LMFDB - Embedded newform 2106.2.b.d.649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2106,2,Mod(649,2106)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2106, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2106.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.b (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:	$16.8164946657$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 $ x^14 + 34x^12 + 435x^10 + 2617x^8 + 7651x^6 + 10260x^4 + 5589x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$-3.24018i$ of defining polynomial
Character	$\chi$	$=$	2106.649
Dual form			2106.2.b.d.649.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -0.633132i q^{5} +2.48683i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -0.633132i q^{5} +2.48683i q^{7} +1.00000i q^{8} -0.633132 q^{10} -4.86004i q^{11} +(-2.81615 - 2.25152i) q^{13} +2.48683 q^{14} +1.00000 q^{16} +6.27837 q^{17} +4.86004i q^{19} +0.633132i q^{20} -4.86004 q^{22} -3.54768 q^{23} +4.59914 q^{25} +(-2.25152 + 2.81615i) q^{26} -2.48683i q^{28} +0.830894 q^{29} -4.31236i q^{31} -1.00000i q^{32} -6.27837i q^{34} +1.57449 q^{35} -7.81310i q^{37} +4.86004 q^{38} +0.633132 q^{40} -0.0783645i q^{41} -9.68740 q^{43} +4.86004i q^{44} +3.54768i q^{46} +4.96601i q^{47} +0.815659 q^{49} -4.59914i q^{50} +(2.81615 + 2.25152i) q^{52} +6.36026 q^{53} -3.07704 q^{55} -2.48683 q^{56} -0.830894i q^{58} -9.08742i q^{59} -10.5707 q^{61} -4.31236 q^{62} -1.00000 q^{64} +(-1.42551 + 1.78299i) q^{65} -7.88457i q^{67} -6.27837 q^{68} -1.57449i q^{70} -12.3085i q^{71} -1.05367i q^{73} -7.81310 q^{74} -4.86004i q^{76} +12.0861 q^{77} +3.37676 q^{79} -0.633132i q^{80} -0.0783645 q^{82} -15.2137i q^{83} -3.97504i q^{85} +9.68740i q^{86} +4.86004 q^{88} -0.595093i q^{89} +(5.59914 - 7.00329i) q^{91} +3.54768 q^{92} +4.96601 q^{94} +3.07704 q^{95} -17.0630i q^{97} -0.815659i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{4}+O(q^{10}) $ 14 * q - 14 * q^4	$ 14 q - 14 q^{4} - 2 q^{13} + 8 q^{14} + 14 q^{16} - 8 q^{17} - 8 q^{23} - 14 q^{25} - 4 q^{26} - 16 q^{29} + 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} + 60 q^{53} - 8 q^{56} - 28 q^{61} - 34 q^{62} - 14 q^{64} - 8 q^{65} + 8 q^{68} + 16 q^{74} - 24 q^{77} - 28 q^{79} - 24 q^{82} + 8 q^{92}+O(q^{100}) $ 14 * q - 14 * q^4 - 2 * q^13 + 8 * q^14 + 14 * q^16 - 8 * q^17 - 8 * q^23 - 14 * q^25 - 4 * q^26 - 16 * q^29 + 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 + 60 * q^53 - 8 * q^56 - 28 * q^61 - 34 * q^62 - 14 * q^64 - 8 * q^65 + 8 * q^68 + 16 * q^74 - 24 * q^77 - 28 * q^79 - 24 * q^82 + 8 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$.

$n$	$1379$	$1783$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	0.633132i		−	0.283145i	−0.989928	−	0.141573i	$-0.954784\pi$
$5$		−	0.633132i		−	0.283145i	0.989928	−	0.141573i	$-0.0452159\pi$
$6$		0			0
$7$			2.48683i			0.939935i	0.882684	+	0.469967i	$0.155734\pi$
$7$			2.48683i			0.939935i	−0.882684	+	0.469967i	$0.844266\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−0.633132			−0.200214
$11$		−	4.86004i		−	1.46536i	−0.680575	−	0.732678i	$-0.738270\pi$
$11$		−	4.86004i		−	1.46536i	0.680575	−	0.732678i	$-0.261730\pi$
$12$		0			0
$13$	−2.81615	−	2.25152i	−0.781058	−	0.624458i
$13$	−2.81615	−	2.25152i	−0.781058	−	0.624458i
$14$	2.48683			0.664634
$15$		0			0
$16$	1.00000			0.250000
$17$	6.27837			1.52273			0.761364	−	0.648325i	$-0.224530\pi$
$17$	6.27837			1.52273			0.761364	+	0.648325i	$0.224530\pi$
$18$		0			0
$19$			4.86004i			1.11497i	0.830187	+	0.557484i	$0.188233\pi$
$19$			4.86004i			1.11497i	−0.830187	+	0.557484i	$0.811767\pi$
$20$			0.633132i			0.141573i
$21$		0			0
$22$	−4.86004			−1.03616
$23$	−3.54768			−0.739742			−0.369871	−	0.929083i	$-0.620598\pi$
$23$	−3.54768			−0.739742			−0.369871	+	0.929083i	$0.620598\pi$
$24$		0			0
$25$	4.59914			0.919829
$26$	−2.25152	+	2.81615i	−0.441559	+	0.552292i
$27$		0			0
$28$		−	2.48683i		−	0.469967i
$29$	0.830894			0.154293			0.0771466	−	0.997020i	$-0.475419\pi$
$29$	0.830894			0.154293			0.0771466	+	0.997020i	$0.475419\pi$
$30$		0			0
$31$		−	4.31236i		−	0.774522i	−0.921970	−	0.387261i	$-0.873421\pi$
$31$		−	4.31236i		−	0.774522i	0.921970	−	0.387261i	$-0.126579\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	6.27837i		−	1.07673i
$35$	1.57449			0.266138
$36$		0			0
$37$		−	7.81310i		−	1.28447i	−0.766509	−	0.642233i	$-0.778008\pi$
$37$		−	7.81310i		−	1.28447i	0.766509	−	0.642233i	$-0.221992\pi$
$38$	4.86004			0.788402
$39$		0			0
$40$	0.633132			0.100107
$41$		−	0.0783645i		−	0.0122385i	−0.999981	−	0.00611924i	$-0.998052\pi$
$41$		−	0.0783645i		−	0.0122385i	0.999981	−	0.00611924i	$-0.00194783\pi$
$42$		0			0
$43$	−9.68740			−1.47732			−0.738658	−	0.674081i	$-0.764540\pi$
$43$	−9.68740			−1.47732			−0.738658	+	0.674081i	$0.764540\pi$
$44$			4.86004i			0.732678i
$45$		0			0
$46$			3.54768i			0.523077i
$47$			4.96601i			0.724367i	0.932107	+	0.362184i	$0.117969\pi$
$47$			4.96601i			0.724367i	−0.932107	+	0.362184i	$0.882031\pi$
$48$		0			0
$49$	0.815659			0.116523
$50$		−	4.59914i		−	0.650417i
$51$		0			0
$52$	2.81615	+	2.25152i	0.390529	+	0.312229i
$53$	6.36026			0.873649			0.436824	−	0.899547i	$-0.356103\pi$
$53$	6.36026			0.873649			0.436824	+	0.899547i	$0.356103\pi$
$54$		0			0
$55$	−3.07704			−0.414909
$56$	−2.48683			−0.332317
$57$		0			0
$58$		−	0.830894i		−	0.109102i
$59$		−	9.08742i		−	1.18308i	−0.806275	−	0.591541i	$-0.798520\pi$
$59$		−	9.08742i		−	1.18308i	0.806275	−	0.591541i	$-0.201480\pi$
$60$		0			0
$61$	−10.5707			−1.35344			−0.676719	−	0.736241i	$-0.736599\pi$
$61$	−10.5707			−1.35344			−0.676719	+	0.736241i	$0.736599\pi$
$62$	−4.31236			−0.547670
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−1.42551	+	1.78299i	−0.176812	+	0.221153i
$66$		0			0
$67$		−	7.88457i		−	0.963253i	−0.876377	−	0.481627i	$-0.840046\pi$
$67$		−	7.88457i		−	0.963253i	0.876377	−	0.481627i	$-0.159954\pi$
$68$	−6.27837			−0.761364
$69$		0			0
$70$		−	1.57449i		−	0.188188i
$71$		−	12.3085i		−	1.46075i	−0.683048	−	0.730374i	$-0.739346\pi$
$71$		−	12.3085i		−	1.46075i	0.683048	−	0.730374i	$-0.260654\pi$
$72$		0			0
$73$		−	1.05367i		−	0.123323i	−0.998097	−	0.0616615i	$-0.980360\pi$
$73$		−	1.05367i		−	0.123323i	0.998097	−	0.0616615i	$-0.0196399\pi$
$74$	−7.81310			−0.908255
$75$		0			0
$76$		−	4.86004i		−	0.557484i
$77$	12.0861			1.37734
$78$		0			0
$79$	3.37676			0.379916			0.189958	−	0.981792i	$-0.439165\pi$
$79$	3.37676			0.379916			0.189958	+	0.981792i	$0.439165\pi$
$80$		−	0.633132i		−	0.0707863i
$81$		0			0
$82$	−0.0783645			−0.00865391
$83$		−	15.2137i		−	1.66992i	−0.550309	−	0.834961i	$-0.685490\pi$
$83$		−	15.2137i		−	1.66992i	0.550309	−	0.834961i	$-0.314510\pi$
$84$		0			0
$85$		−	3.97504i		−	0.431153i
$86$			9.68740i			1.04462i
$87$		0			0
$88$	4.86004			0.518082
$89$		−	0.595093i		−	0.0630797i	−0.999502	−	0.0315398i	$-0.989959\pi$
$89$		−	0.595093i		−	0.0630797i	0.999502	−	0.0315398i	$-0.0100411\pi$
$90$		0			0
$91$	5.59914	−	7.00329i	0.586950	−	0.734144i
$92$	3.54768			0.369871
$93$		0			0
$94$	4.96601			0.512205
$95$	3.07704			0.315698
$96$		0			0
$97$		−	17.0630i		−	1.73249i	−0.499622	−	0.866244i	$-0.666528\pi$
$97$		−	17.0630i		−	1.73249i	0.499622	−	0.866244i	$-0.333472\pi$
$98$		−	0.815659i		−	0.0823940i
$99$		0			0
$100$	−4.59914			−0.459914
$101$	5.07524			0.505005			0.252503	−	0.967596i	$-0.418746\pi$
$101$	5.07524			0.505005			0.252503	+	0.967596i	$0.418746\pi$
$102$		0			0
$103$	−1.07704			−0.106124			−0.0530621	−	0.998591i	$-0.516898\pi$
$103$	−1.07704			−0.106124			−0.0530621	+	0.998591i	$0.516898\pi$
$104$	2.25152	−	2.81615i	0.220779	−	0.276146i
$105$		0			0
$106$		−	6.36026i		−	0.617763i
$107$	9.16763			0.886268			0.443134	−	0.896455i	$-0.353867\pi$
$107$	9.16763			0.886268			0.443134	+	0.896455i	$0.353867\pi$
$108$		0			0
$109$		−	9.90028i		−	0.948275i	−0.880451	−	0.474138i	$-0.842760\pi$
$109$		−	9.90028i		−	0.948275i	0.880451	−	0.474138i	$-0.157240\pi$
$110$			3.07704i			0.293385i
$111$		0			0
$112$			2.48683i			0.234984i
$113$	−6.62175			−0.622922			−0.311461	−	0.950259i	$-0.600818\pi$
$113$	−6.62175			−0.622922			−0.311461	+	0.950259i	$0.600818\pi$
$114$		0			0
$115$			2.24615i			0.209455i
$116$	−0.830894			−0.0771466
$117$		0			0
$118$	−9.08742			−0.836565
$119$			15.6133i			1.43126i
$120$		0			0
$121$	−12.6199			−1.14727
$122$			10.5707i			0.957026i
$123$		0			0
$124$			4.31236i			0.387261i
$125$		−	6.07752i		−	0.543590i
$126$		0			0
$127$	−17.8088			−1.58028			−0.790138	−	0.612929i	$-0.789991\pi$
$127$	−17.8088			−1.58028			−0.790138	+	0.612929i	$0.789991\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	1.78299	+	1.42551i	0.156379	+	0.125025i
$131$	12.0556			1.05330			0.526651	−	0.850082i	$-0.323448\pi$
$131$	12.0556			1.05330			0.526651	+	0.850082i	$0.323448\pi$
$132$		0			0
$133$	−12.0861			−1.04800
$134$	−7.88457			−0.681123
$135$		0			0
$136$			6.27837i			0.538366i
$137$			0.304776i			0.0260387i	0.999915	+	0.0130194i	$0.00414431\pi$
$137$			0.304776i			0.0260387i	−0.999915	+	0.0130194i	$0.995856\pi$
$138$		0			0
$139$	8.18218			0.694003			0.347002	−	0.937864i	$-0.387200\pi$
$139$	8.18218			0.694003			0.347002	+	0.937864i	$0.387200\pi$
$140$	−1.57449			−0.133069
$141$		0			0
$142$	−12.3085			−1.03290
$143$	−10.9424	+	13.6866i	−0.915053	+	1.14453i
$144$		0			0
$145$		−	0.526066i		−	0.0436874i
$146$	−1.05367			−0.0872025
$147$		0			0
$148$			7.81310i			0.642233i
$149$			18.4642i			1.51265i	0.654199	+	0.756323i	$0.273006\pi$
$149$			18.4642i			1.51265i	−0.654199	+	0.756323i	$0.726994\pi$
$150$		0			0
$151$			5.33019i			0.433765i	0.976198	+	0.216882i	$0.0695888\pi$
$151$			5.33019i			0.433765i	−0.976198	+	0.216882i	$0.930411\pi$
$152$	−4.86004			−0.394201
$153$		0			0
$154$		−	12.0861i		−	0.973926i
$155$	−2.73029			−0.219302
$156$		0			0
$157$	−2.93840			−0.234510			−0.117255	−	0.993102i	$-0.537409\pi$
$157$	−2.93840			−0.234510			−0.117255	+	0.993102i	$0.537409\pi$
$158$		−	3.37676i		−	0.268641i
$159$		0			0
$160$	−0.633132			−0.0500535
$161$		−	8.82249i		−	0.695310i
$162$		0			0
$163$		−	7.41051i		−	0.580436i	−0.956961	−	0.290218i	$-0.906272\pi$
$163$		−	7.41051i		−	0.580436i	0.956961	−	0.290218i	$-0.0937278\pi$
$164$			0.0783645i			0.00611924i
$165$		0			0
$166$	−15.2137			−1.18081
$167$			0.0891003i			0.00689479i	0.999994	+	0.00344739i	$0.00109734\pi$
$167$			0.0891003i			0.00689479i	−0.999994	+	0.00344739i	$0.998903\pi$
$168$		0			0
$169$	2.86136	+	12.6812i	0.220104	+	0.975476i
$170$	−3.97504			−0.304871
$171$		0			0
$172$	9.68740			0.738658
$173$	7.59838			0.577694			0.288847	−	0.957375i	$-0.406728\pi$
$173$	7.59838			0.577694			0.288847	+	0.957375i	$0.406728\pi$
$174$		0			0
$175$			11.4373i			0.864579i
$176$		−	4.86004i		−	0.366339i
$177$		0			0
$178$	−0.595093			−0.0446041
$179$	−23.7909			−1.77822			−0.889109	−	0.457695i	$-0.848675\pi$
$179$	−23.7909			−1.77822			−0.889109	+	0.457695i	$0.848675\pi$
$180$		0			0
$181$	−1.63229			−0.121327			−0.0606637	−	0.998158i	$-0.519322\pi$
$181$	−1.63229			−0.121327			−0.0606637	+	0.998158i	$0.519322\pi$
$182$	−7.00329	−	5.59914i	−0.519118	−	0.415036i
$183$		0			0
$184$		−	3.54768i		−	0.261538i
$185$	−4.94673			−0.363690
$186$		0			0
$187$		−	30.5131i		−	2.23134i
$188$		−	4.96601i		−	0.362184i
$189$		0			0
$190$		−	3.07704i		−	0.223232i
$191$	3.16633			0.229108			0.114554	−	0.993417i	$-0.463456\pi$
$191$	3.16633			0.229108			0.114554	+	0.993417i	$0.463456\pi$
$192$		0			0
$193$			16.4614i			1.18492i	0.805600	+	0.592459i	$0.201843\pi$
$193$			16.4614i			1.18492i	−0.805600	+	0.592459i	$0.798157\pi$
$194$	−17.0630			−1.22505
$195$		0			0
$196$	−0.815659			−0.0582614
$197$		−	11.4351i		−	0.814719i	−0.913268	−	0.407360i	$-0.866450\pi$
$197$		−	11.4351i		−	0.814719i	0.913268	−	0.407360i	$-0.133550\pi$
$198$		0			0
$199$	−13.1983			−0.935602			−0.467801	−	0.883834i	$-0.654954\pi$
$199$	−13.1983			−0.935602			−0.467801	+	0.883834i	$0.654954\pi$
$200$			4.59914i			0.325209i
$201$		0			0
$202$		−	5.07524i		−	0.357093i
$203$			2.06630i			0.145026i
$204$		0			0
$205$	−0.0496151			−0.00346527
$206$			1.07704i			0.0750412i
$207$		0			0
$208$	−2.81615	−	2.25152i	−0.195265	−	0.156115i
$209$	23.6199			1.63383
$210$		0			0
$211$	9.27713			0.638664			0.319332	−	0.947643i	$-0.396541\pi$
$211$	9.27713			0.638664			0.319332	+	0.947643i	$0.396541\pi$
$212$	−6.36026			−0.436824
$213$		0			0
$214$		−	9.16763i		−	0.626686i
$215$			6.13341i			0.418295i
$216$		0			0
$217$	10.7241			0.728000
$218$	−9.90028			−0.670532
$219$		0			0
$220$	3.07704			0.207454
$221$	−17.6808	−	14.1358i	−1.18934	−	0.950880i
$222$		0			0
$223$		−	14.0125i		−	0.938344i	−0.883107	−	0.469172i	$-0.844552\pi$
$223$		−	14.0125i		−	0.938344i	0.883107	−	0.469172i	$-0.155448\pi$
$224$	2.48683			0.166159
$225$		0			0
$226$			6.62175i			0.440472i
$227$			2.77227i			0.184002i	0.995759	+	0.0920010i	$0.0293263\pi$
$227$			2.77227i			0.184002i	−0.995759	+	0.0920010i	$0.970674\pi$
$228$		0			0
$229$			16.6468i			1.10005i	0.835148	+	0.550025i	$0.185382\pi$
$229$			16.6468i			1.10005i	−0.835148	+	0.550025i	$0.814618\pi$
$230$	2.24615			0.148107
$231$		0			0
$232$			0.830894i			0.0545509i
$233$	5.45324			0.357254			0.178627	−	0.983917i	$-0.442835\pi$
$233$	5.45324			0.357254			0.178627	+	0.983917i	$0.442835\pi$
$234$		0			0
$235$	3.14414			0.205101
$236$			9.08742i			0.591541i
$237$		0			0
$238$	15.6133			1.01206
$239$			13.6254i			0.881357i	0.897665	+	0.440678i	$0.145262\pi$
$239$			13.6254i			0.881357i	−0.897665	+	0.440678i	$0.854738\pi$
$240$		0			0
$241$			18.9117i			1.21821i	0.793090	+	0.609105i	$0.208471\pi$
$241$			18.9117i			1.21821i	−0.793090	+	0.609105i	$0.791529\pi$
$242$			12.6199i			0.811241i
$243$		0			0
$244$	10.5707			0.676719
$245$		−	0.516420i		−	0.0329929i
$246$		0			0
$247$	10.9424	−	13.6866i	0.696251	−	0.870856i
$248$	4.31236			0.273835
$249$		0			0
$250$	−6.07752			−0.384376
$251$	2.07524			0.130988			0.0654940	−	0.997853i	$-0.479138\pi$
$251$	2.07524			0.130988			0.0654940	+	0.997853i	$0.479138\pi$
$252$		0			0
$253$			17.2419i			1.08399i
$254$			17.8088i			1.11742i
$255$		0			0
$256$	1.00000			0.0625000
$257$	25.0018			1.55957			0.779785	−	0.626047i	$-0.215328\pi$
$257$	25.0018			1.55957			0.779785	+	0.626047i	$0.215328\pi$
$258$		0			0
$259$	19.4299			1.20731
$260$	1.42551	−	1.78299i	0.0884061	−	0.110576i
$261$		0			0
$262$		−	12.0556i		−	0.744797i
$263$	−25.2534			−1.55719			−0.778596	−	0.627526i	$-0.784068\pi$
$263$	−25.2534			−1.55719			−0.778596	+	0.627526i	$0.784068\pi$
$264$		0			0
$265$		−	4.02688i		−	0.247369i
$266$			12.0861i			0.741046i
$267$		0			0
$268$			7.88457i			0.481627i
$269$	23.7793			1.44985			0.724925	−	0.688828i	$-0.241874\pi$
$269$	23.7793			1.44985			0.724925	+	0.688828i	$0.241874\pi$
$270$		0			0
$271$			14.2050i			0.862894i	0.902138	+	0.431447i	$0.141997\pi$
$271$			14.2050i			0.862894i	−0.902138	+	0.431447i	$0.858003\pi$
$272$	6.27837			0.380682
$273$		0			0
$274$	0.304776			0.0184122
$275$		−	22.3520i		−	1.34788i
$276$		0			0
$277$	8.13571			0.488828			0.244414	−	0.969671i	$-0.421404\pi$
$277$	8.13571			0.488828			0.244414	+	0.969671i	$0.421404\pi$
$278$		−	8.18218i		−	0.490735i
$279$		0			0
$280$			1.57449i			0.0940940i
$281$		−	33.0949i		−	1.97428i	−0.159873	−	0.987138i	$-0.551109\pi$
$281$		−	33.0949i		−	1.97428i	0.159873	−	0.987138i	$-0.448891\pi$
$282$		0			0
$283$	−11.8429			−0.703989			−0.351994	−	0.936002i	$-0.614496\pi$
$283$	−11.8429			−0.703989			−0.351994	+	0.936002i	$0.614496\pi$
$284$			12.3085i			0.730374i
$285$		0			0
$286$	13.6866	+	10.9424i	0.809304	+	0.647040i
$287$	0.194879			0.0115034
$288$		0			0
$289$	22.4179			1.31870
$290$	−0.526066			−0.0308916
$291$		0			0
$292$			1.05367i			0.0616615i
$293$		−	14.8818i		−	0.869405i	−0.900574	−	0.434702i	$-0.856854\pi$
$293$		−	14.8818i		−	0.869405i	0.900574	−	0.434702i	$-0.143146\pi$
$294$		0			0
$295$	−5.75354			−0.334984
$296$	7.81310			0.454127
$297$		0			0
$298$	18.4642			1.06960
$299$	9.99078	+	7.98766i	0.577782	+	0.461938i
$300$		0			0
$301$		−	24.0910i		−	1.38858i
$302$	5.33019			0.306718
$303$		0			0
$304$			4.86004i			0.278742i
$305$			6.69264i			0.383220i
$306$		0			0
$307$			23.7291i			1.35429i	0.735849	+	0.677145i	$0.236783\pi$
$307$			23.7291i			1.35429i	−0.735849	+	0.677145i	$0.763217\pi$
$308$	−12.0861			−0.688669
$309$		0			0
$310$			2.73029i			0.155070i
$311$	8.77026			0.497316			0.248658	−	0.968591i	$-0.420011\pi$
$311$	8.77026			0.497316			0.248658	+	0.968591i	$0.420011\pi$
$312$		0			0
$313$	10.2812			0.581129			0.290565	−	0.956855i	$-0.406157\pi$
$313$	10.2812			0.581129			0.290565	+	0.956855i	$0.406157\pi$
$314$			2.93840i			0.165824i
$315$		0			0
$316$	−3.37676			−0.189958
$317$		−	5.80916i		−	0.326275i	−0.986603	−	0.163137i	$-0.947839\pi$
$317$		−	5.80916i		−	0.326275i	0.986603	−	0.163137i	$-0.0521614\pi$
$318$		0			0
$319$		−	4.03818i		−	0.226094i
$320$			0.633132i			0.0353932i
$321$		0			0
$322$	−8.82249			−0.491658
$323$			30.5131i			1.69779i
$324$		0			0
$325$	−12.9519	−	10.3550i	−0.718440	−	0.574394i
$326$	−7.41051			−0.410430
$327$		0			0
$328$	0.0783645			0.00432696
$329$	−12.3496			−0.680858
$330$		0			0
$331$			10.7007i			0.588162i	0.955781	+	0.294081i	$0.0950135\pi$
$331$			10.7007i			0.588162i	−0.955781	+	0.294081i	$0.904986\pi$
$332$			15.2137i			0.834961i
$333$		0			0
$334$	0.0891003			0.00487535
$335$	−4.99197			−0.272741
$336$		0			0
$337$	1.95451			0.106469			0.0532344	−	0.998582i	$-0.483047\pi$
$337$	1.95451			0.106469			0.0532344	+	0.998582i	$0.483047\pi$
$338$	12.6812	−	2.86136i	0.689766	−	0.155637i
$339$		0			0
$340$			3.97504i			0.215577i
$341$	−20.9582			−1.13495
$342$		0			0
$343$			19.4362i			1.04946i
$344$		−	9.68740i		−	0.522310i
$345$		0			0
$346$		−	7.59838i		−	0.408491i
$347$	−4.34070			−0.233021			−0.116510	−	0.993189i	$-0.537171\pi$
$347$	−4.34070			−0.233021			−0.116510	+	0.993189i	$0.537171\pi$
$348$		0			0
$349$			27.9682i			1.49710i	0.663076	+	0.748552i	$0.269251\pi$
$349$			27.9682i			1.49710i	−0.663076	+	0.748552i	$0.730749\pi$
$350$	11.4373			0.611350
$351$		0			0
$352$	−4.86004			−0.259041
$353$		−	16.8078i		−	0.894588i	−0.894387	−	0.447294i	$-0.852388\pi$
$353$		−	16.8078i		−	0.894588i	0.894387	−	0.447294i	$-0.147612\pi$
$354$		0			0
$355$	−7.79289			−0.413604
$356$			0.595093i			0.0315398i
$357$		0			0
$358$			23.7909i			1.25739i
$359$		−	20.0302i		−	1.05715i	−0.848886	−	0.528576i	$-0.822726\pi$
$359$		−	20.0302i		−	1.05715i	0.848886	−	0.528576i	$-0.177274\pi$
$360$		0			0
$361$	−4.61995			−0.243155
$362$			1.63229i			0.0857914i
$363$		0			0
$364$	−5.59914	+	7.00329i	−0.293475	+	0.367072i
$365$	−0.667114			−0.0349183
$366$		0			0
$367$	33.7382			1.76112			0.880560	−	0.473935i	$-0.157167\pi$
$367$	33.7382			1.76112			0.880560	+	0.473935i	$0.157167\pi$
$368$	−3.54768			−0.184936
$369$		0			0
$370$			4.94673i			0.257168i
$371$			15.8169i			0.821173i
$372$		0			0
$373$	13.9629			0.722972			0.361486	−	0.932378i	$-0.382270\pi$
$373$	13.9629			0.722972			0.361486	+	0.932378i	$0.382270\pi$
$374$	−30.5131			−1.57779
$375$		0			0
$376$	−4.96601			−0.256103
$377$	−2.33992	−	1.87077i	−0.120512	−	0.0963496i
$378$		0			0
$379$		−	29.7828i		−	1.52984i	−0.644124	−	0.764921i	$-0.722778\pi$
$379$		−	29.7828i		−	1.52984i	0.644124	−	0.764921i	$-0.277222\pi$
$380$	−3.07704			−0.157849
$381$		0			0
$382$		−	3.16633i		−	0.162004i
$383$			2.38816i			0.122029i	0.998137	+	0.0610146i	$0.0194336\pi$
$383$			2.38816i			0.122029i	−0.998137	+	0.0610146i	$0.980566\pi$
$384$		0			0
$385$		−	7.65210i		−	0.389987i
$386$	16.4614			0.837864
$387$		0			0
$388$			17.0630i			0.866244i
$389$	−24.6725			−1.25095			−0.625473	−	0.780246i	$-0.715094\pi$
$389$	−24.6725			−1.25095			−0.625473	+	0.780246i	$0.715094\pi$
$390$		0			0
$391$	−22.2736			−1.12643
$392$			0.815659i			0.0411970i
$393$		0			0
$394$	−11.4351			−0.576093
$395$		−	2.13794i		−	0.107571i
$396$		0			0
$397$			9.36279i			0.469905i	0.972007	+	0.234952i	$0.0754934\pi$
$397$			9.36279i			0.469905i	−0.972007	+	0.234952i	$0.924507\pi$
$398$			13.1983i			0.661570i
$399$		0			0
$400$	4.59914			0.229957
$401$		−	6.92606i		−	0.345871i	−0.984933	−	0.172935i	$-0.944675\pi$
$401$		−	6.92606i		−	0.345871i	0.984933	−	0.172935i	$-0.0553252\pi$
$402$		0			0
$403$	−9.70934	+	12.1442i	−0.483656	+	0.604947i
$404$	−5.07524			−0.252503
$405$		0			0
$406$	2.06630			0.102549
$407$	−37.9720			−1.88220
$408$		0			0
$409$			18.5920i			0.919314i	0.888096	+	0.459657i	$0.152028\pi$
$409$			18.5920i			0.919314i	−0.888096	+	0.459657i	$0.847972\pi$
$410$			0.0496151i			0.00245031i
$411$		0			0
$412$	1.07704			0.0530621
$413$	22.5989			1.11202
$414$		0			0
$415$	−9.63229			−0.472831
$416$	−2.25152	+	2.81615i	−0.110390	+	0.138073i
$417$		0			0
$418$		−	23.6199i		−	1.15529i
$419$	0.911227			0.0445164			0.0222582	−	0.999752i	$-0.492914\pi$
$419$	0.911227			0.0445164			0.0222582	+	0.999752i	$0.492914\pi$
$420$		0			0
$421$		−	39.3144i		−	1.91607i	−0.286659	−	0.958033i	$-0.592545\pi$
$421$		−	39.3144i		−	1.91607i	0.286659	−	0.958033i	$-0.407455\pi$
$422$		−	9.27713i		−	0.451604i
$423$		0			0
$424$			6.36026i			0.308881i
$425$	28.8751			1.40065
$426$		0			0
$427$		−	26.2876i		−	1.27214i
$428$	−9.16763			−0.443134
$429$		0			0
$430$	6.13341			0.295779
$431$			38.0100i			1.83088i	0.402457	+	0.915439i	$0.368156\pi$
$431$			38.0100i			1.83088i	−0.402457	+	0.915439i	$0.631844\pi$
$432$		0			0
$433$	37.5218			1.80318			0.901592	−	0.432587i	$-0.142399\pi$
$433$	37.5218			1.80318			0.901592	+	0.432587i	$0.142399\pi$
$434$		−	10.7241i		−	0.514774i
$435$		0			0
$436$			9.90028i			0.474138i
$437$		−	17.2419i		−	0.824790i
$438$		0			0
$439$	10.8225			0.516529			0.258265	−	0.966074i	$-0.416849\pi$
$439$	10.8225			0.516529			0.258265	+	0.966074i	$0.416849\pi$
$440$		−	3.07704i		−	0.146692i
$441$		0			0
$442$	−14.1358	+	17.6808i	−0.672373	+	0.840990i
$443$	21.0621			1.00069			0.500346	−	0.865826i	$-0.333206\pi$
$443$	21.0621			1.00069			0.500346	+	0.865826i	$0.333206\pi$
$444$		0			0
$445$	−0.376772			−0.0178607
$446$	−14.0125			−0.663509
$447$		0			0
$448$		−	2.48683i		−	0.117492i
$449$			12.0811i			0.570140i	0.958507	+	0.285070i	$0.0920169\pi$
$449$			12.0811i			0.570140i	−0.958507	+	0.285070i	$0.907983\pi$
$450$		0			0
$451$	−0.380854			−0.0179337
$452$	6.62175			0.311461
$453$		0			0
$454$	2.77227			0.130109
$455$	−4.43400	−	3.54500i	−0.207869	−	0.166192i
$456$		0			0
$457$			28.4505i			1.33086i	0.746461	+	0.665429i	$0.231751\pi$
$457$			28.4505i			1.33086i	−0.746461	+	0.665429i	$0.768249\pi$
$458$	16.6468			0.777853
$459$		0			0
$460$		−	2.24615i		−	0.104727i
$461$		−	17.2904i		−	0.805296i	−0.915355	−	0.402648i	$-0.868090\pi$
$461$		−	17.2904i		−	0.805296i	0.915355	−	0.402648i	$-0.131910\pi$
$462$		0			0
$463$			23.7642i			1.10442i	0.833706	+	0.552208i	$0.186215\pi$
$463$			23.7642i			1.10442i	−0.833706	+	0.552208i	$0.813785\pi$
$464$	0.830894			0.0385733
$465$		0			0
$466$		−	5.45324i		−	0.252616i
$467$	−30.5214			−1.41236			−0.706182	−	0.708030i	$-0.749584\pi$
$467$	−30.5214			−1.41236			−0.706182	+	0.708030i	$0.749584\pi$
$468$		0			0
$469$	19.6076			0.905395
$470$		−	3.14414i		−	0.145028i
$471$		0			0
$472$	9.08742			0.418282
$473$			47.0811i			2.16479i
$474$		0			0
$475$			22.3520i			1.02558i
$476$		−	15.6133i		−	0.715632i
$477$		0			0
$478$	13.6254			0.623213
$479$			22.8134i			1.04237i	0.853444	+	0.521185i	$0.174510\pi$
$479$			22.8134i			1.04237i	−0.853444	+	0.521185i	$0.825490\pi$
$480$		0			0
$481$	−17.5913	+	22.0028i	−0.802095	+	1.00324i
$482$	18.9117			0.861405
$483$		0			0
$484$	12.6199			0.573634
$485$	−10.8031			−0.490545
$486$		0			0
$487$			21.9090i			0.992792i	0.868096	+	0.496396i	$0.165344\pi$
$487$			21.9090i			0.992792i	−0.868096	+	0.496396i	$0.834656\pi$
$488$		−	10.5707i		−	0.478513i
$489$		0			0
$490$	−0.516420			−0.0233295
$491$	25.6660			1.15829			0.579145	−	0.815224i	$-0.303386\pi$
$491$	25.6660			1.15829			0.579145	+	0.815224i	$0.303386\pi$
$492$		0			0
$493$	5.21666			0.234947
$494$	−13.6866	−	10.9424i	−0.615788	−	0.492324i
$495$		0			0
$496$		−	4.31236i		−	0.193630i
$497$	30.6091			1.37301
$498$		0			0
$499$			34.6451i			1.55093i	0.631392	+	0.775464i	$0.282484\pi$
$499$			34.6451i			1.55093i	−0.631392	+	0.775464i	$0.717516\pi$
$500$			6.07752i			0.271795i
$501$		0			0
$502$		−	2.07524i		−	0.0926226i
$503$	16.8864			0.752928			0.376464	−	0.926431i	$-0.377140\pi$
$503$	16.8864			0.752928			0.376464	+	0.926431i	$0.377140\pi$
$504$		0			0
$505$		−	3.21330i		−	0.142990i
$506$	17.2419			0.766494
$507$		0			0
$508$	17.8088			0.790138
$509$		−	11.6183i		−	0.514971i	−0.966282	−	0.257485i	$-0.917106\pi$
$509$		−	11.6183i		−	0.514971i	0.966282	−	0.257485i	$-0.0828939\pi$
$510$		0			0
$511$	2.62031			0.115916
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	25.0018i		−	1.10278i
$515$			0.681911i			0.0300486i
$516$		0			0
$517$	24.1350			1.06146
$518$		−	19.4299i		−	0.853700i
$519$		0			0
$520$	−1.78299	−	1.42551i	−0.0781894	−	0.0625126i
$521$	32.5005			1.42387			0.711937	−	0.702243i	$-0.247818\pi$
$521$	32.5005			1.42387			0.711937	+	0.702243i	$0.247818\pi$
$522$		0			0
$523$	−23.8182			−1.04150			−0.520749	−	0.853710i	$-0.674347\pi$
$523$	−23.8182			−1.04150			−0.520749	+	0.853710i	$0.674347\pi$
$524$	−12.0556			−0.526651
$525$		0			0
$526$			25.2534i			1.10110i
$527$		−	27.0746i		−	1.17939i
$528$		0			0
$529$	−10.4140			−0.452781
$530$	−4.02688			−0.174917
$531$		0			0
$532$	12.0861			0.523999
$533$	−0.176439	+	0.220686i	−0.00764242	+	0.00955897i
$534$		0			0
$535$		−	5.80432i		−	0.250943i
$536$	7.88457			0.340561
$537$		0			0
$538$		−	23.7793i		−	1.02520i
$539$		−	3.96413i		−	0.170747i
$540$		0			0
$541$			32.5049i			1.39750i	0.715368	+	0.698748i	$0.246259\pi$
$541$			32.5049i			1.39750i	−0.715368	+	0.698748i	$0.753741\pi$
$542$	14.2050			0.610158
$543$		0			0
$544$		−	6.27837i		−	0.269183i
$545$	−6.26819			−0.268500
$546$		0			0
$547$	−9.56481			−0.408962			−0.204481	−	0.978871i	$-0.565551\pi$
$547$	−9.56481			−0.408962			−0.204481	+	0.978871i	$0.565551\pi$
$548$		−	0.304776i		−	0.0130194i
$549$		0			0
$550$	−22.3520			−0.953093
$551$			4.03818i			0.172032i
$552$		0			0
$553$			8.39745i			0.357096i
$554$		−	8.13571i		−	0.345653i
$555$		0			0
$556$	−8.18218			−0.347002
$557$		−	34.6218i		−	1.46697i	−0.679703	−	0.733487i	$-0.737891\pi$
$557$		−	34.6218i		−	1.46697i	0.679703	−	0.733487i	$-0.262109\pi$
$558$		0			0
$559$	27.2811	+	21.8113i	1.15387	+	0.922521i
$560$	1.57449			0.0665345
$561$		0			0
$562$	−33.0949			−1.39602
$563$	−6.85514			−0.288910			−0.144455	−	0.989511i	$-0.546143\pi$
$563$	−6.85514			−0.288910			−0.144455	+	0.989511i	$0.546143\pi$
$564$		0			0
$565$			4.19244i			0.176377i
$566$			11.8429i			0.497795i
$567$		0			0
$568$	12.3085			0.516452
$569$	21.3233			0.893921			0.446960	−	0.894554i	$-0.352507\pi$
$569$	21.3233			0.893921			0.446960	+	0.894554i	$0.352507\pi$
$570$		0			0
$571$	9.44758			0.395369			0.197685	−	0.980266i	$-0.436658\pi$
$571$	9.44758			0.395369			0.197685	+	0.980266i	$0.436658\pi$
$572$	10.9424	−	13.6866i	0.457527	−	0.572264i
$573$		0			0
$574$		−	0.194879i		−	0.00813411i
$575$	−16.3163			−0.680436
$576$		0			0
$577$		−	25.5692i		−	1.06446i	−0.846599	−	0.532231i	$-0.821354\pi$
$577$		−	25.5692i		−	1.06446i	0.846599	−	0.532231i	$-0.178646\pi$
$578$		−	22.4179i		−	0.932462i
$579$		0			0
$580$			0.526066i			0.0218437i
$581$	37.8340			1.56962
$582$		0			0
$583$		−	30.9111i		−	1.28021i
$584$	1.05367			0.0436013
$585$		0			0
$586$	−14.8818			−0.614762
$587$		−	7.76028i		−	0.320301i	−0.987093	−	0.160151i	$-0.948802\pi$
$587$		−	7.76028i		−	0.320301i	0.987093	−	0.160151i	$-0.0511980\pi$
$588$		0			0
$589$	20.9582			0.863568
$590$			5.75354i			0.236869i
$591$		0			0
$592$		−	7.81310i		−	0.321117i
$593$			4.65405i			0.191119i	0.995424	+	0.0955594i	$0.0304640\pi$
$593$			4.65405i			0.191119i	−0.995424	+	0.0955594i	$0.969536\pi$
$594$		0			0
$595$	9.88525			0.405256
$596$		−	18.4642i		−	0.756323i
$597$		0			0
$598$	7.98766	−	9.99078i	0.326640	−	0.408554i
$599$	−26.5515			−1.08487			−0.542433	−	0.840099i	$-0.682497\pi$
$599$	−26.5515			−1.08487			−0.542433	+	0.840099i	$0.682497\pi$
$600$		0			0
$601$	−33.5941			−1.37033			−0.685165	−	0.728388i	$-0.740270\pi$
$601$	−33.5941			−1.37033			−0.685165	+	0.728388i	$0.740270\pi$
$602$	−24.0910			−0.981874
$603$		0			0
$604$		−	5.33019i		−	0.216882i
$605$			7.99009i			0.324843i
$606$		0			0
$607$	−14.3516			−0.582515			−0.291258	−	0.956645i	$-0.594074\pi$
$607$	−14.3516			−0.582515			−0.291258	+	0.956645i	$0.594074\pi$
$608$	4.86004			0.197100
$609$		0			0
$610$	6.69264			0.270977
$611$	11.1811	−	13.9850i	0.452337	−	0.565773i
$612$		0			0
$613$		−	32.5357i		−	1.31410i	−0.753845	−	0.657052i	$-0.771803\pi$
$613$		−	32.5357i		−	1.31410i	0.753845	−	0.657052i	$-0.228197\pi$
$614$	23.7291			0.957628
$615$		0			0
$616$			12.0861i			0.486963i
$617$			20.1146i			0.809782i	0.914365	+	0.404891i	$0.132690\pi$
$617$			20.1146i			0.809782i	−0.914365	+	0.404891i	$0.867310\pi$
$618$		0			0
$619$			14.0876i			0.566228i	0.959086	+	0.283114i	$0.0913674\pi$
$619$			14.0876i			0.566228i	−0.959086	+	0.283114i	$0.908633\pi$
$620$	2.73029			0.109651
$621$		0			0
$622$		−	8.77026i		−	0.351655i
$623$	1.47990			0.0592908
$624$		0			0
$625$	19.1478			0.765914
$626$		−	10.2812i		−	0.410920i
$627$		0			0
$628$	2.93840			0.117255
$629$		−	49.0535i		−	1.95589i
$630$		0			0
$631$		−	15.6969i		−	0.624885i	−0.949937	−	0.312443i	$-0.898853\pi$
$631$		−	15.6969i		−	0.624885i	0.949937	−	0.312443i	$-0.101147\pi$
$632$			3.37676i			0.134320i
$633$		0			0
$634$	−5.80916			−0.230711
$635$			11.2753i			0.447448i
$636$		0			0
$637$	−2.29702	−	1.83647i	−0.0910111	−	0.0727636i
$638$	−4.03818			−0.159873
$639$		0			0
$640$	0.633132			0.0250267
$641$	12.8761			0.508575			0.254287	−	0.967129i	$-0.418159\pi$
$641$	12.8761			0.508575			0.254287	+	0.967129i	$0.418159\pi$
$642$		0			0
$643$			33.9208i			1.33770i	0.743395	+	0.668852i	$0.233214\pi$
$643$			33.9208i			1.33770i	−0.743395	+	0.668852i	$0.766786\pi$
$644$			8.82249i			0.347655i
$645$		0			0
$646$	30.5131			1.20052
$647$	4.53172			0.178160			0.0890802	−	0.996024i	$-0.471607\pi$
$647$	4.53172			0.178160			0.0890802	+	0.996024i	$0.471607\pi$
$648$		0			0
$649$	−44.1652			−1.73364
$650$	−10.3550	+	12.9519i	−0.406158	+	0.508014i
$651$		0			0
$652$			7.41051i			0.290218i
$653$	−4.78297			−0.187172			−0.0935861	−	0.995611i	$-0.529833\pi$
$653$	−4.78297			−0.187172			−0.0935861	+	0.995611i	$0.529833\pi$
$654$		0			0
$655$		−	7.63278i		−	0.298237i
$656$		−	0.0783645i		−	0.00305962i
$657$		0			0
$658$			12.3496i			0.481439i
$659$	19.7652			0.769943			0.384972	−	0.922928i	$-0.374211\pi$
$659$	19.7652			0.769943			0.384972	+	0.922928i	$0.374211\pi$
$660$		0			0
$661$		−	27.5334i		−	1.07093i	−0.844559	−	0.535463i	$-0.820137\pi$
$661$		−	27.5334i		−	1.07093i	0.844559	−	0.535463i	$-0.179863\pi$
$662$	10.7007			0.415893
$663$		0			0
$664$	15.2137			0.590407
$665$			7.65210i			0.296736i
$666$		0			0
$667$	−2.94775			−0.114137
$668$		−	0.0891003i		−	0.00344739i
$669$		0			0
$670$			4.99197i			0.192857i
$671$			51.3739i			1.98327i
$672$		0			0
$673$	43.5650			1.67931			0.839653	−	0.543123i	$-0.182758\pi$
$673$	43.5650			1.67931			0.839653	+	0.543123i	$0.182758\pi$
$674$		−	1.95451i		−	0.0752848i
$675$		0			0
$676$	−2.86136	−	12.6812i	−0.110052	−	0.487738i
$677$	−48.3532			−1.85836			−0.929181	−	0.369624i	$-0.879487\pi$
$677$	−48.3532			−1.85836			−0.929181	+	0.369624i	$0.879487\pi$
$678$		0			0
$679$	42.4329			1.62842
$680$	3.97504			0.152436
$681$		0			0
$682$			20.9582i			0.802531i
$683$		−	25.6842i		−	0.982779i	−0.870940	−	0.491390i	$-0.836489\pi$
$683$		−	25.6842i		−	0.982779i	0.870940	−	0.491390i	$-0.163511\pi$
$684$		0			0
$685$	0.192963			0.00737275
$686$	19.4362			0.742079
$687$		0			0
$688$	−9.68740			−0.369329
$689$	−17.9114	−	14.3202i	−0.682371	−	0.545557i
$690$		0			0
$691$		−	4.22345i		−	0.160668i	−0.996768	−	0.0803338i	$-0.974401\pi$
$691$		−	4.22345i		−	0.160668i	0.996768	−	0.0803338i	$-0.0255986\pi$
$692$	−7.59838			−0.288847
$693$		0			0
$694$			4.34070i			0.164771i
$695$		−	5.18040i		−	0.196504i
$696$		0			0
$697$		−	0.492001i		−	0.0186359i
$698$	27.9682			1.05861
$699$		0			0
$700$		−	11.4373i		−	0.432290i
$701$	7.76144			0.293145			0.146573	−	0.989200i	$-0.453176\pi$
$701$	7.76144			0.293145			0.146573	+	0.989200i	$0.453176\pi$
$702$		0			0
$703$	37.9720			1.43214
$704$			4.86004i			0.183169i
$705$		0			0
$706$	−16.8078			−0.632569
$707$			12.6213i			0.474672i
$708$		0			0
$709$			42.8802i			1.61040i	0.593003	+	0.805200i	$0.297942\pi$
$709$			42.8802i			1.61040i	−0.593003	+	0.805200i	$0.702058\pi$
$710$			7.79289i			0.292462i
$711$		0			0
$712$	0.595093			0.0223020
$713$			15.2989i			0.572947i
$714$		0			0
$715$	8.66540	+	6.92801i	0.324068	+	0.259093i
$716$	23.7909			0.889109
$717$		0			0
$718$	−20.0302			−0.747520
$719$	0.181195			0.00675742			0.00337871	−	0.999994i	$-0.498925\pi$
$719$	0.181195			0.00675742			0.00337871	+	0.999994i	$0.498925\pi$
$720$		0			0
$721$		−	2.67843i		−	0.0997499i
$722$			4.61995i			0.171937i
$723$		0			0
$724$	1.63229			0.0606637
$725$	3.82140			0.141923
$726$		0			0
$727$	−32.3387			−1.19938			−0.599688	−	0.800234i	$-0.704709\pi$
$727$	−32.3387			−1.19938			−0.599688	+	0.800234i	$0.704709\pi$
$728$	7.00329	+	5.59914i	0.259559	+	0.207518i
$729$		0			0
$730$			0.667114i			0.0246910i
$731$	−60.8211			−2.24955
$732$		0			0
$733$		−	46.2501i		−	1.70829i	−0.520038	−	0.854143i	$-0.674082\pi$
$733$		−	46.2501i		−	1.70829i	0.520038	−	0.854143i	$-0.325918\pi$
$734$		−	33.7382i		−	1.24530i
$735$		0			0
$736$			3.54768i			0.130769i
$737$	−38.3193			−1.41151
$738$		0			0
$739$		−	8.94056i		−	0.328884i	−0.986387	−	0.164442i	$-0.947418\pi$
$739$		−	8.94056i		−	0.328884i	0.986387	−	0.164442i	$-0.0525823\pi$
$740$	4.94673			0.181845
$741$		0			0
$742$	15.8169			0.580657
$743$			15.7969i			0.579533i	0.957097	+	0.289767i	$0.0935777\pi$
$743$			15.7969i			0.579533i	−0.957097	+	0.289767i	$0.906422\pi$
$744$		0			0
$745$	11.6903			0.428298
$746$		−	13.9629i		−	0.511218i
$747$		0			0
$748$			30.5131i			1.11567i
$749$			22.7984i			0.833034i
$750$		0			0
$751$	47.5923			1.73667			0.868333	−	0.495981i	$-0.165192\pi$
$751$	47.5923			1.73667			0.868333	+	0.495981i	$0.165192\pi$
$752$			4.96601i			0.181092i
$753$		0			0
$754$	−1.87077	+	2.33992i	−0.0681295	+	0.0852149i
$755$	3.37471			0.122818
$756$		0			0
$757$	−36.6710			−1.33283			−0.666414	−	0.745582i	$-0.732172\pi$
$757$	−36.6710			−1.33283			−0.666414	+	0.745582i	$0.732172\pi$
$758$	−29.7828			−1.08176
$759$		0			0
$760$			3.07704i			0.111616i
$761$		−	22.9830i		−	0.833132i	−0.909105	−	0.416566i	$-0.863233\pi$
$761$		−	22.9830i		−	0.833132i	0.909105	−	0.416566i	$-0.136767\pi$
$762$		0			0
$763$	24.6204			0.891317
$764$	−3.16633			−0.114554
$765$		0			0
$766$	2.38816			0.0862876
$767$	−20.4605	+	25.5915i	−0.738785	+	0.924056i
$768$		0			0
$769$			1.72380i			0.0621619i	0.999517	+	0.0310810i	$0.00989497\pi$
$769$			1.72380i			0.0621619i	−0.999517	+	0.0310810i	$0.990105\pi$
$770$	−7.65210			−0.275762
$771$		0			0
$772$		−	16.4614i		−	0.592459i
$773$			7.86189i			0.282772i	0.989955	+	0.141386i	$0.0451559\pi$
$773$			7.86189i			0.282772i	−0.989955	+	0.141386i	$0.954844\pi$
$774$		0			0
$775$		−	19.8331i		−	0.712428i
$776$	17.0630			0.612527
$777$		0			0
$778$			24.6725i			0.884552i
$779$	0.380854			0.0136455
$780$		0			0
$781$	−59.8197			−2.14052
$782$			22.2736i			0.796504i
$783$		0			0
$784$	0.815659			0.0291307
$785$			1.86040i			0.0664003i
$786$		0			0
$787$		−	26.2708i		−	0.936452i	−0.883609	−	0.468226i	$-0.844893\pi$
$787$		−	26.2708i		−	0.936452i	0.883609	−	0.468226i	$-0.155107\pi$
$788$			11.4351i			0.407360i
$789$		0			0
$790$	−2.13794			−0.0760644
$791$		−	16.4672i		−	0.585506i
$792$		0			0
$793$	29.7686	+	23.8001i	1.05711	+	0.845166i
$794$	9.36279			0.332273
$795$		0			0
$796$	13.1983			0.467801
$797$	−10.7639			−0.381276			−0.190638	−	0.981660i	$-0.561056\pi$
$797$	−10.7639			−0.381276			−0.190638	+	0.981660i	$0.561056\pi$
$798$		0			0
$799$			31.1785i			1.10301i
$800$		−	4.59914i		−	0.162604i
$801$		0			0
$802$	−6.92606			−0.244568
$803$	−5.12088			−0.180712
$804$		0			0
$805$	−5.58580			−0.196874
$806$	12.1442	+	9.70934i	0.427762	+	0.341997i
$807$		0			0
$808$			5.07524i			0.178546i
$809$	−2.05900			−0.0723905			−0.0361952	−	0.999345i	$-0.511524\pi$
$809$	−2.05900			−0.0723905			−0.0361952	+	0.999345i	$0.511524\pi$
$810$		0			0
$811$			12.5483i			0.440630i	0.975429	+	0.220315i	$0.0707086\pi$
$811$			12.5483i			0.440630i	−0.975429	+	0.220315i	$0.929291\pi$
$812$		−	2.06630i		−	0.0725128i
$813$		0			0
$814$			37.9720i			1.33092i
$815$	−4.69183			−0.164348
$816$		0			0
$817$		−	47.0811i		−	1.64716i
$818$	18.5920			0.650053
$819$		0			0
$820$	0.0496151			0.00173263
$821$			27.4251i			0.957144i	0.878048	+	0.478572i	$0.158845\pi$
$821$			27.4251i			0.957144i	−0.878048	+	0.478572i	$0.841155\pi$
$822$		0			0
$823$	−3.94117			−0.137380			−0.0686902	−	0.997638i	$-0.521882\pi$
$823$	−3.94117			−0.137380			−0.0686902	+	0.997638i	$0.521882\pi$
$824$		−	1.07704i		−	0.0375206i
$825$		0			0
$826$		−	22.5989i		−	0.786316i
$827$			31.5551i			1.09728i	0.836060	+	0.548639i	$0.184854\pi$
$827$			31.5551i			1.09728i	−0.836060	+	0.548639i	$0.815146\pi$
$828$		0			0
$829$	−48.2865			−1.67706			−0.838529	−	0.544857i	$-0.816584\pi$
$829$	−48.2865			−1.67706			−0.838529	+	0.544857i	$0.816584\pi$
$830$			9.63229i			0.334342i
$831$		0			0
$832$	2.81615	+	2.25152i	0.0976323	+	0.0780573i
$833$	5.12101			0.177432
$834$		0			0
$835$	0.0564122			0.00195223
$836$	−23.6199			−0.816913
$837$		0			0
$838$		−	0.911227i		−	0.0314778i
$839$		−	21.7161i		−	0.749723i	−0.927081	−	0.374862i	$-0.877690\pi$
$839$		−	21.7161i		−	0.749723i	0.927081	−	0.374862i	$-0.122310\pi$
$840$		0			0
$841$	−28.3096			−0.976194
$842$	−39.3144			−1.35486
$843$		0			0
$844$	−9.27713			−0.319332
$845$	8.02887	−	1.81162i	0.276201	−	0.0623215i
$846$		0			0
$847$		−	31.3837i		−	1.07836i
$848$	6.36026			0.218412
$849$		0			0
$850$		−	28.8751i		−	0.990408i
$851$			27.7184i			0.950174i
$852$		0			0
$853$			14.8369i			0.508006i	0.967203	+	0.254003i	$0.0817473\pi$
$853$			14.8369i			0.508006i	−0.967203	+	0.254003i	$0.918253\pi$
$854$	−26.2876			−0.899541
$855$		0			0
$856$			9.16763i			0.313343i
$857$	−8.53007			−0.291382			−0.145691	−	0.989330i	$-0.546540\pi$
$857$	−8.53007			−0.291382			−0.145691	+	0.989330i	$0.546540\pi$
$858$		0			0
$859$	−20.2841			−0.692084			−0.346042	−	0.938219i	$-0.612475\pi$
$859$	−20.2841			−0.692084			−0.346042	+	0.938219i	$0.612475\pi$
$860$		−	6.13341i		−	0.209147i
$861$		0			0
$862$	38.0100			1.29463
$863$			17.3161i			0.589447i	0.955583	+	0.294723i	$0.0952275\pi$
$863$			17.3161i			0.589447i	−0.955583	+	0.294723i	$0.904772\pi$
$864$		0			0
$865$		−	4.81078i		−	0.163571i
$866$		−	37.5218i		−	1.27504i
$867$		0			0
$868$	−10.7241			−0.364000
$869$		−	16.4112i		−	0.556712i
$870$		0			0
$871$	−17.7522	+	22.2041i	−0.601511	+	0.752357i
$872$	9.90028			0.335266
$873$		0			0
$874$	−17.2419			−0.583214
$875$	15.1138			0.510939
$876$		0			0
$877$			8.67389i			0.292897i	0.989218	+	0.146448i	$0.0467842\pi$
$877$			8.67389i			0.292897i	−0.989218	+	0.146448i	$0.953216\pi$
$878$		−	10.8225i		−	0.365241i
$879$		0			0
$880$	−3.07704			−0.103727
$881$	−2.08434			−0.0702232			−0.0351116	−	0.999383i	$-0.511179\pi$
$881$	−2.08434			−0.0702232			−0.0351116	+	0.999383i	$0.511179\pi$
$882$		0			0
$883$	22.5903			0.760224			0.380112	−	0.924941i	$-0.375885\pi$
$883$	22.5903			0.760224			0.380112	+	0.924941i	$0.375885\pi$
$884$	17.6808	+	14.1358i	0.594670	+	0.475440i
$885$		0			0
$886$		−	21.0621i		−	0.707596i
$887$	−52.8818			−1.77560			−0.887799	−	0.460232i	$-0.847766\pi$
$887$	−52.8818			−1.77560			−0.887799	+	0.460232i	$0.847766\pi$
$888$		0			0
$889$		−	44.2876i		−	1.48536i
$890$			0.376772i			0.0126294i
$891$		0			0
$892$			14.0125i			0.469172i
$893$	−24.1350			−0.807647
$894$		0			0
$895$			15.0628i			0.503494i
$896$	−2.48683			−0.0830793
$897$		0			0
$898$	12.0811			0.403150
$899$		−	3.58311i		−	0.119503i
$900$		0			0
$901$	39.9320			1.33033
$902$			0.380854i			0.0126811i
$903$		0			0
$904$		−	6.62175i		−	0.220236i
$905$			1.03346i			0.0343532i
$906$		0			0
$907$	−55.6901			−1.84916			−0.924580	−	0.380989i	$-0.875584\pi$
$907$	−55.6901			−1.84916			−0.924580	+	0.380989i	$0.875584\pi$
$908$		−	2.77227i		−	0.0920010i
$909$		0			0
$910$	−3.54500	+	4.43400i	−0.117516	+	0.146986i
$911$	38.4930			1.27533			0.637665	−	0.770314i	$-0.279900\pi$
$911$	38.4930			1.27533			0.637665	+	0.770314i	$0.279900\pi$
$912$		0			0
$913$	−73.9392			−2.44703
$914$	28.4505			0.941059
$915$		0			0
$916$		−	16.6468i		−	0.550025i
$917$			29.9802i			0.990035i
$918$		0			0
$919$	−26.0262			−0.858525			−0.429263	−	0.903180i	$-0.641227\pi$
$919$	−26.0262			−0.858525			−0.429263	+	0.903180i	$0.641227\pi$
$920$	−2.24615			−0.0740534
$921$		0			0
$922$	−17.2904			−0.569430
$923$	−27.7127	+	34.6625i	−0.912176	+	1.14093i
$924$		0			0
$925$		−	35.9336i		−	1.18149i
$926$	23.7642			0.780940
$927$		0			0
$928$		−	0.830894i		−	0.0272754i
$929$			1.95987i			0.0643012i	0.999483	+	0.0321506i	$0.0102356\pi$
$929$			1.95987i			0.0643012i	−0.999483	+	0.0321506i	$0.989764\pi$
$930$		0			0
$931$			3.96413i			0.129919i
$932$	−5.45324			−0.178627
$933$		0			0
$934$			30.5214i			0.998692i
$935$	−19.3188			−0.631793
$936$		0			0
$937$	−42.2865			−1.38144			−0.690720	−	0.723122i	$-0.742706\pi$
$937$	−42.2865			−1.38144			−0.690720	+	0.723122i	$0.742706\pi$
$938$		−	19.6076i		−	0.640211i
$939$		0			0
$940$	−3.14414			−0.102551
$941$		−	57.8444i		−	1.88567i	−0.333257	−	0.942836i	$-0.608148\pi$
$941$		−	57.8444i		−	1.88567i	0.333257	−	0.942836i	$-0.391852\pi$
$942$		0			0
$943$			0.278012i			0.00905332i
$944$		−	9.08742i		−	0.295770i
$945$		0			0
$946$	47.0811			1.53074
$947$		−	17.0945i		−	0.555498i	−0.960654	−	0.277749i	$-0.910412\pi$
$947$		−	17.0945i		−	0.555498i	0.960654	−	0.277749i	$-0.0895883\pi$
$948$		0			0
$949$	−2.37236	+	2.96729i	−0.0770100	+	0.0963225i
$950$	22.3520			0.725195
$951$		0			0
$952$	−15.6133			−0.506029
$953$	−19.8333			−0.642465			−0.321233	−	0.947000i	$-0.604097\pi$
$953$	−19.8333			−0.642465			−0.321233	+	0.947000i	$0.604097\pi$
$954$		0			0
$955$		−	2.00470i		−	0.0648707i
$956$		−	13.6254i		−	0.440678i
$957$		0			0
$958$	22.8134			0.737067
$959$	−0.757926			−0.0244747
$960$		0			0
$961$	12.4036			0.400116
$962$	22.0028	+	17.5913i	0.709400	+	0.567167i
$963$		0			0
$964$		−	18.9117i		−	0.609105i
$965$	10.4223			0.335504
$966$		0			0
$967$		−	21.9943i		−	0.707288i	−0.935380	−	0.353644i	$-0.884942\pi$
$967$		−	21.9943i		−	0.707288i	0.935380	−	0.353644i	$-0.115058\pi$
$968$		−	12.6199i		−	0.405620i
$969$		0			0
$970$			10.8031i			0.346868i
$971$	16.0662			0.515589			0.257794	−	0.966200i	$-0.417004\pi$
$971$	16.0662			0.515589			0.257794	+	0.966200i	$0.417004\pi$
$972$		0			0
$973$			20.3477i			0.652318i
$974$	21.9090			0.702010
$975$		0			0
$976$	−10.5707			−0.338360
$977$		−	1.39985i		−	0.0447852i	−0.999749	−	0.0223926i	$-0.992872\pi$
$977$		−	1.39985i		−	0.0447852i	0.999749	−	0.0223926i	$-0.00712839\pi$
$978$		0			0
$979$	−2.89217			−0.0924342
$980$			0.516420i			0.0164964i
$981$		0			0
$982$		−	25.6660i		−	0.819035i
$983$			13.2534i			0.422718i	0.977408	+	0.211359i	$0.0677890\pi$
$983$			13.2534i			0.422718i	−0.977408	+	0.211359i	$0.932211\pi$
$984$		0			0
$985$	−7.23994			−0.230684
$986$		−	5.21666i		−	0.166132i
$987$		0			0
$988$	−10.9424	+	13.6866i	−0.348126	+	0.435428i
$989$	34.3678			1.09283
$990$		0			0
$991$	5.66549			0.179970			0.0899851	−	0.995943i	$-0.471318\pi$
$991$	5.66549			0.179970			0.0899851	+	0.995943i	$0.471318\pi$
$992$	−4.31236			−0.136917
$993$		0			0
$994$		−	30.6091i		−	0.970863i
$995$			8.35626i			0.264911i
$996$		0			0
$997$	5.65761			0.179178			0.0895891	−	0.995979i	$-0.471445\pi$
$997$	5.65761			0.179178			0.0895891	+	0.995979i	$0.471445\pi$
$998$	34.6451			1.09667
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.b.d.649.3		14
3.2	odd	2		2106.2.b.c.649.12		14
9.2	odd	6		234.2.t.a.103.8	yes	28
9.4	even	3		702.2.t.a.181.10		28
9.5	odd	6		234.2.t.a.25.1	✓	28
9.7	even	3		702.2.t.a.415.5		28
13.12	even	2	inner	2106.2.b.d.649.12		14
39.38	odd	2		2106.2.b.c.649.3		14
117.25	even	6		702.2.t.a.415.10		28
117.38	odd	6		234.2.t.a.103.1	yes	28
117.77	odd	6		234.2.t.a.25.8	yes	28
117.103	even	6		702.2.t.a.181.5		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.t.a.25.1	✓	28	9.5	odd	6
234.2.t.a.25.8	yes	28	117.77	odd	6
234.2.t.a.103.1	yes	28	117.38	odd	6
234.2.t.a.103.8	yes	28	9.2	odd	6
702.2.t.a.181.5		28	117.103	even	6
702.2.t.a.181.10		28	9.4	even	3
702.2.t.a.415.5		28	9.7	even	3
702.2.t.a.415.10		28	117.25	even	6
2106.2.b.c.649.3		14	39.38	odd	2
2106.2.b.c.649.12		14	3.2	odd	2
2106.2.b.d.649.3		14	1.1	even	1	trivial
2106.2.b.d.649.12		14	13.12	even	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:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.633132i q^{5} +2.48683i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.633132i q^{5} +2.48683i q^{7} +1.00000i q^{8} -0.633132 q^{10} -4.86004i q^{11} +(-2.81615 - 2.25152i) q^{13} +2.48683 q^{14} +1.00000 q^{16} +6.27837 q^{17} +4.86004i q^{19} +0.633132i q^{20} -4.86004 q^{22} -3.54768 q^{23} +4.59914 q^{25} +(-2.25152 + 2.81615i) q^{26} -2.48683i q^{28} +0.830894 q^{29} -4.31236i q^{31} -1.00000i q^{32} -6.27837i q^{34} +1.57449 q^{35} -7.81310i q^{37} +4.86004 q^{38} +0.633132 q^{40} -0.0783645i q^{41} -9.68740 q^{43} +4.86004i q^{44} +3.54768i q^{46} +4.96601i q^{47} +0.815659 q^{49} -4.59914i q^{50} +(2.81615 + 2.25152i) q^{52} +6.36026 q^{53} -3.07704 q^{55} -2.48683 q^{56} -0.830894i q^{58} -9.08742i q^{59} -10.5707 q^{61} -4.31236 q^{62} -1.00000 q^{64} +(-1.42551 + 1.78299i) q^{65} -7.88457i q^{67} -6.27837 q^{68} -1.57449i q^{70} -12.3085i q^{71} -1.05367i q^{73} -7.81310 q^{74} -4.86004i q^{76} +12.0861 q^{77} +3.37676 q^{79} -0.633132i q^{80} -0.0783645 q^{82} -15.2137i q^{83} -3.97504i q^{85} +9.68740i q^{86} +4.86004 q^{88} -0.595093i q^{89} +(5.59914 - 7.00329i) q^{91} +3.54768 q^{92} +4.96601 q^{94} +3.07704 q^{95} -17.0630i q^{97} -0.815659i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4}+O(q^{10}) \) 14 * q - 14 * q^4	\( 14 q - 14 q^{4} - 2 q^{13} + 8 q^{14} + 14 q^{16} - 8 q^{17} - 8 q^{23} - 14 q^{25} - 4 q^{26} - 16 q^{29} + 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} + 60 q^{53} - 8 q^{56} - 28 q^{61} - 34 q^{62} - 14 q^{64} - 8 q^{65} + 8 q^{68} + 16 q^{74} - 24 q^{77} - 28 q^{79} - 24 q^{82} + 8 q^{92}+O(q^{100}) \) 14 * q - 14 * q^4 - 2 * q^13 + 8 * q^14 + 14 * q^16 - 8 * q^17 - 8 * q^23 - 14 * q^25 - 4 * q^26 - 16 * q^29 + 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 + 60 * q^53 - 8 * q^56 - 28 * q^61 - 34 * q^62 - 14 * q^64 - 8 * q^65 + 8 * q^68 + 16 * q^74 - 24 * q^77 - 28 * q^79 - 24 * q^82 + 8 * q^92

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