Embedded newform 245.2.p.b.239.2

• Label: 245.2.p.b.239.2
• Level: $245$
• Weight: $2$
• Character: 245.239
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			239.2
Character	\(\chi\)	\(=\)	245.239
Dual form			245.2.p.b.204.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89092 - 1.50795i) q^{2} +(-0.707804 + 1.46977i) q^{3} +(0.856592 + 3.75298i) q^{4} +(-0.877818 - 2.05656i) q^{5} +(3.55474 - 1.71187i) q^{6} +(-2.64485 - 0.0690833i) q^{7} +(1.94082 - 4.03015i) q^{8} +(0.211233 + 0.264877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 16 q^{4} - q^{5} - 34 q^{6} + 26 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89092	−	1.50795i	−1.33708	−	1.06629i	−0.991802	−	0.127784i	\(-0.959213\pi\)
							−0.345277	−	0.938501i	\(-0.612215\pi\)
\(3\)	−0.707804	+	1.46977i	−0.408651	+	0.848572i	0.590487	+	0.807047i	\(0.298936\pi\)
							−0.999137	+	0.0415246i	\(0.986778\pi\)
\(5\)	−0.877818	−	2.05656i	−0.392572	−	0.919721i
							\(7\)	−2.64485	−	0.0690833i	−0.999659	−	0.0261110i
\(11\)	0.681836	−	0.854995i	0.205581	−	0.257791i								−0.668343	−	0.743854i	\(-0.732996\pi\)
							0.873924	+	0.486063i	\(0.161567\pi\)
\(13\)	1.98986	+	1.58686i	0.551888	+	0.440116i	0.859309	−	0.511458i	\(-0.170894\pi\)
							−0.307420	+	0.951574i	\(0.599466\pi\)
\(17\)	7.66411	+	1.74928i	1.85882	+	0.424264i	0.996609	−	0.0822841i	\(-0.0262215\pi\)
							0.862212	+	0.506548i	\(0.169079\pi\)
\(19\)	−0.108275			−0.0248399			−0.0124199	−	0.999923i	\(-0.503953\pi\)
							−0.0124199	+	0.999923i	\(0.503953\pi\)
\(23\)	6.13072	−	1.39930i	1.27834	−	0.291774i	0.471146	−	0.882055i	\(-0.343840\pi\)
							0.807197	+	0.590282i	\(0.200983\pi\)
\(29\)	0.888541	−	3.89295i	0.164998	−	0.722903i	−0.822950	−	0.568114i	\(-0.807673\pi\)
							0.987948	−	0.154789i	\(-0.0494697\pi\)
\(31\)	2.15548			0.387135			0.193568	−	0.981087i	\(-0.437994\pi\)
							0.193568	+	0.981087i	\(0.437994\pi\)
\(37\)	8.68228	+	1.98167i	1.42736	+	0.325785i	0.865276	−	0.501296i	\(-0.167143\pi\)
							0.562082	+	0.827081i	\(0.310000\pi\)
\(41\)	4.35199	+	2.09581i	0.679666	+	0.327310i	0.741669	−	0.670766i	\(-0.234035\pi\)
							−0.0620029	+	0.998076i	\(0.519749\pi\)
\(43\)	1.18098	+	2.45234i	0.180098	+	0.373978i	0.971400	−	0.237447i	\(-0.0763107\pi\)
							−0.791302	+	0.611425i	\(0.790596\pi\)
\(47\)	−4.35921	−	3.47635i	−0.635856	−	0.507078i	0.251684	−	0.967810i	\(-0.419016\pi\)
							−0.887540	+	0.460731i	\(0.847587\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.p.b.239.2	yes	144
5.4	even	2	inner	245.2.p.b.239.23	yes	144
49.8	even	7	inner	245.2.p.b.204.23	yes	144
245.204	even	14	inner	245.2.p.b.204.2	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.p.b.204.2	✓	144	245.204	even	14	inner
245.2.p.b.204.23	yes	144	49.8	even	7	inner
245.2.p.b.239.2	yes	144	1.1	even	1	trivial
245.2.p.b.239.23	yes	144	5.4	even	2	inner