Embedded newform 33.7.h.a.20.10

• Label: 33.7.h.a.20.10
• Level: $33$
• Weight: $7$
• Character: 33.20
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $88$
• 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			20.10
Character	\(\chi\)	\(=\)	33.20
Dual form			33.7.h.a.5.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.46608 - 1.12620i) q^{2} +(-11.0612 - 24.6303i) q^{3} +(-41.0317 - 29.8113i) q^{4} +(207.457 - 67.4069i) q^{5} +(10.6004 + 97.8275i) q^{6} +(-309.326 - 224.739i) q^{7} +(245.744 + 338.237i) q^{8} +(-484.301 + 544.880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 33 q^{3} + 634 q^{4} + 109 q^{6} + 402 q^{7} - 2471 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.46608	−	1.12620i	−0.433260	−	0.140775i	0.0842641	−	0.996443i	\(-0.473146\pi\)
							−0.517524	+	0.855669i	\(0.673146\pi\)
\(3\)	−11.0612	−	24.6303i	−0.409673	−	0.912232i
							\(5\)	207.457	−	67.4069i	1.65966	−	0.539255i	0.678856	−	0.734271i	\(-0.262476\pi\)
0.980800	+	0.195016i	\(0.0624759\pi\)
\(7\)	−309.326	−	224.739i	−0.901826	−	0.655215i	0.0371081	−	0.999311i	\(-0.488185\pi\)
							−0.938934	+	0.344096i	\(0.888185\pi\)
\(11\)	−1315.45	−	202.875i	−0.988315	−	0.152423i
							\(13\)	−49.3913	+	152.011i	−0.0224812	+	0.0691902i	−0.961668	−	0.274217i	\(-0.911581\pi\)
0.939186	+	0.343408i	\(0.111581\pi\)
\(17\)	2813.90	−	914.290i	0.572745	−	0.186096i	−0.00830293	−	0.999966i	\(-0.502643\pi\)
							0.581048	+	0.813869i	\(0.302643\pi\)
\(19\)	−8908.01	+	6472.05i	−1.29873	+	0.943585i	−0.999942	−	0.0107333i	\(-0.996583\pi\)
							−0.298791	+	0.954318i	\(0.596583\pi\)
\(23\)		−	3868.12i		−	0.317919i	−0.987285	−	0.158960i	\(-0.949186\pi\)
							0.987285	−	0.158960i	\(-0.0508139\pi\)
\(29\)	−650.406	+	895.207i	−0.0266680	+	0.0367054i	−0.822143	−	0.569281i	\(-0.807222\pi\)
							0.795475	+	0.605987i	\(0.207222\pi\)
\(31\)	9410.07	−	28961.2i	0.315870	−	0.972147i	−0.659525	−	0.751682i	\(-0.729243\pi\)
							0.975395	−	0.220465i	\(-0.0707573\pi\)
\(37\)	−32265.8	−	23442.5i	−0.636997	−	0.462805i	0.221820	−	0.975088i	\(-0.428800\pi\)
							−0.858817	+	0.512282i	\(0.828800\pi\)
\(41\)	3524.47	+	4851.01i	0.0511378	+	0.0703851i	0.833820	−	0.552036i	\(-0.186149\pi\)
							−0.782682	+	0.622421i	\(0.786149\pi\)
\(43\)	−88867.7			−1.11773			−0.558867	−	0.829257i	\(-0.688764\pi\)
							−0.558867	+	0.829257i	\(0.688764\pi\)
\(47\)	−62872.8	−	86536.9i	−0.605576	−	0.833504i	0.390628	−	0.920549i	\(-0.372258\pi\)
							−0.996204	+	0.0870442i	\(0.972258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.h.a.20.10	yes	88
3.2	odd	2	inner	33.7.h.a.20.13	yes	88
11.5	even	5	inner	33.7.h.a.5.13	yes	88
33.5	odd	10	inner	33.7.h.a.5.10	✓	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.h.a.5.10	✓	88	33.5	odd	10	inner
33.7.h.a.5.13	yes	88	11.5	even	5	inner
33.7.h.a.20.10	yes	88	1.1	even	1	trivial
33.7.h.a.20.13	yes	88	3.2	odd	2	inner