Embedded newform 105.4.u.a.52.5

• Label: 105.4.u.a.52.5
• 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			52.5
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14346 + 4.26746i) q^{2} +(2.89778 - 0.776457i) q^{3} +(-9.97548 - 5.75935i) q^{4} +(9.24082 - 6.29343i) q^{5} +13.2540i q^{6} +(15.1987 + 10.5830i) q^{7} +(10.9924 - 10.9924i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14346	+	4.26746i	−0.404275	+	1.50877i	0.401115	+	0.916028i	\(0.368623\pi\)
							−0.805389	+	0.592746i	\(0.798044\pi\)
\(3\)	2.89778	−	0.776457i	0.557678	−	0.149429i
							\(5\)	9.24082	−	6.29343i	0.826524	−	0.562902i
\(7\)	15.1987	+	10.5830i	0.820650	+	0.571431i
							\(11\)	−12.0763	+	20.9168i	−0.331013	+	0.573331i	−0.982711	−	0.185147i	\(-0.940724\pi\)
0.651698	+	0.758479i	\(0.274057\pi\)
\(13\)	60.4586	+	60.4586i	1.28986	+	1.28986i	0.934870	+	0.354990i	\(0.115516\pi\)
							0.354990	+	0.934870i	\(0.384484\pi\)
\(17\)	−6.88022	−	25.6773i	−0.0981587	−	0.366333i	0.899321	−	0.437290i	\(-0.144062\pi\)
							−0.997479	+	0.0709564i	\(0.977395\pi\)
\(19\)	3.56097	+	6.16778i	0.0429970	+	0.0744729i	0.886723	−	0.462301i	\(-0.152976\pi\)
							−0.843726	+	0.536774i	\(0.819643\pi\)
\(23\)	−102.631	−	27.4998i	−0.930433	−	0.249309i	−0.238394	−	0.971169i	\(-0.576621\pi\)
							−0.692039	+	0.721860i	\(0.743287\pi\)
\(29\)			131.127i			0.839642i	0.907607	+	0.419821i	\(0.137907\pi\)
							−0.907607	+	0.419821i	\(0.862093\pi\)
\(31\)	−7.79424	−	4.50001i	−0.0451576	−	0.0260718i	0.477251	−	0.878767i	\(-0.341633\pi\)
							−0.522409	+	0.852695i	\(0.674967\pi\)
\(37\)	6.14633	−	22.9384i	0.0273095	−	0.101920i	−0.950926	−	0.309419i	\(-0.899866\pi\)
							0.978235	+	0.207498i	\(0.0665322\pi\)
\(41\)			218.725i			0.833148i	0.909102	+	0.416574i	\(0.136769\pi\)
							−0.909102	+	0.416574i	\(0.863231\pi\)
\(43\)	266.495	−	266.495i	0.945120	−	0.945120i	−0.0534507	−	0.998570i	\(-0.517022\pi\)
							0.998570	+	0.0534507i	\(0.0170220\pi\)
\(47\)	−362.732	−	97.1939i	−1.12574	−	0.301642i	−0.352538	−	0.935797i	\(-0.614681\pi\)
							−0.773206	+	0.634155i	\(0.781348\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.5	✓	96
5.3	odd	4	inner	105.4.u.a.73.20	yes	96
7.5	odd	6	inner	105.4.u.a.82.20	yes	96
35.33	even	12	inner	105.4.u.a.103.5	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.5	✓	96	1.1	even	1	trivial
105.4.u.a.73.20	yes	96	5.3	odd	4	inner
105.4.u.a.82.20	yes	96	7.5	odd	6	inner
105.4.u.a.103.5	yes	96	35.33	even	12	inner