Embedded newform 49.6.g.a.11.8

• Label: 49.6.g.a.11.8
• Level: $49$
• Weight: $6$
• Character: 49.11
• 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			11.8
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.56941 + 2.38407i) q^{2} +(-23.2656 + 3.50672i) q^{3} +(-1.47325 + 19.6592i) q^{4} +(-29.1617 + 74.3029i) q^{5} +(51.4187 - 64.4770i) q^{6} +(10.1504 + 129.244i) q^{7} +(-113.016 - 141.718i) q^{8} +(296.786 - 91.5465i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.56941	+	2.38407i	−0.454213	+	0.421448i	−0.873814	−	0.486261i	\(-0.838360\pi\)
							0.419601	+	0.907709i	\(0.362170\pi\)
\(3\)	−23.2656	+	3.50672i	−1.49249	+	0.224956i	−0.844031	−	0.536294i	\(-0.819824\pi\)
							−0.648457	+	0.761251i	\(0.724586\pi\)
\(5\)	−29.1617	+	74.3029i	−0.521661	+	1.32917i	0.390976	+	0.920401i	\(0.372137\pi\)
							−0.912637	+	0.408770i	\(0.865958\pi\)
\(7\)	10.1504	+	129.244i	0.0782961	+	0.996930i
							\(11\)	−251.192	−	77.4826i	−0.625929	−	0.193073i	−0.0344660	−	0.999406i	\(-0.510973\pi\)
−0.591463	+	0.806332i	\(0.701449\pi\)
\(13\)	47.4978	+	208.102i	0.0779499	+	0.341521i	0.998832	−	0.0483207i	\(-0.0153870\pi\)
							−0.920882	+	0.389841i	\(0.872530\pi\)
\(17\)	510.544	−	348.083i	0.428461	−	0.292120i	−0.329825	−	0.944042i	\(-0.606990\pi\)
							0.758286	+	0.651923i	\(0.226037\pi\)
\(19\)	788.967	+	1366.53i	0.501389	+	0.868432i	0.999999	+	0.00160494i	\(0.000510868\pi\)
							−0.498609	+	0.866827i	\(0.666156\pi\)
\(23\)	3408.08	+	2323.59i	1.34335	+	0.915883i	0.999730	−	0.0232238i	\(-0.00739303\pi\)
							0.343624	+	0.939107i	\(0.388345\pi\)
\(29\)	−6054.01	−	2915.46i	−1.33674	−	0.643742i	−0.377419	−	0.926043i	\(-0.623188\pi\)
							−0.959326	+	0.282300i	\(0.908903\pi\)
\(31\)	−3608.38	+	6249.89i	−0.674385	+	1.16807i	0.302263	+	0.953224i	\(0.402258\pi\)
							−0.976648	+	0.214844i	\(0.931076\pi\)
\(37\)	−329.925	−	4402.54i	−0.0396196	−	0.528687i	−0.981339	−	0.192287i	\(-0.938410\pi\)
							0.941719	−	0.336400i	\(-0.109209\pi\)
\(41\)	10283.0	+	12894.4i	0.955341	+	1.19796i	0.980148	+	0.198267i	\(0.0635314\pi\)
							−0.0248072	+	0.999692i	\(0.507897\pi\)
\(43\)	−5669.02	+	7108.72i	−0.467559	+	0.586301i	−0.958571	−	0.284852i	\(-0.908056\pi\)
							0.491012	+	0.871153i	\(0.336627\pi\)
\(47\)	1331.03	−	1235.01i	0.0878907	−	0.0815506i	−0.635004	−	0.772509i	\(-0.719002\pi\)
							0.722895	+	0.690958i	\(0.242811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.8	yes	264
49.9	even	21	inner	49.6.g.a.9.8	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.8	✓	264	49.9	even	21	inner
49.6.g.a.11.8	yes	264	1.1	even	1	trivial