LMFDB - Embedded newform 966.2.g.c.643.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(643,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("966.643");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.g (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:	$7.71354883526$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			643.3
Root			$1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	966.643
Dual form			966.2.g.c.643.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.64575i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.64575i q^{7} -1.00000 q^{8} -1.00000 q^{9} +5.29150i q^{11} +1.00000i q^{12} -2.00000i q^{13} +2.64575i q^{14} +1.00000 q^{16} -5.29150 q^{17} +1.00000 q^{18} -5.29150 q^{19} +2.64575 q^{21} -5.29150i q^{22} +(4.00000 + 2.64575i) q^{23} -1.00000i q^{24} -5.00000 q^{25} +2.00000i q^{26} -1.00000i q^{27} -2.64575i q^{28} +6.00000 q^{29} +10.0000i q^{31} -1.00000 q^{32} -5.29150 q^{33} +5.29150 q^{34} -1.00000 q^{36} +5.29150 q^{38} +2.00000 q^{39} +12.0000i q^{41} -2.64575 q^{42} +5.29150i q^{43} +5.29150i q^{44} +(-4.00000 - 2.64575i) q^{46} -6.00000i q^{47} +1.00000i q^{48} -7.00000 q^{49} +5.00000 q^{50} -5.29150i q^{51} -2.00000i q^{52} +1.00000i q^{54} +2.64575i q^{56} -5.29150i q^{57} -6.00000 q^{58} -10.5830 q^{61} -10.0000i q^{62} +2.64575i q^{63} +1.00000 q^{64} +5.29150 q^{66} -5.29150i q^{67} -5.29150 q^{68} +(-2.64575 + 4.00000i) q^{69} -8.00000 q^{71} +1.00000 q^{72} +4.00000i q^{73} -5.00000i q^{75} -5.29150 q^{76} +14.0000 q^{77} -2.00000 q^{78} -5.29150i q^{79} +1.00000 q^{81} -12.0000i q^{82} -15.8745 q^{83} +2.64575 q^{84} -5.29150i q^{86} +6.00000i q^{87} -5.29150i q^{88} -5.29150 q^{89} -5.29150 q^{91} +(4.00000 + 2.64575i) q^{92} -10.0000 q^{93} +6.00000i q^{94} -1.00000i q^{96} -5.29150 q^{97} +7.00000 q^{98} -5.29150i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9} + 4 q^{16} + 4 q^{18} + 16 q^{23} - 20 q^{25} + 24 q^{29} - 4 q^{32} - 4 q^{36} + 8 q^{39} - 16 q^{46} - 28 q^{49} + 20 q^{50} - 24 q^{58} + 4 q^{64} - 32 q^{71} + 4 q^{72} + 56 q^{77} - 8 q^{78} + 4 q^{81} + 16 q^{92} - 40 q^{93} + 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9 + 4 * q^16 + 4 * q^18 + 16 * q^23 - 20 * q^25 + 24 * q^29 - 4 * q^32 - 4 * q^36 + 8 * q^39 - 16 * q^46 - 28 * q^49 + 20 * q^50 - 24 * q^58 + 4 * q^64 - 32 * q^71 + 4 * q^72 + 56 * q^77 - 8 * q^78 + 4 * q^81 + 16 * q^92 - 40 * q^93 + 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$.

$n$	$323$	$829$	$925$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.00000			0.500000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		−	1.00000i		−	0.408248i
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$	−1.00000			−0.353553
$9$	−1.00000			−0.333333
$10$		0			0
$11$			5.29150i			1.59545i	0.603023	+	0.797724i	$0.293963\pi$
$11$			5.29150i			1.59545i	−0.603023	+	0.797724i	$0.706037\pi$
$12$			1.00000i			0.288675i
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$			2.64575i			0.707107i
$15$		0			0
$16$	1.00000			0.250000
$17$	−5.29150			−1.28338			−0.641689	−	0.766965i	$-0.721766\pi$
$17$	−5.29150			−1.28338			−0.641689	+	0.766965i	$0.721766\pi$
$18$	1.00000			0.235702
$19$	−5.29150			−1.21395			−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150			−1.21395			−0.606977	+	0.794719i	$0.707618\pi$
$20$		0			0
$21$	2.64575			0.577350
$22$		−	5.29150i		−	1.12815i
$23$	4.00000	+	2.64575i	0.834058	+	0.551677i
$23$	4.00000	+	2.64575i	0.834058	+	0.551677i
$24$		−	1.00000i		−	0.204124i
$25$	−5.00000			−1.00000
$26$			2.00000i			0.392232i
$27$		−	1.00000i		−	0.192450i
$28$		−	2.64575i		−	0.500000i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$	−1.00000			−0.176777
$33$	−5.29150			−0.921132
$34$	5.29150			0.907485
$35$		0			0
$36$	−1.00000			−0.166667
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	5.29150			0.858395
$39$	2.00000			0.320256
$40$		0			0
$41$			12.0000i			1.87409i	0.349215	+	0.937043i	$0.386448\pi$
$41$			12.0000i			1.87409i	−0.349215	+	0.937043i	$0.613552\pi$
$42$	−2.64575			−0.408248
$43$			5.29150i			0.806947i	0.914991	+	0.403473i	$0.132197\pi$
$43$			5.29150i			0.806947i	−0.914991	+	0.403473i	$0.867803\pi$
$44$			5.29150i			0.797724i
$45$		0			0
$46$	−4.00000	−	2.64575i	−0.589768	−	0.390095i
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$			1.00000i			0.144338i
$49$	−7.00000			−1.00000
$50$	5.00000			0.707107
$51$		−	5.29150i		−	0.740959i
$52$		−	2.00000i		−	0.277350i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$			1.00000i			0.136083i
$55$		0			0
$56$			2.64575i			0.353553i
$57$		−	5.29150i		−	0.700877i
$58$	−6.00000			−0.787839
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−10.5830			−1.35501			−0.677507	−	0.735516i	$-0.736940\pi$
$61$	−10.5830			−1.35501			−0.677507	+	0.735516i	$0.736940\pi$
$62$		−	10.0000i		−	1.27000i
$63$			2.64575i			0.333333i
$64$	1.00000			0.125000
$65$		0			0
$66$	5.29150			0.651339
$67$		−	5.29150i		−	0.646460i	−0.946320	−	0.323230i	$-0.895231\pi$
$67$		−	5.29150i		−	0.646460i	0.946320	−	0.323230i	$-0.104769\pi$
$68$	−5.29150			−0.641689
$69$	−2.64575	+	4.00000i	−0.318511	+	0.481543i
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000			0.117851
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$		−	5.00000i		−	0.577350i
$76$	−5.29150			−0.606977
$77$	14.0000			1.59545
$78$	−2.00000			−0.226455
$79$		−	5.29150i		−	0.595341i	−0.954669	−	0.297670i	$-0.903790\pi$
$79$		−	5.29150i		−	0.595341i	0.954669	−	0.297670i	$-0.0962096\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		−	12.0000i		−	1.32518i
$83$	−15.8745			−1.74245			−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745			−1.74245			−0.871227	+	0.490881i	$0.836675\pi$
$84$	2.64575			0.288675
$85$		0			0
$86$		−	5.29150i		−	0.570597i
$87$			6.00000i			0.643268i
$88$		−	5.29150i		−	0.564076i
$89$	−5.29150			−0.560898			−0.280449	−	0.959869i	$-0.590483\pi$
$89$	−5.29150			−0.560898			−0.280449	+	0.959869i	$0.590483\pi$
$90$		0			0
$91$	−5.29150			−0.554700
$92$	4.00000	+	2.64575i	0.417029	+	0.275839i
$93$	−10.0000			−1.03695
$94$			6.00000i			0.618853i
$95$		0			0
$96$		−	1.00000i		−	0.102062i
$97$	−5.29150			−0.537271			−0.268635	−	0.963242i	$-0.586573\pi$
$97$	−5.29150			−0.537271			−0.268635	+	0.963242i	$0.586573\pi$
$98$	7.00000			0.707107
$99$		−	5.29150i		−	0.531816i
$100$	−5.00000			−0.500000
$101$			14.0000i			1.39305i	0.717532	+	0.696526i	$0.245272\pi$
$101$			14.0000i			1.39305i	−0.717532	+	0.696526i	$0.754728\pi$
$102$			5.29150i			0.523937i
$103$	10.5830			1.04277			0.521387	−	0.853320i	$-0.325415\pi$
$103$	10.5830			1.04277			0.521387	+	0.853320i	$0.325415\pi$
$104$			2.00000i			0.196116i
$105$		0			0
$106$		0			0
$107$			15.8745i			1.53465i	0.641260	+	0.767323i	$0.278412\pi$
$107$			15.8745i			1.53465i	−0.641260	+	0.767323i	$0.721588\pi$
$108$		−	1.00000i		−	0.0962250i
$109$		−	10.5830i		−	1.01367i	−0.862044	−	0.506834i	$-0.830816\pi$
$109$		−	10.5830i		−	1.01367i	0.862044	−	0.506834i	$-0.169184\pi$
$110$		0			0
$111$		0			0
$112$		−	2.64575i		−	0.250000i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$			5.29150i			0.495595i
$115$		0			0
$116$	6.00000			0.557086
$117$			2.00000i			0.184900i
$118$		0			0
$119$			14.0000i			1.28338i
$120$		0			0
$121$	−17.0000			−1.54545
$122$	10.5830			0.958140
$123$	−12.0000			−1.08200
$124$			10.0000i			0.898027i
$125$		0			0
$126$		−	2.64575i		−	0.235702i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000			−0.0883883
$129$	−5.29150			−0.465891
$130$		0			0
$131$		−	8.00000i		−	0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$		−	8.00000i		−	0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	−5.29150			−0.460566
$133$			14.0000i			1.21395i
$134$			5.29150i			0.457116i
$135$		0			0
$136$	5.29150			0.453743
$137$			21.1660i			1.80833i	0.427179	+	0.904167i	$0.359507\pi$
$137$			21.1660i			1.80833i	−0.427179	+	0.904167i	$0.640493\pi$
$138$	2.64575	−	4.00000i	0.225221	−	0.340503i
$139$		−	12.0000i		−	1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$		−	12.0000i		−	1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$	8.00000			0.671345
$143$	10.5830			0.884995
$144$	−1.00000			−0.0833333
$145$		0			0
$146$		−	4.00000i		−	0.331042i
$147$		−	7.00000i		−	0.577350i
$148$		0			0
$149$		−	10.5830i		−	0.866994i	−0.901155	−	0.433497i	$-0.857280\pi$
$149$		−	10.5830i		−	0.866994i	0.901155	−	0.433497i	$-0.142720\pi$
$150$			5.00000i			0.408248i
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$	5.29150			0.429198
$153$	5.29150			0.427793
$154$	−14.0000			−1.12815
$155$		0			0
$156$	2.00000			0.160128
$157$	−10.5830			−0.844616			−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830			−0.844616			−0.422308	+	0.906452i	$0.638780\pi$
$158$			5.29150i			0.420969i
$159$		0			0
$160$		0			0
$161$	7.00000	−	10.5830i	0.551677	−	0.834058i
$162$	−1.00000			−0.0785674
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$			12.0000i			0.937043i
$165$		0			0
$166$	15.8745			1.23210
$167$		−	14.0000i		−	1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$		−	14.0000i		−	1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	−2.64575			−0.204124
$169$	9.00000			0.692308
$170$		0			0
$171$	5.29150			0.404651
$172$			5.29150i			0.403473i
$173$		−	14.0000i		−	1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$		−	14.0000i		−	1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$		−	6.00000i		−	0.454859i
$175$			13.2288i			1.00000i
$176$			5.29150i			0.398862i
$177$		0			0
$178$	5.29150			0.396615
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	5.29150			0.392232
$183$		−	10.5830i		−	0.782318i
$184$	−4.00000	−	2.64575i	−0.294884	−	0.195047i
$185$		0			0
$186$	10.0000			0.733236
$187$		−	28.0000i		−	2.04756i
$188$		−	6.00000i		−	0.437595i
$189$	−2.64575			−0.192450
$190$		0			0
$191$			15.8745i			1.14864i	0.818631	+	0.574320i	$0.194733\pi$
$191$			15.8745i			1.14864i	−0.818631	+	0.574320i	$0.805267\pi$
$192$			1.00000i			0.0721688i
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$	5.29150			0.379908
$195$		0			0
$196$	−7.00000			−0.500000
$197$	−22.0000			−1.56744			−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000			−1.56744			−0.783718	+	0.621117i	$0.786679\pi$
$198$			5.29150i			0.376051i
$199$	21.1660			1.50042			0.750209	−	0.661200i	$-0.229953\pi$
$199$	21.1660			1.50042			0.750209	+	0.661200i	$0.229953\pi$
$200$	5.00000			0.353553
$201$	5.29150			0.373234
$202$		−	14.0000i		−	0.985037i
$203$		−	15.8745i		−	1.11417i
$204$		−	5.29150i		−	0.370479i
$205$		0			0
$206$	−10.5830			−0.737353
$207$	−4.00000	−	2.64575i	−0.278019	−	0.183892i
$208$		−	2.00000i		−	0.138675i
$209$		−	28.0000i		−	1.93680i
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		−	8.00000i		−	0.548151i
$214$		−	15.8745i		−	1.08516i
$215$		0			0
$216$			1.00000i			0.0680414i
$217$	26.4575			1.79605
$218$			10.5830i			0.716772i
$219$	−4.00000			−0.270295
$220$		0			0
$221$			10.5830i			0.711890i
$222$		0			0
$223$			14.0000i			0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$			2.64575i			0.176777i
$225$	5.00000			0.333333
$226$		0			0
$227$	26.4575			1.75605			0.878023	−	0.478618i	$-0.158862\pi$
$227$	26.4575			1.75605			0.878023	+	0.478618i	$0.158862\pi$
$228$		−	5.29150i		−	0.350438i
$229$	−21.1660			−1.39869			−0.699345	−	0.714785i	$-0.746525\pi$
$229$	−21.1660			−1.39869			−0.699345	+	0.714785i	$0.746525\pi$
$230$		0			0
$231$			14.0000i			0.921132i
$232$	−6.00000			−0.393919
$233$	10.0000			0.655122			0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000			0.655122			0.327561	+	0.944830i	$0.393773\pi$
$234$		−	2.00000i		−	0.130744i
$235$		0			0
$236$		0			0
$237$	5.29150			0.343720
$238$		−	14.0000i		−	0.907485i
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	15.8745			1.02257			0.511283	−	0.859412i	$-0.329170\pi$
$241$	15.8745			1.02257			0.511283	+	0.859412i	$0.329170\pi$
$242$	17.0000			1.09280
$243$			1.00000i			0.0641500i
$244$	−10.5830			−0.677507
$245$		0			0
$246$	12.0000			0.765092
$247$			10.5830i			0.673380i
$248$		−	10.0000i		−	0.635001i
$249$		−	15.8745i		−	1.00601i
$250$		0			0
$251$	15.8745			1.00199			0.500995	−	0.865450i	$-0.332967\pi$
$251$	15.8745			1.00199			0.500995	+	0.865450i	$0.332967\pi$
$252$			2.64575i			0.166667i
$253$	−14.0000	+	21.1660i	−0.880172	+	1.33070i
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	5.29150			0.329435
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$			8.00000i			0.494242i
$263$		−	5.29150i		−	0.326288i	−0.986602	−	0.163144i	$-0.947836\pi$
$263$		−	5.29150i		−	0.326288i	0.986602	−	0.163144i	$-0.0521635\pi$
$264$	5.29150			0.325669
$265$		0			0
$266$		−	14.0000i		−	0.858395i
$267$		−	5.29150i		−	0.323835i
$268$		−	5.29150i		−	0.323230i
$269$			18.0000i			1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$			18.0000i			1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$		0			0
$271$		−	14.0000i		−	0.850439i	−0.905090	−	0.425220i	$-0.860197\pi$
$271$		−	14.0000i		−	0.850439i	0.905090	−	0.425220i	$-0.139803\pi$
$272$	−5.29150			−0.320844
$273$		−	5.29150i		−	0.320256i
$274$		−	21.1660i		−	1.27869i
$275$		−	26.4575i		−	1.59545i
$276$	−2.64575	+	4.00000i	−0.159256	+	0.240772i
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$			12.0000i			0.719712i
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−6.00000			−0.357295
$283$	26.4575			1.57274			0.786368	−	0.617758i	$-0.211959\pi$
$283$	26.4575			1.57274			0.786368	+	0.617758i	$0.211959\pi$
$284$	−8.00000			−0.474713
$285$		0			0
$286$	−10.5830			−0.625786
$287$	31.7490			1.87409
$288$	1.00000			0.0589256
$289$	11.0000			0.647059
$290$		0			0
$291$		−	5.29150i		−	0.310193i
$292$			4.00000i			0.234082i
$293$	10.5830			0.618266			0.309133	−	0.951019i	$-0.399961\pi$
$293$	10.5830			0.618266			0.309133	+	0.951019i	$0.399961\pi$
$294$			7.00000i			0.408248i
$295$		0			0
$296$		0			0
$297$	5.29150			0.307044
$298$			10.5830i			0.613057i
$299$	5.29150	−	8.00000i	0.306015	−	0.462652i
$300$		−	5.00000i		−	0.288675i
$301$	14.0000			0.806947
$302$	−16.0000			−0.920697
$303$	−14.0000			−0.804279
$304$	−5.29150			−0.303488
$305$		0			0
$306$	−5.29150			−0.302495
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	14.0000			0.797724
$309$			10.5830i			0.602046i
$310$		0			0
$311$			10.0000i			0.567048i	0.958965	+	0.283524i	$0.0915036\pi$
$311$			10.0000i			0.567048i	−0.958965	+	0.283524i	$0.908496\pi$
$312$	−2.00000			−0.113228
$313$	−5.29150			−0.299093			−0.149547	−	0.988755i	$-0.547781\pi$
$313$	−5.29150			−0.299093			−0.149547	+	0.988755i	$0.547781\pi$
$314$	10.5830			0.597234
$315$		0			0
$316$		−	5.29150i		−	0.297670i
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		0			0
$319$			31.7490i			1.77760i
$320$		0			0
$321$	−15.8745			−0.886029
$322$	−7.00000	+	10.5830i	−0.390095	+	0.589768i
$323$	28.0000			1.55796
$324$	1.00000			0.0555556
$325$			10.0000i			0.554700i
$326$	−24.0000			−1.32924
$327$	10.5830			0.585242
$328$		−	12.0000i		−	0.662589i
$329$	−15.8745			−0.875190
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$	−15.8745			−0.871227
$333$		0			0
$334$			14.0000i			0.766046i
$335$		0			0
$336$	2.64575			0.144338
$337$		−	21.1660i		−	1.15299i	−0.817102	−	0.576493i	$-0.804421\pi$
$337$		−	21.1660i		−	1.15299i	0.817102	−	0.576493i	$-0.195579\pi$
$338$	−9.00000			−0.489535
$339$		0			0
$340$		0			0
$341$	−52.9150			−2.86551
$342$	−5.29150			−0.286132
$343$			18.5203i			1.00000i
$344$		−	5.29150i		−	0.285299i
$345$		0			0
$346$			14.0000i			0.752645i
$347$	−32.0000			−1.71785			−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000			−1.71785			−0.858925	+	0.512101i	$0.828867\pi$
$348$			6.00000i			0.321634i
$349$		−	30.0000i		−	1.60586i	−0.596071	−	0.802932i	$-0.703272\pi$
$349$		−	30.0000i		−	1.60586i	0.596071	−	0.802932i	$-0.296728\pi$
$350$		−	13.2288i		−	0.707107i
$351$	−2.00000			−0.106752
$352$		−	5.29150i		−	0.282038i
$353$		−	4.00000i		−	0.212899i	−0.994318	−	0.106449i	$-0.966052\pi$
$353$		−	4.00000i		−	0.212899i	0.994318	−	0.106449i	$-0.0339482\pi$
$354$		0			0
$355$		0			0
$356$	−5.29150			−0.280449
$357$	−14.0000			−0.740959
$358$	12.0000			0.634220
$359$			5.29150i			0.279275i	0.990203	+	0.139637i	$0.0445937\pi$
$359$			5.29150i			0.279275i	−0.990203	+	0.139637i	$0.955406\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$		−	17.0000i		−	0.892269i
$364$	−5.29150			−0.277350
$365$		0			0
$366$			10.5830i			0.553183i
$367$	21.1660			1.10486			0.552428	−	0.833560i	$-0.313701\pi$
$367$	21.1660			1.10486			0.552428	+	0.833560i	$0.313701\pi$
$368$	4.00000	+	2.64575i	0.208514	+	0.137919i
$369$		−	12.0000i		−	0.624695i
$370$		0			0
$371$		0			0
$372$	−10.0000			−0.518476
$373$		−	21.1660i		−	1.09593i	−0.836500	−	0.547967i	$-0.815402\pi$
$373$		−	21.1660i		−	1.09593i	0.836500	−	0.547967i	$-0.184598\pi$
$374$			28.0000i			1.44785i
$375$		0			0
$376$			6.00000i			0.309426i
$377$		−	12.0000i		−	0.618031i
$378$	2.64575			0.136083
$379$			15.8745i			0.815419i	0.913112	+	0.407709i	$0.133672\pi$
$379$			15.8745i			0.815419i	−0.913112	+	0.407709i	$0.866328\pi$
$380$		0			0
$381$			8.00000i			0.409852i
$382$		−	15.8745i		−	0.812210i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		−	1.00000i		−	0.0510310i
$385$		0			0
$386$	2.00000			0.101797
$387$		−	5.29150i		−	0.268982i
$388$	−5.29150			−0.268635
$389$		−	21.1660i		−	1.07316i	−0.843850	−	0.536580i	$-0.819716\pi$
$389$		−	21.1660i		−	1.07316i	0.843850	−	0.536580i	$-0.180284\pi$
$390$		0			0
$391$	−21.1660	−	14.0000i	−1.07041	−	0.708010i
$392$	7.00000			0.353553
$393$	8.00000			0.403547
$394$	22.0000			1.10834
$395$		0			0
$396$		−	5.29150i		−	0.265908i
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	−21.1660			−1.06096
$399$	−14.0000			−0.700877
$400$	−5.00000			−0.250000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	−5.29150			−0.263916
$403$	20.0000			0.996271
$404$			14.0000i			0.696526i
$405$		0			0
$406$			15.8745i			0.787839i
$407$		0			0
$408$			5.29150i			0.261968i
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		0			0
$411$	−21.1660			−1.04404
$412$	10.5830			0.521387
$413$		0			0
$414$	4.00000	+	2.64575i	0.196589	+	0.130032i
$415$		0			0
$416$			2.00000i			0.0980581i
$417$	12.0000			0.587643
$418$			28.0000i			1.36952i
$419$	−5.29150			−0.258507			−0.129253	−	0.991612i	$-0.541258\pi$
$419$	−5.29150			−0.258507			−0.129253	+	0.991612i	$0.541258\pi$
$420$		0			0
$421$			31.7490i			1.54735i	0.633581	+	0.773676i	$0.281584\pi$
$421$			31.7490i			1.54735i	−0.633581	+	0.773676i	$0.718416\pi$
$422$	16.0000			0.778868
$423$			6.00000i			0.291730i
$424$		0			0
$425$	26.4575			1.28338
$426$			8.00000i			0.387601i
$427$			28.0000i			1.35501i
$428$			15.8745i			0.767323i
$429$			10.5830i			0.510952i
$430$		0			0
$431$		−	15.8745i		−	0.764648i	−0.924028	−	0.382324i	$-0.875124\pi$
$431$		−	15.8745i		−	0.764648i	0.924028	−	0.382324i	$-0.124876\pi$
$432$		−	1.00000i		−	0.0481125i
$433$	−37.0405			−1.78005			−0.890027	−	0.455908i	$-0.849315\pi$
$433$	−37.0405			−1.78005			−0.890027	+	0.455908i	$0.849315\pi$
$434$	−26.4575			−1.27000
$435$		0			0
$436$		−	10.5830i		−	0.506834i
$437$	−21.1660	−	14.0000i	−1.01251	−	0.669711i
$438$	4.00000			0.191127
$439$		−	6.00000i		−	0.286364i	−0.989696	−	0.143182i	$-0.954267\pi$
$439$		−	6.00000i		−	0.286364i	0.989696	−	0.143182i	$-0.0457335\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		−	10.5830i		−	0.503382i
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$		0			0
$446$		−	14.0000i		−	0.662919i
$447$	10.5830			0.500559
$448$		−	2.64575i		−	0.125000i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−5.00000			−0.235702
$451$	−63.4980			−2.99001
$452$		0			0
$453$			16.0000i			0.751746i
$454$	−26.4575			−1.24171
$455$		0			0
$456$			5.29150i			0.247797i
$457$			21.1660i			0.990104i	0.868863	+	0.495052i	$0.164851\pi$
$457$			21.1660i			0.990104i	−0.868863	+	0.495052i	$0.835149\pi$
$458$	21.1660			0.989023
$459$			5.29150i			0.246986i
$460$		0			0
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		−	14.0000i		−	0.651339i
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$	6.00000			0.278543
$465$		0			0
$466$	−10.0000			−0.463241
$467$	15.8745			0.734585			0.367292	−	0.930106i	$-0.380285\pi$
$467$	15.8745			0.734585			0.367292	+	0.930106i	$0.380285\pi$
$468$			2.00000i			0.0924500i
$469$	−14.0000			−0.646460
$470$		0			0
$471$		−	10.5830i		−	0.487639i
$472$		0			0
$473$	−28.0000			−1.28744
$474$	−5.29150			−0.243047
$475$	26.4575			1.21395
$476$			14.0000i			0.641689i
$477$		0			0
$478$	−8.00000			−0.365911
$479$	−21.1660			−0.967100			−0.483550	−	0.875317i	$-0.660653\pi$
$479$	−21.1660			−0.967100			−0.483550	+	0.875317i	$0.660653\pi$
$480$		0			0
$481$		0			0
$482$	−15.8745			−0.723064
$483$	10.5830	+	7.00000i	0.481543	+	0.318511i
$484$	−17.0000			−0.772727
$485$		0			0
$486$		−	1.00000i		−	0.0453609i
$487$	40.0000			1.81257			0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000			1.81257			0.906287	+	0.422664i	$0.138905\pi$
$488$	10.5830			0.479070
$489$			24.0000i			1.08532i
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$	−12.0000			−0.541002
$493$	−31.7490			−1.42990
$494$		−	10.5830i		−	0.476152i
$495$		0			0
$496$			10.0000i			0.449013i
$497$			21.1660i			0.949425i
$498$			15.8745i			0.711354i
$499$	−24.0000			−1.07439			−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000			−1.07439			−0.537194	+	0.843459i	$0.680516\pi$
$500$		0			0
$501$	14.0000			0.625474
$502$	−15.8745			−0.708514
$503$	−31.7490			−1.41562			−0.707809	−	0.706404i	$-0.750316\pi$
$503$	−31.7490			−1.41562			−0.707809	+	0.706404i	$0.750316\pi$
$504$		−	2.64575i		−	0.117851i
$505$		0			0
$506$	14.0000	−	21.1660i	0.622376	−	0.940944i
$507$			9.00000i			0.399704i
$508$	8.00000			0.354943
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$		0			0
$511$	10.5830			0.468165
$512$	−1.00000			−0.0441942
$513$			5.29150i			0.233626i
$514$		−	8.00000i		−	0.352865i
$515$		0			0
$516$	−5.29150			−0.232945
$517$	31.7490			1.39632
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$	37.0405			1.62277			0.811387	−	0.584509i	$-0.198713\pi$
$521$	37.0405			1.62277			0.811387	+	0.584509i	$0.198713\pi$
$522$	6.00000			0.262613
$523$	37.0405			1.61967			0.809834	−	0.586659i	$-0.199557\pi$
$523$	37.0405			1.61967			0.809834	+	0.586659i	$0.199557\pi$
$524$		−	8.00000i		−	0.349482i
$525$	−13.2288			−0.577350
$526$			5.29150i			0.230720i
$527$		−	52.9150i		−	2.30501i
$528$	−5.29150			−0.230283
$529$	9.00000	+	21.1660i	0.391304	+	0.920261i
$530$		0			0
$531$		0			0
$532$			14.0000i			0.606977i
$533$	24.0000			1.03956
$534$			5.29150i			0.228986i
$535$		0			0
$536$			5.29150i			0.228558i
$537$		−	12.0000i		−	0.517838i
$538$		−	18.0000i		−	0.776035i
$539$		−	37.0405i		−	1.59545i
$540$		0			0
$541$	18.0000			0.773880			0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000			0.773880			0.386940	+	0.922105i	$0.373532\pi$
$542$			14.0000i			0.601351i
$543$		0			0
$544$	5.29150			0.226871
$545$		0			0
$546$			5.29150i			0.226455i
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$			21.1660i			0.904167i
$549$	10.5830			0.451672
$550$			26.4575i			1.12815i
$551$	−31.7490			−1.35255
$552$	2.64575	−	4.00000i	0.112611	−	0.170251i
$553$	−14.0000			−0.595341
$554$	−2.00000			−0.0849719
$555$		0			0
$556$		−	12.0000i		−	0.508913i
$557$			31.7490i			1.34525i	0.739984	+	0.672624i	$0.234833\pi$
$557$			31.7490i			1.34525i	−0.739984	+	0.672624i	$0.765167\pi$
$558$			10.0000i			0.423334i
$559$	10.5830			0.447613
$560$		0			0
$561$	28.0000			1.18216
$562$		0			0
$563$	5.29150			0.223010			0.111505	−	0.993764i	$-0.464433\pi$
$563$	5.29150			0.223010			0.111505	+	0.993764i	$0.464433\pi$
$564$	6.00000			0.252646
$565$		0			0
$566$	−26.4575			−1.11209
$567$		−	2.64575i		−	0.111111i
$568$	8.00000			0.335673
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	15.8745i		−	0.664327i	−0.943222	−	0.332164i	$-0.892221\pi$
$571$		−	15.8745i		−	0.664327i	0.943222	−	0.332164i	$-0.107779\pi$
$572$	10.5830			0.442498
$573$	−15.8745			−0.663167
$574$	−31.7490			−1.32518
$575$	−20.0000	−	13.2288i	−0.834058	−	0.551677i
$576$	−1.00000			−0.0416667
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$	−11.0000			−0.457540
$579$		−	2.00000i		−	0.0831172i
$580$		0			0
$581$			42.0000i			1.74245i
$582$			5.29150i			0.219340i
$583$		0			0
$584$		−	4.00000i		−	0.165521i
$585$		0			0
$586$	−10.5830			−0.437180
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		−	7.00000i		−	0.288675i
$589$		−	52.9150i		−	2.18033i
$590$		0			0
$591$		−	22.0000i		−	0.904959i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−5.29150			−0.217113
$595$		0			0
$596$		−	10.5830i		−	0.433497i
$597$			21.1660i			0.866267i
$598$	−5.29150	+	8.00000i	−0.216386	+	0.327144i
$599$	−32.0000			−1.30748			−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000			−1.30748			−0.653742	+	0.756717i	$0.726802\pi$
$600$			5.00000i			0.204124i
$601$			44.0000i			1.79480i	0.441221	+	0.897399i	$0.354546\pi$
$601$			44.0000i			1.79480i	−0.441221	+	0.897399i	$0.645454\pi$
$602$	−14.0000			−0.570597
$603$			5.29150i			0.215487i
$604$	16.0000			0.651031
$605$		0			0
$606$	14.0000			0.568711
$607$		−	42.0000i		−	1.70473i	−0.522949	−	0.852364i	$-0.675168\pi$
$607$		−	42.0000i		−	1.70473i	0.522949	−	0.852364i	$-0.324832\pi$
$608$	5.29150			0.214599
$609$	15.8745			0.643268
$610$		0			0
$611$	−12.0000			−0.485468
$612$	5.29150			0.213896
$613$			10.5830i			0.427444i	0.976895	+	0.213722i	$0.0685586\pi$
$613$			10.5830i			0.427444i	−0.976895	+	0.213722i	$0.931441\pi$
$614$		0			0
$615$		0			0
$616$	−14.0000			−0.564076
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		−	10.5830i		−	0.425711i
$619$	−5.29150			−0.212683			−0.106342	−	0.994330i	$-0.533914\pi$
$619$	−5.29150			−0.212683			−0.106342	+	0.994330i	$0.533914\pi$
$620$		0			0
$621$	2.64575	−	4.00000i	0.106170	−	0.160514i
$622$		−	10.0000i		−	0.400963i
$623$			14.0000i			0.560898i
$624$	2.00000			0.0800641
$625$	25.0000			1.00000
$626$	5.29150			0.211491
$627$	28.0000			1.11821
$628$	−10.5830			−0.422308
$629$		0			0
$630$		0			0
$631$			37.0405i			1.47456i	0.675587	+	0.737280i	$0.263890\pi$
$631$			37.0405i			1.47456i	−0.675587	+	0.737280i	$0.736110\pi$
$632$			5.29150i			0.210485i
$633$		−	16.0000i		−	0.635943i
$634$	−6.00000			−0.238290
$635$		0			0
$636$		0			0
$637$			14.0000i			0.554700i
$638$		−	31.7490i		−	1.25696i
$639$	8.00000			0.316475
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	15.8745			0.626517
$643$	−15.8745			−0.626029			−0.313015	−	0.949748i	$-0.601339\pi$
$643$	−15.8745			−0.626029			−0.313015	+	0.949748i	$0.601339\pi$
$644$	7.00000	−	10.5830i	0.275839	−	0.417029i
$645$		0			0
$646$	−28.0000			−1.10165
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$	−1.00000			−0.0392837
$649$		0			0
$650$		−	10.0000i		−	0.392232i
$651$			26.4575i			1.03695i
$652$	24.0000			0.939913
$653$	−38.0000			−1.48705			−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000			−1.48705			−0.743527	+	0.668705i	$0.766849\pi$
$654$	−10.5830			−0.413828
$655$		0			0
$656$			12.0000i			0.468521i
$657$		−	4.00000i		−	0.156055i
$658$	15.8745			0.618853
$659$			15.8745i			0.618383i	0.951000	+	0.309192i	$0.100058\pi$
$659$			15.8745i			0.618383i	−0.951000	+	0.309192i	$0.899942\pi$
$660$		0			0
$661$	−10.5830			−0.411631			−0.205816	−	0.978591i	$-0.565985\pi$
$661$	−10.5830			−0.411631			−0.205816	+	0.978591i	$0.565985\pi$
$662$	−4.00000			−0.155464
$663$	−10.5830			−0.411010
$664$	15.8745			0.616050
$665$		0			0
$666$		0			0
$667$	24.0000	+	15.8745i	0.929284	+	0.614663i
$668$		−	14.0000i		−	0.541676i
$669$	−14.0000			−0.541271
$670$		0			0
$671$		−	56.0000i		−	2.16186i
$672$	−2.64575			−0.102062
$673$	−22.0000			−0.848038			−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000			−0.848038			−0.424019	+	0.905653i	$0.639381\pi$
$674$			21.1660i			0.815284i
$675$			5.00000i			0.192450i
$676$	9.00000			0.346154
$677$	31.7490			1.22021			0.610107	−	0.792319i	$-0.291126\pi$
$677$	31.7490			1.22021			0.610107	+	0.792319i	$0.291126\pi$
$678$		0			0
$679$			14.0000i			0.537271i
$680$		0			0
$681$			26.4575i			1.01385i
$682$	52.9150			2.02622
$683$	−16.0000			−0.612223			−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000			−0.612223			−0.306111	+	0.951996i	$0.599028\pi$
$684$	5.29150			0.202326
$685$		0			0
$686$		−	18.5203i		−	0.707107i
$687$		−	21.1660i		−	0.807534i
$688$			5.29150i			0.201737i
$689$		0			0
$690$		0			0
$691$			28.0000i			1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$			28.0000i			1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$		−	14.0000i		−	0.532200i
$693$	−14.0000			−0.531816
$694$	32.0000			1.21470
$695$		0			0
$696$		−	6.00000i		−	0.227429i
$697$		−	63.4980i		−	2.40516i
$698$			30.0000i			1.13552i
$699$			10.0000i			0.378235i
$700$			13.2288i			0.500000i
$701$		−	31.7490i		−	1.19914i	−0.800321	−	0.599572i	$-0.795338\pi$
$701$		−	31.7490i		−	1.19914i	0.800321	−	0.599572i	$-0.204662\pi$
$702$	2.00000			0.0754851
$703$		0			0
$704$			5.29150i			0.199431i
$705$		0			0
$706$			4.00000i			0.150542i
$707$	37.0405			1.39305
$708$		0			0
$709$		−	21.1660i		−	0.794906i	−0.917622	−	0.397453i	$-0.869894\pi$
$709$		−	21.1660i		−	0.794906i	0.917622	−	0.397453i	$-0.130106\pi$
$710$		0			0
$711$			5.29150i			0.198447i
$712$	5.29150			0.198307
$713$	−26.4575	+	40.0000i	−0.990842	+	1.49801i
$714$	14.0000			0.523937
$715$		0			0
$716$	−12.0000			−0.448461
$717$			8.00000i			0.298765i
$718$		−	5.29150i		−	0.197477i
$719$		−	42.0000i		−	1.56634i	−0.621810	−	0.783168i	$-0.713603\pi$
$719$		−	42.0000i		−	1.56634i	0.621810	−	0.783168i	$-0.286397\pi$
$720$		0			0
$721$		−	28.0000i		−	1.04277i
$722$	−9.00000			−0.334945
$723$			15.8745i			0.590379i
$724$		0			0
$725$	−30.0000			−1.11417
$726$			17.0000i			0.630929i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	5.29150			0.196116
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	28.0000i		−	1.03562i
$732$		−	10.5830i		−	0.391159i
$733$	−10.5830			−0.390892			−0.195446	−	0.980714i	$-0.562615\pi$
$733$	−10.5830			−0.390892			−0.195446	+	0.980714i	$0.562615\pi$
$734$	−21.1660			−0.781252
$735$		0			0
$736$	−4.00000	−	2.64575i	−0.147442	−	0.0975237i
$737$	28.0000			1.03139
$738$			12.0000i			0.441726i
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$	−10.5830			−0.388776
$742$		0			0
$743$		−	15.8745i		−	0.582379i	−0.956665	−	0.291190i	$-0.905949\pi$
$743$		−	15.8745i		−	0.582379i	0.956665	−	0.291190i	$-0.0940511\pi$
$744$	10.0000			0.366618
$745$		0			0
$746$			21.1660i			0.774943i
$747$	15.8745			0.580818
$748$		−	28.0000i		−	1.02378i
$749$	42.0000			1.53465
$750$		0			0
$751$		−	15.8745i		−	0.579269i	−0.957137	−	0.289635i	$-0.906466\pi$
$751$		−	15.8745i		−	0.579269i	0.957137	−	0.289635i	$-0.0935338\pi$
$752$		−	6.00000i		−	0.218797i
$753$			15.8745i			0.578499i
$754$			12.0000i			0.437014i
$755$		0			0
$756$	−2.64575			−0.0962250
$757$		−	10.5830i		−	0.384646i	−0.981332	−	0.192323i	$-0.938398\pi$
$757$		−	10.5830i		−	0.384646i	0.981332	−	0.192323i	$-0.0616021\pi$
$758$		−	15.8745i		−	0.576588i
$759$	−21.1660	−	14.0000i	−0.768278	−	0.508168i
$760$		0			0
$761$			8.00000i			0.290000i	0.989432	+	0.145000i	$0.0463182\pi$
$761$			8.00000i			0.290000i	−0.989432	+	0.145000i	$0.953682\pi$
$762$		−	8.00000i		−	0.289809i
$763$	−28.0000			−1.01367
$764$			15.8745i			0.574320i
$765$		0			0
$766$		0			0
$767$		0			0
$768$			1.00000i			0.0360844i
$769$	5.29150			0.190816			0.0954082	−	0.995438i	$-0.469584\pi$
$769$	5.29150			0.190816			0.0954082	+	0.995438i	$0.469584\pi$
$770$		0			0
$771$	−8.00000			−0.288113
$772$	−2.00000			−0.0719816
$773$	−52.9150			−1.90322			−0.951611	−	0.307306i	$-0.900572\pi$
$773$	−52.9150			−1.90322			−0.951611	+	0.307306i	$0.900572\pi$
$774$			5.29150i			0.190199i
$775$		−	50.0000i		−	1.79605i
$776$	5.29150			0.189954
$777$		0			0
$778$			21.1660i			0.758838i
$779$		−	63.4980i		−	2.27505i
$780$		0			0
$781$		−	42.3320i		−	1.51476i
$782$	21.1660	+	14.0000i	0.756895	+	0.500639i
$783$		−	6.00000i		−	0.214423i
$784$	−7.00000			−0.250000
$785$		0			0
$786$	−8.00000			−0.285351
$787$	37.0405			1.32035			0.660176	−	0.751111i	$-0.270482\pi$
$787$	37.0405			1.32035			0.660176	+	0.751111i	$0.270482\pi$
$788$	−22.0000			−0.783718
$789$	5.29150			0.188382
$790$		0			0
$791$		0			0
$792$			5.29150i			0.188025i
$793$			21.1660i			0.751627i
$794$		−	22.0000i		−	0.780751i
$795$		0			0
$796$	21.1660			0.750209
$797$	21.1660			0.749739			0.374869	−	0.927078i	$-0.377688\pi$
$797$	21.1660			0.749739			0.374869	+	0.927078i	$0.377688\pi$
$798$	14.0000			0.495595
$799$			31.7490i			1.12320i
$800$	5.00000			0.176777
$801$	5.29150			0.186966
$802$		0			0
$803$	−21.1660			−0.746932
$804$	5.29150			0.186617
$805$		0			0
$806$	−20.0000			−0.704470
$807$	−18.0000			−0.633630
$808$		−	14.0000i		−	0.492518i
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$		−	16.0000i		−	0.561836i	−0.959732	−	0.280918i	$-0.909361\pi$
$811$		−	16.0000i		−	0.561836i	0.959732	−	0.280918i	$-0.0906389\pi$
$812$		−	15.8745i		−	0.557086i
$813$	14.0000			0.491001
$814$		0			0
$815$		0			0
$816$		−	5.29150i		−	0.185240i
$817$		−	28.0000i		−	0.979596i
$818$		−	4.00000i		−	0.139857i
$819$	5.29150			0.184900
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$	21.1660			0.738249
$823$	40.0000			1.39431			0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000			1.39431			0.697156	+	0.716919i	$0.254448\pi$
$824$	−10.5830			−0.368676
$825$	26.4575			0.921132
$826$		0			0
$827$			15.8745i			0.552011i	0.961156	+	0.276005i	$0.0890108\pi$
$827$			15.8745i			0.552011i	−0.961156	+	0.276005i	$0.910989\pi$
$828$	−4.00000	−	2.64575i	−0.139010	−	0.0919462i
$829$			14.0000i			0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$			14.0000i			0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$		0			0
$831$			2.00000i			0.0693792i
$832$		−	2.00000i		−	0.0693375i
$833$	37.0405			1.28338
$834$	−12.0000			−0.415526
$835$		0			0
$836$		−	28.0000i		−	0.968400i
$837$	10.0000			0.345651
$838$	5.29150			0.182792
$839$	42.3320			1.46146			0.730732	−	0.682665i	$-0.239179\pi$
$839$	42.3320			1.46146			0.730732	+	0.682665i	$0.239179\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	31.7490i		−	1.09414i
$843$		0			0
$844$	−16.0000			−0.550743
$845$		0			0
$846$		−	6.00000i		−	0.206284i
$847$			44.9778i			1.54545i
$848$		0			0
$849$			26.4575i			0.908019i
$850$	−26.4575			−0.907485
$851$		0			0
$852$		−	8.00000i		−	0.274075i
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		−	28.0000i		−	0.958140i
$855$		0			0
$856$		−	15.8745i		−	0.542580i
$857$		−	24.0000i		−	0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$		−	24.0000i		−	0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$		−	10.5830i		−	0.361298i
$859$			48.0000i			1.63774i	0.573980	+	0.818869i	$0.305399\pi$
$859$			48.0000i			1.63774i	−0.573980	+	0.818869i	$0.694601\pi$
$860$		0			0
$861$			31.7490i			1.08200i
$862$			15.8745i			0.540688i
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$			1.00000i			0.0340207i
$865$		0			0
$866$	37.0405			1.25869
$867$			11.0000i			0.373580i
$868$	26.4575			0.898027
$869$	28.0000			0.949835
$870$		0			0
$871$	−10.5830			−0.358591
$872$			10.5830i			0.358386i
$873$	5.29150			0.179090
$874$	21.1660	+	14.0000i	0.715951	+	0.473557i
$875$		0			0
$876$	−4.00000			−0.135147
$877$	−6.00000			−0.202606			−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000			−0.202606			−0.101303	+	0.994856i	$0.532301\pi$
$878$			6.00000i			0.202490i
$879$			10.5830i			0.356956i
$880$		0			0
$881$	47.6235			1.60448			0.802239	−	0.597003i	$-0.203642\pi$
$881$	47.6235			1.60448			0.802239	+	0.597003i	$0.203642\pi$
$882$	−7.00000			−0.235702
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$			10.5830i			0.355945i
$885$		0			0
$886$	−24.0000			−0.806296
$887$			34.0000i			1.14161i	0.821086	+	0.570804i	$0.193368\pi$
$887$			34.0000i			1.14161i	−0.821086	+	0.570804i	$0.806632\pi$
$888$		0			0
$889$		−	21.1660i		−	0.709885i
$890$		0			0
$891$			5.29150i			0.177272i
$892$			14.0000i			0.468755i
$893$			31.7490i			1.06244i
$894$	−10.5830			−0.353949
$895$		0			0
$896$			2.64575i			0.0883883i
$897$	8.00000	+	5.29150i	0.267112	+	0.176678i
$898$	30.0000			1.00111
$899$			60.0000i			2.00111i
$900$	5.00000			0.166667
$901$		0			0
$902$	63.4980			2.11425
$903$			14.0000i			0.465891i
$904$		0			0
$905$		0			0
$906$		−	16.0000i		−	0.531564i
$907$		−	37.0405i		−	1.22991i	−0.788562	−	0.614955i	$-0.789174\pi$
$907$		−	37.0405i		−	1.22991i	0.788562	−	0.614955i	$-0.210826\pi$
$908$	26.4575			0.878023
$909$		−	14.0000i		−	0.464351i
$910$		0			0
$911$			26.4575i			0.876577i	0.898834	+	0.438288i	$0.144415\pi$
$911$			26.4575i			0.876577i	−0.898834	+	0.438288i	$0.855585\pi$
$912$		−	5.29150i		−	0.175219i
$913$		−	84.0000i		−	2.77999i
$914$		−	21.1660i		−	0.700109i
$915$		0			0
$916$	−21.1660			−0.699345
$917$	−21.1660			−0.698963
$918$		−	5.29150i		−	0.174646i
$919$			15.8745i			0.523652i	0.965115	+	0.261826i	$0.0843246\pi$
$919$			15.8745i			0.523652i	−0.965115	+	0.261826i	$0.915675\pi$
$920$		0			0
$921$		0			0
$922$		−	30.0000i		−	0.987997i
$923$			16.0000i			0.526646i
$924$			14.0000i			0.460566i
$925$		0			0
$926$	16.0000			0.525793
$927$	−10.5830			−0.347591
$928$	−6.00000			−0.196960
$929$			20.0000i			0.656179i	0.944647	+	0.328089i	$0.106405\pi$
$929$			20.0000i			0.656179i	−0.944647	+	0.328089i	$0.893595\pi$
$930$		0			0
$931$	37.0405			1.21395
$932$	10.0000			0.327561
$933$	−10.0000			−0.327385
$934$	−15.8745			−0.519430
$935$		0			0
$936$		−	2.00000i		−	0.0653720i
$937$	47.6235			1.55579			0.777896	−	0.628393i	$-0.216287\pi$
$937$	47.6235			1.55579			0.777896	+	0.628393i	$0.216287\pi$
$938$	14.0000			0.457116
$939$		−	5.29150i		−	0.172682i
$940$		0			0
$941$	−42.3320			−1.37998			−0.689992	−	0.723817i	$-0.742386\pi$
$941$	−42.3320			−1.37998			−0.689992	+	0.723817i	$0.742386\pi$
$942$			10.5830i			0.344813i
$943$	−31.7490	+	48.0000i	−1.03389	+	1.56310i
$944$		0			0
$945$		0			0
$946$	28.0000			0.910359
$947$	−24.0000			−0.779895			−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000			−0.779895			−0.389948	+	0.920837i	$0.627507\pi$
$948$	5.29150			0.171860
$949$	8.00000			0.259691
$950$	−26.4575			−0.858395
$951$			6.00000i			0.194563i
$952$		−	14.0000i		−	0.453743i
$953$		−	42.3320i		−	1.37127i	−0.727946	−	0.685634i	$-0.759525\pi$
$953$		−	42.3320i		−	1.37127i	0.727946	−	0.685634i	$-0.240475\pi$
$954$		0			0
$955$		0			0
$956$	8.00000			0.258738
$957$	−31.7490			−1.02630
$958$	21.1660			0.683843
$959$	56.0000			1.80833
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$		−	15.8745i		−	0.511549i
$964$	15.8745			0.511283
$965$		0			0
$966$	−10.5830	−	7.00000i	−0.340503	−	0.225221i
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$	17.0000			0.546401
$969$			28.0000i			0.899490i
$970$		0			0
$971$	−26.4575			−0.849062			−0.424531	−	0.905413i	$-0.639561\pi$
$971$	−26.4575			−0.849062			−0.424531	+	0.905413i	$0.639561\pi$
$972$			1.00000i			0.0320750i
$973$	−31.7490			−1.01783
$974$	−40.0000			−1.28168
$975$	−10.0000			−0.320256
$976$	−10.5830			−0.338754
$977$			42.3320i			1.35432i	0.735835	+	0.677161i	$0.236790\pi$
$977$			42.3320i			1.35432i	−0.735835	+	0.677161i	$0.763210\pi$
$978$		−	24.0000i		−	0.767435i
$979$		−	28.0000i		−	0.894884i
$980$		0			0
$981$			10.5830i			0.337889i
$982$	8.00000			0.255290
$983$	31.7490			1.01264			0.506318	−	0.862347i	$-0.331006\pi$
$983$	31.7490			1.01264			0.506318	+	0.862347i	$0.331006\pi$
$984$	12.0000			0.382546
$985$		0			0
$986$	31.7490			1.01109
$987$		−	15.8745i		−	0.505291i
$988$			10.5830i			0.336690i
$989$	−14.0000	+	21.1660i	−0.445174	+	0.673040i
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		−	10.0000i		−	0.317500i
$993$			4.00000i			0.126936i
$994$		−	21.1660i		−	0.671345i
$995$		0			0
$996$		−	15.8745i		−	0.503003i
$997$		−	42.0000i		−	1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$		−	42.0000i		−	1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	24.0000			0.759707
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.g.c.643.3	yes	4
3.2	odd	2		2898.2.g.c.2575.1		4
7.6	odd	2	inner	966.2.g.c.643.1	✓	4
21.20	even	2		2898.2.g.c.2575.2		4
23.22	odd	2	inner	966.2.g.c.643.4	yes	4
69.68	even	2		2898.2.g.c.2575.3		4
161.160	even	2	inner	966.2.g.c.643.2	yes	4
483.482	odd	2		2898.2.g.c.2575.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.g.c.643.1	✓	4	7.6	odd	2	inner
966.2.g.c.643.2	yes	4	161.160	even	2	inner
966.2.g.c.643.3	yes	4	1.1	even	1	trivial
966.2.g.c.643.4	yes	4	23.22	odd	2	inner
2898.2.g.c.2575.1		4	3.2	odd	2
2898.2.g.c.2575.2		4	21.20	even	2
2898.2.g.c.2575.3		4	69.68	even	2
2898.2.g.c.2575.4		4	483.482	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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.64575i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.64575i q^{7} -1.00000 q^{8} -1.00000 q^{9} +5.29150i q^{11} +1.00000i q^{12} -2.00000i q^{13} +2.64575i q^{14} +1.00000 q^{16} -5.29150 q^{17} +1.00000 q^{18} -5.29150 q^{19} +2.64575 q^{21} -5.29150i q^{22} +(4.00000 + 2.64575i) q^{23} -1.00000i q^{24} -5.00000 q^{25} +2.00000i q^{26} -1.00000i q^{27} -2.64575i q^{28} +6.00000 q^{29} +10.0000i q^{31} -1.00000 q^{32} -5.29150 q^{33} +5.29150 q^{34} -1.00000 q^{36} +5.29150 q^{38} +2.00000 q^{39} +12.0000i q^{41} -2.64575 q^{42} +5.29150i q^{43} +5.29150i q^{44} +(-4.00000 - 2.64575i) q^{46} -6.00000i q^{47} +1.00000i q^{48} -7.00000 q^{49} +5.00000 q^{50} -5.29150i q^{51} -2.00000i q^{52} +1.00000i q^{54} +2.64575i q^{56} -5.29150i q^{57} -6.00000 q^{58} -10.5830 q^{61} -10.0000i q^{62} +2.64575i q^{63} +1.00000 q^{64} +5.29150 q^{66} -5.29150i q^{67} -5.29150 q^{68} +(-2.64575 + 4.00000i) q^{69} -8.00000 q^{71} +1.00000 q^{72} +4.00000i q^{73} -5.00000i q^{75} -5.29150 q^{76} +14.0000 q^{77} -2.00000 q^{78} -5.29150i q^{79} +1.00000 q^{81} -12.0000i q^{82} -15.8745 q^{83} +2.64575 q^{84} -5.29150i q^{86} +6.00000i q^{87} -5.29150i q^{88} -5.29150 q^{89} -5.29150 q^{91} +(4.00000 + 2.64575i) q^{92} -10.0000 q^{93} +6.00000i q^{94} -1.00000i q^{96} -5.29150 q^{97} +7.00000 q^{98} -5.29150i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9} + 4 q^{16} + 4 q^{18} + 16 q^{23} - 20 q^{25} + 24 q^{29} - 4 q^{32} - 4 q^{36} + 8 q^{39} - 16 q^{46} - 28 q^{49} + 20 q^{50} - 24 q^{58} + 4 q^{64} - 32 q^{71} + 4 q^{72} + 56 q^{77} - 8 q^{78} + 4 q^{81} + 16 q^{92} - 40 q^{93} + 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9 + 4 * q^16 + 4 * q^18 + 16 * q^23 - 20 * q^25 + 24 * q^29 - 4 * q^32 - 4 * q^36 + 8 * q^39 - 16 * q^46 - 28 * q^49 + 20 * q^50 - 24 * q^58 + 4 * q^64 - 32 * q^71 + 4 * q^72 + 56 * q^77 - 8 * q^78 + 4 * q^81 + 16 * q^92 - 40 * q^93 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more