LMFDB - Embedded newform 1764.4.t.b.1097.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(521,1764)]

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, 3, 1]))

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

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

chi := DirichletCharacter("1764.521");

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.t (of order $6$, 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:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1097.7
Root			$-8.00527 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.1097
Dual form			1764.4.t.b.521.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.36813 - 7.56582i) q^{5} +O(q^{10})$	$q+(4.36813 - 7.56582i) q^{5} +(7.60916 - 4.39315i) q^{11} +11.8322i q^{13} +(-22.2920 - 38.6108i) q^{17} +(-10.0856 - 5.82291i) q^{19} +(123.521 + 71.3151i) q^{23} +(24.3389 + 42.1563i) q^{25} +234.018i q^{29} +(-252.809 + 145.960i) q^{31} +(44.4515 - 76.9923i) q^{37} -145.961 q^{41} +144.633 q^{43} +(120.183 - 208.164i) q^{47} +(263.538 - 152.154i) q^{53} -76.7594i q^{55} +(-3.54196 - 6.13486i) q^{59} +(149.810 + 86.4927i) q^{61} +(89.5199 + 51.6844i) q^{65} +(243.280 + 421.373i) q^{67} +653.710i q^{71} +(99.0378 - 57.1795i) q^{73} +(-147.307 + 255.144i) q^{79} +877.193 q^{83} -389.496 q^{85} +(-710.379 + 1230.41i) q^{89} +(-88.1102 + 50.8704i) q^{95} -738.981i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	4.36813	−	7.56582i	0.390697	−	0.676707i	−0.601844	−	0.798613i	$-0.705567\pi$
$5$	4.36813	−	7.56582i	0.390697	−	0.676707i	0.992542	+	0.121906i	$0.0389006\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.60916	−	4.39315i	0.208568	−	0.120417i	−0.392078	−	0.919932i	$-0.628244\pi$
$11$	7.60916	−	4.39315i	0.208568	−	0.120417i	0.600646	+	0.799515i	$0.294910\pi$
$12$		0			0
$13$			11.8322i			0.252435i	0.992003	+	0.126217i	$0.0402837\pi$
$13$			11.8322i			0.252435i	−0.992003	+	0.126217i	$0.959716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.2920	−	38.6108i	−0.318035	−	0.550853i	0.662043	−	0.749466i	$-0.269690\pi$
$17$	−22.2920	−	38.6108i	−0.318035	−	0.550853i	−0.980078	+	0.198613i	$0.936356\pi$
$18$		0			0
$19$	−10.0856	−	5.82291i	−0.121778	−	0.0703088i	0.437873	−	0.899037i	$-0.355732\pi$
$19$	−10.0856	−	5.82291i	−0.121778	−	0.0703088i	−0.559652	+	0.828728i	$0.689065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	123.521	+	71.3151i	1.11982	+	0.646531i	0.941356	−	0.337414i	$-0.109552\pi$
$23$	123.521	+	71.3151i	1.11982	+	0.646531i	0.178469	+	0.983946i	$0.442886\pi$
$24$		0			0
$25$	24.3389	+	42.1563i	0.194711	+	0.337250i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			234.018i			1.49848i	0.662298	+	0.749241i	$0.269581\pi$
$29$			234.018i			1.49848i	−0.662298	+	0.749241i	$0.730419\pi$
$30$		0			0
$31$	−252.809	+	145.960i	−1.46471	+	0.845649i	−0.999223	−	0.0394074i	$-0.987453\pi$
$31$	−252.809	+	145.960i	−1.46471	+	0.845649i	−0.465484	+	0.885056i	$0.654120\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	44.4515	−	76.9923i	0.197508	−	0.342093i	−0.750212	−	0.661197i	$-0.770049\pi$
$37$	44.4515	−	76.9923i	0.197508	−	0.342093i	0.947720	+	0.319104i	$0.103382\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−145.961			−0.555984			−0.277992	−	0.960583i	$-0.589669\pi$
$41$	−145.961			−0.555984			−0.277992	+	0.960583i	$0.589669\pi$
$42$		0			0
$43$	144.633			0.512938			0.256469	−	0.966552i	$-0.417441\pi$
$43$	144.633			0.512938			0.256469	+	0.966552i	$0.417441\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	120.183	−	208.164i	0.372991	−	0.646039i	−0.617033	−	0.786937i	$-0.711666\pi$
$47$	120.183	−	208.164i	0.372991	−	0.646039i	0.990024	+	0.140898i	$0.0449990\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	263.538	−	152.154i	0.683015	−	0.394339i	−0.117975	−	0.993017i	$-0.537640\pi$
$53$	263.538	−	152.154i	0.683015	−	0.394339i	0.800990	+	0.598678i	$0.204307\pi$
$54$		0			0
$55$		−	76.7594i		−	0.188186i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.54196	−	6.13486i	−0.00781566	−	0.0135371i	0.862091	−	0.506753i	$-0.169154\pi$
$59$	−3.54196	−	6.13486i	−0.00781566	−	0.0135371i	−0.869907	+	0.493216i	$0.835821\pi$
$60$		0			0
$61$	149.810	+	86.4927i	0.314445	+	0.181545i	0.648914	−	0.760862i	$-0.275223\pi$
$61$	149.810	+	86.4927i	0.314445	+	0.181545i	−0.334469	+	0.942407i	$0.608557\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	89.5199	+	51.6844i	0.170824	+	0.0986255i
$66$		0			0
$67$	243.280	+	421.373i	0.443603	+	0.768342i	0.997954	−	0.0639408i	$-0.0203669\pi$
$67$	243.280	+	421.373i	0.443603	+	0.768342i	−0.554351	+	0.832283i	$0.687034\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			653.710i			1.09269i	0.837560	+	0.546345i	$0.183981\pi$
$71$			653.710i			1.09269i	−0.837560	+	0.546345i	$0.816019\pi$
$72$		0			0
$73$	99.0378	−	57.1795i	0.158788	−	0.0916761i	−0.418501	−	0.908217i	$-0.637444\pi$
$73$	99.0378	−	57.1795i	0.158788	−	0.0916761i	0.577288	+	0.816540i	$0.304111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−147.307	+	255.144i	−0.209789	+	0.363366i	−0.951648	−	0.307190i	$-0.900611\pi$
$79$	−147.307	+	255.144i	−0.209789	+	0.363366i	0.741859	+	0.670556i	$0.233945\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	877.193			1.16005			0.580027	−	0.814597i	$-0.303042\pi$
$83$	877.193			1.16005			0.580027	+	0.814597i	$0.303042\pi$
$84$		0			0
$85$	−389.496			−0.497021
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−710.379	+	1230.41i	−0.846068	+	1.46543i	0.0386225	+	0.999254i	$0.487703\pi$
$89$	−710.379	+	1230.41i	−0.846068	+	1.46543i	−0.884691	+	0.466179i	$0.845630\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−88.1102	+	50.8704i	−0.0951570	+	0.0549389i
$96$		0			0
$97$		−	738.981i		−	0.773527i	−0.922179	−	0.386764i	$-0.873593\pi$
$97$		−	738.981i		−	0.773527i	0.922179	−	0.386764i	$-0.126407\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	831.442	+	1440.10i	0.819125	+	1.41877i	0.906328	+	0.422575i	$0.138874\pi$
$101$	831.442	+	1440.10i	0.819125	+	1.41877i	−0.0872031	+	0.996191i	$0.527793\pi$
$102$		0			0
$103$	394.807	+	227.942i	0.377684	+	0.218056i	0.676810	−	0.736158i	$-0.263362\pi$
$103$	394.807	+	227.942i	0.377684	+	0.218056i	−0.299126	+	0.954214i	$0.596695\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1279.07	−	738.474i	−1.15563	−	0.667205i	−0.205380	−	0.978682i	$-0.565843\pi$
$107$	−1279.07	−	738.474i	−1.15563	−	0.667205i	−0.950254	+	0.311477i	$0.899176\pi$
$108$		0			0
$109$	−784.202	−	1358.28i	−0.689110	−	1.19357i	−0.972126	−	0.234458i	$-0.924668\pi$
$109$	−784.202	−	1358.28i	−0.689110	−	1.19357i	0.283016	−	0.959115i	$-0.408665\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1077.27i		−	0.896820i	−0.893828	−	0.448410i	$-0.851990\pi$
$113$		−	1077.27i		−	0.896820i	0.893828	−	0.448410i	$-0.148010\pi$
$114$		0			0
$115$	1079.11	−	623.026i	0.875025	−	0.505196i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−626.900	+	1085.82i	−0.471000	+	0.815795i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1517.29			1.08569
$126$		0			0
$127$	1518.00			1.06063			0.530317	−	0.847799i	$-0.322073\pi$
$127$	1518.00			1.06063			0.530317	+	0.847799i	$0.322073\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−201.672	+	349.306i	−0.134505	+	0.232969i	−0.925408	−	0.378972i	$-0.876278\pi$
$131$	−201.672	+	349.306i	−0.134505	+	0.232969i	0.790903	+	0.611941i	$0.209611\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	467.650	−	269.998i	0.291636	−	0.168376i	−0.347044	−	0.937849i	$-0.612814\pi$
$137$	467.650	−	269.998i	0.291636	−	0.168376i	0.638679	+	0.769473i	$0.279481\pi$
$138$		0			0
$139$		−	2042.22i		−	1.24618i	−0.782151	−	0.623089i	$-0.785877\pi$
$139$		−	2042.22i		−	1.24618i	0.782151	−	0.623089i	$-0.214123\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	51.9805	+	90.0328i	0.0303974	+	0.0526498i
$144$		0			0
$145$	1770.53	+	1022.22i	1.01403	+	0.585452i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−180.517	−	104.222i	−0.0992521	−	0.0573032i	0.449552	−	0.893254i	$-0.351584\pi$
$149$	−180.517	−	104.222i	−0.0992521	−	0.0573032i	−0.548804	+	0.835951i	$0.684917\pi$
$150$		0			0
$151$	230.045	+	398.449i	0.123979	+	0.214737i	0.921333	−	0.388774i	$-0.127101\pi$
$151$	230.045	+	398.449i	0.123979	+	0.214737i	−0.797355	+	0.603511i	$0.793768\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2550.28i			1.32157i
$156$		0			0
$157$	2330.15	−	1345.31i	1.18450	−	0.683869i	0.227446	−	0.973791i	$-0.426963\pi$
$157$	2330.15	−	1345.31i	1.18450	−	0.683869i	0.957050	+	0.289921i	$0.0936292\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1137.44	−	1970.10i	0.546571	−	0.946689i	−0.451935	−	0.892051i	$-0.649266\pi$
$163$	1137.44	−	1970.10i	0.546571	−	0.946689i	0.998506	−	0.0546381i	$-0.0174005\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3214.68			1.48958			0.744789	−	0.667300i	$-0.232550\pi$
$167$	3214.68			1.48958			0.744789	+	0.667300i	$0.232550\pi$
$168$		0			0
$169$	2057.00			0.936277
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1782.11	+	3086.71i	−0.783187	+	1.35652i	0.146889	+	0.989153i	$0.453074\pi$
$173$	−1782.11	+	3086.71i	−0.783187	+	1.35652i	−0.930076	+	0.367367i	$0.880259\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−302.852	+	174.852i	−0.126459	+	0.0730113i	−0.561895	−	0.827208i	$-0.689928\pi$
$179$	−302.852	+	174.852i	−0.126459	+	0.0730113i	0.435436	+	0.900220i	$0.356594\pi$
$180$		0			0
$181$			1664.25i			0.683439i	0.939802	+	0.341720i	$0.111009\pi$
$181$			1664.25i			0.683439i	−0.939802	+	0.341720i	$0.888991\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−388.340	−	672.624i	−0.154331	−	0.267310i
$186$		0			0
$187$	−339.246	−	195.864i	−0.132664	−	0.0765935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−213.721	−	123.392i	−0.0809650	−	0.0467452i	0.458971	−	0.888451i	$-0.348218\pi$
$191$	−213.721	−	123.392i	−0.0809650	−	0.0467452i	−0.539936	+	0.841706i	$0.681552\pi$
$192$		0			0
$193$	−1274.22	−	2207.01i	−0.475235	−	0.823130i	0.524363	−	0.851495i	$-0.324303\pi$
$193$	−1274.22	−	2207.01i	−0.475235	−	0.823130i	−0.999598	+	0.0283643i	$0.990970\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4579.00i			1.65604i	0.560697	+	0.828021i	$0.310533\pi$
$197$			4579.00i			1.65604i	−0.560697	+	0.828021i	$0.689467\pi$
$198$		0			0
$199$	−2151.47	+	1242.15i	−0.766402	+	0.442482i	−0.831590	−	0.555391i	$-0.812569\pi$
$199$	−2151.47	+	1242.15i	−0.766402	+	0.442482i	0.0651876	+	0.997873i	$0.479235\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−637.578	+	1104.32i	−0.217221	+	0.376238i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−102.324			−0.0338655
$210$		0			0
$211$	5736.87			1.87177			0.935883	−	0.352311i	$-0.114604\pi$
$211$	5736.87			1.87177			0.935883	+	0.352311i	$0.114604\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	631.775	−	1094.27i	0.200403	−	0.347109i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	456.849	−	263.762i	0.139054	−	0.0802830i
$222$		0			0
$223$		−	5391.46i		−	1.61901i	−0.587115	−	0.809504i	$-0.699736\pi$
$223$		−	5391.46i		−	1.61901i	0.587115	−	0.809504i	$-0.300264\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−747.447	−	1294.62i	−0.218545	−	0.378532i	0.735818	−	0.677179i	$-0.236798\pi$
$227$	−747.447	−	1294.62i	−0.218545	−	0.378532i	−0.954364	+	0.298647i	$0.903465\pi$
$228$		0			0
$229$	−693.480	−	400.381i	−0.200115	−	0.115537i	0.396594	−	0.917994i	$-0.370192\pi$
$229$	−693.480	−	400.381i	−0.200115	−	0.115537i	−0.596709	+	0.802457i	$0.703525\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2273.75	+	1312.75i	0.639305	+	0.369103i	0.784347	−	0.620323i	$-0.212998\pi$
$233$	2273.75	+	1312.75i	0.639305	+	0.369103i	−0.145042	+	0.989426i	$0.546332\pi$
$234$		0			0
$235$	−1049.95	−	1818.57i	−0.291453	−	0.504811i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6283.85i			1.70070i	0.526214	+	0.850352i	$0.323611\pi$
$239$			6283.85i			1.70070i	−0.526214	+	0.850352i	$0.676389\pi$
$240$		0			0
$241$	811.225	−	468.361i	0.216828	−	0.125186i	−0.387653	−	0.921806i	$-0.626714\pi$
$241$	811.225	−	468.361i	0.216828	−	0.125186i	0.604481	+	0.796620i	$0.293381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	68.8976	−	119.334i	0.0177484	−	0.0307411i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2831.45			0.712031			0.356015	−	0.934480i	$-0.384135\pi$
$251$	2831.45			0.712031			0.356015	+	0.934480i	$0.384135\pi$
$252$		0			0
$253$	1253.19			0.311413
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2844.65	+	4927.08i	−0.690445	+	1.19589i	0.281247	+	0.959636i	$0.409252\pi$
$257$	−2844.65	+	4927.08i	−0.690445	+	1.19589i	−0.971692	+	0.236251i	$0.924081\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3479.60	−	2008.95i	0.815823	−	0.471016i	−0.0331509	−	0.999450i	$-0.510554\pi$
$263$	3479.60	−	2008.95i	0.815823	−	0.471016i	0.848974	+	0.528435i	$0.177221\pi$
$264$		0			0
$265$		−	2658.51i		−	0.616268i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1908.81	+	3306.16i	0.432648	+	0.749368i	0.997100	−	0.0760977i	$-0.0242461\pi$
$269$	1908.81	+	3306.16i	0.432648	+	0.749368i	−0.564453	+	0.825465i	$0.690913\pi$
$270$		0			0
$271$	6193.16	+	3575.62i	1.38822	+	0.801489i	0.993115	−	0.117147i	$-0.0373747\pi$
$271$	6193.16	+	3575.62i	1.38822	+	0.801489i	0.395105	+	0.918636i	$0.370708\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	370.398	+	213.849i	0.0812212	+	0.0468931i
$276$		0			0
$277$	2741.91	+	4749.12i	0.594748	+	1.03013i	0.993582	+	0.113111i	$0.0360815\pi$
$277$	2741.91	+	4749.12i	0.594748	+	1.03013i	−0.398834	+	0.917023i	$0.630585\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5525.72i			1.17308i	0.809919	+	0.586542i	$0.199511\pi$
$281$			5525.72i			1.17308i	−0.809919	+	0.586542i	$0.800489\pi$
$282$		0			0
$283$	−217.828	+	125.763i	−0.0457545	+	0.0264164i	−0.522703	−	0.852515i	$-0.675076\pi$
$283$	−217.828	+	125.763i	−0.0457545	+	0.0264164i	0.476948	+	0.878931i	$0.341743\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1462.64	−	2533.36i	0.297707	−	0.515645i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9296.44			1.85360			0.926798	−	0.375560i	$-0.122550\pi$
$293$	9296.44			1.85360			0.926798	+	0.375560i	$0.122550\pi$
$294$		0			0
$295$	−61.8869			−0.0122142
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−843.811	+	1461.52i	−0.163207	+	0.282683i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1308.78	−	755.622i	0.245706	−	0.141858i
$306$		0			0
$307$			1498.96i			0.278666i	0.990246	+	0.139333i	$0.0444958\pi$
$307$			1498.96i			0.278666i	−0.990246	+	0.139333i	$0.955504\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3129.10	+	5419.77i	0.570531	+	0.988189i	0.996511	+	0.0834566i	$0.0265960\pi$
$311$	3129.10	+	5419.77i	0.570531	+	0.988189i	−0.425980	+	0.904733i	$0.640071\pi$
$312$		0			0
$313$	1843.53	+	1064.36i	0.332915	+	0.192208i	0.657134	−	0.753773i	$-0.271768\pi$
$313$	1843.53	+	1064.36i	0.332915	+	0.192208i	−0.324220	+	0.945982i	$0.605102\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4547.45	−	2625.47i	−0.805710	−	0.465177i	0.0397541	−	0.999209i	$-0.487343\pi$
$317$	−4547.45	−	2625.47i	−0.805710	−	0.465177i	−0.845464	+	0.534033i	$0.820676\pi$
$318$		0			0
$319$	1028.07	+	1780.68i	0.180442	+	0.312535i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			519.216i			0.0894427i
$324$		0			0
$325$	−498.799	+	287.982i	−0.0851336	+	0.0491519i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4401.99	+	7624.47i	−0.730983	+	1.26610i	0.225481	+	0.974248i	$0.427605\pi$
$331$	−4401.99	+	7624.47i	−0.730983	+	1.26610i	−0.956464	+	0.291852i	$0.905729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4250.71			0.693257
$336$		0			0
$337$	−1428.63			−0.230927			−0.115463	−	0.993312i	$-0.536835\pi$
$337$	−1428.63			−0.230927			−0.115463	+	0.993312i	$0.536835\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1282.45	+	2221.26i	−0.203661	+	0.352751i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7896.72	−	4559.17i	1.22167	−	0.705329i	0.256393	−	0.966573i	$-0.417466\pi$
$347$	7896.72	−	4559.17i	1.22167	−	0.705329i	0.965273	+	0.261244i	$0.0841327\pi$
$348$		0			0
$349$		−	11899.1i		−	1.82506i	−0.409011	−	0.912530i	$-0.634126\pi$
$349$		−	11899.1i		−	1.82506i	0.409011	−	0.912530i	$-0.365874\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4578.82	+	7930.75i	0.690385	+	1.19578i	0.971712	+	0.236170i	$0.0758923\pi$
$353$	4578.82	+	7930.75i	0.690385	+	1.19578i	−0.281327	+	0.959612i	$0.590774\pi$
$354$		0			0
$355$	4945.85	+	2855.49i	0.739432	+	0.426911i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6729.83	−	3885.47i	−0.989379	−	0.571218i	−0.0842902	−	0.996441i	$-0.526862\pi$
$359$	−6729.83	−	3885.47i	−0.989379	−	0.571218i	−0.905089	+	0.425223i	$0.860196\pi$
$360$		0			0
$361$	−3361.69	−	5822.61i	−0.490113	−	0.848901i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	999.070i		−	0.143270i
$366$		0			0
$367$	5456.63	−	3150.39i	0.776114	−	0.448090i	−0.0589371	−	0.998262i	$-0.518771\pi$
$367$	5456.63	−	3150.39i	0.776114	−	0.448090i	0.835051	+	0.550172i	$0.185438\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5051.76	+	8749.90i	−0.701260	+	1.21462i	0.266764	+	0.963762i	$0.414046\pi$
$373$	−5051.76	+	8749.90i	−0.701260	+	1.21462i	−0.968024	+	0.250857i	$0.919288\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2768.93			−0.378269
$378$		0			0
$379$	1122.50			0.152134			0.0760671	−	0.997103i	$-0.475764\pi$
$379$	1122.50			0.152134			0.0760671	+	0.997103i	$0.475764\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5313.98	+	9204.08i	−0.708960	+	1.22795i	0.256283	+	0.966602i	$0.417502\pi$
$383$	−5313.98	+	9204.08i	−0.708960	+	1.22795i	−0.965243	+	0.261353i	$0.915831\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11571.8	+	6680.96i	−1.50826	+	0.870792i	−0.508301	+	0.861179i	$0.669726\pi$
$389$	−11571.8	+	6680.96i	−1.50826	+	0.870792i	−0.999954	+	0.00961230i	$0.996940\pi$
$390$		0			0
$391$		−	6359.01i		−	0.822478i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1286.91	+	2229.00i	0.163928	+	0.283932i
$396$		0			0
$397$	2304.31	+	1330.40i	0.291310	+	0.168188i	0.638533	−	0.769595i	$-0.279542\pi$
$397$	2304.31	+	1330.40i	0.291310	+	0.168188i	−0.347222	+	0.937783i	$0.612875\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5288.60	+	3053.37i	0.658603	+	0.380245i	0.791745	−	0.610852i	$-0.209173\pi$
$401$	5288.60	+	3053.37i	0.658603	+	0.380245i	−0.133141	+	0.991097i	$0.542506\pi$
$402$		0			0
$403$	−1727.02	−	2991.28i	−0.213471	−	0.369743i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	781.129i		−	0.0951330i
$408$		0			0
$409$	1998.03	−	1153.56i	0.241556	−	0.139462i	−0.374336	−	0.927293i	$-0.622129\pi$
$409$	1998.03	−	1153.56i	0.241556	−	0.139462i	0.615892	+	0.787831i	$0.288796\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3831.69	−	6636.69i	0.453230	−	0.785017i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11116.9			1.29617			0.648083	−	0.761570i	$-0.275571\pi$
$419$	11116.9			1.29617			0.648083	+	0.761570i	$0.275571\pi$
$420$		0			0
$421$	4188.14			0.484840			0.242420	−	0.970171i	$-0.422059\pi$
$421$	4188.14			0.484840			0.242420	+	0.970171i	$0.422059\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1085.12	−	1879.49i	0.123850	−	0.214515i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4253.15	−	2455.56i	0.475329	−	0.274432i	−0.243139	−	0.969992i	$-0.578177\pi$
$431$	4253.15	−	2455.56i	0.475329	−	0.274432i	0.718468	+	0.695560i	$0.244844\pi$
$432$		0			0
$433$			14412.6i			1.59960i	0.600268	+	0.799799i	$0.295061\pi$
$433$			14412.6i			1.59960i	−0.600268	+	0.799799i	$0.704939\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−830.522	−	1438.51i	−0.0909137	−	0.157467i
$438$		0			0
$439$	−7721.97	−	4458.28i	−0.839520	−	0.484697i	0.0175809	−	0.999845i	$-0.494404\pi$
$439$	−7721.97	−	4458.28i	−0.839520	−	0.484697i	−0.857101	+	0.515148i	$0.827737\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1022.14	−	590.133i	−0.109624	−	0.0632913i	0.444186	−	0.895935i	$-0.353493\pi$
$443$	−1022.14	−	590.133i	−0.109624	−	0.0632913i	−0.553809	+	0.832643i	$0.686826\pi$
$444$		0			0
$445$	6206.05	+	10749.2i	0.661113	+	1.14508i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			12536.1i			1.31763i	0.752307	+	0.658813i	$0.228941\pi$
$449$			12536.1i			1.31763i	−0.752307	+	0.658813i	$0.771059\pi$
$450$		0			0
$451$	−1110.64	+	641.230i	−0.115960	+	0.0669498i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5051.92	−	8750.18i	0.517109	−	0.895659i	−0.482694	−	0.875789i	$-0.660342\pi$
$457$	5051.92	−	8750.18i	0.517109	−	0.895659i	0.999803	−	0.0198696i	$-0.00632509\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	418.675			0.0422985			0.0211493	−	0.999776i	$-0.493267\pi$
$461$	418.675			0.0422985			0.0211493	+	0.999776i	$0.493267\pi$
$462$		0			0
$463$	−2178.42			−0.218661			−0.109330	−	0.994005i	$-0.534871\pi$
$463$	−2178.42			−0.218661			−0.109330	+	0.994005i	$0.534871\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1929.85	+	3342.59i	−0.191226	+	0.331213i	−0.945657	−	0.325166i	$-0.894580\pi$
$467$	−1929.85	+	3342.59i	−0.191226	+	0.331213i	0.754431	+	0.656380i	$0.227913\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1100.54	−	635.394i	0.106982	−	0.0617663i
$474$		0			0
$475$		−	566.894i		−	0.0547597i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8106.86	−	14041.5i	−0.773302	−	1.33940i	−0.935744	−	0.352680i	$-0.885270\pi$
$479$	−8106.86	−	14041.5i	−0.773302	−	1.33940i	0.162442	−	0.986718i	$-0.448063\pi$
$480$		0			0
$481$	910.985	+	525.957i	0.0863562	+	0.0498578i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5591.00	−	3227.96i	−0.523452	−	0.302215i
$486$		0			0
$487$	−9902.83	−	17152.2i	−0.921437	−	1.59598i	−0.797193	−	0.603725i	$-0.793683\pi$
$487$	−9902.83	−	17152.2i	−0.921437	−	1.59598i	−0.124244	−	0.992252i	$-0.539651\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11104.5i		−	1.02065i	−0.859983	−	0.510323i	$-0.829526\pi$
$491$		−	11104.5i		−	1.02065i	0.859983	−	0.510323i	$-0.170474\pi$
$492$		0			0
$493$	9035.61	−	5216.71i	0.825443	−	0.476570i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−435.024	+	753.483i	−0.0390267	+	0.0675963i	−0.884879	−	0.465821i	$-0.845759\pi$
$499$	−435.024	+	753.483i	−0.0390267	+	0.0675963i	0.845852	+	0.533417i	$0.179092\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12435.5			−1.10233			−0.551166	−	0.834396i	$-0.685817\pi$
$503$	−12435.5			−1.10233			−0.551166	+	0.834396i	$0.685817\pi$
$504$		0			0
$505$	14527.4			1.28012
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9850.04	−	17060.8i	0.857751	−	1.48567i	−0.0163174	−	0.999867i	$-0.505194\pi$
$509$	9850.04	−	17060.8i	0.857751	−	1.48567i	0.874069	−	0.485802i	$-0.161472\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3449.13	−	1991.36i	0.295120	−	0.170388i
$516$		0			0
$517$		−	2111.94i		−	0.179657i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4813.26	−	8336.82i	−0.404747	−	0.701042i	0.589545	−	0.807735i	$-0.299307\pi$
$521$	−4813.26	−	8336.82i	−0.404747	−	0.701042i	−0.994292	+	0.106694i	$0.965974\pi$
$522$		0			0
$523$	−1451.42	−	837.977i	−0.121350	−	0.0700615i	0.438096	−	0.898928i	$-0.355653\pi$
$523$	−1451.42	−	837.977i	−0.121350	−	0.0700615i	−0.559446	+	0.828867i	$0.688986\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11271.2	+	6507.45i	0.931656	+	0.537892i
$528$		0			0
$529$	4088.17	+	7080.92i	0.336005	+	0.581978i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1727.04i		−	0.140350i
$534$		0			0
$535$	−11174.3	+	6451.50i	−0.903006	+	0.521351i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4323.06	−	7487.76i	0.343554	−	0.595054i	−0.641536	−	0.767093i	$-0.721702\pi$
$541$	4323.06	−	7487.76i	0.343554	−	0.595054i	0.985090	+	0.172040i	$0.0550357\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13702.0			−1.07693
$546$		0			0
$547$	183.297			0.0143276			0.00716382	−	0.999974i	$-0.497720\pi$
$547$	183.297			0.0143276			0.00716382	+	0.999974i	$0.497720\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1362.66	−	2360.20i	0.105356	−	0.182483i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18922.7	+	10925.0i	−1.43946	+	0.831072i	−0.997812	−	0.0661167i	$-0.978939\pi$
$557$	−18922.7	+	10925.0i	−1.43946	+	0.831072i	−0.441647	+	0.897189i	$0.645606\pi$
$558$		0			0
$559$			1711.32i			0.129483i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7624.11	−	13205.3i	−0.570724	−	0.988523i	−0.996492	−	0.0836908i	$-0.973329\pi$
$563$	−7624.11	−	13205.3i	−0.570724	−	0.988523i	0.425767	−	0.904833i	$-0.360004\pi$
$564$		0			0
$565$	−8150.40	−	4705.64i	−0.606885	−	0.350385i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10861.9	+	6271.10i	0.800269	+	0.462035i	0.843565	−	0.537027i	$-0.180453\pi$
$569$	10861.9	+	6271.10i	0.800269	+	0.462035i	−0.0432965	+	0.999062i	$0.513786\pi$
$570$		0			0
$571$	−5721.40	−	9909.76i	−0.419323	−	0.726288i	0.576549	−	0.817063i	$-0.304399\pi$
$571$	−5721.40	−	9909.76i	−0.419323	−	0.726288i	−0.995871	+	0.0907745i	$0.971066\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6942.93i			0.503548i
$576$		0			0
$577$	−9599.35	+	5542.18i	−0.692593	+	0.399869i	−0.804583	−	0.593841i	$-0.797611\pi$
$577$	−9599.35	+	5542.18i	−0.692593	+	0.399869i	0.111990	+	0.993709i	$0.464278\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1336.87	−	2315.53i	0.0949700	−	0.164493i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1493.49			−0.105014			−0.0525068	−	0.998621i	$-0.516721\pi$
$587$	−1493.49			−0.105014			−0.0525068	+	0.998621i	$0.516721\pi$
$588$		0			0
$589$	3399.64			0.237826
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2832.72	+	4906.42i	−0.196165	+	0.339768i	−0.947282	−	0.320401i	$-0.896182\pi$
$593$	−2832.72	+	4906.42i	−0.196165	+	0.339768i	0.751117	+	0.660170i	$0.229516\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4487.18	+	2590.68i	−0.306079	+	0.176715i	−0.645171	−	0.764039i	$-0.723214\pi$
$599$	−4487.18	+	2590.68i	−0.306079	+	0.176715i	0.339091	+	0.940753i	$0.389880\pi$
$600$		0			0
$601$			13911.4i			0.944186i	0.881549	+	0.472093i	$0.156501\pi$
$601$			13911.4i			0.944186i	−0.881549	+	0.472093i	$0.843499\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5476.76	+	9486.03i	0.368036	+	0.637458i
$606$		0			0
$607$	−6537.98	−	3774.70i	−0.437180	−	0.252406i	0.265221	−	0.964188i	$-0.414555\pi$
$607$	−6537.98	−	3774.70i	−0.437180	−	0.252406i	−0.702401	+	0.711782i	$0.747889\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2463.03	+	1422.03i	0.163083	+	0.0941557i
$612$		0			0
$613$	6443.77	+	11160.9i	0.424570	+	0.735377i	0.996380	−	0.0850093i	$-0.0270920\pi$
$613$	6443.77	+	11160.9i	0.424570	+	0.735377i	−0.571810	+	0.820386i	$0.693759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22388.7i		−	1.46084i	−0.683000	−	0.730419i	$-0.739325\pi$
$617$		−	22388.7i		−	1.46084i	0.683000	−	0.730419i	$-0.260675\pi$
$618$		0			0
$619$	−4791.17	+	2766.18i	−0.311104	+	0.179616i	−0.647420	−	0.762133i	$-0.724152\pi$
$619$	−4791.17	+	2766.18i	−0.311104	+	0.179616i	0.336316	+	0.941749i	$0.390819\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	3585.37	−	6210.04i	0.229463	−	0.397442i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3963.65			−0.251257
$630$		0			0
$631$	−24188.4			−1.52603			−0.763015	−	0.646381i	$-0.776282\pi$
$631$	−24188.4			−1.52603			−0.763015	+	0.646381i	$0.776282\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6630.81	−	11484.9i	0.414387	−	0.717739i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7955.44	−	4593.08i	0.490205	−	0.283020i	−0.234455	−	0.972127i	$-0.575330\pi$
$641$	7955.44	−	4593.08i	0.490205	−	0.283020i	0.724659	+	0.689107i	$0.241997\pi$
$642$		0			0
$643$			21051.2i			1.29110i	0.763718	+	0.645550i	$0.223372\pi$
$643$			21051.2i			1.29110i	−0.763718	+	0.645550i	$0.776628\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1834.16	+	3176.85i	0.111450	+	0.193037i	0.916355	−	0.400367i	$-0.131117\pi$
$647$	1834.16	+	3176.85i	0.111450	+	0.193037i	−0.804905	+	0.593403i	$0.797784\pi$
$648$		0			0
$649$	−53.9027	−	31.1207i	−0.00326019	−	0.00188227i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6653.57	−	3841.44i	−0.398736	−	0.230210i	0.287203	−	0.957870i	$-0.407275\pi$
$653$	−6653.57	−	3841.44i	−0.398736	−	0.230210i	−0.685938	+	0.727660i	$0.740608\pi$
$654$		0			0
$655$	1761.85	+	3051.62i	0.105101	+	0.182041i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			529.107i			0.0312763i	0.999878	+	0.0156382i	$0.00497798\pi$
$659$			529.107i			0.0312763i	−0.999878	+	0.0156382i	$0.995022\pi$
$660$		0			0
$661$	11918.0	−	6880.84i	0.701294	−	0.404892i	−0.106535	−	0.994309i	$-0.533976\pi$
$661$	11918.0	−	6880.84i	0.701294	−	0.404892i	0.807829	+	0.589417i	$0.200642\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−16689.0	+	28906.1i	−0.968815	+	1.67804i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1519.90			0.0874443
$672$		0			0
$673$	−31045.7			−1.77819			−0.889095	−	0.457722i	$-0.848666\pi$
$673$	−31045.7			−1.77819			−0.889095	+	0.457722i	$0.848666\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2924.24	−	5064.94i	0.166009	−	0.287535i	−0.771004	−	0.636830i	$-0.780245\pi$
$677$	2924.24	−	5064.94i	0.166009	−	0.287535i	0.937013	+	0.349295i	$0.113579\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11813.9	−	6820.78i	0.661856	−	0.382123i	−0.131127	−	0.991366i	$-0.541860\pi$
$683$	11813.9	−	6820.78i	0.661856	−	0.382123i	0.792984	+	0.609242i	$0.208526\pi$
$684$		0			0
$685$		−	4717.54i		−	0.263136i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1800.31	+	3118.23i	0.0995447	+	0.172417i
$690$		0			0
$691$	−12034.8	−	6948.27i	−0.662552	−	0.382525i	0.130696	−	0.991422i	$-0.458279\pi$
$691$	−12034.8	−	6948.27i	−0.662552	−	0.382525i	−0.793249	+	0.608898i	$0.791612\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15451.1	−	8920.68i	−0.843298	−	0.486879i
$696$		0			0
$697$	3253.76	+	5635.69i	0.176822	+	0.306265i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30902.8i		−	1.66503i	−0.554005	−	0.832513i	$-0.686901\pi$
$701$		−	30902.8i		−	1.66503i	0.554005	−	0.832513i	$-0.313099\pi$
$702$		0			0
$703$	−896.638	+	517.674i	−0.0481044	+	0.0277731i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2160.65	−	3742.35i	0.114450	−	0.198233i	−0.803110	−	0.595831i	$-0.796823\pi$
$709$	2160.65	−	3742.35i	0.114450	−	0.198233i	0.917560	+	0.397598i	$0.130156\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−41636.5			−2.18695
$714$		0			0
$715$	908.229			0.0475047
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16131.0	−	27939.7i	0.836696	−	1.44920i	−0.0559456	−	0.998434i	$-0.517817\pi$
$719$	16131.0	−	27939.7i	0.836696	−	1.44920i	0.892642	−	0.450767i	$-0.148849\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9865.30	+	5695.74i	−0.505363	+	0.291771i
$726$		0			0
$727$		−	12056.4i		−	0.615056i	−0.951539	−	0.307528i	$-0.900498\pi$
$727$		−	12056.4i		−	0.615056i	0.951539	−	0.307528i	$-0.0995017\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3224.15	−	5584.40i	−0.163132	−	0.282553i
$732$		0			0
$733$	22122.9	+	12772.7i	1.11477	+	0.643615i	0.940061	−	0.341005i	$-0.110767\pi$
$733$	22122.9	+	12772.7i	1.11477	+	0.643615i	0.174712	+	0.984620i	$0.444101\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3702.31	+	2137.53i	0.185043	+	0.106834i
$738$		0			0
$739$	12402.9	+	21482.4i	0.617384	+	1.06934i	0.989961	+	0.141340i	$0.0451410\pi$
$739$	12402.9	+	21482.4i	0.617384	+	1.06934i	−0.372577	+	0.928001i	$0.621526\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	23318.4i		−	1.15137i	−0.817671	−	0.575686i	$-0.804735\pi$
$743$		−	23318.4i		−	1.15137i	0.817671	−	0.575686i	$-0.195265\pi$
$744$		0			0
$745$	−1577.05	+	910.508i	−0.0775550	+	0.0447764i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	155.921	−	270.063i	0.00757607	−	0.0131221i	−0.862213	−	0.506547i	$-0.830922\pi$
$751$	155.921	−	270.063i	0.00757607	−	0.0131221i	0.869789	+	0.493425i	$0.164255\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4019.46			0.193752
$756$		0			0
$757$	−26444.1			−1.26965			−0.634825	−	0.772656i	$-0.718928\pi$
$757$	−26444.1			−1.26965			−0.634825	+	0.772656i	$0.718928\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10265.6	+	17780.5i	−0.488997	+	0.846968i	−0.999920	−	0.0126584i	$-0.995971\pi$
$761$	−10265.6	+	17780.5i	−0.488997	+	0.846968i	0.510922	+	0.859627i	$0.329304\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	72.5886	−	41.9090i	0.00341724	−	0.00197294i
$768$		0			0
$769$			19950.3i			0.935533i	0.883852	+	0.467767i	$0.154941\pi$
$769$			19950.3i			0.935533i	−0.883852	+	0.467767i	$0.845059\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19809.6	+	34311.3i	0.921737	+	1.59650i	0.796726	+	0.604340i	$0.206563\pi$
$773$	19809.6	+	34311.3i	0.921737	+	1.59650i	0.125011	+	0.992155i	$0.460103\pi$
$774$		0			0
$775$	−12306.2	−	7105.00i	−0.570390	−	0.329315i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1472.10	+	849.920i	0.0677068	+	0.0390906i
$780$		0			0
$781$	2871.84	+	4974.18i	0.131578	+	0.227900i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	23506.0i		−	1.06874i
$786$		0			0
$787$	−28877.8	+	16672.6i	−1.30798	+	0.755163i	−0.981759	−	0.190130i	$-0.939109\pi$
$787$	−28877.8	+	16672.6i	−1.30798	+	0.755163i	−0.326222	+	0.945293i	$0.605776\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1023.39	+	1772.57i	−0.0458283	+	0.0793769i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15138.0			−0.672793			−0.336397	−	0.941720i	$-0.609208\pi$
$797$	−15138.0			−0.672793			−0.336397	+	0.941720i	$0.609208\pi$
$798$		0			0
$799$	−10716.5			−0.474496
$800$		0			0
$801$		0			0
$802$		0			0
$803$	502.396	−	870.176i	0.0220787	−	0.0382414i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16961.8	+	9792.91i	−0.737139	+	0.425588i	−0.821028	−	0.570888i	$-0.806599\pi$
$809$	−16961.8	+	9792.91i	−0.737139	+	0.425588i	0.0838890	+	0.996475i	$0.473266\pi$
$810$		0			0
$811$		−	11787.4i		−	0.510371i	−0.966892	−	0.255186i	$-0.917863\pi$
$811$		−	11787.4i		−	0.510371i	0.966892	−	0.255186i	$-0.0821365\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9936.96	−	17211.3i	−0.427088	−	0.739737i
$816$		0			0
$817$	−1458.71	−	842.185i	−0.0624648	−	0.0360640i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	798.937	+	461.266i	0.0339623	+	0.0196082i	0.516885	−	0.856055i	$-0.327091\pi$
$821$	798.937	+	461.266i	0.0339623	+	0.0196082i	−0.482923	+	0.875663i	$0.660425\pi$
$822$		0			0
$823$	−11788.0	−	20417.4i	−0.499275	−	0.864770i	0.500725	−	0.865607i	$-0.333067\pi$
$823$	−11788.0	−	20417.4i	−0.499275	−	0.864770i	−1.00000		0.000836881i	$0.999734\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	9010.27i		−	0.378861i	−0.981894	−	0.189430i	$-0.939336\pi$
$827$		−	9010.27i		−	0.378861i	0.981894	−	0.189430i	$-0.0606641\pi$
$828$		0			0
$829$	1211.73	−	699.593i	0.0507661	−	0.0293098i	−0.474402	−	0.880308i	$-0.657336\pi$
$829$	1211.73	−	699.593i	0.0507661	−	0.0293098i	0.525168	+	0.850998i	$0.324002\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	14042.1	−	24321.7i	0.581974	−	1.00801i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10339.8			0.425472			0.212736	−	0.977110i	$-0.431763\pi$
$839$	10339.8			0.425472			0.212736	+	0.977110i	$0.431763\pi$
$840$		0			0
$841$	−30375.2			−1.24545
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8985.24	−	15562.9i	0.365801	−	0.633585i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10981.4	−	6340.12i	0.442348	−	0.255390i
$852$		0			0
$853$			39316.9i			1.57818i	0.614280	+	0.789088i	$0.289447\pi$
$853$			39316.9i			1.57818i	−0.614280	+	0.789088i	$0.710553\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−179.865	−	311.534i	−0.00716926	−	0.0124175i	0.862419	−	0.506196i	$-0.168949\pi$
$857$	−179.865	−	311.534i	−0.00716926	−	0.0124175i	−0.869588	+	0.493778i	$0.835615\pi$
$858$		0			0
$859$	−32209.0	−	18595.9i	−1.27934	−	0.738630i	−0.302617	−	0.953112i	$-0.597860\pi$
$859$	−32209.0	−	18595.9i	−1.27934	−	0.738630i	−0.976728	+	0.214482i	$0.931194\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11006.3	+	6354.50i	0.434136	+	0.250649i	0.701107	−	0.713056i	$-0.252689\pi$
$863$	11006.3	+	6354.50i	0.434136	+	0.250649i	−0.266971	+	0.963705i	$0.586023\pi$
$864$		0			0
$865$	15569.0	+	26966.2i	0.611978	+	1.05998i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			2588.57i			0.101049i
$870$		0			0
$871$	−4985.75	+	2878.53i	−0.193956	+	0.111981i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7481.41	+	12958.2i	−0.288061	+	0.498936i	−0.973347	−	0.229338i	$-0.926344\pi$
$877$	−7481.41	+	12958.2i	−0.288061	+	0.498936i	0.685286	+	0.728274i	$0.259677\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−48954.8			−1.87211			−0.936056	−	0.351851i	$-0.885552\pi$
$881$	−48954.8			−1.87211			−0.936056	+	0.351851i	$0.885552\pi$
$882$		0			0
$883$	4761.13			0.181455			0.0907276	−	0.995876i	$-0.471081\pi$
$883$	4761.13			0.181455			0.0907276	+	0.995876i	$0.471081\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19477.6	−	33736.1i	0.737308	−	1.27706i	−0.216395	−	0.976306i	$-0.569430\pi$
$887$	19477.6	−	33736.1i	0.737308	−	1.27706i	0.953703	−	0.300749i	$-0.0972367\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2424.24	+	1399.64i	−0.0908444	+	0.0524491i
$894$		0			0
$895$			3055.10i			0.114101i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−34157.1	−	59161.8i	−1.26719	−	2.19484i
$900$		0			0
$901$	−11749.6	−	6783.62i	−0.434445	−	0.250827i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12591.4	+	7269.64i	0.462488	+	0.267018i
$906$		0			0
$907$	−7610.74	−	13182.2i	−0.278622	−	0.482588i	0.692420	−	0.721494i	$-0.256544\pi$
$907$	−7610.74	−	13182.2i	−0.278622	−	0.482588i	−0.971043	+	0.238906i	$0.923211\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			6441.70i			0.234273i	0.993116	+	0.117137i	$0.0373716\pi$
$911$			6441.70i			0.234273i	−0.993116	+	0.117137i	$0.962628\pi$
$912$		0			0
$913$	6674.70	−	3853.64i	0.241950	−	0.139690i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−9334.12	+	16167.2i	−0.335042	+	0.580311i	−0.983493	−	0.180946i	$-0.942084\pi$
$919$	−9334.12	+	16167.2i	−0.335042	+	0.580311i	0.648451	+	0.761257i	$0.275417\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7734.79			−0.275833
$924$		0			0
$925$	4327.61			0.153828
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11274.5	−	19527.9i	0.398173	−	0.689656i	−0.595327	−	0.803483i	$-0.702978\pi$
$929$	11274.5	−	19527.9i	0.398173	−	0.689656i	0.993501	+	0.113827i	$0.0363109\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2963.74	+	1711.12i	−0.103663	+	0.0598497i
$936$		0			0
$937$			22939.2i			0.799779i	0.916563	+	0.399889i	$0.130951\pi$
$937$			22939.2i			0.799779i	−0.916563	+	0.399889i	$0.869049\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1808.45	−	3132.32i	−0.0626501	−	0.108513i	0.832999	−	0.553274i	$-0.186622\pi$
$941$	−1808.45	−	3132.32i	−0.0626501	−	0.108513i	−0.895649	+	0.444761i	$0.853289\pi$
$942$		0			0
$943$	−18029.3	−	10409.2i	−0.622604	−	0.359461i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12567.3	+	7255.75i	0.431239	+	0.248976i	0.699874	−	0.714266i	$-0.253239\pi$
$947$	12567.3	+	7255.75i	0.431239	+	0.248976i	−0.268635	+	0.963242i	$0.586573\pi$
$948$		0			0
$949$	676.557	+	1171.83i	0.0231422	+	0.0400835i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31649.8i		−	1.07580i	−0.843008	−	0.537901i	$-0.819218\pi$
$953$		−	31649.8i		−	1.07580i	0.843008	−	0.537901i	$-0.180782\pi$
$954$		0			0
$955$	−1867.12	+	1077.98i	−0.0632656	+	0.0365264i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	27712.9	−	48000.2i	0.930245	−	1.61123i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22263.8			−0.742691
$966$		0			0
$967$	26409.0			0.878238			0.439119	−	0.898429i	$-0.355291\pi$
$967$	26409.0			0.878238			0.439119	+	0.898429i	$0.355291\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4525.04	−	7837.61i	0.149553	−	0.259033i	−0.781510	−	0.623893i	$-0.785550\pi$
$971$	4525.04	−	7837.61i	0.149553	−	0.259033i	0.931062	+	0.364861i	$0.118883\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17401.7	+	10046.8i	−0.569834	+	0.328994i	−0.757083	−	0.653319i	$-0.773376\pi$
$977$	−17401.7	+	10046.8i	−0.569834	+	0.328994i	0.187249	+	0.982313i	$0.440043\pi$
$978$		0			0
$979$			12483.2i			0.407523i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12838.6	−	22237.2i	−0.416571	−	0.721522i	0.579021	−	0.815313i	$-0.303435\pi$
$983$	−12838.6	−	22237.2i	−0.416571	−	0.721522i	−0.995592	+	0.0937907i	$0.970102\pi$
$984$		0			0
$985$	34643.9	+	20001.7i	1.12066	+	0.647011i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17865.3	+	10314.5i	0.574400	+	0.331630i
$990$		0			0
$991$	−2851.55	−	4939.03i	−0.0914051	−	0.158318i	0.816697	−	0.577066i	$-0.195802\pi$
$991$	−2851.55	−	4939.03i	−0.0914051	−	0.158318i	−0.908103	+	0.418748i	$0.862469\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			21703.6i			0.691507i
$996$		0			0
$997$	8147.55	−	4703.99i	0.258812	−	0.149425i	−0.364981	−	0.931015i	$-0.618924\pi$
$997$	8147.55	−	4703.99i	0.258812	−	0.149425i	0.623792	+	0.781590i	$0.285591\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.b.1097.7		16
3.2	odd	2	inner	1764.4.t.b.1097.2		16
7.2	even	3		1764.4.f.a.881.4		16
7.3	odd	6	inner	1764.4.t.b.521.2		16
7.4	even	3		252.4.t.a.17.7	yes	16
7.5	odd	6		1764.4.f.a.881.14		16
7.6	odd	2		252.4.t.a.89.2	yes	16
21.2	odd	6		1764.4.f.a.881.13		16
21.5	even	6		1764.4.f.a.881.3		16
21.11	odd	6		252.4.t.a.17.2	✓	16
21.17	even	6	inner	1764.4.t.b.521.7		16
21.20	even	2		252.4.t.a.89.7	yes	16
28.11	odd	6		1008.4.bt.b.17.7		16
28.27	even	2		1008.4.bt.b.593.2		16
84.11	even	6		1008.4.bt.b.17.2		16
84.83	odd	2		1008.4.bt.b.593.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.2	✓	16	21.11	odd	6
252.4.t.a.17.7	yes	16	7.4	even	3
252.4.t.a.89.2	yes	16	7.6	odd	2
252.4.t.a.89.7	yes	16	21.20	even	2
1008.4.bt.b.17.2		16	84.11	even	6
1008.4.bt.b.17.7		16	28.11	odd	6
1008.4.bt.b.593.2		16	28.27	even	2
1008.4.bt.b.593.7		16	84.83	odd	2
1764.4.f.a.881.3		16	21.5	even	6
1764.4.f.a.881.4		16	7.2	even	3
1764.4.f.a.881.13		16	21.2	odd	6
1764.4.f.a.881.14		16	7.5	odd	6
1764.4.t.b.521.2		16	7.3	odd	6	inner
1764.4.t.b.521.7		16	21.17	even	6	inner
1764.4.t.b.1097.2		16	3.2	odd	2	inner
1764.4.t.b.1097.7		16	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.36813 - 7.56582i) q^{5} +O(q^{10})\)	\(q+(4.36813 - 7.56582i) q^{5} +(7.60916 - 4.39315i) q^{11} +11.8322i q^{13} +(-22.2920 - 38.6108i) q^{17} +(-10.0856 - 5.82291i) q^{19} +(123.521 + 71.3151i) q^{23} +(24.3389 + 42.1563i) q^{25} +234.018i q^{29} +(-252.809 + 145.960i) q^{31} +(44.4515 - 76.9923i) q^{37} -145.961 q^{41} +144.633 q^{43} +(120.183 - 208.164i) q^{47} +(263.538 - 152.154i) q^{53} -76.7594i q^{55} +(-3.54196 - 6.13486i) q^{59} +(149.810 + 86.4927i) q^{61} +(89.5199 + 51.6844i) q^{65} +(243.280 + 421.373i) q^{67} +653.710i q^{71} +(99.0378 - 57.1795i) q^{73} +(-147.307 + 255.144i) q^{79} +877.193 q^{83} -389.496 q^{85} +(-710.379 + 1230.41i) q^{89} +(-88.1102 + 50.8704i) q^{95} -738.981i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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