LMFDB - Embedded newform 4016.2.a.l.1.2

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			1.2
Root			$-2.63379$ 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)

$f(q)$	$=$	$q-2.63379 q^{3} -2.05129 q^{5} +4.98850 q^{7} +3.93685 q^{9} +O(q^{10})$	$q-2.63379 q^{3} -2.05129 q^{5} +4.98850 q^{7} +3.93685 q^{9} +0.504257 q^{11} +4.69381 q^{13} +5.40267 q^{15} +5.45248 q^{17} +5.31235 q^{19} -13.1387 q^{21} +4.77559 q^{23} -0.792198 q^{25} -2.46746 q^{27} -5.28107 q^{29} +9.88935 q^{31} -1.32811 q^{33} -10.2329 q^{35} +7.71429 q^{37} -12.3625 q^{39} -4.02525 q^{41} +7.33084 q^{43} -8.07562 q^{45} +8.07643 q^{47} +17.8852 q^{49} -14.3607 q^{51} -7.42905 q^{53} -1.03438 q^{55} -13.9916 q^{57} -8.76097 q^{59} -8.02599 q^{61} +19.6390 q^{63} -9.62837 q^{65} +7.49585 q^{67} -12.5779 q^{69} +0.654390 q^{71} -8.43587 q^{73} +2.08648 q^{75} +2.51549 q^{77} -14.0954 q^{79} -5.31178 q^{81} -11.4288 q^{83} -11.1846 q^{85} +13.9092 q^{87} -11.9299 q^{89} +23.4151 q^{91} -26.0465 q^{93} -10.8972 q^{95} +7.92112 q^{97} +1.98518 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.63379	−1.52062	−0.760309	−	0.649561i	$-0.774953\pi$
$3$	−2.63379	−1.52062	−0.760309	+	0.649561i	$0.774953\pi$
$4$	0	0
$5$	−2.05129	−0.917366	−0.458683	−	0.888600i	$-0.651679\pi$
$5$	−2.05129	−0.917366	−0.458683	+	0.888600i	$0.651679\pi$
$6$	0	0
$7$	4.98850	1.88548	0.942738	−	0.333533i	$-0.108241\pi$
$7$	4.98850	1.88548	0.942738	+	0.333533i	$0.108241\pi$
$8$	0	0
$9$	3.93685	1.31228
$10$	0	0
$11$	0.504257	0.152039	0.0760195	−	0.997106i	$-0.475779\pi$
$11$	0.504257	0.152039	0.0760195	+	0.997106i	$0.475779\pi$
$12$	0	0
$13$	4.69381	1.30183	0.650914	−	0.759151i	$-0.274386\pi$
$13$	4.69381	1.30183	0.650914	+	0.759151i	$0.274386\pi$
$14$	0	0
$15$	5.40267	1.39496
$16$	0	0
$17$	5.45248	1.32242	0.661211	−	0.750200i	$-0.270043\pi$
$17$	5.45248	1.32242	0.661211	+	0.750200i	$0.270043\pi$
$18$	0	0
$19$	5.31235	1.21874	0.609369	−	0.792887i	$-0.291423\pi$
$19$	5.31235	1.21874	0.609369	+	0.792887i	$0.291423\pi$
$20$	0	0
$21$	−13.1387	−2.86709
$22$	0	0
$23$	4.77559	0.995779	0.497890	−	0.867240i	$-0.334108\pi$
$23$	4.77559	0.995779	0.497890	+	0.867240i	$0.334108\pi$
$24$	0	0
$25$	−0.792198	−0.158440
$26$	0	0
$27$	−2.46746	−0.474862
$28$	0	0
$29$	−5.28107	−0.980670	−0.490335	−	0.871534i	$-0.663126\pi$
$29$	−5.28107	−0.980670	−0.490335	+	0.871534i	$0.663126\pi$
$30$	0	0
$31$	9.88935	1.77618	0.888089	−	0.459671i	$-0.152033\pi$
$31$	9.88935	1.77618	0.888089	+	0.459671i	$0.152033\pi$
$32$	0	0
$33$	−1.32811	−0.231194
$34$	0	0
$35$	−10.2329	−1.72967
$36$	0	0
$37$	7.71429	1.26822	0.634111	−	0.773242i	$-0.281366\pi$
$37$	7.71429	1.26822	0.634111	+	0.773242i	$0.281366\pi$
$38$	0	0
$39$	−12.3625	−1.97958
$40$	0	0
$41$	−4.02525	−0.628638	−0.314319	−	0.949317i	$-0.601776\pi$
$41$	−4.02525	−0.628638	−0.314319	+	0.949317i	$0.601776\pi$
$42$	0	0
$43$	7.33084	1.11794	0.558971	−	0.829187i	$-0.311196\pi$
$43$	7.33084	1.11794	0.558971	+	0.829187i	$0.311196\pi$
$44$	0	0
$45$	−8.07562	−1.20384
$46$	0	0
$47$	8.07643	1.17807	0.589035	−	0.808108i	$-0.299508\pi$
$47$	8.07643	1.17807	0.589035	+	0.808108i	$0.299508\pi$
$48$	0	0
$49$	17.8852	2.55502
$50$	0	0
$51$	−14.3607	−2.01090
$52$	0	0
$53$	−7.42905	−1.02046	−0.510229	−	0.860039i	$-0.670439\pi$
$53$	−7.42905	−1.02046	−0.510229	+	0.860039i	$0.670439\pi$
$54$	0	0
$55$	−1.03438	−0.139475
$56$	0	0
$57$	−13.9916	−1.85324
$58$	0	0
$59$	−8.76097	−1.14058	−0.570291	−	0.821443i	$-0.693169\pi$
$59$	−8.76097	−1.14058	−0.570291	+	0.821443i	$0.693169\pi$
$60$	0	0
$61$	−8.02599	−1.02762	−0.513812	−	0.857903i	$-0.671767\pi$
$61$	−8.02599	−1.02762	−0.513812	+	0.857903i	$0.671767\pi$
$62$	0	0
$63$	19.6390	2.47428
$64$	0	0
$65$	−9.62837	−1.19425
$66$	0	0
$67$	7.49585	0.915764	0.457882	−	0.889013i	$-0.348608\pi$
$67$	7.49585	0.915764	0.457882	+	0.889013i	$0.348608\pi$
$68$	0	0
$69$	−12.5779	−1.51420
$70$	0	0
$71$	0.654390	0.0776618	0.0388309	−	0.999246i	$-0.487637\pi$
$71$	0.654390	0.0776618	0.0388309	+	0.999246i	$0.487637\pi$
$72$	0	0
$73$	−8.43587	−0.987344	−0.493672	−	0.869648i	$-0.664346\pi$
$73$	−8.43587	−0.987344	−0.493672	+	0.869648i	$0.664346\pi$
$74$	0	0
$75$	2.08648	0.240926
$76$	0	0
$77$	2.51549	0.286666
$78$	0	0
$79$	−14.0954	−1.58586	−0.792929	−	0.609314i	$-0.791445\pi$
$79$	−14.0954	−1.58586	−0.792929	+	0.609314i	$0.791445\pi$
$80$	0	0
$81$	−5.31178	−0.590198
$82$	0	0
$83$	−11.4288	−1.25447	−0.627235	−	0.778830i	$-0.715814\pi$
$83$	−11.4288	−1.25447	−0.627235	+	0.778830i	$0.715814\pi$
$84$	0	0
$85$	−11.1846	−1.21314
$86$	0	0
$87$	13.9092	1.49123
$88$	0	0
$89$	−11.9299	−1.26456	−0.632281	−	0.774739i	$-0.717881\pi$
$89$	−11.9299	−1.26456	−0.632281	+	0.774739i	$0.717881\pi$
$90$	0	0
$91$	23.4151	2.45457
$92$	0	0
$93$	−26.0465	−2.70089
$94$	0	0
$95$	−10.8972	−1.11803
$96$	0	0
$97$	7.92112	0.804268	0.402134	−	0.915581i	$-0.368269\pi$
$97$	7.92112	0.804268	0.402134	+	0.915581i	$0.368269\pi$
$98$	0	0
$99$	1.98518	0.199518
$100$	0	0
$101$	−0.732173	−0.0728539	−0.0364270	−	0.999336i	$-0.511598\pi$
$101$	−0.732173	−0.0728539	−0.0364270	+	0.999336i	$0.511598\pi$
$102$	0	0
$103$	5.65694	0.557395	0.278697	−	0.960379i	$-0.410097\pi$
$103$	5.65694	0.557395	0.278697	+	0.960379i	$0.410097\pi$
$104$	0	0
$105$	26.9512	2.63017
$106$	0	0
$107$	−2.89857	−0.280215	−0.140108	−	0.990136i	$-0.544745\pi$
$107$	−2.89857	−0.280215	−0.140108	+	0.990136i	$0.544745\pi$
$108$	0	0
$109$	−2.07840	−0.199074	−0.0995372	−	0.995034i	$-0.531736\pi$
$109$	−2.07840	−0.199074	−0.0995372	+	0.995034i	$0.531736\pi$
$110$	0	0
$111$	−20.3178	−1.92848
$112$	0	0
$113$	16.6507	1.56637	0.783184	−	0.621790i	$-0.213594\pi$
$113$	16.6507	1.56637	0.783184	+	0.621790i	$0.213594\pi$
$114$	0	0
$115$	−9.79613	−0.913494
$116$	0	0
$117$	18.4788	1.70837
$118$	0	0
$119$	27.1997	2.49339
$120$	0	0
$121$	−10.7457	−0.976884
$122$	0	0
$123$	10.6017	0.955919
$124$	0	0
$125$	11.8815	1.06271
$126$	0	0
$127$	−3.16921	−0.281222	−0.140611	−	0.990065i	$-0.544907\pi$
$127$	−3.16921	−0.281222	−0.140611	+	0.990065i	$0.544907\pi$
$128$	0	0
$129$	−19.3079	−1.69997
$130$	0	0
$131$	4.66204	0.407324	0.203662	−	0.979041i	$-0.434716\pi$
$131$	4.66204	0.407324	0.203662	+	0.979041i	$0.434716\pi$
$132$	0	0
$133$	26.5007	2.29790
$134$	0	0
$135$	5.06147	0.435622
$136$	0	0
$137$	−1.23759	−0.105735	−0.0528673	−	0.998602i	$-0.516836\pi$
$137$	−1.23759	−0.105735	−0.0528673	+	0.998602i	$0.516836\pi$
$138$	0	0
$139$	−7.53875	−0.639428	−0.319714	−	0.947514i	$-0.603587\pi$
$139$	−7.53875	−0.639428	−0.319714	+	0.947514i	$0.603587\pi$
$140$	0	0
$141$	−21.2716	−1.79139
$142$	0	0
$143$	2.36688	0.197929
$144$	0	0
$145$	10.8330	0.899634
$146$	0	0
$147$	−47.1057	−3.88522
$148$	0	0
$149$	8.09426	0.663108	0.331554	−	0.943436i	$-0.392427\pi$
$149$	8.09426	0.663108	0.331554	+	0.943436i	$0.392427\pi$
$150$	0	0
$151$	14.9560	1.21710	0.608550	−	0.793515i	$-0.291751\pi$
$151$	14.9560	1.21710	0.608550	+	0.793515i	$0.291751\pi$
$152$	0	0
$153$	21.4656	1.73539
$154$	0	0
$155$	−20.2859	−1.62941
$156$	0	0
$157$	22.5946	1.80325	0.901624	−	0.432521i	$-0.142376\pi$
$157$	22.5946	1.80325	0.901624	+	0.432521i	$0.142376\pi$
$158$	0	0
$159$	19.5665	1.55173
$160$	0	0
$161$	23.8230	1.87752
$162$	0	0
$163$	−19.8892	−1.55784	−0.778921	−	0.627122i	$-0.784233\pi$
$163$	−19.8892	−1.55784	−0.778921	+	0.627122i	$0.784233\pi$
$164$	0	0
$165$	2.72433	0.212089
$166$	0	0
$167$	14.4722	1.11989	0.559947	−	0.828528i	$-0.310822\pi$
$167$	14.4722	1.11989	0.559947	+	0.828528i	$0.310822\pi$
$168$	0	0
$169$	9.03184	0.694757
$170$	0	0
$171$	20.9139	1.59933
$172$	0	0
$173$	−0.440563	−0.0334954	−0.0167477	−	0.999860i	$-0.505331\pi$
$173$	−0.440563	−0.0334954	−0.0167477	+	0.999860i	$0.505331\pi$
$174$	0	0
$175$	−3.95188	−0.298734
$176$	0	0
$177$	23.0746	1.73439
$178$	0	0
$179$	24.3916	1.82311	0.911557	−	0.411174i	$-0.134881\pi$
$179$	24.3916	1.82311	0.911557	+	0.411174i	$0.134881\pi$
$180$	0	0
$181$	22.9366	1.70486	0.852431	−	0.522840i	$-0.175127\pi$
$181$	22.9366	1.70486	0.852431	+	0.522840i	$0.175127\pi$
$182$	0	0
$183$	21.1388	1.56262
$184$	0	0
$185$	−15.8243	−1.16342
$186$	0	0
$187$	2.74945	0.201060
$188$	0	0
$189$	−12.3089	−0.895341
$190$	0	0
$191$	4.45146	0.322096	0.161048	−	0.986947i	$-0.448513\pi$
$191$	4.45146	0.322096	0.161048	+	0.986947i	$0.448513\pi$
$192$	0	0
$193$	−17.0710	−1.22880	−0.614400	−	0.788994i	$-0.710602\pi$
$193$	−17.0710	−1.22880	−0.614400	+	0.788994i	$0.710602\pi$
$194$	0	0
$195$	25.3591	1.81600
$196$	0	0
$197$	−4.78711	−0.341067	−0.170534	−	0.985352i	$-0.554549\pi$
$197$	−4.78711	−0.341067	−0.170534	+	0.985352i	$0.554549\pi$
$198$	0	0
$199$	−24.0781	−1.70685	−0.853425	−	0.521215i	$-0.825479\pi$
$199$	−24.0781	−1.70685	−0.853425	+	0.521215i	$0.825479\pi$
$200$	0	0
$201$	−19.7425	−1.39253
$202$	0	0
$203$	−26.3446	−1.84903
$204$	0	0
$205$	8.25696	0.576691
$206$	0	0
$207$	18.8008	1.30674
$208$	0	0
$209$	2.67879	0.185296
$210$	0	0
$211$	−16.1062	−1.10880	−0.554400	−	0.832250i	$-0.687052\pi$
$211$	−16.1062	−1.10880	−0.554400	+	0.832250i	$0.687052\pi$
$212$	0	0
$213$	−1.72353	−0.118094
$214$	0	0
$215$	−15.0377	−1.02556
$216$	0	0
$217$	49.3330	3.34894
$218$	0	0
$219$	22.2183	1.50137
$220$	0	0
$221$	25.5929	1.72157
$222$	0	0
$223$	−12.9327	−0.866040	−0.433020	−	0.901384i	$-0.642552\pi$
$223$	−12.9327	−0.866040	−0.433020	+	0.901384i	$0.642552\pi$
$224$	0	0
$225$	−3.11876	−0.207917
$226$	0	0
$227$	−2.20043	−0.146048	−0.0730239	−	0.997330i	$-0.523265\pi$
$227$	−2.20043	−0.146048	−0.0730239	+	0.997330i	$0.523265\pi$
$228$	0	0
$229$	−17.6297	−1.16500	−0.582501	−	0.812830i	$-0.697926\pi$
$229$	−17.6297	−1.16500	−0.582501	+	0.812830i	$0.697926\pi$
$230$	0	0
$231$	−6.62526	−0.435910
$232$	0	0
$233$	8.23309	0.539368	0.269684	−	0.962949i	$-0.413081\pi$
$233$	8.23309	0.539368	0.269684	+	0.962949i	$0.413081\pi$
$234$	0	0
$235$	−16.5671	−1.08072
$236$	0	0
$237$	37.1244	2.41149
$238$	0	0
$239$	−1.99841	−0.129266	−0.0646331	−	0.997909i	$-0.520588\pi$
$239$	−1.99841	−0.129266	−0.0646331	+	0.997909i	$0.520588\pi$
$240$	0	0
$241$	−8.43753	−0.543509	−0.271754	−	0.962367i	$-0.587604\pi$
$241$	−8.43753	−0.543509	−0.271754	+	0.962367i	$0.587604\pi$
$242$	0	0
$243$	21.3925	1.37233
$244$	0	0
$245$	−36.6877	−2.34389
$246$	0	0
$247$	24.9352	1.58659
$248$	0	0
$249$	30.1010	1.90757
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	2.40812	0.151397
$254$	0	0
$255$	29.4580	1.84473
$256$	0	0
$257$	12.7245	0.793729	0.396865	−	0.917877i	$-0.370098\pi$
$257$	12.7245	0.793729	0.396865	+	0.917877i	$0.370098\pi$
$258$	0	0
$259$	38.4827	2.39120
$260$	0	0
$261$	−20.7908	−1.28692
$262$	0	0
$263$	−14.7861	−0.911749	−0.455875	−	0.890044i	$-0.650673\pi$
$263$	−14.7861	−0.911749	−0.455875	+	0.890044i	$0.650673\pi$
$264$	0	0
$265$	15.2391	0.936133
$266$	0	0
$267$	31.4207	1.92292
$268$	0	0
$269$	−24.7023	−1.50613	−0.753064	−	0.657948i	$-0.771425\pi$
$269$	−24.7023	−1.50613	−0.753064	+	0.657948i	$0.771425\pi$
$270$	0	0
$271$	−10.0066	−0.607856	−0.303928	−	0.952695i	$-0.598298\pi$
$271$	−10.0066	−0.607856	−0.303928	+	0.952695i	$0.598298\pi$
$272$	0	0
$273$	−61.6704	−3.73246
$274$	0	0
$275$	−0.399471	−0.0240890
$276$	0	0
$277$	−7.12611	−0.428166	−0.214083	−	0.976815i	$-0.568676\pi$
$277$	−7.12611	−0.428166	−0.214083	+	0.976815i	$0.568676\pi$
$278$	0	0
$279$	38.9328	2.33085
$280$	0	0
$281$	12.5505	0.748699	0.374350	−	0.927288i	$-0.377866\pi$
$281$	12.5505	0.748699	0.374350	+	0.927288i	$0.377866\pi$
$282$	0	0
$283$	−10.0645	−0.598270	−0.299135	−	0.954211i	$-0.596698\pi$
$283$	−10.0645	−0.598270	−0.299135	+	0.954211i	$0.596698\pi$
$284$	0	0
$285$	28.7009	1.70010
$286$	0	0
$287$	−20.0800	−1.18528
$288$	0	0
$289$	12.7296	0.748798
$290$	0	0
$291$	−20.8626	−1.22298
$292$	0	0
$293$	−17.5749	−1.02674	−0.513370	−	0.858168i	$-0.671603\pi$
$293$	−17.5749	−1.02674	−0.513370	+	0.858168i	$0.671603\pi$
$294$	0	0
$295$	17.9713	1.04633
$296$	0	0
$297$	−1.24423	−0.0721976
$298$	0	0
$299$	22.4157	1.29633
$300$	0	0
$301$	36.5699	2.10786
$302$	0	0
$303$	1.92839	0.110783
$304$	0	0
$305$	16.4637	0.942707
$306$	0	0
$307$	−16.7038	−0.953337	−0.476668	−	0.879083i	$-0.658156\pi$
$307$	−16.7038	−0.953337	−0.476668	+	0.879083i	$0.658156\pi$
$308$	0	0
$309$	−14.8992	−0.847585
$310$	0	0
$311$	8.91764	0.505673	0.252836	−	0.967509i	$-0.418637\pi$
$311$	8.91764	0.505673	0.252836	+	0.967509i	$0.418637\pi$
$312$	0	0
$313$	−17.7910	−1.00561	−0.502803	−	0.864401i	$-0.667698\pi$
$313$	−17.7910	−1.00561	−0.502803	+	0.864401i	$0.667698\pi$
$314$	0	0
$315$	−40.2853	−2.26982
$316$	0	0
$317$	−7.72756	−0.434023	−0.217012	−	0.976169i	$-0.569631\pi$
$317$	−7.72756	−0.434023	−0.217012	+	0.976169i	$0.569631\pi$
$318$	0	0
$319$	−2.66301	−0.149100
$320$	0	0
$321$	7.63421	0.426100
$322$	0	0
$323$	28.9655	1.61168
$324$	0	0
$325$	−3.71843	−0.206261
$326$	0	0
$327$	5.47406	0.302716
$328$	0	0
$329$	40.2893	2.22122
$330$	0	0
$331$	21.6525	1.19013	0.595064	−	0.803678i	$-0.297127\pi$
$331$	21.6525	1.19013	0.595064	+	0.803678i	$0.297127\pi$
$332$	0	0
$333$	30.3700	1.66426
$334$	0	0
$335$	−15.3762	−0.840091
$336$	0	0
$337$	−5.64597	−0.307556	−0.153778	−	0.988105i	$-0.549144\pi$
$337$	−5.64597	−0.307556	−0.153778	+	0.988105i	$0.549144\pi$
$338$	0	0
$339$	−43.8545	−2.38185
$340$	0	0
$341$	4.98677	0.270049
$342$	0	0
$343$	54.3006	2.93196
$344$	0	0
$345$	25.8010	1.38908
$346$	0	0
$347$	−6.67121	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$347$	−6.67121	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$348$	0	0
$349$	−13.6537	−0.730866	−0.365433	−	0.930838i	$-0.619079\pi$
$349$	−13.6537	−0.730866	−0.365433	+	0.930838i	$0.619079\pi$
$350$	0	0
$351$	−11.5818	−0.618189
$352$	0	0
$353$	−34.7772	−1.85100	−0.925502	−	0.378743i	$-0.876356\pi$
$353$	−34.7772	−1.85100	−0.925502	+	0.378743i	$0.876356\pi$
$354$	0	0
$355$	−1.34235	−0.0712443
$356$	0	0
$357$	−71.6383	−3.79150
$358$	0	0
$359$	26.0446	1.37458	0.687290	−	0.726383i	$-0.258800\pi$
$359$	26.0446	1.37458	0.687290	+	0.726383i	$0.258800\pi$
$360$	0	0
$361$	9.22111	0.485322
$362$	0	0
$363$	28.3020	1.48547
$364$	0	0
$365$	17.3044	0.905756
$366$	0	0
$367$	17.3520	0.905769	0.452885	−	0.891569i	$-0.350395\pi$
$367$	17.3520	0.905769	0.452885	+	0.891569i	$0.350395\pi$
$368$	0	0
$369$	−15.8468	−0.824950
$370$	0	0
$371$	−37.0598	−1.92405
$372$	0	0
$373$	31.1518	1.61298	0.806491	−	0.591247i	$-0.201364\pi$
$373$	31.1518	1.61298	0.806491	+	0.591247i	$0.201364\pi$
$374$	0	0
$375$	−31.2934	−1.61598
$376$	0	0
$377$	−24.7883	−1.27666
$378$	0	0
$379$	−8.32284	−0.427516	−0.213758	−	0.976887i	$-0.568570\pi$
$379$	−8.32284	−0.427516	−0.213758	+	0.976887i	$0.568570\pi$
$380$	0	0
$381$	8.34703	0.427632
$382$	0	0
$383$	12.9642	0.662438	0.331219	−	0.943554i	$-0.392540\pi$
$383$	12.9642	0.662438	0.331219	+	0.943554i	$0.392540\pi$
$384$	0	0
$385$	−5.16000	−0.262978
$386$	0	0
$387$	28.8604	1.46706
$388$	0	0
$389$	−13.2655	−0.672588	−0.336294	−	0.941757i	$-0.609174\pi$
$389$	−13.2655	−0.672588	−0.336294	+	0.941757i	$0.609174\pi$
$390$	0	0
$391$	26.0388	1.31684
$392$	0	0
$393$	−12.2788	−0.619385
$394$	0	0
$395$	28.9138	1.45481
$396$	0	0
$397$	24.1793	1.21352	0.606762	−	0.794884i	$-0.292468\pi$
$397$	24.1793	1.21352	0.606762	+	0.794884i	$0.292468\pi$
$398$	0	0
$399$	−69.7972	−3.49423
$400$	0	0
$401$	27.5226	1.37441	0.687207	−	0.726461i	$-0.258836\pi$
$401$	27.5226	1.37441	0.687207	+	0.726461i	$0.258836\pi$
$402$	0	0
$403$	46.4187	2.31228
$404$	0	0
$405$	10.8960	0.541427
$406$	0	0
$407$	3.88998	0.192819
$408$	0	0
$409$	−3.45814	−0.170994	−0.0854971	−	0.996338i	$-0.527248\pi$
$409$	−3.45814	−0.170994	−0.0854971	+	0.996338i	$0.527248\pi$
$410$	0	0
$411$	3.25956	0.160782
$412$	0	0
$413$	−43.7041	−2.15054
$414$	0	0
$415$	23.4438	1.15081
$416$	0	0
$417$	19.8555	0.972327
$418$	0	0
$419$	0.821777	0.0401464	0.0200732	−	0.999799i	$-0.493610\pi$
$419$	0.821777	0.0401464	0.0200732	+	0.999799i	$0.493610\pi$
$420$	0	0
$421$	13.4416	0.655102	0.327551	−	0.944833i	$-0.393777\pi$
$421$	13.4416	0.655102	0.327551	+	0.944833i	$0.393777\pi$
$422$	0	0
$423$	31.7957	1.54596
$424$	0	0
$425$	−4.31945	−0.209524
$426$	0	0
$427$	−40.0377	−1.93756
$428$	0	0
$429$	−6.23387	−0.300974
$430$	0	0
$431$	8.09923	0.390126	0.195063	−	0.980791i	$-0.437509\pi$
$431$	8.09923	0.390126	0.195063	+	0.980791i	$0.437509\pi$
$432$	0	0
$433$	−11.3542	−0.545646	−0.272823	−	0.962064i	$-0.587957\pi$
$433$	−11.3542	−0.545646	−0.272823	+	0.962064i	$0.587957\pi$
$434$	0	0
$435$	−28.5319	−1.36800
$436$	0	0
$437$	25.3696	1.21359
$438$	0	0
$439$	−1.81101	−0.0864350	−0.0432175	−	0.999066i	$-0.513761\pi$
$439$	−1.81101	−0.0864350	−0.0432175	+	0.999066i	$0.513761\pi$
$440$	0	0
$441$	70.4111	3.35291
$442$	0	0
$443$	38.4884	1.82864	0.914319	−	0.404996i	$-0.132727\pi$
$443$	38.4884	1.82864	0.914319	+	0.404996i	$0.132727\pi$
$444$	0	0
$445$	24.4716	1.16007
$446$	0	0
$447$	−21.3186	−1.00833
$448$	0	0
$449$	−6.65117	−0.313888	−0.156944	−	0.987608i	$-0.550164\pi$
$449$	−6.65117	−0.313888	−0.156944	+	0.987608i	$0.550164\pi$
$450$	0	0
$451$	−2.02976	−0.0955775
$452$	0	0
$453$	−39.3909	−1.85075
$454$	0	0
$455$	−48.0312	−2.25174
$456$	0	0
$457$	15.6248	0.730897	0.365449	−	0.930831i	$-0.380916\pi$
$457$	15.6248	0.730897	0.365449	+	0.930831i	$0.380916\pi$
$458$	0	0
$459$	−13.4538	−0.627968
$460$	0	0
$461$	−42.0880	−1.96023	−0.980116	−	0.198423i	$-0.936418\pi$
$461$	−42.0880	−1.96023	−0.980116	+	0.198423i	$0.936418\pi$
$462$	0	0
$463$	−34.3098	−1.59451	−0.797255	−	0.603643i	$-0.793715\pi$
$463$	−34.3098	−1.59451	−0.797255	+	0.603643i	$0.793715\pi$
$464$	0	0
$465$	53.4289	2.47771
$466$	0	0
$467$	3.39652	0.157172	0.0785862	−	0.996907i	$-0.474959\pi$
$467$	3.39652	0.157172	0.0785862	+	0.996907i	$0.474959\pi$
$468$	0	0
$469$	37.3931	1.72665
$470$	0	0
$471$	−59.5095	−2.74205
$472$	0	0
$473$	3.69663	0.169971
$474$	0	0
$475$	−4.20844	−0.193096
$476$	0	0
$477$	−29.2470	−1.33913
$478$	0	0
$479$	−14.8466	−0.678360	−0.339180	−	0.940722i	$-0.610150\pi$
$479$	−14.8466	−0.678360	−0.339180	+	0.940722i	$0.610150\pi$
$480$	0	0
$481$	36.2094	1.65101
$482$	0	0
$483$	−62.7449	−2.85499
$484$	0	0
$485$	−16.2485	−0.737808
$486$	0	0
$487$	−18.4603	−0.836516	−0.418258	−	0.908328i	$-0.637359\pi$
$487$	−18.4603	−0.836516	−0.418258	+	0.908328i	$0.637359\pi$
$488$	0	0
$489$	52.3840	2.36889
$490$	0	0
$491$	−34.7063	−1.56627	−0.783137	−	0.621849i	$-0.786382\pi$
$491$	−34.7063	−1.56627	−0.783137	+	0.621849i	$0.786382\pi$
$492$	0	0
$493$	−28.7949	−1.29686
$494$	0	0
$495$	−4.07219	−0.183031
$496$	0	0
$497$	3.26443	0.146430
$498$	0	0
$499$	5.98899	0.268104	0.134052	−	0.990974i	$-0.457201\pi$
$499$	5.98899	0.268104	0.134052	+	0.990974i	$0.457201\pi$
$500$	0	0
$501$	−38.1168	−1.70293
$502$	0	0
$503$	−22.1063	−0.985672	−0.492836	−	0.870122i	$-0.664040\pi$
$503$	−22.1063	−0.985672	−0.492836	+	0.870122i	$0.664040\pi$
$504$	0	0
$505$	1.50190	0.0668337
$506$	0	0
$507$	−23.7880	−1.05646
$508$	0	0
$509$	−26.0949	−1.15664	−0.578318	−	0.815812i	$-0.696291\pi$
$509$	−26.0949	−1.15664	−0.578318	+	0.815812i	$0.696291\pi$
$510$	0	0
$511$	−42.0824	−1.86161
$512$	0	0
$513$	−13.1080	−0.578732
$514$	0	0
$515$	−11.6040	−0.511335
$516$	0	0
$517$	4.07260	0.179113
$518$	0	0
$519$	1.16035	0.0509338
$520$	0	0
$521$	−1.18147	−0.0517613	−0.0258807	−	0.999665i	$-0.508239\pi$
$521$	−1.18147	−0.0517613	−0.0258807	+	0.999665i	$0.508239\pi$
$522$	0	0
$523$	−0.247901	−0.0108399	−0.00541997	−	0.999985i	$-0.501725\pi$
$523$	−0.247901	−0.0108399	−0.00541997	+	0.999985i	$0.501725\pi$
$524$	0	0
$525$	10.4084	0.454261
$526$	0	0
$527$	53.9215	2.34886
$528$	0	0
$529$	−0.193739	−0.00842342
$530$	0	0
$531$	−34.4906	−1.49676
$532$	0	0
$533$	−18.8937	−0.818379
$534$	0	0
$535$	5.94581	0.257060
$536$	0	0
$537$	−64.2423	−2.77226
$538$	0	0
$539$	9.01871	0.388463
$540$	0	0
$541$	26.7975	1.15212	0.576058	−	0.817409i	$-0.304590\pi$
$541$	26.7975	1.15212	0.576058	+	0.817409i	$0.304590\pi$
$542$	0	0
$543$	−60.4101	−2.59245
$544$	0	0
$545$	4.26340	0.182624
$546$	0	0
$547$	−14.6713	−0.627301	−0.313650	−	0.949539i	$-0.601552\pi$
$547$	−14.6713	−0.627301	−0.313650	+	0.949539i	$0.601552\pi$
$548$	0	0
$549$	−31.5971	−1.34853
$550$	0	0
$551$	−28.0549	−1.19518
$552$	0	0
$553$	−70.3150	−2.99010
$554$	0	0
$555$	41.6778	1.76912
$556$	0	0
$557$	15.6376	0.662586	0.331293	−	0.943528i	$-0.392515\pi$
$557$	15.6376	0.662586	0.331293	+	0.943528i	$0.392515\pi$
$558$	0	0
$559$	34.4096	1.45537
$560$	0	0
$561$	−7.24147	−0.305735
$562$	0	0
$563$	−16.1006	−0.678559	−0.339279	−	0.940686i	$-0.610183\pi$
$563$	−16.1006	−0.678559	−0.339279	+	0.940686i	$0.610183\pi$
$564$	0	0
$565$	−34.1555	−1.43693
$566$	0	0
$567$	−26.4978	−1.11280
$568$	0	0
$569$	−33.8521	−1.41915	−0.709577	−	0.704628i	$-0.751114\pi$
$569$	−33.8521	−1.41915	−0.709577	+	0.704628i	$0.751114\pi$
$570$	0	0
$571$	17.4415	0.729904	0.364952	−	0.931026i	$-0.381085\pi$
$571$	17.4415	0.729904	0.364952	+	0.931026i	$0.381085\pi$
$572$	0	0
$573$	−11.7242	−0.489786
$574$	0	0
$575$	−3.78321	−0.157771
$576$	0	0
$577$	−13.5943	−0.565937	−0.282968	−	0.959129i	$-0.591319\pi$
$577$	−13.5943	−0.565937	−0.282968	+	0.959129i	$0.591319\pi$
$578$	0	0
$579$	44.9615	1.86854
$580$	0	0
$581$	−57.0125	−2.36528
$582$	0	0
$583$	−3.74615	−0.155149
$584$	0	0
$585$	−37.9054	−1.56720
$586$	0	0
$587$	37.3385	1.54113	0.770563	−	0.637364i	$-0.219975\pi$
$587$	37.3385	1.54113	0.770563	+	0.637364i	$0.219975\pi$
$588$	0	0
$589$	52.5357	2.16470
$590$	0	0
$591$	12.6082	0.518633
$592$	0	0
$593$	−30.2354	−1.24162	−0.620810	−	0.783961i	$-0.713196\pi$
$593$	−30.2354	−1.24162	−0.620810	+	0.783961i	$0.713196\pi$
$594$	0	0
$595$	−55.7946	−2.28736
$596$	0	0
$597$	63.4166	2.59547
$598$	0	0
$599$	−30.0049	−1.22597	−0.612983	−	0.790096i	$-0.710030\pi$
$599$	−30.0049	−1.22597	−0.612983	+	0.790096i	$0.710030\pi$
$600$	0	0
$601$	−20.5063	−0.836469	−0.418234	−	0.908339i	$-0.637351\pi$
$601$	−20.5063	−0.836469	−0.418234	+	0.908339i	$0.637351\pi$
$602$	0	0
$603$	29.5100	1.20174
$604$	0	0
$605$	22.0426	0.896160
$606$	0	0
$607$	16.4809	0.668938	0.334469	−	0.942407i	$-0.391443\pi$
$607$	16.4809	0.668938	0.334469	+	0.942407i	$0.391443\pi$
$608$	0	0
$609$	69.3862	2.81167
$610$	0	0
$611$	37.9092	1.53364
$612$	0	0
$613$	12.1976	0.492658	0.246329	−	0.969186i	$-0.420776\pi$
$613$	12.1976	0.492658	0.246329	+	0.969186i	$0.420776\pi$
$614$	0	0
$615$	−21.7471	−0.876927
$616$	0	0
$617$	−18.3358	−0.738171	−0.369085	−	0.929395i	$-0.620329\pi$
$617$	−18.3358	−0.738171	−0.369085	+	0.929395i	$0.620329\pi$
$618$	0	0
$619$	16.7777	0.674351	0.337175	−	0.941442i	$-0.390528\pi$
$619$	16.7777	0.674351	0.337175	+	0.941442i	$0.390528\pi$
$620$	0	0
$621$	−11.7836	−0.472858
$622$	0	0
$623$	−59.5121	−2.38430
$624$	0	0
$625$	−20.4114	−0.816457
$626$	0	0
$627$	−7.05537	−0.281764
$628$	0	0
$629$	42.0620	1.67712
$630$	0	0
$631$	−24.7690	−0.986040	−0.493020	−	0.870018i	$-0.664107\pi$
$631$	−24.7690	−0.986040	−0.493020	+	0.870018i	$0.664107\pi$
$632$	0	0
$633$	42.4205	1.68606
$634$	0	0
$635$	6.50098	0.257984
$636$	0	0
$637$	83.9495	3.32620
$638$	0	0
$639$	2.57623	0.101914
$640$	0	0
$641$	−0.122148	−0.00482455	−0.00241227	−	0.999997i	$-0.500768\pi$
$641$	−0.122148	−0.00482455	−0.00241227	+	0.999997i	$0.500768\pi$
$642$	0	0
$643$	39.6500	1.56364	0.781821	−	0.623503i	$-0.214291\pi$
$643$	39.6500	1.56364	0.781821	+	0.623503i	$0.214291\pi$
$644$	0	0
$645$	39.6061	1.55949
$646$	0	0
$647$	−1.60138	−0.0629567	−0.0314783	−	0.999504i	$-0.510022\pi$
$647$	−1.60138	−0.0629567	−0.0314783	+	0.999504i	$0.510022\pi$
$648$	0	0
$649$	−4.41778	−0.173413
$650$	0	0
$651$	−129.933	−5.09247
$652$	0	0
$653$	−38.2751	−1.49782	−0.748911	−	0.662671i	$-0.769423\pi$
$653$	−38.2751	−1.49782	−0.748911	+	0.662671i	$0.769423\pi$
$654$	0	0
$655$	−9.56321	−0.373665
$656$	0	0
$657$	−33.2107	−1.29567
$658$	0	0
$659$	45.3710	1.76740	0.883702	−	0.468051i	$-0.155044\pi$
$659$	45.3710	1.76740	0.883702	+	0.468051i	$0.155044\pi$
$660$	0	0
$661$	−17.0692	−0.663915	−0.331957	−	0.943294i	$-0.607709\pi$
$661$	−17.0692	−0.663915	−0.331957	+	0.943294i	$0.607709\pi$
$662$	0	0
$663$	−67.4063	−2.61784
$664$	0	0
$665$	−54.3607	−2.10802
$666$	0	0
$667$	−25.2202	−0.976531
$668$	0	0
$669$	34.0621	1.31692
$670$	0	0
$671$	−4.04716	−0.156239
$672$	0	0
$673$	10.3587	0.399298	0.199649	−	0.979867i	$-0.436020\pi$
$673$	10.3587	0.399298	0.199649	+	0.979867i	$0.436020\pi$
$674$	0	0
$675$	1.95471	0.0752370
$676$	0	0
$677$	9.64087	0.370529	0.185264	−	0.982689i	$-0.440686\pi$
$677$	9.64087	0.370529	0.185264	+	0.982689i	$0.440686\pi$
$678$	0	0
$679$	39.5145	1.51643
$680$	0	0
$681$	5.79547	0.222083
$682$	0	0
$683$	43.4811	1.66376	0.831880	−	0.554956i	$-0.187265\pi$
$683$	43.4811	1.66376	0.831880	+	0.554956i	$0.187265\pi$
$684$	0	0
$685$	2.53866	0.0969973
$686$	0	0
$687$	46.4329	1.77152
$688$	0	0
$689$	−34.8705	−1.32846
$690$	0	0
$691$	−14.1103	−0.536781	−0.268390	−	0.963310i	$-0.586492\pi$
$691$	−14.1103	−0.536781	−0.268390	+	0.963310i	$0.586492\pi$
$692$	0	0
$693$	9.90308	0.376187
$694$	0	0
$695$	15.4642	0.586590
$696$	0	0
$697$	−21.9476	−0.831324
$698$	0	0
$699$	−21.6842	−0.820173
$700$	0	0
$701$	14.6864	0.554698	0.277349	−	0.960769i	$-0.410544\pi$
$701$	14.6864	0.554698	0.277349	+	0.960769i	$0.410544\pi$
$702$	0	0
$703$	40.9810	1.54563
$704$	0	0
$705$	43.6343	1.64336
$706$	0	0
$707$	−3.65245	−0.137364
$708$	0	0
$709$	2.08962	0.0784772	0.0392386	−	0.999230i	$-0.487507\pi$
$709$	2.08962	0.0784772	0.0392386	+	0.999230i	$0.487507\pi$
$710$	0	0
$711$	−55.4915	−2.08109
$712$	0	0
$713$	47.2275	1.76868
$714$	0	0
$715$	−4.85517	−0.181573
$716$	0	0
$717$	5.26338	0.196565
$718$	0	0
$719$	−28.6414	−1.06814	−0.534072	−	0.845439i	$-0.679339\pi$
$719$	−28.6414	−1.06814	−0.534072	+	0.845439i	$0.679339\pi$
$720$	0	0
$721$	28.2196	1.05095
$722$	0	0
$723$	22.2227	0.826470
$724$	0	0
$725$	4.18365	0.155377
$726$	0	0
$727$	−13.3742	−0.496023	−0.248012	−	0.968757i	$-0.579777\pi$
$727$	−13.3742	−0.496023	−0.248012	+	0.968757i	$0.579777\pi$
$728$	0	0
$729$	−40.4079	−1.49659
$730$	0	0
$731$	39.9713	1.47839
$732$	0	0
$733$	2.07522	0.0766501	0.0383250	−	0.999265i	$-0.487798\pi$
$733$	2.07522	0.0766501	0.0383250	+	0.999265i	$0.487798\pi$
$734$	0	0
$735$	96.6277	3.56416
$736$	0	0
$737$	3.77983	0.139232
$738$	0	0
$739$	22.5122	0.828124	0.414062	−	0.910249i	$-0.364110\pi$
$739$	22.5122	0.828124	0.414062	+	0.910249i	$0.364110\pi$
$740$	0	0
$741$	−65.6740	−2.41259
$742$	0	0
$743$	5.33878	0.195861	0.0979304	−	0.995193i	$-0.468778\pi$
$743$	5.33878	0.195861	0.0979304	+	0.995193i	$0.468778\pi$
$744$	0	0
$745$	−16.6037	−0.608312
$746$	0	0
$747$	−44.9933	−1.64622
$748$	0	0
$749$	−14.4595	−0.528339
$750$	0	0
$751$	−38.3692	−1.40011	−0.700055	−	0.714089i	$-0.746841\pi$
$751$	−38.3692	−1.40011	−0.700055	+	0.714089i	$0.746841\pi$
$752$	0	0
$753$	−2.63379	−0.0959806
$754$	0	0
$755$	−30.6791	−1.11653
$756$	0	0
$757$	−16.0487	−0.583300	−0.291650	−	0.956525i	$-0.594204\pi$
$757$	−16.0487	−0.583300	−0.291650	+	0.956525i	$0.594204\pi$
$758$	0	0
$759$	−6.34249	−0.230218
$760$	0	0
$761$	41.8530	1.51717	0.758586	−	0.651573i	$-0.225891\pi$
$761$	41.8530	1.51717	0.758586	+	0.651573i	$0.225891\pi$
$762$	0	0
$763$	−10.3681	−0.375350
$764$	0	0
$765$	−44.0322	−1.59199
$766$	0	0
$767$	−41.1223	−1.48484
$768$	0	0
$769$	49.9999	1.80304	0.901521	−	0.432735i	$-0.142451\pi$
$769$	49.9999	1.80304	0.901521	+	0.432735i	$0.142451\pi$
$770$	0	0
$771$	−33.5135	−1.20696
$772$	0	0
$773$	−25.0716	−0.901762	−0.450881	−	0.892584i	$-0.648890\pi$
$773$	−25.0716	−0.901762	−0.450881	+	0.892584i	$0.648890\pi$
$774$	0	0
$775$	−7.83432	−0.281417
$776$	0	0
$777$	−101.355	−3.63611
$778$	0	0
$779$	−21.3835	−0.766145
$780$	0	0
$781$	0.329981	0.0118076
$782$	0	0
$783$	13.0308	0.465683
$784$	0	0
$785$	−46.3482	−1.65424
$786$	0	0
$787$	−0.223468	−0.00796576	−0.00398288	−	0.999992i	$-0.501268\pi$
$787$	−0.223468	−0.00796576	−0.00398288	+	0.999992i	$0.501268\pi$
$788$	0	0
$789$	38.9434	1.38642
$790$	0	0
$791$	83.0622	2.95335
$792$	0	0
$793$	−37.6725	−1.33779
$794$	0	0
$795$	−40.1367	−1.42350
$796$	0	0
$797$	44.7376	1.58469	0.792344	−	0.610075i	$-0.208861\pi$
$797$	44.7376	1.58469	0.792344	+	0.610075i	$0.208861\pi$
$798$	0	0
$799$	44.0366	1.55790
$800$	0	0
$801$	−46.9660	−1.65946
$802$	0	0
$803$	−4.25384	−0.150115
$804$	0	0
$805$	−48.8680	−1.72237
$806$	0	0
$807$	65.0607	2.29025
$808$	0	0
$809$	38.6083	1.35740	0.678698	−	0.734417i	$-0.262544\pi$
$809$	38.6083	1.35740	0.678698	+	0.734417i	$0.262544\pi$
$810$	0	0
$811$	−32.3994	−1.13770	−0.568848	−	0.822443i	$-0.692611\pi$
$811$	−32.3994	−1.13770	−0.568848	+	0.822443i	$0.692611\pi$
$812$	0	0
$813$	26.3552	0.924317
$814$	0	0
$815$	40.7986	1.42911
$816$	0	0
$817$	38.9440	1.36248
$818$	0	0
$819$	92.1816	3.22108
$820$	0	0
$821$	−11.6599	−0.406932	−0.203466	−	0.979082i	$-0.565221\pi$
$821$	−11.6599	−0.406932	−0.203466	+	0.979082i	$0.565221\pi$
$822$	0	0
$823$	−9.70157	−0.338176	−0.169088	−	0.985601i	$-0.554082\pi$
$823$	−9.70157	−0.338176	−0.169088	+	0.985601i	$0.554082\pi$
$824$	0	0
$825$	1.05212	0.0366302
$826$	0	0
$827$	10.0275	0.348691	0.174346	−	0.984685i	$-0.444219\pi$
$827$	10.0275	0.348691	0.174346	+	0.984685i	$0.444219\pi$
$828$	0	0
$829$	−40.9190	−1.42118	−0.710588	−	0.703608i	$-0.751571\pi$
$829$	−40.9190	−1.42118	−0.710588	+	0.703608i	$0.751571\pi$
$830$	0	0
$831$	18.7687	0.651078
$832$	0	0
$833$	97.5185	3.37882
$834$	0	0
$835$	−29.6868	−1.02735
$836$	0	0
$837$	−24.4015	−0.843440
$838$	0	0
$839$	−26.8033	−0.925352	−0.462676	−	0.886528i	$-0.653111\pi$
$839$	−26.8033	−0.925352	−0.462676	+	0.886528i	$0.653111\pi$
$840$	0	0
$841$	−1.11029	−0.0382859
$842$	0	0
$843$	−33.0553	−1.13849
$844$	0	0
$845$	−18.5269	−0.637346
$846$	0	0
$847$	−53.6051	−1.84189
$848$	0	0
$849$	26.5077	0.909741
$850$	0	0
$851$	36.8403	1.26287
$852$	0	0
$853$	7.57775	0.259457	0.129729	−	0.991550i	$-0.458589\pi$
$853$	7.57775	0.259457	0.129729	+	0.991550i	$0.458589\pi$
$854$	0	0
$855$	−42.9006	−1.46717
$856$	0	0
$857$	39.1653	1.33786	0.668931	−	0.743325i	$-0.266752\pi$
$857$	39.1653	1.33786	0.668931	+	0.743325i	$0.266752\pi$
$858$	0	0
$859$	35.4586	1.20983	0.604917	−	0.796289i	$-0.293206\pi$
$859$	35.4586	1.20983	0.604917	+	0.796289i	$0.293206\pi$
$860$	0	0
$861$	52.8864	1.80236
$862$	0	0
$863$	47.0095	1.60022	0.800111	−	0.599852i	$-0.204774\pi$
$863$	47.0095	1.60022	0.800111	+	0.599852i	$0.204774\pi$
$864$	0	0
$865$	0.903724	0.0307275
$866$	0	0
$867$	−33.5270	−1.13864
$868$	0	0
$869$	−7.10771	−0.241112
$870$	0	0
$871$	35.1841	1.19217
$872$	0	0
$873$	31.1842	1.05543
$874$	0	0
$875$	59.2709	2.00372
$876$	0	0
$877$	−27.5931	−0.931751	−0.465876	−	0.884850i	$-0.654261\pi$
$877$	−27.5931	−0.931751	−0.465876	+	0.884850i	$0.654261\pi$
$878$	0	0
$879$	46.2887	1.56128
$880$	0	0
$881$	−27.1155	−0.913545	−0.456772	−	0.889584i	$-0.650995\pi$
$881$	−27.1155	−0.913545	−0.456772	+	0.889584i	$0.650995\pi$
$882$	0	0
$883$	3.75678	0.126426	0.0632129	−	0.998000i	$-0.479865\pi$
$883$	3.75678	0.126426	0.0632129	+	0.998000i	$0.479865\pi$
$884$	0	0
$885$	−47.3327	−1.59107
$886$	0	0
$887$	28.4388	0.954882	0.477441	−	0.878664i	$-0.341564\pi$
$887$	28.4388	0.954882	0.477441	+	0.878664i	$0.341564\pi$
$888$	0	0
$889$	−15.8096	−0.530238
$890$	0	0
$891$	−2.67850	−0.0897331
$892$	0	0
$893$	42.9049	1.43576
$894$	0	0
$895$	−50.0343	−1.67246
$896$	0	0
$897$	−59.0382	−1.97123
$898$	0	0
$899$	−52.2263	−1.74185
$900$	0	0
$901$	−40.5067	−1.34948
$902$	0	0
$903$	−96.3175	−3.20524
$904$	0	0
$905$	−47.0496	−1.56398
$906$	0	0
$907$	31.1008	1.03268	0.516342	−	0.856382i	$-0.327293\pi$
$907$	31.1008	1.03268	0.516342	+	0.856382i	$0.327293\pi$
$908$	0	0
$909$	−2.88245	−0.0956049
$910$	0	0
$911$	33.4049	1.10675	0.553377	−	0.832931i	$-0.313339\pi$
$911$	33.4049	1.10675	0.553377	+	0.832931i	$0.313339\pi$
$912$	0	0
$913$	−5.76303	−0.190729
$914$	0	0
$915$	−43.3618	−1.43350
$916$	0	0
$917$	23.2566	0.768000
$918$	0	0
$919$	−3.28179	−0.108256	−0.0541282	−	0.998534i	$-0.517238\pi$
$919$	−3.28179	−0.108256	−0.0541282	+	0.998534i	$0.517238\pi$
$920$	0	0
$921$	43.9943	1.44966
$922$	0	0
$923$	3.07158	0.101102
$924$	0	0
$925$	−6.11124	−0.200936
$926$	0	0
$927$	22.2705	0.731459
$928$	0	0
$929$	9.17294	0.300954	0.150477	−	0.988613i	$-0.451919\pi$
$929$	9.17294	0.300954	0.150477	+	0.988613i	$0.451919\pi$
$930$	0	0
$931$	95.0123	3.11390
$932$	0	0
$933$	−23.4872	−0.768936
$934$	0	0
$935$	−5.63993	−0.184445
$936$	0	0
$937$	−40.8952	−1.33599	−0.667994	−	0.744167i	$-0.732847\pi$
$937$	−40.8952	−1.33599	−0.667994	+	0.744167i	$0.732847\pi$
$938$	0	0
$939$	46.8578	1.52914
$940$	0	0
$941$	−44.4477	−1.44895	−0.724477	−	0.689299i	$-0.757919\pi$
$941$	−44.4477	−1.44895	−0.724477	+	0.689299i	$0.757919\pi$
$942$	0	0
$943$	−19.2229	−0.625985
$944$	0	0
$945$	25.2492	0.821356
$946$	0	0
$947$	18.7504	0.609306	0.304653	−	0.952463i	$-0.401460\pi$
$947$	18.7504	0.609306	0.304653	+	0.952463i	$0.401460\pi$
$948$	0	0
$949$	−39.5964	−1.28535
$950$	0	0
$951$	20.3528	0.659984
$952$	0	0
$953$	−19.3771	−0.627686	−0.313843	−	0.949475i	$-0.601617\pi$
$953$	−19.3771	−0.627686	−0.313843	+	0.949475i	$0.601617\pi$
$954$	0	0
$955$	−9.13124	−0.295480
$956$	0	0
$957$	7.01382	0.226725
$958$	0	0
$959$	−6.17373	−0.199360
$960$	0	0
$961$	66.7992	2.15481
$962$	0	0
$963$	−11.4112	−0.367721
$964$	0	0
$965$	35.0177	1.12726
$966$	0	0
$967$	−49.2259	−1.58300	−0.791499	−	0.611170i	$-0.790699\pi$
$967$	−49.2259	−1.58300	−0.791499	+	0.611170i	$0.790699\pi$
$968$	0	0
$969$	−76.2891	−2.45076
$970$	0	0
$971$	7.34917	0.235846	0.117923	−	0.993023i	$-0.462376\pi$
$971$	7.34917	0.235846	0.117923	+	0.993023i	$0.462376\pi$
$972$	0	0
$973$	−37.6071	−1.20563
$974$	0	0
$975$	9.79355	0.313645
$976$	0	0
$977$	−21.8769	−0.699904	−0.349952	−	0.936768i	$-0.613802\pi$
$977$	−21.8769	−0.699904	−0.349952	+	0.936768i	$0.613802\pi$
$978$	0	0
$979$	−6.01571	−0.192263
$980$	0	0
$981$	−8.18233	−0.261242
$982$	0	0
$983$	−26.7306	−0.852574	−0.426287	−	0.904588i	$-0.640179\pi$
$983$	−26.7306	−0.852574	−0.426287	+	0.904588i	$0.640179\pi$
$984$	0	0
$985$	9.81976	0.312883
$986$	0	0
$987$	−106.114	−3.37763
$988$	0	0
$989$	35.0091	1.11322
$990$	0	0
$991$	−20.2092	−0.641968	−0.320984	−	0.947085i	$-0.604014\pi$
$991$	−20.2092	−0.641968	−0.320984	+	0.947085i	$0.604014\pi$
$992$	0	0
$993$	−57.0281	−1.80973
$994$	0	0
$995$	49.3912	1.56581
$996$	0	0
$997$	−17.5018	−0.554287	−0.277144	−	0.960829i	$-0.589388\pi$
$997$	−17.5018	−0.554287	−0.277144	+	0.960829i	$0.589388\pi$
$998$	0	0
$999$	−19.0347	−0.602230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.2		19
4.3	odd	2		2008.2.a.c.1.18	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.18	✓	19	4.3	odd	2
4016.2.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q-2.63379 q^{3} -2.05129 q^{5} +4.98850 q^{7} +3.93685 q^{9} +O(q^{10})\)	\(q-2.63379 q^{3} -2.05129 q^{5} +4.98850 q^{7} +3.93685 q^{9} +0.504257 q^{11} +4.69381 q^{13} +5.40267 q^{15} +5.45248 q^{17} +5.31235 q^{19} -13.1387 q^{21} +4.77559 q^{23} -0.792198 q^{25} -2.46746 q^{27} -5.28107 q^{29} +9.88935 q^{31} -1.32811 q^{33} -10.2329 q^{35} +7.71429 q^{37} -12.3625 q^{39} -4.02525 q^{41} +7.33084 q^{43} -8.07562 q^{45} +8.07643 q^{47} +17.8852 q^{49} -14.3607 q^{51} -7.42905 q^{53} -1.03438 q^{55} -13.9916 q^{57} -8.76097 q^{59} -8.02599 q^{61} +19.6390 q^{63} -9.62837 q^{65} +7.49585 q^{67} -12.5779 q^{69} +0.654390 q^{71} -8.43587 q^{73} +2.08648 q^{75} +2.51549 q^{77} -14.0954 q^{79} -5.31178 q^{81} -11.4288 q^{83} -11.1846 q^{85} +13.9092 q^{87} -11.9299 q^{89} +23.4151 q^{91} -26.0465 q^{93} -10.8972 q^{95} +7.92112 q^{97} +1.98518 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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Modular forms

Varieties

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