LMFDB - Embedded newform 2268.2.i.i.2053.2

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:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 10x^{2} + 100 $ x^4 + 10*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2053.2
Root			$1.58114 + 2.73861i$ of defining polynomial
Character	$\chi$	$=$	2268.2053
Dual form			2268.2.i.i.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 + 2.73861i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(1.58114 + 2.73861i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(3.16228 - 5.47723i) q^{11} +(-1.58114 - 2.73861i) q^{17} +(3.50000 - 6.06218i) q^{19} +(-1.58114 - 2.73861i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(1.58114 + 2.73861i) q^{29} +3.00000 q^{31} +(-6.32456 - 5.47723i) q^{35} +(2.00000 - 3.46410i) q^{37} +(4.74342 - 8.21584i) q^{41} +(-2.50000 - 4.33013i) q^{43} -9.48683 q^{47} +(5.50000 - 4.33013i) q^{49} +(4.74342 + 8.21584i) q^{53} +20.0000 q^{55} -12.6491 q^{59} -3.00000 q^{61} +10.0000 q^{67} +12.6491 q^{71} +(2.50000 + 4.33013i) q^{73} +(-3.16228 + 16.4317i) q^{77} +12.0000 q^{79} +(3.16228 + 5.47723i) q^{83} +(5.00000 - 8.66025i) q^{85} +(-4.74342 + 8.21584i) q^{89} +22.1359 q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{7}+O(q^{10}) $ 4 * q - 10 * q^7	$ 4 q - 10 q^{7} + 14 q^{19} - 10 q^{25} + 12 q^{31} + 8 q^{37} - 10 q^{43} + 22 q^{49} + 80 q^{55} - 12 q^{61} + 40 q^{67} + 10 q^{73} + 48 q^{79} + 20 q^{85} + 10 q^{97}+O(q^{100}) $ 4 * q - 10 * q^7 + 14 * q^19 - 10 * q^25 + 12 * q^31 + 8 * q^37 - 10 * q^43 + 22 * q^49 + 80 * q^55 - 12 * q^61 + 40 * q^67 + 10 * q^73 + 48 * q^79 + 20 * q^85 + 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
$5$	1.58114	+	2.73861i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$5$	1.58114	+	2.73861i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$6$		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
$11$	3.16228	−	5.47723i	0.953463	−	1.65145i	0.215615	−	0.976478i	$-0.430824\pi$
$11$	3.16228	−	5.47723i	0.953463	−	1.65145i	0.737848	−	0.674967i	$-0.235842\pi$
$12$		0			0
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.58114	−	2.73861i	−0.383482	−	0.664211i	0.608075	−	0.793880i	$-0.291942\pi$
$17$	−1.58114	−	2.73861i	−0.383482	−	0.664211i	−0.991557	+	0.129668i	$0.958609\pi$
$18$		0			0
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.58114	−	2.73861i	−0.329690	−	0.571040i	0.652760	−	0.757565i	$-0.273611\pi$
$23$	−1.58114	−	2.73861i	−0.329690	−	0.571040i	−0.982450	+	0.186524i	$0.940278\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.58114	+	2.73861i	0.293610	+	0.508548i	0.974661	−	0.223689i	$-0.0718099\pi$
$29$	1.58114	+	2.73861i	0.293610	+	0.508548i	−0.681051	+	0.732236i	$0.738477\pi$
$30$		0			0
$31$	3.00000			0.538816			0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000			0.538816			0.269408	+	0.963026i	$0.413172\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.32456	−	5.47723i	−1.06904	−	0.925820i
$36$		0			0
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	−0.653476	−	0.756948i	$-0.726690\pi$
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	0.982274	+	0.187453i	$0.0600231\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.74342	−	8.21584i	0.740797	−	1.28310i	−0.211336	−	0.977414i	$-0.567781\pi$
$41$	4.74342	−	8.21584i	0.740797	−	1.28310i	0.952133	−	0.305685i	$-0.0988854\pi$
$42$		0			0
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.48683			−1.38380			−0.691898	−	0.721995i	$-0.743225\pi$
$47$	−9.48683			−1.38380			−0.691898	+	0.721995i	$0.743225\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.74342	+	8.21584i	0.651558	+	1.12853i	0.982745	+	0.184967i	$0.0592178\pi$
$53$	4.74342	+	8.21584i	0.651558	+	1.12853i	−0.331186	+	0.943565i	$0.607449\pi$
$54$		0			0
$55$	20.0000			2.69680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.6491			−1.64677			−0.823387	−	0.567480i	$-0.807918\pi$
$59$	−12.6491			−1.64677			−0.823387	+	0.567480i	$0.807918\pi$
$60$		0			0
$61$	−3.00000			−0.384111			−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000			−0.384111			−0.192055	+	0.981384i	$0.561515\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	10.0000			1.22169			0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000			1.22169			0.610847	+	0.791748i	$0.290829\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.6491			1.50117			0.750587	−	0.660772i	$-0.229771\pi$
$71$	12.6491			1.50117			0.750587	+	0.660772i	$0.229771\pi$
$72$		0			0
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	0.974424	−	0.224716i	$-0.0721453\pi$
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	−0.681822	+	0.731519i	$0.738812\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.16228	+	16.4317i	−0.360375	+	1.87256i
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.16228	+	5.47723i	0.347105	+	0.601204i	0.985734	−	0.168311i	$-0.0538314\pi$
$83$	3.16228	+	5.47723i	0.347105	+	0.601204i	−0.638629	+	0.769515i	$0.720498\pi$
$84$		0			0
$85$	5.00000	−	8.66025i	0.542326	−	0.939336i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.74342	+	8.21584i	−0.502801	+	0.870877i	0.497194	+	0.867640i	$0.334364\pi$
$89$	−4.74342	+	8.21584i	−0.502801	+	0.870877i	−0.999995	+	0.00323751i	$0.998969\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	22.1359			2.27110
$96$		0			0
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.16228	+	5.47723i	−0.314658	+	0.545004i	−0.979365	−	0.202100i	$-0.935223\pi$
$101$	−3.16228	+	5.47723i	−0.314658	+	0.545004i	0.664706	+	0.747105i	$0.268557\pi$
$102$		0			0
$103$	−1.00000	−	1.73205i	−0.0985329	−	0.170664i	0.812545	−	0.582899i	$-0.198082\pi$
$103$	−1.00000	−	1.73205i	−0.0985329	−	0.170664i	−0.911078	+	0.412235i	$0.864748\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	0.648292	−	0.761392i	$-0.275484\pi$
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	−0.983531	+	0.180741i	$0.942150\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.16228	+	5.47723i	−0.297482	+	0.515254i	−0.975559	−	0.219737i	$-0.929480\pi$
$113$	−3.16228	+	5.47723i	−0.297482	+	0.515254i	0.678077	+	0.734991i	$0.262814\pi$
$114$		0			0
$115$	5.00000	−	8.66025i	0.466252	−	0.807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.32456	+	5.47723i	0.579771	+	0.502096i
$120$		0			0
$121$	−14.5000	−	25.1147i	−1.31818	−	2.28316i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	21.0000			1.86345			0.931724	−	0.363166i	$-0.118304\pi$
$127$	21.0000			1.86345			0.931724	+	0.363166i	$0.118304\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.90569	−	13.6931i	−0.690724	−	1.19637i	−0.971601	−	0.236625i	$-0.923959\pi$
$131$	−7.90569	−	13.6931i	−0.690724	−	1.19637i	0.280877	−	0.959744i	$-0.409375\pi$
$132$		0			0
$133$	−3.50000	+	18.1865i	−0.303488	+	1.57697i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.58114	−	2.73861i	0.135086	−	0.233975i	−0.790544	−	0.612405i	$-0.790202\pi$
$137$	1.58114	−	2.73861i	0.135086	−	0.233975i	0.925630	+	0.378429i	$0.123536\pi$
$138$		0			0
$139$	1.00000	−	1.73205i	0.0848189	−	0.146911i	−0.820495	−	0.571654i	$-0.806302\pi$
$139$	1.00000	−	1.73205i	0.0848189	−	0.146911i	0.905314	+	0.424743i	$0.139635\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−5.00000	+	8.66025i	−0.415227	+	0.719195i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.74342	−	8.21584i	−0.388596	−	0.673068i	0.603665	−	0.797238i	$-0.293706\pi$
$149$	−4.74342	−	8.21584i	−0.388596	−	0.673068i	−0.992261	+	0.124170i	$0.960373\pi$
$150$		0			0
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	0.971274	−	0.237964i	$-0.0764802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.74342	+	8.21584i	0.381000	+	0.659912i
$156$		0			0
$157$	−16.0000			−1.27694			−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000			−1.27694			−0.638470	+	0.769647i	$0.720432\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.32456	+	5.47723i	0.498445	+	0.431666i
$162$		0			0
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	0.313304	+	0.949653i	$0.398564\pi$
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.16228	+	5.47723i	−0.244704	+	0.423840i	−0.962048	−	0.272879i	$-0.912024\pi$
$167$	−3.16228	+	5.47723i	−0.244704	+	0.423840i	0.717344	+	0.696719i	$0.245358\pi$
$168$		0			0
$169$	6.50000	+	11.2583i	0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.16228			−0.240424			−0.120212	−	0.992748i	$-0.538357\pi$
$173$	−3.16228			−0.240424			−0.120212	+	0.992748i	$0.538357\pi$
$174$		0			0
$175$	2.50000	−	12.9904i	0.188982	−	0.981981i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.74342	−	8.21584i	−0.354540	−	0.614081i	0.632499	−	0.774561i	$-0.282029\pi$
$179$	−4.74342	−	8.21584i	−0.354540	−	0.614081i	−0.987039	+	0.160480i	$0.948696\pi$
$180$		0			0
$181$	21.0000			1.56092			0.780459	−	0.625207i	$-0.214986\pi$
$181$	21.0000			1.56092			0.780459	+	0.625207i	$0.214986\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.6491			0.929981
$186$		0			0
$187$	−20.0000			−1.46254
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.48683			−0.686443			−0.343222	−	0.939254i	$-0.611518\pi$
$191$	−9.48683			−0.686443			−0.343222	+	0.939254i	$0.611518\pi$
$192$		0			0
$193$	8.00000			0.575853			0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000			0.575853			0.287926	+	0.957653i	$0.407034\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−25.2982			−1.80242			−0.901212	−	0.433379i	$-0.857321\pi$
$197$	−25.2982			−1.80242			−0.901212	+	0.433379i	$0.857321\pi$
$198$		0			0
$199$	6.50000	+	11.2583i	0.460773	+	0.798082i	0.999000	−	0.0447181i	$-0.0142390\pi$
$199$	6.50000	+	11.2583i	0.460773	+	0.798082i	−0.538227	+	0.842800i	$0.680906\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.32456	−	5.47723i	−0.443897	−	0.384426i
$204$		0			0
$205$	30.0000			2.09529
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−22.1359	−	38.3406i	−1.53118	−	2.65207i
$210$		0			0
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	−0.848301	−	0.529514i	$-0.822374\pi$
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	0.882723	+	0.469894i	$0.155708\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.90569	−	13.6931i	0.539164	−	0.933859i
$216$		0			0
$217$	−7.50000	+	2.59808i	−0.509133	+	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.00000	−	1.73205i	−0.0669650	−	0.115987i	0.830599	−	0.556871i	$-0.187998\pi$
$223$	−1.00000	−	1.73205i	−0.0669650	−	0.115987i	−0.897564	+	0.440884i	$0.854665\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.74342	−	8.21584i	0.314832	−	0.545304i	−0.664570	−	0.747226i	$-0.731385\pi$
$227$	4.74342	−	8.21584i	0.314832	−	0.545304i	0.979402	+	0.201922i	$0.0647186\pi$
$228$		0			0
$229$	−6.50000	−	11.2583i	−0.429532	−	0.743971i	0.567300	−	0.823511i	$-0.307988\pi$
$229$	−6.50000	−	11.2583i	−0.429532	−	0.743971i	−0.996832	+	0.0795401i	$0.974655\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.90569	−	13.6931i	0.517919	−	0.897062i	−0.481864	−	0.876246i	$-0.660040\pi$
$233$	7.90569	−	13.6931i	0.517919	−	0.897062i	0.999783	−	0.0208165i	$-0.00662657\pi$
$234$		0			0
$235$	−15.0000	−	25.9808i	−0.978492	−	1.69480i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.74342	+	8.21584i	−0.306826	+	0.531438i	−0.977666	−	0.210164i	$-0.932600\pi$
$239$	−4.74342	+	8.21584i	−0.306826	+	0.531438i	0.670840	+	0.741602i	$0.265934\pi$
$240$		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
$245$	20.5548	+	8.21584i	1.31320	+	0.524891i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.48683			0.598804			0.299402	−	0.954127i	$-0.403213\pi$
$251$	9.48683			0.598804			0.299402	+	0.954127i	$0.403213\pi$
$252$		0			0
$253$	−20.0000			−1.25739
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.48683	−	16.4317i	−0.591772	−	1.02498i	−0.993994	−	0.109437i	$-0.965095\pi$
$257$	−9.48683	−	16.4317i	−0.591772	−	1.02498i	0.402221	−	0.915542i	$-0.368238\pi$
$258$		0			0
$259$	−2.00000	+	10.3923i	−0.124274	+	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.48683	−	16.4317i	0.584983	−	1.01322i	−0.409894	−	0.912133i	$-0.634434\pi$
$263$	9.48683	−	16.4317i	0.584983	−	1.01322i	0.994878	−	0.101088i	$-0.0322322\pi$
$264$		0			0
$265$	−15.0000	+	25.9808i	−0.921443	+	1.59599i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.16228	−	5.47723i	−0.192807	−	0.333952i	0.753372	−	0.657594i	$-0.228426\pi$
$269$	−3.16228	−	5.47723i	−0.192807	−	0.333952i	−0.946180	+	0.323642i	$0.895093\pi$
$270$		0			0
$271$	−4.50000	+	7.79423i	−0.273356	+	0.473466i	−0.969719	−	0.244224i	$-0.921467\pi$
$271$	−4.50000	+	7.79423i	−0.273356	+	0.473466i	0.696363	+	0.717689i	$0.254800\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.8114	+	27.3861i	0.953463	+	1.65145i
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.16228	+	5.47723i	0.188646	+	0.326744i	0.944799	−	0.327651i	$-0.106257\pi$
$281$	3.16228	+	5.47723i	0.188646	+	0.326744i	−0.756153	+	0.654395i	$0.772924\pi$
$282$		0			0
$283$	13.0000			0.772770			0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000			0.772770			0.386385	+	0.922338i	$0.373724\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.74342	+	24.6475i	−0.279995	+	1.45490i
$288$		0			0
$289$	3.50000	−	6.06218i	0.205882	−	0.356599i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$	−20.0000	−	34.6410i	−1.16445	−	2.01688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	10.0000	+	8.66025i	0.576390	+	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.74342	−	8.21584i	−0.271607	−	0.470438i
$306$		0			0
$307$	25.0000			1.42683			0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000			1.42683			0.713413	+	0.700744i	$0.247149\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.32456			0.358633			0.179316	−	0.983791i	$-0.442611\pi$
$311$	6.32456			0.358633			0.179316	+	0.983791i	$0.442611\pi$
$312$		0			0
$313$	−17.0000			−0.960897			−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000			−0.960897			−0.480448	+	0.877023i	$0.659526\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.9737			1.06567			0.532834	−	0.846220i	$-0.321127\pi$
$317$	18.9737			1.06567			0.532834	+	0.846220i	$0.321127\pi$
$318$		0			0
$319$	20.0000			1.11979
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−22.1359			−1.23168
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	23.7171	−	8.21584i	1.30757	−	0.452954i
$330$		0			0
$331$	17.0000			0.934405			0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000			0.934405			0.467202	+	0.884150i	$0.345262\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.8114	+	27.3861i	0.863868	+	1.49626i
$336$		0			0
$337$	−10.5000	+	18.1865i	−0.571971	+	0.990684i	0.424392	+	0.905479i	$0.360488\pi$
$337$	−10.5000	+	18.1865i	−0.571971	+	0.990684i	−0.996363	+	0.0852050i	$0.972845\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.48683	−	16.4317i	0.513741	−	0.889825i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	8.50000	+	14.7224i	0.454995	+	0.788074i	0.998688	−	0.0512103i	$-0.0163079\pi$
$349$	8.50000	+	14.7224i	0.454995	+	0.788074i	−0.543693	+	0.839284i	$0.682975\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.48683	+	16.4317i	−0.504933	+	0.874570i	0.495051	+	0.868864i	$0.335150\pi$
$353$	−9.48683	+	16.4317i	−0.504933	+	0.874570i	−0.999984	+	0.00570573i	$0.998184\pi$
$354$		0			0
$355$	20.0000	+	34.6410i	1.06149	+	1.83855i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.6491	+	21.9089i	−0.667595	+	1.15631i	0.310980	+	0.950416i	$0.399343\pi$
$359$	−12.6491	+	21.9089i	−0.667595	+	1.15631i	−0.978575	+	0.205891i	$0.933991\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.90569	+	13.6931i	−0.413803	+	0.716728i
$366$		0			0
$367$	9.50000	−	16.4545i	0.495896	−	0.858917i	−0.504093	−	0.863649i	$-0.668173\pi$
$367$	9.50000	−	16.4545i	0.495896	−	0.858917i	0.999989	+	0.00473247i	$0.00150640\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.9737	−	16.4317i	−0.985064	−	0.853090i
$372$		0			0
$373$	8.50000	+	14.7224i	0.440113	+	0.762299i	0.997697	−	0.0678218i	$-0.0216049\pi$
$373$	8.50000	+	14.7224i	0.440113	+	0.762299i	−0.557584	+	0.830120i	$0.688272\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.32456	−	10.9545i	−0.323170	−	0.559746i	0.657971	−	0.753044i	$-0.271415\pi$
$383$	−6.32456	−	10.9545i	−0.323170	−	0.559746i	−0.981140	+	0.193297i	$0.938082\pi$
$384$		0			0
$385$	−50.0000	+	17.3205i	−2.54824	+	0.882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.3925	+	30.1247i	−0.881836	+	1.52738i	−0.0325386	+	0.999470i	$0.510359\pi$
$389$	−17.3925	+	30.1247i	−0.881836	+	1.52738i	−0.849298	+	0.527915i	$0.822974\pi$
$390$		0			0
$391$	−5.00000	+	8.66025i	−0.252861	+	0.437968i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	18.9737	+	32.8634i	0.954669	+	1.65353i
$396$		0			0
$397$	7.50000	−	12.9904i	0.376414	−	0.651969i	−0.614123	−	0.789210i	$-0.710490\pi$
$397$	7.50000	−	12.9904i	0.376414	−	0.651969i	0.990538	+	0.137241i	$0.0438236\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.48683	−	16.4317i	−0.473750	−	0.820559i	0.525799	−	0.850609i	$-0.323767\pi$
$401$	−9.48683	−	16.4317i	−0.473750	−	0.820559i	−0.999548	+	0.0300503i	$0.990433\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.6491	−	21.9089i	−0.626993	−	1.08598i
$408$		0			0
$409$	−16.0000			−0.791149			−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000			−0.791149			−0.395575	+	0.918434i	$0.629455\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	31.6228	−	10.9545i	1.55606	−	0.539033i
$414$		0			0
$415$	−10.0000	+	17.3205i	−0.490881	+	0.850230i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.74342	+	8.21584i	−0.231731	+	0.401370i	−0.958318	−	0.285705i	$-0.907772\pi$
$419$	−4.74342	+	8.21584i	−0.231731	+	0.401370i	0.726587	+	0.687075i	$0.241106\pi$
$420$		0			0
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	0.827155	−	0.561973i	$-0.189958\pi$
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	−0.900261	+	0.435351i	$0.856624\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.8114			0.766965
$426$		0			0
$427$	7.50000	−	2.59808i	0.362950	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.1359			−1.05891
$438$		0			0
$439$	−18.0000			−0.859093			−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000			−0.859093			−0.429547	+	0.903045i	$0.641327\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−34.7851			−1.65269			−0.826344	−	0.563166i	$-0.809583\pi$
$443$	−34.7851			−1.65269			−0.826344	+	0.563166i	$0.809583\pi$
$444$		0			0
$445$	−30.0000			−1.42214
$446$		0			0
$447$		0			0
$448$		0			0
$449$	34.7851			1.64161			0.820804	−	0.571210i	$-0.193526\pi$
$449$	34.7851			1.64161			0.820804	+	0.571210i	$0.193526\pi$
$450$		0			0
$451$	−30.0000	−	51.9615i	−1.41264	−	2.44677i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.0000			−1.35656			−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000			−1.35656			−0.678281	+	0.734802i	$0.737275\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.32456	+	10.9545i	0.294564	+	0.510200i	0.974883	−	0.222716i	$-0.0714923\pi$
$461$	6.32456	+	10.9545i	0.294564	+	0.510200i	−0.680319	+	0.732916i	$0.738159\pi$
$462$		0			0
$463$	−12.5000	+	21.6506i	−0.580924	+	1.00619i	0.414446	+	0.910074i	$0.363975\pi$
$463$	−12.5000	+	21.6506i	−0.580924	+	1.00619i	−0.995370	+	0.0961164i	$0.969358\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.16228	+	5.47723i	−0.146333	+	0.253456i	−0.929869	−	0.367890i	$-0.880080\pi$
$467$	−3.16228	+	5.47723i	−0.146333	+	0.253456i	0.783537	+	0.621346i	$0.213414\pi$
$468$		0			0
$469$	−25.0000	+	8.66025i	−1.15439	+	0.399893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−31.6228			−1.45402
$474$		0			0
$475$	17.5000	+	30.3109i	0.802955	+	1.39076i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.0680	+	19.1703i	−0.505709	+	0.875913i	0.494270	+	0.869309i	$0.335436\pi$
$479$	−11.0680	+	19.1703i	−0.505709	+	0.875913i	−0.999978	+	0.00660425i	$0.997898\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.90569	+	13.6931i	−0.358979	+	0.621770i
$486$		0			0
$487$	0.500000	+	0.866025i	0.0226572	+	0.0392434i	0.877132	−	0.480250i	$-0.159454\pi$
$487$	0.500000	+	0.866025i	0.0226572	+	0.0392434i	−0.854475	+	0.519493i	$0.826121\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.48683	−	16.4317i	0.428135	−	0.741551i	−0.568573	−	0.822633i	$-0.692504\pi$
$491$	9.48683	−	16.4317i	0.428135	−	0.741551i	0.996707	+	0.0810819i	$0.0258375\pi$
$492$		0			0
$493$	5.00000	−	8.66025i	0.225189	−	0.390038i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.6228	+	10.9545i	−1.41848	+	0.491374i
$498$		0			0
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	0.776989	−	0.629515i	$-0.216746\pi$
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	−0.933670	+	0.358134i	$0.883413\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.4605			−1.26899			−0.634495	−	0.772927i	$-0.718792\pi$
$503$	−28.4605			−1.26899			−0.634495	+	0.772927i	$0.718792\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.9737	+	32.8634i	0.840993	+	1.45664i	0.889056	+	0.457798i	$0.151362\pi$
$509$	18.9737	+	32.8634i	0.840993	+	1.45664i	−0.0480632	+	0.998844i	$0.515305\pi$
$510$		0			0
$511$	−10.0000	−	8.66025i	−0.442374	−	0.383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.16228	−	5.47723i	0.139347	−	0.241355i
$516$		0			0
$517$	−30.0000	+	51.9615i	−1.31940	+	2.28527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−16.0000	+	27.7128i	−0.699631	+	1.21180i	0.268963	+	0.963150i	$0.413319\pi$
$523$	−16.0000	+	27.7128i	−0.699631	+	1.21180i	−0.968594	+	0.248646i	$0.920014\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.74342	−	8.21584i	−0.206626	−	0.357888i
$528$		0			0
$529$	6.50000	−	11.2583i	0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.32456	−	43.8178i	−0.272418	−	1.88737i
$540$		0			0
$541$	10.5000	−	18.1865i	0.451430	−	0.781900i	−0.547045	−	0.837103i	$-0.684247\pi$
$541$	10.5000	−	18.1865i	0.451430	−	0.781900i	0.998475	+	0.0552031i	$0.0175806\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.0680	−	19.1703i	0.474100	−	0.821165i
$546$		0			0
$547$	−10.5000	−	18.1865i	−0.448948	−	0.777600i	0.549370	−	0.835579i	$-0.314868\pi$
$547$	−10.5000	−	18.1865i	−0.448948	−	0.777600i	−0.998318	+	0.0579790i	$0.981534\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.1359			0.943023
$552$		0			0
$553$	−30.0000	+	10.3923i	−1.27573	+	0.441926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.58114	+	2.73861i	0.0669950	+	0.116039i	0.897577	−	0.440857i	$-0.145326\pi$
$557$	1.58114	+	2.73861i	0.0669950	+	0.116039i	−0.830582	+	0.556896i	$0.811992\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	41.1096			1.73256			0.866282	−	0.499556i	$-0.166503\pi$
$563$	41.1096			1.73256			0.866282	+	0.499556i	$0.166503\pi$
$564$		0			0
$565$	−20.0000			−0.841406
$566$		0			0
$567$		0			0
$568$		0			0
$569$	37.9473			1.59083			0.795417	−	0.606062i	$-0.207252\pi$
$569$	37.9473			1.59083			0.795417	+	0.606062i	$0.207252\pi$
$570$		0			0
$571$	39.0000			1.63210			0.816050	−	0.577982i	$-0.196160\pi$
$571$	39.0000			1.63210			0.816050	+	0.577982i	$0.196160\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	15.8114			0.659380
$576$		0			0
$577$	22.0000	+	38.1051i	0.915872	+	1.58634i	0.805620	+	0.592433i	$0.201833\pi$
$577$	22.0000	+	38.1051i	0.915872	+	1.58634i	0.110252	+	0.993904i	$0.464834\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.6491	−	10.9545i	−0.524774	−	0.454467i
$582$		0			0
$583$	60.0000			2.48495
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.3925	+	30.1247i	0.717866	+	1.24338i	0.961844	+	0.273600i	$0.0882145\pi$
$587$	17.3925	+	30.1247i	0.717866	+	1.24338i	−0.243977	+	0.969781i	$0.578452\pi$
$588$		0			0
$589$	10.5000	−	18.1865i	0.432645	−	0.749363i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.48683	+	16.4317i	−0.389578	+	0.674768i	−0.992393	−	0.123113i	$-0.960712\pi$
$593$	−9.48683	+	16.4317i	−0.389578	+	0.674768i	0.602815	+	0.797881i	$0.294046\pi$
$594$		0			0
$595$	−5.00000	+	25.9808i	−0.204980	+	1.06511i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.4605			−1.16286			−0.581432	−	0.813595i	$-0.697507\pi$
$599$	−28.4605			−1.16286			−0.581432	+	0.813595i	$0.697507\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	45.8530	−	79.4198i	1.86419	−	3.22887i
$606$		0			0
$607$	−7.50000	−	12.9904i	−0.304416	−	0.527263i	0.672715	−	0.739901i	$-0.265128\pi$
$607$	−7.50000	−	12.9904i	−0.304416	−	0.527263i	−0.977131	+	0.212638i	$0.931794\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.5000	−	28.5788i	−0.666429	−	1.15429i	−0.978896	−	0.204360i	$-0.934489\pi$
$613$	−16.5000	−	28.5788i	−0.666429	−	1.15429i	0.312467	−	0.949929i	$-0.398845\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.6491	−	21.9089i	0.509234	−	0.882019i	−0.490709	−	0.871324i	$-0.663262\pi$
$617$	12.6491	−	21.9089i	0.509234	−	0.882019i	0.999943	−	0.0106956i	$-0.00340456\pi$
$618$		0			0
$619$	−11.0000	+	19.0526i	−0.442127	+	0.765787i	−0.997847	−	0.0655827i	$-0.979109\pi$
$619$	−11.0000	+	19.0526i	−0.442127	+	0.765787i	0.555720	+	0.831370i	$0.312443\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.74342	−	24.6475i	0.190041	−	0.987482i
$624$		0			0
$625$	12.5000	+	21.6506i	0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.6491			−0.504353
$630$		0			0
$631$	−31.0000			−1.23409			−0.617045	−	0.786928i	$-0.711670\pi$
$631$	−31.0000			−1.23409			−0.617045	+	0.786928i	$0.711670\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	33.2039	+	57.5109i	1.31766	+	2.28225i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.58114	+	2.73861i	−0.0624512	+	0.108169i	−0.895561	−	0.444940i	$-0.853225\pi$
$641$	−1.58114	+	2.73861i	−0.0624512	+	0.108169i	0.833109	+	0.553108i	$0.186558\pi$
$642$		0			0
$643$	18.5000	−	32.0429i	0.729569	−	1.26365i	−0.227497	−	0.973779i	$-0.573054\pi$
$643$	18.5000	−	32.0429i	0.729569	−	1.26365i	0.957066	−	0.289871i	$-0.0936125\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.9737	−	32.8634i	−0.745932	−	1.29199i	−0.949758	−	0.312985i	$-0.898671\pi$
$647$	−18.9737	−	32.8634i	−0.745932	−	1.29199i	0.203826	−	0.979007i	$-0.434662\pi$
$648$		0			0
$649$	−40.0000	+	69.2820i	−1.57014	+	2.71956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.2302	+	24.6475i	0.556873	+	0.964532i	0.997755	+	0.0669670i	$0.0213322\pi$
$653$	14.2302	+	24.6475i	0.556873	+	0.964532i	−0.440882	+	0.897565i	$0.645334\pi$
$654$		0			0
$655$	25.0000	−	43.3013i	0.976831	−	1.69192i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.58114	−	2.73861i	−0.0615924	−	0.106681i	0.833585	−	0.552391i	$-0.186285\pi$
$659$	−1.58114	−	2.73861i	−0.0615924	−	0.106681i	−0.895177	+	0.445710i	$0.852951\pi$
$660$		0			0
$661$	−17.0000			−0.661223			−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000			−0.661223			−0.330612	+	0.943767i	$0.607255\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−55.3399	+	19.1703i	−2.14599	+	0.743392i
$666$		0			0
$667$	5.00000	−	8.66025i	0.193601	−	0.335326i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.48683	+	16.4317i	−0.366235	+	0.634338i
$672$		0			0
$673$	−2.50000	−	4.33013i	−0.0963679	−	0.166914i	0.813811	−	0.581130i	$-0.197389\pi$
$673$	−2.50000	−	4.33013i	−0.0963679	−	0.166914i	−0.910179	+	0.414216i	$0.864056\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.9473			−1.45843			−0.729217	−	0.684282i	$-0.760116\pi$
$677$	−37.9473			−1.45843			−0.729217	+	0.684282i	$0.760116\pi$
$678$		0			0
$679$	−10.0000	−	8.66025i	−0.383765	−	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.48683	+	16.4317i	0.363004	+	0.628741i	0.988454	−	0.151524i	$-0.0484179\pi$
$683$	9.48683	+	16.4317i	0.363004	+	0.628741i	−0.625450	+	0.780264i	$0.715085\pi$
$684$		0			0
$685$	10.0000			0.382080
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	11.0000			0.418460			0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000			0.418460			0.209230	+	0.977866i	$0.432904\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.32456			0.239904
$696$		0			0
$697$	−30.0000			−1.13633
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.16228			−0.119438			−0.0597188	−	0.998215i	$-0.519020\pi$
$701$	−3.16228			−0.119438			−0.0597188	+	0.998215i	$0.519020\pi$
$702$		0			0
$703$	−14.0000	−	24.2487i	−0.528020	−	0.914557i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.16228	−	16.4317i	0.118930	−	0.617977i
$708$		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
$713$	−4.74342	−	8.21584i	−0.177642	−	0.307686i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	4.00000	+	3.46410i	0.148968	+	0.129010i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.8114			−0.587220
$726$		0			0
$727$	14.5000	+	25.1147i	0.537775	+	0.931454i	0.999023	+	0.0441829i	$0.0140684\pi$
$727$	14.5000	+	25.1147i	0.537775	+	0.931454i	−0.461248	+	0.887271i	$0.652598\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.90569	+	13.6931i	−0.292403	+	0.506456i
$732$		0			0
$733$	−1.50000	−	2.59808i	−0.0554038	−	0.0959621i	0.836993	−	0.547213i	$-0.184311\pi$
$733$	−1.50000	−	2.59808i	−0.0554038	−	0.0959621i	−0.892397	+	0.451251i	$0.850978\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	31.6228	−	54.7723i	1.16484	−	2.01756i
$738$		0			0
$739$	−1.50000	−	2.59808i	−0.0551784	−	0.0955718i	0.837117	−	0.547024i	$-0.184239\pi$
$739$	−1.50000	−	2.59808i	−0.0551784	−	0.0955718i	−0.892295	+	0.451452i	$0.850906\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.2302	−	24.6475i	0.522057	−	0.904230i	−0.477614	−	0.878570i	$-0.658498\pi$
$743$	14.2302	−	24.6475i	0.522057	−	0.904230i	0.999671	−	0.0256596i	$-0.00816860\pi$
$744$		0			0
$745$	15.0000	−	25.9808i	0.549557	−	0.951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.50000	−	2.59808i	−0.0547358	−	0.0948051i	0.837359	−	0.546653i	$-0.184098\pi$
$751$	−1.50000	−	2.59808i	−0.0547358	−	0.0948051i	−0.892095	+	0.451848i	$0.850765\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	53.7587			1.95648
$756$		0			0
$757$	15.0000			0.545184			0.272592	−	0.962130i	$-0.412119\pi$
$757$	15.0000			0.545184			0.272592	+	0.962130i	$0.412119\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.6491	+	21.9089i	0.458530	+	0.794197i	0.998884	−	0.0472409i	$-0.0150429\pi$
$761$	12.6491	+	21.9089i	0.458530	+	0.794197i	−0.540354	+	0.841438i	$0.681710\pi$
$762$		0			0
$763$	14.0000	+	12.1244i	0.506834	+	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	11.5000	−	19.9186i	0.414701	−	0.718283i	−0.580696	−	0.814120i	$-0.697220\pi$
$769$	11.5000	−	19.9186i	0.414701	−	0.718283i	0.995397	+	0.0958377i	$0.0305530\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−17.3925	−	30.1247i	−0.625566	−	1.08351i	−0.988431	−	0.151670i	$-0.951535\pi$
$773$	−17.3925	−	30.1247i	−0.625566	−	1.08351i	0.362865	−	0.931842i	$-0.381798\pi$
$774$		0			0
$775$	−7.50000	+	12.9904i	−0.269408	+	0.466628i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−33.2039	−	57.5109i	−1.18965	−	2.06054i
$780$		0			0
$781$	40.0000	−	69.2820i	1.43131	−	2.47911i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−25.2982	−	43.8178i	−0.902932	−	1.56392i
$786$		0			0
$787$	15.0000			0.534692			0.267346	−	0.963601i	$-0.413853\pi$
$787$	15.0000			0.534692			0.267346	+	0.963601i	$0.413853\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.16228	−	16.4317i	0.112438	−	0.584243i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.48683	+	16.4317i	−0.336041	+	0.582040i	−0.983684	−	0.179904i	$-0.942421\pi$
$797$	−9.48683	+	16.4317i	−0.336041	+	0.582040i	0.647643	+	0.761944i	$0.275755\pi$
$798$		0			0
$799$	15.0000	+	25.9808i	0.530662	+	0.919133i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	31.6228			1.11594
$804$		0			0
$805$	−5.00000	+	25.9808i	−0.176227	+	0.915702i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.8114	+	27.3861i	0.555899	+	0.962845i	0.997833	+	0.0657975i	$0.0209591\pi$
$809$	15.8114	+	27.3861i	0.555899	+	0.962845i	−0.441934	+	0.897047i	$0.645708\pi$
$810$		0			0
$811$	−14.0000			−0.491606			−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000			−0.491606			−0.245803	+	0.969320i	$0.579052\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−53.7587			−1.88309
$816$		0			0
$817$	−35.0000			−1.22449
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.2982			−0.882914			−0.441457	−	0.897282i	$-0.645538\pi$
$821$	−25.2982			−0.882914			−0.441457	+	0.897282i	$0.645538\pi$
$822$		0			0
$823$	−23.0000			−0.801730			−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000			−0.801730			−0.400865	+	0.916137i	$0.631290\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.9737			0.659779			0.329890	−	0.944020i	$-0.392989\pi$
$827$	18.9737			0.659779			0.329890	+	0.944020i	$0.392989\pi$
$828$		0			0
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	0.945589	−	0.325362i	$-0.105486\pi$
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	−0.754567	+	0.656223i	$0.772153\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20.5548	−	8.21584i	−0.712182	−	0.284662i
$834$		0			0
$835$	−20.0000			−0.692129
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.8114	−	27.3861i	−0.545870	−	0.945474i	−0.998552	−	0.0538020i	$-0.982866\pi$
$839$	−15.8114	−	27.3861i	−0.545870	−	0.945474i	0.452682	−	0.891672i	$-0.350467\pi$
$840$		0			0
$841$	9.50000	−	16.4545i	0.327586	−	0.567396i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−20.5548	+	35.6020i	−0.707107	+	1.22474i
$846$		0			0
$847$	58.0000	+	50.2295i	1.99290	+	1.72591i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.6491			−0.433606
$852$		0			0
$853$	21.5000	+	37.2391i	0.736146	+	1.27504i	0.954219	+	0.299110i	$0.0966897\pi$
$853$	21.5000	+	37.2391i	0.736146	+	1.27504i	−0.218073	+	0.975933i	$0.569977\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18.9737	+	32.8634i	−0.648128	+	1.12259i	0.335441	+	0.942061i	$0.391115\pi$
$857$	−18.9737	+	32.8634i	−0.648128	+	1.12259i	−0.983570	+	0.180530i	$0.942219\pi$
$858$		0			0
$859$	−8.50000	−	14.7224i	−0.290016	−	0.502323i	0.683797	−	0.729672i	$-0.260327\pi$
$859$	−8.50000	−	14.7224i	−0.290016	−	0.502323i	−0.973813	+	0.227349i	$0.926994\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.90569	+	13.6931i	−0.269113	+	0.466117i	−0.968633	−	0.248496i	$-0.920064\pi$
$863$	−7.90569	+	13.6931i	−0.269113	+	0.466117i	0.699520	+	0.714613i	$0.253397\pi$
$864$		0			0
$865$	−5.00000	−	8.66025i	−0.170005	−	0.294457i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	37.9473	−	65.7267i	1.28728	−	2.22963i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.50000	+	4.33013i	0.0844190	+	0.146218i	0.905143	−	0.425106i	$-0.139763\pi$
$877$	2.50000	+	4.33013i	0.0844190	+	0.146218i	−0.820724	+	0.571324i	$0.806430\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−47.4342			−1.59810			−0.799049	−	0.601266i	$-0.794663\pi$
$881$	−47.4342			−1.59810			−0.799049	+	0.601266i	$0.794663\pi$
$882$		0			0
$883$	23.0000			0.774012			0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000			0.774012			0.387006	+	0.922077i	$0.373509\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−52.5000	+	18.1865i	−1.76079	+	0.609957i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.2039	+	57.5109i	−1.11113	+	1.92453i
$894$		0			0
$895$	15.0000	−	25.9808i	0.501395	−	0.868441i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.74342	+	8.21584i	0.158202	+	0.274014i
$900$		0			0
$901$	15.0000	−	25.9808i	0.499722	−	0.865545i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	33.2039	+	57.5109i	1.10374	+	1.91173i
$906$		0			0
$907$	12.0000	−	20.7846i	0.398453	−	0.690142i	−0.595082	−	0.803665i	$-0.702880\pi$
$907$	12.0000	−	20.7846i	0.398453	−	0.690142i	0.993535	+	0.113523i	$0.0362137\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	23.7171	+	41.0792i	0.785782	+	1.36101i	0.928531	+	0.371255i	$0.121073\pi$
$911$	23.7171	+	41.0792i	0.785782	+	1.36101i	−0.142749	+	0.989759i	$0.545594\pi$
$912$		0			0
$913$	40.0000			1.32381
$914$		0			0
$915$		0			0
$916$		0			0
$917$	31.6228	+	27.3861i	1.04428	+	0.904370i
$918$		0			0
$919$	5.50000	−	9.52628i	0.181428	−	0.314243i	−0.760939	−	0.648824i	$-0.775261\pi$
$919$	5.50000	−	9.52628i	0.181428	−	0.314243i	0.942367	+	0.334581i	$0.108595\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	10.0000	+	17.3205i	0.328798	+	0.569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.4605			0.933759			0.466879	−	0.884321i	$-0.345378\pi$
$929$	28.4605			0.933759			0.466879	+	0.884321i	$0.345378\pi$
$930$		0			0
$931$	−7.00000	−	48.4974i	−0.229416	−	1.58944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.6228	−	54.7723i	−1.03418	−	1.79124i
$936$		0			0
$937$	44.0000			1.43742			0.718709	−	0.695311i	$-0.244734\pi$
$937$	44.0000			1.43742			0.718709	+	0.695311i	$0.244734\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.2982			0.824698			0.412349	−	0.911026i	$-0.364708\pi$
$941$	25.2982			0.824698			0.412349	+	0.911026i	$0.364708\pi$
$942$		0			0
$943$	−30.0000			−0.976934
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34.7851			1.13036			0.565181	−	0.824967i	$-0.308806\pi$
$947$	34.7851			1.13036			0.565181	+	0.824967i	$0.308806\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.16228			0.102436			0.0512181	−	0.998687i	$-0.483690\pi$
$953$	3.16228			0.102436			0.0512181	+	0.998687i	$0.483690\pi$
$954$		0			0
$955$	−15.0000	−	25.9808i	−0.485389	−	0.840718i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.58114	+	8.21584i	−0.0510576	+	0.265303i
$960$		0			0
$961$	−22.0000			−0.709677
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.6491	+	21.9089i	0.407189	+	0.705273i
$966$		0			0
$967$	−13.0000	+	22.5167i	−0.418052	+	0.724087i	−0.995743	−	0.0921681i	$-0.970620\pi$
$967$	−13.0000	+	22.5167i	−0.418052	+	0.724087i	0.577692	+	0.816255i	$0.303954\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23.7171	+	41.0792i	−0.761117	+	1.31829i	0.181158	+	0.983454i	$0.442016\pi$
$971$	−23.7171	+	41.0792i	−0.761117	+	1.31829i	−0.942275	+	0.334840i	$0.891318\pi$
$972$		0			0
$973$	−1.00000	+	5.19615i	−0.0320585	+	0.166581i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	37.9473			1.21404			0.607021	−	0.794686i	$-0.292364\pi$
$977$	37.9473			1.21404			0.607021	+	0.794686i	$0.292364\pi$
$978$		0			0
$979$	30.0000	+	51.9615i	0.958804	+	1.66070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−40.0000	−	69.2820i	−1.27451	−	2.20751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.90569	+	13.6931i	−0.251386	+	0.435414i
$990$		0			0
$991$	−7.00000	−	12.1244i	−0.222362	−	0.385143i	0.733163	−	0.680053i	$-0.238043\pi$
$991$	−7.00000	−	12.1244i	−0.222362	−	0.385143i	−0.955525	+	0.294911i	$0.904710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.5548	+	35.6020i	−0.651631	+	1.12866i
$996$		0			0
$997$	−20.5000	+	35.5070i	−0.649242	+	1.12452i	0.334063	+	0.942551i	$0.391580\pi$
$997$	−20.5000	+	35.5070i	−0.649242	+	1.12452i	−0.983304	+	0.181968i	$0.941753\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.i.i.2053.2		4
3.2	odd	2	inner	2268.2.i.i.2053.1		4
7.4	even	3		2268.2.l.i.109.1		4
9.2	odd	6		2268.2.l.i.541.2		4
9.4	even	3		756.2.k.d.541.2	yes	4
9.5	odd	6		756.2.k.d.541.1	yes	4
9.7	even	3		2268.2.l.i.541.1		4
21.11	odd	6		2268.2.l.i.109.2		4
63.4	even	3		756.2.k.d.109.2	yes	4
63.5	even	6		5292.2.a.r.1.1		2
63.11	odd	6	inner	2268.2.i.i.865.1		4
63.23	odd	6		5292.2.a.q.1.2		2
63.25	even	3	inner	2268.2.i.i.865.2		4
63.32	odd	6		756.2.k.d.109.1	✓	4
63.40	odd	6		5292.2.a.r.1.2		2
63.58	even	3		5292.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.d.109.1	✓	4	63.32	odd	6
756.2.k.d.109.2	yes	4	63.4	even	3
756.2.k.d.541.1	yes	4	9.5	odd	6
756.2.k.d.541.2	yes	4	9.4	even	3
2268.2.i.i.865.1		4	63.11	odd	6	inner
2268.2.i.i.865.2		4	63.25	even	3	inner
2268.2.i.i.2053.1		4	3.2	odd	2	inner
2268.2.i.i.2053.2		4	1.1	even	1	trivial
2268.2.l.i.109.1		4	7.4	even	3
2268.2.l.i.109.2		4	21.11	odd	6
2268.2.l.i.541.1		4	9.7	even	3
2268.2.l.i.541.2		4	9.2	odd	6
5292.2.a.q.1.1		2	63.58	even	3
5292.2.a.q.1.2		2	63.23	odd	6
5292.2.a.r.1.1		2	63.5	even	6
5292.2.a.r.1.2		2	63.40	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)

\(f(q)\)	\(=\)	\(q+(1.58114 + 2.73861i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(1.58114 + 2.73861i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(3.16228 - 5.47723i) q^{11} +(-1.58114 - 2.73861i) q^{17} +(3.50000 - 6.06218i) q^{19} +(-1.58114 - 2.73861i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(1.58114 + 2.73861i) q^{29} +3.00000 q^{31} +(-6.32456 - 5.47723i) q^{35} +(2.00000 - 3.46410i) q^{37} +(4.74342 - 8.21584i) q^{41} +(-2.50000 - 4.33013i) q^{43} -9.48683 q^{47} +(5.50000 - 4.33013i) q^{49} +(4.74342 + 8.21584i) q^{53} +20.0000 q^{55} -12.6491 q^{59} -3.00000 q^{61} +10.0000 q^{67} +12.6491 q^{71} +(2.50000 + 4.33013i) q^{73} +(-3.16228 + 16.4317i) q^{77} +12.0000 q^{79} +(3.16228 + 5.47723i) q^{83} +(5.00000 - 8.66025i) q^{85} +(-4.74342 + 8.21584i) q^{89} +22.1359 q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{7}+O(q^{10}) \) 4 * q - 10 * q^7	\( 4 q - 10 q^{7} + 14 q^{19} - 10 q^{25} + 12 q^{31} + 8 q^{37} - 10 q^{43} + 22 q^{49} + 80 q^{55} - 12 q^{61} + 40 q^{67} + 10 q^{73} + 48 q^{79} + 20 q^{85} + 10 q^{97}+O(q^{100}) \) 4 * q - 10 * q^7 + 14 * q^19 - 10 * q^25 + 12 * q^31 + 8 * q^37 - 10 * q^43 + 22 * q^49 + 80 * q^55 - 12 * q^61 + 40 * q^67 + 10 * q^73 + 48 * q^79 + 20 * q^85 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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