Embedded newform 33.3.g.a.7.1

• Label: 33.3.g.a.7.1
• Level: $33$
• Weight: $3$
• Character: 33.7
• Analytic conductor: $0.899$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.899184872389\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 3*x^14 - 4*x^13 + 77*x^12 + 88*x^11 - 577*x^10 + 578*x^9 + 1520*x^8 + 1868*x^7 - 1619*x^6 - 16804*x^5 + 32427*x^4 + 43316*x^3 - 71672*x^2 + 83521
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			7.1
Root			\(2.24350 - 2.23726i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.7
Dual form			33.3.g.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.577539 + 0.794915i) q^{2} +(-0.535233 + 1.64728i) q^{3} +(0.937730 + 2.88604i) q^{4} +(-0.321645 + 0.233689i) q^{5} +(-1.00033 - 1.37683i) q^{6} +(6.87311 - 2.23321i) q^{7} +(-6.57364 - 2.13591i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.577539	+	0.794915i	−0.288770	+	0.397457i	−0.928614	−	0.371047i	\(-0.878999\pi\)
							0.639844	+	0.768505i	\(0.278999\pi\)
\(3\)	−0.535233	+	1.64728i	−0.178411	+	0.549093i
							\(5\)	−0.321645	+	0.233689i	−0.0643291	+	0.0467378i	−0.619485	−	0.785008i	\(-0.712659\pi\)
0.555156	+	0.831746i	\(0.312659\pi\)
\(7\)	6.87311	−	2.23321i	0.981872	−	0.319030i	0.226273	−	0.974064i	\(-0.427346\pi\)
							0.755599	+	0.655034i	\(0.227346\pi\)
\(11\)	−0.495253	−	10.9888i	−0.0450230	−	0.998986i
							\(13\)	7.14728	−	9.83739i	0.549791	−	0.756722i	−0.440193	−	0.897903i	\(-0.645090\pi\)
0.989984	+	0.141181i	\(0.0450899\pi\)
\(17\)	12.2903	+	16.9162i	0.722960	+	0.995069i	0.999420	+	0.0340440i	\(0.0108386\pi\)
							−0.276460	+	0.961025i	\(0.589161\pi\)
\(19\)	−10.5689	−	3.43403i	−0.556256	−	0.180739i	0.0173798	−	0.999849i	\(-0.494468\pi\)
							−0.573636	+	0.819110i	\(0.694468\pi\)
\(23\)	−5.92990			−0.257822			−0.128911	−	0.991656i	\(-0.541148\pi\)
							−0.128911	+	0.991656i	\(0.541148\pi\)
\(29\)	23.7332	−	7.71140i	0.818388	−	0.265910i	0.130242	−	0.991482i	\(-0.458425\pi\)
							0.688146	+	0.725572i	\(0.258425\pi\)
\(31\)	−48.2383	−	35.0472i	−1.55607	−	1.13055i	−0.939137	−	0.343542i	\(-0.888373\pi\)
							−0.616937	−	0.787012i	\(-0.711627\pi\)
\(37\)	−1.84391	−	5.67498i	−0.0498355	−	0.153378i	0.923042	−	0.384700i	\(-0.125695\pi\)
							−0.972877	+	0.231322i	\(0.925695\pi\)
\(41\)	−49.4926	−	16.0811i	−1.20714	−	0.392223i	−0.364755	−	0.931104i	\(-0.618847\pi\)
							−0.842382	+	0.538881i	\(0.818847\pi\)
\(43\)		−	17.6439i		−	0.410323i	−0.978728	−	0.205161i	\(-0.934228\pi\)
							0.978728	−	0.205161i	\(-0.0657719\pi\)
\(47\)	−17.1074	+	52.6513i	−0.363988	+	1.12024i	0.586624	+	0.809860i	\(0.300457\pi\)
							−0.950612	+	0.310381i	\(0.899543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.7.1	✓	16
3.2	odd	2		99.3.k.c.73.4		16
4.3	odd	2		528.3.bf.b.337.3		16
11.2	odd	10		363.3.g.g.94.1		16
11.3	even	5		363.3.g.f.118.4		16
11.4	even	5		363.3.g.g.112.1		16
11.5	even	5		363.3.c.e.241.7		16
11.6	odd	10		363.3.c.e.241.10		16
11.7	odd	10		363.3.g.a.112.4		16
11.8	odd	10	inner	33.3.g.a.19.1	yes	16
11.9	even	5		363.3.g.a.94.4		16
11.10	odd	2		363.3.g.f.40.4		16
33.5	odd	10		1089.3.c.m.604.10		16
33.8	even	10		99.3.k.c.19.4		16
33.17	even	10		1089.3.c.m.604.7		16
44.19	even	10		528.3.bf.b.481.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.1	✓	16	1.1	even	1	trivial
33.3.g.a.19.1	yes	16	11.8	odd	10	inner
99.3.k.c.19.4		16	33.8	even	10
99.3.k.c.73.4		16	3.2	odd	2
363.3.c.e.241.7		16	11.5	even	5
363.3.c.e.241.10		16	11.6	odd	10
363.3.g.a.94.4		16	11.9	even	5
363.3.g.a.112.4		16	11.7	odd	10
363.3.g.f.40.4		16	11.10	odd	2
363.3.g.f.118.4		16	11.3	even	5
363.3.g.g.94.1		16	11.2	odd	10
363.3.g.g.112.1		16	11.4	even	5
528.3.bf.b.337.3		16	4.3	odd	2
528.3.bf.b.481.3		16	44.19	even	10
1089.3.c.m.604.7		16	33.17	even	10
1089.3.c.m.604.10		16	33.5	odd	10