LMFDB - Embedded newform 1764.3.z.h.901.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 17x^{2} + 16x + 256 $ x^4 - x^3 + 17x^2 + 16x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.2
Root			$2.26556 + 3.92407i$ of defining polynomial
Character	$\chi$	$=$	1764.901
Dual form			1764.3.z.h.325.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.79669 + 2.19202i) q^{5} +(-9.79669 - 16.9684i) q^{11} +6.11609i q^{13} +(7.59339 - 4.38404i) q^{17} +(26.2967 + 15.1824i) q^{19} +(-12.0000 + 20.7846i) q^{23} +(-2.89008 - 5.00577i) q^{25} -13.5934 q^{29} +(24.2802 - 14.0182i) q^{31} +(24.2967 - 42.0831i) q^{37} +7.14387i q^{41} +53.7802 q^{43} +(-34.5934 - 19.9725i) q^{47} +(-30.7967 - 53.3414i) q^{53} -85.8983i q^{55} +(66.7967 - 38.5651i) q^{59} +(-0.373546 - 0.215667i) q^{61} +(-13.4066 + 23.2209i) q^{65} +(28.4835 + 49.3348i) q^{67} +123.560 q^{71} +(-31.0769 + 17.9422i) q^{73} +(26.0934 - 45.1951i) q^{79} -136.883i q^{83} +38.4397 q^{85} +(13.2198 + 7.63248i) q^{89} +(66.5603 + 115.286i) q^{95} +34.1524i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{5} - 15 q^{11} - 18 q^{17} + 81 q^{19} - 48 q^{23} + 61 q^{25} - 6 q^{29} - 48 q^{31} + 73 q^{37} + 70 q^{43} - 90 q^{47} - 99 q^{53} + 243 q^{59} + 192 q^{61} - 102 q^{65} - 7 q^{67} + 204 q^{71}+ \cdots - 24 q^{95}+O(q^{100}) $ 4 * q - 9 * q^5 - 15 * q^11 - 18 * q^17 + 81 * q^19 - 48 * q^23 + 61 * q^25 - 6 * q^29 - 48 * q^31 + 73 * q^37 + 70 * q^43 - 90 * q^47 - 99 * q^53 + 243 * q^59 + 192 * q^61 - 102 * q^65 - 7 * q^67 + 204 * q^71 + 45 * q^73 + 56 * q^79 + 444 * q^85 + 198 * q^89 - 24 * q^95

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
$5$	3.79669	+	2.19202i	0.759339	+	0.438404i	0.829058	−	0.559162i	$-0.188877\pi$
$5$	3.79669	+	2.19202i	0.759339	+	0.438404i	−0.0697196	+	0.997567i	$0.522210\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.79669	−	16.9684i	−0.890608	−	1.54258i	−0.839147	−	0.543904i	$-0.816946\pi$
$11$	−9.79669	−	16.9684i	−0.890608	−	1.54258i	−0.0514611	−	0.998675i	$-0.516388\pi$
$12$		0			0
$13$			6.11609i			0.470469i	0.971939	+	0.235234i	$0.0755858\pi$
$13$			6.11609i			0.470469i	−0.971939	+	0.235234i	$0.924414\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.59339	−	4.38404i	0.446670	−	0.257885i	−0.259753	−	0.965675i	$-0.583641\pi$
$17$	7.59339	−	4.38404i	0.446670	−	0.257885i	0.706423	+	0.707790i	$0.250308\pi$
$18$		0			0
$19$	26.2967	+	15.1824i	1.38404	+	0.799074i	0.992635	−	0.121146i	$-0.0386569\pi$
$19$	26.2967	+	15.1824i	1.38404	+	0.799074i	0.391402	+	0.920220i	$0.371990\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.0000	+	20.7846i	−0.521739	+	0.903679i	0.477941	+	0.878392i	$0.341383\pi$
$23$	−12.0000	+	20.7846i	−0.521739	+	0.903679i	−0.999680	+	0.0252868i	$0.991950\pi$
$24$		0			0
$25$	−2.89008	−	5.00577i	−0.115603	−	0.200231i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−13.5934			−0.468737			−0.234369	−	0.972148i	$-0.575302\pi$
$29$	−13.5934			−0.468737			−0.234369	+	0.972148i	$0.575302\pi$
$30$		0			0
$31$	24.2802	−	14.0182i	0.783231	−	0.452199i	−0.0543432	−	0.998522i	$-0.517306\pi$
$31$	24.2802	−	14.0182i	0.783231	−	0.452199i	0.837574	+	0.546324i	$0.183973\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	24.2967	−	42.0831i	0.656667	−	1.13738i	−0.324806	−	0.945781i	$-0.605299\pi$
$37$	24.2967	−	42.0831i	0.656667	−	1.13738i	0.981473	−	0.191600i	$-0.0613678\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.14387i			0.174241i	0.996198	+	0.0871204i	$0.0277665\pi$
$41$			7.14387i			0.174241i	−0.996198	+	0.0871204i	$0.972234\pi$
$42$		0			0
$43$	53.7802			1.25070			0.625351	−	0.780344i	$-0.284956\pi$
$43$	53.7802			1.25070			0.625351	+	0.780344i	$0.284956\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−34.5934	−	19.9725i	−0.736030	−	0.424947i	0.0845944	−	0.996415i	$-0.473041\pi$
$47$	−34.5934	−	19.9725i	−0.736030	−	0.424947i	−0.820624	+	0.571469i	$0.806374\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−30.7967	−	53.3414i	−0.581070	−	1.00644i	−0.995353	−	0.0962942i	$-0.969301\pi$
$53$	−30.7967	−	53.3414i	−0.581070	−	1.00644i	0.414283	−	0.910148i	$-0.364032\pi$
$54$		0			0
$55$		−	85.8983i		−	1.56179i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	66.7967	−	38.5651i	1.13215	−	0.653646i	0.187673	−	0.982232i	$-0.439905\pi$
$59$	66.7967	−	38.5651i	1.13215	−	0.653646i	0.944474	+	0.328586i	$0.106572\pi$
$60$		0			0
$61$	−0.373546	−	0.215667i	−0.00612371	−	0.00353553i	0.496935	−	0.867788i	$-0.334459\pi$
$61$	−0.373546	−	0.215667i	−0.00612371	−	0.00353553i	−0.503059	+	0.864252i	$0.667792\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−13.4066	+	23.2209i	−0.206256	+	0.357245i
$66$		0			0
$67$	28.4835	+	49.3348i	0.425126	+	0.736340i	0.996432	−	0.0843968i	$-0.0268963\pi$
$67$	28.4835	+	49.3348i	0.425126	+	0.736340i	−0.571306	+	0.820737i	$0.693563\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	123.560			1.74029			0.870143	−	0.492799i	$-0.164026\pi$
$71$	123.560			1.74029			0.870143	+	0.492799i	$0.164026\pi$
$72$		0			0
$73$	−31.0769	+	17.9422i	−0.425710	+	0.245784i	−0.697517	−	0.716568i	$-0.745712\pi$
$73$	−31.0769	+	17.9422i	−0.425710	+	0.245784i	0.271807	+	0.962352i	$0.412379\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	26.0934	−	45.1951i	0.330296	−	0.572090i	−0.652274	−	0.757983i	$-0.726185\pi$
$79$	26.0934	−	45.1951i	0.330296	−	0.572090i	0.982570	+	0.185894i	$0.0595181\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	136.883i		−	1.64919i	−0.565725	−	0.824594i	$-0.691404\pi$
$83$		−	136.883i		−	1.64919i	0.565725	−	0.824594i	$-0.308596\pi$
$84$		0			0
$85$	38.4397			0.452232
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.2198	+	7.63248i	0.148538	+	0.0857582i	0.572427	−	0.819956i	$-0.306002\pi$
$89$	13.2198	+	7.63248i	0.148538	+	0.0857582i	−0.423889	+	0.905714i	$0.639335\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	66.5603	+	115.286i	0.700635	+	1.21354i
$96$		0			0
$97$			34.1524i			0.352087i	0.984382	+	0.176043i	$0.0563299\pi$
$97$			34.1524i			0.352087i	−0.984382	+	0.176043i	$0.943670\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−113.340	+	65.4372i	−1.12218	+	0.647893i	−0.941957	−	0.335733i	$-0.891016\pi$
$101$	−113.340	+	65.4372i	−1.12218	+	0.647893i	−0.180226	+	0.983625i	$0.557683\pi$
$102$		0			0
$103$	140.857	+	81.3238i	1.36754	+	0.789552i	0.990614	−	0.136690i	$-0.0436464\pi$
$103$	140.857	+	81.3238i	1.36754	+	0.789552i	0.376930	+	0.926242i	$0.376980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−51.9835	+	90.0380i	−0.485827	+	0.841477i	−0.999867	−	0.0162892i	$-0.994815\pi$
$107$	−51.9835	+	90.0380i	−0.485827	+	0.841477i	0.514041	+	0.857766i	$0.328148\pi$
$108$		0			0
$109$	7.89008	+	13.6660i	0.0723861	+	0.125376i	0.899947	−	0.436000i	$-0.143605\pi$
$109$	7.89008	+	13.6660i	0.0723861	+	0.125376i	−0.827561	+	0.561377i	$0.810272\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	59.6265			0.527668			0.263834	−	0.964568i	$-0.415013\pi$
$113$	59.6265			0.527668			0.263834	+	0.964568i	$0.415013\pi$
$114$		0			0
$115$	−91.1206	+	52.6085i	−0.792353	+	0.457465i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−131.450	+	227.679i	−1.08637	+	1.88164i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	134.942i		−	1.07953i
$126$		0			0
$127$	−25.0000			−0.196850			−0.0984252	−	0.995144i	$-0.531381\pi$
$127$	−25.0000			−0.196850			−0.0984252	+	0.995144i	$0.531381\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	196.170	+	113.259i	1.49748	+	0.864572i	0.999996	−	0.00289939i	$-0.000922907\pi$
$131$	196.170	+	113.259i	1.49748	+	0.864572i	0.497487	+	0.867471i	$0.334256\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−123.560	−	214.013i	−0.901900	−	1.56214i	−0.825025	−	0.565096i	$-0.808839\pi$
$137$	−123.560	−	214.013i	−0.901900	−	1.56214i	−0.0768751	−	0.997041i	$-0.524494\pi$
$138$		0			0
$139$		−	147.383i		−	1.06031i	−0.847902	−	0.530154i	$-0.822134\pi$
$139$		−	147.383i		−	1.06031i	0.847902	−	0.530154i	$-0.177866\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	103.780	−	59.9175i	0.725735	−	0.419004i
$144$		0			0
$145$	−51.6099	−	29.7970i	−0.355930	−	0.205497i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	64.4066	−	111.556i	0.432259	−	0.748695i	−0.564808	−	0.825222i	$-0.691050\pi$
$149$	64.4066	−	111.556i	0.432259	−	0.748695i	0.997067	+	0.0765273i	$0.0243832\pi$
$150$		0			0
$151$	−31.0165	−	53.7222i	−0.205408	−	0.355776i	0.744855	−	0.667226i	$-0.232519\pi$
$151$	−31.0165	−	53.7222i	−0.205408	−	0.355776i	−0.950262	+	0.311450i	$0.899185\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	122.912			0.792983
$156$		0			0
$157$	217.121	−	125.355i	1.38293	−	0.798437i	0.390428	−	0.920633i	$-0.372327\pi$
$157$	217.121	−	125.355i	1.38293	−	0.798437i	0.992506	+	0.122196i	$0.0389937\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	48.9669	−	84.8132i	0.300411	−	0.520326i	−0.675818	−	0.737068i	$-0.736210\pi$
$163$	48.9669	−	84.8132i	0.300411	−	0.520326i	0.976229	+	0.216742i	$0.0695430\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			71.7689i			0.429754i	0.976641	+	0.214877i	$0.0689351\pi$
$167$			71.7689i			0.429754i	−0.976641	+	0.214877i	$0.931065\pi$
$168$		0			0
$169$	131.593			0.778659
$170$		0			0
$171$		0			0
$172$		0			0
$173$	261.560	+	151.012i	1.51191	+	0.872901i	0.999903	+	0.0139196i	$0.00443087\pi$
$173$	261.560	+	151.012i	1.51191	+	0.872901i	0.512006	+	0.858982i	$0.328902\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	69.0000	+	119.512i	0.385475	+	0.667662i	0.991835	−	0.127528i	$-0.0407043\pi$
$179$	69.0000	+	119.512i	0.385475	+	0.667662i	−0.606360	+	0.795190i	$0.707371\pi$
$180$		0			0
$181$			82.9733i			0.458416i	0.973377	+	0.229208i	$0.0736136\pi$
$181$			82.9733i			0.458416i	−0.973377	+	0.229208i	$0.926386\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	184.494	−	106.518i	0.997266	−	0.575772i
$186$		0			0
$187$	−148.780	−	85.8983i	−0.795616	−	0.459349i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	86.1537	−	149.223i	0.451067	−	0.781270i	−0.547386	−	0.836880i	$-0.684377\pi$
$191$	86.1537	−	149.223i	0.451067	−	0.781270i	0.998453	+	0.0556101i	$0.0177104\pi$
$192$		0			0
$193$	−137.654	−	238.423i	−0.713232	−	1.23535i	−0.963638	−	0.267213i	$-0.913897\pi$
$193$	−137.654	−	238.423i	−0.713232	−	1.23535i	0.250406	−	0.968141i	$-0.419436\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	154.307			0.783286			0.391643	−	0.920117i	$-0.371907\pi$
$197$	154.307			0.783286			0.391643	+	0.920117i	$0.371907\pi$
$198$		0			0
$199$	162.747	−	93.9621i	0.817825	−	0.472171i	−0.0318411	−	0.999493i	$-0.510137\pi$
$199$	162.747	−	93.9621i	0.817825	−	0.472171i	0.849666	+	0.527322i	$0.176804\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−15.6595	+	27.1231i	−0.0763879	+	0.132308i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	594.949i		−	2.84665i
$210$		0			0
$211$	−348.307			−1.65075			−0.825373	−	0.564588i	$-0.809035\pi$
$211$	−348.307			−1.65075			−0.825373	+	0.564588i	$0.809035\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	204.187	+	117.887i	0.949706	+	0.548313i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	26.8132	+	46.4419i	0.121327	+	0.210144i
$222$		0			0
$223$			30.3581i			0.136135i	0.997681	+	0.0680675i	$0.0216833\pi$
$223$			30.3581i			0.136135i	−0.997681	+	0.0680675i	$0.978317\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	55.8298	−	32.2333i	0.245946	−	0.141997i	−0.371961	−	0.928249i	$-0.621314\pi$
$227$	55.8298	−	32.2333i	0.245946	−	0.141997i	0.617907	+	0.786252i	$0.287981\pi$
$228$		0			0
$229$	−8.67024	−	5.00577i	−0.0378613	−	0.0218592i	0.480950	−	0.876748i	$-0.340292\pi$
$229$	−8.67024	−	5.00577i	−0.0378613	−	0.0218592i	−0.518811	+	0.854889i	$0.673625\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.2529	−	24.6868i	0.0611713	−	0.105952i	−0.833818	−	0.552040i	$-0.813850\pi$
$233$	14.2529	−	24.6868i	0.0611713	−	0.105952i	0.894989	+	0.446088i	$0.147183\pi$
$234$		0			0
$235$	−87.5603	−	151.659i	−0.372597	−	0.645357i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	256.307			1.07242			0.536208	−	0.844086i	$-0.319856\pi$
$239$	256.307			1.07242			0.536208	+	0.844086i	$0.319856\pi$
$240$		0			0
$241$	−186.797	+	107.847i	−0.775090	+	0.447498i	−0.834687	−	0.550724i	$-0.814352\pi$
$241$	−186.797	+	107.847i	−0.775090	+	0.447498i	0.0595974	+	0.998222i	$0.481018\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−92.8570	+	160.833i	−0.375939	+	0.651146i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			371.680i			1.48080i	0.672168	+	0.740398i	$0.265363\pi$
$251$			371.680i			1.48080i	−0.672168	+	0.740398i	$0.734637\pi$
$252$		0			0
$253$	470.241			1.85866
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−64.1206	−	37.0201i	−0.249497	−	0.144047i	0.370037	−	0.929017i	$-0.379345\pi$
$257$	−64.1206	−	37.0201i	−0.249497	−	0.144047i	−0.619534	+	0.784970i	$0.712678\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.65952	+	16.7308i	0.0367282	+	0.0636151i	0.883805	−	0.467855i	$-0.154973\pi$
$263$	9.65952	+	16.7308i	0.0367282	+	0.0636151i	−0.847077	+	0.531470i	$0.821640\pi$
$264$		0			0
$265$		−	270.028i		−	1.01897i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−180.983	+	104.491i	−0.672801	+	0.388442i	−0.797137	−	0.603798i	$-0.793653\pi$
$269$	−180.983	+	104.491i	−0.672801	+	0.388442i	0.124336	+	0.992240i	$0.460320\pi$
$270$		0			0
$271$	−106.357	−	61.4053i	−0.392461	−	0.226588i	0.290765	−	0.956795i	$-0.406090\pi$
$271$	−106.357	−	61.4053i	−0.392461	−	0.226588i	−0.683226	+	0.730207i	$0.739424\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−56.6265	+	98.0799i	−0.205914	+	0.356654i
$276$		0			0
$277$	−13.8901	−	24.0583i	−0.0501447	−	0.0868532i	0.839864	−	0.542798i	$-0.182635\pi$
$277$	−13.8901	−	24.0583i	−0.0501447	−	0.0868532i	−0.890008	+	0.455944i	$0.849302\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−61.4942			−0.218841			−0.109420	−	0.993996i	$-0.534899\pi$
$281$	−61.4942			−0.218841			−0.109420	+	0.993996i	$0.534899\pi$
$282$		0			0
$283$	38.0107	−	21.9455i	0.134313	−	0.0775459i	−0.431338	−	0.902191i	$-0.641958\pi$
$283$	38.0107	−	21.9455i	0.134313	−	0.0775459i	0.565651	+	0.824645i	$0.308625\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−106.060	+	183.702i	−0.366991	+	0.635647i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	223.600i		−	0.763139i	−0.924340	−	0.381569i	$-0.875384\pi$
$293$		−	223.600i		−	0.763139i	0.924340	−	0.381569i	$-0.124616\pi$
$294$		0			0
$295$	338.142			1.14624
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−127.121	−	73.3931i	−0.425153	−	0.245462i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.945494	−	1.63764i	−0.00309998	−	0.00536933i
$306$		0			0
$307$			66.8457i			0.217738i	0.994056	+	0.108869i	$0.0347230\pi$
$307$			66.8457i			0.217738i	−0.994056	+	0.108869i	$0.965277\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−25.3074	+	14.6112i	−0.0813743	+	0.0469815i	−0.540135	−	0.841578i	$-0.681627\pi$
$311$	−25.3074	+	14.6112i	−0.0813743	+	0.0469815i	0.458761	+	0.888560i	$0.348294\pi$
$312$		0			0
$313$	214.874	+	124.057i	0.686497	+	0.396349i	0.802298	−	0.596923i	$-0.203610\pi$
$313$	214.874	+	124.057i	0.686497	+	0.396349i	−0.115801	+	0.993272i	$0.536944\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.32395	−	10.9534i	0.0199494	−	0.0345533i	−0.855878	−	0.517177i	$-0.826983\pi$
$317$	6.32395	−	10.9534i	0.0199494	−	0.0345533i	0.875828	+	0.482624i	$0.160316\pi$
$318$		0			0
$319$	133.170	+	230.658i	0.417462	+	0.723065i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	266.241			0.824276
$324$		0			0
$325$	30.6157	−	17.6760i	0.0942023	−	0.0543877i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	251.450	−	435.525i	0.759669	−	1.31579i	−0.183351	−	0.983048i	$-0.558694\pi$
$331$	251.450	−	435.525i	0.759669	−	1.31579i	0.943019	−	0.332738i	$-0.107972\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			249.746i			0.745509i
$336$		0			0
$337$	−123.681			−0.367006			−0.183503	−	0.983019i	$-0.558744\pi$
$337$	−123.681			−0.367006			−0.183503	+	0.983019i	$0.558744\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−475.731	−	274.663i	−1.39510	−	0.805464i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−113.154	−	195.988i	−0.326091	−	0.564807i	0.655641	−	0.755073i	$-0.272398\pi$
$347$	−113.154	−	195.988i	−0.326091	−	0.564807i	−0.981733	+	0.190266i	$0.939065\pi$
$348$		0			0
$349$			155.669i			0.446043i	0.974814	+	0.223021i	$0.0715920\pi$
$349$			155.669i			0.446043i	−0.974814	+	0.223021i	$0.928408\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−55.4066	+	31.9890i	−0.156959	+	0.0906205i	−0.576422	−	0.817152i	$-0.695552\pi$
$353$	−55.4066	+	31.9890i	−0.156959	+	0.0906205i	0.419463	+	0.907772i	$0.362218\pi$
$354$		0			0
$355$	469.121	+	270.847i	1.32147	+	0.762949i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−151.494	+	262.396i	−0.421989	+	0.730907i	−0.996134	−	0.0878469i	$-0.972001\pi$
$359$	−151.494	+	262.396i	−0.421989	+	0.730907i	0.574145	+	0.818754i	$0.305335\pi$
$360$		0			0
$361$	280.511	+	485.859i	0.777038	+	1.34587i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−157.319			−0.431011
$366$		0			0
$367$	0.00581404	−	0.00335674i	1.58421e−5	−	9.14643e-6i	−0.499992	−	0.866030i	$-0.666664\pi$
$367$	0.00581404	−	0.00335674i	1.58421e−5	−	9.14643e-6i	0.500008	+	0.866021i	$0.333330\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	62.8901	−	108.929i	0.168606	−	0.292034i	−0.769324	−	0.638859i	$-0.779407\pi$
$373$	62.8901	−	108.929i	0.168606	−	0.292034i	0.937930	+	0.346825i	$0.112740\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	83.1384i		−	0.220526i
$378$		0			0
$379$	−485.076			−1.27988			−0.639942	−	0.768423i	$-0.721042\pi$
$379$	−485.076			−1.27988			−0.639942	+	0.768423i	$0.721042\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−225.286	−	130.069i	−0.588214	−	0.339606i	0.176177	−	0.984359i	$-0.443627\pi$
$383$	−225.286	−	130.069i	−0.588214	−	0.339606i	−0.764391	+	0.644753i	$0.776960\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	42.9339	+	74.3636i	0.110370	+	0.191166i	0.915919	−	0.401362i	$-0.131463\pi$
$389$	42.9339	+	74.3636i	0.110370	+	0.191166i	−0.805550	+	0.592528i	$0.798130\pi$
$390$		0			0
$391$			210.434i			0.538195i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	198.137	−	114.395i	0.501613	−	0.289606i
$396$		0			0
$397$	−401.670	−	231.904i	−1.01176	−	0.584142i	−0.100056	−	0.994982i	$-0.531902\pi$
$397$	−401.670	−	231.904i	−1.01176	−	0.584142i	−0.911708	+	0.410840i	$0.865236\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−231.934	+	401.721i	−0.578389	+	1.00180i	0.417276	+	0.908780i	$0.362985\pi$
$401$	−231.934	+	401.721i	−0.578389	+	1.00180i	−0.995664	+	0.0930186i	$0.970348\pi$
$402$		0			0
$403$	85.7364	+	148.500i	0.212745	+	0.368486i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−952.109			−2.33933
$408$		0			0
$409$	−403.687	+	233.069i	−0.987009	+	0.569850i	−0.904379	−	0.426730i	$-0.859665\pi$
$409$	−403.687	+	233.069i	−0.987009	+	0.569850i	−0.0826303	+	0.996580i	$0.526332\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	300.050	−	519.701i	0.723011	−	1.25229i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			89.9655i			0.214715i	0.994220	+	0.107357i	$0.0342389\pi$
$419$			89.9655i			0.214715i	−0.994220	+	0.107357i	$0.965761\pi$
$420$		0			0
$421$	−413.582			−0.982379			−0.491190	−	0.871053i	$-0.663438\pi$
$421$	−413.582			−0.982379			−0.491190	+	0.871053i	$0.663438\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−43.8910	−	25.3405i	−0.103273	−	0.0596246i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	35.0661	+	60.7363i	0.0813599	+	0.140920i	0.903834	−	0.427883i	$-0.140740\pi$
$431$	35.0661	+	60.7363i	0.0813599	+	0.140920i	−0.822474	+	0.568802i	$0.807407\pi$
$432$		0			0
$433$		−	431.223i		−	0.995897i	−0.867207	−	0.497948i	$-0.834087\pi$
$433$		−	431.223i		−	0.995897i	0.867207	−	0.497948i	$-0.165913\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−631.121	+	364.378i	−1.44421	+	0.833816i
$438$		0			0
$439$	81.7091	+	47.1748i	0.186126	+	0.107460i	0.590168	−	0.807281i	$-0.299062\pi$
$439$	81.7091	+	47.1748i	0.186126	+	0.107460i	−0.404042	+	0.914740i	$0.632395\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	206.291	−	357.306i	0.465668	−	0.806560i	−0.533564	−	0.845760i	$-0.679147\pi$
$443$	206.291	−	357.306i	0.465668	−	0.806560i	0.999231	+	0.0391995i	$0.0124808\pi$
$444$		0			0
$445$	33.4611	+	57.9564i	0.0751935	+	0.130239i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−617.802			−1.37595			−0.687975	−	0.725734i	$-0.741500\pi$
$449$	−617.802			−1.37595			−0.687975	+	0.725734i	$0.741500\pi$
$450$		0			0
$451$	121.220	−	69.9863i	0.268780	−	0.155180i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.50000	−	9.52628i	0.0120350	−	0.0208453i	−0.859945	−	0.510386i	$-0.829502\pi$
$457$	5.50000	−	9.52628i	0.0120350	−	0.0208453i	0.871980	+	0.489541i	$0.162836\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			859.960i			1.86542i	0.360624	+	0.932711i	$0.382564\pi$
$461$			859.960i			1.86542i	−0.360624	+	0.932711i	$0.617436\pi$
$462$		0			0
$463$	397.340			0.858187			0.429093	−	0.903260i	$-0.358833\pi$
$463$	397.340			0.858187			0.429093	+	0.903260i	$0.358833\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	257.340	+	148.576i	0.551050	+	0.318149i	0.749545	−	0.661953i	$-0.230272\pi$
$467$	257.340	+	148.576i	0.551050	+	0.318149i	−0.198495	+	0.980102i	$0.563605\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−526.868	−	912.562i	−1.11389	−	1.92931i
$474$		0			0
$475$		−	175.513i		−	0.369502i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	696.648	−	402.210i	1.45438	−	0.839687i	0.455654	−	0.890157i	$-0.349405\pi$
$479$	696.648	−	402.210i	1.45438	−	0.839687i	0.998726	+	0.0504704i	$0.0160720\pi$
$480$		0			0
$481$	257.384	+	148.601i	0.535102	+	0.308942i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−74.8628	+	129.666i	−0.154356	+	0.267353i
$486$		0			0
$487$	−30.4008	−	52.6557i	−0.0624246	−	0.108123i	0.833124	−	0.553086i	$-0.186550\pi$
$487$	−30.4008	−	52.6557i	−0.0624246	−	0.108123i	−0.895549	+	0.444964i	$0.853217\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−514.955			−1.04879			−0.524394	−	0.851475i	$-0.675708\pi$
$491$	−514.955			−1.04879			−0.524394	+	0.851475i	$0.675708\pi$
$492$		0			0
$493$	−103.220	+	59.5940i	−0.209371	+	0.120880i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−444.450	+	769.811i	−0.890682	+	1.54271i	−0.0516230	+	0.998667i	$0.516439\pi$
$499$	−444.450	+	769.811i	−0.890682	+	1.54271i	−0.839059	+	0.544040i	$0.816894\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			822.920i			1.63602i	0.575202	+	0.818012i	$0.304924\pi$
$503$			822.920i			1.63602i	−0.575202	+	0.818012i	$0.695076\pi$
$504$		0			0
$505$	−573.759			−1.13616
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−492.038	−	284.078i	−0.966676	−	0.558111i	−0.0684546	−	0.997654i	$-0.521807\pi$
$509$	−492.038	−	284.078i	−0.966676	−	0.558111i	−0.898221	+	0.439544i	$0.855140\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	356.527	+	617.523i	0.692286	+	1.19907i
$516$		0			0
$517$			782.658i			1.51385i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	118.407	−	68.3621i	0.227268	−	0.131213i	−0.382043	−	0.924145i	$-0.624779\pi$
$521$	118.407	−	68.3621i	0.227268	−	0.131213i	0.609311	+	0.792931i	$0.291446\pi$
$522$		0			0
$523$	−12.6157	−	7.28370i	−0.0241219	−	0.0139268i	0.487891	−	0.872905i	$-0.337767\pi$
$523$	−12.6157	−	7.28370i	−0.0241219	−	0.0139268i	−0.512012	+	0.858978i	$0.671100\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	122.912	−	212.891i	0.233230	−	0.403967i
$528$		0			0
$529$	−23.5000	−	40.7032i	−0.0444234	−	0.0769437i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−43.6926			−0.0819748
$534$		0			0
$535$	−394.731	+	227.898i	−0.737814	+	0.425977i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	462.450	−	800.988i	0.854807	−	1.48057i	−0.0220176	−	0.999758i	$-0.507009\pi$
$541$	462.450	−	800.988i	0.854807	−	1.48057i	0.876824	−	0.480811i	$-0.159658\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			69.1809i			0.126937i
$546$		0			0
$547$	103.626			0.189445			0.0947225	−	0.995504i	$-0.469804\pi$
$547$	103.626			0.189445			0.0947225	+	0.995504i	$0.469804\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−357.461	−	206.380i	−0.648750	−	0.374556i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	206.203	+	357.155i	0.370203	+	0.641211i	0.989597	−	0.143870i	$-0.0459546\pi$
$557$	206.203	+	357.155i	0.370203	+	0.641211i	−0.619393	+	0.785081i	$0.712621\pi$
$558$		0			0
$559$			328.925i			0.588416i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−594.412	+	343.184i	−1.05579	+	0.609562i	−0.924266	−	0.381750i	$-0.875322\pi$
$563$	−594.412	+	343.184i	−1.05579	+	0.609562i	−0.131528	+	0.991313i	$0.541988\pi$
$564$		0			0
$565$	226.383	+	130.702i	0.400679	+	0.231332i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−73.3074	+	126.972i	−0.128836	+	0.223150i	−0.923226	−	0.384258i	$-0.874457\pi$
$569$	−73.3074	+	126.972i	−0.128836	+	0.223150i	0.794390	+	0.607408i	$0.207791\pi$
$570$		0			0
$571$	−178.131	−	308.533i	−0.311964	−	0.540337i	0.666824	−	0.745216i	$-0.267653\pi$
$571$	−178.131	−	308.533i	−0.311964	−	0.540337i	−0.978787	+	0.204878i	$0.934320\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	138.724			0.241259
$576$		0			0
$577$	148.115	−	85.5141i	0.256698	−	0.148205i	−0.366129	−	0.930564i	$-0.619317\pi$
$577$	148.115	−	85.5141i	0.256698	−	0.148205i	0.622827	+	0.782359i	$0.285984\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−603.412	+	1045.14i	−1.03501	+	1.79269i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.8384i		−	0.0423142i	−0.999776	−	0.0211571i	$-0.993265\pi$
$587$		−	24.8384i		−	0.0423142i	0.999776	−	0.0211571i	$-0.00673502\pi$
$588$		0			0
$589$	851.317			1.44536
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−117.846	−	68.0386i	−0.198729	−	0.114736i	0.397333	−	0.917674i	$-0.369936\pi$
$593$	−117.846	−	68.0386i	−0.198729	−	0.114736i	−0.596062	+	0.802938i	$0.703269\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	494.615	+	856.698i	0.825734	+	1.43021i	0.901357	+	0.433077i	$0.142572\pi$
$599$	494.615	+	856.698i	0.825734	+	1.43021i	−0.0756226	+	0.997137i	$0.524094\pi$
$600$		0			0
$601$		−	78.8184i		−	0.131145i	−0.997848	−	0.0655727i	$-0.979113\pi$
$601$		−	78.8184i		−	0.131145i	0.997848	−	0.0655727i	$-0.0208874\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−998.154	+	576.284i	−1.64984	+	0.952536i
$606$		0			0
$607$	−727.850	−	420.225i	−1.19909	−	0.692297i	−0.238741	−	0.971083i	$-0.576735\pi$
$607$	−727.850	−	420.225i	−1.19909	−	0.692297i	−0.960353	+	0.278786i	$0.910068\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	122.154	−	211.576i	0.199924	−	0.346279i
$612$		0			0
$613$	−423.714	−	733.894i	−0.691214	−	1.19722i	−0.971440	−	0.237283i	$-0.923743\pi$
$613$	−423.714	−	733.894i	−0.691214	−	1.19722i	0.280227	−	0.959934i	$-0.409590\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	649.669			1.05295			0.526474	−	0.850191i	$-0.323514\pi$
$617$	649.669			1.05295			0.526474	+	0.850191i	$0.323514\pi$
$618$		0			0
$619$	422.659	−	244.022i	0.682809	−	0.394220i	−0.118104	−	0.993001i	$-0.537682\pi$
$619$	422.659	−	244.022i	0.682809	−	0.394220i	0.800912	+	0.598781i	$0.204348\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	223.543	−	387.188i	0.357669	−	0.619500i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	426.071i		−	0.677378i
$630$		0			0
$631$	−820.473			−1.30027			−0.650137	−	0.759817i	$-0.725288\pi$
$631$	−820.473			−1.30027			−0.650137	+	0.759817i	$0.725288\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−94.9173	−	54.8005i	−0.149476	−	0.0863001i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.93387	−	17.2060i	−0.0154975	−	0.0268424i	0.858173	−	0.513361i	$-0.171600\pi$
$641$	−9.93387	−	17.2060i	−0.0154975	−	0.0268424i	−0.873670	+	0.486519i	$0.838267\pi$
$642$		0			0
$643$			1013.95i			1.57691i	0.615092	+	0.788456i	$0.289119\pi$
$643$			1013.95i			1.57691i	−0.615092	+	0.788456i	$0.710881\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	115.582	−	66.7312i	0.178643	−	0.103139i	−0.408012	−	0.912976i	$-0.633778\pi$
$647$	115.582	−	66.7312i	0.178643	−	0.103139i	0.586655	+	0.809837i	$0.300445\pi$
$648$		0			0
$649$	−1308.77	−	755.621i	−2.01660	−	1.16428i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	302.378	−	523.735i	0.463060	−	0.802044i	−0.536051	−	0.844185i	$-0.680085\pi$
$653$	302.378	−	523.735i	0.463060	−	0.802044i	0.999112	+	0.0421413i	$0.0134180\pi$
$654$		0			0
$655$	496.532	+	860.019i	0.758064	+	1.31301i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−335.451			−0.509031			−0.254515	−	0.967069i	$-0.581916\pi$
$659$	−335.451			−0.509031			−0.254515	+	0.967069i	$0.581916\pi$
$660$		0			0
$661$	−608.186	+	351.136i	−0.920100	+	0.531220i	−0.883667	−	0.468116i	$-0.844933\pi$
$661$	−608.186	+	351.136i	−0.920100	+	0.531220i	−0.0364328	+	0.999336i	$0.511599\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	163.121	−	282.533i	0.244559	−	0.423588i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			8.45130i			0.0125951i
$672$		0			0
$673$	−1119.48			−1.66342			−0.831711	−	0.555209i	$-0.812638\pi$
$673$	−1119.48			−1.66342			−0.831711	+	0.555209i	$0.812638\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	225.709	+	130.313i	0.333396	+	0.192486i	0.657348	−	0.753587i	$-0.271678\pi$
$677$	225.709	+	130.313i	0.333396	+	0.192486i	−0.323952	+	0.946074i	$0.605012\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−100.258	−	173.652i	−0.146790	−	0.254248i	0.783249	−	0.621708i	$-0.213561\pi$
$683$	−100.258	−	173.652i	−0.146790	−	0.254248i	−0.930039	+	0.367460i	$0.880228\pi$
$684$		0			0
$685$		−	1083.39i		−	1.58159i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	326.241	−	188.355i	0.473500	−	0.273375i
$690$		0			0
$691$	−860.933	−	497.060i	−1.24592	−	0.719334i	−0.275629	−	0.961264i	$-0.588886\pi$
$691$	−860.933	−	497.060i	−1.24592	−	0.719334i	−0.970294	+	0.241930i	$0.922220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	323.066	−	559.567i	0.464843	−	0.805132i
$696$		0			0
$697$	31.3190	+	54.2462i	0.0449341	+	0.0778281i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	148.231			0.211457			0.105729	−	0.994395i	$-0.466283\pi$
$701$	148.231			0.211457			0.105729	+	0.994395i	$0.466283\pi$
$702$		0			0
$703$	1277.85	−	737.764i	1.81770	−	1.04945i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−355.988	+	616.590i	−0.502099	+	0.869661i	0.497898	+	0.867236i	$0.334105\pi$
$709$	−355.988	+	616.590i	−0.502099	+	0.869661i	−0.999997	+	0.00242571i	$0.999228\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			672.872i			0.943719i
$714$		0			0
$715$	525.362			0.734772
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−649.109	−	374.763i	−0.902794	−	0.521228i	−0.0246885	−	0.999695i	$-0.507859\pi$
$719$	−649.109	−	374.763i	−0.902794	−	0.521228i	−0.878106	+	0.478467i	$0.841193\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	39.2860	+	68.0453i	0.0541876	+	0.0938556i
$726$		0			0
$727$			865.702i			1.19079i	0.803434	+	0.595393i	$0.203004\pi$
$727$			865.702i			1.19079i	−0.803434	+	0.595393i	$0.796996\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	408.374	−	235.775i	0.558651	−	0.322537i
$732$		0			0
$733$	−83.5710	−	48.2498i	−0.114012	−	0.0658251i	0.441909	−	0.897060i	$-0.354301\pi$
$733$	−83.5710	−	48.2498i	−0.114012	−	0.0658251i	−0.555922	+	0.831235i	$0.687635\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	558.088	−	966.636i	0.757242	−	1.31158i
$738$		0			0
$739$	210.450	+	364.511i	0.284777	+	0.493249i	0.972555	−	0.232673i	$-0.0747470\pi$
$739$	210.450	+	364.511i	0.284777	+	0.493249i	−0.687778	+	0.725921i	$0.741414\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−920.043			−1.23828			−0.619141	−	0.785280i	$-0.712519\pi$
$743$	−920.043			−1.23828			−0.619141	+	0.785280i	$0.712519\pi$
$744$		0			0
$745$	489.064	−	282.361i	0.656462	−	0.379009i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	160.181	−	277.442i	0.213290	−	0.369430i	−0.739452	−	0.673209i	$-0.764915\pi$
$751$	160.181	−	277.442i	0.213290	−	0.369430i	0.952742	+	0.303780i	$0.0982487\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	271.956i		−	0.360206i
$756$		0			0
$757$	330.813			0.437006			0.218503	−	0.975836i	$-0.429883\pi$
$757$	330.813			0.437006			0.218503	+	0.975836i	$0.429883\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	230.615	+	133.146i	0.303042	+	0.174961i	0.643809	−	0.765187i	$-0.277353\pi$
$761$	230.615	+	133.146i	0.303042	+	0.174961i	−0.340767	+	0.940148i	$0.610687\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	235.868	+	408.535i	0.307520	+	0.532640i
$768$		0			0
$769$			241.301i			0.313785i	0.987616	+	0.156893i	$0.0501477\pi$
$769$			241.301i			0.313785i	−0.987616	+	0.156893i	$0.949852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−805.208	+	464.887i	−1.04167	+	0.601406i	−0.920305	−	0.391202i	$-0.872059\pi$
$773$	−805.208	+	464.887i	−1.04167	+	0.601406i	−0.121362	+	0.992608i	$0.538726\pi$
$774$		0			0
$775$	−140.343	−	81.0272i	−0.181088	−	0.104551i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−108.461	+	187.860i	−0.139231	+	0.241156i
$780$		0			0
$781$	−1210.48	−	2096.62i	−1.54991	−	2.68453i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1099.12			1.40015
$786$		0			0
$787$	−961.097	+	554.890i	−1.22122	+	0.705070i	−0.965177	−	0.261597i	$-0.915751\pi$
$787$	−961.097	+	554.890i	−1.22122	+	0.705070i	−0.256039	+	0.966666i	$0.582418\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.31904	−	2.28465i	0.00166336	−	0.00288102i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			231.047i			0.289896i	0.989439	+	0.144948i	$0.0463014\pi$
$797$			231.047i			0.289896i	−0.989439	+	0.144948i	$0.953699\pi$
$798$		0			0
$799$	−350.241			−0.438350
$800$		0			0
$801$		0			0
$802$		0			0
$803$	608.901	+	351.549i	0.758282	+	0.437795i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−159.473	−	276.215i	−0.197123	−	0.341428i	0.750471	−	0.660903i	$-0.229827\pi$
$809$	−159.473	−	276.215i	−0.197123	−	0.341428i	−0.947595	+	0.319476i	$0.896493\pi$
$810$		0			0
$811$		−	196.807i		−	0.242672i	−0.992612	−	0.121336i	$-0.961282\pi$
$811$		−	196.807i		−	0.242672i	0.992612	−	0.121336i	$-0.0387178\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	371.825	−	214.673i	0.456227	−	0.263403i
$816$		0			0
$817$	1414.24	+	816.512i	1.73102	+	0.999403i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−692.005	+	1198.59i	−0.842881	+	1.45991i	0.0445687	+	0.999006i	$0.485809\pi$
$821$	−692.005	+	1198.59i	−0.842881	+	1.45991i	−0.887449	+	0.460906i	$0.847525\pi$
$822$		0			0
$823$	−138.253	−	239.461i	−0.167987	−	0.290961i	0.769725	−	0.638375i	$-0.220393\pi$
$823$	−138.253	−	239.461i	−0.167987	−	0.290961i	−0.937712	+	0.347414i	$0.887060\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1467.81			−1.77486			−0.887431	−	0.460940i	$-0.847512\pi$
$827$	−1467.81			−1.77486			−0.887431	+	0.460940i	$0.847512\pi$
$828$		0			0
$829$	−68.1859	+	39.3671i	−0.0822507	+	0.0474875i	−0.540561	−	0.841305i	$-0.681788\pi$
$829$	−68.1859	+	39.3671i	−0.0822507	+	0.0474875i	0.458311	+	0.888792i	$0.348455\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−157.319	+	272.485i	−0.188406	+	0.326329i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	591.398i		−	0.704884i	−0.935834	−	0.352442i	$-0.885351\pi$
$839$		−	591.398i		−	0.704884i	0.935834	−	0.352442i	$-0.114649\pi$
$840$		0			0
$841$	−656.220			−0.780285
$842$		0			0
$843$		0			0
$844$		0			0
$845$	499.620	+	288.456i	0.591266	+	0.341368i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	583.121	+	1009.99i	0.685218	+	1.18683i
$852$		0			0
$853$		−	1048.80i		−	1.22954i	−0.788707	−	0.614770i	$-0.789249\pi$
$853$		−	1048.80i		−	1.22954i	0.788707	−	0.614770i	$-0.210751\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	92.8132	−	53.5857i	0.108300	−	0.0625271i	−0.444872	−	0.895594i	$-0.646751\pi$
$857$	92.8132	−	53.5857i	0.108300	−	0.0625271i	0.553172	+	0.833067i	$0.313417\pi$
$858$		0			0
$859$	1304.52	+	753.162i	1.51864	+	0.876790i	0.999759	+	0.0219460i	$0.00698618\pi$
$859$	1304.52	+	753.162i	1.51864	+	0.876790i	0.518885	+	0.854844i	$0.326347\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	187.681	−	325.073i	0.217475	−	0.376678i	−0.736560	−	0.676372i	$-0.763551\pi$
$863$	187.681	−	325.073i	0.217475	−	0.376678i	0.954035	+	0.299694i	$0.0968846\pi$
$864$		0			0
$865$	662.043	+	1146.69i	0.765367	+	1.32566i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1022.52			−1.17666
$870$		0			0
$871$	−301.736	+	174.208i	−0.346425	+	0.200009i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−704.395	+	1220.05i	−0.803187	+	1.39116i	0.114321	+	0.993444i	$0.463531\pi$
$877$	−704.395	+	1220.05i	−0.803187	+	1.39116i	−0.917508	+	0.397717i	$0.869803\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1236.04i			1.40299i	0.712673	+	0.701497i	$0.247485\pi$
$881$			1236.04i			1.40299i	−0.712673	+	0.701497i	$0.752515\pi$
$882$		0			0
$883$	−72.9437			−0.0826089			−0.0413045	−	0.999147i	$-0.513151\pi$
$883$	−72.9437			−0.0826089			−0.0413045	+	0.999147i	$0.513151\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1249.01	+	721.116i	1.40813	+	0.812983i	0.995208	−	0.0977838i	$-0.0311754\pi$
$887$	1249.01	+	721.116i	1.40813	+	0.812983i	0.412921	+	0.910767i	$0.364509\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−606.461	−	1050.42i	−0.679128	−	1.17628i
$894$		0			0
$895$			604.998i			0.675975i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−330.050	+	190.554i	−0.367130	+	0.211962i
$900$		0			0
$901$	−467.702	−	270.028i	−0.519093	−	0.299698i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−181.879	+	315.024i	−0.200972	+	0.348093i
$906$		0			0
$907$	604.878	+	1047.68i	0.666900	+	1.15510i	0.978766	+	0.204979i	$0.0657126\pi$
$907$	604.878	+	1047.68i	0.666900	+	1.15510i	−0.311866	+	0.950126i	$0.600954\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1328.22			1.45798			0.728989	−	0.684525i	$-0.239991\pi$
$911$	1328.22			1.45798			0.728989	+	0.684525i	$0.239991\pi$
$912$		0			0
$913$	−2322.67	+	1341.00i	−2.54400	+	1.46878i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	244.429	−	423.363i	0.265973	−	0.460678i	−0.701845	−	0.712329i	$-0.747640\pi$
$919$	244.429	−	423.363i	0.265973	−	0.460678i	0.967818	+	0.251651i	$0.0809735\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			755.707i			0.818750i
$924$		0			0
$925$	−280.878			−0.303651
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−824.055	−	475.768i	−0.887034	−	0.512129i	−0.0140628	−	0.999901i	$-0.504476\pi$
$929$	−824.055	−	475.768i	−0.887034	−	0.512129i	−0.872971	+	0.487772i	$0.837810\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−376.582	−	652.259i	−0.402761	−	0.697603i
$936$		0			0
$937$			309.707i			0.330530i	0.986249	+	0.165265i	$0.0528480\pi$
$937$			309.707i			0.330530i	−0.986249	+	0.165265i	$0.947152\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1015.42	−	586.254i	1.07909	−	0.623011i	0.148437	−	0.988922i	$-0.452576\pi$
$941$	1015.42	−	586.254i	1.07909	−	0.623011i	0.930650	+	0.365910i	$0.119242\pi$
$942$		0			0
$943$	−148.483	−	85.7264i	−0.157458	−	0.0909082i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−362.428	+	627.744i	−0.382712	+	0.662876i	−0.991449	−	0.130496i	$-0.958343\pi$
$947$	−362.428	+	627.744i	−0.382712	+	0.662876i	0.608737	+	0.793372i	$0.291676\pi$
$948$		0			0
$949$	−109.736	−	190.069i	−0.115634	−	0.200283i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27.5371			−0.0288951			−0.0144476	−	0.999896i	$-0.504599\pi$
$953$	−27.5371			−0.0288951			−0.0144476	+	0.999896i	$0.504599\pi$
$954$		0			0
$955$	654.198	−	377.702i	0.685025	−	0.395499i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−87.4826	+	151.524i	−0.0910328	+	0.157674i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1206.96i		−	1.25074i
$966$		0			0
$967$	1279.81			1.32349			0.661744	−	0.749730i	$-0.269817\pi$
$967$	1279.81			1.32349			0.661744	+	0.749730i	$0.269817\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	963.686	+	556.384i	0.992467	+	0.573001i	0.906011	−	0.423254i	$-0.139112\pi$
$971$	963.686	+	556.384i	0.992467	+	0.573001i	0.0864565	+	0.996256i	$0.472446\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−217.879	−	377.378i	−0.223009	−	0.386262i	0.732712	−	0.680539i	$-0.238254\pi$
$977$	−217.879	−	377.378i	−0.223009	−	0.386262i	−0.955720	+	0.294277i	$0.904921\pi$
$978$		0			0
$979$		−	299.092i		−	0.305508i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−243.835	+	140.778i	−0.248052	+	0.143213i	−0.618872	−	0.785492i	$-0.712410\pi$
$983$	−243.835	+	140.778i	−0.248052	+	0.143213i	0.370820	+	0.928705i	$0.379077\pi$
$984$		0			0
$985$	585.858	+	338.245i	0.594780	+	0.343396i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−645.362	+	1117.80i	−0.652540	+	1.13023i
$990$		0			0
$991$	308.786	+	534.833i	0.311590	+	0.539690i	0.978707	−	0.205263i	$-0.0658050\pi$
$991$	308.786	+	534.833i	0.311590	+	0.539690i	−0.667117	+	0.744953i	$0.732472\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	823.868			0.828008
$996$		0			0
$997$	1303.91	−	752.814i	1.30784	−	0.755079i	0.326101	−	0.945335i	$-0.394265\pi$
$997$	1303.91	−	752.814i	1.30784	−	0.755079i	0.981734	+	0.190256i	$0.0609319\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.h.901.2		4
3.2	odd	2		588.3.m.d.313.1		4
7.2	even	3		1764.3.d.f.685.2		4
7.3	odd	6	inner	1764.3.z.h.325.2		4
7.4	even	3		252.3.z.e.73.1		4
7.5	odd	6		1764.3.d.f.685.3		4
7.6	odd	2		252.3.z.e.145.1		4
21.2	odd	6		588.3.d.b.97.4		4
21.5	even	6		588.3.d.b.97.1		4
21.11	odd	6		84.3.m.b.73.2	yes	4
21.17	even	6		588.3.m.d.325.1		4
21.20	even	2		84.3.m.b.61.2	✓	4
28.11	odd	6		1008.3.cg.m.577.1		4
28.27	even	2		1008.3.cg.m.145.1		4
84.11	even	6		336.3.bh.f.241.2		4
84.23	even	6		2352.3.f.f.97.2		4
84.47	odd	6		2352.3.f.f.97.3		4
84.83	odd	2		336.3.bh.f.145.2		4
105.32	even	12		2100.3.be.d.1249.2		8
105.53	even	12		2100.3.be.d.1249.3		8
105.62	odd	4		2100.3.be.d.649.3		8
105.74	odd	6		2100.3.bd.f.1501.2		4
105.83	odd	4		2100.3.be.d.649.2		8
105.104	even	2		2100.3.bd.f.901.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.m.b.61.2	✓	4	21.20	even	2
84.3.m.b.73.2	yes	4	21.11	odd	6
252.3.z.e.73.1		4	7.4	even	3
252.3.z.e.145.1		4	7.6	odd	2
336.3.bh.f.145.2		4	84.83	odd	2
336.3.bh.f.241.2		4	84.11	even	6
588.3.d.b.97.1		4	21.5	even	6
588.3.d.b.97.4		4	21.2	odd	6
588.3.m.d.313.1		4	3.2	odd	2
588.3.m.d.325.1		4	21.17	even	6
1008.3.cg.m.145.1		4	28.27	even	2
1008.3.cg.m.577.1		4	28.11	odd	6
1764.3.d.f.685.2		4	7.2	even	3
1764.3.d.f.685.3		4	7.5	odd	6
1764.3.z.h.325.2		4	7.3	odd	6	inner
1764.3.z.h.901.2		4	1.1	even	1	trivial
2100.3.bd.f.901.1		4	105.104	even	2
2100.3.bd.f.1501.2		4	105.74	odd	6
2100.3.be.d.649.2		8	105.83	odd	4
2100.3.be.d.649.3		8	105.62	odd	4
2100.3.be.d.1249.2		8	105.32	even	12
2100.3.be.d.1249.3		8	105.53	even	12
2352.3.f.f.97.2		4	84.23	even	6
2352.3.f.f.97.3		4	84.47	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.79669 + 2.19202i) q^{5} +(-9.79669 - 16.9684i) q^{11} +6.11609i q^{13} +(7.59339 - 4.38404i) q^{17} +(26.2967 + 15.1824i) q^{19} +(-12.0000 + 20.7846i) q^{23} +(-2.89008 - 5.00577i) q^{25} -13.5934 q^{29} +(24.2802 - 14.0182i) q^{31} +(24.2967 - 42.0831i) q^{37} +7.14387i q^{41} +53.7802 q^{43} +(-34.5934 - 19.9725i) q^{47} +(-30.7967 - 53.3414i) q^{53} -85.8983i q^{55} +(66.7967 - 38.5651i) q^{59} +(-0.373546 - 0.215667i) q^{61} +(-13.4066 + 23.2209i) q^{65} +(28.4835 + 49.3348i) q^{67} +123.560 q^{71} +(-31.0769 + 17.9422i) q^{73} +(26.0934 - 45.1951i) q^{79} -136.883i q^{83} +38.4397 q^{85} +(13.2198 + 7.63248i) q^{89} +(66.5603 + 115.286i) q^{95} +34.1524i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{5} - 15 q^{11} - 18 q^{17} + 81 q^{19} - 48 q^{23} + 61 q^{25} - 6 q^{29} - 48 q^{31} + 73 q^{37} + 70 q^{43} - 90 q^{47} - 99 q^{53} + 243 q^{59} + 192 q^{61} - 102 q^{65} - 7 q^{67} + 204 q^{71}+ \cdots - 24 q^{95}+O(q^{100}) \) 4 * q - 9 * q^5 - 15 * q^11 - 18 * q^17 + 81 * q^19 - 48 * q^23 + 61 * q^25 - 6 * q^29 - 48 * q^31 + 73 * q^37 + 70 * q^43 - 90 * q^47 - 99 * q^53 + 243 * q^59 + 192 * q^61 - 102 * q^65 - 7 * q^67 + 204 * q^71 + 45 * q^73 + 56 * q^79 + 444 * q^85 + 198 * q^89 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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