Embedded newform 161.2.m.a.100.12

• Label: 161.2.m.a.100.12
• Level: $161$
• Weight: $2$
• Character: 161.100
• Analytic conductor: $1.286$
• Analytic rank: $0$
• Dimension: $280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	161.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(14\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			100.12
Character	\(\chi\)	\(=\)	161.100
Dual form			161.2.m.a.95.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.42294 + 1.11901i) q^{2} +(-1.29961 + 1.82504i) q^{3} +(0.301047 + 1.24093i) q^{4} +(-1.36031 + 0.262179i) q^{5} +(-3.89149 + 1.14265i) q^{6} +(0.698815 + 2.55180i) q^{7} +(0.543747 - 1.19064i) q^{8} +(-0.660594 - 1.90866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 11 q^{2} - 9 q^{3} + q^{4} - 11 q^{5} - 52 q^{6} - 20 q^{7} - 32 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.42294	+	1.11901i	1.00617	+	0.791259i	0.978055	−	0.208346i	\(-0.0668079\pi\)
							0.0281125	+	0.999605i	\(0.491050\pi\)
\(3\)	−1.29961	+	1.82504i	−0.750327	+	1.05369i	0.246101	+	0.969244i	\(0.420851\pi\)
							−0.996428	+	0.0844431i	\(0.973089\pi\)
\(5\)	−1.36031	+	0.262179i	−0.608351	+	0.117250i	−0.484117	−	0.875003i	\(-0.660859\pi\)
							−0.124233	+	0.992253i	\(0.539647\pi\)
\(7\)	0.698815	+	2.55180i	0.264127	+	0.964488i
							\(11\)	0.645237	−	0.507420i	0.194546	−	0.152993i	−0.516134	−	0.856508i	\(-0.672630\pi\)
0.710681	+	0.703515i	\(0.248387\pi\)
\(13\)	−1.14131	+	0.733476i	−0.316543	+	0.203430i	−0.689258	−	0.724516i	\(-0.742063\pi\)
							0.372715	+	0.927946i	\(0.378427\pi\)
\(17\)	2.61655	+	2.49487i	0.634606	+	0.605095i	0.937524	−	0.347921i	\(-0.113112\pi\)
							−0.302918	+	0.953017i	\(0.597961\pi\)
\(19\)	4.29096	−	4.09142i	0.984413	−	0.938636i	−0.0137078	−	0.999906i	\(-0.504363\pi\)
							0.998121	+	0.0612697i	\(0.0195150\pi\)
\(23\)	4.02234	−	2.61166i	0.838716	−	0.544569i
							\(29\)	−2.62141	+	0.769716i	−0.486784	+	0.142933i	−0.515910	−	0.856643i	\(-0.672546\pi\)
0.0291254	+	0.999576i	\(0.490728\pi\)
\(31\)	5.58598	−	0.533397i	1.00327	−	0.0958009i	0.419531	−	0.907741i	\(-0.362194\pi\)
							0.583741	+	0.811940i	\(0.301588\pi\)
\(37\)	1.75896	+	5.08219i	0.289172	+	0.835507i	0.992235	+	0.124379i	\(0.0396940\pi\)
							−0.703063	+	0.711128i	\(0.748185\pi\)
\(41\)	−7.47061	−	8.62155i	−1.16671	−	1.34646i	−0.926750	−	0.375678i	\(-0.877410\pi\)
							−0.239963	−	0.970782i	\(-0.577135\pi\)
\(43\)	−0.151450	−	0.331630i	−0.0230959	−	0.0505730i	0.897732	−	0.440541i	\(-0.145213\pi\)
							−0.920828	+	0.389968i	\(0.872486\pi\)
\(47\)	−0.241448	+	0.418201i	−0.0352188	+	0.0610008i	−0.883098	−	0.469189i	\(-0.844546\pi\)
							0.847879	+	0.530190i	\(0.177879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.m.a.100.12	yes	280
7.4	even	3	inner	161.2.m.a.123.3	yes	280
23.3	even	11	inner	161.2.m.a.72.3	✓	280
161.95	even	33	inner	161.2.m.a.95.12	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.m.a.72.3	✓	280	23.3	even	11	inner
161.2.m.a.95.12	yes	280	161.95	even	33	inner
161.2.m.a.100.12	yes	280	1.1	even	1	trivial
161.2.m.a.123.3	yes	280	7.4	even	3	inner