LMFDB - Embedded newform 1764.3.z.k.325.2

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 7^{4} $
Twist minimal:	no (minimal twist has level 196)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			325.2
Root			$-1.60021 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1764.325
Dual form			1764.3.z.k.901.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.91434 + 2.83730i) q^{5} +O(q^{10})$	$q+(-4.91434 + 2.83730i) q^{5} +(-7.94975 + 13.7694i) q^{11} +20.1940i q^{13} +(-0.823656 - 0.475538i) q^{17} +(27.5918 - 15.9301i) q^{19} +(13.0000 + 22.5167i) q^{23} +(3.60051 - 6.23626i) q^{25} -27.7990 q^{29} +(13.1233 + 7.57675i) q^{31} +(16.0000 + 27.7128i) q^{37} -17.3408i q^{41} -59.2965 q^{43} +(66.1105 - 38.1689i) q^{47} +(12.8995 - 22.3426i) q^{53} -90.2232i q^{55} +(-59.2743 - 34.2220i) q^{59} +(-46.9746 + 27.1208i) q^{61} +(-57.2965 - 99.2404i) q^{65} +(-43.6985 + 75.6880i) q^{67} -16.4020 q^{71} +(-60.9216 - 35.1731i) q^{73} +(-20.1005 - 34.8151i) q^{79} +71.5505i q^{83} +5.39697 q^{85} +(66.9617 - 38.6604i) q^{89} +(-90.3970 + 156.572i) q^{95} +128.328i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{11} + 104 q^{23} + 108 q^{25} - 64 q^{29} + 128 q^{37} + 80 q^{43} + 24 q^{53} + 96 q^{65} - 112 q^{67} - 448 q^{71} - 240 q^{79} - 432 q^{85} - 248 q^{95}+O(q^{100}) $ 8 * q - 24 * q^11 + 104 * q^23 + 108 * q^25 - 64 * q^29 + 128 * q^37 + 80 * q^43 + 24 * q^53 + 96 * q^65 - 112 * q^67 - 448 * q^71 - 240 * q^79 - 432 * q^85 - 248 * q^95

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
$5$	−4.91434	+	2.83730i	−0.982868	+	0.567459i	−0.903135	−	0.429357i	$-0.858740\pi$
$5$	−4.91434	+	2.83730i	−0.982868	+	0.567459i	−0.0797335	+	0.996816i	$0.525407\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.94975	+	13.7694i	−0.722704	+	1.25176i	0.237208	+	0.971459i	$0.423768\pi$
$11$	−7.94975	+	13.7694i	−0.722704	+	1.25176i	−0.959912	+	0.280302i	$0.909566\pi$
$12$		0			0
$13$			20.1940i			1.55339i	0.629879	+	0.776694i	$0.283105\pi$
$13$			20.1940i			1.55339i	−0.629879	+	0.776694i	$0.716895\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.823656	−	0.475538i	−0.0484504	−	0.0279728i	0.475579	−	0.879673i	$-0.342239\pi$
$17$	−0.823656	−	0.475538i	−0.0484504	−	0.0279728i	−0.524030	+	0.851700i	$0.675572\pi$
$18$		0			0
$19$	27.5918	−	15.9301i	1.45220	−	0.838428i	0.453593	−	0.891209i	$-0.350142\pi$
$19$	27.5918	−	15.9301i	1.45220	−	0.838428i	0.998606	+	0.0527814i	$0.0168087\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	13.0000	+	22.5167i	0.565217	+	0.978985i	0.997029	+	0.0770216i	$0.0245410\pi$
$23$	13.0000	+	22.5167i	0.565217	+	0.978985i	−0.431812	+	0.901964i	$0.642126\pi$
$24$		0			0
$25$	3.60051	−	6.23626i	0.144020	−	0.249450i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−27.7990			−0.958586			−0.479293	−	0.877655i	$-0.659107\pi$
$29$	−27.7990			−0.958586			−0.479293	+	0.877655i	$0.659107\pi$
$30$		0			0
$31$	13.1233	+	7.57675i	0.423333	+	0.244411i	0.696502	−	0.717555i	$-0.254739\pi$
$31$	13.1233	+	7.57675i	0.423333	+	0.244411i	−0.273170	+	0.961966i	$0.588072\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	16.0000	+	27.7128i	0.432432	+	0.748995i	0.997082	−	0.0763357i	$-0.0243221\pi$
$37$	16.0000	+	27.7128i	0.432432	+	0.748995i	−0.564650	+	0.825331i	$0.690989\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	17.3408i		−	0.422946i	−0.977384	−	0.211473i	$-0.932174\pi$
$41$		−	17.3408i		−	0.422946i	0.977384	−	0.211473i	$-0.0678261\pi$
$42$		0			0
$43$	−59.2965			−1.37899			−0.689494	−	0.724292i	$-0.742167\pi$
$43$	−59.2965			−1.37899			−0.689494	+	0.724292i	$0.742167\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	66.1105	−	38.1689i	1.40661	−	0.812104i	0.411546	−	0.911389i	$-0.364989\pi$
$47$	66.1105	−	38.1689i	1.40661	−	0.812104i	0.995059	+	0.0992848i	$0.0316555\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.8995	−	22.3426i	0.243387	−	0.421558i	−0.718290	−	0.695744i	$-0.755075\pi$
$53$	12.8995	−	22.3426i	0.243387	−	0.421558i	0.961677	+	0.274186i	$0.0884083\pi$
$54$		0			0
$55$		−	90.2232i		−	1.64042i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−59.2743	−	34.2220i	−1.00465	−	0.580034i	−0.0950280	−	0.995475i	$-0.530294\pi$
$59$	−59.2743	−	34.2220i	−1.00465	−	0.580034i	−0.909620	+	0.415441i	$0.863627\pi$
$60$		0			0
$61$	−46.9746	+	27.1208i	−0.770075	+	0.444603i	−0.832902	−	0.553421i	$-0.813322\pi$
$61$	−46.9746	+	27.1208i	−0.770075	+	0.444603i	0.0628261	+	0.998024i	$0.479989\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−57.2965	−	99.2404i	−0.881484	−	1.52678i
$66$		0			0
$67$	−43.6985	+	75.6880i	−0.652216	+	1.12967i	0.330368	+	0.943852i	$0.392827\pi$
$67$	−43.6985	+	75.6880i	−0.652216	+	1.12967i	−0.982584	+	0.185819i	$0.940506\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−16.4020			−0.231014			−0.115507	−	0.993307i	$-0.536849\pi$
$71$	−16.4020			−0.231014			−0.115507	+	0.993307i	$0.536849\pi$
$72$		0			0
$73$	−60.9216	−	35.1731i	−0.834542	−	0.481823i	0.0208633	−	0.999782i	$-0.493359\pi$
$73$	−60.9216	−	35.1731i	−0.834542	−	0.481823i	−0.855405	+	0.517959i	$0.826692\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−20.1005	−	34.8151i	−0.254437	−	0.440697i	0.710306	−	0.703893i	$-0.248557\pi$
$79$	−20.1005	−	34.8151i	−0.254437	−	0.440697i	−0.964742	+	0.263196i	$0.915223\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			71.5505i			0.862055i	0.902339	+	0.431027i	$0.141849\pi$
$83$			71.5505i			0.862055i	−0.902339	+	0.431027i	$0.858151\pi$
$84$		0			0
$85$	5.39697			0.0634938
$86$		0			0
$87$		0			0
$88$		0			0
$89$	66.9617	−	38.6604i	0.752379	−	0.434386i	−0.0741740	−	0.997245i	$-0.523632\pi$
$89$	66.9617	−	38.6604i	0.752379	−	0.434386i	0.826553	+	0.562859i	$0.190299\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−90.3970	+	156.572i	−0.951547	+	1.64813i
$96$		0			0
$97$			128.328i			1.32297i	0.749957	+	0.661486i	$0.230074\pi$
$97$			128.328i			1.32297i	−0.749957	+	0.661486i	$0.769926\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−86.8384	−	50.1362i	−0.859787	−	0.496398i	0.00415427	−	0.999991i	$-0.498678\pi$
$101$	−86.8384	−	50.1362i	−0.859787	−	0.496398i	−0.863941	+	0.503593i	$0.832011\pi$
$102$		0			0
$103$	42.6094	−	24.6005i	0.413683	−	0.238840i	−0.278688	−	0.960382i	$-0.589900\pi$
$103$	42.6094	−	24.6005i	0.413683	−	0.238840i	0.692371	+	0.721542i	$0.256566\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.7990	+	25.6326i	0.138308	+	0.239557i	0.926856	−	0.375416i	$-0.122500\pi$
$107$	14.7990	+	25.6326i	0.138308	+	0.239557i	−0.788548	+	0.614973i	$0.789167\pi$
$108$		0			0
$109$	5.59798	−	9.69599i	0.0513576	−	0.0889540i	−0.839204	−	0.543817i	$-0.816979\pi$
$109$	5.59798	−	9.69599i	0.0513576	−	0.0889540i	0.890561	+	0.454863i	$0.150312\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−133.698			−1.18317			−0.591586	−	0.806242i	$-0.701498\pi$
$113$	−133.698			−1.18317			−0.591586	+	0.806242i	$0.701498\pi$
$114$		0			0
$115$	−127.773	−	73.7697i	−1.11107	−	0.641476i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−65.8970	−	114.137i	−0.544603	−	0.943280i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	101.002i		−	0.808016i
$126$		0			0
$127$	120.995			0.952716			0.476358	−	0.879251i	$-0.341957\pi$
$127$	120.995			0.952716			0.476358	+	0.879251i	$0.341957\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−110.037	+	63.5301i	−0.839980	+	0.484963i	−0.857257	−	0.514888i	$-0.827833\pi$
$131$	−110.037	+	63.5301i	−0.839980	+	0.484963i	0.0172774	+	0.999851i	$0.494500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−74.9497	+	129.817i	−0.547078	+	0.947568i	0.451394	+	0.892325i	$0.350927\pi$
$137$	−74.9497	+	129.817i	−0.547078	+	0.947568i	−0.998473	+	0.0552432i	$0.982407\pi$
$138$		0			0
$139$		−	14.2661i		−	0.102634i	−0.998682	−	0.0513171i	$-0.983658\pi$
$139$		−	14.2661i		−	0.102634i	0.998682	−	0.0513171i	$-0.0163419\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−278.059	−	160.537i	−1.94447	−	1.12264i
$144$		0			0
$145$	136.614	−	78.8740i	0.942164	−	0.543958i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	25.3015	+	43.8235i	0.169809	+	0.294118i	0.938353	−	0.345680i	$-0.112352\pi$
$149$	25.3015	+	43.8235i	0.169809	+	0.294118i	−0.768544	+	0.639797i	$0.779018\pi$
$150$		0			0
$151$	20.8995	−	36.1990i	0.138407	−	0.239728i	−0.788487	−	0.615052i	$-0.789135\pi$
$151$	20.8995	−	36.1990i	0.138407	−	0.239728i	0.926894	+	0.375324i	$0.122468\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−85.9899			−0.554774
$156$		0			0
$157$	−124.067	−	71.6302i	−0.790237	−	0.456243i	0.0498093	−	0.998759i	$-0.484139\pi$
$157$	−124.067	−	71.6302i	−0.790237	−	0.456243i	−0.840046	+	0.542515i	$0.817472\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−57.1457	−	98.9793i	−0.350587	−	0.607235i	0.635765	−	0.771882i	$-0.280685\pi$
$163$	−57.1457	−	98.9793i	−0.350587	−	0.607235i	−0.986352	+	0.164648i	$0.947351\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	290.895i		−	1.74189i	−0.491382	−	0.870944i	$-0.663508\pi$
$167$		−	290.895i		−	1.74189i	0.491382	−	0.870944i	$-0.336492\pi$
$168$		0			0
$169$	−238.799			−1.41301
$170$		0			0
$171$		0			0
$172$		0			0
$173$	184.027	−	106.248i	1.06374	−	0.614151i	0.137276	−	0.990533i	$-0.456165\pi$
$173$	184.027	−	106.248i	1.06374	−	0.614151i	0.926465	+	0.376382i	$0.122832\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.8040	−	20.4452i	0.0659444	−	0.114219i	−0.831168	−	0.556021i	$-0.812327\pi$
$179$	11.8040	−	20.4452i	0.0659444	−	0.114219i	0.897113	+	0.441802i	$0.145661\pi$
$180$		0			0
$181$		−	156.384i		−	0.864002i	−0.901873	−	0.432001i	$-0.857808\pi$
$181$		−	156.384i		−	0.864002i	0.901873	−	0.432001i	$-0.142192\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−157.259	−	90.7935i	−0.850048	−	0.490776i
$186$		0			0
$187$	13.0957	−	7.56081i	0.0700306	−	0.0404322i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−122.794	−	212.685i	−0.642900	−	1.11354i	−0.984782	−	0.173793i	$-0.944398\pi$
$191$	−122.794	−	212.685i	−0.642900	−	1.11354i	0.341882	−	0.939743i	$-0.388936\pi$
$192$		0			0
$193$	−28.7487	+	49.7943i	−0.148957	+	0.258001i	−0.930842	−	0.365421i	$-0.880925\pi$
$193$	−28.7487	+	49.7943i	−0.148957	+	0.258001i	0.781885	+	0.623423i	$0.214258\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−174.402			−0.885289			−0.442645	−	0.896697i	$-0.645960\pi$
$197$	−174.402			−0.885289			−0.442645	+	0.896697i	$0.645960\pi$
$198$		0			0
$199$	145.289	+	83.8827i	0.730096	+	0.421521i	0.818457	−	0.574568i	$-0.194830\pi$
$199$	145.289	+	83.8827i	0.730096	+	0.421521i	−0.0883615	+	0.996088i	$0.528163\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	49.2010	+	85.2186i	0.240005	+	0.415701i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			506.562i			2.42374i
$210$		0			0
$211$	−81.7889			−0.387625			−0.193813	−	0.981039i	$-0.562085\pi$
$211$	−81.7889			−0.387625			−0.193813	+	0.981039i	$0.562085\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	291.403	−	168.242i	1.35536	−	0.782519i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.60303	−	16.6329i	0.0434526	−	0.0752622i
$222$		0			0
$223$		−	91.4275i		−	0.409989i	−0.978763	−	0.204994i	$-0.934282\pi$
$223$		−	91.4275i		−	0.409989i	0.978763	−	0.204994i	$-0.0657176\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	318.308	+	183.775i	1.40224	+	0.809582i	0.994622	−	0.103572i	$-0.0330272\pi$
$227$	318.308	+	183.775i	1.40224	+	0.809582i	0.407615	+	0.913154i	$0.366361\pi$
$228$		0			0
$229$	10.4606	−	6.03941i	0.0456793	−	0.0263730i	−0.476986	−	0.878911i	$-0.658271\pi$
$229$	10.4606	−	6.03941i	0.0456793	−	0.0263730i	0.522666	+	0.852538i	$0.324938\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.849242	−	1.47093i	−0.00364482	−	0.00631301i	0.864197	−	0.503153i	$-0.167827\pi$
$233$	−0.849242	−	1.47093i	−0.00364482	−	0.00631301i	−0.867842	+	0.496840i	$0.834494\pi$
$234$		0			0
$235$	−216.593	+	375.150i	−0.921672	+	1.59638i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−201.397			−0.842665			−0.421333	−	0.906906i	$-0.638437\pi$
$239$	−201.397			−0.842665			−0.421333	+	0.906906i	$0.638437\pi$
$240$		0			0
$241$	84.3675	+	48.7096i	0.350072	+	0.202114i	0.664717	−	0.747095i	$-0.268552\pi$
$241$	84.3675	+	48.7096i	0.350072	+	0.202114i	−0.314645	+	0.949210i	$0.601885\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	321.693	+	557.189i	1.30240	+	2.25583i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	77.2251i		−	0.307670i	−0.988097	−	0.153835i	$-0.950838\pi$
$251$		−	77.2251i		−	0.307670i	0.988097	−	0.153835i	$-0.0491624\pi$
$252$		0			0
$253$	−413.387			−1.63394
$254$		0			0
$255$		0			0
$256$		0			0
$257$	314.958	−	181.841i	1.22552	−	0.707553i	0.259429	−	0.965762i	$-0.416466\pi$
$257$	314.958	−	181.841i	1.22552	−	0.707553i	0.966089	+	0.258209i	$0.0831325\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−205.296	+	355.584i	−0.780595	+	1.35203i	0.151001	+	0.988534i	$0.451750\pi$
$263$	−205.296	+	355.584i	−0.780595	+	1.35203i	−0.931596	+	0.363496i	$0.881583\pi$
$264$		0			0
$265$			146.399i			0.552448i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−266.611	−	153.928i	−0.991118	−	0.572222i	−0.0855095	−	0.996337i	$-0.527252\pi$
$269$	−266.611	−	153.928i	−0.991118	−	0.572222i	−0.905608	+	0.424115i	$0.860585\pi$
$270$		0			0
$271$	−154.513	+	89.2084i	−0.570160	+	0.329182i	−0.757213	−	0.653168i	$-0.773440\pi$
$271$	−154.513	+	89.2084i	−0.570160	+	0.329182i	0.187053	+	0.982350i	$0.440106\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	57.2462	+	99.1533i	0.208168	+	0.360558i
$276$		0			0
$277$	−48.0955	+	83.3038i	−0.173630	+	0.300736i	−0.939686	−	0.342038i	$-0.888883\pi$
$277$	−48.0955	+	83.3038i	−0.173630	+	0.300736i	0.766056	+	0.642773i	$0.222216\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	155.106			0.551977			0.275989	−	0.961161i	$-0.410995\pi$
$281$	155.106			0.551977			0.275989	+	0.961161i	$0.410995\pi$
$282$		0			0
$283$	−271.278	−	156.622i	−0.958580	−	0.553436i	−0.0628440	−	0.998023i	$-0.520017\pi$
$283$	−271.278	−	156.622i	−0.958580	−	0.553436i	−0.895736	+	0.444587i	$0.853350\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−144.048	−	249.498i	−0.498435	−	0.863315i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			202.543i			0.691271i	0.938369	+	0.345636i	$0.112337\pi$
$293$			202.543i			0.691271i	−0.938369	+	0.345636i	$0.887663\pi$
$294$		0			0
$295$	388.392			1.31658
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−454.702	+	262.522i	−1.52074	+	0.878001i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	153.899	−	266.562i	0.504589	−	0.873973i
$306$		0			0
$307$			378.772i			1.23378i	0.787047	+	0.616892i	$0.211609\pi$
$307$			378.772i			1.23378i	−0.787047	+	0.616892i	$0.788391\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−178.235	−	102.904i	−0.573104	−	0.330882i	0.185284	−	0.982685i	$-0.440679\pi$
$311$	−178.235	−	102.904i	−0.573104	−	0.330882i	−0.758388	+	0.651803i	$0.774013\pi$
$312$		0			0
$313$	391.419	−	225.986i	1.25054	−	0.721999i	0.279322	−	0.960197i	$-0.409890\pi$
$313$	391.419	−	225.986i	1.25054	−	0.721999i	0.971217	+	0.238198i	$0.0765569\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−41.6934	−	72.2151i	−0.131525	−	0.227808i	0.792740	−	0.609560i	$-0.208654\pi$
$317$	−41.6934	−	72.2151i	−0.131525	−	0.227808i	−0.924265	+	0.381752i	$0.875321\pi$
$318$		0			0
$319$	220.995	−	382.774i	0.692774	−	1.19992i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−30.3015			−0.0938127
$324$		0			0
$325$	125.935	+	72.7087i	0.387493	+	0.223719i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−76.7437	−	132.924i	−0.231854	−	0.401583i	0.726500	−	0.687167i	$-0.241146\pi$
$331$	−76.7437	−	132.924i	−0.231854	−	0.401583i	−0.958354	+	0.285584i	$0.907813\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	495.942i		−	1.48042i
$336$		0			0
$337$	519.377			1.54118			0.770589	−	0.637333i	$-0.219962\pi$
$337$	519.377			1.54118			0.770589	+	0.637333i	$0.219962\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−208.654	+	120.466i	−0.611888	+	0.353274i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	42.2412	−	73.1638i	0.121732	−	0.210847i	−0.798719	−	0.601705i	$-0.794488\pi$
$347$	42.2412	−	73.1638i	0.121732	−	0.210847i	0.920451	+	0.390858i	$0.127822\pi$
$348$		0			0
$349$		−	187.959i		−	0.538565i	−0.963061	−	0.269283i	$-0.913213\pi$
$349$		−	187.959i		−	0.538565i	0.963061	−	0.269283i	$-0.0867866\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	98.0951	+	56.6352i	0.277890	+	0.160440i	0.632468	−	0.774587i	$-0.282042\pi$
$353$	98.0951	+	56.6352i	0.277890	+	0.160440i	−0.354578	+	0.935026i	$0.615375\pi$
$354$		0			0
$355$	80.6051	−	46.5374i	0.227057	−	0.131091i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	330.588	+	572.595i	0.920858	+	1.59497i	0.798091	+	0.602537i	$0.205843\pi$
$359$	330.588	+	572.595i	0.920858	+	1.59497i	0.122766	+	0.992436i	$0.460823\pi$
$360$		0			0
$361$	327.038	−	566.446i	0.905921	−	1.56910i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	399.186			1.09366
$366$		0			0
$367$	−76.1047	−	43.9391i	−0.207370	−	0.119725i	0.392719	−	0.919659i	$-0.371535\pi$
$367$	−76.1047	−	43.9391i	−0.207370	−	0.119725i	−0.600088	+	0.799934i	$0.704868\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−213.101	−	369.101i	−0.571315	−	0.989547i	−0.996431	−	0.0844079i	$-0.973100\pi$
$373$	−213.101	−	369.101i	−0.571315	−	0.989547i	0.425116	−	0.905139i	$-0.360233\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	561.374i		−	1.48905i
$378$		0			0
$379$	−719.879			−1.89942			−0.949709	−	0.313134i	$-0.898621\pi$
$379$	−719.879			−1.89942			−0.949709	+	0.313134i	$0.898621\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−209.148	+	120.752i	−0.546078	+	0.315278i	−0.747539	−	0.664218i	$-0.768765\pi$
$383$	−209.148	+	120.752i	−0.546078	+	0.315278i	0.201461	+	0.979497i	$0.435431\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−132.704	+	229.849i	−0.341140	+	0.590872i	−0.984645	−	0.174570i	$-0.944146\pi$
$389$	−132.704	+	229.849i	−0.341140	+	0.590872i	0.643505	+	0.765442i	$0.277480\pi$
$390$		0			0
$391$		−	24.7280i		−	0.0632429i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	197.562	+	114.062i	0.500156	+	0.288765i
$396$		0			0
$397$	−156.848	+	90.5561i	−0.395083	+	0.228101i	−0.684360	−	0.729144i	$-0.739918\pi$
$397$	−156.848	+	90.5561i	−0.395083	+	0.228101i	0.289277	+	0.957245i	$0.406585\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−210.296	−	364.244i	−0.524430	−	0.908340i	−0.999595	−	0.0284430i	$-0.990945\pi$
$401$	−210.296	−	364.244i	−0.524430	−	0.908340i	0.475165	−	0.879897i	$-0.342388\pi$
$402$		0			0
$403$	−153.005	+	265.013i	−0.379665	+	0.657599i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−508.784			−1.25008
$408$		0			0
$409$	126.566	+	73.0727i	0.309452	+	0.178662i	0.646681	−	0.762761i	$-0.276156\pi$
$409$	126.566	+	73.0727i	0.309452	+	0.178662i	−0.337229	+	0.941422i	$0.609490\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−203.010	−	351.624i	−0.489181	−	0.847286i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			521.905i			1.24560i	0.782383	+	0.622798i	$0.214004\pi$
$419$			521.905i			1.24560i	−0.782383	+	0.622798i	$0.785996\pi$
$420$		0			0
$421$	746.181			1.77240			0.886200	−	0.463302i	$-0.153335\pi$
$421$	746.181			1.77240			0.886200	+	0.463302i	$0.153335\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.93116	+	3.42435i	−0.0139557	+	0.00805730i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	124.296	−	215.288i	0.288391	−	0.499508i	−0.685035	−	0.728510i	$-0.740213\pi$
$431$	124.296	−	215.288i	0.288391	−	0.499508i	0.973426	+	0.229003i	$0.0735464\pi$
$432$		0			0
$433$		−	494.357i		−	1.14170i	−0.821054	−	0.570851i	$-0.806613\pi$
$433$		−	494.357i		−	1.14170i	0.821054	−	0.570851i	$-0.193387\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	717.386	+	414.183i	1.64162	+	0.947788i
$438$		0			0
$439$	−567.484	+	327.637i	−1.29267	+	0.746325i	−0.979127	−	0.203247i	$-0.934850\pi$
$439$	−567.484	+	327.637i	−1.29267	+	0.746325i	−0.313546	+	0.949573i	$0.601517\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.6985	−	34.1188i	−0.0444661	−	0.0770176i	0.842936	−	0.538014i	$-0.180825\pi$
$443$	−19.6985	−	34.1188i	−0.0444661	−	0.0770176i	−0.887402	+	0.460997i	$0.847492\pi$
$444$		0			0
$445$	−219.382	+	379.980i	−0.492993	+	0.853889i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	700.362			1.55983			0.779913	−	0.625888i	$-0.215263\pi$
$449$	700.362			1.55983			0.779913	+	0.625888i	$0.215263\pi$
$450$		0			0
$451$	238.772	+	137.855i	0.529428	+	0.305665i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	176.739	+	306.120i	0.386737	+	0.669847i	0.992008	−	0.126171i	$-0.0402689\pi$
$457$	176.739	+	306.120i	0.386737	+	0.669847i	−0.605272	+	0.796019i	$0.706936\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			382.482i			0.829680i	0.909894	+	0.414840i	$0.136162\pi$
$461$			382.482i			0.829680i	−0.909894	+	0.414840i	$0.863838\pi$
$462$		0			0
$463$	−309.005			−0.667398			−0.333699	−	0.942680i	$-0.608297\pi$
$463$	−309.005			−0.667398			−0.333699	+	0.942680i	$0.608297\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−398.612	+	230.139i	−0.853559	+	0.492803i	−0.861850	−	0.507163i	$-0.830694\pi$
$467$	−398.612	+	230.139i	−0.853559	+	0.492803i	0.00829089	+	0.999966i	$0.497361\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	471.392	−	816.475i	0.996600	−	1.72616i
$474$		0			0
$475$		−	229.426i		−	0.483002i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.9577	+	7.48116i	0.0270517	+	0.0156183i	0.513465	−	0.858111i	$-0.328362\pi$
$479$	12.9577	+	7.48116i	0.0270517	+	0.0156183i	−0.486413	+	0.873729i	$0.661695\pi$
$480$		0			0
$481$	−559.633	+	323.105i	−1.16348	+	0.671735i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−364.106	−	630.649i	−0.750733	−	1.30031i
$486$		0			0
$487$	218.688	−	378.779i	0.449052	−	0.777781i	−0.549272	−	0.835643i	$-0.685095\pi$
$487$	218.688	−	378.779i	0.449052	−	0.777781i	0.998325	+	0.0578622i	$0.0184284\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	816.583			1.66310			0.831551	−	0.555449i	$-0.187454\pi$
$491$	816.583			1.66310			0.831551	+	0.555449i	$0.187454\pi$
$492$		0			0
$493$	22.8968	+	13.2195i	0.0464438	+	0.0268144i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	124.503	+	215.645i	0.249504	+	0.432154i	0.963388	−	0.268110i	$-0.0863991\pi$
$499$	124.503	+	215.645i	0.249504	+	0.432154i	−0.713884	+	0.700264i	$0.753066\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			423.536i			0.842020i	0.907056	+	0.421010i	$0.138324\pi$
$503$			423.536i			0.842020i	−0.907056	+	0.421010i	$0.861676\pi$
$504$		0			0
$505$	569.005			1.12674
$506$		0			0
$507$		0			0
$508$		0			0
$509$	671.289	−	387.569i	1.31884	−	0.761432i	0.335297	−	0.942112i	$-0.391163\pi$
$509$	671.289	−	387.569i	1.31884	−	0.761432i	0.983542	+	0.180680i	$0.0578299\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−139.598	+	241.791i	−0.271064	+	0.469497i
$516$		0			0
$517$			1213.73i			2.34764i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	441.194	+	254.723i	0.846821	+	0.488913i	0.859577	−	0.511006i	$-0.170727\pi$
$521$	441.194	+	254.723i	0.846821	+	0.488913i	−0.0127557	+	0.999919i	$0.504060\pi$
$522$		0			0
$523$	−457.391	+	264.075i	−0.874553	+	0.504923i	−0.868859	−	0.495060i	$-0.835146\pi$
$523$	−457.391	+	264.075i	−0.874553	+	0.504923i	−0.00569433	+	0.999984i	$0.501813\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.20606	−	12.4813i	−0.0136737	−	0.0236836i
$528$		0			0
$529$	−73.5000	+	127.306i	−0.138941	+	0.240654i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	350.181			0.657000
$534$		0			0
$535$	−145.455	−	83.9783i	−0.271878	−	0.156969i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	264.799	+	458.645i	0.489462	+	0.847773i	0.999926	−	0.0121257i	$-0.00385982\pi$
$541$	264.799	+	458.645i	0.489462	+	0.847773i	−0.510464	+	0.859899i	$0.670526\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			63.5325i			0.116573i
$546$		0			0
$547$	16.9045			0.0309041			0.0154521	−	0.999881i	$-0.495081\pi$
$547$	16.9045			0.0309041			0.0154521	+	0.999881i	$0.495081\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−767.024	+	442.841i	−1.39206	+	0.803705i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	446.980	−	774.192i	0.802477	−	1.38993i	−0.115504	−	0.993307i	$-0.536848\pi$
$557$	446.980	−	774.192i	0.802477	−	1.38993i	0.917981	−	0.396624i	$-0.129818\pi$
$558$		0			0
$559$		−	1197.43i		−	2.14210i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−84.1219	−	48.5678i	−0.149417	−	0.0862661i	0.423427	−	0.905930i	$-0.360827\pi$
$563$	−84.1219	−	48.5678i	−0.149417	−	0.0862661i	−0.572845	+	0.819664i	$0.694160\pi$
$564$		0			0
$565$	657.040	−	379.342i	1.16290	−	0.671402i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	380.090	+	658.336i	0.667997	+	1.15701i	0.978463	+	0.206421i	$0.0661815\pi$
$569$	380.090	+	658.336i	0.667997	+	1.15701i	−0.310466	+	0.950584i	$0.600485\pi$
$570$		0			0
$571$	−169.146	+	292.969i	−0.296227	+	0.513080i	−0.975270	−	0.221019i	$-0.929062\pi$
$571$	−169.146	+	292.969i	−0.296227	+	0.513080i	0.679042	+	0.734099i	$0.262395\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	187.226			0.325611
$576$		0			0
$577$	209.039	+	120.689i	0.362286	+	0.209166i	0.670083	−	0.742286i	$-0.266258\pi$
$577$	209.039	+	120.689i	0.362286	+	0.209166i	−0.307797	+	0.951452i	$0.599592\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	205.095	+	355.236i	0.351793	+	0.609324i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	405.338i		−	0.690525i	−0.938506	−	0.345263i	$-0.887790\pi$
$587$		−	405.338i		−	0.690525i	0.938506	−	0.345263i	$-0.112210\pi$
$588$		0			0
$589$	482.794			0.819684
$590$		0			0
$591$		0			0
$592$		0			0
$593$	128.459	−	74.1656i	0.216625	−	0.125068i	−0.387762	−	0.921760i	$-0.626752\pi$
$593$	128.459	−	74.1656i	0.216625	−	0.125068i	0.604387	+	0.796691i	$0.293418\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	200.894	−	347.959i	0.335383	−	0.580900i	−0.648175	−	0.761491i	$-0.724468\pi$
$599$	200.894	−	347.959i	0.335383	−	0.580900i	0.983558	+	0.180591i	$0.0578009\pi$
$600$		0			0
$601$			782.716i			1.30236i	0.758925	+	0.651178i	$0.225725\pi$
$601$			782.716i			1.30236i	−0.758925	+	0.651178i	$0.774275\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	647.680	+	373.939i	1.07055	+	0.618080i
$606$		0			0
$607$	113.772	−	65.6864i	0.187433	−	0.108215i	−0.403347	−	0.915047i	$-0.632153\pi$
$607$	113.772	−	65.6864i	0.187433	−	0.108215i	0.590781	+	0.806832i	$0.298820\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	770.784	+	1335.04i	1.26151	+	2.18500i
$612$		0			0
$613$	500.382	−	866.687i	0.816284	−	1.41384i	−0.0921190	−	0.995748i	$-0.529364\pi$
$613$	500.382	−	866.687i	0.816284	−	1.41384i	0.908403	−	0.418097i	$-0.137303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−738.563			−1.19702			−0.598511	−	0.801115i	$-0.704241\pi$
$617$	−738.563			−1.19702			−0.598511	+	0.801115i	$0.704241\pi$
$618$		0			0
$619$	826.105	+	476.952i	1.33458	+	0.770520i	0.985998	−	0.166758i	$-0.0533300\pi$
$619$	826.105	+	476.952i	1.33458	+	0.770520i	0.348582	+	0.937278i	$0.386663\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	376.585	+	652.265i	0.602537	+	1.04362i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	30.4344i		−	0.0483854i
$630$		0			0
$631$	−743.176			−1.17777			−0.588887	−	0.808215i	$-0.700434\pi$
$631$	−743.176			−1.17777			−0.588887	+	0.808215i	$0.700434\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−594.611	+	343.299i	−0.936395	+	0.540628i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	119.709	−	207.341i	0.186753	−	0.323465i	−0.757413	−	0.652936i	$-0.773537\pi$
$641$	119.709	−	207.341i	0.186753	−	0.323465i	0.944166	+	0.329471i	$0.106870\pi$
$642$		0			0
$643$			812.453i			1.26353i	0.775158	+	0.631767i	$0.217670\pi$
$643$			812.453i			1.26353i	−0.775158	+	0.631767i	$0.782330\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	617.342	+	356.422i	0.954160	+	0.550885i	0.894371	−	0.447326i	$-0.147624\pi$
$647$	617.342	+	356.422i	0.954160	+	0.550885i	0.0597896	+	0.998211i	$0.480957\pi$
$648$		0			0
$649$	942.431	−	544.113i	1.45213	−	0.838386i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	148.291	+	256.848i	0.227093	+	0.393336i	0.956945	−	0.290269i	$-0.0937447\pi$
$653$	148.291	+	256.848i	0.227093	+	0.393336i	−0.729853	+	0.683604i	$0.760411\pi$
$654$		0			0
$655$	360.508	−	624.417i	0.550393	−	0.953309i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	617.709			0.937342			0.468671	−	0.883373i	$-0.344733\pi$
$659$	617.709			0.937342			0.468671	+	0.883373i	$0.344733\pi$
$660$		0			0
$661$	281.544	+	162.550i	0.425937	+	0.245915i	0.697614	−	0.716474i	$-0.254245\pi$
$661$	281.544	+	162.550i	0.425937	+	0.245915i	−0.271677	+	0.962388i	$0.587578\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−361.387	−	625.940i	−0.541809	−	0.938441i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	862.414i		−	1.28527i
$672$		0			0
$673$	−793.899			−1.17964			−0.589821	−	0.807534i	$-0.700802\pi$
$673$	−793.899			−1.17964			−0.589821	+	0.807534i	$0.700802\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−297.029	+	171.490i	−0.438743	+	0.253309i	−0.703064	−	0.711126i	$-0.748185\pi$
$677$	−297.029	+	171.490i	−0.438743	+	0.253309i	0.264321	+	0.964435i	$0.414852\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−596.080	+	1032.44i	−0.872738	+	1.51163i	−0.0135858	+	0.999908i	$0.504325\pi$
$683$	−596.080	+	1032.44i	−0.872738	+	1.51163i	−0.859153	+	0.511719i	$0.829009\pi$
$684$		0			0
$685$		−	850.619i		−	1.24178i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	451.187	+	260.493i	0.654843	+	0.378074i
$690$		0			0
$691$	−271.032	+	156.481i	−0.392232	+	0.226455i	−0.683127	−	0.730300i	$-0.739380\pi$
$691$	−271.032	+	156.481i	−0.392232	+	0.226455i	0.290895	+	0.956755i	$0.406047\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	40.4773	+	70.1087i	0.0582407	+	0.100876i
$696$		0			0
$697$	−8.24621	+	14.2829i	−0.0118310	+	0.0204919i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−414.010			−0.590599			−0.295300	−	0.955405i	$-0.595419\pi$
$701$	−414.010			−0.590599			−0.295300	+	0.955405i	$0.595419\pi$
$702$		0			0
$703$	882.937	+	509.764i	1.25596	+	0.725127i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−561.980	−	973.378i	−0.792637	−	1.37289i	−0.924329	−	0.381597i	$-0.875374\pi$
$709$	−561.980	−	973.378i	−0.792637	−	1.37289i	0.131691	−	0.991291i	$-0.457959\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			393.991i			0.552582i
$714$		0			0
$715$	1821.97			2.54821
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−940.262	+	542.860i	−1.30774	+	0.755021i	−0.981718	−	0.190342i	$-0.939040\pi$
$719$	−940.262	+	542.860i	−1.30774	+	0.755021i	−0.326018	+	0.945364i	$0.605707\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−100.090	+	173.362i	−0.138056	+	0.239120i
$726$		0			0
$727$			270.606i			0.372223i	0.982529	+	0.186111i	$0.0595885\pi$
$727$			270.606i			0.372223i	−0.982529	+	0.186111i	$0.940412\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	48.8399	+	28.1977i	0.0668124	+	0.0385742i
$732$		0			0
$733$	266.994	−	154.149i	0.364249	−	0.210299i	−0.306694	−	0.951808i	$-0.599223\pi$
$733$	266.994	−	154.149i	0.364249	−	0.210299i	0.670943	+	0.741509i	$0.265890\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−694.784	−	1203.40i	−0.942719	−	1.63284i
$738$		0			0
$739$	−136.261	+	236.012i	−0.184386	+	0.319366i	−0.943370	−	0.331744i	$-0.892363\pi$
$739$	−136.261	+	236.012i	−0.184386	+	0.319366i	0.758983	+	0.651110i	$0.225696\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	225.196			0.303090			0.151545	−	0.988450i	$-0.451575\pi$
$743$	225.196			0.303090			0.151545	+	0.988450i	$0.451575\pi$
$744$		0			0
$745$	−248.681	−	143.576i	−0.333799	−	0.192719i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	255.693	+	442.874i	0.340471	+	0.589712i	0.984520	−	0.175271i	$-0.0560803\pi$
$751$	255.693	+	442.874i	0.340471	+	0.589712i	−0.644050	+	0.764984i	$0.722747\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			237.192i			0.314162i
$756$		0			0
$757$	−779.598			−1.02985			−0.514926	−	0.857235i	$-0.672181\pi$
$757$	−779.598			−1.02985			−0.514926	+	0.857235i	$0.672181\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−168.899	+	97.5140i	−0.221944	+	0.128139i	−0.606850	−	0.794816i	$-0.707567\pi$
$761$	−168.899	+	97.5140i	−0.221944	+	0.128139i	0.384906	+	0.922956i	$0.374234\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	691.080	−	1196.99i	0.901017	−	1.56061i
$768$		0			0
$769$			73.8956i			0.0960931i	0.998845	+	0.0480465i	$0.0152996\pi$
$769$			73.8956i			0.0960931i	−0.998845	+	0.0480465i	$0.984700\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−161.020	−	92.9649i	−0.208305	−	0.120265i	0.392218	−	0.919872i	$-0.371708\pi$
$773$	−161.020	−	92.9649i	−0.208305	−	0.120265i	−0.600523	+	0.799607i	$0.705041\pi$
$774$		0			0
$775$	94.5011	−	54.5602i	0.121937	−	0.0704003i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−276.241	−	478.464i	−0.354610	−	0.614202i
$780$		0			0
$781$	130.392	−	225.845i	0.166955	−	0.289175i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	812.944			1.03560
$786$		0			0
$787$	−618.085	−	356.852i	−0.785369	−	0.453433i	0.0529606	−	0.998597i	$-0.483134\pi$
$787$	−618.085	−	356.852i	−0.785369	−	0.453433i	−0.838330	+	0.545164i	$0.816468\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−547.678	−	948.607i	−0.690641	−	1.19623i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			908.540i			1.13995i	0.821662	+	0.569975i	$0.193047\pi$
$797$			908.540i			1.13995i	−0.821662	+	0.569975i	$0.806953\pi$
$798$		0			0
$799$	−72.6030			−0.0908674
$800$		0			0
$801$		0			0
$802$		0			0
$803$	968.622	−	559.234i	1.20625	−	0.696431i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−160.055	+	277.224i	−0.197843	+	0.342675i	−0.947829	−	0.318779i	$-0.896727\pi$
$809$	−160.055	+	277.224i	−0.197843	+	0.342675i	0.749986	+	0.661454i	$0.230060\pi$
$810$		0			0
$811$		−	839.749i		−	1.03545i	−0.855547	−	0.517724i	$-0.826779\pi$
$811$		−	839.749i		−	1.03545i	0.855547	−	0.517724i	$-0.173221\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	561.667	+	324.279i	0.689162	+	0.397888i
$816$		0			0
$817$	−1636.10	+	944.600i	−2.00256	+	1.15618i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	64.9798	+	112.548i	0.0791471	+	0.137087i	0.902882	−	0.429888i	$-0.141447\pi$
$821$	64.9798	+	112.548i	0.0791471	+	0.137087i	−0.823735	+	0.566975i	$0.808114\pi$
$822$		0			0
$823$	−400.482	+	693.656i	−0.486613	+	0.842838i	−0.999882	−	0.0153898i	$-0.995101\pi$
$823$	−400.482	+	693.656i	−0.486613	+	0.842838i	0.513269	+	0.858228i	$0.328434\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−289.156			−0.349644			−0.174822	−	0.984600i	$-0.555935\pi$
$827$	−289.156			−0.349644			−0.174822	+	0.984600i	$0.555935\pi$
$828$		0			0
$829$	274.133	+	158.270i	0.330679	+	0.190917i	0.656142	−	0.754637i	$-0.272187\pi$
$829$	274.133	+	158.270i	0.330679	+	0.190917i	−0.325464	+	0.945555i	$0.605520\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	825.357	+	1429.56i	0.988451	+	1.71205i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	802.370i		−	0.956341i	−0.878267	−	0.478171i	$-0.841300\pi$
$839$		−	802.370i		−	0.956341i	0.878267	−	0.478171i	$-0.158700\pi$
$840$		0			0
$841$	−68.2162			−0.0811132
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1173.54	−	677.544i	1.38880	−	0.801827i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−416.000	+	720.533i	−0.488837	+	0.846690i
$852$		0			0
$853$			235.386i			0.275950i	0.990436	+	0.137975i	$0.0440594\pi$
$853$			235.386i			0.275950i	−0.990436	+	0.137975i	$0.955941\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1226.53	−	708.138i	−1.43119	−	0.826299i	−0.433980	−	0.900923i	$-0.642891\pi$
$857$	−1226.53	−	708.138i	−1.43119	−	0.826299i	−0.997212	+	0.0746237i	$0.976224\pi$
$858$		0			0
$859$	−1117.26	+	645.052i	−1.30065	+	0.750933i	−0.980516	−	0.196438i	$-0.937063\pi$
$859$	−1117.26	+	645.052i	−1.30065	+	0.750933i	−0.320138	+	0.947371i	$0.603729\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−186.080	−	322.301i	−0.215620	−	0.373465i	0.737844	−	0.674971i	$-0.235844\pi$
$863$	−186.080	−	322.301i	−0.215620	−	0.373465i	−0.953464	+	0.301506i	$0.902511\pi$
$864$		0			0
$865$	−602.915	+	1044.28i	−0.697011	+	1.20726i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	639.176			0.735530
$870$		0			0
$871$	−1528.45	−	882.449i	−1.75482	−	1.01314i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	389.296	+	674.281i	0.443896	+	0.768850i	0.997975	−	0.0636144i	$-0.0202628\pi$
$877$	389.296	+	674.281i	0.443896	+	0.768850i	−0.554079	+	0.832464i	$0.686929\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1206.76i			1.36976i	0.728657	+	0.684879i	$0.240145\pi$
$881$			1206.76i			1.36976i	−0.728657	+	0.684879i	$0.759855\pi$
$882$		0			0
$883$	1132.77			1.28287			0.641435	−	0.767178i	$-0.278340\pi$
$883$	1132.77			1.28287			0.641435	+	0.767178i	$0.278340\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−631.180	+	364.412i	−0.711589	+	0.410836i	−0.811649	−	0.584145i	$-0.801430\pi$
$887$	−631.180	+	364.412i	−0.711589	+	0.410836i	0.100060	+	0.994981i	$0.468097\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1216.07	−	2106.30i	1.36178	−	2.35867i
$894$		0			0
$895$			133.966i			0.149683i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−364.815	−	210.626i	−0.405801	−	0.234289i
$900$		0			0
$901$	−21.2495	+	12.2684i	−0.0235843	+	0.0136164i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	443.709	+	768.526i	0.490286	+	0.849200i
$906$		0			0
$907$	4.80909	−	8.32959i	0.00530220	−	0.00918367i	−0.863362	−	0.504585i	$-0.831646\pi$
$907$	4.80909	−	8.32959i	0.00530220	−	0.00918367i	0.868664	+	0.495401i	$0.164979\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1015.41			−1.11461			−0.557304	−	0.830309i	$-0.688164\pi$
$911$	−1015.41			−1.11461			−0.557304	+	0.830309i	$0.688164\pi$
$912$		0			0
$913$	−985.206	−	568.809i	−1.07909	−	0.623011i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−241.508	−	418.303i	−0.262794	−	0.455172i	0.704189	−	0.710012i	$-0.251311\pi$
$919$	−241.508	−	418.303i	−0.262794	−	0.455172i	−0.966983	+	0.254840i	$0.917977\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	331.223i		−	0.358855i
$924$		0			0
$925$	230.432			0.249116
$926$		0			0
$927$		0			0
$928$		0			0
$929$	816.334	−	471.311i	0.878723	−	0.507331i	0.00848612	−	0.999964i	$-0.497299\pi$
$929$	816.334	−	471.311i	0.878723	−	0.507331i	0.870237	+	0.492633i	$0.163965\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−42.9045	+	74.3129i	−0.0458872	+	0.0794790i
$936$		0			0
$937$		−	1489.44i		−	1.58958i	−0.606882	−	0.794792i	$-0.707580\pi$
$937$		−	1489.44i		−	1.58958i	0.606882	−	0.794792i	$-0.292420\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−951.382	−	549.281i	−1.01103	−	0.583720i	−0.0995389	−	0.995034i	$-0.531737\pi$
$941$	−951.382	−	549.281i	−1.01103	−	0.583720i	−0.911494	+	0.411314i	$0.865070\pi$
$942$		0			0
$943$	390.457	−	225.430i	0.414058	−	0.239057i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	152.452	+	264.055i	0.160984	+	0.278833i	0.935222	−	0.354062i	$-0.115200\pi$
$947$	152.452	+	264.055i	0.160984	+	0.278833i	−0.774238	+	0.632895i	$0.781866\pi$
$948$		0			0
$949$	710.286	−	1230.25i	0.748458	−	1.29637i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	803.578			0.843209			0.421604	−	0.906780i	$-0.361467\pi$
$953$	803.578			0.843209			0.421604	+	0.906780i	$0.361467\pi$
$954$		0			0
$955$	1206.90	+	696.806i	1.26377	+	0.729639i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−365.686	−	633.386i	−0.380526	−	0.659091i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	326.275i		−	0.338109i
$966$		0			0
$967$	−93.6182			−0.0968130			−0.0484065	−	0.998828i	$-0.515414\pi$
$967$	−93.6182			−0.0968130			−0.0484065	+	0.998828i	$0.515414\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	363.252	−	209.723i	0.374100	−	0.215987i	−0.301148	−	0.953577i	$-0.597370\pi$
$971$	363.252	−	209.723i	0.374100	−	0.215987i	0.675248	+	0.737590i	$0.264036\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−839.824	+	1454.62i	−0.859595	+	1.48886i	0.0127213	+	0.999919i	$0.495951\pi$
$977$	−839.824	+	1454.62i	−0.859595	+	1.48886i	−0.872316	+	0.488943i	$0.837383\pi$
$978$		0			0
$979$			1229.36i			1.25573i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	230.673	+	133.179i	0.234663	+	0.135483i	0.612721	−	0.790299i	$-0.290075\pi$
$983$	230.673	+	133.179i	0.234663	+	0.135483i	−0.378059	+	0.925782i	$0.623408\pi$
$984$		0			0
$985$	857.071	−	494.830i	0.870123	−	0.502366i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−770.854	−	1335.16i	−0.779428	−	1.35001i
$990$		0			0
$991$	−447.307	+	774.758i	−0.451369	+	0.781794i	−0.998471	−	0.0552718i	$-0.982397\pi$
$991$	−447.307	+	774.758i	−0.451369	+	0.781794i	0.547102	+	0.837066i	$0.315731\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−952.000			−0.956784
$996$		0			0
$997$	−553.290	−	319.442i	−0.554955	−	0.320403i	0.196163	−	0.980571i	$-0.437152\pi$
$997$	−553.290	−	319.442i	−0.554955	−	0.320403i	−0.751118	+	0.660168i	$0.770485\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.k.325.2		8
3.2	odd	2		196.3.h.c.129.3		8
7.2	even	3	inner	1764.3.z.k.901.3		8
7.3	odd	6		1764.3.d.e.685.3		4
7.4	even	3		1764.3.d.e.685.2		4
7.5	odd	6	inner	1764.3.z.k.901.2		8
7.6	odd	2	inner	1764.3.z.k.325.3		8
12.11	even	2		784.3.s.g.129.2		8
21.2	odd	6		196.3.h.c.117.2		8
21.5	even	6		196.3.h.c.117.3		8
21.11	odd	6		196.3.b.b.97.2	✓	4
21.17	even	6		196.3.b.b.97.3	yes	4
21.20	even	2		196.3.h.c.129.2		8
84.11	even	6		784.3.c.d.97.3		4
84.23	even	6		784.3.s.g.705.3		8
84.47	odd	6		784.3.s.g.705.2		8
84.59	odd	6		784.3.c.d.97.2		4
84.83	odd	2		784.3.s.g.129.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.3.b.b.97.2	✓	4	21.11	odd	6
196.3.b.b.97.3	yes	4	21.17	even	6
196.3.h.c.117.2		8	21.2	odd	6
196.3.h.c.117.3		8	21.5	even	6
196.3.h.c.129.2		8	21.20	even	2
196.3.h.c.129.3		8	3.2	odd	2
784.3.c.d.97.2		4	84.59	odd	6
784.3.c.d.97.3		4	84.11	even	6
784.3.s.g.129.2		8	12.11	even	2
784.3.s.g.129.3		8	84.83	odd	2
784.3.s.g.705.2		8	84.47	odd	6
784.3.s.g.705.3		8	84.23	even	6
1764.3.d.e.685.2		4	7.4	even	3
1764.3.d.e.685.3		4	7.3	odd	6
1764.3.z.k.325.2		8	1.1	even	1	trivial
1764.3.z.k.325.3		8	7.6	odd	2	inner
1764.3.z.k.901.2		8	7.5	odd	6	inner
1764.3.z.k.901.3		8	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.91434 + 2.83730i) q^{5} +O(q^{10})\)	\(q+(-4.91434 + 2.83730i) q^{5} +(-7.94975 + 13.7694i) q^{11} +20.1940i q^{13} +(-0.823656 - 0.475538i) q^{17} +(27.5918 - 15.9301i) q^{19} +(13.0000 + 22.5167i) q^{23} +(3.60051 - 6.23626i) q^{25} -27.7990 q^{29} +(13.1233 + 7.57675i) q^{31} +(16.0000 + 27.7128i) q^{37} -17.3408i q^{41} -59.2965 q^{43} +(66.1105 - 38.1689i) q^{47} +(12.8995 - 22.3426i) q^{53} -90.2232i q^{55} +(-59.2743 - 34.2220i) q^{59} +(-46.9746 + 27.1208i) q^{61} +(-57.2965 - 99.2404i) q^{65} +(-43.6985 + 75.6880i) q^{67} -16.4020 q^{71} +(-60.9216 - 35.1731i) q^{73} +(-20.1005 - 34.8151i) q^{79} +71.5505i q^{83} +5.39697 q^{85} +(66.9617 - 38.6604i) q^{89} +(-90.3970 + 156.572i) q^{95} +128.328i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{11} + 104 q^{23} + 108 q^{25} - 64 q^{29} + 128 q^{37} + 80 q^{43} + 24 q^{53} + 96 q^{65} - 112 q^{67} - 448 q^{71} - 240 q^{79} - 432 q^{85} - 248 q^{95}+O(q^{100}) \) 8 * q - 24 * q^11 + 104 * q^23 + 108 * q^25 - 64 * q^29 + 128 * q^37 + 80 * q^43 + 24 * q^53 + 96 * q^65 - 112 * q^67 - 448 * q^71 - 240 * q^79 - 432 * q^85 - 248 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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