Embedded newform 33.7.g.a.7.7

• Label: 33.7.g.a.7.7
• Level: $33$
• Weight: $7$
• Character: 33.7
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	33.7
Dual form			33.7.g.a.19.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.37250 - 3.26547i) q^{2} +(4.81710 - 14.8255i) q^{3} +(14.7426 + 45.3730i) q^{4} +(26.5586 - 19.2960i) q^{5} +(-36.9836 - 50.9036i) q^{6} +(457.293 - 148.583i) q^{7} +(428.823 + 139.333i) q^{8} +(-196.591 - 142.832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.37250	−	3.26547i	0.296562	−	0.408183i	−0.634569	−	0.772866i	\(-0.718823\pi\)
							0.931132	+	0.364683i	\(0.118823\pi\)
\(3\)	4.81710	−	14.8255i	0.178411	−	0.549093i
							\(5\)	26.5586	−	19.2960i	0.212469	−	0.154368i	−0.476461	−	0.879196i	\(-0.658081\pi\)
0.688930	+	0.724828i	\(0.258081\pi\)
\(7\)	457.293	−	148.583i	1.33321	−	0.433188i	0.446202	−	0.894932i	\(-0.352776\pi\)
							0.887013	+	0.461745i	\(0.152776\pi\)
\(11\)	−1202.12	−	571.378i	−0.903169	−	0.429285i
							\(13\)	2080.68	−	2863.81i	0.947055	−	1.30351i	−0.00576797	−	0.999983i	\(-0.501836\pi\)
0.952823	−	0.303526i	\(-0.0981640\pi\)
\(17\)	1764.20	+	2428.22i	0.359089	+	0.494243i	0.949895	−	0.312571i	\(-0.101190\pi\)
							−0.590806	+	0.806814i	\(0.701190\pi\)
\(19\)	4849.47	+	1575.69i	0.707023	+	0.229726i	0.640388	−	0.768052i	\(-0.278774\pi\)
							0.0666351	+	0.997777i	\(0.478774\pi\)
\(23\)	−1316.98			−0.108242			−0.0541210	−	0.998534i	\(-0.517236\pi\)
							−0.0541210	+	0.998534i	\(0.517236\pi\)
\(29\)	−29198.9	+	9487.31i	−1.19722	+	0.388999i	−0.838736	−	0.544538i	\(-0.816705\pi\)
							−0.358481	+	0.933537i	\(0.616705\pi\)
\(31\)	−11673.8	−	8481.52i	−0.391857	−	0.284701i	0.374359	−	0.927284i	\(-0.377863\pi\)
							−0.766216	+	0.642583i	\(0.777863\pi\)
\(37\)	20051.5	+	61712.1i	0.395860	+	1.21833i	0.928290	+	0.371857i	\(0.121279\pi\)
							−0.532431	+	0.846474i	\(0.678721\pi\)
\(41\)	−78747.2	−	25586.5i	−1.14257	−	0.371244i	−0.324231	−	0.945978i	\(-0.605106\pi\)
							−0.818341	+	0.574734i	\(0.805106\pi\)
\(43\)			106491.i			1.33939i	0.742638	+	0.669693i	\(0.233574\pi\)
							−0.742638	+	0.669693i	\(0.766426\pi\)
\(47\)	59354.1	−	182673.i	0.571686	−	1.75947i	−0.0755105	−	0.997145i	\(-0.524059\pi\)
							0.647196	−	0.762323i	\(-0.275941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.7.7	✓	48
3.2	odd	2		99.7.k.b.73.6		48
11.8	odd	10	inner	33.7.g.a.19.7	yes	48
33.8	even	10		99.7.k.b.19.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.7.7	✓	48	1.1	even	1	trivial
33.7.g.a.19.7	yes	48	11.8	odd	10	inner
99.7.k.b.19.6		48	33.8	even	10
99.7.k.b.73.6		48	3.2	odd	2