LMFDB - Embedded newform 1764.4.k.z.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.z.1549.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.41238 + 11.1066i) q^{5} +O(q^{10})$	$q+(6.41238 + 11.1066i) q^{5} +(-18.4124 + 31.8912i) q^{11} -87.1238 q^{13} +(-51.2990 + 88.8525i) q^{17} +(47.9124 + 82.9867i) q^{19} +(-48.0000 - 83.1384i) q^{23} +(-19.7371 + 34.1857i) q^{25} +212.021 q^{29} +(-79.6238 + 137.912i) q^{31} +(-64.3351 - 111.432i) q^{37} -298.042 q^{41} -33.3297 q^{43} +(-135.598 - 234.863i) q^{47} +(224.134 - 388.212i) q^{53} -472.268 q^{55} +(334.237 - 578.916i) q^{59} +(-121.846 - 211.043i) q^{61} +(-558.670 - 967.645i) q^{65} +(167.789 - 290.618i) q^{67} +339.608 q^{71} +(-459.160 + 795.288i) q^{73} +(68.1495 + 118.038i) q^{79} +287.464 q^{83} -1315.79 q^{85} +(-80.9277 - 140.171i) q^{89} +(-614.464 + 1064.28i) q^{95} -182.680 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5}+O(q^{10}) $ 4 * q + 3 * q^5	$ 4 q + 3 q^{5} - 51 q^{11} - 122 q^{13} - 24 q^{17} + 169 q^{19} - 192 q^{23} - 11 q^{25} + 78 q^{29} - 92 q^{31} + 173 q^{37} + 348 q^{41} - 994 q^{43} - 180 q^{47} + 285 q^{53} - 666 q^{55} + 1269 q^{59} + 328 q^{61} - 1374 q^{65} + 875 q^{67} + 2808 q^{71} - 1361 q^{73} + 182 q^{79} - 798 q^{83} - 4176 q^{85} - 822 q^{89} - 510 q^{95} - 1682 q^{97}+O(q^{100}) $ 4 * q + 3 * q^5 - 51 * q^11 - 122 * q^13 - 24 * q^17 + 169 * q^19 - 192 * q^23 - 11 * q^25 + 78 * q^29 - 92 * q^31 + 173 * q^37 + 348 * q^41 - 994 * q^43 - 180 * q^47 + 285 * q^53 - 666 * q^55 + 1269 * q^59 + 328 * q^61 - 1374 * q^65 + 875 * q^67 + 2808 * q^71 - 1361 * q^73 + 182 * q^79 - 798 * q^83 - 4176 * q^85 - 822 * q^89 - 510 * q^95 - 1682 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.41238	+	11.1066i	0.573540	+	0.993401i	0.996199	+	0.0871118i	$0.0277637\pi$
$5$	6.41238	+	11.1066i	0.573540	+	0.993401i	−0.422658	+	0.906289i	$0.638903\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−18.4124	+	31.8912i	−0.504685	+	0.874141i	0.495300	+	0.868722i	$0.335058\pi$
$11$	−18.4124	+	31.8912i	−0.504685	+	0.874141i	−0.999985	+	0.00541879i	$0.998275\pi$
$12$		0			0
$13$	−87.1238			−1.85875			−0.929376	−	0.369134i	$-0.879654\pi$
$13$	−87.1238			−1.85875			−0.929376	+	0.369134i	$0.879654\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−51.2990	+	88.8525i	−0.731873	+	1.26764i	0.224209	+	0.974541i	$0.428020\pi$
$17$	−51.2990	+	88.8525i	−0.731873	+	1.26764i	−0.956082	+	0.293100i	$0.905313\pi$
$18$		0			0
$19$	47.9124	+	82.9867i	0.578519	+	1.00202i	0.995650	+	0.0931772i	$0.0297023\pi$
$19$	47.9124	+	82.9867i	0.578519	+	1.00202i	−0.417131	+	0.908846i	$0.636964\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−48.0000	−	83.1384i	−0.435161	−	0.753720i	0.562148	−	0.827037i	$-0.309975\pi$
$23$	−48.0000	−	83.1384i	−0.435161	−	0.753720i	−0.997309	+	0.0733164i	$0.976642\pi$
$24$		0			0
$25$	−19.7371	+	34.1857i	−0.157897	+	0.273486i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	212.021			1.35763			0.678815	−	0.734309i	$-0.262494\pi$
$29$	212.021			1.35763			0.678815	+	0.734309i	$0.262494\pi$
$30$		0			0
$31$	−79.6238	+	137.912i	−0.461318	+	0.799026i	−0.999027	−	0.0441046i	$-0.985956\pi$
$31$	−79.6238	+	137.912i	−0.461318	+	0.799026i	0.537709	+	0.843130i	$0.319290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−64.3351	−	111.432i	−0.285855	−	0.495115i	0.686961	−	0.726694i	$-0.258944\pi$
$37$	−64.3351	−	111.432i	−0.285855	−	0.495115i	−0.972816	+	0.231579i	$0.925611\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−298.042			−1.13527			−0.567637	−	0.823279i	$-0.692142\pi$
$41$	−298.042			−1.13527			−0.567637	+	0.823279i	$0.692142\pi$
$42$		0			0
$43$	−33.3297			−0.118203			−0.0591016	−	0.998252i	$-0.518824\pi$
$43$	−33.3297			−0.118203			−0.0591016	+	0.998252i	$0.518824\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−135.598	−	234.863i	−0.420830	−	0.728899i	0.575191	−	0.818019i	$-0.304928\pi$
$47$	−135.598	−	234.863i	−0.420830	−	0.728899i	−0.996021	+	0.0891205i	$0.971594\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	224.134	−	388.212i	0.580890	−	1.00613i	−0.414484	−	0.910057i	$-0.636038\pi$
$53$	224.134	−	388.212i	0.580890	−	1.00613i	0.995374	−	0.0960750i	$-0.0306289\pi$
$54$		0			0
$55$	−472.268			−1.15783
$56$		0			0
$57$		0			0
$58$		0			0
$59$	334.237	−	578.916i	0.737525	−	1.27743i	−0.216082	−	0.976375i	$-0.569328\pi$
$59$	334.237	−	578.916i	0.737525	−	1.27743i	0.953607	−	0.301055i	$-0.0973389\pi$
$60$		0			0
$61$	−121.846	−	211.043i	−0.255750	−	0.442971i	0.709349	−	0.704857i	$-0.248989\pi$
$61$	−121.846	−	211.043i	−0.255750	−	0.442971i	−0.965099	+	0.261886i	$0.915656\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−558.670	−	967.645i	−1.06607	−	1.84649i
$66$		0			0
$67$	167.789	−	290.618i	0.305950	−	0.529921i	−0.671522	−	0.740984i	$-0.734359\pi$
$67$	167.789	−	290.618i	0.305950	−	0.529921i	0.977472	+	0.211063i	$0.0676927\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	339.608			0.567663			0.283831	−	0.958874i	$-0.408394\pi$
$71$	339.608			0.567663			0.283831	+	0.958874i	$0.408394\pi$
$72$		0			0
$73$	−459.160	+	795.288i	−0.736173	+	1.27509i	0.218034	+	0.975941i	$0.430036\pi$
$73$	−459.160	+	795.288i	−0.736173	+	1.27509i	−0.954207	+	0.299147i	$0.903298\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	68.1495	+	118.038i	0.0970559	+	0.168106i	0.910465	−	0.413587i	$-0.135724\pi$
$79$	68.1495	+	118.038i	0.0970559	+	0.168106i	−0.813409	+	0.581692i	$0.802391\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	287.464			0.380160			0.190080	−	0.981769i	$-0.439125\pi$
$83$	287.464			0.380160			0.190080	+	0.981769i	$0.439125\pi$
$84$		0			0
$85$	−1315.79			−1.67903
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−80.9277	−	140.171i	−0.0963856	−	0.166945i	0.813800	−	0.581144i	$-0.197395\pi$
$89$	−80.9277	−	140.171i	−0.0963856	−	0.166945i	−0.910186	+	0.414200i	$0.864062\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−614.464	+	1064.28i	−0.663607	+	1.14940i
$96$		0			0
$97$	−182.680			−0.191220			−0.0956101	−	0.995419i	$-0.530480\pi$
$97$	−182.680			−0.191220			−0.0956101	+	0.995419i	$0.530480\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	766.051	−	1326.84i	0.754703	−	1.30718i	−0.190819	−	0.981625i	$-0.561114\pi$
$101$	766.051	−	1326.84i	0.754703	−	1.30718i	0.945522	−	0.325558i	$-0.105552\pi$
$102$		0			0
$103$	243.954	+	422.541i	0.233374	+	0.404215i	0.958799	−	0.284086i	$-0.0916901\pi$
$103$	243.954	+	422.541i	0.233374	+	0.404215i	−0.725425	+	0.688301i	$0.758357\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	246.176	+	426.389i	0.222418	+	0.385239i	0.955542	−	0.294856i	$-0.0952718\pi$
$107$	246.176	+	426.389i	0.222418	+	0.385239i	−0.733124	+	0.680095i	$0.761938\pi$
$108$		0			0
$109$	−424.036	+	734.452i	−0.372617	+	0.645392i	−0.989967	−	0.141296i	$-0.954873\pi$
$109$	−424.036	+	734.452i	−0.372617	+	0.645392i	0.617350	+	0.786689i	$0.288206\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	736.350			0.613009			0.306505	−	0.951869i	$-0.400841\pi$
$113$	736.350			0.613009			0.306505	+	0.951869i	$0.400841\pi$
$114$		0			0
$115$	615.588	−	1066.23i	0.499164	−	0.864578i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−12.5312	−	21.7046i	−0.00941485	−	0.0163070i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1096.85			0.784839
$126$		0			0
$127$	−2511.37			−1.75471			−0.877355	−	0.479841i	$-0.840694\pi$
$127$	−2511.37			−1.75471			−0.877355	+	0.479841i	$0.840694\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	339.711	+	588.397i	0.226570	+	0.392431i	0.956789	−	0.290782i	$-0.0939153\pi$
$131$	339.711	+	588.397i	0.226570	+	0.392431i	−0.730219	+	0.683213i	$0.760582\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−82.2683	+	142.493i	−0.0513040	+	0.0888612i	−0.890537	−	0.454911i	$-0.849671\pi$
$137$	−82.2683	+	142.493i	−0.0513040	+	0.0888612i	0.839233	+	0.543772i	$0.183004\pi$
$138$		0			0
$139$	521.991			0.318523			0.159261	−	0.987236i	$-0.449089\pi$
$139$	521.991			0.318523			0.159261	+	0.987236i	$0.449089\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1604.16	−	2778.48i	0.938085	−	1.62481i
$144$		0			0
$145$	1359.56	+	2354.82i	0.778656	+	1.34867i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1206.06	−	2088.96i	−0.663117	−	1.14855i	−0.979792	−	0.200019i	$-0.935900\pi$
$149$	−1206.06	−	2088.96i	−0.663117	−	1.14855i	0.316675	−	0.948534i	$-0.397434\pi$
$150$		0			0
$151$	787.289	−	1363.62i	0.424296	−	0.734902i	−0.572059	−	0.820213i	$-0.693855\pi$
$151$	787.289	−	1363.62i	0.424296	−	0.734902i	0.996354	+	0.0853111i	$0.0271884\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2042.31			−1.05834
$156$		0			0
$157$	1039.37	−	1800.24i	0.528349	−	0.915128i	−0.471104	−	0.882078i	$-0.656144\pi$
$157$	1039.37	−	1800.24i	0.528349	−	0.915128i	0.999454	−	0.0330505i	$-0.0105222\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1589.71	+	2753.46i	0.763901	+	1.32312i	0.940826	+	0.338891i	$0.110052\pi$
$163$	1589.71	+	2753.46i	0.763901	+	1.32312i	−0.176924	+	0.984224i	$0.556615\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2979.28			−1.38050			−0.690250	−	0.723571i	$-0.742500\pi$
$167$	−2979.28			−1.38050			−0.690250	+	0.723571i	$0.742500\pi$
$168$		0			0
$169$	5393.55			2.45496
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.18518	−	14.1772i	−0.00359716	−	0.00623046i	0.864221	−	0.503112i	$-0.167812\pi$
$173$	−8.18518	−	14.1772i	−0.00359716	−	0.00623046i	−0.867818	+	0.496882i	$0.834478\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1349.15	−	2336.79i	0.563351	−	0.975753i	−0.433850	−	0.900985i	$-0.642845\pi$
$179$	1349.15	−	2336.79i	0.563351	−	0.975753i	0.997201	−	0.0747677i	$-0.0238215\pi$
$180$		0			0
$181$	−31.3297			−0.0128659			−0.00643293	−	0.999979i	$-0.502048\pi$
$181$	−31.3297			−0.0128659			−0.00643293	+	0.999979i	$0.502048\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	825.082	−	1429.08i	0.327899	−	0.567937i
$186$		0			0
$187$	−1889.07	−	3271.97i	−0.738731	−	1.27952i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−772.587	−	1338.16i	−0.292683	−	0.506942i	0.681760	−	0.731576i	$-0.261215\pi$
$191$	−772.587	−	1338.16i	−0.292683	−	0.506942i	−0.974443	+	0.224634i	$0.927881\pi$
$192$		0			0
$193$	915.099	−	1585.00i	0.341297	−	0.591143i	−0.643377	−	0.765549i	$-0.722467\pi$
$193$	915.099	−	1585.00i	0.341297	−	0.591143i	0.984674	+	0.174406i	$0.0558006\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4728.45			−1.71009			−0.855047	−	0.518551i	$-0.826472\pi$
$197$	−4728.45			−1.71009			−0.855047	+	0.518551i	$0.826472\pi$
$198$		0			0
$199$	−164.125	+	284.272i	−0.0584648	+	0.101264i	−0.893776	−	0.448513i	$-0.851954\pi$
$199$	−164.125	+	284.272i	−0.0584648	+	0.101264i	0.835312	+	0.549777i	$0.185287\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1911.15	−	3310.22i	−0.651126	−	1.12778i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3528.72			−1.16788
$210$		0			0
$211$	−4935.76			−1.61039			−0.805193	−	0.593013i	$-0.797938\pi$
$211$	−4935.76			−1.61039			−0.805193	+	0.593013i	$0.797938\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−213.723	−	370.179i	−0.0677943	−	0.117423i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4469.36	−	7741.16i	1.36037	−	2.35623i
$222$		0			0
$223$	−3446.00			−1.03480			−0.517402	−	0.855742i	$-0.673101\pi$
$223$	−3446.00			−1.03480			−0.517402	+	0.855742i	$0.673101\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2862.37	+	4957.77i	−0.836927	+	1.44960i	0.0555255	+	0.998457i	$0.482317\pi$
$227$	−2862.37	+	4957.77i	−0.836927	+	1.44960i	−0.892452	+	0.451142i	$0.851017\pi$
$228$		0			0
$229$	1508.57	+	2612.93i	0.435325	+	0.754004i	0.997322	−	0.0731345i	$-0.0233002\pi$
$229$	1508.57	+	2612.93i	0.435325	+	0.754004i	−0.561997	+	0.827139i	$0.689967\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	190.856	+	330.573i	0.0536627	+	0.0929466i	0.891609	−	0.452806i	$-0.149577\pi$
$233$	190.856	+	330.573i	0.0536627	+	0.0929466i	−0.837946	+	0.545753i	$0.816244\pi$
$234$		0			0
$235$	1739.01	−	3012.06i	0.482726	−	0.836106i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1377.38			0.372785			0.186392	−	0.982475i	$-0.440320\pi$
$239$	1377.38			0.372785			0.186392	+	0.982475i	$0.440320\pi$
$240$		0			0
$241$	2903.36	−	5028.77i	0.776025	−	1.34411i	−0.158192	−	0.987408i	$-0.550566\pi$
$241$	2903.36	−	5028.77i	0.776025	−	1.34411i	0.934217	−	0.356706i	$-0.116100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4174.31	−	7230.11i	−1.07532	−	1.86251i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4348.52			−1.09353			−0.546765	−	0.837286i	$-0.684141\pi$
$251$	−4348.52			−1.09353			−0.546765	+	0.837286i	$0.684141\pi$
$252$		0			0
$253$	3535.18			0.878477
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2345.56	+	4062.62i	0.569307	+	0.986068i	0.996635	+	0.0819713i	$0.0261216\pi$
$257$	2345.56	+	4062.62i	0.569307	+	0.986068i	−0.427328	+	0.904097i	$0.640545\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3790.81	+	6565.87i	−0.888788	+	1.53943i	−0.0474778	+	0.998872i	$0.515118\pi$
$263$	−3790.81	+	6565.87i	−0.888788	+	1.53943i	−0.841310	+	0.540553i	$0.818215\pi$
$264$		0			0
$265$	5748.93			1.33266
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1800.95	−	3119.33i	0.408200	−	0.707023i	−0.586488	−	0.809958i	$-0.699490\pi$
$269$	1800.95	−	3119.33i	0.408200	−	0.707023i	0.994688	+	0.102935i	$0.0328233\pi$
$270$		0			0
$271$	−1918.13	−	3322.31i	−0.429957	−	0.744707i	0.566912	−	0.823778i	$-0.308138\pi$
$271$	−1918.13	−	3322.31i	−0.429957	−	0.744707i	−0.996869	+	0.0790710i	$0.974805\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−726.815	−	1258.88i	−0.159377	−	0.276048i
$276$		0			0
$277$	−2701.52	+	4679.17i	−0.585988	+	1.01496i	0.408763	+	0.912640i	$0.365960\pi$
$277$	−2701.52	+	4679.17i	−0.585988	+	1.01496i	−0.994751	+	0.102321i	$0.967373\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−150.842			−0.0320230			−0.0160115	−	0.999872i	$-0.505097\pi$
$281$	−150.842			−0.0320230			−0.0160115	+	0.999872i	$0.505097\pi$
$282$		0			0
$283$	908.571	−	1573.69i	0.190844	−	0.330552i	−0.754686	−	0.656086i	$-0.772211\pi$
$283$	908.571	−	1573.69i	0.190844	−	0.330552i	0.945530	+	0.325534i	$0.105544\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2806.68	−	4861.31i	−0.571275	−	0.989478i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2817.59			−0.561792			−0.280896	−	0.959738i	$-0.590632\pi$
$293$	−2817.59			−0.561792			−0.280896	+	0.959738i	$0.590632\pi$
$294$		0			0
$295$	8573.02			1.69200
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4181.94	+	7243.33i	0.808856	+	1.40098i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1562.64	−	2706.57i	0.293365	−	0.508124i
$306$		0			0
$307$	−8589.21			−1.59678			−0.798391	−	0.602139i	$-0.794315\pi$
$307$	−8589.21			−1.59678			−0.798391	+	0.602139i	$0.794315\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2999.15	−	5194.67i	0.546836	−	0.947147i	−0.451653	−	0.892194i	$-0.649166\pi$
$311$	2999.15	−	5194.67i	0.546836	−	0.947147i	0.998489	−	0.0549538i	$-0.0175011\pi$
$312$		0			0
$313$	−2481.64	−	4298.32i	−0.448148	−	0.776216i	0.550117	−	0.835087i	$-0.314583\pi$
$313$	−2481.64	−	4298.32i	−0.448148	−	0.776216i	−0.998266	+	0.0588717i	$0.981250\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1952.30	−	3381.49i	−0.345906	−	0.599127i	0.639612	−	0.768698i	$-0.279095\pi$
$317$	−1952.30	−	3381.49i	−0.345906	−	0.599127i	−0.985518	+	0.169571i	$0.945762\pi$
$318$		0			0
$319$	−3903.81	+	6761.59i	−0.685176	+	1.18676i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9831.43			−1.69361
$324$		0			0
$325$	1719.57	−	2978.39i	0.293491	−	0.508342i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1224.86	+	2121.52i	0.203397	+	0.352295i	0.949621	−	0.313401i	$-0.101468\pi$
$331$	1224.86	+	2121.52i	0.203397	+	0.352295i	−0.746224	+	0.665695i	$0.768135\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4303.69			0.701898
$336$		0			0
$337$	−1770.59			−0.286203			−0.143101	−	0.989708i	$-0.545707\pi$
$337$	−1770.59			−0.286203			−0.143101	+	0.989708i	$0.545707\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2932.13	−	5078.59i	−0.465641	−	0.806513i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2017.36	+	3494.17i	−0.312097	+	0.540567i	−0.978816	−	0.204741i	$-0.934365\pi$
$347$	−2017.36	+	3494.17i	−0.312097	+	0.540567i	0.666719	+	0.745309i	$0.267698\pi$
$348$		0			0
$349$	−6791.53			−1.04167			−0.520834	−	0.853658i	$-0.674379\pi$
$349$	−6791.53			−1.04167			−0.520834	+	0.853658i	$0.674379\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5078.40	−	8796.05i	0.765712	−	1.32625i	−0.174158	−	0.984718i	$-0.555720\pi$
$353$	5078.40	−	8796.05i	0.765712	−	1.32625i	0.939870	−	0.341534i	$-0.110946\pi$
$354$		0			0
$355$	2177.69	+	3771.88i	0.325577	+	0.563917i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6421.36	−	11122.1i	−0.944029	−	1.63511i	−0.757683	−	0.652623i	$-0.773669\pi$
$359$	−6421.36	−	11122.1i	−0.944029	−	1.63511i	−0.186347	−	0.982484i	$-0.559665\pi$
$360$		0			0
$361$	−1161.69	+	2012.11i	−0.169367	+	0.293353i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11777.2			−1.68890
$366$		0			0
$367$	−957.408	+	1658.28i	−0.136175	+	0.235862i	−0.926046	−	0.377411i	$-0.876814\pi$
$367$	−957.408	+	1658.28i	−0.136175	+	0.235862i	0.789871	+	0.613274i	$0.210148\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2857.43	−	4949.21i	−0.396654	−	0.687026i	0.596656	−	0.802497i	$-0.296496\pi$
$373$	−2857.43	−	4949.21i	−0.396654	−	0.687026i	−0.993311	+	0.115471i	$0.963162\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18472.0			−2.52350
$378$		0			0
$379$	−11570.3			−1.56815			−0.784075	−	0.620666i	$-0.786862\pi$
$379$	−11570.3			−1.56815			−0.784075	+	0.620666i	$0.786862\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3059.01	−	5298.36i	−0.408115	−	0.706876i	0.586563	−	0.809903i	$-0.300481\pi$
$383$	−3059.01	−	5298.36i	−0.408115	−	0.706876i	−0.994679	+	0.103027i	$0.967147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1629.23	+	2821.91i	−0.212353	+	0.367806i	−0.952450	−	0.304694i	$-0.901446\pi$
$389$	−1629.23	+	2821.91i	−0.212353	+	0.367806i	0.740098	+	0.672499i	$0.234779\pi$
$390$		0			0
$391$	9849.41			1.27393
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−874.000	+	1513.81i	−0.111331	+	0.192831i
$396$		0			0
$397$	361.963	+	626.938i	0.0457592	+	0.0792572i	0.887998	−	0.459848i	$-0.152096\pi$
$397$	361.963	+	626.938i	0.0457592	+	0.0792572i	−0.842239	+	0.539105i	$0.818763\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5514.91	+	9552.11i	0.686787	+	1.18955i	0.972872	+	0.231346i	$0.0743128\pi$
$401$	5514.91	+	9552.11i	0.686787	+	1.18955i	−0.286085	+	0.958204i	$0.592354\pi$
$402$		0			0
$403$	6937.12	−	12015.4i	0.857475	−	1.48519i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4738.25			0.577067
$408$		0			0
$409$	1341.78	−	2324.03i	0.162217	−	0.280968i	−0.773447	−	0.633861i	$-0.781469\pi$
$409$	1341.78	−	2324.03i	0.162217	−	0.280968i	0.935663	+	0.352894i	$0.114802\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1843.33	+	3192.74i	0.218037	+	0.377652i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10024.9			1.16885			0.584427	−	0.811446i	$-0.301319\pi$
$419$	10024.9			1.16885			0.584427	+	0.811446i	$0.301319\pi$
$420$		0			0
$421$	−5560.68			−0.643731			−0.321866	−	0.946785i	$-0.604310\pi$
$421$	−5560.68			−0.643731			−0.321866	+	0.946785i	$0.604310\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2024.99	−	3507.39i	−0.231121	−	0.400313i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5763.11	−	9982.00i	0.644081	−	1.11558i	−0.340431	−	0.940269i	$-0.610573\pi$
$431$	5763.11	−	9982.00i	0.644081	−	1.11558i	0.984513	−	0.175312i	$-0.0560936\pi$
$432$		0			0
$433$	−2228.79			−0.247365			−0.123683	−	0.992322i	$-0.539470\pi$
$433$	−2228.79			−0.247365			−0.123683	+	0.992322i	$0.539470\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4599.59	−	7966.72i	0.503497	−	0.872082i
$438$		0			0
$439$	2304.63	+	3991.74i	0.250556	+	0.433975i	0.963679	−	0.267063i	$-0.0860533\pi$
$439$	2304.63	+	3991.74i	0.250556	+	0.433975i	−0.713123	+	0.701039i	$0.752720\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	531.662	+	920.865i	0.0570203	+	0.0987621i	0.893127	−	0.449805i	$-0.148507\pi$
$443$	531.662	+	920.865i	0.0570203	+	0.0987621i	−0.836106	+	0.548568i	$0.815173\pi$
$444$		0			0
$445$	1037.88	−	1797.66i	0.110562	−	0.191499i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12265.9			1.28923			0.644613	−	0.764509i	$-0.277018\pi$
$449$	12265.9			1.28923			0.644613	+	0.764509i	$0.277018\pi$
$450$		0			0
$451$	5487.65	−	9504.89i	0.572957	−	0.992390i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8895.59	+	15407.6i	0.910543	+	1.57711i	0.813299	+	0.581846i	$0.197669\pi$
$457$	8895.59	+	15407.6i	0.910543	+	1.57711i	0.0972436	+	0.995261i	$0.468997\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15368.9			1.55272			0.776358	−	0.630293i	$-0.217065\pi$
$461$	15368.9			1.55272			0.776358	+	0.630293i	$0.217065\pi$
$462$		0			0
$463$	−4104.98			−0.412040			−0.206020	−	0.978548i	$-0.566051\pi$
$463$	−4104.98			−0.412040			−0.206020	+	0.978548i	$0.566051\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1903.68	−	3297.27i	−0.188634	−	0.326723i	0.756161	−	0.654385i	$-0.227073\pi$
$467$	−1903.68	−	3297.27i	−0.188634	−	0.326723i	−0.944795	+	0.327662i	$0.893739\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	613.679	−	1062.92i	0.0596554	−	0.103326i
$474$		0			0
$475$	−3782.61			−0.365385
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4937.33	+	8551.70i	−0.470965	+	0.815735i	−0.999448	−	0.0332085i	$-0.989427\pi$
$479$	−4937.33	+	8551.70i	−0.470965	+	0.815735i	0.528484	+	0.848943i	$0.322761\pi$
$480$		0			0
$481$	5605.12	+	9708.35i	0.531334	+	0.920297i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1171.41	−	2028.95i	−0.109673	−	0.189958i
$486$		0			0
$487$	3381.86	−	5857.56i	0.314675	−	0.545033i	−0.664693	−	0.747116i	$-0.731438\pi$
$487$	3381.86	−	5857.56i	0.314675	−	0.545033i	0.979368	+	0.202083i	$0.0647711\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5574.29			0.512351			0.256175	−	0.966630i	$-0.417538\pi$
$491$	5574.29			0.512351			0.256175	+	0.966630i	$0.417538\pi$
$492$		0			0
$493$	−10876.5	+	18838.6i	−0.993612	+	1.72099i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2893.73	−	5012.09i	−0.259601	−	0.449643i	0.706534	−	0.707679i	$-0.250258\pi$
$499$	−2893.73	−	5012.09i	−0.259601	−	0.449643i	−0.966135	+	0.258037i	$0.916925\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8296.10			0.735397			0.367699	−	0.929945i	$-0.380146\pi$
$503$	8296.10			0.735397			0.367699	+	0.929945i	$0.380146\pi$
$504$		0			0
$505$	19648.8			1.73141
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4880.19	+	8452.74i	0.424972	+	0.736073i	0.996418	−	0.0845670i	$-0.0269507\pi$
$509$	4880.19	+	8452.74i	0.424972	+	0.736073i	−0.571446	+	0.820640i	$0.693617\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3128.65	+	5418.98i	−0.267699	+	0.463667i
$516$		0			0
$517$	9986.73			0.849547
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3268.99	−	5662.06i	0.274889	−	0.476121i	−0.695218	−	0.718799i	$-0.744692\pi$
$521$	3268.99	−	5662.06i	0.274889	−	0.476121i	0.970107	+	0.242677i	$0.0780256\pi$
$522$		0			0
$523$	−5171.92	−	8958.02i	−0.432413	−	0.748962i	0.564667	−	0.825319i	$-0.309004\pi$
$523$	−5171.92	−	8958.02i	−0.432413	−	0.748962i	−0.997081	+	0.0763570i	$0.975671\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8169.24	−	14149.5i	−0.675252	−	1.16957i
$528$		0			0
$529$	1475.50	−	2555.64i	0.121271	−	0.210047i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	25966.5			2.11020
$534$		0			0
$535$	−3157.14	+	5468.33i	−0.255131	+	0.441900i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3421.60	+	5926.38i	0.271915	+	0.470970i	0.969352	−	0.245676i	$-0.0790098\pi$
$541$	3421.60	+	5926.38i	0.271915	+	0.470970i	−0.697437	+	0.716646i	$0.745676\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10876.3			−0.854844
$546$		0			0
$547$	−18402.1			−1.43842			−0.719211	−	0.694791i	$-0.755497\pi$
$547$	−18402.1			−1.43842			−0.719211	+	0.694791i	$0.755497\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10158.4	+	17594.9i	0.785414	+	1.36038i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	323.239	−	559.866i	0.0245890	−	0.0425894i	−0.853469	−	0.521144i	$-0.825506\pi$
$557$	323.239	−	559.866i	0.0245890	−	0.0425894i	0.878058	+	0.478554i	$0.158839\pi$
$558$		0			0
$559$	2903.81			0.219710
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3087.40	+	5347.53i	−0.231116	+	0.400305i	−0.958137	−	0.286311i	$-0.907571\pi$
$563$	−3087.40	+	5347.53i	−0.231116	+	0.400305i	0.727021	+	0.686616i	$0.240904\pi$
$564$		0			0
$565$	4721.76	+	8178.32i	0.351585	+	0.608964i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3418.59	−	5921.17i	−0.251871	−	0.436253i	0.712170	−	0.702007i	$-0.247713\pi$
$569$	−3418.59	−	5921.17i	−0.251871	−	0.436253i	−0.964041	+	0.265754i	$0.914379\pi$
$570$		0			0
$571$	2942.77	−	5097.03i	0.215676	−	0.373562i	−0.737805	−	0.675014i	$-0.764138\pi$
$571$	2942.77	−	5097.03i	0.215676	−	0.373562i	0.953482	+	0.301451i	$0.0974711\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3789.53			0.274842
$576$		0			0
$577$	−6001.62	+	10395.1i	−0.433017	+	0.750007i	−0.997131	−	0.0756896i	$-0.975884\pi$
$577$	−6001.62	+	10395.1i	−0.433017	+	0.750007i	0.564115	+	0.825696i	$0.309218\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8253.68	+	14295.8i	0.586334	+	1.01556i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12719.4			−0.894358			−0.447179	−	0.894445i	$-0.647571\pi$
$587$	−12719.4			−0.894358			−0.447179	+	0.894445i	$0.647571\pi$
$588$		0			0
$589$	−15259.9			−1.06752
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9580.20	−	16593.4i	−0.663426	−	1.14909i	−0.979709	−	0.200423i	$-0.935768\pi$
$593$	−9580.20	−	16593.4i	−0.663426	−	1.14909i	0.316283	−	0.948665i	$-0.397565\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	533.983	−	924.885i	0.0364240	−	0.0630881i	−0.847239	−	0.531212i	$-0.821737\pi$
$599$	533.983	−	924.885i	0.0364240	−	0.0630881i	0.883663	+	0.468124i	$0.155070\pi$
$600$		0			0
$601$	−7554.96			−0.512768			−0.256384	−	0.966575i	$-0.582531\pi$
$601$	−7554.96			−0.512768			−0.256384	+	0.966575i	$0.582531\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	160.709	−	278.356i	0.0107996	−	0.0187054i
$606$		0			0
$607$	−5888.71	−	10199.5i	−0.393765	−	0.682020i	0.599178	−	0.800616i	$-0.295494\pi$
$607$	−5888.71	−	10199.5i	−0.393765	−	0.682020i	−0.992943	+	0.118595i	$0.962161\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11813.8	+	20462.1i	0.782219	+	1.35484i
$612$		0			0
$613$	−13426.5	+	23255.3i	−0.884649	+	1.53226i	−0.0385337	+	0.999257i	$0.512269\pi$
$613$	−13426.5	+	23255.3i	−0.884649	+	1.53226i	−0.846115	+	0.533000i	$0.821065\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6816.72			0.444782			0.222391	−	0.974958i	$-0.428614\pi$
$617$	6816.72			0.444782			0.222391	+	0.974958i	$0.428614\pi$
$618$		0			0
$619$	−8356.74	+	14474.3i	−0.542627	+	0.939857i	0.456126	+	0.889915i	$0.349237\pi$
$619$	−8356.74	+	14474.3i	−0.542627	+	0.939857i	−0.998752	+	0.0499414i	$0.984097\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9500.53	+	16455.4i	0.608034	+	1.05315i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13201.3			0.836838
$630$		0			0
$631$	592.225			0.0373631			0.0186815	−	0.999825i	$-0.494053\pi$
$631$	592.225			0.0373631			0.0186815	+	0.999825i	$0.494053\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16103.9	−	27892.7i	−1.00640	−	1.74313i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3481.96	−	6030.92i	0.214554	−	0.371618i	−0.738581	−	0.674165i	$-0.764504\pi$
$641$	3481.96	−	6030.92i	0.214554	−	0.371618i	0.953134	+	0.302547i	$0.0978369\pi$
$642$		0			0
$643$	−5466.06			−0.335242			−0.167621	−	0.985852i	$-0.553608\pi$
$643$	−5466.06			−0.335242			−0.167621	+	0.985852i	$0.553608\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	618.633	−	1071.50i	0.0375904	−	0.0651085i	−0.846618	−	0.532201i	$-0.821365\pi$
$647$	618.633	−	1071.50i	0.0375904	−	0.0651085i	0.884209	+	0.467092i	$0.154698\pi$
$648$		0			0
$649$	12308.2	+	21318.4i	0.744436	+	1.28940i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13575.9	+	23514.2i	0.813578	+	1.40916i	0.910344	+	0.413852i	$0.135817\pi$
$653$	13575.9	+	23514.2i	0.813578	+	1.40916i	−0.0967657	+	0.995307i	$0.530850\pi$
$654$		0			0
$655$	−4356.71	+	7546.05i	−0.259895	+	0.450151i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5900.66			−0.348797			−0.174398	−	0.984675i	$-0.555798\pi$
$659$	−5900.66			−0.348797			−0.174398	+	0.984675i	$0.555798\pi$
$660$		0			0
$661$	1809.49	−	3134.13i	0.106477	−	0.184423i	−0.807864	−	0.589369i	$-0.799376\pi$
$661$	1809.49	−	3134.13i	0.106477	−	0.184423i	0.914341	+	0.404946i	$0.132710\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−10177.0	−	17627.1i	−0.590787	−	1.02327i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8973.86			0.516292
$672$		0			0
$673$	−13952.5			−0.799151			−0.399576	−	0.916700i	$-0.630842\pi$
$673$	−13952.5			−0.799151			−0.399576	+	0.916700i	$0.630842\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14459.4	+	25044.5i	0.820859	+	1.42177i	0.905044	+	0.425319i	$0.139838\pi$
$677$	14459.4	+	25044.5i	0.820859	+	1.42177i	−0.0841848	+	0.996450i	$0.526829\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5565.06	+	9638.97i	−0.311773	+	0.540007i	−0.978746	−	0.205075i	$-0.934256\pi$
$683$	−5565.06	+	9638.97i	−0.311773	+	0.540007i	0.666973	+	0.745082i	$0.267590\pi$
$684$		0			0
$685$	−2110.14			−0.117700
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19527.4	+	33822.5i	−1.07973	+	1.87015i
$690$		0			0
$691$	−7012.64	−	12146.2i	−0.386069	−	0.668690i	0.605848	−	0.795580i	$-0.292834\pi$
$691$	−7012.64	−	12146.2i	−0.386069	−	0.668690i	−0.991917	+	0.126890i	$0.959501\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3347.20	+	5797.52i	0.182686	+	0.316421i
$696$		0			0
$697$	15289.2	−	26481.7i	0.830877	−	1.43912i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30366.7			−1.63614			−0.818070	−	0.575119i	$-0.804956\pi$
$701$	−30366.7			−1.63614			−0.818070	+	0.575119i	$0.804956\pi$
$702$		0			0
$703$	6164.90	−	10677.9i	0.330745	−	0.572867i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8445.48	−	14628.0i	−0.447358	−	0.774847i	0.550855	−	0.834601i	$-0.314302\pi$
$709$	−8445.48	−	14628.0i	−0.447358	−	0.774847i	−0.998213	+	0.0597542i	$0.980968\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15287.8			0.802989
$714$		0			0
$715$	41145.8			2.15212
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5503.61	−	9532.54i	−0.285466	−	0.494442i	0.687256	−	0.726415i	$-0.258815\pi$
$719$	−5503.61	−	9532.54i	−0.285466	−	0.494442i	−0.972722	+	0.231973i	$0.925482\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4184.68	+	7248.08i	−0.214366	+	0.371292i
$726$		0			0
$727$	14618.0			0.745737			0.372869	−	0.927884i	$-0.378374\pi$
$727$	14618.0			0.745737			0.372869	+	0.927884i	$0.378374\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1709.78	−	2961.43i	0.0865096	−	0.149839i
$732$		0			0
$733$	−14261.1	−	24701.0i	−0.718617	−	1.24468i	−0.961548	−	0.274638i	$-0.911442\pi$
$733$	−14261.1	−	24701.0i	−0.718617	−	1.24468i	0.242930	−	0.970044i	$-0.421891\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6178.77	+	10702.0i	0.308817	+	0.534887i
$738$		0			0
$739$	−10119.7	+	17527.9i	−0.503735	+	0.872495i	0.496256	+	0.868176i	$0.334708\pi$
$739$	−10119.7	+	17527.9i	−0.503735	+	0.872495i	−0.999991	+	0.00431822i	$0.998625\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13977.6			−0.690159			−0.345079	−	0.938573i	$-0.612148\pi$
$743$	−13977.6			−0.690159			−0.345079	+	0.938573i	$0.612148\pi$
$744$		0			0
$745$	15467.4	−	26790.4i	0.760649	−	1.31748i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7513.90	−	13014.5i	−0.365095	−	0.632363i	0.623697	−	0.781667i	$-0.285630\pi$
$751$	−7513.90	−	13014.5i	−0.365095	−	0.632363i	−0.988791	+	0.149304i	$0.952297\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20193.6			0.973403
$756$		0			0
$757$	20769.4			0.997196			0.498598	−	0.866833i	$-0.333848\pi$
$757$	20769.4			0.997196			0.498598	+	0.866833i	$0.333848\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5605.57	−	9709.13i	−0.267019	−	0.462491i	0.701071	−	0.713091i	$-0.252705\pi$
$761$	−5605.57	−	9709.13i	−0.267019	−	0.462491i	−0.968091	+	0.250600i	$0.919372\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−29120.0	+	50437.3i	−1.37088	+	2.37443i
$768$		0			0
$769$	−4305.86			−0.201916			−0.100958	−	0.994891i	$-0.532191\pi$
$769$	−4305.86			−0.201916			−0.100958	+	0.994891i	$0.532191\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8320.24	+	14411.1i	−0.387139	+	0.670544i	−0.992063	−	0.125739i	$-0.959870\pi$
$773$	−8320.24	+	14411.1i	−0.387139	+	0.670544i	0.604925	+	0.796283i	$0.293203\pi$
$774$		0			0
$775$	−3143.09	−	5443.99i	−0.145681	−	0.252328i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14279.9	−	24733.5i	−0.656778	−	1.13757i
$780$		0			0
$781$	−6252.99	+	10830.5i	−0.286491	+	0.496217i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	26659.4			1.21212
$786$		0			0
$787$	−4883.53	+	8458.51i	−0.221193	+	0.383118i	−0.955171	−	0.296056i	$-0.904328\pi$
$787$	−4883.53	+	8458.51i	−0.221193	+	0.383118i	0.733977	+	0.679174i	$0.237662\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	10615.6	+	18386.8i	0.475375	+	0.823374i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23118.5			1.02748			0.513738	−	0.857947i	$-0.328260\pi$
$797$	23118.5			1.02748			0.513738	+	0.857947i	$0.328260\pi$
$798$		0			0
$799$	27824.2			1.23198
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16908.4	−	29286.3i	−0.743071	−	1.28704i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5230.33	−	9059.20i	0.227304	−	0.393701i	−0.729705	−	0.683763i	$-0.760342\pi$
$809$	5230.33	−	9059.20i	0.227304	−	0.393701i	0.957008	+	0.290061i	$0.0936757\pi$
$810$		0			0
$811$	9167.55			0.396937			0.198469	−	0.980107i	$-0.436403\pi$
$811$	9167.55			0.396937			0.198469	+	0.980107i	$0.436403\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20387.7	+	35312.5i	−0.876256	+	1.51772i
$816$		0			0
$817$	−1596.91	−	2765.92i	−0.0683827	−	0.118442i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14413.7	+	24965.2i	0.612717	+	1.06126i	0.990780	+	0.135477i	$0.0432567\pi$
$821$	14413.7	+	24965.2i	0.612717	+	1.06126i	−0.378064	+	0.925780i	$0.623410\pi$
$822$		0			0
$823$	−21397.2	+	37061.0i	−0.906268	+	1.56970i	−0.0870618	+	0.996203i	$0.527748\pi$
$823$	−21397.2	+	37061.0i	−0.906268	+	1.56970i	−0.819206	+	0.573499i	$0.805586\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6028.87			0.253500			0.126750	−	0.991935i	$-0.459545\pi$
$827$	6028.87			0.253500			0.126750	+	0.991935i	$0.459545\pi$
$828$		0			0
$829$	−9703.71	+	16807.3i	−0.406542	+	0.704152i	−0.994500	−	0.104740i	$-0.966599\pi$
$829$	−9703.71	+	16807.3i	−0.406542	+	0.704152i	0.587957	+	0.808892i	$0.299932\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−19104.3	−	33089.5i	−0.791773	−	1.37139i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10599.5			−0.436156			−0.218078	−	0.975931i	$-0.569979\pi$
$839$	−10599.5			−0.436156			−0.218078	+	0.975931i	$0.569979\pi$
$840$		0			0
$841$	20563.8			0.843159
$842$		0			0
$843$		0			0
$844$		0			0
$845$	34585.5	+	59903.8i	1.40802	+	2.43876i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6176.17	+	10697.4i	−0.248786	+	0.430909i
$852$		0			0
$853$	34766.1			1.39551			0.697754	−	0.716338i	$-0.254183\pi$
$853$	34766.1			1.39551			0.697754	+	0.716338i	$0.254183\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15002.0	+	25984.2i	−0.597968	+	1.03571i	0.395153	+	0.918615i	$0.370692\pi$
$857$	−15002.0	+	25984.2i	−0.597968	+	1.03571i	−0.993121	+	0.117096i	$0.962642\pi$
$858$		0			0
$859$	−13065.3	−	22629.8i	−0.518955	−	0.898857i	−0.999757	−	0.0220275i	$-0.992988\pi$
$859$	−13065.3	−	22629.8i	−0.518955	−	0.898857i	0.480802	−	0.876829i	$-0.340345\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22334.2	+	38683.9i	0.880955	+	1.52586i	0.850281	+	0.526329i	$0.176432\pi$
$863$	22334.2	+	38683.9i	0.880955	+	1.52586i	0.0306737	+	0.999529i	$0.490235\pi$
$864$		0			0
$865$	104.973	−	181.818i	0.00412623	−	0.00714684i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5019.18			−0.195931
$870$		0			0
$871$	−14618.4	+	25319.8i	−0.568685	+	0.984992i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1839.28	−	3185.72i	−0.0708187	−	0.122662i	0.828442	−	0.560075i	$-0.189228\pi$
$877$	−1839.28	−	3185.72i	−0.0708187	−	0.122662i	−0.899260	+	0.437414i	$0.855895\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−33443.6			−1.27894			−0.639468	−	0.768817i	$-0.720846\pi$
$881$	−33443.6			−1.27894			−0.639468	+	0.768817i	$0.720846\pi$
$882$		0			0
$883$	−21095.4			−0.803983			−0.401991	−	0.915644i	$-0.631682\pi$
$883$	−21095.4			−0.803983			−0.401991	+	0.915644i	$0.631682\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7226.65	−	12516.9i	−0.273560	−	0.473819i	0.696211	−	0.717837i	$-0.254868\pi$
$887$	−7226.65	−	12516.9i	−0.273560	−	0.473819i	−0.969771	+	0.244018i	$0.921534\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12993.6	−	22505.7i	0.486916	−	0.843363i
$894$		0			0
$895$	34604.9			1.29242
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16881.9	+	29240.3i	−0.626299	+	1.08478i
$900$		0			0
$901$	22995.7	+	39829.8i	0.850276	+	1.47272i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−200.898	−	347.965i	−0.00737909	−	0.0127810i
$906$		0			0
$907$	2602.04	−	4506.87i	0.0952584	−	0.164992i	−0.814458	−	0.580223i	$-0.802966\pi$
$907$	2602.04	−	4506.87i	0.0952584	−	0.164992i	0.909716	+	0.415230i	$0.136299\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−31584.8			−1.14868			−0.574342	−	0.818616i	$-0.694742\pi$
$911$	−31584.8			−1.14868			−0.574342	+	0.818616i	$0.694742\pi$
$912$		0			0
$913$	−5292.90	+	9167.57i	−0.191861	+	0.332314i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	23315.3	+	40383.3i	0.836889	+	1.44953i	0.892483	+	0.451081i	$0.148961\pi$
$919$	23315.3	+	40383.3i	0.836889	+	1.44953i	−0.0555943	+	0.998453i	$0.517705\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−29587.9			−1.05514
$924$		0			0
$925$	5079.16			0.180543
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19246.3	+	33335.6i	0.679711	+	1.17729i	0.975068	+	0.221907i	$0.0712280\pi$
$929$	19246.3	+	33335.6i	0.679711	+	1.17729i	−0.295357	+	0.955387i	$0.595439\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24226.9	−	41962.2i	0.847384	−	1.46771i
$936$		0			0
$937$	−39893.8			−1.39090			−0.695450	−	0.718574i	$-0.744795\pi$
$937$	−39893.8			−1.39090			−0.695450	+	0.718574i	$0.744795\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16315.4	−	28259.0i	0.565213	−	0.978977i	−0.431817	−	0.901961i	$-0.642127\pi$
$941$	16315.4	−	28259.0i	0.565213	−	0.978977i	0.997030	−	0.0770161i	$-0.0245393\pi$
$942$		0			0
$943$	14306.0	+	24778.7i	0.494027	+	0.855680i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15014.7	+	26006.1i	0.515217	+	0.892382i	0.999844	+	0.0176614i	$0.00562208\pi$
$947$	15014.7	+	26006.1i	0.515217	+	0.892382i	−0.484627	+	0.874721i	$0.661045\pi$
$948$		0			0
$949$	40003.7	−	69288.5i	1.36836	−	2.37007i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	34963.6			1.18844			0.594220	−	0.804303i	$-0.297461\pi$
$953$	34963.6			1.18844			0.594220	+	0.804303i	$0.297461\pi$
$954$		0			0
$955$	9908.24	−	17161.6i	0.335731	−	0.581503i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2215.61	+	3837.56i	0.0743719	+	0.128816i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	23471.8			0.782990
$966$		0			0
$967$	20520.5			0.682414			0.341207	−	0.939988i	$-0.389164\pi$
$967$	20520.5			0.682414			0.341207	+	0.939988i	$0.389164\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19887.6	−	34446.4i	−0.657286	−	1.13845i	−0.981315	−	0.192406i	$-0.938371\pi$
$971$	−19887.6	−	34446.4i	−0.657286	−	1.13845i	0.324029	−	0.946047i	$-0.394962\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3586.02	+	6211.17i	−0.117428	+	0.203391i	−0.918748	−	0.394845i	$-0.870798\pi$
$977$	−3586.02	+	6211.17i	−0.117428	+	0.203391i	0.801320	+	0.598236i	$0.204132\pi$
$978$		0			0
$979$	5960.29			0.194578
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24174.1	+	41870.8i	−0.784369	+	1.35857i	0.145006	+	0.989431i	$0.453680\pi$
$983$	−24174.1	+	41870.8i	−0.784369	+	1.35857i	−0.929375	+	0.369137i	$0.879653\pi$
$984$		0			0
$985$	−30320.6	−	52516.9i	−0.980807	−	1.69881i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1599.83	+	2770.98i	0.0514373	+	0.0890921i
$990$		0			0
$991$	−18472.3	+	31994.9i	−0.592121	+	1.02558i	0.401826	+	0.915716i	$0.368376\pi$
$991$	−18472.3	+	31994.9i	−0.592121	+	1.02558i	−0.993946	+	0.109867i	$0.964958\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4209.72			−0.134128
$996$		0			0
$997$	28294.8	−	49008.0i	0.898802	−	1.55677i	0.0697741	−	0.997563i	$-0.477772\pi$
$997$	28294.8	−	49008.0i	0.898802	−	1.55677i	0.829028	−	0.559208i	$-0.188895\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.z.361.2		4
3.2	odd	2		588.4.i.i.361.1		4
7.2	even	3	inner	1764.4.k.z.1549.2		4
7.3	odd	6		1764.4.a.x.1.2		2
7.4	even	3		1764.4.a.p.1.1		2
7.5	odd	6		252.4.k.d.37.1		4
7.6	odd	2		252.4.k.d.109.1		4
21.2	odd	6		588.4.i.i.373.1		4
21.5	even	6		84.4.i.b.37.2	yes	4
21.11	odd	6		588.4.a.h.1.2		2
21.17	even	6		588.4.a.g.1.1		2
21.20	even	2		84.4.i.b.25.2	✓	4
84.11	even	6		2352.4.a.bp.1.2		2
84.47	odd	6		336.4.q.h.289.2		4
84.59	odd	6		2352.4.a.cb.1.1		2
84.83	odd	2		336.4.q.h.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.i.b.25.2	✓	4	21.20	even	2
84.4.i.b.37.2	yes	4	21.5	even	6
252.4.k.d.37.1		4	7.5	odd	6
252.4.k.d.109.1		4	7.6	odd	2
336.4.q.h.193.2		4	84.83	odd	2
336.4.q.h.289.2		4	84.47	odd	6
588.4.a.g.1.1		2	21.17	even	6
588.4.a.h.1.2		2	21.11	odd	6
588.4.i.i.361.1		4	3.2	odd	2
588.4.i.i.373.1		4	21.2	odd	6
1764.4.a.p.1.1		2	7.4	even	3
1764.4.a.x.1.2		2	7.3	odd	6
1764.4.k.z.361.2		4	1.1	even	1	trivial
1764.4.k.z.1549.2		4	7.2	even	3	inner
2352.4.a.bp.1.2		2	84.11	even	6
2352.4.a.cb.1.1		2	84.59	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.41238 + 11.1066i) q^{5} +O(q^{10})\)	\(q+(6.41238 + 11.1066i) q^{5} +(-18.4124 + 31.8912i) q^{11} -87.1238 q^{13} +(-51.2990 + 88.8525i) q^{17} +(47.9124 + 82.9867i) q^{19} +(-48.0000 - 83.1384i) q^{23} +(-19.7371 + 34.1857i) q^{25} +212.021 q^{29} +(-79.6238 + 137.912i) q^{31} +(-64.3351 - 111.432i) q^{37} -298.042 q^{41} -33.3297 q^{43} +(-135.598 - 234.863i) q^{47} +(224.134 - 388.212i) q^{53} -472.268 q^{55} +(334.237 - 578.916i) q^{59} +(-121.846 - 211.043i) q^{61} +(-558.670 - 967.645i) q^{65} +(167.789 - 290.618i) q^{67} +339.608 q^{71} +(-459.160 + 795.288i) q^{73} +(68.1495 + 118.038i) q^{79} +287.464 q^{83} -1315.79 q^{85} +(-80.9277 - 140.171i) q^{89} +(-614.464 + 1064.28i) q^{95} -182.680 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5}+O(q^{10}) \) 4 * q + 3 * q^5	\( 4 q + 3 q^{5} - 51 q^{11} - 122 q^{13} - 24 q^{17} + 169 q^{19} - 192 q^{23} - 11 q^{25} + 78 q^{29} - 92 q^{31} + 173 q^{37} + 348 q^{41} - 994 q^{43} - 180 q^{47} + 285 q^{53} - 666 q^{55} + 1269 q^{59} + 328 q^{61} - 1374 q^{65} + 875 q^{67} + 2808 q^{71} - 1361 q^{73} + 182 q^{79} - 798 q^{83} - 4176 q^{85} - 822 q^{89} - 510 q^{95} - 1682 q^{97}+O(q^{100}) \) 4 * q + 3 * q^5 - 51 * q^11 - 122 * q^13 - 24 * q^17 + 169 * q^19 - 192 * q^23 - 11 * q^25 + 78 * q^29 - 92 * q^31 + 173 * q^37 + 348 * q^41 - 994 * q^43 - 180 * q^47 + 285 * q^53 - 666 * q^55 + 1269 * q^59 + 328 * q^61 - 1374 * q^65 + 875 * q^67 + 2808 * q^71 - 1361 * q^73 + 182 * q^79 - 798 * q^83 - 4176 * q^85 - 822 * q^89 - 510 * q^95 - 1682 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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