LMFDB - Embedded newform 4016.2.a.l.1.17

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			1.17
Root			$2.79120$ 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)

$f(q)$	$=$	$q+2.79120 q^{3} -3.17175 q^{5} +1.46800 q^{7} +4.79080 q^{9} +O(q^{10})$	$q+2.79120 q^{3} -3.17175 q^{5} +1.46800 q^{7} +4.79080 q^{9} +4.08094 q^{11} +4.91264 q^{13} -8.85298 q^{15} +3.89545 q^{17} +4.08766 q^{19} +4.09748 q^{21} -7.08850 q^{23} +5.05997 q^{25} +4.99849 q^{27} -1.48341 q^{29} -9.24522 q^{31} +11.3907 q^{33} -4.65612 q^{35} +2.10496 q^{37} +13.7122 q^{39} -6.48549 q^{41} +5.95354 q^{43} -15.1952 q^{45} +8.02766 q^{47} -4.84497 q^{49} +10.8730 q^{51} +4.69183 q^{53} -12.9437 q^{55} +11.4095 q^{57} -5.12725 q^{59} +13.1329 q^{61} +7.03290 q^{63} -15.5816 q^{65} +1.10218 q^{67} -19.7854 q^{69} -1.84841 q^{71} +3.34426 q^{73} +14.1234 q^{75} +5.99082 q^{77} -2.16088 q^{79} -0.420621 q^{81} +16.9630 q^{83} -12.3554 q^{85} -4.14049 q^{87} +16.7682 q^{89} +7.21176 q^{91} -25.8053 q^{93} -12.9650 q^{95} +4.28380 q^{97} +19.5510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.79120	1.61150	0.805750	−	0.592255i	$-0.201762\pi$
$3$	2.79120	1.61150	0.805750	+	0.592255i	$0.201762\pi$
$4$	0	0
$5$	−3.17175	−1.41845	−0.709224	−	0.704983i	$-0.750954\pi$
$5$	−3.17175	−1.41845	−0.709224	+	0.704983i	$0.750954\pi$
$6$	0	0
$7$	1.46800	0.554852	0.277426	−	0.960747i	$-0.410519\pi$
$7$	1.46800	0.554852	0.277426	+	0.960747i	$0.410519\pi$
$8$	0	0
$9$	4.79080	1.59693
$10$	0	0
$11$	4.08094	1.23045	0.615225	−	0.788352i	$-0.289065\pi$
$11$	4.08094	1.23045	0.615225	+	0.788352i	$0.289065\pi$
$12$	0	0
$13$	4.91264	1.36252	0.681260	−	0.732041i	$-0.261432\pi$
$13$	4.91264	1.36252	0.681260	+	0.732041i	$0.261432\pi$
$14$	0	0
$15$	−8.85298	−2.28583
$16$	0	0
$17$	3.89545	0.944785	0.472392	−	0.881388i	$-0.343391\pi$
$17$	3.89545	0.944785	0.472392	+	0.881388i	$0.343391\pi$
$18$	0	0
$19$	4.08766	0.937774	0.468887	−	0.883258i	$-0.344655\pi$
$19$	4.08766	0.937774	0.468887	+	0.883258i	$0.344655\pi$
$20$	0	0
$21$	4.09748	0.894144
$22$	0	0
$23$	−7.08850	−1.47805	−0.739027	−	0.673676i	$-0.764714\pi$
$23$	−7.08850	−1.47805	−0.739027	+	0.673676i	$0.764714\pi$
$24$	0	0
$25$	5.05997	1.01199
$26$	0	0
$27$	4.99849	0.961960
$28$	0	0
$29$	−1.48341	−0.275462	−0.137731	−	0.990470i	$-0.543981\pi$
$29$	−1.48341	−0.275462	−0.137731	+	0.990470i	$0.543981\pi$
$30$	0	0
$31$	−9.24522	−1.66049	−0.830246	−	0.557398i	$-0.811800\pi$
$31$	−9.24522	−1.66049	−0.830246	+	0.557398i	$0.811800\pi$
$32$	0	0
$33$	11.3907	1.98287
$34$	0	0
$35$	−4.65612	−0.787029
$36$	0	0
$37$	2.10496	0.346053	0.173027	−	0.984917i	$-0.444645\pi$
$37$	2.10496	0.346053	0.173027	+	0.984917i	$0.444645\pi$
$38$	0	0
$39$	13.7122	2.19570
$40$	0	0
$41$	−6.48549	−1.01286	−0.506431	−	0.862280i	$-0.669036\pi$
$41$	−6.48549	−1.01286	−0.506431	+	0.862280i	$0.669036\pi$
$42$	0	0
$43$	5.95354	0.907906	0.453953	−	0.891026i	$-0.350013\pi$
$43$	5.95354	0.907906	0.453953	+	0.891026i	$0.350013\pi$
$44$	0	0
$45$	−15.1952	−2.26517
$46$	0	0
$47$	8.02766	1.17096	0.585478	−	0.810689i	$-0.300907\pi$
$47$	8.02766	1.17096	0.585478	+	0.810689i	$0.300907\pi$
$48$	0	0
$49$	−4.84497	−0.692139
$50$	0	0
$51$	10.8730	1.52252
$52$	0	0
$53$	4.69183	0.644472	0.322236	−	0.946659i	$-0.395566\pi$
$53$	4.69183	0.644472	0.322236	+	0.946659i	$0.395566\pi$
$54$	0	0
$55$	−12.9437	−1.74533
$56$	0	0
$57$	11.4095	1.51122
$58$	0	0
$59$	−5.12725	−0.667511	−0.333755	−	0.942660i	$-0.608316\pi$
$59$	−5.12725	−0.667511	−0.333755	+	0.942660i	$0.608316\pi$
$60$	0	0
$61$	13.1329	1.68150	0.840749	−	0.541425i	$-0.182115\pi$
$61$	13.1329	1.68150	0.840749	+	0.541425i	$0.182115\pi$
$62$	0	0
$63$	7.03290	0.886062
$64$	0	0
$65$	−15.5816	−1.93266
$66$	0	0
$67$	1.10218	0.134652	0.0673261	−	0.997731i	$-0.478553\pi$
$67$	1.10218	0.134652	0.0673261	+	0.997731i	$0.478553\pi$
$68$	0	0
$69$	−19.7854	−2.38188
$70$	0	0
$71$	−1.84841	−0.219366	−0.109683	−	0.993967i	$-0.534984\pi$
$71$	−1.84841	−0.219366	−0.109683	+	0.993967i	$0.534984\pi$
$72$	0	0
$73$	3.34426	0.391416	0.195708	−	0.980662i	$-0.437300\pi$
$73$	3.34426	0.391416	0.195708	+	0.980662i	$0.437300\pi$
$74$	0	0
$75$	14.1234	1.63083
$76$	0	0
$77$	5.99082	0.682718
$78$	0	0
$79$	−2.16088	−0.243118	−0.121559	−	0.992584i	$-0.538789\pi$
$79$	−2.16088	−0.243118	−0.121559	+	0.992584i	$0.538789\pi$
$80$	0	0
$81$	−0.420621	−0.0467357
$82$	0	0
$83$	16.9630	1.86193	0.930964	−	0.365111i	$-0.118969\pi$
$83$	16.9630	1.86193	0.930964	+	0.365111i	$0.118969\pi$
$84$	0	0
$85$	−12.3554	−1.34013
$86$	0	0
$87$	−4.14049	−0.443907
$88$	0	0
$89$	16.7682	1.77743	0.888714	−	0.458462i	$-0.151600\pi$
$89$	16.7682	1.77743	0.888714	+	0.458462i	$0.151600\pi$
$90$	0	0
$91$	7.21176	0.755997
$92$	0	0
$93$	−25.8053	−2.67588
$94$	0	0
$95$	−12.9650	−1.33018
$96$	0	0
$97$	4.28380	0.434954	0.217477	−	0.976065i	$-0.430217\pi$
$97$	4.28380	0.434954	0.217477	+	0.976065i	$0.430217\pi$
$98$	0	0
$99$	19.5510	1.96495
$100$	0	0
$101$	−17.3450	−1.72589	−0.862946	−	0.505296i	$-0.831383\pi$
$101$	−17.3450	−1.72589	−0.862946	+	0.505296i	$0.831383\pi$
$102$	0	0
$103$	16.0473	1.58119	0.790596	−	0.612338i	$-0.209771\pi$
$103$	16.0473	1.58119	0.790596	+	0.612338i	$0.209771\pi$
$104$	0	0
$105$	−12.9962	−1.26830
$106$	0	0
$107$	3.21228	0.310543	0.155272	−	0.987872i	$-0.450375\pi$
$107$	3.21228	0.310543	0.155272	+	0.987872i	$0.450375\pi$
$108$	0	0
$109$	−6.83095	−0.654286	−0.327143	−	0.944975i	$-0.606086\pi$
$109$	−6.83095	−0.654286	−0.327143	+	0.944975i	$0.606086\pi$
$110$	0	0
$111$	5.87537	0.557665
$112$	0	0
$113$	−10.7554	−1.01179	−0.505893	−	0.862596i	$-0.668837\pi$
$113$	−10.7554	−1.01179	−0.505893	+	0.862596i	$0.668837\pi$
$114$	0	0
$115$	22.4829	2.09654
$116$	0	0
$117$	23.5355	2.17586
$118$	0	0
$119$	5.71852	0.524216
$120$	0	0
$121$	5.65407	0.514007
$122$	0	0
$123$	−18.1023	−1.63223
$124$	0	0
$125$	−0.190205	−0.0170125
$126$	0	0
$127$	8.49970	0.754227	0.377113	−	0.926167i	$-0.376917\pi$
$127$	8.49970	0.754227	0.377113	+	0.926167i	$0.376917\pi$
$128$	0	0
$129$	16.6175	1.46309
$130$	0	0
$131$	−20.0193	−1.74909	−0.874546	−	0.484943i	$-0.838840\pi$
$131$	−20.0193	−1.74909	−0.874546	+	0.484943i	$0.838840\pi$
$132$	0	0
$133$	6.00069	0.520326
$134$	0	0
$135$	−15.8539	−1.36449
$136$	0	0
$137$	−7.59841	−0.649176	−0.324588	−	0.945856i	$-0.605226\pi$
$137$	−7.59841	−0.649176	−0.324588	+	0.945856i	$0.605226\pi$
$138$	0	0
$139$	−6.98017	−0.592050	−0.296025	−	0.955180i	$-0.595661\pi$
$139$	−6.98017	−0.592050	−0.296025	+	0.955180i	$0.595661\pi$
$140$	0	0
$141$	22.4068	1.88699
$142$	0	0
$143$	20.0482	1.67651
$144$	0	0
$145$	4.70499	0.390729
$146$	0	0
$147$	−13.5233	−1.11538
$148$	0	0
$149$	−22.9987	−1.88412	−0.942062	−	0.335438i	$-0.891116\pi$
$149$	−22.9987	−1.88412	−0.942062	+	0.335438i	$0.891116\pi$
$150$	0	0
$151$	−5.04114	−0.410242	−0.205121	−	0.978737i	$-0.565759\pi$
$151$	−5.04114	−0.410242	−0.205121	+	0.978737i	$0.565759\pi$
$152$	0	0
$153$	18.6623	1.50876
$154$	0	0
$155$	29.3235	2.35532
$156$	0	0
$157$	24.5782	1.96155	0.980776	−	0.195136i	$-0.0625149\pi$
$157$	24.5782	1.96155	0.980776	+	0.195136i	$0.0625149\pi$
$158$	0	0
$159$	13.0958	1.03857
$160$	0	0
$161$	−10.4059	−0.820101
$162$	0	0
$163$	−12.2086	−0.956255	−0.478128	−	0.878290i	$-0.658684\pi$
$163$	−12.2086	−0.956255	−0.478128	+	0.878290i	$0.658684\pi$
$164$	0	0
$165$	−36.1285	−2.81260
$166$	0	0
$167$	23.9890	1.85633	0.928164	−	0.372172i	$-0.121387\pi$
$167$	23.9890	1.85633	0.928164	+	0.372172i	$0.121387\pi$
$168$	0	0
$169$	11.1340	0.856463
$170$	0	0
$171$	19.5832	1.49756
$172$	0	0
$173$	−9.65407	−0.733985	−0.366993	−	0.930224i	$-0.619613\pi$
$173$	−9.65407	−0.733985	−0.366993	+	0.930224i	$0.619613\pi$
$174$	0	0
$175$	7.42804	0.561507
$176$	0	0
$177$	−14.3112	−1.07569
$178$	0	0
$179$	−5.76333	−0.430772	−0.215386	−	0.976529i	$-0.569101\pi$
$179$	−5.76333	−0.430772	−0.215386	+	0.976529i	$0.569101\pi$
$180$	0	0
$181$	1.12273	0.0834521	0.0417260	−	0.999129i	$-0.486714\pi$
$181$	1.12273	0.0834521	0.0417260	+	0.999129i	$0.486714\pi$
$182$	0	0
$183$	36.6566	2.70973
$184$	0	0
$185$	−6.67640	−0.490859
$186$	0	0
$187$	15.8971	1.16251
$188$	0	0
$189$	7.33778	0.533745
$190$	0	0
$191$	18.3435	1.32729	0.663646	−	0.748047i	$-0.269008\pi$
$191$	18.3435	1.32729	0.663646	+	0.748047i	$0.269008\pi$
$192$	0	0
$193$	19.3978	1.39628	0.698142	−	0.715959i	$-0.254010\pi$
$193$	19.3978	1.39628	0.698142	+	0.715959i	$0.254010\pi$
$194$	0	0
$195$	−43.4915	−3.11449
$196$	0	0
$197$	−7.89401	−0.562425	−0.281212	−	0.959646i	$-0.590737\pi$
$197$	−7.89401	−0.562425	−0.281212	+	0.959646i	$0.590737\pi$
$198$	0	0
$199$	−11.2739	−0.799183	−0.399592	−	0.916693i	$-0.630848\pi$
$199$	−11.2739	−0.799183	−0.399592	+	0.916693i	$0.630848\pi$
$200$	0	0
$201$	3.07639	0.216992
$202$	0	0
$203$	−2.17764	−0.152841
$204$	0	0
$205$	20.5703	1.43669
$206$	0	0
$207$	−33.9596	−2.36035
$208$	0	0
$209$	16.6815	1.15388
$210$	0	0
$211$	13.5717	0.934313	0.467157	−	0.884175i	$-0.345278\pi$
$211$	13.5717	0.934313	0.467157	+	0.884175i	$0.345278\pi$
$212$	0	0
$213$	−5.15929	−0.353509
$214$	0	0
$215$	−18.8831	−1.28782
$216$	0	0
$217$	−13.5720	−0.921327
$218$	0	0
$219$	9.33449	0.630767
$220$	0	0
$221$	19.1369	1.28729
$222$	0	0
$223$	13.4756	0.902394	0.451197	−	0.892424i	$-0.350997\pi$
$223$	13.4756	0.902394	0.451197	+	0.892424i	$0.350997\pi$
$224$	0	0
$225$	24.2413	1.61609
$226$	0	0
$227$	−0.297235	−0.0197282	−0.00986410	−	0.999951i	$-0.503140\pi$
$227$	−0.297235	−0.0197282	−0.00986410	+	0.999951i	$0.503140\pi$
$228$	0	0
$229$	−0.384500	−0.0254085	−0.0127042	−	0.999919i	$-0.504044\pi$
$229$	−0.384500	−0.0254085	−0.0127042	+	0.999919i	$0.504044\pi$
$230$	0	0
$231$	16.7216	1.10020
$232$	0	0
$233$	12.6255	0.827124	0.413562	−	0.910476i	$-0.364284\pi$
$233$	12.6255	0.827124	0.413562	+	0.910476i	$0.364284\pi$
$234$	0	0
$235$	−25.4617	−1.66094
$236$	0	0
$237$	−6.03145	−0.391785
$238$	0	0
$239$	0.677340	0.0438135	0.0219067	−	0.999760i	$-0.493026\pi$
$239$	0.677340	0.0438135	0.0219067	+	0.999760i	$0.493026\pi$
$240$	0	0
$241$	4.46115	0.287368	0.143684	−	0.989624i	$-0.454105\pi$
$241$	4.46115	0.287368	0.143684	+	0.989624i	$0.454105\pi$
$242$	0	0
$243$	−16.1695	−1.03727
$244$	0	0
$245$	15.3670	0.981763
$246$	0	0
$247$	20.0812	1.27774
$248$	0	0
$249$	47.3471	3.00050
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−28.9277	−1.81867
$254$	0	0
$255$	−34.4863	−2.15962
$256$	0	0
$257$	8.59939	0.536415	0.268208	−	0.963361i	$-0.413569\pi$
$257$	8.59939	0.536415	0.268208	+	0.963361i	$0.413569\pi$
$258$	0	0
$259$	3.09008	0.192008
$260$	0	0
$261$	−7.10672	−0.439895
$262$	0	0
$263$	18.7106	1.15375	0.576873	−	0.816834i	$-0.304273\pi$
$263$	18.7106	1.15375	0.576873	+	0.816834i	$0.304273\pi$
$264$	0	0
$265$	−14.8813	−0.914149
$266$	0	0
$267$	46.8035	2.86433
$268$	0	0
$269$	−18.7060	−1.14053	−0.570264	−	0.821462i	$-0.693159\pi$
$269$	−18.7060	−1.14053	−0.570264	+	0.821462i	$0.693159\pi$
$270$	0	0
$271$	−11.5613	−0.702301	−0.351150	−	0.936319i	$-0.614209\pi$
$271$	−11.5613	−0.702301	−0.351150	+	0.936319i	$0.614209\pi$
$272$	0	0
$273$	20.1295	1.21829
$274$	0	0
$275$	20.6494	1.24521
$276$	0	0
$277$	−6.30199	−0.378650	−0.189325	−	0.981915i	$-0.560630\pi$
$277$	−6.30199	−0.378650	−0.189325	+	0.981915i	$0.560630\pi$
$278$	0	0
$279$	−44.2920	−2.65169
$280$	0	0
$281$	18.5548	1.10689	0.553443	−	0.832887i	$-0.313314\pi$
$281$	18.5548	1.10689	0.553443	+	0.832887i	$0.313314\pi$
$282$	0	0
$283$	−8.68523	−0.516283	−0.258142	−	0.966107i	$-0.583110\pi$
$283$	−8.68523	−0.516283	−0.258142	+	0.966107i	$0.583110\pi$
$284$	0	0
$285$	−36.1880	−2.14359
$286$	0	0
$287$	−9.52070	−0.561989
$288$	0	0
$289$	−1.82548	−0.107381
$290$	0	0
$291$	11.9569	0.700928
$292$	0	0
$293$	−3.93209	−0.229715	−0.114857	−	0.993382i	$-0.536641\pi$
$293$	−3.93209	−0.229715	−0.114857	+	0.993382i	$0.536641\pi$
$294$	0	0
$295$	16.2623	0.946829
$296$	0	0
$297$	20.3985	1.18364
$298$	0	0
$299$	−34.8232	−2.01388
$300$	0	0
$301$	8.73980	0.503754
$302$	0	0
$303$	−48.4134	−2.78128
$304$	0	0
$305$	−41.6543	−2.38512
$306$	0	0
$307$	−0.634905	−0.0362359	−0.0181180	−	0.999836i	$-0.505767\pi$
$307$	−0.634905	−0.0362359	−0.0181180	+	0.999836i	$0.505767\pi$
$308$	0	0
$309$	44.7914	2.54809
$310$	0	0
$311$	5.10579	0.289523	0.144761	−	0.989467i	$-0.453759\pi$
$311$	5.10579	0.289523	0.144761	+	0.989467i	$0.453759\pi$
$312$	0	0
$313$	9.40242	0.531456	0.265728	−	0.964048i	$-0.414388\pi$
$313$	9.40242	0.531456	0.265728	+	0.964048i	$0.414388\pi$
$314$	0	0
$315$	−22.3066	−1.25683
$316$	0	0
$317$	−32.0359	−1.79931	−0.899657	−	0.436597i	$-0.856184\pi$
$317$	−32.0359	−1.79931	−0.899657	+	0.436597i	$0.856184\pi$
$318$	0	0
$319$	−6.05370	−0.338942
$320$	0	0
$321$	8.96612	0.500440
$322$	0	0
$323$	15.9233	0.885995
$324$	0	0
$325$	24.8578	1.37886
$326$	0	0
$327$	−19.0665	−1.05438
$328$	0	0
$329$	11.7846	0.649707
$330$	0	0
$331$	−0.971344	−0.0533899	−0.0266949	−	0.999644i	$-0.508498\pi$
$331$	−0.971344	−0.0533899	−0.0266949	+	0.999644i	$0.508498\pi$
$332$	0	0
$333$	10.0845	0.552625
$334$	0	0
$335$	−3.49582	−0.190997
$336$	0	0
$337$	−34.6653	−1.88834	−0.944170	−	0.329459i	$-0.893134\pi$
$337$	−34.6653	−1.88834	−0.944170	+	0.329459i	$0.893134\pi$
$338$	0	0
$339$	−30.0206	−1.63049
$340$	0	0
$341$	−37.7292	−2.04315
$342$	0	0
$343$	−17.3884	−0.938887
$344$	0	0
$345$	62.7543	3.37858
$346$	0	0
$347$	−11.9536	−0.641706	−0.320853	−	0.947129i	$-0.603969\pi$
$347$	−11.9536	−0.641706	−0.320853	+	0.947129i	$0.603969\pi$
$348$	0	0
$349$	5.37480	0.287707	0.143853	−	0.989599i	$-0.454051\pi$
$349$	5.37480	0.287707	0.143853	+	0.989599i	$0.454051\pi$
$350$	0	0
$351$	24.5558	1.31069
$352$	0	0
$353$	−17.8467	−0.949882	−0.474941	−	0.880018i	$-0.657531\pi$
$353$	−17.8467	−0.949882	−0.474941	+	0.880018i	$0.657531\pi$
$354$	0	0
$355$	5.86270	0.311160
$356$	0	0
$357$	15.9615	0.844774
$358$	0	0
$359$	−23.8181	−1.25707	−0.628535	−	0.777782i	$-0.716345\pi$
$359$	−23.8181	−1.25707	−0.628535	+	0.777782i	$0.716345\pi$
$360$	0	0
$361$	−2.29101	−0.120579
$362$	0	0
$363$	15.7817	0.828322
$364$	0	0
$365$	−10.6071	−0.555203
$366$	0	0
$367$	20.8081	1.08618	0.543088	−	0.839676i	$-0.317255\pi$
$367$	20.8081	1.08618	0.543088	+	0.839676i	$0.317255\pi$
$368$	0	0
$369$	−31.0707	−1.61747
$370$	0	0
$371$	6.88760	0.357586
$372$	0	0
$373$	−0.0643374	−0.00333126	−0.00166563	−	0.999999i	$-0.500530\pi$
$373$	−0.0643374	−0.00333126	−0.00166563	+	0.999999i	$0.500530\pi$
$374$	0	0
$375$	−0.530901	−0.0274156
$376$	0	0
$377$	−7.28745	−0.375323
$378$	0	0
$379$	11.1653	0.573521	0.286761	−	0.958002i	$-0.407422\pi$
$379$	11.1653	0.573521	0.286761	+	0.958002i	$0.407422\pi$
$380$	0	0
$381$	23.7244	1.21544
$382$	0	0
$383$	−36.9737	−1.88927	−0.944634	−	0.328127i	$-0.893582\pi$
$383$	−36.9737	−1.88927	−0.944634	+	0.328127i	$0.893582\pi$
$384$	0	0
$385$	−19.0014	−0.968399
$386$	0	0
$387$	28.5222	1.44987
$388$	0	0
$389$	26.3286	1.33491	0.667456	−	0.744649i	$-0.267383\pi$
$389$	26.3286	1.33491	0.667456	+	0.744649i	$0.267383\pi$
$390$	0	0
$391$	−27.6129	−1.39644
$392$	0	0
$393$	−55.8778	−2.81866
$394$	0	0
$395$	6.85376	0.344850
$396$	0	0
$397$	2.36991	0.118942	0.0594712	−	0.998230i	$-0.481059\pi$
$397$	2.36991	0.118942	0.0594712	+	0.998230i	$0.481059\pi$
$398$	0	0
$399$	16.7491	0.838506
$400$	0	0
$401$	−24.6032	−1.22863	−0.614313	−	0.789062i	$-0.710567\pi$
$401$	−24.6032	−1.22863	−0.614313	+	0.789062i	$0.710567\pi$
$402$	0	0
$403$	−45.4184	−2.26245
$404$	0	0
$405$	1.33410	0.0662921
$406$	0	0
$407$	8.59022	0.425801
$408$	0	0
$409$	−20.2680	−1.00219	−0.501094	−	0.865393i	$-0.667069\pi$
$409$	−20.2680	−1.00219	−0.501094	+	0.865393i	$0.667069\pi$
$410$	0	0
$411$	−21.2087	−1.04615
$412$	0	0
$413$	−7.52680	−0.370370
$414$	0	0
$415$	−53.8022	−2.64105
$416$	0	0
$417$	−19.4830	−0.954089
$418$	0	0
$419$	−27.3805	−1.33763	−0.668813	−	0.743431i	$-0.733197\pi$
$419$	−27.3805	−1.33763	−0.668813	+	0.743431i	$0.733197\pi$
$420$	0	0
$421$	−21.8011	−1.06252	−0.531261	−	0.847208i	$-0.678282\pi$
$421$	−21.8011	−1.06252	−0.531261	+	0.847208i	$0.678282\pi$
$422$	0	0
$423$	38.4589	1.86994
$424$	0	0
$425$	19.7108	0.956116
$426$	0	0
$427$	19.2791	0.932982
$428$	0	0
$429$	55.9585	2.70170
$430$	0	0
$431$	−29.3456	−1.41353	−0.706764	−	0.707450i	$-0.749846\pi$
$431$	−29.3456	−1.41353	−0.706764	+	0.707450i	$0.749846\pi$
$432$	0	0
$433$	−11.0759	−0.532275	−0.266137	−	0.963935i	$-0.585747\pi$
$433$	−11.0759	−0.532275	−0.266137	+	0.963935i	$0.585747\pi$
$434$	0	0
$435$	13.1326	0.629659
$436$	0	0
$437$	−28.9754	−1.38608
$438$	0	0
$439$	−9.79909	−0.467685	−0.233842	−	0.972275i	$-0.575130\pi$
$439$	−9.79909	−0.467685	−0.233842	+	0.972275i	$0.575130\pi$
$440$	0	0
$441$	−23.2113	−1.10530
$442$	0	0
$443$	14.5396	0.690797	0.345399	−	0.938456i	$-0.387744\pi$
$443$	14.5396	0.690797	0.345399	+	0.938456i	$0.387744\pi$
$444$	0	0
$445$	−53.1845	−2.52119
$446$	0	0
$447$	−64.1939	−3.03627
$448$	0	0
$449$	−24.3827	−1.15069	−0.575345	−	0.817911i	$-0.695132\pi$
$449$	−24.3827	−1.15069	−0.575345	+	0.817911i	$0.695132\pi$
$450$	0	0
$451$	−26.4669	−1.24628
$452$	0	0
$453$	−14.0708	−0.661106
$454$	0	0
$455$	−22.8739	−1.07234
$456$	0	0
$457$	−8.41711	−0.393736	−0.196868	−	0.980430i	$-0.563077\pi$
$457$	−8.41711	−0.393736	−0.196868	+	0.980430i	$0.563077\pi$
$458$	0	0
$459$	19.4714	0.908845
$460$	0	0
$461$	−22.7563	−1.05987	−0.529934	−	0.848039i	$-0.677783\pi$
$461$	−22.7563	−1.05987	−0.529934	+	0.848039i	$0.677783\pi$
$462$	0	0
$463$	0.617455	0.0286955	0.0143478	−	0.999897i	$-0.495433\pi$
$463$	0.617455	0.0286955	0.0143478	+	0.999897i	$0.495433\pi$
$464$	0	0
$465$	81.8478	3.79560
$466$	0	0
$467$	4.89726	0.226618	0.113309	−	0.993560i	$-0.463855\pi$
$467$	4.89726	0.226618	0.113309	+	0.993560i	$0.463855\pi$
$468$	0	0
$469$	1.61799	0.0747120
$470$	0	0
$471$	68.6026	3.16104
$472$	0	0
$473$	24.2960	1.11713
$474$	0	0
$475$	20.6834	0.949022
$476$	0	0
$477$	22.4776	1.02918
$478$	0	0
$479$	−20.9845	−0.958808	−0.479404	−	0.877594i	$-0.659147\pi$
$479$	−20.9845	−0.958808	−0.479404	+	0.877594i	$0.659147\pi$
$480$	0	0
$481$	10.3409	0.471505
$482$	0	0
$483$	−29.0450	−1.32159
$484$	0	0
$485$	−13.5871	−0.616959
$486$	0	0
$487$	29.1421	1.32056	0.660278	−	0.751021i	$-0.270438\pi$
$487$	29.1421	1.32056	0.660278	+	0.751021i	$0.270438\pi$
$488$	0	0
$489$	−34.0768	−1.54101
$490$	0	0
$491$	10.1738	0.459138	0.229569	−	0.973292i	$-0.426268\pi$
$491$	10.1738	0.459138	0.229569	+	0.973292i	$0.426268\pi$
$492$	0	0
$493$	−5.77854	−0.260252
$494$	0	0
$495$	−62.0107	−2.78717
$496$	0	0
$497$	−2.71347	−0.121716
$498$	0	0
$499$	43.8850	1.96456	0.982282	−	0.187408i	$-0.0600087\pi$
$499$	43.8850	1.96456	0.982282	+	0.187408i	$0.0600087\pi$
$500$	0	0
$501$	66.9582	2.99147
$502$	0	0
$503$	17.2432	0.768838	0.384419	−	0.923159i	$-0.374402\pi$
$503$	17.2432	0.768838	0.384419	+	0.923159i	$0.374402\pi$
$504$	0	0
$505$	55.0139	2.44809
$506$	0	0
$507$	31.0773	1.38019
$508$	0	0
$509$	−13.2130	−0.585656	−0.292828	−	0.956165i	$-0.594596\pi$
$509$	−13.2130	−0.585656	−0.292828	+	0.956165i	$0.594596\pi$
$510$	0	0
$511$	4.90937	0.217178
$512$	0	0
$513$	20.4321	0.902101
$514$	0	0
$515$	−50.8981	−2.24284
$516$	0	0
$517$	32.7604	1.44080
$518$	0	0
$519$	−26.9464	−1.18282
$520$	0	0
$521$	23.5825	1.03317	0.516584	−	0.856236i	$-0.327203\pi$
$521$	23.5825	1.03317	0.516584	+	0.856236i	$0.327203\pi$
$522$	0	0
$523$	−18.7099	−0.818129	−0.409064	−	0.912505i	$-0.634145\pi$
$523$	−18.7099	−0.818129	−0.409064	+	0.912505i	$0.634145\pi$
$524$	0	0
$525$	20.7331	0.904868
$526$	0	0
$527$	−36.0143	−1.56881
$528$	0	0
$529$	27.2468	1.18464
$530$	0	0
$531$	−24.5636	−1.06597
$532$	0	0
$533$	−31.8609	−1.38005
$534$	0	0
$535$	−10.1885	−0.440489
$536$	0	0
$537$	−16.0866	−0.694189
$538$	0	0
$539$	−19.7721	−0.851643
$540$	0	0
$541$	2.54093	0.109243	0.0546216	−	0.998507i	$-0.482605\pi$
$541$	2.54093	0.109243	0.0546216	+	0.998507i	$0.482605\pi$
$542$	0	0
$543$	3.13377	0.134483
$544$	0	0
$545$	21.6660	0.928071
$546$	0	0
$547$	−4.66253	−0.199355	−0.0996777	−	0.995020i	$-0.531781\pi$
$547$	−4.66253	−0.199355	−0.0996777	+	0.995020i	$0.531781\pi$
$548$	0	0
$549$	62.9172	2.68524
$550$	0	0
$551$	−6.06368	−0.258321
$552$	0	0
$553$	−3.17217	−0.134894
$554$	0	0
$555$	−18.6352	−0.791019
$556$	0	0
$557$	−22.8438	−0.967922	−0.483961	−	0.875090i	$-0.660802\pi$
$557$	−22.8438	−0.967922	−0.483961	+	0.875090i	$0.660802\pi$
$558$	0	0
$559$	29.2476	1.23704
$560$	0	0
$561$	44.3720	1.87339
$562$	0	0
$563$	24.3924	1.02802	0.514009	−	0.857785i	$-0.328160\pi$
$563$	24.3924	1.02802	0.514009	+	0.857785i	$0.328160\pi$
$564$	0	0
$565$	34.1135	1.43517
$566$	0	0
$567$	−0.617472	−0.0259314
$568$	0	0
$569$	23.2727	0.975642	0.487821	−	0.872944i	$-0.337792\pi$
$569$	23.2727	0.975642	0.487821	+	0.872944i	$0.337792\pi$
$570$	0	0
$571$	1.17200	0.0490466	0.0245233	−	0.999699i	$-0.492193\pi$
$571$	1.17200	0.0490466	0.0245233	+	0.999699i	$0.492193\pi$
$572$	0	0
$573$	51.2005	2.13893
$574$	0	0
$575$	−35.8676	−1.49578
$576$	0	0
$577$	−5.05974	−0.210640	−0.105320	−	0.994438i	$-0.533587\pi$
$577$	−5.05974	−0.210640	−0.105320	+	0.994438i	$0.533587\pi$
$578$	0	0
$579$	54.1432	2.25011
$580$	0	0
$581$	24.9017	1.03309
$582$	0	0
$583$	19.1471	0.792990
$584$	0	0
$585$	−74.6486	−3.08634
$586$	0	0
$587$	43.2444	1.78489	0.892443	−	0.451160i	$-0.148990\pi$
$587$	43.2444	1.78489	0.892443	+	0.451160i	$0.148990\pi$
$588$	0	0
$589$	−37.7914	−1.55717
$590$	0	0
$591$	−22.0338	−0.906348
$592$	0	0
$593$	−6.21061	−0.255039	−0.127520	−	0.991836i	$-0.540702\pi$
$593$	−6.21061	−0.255039	−0.127520	+	0.991836i	$0.540702\pi$
$594$	0	0
$595$	−18.1377	−0.743573
$596$	0	0
$597$	−31.4676	−1.28788
$598$	0	0
$599$	33.6450	1.37470	0.687348	−	0.726328i	$-0.258775\pi$
$599$	33.6450	1.37470	0.687348	+	0.726328i	$0.258775\pi$
$600$	0	0
$601$	3.77142	0.153839	0.0769197	−	0.997037i	$-0.475491\pi$
$601$	3.77142	0.153839	0.0769197	+	0.997037i	$0.475491\pi$
$602$	0	0
$603$	5.28030	0.215031
$604$	0	0
$605$	−17.9333	−0.729092
$606$	0	0
$607$	0.711111	0.0288631	0.0144316	−	0.999896i	$-0.495406\pi$
$607$	0.711111	0.0288631	0.0144316	+	0.999896i	$0.495406\pi$
$608$	0	0
$609$	−6.07824	−0.246303
$610$	0	0
$611$	39.4370	1.59545
$612$	0	0
$613$	18.5045	0.747390	0.373695	−	0.927552i	$-0.378091\pi$
$613$	18.5045	0.747390	0.373695	+	0.927552i	$0.378091\pi$
$614$	0	0
$615$	57.4159	2.31523
$616$	0	0
$617$	−30.1609	−1.21423	−0.607117	−	0.794612i	$-0.707674\pi$
$617$	−30.1609	−1.21423	−0.607117	+	0.794612i	$0.707674\pi$
$618$	0	0
$619$	3.46569	0.139298	0.0696489	−	0.997572i	$-0.477812\pi$
$619$	3.46569	0.139298	0.0696489	+	0.997572i	$0.477812\pi$
$620$	0	0
$621$	−35.4318	−1.42183
$622$	0	0
$623$	24.6158	0.986209
$624$	0	0
$625$	−24.6966	−0.987862
$626$	0	0
$627$	46.5614	1.85949
$628$	0	0
$629$	8.19977	0.326946
$630$	0	0
$631$	14.1645	0.563881	0.281941	−	0.959432i	$-0.409022\pi$
$631$	14.1645	0.563881	0.281941	+	0.959432i	$0.409022\pi$
$632$	0	0
$633$	37.8813	1.50565
$634$	0	0
$635$	−26.9589	−1.06983
$636$	0	0
$637$	−23.8016	−0.943054
$638$	0	0
$639$	−8.85539	−0.350314
$640$	0	0
$641$	8.27550	0.326863	0.163431	−	0.986555i	$-0.447744\pi$
$641$	8.27550	0.326863	0.163431	+	0.986555i	$0.447744\pi$
$642$	0	0
$643$	−24.2028	−0.954464	−0.477232	−	0.878777i	$-0.658360\pi$
$643$	−24.2028	−0.954464	−0.477232	+	0.878777i	$0.658360\pi$
$644$	0	0
$645$	−52.7066	−2.07532
$646$	0	0
$647$	−27.9762	−1.09986	−0.549930	−	0.835211i	$-0.685346\pi$
$647$	−27.9762	−1.09986	−0.549930	+	0.835211i	$0.685346\pi$
$648$	0	0
$649$	−20.9240	−0.821338
$650$	0	0
$651$	−37.8822	−1.48472
$652$	0	0
$653$	−34.0053	−1.33073	−0.665366	−	0.746517i	$-0.731724\pi$
$653$	−34.0053	−1.33073	−0.665366	+	0.746517i	$0.731724\pi$
$654$	0	0
$655$	63.4960	2.48099
$656$	0	0
$657$	16.0217	0.625065
$658$	0	0
$659$	−28.0684	−1.09339	−0.546695	−	0.837332i	$-0.684114\pi$
$659$	−28.0684	−1.09339	−0.546695	+	0.837332i	$0.684114\pi$
$660$	0	0
$661$	−2.45443	−0.0954661	−0.0477330	−	0.998860i	$-0.515200\pi$
$661$	−2.45443	−0.0954661	−0.0477330	+	0.998860i	$0.515200\pi$
$662$	0	0
$663$	53.4150	2.07447
$664$	0	0
$665$	−19.0327	−0.738055
$666$	0	0
$667$	10.5151	0.407148
$668$	0	0
$669$	37.6132	1.45421
$670$	0	0
$671$	53.5946	2.06900
$672$	0	0
$673$	−44.7882	−1.72646	−0.863230	−	0.504812i	$-0.831562\pi$
$673$	−44.7882	−1.72646	−0.863230	+	0.504812i	$0.831562\pi$
$674$	0	0
$675$	25.2922	0.973497
$676$	0	0
$677$	−17.1461	−0.658979	−0.329490	−	0.944159i	$-0.606877\pi$
$677$	−17.1461	−0.658979	−0.329490	+	0.944159i	$0.606877\pi$
$678$	0	0
$679$	6.28862	0.241335
$680$	0	0
$681$	−0.829643	−0.0317920
$682$	0	0
$683$	31.1697	1.19268	0.596338	−	0.802734i	$-0.296622\pi$
$683$	31.1697	1.19268	0.596338	+	0.802734i	$0.296622\pi$
$684$	0	0
$685$	24.1002	0.920822
$686$	0	0
$687$	−1.07322	−0.0409457
$688$	0	0
$689$	23.0492	0.878106
$690$	0	0
$691$	−38.8545	−1.47809	−0.739047	−	0.673654i	$-0.764724\pi$
$691$	−38.8545	−1.47809	−0.739047	+	0.673654i	$0.764724\pi$
$692$	0	0
$693$	28.7008	1.09025
$694$	0	0
$695$	22.1393	0.839792
$696$	0	0
$697$	−25.2639	−0.956937
$698$	0	0
$699$	35.2403	1.33291
$700$	0	0
$701$	16.8935	0.638060	0.319030	−	0.947745i	$-0.396643\pi$
$701$	16.8935	0.638060	0.319030	+	0.947745i	$0.396643\pi$
$702$	0	0
$703$	8.60437	0.324520
$704$	0	0
$705$	−71.0687	−2.67660
$706$	0	0
$707$	−25.4625	−0.957615
$708$	0	0
$709$	−2.60563	−0.0978563	−0.0489282	−	0.998802i	$-0.515581\pi$
$709$	−2.60563	−0.0978563	−0.0489282	+	0.998802i	$0.515581\pi$
$710$	0	0
$711$	−10.3523	−0.388243
$712$	0	0
$713$	65.5347	2.45430
$714$	0	0
$715$	−63.5877	−2.37805
$716$	0	0
$717$	1.89059	0.0706054
$718$	0	0
$719$	23.5927	0.879860	0.439930	−	0.898032i	$-0.355003\pi$
$719$	23.5927	0.879860	0.439930	+	0.898032i	$0.355003\pi$
$720$	0	0
$721$	23.5575	0.877328
$722$	0	0
$723$	12.4520	0.463093
$724$	0	0
$725$	−7.50600	−0.278766
$726$	0	0
$727$	18.9206	0.701725	0.350863	−	0.936427i	$-0.385888\pi$
$727$	18.9206	0.701725	0.350863	+	0.936427i	$0.385888\pi$
$728$	0	0
$729$	−43.8705	−1.62483
$730$	0	0
$731$	23.1917	0.857776
$732$	0	0
$733$	28.3637	1.04764	0.523819	−	0.851830i	$-0.324507\pi$
$733$	28.3637	1.04764	0.523819	+	0.851830i	$0.324507\pi$
$734$	0	0
$735$	42.8925	1.58211
$736$	0	0
$737$	4.49791	0.165683
$738$	0	0
$739$	−28.1496	−1.03550	−0.517749	−	0.855533i	$-0.673230\pi$
$739$	−28.1496	−1.03550	−0.517749	+	0.855533i	$0.673230\pi$
$740$	0	0
$741$	56.0507	2.05907
$742$	0	0
$743$	−39.0165	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$743$	−39.0165	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$744$	0	0
$745$	72.9459	2.67253
$746$	0	0
$747$	81.2662	2.97338
$748$	0	0
$749$	4.71563	0.172305
$750$	0	0
$751$	48.6404	1.77491	0.887457	−	0.460890i	$-0.152470\pi$
$751$	48.6404	1.77491	0.887457	+	0.460890i	$0.152470\pi$
$752$	0	0
$753$	2.79120	0.101717
$754$	0	0
$755$	15.9892	0.581907
$756$	0	0
$757$	−11.7938	−0.428653	−0.214326	−	0.976762i	$-0.568756\pi$
$757$	−11.7938	−0.428653	−0.214326	+	0.976762i	$0.568756\pi$
$758$	0	0
$759$	−80.7431	−2.93079
$760$	0	0
$761$	−45.2260	−1.63944	−0.819720	−	0.572765i	$-0.805871\pi$
$761$	−45.2260	−1.63944	−0.819720	+	0.572765i	$0.805871\pi$
$762$	0	0
$763$	−10.0278	−0.363032
$764$	0	0
$765$	−59.1921	−2.14010
$766$	0	0
$767$	−25.1883	−0.909497
$768$	0	0
$769$	−40.0441	−1.44403	−0.722013	−	0.691879i	$-0.756783\pi$
$769$	−40.0441	−1.44403	−0.722013	+	0.691879i	$0.756783\pi$
$770$	0	0
$771$	24.0026	0.864434
$772$	0	0
$773$	−25.7009	−0.924396	−0.462198	−	0.886777i	$-0.652939\pi$
$773$	−25.7009	−0.924396	−0.462198	+	0.886777i	$0.652939\pi$
$774$	0	0
$775$	−46.7805	−1.68041
$776$	0	0
$777$	8.62504	0.309422
$778$	0	0
$779$	−26.5105	−0.949837
$780$	0	0
$781$	−7.54327	−0.269919
$782$	0	0
$783$	−7.41480	−0.264983
$784$	0	0
$785$	−77.9557	−2.78236
$786$	0	0
$787$	1.09064	0.0388772	0.0194386	−	0.999811i	$-0.493812\pi$
$787$	1.09064	0.0388772	0.0194386	+	0.999811i	$0.493812\pi$
$788$	0	0
$789$	52.2251	1.85926
$790$	0	0
$791$	−15.7890	−0.561392
$792$	0	0
$793$	64.5173	2.29108
$794$	0	0
$795$	−41.5366	−1.47315
$796$	0	0
$797$	−48.0621	−1.70245	−0.851224	−	0.524803i	$-0.824139\pi$
$797$	−48.0621	−1.70245	−0.851224	+	0.524803i	$0.824139\pi$
$798$	0	0
$799$	31.2713	1.10630
$800$	0	0
$801$	80.3332	2.83843
$802$	0	0
$803$	13.6477	0.481617
$804$	0	0
$805$	33.0049	1.16327
$806$	0	0
$807$	−52.2123	−1.83796
$808$	0	0
$809$	25.4345	0.894228	0.447114	−	0.894477i	$-0.352452\pi$
$809$	25.4345	0.894228	0.447114	+	0.894477i	$0.352452\pi$
$810$	0	0
$811$	21.3254	0.748836	0.374418	−	0.927260i	$-0.377843\pi$
$811$	21.3254	0.748836	0.374418	+	0.927260i	$0.377843\pi$
$812$	0	0
$813$	−32.2700	−1.13176
$814$	0	0
$815$	38.7227	1.35640
$816$	0	0
$817$	24.3361	0.851411
$818$	0	0
$819$	34.5501	1.20728
$820$	0	0
$821$	−47.6299	−1.66229	−0.831147	−	0.556053i	$-0.812315\pi$
$821$	−47.6299	−1.66229	−0.831147	+	0.556053i	$0.812315\pi$
$822$	0	0
$823$	8.85160	0.308548	0.154274	−	0.988028i	$-0.450696\pi$
$823$	8.85160	0.308548	0.154274	+	0.988028i	$0.450696\pi$
$824$	0	0
$825$	57.6367	2.00665
$826$	0	0
$827$	7.61153	0.264679	0.132339	−	0.991204i	$-0.457751\pi$
$827$	7.61153	0.264679	0.132339	+	0.991204i	$0.457751\pi$
$828$	0	0
$829$	−9.08423	−0.315508	−0.157754	−	0.987478i	$-0.550425\pi$
$829$	−9.08423	−0.315508	−0.157754	+	0.987478i	$0.550425\pi$
$830$	0	0
$831$	−17.5901	−0.610194
$832$	0	0
$833$	−18.8733	−0.653923
$834$	0	0
$835$	−76.0871	−2.63310
$836$	0	0
$837$	−46.2121	−1.59733
$838$	0	0
$839$	−33.0186	−1.13993	−0.569965	−	0.821669i	$-0.693043\pi$
$839$	−33.0186	−1.13993	−0.569965	+	0.821669i	$0.693043\pi$
$840$	0	0
$841$	−26.7995	−0.924121
$842$	0	0
$843$	51.7901	1.78375
$844$	0	0
$845$	−35.3143	−1.21485
$846$	0	0
$847$	8.30018	0.285198
$848$	0	0
$849$	−24.2422	−0.831991
$850$	0	0
$851$	−14.9210	−0.511486
$852$	0	0
$853$	−5.82057	−0.199293	−0.0996463	−	0.995023i	$-0.531771\pi$
$853$	−5.82057	−0.199293	−0.0996463	+	0.995023i	$0.531771\pi$
$854$	0	0
$855$	−62.1129	−2.12422
$856$	0	0
$857$	−30.9703	−1.05793	−0.528963	−	0.848645i	$-0.677419\pi$
$857$	−30.9703	−1.05793	−0.528963	+	0.848645i	$0.677419\pi$
$858$	0	0
$859$	−40.9710	−1.39791	−0.698955	−	0.715165i	$-0.746351\pi$
$859$	−40.9710	−1.39791	−0.698955	+	0.715165i	$0.746351\pi$
$860$	0	0
$861$	−26.5742	−0.905645
$862$	0	0
$863$	39.1721	1.33343	0.666717	−	0.745311i	$-0.267699\pi$
$863$	39.1721	1.33343	0.666717	+	0.745311i	$0.267699\pi$
$864$	0	0
$865$	30.6202	1.04112
$866$	0	0
$867$	−5.09529	−0.173045
$868$	0	0
$869$	−8.81842	−0.299144
$870$	0	0
$871$	5.41459	0.183466
$872$	0	0
$873$	20.5228	0.694593
$874$	0	0
$875$	−0.279221	−0.00943940
$876$	0	0
$877$	−1.81377	−0.0612468	−0.0306234	−	0.999531i	$-0.509749\pi$
$877$	−1.81377	−0.0612468	−0.0306234	+	0.999531i	$0.509749\pi$
$878$	0	0
$879$	−10.9752	−0.370186
$880$	0	0
$881$	42.6137	1.43569	0.717846	−	0.696202i	$-0.245128\pi$
$881$	42.6137	1.43569	0.717846	+	0.696202i	$0.245128\pi$
$882$	0	0
$883$	48.0758	1.61788	0.808940	−	0.587891i	$-0.200042\pi$
$883$	48.0758	1.61788	0.808940	+	0.587891i	$0.200042\pi$
$884$	0	0
$885$	45.3914	1.52582
$886$	0	0
$887$	19.5462	0.656298	0.328149	−	0.944626i	$-0.393575\pi$
$887$	19.5462	0.656298	0.328149	+	0.944626i	$0.393575\pi$
$888$	0	0
$889$	12.4776	0.418484
$890$	0	0
$891$	−1.71653	−0.0575059
$892$	0	0
$893$	32.8144	1.09809
$894$	0	0
$895$	18.2798	0.611027
$896$	0	0
$897$	−97.1986	−3.24537
$898$	0	0
$899$	13.7144	0.457402
$900$	0	0
$901$	18.2768	0.608887
$902$	0	0
$903$	24.3945	0.811799
$904$	0	0
$905$	−3.56102	−0.118372
$906$	0	0
$907$	−39.5318	−1.31263	−0.656316	−	0.754486i	$-0.727886\pi$
$907$	−39.5318	−1.31263	−0.656316	+	0.754486i	$0.727886\pi$
$908$	0	0
$909$	−83.0965	−2.75614
$910$	0	0
$911$	−28.5331	−0.945343	−0.472671	−	0.881239i	$-0.656710\pi$
$911$	−28.5331	−0.945343	−0.472671	+	0.881239i	$0.656710\pi$
$912$	0	0
$913$	69.2249	2.29101
$914$	0	0
$915$	−116.265	−3.84362
$916$	0	0
$917$	−29.3883	−0.970487
$918$	0	0
$919$	26.2971	0.867460	0.433730	−	0.901043i	$-0.357197\pi$
$919$	26.2971	0.867460	0.433730	+	0.901043i	$0.357197\pi$
$920$	0	0
$921$	−1.77215	−0.0583942
$922$	0	0
$923$	−9.08059	−0.298891
$924$	0	0
$925$	10.6510	0.350204
$926$	0	0
$927$	76.8797	2.52506
$928$	0	0
$929$	−55.7454	−1.82895	−0.914474	−	0.404645i	$-0.867395\pi$
$929$	−55.7454	−1.82895	−0.914474	+	0.404645i	$0.867395\pi$
$930$	0	0
$931$	−19.8046	−0.649070
$932$	0	0
$933$	14.2513	0.466566
$934$	0	0
$935$	−50.4215	−1.64896
$936$	0	0
$937$	−19.9232	−0.650861	−0.325431	−	0.945566i	$-0.605509\pi$
$937$	−19.9232	−0.650861	−0.325431	+	0.945566i	$0.605509\pi$
$938$	0	0
$939$	26.2440	0.856442
$940$	0	0
$941$	60.0379	1.95718	0.978590	−	0.205819i	$-0.0659857\pi$
$941$	60.0379	1.95718	0.978590	+	0.205819i	$0.0659857\pi$
$942$	0	0
$943$	45.9724	1.49707
$944$	0	0
$945$	−23.2736	−0.757090
$946$	0	0
$947$	18.9330	0.615240	0.307620	−	0.951509i	$-0.400467\pi$
$947$	18.9330	0.615240	0.307620	+	0.951509i	$0.400467\pi$
$948$	0	0
$949$	16.4291	0.533312
$950$	0	0
$951$	−89.4186	−2.89960
$952$	0	0
$953$	16.9007	0.547469	0.273734	−	0.961805i	$-0.411741\pi$
$953$	16.9007	0.547469	0.273734	+	0.961805i	$0.411741\pi$
$954$	0	0
$955$	−58.1810	−1.88269
$956$	0	0
$957$	−16.8971	−0.546206
$958$	0	0
$959$	−11.1545	−0.360197
$960$	0	0
$961$	54.4741	1.75723
$962$	0	0
$963$	15.3894	0.495917
$964$	0	0
$965$	−61.5249	−1.98056
$966$	0	0
$967$	−50.1104	−1.61144	−0.805722	−	0.592295i	$-0.798222\pi$
$967$	−50.1104	−1.61144	−0.805722	+	0.592295i	$0.798222\pi$
$968$	0	0
$969$	44.4451	1.42778
$970$	0	0
$971$	−12.4415	−0.399266	−0.199633	−	0.979871i	$-0.563975\pi$
$971$	−12.4415	−0.399266	−0.199633	+	0.979871i	$0.563975\pi$
$972$	0	0
$973$	−10.2469	−0.328500
$974$	0	0
$975$	69.3831	2.22204
$976$	0	0
$977$	−2.63112	−0.0841771	−0.0420885	−	0.999114i	$-0.513401\pi$
$977$	−2.63112	−0.0841771	−0.0420885	+	0.999114i	$0.513401\pi$
$978$	0	0
$979$	68.4301	2.18704
$980$	0	0
$981$	−32.7257	−1.04485
$982$	0	0
$983$	45.8223	1.46150	0.730752	−	0.682643i	$-0.239170\pi$
$983$	45.8223	1.46150	0.730752	+	0.682643i	$0.239170\pi$
$984$	0	0
$985$	25.0378	0.797770
$986$	0	0
$987$	32.8932	1.04700
$988$	0	0
$989$	−42.2017	−1.34193
$990$	0	0
$991$	6.21701	0.197490	0.0987449	−	0.995113i	$-0.468517\pi$
$991$	6.21701	0.197490	0.0987449	+	0.995113i	$0.468517\pi$
$992$	0	0
$993$	−2.71122	−0.0860378
$994$	0	0
$995$	35.7578	1.13360
$996$	0	0
$997$	21.7055	0.687421	0.343711	−	0.939076i	$-0.388316\pi$
$997$	21.7055	0.687421	0.343711	+	0.939076i	$0.388316\pi$
$998$	0	0
$999$	10.5216	0.332889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.17		19
4.3	odd	2		2008.2.a.c.1.3	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.3	✓	19	4.3	odd	2
4016.2.a.l.1.17		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)

\(f(q)\)	\(=\)	\(q+2.79120 q^{3} -3.17175 q^{5} +1.46800 q^{7} +4.79080 q^{9} +O(q^{10})\)	\(q+2.79120 q^{3} -3.17175 q^{5} +1.46800 q^{7} +4.79080 q^{9} +4.08094 q^{11} +4.91264 q^{13} -8.85298 q^{15} +3.89545 q^{17} +4.08766 q^{19} +4.09748 q^{21} -7.08850 q^{23} +5.05997 q^{25} +4.99849 q^{27} -1.48341 q^{29} -9.24522 q^{31} +11.3907 q^{33} -4.65612 q^{35} +2.10496 q^{37} +13.7122 q^{39} -6.48549 q^{41} +5.95354 q^{43} -15.1952 q^{45} +8.02766 q^{47} -4.84497 q^{49} +10.8730 q^{51} +4.69183 q^{53} -12.9437 q^{55} +11.4095 q^{57} -5.12725 q^{59} +13.1329 q^{61} +7.03290 q^{63} -15.5816 q^{65} +1.10218 q^{67} -19.7854 q^{69} -1.84841 q^{71} +3.34426 q^{73} +14.1234 q^{75} +5.99082 q^{77} -2.16088 q^{79} -0.420621 q^{81} +16.9630 q^{83} -12.3554 q^{85} -4.14049 q^{87} +16.7682 q^{89} +7.21176 q^{91} -25.8053 q^{93} -12.9650 q^{95} +4.28380 q^{97} +19.5510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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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Database

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