Embedded newform 242.2.g.a.25.4

• Label: 242.2.g.a.25.4
• Level: $242$
• Weight: $2$
• Character: 242.25
• Analytic conductor: $1.932$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.93237972891\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(5\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			25.4
Character	\(\chi\)	\(=\)	242.25
Dual form			242.2.g.a.213.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.466667 + 0.884433i) q^{2} +(0.313512 + 0.964891i) q^{3} +(-0.564443 + 0.825472i) q^{4} +(-0.337582 + 1.66603i) q^{5} +(-0.707075 + 0.727563i) q^{6} +(0.220471 + 0.838759i) q^{7} +(-0.993482 - 0.113991i) q^{8} +(1.59433 - 1.15835i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{19}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.466667	+	0.884433i	0.329984	+	0.625389i
							\(3\)	0.313512	+	0.964891i	0.181006	+	0.557080i	0.999857	−	0.0169270i	\(-0.00538828\pi\)
−0.818851	+	0.574007i	\(0.805388\pi\)
\(5\)	−0.337582	+	1.66603i	−0.150971	+	0.745073i	0.831034	+	0.556221i	\(0.187749\pi\)
							−0.982006	+	0.188852i	\(0.939523\pi\)
\(7\)	0.220471	+	0.838759i	0.0833303	+	0.317021i	0.995734	−	0.0922702i	\(-0.0294123\pi\)
							−0.912404	+	0.409291i	\(0.865776\pi\)
\(11\)	−2.87440	+	1.65464i	−0.866664	+	0.498892i
							\(13\)	0.0852021	−	0.0304001i	0.0236308	−	0.00843146i	−0.324208	−	0.945986i	\(-0.605098\pi\)
0.347838	+	0.937555i	\(0.386916\pi\)
\(17\)	0.630678	−	0.146658i	0.152962	−	0.0355697i	−0.149317	−	0.988789i	\(-0.547707\pi\)
							0.302278	+	0.953220i	\(0.402253\pi\)
\(19\)	−0.240951	+	0.399572i	−0.0552779	+	0.0916680i	−0.882761	−	0.469823i	\(-0.844318\pi\)
							0.827483	+	0.561491i	\(0.189772\pi\)
\(23\)	0.347545	+	0.223354i	0.0724682	+	0.0465725i	0.576373	−	0.817187i	\(-0.304467\pi\)
							−0.503905	+	0.863759i	\(0.668104\pi\)
\(29\)	1.93089	−	0.816002i	0.358558	−	0.151528i	−0.202547	−	0.979273i	\(-0.564922\pi\)
							0.561105	+	0.827745i	\(0.310376\pi\)
\(31\)	6.24082	−	8.09324i	1.12088	−	1.45359i	0.245991	−	0.969272i	\(-0.420887\pi\)
							0.874893	−	0.484316i	\(-0.160931\pi\)
\(37\)	0.0301724	−	1.05617i	0.00496032	−	0.173634i	−0.993335	−	0.115265i	\(-0.963228\pi\)
							0.998295	−	0.0583688i	\(-0.0185899\pi\)
\(41\)	−1.67922	+	0.290601i	−0.262251	+	0.0453843i	−0.300131	−	0.953898i	\(-0.597030\pi\)
							0.0378802	+	0.999282i	\(0.487939\pi\)
\(43\)	−1.48460	+	3.25081i	−0.226399	+	0.495744i	−0.988408	−	0.151822i	\(-0.951486\pi\)
							0.762009	+	0.647567i	\(0.224213\pi\)
\(47\)	−1.18482	+	0.582539i	−0.172824	+	0.0849720i	−0.525583	−	0.850742i	\(-0.676153\pi\)
							0.352759	+	0.935714i	\(0.385244\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.a.25.4	✓	200
121.92	even	55	inner	242.2.g.a.213.4	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.a.25.4	✓	200	1.1	even	1	trivial
242.2.g.a.213.4	yes	200	121.92	even	55	inner