LMFDB - Embedded newform 2268.2.l.a.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(109,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.109"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.l (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-6,0,5,0,0,0,6,0,-2,0,0,0,3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.109
Dual form			2268.2.l.a.541.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{5} +(2.50000 + 0.866025i) q^{7} +3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(1.50000 - 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} -3.00000 q^{23} +4.00000 q^{25} +(-3.00000 - 5.19615i) q^{29} +(3.50000 + 6.06218i) q^{31} +(-7.50000 - 2.59808i) q^{35} +(0.500000 + 0.866025i) q^{37} +(3.00000 - 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-4.50000 + 7.79423i) q^{47} +(5.50000 + 4.33013i) q^{49} +(1.50000 - 2.59808i) q^{53} -9.00000 q^{55} +(4.50000 + 7.79423i) q^{59} +(0.500000 - 0.866025i) q^{61} +(3.00000 - 5.19615i) q^{65} +(3.50000 + 6.06218i) q^{67} +(0.500000 - 0.866025i) q^{73} +(7.50000 + 2.59808i) q^{77} +(6.50000 - 11.2583i) q^{79} +(6.00000 + 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(7.50000 + 12.9904i) q^{89} +(-4.00000 + 3.46410i) q^{91} +(-1.50000 - 2.59808i) q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 5 q^{7} + 6 q^{11} - 2 q^{13} + 3 q^{17} + q^{19} - 6 q^{23} + 8 q^{25} - 6 q^{29} + 7 q^{31} - 15 q^{35} + q^{37} + 6 q^{41} + 4 q^{43} - 9 q^{47} + 11 q^{49} + 3 q^{53} - 18 q^{55} + 9 q^{59}+ \cdots + 10 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 5 * q^7 + 6 * q^11 - 2 * q^13 + 3 * q^17 + q^19 - 6 * q^23 + 8 * q^25 - 6 * q^29 + 7 * q^31 - 15 * q^35 + q^37 + 6 * q^41 + 4 * q^43 - 9 * q^47 + 11 * q^49 + 3 * q^53 - 18 * q^55 + 9 * q^59 + q^61 + 6 * q^65 + 7 * q^67 + q^73 + 15 * q^77 + 13 * q^79 + 12 * q^83 - 9 * q^85 + 15 * q^89 - 8 * q^91 - 3 * q^95 + 10 * q^97

Character values

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

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−3.00000			−1.34164			−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000			−1.34164			−0.670820	+	0.741620i	$0.734058\pi$
$6$		0			0
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		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
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000			−0.625543			−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000			−0.625543			−0.312772	+	0.949828i	$0.601257\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	0.440652	−	0.897678i	$-0.354747\pi$
$30$		0			0
$31$	3.50000	+	6.06218i	0.628619	+	1.08880i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	3.50000	+	6.06218i	0.628619	+	1.08880i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.50000	−	2.59808i	−1.26773	−	0.439155i
$36$		0			0
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		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
$47$	−4.50000	+	7.79423i	−0.656392	+	1.13691i	0.325150	+	0.945662i	$0.394585\pi$
$47$	−4.50000	+	7.79423i	−0.656392	+	1.13691i	−0.981543	+	0.191243i	$0.938748\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.50000	−	2.59808i	0.206041	−	0.356873i	−0.744423	−	0.667708i	$-0.767275\pi$
$53$	1.50000	−	2.59808i	0.206041	−	0.356873i	0.950464	+	0.310835i	$0.100609\pi$
$54$		0			0
$55$	−9.00000			−1.21356
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	0.994769	+	0.102151i	$0.0325726\pi$
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	−0.408919	+	0.912571i	$0.634094\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	−	5.19615i	0.372104	−	0.644503i
$66$		0			0
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	0.996659	−	0.0816792i	$-0.0260283\pi$
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	−0.569066	+	0.822292i	$0.692695\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	0.500000	−	0.866025i	0.0585206	−	0.101361i	−0.835281	−	0.549823i	$-0.814695\pi$
$73$	0.500000	−	0.866025i	0.0585206	−	0.101361i	0.893801	+	0.448463i	$0.148028\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.50000	+	2.59808i	0.854704	+	0.296078i
$78$		0			0
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	0.922840	+	0.385183i	$0.125862\pi$
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	−0.127842	+	0.991795i	$0.540805\pi$
$90$		0			0
$91$	−4.00000	+	3.46410i	−0.419314	+	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.50000	−	2.59808i	−0.153897	−	0.266557i
$96$		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
$101$	−15.0000			−1.49256			−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000			−1.49256			−0.746278	+	0.665635i	$0.768161\pi$
$102$		0			0
$103$	11.0000			1.08386			0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000			1.08386			0.541931	+	0.840423i	$0.317693\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.50000	+	12.9904i	0.725052	+	1.25583i	0.958952	+	0.283567i	$0.0915178\pi$
$107$	7.50000	+	12.9904i	0.725052	+	1.25583i	−0.233900	+	0.972261i	$0.575149\pi$
$108$		0			0
$109$	0.500000	−	0.866025i	0.0478913	−	0.0829502i	−0.841086	−	0.540901i	$-0.818083\pi$
$109$	0.500000	−	0.866025i	0.0478913	−	0.0829502i	0.888977	+	0.457951i	$0.151417\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	9.00000			0.839254
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	−	5.19615i	0.550019	−	0.476331i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.00000			−0.262111			−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000			−0.262111			−0.131056	+	0.991375i	$0.541837\pi$
$132$		0			0
$133$	0.500000	+	2.59808i	0.0433555	+	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	21.0000			1.79415			0.897076	−	0.441877i	$-0.145687\pi$
$137$	21.0000			1.79415			0.897076	+	0.441877i	$0.145687\pi$
$138$		0			0
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	0.0346338	+	0.999400i	$0.488974\pi$
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	−0.882823	+	0.469706i	$0.844360\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.00000	+	5.19615i	−0.250873	+	0.434524i
$144$		0			0
$145$	9.00000	+	15.5885i	0.747409	+	1.29455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000			−0.245770			−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000			−0.245770			−0.122885	+	0.992421i	$0.539215\pi$
$150$		0			0
$151$	17.0000			1.38344			0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000			1.38344			0.691720	+	0.722166i	$0.256853\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.5000	−	18.1865i	−0.843380	−	1.46078i
$156$		0			0
$157$	6.50000	+	11.2583i	0.518756	+	0.898513i	0.999762	+	0.0217953i	$0.00693820\pi$
$157$	6.50000	+	11.2583i	0.518756	+	0.898513i	−0.481006	+	0.876717i	$0.659728\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.50000	−	2.59808i	−0.591083	−	0.204757i
$162$		0			0
$163$	−5.50000	−	9.52628i	−0.430793	−	0.746156i	0.566149	−	0.824303i	$-0.308433\pi$
$163$	−5.50000	−	9.52628i	−0.430793	−	0.746156i	−0.996942	+	0.0781474i	$0.975100\pi$
$164$		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
$173$	−4.50000	+	7.79423i	−0.342129	+	0.592584i	−0.984828	−	0.173534i	$-0.944481\pi$
$173$	−4.50000	+	7.79423i	−0.342129	+	0.592584i	0.642699	+	0.766119i	$0.277815\pi$
$174$		0			0
$175$	10.0000	+	3.46410i	0.755929	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.5000	−	18.1865i	0.784807	−	1.35933i	−0.144308	−	0.989533i	$-0.546095\pi$
$179$	10.5000	−	18.1865i	0.784807	−	1.35933i	0.929114	−	0.369792i	$-0.120571\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.50000	−	2.59808i	−0.110282	−	0.191014i
$186$		0			0
$187$	4.50000	−	7.79423i	0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.50000	+	7.79423i	−0.325609	+	0.563971i	−0.981635	−	0.190767i	$-0.938902\pi$
$191$	−4.50000	+	7.79423i	−0.325609	+	0.563971i	0.656027	+	0.754738i	$0.272236\pi$
$192$		0			0
$193$	−5.50000	−	9.52628i	−0.395899	−	0.685717i	0.597317	−	0.802005i	$-0.296234\pi$
$193$	−5.50000	−	9.52628i	−0.395899	−	0.685717i	−0.993215	+	0.116289i	$0.962900\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	−0.714893	−	0.699234i	$-0.753524\pi$
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	0.963001	+	0.269498i	$0.0868577\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.00000	−	15.5885i	−0.210559	−	1.09410i
$204$		0			0
$205$	−9.00000	+	15.5885i	−0.628587	+	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.50000	+	2.59808i	0.103757	+	0.179713i
$210$		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
$215$	−6.00000	−	10.3923i	−0.409197	−	0.708749i
$216$		0			0
$217$	3.50000	+	18.1865i	0.237595	+	1.23458i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		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
$227$	3.00000			0.199117			0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000			0.199117			0.0995585	+	0.995032i	$0.468257\pi$
$228$		0			0
$229$	11.0000			0.726900			0.363450	−	0.931614i	$-0.381599\pi$
$229$	11.0000			0.726900			0.363450	+	0.931614i	$0.381599\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.5000	−	18.1865i	−0.687878	−	1.19144i	−0.972523	−	0.232806i	$-0.925209\pi$
$233$	−10.5000	−	18.1865i	−0.687878	−	1.19144i	0.284645	−	0.958633i	$-0.408124\pi$
$234$		0			0
$235$	13.5000	−	23.3827i	0.880643	−	1.52532i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	−0.992195	−	0.124696i	$-0.960204\pi$
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	0.604087	+	0.796918i	$0.293538\pi$
$240$		0			0
$241$	−1.00000			−0.0644157			−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000			−0.0644157			−0.0322078	+	0.999481i	$0.510254\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−16.5000	−	12.9904i	−1.05415	−	0.829925i
$246$		0			0
$247$	−2.00000			−0.127257
$248$		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
$253$	−9.00000			−0.565825
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000			−0.187135			−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000			−0.187135			−0.0935674	+	0.995613i	$0.529827\pi$
$258$		0			0
$259$	0.500000	+	2.59808i	0.0310685	+	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000			0.184988			0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000			0.184988			0.0924940	+	0.995713i	$0.470516\pi$
$264$		0			0
$265$	−4.50000	+	7.79423i	−0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.50000	−	2.59808i	0.0914566	−	0.158408i	−0.816668	−	0.577108i	$-0.804181\pi$
$269$	1.50000	−	2.59808i	0.0914566	−	0.158408i	0.908124	+	0.418701i	$0.137514\pi$
$270$		0			0
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	0.649211	−	0.760609i	$-0.275099\pi$
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	−0.983312	+	0.181928i	$0.941766\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000			0.723627
$276$		0			0
$277$	−13.0000			−0.781094			−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000			−0.781094			−0.390547	+	0.920583i	$0.627714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.0000	+	25.9808i	0.894825	+	1.54988i	0.834021	+	0.551733i	$0.186033\pi$
$281$	15.0000	+	25.9808i	0.894825	+	1.54988i	0.0608039	+	0.998150i	$0.480634\pi$
$282$		0			0
$283$	−14.5000	−	25.1147i	−0.861936	−	1.49292i	−0.870058	−	0.492949i	$-0.835919\pi$
$283$	−14.5000	−	25.1147i	−0.861936	−	1.49292i	0.00812260	−	0.999967i	$-0.497414\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	−	10.3923i	0.708338	−	0.613438i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.00000	−	5.19615i	0.175262	−	0.303562i	−0.764990	−	0.644042i	$-0.777256\pi$
$293$	3.00000	−	5.19615i	0.175262	−	0.303562i	0.940252	+	0.340480i	$0.110589\pi$
$294$		0			0
$295$	−13.5000	−	23.3827i	−0.786000	−	1.36139i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.00000	−	5.19615i	0.173494	−	0.300501i
$300$		0			0
$301$	2.00000	+	10.3923i	0.115278	+	0.599002i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.50000	+	2.59808i	−0.0858898	+	0.148765i
$306$		0			0
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.5000	−	23.3827i	−0.765515	−	1.32591i	−0.939974	−	0.341246i	$-0.889151\pi$
$311$	−13.5000	−	23.3827i	−0.765515	−	1.32591i	0.174459	−	0.984664i	$-0.444182\pi$
$312$		0			0
$313$	−11.5000	+	19.9186i	−0.650018	+	1.12586i	0.333099	+	0.942892i	$0.391906\pi$
$313$	−11.5000	+	19.9186i	−0.650018	+	1.12586i	−0.983118	+	0.182973i	$0.941428\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.50000	+	7.79423i	−0.252745	+	0.437767i	−0.964281	−	0.264883i	$-0.914667\pi$
$317$	−4.50000	+	7.79423i	−0.252745	+	0.437767i	0.711535	+	0.702650i	$0.248000\pi$
$318$		0			0
$319$	−9.00000	−	15.5885i	−0.503903	−	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.00000			0.166924
$324$		0			0
$325$	−4.00000	+	6.92820i	−0.221880	+	0.384308i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.0000	+	15.5885i	−0.992372	+	0.859419i
$330$		0			0
$331$	6.50000	−	11.2583i	0.357272	−	0.618814i	−0.630232	−	0.776407i	$-0.717040\pi$
$331$	6.50000	−	11.2583i	0.357272	−	0.618814i	0.987504	+	0.157593i	$0.0503735\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.5000	−	18.1865i	−0.573676	−	0.993636i
$336$		0			0
$337$	17.0000	−	29.4449i	0.926049	−	1.60396i	0.136184	−	0.990684i	$-0.456516\pi$
$337$	17.0000	−	29.4449i	0.926049	−	1.60396i	0.789865	−	0.613280i	$-0.210150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.5000	+	18.1865i	0.568607	+	0.984856i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.50000	+	7.79423i	0.241573	+	0.418416i	0.961162	−	0.275983i	$-0.0890035\pi$
$347$	4.50000	+	7.79423i	0.241573	+	0.418416i	−0.719590	+	0.694399i	$0.755670\pi$
$348$		0			0
$349$	−13.0000	−	22.5167i	−0.695874	−	1.20529i	−0.969885	−	0.243563i	$-0.921684\pi$
$349$	−13.0000	−	22.5167i	−0.695874	−	1.20529i	0.274011	−	0.961727i	$-0.411649\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.0000			1.11772			0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000			1.11772			0.558859	+	0.829263i	$0.311239\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.50000	+	12.9904i	0.395835	+	0.685606i	0.993207	−	0.116358i	$-0.0371219\pi$
$359$	7.50000	+	12.9904i	0.395835	+	0.685606i	−0.597372	+	0.801964i	$0.703789\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.50000	+	2.59808i	−0.0785136	+	0.135990i
$366$		0			0
$367$	5.00000			0.260998			0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000			0.260998			0.130499	+	0.991448i	$0.458342\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000	−	5.19615i	0.311504	−	0.269771i
$372$		0			0
$373$	−25.0000			−1.29445			−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000			−1.29445			−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	33.0000			1.68622			0.843111	−	0.537740i	$-0.180722\pi$
$383$	33.0000			1.68622			0.843111	+	0.537740i	$0.180722\pi$
$384$		0			0
$385$	−22.5000	−	7.79423i	−1.14671	−	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.0000			−0.760530			−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000			−0.760530			−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$	−4.50000	+	7.79423i	−0.227575	+	0.394171i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−19.5000	+	33.7750i	−0.981151	+	1.69940i
$396$		0			0
$397$	18.5000	+	32.0429i	0.928488	+	1.60819i	0.785853	+	0.618414i	$0.212224\pi$
$397$	18.5000	+	32.0429i	0.928488	+	1.60819i	0.142636	+	0.989775i	$0.454442\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000			−0.149813			−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000			−0.149813			−0.0749064	+	0.997191i	$0.523866\pi$
$402$		0			0
$403$	−14.0000			−0.697390
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.50000	+	2.59808i	0.0743522	+	0.128782i
$408$		0			0
$409$	−5.50000	−	9.52628i	−0.271957	−	0.471044i	0.697406	−	0.716677i	$-0.254338\pi$
$409$	−5.50000	−	9.52628i	−0.271957	−	0.471044i	−0.969363	+	0.245633i	$0.921004\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.50000	+	23.3827i	0.221431	+	1.15059i
$414$		0			0
$415$	−18.0000	−	31.1769i	−0.883585	−	1.53041i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$		0			0
$427$	2.00000	−	1.73205i	0.0967868	−	0.0838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.50000	+	12.9904i	−0.361262	+	0.625725i	−0.988169	−	0.153370i	$-0.950987\pi$
$431$	−7.50000	+	12.9904i	−0.361262	+	0.625725i	0.626907	+	0.779094i	$0.284321\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.50000	−	2.59808i	−0.0717547	−	0.124283i
$438$		0			0
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	−0.853847	−	0.520524i	$-0.825737\pi$
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	0.877711	+	0.479191i	$0.159070\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.50000	+	7.79423i	−0.213801	+	0.370315i	−0.952901	−	0.303281i	$-0.901918\pi$
$443$	−4.50000	+	7.79423i	−0.213801	+	0.370315i	0.739100	+	0.673596i	$0.235251\pi$
$444$		0			0
$445$	−22.5000	−	38.9711i	−1.06660	−	1.84741i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.0000	−	10.3923i	0.562569	−	0.487199i
$456$		0			0
$457$	−11.5000	+	19.9186i	−0.537947	+	0.931752i	0.461067	+	0.887365i	$0.347467\pi$
$457$	−11.5000	+	19.9186i	−0.537947	+	0.931752i	−0.999014	+	0.0443868i	$0.985867\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	0.787668	−	0.616100i	$-0.211288\pi$
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	−0.927392	+	0.374091i	$0.877955\pi$
$462$		0			0
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	−0.618050	−	0.786139i	$-0.712077\pi$
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	0.989841	+	0.142177i	$0.0454103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.5000	−	18.1865i	−0.485882	−	0.841572i	0.513986	−	0.857798i	$-0.328168\pi$
$467$	−10.5000	−	18.1865i	−0.485882	−	0.841572i	−0.999868	+	0.0162260i	$0.994835\pi$
$468$		0			0
$469$	3.50000	+	18.1865i	0.161615	+	0.839776i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000	+	10.3923i	0.275880	+	0.477839i
$474$		0			0
$475$	2.00000	+	3.46410i	0.0917663	+	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.00000			0.137073			0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000			0.137073			0.0685367	+	0.997649i	$0.478167\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.0000	−	25.9808i	−0.681115	−	1.17973i
$486$		0			0
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	−0.457263	−	0.889332i	$-0.651170\pi$
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	0.998815	−	0.0486647i	$-0.0154966\pi$
$492$		0			0
$493$	−18.0000			−0.810679
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	11.0000			0.492428			0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000			0.492428			0.246214	+	0.969216i	$0.420813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	45.0000			2.00247
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.00000			−0.132973			−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000			−0.132973			−0.0664863	+	0.997787i	$0.521179\pi$
$510$		0			0
$511$	2.00000	−	1.73205i	0.0884748	−	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−33.0000			−1.45415
$516$		0			0
$517$	−13.5000	+	23.3827i	−0.593729	+	1.02837i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19.5000	−	33.7750i	0.854311	−	1.47971i	−0.0229727	−	0.999736i	$-0.507313\pi$
$521$	19.5000	−	33.7750i	0.854311	−	1.47971i	0.877283	−	0.479973i	$-0.159354\pi$
$522$		0			0
$523$	0.500000	+	0.866025i	0.0218635	+	0.0378686i	0.876750	−	0.480946i	$-0.159707\pi$
$523$	0.500000	+	0.866025i	0.0218635	+	0.0378686i	−0.854887	+	0.518815i	$0.826373\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.0000			0.914774
$528$		0			0
$529$	−14.0000			−0.608696
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.00000	+	10.3923i	0.259889	+	0.450141i
$534$		0			0
$535$	−22.5000	−	38.9711i	−0.972760	−	1.68487i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.5000	+	12.9904i	0.710705	+	0.559535i
$540$		0			0
$541$	−17.5000	−	30.3109i	−0.752384	−	1.30317i	−0.946664	−	0.322221i	$-0.895571\pi$
$541$	−17.5000	−	30.3109i	−0.752384	−	1.30317i	0.194281	−	0.980946i	$-0.437763\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.50000	+	2.59808i	−0.0642529	+	0.111289i
$546$		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
$551$	3.00000	−	5.19615i	0.127804	−	0.221364i
$552$		0			0
$553$	26.0000	−	22.5167i	1.10563	−	0.957506i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.5000	+	28.5788i	−0.699127	+	1.21092i	0.269642	+	0.962961i	$0.413095\pi$
$557$	−16.5000	+	28.5788i	−0.699127	+	1.21092i	−0.968769	+	0.247964i	$0.920239\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	0.945134	−	0.326682i	$-0.105931\pi$
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	−0.755482	+	0.655169i	$0.772597\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.50000	+	7.79423i	−0.188650	+	0.326751i	−0.944800	−	0.327647i	$-0.893744\pi$
$569$	−4.50000	+	7.79423i	−0.188650	+	0.326751i	0.756151	+	0.654398i	$0.227078\pi$
$570$		0			0
$571$	−14.5000	−	25.1147i	−0.606806	−	1.05102i	−0.991763	−	0.128085i	$-0.959117\pi$
$571$	−14.5000	−	25.1147i	−0.606806	−	1.05102i	0.384957	−	0.922934i	$-0.374216\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	0.500000	−	0.866025i	0.0208153	−	0.0360531i	−0.855430	−	0.517918i	$-0.826707\pi$
$577$	0.500000	−	0.866025i	0.0208153	−	0.0360531i	0.876245	+	0.481865i	$0.160040\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.00000	+	31.1769i	0.248922	+	1.29344i
$582$		0			0
$583$	4.50000	−	7.79423i	0.186371	−	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.00000	+	10.3923i	0.247647	+	0.428936i	0.962872	−	0.269957i	$-0.0870095\pi$
$587$	6.00000	+	10.3923i	0.247647	+	0.428936i	−0.715226	+	0.698893i	$0.753676\pi$
$588$		0			0
$589$	−3.50000	+	6.06218i	−0.144215	+	0.249788i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	0.565792	−	0.824548i	$-0.308570\pi$
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	−0.996976	+	0.0777165i	$0.975237\pi$
$594$		0			0
$595$	−18.0000	+	15.5885i	−0.737928	+	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.5000	−	23.3827i	−0.551595	−	0.955391i	−0.998160	−	0.0606393i	$-0.980686\pi$
$599$	−13.5000	−	23.3827i	−0.551595	−	0.955391i	0.446565	−	0.894751i	$-0.352647\pi$
$600$		0			0
$601$	−7.00000	−	12.1244i	−0.285536	−	0.494563i	0.687203	−	0.726465i	$-0.258838\pi$
$601$	−7.00000	−	12.1244i	−0.285536	−	0.494563i	−0.972739	+	0.231903i	$0.925505\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.00000			0.243935
$606$		0			0
$607$	47.0000			1.90767			0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000			1.90767			0.953836	+	0.300329i	$0.0970966\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.00000	−	15.5885i	−0.364101	−	0.630641i
$612$		0			0
$613$	12.5000	−	21.6506i	0.504870	−	0.874461i	−0.495114	−	0.868828i	$-0.664874\pi$
$613$	12.5000	−	21.6506i	0.504870	−	0.874461i	0.999984	−	0.00563283i	$-0.00179300\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	−	5.19615i	0.120775	−	0.209189i	−0.799298	−	0.600935i	$-0.794795\pi$
$617$	3.00000	−	5.19615i	0.120775	−	0.209189i	0.920074	+	0.391745i	$0.128129\pi$
$618$		0			0
$619$	−31.0000			−1.24600			−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000			−1.24600			−0.622998	+	0.782224i	$0.714085\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.50000	+	38.9711i	0.300481	+	1.56135i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.00000			0.119618
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$	−13.0000	+	5.19615i	−0.515079	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.0000			−0.592464			−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000			−0.592464			−0.296232	+	0.955116i	$0.595730\pi$
$642$		0			0
$643$	−10.0000	+	17.3205i	−0.394362	+	0.683054i	−0.993019	−	0.117951i	$-0.962368\pi$
$643$	−10.0000	+	17.3205i	−0.394362	+	0.683054i	0.598658	+	0.801005i	$0.295701\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.5000	−	18.1865i	0.412798	−	0.714986i	−0.582397	−	0.812905i	$-0.697885\pi$
$647$	10.5000	−	18.1865i	0.412798	−	0.714986i	0.995194	+	0.0979182i	$0.0312184\pi$
$648$		0			0
$649$	13.5000	+	23.3827i	0.529921	+	0.917851i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−39.0000			−1.52619			−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000			−1.52619			−0.763094	+	0.646288i	$0.776321\pi$
$654$		0			0
$655$	9.00000			0.351659
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	0.958902	−	0.283738i	$-0.0915745\pi$
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	−0.725175	+	0.688565i	$0.758241\pi$
$660$		0			0
$661$	−5.50000	−	9.52628i	−0.213925	−	0.370529i	0.739014	−	0.673690i	$-0.235292\pi$
$661$	−5.50000	−	9.52628i	−0.213925	−	0.370529i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.50000	−	7.79423i	−0.0581675	−	0.302247i
$666$		0			0
$667$	9.00000	+	15.5885i	0.348481	+	0.603587i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.50000	−	2.59808i	0.0579069	−	0.100298i
$672$		0			0
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	0.698988	−	0.715134i	$-0.253634\pi$
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	−0.968818	+	0.247774i	$0.920301\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.5000	−	23.3827i	0.518847	−	0.898670i	−0.480913	−	0.876768i	$-0.659695\pi$
$677$	13.5000	−	23.3827i	0.518847	−	0.898670i	0.999760	−	0.0219013i	$-0.00697196\pi$
$678$		0			0
$679$	5.00000	+	25.9808i	0.191882	+	0.997050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.5000	−	18.1865i	0.401771	−	0.695888i	−0.592168	−	0.805814i	$-0.701728\pi$
$683$	10.5000	−	18.1865i	0.401771	−	0.695888i	0.993940	+	0.109926i	$0.0350613\pi$
$684$		0			0
$685$	−63.0000			−2.40711
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	6.50000	−	11.2583i	0.247272	−	0.428287i	−0.715496	−	0.698617i	$-0.753799\pi$
$691$	6.50000	−	11.2583i	0.247272	−	0.428287i	0.962768	+	0.270330i	$0.0871327\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	30.0000	−	51.9615i	1.13796	−	1.97101i
$696$		0			0
$697$	−9.00000	−	15.5885i	−0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	−0.500000	+	0.866025i	−0.0188579	+	0.0326628i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−37.5000	−	12.9904i	−1.41033	−	0.488554i
$708$		0			0
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	−0.856484	−	0.516174i	$-0.827356\pi$
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	0.875262	+	0.483650i	$0.160689\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.5000	−	18.1865i	−0.393228	−	0.681091i
$714$		0			0
$715$	9.00000	−	15.5885i	0.336581	−	0.582975i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.5000	−	18.1865i	−0.391584	−	0.678243i	0.601075	−	0.799193i	$-0.294739\pi$
$719$	−10.5000	−	18.1865i	−0.391584	−	0.678243i	−0.992659	+	0.120950i	$0.961406\pi$
$720$		0			0
$721$	27.5000	+	9.52628i	1.02415	+	0.354777i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.0000	−	20.7846i	−0.445669	−	0.771921i
$726$		0			0
$727$	−16.0000	−	27.7128i	−0.593407	−	1.02781i	−0.993770	−	0.111454i	$-0.964449\pi$
$727$	−16.0000	−	27.7128i	−0.593407	−	1.02781i	0.400362	−	0.916357i	$-0.368884\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000			0.443836
$732$		0			0
$733$	−25.0000			−0.923396			−0.461698	−	0.887037i	$-0.652760\pi$
$733$	−25.0000			−0.923396			−0.461698	+	0.887037i	$0.652760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.5000	+	18.1865i	0.386772	+	0.669910i
$738$		0			0
$739$	9.50000	−	16.4545i	0.349463	−	0.605288i	−0.636691	−	0.771119i	$-0.719697\pi$
$739$	9.50000	−	16.4545i	0.349463	−	0.605288i	0.986154	+	0.165831i	$0.0530307\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0000	+	41.5692i	−0.880475	+	1.52503i	−0.0296605	+	0.999560i	$0.509443\pi$
$743$	−24.0000	+	41.5692i	−0.880475	+	1.52503i	−0.850814	+	0.525467i	$0.823891\pi$
$744$		0			0
$745$	9.00000			0.329734
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.50000	+	38.9711i	0.274044	+	1.42397i
$750$		0			0
$751$	−25.0000			−0.912263			−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000			−0.912263			−0.456131	+	0.889912i	$0.650765\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−51.0000			−1.85608
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000			−0.108750			−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000			−0.108750			−0.0543750	+	0.998521i	$0.517317\pi$
$762$		0			0
$763$	2.00000	−	1.73205i	0.0724049	−	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000			−0.649942
$768$		0			0
$769$	17.0000	−	29.4449i	0.613036	−	1.06181i	−0.377690	−	0.925932i	$-0.623282\pi$
$769$	17.0000	−	29.4449i	0.613036	−	1.06181i	0.990726	−	0.135877i	$-0.0433852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.5000	+	28.5788i	−0.593464	+	1.02791i	0.400298	+	0.916385i	$0.368907\pi$
$773$	−16.5000	+	28.5788i	−0.593464	+	1.02791i	−0.993762	+	0.111524i	$0.964427\pi$
$774$		0			0
$775$	14.0000	+	24.2487i	0.502895	+	0.871039i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.00000			0.214972
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−19.5000	−	33.7750i	−0.695985	−	1.20548i
$786$		0			0
$787$	15.5000	+	26.8468i	0.552515	+	0.956985i	0.998092	+	0.0617409i	$0.0196653\pi$
$787$	15.5000	+	26.8468i	0.552515	+	0.956985i	−0.445577	+	0.895244i	$0.647001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000	−	10.3923i	0.426671	−	0.369508i
$792$		0			0
$793$	1.00000	+	1.73205i	0.0355110	+	0.0615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.0000	−	36.3731i	0.743858	−	1.28840i	−0.206868	−	0.978369i	$-0.566327\pi$
$797$	21.0000	−	36.3731i	0.743858	−	1.28840i	0.950726	−	0.310031i	$-0.100340\pi$
$798$		0			0
$799$	13.5000	+	23.3827i	0.477596	+	0.827220i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.50000	−	2.59808i	0.0529339	−	0.0916841i
$804$		0			0
$805$	22.5000	+	7.79423i	0.793021	+	0.274710i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.5000	+	28.5788i	−0.580109	+	1.00478i	0.415357	+	0.909659i	$0.363657\pi$
$809$	−16.5000	+	28.5788i	−0.580109	+	1.00478i	−0.995466	+	0.0951198i	$0.969677\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.5000	+	28.5788i	0.577970	+	1.00107i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.5000	−	23.3827i	0.471153	−	0.816061i	−0.528302	−	0.849056i	$-0.677171\pi$
$821$	13.5000	−	23.3827i	0.471153	−	0.816061i	0.999456	+	0.0329950i	$0.0105045\pi$
$822$		0			0
$823$	−2.50000	−	4.33013i	−0.0871445	−	0.150939i	0.819159	−	0.573567i	$-0.194441\pi$
$823$	−2.50000	−	4.33013i	−0.0871445	−	0.150939i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	0.500000	−	0.866025i	0.0173657	−	0.0300783i	−0.857212	−	0.514964i	$-0.827805\pi$
$829$	0.500000	−	0.866025i	0.0173657	−	0.0300783i	0.874578	+	0.484885i	$0.161139\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19.5000	−	7.79423i	0.675635	−	0.270054i
$834$		0			0
$835$	18.0000	−	31.1769i	0.622916	−	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	0.581067	−	0.813856i	$-0.302635\pi$
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	−0.995353	+	0.0962912i	$0.969302\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.5000	−	23.3827i	−0.464414	−	0.804389i
$846$		0			0
$847$	−5.00000	−	1.73205i	−0.171802	−	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.50000	−	2.59808i	−0.0514193	−	0.0890609i
$852$		0			0
$853$	11.0000	+	19.0526i	0.376633	+	0.652347i	0.990570	−	0.137008i	$-0.0437485\pi$
$853$	11.0000	+	19.0526i	0.376633	+	0.652347i	−0.613937	+	0.789355i	$0.710415\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.0000			−0.512390			−0.256195	−	0.966625i	$-0.582469\pi$
$857$	−15.0000			−0.512390			−0.256195	+	0.966625i	$0.582469\pi$
$858$		0			0
$859$	23.0000			0.784750			0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000			0.784750			0.392375	+	0.919805i	$0.371654\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25.5000	+	44.1673i	0.868030	+	1.50347i	0.864007	+	0.503480i	$0.167947\pi$
$863$	25.5000	+	44.1673i	0.868030	+	1.50347i	0.00402340	+	0.999992i	$0.498719\pi$
$864$		0			0
$865$	13.5000	−	23.3827i	0.459014	−	0.795035i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	19.5000	−	33.7750i	0.661492	−	1.14574i
$870$		0			0
$871$	−14.0000			−0.474372
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.50000	+	2.59808i	0.253546	+	0.0878310i
$876$		0			0
$877$	−13.0000			−0.438979			−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000			−0.438979			−0.219489	+	0.975615i	$0.570439\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−39.0000			−1.30949			−0.654746	−	0.755849i	$-0.727224\pi$
$887$	−39.0000			−1.30949			−0.654746	+	0.755849i	$0.727224\pi$
$888$		0			0
$889$	20.0000	+	6.92820i	0.670778	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.00000			−0.301174
$894$		0			0
$895$	−31.5000	+	54.5596i	−1.05293	+	1.82373i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.0000	−	36.3731i	0.700389	−	1.21311i
$900$		0			0
$901$	−4.50000	−	7.79423i	−0.149917	−	0.259663i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	30.0000			0.997234
$906$		0			0
$907$	41.0000			1.36138			0.680691	−	0.732570i	$-0.261680\pi$
$907$	41.0000			1.36138			0.680691	+	0.732570i	$0.261680\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.0000	+	41.5692i	0.795155	+	1.37725i	0.922740	+	0.385422i	$0.125944\pi$
$911$	24.0000	+	41.5692i	0.795155	+	1.37725i	−0.127585	+	0.991828i	$0.540723\pi$
$912$		0			0
$913$	18.0000	+	31.1769i	0.595713	+	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.50000	−	2.59808i	−0.247672	−	0.0857960i
$918$		0			0
$919$	0.500000	+	0.866025i	0.0164935	+	0.0285675i	0.874154	−	0.485648i	$-0.161416\pi$
$919$	0.500000	+	0.866025i	0.0164935	+	0.0285675i	−0.857661	+	0.514216i	$0.828083\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.00000	+	3.46410i	0.0657596	+	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.50000	+	7.79423i	−0.147640	+	0.255720i	−0.930355	−	0.366660i	$-0.880501\pi$
$929$	−4.50000	+	7.79423i	−0.147640	+	0.255720i	0.782715	+	0.622381i	$0.213834\pi$
$930$		0			0
$931$	−1.00000	+	6.92820i	−0.0327737	+	0.227063i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−13.5000	+	23.3827i	−0.441497	+	0.764696i
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.5000	+	23.3827i	0.440087	+	0.762254i	0.997695	−	0.0678506i	$-0.0216141\pi$
$941$	13.5000	+	23.3827i	0.440087	+	0.762254i	−0.557608	+	0.830104i	$0.688281\pi$
$942$		0			0
$943$	−9.00000	+	15.5885i	−0.293080	+	0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.5000	+	38.9711i	−0.731152	+	1.26639i	0.225240	+	0.974303i	$0.427684\pi$
$947$	−22.5000	+	38.9711i	−0.731152	+	1.26639i	−0.956391	+	0.292089i	$0.905650\pi$
$948$		0			0
$949$	1.00000	+	1.73205i	0.0324614	+	0.0562247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$	13.5000	−	23.3827i	0.436850	−	0.756646i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	52.5000	+	18.1865i	1.69531	+	0.587274i
$960$		0			0
$961$	−9.00000	+	15.5885i	−0.290323	+	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.5000	+	28.5788i	0.531154	+	0.919985i
$966$		0			0
$967$	8.00000	−	13.8564i	0.257263	−	0.445592i	−0.708245	−	0.705967i	$-0.750513\pi$
$967$	8.00000	−	13.8564i	0.257263	−	0.445592i	0.965508	+	0.260375i	$0.0838461\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25.5000	+	44.1673i	0.818334	+	1.41740i	0.906909	+	0.421326i	$0.138435\pi$
$971$	25.5000	+	44.1673i	0.818334	+	1.41740i	−0.0885751	+	0.996070i	$0.528231\pi$
$972$		0			0
$973$	−40.0000	+	34.6410i	−1.28234	+	1.11054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13.5000	+	23.3827i	0.431903	+	0.748078i	0.997037	−	0.0769208i	$-0.0245089\pi$
$977$	13.5000	+	23.3827i	0.431903	+	0.748078i	−0.565134	+	0.824999i	$0.691176\pi$
$978$		0			0
$979$	22.5000	+	38.9711i	0.719103	+	1.24552i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−33.0000			−1.05254			−0.526268	−	0.850319i	$-0.676409\pi$
$983$	−33.0000			−1.05254			−0.526268	+	0.850319i	$0.676409\pi$
$984$		0			0
$985$	54.0000			1.72058
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.00000	−	10.3923i	−0.190789	−	0.330456i
$990$		0			0
$991$	9.50000	−	16.4545i	0.301777	−	0.522694i	−0.674761	−	0.738036i	$-0.735753\pi$
$991$	9.50000	−	16.4545i	0.301777	−	0.522694i	0.976539	+	0.215342i	$0.0690867\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.5000	+	18.1865i	−0.332872	+	0.576552i
$996$		0			0
$997$	59.0000			1.86855			0.934274	−	0.356555i	$-0.116049\pi$
$997$	59.0000			1.86855			0.934274	+	0.356555i	$0.116049\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.l.a.109.1		2
3.2	odd	2		2268.2.l.h.109.1		2
7.2	even	3		2268.2.i.h.2053.1		2
9.2	odd	6		2268.2.i.a.865.1		2
9.4	even	3		252.2.k.c.109.1		2
9.5	odd	6		28.2.e.a.25.1	yes	2
9.7	even	3		2268.2.i.h.865.1		2
21.2	odd	6		2268.2.i.a.2053.1		2
36.23	even	6		112.2.i.b.81.1		2
36.31	odd	6		1008.2.s.p.865.1		2
45.14	odd	6		700.2.i.c.501.1		2
45.23	even	12		700.2.r.b.249.1		4
45.32	even	12		700.2.r.b.249.2		4
63.2	odd	6		2268.2.l.h.541.1		2
63.4	even	3		1764.2.a.a.1.1		1
63.5	even	6		196.2.e.a.177.1		2
63.13	odd	6		1764.2.k.b.361.1		2
63.16	even	3	inner	2268.2.l.a.541.1		2
63.23	odd	6		28.2.e.a.9.1	✓	2
63.31	odd	6		1764.2.a.j.1.1		1
63.32	odd	6		196.2.a.b.1.1		1
63.40	odd	6		1764.2.k.b.1549.1		2
63.41	even	6		196.2.e.a.165.1		2
63.58	even	3		252.2.k.c.37.1		2
63.59	even	6		196.2.a.a.1.1		1
72.5	odd	6		448.2.i.e.193.1		2
72.59	even	6		448.2.i.c.193.1		2
252.23	even	6		112.2.i.b.65.1		2
252.31	even	6		7056.2.a.bw.1.1		1
252.59	odd	6		784.2.a.g.1.1		1
252.67	odd	6		7056.2.a.f.1.1		1
252.95	even	6		784.2.a.d.1.1		1
252.131	odd	6		784.2.i.d.177.1		2
252.167	odd	6		784.2.i.d.753.1		2
252.247	odd	6		1008.2.s.p.289.1		2
315.23	even	12		700.2.r.b.149.2		4
315.32	even	12		4900.2.e.i.2549.1		2
315.59	even	6		4900.2.a.n.1.1		1
315.122	odd	12		4900.2.e.h.2549.2		2
315.149	odd	6		700.2.i.c.401.1		2
315.158	even	12		4900.2.e.i.2549.2		2
315.212	even	12		700.2.r.b.149.1		4
315.248	odd	12		4900.2.e.h.2549.1		2
315.284	odd	6		4900.2.a.g.1.1		1
504.59	odd	6		3136.2.a.k.1.1		1
504.149	odd	6		448.2.i.e.65.1		2
504.221	odd	6		3136.2.a.h.1.1		1
504.275	even	6		448.2.i.c.65.1		2
504.347	even	6		3136.2.a.s.1.1		1
504.437	even	6		3136.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	63.23	odd	6
28.2.e.a.25.1	yes	2	9.5	odd	6
112.2.i.b.65.1		2	252.23	even	6
112.2.i.b.81.1		2	36.23	even	6
196.2.a.a.1.1		1	63.59	even	6
196.2.a.b.1.1		1	63.32	odd	6
196.2.e.a.165.1		2	63.41	even	6
196.2.e.a.177.1		2	63.5	even	6
252.2.k.c.37.1		2	63.58	even	3
252.2.k.c.109.1		2	9.4	even	3
448.2.i.c.65.1		2	504.275	even	6
448.2.i.c.193.1		2	72.59	even	6
448.2.i.e.65.1		2	504.149	odd	6
448.2.i.e.193.1		2	72.5	odd	6
700.2.i.c.401.1		2	315.149	odd	6
700.2.i.c.501.1		2	45.14	odd	6
700.2.r.b.149.1		4	315.212	even	12
700.2.r.b.149.2		4	315.23	even	12
700.2.r.b.249.1		4	45.23	even	12
700.2.r.b.249.2		4	45.32	even	12
784.2.a.d.1.1		1	252.95	even	6
784.2.a.g.1.1		1	252.59	odd	6
784.2.i.d.177.1		2	252.131	odd	6
784.2.i.d.753.1		2	252.167	odd	6
1008.2.s.p.289.1		2	252.247	odd	6
1008.2.s.p.865.1		2	36.31	odd	6
1764.2.a.a.1.1		1	63.4	even	3
1764.2.a.j.1.1		1	63.31	odd	6
1764.2.k.b.361.1		2	63.13	odd	6
1764.2.k.b.1549.1		2	63.40	odd	6
2268.2.i.a.865.1		2	9.2	odd	6
2268.2.i.a.2053.1		2	21.2	odd	6
2268.2.i.h.865.1		2	9.7	even	3
2268.2.i.h.2053.1		2	7.2	even	3
2268.2.l.a.109.1		2	1.1	even	1	trivial
2268.2.l.a.541.1		2	63.16	even	3	inner
2268.2.l.h.109.1		2	3.2	odd	2
2268.2.l.h.541.1		2	63.2	odd	6
3136.2.a.h.1.1		1	504.221	odd	6
3136.2.a.k.1.1		1	504.59	odd	6
3136.2.a.s.1.1		1	504.347	even	6
3136.2.a.v.1.1		1	504.437	even	6
4900.2.a.g.1.1		1	315.284	odd	6
4900.2.a.n.1.1		1	315.59	even	6
4900.2.e.h.2549.1		2	315.248	odd	12
4900.2.e.h.2549.2		2	315.122	odd	12
4900.2.e.i.2549.1		2	315.32	even	12
4900.2.e.i.2549.2		2	315.158	even	12
7056.2.a.f.1.1		1	252.67	odd	6
7056.2.a.bw.1.1		1	252.31	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{5} +(2.50000 + 0.866025i) q^{7} +3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(1.50000 - 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} -3.00000 q^{23} +4.00000 q^{25} +(-3.00000 - 5.19615i) q^{29} +(3.50000 + 6.06218i) q^{31} +(-7.50000 - 2.59808i) q^{35} +(0.500000 + 0.866025i) q^{37} +(3.00000 - 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-4.50000 + 7.79423i) q^{47} +(5.50000 + 4.33013i) q^{49} +(1.50000 - 2.59808i) q^{53} -9.00000 q^{55} +(4.50000 + 7.79423i) q^{59} +(0.500000 - 0.866025i) q^{61} +(3.00000 - 5.19615i) q^{65} +(3.50000 + 6.06218i) q^{67} +(0.500000 - 0.866025i) q^{73} +(7.50000 + 2.59808i) q^{77} +(6.50000 - 11.2583i) q^{79} +(6.00000 + 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(7.50000 + 12.9904i) q^{89} +(-4.00000 + 3.46410i) q^{91} +(-1.50000 - 2.59808i) q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 5 q^{7} + 6 q^{11} - 2 q^{13} + 3 q^{17} + q^{19} - 6 q^{23} + 8 q^{25} - 6 q^{29} + 7 q^{31} - 15 q^{35} + q^{37} + 6 q^{41} + 4 q^{43} - 9 q^{47} + 11 q^{49} + 3 q^{53} - 18 q^{55} + 9 q^{59}+ \cdots + 10 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 5 * q^7 + 6 * q^11 - 2 * q^13 + 3 * q^17 + q^19 - 6 * q^23 + 8 * q^25 - 6 * q^29 + 7 * q^31 - 15 * q^35 + q^37 + 6 * q^41 + 4 * q^43 - 9 * q^47 + 11 * q^49 + 3 * q^53 - 18 * q^55 + 9 * q^59 + q^61 + 6 * q^65 + 7 * q^67 + q^73 + 15 * q^77 + 13 * q^79 + 12 * q^83 - 9 * q^85 + 15 * q^89 - 8 * q^91 - 3 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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