Embedded newform 245.2.x.a.103.10

• Label: 245.2.x.a.103.10
• 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			103.10
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.594785 + 0.805906i) q^{2} +(-2.97522 + 0.562942i) q^{3} +(0.293796 + 0.952464i) q^{4} +(0.717276 - 2.11790i) q^{5} +(1.31594 - 2.73257i) q^{6} +(-0.775557 - 2.52953i) q^{7} +(-2.83318 - 0.991371i) q^{8} +(5.74240 - 2.25373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.594785	+	0.805906i	−0.420577	+	0.569861i	−0.962746	−	0.270408i	\(-0.912841\pi\)
							0.542169	+	0.840269i	\(0.317603\pi\)
\(3\)	−2.97522	+	0.562942i	−1.71774	+	0.325015i	−0.949174	−	0.314752i	\(-0.898079\pi\)
							−0.768569	+	0.639766i	\(0.779031\pi\)
\(5\)	0.717276	−	2.11790i	0.320775	−	0.947155i
							\(7\)	−0.775557	−	2.52953i	−0.293133	−	0.956072i
\(11\)	−0.863754	+	2.20081i	−0.260431	+	0.663569i								−0.999954	−	0.00956290i	\(-0.996956\pi\)
							0.739523	+	0.673132i	\(0.235051\pi\)
\(13\)	4.34604	+	0.489681i	1.20537	+	0.135813i	0.691708	−	0.722178i	\(-0.256859\pi\)
							0.513666	+	0.857991i	\(0.328287\pi\)
\(17\)	3.10959	+	0.116353i	0.754187	+	0.0282197i	0.411720	−	0.911310i	\(-0.364928\pi\)
							0.342467	+	0.939530i	\(0.388738\pi\)
\(19\)	2.77390	−	4.80453i	0.636376	−	1.10224i	−0.349846	−	0.936807i	\(-0.613766\pi\)
							0.986222	−	0.165428i	\(-0.0529007\pi\)
\(23\)	0.0498992	+	1.33358i	0.0104047	+	0.278071i	0.994784	+	0.102008i	\(0.0325268\pi\)
							−0.984379	+	0.176063i	\(0.943664\pi\)
\(29\)	0.416686	−	0.0951059i	0.0773767	−	0.0176607i	−0.183657	−	0.982990i	\(-0.558794\pi\)
							0.261034	+	0.965330i	\(0.415937\pi\)
\(31\)	6.63674	−	3.83173i	1.19199	−	0.688198i	0.233236	−	0.972420i	\(-0.425069\pi\)
							0.958758	+	0.284222i	\(0.0917353\pi\)
\(37\)	5.46733	−	2.88957i	0.898823	−	0.475042i	0.0470142	−	0.998894i	\(-0.485029\pi\)
							0.851809	+	0.523852i	\(0.175506\pi\)
\(41\)	−2.54237	−	5.27929i	−0.397051	−	0.824486i	−0.999650	−	0.0264416i	\(-0.991582\pi\)
							0.602599	−	0.798044i	\(-0.294132\pi\)
\(43\)	0.467360	+	1.33564i	0.0712717	+	0.203683i	0.974009	−	0.226510i	\(-0.0727314\pi\)
							−0.902737	+	0.430192i	\(0.858446\pi\)
\(47\)	−9.49073	−	7.00448i	−1.38437	−	1.02171i	−0.994921	−	0.100658i	\(-0.967905\pi\)
							−0.389444	−	0.921050i	\(-0.627333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.10	✓	624
5.2	odd	4	inner	245.2.x.a.152.10	yes	624
49.10	odd	42	inner	245.2.x.a.108.10	yes	624
245.157	even	84	inner	245.2.x.a.157.10	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.10	✓	624	1.1	even	1	trivial
245.2.x.a.108.10	yes	624	49.10	odd	42	inner
245.2.x.a.152.10	yes	624	5.2	odd	4	inner
245.2.x.a.157.10	yes	624	245.157	even	84	inner