Embedded newform 115.4.j.a.4.16

• Label: 115.4.j.a.4.16
• Level: $115$
• Weight: $4$
• Character: 115.4
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $340$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(340\)
Relative dimension:	\(34\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	115.4
Dual form			115.4.j.a.29.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.355756 + 0.162468i) q^{2} +(-1.93102 - 6.57646i) q^{3} +(-5.13872 + 5.93040i) q^{4} +(8.88702 + 6.78387i) q^{5} +(1.75544 + 2.02588i) q^{6} +(14.3430 - 2.06222i) q^{7} +(1.74611 - 5.94670i) q^{8} +(-16.8072 + 10.8013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 122 q^{4} - 25 q^{5} - 46 q^{6} + 288 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.355756	+	0.162468i	−0.125779	+	0.0574411i	−0.477310	−	0.878735i	\(-0.658388\pi\)
							0.351532	+	0.936176i	\(0.385661\pi\)
\(3\)	−1.93102	−	6.57646i	−0.371626	−	1.26564i	−0.907038	−	0.421050i	\(-0.861662\pi\)
							0.535412	−	0.844591i	\(-0.320156\pi\)
\(5\)	8.88702	+	6.78387i	0.794879	+	0.606768i
							\(7\)	14.3430	−	2.06222i	0.774451	−	0.111349i	0.256253	−	0.966610i	\(-0.417512\pi\)
0.518198	+	0.855261i	\(0.326603\pi\)
\(11\)	9.53149	−	20.8710i	0.261259	−	0.572078i	−0.732859	−	0.680381i	\(-0.761814\pi\)
							0.994118	+	0.108303i	\(0.0345417\pi\)
\(13\)	1.33684	+	0.192208i	0.0285209	+	0.00410069i	0.156561	−	0.987668i	\(-0.449959\pi\)
							−0.128040	+	0.991769i	\(0.540868\pi\)
\(17\)	61.9383	−	53.6698i	0.883661	−	0.765697i	−0.0894601	−	0.995990i	\(-0.528514\pi\)
							0.973121	+	0.230294i	\(0.0739687\pi\)
\(19\)	37.1196	−	42.8383i	0.448201	−	0.517251i	−0.486019	−	0.873948i	\(-0.661552\pi\)
							0.934220	+	0.356697i	\(0.116097\pi\)
\(23\)	31.6255	−	105.673i	0.286712	−	0.958017i
							\(29\)	121.293	+	139.980i	0.776677	+	0.896333i	0.996865	−	0.0791243i	\(-0.0252124\pi\)
−0.220188	+	0.975458i	\(0.570667\pi\)
\(31\)	101.808	+	29.8936i	0.589849	+	0.173195i	0.563018	−	0.826445i	\(-0.309640\pi\)
							0.0268309	+	0.999640i	\(0.491458\pi\)
\(37\)	52.6047	+	81.8546i	0.233734	+	0.363697i	0.938231	−	0.346010i	\(-0.112464\pi\)
							−0.704497	+	0.709707i	\(0.748827\pi\)
\(41\)	75.2556	+	48.3638i	0.286657	+	0.184223i	0.676065	−	0.736842i	\(-0.263684\pi\)
							−0.389408	+	0.921066i	\(0.627320\pi\)
\(43\)	11.5261	+	39.2544i	0.0408772	+	0.139215i	0.977403	−	0.211383i	\(-0.0677966\pi\)
							−0.936526	+	0.350598i	\(0.885978\pi\)
\(47\)			52.6140i			0.163288i	0.996662	+	0.0816440i	\(0.0260171\pi\)
							−0.996662	+	0.0816440i	\(0.973983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.j.a.4.16	✓	340
5.4	even	2	inner	115.4.j.a.4.19	yes	340
23.6	even	11	inner	115.4.j.a.29.19	yes	340
115.29	even	22	inner	115.4.j.a.29.16	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.j.a.4.16	✓	340	1.1	even	1	trivial
115.4.j.a.4.19	yes	340	5.4	even	2	inner
115.4.j.a.29.16	yes	340	115.29	even	22	inner
115.4.j.a.29.19	yes	340	23.6	even	11	inner