Embedded newform 45.4.e.c.31.4

• Label: 45.4.e.c.31.4
• Level: $45$
• Weight: $4$
• Character: 45.31
• Analytic conductor: $2.655$
• Analytic rank: $0$
• Dimension: $14$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.65508595026\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	x^14 - 2*x^13 + 48*x^12 - 60*x^11 + 1605*x^10 - 1800*x^9 + 23232*x^8 - 2346*x^7 + 209529*x^6 - 55412*x^5 + 765088*x^4 + 276096*x^3 + 1572480*x^2 - 345600*x + 82944
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			31.4
Root			\(0.112625 - 0.195072i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.31
Dual form			45.4.e.c.16.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.112625 + 0.195072i) q^{2} +(2.06755 + 4.76710i) q^{3} +(3.97463 - 6.88426i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-0.697071 + 0.940215i) q^{6} +(15.5970 + 27.0148i) q^{7} +3.59257 q^{8} +(-18.4505 + 19.7124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} - 5 q^{3} - 36 q^{4} + 35 q^{5} - 31 q^{6} - 22 q^{7} - 36 q^{8} + 17 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.112625	+	0.195072i	0.0398189	+	0.0689684i	0.885248	−	0.465119i	\(-0.153989\pi\)
							−0.845429	+	0.534088i	\(0.820655\pi\)
\(3\)	2.06755	+	4.76710i	0.397899	+	0.917429i
							\(5\)	2.50000	−	4.33013i	0.223607	−	0.387298i
\(7\)	15.5970	+	27.0148i	0.842159	+	1.45866i								0.888066	+	0.459717i	\(0.152049\pi\)
							−0.0459062	+	0.998946i	\(0.514618\pi\)
\(11\)	−9.06424	−	15.6997i	−0.248452	−	0.430331i	0.714645	−	0.699488i	\(-0.246588\pi\)
							−0.963096	+	0.269156i	\(0.913255\pi\)
\(13\)	25.0781	−	43.4366i	0.535032	−	0.926702i	−0.464130	−	0.885767i	\(-0.653633\pi\)
							0.999162	−	0.0409353i	\(-0.0130337\pi\)
\(17\)	−131.631			−1.87795			−0.938977	−	0.343981i	\(-0.888224\pi\)
							−0.938977	+	0.343981i	\(0.888224\pi\)
\(19\)	23.2428			0.280646			0.140323	−	0.990106i	\(-0.455186\pi\)
							0.140323	+	0.990106i	\(0.455186\pi\)
\(23\)	16.4928	−	28.5664i	0.149521	−	0.258978i	−0.781529	−	0.623868i	\(-0.785560\pi\)
							0.931051	+	0.364890i	\(0.118893\pi\)
\(29\)	−62.9087	−	108.961i	−0.402823	−	0.697709i	0.591243	−	0.806494i	\(-0.298637\pi\)
							−0.994065	+	0.108784i	\(0.965304\pi\)
\(31\)	−62.5563	+	108.351i	−0.362434	+	0.627754i	−0.988361	−	0.152128i	\(-0.951387\pi\)
							0.625927	+	0.779882i	\(0.284721\pi\)
\(37\)	99.9894			0.444274			0.222137	−	0.975015i	\(-0.428697\pi\)
							0.222137	+	0.975015i	\(0.428697\pi\)
\(41\)	−122.663	+	212.458i	−0.467237	+	0.809278i	−0.999299	−	0.0374272i	\(-0.988084\pi\)
							0.532063	+	0.846705i	\(0.321417\pi\)
\(43\)	−69.5882	−	120.530i	−0.246793	−	0.427458i	0.715841	−	0.698263i	\(-0.246043\pi\)
							−0.962634	+	0.270805i	\(0.912710\pi\)
\(47\)	236.480	+	409.596i	0.733919	+	1.27119i	0.955196	+	0.295975i	\(0.0956444\pi\)
							−0.221276	+	0.975211i	\(0.571022\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.e.c.31.4	yes	14
3.2	odd	2		135.4.e.c.91.4		14
5.2	odd	4		225.4.k.d.49.7		28
5.3	odd	4		225.4.k.d.49.8		28
5.4	even	2		225.4.e.d.76.4		14
9.2	odd	6		135.4.e.c.46.4		14
9.4	even	3		405.4.a.m.1.4		7
9.5	odd	6		405.4.a.n.1.4		7
9.7	even	3	inner	45.4.e.c.16.4	✓	14
45.4	even	6		2025.4.a.bb.1.4		7
45.7	odd	12		225.4.k.d.124.8		28
45.14	odd	6		2025.4.a.ba.1.4		7
45.34	even	6		225.4.e.d.151.4		14
45.43	odd	12		225.4.k.d.124.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.4	✓	14	9.7	even	3	inner
45.4.e.c.31.4	yes	14	1.1	even	1	trivial
135.4.e.c.46.4		14	9.2	odd	6
135.4.e.c.91.4		14	3.2	odd	2
225.4.e.d.76.4		14	5.4	even	2
225.4.e.d.151.4		14	45.34	even	6
225.4.k.d.49.7		28	5.2	odd	4
225.4.k.d.49.8		28	5.3	odd	4
225.4.k.d.124.7		28	45.43	odd	12
225.4.k.d.124.8		28	45.7	odd	12
405.4.a.m.1.4		7	9.4	even	3
405.4.a.n.1.4		7	9.5	odd	6
2025.4.a.ba.1.4		7	45.14	odd	6
2025.4.a.bb.1.4		7	45.4	even	6