Embedded newform 50.7.f.b.13.3

• Label: 50.7.f.b.13.3
• Level: $50$
• Weight: $7$
• Character: 50.13
• Analytic conductor: $11.503$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	50.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5027041810\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	50.13
Dual form			50.7.f.b.27.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.58721 + 0.884927i) q^{2} +(-8.42605 + 16.5371i) q^{3} +(30.4338 - 9.88854i) q^{4} +(-50.0727 + 114.533i) q^{5} +(32.4440 - 99.8524i) q^{6} +(-109.038 + 109.038i) q^{7} +(-161.289 + 82.1811i) q^{8} +(226.020 + 311.089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{2} + 32 q^{3} - 330 q^{5} + 256 q^{6} + 348 q^{7} + 2048 q^{8} - 760 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.58721	+	0.884927i	−0.698401	+	0.110616i
							\(3\)	−8.42605	+	16.5371i	−0.312076	+	0.612483i	−0.992763	−	0.120091i	\(-0.961681\pi\)
0.680687	+	0.732574i	\(0.261681\pi\)
\(5\)	−50.0727	+	114.533i	−0.400582	+	0.916261i
							\(7\)	−109.038	+	109.038i	−0.317896	+	0.317896i	−0.847959	−	0.530063i	\(-0.822168\pi\)
0.530063	+	0.847959i	\(0.322168\pi\)
\(11\)	744.683	+	541.044i	0.559491	+	0.406494i	0.831273	−	0.555865i	\(-0.187613\pi\)
							−0.271782	+	0.962359i	\(0.587613\pi\)
\(13\)	−1597.42	−	253.007i	−0.727093	−	0.115160i	−0.218095	−	0.975928i	\(-0.569984\pi\)
							−0.508998	+	0.860767i	\(0.669984\pi\)
\(17\)	−2607.55	−	5117.61i	−0.530746	−	1.04165i	−0.988308	−	0.152470i	\(-0.951277\pi\)
							0.457562	−	0.889177i	\(-0.348723\pi\)
\(19\)	−5386.00	−	1750.02i	−0.785246	−	0.255142i	−0.111167	−	0.993802i	\(-0.535459\pi\)
							−0.674078	+	0.738660i	\(0.735459\pi\)
\(23\)	−1883.15	−	11889.7i	−0.154775	−	0.977210i	−0.935754	−	0.352653i	\(-0.885280\pi\)
							0.780979	−	0.624557i	\(-0.214720\pi\)
\(29\)	10963.4	−	3562.22i	0.449522	−	0.146059i	−0.0755026	−	0.997146i	\(-0.524056\pi\)
							0.525025	+	0.851087i	\(0.324056\pi\)
\(31\)	−4601.16	+	14160.9i	−0.154448	+	0.475342i	−0.998105	−	0.0615410i	\(-0.980398\pi\)
							0.843656	+	0.536883i	\(0.180398\pi\)
\(37\)	−1976.58	+	12479.6i	−0.0390219	+	0.246375i	−0.999487	−	0.0320256i	\(-0.989804\pi\)
							0.960465	+	0.278400i	\(0.0898042\pi\)
\(41\)	65355.4	−	47483.5i	0.948265	−	0.688955i	−0.00213088	−	0.999998i	\(-0.500678\pi\)
							0.950396	+	0.311043i	\(0.100678\pi\)
\(43\)	55413.4	+	55413.4i	0.696963	+	0.696963i	0.963754	−	0.266791i	\(-0.0859635\pi\)
							−0.266791	+	0.963754i	\(0.585963\pi\)
\(47\)	−74085.6	−	37748.5i	−0.713576	−	0.363585i	0.0592157	−	0.998245i	\(-0.481140\pi\)
							−0.772791	+	0.634660i	\(0.781140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.7.f.b.13.3	✓	64
25.2	odd	20	inner	50.7.f.b.27.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.7.f.b.13.3	✓	64	1.1	even	1	trivial
50.7.f.b.27.3	yes	64	25.2	odd	20	inner