Embedded newform 114.4.e.c.49.1

• Label: 114.4.e.c.49.1
• Level: $114$
• Weight: $4$
• Character: 114.49
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-10})\)
Defining polynomial:	\( x^{4} - 10x^{2} + 100 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.1
Root			\(2.73861 - 1.58114i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.49
Dual form			114.4.e.c.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-7.47723 + 12.9509i) q^{5} +(-3.00000 - 5.19615i) q^{6} -1.95445 q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 8 q^{5} - 12 q^{6} + 36 q^{7} + 32 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.353553	+	0.612372i
							\(3\)	−1.50000	+	2.59808i	−0.288675	+	0.500000i
\(5\)	−7.47723	+	12.9509i	−0.668783	+	1.15837i								0.309461	+	0.950912i	\(0.399851\pi\)
							−0.978245	+	0.207455i	\(0.933482\pi\)
\(7\)	−1.95445			−0.105530			−0.0527652	−	0.998607i	\(-0.516803\pi\)
							−0.0527652	+	0.998607i	\(0.516803\pi\)
\(11\)	−0.954451			−0.0261616			−0.0130808	−	0.999914i	\(-0.504164\pi\)
							−0.0130808	+	0.999914i	\(0.504164\pi\)
\(13\)	−24.4089	−	42.2775i	−0.520755	−	0.901974i	−0.999709	−	0.0241337i	\(-0.992317\pi\)
							0.478954	−	0.877840i	\(-0.341016\pi\)
\(17\)	−6.00000	+	10.3923i	−0.0856008	+	0.148265i	−0.905647	−	0.424032i	\(-0.860614\pi\)
							0.820046	+	0.572297i	\(0.193948\pi\)
\(19\)	−36.8634	−	74.1626i	−0.445107	−	0.895477i
							\(23\)	−25.4772	−	44.1278i	−0.230973	−	0.400056i	0.727122	−	0.686508i	\(-0.240857\pi\)
−0.958095	+	0.286452i	\(0.907524\pi\)
\(29\)	−26.9545	−	46.6865i	−0.172597	−	0.298947i	0.766730	−	0.641970i	\(-0.221883\pi\)
							−0.939327	+	0.343023i	\(0.888549\pi\)
\(31\)	−11.4990			−0.0666218			−0.0333109	−	0.999445i	\(-0.510605\pi\)
							−0.0333109	+	0.999445i	\(0.510605\pi\)
\(37\)	−176.089			−0.782402			−0.391201	−	0.920305i	\(-0.627940\pi\)
							−0.391201	+	0.920305i	\(0.627940\pi\)
\(41\)	−126.863	+	219.734i	−0.483237	+	0.836991i	−0.999815	−	0.0192490i	\(-0.993872\pi\)
							0.516577	+	0.856240i	\(0.327206\pi\)
\(43\)	−244.248	+	423.051i	−0.866222	+	1.50034i	−0.000392978		1.00000i	\(0.500125\pi\)
							−0.865829	+	0.500340i	\(0.833208\pi\)
\(47\)	175.998	+	304.837i	0.546211	+	0.946066i	0.998530	+	0.0542093i	\(0.0172638\pi\)
							−0.452318	+	0.891857i	\(0.649403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.e.c.49.1	yes	4
3.2	odd	2		342.4.g.e.163.2		4
19.7	even	3	inner	114.4.e.c.7.1	✓	4
19.8	odd	6		2166.4.a.k.1.2		2
19.11	even	3		2166.4.a.q.1.2		2
57.26	odd	6		342.4.g.e.235.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.c.7.1	✓	4	19.7	even	3	inner
114.4.e.c.49.1	yes	4	1.1	even	1	trivial
342.4.g.e.163.2		4	3.2	odd	2
342.4.g.e.235.2		4	57.26	odd	6
2166.4.a.k.1.2		2	19.8	odd	6
2166.4.a.q.1.2		2	19.11	even	3