LMFDB - Embedded newform 1764.4.t.c.521.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.t (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.13
Character	$\chi$	$=$	1764.521
Dual form			1764.4.t.c.1097.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.582251 + 1.00849i) q^{5} +O(q^{10})$	$q+(0.582251 + 1.00849i) q^{5} +(-56.8211 - 32.8057i) q^{11} +57.5711i q^{13} +(27.6457 - 47.8837i) q^{17} +(106.374 - 61.4153i) q^{19} +(-47.1869 + 27.2434i) q^{23} +(61.8220 - 107.079i) q^{25} +105.124i q^{29} +(84.6938 + 48.8980i) q^{31} +(65.0046 + 112.591i) q^{37} +69.2322 q^{41} -87.1042 q^{43} +(-118.347 - 204.983i) q^{47} +(357.505 + 206.406i) q^{53} -76.4046i q^{55} +(-21.1051 + 36.5551i) q^{59} +(-731.273 + 422.201i) q^{61} +(-58.0598 + 33.5208i) q^{65} +(-321.867 + 557.490i) q^{67} -749.296i q^{71} +(-691.161 - 399.042i) q^{73} +(-179.996 - 311.762i) q^{79} -1304.90 q^{83} +64.3869 q^{85} +(54.8745 + 95.0454i) q^{89} +(123.873 + 71.5182i) q^{95} +781.538i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 888 q^{25} - 864 q^{37} - 2496 q^{43} - 1056 q^{67} - 16128 q^{85}+O(q^{100}) $ 48 * q - 888 * q^25 - 864 * q^37 - 2496 * q^43 - 1056 * q^67 - 16128 * q^85

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	0.582251	+	1.00849i	0.0520781	+	0.0902020i	0.890889	−	0.454221i	$-0.150082\pi$
$5$	0.582251	+	1.00849i	0.0520781	+	0.0902020i	−0.838811	+	0.544422i	$0.816749\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−56.8211	−	32.8057i	−1.55747	−	0.899208i	−0.997498	−	0.0706989i	$-0.977477\pi$
$11$	−56.8211	−	32.8057i	−1.55747	−	0.899208i	−0.559976	−	0.828509i	$-0.689190\pi$
$12$		0			0
$13$			57.5711i			1.22826i	0.789206	+	0.614129i	$0.210493\pi$
$13$			57.5711i			1.22826i	−0.789206	+	0.614129i	$0.789507\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	27.6457	−	47.8837i	0.394415	−	0.683147i	−0.598611	−	0.801040i	$-0.704281\pi$
$17$	27.6457	−	47.8837i	0.394415	−	0.683147i	0.993026	+	0.117893i	$0.0376139\pi$
$18$		0			0
$19$	106.374	−	61.4153i	1.28442	−	0.741559i	0.306766	−	0.951785i	$-0.400753\pi$
$19$	106.374	−	61.4153i	1.28442	−	0.741559i	0.977653	+	0.210226i	$0.0674199\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−47.1869	+	27.2434i	−0.427789	+	0.246984i	−0.698404	−	0.715703i	$-0.746106\pi$
$23$	−47.1869	+	27.2434i	−0.427789	+	0.246984i	0.270615	+	0.962688i	$0.412773\pi$
$24$		0			0
$25$	61.8220	−	107.079i	0.494576	−	0.856630i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			105.124i			0.673138i	0.941659	+	0.336569i	$0.109267\pi$
$29$			105.124i			0.673138i	−0.941659	+	0.336569i	$0.890733\pi$
$30$		0			0
$31$	84.6938	+	48.8980i	0.490692	+	0.283301i	0.724861	−	0.688895i	$-0.241904\pi$
$31$	84.6938	+	48.8980i	0.490692	+	0.283301i	−0.234170	+	0.972196i	$0.575237\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	65.0046	+	112.591i	0.288829	+	0.500267i	0.973531	−	0.228557i	$-0.0734006\pi$
$37$	65.0046	+	112.591i	0.288829	+	0.500267i	−0.684701	+	0.728824i	$0.740067\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	69.2322			0.263713			0.131857	−	0.991269i	$-0.457906\pi$
$41$	69.2322			0.263713			0.131857	+	0.991269i	$0.457906\pi$
$42$		0			0
$43$	−87.1042			−0.308913			−0.154457	−	0.988000i	$-0.549363\pi$
$43$	−87.1042			−0.308913			−0.154457	+	0.988000i	$0.549363\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−118.347	−	204.983i	−0.367291	−	0.636166i	0.621850	−	0.783136i	$-0.286381\pi$
$47$	−118.347	−	204.983i	−0.367291	−	0.636166i	−0.989141	+	0.146970i	$0.953048\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	357.505	+	206.406i	0.926549	+	0.534943i	0.885719	−	0.464223i	$-0.153666\pi$
$53$	357.505	+	206.406i	0.926549	+	0.534943i	0.0408306	+	0.999166i	$0.487000\pi$
$54$		0			0
$55$		−	76.4046i		−	0.187316i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−21.1051	+	36.5551i	−0.0465703	+	0.0806622i	−0.888371	−	0.459126i	$-0.848162\pi$
$59$	−21.1051	+	36.5551i	−0.0465703	+	0.0806622i	0.841801	+	0.539789i	$0.181496\pi$
$60$		0			0
$61$	−731.273	+	422.201i	−1.53492	+	0.886185i	−0.535792	+	0.844350i	$0.679987\pi$
$61$	−731.273	+	422.201i	−1.53492	+	0.886185i	−0.999125	+	0.0418348i	$0.986680\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−58.0598	+	33.5208i	−0.110791	+	0.0639654i
$66$		0			0
$67$	−321.867	+	557.490i	−0.586900	+	1.01654i	0.407735	+	0.913100i	$0.366319\pi$
$67$	−321.867	+	557.490i	−0.586900	+	1.01654i	−0.994636	+	0.103441i	$0.967015\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	749.296i		−	1.25247i	−0.779636	−	0.626233i	$-0.784596\pi$
$71$		−	749.296i		−	1.25247i	0.779636	−	0.626233i	$-0.215404\pi$
$72$		0			0
$73$	−691.161	−	399.042i	−1.10814	−	0.639786i	−0.169794	−	0.985480i	$-0.554310\pi$
$73$	−691.161	−	399.042i	−1.10814	−	0.639786i	−0.938347	+	0.345694i	$0.887644\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−179.996	−	311.762i	−0.256343	−	0.443999i	0.708916	−	0.705292i	$-0.249184\pi$
$79$	−179.996	−	311.762i	−0.256343	−	0.443999i	−0.965259	+	0.261293i	$0.915851\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1304.90			−1.72568			−0.862840	−	0.505477i	$-0.831317\pi$
$83$	−1304.90			−1.72568			−0.862840	+	0.505477i	$0.831317\pi$
$84$		0			0
$85$	64.3869			0.0821616
$86$		0			0
$87$		0			0
$88$		0			0
$89$	54.8745	+	95.0454i	0.0653560	+	0.113200i	0.896852	−	0.442331i	$-0.145848\pi$
$89$	54.8745	+	95.0454i	0.0653560	+	0.113200i	−0.831496	+	0.555531i	$0.812515\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	123.873	+	71.5182i	0.133780	+	0.0772380i
$96$		0			0
$97$			781.538i			0.818074i	0.912518	+	0.409037i	$0.134135\pi$
$97$			781.538i			0.818074i	−0.912518	+	0.409037i	$0.865865\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	62.2697	−	107.854i	0.0613472	−	0.106257i	−0.833721	−	0.552187i	$-0.813794\pi$
$101$	62.2697	−	107.854i	0.0613472	−	0.106257i	0.895068	+	0.445930i	$0.147127\pi$
$102$		0			0
$103$	369.676	−	213.433i	0.353644	−	0.204176i	−0.312645	−	0.949870i	$-0.601215\pi$
$103$	369.676	−	213.433i	0.353644	−	0.204176i	0.666289	+	0.745694i	$0.267882\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1743.21	+	1006.45i	−1.57498	+	0.909316i	−0.579437	+	0.815017i	$0.696727\pi$
$107$	−1743.21	+	1006.45i	−1.57498	+	0.909316i	−0.995544	+	0.0942984i	$0.969939\pi$
$108$		0			0
$109$	502.580	−	870.495i	0.441637	−	0.764938i	−0.556174	−	0.831066i	$-0.687731\pi$
$109$	502.580	−	870.495i	0.441637	−	0.764938i	0.997811	+	0.0661278i	$0.0210645\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1151.55i			0.958660i	0.877635	+	0.479330i	$0.159120\pi$
$113$			1151.55i			0.958660i	−0.877635	+	0.479330i	$0.840880\pi$
$114$		0			0
$115$	−54.9492	−	31.7250i	−0.0445569	−	0.0257249i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1486.93	+	2575.43i	1.11715	+	1.93496i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	289.546			0.207183
$126$		0			0
$127$	66.4170			0.0464060			0.0232030	−	0.999731i	$-0.492614\pi$
$127$	66.4170			0.0464060			0.0232030	+	0.999731i	$0.492614\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−928.121	−	1607.55i	−0.619010	−	1.07216i	−0.989667	−	0.143387i	$-0.954201\pi$
$131$	−928.121	−	1607.55i	−0.619010	−	1.07216i	0.370657	−	0.928770i	$-0.379133\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−688.705	−	397.624i	−0.429489	−	0.247966i	0.269640	−	0.962961i	$-0.413095\pi$
$137$	−688.705	−	397.624i	−0.429489	−	0.247966i	−0.699129	+	0.714995i	$0.746429\pi$
$138$		0			0
$139$			1076.39i			0.656820i	0.944535	+	0.328410i	$0.106513\pi$
$139$			1076.39i			0.656820i	−0.944535	+	0.328410i	$0.893487\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1888.66	−	3271.25i	1.10446	−	1.91298i
$144$		0			0
$145$	−106.016	+	61.2085i	−0.0607184	+	0.0350558i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1384.55	+	799.372i	−0.761255	+	0.439511i	−0.829746	−	0.558141i	$-0.811515\pi$
$149$	−1384.55	+	799.372i	−0.761255	+	0.439511i	0.0684912	+	0.997652i	$0.478181\pi$
$150$		0			0
$151$	1130.03	−	1957.27i	0.609011	−	1.05484i	−0.382393	−	0.924000i	$-0.624900\pi$
$151$	1130.03	−	1957.27i	0.609011	−	1.05484i	0.991404	−	0.130838i	$-0.0417667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			113.884i			0.0590152i
$156$		0			0
$157$	1162.64	+	671.251i	0.591011	+	0.341221i	0.765497	−	0.643439i	$-0.222493\pi$
$157$	1162.64	+	671.251i	0.591011	+	0.341221i	−0.174486	+	0.984660i	$0.555826\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	522.209	+	904.492i	0.250936	+	0.434634i	0.963784	−	0.266685i	$-0.0859284\pi$
$163$	522.209	+	904.492i	0.250936	+	0.434634i	−0.712848	+	0.701319i	$0.752595\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2538.50			−1.17626			−0.588129	−	0.808767i	$-0.700135\pi$
$167$	−2538.50			−1.17626			−0.588129	+	0.808767i	$0.700135\pi$
$168$		0			0
$169$	−1117.43			−0.508618
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1962.35	−	3398.89i	−0.862396	−	1.49371i	−0.869610	−	0.493740i	$-0.835630\pi$
$173$	−1962.35	−	3398.89i	−0.862396	−	1.49371i	0.00721379	−	0.999974i	$-0.497704\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1697.65	+	980.141i	0.708875	+	0.409269i	0.810644	−	0.585539i	$-0.199117\pi$
$179$	1697.65	+	980.141i	0.708875	+	0.409269i	−0.101770	+	0.994808i	$0.532450\pi$
$180$		0			0
$181$			2865.74i			1.17685i	0.808553	+	0.588423i	$0.200251\pi$
$181$			2865.74i			1.17685i	−0.808553	+	0.588423i	$0.799749\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−75.6980	+	131.113i	−0.0300834	+	0.0521060i
$186$		0			0
$187$	−3141.71	+	1813.87i	−1.22858	+	0.709322i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1747.94	+	1009.17i	−0.662181	+	0.382311i	−0.793108	−	0.609082i	$-0.791538\pi$
$191$	−1747.94	+	1009.17i	−0.662181	+	0.382311i	0.130926	+	0.991392i	$0.458205\pi$
$192$		0			0
$193$	−735.941	+	1274.69i	−0.274478	+	0.475409i	−0.970003	−	0.243092i	$-0.921838\pi$
$193$	−735.941	+	1274.69i	−0.274478	+	0.475409i	0.695526	+	0.718501i	$0.255172\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5503.83i		−	1.99051i	−0.0972758	−	0.995257i	$-0.531013\pi$
$197$		−	5503.83i		−	1.99051i	0.0972758	−	0.995257i	$-0.468987\pi$
$198$		0			0
$199$	−4060.62	−	2344.40i	−1.44648	−	0.835126i	−0.448211	−	0.893928i	$-0.647939\pi$
$199$	−4060.62	−	2344.40i	−1.44648	−	0.835126i	−0.998270	+	0.0588015i	$0.981272\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	40.3105	+	69.8198i	0.0137337	+	0.0237875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8059.08			−2.66726
$210$		0			0
$211$	−1542.29			−0.503202			−0.251601	−	0.967831i	$-0.580957\pi$
$211$	−1542.29			−0.503202			−0.251601	+	0.967831i	$0.580957\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−50.7165	−	87.8436i	−0.0160876	−	0.0278646i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2756.72	+	1591.59i	0.839081	+	0.484444i
$222$		0			0
$223$			544.576i			0.163531i	0.996652	+	0.0817657i	$0.0260559\pi$
$223$			544.576i			0.163531i	−0.996652	+	0.0817657i	$0.973944\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−837.169	+	1450.02i	−0.244779	+	0.423970i	−0.962070	−	0.272804i	$-0.912049\pi$
$227$	−837.169	+	1450.02i	−0.244779	+	0.423970i	0.717290	+	0.696774i	$0.245382\pi$
$228$		0			0
$229$	−4036.07	+	2330.22i	−1.16468	+	0.672426i	−0.952420	−	0.304788i	$-0.901414\pi$
$229$	−4036.07	+	2330.22i	−1.16468	+	0.672426i	−0.212255	+	0.977214i	$0.568081\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5615.51	+	3242.12i	−1.57890	+	0.911580i	−0.583889	+	0.811833i	$0.698470\pi$
$233$	−5615.51	+	3242.12i	−1.57890	+	0.911580i	−0.995013	+	0.0997461i	$0.968197\pi$
$234$		0			0
$235$	137.815	−	238.703i	0.0382556	−	0.0662607i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			890.490i			0.241008i	0.992713	+	0.120504i	$0.0384511\pi$
$239$			890.490i			0.241008i	−0.992713	+	0.120504i	$0.961549\pi$
$240$		0			0
$241$	−4304.74	−	2485.34i	−1.15059	−	0.664294i	−0.201560	−	0.979476i	$-0.564601\pi$
$241$	−4304.74	−	2485.34i	−1.15059	−	0.664294i	−0.949031	+	0.315182i	$0.897934\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3535.75	+	6124.09i	0.910826	+	1.57760i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2469.89			0.621107			0.310554	−	0.950556i	$-0.399486\pi$
$251$	2469.89			0.621107			0.310554	+	0.950556i	$0.399486\pi$
$252$		0			0
$253$	3574.95			0.888360
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1756.91	−	3043.06i	−0.426432	−	0.738602i	0.570121	−	0.821561i	$-0.306896\pi$
$257$	−1756.91	−	3043.06i	−0.426432	−	0.738602i	−0.996553	+	0.0829590i	$0.973563\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2447.74	+	1413.20i	0.573895	+	0.331338i	0.758703	−	0.651436i	$-0.225833\pi$
$263$	2447.74	+	1413.20i	0.573895	+	0.331338i	−0.184809	+	0.982775i	$0.559167\pi$
$264$		0			0
$265$			480.720i			0.111435i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4004.09	−	6935.28i	0.907560	−	1.57194i	0.0901161	−	0.995931i	$-0.471276\pi$
$269$	4004.09	−	6935.28i	0.907560	−	1.57194i	0.817444	−	0.576008i	$-0.195390\pi$
$270$		0			0
$271$	−2734.34	+	1578.67i	−0.612912	+	0.353865i	−0.774104	−	0.633058i	$-0.781799\pi$
$271$	−2734.34	+	1578.67i	−0.612912	+	0.353865i	0.161192	+	0.986923i	$0.448466\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7025.59	+	4056.22i	−1.54058	+	0.889453i
$276$		0			0
$277$	−1127.81	+	1953.42i	−0.244633	+	0.423717i	−0.962028	−	0.272949i	$-0.912001\pi$
$277$	−1127.81	+	1953.42i	−0.244633	+	0.423717i	0.717395	+	0.696666i	$0.245334\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7277.95i			1.54507i	0.634970	+	0.772537i	$0.281013\pi$
$281$			7277.95i			1.54507i	−0.634970	+	0.772537i	$0.718987\pi$
$282$		0			0
$283$	12.9560	+	7.48013i	0.00272139	+	0.00157119i	0.501360	−	0.865239i	$-0.332833\pi$
$283$	12.9560	+	7.48013i	0.00272139	+	0.00157119i	−0.498639	+	0.866810i	$0.666167\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	927.935	+	1607.23i	0.188873	+	0.327138i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8446.59			−1.68415			−0.842074	−	0.539362i	$-0.818665\pi$
$293$	−8446.59			−1.68415			−0.842074	+	0.539362i	$0.818665\pi$
$294$		0			0
$295$	−49.1539			−0.00970118
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1568.43	−	2716.60i	−0.303360	−	0.525435i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−851.569	−	491.654i	−0.159871	−	0.0923017i
$306$		0			0
$307$			7870.82i			1.46323i	0.681719	+	0.731615i	$0.261233\pi$
$307$			7870.82i			1.46323i	−0.681719	+	0.731615i	$0.738767\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3030.67	−	5249.27i	0.552584	−	0.957103i	−0.445504	−	0.895280i	$-0.646975\pi$
$311$	3030.67	−	5249.27i	0.552584	−	0.957103i	0.998087	−	0.0618227i	$-0.0196913\pi$
$312$		0			0
$313$	4885.90	−	2820.87i	0.882323	−	0.509410i	0.0108997	−	0.999941i	$-0.496530\pi$
$313$	4885.90	−	2820.87i	0.882323	−	0.509410i	0.871424	+	0.490531i	$0.163197\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3888.70	+	2245.14i	−0.688994	+	0.397791i	−0.803235	−	0.595662i	$-0.796890\pi$
$317$	−3888.70	+	2245.14i	−0.688994	+	0.397791i	0.114241	+	0.993453i	$0.463556\pi$
$318$		0			0
$319$	3448.66	−	5973.25i	0.605291	−	1.04839i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6791.46i		−	1.16993i
$324$		0			0
$325$	6164.65	+	3559.16i	1.05216	+	0.607467i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5457.22	+	9452.19i	0.906212	+	1.56961i	0.819282	+	0.573391i	$0.194372\pi$
$331$	5457.22	+	9452.19i	0.906212	+	1.56961i	0.0869300	+	0.996214i	$0.472294\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−749.630			−0.122259
$336$		0			0
$337$	−6590.94			−1.06537			−0.532687	−	0.846312i	$-0.678818\pi$
$337$	−6590.94			−1.06537			−0.532687	+	0.846312i	$0.678818\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3208.26	−	5556.87i	−0.509493	−	0.882468i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4612.55	−	2663.05i	−0.713586	−	0.411989i	0.0988012	−	0.995107i	$-0.468499\pi$
$347$	−4612.55	−	2663.05i	−0.713586	−	0.411989i	−0.812388	+	0.583118i	$0.801833\pi$
$348$		0			0
$349$			8102.39i			1.24272i	0.783523	+	0.621362i	$0.213420\pi$
$349$			8102.39i			1.24272i	−0.783523	+	0.621362i	$0.786580\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	277.427	−	480.519i	0.0418300	−	0.0724516i	−0.844352	−	0.535788i	$-0.820015\pi$
$353$	277.427	−	480.519i	0.0418300	−	0.0724516i	0.886182	+	0.463337i	$0.153348\pi$
$354$		0			0
$355$	755.657	−	436.279i	0.112975	−	0.0652261i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	349.424	−	201.740i	0.0513702	−	0.0296586i	−0.474095	−	0.880474i	$-0.657225\pi$
$359$	349.424	−	201.740i	0.0513702	−	0.0296586i	0.525465	+	0.850815i	$0.323891\pi$
$360$		0			0
$361$	4114.17	−	7125.95i	0.599821	−	1.03892i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	929.371i		−	0.133275i
$366$		0			0
$367$	−5617.83	−	3243.45i	−0.799041	−	0.461327i	0.0440946	−	0.999027i	$-0.485960\pi$
$367$	−5617.83	−	3243.45i	−0.799041	−	0.461327i	−0.843136	+	0.537701i	$0.819293\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−3622.90	−	6275.05i	−0.502914	−	0.871072i	−0.999994	−	0.00336765i	$-0.998928\pi$
$373$	−3622.90	−	6275.05i	−0.502914	−	0.871072i	0.497081	−	0.867704i	$-0.334405\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6052.09			−0.826787
$378$		0			0
$379$	−8439.30			−1.14379			−0.571897	−	0.820326i	$-0.693792\pi$
$379$	−8439.30			−1.14379			−0.571897	+	0.820326i	$0.693792\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3154.57	−	5463.87i	−0.420864	−	0.728957i	0.575160	−	0.818041i	$-0.304940\pi$
$383$	−3154.57	−	5463.87i	−0.420864	−	0.728957i	−0.996024	+	0.0890832i	$0.971606\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2981.67	+	1721.47i	0.388630	+	0.224375i	0.681566	−	0.731756i	$-0.261299\pi$
$389$	2981.67	+	1721.47i	0.388630	+	0.224375i	−0.292937	+	0.956132i	$0.594632\pi$
$390$		0			0
$391$			3012.64i			0.389657i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	209.605	−	363.047i	0.0266997	−	0.0462453i
$396$		0			0
$397$	−8919.34	+	5149.58i	−1.12758	+	0.651008i	−0.943325	−	0.331871i	$-0.892320\pi$
$397$	−8919.34	+	5149.58i	−1.12758	+	0.651008i	−0.184254	+	0.982879i	$0.558987\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−120.856	+	69.7764i	−0.0150506	+	0.00868944i	−0.507506	−	0.861648i	$-0.669433\pi$
$401$	−120.856	+	69.7764i	−0.0150506	+	0.00868944i	0.492456	+	0.870337i	$0.336099\pi$
$402$		0			0
$403$	−2815.11	+	4875.91i	−0.347967	+	0.602696i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8530.08i		−	1.03887i
$408$		0			0
$409$	−4385.88	−	2532.19i	−0.530239	−	0.306134i	0.210874	−	0.977513i	$-0.432369\pi$
$409$	−4385.88	−	2532.19i	−0.530239	−	0.306134i	−0.741114	+	0.671379i	$0.765702\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−759.780	−	1315.98i	−0.0898702	−	0.155660i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1888.80			−0.220224			−0.110112	−	0.993919i	$-0.535121\pi$
$419$	−1888.80			−0.220224			−0.110112	+	0.993919i	$0.535121\pi$
$420$		0			0
$421$	12973.6			1.50188			0.750941	−	0.660369i	$-0.229600\pi$
$421$	12973.6			1.50188			0.750941	+	0.660369i	$0.229600\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3418.22	−	5920.53i	−0.390136	−	0.675736i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7994.07	+	4615.38i	0.893413	+	0.515812i	0.875057	−	0.484019i	$-0.160824\pi$
$431$	7994.07	+	4615.38i	0.893413	+	0.515812i	0.0183555	+	0.999832i	$0.494157\pi$
$432$		0			0
$433$		−	9931.42i		−	1.10225i	−0.834423	−	0.551124i	$-0.814199\pi$
$433$		−	9931.42i		−	1.10225i	0.834423	−	0.551124i	$-0.185801\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3346.32	+	5795.99i	−0.366307	+	0.634462i
$438$		0			0
$439$	−5684.09	+	3281.71i	−0.617966	+	0.356783i	−0.776077	−	0.630639i	$-0.782793\pi$
$439$	−5684.09	+	3281.71i	−0.617966	+	0.356783i	0.158111	+	0.987421i	$0.449460\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7539.39	+	4352.87i	−0.808595	+	0.466842i	−0.846468	−	0.532440i	$-0.821275\pi$
$443$	−7539.39	+	4352.87i	−0.808595	+	0.466842i	0.0378730	+	0.999283i	$0.487942\pi$
$444$		0			0
$445$	−63.9014	+	110.681i	−0.00680723	+	0.0117905i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			11408.9i			1.19915i	0.800317	+	0.599577i	$0.204665\pi$
$449$			11408.9i			1.19915i	−0.800317	+	0.599577i	$0.795335\pi$
$450$		0			0
$451$	−3933.85	−	2271.21i	−0.410727	−	0.237133i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4340.84	+	7518.55i	0.444323	+	0.769591i	0.998005	−	0.0631379i	$-0.0201108\pi$
$457$	4340.84	+	7518.55i	0.444323	+	0.769591i	−0.553681	+	0.832729i	$0.686777\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10427.6			1.05349			0.526746	−	0.850023i	$-0.323412\pi$
$461$	10427.6			1.05349			0.526746	+	0.850023i	$0.323412\pi$
$462$		0			0
$463$	4918.91			0.493738			0.246869	−	0.969049i	$-0.420598\pi$
$463$	4918.91			0.493738			0.246869	+	0.969049i	$0.420598\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5656.62	+	9797.55i	0.560508	+	0.970828i	0.997452	+	0.0713395i	$0.0227274\pi$
$467$	5656.62	+	9797.55i	0.560508	+	0.970828i	−0.436944	+	0.899489i	$0.643939\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4949.36	+	2857.51i	0.481124	+	0.277777i
$474$		0			0
$475$		−	15187.3i		−	1.46703i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5574.32	−	9655.01i	0.531727	−	0.920978i	−0.467587	−	0.883947i	$-0.654877\pi$
$479$	5574.32	−	9655.01i	0.531727	−	0.920978i	0.999314	−	0.0370311i	$-0.0117901\pi$
$480$		0			0
$481$	−6482.00	+	3742.39i	−0.614457	+	0.354757i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−788.172	+	455.051i	−0.0737918	+	0.0426037i
$486$		0			0
$487$	−6733.48	+	11662.7i	−0.626536	+	1.08519i	0.361705	+	0.932293i	$0.382195\pi$
$487$	−6733.48	+	11662.7i	−0.626536	+	1.08519i	−0.988242	+	0.152900i	$0.951139\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	5130.89i		−	0.471597i	−0.971802	−	0.235798i	$-0.924229\pi$
$491$		−	5130.89i		−	0.471597i	0.971802	−	0.235798i	$-0.0757705\pi$
$492$		0			0
$493$	5033.72	+	2906.22i	0.459852	+	0.265496i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3968.71	−	6874.01i	−0.356040	−	0.616679i	0.631256	−	0.775575i	$-0.282540\pi$
$499$	−3968.71	−	6874.01i	−0.356040	−	0.616679i	−0.987295	+	0.158896i	$0.949207\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14153.2			−1.25459			−0.627294	−	0.778782i	$-0.715838\pi$
$503$	−14153.2			−1.25459			−0.627294	+	0.778782i	$0.715838\pi$
$504$		0			0
$505$	145.027			0.0127794
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8850.32	+	15329.2i	0.770694	+	1.33488i	0.937183	+	0.348838i	$0.113424\pi$
$509$	8850.32	+	15329.2i	0.770694	+	1.33488i	−0.166489	+	0.986043i	$0.553243\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	430.489	+	248.543i	0.0368342	+	0.0212662i
$516$		0			0
$517$			15529.8i			1.32108i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11420.4	+	19780.7i	−0.960339	+	1.66336i	−0.238690	+	0.971096i	$0.576718\pi$
$521$	−11420.4	+	19780.7i	−0.960339	+	1.66336i	−0.721648	+	0.692260i	$0.756615\pi$
$522$		0			0
$523$	17908.1	−	10339.2i	1.49726	−	0.864443i	0.497264	−	0.867599i	$-0.334338\pi$
$523$	17908.1	−	10339.2i	1.49726	−	0.864443i	0.999995	+	0.00315640i	$0.00100472\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4682.83	−	2703.63i	0.387073	−	0.223476i
$528$		0			0
$529$	−4599.10	+	7965.87i	−0.377998	+	0.654711i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3985.77i			0.323908i
$534$		0			0
$535$	−2029.98	−	1172.01i	−0.164044	−	0.0947109i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	8996.90	+	15583.1i	0.714985	+	1.23839i	0.962965	+	0.269625i	$0.0868998\pi$
$541$	8996.90	+	15583.1i	0.714985	+	1.23839i	−0.247980	+	0.968765i	$0.579767\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1170.51			0.0919986
$546$		0			0
$547$	−11216.0			−0.876712			−0.438356	−	0.898801i	$-0.644439\pi$
$547$	−11216.0			−0.876712			−0.438356	+	0.898801i	$0.644439\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6456.21	+	11182.5i	0.499172	+	0.864591i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18757.9	−	10829.9i	−1.42692	−	0.823834i	−0.430046	−	0.902807i	$-0.641503\pi$
$557$	−18757.9	−	10829.9i	−1.42692	−	0.823834i	−0.996877	+	0.0789726i	$0.974836\pi$
$558$		0			0
$559$		−	5014.68i		−	0.379425i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9238.84	−	16002.1i	0.691600	−	1.19789i	−0.279714	−	0.960083i	$-0.590240\pi$
$563$	9238.84	−	16002.1i	0.691600	−	1.19789i	0.971314	−	0.237802i	$-0.0764270\pi$
$564$		0			0
$565$	−1161.32	+	670.491i	−0.0864730	+	0.0499252i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14511.3	−	8378.12i	1.06915	−	0.617274i	0.141201	−	0.989981i	$-0.454904\pi$
$569$	14511.3	−	8378.12i	1.06915	−	0.617274i	0.927949	+	0.372707i	$0.121570\pi$
$570$		0			0
$571$	10045.3	−	17399.0i	0.736224	−	1.27518i	−0.217961	−	0.975958i	$-0.569941\pi$
$571$	10045.3	−	17399.0i	0.736224	−	1.27518i	0.954184	−	0.299219i	$-0.0967262\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6736.95i			0.488609i
$576$		0			0
$577$	7839.05	+	4525.88i	0.565587	+	0.326542i	0.755385	−	0.655281i	$-0.227450\pi$
$577$	7839.05	+	4525.88i	0.565587	+	0.326542i	−0.189798	+	0.981823i	$0.560783\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−13542.6	−	23456.4i	−0.962051	−	1.66632i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15886.2			1.11703			0.558513	−	0.829496i	$-0.311372\pi$
$587$	15886.2			1.11703			0.558513	+	0.829496i	$0.311372\pi$
$588$		0			0
$589$	12012.3			0.840338
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12975.8	−	22474.7i	−0.898570	−	1.55637i	−0.829324	−	0.558769i	$-0.811274\pi$
$593$	−12975.8	−	22474.7i	−0.898570	−	1.55637i	−0.0692459	−	0.997600i	$-0.522059\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13787.3	−	7960.08i	−0.940455	−	0.542972i	−0.0503519	−	0.998732i	$-0.516034\pi$
$599$	−13787.3	−	7960.08i	−0.940455	−	0.542972i	−0.890103	+	0.455760i	$0.849368\pi$
$600$		0			0
$601$			5872.84i			0.398599i	0.979939	+	0.199300i	$0.0638667\pi$
$601$			5872.84i			0.398599i	−0.979939	+	0.199300i	$0.936133\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1731.53	+	2999.10i	−0.116358	+	0.201538i
$606$		0			0
$607$	23980.6	−	13845.2i	1.60353	−	0.925799i	0.612758	−	0.790271i	$-0.290060\pi$
$607$	23980.6	−	13845.2i	1.60353	−	0.925799i	0.990774	−	0.135528i	$-0.0432731\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11801.1	−	6813.36i	0.781376	−	0.451128i
$612$		0			0
$613$	4006.36	−	6939.22i	0.263973	−	0.457214i	−0.703321	−	0.710872i	$-0.748300\pi$
$613$	4006.36	−	6939.22i	0.263973	−	0.457214i	0.967294	+	0.253658i	$0.0816338\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1546.21i			0.100888i	0.998727	+	0.0504442i	$0.0160637\pi$
$617$			1546.21i			0.100888i	−0.998727	+	0.0504442i	$0.983936\pi$
$618$		0			0
$619$	9942.26	+	5740.16i	0.645578	+	0.372725i	0.786760	−	0.617259i	$-0.211757\pi$
$619$	9942.26	+	5740.16i	0.645578	+	0.372725i	−0.141182	+	0.989984i	$0.545090\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7559.16	−	13092.8i	−0.483786	−	0.837942i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7188.38			0.455675
$630$		0			0
$631$	−12482.7			−0.787524			−0.393762	−	0.919212i	$-0.628827\pi$
$631$	−12482.7			−0.787524			−0.393762	+	0.919212i	$0.628827\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	38.6714	+	66.9808i	0.00241674	+	0.00418591i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3612.00	−	2085.39i	−0.222567	−	0.128499i	0.384571	−	0.923095i	$-0.374349\pi$
$641$	−3612.00	−	2085.39i	−0.222567	−	0.128499i	−0.607138	+	0.794596i	$0.707683\pi$
$642$		0			0
$643$		−	18330.0i		−	1.12421i	−0.827066	−	0.562105i	$-0.809992\pi$
$643$		−	18330.0i		−	1.12421i	0.827066	−	0.562105i	$-0.190008\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5049.66	+	8746.26i	−0.306835	+	0.531455i	−0.977668	−	0.210154i	$-0.932603\pi$
$647$	−5049.66	+	8746.26i	−0.306835	+	0.531455i	0.670833	+	0.741609i	$0.265937\pi$
$648$		0			0
$649$	2398.43	−	1384.73i	0.145064	−	0.0837528i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12968.1	−	7487.16i	0.777155	−	0.448691i	−0.0582660	−	0.998301i	$-0.518557\pi$
$653$	12968.1	−	7487.16i	0.777155	−	0.448691i	0.835421	+	0.549610i	$0.185224\pi$
$654$		0			0
$655$	1080.80	−	1872.00i	0.0644738	−	0.111672i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	25422.5i		−	1.50276i	−0.659870	−	0.751380i	$-0.729389\pi$
$659$		−	25422.5i		−	1.50276i	0.659870	−	0.751380i	$-0.270611\pi$
$660$		0			0
$661$	5789.29	+	3342.45i	0.340661	+	0.196681i	0.660564	−	0.750769i	$-0.270317\pi$
$661$	5789.29	+	3342.45i	0.340661	+	0.196681i	−0.319903	+	0.947450i	$0.603650\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2863.93	−	4960.47i	−0.166254	−	0.287961i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	55402.3			3.18746
$672$		0			0
$673$	4182.44			0.239556			0.119778	−	0.992801i	$-0.461782\pi$
$673$	4182.44			0.239556			0.119778	+	0.992801i	$0.461782\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6115.84	+	10593.0i	0.347195	+	0.601359i	0.985750	−	0.168217i	$-0.0538009\pi$
$677$	6115.84	+	10593.0i	0.347195	+	0.601359i	−0.638555	+	0.769576i	$0.720468\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14005.9	+	8086.33i	0.784659	+	0.453023i	0.838079	−	0.545549i	$-0.183679\pi$
$683$	14005.9	+	8086.33i	0.784659	+	0.453023i	−0.0534198	+	0.998572i	$0.517012\pi$
$684$		0			0
$685$		−	926.068i		−	0.0516544i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11883.0	+	20582.0i	−0.657049	+	1.13804i
$690$		0			0
$691$	−18032.3	+	10410.9i	−0.992734	+	0.573155i	−0.906090	−	0.423084i	$-0.860947\pi$
$691$	−18032.3	+	10410.9i	−0.992734	+	0.573155i	−0.0866437	+	0.996239i	$0.527614\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1085.52	+	626.727i	−0.0592464	+	0.0342059i
$696$		0			0
$697$	1913.97	−	3315.09i	0.104013	−	0.180155i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			35212.3i			1.89722i	0.316451	+	0.948609i	$0.397509\pi$
$701$			35212.3i			1.89722i	−0.316451	+	0.948609i	$0.602491\pi$
$702$		0			0
$703$	13829.6	+	7984.55i	0.741956	+	0.428368i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−10423.6	−	18054.2i	−0.552138	−	0.956331i	−0.998120	−	0.0612888i	$-0.980479\pi$
$709$	−10423.6	−	18054.2i	−0.552138	−	0.956331i	0.445982	−	0.895042i	$-0.352854\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5328.58			−0.279883
$714$		0			0
$715$	4398.70			0.230073
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5446.98	+	9434.44i	0.282528	+	0.489354i	0.972007	−	0.234953i	$-0.0754935\pi$
$719$	5446.98	+	9434.44i	0.282528	+	0.489354i	−0.689478	+	0.724306i	$0.742160\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11256.5	+	6498.96i	0.576630	+	0.332918i
$726$		0			0
$727$		−	31196.0i		−	1.59147i	−0.605646	−	0.795734i	$-0.707085\pi$
$727$		−	31196.0i		−	1.59147i	0.605646	−	0.795734i	$-0.292915\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2408.05	+	4170.87i	−0.121840	+	0.211033i
$732$		0			0
$733$	15525.3	−	8963.52i	0.782318	−	0.451672i	−0.0549330	−	0.998490i	$-0.517495\pi$
$733$	15525.3	−	8963.52i	0.782318	−	0.451672i	0.837251	+	0.546818i	$0.184161\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36577.7	−	21118.1i	1.82816	−	1.05549i
$738$		0			0
$739$	−7530.33	+	13042.9i	−0.374841	+	0.649244i	−0.990303	−	0.138922i	$-0.955636\pi$
$739$	−7530.33	+	13042.9i	−0.374841	+	0.649244i	0.615462	+	0.788167i	$0.288969\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	8750.44i		−	0.432062i	−0.976386	−	0.216031i	$-0.930689\pi$
$743$		−	8750.44i		−	0.432062i	0.976386	−	0.216031i	$-0.0693113\pi$
$744$		0			0
$745$	−1612.31	−	930.870i	−0.0792895	−	0.0457778i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	11415.7	+	19772.5i	0.554678	+	0.960730i	0.997929	+	0.0643326i	$0.0204918\pi$
$751$	11415.7	+	19772.5i	0.554678	+	0.960730i	−0.443251	+	0.896398i	$0.646175\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2631.85			0.126865
$756$		0			0
$757$	12087.5			0.580353			0.290176	−	0.956973i	$-0.406286\pi$
$757$	12087.5			0.580353			0.290176	+	0.956973i	$0.406286\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18359.5	+	31799.5i	0.874547	+	1.51476i	0.857244	+	0.514910i	$0.172175\pi$
$761$	18359.5	+	31799.5i	0.874547	+	1.51476i	0.0173026	+	0.999850i	$0.494492\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2104.52	−	1215.04i	−0.0990739	−	0.0572004i
$768$		0			0
$769$		−	2652.68i		−	0.124393i	−0.998064	−	0.0621964i	$-0.980189\pi$
$769$		−	2652.68i		−	0.124393i	0.998064	−	0.0621964i	$-0.0198105\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12335.0	+	21364.8i	−0.573944	+	0.994100i	0.422211	+	0.906497i	$0.361254\pi$
$773$	−12335.0	+	21364.8i	−0.573944	+	0.994100i	−0.996155	+	0.0876029i	$0.972079\pi$
$774$		0			0
$775$	10471.9	−	6045.94i	0.485369	−	0.280228i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7364.53	−	4251.91i	0.338718	−	0.195559i
$780$		0			0
$781$	−24581.2	+	42575.8i	−1.12623	+	1.95068i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1563.35i			0.0710805i
$786$		0			0
$787$	−14812.4	−	8551.94i	−0.670909	−	0.387349i	0.125512	−	0.992092i	$-0.459943\pi$
$787$	−14812.4	−	8551.94i	−0.670909	−	0.387349i	−0.796421	+	0.604743i	$0.793276\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−24306.6	−	42100.2i	−1.08846	−	1.88527i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26143.0			−1.16190			−0.580948	−	0.813940i	$-0.697318\pi$
$797$	−26143.0			−1.16190			−0.580948	+	0.813940i	$0.697318\pi$
$798$		0			0
$799$	−13087.1			−0.579460
$800$		0			0
$801$		0			0
$802$		0			0
$803$	26181.7	+	45348.0i	1.15060	+	1.99290i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21245.1	−	12265.8i	−0.923283	−	0.533058i	−0.0386023	−	0.999255i	$-0.512291\pi$
$809$	−21245.1	−	12265.8i	−0.923283	−	0.533058i	−0.884681	+	0.466197i	$0.845624\pi$
$810$		0			0
$811$		−	8145.19i		−	0.352671i	−0.984330	−	0.176336i	$-0.943576\pi$
$811$		−	8145.19i		−	0.352671i	0.984330	−	0.176336i	$-0.0564244\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−608.113	+	1053.28i	−0.0261365	+	0.0452698i
$816$		0			0
$817$	−9265.65	+	5349.53i	−0.396774	+	0.229077i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20871.3	−	12050.0i	0.887225	−	0.512240i	0.0141913	−	0.999899i	$-0.495483\pi$
$821$	20871.3	−	12050.0i	0.887225	−	0.512240i	0.873034	+	0.487660i	$0.162149\pi$
$822$		0			0
$823$	−16435.8	+	28467.6i	−0.696131	+	1.20573i	0.273667	+	0.961824i	$0.411763\pi$
$823$	−16435.8	+	28467.6i	−0.696131	+	1.20573i	−0.969798	+	0.243909i	$0.921570\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			5891.21i			0.247712i	0.992300	+	0.123856i	$0.0395260\pi$
$827$			5891.21i			0.247712i	−0.992300	+	0.123856i	$0.960474\pi$
$828$		0			0
$829$	9443.28	+	5452.08i	0.395632	+	0.228418i	0.684597	−	0.728921i	$-0.259978\pi$
$829$	9443.28	+	5452.08i	0.395632	+	0.228418i	−0.288966	+	0.957339i	$0.593311\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1478.05	−	2560.05i	−0.0612574	−	0.106101i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	38021.0			1.56452			0.782259	−	0.622953i	$-0.214067\pi$
$839$	38021.0			1.56452			0.782259	+	0.622953i	$0.214067\pi$
$840$		0			0
$841$	13338.0			0.546885
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−650.627	−	1126.92i	−0.0264879	−	0.0458783i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6134.73	−	3541.89i	−0.247116	−	0.142673i
$852$		0			0
$853$		−	30803.2i		−	1.23644i	−0.786006	−	0.618218i	$-0.787855\pi$
$853$		−	30803.2i		−	1.23644i	0.786006	−	0.618218i	$-0.212145\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2623.05	−	4543.25i	0.104553	−	0.181090i	−0.809003	−	0.587805i	$-0.799992\pi$
$857$	2623.05	−	4543.25i	0.104553	−	0.181090i	0.913555	+	0.406714i	$0.133326\pi$
$858$		0			0
$859$	26498.6	−	15299.0i	1.05253	−	0.607677i	0.129171	−	0.991622i	$-0.458768\pi$
$859$	26498.6	−	15299.0i	1.05253	−	0.607677i	0.923356	+	0.383946i	$0.125435\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2009.18	+	1160.00i	−0.0792507	+	0.0457554i	−0.539102	−	0.842241i	$-0.681236\pi$
$863$	−2009.18	+	1160.00i	−0.0792507	+	0.0457554i	0.459851	+	0.887996i	$0.347903\pi$
$864$		0			0
$865$	2285.16	−	3958.01i	0.0898239	−	0.155580i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			23619.5i			0.922022i
$870$		0			0
$871$	−32095.3	−	18530.3i	−1.24858	−	0.720865i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14115.0	−	24447.9i	−0.543477	−	0.941329i	−0.998701	−	0.0509526i	$-0.983774\pi$
$877$	−14115.0	−	24447.9i	−0.543477	−	0.941329i	0.455224	−	0.890377i	$-0.349559\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16404.3			0.627325			0.313663	−	0.949535i	$-0.398444\pi$
$881$	16404.3			0.627325			0.313663	+	0.949535i	$0.398444\pi$
$882$		0			0
$883$	12816.6			0.488463			0.244232	−	0.969717i	$-0.421464\pi$
$883$	12816.6			0.488463			0.244232	+	0.969717i	$0.421464\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2424.10	+	4198.67i	0.0917625	+	0.158937i	0.908253	−	0.418422i	$-0.137417\pi$
$887$	2424.10	+	4198.67i	0.0917625	+	0.158937i	−0.816490	+	0.577359i	$0.804083\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25178.1	−	14536.6i	−0.943510	−	0.544736i
$894$		0			0
$895$			2282.75i			0.0852558i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5140.34	+	8903.33i	−0.190701	+	0.330303i
$900$		0			0
$901$	19766.9	−	11412.4i	0.730890	−	0.421980i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2890.07	+	1668.58i	−0.106154	+	0.0612879i
$906$		0			0
$907$	17965.6	−	31117.4i	0.657706	−	1.13918i	−0.323503	−	0.946227i	$-0.604860\pi$
$907$	17965.6	−	31117.4i	0.657706	−	1.13918i	0.981208	−	0.192952i	$-0.0618062\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	26942.0i		−	0.979834i	−0.871769	−	0.489917i	$-0.837027\pi$
$911$		−	26942.0i		−	0.979834i	0.871769	−	0.489917i	$-0.162973\pi$
$912$		0			0
$913$	74145.9	+	42808.2i	2.68770	+	1.55175i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	9048.97	+	15673.3i	0.324807	+	0.562583i	0.981473	−	0.191599i	$-0.0613672\pi$
$919$	9048.97	+	15673.3i	0.324807	+	0.562583i	−0.656666	+	0.754181i	$0.728034\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	43137.8			1.53835
$924$		0			0
$925$	16074.8			0.571392
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13518.0	−	23413.9i	−0.477407	−	0.826894i	0.522257	−	0.852788i	$-0.325090\pi$
$929$	−13518.0	−	23413.9i	−0.477407	−	0.826894i	−0.999665	+	0.0258940i	$0.991757\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3658.53	−	2112.26i	−0.127965	−	0.0738804i
$936$		0			0
$937$			22915.3i			0.798942i	0.916746	+	0.399471i	$0.130806\pi$
$937$			22915.3i			0.798942i	−0.916746	+	0.399471i	$0.869194\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14988.4	+	25960.7i	−0.519244	+	0.899356i	0.480506	+	0.876991i	$0.340453\pi$
$941$	−14988.4	+	25960.7i	−0.519244	+	0.899356i	−0.999750	+	0.0223651i	$0.992880\pi$
$942$		0			0
$943$	−3266.85	+	1886.12i	−0.112814	+	0.0651330i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	42391.6	−	24474.8i	1.45464	−	0.839835i	0.455898	−	0.890032i	$-0.349318\pi$
$947$	42391.6	−	24474.8i	1.45464	−	0.839835i	0.998739	+	0.0501965i	$0.0159848\pi$
$948$		0			0
$949$	22973.3	−	39790.9i	0.785822	−	1.36108i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			10983.8i			0.373349i	0.982422	+	0.186674i	$0.0597709\pi$
$953$			10983.8i			0.373349i	−0.982422	+	0.186674i	$0.940229\pi$
$954$		0			0
$955$	−2035.48	−	1175.19i	−0.0689703	−	0.0398200i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−10113.5	−	17517.1i	−0.339481	−	0.587998i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1714.01			−0.0571771
$966$		0			0
$967$	−45637.1			−1.51767			−0.758836	−	0.651281i	$-0.774232\pi$
$967$	−45637.1			−1.51767			−0.758836	+	0.651281i	$0.774232\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14777.0	+	25594.5i	0.488380	+	0.845899i	0.999911	−	0.0133661i	$-0.00425469\pi$
$971$	14777.0	+	25594.5i	0.488380	+	0.845899i	−0.511531	+	0.859265i	$0.670921\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	17536.4	+	10124.6i	0.574246	+	0.331541i	0.758844	−	0.651273i	$-0.225765\pi$
$977$	17536.4	+	10124.6i	0.574246	+	0.331541i	−0.184597	+	0.982814i	$0.559098\pi$
$978$		0			0
$979$		−	7200.78i		−	0.235074i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−110.819	+	191.945i	−0.00359571	+	0.00622796i	−0.867818	−	0.496883i	$-0.834478\pi$
$983$	−110.819	+	191.945i	−0.00359571	+	0.00622796i	0.864222	+	0.503111i	$0.167811\pi$
$984$		0			0
$985$	5550.55	−	3204.61i	0.179548	−	0.103662i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4110.18	−	2373.01i	0.132150	−	0.0762966i
$990$		0			0
$991$	23116.6	−	40039.1i	0.740991	−	1.28343i	−0.211054	−	0.977474i	$-0.567690\pi$
$991$	23116.6	−	40039.1i	0.740991	−	1.28343i	0.952045	−	0.305959i	$-0.0989770\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5460.12i		−	0.173967i
$996$		0			0
$997$	756.084	+	436.525i	0.0240175	+	0.0138665i	0.511961	−	0.859009i	$-0.328919\pi$
$997$	756.084	+	436.525i	0.0240175	+	0.0138665i	−0.487943	+	0.872875i	$0.662253\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.c.521.13		48
3.2	odd	2	inner	1764.4.t.c.521.12		48
7.2	even	3	inner	1764.4.t.c.1097.14		48
7.3	odd	6		1764.4.f.b.881.13	yes	24
7.4	even	3		1764.4.f.b.881.11	✓	24
7.5	odd	6	inner	1764.4.t.c.1097.12		48
7.6	odd	2	inner	1764.4.t.c.521.11		48
21.2	odd	6	inner	1764.4.t.c.1097.11		48
21.5	even	6	inner	1764.4.t.c.1097.13		48
21.11	odd	6		1764.4.f.b.881.14	yes	24
21.17	even	6		1764.4.f.b.881.12	yes	24
21.20	even	2	inner	1764.4.t.c.521.14		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.f.b.881.11	✓	24	7.4	even	3
1764.4.f.b.881.12	yes	24	21.17	even	6
1764.4.f.b.881.13	yes	24	7.3	odd	6
1764.4.f.b.881.14	yes	24	21.11	odd	6
1764.4.t.c.521.11		48	7.6	odd	2	inner
1764.4.t.c.521.12		48	3.2	odd	2	inner
1764.4.t.c.521.13		48	1.1	even	1	trivial
1764.4.t.c.521.14		48	21.20	even	2	inner
1764.4.t.c.1097.11		48	21.2	odd	6	inner
1764.4.t.c.1097.12		48	7.5	odd	6	inner
1764.4.t.c.1097.13		48	21.5	even	6	inner
1764.4.t.c.1097.14		48	7.2	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.582251 + 1.00849i) q^{5} +O(q^{10})\)	\(q+(0.582251 + 1.00849i) q^{5} +(-56.8211 - 32.8057i) q^{11} +57.5711i q^{13} +(27.6457 - 47.8837i) q^{17} +(106.374 - 61.4153i) q^{19} +(-47.1869 + 27.2434i) q^{23} +(61.8220 - 107.079i) q^{25} +105.124i q^{29} +(84.6938 + 48.8980i) q^{31} +(65.0046 + 112.591i) q^{37} +69.2322 q^{41} -87.1042 q^{43} +(-118.347 - 204.983i) q^{47} +(357.505 + 206.406i) q^{53} -76.4046i q^{55} +(-21.1051 + 36.5551i) q^{59} +(-731.273 + 422.201i) q^{61} +(-58.0598 + 33.5208i) q^{65} +(-321.867 + 557.490i) q^{67} -749.296i q^{71} +(-691.161 - 399.042i) q^{73} +(-179.996 - 311.762i) q^{79} -1304.90 q^{83} +64.3869 q^{85} +(54.8745 + 95.0454i) q^{89} +(123.873 + 71.5182i) q^{95} +781.538i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 888 q^{25} - 864 q^{37} - 2496 q^{43} - 1056 q^{67} - 16128 q^{85}+O(q^{100}) \) 48 * q - 888 * q^25 - 864 * q^37 - 2496 * q^43 - 1056 * q^67 - 16128 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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