LMFDB - Embedded newform 2268.2.i.a.2053.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(865,2268)]

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, 4, 4]))

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

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

chi := DirichletCharacter("2268.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.i (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:	$1$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2053.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.2053
Dual form			2268.2.i.a.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{5} +(-0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(-1.50000 - 2.59808i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{11} +(-1.00000 + 1.73205i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} +(-1.50000 - 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(3.00000 + 5.19615i) q^{29} -7.00000 q^{31} +(7.50000 - 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} +(-3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} -9.00000 q^{47} +(-6.50000 - 2.59808i) q^{49} +(-1.50000 - 2.59808i) q^{53} -9.00000 q^{55} +9.00000 q^{59} -1.00000 q^{61} +6.00000 q^{65} -7.00000 q^{67} +(0.500000 + 0.866025i) q^{73} +(6.00000 + 5.19615i) q^{77} -13.0000 q^{79} +(-6.00000 - 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(-4.00000 - 3.46410i) q^{91} -3.00000 q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} - q^{7}+O(q^{10}) $ 2 * q - 3 * q^5 - q^7	$ 2 q - 3 q^{5} - q^{7} + 3 q^{11} - 2 q^{13} - 3 q^{17} + q^{19} - 3 q^{23} - 4 q^{25} + 6 q^{29} - 14 q^{31} + 15 q^{35} + q^{37} - 6 q^{41} + 4 q^{43} - 18 q^{47} - 13 q^{49} - 3 q^{53} - 18 q^{55} + 18 q^{59} - 2 q^{61} + 12 q^{65} - 14 q^{67} + q^{73} + 12 q^{77} - 26 q^{79} - 12 q^{83} - 9 q^{85} - 15 q^{89} - 8 q^{91} - 6 q^{95} + 10 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 - q^7 + 3 * q^11 - 2 * q^13 - 3 * q^17 + q^19 - 3 * q^23 - 4 * q^25 + 6 * q^29 - 14 * q^31 + 15 * q^35 + q^37 - 6 * q^41 + 4 * q^43 - 18 * q^47 - 13 * q^49 - 3 * q^53 - 18 * q^55 + 18 * q^59 - 2 * q^61 + 12 * q^65 - 14 * q^67 + q^73 + 12 * q^77 - 26 * q^79 - 12 * q^83 - 9 * q^85 - 15 * q^89 - 8 * q^91 - 6 * q^95 + 10 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	0.306851	−	0.951757i	$-0.400725\pi$
$6$		0			0
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		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	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	0.666190	−	0.745782i	$-0.267924\pi$
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	−0.978961	+	0.204046i	$0.934591\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		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
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		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	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		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
$47$	−9.00000			−1.31278			−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000			−1.31278			−0.656392	+	0.754420i	$0.727918\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.50000	−	2.59808i	−0.206041	−	0.356873i	0.744423	−	0.667708i	$-0.232725\pi$
$53$	−1.50000	−	2.59808i	−0.206041	−	0.356873i	−0.950464	+	0.310835i	$0.899391\pi$
$54$		0			0
$55$	−9.00000			−1.21356
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.00000			1.17170			0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000			1.17170			0.585850	+	0.810419i	$0.300761\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$	−7.00000			−0.855186			−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000			−0.855186			−0.427593	+	0.903971i	$0.640638\pi$
$68$		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	0.893801	−	0.448463i	$-0.148028\pi$
$73$	0.500000	+	0.866025i	0.0585206	+	0.101361i	−0.835281	+	0.549823i	$0.814695\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000	+	5.19615i	0.683763	+	0.592157i
$78$		0			0
$79$	−13.0000			−1.46261			−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000			−1.46261			−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
$85$	−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	0.127842	+	0.991795i	$0.459195\pi$
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	−0.922840	+	0.385183i	$0.874138\pi$
$90$		0			0
$91$	−4.00000	−	3.46410i	−0.419314	−	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000			−0.307794
$96$		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
$101$	−7.50000	+	12.9904i	−0.746278	+	1.29259i	0.203317	+	0.979113i	$0.434828\pi$
$101$	−7.50000	+	12.9904i	−0.746278	+	1.29259i	−0.949595	+	0.313478i	$0.898506\pi$
$102$		0			0
$103$	−5.50000	−	9.52628i	−0.541931	−	0.938652i	−0.998793	−	0.0491146i	$-0.984360\pi$
$103$	−5.50000	−	9.52628i	−0.541931	−	0.938652i	0.456862	−	0.889538i	$-0.348973\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.50000	+	12.9904i	−0.725052	+	1.25583i	0.233900	+	0.972261i	$0.424851\pi$
$107$	−7.50000	+	12.9904i	−0.725052	+	1.25583i	−0.958952	+	0.283567i	$0.908482\pi$
$108$		0			0
$109$	0.500000	+	0.866025i	0.0478913	+	0.0829502i	0.888977	−	0.457951i	$-0.151417\pi$
$109$	0.500000	+	0.866025i	0.0478913	+	0.0829502i	−0.841086	+	0.540901i	$0.818083\pi$
$110$		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
$115$	−4.50000	+	7.79423i	−0.419627	+	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.50000	−	2.59808i	0.687524	−	0.238165i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		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
$131$	−1.50000	−	2.59808i	−0.131056	−	0.226995i	0.793028	−	0.609185i	$-0.208503\pi$
$131$	−1.50000	−	2.59808i	−0.131056	−	0.226995i	−0.924084	+	0.382190i	$0.875170\pi$
$132$		0			0
$133$	2.00000	+	1.73205i	0.173422	+	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.5000	−	18.1865i	0.897076	−	1.55378i	0.0658609	−	0.997829i	$-0.479021\pi$
$137$	10.5000	−	18.1865i	0.897076	−	1.55378i	0.831215	−	0.555952i	$-0.187646\pi$
$138$		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
$149$	−1.50000	−	2.59808i	−0.122885	−	0.212843i	0.798019	−	0.602632i	$-0.205881\pi$
$149$	−1.50000	−	2.59808i	−0.122885	−	0.212843i	−0.920904	+	0.389789i	$0.872548\pi$
$150$		0			0
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	0.279554	+	0.960130i	$0.409814\pi$
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	−0.971274	+	0.237964i	$0.923520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.5000	+	18.1865i	0.843380	+	1.46078i
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$		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	−0.996942	−	0.0781474i	$-0.975100\pi$
$163$	−5.50000	+	9.52628i	−0.430793	+	0.746156i	0.566149	+	0.824303i	$0.308433\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$	−8.00000	−	6.92820i	−0.604743	−	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	0.144308	−	0.989533i	$-0.453905\pi$
$180$		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
$185$	−3.00000			−0.220564
$186$		0			0
$187$	−9.00000			−0.658145
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.00000			−0.651217			−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000			−0.651217			−0.325609	+	0.945505i	$0.605569\pi$
$192$		0			0
$193$	11.0000			0.791797			0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000			0.791797			0.395899	+	0.918294i	$0.370433\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	3.50000	+	6.06218i	0.248108	+	0.429736i	0.963001	−	0.269498i	$-0.0868577\pi$
$199$	3.50000	+	6.06218i	0.248108	+	0.429736i	−0.714893	+	0.699234i	$0.753524\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−15.0000	+	5.19615i	−1.05279	+	0.364698i
$204$		0			0
$205$	18.0000			1.25717
$206$		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
$221$	6.00000			0.403604
$222$		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
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	−0.811943	−	0.583736i	$-0.801590\pi$
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	0.911502	+	0.411296i	$0.134924\pi$
$228$		0			0
$229$	−5.50000	−	9.52628i	−0.363450	−	0.629514i	0.625076	−	0.780564i	$-0.285068\pi$
$229$	−5.50000	−	9.52628i	−0.363450	−	0.629514i	−0.988526	+	0.151050i	$0.951735\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.5000	−	18.1865i	0.687878	−	1.19144i	−0.284645	−	0.958633i	$-0.591876\pi$
$233$	10.5000	−	18.1865i	0.687878	−	1.19144i	0.972523	−	0.232806i	$-0.0747909\pi$
$234$		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	−0.604087	−	0.796918i	$-0.706462\pi$
$239$	6.00000	−	10.3923i	0.388108	−	0.672222i	0.992195	+	0.124696i	$0.0397955\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.00000	+	20.7846i	0.191663	+	1.32788i
$246$		0			0
$247$	1.00000	+	1.73205i	0.0636285	+	0.110208i
$248$		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
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	0.815442	−	0.578838i	$-0.196494\pi$
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	−0.909010	+	0.416775i	$0.863160\pi$
$258$		0			0
$259$	2.00000	+	1.73205i	0.124274	+	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	−0.816066	−	0.577959i	$-0.803849\pi$
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	0.908560	+	0.417755i	$0.137183\pi$
$264$		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	$-0.195819\pi$
$269$	−1.50000	−	2.59808i	−0.0914566	−	0.158408i	−0.908124	+	0.418701i	$0.862486\pi$
$270$		0			0
$271$	−5.50000	+	9.52628i	−0.334101	+	0.578680i	−0.983312	−	0.181928i	$-0.941766\pi$
$271$	−5.50000	+	9.52628i	−0.334101	+	0.578680i	0.649211	+	0.760609i	$0.275099\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	+	10.3923i	0.361814	+	0.626680i
$276$		0			0
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	−0.601975	−	0.798515i	$-0.705619\pi$
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	0.992522	+	0.122068i	$0.0389525\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.0000	−	25.9808i	−0.894825	−	1.54988i	−0.834021	−	0.551733i	$-0.813967\pi$
$281$	−15.0000	−	25.9808i	−0.894825	−	1.54988i	−0.0608039	−	0.998150i	$-0.519366\pi$
$282$		0			0
$283$	29.0000			1.72387			0.861936	−	0.507018i	$-0.169252\pi$
$283$	29.0000			1.72387			0.861936	+	0.507018i	$0.169252\pi$
$284$		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	−0.940252	−	0.340480i	$-0.889411\pi$
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	0.764990	+	0.644042i	$0.222744\pi$
$294$		0			0
$295$	−13.5000	−	23.3827i	−0.786000	−	1.36139i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000			0.346989
$300$		0			0
$301$	−10.0000	+	3.46410i	−0.576390	+	0.199667i
$302$		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
$311$	−27.0000			−1.53103			−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000			−1.53103			−0.765515	+	0.643418i	$0.777516\pi$
$312$		0			0
$313$	23.0000			1.30004			0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000			1.30004			0.650018	+	0.759918i	$0.274761\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000			−0.505490			−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000			−0.505490			−0.252745	+	0.967533i	$0.581333\pi$
$318$		0			0
$319$	18.0000			1.00781
$320$		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
$329$	4.50000	−	23.3827i	0.248093	−	1.28913i
$330$		0			0
$331$	−13.0000			−0.714545			−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000			−0.714545			−0.357272	+	0.934000i	$0.616293\pi$
$332$		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
$347$	9.00000			0.483145			0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000			0.483145			0.241573	+	0.970383i	$0.422337\pi$
$348$		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
$353$	10.5000	−	18.1865i	0.558859	−	0.967972i	−0.438733	−	0.898617i	$-0.644573\pi$
$353$	10.5000	−	18.1865i	0.558859	−	0.967972i	0.997592	−	0.0693543i	$-0.0220939\pi$
$354$		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	$-0.962878\pi$
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	0.597372	+	0.801964i	$0.296211\pi$
$360$		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
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	−0.923869	−	0.382709i	$-0.874991\pi$
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	0.793370	+	0.608740i	$0.208325\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.50000	−	2.59808i	0.389381	−	0.134885i
$372$		0			0
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	−0.336557	+	0.941663i	$0.609263\pi$
$374$		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
$383$	16.5000	+	28.5788i	0.843111	+	1.46031i	0.887252	+	0.461285i	$0.152611\pi$
$383$	16.5000	+	28.5788i	0.843111	+	1.46031i	−0.0441413	+	0.999025i	$0.514055\pi$
$384$		0			0
$385$	4.50000	−	23.3827i	0.229341	−	1.19169i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.50000	+	12.9904i	−0.380265	+	0.658638i	−0.991100	−	0.133120i	$-0.957501\pi$
$389$	−7.50000	+	12.9904i	−0.380265	+	0.658638i	0.610835	+	0.791758i	$0.290834\pi$
$390$		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	0.142636	−	0.989775i	$-0.454442\pi$
$397$	18.5000	−	32.0429i	0.928488	−	1.60819i	0.785853	−	0.618414i	$-0.212224\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$	7.00000	−	12.1244i	0.348695	−	0.603957i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.50000	−	2.59808i	−0.0743522	−	0.128782i
$408$		0			0
$409$	11.0000			0.543915			0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000			0.543915			0.271957	+	0.962309i	$0.412329\pi$
$410$		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	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		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
$425$	12.0000			0.582086
$426$		0			0
$427$	0.500000	−	2.59808i	0.0241967	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.50000	+	12.9904i	0.361262	+	0.625725i	0.988169	−	0.153370i	$-0.0490126\pi$
$431$	7.50000	+	12.9904i	0.361262	+	0.625725i	−0.626907	+	0.779094i	$0.715679\pi$
$432$		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
$437$	−3.00000			−0.143509
$438$		0			0
$439$	−1.00000			−0.0477274			−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000			−0.0477274			−0.0238637	+	0.999715i	$0.507597\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.00000			−0.427603			−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000			−0.427603			−0.213801	+	0.976877i	$0.568585\pi$
$444$		0			0
$445$	45.0000			2.13320
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	9.00000	+	15.5885i	0.423793	+	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.00000	+	15.5885i	−0.140642	+	0.730798i
$456$		0			0
$457$	23.0000			1.07589			0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000			1.07589			0.537947	+	0.842978i	$0.319200\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.00000	+	5.19615i	0.139724	+	0.242009i	0.927392	−	0.374091i	$-0.122045\pi$
$461$	3.00000	+	5.19615i	0.139724	+	0.242009i	−0.787668	+	0.616100i	$0.788712\pi$
$462$		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	$-0.671832\pi$
$467$	10.5000	−	18.1865i	0.485882	−	0.841572i	0.999868	+	0.0162260i	$0.00516512\pi$
$468$		0			0
$469$	3.50000	−	18.1865i	0.161615	−	0.839776i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	2.00000	+	3.46410i	0.0917663	+	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.50000	−	2.59808i	0.0685367	−	0.118709i	−0.829721	−	0.558179i	$-0.811500\pi$
$479$	1.50000	−	2.59808i	0.0685367	−	0.118709i	0.898257	+	0.439470i	$0.144834\pi$
$480$		0			0
$481$	1.00000	+	1.73205i	0.0455961	+	0.0789747i
$482$		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	0.996915	−	0.0784867i	$-0.0250088\pi$
$487$	9.50000	+	16.4545i	0.430486	+	0.745624i	−0.566429	+	0.824110i	$0.691675\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.0000	+	20.7846i	−0.541552	+	0.937996i	0.457263	+	0.889332i	$0.348830\pi$
$491$	−12.0000	+	20.7846i	−0.541552	+	0.937996i	−0.998815	+	0.0486647i	$0.984503\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−5.50000	−	9.52628i	−0.246214	−	0.426455i	0.716258	−	0.697835i	$-0.245853\pi$
$499$	−5.50000	−	9.52628i	−0.246214	−	0.426455i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		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
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	0.830866	−	0.556473i	$-0.187846\pi$
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	−0.897352	+	0.441315i	$0.854512\pi$
$510$		0			0
$511$	−2.50000	+	0.866025i	−0.110593	+	0.0383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.5000	+	28.5788i	−0.727077	+	1.25933i
$516$		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	−0.877283	−	0.479973i	$-0.840646\pi$
$521$	−19.5000	−	33.7750i	−0.854311	−	1.47971i	0.0229727	−	0.999736i	$-0.492687\pi$
$522$		0			0
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	−0.854887	−	0.518815i	$-0.826373\pi$
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	0.876750	+	0.480946i	$0.159707\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.5000	+	18.1865i	0.457387	+	0.792218i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.00000	−	10.3923i	−0.259889	−	0.450141i
$534$		0			0
$535$	45.0000			1.94552
$536$		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	0.194281	+	0.980946i	$0.437763\pi$
$541$	−17.5000	+	30.3109i	−0.752384	+	1.30317i	−0.946664	+	0.322221i	$0.895571\pi$
$542$		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
$551$	6.00000			0.255609
$552$		0			0
$553$	6.50000	−	33.7750i	0.276408	−	1.43626i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.5000	+	28.5788i	0.699127	+	1.21092i	0.968769	+	0.247964i	$0.0797613\pi$
$557$	16.5000	+	28.5788i	0.699127	+	1.21092i	−0.269642	+	0.962961i	$0.586905\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.00000			0.379305			0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000			0.379305			0.189652	+	0.981851i	$0.439264\pi$
$564$		0			0
$565$	18.0000			0.757266
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.00000			−0.377300			−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000			−0.377300			−0.188650	+	0.982044i	$0.560411\pi$
$570$		0			0
$571$	29.0000			1.21361			0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000			1.21361			0.606806	+	0.794850i	$0.292450\pi$
$572$		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	0.876245	−	0.481865i	$-0.160040\pi$
$577$	0.500000	+	0.866025i	0.0208153	+	0.0360531i	−0.855430	+	0.517918i	$0.826707\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	30.0000	−	10.3923i	1.24461	−	0.431145i
$582$		0			0
$583$	−9.00000			−0.372742
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.00000	−	10.3923i	−0.247647	−	0.428936i	0.715226	−	0.698893i	$-0.246324\pi$
$587$	−6.00000	−	10.3923i	−0.247647	−	0.428936i	−0.962872	+	0.269957i	$0.912990\pi$
$588$		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	$-0.691430\pi$
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	0.996976	+	0.0777165i	$0.0247629\pi$
$594$		0			0
$595$	−18.0000	−	15.5885i	−0.737928	−	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−27.0000			−1.10319			−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000			−1.10319			−0.551595	+	0.834112i	$0.685981\pi$
$600$		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
$605$	3.00000	−	5.19615i	0.121967	−	0.211254i
$606$		0			0
$607$	−23.5000	−	40.7032i	−0.953836	−	1.65209i	−0.737011	−	0.675881i	$-0.763763\pi$
$607$	−23.5000	−	40.7032i	−0.953836	−	1.65209i	−0.216825	−	0.976210i	$-0.569570\pi$
$608$		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	0.999984	+	0.00563283i	$0.00179300\pi$
$613$	12.5000	+	21.6506i	0.504870	+	0.874461i	−0.495114	+	0.868828i	$0.664874\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	+	5.19615i	−0.120775	+	0.209189i	−0.920074	−	0.391745i	$-0.871871\pi$
$617$	−3.00000	+	5.19615i	−0.120775	+	0.209189i	0.799298	+	0.600935i	$0.205205\pi$
$618$		0			0
$619$	15.5000	−	26.8468i	0.622998	−	1.07906i	−0.365927	−	0.930644i	$-0.619248\pi$
$619$	15.5000	−	26.8468i	0.622998	−	1.07906i	0.988924	−	0.148420i	$-0.0474187\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.0000	−	25.9808i	−1.20192	−	1.04090i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		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
$635$	−12.0000	−	20.7846i	−0.476205	−	0.824812i
$636$		0			0
$637$	11.0000	−	8.66025i	0.435836	−	0.343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.50000	+	12.9904i	−0.296232	+	0.513089i	−0.975271	−	0.221013i	$-0.929064\pi$
$641$	−7.50000	+	12.9904i	−0.296232	+	0.513089i	0.679039	+	0.734103i	$0.262397\pi$
$642$		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	$-0.302115\pi$
$647$	−10.5000	−	18.1865i	−0.412798	−	0.714986i	−0.995194	+	0.0979182i	$0.968782\pi$
$648$		0			0
$649$	13.5000	−	23.3827i	0.529921	−	0.917851i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.5000	−	33.7750i	−0.763094	−	1.32172i	−0.941248	−	0.337715i	$-0.890346\pi$
$653$	−19.5000	−	33.7750i	−0.763094	−	1.32172i	0.178154	−	0.984003i	$-0.442987\pi$
$654$		0			0
$655$	−4.50000	+	7.79423i	−0.175830	+	0.304546i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	11.0000			0.427850			0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000			0.427850			0.213925	+	0.976850i	$0.431375\pi$
$662$		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
$677$	27.0000			1.03769			0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000			1.03769			0.518847	+	0.854867i	$0.326361\pi$
$678$		0			0
$679$	−25.0000	+	8.66025i	−0.959412	+	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.5000	−	18.1865i	−0.401771	−	0.695888i	0.592168	−	0.805814i	$-0.298272\pi$
$683$	−10.5000	−	18.1865i	−0.401771	−	0.695888i	−0.993940	+	0.109926i	$0.964939\pi$
$684$		0			0
$685$	−63.0000			−2.40711
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000			0.228582
$690$		0			0
$691$	−13.0000			−0.494543			−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000			−0.494543			−0.247272	+	0.968946i	$0.579534\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	60.0000			2.27593
$696$		0			0
$697$	18.0000			0.681799
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−0.500000	−	0.866025i	−0.0188579	−	0.0326628i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−30.0000	−	25.9808i	−1.12827	−	0.977107i
$708$		0			0
$709$	−1.00000			−0.0375558			−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000			−0.0375558			−0.0187779	+	0.999824i	$0.505978\pi$
$710$		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	$-0.705261\pi$
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	0.992659	+	0.120950i	$0.0385939\pi$
$720$		0			0
$721$	27.5000	−	9.52628i	1.02415	−	0.354777i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−24.0000			−0.891338
$726$		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
$731$	6.00000	−	10.3923i	0.221918	−	0.384373i
$732$		0			0
$733$	12.5000	+	21.6506i	0.461698	+	0.799684i	0.999046	−	0.0436764i	$-0.0139070\pi$
$733$	12.5000	+	21.6506i	0.461698	+	0.799684i	−0.537348	+	0.843361i	$0.680574\pi$
$734$		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	0.986154	−	0.165831i	$-0.0530307\pi$
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	−0.636691	+	0.771119i	$0.719697\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000	−	41.5692i	0.880475	−	1.52503i	0.0296605	−	0.999560i	$-0.490557\pi$
$743$	24.0000	−	41.5692i	0.880475	−	1.52503i	0.850814	−	0.525467i	$-0.176109\pi$
$744$		0			0
$745$	−4.50000	+	7.79423i	−0.164867	+	0.285558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.0000	−	25.9808i	−1.09618	−	0.949316i
$750$		0			0
$751$	12.5000	+	21.6506i	0.456131	+	0.790043i	0.998752	−	0.0499348i	$-0.0159013\pi$
$751$	12.5000	+	21.6506i	0.456131	+	0.790043i	−0.542621	+	0.839978i	$0.682568\pi$
$752$		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
$761$	−1.50000	−	2.59808i	−0.0543750	−	0.0941802i	0.837557	−	0.546350i	$-0.183983\pi$
$761$	−1.50000	−	2.59808i	−0.0543750	−	0.0941802i	−0.891932	+	0.452170i	$0.850650\pi$
$762$		0			0
$763$	−2.50000	+	0.866025i	−0.0905061	+	0.0313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.00000	+	15.5885i	−0.324971	+	0.562867i
$768$		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	0.993762	+	0.111524i	$0.0355733\pi$
$773$	16.5000	+	28.5788i	0.593464	+	1.02791i	−0.400298	+	0.916385i	$0.631093\pi$
$774$		0			0
$775$	14.0000	−	24.2487i	0.502895	−	0.871039i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$		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
$787$	−31.0000			−1.10503			−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000			−1.10503			−0.552515	+	0.833503i	$0.686332\pi$
$788$		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	$0.433673\pi$
$797$	−21.0000	+	36.3731i	−0.743858	+	1.28840i	−0.950726	+	0.310031i	$0.899660\pi$
$798$		0			0
$799$	13.5000	+	23.3827i	0.477596	+	0.827220i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000			0.105868
$804$		0			0
$805$	−18.0000	−	15.5885i	−0.634417	−	0.549421i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.5000	+	28.5788i	0.580109	+	1.00478i	0.995466	+	0.0951198i	$0.0303234\pi$
$809$	16.5000	+	28.5788i	0.580109	+	1.00478i	−0.415357	+	0.909659i	$0.636343\pi$
$810$		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
$815$	33.0000			1.15594
$816$		0			0
$817$	4.00000			0.139942
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.0000			0.942306			0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000			0.942306			0.471153	+	0.882051i	$0.343838\pi$
$822$		0			0
$823$	5.00000			0.174289			0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000			0.174289			0.0871445	+	0.996196i	$0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	0.500000	+	0.866025i	0.0173657	+	0.0300783i	0.874578	−	0.484885i	$-0.161139\pi$
$829$	0.500000	+	0.866025i	0.0173657	+	0.0300783i	−0.857212	+	0.514964i	$0.827805\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.00000	+	20.7846i	0.103944	+	0.720144i
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	0.995353	−	0.0962912i	$-0.0306980\pi$
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	−0.581067	+	0.813856i	$0.697365\pi$
$840$		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
$851$	−3.00000			−0.102839
$852$		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
$857$	−7.50000	+	12.9904i	−0.256195	+	0.443743i	−0.965219	−	0.261441i	$-0.915802\pi$
$857$	−7.50000	+	12.9904i	−0.256195	+	0.443743i	0.709024	+	0.705184i	$0.249136\pi$
$858$		0			0
$859$	−11.5000	−	19.9186i	−0.392375	−	0.679613i	0.600387	−	0.799709i	$-0.295013\pi$
$859$	−11.5000	−	19.9186i	−0.392375	−	0.679613i	−0.992762	+	0.120096i	$0.961680\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.5000	+	44.1673i	−0.868030	+	1.50347i	−0.00402340	+	0.999992i	$0.501281\pi$
$863$	−25.5000	+	44.1673i	−0.868030	+	1.50347i	−0.864007	+	0.503480i	$0.832053\pi$
$864$		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
$871$	7.00000	−	12.1244i	0.237186	−	0.410818i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.50000	−	7.79423i	0.0507093	−	0.263493i
$876$		0			0
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	0.954652	−	0.297724i	$-0.0962275\pi$
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	−0.735163	+	0.677891i	$0.762894\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		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
$887$	−19.5000	−	33.7750i	−0.654746	−	1.13405i	−0.981957	−	0.189102i	$-0.939442\pi$
$887$	−19.5000	−	33.7750i	−0.654746	−	1.13405i	0.327212	−	0.944951i	$-0.393891\pi$
$888$		0			0
$889$	−4.00000	+	20.7846i	−0.134156	+	0.697093i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.50000	+	7.79423i	−0.150587	+	0.260824i
$894$		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
$905$	15.0000	+	25.9808i	0.498617	+	0.863630i
$906$		0			0
$907$	−20.5000	+	35.5070i	−0.680691	+	1.17899i	0.294079	+	0.955781i	$0.404987\pi$
$907$	−20.5000	+	35.5070i	−0.680691	+	1.17899i	−0.974770	+	0.223211i	$0.928346\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	−0.922740	−	0.385422i	$-0.874056\pi$
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	0.127585	−	0.991828i	$-0.459277\pi$
$912$		0			0
$913$	−36.0000			−1.19143
$914$		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	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	0.500000	−	0.866025i	0.0164935	−	0.0285675i	0.874154	+	0.485648i	$0.161416\pi$
$920$		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
$929$	−9.00000			−0.295280			−0.147640	−	0.989041i	$-0.547168\pi$
$929$	−9.00000			−0.295280			−0.147640	+	0.989041i	$0.547168\pi$
$930$		0			0
$931$	−5.50000	+	4.33013i	−0.180255	+	0.141914i
$932$		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
$941$	27.0000			0.880175			0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000			0.880175			0.440087	+	0.897955i	$0.354947\pi$
$942$		0			0
$943$	18.0000			0.586161
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−45.0000			−1.46230			−0.731152	−	0.682215i	$-0.761017\pi$
$947$	−45.0000			−1.46230			−0.731152	+	0.682215i	$0.761017\pi$
$948$		0			0
$949$	−2.00000			−0.0649227
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18.0000			−0.583077			−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000			−0.583077			−0.291539	+	0.956559i	$0.594167\pi$
$954$		0			0
$955$	13.5000	+	23.3827i	0.436850	+	0.756646i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	42.0000	+	36.3731i	1.35625	+	1.17455i
$960$		0			0
$961$	18.0000			0.580645
$962$		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	0.0885751	+	0.996070i	$0.471769\pi$
$971$	−25.5000	+	44.1673i	−0.818334	+	1.41740i	−0.906909	+	0.421326i	$0.861565\pi$
$972$		0			0
$973$	−40.0000	−	34.6410i	−1.28234	−	1.11054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.0000			0.863807			0.431903	−	0.901920i	$-0.357842\pi$
$977$	27.0000			0.863807			0.431903	+	0.901920i	$0.357842\pi$
$978$		0			0
$979$	22.5000	+	38.9711i	0.719103	+	1.24552i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.5000	+	28.5788i	−0.526268	+	0.911523i	0.473263	+	0.880921i	$0.343076\pi$
$983$	−16.5000	+	28.5788i	−0.526268	+	0.911523i	−0.999532	+	0.0306024i	$0.990257\pi$
$984$		0			0
$985$	−27.0000	−	46.7654i	−0.860292	−	1.49007i
$986$		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	0.976539	−	0.215342i	$-0.0690867\pi$
$991$	9.50000	+	16.4545i	0.301777	+	0.522694i	−0.674761	+	0.738036i	$0.735753\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.5000	−	18.1865i	0.332872	−	0.576552i
$996$		0			0
$997$	−29.5000	+	51.0955i	−0.934274	+	1.61821i	−0.158352	+	0.987383i	$0.550618\pi$
$997$	−29.5000	+	51.0955i	−0.934274	+	1.61821i	−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.i.a.2053.1		2
3.2	odd	2		2268.2.i.h.2053.1		2
7.4	even	3		2268.2.l.h.109.1		2
9.2	odd	6		2268.2.l.a.541.1		2
9.4	even	3		28.2.e.a.9.1	✓	2
9.5	odd	6		252.2.k.c.37.1		2
9.7	even	3		2268.2.l.h.541.1		2
21.11	odd	6		2268.2.l.a.109.1		2
36.23	even	6		1008.2.s.p.289.1		2
36.31	odd	6		112.2.i.b.65.1		2
45.4	even	6		700.2.i.c.401.1		2
45.13	odd	12		700.2.r.b.149.2		4
45.22	odd	12		700.2.r.b.149.1		4
63.4	even	3		28.2.e.a.25.1	yes	2
63.5	even	6		1764.2.a.j.1.1		1
63.11	odd	6		2268.2.i.h.865.1		2
63.13	odd	6		196.2.e.a.177.1		2
63.23	odd	6		1764.2.a.a.1.1		1
63.25	even	3	inner	2268.2.i.a.865.1		2
63.31	odd	6		196.2.e.a.165.1		2
63.32	odd	6		252.2.k.c.109.1		2
63.40	odd	6		196.2.a.a.1.1		1
63.41	even	6		1764.2.k.b.1549.1		2
63.58	even	3		196.2.a.b.1.1		1
63.59	even	6		1764.2.k.b.361.1		2
72.13	even	6		448.2.i.e.65.1		2
72.67	odd	6		448.2.i.c.65.1		2
252.23	even	6		7056.2.a.f.1.1		1
252.31	even	6		784.2.i.d.753.1		2
252.67	odd	6		112.2.i.b.81.1		2
252.95	even	6		1008.2.s.p.865.1		2
252.103	even	6		784.2.a.g.1.1		1
252.131	odd	6		7056.2.a.bw.1.1		1
252.139	even	6		784.2.i.d.177.1		2
252.247	odd	6		784.2.a.d.1.1		1
315.4	even	6		700.2.i.c.501.1		2
315.58	odd	12		4900.2.e.i.2549.2		2
315.67	odd	12		700.2.r.b.249.2		4
315.103	even	12		4900.2.e.h.2549.1		2
315.184	even	6		4900.2.a.g.1.1		1
315.193	odd	12		700.2.r.b.249.1		4
315.229	odd	6		4900.2.a.n.1.1		1
315.247	odd	12		4900.2.e.i.2549.1		2
315.292	even	12		4900.2.e.h.2549.2		2
504.67	odd	6		448.2.i.c.193.1		2
504.229	odd	6		3136.2.a.v.1.1		1
504.355	even	6		3136.2.a.k.1.1		1
504.373	even	6		3136.2.a.h.1.1		1
504.445	even	6		448.2.i.e.193.1		2
504.499	odd	6		3136.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	9.4	even	3
28.2.e.a.25.1	yes	2	63.4	even	3
112.2.i.b.65.1		2	36.31	odd	6
112.2.i.b.81.1		2	252.67	odd	6
196.2.a.a.1.1		1	63.40	odd	6
196.2.a.b.1.1		1	63.58	even	3
196.2.e.a.165.1		2	63.31	odd	6
196.2.e.a.177.1		2	63.13	odd	6
252.2.k.c.37.1		2	9.5	odd	6
252.2.k.c.109.1		2	63.32	odd	6
448.2.i.c.65.1		2	72.67	odd	6
448.2.i.c.193.1		2	504.67	odd	6
448.2.i.e.65.1		2	72.13	even	6
448.2.i.e.193.1		2	504.445	even	6
700.2.i.c.401.1		2	45.4	even	6
700.2.i.c.501.1		2	315.4	even	6
700.2.r.b.149.1		4	45.22	odd	12
700.2.r.b.149.2		4	45.13	odd	12
700.2.r.b.249.1		4	315.193	odd	12
700.2.r.b.249.2		4	315.67	odd	12
784.2.a.d.1.1		1	252.247	odd	6
784.2.a.g.1.1		1	252.103	even	6
784.2.i.d.177.1		2	252.139	even	6
784.2.i.d.753.1		2	252.31	even	6
1008.2.s.p.289.1		2	36.23	even	6
1008.2.s.p.865.1		2	252.95	even	6
1764.2.a.a.1.1		1	63.23	odd	6
1764.2.a.j.1.1		1	63.5	even	6
1764.2.k.b.361.1		2	63.59	even	6
1764.2.k.b.1549.1		2	63.41	even	6
2268.2.i.a.865.1		2	63.25	even	3	inner
2268.2.i.a.2053.1		2	1.1	even	1	trivial
2268.2.i.h.865.1		2	63.11	odd	6
2268.2.i.h.2053.1		2	3.2	odd	2
2268.2.l.a.109.1		2	21.11	odd	6
2268.2.l.a.541.1		2	9.2	odd	6
2268.2.l.h.109.1		2	7.4	even	3
2268.2.l.h.541.1		2	9.7	even	3
3136.2.a.h.1.1		1	504.373	even	6
3136.2.a.k.1.1		1	504.355	even	6
3136.2.a.s.1.1		1	504.499	odd	6
3136.2.a.v.1.1		1	504.229	odd	6
4900.2.a.g.1.1		1	315.184	even	6
4900.2.a.n.1.1		1	315.229	odd	6
4900.2.e.h.2549.1		2	315.103	even	12
4900.2.e.h.2549.2		2	315.292	even	12
4900.2.e.i.2549.1		2	315.247	odd	12
4900.2.e.i.2549.2		2	315.58	odd	12
7056.2.a.f.1.1		1	252.23	even	6
7056.2.a.bw.1.1		1	252.131	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{5} +(-0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{11} +(-1.00000 + 1.73205i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} +(-1.50000 - 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(3.00000 + 5.19615i) q^{29} -7.00000 q^{31} +(7.50000 - 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} +(-3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} -9.00000 q^{47} +(-6.50000 - 2.59808i) q^{49} +(-1.50000 - 2.59808i) q^{53} -9.00000 q^{55} +9.00000 q^{59} -1.00000 q^{61} +6.00000 q^{65} -7.00000 q^{67} +(0.500000 + 0.866025i) q^{73} +(6.00000 + 5.19615i) q^{77} -13.0000 q^{79} +(-6.00000 - 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(-4.00000 - 3.46410i) q^{91} -3.00000 q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} - q^{7}+O(q^{10}) \) 2 * q - 3 * q^5 - q^7	\( 2 q - 3 q^{5} - q^{7} + 3 q^{11} - 2 q^{13} - 3 q^{17} + q^{19} - 3 q^{23} - 4 q^{25} + 6 q^{29} - 14 q^{31} + 15 q^{35} + q^{37} - 6 q^{41} + 4 q^{43} - 18 q^{47} - 13 q^{49} - 3 q^{53} - 18 q^{55} + 18 q^{59} - 2 q^{61} + 12 q^{65} - 14 q^{67} + q^{73} + 12 q^{77} - 26 q^{79} - 12 q^{83} - 9 q^{85} - 15 q^{89} - 8 q^{91} - 6 q^{95} + 10 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 - q^7 + 3 * q^11 - 2 * q^13 - 3 * q^17 + q^19 - 3 * q^23 - 4 * q^25 + 6 * q^29 - 14 * q^31 + 15 * q^35 + q^37 - 6 * q^41 + 4 * q^43 - 18 * q^47 - 13 * q^49 - 3 * q^53 - 18 * q^55 + 18 * q^59 - 2 * q^61 + 12 * q^65 - 14 * q^67 + q^73 + 12 * q^77 - 26 * q^79 - 12 * q^83 - 9 * q^85 - 15 * q^89 - 8 * q^91 - 6 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more