Embedded newform 105.3.v.a.37.9

• Label: 105.3.v.a.37.9
• Level: $105$
• Weight: $3$
• Character: 105.37
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.9
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.232416 - 0.0622758i) q^{2} +(1.67303 + 0.448288i) q^{3} +(-3.41396 + 1.97105i) q^{4} +(-2.65242 + 4.23847i) q^{5} +0.416757 q^{6} +(-4.06422 + 5.69931i) q^{7} +(-1.35127 + 1.35127i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.232416	−	0.0622758i	0.116208	−	0.0311379i	−0.200246	−	0.979746i	\(-0.564174\pi\)
							0.316454	+	0.948608i	\(0.397508\pi\)
\(3\)	1.67303	+	0.448288i	0.557678	+	0.149429i
							\(5\)	−2.65242	+	4.23847i	−0.530485	+	0.847694i
\(7\)	−4.06422	+	5.69931i	−0.580603	+	0.814187i
							\(11\)	2.22825	+	3.85944i	0.202568	+	0.350858i	0.949355	−	0.314205i	\(-0.101738\pi\)
−0.746787	+	0.665063i	\(0.768405\pi\)
\(13\)	10.0930	−	10.0930i	0.776385	−	0.776385i	−0.202829	−	0.979214i	\(-0.565014\pi\)
							0.979214	+	0.202829i	\(0.0650137\pi\)
\(17\)	−2.55748	+	9.54465i	−0.150440	+	0.561450i	0.849013	+	0.528372i	\(0.177198\pi\)
							−0.999453	+	0.0330776i	\(0.989469\pi\)
\(19\)	11.1120	+	6.41554i	0.584845	+	0.337660i	0.763056	−	0.646332i	\(-0.223698\pi\)
							−0.178212	+	0.983992i	\(0.557031\pi\)
\(23\)	3.20215	+	11.9506i	0.139224	+	0.519590i	0.999945	+	0.0105132i	\(0.00334653\pi\)
							−0.860721	+	0.509077i	\(0.829987\pi\)
\(29\)			36.7743i			1.26808i	0.773301	+	0.634040i	\(0.218604\pi\)
							−0.773301	+	0.634040i	\(0.781396\pi\)
\(31\)	−8.59913	−	14.8941i	−0.277391	−	0.480456i	0.693344	−	0.720606i	\(-0.256136\pi\)
							−0.970736	+	0.240151i	\(0.922803\pi\)
\(37\)	54.2699	−	14.5416i	1.46675	−	0.393016i	0.564937	−	0.825134i	\(-0.308900\pi\)
							0.901817	+	0.432118i	\(0.142234\pi\)
\(41\)	−46.8347			−1.14231			−0.571155	−	0.820843i	\(-0.693504\pi\)
							−0.571155	+	0.820843i	\(0.693504\pi\)
\(43\)	25.0932	−	25.0932i	0.583563	−	0.583563i	−0.352317	−	0.935881i	\(-0.614606\pi\)
							0.935881	+	0.352317i	\(0.114606\pi\)
\(47\)	44.9155	−	12.0351i	0.955649	−	0.256065i	0.252891	−	0.967495i	\(-0.418618\pi\)
							0.702758	+	0.711429i	\(0.251952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.9	✓	64
3.2	odd	2		315.3.ca.b.37.8		64
5.3	odd	4	inner	105.3.v.a.58.8	yes	64
7.4	even	3	inner	105.3.v.a.67.8	yes	64
15.8	even	4		315.3.ca.b.163.9		64
21.11	odd	6		315.3.ca.b.172.9		64
35.18	odd	12	inner	105.3.v.a.88.9	yes	64
105.53	even	12		315.3.ca.b.298.8		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.9	✓	64	1.1	even	1	trivial
105.3.v.a.58.8	yes	64	5.3	odd	4	inner
105.3.v.a.67.8	yes	64	7.4	even	3	inner
105.3.v.a.88.9	yes	64	35.18	odd	12	inner
315.3.ca.b.37.8		64	3.2	odd	2
315.3.ca.b.163.9		64	15.8	even	4
315.3.ca.b.172.9		64	21.11	odd	6
315.3.ca.b.298.8		64	105.53	even	12