LMFDB - Embedded newform 1386.2.c.a.197.12

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			197.12
Root			$-1.26053 - 1.18788i$ of defining polynomial
Character	$\chi$	$=$	1386.197
Dual form			1386.2.c.a.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.46258i q^{5} -1.00000i q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.46258i q^{5} -1.00000i q^{7} -1.00000 q^{8} -3.46258i q^{10} +(0.748459 + 3.23107i) q^{11} +2.27303i q^{13} +1.00000i q^{14} +1.00000 q^{16} +1.31729 q^{17} -1.04426i q^{19} +3.46258i q^{20} +(-0.748459 - 3.23107i) q^{22} +2.56361i q^{23} -6.98945 q^{25} -2.27303i q^{26} -1.00000i q^{28} +2.36395 q^{29} +2.87098 q^{31} -1.00000 q^{32} -1.31729 q^{34} +3.46258 q^{35} -5.87054 q^{37} +1.04426i q^{38} -3.46258i q^{40} -3.37407 q^{41} +4.69219i q^{43} +(0.748459 + 3.23107i) q^{44} -2.56361i q^{46} +12.8461i q^{47} -1.00000 q^{49} +6.98945 q^{50} +2.27303i q^{52} -3.57996i q^{53} +(-11.1878 + 2.59160i) q^{55} +1.00000i q^{56} -2.36395 q^{58} -4.94539i q^{59} -9.19819i q^{61} -2.87098 q^{62} +1.00000 q^{64} -7.87054 q^{65} -5.35487 q^{67} +1.31729 q^{68} -3.46258 q^{70} -4.36154i q^{71} -14.2217i q^{73} +5.87054 q^{74} -1.04426i q^{76} +(3.23107 - 0.748459i) q^{77} +16.8261i q^{79} +3.46258i q^{80} +3.37407 q^{82} -14.8133 q^{83} +4.56121i q^{85} -4.69219i q^{86} +(-0.748459 - 3.23107i) q^{88} +6.49012i q^{89} +2.27303 q^{91} +2.56361i q^{92} -12.8461i q^{94} +3.61582 q^{95} +10.1315 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			3.46258i			1.54851i	0.632873	+	0.774256i	$0.281876\pi$
$5$			3.46258i			1.54851i	−0.632873	+	0.774256i	$0.718124\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$	−1.00000			−0.353553
$9$		0			0
$10$		−	3.46258i		−	1.09496i
$11$	0.748459	+	3.23107i	0.225669	+	0.974204i
$11$	0.748459	+	3.23107i	0.225669	+	0.974204i
$12$		0			0
$13$			2.27303i			0.630425i	0.949021	+	0.315213i	$0.102076\pi$
$13$			2.27303i			0.630425i	−0.949021	+	0.315213i	$0.897924\pi$
$14$			1.00000i			0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.31729			0.319489			0.159744	−	0.987158i	$-0.448933\pi$
$17$	1.31729			0.319489			0.159744	+	0.987158i	$0.448933\pi$
$18$		0			0
$19$		−	1.04426i		−	0.239569i	−0.992800	−	0.119784i	$-0.961780\pi$
$19$		−	1.04426i		−	0.239569i	0.992800	−	0.119784i	$-0.0382203\pi$
$20$			3.46258i			0.774256i
$21$		0			0
$22$	−0.748459	−	3.23107i	−0.159572	−	0.688866i
$23$			2.56361i			0.534550i	0.963620	+	0.267275i	$0.0861232\pi$
$23$			2.56361i			0.534550i	−0.963620	+	0.267275i	$0.913877\pi$
$24$		0			0
$25$	−6.98945			−1.39789
$26$		−	2.27303i		−	0.445778i
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	2.36395			0.438974			0.219487	−	0.975615i	$-0.429562\pi$
$29$	2.36395			0.438974			0.219487	+	0.975615i	$0.429562\pi$
$30$		0			0
$31$	2.87098			0.515644			0.257822	−	0.966192i	$-0.416995\pi$
$31$	2.87098			0.515644			0.257822	+	0.966192i	$0.416995\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−1.31729			−0.225913
$35$	3.46258			0.585282
$36$		0			0
$37$	−5.87054			−0.965111			−0.482556	−	0.875865i	$-0.660291\pi$
$37$	−5.87054			−0.965111			−0.482556	+	0.875865i	$0.660291\pi$
$38$			1.04426i			0.169401i
$39$		0			0
$40$		−	3.46258i		−	0.547482i
$41$	−3.37407			−0.526940			−0.263470	−	0.964668i	$-0.584867\pi$
$41$	−3.37407			−0.526940			−0.263470	+	0.964668i	$0.584867\pi$
$42$		0			0
$43$			4.69219i			0.715553i	0.933807	+	0.357776i	$0.116465\pi$
$43$			4.69219i			0.715553i	−0.933807	+	0.357776i	$0.883535\pi$
$44$	0.748459	+	3.23107i	0.112834	+	0.487102i
$45$		0			0
$46$		−	2.56361i		−	0.377984i
$47$			12.8461i			1.87380i	0.349598	+	0.936900i	$0.386318\pi$
$47$			12.8461i			1.87380i	−0.349598	+	0.936900i	$0.613682\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	6.98945			0.988457
$51$		0			0
$52$			2.27303i			0.315213i
$53$		−	3.57996i		−	0.491745i	−0.969302	−	0.245873i	$-0.920925\pi$
$53$		−	3.57996i		−	0.491745i	0.969302	−	0.245873i	$-0.0790745\pi$
$54$		0			0
$55$	−11.1878	+	2.59160i	−1.50857	+	0.349451i
$56$			1.00000i			0.133631i
$57$		0			0
$58$	−2.36395			−0.310402
$59$		−	4.94539i		−	0.643835i	−0.946768	−	0.321917i	$-0.895673\pi$
$59$		−	4.94539i		−	0.643835i	0.946768	−	0.321917i	$-0.104327\pi$
$60$		0			0
$61$		−	9.19819i		−	1.17771i	−0.808240	−	0.588853i	$-0.799579\pi$
$61$		−	9.19819i		−	1.17771i	0.808240	−	0.588853i	$-0.200421\pi$
$62$	−2.87098			−0.364615
$63$		0			0
$64$	1.00000			0.125000
$65$	−7.87054			−0.976221
$66$		0			0
$67$	−5.35487			−0.654202			−0.327101	−	0.944989i	$-0.606072\pi$
$67$	−5.35487			−0.654202			−0.327101	+	0.944989i	$0.606072\pi$
$68$	1.31729			0.159744
$69$		0			0
$70$	−3.46258			−0.413857
$71$		−	4.36154i		−	0.517620i	−0.965928	−	0.258810i	$-0.916670\pi$
$71$		−	4.36154i		−	0.517620i	0.965928	−	0.258810i	$-0.0833303\pi$
$72$		0			0
$73$		−	14.2217i		−	1.66453i	−0.554380	−	0.832264i	$-0.687044\pi$
$73$		−	14.2217i		−	1.66453i	0.554380	−	0.832264i	$-0.312956\pi$
$74$	5.87054			0.682437
$75$		0			0
$76$		−	1.04426i		−	0.119784i
$77$	3.23107	−	0.748459i	0.368215	−	0.0852948i
$78$		0			0
$79$			16.8261i			1.89308i	0.322582	+	0.946541i	$0.395449\pi$
$79$			16.8261i			1.89308i	−0.322582	+	0.946541i	$0.604551\pi$
$80$			3.46258i			0.387128i
$81$		0			0
$82$	3.37407			0.372603
$83$	−14.8133			−1.62597			−0.812987	−	0.582282i	$-0.802160\pi$
$83$	−14.8133			−1.62597			−0.812987	+	0.582282i	$0.802160\pi$
$84$		0			0
$85$			4.56121i			0.494732i
$86$		−	4.69219i		−	0.505972i
$87$		0			0
$88$	−0.748459	−	3.23107i	−0.0797859	−	0.344433i
$89$			6.49012i			0.687952i	0.938979	+	0.343976i	$0.111774\pi$
$89$			6.49012i			0.687952i	−0.938979	+	0.343976i	$0.888226\pi$
$90$		0			0
$91$	2.27303			0.238278
$92$			2.56361i			0.267275i
$93$		0			0
$94$		−	12.8461i		−	1.32498i
$95$	3.61582			0.370975
$96$		0			0
$97$	10.1315			1.02870			0.514348	−	0.857581i	$-0.328034\pi$
$97$	10.1315			1.02870			0.514348	+	0.857581i	$0.328034\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	−6.98945			−0.698945
$101$	−14.3303			−1.42591			−0.712957	−	0.701208i	$-0.752644\pi$
$101$	−14.3303			−1.42591			−0.712957	+	0.701208i	$0.752644\pi$
$102$		0			0
$103$	4.38130			0.431703			0.215851	−	0.976426i	$-0.430747\pi$
$103$	4.38130			0.431703			0.215851	+	0.976426i	$0.430747\pi$
$104$		−	2.27303i		−	0.222889i
$105$		0			0
$106$			3.57996i			0.347716i
$107$	−9.99912			−0.966652			−0.483326	−	0.875441i	$-0.660571\pi$
$107$	−9.99912			−0.966652			−0.483326	+	0.875441i	$0.660571\pi$
$108$		0			0
$109$		−	7.40600i		−	0.709366i	−0.934987	−	0.354683i	$-0.884589\pi$
$109$		−	7.40600i		−	0.709366i	0.934987	−	0.354683i	$-0.115411\pi$
$110$	11.1878	−	2.59160i	1.06672	−	0.247099i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$			11.7261i			1.10310i	0.834143	+	0.551549i	$0.185963\pi$
$113$			11.7261i			1.10310i	−0.834143	+	0.551549i	$0.814037\pi$
$114$		0			0
$115$	−8.87671			−0.827757
$116$	2.36395			0.219487
$117$		0			0
$118$			4.94539i			0.455260i
$119$		−	1.31729i		−	0.120755i
$120$		0			0
$121$	−9.87962	+	4.83664i	−0.898147	+	0.439695i
$122$			9.19819i			0.832765i
$123$		0			0
$124$	2.87098			0.257822
$125$		−	6.88861i		−	0.616136i
$126$		0			0
$127$			20.6280i			1.83043i	0.402960	+	0.915217i	$0.367981\pi$
$127$			20.6280i			1.83043i	−0.402960	+	0.915217i	$0.632019\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	7.87054			0.690292
$131$	0.323452			0.0282602			0.0141301	−	0.999900i	$-0.495502\pi$
$131$	0.323452			0.0282602			0.0141301	+	0.999900i	$0.495502\pi$
$132$		0			0
$133$	−1.04426			−0.0905485
$134$	5.35487			0.462591
$135$		0			0
$136$	−1.31729			−0.112956
$137$			9.70127i			0.828835i	0.910087	+	0.414418i	$0.136015\pi$
$137$			9.70127i			0.828835i	−0.910087	+	0.414418i	$0.863985\pi$
$138$		0			0
$139$			8.40529i			0.712927i	0.934309	+	0.356464i	$0.116018\pi$
$139$			8.40529i			0.712927i	−0.934309	+	0.356464i	$0.883982\pi$
$140$	3.46258			0.292641
$141$		0			0
$142$			4.36154i			0.366013i
$143$	−7.34432	+	1.70127i	−0.614163	+	0.142267i
$144$		0			0
$145$			8.18536i			0.679757i
$146$			14.2217i			1.17700i
$147$		0			0
$148$	−5.87054			−0.482556
$149$	−10.8796			−0.891293			−0.445647	−	0.895209i	$-0.647026\pi$
$149$	−10.8796			−0.891293			−0.445647	+	0.895209i	$0.647026\pi$
$150$		0			0
$151$		−	0.611828i		−	0.0497898i	−0.999690	−	0.0248949i	$-0.992075\pi$
$151$		−	0.611828i		−	0.0497898i	0.999690	−	0.0248949i	$-0.00792512\pi$
$152$			1.04426i			0.0847004i
$153$		0			0
$154$	−3.23107	+	0.748459i	−0.260367	+	0.0603125i
$155$			9.94100i			0.798480i
$156$		0			0
$157$	9.65728			0.770735			0.385367	−	0.922763i	$-0.374075\pi$
$157$	9.65728			0.770735			0.385367	+	0.922763i	$0.374075\pi$
$158$		−	16.8261i		−	1.33861i
$159$		0			0
$160$		−	3.46258i		−	0.273741i
$161$	2.56361			0.202041
$162$		0			0
$163$	10.6375			0.833192			0.416596	−	0.909092i	$-0.363223\pi$
$163$	10.6375			0.833192			0.416596	+	0.909092i	$0.363223\pi$
$164$	−3.37407			−0.263470
$165$		0			0
$166$	14.8133			1.14974
$167$	20.2471			1.56677			0.783383	−	0.621539i	$-0.213492\pi$
$167$	20.2471			1.56677			0.783383	+	0.621539i	$0.213492\pi$
$168$		0			0
$169$	7.83333			0.602564
$170$		−	4.56121i		−	0.349829i
$171$		0			0
$172$			4.69219i			0.357776i
$173$	−5.62550			−0.427699			−0.213849	−	0.976867i	$-0.568600\pi$
$173$	−5.62550			−0.427699			−0.213849	+	0.976867i	$0.568600\pi$
$174$		0			0
$175$			6.98945i			0.528352i
$176$	0.748459	+	3.23107i	0.0564172	+	0.243551i
$177$		0			0
$178$		−	6.49012i		−	0.486455i
$179$			1.56278i			0.116808i	0.998293	+	0.0584039i	$0.0186011\pi$
$179$			1.56278i			0.116808i	−0.998293	+	0.0584039i	$0.981399\pi$
$180$		0			0
$181$	−4.49648			−0.334221			−0.167110	−	0.985938i	$-0.553444\pi$
$181$	−4.49648			−0.334221			−0.167110	+	0.985938i	$0.553444\pi$
$182$	−2.27303			−0.168488
$183$		0			0
$184$		−	2.56361i		−	0.188992i
$185$		−	20.3272i		−	1.49449i
$186$		0			0
$187$	0.985934	+	4.25624i	0.0720986	+	0.311247i
$188$			12.8461i			0.936900i
$189$		0			0
$190$	−3.61582			−0.262319
$191$		−	12.9828i		−	0.939401i	−0.882826	−	0.469700i	$-0.844362\pi$
$191$		−	12.9828i		−	0.939401i	0.882826	−	0.469700i	$-0.155638\pi$
$192$		0			0
$193$			16.4642i			1.18512i	0.805527	+	0.592559i	$0.201882\pi$
$193$			16.4642i			1.18512i	−0.805527	+	0.592559i	$0.798118\pi$
$194$	−10.1315			−0.727399
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	−19.6756			−1.40183			−0.700914	−	0.713246i	$-0.747224\pi$
$197$	−19.6756			−1.40183			−0.700914	+	0.713246i	$0.747224\pi$
$198$		0			0
$199$	0.145994			0.0103493			0.00517464	−	0.999987i	$-0.498353\pi$
$199$	0.145994			0.0103493			0.00517464	+	0.999987i	$0.498353\pi$
$200$	6.98945			0.494228
$201$		0			0
$202$	14.3303			1.00827
$203$		−	2.36395i		−	0.165917i
$204$		0			0
$205$		−	11.6830i		−	0.815974i
$206$	−4.38130			−0.305260
$207$		0			0
$208$			2.27303i			0.157606i
$209$	3.37407	−	0.781583i	0.233389	−	0.0540632i
$210$		0			0
$211$			1.18091i			0.0812972i	0.999174	+	0.0406486i	$0.0129424\pi$
$211$			1.18091i			0.0812972i	−0.999174	+	0.0406486i	$0.987058\pi$
$212$		−	3.57996i		−	0.245873i
$213$		0			0
$214$	9.99912			0.683526
$215$	−16.2471			−1.10804
$216$		0			0
$217$		−	2.87098i		−	0.194895i
$218$			7.40600i			0.501597i
$219$		0			0
$220$	−11.1878	+	2.59160i	−0.754283	+	0.174725i
$221$			2.99423i			0.201414i
$222$		0			0
$223$	−0.848717			−0.0568343			−0.0284172	−	0.999596i	$-0.509047\pi$
$223$	−0.848717			−0.0568343			−0.0284172	+	0.999596i	$0.509047\pi$
$224$			1.00000i			0.0668153i
$225$		0			0
$226$		−	11.7261i		−	0.780008i
$227$	−13.5749			−0.900995			−0.450498	−	0.892778i	$-0.648753\pi$
$227$	−13.5749			−0.900995			−0.450498	+	0.892778i	$0.648753\pi$
$228$		0			0
$229$	15.5462			1.02732			0.513661	−	0.857993i	$-0.328289\pi$
$229$	15.5462			1.02732			0.513661	+	0.857993i	$0.328289\pi$
$230$	8.87671			0.585313
$231$		0			0
$232$	−2.36395			−0.155201
$233$	6.30344			0.412952			0.206476	−	0.978452i	$-0.433800\pi$
$233$	6.30344			0.412952			0.206476	+	0.978452i	$0.433800\pi$
$234$		0			0
$235$	−44.4807			−2.90160
$236$		−	4.94539i		−	0.321917i
$237$		0			0
$238$			1.31729i			0.0853870i
$239$	29.2292			1.89068			0.945341	−	0.326084i	$-0.105729\pi$
$239$	29.2292			1.89068			0.945341	+	0.326084i	$0.105729\pi$
$240$		0			0
$241$		−	24.6488i		−	1.58777i	−0.608068	−	0.793885i	$-0.708055\pi$
$241$		−	24.6488i		−	1.58777i	0.608068	−	0.793885i	$-0.291945\pi$
$242$	9.87962	−	4.83664i	0.635086	−	0.310911i
$243$		0			0
$244$		−	9.19819i		−	0.588853i
$245$		−	3.46258i		−	0.221216i
$246$		0			0
$247$	2.37363			0.151030
$248$	−2.87098			−0.182308
$249$		0			0
$250$			6.88861i			0.435674i
$251$		−	23.6799i		−	1.49466i	−0.664452	−	0.747331i	$-0.731335\pi$
$251$		−	23.6799i		−	1.49466i	0.664452	−	0.747331i	$-0.268665\pi$
$252$		0			0
$253$	−8.28321	+	1.91876i	−0.520761	+	0.120631i
$254$		−	20.6280i		−	1.29431i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	20.2929i		−	1.26583i	−0.774220	−	0.632917i	$-0.781857\pi$
$257$		−	20.2929i		−	1.26583i	0.774220	−	0.632917i	$-0.218143\pi$
$258$		0			0
$259$			5.87054i			0.364778i
$260$	−7.87054			−0.488110
$261$		0			0
$262$	−0.323452			−0.0199830
$263$	8.70007			0.536469			0.268235	−	0.963354i	$-0.413560\pi$
$263$	8.70007			0.536469			0.268235	+	0.963354i	$0.413560\pi$
$264$		0			0
$265$	12.3959			0.761474
$266$	1.04426			0.0640275
$267$		0			0
$268$	−5.35487			−0.327101
$269$		−	21.9584i		−	1.33882i	−0.742891	−	0.669412i	$-0.766546\pi$
$269$		−	21.9584i		−	1.33882i	0.742891	−	0.669412i	$-0.233454\pi$
$270$		0			0
$271$			9.56013i			0.580736i	0.956915	+	0.290368i	$0.0937778\pi$
$271$			9.56013i			0.580736i	−0.956915	+	0.290368i	$0.906222\pi$
$272$	1.31729			0.0798722
$273$		0			0
$274$		−	9.70127i		−	0.586075i
$275$	−5.23131	−	22.5834i	−0.315460	−	1.36183i
$276$		0			0
$277$			2.85994i			0.171837i	0.996302	+	0.0859185i	$0.0273825\pi$
$277$			2.85994i			0.171837i	−0.996302	+	0.0859185i	$0.972618\pi$
$278$		−	8.40529i		−	0.504116i
$279$		0			0
$280$	−3.46258			−0.206929
$281$	17.2272			1.02769			0.513844	−	0.857884i	$-0.328221\pi$
$281$	17.2272			1.02769			0.513844	+	0.857884i	$0.328221\pi$
$282$		0			0
$283$			2.40616i			0.143032i	0.997439	+	0.0715158i	$0.0227836\pi$
$283$			2.40616i			0.143032i	−0.997439	+	0.0715158i	$0.977216\pi$
$284$		−	4.36154i		−	0.258810i
$285$		0			0
$286$	7.34432	−	1.70127i	0.434279	−	0.100598i
$287$			3.37407i			0.199165i
$288$		0			0
$289$	−15.2648			−0.897927
$290$		−	8.18536i		−	0.480661i
$291$		0			0
$292$		−	14.2217i		−	0.832264i
$293$	26.7109			1.56047			0.780234	−	0.625488i	$-0.215100\pi$
$293$	26.7109			1.56047			0.780234	+	0.625488i	$0.215100\pi$
$294$		0			0
$295$	17.1238			0.996986
$296$	5.87054			0.341218
$297$		0			0
$298$	10.8796			0.630239
$299$	−5.82717			−0.336994
$300$		0			0
$301$	4.69219			0.270453
$302$			0.611828i			0.0352067i
$303$		0			0
$304$		−	1.04426i		−	0.0598922i
$305$	31.8494			1.82369
$306$		0			0
$307$			1.98925i			0.113532i	0.998388	+	0.0567661i	$0.0180789\pi$
$307$			1.98925i			0.113532i	−0.998388	+	0.0567661i	$0.981921\pi$
$308$	3.23107	−	0.748459i	0.184107	−	0.0426474i
$309$		0			0
$310$		−	9.94100i		−	0.564611i
$311$			4.10821i			0.232955i	0.993193	+	0.116478i	$0.0371603\pi$
$311$			4.10821i			0.232955i	−0.993193	+	0.116478i	$0.962840\pi$
$312$		0			0
$313$	19.1917			1.08478			0.542390	−	0.840127i	$-0.317520\pi$
$313$	19.1917			1.08478			0.542390	+	0.840127i	$0.317520\pi$
$314$	−9.65728			−0.544992
$315$		0			0
$316$			16.8261i			0.946541i
$317$			24.0173i			1.34894i	0.738300	+	0.674472i	$0.235629\pi$
$317$			24.0173i			1.34894i	−0.738300	+	0.674472i	$0.764371\pi$
$318$		0			0
$319$	1.76932	+	7.63808i	0.0990628	+	0.427651i
$320$			3.46258i			0.193564i
$321$		0			0
$322$	−2.56361			−0.142865
$323$		−	1.37558i		−	0.0765396i
$324$		0			0
$325$		−	15.8872i		−	0.881264i
$326$	−10.6375			−0.589156
$327$		0			0
$328$	3.37407			0.186302
$329$	12.8461			0.708230
$330$		0			0
$331$	31.2497			1.71764			0.858820	−	0.512278i	$-0.171198\pi$
$331$	31.2497			1.71764			0.858820	+	0.512278i	$0.171198\pi$
$332$	−14.8133			−0.812987
$333$		0			0
$334$	−20.2471			−1.10787
$335$		−	18.5417i		−	1.01304i
$336$		0			0
$337$			6.50551i			0.354378i	0.984177	+	0.177189i	$0.0567004\pi$
$337$			6.50551i			0.354378i	−0.984177	+	0.177189i	$0.943300\pi$
$338$	−7.83333			−0.426077
$339$		0			0
$340$			4.56121i			0.247366i
$341$	2.14881	+	9.27634i	0.116365	+	0.502342i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		−	4.69219i		−	0.252986i
$345$		0			0
$346$	5.62550			0.302429
$347$	−25.1628			−1.35081			−0.675406	−	0.737446i	$-0.736031\pi$
$347$	−25.1628			−1.35081			−0.675406	+	0.737446i	$0.736031\pi$
$348$		0			0
$349$		−	12.5834i		−	0.673576i	−0.941580	−	0.336788i	$-0.890659\pi$
$349$		−	12.5834i		−	0.673576i	0.941580	−	0.336788i	$-0.109341\pi$
$350$		−	6.98945i		−	0.373602i
$351$		0			0
$352$	−0.748459	−	3.23107i	−0.0398930	−	0.172217i
$353$		−	13.1792i		−	0.701456i	−0.936477	−	0.350728i	$-0.885934\pi$
$353$		−	13.1792i		−	0.701456i	0.936477	−	0.350728i	$-0.114066\pi$
$354$		0			0
$355$	15.1022			0.801541
$356$			6.49012i			0.343976i
$357$		0			0
$358$		−	1.56278i		−	0.0825955i
$359$	17.9186			0.945708			0.472854	−	0.881141i	$-0.343224\pi$
$359$	17.9186			0.945708			0.472854	+	0.881141i	$0.343224\pi$
$360$		0			0
$361$	17.9095			0.942607
$362$	4.49648			0.236330
$363$		0			0
$364$	2.27303			0.119139
$365$	49.2438			2.57754
$366$		0			0
$367$	−5.09764			−0.266094			−0.133047	−	0.991110i	$-0.542476\pi$
$367$	−5.09764			−0.266094			−0.133047	+	0.991110i	$0.542476\pi$
$368$			2.56361i			0.133638i
$369$		0			0
$370$			20.3272i			1.05676i
$371$	−3.57996			−0.185862
$372$		0			0
$373$		−	23.7658i		−	1.23054i	−0.788315	−	0.615272i	$-0.789046\pi$
$373$		−	23.7658i		−	1.23054i	0.788315	−	0.615272i	$-0.210954\pi$
$374$	−0.985934	−	4.25624i	−0.0509814	−	0.220085i
$375$		0			0
$376$		−	12.8461i		−	0.662488i
$377$			5.37333i			0.276740i
$378$		0			0
$379$	10.5912			0.544033			0.272017	−	0.962293i	$-0.412309\pi$
$379$	10.5912			0.544033			0.272017	+	0.962293i	$0.412309\pi$
$380$	3.61582			0.185488
$381$		0			0
$382$			12.9828i			0.664257i
$383$			32.0505i			1.63770i	0.574005	+	0.818852i	$0.305389\pi$
$383$			32.0505i			1.63770i	−0.574005	+	0.818852i	$0.694611\pi$
$384$		0			0
$385$	2.59160	+	11.1878i	0.132080	+	0.570185i
$386$		−	16.4642i		−	0.838004i
$387$		0			0
$388$	10.1315			0.514348
$389$		−	29.8275i		−	1.51232i	−0.654389	−	0.756158i	$-0.727074\pi$
$389$		−	29.8275i		−	1.51232i	0.654389	−	0.756158i	$-0.272926\pi$
$390$		0			0
$391$			3.37701i			0.170783i
$392$	1.00000			0.0505076
$393$		0			0
$394$	19.6756			0.991242
$395$	−58.2616			−2.93146
$396$		0			0
$397$	15.4569			0.775759			0.387880	−	0.921710i	$-0.373208\pi$
$397$	15.4569			0.775759			0.387880	+	0.921710i	$0.373208\pi$
$398$	−0.145994			−0.00731804
$399$		0			0
$400$	−6.98945			−0.349472
$401$			20.9581i			1.04660i	0.852150	+	0.523298i	$0.175299\pi$
$401$			20.9581i			1.04660i	−0.852150	+	0.523298i	$0.824701\pi$
$402$		0			0
$403$			6.52583i			0.325075i
$404$	−14.3303			−0.712957
$405$		0			0
$406$			2.36395i			0.117321i
$407$	−4.39386	−	18.9681i	−0.217795	−	0.940215i
$408$		0			0
$409$			26.2918i			1.30005i	0.759914	+	0.650024i	$0.225241\pi$
$409$			26.2918i			1.30005i	−0.759914	+	0.650024i	$0.774759\pi$
$410$			11.6830i			0.576980i
$411$		0			0
$412$	4.38130			0.215851
$413$	−4.94539			−0.243347
$414$		0			0
$415$		−	51.2923i		−	2.51784i
$416$		−	2.27303i		−	0.111444i
$417$		0			0
$418$	−3.37407	+	0.781583i	−0.165031	+	0.0382285i
$419$			9.81904i			0.479691i	0.970811	+	0.239846i	$0.0770969\pi$
$419$			9.81904i			0.479691i	−0.970811	+	0.239846i	$0.922903\pi$
$420$		0			0
$421$	36.8547			1.79619			0.898095	−	0.439802i	$-0.144951\pi$
$421$	36.8547			1.79619			0.898095	+	0.439802i	$0.144951\pi$
$422$		−	1.18091i		−	0.0574858i
$423$		0			0
$424$			3.57996i			0.173858i
$425$	−9.20710			−0.446610
$426$		0			0
$427$	−9.19819			−0.445131
$428$	−9.99912			−0.483326
$429$		0			0
$430$	16.2471			0.783504
$431$	32.4663			1.56385			0.781924	−	0.623374i	$-0.214238\pi$
$431$	32.4663			1.56385			0.781924	+	0.623374i	$0.214238\pi$
$432$		0			0
$433$	−13.4635			−0.647013			−0.323507	−	0.946226i	$-0.604862\pi$
$433$	−13.4635			−0.647013			−0.323507	+	0.946226i	$0.604862\pi$
$434$			2.87098i			0.137812i
$435$		0			0
$436$		−	7.40600i		−	0.354683i
$437$	2.67707			0.128062
$438$		0			0
$439$			20.6063i			0.983487i	0.870740	+	0.491744i	$0.163640\pi$
$439$			20.6063i			0.983487i	−0.870740	+	0.491744i	$0.836360\pi$
$440$	11.1878	−	2.59160i	0.533359	−	0.123549i
$441$		0			0
$442$		−	2.99423i		−	0.142421i
$443$		−	27.8376i		−	1.32260i	−0.750120	−	0.661302i	$-0.770004\pi$
$443$		−	27.8376i		−	1.32260i	0.750120	−	0.661302i	$-0.229996\pi$
$444$		0			0
$445$	−22.4726			−1.06530
$446$	0.848717			0.0401879
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$			29.7028i			1.40176i	0.713278	+	0.700881i	$0.247210\pi$
$449$			29.7028i			1.40176i	−0.713278	+	0.700881i	$0.752790\pi$
$450$		0			0
$451$	−2.52535	−	10.9018i	−0.118914	−	0.513348i
$452$			11.7261i			0.551549i
$453$		0			0
$454$	13.5749			0.637100
$455$			7.87054i			0.368977i
$456$		0			0
$457$		−	28.4162i		−	1.32925i	−0.747175	−	0.664627i	$-0.768590\pi$
$457$		−	28.4162i		−	1.32925i	0.747175	−	0.664627i	$-0.231410\pi$
$458$	−15.5462			−0.726427
$459$		0			0
$460$	−8.87671			−0.413879
$461$	33.9346			1.58049			0.790247	−	0.612789i	$-0.209952\pi$
$461$	33.9346			1.58049			0.790247	+	0.612789i	$0.209952\pi$
$462$		0			0
$463$	19.7642			0.918522			0.459261	−	0.888301i	$-0.348114\pi$
$463$	19.7642			0.918522			0.459261	+	0.888301i	$0.348114\pi$
$464$	2.36395			0.109744
$465$		0			0
$466$	−6.30344			−0.292001
$467$		−	7.14096i		−	0.330444i	−0.986256	−	0.165222i	$-0.947166\pi$
$467$		−	7.14096i		−	0.330444i	0.986256	−	0.165222i	$-0.0528341\pi$
$468$		0			0
$469$			5.35487i			0.247265i
$470$	44.4807			2.05174
$471$		0			0
$472$			4.94539i			0.227630i
$473$	−15.1608	+	3.51191i	−0.697094	+	0.161478i
$474$		0			0
$475$			7.29877i			0.334891i
$476$		−	1.31729i		−	0.0603777i
$477$		0			0
$478$	−29.2292			−1.33691
$479$	−27.5712			−1.25976			−0.629879	−	0.776693i	$-0.716896\pi$
$479$	−27.5712			−1.25976			−0.629879	+	0.776693i	$0.716896\pi$
$480$		0			0
$481$		−	13.3439i		−	0.608430i
$482$			24.6488i			1.12272i
$483$		0			0
$484$	−9.87962	+	4.83664i	−0.449074	+	0.219847i
$485$			35.0811i			1.59295i
$486$		0			0
$487$	−21.9780			−0.995919			−0.497959	−	0.867200i	$-0.665917\pi$
$487$	−21.9780			−0.995919			−0.497959	+	0.867200i	$0.665917\pi$
$488$			9.19819i			0.416382i
$489$		0			0
$490$			3.46258i			0.156423i
$491$	−13.5474			−0.611386			−0.305693	−	0.952130i	$-0.598888\pi$
$491$	−13.5474			−0.611386			−0.305693	+	0.952130i	$0.598888\pi$
$492$		0			0
$493$	3.11400			0.140247
$494$	−2.37363			−0.106794
$495$		0			0
$496$	2.87098			0.128911
$497$	−4.36154			−0.195642
$498$		0			0
$499$	12.1709			0.544844			0.272422	−	0.962178i	$-0.412175\pi$
$499$	12.1709			0.544844			0.272422	+	0.962178i	$0.412175\pi$
$500$		−	6.88861i		−	0.308068i
$501$		0			0
$502$			23.6799i			1.05689i
$503$	−15.4555			−0.689125			−0.344563	−	0.938763i	$-0.611973\pi$
$503$	−15.4555			−0.689125			−0.344563	+	0.938763i	$0.611973\pi$
$504$		0			0
$505$		−	49.6196i		−	2.20804i
$506$	8.28321	−	1.91876i	0.368234	−	0.0852992i
$507$		0			0
$508$			20.6280i			0.915217i
$509$		−	12.8104i		−	0.567812i	−0.958852	−	0.283906i	$-0.908370\pi$
$509$		−	12.8104i		−	0.567812i	0.958852	−	0.283906i	$-0.0916303\pi$
$510$		0			0
$511$	−14.2217			−0.629132
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			20.2929i			0.895080i
$515$			15.1706i			0.668497i
$516$		0			0
$517$	−41.5067	+	9.61479i	−1.82546	+	0.422858i
$518$		−	5.87054i		−	0.257937i
$519$		0			0
$520$	7.87054			0.345146
$521$		−	2.11230i		−	0.0925416i	−0.998929	−	0.0462708i	$-0.985266\pi$
$521$		−	2.11230i		−	0.0925416i	0.998929	−	0.0462708i	$-0.0147337\pi$
$522$		0			0
$523$		−	22.3355i		−	0.976661i	−0.872659	−	0.488331i	$-0.837606\pi$
$523$		−	22.3355i		−	0.976661i	0.872659	−	0.488331i	$-0.162394\pi$
$524$	0.323452			0.0141301
$525$		0			0
$526$	−8.70007			−0.379341
$527$	3.78191			0.164742
$528$		0			0
$529$	16.4279			0.714256
$530$	−12.3959			−0.538443
$531$		0			0
$532$	−1.04426			−0.0452743
$533$		−	7.66935i		−	0.332196i
$534$		0			0
$535$		−	34.6227i		−	1.49687i
$536$	5.35487			0.231295
$537$		0			0
$538$			21.9584i			0.946692i
$539$	−0.748459	−	3.23107i	−0.0322384	−	0.139172i
$540$		0			0
$541$		−	13.4500i		−	0.578260i	−0.957290	−	0.289130i	$-0.906634\pi$
$541$		−	13.4500i		−	0.578260i	0.957290	−	0.289130i	$-0.0933660\pi$
$542$		−	9.56013i		−	0.410643i
$543$		0			0
$544$	−1.31729			−0.0564782
$545$	25.6438			1.09846
$546$		0			0
$547$			36.9864i			1.58143i	0.612187	+	0.790713i	$0.290290\pi$
$547$			36.9864i			1.58143i	−0.612187	+	0.790713i	$0.709710\pi$
$548$			9.70127i			0.414418i
$549$		0			0
$550$	5.23131	+	22.5834i	0.223064	+	0.962959i
$551$		−	2.46857i		−	0.105165i
$552$		0			0
$553$	16.8261			0.715518
$554$		−	2.85994i		−	0.121507i
$555$		0			0
$556$			8.40529i			0.356464i
$557$	39.0959			1.65655			0.828273	−	0.560324i	$-0.189323\pi$
$557$	39.0959			1.65655			0.828273	+	0.560324i	$0.189323\pi$
$558$		0			0
$559$	−10.6655			−0.451102
$560$	3.46258			0.146321
$561$		0			0
$562$	−17.2272			−0.726685
$563$	4.57303			0.192730			0.0963651	−	0.995346i	$-0.469278\pi$
$563$	4.57303			0.192730			0.0963651	+	0.995346i	$0.469278\pi$
$564$		0			0
$565$	−40.6025			−1.70816
$566$		−	2.40616i		−	0.101139i
$567$		0			0
$568$			4.36154i			0.183006i
$569$	−16.5976			−0.695806			−0.347903	−	0.937531i	$-0.613106\pi$
$569$	−16.5976			−0.695806			−0.347903	+	0.937531i	$0.613106\pi$
$570$		0			0
$571$			6.09248i			0.254962i	0.991841	+	0.127481i	$0.0406892\pi$
$571$			6.09248i			0.254962i	−0.991841	+	0.127481i	$0.959311\pi$
$572$	−7.34432	+	1.70127i	−0.307081	+	0.0711336i
$573$		0			0
$574$		−	3.37407i		−	0.140831i
$575$		−	17.9182i		−	0.747242i
$576$		0			0
$577$	−38.8770			−1.61847			−0.809235	−	0.587485i	$-0.800118\pi$
$577$	−38.8770			−1.61847			−0.809235	+	0.587485i	$0.800118\pi$
$578$	15.2648			0.634930
$579$		0			0
$580$			8.18536i			0.339878i
$581$			14.8133i			0.614560i
$582$		0			0
$583$	11.5671	−	2.67945i	0.479060	−	0.110972i
$584$			14.2217i			0.588499i
$585$		0			0
$586$	−26.7109			−1.10342
$587$			15.0097i			0.619517i	0.950815	+	0.309758i	$0.100248\pi$
$587$			15.0097i			0.619517i	−0.950815	+	0.309758i	$0.899752\pi$
$588$		0			0
$589$		−	2.99804i		−	0.123532i
$590$	−17.1238			−0.704975
$591$		0			0
$592$	−5.87054			−0.241278
$593$	18.2288			0.748567			0.374283	−	0.927314i	$-0.377889\pi$
$593$	18.2288			0.748567			0.374283	+	0.927314i	$0.377889\pi$
$594$		0			0
$595$	4.56121			0.186991
$596$	−10.8796			−0.445647
$597$		0			0
$598$	5.82717			0.238291
$599$		−	4.70809i		−	0.192367i	−0.995364	−	0.0961837i	$-0.969336\pi$
$599$		−	4.70809i		−	0.192367i	0.995364	−	0.0961837i	$-0.0306636\pi$
$600$		0			0
$601$		−	10.5638i		−	0.430904i	−0.976514	−	0.215452i	$-0.930877\pi$
$601$		−	10.5638i		−	0.430904i	0.976514	−	0.215452i	$-0.0691225\pi$
$602$	−4.69219			−0.191239
$603$		0			0
$604$		−	0.611828i		−	0.0248949i
$605$	−16.7473	−	34.2090i	−0.680873	−	1.39079i
$606$		0			0
$607$		−	15.8974i		−	0.645256i	−0.946526	−	0.322628i	$-0.895434\pi$
$607$		−	15.8974i		−	0.645256i	0.946526	−	0.322628i	$-0.104566\pi$
$608$			1.04426i			0.0423502i
$609$		0			0
$610$	−31.8494			−1.28955
$611$	−29.1996			−1.18129
$612$		0			0
$613$			15.6877i			0.633619i	0.948489	+	0.316810i	$0.102612\pi$
$613$			15.6877i			0.633619i	−0.948489	+	0.316810i	$0.897388\pi$
$614$		−	1.98925i		−	0.0802795i
$615$		0			0
$616$	−3.23107	+	0.748459i	−0.130183	+	0.0301563i
$617$			39.8573i			1.60459i	0.596924	+	0.802297i	$0.296389\pi$
$617$			39.8573i			1.60459i	−0.596924	+	0.802297i	$0.703611\pi$
$618$		0			0
$619$	10.5782			0.425174			0.212587	−	0.977142i	$-0.431811\pi$
$619$	10.5782			0.425174			0.212587	+	0.977142i	$0.431811\pi$
$620$			9.94100i			0.399240i
$621$		0			0
$622$		−	4.10821i		−	0.164724i
$623$	6.49012			0.260021
$624$		0			0
$625$	−11.0949			−0.443795
$626$	−19.1917			−0.767055
$627$		0			0
$628$	9.65728			0.385367
$629$	−7.73319			−0.308342
$630$		0			0
$631$	−40.7167			−1.62091			−0.810454	−	0.585803i	$-0.800779\pi$
$631$	−40.7167			−1.62091			−0.810454	+	0.585803i	$0.800779\pi$
$632$		−	16.8261i		−	0.669306i
$633$		0			0
$634$		−	24.0173i		−	0.953848i
$635$	−71.4259			−2.83445
$636$		0			0
$637$		−	2.27303i		−	0.0900607i
$638$	−1.76932	−	7.63808i	−0.0700480	−	0.302395i
$639$		0			0
$640$		−	3.46258i		−	0.136870i
$641$		−	19.0991i		−	0.754370i	−0.926138	−	0.377185i	$-0.876892\pi$
$641$		−	19.0991i		−	0.754370i	0.926138	−	0.377185i	$-0.123108\pi$
$642$		0			0
$643$	−34.6786			−1.36759			−0.683794	−	0.729675i	$-0.739671\pi$
$643$	−34.6786			−1.36759			−0.683794	+	0.729675i	$0.739671\pi$
$644$	2.56361			0.101020
$645$		0			0
$646$			1.37558i			0.0541217i
$647$			46.4006i			1.82419i	0.409974	+	0.912097i	$0.365538\pi$
$647$			46.4006i			1.82419i	−0.409974	+	0.912097i	$0.634462\pi$
$648$		0			0
$649$	15.9789	−	3.70142i	0.627226	−	0.145293i
$650$			15.8872i			0.623148i
$651$		0			0
$652$	10.6375			0.416596
$653$			27.5505i			1.07813i	0.842263	+	0.539066i	$0.181223\pi$
$653$			27.5505i			1.07813i	−0.842263	+	0.539066i	$0.818777\pi$
$654$		0			0
$655$			1.11998i			0.0437612i
$656$	−3.37407			−0.131735
$657$		0			0
$658$	−12.8461			−0.500794
$659$	−32.8770			−1.28071			−0.640353	−	0.768081i	$-0.721212\pi$
$659$	−32.8770			−1.28071			−0.640353	+	0.768081i	$0.721212\pi$
$660$		0			0
$661$	34.2271			1.33128			0.665639	−	0.746274i	$-0.268159\pi$
$661$	34.2271			1.33128			0.665639	+	0.746274i	$0.268159\pi$
$662$	−31.2497			−1.21455
$663$		0			0
$664$	14.8133			0.574869
$665$		−	3.61582i		−	0.140215i
$666$		0			0
$667$			6.06025i			0.234654i
$668$	20.2471			0.783383
$669$		0			0
$670$			18.5417i			0.716327i
$671$	29.7200	−	6.88446i	1.14733	−	0.265772i
$672$		0			0
$673$			40.7662i			1.57142i	0.618594	+	0.785711i	$0.287703\pi$
$673$			40.7662i			1.57142i	−0.618594	+	0.785711i	$0.712297\pi$
$674$		−	6.50551i		−	0.250583i
$675$		0			0
$676$	7.83333			0.301282
$677$	−0.369838			−0.0142140			−0.00710701	−	0.999975i	$-0.502262\pi$
$677$	−0.369838			−0.0142140			−0.00710701	+	0.999975i	$0.502262\pi$
$678$		0			0
$679$		−	10.1315i		−	0.388811i
$680$		−	4.56121i		−	0.174914i
$681$		0			0
$682$	−2.14881	−	9.27634i	−0.0822822	−	0.355210i
$683$		−	2.23908i		−	0.0856762i	−0.999082	−	0.0428381i	$-0.986360\pi$
$683$		−	2.23908i		−	0.0856762i	0.999082	−	0.0428381i	$-0.0136400\pi$
$684$		0			0
$685$	−33.5914			−1.28346
$686$		−	1.00000i		−	0.0381802i
$687$		0			0
$688$			4.69219i			0.178888i
$689$	8.13736			0.310009
$690$		0			0
$691$	−11.4747			−0.436516			−0.218258	−	0.975891i	$-0.570037\pi$
$691$	−11.4747			−0.436516			−0.218258	+	0.975891i	$0.570037\pi$
$692$	−5.62550			−0.213849
$693$		0			0
$694$	25.1628			0.955168
$695$	−29.1040			−1.10398
$696$		0			0
$697$	−4.44461			−0.168352
$698$			12.5834i			0.476291i
$699$		0			0
$700$			6.98945i			0.264176i
$701$	31.9915			1.20830			0.604151	−	0.796870i	$-0.293512\pi$
$701$	31.9915			1.20830			0.604151	+	0.796870i	$0.293512\pi$
$702$		0			0
$703$			6.13035i			0.231211i
$704$	0.748459	+	3.23107i	0.0282086	+	0.121776i
$705$		0			0
$706$			13.1792i			0.496004i
$707$			14.3303i			0.538945i
$708$		0			0
$709$	33.3144			1.25115			0.625575	−	0.780164i	$-0.284864\pi$
$709$	33.3144			1.25115			0.625575	+	0.780164i	$0.284864\pi$
$710$	−15.1022			−0.566775
$711$		0			0
$712$		−	6.49012i		−	0.243228i
$713$			7.36009i			0.275637i
$714$		0			0
$715$	−5.89077	−	25.4303i	−0.220302	−	0.951038i
$716$			1.56278i			0.0584039i
$717$		0			0
$718$	−17.9186			−0.668716
$719$		−	6.62847i		−	0.247200i	−0.992332	−	0.123600i	$-0.960556\pi$
$719$		−	6.62847i		−	0.247200i	0.992332	−	0.123600i	$-0.0394440\pi$
$720$		0			0
$721$		−	4.38130i		−	0.163168i
$722$	−17.9095			−0.666524
$723$		0			0
$724$	−4.49648			−0.167110
$725$	−16.5227			−0.613637
$726$		0			0
$727$	−2.80155			−0.103904			−0.0519520	−	0.998650i	$-0.516544\pi$
$727$	−2.80155			−0.103904			−0.0519520	+	0.998650i	$0.516544\pi$
$728$	−2.27303			−0.0842441
$729$		0			0
$730$	−49.2438			−1.82260
$731$			6.18096i			0.228611i
$732$		0			0
$733$			16.7844i			0.619945i	0.950746	+	0.309972i	$0.100320\pi$
$733$			16.7844i			0.619945i	−0.950746	+	0.309972i	$0.899680\pi$
$734$	5.09764			0.188157
$735$		0			0
$736$		−	2.56361i		−	0.0944960i
$737$	−4.00790	−	17.3020i	−0.147633	−	0.637326i
$738$		0			0
$739$			16.2365i			0.597269i	0.954368	+	0.298635i	$0.0965312\pi$
$739$			16.2365i			0.597269i	−0.954368	+	0.298635i	$0.903469\pi$
$740$		−	20.3272i		−	0.747243i
$741$		0			0
$742$	3.57996			0.131424
$743$	38.8547			1.42544			0.712721	−	0.701448i	$-0.247463\pi$
$743$	38.8547			1.42544			0.712721	+	0.701448i	$0.247463\pi$
$744$		0			0
$745$		−	37.6715i		−	1.38018i
$746$			23.7658i			0.870126i
$747$		0			0
$748$	0.985934	+	4.25624i	0.0360493	+	0.155624i
$749$			9.99912i			0.365360i
$750$		0			0
$751$	36.0367			1.31500			0.657500	−	0.753455i	$-0.271614\pi$
$751$	36.0367			1.31500			0.657500	+	0.753455i	$0.271614\pi$
$752$			12.8461i			0.468450i
$753$		0			0
$754$		−	5.37333i		−	0.195685i
$755$	2.11850			0.0771001
$756$		0			0
$757$	−35.4893			−1.28988			−0.644940	−	0.764233i	$-0.723118\pi$
$757$	−35.4893			−1.28988			−0.644940	+	0.764233i	$0.723118\pi$
$758$	−10.5912			−0.384690
$759$		0			0
$760$	−3.61582			−0.131160
$761$	18.9936			0.688517			0.344259	−	0.938875i	$-0.388130\pi$
$761$	18.9936			0.688517			0.344259	+	0.938875i	$0.388130\pi$
$762$		0			0
$763$	−7.40600			−0.268115
$764$		−	12.9828i		−	0.469700i
$765$		0			0
$766$		−	32.0505i		−	1.15803i
$767$	11.2410			0.405889
$768$		0			0
$769$			9.08915i			0.327763i	0.986480	+	0.163881i	$0.0524015\pi$
$769$			9.08915i			0.327763i	−0.986480	+	0.163881i	$0.947599\pi$
$770$	−2.59160	−	11.1878i	−0.0933946	−	0.403181i
$771$		0			0
$772$			16.4642i			0.592559i
$773$			29.4834i			1.06044i	0.847859	+	0.530221i	$0.177891\pi$
$773$			29.4834i			1.06044i	−0.847859	+	0.530221i	$0.822109\pi$
$774$		0			0
$775$	−20.0666			−0.720813
$776$	−10.1315			−0.363699
$777$		0			0
$778$			29.8275i			1.06937i
$779$			3.52339i			0.126239i
$780$		0			0
$781$	14.0924	−	3.26443i	0.504267	−	0.116811i
$782$		−	3.37701i		−	0.120762i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			33.4391i			1.19349i
$786$		0			0
$787$			24.5920i			0.876611i	0.898826	+	0.438305i	$0.144421\pi$
$787$			24.5920i			0.876611i	−0.898826	+	0.438305i	$0.855579\pi$
$788$	−19.6756			−0.700914
$789$		0			0
$790$	58.2616			2.07286
$791$	11.7261			0.416932
$792$		0			0
$793$	20.9078			0.742456
$794$	−15.4569			−0.548545
$795$		0			0
$796$	0.145994			0.00517464
$797$			17.1948i			0.609071i	0.952501	+	0.304536i	$0.0985013\pi$
$797$			17.1948i			0.609071i	−0.952501	+	0.304536i	$0.901499\pi$
$798$		0			0
$799$			16.9220i			0.598658i
$800$	6.98945			0.247114
$801$		0			0
$802$		−	20.9581i		−	0.740056i
$803$	45.9514	−	10.6444i	1.62159	−	0.375632i
$804$		0			0
$805$			8.87671i			0.312863i
$806$		−	6.52583i		−	0.229862i
$807$		0			0
$808$	14.3303			0.504137
$809$	41.7992			1.46958			0.734791	−	0.678294i	$-0.237280\pi$
$809$	41.7992			1.46958			0.734791	+	0.678294i	$0.237280\pi$
$810$		0			0
$811$		−	23.1697i		−	0.813599i	−0.913518	−	0.406799i	$-0.866645\pi$
$811$		−	23.1697i		−	0.813599i	0.913518	−	0.406799i	$-0.133355\pi$
$812$		−	2.36395i		−	0.0829584i
$813$		0			0
$814$	4.39386	+	18.9681i	0.154005	+	0.664833i
$815$			36.8331i			1.29021i
$816$		0			0
$817$	4.89985			0.171424
$818$		−	26.2918i		−	0.919273i
$819$		0			0
$820$		−	11.6830i		−	0.407987i
$821$	21.2510			0.741665			0.370832	−	0.928700i	$-0.379072\pi$
$821$	21.2510			0.741665			0.370832	+	0.928700i	$0.379072\pi$
$822$		0			0
$823$	18.9574			0.660812			0.330406	−	0.943839i	$-0.392814\pi$
$823$	18.9574			0.660812			0.330406	+	0.943839i	$0.392814\pi$
$824$	−4.38130			−0.152630
$825$		0			0
$826$	4.94539			0.172072
$827$	22.5089			0.782711			0.391356	−	0.920240i	$-0.372006\pi$
$827$	22.5089			0.782711			0.391356	+	0.920240i	$0.372006\pi$
$828$		0			0
$829$	−21.5978			−0.750122			−0.375061	−	0.927000i	$-0.622378\pi$
$829$	−21.5978			−0.750122			−0.375061	+	0.927000i	$0.622378\pi$
$830$			51.2923i			1.78038i
$831$		0			0
$832$			2.27303i			0.0788031i
$833$	−1.31729			−0.0456413
$834$		0			0
$835$			70.1071i			2.42616i
$836$	3.37407	−	0.781583i	0.116694	−	0.0270316i
$837$		0			0
$838$		−	9.81904i		−	0.339193i
$839$		−	5.88914i		−	0.203316i	−0.994819	−	0.101658i	$-0.967585\pi$
$839$		−	5.88914i		−	0.203316i	0.994819	−	0.101658i	$-0.0324147\pi$
$840$		0			0
$841$	−23.4117			−0.807302
$842$	−36.8547			−1.27010
$843$		0			0
$844$			1.18091i			0.0406486i
$845$			27.1235i			0.933078i
$846$		0			0
$847$	4.83664	+	9.87962i	0.166189	+	0.339468i
$848$		−	3.57996i		−	0.122936i
$849$		0			0
$850$	9.20710			0.315801
$851$		−	15.0498i		−	0.515900i
$852$		0			0
$853$		−	55.7205i		−	1.90783i	−0.300071	−	0.953917i	$-0.597010\pi$
$853$		−	55.7205i		−	1.90783i	0.300071	−	0.953917i	$-0.402990\pi$
$854$	9.19819			0.314755
$855$		0			0
$856$	9.99912			0.341763
$857$	−32.2125			−1.10036			−0.550179	−	0.835047i	$-0.685440\pi$
$857$	−32.2125			−1.10036			−0.550179	+	0.835047i	$0.685440\pi$
$858$		0			0
$859$	27.3955			0.934724			0.467362	−	0.884066i	$-0.345204\pi$
$859$	27.3955			0.934724			0.467362	+	0.884066i	$0.345204\pi$
$860$	−16.2471			−0.554021
$861$		0			0
$862$	−32.4663			−1.10581
$863$			21.8235i			0.742880i	0.928457	+	0.371440i	$0.121136\pi$
$863$			21.8235i			0.742880i	−0.928457	+	0.371440i	$0.878864\pi$
$864$		0			0
$865$		−	19.4787i		−	0.662296i
$866$	13.4635			0.457508
$867$		0			0
$868$		−	2.87098i		−	0.0974475i
$869$	−54.3663	+	12.5936i	−1.84425	+	0.427210i
$870$		0			0
$871$		−	12.1718i		−	0.412425i
$872$			7.40600i			0.250799i
$873$		0			0
$874$	−2.67707			−0.0905532
$875$	−6.88861			−0.232877
$876$		0			0
$877$			21.3061i			0.719458i	0.933057	+	0.359729i	$0.117131\pi$
$877$			21.3061i			0.719458i	−0.933057	+	0.359729i	$0.882869\pi$
$878$		−	20.6063i		−	0.695430i
$879$		0			0
$880$	−11.1878	+	2.59160i	−0.377142	+	0.0873627i
$881$			5.80668i			0.195632i	0.995205	+	0.0978160i	$0.0311857\pi$
$881$			5.80668i			0.195632i	−0.995205	+	0.0978160i	$0.968814\pi$
$882$		0			0
$883$	−45.2333			−1.52222			−0.761111	−	0.648622i	$-0.775346\pi$
$883$	−45.2333			−1.52222			−0.761111	+	0.648622i	$0.775346\pi$
$884$			2.99423i			0.100707i
$885$		0			0
$886$			27.8376i			0.935222i
$887$	30.3797			1.02005			0.510026	−	0.860159i	$-0.329636\pi$
$887$	30.3797			1.02005			0.510026	+	0.860159i	$0.329636\pi$
$888$		0			0
$889$	20.6280			0.691839
$890$	22.4726			0.753282
$891$		0			0
$892$	−0.848717			−0.0284172
$893$	13.4146			0.448904
$894$		0			0
$895$	−5.41125			−0.180878
$896$			1.00000i			0.0334077i
$897$		0			0
$898$		−	29.7028i		−	0.991196i
$899$	6.78686			0.226354
$900$		0			0
$901$		−	4.71583i		−	0.157107i
$902$	2.52535	+	10.9018i	0.0840849	+	0.362992i
$903$		0			0
$904$		−	11.7261i		−	0.390004i
$905$		−	15.5694i		−	0.517545i
$906$		0			0
$907$	22.3458			0.741978			0.370989	−	0.928637i	$-0.379019\pi$
$907$	22.3458			0.741978			0.370989	+	0.928637i	$0.379019\pi$
$908$	−13.5749			−0.450498
$909$		0			0
$910$		−	7.87054i		−	0.260906i
$911$			50.1528i			1.66164i	0.556543	+	0.830819i	$0.312128\pi$
$911$			50.1528i			1.66164i	−0.556543	+	0.830819i	$0.687872\pi$
$912$		0			0
$913$	−11.0872	−	47.8629i	−0.366931	−	1.58403i
$914$			28.4162i			0.939925i
$915$		0			0
$916$	15.5462			0.513661
$917$		−	0.323452i		−	0.0106813i
$918$		0			0
$919$			59.7758i			1.97182i	0.167270	+	0.985911i	$0.446505\pi$
$919$			59.7758i			1.97182i	−0.167270	+	0.985911i	$0.553495\pi$
$920$	8.87671			0.292656
$921$		0			0
$922$	−33.9346			−1.11758
$923$	9.91392			0.326321
$924$		0			0
$925$	41.0318			1.34912
$926$	−19.7642			−0.649493
$927$		0			0
$928$	−2.36395			−0.0776004
$929$		−	26.9018i		−	0.882618i	−0.897355	−	0.441309i	$-0.854514\pi$
$929$		−	26.9018i		−	0.882618i	0.897355	−	0.441309i	$-0.145486\pi$
$930$		0			0
$931$			1.04426i			0.0342241i
$932$	6.30344			0.206476
$933$		0			0
$934$			7.14096i			0.233659i
$935$	−14.7376	+	3.41387i	−0.481970	+	0.111646i
$936$		0			0
$937$			11.5506i			0.377343i	0.982040	+	0.188672i	$0.0604182\pi$
$937$			11.5506i			0.377343i	−0.982040	+	0.188672i	$0.939582\pi$
$938$		−	5.35487i		−	0.174843i
$939$		0			0
$940$	−44.4807			−1.45080
$941$	33.0556			1.07758			0.538792	−	0.842439i	$-0.318881\pi$
$941$	33.0556			1.07758			0.538792	+	0.842439i	$0.318881\pi$
$942$		0			0
$943$		−	8.64980i		−	0.281676i
$944$		−	4.94539i		−	0.160959i
$945$		0			0
$946$	15.1608	−	3.51191i	0.492920	−	0.114182i
$947$		−	8.30344i		−	0.269826i	−0.990857	−	0.134913i	$-0.956925\pi$
$947$		−	8.30344i		−	0.269826i	0.990857	−	0.134913i	$-0.0430754\pi$
$948$		0			0
$949$	32.3264			1.04936
$950$		−	7.29877i		−	0.236803i
$951$		0			0
$952$			1.31729i			0.0426935i
$953$	−54.9764			−1.78086			−0.890429	−	0.455121i	$-0.849596\pi$
$953$	−54.9764			−1.78086			−0.890429	+	0.455121i	$0.849596\pi$
$954$		0			0
$955$	44.9539			1.45467
$956$	29.2292			0.945341
$957$		0			0
$958$	27.5712			0.890784
$959$	9.70127			0.313270
$960$		0			0
$961$	−22.7575			−0.734112
$962$			13.3439i			0.430225i
$963$		0			0
$964$		−	24.6488i		−	0.793885i
$965$	−57.0085			−1.83517
$966$		0			0
$967$		−	27.2902i		−	0.877593i	−0.898587	−	0.438796i	$-0.855405\pi$
$967$		−	27.2902i		−	0.877593i	0.898587	−	0.438796i	$-0.144595\pi$
$968$	9.87962	−	4.83664i	0.317543	−	0.155456i
$969$		0			0
$970$		−	35.0811i		−	1.12639i
$971$			10.4192i			0.334367i	0.985926	+	0.167183i	$0.0534672\pi$
$971$			10.4192i			0.334367i	−0.985926	+	0.167183i	$0.946533\pi$
$972$		0			0
$973$	8.40529			0.269461
$974$	21.9780			0.704221
$975$		0			0
$976$		−	9.19819i		−	0.294427i
$977$		−	6.13036i		−	0.196128i	−0.995180	−	0.0980638i	$-0.968735\pi$
$977$		−	6.13036i		−	0.196128i	0.995180	−	0.0980638i	$-0.0312649\pi$
$978$		0			0
$979$	−20.9700	+	4.85759i	−0.670205	+	0.155249i
$980$		−	3.46258i		−	0.110608i
$981$		0			0
$982$	13.5474			0.432315
$983$		−	19.1358i		−	0.610338i	−0.952298	−	0.305169i	$-0.901287\pi$
$983$		−	19.1358i		−	0.610338i	0.952298	−	0.305169i	$-0.0987130\pi$
$984$		0			0
$985$		−	68.1283i		−	2.17075i
$986$	−3.11400			−0.0991699
$987$		0			0
$988$	2.37363			0.0755151
$989$	−12.0290			−0.382499
$990$		0			0
$991$	−3.46565			−0.110090			−0.0550451	−	0.998484i	$-0.517530\pi$
$991$	−3.46565			−0.110090			−0.0550451	+	0.998484i	$0.517530\pi$
$992$	−2.87098			−0.0911538
$993$		0			0
$994$	4.36154			0.138340
$995$			0.505517i			0.0160260i
$996$		0			0
$997$		−	38.3027i		−	1.21306i	−0.795061	−	0.606529i	$-0.792561\pi$
$997$		−	38.3027i		−	1.21306i	0.795061	−	0.606529i	$-0.207439\pi$
$998$	−12.1709			−0.385263
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.c.a.197.12	yes	12
3.2	odd	2		1386.2.c.b.197.1	yes	12
11.10	odd	2		1386.2.c.b.197.12	yes	12
33.32	even	2	inner	1386.2.c.a.197.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.c.a.197.1	✓	12	33.32	even	2	inner
1386.2.c.a.197.12	yes	12	1.1	even	1	trivial
1386.2.c.b.197.1	yes	12	3.2	odd	2
1386.2.c.b.197.12	yes	12	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.46258i q^{5} -1.00000i q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.46258i q^{5} -1.00000i q^{7} -1.00000 q^{8} -3.46258i q^{10} +(0.748459 + 3.23107i) q^{11} +2.27303i q^{13} +1.00000i q^{14} +1.00000 q^{16} +1.31729 q^{17} -1.04426i q^{19} +3.46258i q^{20} +(-0.748459 - 3.23107i) q^{22} +2.56361i q^{23} -6.98945 q^{25} -2.27303i q^{26} -1.00000i q^{28} +2.36395 q^{29} +2.87098 q^{31} -1.00000 q^{32} -1.31729 q^{34} +3.46258 q^{35} -5.87054 q^{37} +1.04426i q^{38} -3.46258i q^{40} -3.37407 q^{41} +4.69219i q^{43} +(0.748459 + 3.23107i) q^{44} -2.56361i q^{46} +12.8461i q^{47} -1.00000 q^{49} +6.98945 q^{50} +2.27303i q^{52} -3.57996i q^{53} +(-11.1878 + 2.59160i) q^{55} +1.00000i q^{56} -2.36395 q^{58} -4.94539i q^{59} -9.19819i q^{61} -2.87098 q^{62} +1.00000 q^{64} -7.87054 q^{65} -5.35487 q^{67} +1.31729 q^{68} -3.46258 q^{70} -4.36154i q^{71} -14.2217i q^{73} +5.87054 q^{74} -1.04426i q^{76} +(3.23107 - 0.748459i) q^{77} +16.8261i q^{79} +3.46258i q^{80} +3.37407 q^{82} -14.8133 q^{83} +4.56121i q^{85} -4.69219i q^{86} +(-0.748459 - 3.23107i) q^{88} +6.49012i q^{89} +2.27303 q^{91} +2.56361i q^{92} -12.8461i q^{94} +3.61582 q^{95} +10.1315 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) 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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