Embedded newform 242.2.g.b.15.6

• Label: 242.2.g.b.15.6
• Level: $242$
• Weight: $2$
• Character: 242.15
• Analytic conductor: $1.932$
• Analytic rank: $0$
• Dimension: $240$
• 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:	\(240\)
Relative dimension:	\(6\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			15.6
Character	\(\chi\)	\(=\)	242.15
Dual form			242.2.g.b.113.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0855750 + 0.996332i) q^{2} +(1.02153 - 3.14394i) q^{3} +(-0.985354 + 0.170522i) q^{4} +(2.66237 - 0.152240i) q^{5} +(3.21982 + 0.748737i) q^{6} +(1.76910 + 2.58723i) q^{7} +(-0.254218 - 0.967147i) q^{8} +(-6.41377 - 4.65988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{26}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0855750	+	0.996332i	0.0605107	+	0.704513i
							\(3\)	1.02153	−	3.14394i	0.589779	−	1.81515i	0.0106118	−	0.999944i	\(-0.496622\pi\)
0.579167	−	0.815209i	\(-0.303378\pi\)
\(5\)	2.66237	−	0.152240i	1.19065	−	0.0680836i	0.549494	−	0.835497i	\(-0.314820\pi\)
							0.641152	+	0.767414i	\(0.278457\pi\)
\(7\)	1.76910	+	2.58723i	0.668657	+	0.977879i	0.999416	+	0.0341641i	\(0.0108769\pi\)
							−0.330760	+	0.943715i	\(0.607305\pi\)
\(11\)	−1.53345	−	2.94084i	−0.462353	−	0.886696i
							\(13\)	−3.34895	+	3.44598i	−0.928831	+	0.955744i	−0.999127	−	0.0417821i	\(-0.986696\pi\)
0.0702962	+	0.997526i	\(0.477606\pi\)
\(17\)	3.18176	+	1.79682i	0.771690	+	0.435793i	0.826665	−	0.562694i	\(-0.190235\pi\)
							−0.0549751	+	0.998488i	\(0.517508\pi\)
\(19\)	−0.392925	−	0.360624i	−0.0901433	−	0.0827327i	0.629837	−	0.776728i	\(-0.283122\pi\)
							−0.719980	+	0.693995i	\(0.755849\pi\)
\(23\)	3.16628	+	0.929705i	0.660216	+	0.193857i	0.594642	−	0.803991i	\(-0.297294\pi\)
							0.0655742	+	0.997848i	\(0.479112\pi\)
\(29\)	−0.890456	−	0.102170i	−0.165353	−	0.0189725i	0.0309193	−	0.999522i	\(-0.490157\pi\)
							−0.196273	+	0.980549i	\(0.562884\pi\)
\(31\)	2.60816	−	4.94301i	0.468439	−	0.887791i	−0.530750	−	0.847528i	\(-0.678090\pi\)
							0.999189	−	0.0402624i	\(-0.0128194\pi\)
\(37\)	−4.76867	+	2.34460i	−0.783965	+	0.385450i	−0.788887	−	0.614538i	\(-0.789342\pi\)
							0.00492202	+	0.999988i	\(0.498433\pi\)
\(41\)	−10.7082	+	4.52530i	−1.67233	+	0.706734i	−0.999455	−	0.0330197i	\(-0.989488\pi\)
							−0.672878	+	0.739753i	\(0.734942\pi\)
\(43\)	0.173988	−	0.111816i	0.0265330	−	0.0170517i	−0.527307	−	0.849675i	\(-0.676798\pi\)
							0.553840	+	0.832623i	\(0.313162\pi\)
\(47\)	5.67116	+	9.40456i	0.827223	+	1.37180i	0.926328	+	0.376719i	\(0.122948\pi\)
							−0.0991042	+	0.995077i	\(0.531598\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.b.15.6	✓	240
121.113	even	55	inner	242.2.g.b.113.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.b.15.6	✓	240	1.1	even	1	trivial
242.2.g.b.113.6	yes	240	121.113	even	55	inner