Embedded newform 67.2.e.c.22.1

• Label: 67.2.e.c.22.1
• Level: $67$
• Weight: $2$
• Character: 67.22
• Analytic conductor: $0.535$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.534997693543\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{11})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 7*x^19 + 39*x^18 - 148*x^17 + 492*x^16 - 1282*x^15 + 2921*x^14 - 4316*x^13 + 2696*x^12 + 9361*x^11 - 20998*x^10 + 10813*x^9 + 44155*x^8 - 46933*x^7 - 36976*x^6 - 49131*x^5 + 105828*x^4 + 108234*x^3 + 143215*x^2 + 41004*x + 4489
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.1
Root			\(-0.165981 + 0.106669i\) of defining polynomial
Character	\(\chi\)	\(=\)	67.22
Dual form			67.2.e.c.64.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25667 - 0.368991i) q^{2} +(-0.385764 - 2.68305i) q^{3} +(-0.239446 + 0.153882i) q^{4} +(-1.35049 + 2.95717i) q^{5} +(-1.47480 - 3.22936i) q^{6} +(4.00914 - 1.17719i) q^{7} +(-1.95949 + 2.26138i) q^{8} +(-4.17146 + 1.22485i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} - 4 q^{3} - 4 q^{4} + q^{5} + 3 q^{6} - 8 q^{7} - 22 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.25667	−	0.368991i	0.888599	−	0.260916i	0.194592	−	0.980884i	\(-0.437662\pi\)
							0.694007	+	0.719968i	\(0.255844\pi\)
\(3\)	−0.385764	−	2.68305i	−0.222721	−	1.54906i	−0.727681	−	0.685916i	\(-0.759402\pi\)
							0.504960	−	0.863143i	\(-0.331507\pi\)
\(5\)	−1.35049	+	2.95717i	−0.603959	+	1.32248i	0.322672	+	0.946511i	\(0.395419\pi\)
							−0.926630	+	0.375974i	\(0.877308\pi\)
\(7\)	4.00914	−	1.17719i	1.51531	−	0.444936i	0.584794	−	0.811182i	\(-0.301175\pi\)
							0.930518	+	0.366246i	\(0.119357\pi\)
\(11\)	−0.0622997	+	0.136417i	−0.0187841	+	0.0411314i	−0.918792	−	0.394743i	\(-0.870834\pi\)
							0.900007	+	0.435875i	\(0.143561\pi\)
\(13\)	−0.628542	−	0.725376i	−0.174326	−	0.201183i	0.661862	−	0.749626i	\(-0.269766\pi\)
							−0.836188	+	0.548442i	\(0.815221\pi\)
\(17\)	−0.311139	−	0.199957i	−0.0754622	−	0.0484966i	0.502366	−	0.864655i	\(-0.332463\pi\)
							−0.577828	+	0.816159i	\(0.696100\pi\)
\(19\)	−7.59646	−	2.23052i	−1.74275	−	0.511717i	−0.753435	−	0.657523i	\(-0.771604\pi\)
							−0.989313	+	0.145806i	\(0.953423\pi\)
\(23\)	0.206011	+	1.43284i	0.0429563	+	0.298768i	0.999962	+	0.00869922i	\(0.00276908\pi\)
							−0.957006	+	0.290069i	\(0.906322\pi\)
\(29\)	1.51049			0.280490			0.140245	−	0.990117i	\(-0.455211\pi\)
							0.140245	+	0.990117i	\(0.455211\pi\)
\(31\)	3.08248	−	3.55737i	0.553630	−	0.638923i	−0.408095	−	0.912939i	\(-0.633807\pi\)
							0.961725	+	0.274016i	\(0.0883522\pi\)
\(37\)	4.71809			0.775648			0.387824	−	0.921733i	\(-0.373227\pi\)
							0.387824	+	0.921733i	\(0.373227\pi\)
\(41\)	3.83945	+	2.46746i	0.599621	+	0.385353i	0.804952	−	0.593339i	\(-0.202191\pi\)
							−0.205331	+	0.978693i	\(0.565827\pi\)
\(43\)	6.83054	+	4.38972i	1.04165	+	0.669426i	0.945394	−	0.325931i	\(-0.105678\pi\)
							0.0962538	+	0.995357i	\(0.469314\pi\)
\(47\)	−0.238631	−	1.65971i	−0.0348079	−	0.242094i	0.964988	−	0.262294i	\(-0.0844789\pi\)
							−0.999796	+	0.0201996i	\(0.993570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.e.c.22.1	✓	20
3.2	odd	2		603.2.u.c.424.2		20
67.8	odd	22		4489.2.a.m.1.7		10
67.59	even	11		4489.2.a.l.1.4		10
67.64	even	11	inner	67.2.e.c.64.1	yes	20
201.131	odd	22		603.2.u.c.64.2		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.c.22.1	✓	20	1.1	even	1	trivial
67.2.e.c.64.1	yes	20	67.64	even	11	inner
603.2.u.c.64.2		20	201.131	odd	22
603.2.u.c.424.2		20	3.2	odd	2
4489.2.a.l.1.4		10	67.59	even	11
4489.2.a.m.1.7		10	67.8	odd	22