Embedded newform 75.4.g.b.46.6

• Label: 75.4.g.b.46.6
• 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			46.6
Character	\(\chi\)	\(=\)	75.46
Dual form			75.4.g.b.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31310 - 4.04131i) q^{2} +(-2.42705 - 1.76336i) q^{3} +(-8.13578 - 5.91099i) q^{4} +(-8.20823 + 7.59111i) q^{5} +(-10.3132 + 7.49299i) q^{6} -28.2853 q^{7} +(-7.06929 + 5.13614i) q^{8} +(2.78115 + 8.55951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31310	−	4.04131i	0.464251	−	1.42882i	−0.395671	−	0.918392i	\(-0.629488\pi\)
							0.859922	−	0.510425i	\(-0.170512\pi\)
\(3\)	−2.42705	−	1.76336i	−0.467086	−	0.339358i
							\(5\)	−8.20823	+	7.59111i	−0.734166	+	0.678970i
\(7\)	−28.2853			−1.52726										−0.763631	−	0.645652i	\(-0.776586\pi\)
							−0.763631	+	0.645652i	\(0.776586\pi\)
\(11\)	15.8151	−	48.6738i	0.433493	−	1.33415i	−0.461130	−	0.887333i	\(-0.652556\pi\)
							0.894623	−	0.446822i	\(-0.147444\pi\)
\(13\)	−7.92908	−	24.4032i	−0.169164	−	0.520633i	0.830155	−	0.557533i	\(-0.188252\pi\)
							−0.999319	+	0.0368996i	\(0.988252\pi\)
\(17\)	−75.4922	+	54.8483i	−1.07703	+	0.782510i	−0.977163	−	0.212490i	\(-0.931843\pi\)
							−0.0998698	+	0.995001i	\(0.531843\pi\)
\(19\)	95.3899	−	69.3048i	1.15179	−	0.836822i	0.163069	−	0.986615i	\(-0.447861\pi\)
							0.988717	+	0.149793i	\(0.0478606\pi\)
\(23\)	−29.1971	+	89.8595i	−0.264696	+	0.814652i	0.727067	+	0.686567i	\(0.240883\pi\)
							−0.991763	+	0.128085i	\(0.959117\pi\)
\(29\)	−46.7041	−	33.9325i	−0.299059	−	0.217279i	0.428128	−	0.903718i	\(-0.359173\pi\)
							−0.727188	+	0.686438i	\(0.759173\pi\)
\(31\)	32.7921	−	23.8248i	0.189988	−	0.138034i	−0.488725	−	0.872438i	\(-0.662538\pi\)
							0.678713	+	0.734403i	\(0.262538\pi\)
\(37\)	−126.405	−	389.034i	−0.561645	−	1.72856i	−0.677716	−	0.735323i	\(-0.737030\pi\)
							0.116072	−	0.993241i	\(-0.462970\pi\)
\(41\)	31.4511	+	96.7965i	0.119801	+	0.368709i	0.992918	−	0.118801i	\(-0.0379051\pi\)
							−0.873117	+	0.487510i	\(0.837905\pi\)
\(43\)	−1.56191			−0.00553930			−0.00276965	−	0.999996i	\(-0.500882\pi\)
							−0.00276965	+	0.999996i	\(0.500882\pi\)
\(47\)	265.694	+	193.038i	0.824584	+	0.599095i	0.918022	−	0.396530i	\(-0.129786\pi\)
							−0.0934379	+	0.995625i	\(0.529786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.46.6	yes	28
3.2	odd	2		225.4.h.a.46.2		28
25.6	even	5	inner	75.4.g.b.31.6	✓	28
25.9	even	10		1875.4.a.f.1.3		14
25.16	even	5		1875.4.a.g.1.12		14
75.56	odd	10		225.4.h.a.181.2		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.6	✓	28	25.6	even	5	inner
75.4.g.b.46.6	yes	28	1.1	even	1	trivial
225.4.h.a.46.2		28	3.2	odd	2
225.4.h.a.181.2		28	75.56	odd	10
1875.4.a.f.1.3		14	25.9	even	10
1875.4.a.g.1.12		14	25.16	even	5