Embedded newform 245.2.x.a.103.22

• Label: 245.2.x.a.103.22
• Level: $245$
• Weight: $2$
• Character: 245.103
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.22
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03775 - 1.40610i) q^{2} +(-2.88002 + 0.544928i) q^{3} +(-0.310689 - 1.00723i) q^{4} +(-0.873838 + 2.05825i) q^{5} +(-2.22252 + 4.61510i) q^{6} +(2.39810 - 1.11765i) q^{7} +(1.56035 + 0.545990i) q^{8} +(5.20492 - 2.04278i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03775	−	1.40610i	0.733801	−	0.994266i	−0.265742	−	0.964044i	\(-0.585617\pi\)
							0.999543	−	0.0302213i	\(-0.00962122\pi\)
\(3\)	−2.88002	+	0.544928i	−1.66278	+	0.314615i	−0.930473	−	0.366362i	\(-0.880603\pi\)
							−0.732306	+	0.680976i	\(0.761556\pi\)
\(5\)	−0.873838	+	2.05825i	−0.390792	+	0.920479i
							\(7\)	2.39810	−	1.11765i	0.906395	−	0.422431i
\(11\)	−1.94698	+	4.96081i	−0.587035	+	1.49574i								0.260753	+	0.965406i	\(0.416029\pi\)
							−0.847788	+	0.530336i	\(0.822066\pi\)
\(13\)	4.91824	+	0.554153i	1.36407	+	0.153694i	0.763465	−	0.645849i	\(-0.223496\pi\)
							0.600609	+	0.799543i	\(0.294925\pi\)
\(17\)	−1.15367	−	0.0431672i	−0.279805	−	0.0104696i	−0.102880	−	0.994694i	\(-0.532806\pi\)
							−0.176926	+	0.984224i	\(0.556615\pi\)
\(19\)	−3.19588	+	5.53542i	−0.733184	+	1.26991i	0.222331	+	0.974971i	\(0.428633\pi\)
							−0.955515	+	0.294941i	\(0.904700\pi\)
\(23\)	0.0326720	+	0.873178i	0.00681258	+	0.182070i	0.998552	+	0.0537994i	\(0.0171331\pi\)
							−0.991739	+	0.128271i	\(0.959057\pi\)
\(29\)	−0.772986	+	0.176429i	−0.143540	+	0.0327620i	−0.293687	−	0.955902i	\(-0.594882\pi\)
							0.150147	+	0.988664i	\(0.452025\pi\)
\(31\)	0.352872	−	0.203731i	0.0633777	−	0.0365911i	−0.467976	−	0.883741i	\(-0.655017\pi\)
							0.531354	+	0.847150i	\(0.321683\pi\)
\(37\)	2.19991	−	1.16268i	0.361662	−	0.191144i	−0.276559	−	0.960997i	\(-0.589194\pi\)
							0.638222	+	0.769853i	\(0.279670\pi\)
\(41\)	−0.961329	−	1.99622i	−0.150134	−	0.311757i	0.812315	−	0.583219i	\(-0.198207\pi\)
							−0.962449	+	0.271462i	\(0.912493\pi\)
\(43\)	−1.49152	−	4.26252i	−0.227455	−	0.650028i	−0.999888	−	0.0149328i	\(-0.995247\pi\)
							0.772434	−	0.635095i	\(-0.219039\pi\)
\(47\)	2.68451	+	1.98126i	0.391576	+	0.288996i	0.771759	−	0.635915i	\(-0.219377\pi\)
							−0.380183	+	0.924911i	\(0.624139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.22	✓	624
5.2	odd	4	inner	245.2.x.a.152.22	yes	624
49.10	odd	42	inner	245.2.x.a.108.22	yes	624
245.157	even	84	inner	245.2.x.a.157.22	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.22	✓	624	1.1	even	1	trivial
245.2.x.a.108.22	yes	624	49.10	odd	42	inner
245.2.x.a.152.22	yes	624	5.2	odd	4	inner
245.2.x.a.157.22	yes	624	245.157	even	84	inner