Embedded newform 245.2.x.a.213.7

• Label: 245.2.x.a.213.7
• Level: $245$
• Weight: $2$
• Character: 245.213
• 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			213.7
Character	\(\chi\)	\(=\)	245.213
Dual form			245.2.x.a.222.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17337 - 1.00976i) q^{2} +(-2.21208 + 1.16912i) q^{3} +(0.0590821 + 0.391984i) q^{4} +(1.97412 - 1.05016i) q^{5} +(3.77611 + 0.861873i) q^{6} +(-1.58937 + 2.11516i) q^{7} +(-1.32072 + 2.10192i) q^{8} +(1.83649 - 2.69364i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17337	−	1.00976i	−0.829695	−	0.714011i	0.131070	−	0.991373i	\(-0.458159\pi\)
							−0.960765	+	0.277362i	\(0.910540\pi\)
\(3\)	−2.21208	+	1.16912i	−1.27714	+	0.674990i	−0.961739	−	0.273967i	\(-0.911664\pi\)
							−0.315405	+	0.948957i	\(0.602140\pi\)
\(5\)	1.97412	−	1.05016i	0.882854	−	0.469647i
							\(7\)	−1.58937	+	2.11516i	−0.600726	+	0.799455i
\(11\)	1.46728	−	1.00037i	0.442401	−	0.301624i								−0.321556	−	0.946890i	\(-0.604206\pi\)
							0.763958	+	0.645266i	\(0.223254\pi\)
\(13\)	3.17186	−	1.10988i	0.879715	−	0.307826i	0.147633	−	0.989042i	\(-0.452834\pi\)
							0.732082	+	0.681216i	\(0.238549\pi\)
\(17\)	2.09912	−	4.81123i	0.509111	−	1.16690i	−0.452322	−	0.891855i	\(-0.649404\pi\)
							0.961433	−	0.275040i	\(-0.0886912\pi\)
\(19\)	−1.96132	−	3.39710i	−0.449957	−	0.779349i	0.548425	−	0.836199i	\(-0.315227\pi\)
							−0.998383	+	0.0568506i	\(0.981894\pi\)
\(23\)	4.65740	−	2.03200i	0.971135	−	0.423702i	0.146402	−	0.989225i	\(-0.453231\pi\)
							0.824732	+	0.565523i	\(0.191326\pi\)
\(29\)	6.10394	+	4.86773i	1.13347	+	0.903914i	0.996241	−	0.0866247i	\(-0.0276081\pi\)
							0.137232	+	0.990539i	\(0.456180\pi\)
\(31\)	−1.69089	−	0.976237i	−0.303693	−	0.175337i	0.340408	−	0.940278i	\(-0.389435\pi\)
							−0.644101	+	0.764941i	\(0.722768\pi\)
\(37\)	−2.71428	+	2.00323i	−0.446225	+	0.329329i	−0.793646	−	0.608380i	\(-0.791820\pi\)
							0.347420	+	0.937709i	\(0.387058\pi\)
\(41\)	3.87462	−	0.884358i	0.605115	−	0.138113i	0.0910231	−	0.995849i	\(-0.470986\pi\)
							0.514092	+	0.857735i	\(0.328129\pi\)
\(43\)	1.45742	−	0.915757i	0.222254	−	0.139652i	−0.416303	−	0.909226i	\(-0.636674\pi\)
							0.638557	+	0.769574i	\(0.279532\pi\)
\(47\)	−1.86490	+	2.16705i	−0.272024	+	0.316097i	−0.877323	−	0.479901i	\(-0.840673\pi\)
							0.605299	+	0.795998i	\(0.293054\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.213.7	yes	624
5.2	odd	4	inner	245.2.x.a.17.20	✓	624
49.26	odd	42	inner	245.2.x.a.173.20	yes	624
245.222	even	84	inner	245.2.x.a.222.7	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.17.20	✓	624	5.2	odd	4	inner
245.2.x.a.173.20	yes	624	49.26	odd	42	inner
245.2.x.a.213.7	yes	624	1.1	even	1	trivial
245.2.x.a.222.7	yes	624	245.222	even	84	inner