LMFDB - Embedded newform 4016.2.a.l.1.13

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			1.13
Root			$1.56031$ 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)

$f(q)$	$=$	$q+1.56031 q^{3} +3.44545 q^{5} +1.77285 q^{7} -0.565441 q^{9} +O(q^{10})$	$q+1.56031 q^{3} +3.44545 q^{5} +1.77285 q^{7} -0.565441 q^{9} +0.674515 q^{11} -2.56431 q^{13} +5.37596 q^{15} +7.81784 q^{17} +3.57546 q^{19} +2.76620 q^{21} +4.75952 q^{23} +6.87111 q^{25} -5.56318 q^{27} -8.77151 q^{29} +4.12516 q^{31} +1.05245 q^{33} +6.10827 q^{35} -8.08708 q^{37} -4.00110 q^{39} +2.30855 q^{41} -3.23776 q^{43} -1.94820 q^{45} +10.7775 q^{47} -3.85699 q^{49} +12.1982 q^{51} +11.7741 q^{53} +2.32400 q^{55} +5.57882 q^{57} -13.3529 q^{59} -11.0528 q^{61} -1.00244 q^{63} -8.83518 q^{65} +6.36140 q^{67} +7.42631 q^{69} -5.50382 q^{71} +0.424653 q^{73} +10.7210 q^{75} +1.19582 q^{77} +3.50836 q^{79} -6.98395 q^{81} -1.14460 q^{83} +26.9360 q^{85} -13.6863 q^{87} +9.02905 q^{89} -4.54614 q^{91} +6.43651 q^{93} +12.3191 q^{95} -15.7420 q^{97} -0.381398 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.56031	0.900844	0.450422	−	0.892816i	$-0.351274\pi$
$3$	1.56031	0.900844	0.450422	+	0.892816i	$0.351274\pi$
$4$	0	0
$5$	3.44545	1.54085	0.770426	−	0.637530i	$-0.220044\pi$
$5$	3.44545	1.54085	0.770426	+	0.637530i	$0.220044\pi$
$6$	0	0
$7$	1.77285	0.670076	0.335038	−	0.942205i	$-0.391251\pi$
$7$	1.77285	0.670076	0.335038	+	0.942205i	$0.391251\pi$
$8$	0	0
$9$	−0.565441	−0.188480
$10$	0	0
$11$	0.674515	0.203374	0.101687	−	0.994816i	$-0.467576\pi$
$11$	0.674515	0.203374	0.101687	+	0.994816i	$0.467576\pi$
$12$	0	0
$13$	−2.56431	−0.711210	−0.355605	−	0.934636i	$-0.615725\pi$
$13$	−2.56431	−0.711210	−0.355605	+	0.934636i	$0.615725\pi$
$14$	0	0
$15$	5.37596	1.38807
$16$	0	0
$17$	7.81784	1.89611	0.948053	−	0.318113i	$-0.103049\pi$
$17$	7.81784	1.89611	0.948053	+	0.318113i	$0.103049\pi$
$18$	0	0
$19$	3.57546	0.820267	0.410133	−	0.912026i	$-0.365482\pi$
$19$	3.57546	0.820267	0.410133	+	0.912026i	$0.365482\pi$
$20$	0	0
$21$	2.76620	0.603634
$22$	0	0
$23$	4.75952	0.992428	0.496214	−	0.868200i	$-0.334723\pi$
$23$	4.75952	0.992428	0.496214	+	0.868200i	$0.334723\pi$
$24$	0	0
$25$	6.87111	1.37422
$26$	0	0
$27$	−5.56318	−1.07064
$28$	0	0
$29$	−8.77151	−1.62883	−0.814415	−	0.580283i	$-0.802942\pi$
$29$	−8.77151	−1.62883	−0.814415	+	0.580283i	$0.802942\pi$
$30$	0	0
$31$	4.12516	0.740900	0.370450	−	0.928852i	$-0.379204\pi$
$31$	4.12516	0.740900	0.370450	+	0.928852i	$0.379204\pi$
$32$	0	0
$33$	1.05245	0.183208
$34$	0	0
$35$	6.10827	1.03249
$36$	0	0
$37$	−8.08708	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$37$	−8.08708	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$38$	0	0
$39$	−4.00110	−0.640689
$40$	0	0
$41$	2.30855	0.360535	0.180268	−	0.983618i	$-0.442304\pi$
$41$	2.30855	0.360535	0.180268	+	0.983618i	$0.442304\pi$
$42$	0	0
$43$	−3.23776	−0.493754	−0.246877	−	0.969047i	$-0.579404\pi$
$43$	−3.23776	−0.493754	−0.246877	+	0.969047i	$0.579404\pi$
$44$	0	0
$45$	−1.94820	−0.290420
$46$	0	0
$47$	10.7775	1.57206	0.786032	−	0.618186i	$-0.212132\pi$
$47$	10.7775	1.57206	0.786032	+	0.618186i	$0.212132\pi$
$48$	0	0
$49$	−3.85699	−0.550999
$50$	0	0
$51$	12.1982	1.70810
$52$	0	0
$53$	11.7741	1.61729	0.808647	−	0.588294i	$-0.200200\pi$
$53$	11.7741	1.61729	0.808647	+	0.588294i	$0.200200\pi$
$54$	0	0
$55$	2.32400	0.313369
$56$	0	0
$57$	5.57882	0.738932
$58$	0	0
$59$	−13.3529	−1.73841	−0.869203	−	0.494456i	$-0.835367\pi$
$59$	−13.3529	−1.73841	−0.869203	+	0.494456i	$0.835367\pi$
$60$	0	0
$61$	−11.0528	−1.41516	−0.707582	−	0.706631i	$-0.750214\pi$
$61$	−11.0528	−1.41516	−0.707582	+	0.706631i	$0.750214\pi$
$62$	0	0
$63$	−1.00244	−0.126296
$64$	0	0
$65$	−8.83518	−1.09587
$66$	0	0
$67$	6.36140	0.777169	0.388584	−	0.921413i	$-0.372964\pi$
$67$	6.36140	0.777169	0.388584	+	0.921413i	$0.372964\pi$
$68$	0	0
$69$	7.42631	0.894022
$70$	0	0
$71$	−5.50382	−0.653183	−0.326592	−	0.945166i	$-0.605900\pi$
$71$	−5.50382	−0.653183	−0.326592	+	0.945166i	$0.605900\pi$
$72$	0	0
$73$	0.424653	0.0497019	0.0248510	−	0.999691i	$-0.492089\pi$
$73$	0.424653	0.0497019	0.0248510	+	0.999691i	$0.492089\pi$
$74$	0	0
$75$	10.7210	1.23796
$76$	0	0
$77$	1.19582	0.136276
$78$	0	0
$79$	3.50836	0.394722	0.197361	−	0.980331i	$-0.436763\pi$
$79$	3.50836	0.394722	0.197361	+	0.980331i	$0.436763\pi$
$80$	0	0
$81$	−6.98395	−0.775995
$82$	0	0
$83$	−1.14460	−0.125636	−0.0628182	−	0.998025i	$-0.520009\pi$
$83$	−1.14460	−0.125636	−0.0628182	+	0.998025i	$0.520009\pi$
$84$	0	0
$85$	26.9360	2.92162
$86$	0	0
$87$	−13.6863	−1.46732
$88$	0	0
$89$	9.02905	0.957077	0.478539	−	0.878066i	$-0.341167\pi$
$89$	9.02905	0.957077	0.478539	+	0.878066i	$0.341167\pi$
$90$	0	0
$91$	−4.54614	−0.476565
$92$	0	0
$93$	6.43651	0.667435
$94$	0	0
$95$	12.3191	1.26391
$96$	0	0
$97$	−15.7420	−1.59836	−0.799181	−	0.601091i	$-0.794733\pi$
$97$	−15.7420	−1.59836	−0.799181	+	0.601091i	$0.794733\pi$
$98$	0	0
$99$	−0.381398	−0.0383319
$100$	0	0
$101$	−6.88004	−0.684590	−0.342295	−	0.939593i	$-0.611204\pi$
$101$	−6.88004	−0.684590	−0.342295	+	0.939593i	$0.611204\pi$
$102$	0	0
$103$	17.8301	1.75686	0.878428	−	0.477875i	$-0.158593\pi$
$103$	17.8301	1.75686	0.878428	+	0.477875i	$0.158593\pi$
$104$	0	0
$105$	9.53079	0.930109
$106$	0	0
$107$	8.84303	0.854888	0.427444	−	0.904042i	$-0.359414\pi$
$107$	8.84303	0.854888	0.427444	+	0.904042i	$0.359414\pi$
$108$	0	0
$109$	−3.22659	−0.309051	−0.154525	−	0.987989i	$-0.549385\pi$
$109$	−3.22659	−0.309051	−0.154525	+	0.987989i	$0.549385\pi$
$110$	0	0
$111$	−12.6183	−1.19768
$112$	0	0
$113$	19.6345	1.84705	0.923527	−	0.383533i	$-0.125293\pi$
$113$	19.6345	1.84705	0.923527	+	0.383533i	$0.125293\pi$
$114$	0	0
$115$	16.3987	1.52918
$116$	0	0
$117$	1.44996	0.134049
$118$	0	0
$119$	13.8599	1.27053
$120$	0	0
$121$	−10.5450	−0.958639
$122$	0	0
$123$	3.60205	0.324786
$124$	0	0
$125$	6.44682	0.576621
$126$	0	0
$127$	12.5748	1.11583	0.557917	−	0.829896i	$-0.311601\pi$
$127$	12.5748	1.11583	0.557917	+	0.829896i	$0.311601\pi$
$128$	0	0
$129$	−5.05190	−0.444795
$130$	0	0
$131$	5.01581	0.438233	0.219117	−	0.975699i	$-0.429682\pi$
$131$	5.01581	0.438233	0.219117	+	0.975699i	$0.429682\pi$
$132$	0	0
$133$	6.33876	0.549641
$134$	0	0
$135$	−19.1677	−1.64969
$136$	0	0
$137$	−16.0847	−1.37420	−0.687102	−	0.726561i	$-0.741118\pi$
$137$	−16.0847	−1.37420	−0.687102	+	0.726561i	$0.741118\pi$
$138$	0	0
$139$	17.6599	1.49789	0.748946	−	0.662631i	$-0.230560\pi$
$139$	17.6599	1.49789	0.748946	+	0.662631i	$0.230560\pi$
$140$	0	0
$141$	16.8163	1.41618
$142$	0	0
$143$	−1.72966	−0.144642
$144$	0	0
$145$	−30.2218	−2.50978
$146$	0	0
$147$	−6.01809	−0.496364
$148$	0	0
$149$	6.15347	0.504112	0.252056	−	0.967713i	$-0.418893\pi$
$149$	6.15347	0.504112	0.252056	+	0.967713i	$0.418893\pi$
$150$	0	0
$151$	−19.5774	−1.59319	−0.796595	−	0.604514i	$-0.793368\pi$
$151$	−19.5774	−1.59319	−0.796595	+	0.604514i	$0.793368\pi$
$152$	0	0
$153$	−4.42053	−0.357378
$154$	0	0
$155$	14.2130	1.14162
$156$	0	0
$157$	−17.0929	−1.36416	−0.682081	−	0.731277i	$-0.738925\pi$
$157$	−17.0929	−1.36416	−0.682081	+	0.731277i	$0.738925\pi$
$158$	0	0
$159$	18.3712	1.45693
$160$	0	0
$161$	8.43792	0.665002
$162$	0	0
$163$	−17.1906	−1.34647	−0.673236	−	0.739427i	$-0.735096\pi$
$163$	−17.1906	−1.34647	−0.673236	+	0.739427i	$0.735096\pi$
$164$	0	0
$165$	3.62616	0.282296
$166$	0	0
$167$	6.06471	0.469301	0.234651	−	0.972080i	$-0.424605\pi$
$167$	6.06471	0.469301	0.234651	+	0.972080i	$0.424605\pi$
$168$	0	0
$169$	−6.42434	−0.494180
$170$	0	0
$171$	−2.02171	−0.154604
$172$	0	0
$173$	−1.24399	−0.0945785	−0.0472892	−	0.998881i	$-0.515058\pi$
$173$	−1.24399	−0.0945785	−0.0472892	+	0.998881i	$0.515058\pi$
$174$	0	0
$175$	12.1815	0.920833
$176$	0	0
$177$	−20.8347	−1.56603
$178$	0	0
$179$	9.68619	0.723980	0.361990	−	0.932182i	$-0.382097\pi$
$179$	9.68619	0.723980	0.361990	+	0.932182i	$0.382097\pi$
$180$	0	0
$181$	−18.3993	−1.36761	−0.683804	−	0.729665i	$-0.739676\pi$
$181$	−18.3993	−1.36761	−0.683804	+	0.729665i	$0.739676\pi$
$182$	0	0
$183$	−17.2457	−1.27484
$184$	0	0
$185$	−27.8636	−2.04857
$186$	0	0
$187$	5.27325	0.385618
$188$	0	0
$189$	−9.86271	−0.717407
$190$	0	0
$191$	−20.3427	−1.47195	−0.735974	−	0.677010i	$-0.763275\pi$
$191$	−20.3427	−1.47195	−0.735974	+	0.677010i	$0.763275\pi$
$192$	0	0
$193$	21.2619	1.53046	0.765232	−	0.643755i	$-0.222624\pi$
$193$	21.2619	1.53046	0.765232	+	0.643755i	$0.222624\pi$
$194$	0	0
$195$	−13.7856	−0.987207
$196$	0	0
$197$	−1.66392	−0.118549	−0.0592746	−	0.998242i	$-0.518879\pi$
$197$	−1.66392	−0.118549	−0.0592746	+	0.998242i	$0.518879\pi$
$198$	0	0
$199$	−12.1317	−0.859994	−0.429997	−	0.902830i	$-0.641485\pi$
$199$	−12.1317	−0.859994	−0.429997	+	0.902830i	$0.641485\pi$
$200$	0	0
$201$	9.92574	0.700108
$202$	0	0
$203$	−15.5506	−1.09144
$204$	0	0
$205$	7.95399	0.555531
$206$	0	0
$207$	−2.69122	−0.187053
$208$	0	0
$209$	2.41170	0.166821
$210$	0	0
$211$	3.32024	0.228575	0.114287	−	0.993448i	$-0.463542\pi$
$211$	3.32024	0.228575	0.114287	+	0.993448i	$0.463542\pi$
$212$	0	0
$213$	−8.58765	−0.588416
$214$	0	0
$215$	−11.1555	−0.760801
$216$	0	0
$217$	7.31330	0.496459
$218$	0	0
$219$	0.662590	0.0447737
$220$	0	0
$221$	−20.0473	−1.34853
$222$	0	0
$223$	20.3810	1.36481	0.682405	−	0.730974i	$-0.260934\pi$
$223$	20.3810	1.36481	0.682405	+	0.730974i	$0.260934\pi$
$224$	0	0
$225$	−3.88521	−0.259014
$226$	0	0
$227$	−0.244815	−0.0162489	−0.00812447	−	0.999967i	$-0.502586\pi$
$227$	−0.244815	−0.0162489	−0.00812447	+	0.999967i	$0.502586\pi$
$228$	0	0
$229$	−15.9071	−1.05117	−0.525586	−	0.850740i	$-0.676154\pi$
$229$	−15.9071	−1.05117	−0.525586	+	0.850740i	$0.676154\pi$
$230$	0	0
$231$	1.86584	0.122763
$232$	0	0
$233$	2.81731	0.184568	0.0922840	−	0.995733i	$-0.470583\pi$
$233$	2.81731	0.184568	0.0922840	+	0.995733i	$0.470583\pi$
$234$	0	0
$235$	37.1334	2.42232
$236$	0	0
$237$	5.47412	0.355582
$238$	0	0
$239$	−17.2865	−1.11817	−0.559086	−	0.829110i	$-0.688848\pi$
$239$	−17.2865	−1.11817	−0.559086	+	0.829110i	$0.688848\pi$
$240$	0	0
$241$	21.7491	1.40098	0.700492	−	0.713660i	$-0.252964\pi$
$241$	21.7491	1.40098	0.700492	+	0.713660i	$0.252964\pi$
$242$	0	0
$243$	5.79243	0.371585
$244$	0	0
$245$	−13.2891	−0.849007
$246$	0	0
$247$	−9.16857	−0.583382
$248$	0	0
$249$	−1.78593	−0.113179
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	3.21036	0.201834
$254$	0	0
$255$	42.0284	2.63192
$256$	0	0
$257$	−15.9843	−0.997075	−0.498537	−	0.866868i	$-0.666129\pi$
$257$	−15.9843	−0.997075	−0.498537	+	0.866868i	$0.666129\pi$
$258$	0	0
$259$	−14.3372	−0.890871
$260$	0	0
$261$	4.95977	0.307002
$262$	0	0
$263$	23.4440	1.44562	0.722809	−	0.691048i	$-0.242851\pi$
$263$	23.4440	1.44562	0.722809	+	0.691048i	$0.242851\pi$
$264$	0	0
$265$	40.5670	2.49201
$266$	0	0
$267$	14.0881	0.862177
$268$	0	0
$269$	31.8876	1.94422	0.972112	−	0.234516i	$-0.0753507\pi$
$269$	31.8876	1.94422	0.972112	+	0.234516i	$0.0753507\pi$
$270$	0	0
$271$	−17.9436	−1.09000	−0.544999	−	0.838437i	$-0.683470\pi$
$271$	−17.9436	−1.09000	−0.544999	+	0.838437i	$0.683470\pi$
$272$	0	0
$273$	−7.09337	−0.429310
$274$	0	0
$275$	4.63467	0.279481
$276$	0	0
$277$	−22.3208	−1.34112	−0.670562	−	0.741853i	$-0.733947\pi$
$277$	−22.3208	−1.34112	−0.670562	+	0.741853i	$0.733947\pi$
$278$	0	0
$279$	−2.33253	−0.139645
$280$	0	0
$281$	−17.9684	−1.07191	−0.535953	−	0.844248i	$-0.680048\pi$
$281$	−17.9684	−1.07191	−0.535953	+	0.844248i	$0.680048\pi$
$282$	0	0
$283$	−18.1918	−1.08139	−0.540695	−	0.841219i	$-0.681839\pi$
$283$	−18.1918	−1.08139	−0.540695	+	0.841219i	$0.681839\pi$
$284$	0	0
$285$	19.2215	1.13858
$286$	0	0
$287$	4.09272	0.241586
$288$	0	0
$289$	44.1187	2.59522
$290$	0	0
$291$	−24.5624	−1.43987
$292$	0	0
$293$	17.7971	1.03972	0.519860	−	0.854251i	$-0.325984\pi$
$293$	17.7971	1.03972	0.519860	+	0.854251i	$0.325984\pi$
$294$	0	0
$295$	−46.0069	−2.67862
$296$	0	0
$297$	−3.75245	−0.217739
$298$	0	0
$299$	−12.2049	−0.705825
$300$	0	0
$301$	−5.74007	−0.330852
$302$	0	0
$303$	−10.7350	−0.616708
$304$	0	0
$305$	−38.0818	−2.18056
$306$	0	0
$307$	1.18725	0.0677598	0.0338799	−	0.999426i	$-0.489214\pi$
$307$	1.18725	0.0677598	0.0338799	+	0.999426i	$0.489214\pi$
$308$	0	0
$309$	27.8205	1.58265
$310$	0	0
$311$	7.66056	0.434391	0.217195	−	0.976128i	$-0.430309\pi$
$311$	7.66056	0.434391	0.217195	+	0.976128i	$0.430309\pi$
$312$	0	0
$313$	2.90608	0.164262	0.0821308	−	0.996622i	$-0.473827\pi$
$313$	2.90608	0.164262	0.0821308	+	0.996622i	$0.473827\pi$
$314$	0	0
$315$	−3.45387	−0.194603
$316$	0	0
$317$	−13.2436	−0.743837	−0.371919	−	0.928265i	$-0.621300\pi$
$317$	−13.2436	−0.743837	−0.371919	+	0.928265i	$0.621300\pi$
$318$	0	0
$319$	−5.91651	−0.331261
$320$	0	0
$321$	13.7978	0.770121
$322$	0	0
$323$	27.9524	1.55531
$324$	0	0
$325$	−17.6196	−0.977361
$326$	0	0
$327$	−5.03447	−0.278407
$328$	0	0
$329$	19.1070	1.05340
$330$	0	0
$331$	−5.08738	−0.279628	−0.139814	−	0.990178i	$-0.544650\pi$
$331$	−5.08738	−0.279628	−0.139814	+	0.990178i	$0.544650\pi$
$332$	0	0
$333$	4.57276	0.250586
$334$	0	0
$335$	21.9179	1.19750
$336$	0	0
$337$	17.5913	0.958258	0.479129	−	0.877744i	$-0.340953\pi$
$337$	17.5913	0.958258	0.479129	+	0.877744i	$0.340953\pi$
$338$	0	0
$339$	30.6358	1.66391
$340$	0	0
$341$	2.78248	0.150680
$342$	0	0
$343$	−19.2479	−1.03929
$344$	0	0
$345$	25.5870	1.37756
$346$	0	0
$347$	16.8192	0.902902	0.451451	−	0.892296i	$-0.350907\pi$
$347$	16.8192	0.902902	0.451451	+	0.892296i	$0.350907\pi$
$348$	0	0
$349$	19.6946	1.05423	0.527114	−	0.849795i	$-0.323274\pi$
$349$	19.6946	1.05423	0.527114	+	0.849795i	$0.323274\pi$
$350$	0	0
$351$	14.2657	0.761447
$352$	0	0
$353$	9.30887	0.495461	0.247731	−	0.968829i	$-0.420315\pi$
$353$	9.30887	0.495461	0.247731	+	0.968829i	$0.420315\pi$
$354$	0	0
$355$	−18.9631	−1.00646
$356$	0	0
$357$	21.6257	1.14455
$358$	0	0
$359$	−20.5168	−1.08284	−0.541418	−	0.840754i	$-0.682112\pi$
$359$	−20.5168	−1.08284	−0.541418	+	0.840754i	$0.682112\pi$
$360$	0	0
$361$	−6.21609	−0.327163
$362$	0	0
$363$	−16.4535	−0.863584
$364$	0	0
$365$	1.46312	0.0765833
$366$	0	0
$367$	−30.2281	−1.57789	−0.788947	−	0.614461i	$-0.789374\pi$
$367$	−30.2281	−1.57789	−0.788947	+	0.614461i	$0.789374\pi$
$368$	0	0
$369$	−1.30535	−0.0679537
$370$	0	0
$371$	20.8737	1.08371
$372$	0	0
$373$	−20.5593	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$373$	−20.5593	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$374$	0	0
$375$	10.0590	0.519446
$376$	0	0
$377$	22.4928	1.15844
$378$	0	0
$379$	10.8837	0.559056	0.279528	−	0.960137i	$-0.409822\pi$
$379$	10.8837	0.559056	0.279528	+	0.960137i	$0.409822\pi$
$380$	0	0
$381$	19.6206	1.00519
$382$	0	0
$383$	−7.96674	−0.407082	−0.203541	−	0.979066i	$-0.565245\pi$
$383$	−7.96674	−0.407082	−0.203541	+	0.979066i	$0.565245\pi$
$384$	0	0
$385$	4.12012	0.209981
$386$	0	0
$387$	1.83076	0.0930628
$388$	0	0
$389$	−0.965699	−0.0489629	−0.0244814	−	0.999700i	$-0.507793\pi$
$389$	−0.965699	−0.0489629	−0.0244814	+	0.999700i	$0.507793\pi$
$390$	0	0
$391$	37.2091	1.88175
$392$	0	0
$393$	7.82620	0.394780
$394$	0	0
$395$	12.0879	0.608207
$396$	0	0
$397$	−0.901626	−0.0452513	−0.0226257	−	0.999744i	$-0.507203\pi$
$397$	−0.901626	−0.0452513	−0.0226257	+	0.999744i	$0.507203\pi$
$398$	0	0
$399$	9.89042	0.495140
$400$	0	0
$401$	29.1871	1.45753	0.728766	−	0.684763i	$-0.240094\pi$
$401$	29.1871	1.45753	0.728766	+	0.684763i	$0.240094\pi$
$402$	0	0
$403$	−10.5782	−0.526936
$404$	0	0
$405$	−24.0629	−1.19569
$406$	0	0
$407$	−5.45485	−0.270387
$408$	0	0
$409$	−3.75999	−0.185920	−0.0929599	−	0.995670i	$-0.529633\pi$
$409$	−3.75999	−0.185920	−0.0929599	+	0.995670i	$0.529633\pi$
$410$	0	0
$411$	−25.0970	−1.23794
$412$	0	0
$413$	−23.6728	−1.16486
$414$	0	0
$415$	−3.94367	−0.193587
$416$	0	0
$417$	27.5548	1.34937
$418$	0	0
$419$	13.3008	0.649785	0.324892	−	0.945751i	$-0.394672\pi$
$419$	13.3008	0.649785	0.324892	+	0.945751i	$0.394672\pi$
$420$	0	0
$421$	10.9188	0.532149	0.266074	−	0.963952i	$-0.414273\pi$
$421$	10.9188	0.532149	0.266074	+	0.963952i	$0.414273\pi$
$422$	0	0
$423$	−6.09405	−0.296303
$424$	0	0
$425$	53.7173	2.60567
$426$	0	0
$427$	−19.5950	−0.948267
$428$	0	0
$429$	−2.69880	−0.130299
$430$	0	0
$431$	−19.2577	−0.927612	−0.463806	−	0.885937i	$-0.653517\pi$
$431$	−19.2577	−0.927612	−0.463806	+	0.885937i	$0.653517\pi$
$432$	0	0
$433$	9.00179	0.432599	0.216299	−	0.976327i	$-0.430601\pi$
$433$	9.00179	0.432599	0.216299	+	0.976327i	$0.430601\pi$
$434$	0	0
$435$	−47.1553	−2.26092
$436$	0	0
$437$	17.0175	0.814055
$438$	0	0
$439$	−36.7393	−1.75347	−0.876735	−	0.480973i	$-0.840283\pi$
$439$	−36.7393	−1.75347	−0.876735	+	0.480973i	$0.840283\pi$
$440$	0	0
$441$	2.18090	0.103852
$442$	0	0
$443$	−19.0488	−0.905036	−0.452518	−	0.891755i	$-0.649474\pi$
$443$	−19.0488	−0.905036	−0.452518	+	0.891755i	$0.649474\pi$
$444$	0	0
$445$	31.1091	1.47471
$446$	0	0
$447$	9.60131	0.454126
$448$	0	0
$449$	−21.5432	−1.01668	−0.508342	−	0.861155i	$-0.669742\pi$
$449$	−21.5432	−1.01668	−0.508342	+	0.861155i	$0.669742\pi$
$450$	0	0
$451$	1.55715	0.0733234
$452$	0	0
$453$	−30.5468	−1.43522
$454$	0	0
$455$	−15.6635	−0.734315
$456$	0	0
$457$	17.8582	0.835372	0.417686	−	0.908591i	$-0.362841\pi$
$457$	17.8582	0.835372	0.417686	+	0.908591i	$0.362841\pi$
$458$	0	0
$459$	−43.4921	−2.03004
$460$	0	0
$461$	13.5424	0.630731	0.315366	−	0.948970i	$-0.397873\pi$
$461$	13.5424	0.630731	0.315366	+	0.948970i	$0.397873\pi$
$462$	0	0
$463$	40.7686	1.89468	0.947338	−	0.320236i	$-0.103762\pi$
$463$	40.7686	1.89468	0.947338	+	0.320236i	$0.103762\pi$
$464$	0	0
$465$	22.1767	1.02842
$466$	0	0
$467$	−29.2711	−1.35451	−0.677253	−	0.735750i	$-0.736830\pi$
$467$	−29.2711	−1.35451	−0.677253	+	0.735750i	$0.736830\pi$
$468$	0	0
$469$	11.2778	0.520762
$470$	0	0
$471$	−26.6702	−1.22890
$472$	0	0
$473$	−2.18392	−0.100417
$474$	0	0
$475$	24.5674	1.12723
$476$	0	0
$477$	−6.65754	−0.304828
$478$	0	0
$479$	−18.3622	−0.838992	−0.419496	−	0.907757i	$-0.637793\pi$
$479$	−18.3622	−0.838992	−0.419496	+	0.907757i	$0.637793\pi$
$480$	0	0
$481$	20.7377	0.945560
$482$	0	0
$483$	13.1658	0.599063
$484$	0	0
$485$	−54.2384	−2.46284
$486$	0	0
$487$	6.04569	0.273956	0.136978	−	0.990574i	$-0.456261\pi$
$487$	6.04569	0.273956	0.136978	+	0.990574i	$0.456261\pi$
$488$	0	0
$489$	−26.8226	−1.21296
$490$	0	0
$491$	0.402699	0.0181735	0.00908677	−	0.999959i	$-0.497108\pi$
$491$	0.402699	0.0181735	0.00908677	+	0.999959i	$0.497108\pi$
$492$	0	0
$493$	−68.5743	−3.08843
$494$	0	0
$495$	−1.31409	−0.0590638
$496$	0	0
$497$	−9.75746	−0.437682
$498$	0	0
$499$	−36.6354	−1.64003	−0.820014	−	0.572344i	$-0.806034\pi$
$499$	−36.6354	−1.64003	−0.820014	+	0.572344i	$0.806034\pi$
$500$	0	0
$501$	9.46281	0.422767
$502$	0	0
$503$	6.64836	0.296436	0.148218	−	0.988955i	$-0.452646\pi$
$503$	6.64836	0.296436	0.148218	+	0.988955i	$0.452646\pi$
$504$	0	0
$505$	−23.7048	−1.05485
$506$	0	0
$507$	−10.0239	−0.445179
$508$	0	0
$509$	−14.5265	−0.643877	−0.321939	−	0.946761i	$-0.604334\pi$
$509$	−14.5265	−0.643877	−0.321939	+	0.946761i	$0.604334\pi$
$510$	0	0
$511$	0.752848	0.0333041
$512$	0	0
$513$	−19.8909	−0.878206
$514$	0	0
$515$	61.4328	2.70705
$516$	0	0
$517$	7.26960	0.319717
$518$	0	0
$519$	−1.94100	−0.0852005
$520$	0	0
$521$	18.8124	0.824184	0.412092	−	0.911142i	$-0.364798\pi$
$521$	18.8124	0.824184	0.412092	+	0.911142i	$0.364798\pi$
$522$	0	0
$523$	22.4015	0.979551	0.489775	−	0.871849i	$-0.337079\pi$
$523$	22.4015	0.979551	0.489775	+	0.871849i	$0.337079\pi$
$524$	0	0
$525$	19.0068	0.829527
$526$	0	0
$527$	32.2498	1.40482
$528$	0	0
$529$	−0.347014	−0.0150876
$530$	0	0
$531$	7.55030	0.327655
$532$	0	0
$533$	−5.91983	−0.256416
$534$	0	0
$535$	30.4682	1.31726
$536$	0	0
$537$	15.1134	0.652193
$538$	0	0
$539$	−2.60160	−0.112059
$540$	0	0
$541$	−33.5516	−1.44250	−0.721249	−	0.692676i	$-0.756431\pi$
$541$	−33.5516	−1.44250	−0.721249	+	0.692676i	$0.756431\pi$
$542$	0	0
$543$	−28.7086	−1.23200
$544$	0	0
$545$	−11.1170	−0.476201
$546$	0	0
$547$	18.4716	0.789787	0.394894	−	0.918727i	$-0.370781\pi$
$547$	18.4716	0.789787	0.394894	+	0.918727i	$0.370781\pi$
$548$	0	0
$549$	6.24970	0.266731
$550$	0	0
$551$	−31.3622	−1.33607
$552$	0	0
$553$	6.21981	0.264493
$554$	0	0
$555$	−43.4758	−1.84545
$556$	0	0
$557$	4.89316	0.207330	0.103665	−	0.994612i	$-0.466943\pi$
$557$	4.89316	0.207330	0.103665	+	0.994612i	$0.466943\pi$
$558$	0	0
$559$	8.30260	0.351163
$560$	0	0
$561$	8.22789	0.347382
$562$	0	0
$563$	19.2433	0.811008	0.405504	−	0.914093i	$-0.367096\pi$
$563$	19.2433	0.811008	0.405504	+	0.914093i	$0.367096\pi$
$564$	0	0
$565$	67.6495	2.84604
$566$	0	0
$567$	−12.3815	−0.519975
$568$	0	0
$569$	−10.5762	−0.443379	−0.221689	−	0.975117i	$-0.571157\pi$
$569$	−10.5762	−0.443379	−0.221689	+	0.975117i	$0.571157\pi$
$570$	0	0
$571$	−11.3506	−0.475009	−0.237504	−	0.971386i	$-0.576329\pi$
$571$	−11.3506	−0.475009	−0.237504	+	0.971386i	$0.576329\pi$
$572$	0	0
$573$	−31.7409	−1.32600
$574$	0	0
$575$	32.7032	1.36382
$576$	0	0
$577$	−7.97543	−0.332021	−0.166011	−	0.986124i	$-0.553089\pi$
$577$	−7.97543	−0.332021	−0.166011	+	0.986124i	$0.553089\pi$
$578$	0	0
$579$	33.1751	1.37871
$580$	0	0
$581$	−2.02921	−0.0841859
$582$	0	0
$583$	7.94179	0.328915
$584$	0	0
$585$	4.99577	0.206550
$586$	0	0
$587$	33.6288	1.38801	0.694005	−	0.719970i	$-0.255845\pi$
$587$	33.6288	1.38801	0.694005	+	0.719970i	$0.255845\pi$
$588$	0	0
$589$	14.7493	0.607735
$590$	0	0
$591$	−2.59622	−0.106794
$592$	0	0
$593$	0.0764533	0.00313956	0.00156978	−	0.999999i	$-0.499500\pi$
$593$	0.0764533	0.00313956	0.00156978	+	0.999999i	$0.499500\pi$
$594$	0	0
$595$	47.7535	1.95770
$596$	0	0
$597$	−18.9292	−0.774721
$598$	0	0
$599$	−35.7483	−1.46064	−0.730319	−	0.683107i	$-0.760628\pi$
$599$	−35.7483	−1.46064	−0.730319	+	0.683107i	$0.760628\pi$
$600$	0	0
$601$	−3.22081	−0.131380	−0.0656898	−	0.997840i	$-0.520925\pi$
$601$	−3.22081	−0.131380	−0.0656898	+	0.997840i	$0.520925\pi$
$602$	0	0
$603$	−3.59699	−0.146481
$604$	0	0
$605$	−36.3324	−1.47712
$606$	0	0
$607$	4.83466	0.196233	0.0981164	−	0.995175i	$-0.468718\pi$
$607$	4.83466	0.196233	0.0981164	+	0.995175i	$0.468718\pi$
$608$	0	0
$609$	−24.2637	−0.983216
$610$	0	0
$611$	−27.6369	−1.11807
$612$	0	0
$613$	14.9027	0.601913	0.300957	−	0.953638i	$-0.402694\pi$
$613$	14.9027	0.601913	0.300957	+	0.953638i	$0.402694\pi$
$614$	0	0
$615$	12.4107	0.500447
$616$	0	0
$617$	−10.8817	−0.438083	−0.219041	−	0.975716i	$-0.570293\pi$
$617$	−10.8817	−0.438083	−0.219041	+	0.975716i	$0.570293\pi$
$618$	0	0
$619$	33.5017	1.34655	0.673273	−	0.739394i	$-0.264888\pi$
$619$	33.5017	1.34655	0.673273	+	0.739394i	$0.264888\pi$
$620$	0	0
$621$	−26.4781	−1.06253
$622$	0	0
$623$	16.0072	0.641314
$624$	0	0
$625$	−12.1434	−0.485735
$626$	0	0
$627$	3.76299	0.150279
$628$	0	0
$629$	−63.2235	−2.52089
$630$	0	0
$631$	−14.9238	−0.594108	−0.297054	−	0.954861i	$-0.596004\pi$
$631$	−14.9238	−0.594108	−0.297054	+	0.954861i	$0.596004\pi$
$632$	0	0
$633$	5.18060	0.205910
$634$	0	0
$635$	43.3259	1.71934
$636$	0	0
$637$	9.89050	0.391876
$638$	0	0
$639$	3.11208	0.123112
$640$	0	0
$641$	20.9037	0.825645	0.412822	−	0.910812i	$-0.364543\pi$
$641$	20.9037	0.825645	0.412822	+	0.910812i	$0.364543\pi$
$642$	0	0
$643$	28.6225	1.12876	0.564381	−	0.825515i	$-0.309115\pi$
$643$	28.6225	1.12876	0.564381	+	0.825515i	$0.309115\pi$
$644$	0	0
$645$	−17.4061	−0.685363
$646$	0	0
$647$	14.8453	0.583628	0.291814	−	0.956475i	$-0.405741\pi$
$647$	14.8453	0.583628	0.291814	+	0.956475i	$0.405741\pi$
$648$	0	0
$649$	−9.00675	−0.353546
$650$	0	0
$651$	11.4110	0.447232
$652$	0	0
$653$	−26.3620	−1.03163	−0.515813	−	0.856701i	$-0.672510\pi$
$653$	−26.3620	−1.03163	−0.515813	+	0.856701i	$0.672510\pi$
$654$	0	0
$655$	17.2817	0.675252
$656$	0	0
$657$	−0.240116	−0.00936783
$658$	0	0
$659$	−29.5135	−1.14968	−0.574842	−	0.818265i	$-0.694936\pi$
$659$	−29.5135	−1.14968	−0.574842	+	0.818265i	$0.694936\pi$
$660$	0	0
$661$	−5.44510	−0.211790	−0.105895	−	0.994377i	$-0.533771\pi$
$661$	−5.44510	−0.211790	−0.105895	+	0.994377i	$0.533771\pi$
$662$	0	0
$663$	−31.2800	−1.21482
$664$	0	0
$665$	21.8399	0.846914
$666$	0	0
$667$	−41.7482	−1.61649
$668$	0	0
$669$	31.8006	1.22948
$670$	0	0
$671$	−7.45527	−0.287807
$672$	0	0
$673$	30.7641	1.18587	0.592935	−	0.805250i	$-0.297969\pi$
$673$	30.7641	1.18587	0.592935	+	0.805250i	$0.297969\pi$
$674$	0	0
$675$	−38.2253	−1.47129
$676$	0	0
$677$	6.83168	0.262563	0.131281	−	0.991345i	$-0.458091\pi$
$677$	6.83168	0.262563	0.131281	+	0.991345i	$0.458091\pi$
$678$	0	0
$679$	−27.9083	−1.07102
$680$	0	0
$681$	−0.381987	−0.0146378
$682$	0	0
$683$	1.52389	0.0583101	0.0291550	−	0.999575i	$-0.490718\pi$
$683$	1.52389	0.0583101	0.0291550	+	0.999575i	$0.490718\pi$
$684$	0	0
$685$	−55.4188	−2.11744
$686$	0	0
$687$	−24.8200	−0.946942
$688$	0	0
$689$	−30.1923	−1.15024
$690$	0	0
$691$	−38.1831	−1.45255	−0.726276	−	0.687403i	$-0.758751\pi$
$691$	−38.1831	−1.45255	−0.726276	+	0.687403i	$0.758751\pi$
$692$	0	0
$693$	−0.676163	−0.0256853
$694$	0	0
$695$	60.8462	2.30803
$696$	0	0
$697$	18.0479	0.683613
$698$	0	0
$699$	4.39587	0.166267
$700$	0	0
$701$	−45.4697	−1.71737	−0.858683	−	0.512507i	$-0.828717\pi$
$701$	−45.4697	−1.71737	−0.858683	+	0.512507i	$0.828717\pi$
$702$	0	0
$703$	−28.9150	−1.09055
$704$	0	0
$705$	57.9395	2.18213
$706$	0	0
$707$	−12.1973	−0.458727
$708$	0	0
$709$	−31.3261	−1.17648	−0.588238	−	0.808688i	$-0.700178\pi$
$709$	−31.3261	−1.17648	−0.588238	+	0.808688i	$0.700178\pi$
$710$	0	0
$711$	−1.98377	−0.0743972
$712$	0	0
$713$	19.6337	0.735289
$714$	0	0
$715$	−5.95946	−0.222871
$716$	0	0
$717$	−26.9723	−1.00730
$718$	0	0
$719$	−41.3327	−1.54145	−0.770725	−	0.637167i	$-0.780106\pi$
$719$	−41.3327	−1.54145	−0.770725	+	0.637167i	$0.780106\pi$
$720$	0	0
$721$	31.6102	1.17723
$722$	0	0
$723$	33.9353	1.26207
$724$	0	0
$725$	−60.2701	−2.23837
$726$	0	0
$727$	−7.66463	−0.284265	−0.142133	−	0.989848i	$-0.545396\pi$
$727$	−7.66463	−0.284265	−0.142133	+	0.989848i	$0.545396\pi$
$728$	0	0
$729$	29.9898	1.11073
$730$	0	0
$731$	−25.3123	−0.936209
$732$	0	0
$733$	13.3135	0.491746	0.245873	−	0.969302i	$-0.420925\pi$
$733$	13.3135	0.491746	0.245873	+	0.969302i	$0.420925\pi$
$734$	0	0
$735$	−20.7350	−0.764823
$736$	0	0
$737$	4.29086	0.158056
$738$	0	0
$739$	20.5478	0.755863	0.377931	−	0.925834i	$-0.376635\pi$
$739$	20.5478	0.755863	0.377931	+	0.925834i	$0.376635\pi$
$740$	0	0
$741$	−14.3058	−0.525536
$742$	0	0
$743$	−38.2897	−1.40471	−0.702356	−	0.711826i	$-0.747869\pi$
$743$	−38.2897	−1.40471	−0.702356	+	0.711826i	$0.747869\pi$
$744$	0	0
$745$	21.2015	0.776762
$746$	0	0
$747$	0.647204	0.0236800
$748$	0	0
$749$	15.6774	0.572840
$750$	0	0
$751$	−10.5542	−0.385127	−0.192563	−	0.981285i	$-0.561680\pi$
$751$	−10.5542	−0.385127	−0.192563	+	0.981285i	$0.561680\pi$
$752$	0	0
$753$	1.56031	0.0568608
$754$	0	0
$755$	−67.4531	−2.45487
$756$	0	0
$757$	−7.83636	−0.284817	−0.142409	−	0.989808i	$-0.545485\pi$
$757$	−7.83636	−0.284817	−0.142409	+	0.989808i	$0.545485\pi$
$758$	0	0
$759$	5.00915	0.181821
$760$	0	0
$761$	−11.5454	−0.418521	−0.209260	−	0.977860i	$-0.567106\pi$
$761$	−11.5454	−0.418521	−0.209260	+	0.977860i	$0.567106\pi$
$762$	0	0
$763$	−5.72026	−0.207087
$764$	0	0
$765$	−15.2307	−0.550667
$766$	0	0
$767$	34.2410	1.23637
$768$	0	0
$769$	36.9335	1.33186	0.665928	−	0.746016i	$-0.268036\pi$
$769$	36.9335	1.33186	0.665928	+	0.746016i	$0.268036\pi$
$770$	0	0
$771$	−24.9405	−0.898209
$772$	0	0
$773$	−5.46395	−0.196525	−0.0982623	−	0.995161i	$-0.531328\pi$
$773$	−5.46395	−0.196525	−0.0982623	+	0.995161i	$0.531328\pi$
$774$	0	0
$775$	28.3444	1.01816
$776$	0	0
$777$	−22.3705	−0.802535
$778$	0	0
$779$	8.25413	0.295735
$780$	0	0
$781$	−3.71241	−0.132840
$782$	0	0
$783$	48.7975	1.74388
$784$	0	0
$785$	−58.8927	−2.10197
$786$	0	0
$787$	−13.1937	−0.470303	−0.235152	−	0.971959i	$-0.575559\pi$
$787$	−13.1937	−0.470303	−0.235152	+	0.971959i	$0.575559\pi$
$788$	0	0
$789$	36.5798	1.30228
$790$	0	0
$791$	34.8090	1.23767
$792$	0	0
$793$	28.3427	1.00648
$794$	0	0
$795$	63.2969	2.24491
$796$	0	0
$797$	16.2904	0.577037	0.288518	−	0.957474i	$-0.406837\pi$
$797$	16.2904	0.577037	0.288518	+	0.957474i	$0.406837\pi$
$798$	0	0
$799$	84.2570	2.98080
$800$	0	0
$801$	−5.10539	−0.180390
$802$	0	0
$803$	0.286435	0.0101081
$804$	0	0
$805$	29.0724	1.02467
$806$	0	0
$807$	49.7545	1.75144
$808$	0	0
$809$	20.5366	0.722028	0.361014	−	0.932560i	$-0.382431\pi$
$809$	20.5366	0.722028	0.361014	+	0.932560i	$0.382431\pi$
$810$	0	0
$811$	−25.5657	−0.897734	−0.448867	−	0.893599i	$-0.648172\pi$
$811$	−25.5657	−0.897734	−0.448867	+	0.893599i	$0.648172\pi$
$812$	0	0
$813$	−27.9976	−0.981918
$814$	0	0
$815$	−59.2293	−2.07471
$816$	0	0
$817$	−11.5765	−0.405010
$818$	0	0
$819$	2.57057	0.0898230
$820$	0	0
$821$	4.73559	0.165273	0.0826366	−	0.996580i	$-0.473666\pi$
$821$	4.73559	0.165273	0.0826366	+	0.996580i	$0.473666\pi$
$822$	0	0
$823$	25.4449	0.886955	0.443477	−	0.896286i	$-0.353745\pi$
$823$	25.4449	0.886955	0.443477	+	0.896286i	$0.353745\pi$
$824$	0	0
$825$	7.23150	0.251769
$826$	0	0
$827$	−34.0190	−1.18296	−0.591479	−	0.806321i	$-0.701456\pi$
$827$	−34.0190	−1.18296	−0.591479	+	0.806321i	$0.701456\pi$
$828$	0	0
$829$	17.1136	0.594380	0.297190	−	0.954818i	$-0.403950\pi$
$829$	17.1136	0.594380	0.297190	+	0.954818i	$0.403950\pi$
$830$	0	0
$831$	−34.8273	−1.20814
$832$	0	0
$833$	−30.1534	−1.04475
$834$	0	0
$835$	20.8956	0.723123
$836$	0	0
$837$	−22.9490	−0.793233
$838$	0	0
$839$	−20.4173	−0.704885	−0.352442	−	0.935834i	$-0.614649\pi$
$839$	−20.4173	−0.704885	−0.352442	+	0.935834i	$0.614649\pi$
$840$	0	0
$841$	47.9394	1.65308
$842$	0	0
$843$	−28.0363	−0.965621
$844$	0	0
$845$	−22.1347	−0.761458
$846$	0	0
$847$	−18.6948	−0.642361
$848$	0	0
$849$	−28.3848	−0.974163
$850$	0	0
$851$	−38.4906	−1.31944
$852$	0	0
$853$	−37.7375	−1.29211	−0.646054	−	0.763291i	$-0.723582\pi$
$853$	−37.7375	−1.29211	−0.646054	+	0.763291i	$0.723582\pi$
$854$	0	0
$855$	−6.96570	−0.238222
$856$	0	0
$857$	4.51925	0.154375	0.0771874	−	0.997017i	$-0.475406\pi$
$857$	4.51925	0.154375	0.0771874	+	0.997017i	$0.475406\pi$
$858$	0	0
$859$	45.0443	1.53689	0.768446	−	0.639915i	$-0.221030\pi$
$859$	45.0443	1.53689	0.768446	+	0.639915i	$0.221030\pi$
$860$	0	0
$861$	6.38591	0.217631
$862$	0	0
$863$	−31.0023	−1.05533	−0.527665	−	0.849452i	$-0.676932\pi$
$863$	−31.0023	−1.05533	−0.527665	+	0.849452i	$0.676932\pi$
$864$	0	0
$865$	−4.28609	−0.145731
$866$	0	0
$867$	68.8387	2.33789
$868$	0	0
$869$	2.36644	0.0802760
$870$	0	0
$871$	−16.3126	−0.552730
$872$	0	0
$873$	8.90119	0.301260
$874$	0	0
$875$	11.4293	0.386380
$876$	0	0
$877$	26.3271	0.889004	0.444502	−	0.895778i	$-0.353381\pi$
$877$	26.3271	0.889004	0.444502	+	0.895778i	$0.353381\pi$
$878$	0	0
$879$	27.7690	0.936626
$880$	0	0
$881$	−9.16807	−0.308880	−0.154440	−	0.988002i	$-0.549357\pi$
$881$	−9.16807	−0.308880	−0.154440	+	0.988002i	$0.549357\pi$
$882$	0	0
$883$	−41.9421	−1.41146	−0.705732	−	0.708479i	$-0.749382\pi$
$883$	−41.9421	−1.41146	−0.705732	+	0.708479i	$0.749382\pi$
$884$	0	0
$885$	−71.7849	−2.41302
$886$	0	0
$887$	47.7785	1.60424	0.802122	−	0.597160i	$-0.203704\pi$
$887$	47.7785	1.60424	0.802122	+	0.597160i	$0.203704\pi$
$888$	0	0
$889$	22.2933	0.747694
$890$	0	0
$891$	−4.71078	−0.157817
$892$	0	0
$893$	38.5346	1.28951
$894$	0	0
$895$	33.3733	1.11555
$896$	0	0
$897$	−19.0433	−0.635838
$898$	0	0
$899$	−36.1839	−1.20680
$900$	0	0
$901$	92.0479	3.06656
$902$	0	0
$903$	−8.95628	−0.298046
$904$	0	0
$905$	−63.3938	−2.10728
$906$	0	0
$907$	−32.9952	−1.09559	−0.547793	−	0.836614i	$-0.684532\pi$
$907$	−32.9952	−1.09559	−0.547793	+	0.836614i	$0.684532\pi$
$908$	0	0
$909$	3.89026	0.129032
$910$	0	0
$911$	37.4196	1.23977	0.619884	−	0.784694i	$-0.287180\pi$
$911$	37.4196	1.23977	0.619884	+	0.784694i	$0.287180\pi$
$912$	0	0
$913$	−0.772051	−0.0255511
$914$	0	0
$915$	−59.4193	−1.96434
$916$	0	0
$917$	8.89229	0.293649
$918$	0	0
$919$	−53.7728	−1.77380	−0.886900	−	0.461961i	$-0.847146\pi$
$919$	−53.7728	−1.77380	−0.886900	+	0.461961i	$0.847146\pi$
$920$	0	0
$921$	1.85247	0.0610410
$922$	0	0
$923$	14.1135	0.464551
$924$	0	0
$925$	−55.5672	−1.82704
$926$	0	0
$927$	−10.0819	−0.331133
$928$	0	0
$929$	−44.7698	−1.46885	−0.734425	−	0.678690i	$-0.762548\pi$
$929$	−44.7698	−1.46885	−0.734425	+	0.678690i	$0.762548\pi$
$930$	0	0
$931$	−13.7905	−0.451966
$932$	0	0
$933$	11.9528	0.391318
$934$	0	0
$935$	18.1687	0.594180
$936$	0	0
$937$	5.60717	0.183178	0.0915891	−	0.995797i	$-0.470805\pi$
$937$	5.60717	0.183178	0.0915891	+	0.995797i	$0.470805\pi$
$938$	0	0
$939$	4.53438	0.147974
$940$	0	0
$941$	−21.9944	−0.716998	−0.358499	−	0.933530i	$-0.616711\pi$
$941$	−21.9944	−0.716998	−0.358499	+	0.933530i	$0.616711\pi$
$942$	0	0
$943$	10.9876	0.357805
$944$	0	0
$945$	−33.9815	−1.10542
$946$	0	0
$947$	26.0002	0.844892	0.422446	−	0.906388i	$-0.361172\pi$
$947$	26.0002	0.844892	0.422446	+	0.906388i	$0.361172\pi$
$948$	0	0
$949$	−1.08894	−0.0353485
$950$	0	0
$951$	−20.6642	−0.670081
$952$	0	0
$953$	−8.62670	−0.279446	−0.139723	−	0.990191i	$-0.544621\pi$
$953$	−8.62670	−0.279446	−0.139723	+	0.990191i	$0.544621\pi$
$954$	0	0
$955$	−70.0898	−2.26805
$956$	0	0
$957$	−9.23158	−0.298415
$958$	0	0
$959$	−28.5157	−0.920821
$960$	0	0
$961$	−13.9831	−0.451068
$962$	0	0
$963$	−5.00021	−0.161129
$964$	0	0
$965$	73.2567	2.35822
$966$	0	0
$967$	−32.5529	−1.04683	−0.523415	−	0.852078i	$-0.675342\pi$
$967$	−32.5529	−1.04683	−0.523415	+	0.852078i	$0.675342\pi$
$968$	0	0
$969$	43.6143	1.40109
$970$	0	0
$971$	11.7797	0.378029	0.189015	−	0.981974i	$-0.439471\pi$
$971$	11.7797	0.378029	0.189015	+	0.981974i	$0.439471\pi$
$972$	0	0
$973$	31.3084	1.00370
$974$	0	0
$975$	−27.4920	−0.880450
$976$	0	0
$977$	4.47319	0.143110	0.0715551	−	0.997437i	$-0.477204\pi$
$977$	4.47319	0.143110	0.0715551	+	0.997437i	$0.477204\pi$
$978$	0	0
$979$	6.09023	0.194644
$980$	0	0
$981$	1.82444	0.0582500
$982$	0	0
$983$	−14.5080	−0.462733	−0.231367	−	0.972867i	$-0.574320\pi$
$983$	−14.5080	−0.462733	−0.231367	+	0.972867i	$0.574320\pi$
$984$	0	0
$985$	−5.73295	−0.182667
$986$	0	0
$987$	29.8128	0.948950
$988$	0	0
$989$	−15.4102	−0.490015
$990$	0	0
$991$	−11.4226	−0.362851	−0.181425	−	0.983405i	$-0.558071\pi$
$991$	−11.4226	−0.362851	−0.181425	+	0.983405i	$0.558071\pi$
$992$	0	0
$993$	−7.93788	−0.251901
$994$	0	0
$995$	−41.7992	−1.32512
$996$	0	0
$997$	−46.0716	−1.45910	−0.729551	−	0.683926i	$-0.760271\pi$
$997$	−46.0716	−1.45910	−0.729551	+	0.683926i	$0.760271\pi$
$998$	0	0
$999$	44.9899	1.42342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.13		19
4.3	odd	2		2008.2.a.c.1.7	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.7	✓	19	4.3	odd	2
4016.2.a.l.1.13		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)

\(f(q)\)	\(=\)	\(q+1.56031 q^{3} +3.44545 q^{5} +1.77285 q^{7} -0.565441 q^{9} +O(q^{10})\)	\(q+1.56031 q^{3} +3.44545 q^{5} +1.77285 q^{7} -0.565441 q^{9} +0.674515 q^{11} -2.56431 q^{13} +5.37596 q^{15} +7.81784 q^{17} +3.57546 q^{19} +2.76620 q^{21} +4.75952 q^{23} +6.87111 q^{25} -5.56318 q^{27} -8.77151 q^{29} +4.12516 q^{31} +1.05245 q^{33} +6.10827 q^{35} -8.08708 q^{37} -4.00110 q^{39} +2.30855 q^{41} -3.23776 q^{43} -1.94820 q^{45} +10.7775 q^{47} -3.85699 q^{49} +12.1982 q^{51} +11.7741 q^{53} +2.32400 q^{55} +5.57882 q^{57} -13.3529 q^{59} -11.0528 q^{61} -1.00244 q^{63} -8.83518 q^{65} +6.36140 q^{67} +7.42631 q^{69} -5.50382 q^{71} +0.424653 q^{73} +10.7210 q^{75} +1.19582 q^{77} +3.50836 q^{79} -6.98395 q^{81} -1.14460 q^{83} +26.9360 q^{85} -13.6863 q^{87} +9.02905 q^{89} -4.54614 q^{91} +6.43651 q^{93} +12.3191 q^{95} -15.7420 q^{97} -0.381398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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Groups

Database

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