Embedded newform 33.7.g.a.13.7

• Label: 33.7.g.a.13.7
• Level: $33$
• Weight: $7$
• Character: 33.13
• 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			13.7
Character	\(\chi\)	\(=\)	33.13
Dual form			33.7.g.a.28.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.674572 - 0.219182i) q^{2} +(12.6113 - 9.16267i) q^{3} +(-51.3701 - 37.3225i) q^{4} +(-8.69665 - 26.7655i) q^{5} +(-10.5155 + 3.41670i) q^{6} +(-301.986 + 415.648i) q^{7} +(53.1545 + 73.1609i) q^{8} +(75.0911 - 231.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.674572	−	0.219182i	−0.0843215	−	0.0273977i	0.266552	−	0.963820i	\(-0.414115\pi\)
							−0.350874	+	0.936423i	\(0.614115\pi\)
\(3\)	12.6113	−	9.16267i	0.467086	−	0.339358i
							\(5\)	−8.69665	−	26.7655i	−0.0695732	−	0.214124i	0.910225	−	0.414115i	\(-0.135909\pi\)
−0.979798	+	0.199991i	\(0.935909\pi\)
\(7\)	−301.986	+	415.648i	−0.880425	+	1.21180i	0.0958775	+	0.995393i	\(0.469434\pi\)
							−0.976303	+	0.216408i	\(0.930566\pi\)
\(11\)	−1328.85	+	75.6743i	−0.998382	+	0.0568552i
							\(13\)	−212.478	−	69.0384i	−0.0967130	−	0.0314240i	0.260261	−	0.965538i	\(-0.416191\pi\)
−0.356974	+	0.934114i	\(0.616191\pi\)
\(17\)	−7428.95	+	2413.81i	−1.51210	+	0.491311i	−0.943520	−	0.331316i	\(-0.892507\pi\)
							−0.568581	+	0.822627i	\(0.692507\pi\)
\(19\)	−7547.57	−	10388.3i	−1.10039	−	1.51456i	−0.834868	−	0.550451i	\(-0.814456\pi\)
							−0.265521	−	0.964105i	\(-0.585544\pi\)
\(23\)	17163.7			1.41068			0.705338	−	0.708871i	\(-0.250795\pi\)
							0.705338	+	0.708871i	\(0.250795\pi\)
\(29\)	−14052.5	+	19341.6i	−0.576181	+	0.793046i	−0.993270	−	0.115820i	\(-0.963050\pi\)
							0.417089	+	0.908866i	\(0.363050\pi\)
\(31\)	9594.33	−	29528.3i	0.322055	−	0.991182i	−0.650698	−	0.759337i	\(-0.725524\pi\)
							0.972753	−	0.231845i	\(-0.0744764\pi\)
\(37\)	31862.0	+	23149.1i	0.629026	+	0.457014i	0.856063	−	0.516872i	\(-0.172904\pi\)
							−0.227037	+	0.973886i	\(0.572904\pi\)
\(41\)	−50088.3	−	68940.6i	−0.726749	−	1.00028i	−0.999273	−	0.0381355i	\(-0.987858\pi\)
							0.272523	−	0.962149i	\(-0.412142\pi\)
\(43\)		−	32717.8i		−	0.411508i	−0.978604	−	0.205754i	\(-0.934035\pi\)
							0.978604	−	0.205754i	\(-0.0659647\pi\)
\(47\)	−130381.	+	94727.0i	−1.25580	+	0.912389i	−0.998543	−	0.0539548i	\(-0.982817\pi\)
							−0.257253	+	0.966344i	\(0.582817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.13.7	✓	48
3.2	odd	2		99.7.k.b.46.6		48
11.6	odd	10	inner	33.7.g.a.28.7	yes	48
33.17	even	10		99.7.k.b.28.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.13.7	✓	48	1.1	even	1	trivial
33.7.g.a.28.7	yes	48	11.6	odd	10	inner
99.7.k.b.28.6		48	33.17	even	10
99.7.k.b.46.6		48	3.2	odd	2