Embedded newform 245.2.x.a.103.24

• Label: 245.2.x.a.103.24
• 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			103.24
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37582 - 1.86418i) q^{2} +(1.07073 - 0.202593i) q^{3} +(-0.992748 - 3.21841i) q^{4} +(1.81615 - 1.30445i) q^{5} +(1.09546 - 2.27476i) q^{6} +(-0.703019 + 2.55064i) q^{7} +(-2.99175 - 1.04686i) q^{8} +(-1.68721 + 0.662180i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37582	−	1.86418i	0.972854	−	1.31817i	0.0249255	−	0.999689i	\(-0.492065\pi\)
							0.947929	−	0.318482i	\(-0.103173\pi\)
\(3\)	1.07073	−	0.202593i	0.618185	−	0.116967i	0.132619	−	0.991167i	\(-0.457661\pi\)
							0.485566	+	0.874200i	\(0.338614\pi\)
\(5\)	1.81615	−	1.30445i	0.812207	−	0.583370i
							\(7\)	−0.703019	+	2.55064i	−0.265716	+	0.964051i
\(11\)	−1.61116	+	4.10516i	−0.485782	+	1.23775i								0.452464	+	0.891783i	\(0.350545\pi\)
							−0.938246	+	0.345969i	\(0.887550\pi\)
\(13\)	0.926254	+	0.104364i	0.256897	+	0.0289453i	0.239473	−	0.970903i	\(-0.423025\pi\)
							0.0174231	+	0.999848i	\(0.494454\pi\)
\(17\)	−5.42609	−	0.203030i	−1.31602	−	0.0492420i	−0.629870	−	0.776701i	\(-0.716892\pi\)
							−0.686151	+	0.727459i	\(0.740701\pi\)
\(19\)	−1.91951	+	3.32470i	−0.440367	+	0.762737i	−0.997717	−	0.0675403i	\(-0.978485\pi\)
							0.557350	+	0.830278i	\(0.311818\pi\)
\(23\)	−0.310675	−	8.30298i	−0.0647803	−	1.73129i	−0.531650	−	0.846964i	\(-0.678428\pi\)
							0.466870	−	0.884326i	\(-0.345382\pi\)
\(29\)	4.98928	−	1.13877i	0.926487	−	0.211465i	0.267447	−	0.963573i	\(-0.413820\pi\)
							0.659040	+	0.752108i	\(0.270963\pi\)
\(31\)	−1.46431	+	0.845422i	−0.262999	+	0.151842i	−0.625702	−	0.780062i	\(-0.715187\pi\)
							0.362703	+	0.931905i	\(0.381854\pi\)
\(37\)	3.32457	−	1.75709i	0.546556	−	0.288863i	−0.171108	−	0.985252i	\(-0.554735\pi\)
							0.717665	+	0.696389i	\(0.245211\pi\)
\(41\)	−3.41460	−	7.09049i	−0.533271	−	1.10735i	−0.977403	−	0.211386i	\(-0.932202\pi\)
							0.444132	−	0.895961i	\(-0.353512\pi\)
\(43\)	2.65863	+	7.59792i	0.405437	+	1.15867i	0.947893	+	0.318588i	\(0.103209\pi\)
							−0.542456	+	0.840084i	\(0.682506\pi\)
\(47\)	3.61690	+	2.66940i	0.527579	+	0.389371i	0.824789	−	0.565441i	\(-0.191294\pi\)
							−0.297209	+	0.954812i	\(0.596056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.24	✓	624
5.2	odd	4	inner	245.2.x.a.152.24	yes	624
49.10	odd	42	inner	245.2.x.a.108.24	yes	624
245.157	even	84	inner	245.2.x.a.157.24	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.24	✓	624	1.1	even	1	trivial
245.2.x.a.108.24	yes	624	49.10	odd	42	inner
245.2.x.a.152.24	yes	624	5.2	odd	4	inner
245.2.x.a.157.24	yes	624	245.157	even	84	inner