LMFDB - Embedded newform 2268.2.j.f.757.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(757,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2268.757");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.j (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:	$18.1100711784$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			757.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.757
Dual form			2268.2.j.f.1513.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-1.00000 + 1.73205i) q^{13} -4.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +2.00000 q^{37} +(6.00000 - 10.3923i) q^{41} +(2.00000 + 3.46410i) q^{43} +(6.00000 + 10.3923i) q^{47} +(-0.500000 + 0.866025i) q^{49} +6.00000 q^{53} +(5.00000 + 8.66025i) q^{61} +(-4.00000 + 6.92820i) q^{67} -6.00000 q^{71} -10.0000 q^{73} +(-3.00000 + 5.19615i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-6.00000 - 10.3923i) q^{83} -12.0000 q^{89} +2.00000 q^{91} +(5.00000 + 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{7}+O(q^{10}) $ 2 * q - q^7	$ 2 q - q^{7} - 6 q^{11} - 2 q^{13} - 8 q^{19} - 6 q^{23} + 5 q^{25} + 6 q^{29} - 8 q^{31} + 4 q^{37} + 12 q^{41} + 4 q^{43} + 12 q^{47} - q^{49} + 12 q^{53} + 10 q^{61} - 8 q^{67} - 12 q^{71} - 20 q^{73} - 6 q^{77} + 4 q^{79} - 12 q^{83} - 24 q^{89} + 4 q^{91} + 10 q^{97}+O(q^{100}) $ 2 * q - q^7 - 6 * q^11 - 2 * q^13 - 8 * q^19 - 6 * q^23 + 5 * q^25 + 6 * q^29 - 8 * q^31 + 4 * q^37 + 12 * q^41 + 4 * q^43 + 12 * q^47 - q^49 + 12 * q^53 + 10 * q^61 - 8 * q^67 - 12 * q^71 - 20 * q^73 - 6 * q^77 + 4 * q^79 - 12 * q^83 - 24 * q^89 + 4 * q^91 + 10 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$8$		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
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		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
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	$0.0214140\pi$
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	−0.440652	+	0.897678i	$0.645253\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000	−	10.3923i	0.937043	−	1.62301i	0.166092	−	0.986110i	$-0.446885\pi$
$41$	6.00000	−	10.3923i	0.937043	−	1.62301i	0.770950	−	0.636895i	$-0.219782\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.856560	+	0.516047i	$0.172597\pi$
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.0186297	+	0.999826i	$0.494070\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	−0.999915	−	0.0130248i	$-0.995854\pi$
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	0.511237	+	0.859440i	$0.329187\pi$
$68$		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
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.00000	+	5.19615i	−0.341882	+	0.592157i
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.0000			−1.27200			−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000			−1.27200			−0.635999	+	0.771690i	$0.719412\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	0.999961	+	0.00888289i	$0.00282755\pi$
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	−0.492287	+	0.870433i	$0.663839\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	−0.971930	−	0.235269i	$-0.924403\pi$
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	0.689714	+	0.724082i	$0.257736\pi$
$114$		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	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		0			0
$133$	2.00000	+	3.46410i	0.173422	+	0.300376i
$134$		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
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000			1.00349
$144$		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	0.438948	+	0.898513i	$0.355351\pi$
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	−0.999169	−	0.0407502i	$-0.987025\pi$
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	0.534875	+	0.844931i	$0.320359\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.00000	−	10.3923i	−0.456172	−	0.790112i	0.542583	−	0.840002i	$-0.317446\pi$
$173$	−6.00000	−	10.3923i	−0.456172	−	0.790112i	−0.998755	+	0.0498898i	$0.984113\pi$
$174$		0			0
$175$	2.50000	−	4.33013i	0.188982	−	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.00000			−0.448461			−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000			−0.448461			−0.224231	+	0.974536i	$0.571987\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		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	0.953912	−	0.300088i	$-0.0970159\pi$
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	−0.736839	+	0.676068i	$0.763683\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$		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
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.00000	−	5.19615i	0.210559	−	0.364698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000	+	20.7846i	0.830057	+	1.43770i
$210$		0			0
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	−0.788935	−	0.614477i	$-0.789367\pi$
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	0.926620	+	0.375999i	$0.122700\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000	+	20.7846i	0.796468	+	1.37952i	0.921903	+	0.387421i	$0.126634\pi$
$227$	12.0000	+	20.7846i	0.796468	+	1.37952i	−0.125435	+	0.992102i	$0.540033\pi$
$228$		0			0
$229$	−1.00000	+	1.73205i	−0.0660819	+	0.114457i	−0.897173	−	0.441679i	$-0.854383\pi$
$229$	−1.00000	+	1.73205i	−0.0660819	+	0.114457i	0.831092	+	0.556136i	$0.187717\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.00000	−	15.5885i	0.582162	−	1.00833i	−0.413061	−	0.910703i	$-0.635540\pi$
$239$	9.00000	−	15.5885i	0.582162	−	1.00833i	0.995223	−	0.0976302i	$-0.0311262\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$		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	−0.990217	−	0.139533i	$-0.955440\pi$
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	0.615948	+	0.787787i	$0.288773\pi$
$258$		0			0
$259$	−1.00000	−	1.73205i	−0.0621370	−	0.107624i
$260$		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
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		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
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	−10.0000	+	17.3205i	−0.594438	+	1.02960i	0.399188	+	0.916869i	$0.369292\pi$
$283$	−10.0000	+	17.3205i	−0.594438	+	1.02960i	−0.993626	+	0.112728i	$0.964041\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.00000	+	10.3923i	−0.350524	+	0.607125i	−0.986341	−	0.164714i	$-0.947330\pi$
$293$	−6.00000	+	10.3923i	−0.350524	+	0.607125i	0.635818	+	0.771839i	$0.280663\pi$
$294$		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
$301$	2.00000	−	3.46410i	0.115278	−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.00000	+	10.3923i	−0.340229	+	0.589294i	−0.984475	−	0.175525i	$-0.943838\pi$
$311$	−6.00000	+	10.3923i	−0.340229	+	0.589294i	0.644246	+	0.764818i	$0.277171\pi$
$312$		0			0
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	0.220006	−	0.975499i	$-0.429392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.0000	−	25.9808i	−0.842484	−	1.45922i	−0.887788	−	0.460252i	$-0.847759\pi$
$317$	−15.0000	−	25.9808i	−0.842484	−	1.45922i	0.0453045	−	0.998973i	$-0.485574\pi$
$318$		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
$325$	−10.0000			−0.554700
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000	−	10.3923i	0.330791	−	0.572946i
$330$		0			0
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	−0.998298	−	0.0583130i	$-0.981428\pi$
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	0.448649	−	0.893708i	$-0.351905\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.0000	−	19.0526i	0.599208	−	1.03786i	−0.393730	−	0.919226i	$-0.628816\pi$
$337$	11.0000	−	19.0526i	0.599208	−	1.03786i	0.992938	−	0.118633i	$-0.0378512\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	48.0000			2.59935
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	−0.999813	−	0.0193540i	$-0.993839\pi$
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	0.516667	+	0.856186i	$0.327172\pi$
$348$		0			0
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	0.968253	−	0.249973i	$-0.0804216\pi$
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	−0.700609	+	0.713545i	$0.747088\pi$
$350$		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
$359$	−6.00000			−0.316668			−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000			−0.316668			−0.158334	+	0.987386i	$0.550612\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		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	$-0.852022\pi$
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	0.0586798	−	0.998277i	$-0.481311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.00000	−	5.19615i	−0.155752	−	0.269771i
$372$		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	$0.376773\pi$
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	−0.990702	+	0.136047i	$0.956560\pi$
$384$		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
$397$	14.0000			0.702640			0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000			0.702640			0.351320	+	0.936255i	$0.385733\pi$
$398$		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
$407$	−6.00000	−	10.3923i	−0.297409	−	0.515127i
$408$		0			0
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	−0.985558	−	0.169338i	$-0.945837\pi$
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	0.639430	+	0.768849i	$0.279170\pi$
$410$		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
$419$	−12.0000	+	20.7846i	−0.586238	+	1.01539i	0.408481	+	0.912767i	$0.366058\pi$
$419$	−12.0000	+	20.7846i	−0.586238	+	1.01539i	−0.994720	+	0.102628i	$0.967275\pi$
$420$		0			0
$421$	−13.0000	−	22.5167i	−0.633581	−	1.09739i	−0.986814	−	0.161859i	$-0.948251\pi$
$421$	−13.0000	−	22.5167i	−0.633581	−	1.09739i	0.353233	−	0.935536i	$-0.385082\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	5.00000	−	8.66025i	0.241967	−	0.419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		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	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.0000	+	25.9808i	0.712672	+	1.23438i	0.963851	+	0.266443i	$0.0858483\pi$
$443$	15.0000	+	25.9808i	0.712672	+	1.23438i	−0.251179	+	0.967941i	$0.580818\pi$
$444$		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
$451$	−72.0000			−3.39035
$452$		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
$461$	6.00000	+	10.3923i	0.279448	+	0.484018i	0.971248	−	0.238071i	$-0.0765153\pi$
$461$	6.00000	+	10.3923i	0.279448	+	0.484018i	−0.691800	+	0.722089i	$0.743182\pi$
$462$		0			0
$463$	14.0000	−	24.2487i	0.650635	−	1.12693i	−0.332334	−	0.943162i	$-0.607836\pi$
$463$	14.0000	−	24.2487i	0.650635	−	1.12693i	0.982969	−	0.183771i	$-0.0588306\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000	−	20.7846i	0.551761	−	0.955677i
$474$		0			0
$475$	−10.0000	−	17.3205i	−0.458831	−	0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	0.695773	−	0.718262i	$-0.255062\pi$
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	−0.969920	+	0.243426i	$0.921729\pi$
$480$		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
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.00000	+	15.5885i	−0.406164	+	0.703497i	−0.994456	−	0.105151i	$-0.966467\pi$
$491$	−9.00000	+	15.5885i	−0.406164	+	0.703497i	0.588292	+	0.808649i	$0.299801\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.00000	+	5.19615i	0.134568	+	0.233079i
$498$		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	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		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	$-0.654815\pi$
$509$	12.0000	−	20.7846i	0.531891	−	0.921262i	0.999307	−	0.0372243i	$-0.0118516\pi$
$510$		0			0
$511$	5.00000	+	8.66025i	0.221187	+	0.383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	36.0000	−	62.3538i	1.58328	−	2.74232i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		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
$533$	12.0000	+	20.7846i	0.519778	+	0.900281i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	0.767752	−	0.640747i	$-0.221375\pi$
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	−0.938779	+	0.344519i	$0.888042\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000	−	20.7846i	−0.511217	−	0.885454i
$552$		0			0
$553$	2.00000	−	3.46410i	0.0850487	−	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−42.0000			−1.77960			−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000			−1.77960			−0.889799	+	0.456354i	$0.849155\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	−0.0554391	−	0.998462i	$-0.517656\pi$
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	0.892413	−	0.451219i	$-0.149011\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−30.0000			−1.25109
$576$		0			0
$577$	−22.0000			−0.915872			−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000			−0.915872			−0.457936	+	0.888985i	$0.651411\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000	+	10.3923i	−0.248922	+	0.431145i
$582$		0			0
$583$	−18.0000	−	31.1769i	−0.745484	−	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000	−	20.7846i	−0.495293	−	0.857873i	0.504692	−	0.863299i	$-0.331606\pi$
$587$	−12.0000	−	20.7846i	−0.495293	−	0.857873i	−0.999985	+	0.00542667i	$0.998273\pi$
$588$		0			0
$589$	16.0000	−	27.7128i	0.659269	−	1.14189i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−48.0000			−1.97112			−0.985562	−	0.169316i	$-0.945844\pi$
$593$	−48.0000			−1.97112			−0.985562	+	0.169316i	$0.945844\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	−0.377869	−	0.925859i	$-0.623343\pi$
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	0.990752	−	0.135686i	$-0.0433238\pi$
$600$		0			0
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	0.844909	−	0.534910i	$-0.179654\pi$
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	−0.885700	+	0.464258i	$0.846321\pi$
$602$		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	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000			−0.970936
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.00000	+	15.5885i	−0.362326	+	0.627568i	−0.988343	−	0.152242i	$-0.951351\pi$
$617$	−9.00000	+	15.5885i	−0.362326	+	0.627568i	0.626017	+	0.779809i	$0.284684\pi$
$618$		0			0
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	0.592025	−	0.805919i	$-0.298329\pi$
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	−0.993959	+	0.109749i	$0.964995\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000	+	10.3923i	0.240385	+	0.416359i
$624$		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
$637$	−1.00000	−	1.73205i	−0.0396214	−	0.0686264i
$638$		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
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	−0.823891	−	0.566748i	$-0.808201\pi$
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	0.902764	+	0.430137i	$0.141535\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	36.0000			1.41531			0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000			1.41531			0.707653	+	0.706560i	$0.249754\pi$
$648$		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
$659$	−3.00000	−	5.19615i	−0.116863	−	0.202413i	0.801660	−	0.597781i	$-0.203951\pi$
$659$	−3.00000	−	5.19615i	−0.116863	−	0.202413i	−0.918523	+	0.395367i	$0.870617\pi$
$660$		0			0
$661$	−19.0000	+	32.9090i	−0.739014	+	1.28001i	0.213925	+	0.976850i	$0.431375\pi$
$661$	−19.0000	+	32.9090i	−0.739014	+	1.28001i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−36.0000			−1.39393
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30.0000	−	51.9615i	1.15814	−	2.00595i
$672$		0			0
$673$	17.0000	+	29.4449i	0.655302	+	1.13502i	0.981818	+	0.189824i	$0.0607919\pi$
$673$	17.0000	+	29.4449i	0.655302	+	1.13502i	−0.326516	+	0.945192i	$0.605875\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	0.957984	−	0.286820i	$-0.0925982\pi$
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	−0.727386	+	0.686229i	$0.759265\pi$
$678$		0			0
$679$	5.00000	−	8.66025i	0.191882	−	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	42.0000			1.60709			0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000			1.60709			0.803543	+	0.595247i	$0.202946\pi$
$684$		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	$-0.142425\pi$
$691$	2.00000	+	3.46410i	0.0760836	+	0.131781i	−0.825473	+	0.564441i	$0.809092\pi$
$692$		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
$703$	−8.00000			−0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.00000	+	10.3923i	−0.225653	+	0.390843i
$708$		0			0
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	0.944509	−	0.328484i	$-0.106538\pi$
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	−0.756730	+	0.653727i	$0.773204\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−24.0000	−	41.5692i	−0.898807	−	1.55678i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0000	+	25.9808i	−0.557086	+	0.964901i
$726$		0			0
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	0.782267	−	0.622944i	$-0.214063\pi$
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	−0.930618	+	0.365991i	$0.880730\pi$
$728$		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	$-0.510222\pi$
$733$	23.0000	−	39.8372i	0.849524	−	1.47142i	0.881633	−	0.471935i	$-0.156444\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	48.0000			1.76810
$738$		0			0
$739$	32.0000			1.17714			0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000			1.17714			0.588570	+	0.808447i	$0.299691\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.00000	−	5.19615i	0.110059	−	0.190628i	−0.805735	−	0.592277i	$-0.798229\pi$
$743$	3.00000	−	5.19615i	0.110059	−	0.190628i	0.915794	+	0.401648i	$0.131563\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.00000	−	5.19615i	−0.109618	−	0.189863i
$750$		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	$0.392946\pi$
$761$	−18.0000	+	31.1769i	−0.652499	+	1.13016i	−0.982514	+	0.186187i	$0.940387\pi$
$762$		0			0
$763$	−7.00000	−	12.1244i	−0.253417	−	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−13.0000	+	22.5167i	−0.468792	+	0.811972i	−0.999364	−	0.0356685i	$-0.988644\pi$
$769$	−13.0000	+	22.5167i	−0.468792	+	0.811972i	0.530572	+	0.847640i	$0.321977\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.0000			−0.431610			−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000			−0.431610			−0.215805	+	0.976436i	$0.569238\pi$
$774$		0			0
$775$	−40.0000			−1.43684
$776$		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	$-0.810621\pi$
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	0.899468	+	0.436987i	$0.143954\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$	−20.0000			−0.710221
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000	−	31.1769i	0.637593	−	1.10434i	−0.348367	−	0.937358i	$-0.613264\pi$
$797$	18.0000	−	31.1769i	0.637593	−	1.10434i	0.985959	−	0.166985i	$-0.0534030\pi$
$798$		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
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		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	0.979246	−	0.202674i	$-0.0649632\pi$
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	−0.665144	+	0.746715i	$0.731630\pi$
$822$		0			0
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	−0.829060	−	0.559159i	$-0.811124\pi$
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	0.898776	+	0.438408i	$0.144457\pi$
$824$		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
$829$	−22.0000			−0.764092			−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000			−0.764092			−0.382046	+	0.924143i	$0.624780\pi$
$830$		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
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	0.950813	−	0.309764i	$-0.100250\pi$
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	−0.743670	+	0.668546i	$0.766917\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	25.0000			0.859010
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	+	10.3923i	−0.205677	+	0.356244i
$852$		0			0
$853$	−7.00000	−	12.1244i	−0.239675	−	0.415130i	0.720946	−	0.692992i	$-0.243708\pi$
$853$	−7.00000	−	12.1244i	−0.239675	−	0.415130i	−0.960621	+	0.277862i	$0.910374\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00000	+	10.3923i	0.204956	+	0.354994i	0.950119	−	0.311888i	$-0.100962\pi$
$857$	6.00000	+	10.3923i	0.204956	+	0.354994i	−0.745163	+	0.666883i	$0.767628\pi$
$858$		0			0
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	−0.829887	−	0.557931i	$-0.811595\pi$
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	0.898126	+	0.439738i	$0.144929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.00000			0.204242			0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000			0.204242			0.102121	+	0.994772i	$0.467437\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.0000	−	20.7846i	0.407072	−	0.705070i
$870$		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	$-0.632481\pi$
$881$	−24.0000			−0.808581			−0.404290	+	0.914631i	$0.632481\pi$
$882$		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	$-0.626754\pi$
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	0.992149	−	0.125061i	$-0.0399128\pi$
$888$		0			0
$889$	2.00000	+	3.46410i	0.0670778	+	0.116182i
$890$		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
$899$	−48.0000			−1.60089
$900$		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
$911$	27.0000	+	46.7654i	0.894550	+	1.54941i	0.834361	+	0.551219i	$0.185837\pi$
$911$	27.0000	+	46.7654i	0.894550	+	1.54941i	0.0601892	+	0.998187i	$0.480830\pi$
$912$		0			0
$913$	−36.0000	+	62.3538i	−1.19143	+	2.06361i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.0000			−0.396275
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.00000	−	10.3923i	0.197492	−	0.342067i
$924$		0			0
$925$	5.00000	+	8.66025i	0.164399	+	0.284747i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.00000	+	10.3923i	0.196854	+	0.340960i	0.947507	−	0.319736i	$-0.103594\pi$
$929$	6.00000	+	10.3923i	0.196854	+	0.340960i	−0.750653	+	0.660697i	$0.770261\pi$
$930$		0			0
$931$	2.00000	−	3.46410i	0.0655474	−	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.0000	+	20.7846i	−0.391189	+	0.677559i	−0.992607	−	0.121376i	$-0.961269\pi$
$941$	−12.0000	+	20.7846i	−0.391189	+	0.677559i	0.601418	+	0.798935i	$0.294603\pi$
$942$		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	0.512461	−	0.858710i	$-0.328734\pi$
$947$	−15.0000	−	25.9808i	−0.487435	−	0.844261i	−0.999896	+	0.0144491i	$0.995401\pi$
$948$		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
$959$	3.00000	−	5.19615i	0.0968751	−	0.167793i
$960$		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
$967$	14.0000	−	24.2487i	0.450210	−	0.779786i	−0.548189	−	0.836354i	$-0.684683\pi$
$967$	14.0000	−	24.2487i	0.450210	−	0.779786i	0.998399	+	0.0565684i	$0.0180159\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$	−4.00000			−0.128234
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.00000	−	5.19615i	0.0959785	−	0.166240i	−0.814038	−	0.580812i	$-0.802735\pi$
$977$	3.00000	−	5.19615i	0.0959785	−	0.166240i	0.910017	+	0.414572i	$0.136069\pi$
$978$		0			0
$979$	36.0000	+	62.3538i	1.15056	+	1.99284i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.0000	−	41.5692i	−0.765481	−	1.32585i	−0.939992	−	0.341197i	$-0.889168\pi$
$983$	−24.0000	−	41.5692i	−0.765481	−	1.32585i	0.174511	−	0.984655i	$-0.444166\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.0000			−0.763156
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.j.f.757.1		2
3.2	odd	2		2268.2.j.i.757.1		2
9.2	odd	6		2268.2.j.i.1513.1		2
9.4	even	3		252.2.a.b.1.1		1
9.5	odd	6		84.2.a.b.1.1	✓	1
9.7	even	3	inner	2268.2.j.f.1513.1		2
36.23	even	6		336.2.a.b.1.1		1
36.31	odd	6		1008.2.a.g.1.1		1
45.4	even	6		6300.2.a.p.1.1		1
45.13	odd	12		6300.2.k.r.6049.1		2
45.14	odd	6		2100.2.a.a.1.1		1
45.22	odd	12		6300.2.k.r.6049.2		2
45.23	even	12		2100.2.k.a.1849.2		2
45.32	even	12		2100.2.k.a.1849.1		2
63.4	even	3		1764.2.k.e.1549.1		2
63.5	even	6		588.2.i.f.361.1		2
63.13	odd	6		1764.2.a.g.1.1		1
63.23	odd	6		588.2.i.c.361.1		2
63.31	odd	6		1764.2.k.d.1549.1		2
63.32	odd	6		588.2.i.c.373.1		2
63.40	odd	6		1764.2.k.d.361.1		2
63.41	even	6		588.2.a.c.1.1		1
63.58	even	3		1764.2.k.e.361.1		2
63.59	even	6		588.2.i.f.373.1		2
72.5	odd	6		1344.2.a.f.1.1		1
72.13	even	6		4032.2.a.u.1.1		1
72.59	even	6		1344.2.a.o.1.1		1
72.67	odd	6		4032.2.a.t.1.1		1
144.5	odd	12		5376.2.c.i.2689.1		2
144.59	even	12		5376.2.c.x.2689.2		2
144.77	odd	12		5376.2.c.i.2689.2		2
144.131	even	12		5376.2.c.x.2689.1		2
180.59	even	6		8400.2.a.ct.1.1		1
252.23	even	6		2352.2.q.s.1537.1		2
252.59	odd	6		2352.2.q.g.961.1		2
252.95	even	6		2352.2.q.s.961.1		2
252.131	odd	6		2352.2.q.g.1537.1		2
252.139	even	6		7056.2.a.x.1.1		1
252.167	odd	6		2352.2.a.s.1.1		1
504.293	even	6		9408.2.a.co.1.1		1
504.419	odd	6		9408.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.b.1.1	✓	1	9.5	odd	6
252.2.a.b.1.1		1	9.4	even	3
336.2.a.b.1.1		1	36.23	even	6
588.2.a.c.1.1		1	63.41	even	6
588.2.i.c.361.1		2	63.23	odd	6
588.2.i.c.373.1		2	63.32	odd	6
588.2.i.f.361.1		2	63.5	even	6
588.2.i.f.373.1		2	63.59	even	6
1008.2.a.g.1.1		1	36.31	odd	6
1344.2.a.f.1.1		1	72.5	odd	6
1344.2.a.o.1.1		1	72.59	even	6
1764.2.a.g.1.1		1	63.13	odd	6
1764.2.k.d.361.1		2	63.40	odd	6
1764.2.k.d.1549.1		2	63.31	odd	6
1764.2.k.e.361.1		2	63.58	even	3
1764.2.k.e.1549.1		2	63.4	even	3
2100.2.a.a.1.1		1	45.14	odd	6
2100.2.k.a.1849.1		2	45.32	even	12
2100.2.k.a.1849.2		2	45.23	even	12
2268.2.j.f.757.1		2	1.1	even	1	trivial
2268.2.j.f.1513.1		2	9.7	even	3	inner
2268.2.j.i.757.1		2	3.2	odd	2
2268.2.j.i.1513.1		2	9.2	odd	6
2352.2.a.s.1.1		1	252.167	odd	6
2352.2.q.g.961.1		2	252.59	odd	6
2352.2.q.g.1537.1		2	252.131	odd	6
2352.2.q.s.961.1		2	252.95	even	6
2352.2.q.s.1537.1		2	252.23	even	6
4032.2.a.t.1.1		1	72.67	odd	6
4032.2.a.u.1.1		1	72.13	even	6
5376.2.c.i.2689.1		2	144.5	odd	12
5376.2.c.i.2689.2		2	144.77	odd	12
5376.2.c.x.2689.1		2	144.131	even	12
5376.2.c.x.2689.2		2	144.59	even	12
6300.2.a.p.1.1		1	45.4	even	6
6300.2.k.r.6049.1		2	45.13	odd	12
6300.2.k.r.6049.2		2	45.22	odd	12
7056.2.a.x.1.1		1	252.139	even	6
8400.2.a.ct.1.1		1	180.59	even	6
9408.2.a.r.1.1		1	504.419	odd	6
9408.2.a.co.1.1		1	504.293	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-1.00000 + 1.73205i) q^{13} -4.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +2.00000 q^{37} +(6.00000 - 10.3923i) q^{41} +(2.00000 + 3.46410i) q^{43} +(6.00000 + 10.3923i) q^{47} +(-0.500000 + 0.866025i) q^{49} +6.00000 q^{53} +(5.00000 + 8.66025i) q^{61} +(-4.00000 + 6.92820i) q^{67} -6.00000 q^{71} -10.0000 q^{73} +(-3.00000 + 5.19615i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-6.00000 - 10.3923i) q^{83} -12.0000 q^{89} +2.00000 q^{91} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7}+O(q^{10}) \) 2 * q - q^7	\( 2 q - q^{7} - 6 q^{11} - 2 q^{13} - 8 q^{19} - 6 q^{23} + 5 q^{25} + 6 q^{29} - 8 q^{31} + 4 q^{37} + 12 q^{41} + 4 q^{43} + 12 q^{47} - q^{49} + 12 q^{53} + 10 q^{61} - 8 q^{67} - 12 q^{71} - 20 q^{73} - 6 q^{77} + 4 q^{79} - 12 q^{83} - 24 q^{89} + 4 q^{91} + 10 q^{97}+O(q^{100}) \) 2 * q - q^7 - 6 * q^11 - 2 * q^13 - 8 * q^19 - 6 * q^23 + 5 * q^25 + 6 * q^29 - 8 * q^31 + 4 * q^37 + 12 * q^41 + 4 * q^43 + 12 * q^47 - q^49 + 12 * q^53 + 10 * q^61 - 8 * q^67 - 12 * q^71 - 20 * q^73 - 6 * q^77 + 4 * q^79 - 12 * q^83 - 24 * q^89 + 4 * q^91 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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