Embedded newform 245.2.x.a.187.18

• Label: 245.2.x.a.187.18
• Level: $245$
• Weight: $2$
• Character: 245.187
• 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			187.18
Character	\(\chi\)	\(=\)	245.187
Dual form			245.2.x.a.38.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0440656 + 1.17768i) q^{2} +(0.970239 + 1.31463i) q^{3} +(0.609421 - 0.0456698i) q^{4} +(-1.66230 + 1.49558i) q^{5} +(-1.50545 + 1.20056i) q^{6} +(2.17259 - 1.50992i) q^{7} +(0.344540 + 3.05788i) q^{8} +(0.0973847 - 0.315713i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0440656	+	1.17768i	0.0311591	+	0.832745i	0.924004	+	0.382382i	\(0.124896\pi\)
							−0.892845	+	0.450363i	\(0.851294\pi\)
\(3\)	0.970239	+	1.31463i	0.560168	+	0.759000i	0.989561	−	0.144117i	\(-0.0460343\pi\)
							−0.429393	+	0.903118i	\(0.641272\pi\)
\(5\)	−1.66230	+	1.49558i	−0.743405	+	0.668842i
							\(7\)	2.17259	−	1.50992i	0.821160	−	0.570698i
\(11\)	1.33225	−	0.410944i	0.401688	−	0.123904i								−0.0873271	−	0.996180i	\(-0.527833\pi\)
							0.489015	+	0.872275i	\(0.337356\pi\)
\(13\)	−4.03434	+	2.53494i	−1.11892	+	0.703066i	−0.958815	−	0.284033i	\(-0.908327\pi\)
							−0.160109	+	0.987099i	\(0.551185\pi\)
\(17\)	−4.55543	+	0.861934i	−1.10485	+	0.209050i	−0.706173	−	0.708040i	\(-0.749580\pi\)
							−0.398682	+	0.917089i	\(0.630532\pi\)
\(19\)	3.94071	−	6.82552i	0.904062	−	1.56588i	0.0818895	−	0.996641i	\(-0.473905\pi\)
							0.822172	−	0.569239i	\(-0.192762\pi\)
\(23\)	0.467585	−	2.47124i	0.0974981	−	0.515290i	−0.899437	−	0.437051i	\(-0.856023\pi\)
							0.996935	−	0.0782386i	\(-0.0249296\pi\)
\(29\)	0.988910	+	2.05349i	0.183636	+	0.381324i	0.972382	−	0.233395i	\(-0.0749836\pi\)
							−0.788746	+	0.614719i	\(0.789269\pi\)
\(31\)	−6.23542	+	3.60002i	−1.11991	+	0.646582i	−0.941379	−	0.337351i	\(-0.890469\pi\)
							−0.178535	+	0.983934i	\(0.557136\pi\)
\(37\)	0.287323	−	0.247262i	0.0472357	−	0.0406496i	−0.628353	−	0.777928i	\(-0.716271\pi\)
							0.675589	+	0.737279i	\(0.263890\pi\)
\(41\)	−2.88812	−	2.30320i	−0.451048	−	0.359699i	0.371463	−	0.928448i	\(-0.378856\pi\)
							−0.822511	+	0.568749i	\(0.807428\pi\)
\(43\)	1.56036	+	0.175811i	0.237953	+	0.0268109i	0.230136	−	0.973159i	\(-0.426083\pi\)
							0.00781756	+	0.999969i	\(0.497512\pi\)
\(47\)	11.5449	−	0.431981i	1.68400	−	0.0630109i	0.821863	−	0.569685i	\(-0.192935\pi\)
							0.862138	+	0.506674i	\(0.169125\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.187.18	yes	624
5.3	odd	4	inner	245.2.x.a.138.18	yes	624
49.38	odd	42	inner	245.2.x.a.87.18	yes	624
245.38	even	84	inner	245.2.x.a.38.18	✓	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.38.18	✓	624	245.38	even	84	inner
245.2.x.a.87.18	yes	624	49.38	odd	42	inner
245.2.x.a.138.18	yes	624	5.3	odd	4	inner
245.2.x.a.187.18	yes	624	1.1	even	1	trivial