Embedded newform 245.4.x.a.103.26

• Label: 245.4.x.a.103.26
• 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			103.26
Character	\(\chi\)	\(=\)	245.103
Dual form			245.4.x.a.157.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44227 + 1.95420i) q^{2} +(2.08813 - 0.395096i) q^{3} +(0.619267 + 2.00761i) q^{4} +(2.29020 - 10.9433i) q^{5} +(-2.23955 + 4.65048i) q^{6} +(6.13252 + 17.4755i) q^{7} +(-23.1564 - 8.10279i) q^{8} +(-20.9294 + 8.21418i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1968 q - 26 q^{2} - 22 q^{3} + 2 q^{5} - 112 q^{6} - 32 q^{7} + 96 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44227	+	1.95420i	−0.509919	+	0.690915i	−0.981621	−	0.190843i	\(-0.938878\pi\)
							0.471702	+	0.881758i	\(0.343640\pi\)
\(3\)	2.08813	−	0.395096i	0.401862	−	0.0760363i	0.0189481	−	0.999820i	\(-0.493968\pi\)
							0.382914	+	0.923784i	\(0.374921\pi\)
\(5\)	2.29020	−	10.9433i	0.204842	−	0.978795i
							\(7\)	6.13252	+	17.4755i	0.331125	+	0.943587i
\(11\)	8.20233	−	20.8992i	0.224827	−	0.572850i								−0.773299	−	0.634041i	\(-0.781395\pi\)
							0.998126	+	0.0611918i	\(0.0194901\pi\)
\(13\)	−33.4714	−	3.77132i	−0.714099	−	0.0804596i	−0.252562	−	0.967581i	\(-0.581273\pi\)
							−0.461537	+	0.887121i	\(0.652702\pi\)
\(17\)	−5.07721	−	0.189976i	−0.0724355	−	0.00271034i	0.00114889	−	0.999999i	\(-0.499634\pi\)
							−0.0735844	+	0.997289i	\(0.523444\pi\)
\(19\)	−67.0465	+	116.128i	−0.809554	+	1.40219i	0.103619	+	0.994617i	\(0.466958\pi\)
							−0.913173	+	0.407572i	\(0.866376\pi\)
\(23\)	6.33701	+	169.360i	0.0574504	+	1.53539i	0.668447	+	0.743760i	\(0.266960\pi\)
							−0.610996	+	0.791634i	\(0.709231\pi\)
\(29\)	38.5046	−	8.78842i	0.246556	−	0.0562748i	−0.0974569	−	0.995240i	\(-0.531071\pi\)
							0.344013	+	0.938965i	\(0.388214\pi\)
\(31\)	−141.680	+	81.7991i	−0.820856	+	0.473921i	−0.850711	−	0.525633i	\(-0.823829\pi\)
							0.0298558	+	0.999554i	\(0.490495\pi\)
\(37\)	46.4979	−	24.5749i	0.206600	−	0.109191i	−0.360768	−	0.932656i	\(-0.617485\pi\)
							0.567368	+	0.823464i	\(0.307962\pi\)
\(41\)	−195.282	−	405.508i	−0.743853	−	1.54463i	−0.835906	−	0.548873i	\(-0.815057\pi\)
							0.0920529	−	0.995754i	\(-0.470657\pi\)
\(43\)	76.0718	+	217.401i	0.269787	+	0.771007i	0.996463	+	0.0840274i	\(0.0267783\pi\)
							−0.726676	+	0.686980i	\(0.758936\pi\)
\(47\)	−416.601	−	307.465i	−1.29292	−	0.954222i	−0.292926	−	0.956135i	\(-0.594629\pi\)
							−0.999998	+	0.00191356i	\(0.999391\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.x.a.103.26	✓	1968
5.2	odd	4	inner	245.4.x.a.152.26	yes	1968
49.10	odd	42	inner	245.4.x.a.108.26	yes	1968
245.157	even	84	inner	245.4.x.a.157.26	yes	1968

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.x.a.103.26	✓	1968	1.1	even	1	trivial
245.4.x.a.108.26	yes	1968	49.10	odd	42	inner
245.4.x.a.152.26	yes	1968	5.2	odd	4	inner
245.4.x.a.157.26	yes	1968	245.157	even	84	inner