Embedded newform 75.4.g.b.46.5

• Label: 75.4.g.b.46.5
• Level: $75$
• Weight: $4$
• Character: 75.46
• 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			46.5
Character	\(\chi\)	\(=\)	75.46
Dual form			75.4.g.b.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.536885 - 1.65236i) q^{2} +(-2.42705 - 1.76336i) q^{3} +(4.03008 + 2.92802i) q^{4} +(8.44668 + 7.32487i) q^{5} +(-4.21675 + 3.06365i) q^{6} +7.66213 q^{7} +(18.2465 - 13.2569i) q^{8} +(2.78115 + 8.55951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.536885	−	1.65236i	0.189818	−	0.584199i	−0.810180	−	0.586181i	\(-0.800631\pi\)
							0.999998	+	0.00198195i	\(0.000630875\pi\)
\(3\)	−2.42705	−	1.76336i	−0.467086	−	0.339358i
							\(5\)	8.44668	+	7.32487i	0.755494	+	0.655156i
\(7\)	7.66213			0.413716										0.206858	−	0.978371i	\(-0.433676\pi\)
							0.206858	+	0.978371i	\(0.433676\pi\)
\(11\)	6.32569	−	19.4685i	0.173388	−	0.533633i	−0.826168	−	0.563424i	\(-0.809484\pi\)
							0.999556	+	0.0297901i	\(0.00948390\pi\)
\(13\)	−4.57589	−	14.0831i	−0.0976248	−	0.300458i	0.890304	−	0.455367i	\(-0.150492\pi\)
							−0.987929	+	0.154908i	\(0.950492\pi\)
\(17\)	88.1826	−	64.0684i	1.25808	−	0.914051i	0.259422	−	0.965764i	\(-0.416468\pi\)
							0.998662	+	0.0517127i	\(0.0164680\pi\)
\(19\)	−79.1097	+	57.4766i	−0.955211	+	0.694001i	−0.952034	−	0.305994i	\(-0.901011\pi\)
							−0.00317732	+	0.999995i	\(0.501011\pi\)
\(23\)	−24.8979	+	76.6279i	−0.225721	+	0.694697i	0.772497	+	0.635018i	\(0.219008\pi\)
							−0.998218	+	0.0596783i	\(0.980992\pi\)
\(29\)	−47.7769	−	34.7119i	−0.305929	−	0.222271i	0.424219	−	0.905560i	\(-0.360549\pi\)
							−0.730148	+	0.683289i	\(0.760549\pi\)
\(31\)	−142.644	+	103.637i	−0.826439	+	0.600443i	−0.918550	−	0.395306i	\(-0.870639\pi\)
							0.0921106	+	0.995749i	\(0.470639\pi\)
\(37\)	−71.2396	−	219.253i	−0.316533	−	0.974189i	−0.975119	−	0.221683i	\(-0.928845\pi\)
							0.658586	−	0.752506i	\(-0.271155\pi\)
\(41\)	55.4575	+	170.681i	0.211244	+	0.650143i	0.999399	+	0.0346656i	\(0.0110366\pi\)
							−0.788155	+	0.615477i	\(0.788963\pi\)
\(43\)	−407.459			−1.44505			−0.722523	−	0.691347i	\(-0.757018\pi\)
							−0.722523	+	0.691347i	\(0.757018\pi\)
\(47\)	−386.035	−	280.471i	−1.19806	−	0.870445i	−0.203972	−	0.978977i	\(-0.565385\pi\)
							−0.994093	+	0.108532i	\(0.965385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.46.5	yes	28
3.2	odd	2		225.4.h.a.46.3		28
25.6	even	5	inner	75.4.g.b.31.5	✓	28
25.9	even	10		1875.4.a.f.1.5		14
25.16	even	5		1875.4.a.g.1.10		14
75.56	odd	10		225.4.h.a.181.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.5	✓	28	25.6	even	5	inner
75.4.g.b.46.5	yes	28	1.1	even	1	trivial
225.4.h.a.46.3		28	3.2	odd	2
225.4.h.a.181.3		28	75.56	odd	10
1875.4.a.f.1.5		14	25.9	even	10
1875.4.a.g.1.10		14	25.16	even	5