Embedded newform 245.4.x.a.103.15

• Label: 245.4.x.a.103.15
• Level: $245$
• Weight: $4$
• Character: 245.103
• Analytic conductor: $14.455$
• Analytic rank: $0$
• Dimension: $1968$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.4554679514\)
Analytic rank:	\(0\)
Dimension:	\(1968\)
Relative dimension:	\(82\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.15
Character	\(\chi\)	\(=\)	245.103
Dual form			245.4.x.a.157.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.39144 + 3.24028i) q^{2} +(5.84824 - 1.10655i) q^{3} +(-2.42242 - 7.85330i) q^{4} +(-5.82711 + 9.54174i) q^{5} +(-10.4002 + 21.5962i) q^{6} +(-10.1877 + 15.4664i) q^{7} +(0.830281 + 0.290528i) q^{8} +(7.84390 - 3.07850i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1968 q - 26 q^{2} - 22 q^{3} + 2 q^{5} - 112 q^{6} - 32 q^{7} + 96 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39144	+	3.24028i	−0.845501	+	1.14561i	0.142289	+	0.989825i	\(0.454554\pi\)
							−0.987790	+	0.155788i	\(0.950208\pi\)
\(3\)	5.84824	−	1.10655i	1.12549	−	0.212955i	0.410348	−	0.911929i	\(-0.365407\pi\)
							0.715147	+	0.698974i	\(0.246360\pi\)
\(5\)	−5.82711	+	9.54174i	−0.521193	+	0.853439i
							\(7\)	−10.1877	+	15.4664i	−0.550085	+	0.835108i
\(11\)	−11.1006	+	28.2838i	−0.304268	+	0.775261i								0.694193	+	0.719789i	\(0.255762\pi\)
							−0.998460	+	0.0554720i	\(0.982334\pi\)
\(13\)	24.1694	+	2.72323i	0.515644	+	0.0580992i	0.365952	−	0.930634i	\(-0.380743\pi\)
							0.149692	+	0.988733i	\(0.452172\pi\)
\(17\)	2.72369	+	0.101913i	0.0388583	+	0.00145398i	0.0568133	−	0.998385i	\(-0.481906\pi\)
							−0.0179550	+	0.999839i	\(0.505716\pi\)
\(19\)	51.1819	−	88.6496i	0.617996	−	1.07040i	−0.371855	−	0.928291i	\(-0.621278\pi\)
							0.989851	−	0.142110i	\(-0.0453886\pi\)
\(23\)	−3.68024	−	98.3564i	−0.0333645	−	0.891684i	−0.911013	−	0.412379i	\(-0.864698\pi\)
							0.877648	−	0.479305i	\(-0.159111\pi\)
\(29\)	−233.101	+	53.2037i	−1.49261	+	0.340679i	−0.889484	−	0.456966i	\(-0.848936\pi\)
							−0.603127	+	0.797645i	\(0.706079\pi\)
\(31\)	−166.636	+	96.2074i	−0.965442	+	0.557398i	−0.897844	−	0.440314i	\(-0.854867\pi\)
							−0.0675987	+	0.997713i	\(0.521534\pi\)
\(37\)	57.3518	−	30.3113i	0.254826	−	0.134680i	−0.334978	−	0.942226i	\(-0.608729\pi\)
							0.589804	+	0.807546i	\(0.299205\pi\)
\(41\)	−154.919	−	321.692i	−0.590103	−	1.22536i	−0.955633	−	0.294561i	\(-0.904827\pi\)
							0.365530	−	0.930800i	\(-0.380888\pi\)
\(43\)	112.113	+	320.401i	0.397606	+	1.13629i	0.952544	+	0.304402i	\(0.0984565\pi\)
							−0.554937	+	0.831892i	\(0.687258\pi\)
\(47\)	−414.225	−	305.712i	−1.28555	−	0.948781i	−0.285598	−	0.958350i	\(-0.592192\pi\)
							−0.999954	+	0.00956892i	\(0.996954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.x.a.103.15	✓	1968
5.2	odd	4	inner	245.4.x.a.152.15	yes	1968
49.10	odd	42	inner	245.4.x.a.108.15	yes	1968
245.157	even	84	inner	245.4.x.a.157.15	yes	1968

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.x.a.103.15	✓	1968	1.1	even	1	trivial
245.4.x.a.108.15	yes	1968	49.10	odd	42	inner
245.4.x.a.152.15	yes	1968	5.2	odd	4	inner
245.4.x.a.157.15	yes	1968	245.157	even	84	inner