Embedded newform 33.8.f.a.2.2

• Label: 33.8.f.a.2.2
• Level: $33$
• Weight: $8$
• Character: 33.2
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	33.2
Dual form			33.8.f.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.54483 + 20.1429i) q^{2} +(34.8194 - 31.2188i) q^{3} +(-259.349 - 188.428i) q^{4} +(-311.568 + 101.235i) q^{5} +(400.951 + 905.686i) q^{6} +(7.55299 - 10.3958i) q^{7} +(3299.65 - 2397.34i) q^{8} +(237.777 - 2174.04i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.54483	+	20.1429i	−0.578487	+	1.78040i	0.0454984	+	0.998964i	\(0.485512\pi\)
							−0.623986	+	0.781436i	\(0.714488\pi\)
\(3\)	34.8194	−	31.2188i	0.744554	−	0.667562i
							\(5\)	−311.568	+	101.235i	−1.11470	+	0.362188i	−0.807743	−	0.589535i	\(-0.799311\pi\)
−0.306957	+	0.951723i	\(0.599311\pi\)
\(7\)	7.55299	−	10.3958i	0.00832292	−	0.0114555i	−0.804835	−	0.593498i	\(-0.797746\pi\)
							0.813158	+	0.582043i	\(0.197746\pi\)
\(11\)	4077.15	−	1692.35i	0.923596	−	0.383368i
							\(13\)	11822.2	+	3841.28i	1.49244	+	0.484924i	0.937804	−	0.347166i	\(-0.112856\pi\)
0.554640	+	0.832091i	\(0.312856\pi\)
\(17\)	−9142.45	−	28137.6i	−0.451327	−	1.38904i	−0.875394	−	0.483411i	\(-0.839398\pi\)
							0.424066	−	0.905631i	\(-0.360602\pi\)
\(19\)	−11899.8	−	16378.6i	−0.398017	−	0.547823i	0.562228	−	0.826982i	\(-0.309944\pi\)
							−0.960245	+	0.279159i	\(0.909944\pi\)
\(23\)		−	25001.6i		−	0.428471i	−0.976782	−	0.214235i	\(-0.931274\pi\)
							0.976782	−	0.214235i	\(-0.0687259\pi\)
\(29\)	−7951.91	−	5777.40i	−0.0605450	−	0.0439885i	0.557101	−	0.830445i	\(-0.311914\pi\)
							−0.617646	+	0.786456i	\(0.711914\pi\)
\(31\)	62537.4	−	192470.i	0.377028	−	1.16037i	−0.565072	−	0.825042i	\(-0.691151\pi\)
							0.942100	−	0.335332i	\(-0.108849\pi\)
\(37\)	−100462.	−	72990.0i	−0.326059	−	0.236896i	0.412697	−	0.910868i	\(-0.364587\pi\)
							−0.738757	+	0.673972i	\(0.764587\pi\)
\(41\)	−101274.	+	73580.0i	−0.229485	+	0.166731i	−0.696586	−	0.717473i	\(-0.745299\pi\)
							0.467101	+	0.884204i	\(0.345299\pi\)
\(43\)		−	629948.i		−	1.20827i	−0.796880	−	0.604137i	\(-0.793518\pi\)
							0.796880	−	0.604137i	\(-0.206482\pi\)
\(47\)	148190.	+	203966.i	0.208198	+	0.286560i	0.900327	−	0.435213i	\(-0.143327\pi\)
							−0.692129	+	0.721774i	\(0.743327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.f.a.2.2	✓	104
3.2	odd	2	inner	33.8.f.a.2.25	yes	104
11.6	odd	10	inner	33.8.f.a.17.25	yes	104
33.17	even	10	inner	33.8.f.a.17.2	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.f.a.2.2	✓	104	1.1	even	1	trivial
33.8.f.a.2.25	yes	104	3.2	odd	2	inner
33.8.f.a.17.2	yes	104	33.17	even	10	inner
33.8.f.a.17.25	yes	104	11.6	odd	10	inner