Embedded newform 45.3.h.a.29.7

• Label: 45.3.h.a.29.7
• Level: $45$
• Weight: $3$
• Character: 45.29
• Analytic conductor: $1.226$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	45.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.22616118962\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 3*x^18 - 19*x^16 + 66*x^14 + 109*x^12 - 813*x^10 + 981*x^8 + 5346*x^6 - 13851*x^4 - 19683*x^2 + 59049
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.7
Root			\(-1.72212 - 0.185238i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.29
Dual form			45.3.h.a.14.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.668935 - 1.15863i) q^{2} +(-0.320841 + 2.98279i) q^{3} +(1.10505 + 1.91401i) q^{4} +(4.41869 - 2.33992i) q^{5} +(3.24133 + 2.36703i) q^{6} +(-7.10792 - 4.10376i) q^{7} +8.30831 q^{8} +(-8.79412 - 1.91401i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.668935	−	1.15863i	0.334467	−	0.579314i	−0.648915	−	0.760861i	\(-0.724777\pi\)
							0.983382	+	0.181547i	\(0.0581103\pi\)
\(3\)	−0.320841	+	2.98279i	−0.106947	+	0.994265i
							\(5\)	4.41869	−	2.33992i	0.883737	−	0.467983i
\(7\)	−7.10792	−	4.10376i	−1.01542	−	0.586251i								−0.102644	−	0.994718i	\(-0.532730\pi\)
							−0.912773	+	0.408467i	\(0.866064\pi\)
\(11\)	−5.67242	−	3.27497i	−0.515675	−	0.297725i	0.219489	−	0.975615i	\(-0.429561\pi\)
							−0.735163	+	0.677890i	\(0.762894\pi\)
\(13\)	−1.29771	+	0.749233i	−0.0998238	+	0.0576333i	−0.549081	−	0.835769i	\(-0.685022\pi\)
							0.449257	+	0.893403i	\(0.351689\pi\)
\(17\)	15.1237			0.889630			0.444815	−	0.895622i	\(-0.353269\pi\)
							0.444815	+	0.895622i	\(0.353269\pi\)
\(19\)	−25.9980			−1.36831			−0.684157	−	0.729334i	\(-0.739830\pi\)
							−0.684157	+	0.729334i	\(0.739830\pi\)
\(23\)	−11.6053	−	20.1010i	−0.504580	−	0.873958i	−0.999986	−	0.00529637i	\(-0.998314\pi\)
							0.495406	−	0.868661i	\(-0.335019\pi\)
\(29\)	6.96344	+	4.02034i	0.240119	+	0.138633i	0.615231	−	0.788347i	\(-0.289063\pi\)
							−0.375113	+	0.926979i	\(0.622396\pi\)
\(31\)	22.5107	+	38.9897i	0.726152	+	1.25773i	0.958498	+	0.285099i	\(0.0920264\pi\)
							−0.232346	+	0.972633i	\(0.574640\pi\)
\(37\)			62.8487i			1.69861i	0.527900	+	0.849307i	\(0.322980\pi\)
							−0.527900	+	0.849307i	\(0.677020\pi\)
\(41\)	9.97361	−	5.75827i	0.243259	−	0.140446i	−0.373415	−	0.927664i	\(-0.621813\pi\)
							0.616674	+	0.787219i	\(0.288480\pi\)
\(43\)	36.9366	+	21.3253i	0.858990	+	0.495938i	0.863674	−	0.504051i	\(-0.168158\pi\)
							−0.00468401	+	0.999989i	\(0.501491\pi\)
\(47\)	8.25020	−	14.2898i	0.175536	−	0.304037i	−0.764811	−	0.644255i	\(-0.777167\pi\)
							0.940347	+	0.340218i	\(0.110501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.h.a.29.7	yes	20
3.2	odd	2		135.3.h.a.89.4		20
5.2	odd	4		225.3.j.e.101.7		20
5.3	odd	4		225.3.j.e.101.4		20
5.4	even	2	inner	45.3.h.a.29.4	yes	20
9.2	odd	6		405.3.d.a.404.13		20
9.4	even	3		135.3.h.a.44.7		20
9.5	odd	6	inner	45.3.h.a.14.4	✓	20
9.7	even	3		405.3.d.a.404.8		20
15.2	even	4		675.3.j.e.251.4		20
15.8	even	4		675.3.j.e.251.7		20
15.14	odd	2		135.3.h.a.89.7		20
45.4	even	6		135.3.h.a.44.4		20
45.13	odd	12		675.3.j.e.476.7		20
45.14	odd	6	inner	45.3.h.a.14.7	yes	20
45.22	odd	12		675.3.j.e.476.4		20
45.23	even	12		225.3.j.e.176.4		20
45.29	odd	6		405.3.d.a.404.7		20
45.32	even	12		225.3.j.e.176.7		20
45.34	even	6		405.3.d.a.404.14		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.4	✓	20	9.5	odd	6	inner
45.3.h.a.14.7	yes	20	45.14	odd	6	inner
45.3.h.a.29.4	yes	20	5.4	even	2	inner
45.3.h.a.29.7	yes	20	1.1	even	1	trivial
135.3.h.a.44.4		20	45.4	even	6
135.3.h.a.44.7		20	9.4	even	3
135.3.h.a.89.4		20	3.2	odd	2
135.3.h.a.89.7		20	15.14	odd	2
225.3.j.e.101.4		20	5.3	odd	4
225.3.j.e.101.7		20	5.2	odd	4
225.3.j.e.176.4		20	45.23	even	12
225.3.j.e.176.7		20	45.32	even	12
405.3.d.a.404.7		20	45.29	odd	6
405.3.d.a.404.8		20	9.7	even	3
405.3.d.a.404.13		20	9.2	odd	6
405.3.d.a.404.14		20	45.34	even	6
675.3.j.e.251.4		20	15.2	even	4
675.3.j.e.251.7		20	15.8	even	4
675.3.j.e.476.4		20	45.22	odd	12
675.3.j.e.476.7		20	45.13	odd	12