LMFDB - Embedded newform 1380.2.i.a.1241.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1241,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1380.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.i (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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + \cdots + 6561 $ x^16 - x^14 - 4x^13 - 2x^12 + 12x^11 + 17x^10 + 16x^9 - 110x^8 + 48x^7 + 153x^6 + 324x^5 - 162x^4 - 972x^3 - 729x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.15
Root			$-1.73058 - 0.0713366i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.a.1241.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73058 - 0.0713366i) q^{3} -1.00000 q^{5} +2.28380i q^{7} +(2.98982 - 0.246908i) q^{9} +O(q^{10})$	\(q+(1.73058 - 0.0713366i) q^{3} -1.00000 q^{5} +2.28380i q^{7} +(2.98982 - 0.246908i) q^{9} +1.33823 q^{11} +1.41470 q^{13} +(-1.73058 + 0.0713366i) q^{15} +0.879796 q^{17} +6.12613i q^{19} +(0.162918 + 3.95230i) q^{21} +(-3.55748 + 3.21626i) q^{23} +1.00000 q^{25} +(5.15652 - 0.640577i) q^{27} -5.48271i q^{29} +2.33732 q^{31} +(2.31591 - 0.0954645i) q^{33} -2.28380i q^{35} -4.22327i q^{37} +(2.44825 - 0.100920i) q^{39} +11.5789i q^{41} +7.56828i q^{43} +(-2.98982 + 0.246908i) q^{45} -7.64098i q^{47} +1.78427 q^{49} +(1.52256 - 0.0627616i) q^{51} +7.85277 q^{53} -1.33823 q^{55} +(0.437017 + 10.6018i) q^{57} -0.0739530i q^{59} +3.35641i q^{61} +(0.563887 + 6.82815i) q^{63} -1.41470 q^{65} -0.0659812i q^{67} +(-5.92708 + 5.81977i) q^{69} +11.6042i q^{71} +3.54295 q^{73} +(1.73058 - 0.0713366i) q^{75} +3.05624i q^{77} -4.00731i q^{79} +(8.87807 - 1.47642i) q^{81} +4.64098 q^{83} -0.879796 q^{85} +(-0.391118 - 9.48827i) q^{87} +0.872226 q^{89} +3.23089i q^{91} +(4.04493 - 0.166737i) q^{93} -6.12613i q^{95} -2.86737i q^{97} +(4.00106 - 0.330418i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 + 2 * q^9	$ 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 + 2 * q^9 - 4 * q^21 + 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 + 2 * q^33 - 14 * q^39 - 2 * q^45 + 8 * q^49 - 2 * q^51 + 4 * q^53 - 2 * q^57 + 30 * q^63 - 4 * q^73 + 10 * q^81 - 4 * q^83 - 10 * q^87 - 28 * q^89 - 10 * q^93 - 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.73058	−	0.0713366i	0.999151	−	0.0411862i
$3$	1.73058	−	0.0713366i	0.999151	−	0.0411862i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			2.28380i			0.863194i	0.902066	+	0.431597i	$0.142050\pi$
$7$			2.28380i			0.863194i	−0.902066	+	0.431597i	$0.857950\pi$
$8$		0			0
$9$	2.98982	−	0.246908i	0.996607	−	0.0823025i
$10$		0			0
$11$	1.33823			0.403490			0.201745	−	0.979438i	$-0.435339\pi$
$11$	1.33823			0.403490			0.201745	+	0.979438i	$0.435339\pi$
$12$		0			0
$13$	1.41470			0.392367			0.196184	−	0.980567i	$-0.437145\pi$
$13$	1.41470			0.392367			0.196184	+	0.980567i	$0.437145\pi$
$14$		0			0
$15$	−1.73058	+	0.0713366i	−0.446834	+	0.0184190i
$16$		0			0
$17$	0.879796			0.213382			0.106691	−	0.994292i	$-0.465974\pi$
$17$	0.879796			0.213382			0.106691	+	0.994292i	$0.465974\pi$
$18$		0			0
$19$			6.12613i			1.40543i	0.711472	+	0.702715i	$0.248029\pi$
$19$			6.12613i			1.40543i	−0.711472	+	0.702715i	$0.751971\pi$
$20$		0			0
$21$	0.162918	+	3.95230i	0.0355517	+	0.862462i
$22$		0			0
$23$	−3.55748	+	3.21626i	−0.741787	+	0.670636i
$23$	−3.55748	+	3.21626i	−0.741787	+	0.670636i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	5.15652	−	0.640577i	0.992372	−	0.123279i
$28$		0			0
$29$		−	5.48271i		−	1.01811i	−0.860733	−	0.509057i	$-0.829994\pi$
$29$		−	5.48271i		−	1.01811i	0.860733	−	0.509057i	$-0.170006\pi$
$30$		0			0
$31$	2.33732			0.419795			0.209898	−	0.977723i	$-0.432687\pi$
$31$	2.33732			0.419795			0.209898	+	0.977723i	$0.432687\pi$
$32$		0			0
$33$	2.31591	−	0.0954645i	0.403148	−	0.0166182i
$34$		0			0
$35$		−	2.28380i		−	0.386032i
$36$		0			0
$37$		−	4.22327i		−	0.694301i	−0.937809	−	0.347151i	$-0.887149\pi$
$37$		−	4.22327i		−	0.694301i	0.937809	−	0.347151i	$-0.112851\pi$
$38$		0			0
$39$	2.44825	−	0.100920i	0.392034	−	0.0161601i
$40$		0			0
$41$			11.5789i			1.80832i	0.427193	+	0.904161i	$0.359503\pi$
$41$			11.5789i			1.80832i	−0.427193	+	0.904161i	$0.640497\pi$
$42$		0			0
$43$			7.56828i			1.15415i	0.816690	+	0.577076i	$0.195806\pi$
$43$			7.56828i			1.15415i	−0.816690	+	0.577076i	$0.804194\pi$
$44$		0			0
$45$	−2.98982	+	0.246908i	−0.445696	+	0.0368068i
$46$		0			0
$47$		−	7.64098i		−	1.11455i	−0.830327	−	0.557276i	$-0.811847\pi$
$47$		−	7.64098i		−	1.11455i	0.830327	−	0.557276i	$-0.188153\pi$
$48$		0			0
$49$	1.78427			0.254895
$50$		0			0
$51$	1.52256	−	0.0627616i	0.213201	−	0.00878839i
$52$		0			0
$53$	7.85277			1.07866			0.539331	−	0.842094i	$-0.318677\pi$
$53$	7.85277			1.07866			0.539331	+	0.842094i	$0.318677\pi$
$54$		0			0
$55$	−1.33823			−0.180446
$56$		0			0
$57$	0.437017	+	10.6018i	0.0578843	+	1.40424i
$58$		0			0
$59$		−	0.0739530i		−	0.00962786i	−0.999988	−	0.00481393i	$-0.998468\pi$
$59$		−	0.0739530i		−	0.00962786i	0.999988	−	0.00481393i	$-0.00153233\pi$
$60$		0			0
$61$			3.35641i			0.429744i	0.976642	+	0.214872i	$0.0689335\pi$
$61$			3.35641i			0.429744i	−0.976642	+	0.214872i	$0.931066\pi$
$62$		0			0
$63$	0.563887	+	6.82815i	0.0710431	+	0.860266i
$64$		0			0
$65$	−1.41470			−0.175472
$66$		0			0
$67$		−	0.0659812i		−	0.00806089i	−0.999992	−	0.00403044i	$-0.998717\pi$
$67$		−	0.0659812i		−	0.00806089i	0.999992	−	0.00403044i	$-0.00128293\pi$
$68$		0			0
$69$	−5.92708	+	5.81977i	−0.713537	+	0.700618i
$70$		0			0
$71$			11.6042i			1.37717i	0.725155	+	0.688586i	$0.241768\pi$
$71$			11.6042i			1.37717i	−0.725155	+	0.688586i	$0.758232\pi$
$72$		0			0
$73$	3.54295			0.414671			0.207336	−	0.978270i	$-0.433521\pi$
$73$	3.54295			0.414671			0.207336	+	0.978270i	$0.433521\pi$
$74$		0			0
$75$	1.73058	−	0.0713366i	0.199830	−	0.00823724i
$76$		0			0
$77$			3.05624i			0.348291i
$78$		0			0
$79$		−	4.00731i		−	0.450858i	−0.974260	−	0.225429i	$-0.927622\pi$
$79$		−	4.00731i		−	0.450858i	0.974260	−	0.225429i	$-0.0723783\pi$
$80$		0			0
$81$	8.87807	−	1.47642i	0.986453	−	0.164047i
$82$		0			0
$83$	4.64098			0.509413			0.254707	−	0.967018i	$-0.418021\pi$
$83$	4.64098			0.509413			0.254707	+	0.967018i	$0.418021\pi$
$84$		0			0
$85$	−0.879796			−0.0954272
$86$		0			0
$87$	−0.391118	−	9.48827i	−0.0419322	−	1.01725i
$88$		0			0
$89$	0.872226			0.0924558			0.0462279	−	0.998931i	$-0.485280\pi$
$89$	0.872226			0.0924558			0.0462279	+	0.998931i	$0.485280\pi$
$90$		0			0
$91$			3.23089i			0.338689i
$92$		0			0
$93$	4.04493	−	0.166737i	0.419439	−	0.0172898i
$94$		0			0
$95$		−	6.12613i		−	0.628527i
$96$		0			0
$97$		−	2.86737i		−	0.291138i	−0.989348	−	0.145569i	$-0.953499\pi$
$97$		−	2.86737i		−	0.291138i	0.989348	−	0.145569i	$-0.0465012\pi$
$98$		0			0
$99$	4.00106	−	0.330418i	0.402121	−	0.0332083i
$100$		0			0
$101$		−	8.62607i		−	0.858326i	−0.903227	−	0.429163i	$-0.858809\pi$
$101$		−	8.62607i		−	0.858326i	0.903227	−	0.429163i	$-0.141191\pi$
$102$		0			0
$103$		−	14.7628i		−	1.45462i	−0.686308	−	0.727311i	$-0.740770\pi$
$103$		−	14.7628i		−	1.45462i	0.686308	−	0.727311i	$-0.259230\pi$
$104$		0			0
$105$	−0.162918	−	3.95230i	−0.0158992	−	0.385705i
$106$		0			0
$107$	−13.8861			−1.34242			−0.671212	−	0.741265i	$-0.734226\pi$
$107$	−13.8861			−1.34242			−0.671212	+	0.741265i	$0.734226\pi$
$108$		0			0
$109$		−	10.7781i		−	1.03235i	−0.856482	−	0.516177i	$-0.827354\pi$
$109$		−	10.7781i		−	1.03235i	0.856482	−	0.516177i	$-0.172646\pi$
$110$		0			0
$111$	−0.301274	−	7.30871i	−0.0285956	−	0.693712i
$112$		0			0
$113$	13.9011			1.30770			0.653851	−	0.756623i	$-0.273152\pi$
$113$	13.9011			1.30770			0.653851	+	0.756623i	$0.273152\pi$
$114$		0			0
$115$	3.55748	−	3.21626i	0.331737	−	0.299917i
$116$		0			0
$117$	4.22970	−	0.349300i	0.391036	−	0.0322928i
$118$		0			0
$119$			2.00928i			0.184190i
$120$		0			0
$121$	−9.20915			−0.837196
$122$		0			0
$123$	0.826000	+	20.0382i	0.0744779	+	1.80679i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−4.10607			−0.364355			−0.182177	−	0.983266i	$-0.558315\pi$
$127$	−4.10607			−0.364355			−0.182177	+	0.983266i	$0.558315\pi$
$128$		0			0
$129$	0.539895	+	13.0975i	0.0475351	+	1.15317i
$130$		0			0
$131$			2.97818i			0.260205i	0.991501	+	0.130103i	$0.0415306\pi$
$131$			2.97818i			0.260205i	−0.991501	+	0.130103i	$0.958469\pi$
$132$		0			0
$133$	−13.9908			−1.21316
$134$		0			0
$135$	−5.15652	+	0.640577i	−0.443802	+	0.0551321i
$136$		0			0
$137$	−17.6069			−1.50426			−0.752130	−	0.659015i	$-0.770973\pi$
$137$	−17.6069			−1.50426			−0.752130	+	0.659015i	$0.770973\pi$
$138$		0			0
$139$	−1.91800			−0.162683			−0.0813415	−	0.996686i	$-0.525920\pi$
$139$	−1.91800			−0.162683			−0.0813415	+	0.996686i	$0.525920\pi$
$140$		0			0
$141$	−0.545082	−	13.2233i	−0.0459042	−	1.11361i
$142$		0			0
$143$	1.89319			0.158316
$144$		0			0
$145$			5.48271i			0.455314i
$146$		0			0
$147$	3.08782	−	0.127284i	0.254679	−	0.0104982i
$148$		0			0
$149$	−3.29540			−0.269970			−0.134985	−	0.990848i	$-0.543099\pi$
$149$	−3.29540			−0.269970			−0.134985	+	0.990848i	$0.543099\pi$
$150$		0			0
$151$	−6.71760			−0.546670			−0.273335	−	0.961919i	$-0.588127\pi$
$151$	−6.71760			−0.546670			−0.273335	+	0.961919i	$0.588127\pi$
$152$		0			0
$153$	2.63043	−	0.217228i	0.212658	−	0.0175619i
$154$		0			0
$155$	−2.33732			−0.187738
$156$		0			0
$157$		−	6.55688i		−	0.523296i	−0.965163	−	0.261648i	$-0.915734\pi$
$157$		−	6.55688i		−	0.523296i	0.965163	−	0.261648i	$-0.0842659\pi$
$158$		0			0
$159$	13.5899	−	0.560190i	1.07775	−	0.0444260i
$160$		0			0
$161$	−7.34528	−	8.12458i	−0.578889	−	0.640306i
$162$		0			0
$163$	8.68400			0.680184			0.340092	−	0.940392i	$-0.389542\pi$
$163$	8.68400			0.680184			0.340092	+	0.940392i	$0.389542\pi$
$164$		0			0
$165$	−2.31591	+	0.0954645i	−0.180293	+	0.00743190i
$166$		0			0
$167$		−	17.4517i		−	1.35046i	−0.737609	−	0.675228i	$-0.764045\pi$
$167$		−	17.4517i		−	1.35046i	0.737609	−	0.675228i	$-0.235955\pi$
$168$		0			0
$169$	−10.9986			−0.846048
$170$		0			0
$171$	1.51259	+	18.3160i	0.115670	+	1.40066i
$172$		0			0
$173$			14.2296i			1.08186i	0.841068	+	0.540929i	$0.181927\pi$
$173$			14.2296i			1.08186i	−0.841068	+	0.540929i	$0.818073\pi$
$174$		0			0
$175$			2.28380i			0.172639i
$176$		0			0
$177$	−0.00527556	−	0.127982i	−0.000396535	−	0.00961969i
$178$		0			0
$179$		−	6.63153i		−	0.495663i	−0.968803	−	0.247832i	$-0.920282\pi$
$179$		−	6.63153i		−	0.495663i	0.968803	−	0.247832i	$-0.0797180\pi$
$180$		0			0
$181$		−	21.7518i		−	1.61680i	−0.588633	−	0.808400i	$-0.700334\pi$
$181$		−	21.7518i		−	1.61680i	0.588633	−	0.808400i	$-0.299666\pi$
$182$		0			0
$183$	0.239435	+	5.80854i	0.0176995	+	0.429380i
$184$		0			0
$185$			4.22327i			0.310501i
$186$		0			0
$187$	1.17737			0.0860975
$188$		0			0
$189$	1.46295	+	11.7764i	0.106414	+	0.856610i
$190$		0			0
$191$	15.2529			1.10366			0.551830	−	0.833957i	$-0.313930\pi$
$191$	15.2529			1.10366			0.551830	+	0.833957i	$0.313930\pi$
$192$		0			0
$193$	−14.2057			−1.02255			−0.511273	−	0.859418i	$-0.670826\pi$
$193$	−14.2057			−1.02255			−0.511273	+	0.859418i	$0.670826\pi$
$194$		0			0
$195$	−2.44825	+	0.100920i	−0.175323	+	0.00722702i
$196$		0			0
$197$		−	4.59137i		−	0.327122i	−0.986533	−	0.163561i	$-0.947702\pi$
$197$		−	4.59137i		−	0.327122i	0.986533	−	0.163561i	$-0.0522980\pi$
$198$		0			0
$199$			1.26901i			0.0899576i	0.998988	+	0.0449788i	$0.0143220\pi$
$199$			1.26901i			0.0899576i	−0.998988	+	0.0449788i	$0.985678\pi$
$200$		0			0
$201$	−0.00470688	−	0.114186i	−0.000331997	−	0.00805405i
$202$		0			0
$203$	12.5214			0.878830
$204$		0			0
$205$		−	11.5789i		−	0.808706i
$206$		0			0
$207$	−9.84213	+	10.4944i	−0.684075	+	0.729411i
$208$		0			0
$209$			8.19814i			0.567077i
$210$		0			0
$211$	14.2956			0.984148			0.492074	−	0.870553i	$-0.336239\pi$
$211$	14.2956			0.984148			0.492074	+	0.870553i	$0.336239\pi$
$212$		0			0
$213$	0.827808	+	20.0821i	0.0567205	+	1.37600i
$214$		0			0
$215$		−	7.56828i		−	0.516152i
$216$		0			0
$217$			5.33797i			0.362365i
$218$		0			0
$219$	6.13137	−	0.252742i	0.414320	−	0.0170787i
$220$		0			0
$221$	1.24465			0.0837240
$222$		0			0
$223$	−7.22989			−0.484149			−0.242075	−	0.970258i	$-0.577828\pi$
$223$	−7.22989			−0.484149			−0.242075	+	0.970258i	$0.577828\pi$
$224$		0			0
$225$	2.98982	−	0.246908i	0.199321	−	0.0164605i
$226$		0			0
$227$	−25.3648			−1.68352			−0.841759	−	0.539853i	$-0.818480\pi$
$227$	−25.3648			−1.68352			−0.841759	+	0.539853i	$0.818480\pi$
$228$		0			0
$229$		−	17.6165i		−	1.16413i	−0.813141	−	0.582067i	$-0.802244\pi$
$229$		−	17.6165i		−	1.16413i	0.813141	−	0.582067i	$-0.197756\pi$
$230$		0			0
$231$	0.218022	+	5.28907i	0.0143448	+	0.347995i
$232$		0			0
$233$		−	3.12892i		−	0.204982i	−0.994734	−	0.102491i	$-0.967319\pi$
$233$		−	3.12892i		−	0.204982i	0.994734	−	0.102491i	$-0.0326813\pi$
$234$		0			0
$235$			7.64098i			0.498443i
$236$		0			0
$237$	−0.285868	−	6.93498i	−0.0185691	−	0.450475i
$238$		0			0
$239$			16.4802i			1.06602i	0.846110	+	0.533008i	$0.178939\pi$
$239$			16.4802i			1.06602i	−0.846110	+	0.533008i	$0.821061\pi$
$240$		0			0
$241$		−	2.20436i		−	0.141996i	−0.997476	−	0.0709978i	$-0.977382\pi$
$241$		−	2.20436i		−	0.141996i	0.997476	−	0.0709978i	$-0.0226183\pi$
$242$		0			0
$243$	15.2589	−	3.18840i	0.978859	−	0.204536i
$244$		0			0
$245$	−1.78427			−0.113993
$246$		0			0
$247$			8.66663i			0.551444i
$248$		0			0
$249$	8.03158	−	0.331071i	0.508981	−	0.0209808i
$250$		0			0
$251$	30.1099			1.90052			0.950261	−	0.311456i	$-0.100817\pi$
$251$	30.1099			1.90052			0.950261	+	0.311456i	$0.100817\pi$
$252$		0			0
$253$	−4.76072	+	4.30408i	−0.299304	+	0.270595i
$254$		0			0
$255$	−1.52256	+	0.0627616i	−0.0953463	+	0.00393029i
$256$		0			0
$257$		−	25.4765i		−	1.58918i	−0.607146	−	0.794590i	$-0.707686\pi$
$257$		−	25.4765i		−	1.58918i	0.607146	−	0.794590i	$-0.292314\pi$
$258$		0			0
$259$	9.64509			0.599317
$260$		0			0
$261$	−1.35372	−	16.3923i	−0.0837933	−	1.01466i
$262$		0			0
$263$	−7.64208			−0.471231			−0.235615	−	0.971846i	$-0.575711\pi$
$263$	−7.64208			−0.471231			−0.235615	+	0.971846i	$0.575711\pi$
$264$		0			0
$265$	−7.85277			−0.482392
$266$		0			0
$267$	1.50946	−	0.0622216i	0.0923773	−	0.00380790i
$268$		0			0
$269$			23.6190i			1.44007i	0.693936	+	0.720037i	$0.255875\pi$
$269$			23.6190i			1.44007i	−0.693936	+	0.720037i	$0.744125\pi$
$270$		0			0
$271$	−5.02833			−0.305449			−0.152725	−	0.988269i	$-0.548805\pi$
$271$	−5.02833			−0.305449			−0.152725	+	0.988269i	$0.548805\pi$
$272$		0			0
$273$	0.230481	+	5.59131i	0.0139493	+	0.338402i
$274$		0			0
$275$	1.33823			0.0806981
$276$		0			0
$277$	−27.4569			−1.64972			−0.824861	−	0.565335i	$-0.808747\pi$
$277$	−27.4569			−1.64972			−0.824861	+	0.565335i	$0.808747\pi$
$278$		0			0
$279$	6.98818	−	0.577102i	0.418371	−	0.0345502i
$280$		0			0
$281$	−2.90648			−0.173386			−0.0866929	−	0.996235i	$-0.527630\pi$
$281$	−2.90648			−0.173386			−0.0866929	+	0.996235i	$0.527630\pi$
$282$		0			0
$283$		−	2.40824i		−	0.143155i	−0.997435	−	0.0715775i	$-0.977197\pi$
$283$		−	2.40824i		−	0.143155i	0.997435	−	0.0715775i	$-0.0228033\pi$
$284$		0			0
$285$	−0.437017	−	10.6018i	−0.0258867	−	0.627994i
$286$		0			0
$287$	−26.4439			−1.56093
$288$		0			0
$289$	−16.2260			−0.954468
$290$		0			0
$291$	−0.204549	−	4.96222i	−0.0119909	−	0.290891i
$292$		0			0
$293$	13.2379			0.773367			0.386683	−	0.922213i	$-0.373621\pi$
$293$	13.2379			0.773367			0.386683	+	0.922213i	$0.373621\pi$
$294$		0			0
$295$			0.0739530i			0.00430571i
$296$		0			0
$297$	6.90058	−	0.857237i	0.400413	−	0.0497420i
$298$		0			0
$299$	−5.03277	+	4.55004i	−0.291053	+	0.263135i
$300$		0			0
$301$	−17.2844			−0.996258
$302$		0			0
$303$	−0.615354	−	14.9281i	−0.0353512	−	0.857597i
$304$		0			0
$305$		−	3.35641i		−	0.192188i
$306$		0			0
$307$	11.2612			0.642708			0.321354	−	0.946959i	$-0.395862\pi$
$307$	11.2612			0.642708			0.321354	+	0.946959i	$0.395862\pi$
$308$		0			0
$309$	−1.05313	−	25.5482i	−0.0599104	−	1.45339i
$310$		0			0
$311$			23.5896i			1.33765i	0.743422	+	0.668823i	$0.233201\pi$
$311$			23.5896i			1.33765i	−0.743422	+	0.668823i	$0.766799\pi$
$312$		0			0
$313$		−	21.6924i		−	1.22613i	−0.790033	−	0.613064i	$-0.789937\pi$
$313$		−	21.6924i		−	1.22613i	0.790033	−	0.613064i	$-0.210063\pi$
$314$		0			0
$315$	−0.563887	−	6.82815i	−0.0317714	−	0.384723i
$316$		0			0
$317$			1.17690i			0.0661016i	0.999454	+	0.0330508i	$0.0105223\pi$
$317$			1.17690i			0.0661016i	−0.999454	+	0.0330508i	$0.989478\pi$
$318$		0			0
$319$		−	7.33710i		−	0.410799i
$320$		0			0
$321$	−24.0311	+	0.990590i	−1.34129	+	0.0552894i
$322$		0			0
$323$			5.38974i			0.299893i
$324$		0			0
$325$	1.41470			0.0784734
$326$		0			0
$327$	−0.768873	−	18.6524i	−0.0425188	−	1.03148i
$328$		0			0
$329$	17.4505			0.962075
$330$		0			0
$331$	2.08358			0.114524			0.0572620	−	0.998359i	$-0.481763\pi$
$331$	2.08358			0.114524			0.0572620	+	0.998359i	$0.481763\pi$
$332$		0			0
$333$	−1.04276	−	12.6268i	−0.0571427	−	0.691946i
$334$		0			0
$335$			0.0659812i			0.00360494i
$336$		0			0
$337$		−	28.8944i		−	1.57398i	−0.616967	−	0.786989i	$-0.711639\pi$
$337$		−	28.8944i		−	1.57398i	0.616967	−	0.786989i	$-0.288361\pi$
$338$		0			0
$339$	24.0569	−	0.991655i	1.30659	−	0.0538593i
$340$		0			0
$341$	3.12787			0.169383
$342$		0			0
$343$			20.0615i			1.08322i
$344$		0			0
$345$	5.92708	−	5.81977i	0.319103	−	0.313326i
$346$		0			0
$347$		−	18.5703i		−	0.996908i	−0.866916	−	0.498454i	$-0.833901\pi$
$347$		−	18.5703i		−	0.996908i	0.866916	−	0.498454i	$-0.166099\pi$
$348$		0			0
$349$	−24.7383			−1.32421			−0.662106	−	0.749411i	$-0.730337\pi$
$349$	−24.7383			−1.32421			−0.662106	+	0.749411i	$0.730337\pi$
$350$		0			0
$351$	7.29492	−	0.906225i	0.389374	−	0.0483707i
$352$		0			0
$353$		−	18.9048i		−	1.00620i	−0.864228	−	0.503101i	$-0.832192\pi$
$353$		−	18.9048i		−	1.00620i	0.864228	−	0.503101i	$-0.167808\pi$
$354$		0			0
$355$		−	11.6042i		−	0.615890i
$356$		0			0
$357$	0.143335	+	3.47721i	0.00758609	+	0.184034i
$358$		0			0
$359$	11.4565			0.604652			0.302326	−	0.953205i	$-0.402237\pi$
$359$	11.4565			0.604652			0.302326	+	0.953205i	$0.402237\pi$
$360$		0			0
$361$	−18.5294			−0.975232
$362$		0			0
$363$	−15.9372	+	0.656950i	−0.836485	+	0.0344809i
$364$		0			0
$365$	−3.54295			−0.185447
$366$		0			0
$367$		−	14.8499i		−	0.775156i	−0.921837	−	0.387578i	$-0.873312\pi$
$367$		−	14.8499i		−	0.775156i	0.921837	−	0.387578i	$-0.126688\pi$
$368$		0			0
$369$	2.85892	+	34.6189i	0.148829	+	1.80219i
$370$		0			0
$371$			17.9341i			0.931095i
$372$		0			0
$373$		−	20.4430i		−	1.05850i	−0.848467	−	0.529249i	$-0.822474\pi$
$373$		−	20.4430i		−	1.05850i	0.848467	−	0.529249i	$-0.177526\pi$
$374$		0			0
$375$	−1.73058	+	0.0713366i	−0.0893668	+	0.00368381i
$376$		0			0
$377$		−	7.75639i		−	0.399474i
$378$		0			0
$379$			34.2485i			1.75923i	0.475687	+	0.879615i	$0.342200\pi$
$379$			34.2485i			1.75923i	−0.475687	+	0.879615i	$0.657800\pi$
$380$		0			0
$381$	−7.10589	+	0.292913i	−0.364046	+	0.0150064i
$382$		0			0
$383$	−22.7322			−1.16156			−0.580779	−	0.814061i	$-0.697252\pi$
$383$	−22.7322			−1.16156			−0.580779	+	0.814061i	$0.697252\pi$
$384$		0			0
$385$		−	3.05624i		−	0.155760i
$386$		0			0
$387$	1.86867	+	22.6278i	0.0949896	+	1.15024i
$388$		0			0
$389$	24.4148			1.23788			0.618940	−	0.785438i	$-0.287562\pi$
$389$	24.4148			1.23788			0.618940	+	0.785438i	$0.287562\pi$
$390$		0			0
$391$	−3.12986	+	2.82965i	−0.158284	+	0.143101i
$392$		0			0
$393$	0.212453	+	5.15399i	0.0107169	+	0.259984i
$394$		0			0
$395$			4.00731i			0.201630i
$396$		0			0
$397$	−23.6505			−1.18699			−0.593493	−	0.804839i	$-0.702251\pi$
$397$	−23.6505			−1.18699			−0.593493	+	0.804839i	$0.702251\pi$
$398$		0			0
$399$	−24.2123	+	0.998058i	−1.21213	+	0.0499654i
$400$		0			0
$401$	−15.4865			−0.773361			−0.386680	−	0.922214i	$-0.626378\pi$
$401$	−15.4865			−0.773361			−0.386680	+	0.922214i	$0.626378\pi$
$402$		0			0
$403$	3.30661			0.164714
$404$		0			0
$405$	−8.87807	+	1.47642i	−0.441155	+	0.0733639i
$406$		0			0
$407$		−	5.65169i		−	0.280144i
$408$		0			0
$409$	2.06826			0.102269			0.0511344	−	0.998692i	$-0.483716\pi$
$409$	2.06826			0.102269			0.0511344	+	0.998692i	$0.483716\pi$
$410$		0			0
$411$	−30.4702	+	1.25602i	−1.50298	+	0.0619548i
$412$		0			0
$413$	0.168894			0.00831072
$414$		0			0
$415$	−4.64098			−0.227817
$416$		0			0
$417$	−3.31926	+	0.136824i	−0.162545	+	0.00670030i
$418$		0			0
$419$	7.26438			0.354888			0.177444	−	0.984131i	$-0.443217\pi$
$419$	7.26438			0.354888			0.177444	+	0.984131i	$0.443217\pi$
$420$		0			0
$421$		−	27.7739i		−	1.35362i	−0.736159	−	0.676809i	$-0.763362\pi$
$421$		−	27.7739i		−	1.35362i	0.736159	−	0.676809i	$-0.236638\pi$
$422$		0			0
$423$	−1.88662	−	22.8452i	−0.0917304	−	1.11077i
$424$		0			0
$425$	0.879796			0.0426764
$426$		0			0
$427$	−7.66537			−0.370953
$428$		0			0
$429$	3.27632	−	0.135054i	0.158182	−	0.00652045i
$430$		0			0
$431$	30.2013			1.45475			0.727373	−	0.686243i	$-0.240741\pi$
$431$	30.2013			1.45475			0.727373	+	0.686243i	$0.240741\pi$
$432$		0			0
$433$		−	10.9727i		−	0.527316i	−0.964616	−	0.263658i	$-0.915071\pi$
$433$		−	10.9727i		−	0.527316i	0.964616	−	0.263658i	$-0.0849291\pi$
$434$		0			0
$435$	0.391118	+	9.48827i	0.0187527	+	0.454928i
$436$		0			0
$437$	−19.7032	−	21.7936i	−0.942531	−	1.04253i
$438$		0			0
$439$	−10.8111			−0.515986			−0.257993	−	0.966147i	$-0.583061\pi$
$439$	−10.8111			−0.515986			−0.257993	+	0.966147i	$0.583061\pi$
$440$		0			0
$441$	5.33464	−	0.440549i	0.254031	−	0.0209785i
$442$		0			0
$443$		−	23.5022i		−	1.11662i	−0.829631	−	0.558312i	$-0.811449\pi$
$443$		−	23.5022i		−	1.11662i	0.829631	−	0.558312i	$-0.188551\pi$
$444$		0			0
$445$	−0.872226			−0.0413475
$446$		0			0
$447$	−5.70296	+	0.235083i	−0.269741	+	0.0111190i
$448$		0			0
$449$			12.2274i			0.577047i	0.957473	+	0.288523i	$0.0931643\pi$
$449$			12.2274i			0.577047i	−0.957473	+	0.288523i	$0.906836\pi$
$450$		0			0
$451$			15.4952i			0.729640i
$452$		0			0
$453$	−11.6253	+	0.479211i	−0.546206	+	0.0225153i
$454$		0			0
$455$		−	3.23089i		−	0.151466i
$456$		0			0
$457$			16.6895i			0.780701i	0.920666	+	0.390350i	$0.127646\pi$
$457$			16.6895i			0.780701i	−0.920666	+	0.390350i	$0.872354\pi$
$458$		0			0
$459$	4.53668	−	0.563577i	0.211754	−	0.0263055i
$460$		0			0
$461$		−	15.0689i		−	0.701826i	−0.936408	−	0.350913i	$-0.885871\pi$
$461$		−	15.0689i		−	0.701826i	0.936408	−	0.350913i	$-0.114129\pi$
$462$		0			0
$463$	−0.628181			−0.0291941			−0.0145970	−	0.999893i	$-0.504647\pi$
$463$	−0.628181			−0.0291941			−0.0145970	+	0.999893i	$0.504647\pi$
$464$		0			0
$465$	−4.04493	+	0.166737i	−0.187579	+	0.00773223i
$466$		0			0
$467$	0.423673			0.0196052			0.00980262	−	0.999952i	$-0.496880\pi$
$467$	0.423673			0.0196052			0.00980262	+	0.999952i	$0.496880\pi$
$468$		0			0
$469$	0.150688			0.00695811
$470$		0			0
$471$	−0.467745	−	11.3472i	−0.0215526	−	0.522852i
$472$		0			0
$473$			10.1281i			0.465689i
$474$		0			0
$475$			6.12613i			0.281086i
$476$		0			0
$477$	23.4784	−	1.93891i	1.07500	−	0.0887766i
$478$		0			0
$479$	23.7473			1.08504			0.542521	−	0.840042i	$-0.317470\pi$
$479$	23.7473			1.08504			0.542521	+	0.840042i	$0.317470\pi$
$480$		0			0
$481$		−	5.97466i		−	0.272421i
$482$		0			0
$483$	−13.2912	−	13.5363i	−0.604770	−	0.615921i
$484$		0			0
$485$			2.86737i			0.130201i
$486$		0			0
$487$	11.6188			0.526497			0.263248	−	0.964728i	$-0.415206\pi$
$487$	11.6188			0.526497			0.263248	+	0.964728i	$0.415206\pi$
$488$		0			0
$489$	15.0284	−	0.619487i	0.679607	−	0.0280142i
$490$		0			0
$491$			32.2065i			1.45346i	0.686923	+	0.726730i	$0.258961\pi$
$491$			32.2065i			1.45346i	−0.686923	+	0.726730i	$0.741039\pi$
$492$		0			0
$493$		−	4.82366i		−	0.217247i
$494$		0			0
$495$	−4.00106	+	0.330418i	−0.179834	+	0.0148512i
$496$		0			0
$497$	−26.5018			−1.18877
$498$		0			0
$499$	39.9184			1.78699			0.893495	−	0.449072i	$-0.148245\pi$
$499$	39.9184			1.78699			0.893495	+	0.449072i	$0.148245\pi$
$500$		0			0
$501$	−1.24495	−	30.2017i	−0.0556202	−	1.34931i
$502$		0			0
$503$	−37.4299			−1.66892			−0.834459	−	0.551070i	$-0.814220\pi$
$503$	−37.4299			−1.66892			−0.834459	+	0.551070i	$0.814220\pi$
$504$		0			0
$505$			8.62607i			0.383855i
$506$		0			0
$507$	−19.0340	+	0.784605i	−0.845330	+	0.0348455i
$508$		0			0
$509$			7.48546i			0.331787i	0.986144	+	0.165894i	$0.0530509\pi$
$509$			7.48546i			0.331787i	−0.986144	+	0.165894i	$0.946949\pi$
$510$		0			0
$511$			8.09139i			0.357942i
$512$		0			0
$513$	3.92426	+	31.5895i	0.173260	+	1.39471i
$514$		0			0
$515$			14.7628i			0.650527i
$516$		0			0
$517$		−	10.2254i		−	0.449711i
$518$		0			0
$519$	1.01509	+	24.6255i	0.0445576	+	1.08094i
$520$		0			0
$521$	−9.17799			−0.402095			−0.201048	−	0.979581i	$-0.564435\pi$
$521$	−9.17799			−0.402095			−0.201048	+	0.979581i	$0.564435\pi$
$522$		0			0
$523$			0.656371i			0.0287011i	0.999897	+	0.0143506i	$0.00456808\pi$
$523$			0.656371i			0.0287011i	−0.999897	+	0.0143506i	$0.995432\pi$
$524$		0			0
$525$	0.162918	+	3.95230i	0.00711034	+	0.172492i
$526$		0			0
$527$	2.05637			0.0895767
$528$		0			0
$529$	2.31140	−	22.8836i	0.100496	−	0.994938i
$530$		0			0
$531$	−0.0182596	−	0.221106i	−0.000792397	−	0.00959520i
$532$		0			0
$533$			16.3807i			0.709526i
$534$		0			0
$535$	13.8861			0.600350
$536$		0			0
$537$	−0.473071	−	11.4764i	−0.0204145	−	0.495243i
$538$		0			0
$539$	2.38775			0.102848
$540$		0			0
$541$	18.7680			0.806899			0.403450	−	0.915002i	$-0.367811\pi$
$541$	18.7680			0.806899			0.403450	+	0.915002i	$0.367811\pi$
$542$		0			0
$543$	−1.55170	−	37.6433i	−0.0665899	−	1.61543i
$544$		0			0
$545$			10.7781i			0.461683i
$546$		0			0
$547$	28.2966			1.20988			0.604938	−	0.796273i	$-0.293198\pi$
$547$	28.2966			1.20988			0.604938	+	0.796273i	$0.293198\pi$
$548$		0			0
$549$	0.828723	+	10.0351i	0.0353691	+	0.428286i
$550$		0			0
$551$	33.5878			1.43089
$552$		0			0
$553$	9.15189			0.389178
$554$		0			0
$555$	0.301274	+	7.30871i	0.0127884	+	0.310237i
$556$		0			0
$557$	−16.4010			−0.694932			−0.347466	−	0.937692i	$-0.612958\pi$
$557$	−16.4010			−0.694932			−0.347466	+	0.937692i	$0.612958\pi$
$558$		0			0
$559$			10.7068i			0.452851i
$560$		0			0
$561$	2.03753	−	0.0839893i	0.0860244	−	0.00354603i
$562$		0			0
$563$	−12.7782			−0.538536			−0.269268	−	0.963065i	$-0.586782\pi$
$563$	−12.7782			−0.538536			−0.269268	+	0.963065i	$0.586782\pi$
$564$		0			0
$565$	−13.9011			−0.584822
$566$		0			0
$567$	3.37184	+	20.2757i	0.141604	+	0.851500i
$568$		0			0
$569$	40.2134			1.68583			0.842917	−	0.538044i	$-0.180837\pi$
$569$	40.2134			1.68583			0.842917	+	0.538044i	$0.180837\pi$
$570$		0			0
$571$		−	12.1298i		−	0.507615i	−0.967255	−	0.253807i	$-0.918317\pi$
$571$		−	12.1298i		−	0.507615i	0.967255	−	0.253807i	$-0.0816830\pi$
$572$		0			0
$573$	26.3963	−	1.08809i	1.10272	−	0.0454556i
$574$		0			0
$575$	−3.55748	+	3.21626i	−0.148357	+	0.134127i
$576$		0			0
$577$	42.8160			1.78245			0.891227	−	0.453557i	$-0.149845\pi$
$577$	42.8160			1.78245			0.891227	+	0.453557i	$0.149845\pi$
$578$		0			0
$579$	−24.5840	+	1.01338i	−1.02168	+	0.0421148i
$580$		0			0
$581$			10.5990i			0.439723i
$582$		0			0
$583$	10.5088			0.435229
$584$		0			0
$585$	−4.22970	+	0.349300i	−0.174877	+	0.0144418i
$586$		0			0
$587$		−	23.1974i		−	0.957458i	−0.877963	−	0.478729i	$-0.841098\pi$
$587$		−	23.1974i		−	0.957458i	0.877963	−	0.478729i	$-0.158902\pi$
$588$		0			0
$589$			14.3187i			0.589993i
$590$		0			0
$591$	−0.327533	−	7.94575i	−0.0134729	−	0.326844i
$592$		0			0
$593$			6.46649i			0.265547i	0.991146	+	0.132773i	$0.0423883\pi$
$593$			6.46649i			0.265547i	−0.991146	+	0.132773i	$0.957612\pi$
$594$		0			0
$595$		−	2.00928i		−	0.0823723i
$596$		0			0
$597$	0.0905267	+	2.19612i	0.00370501	+	0.0898813i
$598$		0			0
$599$		−	19.9481i		−	0.815059i	−0.913192	−	0.407529i	$-0.866390\pi$
$599$		−	19.9481i		−	0.815059i	0.913192	−	0.407529i	$-0.133610\pi$
$600$		0			0
$601$	4.15147			0.169342			0.0846710	−	0.996409i	$-0.473016\pi$
$601$	4.15147			0.169342			0.0846710	+	0.996409i	$0.473016\pi$
$602$		0			0
$603$	−0.0162913	−	0.197272i	−0.000663431	−	0.00803354i
$604$		0			0
$605$	9.20915			0.374405
$606$		0			0
$607$	−0.310677			−0.0126100			−0.00630501	−	0.999980i	$-0.502007\pi$
$607$	−0.310677			−0.0126100			−0.00630501	+	0.999980i	$0.502007\pi$
$608$		0			0
$609$	21.6693	−	0.893234i	0.878084	−	0.0361957i
$610$		0			0
$611$		−	10.8097i		−	0.437314i
$612$		0			0
$613$			44.7842i			1.80882i	0.426668	+	0.904408i	$0.359687\pi$
$613$			44.7842i			1.80882i	−0.426668	+	0.904408i	$0.640313\pi$
$614$		0			0
$615$	−0.826000	−	20.0382i	−0.0333075	−	0.808020i
$616$		0			0
$617$	28.0079			1.12756			0.563779	−	0.825926i	$-0.309347\pi$
$617$	28.0079			1.12756			0.563779	+	0.825926i	$0.309347\pi$
$618$		0			0
$619$		−	21.5032i		−	0.864286i	−0.901805	−	0.432143i	$-0.857758\pi$
$619$		−	21.5032i		−	0.864286i	0.901805	−	0.432143i	$-0.142242\pi$
$620$		0			0
$621$	−16.2840	+	18.8635i	−0.653453	+	0.756967i
$622$		0			0
$623$			1.99199i			0.0798073i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	0.584827	+	14.1875i	0.0233558	+	0.566596i
$628$		0			0
$629$		−	3.71561i		−	0.148151i
$630$		0			0
$631$			29.5221i			1.17526i	0.809131	+	0.587628i	$0.199938\pi$
$631$			29.5221i			1.17526i	−0.809131	+	0.587628i	$0.800062\pi$
$632$		0			0
$633$	24.7397	−	1.01980i	0.983313	−	0.0405333i
$634$		0			0
$635$	4.10607			0.162944
$636$		0			0
$637$	2.52420			0.100013
$638$		0			0
$639$	2.86518	+	34.6946i	0.113345	+	1.37250i
$640$		0			0
$641$	−27.9460			−1.10380			−0.551900	−	0.833910i	$-0.686097\pi$
$641$	−27.9460			−1.10380			−0.551900	+	0.833910i	$0.686097\pi$
$642$		0			0
$643$			32.6165i			1.28627i	0.765752	+	0.643135i	$0.222367\pi$
$643$			32.6165i			1.28627i	−0.765752	+	0.643135i	$0.777633\pi$
$644$		0			0
$645$	−0.539895	−	13.0975i	−0.0212584	−	0.515714i
$646$		0			0
$647$		−	39.7216i		−	1.56162i	−0.624771	−	0.780808i	$-0.714808\pi$
$647$		−	39.7216i		−	1.56162i	0.624771	−	0.780808i	$-0.285192\pi$
$648$		0			0
$649$		−	0.0989658i		−	0.00388475i
$650$		0			0
$651$	0.380793	+	9.23779i	0.0149244	+	0.362058i
$652$		0			0
$653$			22.9025i			0.896244i	0.893972	+	0.448122i	$0.147907\pi$
$653$			22.9025i			0.896244i	−0.893972	+	0.448122i	$0.852093\pi$
$654$		0			0
$655$		−	2.97818i		−	0.116367i
$656$		0			0
$657$	10.5928	−	0.874782i	0.413265	−	0.0341285i
$658$		0			0
$659$	−26.8600			−1.04632			−0.523159	−	0.852235i	$-0.675247\pi$
$659$	−26.8600			−1.04632			−0.523159	+	0.852235i	$0.675247\pi$
$660$		0			0
$661$			41.3829i			1.60961i	0.593540	+	0.804805i	$0.297730\pi$
$661$			41.3829i			1.60961i	−0.593540	+	0.804805i	$0.702270\pi$
$662$		0			0
$663$	2.15396	−	0.0887889i	0.0836529	−	0.00344827i
$664$		0			0
$665$	13.9908			0.542541
$666$		0			0
$667$	17.6338	+	19.5047i	0.682783	+	0.755223i
$668$		0			0
$669$	−12.5119	+	0.515756i	−0.483739	+	0.0199403i
$670$		0			0
$671$			4.49164i			0.173398i
$672$		0			0
$673$	2.33943			0.0901783			0.0450892	−	0.998983i	$-0.485643\pi$
$673$	2.33943			0.0901783			0.0450892	+	0.998983i	$0.485643\pi$
$674$		0			0
$675$	5.15652	−	0.640577i	0.198474	−	0.0246558i
$676$		0			0
$677$	−40.9779			−1.57491			−0.787453	−	0.616374i	$-0.788601\pi$
$677$	−40.9779			−1.57491			−0.787453	+	0.616374i	$0.788601\pi$
$678$		0			0
$679$	6.54850			0.251308
$680$		0			0
$681$	−43.8958	+	1.80944i	−1.68209	+	0.0693377i
$682$		0			0
$683$		−	4.03409i		−	0.154360i	−0.997017	−	0.0771801i	$-0.975408\pi$
$683$		−	4.03409i		−	0.154360i	0.997017	−	0.0771801i	$-0.0245916\pi$
$684$		0			0
$685$	17.6069			0.672726
$686$		0			0
$687$	−1.25670	−	30.4869i	−0.0479463	−	1.16315i
$688$		0			0
$689$	11.1093			0.423231
$690$		0			0
$691$	16.7654			0.637785			0.318893	−	0.947791i	$-0.396689\pi$
$691$	16.7654			0.637785			0.318893	+	0.947791i	$0.396689\pi$
$692$		0			0
$693$	0.754608	+	9.13761i	0.0286652	+	0.347109i
$694$		0			0
$695$	1.91800			0.0727541
$696$		0			0
$697$			10.1871i			0.385863i
$698$		0			0
$699$	−0.223207	−	5.41485i	−0.00844245	−	0.204808i
$700$		0			0
$701$	14.9555			0.564861			0.282430	−	0.959288i	$-0.408859\pi$
$701$	14.9555			0.564861			0.282430	+	0.959288i	$0.408859\pi$
$702$		0			0
$703$	25.8723			0.975791
$704$		0			0
$705$	0.545082	+	13.2233i	0.0205290	+	0.498020i
$706$		0			0
$707$	19.7002			0.740902
$708$		0			0
$709$			37.3019i			1.40090i	0.713700	+	0.700451i	$0.247018\pi$
$709$			37.3019i			1.40090i	−0.713700	+	0.700451i	$0.752982\pi$
$710$		0			0
$711$	−0.989435	−	11.9811i	−0.0371067	−	0.449328i
$712$		0			0
$713$	−8.31499	+	7.51743i	−0.311399	+	0.281530i
$714$		0			0
$715$	−1.89319			−0.0708012
$716$		0			0
$717$	1.17564	+	28.5204i	0.0439052	+	1.06511i
$718$		0			0
$719$			25.9914i			0.969315i	0.874704	+	0.484657i	$0.161056\pi$
$719$			25.9914i			0.969315i	−0.874704	+	0.484657i	$0.838944\pi$
$720$		0			0
$721$	33.7153			1.25562
$722$		0			0
$723$	−0.157252	−	3.81483i	−0.00584826	−	0.141875i
$724$		0			0
$725$		−	5.48271i		−	0.203623i
$726$		0			0
$727$			37.8275i			1.40294i	0.712698	+	0.701471i	$0.247473\pi$
$727$			37.8275i			1.40294i	−0.712698	+	0.701471i	$0.752527\pi$
$728$		0			0
$729$	26.1793	−	6.60630i	0.969604	−	0.244678i
$730$		0			0
$731$			6.65854i			0.246275i
$732$		0			0
$733$			17.2221i			0.636114i	0.948072	+	0.318057i	$0.103030\pi$
$733$			17.2221i			0.636114i	−0.948072	+	0.318057i	$0.896970\pi$
$734$		0			0
$735$	−3.08782	+	0.127284i	−0.113896	+	0.00469492i
$736$		0			0
$737$		−	0.0882978i		−	0.00325249i
$738$		0			0
$739$	46.3942			1.70664			0.853319	−	0.521390i	$-0.174586\pi$
$739$	46.3942			1.70664			0.853319	+	0.521390i	$0.174586\pi$
$740$		0			0
$741$	0.618248	+	14.9983i	0.0227119	+	0.550976i
$742$		0			0
$743$	6.22529			0.228384			0.114192	−	0.993459i	$-0.463572\pi$
$743$	6.22529			0.228384			0.114192	+	0.993459i	$0.463572\pi$
$744$		0			0
$745$	3.29540			0.120734
$746$		0			0
$747$	13.8757	−	1.14589i	0.507685	−	0.0419260i
$748$		0			0
$749$		−	31.7131i		−	1.15877i
$750$		0			0
$751$		−	25.5297i		−	0.931590i	−0.884893	−	0.465795i	$-0.845768\pi$
$751$		−	25.5297i		−	0.931590i	0.884893	−	0.465795i	$-0.154232\pi$
$752$		0			0
$753$	52.1077	−	2.14794i	1.89891	−	0.0782753i
$754$		0			0
$755$	6.71760			0.244478
$756$		0			0
$757$			27.8914i			1.01373i	0.862025	+	0.506865i	$0.169196\pi$
$757$			27.8914i			1.01373i	−0.862025	+	0.506865i	$0.830804\pi$
$758$		0			0
$759$	−7.93177	+	7.78817i	−0.287905	+	0.282693i
$760$		0			0
$761$		−	48.7907i		−	1.76866i	−0.466862	−	0.884330i	$-0.654616\pi$
$761$		−	48.7907i		−	1.76866i	0.466862	−	0.884330i	$-0.345384\pi$
$762$		0			0
$763$	24.6150			0.891123
$764$		0			0
$765$	−2.63043	+	0.217228i	−0.0951035	+	0.00785390i
$766$		0			0
$767$		−	0.104621i		−	0.00377766i
$768$		0			0
$769$			32.1078i			1.15784i	0.815386	+	0.578918i	$0.196525\pi$
$769$			32.1078i			1.15784i	−0.815386	+	0.578918i	$0.803475\pi$
$770$		0			0
$771$	−1.81741	−	44.0892i	−0.0654523	−	1.58783i
$772$		0			0
$773$	−1.48437			−0.0533891			−0.0266945	−	0.999644i	$-0.508498\pi$
$773$	−1.48437			−0.0533891			−0.0266945	+	0.999644i	$0.508498\pi$
$774$		0			0
$775$	2.33732			0.0839591
$776$		0			0
$777$	16.6916	−	0.688048i	0.598808	−	0.0246836i
$778$		0			0
$779$	−70.9338			−2.54147
$780$		0			0
$781$			15.5291i			0.555675i
$782$		0			0
$783$	−3.51210	−	28.2717i	−0.125512	−	1.01035i
$784$		0			0
$785$			6.55688i			0.234025i
$786$		0			0
$787$			33.4976i			1.19406i	0.802219	+	0.597030i	$0.203653\pi$
$787$			33.4976i			1.19406i	−0.802219	+	0.597030i	$0.796347\pi$
$788$		0			0
$789$	−13.2252	+	0.545160i	−0.470831	+	0.0194082i
$790$		0			0
$791$			31.7472i			1.12880i
$792$		0			0
$793$			4.74831i			0.168618i
$794$		0			0
$795$	−13.5899	+	0.560190i	−0.481983	+	0.0198679i
$796$		0			0
$797$	−25.1037			−0.889219			−0.444610	−	0.895724i	$-0.646658\pi$
$797$	−25.1037			−0.889219			−0.444610	+	0.895724i	$0.646658\pi$
$798$		0			0
$799$		−	6.72250i		−	0.237825i
$800$		0			0
$801$	2.60780	−	0.215359i	0.0921421	−	0.00760934i
$802$		0			0
$803$	4.74127			0.167316
$804$		0			0
$805$	7.34528	+	8.12458i	0.258887	+	0.286354i
$806$		0			0
$807$	1.68490	+	40.8745i	0.0593112	+	1.43885i
$808$		0			0
$809$			20.1813i			0.709537i	0.934954	+	0.354769i	$0.115440\pi$
$809$			20.1813i			0.709537i	−0.934954	+	0.354769i	$0.884560\pi$
$810$		0			0
$811$	38.2871			1.34444			0.672222	−	0.740350i	$-0.265340\pi$
$811$	38.2871			1.34444			0.672222	+	0.740350i	$0.265340\pi$
$812$		0			0
$813$	−8.70194	+	0.358704i	−0.305190	+	0.0125803i
$814$		0			0
$815$	−8.68400			−0.304187
$816$		0			0
$817$	−46.3642			−1.62208
$818$		0			0
$819$	0.797731	+	9.65978i	0.0278750	+	0.337540i
$820$		0			0
$821$		−	2.31391i		−	0.0807561i	−0.999184	−	0.0403780i	$-0.987144\pi$
$821$		−	2.31391i		−	0.0807561i	0.999184	−	0.0403780i	$-0.0128562\pi$
$822$		0			0
$823$	−18.3537			−0.639771			−0.319885	−	0.947456i	$-0.603644\pi$
$823$	−18.3537			−0.639771			−0.319885	+	0.947456i	$0.603644\pi$
$824$		0			0
$825$	2.31591	−	0.0954645i	0.0806296	−	0.00332365i
$826$		0			0
$827$	27.0314			0.939973			0.469987	−	0.882673i	$-0.344259\pi$
$827$	27.0314			0.939973			0.469987	+	0.882673i	$0.344259\pi$
$828$		0			0
$829$	42.2676			1.46801			0.734007	−	0.679142i	$-0.237648\pi$
$829$	42.2676			1.46801			0.734007	+	0.679142i	$0.237648\pi$
$830$		0			0
$831$	−47.5163	+	1.95868i	−1.64832	+	0.0679458i
$832$		0			0
$833$	1.56979			0.0543900
$834$		0			0
$835$			17.4517i			0.603943i
$836$		0			0
$837$	12.0524	−	1.49724i	0.416593	−	0.0517520i
$838$		0			0
$839$	−35.5407			−1.22700			−0.613501	−	0.789694i	$-0.710239\pi$
$839$	−35.5407			−1.22700			−0.613501	+	0.789694i	$0.710239\pi$
$840$		0			0
$841$	−1.06010			−0.0365550
$842$		0			0
$843$	−5.02989	+	0.207338i	−0.173239	+	0.00714111i
$844$		0			0
$845$	10.9986			0.378364
$846$		0			0
$847$		−	21.0318i		−	0.722663i
$848$		0			0
$849$	−0.171796	−	4.16766i	−0.00589601	−	0.143034i
$850$		0			0
$851$	13.5831	+	15.0242i	0.465623	+	0.515023i
$852$		0			0
$853$	9.45457			0.323718			0.161859	−	0.986814i	$-0.448251\pi$
$853$	9.45457			0.323718			0.161859	+	0.986814i	$0.448251\pi$
$854$		0			0
$855$	−1.51259	−	18.3160i	−0.0517294	−	0.626395i
$856$		0			0
$857$			19.1879i			0.655445i	0.944774	+	0.327722i	$0.106281\pi$
$857$			19.1879i			0.655445i	−0.944774	+	0.327722i	$0.893719\pi$
$858$		0			0
$859$	−44.6165			−1.52230			−0.761148	−	0.648578i	$-0.775364\pi$
$859$	−44.6165			−1.52230			−0.761148	+	0.648578i	$0.775364\pi$
$860$		0			0
$861$	−45.7633	+	1.88642i	−1.55961	+	0.0642889i
$862$		0			0
$863$		−	33.8113i		−	1.15095i	−0.817819	−	0.575476i	$-0.804817\pi$
$863$		−	33.8113i		−	1.15095i	0.817819	−	0.575476i	$-0.195183\pi$
$864$		0			0
$865$		−	14.2296i		−	0.483822i
$866$		0			0
$867$	−28.0803	+	1.15750i	−0.953658	+	0.0393109i
$868$		0			0
$869$		−	5.36269i		−	0.181917i
$870$		0			0
$871$		−	0.0933436i		−	0.00316283i
$872$		0			0
$873$	−0.707976	−	8.57294i	−0.0239614	−	0.290150i
$874$		0			0
$875$		−	2.28380i		−	0.0772065i
$876$		0			0
$877$	19.7225			0.665983			0.332991	−	0.942930i	$-0.391942\pi$
$877$	19.7225			0.665983			0.332991	+	0.942930i	$0.391942\pi$
$878$		0			0
$879$	22.9093	−	0.944347i	0.772710	−	0.0318520i
$880$		0			0
$881$	0.901957			0.0303877			0.0151939	−	0.999885i	$-0.495163\pi$
$881$	0.901957			0.0303877			0.0151939	+	0.999885i	$0.495163\pi$
$882$		0			0
$883$	18.4568			0.621122			0.310561	−	0.950554i	$-0.399483\pi$
$883$	18.4568			0.621122			0.310561	+	0.950554i	$0.399483\pi$
$884$		0			0
$885$	0.00527556	+	0.127982i	0.000177336	+	0.00430206i
$886$		0			0
$887$		−	14.9686i		−	0.502596i	−0.967910	−	0.251298i	$-0.919143\pi$
$887$		−	14.9686i		−	0.502596i	0.967910	−	0.251298i	$-0.0808574\pi$
$888$		0			0
$889$		−	9.37744i		−	0.314509i
$890$		0			0
$891$	11.8809	−	1.97578i	0.398024	−	0.0661912i
$892$		0			0
$893$	46.8096			1.56642
$894$		0			0
$895$			6.63153i			0.221667i
$896$		0			0
$897$	−8.38504	+	8.23323i	−0.279968	+	0.274899i
$898$		0			0
$899$		−	12.8149i		−	0.427399i
$900$		0			0
$901$	6.90884			0.230167
$902$		0			0
$903$	−29.9121	+	1.23301i	−0.995412	+	0.0410321i
$904$		0			0
$905$			21.7518i			0.723055i
$906$		0			0
$907$		−	44.1167i		−	1.46487i	−0.680836	−	0.732436i	$-0.738383\pi$
$907$		−	44.1167i		−	1.46487i	0.680836	−	0.732436i	$-0.261617\pi$
$908$		0			0
$909$	−2.12984	−	25.7904i	−0.0706424	−	0.855414i
$910$		0			0
$911$	7.94859			0.263349			0.131674	−	0.991293i	$-0.457965\pi$
$911$	7.94859			0.263349			0.131674	+	0.991293i	$0.457965\pi$
$912$		0			0
$913$	6.21067			0.205543
$914$		0			0
$915$	−0.239435	−	5.80854i	−0.00791548	−	0.192024i
$916$		0			0
$917$	−6.80157			−0.224608
$918$		0			0
$919$			17.4405i			0.575309i	0.957734	+	0.287654i	$0.0928754\pi$
$919$			17.4405i			0.575309i	−0.957734	+	0.287654i	$0.907125\pi$
$920$		0			0
$921$	19.4883	−	0.803332i	0.642163	−	0.0264707i
$922$		0			0
$923$			16.4165i			0.540357i
$924$		0			0
$925$		−	4.22327i		−	0.138860i
$926$		0			0
$927$	−3.64505	−	44.1382i	−0.119719	−	1.44969i
$928$		0			0
$929$		−	25.7482i		−	0.844772i	−0.906416	−	0.422386i	$-0.861193\pi$
$929$		−	25.7482i		−	0.844772i	0.906416	−	0.422386i	$-0.138807\pi$
$930$		0			0
$931$			10.9306i			0.358237i
$932$		0			0
$933$	1.68280	+	40.8238i	0.0550925	+	1.33651i
$934$		0			0
$935$	−1.17737			−0.0385040
$936$		0			0
$937$			9.17729i			0.299809i	0.988700	+	0.149905i	$0.0478966\pi$
$937$			9.17729i			0.299809i	−0.988700	+	0.149905i	$0.952103\pi$
$938$		0			0
$939$	−1.54746	−	37.5405i	−0.0504996	−	1.22509i
$940$		0			0
$941$	54.3625			1.77217			0.886083	−	0.463527i	$-0.153416\pi$
$941$	54.3625			1.77217			0.886083	+	0.463527i	$0.153416\pi$
$942$		0			0
$943$	−37.2407	−	41.1918i	−1.21272	−	1.34139i
$944$		0			0
$945$	−1.46295	−	11.7764i	−0.0475897	−	0.383088i
$946$		0			0
$947$			46.4660i			1.50994i	0.655759	+	0.754971i	$0.272349\pi$
$947$			46.4660i			1.50994i	−0.655759	+	0.754971i	$0.727651\pi$
$948$		0			0
$949$	5.01222			0.162703
$950$		0			0
$951$	0.0839564	+	2.03673i	0.00272247	+	0.0660455i
$952$		0			0
$953$	−37.3108			−1.20862			−0.604308	−	0.796750i	$-0.706551\pi$
$953$	−37.3108			−1.20862			−0.604308	+	0.796750i	$0.706551\pi$
$954$		0			0
$955$	−15.2529			−0.493572
$956$		0			0
$957$	−0.523404	−	12.6975i	−0.0169193	−	0.410450i
$958$		0			0
$959$		−	40.2106i		−	1.29847i
$960$		0			0
$961$	−25.5369			−0.823772
$962$		0			0
$963$	−41.5171	+	3.42859i	−1.33787	+	0.110485i
$964$		0			0
$965$	14.2057			0.457296
$966$		0			0
$967$	−15.8381			−0.509317			−0.254659	−	0.967031i	$-0.581963\pi$
$967$	−15.8381			−0.509317			−0.254659	+	0.967031i	$0.581963\pi$
$968$		0			0
$969$	0.384486	+	9.32738i	0.0123515	+	0.299639i
$970$		0			0
$971$	34.2142			1.09799			0.548993	−	0.835827i	$-0.315012\pi$
$971$	34.2142			1.09799			0.548993	+	0.835827i	$0.315012\pi$
$972$		0			0
$973$		−	4.38033i		−	0.140427i
$974$		0			0
$975$	2.44825	−	0.100920i	0.0784068	−	0.00323202i
$976$		0			0
$977$	−20.5049			−0.656010			−0.328005	−	0.944676i	$-0.606376\pi$
$977$	−20.5049			−0.656010			−0.328005	+	0.944676i	$0.606376\pi$
$978$		0			0
$979$	1.16724			0.0373050
$980$		0			0
$981$	−2.66119	−	32.2246i	−0.0849654	−	1.02885i
$982$		0			0
$983$	−8.56736			−0.273256			−0.136628	−	0.990622i	$-0.543627\pi$
$983$	−8.56736			−0.273256			−0.136628	+	0.990622i	$0.543627\pi$
$984$		0			0
$985$			4.59137i			0.146293i
$986$		0			0
$987$	30.1994	−	1.24486i	0.961259	−	0.0396242i
$988$		0			0
$989$	−24.3415	−	26.9240i	−0.774015	−	0.856135i
$990$		0			0
$991$	−8.43728			−0.268019			−0.134010	−	0.990980i	$-0.542785\pi$
$991$	−8.43728			−0.268019			−0.134010	+	0.990980i	$0.542785\pi$
$992$		0			0
$993$	3.60581	−	0.148636i	0.114427	−	0.00471681i
$994$		0			0
$995$		−	1.26901i		−	0.0402303i
$996$		0			0
$997$	31.2340			0.989192			0.494596	−	0.869123i	$-0.335316\pi$
$997$	31.2340			0.989192			0.494596	+	0.869123i	$0.335316\pi$
$998$		0			0
$999$	−2.70533	−	21.7774i	−0.0855929	−	0.689005i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.a.1241.15	✓	16
3.2	odd	2		1380.2.i.b.1241.16	yes	16
23.22	odd	2		1380.2.i.b.1241.15	yes	16
69.68	even	2	inner	1380.2.i.a.1241.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.15	✓	16	1.1	even	1	trivial
1380.2.i.a.1241.16	yes	16	69.68	even	2	inner
1380.2.i.b.1241.15	yes	16	23.22	odd	2
1380.2.i.b.1241.16	yes	16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73058 - 0.0713366i) q^{3} -1.00000 q^{5} +2.28380i q^{7} +(2.98982 - 0.246908i) q^{9} +O(q^{10})\)	\(q+(1.73058 - 0.0713366i) q^{3} -1.00000 q^{5} +2.28380i q^{7} +(2.98982 - 0.246908i) q^{9} +1.33823 q^{11} +1.41470 q^{13} +(-1.73058 + 0.0713366i) q^{15} +0.879796 q^{17} +6.12613i q^{19} +(0.162918 + 3.95230i) q^{21} +(-3.55748 + 3.21626i) q^{23} +1.00000 q^{25} +(5.15652 - 0.640577i) q^{27} -5.48271i q^{29} +2.33732 q^{31} +(2.31591 - 0.0954645i) q^{33} -2.28380i q^{35} -4.22327i q^{37} +(2.44825 - 0.100920i) q^{39} +11.5789i q^{41} +7.56828i q^{43} +(-2.98982 + 0.246908i) q^{45} -7.64098i q^{47} +1.78427 q^{49} +(1.52256 - 0.0627616i) q^{51} +7.85277 q^{53} -1.33823 q^{55} +(0.437017 + 10.6018i) q^{57} -0.0739530i q^{59} +3.35641i q^{61} +(0.563887 + 6.82815i) q^{63} -1.41470 q^{65} -0.0659812i q^{67} +(-5.92708 + 5.81977i) q^{69} +11.6042i q^{71} +3.54295 q^{73} +(1.73058 - 0.0713366i) q^{75} +3.05624i q^{77} -4.00731i q^{79} +(8.87807 - 1.47642i) q^{81} +4.64098 q^{83} -0.879796 q^{85} +(-0.391118 - 9.48827i) q^{87} +0.872226 q^{89} +3.23089i q^{91} +(4.04493 - 0.166737i) q^{93} -6.12613i q^{95} -2.86737i q^{97} +(4.00106 - 0.330418i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 + 2 * q^9	\( 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 + 2 * q^9 - 4 * q^21 + 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 + 2 * q^33 - 14 * q^39 - 2 * q^45 + 8 * q^49 - 2 * q^51 + 4 * q^53 - 2 * q^57 + 30 * q^63 - 4 * q^73 + 10 * q^81 - 4 * q^83 - 10 * q^87 - 28 * q^89 - 10 * q^93 - 26 * q^99

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