LMFDB - Embedded newform 1080.2.q.d.721.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,2,Mod(361,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1080.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.3
Root			$-1.62241 - 0.606458i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.2.q.d.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(2.62241 - 4.54214i) q^{7} +(1.33641 - 2.31473i) q^{11} +(-1.90841 - 3.30545i) q^{13} -3.52884 q^{17} -4.67282 q^{19} +(-2.47842 - 4.29275i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.928007 + 1.60735i) q^{29} +(4.33641 + 7.51089i) q^{31} +5.24482 q^{35} -2.67282 q^{37} +(-1.83641 - 3.18076i) q^{41} +(-1.76442 + 3.05606i) q^{43} +(4.63164 - 8.02224i) q^{47} +(-10.2541 - 17.7605i) q^{49} +2.85601 q^{53} +2.67282 q^{55} +(2.10083 + 3.63875i) q^{59} +(3.98040 - 6.89425i) q^{61} +(1.90841 - 3.30545i) q^{65} +(0.429983 + 0.744753i) q^{67} +15.1625 q^{71} +6.28797 q^{73} +(-7.00924 - 12.1404i) q^{77} +(-2.81681 + 4.87886i) q^{79} +(-1.94958 + 3.37678i) q^{83} +(-1.76442 - 3.05606i) q^{85} +11.0000 q^{89} -20.0185 q^{91} +(-2.33641 - 4.04678i) q^{95} +(1.91764 - 3.32145i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} + 5 q^{7} - 2 q^{11} - 4 q^{17} - 8 q^{19} - 7 q^{23} - 3 q^{25} - 7 q^{29} + 16 q^{31} + 10 q^{35} + 4 q^{37} - q^{41} - 2 q^{43} - 13 q^{47} - 10 q^{49} + 20 q^{53} - 4 q^{55} - 6 q^{59}+ \cdots - 30 q^{97}+O(q^{100}) $ 6 * q + 3 * q^5 + 5 * q^7 - 2 * q^11 - 4 * q^17 - 8 * q^19 - 7 * q^23 - 3 * q^25 - 7 * q^29 + 16 * q^31 + 10 * q^35 + 4 * q^37 - q^41 - 2 * q^43 - 13 * q^47 - 10 * q^49 + 20 * q^53 - 4 * q^55 - 6 * q^59 + 11 * q^61 - q^67 + 28 * q^71 + 32 * q^73 - 12 * q^77 + 6 * q^79 - 21 * q^83 - 2 * q^85 + 66 * q^89 - 60 * q^91 - 4 * q^95 - 30 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	2.62241	−	4.54214i	0.991177	−	1.71677i	0.380803	−	0.924656i	$-0.375648\pi$
$7$	2.62241	−	4.54214i	0.991177	−	1.71677i	0.610374	−	0.792113i	$-0.291019\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	−0.591136	−	0.806572i	$-0.701321\pi$
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	0.994080	+	0.108653i	$0.0346538\pi$
$12$		0			0
$13$	−1.90841	−	3.30545i	−0.529296	−	0.916768i	−0.999416	−	0.0341656i	$-0.989123\pi$
$13$	−1.90841	−	3.30545i	−0.529296	−	0.916768i	0.470120	−	0.882603i	$-0.344211\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.52884			−0.855869			−0.427934	−	0.903810i	$-0.640759\pi$
$17$	−3.52884			−0.855869			−0.427934	+	0.903810i	$0.640759\pi$
$18$		0			0
$19$	−4.67282			−1.07202			−0.536010	−	0.844212i	$-0.680069\pi$
$19$	−4.67282			−1.07202			−0.536010	+	0.844212i	$0.680069\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.47842	−	4.29275i	−0.516787	−	0.895101i	−0.999810	−	0.0194933i	$-0.993795\pi$
$23$	−2.47842	−	4.29275i	−0.516787	−	0.895101i	0.483023	−	0.875608i	$-0.339539\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.928007	+	1.60735i	−0.172327	+	0.298478i	−0.939233	−	0.343281i	$-0.888462\pi$
$29$	−0.928007	+	1.60735i	−0.172327	+	0.298478i	0.766906	+	0.641759i	$0.221795\pi$
$30$		0			0
$31$	4.33641	+	7.51089i	0.778843	+	1.34899i	0.932609	+	0.360888i	$0.117526\pi$
$31$	4.33641	+	7.51089i	0.778843	+	1.34899i	−0.153767	+	0.988107i	$0.549140\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.24482			0.886536
$36$		0			0
$37$	−2.67282			−0.439410			−0.219705	−	0.975566i	$-0.570509\pi$
$37$	−2.67282			−0.439410			−0.219705	+	0.975566i	$0.570509\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.83641	−	3.18076i	−0.286799	−	0.496751i	0.686245	−	0.727371i	$-0.259258\pi$
$41$	−1.83641	−	3.18076i	−0.286799	−	0.496751i	−0.973044	+	0.230620i	$0.925925\pi$
$42$		0			0
$43$	−1.76442	+	3.05606i	−0.269071	+	0.466045i	−0.968622	−	0.248538i	$-0.920050\pi$
$43$	−1.76442	+	3.05606i	−0.269071	+	0.466045i	0.699551	+	0.714583i	$0.253383\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.63164	−	8.02224i	0.675595	−	1.17016i	−0.300700	−	0.953719i	$-0.597220\pi$
$47$	4.63164	−	8.02224i	0.675595	−	1.17016i	0.976295	−	0.216446i	$-0.0694464\pi$
$48$		0			0
$49$	−10.2541	−	17.7605i	−1.46486	−	2.53722i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.85601			0.392304			0.196152	−	0.980574i	$-0.437155\pi$
$53$	2.85601			0.392304			0.196152	+	0.980574i	$0.437155\pi$
$54$		0			0
$55$	2.67282			0.360403
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.10083	+	3.63875i	0.273505	+	0.473724i	0.969757	−	0.244073i	$-0.0784837\pi$
$59$	2.10083	+	3.63875i	0.273505	+	0.473724i	−0.696252	+	0.717797i	$0.745150\pi$
$60$		0			0
$61$	3.98040	−	6.89425i	0.509638	−	0.882719i	−0.490300	−	0.871554i	$-0.663113\pi$
$61$	3.98040	−	6.89425i	0.509638	−	0.882719i	0.999938	−	0.0111647i	$-0.00355392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.90841	−	3.30545i	0.236709	−	0.409991i
$66$		0			0
$67$	0.429983	+	0.744753i	0.0525308	+	0.0909860i	0.891095	−	0.453817i	$-0.149938\pi$
$67$	0.429983	+	0.744753i	0.0525308	+	0.0909860i	−0.838564	+	0.544803i	$0.816605\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.1625			1.79945			0.899726	−	0.436454i	$-0.143766\pi$
$71$	15.1625			1.79945			0.899726	+	0.436454i	$0.143766\pi$
$72$		0			0
$73$	6.28797			0.735952			0.367976	−	0.929835i	$-0.380051\pi$
$73$	6.28797			0.735952			0.367976	+	0.929835i	$0.380051\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.00924	−	12.1404i	−0.798777	−	1.38352i
$78$		0			0
$79$	−2.81681	+	4.87886i	−0.316916	+	0.548914i	−0.979843	−	0.199770i	$-0.935981\pi$
$79$	−2.81681	+	4.87886i	−0.316916	+	0.548914i	0.662927	+	0.748684i	$0.269314\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.94958	+	3.37678i	−0.213995	+	0.370650i	−0.952961	−	0.303093i	$-0.901981\pi$
$83$	−1.94958	+	3.37678i	−0.213995	+	0.370650i	0.738966	+	0.673742i	$0.235314\pi$
$84$		0			0
$85$	−1.76442	−	3.05606i	−0.191378	−	0.331477i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.0000			1.16600			0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000			1.16600			0.582999	+	0.812473i	$0.301879\pi$
$90$		0			0
$91$	−20.0185			−2.09851
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.33641	−	4.04678i	−0.239711	−	0.415191i
$96$		0			0
$97$	1.91764	−	3.32145i	0.194707	−	0.337242i	−0.752097	−	0.659052i	$-0.770958\pi$
$97$	1.91764	−	3.32145i	0.194707	−	0.337242i	0.946804	+	0.321810i	$0.104291\pi$
$98$		0			0
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.q.d.721.3		6
3.2	odd	2		360.2.q.d.241.3	yes	6
4.3	odd	2		2160.2.q.j.721.1		6
9.2	odd	6		3240.2.a.r.1.1		3
9.4	even	3	inner	1080.2.q.d.361.3		6
9.5	odd	6		360.2.q.d.121.3	✓	6
9.7	even	3		3240.2.a.q.1.1		3
12.11	even	2		720.2.q.j.241.1		6
36.7	odd	6		6480.2.a.bu.1.3		3
36.11	even	6		6480.2.a.bx.1.3		3
36.23	even	6		720.2.q.j.481.1		6
36.31	odd	6		2160.2.q.j.1441.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.d.121.3	✓	6	9.5	odd	6
360.2.q.d.241.3	yes	6	3.2	odd	2
720.2.q.j.241.1		6	12.11	even	2
720.2.q.j.481.1		6	36.23	even	6
1080.2.q.d.361.3		6	9.4	even	3	inner
1080.2.q.d.721.3		6	1.1	even	1	trivial
2160.2.q.j.721.1		6	4.3	odd	2
2160.2.q.j.1441.1		6	36.31	odd	6
3240.2.a.q.1.1		3	9.7	even	3
3240.2.a.r.1.1		3	9.2	odd	6
6480.2.a.bu.1.3		3	36.7	odd	6
6480.2.a.bx.1.3		3	36.11	even	6

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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(2.62241 - 4.54214i) q^{7} +(1.33641 - 2.31473i) q^{11} +(-1.90841 - 3.30545i) q^{13} -3.52884 q^{17} -4.67282 q^{19} +(-2.47842 - 4.29275i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.928007 + 1.60735i) q^{29} +(4.33641 + 7.51089i) q^{31} +5.24482 q^{35} -2.67282 q^{37} +(-1.83641 - 3.18076i) q^{41} +(-1.76442 + 3.05606i) q^{43} +(4.63164 - 8.02224i) q^{47} +(-10.2541 - 17.7605i) q^{49} +2.85601 q^{53} +2.67282 q^{55} +(2.10083 + 3.63875i) q^{59} +(3.98040 - 6.89425i) q^{61} +(1.90841 - 3.30545i) q^{65} +(0.429983 + 0.744753i) q^{67} +15.1625 q^{71} +6.28797 q^{73} +(-7.00924 - 12.1404i) q^{77} +(-2.81681 + 4.87886i) q^{79} +(-1.94958 + 3.37678i) q^{83} +(-1.76442 - 3.05606i) q^{85} +11.0000 q^{89} -20.0185 q^{91} +(-2.33641 - 4.04678i) q^{95} +(1.91764 - 3.32145i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} + 5 q^{7} - 2 q^{11} - 4 q^{17} - 8 q^{19} - 7 q^{23} - 3 q^{25} - 7 q^{29} + 16 q^{31} + 10 q^{35} + 4 q^{37} - q^{41} - 2 q^{43} - 13 q^{47} - 10 q^{49} + 20 q^{53} - 4 q^{55} - 6 q^{59}+ \cdots - 30 q^{97}+O(q^{100}) \) 6 * q + 3 * q^5 + 5 * q^7 - 2 * q^11 - 4 * q^17 - 8 * q^19 - 7 * q^23 - 3 * q^25 - 7 * q^29 + 16 * q^31 + 10 * q^35 + 4 * q^37 - q^41 - 2 * q^43 - 13 * q^47 - 10 * q^49 + 20 * q^53 - 4 * q^55 - 6 * q^59 + 11 * q^61 - q^67 + 28 * q^71 + 32 * q^73 - 12 * q^77 + 6 * q^79 - 21 * q^83 - 2 * q^85 + 66 * q^89 - 60 * q^91 - 4 * q^95 - 30 * q^97

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