Embedded newform 49.6.g.a.2.5

• Label: 49.6.g.a.2.5
• Level: $49$
• Weight: $6$
• Character: 49.2
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.56205 - 2.33258i) q^{2} +(-5.53040 + 14.0912i) q^{3} +(25.3041 + 17.2520i) q^{4} +(-100.121 + 15.0908i) q^{5} +(74.6902 - 93.6586i) q^{6} +(-91.8316 + 91.5093i) q^{7} +(6.78132 + 8.50350i) q^{8} +(10.1540 + 9.42155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.56205	−	2.33258i	−1.33679	−	0.412347i	−0.457748	−	0.889082i	\(-0.651344\pi\)
							−0.879047	+	0.476735i	\(0.841820\pi\)
\(3\)	−5.53040	+	14.0912i	−0.354776	+	0.903953i	0.636395	+	0.771363i	\(0.280425\pi\)
							−0.991171	+	0.132590i	\(0.957671\pi\)
\(5\)	−100.121	+	15.0908i	−1.79102	+	0.269953i	−0.958717	−	0.284363i	\(-0.908218\pi\)
							−0.832306	+	0.554316i	\(0.812980\pi\)
\(7\)	−91.8316	+	91.5093i	−0.708349	+	0.705863i
							\(11\)	−221.256	+	205.295i	−0.551331	+	0.511560i	−0.905913	−	0.423465i	\(-0.860814\pi\)
0.354582	+	0.935025i	\(0.384623\pi\)
\(13\)	−175.181	−	767.518i	−0.287494	−	1.25959i	−0.887952	−	0.459937i	\(-0.847872\pi\)
							0.600458	−	0.799656i	\(-0.294985\pi\)
\(17\)	53.0859	+	708.383i	0.0445510	+	0.594492i	0.974148	+	0.225913i	\(0.0725364\pi\)
							−0.929597	+	0.368579i	\(0.879845\pi\)
\(19\)	713.717	−	1236.19i	0.453568	−	0.785603i	−0.545037	−	0.838412i	\(-0.683484\pi\)
							0.998605	+	0.0528096i	\(0.0168176\pi\)
\(23\)	−89.7245	+	1197.29i	−0.0353665	+	0.471933i	0.951200	+	0.308574i	\(0.0998518\pi\)
							−0.986567	+	0.163359i	\(0.947767\pi\)
\(29\)	−390.631	−	188.118i	−0.0862525	−	0.0415370i	0.390260	−	0.920705i	\(-0.372385\pi\)
							−0.476513	+	0.879168i	\(0.658099\pi\)
\(31\)	2717.30	+	4706.50i	0.507847	+	0.879617i	0.999959	+	0.00908502i	\(0.00289189\pi\)
							−0.492112	+	0.870532i	\(0.663775\pi\)
\(37\)	3780.27	−	2577.34i	0.453961	−	0.309505i	−0.314666	−	0.949202i	\(-0.601893\pi\)
							0.768627	+	0.639697i	\(0.220940\pi\)
\(41\)	−3800.23	−	4765.34i	−0.353062	−	0.442725i	0.573309	−	0.819340i	\(-0.305660\pi\)
							−0.926370	+	0.376614i	\(0.877088\pi\)
\(43\)	−14429.9	+	18094.5i	−1.19012	+	1.49237i	−0.361670	+	0.932306i	\(0.617793\pi\)
							−0.828452	+	0.560060i	\(0.810778\pi\)
\(47\)	−1344.66	−	414.772i	−0.0887906	−	0.0273883i	0.250042	−	0.968235i	\(-0.419556\pi\)
							−0.338832	+	0.940847i	\(0.610032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.5	✓	264
49.25	even	21	inner	49.6.g.a.25.5	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.5	✓	264	1.1	even	1	trivial
49.6.g.a.25.5	yes	264	49.25	even	21	inner