LMFDB - Embedded newform 2268.2.l.i.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(109,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 2, 4]))

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

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

chi := DirichletCharacter("2268.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: 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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$1.58114 - 2.73861i$ of defining polynomial
Character	$\chi$	$=$	2268.109
Dual form			2268.2.l.i.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16228 q^{5} +(2.00000 - 1.73205i) q^{7} +O(q^{10})$	$q-3.16228 q^{5} +(2.00000 - 1.73205i) q^{7} -6.32456 q^{11} +(-1.58114 + 2.73861i) q^{17} +(3.50000 + 6.06218i) q^{19} +3.16228 q^{23} +5.00000 q^{25} +(1.58114 + 2.73861i) q^{29} +(-1.50000 - 2.59808i) q^{31} +(-6.32456 + 5.47723i) q^{35} +(2.00000 + 3.46410i) q^{37} +(4.74342 - 8.21584i) q^{41} +(-2.50000 - 4.33013i) q^{43} +(4.74342 - 8.21584i) q^{47} +(1.00000 - 6.92820i) q^{49} +(4.74342 - 8.21584i) q^{53} +20.0000 q^{55} +(6.32456 + 10.9545i) q^{59} +(1.50000 - 2.59808i) q^{61} +(-5.00000 - 8.66025i) q^{67} +12.6491 q^{71} +(2.50000 - 4.33013i) q^{73} +(-12.6491 + 10.9545i) q^{77} +(-6.00000 + 10.3923i) q^{79} +(3.16228 + 5.47723i) q^{83} +(5.00000 - 8.66025i) q^{85} +(-4.74342 - 8.21584i) q^{89} +(-11.0680 - 19.1703i) q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 14 q^{19} + 20 q^{25} - 6 q^{31} + 8 q^{37} - 10 q^{43} + 4 q^{49} + 80 q^{55} + 6 q^{61} - 20 q^{67} + 10 q^{73} - 24 q^{79} + 20 q^{85} + 10 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 + 14 * q^19 + 20 * q^25 - 6 * q^31 + 8 * q^37 - 10 * q^43 + 4 * q^49 + 80 * q^55 + 6 * q^61 - 20 * q^67 + 10 * q^73 - 24 * q^79 + 20 * q^85 + 10 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.16228			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$5$	−3.16228			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.32456			−1.90693			−0.953463	−	0.301511i	$-0.902509\pi$
$11$	−6.32456			−1.90693			−0.953463	+	0.301511i	$0.902509\pi$
$12$		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	−0.991557	−	0.129668i	$-0.958609\pi$
$17$	−1.58114	+	2.73861i	−0.383482	+	0.664211i	0.608075	+	0.793880i	$0.291942\pi$
$18$		0			0
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.16228			0.659380			0.329690	−	0.944089i	$-0.393056\pi$
$23$	3.16228			0.659380			0.329690	+	0.944089i	$0.393056\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		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
$31$	−1.50000	−	2.59808i	−0.269408	−	0.466628i	0.699301	−	0.714827i	$-0.253495\pi$
$31$	−1.50000	−	2.59808i	−0.269408	−	0.466628i	−0.968709	+	0.248199i	$0.920161\pi$
$32$		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	0.982274	−	0.187453i	$-0.0600231\pi$
$37$	2.00000	+	3.46410i	0.328798	+	0.569495i	−0.653476	+	0.756948i	$0.726690\pi$
$38$		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
$47$	4.74342	−	8.21584i	0.691898	−	1.19840i	−0.279317	−	0.960199i	$-0.590108\pi$
$47$	4.74342	−	8.21584i	0.691898	−	1.19840i	0.971215	−	0.238204i	$-0.0765587\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.74342	−	8.21584i	0.651558	−	1.12853i	−0.331186	−	0.943565i	$-0.607449\pi$
$53$	4.74342	−	8.21584i	0.651558	−	1.12853i	0.982745	−	0.184967i	$-0.0592178\pi$
$54$		0			0
$55$	20.0000			2.69680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.32456	+	10.9545i	0.823387	+	1.42615i	0.903146	+	0.429334i	$0.141252\pi$
$59$	6.32456	+	10.9545i	0.823387	+	1.42615i	−0.0797589	+	0.996814i	$0.525415\pi$
$60$		0			0
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	−0.753876	−	0.657017i	$-0.771818\pi$
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	0.945931	+	0.324367i	$0.105151\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	−0.991098	−	0.133135i	$-0.957496\pi$
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	0.380251	−	0.924883i	$-0.375838\pi$
$68$		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	−0.681822	−	0.731519i	$-0.738812\pi$
$73$	2.50000	−	4.33013i	0.292603	−	0.506803i	0.974424	+	0.224716i	$0.0721453\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.6491	+	10.9545i	−1.44150	+	1.24838i
$78$		0			0
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	0.301401	+	0.953498i	$0.402546\pi$
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	−0.976453	+	0.215728i	$0.930788\pi$
$80$		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	−0.999995	−	0.00323751i	$-0.998969\pi$
$89$	−4.74342	−	8.21584i	−0.502801	−	0.870877i	0.497194	−	0.867640i	$-0.334364\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.0680	−	19.1703i	−1.13555	−	1.96683i
$96$		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
$101$	6.32456			0.629317			0.314658	−	0.949205i	$-0.398110\pi$
$101$	6.32456			0.629317			0.314658	+	0.949205i	$0.398110\pi$
$102$		0			0
$103$	2.00000			0.197066			0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000			0.197066			0.0985329	+	0.995134i	$0.468585\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	−0.983531	−	0.180741i	$-0.942150\pi$
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	0.648292	+	0.761392i	$0.275484\pi$
$110$		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
$115$	−10.0000			−0.932505
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.58114	+	8.21584i	0.144943	+	0.753145i
$120$		0			0
$121$	29.0000			2.63636
$122$		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
$131$	15.8114			1.38145			0.690724	−	0.723119i	$-0.257292\pi$
$131$	15.8114			1.38145			0.690724	+	0.723119i	$0.257292\pi$
$132$		0			0
$133$	17.5000	+	6.06218i	1.51744	+	0.525657i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.16228			−0.270172			−0.135086	−	0.990834i	$-0.543131\pi$
$137$	−3.16228			−0.270172			−0.135086	+	0.990834i	$0.543131\pi$
$138$		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
$149$	9.48683			0.777192			0.388596	−	0.921408i	$-0.372960\pi$
$149$	9.48683			0.777192			0.388596	+	0.921408i	$0.372960\pi$
$150$		0			0
$151$	−17.0000			−1.38344			−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000			−1.38344			−0.691720	+	0.722166i	$0.743147\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.74342	+	8.21584i	0.381000	+	0.659912i
$156$		0			0
$157$	8.00000	+	13.8564i	0.638470	+	1.10586i	0.985769	+	0.168107i	$0.0537655\pi$
$157$	8.00000	+	13.8564i	0.638470	+	1.10586i	−0.347299	+	0.937754i	$0.612901\pi$
$158$		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	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−8.50000	−	14.7224i	−0.665771	−	1.15315i	0.313304	−	0.949653i	$-0.398564\pi$
$164$		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
$173$	1.58114	−	2.73861i	0.120212	−	0.208213i	−0.799639	−	0.600481i	$-0.794976\pi$
$173$	1.58114	−	2.73861i	0.120212	−	0.208213i	0.919851	+	0.392268i	$0.128309\pi$
$174$		0			0
$175$	10.0000	−	8.66025i	0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.74342	+	8.21584i	−0.354540	+	0.614081i	−0.987039	−	0.160480i	$-0.948696\pi$
$179$	−4.74342	+	8.21584i	−0.354540	+	0.614081i	0.632499	+	0.774561i	$0.282029\pi$
$180$		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
$185$	−6.32456	−	10.9545i	−0.464991	−	0.805387i
$186$		0			0
$187$	10.0000	−	17.3205i	0.731272	−	1.26660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.74342	−	8.21584i	0.343222	−	0.594477i	−0.641807	−	0.766866i	$-0.721815\pi$
$191$	4.74342	−	8.21584i	0.343222	−	0.594477i	0.985029	+	0.172389i	$0.0551485\pi$
$192$		0			0
$193$	−4.00000	−	6.92820i	−0.287926	−	0.498703i	0.685388	−	0.728178i	$-0.259632\pi$
$193$	−4.00000	−	6.92820i	−0.287926	−	0.498703i	−0.973315	+	0.229475i	$0.926299\pi$
$194$		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	−0.538227	−	0.842800i	$-0.680906\pi$
$199$	6.50000	−	11.2583i	0.460773	−	0.798082i	0.999000	+	0.0447181i	$0.0142390\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.90569	+	2.73861i	0.554871	+	0.192213i
$204$		0			0
$205$	−15.0000	+	25.9808i	−1.04765	+	1.81458i
$206$		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
$227$	−9.48683			−0.629663			−0.314832	−	0.949148i	$-0.601948\pi$
$227$	−9.48683			−0.629663			−0.314832	+	0.949148i	$0.601948\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.90569	+	13.6931i	0.517919	+	0.897062i	0.999783	+	0.0208165i	$0.00662657\pi$
$233$	7.90569	+	13.6931i	0.517919	+	0.897062i	−0.481864	+	0.876246i	$0.660040\pi$
$234$		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
$241$	−1.00000			−0.0644157			−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000			−0.0644157			−0.0322078	+	0.999481i	$0.510254\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.16228	+	21.9089i	−0.202031	+	1.39971i
$246$		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
$257$	18.9737			1.18354			0.591772	−	0.806105i	$-0.298428\pi$
$257$	18.9737			1.18354			0.591772	+	0.806105i	$0.298428\pi$
$258$		0			0
$259$	10.0000	+	3.46410i	0.621370	+	0.215249i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.9737			−1.16997			−0.584983	−	0.811045i	$-0.698899\pi$
$263$	−18.9737			−1.16997			−0.584983	+	0.811045i	$0.698899\pi$
$264$		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	−0.946180	−	0.323642i	$-0.895093\pi$
$269$	−3.16228	+	5.47723i	−0.192807	+	0.333952i	0.753372	+	0.657594i	$0.228426\pi$
$270$		0			0
$271$	−4.50000	−	7.79423i	−0.273356	−	0.473466i	0.696363	−	0.717689i	$-0.254800\pi$
$271$	−4.50000	−	7.79423i	−0.273356	−	0.473466i	−0.969719	+	0.244224i	$0.921467\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−31.6228			−1.90693
$276$		0			0
$277$	−1.00000			−0.0600842			−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000			−0.0600842			−0.0300421	+	0.999549i	$0.509564\pi$
$278$		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
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	0.605575	−	0.795788i	$-0.292943\pi$
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	−0.991960	+	0.126550i	$0.959610\pi$
$284$		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
$301$	−12.5000	−	4.33013i	−0.720488	−	0.249584i
$302$		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
$311$	−3.16228	−	5.47723i	−0.179316	−	0.310585i	0.762330	−	0.647188i	$-0.224055\pi$
$311$	−3.16228	−	5.47723i	−0.179316	−	0.310585i	−0.941647	+	0.336603i	$0.890722\pi$
$312$		0			0
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	−0.519300	−	0.854592i	$-0.673807\pi$
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	0.999748	+	0.0224310i	$0.00714060\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.48683	+	16.4317i	−0.532834	+	0.922895i	0.466431	+	0.884557i	$0.345540\pi$
$317$	−9.48683	+	16.4317i	−0.532834	+	0.922895i	−0.999265	+	0.0383374i	$0.987794\pi$
$318$		0			0
$319$	−10.0000	−	17.3205i	−0.559893	−	0.969762i
$320$		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
$329$	−4.74342	−	24.6475i	−0.261513	−	1.35886i
$330$		0			0
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	−0.999298	−	0.0374662i	$-0.988071\pi$
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	0.532096	+	0.846684i	$0.321405\pi$
$332$		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		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		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
$353$	18.9737			1.00987			0.504933	−	0.863158i	$-0.331517\pi$
$353$	18.9737			1.00987			0.504933	+	0.863158i	$0.331517\pi$
$354$		0			0
$355$	−40.0000			−2.12298
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.6491	−	21.9089i	−0.667595	−	1.15631i	−0.978575	−	0.205891i	$-0.933991\pi$
$359$	−12.6491	−	21.9089i	−0.667595	−	1.15631i	0.310980	−	0.950416i	$-0.399343\pi$
$360$		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
$367$	−19.0000			−0.991792			−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000			−0.991792			−0.495896	+	0.868382i	$0.665160\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.74342	−	24.6475i	−0.246266	−	1.27964i
$372$		0			0
$373$	−17.0000			−0.880227			−0.440113	−	0.897942i	$-0.645062\pi$
$373$	−17.0000			−0.880227			−0.440113	+	0.897942i	$0.645062\pi$
$374$		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
$383$	12.6491			0.646339			0.323170	−	0.946341i	$-0.395252\pi$
$383$	12.6491			0.646339			0.323170	+	0.946341i	$0.395252\pi$
$384$		0			0
$385$	40.0000	−	34.6410i	2.03859	−	1.76547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	34.7851			1.76367			0.881836	−	0.471556i	$-0.156307\pi$
$389$	34.7851			1.76367			0.881836	+	0.471556i	$0.156307\pi$
$390$		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	0.990538	−	0.137241i	$-0.0438236\pi$
$397$	7.50000	+	12.9904i	0.376414	+	0.651969i	−0.614123	+	0.789210i	$0.710490\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.9737			0.947500			0.473750	−	0.880659i	$-0.342900\pi$
$401$	18.9737			0.947500			0.473750	+	0.880659i	$0.342900\pi$
$402$		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
$409$	8.00000	+	13.8564i	0.395575	+	0.685155i	0.993174	−	0.116639i	$-0.0372122\pi$
$409$	8.00000	+	13.8564i	0.395575	+	0.685155i	−0.597600	+	0.801795i	$0.703879\pi$
$410$		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
$425$	−7.90569	+	13.6931i	−0.383482	+	0.664211i
$426$		0			0
$427$	−1.50000	−	7.79423i	−0.0725901	−	0.377189i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		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
$437$	11.0680	+	19.1703i	0.529453	+	0.917039i
$438$		0			0
$439$	9.00000	−	15.5885i	0.429547	−	0.743996i	−0.567286	−	0.823521i	$-0.692007\pi$
$439$	9.00000	−	15.5885i	0.429547	−	0.743996i	0.996833	+	0.0795241i	$0.0253401\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.3925	−	30.1247i	0.826344	−	1.43127i	−0.0745440	−	0.997218i	$-0.523750\pi$
$443$	17.3925	−	30.1247i	0.826344	−	1.43127i	0.900888	−	0.434052i	$-0.142917\pi$
$444$		0			0
$445$	15.0000	+	25.9808i	0.711068	+	1.23161i
$446$		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
$457$	14.5000	−	25.1147i	0.678281	−	1.17482i	−0.297217	−	0.954810i	$-0.596058\pi$
$457$	14.5000	−	25.1147i	0.678281	−	1.17482i	0.975498	−	0.220008i	$-0.0706083\pi$
$458$		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	0.783537	−	0.621346i	$-0.213414\pi$
$467$	−3.16228	−	5.47723i	−0.146333	−	0.253456i	−0.929869	+	0.367890i	$0.880080\pi$
$468$		0			0
$469$	−25.0000	−	8.66025i	−1.15439	−	0.399893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	15.8114	+	27.3861i	0.727008	+	1.25922i
$474$		0			0
$475$	17.5000	+	30.3109i	0.802955	+	1.39076i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	22.1359			1.01142			0.505709	−	0.862704i	$-0.331231\pi$
$479$	22.1359			1.01142			0.505709	+	0.862704i	$0.331231\pi$
$480$		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	−0.854475	−	0.519493i	$-0.826121\pi$
$487$	0.500000	−	0.866025i	0.0226572	−	0.0392434i	0.877132	+	0.480250i	$0.159454\pi$
$488$		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
$493$	−10.0000			−0.450377
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.2982	−	21.9089i	1.13478	−	0.982749i
$498$		0			0
$499$	7.00000			0.313363			0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000			0.313363			0.156682	+	0.987649i	$0.449920\pi$
$500$		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
$509$	−37.9473			−1.68199			−0.840993	−	0.541046i	$-0.818028\pi$
$509$	−37.9473			−1.68199			−0.840993	+	0.541046i	$0.818028\pi$
$510$		0			0
$511$	−2.50000	−	12.9904i	−0.110593	−	0.574661i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.32456			−0.278693
$516$		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	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−16.0000	−	27.7128i	−0.699631	−	1.21180i	−0.968594	−	0.248646i	$-0.920014\pi$
$523$	−16.0000	−	27.7128i	−0.699631	−	1.21180i	0.268963	−	0.963150i	$-0.413319\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.48683			0.413253
$528$		0			0
$529$	−13.0000			−0.565217
$530$		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	0.998475	−	0.0552031i	$-0.0175806\pi$
$541$	10.5000	+	18.1865i	0.451430	+	0.781900i	−0.547045	+	0.837103i	$0.684247\pi$
$542$		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
$551$	−11.0680	+	19.1703i	−0.471511	+	0.816682i
$552$		0			0
$553$	6.00000	+	31.1769i	0.255146	+	1.32578i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.58114	−	2.73861i	0.0669950	−	0.116039i	−0.830582	−	0.556896i	$-0.811992\pi$
$557$	1.58114	−	2.73861i	0.0669950	−	0.116039i	0.897577	+	0.440857i	$0.145326\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.5548	−	35.6020i	−0.866282	−	1.50044i	−0.865769	−	0.500444i	$-0.833170\pi$
$563$	−20.5548	−	35.6020i	−0.866282	−	1.50044i	−0.000512896	−	1.00000i	$-0.500163\pi$
$564$		0			0
$565$	10.0000	−	17.3205i	0.420703	−	0.728679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.9737	+	32.8634i	−0.795417	+	1.37770i	0.127156	+	0.991883i	$0.459415\pi$
$569$	−18.9737	+	32.8634i	−0.795417	+	1.37770i	−0.922574	+	0.385821i	$0.873918\pi$
$570$		0			0
$571$	−19.5000	−	33.7750i	−0.816050	−	1.41344i	−0.908572	−	0.417729i	$-0.862826\pi$
$571$	−19.5000	−	33.7750i	−0.816050	−	1.41344i	0.0925222	−	0.995711i	$-0.470507\pi$
$572$		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	0.110252	−	0.993904i	$-0.464834\pi$
$577$	22.0000	−	38.1051i	0.915872	−	1.58634i	0.805620	−	0.592433i	$-0.201833\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.8114	+	5.47723i	0.655967	+	0.227234i
$582$		0			0
$583$	−30.0000	+	51.9615i	−1.24247	+	2.15203i
$584$		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	0.602815	−	0.797881i	$-0.294046\pi$
$593$	−9.48683	−	16.4317i	−0.389578	−	0.674768i	−0.992393	+	0.123113i	$0.960712\pi$
$594$		0			0
$595$	−5.00000	−	25.9808i	−0.204980	−	1.06511i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.2302	+	24.6475i	0.581432	+	1.00707i	0.995310	+	0.0967377i	$0.0308408\pi$
$599$	14.2302	+	24.6475i	0.581432	+	1.00707i	−0.413878	+	0.910333i	$0.635826\pi$
$600$		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
$605$	−91.7061			−3.72838
$606$		0			0
$607$	15.0000			0.608831			0.304416	−	0.952539i	$-0.401539\pi$
$607$	15.0000			0.608831			0.304416	+	0.952539i	$0.401539\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.5000	+	28.5788i	−0.666429	+	1.15429i	0.312467	+	0.949929i	$0.398845\pi$
$613$	−16.5000	+	28.5788i	−0.666429	+	1.15429i	−0.978896	+	0.204360i	$0.934489\pi$
$614$		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
$619$	22.0000			0.884255			0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000			0.884255			0.442127	+	0.896952i	$0.354224\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.7171	−	8.21584i	−0.950205	−	0.329161i
$624$		0			0
$625$	−25.0000			−1.00000
$626$		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
$635$	−66.4078			−2.63531
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.16228			0.124902			0.0624512	−	0.998048i	$-0.480108\pi$
$641$	3.16228			0.124902			0.0624512	+	0.998048i	$0.480108\pi$
$642$		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	0.203826	+	0.979007i	$0.434662\pi$
$647$	−18.9737	+	32.8634i	−0.745932	+	1.29199i	−0.949758	+	0.312985i	$0.898671\pi$
$648$		0			0
$649$	−40.0000	−	69.2820i	−1.57014	−	2.71956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.4605			−1.11375			−0.556873	−	0.830598i	$-0.687999\pi$
$653$	−28.4605			−1.11375			−0.556873	+	0.830598i	$0.687999\pi$
$654$		0			0
$655$	−50.0000			−1.95366
$656$		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
$661$	8.50000	+	14.7224i	0.330612	+	0.572636i	0.982632	−	0.185565i	$-0.0594116\pi$
$661$	8.50000	+	14.7224i	0.330612	+	0.572636i	−0.652020	+	0.758202i	$0.726078\pi$
$662$		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
$677$	18.9737	−	32.8634i	0.729217	−	1.26304i	−0.227997	−	0.973662i	$-0.573218\pi$
$677$	18.9737	−	32.8634i	0.729217	−	1.26304i	0.957214	−	0.289379i	$-0.0934489\pi$
$678$		0			0
$679$	12.5000	+	4.33013i	0.479706	+	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.48683	−	16.4317i	0.363004	−	0.628741i	−0.625450	−	0.780264i	$-0.715085\pi$
$683$	9.48683	−	16.4317i	0.363004	−	0.628741i	0.988454	+	0.151524i	$0.0484179\pi$
$684$		0			0
$685$	10.0000			0.382080
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−5.50000	+	9.52628i	−0.209230	+	0.362397i	−0.951472	−	0.307735i	$-0.900429\pi$
$691$	−5.50000	+	9.52628i	−0.209230	+	0.362397i	0.742242	+	0.670132i	$0.233762\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.16228	+	5.47723i	−0.119952	+	0.207763i
$696$		0			0
$697$	15.0000	+	25.9808i	0.568166	+	0.984092i
$698$		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
$707$	12.6491	−	10.9545i	0.475719	−	0.411985i
$708$		0			0
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	−0.856484	−	0.516174i	$-0.827356\pi$
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	0.875262	+	0.483650i	$0.160689\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−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	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	4.00000	−	3.46410i	0.148968	−	0.129010i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.90569	+	13.6931i	0.293610	+	0.508548i
$726$		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
$731$	15.8114			0.584805
$732$		0			0
$733$	3.00000			0.110808			0.0554038	−	0.998464i	$-0.482355\pi$
$733$	3.00000			0.110808			0.0554038	+	0.998464i	$0.482355\pi$
$734$		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	−0.892295	−	0.451452i	$-0.850906\pi$
$739$	−1.50000	+	2.59808i	−0.0551784	+	0.0955718i	0.837117	+	0.547024i	$0.184239\pi$
$740$		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
$745$	−30.0000			−1.09911
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3.00000			0.109472			0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000			0.109472			0.0547358	+	0.998501i	$0.482568\pi$
$752$		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
$761$	−25.2982			−0.917060			−0.458530	−	0.888679i	$-0.651624\pi$
$761$	−25.2982			−0.917060			−0.458530	+	0.888679i	$0.651624\pi$
$762$		0			0
$763$	3.50000	+	18.1865i	0.126709	+	0.658397i
$764$		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	0.362865	+	0.931842i	$0.381798\pi$
$773$	−17.3925	+	30.1247i	−0.625566	+	1.08351i	−0.988431	+	0.151670i	$0.951535\pi$
$774$		0			0
$775$	−7.50000	−	12.9904i	−0.269408	−	0.466628i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	66.4078			2.37931
$780$		0			0
$781$	−80.0000			−2.86263
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−25.2982	−	43.8178i	−0.902932	−	1.56392i
$786$		0			0
$787$	−7.50000	−	12.9904i	−0.267346	−	0.463057i	0.700830	−	0.713329i	$-0.252813\pi$
$787$	−7.50000	−	12.9904i	−0.267346	−	0.463057i	−0.968176	+	0.250272i	$0.919480\pi$
$788$		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
$803$	−15.8114	+	27.3861i	−0.557972	+	0.966435i
$804$		0			0
$805$	−20.0000	+	17.3205i	−0.704907	+	0.610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.8114	−	27.3861i	0.555899	−	0.962845i	−0.441934	−	0.897047i	$-0.645708\pi$
$809$	15.8114	−	27.3861i	0.555899	−	0.962845i	0.997833	−	0.0657975i	$-0.0209591\pi$
$810$		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
$815$	26.8794	+	46.5564i	0.941543	+	1.63080i
$816$		0			0
$817$	17.5000	−	30.3109i	0.612247	−	1.06044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.6491	−	21.9089i	0.441457	−	0.764626i	−0.556341	−	0.830954i	$-0.687795\pi$
$821$	12.6491	−	21.9089i	0.441457	−	0.764626i	0.997798	+	0.0663282i	$0.0211284\pi$
$822$		0			0
$823$	11.5000	+	19.9186i	0.400865	+	0.694318i	0.993831	−	0.110910i	$-0.0353764\pi$
$823$	11.5000	+	19.9186i	0.400865	+	0.694318i	−0.592966	+	0.805228i	$0.702043\pi$
$824$		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	−0.754567	−	0.656223i	$-0.772153\pi$
$829$	5.50000	−	9.52628i	0.191023	−	0.330861i	0.945589	+	0.325362i	$0.105486\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.3925	+	13.6931i	0.602615	+	0.474437i
$834$		0			0
$835$	10.0000	−	17.3205i	0.346064	−	0.599401i
$836$		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
$851$	6.32456	+	10.9545i	0.216803	+	0.375514i
$852$		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
$857$	37.9473			1.29626			0.648128	−	0.761531i	$-0.275552\pi$
$857$	37.9473			1.29626			0.648128	+	0.761531i	$0.275552\pi$
$858$		0			0
$859$	17.0000			0.580033			0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000			0.580033			0.290016	+	0.957022i	$0.406339\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.90569	−	13.6931i	−0.269113	−	0.466117i	0.699520	−	0.714613i	$-0.253397\pi$
$863$	−7.90569	−	13.6931i	−0.269113	−	0.466117i	−0.968633	+	0.248496i	$0.920064\pi$
$864$		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
$877$	−5.00000			−0.168838			−0.0844190	−	0.996430i	$-0.526903\pi$
$877$	−5.00000			−0.168838			−0.0844190	+	0.996430i	$0.526903\pi$
$878$		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			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	42.0000	−	36.3731i	1.40863	−	1.21991i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	66.4078			2.22225
$894$		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
$905$	−66.4078			−2.20747
$906$		0			0
$907$	−24.0000			−0.796907			−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000			−0.796907			−0.398453	+	0.917189i	$0.630453\pi$
$908$		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
$913$	−20.0000	−	34.6410i	−0.661903	−	1.14645i
$914$		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	0.942367	−	0.334581i	$-0.108595\pi$
$919$	5.50000	+	9.52628i	0.181428	+	0.314243i	−0.760939	+	0.648824i	$0.775261\pi$
$920$		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
$929$	−14.2302	+	24.6475i	−0.466879	+	0.808659i	−0.999284	−	0.0378311i	$-0.987955\pi$
$929$	−14.2302	+	24.6475i	−0.466879	+	0.808659i	0.532405	+	0.846490i	$0.321288\pi$
$930$		0			0
$931$	45.5000	−	18.1865i	1.49120	−	0.596040i
$932$		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
$941$	−12.6491	−	21.9089i	−0.412349	−	0.714210i	0.582797	−	0.812618i	$-0.301958\pi$
$941$	−12.6491	−	21.9089i	−0.412349	−	0.714210i	−0.995146	+	0.0984080i	$0.968625\pi$
$942$		0			0
$943$	15.0000	−	25.9808i	0.488467	−	0.846050i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.3925	+	30.1247i	−0.565181	+	0.978923i	0.431852	+	0.901945i	$0.357860\pi$
$947$	−17.3925	+	30.1247i	−0.565181	+	0.978923i	−0.997033	+	0.0769779i	$0.975473\pi$
$948$		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
$959$	−6.32456	+	5.47723i	−0.204231	+	0.176869i
$960$		0			0
$961$	11.0000	−	19.0526i	0.354839	−	0.614599i
$962$		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	−0.942275	−	0.334840i	$-0.891318\pi$
$971$	−23.7171	−	41.0792i	−0.761117	−	1.31829i	0.181158	−	0.983454i	$-0.442016\pi$
$972$		0			0
$973$	−1.00000	−	5.19615i	−0.0320585	−	0.166581i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.9737	−	32.8634i	−0.607021	−	1.05139i	−0.991729	−	0.128353i	$-0.959031\pi$
$977$	−18.9737	−	32.8634i	−0.607021	−	1.05139i	0.384707	−	0.923039i	$-0.374302\pi$
$978$		0			0
$979$	30.0000	+	51.9615i	0.958804	+	1.66070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	80.0000			2.54901
$986$		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	−0.955525	−	0.294911i	$-0.904710\pi$
$991$	−7.00000	+	12.1244i	−0.222362	+	0.385143i	0.733163	+	0.680053i	$0.238043\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.5548	+	35.6020i	−0.651631	+	1.12866i
$996$		0			0
$997$	41.0000			1.29848			0.649242	−	0.760582i	$-0.275086\pi$
$997$	41.0000			1.29848			0.649242	+	0.760582i	$0.275086\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.d.109.1	✓	4	9.5	odd	6
756.2.k.d.109.2	yes	4	9.4	even	3
756.2.k.d.541.1	yes	4	63.23	odd	6
756.2.k.d.541.2	yes	4	63.58	even	3
2268.2.i.i.865.1		4	9.2	odd	6
2268.2.i.i.865.2		4	9.7	even	3
2268.2.i.i.2053.1		4	21.2	odd	6
2268.2.i.i.2053.2		4	7.2	even	3
2268.2.l.i.109.1		4	1.1	even	1	trivial
2268.2.l.i.109.2		4	3.2	odd	2	inner
2268.2.l.i.541.1		4	63.16	even	3	inner
2268.2.l.i.541.2		4	63.2	odd	6	inner
5292.2.a.q.1.1		2	63.4	even	3
5292.2.a.q.1.2		2	63.32	odd	6
5292.2.a.r.1.1		2	63.59	even	6
5292.2.a.r.1.2		2	63.31	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.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.16228 q^{5} +(2.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q-3.16228 q^{5} +(2.00000 - 1.73205i) q^{7} -6.32456 q^{11} +(-1.58114 + 2.73861i) q^{17} +(3.50000 + 6.06218i) q^{19} +3.16228 q^{23} +5.00000 q^{25} +(1.58114 + 2.73861i) q^{29} +(-1.50000 - 2.59808i) q^{31} +(-6.32456 + 5.47723i) q^{35} +(2.00000 + 3.46410i) q^{37} +(4.74342 - 8.21584i) q^{41} +(-2.50000 - 4.33013i) q^{43} +(4.74342 - 8.21584i) q^{47} +(1.00000 - 6.92820i) q^{49} +(4.74342 - 8.21584i) q^{53} +20.0000 q^{55} +(6.32456 + 10.9545i) q^{59} +(1.50000 - 2.59808i) q^{61} +(-5.00000 - 8.66025i) q^{67} +12.6491 q^{71} +(2.50000 - 4.33013i) q^{73} +(-12.6491 + 10.9545i) q^{77} +(-6.00000 + 10.3923i) q^{79} +(3.16228 + 5.47723i) q^{83} +(5.00000 - 8.66025i) q^{85} +(-4.74342 - 8.21584i) q^{89} +(-11.0680 - 19.1703i) q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 14 q^{19} + 20 q^{25} - 6 q^{31} + 8 q^{37} - 10 q^{43} + 4 q^{49} + 80 q^{55} + 6 q^{61} - 20 q^{67} + 10 q^{73} - 24 q^{79} + 20 q^{85} + 10 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 + 14 * q^19 + 20 * q^25 - 6 * q^31 + 8 * q^37 - 10 * q^43 + 4 * q^49 + 80 * q^55 + 6 * q^61 - 20 * q^67 + 10 * q^73 - 24 * q^79 + 20 * q^85 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more