LMFDB - Embedded newform 1386.2.e.d.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:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			307.7
Root			$-0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1386.307
Dual form			1386.2.e.d.307.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.41421i q^{5} +(-2.12132 - 1.58114i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +1.41421i q^{5} +(-2.12132 - 1.58114i) q^{7} -1.00000i q^{8} -1.41421 q^{10} +(3.16228 - 1.00000i) q^{11} -1.41421 q^{13} +(1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214 q^{17} +2.82843 q^{19} -1.41421i q^{20} +(1.00000 + 3.16228i) q^{22} -3.16228 q^{23} +3.00000 q^{25} -1.41421i q^{26} +(2.12132 + 1.58114i) q^{28} -6.00000i q^{29} +8.94427i q^{31} +1.00000i q^{32} +4.47214i q^{34} +(2.23607 - 3.00000i) q^{35} +2.00000 q^{37} +2.82843i q^{38} +1.41421 q^{40} +8.94427 q^{41} +6.32456i q^{43} +(-3.16228 + 1.00000i) q^{44} -3.16228i q^{46} +7.07107i q^{47} +(2.00000 + 6.70820i) q^{49} +3.00000i q^{50} +1.41421 q^{52} +3.16228 q^{53} +(1.41421 + 4.47214i) q^{55} +(-1.58114 + 2.12132i) q^{56} +6.00000 q^{58} +2.82843i q^{59} -12.7279 q^{61} -8.94427 q^{62} -1.00000 q^{64} -2.00000i q^{65} +12.0000 q^{67} -4.47214 q^{68} +(3.00000 + 2.23607i) q^{70} +3.16228 q^{71} +2.00000i q^{74} -2.82843 q^{76} +(-8.28934 - 2.87868i) q^{77} -3.16228i q^{79} +1.41421i q^{80} +8.94427i q^{82} +4.47214 q^{83} +6.32456i q^{85} -6.32456 q^{86} +(-1.00000 - 3.16228i) q^{88} -16.9706i q^{89} +(3.00000 + 2.23607i) q^{91} +3.16228 q^{92} -7.07107 q^{94} +4.00000i q^{95} +4.47214i q^{97} +(-6.70820 + 2.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} + 8 q^{22} + 24 q^{25} + 16 q^{37} + 16 q^{49} + 48 q^{58} - 8 q^{64} + 96 q^{67} + 24 q^{70} - 8 q^{88} + 24 q^{91}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^22 + 24 * q^25 + 16 * q^37 + 16 * q^49 + 48 * q^58 - 8 * q^64 + 96 * q^67 + 24 * q^70 - 8 * q^88 + 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	−2.12132	−	1.58114i	−0.801784	−	0.597614i
$7$	−2.12132	−	1.58114i	−0.801784	−	0.597614i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.41421			−0.447214
$11$	3.16228	−	1.00000i	0.953463	−	0.301511i
$11$	3.16228	−	1.00000i	0.953463	−	0.301511i
$12$		0			0
$13$	−1.41421			−0.392232			−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421			−0.392232			−0.196116	+	0.980581i	$0.562833\pi$
$14$	1.58114	−	2.12132i	0.422577	−	0.566947i
$15$		0			0
$16$	1.00000			0.250000
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$	2.82843			0.648886			0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843			0.648886			0.324443	+	0.945905i	$0.394823\pi$
$20$		−	1.41421i		−	0.316228i
$21$		0			0
$22$	1.00000	+	3.16228i	0.213201	+	0.674200i
$23$	−3.16228			−0.659380			−0.329690	−	0.944089i	$-0.606944\pi$
$23$	−3.16228			−0.659380			−0.329690	+	0.944089i	$0.606944\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		−	1.41421i		−	0.277350i
$27$		0			0
$28$	2.12132	+	1.58114i	0.400892	+	0.298807i
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$			8.94427i			1.60644i	0.595683	+	0.803219i	$0.296881\pi$
$31$			8.94427i			1.60644i	−0.595683	+	0.803219i	$0.703119\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			4.47214i			0.766965i
$35$	2.23607	−	3.00000i	0.377964	−	0.507093i
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$			2.82843i			0.458831i
$39$		0			0
$40$	1.41421			0.223607
$41$	8.94427			1.39686			0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427			1.39686			0.698430	+	0.715678i	$0.253882\pi$
$42$		0			0
$43$			6.32456i			0.964486i	0.876038	+	0.482243i	$0.160178\pi$
$43$			6.32456i			0.964486i	−0.876038	+	0.482243i	$0.839822\pi$
$44$	−3.16228	+	1.00000i	−0.476731	+	0.150756i
$45$		0			0
$46$		−	3.16228i		−	0.466252i
$47$			7.07107i			1.03142i	0.856763	+	0.515711i	$0.172472\pi$
$47$			7.07107i			1.03142i	−0.856763	+	0.515711i	$0.827528\pi$
$48$		0			0
$49$	2.00000	+	6.70820i	0.285714	+	0.958315i
$50$			3.00000i			0.424264i
$51$		0			0
$52$	1.41421			0.196116
$53$	3.16228			0.434372			0.217186	−	0.976130i	$-0.430312\pi$
$53$	3.16228			0.434372			0.217186	+	0.976130i	$0.430312\pi$
$54$		0			0
$55$	1.41421	+	4.47214i	0.190693	+	0.603023i
$56$	−1.58114	+	2.12132i	−0.211289	+	0.283473i
$57$		0			0
$58$	6.00000			0.787839
$59$			2.82843i			0.368230i	0.982905	+	0.184115i	$0.0589419\pi$
$59$			2.82843i			0.368230i	−0.982905	+	0.184115i	$0.941058\pi$
$60$		0			0
$61$	−12.7279			−1.62964			−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279			−1.62964			−0.814822	+	0.579712i	$0.803165\pi$
$62$	−8.94427			−1.13592
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$	−4.47214			−0.542326
$69$		0			0
$70$	3.00000	+	2.23607i	0.358569	+	0.267261i
$71$	3.16228			0.375293			0.187647	−	0.982237i	$-0.439914\pi$
$71$	3.16228			0.375293			0.187647	+	0.982237i	$0.439914\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$			2.00000i			0.232495i
$75$		0			0
$76$	−2.82843			−0.324443
$77$	−8.28934	−	2.87868i	−0.944658	−	0.328056i
$78$		0			0
$79$		−	3.16228i		−	0.355784i	−0.984050	−	0.177892i	$-0.943072\pi$
$79$		−	3.16228i		−	0.355784i	0.984050	−	0.177892i	$-0.0569278\pi$
$80$			1.41421i			0.158114i
$81$		0			0
$82$			8.94427i			0.987730i
$83$	4.47214			0.490881			0.245440	−	0.969412i	$-0.421067\pi$
$83$	4.47214			0.490881			0.245440	+	0.969412i	$0.421067\pi$
$84$		0			0
$85$			6.32456i			0.685994i
$86$	−6.32456			−0.681994
$87$		0			0
$88$	−1.00000	−	3.16228i	−0.106600	−	0.337100i
$89$		−	16.9706i		−	1.79888i	−0.437048	−	0.899438i	$-0.643976\pi$
$89$		−	16.9706i		−	1.79888i	0.437048	−	0.899438i	$-0.356024\pi$
$90$		0			0
$91$	3.00000	+	2.23607i	0.314485	+	0.234404i
$92$	3.16228			0.329690
$93$		0			0
$94$	−7.07107			−0.729325
$95$			4.00000i			0.410391i
$96$		0			0
$97$			4.47214i			0.454077i	0.973886	+	0.227038i	$0.0729043\pi$
$97$			4.47214i			0.454077i	−0.973886	+	0.227038i	$0.927096\pi$
$98$	−6.70820	+	2.00000i	−0.677631	+	0.202031i
$99$		0			0
$100$	−3.00000			−0.300000
$101$	13.4164			1.33498			0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164			1.33498			0.667491	+	0.744618i	$0.267368\pi$
$102$		0			0
$103$			8.94427i			0.881305i	0.897678	+	0.440653i	$0.145253\pi$
$103$			8.94427i			0.881305i	−0.897678	+	0.440653i	$0.854747\pi$
$104$			1.41421i			0.138675i
$105$		0			0
$106$			3.16228i			0.307148i
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$		−	3.16228i		−	0.302891i	−0.988466	−	0.151446i	$-0.951607\pi$
$109$		−	3.16228i		−	0.302891i	0.988466	−	0.151446i	$-0.0483928\pi$
$110$	−4.47214	+	1.41421i	−0.426401	+	0.134840i
$111$		0			0
$112$	−2.12132	−	1.58114i	−0.200446	−	0.149404i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		−	4.47214i		−	0.417029i
$116$			6.00000i			0.557086i
$117$		0			0
$118$	−2.82843			−0.260378
$119$	−9.48683	−	7.07107i	−0.869657	−	0.648204i
$120$		0			0
$121$	9.00000	−	6.32456i	0.818182	−	0.574960i
$122$		−	12.7279i		−	1.15233i
$123$		0			0
$124$		−	8.94427i		−	0.803219i
$125$			11.3137i			1.01193i
$126$		0			0
$127$			22.1359i			1.96425i	0.188237	+	0.982124i	$0.439723\pi$
$127$			22.1359i			1.96425i	−0.188237	+	0.982124i	$0.560277\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	2.00000			0.175412
$131$	−4.47214			−0.390732			−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214			−0.390732			−0.195366	+	0.980730i	$0.562590\pi$
$132$		0			0
$133$	−6.00000	−	4.47214i	−0.520266	−	0.387783i
$134$			12.0000i			1.03664i
$135$		0			0
$136$		−	4.47214i		−	0.383482i
$137$	18.9737			1.62103			0.810515	−	0.585718i	$-0.199187\pi$
$137$	18.9737			1.62103			0.810515	+	0.585718i	$0.199187\pi$
$138$		0			0
$139$	2.82843			0.239904			0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843			0.239904			0.119952	+	0.992780i	$0.461726\pi$
$140$	−2.23607	+	3.00000i	−0.188982	+	0.253546i
$141$		0			0
$142$			3.16228i			0.265372i
$143$	−4.47214	+	1.41421i	−0.373979	+	0.118262i
$144$		0			0
$145$	8.48528			0.704664
$146$		0			0
$147$		0			0
$148$	−2.00000			−0.164399
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$		−	15.8114i		−	1.28671i	−0.765567	−	0.643356i	$-0.777541\pi$
$151$		−	15.8114i		−	1.28671i	0.765567	−	0.643356i	$-0.222459\pi$
$152$		−	2.82843i		−	0.229416i
$153$		0			0
$154$	2.87868	−	8.28934i	0.231971	−	0.667974i
$155$	−12.6491			−1.01600
$156$		0			0
$157$			13.4164i			1.07075i	0.844616	+	0.535373i	$0.179829\pi$
$157$			13.4164i			1.07075i	−0.844616	+	0.535373i	$0.820171\pi$
$158$	3.16228			0.251577
$159$		0			0
$160$	−1.41421			−0.111803
$161$	6.70820	+	5.00000i	0.528681	+	0.394055i
$162$		0			0
$163$	−14.0000			−1.09656			−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000			−1.09656			−0.548282	+	0.836293i	$0.684718\pi$
$164$	−8.94427			−0.698430
$165$		0			0
$166$			4.47214i			0.347105i
$167$	−17.8885			−1.38426			−0.692129	−	0.721774i	$-0.743327\pi$
$167$	−17.8885			−1.38426			−0.692129	+	0.721774i	$0.743327\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$	−6.32456			−0.485071
$171$		0			0
$172$		−	6.32456i		−	0.482243i
$173$	13.4164			1.02003			0.510015	−	0.860165i	$-0.329640\pi$
$173$	13.4164			1.02003			0.510015	+	0.860165i	$0.329640\pi$
$174$		0			0
$175$	−6.36396	−	4.74342i	−0.481070	−	0.358569i
$176$	3.16228	−	1.00000i	0.238366	−	0.0753778i
$177$		0			0
$178$	16.9706			1.27200
$179$	12.6491			0.945439			0.472719	−	0.881213i	$-0.343272\pi$
$179$	12.6491			0.945439			0.472719	+	0.881213i	$0.343272\pi$
$180$		0			0
$181$		−	13.4164i		−	0.997234i	−0.866822	−	0.498617i	$-0.833841\pi$
$181$		−	13.4164i		−	0.997234i	0.866822	−	0.498617i	$-0.166159\pi$
$182$	−2.23607	+	3.00000i	−0.165748	+	0.222375i
$183$		0			0
$184$			3.16228i			0.233126i
$185$			2.82843i			0.207950i
$186$		0			0
$187$	14.1421	−	4.47214i	1.03418	−	0.327035i
$188$		−	7.07107i		−	0.515711i
$189$		0			0
$190$	−4.00000			−0.290191
$191$	9.48683			0.686443			0.343222	−	0.939254i	$-0.388482\pi$
$191$	9.48683			0.686443			0.343222	+	0.939254i	$0.388482\pi$
$192$		0			0
$193$		−	6.32456i		−	0.455251i	−0.973749	−	0.227626i	$-0.926904\pi$
$193$		−	6.32456i		−	0.455251i	0.973749	−	0.227626i	$-0.0730963\pi$
$194$	−4.47214			−0.321081
$195$		0			0
$196$	−2.00000	−	6.70820i	−0.142857	−	0.479157i
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$		−	17.8885i		−	1.26809i	−0.773298	−	0.634043i	$-0.781394\pi$
$199$		−	17.8885i		−	1.26809i	0.773298	−	0.634043i	$-0.218606\pi$
$200$		−	3.00000i		−	0.212132i
$201$		0			0
$202$			13.4164i			0.943975i
$203$	−9.48683	+	12.7279i	−0.665845	+	0.893325i
$204$		0			0
$205$			12.6491i			0.883452i
$206$	−8.94427			−0.623177
$207$		0			0
$208$	−1.41421			−0.0980581
$209$	8.94427	−	2.82843i	0.618688	−	0.195646i
$210$		0			0
$211$			18.9737i			1.30620i	0.757271	+	0.653101i	$0.226532\pi$
$211$			18.9737i			1.30620i	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−3.16228			−0.217186
$213$		0			0
$214$	−2.00000			−0.136717
$215$	−8.94427			−0.609994
$216$		0			0
$217$	14.1421	−	18.9737i	0.960031	−	1.28802i
$218$	3.16228			0.214176
$219$		0			0
$220$	−1.41421	−	4.47214i	−0.0953463	−	0.301511i
$221$	−6.32456			−0.425436
$222$		0			0
$223$		−	26.8328i		−	1.79686i	−0.439119	−	0.898429i	$-0.644709\pi$
$223$		−	26.8328i		−	1.79686i	0.439119	−	0.898429i	$-0.355291\pi$
$224$	1.58114	−	2.12132i	0.105644	−	0.141737i
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	4.47214i		−	0.295527i	−0.989023	−	0.147764i	$-0.952793\pi$
$229$		−	4.47214i		−	0.295527i	0.989023	−	0.147764i	$-0.0472075\pi$
$230$	4.47214			0.294884
$231$		0			0
$232$	−6.00000			−0.393919
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	−10.0000			−0.652328
$236$		−	2.82843i		−	0.184115i
$237$		0			0
$238$	7.07107	−	9.48683i	0.458349	−	0.614940i
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		0			0
$241$	−8.48528			−0.546585			−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528			−0.546585			−0.273293	+	0.961931i	$0.588113\pi$
$242$	6.32456	+	9.00000i	0.406558	+	0.578542i
$243$		0			0
$244$	12.7279			0.814822
$245$	−9.48683	+	2.82843i	−0.606092	+	0.180702i
$246$		0			0
$247$	−4.00000			−0.254514
$248$	8.94427			0.567962
$249$		0			0
$250$	−11.3137			−0.715542
$251$		−	19.7990i		−	1.24970i	−0.780744	−	0.624851i	$-0.785160\pi$
$251$		−	19.7990i		−	1.24970i	0.780744	−	0.624851i	$-0.214840\pi$
$252$		0			0
$253$	−10.0000	+	3.16228i	−0.628695	+	0.198811i
$254$	−22.1359			−1.38893
$255$		0			0
$256$	1.00000			0.0625000
$257$			16.9706i			1.05859i	0.848436	+	0.529297i	$0.177544\pi$
$257$			16.9706i			1.05859i	−0.848436	+	0.529297i	$0.822456\pi$
$258$		0			0
$259$	−4.24264	−	3.16228i	−0.263625	−	0.196494i
$260$			2.00000i			0.124035i
$261$		0			0
$262$		−	4.47214i		−	0.276289i
$263$		−	4.00000i		−	0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$		−	4.00000i		−	0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$		0			0
$265$			4.47214i			0.274721i
$266$	4.47214	−	6.00000i	0.274204	−	0.367884i
$267$		0			0
$268$	−12.0000			−0.733017
$269$			18.3848i			1.12094i	0.828175	+	0.560470i	$0.189379\pi$
$269$			18.3848i			1.12094i	−0.828175	+	0.560470i	$0.810621\pi$
$270$		0			0
$271$	−12.7279			−0.773166			−0.386583	−	0.922255i	$-0.626345\pi$
$271$	−12.7279			−0.773166			−0.386583	+	0.922255i	$0.626345\pi$
$272$	4.47214			0.271163
$273$		0			0
$274$			18.9737i			1.14624i
$275$	9.48683	−	3.00000i	0.572078	−	0.180907i
$276$		0			0
$277$		−	22.1359i		−	1.33002i	−0.746834	−	0.665010i	$-0.768427\pi$
$277$		−	22.1359i		−	1.33002i	0.746834	−	0.665010i	$-0.231573\pi$
$278$			2.82843i			0.169638i
$279$		0			0
$280$	−3.00000	−	2.23607i	−0.179284	−	0.133631i
$281$		−	20.0000i		−	1.19310i	−0.802576	−	0.596550i	$-0.796538\pi$
$281$		−	20.0000i		−	1.19310i	0.802576	−	0.596550i	$-0.203462\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	−3.16228			−0.187647
$285$		0			0
$286$	−1.41421	−	4.47214i	−0.0836242	−	0.264443i
$287$	−18.9737	−	14.1421i	−1.11998	−	0.834784i
$288$		0			0
$289$	3.00000			0.176471
$290$			8.48528i			0.498273i
$291$		0			0
$292$		0			0
$293$	4.47214			0.261265			0.130632	−	0.991431i	$-0.458299\pi$
$293$	4.47214			0.261265			0.130632	+	0.991431i	$0.458299\pi$
$294$		0			0
$295$	−4.00000			−0.232889
$296$		−	2.00000i		−	0.116248i
$297$		0			0
$298$	10.0000			0.579284
$299$	4.47214			0.258630
$300$		0			0
$301$	10.0000	−	13.4164i	0.576390	−	0.773309i
$302$	15.8114			0.909843
$303$		0			0
$304$	2.82843			0.162221
$305$		−	18.0000i		−	1.03068i
$306$		0			0
$307$	−14.1421			−0.807134			−0.403567	−	0.914950i	$-0.632230\pi$
$307$	−14.1421			−0.807134			−0.403567	+	0.914950i	$0.632230\pi$
$308$	8.28934	+	2.87868i	0.472329	+	0.164028i
$309$		0			0
$310$		−	12.6491i		−	0.718421i
$311$		−	15.5563i		−	0.882120i	−0.897478	−	0.441060i	$-0.854603\pi$
$311$		−	15.5563i		−	0.882120i	0.897478	−	0.441060i	$-0.145397\pi$
$312$		0			0
$313$		−	26.8328i		−	1.51668i	−0.651859	−	0.758340i	$-0.726011\pi$
$313$		−	26.8328i		−	1.51668i	0.651859	−	0.758340i	$-0.273989\pi$
$314$	−13.4164			−0.757132
$315$		0			0
$316$			3.16228i			0.177892i
$317$	−9.48683			−0.532834			−0.266417	−	0.963858i	$-0.585840\pi$
$317$	−9.48683			−0.532834			−0.266417	+	0.963858i	$0.585840\pi$
$318$		0			0
$319$	−6.00000	−	18.9737i	−0.335936	−	1.06232i
$320$		−	1.41421i		−	0.0790569i
$321$		0			0
$322$	−5.00000	+	6.70820i	−0.278639	+	0.373834i
$323$	12.6491			0.703815
$324$		0			0
$325$	−4.24264			−0.235339
$326$		−	14.0000i		−	0.775388i
$327$		0			0
$328$		−	8.94427i		−	0.493865i
$329$	11.1803	−	15.0000i	0.616392	−	0.826977i
$330$		0			0
$331$	−10.0000			−0.549650			−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000			−0.549650			−0.274825	+	0.961494i	$0.588620\pi$
$332$	−4.47214			−0.245440
$333$		0			0
$334$		−	17.8885i		−	0.978818i
$335$			16.9706i			0.927201i
$336$		0			0
$337$			25.2982i			1.37808i	0.724722	+	0.689041i	$0.241968\pi$
$337$			25.2982i			1.37808i	−0.724722	+	0.689041i	$0.758032\pi$
$338$		−	11.0000i		−	0.598321i
$339$		0			0
$340$		−	6.32456i		−	0.342997i
$341$	8.94427	+	28.2843i	0.484359	+	1.53168i
$342$		0			0
$343$	6.36396	−	17.3925i	0.343622	−	0.939108i
$344$	6.32456			0.340997
$345$		0			0
$346$			13.4164i			0.721271i
$347$			28.0000i			1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$			28.0000i			1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$		0			0
$349$	32.5269			1.74113			0.870563	−	0.492057i	$-0.163755\pi$
$349$	32.5269			1.74113			0.870563	+	0.492057i	$0.163755\pi$
$350$	4.74342	−	6.36396i	0.253546	−	0.340168i
$351$		0			0
$352$	1.00000	+	3.16228i	0.0533002	+	0.168550i
$353$		−	28.2843i		−	1.50542i	−0.658352	−	0.752710i	$-0.728746\pi$
$353$		−	28.2843i		−	1.50542i	0.658352	−	0.752710i	$-0.271254\pi$
$354$		0			0
$355$			4.47214i			0.237356i
$356$			16.9706i			0.899438i
$357$		0			0
$358$			12.6491i			0.668526i
$359$			20.0000i			1.05556i	0.849381	+	0.527780i	$0.176975\pi$
$359$			20.0000i			1.05556i	−0.849381	+	0.527780i	$0.823025\pi$
$360$		0			0
$361$	−11.0000			−0.578947
$362$	13.4164			0.705151
$363$		0			0
$364$	−3.00000	−	2.23607i	−0.157243	−	0.117202i
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−3.16228			−0.164845
$369$		0			0
$370$	−2.82843			−0.147043
$371$	−6.70820	−	5.00000i	−0.348273	−	0.259587i
$372$		0			0
$373$			15.8114i			0.818683i	0.912381	+	0.409341i	$0.134241\pi$
$373$			15.8114i			0.818683i	−0.912381	+	0.409341i	$0.865759\pi$
$374$	4.47214	+	14.1421i	0.231249	+	0.731272i
$375$		0			0
$376$	7.07107			0.364662
$377$			8.48528i			0.437014i
$378$		0			0
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$		−	4.00000i		−	0.205196i
$381$		0			0
$382$			9.48683i			0.485389i
$383$		−	15.5563i		−	0.794892i	−0.917625	−	0.397446i	$-0.869897\pi$
$383$		−	15.5563i		−	0.794892i	0.917625	−	0.397446i	$-0.130103\pi$
$384$		0			0
$385$	4.07107	−	11.7229i	0.207481	−	0.597454i
$386$	6.32456			0.321911
$387$		0			0
$388$		−	4.47214i		−	0.227038i
$389$	34.7851			1.76367			0.881836	−	0.471556i	$-0.156307\pi$
$389$	34.7851			1.76367			0.881836	+	0.471556i	$0.156307\pi$
$390$		0			0
$391$	−14.1421			−0.715199
$392$	6.70820	−	2.00000i	0.338815	−	0.101015i
$393$		0			0
$394$	−2.00000			−0.100759
$395$	4.47214			0.225018
$396$		0			0
$397$			4.47214i			0.224450i	0.993683	+	0.112225i	$0.0357978\pi$
$397$			4.47214i			0.224450i	−0.993683	+	0.112225i	$0.964202\pi$
$398$	17.8885			0.896672
$399$		0			0
$400$	3.00000			0.150000
$401$	18.9737			0.947500			0.473750	−	0.880659i	$-0.342900\pi$
$401$	18.9737			0.947500			0.473750	+	0.880659i	$0.342900\pi$
$402$		0			0
$403$		−	12.6491i		−	0.630097i
$404$	−13.4164			−0.667491
$405$		0			0
$406$	−12.7279	−	9.48683i	−0.631676	−	0.470824i
$407$	6.32456	−	2.00000i	0.313497	−	0.0991363i
$408$		0			0
$409$	−2.82843			−0.139857			−0.0699284	−	0.997552i	$-0.522277\pi$
$409$	−2.82843			−0.139857			−0.0699284	+	0.997552i	$0.522277\pi$
$410$	−12.6491			−0.624695
$411$		0			0
$412$		−	8.94427i		−	0.440653i
$413$	4.47214	−	6.00000i	0.220059	−	0.295241i
$414$		0			0
$415$			6.32456i			0.310460i
$416$		−	1.41421i		−	0.0693375i
$417$		0			0
$418$	2.82843	+	8.94427i	0.138343	+	0.437479i
$419$		−	31.1127i		−	1.51995i	−0.649950	−	0.759977i	$-0.725210\pi$
$419$		−	31.1127i		−	1.51995i	0.649950	−	0.759977i	$-0.274790\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$	−18.9737			−0.923624
$423$		0			0
$424$		−	3.16228i		−	0.153574i
$425$	13.4164			0.650791
$426$		0			0
$427$	27.0000	+	20.1246i	1.30662	+	0.973898i
$428$		−	2.00000i		−	0.0966736i
$429$		0			0
$430$		−	8.94427i		−	0.431331i
$431$			8.00000i			0.385346i	0.981263	+	0.192673i	$0.0617157\pi$
$431$			8.00000i			0.385346i	−0.981263	+	0.192673i	$0.938284\pi$
$432$		0			0
$433$		−	8.94427i		−	0.429834i	−0.976632	−	0.214917i	$-0.931052\pi$
$433$		−	8.94427i		−	0.429834i	0.976632	−	0.214917i	$-0.0689481\pi$
$434$	18.9737	+	14.1421i	0.910765	+	0.678844i
$435$		0			0
$436$			3.16228i			0.151446i
$437$	−8.94427			−0.427863
$438$		0			0
$439$	32.5269			1.55242			0.776212	−	0.630471i	$-0.217138\pi$
$439$	32.5269			1.55242			0.776212	+	0.630471i	$0.217138\pi$
$440$	4.47214	−	1.41421i	0.213201	−	0.0674200i
$441$		0			0
$442$		−	6.32456i		−	0.300828i
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	24.0000			1.13771
$446$	26.8328			1.27057
$447$		0			0
$448$	2.12132	+	1.58114i	0.100223	+	0.0747018i
$449$	−31.6228			−1.49237			−0.746186	−	0.665738i	$-0.768117\pi$
$449$	−31.6228			−1.49237			−0.746186	+	0.665738i	$0.768117\pi$
$450$		0			0
$451$	28.2843	−	8.94427i	1.33185	−	0.421169i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.16228	+	4.24264i	−0.148250	+	0.198898i
$456$		0			0
$457$			18.9737i			0.887551i	0.896138	+	0.443775i	$0.146361\pi$
$457$			18.9737i			0.887551i	−0.896138	+	0.443775i	$0.853639\pi$
$458$	4.47214			0.208969
$459$		0			0
$460$			4.47214i			0.208514i
$461$	4.47214			0.208288			0.104144	−	0.994562i	$-0.466790\pi$
$461$	4.47214			0.208288			0.104144	+	0.994562i	$0.466790\pi$
$462$		0			0
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$		−	6.00000i		−	0.278543i
$465$		0			0
$466$	6.00000			0.277945
$467$		−	16.9706i		−	0.785304i	−0.919687	−	0.392652i	$-0.871558\pi$
$467$		−	16.9706i		−	0.785304i	0.919687	−	0.392652i	$-0.128442\pi$
$468$		0			0
$469$	−25.4558	−	18.9737i	−1.17544	−	0.876122i
$470$		−	10.0000i		−	0.461266i
$471$		0			0
$472$	2.82843			0.130189
$473$	6.32456	+	20.0000i	0.290803	+	0.919601i
$474$		0			0
$475$	8.48528			0.389331
$476$	9.48683	+	7.07107i	0.434828	+	0.324102i
$477$		0			0
$478$	−16.0000			−0.731823
$479$	−35.7771			−1.63470			−0.817348	−	0.576144i	$-0.804557\pi$
$479$	−35.7771			−1.63470			−0.817348	+	0.576144i	$0.804557\pi$
$480$		0			0
$481$	−2.82843			−0.128965
$482$		−	8.48528i		−	0.386494i
$483$		0			0
$484$	−9.00000	+	6.32456i	−0.409091	+	0.287480i
$485$	−6.32456			−0.287183
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$			12.7279i			0.576166i
$489$		0			0
$490$	−2.82843	−	9.48683i	−0.127775	−	0.428571i
$491$		−	18.0000i		−	0.812329i	−0.913800	−	0.406164i	$-0.866866\pi$
$491$		−	18.0000i		−	0.812329i	0.913800	−	0.406164i	$-0.133134\pi$
$492$		0			0
$493$		−	26.8328i		−	1.20849i
$494$		−	4.00000i		−	0.179969i
$495$		0			0
$496$			8.94427i			0.401610i
$497$	−6.70820	−	5.00000i	−0.300904	−	0.224281i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		−	11.3137i		−	0.505964i
$501$		0			0
$502$	19.7990			0.883672
$503$	−35.7771			−1.59522			−0.797611	−	0.603173i	$-0.793903\pi$
$503$	−35.7771			−1.59522			−0.797611	+	0.603173i	$0.793903\pi$
$504$		0			0
$505$			18.9737i			0.844317i
$506$	−3.16228	−	10.0000i	−0.140580	−	0.444554i
$507$		0			0
$508$		−	22.1359i		−	0.982124i
$509$		−	9.89949i		−	0.438787i	−0.975636	−	0.219394i	$-0.929592\pi$
$509$		−	9.89949i		−	0.438787i	0.975636	−	0.219394i	$-0.0704079\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−16.9706			−0.748539
$515$	−12.6491			−0.557386
$516$		0			0
$517$	7.07107	+	22.3607i	0.310985	+	0.983422i
$518$	3.16228	−	4.24264i	0.138943	−	0.186411i
$519$		0			0
$520$	−2.00000			−0.0877058
$521$			19.7990i			0.867409i	0.901055	+	0.433705i	$0.142794\pi$
$521$			19.7990i			0.867409i	−0.901055	+	0.433705i	$0.857206\pi$
$522$		0			0
$523$	−22.6274			−0.989428			−0.494714	−	0.869056i	$-0.664727\pi$
$523$	−22.6274			−0.989428			−0.494714	+	0.869056i	$0.664727\pi$
$524$	4.47214			0.195366
$525$		0			0
$526$	4.00000			0.174408
$527$			40.0000i			1.74243i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	−4.47214			−0.194257
$531$		0			0
$532$	6.00000	+	4.47214i	0.260133	+	0.193892i
$533$	−12.6491			−0.547894
$534$		0			0
$535$	−2.82843			−0.122284
$536$		−	12.0000i		−	0.518321i
$537$		0			0
$538$	−18.3848			−0.792624
$539$	13.0328	+	19.2132i	0.561361	+	0.827571i
$540$		0			0
$541$			22.1359i			0.951699i	0.879527	+	0.475849i	$0.157859\pi$
$541$			22.1359i			0.951699i	−0.879527	+	0.475849i	$0.842141\pi$
$542$		−	12.7279i		−	0.546711i
$543$		0			0
$544$			4.47214i			0.191741i
$545$	4.47214			0.191565
$546$		0			0
$547$			12.6491i			0.540837i	0.962743	+	0.270418i	$0.0871621\pi$
$547$			12.6491i			0.540837i	−0.962743	+	0.270418i	$0.912838\pi$
$548$	−18.9737			−0.810515
$549$		0			0
$550$	3.00000	+	9.48683i	0.127920	+	0.404520i
$551$		−	16.9706i		−	0.722970i
$552$		0			0
$553$	−5.00000	+	6.70820i	−0.212622	+	0.285262i
$554$	22.1359			0.940466
$555$		0			0
$556$	−2.82843			−0.119952
$557$		−	2.00000i		−	0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$		−	2.00000i		−	0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$		0			0
$559$		−	8.94427i		−	0.378302i
$560$	2.23607	−	3.00000i	0.0944911	−	0.126773i
$561$		0			0
$562$	20.0000			0.843649
$563$	−35.7771			−1.50782			−0.753912	−	0.656975i	$-0.771836\pi$
$563$	−35.7771			−1.50782			−0.753912	+	0.656975i	$0.771836\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		−	3.16228i		−	0.132686i
$569$			24.0000i			1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$			24.0000i			1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$		0			0
$571$		−	12.6491i		−	0.529349i	−0.964338	−	0.264674i	$-0.914736\pi$
$571$		−	12.6491i		−	0.529349i	0.964338	−	0.264674i	$-0.0852645\pi$
$572$	4.47214	−	1.41421i	0.186989	−	0.0591312i
$573$		0			0
$574$	14.1421	−	18.9737i	0.590281	−	0.791946i
$575$	−9.48683			−0.395628
$576$		0			0
$577$			31.3050i			1.30324i	0.758545	+	0.651621i	$0.225911\pi$
$577$			31.3050i			1.30324i	−0.758545	+	0.651621i	$0.774089\pi$
$578$			3.00000i			0.124784i
$579$		0			0
$580$	−8.48528			−0.352332
$581$	−9.48683	−	7.07107i	−0.393580	−	0.293357i
$582$		0			0
$583$	10.0000	−	3.16228i	0.414158	−	0.130968i
$584$		0			0
$585$		0			0
$586$			4.47214i			0.184742i
$587$			42.4264i			1.75113i	0.483105	+	0.875563i	$0.339509\pi$
$587$			42.4264i			1.75113i	−0.483105	+	0.875563i	$0.660491\pi$
$588$		0			0
$589$			25.2982i			1.04240i
$590$		−	4.00000i		−	0.164677i
$591$		0			0
$592$	2.00000			0.0821995
$593$	−4.47214			−0.183649			−0.0918243	−	0.995775i	$-0.529270\pi$
$593$	−4.47214			−0.183649			−0.0918243	+	0.995775i	$0.529270\pi$
$594$		0			0
$595$	10.0000	−	13.4164i	0.409960	−	0.550019i
$596$			10.0000i			0.409616i
$597$		0			0
$598$			4.47214i			0.182879i
$599$	−9.48683			−0.387621			−0.193811	−	0.981039i	$-0.562085\pi$
$599$	−9.48683			−0.387621			−0.193811	+	0.981039i	$0.562085\pi$
$600$		0			0
$601$	−33.9411			−1.38449			−0.692244	−	0.721664i	$-0.743378\pi$
$601$	−33.9411			−1.38449			−0.692244	+	0.721664i	$0.743378\pi$
$602$	13.4164	+	10.0000i	0.546812	+	0.407570i
$603$		0			0
$604$			15.8114i			0.643356i
$605$	8.94427	+	12.7279i	0.363636	+	0.517464i
$606$		0			0
$607$	−38.1838			−1.54983			−0.774916	−	0.632065i	$-0.782208\pi$
$607$	−38.1838			−1.54983			−0.774916	+	0.632065i	$0.782208\pi$
$608$			2.82843i			0.114708i
$609$		0			0
$610$	18.0000			0.728799
$611$		−	10.0000i		−	0.404557i
$612$		0			0
$613$		−	28.4605i		−	1.14951i	−0.818326	−	0.574754i	$-0.805098\pi$
$613$		−	28.4605i		−	1.14951i	0.818326	−	0.574754i	$-0.194902\pi$
$614$		−	14.1421i		−	0.570730i
$615$		0			0
$616$	−2.87868	+	8.28934i	−0.115985	+	0.333987i
$617$	−37.9473			−1.52770			−0.763851	−	0.645393i	$-0.776694\pi$
$617$	−37.9473			−1.52770			−0.763851	+	0.645393i	$0.776694\pi$
$618$		0			0
$619$		−	4.47214i		−	0.179750i	−0.995953	−	0.0898752i	$-0.971353\pi$
$619$		−	4.47214i		−	0.179750i	0.995953	−	0.0898752i	$-0.0286468\pi$
$620$	12.6491			0.508001
$621$		0			0
$622$	15.5563			0.623753
$623$	−26.8328	+	36.0000i	−1.07503	+	1.44231i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$	26.8328			1.07246
$627$		0			0
$628$		−	13.4164i		−	0.535373i
$629$	8.94427			0.356631
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$	−3.16228			−0.125789
$633$		0			0
$634$		−	9.48683i		−	0.376770i
$635$	−31.3050			−1.24230
$636$		0			0
$637$	−2.82843	−	9.48683i	−0.112066	−	0.375882i
$638$	18.9737	−	6.00000i	0.751175	−	0.237542i
$639$		0			0
$640$	1.41421			0.0559017
$641$	6.32456			0.249805			0.124902	−	0.992169i	$-0.460138\pi$
$641$	6.32456			0.249805			0.124902	+	0.992169i	$0.460138\pi$
$642$		0			0
$643$			31.3050i			1.23455i	0.786749	+	0.617273i	$0.211763\pi$
$643$			31.3050i			1.23455i	−0.786749	+	0.617273i	$0.788237\pi$
$644$	−6.70820	−	5.00000i	−0.264340	−	0.197028i
$645$		0			0
$646$			12.6491i			0.497673i
$647$		−	32.5269i		−	1.27876i	−0.768889	−	0.639382i	$-0.779190\pi$
$647$		−	32.5269i		−	1.27876i	0.768889	−	0.639382i	$-0.220810\pi$
$648$		0			0
$649$	2.82843	+	8.94427i	0.111025	+	0.351093i
$650$		−	4.24264i		−	0.166410i
$651$		0			0
$652$	14.0000			0.548282
$653$	47.4342			1.85624			0.928121	−	0.372278i	$-0.121423\pi$
$653$	47.4342			1.85624			0.928121	+	0.372278i	$0.121423\pi$
$654$		0			0
$655$		−	6.32456i		−	0.247121i
$656$	8.94427			0.349215
$657$		0			0
$658$	15.0000	+	11.1803i	0.584761	+	0.435855i
$659$			30.0000i			1.16863i	0.811525	+	0.584317i	$0.198638\pi$
$659$			30.0000i			1.16863i	−0.811525	+	0.584317i	$0.801362\pi$
$660$		0			0
$661$			40.2492i			1.56551i	0.622328	+	0.782757i	$0.286187\pi$
$661$			40.2492i			1.56551i	−0.622328	+	0.782757i	$0.713813\pi$
$662$		−	10.0000i		−	0.388661i
$663$		0			0
$664$		−	4.47214i		−	0.173553i
$665$	6.32456	−	8.48528i	0.245256	−	0.329045i
$666$		0			0
$667$			18.9737i			0.734663i
$668$	17.8885			0.692129
$669$		0			0
$670$	−16.9706			−0.655630
$671$	−40.2492	+	12.7279i	−1.55380	+	0.491356i
$672$		0			0
$673$			18.9737i			0.731381i	0.930736	+	0.365691i	$0.119167\pi$
$673$			18.9737i			0.731381i	−0.930736	+	0.365691i	$0.880833\pi$
$674$	−25.2982			−0.974451
$675$		0			0
$676$	11.0000			0.423077
$677$	−31.3050			−1.20315			−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050			−1.20315			−0.601574	+	0.798817i	$0.705459\pi$
$678$		0			0
$679$	7.07107	−	9.48683i	0.271363	−	0.364071i
$680$	6.32456			0.242536
$681$		0			0
$682$	−28.2843	+	8.94427i	−1.08306	+	0.342494i
$683$	−37.9473			−1.45201			−0.726007	−	0.687687i	$-0.758626\pi$
$683$	−37.9473			−1.45201			−0.726007	+	0.687687i	$0.758626\pi$
$684$		0			0
$685$			26.8328i			1.02523i
$686$	17.3925	+	6.36396i	0.664050	+	0.242977i
$687$		0			0
$688$			6.32456i			0.241121i
$689$	−4.47214			−0.170375
$690$		0			0
$691$			13.4164i			0.510384i	0.966890	+	0.255192i	$0.0821387\pi$
$691$			13.4164i			0.510384i	−0.966890	+	0.255192i	$0.917861\pi$
$692$	−13.4164			−0.510015
$693$		0			0
$694$	−28.0000			−1.06287
$695$			4.00000i			0.151729i
$696$		0			0
$697$	40.0000			1.51511
$698$			32.5269i			1.23116i
$699$		0			0
$700$	6.36396	+	4.74342i	0.240535	+	0.179284i
$701$		−	18.0000i		−	0.679851i	−0.940452	−	0.339925i	$-0.889598\pi$
$701$		−	18.0000i		−	0.679851i	0.940452	−	0.339925i	$-0.110402\pi$
$702$		0			0
$703$	5.65685			0.213352
$704$	−3.16228	+	1.00000i	−0.119183	+	0.0376889i
$705$		0			0
$706$	28.2843			1.06449
$707$	−28.4605	−	21.2132i	−1.07037	−	0.797805i
$708$		0			0
$709$	−14.0000			−0.525781			−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000			−0.525781			−0.262891	+	0.964826i	$0.584676\pi$
$710$	−4.47214			−0.167836
$711$		0			0
$712$	−16.9706			−0.635999
$713$		−	28.2843i		−	1.05925i
$714$		0			0
$715$	−2.00000	−	6.32456i	−0.0747958	−	0.236525i
$716$	−12.6491			−0.472719
$717$		0			0
$718$	−20.0000			−0.746393
$719$			18.3848i			0.685636i	0.939402	+	0.342818i	$0.111381\pi$
$719$			18.3848i			0.685636i	−0.939402	+	0.342818i	$0.888619\pi$
$720$		0			0
$721$	14.1421	−	18.9737i	0.526681	−	0.706616i
$722$		−	11.0000i		−	0.409378i
$723$		0			0
$724$			13.4164i			0.498617i
$725$		−	18.0000i		−	0.668503i
$726$		0			0
$727$			26.8328i			0.995174i	0.867414	+	0.497587i	$0.165780\pi$
$727$			26.8328i			0.995174i	−0.867414	+	0.497587i	$0.834220\pi$
$728$	2.23607	−	3.00000i	0.0828742	−	0.111187i
$729$		0			0
$730$		0			0
$731$			28.2843i			1.04613i
$732$		0			0
$733$	21.2132			0.783528			0.391764	−	0.920066i	$-0.371865\pi$
$733$	21.2132			0.783528			0.391764	+	0.920066i	$0.371865\pi$
$734$		0			0
$735$		0			0
$736$		−	3.16228i		−	0.116563i
$737$	37.9473	−	12.0000i	1.39781	−	0.442026i
$738$		0			0
$739$			18.9737i			0.697958i	0.937131	+	0.348979i	$0.113471\pi$
$739$			18.9737i			0.697958i	−0.937131	+	0.348979i	$0.886529\pi$
$740$		−	2.82843i		−	0.103975i
$741$		0			0
$742$	5.00000	−	6.70820i	0.183556	−	0.246266i
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$		0			0
$745$	14.1421			0.518128
$746$	−15.8114			−0.578896
$747$		0			0
$748$	−14.1421	+	4.47214i	−0.517088	+	0.163517i
$749$	3.16228	−	4.24264i	0.115547	−	0.155023i
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$			7.07107i			0.257855i
$753$		0			0
$754$	−8.48528			−0.309016
$755$	22.3607			0.813788
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		−	10.0000i		−	0.363216i
$759$		0			0
$760$	4.00000			0.145095
$761$	22.3607			0.810574			0.405287	−	0.914190i	$-0.367172\pi$
$761$	22.3607			0.810574			0.405287	+	0.914190i	$0.367172\pi$
$762$		0			0
$763$	−5.00000	+	6.70820i	−0.181012	+	0.242853i
$764$	−9.48683			−0.343222
$765$		0			0
$766$	15.5563			0.562074
$767$		−	4.00000i		−	0.144432i
$768$		0			0
$769$	−25.4558			−0.917961			−0.458981	−	0.888446i	$-0.651785\pi$
$769$	−25.4558			−0.917961			−0.458981	+	0.888446i	$0.651785\pi$
$770$	11.7229	+	4.07107i	0.422464	+	0.146711i
$771$		0			0
$772$			6.32456i			0.227626i
$773$		−	15.5563i		−	0.559523i	−0.960070	−	0.279761i	$-0.909745\pi$
$773$		−	15.5563i		−	0.559523i	0.960070	−	0.279761i	$-0.0902554\pi$
$774$		0			0
$775$			26.8328i			0.963863i
$776$	4.47214			0.160540
$777$		0			0
$778$			34.7851i			1.24710i
$779$	25.2982			0.906403
$780$		0			0
$781$	10.0000	−	3.16228i	0.357828	−	0.113155i
$782$		−	14.1421i		−	0.505722i
$783$		0			0
$784$	2.00000	+	6.70820i	0.0714286	+	0.239579i
$785$	−18.9737			−0.677199
$786$		0			0
$787$	−28.2843			−1.00823			−0.504113	−	0.863638i	$-0.668180\pi$
$787$	−28.2843			−1.00823			−0.504113	+	0.863638i	$0.668180\pi$
$788$		−	2.00000i		−	0.0712470i
$789$		0			0
$790$			4.47214i			0.159111i
$791$		0			0
$792$		0			0
$793$	18.0000			0.639199
$794$	−4.47214			−0.158710
$795$		0			0
$796$			17.8885i			0.634043i
$797$		−	21.2132i		−	0.751410i	−0.926739	−	0.375705i	$-0.877401\pi$
$797$		−	21.2132i		−	0.751410i	0.926739	−	0.375705i	$-0.122599\pi$
$798$		0			0
$799$			31.6228i			1.11873i
$800$			3.00000i			0.106066i
$801$		0			0
$802$			18.9737i			0.669983i
$803$		0			0
$804$		0			0
$805$	−7.07107	+	9.48683i	−0.249222	+	0.334367i
$806$	12.6491			0.445546
$807$		0			0
$808$		−	13.4164i		−	0.471988i
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$		0			0
$811$	33.9411			1.19183			0.595917	−	0.803046i	$-0.296789\pi$
$811$	33.9411			1.19183			0.595917	+	0.803046i	$0.296789\pi$
$812$	9.48683	−	12.7279i	0.332923	−	0.446663i
$813$		0			0
$814$	2.00000	+	6.32456i	0.0701000	+	0.221676i
$815$		−	19.7990i		−	0.693528i
$816$		0			0
$817$			17.8885i			0.625841i
$818$		−	2.82843i		−	0.0988936i
$819$		0			0
$820$		−	12.6491i		−	0.441726i
$821$			10.0000i			0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$			10.0000i			0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$		0			0
$823$	−4.00000			−0.139431			−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000			−0.139431			−0.0697156	+	0.997567i	$0.522209\pi$
$824$	8.94427			0.311588
$825$		0			0
$826$	6.00000	+	4.47214i	0.208767	+	0.155606i
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$		−	4.47214i		−	0.155324i	−0.996980	−	0.0776619i	$-0.975255\pi$
$829$		−	4.47214i		−	0.155324i	0.996980	−	0.0776619i	$-0.0247455\pi$
$830$	−6.32456			−0.219529
$831$		0			0
$832$	1.41421			0.0490290
$833$	8.94427	+	30.0000i	0.309901	+	1.03944i
$834$		0			0
$835$		−	25.2982i		−	0.875481i
$836$	−8.94427	+	2.82843i	−0.309344	+	0.0978232i
$837$		0			0
$838$	31.1127			1.07477
$839$		−	32.5269i		−	1.12295i	−0.827492	−	0.561477i	$-0.810233\pi$
$839$		−	32.5269i		−	1.12295i	0.827492	−	0.561477i	$-0.189767\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		−	30.0000i		−	1.03387i
$843$		0			0
$844$		−	18.9737i		−	0.653101i
$845$		−	15.5563i		−	0.535155i
$846$		0			0
$847$	−29.0919	−	0.813842i	−0.999609	−	0.0279639i
$848$	3.16228			0.108593
$849$		0			0
$850$			13.4164i			0.460179i
$851$	−6.32456			−0.216803
$852$		0			0
$853$	15.5563			0.532639			0.266320	−	0.963885i	$-0.414192\pi$
$853$	15.5563			0.532639			0.266320	+	0.963885i	$0.414192\pi$
$854$	−20.1246	+	27.0000i	−0.688650	+	0.923921i
$855$		0			0
$856$	2.00000			0.0683586
$857$	−8.94427			−0.305531			−0.152765	−	0.988263i	$-0.548818\pi$
$857$	−8.94427			−0.305531			−0.152765	+	0.988263i	$0.548818\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	8.94427			0.304997
$861$		0			0
$862$	−8.00000			−0.272481
$863$	28.4605			0.968807			0.484403	−	0.874845i	$-0.339037\pi$
$863$	28.4605			0.968807			0.484403	+	0.874845i	$0.339037\pi$
$864$		0			0
$865$			18.9737i			0.645124i
$866$	8.94427			0.303939
$867$		0			0
$868$	−14.1421	+	18.9737i	−0.480015	+	0.644008i
$869$	−3.16228	−	10.0000i	−0.107273	−	0.339227i
$870$		0			0
$871$	−16.9706			−0.575026
$872$	−3.16228			−0.107088
$873$		0			0
$874$		−	8.94427i		−	0.302545i
$875$	17.8885	−	24.0000i	0.604743	−	0.811348i
$876$		0			0
$877$		−	41.1096i		−	1.38817i	−0.719892	−	0.694086i	$-0.755809\pi$
$877$		−	41.1096i		−	1.38817i	0.719892	−	0.694086i	$-0.244191\pi$
$878$			32.5269i			1.09773i
$879$		0			0
$880$	1.41421	+	4.47214i	0.0476731	+	0.150756i
$881$		−	5.65685i		−	0.190584i	−0.995449	−	0.0952921i	$-0.969621\pi$
$881$		−	5.65685i		−	0.190584i	0.995449	−	0.0952921i	$-0.0303785\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$	6.32456			0.212718
$885$		0			0
$886$		0			0
$887$	−44.7214			−1.50160			−0.750798	−	0.660532i	$-0.770331\pi$
$887$	−44.7214			−1.50160			−0.750798	+	0.660532i	$0.770331\pi$
$888$		0			0
$889$	35.0000	−	46.9574i	1.17386	−	1.57490i
$890$			24.0000i			0.804482i
$891$		0			0
$892$			26.8328i			0.898429i
$893$			20.0000i			0.669274i
$894$		0			0
$895$			17.8885i			0.597948i
$896$	−1.58114	+	2.12132i	−0.0528221	+	0.0708683i
$897$		0			0
$898$		−	31.6228i		−	1.05527i
$899$	53.6656			1.78985
$900$		0			0
$901$	14.1421			0.471143
$902$	8.94427	+	28.2843i	0.297812	+	0.941763i
$903$		0			0
$904$		0			0
$905$	18.9737			0.630706
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$		0			0
$909$		0			0
$910$	−4.24264	−	3.16228i	−0.140642	−	0.104828i
$911$	−28.4605			−0.942938			−0.471469	−	0.881883i	$-0.656276\pi$
$911$	−28.4605			−0.942938			−0.471469	+	0.881883i	$0.656276\pi$
$912$		0			0
$913$	14.1421	−	4.47214i	0.468036	−	0.148006i
$914$	−18.9737			−0.627593
$915$		0			0
$916$			4.47214i			0.147764i
$917$	9.48683	+	7.07107i	0.313283	+	0.233507i
$918$		0			0
$919$			22.1359i			0.730197i	0.930969	+	0.365099i	$0.118965\pi$
$919$			22.1359i			0.730197i	−0.930969	+	0.365099i	$0.881035\pi$
$920$	−4.47214			−0.147442
$921$		0			0
$922$			4.47214i			0.147282i
$923$	−4.47214			−0.147202
$924$		0			0
$925$	6.00000			0.197279
$926$		−	4.00000i		−	0.131448i
$927$		0			0
$928$	6.00000			0.196960
$929$			31.1127i			1.02077i	0.859945	+	0.510387i	$0.170498\pi$
$929$			31.1127i			1.02077i	−0.859945	+	0.510387i	$0.829502\pi$
$930$		0			0
$931$	5.65685	+	18.9737i	0.185396	+	0.621837i
$932$			6.00000i			0.196537i
$933$		0			0
$934$	16.9706			0.555294
$935$	6.32456	+	20.0000i	0.206835	+	0.654070i
$936$		0			0
$937$	−45.2548			−1.47841			−0.739205	−	0.673480i	$-0.764799\pi$
$937$	−45.2548			−1.47841			−0.739205	+	0.673480i	$0.764799\pi$
$938$	18.9737	−	25.4558i	0.619512	−	0.831163i
$939$		0			0
$940$	10.0000			0.326164
$941$	31.3050			1.02051			0.510256	−	0.860022i	$-0.329551\pi$
$941$	31.3050			1.02051			0.510256	+	0.860022i	$0.329551\pi$
$942$		0			0
$943$	−28.2843			−0.921063
$944$			2.82843i			0.0920575i
$945$		0			0
$946$	−20.0000	+	6.32456i	−0.650256	+	0.205629i
$947$	31.6228			1.02760			0.513801	−	0.857909i	$-0.328237\pi$
$947$	31.6228			1.02760			0.513801	+	0.857909i	$0.328237\pi$
$948$		0			0
$949$		0			0
$950$			8.48528i			0.275299i
$951$		0			0
$952$	−7.07107	+	9.48683i	−0.229175	+	0.307470i
$953$		−	54.0000i		−	1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$		−	54.0000i		−	1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$		0			0
$955$			13.4164i			0.434145i
$956$		−	16.0000i		−	0.517477i
$957$		0			0
$958$		−	35.7771i		−	1.15591i
$959$	−40.2492	−	30.0000i	−1.29972	−	0.968751i
$960$		0			0
$961$	−49.0000			−1.58065
$962$		−	2.82843i		−	0.0911922i
$963$		0			0
$964$	8.48528			0.273293
$965$	8.94427			0.287926
$966$		0			0
$967$			9.48683i			0.305076i	0.988298	+	0.152538i	$0.0487446\pi$
$967$			9.48683i			0.305076i	−0.988298	+	0.152538i	$0.951255\pi$
$968$	−6.32456	−	9.00000i	−0.203279	−	0.289271i
$969$		0			0
$970$		−	6.32456i		−	0.203069i
$971$			19.7990i			0.635380i	0.948195	+	0.317690i	$0.102907\pi$
$971$			19.7990i			0.635380i	−0.948195	+	0.317690i	$0.897093\pi$
$972$		0			0
$973$	−6.00000	−	4.47214i	−0.192351	−	0.143370i
$974$			12.0000i			0.384505i
$975$		0			0
$976$	−12.7279			−0.407411
$977$	−18.9737			−0.607021			−0.303511	−	0.952828i	$-0.598159\pi$
$977$	−18.9737			−0.607021			−0.303511	+	0.952828i	$0.598159\pi$
$978$		0			0
$979$	−16.9706	−	53.6656i	−0.542382	−	1.71516i
$980$	9.48683	−	2.82843i	0.303046	−	0.0903508i
$981$		0			0
$982$	18.0000			0.574403
$983$			41.0122i			1.30809i	0.756457	+	0.654043i	$0.226928\pi$
$983$			41.0122i			1.30809i	−0.756457	+	0.654043i	$0.773072\pi$
$984$		0			0
$985$	−2.82843			−0.0901212
$986$	26.8328			0.854531
$987$		0			0
$988$	4.00000			0.127257
$989$		−	20.0000i		−	0.635963i
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	−8.94427			−0.283981
$993$		0			0
$994$	5.00000	−	6.70820i	0.158590	−	0.212771i
$995$	25.2982			0.802008
$996$		0			0
$997$	−21.2132			−0.671829			−0.335914	−	0.941893i	$-0.609045\pi$
$997$	−21.2132			−0.671829			−0.335914	+	0.941893i	$0.609045\pi$
$998$		−	10.0000i		−	0.316544i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.e.d.307.7	yes	8
3.2	odd	2	inner	1386.2.e.d.307.1	✓	8
7.6	odd	2	inner	1386.2.e.d.307.5	yes	8
11.10	odd	2	inner	1386.2.e.d.307.4	yes	8
21.20	even	2	inner	1386.2.e.d.307.3	yes	8
33.32	even	2	inner	1386.2.e.d.307.6	yes	8
77.76	even	2	inner	1386.2.e.d.307.2	yes	8
231.230	odd	2	inner	1386.2.e.d.307.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.e.d.307.1	✓	8	3.2	odd	2	inner
1386.2.e.d.307.2	yes	8	77.76	even	2	inner
1386.2.e.d.307.3	yes	8	21.20	even	2	inner
1386.2.e.d.307.4	yes	8	11.10	odd	2	inner
1386.2.e.d.307.5	yes	8	7.6	odd	2	inner
1386.2.e.d.307.6	yes	8	33.32	even	2	inner
1386.2.e.d.307.7	yes	8	1.1	even	1	trivial
1386.2.e.d.307.8	yes	8	231.230	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.41421i q^{5} +(-2.12132 - 1.58114i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.41421i q^{5} +(-2.12132 - 1.58114i) q^{7} -1.00000i q^{8} -1.41421 q^{10} +(3.16228 - 1.00000i) q^{11} -1.41421 q^{13} +(1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214 q^{17} +2.82843 q^{19} -1.41421i q^{20} +(1.00000 + 3.16228i) q^{22} -3.16228 q^{23} +3.00000 q^{25} -1.41421i q^{26} +(2.12132 + 1.58114i) q^{28} -6.00000i q^{29} +8.94427i q^{31} +1.00000i q^{32} +4.47214i q^{34} +(2.23607 - 3.00000i) q^{35} +2.00000 q^{37} +2.82843i q^{38} +1.41421 q^{40} +8.94427 q^{41} +6.32456i q^{43} +(-3.16228 + 1.00000i) q^{44} -3.16228i q^{46} +7.07107i q^{47} +(2.00000 + 6.70820i) q^{49} +3.00000i q^{50} +1.41421 q^{52} +3.16228 q^{53} +(1.41421 + 4.47214i) q^{55} +(-1.58114 + 2.12132i) q^{56} +6.00000 q^{58} +2.82843i q^{59} -12.7279 q^{61} -8.94427 q^{62} -1.00000 q^{64} -2.00000i q^{65} +12.0000 q^{67} -4.47214 q^{68} +(3.00000 + 2.23607i) q^{70} +3.16228 q^{71} +2.00000i q^{74} -2.82843 q^{76} +(-8.28934 - 2.87868i) q^{77} -3.16228i q^{79} +1.41421i q^{80} +8.94427i q^{82} +4.47214 q^{83} +6.32456i q^{85} -6.32456 q^{86} +(-1.00000 - 3.16228i) q^{88} -16.9706i q^{89} +(3.00000 + 2.23607i) q^{91} +3.16228 q^{92} -7.07107 q^{94} +4.00000i q^{95} +4.47214i q^{97} +(-6.70820 + 2.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{22} + 24 q^{25} + 16 q^{37} + 16 q^{49} + 48 q^{58} - 8 q^{64} + 96 q^{67} + 24 q^{70} - 8 q^{88} + 24 q^{91}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^22 + 24 * q^25 + 16 * q^37 + 16 * q^49 + 48 * q^58 - 8 * q^64 + 96 * q^67 + 24 * q^70 - 8 * q^88 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more