Embedded newform 33.7.h.a.5.7

• Label: 33.7.h.a.5.7
• Level: $33$
• Weight: $7$
• Character: 33.5
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	33.5
Dual form			33.7.h.a.20.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.30214 + 2.04769i) q^{2} +(25.8420 - 7.82243i) q^{3} +(-16.2532 + 11.8086i) q^{4} +(66.8441 + 21.7190i) q^{5} +(-146.842 + 102.214i) q^{6} +(-126.894 + 92.1938i) q^{7} +(327.525 - 450.800i) q^{8} +(606.619 - 404.295i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 33 q^{3} + 634 q^{4} + 109 q^{6} + 402 q^{7} - 2471 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.30214	+	2.04769i	−0.787767	+	0.255961i	−0.675153	−	0.737678i	\(-0.735922\pi\)
							−0.112614	+	0.993639i	\(0.535922\pi\)
\(3\)	25.8420	−	7.82243i	0.957112	−	0.289720i
							\(5\)	66.8441	+	21.7190i	0.534753	+	0.173752i	0.563930	−	0.825823i	\(-0.309289\pi\)
−0.0291773	+	0.999574i	\(0.509289\pi\)
\(7\)	−126.894	+	92.1938i	−0.369953	+	0.268786i	−0.757191	−	0.653193i	\(-0.773429\pi\)
							0.387239	+	0.921980i	\(0.373429\pi\)
\(11\)	231.398	+	1310.73i	0.173852	+	0.984772i
							\(13\)	1250.80	+	3849.56i	0.569321	+	1.75219i	0.654750	+	0.755846i	\(0.272774\pi\)
−0.0854288	+	0.996344i	\(0.527226\pi\)
\(17\)	648.460	+	210.698i	0.131989	+	0.0428857i	0.374266	−	0.927321i	\(-0.377895\pi\)
							−0.242278	+	0.970207i	\(0.577895\pi\)
\(19\)	9251.30	+	6721.47i	1.34878	+	0.979948i	0.999071	+	0.0430912i	\(0.0137206\pi\)
							0.349712	+	0.936857i	\(0.386279\pi\)
\(23\)		−	570.368i		−	0.0468782i	−0.999725	−	0.0234391i	\(-0.992538\pi\)
							0.999725	−	0.0234391i	\(-0.00746159\pi\)
\(29\)	−14380.9	−	19793.6i	−0.589647	−	0.811579i	0.405065	−	0.914288i	\(-0.367249\pi\)
							−0.994711	+	0.102709i	\(0.967249\pi\)
\(31\)	−455.411	−	1401.61i	−0.0152869	−	0.0470481i	0.943122	−	0.332447i	\(-0.107874\pi\)
							−0.958409	+	0.285398i	\(0.907874\pi\)
\(37\)	40401.2	−	29353.2i	0.797607	−	0.579495i	−0.112604	−	0.993640i	\(-0.535919\pi\)
							0.910211	+	0.414145i	\(0.135919\pi\)
\(41\)	−14168.7	+	19501.6i	−0.205579	+	0.282956i	−0.899340	−	0.437250i	\(-0.855953\pi\)
							0.693761	+	0.720206i	\(0.255953\pi\)
\(43\)	−90058.1			−1.13271			−0.566353	−	0.824163i	\(-0.691646\pi\)
							−0.566353	+	0.824163i	\(0.691646\pi\)
\(47\)	82389.5	−	113399.i	0.793558	−	1.09224i	−0.200098	−	0.979776i	\(-0.564126\pi\)
							0.993656	−	0.112463i	\(-0.0358739\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.h.a.5.7	✓	88
3.2	odd	2	inner	33.7.h.a.5.16	yes	88
11.9	even	5	inner	33.7.h.a.20.16	yes	88
33.20	odd	10	inner	33.7.h.a.20.7	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.h.a.5.7	✓	88	1.1	even	1	trivial
33.7.h.a.5.16	yes	88	3.2	odd	2	inner
33.7.h.a.20.7	yes	88	33.20	odd	10	inner
33.7.h.a.20.16	yes	88	11.9	even	5	inner