LMFDB - Embedded newform 2100.2.x.c.1357.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1357,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("2100.1357");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.x (of order $4$, degree $2$, 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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.478584585616890104119296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 31x^{12} + 336x^{8} - 19375x^{4} + 390625 $ x^16 - 31x^12 + 336x^8 - 19375*x^4 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1357.7
Root			$2.23460 + 0.0811201i$ of defining polynomial
Character	$\chi$	$=$	2100.1357
Dual form			2100.2.x.c.1693.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{3} +(1.71696 - 2.01297i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{3} +(1.71696 - 2.01297i) q^{7} +1.00000i q^{9} -4.27492 q^{11} +(-2.31572 - 2.31572i) q^{13} +(-1.41421 + 1.41421i) q^{17} +6.50958 q^{19} +(2.63746 - 0.209313i) q^{21} +(3.37822 - 3.37822i) q^{23} +(-0.707107 + 0.707107i) q^{27} -8.27492i q^{29} +3.04547i q^{31} +(-3.02282 - 3.02282i) q^{33} +(7.68517 + 7.68517i) q^{37} -3.27492i q^{39} -12.1819i q^{41} +(5.53170 - 5.53170i) q^{43} +(-1.41421 + 1.41421i) q^{47} +(-1.10411 - 6.91238i) q^{49} -2.00000 q^{51} +(8.61390 - 8.61390i) q^{53} +(4.60297 + 4.60297i) q^{57} -8.71780 q^{59} -3.04547i q^{61} +(2.01297 + 1.71696i) q^{63} +(-3.08221 - 3.08221i) q^{67} +4.77753 q^{69} +1.72508 q^{71} +(7.07107 + 7.07107i) q^{73} +(-7.33985 + 8.60529i) q^{77} -10.8248i q^{79} -1.00000 q^{81} +(1.02542 + 1.02542i) q^{83} +(5.85125 - 5.85125i) q^{87} -7.88054 q^{89} +(-8.63746 + 0.685484i) q^{91} +(-2.15348 + 2.15348i) q^{93} +(-1.92692 + 1.92692i) q^{97} -4.27492i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{11} + 12 q^{21} - 32 q^{51} + 88 q^{71} - 16 q^{81} - 108 q^{91}+O(q^{100}) $ 16 * q - 8 * q^11 + 12 * q^21 - 32 * q^51 + 88 * q^71 - 16 * q^81 - 108 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

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
$3$	0.707107	+	0.707107i	0.408248	+	0.408248i
$3$	0.707107	+	0.707107i	0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.71696	−	2.01297i	0.648949	−	0.760832i
$7$	1.71696	−	2.01297i	0.648949	−	0.760832i
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−4.27492			−1.28894			−0.644468	−	0.764631i	$-0.722921\pi$
$11$	−4.27492			−1.28894			−0.644468	+	0.764631i	$0.722921\pi$
$12$		0			0
$13$	−2.31572	−	2.31572i	−0.642264	−	0.642264i	0.308847	−	0.951112i	$-0.400057\pi$
$13$	−2.31572	−	2.31572i	−0.642264	−	0.642264i	−0.951112	+	0.308847i	$0.900057\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.41421	+	1.41421i	−0.342997	+	0.342997i	−0.857493	−	0.514496i	$-0.827979\pi$
$17$	−1.41421	+	1.41421i	−0.342997	+	0.342997i	0.514496	+	0.857493i	$0.327979\pi$
$18$		0			0
$19$	6.50958			1.49340			0.746700	−	0.665161i	$-0.231637\pi$
$19$	6.50958			1.49340			0.746700	+	0.665161i	$0.231637\pi$
$20$		0			0
$21$	2.63746	−	0.209313i	0.575541	−	0.0456759i
$22$		0			0
$23$	3.37822	−	3.37822i	0.704408	−	0.704408i	−0.260946	−	0.965353i	$-0.584034\pi$
$23$	3.37822	−	3.37822i	0.704408	−	0.704408i	0.965353	+	0.260946i	$0.0840344\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.707107	+	0.707107i	−0.136083	+	0.136083i
$28$		0			0
$29$		−	8.27492i		−	1.53661i	−0.640082	−	0.768307i	$-0.721100\pi$
$29$		−	8.27492i		−	1.53661i	0.640082	−	0.768307i	$-0.278900\pi$
$30$		0			0
$31$			3.04547i			0.546983i	0.961874	+	0.273492i	$0.0881786\pi$
$31$			3.04547i			0.546983i	−0.961874	+	0.273492i	$0.911821\pi$
$32$		0			0
$33$	−3.02282	−	3.02282i	−0.526206	−	0.526206i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.68517	+	7.68517i	1.26343	+	1.26343i	0.949417	+	0.314017i	$0.101675\pi$
$37$	7.68517	+	7.68517i	1.26343	+	1.26343i	0.314017	+	0.949417i	$0.398325\pi$
$38$		0			0
$39$		−	3.27492i		−	0.524406i
$40$		0			0
$41$		−	12.1819i		−	1.90249i	−0.308432	−	0.951247i	$-0.599804\pi$
$41$		−	12.1819i		−	1.90249i	0.308432	−	0.951247i	$-0.400196\pi$
$42$		0			0
$43$	5.53170	−	5.53170i	0.843576	−	0.843576i	−0.145746	−	0.989322i	$-0.546558\pi$
$43$	5.53170	−	5.53170i	0.843576	−	0.843576i	0.989322	+	0.145746i	$0.0465583\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.41421	+	1.41421i	−0.206284	+	0.206284i	−0.802686	−	0.596402i	$-0.796597\pi$
$47$	−1.41421	+	1.41421i	−0.206284	+	0.206284i	0.596402	+	0.802686i	$0.296597\pi$
$48$		0			0
$49$	−1.10411	−	6.91238i	−0.157730	−	0.987482i
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	8.61390	−	8.61390i	1.18321	−	1.18321i	0.204303	−	0.978908i	$-0.434507\pi$
$53$	8.61390	−	8.61390i	1.18321	−	1.18321i	0.978908	−	0.204303i	$-0.0654928\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.60297	+	4.60297i	0.609678	+	0.609678i
$58$		0			0
$59$	−8.71780			−1.13496			−0.567480	−	0.823387i	$-0.692082\pi$
$59$	−8.71780			−1.13496			−0.567480	+	0.823387i	$0.692082\pi$
$60$		0			0
$61$		−	3.04547i		−	0.389933i	−0.980810	−	0.194967i	$-0.937540\pi$
$61$		−	3.04547i		−	0.389933i	0.980810	−	0.194967i	$-0.0624598\pi$
$62$		0			0
$63$	2.01297	+	1.71696i	0.253611	+	0.216316i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.08221	−	3.08221i	−0.376552	−	0.376552i	0.493305	−	0.869856i	$-0.335789\pi$
$67$	−3.08221	−	3.08221i	−0.376552	−	0.376552i	−0.869856	+	0.493305i	$0.835789\pi$
$68$		0			0
$69$	4.77753			0.575147
$70$		0			0
$71$	1.72508			0.204730			0.102365	−	0.994747i	$-0.467359\pi$
$71$	1.72508			0.204730			0.102365	+	0.994747i	$0.467359\pi$
$72$		0			0
$73$	7.07107	+	7.07107i	0.827606	+	0.827606i	0.987185	−	0.159579i	$-0.0510137\pi$
$73$	7.07107	+	7.07107i	0.827606	+	0.827606i	−0.159579	+	0.987185i	$0.551014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.33985	+	8.60529i	−0.836454	+	0.980664i
$78$		0			0
$79$		−	10.8248i		−	1.21788i	−0.793216	−	0.608940i	$-0.791595\pi$
$79$		−	10.8248i		−	1.21788i	0.793216	−	0.608940i	$-0.208405\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	1.02542	+	1.02542i	0.112555	+	0.112555i	0.761141	−	0.648586i	$-0.224639\pi$
$83$	1.02542	+	1.02542i	0.112555	+	0.112555i	−0.648586	+	0.761141i	$0.724639\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	5.85125	−	5.85125i	0.627320	−	0.627320i
$88$		0			0
$89$	−7.88054			−0.835336			−0.417668	−	0.908600i	$-0.637152\pi$
$89$	−7.88054			−0.835336			−0.417668	+	0.908600i	$0.637152\pi$
$90$		0			0
$91$	−8.63746	+	0.685484i	−0.905452	+	0.0718582i
$92$		0			0
$93$	−2.15348	+	2.15348i	−0.223305	+	0.223305i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.92692	+	1.92692i	−0.195650	+	0.195650i	−0.798132	−	0.602483i	$-0.794178\pi$
$97$	−1.92692	+	1.92692i	−0.195650	+	0.195650i	0.602483	+	0.798132i	$0.294178\pi$
$98$		0			0
$99$		−	4.27492i		−	0.429645i
$100$		0			0
$101$			16.4833i			1.64015i	0.572260	+	0.820073i	$0.306067\pi$
$101$			16.4833i			1.64015i	−0.572260	+	0.820073i	$0.693933\pi$
$102$		0			0
$103$	−6.04565	−	6.04565i	−0.595695	−	0.595695i	0.343469	−	0.939164i	$-0.388398\pi$
$103$	−6.04565	−	6.04565i	−0.595695	−	0.595695i	−0.939164	+	0.343469i	$0.888398\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.85746	+	1.85746i	0.179568	+	0.179568i	0.791167	−	0.611600i	$-0.209474\pi$
$107$	1.85746	+	1.85746i	0.179568	+	0.179568i	−0.611600	+	0.791167i	$0.709474\pi$
$108$		0			0
$109$			1.00000i			0.0957826i	0.998853	+	0.0478913i	$0.0152501\pi$
$109$			1.00000i			0.0957826i	−0.998853	+	0.0478913i	$0.984750\pi$
$110$		0			0
$111$			10.8685i			1.03159i
$112$		0			0
$113$	−2.11279	+	2.11279i	−0.198754	+	0.198754i	−0.799466	−	0.600712i	$-0.794884\pi$
$113$	−2.11279	+	2.11279i	−0.198754	+	0.198754i	0.600712	+	0.799466i	$0.294884\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.31572	−	2.31572i	0.214088	−	0.214088i
$118$		0			0
$119$	0.418627	+	5.27492i	0.0383755	+	0.483551i
$120$		0			0
$121$	7.27492			0.661356
$122$		0			0
$123$	8.61390	−	8.61390i	0.776690	−	0.776690i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.41974	+	6.41974i	0.569660	+	0.569660i	0.932033	−	0.362373i	$-0.118033\pi$
$127$	6.41974	+	6.41974i	0.569660	+	0.569660i	−0.362373	+	0.932033i	$0.618033\pi$
$128$		0			0
$129$	7.82300			0.688777
$130$		0			0
$131$		−	11.2296i		−	0.981131i	−0.871404	−	0.490566i	$-0.836790\pi$
$131$		−	11.2296i		−	0.981131i	0.871404	−	0.490566i	$-0.163210\pi$
$132$		0			0
$133$	11.1767	−	13.1036i	0.969140	−	1.13623i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.12290	+	3.12290i	0.266807	+	0.266807i	0.827812	−	0.561005i	$-0.189585\pi$
$137$	3.12290	+	3.12290i	0.266807	+	0.266807i	−0.561005	+	0.827812i	$0.689585\pi$
$138$		0			0
$139$	3.46410			0.293821			0.146911	−	0.989150i	$-0.453067\pi$
$139$	3.46410			0.293821			0.146911	+	0.989150i	$0.453067\pi$
$140$		0			0
$141$	−2.00000			−0.168430
$142$		0			0
$143$	9.89949	+	9.89949i	0.827837	+	0.827837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.10706	−	5.66851i	0.338745	−	0.467531i
$148$		0			0
$149$		−	15.7251i		−	1.28825i	−0.764921	−	0.644124i	$-0.777222\pi$
$149$		−	15.7251i		−	1.28825i	0.764921	−	0.644124i	$-0.222778\pi$
$150$		0			0
$151$	−11.0000			−0.895167			−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000			−0.895167			−0.447584	+	0.894242i	$0.647715\pi$
$152$		0			0
$153$	−1.41421	−	1.41421i	−0.114332	−	0.114332i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.36136	−	8.36136i	0.667309	−	0.667309i	−0.289783	−	0.957092i	$-0.593583\pi$
$157$	8.36136	−	8.36136i	0.667309	−	0.667309i	0.957092	+	0.289783i	$0.0935831\pi$
$158$		0			0
$159$	12.1819			0.966087
$160$		0			0
$161$	−1.00000	−	12.6005i	−0.0788110	−	0.993061i
$162$		0			0
$163$	12.0328	−	12.0328i	0.942483	−	0.942483i	−0.0559508	−	0.998434i	$-0.517819\pi$
$163$	12.0328	−	12.0328i	0.942483	−	0.942483i	0.998434	+	0.0559508i	$0.0178190\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.21722	+	3.21722i	−0.248956	+	0.248956i	−0.820542	−	0.571586i	$-0.806328\pi$
$167$	−3.21722	+	3.21722i	−0.248956	+	0.248956i	0.571586	+	0.820542i	$0.306328\pi$
$168$		0			0
$169$		−	2.27492i		−	0.174994i
$170$		0			0
$171$			6.50958i			0.497800i
$172$		0			0
$173$	7.84865	+	7.84865i	0.596722	+	0.596722i	0.939439	−	0.342717i	$-0.111347\pi$
$173$	7.84865	+	7.84865i	0.596722	+	0.596722i	−0.342717	+	0.939439i	$0.611347\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.16441	−	6.16441i	−0.463346	−	0.463346i
$178$		0			0
$179$		−	22.5498i		−	1.68545i	−0.538341	−	0.842727i	$-0.680949\pi$
$179$		−	22.5498i		−	1.68545i	0.538341	−	0.842727i	$-0.319051\pi$
$180$		0			0
$181$			18.6915i			1.38933i	0.719335	+	0.694663i	$0.244447\pi$
$181$			18.6915i			1.38933i	−0.719335	+	0.694663i	$0.755553\pi$
$182$		0			0
$183$	2.15348	−	2.15348i	0.159190	−	0.159190i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	6.04565	−	6.04565i	0.442101	−	0.442101i
$188$		0			0
$189$	0.209313	+	2.63746i	0.0152253	+	0.191847i
$190$		0			0
$191$	−2.54983			−0.184500			−0.0922498	−	0.995736i	$-0.529406\pi$
$191$	−2.54983			−0.184500			−0.0922498	+	0.995736i	$0.529406\pi$
$192$		0			0
$193$	−11.1041	+	11.1041i	−0.799289	+	0.799289i	−0.982983	−	0.183694i	$-0.941194\pi$
$193$	−11.1041	+	11.1041i	−0.799289	+	0.799289i	0.183694	+	0.982983i	$0.441194\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.7070	+	15.7070i	1.11908	+	1.11908i	0.991877	+	0.127204i	$0.0406004\pi$
$197$	15.7070	+	15.7070i	1.11908	+	1.11908i	0.127204	+	0.991877i	$0.459400\pi$
$198$		0			0
$199$	−10.8109			−0.766367			−0.383183	−	0.923672i	$-0.625172\pi$
$199$	−10.8109			−0.766367			−0.383183	+	0.923672i	$0.625172\pi$
$200$		0			0
$201$		−	4.35890i		−	0.307453i
$202$		0			0
$203$	−16.6572	−	14.2077i	−1.16910	−	0.997184i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.37822	+	3.37822i	0.234803	+	0.234803i
$208$		0			0
$209$	−27.8279			−1.92490
$210$		0			0
$211$	9.27492			0.638512			0.319256	−	0.947669i	$-0.396567\pi$
$211$	9.27492			0.638512			0.319256	+	0.947669i	$0.396567\pi$
$212$		0			0
$213$	1.21982	+	1.21982i	0.0835805	+	0.0835805i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.13045	+	5.22895i	0.416162	+	0.354964i
$218$		0			0
$219$			10.0000i			0.675737i
$220$		0			0
$221$	6.54983			0.440590
$222$		0			0
$223$	12.6040	+	12.6040i	0.844026	+	0.844026i	0.989380	−	0.145353i	$-0.0464319\pi$
$223$	12.6040	+	12.6040i	0.844026	+	0.844026i	−0.145353	+	0.989380i	$0.546432\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.1369	+	18.1369i	−1.20379	+	1.20379i	−0.230786	+	0.973004i	$0.574130\pi$
$227$	−18.1369	+	18.1369i	−1.20379	+	1.20379i	−0.973004	+	0.230786i	$0.925870\pi$
$228$		0			0
$229$	16.9019			1.11691			0.558454	−	0.829536i	$-0.311395\pi$
$229$	16.9019			1.11691			0.558454	+	0.829536i	$0.311395\pi$
$230$		0			0
$231$	−11.2749	+	0.894797i	−0.741835	+	0.0588733i
$232$		0			0
$233$	−2.78619	+	2.78619i	−0.182530	+	0.182530i	−0.792457	−	0.609928i	$-0.791199\pi$
$233$	−2.78619	+	2.78619i	−0.182530	+	0.182530i	0.609928	+	0.792457i	$0.291199\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	7.65426	−	7.65426i	0.497197	−	0.497197i
$238$		0			0
$239$			21.0997i			1.36482i	0.730968	+	0.682412i	$0.239069\pi$
$239$			21.0997i			1.36482i	−0.730968	+	0.682412i	$0.760931\pi$
$240$		0			0
$241$			15.1123i			0.973468i	0.873550	+	0.486734i	$0.161812\pi$
$241$			15.1123i			0.973468i	−0.873550	+	0.486734i	$0.838188\pi$
$242$		0			0
$243$	−0.707107	−	0.707107i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−15.0743	−	15.0743i	−0.959157	−	0.959157i
$248$		0			0
$249$			1.45017i			0.0919005i
$250$		0			0
$251$		−	2.62685i		−	0.165805i	−0.996558	−	0.0829026i	$-0.973581\pi$
$251$		−	2.62685i		−	0.165805i	0.996558	−	0.0829026i	$-0.0264190\pi$
$252$		0			0
$253$	−14.4416	+	14.4416i	−0.907937	+	0.907937i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.7481	+	17.7481i	−1.10710	+	1.10710i	−0.113569	+	0.993530i	$0.536228\pi$
$257$	−17.7481	+	17.7481i	−1.10710	+	1.10710i	−0.993530	+	0.113569i	$0.963772\pi$
$258$		0			0
$259$	28.6652	−	2.27492i	1.78117	−	0.141356i
$260$		0			0
$261$	8.27492			0.512205
$262$		0			0
$263$	3.97025	−	3.97025i	0.244816	−	0.244816i	−0.574023	−	0.818839i	$-0.694618\pi$
$263$	3.97025	−	3.97025i	0.244816	−	0.244816i	0.818839	+	0.574023i	$0.194618\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.57239	−	5.57239i	−0.341025	−	0.341025i
$268$		0			0
$269$	−26.8756			−1.63863			−0.819316	−	0.573342i	$-0.805647\pi$
$269$	−26.8756			−1.63863			−0.819316	+	0.573342i	$0.805647\pi$
$270$		0			0
$271$		−	5.13861i		−	0.312148i	−0.987745	−	0.156074i	$-0.950116\pi$
$271$		−	5.13861i		−	0.312148i	0.987745	−	0.156074i	$-0.0498839\pi$
$272$		0			0
$273$	−6.59232	−	5.62290i	−0.398985	−	0.340313i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.05246	−	7.05246i	−0.423741	−	0.423741i	0.462749	−	0.886490i	$-0.346863\pi$
$277$	−7.05246	−	7.05246i	−0.423741	−	0.423741i	−0.886490	+	0.462749i	$0.846863\pi$
$278$		0			0
$279$	−3.04547			−0.182328
$280$		0			0
$281$	20.2749			1.20950			0.604750	−	0.796415i	$-0.293273\pi$
$281$	20.2749			1.20950			0.604750	+	0.796415i	$0.293273\pi$
$282$		0			0
$283$	2.95235	+	2.95235i	0.175499	+	0.175499i	0.789390	−	0.613892i	$-0.210397\pi$
$283$	2.95235	+	2.95235i	0.175499	+	0.175499i	−0.613892	+	0.789390i	$0.710397\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.5218	−	20.9158i	−1.44748	−	1.23462i
$288$		0			0
$289$			13.0000i			0.764706i
$290$		0			0
$291$	−2.72508			−0.159747
$292$		0			0
$293$	10.6771	+	10.6771i	0.623762	+	0.623762i	0.946491	−	0.322730i	$-0.104600\pi$
$293$	10.6771	+	10.6771i	0.623762	+	0.623762i	−0.322730	+	0.946491i	$0.604600\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.02282	−	3.02282i	0.175402	−	0.175402i
$298$		0			0
$299$	−15.6460			−0.904832
$300$		0			0
$301$	−1.63746	−	20.6328i	−0.0943815	−	1.18926i
$302$		0			0
$303$	−11.6554	+	11.6554i	−0.669586	+	0.669586i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	16.4579	−	16.4579i	0.939299	−	0.939299i	−0.0589611	−	0.998260i	$-0.518779\pi$
$307$	16.4579	−	16.4579i	0.939299	−	0.939299i	0.998260	+	0.0589611i	$0.0187788\pi$
$308$		0			0
$309$		−	8.54983i		−	0.486383i
$310$		0			0
$311$		−	19.2252i		−	1.09016i	−0.838384	−	0.545080i	$-0.816499\pi$
$311$		−	19.2252i		−	1.09016i	0.838384	−	0.545080i	$-0.183501\pi$
$312$		0			0
$313$	−24.9431	−	24.9431i	−1.40987	−	1.40987i	−0.760317	−	0.649553i	$-0.774956\pi$
$313$	−24.9431	−	24.9431i	−1.40987	−	1.40987i	−0.649553	−	0.760317i	$-0.725044\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.7267	+	10.7267i	0.602471	+	0.602471i	0.940968	−	0.338497i	$-0.109918\pi$
$317$	10.7267	+	10.7267i	0.602471	+	0.602471i	−0.338497	+	0.940968i	$0.609918\pi$
$318$		0			0
$319$			35.3746i			1.98060i
$320$		0			0
$321$			2.62685i			0.146616i
$322$		0			0
$323$	−9.20593	+	9.20593i	−0.512232	+	0.512232i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.707107	+	0.707107i	−0.0391031	+	0.0391031i
$328$		0			0
$329$	0.418627	+	5.27492i	0.0230796	+	0.290816i
$330$		0			0
$331$	−27.3746			−1.50464			−0.752322	−	0.658796i	$-0.771066\pi$
$331$	−27.3746			−1.50464			−0.752322	+	0.658796i	$0.771066\pi$
$332$		0			0
$333$	−7.68517	+	7.68517i	−0.421145	+	0.421145i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.33753	+	3.33753i	0.181807	+	0.181807i	0.792143	−	0.610336i	$-0.208966\pi$
$337$	3.33753	+	3.33753i	0.181807	+	0.181807i	−0.610336	+	0.792143i	$0.708966\pi$
$338$		0			0
$339$	−2.98793			−0.162282
$340$		0			0
$341$		−	13.0192i		−	0.705027i
$342$		0			0
$343$	−15.8101	−	9.64572i	−0.853667	−	0.520820i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.41974	−	6.41974i	−0.344630	−	0.344630i	0.513475	−	0.858105i	$-0.328358\pi$
$347$	−6.41974	−	6.41974i	−0.344630	−	0.344630i	−0.858105	+	0.513475i	$0.828358\pi$
$348$		0			0
$349$	7.88054			0.421836			0.210918	−	0.977504i	$-0.432355\pi$
$349$	7.88054			0.421836			0.210918	+	0.977504i	$0.432355\pi$
$350$		0			0
$351$	3.27492			0.174802
$352$		0			0
$353$	15.9451	+	15.9451i	0.848674	+	0.848674i	0.989968	−	0.141294i	$-0.0451261\pi$
$353$	15.9451	+	15.9451i	0.848674	+	0.848674i	−0.141294	+	0.989968i	$0.545126\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.43392	+	4.02594i	−0.181742	+	0.213076i
$358$		0			0
$359$			11.3746i			0.600328i	0.953888	+	0.300164i	$0.0970414\pi$
$359$			11.3746i			0.600328i	−0.953888	+	0.300164i	$0.902959\pi$
$360$		0			0
$361$	23.3746			1.23024
$362$		0			0
$363$	5.14414	+	5.14414i	0.269998	+	0.269998i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.95235	−	2.95235i	0.154111	−	0.154111i	−0.625840	−	0.779951i	$-0.715244\pi$
$367$	2.95235	−	2.95235i	0.154111	−	0.154111i	0.779951	+	0.625840i	$0.215244\pi$
$368$		0			0
$369$	12.1819			0.634164
$370$		0			0
$371$	−2.54983	−	32.1293i	−0.132381	−	1.66807i
$372$		0			0
$373$	−17.8605	+	17.8605i	−0.924783	+	0.924783i	−0.997363	−	0.0725797i	$-0.976877\pi$
$373$	−17.8605	+	17.8605i	−0.924783	+	0.924783i	0.0725797	+	0.997363i	$0.476877\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−19.1624	+	19.1624i	−0.986912	+	0.986912i
$378$		0			0
$379$		−	15.0000i		−	0.770498i	−0.922813	−	0.385249i	$-0.874116\pi$
$379$		−	15.0000i		−	0.770498i	0.922813	−	0.385249i	$-0.125884\pi$
$380$		0			0
$381$			9.07888i			0.465125i
$382$		0			0
$383$	−25.8446	−	25.8446i	−1.32060	−	1.32060i	−0.913291	−	0.407309i	$-0.866467\pi$
$383$	−25.8446	−	25.8446i	−1.32060	−	1.32060i	−0.407309	−	0.913291i	$-0.633533\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	5.53170	+	5.53170i	0.281192	+	0.281192i
$388$		0			0
$389$			19.3746i			0.982331i	0.871066	+	0.491165i	$0.163429\pi$
$389$			19.3746i			0.982331i	−0.871066	+	0.491165i	$0.836571\pi$
$390$		0			0
$391$			9.55505i			0.483220i
$392$		0			0
$393$	7.94050	−	7.94050i	0.400545	−	0.400545i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.901503	−	0.901503i	0.0452451	−	0.0452451i	−0.684122	−	0.729367i	$-0.739815\pi$
$397$	0.901503	−	0.901503i	0.0452451	−	0.0452451i	0.729367	+	0.684122i	$0.239815\pi$
$398$		0			0
$399$	17.1687	−	1.36254i	0.859512	−	0.0682124i
$400$		0			0
$401$	6.27492			0.313354			0.156677	−	0.987650i	$-0.449922\pi$
$401$	6.27492			0.313354			0.156677	+	0.987650i	$0.449922\pi$
$402$		0			0
$403$	7.05246	−	7.05246i	0.351308	−	0.351308i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−32.8535	−	32.8535i	−1.62849	−	1.62849i
$408$		0			0
$409$	16.1797			0.800035			0.400018	−	0.916507i	$-0.369004\pi$
$409$	16.1797			0.800035			0.400018	+	0.916507i	$0.369004\pi$
$410$		0			0
$411$			4.41644i			0.217847i
$412$		0			0
$413$	−14.9681	+	17.5487i	−0.736532	+	0.863514i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.44949	+	2.44949i	0.119952	+	0.119952i
$418$		0			0
$419$	5.25370			0.256660			0.128330	−	0.991732i	$-0.459038\pi$
$419$	5.25370			0.256660			0.128330	+	0.991732i	$0.459038\pi$
$420$		0			0
$421$	8.27492			0.403295			0.201647	−	0.979458i	$-0.435370\pi$
$421$	8.27492			0.403295			0.201647	+	0.979458i	$0.435370\pi$
$422$		0			0
$423$	−1.41421	−	1.41421i	−0.0687614	−	0.0687614i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.13045	−	5.22895i	−0.296674	−	0.253047i
$428$		0			0
$429$			14.0000i			0.675926i
$430$		0			0
$431$	−19.6495			−0.946483			−0.473242	−	0.880933i	$-0.656916\pi$
$431$	−19.6495			−0.946483			−0.473242	+	0.880933i	$0.656916\pi$
$432$		0			0
$433$	0.901503	+	0.901503i	0.0433235	+	0.0433235i	0.728437	−	0.685113i	$-0.240247\pi$
$433$	0.901503	+	0.901503i	0.0433235	+	0.0433235i	−0.685113	+	0.728437i	$0.740247\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.9908	−	21.9908i	1.05196	−	1.05196i
$438$		0			0
$439$	−2.93039			−0.139860			−0.0699299	−	0.997552i	$-0.522278\pi$
$439$	−2.93039			−0.139860			−0.0699299	+	0.997552i	$0.522278\pi$
$440$		0			0
$441$	6.91238	−	1.10411i	0.329161	−	0.0525767i
$442$		0			0
$443$	12.8395	−	12.8395i	0.610022	−	0.610022i	−0.332930	−	0.942952i	$-0.608037\pi$
$443$	12.8395	−	12.8395i	0.610022	−	0.610022i	0.942952	+	0.332930i	$0.108037\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	11.1193	−	11.1193i	0.525925	−	0.525925i
$448$		0			0
$449$			14.8248i			0.699623i	0.936820	+	0.349812i	$0.113754\pi$
$449$			14.8248i			0.699623i	−0.936820	+	0.349812i	$0.886246\pi$
$450$		0			0
$451$			52.0766i			2.45219i
$452$		0			0
$453$	−7.77817	−	7.77817i	−0.365451	−	0.365451i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.54263	+	9.54263i	0.446386	+	0.446386i	0.894151	−	0.447765i	$-0.147780\pi$
$457$	9.54263	+	9.54263i	0.446386	+	0.446386i	−0.447765	+	0.894151i	$0.647780\pi$
$458$		0			0
$459$		−	2.00000i		−	0.0933520i
$460$		0			0
$461$			30.3397i			1.41306i	0.707684	+	0.706529i	$0.249740\pi$
$461$			30.3397i			1.41306i	−0.707684	+	0.706529i	$0.750260\pi$
$462$		0			0
$463$	−9.87934	+	9.87934i	−0.459132	+	0.459132i	−0.898370	−	0.439239i	$-0.855248\pi$
$463$	−9.87934	+	9.87934i	−0.459132	+	0.459132i	0.439239	+	0.898370i	$0.355248\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.2883	+	10.2883i	−0.476085	+	0.476085i	−0.903877	−	0.427792i	$-0.859292\pi$
$467$	−10.2883	+	10.2883i	−0.476085	+	0.476085i	0.427792	+	0.903877i	$0.359292\pi$
$468$		0			0
$469$	−11.4964	+	0.912376i	−0.530855	+	0.0421296i
$470$		0			0
$471$	11.8248			0.544856
$472$		0			0
$473$	−23.6475	+	23.6475i	−1.08732	+	1.08732i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	8.61390	+	8.61390i	0.394404	+	0.394404i
$478$		0			0
$479$	−24.2487			−1.10795			−0.553976	−	0.832533i	$-0.686890\pi$
$479$	−24.2487			−1.10795			−0.553976	+	0.832533i	$0.686890\pi$
$480$		0			0
$481$		−	35.5934i		−	1.62292i
$482$		0			0
$483$	8.20281	−	9.61702i	0.373241	−	0.437590i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−15.4110	−	15.4110i	−0.698341	−	0.698341i	0.265712	−	0.964052i	$-0.414393\pi$
$487$	−15.4110	−	15.4110i	−0.698341	−	0.698341i	−0.964052	+	0.265712i	$0.914393\pi$
$488$		0			0
$489$	17.0170			0.769534
$490$		0			0
$491$	−23.9244			−1.07969			−0.539847	−	0.841763i	$-0.681518\pi$
$491$	−23.9244			−1.07969			−0.539847	+	0.841763i	$0.681518\pi$
$492$		0			0
$493$	11.7025	+	11.7025i	0.527054	+	0.527054i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.96189	−	3.47254i	0.132859	−	0.155765i
$498$		0			0
$499$			31.8248i			1.42467i	0.701839	+	0.712336i	$0.252363\pi$
$499$			31.8248i			1.42467i	−0.701839	+	0.712336i	$0.747637\pi$
$500$		0			0
$501$	−4.54983			−0.203272
$502$		0			0
$503$	−8.48528	−	8.48528i	−0.378340	−	0.378340i	0.492163	−	0.870503i	$-0.336206\pi$
$503$	−8.48528	−	8.48528i	−0.378340	−	0.378340i	−0.870503	+	0.492163i	$0.836206\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.60861	−	1.60861i	0.0714409	−	0.0714409i
$508$		0			0
$509$	17.3205			0.767718			0.383859	−	0.923392i	$-0.374595\pi$
$509$	17.3205			0.767718			0.383859	+	0.923392i	$0.374595\pi$
$510$		0			0
$511$	26.3746	−	2.09313i	1.16674	−	0.0925948i
$512$		0			0
$513$	−4.60297	+	4.60297i	−0.203226	+	0.203226i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.04565	−	6.04565i	0.265887	−	0.265887i
$518$		0			0
$519$			11.0997i			0.487221i
$520$		0			0
$521$			40.7320i			1.78450i	0.451542	+	0.892250i	$0.350874\pi$
$521$			40.7320i			1.78450i	−0.451542	+	0.892250i	$0.649126\pi$
$522$		0			0
$523$	27.9125	+	27.9125i	1.22053	+	1.22053i	0.967444	+	0.253085i	$0.0814452\pi$
$523$	27.9125	+	27.9125i	1.22053	+	1.22053i	0.253085	+	0.967444i	$0.418555\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.30695	−	4.30695i	−0.187614	−	0.187614i
$528$		0			0
$529$			0.175248i			0.00761949i
$530$		0			0
$531$		−	8.71780i		−	0.378320i
$532$		0			0
$533$	−28.2098	+	28.2098i	−1.22190	+	1.22190i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	15.9451	−	15.9451i	0.688084	−	0.688084i
$538$		0			0
$539$	4.71998	+	29.5498i	0.203304	+	1.27280i
$540$		0			0
$541$	27.1993			1.16939			0.584695	−	0.811253i	$-0.301214\pi$
$541$	27.1993			1.16939			0.584695	+	0.811253i	$0.301214\pi$
$542$		0			0
$543$	−13.2169	+	13.2169i	−0.567190	+	0.567190i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.27720	+	8.27720i	0.353908	+	0.353908i	0.861561	−	0.507654i	$-0.169487\pi$
$547$	8.27720	+	8.27720i	0.353908	+	0.353908i	−0.507654	+	0.861561i	$0.669487\pi$
$548$		0			0
$549$	3.04547			0.129978
$550$		0			0
$551$		−	53.8662i		−	2.29478i
$552$		0			0
$553$	−21.7899	−	18.5856i	−0.926602	−	0.790342i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.8081	−	10.8081i	−0.457953	−	0.457953i	0.440030	−	0.897983i	$-0.354968\pi$
$557$	−10.8081	−	10.8081i	−0.457953	−	0.457953i	−0.897983	+	0.440030i	$0.854968\pi$
$558$		0			0
$559$	−25.6197			−1.08360
$560$		0			0
$561$	8.54983			0.360974
$562$		0			0
$563$	30.0873	+	30.0873i	1.26803	+	1.26803i	0.947106	+	0.320922i	$0.103993\pi$
$563$	30.0873	+	30.0873i	1.26803	+	1.26803i	0.320922	+	0.947106i	$0.396007\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.71696	+	2.01297i	−0.0721055	+	0.0845369i
$568$		0			0
$569$		−	19.1752i		−	0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$569$		−	19.1752i		−	0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$570$		0			0
$571$	−12.0997			−0.506355			−0.253178	−	0.967420i	$-0.581476\pi$
$571$	−12.0997			−0.506355			−0.253178	+	0.967420i	$0.581476\pi$
$572$		0			0
$573$	−1.80301	−	1.80301i	−0.0753216	−	0.0753216i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.34114	−	3.34114i	0.139093	−	0.139093i	−0.634132	−	0.773225i	$-0.718642\pi$
$577$	3.34114	−	3.34114i	0.139093	−	0.139093i	0.773225	+	0.634132i	$0.218642\pi$
$578$		0			0
$579$	−15.7035			−0.652617
$580$		0			0
$581$	3.82475	−	0.303539i	0.158677	−	0.0125929i
$582$		0			0
$583$	−36.8237	+	36.8237i	−1.52508	+	1.52508i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.04565	+	6.04565i	−0.249531	+	0.249531i	−0.820778	−	0.571247i	$-0.806460\pi$
$587$	−6.04565	+	6.04565i	−0.249531	+	0.249531i	0.571247	+	0.820778i	$0.306460\pi$
$588$		0			0
$589$			19.8248i			0.816865i
$590$		0			0
$591$			22.2131i			0.913726i
$592$		0			0
$593$	−14.1421	−	14.1421i	−0.580748	−	0.580748i	0.354361	−	0.935109i	$-0.384698\pi$
$593$	−14.1421	−	14.1421i	−0.580748	−	0.580748i	−0.935109	+	0.354361i	$0.884698\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.64448	−	7.64448i	−0.312868	−	0.312868i
$598$		0			0
$599$			15.3746i			0.628189i	0.949392	+	0.314094i	$0.101701\pi$
$599$			15.3746i			0.628189i	−0.949392	+	0.314094i	$0.898299\pi$
$600$		0			0
$601$		−	32.6630i		−	1.33235i	−0.745795	−	0.666175i	$-0.767930\pi$
$601$		−	32.6630i		−	1.33235i	0.745795	−	0.666175i	$-0.232070\pi$
$602$		0			0
$603$	3.08221	−	3.08221i	0.125517	−	0.125517i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−26.6222	+	26.6222i	−1.08056	+	1.08056i	−0.0841057	+	0.996457i	$0.526803\pi$
$607$	−26.6222	+	26.6222i	−1.08056	+	1.08056i	−0.996457	+	0.0841057i	$0.973197\pi$
$608$		0			0
$609$	−1.73205	−	21.8248i	−0.0701862	−	0.884384i
$610$		0			0
$611$	6.54983			0.264978
$612$		0			0
$613$	−8.19582	+	8.19582i	−0.331026	+	0.331026i	−0.852976	−	0.521950i	$-0.825205\pi$
$613$	−8.19582	+	8.19582i	−0.331026	+	0.331026i	0.521950	+	0.852976i	$0.325205\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.2991	−	16.2991i	−0.656176	−	0.656176i	0.298297	−	0.954473i	$-0.403581\pi$
$617$	−16.2991	−	16.2991i	−0.656176	−	0.656176i	−0.954473	+	0.298297i	$0.903581\pi$
$618$		0			0
$619$	−38.6388			−1.55303			−0.776513	−	0.630101i	$-0.783013\pi$
$619$	−38.6388			−1.55303			−0.776513	+	0.630101i	$0.783013\pi$
$620$		0			0
$621$			4.77753i			0.191716i
$622$		0			0
$623$	−13.5306	+	15.8633i	−0.542091	+	0.635550i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−19.6773	−	19.6773i	−0.785836	−	0.785836i
$628$		0			0
$629$	−21.7370			−0.866709
$630$		0			0
$631$	−29.0000			−1.15447			−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000			−1.15447			−0.577236	+	0.816577i	$0.695869\pi$
$632$		0			0
$633$	6.55836	+	6.55836i	0.260671	+	0.260671i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−13.4503	+	18.5639i	−0.532920	+	0.735529i
$638$		0			0
$639$			1.72508i			0.0682432i
$640$		0			0
$641$	−5.72508			−0.226127			−0.113064	−	0.993588i	$-0.536066\pi$
$641$	−5.72508			−0.226127			−0.113064	+	0.993588i	$0.536066\pi$
$642$		0			0
$643$	−21.3542	−	21.3542i	−0.842126	−	0.842126i	0.147009	−	0.989135i	$-0.453035\pi$
$643$	−21.3542	−	21.3542i	−0.842126	−	0.842126i	−0.989135	+	0.147009i	$0.953035\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.60601	+	3.60601i	−0.141767	+	0.141767i	−0.774428	−	0.632662i	$-0.781962\pi$
$647$	−3.60601	+	3.60601i	−0.141767	+	0.141767i	0.632662	+	0.774428i	$0.281962\pi$
$648$		0			0
$649$	37.2679			1.46289
$650$		0			0
$651$	0.637459	+	8.03231i	0.0249840	+	0.314811i
$652$		0			0
$653$	32.5168	−	32.5168i	1.27248	−	1.27248i	0.327697	−	0.944783i	$-0.393728\pi$
$653$	32.5168	−	32.5168i	1.27248	−	1.27248i	0.944783	−	0.327697i	$-0.106272\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7.07107	+	7.07107i	−0.275869	+	0.275869i
$658$		0			0
$659$			2.90033i			0.112981i	0.998403	+	0.0564904i	$0.0179910\pi$
$659$			2.90033i			0.112981i	−0.998403	+	0.0564904i	$0.982009\pi$
$660$		0			0
$661$			25.0860i			0.975731i	0.872919	+	0.487865i	$0.162224\pi$
$661$			25.0860i			0.975731i	−0.872919	+	0.487865i	$0.837776\pi$
$662$		0			0
$663$	4.63143	+	4.63143i	0.179870	+	0.179870i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−27.9545	−	27.9545i	−1.08240	−	1.08240i
$668$		0			0
$669$			17.8248i			0.689145i
$670$		0			0
$671$			13.0192i			0.502599i
$672$		0			0
$673$	31.9247	−	31.9247i	1.23061	−	1.23061i	0.266878	−	0.963730i	$-0.414008\pi$
$673$	31.9247	−	31.9247i	1.23061	−	1.23061i	0.963730	−	0.266878i	$-0.0859922\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.9451	−	15.9451i	0.612822	−	0.612822i	−0.330859	−	0.943680i	$-0.607338\pi$
$677$	15.9451	−	15.9451i	0.612822	−	0.612822i	0.943680	+	0.330859i	$0.107338\pi$
$678$		0			0
$679$	0.570396	+	7.18729i	0.0218898	+	0.275823i
$680$		0			0
$681$	−25.6495			−0.982891
$682$		0			0
$683$	29.1386	−	29.1386i	1.11496	−	1.11496i	0.122485	−	0.992470i	$-0.460914\pi$
$683$	29.1386	−	29.1386i	1.11496	−	1.11496i	0.992470	−	0.122485i	$-0.0390865\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	11.9514	+	11.9514i	0.455976	+	0.455976i
$688$		0			0
$689$	−39.8947			−1.51987
$690$		0			0
$691$		−	4.18627i		−	0.159253i	−0.996825	−	0.0796266i	$-0.974627\pi$
$691$		−	4.18627i		−	0.159253i	0.996825	−	0.0796266i	$-0.0253728\pi$
$692$		0			0
$693$	−8.60529	−	7.33985i	−0.326888	−	0.278818i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	17.2278	+	17.2278i	0.652550	+	0.652550i
$698$		0			0
$699$	−3.94027			−0.149035
$700$		0			0
$701$	45.2990			1.71092			0.855460	−	0.517869i	$-0.173275\pi$
$701$	45.2990			1.71092			0.855460	+	0.517869i	$0.173275\pi$
$702$		0			0
$703$	50.0272	+	50.0272i	1.88681	+	1.88681i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.1803	+	28.3011i	1.24787	+	1.06437i
$708$		0			0
$709$		−	3.27492i		−	0.122992i	−0.998107	−	0.0614960i	$-0.980413\pi$
$709$		−	3.27492i		−	0.122992i	0.998107	−	0.0614960i	$-0.0195872\pi$
$710$		0			0
$711$	10.8248			0.405960
$712$		0			0
$713$	10.2883	+	10.2883i	0.385299	+	0.385299i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−14.9197	+	14.9197i	−0.557187	+	0.557187i
$718$		0			0
$719$	28.7802			1.07332			0.536661	−	0.843798i	$-0.319685\pi$
$719$	28.7802			1.07332			0.536661	+	0.843798i	$0.319685\pi$
$720$		0			0
$721$	−22.5498	+	1.78959i	−0.839800	+	0.0666480i
$722$		0			0
$723$	−10.6860	+	10.6860i	−0.397417	+	0.397417i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−5.92173	+	5.92173i	−0.219625	+	0.219625i	−0.808340	−	0.588716i	$-0.799634\pi$
$727$	−5.92173	+	5.92173i	−0.219625	+	0.219625i	0.588716	+	0.808340i	$0.299634\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$			15.6460i			0.578688i
$732$		0			0
$733$	−27.2588	−	27.2588i	−1.00683	−	1.00683i	−0.999977	−	0.00685205i	$-0.997819\pi$
$733$	−27.2588	−	27.2588i	−1.00683	−	1.00683i	−0.00685205	−	0.999977i	$-0.502181\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.1762	+	13.1762i	0.485351	+	0.485351i
$738$		0			0
$739$		−	51.3746i		−	1.88984i	−0.327295	−	0.944922i	$-0.606137\pi$
$739$		−	51.3746i		−	1.88984i	0.327295	−	0.944922i	$-0.393863\pi$
$740$		0			0
$741$		−	21.3183i		−	0.783148i
$742$		0			0
$743$	1.85746	−	1.85746i	0.0681437	−	0.0681437i	−0.672214	−	0.740357i	$-0.734656\pi$
$743$	1.85746	−	1.85746i	0.0681437	−	0.0681437i	0.740357	+	0.672214i	$0.234656\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.02542	+	1.02542i	−0.0375182	+	0.0375182i
$748$		0			0
$749$	6.92820	−	0.549834i	0.253151	−	0.0200905i
$750$		0			0
$751$	24.0000			0.875772			0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000			0.875772			0.437886	+	0.899030i	$0.355727\pi$
$752$		0			0
$753$	1.85746	−	1.85746i	0.0676897	−	0.0676897i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−15.3297	−	15.3297i	−0.557166	−	0.557166i	0.371334	−	0.928499i	$-0.378901\pi$
$757$	−15.3297	−	15.3297i	−0.557166	−	0.557166i	−0.928499	+	0.371334i	$0.878901\pi$
$758$		0			0
$759$	−20.4235			−0.741327
$760$		0			0
$761$			13.8564i			0.502294i	0.967949	+	0.251147i	$0.0808078\pi$
$761$			13.8564i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$762$		0			0
$763$	2.01297	+	1.71696i	0.0728745	+	0.0621581i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.1879	+	20.1879i	0.728944	+	0.728944i
$768$		0			0
$769$	18.8066			0.678182			0.339091	−	0.940754i	$-0.389881\pi$
$769$	18.8066			0.678182			0.339091	+	0.940754i	$0.389881\pi$
$770$		0			0
$771$	−25.0997			−0.903942
$772$		0			0
$773$	16.9706	+	16.9706i	0.610389	+	0.610389i	0.943047	−	0.332659i	$-0.107946\pi$
$773$	16.9706	+	16.9706i	0.610389	+	0.610389i	−0.332659	+	0.943047i	$0.607946\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	21.8779	+	18.6607i	0.784867	+	0.669449i
$778$		0			0
$779$		−	79.2990i		−	2.84118i
$780$		0			0
$781$	−7.37459			−0.263883
$782$		0			0
$783$	5.85125	+	5.85125i	0.209107	+	0.209107i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	23.6699	−	23.6699i	0.843740	−	0.843740i	−0.145603	−	0.989343i	$-0.546512\pi$
$787$	23.6699	−	23.6699i	0.843740	−	0.843740i	0.989343	+	0.145603i	$0.0465123\pi$
$788$		0			0
$789$	5.61478			0.199891
$790$		0			0
$791$	0.625414	+	7.88054i	0.0222372	+	0.280200i
$792$		0			0
$793$	−7.05246	+	7.05246i	−0.250440	+	0.250440i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.43964	+	2.43964i	−0.0864163	+	0.0864163i	−0.748994	−	0.662577i	$-0.769463\pi$
$797$	−2.43964	+	2.43964i	−0.0864163	+	0.0864163i	0.662577	+	0.748994i	$0.269463\pi$
$798$		0			0
$799$		−	4.00000i		−	0.141510i
$800$		0			0
$801$		−	7.88054i		−	0.278445i
$802$		0			0
$803$	−30.2282	−	30.2282i	−1.06673	−	1.06673i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.0039	−	19.0039i	−0.668969	−	0.668969i
$808$		0			0
$809$		−	14.2749i		−	0.501879i	−0.968003	−	0.250940i	$-0.919260\pi$
$809$		−	14.2749i		−	0.501879i	0.968003	−	0.250940i	$-0.0807396\pi$
$810$		0			0
$811$			23.8301i			0.836787i	0.908266	+	0.418394i	$0.137407\pi$
$811$			23.8301i			0.836787i	−0.908266	+	0.418394i	$0.862593\pi$
$812$		0			0
$813$	3.63354	−	3.63354i	0.127434	−	0.127434i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	36.0090	−	36.0090i	1.25980	−	1.25980i
$818$		0			0
$819$	−0.685484	−	8.63746i	−0.0239527	−	0.301817i
$820$		0			0
$821$	−34.0000			−1.18661			−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000			−1.18661			−0.593304	+	0.804978i	$0.702177\pi$
$822$		0			0
$823$	18.4526	−	18.4526i	0.643216	−	0.643216i	−0.308129	−	0.951345i	$-0.599703\pi$
$823$	18.4526	−	18.4526i	0.643216	−	0.643216i	0.951345	+	0.308129i	$0.0997029\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.03199	+	9.03199i	0.314073	+	0.314073i	0.846485	−	0.532412i	$-0.178714\pi$
$827$	9.03199	+	9.03199i	0.314073	+	0.314073i	−0.532412	+	0.846485i	$0.678714\pi$
$828$		0			0
$829$	−25.3161			−0.879266			−0.439633	−	0.898178i	$-0.644891\pi$
$829$	−25.3161			−0.879266			−0.439633	+	0.898178i	$0.644891\pi$
$830$		0			0
$831$		−	9.97368i		−	0.345983i
$832$		0			0
$833$	11.3370	+	8.21413i	0.392805	+	0.284603i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.15348	−	2.15348i	−0.0744350	−	0.0744350i
$838$		0			0
$839$	33.0816			1.14210			0.571052	−	0.820914i	$-0.306536\pi$
$839$	33.0816			1.14210			0.571052	+	0.820914i	$0.306536\pi$
$840$		0			0
$841$	−39.4743			−1.36118
$842$		0			0
$843$	14.3365	+	14.3365i	0.493776	+	0.493776i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	12.4907	−	14.6442i	0.429186	−	0.503181i
$848$		0			0
$849$			4.17525i			0.143294i
$850$		0			0
$851$	51.9244			1.77995
$852$		0			0
$853$	18.8975	+	18.8975i	0.647038	+	0.647038i	0.952276	−	0.305238i	$-0.0987361\pi$
$853$	18.8975	+	18.8975i	0.647038	+	0.647038i	−0.305238	+	0.952276i	$0.598736\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.0913	−	12.0913i	0.413031	−	0.413031i	−0.469762	−	0.882793i	$-0.655660\pi$
$857$	12.0913	−	12.0913i	0.413031	−	0.413031i	0.882793	+	0.469762i	$0.155660\pi$
$858$		0			0
$859$	−27.1057			−0.924836			−0.462418	−	0.886662i	$-0.653018\pi$
$859$	−27.1057			−0.924836			−0.462418	+	0.886662i	$0.653018\pi$
$860$		0			0
$861$	−2.54983	−	32.1293i	−0.0868981	−	1.09496i
$862$		0			0
$863$	8.19582	−	8.19582i	0.278989	−	0.278989i	−0.553716	−	0.832705i	$-0.686791\pi$
$863$	8.19582	−	8.19582i	0.278989	−	0.278989i	0.832705	+	0.553716i	$0.186791\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−9.19239	+	9.19239i	−0.312190	+	0.312190i
$868$		0			0
$869$			46.2749i			1.56977i
$870$		0			0
$871$			14.2750i			0.483691i
$872$		0			0
$873$	−1.92692	−	1.92692i	−0.0652165	−	0.0652165i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.68434	+	4.68434i	0.158179	+	0.158179i	0.781759	−	0.623580i	$-0.214323\pi$
$877$	4.68434	+	4.68434i	0.158179	+	0.158179i	−0.623580	+	0.781759i	$0.714323\pi$
$878$		0			0
$879$			15.0997i			0.509299i
$880$		0			0
$881$		−	10.2772i		−	0.346248i	−0.984900	−	0.173124i	$-0.944614\pi$
$881$		−	10.2772i		−	0.346248i	0.984900	−	0.173124i	$-0.0553862\pi$
$882$		0			0
$883$	28.3319	−	28.3319i	0.953444	−	0.953444i	−0.0455194	−	0.998963i	$-0.514494\pi$
$883$	28.3319	−	28.3319i	0.953444	−	0.953444i	0.998963	+	0.0455194i	$0.0144943\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.9451	+	15.9451i	−0.535385	+	0.535385i	−0.922170	−	0.386785i	$-0.873586\pi$
$887$	−15.9451	+	15.9451i	−0.535385	+	0.535385i	0.386785	+	0.922170i	$0.373586\pi$
$888$		0			0
$889$	23.9452	−	1.90033i	0.803095	−	0.0637351i
$890$		0			0
$891$	4.27492			0.143215
$892$		0			0
$893$	−9.20593	+	9.20593i	−0.308065	+	0.308065i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−11.0634	−	11.0634i	−0.369396	−	0.369396i
$898$		0			0
$899$	25.2011			0.840502
$900$		0			0
$901$			24.3638i			0.811676i
$902$		0			0
$903$	13.4318	−	15.7475i	0.446981	−	0.524043i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	27.1071	+	27.1071i	0.900078	+	0.900078i	0.995442	−	0.0953644i	$-0.0304016\pi$
$907$	27.1071	+	27.1071i	0.900078	+	0.900078i	−0.0953644	+	0.995442i	$0.530402\pi$
$908$		0			0
$909$	−16.4833			−0.546715
$910$		0			0
$911$	−20.2749			−0.671738			−0.335869	−	0.941909i	$-0.609030\pi$
$911$	−20.2749			−0.671738			−0.335869	+	0.941909i	$0.609030\pi$
$912$		0			0
$913$	−4.38359	−	4.38359i	−0.145076	−	0.145076i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.6048	−	19.2807i	−0.746476	−	0.636704i
$918$		0			0
$919$			54.2990i			1.79116i	0.444902	+	0.895579i	$0.353239\pi$
$919$			54.2990i			1.79116i	−0.444902	+	0.895579i	$0.646761\pi$
$920$		0			0
$921$	23.2749			0.766935
$922$		0			0
$923$	−3.99480	−	3.99480i	−0.131491	−	0.131491i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	6.04565	−	6.04565i	0.198565	−	0.198565i
$928$		0			0
$929$	57.2152			1.87717			0.938585	−	0.345047i	$-0.112137\pi$
$929$	57.2152			1.87717			0.938585	+	0.345047i	$0.112137\pi$
$930$		0			0
$931$	−7.18729	−	44.9966i	−0.235554	−	1.47471i
$932$		0			0
$933$	13.5943	−	13.5943i	0.445056	−	0.445056i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−8.60920	+	8.60920i	−0.281250	+	0.281250i	−0.833608	−	0.552357i	$-0.813728\pi$
$937$	−8.60920	+	8.60920i	−0.281250	+	0.281250i	0.552357	+	0.833608i	$0.313728\pi$
$938$		0			0
$939$		−	35.2749i		−	1.15115i
$940$		0			0
$941$			51.2394i			1.67036i	0.549980	+	0.835178i	$0.314635\pi$
$941$			51.2394i			1.67036i	−0.549980	+	0.835178i	$0.685365\pi$
$942$		0			0
$943$	−41.1531	−	41.1531i	−1.34013	−	1.34013i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34.3742	+	34.3742i	1.11701	+	1.11701i	0.992178	+	0.124835i	$0.0398402\pi$
$947$	34.3742	+	34.3742i	1.11701	+	1.11701i	0.124835	+	0.992178i	$0.460160\pi$
$948$		0			0
$949$		−	32.7492i		−	1.06308i
$950$		0			0
$951$			15.1698i			0.491915i
$952$		0			0
$953$	−30.9960	+	30.9960i	−1.00406	+	1.00406i	−0.00406768	+	0.999992i	$0.501295\pi$
$953$	−30.9960	+	30.9960i	−1.00406	+	1.00406i	−0.999992	+	0.00406768i	$0.998705\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−25.0136	+	25.0136i	−0.808575	+	0.808575i
$958$		0			0
$959$	11.6482	−	0.924421i	0.376140	−	0.0298511i
$960$		0			0
$961$	21.7251			0.700809
$962$		0			0
$963$	−1.85746	+	1.85746i	−0.0598559	+	0.0598559i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.0813793	+	0.0813793i	0.00261698	+	0.00261698i	0.708414	−	0.705797i	$-0.249411\pi$
$967$	0.0813793	+	0.0813793i	0.00261698	+	0.00261698i	−0.705797	+	0.708414i	$0.749411\pi$
$968$		0			0
$969$	−13.0192			−0.418235
$970$		0			0
$971$		−	43.3588i		−	1.39145i	−0.718308	−	0.695725i	$-0.755083\pi$
$971$		−	43.3588i		−	1.39145i	0.718308	−	0.695725i	$-0.244917\pi$
$972$		0			0
$973$	5.94772	−	6.97314i	0.190675	−	0.223549i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.7070	+	15.7070i	0.502513	+	0.502513i	0.912218	−	0.409705i	$-0.134368\pi$
$977$	15.7070	+	15.7070i	0.502513	+	0.502513i	−0.409705	+	0.912218i	$0.634368\pi$
$978$		0			0
$979$	33.6887			1.07669
$980$		0			0
$981$	−1.00000			−0.0319275
$982$		0			0
$983$	−33.4454	−	33.4454i	−1.06674	−	1.06674i	−0.997607	−	0.0691370i	$-0.977975\pi$
$983$	−33.4454	−	33.4454i	−1.06674	−	1.06674i	−0.0691370	−	0.997607i	$-0.522025\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.43392	+	4.02594i	−0.109303	+	0.128147i
$988$		0			0
$989$		−	37.3746i		−	1.18844i
$990$		0			0
$991$	−6.45017			−0.204896			−0.102448	−	0.994738i	$-0.532668\pi$
$991$	−6.45017			−0.204896			−0.102448	+	0.994738i	$0.532668\pi$
$992$		0			0
$993$	−19.3568	−	19.3568i	−0.614268	−	0.614268i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−32.0312	+	32.0312i	−1.01444	+	1.01444i	−0.0145452	+	0.999894i	$0.504630\pi$
$997$	−32.0312	+	32.0312i	−1.01444	+	1.01444i	−0.999894	+	0.0145452i	$0.995370\pi$
$998$		0			0
$999$	−10.8685			−0.343863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.x.c.1357.7	yes	16
5.2	odd	4	inner	2100.2.x.c.1693.8	yes	16
5.3	odd	4	inner	2100.2.x.c.1693.1	yes	16
5.4	even	2	inner	2100.2.x.c.1357.2	yes	16
7.6	odd	2	inner	2100.2.x.c.1357.1	✓	16
35.13	even	4	inner	2100.2.x.c.1693.7	yes	16
35.27	even	4	inner	2100.2.x.c.1693.2	yes	16
35.34	odd	2	inner	2100.2.x.c.1357.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.x.c.1357.1	✓	16	7.6	odd	2	inner
2100.2.x.c.1357.2	yes	16	5.4	even	2	inner
2100.2.x.c.1357.7	yes	16	1.1	even	1	trivial
2100.2.x.c.1357.8	yes	16	35.34	odd	2	inner
2100.2.x.c.1693.1	yes	16	5.3	odd	4	inner
2100.2.x.c.1693.2	yes	16	35.27	even	4	inner
2100.2.x.c.1693.7	yes	16	35.13	even	4	inner
2100.2.x.c.1693.8	yes	16	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{3} +(1.71696 - 2.01297i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{3} +(1.71696 - 2.01297i) q^{7} +1.00000i q^{9} -4.27492 q^{11} +(-2.31572 - 2.31572i) q^{13} +(-1.41421 + 1.41421i) q^{17} +6.50958 q^{19} +(2.63746 - 0.209313i) q^{21} +(3.37822 - 3.37822i) q^{23} +(-0.707107 + 0.707107i) q^{27} -8.27492i q^{29} +3.04547i q^{31} +(-3.02282 - 3.02282i) q^{33} +(7.68517 + 7.68517i) q^{37} -3.27492i q^{39} -12.1819i q^{41} +(5.53170 - 5.53170i) q^{43} +(-1.41421 + 1.41421i) q^{47} +(-1.10411 - 6.91238i) q^{49} -2.00000 q^{51} +(8.61390 - 8.61390i) q^{53} +(4.60297 + 4.60297i) q^{57} -8.71780 q^{59} -3.04547i q^{61} +(2.01297 + 1.71696i) q^{63} +(-3.08221 - 3.08221i) q^{67} +4.77753 q^{69} +1.72508 q^{71} +(7.07107 + 7.07107i) q^{73} +(-7.33985 + 8.60529i) q^{77} -10.8248i q^{79} -1.00000 q^{81} +(1.02542 + 1.02542i) q^{83} +(5.85125 - 5.85125i) q^{87} -7.88054 q^{89} +(-8.63746 + 0.685484i) q^{91} +(-2.15348 + 2.15348i) q^{93} +(-1.92692 + 1.92692i) q^{97} -4.27492i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{11} + 12 q^{21} - 32 q^{51} + 88 q^{71} - 16 q^{81} - 108 q^{91}+O(q^{100}) \) 16 * q - 8 * q^11 + 12 * q^21 - 32 * q^51 + 88 * q^71 - 16 * q^81 - 108 * q^91

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