Embedded newform 245.2.t.a.149.8

• Label: 245.2.t.a.149.8
• Level: $245$
• Weight: $2$
• Character: 245.149
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			149.8
Character	\(\chi\)	\(=\)	245.149
Dual form			245.2.t.a.74.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.443222 + 1.43689i) q^{2} +(1.53230 + 0.601382i) q^{3} +(-0.215728 - 0.147081i) q^{4} +(1.53396 + 1.62695i) q^{5} +(-1.54327 + 1.93520i) q^{6} +(-2.62623 - 0.320787i) q^{7} +(-2.04432 + 1.63029i) q^{8} +(-0.212881 - 0.197525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 50 q^{4} - 12 q^{5} - 10 q^{6} - 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.443222	+	1.43689i	−0.313405	+	1.01603i	0.652642	+	0.757667i	\(0.273661\pi\)
							−0.966047	+	0.258367i	\(0.916815\pi\)
\(3\)	1.53230	+	0.601382i	0.884672	+	0.347208i	0.763795	−	0.645459i	\(-0.223334\pi\)
							0.120878	+	0.992667i	\(0.461429\pi\)
\(5\)	1.53396	+	1.62695i	0.686006	+	0.727596i
							\(7\)	−2.62623	−	0.320787i	−0.992622	−	0.121246i
\(11\)	−0.171375	+	0.159013i	−0.0516715	+	0.0479441i								−0.705586	−	0.708624i	\(-0.749316\pi\)
							0.653914	+	0.756569i	\(0.273126\pi\)
\(13\)	5.60004	−	1.27817i	1.55317	−	0.354502i	0.642059	−	0.766656i	\(-0.278081\pi\)
							0.911114	+	0.412154i	\(0.135223\pi\)
\(17\)	0.840524	−	0.0629886i	0.203857	−	0.0152770i	0.0275899	−	0.999619i	\(-0.491217\pi\)
							0.176267	+	0.984342i	\(0.443598\pi\)
\(19\)	0.837168	−	1.45002i	0.192060	−	0.332657i	−0.753873	−	0.657020i	\(-0.771817\pi\)
							0.945933	+	0.324363i	\(0.105150\pi\)
\(23\)	−2.39667	−	0.179606i	−0.499740	−	0.0374504i	−0.177524	−	0.984116i	\(-0.556809\pi\)
							−0.322216	+	0.946666i	\(0.604428\pi\)
\(29\)	7.41896	+	3.57278i	1.37767	+	0.663449i	0.968500	−	0.249012i	\(-0.0801058\pi\)
							0.409165	+	0.912460i	\(0.365820\pi\)
\(31\)	−1.84350	−	3.19304i	−0.331102	−	0.573486i	0.651626	−	0.758541i	\(-0.274087\pi\)
							−0.982728	+	0.185054i	\(0.940754\pi\)
\(37\)	−6.48867	−	9.51713i	−1.06673	−	1.56461i	−0.799208	−	0.601055i	\(-0.794747\pi\)
							−0.267522	−	0.963552i	\(-0.586205\pi\)
\(41\)	2.71880	+	3.40926i	0.424605	+	0.532438i	0.947413	−	0.320013i	\(-0.103687\pi\)
							−0.522808	+	0.852450i	\(0.675116\pi\)
\(43\)	3.89995	+	3.11011i	0.594737	+	0.474287i	0.873999	−	0.485928i	\(-0.161518\pi\)
							−0.279262	+	0.960215i	\(0.590090\pi\)
\(47\)	−1.90079	+	6.16221i	−0.277259	+	0.898851i	0.704660	+	0.709546i	\(0.251100\pi\)
							−0.981918	+	0.189305i	\(0.939376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.t.a.149.8	yes	312
5.4	even	2	inner	245.2.t.a.149.19	yes	312
49.25	even	21	inner	245.2.t.a.74.19	yes	312
245.74	even	42	inner	245.2.t.a.74.8	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.t.a.74.8	✓	312	245.74	even	42	inner
245.2.t.a.74.19	yes	312	49.25	even	21	inner
245.2.t.a.149.8	yes	312	1.1	even	1	trivial
245.2.t.a.149.19	yes	312	5.4	even	2	inner