Embedded newform 245.2.t.a.179.6

• Label: 245.2.t.a.179.6
• Level: $245$
• Weight: $2$
• Character: 245.179
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			179.6
Character	\(\chi\)	\(=\)	245.179
Dual form			245.2.t.a.219.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66122 + 0.124491i) q^{2} +(0.137023 - 0.444217i) q^{3} +(0.766500 - 0.115531i) q^{4} +(-2.19295 + 0.436988i) q^{5} +(-0.172324 + 0.755001i) q^{6} +(-2.54496 + 0.723316i) q^{7} +(1.98928 - 0.454041i) q^{8} +(2.30016 + 1.56822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 50 q^{4} - 12 q^{5} - 10 q^{6} - 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66122	+	0.124491i	−1.17466	+	0.0880287i	−0.647724	−	0.761875i	\(-0.724279\pi\)
							−0.526938	+	0.849904i	\(0.676660\pi\)
\(3\)	0.137023	−	0.444217i	0.0791101	−	0.256469i	−0.907374	−	0.420323i	\(-0.861917\pi\)
							0.986484	+	0.163855i	\(0.0523928\pi\)
\(5\)	−2.19295	+	0.436988i	−0.980718	+	0.195427i
							\(7\)	−2.54496	+	0.723316i	−0.961904	+	0.273388i
\(11\)	4.31042	−	2.93879i	1.29964	−	0.886080i								0.301905	−	0.953338i	\(-0.402378\pi\)
							0.997736	+	0.0672584i	\(0.0214252\pi\)
\(13\)	1.16765	−	2.42465i	0.323848	−	0.672478i	−0.673951	−	0.738776i	\(-0.735404\pi\)
							0.997800	+	0.0662974i	\(0.0211186\pi\)
\(17\)	1.48809	−	0.584033i	0.360915	−	0.141649i	−0.177953	−	0.984039i	\(-0.556947\pi\)
							0.538868	+	0.842390i	\(0.318852\pi\)
\(19\)	−2.78857	−	4.82995i	−0.639742	−	1.10807i	−0.985489	−	0.169738i	\(-0.945708\pi\)
							0.345747	−	0.938328i	\(-0.387625\pi\)
\(23\)	6.85946	+	2.69214i	1.43030	+	0.561350i	0.949019	−	0.315218i	\(-0.102078\pi\)
							0.481277	+	0.876568i	\(0.340173\pi\)
\(29\)	2.50036	−	3.13535i	0.464305	−	0.582220i	−0.493461	−	0.869768i	\(-0.664269\pi\)
							0.957766	+	0.287548i	\(0.0928399\pi\)
\(31\)	2.40781	−	4.17046i	0.432456	−	0.749036i	−0.564628	−	0.825345i	\(-0.690980\pi\)
							0.997084	+	0.0763095i	\(0.0243137\pi\)
\(37\)	−0.944115	+	6.26379i	−0.155212	+	1.02976i	0.766601	+	0.642124i	\(0.221947\pi\)
							−0.921812	+	0.387637i	\(0.873291\pi\)
\(41\)	1.48461	+	6.50449i	0.231857	+	1.01583i	0.948098	+	0.317978i	\(0.103004\pi\)
							−0.716241	+	0.697853i	\(0.754139\pi\)
\(43\)	−3.88063	−	0.885728i	−0.591790	−	0.135072i	−0.0838706	−	0.996477i	\(-0.526728\pi\)
							−0.507920	+	0.861404i	\(0.669585\pi\)
\(47\)	1.12669	−	0.0844340i	0.164345	−	0.0123160i	0.00769667	−	0.999970i	\(-0.497550\pi\)
							0.156649	+	0.987654i	\(0.449931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.t.a.179.6	✓	312
5.4	even	2	inner	245.2.t.a.179.21	yes	312
49.23	even	21	inner	245.2.t.a.219.21	yes	312
245.219	even	42	inner	245.2.t.a.219.6	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.t.a.179.6	✓	312	1.1	even	1	trivial
245.2.t.a.179.21	yes	312	5.4	even	2	inner
245.2.t.a.219.6	yes	312	245.219	even	42	inner
245.2.t.a.219.21	yes	312	49.23	even	21	inner