LMFDB - Embedded newform 4016.2.a.l.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 $ x^19 - 6x^18 - 21x^17 + 179x^16 + 90x^15 - 2109x^14 + 926x^13 + 12681x^12 - 10845x^11 - 41921x^10 + 43551x^9 + 76260x^8 - 80907x^7 - 72526x^6 + 64793x^5 + 33209x^4 - 13777x^3 - 7607x^2 - 747x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.00508866$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.00508866 q^{3} -3.22565 q^{5} -4.03772 q^{7} -2.99997 q^{9} +O(q^{10})$	$q+0.00508866 q^{3} -3.22565 q^{5} -4.03772 q^{7} -2.99997 q^{9} +0.837612 q^{11} +2.15418 q^{13} -0.0164142 q^{15} -5.70183 q^{17} -3.20566 q^{19} -0.0205466 q^{21} -7.31384 q^{23} +5.40483 q^{25} -0.0305318 q^{27} -10.2145 q^{29} -2.92651 q^{31} +0.00426232 q^{33} +13.0243 q^{35} +1.03673 q^{37} +0.0109619 q^{39} -6.57233 q^{41} +8.92146 q^{43} +9.67687 q^{45} +0.200804 q^{47} +9.30316 q^{49} -0.0290147 q^{51} -4.20465 q^{53} -2.70184 q^{55} -0.0163125 q^{57} +4.19241 q^{59} -11.8442 q^{61} +12.1130 q^{63} -6.94862 q^{65} +5.06597 q^{67} -0.0372177 q^{69} +6.00011 q^{71} -3.36762 q^{73} +0.0275033 q^{75} -3.38204 q^{77} -6.23780 q^{79} +8.99977 q^{81} -14.7434 q^{83} +18.3921 q^{85} -0.0519783 q^{87} -7.88751 q^{89} -8.69795 q^{91} -0.0148920 q^{93} +10.3403 q^{95} +4.91922 q^{97} -2.51281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * 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
$3$	0.00508866	0.00293794	0.00146897	−	0.999999i	$-0.499532\pi$
$3$	0.00508866	0.00293794	0.00146897	+	0.999999i	$0.499532\pi$
$4$	0	0
$5$	−3.22565	−1.44256	−0.721278	−	0.692646i	$-0.756445\pi$
$5$	−3.22565	−1.44256	−0.721278	+	0.692646i	$0.756445\pi$
$6$	0	0
$7$	−4.03772	−1.52611	−0.763057	−	0.646331i	$-0.776302\pi$
$7$	−4.03772	−1.52611	−0.763057	+	0.646331i	$0.776302\pi$
$8$	0	0
$9$	−2.99997	−0.999991
$10$	0	0
$11$	0.837612	0.252550	0.126275	−	0.991995i	$-0.459698\pi$
$11$	0.837612	0.252550	0.126275	+	0.991995i	$0.459698\pi$
$12$	0	0
$13$	2.15418	0.597461	0.298730	−	0.954338i	$-0.403437\pi$
$13$	2.15418	0.597461	0.298730	+	0.954338i	$0.403437\pi$
$14$	0	0
$15$	−0.0164142	−0.00423814
$16$	0	0
$17$	−5.70183	−1.38290	−0.691449	−	0.722426i	$-0.743027\pi$
$17$	−5.70183	−1.38290	−0.691449	+	0.722426i	$0.743027\pi$
$18$	0	0
$19$	−3.20566	−0.735429	−0.367715	−	0.929939i	$-0.619860\pi$
$19$	−3.20566	−0.735429	−0.367715	+	0.929939i	$0.619860\pi$
$20$	0	0
$21$	−0.0205466	−0.00448363
$22$	0	0
$23$	−7.31384	−1.52504	−0.762521	−	0.646964i	$-0.776039\pi$
$23$	−7.31384	−1.52504	−0.762521	+	0.646964i	$0.776039\pi$
$24$	0	0
$25$	5.40483	1.08097
$26$	0	0
$27$	−0.0305318	−0.00587585
$28$	0	0
$29$	−10.2145	−1.89679	−0.948396	−	0.317090i	$-0.897294\pi$
$29$	−10.2145	−1.89679	−0.948396	+	0.317090i	$0.897294\pi$
$30$	0	0
$31$	−2.92651	−0.525617	−0.262809	−	0.964848i	$-0.584649\pi$
$31$	−2.92651	−0.525617	−0.262809	+	0.964848i	$0.584649\pi$
$32$	0	0
$33$	0.00426232	0.000741975 0
$34$	0	0
$35$	13.0243	2.20150
$36$	0	0
$37$	1.03673	0.170438	0.0852189	−	0.996362i	$-0.472841\pi$
$37$	1.03673	0.170438	0.0852189	+	0.996362i	$0.472841\pi$
$38$	0	0
$39$	0.0109619	0.00175530
$40$	0	0
$41$	−6.57233	−1.02643	−0.513213	−	0.858261i	$-0.671545\pi$
$41$	−6.57233	−1.02643	−0.513213	+	0.858261i	$0.671545\pi$
$42$	0	0
$43$	8.92146	1.36051	0.680255	−	0.732976i	$-0.261869\pi$
$43$	8.92146	1.36051	0.680255	+	0.732976i	$0.261869\pi$
$44$	0	0
$45$	9.67687	1.44254
$46$	0	0
$47$	0.200804	0.0292903	0.0146452	−	0.999893i	$-0.495338\pi$
$47$	0.200804	0.0292903	0.0146452	+	0.999893i	$0.495338\pi$
$48$	0	0
$49$	9.30316	1.32902
$50$	0	0
$51$	−0.0290147	−0.00406287
$52$	0	0
$53$	−4.20465	−0.577553	−0.288777	−	0.957397i	$-0.593248\pi$
$53$	−4.20465	−0.577553	−0.288777	+	0.957397i	$0.593248\pi$
$54$	0	0
$55$	−2.70184	−0.364317
$56$	0	0
$57$	−0.0163125	−0.00216065
$58$	0	0
$59$	4.19241	0.545805	0.272903	−	0.962042i	$-0.412016\pi$
$59$	4.19241	0.545805	0.272903	+	0.962042i	$0.412016\pi$
$60$	0	0
$61$	−11.8442	−1.51650	−0.758248	−	0.651966i	$-0.773944\pi$
$61$	−11.8442	−1.51650	−0.758248	+	0.651966i	$0.773944\pi$
$62$	0	0
$63$	12.1130	1.52610
$64$	0	0
$65$	−6.94862	−0.861871
$66$	0	0
$67$	5.06597	0.618907	0.309453	−	0.950915i	$-0.399854\pi$
$67$	5.06597	0.618907	0.309453	+	0.950915i	$0.399854\pi$
$68$	0	0
$69$	−0.0372177	−0.00448048
$70$	0	0
$71$	6.00011	0.712082	0.356041	−	0.934470i	$-0.384126\pi$
$71$	6.00011	0.712082	0.356041	+	0.934470i	$0.384126\pi$
$72$	0	0
$73$	−3.36762	−0.394151	−0.197075	−	0.980388i	$-0.563144\pi$
$73$	−3.36762	−0.394151	−0.197075	+	0.980388i	$0.563144\pi$
$74$	0	0
$75$	0.0275033	0.00317581
$76$	0	0
$77$	−3.38204	−0.385419
$78$	0	0
$79$	−6.23780	−0.701807	−0.350904	−	0.936412i	$-0.614126\pi$
$79$	−6.23780	−0.701807	−0.350904	+	0.936412i	$0.614126\pi$
$80$	0	0
$81$	8.99977	0.999974
$82$	0	0
$83$	−14.7434	−1.61829	−0.809147	−	0.587606i	$-0.800071\pi$
$83$	−14.7434	−1.61829	−0.809147	+	0.587606i	$0.800071\pi$
$84$	0	0
$85$	18.3921	1.99491
$86$	0	0
$87$	−0.0519783	−0.00557266
$88$	0	0
$89$	−7.88751	−0.836074	−0.418037	−	0.908430i	$-0.637282\pi$
$89$	−7.88751	−0.836074	−0.418037	+	0.908430i	$0.637282\pi$
$90$	0	0
$91$	−8.69795	−0.911793
$92$	0	0
$93$	−0.0148920	−0.00154423
$94$	0	0
$95$	10.3403	1.06090
$96$	0	0
$97$	4.91922	0.499471	0.249736	−	0.968314i	$-0.419656\pi$
$97$	4.91922	0.499471	0.249736	+	0.968314i	$0.419656\pi$
$98$	0	0
$99$	−2.51281	−0.252547
$100$	0	0
$101$	5.75877	0.573019	0.286510	−	0.958077i	$-0.407505\pi$
$101$	5.75877	0.573019	0.286510	+	0.958077i	$0.407505\pi$
$102$	0	0
$103$	17.1415	1.68900	0.844501	−	0.535554i	$-0.179897\pi$
$103$	17.1415	1.68900	0.844501	+	0.535554i	$0.179897\pi$
$104$	0	0
$105$	0.0662761	0.00646788
$106$	0	0
$107$	4.53615	0.438526	0.219263	−	0.975666i	$-0.429635\pi$
$107$	4.53615	0.438526	0.219263	+	0.975666i	$0.429635\pi$
$108$	0	0
$109$	−10.6620	−1.02123	−0.510616	−	0.859809i	$-0.670583\pi$
$109$	−10.6620	−1.02123	−0.510616	+	0.859809i	$0.670583\pi$
$110$	0	0
$111$	0.00527558	0.000500736 0
$112$	0	0
$113$	−9.39850	−0.884137	−0.442068	−	0.896981i	$-0.645755\pi$
$113$	−9.39850	−0.884137	−0.442068	+	0.896981i	$0.645755\pi$
$114$	0	0
$115$	23.5919	2.19996
$116$	0	0
$117$	−6.46247	−0.597456
$118$	0	0
$119$	23.0224	2.11046
$120$	0	0
$121$	−10.2984	−0.936219
$122$	0	0
$123$	−0.0334444	−0.00301558
$124$	0	0
$125$	−1.30585	−0.116799
$126$	0	0
$127$	−5.32280	−0.472322	−0.236161	−	0.971714i	$-0.575889\pi$
$127$	−5.32280	−0.472322	−0.236161	+	0.971714i	$0.575889\pi$
$128$	0	0
$129$	0.0453982	0.00399709
$130$	0	0
$131$	18.5048	1.61677	0.808387	−	0.588652i	$-0.200341\pi$
$131$	18.5048	1.61677	0.808387	+	0.588652i	$0.200341\pi$
$132$	0	0
$133$	12.9436	1.12235
$134$	0	0
$135$	0.0984850	0.00847624
$136$	0	0
$137$	9.40457	0.803487	0.401743	−	0.915752i	$-0.368404\pi$
$137$	9.40457	0.803487	0.401743	+	0.915752i	$0.368404\pi$
$138$	0	0
$139$	16.7308	1.41909	0.709543	−	0.704663i	$-0.248902\pi$
$139$	16.7308	1.41909	0.709543	+	0.704663i	$0.248902\pi$
$140$	0	0
$141$	0.00102182	8.60531e−5 0
$142$	0	0
$143$	1.80436	0.150888
$144$	0	0
$145$	32.9485	2.73623
$146$	0	0
$147$	0.0473406	0.00390459
$148$	0	0
$149$	−11.6899	−0.957676	−0.478838	−	0.877903i	$-0.658942\pi$
$149$	−11.6899	−0.957676	−0.478838	+	0.877903i	$0.658942\pi$
$150$	0	0
$151$	14.1595	1.15228	0.576140	−	0.817351i	$-0.304558\pi$
$151$	14.1595	1.15228	0.576140	+	0.817351i	$0.304558\pi$
$152$	0	0
$153$	17.1053	1.38289
$154$	0	0
$155$	9.43992	0.758232
$156$	0	0
$157$	−4.63804	−0.370156	−0.185078	−	0.982724i	$-0.559254\pi$
$157$	−4.63804	−0.370156	−0.185078	+	0.982724i	$0.559254\pi$
$158$	0	0
$159$	−0.0213960	−0.00169682
$160$	0	0
$161$	29.5312	2.32739
$162$	0	0
$163$	−8.81836	−0.690707	−0.345354	−	0.938473i	$-0.612241\pi$
$163$	−8.81836	−0.690707	−0.345354	+	0.938473i	$0.612241\pi$
$164$	0	0
$165$	−0.0137488	−0.00107034
$166$	0	0
$167$	−6.14898	−0.475822	−0.237911	−	0.971287i	$-0.576463\pi$
$167$	−6.14898	−0.475822	−0.237911	+	0.971287i	$0.576463\pi$
$168$	0	0
$169$	−8.35953	−0.643040
$170$	0	0
$171$	9.61690	0.735423
$172$	0	0
$173$	−0.0164601	−0.00125144	−0.000625721	−	1.00000i	$-0.500199\pi$
$173$	−0.0164601	−0.00125144	−0.000625721		1.00000i	$0.500199\pi$
$174$	0	0
$175$	−21.8232	−1.64968
$176$	0	0
$177$	0.0213337	0.00160354
$178$	0	0
$179$	3.87651	0.289744	0.144872	−	0.989450i	$-0.453723\pi$
$179$	3.87651	0.289744	0.144872	+	0.989450i	$0.453723\pi$
$180$	0	0
$181$	−14.0526	−1.04452	−0.522262	−	0.852785i	$-0.674912\pi$
$181$	−14.0526	−1.04452	−0.522262	+	0.852785i	$0.674912\pi$
$182$	0	0
$183$	−0.0602711	−0.00445537
$184$	0	0
$185$	−3.34414	−0.245866
$186$	0	0
$187$	−4.77592	−0.349250
$188$	0	0
$189$	0.123279	0.00896722
$190$	0	0
$191$	−10.0386	−0.726371	−0.363185	−	0.931717i	$-0.618311\pi$
$191$	−10.0386	−0.726371	−0.363185	+	0.931717i	$0.618311\pi$
$192$	0	0
$193$	21.1779	1.52442	0.762208	−	0.647332i	$-0.224115\pi$
$193$	21.1779	1.52442	0.762208	+	0.647332i	$0.224115\pi$
$194$	0	0
$195$	−0.0353592	−0.00253212
$196$	0	0
$197$	14.0266	0.999354	0.499677	−	0.866212i	$-0.333452\pi$
$197$	14.0266	0.999354	0.499677	+	0.866212i	$0.333452\pi$
$198$	0	0
$199$	21.0825	1.49450	0.747248	−	0.664545i	$-0.231375\pi$
$199$	21.0825	1.49450	0.747248	+	0.664545i	$0.231375\pi$
$200$	0	0
$201$	0.0257790	0.00181831
$202$	0	0
$203$	41.2434	2.89472
$204$	0	0
$205$	21.2001	1.48068
$206$	0	0
$207$	21.9413	1.52503
$208$	0	0
$209$	−2.68510	−0.185732
$210$	0	0
$211$	−5.57737	−0.383962	−0.191981	−	0.981399i	$-0.561491\pi$
$211$	−5.57737	−0.383962	−0.191981	+	0.981399i	$0.561491\pi$
$212$	0	0
$213$	0.0305325	0.00209205
$214$	0	0
$215$	−28.7775	−1.96261
$216$	0	0
$217$	11.8164	0.802152
$218$	0	0
$219$	−0.0171367	−0.00115799
$220$	0	0
$221$	−12.2827	−0.826227
$222$	0	0
$223$	−23.5595	−1.57766	−0.788831	−	0.614610i	$-0.789313\pi$
$223$	−23.5595	−1.57766	−0.788831	+	0.614610i	$0.789313\pi$
$224$	0	0
$225$	−16.2144	−1.08096
$226$	0	0
$227$	−24.6444	−1.63571	−0.817853	−	0.575428i	$-0.804836\pi$
$227$	−24.6444	−1.63571	−0.817853	+	0.575428i	$0.804836\pi$
$228$	0	0
$229$	−8.34906	−0.551721	−0.275861	−	0.961198i	$-0.588963\pi$
$229$	−8.34906	−0.551721	−0.275861	+	0.961198i	$0.588963\pi$
$230$	0	0
$231$	−0.0172100	−0.00113234
$232$	0	0
$233$	12.7266	0.833749	0.416875	−	0.908964i	$-0.363125\pi$
$233$	12.7266	0.833749	0.416875	+	0.908964i	$0.363125\pi$
$234$	0	0
$235$	−0.647725	−0.0422529
$236$	0	0
$237$	−0.0317420	−0.00206187
$238$	0	0
$239$	−4.29304	−0.277693	−0.138847	−	0.990314i	$-0.544340\pi$
$239$	−4.29304	−0.277693	−0.138847	+	0.990314i	$0.544340\pi$
$240$	0	0
$241$	14.3683	0.925542	0.462771	−	0.886478i	$-0.346855\pi$
$241$	14.3683	0.925542	0.462771	+	0.886478i	$0.346855\pi$
$242$	0	0
$243$	0.137392	0.00881371
$244$	0	0
$245$	−30.0088	−1.91719
$246$	0	0
$247$	−6.90556	−0.439390
$248$	0	0
$249$	−0.0750239	−0.00475445
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−6.12616	−0.385149
$254$	0	0
$255$	0.0935912	0.00586091
$256$	0	0
$257$	−22.5062	−1.40390	−0.701949	−	0.712227i	$-0.747687\pi$
$257$	−22.5062	−1.40390	−0.701949	+	0.712227i	$0.747687\pi$
$258$	0	0
$259$	−4.18603	−0.260107
$260$	0	0
$261$	30.6433	1.89677
$262$	0	0
$263$	5.32233	0.328189	0.164094	−	0.986445i	$-0.447530\pi$
$263$	5.32233	0.328189	0.164094	+	0.986445i	$0.447530\pi$
$264$	0	0
$265$	13.5627	0.833152
$266$	0	0
$267$	−0.0401369	−0.00245634
$268$	0	0
$269$	2.17800	0.132795	0.0663976	−	0.997793i	$-0.478849\pi$
$269$	2.17800	0.132795	0.0663976	+	0.997793i	$0.478849\pi$
$270$	0	0
$271$	4.02438	0.244464	0.122232	−	0.992502i	$-0.460995\pi$
$271$	4.02438	0.244464	0.122232	+	0.992502i	$0.460995\pi$
$272$	0	0
$273$	−0.0442609	−0.00267879
$274$	0	0
$275$	4.52715	0.272998
$276$	0	0
$277$	−15.3578	−0.922763	−0.461381	−	0.887202i	$-0.652646\pi$
$277$	−15.3578	−0.922763	−0.461381	+	0.887202i	$0.652646\pi$
$278$	0	0
$279$	8.77947	0.525613
$280$	0	0
$281$	29.8573	1.78114	0.890568	−	0.454850i	$-0.150307\pi$
$281$	29.8573	1.78114	0.890568	+	0.454850i	$0.150307\pi$
$282$	0	0
$283$	−8.64598	−0.513950	−0.256975	−	0.966418i	$-0.582726\pi$
$283$	−8.64598	−0.513950	−0.256975	+	0.966418i	$0.582726\pi$
$284$	0	0
$285$	0.0526185	0.00311685
$286$	0	0
$287$	26.5372	1.56644
$288$	0	0
$289$	15.5109	0.912405
$290$	0	0
$291$	0.0250322	0.00146742
$292$	0	0
$293$	−13.2112	−0.771805	−0.385902	−	0.922540i	$-0.626110\pi$
$293$	−13.2112	−0.771805	−0.385902	+	0.922540i	$0.626110\pi$
$294$	0	0
$295$	−13.5233	−0.787354
$296$	0	0
$297$	−0.0255738	−0.00148394
$298$	0	0
$299$	−15.7553	−0.911153
$300$	0	0
$301$	−36.0223	−2.07629
$302$	0	0
$303$	0.0293044	0.00168350
$304$	0	0
$305$	38.2053	2.18763
$306$	0	0
$307$	−22.2553	−1.27017	−0.635087	−	0.772440i	$-0.719036\pi$
$307$	−22.2553	−1.27017	−0.635087	+	0.772440i	$0.719036\pi$
$308$	0	0
$309$	0.0872272	0.00496218
$310$	0	0
$311$	−29.3166	−1.66239	−0.831196	−	0.555980i	$-0.812343\pi$
$311$	−29.3166	−1.66239	−0.831196	+	0.555980i	$0.812343\pi$
$312$	0	0
$313$	20.4937	1.15837	0.579185	−	0.815196i	$-0.303371\pi$
$313$	20.4937	1.15837	0.579185	+	0.815196i	$0.303371\pi$
$314$	0	0
$315$	−39.0725	−2.20148
$316$	0	0
$317$	−30.4893	−1.71245	−0.856226	−	0.516602i	$-0.827197\pi$
$317$	−30.4893	−1.71245	−0.856226	+	0.516602i	$0.827197\pi$
$318$	0	0
$319$	−8.55582	−0.479034
$320$	0	0
$321$	0.0230829	0.00128836
$322$	0	0
$323$	18.2781	1.01702
$324$	0	0
$325$	11.6430	0.645835
$326$	0	0
$327$	−0.0542552	−0.00300032
$328$	0	0
$329$	−0.810791	−0.0447003
$330$	0	0
$331$	4.66546	0.256437	0.128218	−	0.991746i	$-0.459074\pi$
$331$	4.66546	0.256437	0.128218	+	0.991746i	$0.459074\pi$
$332$	0	0
$333$	−3.11017	−0.170436
$334$	0	0
$335$	−16.3411	−0.892807
$336$	0	0
$337$	−21.7495	−1.18477	−0.592385	−	0.805655i	$-0.701814\pi$
$337$	−21.7495	−1.18477	−0.592385	+	0.805655i	$0.701814\pi$
$338$	0	0
$339$	−0.0478258	−0.00259754
$340$	0	0
$341$	−2.45128	−0.132744
$342$	0	0
$343$	−9.29951	−0.502126
$344$	0	0
$345$	0.120051	0.00646334
$346$	0	0
$347$	−0.120134	−0.00644914	−0.00322457	−	0.999995i	$-0.501026\pi$
$347$	−0.120134	−0.00644914	−0.00322457	+	0.999995i	$0.501026\pi$
$348$	0	0
$349$	−10.9551	−0.586415	−0.293208	−	0.956049i	$-0.594723\pi$
$349$	−10.9551	−0.586415	−0.293208	+	0.956049i	$0.594723\pi$
$350$	0	0
$351$	−0.0657709	−0.00351059
$352$	0	0
$353$	16.3214	0.868701	0.434350	−	0.900744i	$-0.356978\pi$
$353$	16.3214	0.868701	0.434350	+	0.900744i	$0.356978\pi$
$354$	0	0
$355$	−19.3543	−1.02722
$356$	0	0
$357$	0.117153	0.00620040
$358$	0	0
$359$	16.2634	0.858351	0.429176	−	0.903221i	$-0.358804\pi$
$359$	16.2634	0.858351	0.429176	+	0.903221i	$0.358804\pi$
$360$	0	0
$361$	−8.72373	−0.459144
$362$	0	0
$363$	−0.0524051	−0.00275055
$364$	0	0
$365$	10.8628	0.568584
$366$	0	0
$367$	−28.5899	−1.49238	−0.746190	−	0.665733i	$-0.768119\pi$
$367$	−28.5899	−1.49238	−0.746190	+	0.665733i	$0.768119\pi$
$368$	0	0
$369$	19.7168	1.02642
$370$	0	0
$371$	16.9772	0.881412
$372$	0	0
$373$	21.2345	1.09948	0.549740	−	0.835336i	$-0.314727\pi$
$373$	21.2345	1.09948	0.549740	+	0.835336i	$0.314727\pi$
$374$	0	0
$375$	−0.00664502	−0.000343147 0
$376$	0	0
$377$	−22.0039	−1.13326
$378$	0	0
$379$	−22.9266	−1.17766	−0.588830	−	0.808257i	$-0.700411\pi$
$379$	−22.9266	−1.17766	−0.588830	+	0.808257i	$0.700411\pi$
$380$	0	0
$381$	−0.0270859	−0.00138765
$382$	0	0
$383$	−13.9353	−0.712059	−0.356030	−	0.934475i	$-0.615870\pi$
$383$	−13.9353	−0.712059	−0.356030	+	0.934475i	$0.615870\pi$
$384$	0	0
$385$	10.9093	0.555989
$386$	0	0
$387$	−26.7641	−1.36050
$388$	0	0
$389$	−22.0424	−1.11760	−0.558798	−	0.829304i	$-0.688737\pi$
$389$	−22.0424	−1.11760	−0.558798	+	0.829304i	$0.688737\pi$
$390$	0	0
$391$	41.7023	2.10898
$392$	0	0
$393$	0.0941647	0.00474998
$394$	0	0
$395$	20.1210	1.01240
$396$	0	0
$397$	−29.4529	−1.47820	−0.739100	−	0.673596i	$-0.764749\pi$
$397$	−29.4529	−1.47820	−0.739100	+	0.673596i	$0.764749\pi$
$398$	0	0
$399$	0.0658653	0.00329739
$400$	0	0
$401$	7.34488	0.366786	0.183393	−	0.983040i	$-0.441292\pi$
$401$	7.34488	0.366786	0.183393	+	0.983040i	$0.441292\pi$
$402$	0	0
$403$	−6.30423	−0.314036
$404$	0	0
$405$	−29.0301	−1.44252
$406$	0	0
$407$	0.868380	0.0430440
$408$	0	0
$409$	19.3108	0.954856	0.477428	−	0.878671i	$-0.341569\pi$
$409$	19.3108	0.954856	0.477428	+	0.878671i	$0.341569\pi$
$410$	0	0
$411$	0.0478567	0.00236059
$412$	0	0
$413$	−16.9278	−0.832961
$414$	0	0
$415$	47.5570	2.33448
$416$	0	0
$417$	0.0851372	0.00416918
$418$	0	0
$419$	−22.7389	−1.11087	−0.555434	−	0.831561i	$-0.687448\pi$
$419$	−22.7389	−1.11087	−0.555434	+	0.831561i	$0.687448\pi$
$420$	0	0
$421$	−4.53304	−0.220927	−0.110463	−	0.993880i	$-0.535234\pi$
$421$	−4.53304	−0.220927	−0.110463	+	0.993880i	$0.535234\pi$
$422$	0	0
$423$	−0.602407	−0.0292901
$424$	0	0
$425$	−30.8174	−1.49487
$426$	0	0
$427$	47.8236	2.31434
$428$	0	0
$429$	0.00918179	0.000443301 0
$430$	0	0
$431$	−7.69148	−0.370486	−0.185243	−	0.982693i	$-0.559307\pi$
$431$	−7.69148	−0.370486	−0.185243	+	0.982693i	$0.559307\pi$
$432$	0	0
$433$	−3.53012	−0.169647	−0.0848234	−	0.996396i	$-0.527033\pi$
$433$	−3.53012	−0.169647	−0.0848234	+	0.996396i	$0.527033\pi$
$434$	0	0
$435$	0.167664	0.00803887
$436$	0	0
$437$	23.4457	1.12156
$438$	0	0
$439$	24.7723	1.18232	0.591159	−	0.806555i	$-0.298671\pi$
$439$	24.7723	1.18232	0.591159	+	0.806555i	$0.298671\pi$
$440$	0	0
$441$	−27.9092	−1.32901
$442$	0	0
$443$	17.1423	0.814457	0.407228	−	0.913326i	$-0.366495\pi$
$443$	17.1423	0.814457	0.407228	+	0.913326i	$0.366495\pi$
$444$	0	0
$445$	25.4424	1.20608
$446$	0	0
$447$	−0.0594860	−0.00281359
$448$	0	0
$449$	34.1551	1.61188	0.805939	−	0.591999i	$-0.201661\pi$
$449$	34.1551	1.61188	0.805939	+	0.591999i	$0.201661\pi$
$450$	0	0
$451$	−5.50506	−0.259223
$452$	0	0
$453$	0.0720526	0.00338533
$454$	0	0
$455$	28.0566	1.31531
$456$	0	0
$457$	−25.3730	−1.18690	−0.593450	−	0.804871i	$-0.702235\pi$
$457$	−25.3730	−1.18690	−0.593450	+	0.804871i	$0.702235\pi$
$458$	0	0
$459$	0.174087	0.00812570
$460$	0	0
$461$	19.1468	0.891753	0.445877	−	0.895094i	$-0.352892\pi$
$461$	19.1468	0.891753	0.445877	+	0.895094i	$0.352892\pi$
$462$	0	0
$463$	26.7961	1.24532	0.622660	−	0.782492i	$-0.286052\pi$
$463$	26.7961	1.24532	0.622660	+	0.782492i	$0.286052\pi$
$464$	0	0
$465$	0.0480365	0.00222764
$466$	0	0
$467$	−33.9372	−1.57043	−0.785214	−	0.619225i	$-0.787447\pi$
$467$	−33.9372	−1.57043	−0.785214	+	0.619225i	$0.787447\pi$
$468$	0	0
$469$	−20.4549	−0.944522
$470$	0	0
$471$	−0.0236014	−0.00108750
$472$	0	0
$473$	7.47272	0.343596
$474$	0	0
$475$	−17.3261	−0.794974
$476$	0	0
$477$	12.6138	0.577548
$478$	0	0
$479$	3.93196	0.179656	0.0898279	−	0.995957i	$-0.471368\pi$
$479$	3.93196	0.179656	0.0898279	+	0.995957i	$0.471368\pi$
$480$	0	0
$481$	2.23330	0.101830
$482$	0	0
$483$	0.150274	0.00683772
$484$	0	0
$485$	−15.8677	−0.720515
$486$	0	0
$487$	−20.5036	−0.929105	−0.464552	−	0.885546i	$-0.653785\pi$
$487$	−20.5036	−0.929105	−0.464552	+	0.885546i	$0.653785\pi$
$488$	0	0
$489$	−0.0448736	−0.00202926
$490$	0	0
$491$	17.4190	0.786108	0.393054	−	0.919515i	$-0.371419\pi$
$491$	17.4190	0.786108	0.393054	+	0.919515i	$0.371419\pi$
$492$	0	0
$493$	58.2415	2.62307
$494$	0	0
$495$	8.10546	0.364314
$496$	0	0
$497$	−24.2268	−1.08672
$498$	0	0
$499$	−11.6633	−0.522119	−0.261060	−	0.965323i	$-0.584072\pi$
$499$	−11.6633	−0.522119	−0.261060	+	0.965323i	$0.584072\pi$
$500$	0	0
$501$	−0.0312900	−0.00139794
$502$	0	0
$503$	−32.0320	−1.42824	−0.714118	−	0.700026i	$-0.753172\pi$
$503$	−32.0320	−1.42824	−0.714118	+	0.700026i	$0.753172\pi$
$504$	0	0
$505$	−18.5758	−0.826612
$506$	0	0
$507$	−0.0425388	−0.00188921
$508$	0	0
$509$	8.13675	0.360655	0.180328	−	0.983607i	$-0.442284\pi$
$509$	8.13675	0.360655	0.180328	+	0.983607i	$0.442284\pi$
$510$	0	0
$511$	13.5975	0.601518
$512$	0	0
$513$	0.0978747	0.00432127
$514$	0	0
$515$	−55.2925	−2.43648
$516$	0	0
$517$	0.168196	0.00739725
$518$	0	0
$519$	−8.37600e−5 0	−3.67666e−6 0
$520$	0	0
$521$	27.5428	1.20667	0.603335	−	0.797488i	$-0.293838\pi$
$521$	27.5428	1.20667	0.603335	+	0.797488i	$0.293838\pi$
$522$	0	0
$523$	4.28916	0.187552	0.0937760	−	0.995593i	$-0.470106\pi$
$523$	4.28916	0.187552	0.0937760	+	0.995593i	$0.470106\pi$
$524$	0	0
$525$	−0.111051	−0.00484665
$526$	0	0
$527$	16.6865	0.726875
$528$	0	0
$529$	30.4923	1.32575
$530$	0	0
$531$	−12.5771	−0.545801
$532$	0	0
$533$	−14.1580	−0.613249
$534$	0	0
$535$	−14.6320	−0.632598
$536$	0	0
$537$	0.0197262	0.000851249 0
$538$	0	0
$539$	7.79244	0.335644
$540$	0	0
$541$	−18.6598	−0.802249	−0.401125	−	0.916023i	$-0.631381\pi$
$541$	−18.6598	−0.802249	−0.401125	+	0.916023i	$0.631381\pi$
$542$	0	0
$543$	−0.0715091	−0.00306875
$544$	0	0
$545$	34.3918	1.47318
$546$	0	0
$547$	38.5017	1.64621	0.823107	−	0.567886i	$-0.192239\pi$
$547$	38.5017	1.64621	0.823107	+	0.567886i	$0.192239\pi$
$548$	0	0
$549$	35.5323	1.51648
$550$	0	0
$551$	32.7443	1.39496
$552$	0	0
$553$	25.1865	1.07104
$554$	0	0
$555$	−0.0170172	−0.000722339 0
$556$	0	0
$557$	−11.5468	−0.489253	−0.244626	−	0.969617i	$-0.578665\pi$
$557$	−11.5468	−0.489253	−0.244626	+	0.969617i	$0.578665\pi$
$558$	0	0
$559$	19.2184	0.812851
$560$	0	0
$561$	−0.0243030	−0.00102608
$562$	0	0
$563$	−36.5478	−1.54031	−0.770154	−	0.637858i	$-0.779820\pi$
$563$	−36.5478	−1.54031	−0.770154	+	0.637858i	$0.779820\pi$
$564$	0	0
$565$	30.3163	1.27542
$566$	0	0
$567$	−36.3385	−1.52607
$568$	0	0
$569$	−0.635721	−0.0266508	−0.0133254	−	0.999911i	$-0.504242\pi$
$569$	−0.635721	−0.0266508	−0.0133254	+	0.999911i	$0.504242\pi$
$570$	0	0
$571$	−3.10569	−0.129969	−0.0649846	−	0.997886i	$-0.520700\pi$
$571$	−3.10569	−0.129969	−0.0649846	+	0.997886i	$0.520700\pi$
$572$	0	0
$573$	−0.0510832	−0.00213403
$574$	0	0
$575$	−39.5301	−1.64852
$576$	0	0
$577$	−36.2808	−1.51039	−0.755195	−	0.655501i	$-0.772458\pi$
$577$	−36.2808	−1.51039	−0.755195	+	0.655501i	$0.772458\pi$
$578$	0	0
$579$	0.107767	0.00447864
$580$	0	0
$581$	59.5295	2.46970
$582$	0	0
$583$	−3.52186	−0.145861
$584$	0	0
$585$	20.8457	0.861863
$586$	0	0
$587$	−0.948654	−0.0391551	−0.0195776	−	0.999808i	$-0.506232\pi$
$587$	−0.948654	−0.0391551	−0.0195776	+	0.999808i	$0.506232\pi$
$588$	0	0
$589$	9.38141	0.386554
$590$	0	0
$591$	0.0713766	0.00293604
$592$	0	0
$593$	−30.0368	−1.23346	−0.616732	−	0.787174i	$-0.711544\pi$
$593$	−30.0368	−1.23346	−0.616732	+	0.787174i	$0.711544\pi$
$594$	0	0
$595$	−74.2622	−3.04445
$596$	0	0
$597$	0.107281	0.00439074
$598$	0	0
$599$	−20.1520	−0.823389	−0.411695	−	0.911322i	$-0.635063\pi$
$599$	−20.1520	−0.823389	−0.411695	+	0.911322i	$0.635063\pi$
$600$	0	0
$601$	−36.6973	−1.49691	−0.748456	−	0.663184i	$-0.769205\pi$
$601$	−36.6973	−1.49691	−0.748456	+	0.663184i	$0.769205\pi$
$602$	0	0
$603$	−15.1978	−0.618901
$604$	0	0
$605$	33.2191	1.35055
$606$	0	0
$607$	−18.5022	−0.750982	−0.375491	−	0.926826i	$-0.622526\pi$
$607$	−18.5022	−0.750982	−0.375491	+	0.926826i	$0.622526\pi$
$608$	0	0
$609$	0.209874	0.00850451
$610$	0	0
$611$	0.432568	0.0174998
$612$	0	0
$613$	47.9329	1.93599	0.967996	−	0.250965i	$-0.0807478\pi$
$613$	47.9329	1.93599	0.967996	+	0.250965i	$0.0807478\pi$
$614$	0	0
$615$	0.107880	0.00435014
$616$	0	0
$617$	−49.6643	−1.99941	−0.999705	−	0.0242771i	$-0.992272\pi$
$617$	−49.6643	−1.99941	−0.999705	+	0.0242771i	$0.992272\pi$
$618$	0	0
$619$	34.0130	1.36710	0.683549	−	0.729904i	$-0.260435\pi$
$619$	34.0130	1.36710	0.683549	+	0.729904i	$0.260435\pi$
$620$	0	0
$621$	0.223305	0.00896092
$622$	0	0
$623$	31.8475	1.27594
$624$	0	0
$625$	−22.8119	−0.912478
$626$	0	0
$627$	−0.0136636	−0.000545670 0
$628$	0	0
$629$	−5.91127	−0.235698
$630$	0	0
$631$	8.04486	0.320261	0.160130	−	0.987096i	$-0.448809\pi$
$631$	8.04486	0.320261	0.160130	+	0.987096i	$0.448809\pi$
$632$	0	0
$633$	−0.0283813	−0.00112806
$634$	0	0
$635$	17.1695	0.681350
$636$	0	0
$637$	20.0406	0.794039
$638$	0	0
$639$	−18.0002	−0.712076
$640$	0	0
$641$	10.0453	0.396764	0.198382	−	0.980125i	$-0.436431\pi$
$641$	10.0453	0.396764	0.198382	+	0.980125i	$0.436431\pi$
$642$	0	0
$643$	−26.8296	−1.05806	−0.529028	−	0.848604i	$-0.677443\pi$
$643$	−26.8296	−1.05806	−0.529028	+	0.848604i	$0.677443\pi$
$644$	0	0
$645$	−0.146439	−0.00576603
$646$	0	0
$647$	−2.58862	−0.101769	−0.0508846	−	0.998705i	$-0.516204\pi$
$647$	−2.58862	−0.101769	−0.0508846	+	0.998705i	$0.516204\pi$
$648$	0	0
$649$	3.51161	0.137843
$650$	0	0
$651$	0.0601298	0.00235667
$652$	0	0
$653$	−8.07411	−0.315964	−0.157982	−	0.987442i	$-0.550499\pi$
$653$	−8.07411	−0.315964	−0.157982	+	0.987442i	$0.550499\pi$
$654$	0	0
$655$	−59.6901	−2.33229
$656$	0	0
$657$	10.1028	0.394147
$658$	0	0
$659$	12.9272	0.503572	0.251786	−	0.967783i	$-0.418982\pi$
$659$	12.9272	0.503572	0.251786	+	0.967783i	$0.418982\pi$
$660$	0	0
$661$	37.2628	1.44936	0.724678	−	0.689087i	$-0.241988\pi$
$661$	37.2628	1.44936	0.724678	+	0.689087i	$0.241988\pi$
$662$	0	0
$663$	−0.0625027	−0.00242740
$664$	0	0
$665$	−41.7514	−1.61905
$666$	0	0
$667$	74.7075	2.89269
$668$	0	0
$669$	−0.119886	−0.00463507
$670$	0	0
$671$	−9.92085	−0.382990
$672$	0	0
$673$	24.8538	0.958044	0.479022	−	0.877803i	$-0.340991\pi$
$673$	24.8538	0.958044	0.479022	+	0.877803i	$0.340991\pi$
$674$	0	0
$675$	−0.165019	−0.00635160
$676$	0	0
$677$	−17.0420	−0.654979	−0.327489	−	0.944855i	$-0.606203\pi$
$677$	−17.0420	−0.654979	−0.327489	+	0.944855i	$0.606203\pi$
$678$	0	0
$679$	−19.8624	−0.762250
$680$	0	0
$681$	−0.125407	−0.00480560
$682$	0	0
$683$	−25.5143	−0.976278	−0.488139	−	0.872766i	$-0.662324\pi$
$683$	−25.5143	−0.976278	−0.488139	+	0.872766i	$0.662324\pi$
$684$	0	0
$685$	−30.3359	−1.15907
$686$	0	0
$687$	−0.0424855	−0.00162092
$688$	0	0
$689$	−9.05755	−0.345065
$690$	0	0
$691$	−33.2382	−1.26444	−0.632221	−	0.774788i	$-0.717856\pi$
$691$	−33.2382	−1.26444	−0.632221	+	0.774788i	$0.717856\pi$
$692$	0	0
$693$	10.1460	0.385416
$694$	0	0
$695$	−53.9676	−2.04711
$696$	0	0
$697$	37.4743	1.41944
$698$	0	0
$699$	0.0647615	0.00244950
$700$	0	0
$701$	−51.2106	−1.93420	−0.967098	−	0.254403i	$-0.918121\pi$
$701$	−51.2106	−1.93420	−0.967098	+	0.254403i	$0.918121\pi$
$702$	0	0
$703$	−3.32341	−0.125345
$704$	0	0
$705$	−0.00329605	−0.000124136 0
$706$	0	0
$707$	−23.2523	−0.874492
$708$	0	0
$709$	−32.9916	−1.23902	−0.619512	−	0.784987i	$-0.712670\pi$
$709$	−32.9916	−1.23902	−0.619512	+	0.784987i	$0.712670\pi$
$710$	0	0
$711$	18.7132	0.701801
$712$	0	0
$713$	21.4041	0.801588
$714$	0	0
$715$	−5.82025	−0.217665
$716$	0	0
$717$	−0.0218458	−0.000815846 0
$718$	0	0
$719$	10.8395	0.404246	0.202123	−	0.979360i	$-0.435216\pi$
$719$	10.8395	0.404246	0.202123	+	0.979360i	$0.435216\pi$
$720$	0	0
$721$	−69.2125	−2.57761
$722$	0	0
$723$	0.0731152	0.00271918
$724$	0	0
$725$	−55.2078	−2.05037
$726$	0	0
$727$	19.2435	0.713703	0.356851	−	0.934161i	$-0.383850\pi$
$727$	19.2435	0.713703	0.356851	+	0.934161i	$0.383850\pi$
$728$	0	0
$729$	−26.9986	−0.999948
$730$	0	0
$731$	−50.8686	−1.88144
$732$	0	0
$733$	38.5648	1.42442	0.712211	−	0.701965i	$-0.247694\pi$
$733$	38.5648	1.42442	0.712211	+	0.701965i	$0.247694\pi$
$734$	0	0
$735$	−0.152704	−0.00563258
$736$	0	0
$737$	4.24332	0.156305
$738$	0	0
$739$	−13.7326	−0.505163	−0.252582	−	0.967576i	$-0.581280\pi$
$739$	−13.7326	−0.505163	−0.252582	+	0.967576i	$0.581280\pi$
$740$	0	0
$741$	−0.0351400	−0.00129090
$742$	0	0
$743$	20.9824	0.769768	0.384884	−	0.922965i	$-0.374241\pi$
$743$	20.9824	0.769768	0.384884	+	0.922965i	$0.374241\pi$
$744$	0	0
$745$	37.7076	1.38150
$746$	0	0
$747$	44.2297	1.61828
$748$	0	0
$749$	−18.3157	−0.669240
$750$	0	0
$751$	3.73296	0.136218	0.0681088	−	0.997678i	$-0.478303\pi$
$751$	3.73296	0.136218	0.0681088	+	0.997678i	$0.478303\pi$
$752$	0	0
$753$	0.00508866	0.000185441 0
$754$	0	0
$755$	−45.6735	−1.66223
$756$	0	0
$757$	−6.06066	−0.220278	−0.110139	−	0.993916i	$-0.535130\pi$
$757$	−6.06066	−0.220278	−0.110139	+	0.993916i	$0.535130\pi$
$758$	0	0
$759$	−0.0311739	−0.00113154
$760$	0	0
$761$	45.4518	1.64762	0.823812	−	0.566862i	$-0.191843\pi$
$761$	45.4518	1.64762	0.823812	+	0.566862i	$0.191843\pi$
$762$	0	0
$763$	43.0500	1.55852
$764$	0	0
$765$	−55.1759	−1.99489
$766$	0	0
$767$	9.03119	0.326097
$768$	0	0
$769$	6.59385	0.237780	0.118890	−	0.992907i	$-0.462066\pi$
$769$	6.59385	0.237780	0.118890	+	0.992907i	$0.462066\pi$
$770$	0	0
$771$	−0.114526	−0.00412457
$772$	0	0
$773$	−20.3663	−0.732525	−0.366263	−	0.930511i	$-0.619363\pi$
$773$	−20.3663	−0.732525	−0.366263	+	0.930511i	$0.619363\pi$
$774$	0	0
$775$	−15.8173	−0.568175
$776$	0	0
$777$	−0.0213013	−0.000764180 0
$778$	0	0
$779$	21.0687	0.754864
$780$	0	0
$781$	5.02576	0.179836
$782$	0	0
$783$	0.311868	0.0111453
$784$	0	0
$785$	14.9607	0.533970
$786$	0	0
$787$	42.1822	1.50363	0.751817	−	0.659372i	$-0.229178\pi$
$787$	42.1822	1.50363	0.751817	+	0.659372i	$0.229178\pi$
$788$	0	0
$789$	0.0270835	0.000964198 0
$790$	0	0
$791$	37.9485	1.34929
$792$	0	0
$793$	−25.5145	−0.906047
$794$	0	0
$795$	0.0690161	0.00244775
$796$	0	0
$797$	−53.4304	−1.89260	−0.946300	−	0.323289i	$-0.895211\pi$
$797$	−53.4304	−1.89260	−0.946300	+	0.323289i	$0.895211\pi$
$798$	0	0
$799$	−1.14495	−0.0405055
$800$	0	0
$801$	23.6623	0.836067
$802$	0	0
$803$	−2.82076	−0.0995425
$804$	0	0
$805$	−95.2575	−3.35738
$806$	0	0
$807$	0.0110831	0.000390144 0
$808$	0	0
$809$	−33.0763	−1.16290	−0.581450	−	0.813582i	$-0.697514\pi$
$809$	−33.0763	−1.16290	−0.581450	+	0.813582i	$0.697514\pi$
$810$	0	0
$811$	18.7805	0.659472	0.329736	−	0.944073i	$-0.393040\pi$
$811$	18.7805	0.659472	0.329736	+	0.944073i	$0.393040\pi$
$812$	0	0
$813$	0.0204787	0.000718220 0
$814$	0	0
$815$	28.4450	0.996384
$816$	0	0
$817$	−28.5992	−1.00056
$818$	0	0
$819$	26.0936	0.911785
$820$	0	0
$821$	48.6583	1.69819	0.849094	−	0.528242i	$-0.177149\pi$
$821$	48.6583	1.69819	0.849094	+	0.528242i	$0.177149\pi$
$822$	0	0
$823$	−9.30773	−0.324447	−0.162224	−	0.986754i	$-0.551867\pi$
$823$	−9.30773	−0.324447	−0.162224	+	0.986754i	$0.551867\pi$
$824$	0	0
$825$	0.0230371	0.000802050 0
$826$	0	0
$827$	−38.0284	−1.32238	−0.661189	−	0.750219i	$-0.729948\pi$
$827$	−38.0284	−1.32238	−0.661189	+	0.750219i	$0.729948\pi$
$828$	0	0
$829$	4.77369	0.165797	0.0828985	−	0.996558i	$-0.473582\pi$
$829$	4.77369	0.165797	0.0828985	+	0.996558i	$0.473582\pi$
$830$	0	0
$831$	−0.0781508	−0.00271102
$832$	0	0
$833$	−53.0451	−1.83790
$834$	0	0
$835$	19.8345	0.686400
$836$	0	0
$837$	0.0893518	0.00308845
$838$	0	0
$839$	35.4834	1.22502	0.612512	−	0.790462i	$-0.290159\pi$
$839$	35.4834	1.22502	0.612512	+	0.790462i	$0.290159\pi$
$840$	0	0
$841$	75.3367	2.59782
$842$	0	0
$843$	0.151934	0.00523287
$844$	0	0
$845$	26.9649	0.927622
$846$	0	0
$847$	41.5821	1.42878
$848$	0	0
$849$	−0.0439964	−0.00150995
$850$	0	0
$851$	−7.58250	−0.259925
$852$	0	0
$853$	0.907849	0.0310842	0.0155421	−	0.999879i	$-0.495053\pi$
$853$	0.907849	0.0310842	0.0155421	+	0.999879i	$0.495053\pi$
$854$	0	0
$855$	−31.0208	−1.06089
$856$	0	0
$857$	−22.0871	−0.754479	−0.377240	−	0.926116i	$-0.623127\pi$
$857$	−22.0871	−0.754479	−0.377240	+	0.926116i	$0.623127\pi$
$858$	0	0
$859$	19.3597	0.660546	0.330273	−	0.943885i	$-0.392859\pi$
$859$	19.3597	0.660546	0.330273	+	0.943885i	$0.392859\pi$
$860$	0	0
$861$	0.135039	0.00460211
$862$	0	0
$863$	7.67955	0.261415	0.130708	−	0.991421i	$-0.458275\pi$
$863$	7.67955	0.261415	0.130708	+	0.991421i	$0.458275\pi$
$864$	0	0
$865$	0.0530947	0.00180527
$866$	0	0
$867$	0.0789296	0.00268059
$868$	0	0
$869$	−5.22485	−0.177241
$870$	0	0
$871$	10.9130	0.369772
$872$	0	0
$873$	−14.7575	−0.499467
$874$	0	0
$875$	5.27265	0.178248
$876$	0	0
$877$	−34.1159	−1.15201	−0.576006	−	0.817445i	$-0.695390\pi$
$877$	−34.1159	−1.15201	−0.576006	+	0.817445i	$0.695390\pi$
$878$	0	0
$879$	−0.0672271	−0.00226752
$880$	0	0
$881$	5.95426	0.200604	0.100302	−	0.994957i	$-0.468019\pi$
$881$	5.95426	0.200604	0.100302	+	0.994957i	$0.468019\pi$
$882$	0	0
$883$	44.9508	1.51271	0.756357	−	0.654159i	$-0.226977\pi$
$883$	44.9508	1.51271	0.756357	+	0.654159i	$0.226977\pi$
$884$	0	0
$885$	−0.0688152	−0.00231320
$886$	0	0
$887$	−7.21616	−0.242295	−0.121147	−	0.992635i	$-0.538657\pi$
$887$	−7.21616	−0.242295	−0.121147	+	0.992635i	$0.538657\pi$
$888$	0	0
$889$	21.4919	0.720817
$890$	0	0
$891$	7.53831	0.252543
$892$	0	0
$893$	−0.643710	−0.0215409
$894$	0	0
$895$	−12.5043	−0.417971
$896$	0	0
$897$	−0.0801734	−0.00267691
$898$	0	0
$899$	29.8930	0.996986
$900$	0	0
$901$	23.9742	0.798696
$902$	0	0
$903$	−0.183305	−0.00610002
$904$	0	0
$905$	45.3289	1.50678
$906$	0	0
$907$	17.3654	0.576609	0.288305	−	0.957539i	$-0.406908\pi$
$907$	17.3654	0.576609	0.288305	+	0.957539i	$0.406908\pi$
$908$	0	0
$909$	−17.2762	−0.573014
$910$	0	0
$911$	40.2152	1.33239	0.666195	−	0.745777i	$-0.267922\pi$
$911$	40.2152	1.33239	0.666195	+	0.745777i	$0.267922\pi$
$912$	0	0
$913$	−12.3492	−0.408699
$914$	0	0
$915$	0.194414	0.00642712
$916$	0	0
$917$	−74.7172	−2.46738
$918$	0	0
$919$	10.4579	0.344975	0.172487	−	0.985012i	$-0.444820\pi$
$919$	10.4579	0.344975	0.172487	+	0.985012i	$0.444820\pi$
$920$	0	0
$921$	−0.113249	−0.00373170
$922$	0	0
$923$	12.9253	0.425441
$924$	0	0
$925$	5.60337	0.184238
$926$	0	0
$927$	−51.4240	−1.68899
$928$	0	0
$929$	−40.5200	−1.32942	−0.664709	−	0.747102i	$-0.731444\pi$
$929$	−40.5200	−1.32942	−0.664709	+	0.747102i	$0.731444\pi$
$930$	0	0
$931$	−29.8228	−0.977402
$932$	0	0
$933$	−0.149182	−0.00488400
$934$	0	0
$935$	15.4055	0.503813
$936$	0	0
$937$	35.5748	1.16218	0.581090	−	0.813839i	$-0.302626\pi$
$937$	35.5748	1.16218	0.581090	+	0.813839i	$0.302626\pi$
$938$	0	0
$939$	0.104285	0.00340322
$940$	0	0
$941$	1.37595	0.0448548	0.0224274	−	0.999748i	$-0.492861\pi$
$941$	1.37595	0.0448548	0.0224274	+	0.999748i	$0.492861\pi$
$942$	0	0
$943$	48.0690	1.56534
$944$	0	0
$945$	−0.397655	−0.0129357
$946$	0	0
$947$	22.8072	0.741134	0.370567	−	0.928806i	$-0.379163\pi$
$947$	22.8072	0.741134	0.370567	+	0.928806i	$0.379163\pi$
$948$	0	0
$949$	−7.25445	−0.235490
$950$	0	0
$951$	−0.155150	−0.00503108
$952$	0	0
$953$	−49.3177	−1.59756	−0.798778	−	0.601626i	$-0.794520\pi$
$953$	−49.3177	−1.59756	−0.798778	+	0.601626i	$0.794520\pi$
$954$	0	0
$955$	32.3812	1.04783
$956$	0	0
$957$	−0.0435376	−0.00140737
$958$	0	0
$959$	−37.9730	−1.22621
$960$	0	0
$961$	−22.4355	−0.723726
$962$	0	0
$963$	−13.6083	−0.438522
$964$	0	0
$965$	−68.3124	−2.19905
$966$	0	0
$967$	−33.8819	−1.08957	−0.544785	−	0.838576i	$-0.683389\pi$
$967$	−33.8819	−1.08957	−0.544785	+	0.838576i	$0.683389\pi$
$968$	0	0
$969$	0.0930112	0.00298795
$970$	0	0
$971$	32.5137	1.04341	0.521707	−	0.853125i	$-0.325295\pi$
$971$	32.5137	1.04341	0.521707	+	0.853125i	$0.325295\pi$
$972$	0	0
$973$	−67.5541	−2.16569
$974$	0	0
$975$	0.0592470	0.00189742
$976$	0	0
$977$	58.1261	1.85962	0.929809	−	0.368041i	$-0.119971\pi$
$977$	58.1261	1.85962	0.929809	+	0.368041i	$0.119971\pi$
$978$	0	0
$979$	−6.60667	−0.211150
$980$	0	0
$981$	31.9857	1.02122
$982$	0	0
$983$	−10.9860	−0.350399	−0.175200	−	0.984533i	$-0.556057\pi$
$983$	−10.9860	−0.350399	−0.175200	+	0.984533i	$0.556057\pi$
$984$	0	0
$985$	−45.2450	−1.44162
$986$	0	0
$987$	−0.00412584	−0.000131327 0
$988$	0	0
$989$	−65.2501	−2.07483
$990$	0	0
$991$	−32.9144	−1.04556	−0.522780	−	0.852468i	$-0.675105\pi$
$991$	−32.9144	−1.04556	−0.522780	+	0.852468i	$0.675105\pi$
$992$	0	0
$993$	0.0237409	0.000753395 0
$994$	0	0
$995$	−68.0047	−2.15589
$996$	0	0
$997$	47.5649	1.50640	0.753198	−	0.657794i	$-0.228510\pi$
$997$	47.5649	1.50640	0.753198	+	0.657794i	$0.228510\pi$
$998$	0	0
$999$	−0.0316533	−0.00100147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.10		19
4.3	odd	2		2008.2.a.c.1.10	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.10	✓	19	4.3	odd	2
4016.2.a.l.1.10		19	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+0.00508866 q^{3} -3.22565 q^{5} -4.03772 q^{7} -2.99997 q^{9} +O(q^{10})\)	\(q+0.00508866 q^{3} -3.22565 q^{5} -4.03772 q^{7} -2.99997 q^{9} +0.837612 q^{11} +2.15418 q^{13} -0.0164142 q^{15} -5.70183 q^{17} -3.20566 q^{19} -0.0205466 q^{21} -7.31384 q^{23} +5.40483 q^{25} -0.0305318 q^{27} -10.2145 q^{29} -2.92651 q^{31} +0.00426232 q^{33} +13.0243 q^{35} +1.03673 q^{37} +0.0109619 q^{39} -6.57233 q^{41} +8.92146 q^{43} +9.67687 q^{45} +0.200804 q^{47} +9.30316 q^{49} -0.0290147 q^{51} -4.20465 q^{53} -2.70184 q^{55} -0.0163125 q^{57} +4.19241 q^{59} -11.8442 q^{61} +12.1130 q^{63} -6.94862 q^{65} +5.06597 q^{67} -0.0372177 q^{69} +6.00011 q^{71} -3.36762 q^{73} +0.0275033 q^{75} -3.38204 q^{77} -6.23780 q^{79} +8.99977 q^{81} -14.7434 q^{83} +18.3921 q^{85} -0.0519783 q^{87} -7.88751 q^{89} -8.69795 q^{91} -0.0148920 q^{93} +10.3403 q^{95} +4.91922 q^{97} -2.51281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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