Embedded newform 13.5.f.a.2.2

• Label: 13.5.f.a.2.2
• Level: $13$
• Weight: $5$
• Character: 13.2
• Analytic conductor: $1.344$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34380952009\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 152*x^14 + 9190*x^12 + 285720*x^10 + 4862025*x^8 + 43573680*x^6 + 169417008*x^4 + 100636992*x^2 + 3779136
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{2}\cdot 13^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.2
Root			\(5.33868i\) of defining polynomial
Character	\(\chi\)	\(=\)	13.2
Dual form			13.5.f.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.41998 - 0.916381i) q^{2} +(-5.16500 - 8.94604i) q^{3} +(-2.99988 - 1.73198i) q^{4} +(8.65321 - 8.65321i) q^{5} +(9.46622 + 35.3284i) q^{6} +(-2.94871 + 0.790103i) q^{7} +(48.7300 + 48.7300i) q^{8} +(-12.8545 + 22.2646i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.41998	−	0.916381i	−0.854996	−	0.229095i	−0.195407	−	0.980722i	\(-0.562603\pi\)
							−0.659589	+	0.751627i	\(0.729269\pi\)
\(3\)	−5.16500	−	8.94604i	−0.573889	−	0.994005i	−0.996161	−	0.0875358i	\(-0.972101\pi\)
							0.422272	−	0.906469i	\(-0.361233\pi\)
\(5\)	8.65321	−	8.65321i	0.346129	−	0.346129i	−0.512537	−	0.858665i	\(-0.671294\pi\)
							0.858665	+	0.512537i	\(0.171294\pi\)
\(7\)	−2.94871	+	0.790103i	−0.0601777	+	0.0161246i	−0.288782	−	0.957395i	\(-0.593250\pi\)
							0.228605	+	0.973519i	\(0.426584\pi\)
\(11\)	53.6740	−	200.314i	0.443587	−	1.65549i	−0.276053	−	0.961142i	\(-0.589027\pi\)
							0.719640	−	0.694347i	\(-0.244307\pi\)
\(13\)	−168.828	−	7.63056i	−0.998980	−	0.0451512i
							\(17\)	419.914	+	242.437i	1.45299	+	0.838884i	0.998650	−	0.0519444i	\(-0.0165419\pi\)
0.454340	+	0.890828i	\(0.349875\pi\)
\(19\)	−4.81436	−	17.9674i	−0.0133362	−	0.0497713i	0.958937	−	0.283619i	\(-0.0915350\pi\)
							−0.972273	+	0.233847i	\(0.924868\pi\)
\(23\)	100.029	−	57.7520i	0.189092	−	0.109172i	−0.402466	−	0.915435i	\(-0.631847\pi\)
							0.591557	+	0.806263i	\(0.298513\pi\)
\(29\)	−59.3250	−	102.754i	−0.0705410	−	0.122181i	0.828598	−	0.559845i	\(-0.189139\pi\)
							−0.899139	+	0.437664i	\(0.855806\pi\)
\(31\)	1056.86	−	1056.86i	1.09975	−	1.09975i	0.105310	−	0.994439i	\(-0.466416\pi\)
							0.994439	−	0.105310i	\(-0.0335836\pi\)
\(37\)	6.33806	−	23.6540i	0.00462970	−	0.0172783i	−0.963572	−	0.267448i	\(-0.913820\pi\)
							0.968202	+	0.250170i	\(0.0804864\pi\)
\(41\)	1705.47	+	456.980i	1.01456	+	0.271850i	0.727533	−	0.686073i	\(-0.240667\pi\)
							0.287025	+	0.957923i	\(0.407334\pi\)
\(43\)	1797.41	+	1037.74i	0.972099	+	0.561242i	0.899876	−	0.436147i	\(-0.143657\pi\)
							0.0722236	+	0.997388i	\(0.476990\pi\)
\(47\)	−507.246	−	507.246i	−0.229627	−	0.229627i	0.582910	−	0.812537i	\(-0.301914\pi\)
							−0.812537	+	0.582910i	\(0.801914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.5.f.a.2.2	✓	16
3.2	odd	2		117.5.bd.c.28.3		16
13.2	odd	12		169.5.d.c.99.3		16
13.3	even	3		169.5.d.d.70.6		16
13.7	odd	12	inner	13.5.f.a.7.2	yes	16
13.10	even	6		169.5.d.c.70.3		16
13.11	odd	12		169.5.d.d.99.6		16
39.20	even	12		117.5.bd.c.46.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.5.f.a.2.2	✓	16	1.1	even	1	trivial
13.5.f.a.7.2	yes	16	13.7	odd	12	inner
117.5.bd.c.28.3		16	3.2	odd	2
117.5.bd.c.46.3		16	39.20	even	12
169.5.d.c.70.3		16	13.10	even	6
169.5.d.c.99.3		16	13.2	odd	12
169.5.d.d.70.6		16	13.3	even	3
169.5.d.d.99.6		16	13.11	odd	12