LMFDB - Embedded newform 1764.2.k.d.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(361,1764)]

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, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

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.k (of order $3$, 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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.2.k.d.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+(-3.00000 - 5.19615i) q^{11} -2.00000 q^{13} +(-2.00000 + 3.46410i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} -6.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-1.00000 + 1.73205i) q^{37} +12.0000 q^{41} -4.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(-3.00000 - 5.19615i) q^{53} +(-5.00000 + 8.66025i) q^{61} +(-4.00000 - 6.92820i) q^{67} -6.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(2.00000 - 3.46410i) q^{79} -12.0000 q^{83} +(-6.00000 + 10.3923i) q^{89} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 6 q^{11} - 4 q^{13} - 4 q^{19} - 6 q^{23} + 5 q^{25} - 12 q^{29} + 8 q^{31} - 2 q^{37} + 24 q^{41} - 8 q^{43} - 12 q^{47} - 6 q^{53} - 10 q^{61} - 8 q^{67} - 12 q^{71} - 10 q^{73} + 4 q^{79} - 24 q^{83} - 12 q^{89} + 20 q^{97}+O(q^{100}) $ 2 * q - 6 * q^11 - 4 * q^13 - 4 * q^19 - 6 * q^23 + 5 * q^25 - 12 * q^29 + 8 * q^31 - 2 * q^37 + 24 * q^41 - 8 * q^43 - 12 * q^47 - 6 * q^53 - 10 * q^61 - 8 * q^67 - 12 * q^71 - 10 * q^73 + 4 * q^79 - 24 * q^83 - 12 * q^89 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.0829925	−	0.996550i	$-0.526448\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	$-0.985065\pi$
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	0.540068	+	0.841621i	$0.318398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	0.961625	+	0.274367i	$0.0884683\pi$
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	−0.243204	+	0.969975i	$0.578198\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	0.772043	+	0.635571i	$0.219235\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	12.0000			1.87409			0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000			1.87409			0.937043	+	0.349215i	$0.113552\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.856560	+	0.516047i	$0.827403\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	0.345207	+	0.938527i	$0.387809\pi$
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	−0.985391	+	0.170305i	$0.945525\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	0.511237	−	0.859440i	$-0.329187\pi$
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	−0.999915	+	0.0130248i	$0.995854\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	0.409644	−	0.912245i	$-0.365653\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	0.350304	+	0.936636i	$0.386078\pi$
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	−0.986303	+	0.164946i	$0.947255\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	0.993258	+	0.115924i	$0.0369830\pi$
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	−0.396236	+	0.918149i	$0.629684\pi$
$102$		0			0
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	−0.598858	−	0.800855i	$-0.704379\pi$
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	0.992990	+	0.118199i	$0.0377120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	0.683793	+	0.729676i	$0.260329\pi$
$108$		0			0
$109$	−7.00000	−	12.1244i	−0.670478	−	1.16130i	−0.977769	−	0.209687i	$-0.932756\pi$
$109$	−7.00000	−	12.1244i	−0.670478	−	1.16130i	0.307290	−	0.951616i	$-0.400578\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−12.5000	+	21.6506i	−1.13636	+	1.96824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	+	10.3923i	0.501745	+	0.869048i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000	−	13.8564i	0.626608	−	1.08532i	−0.361619	−	0.932326i	$-0.617776\pi$
$163$	8.00000	−	13.8564i	0.626608	−	1.08532i	0.988227	−	0.152992i	$-0.0488907\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	0.956088	−	0.293079i	$-0.0946798\pi$
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	−0.731858	+	0.681457i	$0.761346\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	−0.736839	−	0.676068i	$-0.763683\pi$
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	0.953912	+	0.300088i	$0.0970159\pi$
$192$		0			0
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	0.987945	−	0.154805i	$-0.0494748\pi$
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	−0.628037	+	0.778183i	$0.716141\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	−0.996850	−	0.0793045i	$-0.974730\pi$
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	0.429745	−	0.902950i	$-0.358603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	−0.921903	−	0.387421i	$-0.873366\pi$
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	0.125435	−	0.992102i	$-0.459967\pi$
$228$		0			0
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	−0.831092	−	0.556136i	$-0.812283\pi$
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	0.897173	+	0.441679i	$0.145617\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	−0.750867	−	0.660454i	$-0.770364\pi$
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	0.947403	+	0.320043i	$0.103697\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	0.658838	−	0.752285i	$-0.271048\pi$
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	−0.980917	+	0.194429i	$0.937715\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.00000	−	6.92820i	0.254514	−	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	−0.615948	−	0.787787i	$-0.711227\pi$
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	0.990217	+	0.139533i	$0.0445601\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.00000	−	5.19615i	−0.184988	−	0.320408i	0.758585	−	0.651575i	$-0.225891\pi$
$263$	−3.00000	−	5.19615i	−0.184988	−	0.320408i	−0.943572	+	0.331166i	$0.892558\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.0000	−	20.7846i	−0.731653	−	1.26726i	−0.956176	−	0.292791i	$-0.905416\pi$
$269$	−12.0000	−	20.7846i	−0.731653	−	1.26726i	0.224523	−	0.974469i	$-0.427917\pi$
$270$		0			0
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	−0.999870	−	0.0161307i	$-0.994865\pi$
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	0.513905	+	0.857847i	$0.328199\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.0000	−	25.9808i	0.904534	−	1.56670i
$276$		0			0
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	0.980373	+	0.197153i	$0.0631696\pi$
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	−0.319447	+	0.947604i	$0.603497\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	0.993626	+	0.112728i	$0.0359589\pi$
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	−0.399188	+	0.916869i	$0.630708\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.0000			−0.701047			−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	−	10.3923i	0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	0.984475	−	0.175525i	$-0.0561621\pi$
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	−0.644246	+	0.764818i	$0.722829\pi$
$312$		0			0
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	−0.220006	−	0.975499i	$-0.570608\pi$
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	0.954810	−	0.297218i	$-0.0960589\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.0000	+	25.9808i	−0.842484	+	1.45922i	0.0453045	+	0.998973i	$0.485574\pi$
$317$	−15.0000	+	25.9808i	−0.842484	+	1.45922i	−0.887788	+	0.460252i	$0.847759\pi$
$318$		0			0
$319$	18.0000	+	31.1769i	1.00781	+	1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−5.00000	−	8.66025i	−0.277350	−	0.480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	0.448649	+	0.893708i	$0.351905\pi$
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	−0.998298	+	0.0583130i	$0.981428\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.0000	−	41.5692i	1.29967	−	2.25110i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	0.516667	−	0.856186i	$-0.327172\pi$
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	−0.999813	+	0.0193540i	$0.993839\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.00000	−	5.19615i	0.158334	−	0.274242i	−0.775934	−	0.630814i	$-0.782721\pi$
$359$	3.00000	−	5.19615i	0.158334	−	0.274242i	0.934268	+	0.356572i	$0.116054\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	0.893873	+	0.448320i	$0.147978\pi$
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	−0.0586798	+	0.998277i	$0.518689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.0000	+	25.9808i	0.760530	+	1.31728i	0.942578	+	0.333987i	$0.108394\pi$
$389$	15.0000	+	25.9808i	0.760530	+	1.31728i	−0.182047	+	0.983290i	$0.558272\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.00000	−	12.1244i	0.351320	−	0.608504i	−0.635161	−	0.772380i	$-0.719066\pi$
$397$	7.00000	−	12.1244i	0.351320	−	0.608504i	0.986481	+	0.163876i	$0.0523996\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$	−8.00000	−	13.8564i	−0.398508	−	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.00000	+	5.19615i	0.144505	+	0.250290i	0.929188	−	0.369607i	$-0.120508\pi$
$431$	3.00000	+	5.19615i	0.144505	+	0.250290i	−0.784683	+	0.619897i	$0.787174\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	−	20.7846i	−0.574038	−	0.994263i
$438$		0			0
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	−0.754642	−	0.656136i	$-0.772190\pi$
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	0.945552	+	0.325471i	$0.105523\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.0000	−	25.9808i	0.712672	−	1.23438i	−0.251179	−	0.967941i	$-0.580818\pi$
$443$	15.0000	−	25.9808i	0.712672	−	1.23438i	0.963851	−	0.266443i	$-0.0858483\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$	−36.0000	−	62.3538i	−1.69517	−	2.93613i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	0.999857	−	0.0168929i	$-0.00537742\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$	−28.0000			−1.30127			−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000			−1.30127			−0.650635	+	0.759390i	$0.725497\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000	+	20.7846i	0.551761	+	0.955677i
$474$		0			0
$475$	−20.0000			−0.917663
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	0.969920	−	0.243426i	$-0.0782712\pi$
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	−0.695773	+	0.718262i	$0.744938\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−16.0000	−	27.7128i	−0.725029	−	1.25579i	−0.958962	−	0.283535i	$-0.908493\pi$
$487$	−16.0000	−	27.7128i	−0.725029	−	1.25579i	0.233933	−	0.972253i	$-0.424840\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.0000			0.812329			0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000			0.812329			0.406164	+	0.913800i	$0.366866\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.0000	+	20.7846i	−0.531891	+	0.921262i	0.467416	+	0.884037i	$0.345185\pi$
$509$	−12.0000	+	20.7846i	−0.531891	+	0.921262i	−0.999307	+	0.0372243i	$0.988148\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	72.0000			3.16656
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−2.00000	+	3.46410i	−0.0874539	+	0.151475i	−0.906434	−	0.422347i	$-0.861206\pi$
$523$	−2.00000	+	3.46410i	−0.0874539	+	0.151475i	0.818980	+	0.573822i	$0.194540\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0000			−1.03956
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	−0.886721	−	0.462304i	$-0.847023\pi$
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	0.843728	+	0.536771i	$0.180356\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.0000	−	20.7846i	0.511217	−	0.885454i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.0000	+	36.3731i	0.889799	+	1.54118i	0.840113	+	0.542411i	$0.182489\pi$
$557$	21.0000	+	36.3731i	0.889799	+	1.54118i	0.0496855	+	0.998765i	$0.484178\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	0.892413	+	0.451219i	$0.149011\pi$
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	−0.0554391	+	0.998462i	$0.517656\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−30.0000			−1.25109
$576$		0			0
$577$	−11.0000	−	19.0526i	−0.457936	−	0.793168i	0.540916	−	0.841077i	$-0.318078\pi$
$577$	−11.0000	−	19.0526i	−0.457936	−	0.793168i	−0.998852	+	0.0479084i	$0.984744\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−18.0000	+	31.1769i	−0.745484	+	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24.0000			−0.990586			−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000			−0.990586			−0.495293	+	0.868726i	$0.664939\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.0000	+	41.5692i	−0.985562	+	1.70704i	−0.346149	+	0.938179i	$0.612511\pi$
$593$	−24.0000	+	41.5692i	−0.985562	+	1.70704i	−0.639413	+	0.768864i	$0.720822\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	0.990752	+	0.135686i	$0.0433238\pi$
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	−0.377869	+	0.925859i	$0.623343\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	16.0000	−	27.7128i	0.649420	−	1.12483i	−0.333842	−	0.942629i	$-0.608345\pi$
$607$	16.0000	−	27.7128i	0.649420	−	1.12483i	0.983262	−	0.182199i	$-0.0583216\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0000	−	20.7846i	0.485468	−	0.840855i
$612$		0			0
$613$	−19.0000	−	32.9090i	−0.767403	−	1.32918i	−0.938967	−	0.344008i	$-0.888215\pi$
$613$	−19.0000	−	32.9090i	−0.767403	−	1.32918i	0.171564	−	0.985173i	$-0.445118\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	0.993959	−	0.109749i	$-0.0350048\pi$
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	−0.592025	+	0.805919i	$0.701671\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0000	+	31.1769i	0.707653	+	1.22569i	0.965726	+	0.259565i	$0.0835793\pi$
$647$	18.0000	+	31.1769i	0.707653	+	1.22569i	−0.258073	+	0.966126i	$0.583087\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	−0.986634	−	0.162951i	$-0.947899\pi$
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	0.634437	+	0.772975i	$0.281232\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$	19.0000	+	32.9090i	0.739014	+	1.28001i	0.952940	+	0.303160i	$0.0980418\pi$
$661$	19.0000	+	32.9090i	0.739014	+	1.28001i	−0.213925	+	0.976850i	$0.568625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	18.0000	−	31.1769i	0.696963	−	1.20717i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	60.0000			2.31627
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	−0.957984	−	0.286820i	$-0.907402\pi$
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	0.727386	+	0.686229i	$0.240735\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.0000	−	36.3731i	−0.803543	−	1.39178i	−0.917270	−	0.398265i	$-0.869613\pi$
$683$	−21.0000	−	36.3731i	−0.803543	−	1.39178i	0.113728	−	0.993512i	$-0.463721\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$	−2.00000	+	3.46410i	−0.0760836	+	0.131781i	−0.901557	−	0.432660i	$-0.857575\pi$
$691$	−2.00000	+	3.46410i	−0.0760836	+	0.131781i	0.825473	+	0.564441i	$0.190908\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$	−4.00000	−	6.92820i	−0.150863	−	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−48.0000			−1.79761
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0000	−	25.9808i	−0.557086	−	0.964901i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−23.0000	+	39.8372i	−0.849524	+	1.47142i	0.0321090	+	0.999484i	$0.489778\pi$
$733$	−23.0000	+	39.8372i	−0.849524	+	1.47142i	−0.881633	+	0.471935i	$0.843556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.0000	+	41.5692i	−0.884051	+	1.53122i
$738$		0			0
$739$	−16.0000	−	27.7128i	−0.588570	−	1.01943i	−0.994420	−	0.105493i	$-0.966358\pi$
$739$	−16.0000	−	27.7128i	−0.588570	−	1.01943i	0.405851	−	0.913939i	$-0.366975\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.00000			−0.220119			−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000			−0.220119			−0.110059	+	0.993925i	$0.535104\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	0.411170	+	0.911559i	$0.365120\pi$
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	−0.995018	+	0.0996961i	$0.968213\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000	−	31.1769i	0.652499	−	1.13016i	−0.330015	−	0.943976i	$-0.607054\pi$
$761$	18.0000	−	31.1769i	0.652499	−	1.13016i	0.982514	−	0.186187i	$-0.0596129\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.00000	−	10.3923i	−0.215805	−	0.373785i	0.737716	−	0.675111i	$-0.235904\pi$
$773$	−6.00000	−	10.3923i	−0.215805	−	0.373785i	−0.953521	+	0.301326i	$0.902571\pi$
$774$		0			0
$775$	−20.0000	+	34.6410i	−0.718421	+	1.24434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000	+	41.5692i	−0.859889	+	1.48937i
$780$		0			0
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	10.0000	−	17.3205i	0.355110	−	0.615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36.0000			1.27519			0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000			1.27519			0.637593	+	0.770374i	$0.279930\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−30.0000	+	51.9615i	−1.05868	+	1.83368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−27.0000	−	46.7654i	−0.949269	−	1.64418i	−0.746968	−	0.664860i	$-0.768491\pi$
$809$	−27.0000	−	46.7654i	−0.949269	−	1.64418i	−0.202301	−	0.979323i	$-0.564842\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	8.00000	−	13.8564i	0.279885	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.00000	−	15.5885i	0.314102	−	0.544041i	−0.665144	−	0.746715i	$-0.731630\pi$
$821$	9.00000	−	15.5885i	0.314102	−	0.544041i	0.979246	+	0.202674i	$0.0649632\pi$
$822$		0			0
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	0.898776	−	0.438408i	$-0.144457\pi$
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	−0.829060	+	0.559159i	$0.811124\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.00000			0.208640			0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000			0.208640			0.104320	+	0.994544i	$0.466733\pi$
$828$		0			0
$829$	−11.0000	−	19.0526i	−0.382046	−	0.661723i	0.609309	−	0.792933i	$-0.291447\pi$
$829$	−11.0000	−	19.0526i	−0.382046	−	0.661723i	−0.991355	+	0.131210i	$0.958114\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	−	10.3923i	−0.205677	−	0.356244i
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.00000	−	10.3923i	−0.204956	−	0.354994i	0.745163	−	0.666883i	$-0.232372\pi$
$857$	−6.00000	−	10.3923i	−0.204956	−	0.354994i	−0.950119	+	0.311888i	$0.899038\pi$
$858$		0			0
$859$	−2.00000	+	3.46410i	−0.0682391	+	0.118194i	−0.898126	−	0.439738i	$-0.855071\pi$
$859$	−2.00000	+	3.46410i	−0.0682391	+	0.118194i	0.829887	+	0.557931i	$0.188405\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	−0.912558	−	0.408946i	$-0.865896\pi$
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	0.810437	+	0.585826i	$0.199230\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	8.00000	+	13.8564i	0.271070	+	0.469506i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	−0.959671	−	0.281126i	$-0.909292\pi$
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	0.723298	+	0.690536i	$0.242625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.0000			0.808581			0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000			0.808581			0.404290	+	0.914631i	$0.367519\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	0.387768	+	0.921757i	$0.373246\pi$
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	−0.992149	+	0.125061i	$0.960087\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24.0000	−	41.5692i	−0.803129	−	1.39106i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000	−	41.5692i	−0.800445	−	1.38641i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.00000	+	3.46410i	0.0664089	+	0.115024i	0.897318	−	0.441384i	$-0.145512\pi$
$907$	2.00000	+	3.46410i	0.0664089	+	0.115024i	−0.830909	+	0.556408i	$0.812179\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−54.0000			−1.78910			−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000			−1.78910			−0.894550	+	0.446968i	$0.852504\pi$
$912$		0			0
$913$	36.0000	+	62.3538i	1.19143	+	2.06361i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	−0.703378	−	0.710816i	$-0.748326\pi$
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	0.967274	+	0.253735i	$0.0816592\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000			0.394985
$924$		0			0
$925$	−10.0000			−0.328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.00000	+	10.3923i	−0.196854	+	0.340960i	−0.947507	−	0.319736i	$-0.896406\pi$
$929$	−6.00000	+	10.3923i	−0.196854	+	0.340960i	0.750653	+	0.660697i	$0.229739\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.0000	+	20.7846i	0.391189	+	0.677559i	0.992607	−	0.121376i	$-0.0387306\pi$
$941$	12.0000	+	20.7846i	0.391189	+	0.677559i	−0.601418	+	0.798935i	$0.705397\pi$
$942$		0			0
$943$	−36.0000	+	62.3538i	−1.17232	+	2.03052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	−0.999896	−	0.0144491i	$-0.995401\pi$
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	0.512461	+	0.858710i	$0.328734\pi$
$948$		0			0
$949$	10.0000	+	17.3205i	0.324614	+	0.562247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−28.0000			−0.900419			−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000			−0.900419			−0.450210	+	0.892923i	$0.648651\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000	+	10.3923i	−0.192549	+	0.333505i	−0.946094	−	0.323891i	$-0.895009\pi$
$971$	−6.00000	+	10.3923i	−0.192549	+	0.333505i	0.753545	+	0.657396i	$0.228342\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.00000	+	5.19615i	0.0959785	+	0.166240i	0.910017	−	0.414572i	$-0.136069\pi$
$977$	3.00000	+	5.19615i	0.0959785	+	0.166240i	−0.814038	+	0.580812i	$0.802735\pi$
$978$		0			0
$979$	72.0000			2.30113
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	0.939992	+	0.341197i	$0.110832\pi$
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	−0.174511	+	0.984655i	$0.555834\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000	−	20.7846i	0.381578	−	0.660912i
$990$		0			0
$991$	20.0000	+	34.6410i	0.635321	+	1.10041i	0.986447	+	0.164080i	$0.0524655\pi$
$991$	20.0000	+	34.6410i	0.635321	+	1.10041i	−0.351126	+	0.936328i	$0.614201\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	−0.934274	+	0.356555i	$0.883951\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.k.d.1549.1		2
3.2	odd	2		588.2.i.f.373.1		2
7.2	even	3		1764.2.a.g.1.1		1
7.3	odd	6		1764.2.k.e.361.1		2
7.4	even	3	inner	1764.2.k.d.361.1		2
7.5	odd	6		252.2.a.b.1.1		1
7.6	odd	2		1764.2.k.e.1549.1		2
12.11	even	2		2352.2.q.g.961.1		2
21.2	odd	6		588.2.a.c.1.1		1
21.5	even	6		84.2.a.b.1.1	✓	1
21.11	odd	6		588.2.i.f.361.1		2
21.17	even	6		588.2.i.c.361.1		2
21.20	even	2		588.2.i.c.373.1		2
28.19	even	6		1008.2.a.g.1.1		1
28.23	odd	6		7056.2.a.x.1.1		1
35.12	even	12		6300.2.k.r.6049.2		2
35.19	odd	6		6300.2.a.p.1.1		1
35.33	even	12		6300.2.k.r.6049.1		2
56.5	odd	6		4032.2.a.u.1.1		1
56.19	even	6		4032.2.a.t.1.1		1
63.5	even	6		2268.2.j.i.1513.1		2
63.40	odd	6		2268.2.j.f.1513.1		2
63.47	even	6		2268.2.j.i.757.1		2
63.61	odd	6		2268.2.j.f.757.1		2
84.11	even	6		2352.2.q.g.1537.1		2
84.23	even	6		2352.2.a.s.1.1		1
84.47	odd	6		336.2.a.b.1.1		1
84.59	odd	6		2352.2.q.s.1537.1		2
84.83	odd	2		2352.2.q.s.961.1		2
105.47	odd	12		2100.2.k.a.1849.1		2
105.68	odd	12		2100.2.k.a.1849.2		2
105.89	even	6		2100.2.a.a.1.1		1
168.5	even	6		1344.2.a.f.1.1		1
168.107	even	6		9408.2.a.r.1.1		1
168.131	odd	6		1344.2.a.o.1.1		1
168.149	odd	6		9408.2.a.co.1.1		1
336.5	even	12		5376.2.c.i.2689.1		2
336.131	odd	12		5376.2.c.x.2689.1		2
336.173	even	12		5376.2.c.i.2689.2		2
336.299	odd	12		5376.2.c.x.2689.2		2
420.299	odd	6		8400.2.a.ct.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.b.1.1	✓	1	21.5	even	6
252.2.a.b.1.1		1	7.5	odd	6
336.2.a.b.1.1		1	84.47	odd	6
588.2.a.c.1.1		1	21.2	odd	6
588.2.i.c.361.1		2	21.17	even	6
588.2.i.c.373.1		2	21.20	even	2
588.2.i.f.361.1		2	21.11	odd	6
588.2.i.f.373.1		2	3.2	odd	2
1008.2.a.g.1.1		1	28.19	even	6
1344.2.a.f.1.1		1	168.5	even	6
1344.2.a.o.1.1		1	168.131	odd	6
1764.2.a.g.1.1		1	7.2	even	3
1764.2.k.d.361.1		2	7.4	even	3	inner
1764.2.k.d.1549.1		2	1.1	even	1	trivial
1764.2.k.e.361.1		2	7.3	odd	6
1764.2.k.e.1549.1		2	7.6	odd	2
2100.2.a.a.1.1		1	105.89	even	6
2100.2.k.a.1849.1		2	105.47	odd	12
2100.2.k.a.1849.2		2	105.68	odd	12
2268.2.j.f.757.1		2	63.61	odd	6
2268.2.j.f.1513.1		2	63.40	odd	6
2268.2.j.i.757.1		2	63.47	even	6
2268.2.j.i.1513.1		2	63.5	even	6
2352.2.a.s.1.1		1	84.23	even	6
2352.2.q.g.961.1		2	12.11	even	2
2352.2.q.g.1537.1		2	84.11	even	6
2352.2.q.s.961.1		2	84.83	odd	2
2352.2.q.s.1537.1		2	84.59	odd	6
4032.2.a.t.1.1		1	56.19	even	6
4032.2.a.u.1.1		1	56.5	odd	6
5376.2.c.i.2689.1		2	336.5	even	12
5376.2.c.i.2689.2		2	336.173	even	12
5376.2.c.x.2689.1		2	336.131	odd	12
5376.2.c.x.2689.2		2	336.299	odd	12
6300.2.a.p.1.1		1	35.19	odd	6
6300.2.k.r.6049.1		2	35.33	even	12
6300.2.k.r.6049.2		2	35.12	even	12
7056.2.a.x.1.1		1	28.23	odd	6
8400.2.a.ct.1.1		1	420.299	odd	6
9408.2.a.r.1.1		1	168.107	even	6
9408.2.a.co.1.1		1	168.149	odd	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.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+(-3.00000 - 5.19615i) q^{11} -2.00000 q^{13} +(-2.00000 + 3.46410i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} -6.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-1.00000 + 1.73205i) q^{37} +12.0000 q^{41} -4.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(-3.00000 - 5.19615i) q^{53} +(-5.00000 + 8.66025i) q^{61} +(-4.00000 - 6.92820i) q^{67} -6.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(2.00000 - 3.46410i) q^{79} -12.0000 q^{83} +(-6.00000 + 10.3923i) q^{89} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 6 q^{11} - 4 q^{13} - 4 q^{19} - 6 q^{23} + 5 q^{25} - 12 q^{29} + 8 q^{31} - 2 q^{37} + 24 q^{41} - 8 q^{43} - 12 q^{47} - 6 q^{53} - 10 q^{61} - 8 q^{67} - 12 q^{71} - 10 q^{73} + 4 q^{79} - 24 q^{83} - 12 q^{89} + 20 q^{97}+O(q^{100}) \) 2 * q - 6 * q^11 - 4 * q^13 - 4 * q^19 - 6 * q^23 + 5 * q^25 - 12 * q^29 + 8 * q^31 - 2 * q^37 + 24 * q^41 - 8 * q^43 - 12 * q^47 - 6 * q^53 - 10 * q^61 - 8 * q^67 - 12 * q^71 - 10 * q^73 + 4 * q^79 - 24 * q^83 - 12 * q^89 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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