Embedded newform 245.2.t.a.219.13

• Label: 245.2.t.a.219.13
• Level: $245$
• Weight: $2$
• Character: 245.219
• 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			219.13
Character	\(\chi\)	\(=\)	245.219
Dual form			245.2.t.a.179.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.348545 - 0.0261198i) q^{2} +(-0.132268 - 0.428802i) q^{3} +(-1.85686 - 0.279877i) q^{4} +(0.604344 + 2.15285i) q^{5} +(0.0349010 + 0.152911i) q^{6} +(0.493697 - 2.59928i) q^{7} +(1.32141 + 0.301603i) q^{8} +(2.31234 - 1.57653i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 50 q^{4} - 12 q^{5} - 10 q^{6} - 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.348545	−	0.0261198i	−0.246458	−	0.0184695i	−0.0490719	−	0.998795i	\(-0.515626\pi\)
							−0.197387	+	0.980326i	\(0.563245\pi\)
\(3\)	−0.132268	−	0.428802i	−0.0763648	−	0.247569i	0.909339	−	0.416056i	\(-0.136588\pi\)
							−0.985704	+	0.168487i	\(0.946112\pi\)
\(5\)	0.604344	+	2.15285i	0.270271	+	0.962784i
							\(7\)	0.493697	−	2.59928i	0.186600	−	0.982436i
\(11\)	3.52784	+	2.40524i	1.06368	+	0.725207i								0.962943	−	0.269705i	\(-0.0869263\pi\)
							0.100741	+	0.994913i	\(0.467879\pi\)
\(13\)	−1.69814	−	3.52623i	−0.470980	−	0.977999i	−0.992209	−	0.124583i	\(-0.960241\pi\)
							0.521230	−	0.853416i	\(-0.325473\pi\)
\(17\)	5.27256	+	2.06933i	1.27878	+	0.501885i	0.904913	−	0.425596i	\(-0.139936\pi\)
							0.373869	+	0.927481i	\(0.378031\pi\)
\(19\)	1.19716	−	2.07355i	0.274648	−	0.475704i	−0.695398	−	0.718625i	\(-0.744772\pi\)
							0.970046	+	0.242920i	\(0.0781053\pi\)
\(23\)	1.80160	−	0.707075i	0.375659	−	0.147435i	−0.169997	−	0.985445i	\(-0.554376\pi\)
							0.545656	+	0.838009i	\(0.316281\pi\)
\(29\)	−4.46361	−	5.59719i	−0.828871	−	1.03937i	−0.998548	−	0.0538717i	\(-0.982844\pi\)
							0.169677	−	0.985500i	\(-0.445728\pi\)
\(31\)	−0.331576	−	0.574307i	−0.0595529	−	0.103149i	0.834712	−	0.550687i	\(-0.185634\pi\)
							−0.894265	+	0.447538i	\(0.852301\pi\)
\(37\)	1.39298	+	9.24180i	0.229004	+	1.51934i	0.747499	+	0.664263i	\(0.231254\pi\)
							−0.518495	+	0.855081i	\(0.673508\pi\)
\(41\)	−0.619076	+	2.71235i	−0.0966834	+	0.423597i	−0.999985	−	0.00540774i	\(-0.998279\pi\)
							0.903302	+	0.429005i	\(0.141136\pi\)
\(43\)	10.2386	−	2.33689i	1.56137	−	0.356372i	0.647396	−	0.762154i	\(-0.275858\pi\)
							0.913970	+	0.405782i	\(0.133001\pi\)
\(47\)	−8.68294	−	0.650697i	−1.26654	−	0.0949138i	−0.575541	−	0.817773i	\(-0.695209\pi\)
							−0.690996	+	0.722859i	\(0.742828\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.t.a.219.13	yes	312
5.4	even	2	inner	245.2.t.a.219.14	yes	312
49.32	even	21	inner	245.2.t.a.179.14	yes	312
245.179	even	42	inner	245.2.t.a.179.13	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.t.a.179.13	✓	312	245.179	even	42	inner
245.2.t.a.179.14	yes	312	49.32	even	21	inner
245.2.t.a.219.13	yes	312	1.1	even	1	trivial
245.2.t.a.219.14	yes	312	5.4	even	2	inner