LMFDB - Embedded newform 1764.2.t.c.521.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(521,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.t (of order $6$, degree $2$, not 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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.8
Root			$-0.991445 + 0.130526i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.2.t.c.1097.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.923880 + 1.60021i) q^{5} +O(q^{10})$	$q+(0.923880 + 1.60021i) q^{5} +(1.73205 + 1.00000i) q^{11} +4.46088i q^{13} +(-1.14805 + 1.98848i) q^{17} +(-1.32565 + 0.765367i) q^{19} +(-7.64564 + 4.41421i) q^{23} +(0.792893 - 1.37333i) q^{25} +1.17157i q^{29} +(-5.07517 - 2.93015i) q^{31} +(-4.12132 - 7.13834i) q^{37} +11.8519 q^{41} +1.17157 q^{43} +(4.01254 + 6.94993i) q^{47} +(3.25397 + 1.87868i) q^{53} +3.69552i q^{55} +(-4.90923 + 8.50303i) q^{59} +(-10.6523 + 6.15013i) q^{61} +(-7.13834 + 4.12132i) q^{65} +(-6.24264 + 10.8126i) q^{67} -13.3137i q^{71} +(2.37676 + 1.37222i) q^{73} +(-5.65685 - 9.79796i) q^{79} +10.4525 q^{83} -4.24264 q^{85} +(7.23252 + 12.5271i) q^{89} +(-2.44949 - 1.41421i) q^{95} +2.74444i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67}+O(q^{100}) $ 16 * q + 24 * q^25 - 32 * q^37 + 64 * q^43 - 32 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	0.923880	+	1.60021i	0.413171	+	0.715634i	0.995235	−	0.0975096i	$-0.0310877\pi$
$5$	0.923880	+	1.60021i	0.413171	+	0.715634i	−0.582063	+	0.813144i	$0.697754\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	0.737848	−	0.674967i	$-0.235842\pi$
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	−0.215615	+	0.976478i	$0.569176\pi$
$12$		0			0
$13$			4.46088i			1.23723i	0.785695	+	0.618613i	$0.212305\pi$
$13$			4.46088i			1.23723i	−0.785695	+	0.618613i	$0.787695\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.14805	+	1.98848i	−0.278443	+	0.482278i	−0.970998	−	0.239088i	$-0.923152\pi$
$17$	−1.14805	+	1.98848i	−0.278443	+	0.482278i	0.692555	+	0.721365i	$0.256485\pi$
$18$		0			0
$19$	−1.32565	+	0.765367i	−0.304126	+	0.175587i	−0.644295	−	0.764777i	$-0.722849\pi$
$19$	−1.32565	+	0.765367i	−0.304126	+	0.175587i	0.340169	+	0.940364i	$0.389516\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.64564	+	4.41421i	−1.59423	+	0.920427i	−0.601656	+	0.798755i	$0.705492\pi$
$23$	−7.64564	+	4.41421i	−1.59423	+	0.920427i	−0.992570	+	0.121672i	$0.961174\pi$
$24$		0			0
$25$	0.792893	−	1.37333i	0.158579	−	0.274666i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.17157i			0.217556i	0.994066	+	0.108778i	$0.0346937\pi$
$29$			1.17157i			0.217556i	−0.994066	+	0.108778i	$0.965306\pi$
$30$		0			0
$31$	−5.07517	−	2.93015i	−0.911528	−	0.526271i	−0.0306053	−	0.999532i	$-0.509743\pi$
$31$	−5.07517	−	2.93015i	−0.911528	−	0.526271i	−0.880922	+	0.473261i	$0.843077\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.12132	−	7.13834i	−0.677541	−	1.17354i	−0.975719	−	0.219025i	$-0.929712\pi$
$37$	−4.12132	−	7.13834i	−0.677541	−	1.17354i	0.298178	−	0.954510i	$-0.403621\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.8519			1.85096			0.925480	−	0.378798i	$-0.123662\pi$
$41$	11.8519			1.85096			0.925480	+	0.378798i	$0.123662\pi$
$42$		0			0
$43$	1.17157			0.178663			0.0893316	−	0.996002i	$-0.471527\pi$
$43$	1.17157			0.178663			0.0893316	+	0.996002i	$0.471527\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.01254	+	6.94993i	0.585290	+	1.01375i	0.994839	+	0.101464i	$0.0323525\pi$
$47$	4.01254	+	6.94993i	0.585290	+	1.01375i	−0.409550	+	0.912288i	$0.634314\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.25397	+	1.87868i	0.446967	+	0.258056i	0.706548	−	0.707665i	$-0.250251\pi$
$53$	3.25397	+	1.87868i	0.446967	+	0.258056i	−0.259581	+	0.965721i	$0.583585\pi$
$54$		0			0
$55$			3.69552i			0.498304i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.90923	+	8.50303i	−0.639127	+	1.10700i	0.346498	+	0.938051i	$0.387371\pi$
$59$	−4.90923	+	8.50303i	−0.639127	+	1.10700i	−0.985625	+	0.168949i	$0.945963\pi$
$60$		0			0
$61$	−10.6523	+	6.15013i	−1.36389	+	0.787444i	−0.990140	−	0.140085i	$-0.955263\pi$
$61$	−10.6523	+	6.15013i	−1.36389	+	0.787444i	−0.373753	+	0.927528i	$0.621929\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.13834	+	4.12132i	−0.885402	+	0.511187i
$66$		0			0
$67$	−6.24264	+	10.8126i	−0.762660	+	1.32097i	0.178815	+	0.983883i	$0.442774\pi$
$67$	−6.24264	+	10.8126i	−0.762660	+	1.32097i	−0.941475	+	0.337083i	$0.890560\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.3137i		−	1.58005i	−0.613077	−	0.790023i	$-0.710068\pi$
$71$		−	13.3137i		−	1.58005i	0.613077	−	0.790023i	$-0.289932\pi$
$72$		0			0
$73$	2.37676	+	1.37222i	0.278178	+	0.160606i	0.632598	−	0.774480i	$-0.281988\pi$
$73$	2.37676	+	1.37222i	0.278178	+	0.160606i	−0.354420	+	0.935086i	$0.615322\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.65685	−	9.79796i	−0.636446	−	1.10236i	−0.986207	−	0.165518i	$-0.947071\pi$
$79$	−5.65685	−	9.79796i	−0.636446	−	1.10236i	0.349761	−	0.936839i	$-0.386263\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.4525			1.14731			0.573656	−	0.819097i	$-0.305525\pi$
$83$	10.4525			1.14731			0.573656	+	0.819097i	$0.305525\pi$
$84$		0			0
$85$	−4.24264			−0.460179
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.23252	+	12.5271i	0.766646	+	1.32787i	0.939372	+	0.342900i	$0.111409\pi$
$89$	7.23252	+	12.5271i	0.766646	+	1.32787i	−0.172726	+	0.984970i	$0.555258\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.44949	−	1.41421i	−0.251312	−	0.145095i
$96$		0			0
$97$			2.74444i			0.278656i	0.990246	+	0.139328i	$0.0444942\pi$
$97$			2.74444i			0.278656i	−0.990246	+	0.139328i	$0.955506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.37222	+	2.37676i	−0.136541	+	0.236496i	−0.926185	−	0.377069i	$-0.876932\pi$
$101$	−1.37222	+	2.37676i	−0.136541	+	0.236496i	0.789644	+	0.613565i	$0.210265\pi$
$102$		0			0
$103$	0.549104	−	0.317025i	0.0541048	−	0.0312374i	−0.472704	−	0.881221i	$-0.656722\pi$
$103$	0.549104	−	0.317025i	0.0541048	−	0.0312374i	0.526808	+	0.849984i	$0.323389\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.1097	+	6.41421i	−1.07402	+	0.620085i	−0.929277	−	0.369384i	$-0.879569\pi$
$107$	−11.1097	+	6.41421i	−1.07402	+	0.620085i	−0.144743	+	0.989469i	$0.546235\pi$
$108$		0			0
$109$	−1.53553	+	2.65962i	−0.147077	+	0.254746i	−0.930146	−	0.367190i	$-0.880320\pi$
$109$	−1.53553	+	2.65962i	−0.147077	+	0.254746i	0.783069	+	0.621935i	$0.213653\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.4142i			1.26190i	0.775822	+	0.630952i	$0.217335\pi$
$113$			13.4142i			1.26190i	−0.775822	+	0.630952i	$0.782665\pi$
$114$		0			0
$115$	−14.1273	−	8.15640i	−1.31738	−	0.760589i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.50000	−	6.06218i	−0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1689			1.08842
$126$		0			0
$127$	6.82843			0.605925			0.302962	−	0.953002i	$-0.402024\pi$
$127$	6.82843			0.605925			0.302962	+	0.953002i	$0.402024\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.67459	−	9.82868i	−0.495792	−	0.858736i	0.504197	−	0.863589i	$-0.331789\pi$
$131$	−5.67459	−	9.82868i	−0.495792	−	0.858736i	−0.999988	+	0.00485273i	$0.998455\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.47871	+	2.58579i	0.382642	+	0.220919i	0.678967	−	0.734169i	$-0.262428\pi$
$137$	4.47871	+	2.58579i	0.382642	+	0.220919i	−0.296325	+	0.955087i	$0.595761\pi$
$138$		0			0
$139$			6.49435i			0.550844i	0.961323	+	0.275422i	$0.0888176\pi$
$139$			6.49435i			0.550844i	−0.961323	+	0.275422i	$0.911182\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.46088	+	7.72648i	−0.373038	+	0.646121i
$144$		0			0
$145$	−1.87476	+	1.08239i	−0.155690	+	0.0898878i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.12372	+	3.53553i	−0.501675	+	0.289642i	−0.729405	−	0.684082i	$-0.760203\pi$
$149$	−6.12372	+	3.53553i	−0.501675	+	0.289642i	0.227730	+	0.973724i	$0.426870\pi$
$150$		0			0
$151$	9.07107	−	15.7116i	0.738193	−	1.27859i	−0.215115	−	0.976589i	$-0.569013\pi$
$151$	9.07107	−	15.7116i	0.738193	−	1.27859i	0.953308	−	0.301999i	$-0.0976540\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	10.8284i		−	0.869760i
$156$		0			0
$157$	9.71496	+	5.60894i	0.775338	+	0.447642i	0.834776	−	0.550590i	$-0.185597\pi$
$157$	9.71496	+	5.60894i	0.775338	+	0.447642i	−0.0594373	+	0.998232i	$0.518931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.41421	+	2.44949i	0.110770	+	0.191859i	0.916081	−	0.400994i	$-0.131335\pi$
$163$	1.41421	+	2.44949i	0.110770	+	0.191859i	−0.805311	+	0.592852i	$0.798002\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−16.3128			−1.26232			−0.631161	−	0.775652i	$-0.717421\pi$
$167$	−16.3128			−1.26232			−0.631161	+	0.775652i	$0.717421\pi$
$168$		0			0
$169$	−6.89949			−0.530730
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.05728	+	10.4915i	0.460526	+	0.797655i	0.998987	−	0.0449956i	$-0.0143274\pi$
$173$	6.05728	+	10.4915i	0.460526	+	0.797655i	−0.538461	+	0.842650i	$0.680994\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.5300	+	6.65685i	0.861793	+	0.497557i	0.864612	−	0.502439i	$-0.167564\pi$
$179$	11.5300	+	6.65685i	0.861793	+	0.497557i	−0.00281905	+	0.999996i	$0.500897\pi$
$180$		0			0
$181$			14.4650i			1.07518i	0.843207	+	0.537589i	$0.180665\pi$
$181$			14.4650i			1.07518i	−0.843207	+	0.537589i	$0.819335\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.61521	−	13.1899i	0.559881	−	0.969743i
$186$		0			0
$187$	−3.97696	+	2.29610i	−0.290824	+	0.167908i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.60181	−	2.65685i	0.332975	−	0.192243i	−0.324186	−	0.945993i	$-0.605090\pi$
$191$	4.60181	−	2.65685i	0.332975	−	0.192243i	0.657161	+	0.753750i	$0.271757\pi$
$192$		0			0
$193$	−0.828427	+	1.43488i	−0.0596315	+	0.103285i	−0.894300	−	0.447468i	$-0.852326\pi$
$193$	−0.828427	+	1.43488i	−0.0596315	+	0.103285i	0.834669	+	0.550753i	$0.185659\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			11.0711i			0.788781i	0.918943	+	0.394390i	$0.129044\pi$
$197$			11.0711i			0.788781i	−0.918943	+	0.394390i	$0.870956\pi$
$198$		0			0
$199$	−2.65131	−	1.53073i	−0.187946	−	0.108511i	0.403074	−	0.915167i	$-0.367942\pi$
$199$	−2.65131	−	1.53073i	−0.187946	−	0.108511i	−0.591021	+	0.806656i	$0.701275\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	10.9497	+	18.9655i	0.764764	+	1.32461i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.06147			−0.211766
$210$		0			0
$211$	26.6274			1.83311			0.916553	−	0.399912i	$-0.130959\pi$
$211$	26.6274			1.83311			0.916553	+	0.399912i	$0.130959\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.08239	+	1.87476i	0.0738185	+	0.127857i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.87039	−	5.12132i	−0.596687	−	0.344497i
$222$		0			0
$223$		−	21.2764i		−	1.42477i	−0.701786	−	0.712387i	$-0.747614\pi$
$223$		−	21.2764i		−	1.42477i	0.701786	−	0.712387i	$-0.252386\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.93015	+	5.07517i	−0.194481	+	0.336851i	−0.946730	−	0.322028i	$-0.895636\pi$
$227$	−2.93015	+	5.07517i	−0.194481	+	0.336851i	0.752249	+	0.658879i	$0.228969\pi$
$228$		0			0
$229$	11.4289	−	6.59847i	0.755242	−	0.436039i	−0.0723426	−	0.997380i	$-0.523048\pi$
$229$	11.4289	−	6.59847i	0.755242	−	0.436039i	0.827585	+	0.561340i	$0.189714\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0379	−	10.4142i	1.18171	−	0.682258i	0.225297	−	0.974290i	$-0.427665\pi$
$233$	18.0379	−	10.4142i	1.18171	−	0.682258i	0.956408	+	0.292032i	$0.0943315\pi$
$234$		0			0
$235$	−7.41421	+	12.8418i	−0.483650	+	0.837706i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3.17157i		−	0.205152i	−0.994725	−	0.102576i	$-0.967292\pi$
$239$		−	3.17157i		−	0.205152i	0.994725	−	0.102576i	$-0.0327085\pi$
$240$		0			0
$241$	−5.41634	−	3.12713i	−0.348897	−	0.201436i	0.315302	−	0.948991i	$-0.397894\pi$
$241$	−5.41634	−	3.12713i	−0.348897	−	0.201436i	−0.664199	+	0.747555i	$0.731227\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.41421	−	5.91359i	−0.217241	−	0.376273i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.7457			−1.24634			−0.623169	−	0.782088i	$-0.714155\pi$
$251$	−19.7457			−1.24634			−0.623169	+	0.782088i	$0.714155\pi$
$252$		0			0
$253$	−17.6569			−1.11008
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.56001	−	11.3623i	−0.409202	−	0.708759i	0.585598	−	0.810601i	$-0.300860\pi$
$257$	−6.56001	−	11.3623i	−0.409202	−	0.708759i	−0.994801	+	0.101842i	$0.967526\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.22538	−	4.17157i	−0.445536	−	0.257230i	0.260407	−	0.965499i	$-0.416143\pi$
$263$	−7.22538	−	4.17157i	−0.445536	−	0.257230i	−0.705943	+	0.708269i	$0.749477\pi$
$264$		0			0
$265$			6.94269i			0.426486i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.3659	−	21.4184i	0.753964	−	1.30590i	−0.191924	−	0.981410i	$-0.561473\pi$
$269$	12.3659	−	21.4184i	0.753964	−	1.30590i	0.945888	−	0.324493i	$-0.105194\pi$
$270$		0			0
$271$	20.5281	−	11.8519i	1.24700	−	0.719953i	0.276486	−	0.961018i	$-0.410830\pi$
$271$	20.5281	−	11.8519i	1.24700	−	0.719953i	0.970509	+	0.241065i	$0.0774967\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.74666	−	1.58579i	0.165630	−	0.0956265i
$276$		0			0
$277$	10.4853	−	18.1610i	0.630000	−	1.09119i	−0.357552	−	0.933893i	$-0.616388\pi$
$277$	10.4853	−	18.1610i	0.630000	−	1.09119i	0.987551	−	0.157298i	$-0.0502783\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6.48528i			0.386879i	0.981112	+	0.193440i	$0.0619644\pi$
$281$			6.48528i			0.386879i	−0.981112	+	0.193440i	$0.938036\pi$
$282$		0			0
$283$	27.7055	+	15.9958i	1.64692	+	0.950850i	0.978286	+	0.207260i	$0.0664547\pi$
$283$	27.7055	+	15.9958i	1.64692	+	0.950850i	0.668635	+	0.743590i	$0.266879\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.86396	+	10.1567i	0.344939	+	0.597452i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.34211			0.487351			0.243676	−	0.969857i	$-0.421647\pi$
$293$	8.34211			0.487351			0.243676	+	0.969857i	$0.421647\pi$
$294$		0			0
$295$	−18.1421			−1.05628
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.6913	−	34.1063i	−1.13878	−	1.97242i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.6830	−	11.3640i	−1.12704	−	0.650699i
$306$		0			0
$307$		−	28.5587i		−	1.62993i	−0.579510	−	0.814965i	$-0.696756\pi$
$307$		−	28.5587i		−	1.62993i	0.579510	−	0.814965i	$-0.303244\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.9552	+	18.9750i	−0.621215	+	1.07598i	0.368045	+	0.929808i	$0.380027\pi$
$311$	−10.9552	+	18.9750i	−0.621215	+	1.07598i	−0.989260	+	0.146167i	$0.953306\pi$
$312$		0			0
$313$	−19.6379	+	11.3379i	−1.11000	+	0.640857i	−0.938829	−	0.344385i	$-0.888088\pi$
$313$	−19.6379	+	11.3379i	−1.11000	+	0.640857i	−0.171169	+	0.985242i	$0.554754\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.9947	−	12.1213i	1.17918	−	0.680801i	0.223357	−	0.974737i	$-0.428298\pi$
$317$	20.9947	−	12.1213i	1.17918	−	0.680801i	0.955825	+	0.293936i	$0.0949651\pi$
$318$		0			0
$319$	−1.17157	+	2.02922i	−0.0655955	+	0.113615i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	3.51472i		−	0.195564i
$324$		0			0
$325$	6.12627	+	3.53701i	0.339824	+	0.196198i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.34315	+	4.05845i	0.128791	+	0.223072i	0.923208	−	0.384300i	$-0.125557\pi$
$331$	2.34315	+	4.05845i	0.128791	+	0.223072i	−0.794417	+	0.607372i	$0.792224\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−23.0698			−1.26044
$336$		0			0
$337$	−12.9289			−0.704284			−0.352142	−	0.935947i	$-0.614547\pi$
$337$	−12.9289			−0.704284			−0.352142	+	0.935947i	$0.614547\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.86030	−	10.1503i	−0.317353	−	0.549672i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.48617	+	4.89949i	0.455562	+	0.263019i	0.710176	−	0.704024i	$-0.248615\pi$
$347$	8.48617	+	4.89949i	0.455562	+	0.263019i	−0.254615	+	0.967043i	$0.581949\pi$
$348$		0			0
$349$		−	30.0669i		−	1.60944i	−0.593652	−	0.804722i	$-0.702315\pi$
$349$		−	30.0669i		−	1.60944i	0.593652	−	0.804722i	$-0.297685\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.2856	−	28.2075i	0.866796	−	1.50133i	0.00154235	−	0.999999i	$-0.499509\pi$
$353$	16.2856	−	28.2075i	0.866796	−	1.50133i	0.865253	−	0.501335i	$-0.167158\pi$
$354$		0			0
$355$	21.3047	−	12.3003i	1.13074	−	0.652830i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	27.2416	−	15.7279i	1.43775	−	0.830088i	0.440061	−	0.897968i	$-0.354957\pi$
$359$	27.2416	−	15.7279i	1.43775	−	0.830088i	0.997694	+	0.0678799i	$0.0216235\pi$
$360$		0			0
$361$	−8.32843	+	14.4253i	−0.438338	+	0.759224i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			5.07107i			0.265432i
$366$		0			0
$367$	−24.1834	−	13.9623i	−1.26237	−	0.728827i	−0.288833	−	0.957379i	$-0.593267\pi$
$367$	−24.1834	−	13.9623i	−1.26237	−	0.728827i	−0.973532	+	0.228553i	$0.926601\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	7.31371	+	12.6677i	0.378689	+	0.655909i	0.990872	−	0.134807i	$-0.0430415\pi$
$373$	7.31371	+	12.6677i	0.378689	+	0.655909i	−0.612182	+	0.790717i	$0.709708\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.22625			−0.269166
$378$		0			0
$379$	20.2843			1.04193			0.520967	−	0.853577i	$-0.325572\pi$
$379$	20.2843			1.04193			0.520967	+	0.853577i	$0.325572\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.06147	+	5.30262i	0.156434	+	0.270951i	0.933580	−	0.358369i	$-0.116667\pi$
$383$	3.06147	+	5.30262i	0.156434	+	0.270951i	−0.777146	+	0.629320i	$0.783334\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.91359	−	3.41421i	−0.299831	−	0.173107i	0.342536	−	0.939505i	$-0.388714\pi$
$389$	−5.91359	−	3.41421i	−0.299831	−	0.173107i	−0.642367	+	0.766397i	$0.722047\pi$
$390$		0			0
$391$		−	20.2710i		−	1.02515i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.4525	−	18.1043i	0.525923	−	0.910925i
$396$		0			0
$397$	6.90282	−	3.98535i	0.346443	−	0.200019i	−0.316675	−	0.948534i	$-0.602566\pi$
$397$	6.90282	−	3.98535i	0.346443	−	0.200019i	0.663117	+	0.748515i	$0.269233\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	19.4728	−	11.2426i	0.972426	−	0.561431i	0.0724514	−	0.997372i	$-0.476918\pi$
$401$	19.4728	−	11.2426i	0.972426	−	0.561431i	0.899975	+	0.435941i	$0.143584\pi$
$402$		0			0
$403$	13.0711	−	22.6398i	0.651116	−	1.12777i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.4853i		−	0.817145i
$408$		0			0
$409$	24.7796	+	14.3065i	1.22527	+	0.707413i	0.966038	−	0.258401i	$-0.0831956\pi$
$409$	24.7796	+	14.3065i	1.22527	+	0.707413i	0.259237	+	0.965814i	$0.416529\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	9.65685	+	16.7262i	0.474036	+	0.821055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.5194			−0.709321			−0.354661	−	0.934995i	$-0.615404\pi$
$419$	−14.5194			−0.709321			−0.354661	+	0.934995i	$0.615404\pi$
$420$		0			0
$421$	−7.31371			−0.356448			−0.178224	−	0.983990i	$-0.557035\pi$
$421$	−7.31371			−0.356448			−0.178224	+	0.983990i	$0.557035\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.82056	+	3.15331i	0.0883103	+	0.152958i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.25460	−	5.34315i	−0.445778	−	0.257370i	0.260267	−	0.965537i	$-0.416189\pi$
$431$	−9.25460	−	5.34315i	−0.445778	−	0.257370i	−0.706046	+	0.708166i	$0.749523\pi$
$432$		0			0
$433$			19.6913i			0.946303i	0.880981	+	0.473152i	$0.156884\pi$
$433$			19.6913i			0.946303i	−0.880981	+	0.473152i	$0.843116\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.75699	−	11.7034i	0.323230	−	0.559852i
$438$		0			0
$439$	−16.2295	+	9.37011i	−0.774592	+	0.447211i	−0.834510	−	0.550992i	$-0.814249\pi$
$439$	−16.2295	+	9.37011i	−0.774592	+	0.447211i	0.0599181	+	0.998203i	$0.480916\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.79050	−	3.34315i	0.275115	−	0.158838i	−0.356095	−	0.934450i	$-0.615892\pi$
$443$	5.79050	−	3.34315i	0.275115	−	0.158838i	0.631210	+	0.775612i	$0.282559\pi$
$444$		0			0
$445$	−13.3640	+	23.1471i	−0.633513	+	1.09728i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.5563i			0.922921i	0.887161	+	0.461461i	$0.152674\pi$
$449$			19.5563i			0.922921i	−0.887161	+	0.461461i	$0.847326\pi$
$450$		0			0
$451$	20.5281	+	11.8519i	0.966632	+	0.558085i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	0.999857	+	0.0168929i	$0.00537742\pi$
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	−0.485299	+	0.874348i	$0.661289\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−28.1647			−1.31176			−0.655881	−	0.754864i	$-0.727703\pi$
$461$	−28.1647			−1.31176			−0.655881	+	0.754864i	$0.727703\pi$
$462$		0			0
$463$	2.82843			0.131448			0.0657241	−	0.997838i	$-0.479064\pi$
$463$	2.82843			0.131448			0.0657241	+	0.997838i	$0.479064\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.6089	−	32.2316i	−0.861118	−	1.49150i	−0.870851	−	0.491547i	$-0.836432\pi$
$467$	−18.6089	−	32.2316i	−0.861118	−	1.49150i	0.00973373	−	0.999953i	$-0.496902\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.02922	+	1.17157i	0.0933038	+	0.0538690i
$474$		0			0
$475$			2.42742i			0.111378i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.4860	+	21.6263i	−0.570499	+	0.988133i	0.426016	+	0.904716i	$0.359917\pi$
$479$	−12.4860	+	21.6263i	−0.570499	+	0.988133i	−0.996515	+	0.0834170i	$0.973417\pi$
$480$		0			0
$481$	31.8433	−	18.3847i	1.45193	−	0.838272i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.39167	+	2.53553i	−0.199416	+	0.115133i
$486$		0			0
$487$	9.89949	−	17.1464i	0.448589	−	0.776979i	−0.549706	−	0.835359i	$-0.685260\pi$
$487$	9.89949	−	17.1464i	0.448589	−	0.776979i	0.998294	+	0.0583797i	$0.0185934\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.5147i		−	0.790428i	−0.918589	−	0.395214i	$-0.870670\pi$
$491$		−	17.5147i		−	0.790428i	0.918589	−	0.395214i	$-0.129330\pi$
$492$		0			0
$493$	−2.32965	−	1.34502i	−0.104922	−	0.0605769i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.07107	+	1.85514i	0.0479476	+	0.0830476i	0.889003	−	0.457901i	$-0.151399\pi$
$499$	1.07107	+	1.85514i	0.0479476	+	0.0830476i	−0.841056	+	0.540949i	$0.818065\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.2459			−0.546016			−0.273008	−	0.962012i	$-0.588019\pi$
$503$	−12.2459			−0.546016			−0.273008	+	0.962012i	$0.588019\pi$
$504$		0			0
$505$	−5.07107			−0.225660
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.67878	+	4.63979i	0.118735	+	0.205655i	0.919267	−	0.393635i	$-0.128783\pi$
$509$	2.67878	+	4.63979i	0.118735	+	0.205655i	−0.800532	+	0.599291i	$0.795449\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.01461	+	0.585786i	0.0447091	+	0.0258128i
$516$		0			0
$517$			16.0502i			0.705886i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.99789	+	13.8528i	−0.350394	+	0.606900i	−0.986318	−	0.164851i	$-0.947286\pi$
$521$	−7.99789	+	13.8528i	−0.350394	+	0.606900i	0.635925	+	0.771751i	$0.280619\pi$
$522$		0			0
$523$	25.2816	−	14.5964i	1.10549	−	0.638254i	0.167832	−	0.985816i	$-0.446323\pi$
$523$	25.2816	−	14.5964i	1.10549	−	0.638254i	0.937657	+	0.347561i	$0.112990\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11.6531	−	6.72792i	0.507617	−	0.293073i
$528$		0			0
$529$	27.4706	−	47.5804i	1.19437	−	2.06871i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			52.8701i			2.29006i
$534$		0			0
$535$	−20.5281	−	11.8519i	−0.887508	−	0.512403i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.82843	−	17.0233i	−0.422557	−	0.731890i	0.573632	−	0.819113i	$-0.305534\pi$
$541$	−9.82843	−	17.0233i	−0.422557	−	0.731890i	−0.996189	+	0.0872230i	$0.972201\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.67459			−0.243073
$546$		0			0
$547$	16.0000			0.684111			0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000			0.684111			0.342055	+	0.939680i	$0.388877\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.896683	−	1.55310i	−0.0382000	−	0.0661643i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.55860	+	4.36396i	0.320268	+	0.184907i	0.651512	−	0.758638i	$-0.274135\pi$
$557$	7.55860	+	4.36396i	0.320268	+	0.184907i	−0.331244	+	0.943545i	$0.607468\pi$
$558$		0			0
$559$			5.22625i			0.221047i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.4077	−	37.0793i	0.902229	−	1.56271i	0.0776342	−	0.996982i	$-0.475263\pi$
$563$	21.4077	−	37.0793i	0.902229	−	1.56271i	0.824595	−	0.565724i	$-0.191403\pi$
$564$		0			0
$565$	−21.4655	+	12.3931i	−0.903061	+	0.521382i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.2767	+	8.24264i	−0.598509	+	0.345549i	−0.768455	−	0.639904i	$-0.778974\pi$
$569$	−14.2767	+	8.24264i	−0.598509	+	0.345549i	0.169946	+	0.985453i	$0.445641\pi$
$570$		0			0
$571$	−21.3137	+	36.9164i	−0.891951	+	1.54490i	−0.0544175	+	0.998518i	$0.517330\pi$
$571$	−21.3137	+	36.9164i	−0.891951	+	1.54490i	−0.837533	+	0.546386i	$0.816003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			14.0000i			0.583840i
$576$		0			0
$577$	12.7545	+	7.36384i	0.530979	+	0.306561i	0.741415	−	0.671047i	$-0.234155\pi$
$577$	12.7545	+	7.36384i	0.530979	+	0.306561i	−0.210436	+	0.977608i	$0.567488\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.75736	+	6.50794i	0.155614	+	0.269531i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.9414			0.657971			0.328986	−	0.944335i	$-0.393293\pi$
$587$	15.9414			0.657971			0.328986	+	0.944335i	$0.393293\pi$
$588$		0			0
$589$	8.97056			0.369626
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.04473	+	3.54158i	0.0839671	+	0.145435i	0.904951	−	0.425517i	$-0.139908\pi$
$593$	2.04473	+	3.54158i	0.0839671	+	0.145435i	−0.820983	+	0.570952i	$0.806574\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.5300	−	6.65685i	−0.471103	−	0.271992i	0.245598	−	0.969372i	$-0.421016\pi$
$599$	−11.5300	−	6.65685i	−0.471103	−	0.271992i	−0.716702	+	0.697380i	$0.754349\pi$
$600$		0			0
$601$		−	42.0501i		−	1.71526i	−0.514267	−	0.857630i	$-0.671936\pi$
$601$		−	42.0501i		−	1.71526i	0.514267	−	0.857630i	$-0.328064\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.46716	−	11.2014i	0.262927	−	0.455403i
$606$		0			0
$607$	−0.776550	+	0.448342i	−0.0315192	+	0.0181976i	−0.515677	−	0.856783i	$-0.672459\pi$
$607$	−0.776550	+	0.448342i	−0.0315192	+	0.0181976i	0.484158	+	0.874981i	$0.339126\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−31.0028	+	17.8995i	−1.25424	+	0.724136i
$612$		0			0
$613$	1.39340	−	2.41344i	0.0562788	−	0.0974778i	−0.836513	−	0.547946i	$-0.815410\pi$
$613$	1.39340	−	2.41344i	0.0562788	−	0.0974778i	0.892792	+	0.450469i	$0.148743\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	33.4558i		−	1.34688i	−0.739241	−	0.673441i	$-0.764816\pi$
$617$		−	33.4558i		−	1.34688i	0.739241	−	0.673441i	$-0.235184\pi$
$618$		0			0
$619$	−21.8538	−	12.6173i	−0.878378	−	0.507132i	−0.00825456	−	0.999966i	$-0.502628\pi$
$619$	−21.8538	−	12.6173i	−0.878378	−	0.507132i	−0.870123	+	0.492834i	$0.835961\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.27817	+	12.6062i	0.291127	+	0.504247i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.9259			0.754626
$630$		0			0
$631$	4.48528			0.178556			0.0892781	−	0.996007i	$-0.471544\pi$
$631$	4.48528			0.178556			0.0892781	+	0.996007i	$0.471544\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.30864	+	10.9269i	0.250351	+	0.433620i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.7775	+	13.7279i	0.939153	+	0.542220i	0.889695	−	0.456556i	$-0.150917\pi$
$641$	23.7775	+	13.7279i	0.939153	+	0.542220i	0.0494584	+	0.998776i	$0.484250\pi$
$642$		0			0
$643$			47.1451i			1.85922i	0.368546	+	0.929610i	$0.379856\pi$
$643$			47.1451i			1.85922i	−0.368546	+	0.929610i	$0.620144\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.1606	+	31.4550i	−0.713965	+	1.23662i	0.249392	+	0.968403i	$0.419769\pi$
$647$	−18.1606	+	31.4550i	−0.713965	+	1.23662i	−0.963357	+	0.268222i	$0.913564\pi$
$648$		0			0
$649$	−17.0061	+	9.81845i	−0.667546	+	0.385408i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.5680	−	17.0711i	1.15708	−	0.668043i	0.206480	−	0.978451i	$-0.433799\pi$
$653$	29.5680	−	17.0711i	1.15708	−	0.668043i	0.950603	+	0.310408i	$0.100466\pi$
$654$		0			0
$655$	10.4853	−	18.1610i	0.409694	−	0.709611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	30.9706i		−	1.20644i	−0.797574	−	0.603221i	$-0.793884\pi$
$659$		−	30.9706i		−	1.20644i	0.797574	−	0.603221i	$-0.206116\pi$
$660$		0			0
$661$	0.890273	+	0.514000i	0.0346276	+	0.0199923i	0.517214	−	0.855856i	$-0.326969\pi$
$661$	0.890273	+	0.514000i	0.0346276	+	0.0199923i	−0.482586	+	0.875848i	$0.660303\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.17157	−	8.95743i	−0.200244	−	0.346833i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.6005			−0.949693
$672$		0			0
$673$	−28.0416			−1.08093			−0.540463	−	0.841368i	$-0.681751\pi$
$673$	−28.0416			−1.08093			−0.540463	+	0.841368i	$0.681751\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.02928	+	8.71096i	0.193291	+	0.334790i	0.946339	−	0.323176i	$-0.104751\pi$
$677$	5.02928	+	8.71096i	0.193291	+	0.334790i	−0.753048	+	0.657966i	$0.771417\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.2838	+	6.51472i	0.431764	+	0.249279i	0.700098	−	0.714047i	$-0.253140\pi$
$683$	11.2838	+	6.51472i	0.431764	+	0.249279i	−0.268334	+	0.963326i	$0.586473\pi$
$684$		0			0
$685$			9.55582i			0.365109i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.38057	+	14.5156i	−0.319274	+	0.553000i
$690$		0			0
$691$	−27.9330	+	16.1271i	−1.06262	+	0.613504i	−0.926156	−	0.377141i	$-0.876907\pi$
$691$	−27.9330	+	16.1271i	−1.06262	+	0.613504i	−0.136464	+	0.990645i	$0.543574\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.3923	+	6.00000i	−0.394203	+	0.227593i
$696$		0			0
$697$	−13.6066	+	23.5673i	−0.515387	+	0.892676i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	35.4558i		−	1.33915i	−0.742745	−	0.669574i	$-0.766477\pi$
$701$		−	35.4558i		−	1.33915i	0.742745	−	0.669574i	$-0.233523\pi$
$702$		0			0
$703$	10.9269	+	6.30864i	0.412116	+	0.237935i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.4350	−	19.8061i	−0.429452	−	0.743832i	0.567373	−	0.823461i	$-0.307960\pi$
$709$	−11.4350	−	19.8061i	−0.429452	−	0.743832i	−0.996825	+	0.0796290i	$0.974626\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	51.7373			1.93758
$714$		0			0
$715$	−16.4853			−0.616515
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.2513	+	22.9520i	0.494192	+	0.855965i	0.999978	−	0.00669409i	$-0.00213081\pi$
$719$	13.2513	+	22.9520i	0.494192	+	0.855965i	−0.505786	+	0.862659i	$0.668797\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.60896	+	0.928932i	0.0597552	+	0.0344997i
$726$		0			0
$727$		−	25.3434i		−	0.939933i	−0.882684	−	0.469967i	$-0.844266\pi$
$727$		−	25.3434i		−	0.939933i	0.882684	−	0.469967i	$-0.155734\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.34502	+	2.32965i	−0.0497475	+	0.0861653i
$732$		0			0
$733$	−1.37276	+	0.792563i	−0.0507040	+	0.0292740i	−0.525138	−	0.851017i	$-0.675986\pi$
$733$	−1.37276	+	0.792563i	−0.0507040	+	0.0292740i	0.474434	+	0.880291i	$0.342653\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.6251	+	12.4853i	−0.796572	+	0.459901i
$738$		0			0
$739$	−26.7279	+	46.2941i	−0.983203	+	1.70296i	−0.333536	+	0.942737i	$0.608242\pi$
$739$	−26.7279	+	46.2941i	−0.983203	+	1.70296i	−0.649667	+	0.760219i	$0.725091\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.3431i			1.18655i	0.804998	+	0.593277i	$0.202166\pi$
$743$			32.3431i			1.18655i	−0.804998	+	0.593277i	$0.797834\pi$
$744$		0			0
$745$	−11.3152	−	6.53281i	−0.414556	−	0.239344i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.1421	−	17.5667i	−0.370092	−	0.641018i	0.619488	−	0.785006i	$-0.287340\pi$
$751$	−10.1421	−	17.5667i	−0.370092	−	0.641018i	−0.989579	+	0.143989i	$0.954007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	33.5223			1.22000
$756$		0			0
$757$	37.8995			1.37748			0.688740	−	0.725008i	$-0.258164\pi$
$757$	37.8995			1.37748			0.688740	+	0.725008i	$0.258164\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.28692	+	12.6213i	0.264151	+	0.457522i	0.967341	−	0.253480i	$-0.0815751\pi$
$761$	7.28692	+	12.6213i	0.264151	+	0.457522i	−0.703190	+	0.711002i	$0.748242\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−37.9310	−	21.8995i	−1.36961	−	0.790745i
$768$		0			0
$769$			13.5684i			0.489288i	0.969613	+	0.244644i	$0.0786710\pi$
$769$			13.5684i			0.489288i	−0.969613	+	0.244644i	$0.921329\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.11874	+	15.7941i	−0.327978	+	0.568075i	−0.982111	−	0.188305i	$-0.939701\pi$
$773$	−9.11874	+	15.7941i	−0.327978	+	0.568075i	0.654132	+	0.756380i	$0.273034\pi$
$774$		0			0
$775$	−8.04814	+	4.64659i	−0.289098	+	0.166911i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.7116	+	9.07107i	−0.562925	+	0.325005i
$780$		0			0
$781$	13.3137	−	23.0600i	0.476402	−	0.825152i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			20.7279i			0.739811i
$786$		0			0
$787$	3.10620	+	1.79337i	0.110724	+	0.0639266i	0.554339	−	0.832291i	$-0.312971\pi$
$787$	3.10620	+	1.79337i	0.110724	+	0.0639266i	−0.443615	+	0.896217i	$0.646304\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−27.4350	−	47.5189i	−0.974246	−	1.68744i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	30.4064			1.07705			0.538526	−	0.842609i	$-0.318982\pi$
$797$	30.4064			1.07705			0.538526	+	0.842609i	$0.318982\pi$
$798$		0			0
$799$	−18.4264			−0.651879
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.74444	+	4.75351i	0.0968493	+	0.167748i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.3712	−	10.6066i	−0.645896	−	0.372908i	0.140986	−	0.990012i	$-0.454973\pi$
$809$	−18.3712	−	10.6066i	−0.645896	−	0.372908i	−0.786882	+	0.617103i	$0.788306\pi$
$810$		0			0
$811$		−	13.1426i		−	0.461497i	−0.973013	−	0.230749i	$-0.925882\pi$
$811$		−	13.1426i		−	0.461497i	0.973013	−	0.230749i	$-0.0741175\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.61313	+	4.52607i	−0.0915338	+	0.158541i
$816$		0			0
$817$	−1.55310	+	0.896683i	−0.0543361	+	0.0313710i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.9814	+	18.4645i	−1.11616	+	0.644414i	−0.940417	−	0.340022i	$-0.889565\pi$
$821$	−31.9814	+	18.4645i	−1.11616	+	0.644414i	−0.175741	+	0.984436i	$0.556232\pi$
$822$		0			0
$823$	−8.14214	+	14.1026i	−0.283817	+	0.491585i	−0.972322	−	0.233646i	$-0.924934\pi$
$823$	−8.14214	+	14.1026i	−0.283817	+	0.491585i	0.688505	+	0.725232i	$0.258267\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.51472i		−	0.0526719i	−0.999653	−	0.0263360i	$-0.991616\pi$
$827$		−	1.51472i		−	0.0526719i	0.999653	−	0.0263360i	$-0.00838397\pi$
$828$		0			0
$829$	−6.05966	−	3.49854i	−0.210461	−	0.121509i	0.391065	−	0.920363i	$-0.372107\pi$
$829$	−6.05966	−	3.49854i	−0.210461	−	0.121509i	−0.601525	+	0.798854i	$0.705440\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−15.0711	−	26.1039i	−0.521556	−	0.903361i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.6842			0.576003			0.288002	−	0.957630i	$-0.407009\pi$
$839$	16.6842			0.576003			0.288002	+	0.957630i	$0.407009\pi$
$840$		0			0
$841$	27.6274			0.952670
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.37430	−	11.0406i	−0.219283	−	0.379809i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	63.0203	+	36.3848i	2.16031	+	1.24725i
$852$		0			0
$853$			23.7264i			0.812376i	0.913790	+	0.406188i	$0.133142\pi$
$853$			23.7264i			0.812376i	−0.913790	+	0.406188i	$0.866858\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.1124	+	34.8358i	−0.687028	+	1.18997i	0.285767	+	0.958299i	$0.407752\pi$
$857$	−20.1124	+	34.8358i	−0.687028	+	1.18997i	−0.972795	+	0.231668i	$0.925582\pi$
$858$		0			0
$859$	28.8037	−	16.6298i	0.982770	−	0.567403i	0.0796647	−	0.996822i	$-0.474615\pi$
$859$	28.8037	−	16.6298i	0.982770	−	0.567403i	0.903105	+	0.429419i	$0.141282\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.1405	+	18.5563i	−1.09408	+	0.631665i	−0.934659	−	0.355546i	$-0.884295\pi$
$863$	−32.1405	+	18.5563i	−1.09408	+	0.631665i	−0.159418	+	0.987211i	$0.550962\pi$
$864$		0			0
$865$	−11.1924	+	19.3858i	−0.380553	+	0.659136i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	22.6274i		−	0.767583i
$870$		0			0
$871$	−48.2336	−	27.8477i	−1.63433	−	0.943583i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.53553	−	13.0519i	−0.254457	−	0.440732i	0.710291	−	0.703908i	$-0.248563\pi$
$877$	−7.53553	−	13.0519i	−0.254457	−	0.440732i	−0.964748	+	0.263176i	$0.915230\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.8155			−0.566530			−0.283265	−	0.959042i	$-0.591418\pi$
$881$	−16.8155			−0.566530			−0.283265	+	0.959042i	$0.591418\pi$
$882$		0			0
$883$	−50.1421			−1.68742			−0.843709	−	0.536801i	$-0.819632\pi$
$883$	−50.1421			−1.68742			−0.843709	+	0.536801i	$0.819632\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0209	+	41.6054i	0.806542	+	1.39697i	0.915245	+	0.402897i	$0.131997\pi$
$887$	24.0209	+	41.6054i	0.806542	+	1.39697i	−0.108703	+	0.994074i	$0.534670\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.6385	−	6.14214i	−0.356004	−	0.205539i
$894$		0			0
$895$			24.6005i			0.822305i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.43289	−	5.94593i	0.114493	−	0.198308i
$900$		0			0
$901$	−7.47144	+	4.31364i	−0.248910	+	0.143708i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23.1471	+	13.3640i	−0.769434	+	0.444233i
$906$		0			0
$907$	7.75736	−	13.4361i	0.257579	−	0.446140i	−0.708014	−	0.706198i	$-0.750409\pi$
$907$	7.75736	−	13.4361i	0.257579	−	0.446140i	0.965593	+	0.260059i	$0.0837419\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	14.2843i		−	0.473259i	−0.971600	−	0.236630i	$-0.923957\pi$
$911$		−	14.2843i		−	0.473259i	0.971600	−	0.236630i	$-0.0760428\pi$
$912$		0			0
$913$	18.1043	+	10.4525i	0.599164	+	0.345927i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	16.2426	+	28.1331i	0.535795	+	0.928025i	0.999124	+	0.0418384i	$0.0133215\pi$
$919$	16.2426	+	28.1331i	0.535795	+	0.928025i	−0.463329	+	0.886186i	$0.653345\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	59.3909			1.95488
$924$		0			0
$925$	−13.0711			−0.429774
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.792563	+	1.37276i	0.0260032	+	0.0450388i	0.878734	−	0.477312i	$-0.158389\pi$
$929$	0.792563	+	1.37276i	0.0260032	+	0.0450388i	−0.852731	+	0.522350i	$0.825055\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7.34847	−	4.24264i	−0.240321	−	0.138749i
$936$		0			0
$937$			45.1116i			1.47373i	0.676039	+	0.736866i	$0.263695\pi$
$937$			45.1116i			1.47373i	−0.676039	+	0.736866i	$0.736305\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.1564	+	36.6439i	−0.689678	+	1.19456i	0.282264	+	0.959337i	$0.408915\pi$
$941$	−21.1564	+	36.6439i	−0.689678	+	1.19456i	−0.971942	+	0.235221i	$0.924419\pi$
$942$		0			0
$943$	−90.6155	+	52.3169i	−2.95085	+	1.70367i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.74666	+	1.58579i	−0.0892545	+	0.0515311i	−0.543963	−	0.839109i	$-0.683077\pi$
$947$	−2.74666	+	1.58579i	−0.0892545	+	0.0515311i	0.454708	+	0.890640i	$0.349743\pi$
$948$		0			0
$949$	−6.12132	+	10.6024i	−0.198706	+	0.344170i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			5.12994i			0.166175i	0.996542	+	0.0830876i	$0.0264781\pi$
$953$			5.12994i			0.166175i	−0.996542	+	0.0830876i	$0.973522\pi$
$954$		0			0
$955$	8.50303	+	4.90923i	0.275152	+	0.158859i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.67157	+	2.89525i	0.0539217	+	0.0933951i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.06147			−0.0985521
$966$		0			0
$967$	32.4853			1.04466			0.522328	−	0.852745i	$-0.325064\pi$
$967$	32.4853			1.04466			0.522328	+	0.852745i	$0.325064\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.896683	−	1.55310i	−0.0287759	−	0.0498414i	0.851279	−	0.524714i	$-0.175828\pi$
$971$	−0.896683	−	1.55310i	−0.0287759	−	0.0498414i	−0.880055	+	0.474872i	$0.842494\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.2767	+	8.24264i	0.456751	+	0.263705i	0.710677	−	0.703518i	$-0.248389\pi$
$977$	14.2767	+	8.24264i	0.456751	+	0.263705i	−0.253926	+	0.967224i	$0.581722\pi$
$978$		0			0
$979$			28.9301i			0.924610i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	13.6997	−	23.7285i	0.436952	−	0.756823i	−0.560501	−	0.828154i	$-0.689391\pi$
$983$	13.6997	−	23.7285i	0.436952	−	0.756823i	0.997453	+	0.0713309i	$0.0227246\pi$
$984$		0			0
$985$	−17.7160	+	10.2283i	−0.564478	+	0.325902i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.95743	+	5.17157i	−0.284830	+	0.164446i
$990$		0			0
$991$	19.0711	−	33.0321i	0.605812	−	1.04930i	−0.386110	−	0.922453i	$-0.626182\pi$
$991$	19.0711	−	33.0321i	0.605812	−	1.04930i	0.991923	−	0.126845i	$-0.0404851\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.65685i		−	0.179334i
$996$		0			0
$997$	−35.5457	−	20.5223i	−1.12574	−	0.649949i	−0.182883	−	0.983135i	$-0.558543\pi$
$997$	−35.5457	−	20.5223i	−1.12574	−	0.649949i	−0.942861	+	0.333186i	$0.891876\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.t.c.521.8		16
3.2	odd	2	inner	1764.2.t.c.521.1		16
7.2	even	3	inner	1764.2.t.c.1097.7		16
7.3	odd	6		1764.2.f.b.881.7	yes	8
7.4	even	3		1764.2.f.b.881.1	✓	8
7.5	odd	6	inner	1764.2.t.c.1097.1		16
7.6	odd	2	inner	1764.2.t.c.521.2		16
21.2	odd	6	inner	1764.2.t.c.1097.2		16
21.5	even	6	inner	1764.2.t.c.1097.8		16
21.11	odd	6		1764.2.f.b.881.8	yes	8
21.17	even	6		1764.2.f.b.881.2	yes	8
21.20	even	2	inner	1764.2.t.c.521.7		16
28.3	even	6		7056.2.k.e.881.8		8
28.11	odd	6		7056.2.k.e.881.2		8
84.11	even	6		7056.2.k.e.881.7		8
84.59	odd	6		7056.2.k.e.881.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.f.b.881.1	✓	8	7.4	even	3
1764.2.f.b.881.2	yes	8	21.17	even	6
1764.2.f.b.881.7	yes	8	7.3	odd	6
1764.2.f.b.881.8	yes	8	21.11	odd	6
1764.2.t.c.521.1		16	3.2	odd	2	inner
1764.2.t.c.521.2		16	7.6	odd	2	inner
1764.2.t.c.521.7		16	21.20	even	2	inner
1764.2.t.c.521.8		16	1.1	even	1	trivial
1764.2.t.c.1097.1		16	7.5	odd	6	inner
1764.2.t.c.1097.2		16	21.2	odd	6	inner
1764.2.t.c.1097.7		16	7.2	even	3	inner
1764.2.t.c.1097.8		16	21.5	even	6	inner
7056.2.k.e.881.1		8	84.59	odd	6
7056.2.k.e.881.2		8	28.11	odd	6
7056.2.k.e.881.7		8	84.11	even	6
7056.2.k.e.881.8		8	28.3	even	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.923880 + 1.60021i) q^{5} +O(q^{10})\)	\(q+(0.923880 + 1.60021i) q^{5} +(1.73205 + 1.00000i) q^{11} +4.46088i q^{13} +(-1.14805 + 1.98848i) q^{17} +(-1.32565 + 0.765367i) q^{19} +(-7.64564 + 4.41421i) q^{23} +(0.792893 - 1.37333i) q^{25} +1.17157i q^{29} +(-5.07517 - 2.93015i) q^{31} +(-4.12132 - 7.13834i) q^{37} +11.8519 q^{41} +1.17157 q^{43} +(4.01254 + 6.94993i) q^{47} +(3.25397 + 1.87868i) q^{53} +3.69552i q^{55} +(-4.90923 + 8.50303i) q^{59} +(-10.6523 + 6.15013i) q^{61} +(-7.13834 + 4.12132i) q^{65} +(-6.24264 + 10.8126i) q^{67} -13.3137i q^{71} +(2.37676 + 1.37222i) q^{73} +(-5.65685 - 9.79796i) q^{79} +10.4525 q^{83} -4.24264 q^{85} +(7.23252 + 12.5271i) q^{89} +(-2.44949 - 1.41421i) q^{95} +2.74444i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67}+O(q^{100}) \) 16 * q + 24 * q^25 - 32 * q^37 + 64 * q^43 - 32 * q^67

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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