Embedded newform 76.3.g.c.7.5

• Label: 76.3.g.c.7.5
• Level: $76$
• Weight: $3$
• Character: 76.7
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20067 - 1.59950i) q^{2} +(-1.67542 + 0.967303i) q^{3} +(-1.11680 + 3.84093i) q^{4} +(-1.80536 - 3.12697i) q^{5} +(3.55882 + 1.51842i) q^{6} +13.2414i q^{7} +(7.48448 - 2.82535i) q^{8} +(-2.62865 + 4.55296i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20067	−	1.59950i	−0.600333	−	0.799750i
							\(3\)	−1.67542	+	0.967303i	−0.558473	+	0.322434i	−0.752532	−	0.658555i	\(-0.771168\pi\)
0.194060	+	0.980990i	\(0.437834\pi\)
\(5\)	−1.80536	−	3.12697i	−0.361072	−	0.625395i	0.627066	−	0.778966i	\(-0.284256\pi\)
							−0.988138	+	0.153572i	\(0.950922\pi\)
\(7\)			13.2414i			1.89163i	0.324708	+	0.945814i	\(0.394734\pi\)
							−0.324708	+	0.945814i	\(0.605266\pi\)
\(11\)			2.62998i			0.239089i	0.992829	+	0.119544i	\(0.0381434\pi\)
							−0.992829	+	0.119544i	\(0.961857\pi\)
\(13\)	−0.424512	+	0.735277i	−0.0326548	+	0.0565598i	−0.881891	−	0.471453i	\(-0.843730\pi\)
							0.849236	+	0.528013i	\(0.177063\pi\)
\(17\)	6.24458	+	10.8159i	0.367328	+	0.636231i	0.989147	−	0.146930i	\(-0.0469392\pi\)
							−0.621819	+	0.783161i	\(0.713606\pi\)
\(19\)	−18.2707	+	5.21350i	−0.961617	+	0.274395i
							\(23\)	−26.9066	−	15.5345i	−1.16985	−	0.675415i	−0.216208	−	0.976347i	\(-0.569369\pi\)
−0.953645	+	0.300932i	\(0.902702\pi\)
\(29\)	−9.98430	+	17.2933i	−0.344286	+	0.596321i	−0.985224	−	0.171272i	\(-0.945212\pi\)
							0.640938	+	0.767593i	\(0.278546\pi\)
\(31\)			28.0966i			0.906341i	0.891424	+	0.453170i	\(0.149707\pi\)
							−0.891424	+	0.453170i	\(0.850293\pi\)
\(37\)	61.9366			1.67396			0.836982	−	0.547231i	\(-0.184318\pi\)
							0.836982	+	0.547231i	\(0.184318\pi\)
\(41\)	10.4785	+	18.1493i	0.255574	+	0.442667i	0.965051	−	0.262061i	\(-0.0844023\pi\)
							−0.709477	+	0.704728i	\(0.751069\pi\)
\(43\)	23.9849	−	13.8477i	0.557788	−	0.322039i	−0.194469	−	0.980909i	\(-0.562299\pi\)
							0.752257	+	0.658870i	\(0.228965\pi\)
\(47\)	44.2902	+	25.5710i	0.942346	+	0.544064i	0.890695	−	0.454602i	\(-0.150218\pi\)
							0.0516508	+	0.998665i	\(0.483552\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.g.c.7.5	✓	28
4.3	odd	2	inner	76.3.g.c.7.6	yes	28
19.11	even	3	inner	76.3.g.c.11.6	yes	28
76.11	odd	6	inner	76.3.g.c.11.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.g.c.7.5	✓	28	1.1	even	1	trivial
76.3.g.c.7.6	yes	28	4.3	odd	2	inner
76.3.g.c.11.5	yes	28	76.11	odd	6	inner
76.3.g.c.11.6	yes	28	19.11	even	3	inner