LMFDB - Embedded newform 5292.2.i.j.2125.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1549,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("5292.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2125.11
Character	$\chi$	$=$	5292.2125
Dual form			5292.2.i.j.1549.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73981 - 3.01343i) q^{5} +O(q^{10})$	$q+(1.73981 - 3.01343i) q^{5} +(1.25788 + 2.17871i) q^{11} +(-0.292110 - 0.505949i) q^{13} +(0.547519 - 0.948331i) q^{17} +(2.96834 + 5.14132i) q^{19} +(3.19264 - 5.52982i) q^{23} +(-3.55384 - 6.15544i) q^{25} +(-0.918333 + 1.59060i) q^{29} +7.03743 q^{31} +(0.702576 + 1.21690i) q^{37} +(-5.37855 - 9.31593i) q^{41} +(-5.67879 + 9.83596i) q^{43} +7.53129 q^{47} +(5.82285 - 10.0855i) q^{53} +8.75386 q^{55} +4.45551 q^{59} +12.3524 q^{61} -2.03286 q^{65} -12.6707 q^{67} +4.93390 q^{71} +(-4.35558 + 7.54408i) q^{73} -0.560411 q^{79} +(3.68472 - 6.38212i) q^{83} +(-1.90515 - 3.29982i) q^{85} +(6.07256 + 10.5180i) q^{89} +20.6573 q^{95} +(-6.98486 + 12.0981i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} - 72 q^{65} - 24 q^{67} - 48 q^{71} - 24 q^{79} + 12 q^{85} + 64 q^{95}+O(q^{100}) $ 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 - 72 * q^65 - 24 * q^67 - 48 * q^71 - 24 * q^79 + 12 * q^85 + 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	1.73981	−	3.01343i	0.778065	−	1.34765i	−0.154991	−	0.987916i	$-0.549535\pi$
$5$	1.73981	−	3.01343i	0.778065	−	1.34765i	0.933056	−	0.359732i	$-0.117132\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.25788	+	2.17871i	0.379265	+	0.656906i	0.990956	−	0.134191i	$-0.0428435\pi$
$11$	1.25788	+	2.17871i	0.379265	+	0.656906i	−0.611690	+	0.791097i	$0.709510\pi$
$12$		0			0
$13$	−0.292110	−	0.505949i	−0.0810167	−	0.140325i	0.822670	−	0.568519i	$-0.192483\pi$
$13$	−0.292110	−	0.505949i	−0.0810167	−	0.140325i	−0.903687	+	0.428194i	$0.859150\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.547519	−	0.948331i	0.132793	−	0.230004i	−0.791959	−	0.610574i	$-0.790939\pi$
$17$	0.547519	−	0.948331i	0.132793	−	0.230004i	0.924752	+	0.380570i	$0.124272\pi$
$18$		0			0
$19$	2.96834	+	5.14132i	0.680984	+	1.17950i	0.974681	+	0.223601i	$0.0717812\pi$
$19$	2.96834	+	5.14132i	0.680984	+	1.17950i	−0.293696	+	0.955899i	$0.594885\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.19264	−	5.52982i	0.665712	−	1.15305i	−0.313380	−	0.949628i	$-0.601461\pi$
$23$	3.19264	−	5.52982i	0.665712	−	1.15305i	0.979092	−	0.203419i	$-0.0652053\pi$
$24$		0			0
$25$	−3.55384	−	6.15544i	−0.710769	−	1.23109i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.918333	+	1.59060i	−0.170530	+	0.295367i	−0.938605	−	0.344993i	$-0.887881\pi$
$29$	−0.918333	+	1.59060i	−0.170530	+	0.295367i	0.768075	+	0.640360i	$0.221215\pi$
$30$		0			0
$31$	7.03743			1.26396			0.631980	−	0.774985i	$-0.282242\pi$
$31$	7.03743			1.26396			0.631980	+	0.774985i	$0.282242\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.702576	+	1.21690i	0.115503	+	0.200057i	0.917981	−	0.396625i	$-0.129819\pi$
$37$	0.702576	+	1.21690i	0.115503	+	0.200057i	−0.802478	+	0.596682i	$0.796485\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.37855	−	9.31593i	−0.839989	−	1.45490i	−0.889903	−	0.456150i	$-0.849228\pi$
$41$	−5.37855	−	9.31593i	−0.839989	−	1.45490i	0.0499141	−	0.998754i	$-0.484105\pi$
$42$		0			0
$43$	−5.67879	+	9.83596i	−0.866008	+	1.49997i	3.53909e−5		1.00000i	$0.499989\pi$
$43$	−5.67879	+	9.83596i	−0.866008	+	1.49997i	−0.866043	+	0.499969i	$0.833345\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.53129			1.09855			0.549276	−	0.835641i	$-0.314904\pi$
$47$	7.53129			1.09855			0.549276	+	0.835641i	$0.314904\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.82285	−	10.0855i	0.799830	−	1.38535i	−0.119897	−	0.992786i	$-0.538256\pi$
$53$	5.82285	−	10.0855i	0.799830	−	1.38535i	0.919727	−	0.392560i	$-0.128410\pi$
$54$		0			0
$55$	8.75386			1.18037
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.45551			0.580058			0.290029	−	0.957018i	$-0.406335\pi$
$59$	4.45551			0.580058			0.290029	+	0.957018i	$0.406335\pi$
$60$		0			0
$61$	12.3524			1.58157			0.790784	−	0.612095i	$-0.209673\pi$
$61$	12.3524			1.58157			0.790784	+	0.612095i	$0.209673\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.03286			−0.252145
$66$		0			0
$67$	−12.6707			−1.54798			−0.773988	−	0.633201i	$-0.781741\pi$
$67$	−12.6707			−1.54798			−0.773988	+	0.633201i	$0.781741\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.93390			0.585546			0.292773	−	0.956182i	$-0.405422\pi$
$71$	4.93390			0.585546			0.292773	+	0.956182i	$0.405422\pi$
$72$		0			0
$73$	−4.35558	+	7.54408i	−0.509782	+	0.882968i	0.490154	+	0.871636i	$0.336940\pi$
$73$	−4.35558	+	7.54408i	−0.509782	+	0.882968i	−0.999936	+	0.0113320i	$0.996393\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.560411			−0.0630512			−0.0315256	−	0.999503i	$-0.510037\pi$
$79$	−0.560411			−0.0630512			−0.0315256	+	0.999503i	$0.510037\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.68472	−	6.38212i	0.404451	−	0.700529i	−0.589807	−	0.807544i	$-0.700796\pi$
$83$	3.68472	−	6.38212i	0.404451	−	0.700529i	0.994257	+	0.107015i	$0.0341294\pi$
$84$		0			0
$85$	−1.90515	−	3.29982i	−0.206643	−	0.357916i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.07256	+	10.5180i	0.643690	+	1.11490i	0.984602	+	0.174808i	$0.0559306\pi$
$89$	6.07256	+	10.5180i	0.643690	+	1.11490i	−0.340913	+	0.940095i	$0.610736\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	20.6573			2.11940
$96$		0			0
$97$	−6.98486	+	12.0981i	−0.709205	+	1.22838i	0.255947	+	0.966691i	$0.417613\pi$
$97$	−6.98486	+	12.0981i	−0.709205	+	1.22838i	−0.965152	+	0.261688i	$0.915721\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.58825	−	7.94708i	−0.456548	−	0.790764i	0.542228	−	0.840231i	$-0.317581\pi$
$101$	−4.58825	−	7.94708i	−0.456548	−	0.790764i	−0.998776	+	0.0494676i	$0.984248\pi$
$102$		0			0
$103$	0.239538	−	0.414892i	0.0236024	−	0.0408805i	−0.853983	−	0.520301i	$-0.825820\pi$
$103$	0.239538	−	0.414892i	0.0236024	−	0.0408805i	0.877585	+	0.479420i	$0.159153\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.67882	−	4.63986i	−0.258972	−	0.448552i	0.706995	−	0.707219i	$-0.250050\pi$
$107$	−2.67882	−	4.63986i	−0.258972	−	0.448552i	−0.965967	+	0.258666i	$0.916717\pi$
$108$		0			0
$109$	7.87535	−	13.6405i	0.754322	−	1.30652i	−0.191389	−	0.981514i	$-0.561299\pi$
$109$	7.87535	−	13.6405i	0.754322	−	1.30652i	0.945711	−	0.325009i	$-0.105367\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.92483	+	11.9942i	0.651433	+	1.12832i	0.982775	+	0.184804i	$0.0591652\pi$
$113$	6.92483	+	11.9942i	0.651433	+	1.12832i	−0.331342	+	0.943511i	$0.607501\pi$
$114$		0			0
$115$	−11.1092	−	19.2416i	−1.03593	−	1.79429i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.33548	−	4.04516i	0.212316	−	0.367742i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.33394			−0.655968
$126$		0			0
$127$	20.7533			1.84156			0.920780	−	0.390083i	$-0.127554\pi$
$127$	20.7533			1.84156			0.920780	+	0.390083i	$0.127554\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.799491	+	1.38476i	−0.0698518	+	0.120987i	−0.898836	−	0.438285i	$-0.855586\pi$
$131$	−0.799491	+	1.38476i	−0.0698518	+	0.120987i	0.828984	+	0.559272i	$0.188919\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.82610	−	6.62700i	−0.326886	−	0.566182i	0.655007	−	0.755623i	$-0.272666\pi$
$137$	−3.82610	−	6.62700i	−0.326886	−	0.566182i	−0.981892	+	0.189441i	$0.939333\pi$
$138$		0			0
$139$	7.99424	+	13.8464i	0.678062	+	1.17444i	0.975564	+	0.219717i	$0.0705134\pi$
$139$	7.99424	+	13.8464i	0.678062	+	1.17444i	−0.297501	+	0.954721i	$0.596153\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.734878	−	1.27285i	0.0614536	−	0.106441i
$144$		0			0
$145$	3.19544	+	5.53467i	0.265367	+	0.459629i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.43734	+	2.48955i	−0.117752	+	0.203952i	−0.918876	−	0.394546i	$-0.870902\pi$
$149$	−1.43734	+	2.48955i	−0.117752	+	0.203952i	0.801125	+	0.598497i	$0.204235\pi$
$150$		0			0
$151$	−4.58076	−	7.93411i	−0.372777	−	0.645669i	0.617215	−	0.786795i	$-0.288261\pi$
$151$	−4.58076	−	7.93411i	−0.372777	−	0.645669i	−0.989992	+	0.141126i	$0.954928\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.2438	−	21.2068i	0.983443	−	1.70337i
$156$		0			0
$157$	12.7882			1.02061			0.510304	−	0.859994i	$-0.329533\pi$
$157$	12.7882			1.02061			0.510304	+	0.859994i	$0.329533\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−7.18390	−	12.4429i	−0.562686	−	0.974601i	−0.997261	−	0.0739653i	$-0.976435\pi$
$163$	−7.18390	−	12.4429i	−0.562686	−	0.974601i	0.434575	−	0.900636i	$-0.356899\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.38280	−	4.12714i	−0.184387	−	0.319367i	0.758983	−	0.651111i	$-0.225697\pi$
$167$	−2.38280	−	4.12714i	−0.184387	−	0.319367i	−0.943370	+	0.331743i	$0.892363\pi$
$168$		0			0
$169$	6.32934	−	10.9627i	0.486873	−	0.843288i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−25.1167			−1.90958			−0.954792	−	0.297275i	$-0.903922\pi$
$173$	−25.1167			−1.90958			−0.954792	+	0.297275i	$0.903922\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.41168	−	5.90920i	0.255001	−	0.441675i	−0.709895	−	0.704308i	$-0.751258\pi$
$179$	3.41168	−	5.90920i	0.255001	−	0.441675i	0.964896	+	0.262633i	$0.0845909\pi$
$180$		0			0
$181$	−13.4735			−1.00148			−0.500739	−	0.865598i	$-0.666938\pi$
$181$	−13.4735			−1.00148			−0.500739	+	0.865598i	$0.666938\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.88938			0.359475
$186$		0			0
$187$	2.75485			0.201455
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14.2064			−1.02794			−0.513968	−	0.857809i	$-0.671825\pi$
$191$	−14.2064			−1.02794			−0.513968	+	0.857809i	$0.671825\pi$
$192$		0			0
$193$	−6.78520			−0.488410			−0.244205	−	0.969724i	$-0.578527\pi$
$193$	−6.78520			−0.488410			−0.244205	+	0.969724i	$0.578527\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.5287			−1.32011			−0.660057	−	0.751215i	$-0.729468\pi$
$197$	−18.5287			−1.32011			−0.660057	+	0.751215i	$0.729468\pi$
$198$		0			0
$199$	8.39804	−	14.5458i	0.595321	−	1.03113i	−0.398180	−	0.917307i	$-0.630358\pi$
$199$	8.39804	−	14.5458i	0.595321	−	1.03113i	0.993501	−	0.113819i	$-0.0363086\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−37.4305			−2.61426
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.46763	+	12.9343i	−0.516547	+	0.894686i
$210$		0			0
$211$	−10.7912	−	18.6909i	−0.742896	−	1.28673i	−0.951171	−	0.308664i	$-0.900118\pi$
$211$	−10.7912	−	18.6909i	−0.742896	−	1.28673i	0.208275	−	0.978070i	$-0.433215\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.7600	+	34.2253i	1.34762	+	2.33415i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.639743			−0.0430338
$222$		0			0
$223$	−0.495791	+	0.858736i	−0.0332006	+	0.0575052i	−0.882148	−	0.470972i	$-0.843903\pi$
$223$	−0.495791	+	0.858736i	−0.0332006	+	0.0575052i	0.848948	+	0.528477i	$0.177237\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.46567	−	2.53861i	−0.0972799	−	0.168494i	0.813278	−	0.581875i	$-0.197681\pi$
$227$	−1.46567	−	2.53861i	−0.0972799	−	0.168494i	−0.910558	+	0.413382i	$0.864348\pi$
$228$		0			0
$229$	−2.19201	+	3.79667i	−0.144852	+	0.250891i	−0.929318	−	0.369281i	$-0.879604\pi$
$229$	−2.19201	+	3.79667i	−0.144852	+	0.250891i	0.784466	+	0.620172i	$0.212937\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.543158	−	0.940778i	−0.0355835	−	0.0616324i	0.847685	−	0.530500i	$-0.177996\pi$
$233$	−0.543158	−	0.940778i	−0.0355835	−	0.0616324i	−0.883269	+	0.468867i	$0.844662\pi$
$234$		0			0
$235$	13.1030	−	22.6950i	0.854744	−	1.48046i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.91423	−	3.31554i	−0.123821	−	0.214464i	0.797450	−	0.603384i	$-0.206182\pi$
$239$	−1.91423	−	3.31554i	−0.123821	−	0.214464i	−0.921271	+	0.388920i	$0.872848\pi$
$240$		0			0
$241$	−6.46271	−	11.1937i	−0.416300	−	0.721052i	0.579264	−	0.815140i	$-0.303340\pi$
$241$	−6.46271	−	11.1937i	−0.416300	−	0.721052i	−0.995564	+	0.0940877i	$0.970007\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.73416	−	3.00366i	0.110342	−	0.191118i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.6654			1.24127			0.620634	−	0.784101i	$-0.286875\pi$
$251$	19.6654			1.24127			0.620634	+	0.784101i	$0.286875\pi$
$252$		0			0
$253$	16.0638			1.00992
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.6799	+	21.9622i	−0.790948	+	1.36996i	0.134433	+	0.990923i	$0.457079\pi$
$257$	−12.6799	+	21.9622i	−0.790948	+	1.36996i	−0.925381	+	0.379039i	$0.876255\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.43798	+	7.68680i	0.273657	+	0.473988i	0.969796	−	0.243919i	$-0.0784332\pi$
$263$	4.43798	+	7.68680i	0.273657	+	0.473988i	−0.696138	+	0.717908i	$0.745100\pi$
$264$		0			0
$265$	−20.2612	−	35.0935i	−1.24464	−	2.15578i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.15338	−	15.8541i	0.558092	−	0.966643i	−0.439564	−	0.898211i	$-0.644867\pi$
$269$	9.15338	−	15.8541i	0.558092	−	0.966643i	0.997656	−	0.0684319i	$-0.0217996\pi$
$270$		0			0
$271$	−2.34465	−	4.06106i	−0.142427	−	0.246692i	0.785983	−	0.618248i	$-0.212157\pi$
$271$	−2.34465	−	4.06106i	−0.142427	−	0.246692i	−0.928410	+	0.371557i	$0.878824\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.94062	−	15.4856i	0.539140	−	0.933817i
$276$		0			0
$277$	−2.82807	−	4.89836i	−0.169922	−	0.294314i	0.768470	−	0.639886i	$-0.221018\pi$
$277$	−2.82807	−	4.89836i	−0.169922	−	0.294314i	−0.938392	+	0.345572i	$0.887685\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.36370	−	9.29020i	0.319971	−	0.554207i	−0.660510	−	0.750817i	$-0.729660\pi$
$281$	5.36370	−	9.29020i	0.319971	−	0.554207i	0.980482	+	0.196610i	$0.0629933\pi$
$282$		0			0
$283$	−23.8106			−1.41539			−0.707697	−	0.706517i	$-0.750266\pi$
$283$	−23.8106			−1.41539			−0.707697	+	0.706517i	$0.750266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.90045	+	13.6840i	0.464732	+	0.804940i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.3315	+	17.8946i	0.603570	+	1.04541i	0.992276	+	0.124052i	$0.0395889\pi$
$293$	10.3315	+	17.8946i	0.603570	+	1.04541i	−0.388706	+	0.921362i	$0.627078\pi$
$294$		0			0
$295$	7.75171	−	13.4264i	0.451322	−	0.781713i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.73041			−0.215735
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	21.4908	−	37.2232i	1.23056	−	2.13140i
$306$		0			0
$307$	11.9227			0.680464			0.340232	−	0.940342i	$-0.389494\pi$
$307$	11.9227			0.680464			0.340232	+	0.940342i	$0.389494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.13271			0.517868			0.258934	−	0.965895i	$-0.416629\pi$
$311$	9.13271			0.517868			0.258934	+	0.965895i	$0.416629\pi$
$312$		0			0
$313$	13.8396			0.782260			0.391130	−	0.920335i	$-0.372084\pi$
$313$	13.8396			0.782260			0.391130	+	0.920335i	$0.372084\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.83157			−0.102871			−0.0514357	−	0.998676i	$-0.516380\pi$
$317$	−1.83157			−0.102871			−0.0514357	+	0.998676i	$0.516380\pi$
$318$		0			0
$319$	−4.62061			−0.258705
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.50089			0.361719
$324$		0			0
$325$	−2.07623	+	3.59613i	−0.115168	+	0.199477i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.206264			−0.0113373			−0.00566864	−	0.999984i	$-0.501804\pi$
$331$	−0.206264			−0.0113373			−0.00566864	+	0.999984i	$0.501804\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.0446	+	38.1824i	−1.20442	+	2.08613i
$336$		0			0
$337$	0.756536	+	1.31036i	0.0412111	+	0.0713798i	0.885895	−	0.463885i	$-0.153545\pi$
$337$	0.756536	+	1.31036i	0.0412111	+	0.0713798i	−0.844684	+	0.535265i	$0.820212\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.85224	+	15.3325i	0.479376	+	0.830303i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.21813			0.172758			0.0863792	−	0.996262i	$-0.472470\pi$
$347$	3.21813			0.172758			0.0863792	+	0.996262i	$0.472470\pi$
$348$		0			0
$349$	−7.04006	+	12.1937i	−0.376846	+	0.652716i	−0.990601	−	0.136780i	$-0.956325\pi$
$349$	−7.04006	+	12.1937i	−0.376846	+	0.652716i	0.613756	+	0.789496i	$0.289658\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.68465	−	2.91790i	−0.0896648	−	0.155304i	0.817705	−	0.575638i	$-0.195246\pi$
$353$	−1.68465	−	2.91790i	−0.0896648	−	0.155304i	−0.907369	+	0.420334i	$0.861913\pi$
$354$		0			0
$355$	8.58403	−	14.8680i	0.455593	−	0.789110i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.28913	−	12.6251i	−0.384706	−	0.666330i	0.607023	−	0.794685i	$-0.292364\pi$
$359$	−7.28913	−	12.6251i	−0.384706	−	0.666330i	−0.991728	+	0.128355i	$0.959030\pi$
$360$		0			0
$361$	−8.12211	+	14.0679i	−0.427479	+	0.740416i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.1557	+	26.2505i	0.793286	+	1.37401i
$366$		0			0
$367$	0.368367	+	0.638030i	0.0192286	+	0.0333049i	0.875480	−	0.483255i	$-0.160546\pi$
$367$	0.368367	+	0.638030i	0.0192286	+	0.0333049i	−0.856251	+	0.516560i	$0.827212\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−14.4514	+	25.0306i	−0.748267	+	1.29604i	0.200385	+	0.979717i	$0.435781\pi$
$373$	−14.4514	+	25.0306i	−0.748267	+	1.29604i	−0.948653	+	0.316320i	$0.897553\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.07302			0.0552632
$378$		0			0
$379$	−8.88267			−0.456272			−0.228136	−	0.973629i	$-0.573263\pi$
$379$	−8.88267			−0.456272			−0.228136	+	0.973629i	$0.573263\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.70273	−	13.3415i	0.393591	−	0.681720i	−0.599329	−	0.800503i	$-0.704566\pi$
$383$	7.70273	−	13.3415i	0.393591	−	0.681720i	0.992920	+	0.118783i	$0.0378992\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.1570	−	17.5924i	−0.514978	−	0.891968i	−0.999849	−	0.0173821i	$-0.994467\pi$
$389$	−10.1570	−	17.5924i	−0.514978	−	0.891968i	0.484871	−	0.874586i	$-0.338867\pi$
$390$		0			0
$391$	−3.49606	−	6.05536i	−0.176804	−	0.306233i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.975006	+	1.68876i	−0.0490579	+	0.0849708i
$396$		0			0
$397$	4.32895	+	7.49796i	0.217264	+	0.376312i	0.953970	−	0.299901i	$-0.0969535\pi$
$397$	4.32895	+	7.49796i	0.217264	+	0.376312i	−0.736707	+	0.676212i	$0.763620\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.5273	−	30.3582i	0.875272	−	1.51602i	0.0187988	−	0.999823i	$-0.494016\pi$
$401$	17.5273	−	30.3582i	0.875272	−	1.51602i	0.856473	−	0.516192i	$-0.172651\pi$
$402$		0			0
$403$	−2.05570	−	3.56058i	−0.102402	−	0.177365i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.76751	+	3.06142i	−0.0876124	+	0.151749i
$408$		0			0
$409$	−13.2336			−0.654361			−0.327180	−	0.944962i	$-0.606098\pi$
$409$	−13.2336			−0.654361			−0.327180	+	0.944962i	$0.606098\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.8214	−	22.2073i	−0.629377	−	1.09011i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.43952	−	7.68947i	−0.216885	−	0.375655i	0.736969	−	0.675926i	$-0.236256\pi$
$419$	−4.43952	−	7.68947i	−0.216885	−	0.375655i	−0.953854	+	0.300271i	$0.902923\pi$
$420$		0			0
$421$	2.00273	−	3.46884i	0.0976073	−	0.169061i	−0.813087	−	0.582143i	$-0.802214\pi$
$421$	2.00273	−	3.46884i	0.0976073	−	0.169061i	0.910694	+	0.413082i	$0.135548\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.78319			−0.377540
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.48548	+	12.9652i	−0.360563	+	0.624513i	−0.988054	−	0.154111i	$-0.950749\pi$
$431$	−7.48548	+	12.9652i	−0.360563	+	0.624513i	0.627491	+	0.778624i	$0.284082\pi$
$432$		0			0
$433$	−15.3215			−0.736304			−0.368152	−	0.929766i	$-0.620009\pi$
$433$	−15.3215			−0.736304			−0.368152	+	0.929766i	$0.620009\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	37.9074			1.81336
$438$		0			0
$439$	12.0731			0.576220			0.288110	−	0.957597i	$-0.406973\pi$
$439$	12.0731			0.576220			0.288110	+	0.957597i	$0.406973\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0579			−0.572887			−0.286444	−	0.958097i	$-0.592473\pi$
$443$	−12.0579			−0.572887			−0.286444	+	0.958097i	$0.592473\pi$
$444$		0			0
$445$	42.2603			2.00333
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.9502			0.799928			0.399964	−	0.916531i	$-0.369023\pi$
$449$	16.9502			0.799928			0.399964	+	0.916531i	$0.369023\pi$
$450$		0			0
$451$	13.5311	−	23.4366i	0.637157	−	1.10359i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.76646			0.269743			0.134872	−	0.990863i	$-0.456938\pi$
$457$	5.76646			0.269743			0.134872	+	0.990863i	$0.456938\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.9138	−	31.0277i	0.834330	−	1.44510i	−0.0602447	−	0.998184i	$-0.519188\pi$
$461$	17.9138	−	31.0277i	0.834330	−	1.44510i	0.894575	−	0.446918i	$-0.147479\pi$
$462$		0			0
$463$	1.53947	+	2.66645i	0.0715455	+	0.123920i	0.899579	−	0.436758i	$-0.143874\pi$
$463$	1.53947	+	2.66645i	0.0715455	+	0.123920i	−0.828033	+	0.560679i	$0.810540\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.42738	+	2.47230i	0.0660515	+	0.114404i	0.897160	−	0.441706i	$-0.145627\pi$
$467$	1.42738	+	2.47230i	0.0660515	+	0.114404i	−0.831108	+	0.556110i	$0.812293\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.5730			−1.31379
$474$		0			0
$475$	21.0981	−	36.5429i	0.968045	−	1.67670i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.18688	−	3.78779i	−0.0999211	−	0.173068i	0.811731	−	0.584032i	$-0.198526\pi$
$479$	−2.18688	−	3.78779i	−0.0999211	−	0.173068i	−0.911652	+	0.410964i	$0.865192\pi$
$480$		0			0
$481$	0.410459	−	0.710936i	0.0187153	−	0.0324159i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	24.3046	+	42.0968i	1.10361	+	1.91152i
$486$		0			0
$487$	15.2678	−	26.4447i	0.691852	−	1.19832i	−0.279378	−	0.960181i	$-0.590128\pi$
$487$	15.2678	−	26.4447i	0.691852	−	1.19832i	0.971230	−	0.238142i	$-0.0765383\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.3502	+	36.9797i	0.963522	+	1.66887i	0.713534	+	0.700621i	$0.247094\pi$
$491$	21.3502	+	36.9797i	0.963522	+	1.66887i	0.249989	+	0.968249i	$0.419573\pi$
$492$		0			0
$493$	1.00561	+	1.74177i	0.0452904	+	0.0784453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	11.3852	−	19.7197i	0.509670	−	0.882774i	−0.490267	−	0.871572i	$-0.663101\pi$
$499$	11.3852	−	19.7197i	0.509670	−	0.882774i	0.999937	−	0.0112020i	$-0.00356578\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.0843			1.07387			0.536933	−	0.843625i	$-0.319583\pi$
$503$	24.0843			1.07387			0.536933	+	0.843625i	$0.319583\pi$
$504$		0			0
$505$	−31.9306			−1.42089
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.2350	−	21.1917i	0.542307	−	0.939304i	−0.456464	−	0.889742i	$-0.650884\pi$
$509$	12.2350	−	21.1917i	0.542307	−	0.939304i	0.998771	−	0.0495618i	$-0.0157825\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.833498	−	1.44366i	−0.0367283	−	0.0636153i
$516$		0			0
$517$	9.47346	+	16.4085i	0.416642	+	0.721646i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.3743	+	33.5573i	−0.848805	+	1.47017i	0.0334709	+	0.999440i	$0.489344\pi$
$521$	−19.3743	+	33.5573i	−0.848805	+	1.47017i	−0.882276	+	0.470733i	$0.843989\pi$
$522$		0			0
$523$	12.8473	+	22.2521i	0.561771	+	0.973016i	0.997342	+	0.0728616i	$0.0232131\pi$
$523$	12.8473	+	22.2521i	0.561771	+	0.973016i	−0.435571	+	0.900154i	$0.643454\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.85313	−	6.67381i	0.167845	−	0.290716i
$528$		0			0
$529$	−8.88593	−	15.3909i	−0.386345	−	0.669169i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.14226	+	5.44255i	−0.136106	+	0.235743i
$534$		0			0
$535$	−18.6425			−0.805987
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.92878	+	8.53690i	0.211905	+	0.367030i	0.952311	−	0.305130i	$-0.0986999\pi$
$541$	4.92878	+	8.53690i	0.211905	+	0.367030i	−0.740406	+	0.672160i	$0.765367\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−27.4032	−	47.4636i	−1.17382	−	2.03312i
$546$		0			0
$547$	−3.94133	+	6.82659i	−0.168519	+	0.291884i	−0.937899	−	0.346907i	$-0.887232\pi$
$547$	−3.94133	+	6.82659i	−0.168519	+	0.291884i	0.769380	+	0.638791i	$0.220565\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.9037			−0.464514
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.1686	+	17.6125i	−0.430857	+	0.746266i	−0.996947	−	0.0780770i	$-0.975122\pi$
$557$	−10.1686	+	17.6125i	−0.430857	+	0.746266i	0.566090	+	0.824343i	$0.308455\pi$
$558$		0			0
$559$	6.63533			0.280644
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.1820			0.850568			0.425284	−	0.905060i	$-0.360174\pi$
$563$	20.1820			0.850568			0.425284	+	0.905060i	$0.360174\pi$
$564$		0			0
$565$	48.1914			2.02743
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.1049			−1.01053			−0.505266	−	0.862964i	$-0.668606\pi$
$569$	−24.1049			−1.01053			−0.505266	+	0.862964i	$0.668606\pi$
$570$		0			0
$571$	6.45527			0.270144			0.135072	−	0.990836i	$-0.456873\pi$
$571$	6.45527			0.270144			0.135072	+	0.990836i	$0.456873\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−45.3846			−1.89267
$576$		0			0
$577$	−9.20385	+	15.9415i	−0.383161	+	0.663654i	−0.991512	−	0.130014i	$-0.958498\pi$
$577$	−9.20385	+	15.9415i	−0.383161	+	0.663654i	0.608351	+	0.793668i	$0.291831\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	29.2978			1.21339
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.8848	+	39.6376i	−0.944557	+	1.63602i	−0.187921	+	0.982184i	$0.560175\pi$
$587$	−22.8848	+	39.6376i	−0.944557	+	1.63602i	−0.756636	+	0.653837i	$0.773158\pi$
$588$		0			0
$589$	20.8895	+	36.1817i	0.860737	+	1.49084i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.71630	+	15.0971i	0.357935	+	0.619962i	0.987616	−	0.156892i	$-0.0501475\pi$
$593$	8.71630	+	15.0971i	0.357935	+	0.619962i	−0.629680	+	0.776854i	$0.716814\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.44444			−0.140736			−0.0703680	−	0.997521i	$-0.522417\pi$
$599$	−3.44444			−0.140736			−0.0703680	+	0.997521i	$0.522417\pi$
$600$		0			0
$601$	12.1666	−	21.0731i	0.496284	−	0.859590i	−0.503706	−	0.863875i	$-0.668031\pi$
$601$	12.1666	−	21.0731i	0.496284	−	0.859590i	0.999991	+	0.00428500i	$0.00136396\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.12655	−	14.0756i	−0.330391	−	0.572254i
$606$		0			0
$607$	9.96073	−	17.2525i	0.404294	−	0.700257i	−0.589945	−	0.807443i	$-0.700851\pi$
$607$	9.96073	−	17.2525i	0.404294	−	0.700257i	0.994239	+	0.107186i	$0.0341840\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.19997	−	3.81045i	−0.0890011	−	0.154154i
$612$		0			0
$613$	−20.3848	+	35.3075i	−0.823334	+	1.42606i	0.0798515	+	0.996807i	$0.474555\pi$
$613$	−20.3848	+	35.3075i	−0.823334	+	1.42606i	−0.903186	+	0.429250i	$0.858778\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.5453	+	19.9970i	0.464796	+	0.805050i	0.999192	−	0.0401838i	$-0.0127944\pi$
$617$	11.5453	+	19.9970i	0.464796	+	0.805050i	−0.534396	+	0.845234i	$0.679461\pi$
$618$		0			0
$619$	22.5584	+	39.0723i	0.906698	+	1.57045i	0.818621	+	0.574334i	$0.194739\pi$
$619$	22.5584	+	39.0723i	0.906698	+	1.57045i	0.0880774	+	0.996114i	$0.471928\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.00960	−	8.67688i	0.200384	−	0.347075i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.53870			0.0613518
$630$		0			0
$631$	−36.7010			−1.46104			−0.730521	−	0.682890i	$-0.760723\pi$
$631$	−36.7010			−1.46104			−0.730521	+	0.682890i	$0.760723\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	36.1067	−	62.5387i	1.43285	−	2.48177i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.12245	+	12.3365i	0.281320	+	0.487261i	0.971710	−	0.236177i	$-0.0758944\pi$
$641$	7.12245	+	12.3365i	0.281320	+	0.487261i	−0.690390	+	0.723437i	$0.742561\pi$
$642$		0			0
$643$	18.0592	+	31.2795i	0.712187	+	1.23354i	0.964035	+	0.265777i	$0.0856285\pi$
$643$	18.0592	+	31.2795i	0.712187	+	1.23354i	−0.251848	+	0.967767i	$0.581038\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.73327	+	6.46622i	−0.146770	+	0.254213i	−0.930032	−	0.367479i	$-0.880221\pi$
$647$	−3.73327	+	6.46622i	−0.146770	+	0.254213i	0.783262	+	0.621692i	$0.213554\pi$
$648$		0			0
$649$	5.60449	+	9.70727i	0.219996	+	0.381043i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.0545	+	34.7354i	−0.784794	+	1.35930i	0.144329	+	0.989530i	$0.453898\pi$
$653$	−20.0545	+	34.7354i	−0.784794	+	1.35930i	−0.929122	+	0.369773i	$0.879436\pi$
$654$		0			0
$655$	2.78192	+	4.81842i	0.108698	+	0.188271i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.96459	−	6.86688i	0.154439	−	0.267496i	−0.778416	−	0.627749i	$-0.783976\pi$
$659$	3.96459	−	6.86688i	0.154439	−	0.267496i	0.932855	+	0.360253i	$0.117310\pi$
$660$		0			0
$661$	22.1286			0.860704			0.430352	−	0.902661i	$-0.358389\pi$
$661$	22.1286			0.860704			0.430352	+	0.902661i	$0.358389\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	5.86382	+	10.1564i	0.227048	+	0.393259i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.5379	+	26.9124i	0.599834	+	1.03894i
$672$		0			0
$673$	6.60773	−	11.4449i	0.254709	−	0.441169i	−0.710107	−	0.704094i	$-0.751354\pi$
$673$	6.60773	−	11.4449i	0.254709	−	0.441169i	0.964817	+	0.262924i	$0.0846869\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.0210			−0.769471			−0.384736	−	0.923027i	$-0.625707\pi$
$677$	−20.0210			−0.769471			−0.384736	+	0.923027i	$0.625707\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.7716	−	18.6569i	0.412162	−	0.713886i	−0.582964	−	0.812498i	$-0.698107\pi$
$683$	10.7716	−	18.6569i	0.412162	−	0.713886i	0.995126	+	0.0986124i	$0.0314404\pi$
$684$		0			0
$685$	−26.6267			−1.01735
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.80365			−0.259198
$690$		0			0
$691$	−43.7386			−1.66389			−0.831947	−	0.554855i	$-0.812774\pi$
$691$	−43.7386			−1.66389			−0.831947	+	0.554855i	$0.812774\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	55.6337			2.11031
$696$		0			0
$697$	−11.7794			−0.446178
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.2894			−0.615244			−0.307622	−	0.951509i	$-0.599533\pi$
$701$	−16.2894			−0.615244			−0.307622	+	0.951509i	$0.599533\pi$
$702$		0			0
$703$	−4.17097	+	7.22434i	−0.157311	+	0.272471i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	36.8100			1.38243			0.691214	−	0.722650i	$-0.257076\pi$
$709$	36.8100			1.38243			0.691214	+	0.722650i	$0.257076\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	22.4680	−	38.9157i	0.841433	−	1.45741i
$714$		0			0
$715$	−2.55709	−	4.42901i	−0.0956298	−	0.165636i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.8787	+	27.5028i	0.592177	+	1.02568i	0.993939	+	0.109936i	$0.0350646\pi$
$719$	15.8787	+	27.5028i	0.592177	+	1.02568i	−0.401762	+	0.915744i	$0.631602\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.0545			0.484830
$726$		0			0
$727$	−2.83596	+	4.91203i	−0.105180	+	0.182177i	−0.913812	−	0.406138i	$-0.866875\pi$
$727$	−2.83596	+	4.91203i	−0.105180	+	0.182177i	0.808632	+	0.588315i	$0.200209\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.21849	+	10.7707i	0.229999	+	0.398370i
$732$		0			0
$733$	−11.9926	+	20.7719i	−0.442958	+	0.767226i	−0.997907	−	0.0646579i	$-0.979404\pi$
$733$	−11.9926	+	20.7719i	−0.442958	+	0.767226i	0.554949	+	0.831884i	$0.312738\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.9382	−	27.6059i	−0.587093	−	1.01687i
$738$		0			0
$739$	0.162996	−	0.282317i	0.00599590	−	0.0103852i	−0.863012	−	0.505184i	$-0.831425\pi$
$739$	0.162996	−	0.282317i	0.00599590	−	0.0103852i	0.869008	+	0.494798i	$0.164758\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.3464	−	23.1166i	−0.489631	−	0.848066i	0.510298	−	0.859998i	$-0.329535\pi$
$743$	−13.3464	−	23.1166i	−0.489631	−	0.848066i	−0.999929	+	0.0119319i	$0.996202\pi$
$744$		0			0
$745$	5.00139	+	8.66266i	0.183237	+	0.317375i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.6978	+	27.1893i	−0.572820	+	0.992153i	0.423455	+	0.905917i	$0.360817\pi$
$751$	−15.6978	+	27.1893i	−0.572820	+	0.992153i	−0.996275	+	0.0862357i	$0.972516\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.8785			−1.16018
$756$		0			0
$757$	−0.144979			−0.00526933			−0.00263467	−	0.999997i	$-0.500839\pi$
$757$	−0.144979			−0.00526933			−0.00263467	+	0.999997i	$0.500839\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.69005	+	11.5875i	−0.242514	+	0.420047i	−0.961430	−	0.275050i	$-0.911305\pi$
$761$	−6.69005	+	11.5875i	−0.242514	+	0.420047i	0.718916	+	0.695097i	$0.244639\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.30150	−	2.25426i	−0.0469944	−	0.0813966i
$768$		0			0
$769$	5.98750	+	10.3707i	0.215915	+	0.373975i	0.953555	−	0.301218i	$-0.0973933\pi$
$769$	5.98750	+	10.3707i	0.215915	+	0.373975i	−0.737640	+	0.675194i	$0.764060\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.9471	−	24.1571i	0.501642	−	0.868869i	−0.498356	−	0.866972i	$-0.666063\pi$
$773$	13.9471	−	24.1571i	0.501642	−	0.868869i	0.999998	−	0.00189699i	$-0.000603830\pi$
$774$		0			0
$775$	−25.0099	−	43.3185i	−0.898384	−	1.55605i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31.9308	−	55.3057i	1.14404	−	1.98153i
$780$		0			0
$781$	6.20626	+	10.7495i	0.222077	+	0.384649i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.2489	−	38.5363i	0.794099	−	1.37542i
$786$		0			0
$787$	0.104507			0.00372527			0.00186264	−	0.999998i	$-0.499407\pi$
$787$	0.104507			0.00372527			0.00186264	+	0.999998i	$0.499407\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.60827	−	6.24971i	−0.128133	−	0.221934i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17.5235	−	30.3516i	−0.620715	−	1.07511i	−0.989353	−	0.145537i	$-0.953509\pi$
$797$	−17.5235	−	30.3516i	−0.620715	−	1.07511i	0.368638	−	0.929573i	$-0.379824\pi$
$798$		0			0
$799$	4.12353	−	7.14216i	0.145880	−	0.252671i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.9152			−0.773369
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.1259	+	38.3233i	−0.777907	+	1.34737i	0.155239	+	0.987877i	$0.450385\pi$
$809$	−22.1259	+	38.3233i	−0.777907	+	1.34737i	−0.933146	+	0.359497i	$0.882948\pi$
$810$		0			0
$811$	0.903637			0.0317310			0.0158655	−	0.999874i	$-0.494950\pi$
$811$	0.903637			0.0317310			0.0158655	+	0.999874i	$0.494950\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−49.9943			−1.75123
$816$		0			0
$817$	−67.4264			−2.35895
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−46.3565			−1.61785			−0.808927	−	0.587910i	$-0.799951\pi$
$821$	−46.3565			−1.61785			−0.808927	+	0.587910i	$0.799951\pi$
$822$		0			0
$823$	−30.9829			−1.08000			−0.539998	−	0.841666i	$-0.681575\pi$
$823$	−30.9829			−1.08000			−0.539998	+	0.841666i	$0.681575\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.0923			−0.872544			−0.436272	−	0.899815i	$-0.643701\pi$
$827$	−25.0923			−0.872544			−0.436272	+	0.899815i	$0.643701\pi$
$828$		0			0
$829$	21.1853	−	36.6941i	0.735798	−	1.27444i	−0.218575	−	0.975820i	$-0.570141\pi$
$829$	21.1853	−	36.6941i	0.735798	−	1.27444i	0.954373	−	0.298619i	$-0.0965258\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−16.5825			−0.573860
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.36843	+	2.37020i	−0.0472435	+	0.0818282i	−0.888680	−	0.458528i	$-0.848377\pi$
$839$	−1.36843	+	2.37020i	−0.0472435	+	0.0818282i	0.841437	+	0.540356i	$0.181710\pi$
$840$		0			0
$841$	12.8133	+	22.1933i	0.441839	+	0.765287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−22.0237	−	38.1461i	−0.757637	−	1.31227i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.97230			0.307566
$852$		0			0
$853$	−4.59273	+	7.95485i	−0.157252	+	0.272369i	−0.933877	−	0.357595i	$-0.883597\pi$
$853$	−4.59273	+	7.95485i	−0.157252	+	0.272369i	0.776625	+	0.629964i	$0.216930\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.06106	−	10.4981i	−0.207042	−	0.358607i	0.743739	−	0.668470i	$-0.233050\pi$
$857$	−6.06106	−	10.4981i	−0.207042	−	0.358607i	−0.950781	+	0.309862i	$0.899717\pi$
$858$		0			0
$859$	3.41626	−	5.91714i	0.116561	−	0.201890i	−0.801841	−	0.597537i	$-0.796146\pi$
$859$	3.41626	−	5.91714i	0.116561	−	0.201890i	0.918403	+	0.395647i	$0.129480\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.2595	+	45.4827i	0.893882	+	1.54825i	0.835182	+	0.549974i	$0.185362\pi$
$863$	26.2595	+	45.4827i	0.893882	+	1.54825i	0.0587005	+	0.998276i	$0.481304\pi$
$864$		0			0
$865$	−43.6981	+	75.6873i	−1.48578	+	2.57345i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.704930	−	1.22097i	−0.0239131	−	0.0414187i
$870$		0			0
$871$	3.70125	+	6.41074i	0.125412	+	0.217220i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−16.0782	+	27.8482i	−0.542922	+	0.940368i	0.455813	+	0.890076i	$0.349349\pi$
$877$	−16.0782	+	27.8482i	−0.542922	+	0.940368i	−0.998735	+	0.0502923i	$0.983985\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.9101			0.569715			0.284858	−	0.958570i	$-0.408054\pi$
$881$	16.9101			0.569715			0.284858	+	0.958570i	$0.408054\pi$
$882$		0			0
$883$	−13.9999			−0.471135			−0.235567	−	0.971858i	$-0.575695\pi$
$883$	−13.9999			−0.471135			−0.235567	+	0.971858i	$0.575695\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	22.2283	−	38.5005i	0.746352	−	1.29272i	−0.203208	−	0.979136i	$-0.565137\pi$
$887$	22.2283	−	38.5005i	0.746352	−	1.29272i	0.949560	−	0.313585i	$-0.101530\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	22.3555	+	38.7208i	0.748097	+	1.29574i
$894$		0			0
$895$	−11.8713	−	20.5617i	−0.396814	−	0.687303i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.46271	+	11.1937i	−0.215543	+	0.373332i
$900$		0			0
$901$	−6.37624	−	11.0440i	−0.212423	−	0.367928i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23.4413	+	40.6015i	−0.779215	+	1.34964i
$906$		0			0
$907$	8.22392	+	14.2442i	0.273071	+	0.472972i	0.969647	−	0.244511i	$-0.0786273\pi$
$907$	8.22392	+	14.2442i	0.273071	+	0.472972i	−0.696576	+	0.717483i	$0.745294\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.1577	−	43.5745i	0.833513	−	1.44369i	−0.0617228	−	0.998093i	$-0.519659\pi$
$911$	25.1577	−	43.5745i	0.833513	−	1.44369i	0.895236	−	0.445593i	$-0.147007\pi$
$912$		0			0
$913$	18.5397			0.613576
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−29.2722	−	50.7009i	−0.965600	−	1.67247i	−0.707994	−	0.706219i	$-0.750400\pi$
$919$	−29.2722	−	50.7009i	−0.965600	−	1.67247i	−0.257606	−	0.966250i	$-0.582934\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.44124	−	2.49630i	−0.0474391	−	0.0821669i
$924$		0			0
$925$	4.99370	−	8.64933i	0.164192	−	0.284388i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.637344			−0.0209106			−0.0104553	−	0.999945i	$-0.503328\pi$
$929$	−0.637344			−0.0209106			−0.0104553	+	0.999945i	$0.503328\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.79291	−	8.30156i	0.156745	−	0.271490i
$936$		0			0
$937$	19.0780			0.623250			0.311625	−	0.950205i	$-0.399127\pi$
$937$	19.0780			0.623250			0.311625	+	0.950205i	$0.399127\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−29.6306			−0.965931			−0.482965	−	0.875639i	$-0.660440\pi$
$941$	−29.6306			−0.965931			−0.482965	+	0.875639i	$0.660440\pi$
$942$		0			0
$943$	−68.6872			−2.23676
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.3362			−1.11578			−0.557888	−	0.829916i	$-0.688388\pi$
$947$	−34.3362			−1.11578			−0.557888	+	0.829916i	$0.688388\pi$
$948$		0			0
$949$	5.08923			0.165203
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−53.6361			−1.73744			−0.868722	−	0.495301i	$-0.835058\pi$
$953$	−53.6361			−1.73744			−0.868722	+	0.495301i	$0.835058\pi$
$954$		0			0
$955$	−24.7163	+	42.8099i	−0.799800	+	1.38529i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	18.5254			0.597595
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11.8049	+	20.4467i	−0.380014	+	0.658204i
$966$		0			0
$967$	14.5629	+	25.2236i	0.468310	+	0.811136i	0.999344	−	0.0362139i	$-0.0115298\pi$
$967$	14.5629	+	25.2236i	0.468310	+	0.811136i	−0.531034	+	0.847350i	$0.678196\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.9191	−	20.6446i	−0.382503	−	0.662515i	0.608916	−	0.793235i	$-0.291605\pi$
$971$	−11.9191	−	20.6446i	−0.382503	−	0.662515i	−0.991419	+	0.130719i	$0.958271\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38.1683			−1.22111			−0.610556	−	0.791973i	$-0.709054\pi$
$977$	−38.1683			−1.22111			−0.610556	+	0.791973i	$0.709054\pi$
$978$		0			0
$979$	−15.2771	+	26.4607i	−0.488258	+	0.845688i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.78769	+	6.56046i	0.120808	+	0.209246i	0.920087	−	0.391715i	$-0.128118\pi$
$983$	3.78769	+	6.56046i	0.120808	+	0.209246i	−0.799278	+	0.600961i	$0.794785\pi$
$984$		0			0
$985$	−32.2363	+	55.8349i	−1.02713	+	1.77905i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.2607	+	62.8054i	1.15302	+	1.99709i
$990$		0			0
$991$	4.68952	−	8.12248i	0.148967	−	0.258019i	−0.781879	−	0.623431i	$-0.785738\pi$
$991$	4.68952	−	8.12248i	0.148967	−	0.258019i	0.930846	+	0.365411i	$0.119072\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.2219	−	50.6138i	−0.926397	−	1.60457i
$996$		0			0
$997$	4.21829	+	7.30629i	0.133595	+	0.231393i	0.925060	−	0.379822i	$-0.124015\pi$
$997$	4.21829	+	7.30629i	0.133595	+	0.231393i	−0.791465	+	0.611214i	$0.790681\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.i.j.2125.11		24
3.2	odd	2		1764.2.i.j.1537.9		24
7.2	even	3		5292.2.l.j.3313.2		24
7.3	odd	6		5292.2.j.i.3529.2		24
7.4	even	3		5292.2.j.i.3529.11		24
7.5	odd	6		5292.2.l.j.3313.11		24
7.6	odd	2	inner	5292.2.i.j.2125.2		24
9.4	even	3		5292.2.l.j.361.2		24
9.5	odd	6		1764.2.l.j.949.9		24
21.2	odd	6		1764.2.l.j.961.9		24
21.5	even	6		1764.2.l.j.961.4		24
21.11	odd	6		1764.2.j.i.1177.1	yes	24
21.17	even	6		1764.2.j.i.1177.12	yes	24
21.20	even	2		1764.2.i.j.1537.4		24
63.4	even	3		5292.2.j.i.1765.11		24
63.5	even	6		1764.2.i.j.373.4		24
63.13	odd	6		5292.2.l.j.361.11		24
63.23	odd	6		1764.2.i.j.373.9		24
63.31	odd	6		5292.2.j.i.1765.2		24
63.32	odd	6		1764.2.j.i.589.1	✓	24
63.40	odd	6	inner	5292.2.i.j.1549.2		24
63.41	even	6		1764.2.l.j.949.4		24
63.58	even	3	inner	5292.2.i.j.1549.11		24
63.59	even	6		1764.2.j.i.589.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.4		24	63.5	even	6
1764.2.i.j.373.9		24	63.23	odd	6
1764.2.i.j.1537.4		24	21.20	even	2
1764.2.i.j.1537.9		24	3.2	odd	2
1764.2.j.i.589.1	✓	24	63.32	odd	6
1764.2.j.i.589.12	yes	24	63.59	even	6
1764.2.j.i.1177.1	yes	24	21.11	odd	6
1764.2.j.i.1177.12	yes	24	21.17	even	6
1764.2.l.j.949.4		24	63.41	even	6
1764.2.l.j.949.9		24	9.5	odd	6
1764.2.l.j.961.4		24	21.5	even	6
1764.2.l.j.961.9		24	21.2	odd	6
5292.2.i.j.1549.2		24	63.40	odd	6	inner
5292.2.i.j.1549.11		24	63.58	even	3	inner
5292.2.i.j.2125.2		24	7.6	odd	2	inner
5292.2.i.j.2125.11		24	1.1	even	1	trivial
5292.2.j.i.1765.2		24	63.31	odd	6
5292.2.j.i.1765.11		24	63.4	even	3
5292.2.j.i.3529.2		24	7.3	odd	6
5292.2.j.i.3529.11		24	7.4	even	3
5292.2.l.j.361.2		24	9.4	even	3
5292.2.l.j.361.11		24	63.13	odd	6
5292.2.l.j.3313.2		24	7.2	even	3
5292.2.l.j.3313.11		24	7.5	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73981 - 3.01343i) q^{5} +O(q^{10})\)	\(q+(1.73981 - 3.01343i) q^{5} +(1.25788 + 2.17871i) q^{11} +(-0.292110 - 0.505949i) q^{13} +(0.547519 - 0.948331i) q^{17} +(2.96834 + 5.14132i) q^{19} +(3.19264 - 5.52982i) q^{23} +(-3.55384 - 6.15544i) q^{25} +(-0.918333 + 1.59060i) q^{29} +7.03743 q^{31} +(0.702576 + 1.21690i) q^{37} +(-5.37855 - 9.31593i) q^{41} +(-5.67879 + 9.83596i) q^{43} +7.53129 q^{47} +(5.82285 - 10.0855i) q^{53} +8.75386 q^{55} +4.45551 q^{59} +12.3524 q^{61} -2.03286 q^{65} -12.6707 q^{67} +4.93390 q^{71} +(-4.35558 + 7.54408i) q^{73} -0.560411 q^{79} +(3.68472 - 6.38212i) q^{83} +(-1.90515 - 3.29982i) q^{85} +(6.07256 + 10.5180i) q^{89} +20.6573 q^{95} +(-6.98486 + 12.0981i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} - 72 q^{65} - 24 q^{67} - 48 q^{71} - 24 q^{79} + 12 q^{85} + 64 q^{95}+O(q^{100}) \) 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 - 72 * q^65 - 24 * q^67 - 48 * q^71 - 24 * q^79 + 12 * q^85 + 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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