Embedded newform 71.5.f.a.23.11

• Label: 71.5.f.a.23.11
• Level: $71$
• Weight: $5$
• Character: 71.23
• Analytic conductor: $7.339$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	71.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.33926737895\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			23.11
Character	\(\chi\)	\(=\)	71.23
Dual form			71.5.f.a.34.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.288161 - 0.361343i) q^{2} +(2.41740 - 10.5913i) q^{3} +(3.51280 - 15.3906i) q^{4} -24.3752 q^{5} +(-4.52369 + 2.17849i) q^{6} +(-19.5244 - 15.5702i) q^{7} +(-13.2360 + 6.37413i) q^{8} +(-33.3534 - 16.0622i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 11 q^{2} + 20 q^{3} - 187 q^{4} - 18 q^{5} + 93 q^{6} - 7 q^{7} - 123 q^{8} - 329 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.288161	−	0.361343i	−0.0720403	−	0.0903357i	0.744506	−	0.667616i	\(-0.232685\pi\)
							−0.816546	+	0.577281i	\(0.804114\pi\)
\(3\)	2.41740	−	10.5913i	0.268600	−	1.17681i	−0.643044	−	0.765829i	\(-0.722329\pi\)
							0.911643	−	0.410982i	\(-0.134814\pi\)
\(5\)	−24.3752			−0.975007			−0.487504	−	0.873121i	\(-0.662092\pi\)
							−0.487504	+	0.873121i	\(0.662092\pi\)
\(7\)	−19.5244	−	15.5702i	−0.398458	−	0.317760i	0.403678	−	0.914901i	\(-0.367732\pi\)
							−0.802136	+	0.597142i	\(0.796303\pi\)
\(11\)	−25.5227	+	52.9984i	−0.210931	+	0.438003i	−0.979413	−	0.201869i	\(-0.935298\pi\)
							0.768481	+	0.639872i	\(0.221013\pi\)
\(13\)	144.089	+	299.205i	0.852600	+	1.77044i	0.594017	+	0.804452i	\(0.297541\pi\)
							0.258582	+	0.965989i	\(0.416745\pi\)
\(17\)		−	484.568i		−	1.67670i	−0.545129	−	0.838352i	\(-0.683519\pi\)
							0.545129	−	0.838352i	\(-0.316481\pi\)
\(19\)	−65.4925	−	286.941i	−0.181420	−	0.794851i	−0.980955	−	0.194233i	\(-0.937778\pi\)
							0.799536	−	0.600618i	\(-0.205079\pi\)
\(23\)	−673.814	−	537.349i	−1.27375	−	1.01578i	−0.998519	−	0.0544046i	\(-0.982674\pi\)
							−0.275232	−	0.961378i	\(-0.588755\pi\)
\(29\)	253.381	−	1110.13i	0.301285	−	1.32002i	−0.566905	−	0.823783i	\(-0.691859\pi\)
							0.868190	−	0.496232i	\(-0.165284\pi\)
\(31\)	692.425	+	1437.84i	0.720525	+	1.49619i	0.862360	+	0.506296i	\(0.168986\pi\)
							−0.141834	+	0.989890i	\(0.545300\pi\)
\(37\)	−169.718	−	212.820i	−0.123973	−	0.155457i	0.715972	−	0.698129i	\(-0.245984\pi\)
							−0.839944	+	0.542673i	\(0.817412\pi\)
\(41\)	−763.304	−	1585.02i	−0.454077	−	0.942901i	−0.994814	−	0.101709i	\(-0.967569\pi\)
							0.540737	−	0.841192i	\(-0.318145\pi\)
\(43\)	493.281	+	618.555i	0.266783	+	0.334535i	0.897120	−	0.441786i	\(-0.145655\pi\)
							−0.630338	+	0.776321i	\(0.717084\pi\)
\(47\)	−901.555	+	205.774i	−0.408128	+	0.0931526i	−0.421656	−	0.906756i	\(-0.638551\pi\)
							0.0135277	+	0.999908i	\(0.495694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.5.f.a.23.11	✓	138
71.34	odd	14	inner	71.5.f.a.34.11	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.5.f.a.23.11	✓	138	1.1	even	1	trivial
71.5.f.a.34.11	yes	138	71.34	odd	14	inner