Embedded newform 245.2.x.a.108.6

• Label: 245.2.x.a.108.6
• Level: $245$
• Weight: $2$
• Character: 245.108
• 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			108.6
Character	\(\chi\)	\(=\)	245.108
Dual form			245.2.x.a.152.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58305 + 1.16834i) q^{2} +(0.203804 - 1.07713i) q^{3} +(0.551504 - 1.78793i) q^{4} +(1.69056 - 1.46356i) q^{5} +(0.935823 + 1.94326i) q^{6} +(-2.28303 + 1.33709i) q^{7} +(-0.0837905 - 0.239459i) q^{8} +(1.67395 + 0.656978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58305	+	1.16834i	−1.11938	+	0.826142i	−0.986300	−	0.164963i	\(-0.947250\pi\)
							−0.133083	+	0.991105i	\(0.542488\pi\)
\(3\)	0.203804	−	1.07713i	0.117666	−	0.621880i	−0.873236	−	0.487297i	\(-0.837983\pi\)
							0.990902	−	0.134583i	\(-0.0429695\pi\)
\(5\)	1.69056	−	1.46356i	0.756041	−	0.654524i
							\(7\)	−2.28303	+	1.33709i	−0.862902	+	0.505371i
\(11\)	−0.791528	−	2.01678i	−0.238655	−	0.608082i								0.760494	−	0.649345i	\(-0.224957\pi\)
							−0.999148	+	0.0412632i	\(0.986862\pi\)
\(13\)	−0.742121	−	6.58651i	−0.205827	−	1.82677i	−0.494312	−	0.869285i	\(-0.664580\pi\)
							0.288485	−	0.957485i	\(-0.406849\pi\)
\(17\)	0.0422926	+	1.13029i	0.0102575	+	0.274137i	0.994984	+	0.100031i	\(0.0318941\pi\)
							−0.984727	+	0.174106i	\(0.944296\pi\)
\(19\)	−2.02327	−	3.50440i	−0.464169	−	0.803965i	0.534994	−	0.844856i	\(-0.320314\pi\)
							−0.999164	+	0.0408909i	\(0.986980\pi\)
\(23\)	4.83864	+	0.181049i	1.00893	+	0.0377513i	0.536740	−	0.843748i	\(-0.319656\pi\)
							0.472186	+	0.881499i	\(0.343465\pi\)
\(29\)	9.97900	+	2.27764i	1.85305	+	0.422947i	0.995769	−	0.0918954i	\(-0.0292925\pi\)
							0.857285	+	0.514843i	\(0.172150\pi\)
\(31\)	−1.41550	−	0.817241i	−0.254232	−	0.146781i	0.367469	−	0.930036i	\(-0.380225\pi\)
							−0.621700	+	0.783255i	\(0.713558\pi\)
\(37\)	−2.57350	+	4.86930i	−0.423081	+	0.800508i	−0.999866	−	0.0163878i	\(-0.994783\pi\)
							0.576785	+	0.816896i	\(0.304307\pi\)
\(41\)	0.233559	−	0.484991i	0.0364758	−	0.0757428i	−0.881935	−	0.471372i	\(-0.843759\pi\)
							0.918410	+	0.395629i	\(0.129473\pi\)
\(43\)	−5.06899	−	1.77372i	−0.773014	−	0.270489i	−0.0851882	−	0.996365i	\(-0.527149\pi\)
							−0.687826	+	0.725875i	\(0.741435\pi\)
\(47\)	−4.41841	−	5.98674i	−0.644492	−	0.873255i	0.353575	−	0.935406i	\(-0.384966\pi\)
							−0.998067	+	0.0621508i	\(0.980204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.108.6	yes	624
5.2	odd	4	inner	245.2.x.a.157.6	yes	624
49.5	odd	42	inner	245.2.x.a.103.6	✓	624
245.152	even	84	inner	245.2.x.a.152.6	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.6	✓	624	49.5	odd	42	inner
245.2.x.a.108.6	yes	624	1.1	even	1	trivial
245.2.x.a.152.6	yes	624	245.152	even	84	inner
245.2.x.a.157.6	yes	624	5.2	odd	4	inner