Embedded newform 67.2.e.c.64.1

• Label: 67.2.e.c.64.1
• Level: $67$
• Weight: $2$
• Character: 67.64
• 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			64.1
Root			\(-0.165981 - 0.106669i\) of defining polynomial
Character	\(\chi\)	\(=\)	67.64
Dual form			67.2.e.c.22.1

$q$-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 \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)

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	0.694007	−	0.719968i	\(-0.255844\pi\)
							0.194592	+	0.980884i	\(0.437662\pi\)
\(3\)	−0.385764	+	2.68305i	−0.222721	+	1.54906i	0.504960	+	0.863143i	\(0.331507\pi\)
							−0.727681	+	0.685916i	\(0.759402\pi\)
\(5\)	−1.35049	−	2.95717i	−0.603959	−	1.32248i	−0.926630	−	0.375974i	\(-0.877308\pi\)
							0.322672	−	0.946511i	\(-0.395419\pi\)
\(7\)	4.00914	+	1.17719i	1.51531	+	0.444936i	0.930518	−	0.366246i	\(-0.119357\pi\)
							0.584794	+	0.811182i	\(0.301175\pi\)
\(11\)	−0.0622997	−	0.136417i	−0.0187841	−	0.0411314i	0.900007	−	0.435875i	\(-0.143561\pi\)
							−0.918792	+	0.394743i	\(0.870834\pi\)
\(13\)	−0.628542	+	0.725376i	−0.174326	+	0.201183i	−0.836188	−	0.548442i	\(-0.815221\pi\)
							0.661862	+	0.749626i	\(0.269766\pi\)
\(17\)	−0.311139	+	0.199957i	−0.0754622	+	0.0484966i	−0.577828	−	0.816159i	\(-0.696100\pi\)
							0.502366	+	0.864655i	\(0.332463\pi\)
\(19\)	−7.59646	+	2.23052i	−1.74275	+	0.511717i	−0.989313	−	0.145806i	\(-0.953423\pi\)
							−0.753435	+	0.657523i	\(0.771604\pi\)
\(23\)	0.206011	−	1.43284i	0.0429563	−	0.298768i	−0.957006	−	0.290069i	\(-0.906322\pi\)
							0.999962	−	0.00869922i	\(-0.00276908\pi\)
\(29\)	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	0.961725	−	0.274016i	\(-0.0883522\pi\)
							−0.408095	+	0.912939i	\(0.633807\pi\)
\(37\)	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	−0.205331	−	0.978693i	\(-0.565827\pi\)
							0.804952	+	0.593339i	\(0.202191\pi\)
\(43\)	6.83054	−	4.38972i	1.04165	−	0.669426i	0.0962538	−	0.995357i	\(-0.469314\pi\)
							0.945394	+	0.325931i	\(0.105678\pi\)
\(47\)	−0.238631	+	1.65971i	−0.0348079	+	0.242094i	−0.999796	−	0.0201996i	\(-0.993570\pi\)
							0.964988	+	0.262294i	\(0.0844789\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.c.22.1	✓	20	67.22	even	11	inner
67.2.e.c.64.1	yes	20	1.1	even	1	trivial
603.2.u.c.64.2		20	3.2	odd	2
603.2.u.c.424.2		20	201.89	odd	22
4489.2.a.l.1.4		10	67.25	even	11
4489.2.a.m.1.7		10	67.42	odd	22