Embedded newform 33.4.e.c.25.2

• Label: 33.4.e.c.25.2
• Level: $33$
• Weight: $4$
• Character: 33.25
• Analytic conductor: $1.947$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94706303019\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 3*x^11 + 21*x^10 - 26*x^9 + 281*x^8 + 486*x^7 + 3506*x^6 + 15102*x^5 + 46669*x^4 + 41850*x^3 + 16292*x^2 + 616*x + 1936
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			25.2
Root			\(-0.654357 - 0.475418i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.25
Dual form			33.4.e.c.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46337 + 1.06320i) q^{2} +(-0.927051 - 2.85317i) q^{3} +(-1.46107 + 4.49672i) q^{4} +(17.3626 + 12.6147i) q^{5} +(4.39012 + 3.18961i) q^{6} +(-5.78498 + 17.8043i) q^{7} +(-7.11451 - 21.8962i) q^{8} +(-7.28115 + 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 9 q^{3} - 16 q^{4} + 28 q^{5} + 12 q^{7} - 112 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46337	+	1.06320i	−0.517381	+	0.375899i	−0.815616	−	0.578593i	\(-0.803602\pi\)
							0.298235	+	0.954492i	\(0.403602\pi\)
\(3\)	−0.927051	−	2.85317i	−0.178411	−	0.549093i
							\(5\)	17.3626	+	12.6147i	1.55296	+	1.12829i	0.941499	+	0.337016i	\(0.109418\pi\)
0.611460	+	0.791275i	\(0.290582\pi\)
\(7\)	−5.78498	+	17.8043i	−0.312359	+	0.961344i	0.664468	+	0.747317i	\(0.268658\pi\)
							−0.976828	+	0.214027i	\(0.931342\pi\)
\(11\)	30.2087	−	20.4556i	0.828025	−	0.560691i
							\(13\)	−11.2281	+	8.15769i	−0.239547	+	0.174041i	−0.701081	−	0.713081i	\(-0.747299\pi\)
0.461534	+	0.887122i	\(0.347299\pi\)
\(17\)	17.6108	+	12.7950i	0.251249	+	0.182543i	0.706280	−	0.707932i	\(-0.250372\pi\)
							−0.455031	+	0.890476i	\(0.650372\pi\)
\(19\)	−39.0184	−	120.086i	−0.471128	−	1.44998i	−0.851110	−	0.524988i	\(-0.824070\pi\)
							0.379982	−	0.924994i	\(-0.375930\pi\)
\(23\)	1.06620			0.00966600			0.00483300	−	0.999988i	\(-0.498462\pi\)
							0.00483300	+	0.999988i	\(0.498462\pi\)
\(29\)	40.3127	−	124.070i	0.258134	−	0.794454i	−0.735062	−	0.677999i	\(-0.762847\pi\)
							0.993196	−	0.116454i	\(-0.0371528\pi\)
\(31\)	156.035	−	113.366i	0.904021	−	0.656810i	−0.0354747	−	0.999371i	\(-0.511294\pi\)
							0.939496	+	0.342561i	\(0.111294\pi\)
\(37\)	12.8995	−	39.7007i	0.0573155	−	0.176399i	−0.918300	−	0.395885i	\(-0.870438\pi\)
							0.975616	+	0.219486i	\(0.0704380\pi\)
\(41\)	−18.3159	−	56.3705i	−0.0697673	−	0.214722i	0.910094	−	0.414403i	\(-0.136009\pi\)
							−0.979861	+	0.199681i	\(0.936009\pi\)
\(43\)	−172.713			−0.612524			−0.306262	−	0.951947i	\(-0.599078\pi\)
							−0.306262	+	0.951947i	\(0.599078\pi\)
\(47\)	−41.3932	−	127.395i	−0.128464	−	0.395372i	0.866052	−	0.499954i	\(-0.166650\pi\)
							−0.994516	+	0.104581i	\(0.966650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.e.c.25.2	yes	12
3.2	odd	2		99.4.f.d.91.2		12
11.2	odd	10		363.4.a.u.1.3		6
11.4	even	5	inner	33.4.e.c.4.2	✓	12
11.9	even	5		363.4.a.v.1.4		6
33.2	even	10		1089.4.a.bk.1.4		6
33.20	odd	10		1089.4.a.bi.1.3		6
33.26	odd	10		99.4.f.d.37.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.4.2	✓	12	11.4	even	5	inner
33.4.e.c.25.2	yes	12	1.1	even	1	trivial
99.4.f.d.37.2		12	33.26	odd	10
99.4.f.d.91.2		12	3.2	odd	2
363.4.a.u.1.3		6	11.2	odd	10
363.4.a.v.1.4		6	11.9	even	5
1089.4.a.bi.1.3		6	33.20	odd	10
1089.4.a.bk.1.4		6	33.2	even	10