LMFDB - Embedded newform 4016.2.a.l.1.15

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			1.15
Root			$2.43794$ 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)

$f(q)$	$=$	$q+2.43794 q^{3} -3.80787 q^{5} -4.24792 q^{7} +2.94357 q^{9} +O(q^{10})$	$q+2.43794 q^{3} -3.80787 q^{5} -4.24792 q^{7} +2.94357 q^{9} +0.641638 q^{11} -6.42799 q^{13} -9.28338 q^{15} +3.87787 q^{17} -4.45162 q^{19} -10.3562 q^{21} +8.73456 q^{23} +9.49989 q^{25} -0.137572 q^{27} +6.79913 q^{29} -8.39275 q^{31} +1.56428 q^{33} +16.1755 q^{35} +0.0277760 q^{37} -15.6711 q^{39} +5.21393 q^{41} -2.29135 q^{43} -11.2087 q^{45} +10.7373 q^{47} +11.0448 q^{49} +9.45402 q^{51} -10.4461 q^{53} -2.44327 q^{55} -10.8528 q^{57} +2.18891 q^{59} +6.47425 q^{61} -12.5041 q^{63} +24.4770 q^{65} +1.11039 q^{67} +21.2944 q^{69} +16.1635 q^{71} +5.02776 q^{73} +23.1602 q^{75} -2.72563 q^{77} +17.4103 q^{79} -9.16610 q^{81} -11.7246 q^{83} -14.7664 q^{85} +16.5759 q^{87} +5.47066 q^{89} +27.3056 q^{91} -20.4611 q^{93} +16.9512 q^{95} -17.9181 q^{97} +1.88871 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.43794	1.40755	0.703774	−	0.710424i	$-0.251497\pi$
$3$	2.43794	1.40755	0.703774	+	0.710424i	$0.251497\pi$
$4$	0	0
$5$	−3.80787	−1.70293	−0.851466	−	0.524410i	$-0.824286\pi$
$5$	−3.80787	−1.70293	−0.851466	+	0.524410i	$0.824286\pi$
$6$	0	0
$7$	−4.24792	−1.60556	−0.802782	−	0.596273i	$-0.796648\pi$
$7$	−4.24792	−1.60556	−0.802782	+	0.596273i	$0.796648\pi$
$8$	0	0
$9$	2.94357	0.981190
$10$	0	0
$11$	0.641638	0.193461	0.0967305	−	0.995311i	$-0.469161\pi$
$11$	0.641638	0.193461	0.0967305	+	0.995311i	$0.469161\pi$
$12$	0	0
$13$	−6.42799	−1.78280	−0.891402	−	0.453214i	$-0.850277\pi$
$13$	−6.42799	−1.78280	−0.891402	+	0.453214i	$0.850277\pi$
$14$	0	0
$15$	−9.28338	−2.39696
$16$	0	0
$17$	3.87787	0.940521	0.470260	−	0.882528i	$-0.344160\pi$
$17$	3.87787	0.940521	0.470260	+	0.882528i	$0.344160\pi$
$18$	0	0
$19$	−4.45162	−1.02127	−0.510636	−	0.859797i	$-0.670590\pi$
$19$	−4.45162	−1.02127	−0.510636	+	0.859797i	$0.670590\pi$
$20$	0	0
$21$	−10.3562	−2.25991
$22$	0	0
$23$	8.73456	1.82128	0.910640	−	0.413200i	$-0.135589\pi$
$23$	8.73456	1.82128	0.910640	+	0.413200i	$0.135589\pi$
$24$	0	0
$25$	9.49989	1.89998
$26$	0	0
$27$	−0.137572	−0.0264757
$28$	0	0
$29$	6.79913	1.26257	0.631283	−	0.775552i	$-0.282529\pi$
$29$	6.79913	1.26257	0.631283	+	0.775552i	$0.282529\pi$
$30$	0	0
$31$	−8.39275	−1.50738	−0.753691	−	0.657228i	$-0.771729\pi$
$31$	−8.39275	−1.50738	−0.753691	+	0.657228i	$0.771729\pi$
$32$	0	0
$33$	1.56428	0.272306
$34$	0	0
$35$	16.1755	2.73417
$36$	0	0
$37$	0.0277760	0.00456634	0.00228317	−	0.999997i	$-0.499273\pi$
$37$	0.0277760	0.00456634	0.00228317	+	0.999997i	$0.499273\pi$
$38$	0	0
$39$	−15.6711	−2.50938
$40$	0	0
$41$	5.21393	0.814279	0.407140	−	0.913366i	$-0.366526\pi$
$41$	5.21393	0.814279	0.407140	+	0.913366i	$0.366526\pi$
$42$	0	0
$43$	−2.29135	−0.349427	−0.174713	−	0.984619i	$-0.555900\pi$
$43$	−2.29135	−0.349427	−0.174713	+	0.984619i	$0.555900\pi$
$44$	0	0
$45$	−11.2087	−1.67090
$46$	0	0
$47$	10.7373	1.56619	0.783097	−	0.621899i	$-0.213639\pi$
$47$	10.7373	1.56619	0.783097	+	0.621899i	$0.213639\pi$
$48$	0	0
$49$	11.0448	1.57783
$50$	0	0
$51$	9.45402	1.32383
$52$	0	0
$53$	−10.4461	−1.43489	−0.717444	−	0.696616i	$-0.754688\pi$
$53$	−10.4461	−1.43489	−0.717444	+	0.696616i	$0.754688\pi$
$54$	0	0
$55$	−2.44327	−0.329451
$56$	0	0
$57$	−10.8528	−1.43749
$58$	0	0
$59$	2.18891	0.284972	0.142486	−	0.989797i	$-0.454490\pi$
$59$	2.18891	0.284972	0.142486	+	0.989797i	$0.454490\pi$
$60$	0	0
$61$	6.47425	0.828943	0.414471	−	0.910062i	$-0.363966\pi$
$61$	6.47425	0.828943	0.414471	+	0.910062i	$0.363966\pi$
$62$	0	0
$63$	−12.5041	−1.57536
$64$	0	0
$65$	24.4770	3.03599
$66$	0	0
$67$	1.11039	0.135656	0.0678279	−	0.997697i	$-0.478393\pi$
$67$	1.11039	0.135656	0.0678279	+	0.997697i	$0.478393\pi$
$68$	0	0
$69$	21.2944	2.56354
$70$	0	0
$71$	16.1635	1.91825	0.959126	−	0.282980i	$-0.0913231\pi$
$71$	16.1635	1.91825	0.959126	+	0.282980i	$0.0913231\pi$
$72$	0	0
$73$	5.02776	0.588455	0.294227	−	0.955735i	$-0.404938\pi$
$73$	5.02776	0.588455	0.294227	+	0.955735i	$0.404938\pi$
$74$	0	0
$75$	23.1602	2.67431
$76$	0	0
$77$	−2.72563	−0.310614
$78$	0	0
$79$	17.4103	1.95881	0.979405	−	0.201904i	$-0.0647128\pi$
$79$	17.4103	1.95881	0.979405	+	0.201904i	$0.0647128\pi$
$80$	0	0
$81$	−9.16610	−1.01846
$82$	0	0
$83$	−11.7246	−1.28695	−0.643473	−	0.765469i	$-0.722507\pi$
$83$	−11.7246	−1.28695	−0.643473	+	0.765469i	$0.722507\pi$
$84$	0	0
$85$	−14.7664	−1.60164
$86$	0	0
$87$	16.5759	1.77712
$88$	0	0
$89$	5.47066	0.579889	0.289944	−	0.957043i	$-0.406363\pi$
$89$	5.47066	0.579889	0.289944	+	0.957043i	$0.406363\pi$
$90$	0	0
$91$	27.3056	2.86240
$92$	0	0
$93$	−20.4611	−2.12171
$94$	0	0
$95$	16.9512	1.73916
$96$	0	0
$97$	−17.9181	−1.81931	−0.909654	−	0.415366i	$-0.863653\pi$
$97$	−17.9181	−1.81931	−0.909654	+	0.415366i	$0.863653\pi$
$98$	0	0
$99$	1.88871	0.189822
$100$	0	0
$101$	4.32803	0.430655	0.215327	−	0.976542i	$-0.430918\pi$
$101$	4.32803	0.430655	0.215327	+	0.976542i	$0.430918\pi$
$102$	0	0
$103$	−5.13924	−0.506384	−0.253192	−	0.967416i	$-0.581480\pi$
$103$	−5.13924	−0.506384	−0.253192	+	0.967416i	$0.581480\pi$
$104$	0	0
$105$	39.4351	3.84847
$106$	0	0
$107$	−0.952079	−0.0920409	−0.0460205	−	0.998940i	$-0.514654\pi$
$107$	−0.952079	−0.0920409	−0.0460205	+	0.998940i	$0.514654\pi$
$108$	0	0
$109$	−4.66719	−0.447036	−0.223518	−	0.974700i	$-0.571754\pi$
$109$	−4.66719	−0.447036	−0.223518	+	0.974700i	$0.571754\pi$
$110$	0	0
$111$	0.0677163	0.00642734
$112$	0	0
$113$	−4.40581	−0.414463	−0.207232	−	0.978292i	$-0.566445\pi$
$113$	−4.40581	−0.414463	−0.207232	+	0.978292i	$0.566445\pi$
$114$	0	0
$115$	−33.2601	−3.10152
$116$	0	0
$117$	−18.9212	−1.74927
$118$	0	0
$119$	−16.4729	−1.51007
$120$	0	0
$121$	−10.5883	−0.962573
$122$	0	0
$123$	12.7113	1.14614
$124$	0	0
$125$	−17.1350	−1.53260
$126$	0	0
$127$	5.08049	0.450820	0.225410	−	0.974264i	$-0.427628\pi$
$127$	5.08049	0.450820	0.225410	+	0.974264i	$0.427628\pi$
$128$	0	0
$129$	−5.58617	−0.491835
$130$	0	0
$131$	9.51634	0.831447	0.415723	−	0.909491i	$-0.363528\pi$
$131$	9.51634	0.831447	0.415723	+	0.909491i	$0.363528\pi$
$132$	0	0
$133$	18.9101	1.63972
$134$	0	0
$135$	0.523856	0.0450864
$136$	0	0
$137$	−0.708274	−0.0605119	−0.0302560	−	0.999542i	$-0.509632\pi$
$137$	−0.708274	−0.0605119	−0.0302560	+	0.999542i	$0.509632\pi$
$138$	0	0
$139$	21.0943	1.78919	0.894596	−	0.446876i	$-0.147464\pi$
$139$	21.0943	1.78919	0.894596	+	0.446876i	$0.147464\pi$
$140$	0	0
$141$	26.1769	2.20449
$142$	0	0
$143$	−4.12444	−0.344903
$144$	0	0
$145$	−25.8902	−2.15007
$146$	0	0
$147$	26.9267	2.22088
$148$	0	0
$149$	−7.46023	−0.611166	−0.305583	−	0.952165i	$-0.598851\pi$
$149$	−7.46023	−0.611166	−0.305583	+	0.952165i	$0.598851\pi$
$150$	0	0
$151$	3.15002	0.256345	0.128172	−	0.991752i	$-0.459089\pi$
$151$	3.15002	0.256345	0.128172	+	0.991752i	$0.459089\pi$
$152$	0	0
$153$	11.4148	0.922830
$154$	0	0
$155$	31.9585	2.56697
$156$	0	0
$157$	−12.9995	−1.03747	−0.518735	−	0.854935i	$-0.673597\pi$
$157$	−12.9995	−1.03747	−0.518735	+	0.854935i	$0.673597\pi$
$158$	0	0
$159$	−25.4671	−2.01967
$160$	0	0
$161$	−37.1037	−2.92418
$162$	0	0
$163$	14.1084	1.10506	0.552529	−	0.833494i	$-0.313663\pi$
$163$	14.1084	1.10506	0.552529	+	0.833494i	$0.313663\pi$
$164$	0	0
$165$	−5.95657	−0.463718
$166$	0	0
$167$	12.6668	0.980187	0.490094	−	0.871670i	$-0.336963\pi$
$167$	12.6668	0.980187	0.490094	+	0.871670i	$0.336963\pi$
$168$	0	0
$169$	28.3190	2.17839
$170$	0	0
$171$	−13.1037	−1.00206
$172$	0	0
$173$	14.8802	1.13132	0.565661	−	0.824638i	$-0.308621\pi$
$173$	14.8802	1.13132	0.565661	+	0.824638i	$0.308621\pi$
$174$	0	0
$175$	−40.3548	−3.05054
$176$	0	0
$177$	5.33645	0.401112
$178$	0	0
$179$	0.106812	0.00798352	0.00399176	−	0.999992i	$-0.498729\pi$
$179$	0.106812	0.00798352	0.00399176	+	0.999992i	$0.498729\pi$
$180$	0	0
$181$	8.79268	0.653555	0.326777	−	0.945101i	$-0.394037\pi$
$181$	8.79268	0.653555	0.326777	+	0.945101i	$0.394037\pi$
$182$	0	0
$183$	15.7839	1.16678
$184$	0	0
$185$	−0.105767	−0.00777617
$186$	0	0
$187$	2.48819	0.181954
$188$	0	0
$189$	0.584395	0.0425084
$190$	0	0
$191$	7.61264	0.550831	0.275416	−	0.961325i	$-0.411185\pi$
$191$	7.61264	0.550831	0.275416	+	0.961325i	$0.411185\pi$
$192$	0	0
$193$	14.9987	1.07963	0.539815	−	0.841783i	$-0.318494\pi$
$193$	14.9987	1.07963	0.539815	+	0.841783i	$0.318494\pi$
$194$	0	0
$195$	59.6734	4.27330
$196$	0	0
$197$	−7.97986	−0.568542	−0.284271	−	0.958744i	$-0.591751\pi$
$197$	−7.97986	−0.568542	−0.284271	+	0.958744i	$0.591751\pi$
$198$	0	0
$199$	−7.07333	−0.501415	−0.250708	−	0.968063i	$-0.580663\pi$
$199$	−7.07333	−0.501415	−0.250708	+	0.968063i	$0.580663\pi$
$200$	0	0
$201$	2.70707	0.190942
$202$	0	0
$203$	−28.8822	−2.02713
$204$	0	0
$205$	−19.8540	−1.38666
$206$	0	0
$207$	25.7108	1.78702
$208$	0	0
$209$	−2.85633	−0.197577
$210$	0	0
$211$	−5.13086	−0.353223	−0.176612	−	0.984281i	$-0.556514\pi$
$211$	−5.13086	−0.353223	−0.176612	+	0.984281i	$0.556514\pi$
$212$	0	0
$213$	39.4056	2.70003
$214$	0	0
$215$	8.72515	0.595050
$216$	0	0
$217$	35.6517	2.42020
$218$	0	0
$219$	12.2574	0.828278
$220$	0	0
$221$	−24.9269	−1.67676
$222$	0	0
$223$	−1.89736	−0.127056	−0.0635282	−	0.997980i	$-0.520235\pi$
$223$	−1.89736	−0.127056	−0.0635282	+	0.997980i	$0.520235\pi$
$224$	0	0
$225$	27.9636	1.86424
$226$	0	0
$227$	9.69556	0.643517	0.321759	−	0.946822i	$-0.395726\pi$
$227$	9.69556	0.643517	0.321759	+	0.946822i	$0.395726\pi$
$228$	0	0
$229$	4.61645	0.305064	0.152532	−	0.988299i	$-0.451257\pi$
$229$	4.61645	0.305064	0.152532	+	0.988299i	$0.451257\pi$
$230$	0	0
$231$	−6.64493	−0.437204
$232$	0	0
$233$	−21.3771	−1.40046	−0.700231	−	0.713917i	$-0.746920\pi$
$233$	−21.3771	−1.40046	−0.700231	+	0.713917i	$0.746920\pi$
$234$	0	0
$235$	−40.8862	−2.66712
$236$	0	0
$237$	42.4453	2.75712
$238$	0	0
$239$	−27.8271	−1.79998	−0.899992	−	0.435907i	$-0.856428\pi$
$239$	−27.8271	−1.79998	−0.899992	+	0.435907i	$0.856428\pi$
$240$	0	0
$241$	1.24354	0.0801035	0.0400518	−	0.999198i	$-0.487248\pi$
$241$	1.24354	0.0801035	0.0400518	+	0.999198i	$0.487248\pi$
$242$	0	0
$243$	−21.9337	−1.40705
$244$	0	0
$245$	−42.0573	−2.68694
$246$	0	0
$247$	28.6150	1.82073
$248$	0	0
$249$	−28.5840	−1.81144
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	5.60442	0.352347
$254$	0	0
$255$	−35.9997	−2.25439
$256$	0	0
$257$	13.0793	0.815867	0.407933	−	0.913012i	$-0.366250\pi$
$257$	13.0793	0.815867	0.407933	+	0.913012i	$0.366250\pi$
$258$	0	0
$259$	−0.117990	−0.00733155
$260$	0	0
$261$	20.0137	1.23882
$262$	0	0
$263$	6.55939	0.404469	0.202235	−	0.979337i	$-0.435180\pi$
$263$	6.55939	0.404469	0.202235	+	0.979337i	$0.435180\pi$
$264$	0	0
$265$	39.7776	2.44352
$266$	0	0
$267$	13.3372	0.816221
$268$	0	0
$269$	5.73200	0.349486	0.174743	−	0.984614i	$-0.444091\pi$
$269$	5.73200	0.349486	0.174743	+	0.984614i	$0.444091\pi$
$270$	0	0
$271$	−18.2205	−1.10681	−0.553407	−	0.832911i	$-0.686673\pi$
$271$	−18.2205	−1.10681	−0.553407	+	0.832911i	$0.686673\pi$
$272$	0	0
$273$	66.5695	4.02897
$274$	0	0
$275$	6.09549	0.367572
$276$	0	0
$277$	9.13912	0.549116	0.274558	−	0.961571i	$-0.411468\pi$
$277$	9.13912	0.549116	0.274558	+	0.961571i	$0.411468\pi$
$278$	0	0
$279$	−24.7047	−1.47903
$280$	0	0
$281$	−2.51558	−0.150067	−0.0750335	−	0.997181i	$-0.523906\pi$
$281$	−2.51558	−0.150067	−0.0750335	+	0.997181i	$0.523906\pi$
$282$	0	0
$283$	−6.49849	−0.386295	−0.193148	−	0.981170i	$-0.561870\pi$
$283$	−6.49849	−0.386295	−0.193148	+	0.981170i	$0.561870\pi$
$284$	0	0
$285$	41.3261	2.44795
$286$	0	0
$287$	−22.1484	−1.30738
$288$	0	0
$289$	−1.96215	−0.115421
$290$	0	0
$291$	−43.6834	−2.56076
$292$	0	0
$293$	21.1585	1.23610	0.618048	−	0.786141i	$-0.287924\pi$
$293$	21.1585	1.23610	0.618048	+	0.786141i	$0.287924\pi$
$294$	0	0
$295$	−8.33510	−0.485288
$296$	0	0
$297$	−0.0882713	−0.00512202
$298$	0	0
$299$	−56.1456	−3.24698
$300$	0	0
$301$	9.73345	0.561027
$302$	0	0
$303$	10.5515	0.606167
$304$	0	0
$305$	−24.6531	−1.41163
$306$	0	0
$307$	11.3553	0.648082	0.324041	−	0.946043i	$-0.394958\pi$
$307$	11.3553	0.648082	0.324041	+	0.946043i	$0.394958\pi$
$308$	0	0
$309$	−12.5292	−0.712760
$310$	0	0
$311$	27.0513	1.53394	0.766970	−	0.641683i	$-0.221763\pi$
$311$	27.0513	1.53394	0.766970	+	0.641683i	$0.221763\pi$
$312$	0	0
$313$	−19.3601	−1.09430	−0.547150	−	0.837034i	$-0.684287\pi$
$313$	−19.3601	−1.09430	−0.547150	+	0.837034i	$0.684287\pi$
$314$	0	0
$315$	47.6138	2.68274
$316$	0	0
$317$	6.92567	0.388984	0.194492	−	0.980904i	$-0.437694\pi$
$317$	6.92567	0.388984	0.194492	+	0.980904i	$0.437694\pi$
$318$	0	0
$319$	4.36258	0.244257
$320$	0	0
$321$	−2.32111	−0.129552
$322$	0	0
$323$	−17.2628	−0.960528
$324$	0	0
$325$	−61.0652	−3.38729
$326$	0	0
$327$	−11.3783	−0.629224
$328$	0	0
$329$	−45.6112	−2.51462
$330$	0	0
$331$	33.4630	1.83929	0.919647	−	0.392747i	$-0.128475\pi$
$331$	33.4630	1.83929	0.919647	+	0.392747i	$0.128475\pi$
$332$	0	0
$333$	0.0817605	0.00448045
$334$	0	0
$335$	−4.22823	−0.231013
$336$	0	0
$337$	−1.27145	−0.0692604	−0.0346302	−	0.999400i	$-0.511025\pi$
$337$	−1.27145	−0.0692604	−0.0346302	+	0.999400i	$0.511025\pi$
$338$	0	0
$339$	−10.7411	−0.583377
$340$	0	0
$341$	−5.38511	−0.291620
$342$	0	0
$343$	−17.1821	−0.927748
$344$	0	0
$345$	−81.0862	−4.36553
$346$	0	0
$347$	−10.2988	−0.552867	−0.276433	−	0.961033i	$-0.589152\pi$
$347$	−10.2988	−0.552867	−0.276433	+	0.961033i	$0.589152\pi$
$348$	0	0
$349$	−9.17474	−0.491113	−0.245556	−	0.969382i	$-0.578971\pi$
$349$	−9.17474	−0.491113	−0.245556	+	0.969382i	$0.578971\pi$
$350$	0	0
$351$	0.884310	0.0472010
$352$	0	0
$353$	−19.3988	−1.03249	−0.516247	−	0.856440i	$-0.672671\pi$
$353$	−19.3988	−1.03249	−0.516247	+	0.856440i	$0.672671\pi$
$354$	0	0
$355$	−61.5484	−3.26665
$356$	0	0
$357$	−40.1599	−2.12549
$358$	0	0
$359$	21.2448	1.12126	0.560629	−	0.828067i	$-0.310560\pi$
$359$	21.2448	1.12126	0.560629	+	0.828067i	$0.310560\pi$
$360$	0	0
$361$	0.816959	0.0429979
$362$	0	0
$363$	−25.8137	−1.35487
$364$	0	0
$365$	−19.1451	−1.00210
$366$	0	0
$367$	−5.02047	−0.262066	−0.131033	−	0.991378i	$-0.541829\pi$
$367$	−5.02047	−0.262066	−0.131033	+	0.991378i	$0.541829\pi$
$368$	0	0
$369$	15.3476	0.798963
$370$	0	0
$371$	44.3744	2.30380
$372$	0	0
$373$	12.8491	0.665300	0.332650	−	0.943050i	$-0.392057\pi$
$373$	12.8491	0.665300	0.332650	+	0.943050i	$0.392057\pi$
$374$	0	0
$375$	−41.7742	−2.15721
$376$	0	0
$377$	−43.7047	−2.25091
$378$	0	0
$379$	−6.93442	−0.356197	−0.178099	−	0.984013i	$-0.556995\pi$
$379$	−6.93442	−0.356197	−0.178099	+	0.984013i	$0.556995\pi$
$380$	0	0
$381$	12.3859	0.634551
$382$	0	0
$383$	12.1325	0.619942	0.309971	−	0.950746i	$-0.399681\pi$
$383$	12.1325	0.619942	0.309971	+	0.950746i	$0.399681\pi$
$384$	0	0
$385$	10.3788	0.528955
$386$	0	0
$387$	−6.74474	−0.342854
$388$	0	0
$389$	−28.1401	−1.42676	−0.713380	−	0.700777i	$-0.752837\pi$
$389$	−28.1401	−1.42676	−0.713380	+	0.700777i	$0.752837\pi$
$390$	0	0
$391$	33.8714	1.71295
$392$	0	0
$393$	23.2003	1.17030
$394$	0	0
$395$	−66.2962	−3.33572
$396$	0	0
$397$	−17.8618	−0.896457	−0.448229	−	0.893919i	$-0.647945\pi$
$397$	−17.8618	−0.896457	−0.448229	+	0.893919i	$0.647945\pi$
$398$	0	0
$399$	46.1019	2.30798
$400$	0	0
$401$	7.00313	0.349719	0.174860	−	0.984593i	$-0.444053\pi$
$401$	7.00313	0.349719	0.174860	+	0.984593i	$0.444053\pi$
$402$	0	0
$403$	53.9485	2.68737
$404$	0	0
$405$	34.9034	1.73436
$406$	0	0
$407$	0.0178221	0.000883409 0
$408$	0	0
$409$	34.5604	1.70890	0.854450	−	0.519534i	$-0.173894\pi$
$409$	34.5604	1.70890	0.854450	+	0.519534i	$0.173894\pi$
$410$	0	0
$411$	−1.72673	−0.0851734
$412$	0	0
$413$	−9.29833	−0.457541
$414$	0	0
$415$	44.6459	2.19158
$416$	0	0
$417$	51.4266	2.51837
$418$	0	0
$419$	−3.34244	−0.163289	−0.0816443	−	0.996662i	$-0.526017\pi$
$419$	−3.34244	−0.163289	−0.0816443	+	0.996662i	$0.526017\pi$
$420$	0	0
$421$	7.47481	0.364300	0.182150	−	0.983271i	$-0.441694\pi$
$421$	7.47481	0.364300	0.182150	+	0.983271i	$0.441694\pi$
$422$	0	0
$423$	31.6060	1.53673
$424$	0	0
$425$	36.8393	1.78697
$426$	0	0
$427$	−27.5021	−1.33092
$428$	0	0
$429$	−10.0552	−0.485467
$430$	0	0
$431$	0.190315	0.00916717	0.00458359	−	0.999989i	$-0.498541\pi$
$431$	0.190315	0.00916717	0.00458359	+	0.999989i	$0.498541\pi$
$432$	0	0
$433$	−39.3372	−1.89043	−0.945214	−	0.326452i	$-0.894147\pi$
$433$	−39.3372	−1.89043	−0.945214	+	0.326452i	$0.894147\pi$
$434$	0	0
$435$	−63.1189	−3.02632
$436$	0	0
$437$	−38.8830	−1.86002
$438$	0	0
$439$	−29.7102	−1.41799	−0.708995	−	0.705213i	$-0.750851\pi$
$439$	−29.7102	−1.41799	−0.708995	+	0.705213i	$0.750851\pi$
$440$	0	0
$441$	32.5112	1.54815
$442$	0	0
$443$	32.1253	1.52632	0.763160	−	0.646210i	$-0.223647\pi$
$443$	32.1253	1.52632	0.763160	+	0.646210i	$0.223647\pi$
$444$	0	0
$445$	−20.8316	−0.987512
$446$	0	0
$447$	−18.1876	−0.860245
$448$	0	0
$449$	−23.8243	−1.12434	−0.562169	−	0.827023i	$-0.690033\pi$
$449$	−23.8243	−1.12434	−0.562169	+	0.827023i	$0.690033\pi$
$450$	0	0
$451$	3.34546	0.157531
$452$	0	0
$453$	7.67957	0.360818
$454$	0	0
$455$	−103.976	−4.87448
$456$	0	0
$457$	21.2708	0.995005	0.497503	−	0.867462i	$-0.334250\pi$
$457$	21.2708	0.995005	0.497503	+	0.867462i	$0.334250\pi$
$458$	0	0
$459$	−0.533485	−0.0249010
$460$	0	0
$461$	−7.94544	−0.370056	−0.185028	−	0.982733i	$-0.559238\pi$
$461$	−7.94544	−0.370056	−0.185028	+	0.982733i	$0.559238\pi$
$462$	0	0
$463$	10.3986	0.483263	0.241632	−	0.970368i	$-0.422317\pi$
$463$	10.3986	0.483263	0.241632	+	0.970368i	$0.422317\pi$
$464$	0	0
$465$	77.9131	3.61313
$466$	0	0
$467$	3.65933	0.169333	0.0846667	−	0.996409i	$-0.473017\pi$
$467$	3.65933	0.169333	0.0846667	+	0.996409i	$0.473017\pi$
$468$	0	0
$469$	−4.71685	−0.217804
$470$	0	0
$471$	−31.6920	−1.46029
$472$	0	0
$473$	−1.47021	−0.0676005
$474$	0	0
$475$	−42.2899	−1.94040
$476$	0	0
$477$	−30.7490	−1.40790
$478$	0	0
$479$	16.8608	0.770387	0.385194	−	0.922836i	$-0.374135\pi$
$479$	16.8608	0.770387	0.385194	+	0.922836i	$0.374135\pi$
$480$	0	0
$481$	−0.178544	−0.00814089
$482$	0	0
$483$	−90.4567	−4.11592
$484$	0	0
$485$	68.2299	3.09816
$486$	0	0
$487$	0.0783977	0.00355254	0.00177627	−	0.999998i	$-0.499435\pi$
$487$	0.0783977	0.00355254	0.00177627	+	0.999998i	$0.499435\pi$
$488$	0	0
$489$	34.3956	1.55542
$490$	0	0
$491$	4.71643	0.212849	0.106425	−	0.994321i	$-0.466060\pi$
$491$	4.71643	0.212849	0.106425	+	0.994321i	$0.466060\pi$
$492$	0	0
$493$	26.3661	1.18747
$494$	0	0
$495$	−7.19195	−0.323254
$496$	0	0
$497$	−68.6612	−3.07987
$498$	0	0
$499$	−17.3929	−0.778613	−0.389307	−	0.921108i	$-0.627285\pi$
$499$	−17.3929	−0.778613	−0.389307	+	0.921108i	$0.627285\pi$
$500$	0	0
$501$	30.8810	1.37966
$502$	0	0
$503$	38.5140	1.71725	0.858627	−	0.512601i	$-0.171318\pi$
$503$	38.5140	1.71725	0.858627	+	0.512601i	$0.171318\pi$
$504$	0	0
$505$	−16.4806	−0.733376
$506$	0	0
$507$	69.0402	3.06618
$508$	0	0
$509$	−38.7811	−1.71894	−0.859472	−	0.511184i	$-0.829207\pi$
$509$	−38.7811	−1.71894	−0.859472	+	0.511184i	$0.829207\pi$
$510$	0	0
$511$	−21.3575	−0.944802
$512$	0	0
$513$	0.612418	0.0270389
$514$	0	0
$515$	19.5696	0.862338
$516$	0	0
$517$	6.88945	0.302998
$518$	0	0
$519$	36.2772	1.59239
$520$	0	0
$521$	22.6453	0.992110	0.496055	−	0.868291i	$-0.334781\pi$
$521$	22.6453	0.992110	0.496055	+	0.868291i	$0.334781\pi$
$522$	0	0
$523$	13.2687	0.580200	0.290100	−	0.956996i	$-0.406311\pi$
$523$	13.2687	0.580200	0.290100	+	0.956996i	$0.406311\pi$
$524$	0	0
$525$	−98.3827	−4.29377
$526$	0	0
$527$	−32.5460	−1.41772
$528$	0	0
$529$	53.2925	2.31706
$530$	0	0
$531$	6.44322	0.279612
$532$	0	0
$533$	−33.5151	−1.45170
$534$	0	0
$535$	3.62539	0.156739
$536$	0	0
$537$	0.260402	0.0112372
$538$	0	0
$539$	7.08678	0.305249
$540$	0	0
$541$	−11.3206	−0.486710	−0.243355	−	0.969937i	$-0.578248\pi$
$541$	−11.3206	−0.486710	−0.243355	+	0.969937i	$0.578248\pi$
$542$	0	0
$543$	21.4361	0.919909
$544$	0	0
$545$	17.7721	0.761272
$546$	0	0
$547$	−6.04429	−0.258435	−0.129218	−	0.991616i	$-0.541247\pi$
$547$	−6.04429	−0.258435	−0.129218	+	0.991616i	$0.541247\pi$
$548$	0	0
$549$	19.0574	0.813351
$550$	0	0
$551$	−30.2672	−1.28942
$552$	0	0
$553$	−73.9575	−3.14499
$554$	0	0
$555$	−0.257855	−0.0109453
$556$	0	0
$557$	32.0011	1.35593	0.677965	−	0.735094i	$-0.262862\pi$
$557$	32.0011	1.35593	0.677965	+	0.735094i	$0.262862\pi$
$558$	0	0
$559$	14.7287	0.622959
$560$	0	0
$561$	6.06606	0.256109
$562$	0	0
$563$	−21.6139	−0.910918	−0.455459	−	0.890257i	$-0.650525\pi$
$563$	−21.6139	−0.910918	−0.455459	+	0.890257i	$0.650525\pi$
$564$	0	0
$565$	16.7767	0.705803
$566$	0	0
$567$	38.9369	1.63520
$568$	0	0
$569$	28.5590	1.19726	0.598628	−	0.801027i	$-0.295713\pi$
$569$	28.5590	1.19726	0.598628	+	0.801027i	$0.295713\pi$
$570$	0	0
$571$	20.9921	0.878494	0.439247	−	0.898366i	$-0.355245\pi$
$571$	20.9921	0.878494	0.439247	+	0.898366i	$0.355245\pi$
$572$	0	0
$573$	18.5592	0.775321
$574$	0	0
$575$	82.9773	3.46039
$576$	0	0
$577$	−28.9722	−1.20613	−0.603064	−	0.797693i	$-0.706054\pi$
$577$	−28.9722	−1.20613	−0.603064	+	0.797693i	$0.706054\pi$
$578$	0	0
$579$	36.5660	1.51963
$580$	0	0
$581$	49.8053	2.06627
$582$	0	0
$583$	−6.70264	−0.277595
$584$	0	0
$585$	72.0496	2.97889
$586$	0	0
$587$	3.06556	0.126529	0.0632647	−	0.997997i	$-0.479849\pi$
$587$	3.06556	0.126529	0.0632647	+	0.997997i	$0.479849\pi$
$588$	0	0
$589$	37.3614	1.53945
$590$	0	0
$591$	−19.4545	−0.800249
$592$	0	0
$593$	14.7606	0.606145	0.303073	−	0.952967i	$-0.401988\pi$
$593$	14.7606	0.606145	0.303073	+	0.952967i	$0.401988\pi$
$594$	0	0
$595$	62.7266	2.57154
$596$	0	0
$597$	−17.2444	−0.705766
$598$	0	0
$599$	−42.3543	−1.73055	−0.865275	−	0.501297i	$-0.832856\pi$
$599$	−42.3543	−1.73055	−0.865275	+	0.501297i	$0.832856\pi$
$600$	0	0
$601$	13.1840	0.537788	0.268894	−	0.963170i	$-0.413342\pi$
$601$	13.1840	0.537788	0.268894	+	0.963170i	$0.413342\pi$
$602$	0	0
$603$	3.26851	0.133104
$604$	0	0
$605$	40.3189	1.63920
$606$	0	0
$607$	−6.13595	−0.249051	−0.124525	−	0.992216i	$-0.539741\pi$
$607$	−6.13595	−0.249051	−0.124525	+	0.992216i	$0.539741\pi$
$608$	0	0
$609$	−70.4131	−2.85328
$610$	0	0
$611$	−69.0192	−2.79222
$612$	0	0
$613$	25.8501	1.04407	0.522037	−	0.852923i	$-0.325172\pi$
$613$	25.8501	1.04407	0.522037	+	0.852923i	$0.325172\pi$
$614$	0	0
$615$	−48.4029	−1.95179
$616$	0	0
$617$	28.8614	1.16192	0.580959	−	0.813933i	$-0.302678\pi$
$617$	28.8614	1.16192	0.580959	+	0.813933i	$0.302678\pi$
$618$	0	0
$619$	28.1786	1.13259	0.566297	−	0.824201i	$-0.308376\pi$
$619$	28.1786	1.13259	0.566297	+	0.824201i	$0.308376\pi$
$620$	0	0
$621$	−1.20163	−0.0482197
$622$	0	0
$623$	−23.2389	−0.931048
$624$	0	0
$625$	17.7485	0.709939
$626$	0	0
$627$	−6.96357	−0.278098
$628$	0	0
$629$	0.107712	0.00429474
$630$	0	0
$631$	9.97687	0.397173	0.198587	−	0.980083i	$-0.436365\pi$
$631$	9.97687	0.397173	0.198587	+	0.980083i	$0.436365\pi$
$632$	0	0
$633$	−12.5088	−0.497178
$634$	0	0
$635$	−19.3458	−0.767716
$636$	0	0
$637$	−70.9961	−2.81297
$638$	0	0
$639$	47.5783	1.88217
$640$	0	0
$641$	27.8665	1.10066	0.550330	−	0.834947i	$-0.314502\pi$
$641$	27.8665	1.10066	0.550330	+	0.834947i	$0.314502\pi$
$642$	0	0
$643$	10.5363	0.415511	0.207756	−	0.978181i	$-0.433384\pi$
$643$	10.5363	0.415511	0.207756	+	0.978181i	$0.433384\pi$
$644$	0	0
$645$	21.2714	0.837562
$646$	0	0
$647$	33.4947	1.31681	0.658406	−	0.752663i	$-0.271231\pi$
$647$	33.4947	1.31681	0.658406	+	0.752663i	$0.271231\pi$
$648$	0	0
$649$	1.40449	0.0551310
$650$	0	0
$651$	86.9170	3.40654
$652$	0	0
$653$	17.5007	0.684854	0.342427	−	0.939544i	$-0.388751\pi$
$653$	17.5007	0.684854	0.342427	+	0.939544i	$0.388751\pi$
$654$	0	0
$655$	−36.2370	−1.41590
$656$	0	0
$657$	14.7996	0.577386
$658$	0	0
$659$	11.7615	0.458164	0.229082	−	0.973407i	$-0.426428\pi$
$659$	11.7615	0.458164	0.229082	+	0.973407i	$0.426428\pi$
$660$	0	0
$661$	−6.21377	−0.241688	−0.120844	−	0.992672i	$-0.538560\pi$
$661$	−6.21377	−0.241688	−0.120844	+	0.992672i	$0.538560\pi$
$662$	0	0
$663$	−60.7703	−2.36012
$664$	0	0
$665$	−72.0074	−2.79233
$666$	0	0
$667$	59.3874	2.29949
$668$	0	0
$669$	−4.62565	−0.178838
$670$	0	0
$671$	4.15412	0.160368
$672$	0	0
$673$	−12.0624	−0.464972	−0.232486	−	0.972600i	$-0.574686\pi$
$673$	−12.0624	−0.464972	−0.232486	+	0.972600i	$0.574686\pi$
$674$	0	0
$675$	−1.30692	−0.0503033
$676$	0	0
$677$	3.51583	0.135125	0.0675623	−	0.997715i	$-0.478478\pi$
$677$	3.51583	0.135125	0.0675623	+	0.997715i	$0.478478\pi$
$678$	0	0
$679$	76.1147	2.92102
$680$	0	0
$681$	23.6372	0.905781
$682$	0	0
$683$	41.6133	1.59229	0.796144	−	0.605107i	$-0.206870\pi$
$683$	41.6133	1.59229	0.796144	+	0.605107i	$0.206870\pi$
$684$	0	0
$685$	2.69702	0.103048
$686$	0	0
$687$	11.2547	0.429392
$688$	0	0
$689$	67.1477	2.55812
$690$	0	0
$691$	−14.5676	−0.554179	−0.277090	−	0.960844i	$-0.589370\pi$
$691$	−14.5676	−0.554179	−0.277090	+	0.960844i	$0.589370\pi$
$692$	0	0
$693$	−8.02307	−0.304771
$694$	0	0
$695$	−80.3242	−3.04687
$696$	0	0
$697$	20.2189	0.765847
$698$	0	0
$699$	−52.1162	−1.97122
$700$	0	0
$701$	−30.5678	−1.15453	−0.577266	−	0.816556i	$-0.695880\pi$
$701$	−30.5678	−1.15453	−0.577266	+	0.816556i	$0.695880\pi$
$702$	0	0
$703$	−0.123648	−0.00466348
$704$	0	0
$705$	−99.6783	−3.75410
$706$	0	0
$707$	−18.3851	−0.691444
$708$	0	0
$709$	36.7129	1.37878	0.689391	−	0.724390i	$-0.257878\pi$
$709$	36.7129	1.37878	0.689391	+	0.724390i	$0.257878\pi$
$710$	0	0
$711$	51.2484	1.92197
$712$	0	0
$713$	−73.3070	−2.74537
$714$	0	0
$715$	15.7053	0.587346
$716$	0	0
$717$	−67.8408	−2.53356
$718$	0	0
$719$	12.3432	0.460323	0.230161	−	0.973152i	$-0.426075\pi$
$719$	12.3432	0.460323	0.230161	+	0.973152i	$0.426075\pi$
$720$	0	0
$721$	21.8311	0.813032
$722$	0	0
$723$	3.03168	0.112750
$724$	0	0
$725$	64.5910	2.39885
$726$	0	0
$727$	11.9272	0.442355	0.221177	−	0.975234i	$-0.429010\pi$
$727$	11.9272	0.442355	0.221177	+	0.975234i	$0.429010\pi$
$728$	0	0
$729$	−25.9749	−0.962033
$730$	0	0
$731$	−8.88553	−0.328643
$732$	0	0
$733$	10.8020	0.398980	0.199490	−	0.979900i	$-0.436072\pi$
$733$	10.8020	0.398980	0.199490	+	0.979900i	$0.436072\pi$
$734$	0	0
$735$	−102.533	−3.78200
$736$	0	0
$737$	0.712469	0.0262441
$738$	0	0
$739$	25.9137	0.953252	0.476626	−	0.879106i	$-0.341860\pi$
$739$	25.9137	0.953252	0.476626	+	0.879106i	$0.341860\pi$
$740$	0	0
$741$	69.7617	2.56276
$742$	0	0
$743$	41.6086	1.52647	0.763236	−	0.646120i	$-0.223609\pi$
$743$	41.6086	1.52647	0.763236	+	0.646120i	$0.223609\pi$
$744$	0	0
$745$	28.4076	1.04077
$746$	0	0
$747$	−34.5123	−1.26274
$748$	0	0
$749$	4.04435	0.147778
$750$	0	0
$751$	9.99996	0.364904	0.182452	−	0.983215i	$-0.441597\pi$
$751$	9.99996	0.364904	0.182452	+	0.983215i	$0.441597\pi$
$752$	0	0
$753$	2.43794	0.0888436
$754$	0	0
$755$	−11.9949	−0.436538
$756$	0	0
$757$	−29.7707	−1.08203	−0.541017	−	0.841012i	$-0.681961\pi$
$757$	−29.7707	−1.08203	−0.541017	+	0.841012i	$0.681961\pi$
$758$	0	0
$759$	13.6633	0.495945
$760$	0	0
$761$	−13.5068	−0.489619	−0.244810	−	0.969571i	$-0.578725\pi$
$761$	−13.5068	−0.489619	−0.244810	+	0.969571i	$0.578725\pi$
$762$	0	0
$763$	19.8259	0.717744
$764$	0	0
$765$	−43.4660	−1.57152
$766$	0	0
$767$	−14.0703	−0.508049
$768$	0	0
$769$	−44.6238	−1.60918	−0.804588	−	0.593833i	$-0.797614\pi$
$769$	−44.6238	−1.60918	−0.804588	+	0.593833i	$0.797614\pi$
$770$	0	0
$771$	31.8867	1.14837
$772$	0	0
$773$	−13.1116	−0.471592	−0.235796	−	0.971803i	$-0.575770\pi$
$773$	−13.1116	−0.471592	−0.235796	+	0.971803i	$0.575770\pi$
$774$	0	0
$775$	−79.7302	−2.86399
$776$	0	0
$777$	−0.287653	−0.0103195
$778$	0	0
$779$	−23.2105	−0.831601
$780$	0	0
$781$	10.3711	0.371107
$782$	0	0
$783$	−0.935369	−0.0334274
$784$	0	0
$785$	49.5003	1.76674
$786$	0	0
$787$	−12.9633	−0.462093	−0.231047	−	0.972943i	$-0.574215\pi$
$787$	−12.9633	−0.462093	−0.231047	+	0.972943i	$0.574215\pi$
$788$	0	0
$789$	15.9914	0.569309
$790$	0	0
$791$	18.7155	0.665447
$792$	0	0
$793$	−41.6164	−1.47784
$794$	0	0
$795$	96.9755	3.43937
$796$	0	0
$797$	−43.4401	−1.53873	−0.769364	−	0.638811i	$-0.779427\pi$
$797$	−43.4401	−1.53873	−0.769364	+	0.638811i	$0.779427\pi$
$798$	0	0
$799$	41.6378	1.47304
$800$	0	0
$801$	16.1033	0.568981
$802$	0	0
$803$	3.22600	0.113843
$804$	0	0
$805$	141.286	4.97968
$806$	0	0
$807$	13.9743	0.491918
$808$	0	0
$809$	29.2338	1.02781	0.513904	−	0.857848i	$-0.328199\pi$
$809$	29.2338	1.02781	0.513904	+	0.857848i	$0.328199\pi$
$810$	0	0
$811$	13.1613	0.462155	0.231077	−	0.972935i	$-0.425775\pi$
$811$	13.1613	0.462155	0.231077	+	0.972935i	$0.425775\pi$
$812$	0	0
$813$	−44.4205	−1.55789
$814$	0	0
$815$	−53.7231	−1.88184
$816$	0	0
$817$	10.2002	0.356860
$818$	0	0
$819$	80.3759	2.80856
$820$	0	0
$821$	35.0919	1.22472	0.612358	−	0.790581i	$-0.290221\pi$
$821$	35.0919	1.22472	0.612358	+	0.790581i	$0.290221\pi$
$822$	0	0
$823$	−27.0914	−0.944346	−0.472173	−	0.881506i	$-0.656530\pi$
$823$	−27.0914	−0.944346	−0.472173	+	0.881506i	$0.656530\pi$
$824$	0	0
$825$	14.8605	0.517375
$826$	0	0
$827$	−21.0678	−0.732601	−0.366300	−	0.930497i	$-0.619376\pi$
$827$	−21.0678	−0.732601	−0.366300	+	0.930497i	$0.619376\pi$
$828$	0	0
$829$	11.2442	0.390528	0.195264	−	0.980751i	$-0.437444\pi$
$829$	11.2442	0.390528	0.195264	+	0.980751i	$0.437444\pi$
$830$	0	0
$831$	22.2807	0.772907
$832$	0	0
$833$	42.8304	1.48398
$834$	0	0
$835$	−48.2336	−1.66919
$836$	0	0
$837$	1.15461	0.0399090
$838$	0	0
$839$	−15.5106	−0.535485	−0.267743	−	0.963490i	$-0.586278\pi$
$839$	−15.5106	−0.535485	−0.267743	+	0.963490i	$0.586278\pi$
$840$	0	0
$841$	17.2282	0.594074
$842$	0	0
$843$	−6.13284	−0.211226
$844$	0	0
$845$	−107.835	−3.70964
$846$	0	0
$847$	44.9783	1.54547
$848$	0	0
$849$	−15.8430	−0.543729
$850$	0	0
$851$	0.242611	0.00831659
$852$	0	0
$853$	36.9870	1.26641	0.633205	−	0.773984i	$-0.281739\pi$
$853$	36.9870	1.26641	0.633205	+	0.773984i	$0.281739\pi$
$854$	0	0
$855$	49.8971	1.70644
$856$	0	0
$857$	9.22432	0.315097	0.157548	−	0.987511i	$-0.449641\pi$
$857$	9.22432	0.315097	0.157548	+	0.987511i	$0.449641\pi$
$858$	0	0
$859$	−21.0671	−0.718802	−0.359401	−	0.933183i	$-0.617019\pi$
$859$	−21.0671	−0.718802	−0.359401	+	0.933183i	$0.617019\pi$
$860$	0	0
$861$	−53.9965	−1.84020
$862$	0	0
$863$	−2.08972	−0.0711347	−0.0355674	−	0.999367i	$-0.511324\pi$
$863$	−2.08972	−0.0711347	−0.0355674	+	0.999367i	$0.511324\pi$
$864$	0	0
$865$	−56.6620	−1.92657
$866$	0	0
$867$	−4.78362	−0.162460
$868$	0	0
$869$	11.1711	0.378954
$870$	0	0
$871$	−7.13758	−0.241848
$872$	0	0
$873$	−52.7432	−1.78509
$874$	0	0
$875$	72.7882	2.46069
$876$	0	0
$877$	−15.2993	−0.516619	−0.258310	−	0.966062i	$-0.583165\pi$
$877$	−15.2993	−0.516619	−0.258310	+	0.966062i	$0.583165\pi$
$878$	0	0
$879$	51.5833	1.73986
$880$	0	0
$881$	−3.10260	−0.104529	−0.0522646	−	0.998633i	$-0.516644\pi$
$881$	−3.10260	−0.104529	−0.0522646	+	0.998633i	$0.516644\pi$
$882$	0	0
$883$	9.15452	0.308074	0.154037	−	0.988065i	$-0.450772\pi$
$883$	9.15452	0.308074	0.154037	+	0.988065i	$0.450772\pi$
$884$	0	0
$885$	−20.3205	−0.683066
$886$	0	0
$887$	8.30216	0.278759	0.139380	−	0.990239i	$-0.455489\pi$
$887$	8.30216	0.278759	0.139380	+	0.990239i	$0.455489\pi$
$888$	0	0
$889$	−21.5815	−0.723820
$890$	0	0
$891$	−5.88132	−0.197032
$892$	0	0
$893$	−47.7984	−1.59951
$894$	0	0
$895$	−0.406727	−0.0135954
$896$	0	0
$897$	−136.880	−4.57029
$898$	0	0
$899$	−57.0634	−1.90317
$900$	0	0
$901$	−40.5087	−1.34954
$902$	0	0
$903$	23.7296	0.789672
$904$	0	0
$905$	−33.4814	−1.11296
$906$	0	0
$907$	30.7139	1.01984	0.509919	−	0.860223i	$-0.329675\pi$
$907$	30.7139	1.01984	0.509919	+	0.860223i	$0.329675\pi$
$908$	0	0
$909$	12.7399	0.422554
$910$	0	0
$911$	41.3624	1.37040	0.685198	−	0.728357i	$-0.259716\pi$
$911$	41.3624	1.37040	0.685198	+	0.728357i	$0.259716\pi$
$912$	0	0
$913$	−7.52297	−0.248974
$914$	0	0
$915$	−60.1029	−1.98694
$916$	0	0
$917$	−40.4247	−1.33494
$918$	0	0
$919$	−6.48728	−0.213995	−0.106998	−	0.994259i	$-0.534124\pi$
$919$	−6.48728	−0.213995	−0.106998	+	0.994259i	$0.534124\pi$
$920$	0	0
$921$	27.6836	0.912206
$922$	0	0
$923$	−103.899	−3.41986
$924$	0	0
$925$	0.263869	0.00867595
$926$	0	0
$927$	−15.1277	−0.496859
$928$	0	0
$929$	11.5011	0.377340	0.188670	−	0.982041i	$-0.439582\pi$
$929$	11.5011	0.377340	0.188670	+	0.982041i	$0.439582\pi$
$930$	0	0
$931$	−49.1675	−1.61140
$932$	0	0
$933$	65.9496	2.15909
$934$	0	0
$935$	−9.47469	−0.309856
$936$	0	0
$937$	−11.7316	−0.383253	−0.191627	−	0.981468i	$-0.561376\pi$
$937$	−11.7316	−0.383253	−0.191627	+	0.981468i	$0.561376\pi$
$938$	0	0
$939$	−47.1990	−1.54028
$940$	0	0
$941$	3.72635	0.121475	0.0607377	−	0.998154i	$-0.480655\pi$
$941$	3.72635	0.121475	0.0607377	+	0.998154i	$0.480655\pi$
$942$	0	0
$943$	45.5414	1.48303
$944$	0	0
$945$	−2.22530	−0.0723890
$946$	0	0
$947$	−16.7815	−0.545325	−0.272663	−	0.962110i	$-0.587904\pi$
$947$	−16.7815	−0.545325	−0.272663	+	0.962110i	$0.587904\pi$
$948$	0	0
$949$	−32.3184	−1.04910
$950$	0	0
$951$	16.8844	0.547514
$952$	0	0
$953$	−45.1635	−1.46299	−0.731494	−	0.681847i	$-0.761177\pi$
$953$	−45.1635	−1.46299	−0.731494	+	0.681847i	$0.761177\pi$
$954$	0	0
$955$	−28.9880	−0.938028
$956$	0	0
$957$	10.6357	0.343804
$958$	0	0
$959$	3.00869	0.0971557
$960$	0	0
$961$	39.4383	1.27220
$962$	0	0
$963$	−2.80251	−0.0903096
$964$	0	0
$965$	−57.1132	−1.83854
$966$	0	0
$967$	11.9241	0.383455	0.191727	−	0.981448i	$-0.438591\pi$
$967$	11.9241	0.383455	0.191727	+	0.981448i	$0.438591\pi$
$968$	0	0
$969$	−42.0857	−1.35199
$970$	0	0
$971$	−38.9790	−1.25090	−0.625448	−	0.780266i	$-0.715084\pi$
$971$	−38.9790	−1.25090	−0.625448	+	0.780266i	$0.715084\pi$
$972$	0	0
$973$	−89.6067	−2.87266
$974$	0	0
$975$	−148.873	−4.76777
$976$	0	0
$977$	19.0008	0.607890	0.303945	−	0.952690i	$-0.401696\pi$
$977$	19.0008	0.607890	0.303945	+	0.952690i	$0.401696\pi$
$978$	0	0
$979$	3.51018	0.112186
$980$	0	0
$981$	−13.7382	−0.438627
$982$	0	0
$983$	30.7722	0.981479	0.490740	−	0.871306i	$-0.336727\pi$
$983$	30.7722	0.981479	0.490740	+	0.871306i	$0.336727\pi$
$984$	0	0
$985$	30.3863	0.968188
$986$	0	0
$987$	−111.197	−3.53945
$988$	0	0
$989$	−20.0139	−0.636404
$990$	0	0
$991$	16.3172	0.518332	0.259166	−	0.965833i	$-0.416552\pi$
$991$	16.3172	0.518332	0.259166	+	0.965833i	$0.416552\pi$
$992$	0	0
$993$	81.5809	2.58889
$994$	0	0
$995$	26.9343	0.853876
$996$	0	0
$997$	−34.3466	−1.08777	−0.543884	−	0.839160i	$-0.683047\pi$
$997$	−34.3466	−1.08777	−0.543884	+	0.839160i	$0.683047\pi$
$998$	0	0
$999$	−0.00382119	−0.000120897 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.15		19
4.3	odd	2		2008.2.a.c.1.5	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.5	✓	19	4.3	odd	2
4016.2.a.l.1.15		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)

\(f(q)\)	\(=\)	\(q+2.43794 q^{3} -3.80787 q^{5} -4.24792 q^{7} +2.94357 q^{9} +O(q^{10})\)	\(q+2.43794 q^{3} -3.80787 q^{5} -4.24792 q^{7} +2.94357 q^{9} +0.641638 q^{11} -6.42799 q^{13} -9.28338 q^{15} +3.87787 q^{17} -4.45162 q^{19} -10.3562 q^{21} +8.73456 q^{23} +9.49989 q^{25} -0.137572 q^{27} +6.79913 q^{29} -8.39275 q^{31} +1.56428 q^{33} +16.1755 q^{35} +0.0277760 q^{37} -15.6711 q^{39} +5.21393 q^{41} -2.29135 q^{43} -11.2087 q^{45} +10.7373 q^{47} +11.0448 q^{49} +9.45402 q^{51} -10.4461 q^{53} -2.44327 q^{55} -10.8528 q^{57} +2.18891 q^{59} +6.47425 q^{61} -12.5041 q^{63} +24.4770 q^{65} +1.11039 q^{67} +21.2944 q^{69} +16.1635 q^{71} +5.02776 q^{73} +23.1602 q^{75} -2.72563 q^{77} +17.4103 q^{79} -9.16610 q^{81} -11.7246 q^{83} -14.7664 q^{85} +16.5759 q^{87} +5.47066 q^{89} +27.3056 q^{91} -20.4611 q^{93} +16.9512 q^{95} -17.9181 q^{97} +1.88871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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