LMFDB - Embedded newform 2016.2.k.a.1889.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(1889,2016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.k (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 $ x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.14
Root			$-2.13875i$ of defining polynomial
Character	$\chi$	$=$	2016.1889
Dual form			2016.2.k.a.1889.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61313 q^{5} +(-1.25928 + 2.32685i) q^{7} +O(q^{10})$	$q+2.61313 q^{5} +(-1.25928 + 2.32685i) q^{7} +4.29945i q^{11} -1.53073i q^{13} -0.448342 q^{17} +1.92762i q^{19} +0.737669i q^{23} +1.82843 q^{25} +1.41421i q^{29} +6.58132i q^{31} +(-3.29066 + 6.08034i) q^{35} -2.82843 q^{37} -6.94269 q^{41} +9.64212 q^{43} -2.72607 q^{47} +(-3.82843 - 5.86030i) q^{49} +2.58579i q^{53} +11.2350i q^{55} -9.30739 q^{59} +8.28772i q^{61} -4.00000i q^{65} -1.47534 q^{67} -11.4230i q^{71} +11.0866i q^{73} +(-10.0042 - 5.41421i) q^{77} +15.7225 q^{79} -15.8887 q^{83} -1.17157 q^{85} +17.3952 q^{89} +(3.56178 + 1.92762i) q^{91} +5.03712i q^{95} -6.75699i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{25} - 16 q^{49} - 64 q^{85}+O(q^{100}) $ 16 * q - 16 * q^25 - 16 * q^49 - 64 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.61313			1.16863			0.584313	−	0.811529i	$-0.301364\pi$
$5$	2.61313			1.16863			0.584313	+	0.811529i	$0.301364\pi$
$6$		0			0
$7$	−1.25928	+	2.32685i	−0.475963	+	0.879465i
$7$	−1.25928	+	2.32685i	−0.475963	+	0.879465i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.29945i			1.29633i	0.761499	+	0.648167i	$0.224464\pi$
$11$			4.29945i			1.29633i	−0.761499	+	0.648167i	$0.775536\pi$
$12$		0			0
$13$		−	1.53073i		−	0.424549i	−0.977210	−	0.212275i	$-0.931913\pi$
$13$		−	1.53073i		−	0.424549i	0.977210	−	0.212275i	$-0.0680871\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.448342			−0.108739			−0.0543694	−	0.998521i	$-0.517315\pi$
$17$	−0.448342			−0.108739			−0.0543694	+	0.998521i	$0.517315\pi$
$18$		0			0
$19$			1.92762i			0.442227i	0.975248	+	0.221113i	$0.0709691\pi$
$19$			1.92762i			0.442227i	−0.975248	+	0.221113i	$0.929031\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.737669i			0.153815i	0.997038	+	0.0769073i	$0.0245046\pi$
$23$			0.737669i			0.153815i	−0.997038	+	0.0769073i	$0.975495\pi$
$24$		0			0
$25$	1.82843			0.365685
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		0			0
$31$			6.58132i			1.18204i	0.806657	+	0.591020i	$0.201274\pi$
$31$			6.58132i			1.18204i	−0.806657	+	0.591020i	$0.798726\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.29066	+	6.08034i	−0.556223	+	1.02777i
$36$		0			0
$37$	−2.82843			−0.464991			−0.232495	−	0.972598i	$-0.574689\pi$
$37$	−2.82843			−0.464991			−0.232495	+	0.972598i	$0.574689\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.94269			−1.08427			−0.542133	−	0.840292i	$-0.682383\pi$
$41$	−6.94269			−1.08427			−0.542133	+	0.840292i	$0.682383\pi$
$42$		0			0
$43$	9.64212			1.47041			0.735205	−	0.677845i	$-0.237086\pi$
$43$	9.64212			1.47041			0.735205	+	0.677845i	$0.237086\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.72607			−0.397638			−0.198819	−	0.980036i	$-0.563711\pi$
$47$	−2.72607			−0.397638			−0.198819	+	0.980036i	$0.563711\pi$
$48$		0			0
$49$	−3.82843	−	5.86030i	−0.546918	−	0.837186i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.58579i			0.355185i	0.984104	+	0.177593i	$0.0568309\pi$
$53$			2.58579i			0.355185i	−0.984104	+	0.177593i	$0.943169\pi$
$54$		0			0
$55$			11.2350i			1.51493i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.30739			−1.21172			−0.605859	−	0.795572i	$-0.707171\pi$
$59$	−9.30739			−1.21172			−0.605859	+	0.795572i	$0.707171\pi$
$60$		0			0
$61$			8.28772i			1.06113i	0.847643	+	0.530567i	$0.178021\pi$
$61$			8.28772i			1.06113i	−0.847643	+	0.530567i	$0.821979\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4.00000i		−	0.496139i
$66$		0			0
$67$	−1.47534			−0.180241			−0.0901206	−	0.995931i	$-0.528725\pi$
$67$	−1.47534			−0.180241			−0.0901206	+	0.995931i	$0.528725\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.4230i		−	1.35566i	−0.735218	−	0.677831i	$-0.762920\pi$
$71$		−	11.4230i		−	1.35566i	0.735218	−	0.677831i	$-0.237080\pi$
$72$		0			0
$73$			11.0866i			1.29758i	0.760966	+	0.648792i	$0.224725\pi$
$73$			11.0866i			1.29758i	−0.760966	+	0.648792i	$0.775275\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.0042	−	5.41421i	−1.14008	−	0.617007i
$78$		0			0
$79$	15.7225			1.76892			0.884458	−	0.466620i	$-0.154528\pi$
$79$	15.7225			1.76892			0.884458	+	0.466620i	$0.154528\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.8887			−1.74401			−0.872006	−	0.489496i	$-0.837181\pi$
$83$	−15.8887			−1.74401			−0.872006	+	0.489496i	$0.837181\pi$
$84$		0			0
$85$	−1.17157			−0.127075
$86$		0			0
$87$		0			0
$88$		0			0
$89$	17.3952			1.84389			0.921944	−	0.387324i	$-0.126601\pi$
$89$	17.3952			1.84389			0.921944	+	0.387324i	$0.126601\pi$
$90$		0			0
$91$	3.56178	+	1.92762i	0.373376	+	0.202070i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.03712i			0.516798i
$96$		0			0
$97$		−	6.75699i		−	0.686068i	−0.939323	−	0.343034i	$-0.888545\pi$
$97$		−	6.75699i		−	0.686068i	0.939323	−	0.343034i	$-0.111455\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.83938			0.780047			0.390024	−	0.920805i	$-0.372467\pi$
$101$	7.83938			0.780047			0.390024	+	0.920805i	$0.372467\pi$
$102$		0			0
$103$			15.8887i			1.56556i	0.622298	+	0.782780i	$0.286199\pi$
$103$			15.8887i			1.56556i	−0.622298	+	0.782780i	$0.713801\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.737669i		−	0.0713132i	−0.999364	−	0.0356566i	$-0.988648\pi$
$107$		−	0.737669i		−	0.0713132i	0.999364	−	0.0356566i	$-0.0113523\pi$
$108$		0			0
$109$	15.3137			1.46679			0.733394	−	0.679804i	$-0.237935\pi$
$109$	15.3137			1.46679			0.733394	+	0.679804i	$0.237935\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.07107i		−	0.288902i	−0.989512	−	0.144451i	$-0.953858\pi$
$113$		−	3.07107i		−	0.288902i	0.989512	−	0.144451i	$-0.0461416\pi$
$114$		0			0
$115$			1.92762i			0.179752i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.564588	−	1.04322i	0.0517557	−	0.0956320i
$120$		0			0
$121$	−7.48528			−0.680480
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.28772			−0.741276
$126$		0			0
$127$	8.59890			0.763029			0.381515	−	0.924363i	$-0.375403\pi$
$127$	8.59890			0.763029			0.381515	+	0.924363i	$0.375403\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.7439			−1.72504			−0.862518	−	0.506026i	$-0.831114\pi$
$131$	−19.7439			−1.72504			−0.862518	+	0.506026i	$0.831114\pi$
$132$		0			0
$133$	−4.48528	−	2.42742i	−0.388923	−	0.210484i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			19.0711i			1.62935i	0.579917	+	0.814676i	$0.303085\pi$
$137$			19.0711i			1.62935i	−0.579917	+	0.814676i	$0.696915\pi$
$138$		0			0
$139$			3.85525i			0.326998i	0.986544	+	0.163499i	$0.0522780\pi$
$139$			3.85525i			0.326998i	−0.986544	+	0.163499i	$0.947722\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.58132			0.550357
$144$		0			0
$145$			3.69552i			0.306896i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	6.58579i		−	0.539529i	−0.962926	−	0.269764i	$-0.913054\pi$
$149$		−	6.58579i		−	0.539529i	0.962926	−	0.269764i	$-0.0869458\pi$
$150$		0			0
$151$	14.6792			1.19458			0.597290	−	0.802025i	$-0.296244\pi$
$151$	14.6792			1.19458			0.597290	+	0.802025i	$0.296244\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			17.1978i			1.38136i
$156$		0			0
$157$		−	20.0083i		−	1.59684i	−0.602102	−	0.798419i	$-0.705670\pi$
$157$		−	20.0083i		−	1.59684i	0.602102	−	0.798419i	$-0.294330\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.71644	−	0.928932i	−0.135275	−	0.0732101i
$162$		0			0
$163$	−18.6731			−1.46259			−0.731297	−	0.682059i	$-0.761085\pi$
$163$	−18.6731			−1.46259			−0.731297	+	0.682059i	$0.761085\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.4366			0.807605			0.403803	−	0.914846i	$-0.367688\pi$
$167$	10.4366			0.807605			0.403803	+	0.914846i	$0.367688\pi$
$168$		0			0
$169$	10.6569			0.819758
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.10748			0.692429			0.346214	−	0.938155i	$-0.387467\pi$
$173$	9.10748			0.692429			0.346214	+	0.938155i	$0.387467\pi$
$174$		0			0
$175$	−2.30250	+	4.25447i	−0.174053	+	0.321608i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			2.82411i			0.211084i	0.994415	+	0.105542i	$0.0336578\pi$
$179$			2.82411i			0.211084i	−0.994415	+	0.105542i	$0.966342\pi$
$180$		0			0
$181$		−	10.1899i		−	0.757407i	−0.925518	−	0.378704i	$-0.876370\pi$
$181$		−	10.1899i		−	0.757407i	0.925518	−	0.378704i	$-0.123630\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.39104			−0.543400
$186$		0			0
$187$		−	1.92762i		−	0.140962i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			9.33657i			0.675571i	0.941223	+	0.337785i	$0.109678\pi$
$191$			9.33657i			0.675571i	−0.941223	+	0.337785i	$0.890322\pi$
$192$		0			0
$193$	15.7990			1.13724			0.568618	−	0.822602i	$-0.307478\pi$
$193$	15.7990			1.13724			0.568618	+	0.822602i	$0.307478\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			16.7279i			1.19182i	0.803053	+	0.595908i	$0.203208\pi$
$197$			16.7279i			1.19182i	−0.803053	+	0.595908i	$0.796792\pi$
$198$		0			0
$199$			13.9611i			0.989675i	0.868986	+	0.494837i	$0.164772\pi$
$199$			13.9611i			0.989675i	−0.868986	+	0.494837i	$0.835228\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.29066	−	1.78089i	−0.230959	−	0.124994i
$204$		0			0
$205$	−18.1421			−1.26710
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.28772			−0.573274
$210$		0			0
$211$	−19.7164			−1.35733			−0.678665	−	0.734448i	$-0.737441\pi$
$211$	−19.7164			−1.35733			−0.678665	+	0.734448i	$0.737441\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	25.1961			1.71836
$216$		0			0
$217$	−15.3137	−	8.28772i	−1.03956	−	0.562607i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			0.686292i			0.0461650i
$222$		0			0
$223$		−	4.65369i		−	0.311634i	−0.987786	−	0.155817i	$-0.950199\pi$
$223$		−	4.65369i		−	0.311634i	0.987786	−	0.155817i	$-0.0498011\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.0179			−1.12952			−0.564758	−	0.825257i	$-0.691031\pi$
$227$	−17.0179			−1.12952			−0.564758	+	0.825257i	$0.691031\pi$
$228$		0			0
$229$			28.0334i			1.85250i	0.376910	+	0.926250i	$0.376987\pi$
$229$			28.0334i			1.85250i	−0.376910	+	0.926250i	$0.623013\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.0711i		−	0.987338i	−0.869650	−	0.493669i	$-0.835655\pi$
$233$		−	15.0711i		−	0.987338i	0.869650	−	0.493669i	$-0.164345\pi$
$234$		0			0
$235$	−7.12356			−0.464690
$236$		0			0
$237$		0			0
$238$		0			0
$239$			16.4601i			1.06472i	0.846519	+	0.532359i	$0.178694\pi$
$239$			16.4601i			1.06472i	−0.846519	+	0.532359i	$0.821306\pi$
$240$		0			0
$241$			17.2095i			1.10856i	0.832330	+	0.554280i	$0.187006\pi$
$241$			17.2095i			1.10856i	−0.832330	+	0.554280i	$0.812994\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.0042	−	15.3137i	−0.639142	−	0.978357i
$246$		0			0
$247$	2.95068			0.187747
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0335			0.759545			0.379772	−	0.925080i	$-0.376002\pi$
$251$	12.0335			0.759545			0.379772	+	0.925080i	$0.376002\pi$
$252$		0			0
$253$	−3.17157			−0.199395
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.61313			0.163002			0.0815012	−	0.996673i	$-0.474029\pi$
$257$	2.61313			0.163002			0.0815012	+	0.996673i	$0.474029\pi$
$258$		0			0
$259$	3.56178	−	6.58132i	0.221318	−	0.408943i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	18.5466i		−	1.14363i	−0.820382	−	0.571816i	$-0.806239\pi$
$263$		−	18.5466i		−	1.14363i	0.820382	−	0.571816i	$-0.193761\pi$
$264$		0			0
$265$			6.75699i			0.415078i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.0238			1.03796			0.518979	−	0.854787i	$-0.326312\pi$
$269$	17.0238			1.03796			0.518979	+	0.854787i	$0.326312\pi$
$270$		0			0
$271$		−	6.58132i		−	0.399786i	−0.979818	−	0.199893i	$-0.935940\pi$
$271$		−	6.58132i		−	0.399786i	0.979818	−	0.199893i	$-0.0640595\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			7.86123i			0.474050i
$276$		0			0
$277$	27.4558			1.64966			0.824831	−	0.565380i	$-0.191270\pi$
$277$	27.4558			1.64966			0.824831	+	0.565380i	$0.191270\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	16.7279i		−	0.997904i	−0.866630	−	0.498952i	$-0.833718\pi$
$281$		−	16.7279i		−	0.997904i	0.866630	−	0.498952i	$-0.166282\pi$
$282$		0			0
$283$		−	11.2350i		−	0.667852i	−0.942599	−	0.333926i	$-0.891626\pi$
$283$		−	11.2350i		−	0.667852i	0.942599	−	0.333926i	$-0.108374\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.74280	−	16.1546i	0.516071	−	0.953575i
$288$		0			0
$289$	−16.7990			−0.988176
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.7975			0.689219			0.344609	−	0.938746i	$-0.388011\pi$
$293$	11.7975			0.689219			0.344609	+	0.938746i	$0.388011\pi$
$294$		0			0
$295$	−24.3214			−1.41604
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.12918			0.0653019
$300$		0			0
$301$	−12.1421	+	22.4357i	−0.699861	+	1.29317i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			21.6569i			1.24007i
$306$		0			0
$307$		−	24.3976i		−	1.39245i	−0.717825	−	0.696223i	$-0.754862\pi$
$307$		−	24.3976i		−	1.39245i	0.717825	−	0.696223i	$-0.245138\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	17.0179			0.964995			0.482498	−	0.875897i	$-0.339730\pi$
$311$	17.0179			0.964995			0.482498	+	0.875897i	$0.339730\pi$
$312$		0			0
$313$		−	23.4412i		−	1.32498i	−0.749073	−	0.662488i	$-0.769501\pi$
$313$		−	23.4412i		−	1.32498i	0.749073	−	0.662488i	$-0.230499\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.10051i		−	0.342639i	−0.985216	−	0.171319i	$-0.945197\pi$
$317$		−	6.10051i		−	0.342639i	0.985216	−	0.171319i	$-0.0548030\pi$
$318$		0			0
$319$	−6.08034			−0.340434
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	0.864233i		−	0.0480872i
$324$		0			0
$325$		−	2.79884i		−	0.155251i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.43289	−	6.34315i	0.189261	−	0.349709i
$330$		0			0
$331$	12.5928			0.692163			0.346081	−	0.938204i	$-0.387512\pi$
$331$	12.5928			0.692163			0.346081	+	0.938204i	$0.387512\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.85525			−0.210635
$336$		0			0
$337$	21.6569			1.17972			0.589862	−	0.807504i	$-0.299182\pi$
$337$	21.6569			1.17972			0.589862	+	0.807504i	$0.299182\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−28.2960			−1.53232
$342$		0			0
$343$	18.4571	−	1.52840i	0.996589	−	0.0825258i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	21.4973i		−	1.15403i	−0.816732	−	0.577017i	$-0.804217\pi$
$347$		−	21.4973i		−	1.15403i	0.816732	−	0.577017i	$-0.195783\pi$
$348$		0			0
$349$			3.43289i			0.183758i	0.995770	+	0.0918791i	$0.0292873\pi$
$349$			3.43289i			0.183758i	−0.995770	+	0.0918791i	$0.970713\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.83938			−0.417248			−0.208624	−	0.977996i	$-0.566898\pi$
$353$	−7.83938			−0.417248			−0.208624	+	0.977996i	$0.566898\pi$
$354$		0			0
$355$		−	29.8498i		−	1.58426i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	30.0962i		−	1.58841i	−0.607647	−	0.794207i	$-0.707886\pi$
$359$		−	30.0962i		−	1.58841i	0.607647	−	0.794207i	$-0.292114\pi$
$360$		0			0
$361$	15.2843			0.804435
$362$		0			0
$363$		0			0
$364$		0			0
$365$			28.9706i			1.51639i
$366$		0			0
$367$		−	30.9790i		−	1.61709i	−0.588436	−	0.808544i	$-0.700256\pi$
$367$		−	30.9790i		−	1.61709i	0.588436	−	0.808544i	$-0.299744\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.01673	−	3.25623i	−0.312373	−	0.169055i
$372$		0			0
$373$	−14.9706			−0.775146			−0.387573	−	0.921839i	$-0.626687\pi$
$373$	−14.9706			−0.775146			−0.387573	+	0.921839i	$0.626687\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.16478			0.111492
$378$		0			0
$379$	7.55568			0.388109			0.194055	−	0.980991i	$-0.437836\pi$
$379$	7.55568			0.388109			0.194055	+	0.980991i	$0.437836\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.58132			−0.336289			−0.168145	−	0.985762i	$-0.553778\pi$
$383$	−6.58132			−0.336289			−0.168145	+	0.985762i	$0.553778\pi$
$384$		0			0
$385$	−26.1421	−	14.1480i	−1.33233	−	0.721050i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			20.9289i			1.06114i	0.847641	+	0.530569i	$0.178022\pi$
$389$			20.9289i			1.06114i	−0.847641	+	0.530569i	$0.821978\pi$
$390$		0			0
$391$		−	0.330728i		−	0.0167256i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	41.0848			2.06720
$396$		0			0
$397$		−	24.8632i		−	1.24785i	−0.781486	−	0.623923i	$-0.785538\pi$
$397$		−	24.8632i		−	1.24785i	0.781486	−	0.623923i	$-0.214462\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.5858i		−	0.528629i	−0.964437	−	0.264314i	$-0.914854\pi$
$401$		−	10.5858i		−	0.528629i	0.964437	−	0.264314i	$-0.0851457\pi$
$402$		0			0
$403$	10.0742			0.501834
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	12.1607i		−	0.602783i
$408$		0			0
$409$		−	15.9414i		−	0.788251i	−0.919057	−	0.394125i	$-0.871048\pi$
$409$		−	15.9414i		−	0.788251i	0.919057	−	0.394125i	$-0.128952\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.7206	−	21.6569i	0.576733	−	1.06566i
$414$		0			0
$415$	−41.5192			−2.03810
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.1626			−0.643037			−0.321518	−	0.946903i	$-0.604193\pi$
$419$	−13.1626			−0.643037			−0.321518	+	0.946903i	$0.604193\pi$
$420$		0			0
$421$	−9.51472			−0.463719			−0.231860	−	0.972749i	$-0.574481\pi$
$421$	−9.51472			−0.463719			−0.231860	+	0.972749i	$0.574481\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.819760			−0.0397642
$426$		0			0
$427$	−19.2842	−	10.4366i	−0.933230	−	0.505061i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	20.8862i		−	1.00605i	−0.864272	−	0.503025i	$-0.832220\pi$
$431$		−	20.8862i		−	1.00605i	0.864272	−	0.503025i	$-0.167780\pi$
$432$		0			0
$433$			11.7206i			0.563256i	0.959524	+	0.281628i	$0.0908745\pi$
$433$			11.7206i			0.563256i	−0.959524	+	0.281628i	$0.909126\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.42195			−0.0680210
$438$		0			0
$439$		−	13.9611i		−	0.666326i	−0.942869	−	0.333163i	$-0.891884\pi$
$439$		−	13.9611i		−	0.666326i	0.942869	−	0.333163i	$-0.108116\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			0.737669i			0.0350477i	0.999846	+	0.0175239i	$0.00557830\pi$
$443$			0.737669i			0.0350477i	−0.999846	+	0.0175239i	$0.994422\pi$
$444$		0			0
$445$	45.4558			2.15481
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.0416i		−	0.568280i	−0.958783	−	0.284140i	$-0.908292\pi$
$449$		−	12.0416i		−	0.568280i	0.958783	−	0.284140i	$-0.0917080\pi$
$450$		0			0
$451$		−	29.8498i		−	1.40557i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.30739	+	5.03712i	0.436337	+	0.236144i
$456$		0			0
$457$	−8.97056			−0.419625			−0.209813	−	0.977742i	$-0.567285\pi$
$457$	−8.97056			−0.419625			−0.209813	+	0.977742i	$0.567285\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.50981			0.163468			0.0817341	−	0.996654i	$-0.473954\pi$
$461$	3.50981			0.163468			0.0817341	+	0.996654i	$0.473954\pi$
$462$		0			0
$463$	8.59890			0.399625			0.199812	−	0.979834i	$-0.435967\pi$
$463$	8.59890			0.399625			0.199812	+	0.979834i	$0.435967\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.6482			1.41823			0.709115	−	0.705092i	$-0.249095\pi$
$467$	30.6482			1.41823			0.709115	+	0.705092i	$0.249095\pi$
$468$		0			0
$469$	1.85786	−	3.43289i	0.0857882	−	0.158516i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			41.4558i			1.90614i
$474$		0			0
$475$			3.52452i			0.161716i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−26.3253			−1.20283			−0.601416	−	0.798936i	$-0.705397\pi$
$479$	−26.3253			−1.20283			−0.601416	+	0.798936i	$0.705397\pi$
$480$		0			0
$481$			4.32957i			0.197411i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	17.6569i		−	0.801756i
$486$		0			0
$487$	−11.7286			−0.531472			−0.265736	−	0.964046i	$-0.585615\pi$
$487$	−11.7286			−0.531472			−0.265736	+	0.964046i	$0.585615\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.1084i		−	0.997736i	−0.866678	−	0.498868i	$-0.833749\pi$
$491$		−	22.1084i		−	0.997736i	0.866678	−	0.498868i	$-0.166251\pi$
$492$		0			0
$493$		−	0.634051i		−	0.0285562i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	26.5796	+	14.3848i	1.19226	+	0.645245i
$498$		0			0
$499$	14.6792			0.657133			0.328567	−	0.944481i	$-0.393434\pi$
$499$	14.6792			0.657133			0.328567	+	0.944481i	$0.393434\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	34.5035			1.53843			0.769217	−	0.638988i	$-0.220647\pi$
$503$	34.5035			1.53843			0.769217	+	0.638988i	$0.220647\pi$
$504$		0			0
$505$	20.4853			0.911583
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.04601			−0.267985			−0.133992	−	0.990982i	$-0.542780\pi$
$509$	−6.04601			−0.267985			−0.133992	+	0.990982i	$0.542780\pi$
$510$		0			0
$511$	−25.7967	−	13.9611i	−1.14118	−	0.617602i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			41.5192i			1.82955i
$516$		0			0
$517$		−	11.7206i		−	0.515472i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.4567			−0.896223			−0.448111	−	0.893978i	$-0.647903\pi$
$521$	−20.4567			−0.896223			−0.448111	+	0.893978i	$0.647903\pi$
$522$		0			0
$523$			5.45214i			0.238405i	0.992870	+	0.119203i	$0.0380338\pi$
$523$			5.45214i			0.238405i	−0.992870	+	0.119203i	$0.961966\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.95068i		−	0.128534i
$528$		0			0
$529$	22.4558			0.976341
$530$		0			0
$531$		0			0
$532$		0			0
$533$			10.6274i			0.460325i
$534$		0			0
$535$		−	1.92762i		−	0.0833384i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.1961	−	16.4601i	1.08527	−	0.708988i
$540$		0			0
$541$	−33.7990			−1.45313			−0.726566	−	0.687097i	$-0.758885\pi$
$541$	−33.7990			−1.45313			−0.726566	+	0.687097i	$0.758885\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	40.0166			1.71412
$546$		0			0
$547$	−29.9696			−1.28141			−0.640704	−	0.767788i	$-0.721357\pi$
$547$	−29.9696			−1.28141			−0.640704	+	0.767788i	$0.721357\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.72607			−0.116134
$552$		0			0
$553$	−19.7990	+	36.5838i	−0.841939	+	1.55570i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	10.8701i		−	0.460579i	−0.973122	−	0.230290i	$-0.926033\pi$
$557$		−	10.8701i		−	0.460579i	0.973122	−	0.230290i	$-0.0739673\pi$
$558$		0			0
$559$		−	14.7595i		−	0.624261i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.72607			−0.114890			−0.0574451	−	0.998349i	$-0.518295\pi$
$563$	−2.72607			−0.114890			−0.0574451	+	0.998349i	$0.518295\pi$
$564$		0			0
$565$		−	8.02509i		−	0.337618i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	36.7279i		−	1.53971i	−0.638216	−	0.769857i	$-0.720327\pi$
$569$		−	36.7279i		−	1.53971i	0.638216	−	0.769857i	$-0.279673\pi$
$570$		0			0
$571$	−8.59890			−0.359853			−0.179926	−	0.983680i	$-0.557586\pi$
$571$	−8.59890			−0.359853			−0.179926	+	0.983680i	$0.557586\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.34877i			0.0562478i
$576$		0			0
$577$			28.2960i			1.17798i	0.808140	+	0.588990i	$0.200474\pi$
$577$			28.2960i			1.17798i	−0.808140	+	0.588990i	$0.799526\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	20.0083	−	36.9706i	0.830085	−	1.53380i
$582$		0			0
$583$	−11.1175			−0.460438
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.5992			0.974043			0.487021	−	0.873390i	$-0.338083\pi$
$587$	23.5992			0.974043			0.487021	+	0.873390i	$0.338083\pi$
$588$		0			0
$589$	−12.6863			−0.522730
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.3421			−1.41026			−0.705130	−	0.709078i	$-0.749111\pi$
$593$	−34.3421			−1.41026			−0.705130	+	0.709078i	$0.749111\pi$
$594$		0			0
$595$	1.47534	−	2.72607i	0.0604830	−	0.111758i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			45.8186i			1.87210i	0.351869	+	0.936049i	$0.385546\pi$
$599$			45.8186i			1.87210i	−0.351869	+	0.936049i	$0.614454\pi$
$600$		0			0
$601$			28.2960i			1.15422i	0.816667	+	0.577110i	$0.195820\pi$
$601$			28.2960i			1.15422i	−0.816667	+	0.577110i	$0.804180\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.5600			−0.795226
$606$		0			0
$607$		−	21.6716i		−	0.879622i	−0.898090	−	0.439811i	$-0.855045\pi$
$607$		−	21.6716i		−	0.879622i	0.898090	−	0.439811i	$-0.144955\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4.17289i			0.168817i
$612$		0			0
$613$	34.2843			1.38473			0.692364	−	0.721548i	$-0.256569\pi$
$613$	34.2843			1.38473			0.692364	+	0.721548i	$0.256569\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4.04163i		−	0.162710i	−0.996685	−	0.0813550i	$-0.974075\pi$
$617$		−	4.04163i		−	0.162710i	0.996685	−	0.0813550i	$-0.0259247\pi$
$618$		0			0
$619$			46.5369i			1.87048i	0.354018	+	0.935238i	$0.384815\pi$
$619$			46.5369i			1.87048i	−0.354018	+	0.935238i	$0.615185\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−21.9054	+	40.4760i	−0.877622	+	1.62163i
$624$		0			0
$625$	−30.7990			−1.23196
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.26810			0.0505625
$630$		0			0
$631$	35.8709			1.42800			0.714000	−	0.700146i	$-0.246882\pi$
$631$	35.8709			1.42800			0.714000	+	0.700146i	$0.246882\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.4700			0.891695
$636$		0			0
$637$	−8.97056	+	5.86030i	−0.355427	+	0.232194i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			36.2426i			1.43150i	0.698357	+	0.715749i	$0.253915\pi$
$641$			36.2426i			1.43150i	−0.698357	+	0.715749i	$0.746085\pi$
$642$		0			0
$643$		−	12.8319i		−	0.506041i	−0.967461	−	0.253020i	$-0.918576\pi$
$643$		−	12.8319i		−	0.506041i	0.967461	−	0.253020i	$-0.0814240\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.3743			1.31208			0.656039	−	0.754727i	$-0.272230\pi$
$647$	33.3743			1.31208			0.656039	+	0.754727i	$0.272230\pi$
$648$		0			0
$649$		−	40.0166i		−	1.57079i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			8.72792i			0.341550i	0.985310	+	0.170775i	$0.0546271\pi$
$653$			8.72792i			0.341550i	−0.985310	+	0.170775i	$0.945373\pi$
$654$		0			0
$655$	−51.5934			−2.01592
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	20.6330i		−	0.803748i	−0.915695	−	0.401874i	$-0.868359\pi$
$659$		−	20.6330i		−	0.803748i	0.915695	−	0.401874i	$-0.131641\pi$
$660$		0			0
$661$		−	13.1426i		−	0.511186i	−0.966784	−	0.255593i	$-0.917729\pi$
$661$		−	13.1426i		−	0.511186i	0.966784	−	0.255593i	$-0.0822707\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11.7206	−	6.34315i	−0.454506	−	0.245977i
$666$		0			0
$667$	−1.04322			−0.0403937
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−35.6326			−1.37558
$672$		0			0
$673$	22.8284			0.879971			0.439986	−	0.898005i	$-0.354984\pi$
$673$	22.8284			0.879971			0.439986	+	0.898005i	$0.354984\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−34.3421			−1.31987			−0.659936	−	0.751322i	$-0.729417\pi$
$677$	−34.3421			−1.31987			−0.659936	+	0.751322i	$0.729417\pi$
$678$		0			0
$679$	15.7225	+	8.50894i	0.603373	+	0.326543i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	39.3062i		−	1.50401i	−0.659158	−	0.752004i	$-0.729087\pi$
$683$		−	39.3062i		−	1.50401i	0.659158	−	0.752004i	$-0.270913\pi$
$684$		0			0
$685$			49.8351i			1.90410i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.95815			0.150794
$690$		0			0
$691$			27.9222i			1.06221i	0.847306	+	0.531104i	$0.178223\pi$
$691$			27.9222i			1.06221i	−0.847306	+	0.531104i	$0.821777\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			10.0742i			0.382138i
$696$		0			0
$697$	3.11270			0.117902
$698$		0			0
$699$		0			0
$700$		0			0
$701$			32.0416i			1.21020i	0.796151	+	0.605098i	$0.206866\pi$
$701$			32.0416i			1.21020i	−0.796151	+	0.605098i	$0.793134\pi$
$702$		0			0
$703$		−	5.45214i		−	0.205631i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.87197	+	18.2410i	−0.371274	+	0.686024i
$708$		0			0
$709$	−24.2843			−0.912015			−0.456007	−	0.889976i	$-0.650721\pi$
$709$	−24.2843			−0.912015			−0.456007	+	0.889976i	$0.650721\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.85483			−0.181815
$714$		0			0
$715$	17.1978			0.643161
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.8731			−0.778436			−0.389218	−	0.921146i	$-0.627255\pi$
$719$	−20.8731			−0.778436			−0.389218	+	0.921146i	$0.627255\pi$
$720$		0			0
$721$	−36.9706	−	20.0083i	−1.37686	−	0.745149i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2.58579i			0.0960337i
$726$		0			0
$727$		−	26.7930i		−	0.993697i	−0.867837	−	0.496848i	$-0.834491\pi$
$727$		−	26.7930i		−	0.993697i	0.867837	−	0.496848i	$-0.165509\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.32296			−0.159891
$732$		0			0
$733$		−	23.1786i		−	0.856120i	−0.903750	−	0.428060i	$-0.859197\pi$
$733$		−	23.1786i		−	0.856120i	0.903750	−	0.428060i	$-0.140803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.34315i		−	0.233653i
$738$		0			0
$739$	−4.42602			−0.162814			−0.0814068	−	0.996681i	$-0.525941\pi$
$739$	−4.42602			−0.162814			−0.0814068	+	0.996681i	$0.525941\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.86123i		−	0.288401i	−0.989549	−	0.144200i	$-0.953939\pi$
$743$		−	7.86123i		−	0.288401i	0.989549	−	0.144200i	$-0.0460610\pi$
$744$		0			0
$745$		−	17.2095i		−	0.630507i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.71644	+	0.928932i	0.0627175	+	0.0339424i
$750$		0			0
$751$	16.7657			0.611789			0.305894	−	0.952065i	$-0.401045\pi$
$751$	16.7657			0.611789			0.305894	+	0.952065i	$0.401045\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.3587			1.39602
$756$		0			0
$757$	−6.34315			−0.230546			−0.115273	−	0.993334i	$-0.536774\pi$
$757$	−6.34315			−0.230546			−0.115273	+	0.993334i	$0.536774\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.46796			−0.270713			−0.135357	−	0.990797i	$-0.543218\pi$
$761$	−7.46796			−0.270713			−0.135357	+	0.990797i	$0.543218\pi$
$762$		0			0
$763$	−19.2842	+	35.6326i	−0.698137	+	1.28999i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			14.2471i			0.514434i
$768$		0			0
$769$		−	35.1618i		−	1.26797i	−0.773346	−	0.633984i	$-0.781419\pi$
$769$		−	35.1618i		−	1.26797i	0.773346	−	0.633984i	$-0.218581\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.30318			0.190742			0.0953710	−	0.995442i	$-0.469596\pi$
$773$	5.30318			0.190742			0.0953710	+	0.995442i	$0.469596\pi$
$774$		0			0
$775$			12.0335i			0.432254i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	13.3829i		−	0.479492i
$780$		0			0
$781$	49.1127			1.75739
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	52.2843i		−	1.86611i
$786$		0			0
$787$		−	7.71049i		−	0.274849i	−0.990512	−	0.137425i	$-0.956118\pi$
$787$		−	7.71049i		−	0.274849i	0.990512	−	0.137425i	$-0.0438825\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.14590	+	3.86733i	0.254079	+	0.137507i
$792$		0			0
$793$	12.6863			0.450503
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.21080			0.290841			0.145421	−	0.989370i	$-0.453546\pi$
$797$	8.21080			0.290841			0.145421	+	0.989370i	$0.453546\pi$
$798$		0			0
$799$	1.22221			0.0432387
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−47.6661			−1.68210
$804$		0			0
$805$	−4.48528	−	2.42742i	−0.158085	−	0.0855552i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			43.3553i			1.52429i	0.647405	+	0.762146i	$0.275854\pi$
$809$			43.3553i			1.52429i	−0.647405	+	0.762146i	$0.724146\pi$
$810$		0			0
$811$			37.2295i			1.30731i	0.756794	+	0.653653i	$0.226764\pi$
$811$			37.2295i			1.30731i	−0.756794	+	0.653653i	$0.773236\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−48.7953			−1.70922
$816$		0			0
$817$			18.5864i			0.650255i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	10.5858i		−	0.369446i	−0.982791	−	0.184723i	$-0.940861\pi$
$821$		−	10.5858i		−	0.369446i	0.982791	−	0.184723i	$-0.0591389\pi$
$822$		0			0
$823$	50.1181			1.74701			0.873503	−	0.486818i	$-0.161843\pi$
$823$	50.1181			1.74701			0.873503	+	0.486818i	$0.161843\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	38.6951i		−	1.34556i	−0.739843	−	0.672780i	$-0.765100\pi$
$827$		−	38.6951i		−	1.34556i	0.739843	−	0.672780i	$-0.234900\pi$
$828$		0			0
$829$			31.0949i			1.07997i	0.841675	+	0.539985i	$0.181570\pi$
$829$			31.0949i			1.07997i	−0.841675	+	0.539985i	$0.818430\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.71644	+	2.62742i	0.0594712	+	0.0910346i
$834$		0			0
$835$	27.2720			0.943788
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−49.2630			−1.70075			−0.850374	−	0.526179i	$-0.823624\pi$
$839$	−49.2630			−1.70075			−0.850374	+	0.526179i	$0.823624\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	27.8477			0.957990
$846$		0			0
$847$	9.42607	−	17.4171i	0.323883	−	0.598459i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2.08644i		−	0.0715224i
$852$		0			0
$853$		−	8.28772i		−	0.283766i	−0.989883	−	0.141883i	$-0.954684\pi$
$853$		−	8.28772i		−	0.283766i	0.989883	−	0.141883i	$-0.0453157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	36.1354			1.23436			0.617181	−	0.786821i	$-0.288275\pi$
$857$	36.1354			1.23436			0.617181	+	0.786821i	$0.288275\pi$
$858$		0			0
$859$			37.5603i			1.28154i	0.767733	+	0.640770i	$0.221385\pi$
$859$			37.5603i			1.28154i	−0.767733	+	0.640770i	$0.778615\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			4.91056i			0.167157i	0.996501	+	0.0835786i	$0.0266350\pi$
$863$			4.91056i			0.167157i	−0.996501	+	0.0835786i	$0.973365\pi$
$864$		0			0
$865$	23.7990			0.809190
$866$		0			0
$867$		0			0
$868$		0			0
$869$			67.5980i			2.29310i
$870$		0			0
$871$			2.25835i			0.0765213i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.4366	−	19.2842i	0.352820	−	0.651927i
$876$		0			0
$877$	−9.31371			−0.314502			−0.157251	−	0.987559i	$-0.550263\pi$
$877$	−9.31371			−0.314502			−0.157251	+	0.987559i	$0.550263\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	21.3533			0.719413			0.359706	−	0.933066i	$-0.382877\pi$
$881$	21.3533			0.719413			0.359706	+	0.933066i	$0.382877\pi$
$882$		0			0
$883$	31.0128			1.04366			0.521832	−	0.853048i	$-0.325249\pi$
$883$	31.0128			1.04366			0.521832	+	0.853048i	$0.325249\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.17821			−0.274597			−0.137299	−	0.990530i	$-0.543842\pi$
$887$	−8.17821			−0.274597			−0.137299	+	0.990530i	$0.543842\pi$
$888$		0			0
$889$	−10.8284	+	20.0083i	−0.363174	+	0.671058i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.25483i		−	0.175846i
$894$		0			0
$895$			7.37976i			0.246678i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.30739			−0.310419
$900$		0			0
$901$		−	1.15932i		−	0.0386224i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	26.6274i		−	0.885125i
$906$		0			0
$907$	22.6670			0.752647			0.376323	−	0.926488i	$-0.377188\pi$
$907$	22.6670			0.752647			0.376323	+	0.926488i	$0.377188\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			37.2197i			1.23314i	0.787298	+	0.616572i	$0.211479\pi$
$911$			37.2197i			1.23314i	−0.787298	+	0.616572i	$0.788521\pi$
$912$		0			0
$913$		−	68.3127i		−	2.26082i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.8632	−	45.9411i	0.821054	−	1.51711i
$918$		0			0
$919$	16.7657			0.553049			0.276525	−	0.961007i	$-0.410817\pi$
$919$	16.7657			0.553049			0.276525	+	0.961007i	$0.410817\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17.4856			−0.575545
$924$		0			0
$925$	−5.17157			−0.170040
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.5378			−1.00191			−0.500956	−	0.865473i	$-0.667018\pi$
$929$	−30.5378			−1.00191			−0.500956	+	0.865473i	$0.667018\pi$
$930$		0			0
$931$	11.2965	−	7.37976i	0.370226	−	0.241862i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	5.03712i		−	0.164731i
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20.4567			−0.666868			−0.333434	−	0.942773i	$-0.608207\pi$
$941$	−20.4567			−0.666868			−0.333434	+	0.942773i	$0.608207\pi$
$942$		0			0
$943$		−	5.12141i		−	0.166776i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	17.0712i		−	0.554741i	−0.960763	−	0.277370i	$-0.910537\pi$
$947$		−	17.0712i		−	0.554741i	0.960763	−	0.277370i	$-0.0894629\pi$
$948$		0			0
$949$	16.9706			0.550888
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	14.8701i		−	0.481688i	−0.970564	−	0.240844i	$-0.922576\pi$
$953$		−	14.8701i		−	0.481688i	0.970564	−	0.240844i	$-0.0774243\pi$
$954$		0			0
$955$			24.3976i			0.789489i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−44.3754	−	24.0158i	−1.43296	−	0.775511i
$960$		0			0
$961$	−12.3137			−0.397216
$962$		0			0
$963$		0			0
$964$		0			0
$965$	41.2848			1.32900
$966$		0			0
$967$	−15.7225			−0.505600			−0.252800	−	0.967518i	$-0.581352\pi$
$967$	−15.7225			−0.505600			−0.252800	+	0.967518i	$0.581352\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35.6326			1.14351			0.571753	−	0.820426i	$-0.306264\pi$
$971$	35.6326			1.14351			0.571753	+	0.820426i	$0.306264\pi$
$972$		0			0
$973$	−8.97056	−	4.85483i	−0.287583	−	0.155639i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			2.58579i			0.0827266i	0.999144	+	0.0413633i	$0.0131701\pi$
$977$			2.58579i			0.0827266i	−0.999144	+	0.0413633i	$0.986830\pi$
$978$		0			0
$979$			74.7898i			2.39029i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−26.7930			−0.854563			−0.427282	−	0.904119i	$-0.640529\pi$
$983$	−26.7930			−0.854563			−0.427282	+	0.904119i	$0.640529\pi$
$984$		0			0
$985$			43.7122i			1.39279i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7.11270i			0.226171i
$990$		0			0
$991$	25.9757			0.825145			0.412573	−	0.910925i	$-0.364630\pi$
$991$	25.9757			0.825145			0.412573	+	0.910925i	$0.364630\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			36.4821i			1.15656i
$996$		0			0
$997$			41.4386i			1.31237i	0.754599	+	0.656187i	$0.227832\pi$
$997$			41.4386i			1.31237i	−0.754599	+	0.656187i	$0.772168\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.k.a.1889.14	yes	16
3.2	odd	2	inner	2016.2.k.a.1889.2	yes	16
4.3	odd	2	inner	2016.2.k.a.1889.15	yes	16
7.6	odd	2	inner	2016.2.k.a.1889.1	✓	16
8.3	odd	2		4032.2.k.g.3905.3		16
8.5	even	2		4032.2.k.g.3905.2		16
12.11	even	2	inner	2016.2.k.a.1889.3	yes	16
21.20	even	2	inner	2016.2.k.a.1889.13	yes	16
24.5	odd	2		4032.2.k.g.3905.14		16
24.11	even	2		4032.2.k.g.3905.15		16
28.27	even	2	inner	2016.2.k.a.1889.4	yes	16
56.13	odd	2		4032.2.k.g.3905.13		16
56.27	even	2		4032.2.k.g.3905.16		16
84.83	odd	2	inner	2016.2.k.a.1889.16	yes	16
168.83	odd	2		4032.2.k.g.3905.4		16
168.125	even	2		4032.2.k.g.3905.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.k.a.1889.1	✓	16	7.6	odd	2	inner
2016.2.k.a.1889.2	yes	16	3.2	odd	2	inner
2016.2.k.a.1889.3	yes	16	12.11	even	2	inner
2016.2.k.a.1889.4	yes	16	28.27	even	2	inner
2016.2.k.a.1889.13	yes	16	21.20	even	2	inner
2016.2.k.a.1889.14	yes	16	1.1	even	1	trivial
2016.2.k.a.1889.15	yes	16	4.3	odd	2	inner
2016.2.k.a.1889.16	yes	16	84.83	odd	2	inner
4032.2.k.g.3905.1		16	168.125	even	2
4032.2.k.g.3905.2		16	8.5	even	2
4032.2.k.g.3905.3		16	8.3	odd	2
4032.2.k.g.3905.4		16	168.83	odd	2
4032.2.k.g.3905.13		16	56.13	odd	2
4032.2.k.g.3905.14		16	24.5	odd	2
4032.2.k.g.3905.15		16	24.11	even	2
4032.2.k.g.3905.16		16	56.27	even	2

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.61313 q^{5} +(-1.25928 + 2.32685i) q^{7} +O(q^{10})\)	\(q+2.61313 q^{5} +(-1.25928 + 2.32685i) q^{7} +4.29945i q^{11} -1.53073i q^{13} -0.448342 q^{17} +1.92762i q^{19} +0.737669i q^{23} +1.82843 q^{25} +1.41421i q^{29} +6.58132i q^{31} +(-3.29066 + 6.08034i) q^{35} -2.82843 q^{37} -6.94269 q^{41} +9.64212 q^{43} -2.72607 q^{47} +(-3.82843 - 5.86030i) q^{49} +2.58579i q^{53} +11.2350i q^{55} -9.30739 q^{59} +8.28772i q^{61} -4.00000i q^{65} -1.47534 q^{67} -11.4230i q^{71} +11.0866i q^{73} +(-10.0042 - 5.41421i) q^{77} +15.7225 q^{79} -15.8887 q^{83} -1.17157 q^{85} +17.3952 q^{89} +(3.56178 + 1.92762i) q^{91} +5.03712i q^{95} -6.75699i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{25} - 16 q^{49} - 64 q^{85}+O(q^{100}) \) 16 * q - 16 * q^25 - 16 * q^49 - 64 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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