Embedded newform 76.4.f.a.27.15

• Label: 76.4.f.a.27.15
• Level: $76$
• Weight: $4$
• Character: 76.27
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.15
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00695190 - 2.82842i) q^{2} +(-0.676811 + 1.17227i) q^{3} +(-7.99990 + 0.0393258i) q^{4} +(-5.49309 + 9.51431i) q^{5} +(3.32038 + 1.90615i) q^{6} +11.1169i q^{7} +(0.166844 + 22.6268i) q^{8} +(12.5839 + 21.7959i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00695190	−	2.82842i	−0.00245787	−	0.999997i
							\(3\)	−0.676811	+	1.17227i	−0.130252	+	0.225604i	−0.923774	−	0.382939i	\(-0.874912\pi\)
0.793521	+	0.608542i	\(0.208245\pi\)
\(5\)	−5.49309	+	9.51431i	−0.491317	+	0.850986i	−0.999950	−	0.00999751i	\(-0.996818\pi\)
							0.508633	+	0.860983i	\(0.330151\pi\)
\(7\)			11.1169i			0.600258i	0.953899	+	0.300129i	\(0.0970298\pi\)
							−0.953899	+	0.300129i	\(0.902970\pi\)
\(11\)		−	30.2803i		−	0.829988i	−0.909824	−	0.414994i	\(-0.863784\pi\)
							0.909824	−	0.414994i	\(-0.136216\pi\)
\(13\)	−50.5440	+	29.1816i	−1.07834	+	0.622578i	−0.930447	−	0.366426i	\(-0.880581\pi\)
							−0.147890	+	0.989004i	\(0.547248\pi\)
\(17\)	−42.7132	+	73.9814i	−0.609380	+	1.05548i	0.381962	+	0.924178i	\(0.375248\pi\)
							−0.991343	+	0.131300i	\(0.958085\pi\)
\(19\)	71.4136	−	41.9416i	0.862285	−	0.506424i
							\(23\)	−66.3834	+	38.3265i	−0.601821	+	0.347462i	−0.769758	−	0.638336i	\(-0.779623\pi\)
0.167937	+	0.985798i	\(0.446290\pi\)
\(29\)	48.1403	−	27.7938i	0.308256	−	0.177972i	−0.337890	−	0.941186i	\(-0.609713\pi\)
							0.646146	+	0.763214i	\(0.276380\pi\)
\(31\)	−256.345			−1.48519			−0.742594	−	0.669742i	\(-0.766405\pi\)
							−0.742594	+	0.669742i	\(0.766405\pi\)
\(37\)		−	10.8550i		−	0.0482309i	−0.999709	−	0.0241154i	\(-0.992323\pi\)
							0.999709	−	0.0241154i	\(-0.00767693\pi\)
\(41\)	−243.046	−	140.323i	−0.925791	−	0.534506i	−0.0403132	−	0.999187i	\(-0.512836\pi\)
							−0.885478	+	0.464681i	\(0.846169\pi\)
\(43\)	346.008	+	199.768i	1.22711	+	0.708472i	0.966424	−	0.256952i	\(-0.0827182\pi\)
							0.260685	+	0.965424i	\(0.416052\pi\)
\(47\)	96.5469	−	55.7414i	0.299634	−	0.172994i	−0.342644	−	0.939465i	\(-0.611323\pi\)
							0.642279	+	0.766471i	\(0.277989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.15	yes	56
4.3	odd	2	inner	76.4.f.a.27.6	✓	56
19.12	odd	6	inner	76.4.f.a.31.6	yes	56
76.31	even	6	inner	76.4.f.a.31.15	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.6	✓	56	4.3	odd	2	inner
76.4.f.a.27.15	yes	56	1.1	even	1	trivial
76.4.f.a.31.6	yes	56	19.12	odd	6	inner
76.4.f.a.31.15	yes	56	76.31	even	6	inner