Embedded newform 75.4.g.a.61.5

• Label: 75.4.g.a.61.5
• Level: $75$
• Weight: $4$
• Character: 75.61
• Analytic conductor: $4.425$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.42514325043\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.5
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.a.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20373 - 0.874560i) q^{2} +(-0.927051 + 2.85317i) q^{3} +(-1.78803 + 5.50298i) q^{4} +(-10.4360 - 4.01110i) q^{5} +(1.37935 + 4.24521i) q^{6} -27.0465 q^{7} +(6.33866 + 19.5084i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} + 21 q^{3} - 18 q^{4} + 15 q^{5} + 12 q^{6} + 58 q^{7} - 111 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20373	−	0.874560i	0.425583	−	0.309204i	−0.354298	−	0.935133i	\(-0.615280\pi\)
							0.779880	+	0.625929i	\(0.215280\pi\)
\(3\)	−0.927051	+	2.85317i	−0.178411	+	0.549093i
							\(5\)	−10.4360	−	4.01110i	−0.933428	−	0.358764i
\(7\)	−27.0465			−1.46037										−0.730187	−	0.683247i	\(-0.760567\pi\)
							−0.730187	+	0.683247i	\(0.760567\pi\)
\(11\)	10.1294	−	7.35945i	0.277649	−	0.201724i	−0.440243	−	0.897879i	\(-0.645108\pi\)
							0.717891	+	0.696155i	\(0.245108\pi\)
\(13\)	−7.47633	−	5.43187i	−0.159505	−	0.115887i	0.505170	−	0.863020i	\(-0.331430\pi\)
							−0.664675	+	0.747133i	\(0.731430\pi\)
\(17\)	33.8863	+	104.291i	0.483448	+	1.48790i	0.834216	+	0.551439i	\(0.185921\pi\)
							−0.350767	+	0.936463i	\(0.614079\pi\)
\(19\)	27.7618	+	85.4421i	0.335211	+	1.03167i	0.966618	+	0.256221i	\(0.0824774\pi\)
							−0.631408	+	0.775451i	\(0.717523\pi\)
\(23\)	−17.7909	+	12.9258i	−0.161289	+	0.117184i	−0.665502	−	0.746396i	\(-0.731783\pi\)
							0.504213	+	0.863579i	\(0.331783\pi\)
\(29\)	−4.78883	+	14.7385i	−0.0306642	+	0.0943748i	−0.965217	−	0.261449i	\(-0.915800\pi\)
							0.934553	+	0.355824i	\(0.115800\pi\)
\(31\)	−87.3379	−	268.798i	−0.506011	−	1.55734i	−0.799064	−	0.601246i	\(-0.794671\pi\)
							0.293053	−	0.956096i	\(-0.405329\pi\)
\(37\)	−182.090	−	132.296i	−0.809066	−	0.587821i	0.104494	−	0.994526i	\(-0.466678\pi\)
							−0.913560	+	0.406705i	\(0.866678\pi\)
\(41\)	232.213	+	168.713i	0.884526	+	0.642646i	0.934445	−	0.356107i	\(-0.115896\pi\)
							−0.0499187	+	0.998753i	\(0.515896\pi\)
\(43\)	−285.067			−1.01098			−0.505491	−	0.862832i	\(-0.668689\pi\)
							−0.505491	+	0.862832i	\(0.668689\pi\)
\(47\)	10.2380	−	31.5093i	0.0317737	−	0.0977895i	−0.933912	−	0.357503i	\(-0.883628\pi\)
							0.965686	+	0.259714i	\(0.0836282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.a.61.5	yes	28
3.2	odd	2		225.4.h.c.136.3		28
25.4	even	10		1875.4.a.e.1.10		14
25.16	even	5	inner	75.4.g.a.16.5	✓	28
25.21	even	5		1875.4.a.h.1.5		14
75.41	odd	10		225.4.h.c.91.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.a.16.5	✓	28	25.16	even	5	inner
75.4.g.a.61.5	yes	28	1.1	even	1	trivial
225.4.h.c.91.3		28	75.41	odd	10
225.4.h.c.136.3		28	3.2	odd	2
1875.4.a.e.1.10		14	25.4	even	10
1875.4.a.h.1.5		14	25.21	even	5