Embedded newform 105.4.u.a.52.6

• Label: 105.4.u.a.52.6
• Level: $105$
• Weight: $4$
• Character: 105.52
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			52.6
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.871352 + 3.25193i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(-2.88758 - 1.66715i) q^{4} +(6.94823 + 8.75911i) q^{5} -10.0999i q^{6} +(10.8772 + 14.9896i) q^{7} +(-11.1071 + 11.1071i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.871352	+	3.25193i	−0.308069	+	1.14973i	0.622202	+	0.782857i	\(0.286238\pi\)
							−0.930271	+	0.366873i	\(0.880428\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	6.94823	+	8.75911i	0.621469	+	0.783439i
\(7\)	10.8772	+	14.9896i	0.587313	+	0.809360i
							\(11\)	−5.62205	+	9.73767i	−0.154101	+	0.266911i	−0.932731	−	0.360572i	\(-0.882581\pi\)
0.778630	+	0.627483i	\(0.215915\pi\)
\(13\)	−26.7257	−	26.7257i	−0.570182	−	0.570182i	0.361997	−	0.932179i	\(-0.382095\pi\)
							−0.932179	+	0.361997i	\(0.882095\pi\)
\(17\)	15.0592	+	56.2015i	0.214846	+	0.801816i	0.986221	+	0.165435i	\(0.0529027\pi\)
							−0.771375	+	0.636381i	\(0.780431\pi\)
\(19\)	−53.3485	−	92.4023i	−0.644157	−	1.11571i	−0.984495	−	0.175410i	\(-0.943875\pi\)
							0.340338	−	0.940303i	\(-0.389458\pi\)
\(23\)	31.3262	+	8.39383i	0.283998	+	0.0760972i	0.398006	−	0.917383i	\(-0.369702\pi\)
							−0.114008	+	0.993480i	\(0.536369\pi\)
\(29\)		−	30.8851i		−	0.197766i	−0.995099	−	0.0988829i	\(-0.968473\pi\)
							0.995099	−	0.0988829i	\(-0.0315269\pi\)
\(31\)	25.2050	+	14.5521i	0.146030	+	0.0843107i	0.571235	−	0.820787i	\(-0.306465\pi\)
							−0.425205	+	0.905097i	\(0.639798\pi\)
\(37\)	−43.9853	+	164.155i	−0.195436	+	0.729378i	0.796717	+	0.604352i	\(0.206568\pi\)
							−0.992154	+	0.125026i	\(0.960099\pi\)
\(41\)			221.168i			0.842454i	0.906955	+	0.421227i	\(0.138400\pi\)
							−0.906955	+	0.421227i	\(0.861600\pi\)
\(43\)	25.7945	−	25.7945i	0.0914795	−	0.0914795i	−0.659886	−	0.751366i	\(-0.729395\pi\)
							0.751366	+	0.659886i	\(0.229395\pi\)
\(47\)	490.841	+	131.520i	1.52333	+	0.408175i	0.920836	−	0.389949i	\(-0.127508\pi\)
							0.602493	+	0.798124i	\(0.294174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.6	✓	96
5.3	odd	4	inner	105.4.u.a.73.19	yes	96
7.5	odd	6	inner	105.4.u.a.82.19	yes	96
35.33	even	12	inner	105.4.u.a.103.6	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.6	✓	96	1.1	even	1	trivial
105.4.u.a.73.19	yes	96	5.3	odd	4	inner
105.4.u.a.82.19	yes	96	7.5	odd	6	inner
105.4.u.a.103.6	yes	96	35.33	even	12	inner