LMFDB - Embedded newform 1386.2.c.b.197.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(197,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1386.197");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.c (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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 4x^{10} - 8x^{9} + 17x^{8} - 16x^{7} + 88x^{6} - 48x^{5} + 153x^{4} - 216x^{3} + 324x^{2} + 729 $ x^12 + 4x^10 - 8x^9 + 17x^8 - 16x^7 + 88x^6 - 48x^5 + 153x^4 - 216x^3 + 324*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.11
Root			$1.49380 + 0.876678i$ of defining polynomial
Character	$\chi$	$=$	1386.197
Dual form			1386.2.c.b.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.35236i q^{5} +1.00000i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.35236i q^{5} +1.00000i q^{7} +1.00000 q^{8} +3.35236i q^{10} +(-3.06881 - 1.25793i) q^{11} +2.62610i q^{13} +1.00000i q^{14} +1.00000 q^{16} -2.58660 q^{17} +7.21270i q^{19} +3.35236i q^{20} +(-3.06881 - 1.25793i) q^{22} -5.09458i q^{23} -6.23832 q^{25} +2.62610i q^{26} +1.00000i q^{28} +1.96651 q^{29} -4.93541 q^{31} +1.00000 q^{32} -2.58660 q^{34} -3.35236 q^{35} +10.8036 q^{37} +7.21270i q^{38} +3.35236i q^{40} -9.07304 q^{41} +4.02738i q^{43} +(-3.06881 - 1.25793i) q^{44} -5.09458i q^{46} -7.16146i q^{47} -1.00000 q^{49} -6.23832 q^{50} +2.62610i q^{52} +10.3351i q^{53} +(4.21702 - 10.2878i) q^{55} +1.00000i q^{56} +1.96651 q^{58} +11.5083i q^{59} +4.07862i q^{61} -4.93541 q^{62} +1.00000 q^{64} -8.80362 q^{65} -2.06511 q^{67} -2.58660 q^{68} -3.35236 q^{70} -11.7993i q^{71} -5.29996i q^{73} +10.8036 q^{74} +7.21270i q^{76} +(1.25793 - 3.06881i) q^{77} -3.51764i q^{79} +3.35236i q^{80} -9.07304 q^{82} +2.98781 q^{83} -8.67123i q^{85} +4.02738i q^{86} +(-3.06881 - 1.25793i) q^{88} +12.8665i q^{89} -2.62610 q^{91} -5.09458i q^{92} -7.16146i q^{94} -24.1796 q^{95} +17.3108 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} - 4 q^{11} + 12 q^{16} + 16 q^{17} - 4 q^{22} - 4 q^{25} + 16 q^{29} + 12 q^{32} + 16 q^{34} - 8 q^{35} + 24 q^{37} + 16 q^{41} - 4 q^{44} - 12 q^{49} - 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} + 16 q^{68} - 8 q^{70} + 24 q^{74} + 8 q^{77} + 16 q^{82} + 16 q^{83} - 4 q^{88} - 48 q^{95} + 48 q^{97} - 12 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8 - 4 * q^11 + 12 * q^16 + 16 * q^17 - 4 * q^22 - 4 * q^25 + 16 * q^29 + 12 * q^32 + 16 * q^34 - 8 * q^35 + 24 * q^37 + 16 * q^41 - 4 * q^44 - 12 * q^49 - 4 * q^50 - 8 * q^55 + 16 * q^58 + 12 * q^64 - 48 * q^67 + 16 * q^68 - 8 * q^70 + 24 * q^74 + 8 * q^77 + 16 * q^82 + 16 * q^83 - 4 * q^88 - 48 * q^95 + 48 * q^97 - 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			3.35236i			1.49922i	0.661879	+	0.749611i	$0.269759\pi$
$5$			3.35236i			1.49922i	−0.661879	+	0.749611i	$0.730241\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$	1.00000			0.353553
$9$		0			0
$10$			3.35236i			1.06011i
$11$	−3.06881	−	1.25793i	−0.925282	−	0.379279i
$11$	−3.06881	−	1.25793i	−0.925282	−	0.379279i
$12$		0			0
$13$			2.62610i			0.728348i	0.931331	+	0.364174i	$0.118649\pi$
$13$			2.62610i			0.728348i	−0.931331	+	0.364174i	$0.881351\pi$
$14$			1.00000i			0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.58660			−0.627344			−0.313672	−	0.949531i	$-0.601559\pi$
$17$	−2.58660			−0.627344			−0.313672	+	0.949531i	$0.601559\pi$
$18$		0			0
$19$			7.21270i			1.65471i	0.561681	+	0.827354i	$0.310155\pi$
$19$			7.21270i			1.65471i	−0.561681	+	0.827354i	$0.689845\pi$
$20$			3.35236i			0.749611i
$21$		0			0
$22$	−3.06881	−	1.25793i	−0.654273	−	0.268191i
$23$		−	5.09458i		−	1.06229i	−0.847280	−	0.531147i	$-0.821761\pi$
$23$		−	5.09458i		−	1.06229i	0.847280	−	0.531147i	$-0.178239\pi$
$24$		0			0
$25$	−6.23832			−1.24766
$26$			2.62610i			0.515020i
$27$		0			0
$28$			1.00000i			0.188982i
$29$	1.96651			0.365171			0.182586	−	0.983190i	$-0.441553\pi$
$29$	1.96651			0.365171			0.182586	+	0.983190i	$0.441553\pi$
$30$		0			0
$31$	−4.93541			−0.886426			−0.443213	−	0.896416i	$-0.646162\pi$
$31$	−4.93541			−0.886426			−0.443213	+	0.896416i	$0.646162\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	−2.58660			−0.443599
$35$	−3.35236			−0.566652
$36$		0			0
$37$	10.8036			1.77610			0.888052	−	0.459742i	$-0.152058\pi$
$37$	10.8036			1.77610			0.888052	+	0.459742i	$0.152058\pi$
$38$			7.21270i			1.17005i
$39$		0			0
$40$			3.35236i			0.530055i
$41$	−9.07304			−1.41697			−0.708486	−	0.705725i	$-0.750621\pi$
$41$	−9.07304			−1.41697			−0.708486	+	0.705725i	$0.750621\pi$
$42$		0			0
$43$			4.02738i			0.614170i	0.951682	+	0.307085i	$0.0993536\pi$
$43$			4.02738i			0.614170i	−0.951682	+	0.307085i	$0.900646\pi$
$44$	−3.06881	−	1.25793i	−0.462641	−	0.189639i
$45$		0			0
$46$		−	5.09458i		−	0.751155i
$47$		−	7.16146i		−	1.04461i	−0.852760	−	0.522303i	$-0.825073\pi$
$47$		−	7.16146i		−	1.04461i	0.852760	−	0.522303i	$-0.174927\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	−6.23832			−0.882232
$51$		0			0
$52$			2.62610i			0.364174i
$53$			10.3351i			1.41964i	0.704383	+	0.709820i	$0.251224\pi$
$53$			10.3351i			1.41964i	−0.704383	+	0.709820i	$0.748776\pi$
$54$		0			0
$55$	4.21702	−	10.2878i	0.568623	−	1.38720i
$56$			1.00000i			0.133631i
$57$		0			0
$58$	1.96651			0.258215
$59$			11.5083i			1.49826i	0.662424	+	0.749129i	$0.269528\pi$
$59$			11.5083i			1.49826i	−0.662424	+	0.749129i	$0.730472\pi$
$60$		0			0
$61$			4.07862i			0.522214i	0.965310	+	0.261107i	$0.0840876\pi$
$61$			4.07862i			0.522214i	−0.965310	+	0.261107i	$0.915912\pi$
$62$	−4.93541			−0.626798
$63$		0			0
$64$	1.00000			0.125000
$65$	−8.80362			−1.09196
$66$		0			0
$67$	−2.06511			−0.252294			−0.126147	−	0.992012i	$-0.540261\pi$
$67$	−2.06511			−0.252294			−0.126147	+	0.992012i	$0.540261\pi$
$68$	−2.58660			−0.313672
$69$		0			0
$70$	−3.35236			−0.400684
$71$		−	11.7993i		−	1.40032i	−0.713986	−	0.700160i	$-0.753112\pi$
$71$		−	11.7993i		−	1.40032i	0.713986	−	0.700160i	$-0.246888\pi$
$72$		0			0
$73$		−	5.29996i		−	0.620314i	−0.950685	−	0.310157i	$-0.899618\pi$
$73$		−	5.29996i		−	0.620314i	0.950685	−	0.310157i	$-0.100382\pi$
$74$	10.8036			1.25590
$75$		0			0
$76$			7.21270i			0.827354i
$77$	1.25793	−	3.06881i	0.143354	−	0.349724i
$78$		0			0
$79$		−	3.51764i		−	0.395765i	−0.980226	−	0.197883i	$-0.936593\pi$
$79$		−	3.51764i		−	0.395765i	0.980226	−	0.197883i	$-0.0634065\pi$
$80$			3.35236i			0.374805i
$81$		0			0
$82$	−9.07304			−1.00195
$83$	2.98781			0.327955			0.163977	−	0.986464i	$-0.447568\pi$
$83$	2.98781			0.327955			0.163977	+	0.986464i	$0.447568\pi$
$84$		0			0
$85$		−	8.67123i		−	0.940527i
$86$			4.02738i			0.434284i
$87$		0			0
$88$	−3.06881	−	1.25793i	−0.327137	−	0.134095i
$89$			12.8665i			1.36385i	0.731424	+	0.681923i	$0.238856\pi$
$89$			12.8665i			1.36385i	−0.731424	+	0.681923i	$0.761144\pi$
$90$		0			0
$91$	−2.62610			−0.275290
$92$		−	5.09458i		−	0.531147i
$93$		0			0
$94$		−	7.16146i		−	0.738648i
$95$	−24.1796			−2.48077
$96$		0			0
$97$	17.3108			1.75765			0.878825	−	0.477145i	$-0.158328\pi$
$97$	17.3108			1.75765			0.878825	+	0.477145i	$0.158328\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	−6.23832			−0.623832
$101$	17.2412			1.71557			0.857784	−	0.514010i	$-0.171841\pi$
$101$	17.2412			1.71557			0.857784	+	0.514010i	$0.171841\pi$
$102$		0			0
$103$	11.7251			1.15531			0.577653	−	0.816283i	$-0.303969\pi$
$103$	11.7251			1.15531			0.577653	+	0.816283i	$0.303969\pi$
$104$			2.62610i			0.257510i
$105$		0			0
$106$			10.3351i			1.00384i
$107$	−7.73642			−0.747908			−0.373954	−	0.927447i	$-0.621998\pi$
$107$	−7.73642			−0.747908			−0.373954	+	0.927447i	$0.621998\pi$
$108$		0			0
$109$			11.4780i			1.09940i	0.835363	+	0.549698i	$0.185257\pi$
$109$			11.4780i			1.09940i	−0.835363	+	0.549698i	$0.814743\pi$
$110$	4.21702	−	10.2878i	0.402077	−	0.980901i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$		−	1.11033i		−	0.104451i	−0.998635	−	0.0522254i	$-0.983369\pi$
$113$		−	1.11033i		−	0.104451i	0.998635	−	0.0522254i	$-0.0166314\pi$
$114$		0			0
$115$	17.0789			1.59261
$116$	1.96651			0.182586
$117$		0			0
$118$			11.5083i			1.05943i
$119$		−	2.58660i		−	0.237114i
$120$		0			0
$121$	7.83525	+	7.72068i	0.712295	+	0.701880i
$122$			4.07862i			0.369261i
$123$		0			0
$124$	−4.93541			−0.443213
$125$		−	4.15130i		−	0.371304i
$126$		0			0
$127$			18.8565i			1.67325i	0.547779	+	0.836623i	$0.315473\pi$
$127$			18.8565i			1.67325i	−0.547779	+	0.836623i	$0.684527\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	−8.80362			−0.772129
$131$	7.68865			0.671761			0.335880	−	0.941905i	$-0.390966\pi$
$131$	7.68865			0.671761			0.335880	+	0.941905i	$0.390966\pi$
$132$		0			0
$133$	−7.21270			−0.625421
$134$	−2.06511			−0.178399
$135$		0			0
$136$	−2.58660			−0.221799
$137$		−	0.0590051i		−	0.00504115i	−0.999997	−	0.00252057i	$-0.999198\pi$
$137$		−	0.0590051i		−	0.00504115i	0.999997	−	0.00252057i	$-0.000802325\pi$
$138$		0			0
$139$		−	10.2147i		−	0.866399i	−0.901298	−	0.433199i	$-0.857385\pi$
$139$		−	10.2147i		−	0.866399i	0.901298	−	0.433199i	$-0.142615\pi$
$140$	−3.35236			−0.283326
$141$		0			0
$142$		−	11.7993i		−	0.990176i
$143$	3.30343	−	8.05901i	0.276247	−	0.673928i
$144$		0			0
$145$			6.59245i			0.547473i
$146$		−	5.29996i		−	0.438628i
$147$		0			0
$148$	10.8036			0.888052
$149$	−6.83525			−0.559965			−0.279983	−	0.960005i	$-0.590329\pi$
$149$	−6.83525			−0.559965			−0.279983	+	0.960005i	$0.590329\pi$
$150$		0			0
$151$		−	7.92529i		−	0.644951i	−0.946578	−	0.322476i	$-0.895485\pi$
$151$		−	7.92529i		−	0.644951i	0.946578	−	0.322476i	$-0.104515\pi$
$152$			7.21270i			0.585027i
$153$		0			0
$154$	1.25793	−	3.06881i	0.101367	−	0.247292i
$155$		−	16.5453i		−	1.32895i
$156$		0			0
$157$	−4.66443			−0.372262			−0.186131	−	0.982525i	$-0.559595\pi$
$157$	−4.66443			−0.372262			−0.186131	+	0.982525i	$0.559595\pi$
$158$		−	3.51764i		−	0.279848i
$159$		0			0
$160$			3.35236i			0.265027i
$161$	5.09458			0.401509
$162$		0			0
$163$	21.4168			1.67750			0.838748	−	0.544519i	$-0.183288\pi$
$163$	21.4168			1.67750			0.838748	+	0.544519i	$0.183288\pi$
$164$	−9.07304			−0.708486
$165$		0			0
$166$	2.98781			0.231899
$167$	9.50124			0.735228			0.367614	−	0.929978i	$-0.380175\pi$
$167$	9.50124			0.735228			0.367614	+	0.929978i	$0.380175\pi$
$168$		0			0
$169$	6.10361			0.469509
$170$		−	8.67123i		−	0.665053i
$171$		0			0
$172$			4.02738i			0.307085i
$173$	9.20483			0.699830			0.349915	−	0.936781i	$-0.386210\pi$
$173$	9.20483			0.699830			0.349915	+	0.936781i	$0.386210\pi$
$174$		0			0
$175$		−	6.23832i		−	0.471573i
$176$	−3.06881	−	1.25793i	−0.231321	−	0.0948197i
$177$		0			0
$178$			12.8665i			0.964385i
$179$		−	16.7658i		−	1.25313i	−0.779368	−	0.626566i	$-0.784460\pi$
$179$		−	16.7658i		−	1.25313i	0.779368	−	0.626566i	$-0.215540\pi$
$180$		0			0
$181$	−0.269417			−0.0200256			−0.0100128	−	0.999950i	$-0.503187\pi$
$181$	−0.269417			−0.0200256			−0.0100128	+	0.999950i	$0.503187\pi$
$182$	−2.62610			−0.194659
$183$		0			0
$184$		−	5.09458i		−	0.375578i
$185$			36.2176i			2.66277i
$186$		0			0
$187$	7.93781	+	3.25376i	0.580470	+	0.237938i
$188$		−	7.16146i		−	0.522303i
$189$		0			0
$190$	−24.1796			−1.75417
$191$		−	1.50413i		−	0.108835i	−0.998518	−	0.0544175i	$-0.982670\pi$
$191$		−	1.50413i		−	0.108835i	0.998518	−	0.0544175i	$-0.0173302\pi$
$192$		0			0
$193$			2.00864i			0.144585i	0.997383	+	0.0722925i	$0.0230315\pi$
$193$			2.00864i			0.144585i	−0.997383	+	0.0722925i	$0.976968\pi$
$194$	17.3108			1.24285
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	−21.5390			−1.53459			−0.767296	−	0.641294i	$-0.778398\pi$
$197$	−21.5390			−1.53459			−0.767296	+	0.641294i	$0.778398\pi$
$198$		0			0
$199$	9.24124			0.655094			0.327547	−	0.944835i	$-0.393778\pi$
$199$	9.24124			0.655094			0.327547	+	0.944835i	$0.393778\pi$
$200$	−6.23832			−0.441116
$201$		0			0
$202$	17.2412			1.21309
$203$			1.96651i			0.138022i
$204$		0			0
$205$		−	30.4161i		−	2.12435i
$206$	11.7251			0.816924
$207$		0			0
$208$			2.62610i			0.182087i
$209$	9.07304	−	22.1344i	0.627595	−	1.53107i
$210$		0			0
$211$		−	15.8783i		−	1.09311i	−0.837424	−	0.546553i	$-0.815940\pi$
$211$		−	15.8783i		−	1.09311i	0.837424	−	0.546553i	$-0.184060\pi$
$212$			10.3351i			0.709820i
$213$		0			0
$214$	−7.73642			−0.528851
$215$	−13.5012			−0.920777
$216$		0			0
$217$		−	4.93541i		−	0.335038i
$218$			11.4780i			0.777391i
$219$		0			0
$220$	4.21702	−	10.2878i	0.284311	−	0.693602i
$221$		−	6.79267i		−	0.456925i
$222$		0			0
$223$	−18.7704			−1.25696			−0.628481	−	0.777825i	$-0.716323\pi$
$223$	−18.7704			−1.25696			−0.628481	+	0.777825i	$0.716323\pi$
$224$			1.00000i			0.0668153i
$225$		0			0
$226$		−	1.11033i		−	0.0738578i
$227$	−12.6354			−0.838642			−0.419321	−	0.907838i	$-0.637732\pi$
$227$	−12.6354			−0.838642			−0.419321	+	0.907838i	$0.637732\pi$
$228$		0			0
$229$	−10.8514			−0.717080			−0.358540	−	0.933514i	$-0.616725\pi$
$229$	−10.8514			−0.717080			−0.358540	+	0.933514i	$0.616725\pi$
$230$	17.0789			1.12615
$231$		0			0
$232$	1.96651			0.129108
$233$	11.8045			0.773336			0.386668	−	0.922219i	$-0.373626\pi$
$233$	11.8045			0.773336			0.386668	+	0.922219i	$0.373626\pi$
$234$		0			0
$235$	24.0078			1.56610
$236$			11.5083i			0.749129i
$237$		0			0
$238$		−	2.58660i		−	0.167665i
$239$	19.5435			1.26416			0.632082	−	0.774902i	$-0.282201\pi$
$239$	19.5435			1.26416			0.632082	+	0.774902i	$0.282201\pi$
$240$		0			0
$241$		−	19.0766i		−	1.22883i	−0.788983	−	0.614415i	$-0.789392\pi$
$241$		−	19.0766i		−	1.22883i	0.788983	−	0.614415i	$-0.210608\pi$
$242$	7.83525	+	7.72068i	0.503669	+	0.496304i
$243$		0			0
$244$			4.07862i			0.261107i
$245$		−	3.35236i		−	0.214174i
$246$		0			0
$247$	−18.9413			−1.20520
$248$	−4.93541			−0.313399
$249$		0			0
$250$		−	4.15130i		−	0.262551i
$251$		−	2.22760i		−	0.140605i	−0.997526	−	0.0703023i	$-0.977604\pi$
$251$		−	2.22760i		−	0.140605i	0.997526	−	0.0703023i	$-0.0223964\pi$
$252$		0			0
$253$	−6.40861	+	15.6343i	−0.402906	+	0.982922i
$254$			18.8565i			1.18316i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	18.2288i		−	1.13708i	−0.822656	−	0.568540i	$-0.807509\pi$
$257$		−	18.2288i		−	1.13708i	0.822656	−	0.568540i	$-0.192491\pi$
$258$		0			0
$259$			10.8036i			0.671305i
$260$	−8.80362			−0.545978
$261$		0			0
$262$	7.68865			0.475007
$263$	−19.1050			−1.17806			−0.589032	−	0.808110i	$-0.700491\pi$
$263$	−19.1050			−1.17806			−0.589032	+	0.808110i	$0.700491\pi$
$264$		0			0
$265$	−34.6471			−2.12835
$266$	−7.21270			−0.442239
$267$		0			0
$268$	−2.06511			−0.126147
$269$		−	7.14629i		−	0.435717i	−0.975980	−	0.217859i	$-0.930093\pi$
$269$		−	7.14629i		−	0.435717i	0.975980	−	0.217859i	$-0.0699072\pi$
$270$		0			0
$271$			7.19000i			0.436762i	0.975864	+	0.218381i	$0.0700775\pi$
$271$			7.19000i			0.436762i	−0.975864	+	0.218381i	$0.929923\pi$
$272$	−2.58660			−0.156836
$273$		0			0
$274$		−	0.0590051i		−	0.00356463i
$275$	19.1443	+	7.84734i	1.15444	+	0.473213i
$276$		0			0
$277$		−	16.7302i		−	1.00522i	−0.864513	−	0.502611i	$-0.832373\pi$
$277$		−	16.7302i		−	1.00522i	0.864513	−	0.502611i	$-0.167627\pi$
$278$		−	10.2147i		−	0.612637i
$279$		0			0
$280$	−3.35236			−0.200342
$281$	22.0507			1.31543			0.657717	−	0.753265i	$-0.271522\pi$
$281$	22.0507			1.31543			0.657717	+	0.753265i	$0.271522\pi$
$282$		0			0
$283$		−	21.9511i		−	1.30486i	−0.757850	−	0.652429i	$-0.773750\pi$
$283$		−	21.9511i		−	1.30486i	0.757850	−	0.652429i	$-0.226250\pi$
$284$		−	11.7993i		−	0.700160i
$285$		0			0
$286$	3.30343	−	8.05901i	0.195336	−	0.476539i
$287$		−	9.07304i		−	0.535565i
$288$		0			0
$289$	−10.3095			−0.606440
$290$			6.59245i			0.387122i
$291$		0			0
$292$		−	5.29996i		−	0.310157i
$293$	7.09898			0.414727			0.207364	−	0.978264i	$-0.433512\pi$
$293$	7.09898			0.414727			0.207364	+	0.978264i	$0.433512\pi$
$294$		0			0
$295$	−38.5801			−2.24622
$296$	10.8036			0.627948
$297$		0			0
$298$	−6.83525			−0.395955
$299$	13.3789			0.773720
$300$		0			0
$301$	−4.02738			−0.232135
$302$		−	7.92529i		−	0.456049i
$303$		0			0
$304$			7.21270i			0.413677i
$305$	−13.6730			−0.782915
$306$		0			0
$307$		−	10.7723i		−	0.614807i	−0.951579	−	0.307403i	$-0.900540\pi$
$307$		−	10.7723i		−	0.614807i	0.951579	−	0.307403i	$-0.0994601\pi$
$308$	1.25793	−	3.06881i	0.0716770	−	0.174862i
$309$		0			0
$310$		−	16.5453i		−	0.939709i
$311$			27.2967i			1.54785i	0.633275	+	0.773927i	$0.281710\pi$
$311$			27.2967i			1.54785i	−0.633275	+	0.773927i	$0.718290\pi$
$312$		0			0
$313$	6.46924			0.365663			0.182832	−	0.983144i	$-0.441474\pi$
$313$	6.46924			0.365663			0.182832	+	0.983144i	$0.441474\pi$
$314$	−4.66443			−0.263229
$315$		0			0
$316$		−	3.51764i		−	0.197883i
$317$			4.20033i			0.235914i	0.993019	+	0.117957i	$0.0376345\pi$
$317$			4.20033i			0.235914i	−0.993019	+	0.117957i	$0.962365\pi$
$318$		0			0
$319$	−6.03485	−	2.47372i	−0.337887	−	0.138502i
$320$			3.35236i			0.187403i
$321$		0			0
$322$	5.09458			0.283910
$323$		−	18.6564i		−	1.03807i
$324$		0			0
$325$		−	16.3824i		−	0.908734i
$326$	21.4168			1.18617
$327$		0			0
$328$	−9.07304			−0.500975
$329$	7.16146			0.394824
$330$		0			0
$331$	−23.9372			−1.31570			−0.657852	−	0.753147i	$-0.728535\pi$
$331$	−23.9372			−1.31570			−0.657852	+	0.753147i	$0.728535\pi$
$332$	2.98781			0.163977
$333$		0			0
$334$	9.50124			0.519885
$335$		−	6.92300i		−	0.378244i
$336$		0			0
$337$			26.6983i			1.45435i	0.686451	+	0.727176i	$0.259168\pi$
$337$			26.6983i			1.45435i	−0.686451	+	0.727176i	$0.740832\pi$
$338$	6.10361			0.331993
$339$		0			0
$340$		−	8.67123i		−	0.470263i
$341$	15.1459	+	6.20838i	0.820195	+	0.336203i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$			4.02738i			0.217142i
$345$		0			0
$346$	9.20483			0.494855
$347$	5.57336			0.299194			0.149597	−	0.988747i	$-0.452202\pi$
$347$	5.57336			0.299194			0.149597	+	0.988747i	$0.452202\pi$
$348$		0			0
$349$			7.76028i			0.415398i	0.978193	+	0.207699i	$0.0665975\pi$
$349$			7.76028i			0.415398i	−0.978193	+	0.207699i	$0.933402\pi$
$350$		−	6.23832i		−	0.333452i
$351$		0			0
$352$	−3.06881	−	1.25793i	−0.163568	−	0.0670477i
$353$			7.59456i			0.404217i	0.979363	+	0.202109i	$0.0647794\pi$
$353$			7.59456i			0.404217i	−0.979363	+	0.202109i	$0.935221\pi$
$354$		0			0
$355$	39.5555			2.09939
$356$			12.8665i			0.681923i
$357$		0			0
$358$		−	16.7658i		−	0.886099i
$359$	34.0547			1.79734			0.898669	−	0.438628i	$-0.144535\pi$
$359$	34.0547			1.79734			0.898669	+	0.438628i	$0.144535\pi$
$360$		0			0
$361$	−33.0231			−1.73806
$362$	−0.269417			−0.0141603
$363$		0			0
$364$	−2.62610			−0.137645
$365$	17.7674			0.929988
$366$		0			0
$367$	27.5605			1.43865			0.719324	−	0.694675i	$-0.244452\pi$
$367$	27.5605			1.43865			0.719324	+	0.694675i	$0.244452\pi$
$368$		−	5.09458i		−	0.265574i
$369$		0			0
$370$			36.2176i			1.88287i
$371$	−10.3351			−0.536574
$372$		0			0
$373$			16.5994i			0.859485i	0.902952	+	0.429742i	$0.141396\pi$
$373$			16.5994i			0.859485i	−0.902952	+	0.429742i	$0.858604\pi$
$374$	7.93781	+	3.25376i	0.410454	+	0.168248i
$375$		0			0
$376$		−	7.16146i		−	0.369324i
$377$			5.16424i			0.265972i
$378$		0			0
$379$	37.3557			1.91883			0.959417	−	0.281992i	$-0.0909951\pi$
$379$	37.3557			1.91883			0.959417	+	0.281992i	$0.0909951\pi$
$380$	−24.1796			−1.24039
$381$		0			0
$382$		−	1.50413i		−	0.0769580i
$383$		−	10.9409i		−	0.559052i	−0.960138	−	0.279526i	$-0.909823\pi$
$383$		−	10.9409i		−	0.559052i	0.960138	−	0.279526i	$-0.0901773\pi$
$384$		0			0
$385$	10.2878	+	4.21702i	0.524313	+	0.214919i
$386$			2.00864i			0.102237i
$387$		0			0
$388$	17.3108			0.878825
$389$		−	22.9680i		−	1.16452i	−0.813002	−	0.582261i	$-0.802168\pi$
$389$		−	22.9680i		−	1.16452i	0.813002	−	0.582261i	$-0.197832\pi$
$390$		0			0
$391$			13.1777i			0.666424i
$392$	−1.00000			−0.0505076
$393$		0			0
$394$	−21.5390			−1.08512
$395$	11.7924			0.593340
$396$		0			0
$397$	−10.4585			−0.524896			−0.262448	−	0.964946i	$-0.584530\pi$
$397$	−10.4585			−0.524896			−0.262448	+	0.964946i	$0.584530\pi$
$398$	9.24124			0.463222
$399$		0			0
$400$	−6.23832			−0.311916
$401$			34.8379i			1.73972i	0.493296	+	0.869862i	$0.335792\pi$
$401$			34.8379i			1.73972i	−0.493296	+	0.869862i	$0.664208\pi$
$402$		0			0
$403$		−	12.9609i		−	0.645627i
$404$	17.2412			0.857784
$405$		0			0
$406$			1.96651i			0.0975962i
$407$	−33.1543	−	13.5902i	−1.64340	−	0.673639i
$408$		0			0
$409$		−	14.5088i		−	0.717413i	−0.933450	−	0.358706i	$-0.883218\pi$
$409$		−	14.5088i		−	0.717413i	0.933450	−	0.358706i	$-0.116782\pi$
$410$		−	30.4161i		−	1.50214i
$411$		0			0
$412$	11.7251			0.577653
$413$	−11.5083			−0.566289
$414$		0			0
$415$			10.0162i			0.491677i
$416$			2.62610i			0.128755i
$417$		0			0
$418$	9.07304	−	22.1344i	0.443777	−	1.08263i
$419$			26.4172i			1.29057i	0.763943	+	0.645283i	$0.223261\pi$
$419$			26.4172i			1.29057i	−0.763943	+	0.645283i	$0.776739\pi$
$420$		0			0
$421$	−8.33866			−0.406402			−0.203201	−	0.979137i	$-0.565134\pi$
$421$	−8.33866			−0.406402			−0.203201	+	0.979137i	$0.565134\pi$
$422$		−	15.8783i		−	0.772943i
$423$		0			0
$424$			10.3351i			0.501919i
$425$	16.1361			0.782714
$426$		0			0
$427$	−4.07862			−0.197378
$428$	−7.73642			−0.373954
$429$		0			0
$430$	−13.5012			−0.651088
$431$	7.96449			0.383636			0.191818	−	0.981430i	$-0.438562\pi$
$431$	7.96449			0.383636			0.191818	+	0.981430i	$0.438562\pi$
$432$		0			0
$433$	−35.1648			−1.68991			−0.844956	−	0.534836i	$-0.820373\pi$
$433$	−35.1648			−1.68991			−0.844956	+	0.534836i	$0.820373\pi$
$434$		−	4.93541i		−	0.236907i
$435$		0			0
$436$			11.4780i			0.549698i
$437$	36.7457			1.75779
$438$		0			0
$439$			2.39919i			0.114507i	0.998360	+	0.0572536i	$0.0182343\pi$
$439$			2.39919i			0.114507i	−0.998360	+	0.0572536i	$0.981766\pi$
$440$	4.21702	−	10.2878i	0.201039	−	0.490450i
$441$		0			0
$442$		−	6.79267i		−	0.323095i
$443$			15.0747i			0.716221i	0.933679	+	0.358110i	$0.116579\pi$
$443$			15.0747i			0.716221i	−0.933679	+	0.358110i	$0.883421\pi$
$444$		0			0
$445$	−43.1332			−2.04471
$446$	−18.7704			−0.888806
$447$		0			0
$448$			1.00000i			0.0472456i
$449$		−	26.3561i		−	1.24382i	−0.783087	−	0.621912i	$-0.786356\pi$
$449$		−	26.3561i		−	1.24382i	0.783087	−	0.621912i	$-0.213644\pi$
$450$		0			0
$451$	27.8435	+	11.4132i	1.31110	+	0.537427i
$452$		−	1.11033i		−	0.0522254i
$453$		0			0
$454$	−12.6354			−0.593010
$455$		−	8.80362i		−	0.412720i
$456$		0			0
$457$			21.0121i			0.982905i	0.870904	+	0.491453i	$0.163534\pi$
$457$			21.0121i			0.982905i	−0.870904	+	0.491453i	$0.836466\pi$
$458$	−10.8514			−0.507052
$459$		0			0
$460$	17.0789			0.796307
$461$	−38.9227			−1.81281			−0.906406	−	0.422408i	$-0.861185\pi$
$461$	−38.9227			−1.81281			−0.906406	+	0.422408i	$0.861185\pi$
$462$		0			0
$463$	−21.5767			−1.00275			−0.501377	−	0.865229i	$-0.667173\pi$
$463$	−21.5767			−1.00275			−0.501377	+	0.865229i	$0.667173\pi$
$464$	1.96651			0.0912929
$465$		0			0
$466$	11.8045			0.546831
$467$			5.78872i			0.267870i	0.990990	+	0.133935i	$0.0427613\pi$
$467$			5.78872i			0.267870i	−0.990990	+	0.133935i	$0.957239\pi$
$468$		0			0
$469$		−	2.06511i		−	0.0953581i
$470$	24.0078			1.10740
$471$		0			0
$472$			11.5083i			0.529715i
$473$	5.06615	−	12.3593i	0.232942	−	0.568281i
$474$		0			0
$475$		−	44.9951i		−	2.06452i
$476$		−	2.58660i		−	0.118557i
$477$		0			0
$478$	19.5435			0.893899
$479$	10.0153			0.457609			0.228804	−	0.973472i	$-0.426518\pi$
$479$	10.0153			0.457609			0.228804	+	0.973472i	$0.426518\pi$
$480$		0			0
$481$			28.3714i			1.29362i
$482$		−	19.0766i		−	0.868914i
$483$		0			0
$484$	7.83525	+	7.72068i	0.356148	+	0.350940i
$485$			58.0322i			2.63511i
$486$		0			0
$487$	−2.74022			−0.124171			−0.0620856	−	0.998071i	$-0.519775\pi$
$487$	−2.74022			−0.124171			−0.0620856	+	0.998071i	$0.519775\pi$
$488$			4.07862i			0.184631i
$489$		0			0
$490$		−	3.35236i		−	0.151444i
$491$	18.0042			0.812517			0.406259	−	0.913758i	$-0.366833\pi$
$491$	18.0042			0.812517			0.406259	+	0.913758i	$0.366833\pi$
$492$		0			0
$493$	−5.08658			−0.229088
$494$	−18.9413			−0.852207
$495$		0			0
$496$	−4.93541			−0.221607
$497$	11.7993			0.529271
$498$		0			0
$499$	31.6254			1.41575			0.707874	−	0.706338i	$-0.249654\pi$
$499$	31.6254			1.41575			0.707874	+	0.706338i	$0.249654\pi$
$500$		−	4.15130i		−	0.185652i
$501$		0			0
$502$		−	2.22760i		−	0.0994225i
$503$	−32.8122			−1.46302			−0.731511	−	0.681829i	$-0.761185\pi$
$503$	−32.8122			−1.46302			−0.731511	+	0.681829i	$0.761185\pi$
$504$		0			0
$505$			57.7989i			2.57202i
$506$	−6.40861	+	15.6343i	−0.284897	+	0.695031i
$507$		0			0
$508$			18.8565i			0.836623i
$509$			7.25583i			0.321609i	0.986986	+	0.160804i	$0.0514088\pi$
$509$			7.25583i			0.321609i	−0.986986	+	0.160804i	$0.948591\pi$
$510$		0			0
$511$	5.29996			0.234457
$512$	1.00000			0.0441942
$513$		0			0
$514$		−	18.2288i		−	0.804036i
$515$			39.3067i			1.73206i
$516$		0			0
$517$	−9.00858	+	21.9772i	−0.396197	+	0.966556i
$518$			10.8036i			0.474684i
$519$		0			0
$520$	−8.80362			−0.386064
$521$			2.42192i			0.106106i	0.998592	+	0.0530531i	$0.0168952\pi$
$521$			2.42192i			0.106106i	−0.998592	+	0.0530531i	$0.983105\pi$
$522$		0			0
$523$			12.9406i			0.565853i	0.959142	+	0.282926i	$0.0913052\pi$
$523$			12.9406i			0.565853i	−0.959142	+	0.282926i	$0.908695\pi$
$524$	7.68865			0.335880
$525$		0			0
$526$	−19.1050			−0.833016
$527$	12.7660			0.556094
$528$		0			0
$529$	−2.95479			−0.128469
$530$	−34.6471			−1.50497
$531$		0			0
$532$	−7.21270			−0.312710
$533$		−	23.8267i		−	1.03205i
$534$		0			0
$535$		−	25.9353i		−	1.12128i
$536$	−2.06511			−0.0891993
$537$		0			0
$538$		−	7.14629i		−	0.308099i
$539$	3.06881	+	1.25793i	0.132183	+	0.0541827i
$540$		0			0
$541$			42.9544i			1.84675i	0.383896	+	0.923377i	$0.374582\pi$
$541$			42.9544i			1.84675i	−0.383896	+	0.923377i	$0.625418\pi$
$542$			7.19000i			0.308837i
$543$		0			0
$544$	−2.58660			−0.110900
$545$	−38.4785			−1.64824
$546$		0			0
$547$		−	25.5865i		−	1.09400i	−0.837133	−	0.547000i	$-0.815770\pi$
$547$		−	25.5865i		−	1.09400i	0.837133	−	0.547000i	$-0.184230\pi$
$548$		−	0.0590051i		−	0.00252057i
$549$		0			0
$550$	19.1443	+	7.84734i	0.816314	+	0.334612i
$551$			14.1838i			0.604252i
$552$		0			0
$553$	3.51764			0.149585
$554$		−	16.7302i		−	0.710800i
$555$		0			0
$556$		−	10.2147i		−	0.433199i
$557$	−7.11490			−0.301468			−0.150734	−	0.988574i	$-0.548164\pi$
$557$	−7.11490			−0.301468			−0.150734	+	0.988574i	$0.548164\pi$
$558$		0			0
$559$	−10.5763			−0.447330
$560$	−3.35236			−0.141663
$561$		0			0
$562$	22.0507			0.930153
$563$	−0.410169			−0.0172866			−0.00864329	−	0.999963i	$-0.502751\pi$
$563$	−0.410169			−0.0172866			−0.00864329	+	0.999963i	$0.502751\pi$
$564$		0			0
$565$	3.72222			0.156595
$566$		−	21.9511i		−	0.922674i
$567$		0			0
$568$		−	11.7993i		−	0.495088i
$569$	−26.4731			−1.10981			−0.554904	−	0.831914i	$-0.687245\pi$
$569$	−26.4731			−1.10981			−0.554904	+	0.831914i	$0.687245\pi$
$570$		0			0
$571$		−	24.0408i		−	1.00607i	−0.864265	−	0.503037i	$-0.832216\pi$
$571$		−	24.0408i		−	1.00607i	0.864265	−	0.503037i	$-0.167784\pi$
$572$	3.30343	−	8.05901i	0.138124	−	0.336964i
$573$		0			0
$574$		−	9.07304i		−	0.378701i
$575$			31.7817i			1.32539i
$576$		0			0
$577$	32.0445			1.33403			0.667015	−	0.745044i	$-0.267572\pi$
$577$	32.0445			1.33403			0.667015	+	0.745044i	$0.267572\pi$
$578$	−10.3095			−0.428818
$579$		0			0
$580$			6.59245i			0.273736i
$581$			2.98781i			0.123955i
$582$		0			0
$583$	13.0008	−	31.7166i	0.538440	−	1.31357i
$584$		−	5.29996i		−	0.219314i
$585$		0			0
$586$	7.09898			0.293256
$587$		−	16.4754i		−	0.680012i	−0.940423	−	0.340006i	$-0.889571\pi$
$587$		−	16.4754i		−	0.680012i	0.940423	−	0.340006i	$-0.110429\pi$
$588$		0			0
$589$		−	35.5977i		−	1.46678i
$590$	−38.5801			−1.58832
$591$		0			0
$592$	10.8036			0.444026
$593$	39.4117			1.61844			0.809222	−	0.587502i	$-0.199889\pi$
$593$	39.4117			1.61844			0.809222	+	0.587502i	$0.199889\pi$
$594$		0			0
$595$	8.67123			0.355486
$596$	−6.83525			−0.279983
$597$		0			0
$598$	13.3789			0.547103
$599$		−	40.0955i		−	1.63826i	−0.573609	−	0.819130i	$-0.694457\pi$
$599$		−	40.0955i		−	1.63826i	0.573609	−	0.819130i	$-0.305543\pi$
$600$		0			0
$601$		−	5.11101i		−	0.208483i	−0.994552	−	0.104241i	$-0.966759\pi$
$601$		−	5.11101i		−	0.208483i	0.994552	−	0.104241i	$-0.0332414\pi$
$602$	−4.02738			−0.164144
$603$		0			0
$604$		−	7.92529i		−	0.322476i
$605$	−25.8825	+	26.2666i	−1.05227	+	1.06789i
$606$		0			0
$607$		−	13.3477i		−	0.541768i	−0.962612	−	0.270884i	$-0.912684\pi$
$607$		−	13.3477i		−	0.541768i	0.962612	−	0.270884i	$-0.0873159\pi$
$608$			7.21270i			0.292514i
$609$		0			0
$610$	−13.6730			−0.553604
$611$	18.8067			0.760837
$612$		0			0
$613$		−	9.39875i		−	0.379612i	−0.981822	−	0.189806i	$-0.939214\pi$
$613$		−	9.39875i		−	0.379612i	0.981822	−	0.189806i	$-0.0607859\pi$
$614$		−	10.7723i		−	0.434734i
$615$		0			0
$616$	1.25793	−	3.06881i	0.0506833	−	0.123646i
$617$			40.8054i			1.64276i	0.570380	+	0.821381i	$0.306796\pi$
$617$			40.8054i			1.64276i	−0.570380	+	0.821381i	$0.693204\pi$
$618$		0			0
$619$	1.47643			0.0593426			0.0296713	−	0.999560i	$-0.490554\pi$
$619$	1.47643			0.0593426			0.0296713	+	0.999560i	$0.490554\pi$
$620$		−	16.5453i		−	0.664475i
$621$		0			0
$622$			27.2967i			1.09450i
$623$	−12.8665			−0.515486
$624$		0			0
$625$	−17.2750			−0.690998
$626$	6.46924			0.258563
$627$		0			0
$628$	−4.66443			−0.186131
$629$	−27.9447			−1.11423
$630$		0			0
$631$	−23.3883			−0.931073			−0.465537	−	0.885029i	$-0.654139\pi$
$631$	−23.3883			−0.931073			−0.465537	+	0.885029i	$0.654139\pi$
$632$		−	3.51764i		−	0.139924i
$633$		0			0
$634$			4.20033i			0.166817i
$635$	−63.2139			−2.50857
$636$		0			0
$637$		−	2.62610i		−	0.104050i
$638$	−6.03485	−	2.47372i	−0.238922	−	0.0979356i
$639$		0			0
$640$			3.35236i			0.132514i
$641$		−	0.891543i		−	0.0352138i	−0.999845	−	0.0176069i	$-0.994395\pi$
$641$		−	0.891543i		−	0.0352138i	0.999845	−	0.0176069i	$-0.00560475\pi$
$642$		0			0
$643$	1.02877			0.0405708			0.0202854	−	0.999794i	$-0.493543\pi$
$643$	1.02877			0.0405708			0.0202854	+	0.999794i	$0.493543\pi$
$644$	5.09458			0.200755
$645$		0			0
$646$		−	18.6564i		−	0.734026i
$647$			9.15524i			0.359930i	0.983673	+	0.179965i	$0.0575984\pi$
$647$			9.15524i			0.359930i	−0.983673	+	0.179965i	$0.942402\pi$
$648$		0			0
$649$	14.4766	−	35.3170i	0.568258	−	1.38631i
$650$		−	16.3824i		−	0.642572i
$651$		0			0
$652$	21.4168			0.838748
$653$		−	10.5760i		−	0.413872i	−0.978354	−	0.206936i	$-0.933651\pi$
$653$		−	10.5760i		−	0.413872i	0.978354	−	0.206936i	$-0.0663492\pi$
$654$		0			0
$655$			25.7751i			1.00712i
$656$	−9.07304			−0.354243
$657$		0			0
$658$	7.16146			0.279183
$659$	−38.0445			−1.48200			−0.741002	−	0.671503i	$-0.765649\pi$
$659$	−38.0445			−1.48200			−0.741002	+	0.671503i	$0.765649\pi$
$660$		0			0
$661$	−5.61705			−0.218478			−0.109239	−	0.994016i	$-0.534841\pi$
$661$	−5.61705			−0.218478			−0.109239	+	0.994016i	$0.534841\pi$
$662$	−23.9372			−0.930344
$663$		0			0
$664$	2.98781			0.115949
$665$		−	24.1796i		−	0.937644i
$666$		0			0
$667$		−	10.0185i		−	0.387920i
$668$	9.50124			0.367614
$669$		0			0
$670$		−	6.92300i		−	0.267459i
$671$	5.13061	−	12.5165i	0.198065	−	0.483196i
$672$		0			0
$673$			16.7641i			0.646207i	0.946364	+	0.323104i	$0.104726\pi$
$673$			16.7641i			0.646207i	−0.946364	+	0.323104i	$0.895274\pi$
$674$			26.6983i			1.02838i
$675$		0			0
$676$	6.10361			0.234754
$677$	−46.9213			−1.80333			−0.901666	−	0.432433i	$-0.857655\pi$
$677$	−46.9213			−1.80333			−0.901666	+	0.432433i	$0.857655\pi$
$678$		0			0
$679$			17.3108i			0.664329i
$680$		−	8.67123i		−	0.332526i
$681$		0			0
$682$	15.1459	+	6.20838i	0.579965	+	0.237731i
$683$			5.60839i			0.214599i	0.994227	+	0.107299i	$0.0342204\pi$
$683$			5.60839i			0.214599i	−0.994227	+	0.107299i	$0.965780\pi$
$684$		0			0
$685$	0.197806			0.00755780
$686$		−	1.00000i		−	0.0381802i
$687$		0			0
$688$			4.02738i			0.153543i
$689$	−27.1411			−1.03399
$690$		0			0
$691$	−41.8435			−1.59180			−0.795901	−	0.605427i	$-0.793002\pi$
$691$	−41.8435			−1.59180			−0.795901	+	0.605427i	$0.793002\pi$
$692$	9.20483			0.349915
$693$		0			0
$694$	5.57336			0.211562
$695$	34.2433			1.29892
$696$		0			0
$697$	23.4684			0.888928
$698$			7.76028i			0.293731i
$699$		0			0
$700$		−	6.23832i		−	0.235786i
$701$	33.0395			1.24789			0.623943	−	0.781470i	$-0.285530\pi$
$701$	33.0395			1.24789			0.623943	+	0.781470i	$0.285530\pi$
$702$		0			0
$703$			77.9233i			2.93893i
$704$	−3.06881	−	1.25793i	−0.115660	−	0.0474099i
$705$		0			0
$706$			7.59456i			0.285825i
$707$			17.2412i			0.648424i
$708$		0			0
$709$	7.70620			0.289413			0.144706	−	0.989475i	$-0.453776\pi$
$709$	7.70620			0.289413			0.144706	+	0.989475i	$0.453776\pi$
$710$	39.5555			1.48449
$711$		0			0
$712$			12.8665i			0.482193i
$713$			25.1439i			0.941645i
$714$		0			0
$715$	27.0167	+	11.0743i	1.01037	+	0.414156i
$716$		−	16.7658i		−	0.626566i
$717$		0			0
$718$	34.0547			1.27091
$719$			5.15584i			0.192280i	0.995368	+	0.0961402i	$0.0306497\pi$
$719$			5.15584i			0.192280i	−0.995368	+	0.0961402i	$0.969350\pi$
$720$		0			0
$721$			11.7251i			0.436664i
$722$	−33.0231			−1.22899
$723$		0			0
$724$	−0.269417			−0.0100128
$725$	−12.2677			−0.455611
$726$		0			0
$727$	−18.9729			−0.703665			−0.351833	−	0.936063i	$-0.614441\pi$
$727$	−18.9729			−0.703665			−0.351833	+	0.936063i	$0.614441\pi$
$728$	−2.62610			−0.0973296
$729$		0			0
$730$	17.7674			0.657601
$731$		−	10.4172i		−	0.385296i
$732$		0			0
$733$			4.83717i			0.178665i	0.996002	+	0.0893325i	$0.0284734\pi$
$733$			4.83717i			0.178665i	−0.996002	+	0.0893325i	$0.971527\pi$
$734$	27.5605			1.01728
$735$		0			0
$736$		−	5.09458i		−	0.187789i
$737$	6.33745	+	2.59776i	0.233443	+	0.0956897i
$738$		0			0
$739$		−	2.53199i		−	0.0931406i	−0.998915	−	0.0465703i	$-0.985171\pi$
$739$		−	2.53199i		−	0.0931406i	0.998915	−	0.0465703i	$-0.0148291\pi$
$740$			36.2176i			1.33139i
$741$		0			0
$742$	−10.3351			−0.379415
$743$	6.33866			0.232543			0.116272	−	0.993217i	$-0.462906\pi$
$743$	6.33866			0.232543			0.116272	+	0.993217i	$0.462906\pi$
$744$		0			0
$745$		−	22.9142i		−	0.839512i
$746$			16.5994i			0.607747i
$747$		0			0
$748$	7.93781	+	3.25376i	0.290235	+	0.118969i
$749$		−	7.73642i		−	0.282683i
$750$		0			0
$751$	49.8904			1.82053			0.910263	−	0.414031i	$-0.135880\pi$
$751$	49.8904			1.82053			0.910263	+	0.414031i	$0.135880\pi$
$752$		−	7.16146i		−	0.261152i
$753$		0			0
$754$			5.16424i			0.188071i
$755$	26.5684			0.966925
$756$		0			0
$757$	7.16546			0.260433			0.130216	−	0.991486i	$-0.458433\pi$
$757$	7.16546			0.260433			0.130216	+	0.991486i	$0.458433\pi$
$758$	37.3557			1.35682
$759$		0			0
$760$	−24.1796			−0.877085
$761$	41.1465			1.49156			0.745779	−	0.666193i	$-0.232077\pi$
$761$	41.1465			1.49156			0.745779	+	0.666193i	$0.232077\pi$
$762$		0			0
$763$	−11.4780			−0.415533
$764$		−	1.50413i		−	0.0544175i
$765$		0			0
$766$		−	10.9409i		−	0.395309i
$767$	−30.2220			−1.09125
$768$		0			0
$769$		−	41.6306i		−	1.50124i	−0.660736	−	0.750618i	$-0.729756\pi$
$769$		−	41.6306i		−	1.50124i	0.660736	−	0.750618i	$-0.270244\pi$
$770$	10.2878	+	4.21702i	0.370746	+	0.151971i
$771$		0			0
$772$			2.00864i			0.0722925i
$773$			10.0653i			0.362025i	0.983481	+	0.181013i	$0.0579375\pi$
$773$			10.0653i			0.362025i	−0.983481	+	0.181013i	$0.942063\pi$
$774$		0			0
$775$	30.7887			1.10596
$776$	17.3108			0.621423
$777$		0			0
$778$		−	22.9680i		−	0.823442i
$779$		−	65.4411i		−	2.34467i
$780$		0			0
$781$	−14.8426	+	36.2099i	−0.531112	+	1.29569i
$782$			13.1777i			0.471233i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$		−	15.6369i		−	0.558104i
$786$		0			0
$787$			52.3003i			1.86431i	0.362065	+	0.932153i	$0.382072\pi$
$787$			52.3003i			1.86431i	−0.362065	+	0.932153i	$0.617928\pi$
$788$	−21.5390			−0.767296
$789$		0			0
$790$	11.7924			0.419555
$791$	1.11033			0.0394787
$792$		0			0
$793$	−10.7109			−0.380354
$794$	−10.4585			−0.371158
$795$		0			0
$796$	9.24124			0.327547
$797$			19.9738i			0.707509i	0.935338	+	0.353754i	$0.115095\pi$
$797$			19.9738i			0.707509i	−0.935338	+	0.353754i	$0.884905\pi$
$798$		0			0
$799$			18.5239i			0.655327i
$800$	−6.23832			−0.220558
$801$		0			0
$802$			34.8379i			1.23017i
$803$	−6.66696	+	16.2646i	−0.235272	+	0.573965i
$804$		0			0
$805$			17.0789i			0.601952i
$806$		−	12.9609i		−	0.456527i
$807$		0			0
$808$	17.2412			0.606545
$809$	21.1592			0.743916			0.371958	−	0.928249i	$-0.378686\pi$
$809$	21.1592			0.743916			0.371958	+	0.928249i	$0.378686\pi$
$810$		0			0
$811$			25.1236i			0.882209i	0.897456	+	0.441104i	$0.145413\pi$
$811$			25.1236i			0.882209i	−0.897456	+	0.441104i	$0.854587\pi$
$812$			1.96651i			0.0690109i
$813$		0			0
$814$	−33.1543	−	13.5902i	−1.16206	−	0.476335i
$815$			71.7970i			2.51494i
$816$		0			0
$817$	−29.0483			−1.01627
$818$		−	14.5088i		−	0.507287i
$819$		0			0
$820$		−	30.4161i		−	1.06218i
$821$	−28.4097			−0.991504			−0.495752	−	0.868464i	$-0.665107\pi$
$821$	−28.4097			−0.991504			−0.495752	+	0.868464i	$0.665107\pi$
$822$		0			0
$823$	−30.5807			−1.06598			−0.532989	−	0.846122i	$-0.678931\pi$
$823$	−30.5807			−1.06598			−0.532989	+	0.846122i	$0.678931\pi$
$824$	11.7251			0.408462
$825$		0			0
$826$	−11.5083			−0.400427
$827$	−34.3496			−1.19445			−0.597227	−	0.802072i	$-0.703731\pi$
$827$	−34.3496			−1.19445			−0.597227	+	0.802072i	$0.703731\pi$
$828$		0			0
$829$	−24.0885			−0.836629			−0.418314	−	0.908302i	$-0.637379\pi$
$829$	−24.0885			−0.836629			−0.418314	+	0.908302i	$0.637379\pi$
$830$			10.0162i			0.347668i
$831$		0			0
$832$			2.62610i			0.0910435i
$833$	2.58660			0.0896205
$834$		0			0
$835$			31.8516i			1.10227i
$836$	9.07304	−	22.1344i	0.313798	−	0.765536i
$837$		0			0
$838$			26.4172i			0.912569i
$839$		−	47.9574i		−	1.65567i	−0.560969	−	0.827837i	$-0.689571\pi$
$839$		−	47.9574i		−	1.65567i	0.560969	−	0.827837i	$-0.310429\pi$
$840$		0			0
$841$	−25.1328			−0.866650
$842$	−8.33866			−0.287369
$843$		0			0
$844$		−	15.8783i		−	0.546553i
$845$			20.4615i			0.703898i
$846$		0			0
$847$	−7.72068	+	7.83525i	−0.265286	+	0.269222i
$848$			10.3351i			0.354910i
$849$		0			0
$850$	16.1361			0.553463
$851$		−	55.0400i		−	1.88675i
$852$		0			0
$853$		−	13.9062i		−	0.476139i	−0.971248	−	0.238070i	$-0.923485\pi$
$853$		−	13.9062i		−	0.476139i	0.971248	−	0.238070i	$-0.0765146\pi$
$854$	−4.07862			−0.139568
$855$		0			0
$856$	−7.73642			−0.264426
$857$	−44.1782			−1.50910			−0.754549	−	0.656244i	$-0.772144\pi$
$857$	−44.1782			−1.50910			−0.754549	+	0.656244i	$0.772144\pi$
$858$		0			0
$859$	6.11542			0.208656			0.104328	−	0.994543i	$-0.466731\pi$
$859$	6.11542			0.208656			0.104328	+	0.994543i	$0.466731\pi$
$860$	−13.5012			−0.460389
$861$		0			0
$862$	7.96449			0.271272
$863$		−	21.2021i		−	0.721729i	−0.932618	−	0.360864i	$-0.882482\pi$
$863$		−	21.2021i		−	0.721729i	0.932618	−	0.360864i	$-0.117518\pi$
$864$		0			0
$865$			30.8579i			1.04920i
$866$	−35.1648			−1.19495
$867$		0			0
$868$		−	4.93541i		−	0.167519i
$869$	−4.42493	+	10.7950i	−0.150105	+	0.366195i
$870$		0			0
$871$		−	5.42319i		−	0.183758i
$872$			11.4780i			0.388695i
$873$		0			0
$874$	36.7457			1.24294
$875$	4.15130			0.140340
$876$		0			0
$877$			29.6758i			1.00208i	0.865424	+	0.501040i	$0.167049\pi$
$877$			29.6758i			1.00208i	−0.865424	+	0.501040i	$0.832951\pi$
$878$			2.39919i			0.0809688i
$879$		0			0
$880$	4.21702	−	10.2878i	0.142156	−	0.346801i
$881$		−	5.91565i		−	0.199303i	−0.995022	−	0.0996517i	$-0.968227\pi$
$881$		−	5.91565i		−	0.199303i	0.995022	−	0.0996517i	$-0.0317729\pi$
$882$		0			0
$883$	22.5291			0.758163			0.379082	−	0.925363i	$-0.376240\pi$
$883$	22.5291			0.758163			0.379082	+	0.925363i	$0.376240\pi$
$884$		−	6.79267i		−	0.228462i
$885$		0			0
$886$			15.0747i			0.506444i
$887$	19.4196			0.652047			0.326023	−	0.945362i	$-0.394291\pi$
$887$	19.4196			0.652047			0.326023	+	0.945362i	$0.394291\pi$
$888$		0			0
$889$	−18.8565			−0.632427
$890$	−43.1332			−1.44583
$891$		0			0
$892$	−18.7704			−0.628481
$893$	51.6535			1.72852
$894$		0			0
$895$	56.2049			1.87872
$896$			1.00000i			0.0334077i
$897$		0			0
$898$		−	26.3561i		−	0.879516i
$899$	−9.70553			−0.323698
$900$		0			0
$901$		−	26.7329i		−	0.890602i
$902$	27.8435	+	11.4132i	0.927087	+	0.380018i
$903$		0			0
$904$		−	1.11033i		−	0.0369289i
$905$		−	0.903184i		−	0.0300229i
$906$		0			0
$907$	24.7538			0.821935			0.410968	−	0.911650i	$-0.365191\pi$
$907$	24.7538			0.821935			0.410968	+	0.911650i	$0.365191\pi$
$908$	−12.6354			−0.419321
$909$		0			0
$910$		−	8.80362i		−	0.291837i
$911$		−	33.0149i		−	1.09383i	−0.837188	−	0.546915i	$-0.815802\pi$
$911$		−	33.0149i		−	1.09383i	0.837188	−	0.546915i	$-0.184198\pi$
$912$		0			0
$913$	−9.16903	−	3.75844i	−0.303451	−	0.124386i
$914$			21.0121i			0.695019i
$915$		0			0
$916$	−10.8514			−0.358540
$917$			7.68865i			0.253902i
$918$		0			0
$919$		−	4.41886i		−	0.145765i	−0.997341	−	0.0728823i	$-0.976780\pi$
$919$		−	4.41886i		−	0.145765i	0.997341	−	0.0728823i	$-0.0232198\pi$
$920$	17.0789			0.563074
$921$		0			0
$922$	−38.9227			−1.28185
$923$	30.9861			1.01992
$924$		0			0
$925$	−67.3965			−2.21598
$926$	−21.5767			−0.709054
$927$		0			0
$928$	1.96651			0.0645538
$929$		−	24.4841i		−	0.803298i	−0.915794	−	0.401649i	$-0.868437\pi$
$929$		−	24.4841i		−	0.803298i	0.915794	−	0.401649i	$-0.131563\pi$
$930$		0			0
$931$		−	7.21270i		−	0.236387i
$932$	11.8045			0.386668
$933$		0			0
$934$			5.78872i			0.189413i
$935$	−10.9078	+	26.6104i	−0.356722	+	0.870253i
$936$		0			0
$937$			35.7394i			1.16755i	0.811914	+	0.583777i	$0.198426\pi$
$937$			35.7394i			1.16755i	−0.811914	+	0.583777i	$0.801574\pi$
$938$		−	2.06511i		−	0.0674283i
$939$		0			0
$940$	24.0078			0.783048
$941$	28.9361			0.943290			0.471645	−	0.881788i	$-0.343660\pi$
$941$	28.9361			0.943290			0.471645	+	0.881788i	$0.343660\pi$
$942$		0			0
$943$			46.2234i			1.50524i
$944$			11.5083i			0.374565i
$945$		0			0
$946$	5.06615	−	12.3593i	0.164715	−	0.401835i
$947$			9.80446i			0.318602i	0.987230	+	0.159301i	$0.0509241\pi$
$947$			9.80446i			0.318602i	−0.987230	+	0.159301i	$0.949076\pi$
$948$		0			0
$949$	13.9182			0.451805
$950$		−	44.9951i		−	1.45984i
$951$		0			0
$952$		−	2.58660i		−	0.0838323i
$953$	−47.3568			−1.53404			−0.767019	−	0.641624i	$-0.778261\pi$
$953$	−47.3568			−1.53404			−0.767019	+	0.641624i	$0.778261\pi$
$954$		0			0
$955$	5.04238			0.163168
$956$	19.5435			0.632082
$957$		0			0
$958$	10.0153			0.323578
$959$	0.0590051			0.00190538
$960$		0			0
$961$	−6.64170			−0.214248
$962$			28.3714i			0.914730i
$963$		0			0
$964$		−	19.0766i		−	0.614415i
$965$	−6.73368			−0.216765
$966$		0			0
$967$			19.7393i			0.634773i	0.948296	+	0.317387i	$0.102805\pi$
$967$			19.7393i			0.634773i	−0.948296	+	0.317387i	$0.897195\pi$
$968$	7.83525	+	7.72068i	0.251834	+	0.248152i
$969$		0			0
$970$			58.0322i			1.86330i
$971$			6.59871i			0.211763i	0.994379	+	0.105881i	$0.0337664\pi$
$971$			6.59871i			0.211763i	−0.994379	+	0.105881i	$0.966234\pi$
$972$		0			0
$973$	10.2147			0.327468
$974$	−2.74022			−0.0878023
$975$		0			0
$976$			4.07862i			0.130554i
$977$		−	23.7451i		−	0.759674i	−0.925053	−	0.379837i	$-0.875980\pi$
$977$		−	23.7451i		−	0.759674i	0.925053	−	0.379837i	$-0.124020\pi$
$978$		0			0
$979$	16.1851	−	39.4849i	0.517278	−	1.26194i
$980$		−	3.35236i		−	0.107087i
$981$		0			0
$982$	18.0042			0.574536
$983$			21.3664i			0.681481i	0.940157	+	0.340741i	$0.110678\pi$
$983$			21.3664i			0.681481i	−0.940157	+	0.340741i	$0.889322\pi$
$984$		0			0
$985$		−	72.2065i		−	2.30069i
$986$	−5.08658			−0.161990
$987$		0			0
$988$	−18.9413			−0.602602
$989$	20.5178			0.652429
$990$		0			0
$991$	−50.4630			−1.60301			−0.801505	−	0.597988i	$-0.795967\pi$
$991$	−50.4630			−1.60301			−0.801505	+	0.597988i	$0.795967\pi$
$992$	−4.93541			−0.156700
$993$		0			0
$994$	11.7993			0.374251
$995$			30.9800i			0.982131i
$996$		0			0
$997$		−	22.1418i		−	0.701238i	−0.936518	−	0.350619i	$-0.885971\pi$
$997$		−	22.1418i		−	0.701238i	0.936518	−	0.350619i	$-0.114029\pi$
$998$	31.6254			1.00109
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.c.b.197.11	yes	12
3.2	odd	2		1386.2.c.a.197.2	✓	12
11.10	odd	2		1386.2.c.a.197.11	yes	12
33.32	even	2	inner	1386.2.c.b.197.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.c.a.197.2	✓	12	3.2	odd	2
1386.2.c.a.197.11	yes	12	11.10	odd	2
1386.2.c.b.197.2	yes	12	33.32	even	2	inner
1386.2.c.b.197.11	yes	12	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.35236i q^{5} +1.00000i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.35236i q^{5} +1.00000i q^{7} +1.00000 q^{8} +3.35236i q^{10} +(-3.06881 - 1.25793i) q^{11} +2.62610i q^{13} +1.00000i q^{14} +1.00000 q^{16} -2.58660 q^{17} +7.21270i q^{19} +3.35236i q^{20} +(-3.06881 - 1.25793i) q^{22} -5.09458i q^{23} -6.23832 q^{25} +2.62610i q^{26} +1.00000i q^{28} +1.96651 q^{29} -4.93541 q^{31} +1.00000 q^{32} -2.58660 q^{34} -3.35236 q^{35} +10.8036 q^{37} +7.21270i q^{38} +3.35236i q^{40} -9.07304 q^{41} +4.02738i q^{43} +(-3.06881 - 1.25793i) q^{44} -5.09458i q^{46} -7.16146i q^{47} -1.00000 q^{49} -6.23832 q^{50} +2.62610i q^{52} +10.3351i q^{53} +(4.21702 - 10.2878i) q^{55} +1.00000i q^{56} +1.96651 q^{58} +11.5083i q^{59} +4.07862i q^{61} -4.93541 q^{62} +1.00000 q^{64} -8.80362 q^{65} -2.06511 q^{67} -2.58660 q^{68} -3.35236 q^{70} -11.7993i q^{71} -5.29996i q^{73} +10.8036 q^{74} +7.21270i q^{76} +(1.25793 - 3.06881i) q^{77} -3.51764i q^{79} +3.35236i q^{80} -9.07304 q^{82} +2.98781 q^{83} -8.67123i q^{85} +4.02738i q^{86} +(-3.06881 - 1.25793i) q^{88} +12.8665i q^{89} -2.62610 q^{91} -5.09458i q^{92} -7.16146i q^{94} -24.1796 q^{95} +17.3108 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} - 4 q^{11} + 12 q^{16} + 16 q^{17} - 4 q^{22} - 4 q^{25} + 16 q^{29} + 12 q^{32} + 16 q^{34} - 8 q^{35} + 24 q^{37} + 16 q^{41} - 4 q^{44} - 12 q^{49} - 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} + 16 q^{68} - 8 q^{70} + 24 q^{74} + 8 q^{77} + 16 q^{82} + 16 q^{83} - 4 q^{88} - 48 q^{95} + 48 q^{97} - 12 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8 - 4 * q^11 + 12 * q^16 + 16 * q^17 - 4 * q^22 - 4 * q^25 + 16 * q^29 + 12 * q^32 + 16 * q^34 - 8 * q^35 + 24 * q^37 + 16 * q^41 - 4 * q^44 - 12 * q^49 - 4 * q^50 - 8 * q^55 + 16 * q^58 + 12 * q^64 - 48 * q^67 + 16 * q^68 - 8 * q^70 + 24 * q^74 + 8 * q^77 + 16 * q^82 + 16 * q^83 - 4 * q^88 - 48 * q^95 + 48 * q^97 - 12 * q^98

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