LMFDB - Embedded newform 1764.4.k.bb.361.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 27x^{6} + 10x^{5} + 446x^{4} + 62x^{3} + 3061x^{2} + 2142x + 14161 $ x^8 - 2x^7 + 27x^6 + 10x^5 + 446x^4 + 62x^3 + 3061x^2 + 2142*x + 14161
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.3
Root			$2.46576 - 4.27083i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.bb.1549.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.32752 + 9.22754i) q^{5} +O(q^{10})$	$q+(5.32752 + 9.22754i) q^{5} +(-3.32699 + 5.76252i) q^{11} -75.9335 q^{13} +(52.1435 - 90.3152i) q^{17} +(42.7472 + 74.0402i) q^{19} +(-34.3366 - 59.4728i) q^{23} +(5.73498 - 9.93327i) q^{25} -87.7843 q^{29} +(31.3841 - 54.3589i) q^{31} +(-21.1047 - 36.5544i) q^{37} +313.904 q^{41} +306.591 q^{43} +(-107.541 - 186.266i) q^{47} +(262.512 - 454.684i) q^{53} -70.8986 q^{55} +(-180.246 + 312.195i) q^{59} +(400.363 + 693.449i) q^{61} +(-404.537 - 700.679i) q^{65} +(20.1143 - 34.8390i) q^{67} +298.781 q^{71} +(258.563 - 447.844i) q^{73} +(611.233 + 1058.69i) q^{79} +1328.55 q^{83} +1111.18 q^{85} +(319.969 + 554.203i) q^{89} +(-455.473 + 788.902i) q^{95} -1425.65 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{17} + 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} + 48 q^{31} - 256 q^{37} + 2016 q^{41} - 224 q^{43} - 864 q^{47} - 648 q^{53} - 4704 q^{55} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} + 672 q^{73} + 1984 q^{79} + 6240 q^{83} + 1360 q^{85} - 2160 q^{89} - 3744 q^{95} - 4032 q^{97}+O(q^{100}) $ 8 * q - 48 * q^17 + 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 + 48 * q^31 - 256 * q^37 + 2016 * q^41 - 224 * q^43 - 864 * q^47 - 648 * q^53 - 4704 * q^55 - 336 * q^59 + 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 + 672 * q^73 + 1984 * q^79 + 6240 * q^83 + 1360 * q^85 - 2160 * q^89 - 3744 * q^95 - 4032 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	5.32752	+	9.22754i	0.476508	+	0.825336i	0.999638	−	0.0269168i	$-0.00856891\pi$
$5$	5.32752	+	9.22754i	0.476508	+	0.825336i	−0.523129	+	0.852253i	$0.675236\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.32699	+	5.76252i	−0.0911933	+	0.157951i	−0.908014	−	0.418941i	$-0.862401\pi$
$11$	−3.32699	+	5.76252i	−0.0911933	+	0.157951i	0.816820	+	0.576892i	$0.195735\pi$
$12$		0			0
$13$	−75.9335			−1.62001			−0.810006	−	0.586422i	$-0.800536\pi$
$13$	−75.9335			−1.62001			−0.810006	+	0.586422i	$0.800536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	52.1435	−	90.3152i	0.743921	−	1.28851i	−0.206776	−	0.978388i	$-0.566297\pi$
$17$	52.1435	−	90.3152i	0.743921	−	1.28851i	0.950697	−	0.310121i	$-0.100369\pi$
$18$		0			0
$19$	42.7472	+	74.0402i	0.516151	+	0.894000i	0.999824	+	0.0187511i	$0.00596900\pi$
$19$	42.7472	+	74.0402i	0.516151	+	0.894000i	−0.483673	+	0.875249i	$0.660698\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−34.3366	−	59.4728i	−0.311291	−	0.539171i	0.667351	−	0.744743i	$-0.267428\pi$
$23$	−34.3366	−	59.4728i	−0.311291	−	0.539171i	−0.978642	+	0.205572i	$0.934095\pi$
$24$		0			0
$25$	5.73498	−	9.93327i	0.0458798	−	0.0794662i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−87.7843			−0.562108			−0.281054	−	0.959692i	$-0.590684\pi$
$29$	−87.7843			−0.562108			−0.281054	+	0.959692i	$0.590684\pi$
$30$		0			0
$31$	31.3841	−	54.3589i	0.181831	−	0.314940i	−0.760673	−	0.649135i	$-0.775131\pi$
$31$	31.3841	−	54.3589i	0.181831	−	0.314940i	0.942504	+	0.334195i	$0.108464\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−21.1047	−	36.5544i	−0.0937726	−	0.162419i	0.815323	−	0.579006i	$-0.196559\pi$
$37$	−21.1047	−	36.5544i	−0.0937726	−	0.162419i	−0.909096	+	0.416587i	$0.863226\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	313.904			1.19570			0.597848	−	0.801609i	$-0.296023\pi$
$41$	313.904			1.19570			0.597848	+	0.801609i	$0.296023\pi$
$42$		0			0
$43$	306.591			1.08732			0.543659	−	0.839306i	$-0.317038\pi$
$43$	306.591			1.08732			0.543659	+	0.839306i	$0.317038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−107.541	−	186.266i	−0.333754	−	0.578079i	0.649491	−	0.760369i	$-0.274982\pi$
$47$	−107.541	−	186.266i	−0.333754	−	0.578079i	−0.983245	+	0.182291i	$0.941649\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	262.512	−	454.684i	0.680354	−	1.17841i	−0.294519	−	0.955646i	$-0.595159\pi$
$53$	262.512	−	454.684i	0.680354	−	1.17841i	0.974873	−	0.222762i	$-0.0715073\pi$
$54$		0			0
$55$	−70.8986			−0.173818
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−180.246	+	312.195i	−0.397729	+	0.688886i	−0.993445	−	0.114308i	$-0.963535\pi$
$59$	−180.246	+	312.195i	−0.397729	+	0.688886i	0.595717	+	0.803195i	$0.296868\pi$
$60$		0			0
$61$	400.363	+	693.449i	0.840348	+	1.45553i	0.889601	+	0.456739i	$0.150983\pi$
$61$	400.363	+	693.449i	0.840348	+	1.45553i	−0.0492530	+	0.998786i	$0.515684\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−404.537	−	700.679i	−0.771949	−	1.33705i
$66$		0			0
$67$	20.1143	−	34.8390i	0.0366769	−	0.0635262i	−0.847104	−	0.531427i	$-0.821656\pi$
$67$	20.1143	−	34.8390i	0.0366769	−	0.0635262i	0.883781	+	0.467900i	$0.154989\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	298.781			0.499419			0.249709	−	0.968321i	$-0.419665\pi$
$71$	298.781			0.499419			0.249709	+	0.968321i	$0.419665\pi$
$72$		0			0
$73$	258.563	−	447.844i	0.414555	−	0.718030i	−0.580827	−	0.814027i	$-0.697271\pi$
$73$	258.563	−	447.844i	0.414555	−	0.718030i	0.995382	+	0.0959971i	$0.0306040\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	611.233	+	1058.69i	0.870494	+	1.50774i	0.861486	+	0.507781i	$0.169534\pi$
$79$	611.233	+	1058.69i	0.870494	+	1.50774i	0.00900832	+	0.999959i	$0.497133\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1328.55			1.75696			0.878479	−	0.477782i	$-0.158559\pi$
$83$	1328.55			1.75696			0.878479	+	0.477782i	$0.158559\pi$
$84$		0			0
$85$	1111.18			1.41794
$86$		0			0
$87$		0			0
$88$		0			0
$89$	319.969	+	554.203i	0.381086	+	0.660060i	0.991218	−	0.132240i	$-0.0422169\pi$
$89$	319.969	+	554.203i	0.381086	+	0.660060i	−0.610132	+	0.792300i	$0.708884\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−455.473	+	788.902i	−0.491900	+	0.851996i
$96$		0			0
$97$	−1425.65			−1.49230			−0.746149	−	0.665779i	$-0.768099\pi$
$97$	−1425.65			−1.49230			−0.746149	+	0.665779i	$0.768099\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	496.085	−	859.244i	0.488736	−	0.846515i	−0.511180	−	0.859473i	$-0.670792\pi$
$101$	496.085	−	859.244i	0.488736	−	0.846515i	0.999916	+	0.0129585i	$0.00412493\pi$
$102$		0			0
$103$	133.786	+	231.724i	0.127984	+	0.221674i	0.922895	−	0.385051i	$-0.125816\pi$
$103$	133.786	+	231.724i	0.127984	+	0.221674i	−0.794912	+	0.606725i	$0.792483\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−768.312	−	1330.76i	−0.694164	−	1.20233i	−0.970462	−	0.241255i	$-0.922441\pi$
$107$	−768.312	−	1330.76i	−0.694164	−	1.20233i	0.276298	−	0.961072i	$-0.410892\pi$
$108$		0			0
$109$	−499.410	+	865.004i	−0.438852	+	0.760113i	−0.997601	−	0.0692232i	$-0.977948\pi$
$109$	−499.410	+	865.004i	−0.438852	+	0.760113i	0.558750	+	0.829336i	$0.311281\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−939.006			−0.781719			−0.390860	−	0.920450i	$-0.627822\pi$
$113$	−939.006			−0.781719			−0.390860	+	0.920450i	$0.627822\pi$
$114$		0			0
$115$	365.858	−	633.685i	0.296665	−	0.513839i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	643.362	+	1114.34i	0.483368	+	0.837217i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1454.09			1.04046
$126$		0			0
$127$	−1621.27			−1.13279			−0.566397	−	0.824133i	$-0.691663\pi$
$127$	−1621.27			−1.13279			−0.566397	+	0.824133i	$0.691663\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−759.214	−	1315.00i	−0.506357	−	0.877037i	−0.999973	−	0.00735640i	$-0.997658\pi$
$131$	−759.214	−	1315.00i	−0.506357	−	0.877037i	0.493616	−	0.869680i	$-0.335675\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1242.50	+	2152.07i	−0.774844	+	1.34207i	0.160038	+	0.987111i	$0.448838\pi$
$137$	−1242.50	+	2152.07i	−0.774844	+	1.34207i	−0.934882	+	0.354958i	$0.884495\pi$
$138$		0			0
$139$	1655.36			1.01011			0.505057	−	0.863086i	$-0.331471\pi$
$139$	1655.36			1.01011			0.505057	+	0.863086i	$0.331471\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	252.630	−	437.568i	0.147734	−	0.255883i
$144$		0			0
$145$	−467.673	−	810.033i	−0.267849	−	0.463928i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	175.095	+	303.274i	0.0962709	+	0.166746i	0.910138	−	0.414305i	$-0.135975\pi$
$149$	175.095	+	303.274i	0.0962709	+	0.166746i	−0.813867	+	0.581051i	$0.802642\pi$
$150$		0			0
$151$	−1669.07	+	2890.91i	−0.899516	+	1.55801i	−0.0714022	+	0.997448i	$0.522747\pi$
$151$	−1669.07	+	2890.91i	−0.899516	+	1.55801i	−0.828114	+	0.560560i	$0.810586\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	668.799			0.346576
$156$		0			0
$157$	−871.836	+	1510.06i	−0.443185	+	0.767620i	−0.997924	−	0.0644052i	$-0.979485\pi$
$157$	−871.836	+	1510.06i	−0.443185	+	0.767620i	0.554738	+	0.832025i	$0.312818\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1715.78	−	2971.82i	−0.824480	−	1.42804i	−0.902316	−	0.431076i	$-0.858134\pi$
$163$	−1715.78	−	2971.82i	−0.824480	−	1.42804i	0.0778354	−	0.996966i	$-0.475199\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3381.05			1.56667			0.783335	−	0.621600i	$-0.213517\pi$
$167$	3381.05			1.56667			0.783335	+	0.621600i	$0.213517\pi$
$168$		0			0
$169$	3568.89			1.62444
$170$		0			0
$171$		0			0
$172$		0			0
$173$	670.501	+	1161.34i	0.294666	+	0.510377i	0.974907	−	0.222612i	$-0.0714583\pi$
$173$	670.501	+	1161.34i	0.294666	+	0.510377i	−0.680241	+	0.732988i	$0.738125\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	562.574	−	974.407i	0.234909	−	0.406875i	−0.724337	−	0.689446i	$-0.757854\pi$
$179$	562.574	−	974.407i	0.234909	−	0.406875i	0.959246	+	0.282571i	$0.0911874\pi$
$180$		0			0
$181$	3535.04			1.45170			0.725848	−	0.687855i	$-0.241447\pi$
$181$	3535.04			1.45170			0.725848	+	0.687855i	$0.241447\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	224.871	−	389.488i	0.0893668	−	0.154788i
$186$		0			0
$187$	346.962	+	600.956i	0.135681	+	0.235007i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1319.95	+	2286.21i	0.500041	+	0.866097i	1.00000		4.78599e-5i	$1.52343e-5\pi$
$191$	1319.95	+	2286.21i	0.500041	+	0.866097i	−0.499959	+	0.866049i	$0.666651\pi$
$192$		0			0
$193$	−523.526	+	906.774i	−0.195255	+	0.338192i	−0.946984	−	0.321280i	$-0.895887\pi$
$193$	−523.526	+	906.774i	−0.195255	+	0.338192i	0.751729	+	0.659472i	$0.229220\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4585.45			1.65837			0.829187	−	0.558972i	$-0.188804\pi$
$197$	4585.45			1.65837			0.829187	+	0.558972i	$0.188804\pi$
$198$		0			0
$199$	415.371	−	719.444i	0.147964	−	0.256282i	−0.782511	−	0.622637i	$-0.786061\pi$
$199$	415.371	−	719.444i	0.147964	−	0.256282i	0.930475	+	0.366356i	$0.119395\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1672.33	+	2896.56i	0.569759	+	0.986852i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−568.878			−0.188278
$210$		0			0
$211$	−2630.08			−0.858114			−0.429057	−	0.903277i	$-0.641154\pi$
$211$	−2630.08			−0.858114			−0.429057	+	0.903277i	$0.641154\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1633.37	+	2829.08i	0.518116	+	0.897404i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3959.44	+	6857.94i	−1.20516	+	2.08740i
$222$		0			0
$223$	863.988			0.259448			0.129724	−	0.991550i	$-0.458591\pi$
$223$	863.988			0.259448			0.129724	+	0.991550i	$0.458591\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2080.18	−	3602.97i	0.608221	−	1.05347i	−0.383312	−	0.923619i	$-0.625217\pi$
$227$	2080.18	−	3602.97i	0.608221	−	1.05347i	0.991533	−	0.129851i	$-0.0414500\pi$
$228$		0			0
$229$	−90.6052	−	156.933i	−0.0261457	−	0.0452856i	0.852657	−	0.522472i	$-0.174990\pi$
$229$	−90.6052	−	156.933i	−0.0261457	−	0.0452856i	−0.878802	+	0.477186i	$0.841657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1075.22	+	1862.34i	0.302319	+	0.523631i	0.976661	−	0.214788i	$-0.0689060\pi$
$233$	1075.22	+	1862.34i	0.302319	+	0.523631i	−0.674342	+	0.738419i	$0.735573\pi$
$234$		0			0
$235$	1145.85	−	1984.67i	0.318073	−	0.550918i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6939.47			1.87815			0.939073	−	0.343719i	$-0.111687\pi$
$239$	6939.47			1.87815			0.939073	+	0.343719i	$0.111687\pi$
$240$		0			0
$241$	−103.085	+	178.548i	−0.0275530	+	0.0477232i	−0.879473	−	0.475949i	$-0.842105\pi$
$241$	−103.085	+	178.548i	−0.0275530	+	0.0477232i	0.851920	+	0.523672i	$0.175438\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3245.94	−	5622.13i	−0.836171	−	1.44829i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5011.93			1.26036			0.630180	−	0.776449i	$-0.282981\pi$
$251$	5011.93			1.26036			0.630180	+	0.776449i	$0.282981\pi$
$252$		0			0
$253$	456.951			0.113551
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2931.54	+	5077.58i	0.711535	+	1.23241i	0.964281	+	0.264882i	$0.0853329\pi$
$257$	2931.54	+	5077.58i	0.711535	+	1.23241i	−0.252746	+	0.967533i	$0.581334\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2819.78	+	4884.00i	−0.661122	+	1.14510i	0.319200	+	0.947688i	$0.396586\pi$
$263$	−2819.78	+	4884.00i	−0.661122	+	1.14510i	−0.980321	+	0.197409i	$0.936747\pi$
$264$		0			0
$265$	5594.15			1.29678
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1488.32	−	2577.84i	0.337339	−	0.584289i	−0.646592	−	0.762836i	$-0.723806\pi$
$269$	1488.32	−	2577.84i	0.337339	−	0.584289i	0.983931	+	0.178547i	$0.0571397\pi$
$270$		0			0
$271$	−1403.67	−	2431.22i	−0.314637	−	0.544968i	0.664723	−	0.747090i	$-0.268550\pi$
$271$	−1403.67	−	2431.22i	−0.314637	−	0.544968i	−0.979360	+	0.202122i	$0.935216\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	38.1605	+	66.0959i	0.00836787	+	0.0144936i
$276$		0			0
$277$	959.194	−	1661.37i	0.208059	−	0.360369i	−0.743044	−	0.669243i	$-0.766619\pi$
$277$	959.194	−	1661.37i	0.208059	−	0.360369i	0.951103	+	0.308874i	$0.0999520\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5209.50			−1.10595			−0.552977	−	0.833197i	$-0.686508\pi$
$281$	−5209.50			−1.10595			−0.552977	+	0.833197i	$0.686508\pi$
$282$		0			0
$283$	3748.35	−	6492.33i	0.787337	−	1.36371i	−0.140256	−	0.990115i	$-0.544793\pi$
$283$	3748.35	−	6492.33i	0.787337	−	1.36371i	0.927593	−	0.373592i	$-0.121874\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2981.39	−	5163.92i	−0.606837	−	1.05107i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3358.94			−0.669732			−0.334866	−	0.942266i	$-0.608691\pi$
$293$	−3358.94			−0.669732			−0.334866	+	0.942266i	$0.608691\pi$
$294$		0			0
$295$	−3841.05			−0.758084
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2607.30	+	4515.98i	0.504294	+	0.873463i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4265.89	+	7388.73i	−0.800865	+	1.38714i
$306$		0			0
$307$	7330.92			1.36286			0.681429	−	0.731884i	$-0.261359\pi$
$307$	7330.92			1.36286			0.681429	+	0.731884i	$0.261359\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1019.27	−	1765.42i	0.185843	−	0.321890i	−0.758017	−	0.652235i	$-0.773832\pi$
$311$	1019.27	−	1765.42i	0.185843	−	0.321890i	0.943860	+	0.330345i	$0.107165\pi$
$312$		0			0
$313$	2069.44	+	3584.37i	0.373711	+	0.647286i	0.990133	−	0.140130i	$-0.0447520\pi$
$313$	2069.44	+	3584.37i	0.373711	+	0.647286i	−0.616423	+	0.787416i	$0.711419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3776.13	−	6540.44i	−0.669049	−	1.15883i	−0.978171	−	0.207804i	$-0.933368\pi$
$317$	−3776.13	−	6540.44i	−0.669049	−	1.15883i	0.309122	−	0.951022i	$-0.399965\pi$
$318$		0			0
$319$	292.058	−	505.859i	0.0512605	−	0.0887858i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8915.94			1.53590
$324$		0			0
$325$	−435.477	+	754.268i	−0.0743258	+	0.128736i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2788.85	−	4830.43i	−0.463109	−	0.802128i	0.536005	−	0.844215i	$-0.319933\pi$
$331$	−2788.85	−	4830.43i	−0.463109	−	0.802128i	−0.999114	+	0.0420865i	$0.986599\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	428.637			0.0699073
$336$		0			0
$337$	4467.16			0.722083			0.361041	−	0.932550i	$-0.382421\pi$
$337$	4467.16			0.722083			0.361041	+	0.932550i	$0.382421\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	208.830	+	361.704i	0.0331635	+	0.0574409i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4894.21	−	8477.01i	0.757161	−	1.31144i	−0.187132	−	0.982335i	$-0.559919\pi$
$347$	4894.21	−	8477.01i	0.757161	−	1.31144i	0.944293	−	0.329106i	$-0.106747\pi$
$348$		0			0
$349$	4746.95			0.728075			0.364038	−	0.931384i	$-0.381398\pi$
$349$	4746.95			0.728075			0.364038	+	0.931384i	$0.381398\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4717.33	−	8170.65i	0.711269	−	1.23195i	−0.253112	−	0.967437i	$-0.581454\pi$
$353$	4717.33	−	8170.65i	0.711269	−	1.23195i	0.964381	−	0.264517i	$-0.0852127\pi$
$354$		0			0
$355$	1591.76	+	2757.01i	0.237977	+	0.412189i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5653.77	+	9792.62i	0.831183	+	1.43965i	0.897101	+	0.441826i	$0.145669\pi$
$359$	5653.77	+	9792.62i	0.831183	+	1.43965i	−0.0659178	+	0.997825i	$0.520998\pi$
$360$		0			0
$361$	−225.138	+	389.950i	−0.0328237	+	0.0568523i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5510.00			0.790155
$366$		0			0
$367$	3044.80	−	5273.76i	0.433072	−	0.750103i	−0.564064	−	0.825731i	$-0.690763\pi$
$367$	3044.80	−	5273.76i	0.433072	−	0.750103i	0.997136	+	0.0756282i	$0.0240962\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1300.92	+	2253.26i	0.180588	+	0.312787i	0.942081	−	0.335386i	$-0.108867\pi$
$373$	1300.92	+	2253.26i	0.180588	+	0.312787i	−0.761493	+	0.648173i	$0.775533\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6665.77			0.910622
$378$		0			0
$379$	10416.2			1.41173			0.705865	−	0.708347i	$-0.250559\pi$
$379$	10416.2			1.41173			0.705865	+	0.708347i	$0.250559\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	832.230	+	1441.47i	0.111031	+	0.192312i	0.916186	−	0.400753i	$-0.131251\pi$
$383$	832.230	+	1441.47i	0.111031	+	0.192312i	−0.805155	+	0.593064i	$0.797918\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−667.078	+	1155.41i	−0.0869465	+	0.150596i	−0.906219	−	0.422809i	$-0.861044\pi$
$389$	−667.078	+	1155.41i	−0.0869465	+	0.150596i	0.819272	+	0.573404i	$0.194378\pi$
$390$		0			0
$391$	−7161.73			−0.926302
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6512.71	+	11280.4i	−0.829596	+	1.43690i
$396$		0			0
$397$	2222.44	+	3849.38i	0.280960	+	0.486637i	0.971621	−	0.236541i	$-0.0760139\pi$
$397$	2222.44	+	3849.38i	0.280960	+	0.486637i	−0.690662	+	0.723178i	$0.742681\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1324.91	−	2294.81i	−0.164995	−	0.285779i	0.771659	−	0.636037i	$-0.219427\pi$
$401$	−1324.91	−	2294.81i	−0.164995	−	0.285779i	−0.936653	+	0.350258i	$0.886094\pi$
$402$		0			0
$403$	−2383.11	+	4127.66i	−0.294568	+	0.510207i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	280.860			0.0342057
$408$		0			0
$409$	−4796.33	+	8307.49i	−0.579862	+	1.00435i	0.415633	+	0.909532i	$0.363560\pi$
$409$	−4796.33	+	8307.49i	−0.579862	+	1.00435i	−0.995495	+	0.0948173i	$0.969773\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	7077.89	+	12259.3i	0.837205	+	1.45008i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16850.1			1.96463			0.982317	−	0.187224i	$-0.0599491\pi$
$419$	16850.1			1.96463			0.982317	+	0.187224i	$0.0599491\pi$
$420$		0			0
$421$	1691.10			0.195770			0.0978849	−	0.995198i	$-0.468792\pi$
$421$	1691.10			0.195770			0.0978849	+	0.995198i	$0.468792\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−598.084	−	1035.91i	−0.0682619	−	0.118233i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3394.67	+	5879.74i	−0.379386	+	0.657117i	−0.990973	−	0.134061i	$-0.957198\pi$
$431$	−3394.67	+	5879.74i	−0.379386	+	0.657117i	0.611587	+	0.791177i	$0.290532\pi$
$432$		0			0
$433$	−10386.6			−1.15276			−0.576382	−	0.817180i	$-0.695536\pi$
$433$	−10386.6			−1.15276			−0.576382	+	0.817180i	$0.695536\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2935.59	−	5084.58i	0.321346	−	0.556587i
$438$		0			0
$439$	−2106.13	−	3647.92i	−0.228975	−	0.396596i	0.728530	−	0.685014i	$-0.240204\pi$
$439$	−2106.13	−	3647.92i	−0.228975	−	0.396596i	−0.957505	+	0.288418i	$0.906871\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−904.980	−	1567.47i	−0.0970585	−	0.168110i	0.813407	−	0.581694i	$-0.197610\pi$
$443$	−904.980	−	1567.47i	−0.0970585	−	0.168110i	−0.910466	+	0.413584i	$0.864277\pi$
$444$		0			0
$445$	−3409.28	+	5905.05i	−0.363181	+	0.629048i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1768.29			−0.185859			−0.0929297	−	0.995673i	$-0.529623\pi$
$449$	−1768.29			−0.185859			−0.0929297	+	0.995673i	$0.529623\pi$
$450$		0			0
$451$	−1044.36	+	1808.88i	−0.109040	+	0.188862i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2186.30	−	3786.79i	−0.223788	−	0.387611i	0.732167	−	0.681125i	$-0.238509\pi$
$457$	−2186.30	−	3786.79i	−0.223788	−	0.387611i	−0.955955	+	0.293513i	$0.905176\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1164.34			−0.117633			−0.0588165	−	0.998269i	$-0.518733\pi$
$461$	−1164.34			−0.117633			−0.0588165	+	0.998269i	$0.518733\pi$
$462$		0			0
$463$	−14893.9			−1.49498			−0.747491	−	0.664272i	$-0.768742\pi$
$463$	−14893.9			−1.49498			−0.747491	+	0.664272i	$0.768742\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9580.14	−	16593.3i	−0.949285	−	1.64421i	−0.746936	−	0.664896i	$-0.768476\pi$
$467$	−9580.14	−	16593.3i	−0.949285	−	1.64421i	−0.202349	−	0.979314i	$-0.564857\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1020.03	+	1766.74i	−0.0991562	+	0.171744i
$474$		0			0
$475$	980.616			0.0947237
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4703.14	−	8146.08i	0.448626	−	0.777043i	−0.549671	−	0.835381i	$-0.685247\pi$
$479$	4703.14	−	8146.08i	0.448626	−	0.777043i	0.998297	+	0.0583380i	$0.0185801\pi$
$480$		0			0
$481$	1602.55	+	2775.70i	0.151913	+	0.263121i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7595.19	−	13155.3i	−0.711092	−	1.23165i
$486$		0			0
$487$	−6230.39	+	10791.4i	−0.579725	+	1.00411i	0.415786	+	0.909463i	$0.363507\pi$
$487$	−6230.39	+	10791.4i	−0.579725	+	1.00411i	−0.995511	+	0.0946505i	$0.969827\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12868.0			−1.18274			−0.591369	−	0.806401i	$-0.701412\pi$
$491$	−12868.0			−1.18274			−0.591369	+	0.806401i	$0.701412\pi$
$492$		0			0
$493$	−4577.38	+	7928.26i	−0.418164	+	0.724281i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6852.50	+	11868.9i	0.614750	+	1.06478i	0.990428	+	0.138028i	$0.0440763\pi$
$499$	6852.50	+	11868.9i	0.614750	+	1.06478i	−0.375679	+	0.926750i	$0.622590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10126.1			0.897616			0.448808	−	0.893628i	$-0.351849\pi$
$503$	10126.1			0.897616			0.448808	+	0.893628i	$0.351849\pi$
$504$		0			0
$505$	10571.6			0.931546
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3118.03	+	5400.59i	0.271521	+	0.470288i	0.969252	−	0.246072i	$-0.0791400\pi$
$509$	3118.03	+	5400.59i	0.271521	+	0.470288i	−0.697730	+	0.716360i	$0.745807\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1425.50	+	2469.03i	−0.121971	+	0.211259i
$516$		0			0
$517$	1431.15			0.121744
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8508.38	+	14737.0i	−0.715468	+	1.23923i	0.247310	+	0.968936i	$0.420453\pi$
$521$	−8508.38	+	14737.0i	−0.715468	+	1.23923i	−0.962779	+	0.270291i	$0.912880\pi$
$522$		0			0
$523$	−2907.31	−	5035.61i	−0.243074	−	0.421017i	0.718514	−	0.695512i	$-0.244822\pi$
$523$	−2907.31	−	5035.61i	−0.243074	−	0.421017i	−0.961588	+	0.274496i	$0.911489\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3272.96	−	5668.93i	−0.270536	−	0.468582i
$528$		0			0
$529$	3725.49	−	6452.74i	0.306196	−	0.530348i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−23835.8			−1.93704
$534$		0			0
$535$	8186.41	−	14179.3i	0.661550	−	1.14584i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5034.59	+	8720.17i	0.400100	+	0.692993i	0.993738	−	0.111739i	$-0.0356421\pi$
$541$	5034.59	+	8720.17i	0.400100	+	0.692993i	−0.593638	+	0.804732i	$0.702309\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10642.5			−0.836466
$546$		0			0
$547$	−7437.51			−0.581362			−0.290681	−	0.956820i	$-0.593882\pi$
$547$	−7437.51			−0.581362			−0.290681	+	0.956820i	$0.593882\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3752.53	−	6499.57i	−0.290133	−	0.502525i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10893.6	−	18868.2i	0.828680	−	1.43532i	−0.0703944	−	0.997519i	$-0.522426\pi$
$557$	10893.6	−	18868.2i	0.828680	−	1.43532i	0.899074	−	0.437796i	$-0.144241\pi$
$558$		0			0
$559$	−23280.5			−1.76147
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1286.24	+	2227.83i	−0.0962851	+	0.166771i	−0.910144	−	0.414292i	$-0.864029\pi$
$563$	−1286.24	+	2227.83i	−0.0962851	+	0.166771i	0.813859	+	0.581062i	$0.197363\pi$
$564$		0			0
$565$	−5002.58	−	8664.72i	−0.372496	−	0.645181i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8790.29	−	15225.2i	−0.647642	−	1.12175i	−0.983685	−	0.179902i	$-0.942422\pi$
$569$	−8790.29	−	15225.2i	−0.647642	−	1.12175i	0.336043	−	0.941847i	$-0.390911\pi$
$570$		0			0
$571$	−3610.37	+	6253.35i	−0.264605	+	0.458309i	−0.967460	−	0.253024i	$-0.918575\pi$
$571$	−3610.37	+	6253.35i	−0.264605	+	0.458309i	0.702855	+	0.711333i	$0.251908\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−787.679			−0.0571278
$576$		0			0
$577$	5577.54	−	9660.58i	0.402419	−	0.697011i	−0.591598	−	0.806233i	$-0.701503\pi$
$577$	5577.54	−	9660.58i	0.402419	−	0.697011i	0.994017	+	0.109222i	$0.0348360\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1746.75	+	3025.46i	0.124088	+	0.214926i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15216.2			−1.06992			−0.534958	−	0.844879i	$-0.679673\pi$
$587$	−15216.2			−1.06992			−0.534958	+	0.844879i	$0.679673\pi$
$588$		0			0
$589$	5366.33			0.375409
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1921.95	−	3328.91i	−0.133094	−	0.230526i	0.791774	−	0.610815i	$-0.209158\pi$
$593$	−1921.95	−	3328.91i	−0.133094	−	0.230526i	−0.924868	+	0.380289i	$0.875825\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9373.48	−	16235.3i	0.639382	−	1.10744i	−0.346186	−	0.938166i	$-0.612524\pi$
$599$	9373.48	−	16235.3i	0.639382	−	1.10744i	0.985569	−	0.169277i	$-0.0541431\pi$
$600$		0			0
$601$	9864.63			0.669529			0.334764	−	0.942302i	$-0.391343\pi$
$601$	9864.63			0.669529			0.334764	+	0.942302i	$0.391343\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6855.06	+	11873.3i	−0.460657	+	0.797882i
$606$		0			0
$607$	11146.4	+	19306.2i	0.745336	+	1.29096i	0.950038	+	0.312136i	$0.101044\pi$
$607$	11146.4	+	19306.2i	0.745336	+	1.29096i	−0.204701	+	0.978824i	$0.565622\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8165.94	+	14143.8i	0.540685	+	0.936494i
$612$		0			0
$613$	−3023.39	+	5236.67i	−0.199206	+	0.345036i	−0.948271	−	0.317461i	$-0.897170\pi$
$613$	−3023.39	+	5236.67i	−0.199206	+	0.345036i	0.749065	+	0.662497i	$0.230503\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9383.68			0.612273			0.306137	−	0.951988i	$-0.400964\pi$
$617$	9383.68			0.612273			0.306137	+	0.951988i	$0.400964\pi$
$618$		0			0
$619$	−1994.75	+	3455.00i	−0.129525	+	0.224343i	−0.923492	−	0.383617i	$-0.874678\pi$
$619$	−1994.75	+	3455.00i	−0.129525	+	0.224343i	0.793968	+	0.607960i	$0.208012\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7029.85	+	12176.1i	0.449910	+	0.779267i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4401.88			−0.279038
$630$		0			0
$631$	11434.0			0.721363			0.360681	−	0.932689i	$-0.382544\pi$
$631$	11434.0			0.721363			0.360681	+	0.932689i	$0.382544\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8637.37	−	14960.4i	−0.539785	−	0.934935i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7548.63	+	13074.6i	−0.465137	+	0.805641i	−0.999208	−	0.0397989i	$-0.987328\pi$
$641$	−7548.63	+	13074.6i	−0.465137	+	0.805641i	0.534071	+	0.845440i	$0.320662\pi$
$642$		0			0
$643$	−4170.19			−0.255764			−0.127882	−	0.991789i	$-0.540818\pi$
$643$	−4170.19			−0.255764			−0.127882	+	0.991789i	$0.540818\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2281.94	−	3952.44i	0.138659	−	0.240164i	−0.788330	−	0.615252i	$-0.789054\pi$
$647$	2281.94	−	3952.44i	0.138659	−	0.240164i	0.926989	+	0.375088i	$0.122387\pi$
$648$		0			0
$649$	−1199.35	−	2077.34i	−0.0725404	−	0.125644i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1909.52	+	3307.39i	0.114434	+	0.198205i	0.917553	−	0.397613i	$-0.130161\pi$
$653$	1909.52	+	3307.39i	0.114434	+	0.198205i	−0.803119	+	0.595818i	$0.796828\pi$
$654$		0			0
$655$	8089.46	−	14011.4i	0.482567	−	0.835830i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4326.02			0.255717			0.127859	−	0.991792i	$-0.459190\pi$
$659$	4326.02			0.255717			0.127859	+	0.991792i	$0.459190\pi$
$660$		0			0
$661$	−14658.9	+	25389.9i	−0.862579	+	1.49403i	0.00685243	+	0.999977i	$0.497819\pi$
$661$	−14658.9	+	25389.9i	−0.862579	+	1.49403i	−0.869431	+	0.494054i	$0.835515\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	3014.22	+	5220.78i	0.174979	+	0.303072i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5328.02			−0.306536
$672$		0			0
$673$	−5483.56			−0.314080			−0.157040	−	0.987592i	$-0.550195\pi$
$673$	−5483.56			−0.314080			−0.157040	+	0.987592i	$0.550195\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6034.67	+	10452.4i	0.342587	+	0.593378i	0.984912	−	0.173054i	$-0.0553635\pi$
$677$	6034.67	+	10452.4i	0.342587	+	0.593378i	−0.642325	+	0.766432i	$0.722030\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15398.4	−	26670.7i	0.862667	−	1.49418i	−0.00667825	−	0.999978i	$-0.502126\pi$
$683$	15398.4	−	26670.7i	0.862667	−	1.49418i	0.869345	−	0.494205i	$-0.164541\pi$
$684$		0			0
$685$	−26477.7			−1.47688
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19933.4	+	34525.7i	−1.10218	+	1.90903i
$690$		0			0
$691$	−3250.98	−	5630.87i	−0.178977	−	0.309997i	0.762553	−	0.646925i	$-0.223945\pi$
$691$	−3250.98	−	5630.87i	−0.178977	−	0.309997i	−0.941530	+	0.336928i	$0.890612\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8818.98	+	15274.9i	0.481328	+	0.833685i
$696$		0			0
$697$	16368.0	−	28350.3i	0.889504	−	1.54067i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2235.98			0.120473			0.0602367	−	0.998184i	$-0.480814\pi$
$701$	2235.98			0.120473			0.0602367	+	0.998184i	$0.480814\pi$
$702$		0			0
$703$	1804.33	−	3125.19i	0.0968016	−	0.167665i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17782.2	−	30799.7i	−0.941926	−	1.63146i	−0.761792	−	0.647822i	$-0.775680\pi$
$709$	−17782.2	−	30799.7i	−0.941926	−	1.63146i	−0.180134	−	0.983642i	$-0.557653\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4310.50			−0.226409
$714$		0			0
$715$	5383.57			0.281586
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4962.07	+	8594.56i	0.257377	+	0.445790i	0.965538	−	0.260260i	$-0.0838084\pi$
$719$	4962.07	+	8594.56i	0.257377	+	0.445790i	−0.708161	+	0.706051i	$0.750475\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−503.441	+	871.985i	−0.0257894	+	0.0446686i
$726$		0			0
$727$	−880.081			−0.0448974			−0.0224487	−	0.999748i	$-0.507146\pi$
$727$	−880.081			−0.0448974			−0.0224487	+	0.999748i	$0.507146\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	15986.7	−	27689.8i	0.808879	−	1.40102i
$732$		0			0
$733$	−8668.08	−	15013.5i	−0.436784	−	0.756532i	0.560655	−	0.828049i	$-0.310549\pi$
$733$	−8668.08	−	15013.5i	−0.436784	−	0.756532i	−0.997439	+	0.0715172i	$0.977216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	133.840	+	231.818i	0.00668937	+	0.0115863i
$738$		0			0
$739$	12293.2	−	21292.4i	0.611924	−	1.05988i	−0.378992	−	0.925400i	$-0.623729\pi$
$739$	12293.2	−	21292.4i	0.611924	−	1.05988i	0.990916	−	0.134483i	$-0.0429375\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14579.3			−0.719870			−0.359935	−	0.932977i	$-0.617201\pi$
$743$	−14579.3			−0.719870			−0.359935	+	0.932977i	$0.617201\pi$
$744$		0			0
$745$	−1865.65	+	3231.40i	−0.0917478	+	0.158912i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8042.98	−	13930.8i	−0.390802	−	0.676889i	0.601753	−	0.798682i	$-0.294469\pi$
$751$	−8042.98	−	13930.8i	−0.390802	−	0.676889i	−0.992556	+	0.121793i	$0.961136\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−35568.0			−1.71451
$756$		0			0
$757$	−37017.3			−1.77730			−0.888651	−	0.458584i	$-0.848357\pi$
$757$	−37017.3			−1.77730			−0.888651	+	0.458584i	$0.848357\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12303.2	−	21309.8i	−0.586060	−	1.01509i	−0.994742	−	0.102410i	$-0.967345\pi$
$761$	−12303.2	−	21309.8i	−0.586060	−	1.01509i	0.408682	−	0.912677i	$-0.365989\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13686.7	−	23706.0i	0.644325	−	1.11600i
$768$		0			0
$769$	12715.6			0.596275			0.298137	−	0.954523i	$-0.403635\pi$
$769$	12715.6			0.596275			0.298137	+	0.954523i	$0.403635\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1587.11	+	2748.96i	−0.0738480	+	0.127908i	−0.900585	−	0.434681i	$-0.856861\pi$
$773$	−1587.11	+	2748.96i	−0.0738480	+	0.127908i	0.826737	+	0.562589i	$0.190195\pi$
$774$		0			0
$775$	−359.975	−	623.495i	−0.0166847	−	0.0288988i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13418.5	+	23241.5i	0.617160	+	1.06895i
$780$		0			0
$781$	−994.042	+	1721.73i	−0.0455437	+	0.0788840i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−18578.9			−0.844726
$786$		0			0
$787$	−7784.94	+	13483.9i	−0.352609	+	0.610736i	−0.986706	−	0.162517i	$-0.948039\pi$
$787$	−7784.94	+	13483.9i	−0.352609	+	0.610736i	0.634097	+	0.773254i	$0.281372\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−30400.9	−	52656.0i	−1.36137	−	2.35797i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31514.5			−1.40063			−0.700314	−	0.713835i	$-0.746957\pi$
$797$	−31514.5			−1.40063			−0.700314	+	0.713835i	$0.746957\pi$
$798$		0			0
$799$	−22430.2			−0.993146
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1720.48	+	2979.95i	0.0756093	+	0.130959i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5514.54	−	9551.47i	0.239655	−	0.415095i	−0.720960	−	0.692976i	$-0.756299\pi$
$809$	5514.54	−	9551.47i	0.239655	−	0.415095i	0.960615	+	0.277882i	$0.0896323\pi$
$810$		0			0
$811$	20830.5			0.901920			0.450960	−	0.892544i	$-0.351082\pi$
$811$	20830.5			0.901920			0.450960	+	0.892544i	$0.351082\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18281.7	−	31664.9i	0.785743	−	1.36095i
$816$		0			0
$817$	13105.9	+	22700.1i	0.561221	+	0.972063i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4674.53	−	8096.53i	−0.198712	−	0.344179i	0.749399	−	0.662118i	$-0.230342\pi$
$821$	−4674.53	−	8096.53i	−0.198712	−	0.344179i	−0.948111	+	0.317940i	$0.897009\pi$
$822$		0			0
$823$	16445.3	−	28484.1i	0.696533	−	1.20643i	−0.273129	−	0.961978i	$-0.588058\pi$
$823$	16445.3	−	28484.1i	0.696533	−	1.20643i	0.969661	−	0.244452i	$-0.0786082\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13081.2			0.550033			0.275016	−	0.961440i	$-0.411317\pi$
$827$	13081.2			0.550033			0.275016	+	0.961440i	$0.411317\pi$
$828$		0			0
$829$	14395.6	−	24933.9i	0.603112	−	1.04462i	−0.389235	−	0.921138i	$-0.627261\pi$
$829$	14395.6	−	24933.9i	0.603112	−	1.04462i	0.992347	−	0.123482i	$-0.0394060\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	18012.6	+	31198.8i	0.746531	+	1.29303i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36766.8			−1.51291			−0.756455	−	0.654046i	$-0.773070\pi$
$839$	−36766.8			−1.51291			−0.756455	+	0.654046i	$0.773070\pi$
$840$		0			0
$841$	−16682.9			−0.684034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	19013.4	+	32932.1i	0.774058	+	1.34071i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1449.33	+	2510.31i	−0.0583810	+	0.101119i
$852$		0			0
$853$	−5899.55			−0.236808			−0.118404	−	0.992966i	$-0.537778\pi$
$853$	−5899.55			−0.236808			−0.118404	+	0.992966i	$0.537778\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21138.5	+	36613.0i	−0.842564	+	1.45936i	0.0451553	+	0.998980i	$0.485622\pi$
$857$	−21138.5	+	36613.0i	−0.842564	+	1.45936i	−0.887720	+	0.460384i	$0.847712\pi$
$858$		0			0
$859$	−3171.73	−	5493.59i	−0.125981	−	0.218206i	0.796135	−	0.605119i	$-0.206875\pi$
$859$	−3171.73	−	5493.59i	−0.125981	−	0.218206i	−0.922116	+	0.386913i	$0.873541\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3464.89	+	6001.36i	0.136670	+	0.236719i	0.926234	−	0.376949i	$-0.123027\pi$
$863$	3464.89	+	6001.36i	0.136670	+	0.236719i	−0.789564	+	0.613668i	$0.789693\pi$
$864$		0			0
$865$	−7144.22	+	12374.1i	−0.280822	+	0.486397i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8134.27			−0.317533
$870$		0			0
$871$	−1527.35	+	2645.44i	−0.0594170	+	0.102913i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1643.94	−	2847.39i	−0.0632976	−	0.109635i	0.832640	−	0.553815i	$-0.186828\pi$
$877$	−1643.94	−	2847.39i	−0.0632976	−	0.109635i	−0.895938	+	0.444180i	$0.853495\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−46875.9			−1.79261			−0.896304	−	0.443439i	$-0.853758\pi$
$881$	−46875.9			−1.79261			−0.896304	+	0.443439i	$0.853758\pi$
$882$		0			0
$883$	42479.4			1.61897			0.809483	−	0.587144i	$-0.199748\pi$
$883$	42479.4			1.61897			0.809483	+	0.587144i	$0.199748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1840.08	−	3187.11i	−0.0696547	−	0.120645i	0.829095	−	0.559108i	$-0.188856\pi$
$887$	−1840.08	−	3187.11i	−0.0696547	−	0.120645i	−0.898749	+	0.438463i	$0.855523\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9194.12	−	15924.7i	0.344535	−	0.596752i
$894$		0			0
$895$	11988.5			0.447745
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2755.04	+	4771.86i	−0.102209	+	0.177031i
$900$		0			0
$901$	−27376.6	−	47417.6i	−1.01226	−	1.75328i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18833.0	+	32619.7i	0.691745	+	1.19814i
$906$		0			0
$907$	3551.09	−	6150.67i	0.130002	−	0.225171i	−0.793675	−	0.608342i	$-0.791835\pi$
$907$	3551.09	−	6150.67i	0.130002	−	0.225171i	0.923677	+	0.383172i	$0.125168\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−38119.0			−1.38632			−0.693161	−	0.720783i	$-0.743782\pi$
$911$	−38119.0			−1.38632			−0.693161	+	0.720783i	$0.743782\pi$
$912$		0			0
$913$	−4420.08	+	7655.81i	−0.160223	+	0.277514i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−8595.28	−	14887.5i	−0.308522	−	0.534377i	0.669517	−	0.742797i	$-0.266501\pi$
$919$	−8595.28	−	14887.5i	−0.308522	−	0.534377i	−0.978039	+	0.208420i	$0.933168\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−22687.5			−0.809065
$924$		0			0
$925$	−484.139			−0.0172091
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16128.2	−	27934.9i	−0.569591	−	0.986561i	−0.996606	−	0.0823167i	$-0.973768\pi$
$929$	−16128.2	−	27934.9i	−0.569591	−	0.986561i	0.427015	−	0.904245i	$-0.359565\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3696.90	+	6403.22i	−0.129306	+	0.223965i
$936$		0			0
$937$	−26213.6			−0.913940			−0.456970	−	0.889482i	$-0.651065\pi$
$937$	−26213.6			−0.913940			−0.456970	+	0.889482i	$0.651065\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17120.0	+	29652.8i	−0.593089	+	1.02726i	0.400724	+	0.916199i	$0.368759\pi$
$941$	−17120.0	+	29652.8i	−0.593089	+	1.02726i	−0.993813	+	0.111062i	$0.964575\pi$
$942$		0			0
$943$	−10778.4	−	18668.7i	−0.372209	−	0.644685i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18002.3	+	31180.8i	0.617735	+	1.06995i	0.989898	+	0.141781i	$0.0452828\pi$
$947$	18002.3	+	31180.8i	0.617735	+	1.06995i	−0.372163	+	0.928167i	$0.621384\pi$
$948$		0			0
$949$	−19633.6	+	34006.4i	−0.671584	+	1.16322i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29488.1			1.00232			0.501161	−	0.865354i	$-0.332907\pi$
$953$	29488.1			1.00232			0.501161	+	0.865354i	$0.332907\pi$
$954$		0			0
$955$	−14064.1	+	24359.7i	−0.476548	+	0.825405i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	12925.6	+	22387.7i	0.433875	+	0.751494i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11156.4			−0.372163
$966$		0			0
$967$	−56009.0			−1.86260			−0.931298	−	0.364259i	$-0.881322\pi$
$967$	−56009.0			−1.86260			−0.931298	+	0.364259i	$0.881322\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6788.44	−	11757.9i	−0.224358	−	0.388599i	0.731769	−	0.681553i	$-0.238695\pi$
$971$	−6788.44	−	11757.9i	−0.224358	−	0.388599i	−0.956127	+	0.292954i	$0.905362\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15887.6	+	27518.1i	−0.520254	+	0.901107i	0.479468	+	0.877559i	$0.340829\pi$
$977$	−15887.6	+	27518.1i	−0.520254	+	0.901107i	−0.999723	+	0.0235477i	$0.992504\pi$
$978$		0			0
$979$	−4258.14			−0.139010
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16377.7	+	28367.0i	−0.531401	+	0.920414i	0.467927	+	0.883767i	$0.345001\pi$
$983$	−16377.7	+	28367.0i	−0.531401	+	0.920414i	−0.999328	+	0.0366471i	$0.988332\pi$
$984$		0			0
$985$	24429.1	+	42312.4i	0.790228	+	1.36872i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10527.3	−	18233.8i	−0.338472	−	0.586251i
$990$		0			0
$991$	20398.5	−	35331.3i	0.653865	−	1.13253i	−0.328312	−	0.944569i	$-0.606480\pi$
$991$	20398.5	−	35331.3i	0.653865	−	1.13253i	0.982177	−	0.187958i	$-0.0601869\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8851.60			0.282025
$996$		0			0
$997$	−8685.25	+	15043.3i	−0.275892	+	0.477859i	−0.970360	−	0.241665i	$-0.922307\pi$
$997$	−8685.25	+	15043.3i	−0.275892	+	0.477859i	0.694468	+	0.719524i	$0.255640\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.bb.361.3		8
3.2	odd	2		588.4.i.l.361.2		8
7.2	even	3	inner	1764.4.k.bb.1549.3		8
7.3	odd	6		1764.4.a.ba.1.3		4
7.4	even	3		1764.4.a.bc.1.2		4
7.5	odd	6		1764.4.k.bd.1549.2		8
7.6	odd	2		1764.4.k.bd.361.2		8
21.2	odd	6		588.4.i.l.373.2		8
21.5	even	6		588.4.i.k.373.3		8
21.11	odd	6		588.4.a.j.1.3	✓	4
21.17	even	6		588.4.a.k.1.2	yes	4
21.20	even	2		588.4.i.k.361.3		8
84.11	even	6		2352.4.a.cq.1.3		4
84.59	odd	6		2352.4.a.cl.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.3	✓	4	21.11	odd	6
588.4.a.k.1.2	yes	4	21.17	even	6
588.4.i.k.361.3		8	21.20	even	2
588.4.i.k.373.3		8	21.5	even	6
588.4.i.l.361.2		8	3.2	odd	2
588.4.i.l.373.2		8	21.2	odd	6
1764.4.a.ba.1.3		4	7.3	odd	6
1764.4.a.bc.1.2		4	7.4	even	3
1764.4.k.bb.361.3		8	1.1	even	1	trivial
1764.4.k.bb.1549.3		8	7.2	even	3	inner
1764.4.k.bd.361.2		8	7.6	odd	2
1764.4.k.bd.1549.2		8	7.5	odd	6
2352.4.a.cl.1.2		4	84.59	odd	6
2352.4.a.cq.1.3		4	84.11	even	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.32752 + 9.22754i) q^{5} +O(q^{10})\)	\(q+(5.32752 + 9.22754i) q^{5} +(-3.32699 + 5.76252i) q^{11} -75.9335 q^{13} +(52.1435 - 90.3152i) q^{17} +(42.7472 + 74.0402i) q^{19} +(-34.3366 - 59.4728i) q^{23} +(5.73498 - 9.93327i) q^{25} -87.7843 q^{29} +(31.3841 - 54.3589i) q^{31} +(-21.1047 - 36.5544i) q^{37} +313.904 q^{41} +306.591 q^{43} +(-107.541 - 186.266i) q^{47} +(262.512 - 454.684i) q^{53} -70.8986 q^{55} +(-180.246 + 312.195i) q^{59} +(400.363 + 693.449i) q^{61} +(-404.537 - 700.679i) q^{65} +(20.1143 - 34.8390i) q^{67} +298.781 q^{71} +(258.563 - 447.844i) q^{73} +(611.233 + 1058.69i) q^{79} +1328.55 q^{83} +1111.18 q^{85} +(319.969 + 554.203i) q^{89} +(-455.473 + 788.902i) q^{95} -1425.65 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{17} + 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} + 48 q^{31} - 256 q^{37} + 2016 q^{41} - 224 q^{43} - 864 q^{47} - 648 q^{53} - 4704 q^{55} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} + 672 q^{73} + 1984 q^{79} + 6240 q^{83} + 1360 q^{85} - 2160 q^{89} - 3744 q^{95} - 4032 q^{97}+O(q^{100}) \) 8 * q - 48 * q^17 + 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 + 48 * q^31 - 256 * q^37 + 2016 * q^41 - 224 * q^43 - 864 * q^47 - 648 * q^53 - 4704 * q^55 - 336 * q^59 + 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 + 672 * q^73 + 1984 * q^79 + 6240 * q^83 + 1360 * q^85 - 2160 * q^89 - 3744 * q^95 - 4032 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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