	−1.08226
$210$	0	0
$211$	21.9682	1.51236	0.756178	−	0.654366i	$-0.227064\pi$
$211$	21.9682	1.51236	0.756178	+	0.654366i	$0.227064\pi$
$212$	0	0
$213$	38.8865	2.66446
$214$	0	0
$215$	6.00280	0.409388
$216$	0	0
$217$	0.262205	0.0177996
$218$	0	0
$219$	5.48602	0.370711
$220$	0	0
$221$	2.93586	0.197488
$222$	0	0
$223$	−28.1641	−1.88601	−0.943004	−	0.332782i	$-0.892013\pi$
$223$	−28.1641	−1.88601	−0.943004	+	0.332782i	$0.892013\pi$
$224$	0	0
$225$	−23.5671	−1.57114
$226$	0	0
$227$	16.9484	1.12490	0.562452	−	0.826830i	$-0.309858\pi$
$227$	16.9484	1.12490	0.562452	+	0.826830i	$0.309858\pi$
$228$	0	0
$229$	−0.285152	−0.0188434	−0.00942168	−	0.999956i	$-0.502999\pi$
$229$	−0.285152	−0.0188434	−0.00942168	+	0.999956i	$0.502999\pi$
$230$	0	0
$231$	13.5649	0.892504
$232$	0	0
$233$	−4.87105	−0.319113	−0.159557	−	0.987189i	$-0.551006\pi$
$233$	−4.87105	−0.319113	−0.159557	+	0.987189i	$0.551006\pi$
$234$	0	0
$235$	−0.691750	−0.0451248
$236$	0	0
$237$	−13.5649	−0.881134
$238$	0	0
$239$	3.54193	0.229109	0.114554	−	0.993417i	$-0.463456\pi$
$239$	3.54193	0.229109	0.114554	+	0.993417i	$0.463456\pi$
$240$	0	0
$241$	−13.3662	−0.860994	−0.430497	−	0.902592i	$-0.641662\pi$
$241$	−13.3662	−0.860994	−0.430497	+	0.902592i	$0.641662\pi$
$242$	0	0
$243$	9.92961	0.636985
$244$	0	0
$245$	−0.776183	−0.0495885
$246$	0	0
$247$	1.71402	0.109060
$248$	0	0
$249$	52.1593	3.30546
$250$	0	0
$251$	8.71109	0.549839	0.274919	−	0.961467i	$-0.411349\pi$
$251$	8.71109	0.549839	0.274919	+	0.961467i	$0.411349\pi$
$252$	0	0
$253$	4.69175	0.294968
$254$	0	0
$255$	12.8185	0.802725
$256$	0	0
$257$	−13.3731	−0.834189	−0.417094	−	0.908863i	$-0.636952\pi$
$257$	−13.3731	−0.834189	−0.417094	+	0.908863i	$0.636952\pi$
$258$	0	0
$259$	2.62901	0.163359
$260$	0	0
$261$	18.8244	1.16520
$262$	0	0
$263$	5.97786	0.368611	0.184305	−	0.982869i	$-0.440996\pi$
$263$	5.97786	0.368611	0.184305	+	0.982869i	$0.440996\pi$
$264$	0	0
$265$	10.5288	0.646781
$266$	0	0
$267$	−25.4101	−1.55507
$268$	0	0
$269$	−21.6843	−1.32212	−0.661059	−	0.750334i	$-0.729893\pi$
$269$	−21.6843	−1.32212	−0.661059	+	0.750334i	$0.729893\pi$
$270$	0	0
$271$	3.43858	0.208879	0.104439	−	0.994531i	$-0.466695\pi$
$271$	3.43858	0.208879	0.104439	+	0.994531i	$0.466695\pi$
$272$	0	0
$273$	−1.48602	−0.0899382
$274$	0	0
$275$	20.6322	1.24417
$276$	0	0
$277$	3.20293	0.192445	0.0962226	−	0.995360i	$-0.469324\pi$
$277$	3.20293	0.192445	0.0962226	+	0.995360i	$0.469324\pi$
$278$	0	0
$279$	1.40520	0.0841270
$280$	0	0
$281$	−14.8872	−0.888094	−0.444047	−	0.896003i	$-0.646458\pi$
$281$	−14.8872	−0.888094	−0.444047	+	0.896003i	$0.646458\pi$
$282$	0	0
$283$	12.7183	0.756025	0.378012	−	0.925801i	$-0.376608\pi$
$283$	12.7183	0.756025	0.378012	+	0.925801i	$0.376608\pi$
$284$	0	0
$285$	7.48369	0.443295
$286$	0	0
$287$	−0.912918	−0.0538879
$288$	0	0
$289$	15.6275	0.919264
$290$	0	0
$291$	−23.4506	−1.37470
$292$	0	0
$293$	−26.5683	−1.55214	−0.776069	−	0.630648i	$-0.782789\pi$
$293$	−26.5683	−1.55214	−0.776069	+	0.630648i	$0.782789\pi$
$294$	0	0
$295$	3.39754	0.197812
$296$	0	0
$297$	32.0016	1.85692
$298$	0	0
$299$	−0.513978	−0.0297241
$300$	0	0
$301$	−7.73375	−0.445766
$302$	0	0
$303$	36.5052	2.09717
$304$	0	0
$305$	1.09069	0.0624527
$306$	0	0
$307$	−19.8500	−1.13290	−0.566451	−	0.824096i	$-0.691684\pi$
$307$	−19.8500	−1.13290	−0.566451	+	0.824096i	$0.691684\pi$
$308$	0	0
$309$	28.5413	1.62366
$310$	0	0
$311$	9.59701	0.544197	0.272098	−	0.962269i	$-0.412282\pi$
$311$	9.59701	0.544197	0.272098	+	0.962269i	$0.412282\pi$
$312$	0	0
$313$	−25.6244	−1.44838	−0.724188	−	0.689603i	$-0.757785\pi$
$313$	−25.6244	−1.44838	−0.724188	+	0.689603i	$0.757785\pi$
$314$	0	0
$315$	−4.15968	−0.234372
$316$	0	0
$317$	−30.8341	−1.73182	−0.865909	−	0.500201i	$-0.833259\pi$
$317$	−30.8341	−1.73182	−0.865909	+	0.500201i	$0.833259\pi$
$318$	0	0
$319$	−16.4801	−0.922710
$320$	0	0
$321$	41.2596	2.30289
$322$	0	0
$323$	19.0486	1.05989
$324$	0	0
$325$	−2.26024	−0.125375
$326$	0	0
$327$	46.6793	2.58137
$328$	0	0
$329$	0.891220	0.0491345
$330$	0	0
$331$	−10.7972	−0.593466	−0.296733	−	0.954961i	$-0.595897\pi$
$331$	−10.7972	−0.593466	−0.296733	+	0.954961i	$0.595897\pi$
$332$	0	0
$333$	14.0893	0.772088
$334$	0	0
$335$	−4.02061	−0.219669
$336$	0	0
$337$	7.68505	0.418631	0.209316	−	0.977848i	$-0.432876\pi$
$337$	7.68505	0.418631	0.209316	+	0.977848i	$0.432876\pi$
$338$	0	0
$339$	31.3517	1.70279
$340$	0	0
$341$	−1.23020	−0.0666191
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.24412	−0.120819
$346$	0	0
$347$	0.531627	0.0285392	0.0142696	−	0.999898i	$-0.495458\pi$
$347$	0.531627	0.0285392	0.0142696	+	0.999898i	$0.495458\pi$
$348$	0	0
$349$	28.1681	1.50781	0.753903	−	0.656986i	$-0.228169\pi$
$349$	28.1681	1.50781	0.753903	+	0.656986i	$0.228169\pi$
$350$	0	0
$351$	−3.50575	−0.187123
$352$	0	0
$353$	−15.6437	−0.832632	−0.416316	−	0.909220i	$-0.636679\pi$
$353$	−15.6437	−0.832632	−0.416316	+	0.909220i	$0.636679\pi$
$354$	0	0
$355$	10.4395	0.554073
$356$	0	0
$357$	−16.5148	−0.874055
$358$	0	0
$359$	11.1241	0.587106	0.293553	−	0.955943i	$-0.405162\pi$
$359$	11.1241	0.587106	0.293553	+	0.955943i	$0.405162\pi$
$360$	0	0
$361$	−7.87907	−0.414688
$362$	0	0
$363$	−31.8396	−1.67115
$364$	0	0
$365$	1.47279	0.0770893
$366$	0	0
$367$	−10.8635	−0.567070	−0.283535	−	0.958962i	$-0.591507\pi$
$367$	−10.8635	−0.567070	−0.283535	+	0.958962i	$0.591507\pi$
$368$	0	0
$369$	−4.89247	−0.254692
$370$	0	0
$371$	−13.5649	−0.704253
$372$	0	0
$373$	9.71843	0.503201	0.251601	−	0.967831i	$-0.419043\pi$
$373$	9.71843	0.503201	0.251601	+	0.967831i	$0.419043\pi$
$374$	0	0
$375$	−21.0892	−1.08904
$376$	0	0
$377$	1.80539	0.0929821
$378$	0	0
$379$	−34.7936	−1.78722	−0.893612	−	0.448840i	$-0.851837\pi$
$379$	−34.7936	−1.78722	−0.893612	+	0.448840i	$0.851837\pi$
$380$	0	0
$381$	26.1681	1.34063
$382$	0	0
$383$	−8.09651	−0.413712	−0.206856	−	0.978371i	$-0.566323\pi$
$383$	−8.09651	−0.413712	−0.206856	+	0.978371i	$0.566323\pi$
$384$	0	0
$385$	3.64166	0.185596
$386$	0	0
$387$	−41.4463	−2.10683
$388$	0	0
$389$	−11.4447	−0.580270	−0.290135	−	0.956986i	$-0.593700\pi$
$389$	−11.4447	−0.580270	−0.290135	+	0.956986i	$0.593700\pi$
$390$	0	0
$391$	−5.71205	−0.288871
$392$	0	0
$393$	24.5530	1.23854
$394$	0	0
$395$	−3.64166	−0.183232
$396$	0	0
$397$	−7.92114	−0.397551	−0.198775	−	0.980045i	$-0.563696\pi$
$397$	−7.92114	−0.397551	−0.198775	+	0.980045i	$0.563696\pi$
$398$	0	0
$399$	−9.64166	−0.482687
$400$	0	0
$401$	33.4241	1.66912	0.834560	−	0.550917i	$-0.185722\pi$
$401$	33.4241	1.66912	0.834560	+	0.550917i	$0.185722\pi$
$402$	0	0
$403$	0.134768	0.00671325
$404$	0	0
$405$	−2.82768	−0.140508
$406$	0	0
$407$	−12.3347	−0.611407
$408$	0	0
$409$	−37.1527	−1.83708	−0.918542	−	0.395325i	$-0.870632\pi$
$409$	−37.1527	−1.83708	−0.918542	+	0.395325i	$0.870632\pi$
$410$	0	0
$411$	16.1939	0.798786
$412$	0	0
$413$	−4.37724	−0.215390
$414$	0	0
$415$	14.0028	0.687370
$416$	0	0
$417$	−34.2309	−1.67629
$418$	0	0
$419$	−15.1298	−0.739137	−0.369569	−	0.929203i	$-0.620495\pi$
$419$	−15.1298	−0.739137	−0.369569	+	0.929203i	$0.620495\pi$
$420$	0	0
$421$	33.8761	1.65102	0.825511	−	0.564386i	$-0.190887\pi$
$421$	33.8761	1.65102	0.825511	+	0.564386i	$0.190887\pi$
$422$	0	0
$423$	4.77618	0.232226
$424$	0	0
$425$	−25.1190	−1.21845
$426$	0	0
$427$	−1.40520	−0.0680023
$428$	0	0
$429$	6.97204	0.336613
$430$	0	0
$431$	−24.6387	−1.18681	−0.593403	−	0.804906i	$-0.702216\pi$
$431$	−24.6387	−1.18681	−0.593403	+	0.804906i	$0.702216\pi$
$432$	0	0
$433$	−19.4555	−0.934972	−0.467486	−	0.884000i	$-0.654840\pi$
$433$	−19.4555	−0.934972	−0.467486	+	0.884000i	$0.654840\pi$
$434$	0	0
$435$	7.88263	0.377943
$436$	0	0
$437$	−3.33481	−0.159525
$438$	0	0
$439$	30.5245	1.45686	0.728428	−	0.685122i	$-0.240251\pi$
$439$	30.5245	1.45686	0.728428	+	0.685122i	$0.240251\pi$
$440$	0	0
$441$	5.35915	0.255198
$442$	0	0
$443$	−37.9325	−1.80223	−0.901113	−	0.433584i	$-0.857249\pi$
$443$	−37.9325	−1.80223	−0.901113	+	0.433584i	$0.857249\pi$
$444$	0	0
$445$	−6.82164	−0.323377
$446$	0	0
$447$	−52.5583	−2.48592
$448$	0	0
$449$	−13.8327	−0.652804	−0.326402	−	0.945231i	$-0.605836\pi$
$449$	−13.8327	−0.652804	−0.326402	+	0.945231i	$0.605836\pi$
$450$	0	0
$451$	4.28318	0.201687
$452$	0	0
$453$	2.37319	0.111502
$454$	0	0
$455$	−0.398941	−0.0187026
$456$	0	0
$457$	27.2596	1.27515	0.637576	−	0.770387i	$-0.279937\pi$
$457$	27.2596	1.27515	0.637576	+	0.770387i	$0.279937\pi$
$458$	0	0
$459$	−38.9609	−1.81854
$460$	0	0
$461$	23.7611	1.10667	0.553333	−	0.832960i	$-0.313356\pi$
$461$	23.7611	1.10667	0.553333	+	0.832960i	$0.313356\pi$
$462$	0	0
$463$	13.6498	0.634358	0.317179	−	0.948366i	$-0.397264\pi$
$463$	13.6498	0.634358	0.317179	+	0.948366i	$0.397264\pi$
$464$	0	0
$465$	0.588419	0.0272873
$466$	0	0
$467$	0.481014	0.0222587	0.0111293	−	0.999938i	$-0.496457\pi$
$467$	0.481014	0.0222587	0.0111293	+	0.999938i	$0.496457\pi$
$468$	0	0
$469$	5.17998	0.239189
$470$	0	0
$471$	−4.38932	−0.202249
$472$	0	0
$473$	36.2848	1.66838
$474$	0	0
$475$	−14.6649	−0.672874
$476$	0	0
$477$	−72.6963	−3.32853
$478$	0	0
$479$	−16.9484	−0.774391	−0.387196	−	0.921998i	$-0.626556\pi$
$479$	−16.9484	−0.774391	−0.387196	+	0.921998i	$0.626556\pi$
$480$	0	0
$481$	1.35125	0.0616119
$482$	0	0
$483$	2.89122	0.131555
$484$	0	0
$485$	−6.29561	−0.285869
$486$	0	0
$487$	26.0886	1.18219	0.591093	−	0.806603i	$-0.298697\pi$
$487$	26.0886	1.18219	0.591093	+	0.806603i	$0.298697\pi$
$488$	0	0
$489$	−46.9676	−2.12395
$490$	0	0
$491$	30.4263	1.37312	0.686560	−	0.727073i	$-0.259120\pi$
$491$	30.4263	1.37312	0.686560	+	0.727073i	$0.259120\pi$
$492$	0	0
$493$	20.0640	0.903637
$494$	0	0
$495$	19.5162	0.877187
$496$	0	0
$497$	−13.4498	−0.603308
$498$	0	0
$499$	0.378490	0.0169436	0.00847178	−	0.999964i	$-0.497303\pi$
$499$	0.378490	0.0169436	0.00847178	+	0.999964i	$0.497303\pi$
$500$	0	0
$501$	−43.2818	−1.93369
$502$	0	0
$503$	30.5177	1.36072	0.680358	−	0.732880i	$-0.261824\pi$
$503$	30.5177	1.36072	0.680358	+	0.732880i	$0.261824\pi$
$504$	0	0
$505$	9.80025	0.436105
$506$	0	0
$507$	36.8221	1.63533
$508$	0	0
$509$	18.0399	0.799603	0.399802	−	0.916602i	$-0.369079\pi$
$509$	18.0399	0.799603	0.399802	+	0.916602i	$0.369079\pi$
$510$	0	0
$511$	−1.89748	−0.0839394
$512$	0	0
$513$	−22.7461	−1.00427
$514$	0	0
$515$	7.66227	0.337640
$516$	0	0
$517$	−4.18138	−0.183897
$518$	0	0
$519$	−7.55678	−0.331706
$520$	0	0
$521$	43.3219	1.89797	0.948984	−	0.315323i	$-0.102113\pi$
$521$	43.3219	1.89797	0.948984	+	0.315323i	$0.102113\pi$
$522$	0	0
$523$	−17.3835	−0.760127	−0.380064	−	0.924960i	$-0.624098\pi$
$523$	−17.3835	−0.760127	−0.380064	+	0.924960i	$0.624098\pi$
$524$	0	0
$525$	12.7143	0.554896
$526$	0	0
$527$	1.49773	0.0652421
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−23.4583	−1.01800
$532$	0	0
$533$	−0.469220	−0.0203242
$534$	0	0
$535$	11.0766	0.478885
$536$	0	0
$537$	−37.1402	−1.60272
$538$	0	0
$539$	−4.69175	−0.202088
$540$	0	0
$541$	5.87814	0.252721	0.126360	−	0.991984i	$-0.459670\pi$
$541$	5.87814	0.252721	0.126360	+	0.991984i	$0.459670\pi$
$542$	0	0
$543$	−42.6166	−1.82885
$544$	0	0
$545$	12.5316	0.536796
$546$	0	0
$547$	28.4919	1.21823	0.609113	−	0.793083i	$-0.291525\pi$
$547$	28.4919	1.21823	0.609113	+	0.793083i	$0.291525\pi$
$548$	0	0
$549$	−7.53067	−0.321401
$550$	0	0
$551$	11.7138	0.499023
$552$	0	0
$553$	4.69175	0.199513
$554$	0	0
$555$	5.89981	0.250433
$556$	0	0
$557$	−45.6920	−1.93603	−0.968015	−	0.250891i	$-0.919276\pi$
$557$	−45.6920	−1.93603	−0.968015	+	0.250891i	$0.919276\pi$
$558$	0	0
$559$	−3.97497	−0.168123
$560$	0	0
$561$	77.4832	3.27134
$562$	0	0
$563$	−39.9807	−1.68499	−0.842494	−	0.538706i	$-0.818913\pi$
$563$	−39.9807	−1.68499	−0.842494	+	0.538706i	$0.818913\pi$
$564$	0	0
$565$	8.41675	0.354095
$566$	0	0
$567$	3.64306	0.152994
$568$	0	0
$569$	−36.9359	−1.54843	−0.774216	−	0.632921i	$-0.781856\pi$
$569$	−36.9359	−1.54843	−0.774216	+	0.632921i	$0.781856\pi$
$570$	0	0
$571$	7.92852	0.331798	0.165899	−	0.986143i	$-0.446947\pi$
$571$	7.92852	0.331798	0.165899	+	0.986143i	$0.446947\pi$
$572$	0	0
$573$	8.10460	0.338575
$574$	0	0
$575$	4.39754	0.183390
$576$	0	0
$577$	−0.232409	−0.00967532	−0.00483766	−	0.999988i	$-0.501540\pi$
$577$	−0.232409	−0.00967532	−0.00483766	+	0.999988i	$0.501540\pi$
$578$	0	0
$579$	2.89760	0.120420
$580$	0	0
$581$	−18.0406	−0.748450
$582$	0	0
$583$	63.6430	2.63582
$584$	0	0
$585$	−2.13798	−0.0883947
$586$	0	0
$587$	−11.4952	−0.474458	−0.237229	−	0.971454i	$-0.576239\pi$
$587$	−11.4952	−0.474458	−0.237229	+	0.971454i	$0.576239\pi$
$588$	0	0
$589$	0.874403	0.0360292
$590$	0	0
$591$	67.7986	2.78886
$592$	0	0
$593$	−23.3835	−0.960245	−0.480123	−	0.877201i	$-0.659408\pi$
$593$	−23.3835	−0.960245	−0.480123	+	0.877201i	$0.659408\pi$
$594$	0	0
$595$	−4.43359	−0.181760
$596$	0	0
$597$	−30.9971	−1.26863
$598$	0	0
$599$	41.7279	1.70496	0.852478	−	0.522764i	$-0.175099\pi$
$599$	41.7279	1.70496	0.852478	+	0.522764i	$0.175099\pi$
$600$	0	0
$601$	15.9462	0.650458	0.325229	−	0.945635i	$-0.394559\pi$
$601$	15.9462	0.650458	0.325229	+	0.945635i	$0.394559\pi$
$602$	0	0
$603$	27.7603	1.13049
$604$	0	0
$605$	−8.54772	−0.347514
$606$	0	0
$607$	9.97830	0.405007	0.202503	−	0.979282i	$-0.435092\pi$
$607$	9.97830	0.405007	0.202503	+	0.979282i	$0.435092\pi$
$608$	0	0
$609$	−10.1556	−0.411527
$610$	0	0
$611$	0.458067	0.0185314
$612$	0	0
$613$	−6.78978	−0.274237	−0.137118	−	0.990555i	$-0.543784\pi$
$613$	−6.78978	−0.274237	−0.137118	+	0.990555i	$0.543784\pi$
$614$	0	0
$615$	−2.04869	−0.0826113
$616$	0	0
$617$	−35.6055	−1.43342	−0.716711	−	0.697370i	$-0.754353\pi$
$617$	−35.6055	−1.43342	−0.716711	+	0.697370i	$0.754353\pi$
$618$	0	0
$619$	−3.18680	−0.128088	−0.0640442	−	0.997947i	$-0.520400\pi$
$619$	−3.18680	−0.128088	−0.0640442	+	0.997947i	$0.520400\pi$
$620$	0	0
$621$	6.82083	0.273710
$622$	0	0
$623$	8.78870	0.352112
$624$	0	0
$625$	16.3261	0.653042
$626$	0	0
$627$	45.2362	1.80656
$628$	0	0
$629$	15.0171	0.598769
$630$	0	0
$631$	−33.7672	−1.34425	−0.672126	−	0.740437i	$-0.734619\pi$
$631$	−33.7672	−1.34425	−0.672126	+	0.740437i	$0.734619\pi$
$632$	0	0
$633$	−63.5150	−2.52449
$634$	0	0
$635$	7.02515	0.278785
$636$	0	0
$637$	0.513978	0.0203645
$638$	0	0
$639$	−72.0798	−2.85143
$640$	0	0
$641$	−7.83555	−0.309486	−0.154743	−	0.987955i	$-0.549455\pi$
$641$	−7.83555	−0.309486	−0.154743	+	0.987955i	$0.549455\pi$
$642$	0	0
$643$	−22.2273	−0.876558	−0.438279	−	0.898839i	$-0.644412\pi$
$643$	−22.2273	−0.876558	−0.438279	+	0.898839i	$0.644412\pi$
$644$	0	0
$645$	−17.3554	−0.683369
$646$	0	0
$647$	25.9361	1.01965	0.509827	−	0.860277i	$-0.329709\pi$
$647$	25.9361	1.01965	0.509827	+	0.860277i	$0.329709\pi$
$648$	0	0
$649$	20.5369	0.806145
$650$	0	0
$651$	−0.758093	−0.0297120
$652$	0	0
$653$	−1.44543	−0.0565639	−0.0282820	−	0.999600i	$-0.509004\pi$
$653$	−1.44543	−0.0565639	−0.0282820	+	0.999600i	$0.509004\pi$
$654$	0	0
$655$	6.59156	0.257554
$656$	0	0
$657$	−10.1689	−0.396725
$658$	0	0
$659$	35.8553	1.39672	0.698362	−	0.715745i	$-0.253913\pi$
$659$	35.8553	1.39672	0.698362	+	0.715745i	$0.253913\pi$
$660$	0	0
$661$	27.3590	1.06414	0.532071	−	0.846700i	$-0.321414\pi$
$661$	27.3590	1.06414	0.532071	+	0.846700i	$0.321414\pi$
$662$	0	0
$663$	−8.48823	−0.329656
$664$	0	0
$665$	−2.58842	−0.100375
$666$	0	0
$667$	−3.51258	−0.136008
$668$	0	0
$669$	81.4286	3.14821
$670$	0	0
$671$	6.59283	0.254514
$672$	0	0
$673$	13.5170	0.521042	0.260521	−	0.965468i	$-0.416106\pi$
$673$	13.5170	0.521042	0.260521	+	0.965468i	$0.416106\pi$
$674$	0	0
$675$	29.9949	1.15450
$676$	0	0
$677$	−2.48465	−0.0954928	−0.0477464	−	0.998859i	$-0.515204\pi$
$677$	−2.48465	−0.0954928	−0.0477464	+	0.998859i	$0.515204\pi$
$678$	0	0
$679$	8.11099	0.311271
$680$	0	0
$681$	−49.0015	−1.87774
$682$	0	0
$683$	−41.6356	−1.59314	−0.796572	−	0.604544i	$-0.793355\pi$
$683$	−41.6356	−1.59314	−0.796572	+	0.604544i	$0.793355\pi$
$684$	0	0
$685$	4.34745	0.166107
$686$	0	0
$687$	0.824436	0.0314542
$688$	0	0
$689$	−6.97204	−0.265614
$690$	0	0
$691$	−35.4961	−1.35034	−0.675168	−	0.737664i	$-0.735929\pi$
$691$	−35.4961	−1.35034	−0.675168	+	0.737664i	$0.735929\pi$
$692$	0	0
$693$	−25.1438	−0.955134
$694$	0	0
$695$	−9.18969	−0.348585
$696$	0	0
$697$	−5.21463	−0.197518
$698$	0	0
$699$	14.0833	0.532678
$700$	0	0
$701$	37.7323	1.42513	0.712565	−	0.701606i	$-0.247533\pi$
$701$	37.7323	1.42513	0.712565	+	0.701606i	$0.247533\pi$
$702$	0	0
$703$	8.76725	0.330663
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	−12.6262	−0.474858
$708$	0	0
$709$	−39.7351	−1.49228	−0.746142	−	0.665787i	$-0.768096\pi$
$709$	−39.7351	−1.49228	−0.746142	+	0.665787i	$0.768096\pi$
$710$	0	0
$711$	25.1438	0.942966
$712$	0	0
$713$	−0.262205	−0.00981966
$714$	0	0
$715$	1.87173	0.0699987
$716$	0	0
$717$	−10.2405	−0.382439
$718$	0	0
$719$	6.53801	0.243827	0.121913	−	0.992541i	$-0.461097\pi$
$719$	6.53801	0.243827	0.121913	+	0.992541i	$0.461097\pi$
$720$	0	0
$721$	−9.87173	−0.367642
$722$	0	0
$723$	38.6447	1.43721
$724$	0	0
$725$	−15.4467	−0.573676
$726$	0	0
$727$	−25.0280	−0.928235	−0.464118	−	0.885774i	$-0.653629\pi$
$727$	−25.0280	−0.928235	−0.464118	+	0.885774i	$0.653629\pi$
$728$	0	0
$729$	−39.6378	−1.46807
$730$	0	0
$731$	−44.1755	−1.63389
$732$	0	0
$733$	18.1959	0.672079	0.336040	−	0.941848i	$-0.390912\pi$
$733$	18.1959	0.672079	0.336040	+	0.941848i	$0.390912\pi$
$734$	0	0
$735$	2.24412	0.0827754
$736$	0	0
$737$	−24.3032	−0.895219
$738$	0	0
$739$	−6.09001	−0.224025	−0.112012	−	0.993707i	$-0.535730\pi$
$739$	−6.09001	−0.224025	−0.112012	+	0.993707i	$0.535730\pi$
$740$	0	0
$741$	−4.95560	−0.182048
$742$	0	0
$743$	3.59143	0.131757	0.0658785	−	0.997828i	$-0.479015\pi$
$743$	3.59143	0.131757	0.0658785	+	0.997828i	$0.479015\pi$
$744$	0	0
$745$	−14.1099	−0.516947
$746$	0	0
$747$	−96.6823	−3.53742
$748$	0	0
$749$	−14.2707	−0.521439
$750$	0	0
$751$	29.5074	1.07674	0.538371	−	0.842708i	$-0.319040\pi$
$751$	29.5074	1.07674	0.538371	+	0.842708i	$0.319040\pi$
$752$	0	0
$753$	−25.1857	−0.917817
$754$	0	0
$755$	0.637112	0.0231869
$756$	0	0
$757$	6.19389	0.225121	0.112560	−	0.993645i	$-0.464095\pi$
$757$	6.19389	0.225121	0.112560	+	0.993645i	$0.464095\pi$
$758$	0	0
$759$	−13.5649	−0.492374
$760$	0	0
$761$	−30.9964	−1.12362	−0.561808	−	0.827267i	$-0.689894\pi$
$761$	−30.9964	−1.12362	−0.561808	+	0.827267i	$0.689894\pi$
$762$	0	0
$763$	−16.1452	−0.584495
$764$	0	0
$765$	−23.7603	−0.859055
$766$	0	0
$767$	−2.24980	−0.0812357
$768$	0	0
$769$	−4.34548	−0.156702	−0.0783510	−	0.996926i	$-0.524965\pi$
$769$	−4.34548	−0.156702	−0.0783510	+	0.996926i	$0.524965\pi$
$770$	0	0
$771$	38.6645	1.39247
$772$	0	0
$773$	43.7746	1.57446	0.787232	−	0.616657i	$-0.211514\pi$
$773$	43.7746	1.57446	0.787232	+	0.616657i	$0.211514\pi$
$774$	0	0
$775$	−1.15306	−0.0414191
$776$	0	0
$777$	−7.60106	−0.272686
$778$	0	0
$779$	−3.04440	−0.109077
$780$	0	0
$781$	63.1033	2.25801
$782$	0	0
$783$	−23.9587	−0.856214
$784$	0	0
$785$	−1.17836	−0.0420576
$786$	0	0
$787$	42.2779	1.50704	0.753522	−	0.657423i	$-0.228353\pi$
$787$	42.2779	1.50704	0.753522	+	0.657423i	$0.228353\pi$
$788$	0	0
$789$	−17.2833	−0.615302
$790$	0	0
$791$	−10.8438	−0.385560
$792$	0	0
$793$	−0.722240	−0.0256475
$794$	0	0
$795$	−30.4412	−1.07964
$796$	0	0
$797$	12.5055	0.442968	0.221484	−	0.975164i	$-0.428910\pi$
$797$	12.5055	0.442968	0.221484	+	0.975164i	$0.428910\pi$
$798$	0	0
$799$	5.09069	0.180096
$800$	0	0
$801$	47.1000	1.66420
$802$	0	0
$803$	8.90249	0.314162
$804$	0	0
$805$	0.776183	0.0273568
$806$	0	0
$807$	62.6942	2.20694
$808$	0	0
$809$	−6.31936	−0.222177	−0.111088	−	0.993811i	$-0.535434\pi$
$809$	−6.31936	−0.222177	−0.111088	+	0.993811i	$0.535434\pi$
$810$	0	0
$811$	21.3071	0.748193	0.374097	−	0.927390i	$-0.377953\pi$
$811$	21.3071	0.748193	0.374097	+	0.927390i	$0.377953\pi$
$812$	0	0
$813$	−9.94168	−0.348670
$814$	0	0
$815$	−12.6090	−0.441675
$816$	0	0
$817$	−25.7905	−0.902297
$818$	0	0
$819$	2.75448	0.0962495
$820$	0	0
$821$	52.7203	1.83995	0.919976	−	0.391975i	$-0.128208\pi$
$821$	52.7203	1.83995	0.919976	+	0.391975i	$0.128208\pi$
$822$	0	0
$823$	−36.5265	−1.27323	−0.636617	−	0.771180i	$-0.719667\pi$
$823$	−36.5265	−1.27323	−0.636617	+	0.771180i	$0.719667\pi$
$824$	0	0
$825$	−59.6521	−2.07682
$826$	0	0
$827$	2.92055	0.101557	0.0507787	−	0.998710i	$-0.483830\pi$
$827$	2.92055	0.101557	0.0507787	+	0.998710i	$0.483830\pi$
$828$	0	0
$829$	13.4956	0.468721	0.234360	−	0.972150i	$-0.424700\pi$
$829$	13.4956	0.468721	0.234360	+	0.972150i	$0.424700\pi$
$830$	0	0
$831$	−9.26036	−0.321238
$832$	0	0
$833$	5.71205	0.197911
$834$	0	0
$835$	−11.6195	−0.402110
$836$	0	0
$837$	−1.78846	−0.0618181
$838$	0	0
$839$	49.0836	1.69455	0.847276	−	0.531152i	$-0.178241\pi$
$839$	49.0836	1.69455	0.847276	+	0.531152i	$0.178241\pi$
$840$	0	0
$841$	−16.6618	−0.574545
$842$	0	0
$843$	43.0421	1.48245
$844$	0	0
$845$	9.88533	0.340066
$846$	0	0
$847$	11.0125	0.378394
$848$	0	0
$849$	−36.7714	−1.26199
$850$	0	0
$851$	−2.62901	−0.0901215
$852$	0	0
$853$	50.8326	1.74048	0.870238	−	0.492632i	$-0.163965\pi$
$853$	50.8326	1.74048	0.870238	+	0.492632i	$0.163965\pi$
$854$	0	0
$855$	−13.8717	−0.474403
$856$	0	0
$857$	−31.3316	−1.07027	−0.535133	−	0.844768i	$-0.679739\pi$
$857$	−31.3316	−1.07027	−0.535133	+	0.844768i	$0.679739\pi$
$858$	0	0
$859$	−40.3963	−1.37830	−0.689151	−	0.724618i	$-0.742016\pi$
$859$	−40.3963	−1.37830	−0.689151	+	0.724618i	$0.742016\pi$
$860$	0	0
$861$	2.63945	0.0899521
$862$	0	0
$863$	33.2183	1.13077	0.565383	−	0.824829i	$-0.308729\pi$
$863$	33.2183	1.13077	0.565383	+	0.824829i	$0.308729\pi$
$864$	0	0
$865$	−2.02871	−0.0689782
$866$	0	0
$867$	−45.1825	−1.53448
$868$	0	0
$869$	−22.0125	−0.746723
$870$	0	0
$871$	2.66239	0.0902118
$872$	0	0
$873$	43.4680	1.47117
$874$	0	0
$875$	7.29421	0.246589
$876$	0	0
$877$	17.9389	0.605753	0.302876	−	0.953030i	$-0.402053\pi$
$877$	17.9389	0.605753	0.302876	+	0.953030i	$0.402053\pi$
$878$	0	0
$879$	76.8149	2.59090
$880$	0	0
$881$	−13.8520	−0.466684	−0.233342	−	0.972395i	$-0.574966\pi$
$881$	−13.8520	−0.466684	−0.233342	+	0.972395i	$0.574966\pi$
$882$	0	0
$883$	18.6705	0.628312	0.314156	−	0.949371i	$-0.398278\pi$
$883$	18.6705	0.628312	0.314156	+	0.949371i	$0.398278\pi$
$884$	0	0
$885$	−9.82304	−0.330198
$886$	0	0
$887$	−40.2410	−1.35116	−0.675581	−	0.737286i	$-0.736107\pi$
$887$	−40.2410	−1.35116	−0.675581	+	0.737286i	$0.736107\pi$
$888$	0	0
$889$	−9.05090	−0.303558
$890$	0	0
$891$	−17.0923	−0.572614
$892$	0	0
$893$	2.97204	0.0994557
$894$	0	0
$895$	−9.97073	−0.333285
$896$	0	0
$897$	1.48602	0.0496168
$898$	0	0
$899$	0.921016	0.0307176
$900$	0	0
$901$	−77.4832	−2.58134
$902$	0	0
$903$	22.3600	0.744093
$904$	0	0
$905$	−11.4409	−0.380310
$906$	0	0
$907$	−19.6725	−0.653216	−0.326608	−	0.945160i	$-0.605906\pi$
$907$	−19.6725	−0.653216	−0.326608	+	0.945160i	$0.605906\pi$
$908$	0	0
$909$	−67.6658	−2.24433
$910$	0	0
$911$	−30.7140	−1.01760	−0.508800	−	0.860885i	$-0.669911\pi$
$911$	−30.7140	−1.01760	−0.508800	+	0.860885i	$0.669911\pi$
$912$	0	0
$913$	84.6420	2.80124
$914$	0	0
$915$	−3.15343	−0.104249
$916$	0	0
$917$	−8.49228	−0.280440
$918$	0	0
$919$	9.15911	0.302131	0.151066	−	0.988524i	$-0.451730\pi$
$919$	9.15911	0.302131	0.151066	+	0.988524i	$0.451730\pi$
$920$	0	0
$921$	57.3908	1.89109
$922$	0	0
$923$	−6.91292	−0.227541
$924$	0	0
$925$	−11.5612	−0.380130
$926$	0	0
$927$	−52.9041	−1.73760
$928$	0	0
$929$	−31.2286	−1.02458	−0.512289	−	0.858813i	$-0.671202\pi$
$929$	−31.2286	−1.02458	−0.512289	+	0.858813i	$0.671202\pi$
$930$	0	0
$931$	3.33481	0.109294
$932$	0	0
$933$	−27.7471	−0.908398
$934$	0	0
$935$	20.8013	0.680276
$936$	0	0
$937$	−48.7091	−1.59126	−0.795629	−	0.605784i	$-0.792859\pi$
$937$	−48.7091	−1.59126	−0.795629	+	0.605784i	$0.792859\pi$
$938$	0	0
$939$	74.0857	2.41770
$940$	0	0
$941$	59.3715	1.93546	0.967728	−	0.251997i	$-0.0810873\pi$
$941$	59.3715	1.93546	0.967728	+	0.251997i	$0.0810873\pi$
$942$	0	0
$943$	0.912918	0.0297287
$944$	0	0
$945$	5.29421	0.172221
$946$	0	0
$947$	21.6615	0.703906	0.351953	−	0.936018i	$-0.385518\pi$
$947$	21.6615	0.703906	0.351953	+	0.936018i	$0.385518\pi$
$948$	0	0
$949$	−0.975261	−0.0316583
$950$	0	0
$951$	89.1483	2.89083
$952$	0	0
$953$	−22.3291	−0.723310	−0.361655	−	0.932312i	$-0.617788\pi$
$953$	−22.3291	−0.723310	−0.361655	+	0.932312i	$0.617788\pi$
$954$	0	0
$955$	2.17578	0.0704065
$956$	0	0
$957$	47.6477	1.54023
$958$	0	0
$959$	−5.60106	−0.180868
$960$	0	0
$961$	−30.9312	−0.997782
$962$	0	0
$963$	−76.4787	−2.46449
$964$	0	0
$965$	0.777897	0.0250414
$966$	0	0
$967$	−6.81248	−0.219074	−0.109537	−	0.993983i	$-0.534937\pi$
$967$	−6.81248	−0.219074	−0.109537	+	0.993983i	$0.534937\pi$
$968$	0	0
$969$	−55.0736	−1.76922
$970$	0	0
$971$	−11.3058	−0.362822	−0.181411	−	0.983407i	$-0.558066\pi$
$971$	−11.3058	−0.362822	−0.181411	+	0.983407i	$0.558066\pi$
$972$	0	0
$973$	11.8396	0.379560
$974$	0	0
$975$	6.53484	0.209282
$976$	0	0
$977$	19.4550	0.622420	0.311210	−	0.950341i	$-0.399266\pi$
$977$	19.4550	0.622420	0.311210	+	0.950341i	$0.399266\pi$
$978$	0	0
$979$	−41.2344	−1.31786
$980$	0	0
$981$	−86.5246	−2.76252
$982$	0	0
$983$	6.17328	0.196897	0.0984486	−	0.995142i	$-0.468612\pi$
$983$	6.17328	0.196897	0.0984486	+	0.995142i	$0.468612\pi$
$984$	0	0
$985$	18.2014	0.579944
$986$	0	0
$987$	−2.57671	−0.0820177
$988$	0	0
$989$	7.73375	0.245919
$990$	0	0
$991$	19.0073	0.603788	0.301894	−	0.953341i	$-0.402381\pi$
$991$	19.0073	0.603788	0.301894	+	0.953341i	$0.402381\pi$
$992$	0	0
$993$	31.2170	0.990641
$994$	0	0
$995$	−8.32154	−0.263810
$996$	0	0
$997$	15.3142	0.485005	0.242503	−	0.970151i	$-0.422032\pi$
$997$	15.3142	0.485005	0.242503	+	0.970151i	$0.422032\pi$
$998$	0	0
$999$	−17.9321	−0.567345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1288.2.a.n.1.1	✓	4
4.3	odd	2		2576.2.a.ba.1.4		4
7.6	odd	2		9016.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.n.1.1	✓	4	1.1	even	1	trivial
2576.2.a.ba.1.4		4	4.3	odd	2
9016.2.a.bf.1.4		4	7.6	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:	\( N \)	\(=\)	\( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1288.a (trivial)

\(f(q)\)	\(=\)	\(q-2.89122 q^{3} -0.776183 q^{5} +1.00000 q^{7} +5.35915 q^{9} +O(q^{10})\)	\(q-2.89122 q^{3} -0.776183 q^{5} +1.00000 q^{7} +5.35915 q^{9} -4.69175 q^{11} +0.513978 q^{13} +2.24412 q^{15} +5.71205 q^{17} +3.33481 q^{19} -2.89122 q^{21} -1.00000 q^{23} -4.39754 q^{25} -6.82083 q^{27} +3.51258 q^{29} +0.262205 q^{31} +13.5649 q^{33} -0.776183 q^{35} +2.62901 q^{37} -1.48602 q^{39} -0.912918 q^{41} -7.73375 q^{43} -4.15968 q^{45} +0.891220 q^{47} +1.00000 q^{49} -16.5148 q^{51} -13.5649 q^{53} +3.64166 q^{55} -9.64166 q^{57} -4.37724 q^{59} -1.40520 q^{61} +5.35915 q^{63} -0.398941 q^{65} +5.17998 q^{67} +2.89122 q^{69} -13.4498 q^{71} -1.89748 q^{73} +12.7143 q^{75} -4.69175 q^{77} +4.69175 q^{79} +3.64306 q^{81} -18.0406 q^{83} -4.43359 q^{85} -10.1556 q^{87} +8.78870 q^{89} +0.513978 q^{91} -0.758093 q^{93} -2.58842 q^{95} +8.11099 q^{97} -25.1438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 4 * q^7 + q^9	\( 4 q - 3 q^{3} + 4 q^{7} + q^{9} - 10 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 10 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 9 q^{27} - 13 q^{29} + 3 q^{31} + 20 q^{33} - 11 q^{39} + q^{41} - 8 q^{43} + 4 q^{45} - 5 q^{47} + 4 q^{49} - 4 q^{51} - 20 q^{53} - 22 q^{55} - 2 q^{57} - 14 q^{59} + 8 q^{61} + q^{63} - 2 q^{65} - 18 q^{67} + 3 q^{69} - 25 q^{71} + 15 q^{73} - 11 q^{75} - 10 q^{77} + 10 q^{79} - 36 q^{83} - 28 q^{85} + q^{87} + 4 q^{89} - 3 q^{91} + 17 q^{93} - 36 q^{95} + 6 q^{97} - 28 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 4 * q^7 + q^9 - 10 * q^11 - 3 * q^13 - 6 * q^15 - 4 * q^17 - 10 * q^19 - 3 * q^21 - 4 * q^23 + 4 * q^25 - 9 * q^27 - 13 * q^29 + 3 * q^31 + 20 * q^33 - 11 * q^39 + q^41 - 8 * q^43 + 4 * q^45 - 5 * q^47 + 4 * q^49 - 4 * q^51 - 20 * q^53 - 22 * q^55 - 2 * q^57 - 14 * q^59 + 8 * q^61 + q^63 - 2 * q^65 - 18 * q^67 + 3 * q^69 - 25 * q^71 + 15 * q^73 - 11 * q^75 - 10 * q^77 + 10 * q^79 - 36 * q^83 - 28 * q^85 + q^87 + 4 * q^89 - 3 * q^91 + 17 * q^93 - 36 * q^95 + 6 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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