LMFDB - Embedded newform 5292.2.l.j.3313.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(361,5292)]

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, 4]))

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

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

chi := DirichletCharacter("5292.361");

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.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:	$42.2568327497$
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			3313.11
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.j.361.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.47961 q^{5} +O(q^{10})$	$q+3.47961 q^{5} -2.51576 q^{11} +(0.292110 + 0.505949i) q^{13} +(-0.547519 - 0.948331i) q^{17} +(-2.96834 + 5.14132i) q^{19} -6.38528 q^{23} +7.10769 q^{25} +(-0.918333 + 1.59060i) q^{29} +(3.51872 - 6.09459i) q^{31} +(0.702576 - 1.21690i) q^{37} +(5.37855 + 9.31593i) q^{41} +(-5.67879 + 9.83596i) q^{43} +(3.76565 + 6.52229i) q^{47} +(5.82285 + 10.0855i) q^{53} -8.75386 q^{55} +(2.22775 - 3.85858i) q^{59} +(6.17622 + 10.6975i) q^{61} +(1.01643 + 1.76051i) q^{65} +(6.33536 - 10.9732i) q^{67} +4.93390 q^{71} +(4.35558 + 7.54408i) q^{73} +(0.280206 + 0.485330i) q^{79} +(-3.68472 + 6.38212i) q^{83} +(-1.90515 - 3.29982i) q^{85} +(-6.07256 + 10.5180i) q^{89} +(-10.3287 + 17.8898i) q^{95} +(6.98486 - 12.0981i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) $ 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	3.47961			1.55613			0.778065	−	0.628184i	$-0.216202\pi$
$5$	3.47961			1.55613			0.778065	+	0.628184i	$0.216202\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.51576			−0.758530			−0.379265	−	0.925288i	$-0.623823\pi$
$11$	−2.51576			−0.758530			−0.379265	+	0.925288i	$0.623823\pi$
$12$		0			0
$13$	0.292110	+	0.505949i	0.0810167	+	0.140325i	0.903687	−	0.428194i	$-0.140850\pi$
$13$	0.292110	+	0.505949i	0.0810167	+	0.140325i	−0.822670	+	0.568519i	$0.807517\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.547519	−	0.948331i	−0.132793	−	0.230004i	0.791959	−	0.610574i	$-0.209061\pi$
$17$	−0.547519	−	0.948331i	−0.132793	−	0.230004i	−0.924752	+	0.380570i	$0.875728\pi$
$18$		0			0
$19$	−2.96834	+	5.14132i	−0.680984	+	1.17950i	0.293696	+	0.955899i	$0.405115\pi$
$19$	−2.96834	+	5.14132i	−0.680984	+	1.17950i	−0.974681	+	0.223601i	$0.928219\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.38528			−1.33142			−0.665712	−	0.746209i	$-0.731872\pi$
$23$	−6.38528			−1.33142			−0.665712	+	0.746209i	$0.731872\pi$
$24$		0			0
$25$	7.10769			1.42154
$26$		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
$31$	3.51872	−	6.09459i	0.631980	−	1.09462i	−0.355166	−	0.934803i	$-0.615576\pi$
$31$	3.51872	−	6.09459i	0.631980	−	1.09462i	0.987146	−	0.159818i	$-0.0510909\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.702576	−	1.21690i	0.115503	−	0.200057i	−0.802478	−	0.596682i	$-0.796485\pi$
$37$	0.702576	−	1.21690i	0.115503	−	0.200057i	0.917981	+	0.396625i	$0.129819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.37855	+	9.31593i	0.839989	+	1.45490i	0.889903	+	0.456150i	$0.150772\pi$
$41$	5.37855	+	9.31593i	0.839989	+	1.45490i	−0.0499141	+	0.998754i	$0.515895\pi$
$42$		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
$47$	3.76565	+	6.52229i	0.549276	+	0.951374i	0.998324	+	0.0578664i	$0.0184298\pi$
$47$	3.76565	+	6.52229i	0.549276	+	0.951374i	−0.449048	+	0.893507i	$0.648237\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.82285	+	10.0855i	0.799830	+	1.38535i	0.919727	+	0.392560i	$0.128410\pi$
$53$	5.82285	+	10.0855i	0.799830	+	1.38535i	−0.119897	+	0.992786i	$0.538256\pi$
$54$		0			0
$55$	−8.75386			−1.18037
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.22775	−	3.85858i	0.290029	−	0.502345i	−0.683787	−	0.729681i	$-0.739668\pi$
$59$	2.22775	−	3.85858i	0.290029	−	0.502345i	0.973816	+	0.227337i	$0.0730018\pi$
$60$		0			0
$61$	6.17622	+	10.6975i	0.790784	+	1.36968i	0.925482	+	0.378792i	$0.123660\pi$
$61$	6.17622	+	10.6975i	0.790784	+	1.36968i	−0.134698	+	0.990887i	$0.543006\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.01643	+	1.76051i	0.126072	+	0.218364i
$66$		0			0
$67$	6.33536	−	10.9732i	0.773988	−	1.34059i	−0.161374	−	0.986893i	$-0.551593\pi$
$67$	6.33536	−	10.9732i	0.773988	−	1.34059i	0.935362	−	0.353693i	$-0.115074\pi$
$68$		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	0.999936	+	0.0113320i	$0.00360715\pi$
$73$	4.35558	+	7.54408i	0.509782	+	0.882968i	−0.490154	+	0.871636i	$0.663060\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.280206	+	0.485330i	0.0315256	+	0.0546039i	0.881358	−	0.472450i	$-0.156630\pi$
$79$	0.280206	+	0.485330i	0.0315256	+	0.0546039i	−0.849832	+	0.527053i	$0.823297\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.68472	+	6.38212i	−0.404451	+	0.700529i	−0.994257	−	0.107015i	$-0.965871\pi$
$83$	−3.68472	+	6.38212i	−0.404451	+	0.700529i	0.589807	+	0.807544i	$0.299204\pi$
$84$		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	0.340913	+	0.940095i	$0.389264\pi$
$89$	−6.07256	+	10.5180i	−0.643690	+	1.11490i	−0.984602	+	0.174808i	$0.944069\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.3287	+	17.8898i	−1.05970	+	1.83545i
$96$		0			0
$97$	6.98486	−	12.0981i	0.709205	−	1.22838i	−0.255947	−	0.966691i	$-0.582387\pi$
$97$	6.98486	−	12.0981i	0.709205	−	1.22838i	0.965152	−	0.261688i	$-0.0842793\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.17649			−0.913095			−0.456548	−	0.889699i	$-0.650914\pi$
$101$	−9.17649			−0.913095			−0.456548	+	0.889699i	$0.650914\pi$
$102$		0			0
$103$	0.479075			0.0472047			0.0236024	−	0.999721i	$-0.492486\pi$
$103$	0.479075			0.0472047			0.0236024	+	0.999721i	$0.492486\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.67882	+	4.63986i	−0.258972	+	0.448552i	−0.965967	−	0.258666i	$-0.916717\pi$
$107$	−2.67882	+	4.63986i	−0.258972	+	0.448552i	0.706995	+	0.707219i	$0.250050\pi$
$108$		0			0
$109$	7.87535	+	13.6405i	0.754322	+	1.30652i	0.945711	+	0.325009i	$0.105367\pi$
$109$	7.87535	+	13.6405i	0.754322	+	1.30652i	−0.191389	+	0.981514i	$0.561299\pi$
$110$		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
$115$	−22.2183			−2.07187
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.67095			−0.424632
$122$		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
$131$	−1.59898			−0.139704			−0.0698518	−	0.997557i	$-0.522253\pi$
$131$	−1.59898			−0.139704			−0.0698518	+	0.997557i	$0.522253\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.65220			0.653771			0.326886	−	0.945064i	$-0.394001\pi$
$137$	7.65220			0.653771			0.326886	+	0.945064i	$0.394001\pi$
$138$		0			0
$139$	−7.99424	−	13.8464i	−0.678062	−	1.17444i	−0.975564	−	0.219717i	$-0.929487\pi$
$139$	−7.99424	−	13.8464i	−0.678062	−	1.17444i	0.297501	−	0.954721i	$-0.403847\pi$
$140$		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
$149$	2.87468			0.235503			0.117752	−	0.993043i	$-0.462431\pi$
$149$	2.87468			0.235503			0.117752	+	0.993043i	$0.462431\pi$
$150$		0			0
$151$	9.16152			0.745554			0.372777	−	0.927921i	$-0.378406\pi$
$151$	9.16152			0.745554			0.372777	+	0.927921i	$0.378406\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.2438	−	21.2068i	0.983443	−	1.70337i
$156$		0			0
$157$	6.39409	−	11.0749i	0.510304	−	0.883873i	−0.489625	−	0.871933i	$-0.662866\pi$
$157$	6.39409	−	11.0749i	0.510304	−	0.883873i	0.999929	−	0.0119393i	$-0.00380048\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−7.18390	+	12.4429i	−0.562686	+	0.974601i	0.434575	+	0.900636i	$0.356899\pi$
$163$	−7.18390	+	12.4429i	−0.562686	+	0.974601i	−0.997261	+	0.0739653i	$0.976435\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.38280	+	4.12714i	0.184387	+	0.319367i	0.943370	−	0.331743i	$-0.107637\pi$
$167$	2.38280	+	4.12714i	0.184387	+	0.319367i	−0.758983	+	0.651111i	$0.774303\pi$
$168$		0			0
$169$	6.32934	−	10.9627i	0.486873	−	0.843288i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.5583	−	21.7517i	−0.954792	−	1.65375i	−0.734844	−	0.678237i	$-0.762744\pi$
$173$	−12.5583	−	21.7517i	−0.954792	−	1.65375i	−0.219948	−	0.975512i	$-0.570589\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.41168	+	5.90920i	0.255001	+	0.441675i	0.964896	−	0.262633i	$-0.0845909\pi$
$179$	3.41168	+	5.90920i	0.255001	+	0.441675i	−0.709895	+	0.704308i	$0.751258\pi$
$180$		0			0
$181$	13.4735			1.00148			0.500739	−	0.865598i	$-0.333062\pi$
$181$	13.4735			1.00148			0.500739	+	0.865598i	$0.333062\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.44469	−	4.23433i	0.179737	−	0.311314i
$186$		0			0
$187$	1.37743	+	2.38577i	0.100727	+	0.174465i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.10318	+	12.3031i	0.513968	+	0.890218i	0.999869	+	0.0162045i	$0.00515827\pi$
$191$	7.10318	+	12.3031i	0.513968	+	0.890218i	−0.485901	+	0.874014i	$0.661508\pi$
$192$		0			0
$193$	3.39260	−	5.87616i	0.244205	−	0.422975i	−0.717703	−	0.696349i	$-0.754806\pi$
$193$	3.39260	−	5.87616i	0.244205	−	0.422975i	0.961908	+	0.273374i	$0.0881397\pi$
$194$		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	−0.993501	−	0.113819i	$-0.963691\pi$
$199$	−8.39804	−	14.5458i	−0.595321	−	1.03113i	0.398180	−	0.917307i	$-0.369642\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	18.7153	+	32.4158i	1.30713	+	2.26402i
$206$		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
$221$	0.319872	−	0.554034i	0.0215169	−	0.0372683i
$222$		0			0
$223$	0.495791	−	0.858736i	0.0332006	−	0.0575052i	−0.848948	−	0.528477i	$-0.822763\pi$
$223$	0.495791	−	0.858736i	0.0332006	−	0.0575052i	0.882148	+	0.470972i	$0.156097\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.93134			−0.194560			−0.0972799	−	0.995257i	$-0.531014\pi$
$227$	−2.93134			−0.194560			−0.0972799	+	0.995257i	$0.531014\pi$
$228$		0			0
$229$	−4.38401			−0.289704			−0.144852	−	0.989453i	$-0.546271\pi$
$229$	−4.38401			−0.289704			−0.144852	+	0.989453i	$0.546271\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.543158	+	0.940778i	−0.0355835	+	0.0616324i	−0.883269	−	0.468867i	$-0.844662\pi$
$233$	−0.543158	+	0.940778i	−0.0355835	+	0.0616324i	0.847685	+	0.530500i	$0.177996\pi$
$234$		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
$241$	−12.9254			−0.832599			−0.416300	−	0.909227i	$-0.636673\pi$
$241$	−12.9254			−0.832599			−0.416300	+	0.909227i	$0.636673\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.46833			−0.220685
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.6654			−1.24127			−0.620634	−	0.784101i	$-0.713125\pi$
$251$	−19.6654			−1.24127			−0.620634	+	0.784101i	$0.713125\pi$
$252$		0			0
$253$	16.0638			1.00992
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−25.3597			−1.58190			−0.790948	−	0.611884i	$-0.790412\pi$
$257$	−25.3597			−1.58190			−0.790948	+	0.611884i	$0.790412\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.87595			−0.547315			−0.273657	−	0.961827i	$-0.588233\pi$
$263$	−8.87595			−0.547315			−0.273657	+	0.961827i	$0.588233\pi$
$264$		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	−0.997656	−	0.0684319i	$-0.978200\pi$
$269$	−9.15338	−	15.8541i	−0.558092	−	0.966643i	0.439564	−	0.898211i	$-0.355133\pi$
$270$		0			0
$271$	2.34465	−	4.06106i	0.142427	−	0.246692i	−0.785983	−	0.618248i	$-0.787843\pi$
$271$	2.34465	−	4.06106i	0.142427	−	0.246692i	0.928410	+	0.371557i	$0.121176\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−17.8812			−1.07828
$276$		0			0
$277$	5.65614			0.339845			0.169922	−	0.985457i	$-0.445648\pi$
$277$	5.65614			0.339845			0.169922	+	0.985457i	$0.445648\pi$
$278$		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
$283$	−11.9053	+	20.6206i	−0.707697	+	1.22577i	0.258013	+	0.966141i	$0.416932\pi$
$283$	−11.9053	+	20.6206i	−0.707697	+	1.22577i	−0.965710	+	0.259625i	$0.916401\pi$
$284$		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	$-0.960411\pi$
$293$	−10.3315	−	17.8946i	−0.603570	−	1.04541i	0.388706	−	0.921362i	$-0.372922\pi$
$294$		0			0
$295$	7.75171	−	13.4264i	0.451322	−	0.781713i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.86521	−	3.23063i	−0.107868	−	0.186832i
$300$		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	$-0.610506\pi$
$307$	−11.9227			−0.680464			−0.340232	+	0.940342i	$0.610506\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.56635	−	7.90916i	0.258934	−	0.448487i	−0.707022	−	0.707191i	$-0.749962\pi$
$311$	4.56635	−	7.90916i	0.258934	−	0.448487i	0.965957	+	0.258704i	$0.0832954\pi$
$312$		0			0
$313$	6.91980	+	11.9854i	0.391130	+	0.677457i	0.992599	−	0.121439i	$-0.0387509\pi$
$313$	6.91980	+	11.9854i	0.391130	+	0.677457i	−0.601469	+	0.798896i	$0.705418\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.915786	+	1.58619i	0.0514357	+	0.0890892i	0.890597	−	0.454794i	$-0.150287\pi$
$317$	0.915786	+	1.58619i	0.0514357	+	0.0890892i	−0.839161	+	0.543883i	$0.816954\pi$
$318$		0			0
$319$	2.31031	−	4.00157i	0.129352	−	0.224045i
$320$		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
$331$	0.103132	+	0.178630i	0.00566864	+	0.00981838i	0.868846	−	0.495083i	$-0.164862\pi$
$331$	0.103132	+	0.178630i	0.00566864	+	0.00981838i	−0.863177	+	0.504901i	$0.831529\pi$
$332$		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
$347$	−1.60907	+	2.78698i	−0.0863792	+	0.149613i	−0.905978	−	0.423325i	$-0.860863\pi$
$347$	−1.60907	+	2.78698i	−0.0863792	+	0.149613i	0.819599	+	0.572938i	$0.194196\pi$
$348$		0			0
$349$	7.04006	−	12.1937i	0.376846	−	0.652716i	−0.613756	−	0.789496i	$-0.710342\pi$
$349$	7.04006	−	12.1937i	0.376846	−	0.652716i	0.990601	+	0.136780i	$0.0436753\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.36930			−0.179330			−0.0896648	−	0.995972i	$-0.528580\pi$
$353$	−3.36930			−0.179330			−0.0896648	+	0.995972i	$0.528580\pi$
$354$		0			0
$355$	17.1681			0.911186
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.28913	+	12.6251i	−0.384706	+	0.666330i	−0.991728	−	0.128355i	$-0.959030\pi$
$359$	−7.28913	+	12.6251i	−0.384706	+	0.666330i	0.607023	+	0.794685i	$0.292364\pi$
$360$		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
$367$	0.736734			0.0384572			0.0192286	−	0.999815i	$-0.493879\pi$
$367$	0.736734			0.0384572			0.0192286	+	0.999815i	$0.493879\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	28.9029			1.49653			0.748267	−	0.663397i	$-0.230886\pi$
$373$	28.9029			1.49653			0.748267	+	0.663397i	$0.230886\pi$
$374$		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
$383$	15.4055			0.787182			0.393591	−	0.919286i	$-0.371233\pi$
$383$	15.4055			0.787182			0.393591	+	0.919286i	$0.371233\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.3139			1.02996			0.514978	−	0.857204i	$-0.327800\pi$
$389$	20.3139			1.02996			0.514978	+	0.857204i	$0.327800\pi$
$390$		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	$-0.903047\pi$
$397$	−4.32895	+	7.49796i	−0.217264	+	0.376312i	0.736707	+	0.676212i	$0.236380\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−35.0546			−1.75054			−0.875272	−	0.483631i	$-0.839318\pi$
$401$	−35.0546			−1.75054			−0.875272	+	0.483631i	$0.839318\pi$
$402$		0			0
$403$	4.11141			0.204804
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.76751	+	3.06142i	−0.0876124	+	0.151749i
$408$		0			0
$409$	−6.61681	+	11.4607i	−0.327180	+	0.566693i	−0.981951	−	0.189135i	$-0.939432\pi$
$409$	−6.61681	+	11.4607i	−0.327180	+	0.566693i	0.654771	+	0.755827i	$0.272765\pi$
$410$		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	0.953854	−	0.300271i	$-0.0970771\pi$
$419$	4.43952	+	7.68947i	0.216885	+	0.375655i	−0.736969	+	0.675926i	$0.763744\pi$
$420$		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
$425$	−3.89160	−	6.74044i	−0.188770	−	0.326959i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.48548	−	12.9652i	−0.360563	−	0.624513i	0.627491	−	0.778624i	$-0.284082\pi$
$431$	−7.48548	−	12.9652i	−0.360563	−	0.624513i	−0.988054	+	0.154111i	$0.950749\pi$
$432$		0			0
$433$	15.3215			0.736304			0.368152	−	0.929766i	$-0.379991\pi$
$433$	15.3215			0.736304			0.368152	+	0.929766i	$0.379991\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.9537	−	32.8288i	0.906679	−	1.57041i
$438$		0			0
$439$	6.03657	+	10.4556i	0.288110	+	0.499021i	0.973359	−	0.229288i	$-0.0736399\pi$
$439$	6.03657	+	10.4556i	0.288110	+	0.499021i	−0.685249	+	0.728309i	$0.740307\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.02894	+	10.4424i	0.286444	+	0.496135i	0.972958	−	0.230981i	$-0.0741935\pi$
$443$	6.02894	+	10.4424i	0.286444	+	0.496135i	−0.686515	+	0.727116i	$0.740860\pi$
$444$		0			0
$445$	−21.1301	+	36.5985i	−1.00166	+	1.73493i
$446$		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
$457$	−2.88323	−	4.99390i	−0.134872	−	0.233605i	0.790677	−	0.612234i	$-0.209729\pi$
$457$	−2.88323	−	4.99390i	−0.134872	−	0.233605i	−0.925548	+	0.378629i	$0.876396\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.9138	+	31.0277i	−0.834330	+	1.44510i	0.0602447	+	0.998184i	$0.480812\pi$
$461$	−17.9138	+	31.0277i	−0.834330	+	1.44510i	−0.894575	+	0.446918i	$0.852521\pi$
$462$		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	$-0.854373\pi$
$467$	−1.42738	+	2.47230i	−0.0660515	+	0.114404i	0.831108	+	0.556110i	$0.187707\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.2865	−	24.7449i	0.656893	−	1.13777i
$474$		0			0
$475$	−21.0981	+	36.5429i	−0.968045	+	1.67670i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.37376			−0.199842			−0.0999211	−	0.994995i	$-0.531859\pi$
$479$	−4.37376			−0.199842			−0.0999211	+	0.994995i	$0.531859\pi$
$480$		0			0
$481$	0.820918			0.0374307
$482$		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	0.971230	+	0.238142i	$0.0765383\pi$
$487$	15.2678	+	26.4447i	0.691852	+	1.19832i	−0.279378	+	0.960181i	$0.590128\pi$
$488$		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
$493$	2.01122			0.0905808
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−22.7703			−1.01934			−0.509670	−	0.860370i	$-0.670232\pi$
$499$	−22.7703			−1.01934			−0.509670	+	0.860370i	$0.670232\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0843			−1.07387			−0.536933	−	0.843625i	$-0.680417\pi$
$503$	−24.0843			−1.07387			−0.536933	+	0.843625i	$0.680417\pi$
$504$		0			0
$505$	−31.9306			−1.42089
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.4700			1.08461			0.542307	−	0.840180i	$-0.317551\pi$
$509$	24.4700			1.08461			0.542307	+	0.840180i	$0.317551\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.66700			0.0734566
$516$		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	0.882276	+	0.470733i	$0.156011\pi$
$521$	19.3743	+	33.5573i	0.848805	+	1.47017i	−0.0334709	+	0.999440i	$0.510656\pi$
$522$		0			0
$523$	−12.8473	+	22.2521i	−0.561771	+	0.973016i	0.435571	+	0.900154i	$0.356546\pi$
$523$	−12.8473	+	22.2521i	−0.561771	+	0.973016i	−0.997342	+	0.0728616i	$0.976787\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.70626			−0.335690
$528$		0			0
$529$	17.7719			0.772689
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.14226	+	5.44255i	−0.136106	+	0.235743i
$534$		0			0
$535$	−9.32127	+	16.1449i	−0.402993	+	0.698005i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.92878	−	8.53690i	0.211905	−	0.367030i	−0.740406	−	0.672160i	$-0.765367\pi$
$541$	4.92878	−	8.53690i	0.211905	−	0.367030i	0.952311	+	0.305130i	$0.0986999\pi$
$542$		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
$551$	−5.45185	−	9.44289i	−0.232257	−	0.402281i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.1686	−	17.6125i	−0.430857	−	0.746266i	0.566090	−	0.824343i	$-0.308455\pi$
$557$	−10.1686	−	17.6125i	−0.430857	−	0.746266i	−0.996947	+	0.0780770i	$0.975122\pi$
$558$		0			0
$559$	−6.63533			−0.280644
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.0910	−	17.4781i	0.425284	−	0.736614i	−0.571163	−	0.820837i	$-0.693507\pi$
$563$	10.0910	−	17.4781i	0.425284	−	0.736614i	0.996447	+	0.0842230i	$0.0268408\pi$
$564$		0			0
$565$	24.0957	+	41.7350i	1.01371	+	1.75580i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.0525	+	20.8755i	0.505266	+	0.875146i	0.999981	+	0.00609110i	$0.00193887\pi$
$569$	12.0525	+	20.8755i	0.505266	+	0.875146i	−0.494716	+	0.869055i	$0.664728\pi$
$570$		0			0
$571$	−3.22763	+	5.59042i	−0.135072	+	0.233952i	−0.925625	−	0.378442i	$-0.876460\pi$
$571$	−3.22763	+	5.59042i	−0.135072	+	0.233952i	0.790553	+	0.612394i	$0.209793\pi$
$572$		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	$-0.0415021\pi$
$577$	9.20385	+	15.9415i	0.383161	+	0.663654i	−0.608351	+	0.793668i	$0.708169\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−14.6489	−	25.3726i	−0.606695	−	1.05083i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.8848	−	39.6376i	0.944557	−	1.63602i	0.187921	−	0.982184i	$-0.439825\pi$
$587$	22.8848	−	39.6376i	0.944557	−	1.63602i	0.756636	−	0.653837i	$-0.226842\pi$
$588$		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	$-0.949853\pi$
$593$	−8.71630	+	15.0971i	−0.357935	+	0.619962i	0.629680	+	0.776854i	$0.283186\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.72222	−	2.98297i	0.0703680	−	0.121881i	−0.828695	−	0.559701i	$-0.810916\pi$
$599$	1.72222	−	2.98297i	0.0703680	−	0.121881i	0.899063	+	0.437820i	$0.144249\pi$
$600$		0			0
$601$	−12.1666	+	21.0731i	−0.496284	+	0.859590i	−0.999991	−	0.00428500i	$-0.998636\pi$
$601$	−12.1666	+	21.0731i	−0.496284	+	0.859590i	0.503706	+	0.863875i	$0.331969\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.2531			−0.660782
$606$		0			0
$607$	19.9215			0.808587			0.404294	−	0.914629i	$-0.367517\pi$
$607$	19.9215			0.808587			0.404294	+	0.914629i	$0.367517\pi$
$608$		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	−0.903186	−	0.429250i	$-0.858778\pi$
$613$	−20.3848	−	35.3075i	−0.823334	−	1.42606i	0.0798515	−	0.996807i	$-0.474555\pi$
$614$		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
$619$	45.1168			1.81340			0.906698	−	0.421780i	$-0.138594\pi$
$619$	45.1168			1.81340			0.906698	+	0.421780i	$0.138594\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−10.0192			−0.400768
$626$		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
$635$	72.2135			2.86570
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.2449			−0.562640			−0.281320	−	0.959614i	$-0.590772\pi$
$641$	−14.2449			−0.562640			−0.281320	+	0.959614i	$0.590772\pi$
$642$		0			0
$643$	−18.0592	−	31.2795i	−0.712187	−	1.23354i	−0.964035	−	0.265777i	$-0.914372\pi$
$643$	−18.0592	−	31.2795i	−0.712187	−	1.23354i	0.251848	−	0.967767i	$-0.418962\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.73327	+	6.46622i	0.146770	+	0.254213i	0.930032	−	0.367479i	$-0.119779\pi$
$647$	3.73327	+	6.46622i	0.146770	+	0.254213i	−0.783262	+	0.621692i	$0.786446\pi$
$648$		0			0
$649$	−5.60449	+	9.70727i	−0.219996	+	0.381043i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	40.1090			1.56959			0.784794	−	0.619757i	$-0.212769\pi$
$653$	40.1090			1.56959			0.784794	+	0.619757i	$0.212769\pi$
$654$		0			0
$655$	−5.56383			−0.217397
$656$		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
$661$	11.0643	−	19.1640i	0.430352	−	0.745392i	−0.566551	−	0.824026i	$-0.691723\pi$
$661$	11.0643	−	19.1640i	0.430352	−	0.745392i	0.996904	+	0.0786346i	$0.0250560\pi$
$662$		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
$677$	−10.0105	−	17.3387i	−0.384736	−	0.666382i	0.606997	−	0.794704i	$-0.292374\pi$
$677$	−10.0105	−	17.3387i	−0.384736	−	0.666382i	−0.991732	+	0.128323i	$0.959041\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.7716	+	18.6569i	0.412162	+	0.713886i	0.995126	−	0.0986124i	$-0.0314404\pi$
$683$	10.7716	+	18.6569i	0.412162	+	0.713886i	−0.582964	+	0.812498i	$0.698107\pi$
$684$		0			0
$685$	26.6267			1.01735
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.40182	+	5.89213i	−0.129599	+	0.224472i
$690$		0			0
$691$	−21.8693	−	37.8787i	−0.831947	−	1.44097i	−0.896492	−	0.443060i	$-0.853893\pi$
$691$	−21.8693	−	37.8787i	−0.831947	−	1.44097i	0.0645449	−	0.997915i	$-0.479440\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.8168	−	48.1802i	−1.05515	−	1.82758i
$696$		0			0
$697$	5.88972	−	10.2013i	0.223089	−	0.386402i
$698$		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
$709$	−18.4050	−	31.8784i	−0.691214	−	1.19722i	−0.971440	−	0.237284i	$-0.923743\pi$
$709$	−18.4050	−	31.8784i	−0.691214	−	1.19722i	0.280226	−	0.959934i	$-0.409591\pi$
$710$		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	0.401762	+	0.915744i	$0.368398\pi$
$719$	−15.8787	+	27.5028i	−0.592177	+	1.02568i	−0.993939	+	0.109936i	$0.964935\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.52723	+	11.3055i	−0.242415	+	0.419875i
$726$		0			0
$727$	2.83596	−	4.91203i	0.105180	−	0.182177i	−0.808632	−	0.588315i	$-0.799791\pi$
$727$	2.83596	−	4.91203i	0.105180	−	0.182177i	0.913812	+	0.406138i	$0.133125\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.4370			0.459999
$732$		0			0
$733$	−23.9853			−0.885917			−0.442958	−	0.896542i	$-0.646071\pi$
$733$	−23.9853			−0.885917			−0.442958	+	0.896542i	$0.646071\pi$
$734$		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	0.869008	−	0.494798i	$-0.164758\pi$
$739$	0.162996	+	0.282317i	0.00599590	+	0.0103852i	−0.863012	+	0.505184i	$0.831425\pi$
$740$		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
$745$	10.0028			0.366473
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	31.3955			1.14564			0.572820	−	0.819681i	$-0.305850\pi$
$751$	31.3955			1.14564			0.572820	+	0.819681i	$0.305850\pi$
$752$		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
$761$	−13.3801			−0.485028			−0.242514	−	0.970148i	$-0.577972\pi$
$761$	−13.3801			−0.485028			−0.242514	+	0.970148i	$0.577972\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.60300			0.0939887
$768$		0			0
$769$	−5.98750	−	10.3707i	−0.215915	−	0.373975i	0.737640	−	0.675194i	$-0.235940\pi$
$769$	−5.98750	−	10.3707i	−0.215915	−	0.373975i	−0.953555	+	0.301218i	$0.902607\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13.9471	−	24.1571i	−0.501642	−	0.868869i	−0.999998	−	0.00189699i	$-0.999396\pi$
$773$	−13.9471	−	24.1571i	−0.501642	−	0.868869i	0.498356	−	0.866972i	$-0.333937\pi$
$774$		0			0
$775$	25.0099	−	43.3185i	0.898384	−	1.55605i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−63.8615			−2.28808
$780$		0			0
$781$	−12.4125			−0.444155
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.2489	−	38.5363i	0.794099	−	1.37542i
$786$		0			0
$787$	0.0522535	−	0.0905057i	0.00186264	−	0.00322618i	−0.865093	−	0.501612i	$-0.832740\pi$
$787$	0.0522535	−	0.0905057i	0.00186264	−	0.00322618i	0.866955	+	0.498386i	$0.166074\pi$
$788$		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	$0.0464910\pi$
$797$	17.5235	+	30.3516i	0.620715	+	1.07511i	−0.368638	+	0.929573i	$0.620176\pi$
$798$		0			0
$799$	4.12353	−	7.14216i	0.145880	−	0.252671i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.9576	−	18.9791i	−0.386685	−	0.669758i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.1259	−	38.3233i	−0.777907	−	1.34737i	−0.933146	−	0.359497i	$-0.882948\pi$
$809$	−22.1259	−	38.3233i	−0.777907	−	1.34737i	0.155239	−	0.987877i	$-0.450385\pi$
$810$		0			0
$811$	−0.903637			−0.0317310			−0.0158655	−	0.999874i	$-0.505050\pi$
$811$	−0.903637			−0.0317310			−0.0158655	+	0.999874i	$0.505050\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−24.9972	+	43.2964i	−0.875613	+	1.51661i
$816$		0			0
$817$	−33.7132	−	58.3930i	−1.17948	−	2.04291i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.1783	+	40.1459i	0.808927	+	1.40110i	0.913608	+	0.406596i	$0.133284\pi$
$821$	23.1783	+	40.1459i	0.808927	+	1.40110i	−0.104681	+	0.994506i	$0.533382\pi$
$822$		0			0
$823$	15.4915	−	26.8320i	0.539998	−	0.935305i	−0.458905	−	0.888485i	$-0.651758\pi$
$823$	15.4915	−	26.8320i	0.539998	−	0.935305i	0.998903	−	0.0468193i	$-0.0149085\pi$
$824$		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	−0.954373	−	0.298619i	$-0.903474\pi$
$829$	−21.1853	−	36.6941i	−0.735798	−	1.27444i	0.218575	−	0.975820i	$-0.429859\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	8.29123	+	14.3608i	0.286930	+	0.496977i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.36843	−	2.37020i	0.0472435	−	0.0818282i	−0.841437	−	0.540356i	$-0.818290\pi$
$839$	1.36843	−	2.37020i	0.0472435	−	0.0818282i	0.888680	+	0.458528i	$0.151623\pi$
$840$		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
$851$	−4.48615	+	7.77024i	−0.153783	+	0.266360i
$852$		0			0
$853$	4.59273	−	7.95485i	0.157252	−	0.272369i	−0.776625	−	0.629964i	$-0.783070\pi$
$853$	4.59273	−	7.95485i	0.157252	−	0.272369i	0.933877	+	0.357595i	$0.116403\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.1221			−0.414084			−0.207042	−	0.978332i	$-0.566384\pi$
$857$	−12.1221			−0.414084			−0.207042	+	0.978332i	$0.566384\pi$
$858$		0			0
$859$	6.83252			0.233123			0.116561	−	0.993183i	$-0.462813\pi$
$859$	6.83252			0.233123			0.116561	+	0.993183i	$0.462813\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.2595	−	45.4827i	0.893882	−	1.54825i	0.0587005	−	0.998276i	$-0.481304\pi$
$863$	26.2595	−	45.4827i	0.893882	−	1.54825i	0.835182	−	0.549974i	$-0.185362\pi$
$864$		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
$871$	7.40249			0.250824
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	32.1564			1.08584			0.542922	−	0.839783i	$-0.317318\pi$
$877$	32.1564			1.08584			0.542922	+	0.839783i	$0.317318\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.9101			−0.569715			−0.284858	−	0.958570i	$-0.591946\pi$
$881$	−16.9101			−0.569715			−0.284858	+	0.958570i	$0.591946\pi$
$882$		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
$887$	44.4566			1.49270			0.746352	−	0.665551i	$-0.231803\pi$
$887$	44.4566			1.49270			0.746352	+	0.665551i	$0.231803\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−44.7109			−1.49619
$894$		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
$905$	46.8826			1.55843
$906$		0			0
$907$	−16.4478			−0.546142			−0.273071	−	0.961994i	$-0.588039\pi$
$907$	−16.4478			−0.546142			−0.273071	+	0.961994i	$0.588039\pi$
$908$		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
$913$	9.26987	−	16.0559i	0.306788	−	0.531372i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−29.2722	+	50.7009i	−0.965600	+	1.67247i	−0.257606	+	0.966250i	$0.582934\pi$
$919$	−29.2722	+	50.7009i	−0.965600	+	1.67247i	−0.707994	+	0.706219i	$0.750400\pi$
$920$		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
$929$	−0.318672	−	0.551956i	−0.0104553	−	0.0181091i	0.860750	−	0.509027i	$-0.169995\pi$
$929$	−0.318672	−	0.551956i	−0.0104553	−	0.0181091i	−0.871206	+	0.490918i	$0.836661\pi$
$930$		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	$-0.600873\pi$
$937$	−19.0780			−0.623250			−0.311625	+	0.950205i	$0.600873\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.8153	+	25.6609i	−0.482965	+	0.836521i	−0.999809	−	0.0195594i	$-0.993774\pi$
$941$	−14.8153	+	25.6609i	−0.482965	+	0.836521i	0.516843	+	0.856080i	$0.327107\pi$
$942$		0			0
$943$	−34.3436	−	59.4848i	−1.11838	−	1.93709i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.1681	+	29.7360i	0.557888	+	0.966290i	0.997673	+	0.0681867i	$0.0217213\pi$
$947$	17.1681	+	29.7360i	0.557888	+	0.966290i	−0.439785	+	0.898103i	$0.644945\pi$
$948$		0			0
$949$	−2.54461	+	4.40740i	−0.0826017	+	0.143070i
$950$		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
$961$	−9.26272	−	16.0435i	−0.298798	−	0.517533i
$962$		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	$-0.708395\pi$
$971$	11.9191	−	20.6446i	0.382503	−	0.662515i	0.991419	+	0.130719i	$0.0417287\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.0842	−	33.0547i	0.610556	−	1.05751i	−0.380591	−	0.924744i	$-0.624279\pi$
$977$	19.0842	−	33.0547i	0.610556	−	1.05751i	0.991147	−	0.132771i	$-0.0423874\pi$
$978$		0			0
$979$	15.2771	−	26.4607i	0.488258	−	0.845688i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.57537			0.241617			0.120808	−	0.992676i	$-0.461451\pi$
$983$	7.57537			0.241617			0.120808	+	0.992676i	$0.461451\pi$
$984$		0			0
$985$	−64.4726			−2.05427
$986$		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	0.930846	−	0.365411i	$-0.119072\pi$
$991$	4.68952	+	8.12248i	0.148967	+	0.258019i	−0.781879	+	0.623431i	$0.785738\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.2219	−	50.6138i	−0.926397	−	1.60457i
$996$		0			0
$997$	8.43658			0.267189			0.133595	−	0.991036i	$-0.457348\pi$
$997$	8.43658			0.267189			0.133595	+	0.991036i	$0.457348\pi$
$998$		0			0
$999$		0			0

Twists

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

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

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.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+3.47961 q^{5} +O(q^{10})\)	\(q+3.47961 q^{5} -2.51576 q^{11} +(0.292110 + 0.505949i) q^{13} +(-0.547519 - 0.948331i) q^{17} +(-2.96834 + 5.14132i) q^{19} -6.38528 q^{23} +7.10769 q^{25} +(-0.918333 + 1.59060i) q^{29} +(3.51872 - 6.09459i) q^{31} +(0.702576 - 1.21690i) q^{37} +(5.37855 + 9.31593i) q^{41} +(-5.67879 + 9.83596i) q^{43} +(3.76565 + 6.52229i) q^{47} +(5.82285 + 10.0855i) q^{53} -8.75386 q^{55} +(2.22775 - 3.85858i) q^{59} +(6.17622 + 10.6975i) q^{61} +(1.01643 + 1.76051i) q^{65} +(6.33536 - 10.9732i) q^{67} +4.93390 q^{71} +(4.35558 + 7.54408i) q^{73} +(0.280206 + 0.485330i) q^{79} +(-3.68472 + 6.38212i) q^{83} +(-1.90515 - 3.29982i) q^{85} +(-6.07256 + 10.5180i) q^{89} +(-10.3287 + 17.8898i) q^{95} +(6.98486 - 12.0981i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) \) 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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