Embedded newform 116.2.e.c.75.2

• Label: 116.2.e.c.75.2
• Level: $116$
• Weight: $2$
• Character: 116.75
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 6*x^16 + 10*x^15 - 6*x^14 + 6*x^13 + 9*x^12 - 24*x^11 + 56*x^10 - 48*x^9 + 36*x^8 + 48*x^7 - 96*x^6 + 320*x^5 - 384*x^4 + 256*x^3 + 512*x^2 - 1024*x + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			75.2
Root			\(0.471198 - 1.33341i\) of defining polynomial
Character	\(\chi\)	\(=\)	116.75
Dual form			116.2.e.c.99.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33341 - 0.471198i) q^{2} +(2.28786 + 2.28786i) q^{3} +(1.55595 + 1.25660i) q^{4} -3.10840i q^{5} +(-1.97261 - 4.12868i) q^{6} +1.98539i q^{7} +(-1.48260 - 2.40871i) q^{8} +7.46858i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33341	−	0.471198i	−0.942861	−	0.333187i
							\(3\)	2.28786	+	2.28786i	1.32089	+	1.32089i	0.913052	+	0.407842i	\(0.133719\pi\)
0.407842	+	0.913052i	\(0.366281\pi\)
\(5\)		−	3.10840i		−	1.39012i	−0.718953	−	0.695059i	\(-0.755378\pi\)
							0.718953	−	0.695059i	\(-0.244622\pi\)
\(7\)			1.98539i			0.750408i	0.926942	+	0.375204i	\(0.122427\pi\)
							−0.926942	+	0.375204i	\(0.877573\pi\)
\(11\)	−0.576033	−	0.576033i	−0.173680	−	0.173680i	0.614914	−	0.788594i	\(-0.289191\pi\)
							−0.788594	+	0.614914i	\(0.789191\pi\)
\(13\)		−	2.72442i		−	0.755617i	−0.925884	−	0.377809i	\(-0.876678\pi\)
							0.925884	−	0.377809i	\(-0.123322\pi\)
\(17\)	−0.566102	+	0.566102i	−0.137300	+	0.137300i	−0.772416	−	0.635117i	\(-0.780952\pi\)
							0.635117	+	0.772416i	\(0.280952\pi\)
\(19\)	−1.84500	−	1.84500i	−0.423272	−	0.423272i	0.463057	−	0.886329i	\(-0.346753\pi\)
							−0.886329	+	0.463057i	\(0.846753\pi\)
\(23\)			2.59032i			0.540119i	0.962844	+	0.270059i	\(0.0870434\pi\)
							−0.962844	+	0.270059i	\(0.912957\pi\)
\(29\)	1.29052	−	5.22825i	0.239643	−	0.970861i
							\(31\)	−6.23308	−	6.23308i	−1.11949	−	1.11949i	−0.991816	−	0.127678i	\(-0.959248\pi\)
−0.127678	−	0.991816i	\(-0.540752\pi\)
\(37\)	2.93773	+	2.93773i	0.482959	+	0.482959i	0.906075	−	0.423116i	\(-0.139064\pi\)
							−0.423116	+	0.906075i	\(0.639064\pi\)
\(41\)	−6.17806	−	6.17806i	−0.964850	−	0.964850i	0.0345524	−	0.999403i	\(-0.488999\pi\)
							−0.999403	+	0.0345524i	\(0.988999\pi\)
\(43\)	−1.27619	−	1.27619i	−0.194616	−	0.194616i	0.603071	−	0.797687i	\(-0.293944\pi\)
							−0.797687	+	0.603071i	\(0.793944\pi\)
\(47\)	−2.35030	+	2.35030i	−0.342826	+	0.342826i	−0.857429	−	0.514603i	\(-0.827939\pi\)
							0.514603	+	0.857429i	\(0.327939\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.e.c.75.2	✓	20
4.3	odd	2	inner	116.2.e.c.75.6	yes	20
29.12	odd	4	inner	116.2.e.c.99.6	yes	20
116.99	even	4	inner	116.2.e.c.99.2	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.e.c.75.2	✓	20	1.1	even	1	trivial
116.2.e.c.75.6	yes	20	4.3	odd	2	inner
116.2.e.c.99.2	yes	20	116.99	even	4	inner
116.2.e.c.99.6	yes	20	29.12	odd	4	inner