LMFDB - Embedded newform 1764.4.t.c.1097.9

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			1097.9
Character	$\chi$	$=$	1764.1097
Dual form			1764.4.t.c.521.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.27916 + 5.67967i) q^{5} +O(q^{10})$	$q+(-3.27916 + 5.67967i) q^{5} +(34.2427 - 19.7700i) q^{11} -79.8324i q^{13} +(-34.1569 - 59.1615i) q^{17} +(-9.89034 - 5.71019i) q^{19} +(-118.003 - 68.1289i) q^{23} +(40.9942 + 71.0040i) q^{25} +9.32277i q^{29} +(69.9858 - 40.4063i) q^{31} +(-117.636 + 203.751i) q^{37} +50.7669 q^{41} -24.3676 q^{43} +(-114.479 + 198.284i) q^{47} +(-413.578 + 238.780i) q^{53} +259.316i q^{55} +(194.591 + 337.042i) q^{59} +(345.404 + 199.419i) q^{61} +(453.422 + 261.783i) q^{65} +(-377.437 - 653.740i) q^{67} +799.714i q^{71} +(523.232 - 302.088i) q^{73} +(-22.9193 + 39.6975i) q^{79} +45.6855 q^{83} +448.024 q^{85} +(-163.313 + 282.867i) q^{89} +(64.8640 - 37.4493i) q^{95} -109.900i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.27916	+	5.67967i	−0.293297	+	0.508005i	−0.974587	−	0.224008i	$-0.928086\pi$
$5$	−3.27916	+	5.67967i	−0.293297	+	0.508005i	0.681290	+	0.732013i	$0.261419\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	34.2427	−	19.7700i	0.938596	−	0.541899i	0.0490760	−	0.998795i	$-0.484372\pi$
$11$	34.2427	−	19.7700i	0.938596	−	0.541899i	0.889520	+	0.456896i	$0.151039\pi$
$12$		0			0
$13$		−	79.8324i		−	1.70319i	−0.524197	−	0.851597i	$-0.675635\pi$
$13$		−	79.8324i		−	1.70319i	0.524197	−	0.851597i	$-0.324365\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−34.1569	−	59.1615i	−0.487310	−	0.844045i	0.512584	−	0.858637i	$-0.328688\pi$
$17$	−34.1569	−	59.1615i	−0.487310	−	0.844045i	−0.999894	+	0.0145918i	$0.995355\pi$
$18$		0			0
$19$	−9.89034	−	5.71019i	−0.119421	−	0.0689478i	0.439100	−	0.898438i	$-0.355298\pi$
$19$	−9.89034	−	5.71019i	−0.119421	−	0.0689478i	−0.558521	+	0.829491i	$0.688631\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−118.003	−	68.1289i	−1.06979	−	0.617646i	−0.141669	−	0.989914i	$-0.545247\pi$
$23$	−118.003	−	68.1289i	−1.06979	−	0.617646i	−0.928125	+	0.372268i	$0.878580\pi$
$24$		0			0
$25$	40.9942	+	71.0040i	0.327954	+	0.568032i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			9.32277i			0.0596964i	0.999554	+	0.0298482i	$0.00950239\pi$
$29$			9.32277i			0.0596964i	−0.999554	+	0.0298482i	$0.990498\pi$
$30$		0			0
$31$	69.9858	−	40.4063i	0.405478	−	0.234103i	−0.283367	−	0.959012i	$-0.591451\pi$
$31$	69.9858	−	40.4063i	0.405478	−	0.234103i	0.688845	+	0.724909i	$0.258118\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−117.636	+	203.751i	−0.522681	+	0.905309i	0.476971	+	0.878919i	$0.341735\pi$
$37$	−117.636	+	203.751i	−0.522681	+	0.905309i	−0.999652	+	0.0263904i	$0.991599\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	50.7669			0.193377			0.0966886	−	0.995315i	$-0.469175\pi$
$41$	50.7669			0.193377			0.0966886	+	0.995315i	$0.469175\pi$
$42$		0			0
$43$	−24.3676			−0.0864191			−0.0432096	−	0.999066i	$-0.513758\pi$
$43$	−24.3676			−0.0864191			−0.0432096	+	0.999066i	$0.513758\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−114.479	+	198.284i	−0.355287	+	0.615376i	−0.987167	−	0.159691i	$-0.948950\pi$
$47$	−114.479	+	198.284i	−0.355287	+	0.615376i	0.631880	+	0.775066i	$0.282284\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−413.578	+	238.780i	−1.07187	+	0.618847i	−0.928693	−	0.370849i	$-0.879067\pi$
$53$	−413.578	+	238.780i	−1.07187	+	0.618847i	−0.143182	+	0.989696i	$0.545733\pi$
$54$		0			0
$55$			259.316i			0.635749i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	194.591	+	337.042i	0.429384	+	0.743714i	0.996819	−	0.0797038i	$-0.0253975\pi$
$59$	194.591	+	337.042i	0.429384	+	0.743714i	−0.567435	+	0.823418i	$0.692064\pi$
$60$		0			0
$61$	345.404	+	199.419i	0.724992	+	0.418574i	0.816587	−	0.577222i	$-0.195863\pi$
$61$	345.404	+	199.419i	0.724992	+	0.418574i	−0.0915953	+	0.995796i	$0.529197\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	453.422	+	261.783i	0.865232	+	0.499542i
$66$		0			0
$67$	−377.437	−	653.740i	−0.688228	−	1.19205i	−0.972411	−	0.233276i	$-0.925055\pi$
$67$	−377.437	−	653.740i	−0.688228	−	1.19205i	0.284182	−	0.958770i	$-0.408278\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			799.714i			1.33674i	0.743829	+	0.668370i	$0.233008\pi$
$71$			799.714i			1.33674i	−0.743829	+	0.668370i	$0.766992\pi$
$72$		0			0
$73$	523.232	−	302.088i	0.838900	−	0.484339i	−0.0179900	−	0.999838i	$-0.505727\pi$
$73$	523.232	−	302.088i	0.838900	−	0.484339i	0.856890	+	0.515499i	$0.172393\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−22.9193	+	39.6975i	−0.0326408	+	0.0565356i	−0.881884	−	0.471466i	$-0.843725\pi$
$79$	−22.9193	+	39.6975i	−0.0326408	+	0.0565356i	0.849244	+	0.528001i	$0.177058\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	45.6855			0.0604172			0.0302086	−	0.999544i	$-0.490383\pi$
$83$	45.6855			0.0604172			0.0302086	+	0.999544i	$0.490383\pi$
$84$		0			0
$85$	448.024			0.571706
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−163.313	+	282.867i	−0.194508	+	0.336897i	−0.946739	−	0.322002i	$-0.895644\pi$
$89$	−163.313	+	282.867i	−0.194508	+	0.336897i	0.752231	+	0.658899i	$0.228978\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	64.8640	−	37.4493i	0.0700517	−	0.0404444i
$96$		0			0
$97$		−	109.900i		−	0.115038i	−0.998344	−	0.0575191i	$-0.981681\pi$
$97$		−	109.900i		−	0.115038i	0.998344	−	0.0575191i	$-0.0183190\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−988.707	−	1712.49i	−0.974059	−	1.68712i	−0.683005	−	0.730414i	$-0.739327\pi$
$101$	−988.707	−	1712.49i	−0.974059	−	1.68712i	−0.291054	−	0.956707i	$-0.594006\pi$
$102$		0			0
$103$	−1519.02	−	877.006i	−1.45314	−	0.838971i	−0.454481	−	0.890756i	$-0.650175\pi$
$103$	−1519.02	−	877.006i	−1.45314	−	0.838971i	−0.998658	+	0.0517856i	$0.983509\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−628.349	−	362.777i	−0.567708	−	0.327767i	0.188525	−	0.982068i	$-0.439629\pi$
$107$	−628.349	−	362.777i	−0.567708	−	0.327767i	−0.756234	+	0.654302i	$0.772963\pi$
$108$		0			0
$109$	297.498	+	515.281i	0.261423	+	0.452798i	0.966620	−	0.256213i	$-0.0824748\pi$
$109$	297.498	+	515.281i	0.261423	+	0.452798i	−0.705197	+	0.709011i	$0.749142\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			311.261i			0.259123i	0.991571	+	0.129562i	$0.0413570\pi$
$113$			311.261i			0.259123i	−0.991571	+	0.129562i	$0.958643\pi$
$114$		0			0
$115$	773.900	−	446.811i	0.627535	−	0.362308i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	116.207	−	201.277i	0.0873082	−	0.151222i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1357.50			−0.971346
$126$		0			0
$127$	−2397.08			−1.67485			−0.837426	−	0.546551i	$-0.815940\pi$
$127$	−2397.08			−1.67485			−0.837426	+	0.546551i	$0.815940\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−667.091	+	1155.44i	−0.444916	+	0.770618i	−0.998046	−	0.0624773i	$-0.980100\pi$
$131$	−667.091	+	1155.44i	−0.444916	+	0.770618i	0.553130	+	0.833095i	$0.313433\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−983.475	+	567.809i	−0.613313	+	0.354097i	−0.774261	−	0.632866i	$-0.781878\pi$
$137$	−983.475	+	567.809i	−0.613313	+	0.354097i	0.160948	+	0.986963i	$0.448545\pi$
$138$		0			0
$139$		−	1869.35i		−	1.14069i	−0.821404	−	0.570347i	$-0.806809\pi$
$139$		−	1869.35i		−	1.14069i	0.821404	−	0.570347i	$-0.193191\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1578.29	−	2733.67i	−0.922958	−	1.59861i
$144$		0			0
$145$	−52.9503	−	30.5709i	−0.0303261	−	0.0175088i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	846.445	+	488.695i	0.465392	+	0.268694i	0.714309	−	0.699830i	$-0.246741\pi$
$149$	846.445	+	488.695i	0.465392	+	0.268694i	−0.248916	+	0.968525i	$0.580074\pi$
$150$		0			0
$151$	−517.459	−	896.266i	−0.278876	−	0.483027i	0.692230	−	0.721677i	$-0.256628\pi$
$151$	−517.459	−	896.266i	−0.278876	−	0.483027i	−0.971106	+	0.238650i	$0.923295\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			529.995i			0.274647i
$156$		0			0
$157$	54.0241	−	31.1908i	0.0274624	−	0.0158554i	−0.486206	−	0.873844i	$-0.661620\pi$
$157$	54.0241	−	31.1908i	0.0274624	−	0.0158554i	0.513668	+	0.857989i	$0.328286\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−198.891	+	344.489i	−0.0955725	+	0.165536i	−0.909847	−	0.414943i	$-0.863802\pi$
$163$	−198.891	+	344.489i	−0.0955725	+	0.165536i	0.814275	+	0.580479i	$0.197135\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2001.69			0.927519			0.463759	−	0.885961i	$-0.346500\pi$
$167$	2001.69			0.927519			0.463759	+	0.885961i	$0.346500\pi$
$168$		0			0
$169$	−4176.21			−1.90087
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−999.942	+	1731.95i	−0.439446	+	0.761143i	−0.997647	−	0.0685630i	$-0.978159\pi$
$173$	−999.942	+	1731.95i	−0.439446	+	0.761143i	0.558201	+	0.829706i	$0.311492\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3289.27	+	1899.06i	−1.37347	+	0.792975i	−0.991363	−	0.131143i	$-0.958135\pi$
$179$	−3289.27	+	1899.06i	−1.37347	+	0.792975i	−0.382109	+	0.924117i	$0.624802\pi$
$180$		0			0
$181$			2540.89i			1.04344i	0.853116	+	0.521720i	$0.174710\pi$
$181$			2540.89i			1.04344i	−0.853116	+	0.521720i	$0.825290\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−771.493	−	1336.26i	−0.306601	−	0.531049i
$186$		0			0
$187$	−2339.25	−	1350.57i	−0.914774	−	0.528145i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4534.50	+	2618.00i	1.71783	+	0.991789i	0.922858	+	0.385140i	$0.125847\pi$
$191$	4534.50	+	2618.00i	1.71783	+	0.991789i	0.794970	+	0.606649i	$0.207487\pi$
$192$		0			0
$193$	−2557.59	−	4429.88i	−0.953883	−	1.65217i	−0.736903	−	0.675998i	$-0.763713\pi$
$193$	−2557.59	−	4429.88i	−0.953883	−	1.65217i	−0.216980	−	0.976176i	$-0.569621\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3849.94i			1.39237i	0.717862	+	0.696186i	$0.245121\pi$
$197$			3849.94i			1.39237i	−0.717862	+	0.696186i	$0.754879\pi$
$198$		0			0
$199$	2104.82	−	1215.22i	0.749782	−	0.432887i	−0.0758329	−	0.997121i	$-0.524162\pi$
$199$	2104.82	−	1215.22i	0.749782	−	0.432887i	0.825615	+	0.564233i	$0.190828\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−166.473	+	288.340i	−0.0567170	+	0.0982366i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−451.562			−0.149451
$210$		0			0
$211$	4284.81			1.39800			0.699000	−	0.715121i	$-0.253629\pi$
$211$	4284.81			1.39800			0.699000	+	0.715121i	$0.253629\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	79.9053	−	138.400i	0.0253465	−	0.0439014i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4723.00	+	2726.83i	−1.43757	+	0.829983i
$222$		0			0
$223$			1484.34i			0.445735i	0.974849	+	0.222868i	$0.0715418\pi$
$223$			1484.34i			0.445735i	−0.974849	+	0.222868i	$0.928458\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2355.84	−	4080.44i	−0.688823	−	1.19308i	−0.972219	−	0.234073i	$-0.924794\pi$
$227$	−2355.84	−	4080.44i	−0.688823	−	1.19308i	0.283396	−	0.959003i	$-0.408539\pi$
$228$		0			0
$229$	−3382.78	−	1953.05i	−0.976160	−	0.563586i	−0.0750511	−	0.997180i	$-0.523912\pi$
$229$	−3382.78	−	1953.05i	−0.976160	−	0.563586i	−0.901108	+	0.433594i	$0.857245\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2409.55	−	1391.15i	−0.677489	−	0.391148i	0.121419	−	0.992601i	$-0.461255\pi$
$233$	−2409.55	−	1391.15i	−0.677489	−	0.391148i	−0.798908	+	0.601453i	$0.794589\pi$
$234$		0			0
$235$	−750.791	−	1300.41i	−0.208409	−	0.360976i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6237.16i			1.68807i	0.536289	+	0.844034i	$0.319826\pi$
$239$			6237.16i			1.68807i	−0.536289	+	0.844034i	$0.680174\pi$
$240$		0			0
$241$	5102.67	−	2946.03i	1.36387	−	0.787429i	0.373732	−	0.927537i	$-0.378078\pi$
$241$	5102.67	−	2946.03i	1.36387	−	0.787429i	0.990136	+	0.140107i	$0.0447448\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−455.858	+	789.569i	−0.117431	+	0.203397i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4252.57			−1.06940			−0.534701	−	0.845041i	$-0.679576\pi$
$251$	−4252.57			−1.06940			−0.534701	+	0.845041i	$0.679576\pi$
$252$		0			0
$253$	−5387.64			−1.33881
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3188.99	+	5523.49i	−0.774022	+	1.34065i	0.161320	+	0.986902i	$0.448425\pi$
$257$	−3188.99	+	5523.49i	−0.774022	+	1.34065i	−0.935342	+	0.353744i	$0.884908\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2608.95	−	1506.28i	0.611692	−	0.353160i	−0.161936	−	0.986801i	$-0.551774\pi$
$263$	2608.95	−	1506.28i	0.611692	−	0.353160i	0.773627	+	0.633641i	$0.218440\pi$
$264$		0			0
$265$		−	3131.99i		−	0.726024i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2392.65	−	4144.19i	−0.542314	−	0.939315i	−0.998771	−	0.0495695i	$-0.984215\pi$
$269$	−2392.65	−	4144.19i	−0.542314	−	0.939315i	0.456457	−	0.889746i	$-0.349118\pi$
$270$		0			0
$271$	880.131	+	508.144i	0.197285	+	0.113902i	0.595388	−	0.803438i	$-0.296998\pi$
$271$	880.131	+	508.144i	0.197285	+	0.113902i	−0.398104	+	0.917340i	$0.630332\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2807.50	+	1620.91i	0.615632	+	0.355435i
$276$		0			0
$277$	334.462	+	579.305i	0.0725483	+	0.125657i	0.900017	−	0.435854i	$-0.143554\pi$
$277$	334.462	+	579.305i	0.0725483	+	0.125657i	−0.827469	+	0.561511i	$0.810220\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2593.65i			0.550620i	0.961355	+	0.275310i	$0.0887805\pi$
$281$			2593.65i			0.550620i	−0.961355	+	0.275310i	$0.911219\pi$
$282$		0			0
$283$	5494.58	−	3172.30i	1.15413	−	0.666338i	0.204240	−	0.978921i	$-0.434528\pi$
$283$	5494.58	−	3172.30i	1.15413	−	0.666338i	0.949890	+	0.312583i	$0.101194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	123.111	−	213.234i	0.0250582	−	0.0434021i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1039.64			−0.207292			−0.103646	−	0.994614i	$-0.533051\pi$
$293$	−1039.64			−0.207292			−0.103646	+	0.994614i	$0.533051\pi$
$294$		0			0
$295$	−2552.39			−0.503748
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5438.89	+	9420.44i	−1.05197	+	1.82207i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2265.27	+	1307.86i	−0.425276	+	0.245533i
$306$		0			0
$307$		−	3398.31i		−	0.631765i	−0.948798	−	0.315882i	$-0.897699\pi$
$307$		−	3398.31i		−	0.631765i	0.948798	−	0.315882i	$-0.102301\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4756.89	−	8239.17i	−0.867326	−	1.50225i	−0.864719	−	0.502256i	$-0.832503\pi$
$311$	−4756.89	−	8239.17i	−0.867326	−	1.50225i	−0.00260717	−	0.999997i	$-0.500830\pi$
$312$		0			0
$313$	823.587	+	475.498i	0.148728	+	0.0858682i	0.572517	−	0.819893i	$-0.305967\pi$
$313$	823.587	+	475.498i	0.148728	+	0.0858682i	−0.423789	+	0.905761i	$0.639300\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6249.91	−	3608.39i	−1.10735	−	0.639329i	−0.169209	−	0.985580i	$-0.554121\pi$
$317$	−6249.91	−	3608.39i	−1.10735	−	0.639329i	−0.938142	+	0.346251i	$0.887454\pi$
$318$		0			0
$319$	184.311	+	319.237i	0.0323494	+	0.0560308i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			780.170i			0.134396i
$324$		0			0
$325$	5668.42	−	3272.67i	0.967469	−	0.558569i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4365.53	+	7561.32i	−0.724928	+	1.25561i	0.234075	+	0.972219i	$0.424794\pi$
$331$	−4365.53	+	7561.32i	−0.724928	+	1.25561i	−0.959003	+	0.283394i	$0.908540\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4950.71			0.807421
$336$		0			0
$337$	−5333.20			−0.862071			−0.431035	−	0.902335i	$-0.641852\pi$
$337$	−5333.20			−0.862071			−0.431035	+	0.902335i	$0.641852\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1597.67	−	2767.24i	0.253720	−	0.439456i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9172.37	+	5295.67i	−1.41902	+	0.819270i	−0.996213	−	0.0869514i	$-0.972288\pi$
$347$	−9172.37	+	5295.67i	−1.41902	+	0.819270i	−0.422804	+	0.906221i	$0.638954\pi$
$348$		0			0
$349$		−	4590.04i		−	0.704009i	−0.935998	−	0.352005i	$-0.885500\pi$
$349$		−	4590.04i		−	0.704009i	0.935998	−	0.352005i	$-0.114500\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−779.032	−	1349.32i	−0.117461	−	0.203448i	0.801300	−	0.598263i	$-0.204142\pi$
$353$	−779.032	−	1349.32i	−0.117461	−	0.203448i	−0.918761	+	0.394815i	$0.870809\pi$
$354$		0			0
$355$	−4542.11	−	2622.39i	−0.679071	−	0.392062i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	137.428	+	79.3440i	0.0202038	+	0.0116647i	0.510068	−	0.860134i	$-0.329620\pi$
$359$	137.428	+	79.3440i	0.0202038	+	0.0116647i	−0.489864	+	0.871799i	$0.662954\pi$
$360$		0			0
$361$	−3364.29	−	5827.12i	−0.490492	−	0.849558i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3962.39i			0.568221i
$366$		0			0
$367$	−5680.46	+	3279.61i	−0.807950	+	0.466470i	−0.846243	−	0.532797i	$-0.821141\pi$
$367$	−5680.46	+	3279.61i	−0.807950	+	0.466470i	0.0382936	+	0.999267i	$0.487808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2182.12	+	3779.53i	−0.302911	+	0.524656i	−0.976794	−	0.214181i	$-0.931292\pi$
$373$	−2182.12	+	3779.53i	−0.302911	+	0.524656i	0.673883	+	0.738838i	$0.264625\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	744.259			0.101675
$378$		0			0
$379$	2376.69			0.322117			0.161058	−	0.986945i	$-0.448509\pi$
$379$	2376.69			0.322117			0.161058	+	0.986945i	$0.448509\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−647.067	+	1120.75i	−0.0863279	+	0.149524i	−0.905956	−	0.423371i	$-0.860847\pi$
$383$	−647.067	+	1120.75i	−0.0863279	+	0.149524i	0.819628	+	0.572896i	$0.194180\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8314.04	+	4800.11i	−1.08365	+	0.625644i	−0.931878	−	0.362772i	$-0.881830\pi$
$389$	−8314.04	+	4800.11i	−1.08365	+	0.625644i	−0.151769	+	0.988416i	$0.548497\pi$
$390$		0			0
$391$			9308.29i			1.20394i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−150.312	−	260.349i	−0.0191469	−	0.0331635i
$396$		0			0
$397$	−2834.36	−	1636.42i	−0.358318	−	0.206875i	0.310025	−	0.950728i	$-0.399663\pi$
$397$	−2834.36	−	1636.42i	−0.358318	−	0.206875i	−0.668343	+	0.743854i	$0.732996\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12659.5	+	7308.97i	1.57652	+	0.910205i	0.995340	+	0.0964319i	$0.0307430\pi$
$401$	12659.5	+	7308.97i	1.57652	+	0.910205i	0.581182	+	0.813773i	$0.302590\pi$
$402$		0			0
$403$	−3225.73	−	5587.13i	−0.398722	−	0.690607i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			9302.64i			1.13296i
$408$		0			0
$409$	2613.35	−	1508.82i	0.315945	−	0.182411i	−0.333639	−	0.942701i	$-0.608277\pi$
$409$	2613.35	−	1508.82i	0.315945	−	0.182411i	0.649584	+	0.760290i	$0.274943\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−149.810	+	259.479i	−0.0177202	+	0.0306923i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7955.52			0.927572			0.463786	−	0.885947i	$-0.346491\pi$
$419$	7955.52			0.927572			0.463786	+	0.885947i	$0.346491\pi$
$420$		0			0
$421$	4238.67			0.490689			0.245344	−	0.969436i	$-0.421099\pi$
$421$	4238.67			0.490689			0.245344	+	0.969436i	$0.421099\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2800.47	−	4850.56i	0.319630	−	0.553616i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7717.51	−	4455.71i	0.862504	−	0.497967i	−0.00234585	−	0.999997i	$-0.500747\pi$
$431$	7717.51	−	4455.71i	0.862504	−	0.497967i	0.864850	+	0.502030i	$0.167413\pi$
$432$		0			0
$433$		−	427.546i		−	0.0474517i	−0.999719	−	0.0237258i	$-0.992447\pi$
$433$		−	427.546i		−	0.0474517i	0.999719	−	0.0237258i	$-0.00755287\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	778.058	+	1347.64i	0.0851707	+	0.147520i
$438$		0			0
$439$	−5009.56	−	2892.27i	−0.544631	−	0.314443i	0.202323	−	0.979319i	$-0.435151\pi$
$439$	−5009.56	−	2892.27i	−0.544631	−	0.314443i	−0.746954	+	0.664876i	$0.768484\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−175.946	−	101.583i	−0.0188701	−	0.0108947i	0.490535	−	0.871421i	$-0.336801\pi$
$443$	−175.946	−	101.583i	−0.0188701	−	0.0108947i	−0.509405	+	0.860527i	$0.670135\pi$
$444$		0			0
$445$	−1071.06	−	1855.13i	−0.114097	−	0.197622i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			609.140i			0.0640247i	0.999487	+	0.0320124i	$0.0101916\pi$
$449$			609.140i			0.0640247i	−0.999487	+	0.0320124i	$0.989808\pi$
$450$		0			0
$451$	1738.40	−	1003.66i	0.181503	−	0.104791i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6135.29	−	10626.6i	0.628001	−	1.08773i	−0.359951	−	0.932971i	$-0.617207\pi$
$457$	6135.29	−	10626.6i	0.628001	−	1.08773i	0.987952	−	0.154759i	$-0.0494601\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15165.2			−1.53213			−0.766066	−	0.642762i	$-0.777788\pi$
$461$	−15165.2			−1.53213			−0.766066	+	0.642762i	$0.777788\pi$
$462$		0			0
$463$	−14315.9			−1.43697			−0.718485	−	0.695542i	$-0.755164\pi$
$463$	−14315.9			−1.43697			−0.718485	+	0.695542i	$0.755164\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8482.12	+	14691.5i	−0.840483	+	1.45576i	0.0490032	+	0.998799i	$0.484396\pi$
$467$	−8482.12	+	14691.5i	−0.840483	+	1.45576i	−0.889487	+	0.456961i	$0.848938\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−834.411	+	481.748i	−0.0811126	+	0.0468304i
$474$		0			0
$475$		−	936.339i		−	0.0904467i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1799.92	+	3117.55i	0.171692	+	0.297379i	0.939011	−	0.343886i	$-0.111743\pi$
$479$	1799.92	+	3117.55i	0.171692	+	0.297379i	−0.767320	+	0.641265i	$0.778410\pi$
$480$		0			0
$481$	16265.9	+	9391.14i	1.54192	+	0.890226i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	624.199	+	360.381i	0.0584400	+	0.0337404i
$486$		0			0
$487$	2752.29	+	4767.10i	0.256094	+	0.443569i	0.965192	−	0.261542i	$-0.0842309\pi$
$487$	2752.29	+	4767.10i	0.256094	+	0.443569i	−0.709098	+	0.705110i	$0.750898\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	13643.7i		−	1.25404i	−0.779003	−	0.627020i	$-0.784274\pi$
$491$		−	13643.7i		−	1.25404i	0.779003	−	0.627020i	$-0.215726\pi$
$492$		0			0
$493$	551.549	−	318.437i	0.0503865	−	0.0290906i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	8536.31	−	14785.3i	0.765807	−	1.32642i	−0.174011	−	0.984744i	$-0.555673\pi$
$499$	8536.31	−	14785.3i	0.765807	−	1.32642i	0.939819	−	0.341674i	$-0.110994\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7574.12			−0.671398			−0.335699	−	0.941969i	$-0.608972\pi$
$503$	−7574.12			−0.671398			−0.335699	+	0.941969i	$0.608972\pi$
$504$		0			0
$505$	12968.5			1.14275
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8460.09	−	14653.3i	0.736713	−	1.27602i	−0.217255	−	0.976115i	$-0.569710\pi$
$509$	8460.09	−	14653.3i	0.736713	−	1.27602i	0.953968	−	0.299909i	$-0.0969564\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9962.22	−	5751.69i	0.852403	−	0.492135i
$516$		0			0
$517$			9053.02i			0.770119i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10540.2	−	18256.2i	−0.886326	−	1.53516i	−0.844186	−	0.536051i	$-0.819916\pi$
$521$	−10540.2	−	18256.2i	−0.886326	−	1.53516i	−0.0421406	−	0.999112i	$-0.513418\pi$
$522$		0			0
$523$	−15348.8	−	8861.62i	−1.28328	−	0.740902i	−0.305833	−	0.952085i	$-0.598935\pi$
$523$	−15348.8	−	8861.62i	−1.28328	−	0.740902i	−0.977446	+	0.211183i	$0.932268\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4780.99	−	2760.31i	−0.395187	−	0.228161i
$528$		0			0
$529$	3199.60	+	5541.87i	0.262974	+	0.455484i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4052.85i		−	0.329359i
$534$		0			0
$535$	4120.91	−	2379.21i	0.333014	−	0.192266i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	383.271	−	663.844i	0.0304586	−	0.0527558i	−0.850394	−	0.526146i	$-0.823637\pi$
$541$	383.271	−	663.844i	0.0304586	−	0.0527558i	0.880853	+	0.473390i	$0.156970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3902.17			−0.306699
$546$		0			0
$547$	−6579.52			−0.514296			−0.257148	−	0.966372i	$-0.582783\pi$
$547$	−6579.52			−0.514296			−0.257148	+	0.966372i	$0.582783\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	53.2348	−	92.2054i	0.00411593	−	0.00712901i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7811.71	−	4510.09i	0.594242	−	0.343086i	−0.172531	−	0.985004i	$-0.555194\pi$
$557$	7811.71	−	4510.09i	0.594242	−	0.343086i	0.766773	+	0.641918i	$0.221861\pi$
$558$		0			0
$559$			1945.32i			0.147189i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10544.4	+	18263.4i	0.789328	+	1.36716i	0.926379	+	0.376592i	$0.122904\pi$
$563$	10544.4	+	18263.4i	0.789328	+	1.36716i	−0.137051	+	0.990564i	$0.543763\pi$
$564$		0			0
$565$	−1767.86	−	1020.67i	−0.131636	−	0.0760001i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13686.2	−	7901.71i	−1.00835	−	0.582174i	−0.0976451	−	0.995221i	$-0.531131\pi$
$569$	−13686.2	−	7901.71i	−1.00835	−	0.582174i	−0.910709	+	0.413048i	$0.864464\pi$
$570$		0			0
$571$	13091.9	+	22675.8i	0.959505	+	1.66191i	0.723706	+	0.690109i	$0.242437\pi$
$571$	13091.9	+	22675.8i	0.959505	+	1.66191i	0.235799	+	0.971802i	$0.424229\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	11171.6i		−	0.810237i
$576$		0			0
$577$	−20391.8	+	11773.2i	−1.47127	+	0.849439i	−0.999479	−	0.0322714i	$-0.989726\pi$
$577$	−20391.8	+	11773.2i	−1.47127	+	0.849439i	−0.471792	+	0.881710i	$0.656393\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−9441.35	+	16352.9i	−0.670705	+	1.16169i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12296.8			0.864640			0.432320	−	0.901720i	$-0.357695\pi$
$587$	12296.8			0.864640			0.432320	+	0.901720i	$0.357695\pi$
$588$		0			0
$589$	−922.911			−0.0645634
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4871.98	−	8438.52i	0.337383	−	0.584365i	−0.646556	−	0.762866i	$-0.723791\pi$
$593$	4871.98	−	8438.52i	0.337383	−	0.584365i	0.983940	+	0.178501i	$0.0571248\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3700.30	−	2136.37i	0.252404	−	0.145726i	−0.368460	−	0.929643i	$-0.620115\pi$
$599$	3700.30	−	2136.37i	0.252404	−	0.145726i	0.620865	+	0.783918i	$0.286782\pi$
$600$		0			0
$601$		−	28823.5i		−	1.95630i	−0.207903	−	0.978149i	$-0.566664\pi$
$601$		−	28823.5i		−	1.95630i	0.207903	−	0.978149i	$-0.433336\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	762.124	+	1320.04i	0.0512145	+	0.0887060i
$606$		0			0
$607$	7006.35	+	4045.12i	0.468499	+	0.270488i	0.715611	−	0.698499i	$-0.246148\pi$
$607$	7006.35	+	4045.12i	0.468499	+	0.270488i	−0.247112	+	0.968987i	$0.579482\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15829.5	+	9139.14i	1.04810	+	0.605123i
$612$		0			0
$613$	−8498.69	−	14720.2i	−0.559966	−	0.969889i	−0.997499	−	0.0706867i	$-0.977481\pi$
$613$	−8498.69	−	14720.2i	−0.559966	−	0.969889i	0.437533	−	0.899202i	$-0.355852\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4408.19i			0.287629i	0.989605	+	0.143815i	$0.0459369\pi$
$617$			4408.19i			0.287629i	−0.989605	+	0.143815i	$0.954063\pi$
$618$		0			0
$619$	−13169.9	+	7603.66i	−0.855160	+	0.493727i	−0.862389	−	0.506247i	$-0.831032\pi$
$619$	−13169.9	+	7603.66i	−0.855160	+	0.493727i	0.00722832	+	0.999974i	$0.497699\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−672.826	+	1165.37i	−0.0430608	+	0.0745836i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16072.3			1.01883
$630$		0			0
$631$	17710.9			1.11737			0.558685	−	0.829380i	$-0.311306\pi$
$631$	17710.9			1.11737			0.558685	+	0.829380i	$0.311306\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7860.40	−	13614.6i	0.491229	−	0.850834i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9213.97	+	5319.69i	−0.567753	+	0.327793i	−0.756252	−	0.654281i	$-0.772971\pi$
$641$	−9213.97	+	5319.69i	−0.567753	+	0.327793i	0.188498	+	0.982074i	$0.439638\pi$
$642$		0			0
$643$			14402.2i			0.883306i	0.897186	+	0.441653i	$0.145608\pi$
$643$			14402.2i			0.883306i	−0.897186	+	0.441653i	$0.854392\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13769.9	+	23850.2i	0.836710	+	1.44922i	0.892630	+	0.450789i	$0.148857\pi$
$647$	13769.9	+	23850.2i	0.836710	+	1.44922i	−0.0559201	+	0.998435i	$0.517809\pi$
$648$		0			0
$649$	13326.7	+	7694.15i	0.806036	+	0.465365i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13163.2	−	7599.77i	−0.788845	−	0.455440i	0.0507109	−	0.998713i	$-0.483851\pi$
$653$	−13163.2	−	7599.77i	−0.788845	−	0.455440i	−0.839556	+	0.543274i	$0.817185\pi$
$654$		0			0
$655$	−4375.00	−	7577.72i	−0.260985	−	0.452040i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	17308.4i		−	1.02313i	−0.859246	−	0.511563i	$-0.829067\pi$
$659$		−	17308.4i		−	1.02313i	0.859246	−	0.511563i	$-0.170933\pi$
$660$		0			0
$661$	−16408.1	+	9473.20i	−0.965507	+	0.557436i	−0.897863	−	0.440274i	$-0.854881\pi$
$661$	−16408.1	+	9473.20i	−0.965507	+	0.557436i	−0.0676432	+	0.997710i	$0.521548\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	635.151	−	1100.11i	0.0368713	−	0.0638629i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15770.1			0.907299
$672$		0			0
$673$	992.154			0.0568273			0.0284136	−	0.999596i	$-0.490954\pi$
$673$	992.154			0.0568273			0.0284136	+	0.999596i	$0.490954\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10190.7	−	17650.9i	0.578526	−	1.00204i	−0.417123	−	0.908850i	$-0.636962\pi$
$677$	10190.7	−	17650.9i	0.578526	−	1.00204i	0.995649	−	0.0931857i	$-0.0297050\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14531.3	−	8389.63i	0.814090	−	0.470015i	−0.0342842	−	0.999412i	$-0.510915\pi$
$683$	14531.3	−	8389.63i	0.814090	−	0.470015i	0.848374	+	0.529397i	$0.177582\pi$
$684$		0			0
$685$		−	7447.75i		−	0.415422i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	19062.3	+	33017.0i	1.05402	+	1.82561i
$690$		0			0
$691$	13082.6	+	7553.27i	0.720242	+	0.415832i	0.814842	−	0.579684i	$-0.196824\pi$
$691$	13082.6	+	7553.27i	0.720242	+	0.415832i	−0.0945999	+	0.995515i	$0.530157\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10617.3	+	6129.91i	0.579479	+	0.334562i
$696$		0			0
$697$	−1734.04	−	3003.45i	−0.0942346	−	0.163219i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	4771.58i		−	0.257090i	−0.991704	−	0.128545i	$-0.958969\pi$
$701$		−	4771.58i		−	0.257090i	0.991704	−	0.128545i	$-0.0410306\pi$
$702$		0			0
$703$	2326.91	−	1343.44i	0.124838	−	0.0720753i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3189.38	−	5524.17i	0.168942	−	0.292616i	−0.769106	−	0.639121i	$-0.779298\pi$
$709$	3189.38	−	5524.17i	0.168942	−	0.292616i	0.938048	+	0.346505i	$0.112632\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11011.4			−0.578371
$714$		0			0
$715$	20701.8			1.08280
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13420.0	+	23244.1i	−0.696079	+	1.20564i	0.273736	+	0.961805i	$0.411741\pi$
$719$	−13420.0	+	23244.1i	−0.696079	+	1.20564i	−0.969815	+	0.243840i	$0.921593\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−661.955	+	382.180i	−0.0339095	+	0.0195777i
$726$		0			0
$727$			13222.5i			0.674547i	0.941407	+	0.337273i	$0.109505\pi$
$727$			13222.5i			0.674547i	−0.941407	+	0.337273i	$0.890495\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	832.322	+	1441.62i	0.0421129	+	0.0729417i
$732$		0			0
$733$	18543.8	+	10706.2i	0.934420	+	0.539488i	0.888207	−	0.459444i	$-0.151951\pi$
$733$	18543.8	+	10706.2i	0.934420	+	0.539488i	0.0462131	+	0.998932i	$0.485285\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25848.9	−	14923.9i	−1.29194	−	0.745900i
$738$		0			0
$739$	17088.5	+	29598.2i	0.850624	+	1.47332i	0.880646	+	0.473775i	$0.157109\pi$
$739$	17088.5	+	29598.2i	0.850624	+	1.47332i	−0.0300220	+	0.999549i	$0.509558\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	23205.8i		−	1.14581i	−0.819621	−	0.572906i	$-0.805816\pi$
$743$		−	23205.8i		−	1.14581i	0.819621	−	0.572906i	$-0.194184\pi$
$744$		0			0
$745$	−5551.26	+	3205.02i	−0.272996	+	0.157615i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9232.97	+	15992.0i	−0.448623	+	0.777038i	−0.998297	−	0.0583418i	$-0.981419\pi$
$751$	−9232.97	+	15992.0i	−0.448623	+	0.777038i	0.549674	+	0.835379i	$0.314752\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6787.33			0.327174
$756$		0			0
$757$	13606.4			0.653280			0.326640	−	0.945149i	$-0.394084\pi$
$757$	13606.4			0.653280			0.326640	+	0.945149i	$0.394084\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9008.18	−	15602.6i	0.429101	−	0.743225i	−0.567692	−	0.823241i	$-0.692164\pi$
$761$	9008.18	−	15602.6i	0.429101	−	0.743225i	0.996794	+	0.0800155i	$0.0254970\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26906.9	−	15534.7i	1.26669	−	0.731324i
$768$		0			0
$769$			21716.1i			1.01834i	0.860666	+	0.509169i	$0.170047\pi$
$769$			21716.1i			1.01834i	−0.860666	+	0.509169i	$0.829953\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18105.0	+	31358.7i	0.842420	+	1.45911i	0.887843	+	0.460146i	$0.152203\pi$
$773$	18105.0	+	31358.7i	0.842420	+	1.45911i	−0.0454230	+	0.998968i	$0.514464\pi$
$774$		0			0
$775$	5738.02	+	3312.85i	0.265956	+	0.153550i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−502.102	−	289.889i	−0.0230933	−	0.0133329i
$780$		0			0
$781$	15810.4	+	27384.3i	0.724378	+	1.25466i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			409.119i			0.0186014i
$786$		0			0
$787$	13622.5	−	7864.94i	0.617012	−	0.356232i	−0.158693	−	0.987328i	$-0.550728\pi$
$787$	13622.5	−	7864.94i	0.617012	−	0.356232i	0.775705	+	0.631096i	$0.217395\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	15920.1	−	27574.5i	0.712913	−	1.23480i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31098.5			−1.38214			−0.691070	−	0.722787i	$-0.742861\pi$
$797$	−31098.5			−1.38214			−0.691070	+	0.722787i	$0.742861\pi$
$798$		0			0
$799$	15641.0			0.692540
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11944.6	−	20688.6i	0.524926	−	0.909198i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22218.0	−	12827.6i	0.965568	−	0.557471i	0.0676861	−	0.997707i	$-0.478438\pi$
$809$	22218.0	−	12827.6i	0.965568	−	0.557471i	0.897882	+	0.440235i	$0.145105\pi$
$810$		0			0
$811$			17351.0i			0.751263i	0.926769	+	0.375632i	$0.122574\pi$
$811$			17351.0i			0.751263i	−0.926769	+	0.375632i	$0.877426\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1304.39	−	2259.27i	−0.0560623	−	0.0971027i
$816$		0			0
$817$	241.004	+	139.144i	0.0103203	+	0.00595841i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20178.2	−	11649.9i	−0.857765	−	0.495231i	0.00549814	−	0.999985i	$-0.498250\pi$
$821$	−20178.2	−	11649.9i	−0.857765	−	0.495231i	−0.863263	+	0.504754i	$0.831583\pi$
$822$		0			0
$823$	7212.13	+	12491.8i	0.305466	+	0.529083i	0.977365	−	0.211560i	$-0.0678543\pi$
$823$	7212.13	+	12491.8i	0.305466	+	0.529083i	−0.671899	+	0.740643i	$0.734521\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	11516.8i		−	0.484255i	−0.970244	−	0.242127i	$-0.922155\pi$
$827$		−	11516.8i		−	0.484255i	0.970244	−	0.242127i	$-0.0778452\pi$
$828$		0			0
$829$	8122.14	−	4689.32i	0.340282	−	0.196462i	−0.320115	−	0.947379i	$-0.603722\pi$
$829$	8122.14	−	4689.32i	0.340282	−	0.196462i	0.660397	+	0.750917i	$0.270388\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−6563.87	+	11369.0i	−0.272039	+	0.471185i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−31050.7			−1.27770			−0.638848	−	0.769333i	$-0.720589\pi$
$839$	−31050.7			−1.27770			−0.638848	+	0.769333i	$0.720589\pi$
$840$		0			0
$841$	24302.1			0.996436
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13694.5	−	23719.5i	0.557519	−	0.965652i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	27762.7	−	16028.8i	1.11832	−	0.645663i
$852$		0			0
$853$		−	41308.2i		−	1.65811i	−0.559170	−	0.829053i	$-0.688880\pi$
$853$		−	41308.2i		−	1.65811i	0.559170	−	0.829053i	$-0.311120\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13700.1	+	23729.3i	0.546077	+	0.945832i	0.998538	+	0.0540486i	$0.0172126\pi$
$857$	13700.1	+	23729.3i	0.546077	+	0.945832i	−0.452462	+	0.891784i	$0.649454\pi$
$858$		0			0
$859$	−3112.62	−	1797.07i	−0.123633	−	0.0713798i	0.436908	−	0.899506i	$-0.356074\pi$
$859$	−3112.62	−	1797.07i	−0.123633	−	0.0713798i	−0.560541	+	0.828126i	$0.689407\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3029.24	+	1748.93i	0.119486	+	0.0689852i	0.558552	−	0.829470i	$-0.311357\pi$
$863$	3029.24	+	1748.93i	0.119486	+	0.0689852i	−0.439066	+	0.898455i	$0.644691\pi$
$864$		0			0
$865$	−6557.94	−	11358.7i	−0.257776	−	0.446482i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1812.46i			0.0707521i
$870$		0			0
$871$	−52189.6	+	30131.7i	−2.03029	+	1.17219i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	21247.7	−	36802.1i	0.818111	−	1.41701i	−0.0889617	−	0.996035i	$-0.528355\pi$
$877$	21247.7	−	36802.1i	0.818111	−	1.41701i	0.907072	−	0.420974i	$-0.138312\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5554.52			0.212414			0.106207	−	0.994344i	$-0.466129\pi$
$881$	5554.52			0.212414			0.106207	+	0.994344i	$0.466129\pi$
$882$		0			0
$883$	−1152.28			−0.0439154			−0.0219577	−	0.999759i	$-0.506990\pi$
$883$	−1152.28			−0.0439154			−0.0219577	+	0.999759i	$0.506990\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18868.8	+	32681.7i	−0.714264	+	1.23714i	0.248979	+	0.968509i	$0.419905\pi$
$887$	−18868.8	+	32681.7i	−0.714264	+	1.23714i	−0.963243	+	0.268632i	$0.913428\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2264.48	−	1307.40i	0.0848575	−	0.0489925i
$894$		0			0
$895$		−	24909.3i		−	0.930308i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	376.699	+	652.461i	0.0139751	+	0.0242056i
$900$		0			0
$901$	28253.1	+	16311.9i	1.04467	+	0.603141i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14431.4	−	8331.99i	−0.530074	−	0.306038i
$906$		0			0
$907$	14064.8	+	24360.9i	0.514900	+	0.891832i	0.999851	+	0.0172909i	$0.00550414\pi$
$907$	14064.8	+	24360.9i	0.514900	+	0.891832i	−0.484951	+	0.874541i	$0.661163\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	35517.2i		−	1.29170i	−0.763465	−	0.645849i	$-0.776503\pi$
$911$		−	35517.2i		−	1.29170i	0.763465	−	0.645849i	$-0.223497\pi$
$912$		0			0
$913$	1564.39	−	903.202i	0.0567074	−	0.0327400i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−9544.14	+	16530.9i	−0.342581	+	0.593368i	−0.984911	−	0.173060i	$-0.944634\pi$
$919$	−9544.14	+	16530.9i	−0.342581	+	0.593368i	0.642330	+	0.766428i	$0.277968\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	63843.1			2.27673
$924$		0			0
$925$	−19289.5			−0.685660
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22986.0	−	39813.0i	0.811784	−	1.40605i	−0.0998306	−	0.995004i	$-0.531830\pi$
$929$	22986.0	−	39813.0i	0.811784	−	1.40605i	0.911614	−	0.411046i	$-0.134837\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15341.5	−	8857.44i	0.536601	−	0.309807i
$936$		0			0
$937$			10653.2i			0.371423i	0.982604	+	0.185712i	$0.0594591\pi$
$937$			10653.2i			0.371423i	−0.982604	+	0.185712i	$0.940541\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3480.93	+	6029.15i	0.120590	+	0.208868i	0.920000	−	0.391917i	$-0.128188\pi$
$941$	3480.93	+	6029.15i	0.120590	+	0.208868i	−0.799411	+	0.600785i	$0.794855\pi$
$942$		0			0
$943$	−5990.64	−	3458.70i	−0.206874	−	0.119439i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28060.9	+	16200.9i	0.962889	+	0.555924i	0.897061	−	0.441907i	$-0.145698\pi$
$947$	28060.9	+	16200.9i	0.962889	+	0.555924i	0.0658278	+	0.997831i	$0.479031\pi$
$948$		0			0
$949$	−24116.4	−	41770.9i	−0.824924	−	1.42881i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	27432.4i		−	0.932448i	−0.884667	−	0.466224i	$-0.845614\pi$
$953$		−	27432.4i		−	0.932448i	0.884667	−	0.466224i	$-0.154386\pi$
$954$		0			0
$955$	−29738.7	+	17169.7i	−1.00767	+	0.581777i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−11630.2	+	20144.0i	−0.390392	+	0.676178i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	33547.0			1.11908
$966$		0			0
$967$	43449.7			1.44493			0.722465	−	0.691408i	$-0.243009\pi$
$967$	43449.7			1.44493			0.722465	+	0.691408i	$0.243009\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5273.30	−	9133.63i	0.174283	−	0.301866i	−0.765630	−	0.643281i	$-0.777573\pi$
$971$	5273.30	−	9133.63i	0.174283	−	0.301866i	0.939913	+	0.341415i	$0.110906\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30648.2	+	17694.8i	−1.00361	+	0.579433i	−0.909313	−	0.416112i	$-0.863392\pi$
$977$	−30648.2	+	17694.8i	−1.00361	+	0.579433i	−0.0942933	+	0.995544i	$0.530059\pi$
$978$		0			0
$979$			12914.8i			0.421614i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	13990.4	+	24232.1i	0.453942	+	0.786251i	0.998627	−	0.0523900i	$-0.0166839\pi$
$983$	13990.4	+	24232.1i	0.453942	+	0.786251i	−0.544684	+	0.838641i	$0.683351\pi$
$984$		0			0
$985$	−21866.4	−	12624.6i	−0.707332	−	0.408378i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2875.44	+	1660.14i	0.0924507	+	0.0533765i
$990$		0			0
$991$	5288.79	+	9160.46i	0.169530	+	0.293634i	0.938255	−	0.345945i	$-0.112442\pi$
$991$	5288.79	+	9160.46i	0.169530	+	0.293634i	−0.768725	+	0.639580i	$0.779108\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			15939.6i			0.507858i
$996$		0			0
$997$	−23948.3	+	13826.6i	−0.760734	+	0.439210i	−0.829559	−	0.558419i	$-0.811408\pi$
$997$	−23948.3	+	13826.6i	−0.760734	+	0.439210i	0.0688254	+	0.997629i	$0.478075\pi$
$998$		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.1097.9		48
3.2	odd	2	inner	1764.4.t.c.1097.16		48
7.2	even	3		1764.4.f.b.881.15	yes	24
7.3	odd	6	inner	1764.4.t.c.521.16		48
7.4	even	3	inner	1764.4.t.c.521.10		48
7.5	odd	6		1764.4.f.b.881.9	✓	24
7.6	odd	2	inner	1764.4.t.c.1097.15		48
21.2	odd	6		1764.4.f.b.881.10	yes	24
21.5	even	6		1764.4.f.b.881.16	yes	24
21.11	odd	6	inner	1764.4.t.c.521.15		48
21.17	even	6	inner	1764.4.t.c.521.9		48
21.20	even	2	inner	1764.4.t.c.1097.10		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.f.b.881.9	✓	24	7.5	odd	6
1764.4.f.b.881.10	yes	24	21.2	odd	6
1764.4.f.b.881.15	yes	24	7.2	even	3
1764.4.f.b.881.16	yes	24	21.5	even	6
1764.4.t.c.521.9		48	21.17	even	6	inner
1764.4.t.c.521.10		48	7.4	even	3	inner
1764.4.t.c.521.15		48	21.11	odd	6	inner
1764.4.t.c.521.16		48	7.3	odd	6	inner
1764.4.t.c.1097.9		48	1.1	even	1	trivial
1764.4.t.c.1097.10		48	21.20	even	2	inner
1764.4.t.c.1097.15		48	7.6	odd	2	inner
1764.4.t.c.1097.16		48	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-3.27916 + 5.67967i) q^{5} +O(q^{10})\)	\(q+(-3.27916 + 5.67967i) q^{5} +(34.2427 - 19.7700i) q^{11} -79.8324i q^{13} +(-34.1569 - 59.1615i) q^{17} +(-9.89034 - 5.71019i) q^{19} +(-118.003 - 68.1289i) q^{23} +(40.9942 + 71.0040i) q^{25} +9.32277i q^{29} +(69.9858 - 40.4063i) q^{31} +(-117.636 + 203.751i) q^{37} +50.7669 q^{41} -24.3676 q^{43} +(-114.479 + 198.284i) q^{47} +(-413.578 + 238.780i) q^{53} +259.316i q^{55} +(194.591 + 337.042i) q^{59} +(345.404 + 199.419i) q^{61} +(453.422 + 261.783i) q^{65} +(-377.437 - 653.740i) q^{67} +799.714i q^{71} +(523.232 - 302.088i) q^{73} +(-22.9193 + 39.6975i) q^{79} +45.6855 q^{83} +448.024 q^{85} +(-163.313 + 282.867i) q^{89} +(64.8640 - 37.4493i) q^{95} -109.900i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more