LMFDB - Embedded newform 1232.2.e.e.769.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(769,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.e (of order $2$, degree $1$, not 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:	$9.83756952902$
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			769.1
Root			$-1.54779 - 1.54779i$ of defining polynomial
Character	$\chi$	$=$	1232.769
Dual form			1232.2.e.e.769.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.09557i q^{3} +0.646084i q^{5} +(-2.44949 + 1.00000i) q^{7} -6.58258 q^{9} +O(q^{10})$	$q-3.09557i q^{3} +0.646084i q^{5} +(-2.44949 + 1.00000i) q^{7} -6.58258 q^{9} +(-2.79129 + 1.79129i) q^{11} -3.09557 q^{13} +2.00000 q^{15} +3.74166 q^{17} +5.54506 q^{19} +(3.09557 + 7.58258i) q^{21} -4.00000 q^{23} +4.58258 q^{25} +11.0901i q^{27} +7.58258i q^{29} -1.15732i q^{31} +(5.54506 + 8.64064i) q^{33} +(-0.646084 - 1.58258i) q^{35} -5.58258 q^{37} +9.58258i q^{39} +5.03383 q^{41} +11.1652i q^{43} -4.25290i q^{45} +5.03383i q^{47} +(5.00000 - 4.89898i) q^{49} -11.5826i q^{51} -2.41742 q^{53} +(-1.15732 - 1.80341i) q^{55} -17.1652i q^{57} +3.09557i q^{59} -9.28672 q^{61} +(16.1240 - 6.58258i) q^{63} -2.00000i q^{65} -1.58258 q^{67} +12.3823i q^{69} -2.00000 q^{71} -6.32599 q^{73} -14.1857i q^{75} +(5.04594 - 7.17903i) q^{77} -4.00000i q^{79} +14.5826 q^{81} -9.15188 q^{83} +2.41742i q^{85} +23.4724 q^{87} -9.79796i q^{89} +(7.58258 - 3.09557i) q^{91} -3.58258 q^{93} +3.58258i q^{95} +15.9891i q^{97} +(18.3739 - 11.7913i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^9	$ 8 q - 16 q^{9} - 4 q^{11} + 16 q^{15} - 32 q^{23} - 8 q^{37} + 40 q^{49} - 56 q^{53} + 24 q^{67} - 16 q^{71} + 4 q^{77} + 80 q^{81} + 24 q^{91} + 8 q^{93} + 92 q^{99}+O(q^{100}) $ 8 * q - 16 * q^9 - 4 * q^11 + 16 * q^15 - 32 * q^23 - 8 * q^37 + 40 * q^49 - 56 * q^53 + 24 * q^67 - 16 * q^71 + 4 * q^77 + 80 * q^81 + 24 * q^91 + 8 * q^93 + 92 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	3.09557i		−	1.78723i	−0.448834	−	0.893615i	$-0.648161\pi$
$3$		−	3.09557i		−	1.78723i	0.448834	−	0.893615i	$-0.351839\pi$
$4$		0			0
$5$			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
$8$		0			0
$9$	−6.58258			−2.19419
$10$		0			0
$11$	−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	$0.641232\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$		0			0
$17$	3.74166			0.907485			0.453743	−	0.891133i	$-0.350089\pi$
$17$	3.74166			0.907485			0.453743	+	0.891133i	$0.350089\pi$
$18$		0			0
$19$	5.54506			1.27212			0.636062	−	0.771638i	$-0.280562\pi$
$19$	5.54506			1.27212			0.636062	+	0.771638i	$0.280562\pi$
$20$		0			0
$21$	3.09557	+	7.58258i	0.675510	+	1.65465i
$22$		0			0
$23$	−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
$26$		0			0
$27$			11.0901i			2.13430i
$28$		0			0
$29$			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	$-0.0331420\pi$
$32$		0			0
$33$	5.54506	+	8.64064i	0.965272	+	1.50414i
$34$		0			0
$35$	−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	$0.651751\pi$
$38$		0			0
$39$			9.58258i			1.53444i
$40$		0			0
$41$	5.03383			0.786151			0.393076	−	0.919506i	$-0.371411\pi$
$41$	5.03383			0.786151			0.393076	+	0.919506i	$0.371411\pi$
$42$		0			0
$43$			11.1652i			1.70267i	0.524623	+	0.851335i	$0.324206\pi$
$43$			11.1652i			1.70267i	−0.524623	+	0.851335i	$0.675794\pi$
$44$		0			0
$45$		−	4.25290i		−	0.633984i
$46$		0			0
$47$			5.03383i			0.734259i	0.930170	+	0.367129i	$0.119659\pi$
$47$			5.03383i			0.734259i	−0.930170	+	0.367129i	$0.880341\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$		−	11.5826i		−	1.62189i
$52$		0			0
$53$	−2.41742			−0.332059			−0.166029	−	0.986121i	$-0.553095\pi$
$53$	−2.41742			−0.332059			−0.166029	+	0.986121i	$0.553095\pi$
$54$		0			0
$55$	−1.15732	−	1.80341i	−0.156053	−	0.243171i
$56$		0			0
$57$		−	17.1652i		−	2.27358i
$58$		0			0
$59$			3.09557i			0.403009i	0.979488	+	0.201505i	$0.0645831\pi$
$59$			3.09557i			0.403009i	−0.979488	+	0.201505i	$0.935417\pi$
$60$		0			0
$61$	−9.28672			−1.18904			−0.594521	−	0.804080i	$-0.702658\pi$
$61$	−9.28672			−1.18904			−0.594521	+	0.804080i	$0.702658\pi$
$62$		0			0
$63$	16.1240	−	6.58258i	2.03143	−	0.829327i
$64$		0			0
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$	−1.58258			−0.193342			−0.0966712	−	0.995316i	$-0.530820\pi$
$67$	−1.58258			−0.193342			−0.0966712	+	0.995316i	$0.530820\pi$
$68$		0			0
$69$			12.3823i			1.49065i
$70$		0			0
$71$	−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	$0.620711\pi$
$74$		0			0
$75$		−	14.1857i		−	1.63802i
$76$		0			0
$77$	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			0
$81$	14.5826			1.62029
$82$		0			0
$83$	−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
$86$		0			0
$87$	23.4724			2.51651
$88$		0			0
$89$		−	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
$92$		0			0
$93$	−3.58258			−0.371496
$94$		0			0
$95$			3.58258i			0.367565i
$96$		0			0
$97$			15.9891i			1.62345i	0.584041	+	0.811724i	$0.301471\pi$
$97$			15.9891i			1.62345i	−0.584041	+	0.811724i	$0.698529\pi$
$98$		0			0
$99$	18.3739	−	11.7913i	1.84664	−	1.18507i
$100$		0			0
$101$	−0.511238			−0.0508701			−0.0254351	−	0.999676i	$-0.508097\pi$
$101$	−0.511238			−0.0508701			−0.0254351	+	0.999676i	$0.508097\pi$
$102$		0			0
$103$			13.5396i			1.33410i	0.745014	+	0.667049i	$0.232443\pi$
$103$			13.5396i			1.33410i	−0.745014	+	0.667049i	$0.767557\pi$
$104$		0			0
$105$	−4.89898	+	2.00000i	−0.478091	+	0.195180i
$106$		0			0
$107$			8.41742i			0.813743i	0.913485	+	0.406872i	$0.133380\pi$
$107$			8.41742i			0.813743i	−0.913485	+	0.406872i	$0.866620\pi$
$108$		0			0
$109$			1.16515i			0.111601i	0.998442	+	0.0558006i	$0.0177711\pi$
$109$			1.16515i			0.111601i	−0.998442	+	0.0558006i	$0.982229\pi$
$110$		0			0
$111$			17.2813i			1.64027i
$112$		0			0
$113$	−16.7477			−1.57549			−0.787747	−	0.615999i	$-0.788752\pi$
$113$	−16.7477			−1.57549			−0.787747	+	0.615999i	$0.788752\pi$
$114$		0			0
$115$		−	2.58434i		−	0.240991i
$116$		0			0
$117$	20.3768			1.88384
$118$		0			0
$119$	−9.16515	+	3.74166i	−0.840168	+	0.342997i
$120$		0			0
$121$	4.58258	−	10.0000i	0.416598	−	0.909091i
$122$		0			0
$123$		−	15.5826i		−	1.40503i
$124$		0			0
$125$			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	$0.920330\pi$
$128$		0			0
$129$	34.5625			3.04306
$130$		0			0
$131$	−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
$134$		0			0
$135$	−7.16515			−0.616678
$136$		0			0
$137$	15.1652			1.29565			0.647823	−	0.761791i	$-0.275680\pi$
$137$	15.1652			1.29565			0.647823	+	0.761791i	$0.275680\pi$
$138$		0			0
$139$	−3.23042			−0.274001			−0.137000	−	0.990571i	$-0.543746\pi$
$139$	−3.23042			−0.274001			−0.137000	+	0.990571i	$0.543746\pi$
$140$		0			0
$141$	15.5826			1.31229
$142$		0			0
$143$	8.64064	−	5.54506i	0.722566	−	0.463701i
$144$		0			0
$145$	−4.89898			−0.406838
$146$		0			0
$147$	−15.1652	−	15.4779i	−1.25080	−	1.27659i
$148$		0			0
$149$			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	$-0.0465669\pi$
$152$		0			0
$153$	−24.6297			−1.99120
$154$		0			0
$155$	0.747727			0.0600589
$156$		0			0
$157$		−	20.5117i		−	1.63701i	−0.574498	−	0.818506i	$-0.694803\pi$
$157$		−	20.5117i		−	1.63701i	0.574498	−	0.818506i	$-0.305197\pi$
$158$		0			0
$159$			7.48331i			0.593465i
$160$		0			0
$161$	9.79796	−	4.00000i	0.772187	−	0.315244i
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$	−5.58258	+	3.58258i	−0.434603	+	0.278903i
$166$		0			0
$167$	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
$170$		0			0
$171$	−36.5008			−2.79129
$172$		0			0
$173$	−22.9612			−1.74571			−0.872853	−	0.487983i	$-0.837733\pi$
$173$	−22.9612			−1.74571			−0.872853	+	0.487983i	$0.837733\pi$
$174$		0			0
$175$	−11.2250	+	4.58258i	−0.848528	+	0.346410i
$176$		0			0
$177$	9.58258			0.720270
$178$		0			0
$179$	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	$-0.787845\pi$
$181$		−	16.6352i		−	1.23648i	0.785988	−	0.618242i	$-0.212155\pi$
$182$		0			0
$183$			28.7477i			2.12509i
$184$		0			0
$185$		−	3.60681i		−	0.265178i
$186$		0			0
$187$	−10.4440	+	6.70239i	−0.763744	+	0.490127i
$188$		0			0
$189$	−11.0901	−	27.1652i	−0.806688	−	1.97597i
$190$		0			0
$191$	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	$0.261274\pi$
$193$			20.3303i			1.46341i	−0.681623	+	0.731704i	$0.738726\pi$
$194$		0			0
$195$	−6.19115			−0.443357
$196$		0			0
$197$			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	$-0.337814\pi$
$200$		0			0
$201$			4.89898i			0.345547i
$202$		0			0
$203$	−7.58258	−	18.5734i	−0.532192	−	1.30360i
$204$		0			0
$205$			3.25227i			0.227149i
$206$		0			0
$207$	26.3303			1.83008
$208$		0			0
$209$	−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	$-0.120626\pi$
$212$		0			0
$213$			6.19115i			0.424210i
$214$		0			0
$215$	−7.21362			−0.491965
$216$		0			0
$217$	1.15732	+	2.83485i	0.0785641	+	0.192442i
$218$		0			0
$219$			19.5826i			1.32327i
$220$		0			0
$221$	−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	$-0.0679358\pi$
$224$		0			0
$225$	−30.1652			−2.01101
$226$		0			0
$227$	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	$-0.858348\pi$
$229$		−	13.0284i		−	0.860939i	0.902605	−	0.430470i	$-0.141652\pi$
$230$		0			0
$231$	−22.2232	−	15.6201i	−1.46218	−	1.02773i
$232$		0			0
$233$			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
$236$		0			0
$237$	−12.3823			−0.804316
$238$		0			0
$239$		−	17.5826i		−	1.13732i	−0.822572	−	0.568661i	$-0.807462\pi$
$239$		−	17.5826i		−	1.13732i	0.822572	−	0.568661i	$-0.192538\pi$
$240$		0			0
$241$	−25.9219			−1.66978			−0.834889	−	0.550419i	$-0.814468\pi$
$241$	−25.9219			−1.66978			−0.834889	+	0.550419i	$0.814468\pi$
$242$		0			0
$243$		−	11.8711i		−	0.761529i
$244$		0			0
$245$	3.16515	+	3.23042i	0.202214	+	0.206384i
$246$		0			0
$247$	−17.1652			−1.09219
$248$		0			0
$249$			28.3303i			1.79536i
$250$		0			0
$251$		−	2.07310i		−	0.130853i	−0.997857	−	0.0654264i	$-0.979159\pi$
$251$		−	2.07310i		−	0.130853i	0.997857	−	0.0654264i	$-0.0208407\pi$
$252$		0			0
$253$	11.1652	−	7.16515i	0.701947	−	0.450469i
$254$		0			0
$255$	7.48331			0.468623
$256$		0			0
$257$			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
$260$		0			0
$261$		−	49.9129i		−	3.08953i
$262$		0			0
$263$			28.3303i			1.74692i	0.486895	+	0.873461i	$0.338130\pi$
$263$			28.3303i			1.74692i	−0.486895	+	0.873461i	$0.661870\pi$
$264$		0			0
$265$		−	1.56186i		−	0.0959442i
$266$		0			0
$267$	−30.3303			−1.85618
$268$		0			0
$269$			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	$-0.452460\pi$
$271$	4.89898			0.297592			0.148796	+	0.988868i	$0.452460\pi$
$272$		0			0
$273$	−9.58258	−	23.4724i	−0.579964	−	1.42062i
$274$		0			0
$275$	−12.7913	+	8.20871i	−0.771344	+	0.495004i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		0			0
$279$			7.61816i			0.456087i
$280$		0			0
$281$			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	$-0.653601\pi$
$283$	−15.6127			−0.928079			−0.464040	+	0.885814i	$0.653601\pi$
$284$		0			0
$285$	11.0901			0.656922
$286$		0			0
$287$	−12.3303	+	5.03383i	−0.727835	+	0.297137i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	49.4955			2.90147
$292$		0			0
$293$	21.3993			1.25016			0.625081	−	0.780560i	$-0.285066\pi$
$293$	21.3993			1.25016			0.625081	+	0.780560i	$0.285066\pi$
$294$		0			0
$295$	−2.00000			−0.116445
$296$		0			0
$297$	−19.8656	−	30.9557i	−1.15272	−	1.79623i
$298$		0			0
$299$	12.3823			0.716087
$300$		0			0
$301$	−11.1652	−	27.3489i	−0.643549	−	1.57637i
$302$		0			0
$303$			1.58258i			0.0909166i
$304$		0			0
$305$		−	6.00000i		−	0.343559i
$306$		0			0
$307$	−8.12940			−0.463969			−0.231985	−	0.972719i	$-0.574522\pi$
$307$	−8.12940			−0.463969			−0.231985	+	0.972719i	$0.574522\pi$
$308$		0			0
$309$	41.9129			2.38434
$310$		0			0
$311$		−	9.93280i		−	0.563238i	−0.959526	−	0.281619i	$-0.909129\pi$
$311$		−	9.93280i		−	0.563238i	0.959526	−	0.281619i	$-0.0908714\pi$
$312$		0			0
$313$			1.02248i			0.0577938i	0.999582	+	0.0288969i	$0.00919945\pi$
$313$			1.02248i			0.0577938i	−0.999582	+	0.0288969i	$0.990801\pi$
$314$		0			0
$315$	4.25290	+	10.4174i	0.239624	+	0.586955i
$316$		0			0
$317$	−9.16515			−0.514766			−0.257383	−	0.966309i	$-0.582860\pi$
$317$	−9.16515			−0.514766			−0.257383	+	0.966309i	$0.582860\pi$
$318$		0			0
$319$	−13.5826	−	21.1652i	−0.760478	−	1.18502i
$320$		0			0
$321$	26.0568			1.45435
$322$		0			0
$323$	20.7477			1.15443
$324$		0			0
$325$	−14.1857			−0.786881
$326$		0			0
$327$	3.60681			0.199457
$328$		0			0
$329$	−5.03383	−	12.3303i	−0.277524	−	0.679792i
$330$		0			0
$331$	14.4174			0.792453			0.396227	−	0.918153i	$-0.370319\pi$
$331$	14.4174			0.792453			0.396227	+	0.918153i	$0.370319\pi$
$332$		0			0
$333$	36.7477			2.01376
$334$		0			0
$335$		−	1.02248i		−	0.0558639i
$336$		0			0
$337$		−	29.1652i		−	1.58873i	−0.607443	−	0.794364i	$-0.707805\pi$
$337$		−	29.1652i		−	1.58873i	0.607443	−	0.794364i	$-0.292195\pi$
$338$		0			0
$339$			51.8438i			2.81577i
$340$		0			0
$341$	2.07310	+	3.23042i	0.112264	+	0.174937i
$342$		0			0
$343$	−7.34847	+	17.0000i	−0.396780	+	0.917914i
$344$		0			0
$345$	−8.00000			−0.430706
$346$		0			0
$347$		−	0.834849i		−	0.0448170i	−0.999749	−	0.0224085i	$-0.992867\pi$
$347$		−	0.834849i		−	0.0448170i	0.999749	−	0.0224085i	$-0.00713345\pi$
$348$		0			0
$349$	−2.07310			−0.110970			−0.0554852	−	0.998460i	$-0.517671\pi$
$349$	−2.07310			−0.110970			−0.0554852	+	0.998460i	$0.517671\pi$
$350$		0			0
$351$		−	34.3303i		−	1.83242i
$352$		0			0
$353$			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
$356$		0			0
$357$	11.5826	+	28.3714i	0.613015	+	1.50157i
$358$		0			0
$359$		−	0.417424i		−	0.0220308i	−0.999939	−	0.0110154i	$-0.996494\pi$
$359$		−	0.417424i		−	0.0220308i	0.999939	−	0.0110154i	$-0.00350638\pi$
$360$		0			0
$361$	11.7477			0.618301
$362$		0			0
$363$	−30.9557	−	14.1857i	−1.62475	−	0.744556i
$364$		0			0
$365$		−	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	$-0.0118581\pi$
$368$		0			0
$369$	−33.1355			−1.72497
$370$		0			0
$371$	5.92146	−	2.41742i	0.307427	−	0.125506i
$372$		0			0
$373$		−	0.417424i		−	0.0216134i	−0.999942	−	0.0108067i	$-0.996560\pi$
$373$		−	0.417424i		−	0.0216134i	0.999942	−	0.0108067i	$-0.00343995\pi$
$374$		0			0
$375$	19.1652			0.989684
$376$		0			0
$377$		−	23.4724i		−	1.20889i
$378$		0			0
$379$	−14.3303			−0.736098			−0.368049	−	0.929806i	$-0.619974\pi$
$379$	−14.3303			−0.736098			−0.368049	+	0.929806i	$0.619974\pi$
$380$		0			0
$381$	17.2813			0.885346
$382$		0			0
$383$			4.76413i			0.243436i	0.992565	+	0.121718i	$0.0388403\pi$
$383$			4.76413i			0.243436i	−0.992565	+	0.121718i	$0.961160\pi$
$384$		0			0
$385$	4.63825	+	3.26010i	0.236387	+	0.166150i
$386$		0			0
$387$		−	73.4955i		−	3.73598i
$388$		0			0
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	−14.9666			−0.756895
$392$		0			0
$393$			29.1652i			1.47119i
$394$		0			0
$395$	2.58434			0.130032
$396$		0			0
$397$			23.8488i			1.19694i	0.801146	+	0.598469i	$0.204224\pi$
$397$			23.8488i			1.19694i	−0.801146	+	0.598469i	$0.795776\pi$
$398$		0			0
$399$	17.1652	+	42.0459i	0.859332	+	2.10493i
$400$		0			0
$401$	−1.58258			−0.0790301			−0.0395150	−	0.999219i	$-0.512581\pi$
$401$	−1.58258			−0.0790301			−0.0395150	+	0.999219i	$0.512581\pi$
$402$		0			0
$403$			3.58258i			0.178461i
$404$		0			0
$405$			9.42157i			0.468161i
$406$		0			0
$407$	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	$0.421024\pi$
$410$		0			0
$411$		−	46.9448i		−	2.31562i
$412$		0			0
$413$	−3.09557	−	7.58258i	−0.152323	−	0.373114i
$414$		0			0
$415$		−	5.91288i		−	0.290252i
$416$		0			0
$417$			10.0000i			0.489702i
$418$		0			0
$419$		−	5.67991i		−	0.277482i	−0.990329	−	0.138741i	$-0.955694\pi$
$419$		−	5.67991i		−	0.277482i	0.990329	−	0.138741i	$-0.0443055\pi$
$420$		0			0
$421$	−1.58258			−0.0771300			−0.0385650	−	0.999256i	$-0.512279\pi$
$421$	−1.58258			−0.0771300			−0.0385650	+	0.999256i	$0.512279\pi$
$422$		0			0
$423$		−	33.1355i		−	1.61110i
$424$		0			0
$425$	17.1464			0.831724
$426$		0			0
$427$	22.7477	−	9.28672i	1.10084	−	0.449416i
$428$		0			0
$429$	−17.1652	−	26.7477i	−0.828741	−	1.29139i
$430$		0			0
$431$			17.9129i			0.862833i	0.902153	+	0.431416i	$0.141986\pi$
$431$			17.9129i			0.862833i	−0.902153	+	0.431416i	$0.858014\pi$
$432$		0			0
$433$		−	28.1017i		−	1.35048i	−0.737597	−	0.675241i	$-0.764040\pi$
$433$		−	28.1017i		−	1.35048i	0.737597	−	0.675241i	$-0.235960\pi$
$434$		0			0
$435$			15.1652i			0.727113i
$436$		0			0
$437$	−22.1803			−1.06103
$438$		0			0
$439$	−2.31464			−0.110472			−0.0552360	−	0.998473i	$-0.517591\pi$
$439$	−2.31464			−0.110472			−0.0552360	+	0.998473i	$0.517591\pi$
$440$		0			0
$441$	−32.9129	+	32.2479i	−1.56728	+	1.53561i
$442$		0			0
$443$	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
$446$		0			0
$447$	43.3380			2.04982
$448$		0			0
$449$	12.8348			0.605714			0.302857	−	0.953036i	$-0.402060\pi$
$449$	12.8348			0.605714			0.302857	+	0.953036i	$0.402060\pi$
$450$		0			0
$451$	−14.0509	+	9.01703i	−0.661629	+	0.424595i
$452$		0			0
$453$	−11.0901			−0.521060
$454$		0			0
$455$	2.00000	+	4.89898i	0.0937614	+	0.229668i
$456$		0			0
$457$		−	14.8348i		−	0.693945i	−0.937875	−	0.346972i	$-0.887210\pi$
$457$		−	14.8348i		−	0.693945i	0.937875	−	0.346972i	$-0.112790\pi$
$458$		0			0
$459$			41.4955i			1.93684i
$460$		0			0
$461$	26.2983			1.22483			0.612417	−	0.790535i	$-0.290197\pi$
$461$	26.2983			1.22483			0.612417	+	0.790535i	$0.290197\pi$
$462$		0			0
$463$	13.1652			0.611836			0.305918	−	0.952058i	$-0.401037\pi$
$463$	13.1652			0.611836			0.305918	+	0.952058i	$0.401037\pi$
$464$		0			0
$465$		−	2.31464i		−	0.107339i
$466$		0			0
$467$		−	0.511238i		−	0.0236573i	−0.999930	−	0.0118286i	$-0.996235\pi$
$467$		−	0.511238i		−	0.0236573i	0.999930	−	0.0118286i	$-0.00376526\pi$
$468$		0			0
$469$	3.87650	−	1.58258i	0.179000	−	0.0730766i
$470$		0			0
$471$	−63.4955			−2.92572
$472$		0			0
$473$	−20.0000	−	31.1652i	−0.919601	−	1.43298i
$474$		0			0
$475$	25.4107			1.16592
$476$		0			0
$477$	15.9129			0.728601
$478$		0			0
$479$	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
$482$		0			0
$483$	−12.3823	−	30.3303i	−0.563414	−	1.38008i
$484$		0			0
$485$	−10.3303			−0.469075
$486$		0			0
$487$	33.4955			1.51782			0.758912	−	0.651193i	$-0.225731\pi$
$487$	33.4955			1.51782			0.758912	+	0.651193i	$0.225731\pi$
$488$		0			0
$489$			12.3823i			0.559947i
$490$		0			0
$491$		−	21.4955i		−	0.970076i	−0.874493	−	0.485038i	$-0.838806\pi$
$491$		−	21.4955i		−	0.970076i	0.874493	−	0.485038i	$-0.161194\pi$
$492$		0			0
$493$			28.3714i			1.27778i
$494$		0			0
$495$	7.61816	+	11.8711i	0.342411	+	0.533564i
$496$		0			0
$497$	4.89898	−	2.00000i	0.219749	−	0.0897123i
$498$		0			0
$499$	−27.9129			−1.24955			−0.624776	−	0.780804i	$-0.714810\pi$
$499$	−27.9129			−1.24955			−0.624776	+	0.780804i	$0.714810\pi$
$500$		0			0
$501$		−	19.1652i		−	0.856236i
$502$		0			0
$503$	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
$506$		0			0
$507$			10.5789i			0.469825i
$508$		0			0
$509$			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
$512$		0			0
$513$			61.4955i			2.71509i
$514$		0			0
$515$	−8.74773			−0.385471
$516$		0			0
$517$	−9.01703	−	14.0509i	−0.396569	−	0.617956i
$518$		0			0
$519$			71.0780i			3.11998i
$520$		0			0
$521$			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	$-0.698441\pi$
$523$	−26.7028			−1.16763			−0.583817	+	0.811885i	$0.698441\pi$
$524$		0			0
$525$	14.1857	+	34.7477i	0.619115	+	1.51652i
$526$		0			0
$527$		−	4.33030i		−	0.188631i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	20.3768i		−	0.884280i
$532$		0			0
$533$	−15.5826			−0.674956
$534$		0			0
$535$	−5.43836			−0.235121
$536$		0			0
$537$		−	22.1803i		−	0.957149i
$538$		0			0
$539$	−5.18096	+	22.6309i	−0.223160	+	0.974782i
$540$		0			0
$541$			9.25227i			0.397786i	0.980021	+	0.198893i	$0.0637347\pi$
$541$			9.25227i			0.397786i	−0.980021	+	0.198893i	$0.936265\pi$
$542$		0			0
$543$	−51.4955			−2.20988
$544$		0			0
$545$	−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	$0.981291\pi$
$548$		0			0
$549$	61.1305			2.60899
$550$		0			0
$551$			42.0459i			1.79121i
$552$		0			0
$553$	4.00000	+	9.79796i	0.170097	+	0.416652i
$554$		0			0
$555$	−11.1652			−0.473934
$556$		0			0
$557$		−	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			0
$561$	20.7477	+	32.3303i	0.875970	+	1.36499i
$562$		0			0
$563$	−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
$566$		0			0
$567$	−35.7199	+	14.5826i	−1.50009	+	0.612411i
$568$		0			0
$569$			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	$0.976116\pi$
$572$		0			0
$573$		−	55.7203i		−	2.32775i
$574$		0			0
$575$	−18.3303			−0.764426
$576$		0			0
$577$			0.269691i			0.0112274i	0.999984	+	0.00561369i	$0.00178690\pi$
$577$			0.269691i			0.0112274i	−0.999984	+	0.00561369i	$0.998213\pi$
$578$		0			0
$579$	62.9339			2.61545
$580$		0			0
$581$	22.4174	−	9.15188i	0.930031	−	0.379684i
$582$		0			0
$583$	6.74773	−	4.33030i	0.279462	−	0.179343i
$584$		0			0
$585$			13.1652i			0.544312i
$586$		0			0
$587$			17.0397i			0.703305i	0.936131	+	0.351652i	$0.114380\pi$
$587$			17.0397i			0.703305i	−0.936131	+	0.351652i	$0.885620\pi$
$588$		0			0
$589$		−	6.41742i		−	0.264425i
$590$		0			0
$591$	58.0350			2.38724
$592$		0			0
$593$	0.134846			0.00553744			0.00276872	−	0.999996i	$-0.499119\pi$
$593$	0.134846			0.00553744			0.00276872	+	0.999996i	$0.499119\pi$
$594$		0			0
$595$	−2.41742	−	5.92146i	−0.0991047	−	0.242756i
$596$		0			0
$597$	−76.2432			−3.12043
$598$		0			0
$599$	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	$0.509266\pi$
$602$		0			0
$603$	10.4174			0.424230
$604$		0			0
$605$	6.46084	+	2.96073i	0.262670	+	0.120371i
$606$		0			0
$607$	−46.9448			−1.90543			−0.952716	−	0.303862i	$-0.901724\pi$
$607$	−46.9448			−1.90543			−0.952716	+	0.303862i	$0.901724\pi$
$608$		0			0
$609$	−57.4955	+	23.4724i	−2.32983	+	0.951150i
$610$		0			0
$611$		−	15.5826i		−	0.630404i
$612$		0			0
$613$			13.9129i			0.561936i	0.959717	+	0.280968i	$0.0906555\pi$
$613$			13.9129i			0.561936i	−0.959717	+	0.280968i	$0.909345\pi$
$614$		0			0
$615$	10.0677			0.405967
$616$		0			0
$617$	25.9129			1.04321			0.521607	−	0.853186i	$-0.325333\pi$
$617$	25.9129			1.04321			0.521607	+	0.853186i	$0.325333\pi$
$618$		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			0
$621$		−	44.3605i		−	1.78013i
$622$		0			0
$623$	9.79796	+	24.0000i	0.392547	+	0.961540i
$624$		0			0
$625$	18.9129			0.756515
$626$		0			0
$627$	30.7477	+	47.9129i	1.22795	+	1.91346i
$628$		0			0
$629$	−20.8881			−0.832863
$630$		0			0
$631$	5.16515			0.205621			0.102811	−	0.994701i	$-0.467216\pi$
$631$	5.16515			0.205621			0.102811	+	0.994701i	$0.467216\pi$
$632$		0			0
$633$	−33.2704			−1.32238
$634$		0			0
$635$	−3.60681			−0.143132
$636$		0			0
$637$	−15.4779	+	15.1652i	−0.613255	+	0.600865i
$638$		0			0
$639$	13.1652			0.520805
$640$		0			0
$641$	29.9129			1.18149			0.590744	−	0.806859i	$-0.298834\pi$
$641$	29.9129			1.18149			0.590744	+	0.806859i	$0.298834\pi$
$642$		0			0
$643$			16.7700i			0.661346i	0.943745	+	0.330673i	$0.107276\pi$
$643$			16.7700i			0.661346i	−0.943745	+	0.330673i	$0.892724\pi$
$644$		0			0
$645$			22.3303i			0.879255i
$646$		0			0
$647$			44.7650i			1.75990i	0.475071	+	0.879948i	$0.342423\pi$
$647$			44.7650i			1.75990i	−0.475071	+	0.879948i	$0.657577\pi$
$648$		0			0
$649$	−5.54506	−	8.64064i	−0.217663	−	0.339175i
$650$		0			0
$651$	8.77548	−	3.58258i	0.343938	−	0.140412i
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		−	6.08712i		−	0.237844i
$656$		0			0
$657$	41.6413			1.62458
$658$		0			0
$659$			30.3303i			1.18150i	0.806854	+	0.590750i	$0.201168\pi$
$659$			30.3303i			1.18150i	−0.806854	+	0.590750i	$0.798832\pi$
$660$		0			0
$661$			25.4107i			0.988361i	0.869359	+	0.494180i	$0.164532\pi$
$661$			25.4107i			0.988361i	−0.869359	+	0.494180i	$0.835468\pi$
$662$		0			0
$663$			35.8547i			1.39248i
$664$		0			0
$665$	−3.58258	−	8.77548i	−0.138926	−	0.340299i
$666$		0			0
$667$		−	30.3303i		−	1.17439i
$668$		0			0
$669$	−19.5826			−0.757106
$670$		0			0
$671$	25.9219	−	16.6352i	1.00070	−	0.642194i
$672$		0			0
$673$		−	18.3303i		−	0.706581i	−0.935514	−	0.353291i	$-0.885063\pi$
$673$		−	18.3303i		−	0.706581i	0.935514	−	0.353291i	$-0.114937\pi$
$674$		0			0
$675$			50.8213i			1.95611i
$676$		0			0
$677$	31.4670			1.20937			0.604687	−	0.796463i	$-0.293298\pi$
$677$	31.4670			1.20937			0.604687	+	0.796463i	$0.293298\pi$
$678$		0			0
$679$	−15.9891	−	39.1652i	−0.613606	−	1.50302i
$680$		0			0
$681$		−	36.3303i		−	1.39218i
$682$		0			0
$683$	−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
$686$		0			0
$687$	−40.3303			−1.53870
$688$		0			0
$689$	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	$-0.254162\pi$
$692$		0			0
$693$	−33.2153	+	47.2565i	−1.26175	+	1.79513i
$694$		0			0
$695$		−	2.08712i		−	0.0791690i
$696$		0			0
$697$	18.8348			0.713421
$698$		0			0
$699$	27.3489			1.03443
$700$		0			0
$701$			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
$704$		0			0
$705$			10.0677i			0.379170i
$706$		0			0
$707$	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	$0.404932\pi$
$710$		0			0
$711$			26.3303i			0.987464i
$712$		0			0
$713$			4.62929i			0.173368i
$714$		0			0
$715$	3.58258	+	5.58258i	0.133981	+	0.208776i
$716$		0			0
$717$	−54.4282			−2.03266
$718$		0			0
$719$		−	15.8543i		−	0.591264i	−0.955302	−	0.295632i	$-0.904470\pi$
$719$		−	15.8543i		−	0.591264i	0.955302	−	0.295632i	$-0.0955302\pi$
$720$		0			0
$721$	−13.5396	−	33.1652i	−0.504242	−	1.23513i
$722$		0			0
$723$			80.2432i			2.98428i
$724$		0			0
$725$			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	$-0.420396\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$			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	$0.639390\pi$
$734$		0			0
$735$	10.0000	−	9.79796i	0.368856	−	0.361403i
$736$		0			0
$737$	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	$-0.0445382\pi$
$740$		0			0
$741$			53.1360i			1.95200i
$742$		0			0
$743$		−	43.9129i		−	1.61101i	−0.592591	−	0.805504i	$-0.701895\pi$
$743$		−	43.9129i		−	1.61101i	0.592591	−	0.805504i	$-0.298105\pi$
$744$		0			0
$745$	−9.04517			−0.331390
$746$		0			0
$747$	60.2429			2.20417
$748$		0			0
$749$	−8.41742	−	20.6184i	−0.307566	−	0.753380i
$750$		0			0
$751$	−13.4955			−0.492456			−0.246228	−	0.969212i	$-0.579191\pi$
$751$	−13.4955			−0.492456			−0.246228	+	0.969212i	$0.579191\pi$
$752$		0			0
$753$	−6.41742			−0.233864
$754$		0			0
$755$	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	$0.851736\pi$
$758$		0			0
$759$	−22.1803	−	34.5625i	−0.805092	−	1.25454i
$760$		0			0
$761$	20.7532			0.752304			0.376152	−	0.926558i	$-0.377247\pi$
$761$	20.7532			0.752304			0.376152	+	0.926558i	$0.377247\pi$
$762$		0			0
$763$	−1.16515	−	2.85403i	−0.0421813	−	0.103323i
$764$		0			0
$765$		−	15.9129i		−	0.575331i
$766$		0			0
$767$		−	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	$0.319452\pi$
$770$		0			0
$771$	15.1652			0.546160
$772$		0			0
$773$		−	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
$776$		0			0
$777$	−17.2813	−	42.3303i	−0.619962	−	1.51859i
$778$		0			0
$779$	27.9129			1.00008
$780$		0			0
$781$	5.58258	−	3.58258i	0.199760	−	0.128195i
$782$		0			0
$783$	−84.0917			−3.00519
$784$		0			0
$785$	13.2523			0.472994
$786$		0			0
$787$	40.3773			1.43930			0.719648	−	0.694339i	$-0.244303\pi$
$787$	40.3773			1.43930			0.719648	+	0.694339i	$0.244303\pi$
$788$		0			0
$789$	87.6985			3.12215
$790$		0			0
$791$	41.0234	−	16.7477i	1.45862	−	0.595481i
$792$		0			0
$793$	28.7477			1.02086
$794$		0			0
$795$	−4.83485			−0.171474
$796$		0			0
$797$		−	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
$800$		0			0
$801$			64.4958i			2.27885i
$802$		0			0
$803$	17.6577	−	11.3317i	0.623126	−	0.399886i
$804$		0			0
$805$	2.58434	+	6.33030i	0.0910859	+	0.223114i
$806$		0			0
$807$	62.6606			2.20576
$808$		0			0
$809$		−	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	$-0.321381\pi$
$811$	30.3097			1.06432			0.532158	+	0.846645i	$0.321381\pi$
$812$		0			0
$813$		−	15.1652i		−	0.531865i
$814$		0			0
$815$		−	2.58434i		−	0.0905253i
$816$		0			0
$817$			61.9115i			2.16601i
$818$		0			0
$819$	−49.9129	+	20.3768i	−1.74410	+	0.712025i
$820$		0			0
$821$		−	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	$-0.251882\pi$
$823$	40.3303			1.40583			0.702913	+	0.711276i	$0.251882\pi$
$824$		0			0
$825$	25.4107	+	39.5964i	0.884686	+	1.37857i
$826$		0			0
$827$		−	15.1652i		−	0.527344i	−0.964612	−	0.263672i	$-0.915066\pi$
$827$		−	15.1652i		−	0.527344i	0.964612	−	0.263672i	$-0.0849337\pi$
$828$		0			0
$829$		−	4.25290i		−	0.147709i	−0.997269	−	0.0738546i	$-0.976470\pi$
$829$		−	4.25290i		−	0.147709i	0.997269	−	0.0738546i	$-0.0235301\pi$
$830$		0			0
$831$	−6.19115			−0.214769
$832$		0			0
$833$	18.7083	−	18.3303i	0.648204	−	0.635107i
$834$		0			0
$835$			4.00000i			0.138426i
$836$		0			0
$837$	12.8348			0.443637
$838$		0			0
$839$		−	24.6297i		−	0.850313i	−0.905120	−	0.425157i	$-0.860219\pi$
$839$		−	24.6297i		−	0.850313i	0.905120	−	0.425157i	$-0.139781\pi$
$840$		0			0
$841$	−28.4955			−0.982602
$842$		0			0
$843$	22.1803			0.763929
$844$		0			0
$845$		−	2.20794i		−	0.0759555i
$846$		0			0
$847$	−1.22497	+	29.0775i	−0.0420905	+	0.999114i
$848$		0			0
$849$			48.3303i			1.65869i
$850$		0			0
$851$	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	$0.327204\pi$
$854$		0			0
$855$		−	23.5826i		−	0.806507i
$856$		0			0
$857$	21.2926			0.727342			0.363671	−	0.931527i	$-0.381523\pi$
$857$	21.2926			0.727342			0.363671	+	0.931527i	$0.381523\pi$
$858$		0			0
$859$			31.7367i			1.08284i	0.840752	+	0.541421i	$0.182113\pi$
$859$			31.7367i			1.08284i	−0.840752	+	0.541421i	$0.817887\pi$
$860$		0			0
$861$	15.5826	+	38.1694i	0.531053	+	1.30081i
$862$		0			0
$863$	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
$866$		0			0
$867$			9.28672i			0.315394i
$868$		0			0
$869$	7.16515	+	11.1652i	0.243061	+	0.378752i
$870$		0			0
$871$	4.89898			0.165996
$872$		0			0
$873$		−	105.250i		−	3.56216i
$874$		0			0
$875$	−6.19115	−	15.1652i	−0.209299	−	0.512676i
$876$		0			0
$877$			25.1652i			0.849767i	0.905248	+	0.424883i	$0.139685\pi$
$877$			25.1652i			0.849767i	−0.905248	+	0.424883i	$0.860315\pi$
$878$		0			0
$879$		−	66.2432i		−	2.23433i
$880$		0			0
$881$		−	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	$-0.675709\pi$
$883$	−31.1652			−1.04879			−0.524395	+	0.851475i	$0.675709\pi$
$884$		0			0
$885$			6.19115i			0.208113i
$886$		0			0
$887$	−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
$890$		0			0
$891$	−40.7042	+	26.1216i	−1.36364	+	0.875106i
$892$		0			0
$893$			27.9129i			0.934069i
$894$		0			0
$895$			4.62929i			0.154740i
$896$		0			0
$897$		−	38.3303i		−	1.27981i
$898$		0			0
$899$	8.77548			0.292679
$900$		0			0
$901$	−9.04517			−0.301338
$902$		0			0
$903$	−84.6606	+	34.5625i	−2.81733	+	1.15017i
$904$		0			0
$905$	10.7477			0.357267
$906$		0			0
$907$	39.9129			1.32529			0.662643	−	0.748936i	$-0.269435\pi$
$907$	39.9129			1.32529			0.662643	+	0.748936i	$0.269435\pi$
$908$		0			0
$909$	3.36526			0.111619
$910$		0			0
$911$	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
$914$		0			0
$915$	−18.5734			−0.614019
$916$		0			0
$917$	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	$-0.199080\pi$
$920$		0			0
$921$			25.1652i			0.829220i
$922$		0			0
$923$	6.19115			0.203784
$924$		0			0
$925$	−25.5826			−0.841150
$926$		0			0
$927$		−	89.1255i		−	2.92727i
$928$		0			0
$929$			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
$932$		0			0
$933$	−30.7477			−1.00664
$934$		0			0
$935$	−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	$0.761042\pi$
$938$		0			0
$939$	3.16515			0.103291
$940$		0			0
$941$	−31.4670			−1.02579			−0.512897	−	0.858450i	$-0.671428\pi$
$941$	−31.4670			−1.02579			−0.512897	+	0.858450i	$0.671428\pi$
$942$		0			0
$943$	−20.1353			−0.655696
$944$		0			0
$945$	17.5510	−	7.16515i	0.570933	−	0.233082i
$946$		0			0
$947$	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
$950$		0			0
$951$			28.3714i			0.920006i
$952$		0			0
$953$		−	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
$956$		0			0
$957$	−65.5183	+	42.0459i	−2.11791	+	1.35915i
$958$		0			0
$959$	−37.1469	+	15.1652i	−1.19954	+	0.489708i
$960$		0			0
$961$	29.6606			0.956794
$962$		0			0
$963$		−	55.4083i		−	1.78551i
$964$		0			0
$965$	−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	$-0.169546\pi$
$968$		0			0
$969$		−	64.2261i		−	2.06324i
$970$		0			0
$971$			17.7925i			0.570989i	0.958380	+	0.285494i	$0.0921578\pi$
$971$			17.7925i			0.570989i	−0.958380	+	0.285494i	$0.907842\pi$
$972$		0			0
$973$	7.91288	−	3.23042i	0.253675	−	0.103562i
$974$		0			0
$975$			43.9129i			1.40634i
$976$		0			0
$977$	−9.16515			−0.293219			−0.146610	−	0.989194i	$-0.546836\pi$
$977$	−9.16515			−0.293219			−0.146610	+	0.989194i	$0.546836\pi$
$978$		0			0
$979$	17.5510	+	27.3489i	0.560931	+	0.874075i
$980$		0			0
$981$		−	7.66970i		−	0.244875i
$982$		0			0
$983$		−	1.69670i		−	0.0541165i	−0.999634	−	0.0270582i	$-0.991386\pi$
$983$		−	1.69670i		−	0.0541165i	0.999634	−	0.0270582i	$-0.00861395\pi$
$984$		0			0
$985$	−12.1126			−0.385940
$986$		0			0
$987$	−38.1694	+	15.5826i	−1.21494	+	0.495999i
$988$		0			0
$989$		−	44.6606i		−	1.42012i
$990$		0			0
$991$	26.6606			0.846902			0.423451	−	0.905919i	$-0.360819\pi$
$991$	26.6606			0.846902			0.423451	+	0.905919i	$0.360819\pi$
$992$		0			0
$993$		−	44.6302i		−	1.41630i
$994$		0			0
$995$	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	$0.320335\pi$
$998$		0			0
$999$		−	61.9115i		−	1.95879i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.e.e.769.1		8
4.3	odd	2		154.2.c.a.153.8	yes	8
7.6	odd	2	inner	1232.2.e.e.769.8		8
11.10	odd	2	inner	1232.2.e.e.769.2		8
12.11	even	2		1386.2.e.b.307.2		8
28.3	even	6		1078.2.i.b.901.1		16
28.11	odd	6		1078.2.i.b.901.4		16
28.19	even	6		1078.2.i.b.1011.8		16
28.23	odd	6		1078.2.i.b.1011.5		16
28.27	even	2		154.2.c.a.153.5	yes	8
44.43	even	2		154.2.c.a.153.4	yes	8
77.76	even	2	inner	1232.2.e.e.769.7		8
84.83	odd	2		1386.2.e.b.307.3		8
132.131	odd	2		1386.2.e.b.307.6		8
308.87	odd	6		1078.2.i.b.901.5		16
308.131	odd	6		1078.2.i.b.1011.4		16
308.219	even	6		1078.2.i.b.1011.1		16
308.263	even	6		1078.2.i.b.901.8		16
308.307	odd	2		154.2.c.a.153.1	✓	8
924.923	even	2		1386.2.e.b.307.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.c.a.153.1	✓	8	308.307	odd	2
154.2.c.a.153.4	yes	8	44.43	even	2
154.2.c.a.153.5	yes	8	28.27	even	2
154.2.c.a.153.8	yes	8	4.3	odd	2
1078.2.i.b.901.1		16	28.3	even	6
1078.2.i.b.901.4		16	28.11	odd	6
1078.2.i.b.901.5		16	308.87	odd	6
1078.2.i.b.901.8		16	308.263	even	6
1078.2.i.b.1011.1		16	308.219	even	6
1078.2.i.b.1011.4		16	308.131	odd	6
1078.2.i.b.1011.5		16	28.23	odd	6
1078.2.i.b.1011.8		16	28.19	even	6
1232.2.e.e.769.1		8	1.1	even	1	trivial
1232.2.e.e.769.2		8	11.10	odd	2	inner
1232.2.e.e.769.7		8	77.76	even	2	inner
1232.2.e.e.769.8		8	7.6	odd	2	inner
1386.2.e.b.307.2		8	12.11	even	2
1386.2.e.b.307.3		8	84.83	odd	2
1386.2.e.b.307.6		8	132.131	odd	2
1386.2.e.b.307.7		8	924.923	even	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.09557i q^{3} +0.646084i q^{5} +(-2.44949 + 1.00000i) q^{7} -6.58258 q^{9} +O(q^{10})\)	\(q-3.09557i q^{3} +0.646084i q^{5} +(-2.44949 + 1.00000i) q^{7} -6.58258 q^{9} +(-2.79129 + 1.79129i) q^{11} -3.09557 q^{13} +2.00000 q^{15} +3.74166 q^{17} +5.54506 q^{19} +(3.09557 + 7.58258i) q^{21} -4.00000 q^{23} +4.58258 q^{25} +11.0901i q^{27} +7.58258i q^{29} -1.15732i q^{31} +(5.54506 + 8.64064i) q^{33} +(-0.646084 - 1.58258i) q^{35} -5.58258 q^{37} +9.58258i q^{39} +5.03383 q^{41} +11.1652i q^{43} -4.25290i q^{45} +5.03383i q^{47} +(5.00000 - 4.89898i) q^{49} -11.5826i q^{51} -2.41742 q^{53} +(-1.15732 - 1.80341i) q^{55} -17.1652i q^{57} +3.09557i q^{59} -9.28672 q^{61} +(16.1240 - 6.58258i) q^{63} -2.00000i q^{65} -1.58258 q^{67} +12.3823i q^{69} -2.00000 q^{71} -6.32599 q^{73} -14.1857i q^{75} +(5.04594 - 7.17903i) q^{77} -4.00000i q^{79} +14.5826 q^{81} -9.15188 q^{83} +2.41742i q^{85} +23.4724 q^{87} -9.79796i q^{89} +(7.58258 - 3.09557i) q^{91} -3.58258 q^{93} +3.58258i q^{95} +15.9891i q^{97} +(18.3739 - 11.7913i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^9	\( 8 q - 16 q^{9} - 4 q^{11} + 16 q^{15} - 32 q^{23} - 8 q^{37} + 40 q^{49} - 56 q^{53} + 24 q^{67} - 16 q^{71} + 4 q^{77} + 80 q^{81} + 24 q^{91} + 8 q^{93} + 92 q^{99}+O(q^{100}) \) 8 * q - 16 * q^9 - 4 * q^11 + 16 * q^15 - 32 * q^23 - 8 * q^37 + 40 * q^49 - 56 * q^53 + 24 * q^67 - 16 * q^71 + 4 * q^77 + 80 * q^81 + 24 * q^91 + 8 * q^93 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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