Embedded newform 93.2.p.b.86.4

• Label: 93.2.p.b.86.4
• Level: $93$
• Weight: $2$
• Character: 93.86
• Analytic conductor: $0.743$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			86.4
Character	\(\chi\)	\(=\)	93.86
Dual form			93.2.p.b.53.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.304287 + 0.0988687i) q^{2} +(-1.70522 + 0.303660i) q^{3} +(-1.53522 + 1.11540i) q^{4} +(-2.00745 - 1.15900i) q^{5} +(0.488855 - 0.260993i) q^{6} +(-2.85847 - 1.27267i) q^{7} +(0.732987 - 1.00887i) q^{8} +(2.81558 - 1.03562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.304287	+	0.0988687i	−0.215163	+	0.0699107i	−0.414615	−	0.909997i	\(-0.636084\pi\)
							0.199452	+	0.979908i	\(0.436084\pi\)
\(3\)	−1.70522	+	0.303660i	−0.984512	+	0.175318i
							\(5\)	−2.00745	−	1.15900i	−0.897761	−	0.518323i	−0.0212880	−	0.999773i	\(-0.506777\pi\)
−0.876473	+	0.481451i	\(0.840110\pi\)
\(7\)	−2.85847	−	1.27267i	−1.08040	−	0.481025i	−0.212192	−	0.977228i	\(-0.568060\pi\)
							−0.868206	+	0.496203i	\(0.834727\pi\)
\(11\)	0.566026	+	5.38538i	0.170663	+	1.62375i	0.659727	+	0.751505i	\(0.270672\pi\)
							−0.489064	+	0.872248i	\(0.662662\pi\)
\(13\)	−3.21419	+	2.89407i	−0.891456	+	0.802671i	−0.981140	−	0.193300i	\(-0.938081\pi\)
							0.0896837	+	0.995970i	\(0.471414\pi\)
\(17\)	0.332168	−	3.16036i	0.0805625	−	0.766501i	−0.877430	−	0.479706i	\(-0.840743\pi\)
							0.957992	−	0.286795i	\(-0.0925899\pi\)
\(19\)	−1.18996	+	1.32158i	−0.272995	+	0.303192i	−0.864016	−	0.503465i	\(-0.832058\pi\)
							0.591021	+	0.806656i	\(0.298725\pi\)
\(23\)	−0.927923	−	0.674176i	−0.193485	−	0.140575i	0.486825	−	0.873500i	\(-0.338155\pi\)
							−0.680310	+	0.732924i	\(0.738155\pi\)
\(29\)	−1.38522	−	4.26327i	−0.257229	−	0.791670i	−0.993382	−	0.114855i	\(-0.963360\pi\)
							0.736153	−	0.676815i	\(-0.236640\pi\)
\(31\)	−5.14937	+	2.11754i	−0.924855	+	0.380321i
							\(37\)	2.91344	−	1.68208i	0.478967	−	0.276532i	−0.241019	−	0.970520i	\(-0.577482\pi\)
0.719986	+	0.693989i	\(0.244148\pi\)
\(41\)	−1.17169	+	5.51239i	−0.182988	+	0.860890i	0.786851	+	0.617143i	\(0.211710\pi\)
							−0.969839	+	0.243747i	\(0.921623\pi\)
\(43\)	6.01772	+	5.41838i	0.917693	+	0.826295i	0.985197	−	0.171424i	\(-0.0548369\pi\)
							−0.0675041	+	0.997719i	\(0.521504\pi\)
\(47\)	−11.6057	−	3.77091i	−1.69286	−	0.550044i	−0.705525	−	0.708685i	\(-0.749289\pi\)
							−0.987336	+	0.158642i	\(0.949289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.86.4	yes	64
3.2	odd	2	inner	93.2.p.b.86.5	yes	64
31.22	odd	30	inner	93.2.p.b.53.5	yes	64
93.53	even	30	inner	93.2.p.b.53.4	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.53.4	✓	64	93.53	even	30	inner
93.2.p.b.53.5	yes	64	31.22	odd	30	inner
93.2.p.b.86.4	yes	64	1.1	even	1	trivial
93.2.p.b.86.5	yes	64	3.2	odd	2	inner