LMFDB - Embedded newform 2106.2.b.d.649.8

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			649.8
Root			$1.15887i$ of defining polynomial
Character	$\chi$	$=$	2106.649
Dual form			2106.2.b.d.649.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -3.83645i q^{5} +3.36944i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -3.83645i q^{5} +3.36944i q^{7} -1.00000i q^{8} +3.83645 q^{10} -1.99180i q^{11} +(-2.51098 + 2.58747i) q^{13} -3.36944 q^{14} +1.00000 q^{16} -2.60129 q^{17} +1.99180i q^{19} +3.83645i q^{20} +1.99180 q^{22} +4.27239 q^{23} -9.71833 q^{25} +(-2.58747 - 2.51098i) q^{26} -3.36944i q^{28} -8.74939 q^{29} +5.28060i q^{31} +1.00000i q^{32} -2.60129i q^{34} +12.9267 q^{35} -5.08689i q^{37} -1.99180 q^{38} -3.83645 q^{40} +8.24323i q^{41} +10.2637 q^{43} +1.99180i q^{44} +4.27239i q^{46} +4.88188i q^{47} -4.35315 q^{49} -9.71833i q^{50} +(2.51098 - 2.58747i) q^{52} +9.16444 q^{53} -7.64143 q^{55} +3.36944 q^{56} -8.74939i q^{58} +7.31305i q^{59} -11.2734 q^{61} -5.28060 q^{62} -1.00000 q^{64} +(9.92669 + 9.63323i) q^{65} +5.66486i q^{67} +2.60129 q^{68} +12.9267i q^{70} +1.94505i q^{71} +8.41425i q^{73} +5.08689 q^{74} -1.99180i q^{76} +6.71125 q^{77} -5.95213 q^{79} -3.83645i q^{80} -8.24323 q^{82} -2.35164i q^{83} +9.97970i q^{85} +10.2637i q^{86} -1.99180 q^{88} +6.28290i q^{89} +(-8.71833 - 8.46059i) q^{91} -4.27239 q^{92} -4.88188 q^{94} +7.64143 q^{95} +10.3035i q^{97} -4.35315i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	3.83645i		−	1.71571i	−0.513890	−	0.857856i	$-0.671796\pi$
$5$		−	3.83645i		−	1.71571i	0.513890	−	0.857856i	$-0.328204\pi$
$6$		0			0
$7$			3.36944i			1.27353i	0.771058	+	0.636765i	$0.219728\pi$
$7$			3.36944i			1.27353i	−0.771058	+	0.636765i	$0.780272\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	3.83645			1.21319
$11$		−	1.99180i		−	0.600550i	−0.953853	−	0.300275i	$-0.902922\pi$
$11$		−	1.99180i		−	0.600550i	0.953853	−	0.300275i	$-0.0970784\pi$
$12$		0			0
$13$	−2.51098	+	2.58747i	−0.696419	+	0.717635i
$13$	−2.51098	+	2.58747i	−0.696419	+	0.717635i
$14$	−3.36944			−0.900522
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.60129			−0.630905			−0.315452	−	0.948941i	$-0.602156\pi$
$17$	−2.60129			−0.630905			−0.315452	+	0.948941i	$0.602156\pi$
$18$		0			0
$19$			1.99180i			0.456950i	0.973550	+	0.228475i	$0.0733739\pi$
$19$			1.99180i			0.456950i	−0.973550	+	0.228475i	$0.926626\pi$
$20$			3.83645i			0.857856i
$21$		0			0
$22$	1.99180			0.424653
$23$	4.27239			0.890856			0.445428	−	0.895318i	$-0.353052\pi$
$23$	4.27239			0.890856			0.445428	+	0.895318i	$0.353052\pi$
$24$		0			0
$25$	−9.71833			−1.94367
$26$	−2.58747	−	2.51098i	−0.507445	−	0.492443i
$27$		0			0
$28$		−	3.36944i		−	0.636765i
$29$	−8.74939			−1.62472			−0.812360	−	0.583156i	$-0.801818\pi$
$29$	−8.74939			−1.62472			−0.812360	+	0.583156i	$0.801818\pi$
$30$		0			0
$31$			5.28060i			0.948423i	0.880411	+	0.474212i	$0.157267\pi$
$31$			5.28060i			0.948423i	−0.880411	+	0.474212i	$0.842733\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.60129i		−	0.446117i
$35$	12.9267			2.18501
$36$		0			0
$37$		−	5.08689i		−	0.836280i	−0.908383	−	0.418140i	$-0.862682\pi$
$37$		−	5.08689i		−	0.836280i	0.908383	−	0.418140i	$-0.137318\pi$
$38$	−1.99180			−0.323112
$39$		0			0
$40$	−3.83645			−0.606596
$41$			8.24323i			1.28738i	0.765288	+	0.643688i	$0.222597\pi$
$41$			8.24323i			1.28738i	−0.765288	+	0.643688i	$0.777403\pi$
$42$		0			0
$43$	10.2637			1.56520			0.782598	−	0.622527i	$-0.213894\pi$
$43$	10.2637			1.56520			0.782598	+	0.622527i	$0.213894\pi$
$44$			1.99180i			0.300275i
$45$		0			0
$46$			4.27239i			0.629930i
$47$			4.88188i			0.712096i	0.934468	+	0.356048i	$0.115876\pi$
$47$			4.88188i			0.712096i	−0.934468	+	0.356048i	$0.884124\pi$
$48$		0			0
$49$	−4.35315			−0.621878
$50$		−	9.71833i		−	1.37438i
$51$		0			0
$52$	2.51098	−	2.58747i	0.348210	−	0.358817i
$53$	9.16444			1.25883			0.629416	−	0.777068i	$-0.283294\pi$
$53$	9.16444			1.25883			0.629416	+	0.777068i	$0.283294\pi$
$54$		0			0
$55$	−7.64143			−1.03037
$56$	3.36944			0.450261
$57$		0			0
$58$		−	8.74939i		−	1.14885i
$59$			7.31305i			0.952078i	0.879424	+	0.476039i	$0.157928\pi$
$59$			7.31305i			0.952078i	−0.879424	+	0.476039i	$0.842072\pi$
$60$		0			0
$61$	−11.2734			−1.44341			−0.721705	−	0.692201i	$-0.756641\pi$
$61$	−11.2734			−1.44341			−0.721705	+	0.692201i	$0.756641\pi$
$62$	−5.28060			−0.670636
$63$		0			0
$64$	−1.00000			−0.125000
$65$	9.92669	+	9.63323i	1.23125	+	1.19485i
$66$		0			0
$67$			5.66486i			0.692073i	0.938221	+	0.346037i	$0.112473\pi$
$67$			5.66486i			0.692073i	−0.938221	+	0.346037i	$0.887527\pi$
$68$	2.60129			0.315452
$69$		0			0
$70$			12.9267i			1.54504i
$71$			1.94505i			0.230835i	0.993317	+	0.115418i	$0.0368206\pi$
$71$			1.94505i			0.230835i	−0.993317	+	0.115418i	$0.963179\pi$
$72$		0			0
$73$			8.41425i			0.984814i	0.870365	+	0.492407i	$0.163883\pi$
$73$			8.41425i			0.984814i	−0.870365	+	0.492407i	$0.836117\pi$
$74$	5.08689			0.591339
$75$		0			0
$76$		−	1.99180i		−	0.228475i
$77$	6.71125			0.764818
$78$		0			0
$79$	−5.95213			−0.669667			−0.334834	−	0.942277i	$-0.608680\pi$
$79$	−5.95213			−0.669667			−0.334834	+	0.942277i	$0.608680\pi$
$80$		−	3.83645i		−	0.428928i
$81$		0			0
$82$	−8.24323			−0.910313
$83$		−	2.35164i		−	0.258126i	−0.991636	−	0.129063i	$-0.958803\pi$
$83$		−	2.35164i		−	0.258126i	0.991636	−	0.129063i	$-0.0411970\pi$
$84$		0			0
$85$			9.97970i			1.08245i
$86$			10.2637i			1.10676i
$87$		0			0
$88$	−1.99180			−0.212326
$89$			6.28290i			0.665986i	0.942929	+	0.332993i	$0.108059\pi$
$89$			6.28290i			0.665986i	−0.942929	+	0.332993i	$0.891941\pi$
$90$		0			0
$91$	−8.71833	−	8.46059i	−0.913929	−	0.886911i
$92$	−4.27239			−0.445428
$93$		0			0
$94$	−4.88188			−0.503528
$95$	7.64143			0.783994
$96$		0			0
$97$			10.3035i			1.04617i	0.852282	+	0.523083i	$0.175218\pi$
$97$			10.3035i			1.04617i	−0.852282	+	0.523083i	$0.824782\pi$
$98$		−	4.35315i		−	0.439734i
$99$		0			0
$100$	9.71833			0.971833
$101$	2.55921			0.254650			0.127325	−	0.991861i	$-0.459361\pi$
$101$	2.55921			0.254650			0.127325	+	0.991861i	$0.459361\pi$
$102$		0			0
$103$	−5.64143			−0.555867			−0.277933	−	0.960600i	$-0.589649\pi$
$103$	−5.64143			−0.555867			−0.277933	+	0.960600i	$0.589649\pi$
$104$	2.58747	+	2.51098i	0.253722	+	0.246221i
$105$		0			0
$106$			9.16444i			0.890129i
$107$	−18.3051			−1.76962			−0.884812	−	0.465948i	$-0.845713\pi$
$107$	−18.3051			−1.76962			−0.884812	+	0.465948i	$0.845713\pi$
$108$		0			0
$109$			6.74904i			0.646441i	0.946324	+	0.323221i	$0.104766\pi$
$109$			6.74904i			0.646441i	−0.946324	+	0.323221i	$0.895234\pi$
$110$		−	7.64143i		−	0.728582i
$111$		0			0
$112$			3.36944i			0.318382i
$113$	5.95051			0.559777			0.279889	−	0.960032i	$-0.409702\pi$
$113$	5.95051			0.559777			0.279889	+	0.960032i	$0.409702\pi$
$114$		0			0
$115$		−	16.3908i		−	1.52845i
$116$	8.74939			0.812360
$117$		0			0
$118$	−7.31305			−0.673221
$119$		−	8.76489i		−	0.803476i
$120$		0			0
$121$	7.03274			0.639340
$122$		−	11.2734i		−	1.02064i
$123$		0			0
$124$		−	5.28060i		−	0.474212i
$125$			18.1016i			1.61906i
$126$		0			0
$127$	−5.93126			−0.526314			−0.263157	−	0.964753i	$-0.584764\pi$
$127$	−5.93126			−0.526314			−0.263157	+	0.964753i	$0.584764\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−9.63323	+	9.92669i	−0.844890	+	0.870628i
$131$	−13.7649			−1.20265			−0.601324	−	0.799006i	$-0.705360\pi$
$131$	−13.7649			−1.20265			−0.601324	+	0.799006i	$0.705360\pi$
$132$		0			0
$133$	−6.71125			−0.581939
$134$	−5.66486			−0.489370
$135$		0			0
$136$			2.60129i			0.223059i
$137$		−	3.81984i		−	0.326351i	−0.986597	−	0.163175i	$-0.947826\pi$
$137$		−	3.81984i		−	0.326351i	0.986597	−	0.163175i	$-0.0521736\pi$
$138$		0			0
$139$	13.6097			1.15436			0.577182	−	0.816616i	$-0.304152\pi$
$139$	13.6097			1.15436			0.577182	+	0.816616i	$0.304152\pi$
$140$	−12.9267			−1.09250
$141$		0			0
$142$	−1.94505			−0.163225
$143$	5.15372	+	5.00136i	0.430976	+	0.418235i
$144$		0			0
$145$			33.5666i			2.78755i
$146$	−8.41425			−0.696368
$147$		0			0
$148$			5.08689i			0.418140i
$149$			7.79096i			0.638260i	0.947711	+	0.319130i	$0.103391\pi$
$149$			7.79096i			0.638260i	−0.947711	+	0.319130i	$0.896609\pi$
$150$		0			0
$151$			21.3658i			1.73873i	0.494174	+	0.869363i	$0.335471\pi$
$151$			21.3658i			1.73873i	−0.494174	+	0.869363i	$0.664529\pi$
$152$	1.99180			0.161556
$153$		0			0
$154$			6.71125i			0.540808i
$155$	20.2587			1.62722
$156$		0			0
$157$	−4.25143			−0.339301			−0.169651	−	0.985504i	$-0.554264\pi$
$157$	−4.25143			−0.339301			−0.169651	+	0.985504i	$0.554264\pi$
$158$		−	5.95213i		−	0.473526i
$159$		0			0
$160$	3.83645			0.303298
$161$			14.3956i			1.13453i
$162$		0			0
$163$		−	11.0533i		−	0.865764i	−0.901451	−	0.432882i	$-0.857497\pi$
$163$		−	11.0533i		−	0.865764i	0.901451	−	0.432882i	$-0.142503\pi$
$164$		−	8.24323i		−	0.643688i
$165$		0			0
$166$	2.35164			0.182523
$167$			9.40375i			0.727684i	0.931461	+	0.363842i	$0.118535\pi$
$167$			9.40375i			0.727684i	−0.931461	+	0.363842i	$0.881465\pi$
$168$		0			0
$169$	−0.389998	−	12.9941i	−0.0299998	−	0.999550i
$170$	−9.97970			−0.765408
$171$		0			0
$172$	−10.2637			−0.782598
$173$	−2.17769			−0.165567			−0.0827835	−	0.996568i	$-0.526381\pi$
$173$	−2.17769			−0.165567			−0.0827835	+	0.996568i	$0.526381\pi$
$174$		0			0
$175$		−	32.7454i		−	2.47532i
$176$		−	1.99180i		−	0.150137i
$177$		0			0
$178$	−6.28290			−0.470923
$179$	−7.91152			−0.591334			−0.295667	−	0.955291i	$-0.595542\pi$
$179$	−7.91152			−0.591334			−0.295667	+	0.955291i	$0.595542\pi$
$180$		0			0
$181$	−1.02195			−0.0759611			−0.0379806	−	0.999278i	$-0.512092\pi$
$181$	−1.02195			−0.0759611			−0.0379806	+	0.999278i	$0.512092\pi$
$182$	8.46059	−	8.71833i	0.627141	−	0.646246i
$183$		0			0
$184$		−	4.27239i		−	0.314965i
$185$	−19.5156			−1.43481
$186$		0			0
$187$			5.18124i			0.378890i
$188$		−	4.88188i		−	0.356048i
$189$		0			0
$190$			7.64143i			0.554368i
$191$	−4.06108			−0.293849			−0.146925	−	0.989148i	$-0.546938\pi$
$191$	−4.06108			−0.293849			−0.146925	+	0.989148i	$0.546938\pi$
$192$		0			0
$193$			16.6417i			1.19790i	0.800788	+	0.598948i	$0.204414\pi$
$193$			16.6417i			1.19790i	−0.800788	+	0.598948i	$0.795586\pi$
$194$	−10.3035			−0.739751
$195$		0			0
$196$	4.35315			0.310939
$197$		−	20.8461i		−	1.48522i	−0.669723	−	0.742611i	$-0.733587\pi$
$197$		−	20.8461i		−	1.48522i	0.669723	−	0.742611i	$-0.266413\pi$
$198$		0			0
$199$	15.4367			1.09428			0.547138	−	0.837042i	$-0.315717\pi$
$199$	15.4367			1.09428			0.547138	+	0.837042i	$0.315717\pi$
$200$			9.71833i			0.687190i
$201$		0			0
$202$			2.55921i			0.180065i
$203$		−	29.4806i		−	2.06913i
$204$		0			0
$205$	31.6247			2.20877
$206$		−	5.64143i		−	0.393057i
$207$		0			0
$208$	−2.51098	+	2.58747i	−0.174105	+	0.179409i
$209$	3.96726			0.274421
$210$		0			0
$211$	−3.73323			−0.257006			−0.128503	−	0.991709i	$-0.541017\pi$
$211$	−3.73323			−0.257006			−0.128503	+	0.991709i	$0.541017\pi$
$212$	−9.16444			−0.629416
$213$		0			0
$214$		−	18.3051i		−	1.25131i
$215$		−	39.3761i		−	2.68543i
$216$		0			0
$217$	−17.7927			−1.20785
$218$	−6.74904			−0.457103
$219$		0			0
$220$	7.64143			0.515185
$221$	6.53177	−	6.73075i	0.439374	−	0.452759i
$222$		0			0
$223$			2.75301i			0.184355i	0.995743	+	0.0921775i	$0.0293827\pi$
$223$			2.75301i			0.184355i	−0.995743	+	0.0921775i	$0.970617\pi$
$224$	−3.36944			−0.225130
$225$		0			0
$226$			5.95051i			0.395822i
$227$		−	3.82159i		−	0.253648i	−0.991925	−	0.126824i	$-0.959522\pi$
$227$		−	3.82159i		−	0.253648i	0.991925	−	0.126824i	$-0.0404784\pi$
$228$		0			0
$229$		−	10.9742i		−	0.725195i	−0.931946	−	0.362597i	$-0.881890\pi$
$229$		−	10.9742i		−	0.725195i	0.931946	−	0.362597i	$-0.118110\pi$
$230$	16.3908			1.08078
$231$		0			0
$232$			8.74939i			0.574426i
$233$	−2.26098			−0.148122			−0.0740610	−	0.997254i	$-0.523596\pi$
$233$	−2.26098			−0.148122			−0.0740610	+	0.997254i	$0.523596\pi$
$234$		0			0
$235$	18.7291			1.22175
$236$		−	7.31305i		−	0.476039i
$237$		0			0
$238$	8.76489			0.568143
$239$		−	6.30635i		−	0.407924i	−0.978979	−	0.203962i	$-0.934618\pi$
$239$		−	6.30635i		−	0.407924i	0.978979	−	0.203962i	$-0.0653819\pi$
$240$		0			0
$241$		−	19.8311i		−	1.27744i	−0.769441	−	0.638718i	$-0.779465\pi$
$241$		−	19.8311i		−	1.27744i	0.769441	−	0.638718i	$-0.220535\pi$
$242$			7.03274i			0.452082i
$243$		0			0
$244$	11.2734			0.721705
$245$			16.7006i			1.06696i
$246$		0			0
$247$	−5.15372	−	5.00136i	−0.327923	−	0.318229i
$248$	5.28060			0.335318
$249$		0			0
$250$	−18.1016			−1.14485
$251$	−0.440795			−0.0278227			−0.0139114	−	0.999903i	$-0.504428\pi$
$251$	−0.440795			−0.0278227			−0.0139114	+	0.999903i	$0.504428\pi$
$252$		0			0
$253$		−	8.50975i		−	0.535003i
$254$		−	5.93126i		−	0.372160i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−13.5756			−0.846824			−0.423412	−	0.905937i	$-0.639168\pi$
$257$	−13.5756			−0.846824			−0.423412	+	0.905937i	$0.639168\pi$
$258$		0			0
$259$	17.1400			1.06503
$260$	−9.92669	−	9.63323i	−0.615627	−	0.597427i
$261$		0			0
$262$		−	13.7649i		−	0.850400i
$263$	−22.3784			−1.37991			−0.689956	−	0.723851i	$-0.742370\pi$
$263$	−22.3784			−1.37991			−0.689956	+	0.723851i	$0.742370\pi$
$264$		0			0
$265$		−	35.1589i		−	2.15979i
$266$		−	6.71125i		−	0.411493i
$267$		0			0
$268$		−	5.66486i		−	0.346037i
$269$	30.4520			1.85669			0.928345	−	0.371719i	$-0.121232\pi$
$269$	30.4520			1.85669			0.928345	+	0.371719i	$0.121232\pi$
$270$		0			0
$271$		−	2.03437i		−	0.123579i	−0.998089	−	0.0617897i	$-0.980319\pi$
$271$		−	2.03437i		−	0.123579i	0.998089	−	0.0617897i	$-0.0196808\pi$
$272$	−2.60129			−0.157726
$273$		0			0
$274$	3.81984			0.230765
$275$			19.3570i			1.16727i
$276$		0			0
$277$	−28.9135			−1.73724			−0.868621	−	0.495477i	$-0.834993\pi$
$277$	−28.9135			−1.73724			−0.868621	+	0.495477i	$0.834993\pi$
$278$			13.6097i			0.816258i
$279$		0			0
$280$		−	12.9267i		−	0.772518i
$281$		−	20.5110i		−	1.22358i	−0.791020	−	0.611791i	$-0.790450\pi$
$281$		−	20.5110i		−	1.22358i	0.791020	−	0.611791i	$-0.209550\pi$
$282$		0			0
$283$	−1.63998			−0.0974868			−0.0487434	−	0.998811i	$-0.515522\pi$
$283$	−1.63998			−0.0974868			−0.0487434	+	0.998811i	$0.515522\pi$
$284$		−	1.94505i		−	0.115418i
$285$		0			0
$286$	−5.00136	+	5.15372i	−0.295737	+	0.304746i
$287$	−27.7751			−1.63951
$288$		0			0
$289$	−10.2333			−0.601959
$290$	−33.5666			−1.97110
$291$		0			0
$292$		−	8.41425i		−	0.492407i
$293$			12.4220i			0.725703i	0.931847	+	0.362852i	$0.118197\pi$
$293$			12.4220i			0.725703i	−0.931847	+	0.362852i	$0.881803\pi$
$294$		0			0
$295$	28.0561			1.63349
$296$	−5.08689			−0.295669
$297$		0			0
$298$	−7.79096			−0.451318
$299$	−10.7279	+	11.0547i	−0.620409	+	0.639309i
$300$		0			0
$301$			34.5829i			1.99332i
$302$	−21.3658			−1.22947
$303$		0			0
$304$			1.99180i			0.114237i
$305$			43.2498i			2.47647i
$306$		0			0
$307$		−	25.8780i		−	1.47693i	−0.674289	−	0.738467i	$-0.735550\pi$
$307$		−	25.8780i		−	1.47693i	0.674289	−	0.738467i	$-0.264450\pi$
$308$	−6.71125			−0.382409
$309$		0			0
$310$			20.2587i			1.15062i
$311$	25.3822			1.43929			0.719645	−	0.694342i	$-0.244304\pi$
$311$	25.3822			1.43929			0.719645	+	0.694342i	$0.244304\pi$
$312$		0			0
$313$	−10.2000			−0.576535			−0.288268	−	0.957550i	$-0.593079\pi$
$313$	−10.2000			−0.576535			−0.288268	+	0.957550i	$0.593079\pi$
$314$		−	4.25143i		−	0.239922i
$315$		0			0
$316$	5.95213			0.334834
$317$		−	12.1952i		−	0.684950i	−0.939527	−	0.342475i	$-0.888735\pi$
$317$		−	12.1952i		−	0.684950i	0.939527	−	0.342475i	$-0.111265\pi$
$318$		0			0
$319$			17.4270i			0.975726i
$320$			3.83645i			0.214464i
$321$		0			0
$322$	−14.3956			−0.802235
$323$		−	5.18124i		−	0.288292i
$324$		0			0
$325$	24.4025	−	25.1459i	1.35361	−	1.39484i
$326$	11.0533			0.612187
$327$		0			0
$328$	8.24323			0.455156
$329$	−16.4492			−0.906875
$330$		0			0
$331$			22.4765i			1.23542i	0.786405	+	0.617711i	$0.211940\pi$
$331$			22.4765i			1.23542i	−0.786405	+	0.617711i	$0.788060\pi$
$332$			2.35164i			0.129063i
$333$		0			0
$334$	−9.40375			−0.514550
$335$	21.7329			1.18740
$336$		0			0
$337$	−30.7950			−1.67751			−0.838755	−	0.544510i	$-0.816716\pi$
$337$	−30.7950			−1.67751			−0.838755	+	0.544510i	$0.816716\pi$
$338$	12.9941	−	0.389998i	0.706789	−	0.0212131i
$339$		0			0
$340$		−	9.97970i		−	0.541225i
$341$	10.5179			0.569575
$342$		0			0
$343$			8.91842i			0.481550i
$344$		−	10.2637i		−	0.553381i
$345$		0			0
$346$		−	2.17769i		−	0.117074i
$347$	29.0471			1.55933			0.779664	−	0.626198i	$-0.215390\pi$
$347$	29.0471			1.55933			0.779664	+	0.626198i	$0.215390\pi$
$348$		0			0
$349$			12.4672i			0.667352i	0.942688	+	0.333676i	$0.108289\pi$
$349$			12.4672i			0.667352i	−0.942688	+	0.333676i	$0.891711\pi$
$350$	32.7454			1.75031
$351$		0			0
$352$	1.99180			0.106163
$353$			34.4293i			1.83249i	0.400621	+	0.916244i	$0.368795\pi$
$353$			34.4293i			1.83249i	−0.400621	+	0.916244i	$0.631205\pi$
$354$		0			0
$355$	7.46209			0.396047
$356$		−	6.28290i		−	0.332993i
$357$		0			0
$358$		−	7.91152i		−	0.418137i
$359$		−	34.2496i		−	1.80762i	−0.427931	−	0.903812i	$-0.640757\pi$
$359$		−	34.2496i		−	1.80762i	0.427931	−	0.903812i	$-0.359243\pi$
$360$		0			0
$361$	15.0327			0.791197
$362$		−	1.02195i		−	0.0537126i
$363$		0			0
$364$	8.71833	+	8.46059i	0.456965	+	0.443455i
$365$	32.2808			1.68966
$366$		0			0
$367$	−5.03611			−0.262883			−0.131441	−	0.991324i	$-0.541961\pi$
$367$	−5.03611			−0.262883			−0.131441	+	0.991324i	$0.541961\pi$
$368$	4.27239			0.222714
$369$		0			0
$370$		−	19.5156i		−	1.01457i
$371$			30.8790i			1.60316i
$372$		0			0
$373$	11.2141			0.580645			0.290322	−	0.956929i	$-0.406237\pi$
$373$	11.2141			0.580645			0.290322	+	0.956929i	$0.406237\pi$
$374$	−5.18124			−0.267916
$375$		0			0
$376$	4.88188			0.251764
$377$	21.9695	−	22.6388i	1.13149	−	1.16596i
$378$		0			0
$379$		−	5.12623i		−	0.263317i	−0.991295	−	0.131658i	$-0.957970\pi$
$379$		−	5.12623i		−	0.263317i	0.991295	−	0.131658i	$-0.0420302\pi$
$380$	−7.64143			−0.391997
$381$		0			0
$382$		−	4.06108i		−	0.207783i
$383$		−	18.9420i		−	0.967889i	−0.875099	−	0.483944i	$-0.839204\pi$
$383$		−	18.9420i		−	0.967889i	0.875099	−	0.483944i	$-0.160796\pi$
$384$		0			0
$385$		−	25.7474i		−	1.31221i
$386$	−16.6417			−0.847041
$387$		0			0
$388$		−	10.3035i		−	0.523083i
$389$	20.8397			1.05662			0.528308	−	0.849053i	$-0.322827\pi$
$389$	20.8397			1.05662			0.528308	+	0.849053i	$0.322827\pi$
$390$		0			0
$391$	−11.1137			−0.562045
$392$			4.35315i			0.219867i
$393$		0			0
$394$	20.8461			1.05021
$395$			22.8350i			1.14896i
$396$		0			0
$397$		−	2.17152i		−	0.108985i	−0.998514	−	0.0544927i	$-0.982646\pi$
$397$		−	2.17152i		−	0.108985i	0.998514	−	0.0544927i	$-0.0173541\pi$
$398$			15.4367i			0.773770i
$399$		0			0
$400$	−9.71833			−0.485917
$401$		−	10.8033i		−	0.539489i	−0.962932	−	0.269745i	$-0.913061\pi$
$401$		−	10.8033i		−	0.539489i	0.962932	−	0.269745i	$-0.0869392\pi$
$402$		0			0
$403$	−13.6634	−	13.2595i	−0.680622	−	0.660500i
$404$	−2.55921			−0.127325
$405$		0			0
$406$	29.4806			1.46310
$407$	−10.1321			−0.502228
$408$		0			0
$409$			1.55301i			0.0767912i	0.999263	+	0.0383956i	$0.0122247\pi$
$409$			1.55301i			0.0767912i	−0.999263	+	0.0383956i	$0.987775\pi$
$410$			31.6247i			1.56183i
$411$		0			0
$412$	5.64143			0.277933
$413$	−24.6409			−1.21250
$414$		0			0
$415$	−9.02195			−0.442870
$416$	−2.58747	−	2.51098i	−0.126861	−	0.123111i
$417$		0			0
$418$			3.96726i			0.194045i
$419$	−5.62905			−0.274997			−0.137499	−	0.990502i	$-0.543906\pi$
$419$	−5.62905			−0.274997			−0.137499	+	0.990502i	$0.543906\pi$
$420$		0			0
$421$			11.9941i			0.584556i	0.956333	+	0.292278i	$0.0944132\pi$
$421$			11.9941i			0.584556i	−0.956333	+	0.292278i	$0.905587\pi$
$422$		−	3.73323i		−	0.181731i
$423$		0			0
$424$		−	9.16444i		−	0.445064i
$425$	25.2802			1.22627
$426$		0			0
$427$		−	37.9850i		−	1.83822i
$428$	18.3051			0.884812
$429$		0			0
$430$	39.3761			1.89888
$431$		−	0.0127142i		−	0.000612420i	−1.00000		0.000306210i	$-0.999903\pi$
$431$		−	0.0127142i		−	0.000612420i	1.00000		0.000306210i	$-9.74697e-5\pi$
$432$		0			0
$433$	30.8717			1.48360			0.741800	−	0.670621i	$-0.233972\pi$
$433$	30.8717			1.48360			0.741800	+	0.670621i	$0.233972\pi$
$434$		−	17.7927i		−	0.854075i
$435$		0			0
$436$		−	6.74904i		−	0.323221i
$437$			8.50975i			0.407076i
$438$		0			0
$439$	16.3956			0.782519			0.391259	−	0.920280i	$-0.372040\pi$
$439$	16.3956			0.782519			0.391259	+	0.920280i	$0.372040\pi$
$440$			7.64143i			0.364291i
$441$		0			0
$442$	6.73075	+	6.53177i	0.320149	+	0.310685i
$443$	4.04515			0.192191			0.0960955	−	0.995372i	$-0.469365\pi$
$443$	4.04515			0.192191			0.0960955	+	0.995372i	$0.469365\pi$
$444$		0			0
$445$	24.1040			1.14264
$446$	−2.75301			−0.130359
$447$		0			0
$448$		−	3.36944i		−	0.159191i
$449$			15.8943i			0.750099i	0.927005	+	0.375049i	$0.122374\pi$
$449$			15.8943i			0.750099i	−0.927005	+	0.375049i	$0.877626\pi$
$450$		0			0
$451$	16.4189			0.773134
$452$	−5.95051			−0.279889
$453$		0			0
$454$	3.82159			0.179356
$455$	−32.4586	+	33.4474i	−1.52168	+	1.56804i
$456$		0			0
$457$			29.1414i			1.36318i	0.731735	+	0.681589i	$0.238711\pi$
$457$			29.1414i			1.36318i	−0.731735	+	0.681589i	$0.761289\pi$
$458$	10.9742			0.512790
$459$		0			0
$460$			16.3908i			0.764226i
$461$			28.1429i			1.31075i	0.755306	+	0.655373i	$0.227488\pi$
$461$			28.1429i			1.31075i	−0.755306	+	0.655373i	$0.772512\pi$
$462$		0			0
$463$		−	31.8672i		−	1.48100i	−0.672059	−	0.740498i	$-0.734590\pi$
$463$		−	31.8672i		−	1.48100i	0.672059	−	0.740498i	$-0.265410\pi$
$464$	−8.74939			−0.406180
$465$		0			0
$466$		−	2.26098i		−	0.104738i
$467$	−1.94189			−0.0898600			−0.0449300	−	0.998990i	$-0.514306\pi$
$467$	−1.94189			−0.0898600			−0.0449300	+	0.998990i	$0.514306\pi$
$468$		0			0
$469$	−19.0874			−0.881376
$470$			18.7291i			0.863909i
$471$		0			0
$472$	7.31305			0.336610
$473$		−	20.4432i		−	0.939978i
$474$		0			0
$475$		−	19.3570i		−	0.888158i
$476$			8.76489i			0.401738i
$477$		0			0
$478$	6.30635			0.288446
$479$		−	10.9247i		−	0.499164i	−0.968354	−	0.249582i	$-0.919707\pi$
$479$		−	10.9247i		−	0.499164i	0.968354	−	0.249582i	$-0.0802932\pi$
$480$		0			0
$481$	13.1622	+	12.7731i	0.600143	+	0.582401i
$482$	19.8311			0.903284
$483$		0			0
$484$	−7.03274			−0.319670
$485$	39.5290			1.79492
$486$		0			0
$487$			38.3978i			1.73997i	0.493078	+	0.869985i	$0.335872\pi$
$487$			38.3978i			1.73997i	−0.493078	+	0.869985i	$0.664128\pi$
$488$			11.2734i			0.510322i
$489$		0			0
$490$	−16.7006			−0.754457
$491$	−10.5533			−0.476262			−0.238131	−	0.971233i	$-0.576535\pi$
$491$	−10.5533			−0.476262			−0.238131	+	0.971233i	$0.576535\pi$
$492$		0			0
$493$	22.7597			1.02504
$494$	5.00136	−	5.15372i	0.225022	−	0.231877i
$495$		0			0
$496$			5.28060i			0.237106i
$497$	−6.55374			−0.293975
$498$		0			0
$499$		−	0.992867i		−	0.0444468i	−0.999753	−	0.0222234i	$-0.992925\pi$
$499$		−	0.992867i		−	0.0444468i	0.999753	−	0.0222234i	$-0.00707452\pi$
$500$		−	18.1016i		−	0.809530i
$501$		0			0
$502$		−	0.440795i		−	0.0196736i
$503$	−9.23683			−0.411850			−0.205925	−	0.978568i	$-0.566020\pi$
$503$	−9.23683			−0.411850			−0.205925	+	0.978568i	$0.566020\pi$
$504$		0			0
$505$		−	9.81826i		−	0.436907i
$506$	8.50975			0.378304
$507$		0			0
$508$	5.93126			0.263157
$509$			9.64378i			0.427453i	0.976894	+	0.213727i	$0.0685601\pi$
$509$			9.64378i			0.427453i	−0.976894	+	0.213727i	$0.931440\pi$
$510$		0			0
$511$	−28.3513			−1.25419
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	13.5756i		−	0.598795i
$515$			21.6431i			0.953707i
$516$		0			0
$517$	9.72373			0.427649
$518$			17.1400i			0.753088i
$519$		0			0
$520$	9.63323	−	9.92669i	0.422445	−	0.435314i
$521$	18.9584			0.830581			0.415291	−	0.909689i	$-0.363680\pi$
$521$	18.9584			0.830581			0.415291	+	0.909689i	$0.363680\pi$
$522$		0			0
$523$	24.4694			1.06997			0.534986	−	0.844861i	$-0.320317\pi$
$523$	24.4694			1.06997			0.534986	+	0.844861i	$0.320317\pi$
$524$	13.7649			0.601324
$525$		0			0
$526$		−	22.3784i		−	0.975745i
$527$		−	13.7363i		−	0.598365i
$528$		0			0
$529$	−4.74664			−0.206376
$530$	35.1589			1.52720
$531$		0			0
$532$	6.71125			0.290970
$533$	−21.3291	−	20.6986i	−0.923866	−	0.896554i
$534$		0			0
$535$			70.2267i			3.03616i
$536$	5.66486			0.244685
$537$		0			0
$538$			30.4520i			1.31288i
$539$			8.67059i			0.373469i
$540$		0			0
$541$		−	6.77020i		−	0.291074i	−0.989353	−	0.145537i	$-0.953509\pi$
$541$		−	6.77020i		−	0.291074i	0.989353	−	0.145537i	$-0.0464909\pi$
$542$	2.03437			0.0873839
$543$		0			0
$544$		−	2.60129i		−	0.111529i
$545$	25.8924			1.10911
$546$		0			0
$547$	8.47377			0.362312			0.181156	−	0.983454i	$-0.442016\pi$
$547$	8.47377			0.362312			0.181156	+	0.983454i	$0.442016\pi$
$548$			3.81984i			0.163175i
$549$		0			0
$550$	−19.3570			−0.825383
$551$		−	17.4270i		−	0.742416i
$552$		0			0
$553$		−	20.0554i		−	0.852841i
$554$		−	28.9135i		−	1.22842i
$555$		0			0
$556$	−13.6097			−0.577182
$557$		−	4.77737i		−	0.202424i	−0.994865	−	0.101212i	$-0.967728\pi$
$557$		−	4.77737i		−	0.202424i	0.994865	−	0.101212i	$-0.0322720\pi$
$558$		0			0
$559$	−25.7719	+	26.5570i	−1.09003	+	1.12324i
$560$	12.9267			0.546252
$561$		0			0
$562$	20.5110			0.865202
$563$	−9.92833			−0.418429			−0.209215	−	0.977870i	$-0.567091\pi$
$563$	−9.92833			−0.418429			−0.209215	+	0.977870i	$0.567091\pi$
$564$		0			0
$565$		−	22.8288i		−	0.960416i
$566$		−	1.63998i		−	0.0689336i
$567$		0			0
$568$	1.94505			0.0816125
$569$	−29.8614			−1.25186			−0.625929	−	0.779880i	$-0.715280\pi$
$569$	−29.8614			−1.25186			−0.625929	+	0.779880i	$0.715280\pi$
$570$		0			0
$571$	30.1393			1.26129			0.630645	−	0.776072i	$-0.282790\pi$
$571$	30.1393			1.26129			0.630645	+	0.776072i	$0.282790\pi$
$572$	−5.15372	−	5.00136i	−0.215488	−	0.209117i
$573$		0			0
$574$		−	27.7751i		−	1.15931i
$575$	−41.5205			−1.73153
$576$		0			0
$577$			7.87256i			0.327739i	0.986482	+	0.163869i	$0.0523976\pi$
$577$			7.87256i			0.327739i	−0.986482	+	0.163869i	$0.947602\pi$
$578$		−	10.2333i		−	0.425649i
$579$		0			0
$580$		−	33.5666i		−	1.39378i
$581$	7.92372			0.328731
$582$		0			0
$583$		−	18.2537i		−	0.755991i
$584$	8.41425			0.348184
$585$		0			0
$586$	−12.4220			−0.513150
$587$		−	21.5517i		−	0.889534i	−0.895646	−	0.444767i	$-0.853287\pi$
$587$		−	21.5517i		−	0.889534i	0.895646	−	0.444767i	$-0.146713\pi$
$588$		0			0
$589$	−10.5179			−0.433382
$590$			28.0561i			1.15505i
$591$		0			0
$592$		−	5.08689i		−	0.209070i
$593$			8.13186i			0.333935i	0.985962	+	0.166968i	$0.0533975\pi$
$593$			8.13186i			0.333935i	−0.985962	+	0.166968i	$0.946602\pi$
$594$		0			0
$595$	−33.6260			−1.37853
$596$		−	7.79096i		−	0.319130i
$597$		0			0
$598$	−11.0547	−	10.7279i	−0.452060	−	0.438696i
$599$	−46.6038			−1.90418			−0.952090	−	0.305818i	$-0.901070\pi$
$599$	−46.6038			−1.90418			−0.952090	+	0.305818i	$0.901070\pi$
$600$		0			0
$601$	20.7652			0.847030			0.423515	−	0.905889i	$-0.360796\pi$
$601$	20.7652			0.847030			0.423515	+	0.905889i	$0.360796\pi$
$602$	−34.5829			−1.40949
$603$		0			0
$604$		−	21.3658i		−	0.869363i
$605$		−	26.9807i		−	1.09692i
$606$		0			0
$607$	−32.6353			−1.32463			−0.662313	−	0.749228i	$-0.730425\pi$
$607$	−32.6353			−1.32463			−0.662313	+	0.749228i	$0.730425\pi$
$608$	−1.99180			−0.0807781
$609$		0			0
$610$	−43.2498			−1.75113
$611$	−12.6317	−	12.2583i	−0.511025	−	0.495917i
$612$		0			0
$613$		−	12.8107i		−	0.517421i	−0.965955	−	0.258711i	$-0.916702\pi$
$613$		−	12.8107i		−	0.517421i	0.965955	−	0.258711i	$-0.0832976\pi$
$614$	25.8780			1.04435
$615$		0			0
$616$		−	6.71125i		−	0.270404i
$617$		−	21.4793i		−	0.864726i	−0.901700	−	0.432363i	$-0.857680\pi$
$617$		−	21.4793i		−	0.864726i	0.901700	−	0.432363i	$-0.142320\pi$
$618$		0			0
$619$		−	45.7658i		−	1.83948i	−0.392526	−	0.919741i	$-0.628399\pi$
$619$		−	45.7658i		−	1.83948i	0.392526	−	0.919741i	$-0.371601\pi$
$620$	−20.2587			−0.813610
$621$		0			0
$622$			25.3822i			1.01773i
$623$	−21.1699			−0.848153
$624$		0			0
$625$	20.8543			0.834172
$626$		−	10.2000i		−	0.407672i
$627$		0			0
$628$	4.25143			0.169651
$629$			13.2325i			0.527613i
$630$		0			0
$631$		−	10.7134i		−	0.426495i	−0.976998	−	0.213248i	$-0.931596\pi$
$631$		−	10.7134i		−	0.426495i	0.976998	−	0.213248i	$-0.0684041\pi$
$632$			5.95213i			0.236763i
$633$		0			0
$634$	12.1952			0.484333
$635$			22.7550i			0.903003i
$636$		0			0
$637$	10.9306	−	11.2636i	0.433088	−	0.446281i
$638$	−17.4270			−0.689942
$639$		0			0
$640$	−3.83645			−0.151649
$641$	38.1401			1.50644			0.753221	−	0.657767i	$-0.228499\pi$
$641$	38.1401			1.50644			0.753221	+	0.657767i	$0.228499\pi$
$642$		0			0
$643$			8.17018i			0.322201i	0.986938	+	0.161100i	$0.0515042\pi$
$643$			8.17018i			0.322201i	−0.986938	+	0.161100i	$0.948496\pi$
$644$		−	14.3956i		−	0.567266i
$645$		0			0
$646$	5.18124			0.203853
$647$	0.977276			0.0384207			0.0192103	−	0.999815i	$-0.493885\pi$
$647$	0.977276			0.0384207			0.0192103	+	0.999815i	$0.493885\pi$
$648$		0			0
$649$	14.5661			0.571770
$650$	25.1459	+	24.4025i	0.986303	+	0.957145i
$651$		0			0
$652$			11.0533i			0.432882i
$653$	−13.9664			−0.546547			−0.273273	−	0.961936i	$-0.588106\pi$
$653$	−13.9664			−0.546547			−0.273273	+	0.961936i	$0.588106\pi$
$654$		0			0
$655$			52.8084i			2.06340i
$656$			8.24323i			0.321844i
$657$		0			0
$658$		−	16.4492i		−	0.641258i
$659$	18.9688			0.738918			0.369459	−	0.929247i	$-0.379543\pi$
$659$	18.9688			0.738918			0.369459	+	0.929247i	$0.379543\pi$
$660$		0			0
$661$		−	18.4137i		−	0.716209i	−0.933682	−	0.358104i	$-0.883423\pi$
$661$		−	18.4137i		−	0.716209i	0.933682	−	0.358104i	$-0.116577\pi$
$662$	−22.4765			−0.873575
$663$		0			0
$664$	−2.35164			−0.0912614
$665$			25.7474i			0.998440i
$666$		0			0
$667$	−37.3808			−1.44739
$668$		−	9.40375i		−	0.363842i
$669$		0			0
$670$			21.7329i			0.839617i
$671$			22.4543i			0.866839i
$672$		0			0
$673$	−27.9391			−1.07697			−0.538486	−	0.842634i	$-0.681004\pi$
$673$	−27.9391			−1.07697			−0.538486	+	0.842634i	$0.681004\pi$
$674$		−	30.7950i		−	1.18618i
$675$		0			0
$676$	0.389998	+	12.9941i	0.0149999	+	0.499775i
$677$	3.57226			0.137293			0.0686465	−	0.997641i	$-0.478132\pi$
$677$	3.57226			0.137293			0.0686465	+	0.997641i	$0.478132\pi$
$678$		0			0
$679$	−34.7172			−1.33232
$680$	9.97970			0.382704
$681$		0			0
$682$			10.5179i			0.402751i
$683$		−	11.0777i		−	0.423875i	−0.977283	−	0.211938i	$-0.932023\pi$
$683$		−	11.0777i		−	0.423875i	0.977283	−	0.211938i	$-0.0679774\pi$
$684$		0			0
$685$	−14.6546			−0.559924
$686$	−8.91842			−0.340507
$687$		0			0
$688$	10.2637			0.391299
$689$	−23.0117	+	23.7127i	−0.876675	+	0.903382i
$690$		0			0
$691$			19.5506i			0.743741i	0.928285	+	0.371871i	$0.121284\pi$
$691$			19.5506i			0.743741i	−0.928285	+	0.371871i	$0.878716\pi$
$692$	2.17769			0.0827835
$693$		0			0
$694$			29.0471i			1.10261i
$695$		−	52.2131i		−	1.98055i
$696$		0			0
$697$		−	21.4430i		−	0.812212i
$698$	−12.4672			−0.471889
$699$		0			0
$700$			32.7454i			1.23766i
$701$	−46.2105			−1.74535			−0.872673	−	0.488306i	$-0.837615\pi$
$701$	−46.2105			−1.74535			−0.872673	+	0.488306i	$0.837615\pi$
$702$		0			0
$703$	10.1321			0.382138
$704$			1.99180i			0.0750687i
$705$		0			0
$706$	−34.4293			−1.29576
$707$			8.62310i			0.324305i
$708$		0			0
$709$		−	32.4841i		−	1.21997i	−0.792415	−	0.609983i	$-0.791176\pi$
$709$		−	32.4841i		−	1.21997i	0.792415	−	0.609983i	$-0.208824\pi$
$710$			7.46209i			0.280047i
$711$		0			0
$712$	6.28290			0.235462
$713$			22.5608i			0.844908i
$714$		0			0
$715$	19.1874	−	19.7720i	0.717570	−	0.739430i
$716$	7.91152			0.295667
$717$		0			0
$718$	34.2496			1.27818
$719$	−30.0876			−1.12208			−0.561038	−	0.827790i	$-0.689598\pi$
$719$	−30.0876			−1.12208			−0.561038	+	0.827790i	$0.689598\pi$
$720$		0			0
$721$		−	19.0085i		−	0.707913i
$722$			15.0327i			0.559461i
$723$		0			0
$724$	1.02195			0.0379806
$725$	85.0295			3.15791
$726$		0			0
$727$	4.61814			0.171277			0.0856386	−	0.996326i	$-0.472707\pi$
$727$	4.61814			0.171277			0.0856386	+	0.996326i	$0.472707\pi$
$728$	−8.46059	+	8.71833i	−0.313570	+	0.323123i
$729$		0			0
$730$			32.2808i			1.19477i
$731$	−26.6988			−0.987490
$732$		0			0
$733$		−	44.6467i		−	1.64906i	−0.565817	−	0.824531i	$-0.691439\pi$
$733$		−	44.6467i		−	1.64906i	0.565817	−	0.824531i	$-0.308561\pi$
$734$		−	5.03611i		−	0.185886i
$735$		0			0
$736$			4.27239i			0.157483i
$737$	11.2833			0.415624
$738$		0			0
$739$		−	3.94466i		−	0.145107i	−0.997365	−	0.0725534i	$-0.976885\pi$
$739$		−	3.94466i		−	0.145107i	0.997365	−	0.0725534i	$-0.0231148\pi$
$740$	19.5156			0.717407
$741$		0			0
$742$	−30.8790			−1.13361
$743$		−	54.0573i		−	1.98317i	−0.129457	−	0.991585i	$-0.541323\pi$
$743$		−	54.0573i		−	1.98317i	0.129457	−	0.991585i	$-0.458677\pi$
$744$		0			0
$745$	29.8896			1.09507
$746$			11.2141i			0.410578i
$747$		0			0
$748$		−	5.18124i		−	0.189445i
$749$		−	61.6781i		−	2.25367i
$750$		0			0
$751$	3.93260			0.143503			0.0717513	−	0.997423i	$-0.477141\pi$
$751$	3.93260			0.143503			0.0717513	+	0.997423i	$0.477141\pi$
$752$			4.88188i			0.178024i
$753$		0			0
$754$	22.6388	+	21.9695i	0.824456	+	0.800082i
$755$	81.9689			2.98315
$756$		0			0
$757$	−0.241542			−0.00877899			−0.00438949	−	0.999990i	$-0.501397\pi$
$757$	−0.241542			−0.00877899			−0.00438949	+	0.999990i	$0.501397\pi$
$758$	5.12623			0.186193
$759$		0			0
$760$		−	7.64143i		−	0.277184i
$761$			15.7771i			0.571920i	0.958242	+	0.285960i	$0.0923124\pi$
$761$			15.7771i			0.571920i	−0.958242	+	0.285960i	$0.907688\pi$
$762$		0			0
$763$	−22.7405			−0.823262
$764$	4.06108			0.146925
$765$		0			0
$766$	18.9420			0.684401
$767$	−18.9223	−	18.3629i	−0.683245	−	0.663046i
$768$		0			0
$769$			8.13172i			0.293237i	0.989193	+	0.146619i	$0.0468390\pi$
$769$			8.13172i			0.293237i	−0.989193	+	0.146619i	$0.953161\pi$
$770$	25.7474			0.927871
$771$		0			0
$772$		−	16.6417i		−	0.598948i
$773$		−	19.4126i		−	0.698221i	−0.937082	−	0.349110i	$-0.886484\pi$
$773$		−	19.4126i		−	0.698221i	0.937082	−	0.349110i	$-0.113516\pi$
$774$		0			0
$775$		−	51.3186i		−	1.84342i
$776$	10.3035			0.369875
$777$		0			0
$778$			20.8397i			0.747141i
$779$	−16.4189			−0.588266
$780$		0			0
$781$	3.87415			0.138628
$782$		−	11.1137i		−	0.397426i
$783$		0			0
$784$	−4.35315			−0.155469
$785$			16.3104i			0.582143i
$786$		0			0
$787$			41.1928i			1.46837i	0.678952	+	0.734183i	$0.262434\pi$
$787$			41.1928i			1.46837i	−0.678952	+	0.734183i	$0.737566\pi$
$788$			20.8461i			0.742611i
$789$		0			0
$790$	−22.8350			−0.812434
$791$			20.0499i			0.712893i
$792$		0			0
$793$	28.3072	−	29.1695i	1.00522	−	1.03584i
$794$	2.17152			0.0770642
$795$		0			0
$796$	−15.4367			−0.547138
$797$	3.59275			0.127262			0.0636309	−	0.997973i	$-0.479732\pi$
$797$	3.59275			0.127262			0.0636309	+	0.997973i	$0.479732\pi$
$798$		0			0
$799$		−	12.6992i		−	0.449265i
$800$		−	9.71833i		−	0.343595i
$801$		0			0
$802$	10.8033			0.381476
$803$	16.7595			0.591430
$804$		0			0
$805$	55.2279			1.94653
$806$	13.2595	−	13.6634i	0.467044	−	0.481272i
$807$		0			0
$808$		−	2.55921i		−	0.0900325i
$809$	−10.0592			−0.353664			−0.176832	−	0.984241i	$-0.556585\pi$
$809$	−10.0592			−0.353664			−0.176832	+	0.984241i	$0.556585\pi$
$810$		0			0
$811$			29.4252i			1.03326i	0.856209	+	0.516630i	$0.172814\pi$
$811$			29.4252i			1.03326i	−0.856209	+	0.516630i	$0.827186\pi$
$812$			29.4806i			1.03457i
$813$		0			0
$814$		−	10.1321i		−	0.355129i
$815$	−42.4055			−1.48540
$816$		0			0
$817$			20.4432i			0.715216i
$818$	−1.55301			−0.0542996
$819$		0			0
$820$	−31.6247			−1.10438
$821$			10.9344i			0.381612i	0.981628	+	0.190806i	$0.0611101\pi$
$821$			10.9344i			0.381612i	−0.981628	+	0.190806i	$0.938890\pi$
$822$		0			0
$823$	15.7768			0.549945			0.274973	−	0.961452i	$-0.411331\pi$
$823$	15.7768			0.549945			0.274973	+	0.961452i	$0.411331\pi$
$824$			5.64143i			0.196529i
$825$		0			0
$826$		−	24.6409i		−	0.857367i
$827$			20.9583i			0.728793i	0.931244	+	0.364396i	$0.118725\pi$
$827$			20.9583i			0.728793i	−0.931244	+	0.364396i	$0.881275\pi$
$828$		0			0
$829$	−30.6108			−1.06316			−0.531579	−	0.847009i	$-0.678401\pi$
$829$	−30.6108			−1.06316			−0.531579	+	0.847009i	$0.678401\pi$
$830$		−	9.02195i		−	0.313157i
$831$		0			0
$832$	2.51098	−	2.58747i	0.0870524	−	0.0897044i
$833$	11.3238			0.392346
$834$		0			0
$835$	36.0770			1.24850
$836$	−3.96726			−0.137211
$837$		0			0
$838$		−	5.62905i		−	0.194452i
$839$		−	22.7084i		−	0.783982i	−0.919969	−	0.391991i	$-0.871786\pi$
$839$		−	22.7084i		−	0.783982i	0.919969	−	0.391991i	$-0.128214\pi$
$840$		0			0
$841$	47.5518			1.63972
$842$	−11.9941			−0.413343
$843$		0			0
$844$	3.73323			0.128503
$845$	−49.8514	+	1.49621i	−1.71494	+	0.0514710i
$846$		0			0
$847$			23.6964i			0.814218i
$848$	9.16444			0.314708
$849$		0			0
$850$			25.2802i			0.867103i
$851$		−	21.7332i		−	0.745005i
$852$		0			0
$853$		−	14.6122i		−	0.500313i	−0.968205	−	0.250157i	$-0.919518\pi$
$853$		−	14.6122i		−	0.500313i	0.968205	−	0.250157i	$-0.0804821\pi$
$854$	37.9850			1.29982
$855$		0			0
$856$			18.3051i			0.625657i
$857$	−14.2230			−0.485848			−0.242924	−	0.970045i	$-0.578107\pi$
$857$	−14.2230			−0.485848			−0.242924	+	0.970045i	$0.578107\pi$
$858$		0			0
$859$	41.6331			1.42050			0.710252	−	0.703948i	$-0.248581\pi$
$859$	41.6331			1.42050			0.710252	+	0.703948i	$0.248581\pi$
$860$			39.3761i			1.34271i
$861$		0			0
$862$	0.0127142			0.000433047
$863$			13.5656i			0.461780i	0.972980	+	0.230890i	$0.0741638\pi$
$863$			13.5656i			0.461780i	−0.972980	+	0.230890i	$0.925836\pi$
$864$		0			0
$865$			8.35461i			0.284065i
$866$			30.8717i			1.04906i
$867$		0			0
$868$	17.7927			0.603923
$869$			11.8554i			0.402168i
$870$		0			0
$871$	−14.6577	−	14.2243i	−0.496656	−	0.481973i
$872$	6.74904			0.228551
$873$		0			0
$874$	−8.50975			−0.287847
$875$	−60.9924			−2.06192
$876$		0			0
$877$			41.5884i			1.40434i	0.712009	+	0.702170i	$0.247785\pi$
$877$			41.5884i			1.40434i	−0.712009	+	0.702170i	$0.752215\pi$
$878$			16.3956i			0.553324i
$879$		0			0
$880$	−7.64143			−0.257593
$881$	15.0582			0.507323			0.253662	−	0.967293i	$-0.418365\pi$
$881$	15.0582			0.507323			0.253662	+	0.967293i	$0.418365\pi$
$882$		0			0
$883$	21.3359			0.718010			0.359005	−	0.933336i	$-0.383116\pi$
$883$	21.3359			0.718010			0.359005	+	0.933336i	$0.383116\pi$
$884$	−6.53177	+	6.73075i	−0.219687	+	0.226380i
$885$		0			0
$886$			4.04515i			0.135900i
$887$	30.9351			1.03870			0.519350	−	0.854562i	$-0.326174\pi$
$887$	30.9351			1.03870			0.519350	+	0.854562i	$0.326174\pi$
$888$		0			0
$889$		−	19.9850i		−	0.670276i
$890$			24.1040i			0.807968i
$891$		0			0
$892$		−	2.75301i		−	0.0921775i
$893$	−9.72373			−0.325392
$894$		0			0
$895$			30.3521i			1.01456i
$896$	3.36944			0.112565
$897$		0			0
$898$	−15.8943			−0.530400
$899$		−	46.2020i		−	1.54092i
$900$		0			0
$901$	−23.8393			−0.794203
$902$			16.4189i			0.546688i
$903$		0			0
$904$		−	5.95051i		−	0.197911i
$905$			3.92067i			0.130327i
$906$		0			0
$907$	−26.4115			−0.876979			−0.438490	−	0.898736i	$-0.644486\pi$
$907$	−26.4115			−0.876979			−0.438490	+	0.898736i	$0.644486\pi$
$908$			3.82159i			0.126824i
$909$		0			0
$910$	−33.4474	−	32.4586i	−1.10877	−	1.07599i
$911$	−16.8675			−0.558845			−0.279423	−	0.960168i	$-0.590143\pi$
$911$	−16.8675			−0.558845			−0.279423	+	0.960168i	$0.590143\pi$
$912$		0			0
$913$	−4.68400			−0.155018
$914$	−29.1414			−0.963913
$915$		0			0
$916$			10.9742i			0.362597i
$917$		−	46.3801i		−	1.53161i
$918$		0			0
$919$	28.6707			0.945758			0.472879	−	0.881127i	$-0.343215\pi$
$919$	28.6707			0.945758			0.472879	+	0.881127i	$0.343215\pi$
$920$	−16.3908			−0.540389
$921$		0			0
$922$	−28.1429			−0.926837
$923$	−5.03276	−	4.88398i	−0.165655	−	0.160758i
$924$		0			0
$925$			49.4361i			1.62545i
$926$	31.8672			1.04722
$927$		0			0
$928$		−	8.74939i		−	0.287213i
$929$			42.0022i			1.37805i	0.724738	+	0.689024i	$0.241961\pi$
$929$			42.0022i			1.37805i	−0.724738	+	0.689024i	$0.758039\pi$
$930$		0			0
$931$		−	8.67059i		−	0.284167i
$932$	2.26098			0.0740610
$933$		0			0
$934$		−	1.94189i		−	0.0635406i
$935$	19.8776			0.650066
$936$		0			0
$937$	44.9192			1.46745			0.733724	−	0.679448i	$-0.237781\pi$
$937$	44.9192			1.46745			0.733724	+	0.679448i	$0.237781\pi$
$938$		−	19.0874i		−	0.623227i
$939$		0			0
$940$	−18.7291			−0.610876
$941$		−	9.56587i		−	0.311838i	−0.987770	−	0.155919i	$-0.950166\pi$
$941$		−	9.56587i		−	0.311838i	0.987770	−	0.155919i	$-0.0498339\pi$
$942$		0			0
$943$			35.2183i			1.14687i
$944$			7.31305i			0.238020i
$945$		0			0
$946$	20.4432			0.664665
$947$		−	29.2255i		−	0.949701i	−0.880066	−	0.474850i	$-0.842502\pi$
$947$		−	29.2255i		−	0.949701i	0.880066	−	0.474850i	$-0.157498\pi$
$948$		0			0
$949$	−21.7716	−	21.1280i	−0.706737	−	0.685843i
$950$	19.3570			0.628023
$951$		0			0
$952$	−8.76489			−0.284072
$953$	−6.46873			−0.209543			−0.104771	−	0.994496i	$-0.533411\pi$
$953$	−6.46873			−0.209543			−0.104771	+	0.994496i	$0.533411\pi$
$954$		0			0
$955$			15.5801i			0.504161i
$956$			6.30635i			0.203962i
$957$		0			0
$958$	10.9247			0.352962
$959$	12.8707			0.415617
$960$		0			0
$961$	3.11530			0.100494
$962$	−12.7731	+	13.1622i	−0.411820	+	0.424366i
$963$		0			0
$964$			19.8311i			0.638718i
$965$	63.8450			2.05524
$966$		0			0
$967$			48.5056i			1.55984i	0.625881	+	0.779918i	$0.284739\pi$
$967$			48.5056i			1.55984i	−0.625881	+	0.779918i	$0.715261\pi$
$968$		−	7.03274i		−	0.226041i
$969$		0			0
$970$			39.5290i			1.26920i
$971$	4.35773			0.139846			0.0699231	−	0.997552i	$-0.477725\pi$
$971$	4.35773			0.139846			0.0699231	+	0.997552i	$0.477725\pi$
$972$		0			0
$973$			45.8573i			1.47012i
$974$	−38.3978			−1.23034
$975$		0			0
$976$	−11.2734			−0.360852
$977$		−	21.6963i		−	0.694126i	−0.937842	−	0.347063i	$-0.887179\pi$
$977$		−	21.6963i		−	0.694126i	0.937842	−	0.347063i	$-0.112821\pi$
$978$		0			0
$979$	12.5143			0.399958
$980$		−	16.7006i		−	0.533482i
$981$		0			0
$982$		−	10.5533i		−	0.336768i
$983$			28.1144i			0.896709i	0.893856	+	0.448355i	$0.147990\pi$
$983$			28.1144i			0.896709i	−0.893856	+	0.448355i	$0.852010\pi$
$984$		0			0
$985$	−79.9749			−2.54821
$986$			22.7597i			0.724816i
$987$		0			0
$988$	5.15372	+	5.00136i	0.163962	+	0.159114i
$989$	43.8505			1.39436
$990$		0			0
$991$	−42.2239			−1.34129			−0.670643	−	0.741780i	$-0.733982\pi$
$991$	−42.2239			−1.34129			−0.670643	+	0.741780i	$0.733982\pi$
$992$	−5.28060			−0.167659
$993$		0			0
$994$		−	6.55374i		−	0.207872i
$995$		−	59.2219i		−	1.87746i
$996$		0			0
$997$	0.617278			0.0195494			0.00977470	−	0.999952i	$-0.496889\pi$
$997$	0.617278			0.0195494			0.00977470	+	0.999952i	$0.496889\pi$
$998$	0.992867			0.0314287
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.b.d.649.8		14
3.2	odd	2		2106.2.b.c.649.7		14
9.2	odd	6		234.2.t.a.103.3	yes	28
9.4	even	3		702.2.t.a.181.1		28
9.5	odd	6		234.2.t.a.25.10	yes	28
9.7	even	3		702.2.t.a.415.14		28
13.12	even	2	inner	2106.2.b.d.649.7		14
39.38	odd	2		2106.2.b.c.649.8		14
117.25	even	6		702.2.t.a.415.1		28
117.38	odd	6		234.2.t.a.103.10	yes	28
117.77	odd	6		234.2.t.a.25.3	✓	28
117.103	even	6		702.2.t.a.181.14		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.t.a.25.3	✓	28	117.77	odd	6
234.2.t.a.25.10	yes	28	9.5	odd	6
234.2.t.a.103.3	yes	28	9.2	odd	6
234.2.t.a.103.10	yes	28	117.38	odd	6
702.2.t.a.181.1		28	9.4	even	3
702.2.t.a.181.14		28	117.103	even	6
702.2.t.a.415.1		28	117.25	even	6
702.2.t.a.415.14		28	9.7	even	3
2106.2.b.c.649.7		14	3.2	odd	2
2106.2.b.c.649.8		14	39.38	odd	2
2106.2.b.d.649.7		14	13.12	even	2	inner
2106.2.b.d.649.8		14	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -3.83645i q^{5} +3.36944i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -3.83645i q^{5} +3.36944i q^{7} -1.00000i q^{8} +3.83645 q^{10} -1.99180i q^{11} +(-2.51098 + 2.58747i) q^{13} -3.36944 q^{14} +1.00000 q^{16} -2.60129 q^{17} +1.99180i q^{19} +3.83645i q^{20} +1.99180 q^{22} +4.27239 q^{23} -9.71833 q^{25} +(-2.58747 - 2.51098i) q^{26} -3.36944i q^{28} -8.74939 q^{29} +5.28060i q^{31} +1.00000i q^{32} -2.60129i q^{34} +12.9267 q^{35} -5.08689i q^{37} -1.99180 q^{38} -3.83645 q^{40} +8.24323i q^{41} +10.2637 q^{43} +1.99180i q^{44} +4.27239i q^{46} +4.88188i q^{47} -4.35315 q^{49} -9.71833i q^{50} +(2.51098 - 2.58747i) q^{52} +9.16444 q^{53} -7.64143 q^{55} +3.36944 q^{56} -8.74939i q^{58} +7.31305i q^{59} -11.2734 q^{61} -5.28060 q^{62} -1.00000 q^{64} +(9.92669 + 9.63323i) q^{65} +5.66486i q^{67} +2.60129 q^{68} +12.9267i q^{70} +1.94505i q^{71} +8.41425i q^{73} +5.08689 q^{74} -1.99180i q^{76} +6.71125 q^{77} -5.95213 q^{79} -3.83645i q^{80} -8.24323 q^{82} -2.35164i q^{83} +9.97970i q^{85} +10.2637i q^{86} -1.99180 q^{88} +6.28290i q^{89} +(-8.71833 - 8.46059i) q^{91} -4.27239 q^{92} -4.88188 q^{94} +7.64143 q^{95} +10.3035i q^{97} -4.35315i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4}+O(q^{10}) \) 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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