Embedded newform 75.4.g.b.31.6

• Label: 75.4.g.b.31.6
• Level: $75$
• Weight: $4$
• Character: 75.31
• 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			31.6
Character	\(\chi\)	\(=\)	75.31
Dual form			75.4.g.b.46.6

$q$-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 \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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	0.859922	+	0.510425i	\(0.170512\pi\)
							−0.395671	+	0.918392i	\(0.629488\pi\)
\(3\)	−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	0.894623	+	0.446822i	\(0.147444\pi\)
							−0.461130	+	0.887333i	\(0.652556\pi\)
\(13\)	−7.92908	+	24.4032i	−0.169164	+	0.520633i	−0.999319	−	0.0368996i	\(-0.988252\pi\)
							0.830155	+	0.557533i	\(0.188252\pi\)
\(17\)	−75.4922	−	54.8483i	−1.07703	−	0.782510i	−0.0998698	−	0.995001i	\(-0.531843\pi\)
							−0.977163	+	0.212490i	\(0.931843\pi\)
\(19\)	95.3899	+	69.3048i	1.15179	+	0.836822i	0.988717	−	0.149793i	\(-0.0478606\pi\)
							0.163069	+	0.986615i	\(0.447861\pi\)
\(23\)	−29.1971	−	89.8595i	−0.264696	−	0.814652i	−0.991763	−	0.128085i	\(-0.959117\pi\)
							0.727067	−	0.686567i	\(-0.240883\pi\)
\(29\)	−46.7041	+	33.9325i	−0.299059	+	0.217279i	−0.727188	−	0.686438i	\(-0.759173\pi\)
							0.428128	+	0.903718i	\(0.359173\pi\)
\(31\)	32.7921	+	23.8248i	0.189988	+	0.138034i	0.678713	−	0.734403i	\(-0.262538\pi\)
							−0.488725	+	0.872438i	\(0.662538\pi\)
\(37\)	−126.405	+	389.034i	−0.561645	+	1.72856i	0.116072	+	0.993241i	\(0.462970\pi\)
							−0.677716	+	0.735323i	\(0.737030\pi\)
\(41\)	31.4511	−	96.7965i	0.119801	−	0.368709i	−0.873117	−	0.487510i	\(-0.837905\pi\)
							0.992918	+	0.118801i	\(0.0379051\pi\)
\(43\)	−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	−0.0934379	−	0.995625i	\(-0.529786\pi\)
							0.918022	+	0.396530i	\(0.129786\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.6	✓	28	1.1	even	1	trivial
75.4.g.b.46.6	yes	28	25.21	even	5	inner
225.4.h.a.46.2		28	75.71	odd	10
225.4.h.a.181.2		28	3.2	odd	2
1875.4.a.f.1.3		14	25.14	even	10
1875.4.a.g.1.12		14	25.11	even	5