LMFDB - Embedded newform 2160.2.f.n.1729.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(1729,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2160.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.f (of order $2$, degree $1$, 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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.3
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2160.1729
Dual form			2160.2.f.n.1729.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22474 - 0.224745i) q^{5} +1.00000i q^{7} -0.449490 q^{11} +3.89898i q^{13} +4.89898i q^{17} -5.89898 q^{19} +4.44949i q^{23} +(4.89898 - 1.00000i) q^{25} -0.449490 q^{29} -6.00000 q^{31} +(0.224745 + 2.22474i) q^{35} -0.101021i q^{37} -9.34847 q^{41} +6.00000i q^{43} +4.89898i q^{47} +6.00000 q^{49} +4.44949i q^{53} +(-1.00000 + 0.101021i) q^{55} -4.89898 q^{59} +8.79796 q^{61} +(0.876276 + 8.67423i) q^{65} -14.7980i q^{67} +11.5505 q^{71} -3.89898i q^{73} -0.449490i q^{77} -3.89898 q^{79} +7.55051i q^{83} +(1.10102 + 10.8990i) q^{85} +12.0000 q^{89} -3.89898 q^{91} +(-13.1237 + 1.32577i) q^{95} -15.8990i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 8 q^{11} - 4 q^{19} + 8 q^{29} - 24 q^{31} - 4 q^{35} - 8 q^{41} + 24 q^{49} - 4 q^{55} - 4 q^{61} + 28 q^{65} + 56 q^{71} + 4 q^{79} + 24 q^{85} + 48 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) $ 4 * q + 4 * q^5 + 8 * q^11 - 4 * q^19 + 8 * q^29 - 24 * q^31 - 4 * q^35 - 8 * q^41 + 24 * q^49 - 4 * q^55 - 4 * q^61 + 28 * q^65 + 56 * q^71 + 4 * q^79 + 24 * q^85 + 48 * q^89 + 4 * q^91 - 28 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	2.22474	−	0.224745i	0.994936	−	0.100509i
$5$	2.22474	−	0.224745i	0.994936	−	0.100509i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.449490			−0.135526			−0.0677631	−	0.997701i	$-0.521586\pi$
$11$	−0.449490			−0.135526			−0.0677631	+	0.997701i	$0.521586\pi$
$12$		0			0
$13$			3.89898i			1.08138i	0.841221	+	0.540691i	$0.181837\pi$
$13$			3.89898i			1.08138i	−0.841221	+	0.540691i	$0.818163\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$	−5.89898			−1.35332			−0.676659	−	0.736296i	$-0.736573\pi$
$19$	−5.89898			−1.35332			−0.676659	+	0.736296i	$0.736573\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.44949i			0.927783i	0.885892	+	0.463891i	$0.153547\pi$
$23$			4.44949i			0.927783i	−0.885892	+	0.463891i	$0.846453\pi$
$24$		0			0
$25$	4.89898	−	1.00000i	0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.449490			−0.0834681			−0.0417341	−	0.999129i	$-0.513288\pi$
$29$	−0.449490			−0.0834681			−0.0417341	+	0.999129i	$0.513288\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.224745	+	2.22474i	0.0379888	+	0.376051i
$36$		0			0
$37$		−	0.101021i		−	0.0166077i	−0.999966	−	0.00830384i	$-0.997357\pi$
$37$		−	0.101021i		−	0.0166077i	0.999966	−	0.00830384i	$-0.00264322\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.34847			−1.45999			−0.729993	−	0.683455i	$-0.760477\pi$
$41$	−9.34847			−1.45999			−0.729993	+	0.683455i	$0.760477\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.89898i			0.714590i	0.933992	+	0.357295i	$0.116301\pi$
$47$			4.89898i			0.714590i	−0.933992	+	0.357295i	$0.883699\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.44949i			0.611184i	0.952162	+	0.305592i	$0.0988544\pi$
$53$			4.44949i			0.611184i	−0.952162	+	0.305592i	$0.901146\pi$
$54$		0			0
$55$	−1.00000	+	0.101021i	−0.134840	+	0.0136216i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$	8.79796			1.12646			0.563232	−	0.826299i	$-0.309558\pi$
$61$	8.79796			1.12646			0.563232	+	0.826299i	$0.309558\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.876276	+	8.67423i	0.108689	+	1.07591i
$66$		0			0
$67$		−	14.7980i		−	1.80786i	−0.427682	−	0.903929i	$-0.640670\pi$
$67$		−	14.7980i		−	1.80786i	0.427682	−	0.903929i	$-0.359330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.5505			1.37079			0.685397	−	0.728170i	$-0.259629\pi$
$71$	11.5505			1.37079			0.685397	+	0.728170i	$0.259629\pi$
$72$		0			0
$73$		−	3.89898i		−	0.456341i	−0.973621	−	0.228171i	$-0.926726\pi$
$73$		−	3.89898i		−	0.456341i	0.973621	−	0.228171i	$-0.0732744\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.449490i		−	0.0512241i
$78$		0			0
$79$	−3.89898			−0.438669			−0.219335	−	0.975650i	$-0.570389\pi$
$79$	−3.89898			−0.438669			−0.219335	+	0.975650i	$0.570389\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.55051i			0.828776i	0.910100	+	0.414388i	$0.136004\pi$
$83$			7.55051i			0.828776i	−0.910100	+	0.414388i	$0.863996\pi$
$84$		0			0
$85$	1.10102	+	10.8990i	0.119422	+	1.18216i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$	−3.89898			−0.408724
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.1237	+	1.32577i	−1.34647	+	0.136021i
$96$		0			0
$97$		−	15.8990i		−	1.61430i	−0.590349	−	0.807148i	$-0.701010\pi$
$97$		−	15.8990i		−	1.61430i	0.590349	−	0.807148i	$-0.298990\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.10102			−0.706578			−0.353289	−	0.935514i	$-0.614937\pi$
$101$	−7.10102			−0.706578			−0.353289	+	0.935514i	$0.614937\pi$
$102$		0			0
$103$		−	10.7980i		−	1.06395i	−0.846759	−	0.531977i	$-0.821449\pi$
$103$		−	10.7980i		−	1.06395i	0.846759	−	0.531977i	$-0.178551\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.2474i			1.37735i	0.725069	+	0.688676i	$0.241808\pi$
$107$			14.2474i			1.37735i	−0.725069	+	0.688676i	$0.758192\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.3485i			1.63201i	0.578047	+	0.816003i	$0.303815\pi$
$113$			17.3485i			1.63201i	−0.578047	+	0.816003i	$0.696185\pi$
$114$		0			0
$115$	1.00000	+	9.89898i	0.0932505	+	0.923085i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.89898			−0.449089
$120$		0			0
$121$	−10.7980			−0.981633
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.6742	−	3.32577i	0.954733	−	0.297465i
$126$		0			0
$127$			3.79796i			0.337014i	0.985700	+	0.168507i	$0.0538946\pi$
$127$			3.79796i			0.337014i	−0.985700	+	0.168507i	$0.946105\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.10102			0.620419			0.310210	−	0.950668i	$-0.399601\pi$
$131$	7.10102			0.620419			0.310210	+	0.950668i	$0.399601\pi$
$132$		0			0
$133$		−	5.89898i		−	0.511506i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			17.3485i			1.48218i	0.671406	+	0.741090i	$0.265691\pi$
$137$			17.3485i			1.48218i	−0.671406	+	0.741090i	$0.734309\pi$
$138$		0			0
$139$	9.69694			0.822484			0.411242	−	0.911526i	$-0.365095\pi$
$139$	9.69694			0.822484			0.411242	+	0.911526i	$0.365095\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1.75255i		−	0.146556i
$144$		0			0
$145$	−1.00000	+	0.101021i	−0.0830455	+	0.00838930i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.79796			−0.802680			−0.401340	−	0.915929i	$-0.631455\pi$
$149$	−9.79796			−0.802680			−0.401340	+	0.915929i	$0.631455\pi$
$150$		0			0
$151$	6.10102			0.496494			0.248247	−	0.968697i	$-0.420146\pi$
$151$	6.10102			0.496494			0.248247	+	0.968697i	$0.420146\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.3485	+	1.34847i	−1.07217	+	0.108312i
$156$		0			0
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.44949			−0.350669
$162$		0			0
$163$		−	1.00000i		−	0.0783260i	−0.999233	−	0.0391630i	$-0.987531\pi$
$163$		−	1.00000i		−	0.0783260i	0.999233	−	0.0391630i	$-0.0124692\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.20204i			0.170399i	0.996364	+	0.0851995i	$0.0271528\pi$
$167$			2.20204i			0.170399i	−0.996364	+	0.0851995i	$0.972847\pi$
$168$		0			0
$169$	−2.20204			−0.169388
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	16.8990i		−	1.28481i	−0.766367	−	0.642403i	$-0.777938\pi$
$173$		−	16.8990i		−	1.28481i	0.766367	−	0.642403i	$-0.222062\pi$
$174$		0			0
$175$	1.00000	+	4.89898i	0.0755929	+	0.370328i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.34847			0.399763			0.199882	−	0.979820i	$-0.435944\pi$
$179$	5.34847			0.399763			0.199882	+	0.979820i	$0.435944\pi$
$180$		0			0
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.0227038	−	0.224745i	−0.00166922	−	0.0165236i
$186$		0			0
$187$		−	2.20204i		−	0.161029i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	21.7980			1.57724			0.788622	−	0.614878i	$-0.210795\pi$
$191$	21.7980			1.57724			0.788622	+	0.614878i	$0.210795\pi$
$192$		0			0
$193$			23.8990i			1.72029i	0.510053	+	0.860143i	$0.329626\pi$
$193$			23.8990i			1.72029i	−0.510053	+	0.860143i	$0.670374\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	21.6969			1.53806			0.769028	−	0.639216i	$-0.220741\pi$
$199$	21.6969			1.53806			0.769028	+	0.639216i	$0.220741\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	0.449490i		−	0.0315480i
$204$		0			0
$205$	−20.7980	+	2.10102i	−1.45259	+	0.146742i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.65153			0.183410
$210$		0			0
$211$	7.69694			0.529879			0.264940	−	0.964265i	$-0.414648\pi$
$211$	7.69694			0.529879			0.264940	+	0.964265i	$0.414648\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.34847	+	13.3485i	0.0919648	+	0.910358i
$216$		0			0
$217$		−	6.00000i		−	0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−19.1010			−1.28487
$222$		0			0
$223$			25.5959i			1.71403i	0.515292	+	0.857015i	$0.327684\pi$
$223$			25.5959i			1.71403i	−0.515292	+	0.857015i	$0.672316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	19.1464i		−	1.27079i	−0.772186	−	0.635397i	$-0.780837\pi$
$227$		−	19.1464i		−	1.27079i	0.772186	−	0.635397i	$-0.219163\pi$
$228$		0			0
$229$	−9.79796			−0.647467			−0.323734	−	0.946148i	$-0.604938\pi$
$229$	−9.79796			−0.647467			−0.323734	+	0.946148i	$0.604938\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			26.6969i			1.74897i	0.485048	+	0.874487i	$0.338802\pi$
$233$			26.6969i			1.74897i	−0.485048	+	0.874487i	$0.661198\pi$
$234$		0			0
$235$	1.10102	+	10.8990i	0.0718227	+	0.710971i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.3485			−1.38092			−0.690459	−	0.723372i	$-0.742591\pi$
$239$	−21.3485			−1.38092			−0.690459	+	0.723372i	$0.742591\pi$
$240$		0			0
$241$	−14.5959			−0.940206			−0.470103	−	0.882612i	$-0.655783\pi$
$241$	−14.5959			−0.940206			−0.470103	+	0.882612i	$0.655783\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.3485	−	1.34847i	0.852802	−	0.0861505i
$246$		0			0
$247$		−	23.0000i		−	1.46345i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	29.3485			1.85246			0.926229	−	0.376960i	$-0.123031\pi$
$251$	29.3485			1.85246			0.926229	+	0.376960i	$0.123031\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7.10102i			0.442949i	0.975166	+	0.221475i	$0.0710870\pi$
$257$			7.10102i			0.442949i	−0.975166	+	0.221475i	$0.928913\pi$
$258$		0			0
$259$	0.101021			0.00627711
$260$		0			0
$261$		0			0
$262$		0			0
$263$			16.8990i			1.04204i	0.853546	+	0.521018i	$0.174448\pi$
$263$			16.8990i			1.04204i	−0.853546	+	0.521018i	$0.825552\pi$
$264$		0			0
$265$	1.00000	+	9.89898i	0.0614295	+	0.608089i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	24.4949			1.49348			0.746740	−	0.665116i	$-0.231618\pi$
$269$	24.4949			1.49348			0.746740	+	0.665116i	$0.231618\pi$
$270$		0			0
$271$	15.6969			0.953521			0.476761	−	0.879033i	$-0.341811\pi$
$271$	15.6969			0.953521			0.476761	+	0.879033i	$0.341811\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.20204	+	0.449490i	−0.132788	+	0.0271053i
$276$		0			0
$277$		−	21.7980i		−	1.30971i	−0.755753	−	0.654856i	$-0.772729\pi$
$277$		−	21.7980i		−	1.30971i	0.755753	−	0.654856i	$-0.227271\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−26.6969			−1.59261			−0.796303	−	0.604898i	$-0.793214\pi$
$281$	−26.6969			−1.59261			−0.796303	+	0.604898i	$0.793214\pi$
$282$		0			0
$283$			13.5959i			0.808193i	0.914716	+	0.404097i	$0.132414\pi$
$283$			13.5959i			0.808193i	−0.914716	+	0.404097i	$0.867586\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	9.34847i		−	0.551823i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	28.9444i		−	1.69095i	−0.534016	−	0.845475i	$-0.679318\pi$
$293$		−	28.9444i		−	1.69095i	0.534016	−	0.845475i	$-0.320682\pi$
$294$		0			0
$295$	−10.8990	+	1.10102i	−0.634563	+	0.0641039i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.3485			−1.00329
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.5732	−	1.97730i	1.12076	−	0.113220i
$306$		0			0
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.4949			−1.38898			−0.694489	−	0.719503i	$-0.744370\pi$
$311$	−24.4949			−1.38898			−0.694489	+	0.719503i	$0.744370\pi$
$312$		0			0
$313$			14.1010i			0.797037i	0.917160	+	0.398518i	$0.130475\pi$
$313$			14.1010i			0.797037i	−0.917160	+	0.398518i	$0.869525\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	19.5959i		−	1.10062i	−0.834962	−	0.550308i	$-0.814510\pi$
$317$		−	19.5959i		−	1.10062i	0.834962	−	0.550308i	$-0.185490\pi$
$318$		0			0
$319$	0.202041			0.0113121
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	28.8990i		−	1.60798i
$324$		0			0
$325$	3.89898	+	19.1010i	0.216276	+	1.05953i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.89898			−0.270089
$330$		0			0
$331$	12.1010			0.665132			0.332566	−	0.943080i	$-0.392086\pi$
$331$	12.1010			0.665132			0.332566	+	0.943080i	$0.392086\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.32577	−	32.9217i	−0.181706	−	1.79870i
$336$		0			0
$337$		−	20.1010i		−	1.09497i	−0.836815	−	0.547486i	$-0.815585\pi$
$337$		−	20.1010i		−	1.09497i	0.836815	−	0.547486i	$-0.184415\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.69694			0.146047
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.2474i			0.550112i	0.961428	+	0.275056i	$0.0886964\pi$
$347$			10.2474i			0.550112i	−0.961428	+	0.275056i	$0.911304\pi$
$348$		0			0
$349$	−8.79796			−0.470944			−0.235472	−	0.971881i	$-0.575664\pi$
$349$	−8.79796			−0.470944			−0.235472	+	0.971881i	$0.575664\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	14.2020i		−	0.755898i	−0.925826	−	0.377949i	$-0.876629\pi$
$353$		−	14.2020i		−	0.755898i	0.925826	−	0.377949i	$-0.123371\pi$
$354$		0			0
$355$	25.6969	−	2.59592i	1.36385	−	0.137777i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.1464			1.01051			0.505255	−	0.862970i	$-0.331398\pi$
$359$	19.1464			1.01051			0.505255	+	0.862970i	$0.331398\pi$
$360$		0			0
$361$	15.7980			0.831472
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.876276	−	8.67423i	−0.0458664	−	0.454030i
$366$		0			0
$367$		−	22.7980i		−	1.19004i	−0.803709	−	0.595022i	$-0.797143\pi$
$367$		−	22.7980i		−	1.19004i	0.803709	−	0.595022i	$-0.202857\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.44949			−0.231006
$372$		0			0
$373$		−	21.6969i		−	1.12342i	−0.827333	−	0.561712i	$-0.810143\pi$
$373$		−	21.6969i		−	1.12342i	0.827333	−	0.561712i	$-0.189857\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.75255i		−	0.0902610i
$378$		0			0
$379$	−17.8990			−0.919409			−0.459704	−	0.888072i	$-0.652045\pi$
$379$	−17.8990			−0.919409			−0.459704	+	0.888072i	$0.652045\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	29.3939i		−	1.50196i	−0.660327	−	0.750978i	$-0.729582\pi$
$383$		−	29.3939i		−	1.50196i	0.660327	−	0.750978i	$-0.270418\pi$
$384$		0			0
$385$	−0.101021	−	1.00000i	−0.00514848	−	0.0509647i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.59592			0.385128			0.192564	−	0.981284i	$-0.438320\pi$
$389$	7.59592			0.385128			0.192564	+	0.981284i	$0.438320\pi$
$390$		0			0
$391$	−21.7980			−1.10237
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.67423	+	0.876276i	−0.436448	+	0.0440902i
$396$		0			0
$397$		−	31.5959i		−	1.58575i	−0.609382	−	0.792877i	$-0.708582\pi$
$397$		−	31.5959i		−	1.58575i	0.609382	−	0.792877i	$-0.291418\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.3485			−0.866341			−0.433171	−	0.901312i	$-0.642605\pi$
$401$	−17.3485			−0.866341			−0.433171	+	0.901312i	$0.642605\pi$
$402$		0			0
$403$		−	23.3939i		−	1.16533i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.0454077i			0.00225078i
$408$		0			0
$409$	−2.79796			−0.138350			−0.0691751	−	0.997605i	$-0.522037\pi$
$409$	−2.79796			−0.138350			−0.0691751	+	0.997605i	$0.522037\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	4.89898i		−	0.241063i
$414$		0			0
$415$	1.69694	+	16.7980i	0.0832994	+	0.824579i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.59592			0.371085			0.185542	−	0.982636i	$-0.440596\pi$
$419$	7.59592			0.371085			0.185542	+	0.982636i	$0.440596\pi$
$420$		0			0
$421$	28.7980			1.40353			0.701763	−	0.712410i	$-0.252396\pi$
$421$	28.7980			1.40353			0.701763	+	0.712410i	$0.252396\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.89898	+	24.0000i	0.237635	+	1.16417i
$426$		0			0
$427$			8.79796i			0.425763i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.0454			0.965553			0.482777	−	0.875744i	$-0.339628\pi$
$431$	20.0454			0.965553			0.482777	+	0.875744i	$0.339628\pi$
$432$		0			0
$433$			31.5959i			1.51840i	0.650856	+	0.759201i	$0.274410\pi$
$433$			31.5959i			1.51840i	−0.650856	+	0.759201i	$0.725590\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	26.2474i		−	1.25559i
$438$		0			0
$439$	−8.20204			−0.391462			−0.195731	−	0.980658i	$-0.562708\pi$
$439$	−8.20204			−0.391462			−0.195731	+	0.980658i	$0.562708\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			31.1010i			1.47765i	0.673895	+	0.738827i	$0.264620\pi$
$443$			31.1010i			1.47765i	−0.673895	+	0.738827i	$0.735380\pi$
$444$		0			0
$445$	26.6969	−	2.69694i	1.26556	−	0.127847i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.59592			−0.358474			−0.179237	−	0.983806i	$-0.557363\pi$
$449$	−7.59592			−0.358474			−0.179237	+	0.983806i	$0.557363\pi$
$450$		0			0
$451$	4.20204			0.197866
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.67423	+	0.876276i	−0.406654	+	0.0410804i
$456$		0			0
$457$			7.59592i			0.355322i	0.984092	+	0.177661i	$0.0568531\pi$
$457$			7.59592i			0.355322i	−0.984092	+	0.177661i	$0.943147\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.20204			−0.102559			−0.0512796	−	0.998684i	$-0.516330\pi$
$461$	−2.20204			−0.102559			−0.0512796	+	0.998684i	$0.516330\pi$
$462$		0			0
$463$		−	6.59592i		−	0.306538i	−0.988184	−	0.153269i	$-0.951020\pi$
$463$		−	6.59592i		−	0.306538i	0.988184	−	0.153269i	$-0.0489801\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$	14.7980			0.683306
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.69694i		−	0.124005i
$474$		0			0
$475$	−28.8990	+	5.89898i	−1.32598	+	0.270664i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.9444			1.13974			0.569869	−	0.821736i	$-0.306994\pi$
$479$	24.9444			1.13974			0.569869	+	0.821736i	$0.306994\pi$
$480$		0			0
$481$	0.393877			0.0179592
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.57321	−	35.3712i	−0.162251	−	1.60612i
$486$		0			0
$487$		−	40.3939i		−	1.83042i	−0.402976	−	0.915211i	$-0.632024\pi$
$487$		−	40.3939i		−	1.83042i	0.402976	−	0.915211i	$-0.367976\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	28.9444			1.30624			0.653121	−	0.757254i	$-0.273459\pi$
$491$	28.9444			1.30624			0.653121	+	0.757254i	$0.273459\pi$
$492$		0			0
$493$		−	2.20204i		−	0.0991749i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			11.5505i			0.518111i
$498$		0			0
$499$	−21.3939			−0.957721			−0.478861	−	0.877891i	$-0.658950\pi$
$499$	−21.3939			−0.957721			−0.478861	+	0.877891i	$0.658950\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	18.6515i		−	0.831631i	−0.909449	−	0.415815i	$-0.863496\pi$
$503$		−	18.6515i		−	0.831631i	0.909449	−	0.415815i	$-0.136504\pi$
$504$		0			0
$505$	−15.7980	+	1.59592i	−0.703000	+	0.0710174i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.1464			0.671354			0.335677	−	0.941977i	$-0.391035\pi$
$509$	15.1464			0.671354			0.335677	+	0.941977i	$0.391035\pi$
$510$		0			0
$511$	3.89898			0.172481
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.42679	−	24.0227i	−0.106937	−	1.05857i
$516$		0			0
$517$		−	2.20204i		−	0.0968457i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.0454			1.05345			0.526724	−	0.850036i	$-0.323420\pi$
$521$	24.0454			1.05345			0.526724	+	0.850036i	$0.323420\pi$
$522$		0			0
$523$		−	2.79796i		−	0.122346i	−0.998127	−	0.0611731i	$-0.980516\pi$
$523$		−	2.79796i		−	0.122346i	0.998127	−	0.0611731i	$-0.0194842\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	29.3939i		−	1.28042i
$528$		0			0
$529$	3.20204			0.139219
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	36.4495i		−	1.57880i
$534$		0			0
$535$	3.20204	+	31.6969i	0.138436	+	1.37038i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.69694			−0.116165
$540$		0			0
$541$	−0.595918			−0.0256205			−0.0128103	−	0.999918i	$-0.504078\pi$
$541$	−0.595918			−0.0256205			−0.0128103	+	0.999918i	$0.504078\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	11.0000i		−	0.470326i	−0.971956	−	0.235163i	$-0.924438\pi$
$547$		−	11.0000i		−	0.470326i	0.971956	−	0.235163i	$-0.0755624\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.65153			0.112959
$552$		0			0
$553$		−	3.89898i		−	0.165801i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	28.8990i		−	1.22449i	−0.790668	−	0.612245i	$-0.790267\pi$
$557$		−	28.8990i		−	1.22449i	0.790668	−	0.612245i	$-0.209733\pi$
$558$		0			0
$559$	−23.3939			−0.989456
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	3.89898	+	38.5959i	0.164031	+	1.62374i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.89898			−0.205376			−0.102688	−	0.994714i	$-0.532744\pi$
$569$	−4.89898			−0.205376			−0.102688	+	0.994714i	$0.532744\pi$
$570$		0			0
$571$	−39.6969			−1.66127			−0.830633	−	0.556821i	$-0.812021\pi$
$571$	−39.6969			−1.66127			−0.830633	+	0.556821i	$0.812021\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.44949	+	21.7980i	0.185557	+	0.909038i
$576$		0			0
$577$		−	14.1010i		−	0.587033i	−0.955954	−	0.293517i	$-0.905174\pi$
$577$		−	14.1010i		−	0.587033i	0.955954	−	0.293517i	$-0.0948256\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.55051			−0.313248
$582$		0			0
$583$		−	2.00000i		−	0.0828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.9444i		−	1.02957i	−0.857321	−	0.514783i	$-0.827873\pi$
$587$		−	24.9444i		−	1.02957i	0.857321	−	0.514783i	$-0.172127\pi$
$588$		0			0
$589$	35.3939			1.45838
$590$		0			0
$591$		0			0
$592$		0			0
$593$			18.6515i			0.765927i	0.923764	+	0.382963i	$0.125096\pi$
$593$			18.6515i			0.765927i	−0.923764	+	0.382963i	$0.874904\pi$
$594$		0			0
$595$	−10.8990	+	1.10102i	−0.446815	+	0.0451374i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−41.8434			−1.70967			−0.854837	−	0.518897i	$-0.826343\pi$
$599$	−41.8434			−1.70967			−0.854837	+	0.518897i	$0.826343\pi$
$600$		0			0
$601$	43.5959			1.77831			0.889157	−	0.457602i	$-0.151291\pi$
$601$	43.5959			1.77831			0.889157	+	0.457602i	$0.151291\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.0227	+	2.42679i	−0.976662	+	0.0986629i
$606$		0			0
$607$			30.5959i			1.24185i	0.783870	+	0.620925i	$0.213243\pi$
$607$			30.5959i			1.24185i	−0.783870	+	0.620925i	$0.786757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−19.1010			−0.772745
$612$		0			0
$613$		−	23.8990i		−	0.965271i	−0.875821	−	0.482635i	$-0.839680\pi$
$613$		−	23.8990i		−	0.965271i	0.875821	−	0.482635i	$-0.160320\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.0454i			0.484930i	0.970160	+	0.242465i	$0.0779559\pi$
$617$			12.0454i			0.484930i	−0.970160	+	0.242465i	$0.922044\pi$
$618$		0			0
$619$	13.4949			0.542406			0.271203	−	0.962522i	$-0.412579\pi$
$619$	13.4949			0.542406			0.271203	+	0.962522i	$0.412579\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.494897			0.0197329
$630$		0			0
$631$	−25.6969			−1.02298			−0.511489	−	0.859290i	$-0.670906\pi$
$631$	−25.6969			−1.02298			−0.511489	+	0.859290i	$0.670906\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.853572	+	8.44949i	0.0338730	+	0.335308i
$636$		0			0
$637$			23.3939i			0.926899i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	33.7980			1.33494			0.667470	−	0.744637i	$-0.267377\pi$
$641$	33.7980			1.33494			0.667470	+	0.744637i	$0.267377\pi$
$642$		0			0
$643$			39.7980i			1.56948i	0.619826	+	0.784739i	$0.287203\pi$
$643$			39.7980i			1.56948i	−0.619826	+	0.784739i	$0.712797\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			26.2474i			1.03189i	0.856621	+	0.515947i	$0.172560\pi$
$647$			26.2474i			1.03189i	−0.856621	+	0.515947i	$0.827440\pi$
$648$		0			0
$649$	2.20204			0.0864377
$650$		0			0
$651$		0			0
$652$		0			0
$653$			9.79796i			0.383424i	0.981451	+	0.191712i	$0.0614039\pi$
$653$			9.79796i			0.383424i	−0.981451	+	0.191712i	$0.938596\pi$
$654$		0			0
$655$	15.7980	−	1.59592i	0.617277	−	0.0623577i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−33.7980			−1.31658			−0.658291	−	0.752764i	$-0.728720\pi$
$659$	−33.7980			−1.31658			−0.658291	+	0.752764i	$0.728720\pi$
$660$		0			0
$661$	−46.7980			−1.82023			−0.910115	−	0.414356i	$-0.864007\pi$
$661$	−46.7980			−1.82023			−0.910115	+	0.414356i	$0.864007\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.32577	−	13.1237i	−0.0514110	−	0.508916i
$666$		0			0
$667$		−	2.00000i		−	0.0774403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.95459			−0.152665
$672$		0			0
$673$			45.6969i			1.76149i	0.473593	+	0.880744i	$0.342957\pi$
$673$			45.6969i			1.76149i	−0.473593	+	0.880744i	$0.657043\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			14.6969i			0.564849i	0.959289	+	0.282425i	$0.0911387\pi$
$677$			14.6969i			0.564849i	−0.959289	+	0.282425i	$0.908861\pi$
$678$		0			0
$679$	15.8990			0.610147
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	14.6515i		−	0.560625i	−0.959909	−	0.280313i	$-0.909562\pi$
$683$		−	14.6515i		−	0.560625i	0.959909	−	0.280313i	$-0.0904381\pi$
$684$		0			0
$685$	3.89898	+	38.5959i	0.148972	+	1.47467i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.3485			−0.660924
$690$		0			0
$691$	13.5959			0.517213			0.258607	−	0.965983i	$-0.416737\pi$
$691$	13.5959			0.517213			0.258607	+	0.965983i	$0.416737\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.5732	−	2.17934i	0.818319	−	0.0826670i
$696$		0			0
$697$		−	45.7980i		−	1.73472i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.44949			0.168055			0.0840275	−	0.996463i	$-0.473222\pi$
$701$	4.44949			0.168055			0.0840275	+	0.996463i	$0.473222\pi$
$702$		0			0
$703$			0.595918i			0.0224755i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.10102i		−	0.267061i
$708$		0			0
$709$	0.595918			0.0223802			0.0111901	−	0.999937i	$-0.496438\pi$
$709$	0.595918			0.0223802			0.0111901	+	0.999937i	$0.496438\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	26.6969i		−	0.999808i
$714$		0			0
$715$	−0.393877	−	3.89898i	−0.0147302	−	0.145814i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.44949			0.165938			0.0829690	−	0.996552i	$-0.473560\pi$
$719$	4.44949			0.165938			0.0829690	+	0.996552i	$0.473560\pi$
$720$		0			0
$721$	10.7980			0.402137
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.20204	+	0.449490i	−0.0817818	+	0.0166936i
$726$		0			0
$727$		−	39.7980i		−	1.47602i	−0.674787	−	0.738012i	$-0.735765\pi$
$727$		−	39.7980i		−	1.47602i	0.674787	−	0.738012i	$-0.264235\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−29.3939			−1.08717
$732$		0			0
$733$		−	14.2020i		−	0.524564i	−0.964991	−	0.262282i	$-0.915525\pi$
$733$		−	14.2020i		−	0.524564i	0.964991	−	0.262282i	$-0.0844751\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			6.65153i			0.245012i
$738$		0			0
$739$	1.59592			0.0587068			0.0293534	−	0.999569i	$-0.490655\pi$
$739$	1.59592			0.0587068			0.0293534	+	0.999569i	$0.490655\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			19.1010i			0.700748i	0.936610	+	0.350374i	$0.113946\pi$
$743$			19.1010i			0.700748i	−0.936610	+	0.350374i	$0.886054\pi$
$744$		0			0
$745$	−21.7980	+	2.20204i	−0.798615	+	0.0806765i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.2474			−0.520590
$750$		0			0
$751$	−15.8990			−0.580162			−0.290081	−	0.957002i	$-0.593682\pi$
$751$	−15.8990			−0.580162			−0.290081	+	0.957002i	$0.593682\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13.5732	−	1.37117i	0.493980	−	0.0499021i
$756$		0			0
$757$			15.6969i			0.570515i	0.958451	+	0.285257i	$0.0920791\pi$
$757$			15.6969i			0.570515i	−0.958451	+	0.285257i	$0.907921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.6515			0.676117			0.338059	−	0.941125i	$-0.390230\pi$
$761$	18.6515			0.676117			0.338059	+	0.941125i	$0.390230\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	19.1010i		−	0.689698i
$768$		0			0
$769$	16.3939			0.591178			0.295589	−	0.955315i	$-0.404484\pi$
$769$	16.3939			0.591178			0.295589	+	0.955315i	$0.404484\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			26.6515i			0.958589i	0.877654	+	0.479294i	$0.159107\pi$
$773$			26.6515i			0.958589i	−0.877654	+	0.479294i	$0.840893\pi$
$774$		0			0
$775$	−29.3939	+	6.00000i	−1.05586	+	0.215526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	55.1464			1.97583
$780$		0			0
$781$	−5.19184			−0.185778
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.449490	+	4.44949i	0.0160430	+	0.158809i
$786$		0			0
$787$		−	21.2020i		−	0.755771i	−0.925852	−	0.377886i	$-0.876651\pi$
$787$		−	21.2020i		−	0.755771i	0.925852	−	0.377886i	$-0.123349\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.3485			−0.616841
$792$		0			0
$793$			34.3031i			1.21814i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	12.4949i		−	0.442592i	−0.975207	−	0.221296i	$-0.928971\pi$
$797$		−	12.4949i		−	0.442592i	0.975207	−	0.221296i	$-0.0710287\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1.75255i			0.0618462i
$804$		0			0
$805$	−9.89898	+	1.00000i	−0.348893	+	0.0352454i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.4949			−0.439297			−0.219649	−	0.975579i	$-0.570491\pi$
$809$	−12.4949			−0.439297			−0.219649	+	0.975579i	$0.570491\pi$
$810$		0			0
$811$	13.5959			0.477417			0.238709	−	0.971091i	$-0.423276\pi$
$811$	13.5959			0.477417			0.238709	+	0.971091i	$0.423276\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.224745	−	2.22474i	−0.00787247	−	0.0779294i
$816$		0			0
$817$		−	35.3939i		−	1.23828i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.30306			0.324679			0.162340	−	0.986735i	$-0.448096\pi$
$821$	9.30306			0.324679			0.162340	+	0.986735i	$0.448096\pi$
$822$		0			0
$823$			28.7980i			1.00383i	0.864916	+	0.501917i	$0.167372\pi$
$823$			28.7980i			1.00383i	−0.864916	+	0.501917i	$0.832628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			52.8990i			1.83948i	0.392531	+	0.919739i	$0.371600\pi$
$827$			52.8990i			1.83948i	−0.392531	+	0.919739i	$0.628400\pi$
$828$		0			0
$829$	−28.7980			−1.00019			−0.500097	−	0.865969i	$-0.666702\pi$
$829$	−28.7980			−1.00019			−0.500097	+	0.865969i	$0.666702\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			29.3939i			1.01844i
$834$		0			0
$835$	0.494897	+	4.89898i	0.0171266	+	0.169536i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	50.6969			1.75025			0.875126	−	0.483894i	$-0.160778\pi$
$839$	50.6969			1.75025			0.875126	+	0.483894i	$0.160778\pi$
$840$		0			0
$841$	−28.7980			−0.993033
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.89898	+	0.494897i	−0.168530	+	0.0170250i
$846$		0			0
$847$		−	10.7980i		−	0.371022i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.449490			0.0154083
$852$		0			0
$853$		−	4.30306i		−	0.147334i	−0.997283	−	0.0736670i	$-0.976530\pi$
$853$		−	4.30306i		−	0.147334i	0.997283	−	0.0736670i	$-0.0234702\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			4.44949i			0.151992i	0.997108	+	0.0759958i	$0.0242136\pi$
$857$			4.44949i			0.151992i	−0.997108	+	0.0759958i	$0.975786\pi$
$858$		0			0
$859$	45.6969			1.55916			0.779580	−	0.626303i	$-0.215433\pi$
$859$	45.6969			1.55916			0.779580	+	0.626303i	$0.215433\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	41.3939i		−	1.40906i	−0.709672	−	0.704532i	$-0.751157\pi$
$863$		−	41.3939i		−	1.40906i	0.709672	−	0.704532i	$-0.248843\pi$
$864$		0			0
$865$	−3.79796	−	37.5959i	−0.129134	−	1.27830i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.75255			0.0594512
$870$		0			0
$871$	57.6969			1.95499
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.32577	+	10.6742i	0.112431	+	0.360855i
$876$		0			0
$877$		−	19.4949i		−	0.658296i	−0.944278	−	0.329148i	$-0.893239\pi$
$877$		−	19.4949i		−	0.658296i	0.944278	−	0.329148i	$-0.106761\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.9444			−0.840398			−0.420199	−	0.907432i	$-0.638040\pi$
$881$	−24.9444			−0.840398			−0.420199	+	0.907432i	$0.638040\pi$
$882$		0			0
$883$			20.7980i			0.699907i	0.936767	+	0.349954i	$0.113803\pi$
$883$			20.7980i			0.699907i	−0.936767	+	0.349954i	$0.886197\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			43.5959i			1.46381i	0.681408	+	0.731904i	$0.261368\pi$
$887$			43.5959i			1.46381i	−0.681408	+	0.731904i	$0.738632\pi$
$888$		0			0
$889$	−3.79796			−0.127379
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	28.8990i		−	0.967067i
$894$		0			0
$895$	11.8990	−	1.20204i	0.397739	−	0.0401798i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.69694			0.0899479
$900$		0			0
$901$	−21.7980			−0.726195
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.1237	+	1.12372i	−0.369765	+	0.0373539i
$906$		0			0
$907$		−	14.5959i		−	0.484650i	−0.970195	−	0.242325i	$-0.922090\pi$
$907$		−	14.5959i		−	0.484650i	0.970195	−	0.242325i	$-0.0779099\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.69694			0.0893536			0.0446768	−	0.999001i	$-0.485774\pi$
$911$	2.69694			0.0893536			0.0446768	+	0.999001i	$0.485774\pi$
$912$		0			0
$913$		−	3.39388i		−	0.112321i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			7.10102i			0.234496i
$918$		0			0
$919$	47.3939			1.56338			0.781690	−	0.623667i	$-0.214358\pi$
$919$	47.3939			1.56338			0.781690	+	0.623667i	$0.214358\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			45.0352i			1.48235i
$924$		0			0
$925$	−0.101021	−	0.494897i	−0.00332153	−	0.0162721i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.85357			−0.159241			−0.0796203	−	0.996825i	$-0.525371\pi$
$929$	−4.85357			−0.159241			−0.0796203	+	0.996825i	$0.525371\pi$
$930$		0			0
$931$	−35.3939			−1.15999
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.494897	−	4.89898i	−0.0161849	−	0.160214i
$936$		0			0
$937$		−	31.4949i		−	1.02889i	−0.857522	−	0.514447i	$-0.827997\pi$
$937$		−	31.4949i		−	1.02889i	0.857522	−	0.514447i	$-0.172003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	53.3485			1.73911			0.869555	−	0.493836i	$-0.164406\pi$
$941$	53.3485			1.73911			0.869555	+	0.493836i	$0.164406\pi$
$942$		0			0
$943$		−	41.5959i		−	1.35455i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			4.89898i			0.159195i	0.996827	+	0.0795977i	$0.0253636\pi$
$947$			4.89898i			0.159195i	−0.996827	+	0.0795977i	$0.974636\pi$
$948$		0			0
$949$	15.2020			0.493479
$950$		0			0
$951$		0			0
$952$		0			0
$953$			41.3939i			1.34088i	0.741964	+	0.670440i	$0.233894\pi$
$953$			41.3939i			1.34088i	−0.741964	+	0.670440i	$0.766106\pi$
$954$		0			0
$955$	48.4949	−	4.89898i	1.56926	−	0.158527i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17.3485			−0.560211
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.37117	+	53.1691i	0.172904	+	1.71157i
$966$		0			0
$967$			14.5959i			0.469373i	0.972071	+	0.234687i	$0.0754064\pi$
$967$			14.5959i			0.469373i	−0.972071	+	0.234687i	$0.924594\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−58.2929			−1.87071			−0.935353	−	0.353715i	$-0.884918\pi$
$971$	−58.2929			−1.87071			−0.935353	+	0.353715i	$0.884918\pi$
$972$		0			0
$973$			9.69694i			0.310870i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	14.2474i		−	0.455816i	−0.973683	−	0.227908i	$-0.926811\pi$
$977$		−	14.2474i		−	0.455816i	0.973683	−	0.227908i	$-0.0731886\pi$
$978$		0			0
$979$	−5.39388			−0.172389
$980$		0			0
$981$		0			0
$982$		0			0
$983$			55.1464i			1.75890i	0.475993	+	0.879449i	$0.342089\pi$
$983$			55.1464i			1.75890i	−0.475993	+	0.879449i	$0.657911\pi$
$984$		0			0
$985$	−2.69694	−	26.6969i	−0.0859316	−	0.850635i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.6969			−0.848913
$990$		0			0
$991$	2.30306			0.0731592			0.0365796	−	0.999331i	$-0.488354\pi$
$991$	2.30306			0.0731592			0.0365796	+	0.999331i	$0.488354\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	48.2702	−	4.87628i	1.53027	−	0.154588i
$996$		0			0
$997$			38.2020i			1.20987i	0.796275	+	0.604935i	$0.206801\pi$
$997$			38.2020i			1.20987i	−0.796275	+	0.604935i	$0.793199\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.f.n.1729.3		4
3.2	odd	2		2160.2.f.i.1729.2		4
4.3	odd	2		1080.2.f.f.649.3	yes	4
5.4	even	2	inner	2160.2.f.n.1729.4		4
12.11	even	2		1080.2.f.e.649.2	yes	4
15.14	odd	2		2160.2.f.i.1729.1		4
20.3	even	4		5400.2.a.bw.1.2		2
20.7	even	4		5400.2.a.cc.1.2		2
20.19	odd	2		1080.2.f.f.649.4	yes	4
60.23	odd	4		5400.2.a.bz.1.1		2
60.47	odd	4		5400.2.a.cf.1.1		2
60.59	even	2		1080.2.f.e.649.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.f.e.649.1	✓	4	60.59	even	2
1080.2.f.e.649.2	yes	4	12.11	even	2
1080.2.f.f.649.3	yes	4	4.3	odd	2
1080.2.f.f.649.4	yes	4	20.19	odd	2
2160.2.f.i.1729.1		4	15.14	odd	2
2160.2.f.i.1729.2		4	3.2	odd	2
2160.2.f.n.1729.3		4	1.1	even	1	trivial
2160.2.f.n.1729.4		4	5.4	even	2	inner
5400.2.a.bw.1.2		2	20.3	even	4
5400.2.a.bz.1.1		2	60.23	odd	4
5400.2.a.cc.1.2		2	20.7	even	4
5400.2.a.cf.1.1		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.22474 - 0.224745i) q^{5} +1.00000i q^{7} -0.449490 q^{11} +3.89898i q^{13} +4.89898i q^{17} -5.89898 q^{19} +4.44949i q^{23} +(4.89898 - 1.00000i) q^{25} -0.449490 q^{29} -6.00000 q^{31} +(0.224745 + 2.22474i) q^{35} -0.101021i q^{37} -9.34847 q^{41} +6.00000i q^{43} +4.89898i q^{47} +6.00000 q^{49} +4.44949i q^{53} +(-1.00000 + 0.101021i) q^{55} -4.89898 q^{59} +8.79796 q^{61} +(0.876276 + 8.67423i) q^{65} -14.7980i q^{67} +11.5505 q^{71} -3.89898i q^{73} -0.449490i q^{77} -3.89898 q^{79} +7.55051i q^{83} +(1.10102 + 10.8990i) q^{85} +12.0000 q^{89} -3.89898 q^{91} +(-13.1237 + 1.32577i) q^{95} -15.8990i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 8 q^{11} - 4 q^{19} + 8 q^{29} - 24 q^{31} - 4 q^{35} - 8 q^{41} + 24 q^{49} - 4 q^{55} - 4 q^{61} + 28 q^{65} + 56 q^{71} + 4 q^{79} + 24 q^{85} + 48 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) \) 4 * q + 4 * q^5 + 8 * q^11 - 4 * q^19 + 8 * q^29 - 24 * q^31 - 4 * q^35 - 8 * q^41 + 24 * q^49 - 4 * q^55 - 4 * q^61 + 28 * q^65 + 56 * q^71 + 4 * q^79 + 24 * q^85 + 48 * q^89 + 4 * q^91 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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