LMFDB - Embedded newform 1386.2.c.b.197.6

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			197.6
Root			$1.38690 - 1.03755i$ of defining polynomial
Character	$\chi$	$=$	1386.197
Dual form			1386.2.c.b.197.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.494054i q^{5} +1.00000i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.494054i q^{5} +1.00000i q^{7} +1.00000 q^{8} -0.494054i q^{10} +(0.197809 - 3.31072i) q^{11} -5.50379i q^{13} +1.00000i q^{14} +1.00000 q^{16} -2.35484 q^{17} -1.14894i q^{19} -0.494054i q^{20} +(0.197809 - 3.31072i) q^{22} -4.19401i q^{23} +4.75591 q^{25} -5.50379i q^{26} +1.00000i q^{28} -0.175313 q^{29} +1.40822 q^{31} +1.00000 q^{32} -2.35484 q^{34} +0.494054 q^{35} +4.71917 q^{37} -1.14894i q^{38} -0.494054i q^{40} +3.80384 q^{41} +9.72961i q^{43} +(0.197809 - 3.31072i) q^{44} -4.19401i q^{46} -4.06499i q^{47} -1.00000 q^{49} +4.75591 q^{50} -5.50379i q^{52} -2.97863i q^{53} +(-1.63568 - 0.0977284i) q^{55} +1.00000i q^{56} -0.175313 q^{58} -2.26894i q^{59} +4.51568i q^{61} +1.40822 q^{62} +1.00000 q^{64} -2.71917 q^{65} +8.46560 q^{67} -2.35484 q^{68} +0.494054 q^{70} -3.20590i q^{71} -2.39328i q^{73} +4.71917 q^{74} -1.14894i q^{76} +(3.31072 + 0.197809i) q^{77} -1.55387i q^{79} -0.494054i q^{80} +3.80384 q^{82} -4.29555 q^{83} +1.16342i q^{85} +9.72961i q^{86} +(0.197809 - 3.31072i) q^{88} -2.32970i q^{89} +5.50379 q^{91} -4.19401i q^{92} -4.06499i q^{94} -0.567641 q^{95} +10.3141 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	0.494054i		−	0.220948i	−0.993879	−	0.110474i	$-0.964763\pi$
$5$		−	0.494054i		−	0.220948i	0.993879	−	0.110474i	$-0.0352368\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$	1.00000			0.353553
$9$		0			0
$10$		−	0.494054i		−	0.156234i
$11$	0.197809	−	3.31072i	0.0596417	−	0.998220i
$11$	0.197809	−	3.31072i	0.0596417	−	0.998220i
$12$		0			0
$13$		−	5.50379i		−	1.52648i	−0.646117	−	0.763238i	$-0.723608\pi$
$13$		−	5.50379i		−	1.52648i	0.646117	−	0.763238i	$-0.276392\pi$
$14$			1.00000i			0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.35484			−0.571134			−0.285567	−	0.958359i	$-0.592182\pi$
$17$	−2.35484			−0.571134			−0.285567	+	0.958359i	$0.592182\pi$
$18$		0			0
$19$		−	1.14894i		−	0.263586i	−0.991277	−	0.131793i	$-0.957927\pi$
$19$		−	1.14894i		−	0.263586i	0.991277	−	0.131793i	$-0.0420734\pi$
$20$		−	0.494054i		−	0.110474i
$21$		0			0
$22$	0.197809	−	3.31072i	0.0421731	−	0.705848i
$23$		−	4.19401i		−	0.874511i	−0.899337	−	0.437256i	$-0.855951\pi$
$23$		−	4.19401i		−	0.874511i	0.899337	−	0.437256i	$-0.144049\pi$
$24$		0			0
$25$	4.75591			0.951182
$26$		−	5.50379i		−	1.07938i
$27$		0			0
$28$			1.00000i			0.188982i
$29$	−0.175313			−0.0325547			−0.0162774	−	0.999868i	$-0.505181\pi$
$29$	−0.175313			−0.0325547			−0.0162774	+	0.999868i	$0.505181\pi$
$30$		0			0
$31$	1.40822			0.252923			0.126462	−	0.991971i	$-0.459638\pi$
$31$	1.40822			0.252923			0.126462	+	0.991971i	$0.459638\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	−2.35484			−0.403853
$35$	0.494054			0.0835104
$36$		0			0
$37$	4.71917			0.775827			0.387913	−	0.921696i	$-0.373196\pi$
$37$	4.71917			0.775827			0.387913	+	0.921696i	$0.373196\pi$
$38$		−	1.14894i		−	0.186383i
$39$		0			0
$40$		−	0.494054i		−	0.0781168i
$41$	3.80384			0.594059			0.297030	−	0.954868i	$-0.404004\pi$
$41$	3.80384			0.594059			0.297030	+	0.954868i	$0.404004\pi$
$42$		0			0
$43$			9.72961i			1.48375i	0.670537	+	0.741876i	$0.266064\pi$
$43$			9.72961i			1.48375i	−0.670537	+	0.741876i	$0.733936\pi$
$44$	0.197809	−	3.31072i	0.0298209	−	0.499110i
$45$		0			0
$46$		−	4.19401i		−	0.618373i
$47$		−	4.06499i		−	0.592939i	−0.955042	−	0.296470i	$-0.904191\pi$
$47$		−	4.06499i		−	0.592939i	0.955042	−	0.296470i	$-0.0958093\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	4.75591			0.672587
$51$		0			0
$52$		−	5.50379i		−	0.763238i
$53$		−	2.97863i		−	0.409146i	−0.978851	−	0.204573i	$-0.934419\pi$
$53$		−	2.97863i		−	0.409146i	0.978851	−	0.204573i	$-0.0655806\pi$
$54$		0			0
$55$	−1.63568	−	0.0977284i	−0.220554	−	0.0131777i
$56$			1.00000i			0.133631i
$57$		0			0
$58$	−0.175313			−0.0230197
$59$		−	2.26894i		−	0.295391i	−0.989033	−	0.147695i	$-0.952815\pi$
$59$		−	2.26894i		−	0.295391i	0.989033	−	0.147695i	$-0.0471855\pi$
$60$		0			0
$61$			4.51568i			0.578174i	0.957303	+	0.289087i	$0.0933517\pi$
$61$			4.51568i			0.578174i	−0.957303	+	0.289087i	$0.906648\pi$
$62$	1.40822			0.178844
$63$		0			0
$64$	1.00000			0.125000
$65$	−2.71917			−0.337272
$66$		0			0
$67$	8.46560			1.03424			0.517119	−	0.855914i	$-0.327005\pi$
$67$	8.46560			1.03424			0.517119	+	0.855914i	$0.327005\pi$
$68$	−2.35484			−0.285567
$69$		0			0
$70$	0.494054			0.0590508
$71$		−	3.20590i		−	0.380470i	−0.981739	−	0.190235i	$-0.939075\pi$
$71$		−	3.20590i		−	0.380470i	0.981739	−	0.190235i	$-0.0609250\pi$
$72$		0			0
$73$		−	2.39328i		−	0.280112i	−0.990144	−	0.140056i	$-0.955272\pi$
$73$		−	2.39328i		−	0.280112i	0.990144	−	0.140056i	$-0.0447282\pi$
$74$	4.71917			0.548592
$75$		0			0
$76$		−	1.14894i		−	0.131793i
$77$	3.31072	+	0.197809i	0.377292	+	0.0225424i
$78$		0			0
$79$		−	1.55387i		−	0.174824i	−0.996172	−	0.0874121i	$-0.972140\pi$
$79$		−	1.55387i		−	0.174824i	0.996172	−	0.0874121i	$-0.0278597\pi$
$80$		−	0.494054i		−	0.0552369i
$81$		0			0
$82$	3.80384			0.420063
$83$	−4.29555			−0.471498			−0.235749	−	0.971814i	$-0.575754\pi$
$83$	−4.29555			−0.471498			−0.235749	+	0.971814i	$0.575754\pi$
$84$		0			0
$85$			1.16342i			0.126191i
$86$			9.72961i			1.04917i
$87$		0			0
$88$	0.197809	−	3.31072i	0.0210865	−	0.352924i
$89$		−	2.32970i		−	0.246948i	−0.992348	−	0.123474i	$-0.960596\pi$
$89$		−	2.32970i		−	0.246948i	0.992348	−	0.123474i	$-0.0394036\pi$
$90$		0			0
$91$	5.50379			0.576954
$92$		−	4.19401i		−	0.437256i
$93$		0			0
$94$		−	4.06499i		−	0.419271i
$95$	−0.567641			−0.0582387
$96$		0			0
$97$	10.3141			1.04724			0.523618	−	0.851953i	$-0.324582\pi$
$97$	10.3141			1.04724			0.523618	+	0.851953i	$0.324582\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	4.75591			0.475591
$101$	−12.6873			−1.26244			−0.631218	−	0.775606i	$-0.717445\pi$
$101$	−12.6873			−1.26244			−0.631218	+	0.775606i	$0.717445\pi$
$102$		0			0
$103$	−17.9022			−1.76395			−0.881977	−	0.471292i	$-0.843788\pi$
$103$	−17.9022			−1.76395			−0.881977	+	0.471292i	$0.843788\pi$
$104$		−	5.50379i		−	0.539691i
$105$		0			0
$106$		−	2.97863i		−	0.289310i
$107$	−8.25477			−0.798019			−0.399010	−	0.916947i	$-0.630646\pi$
$107$	−8.25477			−0.798019			−0.399010	+	0.916947i	$0.630646\pi$
$108$		0			0
$109$			10.9329i			1.04718i	0.851971	+	0.523589i	$0.175407\pi$
$109$			10.9329i			1.04718i	−0.851971	+	0.523589i	$0.824593\pi$
$110$	−1.63568	−	0.0977284i	−0.155955	−	0.00931804i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$		−	9.75856i		−	0.918009i	−0.888434	−	0.459004i	$-0.848206\pi$
$113$		−	9.75856i		−	0.918009i	0.888434	−	0.459004i	$-0.151794\pi$
$114$		0			0
$115$	−2.07207			−0.193221
$116$	−0.175313			−0.0162774
$117$		0			0
$118$		−	2.26894i		−	0.208873i
$119$		−	2.35484i		−	0.215868i
$120$		0			0
$121$	−10.9217	−	1.30978i	−0.992886	−	0.119071i
$122$			4.51568i			0.408831i
$123$		0			0
$124$	1.40822			0.126462
$125$		−	4.81995i		−	0.431109i
$126$		0			0
$127$			9.36217i			0.830758i	0.909648	+	0.415379i	$0.136351\pi$
$127$			9.36217i			0.830758i	−0.909648	+	0.415379i	$0.863649\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	−2.71917			−0.238487
$131$	−5.14608			−0.449615			−0.224808	−	0.974403i	$-0.572175\pi$
$131$	−5.14608			−0.449615			−0.224808	+	0.974403i	$0.572175\pi$
$132$		0			0
$133$	1.14894			0.0996262
$134$	8.46560			0.731316
$135$		0			0
$136$	−2.35484			−0.201926
$137$			6.91130i			0.590472i	0.955424	+	0.295236i	$0.0953983\pi$
$137$			6.91130i			0.590472i	−0.955424	+	0.295236i	$0.904602\pi$
$138$		0			0
$139$			16.6837i			1.41510i	0.706665	+	0.707549i	$0.250199\pi$
$139$			16.6837i			1.41510i	−0.706665	+	0.707549i	$0.749801\pi$
$140$	0.494054			0.0417552
$141$		0			0
$142$		−	3.20590i		−	0.269033i
$143$	−18.2215	−	1.08870i	−1.52376	−	0.0910417i
$144$		0			0
$145$			0.0866139i			0.00719289i
$146$		−	2.39328i		−	0.198069i
$147$		0			0
$148$	4.71917			0.387913
$149$	11.9217			0.976667			0.488334	−	0.872657i	$-0.337605\pi$
$149$	11.9217			0.976667			0.488334	+	0.872657i	$0.337605\pi$
$150$		0			0
$151$			3.35477i			0.273008i	0.990640	+	0.136504i	$0.0435866\pi$
$151$			3.35477i			0.273008i	−0.990640	+	0.136504i	$0.956413\pi$
$152$		−	1.14894i		−	0.0931917i
$153$		0			0
$154$	3.31072	+	0.197809i	0.266785	+	0.0159399i
$155$		−	0.695736i		−	0.0558828i
$156$		0			0
$157$	15.6890			1.25212			0.626060	−	0.779775i	$-0.284666\pi$
$157$	15.6890			1.25212			0.626060	+	0.779775i	$0.284666\pi$
$158$		−	1.55387i		−	0.123619i
$159$		0			0
$160$		−	0.494054i		−	0.0390584i
$161$	4.19401			0.330534
$162$		0			0
$163$	20.5594			1.61033			0.805167	−	0.593048i	$-0.202076\pi$
$163$	20.5594			1.61033			0.805167	+	0.593048i	$0.202076\pi$
$164$	3.80384			0.297030
$165$		0			0
$166$	−4.29555			−0.333399
$167$	−8.80695			−0.681503			−0.340751	−	0.940153i	$-0.610681\pi$
$167$	−8.80695			−0.681503			−0.340751	+	0.940153i	$0.610681\pi$
$168$		0			0
$169$	−17.2917			−1.33013
$170$			1.16342i			0.0892303i
$171$		0			0
$172$			9.72961i			0.741876i
$173$	−3.93122			−0.298885			−0.149443	−	0.988770i	$-0.547748\pi$
$173$	−3.93122			−0.298885			−0.149443	+	0.988770i	$0.547748\pi$
$174$		0			0
$175$			4.75591i			0.359513i
$176$	0.197809	−	3.31072i	0.0149104	−	0.249555i
$177$		0			0
$178$		−	2.32970i		−	0.174619i
$179$			23.5547i			1.76056i	0.474452	+	0.880281i	$0.342646\pi$
$179$			23.5547i			1.76056i	−0.474452	+	0.880281i	$0.657354\pi$
$180$		0			0
$181$	6.52301			0.484851			0.242426	−	0.970170i	$-0.422057\pi$
$181$	6.52301			0.484851			0.242426	+	0.970170i	$0.422057\pi$
$182$	5.50379			0.407968
$183$		0			0
$184$		−	4.19401i		−	0.309186i
$185$		−	2.33153i		−	0.171417i
$186$		0			0
$187$	−0.465810	+	7.79623i	−0.0340634	+	0.570117i
$188$		−	4.06499i		−	0.296470i
$189$		0			0
$190$	−0.567641			−0.0411810
$191$			11.4274i			0.826858i	0.910536	+	0.413429i	$0.135669\pi$
$191$			11.4274i			0.826858i	−0.910536	+	0.413429i	$0.864331\pi$
$192$		0			0
$193$			7.02654i			0.505781i	0.967495	+	0.252891i	$0.0813813\pi$
$193$			7.02654i			0.505781i	−0.967495	+	0.252891i	$0.918619\pi$
$194$	10.3141			0.740507
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	21.7037			1.54632			0.773162	−	0.634209i	$-0.218674\pi$
$197$	21.7037			1.54632			0.773162	+	0.634209i	$0.218674\pi$
$198$		0			0
$199$	10.9073			0.773196			0.386598	−	0.922248i	$-0.373650\pi$
$199$	10.9073			0.773196			0.386598	+	0.922248i	$0.373650\pi$
$200$	4.75591			0.336294
$201$		0			0
$202$	−12.6873			−0.892677
$203$		−	0.175313i		−	0.0123045i
$204$		0			0
$205$		−	1.87930i		−	0.131256i
$206$	−17.9022			−1.24730
$207$		0			0
$208$		−	5.50379i		−	0.381619i
$209$	−3.80384	−	0.227272i	−0.263117	−	0.0157207i
$210$		0			0
$211$			8.51137i			0.585947i	0.956121	+	0.292973i	$0.0946447\pi$
$211$			8.51137i			0.585947i	−0.956121	+	0.292973i	$0.905355\pi$
$212$		−	2.97863i		−	0.204573i
$213$		0			0
$214$	−8.25477			−0.564285
$215$	4.80695			0.327832
$216$		0			0
$217$			1.40822i			0.0955960i
$218$			10.9329i			0.740467i
$219$		0			0
$220$	−1.63568	−	0.0977284i	−0.110277	−	0.00658885i
$221$			12.9606i			0.871822i
$222$		0			0
$223$	−4.55627			−0.305110			−0.152555	−	0.988295i	$-0.548750\pi$
$223$	−4.55627			−0.305110			−0.152555	+	0.988295i	$0.548750\pi$
$224$			1.00000i			0.0668153i
$225$		0			0
$226$		−	9.75856i		−	0.649130i
$227$	−23.6027			−1.56656			−0.783282	−	0.621667i	$-0.786456\pi$
$227$	−23.6027			−1.56656			−0.783282	+	0.621667i	$0.786456\pi$
$228$		0			0
$229$	−18.1200			−1.19741			−0.598703	−	0.800971i	$-0.704317\pi$
$229$	−18.1200			−1.19741			−0.598703	+	0.800971i	$0.704317\pi$
$230$	−2.07207			−0.136628
$231$		0			0
$232$	−0.175313			−0.0115098
$233$	−17.1422			−1.12303			−0.561513	−	0.827468i	$-0.689780\pi$
$233$	−17.1422			−1.12303			−0.561513	+	0.827468i	$0.689780\pi$
$234$		0			0
$235$	−2.00832			−0.131009
$236$		−	2.26894i		−	0.147695i
$237$		0			0
$238$		−	2.35484i		−	0.152642i
$239$	9.76331			0.631536			0.315768	−	0.948836i	$-0.397738\pi$
$239$	9.76331			0.631536			0.315768	+	0.948836i	$0.397738\pi$
$240$		0			0
$241$		−	22.8693i		−	1.47314i	−0.676360	−	0.736571i	$-0.736444\pi$
$241$		−	22.8693i		−	1.47314i	0.676360	−	0.736571i	$-0.263556\pi$
$242$	−10.9217	−	1.30978i	−0.702076	−	0.0841960i
$243$		0			0
$244$			4.51568i			0.289087i
$245$			0.494054i			0.0315640i
$246$		0			0
$247$	−6.32355			−0.402358
$248$	1.40822			0.0894219
$249$		0			0
$250$		−	4.81995i		−	0.304840i
$251$			17.7124i			1.11800i	0.829168	+	0.559000i	$0.188815\pi$
$251$			17.7124i			1.11800i	−0.829168	+	0.559000i	$0.811185\pi$
$252$		0			0
$253$	−13.8852	−	0.829613i	−0.872954	−	0.0521573i
$254$			9.36217i			0.587435i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	15.0090i		−	0.936237i	−0.883666	−	0.468119i	$-0.844932\pi$
$257$		−	15.0090i		−	0.936237i	0.883666	−	0.468119i	$-0.155068\pi$
$258$		0			0
$259$			4.71917i			0.293235i
$260$	−2.71917			−0.168636
$261$		0			0
$262$	−5.14608			−0.317926
$263$	12.4323			0.766611			0.383305	−	0.923622i	$-0.374786\pi$
$263$	12.4323			0.766611			0.383305	+	0.923622i	$0.374786\pi$
$264$		0			0
$265$	−1.47160			−0.0903999
$266$	1.14894			0.0704463
$267$		0			0
$268$	8.46560			0.517119
$269$			8.77323i			0.534913i	0.963570	+	0.267457i	$0.0861832\pi$
$269$			8.77323i			0.534913i	−0.963570	+	0.267457i	$0.913817\pi$
$270$		0			0
$271$		−	17.4734i		−	1.06143i	−0.847549	−	0.530716i	$-0.821923\pi$
$271$		−	17.4734i		−	1.06143i	0.847549	−	0.530716i	$-0.178077\pi$
$272$	−2.35484			−0.142783
$273$		0			0
$274$			6.91130i			0.417527i
$275$	0.940763	−	15.7455i	0.0567301	−	0.949489i
$276$		0			0
$277$			0.0747207i			0.00448953i	0.999997	+	0.00224477i	$0.000714532\pi$
$277$			0.0747207i			0.00448953i	−0.999997	+	0.00224477i	$0.999285\pi$
$278$			16.6837i			1.00062i
$279$		0			0
$280$	0.494054			0.0295254
$281$	11.3582			0.677574			0.338787	−	0.940863i	$-0.389983\pi$
$281$	11.3582			0.677574			0.338787	+	0.940863i	$0.389983\pi$
$282$		0			0
$283$			4.42898i			0.263275i	0.991298	+	0.131638i	$0.0420235\pi$
$283$			4.42898i			0.263275i	−0.991298	+	0.131638i	$0.957976\pi$
$284$		−	3.20590i		−	0.190235i
$285$		0			0
$286$	−18.2215	−	1.08870i	−1.07746	−	0.0643762i
$287$			3.80384i			0.224533i
$288$		0			0
$289$	−11.4547			−0.673806
$290$			0.0866139i			0.00508614i
$291$		0			0
$292$		−	2.39328i		−	0.140056i
$293$	29.5911			1.72873			0.864364	−	0.502866i	$-0.167721\pi$
$293$	29.5911			1.72873			0.864364	+	0.502866i	$0.167721\pi$
$294$		0			0
$295$	−1.12098			−0.0652659
$296$	4.71917			0.274296
$297$		0			0
$298$	11.9217			0.690608
$299$	−23.0829			−1.33492
$300$		0			0
$301$	−9.72961			−0.560806
$302$			3.35477i			0.193046i
$303$		0			0
$304$		−	1.14894i		−	0.0658965i
$305$	2.23099			0.127746
$306$		0			0
$307$			16.6318i			0.949228i	0.880194	+	0.474614i	$0.157412\pi$
$307$			16.6318i			0.949228i	−0.880194	+	0.474614i	$0.842588\pi$
$308$	3.31072	+	0.197809i	0.188646	+	0.0112712i
$309$		0			0
$310$		−	0.695736i		−	0.0395151i
$311$			22.8518i			1.29581i	0.761722	+	0.647904i	$0.224354\pi$
$311$			22.8518i			1.29581i	−0.761722	+	0.647904i	$0.775646\pi$
$312$		0			0
$313$	−30.0474			−1.69838			−0.849190	−	0.528088i	$-0.822909\pi$
$313$	−30.0474			−1.69838			−0.849190	+	0.528088i	$0.822909\pi$
$314$	15.6890			0.885383
$315$		0			0
$316$		−	1.55387i		−	0.0874121i
$317$			29.0271i			1.63032i	0.579235	+	0.815161i	$0.303351\pi$
$317$			29.0271i			1.63032i	−0.579235	+	0.815161i	$0.696649\pi$
$318$		0			0
$319$	−0.0346784	+	0.580411i	−0.00194162	+	0.0324968i
$320$		−	0.494054i		−	0.0276185i
$321$		0			0
$322$	4.19401			0.233723
$323$			2.70559i			0.150543i
$324$		0			0
$325$		−	26.1755i		−	1.45196i
$326$	20.5594			1.13868
$327$		0			0
$328$	3.80384			0.210032
$329$	4.06499			0.224110
$330$		0			0
$331$	−8.73792			−0.480280			−0.240140	−	0.970738i	$-0.577193\pi$
$331$	−8.73792			−0.480280			−0.240140	+	0.970738i	$0.577193\pi$
$332$	−4.29555			−0.235749
$333$		0			0
$334$	−8.80695			−0.481895
$335$		−	4.18246i		−	0.228512i
$336$		0			0
$337$		−	11.7423i		−	0.639645i	−0.947477	−	0.319823i	$-0.896377\pi$
$337$		−	11.7423i		−	0.639645i	0.947477	−	0.319823i	$-0.103623\pi$
$338$	−17.2917			−0.940545
$339$		0			0
$340$			1.16342i			0.0630953i
$341$	0.278558	−	4.66221i	0.0150848	−	0.252473i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$			9.72961i			0.524586i
$345$		0			0
$346$	−3.93122			−0.211344
$347$	31.8069			1.70749			0.853743	−	0.520695i	$-0.174327\pi$
$347$	31.8069			1.70749			0.853743	+	0.520695i	$0.174327\pi$
$348$		0			0
$349$		−	2.68877i		−	0.143927i	−0.997407	−	0.0719633i	$-0.977074\pi$
$349$		−	2.68877i		−	0.143927i	0.997407	−	0.0719633i	$-0.0229264\pi$
$350$			4.75591i			0.254214i
$351$		0			0
$352$	0.197809	−	3.31072i	0.0105433	−	0.176462i
$353$			33.0209i			1.75753i	0.477258	+	0.878763i	$0.341631\pi$
$353$			33.0209i			1.75753i	−0.477258	+	0.878763i	$0.658369\pi$
$354$		0			0
$355$	−1.58389			−0.0840640
$356$		−	2.32970i		−	0.123474i
$357$		0			0
$358$			23.5547i			1.24491i
$359$	−29.3208			−1.54749			−0.773747	−	0.633494i	$-0.781620\pi$
$359$	−29.3208			−1.54749			−0.773747	+	0.633494i	$0.781620\pi$
$360$		0			0
$361$	17.6799			0.930522
$362$	6.52301			0.342842
$363$		0			0
$364$	5.50379			0.288477
$365$	−1.18241			−0.0618901
$366$		0			0
$367$	−31.7102			−1.65526			−0.827628	−	0.561277i	$-0.810310\pi$
$367$	−31.7102			−1.65526			−0.827628	+	0.561277i	$0.810310\pi$
$368$		−	4.19401i		−	0.218628i
$369$		0			0
$370$		−	2.33153i		−	0.121210i
$371$	2.97863			0.154643
$372$		0			0
$373$		−	14.7481i		−	0.763627i	−0.924239	−	0.381814i	$-0.875300\pi$
$373$		−	14.7481i		−	0.763627i	0.924239	−	0.381814i	$-0.124700\pi$
$374$	−0.465810	+	7.79623i	−0.0240865	+	0.403134i
$375$		0			0
$376$		−	4.06499i		−	0.209636i
$377$			0.964883i			0.0496940i
$378$		0			0
$379$	−5.65408			−0.290431			−0.145215	−	0.989400i	$-0.546387\pi$
$379$	−5.65408			−0.290431			−0.145215	+	0.989400i	$0.546387\pi$
$380$	−0.567641			−0.0291194
$381$		0			0
$382$			11.4274i			0.584677i
$383$			5.75879i			0.294260i	0.989117	+	0.147130i	$0.0470036\pi$
$383$			5.75879i			0.294260i	−0.989117	+	0.147130i	$0.952996\pi$
$384$		0			0
$385$	0.0977284	−	1.63568i	0.00498070	−	0.0833617i
$386$			7.02654i			0.357641i
$387$		0			0
$388$	10.3141			0.523618
$389$		−	22.2885i		−	1.13007i	−0.825065	−	0.565037i	$-0.808862\pi$
$389$		−	22.2885i		−	1.13007i	0.825065	−	0.565037i	$-0.191138\pi$
$390$		0			0
$391$			9.87624i			0.499463i
$392$	−1.00000			−0.0505076
$393$		0			0
$394$	21.7037			1.09342
$395$	−0.767696			−0.0386270
$396$		0			0
$397$	36.2092			1.81729			0.908644	−	0.417571i	$-0.137118\pi$
$397$	36.2092			1.81729			0.908644	+	0.417571i	$0.137118\pi$
$398$	10.9073			0.546732
$399$		0			0
$400$	4.75591			0.237796
$401$			14.7482i			0.736490i	0.929729	+	0.368245i	$0.120041\pi$
$401$			14.7482i			0.736490i	−0.929729	+	0.368245i	$0.879959\pi$
$402$		0			0
$403$		−	7.75053i		−	0.386082i
$404$	−12.6873			−0.631218
$405$		0			0
$406$		−	0.175313i		−	0.00870062i
$407$	0.933495	−	15.6239i	0.0462716	−	0.774446i
$408$		0			0
$409$		−	34.3987i		−	1.70091i	−0.526050	−	0.850453i	$-0.676328\pi$
$409$		−	34.3987i		−	1.70091i	0.526050	−	0.850453i	$-0.323672\pi$
$410$		−	1.87930i		−	0.0928120i
$411$		0			0
$412$	−17.9022			−0.881977
$413$	2.26894			0.111647
$414$		0			0
$415$			2.12223i			0.104176i
$416$		−	5.50379i		−	0.269845i
$417$		0			0
$418$	−3.80384	−	0.227272i	−0.186052	−	0.0111162i
$419$			38.9117i			1.90096i	0.310782	+	0.950481i	$0.399409\pi$
$419$			38.9117i			1.90096i	−0.310782	+	0.950481i	$0.600591\pi$
$420$		0			0
$421$	−11.6945			−0.569956			−0.284978	−	0.958534i	$-0.591986\pi$
$421$	−11.6945			−0.569956			−0.284978	+	0.958534i	$0.591986\pi$
$422$			8.51137i			0.414327i
$423$		0			0
$424$		−	2.97863i		−	0.144655i
$425$	−11.1994			−0.543252
$426$		0			0
$427$	−4.51568			−0.218529
$428$	−8.25477			−0.399010
$429$		0			0
$430$	4.80695			0.231812
$431$	7.16905			0.345321			0.172661	−	0.984981i	$-0.444764\pi$
$431$	7.16905			0.345321			0.172661	+	0.984981i	$0.444764\pi$
$432$		0			0
$433$	19.4899			0.936625			0.468313	−	0.883563i	$-0.344862\pi$
$433$	19.4899			0.936625			0.468313	+	0.883563i	$0.344862\pi$
$434$			1.40822i			0.0675966i
$435$		0			0
$436$			10.9329i			0.523589i
$437$	−4.81868			−0.230509
$438$		0			0
$439$		−	3.30603i		−	0.157788i	−0.996883	−	0.0788941i	$-0.974861\pi$
$439$		−	3.30603i		−	0.157788i	0.996883	−	0.0788941i	$-0.0251389\pi$
$440$	−1.63568	−	0.0977284i	−0.0779777	−	0.00465902i
$441$		0			0
$442$			12.9606i			0.616472i
$443$		−	27.5415i		−	1.30854i	−0.756263	−	0.654268i	$-0.772977\pi$
$443$		−	27.5415i		−	1.30854i	0.756263	−	0.654268i	$-0.227023\pi$
$444$		0			0
$445$	−1.15100			−0.0545626
$446$	−4.55627			−0.215746
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			12.0032i			0.566468i	0.959051	+	0.283234i	$0.0914073\pi$
$449$			12.0032i			0.566468i	−0.959051	+	0.283234i	$0.908593\pi$
$450$		0			0
$451$	0.752434	−	12.5934i	0.0354307	−	0.593002i
$452$		−	9.75856i		−	0.459004i
$453$		0			0
$454$	−23.6027			−1.10773
$455$		−	2.71917i		−	0.127477i
$456$		0			0
$457$			10.5366i			0.492881i	0.969158	+	0.246441i	$0.0792611\pi$
$457$			10.5366i			0.492881i	−0.969158	+	0.246441i	$0.920739\pi$
$458$	−18.1200			−0.846693
$459$		0			0
$460$	−2.07207			−0.0966106
$461$	26.1304			1.21701			0.608506	−	0.793549i	$-0.291769\pi$
$461$	26.1304			1.21701			0.608506	+	0.793549i	$0.291769\pi$
$462$		0			0
$463$	11.6684			0.542276			0.271138	−	0.962541i	$-0.412600\pi$
$463$	11.6684			0.542276			0.271138	+	0.962541i	$0.412600\pi$
$464$	−0.175313			−0.00813868
$465$		0			0
$466$	−17.1422			−0.794099
$467$		−	32.7068i		−	1.51349i	−0.653710	−	0.756745i	$-0.726788\pi$
$467$		−	32.7068i		−	1.51349i	0.653710	−	0.756745i	$-0.273212\pi$
$468$		0			0
$469$			8.46560i			0.390905i
$470$	−2.00832			−0.0926370
$471$		0			0
$472$		−	2.26894i		−	0.104436i
$473$	32.2120	+	1.92461i	1.48111	+	0.0884935i
$474$		0			0
$475$		−	5.46428i		−	0.250718i
$476$		−	2.35484i		−	0.107934i
$477$		0			0
$478$	9.76331			0.446563
$479$	−14.6716			−0.670363			−0.335181	−	0.942154i	$-0.608798\pi$
$479$	−14.6716			−0.670363			−0.335181	+	0.942154i	$0.608798\pi$
$480$		0			0
$481$		−	25.9733i		−	1.18428i
$482$		−	22.8693i		−	1.04167i
$483$		0			0
$484$	−10.9217	−	1.30978i	−0.496443	−	0.0595355i
$485$		−	5.09571i		−	0.231384i
$486$		0			0
$487$	19.7666			0.895710			0.447855	−	0.894106i	$-0.352188\pi$
$487$	19.7666			0.895710			0.447855	+	0.894106i	$0.352188\pi$
$488$			4.51568i			0.204415i
$489$		0			0
$490$			0.494054i			0.0223191i
$491$	25.8600			1.16705			0.583523	−	0.812097i	$-0.301674\pi$
$491$	25.8600			1.16705			0.583523	+	0.812097i	$0.301674\pi$
$492$		0			0
$493$	0.412834			0.0185931
$494$	−6.32355			−0.284510
$495$		0			0
$496$	1.40822			0.0632308
$497$	3.20590			0.143804
$498$		0			0
$499$	−24.1647			−1.08176			−0.540880	−	0.841100i	$-0.681909\pi$
$499$	−24.1647			−1.08176			−0.540880	+	0.841100i	$0.681909\pi$
$500$		−	4.81995i		−	0.215555i
$501$		0			0
$502$			17.7124i			0.790545i
$503$	27.4249			1.22282			0.611408	−	0.791315i	$-0.290603\pi$
$503$	27.4249			1.22282			0.611408	+	0.791315i	$0.290603\pi$
$504$		0			0
$505$			6.26822i			0.278932i
$506$	−13.8852	−	0.829613i	−0.617272	−	0.0368808i
$507$		0			0
$508$			9.36217i			0.415379i
$509$			14.1452i			0.626976i	0.949592	+	0.313488i	$0.101498\pi$
$509$			14.1452i			0.626976i	−0.949592	+	0.313488i	$0.898502\pi$
$510$		0			0
$511$	2.39328			0.105872
$512$	1.00000			0.0441942
$513$		0			0
$514$		−	15.0090i		−	0.662020i
$515$			8.84464i			0.389742i
$516$		0			0
$517$	−13.4580	−	0.804092i	−0.591884	−	0.0353639i
$518$			4.71917i			0.207348i
$519$		0			0
$520$	−2.71917			−0.119243
$521$		−	39.6726i		−	1.73809i	−0.494736	−	0.869043i	$-0.664735\pi$
$521$		−	39.6726i		−	1.73809i	0.494736	−	0.869043i	$-0.335265\pi$
$522$		0			0
$523$			38.1535i			1.66834i	0.551510	+	0.834169i	$0.314052\pi$
$523$			38.1535i			1.66834i	−0.551510	+	0.834169i	$0.685948\pi$
$524$	−5.14608			−0.224808
$525$		0			0
$526$	12.4323			0.542076
$527$	−3.31613			−0.144453
$528$		0			0
$529$	5.41030			0.235230
$530$	−1.47160			−0.0639224
$531$		0			0
$532$	1.14894			0.0498131
$533$		−	20.9355i		−	0.906818i
$534$		0			0
$535$			4.07831i			0.176321i
$536$	8.46560			0.365658
$537$		0			0
$538$			8.77323i			0.378241i
$539$	−0.197809	+	3.31072i	−0.00852024	+	0.142603i
$540$		0			0
$541$		−	43.4728i		−	1.86904i	−0.355909	−	0.934521i	$-0.615829\pi$
$541$		−	43.4728i		−	1.86904i	0.355909	−	0.934521i	$-0.384171\pi$
$542$		−	17.4734i		−	0.750546i
$543$		0			0
$544$	−2.35484			−0.100963
$545$	5.40142			0.231372
$546$		0			0
$547$			21.9894i			0.940200i	0.882613	+	0.470100i	$0.155782\pi$
$547$			21.9894i			0.940200i	−0.882613	+	0.470100i	$0.844218\pi$
$548$			6.91130i			0.295236i
$549$		0			0
$550$	0.940763	−	15.7455i	0.0401143	−	0.671390i
$551$			0.201424i			0.00858097i
$552$		0			0
$553$	1.55387			0.0660773
$554$			0.0747207i			0.00317458i
$555$		0			0
$556$			16.6837i			0.707549i
$557$	−11.7157			−0.496412			−0.248206	−	0.968707i	$-0.579841\pi$
$557$	−11.7157			−0.496412			−0.248206	+	0.968707i	$0.579841\pi$
$558$		0			0
$559$	53.5497			2.26491
$560$	0.494054			0.0208776
$561$		0			0
$562$	11.3582			0.479117
$563$	−28.6890			−1.20910			−0.604549	−	0.796568i	$-0.706647\pi$
$563$	−28.6890			−1.20910			−0.604549	+	0.796568i	$0.706647\pi$
$564$		0			0
$565$	−4.82126			−0.202832
$566$			4.42898i			0.186164i
$567$		0			0
$568$		−	3.20590i		−	0.134517i
$569$	−16.6233			−0.696886			−0.348443	−	0.937330i	$-0.613289\pi$
$569$	−16.6233			−0.696886			−0.348443	+	0.937330i	$0.613289\pi$
$570$		0			0
$571$			25.3677i			1.06160i	0.847496	+	0.530802i	$0.178109\pi$
$571$			25.3677i			1.06160i	−0.847496	+	0.530802i	$0.821891\pi$
$572$	−18.2215	−	1.08870i	−0.761880	−	0.0455208i
$573$		0			0
$574$			3.80384i			0.158769i
$575$		−	19.9463i		−	0.831819i
$576$		0			0
$577$	14.8426			0.617905			0.308952	−	0.951078i	$-0.400022\pi$
$577$	14.8426			0.617905			0.308952	+	0.951078i	$0.400022\pi$
$578$	−11.4547			−0.476453
$579$		0			0
$580$			0.0866139i			0.00359645i
$581$		−	4.29555i		−	0.178209i
$582$		0			0
$583$	−9.86141	−	0.589200i	−0.408418	−	0.0244022i
$584$		−	2.39328i		−	0.0990345i
$585$		0			0
$586$	29.5911			1.22240
$587$		−	15.2765i		−	0.630529i	−0.949004	−	0.315265i	$-0.897907\pi$
$587$		−	15.2765i		−	0.630529i	0.949004	−	0.315265i	$-0.102093\pi$
$588$		0			0
$589$		−	1.61796i		−	0.0666671i
$590$	−1.12098			−0.0461499
$591$		0			0
$592$	4.71917			0.193957
$593$	29.8907			1.22746			0.613732	−	0.789515i	$-0.289668\pi$
$593$	29.8907			1.22746			0.613732	+	0.789515i	$0.289668\pi$
$594$		0			0
$595$	−1.16342			−0.0476956
$596$	11.9217			0.488334
$597$		0			0
$598$	−23.0829			−0.943932
$599$		−	2.09158i		−	0.0854594i	−0.999087	−	0.0427297i	$-0.986395\pi$
$599$		−	2.09158i		−	0.0854594i	0.999087	−	0.0427297i	$-0.0136054\pi$
$600$		0			0
$601$		−	27.8385i		−	1.13555i	−0.823182	−	0.567777i	$-0.807803\pi$
$601$		−	27.8385i		−	1.13555i	0.823182	−	0.567777i	$-0.192197\pi$
$602$	−9.72961			−0.396549
$603$		0			0
$604$			3.35477i			0.136504i
$605$	−0.647103	+	5.39593i	−0.0263085	+	0.219376i
$606$		0			0
$607$		−	23.7941i		−	0.965774i	−0.875683	−	0.482887i	$-0.839588\pi$
$607$		−	23.7941i		−	0.965774i	0.875683	−	0.482887i	$-0.160412\pi$
$608$		−	1.14894i		−	0.0465959i
$609$		0			0
$610$	2.23099			0.0903302
$611$	−22.3728			−0.905108
$612$		0			0
$613$			0.956873i			0.0386477i	0.999813	+	0.0193239i	$0.00615136\pi$
$613$			0.956873i			0.0386477i	−0.999813	+	0.0193239i	$0.993849\pi$
$614$			16.6318i			0.671205i
$615$		0			0
$616$	3.31072	+	0.197809i	0.133393	+	0.00796996i
$617$		−	0.933715i		−	0.0375899i	−0.999823	−	0.0187950i	$-0.994017\pi$
$617$		−	0.933715i		−	0.0375899i	0.999823	−	0.0187950i	$-0.00598298\pi$
$618$		0			0
$619$	−9.62559			−0.386885			−0.193443	−	0.981112i	$-0.561965\pi$
$619$	−9.62559			−0.386885			−0.193443	+	0.981112i	$0.561965\pi$
$620$		−	0.695736i		−	0.0279414i
$621$		0			0
$622$			22.8518i			0.916274i
$623$	2.32970			0.0933377
$624$		0			0
$625$	21.3982			0.855930
$626$	−30.0474			−1.20094
$627$		0			0
$628$	15.6890			0.626060
$629$	−11.1129			−0.443101
$630$		0			0
$631$	−0.108279			−0.00431051			−0.00215526	−	0.999998i	$-0.500686\pi$
$631$	−0.108279			−0.00431051			−0.00215526	+	0.999998i	$0.500686\pi$
$632$		−	1.55387i		−	0.0618097i
$633$		0			0
$634$			29.0271i			1.15281i
$635$	4.62542			0.183554
$636$		0			0
$637$			5.50379i			0.218068i
$638$	−0.0346784	+	0.580411i	−0.00137293	+	0.0229787i
$639$		0			0
$640$		−	0.494054i		−	0.0195292i
$641$		−	7.05766i		−	0.278761i	−0.990239	−	0.139380i	$-0.955489\pi$
$641$		−	7.05766i		−	0.278761i	0.990239	−	0.139380i	$-0.0445111\pi$
$642$		0			0
$643$	12.5649			0.495510			0.247755	−	0.968823i	$-0.420307\pi$
$643$	12.5649			0.495510			0.247755	+	0.968823i	$0.420307\pi$
$644$	4.19401			0.165267
$645$		0			0
$646$			2.70559i			0.106450i
$647$		−	15.4233i		−	0.606352i	−0.952935	−	0.303176i	$-0.901953\pi$
$647$		−	15.4233i		−	0.606352i	0.952935	−	0.303176i	$-0.0980471\pi$
$648$		0			0
$649$	−7.51182	−	0.448817i	−0.294865	−	0.0176176i
$650$		−	26.1755i		−	1.02669i
$651$		0			0
$652$	20.5594			0.805167
$653$		−	24.4316i		−	0.956083i	−0.878337	−	0.478042i	$-0.841347\pi$
$653$		−	24.4316i		−	0.956083i	0.878337	−	0.478042i	$-0.158653\pi$
$654$		0			0
$655$			2.54244i			0.0993415i
$656$	3.80384			0.148515
$657$		0			0
$658$	4.06499			0.158470
$659$	−20.8426			−0.811912			−0.405956	−	0.913893i	$-0.633061\pi$
$659$	−20.8426			−0.811912			−0.405956	+	0.913893i	$0.633061\pi$
$660$		0			0
$661$	28.8031			1.12031			0.560156	−	0.828387i	$-0.310741\pi$
$661$	28.8031			1.12031			0.560156	+	0.828387i	$0.310741\pi$
$662$	−8.73792			−0.339609
$663$		0			0
$664$	−4.29555			−0.166700
$665$		−	0.567641i		−	0.0220122i
$666$		0			0
$667$			0.735262i			0.0284695i
$668$	−8.80695			−0.340751
$669$		0			0
$670$		−	4.18246i		−	0.161583i
$671$	14.9502	+	0.893243i	0.577145	+	0.0344833i
$672$		0			0
$673$			3.39664i			0.130931i	0.997855	+	0.0654654i	$0.0208532\pi$
$673$			3.39664i			0.130931i	−0.997855	+	0.0654654i	$0.979147\pi$
$674$		−	11.7423i		−	0.452298i
$675$		0			0
$676$	−17.2917			−0.665065
$677$	−34.4280			−1.32318			−0.661588	−	0.749868i	$-0.730117\pi$
$677$	−34.4280			−1.32318			−0.661588	+	0.749868i	$0.730117\pi$
$678$		0			0
$679$			10.3141i			0.395818i
$680$			1.16342i			0.0446151i
$681$		0			0
$682$	0.278558	−	4.66221i	0.0106665	−	0.178525i
$683$			34.4708i			1.31899i	0.751710	+	0.659494i	$0.229229\pi$
$683$			34.4708i			1.31899i	−0.751710	+	0.659494i	$0.770771\pi$
$684$		0			0
$685$	3.41456			0.130463
$686$		−	1.00000i		−	0.0381802i
$687$		0			0
$688$			9.72961i			0.370938i
$689$	−16.3937			−0.624552
$690$		0			0
$691$	−14.7524			−0.561209			−0.280605	−	0.959823i	$-0.590535\pi$
$691$	−14.7524			−0.561209			−0.280605	+	0.959823i	$0.590535\pi$
$692$	−3.93122			−0.149443
$693$		0			0
$694$	31.8069			1.20737
$695$	8.24267			0.312662
$696$		0			0
$697$	−8.95744			−0.339287
$698$		−	2.68877i		−	0.101771i
$699$		0			0
$700$			4.75591i			0.179757i
$701$	30.8440			1.16496			0.582481	−	0.812844i	$-0.302082\pi$
$701$	30.8440			1.16496			0.582481	+	0.812844i	$0.302082\pi$
$702$		0			0
$703$		−	5.42207i		−	0.204497i
$704$	0.197809	−	3.31072i	0.00745521	−	0.124777i
$705$		0			0
$706$			33.0209i			1.24276i
$707$		−	12.6873i		−	0.477156i
$708$		0			0
$709$	−31.6627			−1.18912			−0.594558	−	0.804052i	$-0.702673\pi$
$709$	−31.6627			−1.18912			−0.594558	+	0.804052i	$0.702673\pi$
$710$	−1.58389			−0.0594423
$711$		0			0
$712$		−	2.32970i		−	0.0873094i
$713$		−	5.90608i		−	0.221184i
$714$		0			0
$715$	−0.537877	+	9.00241i	−0.0201154	+	0.336671i
$716$			23.5547i			0.880281i
$717$		0			0
$718$	−29.3208			−1.09424
$719$			0.571837i			0.0213259i	0.999943	+	0.0106630i	$0.00339419\pi$
$719$			0.571837i			0.0213259i	−0.999943	+	0.0106630i	$0.996606\pi$
$720$		0			0
$721$		−	17.9022i		−	0.666712i
$722$	17.6799			0.657979
$723$		0			0
$724$	6.52301			0.242426
$725$	−0.833771			−0.0309655
$726$		0			0
$727$	37.9120			1.40608			0.703038	−	0.711152i	$-0.251826\pi$
$727$	37.9120			1.40608			0.703038	+	0.711152i	$0.251826\pi$
$728$	5.50379			0.203984
$729$		0			0
$730$	−1.18241			−0.0437629
$731$		−	22.9117i		−	0.847421i
$732$		0			0
$733$		−	40.9455i		−	1.51236i	−0.654366	−	0.756178i	$-0.727065\pi$
$733$		−	40.9455i		−	1.51236i	0.654366	−	0.756178i	$-0.272935\pi$
$734$	−31.7102			−1.17044
$735$		0			0
$736$		−	4.19401i		−	0.154593i
$737$	1.67457	−	28.0272i	0.0616837	−	1.03240i
$738$		0			0
$739$			11.3263i			0.416643i	0.978060	+	0.208321i	$0.0668000\pi$
$739$			11.3263i			0.416643i	−0.978060	+	0.208321i	$0.933200\pi$
$740$		−	2.33153i		−	0.0857086i
$741$		0			0
$742$	2.97863			0.109349
$743$	9.69453			0.355658			0.177829	−	0.984061i	$-0.443093\pi$
$743$	9.69453			0.355658			0.177829	+	0.984061i	$0.443093\pi$
$744$		0			0
$745$		−	5.88999i		−	0.215792i
$746$		−	14.7481i		−	0.539966i
$747$		0			0
$748$	−0.465810	+	7.79623i	−0.0170317	+	0.285059i
$749$		−	8.25477i		−	0.301623i
$750$		0			0
$751$	−14.5601			−0.531305			−0.265652	−	0.964069i	$-0.585587\pi$
$751$	−14.5601			−0.531305			−0.265652	+	0.964069i	$0.585587\pi$
$752$		−	4.06499i		−	0.148235i
$753$		0			0
$754$			0.964883i			0.0351390i
$755$	1.65744			0.0603204
$756$		0			0
$757$	10.9848			0.399251			0.199625	−	0.979872i	$-0.436027\pi$
$757$	10.9848			0.399251			0.199625	+	0.979872i	$0.436027\pi$
$758$	−5.65408			−0.205365
$759$		0			0
$760$	−0.567641			−0.0205905
$761$	14.6791			0.532117			0.266058	−	0.963957i	$-0.414279\pi$
$761$	14.6791			0.532117			0.266058	+	0.963957i	$0.414279\pi$
$762$		0			0
$763$	−10.9329			−0.395796
$764$			11.4274i			0.413429i
$765$		0			0
$766$			5.75879i			0.208073i
$767$	−12.4878			−0.450907
$768$		0			0
$769$			30.9225i			1.11509i	0.830145	+	0.557547i	$0.188258\pi$
$769$			30.9225i			1.11509i	−0.830145	+	0.557547i	$0.811742\pi$
$770$	0.0977284	−	1.63568i	0.00352189	−	0.0589456i
$771$		0			0
$772$			7.02654i			0.252891i
$773$			7.38246i			0.265528i	0.991148	+	0.132764i	$0.0423853\pi$
$773$			7.38246i			0.265528i	−0.991148	+	0.132764i	$0.957615\pi$
$774$		0			0
$775$	6.69736			0.240576
$776$	10.3141			0.370254
$777$		0			0
$778$		−	22.2885i		−	0.799083i
$779$		−	4.37040i		−	0.156586i
$780$		0			0
$781$	−10.6138	−	0.634156i	−0.379793	−	0.0226919i
$782$			9.87624i			0.353174i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$		−	7.75123i		−	0.276653i
$786$		0			0
$787$			15.5645i			0.554815i	0.960752	+	0.277407i	$0.0894751\pi$
$787$			15.5645i			0.554815i	−0.960752	+	0.277407i	$0.910525\pi$
$788$	21.7037			0.773162
$789$		0			0
$790$	−0.767696			−0.0273134
$791$	9.75856			0.346975
$792$		0			0
$793$	24.8534			0.882569
$794$	36.2092			1.28502
$795$		0			0
$796$	10.9073			0.386598
$797$		−	21.6991i		−	0.768622i	−0.923204	−	0.384311i	$-0.874439\pi$
$797$		−	21.6991i		−	0.768622i	0.923204	−	0.384311i	$-0.125561\pi$
$798$		0			0
$799$			9.57241i			0.338648i
$800$	4.75591			0.168147
$801$		0			0
$802$			14.7482i			0.520777i
$803$	−7.92347	−	0.473412i	−0.279613	−	0.0167063i
$804$		0			0
$805$		−	2.07207i		−	0.0730308i
$806$		−	7.75053i		−	0.273001i
$807$		0			0
$808$	−12.6873			−0.446338
$809$	53.4012			1.87749			0.938744	−	0.344615i	$-0.111991\pi$
$809$	53.4012			1.87749			0.938744	+	0.344615i	$0.111991\pi$
$810$		0			0
$811$			24.4969i			0.860203i	0.902781	+	0.430102i	$0.141522\pi$
$811$			24.4969i			0.860203i	−0.902781	+	0.430102i	$0.858478\pi$
$812$		−	0.175313i		−	0.00615226i
$813$		0			0
$814$	0.933495	−	15.6239i	0.0327190	−	0.547616i
$815$		−	10.1574i		−	0.355800i
$816$		0			0
$817$	11.1788			0.391096
$818$		−	34.3987i		−	1.20272i
$819$		0			0
$820$		−	1.87930i		−	0.0656280i
$821$	−2.13755			−0.0746011			−0.0373006	−	0.999304i	$-0.511876\pi$
$821$	−2.13755			−0.0746011			−0.0373006	+	0.999304i	$0.511876\pi$
$822$		0			0
$823$	18.8542			0.657216			0.328608	−	0.944466i	$-0.393421\pi$
$823$	18.8542			0.657216			0.328608	+	0.944466i	$0.393421\pi$
$824$	−17.9022			−0.623652
$825$		0			0
$826$	2.26894			0.0789465
$827$	−40.0950			−1.39424			−0.697119	−	0.716955i	$-0.745535\pi$
$827$	−40.0950			−1.39424			−0.697119	+	0.716955i	$0.745535\pi$
$828$		0			0
$829$	24.3617			0.846116			0.423058	−	0.906103i	$-0.360957\pi$
$829$	24.3617			0.846116			0.423058	+	0.906103i	$0.360957\pi$
$830$			2.12223i			0.0736638i
$831$		0			0
$832$		−	5.50379i		−	0.190810i
$833$	2.35484			0.0815905
$834$		0			0
$835$			4.35111i			0.150576i
$836$	−3.80384	−	0.227272i	−0.131558	−	0.00786036i
$837$		0			0
$838$			38.9117i			1.34418i
$839$			38.0643i			1.31412i	0.753837	+	0.657062i	$0.228201\pi$
$839$			38.0643i			1.31412i	−0.753837	+	0.657062i	$0.771799\pi$
$840$		0			0
$841$	−28.9693			−0.998940
$842$	−11.6945			−0.403020
$843$		0			0
$844$			8.51137i			0.292973i
$845$			8.54304i			0.293889i
$846$		0			0
$847$	1.30978	−	10.9217i	0.0450046	−	0.375276i
$848$		−	2.97863i		−	0.102287i
$849$		0			0
$850$	−11.1994			−0.384137
$851$		−	19.7922i		−	0.678469i
$852$		0			0
$853$		−	20.7529i		−	0.710565i	−0.934759	−	0.355283i	$-0.884385\pi$
$853$		−	20.7529i		−	0.710565i	0.934759	−	0.355283i	$-0.115615\pi$
$854$	−4.51568			−0.154523
$855$		0			0
$856$	−8.25477			−0.282142
$857$	−4.82559			−0.164839			−0.0824195	−	0.996598i	$-0.526265\pi$
$857$	−4.82559			−0.164839			−0.0824195	+	0.996598i	$0.526265\pi$
$858$		0			0
$859$	25.4365			0.867883			0.433941	−	0.900941i	$-0.357123\pi$
$859$	25.4365			0.867883			0.433941	+	0.900941i	$0.357123\pi$
$860$	4.80695			0.163916
$861$		0			0
$862$	7.16905			0.244179
$863$		−	26.9278i		−	0.916634i	−0.888789	−	0.458317i	$-0.848452\pi$
$863$		−	26.9278i		−	0.916634i	0.888789	−	0.458317i	$-0.151548\pi$
$864$		0			0
$865$			1.94224i			0.0660380i
$866$	19.4899			0.662294
$867$		0			0
$868$			1.40822i			0.0477980i
$869$	−5.14443	−	0.307370i	−0.174513	−	0.0104268i
$870$		0			0
$871$		−	46.5929i		−	1.57874i
$872$			10.9329i			0.370233i
$873$		0			0
$874$	−4.81868			−0.162994
$875$	4.81995			0.162944
$876$		0			0
$877$		−	26.8232i		−	0.905756i	−0.891573	−	0.452878i	$-0.850397\pi$
$877$		−	26.8232i		−	0.905756i	0.891573	−	0.452878i	$-0.149603\pi$
$878$		−	3.30603i		−	0.111573i
$879$		0			0
$880$	−1.63568	−	0.0977284i	−0.0551386	−	0.00329442i
$881$			34.0831i			1.14829i	0.818754	+	0.574144i	$0.194665\pi$
$881$			34.0831i			1.14829i	−0.818754	+	0.574144i	$0.805335\pi$
$882$		0			0
$883$	14.5735			0.490437			0.245219	−	0.969468i	$-0.421140\pi$
$883$	14.5735			0.490437			0.245219	+	0.969468i	$0.421140\pi$
$884$			12.9606i			0.435911i
$885$		0			0
$886$		−	27.5415i		−	0.925275i
$887$	9.53885			0.320283			0.160142	−	0.987094i	$-0.448805\pi$
$887$	9.53885			0.320283			0.160142	+	0.987094i	$0.448805\pi$
$888$		0			0
$889$	−9.36217			−0.313997
$890$	−1.15100			−0.0385816
$891$		0			0
$892$	−4.55627			−0.152555
$893$	−4.67045			−0.156290
$894$		0			0
$895$	11.6373			0.388992
$896$			1.00000i			0.0334077i
$897$		0			0
$898$			12.0032i			0.400553i
$899$	−0.246878			−0.00823385
$900$		0			0
$901$			7.01421i			0.233677i
$902$	0.752434	−	12.5934i	0.0250533	−	0.419316i
$903$		0			0
$904$		−	9.75856i		−	0.324565i
$905$		−	3.22272i		−	0.107127i
$906$		0			0
$907$	4.51317			0.149857			0.0749286	−	0.997189i	$-0.476127\pi$
$907$	4.51317			0.149857			0.0749286	+	0.997189i	$0.476127\pi$
$908$	−23.6027			−0.783282
$909$		0			0
$910$		−	2.71917i		−	0.0901396i
$911$			33.6651i			1.11537i	0.830052	+	0.557687i	$0.188311\pi$
$911$			33.6651i			1.11537i	−0.830052	+	0.557687i	$0.811689\pi$
$912$		0			0
$913$	−0.849699	+	14.2214i	−0.0281209	+	0.470658i
$914$			10.5366i			0.348520i
$915$		0			0
$916$	−18.1200			−0.598703
$917$		−	5.14608i		−	0.169939i
$918$		0			0
$919$		−	40.5018i		−	1.33603i	−0.744147	−	0.668016i	$-0.767144\pi$
$919$		−	40.5018i		−	1.33603i	0.744147	−	0.668016i	$-0.232856\pi$
$920$	−2.07207			−0.0683140
$921$		0			0
$922$	26.1304			0.860558
$923$	−17.6446			−0.580779
$924$		0			0
$925$	22.4439			0.737953
$926$	11.6684			0.383447
$927$		0			0
$928$	−0.175313			−0.00575492
$929$			9.20676i			0.302064i	0.988529	+	0.151032i	$0.0482596\pi$
$929$			9.20676i			0.302064i	−0.988529	+	0.151032i	$0.951740\pi$
$930$		0			0
$931$			1.14894i			0.0376551i
$932$	−17.1422			−0.561513
$933$		0			0
$934$		−	32.7068i		−	1.07020i
$935$	3.85176	+	0.230135i	0.125966	+	0.00752623i
$936$		0			0
$937$			12.2121i			0.398953i	0.979903	+	0.199476i	$0.0639241\pi$
$937$			12.2121i			0.398953i	−0.979903	+	0.199476i	$0.936076\pi$
$938$			8.46560i			0.276412i
$939$		0			0
$940$	−2.00832			−0.0655043
$941$	−23.9397			−0.780412			−0.390206	−	0.920728i	$-0.627596\pi$
$941$	−23.9397			−0.780412			−0.390206	+	0.920728i	$0.627596\pi$
$942$		0			0
$943$		−	15.9533i		−	0.519512i
$944$		−	2.26894i		−	0.0738477i
$945$		0			0
$946$	32.2120	+	1.92461i	1.04730	+	0.0625744i
$947$		−	19.1422i		−	0.622039i	−0.950404	−	0.311020i	$-0.899330\pi$
$947$		−	19.1422i		−	0.622039i	0.950404	−	0.311020i	$-0.100670\pi$
$948$		0			0
$949$	−13.1721			−0.427584
$950$		−	5.46428i		−	0.177285i
$951$		0			0
$952$		−	2.35484i		−	0.0763210i
$953$	32.3634			1.04835			0.524177	−	0.851609i	$-0.324373\pi$
$953$	32.3634			1.04835			0.524177	+	0.851609i	$0.324373\pi$
$954$		0			0
$955$	5.64576			0.182692
$956$	9.76331			0.315768
$957$		0			0
$958$	−14.6716			−0.474018
$959$	−6.91130			−0.223178
$960$		0			0
$961$	−29.0169			−0.936030
$962$		−	25.9733i		−	0.837413i
$963$		0			0
$964$		−	22.8693i		−	0.736571i
$965$	3.47149			0.111751
$966$		0			0
$967$		−	39.0758i		−	1.25659i	−0.777974	−	0.628297i	$-0.783752\pi$
$967$		−	39.0758i		−	1.25659i	0.777974	−	0.628297i	$-0.216248\pi$
$968$	−10.9217	−	1.30978i	−0.351038	−	0.0420980i
$969$		0			0
$970$		−	5.09571i		−	0.163613i
$971$		−	7.23340i		−	0.232131i	−0.993242	−	0.116065i	$-0.962972\pi$
$971$		−	7.23340i		−	0.232131i	0.993242	−	0.116065i	$-0.0370282\pi$
$972$		0			0
$973$	−16.6837			−0.534856
$974$	19.7666			0.633363
$975$		0			0
$976$			4.51568i			0.144543i
$977$		−	39.3512i		−	1.25896i	−0.777018	−	0.629479i	$-0.783268\pi$
$977$		−	39.3512i		−	1.25896i	0.777018	−	0.629479i	$-0.216732\pi$
$978$		0			0
$979$	−7.71300	−	0.460837i	−0.246509	−	0.0147284i
$980$			0.494054i			0.0157820i
$981$		0			0
$982$	25.8600			0.825225
$983$			26.0176i			0.829831i	0.909860	+	0.414916i	$0.136189\pi$
$983$			26.0176i			0.829831i	−0.909860	+	0.414916i	$0.863811\pi$
$984$		0			0
$985$		−	10.7228i		−	0.341657i
$986$	0.412834			0.0131473
$987$		0			0
$988$	−6.32355			−0.201179
$989$	40.8061			1.29756
$990$		0			0
$991$	31.0427			0.986102			0.493051	−	0.870000i	$-0.335882\pi$
$991$	31.0427			0.986102			0.493051	+	0.870000i	$0.335882\pi$
$992$	1.40822			0.0447110
$993$		0			0
$994$	3.20590			0.101685
$995$		−	5.38878i		−	0.170836i
$996$		0			0
$997$			37.1470i			1.17646i	0.808694	+	0.588229i	$0.200175\pi$
$997$			37.1470i			1.17646i	−0.808694	+	0.588229i	$0.799825\pi$
$998$	−24.1647			−0.764919
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.c.b.197.6	yes	12
3.2	odd	2		1386.2.c.a.197.7	yes	12
11.10	odd	2		1386.2.c.a.197.6	✓	12
33.32	even	2	inner	1386.2.c.b.197.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.c.a.197.6	✓	12	11.10	odd	2
1386.2.c.a.197.7	yes	12	3.2	odd	2
1386.2.c.b.197.6	yes	12	1.1	even	1	trivial
1386.2.c.b.197.7	yes	12	33.32	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.494054i q^{5} +1.00000i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.494054i q^{5} +1.00000i q^{7} +1.00000 q^{8} -0.494054i q^{10} +(0.197809 - 3.31072i) q^{11} -5.50379i q^{13} +1.00000i q^{14} +1.00000 q^{16} -2.35484 q^{17} -1.14894i q^{19} -0.494054i q^{20} +(0.197809 - 3.31072i) q^{22} -4.19401i q^{23} +4.75591 q^{25} -5.50379i q^{26} +1.00000i q^{28} -0.175313 q^{29} +1.40822 q^{31} +1.00000 q^{32} -2.35484 q^{34} +0.494054 q^{35} +4.71917 q^{37} -1.14894i q^{38} -0.494054i q^{40} +3.80384 q^{41} +9.72961i q^{43} +(0.197809 - 3.31072i) q^{44} -4.19401i q^{46} -4.06499i q^{47} -1.00000 q^{49} +4.75591 q^{50} -5.50379i q^{52} -2.97863i q^{53} +(-1.63568 - 0.0977284i) q^{55} +1.00000i q^{56} -0.175313 q^{58} -2.26894i q^{59} +4.51568i q^{61} +1.40822 q^{62} +1.00000 q^{64} -2.71917 q^{65} +8.46560 q^{67} -2.35484 q^{68} +0.494054 q^{70} -3.20590i q^{71} -2.39328i q^{73} +4.71917 q^{74} -1.14894i q^{76} +(3.31072 + 0.197809i) q^{77} -1.55387i q^{79} -0.494054i q^{80} +3.80384 q^{82} -4.29555 q^{83} +1.16342i q^{85} +9.72961i q^{86} +(0.197809 - 3.31072i) q^{88} -2.32970i q^{89} +5.50379 q^{91} -4.19401i q^{92} -4.06499i q^{94} -0.567641 q^{95} +10.3141 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) 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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