Embedded newform 245.2.x.a.187.23

• Label: 245.2.x.a.187.23
• 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			187.23
Character	\(\chi\)	\(=\)	245.187
Dual form			245.2.x.a.38.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0743893 + 1.98809i) q^{2} +(-0.153109 - 0.207456i) q^{3} +(-1.95258 + 0.146326i) q^{4} +(1.96283 + 1.07111i) q^{5} +(0.401052 - 0.319828i) q^{6} +(2.51435 + 0.823431i) q^{7} +(0.00934343 + 0.0829252i) q^{8} +(0.864670 - 2.80319i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0743893	+	1.98809i	0.0526011	+	1.40580i	0.736608	+	0.676320i	\(0.236426\pi\)
							−0.684007	+	0.729475i	\(0.739764\pi\)
\(3\)	−0.153109	−	0.207456i	−0.0883977	−	0.119775i	0.758167	−	0.652060i	\(-0.226095\pi\)
							−0.846565	+	0.532286i	\(0.821333\pi\)
\(5\)	1.96283	+	1.07111i	0.877806	+	0.479017i
							\(7\)	2.51435	+	0.823431i	0.950335	+	0.311228i
\(11\)	−2.75296	+	0.849177i	−0.830050	+	0.256036i								−0.680528	−	0.732722i	\(-0.738250\pi\)
							−0.149522	+	0.988758i	\(0.547773\pi\)
\(13\)	3.20533	−	2.01404i	0.888999	−	0.558595i	−0.00832589	−	0.999965i	\(-0.502650\pi\)
							0.897325	+	0.441370i	\(0.145507\pi\)
\(17\)	−6.03309	+	1.14152i	−1.46324	+	0.276860i	−0.855718	−	0.517443i	\(-0.826884\pi\)
							−0.607522	+	0.794303i	\(0.707836\pi\)
\(19\)	−1.91591	+	3.31846i	−0.439541	+	0.761307i	−0.997654	−	0.0684581i	\(-0.978192\pi\)
							0.558113	+	0.829765i	\(0.311525\pi\)
\(23\)	0.465876	−	2.46222i	0.0971419	−	0.513407i	−0.899870	−	0.436158i	\(-0.856339\pi\)
							0.997012	−	0.0772491i	\(-0.0246137\pi\)
\(29\)	−1.94161	−	4.03180i	−0.360549	−	0.748687i	0.639245	−	0.769003i	\(-0.279247\pi\)
							−0.999794	+	0.0203163i	\(0.993533\pi\)
\(31\)	2.00906	−	1.15993i	0.360838	−	0.208330i	−0.308611	−	0.951188i	\(-0.599864\pi\)
							0.669448	+	0.742859i	\(0.266531\pi\)
\(37\)	8.62988	−	7.42661i	1.41874	−	1.22093i	0.479560	−	0.877509i	\(-0.340796\pi\)
							0.939183	−	0.343418i	\(-0.111585\pi\)
\(41\)	−6.23760	−	4.97432i	−0.974150	−	0.776859i	0.000636113	−	1.00000i	\(-0.499798\pi\)
							−0.974786	+	0.223141i	\(0.928369\pi\)
\(43\)	3.59120	+	0.404631i	0.547653	+	0.0617056i	0.381456	−	0.924387i	\(-0.375423\pi\)
							0.166197	+	0.986093i	\(0.446851\pi\)
\(47\)	0.971307	−	0.0363437i	0.141680	−	0.00530128i	0.0335427	−	0.999437i	\(-0.489321\pi\)
							0.108137	+	0.994136i	\(0.465511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.187.23	yes	624
5.3	odd	4	inner	245.2.x.a.138.23	yes	624
49.38	odd	42	inner	245.2.x.a.87.23	yes	624
245.38	even	84	inner	245.2.x.a.38.23	✓	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.38.23	✓	624	245.38	even	84	inner
245.2.x.a.87.23	yes	624	49.38	odd	42	inner
245.2.x.a.138.23	yes	624	5.3	odd	4	inner
245.2.x.a.187.23	yes	624	1.1	even	1	trivial