LMFDB - Embedded newform 1386.2.e.b.307.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(307,1386)]

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([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("1386.307");

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.e (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:	$8$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 23x^{4} + 1$ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			307.7
Root			$0.323042 - 0.323042i$ of defining polynomial
Character	$\chi$	$=$	1386.307
Dual form			1386.2.e.b.307.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.646084i q^{5} +(2.44949 + 1.00000i) q^{7} -1.00000i q^{8} -0.646084 q^{10} +(-2.79129 - 1.79129i) q^{11} -3.09557 q^{13} +(-1.00000 + 2.44949i) q^{14} +1.00000 q^{16} -3.74166 q^{17} -5.54506 q^{19} -0.646084i q^{20} +(1.79129 - 2.79129i) q^{22} -4.00000 q^{23} +4.58258 q^{25} -3.09557i q^{26} +(-2.44949 - 1.00000i) q^{28} +7.58258i q^{29} -1.15732i q^{31} +1.00000i q^{32} -3.74166i q^{34} +(-0.646084 + 1.58258i) q^{35} -5.58258 q^{37} -5.54506i q^{38} +0.646084 q^{40} -5.03383 q^{41} +11.1652i q^{43} +(2.79129 + 1.79129i) q^{44} -4.00000i q^{46} -5.03383i q^{47} +(5.00000 + 4.89898i) q^{49} +4.58258i q^{50} +3.09557 q^{52} +2.41742 q^{53} +(1.15732 - 1.80341i) q^{55} +(1.00000 - 2.44949i) q^{56} -7.58258 q^{58} -3.09557i q^{59} -9.28672 q^{61} +1.15732 q^{62} -1.00000 q^{64} -2.00000i q^{65} +1.58258 q^{67} +3.74166 q^{68} +(-1.58258 - 0.646084i) q^{70} -2.00000 q^{71} -6.32599 q^{73} -5.58258i q^{74} +5.54506 q^{76} +(-5.04594 - 7.17903i) q^{77} -4.00000i q^{79} +0.646084i q^{80} -5.03383i q^{82} -9.15188 q^{83} -2.41742i q^{85} -11.1652 q^{86} +(-1.79129 + 2.79129i) q^{88} -9.79796i q^{89} +(-7.58258 - 3.09557i) q^{91} +4.00000 q^{92} +5.03383 q^{94} -3.58258i q^{95} -15.9891i q^{97} +(-4.89898 + 5.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{4} - 4 q^{11} - 8 q^{14} + 8 q^{16} - 4 q^{22} - 32 q^{23} - 8 q^{37} + 4 q^{44} + 40 q^{49} + 56 q^{53} + 8 q^{56} - 24 q^{58} - 8 q^{64} - 24 q^{67} + 24 q^{70} - 16 q^{71} - 4 q^{77} - 16 q^{86}+ \cdots + 32 q^{92}+O(q^{100})$ 8 * q - 8 * q^4 - 4 * q^11 - 8 * q^14 + 8 * q^16 - 4 * q^22 - 32 * q^23 - 8 * q^37 + 4 * q^44 + 40 * q^49 + 56 * q^53 + 8 * q^56 - 24 * q^58 - 8 * q^64 - 24 * q^67 + 24 * q^70 - 16 * q^71 - 4 * q^77 - 16 * q^86 + 4 * q^88 - 24 * q^91 + 32 * q^92

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			0.646084i			0.288937i	0.989509	+	0.144469i	$0.0461473\pi$
$5$			0.646084i			0.288937i	−0.989509	+	0.144469i	$0.953853\pi$
$6$		0			0
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−0.646084			−0.204310
$11$	−2.79129	−	1.79129i	−0.841605	−	0.540094i
$11$	−2.79129	−	1.79129i	−0.841605	−	0.540094i
$12$		0			0
$13$	−3.09557			−0.858558			−0.429279	−	0.903172i	$-0.641232\pi$
$13$	−3.09557			−0.858558			−0.429279	+	0.903172i	$0.641232\pi$
$14$	−1.00000	+	2.44949i	−0.267261	+	0.654654i
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.74166			−0.907485			−0.453743	−	0.891133i	$-0.649911\pi$
$17$	−3.74166			−0.907485			−0.453743	+	0.891133i	$0.649911\pi$
$18$		0			0
$19$	−5.54506			−1.27212			−0.636062	−	0.771638i	$-0.719438\pi$
$19$	−5.54506			−1.27212			−0.636062	+	0.771638i	$0.719438\pi$
$20$		−	0.646084i		−	0.144469i
$21$		0			0
$22$	1.79129	−	2.79129i	0.381904	−	0.595105i
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	4.58258			0.916515
$26$		−	3.09557i		−	0.607092i
$27$		0			0
$28$	−2.44949	−	1.00000i	−0.462910	−	0.188982i
$29$			7.58258i			1.40805i	0.710176	+	0.704024i	$0.248615\pi$
$29$			7.58258i			1.40805i	−0.710176	+	0.704024i	$0.751385\pi$
$30$		0			0
$31$		−	1.15732i		−	0.207861i	−0.994585	−	0.103931i	$-0.966858\pi$
$31$		−	1.15732i		−	0.207861i	0.994585	−	0.103931i	$-0.0331420\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	3.74166i		−	0.641689i
$35$	−0.646084	+	1.58258i	−0.109208	+	0.267504i
$36$		0			0
$37$	−5.58258			−0.917770			−0.458885	−	0.888496i	$-0.651751\pi$
$37$	−5.58258			−0.917770			−0.458885	+	0.888496i	$0.651751\pi$
$38$		−	5.54506i		−	0.899528i
$39$		0			0
$40$	0.646084			0.102155
$41$	−5.03383			−0.786151			−0.393076	−	0.919506i	$-0.628589\pi$
$41$	−5.03383			−0.786151			−0.393076	+	0.919506i	$0.628589\pi$
$42$		0			0
$43$			11.1652i			1.70267i	0.524623	+	0.851335i	$0.324206\pi$
$43$			11.1652i			1.70267i	−0.524623	+	0.851335i	$0.675794\pi$
$44$	2.79129	+	1.79129i	0.420802	+	0.270047i
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$		−	5.03383i		−	0.734259i	−0.930170	−	0.367129i	$-0.880341\pi$
$47$		−	5.03383i		−	0.734259i	0.930170	−	0.367129i	$-0.119659\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$			4.58258i			0.648074i
$51$		0			0
$52$	3.09557			0.429279
$53$	2.41742			0.332059			0.166029	−	0.986121i	$-0.446905\pi$
$53$	2.41742			0.332059			0.166029	+	0.986121i	$0.446905\pi$
$54$		0			0
$55$	1.15732	−	1.80341i	0.156053	−	0.243171i
$56$	1.00000	−	2.44949i	0.133631	−	0.327327i
$57$		0			0
$58$	−7.58258			−0.995641
$59$		−	3.09557i		−	0.403009i	−0.979488	−	0.201505i	$-0.935417\pi$
$59$		−	3.09557i		−	0.403009i	0.979488	−	0.201505i	$-0.0645831\pi$
$60$		0			0
$61$	−9.28672			−1.18904			−0.594521	−	0.804080i	$-0.702658\pi$
$61$	−9.28672			−1.18904			−0.594521	+	0.804080i	$0.702658\pi$
$62$	1.15732			0.146980
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$	1.58258			0.193342			0.0966712	−	0.995316i	$-0.469180\pi$
$67$	1.58258			0.193342			0.0966712	+	0.995316i	$0.469180\pi$
$68$	3.74166			0.453743
$69$		0			0
$70$	−1.58258	−	0.646084i	−0.189154	−	0.0772218i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	−6.32599			−0.740401			−0.370201	−	0.928952i	$-0.620711\pi$
$73$	−6.32599			−0.740401			−0.370201	+	0.928952i	$0.620711\pi$
$74$		−	5.58258i		−	0.648961i
$75$		0			0
$76$	5.54506			0.636062
$77$	−5.04594	−	7.17903i	−0.575039	−	0.818126i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$			0.646084i			0.0722344i
$81$		0			0
$82$		−	5.03383i		−	0.555893i
$83$	−9.15188			−1.00455			−0.502274	−	0.864708i	$-0.667503\pi$
$83$	−9.15188			−1.00455			−0.502274	+	0.864708i	$0.667503\pi$
$84$		0			0
$85$		−	2.41742i		−	0.262206i
$86$	−11.1652			−1.20397
$87$		0			0
$88$	−1.79129	+	2.79129i	−0.190952	+	0.297552i
$89$		−	9.79796i		−	1.03858i	−0.854598	−	0.519291i	$-0.826196\pi$
$89$		−	9.79796i		−	1.03858i	0.854598	−	0.519291i	$-0.173804\pi$
$90$		0			0
$91$	−7.58258	−	3.09557i	−0.794870	−	0.324504i
$92$	4.00000			0.417029
$93$		0			0
$94$	5.03383			0.519199
$95$		−	3.58258i		−	0.367565i
$96$		0			0
$97$		−	15.9891i		−	1.62345i	−0.584041	−	0.811724i	$-0.698529\pi$
$97$		−	15.9891i		−	1.62345i	0.584041	−	0.811724i	$-0.301471\pi$
$98$	−4.89898	+	5.00000i	−0.494872	+	0.505076i
$99$		0			0
$100$	−4.58258			−0.458258
$101$	0.511238			0.0508701			0.0254351	−	0.999676i	$-0.491903\pi$
$101$	0.511238			0.0508701			0.0254351	+	0.999676i	$0.491903\pi$
$102$		0			0
$103$			13.5396i			1.33410i	0.745014	+	0.667049i	$0.232443\pi$
$103$			13.5396i			1.33410i	−0.745014	+	0.667049i	$0.767557\pi$
$104$			3.09557i			0.303546i
$105$		0			0
$106$			2.41742i			0.234801i
$107$		−	8.41742i		−	0.813743i	−0.913485	−	0.406872i	$-0.866620\pi$
$107$		−	8.41742i		−	0.813743i	0.913485	−	0.406872i	$-0.133380\pi$
$108$		0			0
$109$		−	1.16515i		−	0.111601i	−0.998442	−	0.0558006i	$-0.982229\pi$
$109$		−	1.16515i		−	0.111601i	0.998442	−	0.0558006i	$-0.0177711\pi$
$110$	1.80341	+	1.15732i	0.171948	+	0.110346i
$111$		0			0
$112$	2.44949	+	1.00000i	0.231455	+	0.0944911i
$113$	16.7477			1.57549			0.787747	−	0.615999i	$-0.211248\pi$
$113$	16.7477			1.57549			0.787747	+	0.615999i	$0.211248\pi$
$114$		0			0
$115$		−	2.58434i		−	0.240991i
$116$		−	7.58258i		−	0.704024i
$117$		0			0
$118$	3.09557			0.284971
$119$	−9.16515	−	3.74166i	−0.840168	−	0.342997i
$120$		0			0
$121$	4.58258	+	10.0000i	0.416598	+	0.909091i
$122$		−	9.28672i		−	0.840780i
$123$		0			0
$124$			1.15732i			0.103931i
$125$			6.19115i			0.553753i
$126$		0			0
$127$			5.58258i			0.495373i	0.968840	+	0.247687i	$0.0796704\pi$
$127$			5.58258i			0.495373i	−0.968840	+	0.247687i	$0.920330\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	2.00000			0.175412
$131$	−9.42157			−0.823166			−0.411583	−	0.911372i	$-0.635024\pi$
$131$	−9.42157			−0.823166			−0.411583	+	0.911372i	$0.635024\pi$
$132$		0			0
$133$	−13.5826	−	5.54506i	−1.17776	−	0.480818i
$134$			1.58258i			0.136714i
$135$		0			0
$136$			3.74166i			0.320844i
$137$	−15.1652			−1.29565			−0.647823	−	0.761791i	$-0.724320\pi$
$137$	−15.1652			−1.29565			−0.647823	+	0.761791i	$0.724320\pi$
$138$		0			0
$139$	3.23042			0.274001			0.137000	−	0.990571i	$-0.456254\pi$
$139$	3.23042			0.274001			0.137000	+	0.990571i	$0.456254\pi$
$140$	0.646084	−	1.58258i	0.0546040	−	0.133752i
$141$		0			0
$142$		−	2.00000i		−	0.167836i
$143$	8.64064	+	5.54506i	0.722566	+	0.463701i
$144$		0			0
$145$	−4.89898			−0.406838
$146$		−	6.32599i		−	0.523543i
$147$		0			0
$148$	5.58258			0.458885
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		−	3.58258i		−	0.291546i	−0.989318	−	0.145773i	$-0.953433\pi$
$151$		−	3.58258i		−	0.291546i	0.989318	−	0.145773i	$-0.0465669\pi$
$152$			5.54506i			0.449764i
$153$		0			0
$154$	7.17903	−	5.04594i	0.578503	−	0.406614i
$155$	0.747727			0.0600589
$156$		0			0
$157$			20.5117i			1.63701i	0.574498	+	0.818506i	$0.305197\pi$
$157$			20.5117i			1.63701i	−0.574498	+	0.818506i	$0.694803\pi$
$158$	4.00000			0.318223
$159$		0			0
$160$	−0.646084			−0.0510774
$161$	−9.79796	−	4.00000i	−0.772187	−	0.315244i
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	5.03383			0.393076
$165$		0			0
$166$		−	9.15188i		−	0.710323i
$167$	6.19115			0.479085			0.239543	−	0.970886i	$-0.423002\pi$
$167$	6.19115			0.479085			0.239543	+	0.970886i	$0.423002\pi$
$168$		0			0
$169$	−3.41742			−0.262879
$170$	2.41742			0.185408
$171$		0			0
$172$		−	11.1652i		−	0.851335i
$173$	22.9612			1.74571			0.872853	−	0.487983i	$-0.162267\pi$
$173$	22.9612			1.74571			0.872853	+	0.487983i	$0.162267\pi$
$174$		0			0
$175$	11.2250	+	4.58258i	0.848528	+	0.346410i
$176$	−2.79129	−	1.79129i	−0.210401	−	0.135023i
$177$		0			0
$178$	9.79796			0.734388
$179$	7.16515			0.535549			0.267774	−	0.963482i	$-0.413712\pi$
$179$	7.16515			0.535549			0.267774	+	0.963482i	$0.413712\pi$
$180$		0			0
$181$			16.6352i			1.23648i	0.785988	+	0.618242i	$0.212155\pi$
$181$			16.6352i			1.23648i	−0.785988	+	0.618242i	$0.787845\pi$
$182$	3.09557	−	7.58258i	0.229459	−	0.562058i
$183$		0			0
$184$			4.00000i			0.294884i
$185$		−	3.60681i		−	0.265178i
$186$		0			0
$187$	10.4440	+	6.70239i	0.763744	+	0.490127i
$188$			5.03383i			0.367129i
$189$		0			0
$190$	3.58258			0.259907
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$		−	20.3303i		−	1.46341i	−0.681623	−	0.731704i	$-0.738726\pi$
$193$		−	20.3303i		−	1.46341i	0.681623	−	0.731704i	$-0.261274\pi$
$194$	15.9891			1.14795
$195$		0			0
$196$	−5.00000	−	4.89898i	−0.357143	−	0.349927i
$197$			18.7477i			1.33572i	0.744287	+	0.667860i	$0.232790\pi$
$197$			18.7477i			1.33572i	−0.744287	+	0.667860i	$0.767210\pi$
$198$		0			0
$199$		−	24.6297i		−	1.74596i	−0.487759	−	0.872978i	$-0.662186\pi$
$199$		−	24.6297i		−	1.74596i	0.487759	−	0.872978i	$-0.337814\pi$
$200$		−	4.58258i		−	0.324037i
$201$		0			0
$202$			0.511238i			0.0359706i
$203$	−7.58258	+	18.5734i	−0.532192	+	1.30360i
$204$		0			0
$205$		−	3.25227i		−	0.227149i
$206$	−13.5396			−0.943350
$207$		0			0
$208$	−3.09557			−0.214639
$209$	15.4779	+	9.93280i	1.07063	+	0.687066i
$210$		0			0
$211$		−	10.7477i		−	0.739904i	−0.929051	−	0.369952i	$-0.879374\pi$
$211$		−	10.7477i		−	0.739904i	0.929051	−	0.369952i	$-0.120626\pi$
$212$	−2.41742			−0.166029
$213$		0			0
$214$	8.41742			0.575403
$215$	−7.21362			−0.491965
$216$		0			0
$217$	1.15732	−	2.83485i	0.0785641	−	0.192442i
$218$	1.16515			0.0789140
$219$		0			0
$220$	−1.15732	+	1.80341i	−0.0780266	+	0.121586i
$221$	11.5826			0.779128
$222$		0			0
$223$		−	6.32599i		−	0.423620i	−0.977311	−	0.211810i	$-0.932064\pi$
$223$		−	6.32599i		−	0.423620i	0.977311	−	0.211810i	$-0.0679358\pi$
$224$	−1.00000	+	2.44949i	−0.0668153	+	0.163663i
$225$		0			0
$226$			16.7477i			1.11404i
$227$	11.7362			0.778960			0.389480	−	0.921035i	$-0.372655\pi$
$227$	11.7362			0.778960			0.389480	+	0.921035i	$0.372655\pi$
$228$		0			0
$229$			13.0284i			0.860939i	0.902605	+	0.430470i	$0.141652\pi$
$229$			13.0284i			0.860939i	−0.902605	+	0.430470i	$0.858348\pi$
$230$	2.58434			0.170406
$231$		0			0
$232$	7.58258			0.497820
$233$			8.83485i			0.578790i	0.957210	+	0.289395i	$0.0934541\pi$
$233$			8.83485i			0.578790i	−0.957210	+	0.289395i	$0.906546\pi$
$234$		0			0
$235$	3.25227			0.212155
$236$			3.09557i			0.201505i
$237$		0			0
$238$	3.74166	−	9.16515i	0.242536	−	0.594089i
$239$			17.5826i			1.13732i	0.822572	+	0.568661i	$0.192538\pi$
$239$			17.5826i			1.13732i	−0.822572	+	0.568661i	$0.807462\pi$
$240$		0			0
$241$	−25.9219			−1.66978			−0.834889	−	0.550419i	$-0.814468\pi$
$241$	−25.9219			−1.66978			−0.834889	+	0.550419i	$0.814468\pi$
$242$	−10.0000	+	4.58258i	−0.642824	+	0.294579i
$243$		0			0
$244$	9.28672			0.594521
$245$	−3.16515	+	3.23042i	−0.202214	+	0.206384i
$246$		0			0
$247$	17.1652			1.09219
$248$	−1.15732			−0.0734900
$249$		0			0
$250$	−6.19115			−0.391563
$251$			2.07310i			0.130853i	0.997857	+	0.0654264i	$0.0208407\pi$
$251$			2.07310i			0.130853i	−0.997857	+	0.0654264i	$0.979159\pi$
$252$		0			0
$253$	11.1652	+	7.16515i	0.701947	+	0.450469i
$254$	−5.58258			−0.350282
$255$		0			0
$256$	1.00000			0.0625000
$257$			4.89898i			0.305590i	0.988258	+	0.152795i	$0.0488274\pi$
$257$			4.89898i			0.305590i	−0.988258	+	0.152795i	$0.951173\pi$
$258$		0			0
$259$	−13.6745	−	5.58258i	−0.849690	−	0.346884i
$260$			2.00000i			0.124035i
$261$		0			0
$262$		−	9.42157i		−	0.582066i
$263$		−	28.3303i		−	1.74692i	−0.486895	−	0.873461i	$-0.661870\pi$
$263$		−	28.3303i		−	1.74692i	0.486895	−	0.873461i	$-0.338130\pi$
$264$		0			0
$265$			1.56186i			0.0959442i
$266$	5.54506	−	13.5826i	0.339990	−	0.832801i
$267$		0			0
$268$	−1.58258			−0.0966712
$269$			20.2420i			1.23418i	0.786894	+	0.617088i	$0.211688\pi$
$269$			20.2420i			1.23418i	−0.786894	+	0.617088i	$0.788312\pi$
$270$		0			0
$271$	−4.89898			−0.297592			−0.148796	−	0.988868i	$-0.547540\pi$
$271$	−4.89898			−0.297592			−0.148796	+	0.988868i	$0.547540\pi$
$272$	−3.74166			−0.226871
$273$		0			0
$274$		−	15.1652i		−	0.916160i
$275$	−12.7913	−	8.20871i	−0.771344	−	0.495004i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$			3.23042i			0.193748i
$279$		0			0
$280$	1.58258	+	0.646084i	0.0945770	+	0.0386109i
$281$			7.16515i			0.427437i	0.976895	+	0.213719i	$0.0685575\pi$
$281$			7.16515i			0.427437i	−0.976895	+	0.213719i	$0.931442\pi$
$282$		0			0
$283$	15.6127			0.928079			0.464040	−	0.885814i	$-0.346399\pi$
$283$	15.6127			0.928079			0.464040	+	0.885814i	$0.346399\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	−5.54506	+	8.64064i	−0.327886	+	0.510932i
$287$	−12.3303	−	5.03383i	−0.727835	−	0.297137i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		−	4.89898i		−	0.287678i
$291$		0			0
$292$	6.32599			0.370201
$293$	−21.3993			−1.25016			−0.625081	−	0.780560i	$-0.714934\pi$
$293$	−21.3993			−1.25016			−0.625081	+	0.780560i	$0.714934\pi$
$294$		0			0
$295$	2.00000			0.116445
$296$			5.58258i			0.324481i
$297$		0			0
$298$	−14.0000			−0.810998
$299$	12.3823			0.716087
$300$		0			0
$301$	−11.1652	+	27.3489i	−0.643549	+	1.57637i
$302$	3.58258			0.206154
$303$		0			0
$304$	−5.54506			−0.318031
$305$		−	6.00000i		−	0.343559i
$306$		0			0
$307$	8.12940			0.463969			0.231985	−	0.972719i	$-0.425478\pi$
$307$	8.12940			0.463969			0.231985	+	0.972719i	$0.425478\pi$
$308$	5.04594	+	7.17903i	0.287519	+	0.409063i
$309$		0			0
$310$			0.747727i			0.0424680i
$311$			9.93280i			0.563238i	0.959526	+	0.281619i	$0.0908714\pi$
$311$			9.93280i			0.563238i	−0.959526	+	0.281619i	$0.909129\pi$
$312$		0			0
$313$		−	1.02248i		−	0.0577938i	−0.999582	−	0.0288969i	$-0.990801\pi$
$313$		−	1.02248i		−	0.0577938i	0.999582	−	0.0288969i	$-0.00919945\pi$
$314$	−20.5117			−1.15754
$315$		0			0
$316$			4.00000i			0.225018i
$317$	9.16515			0.514766			0.257383	−	0.966309i	$-0.417140\pi$
$317$	9.16515			0.514766			0.257383	+	0.966309i	$0.417140\pi$
$318$		0			0
$319$	13.5826	−	21.1652i	0.760478	−	1.18502i
$320$		−	0.646084i		−	0.0361172i
$321$		0			0
$322$	4.00000	−	9.79796i	0.222911	−	0.546019i
$323$	20.7477			1.15443
$324$		0			0
$325$	−14.1857			−0.786881
$326$			4.00000i			0.221540i
$327$		0			0
$328$			5.03383i			0.277946i
$329$	5.03383	−	12.3303i	0.277524	−	0.679792i
$330$		0			0
$331$	−14.4174			−0.792453			−0.396227	−	0.918153i	$-0.629681\pi$
$331$	−14.4174			−0.792453			−0.396227	+	0.918153i	$0.629681\pi$
$332$	9.15188			0.502274
$333$		0			0
$334$			6.19115i			0.338764i
$335$			1.02248i			0.0558639i
$336$		0			0
$337$			29.1652i			1.58873i	0.607443	+	0.794364i	$0.292195\pi$
$337$			29.1652i			1.58873i	−0.607443	+	0.794364i	$0.707805\pi$
$338$		−	3.41742i		−	0.185883i
$339$		0			0
$340$			2.41742i			0.131103i
$341$	−2.07310	+	3.23042i	−0.112264	+	0.174937i
$342$		0			0
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$	11.1652			0.601985
$345$		0			0
$346$			22.9612i			1.23440i
$347$			0.834849i			0.0448170i	0.999749	+	0.0224085i	$0.00713345\pi$
$347$			0.834849i			0.0448170i	−0.999749	+	0.0224085i	$0.992867\pi$
$348$		0			0
$349$	−2.07310			−0.110970			−0.0554852	−	0.998460i	$-0.517671\pi$
$349$	−2.07310			−0.110970			−0.0554852	+	0.998460i	$0.517671\pi$
$350$	−4.58258	+	11.2250i	−0.244949	+	0.600000i
$351$		0			0
$352$	1.79129	−	2.79129i	0.0954760	−	0.148776i
$353$			7.48331i			0.398297i	0.979969	+	0.199148i	$0.0638176\pi$
$353$			7.48331i			0.398297i	−0.979969	+	0.199148i	$0.936182\pi$
$354$		0			0
$355$		−	1.29217i		−	0.0685811i
$356$			9.79796i			0.519291i
$357$		0			0
$358$			7.16515i			0.378690i
$359$			0.417424i			0.0220308i	0.999939	+	0.0110154i	$0.00350638\pi$
$359$			0.417424i			0.0220308i	−0.999939	+	0.0110154i	$0.996494\pi$
$360$		0			0
$361$	11.7477			0.618301
$362$	−16.6352			−0.874326
$363$		0			0
$364$	7.58258	+	3.09557i	0.397435	+	0.162252i
$365$		−	4.08712i		−	0.213930i
$366$		0			0
$367$		−	1.42701i		−	0.0744895i	−0.999306	−	0.0372447i	$-0.988142\pi$
$367$		−	1.42701i		−	0.0744895i	0.999306	−	0.0372447i	$-0.0118581\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$	3.60681			0.187509
$371$	5.92146	+	2.41742i	0.307427	+	0.125506i
$372$		0			0
$373$			0.417424i			0.0216134i	0.999942	+	0.0108067i	$0.00343995\pi$
$373$			0.417424i			0.0216134i	−0.999942	+	0.0108067i	$0.996560\pi$
$374$	−6.70239	+	10.4440i	−0.346572	+	0.540049i
$375$		0			0
$376$	−5.03383			−0.259600
$377$		−	23.4724i		−	1.20889i
$378$		0			0
$379$	14.3303			0.736098			0.368049	−	0.929806i	$-0.380026\pi$
$379$	14.3303			0.736098			0.368049	+	0.929806i	$0.380026\pi$
$380$			3.58258i			0.183782i
$381$		0			0
$382$			18.0000i			0.920960i
$383$		−	4.76413i		−	0.243436i	−0.992565	−	0.121718i	$-0.961160\pi$
$383$		−	4.76413i		−	0.243436i	0.992565	−	0.121718i	$-0.0388403\pi$
$384$		0			0
$385$	4.63825	−	3.26010i	0.236387	−	0.166150i
$386$	20.3303			1.03479
$387$		0			0
$388$			15.9891i			0.811724i
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$	14.9666			0.756895
$392$	4.89898	−	5.00000i	0.247436	−	0.252538i
$393$		0			0
$394$	−18.7477			−0.944497
$395$	2.58434			0.130032
$396$		0			0
$397$		−	23.8488i		−	1.19694i	−0.801146	−	0.598469i	$-0.795776\pi$
$397$		−	23.8488i		−	1.19694i	0.801146	−	0.598469i	$-0.204224\pi$
$398$	24.6297			1.23458
$399$		0			0
$400$	4.58258			0.229129
$401$	1.58258			0.0790301			0.0395150	−	0.999219i	$-0.487419\pi$
$401$	1.58258			0.0790301			0.0395150	+	0.999219i	$0.487419\pi$
$402$		0			0
$403$			3.58258i			0.178461i
$404$	−0.511238			−0.0254351
$405$		0			0
$406$	−18.5734	−	7.58258i	−0.921784	−	0.376317i
$407$	15.5826	+	10.0000i	0.772400	+	0.495682i
$408$		0			0
$409$	9.93280			0.491146			0.245573	−	0.969378i	$-0.421024\pi$
$409$	9.93280			0.491146			0.245573	+	0.969378i	$0.421024\pi$
$410$	3.25227			0.160618
$411$		0			0
$412$		−	13.5396i		−	0.667049i
$413$	3.09557	−	7.58258i	0.152323	−	0.373114i
$414$		0			0
$415$		−	5.91288i		−	0.290252i
$416$		−	3.09557i		−	0.151773i
$417$		0			0
$418$	−9.93280	+	15.4779i	−0.485829	+	0.757047i
$419$			5.67991i			0.277482i	0.990329	+	0.138741i	$0.0443055\pi$
$419$			5.67991i			0.277482i	−0.990329	+	0.138741i	$0.955694\pi$
$420$		0			0
$421$	−1.58258			−0.0771300			−0.0385650	−	0.999256i	$-0.512279\pi$
$421$	−1.58258			−0.0771300			−0.0385650	+	0.999256i	$0.512279\pi$
$422$	10.7477			0.523191
$423$		0			0
$424$		−	2.41742i		−	0.117401i
$425$	−17.1464			−0.831724
$426$		0			0
$427$	−22.7477	−	9.28672i	−1.10084	−	0.449416i
$428$			8.41742i			0.406872i
$429$		0			0
$430$		−	7.21362i		−	0.347872i
$431$		−	17.9129i		−	0.862833i	−0.902153	−	0.431416i	$-0.858014\pi$
$431$		−	17.9129i		−	0.862833i	0.902153	−	0.431416i	$-0.141986\pi$
$432$		0			0
$433$			28.1017i			1.35048i	0.737597	+	0.675241i	$0.235960\pi$
$433$			28.1017i			1.35048i	−0.737597	+	0.675241i	$0.764040\pi$
$434$	2.83485	+	1.15732i	0.136077	+	0.0555532i
$435$		0			0
$436$			1.16515i			0.0558006i
$437$	22.1803			1.06103
$438$		0			0
$439$	2.31464			0.110472			0.0552360	−	0.998473i	$-0.482409\pi$
$439$	2.31464			0.110472			0.0552360	+	0.998473i	$0.482409\pi$
$440$	−1.80341	−	1.15732i	−0.0859740	−	0.0551732i
$441$		0			0
$442$			11.5826i			0.550927i
$443$	3.16515			0.150381			0.0751904	−	0.997169i	$-0.476044\pi$
$443$	3.16515			0.150381			0.0751904	+	0.997169i	$0.476044\pi$
$444$		0			0
$445$	6.33030			0.300085
$446$	6.32599			0.299544
$447$		0			0
$448$	−2.44949	−	1.00000i	−0.115728	−	0.0472456i
$449$	−12.8348			−0.605714			−0.302857	−	0.953036i	$-0.597940\pi$
$449$	−12.8348			−0.605714			−0.302857	+	0.953036i	$0.597940\pi$
$450$		0			0
$451$	14.0509	+	9.01703i	0.661629	+	0.424595i
$452$	−16.7477			−0.787747
$453$		0			0
$454$			11.7362i			0.550808i
$455$	2.00000	−	4.89898i	0.0937614	−	0.229668i
$456$		0			0
$457$			14.8348i			0.693945i	0.937875	+	0.346972i	$0.112790\pi$
$457$			14.8348i			0.693945i	−0.937875	+	0.346972i	$0.887210\pi$
$458$	−13.0284			−0.608776
$459$		0			0
$460$			2.58434i			0.120495i
$461$	−26.2983			−1.22483			−0.612417	−	0.790535i	$-0.709803\pi$
$461$	−26.2983			−1.22483			−0.612417	+	0.790535i	$0.709803\pi$
$462$		0			0
$463$	−13.1652			−0.611836			−0.305918	−	0.952058i	$-0.598963\pi$
$463$	−13.1652			−0.611836			−0.305918	+	0.952058i	$0.598963\pi$
$464$			7.58258i			0.352012i
$465$		0			0
$466$	−8.83485			−0.409266
$467$			0.511238i			0.0236573i	0.999930	+	0.0118286i	$0.00376526\pi$
$467$			0.511238i			0.0236573i	−0.999930	+	0.0118286i	$0.996235\pi$
$468$		0			0
$469$	3.87650	+	1.58258i	0.179000	+	0.0730766i
$470$			3.25227i			0.150016i
$471$		0			0
$472$	−3.09557			−0.142485
$473$	20.0000	−	31.1652i	0.919601	−	1.43298i
$474$		0			0
$475$	−25.4107			−1.16592
$476$	9.16515	+	3.74166i	0.420084	+	0.171499i
$477$		0			0
$478$	−17.5826			−0.804208
$479$	6.46084			0.295203			0.147602	−	0.989047i	$-0.452845\pi$
$479$	6.46084			0.295203			0.147602	+	0.989047i	$0.452845\pi$
$480$		0			0
$481$	17.2813			0.787958
$482$		−	25.9219i		−	1.18071i
$483$		0			0
$484$	−4.58258	−	10.0000i	−0.208299	−	0.454545i
$485$	10.3303			0.469075
$486$		0			0
$487$	−33.4955			−1.51782			−0.758912	−	0.651193i	$-0.774269\pi$
$487$	−33.4955			−1.51782			−0.758912	+	0.651193i	$0.774269\pi$
$488$			9.28672i			0.420390i
$489$		0			0
$490$	−3.23042	−	3.16515i	−0.145935	−	0.142987i
$491$			21.4955i			0.970076i	0.874493	+	0.485038i	$0.161194\pi$
$491$			21.4955i			0.970076i	−0.874493	+	0.485038i	$0.838806\pi$
$492$		0			0
$493$		−	28.3714i		−	1.27778i
$494$			17.1652i			0.772297i
$495$		0			0
$496$		−	1.15732i		−	0.0519653i
$497$	−4.89898	−	2.00000i	−0.219749	−	0.0897123i
$498$		0			0
$499$	27.9129			1.24955			0.624776	−	0.780804i	$-0.285190\pi$
$499$	27.9129			1.24955			0.624776	+	0.780804i	$0.285190\pi$
$500$		−	6.19115i		−	0.276877i
$501$		0			0
$502$	−2.07310			−0.0925268
$503$	31.9782			1.42584			0.712919	−	0.701246i	$-0.247373\pi$
$503$	31.9782			1.42584			0.712919	+	0.701246i	$0.247373\pi$
$504$		0			0
$505$			0.330303i			0.0146983i
$506$	−7.16515	+	11.1652i	−0.318530	+	0.496352i
$507$		0			0
$508$		−	5.58258i		−	0.247687i
$509$			17.9274i			0.794616i	0.917685	+	0.397308i	$0.130056\pi$
$509$			17.9274i			0.794616i	−0.917685	+	0.397308i	$0.869944\pi$
$510$		0			0
$511$	−15.4955	−	6.32599i	−0.685479	−	0.279845i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−4.89898			−0.216085
$515$	−8.74773			−0.385471
$516$		0			0
$517$	−9.01703	+	14.0509i	−0.396569	+	0.617956i
$518$	5.58258	−	13.6745i	0.245284	−	0.600821i
$519$		0			0
$520$	−2.00000			−0.0877058
$521$			35.5850i			1.55901i	0.626397	+	0.779504i	$0.284529\pi$
$521$			35.5850i			1.55901i	−0.626397	+	0.779504i	$0.715471\pi$
$522$		0			0
$523$	26.7028			1.16763			0.583817	−	0.811885i	$-0.301559\pi$
$523$	26.7028			1.16763			0.583817	+	0.811885i	$0.301559\pi$
$524$	9.42157			0.411583
$525$		0			0
$526$	28.3303			1.23526
$527$			4.33030i			0.188631i
$528$		0			0
$529$	−7.00000			−0.304348
$530$	−1.56186			−0.0678428
$531$		0			0
$532$	13.5826	+	5.54506i	0.588879	+	0.240409i
$533$	15.5826			0.674956
$534$		0			0
$535$	5.43836			0.235121
$536$		−	1.58258i		−	0.0683569i
$537$		0			0
$538$	−20.2420			−0.872695
$539$	−5.18096	−	22.6309i	−0.223160	−	0.974782i
$540$		0			0
$541$		−	9.25227i		−	0.397786i	−0.980021	−	0.198893i	$-0.936265\pi$
$541$		−	9.25227i		−	0.397786i	0.980021	−	0.198893i	$-0.0637347\pi$
$542$		−	4.89898i		−	0.210429i
$543$		0			0
$544$		−	3.74166i		−	0.160422i
$545$	0.752785			0.0322458
$546$		0			0
$547$			2.74773i			0.117484i	0.998273	+	0.0587422i	$0.0187090\pi$
$547$			2.74773i			0.117484i	−0.998273	+	0.0587422i	$0.981291\pi$
$548$	15.1652			0.647823
$549$		0			0
$550$	8.20871	−	12.7913i	0.350021	−	0.545422i
$551$		−	42.0459i		−	1.79121i
$552$		0			0
$553$	4.00000	−	9.79796i	0.170097	−	0.416652i
$554$	−2.00000			−0.0849719
$555$		0			0
$556$	−3.23042			−0.137000
$557$		−	9.91288i		−	0.420022i	−0.977699	−	0.210011i	$-0.932650\pi$
$557$		−	9.91288i		−	0.420022i	0.977699	−	0.210011i	$-0.0673500\pi$
$558$		0			0
$559$		−	34.5625i		−	1.46184i
$560$	−0.646084	+	1.58258i	−0.0273020	+	0.0668760i
$561$		0			0
$562$	−7.16515			−0.302244
$563$	−42.4223			−1.78788			−0.893942	−	0.448182i	$-0.852072\pi$
$563$	−42.4223			−1.78788			−0.893942	+	0.448182i	$0.852072\pi$
$564$		0			0
$565$			10.8204i			0.455219i
$566$			15.6127i			0.656251i
$567$		0			0
$568$			2.00000i			0.0839181i
$569$			15.4955i			0.649603i	0.945782	+	0.324802i	$0.105298\pi$
$569$			15.4955i			0.649603i	−0.945782	+	0.324802i	$0.894702\pi$
$570$		0			0
$571$			3.58258i			0.149926i	0.997186	+	0.0749631i	$0.0238839\pi$
$571$			3.58258i			0.149926i	−0.997186	+	0.0749631i	$0.976116\pi$
$572$	−8.64064	−	5.54506i	−0.361283	−	0.231851i
$573$		0			0
$574$	5.03383	−	12.3303i	0.210108	−	0.514657i
$575$	−18.3303			−0.764426
$576$		0			0
$577$		−	0.269691i		−	0.0112274i	−0.999984	−	0.00561369i	$-0.998213\pi$
$577$		−	0.269691i		−	0.0112274i	0.999984	−	0.00561369i	$-0.00178690\pi$
$578$		−	3.00000i		−	0.124784i
$579$		0			0
$580$	4.89898			0.203419
$581$	−22.4174	−	9.15188i	−0.930031	−	0.379684i
$582$		0			0
$583$	−6.74773	−	4.33030i	−0.279462	−	0.179343i
$584$			6.32599i			0.261771i
$585$		0			0
$586$		−	21.3993i		−	0.883998i
$587$		−	17.0397i		−	0.703305i	−0.936131	−	0.351652i	$-0.885620\pi$
$587$		−	17.0397i		−	0.703305i	0.936131	−	0.351652i	$-0.114380\pi$
$588$		0			0
$589$			6.41742i			0.264425i
$590$			2.00000i			0.0823387i
$591$		0			0
$592$	−5.58258			−0.229442
$593$	−0.134846			−0.00553744			−0.00276872	−	0.999996i	$-0.500881\pi$
$593$	−0.134846			−0.00553744			−0.00276872	+	0.999996i	$0.500881\pi$
$594$		0			0
$595$	2.41742	−	5.92146i	0.0991047	−	0.242756i
$596$		−	14.0000i		−	0.573462i
$597$		0			0
$598$			12.3823i			0.506350i
$599$	2.83485			0.115829			0.0579144	−	0.998322i	$-0.481555\pi$
$599$	2.83485			0.115829			0.0579144	+	0.998322i	$0.481555\pi$
$600$		0			0
$601$	−1.42701			−0.0582091			−0.0291045	−	0.999576i	$-0.509266\pi$
$601$	−1.42701			−0.0582091			−0.0291045	+	0.999576i	$0.509266\pi$
$602$	−27.3489	−	11.1652i	−1.11466	−	0.455058i
$603$		0			0
$604$			3.58258i			0.145773i
$605$	−6.46084	+	2.96073i	−0.262670	+	0.120371i
$606$		0			0
$607$	46.9448			1.90543			0.952716	−	0.303862i	$-0.0982761\pi$
$607$	46.9448			1.90543			0.952716	+	0.303862i	$0.0982761\pi$
$608$		−	5.54506i		−	0.224882i
$609$		0			0
$610$	6.00000			0.242933
$611$			15.5826i			0.630404i
$612$		0			0
$613$		−	13.9129i		−	0.561936i	−0.959717	−	0.280968i	$-0.909345\pi$
$613$		−	13.9129i		−	0.561936i	0.959717	−	0.280968i	$-0.0906555\pi$
$614$			8.12940i			0.328076i
$615$		0			0
$616$	−7.17903	+	5.04594i	−0.289251	+	0.203307i
$617$	−25.9129			−1.04321			−0.521607	−	0.853186i	$-0.674667\pi$
$617$	−25.9129			−1.04321			−0.521607	+	0.853186i	$0.674667\pi$
$618$		0			0
$619$			0.780929i			0.0313882i	0.999877	+	0.0156941i	$0.00499579\pi$
$619$			0.780929i			0.0313882i	−0.999877	+	0.0156941i	$0.995004\pi$
$620$	−0.747727			−0.0300294
$621$		0			0
$622$	−9.93280			−0.398269
$623$	9.79796	−	24.0000i	0.392547	−	0.961540i
$624$		0			0
$625$	18.9129			0.756515
$626$	1.02248			0.0408664
$627$		0			0
$628$		−	20.5117i		−	0.818506i
$629$	20.8881			0.832863
$630$		0			0
$631$	−5.16515			−0.205621			−0.102811	−	0.994701i	$-0.532784\pi$
$631$	−5.16515			−0.205621			−0.102811	+	0.994701i	$0.532784\pi$
$632$	−4.00000			−0.159111
$633$		0			0
$634$			9.16515i			0.363995i
$635$	−3.60681			−0.143132
$636$		0			0
$637$	−15.4779	−	15.1652i	−0.613255	−	0.600865i
$638$	21.1652	+	13.5826i	0.837936	+	0.537739i
$639$		0			0
$640$	0.646084			0.0255387
$641$	−29.9129			−1.18149			−0.590744	−	0.806859i	$-0.701166\pi$
$641$	−29.9129			−1.18149			−0.590744	+	0.806859i	$0.701166\pi$
$642$		0			0
$643$			16.7700i			0.661346i	0.943745	+	0.330673i	$0.107276\pi$
$643$			16.7700i			0.661346i	−0.943745	+	0.330673i	$0.892724\pi$
$644$	9.79796	+	4.00000i	0.386094	+	0.157622i
$645$		0			0
$646$			20.7477i			0.816308i
$647$		−	44.7650i		−	1.75990i	−0.475071	−	0.879948i	$-0.657577\pi$
$647$		−	44.7650i		−	1.75990i	0.475071	−	0.879948i	$-0.342423\pi$
$648$		0			0
$649$	−5.54506	+	8.64064i	−0.217663	+	0.339175i
$650$		−	14.1857i		−	0.556409i
$651$		0			0
$652$	−4.00000			−0.156652
$653$	6.00000			0.234798			0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000			0.234798			0.117399	+	0.993085i	$0.462544\pi$
$654$		0			0
$655$		−	6.08712i		−	0.237844i
$656$	−5.03383			−0.196538
$657$		0			0
$658$	12.3303	+	5.03383i	0.480685	+	0.196239i
$659$		−	30.3303i		−	1.18150i	−0.806854	−	0.590750i	$-0.798832\pi$
$659$		−	30.3303i		−	1.18150i	0.806854	−	0.590750i	$-0.201168\pi$
$660$		0			0
$661$		−	25.4107i		−	0.988361i	−0.869359	−	0.494180i	$-0.835468\pi$
$661$		−	25.4107i		−	0.988361i	0.869359	−	0.494180i	$-0.164532\pi$
$662$		−	14.4174i		−	0.560349i
$663$		0			0
$664$			9.15188i			0.355162i
$665$	3.58258	−	8.77548i	0.138926	−	0.340299i
$666$		0			0
$667$		−	30.3303i		−	1.17439i
$668$	−6.19115			−0.239543
$669$		0			0
$670$	−1.02248			−0.0395017
$671$	25.9219	+	16.6352i	1.00070	+	0.642194i
$672$		0			0
$673$			18.3303i			0.706581i	0.935514	+	0.353291i	$0.114937\pi$
$673$			18.3303i			0.706581i	−0.935514	+	0.353291i	$0.885063\pi$
$674$	−29.1652			−1.12340
$675$		0			0
$676$	3.41742			0.131439
$677$	−31.4670			−1.20937			−0.604687	−	0.796463i	$-0.706702\pi$
$677$	−31.4670			−1.20937			−0.604687	+	0.796463i	$0.706702\pi$
$678$		0			0
$679$	15.9891	−	39.1652i	0.613606	−	1.50302i
$680$	−2.41742			−0.0927040
$681$		0			0
$682$	−3.23042	−	2.07310i	−0.123699	−	0.0793830i
$683$	−45.4955			−1.74084			−0.870418	−	0.492314i	$-0.836151\pi$
$683$	−45.4955			−1.74084			−0.870418	+	0.492314i	$0.836151\pi$
$684$		0			0
$685$		−	9.79796i		−	0.374361i
$686$	−17.0000	+	7.34847i	−0.649063	+	0.280566i
$687$		0			0
$688$			11.1652i			0.425667i
$689$	−7.48331			−0.285092
$690$		0			0
$691$		−	37.6581i		−	1.43258i	−0.697801	−	0.716291i	$-0.745838\pi$
$691$		−	37.6581i		−	1.43258i	0.697801	−	0.716291i	$-0.254162\pi$
$692$	−22.9612			−0.872853
$693$		0			0
$694$	−0.834849			−0.0316904
$695$			2.08712i			0.0791690i
$696$		0			0
$697$	18.8348			0.713421
$698$		−	2.07310i		−	0.0784679i
$699$		0			0
$700$	−11.2250	−	4.58258i	−0.424264	−	0.173205i
$701$			24.3303i			0.918943i	0.888193	+	0.459471i	$0.151961\pi$
$701$			24.3303i			0.918943i	−0.888193	+	0.459471i	$0.848039\pi$
$702$		0			0
$703$	30.9557			1.16752
$704$	2.79129	+	1.79129i	0.105201	+	0.0675117i
$705$		0			0
$706$	−7.48331			−0.281638
$707$	1.25227	+	0.511238i	0.0470966	+	0.0192271i
$708$		0			0
$709$	15.6697			0.588488			0.294244	−	0.955730i	$-0.404932\pi$
$709$	15.6697			0.588488			0.294244	+	0.955730i	$0.404932\pi$
$710$	1.29217			0.0484942
$711$		0			0
$712$	−9.79796			−0.367194
$713$			4.62929i			0.173368i
$714$		0			0
$715$	−3.58258	+	5.58258i	−0.133981	+	0.208776i
$716$	−7.16515			−0.267774
$717$		0			0
$718$	−0.417424			−0.0155781
$719$			15.8543i			0.591264i	0.955302	+	0.295632i	$0.0955302\pi$
$719$			15.8543i			0.591264i	−0.955302	+	0.295632i	$0.904470\pi$
$720$		0			0
$721$	−13.5396	+	33.1652i	−0.504242	+	1.23513i
$722$			11.7477i			0.437205i
$723$		0			0
$724$		−	16.6352i		−	0.618242i
$725$			34.7477i			1.29050i
$726$		0			0
$727$		−	52.2484i		−	1.93778i	−0.247483	−	0.968892i	$-0.579604\pi$
$727$		−	52.2484i		−	1.93778i	0.247483	−	0.968892i	$-0.420396\pi$
$728$	−3.09557	+	7.58258i	−0.114730	+	0.281029i
$729$		0			0
$730$	4.08712			0.151271
$731$		−	41.7762i		−	1.54515i
$732$		0			0
$733$	−22.9612			−0.848091			−0.424045	−	0.905641i	$-0.639390\pi$
$733$	−22.9612			−0.848091			−0.424045	+	0.905641i	$0.639390\pi$
$734$	1.42701			0.0526720
$735$		0			0
$736$		−	4.00000i		−	0.147442i
$737$	−4.41742	−	2.83485i	−0.162718	−	0.104423i
$738$		0			0
$739$		−	7.58258i		−	0.278930i	−0.990227	−	0.139465i	$-0.955462\pi$
$739$		−	7.58258i		−	0.278930i	0.990227	−	0.139465i	$-0.0445382\pi$
$740$			3.60681i			0.132589i
$741$		0			0
$742$	−2.41742	+	5.92146i	−0.0887464	+	0.217383i
$743$			43.9129i			1.61101i	0.592591	+	0.805504i	$0.298105\pi$
$743$			43.9129i			1.61101i	−0.592591	+	0.805504i	$0.701895\pi$
$744$		0			0
$745$	−9.04517			−0.331390
$746$	−0.417424			−0.0152830
$747$		0			0
$748$	−10.4440	−	6.70239i	−0.381872	−	0.245063i
$749$	8.41742	−	20.6184i	0.307566	−	0.753380i
$750$		0			0
$751$	13.4955			0.492456			0.246228	−	0.969212i	$-0.420809\pi$
$751$	13.4955			0.492456			0.246228	+	0.969212i	$0.420809\pi$
$752$		−	5.03383i		−	0.183565i
$753$		0			0
$754$	23.4724			0.854815
$755$	2.31464			0.0842385
$756$		0			0
$757$	−49.1652			−1.78694			−0.893469	−	0.449125i	$-0.851736\pi$
$757$	−49.1652			−1.78694			−0.893469	+	0.449125i	$0.851736\pi$
$758$			14.3303i			0.520500i
$759$		0			0
$760$	−3.58258			−0.129954
$761$	−20.7532			−0.752304			−0.376152	−	0.926558i	$-0.622753\pi$
$761$	−20.7532			−0.752304			−0.376152	+	0.926558i	$0.622753\pi$
$762$		0			0
$763$	1.16515	−	2.85403i	0.0421813	−	0.103323i
$764$	−18.0000			−0.651217
$765$		0			0
$766$	4.76413			0.172135
$767$			9.58258i			0.346007i
$768$		0			0
$769$	29.7984			1.07456			0.537279	−	0.843404i	$-0.319452\pi$
$769$	29.7984			1.07456			0.537279	+	0.843404i	$0.319452\pi$
$770$	3.26010	+	4.63825i	0.117486	+	0.167151i
$771$		0			0
$772$			20.3303i			0.731704i
$773$		−	51.1977i		−	1.84145i	−0.390207	−	0.920727i	$-0.627596\pi$
$773$		−	51.1977i		−	1.84145i	0.390207	−	0.920727i	$-0.372404\pi$
$774$		0			0
$775$		−	5.30352i		−	0.190508i
$776$	−15.9891			−0.573975
$777$		0			0
$778$		−	10.0000i		−	0.358517i
$779$	27.9129			1.00008
$780$		0			0
$781$	5.58258	+	3.58258i	0.199760	+	0.128195i
$782$			14.9666i			0.535206i
$783$		0			0
$784$	5.00000	+	4.89898i	0.178571	+	0.174964i
$785$	−13.2523			−0.472994
$786$		0			0
$787$	−40.3773			−1.43930			−0.719648	−	0.694339i	$-0.755697\pi$
$787$	−40.3773			−1.43930			−0.719648	+	0.694339i	$0.755697\pi$
$788$		−	18.7477i		−	0.667860i
$789$		0			0
$790$			2.58434i			0.0919465i
$791$	41.0234	+	16.7477i	1.45862	+	0.595481i
$792$		0			0
$793$	28.7477			1.02086
$794$	23.8488			0.846363
$795$		0			0
$796$			24.6297i			0.872978i
$797$		−	36.2311i		−	1.28337i	−0.766968	−	0.641686i	$-0.778235\pi$
$797$		−	36.2311i		−	1.28337i	0.766968	−	0.641686i	$-0.221765\pi$
$798$		0			0
$799$			18.8348i			0.666329i
$800$			4.58258i			0.162019i
$801$		0			0
$802$			1.58258i			0.0558827i
$803$	17.6577	+	11.3317i	0.623126	+	0.399886i
$804$		0			0
$805$	2.58434	−	6.33030i	0.0910859	−	0.223114i
$806$	−3.58258			−0.126191
$807$		0			0
$808$		−	0.511238i		−	0.0179853i
$809$		−	33.1652i		−	1.16602i	−0.812463	−	0.583012i	$-0.801874\pi$
$809$		−	33.1652i		−	1.16602i	0.812463	−	0.583012i	$-0.198126\pi$
$810$		0			0
$811$	−30.3097			−1.06432			−0.532158	−	0.846645i	$-0.678619\pi$
$811$	−30.3097			−1.06432			−0.532158	+	0.846645i	$0.678619\pi$
$812$	7.58258	−	18.5734i	0.266096	−	0.651800i
$813$		0			0
$814$	−10.0000	+	15.5826i	−0.350500	+	0.546169i
$815$			2.58434i			0.0905253i
$816$		0			0
$817$		−	61.9115i		−	2.16601i
$818$			9.93280i			0.347292i
$819$		0			0
$820$			3.25227i			0.113574i
$821$		−	11.4955i		−	0.401194i	−0.979674	−	0.200597i	$-0.935712\pi$
$821$		−	11.4955i		−	0.401194i	0.979674	−	0.200597i	$-0.0642882\pi$
$822$		0			0
$823$	−40.3303			−1.40583			−0.702913	−	0.711276i	$-0.748118\pi$
$823$	−40.3303			−1.40583			−0.702913	+	0.711276i	$0.748118\pi$
$824$	13.5396			0.471675
$825$		0			0
$826$	7.58258	+	3.09557i	0.263832	+	0.107709i
$827$			15.1652i			0.527344i	0.964612	+	0.263672i	$0.0849337\pi$
$827$			15.1652i			0.527344i	−0.964612	+	0.263672i	$0.915066\pi$
$828$		0			0
$829$			4.25290i			0.147709i	0.997269	+	0.0738546i	$0.0235301\pi$
$829$			4.25290i			0.147709i	−0.997269	+	0.0738546i	$0.976470\pi$
$830$	5.91288			0.205239
$831$		0			0
$832$	3.09557			0.107320
$833$	−18.7083	−	18.3303i	−0.648204	−	0.635107i
$834$		0			0
$835$			4.00000i			0.138426i
$836$	−15.4779	−	9.93280i	−0.535313	−	0.343533i
$837$		0			0
$838$	−5.67991			−0.196209
$839$			24.6297i			0.850313i	0.905120	+	0.425157i	$0.139781\pi$
$839$			24.6297i			0.850313i	−0.905120	+	0.425157i	$0.860219\pi$
$840$		0			0
$841$	−28.4955			−0.982602
$842$		−	1.58258i		−	0.0545392i
$843$		0			0
$844$			10.7477i			0.369952i
$845$		−	2.20794i		−	0.0759555i
$846$		0			0
$847$	1.22497	+	29.0775i	0.0420905	+	0.999114i
$848$	2.41742			0.0830147
$849$		0			0
$850$		−	17.1464i		−	0.588118i
$851$	22.3303			0.765473
$852$		0			0
$853$	30.1748			1.03317			0.516583	−	0.856237i	$-0.327204\pi$
$853$	30.1748			1.03317			0.516583	+	0.856237i	$0.327204\pi$
$854$	9.28672	−	22.7477i	0.317785	−	0.778411i
$855$		0			0
$856$	−8.41742			−0.287702
$857$	−21.2926			−0.727342			−0.363671	−	0.931527i	$-0.618477\pi$
$857$	−21.2926			−0.727342			−0.363671	+	0.931527i	$0.618477\pi$
$858$		0			0
$859$			31.7367i			1.08284i	0.840752	+	0.541421i	$0.182113\pi$
$859$			31.7367i			1.08284i	−0.840752	+	0.541421i	$0.817887\pi$
$860$	7.21362			0.245983
$861$		0			0
$862$	17.9129			0.610115
$863$	33.1652			1.12895			0.564477	−	0.825448i	$-0.309078\pi$
$863$	33.1652			1.12895			0.564477	+	0.825448i	$0.309078\pi$
$864$		0			0
$865$			14.8348i			0.504400i
$866$	−28.1017			−0.954935
$867$		0			0
$868$	−1.15732	+	2.83485i	−0.0392821	+	0.0962210i
$869$	−7.16515	+	11.1652i	−0.243061	+	0.378752i
$870$		0			0
$871$	−4.89898			−0.165996
$872$	−1.16515			−0.0394570
$873$		0			0
$874$			22.1803i			0.750258i
$875$	−6.19115	+	15.1652i	−0.209299	+	0.512676i
$876$		0			0
$877$		−	25.1652i		−	0.849767i	−0.905248	−	0.424883i	$-0.860315\pi$
$877$		−	25.1652i		−	0.849767i	0.905248	−	0.424883i	$-0.139685\pi$
$878$			2.31464i			0.0781155i
$879$		0			0
$880$	1.15732	−	1.80341i	0.0390133	−	0.0607928i
$881$		−	22.1803i		−	0.747272i	−0.927575	−	0.373636i	$-0.878111\pi$
$881$		−	22.1803i		−	0.747272i	0.927575	−	0.373636i	$-0.121889\pi$
$882$		0			0
$883$	31.1652			1.04879			0.524395	−	0.851475i	$-0.324291\pi$
$883$	31.1652			1.04879			0.524395	+	0.851475i	$0.324291\pi$
$884$	−11.5826			−0.389564
$885$		0			0
$886$			3.16515i			0.106335i
$887$	−28.9108			−0.970729			−0.485365	−	0.874312i	$-0.661313\pi$
$887$	−28.9108			−0.970729			−0.485365	+	0.874312i	$0.661313\pi$
$888$		0			0
$889$	−5.58258	+	13.6745i	−0.187234	+	0.458627i
$890$			6.33030i			0.212192i
$891$		0			0
$892$			6.32599i			0.211810i
$893$			27.9129i			0.934069i
$894$		0			0
$895$			4.62929i			0.154740i
$896$	1.00000	−	2.44949i	0.0334077	−	0.0818317i
$897$		0			0
$898$		−	12.8348i		−	0.428304i
$899$	8.77548			0.292679
$900$		0			0
$901$	−9.04517			−0.301338
$902$	−9.01703	+	14.0509i	−0.300234	+	0.467842i
$903$		0			0
$904$		−	16.7477i		−	0.557021i
$905$	−10.7477			−0.357267
$906$		0			0
$907$	−39.9129			−1.32529			−0.662643	−	0.748936i	$-0.730565\pi$
$907$	−39.9129			−1.32529			−0.662643	+	0.748936i	$0.730565\pi$
$908$	−11.7362			−0.389480
$909$		0			0
$910$	4.89898	+	2.00000i	0.162400	+	0.0662994i
$911$	15.1652			0.502444			0.251222	−	0.967930i	$-0.419168\pi$
$911$	15.1652			0.502444			0.251222	+	0.967930i	$0.419168\pi$
$912$		0			0
$913$	25.5455	+	16.3936i	0.845433	+	0.542550i
$914$	−14.8348			−0.490693
$915$		0			0
$916$		−	13.0284i		−	0.430470i
$917$	−23.0780	−	9.42157i	−0.762104	−	0.311128i
$918$		0			0
$919$		−	35.4955i		−	1.17089i	−0.810713	−	0.585443i	$-0.800920\pi$
$919$		−	35.4955i		−	1.17089i	0.810713	−	0.585443i	$-0.199080\pi$
$920$	−2.58434			−0.0852030
$921$		0			0
$922$		−	26.2983i		−	0.866088i
$923$	6.19115			0.203784
$924$		0			0
$925$	−25.5826			−0.841150
$926$		−	13.1652i		−	0.432634i
$927$		0			0
$928$	−7.58258			−0.248910
$929$			38.7087i			1.26999i	0.772515	+	0.634996i	$0.218998\pi$
$929$			38.7087i			1.26999i	−0.772515	+	0.634996i	$0.781002\pi$
$930$		0			0
$931$	−27.7253	−	27.1652i	−0.908661	−	0.890302i
$932$		−	8.83485i		−	0.289395i
$933$		0			0
$934$	−0.511238			−0.0167282
$935$	−4.33030	+	6.74773i	−0.141616	+	0.220674i
$936$		0			0
$937$	−44.7650			−1.46241			−0.731205	−	0.682158i	$-0.761042\pi$
$937$	−44.7650			−1.46241			−0.731205	+	0.682158i	$0.761042\pi$
$938$	−1.58258	+	3.87650i	−0.0516729	+	0.126572i
$939$		0			0
$940$	−3.25227			−0.106077
$941$	31.4670			1.02579			0.512897	−	0.858450i	$-0.328572\pi$
$941$	31.4670			1.02579			0.512897	+	0.858450i	$0.328572\pi$
$942$		0			0
$943$	20.1353			0.655696
$944$		−	3.09557i		−	0.100752i
$945$		0			0
$946$	31.1652	+	20.0000i	1.01327	+	0.650256i
$947$	5.58258			0.181409			0.0907047	−	0.995878i	$-0.471088\pi$
$947$	5.58258			0.181409			0.0907047	+	0.995878i	$0.471088\pi$
$948$		0			0
$949$	19.5826			0.635677
$950$		−	25.4107i		−	0.824431i
$951$		0			0
$952$	−3.74166	+	9.16515i	−0.121268	+	0.297044i
$953$		−	18.3303i		−	0.593777i	−0.954912	−	0.296888i	$-0.904051\pi$
$953$		−	18.3303i		−	0.593777i	0.954912	−	0.296888i	$-0.0959489\pi$
$954$		0			0
$955$			11.6295i			0.376322i
$956$		−	17.5826i		−	0.568661i
$957$		0			0
$958$			6.46084i			0.208740i
$959$	−37.1469	−	15.1652i	−1.19954	−	0.489708i
$960$		0			0
$961$	29.6606			0.956794
$962$			17.2813i			0.557171i
$963$		0			0
$964$	25.9219			0.834889
$965$	13.1351			0.422833
$966$		0			0
$967$		−	31.5826i		−	1.01563i	−0.861467	−	0.507814i	$-0.830454\pi$
$967$		−	31.5826i		−	1.01563i	0.861467	−	0.507814i	$-0.169546\pi$
$968$	10.0000	−	4.58258i	0.321412	−	0.147290i
$969$		0			0
$970$			10.3303i			0.331686i
$971$		−	17.7925i		−	0.570989i	−0.958380	−	0.285494i	$-0.907842\pi$
$971$		−	17.7925i		−	0.570989i	0.958380	−	0.285494i	$-0.0921578\pi$
$972$		0			0
$973$	7.91288	+	3.23042i	0.253675	+	0.103562i
$974$		−	33.4955i		−	1.07326i
$975$		0			0
$976$	−9.28672			−0.297261
$977$	9.16515			0.293219			0.146610	−	0.989194i	$-0.453164\pi$
$977$	9.16515			0.293219			0.146610	+	0.989194i	$0.453164\pi$
$978$		0			0
$979$	−17.5510	+	27.3489i	−0.560931	+	0.874075i
$980$	3.16515	−	3.23042i	0.101107	−	0.103192i
$981$		0			0
$982$	−21.4955			−0.685948
$983$			1.69670i			0.0541165i	0.999634	+	0.0270582i	$0.00861395\pi$
$983$			1.69670i			0.0541165i	−0.999634	+	0.0270582i	$0.991386\pi$
$984$		0			0
$985$	−12.1126			−0.385940
$986$	28.3714			0.903529
$987$		0			0
$988$	−17.1652			−0.546096
$989$		−	44.6606i		−	1.42012i
$990$		0			0
$991$	−26.6606			−0.846902			−0.423451	−	0.905919i	$-0.639181\pi$
$991$	−26.6606			−0.846902			−0.423451	+	0.905919i	$0.639181\pi$
$992$	1.15732			0.0367450
$993$		0			0
$994$	2.00000	−	4.89898i	0.0634361	−	0.155386i
$995$	15.9129			0.504472
$996$		0			0
$997$	33.7816			1.06987			0.534937	−	0.844892i	$-0.320335\pi$
$997$	33.7816			1.06987			0.534937	+	0.844892i	$0.320335\pi$
$998$			27.9129i			0.883567i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.e.b.307.7		8
3.2	odd	2		154.2.c.a.153.1	✓	8
7.6	odd	2	inner	1386.2.e.b.307.6		8
11.10	odd	2	inner	1386.2.e.b.307.3		8
12.11	even	2		1232.2.e.e.769.7		8
21.2	odd	6		1078.2.i.b.1011.4		16
21.5	even	6		1078.2.i.b.1011.1		16
21.11	odd	6		1078.2.i.b.901.5		16
21.17	even	6		1078.2.i.b.901.8		16
21.20	even	2		154.2.c.a.153.4	yes	8
33.32	even	2		154.2.c.a.153.5	yes	8
77.76	even	2	inner	1386.2.e.b.307.2		8
84.83	odd	2		1232.2.e.e.769.2		8
132.131	odd	2		1232.2.e.e.769.8		8
231.32	even	6		1078.2.i.b.901.1		16
231.65	even	6		1078.2.i.b.1011.8		16
231.131	odd	6		1078.2.i.b.1011.5		16
231.164	odd	6		1078.2.i.b.901.4		16
231.230	odd	2		154.2.c.a.153.8	yes	8
924.923	even	2		1232.2.e.e.769.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.c.a.153.1	✓	8	3.2	odd	2
154.2.c.a.153.4	yes	8	21.20	even	2
154.2.c.a.153.5	yes	8	33.32	even	2
154.2.c.a.153.8	yes	8	231.230	odd	2
1078.2.i.b.901.1		16	231.32	even	6
1078.2.i.b.901.4		16	231.164	odd	6
1078.2.i.b.901.5		16	21.11	odd	6
1078.2.i.b.901.8		16	21.17	even	6
1078.2.i.b.1011.1		16	21.5	even	6
1078.2.i.b.1011.4		16	21.2	odd	6
1078.2.i.b.1011.5		16	231.131	odd	6
1078.2.i.b.1011.8		16	231.65	even	6
1232.2.e.e.769.1		8	924.923	even	2
1232.2.e.e.769.2		8	84.83	odd	2
1232.2.e.e.769.7		8	12.11	even	2
1232.2.e.e.769.8		8	132.131	odd	2
1386.2.e.b.307.2		8	77.76	even	2	inner
1386.2.e.b.307.3		8	11.10	odd	2	inner
1386.2.e.b.307.6		8	7.6	odd	2	inner
1386.2.e.b.307.7		8	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1386.2.e.b.307.7

Level	$1386$
Weight	$2$
Character	1386.307
Analytic conductor	$11.067$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$