LMFDB - Embedded newform 2160.2.f.n.1729.2

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			1729.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2160.1729
Dual form			2160.2.f.n.1729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.224745 + 2.22474i) q^{5} +1.00000i q^{7} +O(q^{10})$	$q+(-0.224745 + 2.22474i) q^{5} +1.00000i q^{7} +4.44949 q^{11} -5.89898i q^{13} -4.89898i q^{17} +3.89898 q^{19} -0.449490i q^{23} +(-4.89898 - 1.00000i) q^{25} +4.44949 q^{29} -6.00000 q^{31} +(-2.22474 - 0.224745i) q^{35} -9.89898i q^{37} +5.34847 q^{41} +6.00000i q^{43} -4.89898i q^{47} +6.00000 q^{49} -0.449490i q^{53} +(-1.00000 + 9.89898i) q^{55} +4.89898 q^{59} -10.7980 q^{61} +(13.1237 + 1.32577i) q^{65} +4.79796i q^{67} +16.4495 q^{71} +5.89898i q^{73} +4.44949i q^{77} +5.89898 q^{79} +12.4495i q^{83} +(10.8990 + 1.10102i) q^{85} +12.0000 q^{89} +5.89898 q^{91} +(-0.876276 + 8.67423i) q^{95} -6.10102i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 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

Display 100 coefficients 10 coefficients

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
$5$	−0.224745	+	2.22474i	−0.100509	+	0.994936i
$5$	−0.224745	+	2.22474i	−0.100509	+	0.994936i
$6$		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
$11$	4.44949			1.34157			0.670786	−	0.741651i	$-0.265957\pi$
$11$	4.44949			1.34157			0.670786	+	0.741651i	$0.265957\pi$
$12$		0			0
$13$		−	5.89898i		−	1.63608i	−0.575160	−	0.818041i	$-0.695060\pi$
$13$		−	5.89898i		−	1.63608i	0.575160	−	0.818041i	$-0.304940\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$	3.89898			0.894487			0.447244	−	0.894412i	$-0.352406\pi$
$19$	3.89898			0.894487			0.447244	+	0.894412i	$0.352406\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.449490i		−	0.0937251i	−0.998901	−	0.0468625i	$-0.985078\pi$
$23$		−	0.449490i		−	0.0937251i	0.998901	−	0.0468625i	$-0.0149223\pi$
$24$		0			0
$25$	−4.89898	−	1.00000i	−0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.44949			0.826250			0.413125	−	0.910674i	$-0.364437\pi$
$29$	4.44949			0.826250			0.413125	+	0.910674i	$0.364437\pi$
$30$		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
$35$	−2.22474	−	0.224745i	−0.376051	−	0.0379888i
$36$		0			0
$37$		−	9.89898i		−	1.62738i	−0.581298	−	0.813691i	$-0.697455\pi$
$37$		−	9.89898i		−	1.62738i	0.581298	−	0.813691i	$-0.302545\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.34847			0.835291			0.417645	−	0.908610i	$-0.362855\pi$
$41$	5.34847			0.835291			0.417645	+	0.908610i	$0.362855\pi$
$42$		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	$-0.883699\pi$
$47$		−	4.89898i		−	0.714590i	0.933992	−	0.357295i	$-0.116301\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.449490i		−	0.0617422i	−0.999523	−	0.0308711i	$-0.990172\pi$
$53$		−	0.449490i		−	0.0617422i	0.999523	−	0.0308711i	$-0.00982813\pi$
$54$		0			0
$55$	−1.00000	+	9.89898i	−0.134840	+	1.33478i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.89898			0.637793			0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898			0.637793			0.318896	+	0.947790i	$0.396688\pi$
$60$		0			0
$61$	−10.7980			−1.38254			−0.691268	−	0.722598i	$-0.742948\pi$
$61$	−10.7980			−1.38254			−0.691268	+	0.722598i	$0.742948\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.1237	+	1.32577i	1.62780	+	0.164441i
$66$		0			0
$67$			4.79796i			0.586164i	0.956087	+	0.293082i	$0.0946809\pi$
$67$			4.79796i			0.586164i	−0.956087	+	0.293082i	$0.905319\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16.4495			1.95220			0.976098	−	0.217332i	$-0.0697356\pi$
$71$	16.4495			1.95220			0.976098	+	0.217332i	$0.0697356\pi$
$72$		0			0
$73$			5.89898i			0.690423i	0.938525	+	0.345212i	$0.112193\pi$
$73$			5.89898i			0.690423i	−0.938525	+	0.345212i	$0.887807\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.44949i			0.507066i
$78$		0			0
$79$	5.89898			0.663687			0.331844	−	0.943334i	$-0.392329\pi$
$79$	5.89898			0.663687			0.331844	+	0.943334i	$0.392329\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.4495i			1.36651i	0.730180	+	0.683255i	$0.239436\pi$
$83$			12.4495i			1.36651i	−0.730180	+	0.683255i	$0.760564\pi$
$84$		0			0
$85$	10.8990	+	1.10102i	1.18216	+	0.119422i
$86$		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
$91$	5.89898			0.618381
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.876276	+	8.67423i	−0.0899040	+	0.889958i
$96$		0			0
$97$		−	6.10102i		−	0.619465i	−0.950824	−	0.309732i	$-0.899761\pi$
$97$		−	6.10102i		−	0.619465i	0.950824	−	0.309732i	$-0.100239\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16.8990			−1.68151			−0.840756	−	0.541415i	$-0.817889\pi$
$101$	−16.8990			−1.68151			−0.840756	+	0.541415i	$0.817889\pi$
$102$		0			0
$103$			8.79796i			0.866889i	0.901180	+	0.433444i	$0.142702\pi$
$103$			8.79796i			0.866889i	−0.901180	+	0.433444i	$0.857298\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.2474i		−	0.990658i	−0.868705	−	0.495329i	$-0.835047\pi$
$107$		−	10.2474i		−	0.990658i	0.868705	−	0.495329i	$-0.164953\pi$
$108$		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
$113$			2.65153i			0.249435i	0.992192	+	0.124718i	$0.0398025\pi$
$113$			2.65153i			0.249435i	−0.992192	+	0.124718i	$0.960198\pi$
$114$		0			0
$115$	1.00000	+	0.101021i	0.0932505	+	0.00942021i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.89898			0.449089
$120$		0			0
$121$	8.79796			0.799814
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.32577	−	10.6742i	0.297465	−	0.954733i
$126$		0			0
$127$		−	15.7980i		−	1.40184i	−0.713239	−	0.700921i	$-0.752773\pi$
$127$		−	15.7980i		−	1.40184i	0.713239	−	0.700921i	$-0.247227\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	16.8990			1.47647			0.738235	−	0.674543i	$-0.235659\pi$
$131$	16.8990			1.47647			0.738235	+	0.674543i	$0.235659\pi$
$132$		0			0
$133$			3.89898i			0.338084i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2.65153i			0.226536i	0.993564	+	0.113268i	$0.0361318\pi$
$137$			2.65153i			0.226536i	−0.993564	+	0.113268i	$0.963868\pi$
$138$		0			0
$139$	−19.6969			−1.67067			−0.835336	−	0.549739i	$-0.814727\pi$
$139$	−19.6969			−1.67067			−0.835336	+	0.549739i	$0.814727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	26.2474i		−	2.19492i
$144$		0			0
$145$	−1.00000	+	9.89898i	−0.0830455	+	0.822066i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.79796			0.802680			0.401340	−	0.915929i	$-0.368545\pi$
$149$	9.79796			0.802680			0.401340	+	0.915929i	$0.368545\pi$
$150$		0			0
$151$	15.8990			1.29384			0.646920	−	0.762558i	$-0.276057\pi$
$151$	15.8990			1.29384			0.646920	+	0.762558i	$0.276057\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.34847	−	13.3485i	0.108312	−	1.07217i
$156$		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
$161$	0.449490			0.0354248
$162$		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
$167$			21.7980i			1.68678i	0.537304	+	0.843388i	$0.319443\pi$
$167$			21.7980i			1.68678i	−0.537304	+	0.843388i	$0.680557\pi$
$168$		0			0
$169$	−21.7980			−1.67677
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	7.10102i		−	0.539881i	−0.962877	−	0.269940i	$-0.912996\pi$
$173$		−	7.10102i		−	0.539881i	0.962877	−	0.269940i	$-0.0870040\pi$
$174$		0			0
$175$	1.00000	−	4.89898i	0.0755929	−	0.370328i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.34847			−0.698737			−0.349369	−	0.936985i	$-0.613604\pi$
$179$	−9.34847			−0.698737			−0.349369	+	0.936985i	$0.613604\pi$
$180$		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
$185$	22.0227	+	2.22474i	1.61914	+	0.163566i
$186$		0			0
$187$		−	21.7980i		−	1.59402i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.20204			0.159334			0.0796670	−	0.996822i	$-0.474614\pi$
$191$	2.20204			0.159334			0.0796670	+	0.996822i	$0.474614\pi$
$192$		0			0
$193$			14.1010i			1.01501i	0.861648	+	0.507507i	$0.169433\pi$
$193$			14.1010i			1.01501i	−0.861648	+	0.507507i	$0.830567\pi$
$194$		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
$199$	−7.69694			−0.545622			−0.272811	−	0.962068i	$-0.587953\pi$
$199$	−7.69694			−0.545622			−0.272811	+	0.962068i	$0.587953\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			4.44949i			0.312293i
$204$		0			0
$205$	−1.20204	+	11.8990i	−0.0839542	+	0.831061i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.3485			1.20002
$210$		0			0
$211$	−21.6969			−1.49368			−0.746839	−	0.665004i	$-0.768430\pi$
$211$	−21.6969			−1.49368			−0.746839	+	0.665004i	$0.768430\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.3485	−	1.34847i	−0.910358	−	0.0919648i
$216$		0			0
$217$		−	6.00000i		−	0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−28.8990			−1.94396
$222$		0			0
$223$		−	13.5959i		−	0.910450i	−0.890376	−	0.455225i	$-0.849559\pi$
$223$		−	13.5959i		−	0.910450i	0.890376	−	0.455225i	$-0.150441\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			15.1464i			1.00530i	0.864489	+	0.502652i	$0.167642\pi$
$227$			15.1464i			1.00530i	−0.864489	+	0.502652i	$0.832358\pi$
$228$		0			0
$229$	9.79796			0.647467			0.323734	−	0.946148i	$-0.395062\pi$
$229$	9.79796			0.647467			0.323734	+	0.946148i	$0.395062\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.69694i		−	0.176682i	−0.996090	−	0.0883412i	$-0.971843\pi$
$233$		−	2.69694i		−	0.176682i	0.996090	−	0.0883412i	$-0.0281566\pi$
$234$		0			0
$235$	10.8990	+	1.10102i	0.710971	+	0.0718227i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.65153			−0.430252			−0.215126	−	0.976586i	$-0.569016\pi$
$239$	−6.65153			−0.430252			−0.215126	+	0.976586i	$0.569016\pi$
$240$		0			0
$241$	24.5959			1.58436			0.792181	−	0.610286i	$-0.208945\pi$
$241$	24.5959			1.58436			0.792181	+	0.610286i	$0.208945\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.34847	+	13.3485i	−0.0861505	+	0.852802i
$246$		0			0
$247$		−	23.0000i		−	1.46345i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.6515			0.924796			0.462398	−	0.886672i	$-0.346989\pi$
$251$	14.6515			0.924796			0.462398	+	0.886672i	$0.346989\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.8990i			1.05413i	0.849825	+	0.527065i	$0.176707\pi$
$257$			16.8990i			1.05413i	−0.849825	+	0.527065i	$0.823293\pi$
$258$		0			0
$259$	9.89898			0.615093
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7.10102i			0.437868i	0.975740	+	0.218934i	$0.0702579\pi$
$263$			7.10102i			0.437868i	−0.975740	+	0.218934i	$0.929742\pi$
$264$		0			0
$265$	1.00000	+	0.101021i	0.0614295	+	0.00620564i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−24.4949			−1.49348			−0.746740	−	0.665116i	$-0.768382\pi$
$269$	−24.4949			−1.49348			−0.746740	+	0.665116i	$0.768382\pi$
$270$		0			0
$271$	−13.6969			−0.832030			−0.416015	−	0.909358i	$-0.636574\pi$
$271$	−13.6969			−0.832030			−0.416015	+	0.909358i	$0.636574\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.7980	−	4.44949i	−1.31447	−	0.268314i
$276$		0			0
$277$		−	2.20204i		−	0.132308i	−0.997809	−	0.0661539i	$-0.978927\pi$
$277$		−	2.20204i		−	0.132308i	0.997809	−	0.0661539i	$-0.0210728\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.69694			0.160886			0.0804429	−	0.996759i	$-0.474367\pi$
$281$	2.69694			0.160886			0.0804429	+	0.996759i	$0.474367\pi$
$282$		0			0
$283$		−	25.5959i		−	1.52152i	−0.649034	−	0.760760i	$-0.724827\pi$
$283$		−	25.5959i		−	1.52152i	0.649034	−	0.760760i	$-0.275173\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.34847i			0.315710i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.9444i			1.45727i	0.684904	+	0.728633i	$0.259844\pi$
$293$			24.9444i			1.45727i	−0.684904	+	0.728633i	$0.740156\pi$
$294$		0			0
$295$	−1.10102	+	10.8990i	−0.0641039	+	0.634563i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.65153			−0.153342
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.42679	−	24.0227i	0.138957	−	1.37554i
$306$		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	$-0.255630\pi$
$311$	24.4949			1.38898			0.694489	+	0.719503i	$0.255630\pi$
$312$		0			0
$313$			23.8990i			1.35085i	0.737429	+	0.675425i	$0.236040\pi$
$313$			23.8990i			1.35085i	−0.737429	+	0.675425i	$0.763960\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			19.5959i			1.10062i	0.834962	+	0.550308i	$0.185490\pi$
$317$			19.5959i			1.10062i	−0.834962	+	0.550308i	$0.814510\pi$
$318$		0			0
$319$	19.7980			1.10847
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	19.1010i		−	1.06281i
$324$		0			0
$325$	−5.89898	+	28.8990i	−0.327217	+	1.60303i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.89898			0.270089
$330$		0			0
$331$	21.8990			1.20368			0.601838	−	0.798618i	$-0.294435\pi$
$331$	21.8990			1.20368			0.601838	+	0.798618i	$0.294435\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.6742	−	1.07832i	−0.583196	−	0.0589147i
$336$		0			0
$337$		−	29.8990i		−	1.62870i	−0.580373	−	0.814351i	$-0.697093\pi$
$337$		−	29.8990i		−	1.62870i	0.580373	−	0.814351i	$-0.302907\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−26.6969			−1.44572
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	14.2474i		−	0.764843i	−0.923988	−	0.382422i	$-0.875090\pi$
$347$		−	14.2474i		−	0.764843i	0.923988	−	0.382422i	$-0.124910\pi$
$348$		0			0
$349$	10.7980			0.578001			0.289001	−	0.957329i	$-0.406677\pi$
$349$	10.7980			0.578001			0.289001	+	0.957329i	$0.406677\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	33.7980i		−	1.79888i	−0.437040	−	0.899442i	$-0.643973\pi$
$353$		−	33.7980i		−	1.79888i	0.437040	−	0.899442i	$-0.356027\pi$
$354$		0			0
$355$	−3.69694	+	36.5959i	−0.196213	+	1.94231i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.1464			−0.799398			−0.399699	−	0.916646i	$-0.630885\pi$
$359$	−15.1464			−0.799398			−0.399699	+	0.916646i	$0.630885\pi$
$360$		0			0
$361$	−3.79796			−0.199893
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.1237	−	1.32577i	−0.686927	−	0.0693937i
$366$		0			0
$367$		−	3.20204i		−	0.167145i	−0.996502	−	0.0835726i	$-0.973367\pi$
$367$		−	3.20204i		−	0.167145i	0.996502	−	0.0835726i	$-0.0266330\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.449490			0.0233363
$372$		0			0
$373$			7.69694i			0.398532i	0.979945	+	0.199266i	$0.0638558\pi$
$373$			7.69694i			0.398532i	−0.979945	+	0.199266i	$0.936144\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	26.2474i		−	1.35181i
$378$		0			0
$379$	−8.10102			−0.416121			−0.208061	−	0.978116i	$-0.566715\pi$
$379$	−8.10102			−0.416121			−0.208061	+	0.978116i	$0.566715\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			29.3939i			1.50196i	0.660327	+	0.750978i	$0.270418\pi$
$383$			29.3939i			1.50196i	−0.660327	+	0.750978i	$0.729582\pi$
$384$		0			0
$385$	−9.89898	−	1.00000i	−0.504499	−	0.0509647i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−31.5959			−1.60198			−0.800988	−	0.598680i	$-0.795692\pi$
$389$	−31.5959			−1.60198			−0.800988	+	0.598680i	$0.795692\pi$
$390$		0			0
$391$	−2.20204			−0.111362
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.32577	+	13.1237i	−0.0667065	+	0.660326i
$396$		0			0
$397$			7.59592i			0.381228i	0.981665	+	0.190614i	$0.0610479\pi$
$397$			7.59592i			0.381228i	−0.981665	+	0.190614i	$0.938952\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.65153			−0.132411			−0.0662056	−	0.997806i	$-0.521089\pi$
$401$	−2.65153			−0.132411			−0.0662056	+	0.997806i	$0.521089\pi$
$402$		0			0
$403$			35.3939i			1.76309i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	44.0454i		−	2.18325i
$408$		0			0
$409$	16.7980			0.830606			0.415303	−	0.909683i	$-0.363676\pi$
$409$	16.7980			0.830606			0.415303	+	0.909683i	$0.363676\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.89898i			0.241063i
$414$		0			0
$415$	−27.6969	−	2.79796i	−1.35959	−	0.137346i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.5959			−1.54356			−0.771781	−	0.635889i	$-0.780634\pi$
$419$	−31.5959			−1.54356			−0.771781	+	0.635889i	$0.780634\pi$
$420$		0			0
$421$	9.20204			0.448480			0.224240	−	0.974534i	$-0.428010\pi$
$421$	9.20204			0.448480			0.224240	+	0.974534i	$0.428010\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.89898	+	24.0000i	−0.237635	+	1.16417i
$426$		0			0
$427$		−	10.7980i		−	0.522550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.0454			−1.15823			−0.579113	−	0.815247i	$-0.696601\pi$
$431$	−24.0454			−1.15823			−0.579113	+	0.815247i	$0.696601\pi$
$432$		0			0
$433$		−	7.59592i		−	0.365037i	−0.983202	−	0.182518i	$-0.941575\pi$
$433$		−	7.59592i		−	0.365037i	0.983202	−	0.182518i	$-0.0584249\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.75255i		−	0.0838359i
$438$		0			0
$439$	−27.7980			−1.32672			−0.663362	−	0.748299i	$-0.730871\pi$
$439$	−27.7980			−1.32672			−0.663362	+	0.748299i	$0.730871\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			40.8990i			1.94317i	0.236693	+	0.971585i	$0.423937\pi$
$443$			40.8990i			1.94317i	−0.236693	+	0.971585i	$0.576063\pi$
$444$		0			0
$445$	−2.69694	+	26.6969i	−0.127847	+	1.26556i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	31.5959			1.49110			0.745552	−	0.666448i	$-0.232186\pi$
$449$	31.5959			1.49110			0.745552	+	0.666448i	$0.232186\pi$
$450$		0			0
$451$	23.7980			1.12060
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.32577	+	13.1237i	−0.0621528	+	0.615250i
$456$		0			0
$457$		−	31.5959i		−	1.47799i	−0.673708	−	0.738997i	$-0.735300\pi$
$457$		−	31.5959i		−	1.47799i	0.673708	−	0.738997i	$-0.264700\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.7980			−1.01523			−0.507616	−	0.861583i	$-0.669473\pi$
$461$	−21.7980			−1.01523			−0.507616	+	0.861583i	$0.669473\pi$
$462$		0			0
$463$			32.5959i			1.51486i	0.652916	+	0.757430i	$0.273545\pi$
$463$			32.5959i			1.51486i	−0.652916	+	0.757430i	$0.726455\pi$
$464$		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
$469$	−4.79796			−0.221549
$470$		0			0
$471$		0			0
$472$		0			0
$473$			26.6969i			1.22753i
$474$		0			0
$475$	−19.1010	−	3.89898i	−0.876415	−	0.178897i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−28.9444			−1.32250			−0.661251	−	0.750164i	$-0.729974\pi$
$479$	−28.9444			−1.32250			−0.661251	+	0.750164i	$0.729974\pi$
$480$		0			0
$481$	−58.3939			−2.66253
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.5732	+	1.37117i	0.616328	+	0.0622618i
$486$		0			0
$487$			18.3939i			0.833506i	0.909020	+	0.416753i	$0.136832\pi$
$487$			18.3939i			0.833506i	−0.909020	+	0.416753i	$0.863168\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.9444			−1.12572			−0.562862	−	0.826551i	$-0.690300\pi$
$491$	−24.9444			−1.12572			−0.562862	+	0.826551i	$0.690300\pi$
$492$		0			0
$493$		−	21.7980i		−	0.981731i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			16.4495i			0.737860i
$498$		0			0
$499$	37.3939			1.67398			0.836990	−	0.547218i	$-0.184313\pi$
$499$	37.3939			1.67398			0.836990	+	0.547218i	$0.184313\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	33.3485i		−	1.48693i	−0.668772	−	0.743467i	$-0.733180\pi$
$503$		−	33.3485i		−	1.48693i	0.668772	−	0.743467i	$-0.266820\pi$
$504$		0			0
$505$	3.79796	−	37.5959i	0.169007	−	1.67300i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.1464			−0.848651			−0.424325	−	0.905510i	$-0.639489\pi$
$509$	−19.1464			−0.848651			−0.424325	+	0.905510i	$0.639489\pi$
$510$		0			0
$511$	−5.89898			−0.260955
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19.5732	−	1.97730i	−0.862499	−	0.0871301i
$516$		0			0
$517$		−	21.7980i		−	0.958673i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.0454			−0.878205			−0.439103	−	0.898437i	$-0.644704\pi$
$521$	−20.0454			−0.878205			−0.439103	+	0.898437i	$0.644704\pi$
$522$		0			0
$523$			16.7980i			0.734523i	0.930118	+	0.367262i	$0.119705\pi$
$523$			16.7980i			0.734523i	−0.930118	+	0.367262i	$0.880295\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			29.3939i			1.28042i
$528$		0			0
$529$	22.7980			0.991216
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	31.5505i		−	1.36660i
$534$		0			0
$535$	22.7980	+	2.30306i	0.985642	+	0.0995700i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.6969			1.14992
$540$		0			0
$541$	38.5959			1.65937			0.829684	−	0.558233i	$-0.188521\pi$
$541$	38.5959			1.65937			0.829684	+	0.558233i	$0.188521\pi$
$542$		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
$551$	17.3485			0.739070
$552$		0			0
$553$			5.89898i			0.250850i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	19.1010i		−	0.809336i	−0.914464	−	0.404668i	$-0.867387\pi$
$557$		−	19.1010i		−	0.809336i	0.914464	−	0.404668i	$-0.132613\pi$
$558$		0			0
$559$	35.3939			1.49700
$560$		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
$565$	−5.89898	−	0.595918i	−0.248172	−	0.0250705i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.89898			0.205376			0.102688	−	0.994714i	$-0.467256\pi$
$569$	4.89898			0.205376			0.102688	+	0.994714i	$0.467256\pi$
$570$		0			0
$571$	−10.3031			−0.431170			−0.215585	−	0.976485i	$-0.569166\pi$
$571$	−10.3031			−0.431170			−0.215585	+	0.976485i	$0.569166\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.449490	+	2.20204i	−0.0187450	+	0.0918315i
$576$		0			0
$577$		−	23.8990i		−	0.994928i	−0.867485	−	0.497464i	$-0.834265\pi$
$577$		−	23.8990i		−	0.994928i	0.867485	−	0.497464i	$-0.165735\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.4495			−0.516492
$582$		0			0
$583$		−	2.00000i		−	0.0828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			28.9444i			1.19466i	0.801995	+	0.597331i	$0.203772\pi$
$587$			28.9444i			1.19466i	−0.801995	+	0.597331i	$0.796228\pi$
$588$		0			0
$589$	−23.3939			−0.963928
$590$		0			0
$591$		0			0
$592$		0			0
$593$			33.3485i			1.36946i	0.728798	+	0.684729i	$0.240079\pi$
$593$			33.3485i			1.36946i	−0.728798	+	0.684729i	$0.759921\pi$
$594$		0			0
$595$	−1.10102	+	10.8990i	−0.0451374	+	0.446815i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.8434			0.892496			0.446248	−	0.894909i	$-0.352760\pi$
$599$	21.8434			0.892496			0.446248	+	0.894909i	$0.352760\pi$
$600$		0			0
$601$	4.40408			0.179646			0.0898231	−	0.995958i	$-0.471370\pi$
$601$	4.40408			0.179646			0.0898231	+	0.995958i	$0.471370\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.97730	+	19.5732i	−0.0803885	+	0.795764i
$606$		0			0
$607$		−	8.59592i		−	0.348898i	−0.984666	−	0.174449i	$-0.944186\pi$
$607$		−	8.59592i		−	0.348898i	0.984666	−	0.174449i	$-0.0558143\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−28.8990			−1.16913
$612$		0			0
$613$		−	14.1010i		−	0.569535i	−0.958597	−	0.284767i	$-0.908084\pi$
$613$		−	14.1010i		−	0.569535i	0.958597	−	0.284767i	$-0.0919164\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.0454i		−	1.29010i	−0.764141	−	0.645050i	$-0.776837\pi$
$617$		−	32.0454i		−	1.29010i	0.764141	−	0.645050i	$-0.223163\pi$
$618$		0			0
$619$	−35.4949			−1.42666			−0.713330	−	0.700828i	$-0.752814\pi$
$619$	−35.4949			−1.42666			−0.713330	+	0.700828i	$0.752814\pi$
$620$		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
$629$	−48.4949			−1.93362
$630$		0			0
$631$	3.69694			0.147173			0.0735864	−	0.997289i	$-0.476556\pi$
$631$	3.69694			0.147173			0.0735864	+	0.997289i	$0.476556\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	35.1464	+	3.55051i	1.39474	+	0.140898i
$636$		0			0
$637$		−	35.3939i		−	1.40236i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.2020			0.560947			0.280473	−	0.959862i	$-0.409509\pi$
$641$	14.2020			0.560947			0.280473	+	0.959862i	$0.409509\pi$
$642$		0			0
$643$			20.2020i			0.796691i	0.917236	+	0.398345i	$0.130415\pi$
$643$			20.2020i			0.796691i	−0.917236	+	0.398345i	$0.869585\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.75255i			0.0688999i	0.999406	+	0.0344500i	$0.0109679\pi$
$647$			1.75255i			0.0688999i	−0.999406	+	0.0344500i	$0.989032\pi$
$648$		0			0
$649$	21.7980			0.855645
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	9.79796i		−	0.383424i	−0.981451	−	0.191712i	$-0.938596\pi$
$653$		−	9.79796i		−	0.383424i	0.981451	−	0.191712i	$-0.0614039\pi$
$654$		0			0
$655$	−3.79796	+	37.5959i	−0.148399	+	1.46899i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.2020			−0.553233			−0.276616	−	0.960980i	$-0.589213\pi$
$659$	−14.2020			−0.553233			−0.276616	+	0.960980i	$0.589213\pi$
$660$		0			0
$661$	−27.2020			−1.05804			−0.529018	−	0.848610i	$-0.677440\pi$
$661$	−27.2020			−1.05804			−0.529018	+	0.848610i	$0.677440\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.67423	−	0.876276i	−0.336372	−	0.0339805i
$666$		0			0
$667$		−	2.00000i		−	0.0774403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−48.0454			−1.85477
$672$		0			0
$673$			16.3031i			0.628437i	0.949351	+	0.314218i	$0.101742\pi$
$673$			16.3031i			0.628437i	−0.949351	+	0.314218i	$0.898258\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	14.6969i		−	0.564849i	−0.959289	−	0.282425i	$-0.908861\pi$
$677$		−	14.6969i		−	0.564849i	0.959289	−	0.282425i	$-0.0911387\pi$
$678$		0			0
$679$	6.10102			0.234136
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	29.3485i		−	1.12299i	−0.827481	−	0.561494i	$-0.810227\pi$
$683$		−	29.3485i		−	1.12299i	0.827481	−	0.561494i	$-0.189773\pi$
$684$		0			0
$685$	−5.89898	−	0.595918i	−0.225388	−	0.0227689i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.65153			−0.101015
$690$		0			0
$691$	−25.5959			−0.973715			−0.486857	−	0.873482i	$-0.661857\pi$
$691$	−25.5959			−0.973715			−0.486857	+	0.873482i	$0.661857\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.42679	−	43.8207i	0.167918	−	1.66221i
$696$		0			0
$697$		−	26.2020i		−	0.992473i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.449490			−0.0169770			−0.00848850	−	0.999964i	$-0.502702\pi$
$701$	−0.449490			−0.0169770			−0.00848850	+	0.999964i	$0.502702\pi$
$702$		0			0
$703$		−	38.5959i		−	1.45567i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	16.8990i		−	0.635552i
$708$		0			0
$709$	−38.5959			−1.44950			−0.724750	−	0.689012i	$-0.758045\pi$
$709$	−38.5959			−1.44950			−0.724750	+	0.689012i	$0.758045\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.69694i			0.101001i
$714$		0			0
$715$	58.3939	+	5.89898i	2.18381	+	0.220609i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.449490			−0.0167631			−0.00838157	−	0.999965i	$-0.502668\pi$
$719$	−0.449490			−0.0167631			−0.00838157	+	0.999965i	$0.502668\pi$
$720$		0			0
$721$	−8.79796			−0.327653
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.7980	−	4.44949i	−0.809556	−	0.165250i
$726$		0			0
$727$		−	20.2020i		−	0.749252i	−0.927176	−	0.374626i	$-0.877771\pi$
$727$		−	20.2020i		−	0.749252i	0.927176	−	0.374626i	$-0.122229\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	29.3939			1.08717
$732$		0			0
$733$		−	33.7980i		−	1.24836i	−0.781282	−	0.624178i	$-0.785434\pi$
$733$		−	33.7980i		−	1.24836i	0.781282	−	0.624178i	$-0.214566\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			21.3485i			0.786381i
$738$		0			0
$739$	−37.5959			−1.38299			−0.691494	−	0.722382i	$-0.743047\pi$
$739$	−37.5959			−1.38299			−0.691494	+	0.722382i	$0.743047\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			28.8990i			1.06020i	0.847935	+	0.530100i	$0.177846\pi$
$743$			28.8990i			1.06020i	−0.847935	+	0.530100i	$0.822154\pi$
$744$		0			0
$745$	−2.20204	+	21.7980i	−0.0806765	+	0.798615i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.2474			0.374434
$750$		0			0
$751$	−6.10102			−0.222629			−0.111315	−	0.993785i	$-0.535506\pi$
$751$	−6.10102			−0.222629			−0.111315	+	0.993785i	$0.535506\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.57321	+	35.3712i	−0.130043	+	1.28729i
$756$		0			0
$757$		−	13.6969i		−	0.497824i	−0.968526	−	0.248912i	$-0.919927\pi$
$757$		−	13.6969i		−	0.497824i	0.968526	−	0.248912i	$-0.0800729\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.3485			1.20888			0.604441	−	0.796650i	$-0.293397\pi$
$761$	33.3485			1.20888			0.604441	+	0.796650i	$0.293397\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	28.8990i		−	1.04348i
$768$		0			0
$769$	−42.3939			−1.52876			−0.764381	−	0.644765i	$-0.776955\pi$
$769$	−42.3939			−1.52876			−0.764381	+	0.644765i	$0.776955\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			41.3485i			1.48720i	0.668624	+	0.743601i	$0.266884\pi$
$773$			41.3485i			1.48720i	−0.668624	+	0.743601i	$0.733116\pi$
$774$		0			0
$775$	29.3939	+	6.00000i	1.05586	+	0.215526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.8536			0.747157
$780$		0			0
$781$	73.1918			2.61901
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.44949	−	0.449490i	−0.158809	−	0.0160430i
$786$		0			0
$787$		−	40.7980i		−	1.45429i	−0.686484	−	0.727145i	$-0.740847\pi$
$787$		−	40.7980i		−	1.45429i	0.686484	−	0.727145i	$-0.259153\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.65153			−0.0942776
$792$		0			0
$793$			63.6969i			2.26194i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			36.4949i			1.29272i	0.763034	+	0.646358i	$0.223709\pi$
$797$			36.4949i			1.29272i	−0.763034	+	0.646358i	$0.776291\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0

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)

\(f(q)\)	\(=\)	\(q+(-0.224745 + 2.22474i) q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+(-0.224745 + 2.22474i) q^{5} +1.00000i q^{7} +4.44949 q^{11} -5.89898i q^{13} -4.89898i q^{17} +3.89898 q^{19} -0.449490i q^{23} +(-4.89898 - 1.00000i) q^{25} +4.44949 q^{29} -6.00000 q^{31} +(-2.22474 - 0.224745i) q^{35} -9.89898i q^{37} +5.34847 q^{41} +6.00000i q^{43} -4.89898i q^{47} +6.00000 q^{49} -0.449490i q^{53} +(-1.00000 + 9.89898i) q^{55} +4.89898 q^{59} -10.7980 q^{61} +(13.1237 + 1.32577i) q^{65} +4.79796i q^{67} +16.4495 q^{71} +5.89898i q^{73} +4.44949i q^{77} +5.89898 q^{79} +12.4495i q^{83} +(10.8990 + 1.10102i) q^{85} +12.0000 q^{89} +5.89898 q^{91} +(-0.876276 + 8.67423i) q^{95} -6.10102i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} + 8 q^{11} - 4 q^{19} + 8 q^{29} - 24 q^{31} - 4 q^{35} - 8 q^{41} + 24 q^{49} - 4 q^{55} - 4 q^{61} + 28 q^{65} + 56 q^{71} + 4 q^{79} + 24 q^{85} + 48 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) \) 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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