Embedded newform 69.7.f.a.7.15

• Label: 69.7.f.a.7.15
• Level: $69$
• Weight: $7$
• Character: 69.7
• Analytic conductor: $15.874$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8737317698\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.15
Character	\(\chi\)	\(=\)	69.7
Dual form			69.7.f.a.10.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.991787 - 1.14458i) q^{2} +(13.1138 - 8.42776i) q^{3} +(8.78172 + 61.0782i) q^{4} +(-45.4301 + 20.7472i) q^{5} +(3.35987 - 23.3684i) q^{6} +(28.4560 + 96.9124i) q^{7} +(160.160 + 102.928i) q^{8} +(100.946 - 221.041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.991787	−	1.14458i	0.123973	−	0.143073i	−0.690370	−	0.723457i	\(-0.742552\pi\)
							0.814343	+	0.580384i	\(0.197098\pi\)
\(3\)	13.1138	−	8.42776i	0.485698	−	0.312139i
							\(5\)	−45.4301	+	20.7472i	−0.363441	+	0.165978i	−0.588762	−	0.808307i	\(-0.700384\pi\)
0.225321	+	0.974285i	\(0.427657\pi\)
\(7\)	28.4560	+	96.9124i	0.0829622	+	0.282543i	0.990517	−	0.137390i	\(-0.0438715\pi\)
							−0.907555	+	0.419934i	\(0.862053\pi\)
\(11\)	−1340.90	+	1161.90i	−1.00744	+	0.872951i	−0.991912	−	0.126929i	\(-0.959488\pi\)
							−0.0155272	+	0.999879i	\(0.504943\pi\)
\(13\)	−2565.35	−	753.254i	−1.16766	−	0.342856i	−0.360255	−	0.932854i	\(-0.617310\pi\)
							−0.807405	+	0.589998i	\(0.799128\pi\)
\(17\)	2498.34	+	359.207i	0.508516	+	0.0731135i	0.391797	−	0.920052i	\(-0.371853\pi\)
							0.116718	+	0.993165i	\(0.462763\pi\)
\(19\)	−187.909	+	27.0172i	−0.0273960	+	0.00393895i	−0.155999	−	0.987757i	\(-0.549860\pi\)
							0.128603	+	0.991696i	\(0.458951\pi\)
\(23\)	4831.81	+	11166.4i	0.397124	+	0.917765i
							\(29\)	−3403.00	+	23668.4i	−0.139530	+	0.970453i	0.792964	+	0.609268i	\(0.208537\pi\)
−0.932494	+	0.361185i	\(0.882372\pi\)
\(31\)	15560.3	+	10000.0i	0.522316	+	0.335672i	0.775087	−	0.631854i	\(-0.217706\pi\)
							−0.252772	+	0.967526i	\(0.581342\pi\)
\(37\)	−76017.9	−	34716.2i	−1.50076	−	0.685373i	−0.515570	−	0.856848i	\(-0.672420\pi\)
							−0.985189	+	0.171474i	\(0.945147\pi\)
\(41\)	−11665.4	−	25543.7i	−0.169258	−	0.370624i	0.805927	−	0.592015i	\(-0.201667\pi\)
							−0.975185	+	0.221392i	\(0.928940\pi\)
\(43\)	48113.3	+	74865.8i	0.605146	+	0.941625i	0.999741	+	0.0227701i	\(0.00724859\pi\)
							−0.394595	+	0.918855i	\(0.629115\pi\)
\(47\)	145455.			1.40099			0.700496	−	0.713656i	\(-0.252962\pi\)
							0.700496	+	0.713656i	\(0.252962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.7.15	✓	240
23.10	odd	22	inner	69.7.f.a.10.15	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.7.15	✓	240	1.1	even	1	trivial
69.7.f.a.10.15	yes	240	23.10	odd	22	inner