LMFDB - Embedded newform 4016.2.a.l.1.19

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			1.19
Root			$3.26349$ 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)

$f(q)$	$=$	$q+3.26349 q^{3} +2.68811 q^{5} -0.141739 q^{7} +7.65038 q^{9} +O(q^{10})$	$q+3.26349 q^{3} +2.68811 q^{5} -0.141739 q^{7} +7.65038 q^{9} +0.999322 q^{11} -0.730044 q^{13} +8.77263 q^{15} -2.91709 q^{17} +6.46250 q^{19} -0.462566 q^{21} +4.39136 q^{23} +2.22595 q^{25} +15.1765 q^{27} +8.02510 q^{29} -9.19633 q^{31} +3.26128 q^{33} -0.381012 q^{35} -7.19305 q^{37} -2.38249 q^{39} -1.46896 q^{41} -5.90097 q^{43} +20.5651 q^{45} -8.09886 q^{47} -6.97991 q^{49} -9.51990 q^{51} -8.64708 q^{53} +2.68629 q^{55} +21.0903 q^{57} -2.29994 q^{59} -4.13515 q^{61} -1.08436 q^{63} -1.96244 q^{65} +11.7000 q^{67} +14.3312 q^{69} +7.62814 q^{71} -6.13213 q^{73} +7.26436 q^{75} -0.141643 q^{77} -11.2146 q^{79} +26.5772 q^{81} +3.13912 q^{83} -7.84147 q^{85} +26.1898 q^{87} -1.34275 q^{89} +0.103476 q^{91} -30.0121 q^{93} +17.3719 q^{95} +15.4647 q^{97} +7.64519 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.26349	1.88418	0.942089	−	0.335363i	$-0.108859\pi$
$3$	3.26349	1.88418	0.942089	+	0.335363i	$0.108859\pi$
$4$	0	0
$5$	2.68811	1.20216	0.601080	−	0.799189i	$-0.294737\pi$
$5$	2.68811	1.20216	0.601080	+	0.799189i	$0.294737\pi$
$6$	0	0
$7$	−0.141739	−0.0535725	−0.0267862	−	0.999641i	$-0.508527\pi$
$7$	−0.141739	−0.0535725	−0.0267862	+	0.999641i	$0.508527\pi$
$8$	0	0
$9$	7.65038	2.55013
$10$	0	0
$11$	0.999322	0.301307	0.150654	−	0.988587i	$-0.451862\pi$
$11$	0.999322	0.301307	0.150654	+	0.988587i	$0.451862\pi$
$12$	0	0
$13$	−0.730044	−0.202478	−0.101239	−	0.994862i	$-0.532281\pi$
$13$	−0.730044	−0.202478	−0.101239	+	0.994862i	$0.532281\pi$
$14$	0	0
$15$	8.77263	2.26508
$16$	0	0
$17$	−2.91709	−0.707498	−0.353749	−	0.935340i	$-0.615093\pi$
$17$	−2.91709	−0.707498	−0.353749	+	0.935340i	$0.615093\pi$
$18$	0	0
$19$	6.46250	1.48260	0.741300	−	0.671174i	$-0.234210\pi$
$19$	6.46250	1.48260	0.741300	+	0.671174i	$0.234210\pi$
$20$	0	0
$21$	−0.462566	−0.100940
$22$	0	0
$23$	4.39136	0.915662	0.457831	−	0.889039i	$-0.348626\pi$
$23$	4.39136	0.915662	0.457831	+	0.889039i	$0.348626\pi$
$24$	0	0
$25$	2.22595	0.445190
$26$	0	0
$27$	15.1765	2.92071
$28$	0	0
$29$	8.02510	1.49022	0.745112	−	0.666940i	$-0.232396\pi$
$29$	8.02510	1.49022	0.745112	+	0.666940i	$0.232396\pi$
$30$	0	0
$31$	−9.19633	−1.65171	−0.825855	−	0.563883i	$-0.809307\pi$
$31$	−9.19633	−1.65171	−0.825855	+	0.563883i	$0.809307\pi$
$32$	0	0
$33$	3.26128	0.567716
$34$	0	0
$35$	−0.381012	−0.0644027
$36$	0	0
$37$	−7.19305	−1.18253	−0.591265	−	0.806478i	$-0.701371\pi$
$37$	−7.19305	−1.18253	−0.591265	+	0.806478i	$0.701371\pi$
$38$	0	0
$39$	−2.38249	−0.381504
$40$	0	0
$41$	−1.46896	−0.229413	−0.114706	−	0.993399i	$-0.536593\pi$
$41$	−1.46896	−0.229413	−0.114706	+	0.993399i	$0.536593\pi$
$42$	0	0
$43$	−5.90097	−0.899890	−0.449945	−	0.893056i	$-0.648556\pi$
$43$	−5.90097	−0.899890	−0.449945	+	0.893056i	$0.648556\pi$
$44$	0	0
$45$	20.5651	3.06566
$46$	0	0
$47$	−8.09886	−1.18134	−0.590670	−	0.806913i	$-0.701136\pi$
$47$	−8.09886	−1.18134	−0.590670	+	0.806913i	$0.701136\pi$
$48$	0	0
$49$	−6.97991	−0.997130
$50$	0	0
$51$	−9.51990	−1.33305
$52$	0	0
$53$	−8.64708	−1.18777	−0.593884	−	0.804551i	$-0.702406\pi$
$53$	−8.64708	−1.18777	−0.593884	+	0.804551i	$0.702406\pi$
$54$	0	0
$55$	2.68629	0.362219
$56$	0	0
$57$	21.0903	2.79348
$58$	0	0
$59$	−2.29994	−0.299427	−0.149713	−	0.988729i	$-0.547835\pi$
$59$	−2.29994	−0.299427	−0.149713	+	0.988729i	$0.547835\pi$
$60$	0	0
$61$	−4.13515	−0.529451	−0.264726	−	0.964324i	$-0.585281\pi$
$61$	−4.13515	−0.529451	−0.264726	+	0.964324i	$0.585281\pi$
$62$	0	0
$63$	−1.08436	−0.136617
$64$	0	0
$65$	−1.96244	−0.243411
$66$	0	0
$67$	11.7000	1.42939	0.714693	−	0.699438i	$-0.246566\pi$
$67$	11.7000	1.42939	0.714693	+	0.699438i	$0.246566\pi$
$68$	0	0
$69$	14.3312	1.72527
$70$	0	0
$71$	7.62814	0.905294	0.452647	−	0.891690i	$-0.350480\pi$
$71$	7.62814	0.905294	0.452647	+	0.891690i	$0.350480\pi$
$72$	0	0
$73$	−6.13213	−0.717711	−0.358856	−	0.933393i	$-0.616833\pi$
$73$	−6.13213	−0.717711	−0.358856	+	0.933393i	$0.616833\pi$
$74$	0	0
$75$	7.26436	0.838816
$76$	0	0
$77$	−0.141643	−0.0161418
$78$	0	0
$79$	−11.2146	−1.26174	−0.630870	−	0.775889i	$-0.717302\pi$
$79$	−11.2146	−1.26174	−0.630870	+	0.775889i	$0.717302\pi$
$80$	0	0
$81$	26.5772	2.95302
$82$	0	0
$83$	3.13912	0.344563	0.172282	−	0.985048i	$-0.444886\pi$
$83$	3.13912	0.344563	0.172282	+	0.985048i	$0.444886\pi$
$84$	0	0
$85$	−7.84147	−0.850526
$86$	0	0
$87$	26.1898	2.80785
$88$	0	0
$89$	−1.34275	−0.142332	−0.0711658	−	0.997465i	$-0.522672\pi$
$89$	−1.34275	−0.142332	−0.0711658	+	0.997465i	$0.522672\pi$
$90$	0	0
$91$	0.103476	0.0108472
$92$	0	0
$93$	−30.0121	−3.11211
$94$	0	0
$95$	17.3719	1.78232
$96$	0	0
$97$	15.4647	1.57020	0.785101	−	0.619368i	$-0.212611\pi$
$97$	15.4647	1.57020	0.785101	+	0.619368i	$0.212611\pi$
$98$	0	0
$99$	7.64519	0.768371
$100$	0	0
$101$	−2.85622	−0.284204	−0.142102	−	0.989852i	$-0.545386\pi$
$101$	−2.85622	−0.284204	−0.142102	+	0.989852i	$0.545386\pi$
$102$	0	0
$103$	3.76166	0.370647	0.185324	−	0.982678i	$-0.440667\pi$
$103$	3.76166	0.370647	0.185324	+	0.982678i	$0.440667\pi$
$104$	0	0
$105$	−1.24343	−0.121346
$106$	0	0
$107$	6.41398	0.620063	0.310032	−	0.950726i	$-0.399660\pi$
$107$	6.41398	0.620063	0.310032	+	0.950726i	$0.399660\pi$
$108$	0	0
$109$	−6.09282	−0.583586	−0.291793	−	0.956482i	$-0.594252\pi$
$109$	−6.09282	−0.583586	−0.291793	+	0.956482i	$0.594252\pi$
$110$	0	0
$111$	−23.4744	−2.22810
$112$	0	0
$113$	−16.3765	−1.54057	−0.770285	−	0.637700i	$-0.779886\pi$
$113$	−16.3765	−1.54057	−0.770285	+	0.637700i	$0.779886\pi$
$114$	0	0
$115$	11.8045	1.10077
$116$	0	0
$117$	−5.58511	−0.516344
$118$	0	0
$119$	0.413467	0.0379024
$120$	0	0
$121$	−10.0014	−0.909214
$122$	0	0
$123$	−4.79393	−0.432254
$124$	0	0
$125$	−7.45696	−0.666971
$126$	0	0
$127$	7.95388	0.705793	0.352896	−	0.935662i	$-0.385197\pi$
$127$	7.95388	0.705793	0.352896	+	0.935662i	$0.385197\pi$
$128$	0	0
$129$	−19.2578	−1.69555
$130$	0	0
$131$	14.1778	1.23872	0.619358	−	0.785108i	$-0.287393\pi$
$131$	14.1778	1.23872	0.619358	+	0.785108i	$0.287393\pi$
$132$	0	0
$133$	−0.915992	−0.0794266
$134$	0	0
$135$	40.7961	3.51117
$136$	0	0
$137$	1.76452	0.150754	0.0753768	−	0.997155i	$-0.475984\pi$
$137$	1.76452	0.150754	0.0753768	+	0.997155i	$0.475984\pi$
$138$	0	0
$139$	9.92052	0.841447	0.420724	−	0.907189i	$-0.361776\pi$
$139$	9.92052	0.841447	0.420724	+	0.907189i	$0.361776\pi$
$140$	0	0
$141$	−26.4306	−2.22585
$142$	0	0
$143$	−0.729549	−0.0610079
$144$	0	0
$145$	21.5724	1.79149
$146$	0	0
$147$	−22.7789	−1.87877
$148$	0	0
$149$	18.8055	1.54061	0.770303	−	0.637678i	$-0.220105\pi$
$149$	18.8055	1.54061	0.770303	+	0.637678i	$0.220105\pi$
$150$	0	0
$151$	−1.07075	−0.0871363	−0.0435681	−	0.999050i	$-0.513873\pi$
$151$	−1.07075	−0.0871363	−0.0435681	+	0.999050i	$0.513873\pi$
$152$	0	0
$153$	−22.3168	−1.80421
$154$	0	0
$155$	−24.7208	−1.98562
$156$	0	0
$157$	2.60308	0.207749	0.103874	−	0.994590i	$-0.466876\pi$
$157$	2.60308	0.207749	0.103874	+	0.994590i	$0.466876\pi$
$158$	0	0
$159$	−28.2197	−2.23796
$160$	0	0
$161$	−0.622429	−0.0490543
$162$	0	0
$163$	20.7722	1.62701	0.813504	−	0.581560i	$-0.197557\pi$
$163$	20.7722	1.62701	0.813504	+	0.581560i	$0.197557\pi$
$164$	0	0
$165$	8.76669	0.682486
$166$	0	0
$167$	−16.7203	−1.29385	−0.646926	−	0.762553i	$-0.723946\pi$
$167$	−16.7203	−1.29385	−0.646926	+	0.762553i	$0.723946\pi$
$168$	0	0
$169$	−12.4670	−0.959003
$170$	0	0
$171$	49.4406	3.78082
$172$	0	0
$173$	24.8163	1.88675	0.943374	−	0.331730i	$-0.107632\pi$
$173$	24.8163	1.88675	0.943374	+	0.331730i	$0.107632\pi$
$174$	0	0
$175$	−0.315505	−0.0238499
$176$	0	0
$177$	−7.50584	−0.564173
$178$	0	0
$179$	3.24650	0.242655	0.121328	−	0.992613i	$-0.461285\pi$
$179$	3.24650	0.242655	0.121328	+	0.992613i	$0.461285\pi$
$180$	0	0
$181$	−8.42946	−0.626557	−0.313278	−	0.949661i	$-0.601427\pi$
$181$	−8.42946	−0.626557	−0.313278	+	0.949661i	$0.601427\pi$
$182$	0	0
$183$	−13.4950	−0.997580
$184$	0	0
$185$	−19.3357	−1.42159
$186$	0	0
$187$	−2.91511	−0.213174
$188$	0	0
$189$	−2.15111	−0.156470
$190$	0	0
$191$	10.6092	0.767652	0.383826	−	0.923405i	$-0.374606\pi$
$191$	10.6092	0.767652	0.383826	+	0.923405i	$0.374606\pi$
$192$	0	0
$193$	−11.4173	−0.821834	−0.410917	−	0.911673i	$-0.634791\pi$
$193$	−11.4173	−0.821834	−0.410917	+	0.911673i	$0.634791\pi$
$194$	0	0
$195$	−6.40440	−0.458629
$196$	0	0
$197$	−11.3536	−0.808908	−0.404454	−	0.914558i	$-0.632538\pi$
$197$	−11.3536	−0.808908	−0.404454	+	0.914558i	$0.632538\pi$
$198$	0	0
$199$	−2.69213	−0.190840	−0.0954199	−	0.995437i	$-0.530419\pi$
$199$	−2.69213	−0.190840	−0.0954199	+	0.995437i	$0.530419\pi$
$200$	0	0
$201$	38.1829	2.69322
$202$	0	0
$203$	−1.13747	−0.0798350
$204$	0	0
$205$	−3.94872	−0.275791
$206$	0	0
$207$	33.5956	2.33505
$208$	0	0
$209$	6.45812	0.446718
$210$	0	0
$211$	6.38679	0.439684	0.219842	−	0.975535i	$-0.429446\pi$
$211$	6.38679	0.439684	0.219842	+	0.975535i	$0.429446\pi$
$212$	0	0
$213$	24.8944	1.70573
$214$	0	0
$215$	−15.8625	−1.08181
$216$	0	0
$217$	1.30348	0.0884862
$218$	0	0
$219$	−20.0121	−1.35230
$220$	0	0
$221$	2.12960	0.143253
$222$	0	0
$223$	−22.3311	−1.49540	−0.747700	−	0.664037i	$-0.768842\pi$
$223$	−22.3311	−1.49540	−0.747700	+	0.664037i	$0.768842\pi$
$224$	0	0
$225$	17.0293	1.13529
$226$	0	0
$227$	26.7051	1.77248	0.886238	−	0.463229i	$-0.153309\pi$
$227$	26.7051	1.77248	0.886238	+	0.463229i	$0.153309\pi$
$228$	0	0
$229$	9.50338	0.628001	0.314001	−	0.949423i	$-0.398331\pi$
$229$	9.50338	0.628001	0.314001	+	0.949423i	$0.398331\pi$
$230$	0	0
$231$	−0.462252	−0.0304140
$232$	0	0
$233$	9.89503	0.648245	0.324122	−	0.946015i	$-0.394931\pi$
$233$	9.89503	0.648245	0.324122	+	0.946015i	$0.394931\pi$
$234$	0	0
$235$	−21.7706	−1.42016
$236$	0	0
$237$	−36.5987	−2.37734
$238$	0	0
$239$	1.50064	0.0970683	0.0485341	−	0.998822i	$-0.484545\pi$
$239$	1.50064	0.0970683	0.0485341	+	0.998822i	$0.484545\pi$
$240$	0	0
$241$	19.6428	1.26531	0.632653	−	0.774435i	$-0.281966\pi$
$241$	19.6428	1.26531	0.632653	+	0.774435i	$0.281966\pi$
$242$	0	0
$243$	41.2049	2.64330
$244$	0	0
$245$	−18.7628	−1.19871
$246$	0	0
$247$	−4.71791	−0.300193
$248$	0	0
$249$	10.2445	0.649219
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	4.38839	0.275895
$254$	0	0
$255$	−25.5906	−1.60254
$256$	0	0
$257$	−13.1720	−0.821647	−0.410824	−	0.911715i	$-0.634759\pi$
$257$	−13.1720	−0.821647	−0.410824	+	0.911715i	$0.634759\pi$
$258$	0	0
$259$	1.01954	0.0633510
$260$	0	0
$261$	61.3950	3.80026
$262$	0	0
$263$	−3.63534	−0.224165	−0.112082	−	0.993699i	$-0.535752\pi$
$263$	−3.63534	−0.224165	−0.112082	+	0.993699i	$0.535752\pi$
$264$	0	0
$265$	−23.2443	−1.42789
$266$	0	0
$267$	−4.38206	−0.268178
$268$	0	0
$269$	−19.6854	−1.20024	−0.600121	−	0.799909i	$-0.704881\pi$
$269$	−19.6854	−1.20024	−0.600121	+	0.799909i	$0.704881\pi$
$270$	0	0
$271$	−5.52440	−0.335583	−0.167792	−	0.985822i	$-0.553664\pi$
$271$	−5.52440	−0.335583	−0.167792	+	0.985822i	$0.553664\pi$
$272$	0	0
$273$	0.337693	0.0204381
$274$	0	0
$275$	2.22444	0.134139
$276$	0	0
$277$	−26.2575	−1.57766	−0.788830	−	0.614611i	$-0.789313\pi$
$277$	−26.2575	−1.57766	−0.788830	+	0.614611i	$0.789313\pi$
$278$	0	0
$279$	−70.3554	−4.21207
$280$	0	0
$281$	−17.2616	−1.02974	−0.514870	−	0.857268i	$-0.672160\pi$
$281$	−17.2616	−1.02974	−0.514870	+	0.857268i	$0.672160\pi$
$282$	0	0
$283$	8.94805	0.531907	0.265953	−	0.963986i	$-0.414313\pi$
$283$	8.94805	0.531907	0.265953	+	0.963986i	$0.414313\pi$
$284$	0	0
$285$	56.6932	3.35821
$286$	0	0
$287$	0.208209	0.0122902
$288$	0	0
$289$	−8.49058	−0.499446
$290$	0	0
$291$	50.4689	2.95854
$292$	0	0
$293$	−12.7750	−0.746323	−0.373162	−	0.927766i	$-0.621726\pi$
$293$	−12.7750	−0.746323	−0.373162	+	0.927766i	$0.621726\pi$
$294$	0	0
$295$	−6.18250	−0.359959
$296$	0	0
$297$	15.1662	0.880031
$298$	0	0
$299$	−3.20589	−0.185401
$300$	0	0
$301$	0.836400	0.0482093
$302$	0	0
$303$	−9.32125	−0.535492
$304$	0	0
$305$	−11.1157	−0.636485
$306$	0	0
$307$	21.0357	1.20057	0.600286	−	0.799786i	$-0.295054\pi$
$307$	21.0357	1.20057	0.600286	+	0.799786i	$0.295054\pi$
$308$	0	0
$309$	12.2761	0.698365
$310$	0	0
$311$	−8.26243	−0.468519	−0.234260	−	0.972174i	$-0.575267\pi$
$311$	−8.26243	−0.468519	−0.234260	+	0.972174i	$0.575267\pi$
$312$	0	0
$313$	−0.676425	−0.0382338	−0.0191169	−	0.999817i	$-0.506085\pi$
$313$	−0.676425	−0.0382338	−0.0191169	+	0.999817i	$0.506085\pi$
$314$	0	0
$315$	−2.91488	−0.164235
$316$	0	0
$317$	9.32452	0.523717	0.261859	−	0.965106i	$-0.415665\pi$
$317$	9.32452	0.523717	0.261859	+	0.965106i	$0.415665\pi$
$318$	0	0
$319$	8.01966	0.449015
$320$	0	0
$321$	20.9320	1.16831
$322$	0	0
$323$	−18.8517	−1.04894
$324$	0	0
$325$	−1.62504	−0.0901410
$326$	0	0
$327$	−19.8839	−1.09958
$328$	0	0
$329$	1.14793	0.0632873
$330$	0	0
$331$	−12.8284	−0.705114	−0.352557	−	0.935790i	$-0.614688\pi$
$331$	−12.8284	−0.705114	−0.352557	+	0.935790i	$0.614688\pi$
$332$	0	0
$333$	−55.0295	−3.01560
$334$	0	0
$335$	31.4510	1.71835
$336$	0	0
$337$	−20.8406	−1.13526	−0.567630	−	0.823284i	$-0.692140\pi$
$337$	−20.8406	−1.13526	−0.567630	+	0.823284i	$0.692140\pi$
$338$	0	0
$339$	−53.4445	−2.90271
$340$	0	0
$341$	−9.19009	−0.497672
$342$	0	0
$343$	1.98151	0.106991
$344$	0	0
$345$	38.5238	2.07405
$346$	0	0
$347$	32.2244	1.72990	0.864949	−	0.501860i	$-0.167351\pi$
$347$	32.2244	1.72990	0.864949	+	0.501860i	$0.167351\pi$
$348$	0	0
$349$	−9.37339	−0.501746	−0.250873	−	0.968020i	$-0.580718\pi$
$349$	−9.37339	−0.501746	−0.250873	+	0.968020i	$0.580718\pi$
$350$	0	0
$351$	−11.0795	−0.591379
$352$	0	0
$353$	−23.6273	−1.25755	−0.628777	−	0.777586i	$-0.716444\pi$
$353$	−23.6273	−1.25755	−0.628777	+	0.777586i	$0.716444\pi$
$354$	0	0
$355$	20.5053	1.08831
$356$	0	0
$357$	1.34935	0.0714149
$358$	0	0
$359$	−6.59112	−0.347866	−0.173933	−	0.984757i	$-0.555648\pi$
$359$	−6.59112	−0.347866	−0.173933	+	0.984757i	$0.555648\pi$
$360$	0	0
$361$	22.7640	1.19810
$362$	0	0
$363$	−32.6393	−1.71312
$364$	0	0
$365$	−16.4838	−0.862804
$366$	0	0
$367$	−12.1176	−0.632535	−0.316267	−	0.948670i	$-0.602430\pi$
$367$	−12.1176	−0.632535	−0.316267	+	0.948670i	$0.602430\pi$
$368$	0	0
$369$	−11.2381	−0.585031
$370$	0	0
$371$	1.22563	0.0636316
$372$	0	0
$373$	14.1650	0.733437	0.366718	−	0.930332i	$-0.380481\pi$
$373$	14.1650	0.733437	0.366718	+	0.930332i	$0.380481\pi$
$374$	0	0
$375$	−24.3357	−1.25669
$376$	0	0
$377$	−5.85867	−0.301737
$378$	0	0
$379$	−27.6330	−1.41941	−0.709706	−	0.704498i	$-0.751172\pi$
$379$	−27.6330	−1.41941	−0.709706	+	0.704498i	$0.751172\pi$
$380$	0	0
$381$	25.9574	1.32984
$382$	0	0
$383$	−35.1580	−1.79649	−0.898244	−	0.439497i	$-0.855157\pi$
$383$	−35.1580	−1.79649	−0.898244	+	0.439497i	$0.855157\pi$
$384$	0	0
$385$	−0.380753	−0.0194050
$386$	0	0
$387$	−45.1447	−2.29483
$388$	0	0
$389$	37.1574	1.88396	0.941978	−	0.335674i	$-0.108964\pi$
$389$	37.1574	1.88396	0.941978	+	0.335674i	$0.108964\pi$
$390$	0	0
$391$	−12.8100	−0.647830
$392$	0	0
$393$	46.2690	2.33396
$394$	0	0
$395$	−30.1461	−1.51681
$396$	0	0
$397$	−5.29565	−0.265781	−0.132891	−	0.991131i	$-0.542426\pi$
$397$	−5.29565	−0.265781	−0.132891	+	0.991131i	$0.542426\pi$
$398$	0	0
$399$	−2.98933	−0.149654
$400$	0	0
$401$	−20.7481	−1.03611	−0.518054	−	0.855348i	$-0.673343\pi$
$401$	−20.7481	−1.03611	−0.518054	+	0.855348i	$0.673343\pi$
$402$	0	0
$403$	6.71372	0.334434
$404$	0	0
$405$	71.4424	3.55000
$406$	0	0
$407$	−7.18817	−0.356304
$408$	0	0
$409$	12.5710	0.621598	0.310799	−	0.950476i	$-0.399403\pi$
$409$	12.5710	0.621598	0.310799	+	0.950476i	$0.399403\pi$
$410$	0	0
$411$	5.75851	0.284046
$412$	0	0
$413$	0.325992	0.0160410
$414$	0	0
$415$	8.43831	0.414221
$416$	0	0
$417$	32.3755	1.58544
$418$	0	0
$419$	29.3889	1.43574	0.717870	−	0.696177i	$-0.245117\pi$
$419$	29.3889	1.43574	0.717870	+	0.696177i	$0.245117\pi$
$420$	0	0
$421$	13.7743	0.671317	0.335659	−	0.941984i	$-0.391041\pi$
$421$	13.7743	0.671317	0.335659	+	0.941984i	$0.391041\pi$
$422$	0	0
$423$	−61.9593	−3.01257
$424$	0	0
$425$	−6.49329	−0.314971
$426$	0	0
$427$	0.586114	0.0283640
$428$	0	0
$429$	−2.38088	−0.114950
$430$	0	0
$431$	33.0480	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$431$	33.0480	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$432$	0	0
$433$	30.9465	1.48719	0.743596	−	0.668629i	$-0.233119\pi$
$433$	30.9465	1.48719	0.743596	+	0.668629i	$0.233119\pi$
$434$	0	0
$435$	70.4012	3.37548
$436$	0	0
$437$	28.3792	1.35756
$438$	0	0
$439$	5.44865	0.260050	0.130025	−	0.991511i	$-0.458494\pi$
$439$	5.44865	0.260050	0.130025	+	0.991511i	$0.458494\pi$
$440$	0	0
$441$	−53.3990	−2.54281
$442$	0	0
$443$	−20.2264	−0.960987	−0.480494	−	0.876998i	$-0.659543\pi$
$443$	−20.2264	−0.960987	−0.480494	+	0.876998i	$0.659543\pi$
$444$	0	0
$445$	−3.60947	−0.171105
$446$	0	0
$447$	61.3716	2.90278
$448$	0	0
$449$	19.3331	0.912387	0.456194	−	0.889881i	$-0.349212\pi$
$449$	19.3331	0.912387	0.456194	+	0.889881i	$0.349212\pi$
$450$	0	0
$451$	−1.46796	−0.0691236
$452$	0	0
$453$	−3.49438	−0.164180
$454$	0	0
$455$	0.278155	0.0130401
$456$	0	0
$457$	17.0017	0.795304	0.397652	−	0.917536i	$-0.369825\pi$
$457$	17.0017	0.795304	0.397652	+	0.917536i	$0.369825\pi$
$458$	0	0
$459$	−44.2711	−2.06640
$460$	0	0
$461$	−11.0300	−0.513719	−0.256859	−	0.966449i	$-0.582688\pi$
$461$	−11.0300	−0.513719	−0.256859	+	0.966449i	$0.582688\pi$
$462$	0	0
$463$	−8.98953	−0.417779	−0.208890	−	0.977939i	$-0.566985\pi$
$463$	−8.98953	−0.417779	−0.208890	+	0.977939i	$0.566985\pi$
$464$	0	0
$465$	−80.6760	−3.74126
$466$	0	0
$467$	9.36750	0.433476	0.216738	−	0.976230i	$-0.430458\pi$
$467$	9.36750	0.433476	0.216738	+	0.976230i	$0.430458\pi$
$468$	0	0
$469$	−1.65836	−0.0765757
$470$	0	0
$471$	8.49514	0.391435
$472$	0	0
$473$	−5.89697	−0.271143
$474$	0	0
$475$	14.3852	0.660038
$476$	0	0
$477$	−66.1534	−3.02896
$478$	0	0
$479$	−28.5331	−1.30371	−0.651855	−	0.758343i	$-0.726009\pi$
$479$	−28.5331	−1.30371	−0.651855	+	0.758343i	$0.726009\pi$
$480$	0	0
$481$	5.25124	0.239436
$482$	0	0
$483$	−2.03129	−0.0924270
$484$	0	0
$485$	41.5709	1.88764
$486$	0	0
$487$	−29.4743	−1.33561	−0.667803	−	0.744338i	$-0.732765\pi$
$487$	−29.4743	−1.33561	−0.667803	+	0.744338i	$0.732765\pi$
$488$	0	0
$489$	67.7900	3.06557
$490$	0	0
$491$	17.6118	0.794808	0.397404	−	0.917644i	$-0.369911\pi$
$491$	17.6118	0.794808	0.397404	+	0.917644i	$0.369911\pi$
$492$	0	0
$493$	−23.4099	−1.05433
$494$	0	0
$495$	20.5511	0.923705
$496$	0	0
$497$	−1.08121	−0.0484988
$498$	0	0
$499$	−39.2710	−1.75801	−0.879005	−	0.476812i	$-0.841792\pi$
$499$	−39.2710	−1.75801	−0.879005	+	0.476812i	$0.841792\pi$
$500$	0	0
$501$	−54.5664	−2.43785
$502$	0	0
$503$	11.9101	0.531045	0.265522	−	0.964105i	$-0.414456\pi$
$503$	11.9101	0.531045	0.265522	+	0.964105i	$0.414456\pi$
$504$	0	0
$505$	−7.67784	−0.341659
$506$	0	0
$507$	−40.6861	−1.80693
$508$	0	0
$509$	38.1660	1.69168	0.845839	−	0.533438i	$-0.179100\pi$
$509$	38.1660	1.69168	0.845839	+	0.533438i	$0.179100\pi$
$510$	0	0
$511$	0.869164	0.0384496
$512$	0	0
$513$	98.0780	4.33025
$514$	0	0
$515$	10.1118	0.445577
$516$	0	0
$517$	−8.09337	−0.355946
$518$	0	0
$519$	80.9878	3.55497
$520$	0	0
$521$	−13.5735	−0.594666	−0.297333	−	0.954774i	$-0.596097\pi$
$521$	−13.5735	−0.594666	−0.297333	+	0.954774i	$0.596097\pi$
$522$	0	0
$523$	3.54049	0.154815	0.0774075	−	0.997000i	$-0.475336\pi$
$523$	3.54049	0.154815	0.0774075	+	0.997000i	$0.475336\pi$
$524$	0	0
$525$	−1.02965	−0.0449375
$526$	0	0
$527$	26.8265	1.16858
$528$	0	0
$529$	−3.71594	−0.161562
$530$	0	0
$531$	−17.5954	−0.763576
$532$	0	0
$533$	1.07240	0.0464509
$534$	0	0
$535$	17.2415	0.745415
$536$	0	0
$537$	10.5949	0.457205
$538$	0	0
$539$	−6.97518	−0.300442
$540$	0	0
$541$	39.5113	1.69873	0.849363	−	0.527810i	$-0.176987\pi$
$541$	39.5113	1.69873	0.849363	+	0.527810i	$0.176987\pi$
$542$	0	0
$543$	−27.5095	−1.18054
$544$	0	0
$545$	−16.3782	−0.701564
$546$	0	0
$547$	−13.5829	−0.580763	−0.290381	−	0.956911i	$-0.593782\pi$
$547$	−13.5829	−0.580763	−0.290381	+	0.956911i	$0.593782\pi$
$548$	0	0
$549$	−31.6354	−1.35017
$550$	0	0
$551$	51.8622	2.20940
$552$	0	0
$553$	1.58955	0.0675945
$554$	0	0
$555$	−63.1019	−2.67853
$556$	0	0
$557$	−37.7913	−1.60127	−0.800635	−	0.599153i	$-0.795504\pi$
$557$	−37.7913	−1.60127	−0.800635	+	0.599153i	$0.795504\pi$
$558$	0	0
$559$	4.30797	0.182208
$560$	0	0
$561$	−9.51345	−0.401658
$562$	0	0
$563$	−30.9046	−1.30248	−0.651238	−	0.758874i	$-0.725750\pi$
$563$	−30.9046	−1.30248	−0.651238	+	0.758874i	$0.725750\pi$
$564$	0	0
$565$	−44.0218	−1.85201
$566$	0	0
$567$	−3.76703	−0.158200
$568$	0	0
$569$	14.8670	0.623257	0.311628	−	0.950204i	$-0.399126\pi$
$569$	14.8670	0.623257	0.311628	+	0.950204i	$0.399126\pi$
$570$	0	0
$571$	−23.4695	−0.982168	−0.491084	−	0.871112i	$-0.663399\pi$
$571$	−23.4695	−0.982168	−0.491084	+	0.871112i	$0.663399\pi$
$572$	0	0
$573$	34.6229	1.44639
$574$	0	0
$575$	9.77494	0.407643
$576$	0	0
$577$	31.3771	1.30625	0.653123	−	0.757252i	$-0.273458\pi$
$577$	31.3771	1.30625	0.653123	+	0.757252i	$0.273458\pi$
$578$	0	0
$579$	−37.2602	−1.54848
$580$	0	0
$581$	−0.444938	−0.0184591
$582$	0	0
$583$	−8.64122	−0.357883
$584$	0	0
$585$	−15.0134	−0.620728
$586$	0	0
$587$	1.74076	0.0718490	0.0359245	−	0.999355i	$-0.488562\pi$
$587$	1.74076	0.0718490	0.0359245	+	0.999355i	$0.488562\pi$
$588$	0	0
$589$	−59.4313	−2.44882
$590$	0	0
$591$	−37.0523	−1.52413
$592$	0	0
$593$	−35.3470	−1.45153	−0.725764	−	0.687944i	$-0.758514\pi$
$593$	−35.3470	−1.45153	−0.725764	+	0.687944i	$0.758514\pi$
$594$	0	0
$595$	1.11145	0.0455648
$596$	0	0
$597$	−8.78573	−0.359576
$598$	0	0
$599$	46.9478	1.91823	0.959117	−	0.283010i	$-0.0913328\pi$
$599$	46.9478	1.91823	0.959117	+	0.283010i	$0.0913328\pi$
$600$	0	0
$601$	−15.3078	−0.624417	−0.312208	−	0.950014i	$-0.601069\pi$
$601$	−15.3078	−0.624417	−0.312208	+	0.950014i	$0.601069\pi$
$602$	0	0
$603$	89.5096	3.64511
$604$	0	0
$605$	−26.8848	−1.09302
$606$	0	0
$607$	2.39471	0.0971981	0.0485991	−	0.998818i	$-0.484524\pi$
$607$	2.39471	0.0971981	0.0485991	+	0.998818i	$0.484524\pi$
$608$	0	0
$609$	−3.71213	−0.150423
$610$	0	0
$611$	5.91252	0.239195
$612$	0	0
$613$	−8.16630	−0.329834	−0.164917	−	0.986307i	$-0.552736\pi$
$613$	−8.16630	−0.329834	−0.164917	+	0.986307i	$0.552736\pi$
$614$	0	0
$615$	−12.8866	−0.519639
$616$	0	0
$617$	−39.2346	−1.57953	−0.789763	−	0.613413i	$-0.789796\pi$
$617$	−39.2346	−1.57953	−0.789763	+	0.613413i	$0.789796\pi$
$618$	0	0
$619$	−4.83725	−0.194425	−0.0972127	−	0.995264i	$-0.530993\pi$
$619$	−4.83725	−0.194425	−0.0972127	+	0.995264i	$0.530993\pi$
$620$	0	0
$621$	66.6454	2.67439
$622$	0	0
$623$	0.190321	0.00762505
$624$	0	0
$625$	−31.1749	−1.24700
$626$	0	0
$627$	21.0760	0.841696
$628$	0	0
$629$	20.9828	0.836638
$630$	0	0
$631$	19.1069	0.760634	0.380317	−	0.924856i	$-0.375815\pi$
$631$	19.1069	0.760634	0.380317	+	0.924856i	$0.375815\pi$
$632$	0	0
$633$	20.8432	0.828444
$634$	0	0
$635$	21.3809	0.848476
$636$	0	0
$637$	5.09564	0.201897
$638$	0	0
$639$	58.3582	2.30861
$640$	0	0
$641$	21.5542	0.851342	0.425671	−	0.904878i	$-0.360038\pi$
$641$	21.5542	0.851342	0.425671	+	0.904878i	$0.360038\pi$
$642$	0	0
$643$	−8.17855	−0.322530	−0.161265	−	0.986911i	$-0.551557\pi$
$643$	−8.17855	−0.322530	−0.161265	+	0.986911i	$0.551557\pi$
$644$	0	0
$645$	−51.7670	−2.03833
$646$	0	0
$647$	16.1801	0.636104	0.318052	−	0.948073i	$-0.396971\pi$
$647$	16.1801	0.636104	0.318052	+	0.948073i	$0.396971\pi$
$648$	0	0
$649$	−2.29838	−0.0902194
$650$	0	0
$651$	4.25390	0.166724
$652$	0	0
$653$	−18.7787	−0.734867	−0.367433	−	0.930050i	$-0.619763\pi$
$653$	−18.7787	−0.734867	−0.367433	+	0.930050i	$0.619763\pi$
$654$	0	0
$655$	38.1114	1.48914
$656$	0	0
$657$	−46.9131	−1.83025
$658$	0	0
$659$	24.5823	0.957589	0.478795	−	0.877927i	$-0.341074\pi$
$659$	24.5823	0.957589	0.478795	+	0.877927i	$0.341074\pi$
$660$	0	0
$661$	18.7425	0.728999	0.364499	−	0.931204i	$-0.381240\pi$
$661$	18.7425	0.728999	0.364499	+	0.931204i	$0.381240\pi$
$662$	0	0
$663$	6.94994	0.269913
$664$	0	0
$665$	−2.46229	−0.0954835
$666$	0	0
$667$	35.2411	1.36454
$668$	0	0
$669$	−72.8773	−2.81760
$670$	0	0
$671$	−4.13234	−0.159527
$672$	0	0
$673$	27.6082	1.06422	0.532110	−	0.846675i	$-0.321399\pi$
$673$	27.6082	1.06422	0.532110	+	0.846675i	$0.321399\pi$
$674$	0	0
$675$	33.7820	1.30027
$676$	0	0
$677$	14.0727	0.540859	0.270430	−	0.962740i	$-0.412834\pi$
$677$	14.0727	0.540859	0.270430	+	0.962740i	$0.412834\pi$
$678$	0	0
$679$	−2.19196	−0.0841196
$680$	0	0
$681$	87.1517	3.33966
$682$	0	0
$683$	10.5430	0.403415	0.201707	−	0.979446i	$-0.435351\pi$
$683$	10.5430	0.403415	0.201707	+	0.979446i	$0.435351\pi$
$684$	0	0
$685$	4.74324	0.181230
$686$	0	0
$687$	31.0142	1.18327
$688$	0	0
$689$	6.31274	0.240496
$690$	0	0
$691$	0.448588	0.0170651	0.00853255	−	0.999964i	$-0.497284\pi$
$691$	0.448588	0.0170651	0.00853255	+	0.999964i	$0.497284\pi$
$692$	0	0
$693$	−1.08363	−0.0411635
$694$	0	0
$695$	26.6675	1.01155
$696$	0	0
$697$	4.28508	0.162309
$698$	0	0
$699$	32.2923	1.22141
$700$	0	0
$701$	−16.2324	−0.613088	−0.306544	−	0.951857i	$-0.599173\pi$
$701$	−16.2324	−0.613088	−0.306544	+	0.951857i	$0.599173\pi$
$702$	0	0
$703$	−46.4851	−1.75322
$704$	0	0
$705$	−71.0483	−2.67583
$706$	0	0
$707$	0.404839	0.0152255
$708$	0	0
$709$	46.7851	1.75705	0.878525	−	0.477696i	$-0.158528\pi$
$709$	46.7851	1.75705	0.878525	+	0.477696i	$0.158528\pi$
$710$	0	0
$711$	−85.7958	−3.21759
$712$	0	0
$713$	−40.3844	−1.51241
$714$	0	0
$715$	−1.96111	−0.0733413
$716$	0	0
$717$	4.89732	0.182894
$718$	0	0
$719$	−23.3436	−0.870568	−0.435284	−	0.900293i	$-0.643352\pi$
$719$	−23.3436	−0.870568	−0.435284	+	0.900293i	$0.643352\pi$
$720$	0	0
$721$	−0.533175	−0.0198565
$722$	0	0
$723$	64.1042	2.38406
$724$	0	0
$725$	17.8634	0.663432
$726$	0	0
$727$	20.9666	0.777607	0.388803	−	0.921321i	$-0.372889\pi$
$727$	20.9666	0.777607	0.388803	+	0.921321i	$0.372889\pi$
$728$	0	0
$729$	54.7404	2.02742
$730$	0	0
$731$	17.2137	0.636670
$732$	0	0
$733$	3.06303	0.113136	0.0565678	−	0.998399i	$-0.481984\pi$
$733$	3.06303	0.113136	0.0565678	+	0.998399i	$0.481984\pi$
$734$	0	0
$735$	−61.2322	−2.25858
$736$	0	0
$737$	11.6921	0.430684
$738$	0	0
$739$	36.8762	1.35651	0.678256	−	0.734826i	$-0.262736\pi$
$739$	36.8762	1.35651	0.678256	+	0.734826i	$0.262736\pi$
$740$	0	0
$741$	−15.3969	−0.565618
$742$	0	0
$743$	−2.91277	−0.106859	−0.0534296	−	0.998572i	$-0.517015\pi$
$743$	−2.91277	−0.106859	−0.0534296	+	0.998572i	$0.517015\pi$
$744$	0	0
$745$	50.5513	1.85206
$746$	0	0
$747$	24.0155	0.878680
$748$	0	0
$749$	−0.909115	−0.0332183
$750$	0	0
$751$	−25.1532	−0.917852	−0.458926	−	0.888475i	$-0.651766\pi$
$751$	−25.1532	−0.917852	−0.458926	+	0.888475i	$0.651766\pi$
$752$	0	0
$753$	3.26349	0.118928
$754$	0	0
$755$	−2.87829	−0.104752
$756$	0	0
$757$	−26.4798	−0.962423	−0.481212	−	0.876604i	$-0.659803\pi$
$757$	−26.4798	−0.962423	−0.481212	+	0.876604i	$0.659803\pi$
$758$	0	0
$759$	14.3215	0.519836
$760$	0	0
$761$	46.7993	1.69647	0.848237	−	0.529617i	$-0.177664\pi$
$761$	46.7993	1.69647	0.848237	+	0.529617i	$0.177664\pi$
$762$	0	0
$763$	0.863592	0.0312641
$764$	0	0
$765$	−59.9902	−2.16895
$766$	0	0
$767$	1.67906	0.0606272
$768$	0	0
$769$	48.1347	1.73578	0.867890	−	0.496756i	$-0.165475\pi$
$769$	48.1347	1.73578	0.867890	+	0.496756i	$0.165475\pi$
$770$	0	0
$771$	−42.9868	−1.54813
$772$	0	0
$773$	−6.27208	−0.225591	−0.112795	−	0.993618i	$-0.535980\pi$
$773$	−6.27208	−0.225591	−0.112795	+	0.993618i	$0.535980\pi$
$774$	0	0
$775$	−20.4705	−0.735324
$776$	0	0
$777$	3.32726	0.119365
$778$	0	0
$779$	−9.49314	−0.340127
$780$	0	0
$781$	7.62297	0.272771
$782$	0	0
$783$	121.793	4.35251
$784$	0	0
$785$	6.99738	0.249747
$786$	0	0
$787$	−41.7252	−1.48734	−0.743671	−	0.668546i	$-0.766917\pi$
$787$	−41.7252	−1.48734	−0.743671	+	0.668546i	$0.766917\pi$
$788$	0	0
$789$	−11.8639	−0.422366
$790$	0	0
$791$	2.32119	0.0825321
$792$	0	0
$793$	3.01884	0.107202
$794$	0	0
$795$	−75.8576	−2.69039
$796$	0	0
$797$	−23.4451	−0.830469	−0.415234	−	0.909714i	$-0.636300\pi$
$797$	−23.4451	−0.830469	−0.415234	+	0.909714i	$0.636300\pi$
$798$	0	0
$799$	23.6251	0.835796
$800$	0	0
$801$	−10.2726	−0.362963
$802$	0	0
$803$	−6.12797	−0.216251
$804$	0	0
$805$	−1.67316	−0.0589711
$806$	0	0
$807$	−64.2433	−2.26147
$808$	0	0
$809$	26.3071	0.924908	0.462454	−	0.886643i	$-0.346969\pi$
$809$	26.3071	0.924908	0.462454	+	0.886643i	$0.346969\pi$
$810$	0	0
$811$	50.4543	1.77169	0.885845	−	0.463981i	$-0.153579\pi$
$811$	50.4543	1.77169	0.885845	+	0.463981i	$0.153579\pi$
$812$	0	0
$813$	−18.0288	−0.632298
$814$	0	0
$815$	55.8381	1.95592
$816$	0	0
$817$	−38.1350	−1.33418
$818$	0	0
$819$	0.791631	0.0276618
$820$	0	0
$821$	−5.33102	−0.186054	−0.0930269	−	0.995664i	$-0.529654\pi$
$821$	−5.33102	−0.186054	−0.0930269	+	0.995664i	$0.529654\pi$
$822$	0	0
$823$	29.6751	1.03441	0.517204	−	0.855862i	$-0.326973\pi$
$823$	29.6751	1.03441	0.517204	+	0.855862i	$0.326973\pi$
$824$	0	0
$825$	7.25944	0.252741
$826$	0	0
$827$	−20.4713	−0.711857	−0.355929	−	0.934513i	$-0.615835\pi$
$827$	−20.4713	−0.711857	−0.355929	+	0.934513i	$0.615835\pi$
$828$	0	0
$829$	−26.0612	−0.905144	−0.452572	−	0.891728i	$-0.649493\pi$
$829$	−26.0612	−0.905144	−0.452572	+	0.891728i	$0.649493\pi$
$830$	0	0
$831$	−85.6912	−2.97259
$832$	0	0
$833$	20.3610	0.705468
$834$	0	0
$835$	−44.9459	−1.55542
$836$	0	0
$837$	−139.568	−4.82417
$838$	0	0
$839$	−22.5639	−0.778993	−0.389496	−	0.921028i	$-0.627351\pi$
$839$	−22.5639	−0.778993	−0.389496	+	0.921028i	$0.627351\pi$
$840$	0	0
$841$	35.4022	1.22077
$842$	0	0
$843$	−56.3330	−1.94021
$844$	0	0
$845$	−33.5128	−1.15288
$846$	0	0
$847$	1.41759	0.0487089
$848$	0	0
$849$	29.2019	1.00221
$850$	0	0
$851$	−31.5873	−1.08280
$852$	0	0
$853$	−20.9292	−0.716602	−0.358301	−	0.933606i	$-0.616644\pi$
$853$	−20.9292	−0.716602	−0.358301	+	0.933606i	$0.616644\pi$
$854$	0	0
$855$	132.902	4.54515
$856$	0	0
$857$	−5.41194	−0.184868	−0.0924341	−	0.995719i	$-0.529465\pi$
$857$	−5.41194	−0.184868	−0.0924341	+	0.995719i	$0.529465\pi$
$858$	0	0
$859$	−45.0244	−1.53621	−0.768106	−	0.640322i	$-0.778801\pi$
$859$	−45.0244	−1.53621	−0.768106	+	0.640322i	$0.778801\pi$
$860$	0	0
$861$	0.679489	0.0231569
$862$	0	0
$863$	−51.3578	−1.74824	−0.874119	−	0.485711i	$-0.838561\pi$
$863$	−51.3578	−1.74824	−0.874119	+	0.485711i	$0.838561\pi$
$864$	0	0
$865$	66.7090	2.26817
$866$	0	0
$867$	−27.7090	−0.941045
$868$	0	0
$869$	−11.2070	−0.380171
$870$	0	0
$871$	−8.54153	−0.289419
$872$	0	0
$873$	118.311	4.00421
$874$	0	0
$875$	1.05695	0.0357313
$876$	0	0
$877$	55.3245	1.86818	0.934089	−	0.357041i	$-0.116214\pi$
$877$	55.3245	1.86818	0.934089	+	0.357041i	$0.116214\pi$
$878$	0	0
$879$	−41.6911	−1.40621
$880$	0	0
$881$	−33.9256	−1.14298	−0.571491	−	0.820609i	$-0.693635\pi$
$881$	−33.9256	−1.14298	−0.571491	+	0.820609i	$0.693635\pi$
$882$	0	0
$883$	2.84662	0.0957965	0.0478982	−	0.998852i	$-0.484748\pi$
$883$	2.84662	0.0957965	0.0478982	+	0.998852i	$0.484748\pi$
$884$	0	0
$885$	−20.1765	−0.678227
$886$	0	0
$887$	−43.7142	−1.46778	−0.733889	−	0.679270i	$-0.762297\pi$
$887$	−43.7142	−1.46778	−0.733889	+	0.679270i	$0.762297\pi$
$888$	0	0
$889$	−1.12738	−0.0378111
$890$	0	0
$891$	26.5591	0.889765
$892$	0	0
$893$	−52.3389	−1.75145
$894$	0	0
$895$	8.72697	0.291710
$896$	0	0
$897$	−10.4624	−0.349329
$898$	0	0
$899$	−73.8014	−2.46142
$900$	0	0
$901$	25.2243	0.840343
$902$	0	0
$903$	2.72959	0.0908349
$904$	0	0
$905$	−22.6593	−0.753222
$906$	0	0
$907$	9.86517	0.327568	0.163784	−	0.986496i	$-0.447630\pi$
$907$	9.86517	0.327568	0.163784	+	0.986496i	$0.447630\pi$
$908$	0	0
$909$	−21.8512	−0.724757
$910$	0	0
$911$	21.1874	0.701971	0.350986	−	0.936381i	$-0.385847\pi$
$911$	21.1874	0.701971	0.350986	+	0.936381i	$0.385847\pi$
$912$	0	0
$913$	3.13700	0.103819
$914$	0	0
$915$	−36.2761	−1.19925
$916$	0	0
$917$	−2.00955	−0.0663611
$918$	0	0
$919$	−16.6150	−0.548077	−0.274038	−	0.961719i	$-0.588360\pi$
$919$	−16.6150	−0.548077	−0.274038	+	0.961719i	$0.588360\pi$
$920$	0	0
$921$	68.6499	2.26209
$922$	0	0
$923$	−5.56888	−0.183302
$924$	0	0
$925$	−16.0113	−0.526450
$926$	0	0
$927$	28.7781	0.945197
$928$	0	0
$929$	−24.9857	−0.819755	−0.409877	−	0.912141i	$-0.634429\pi$
$929$	−24.9857	−0.819755	−0.409877	+	0.912141i	$0.634429\pi$
$930$	0	0
$931$	−45.1077	−1.47834
$932$	0	0
$933$	−26.9644	−0.882774
$934$	0	0
$935$	−7.83615	−0.256270
$936$	0	0
$937$	−6.31358	−0.206256	−0.103128	−	0.994668i	$-0.532885\pi$
$937$	−6.31358	−0.206256	−0.103128	+	0.994668i	$0.532885\pi$
$938$	0	0
$939$	−2.20751	−0.0720393
$940$	0	0
$941$	−34.2847	−1.11765	−0.558825	−	0.829286i	$-0.688748\pi$
$941$	−34.2847	−1.11765	−0.558825	+	0.829286i	$0.688748\pi$
$942$	0	0
$943$	−6.45072	−0.210064
$944$	0	0
$945$	−5.78241	−0.188102
$946$	0	0
$947$	47.8828	1.55598	0.777990	−	0.628276i	$-0.216239\pi$
$947$	47.8828	1.55598	0.777990	+	0.628276i	$0.216239\pi$
$948$	0	0
$949$	4.47672	0.145320
$950$	0	0
$951$	30.4305	0.986777
$952$	0	0
$953$	−1.49143	−0.0483121	−0.0241561	−	0.999708i	$-0.507690\pi$
$953$	−1.49143	−0.0483121	−0.0241561	+	0.999708i	$0.507690\pi$
$954$	0	0
$955$	28.5186	0.922841
$956$	0	0
$957$	26.1721	0.846024
$958$	0	0
$959$	−0.250103	−0.00807624
$960$	0	0
$961$	53.5724	1.72814
$962$	0	0
$963$	49.0694	1.58124
$964$	0	0
$965$	−30.6909	−0.987976
$966$	0	0
$967$	−22.4586	−0.722218	−0.361109	−	0.932524i	$-0.617602\pi$
$967$	−22.4586	−0.722218	−0.361109	+	0.932524i	$0.617602\pi$
$968$	0	0
$969$	−61.5224	−1.97638
$970$	0	0
$971$	−24.8941	−0.798889	−0.399444	−	0.916757i	$-0.630797\pi$
$971$	−24.8941	−0.798889	−0.399444	+	0.916757i	$0.630797\pi$
$972$	0	0
$973$	−1.40613	−0.0450784
$974$	0	0
$975$	−5.30330	−0.169842
$976$	0	0
$977$	−59.0889	−1.89042	−0.945210	−	0.326463i	$-0.894143\pi$
$977$	−59.0889	−1.89042	−0.945210	+	0.326463i	$0.894143\pi$
$978$	0	0
$979$	−1.34184	−0.0428855
$980$	0	0
$981$	−46.6123	−1.48822
$982$	0	0
$983$	−53.2091	−1.69711	−0.848553	−	0.529111i	$-0.822526\pi$
$983$	−53.2091	−1.69711	−0.848553	+	0.529111i	$0.822526\pi$
$984$	0	0
$985$	−30.5196	−0.972437
$986$	0	0
$987$	3.74625	0.119245
$988$	0	0
$989$	−25.9133	−0.823995
$990$	0	0
$991$	−15.7360	−0.499870	−0.249935	−	0.968263i	$-0.580409\pi$
$991$	−15.7360	−0.499870	−0.249935	+	0.968263i	$0.580409\pi$
$992$	0	0
$993$	−41.8655	−1.32856
$994$	0	0
$995$	−7.23674	−0.229420
$996$	0	0
$997$	60.0706	1.90246	0.951228	−	0.308490i	$-0.0998235\pi$
$997$	60.0706	1.90246	0.951228	+	0.308490i	$0.0998235\pi$
$998$	0	0
$999$	−109.165	−3.45383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.19		19
4.3	odd	2		2008.2.a.c.1.1	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.1	✓	19	4.3	odd	2
4016.2.a.l.1.19		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)

\(f(q)\)	\(=\)	\(q+3.26349 q^{3} +2.68811 q^{5} -0.141739 q^{7} +7.65038 q^{9} +O(q^{10})\)	\(q+3.26349 q^{3} +2.68811 q^{5} -0.141739 q^{7} +7.65038 q^{9} +0.999322 q^{11} -0.730044 q^{13} +8.77263 q^{15} -2.91709 q^{17} +6.46250 q^{19} -0.462566 q^{21} +4.39136 q^{23} +2.22595 q^{25} +15.1765 q^{27} +8.02510 q^{29} -9.19633 q^{31} +3.26128 q^{33} -0.381012 q^{35} -7.19305 q^{37} -2.38249 q^{39} -1.46896 q^{41} -5.90097 q^{43} +20.5651 q^{45} -8.09886 q^{47} -6.97991 q^{49} -9.51990 q^{51} -8.64708 q^{53} +2.68629 q^{55} +21.0903 q^{57} -2.29994 q^{59} -4.13515 q^{61} -1.08436 q^{63} -1.96244 q^{65} +11.7000 q^{67} +14.3312 q^{69} +7.62814 q^{71} -6.13213 q^{73} +7.26436 q^{75} -0.141643 q^{77} -11.2146 q^{79} +26.5772 q^{81} +3.13912 q^{83} -7.84147 q^{85} +26.1898 q^{87} -1.34275 q^{89} +0.103476 q^{91} -30.0121 q^{93} +17.3719 q^{95} +15.4647 q^{97} +7.64519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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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