Embedded newform 5.15.c.a.3.3

• Label: 5.15.c.a.3.3
• Level: $5$
• Weight: $15$
• Character: 5.3
• Analytic conductor: $6.216$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.21644840760\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	x^12 + 19558*x^10 + 126669069*x^8 + 319537876249*x^6 + 246445182947944*x^4 + 802663999329168*x^2 + 494659590011136
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{24}\cdot 3^{8}\cdot 5^{13} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.3
Root			\(63.9679i\) of defining polynomial
Character	\(\chi\)	\(=\)	5.3
Dual form			5.15.c.a.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-51.6522 + 51.6522i) q^{2} +(-332.943 - 332.943i) q^{3} +11048.1i q^{4} +(-48483.9 - 61260.4i) q^{5} +34394.5 q^{6} +(918301. - 918301. i) q^{7} +(-1.41693e6 - 1.41693e6i) q^{8} -4.56127e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 130 q^{2} - 2160 q^{3} + 17240 q^{5} + 306744 q^{6} - 1646400 q^{7} + 5926980 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−51.6522	+	51.6522i	−0.403533	+	0.403533i	−0.879476	−	0.475943i	\(-0.842107\pi\)
							0.475943	+	0.879476i	\(0.342107\pi\)
\(3\)	−332.943	−	332.943i	−0.152237	−	0.152237i	0.626879	−	0.779117i	\(-0.284332\pi\)
							−0.779117	+	0.626879i	\(0.784332\pi\)
\(5\)	−48483.9	−	61260.4i	−0.620593	−	0.784133i
							\(7\)	918301.	−	918301.i	1.11506	−	1.11506i	0.122606	−	0.992455i	\(-0.460875\pi\)
0.992455	−	0.122606i	\(-0.0391253\pi\)
\(11\)	−1.30961e7			−0.672035			−0.336018	−	0.941856i	\(-0.609080\pi\)
							−0.336018	+	0.941856i	\(0.609080\pi\)
\(13\)	1.79002e7	+	1.79002e7i	0.285270	+	0.285270i	0.835206	−	0.549937i	\(-0.185348\pi\)
							−0.549937	+	0.835206i	\(0.685348\pi\)
\(17\)	−2.33228e8	+	2.33228e8i	−0.568378	+	0.568378i	−0.931674	−	0.363296i	\(-0.881652\pi\)
							0.363296	+	0.931674i	\(0.381652\pi\)
\(19\)		−	1.52711e9i		−	1.70842i	−0.519927	−	0.854211i	\(-0.674041\pi\)
							0.519927	−	0.854211i	\(-0.325959\pi\)
\(23\)	−3.77315e8	−	3.77315e8i	−0.110818	−	0.110818i	0.649524	−	0.760341i	\(-0.274968\pi\)
							−0.760341	+	0.649524i	\(0.774968\pi\)
\(29\)		−	8.97688e9i		−	0.520402i	−0.965554	−	0.260201i	\(-0.916211\pi\)
							0.965554	−	0.260201i	\(-0.0837889\pi\)
\(31\)	−1.37675e10			−0.500408			−0.250204	−	0.968193i	\(-0.580498\pi\)
							−0.250204	+	0.968193i	\(0.580498\pi\)
\(37\)	5.04349e10	−	5.04349e10i	0.531274	−	0.531274i	−0.389677	−	0.920952i	\(-0.627413\pi\)
							0.920952	+	0.389677i	\(0.127413\pi\)
\(41\)	−2.66079e11			−1.36623			−0.683115	−	0.730311i	\(-0.739375\pi\)
							−0.683115	+	0.730311i	\(0.739375\pi\)
\(43\)	2.24472e11	+	2.24472e11i	0.825815	+	0.825815i	0.986935	−	0.161120i	\(-0.0515105\pi\)
							−0.161120	+	0.986935i	\(0.551511\pi\)
\(47\)	3.05185e11	−	3.05185e11i	0.602391	−	0.602391i	−0.338555	−	0.940947i	\(-0.609938\pi\)
							0.940947	+	0.338555i	\(0.109938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.15.c.a.3.3	yes	12
3.2	odd	2		45.15.g.a.28.4		12
4.3	odd	2		80.15.p.c.33.3		12
5.2	odd	4	inner	5.15.c.a.2.3	✓	12
5.3	odd	4		25.15.c.b.7.4		12
5.4	even	2		25.15.c.b.18.4		12
15.2	even	4		45.15.g.a.37.4		12
20.7	even	4		80.15.p.c.17.3		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.15.c.a.2.3	✓	12	5.2	odd	4	inner
5.15.c.a.3.3	yes	12	1.1	even	1	trivial
25.15.c.b.7.4		12	5.3	odd	4
25.15.c.b.18.4		12	5.4	even	2
45.15.g.a.28.4		12	3.2	odd	2
45.15.g.a.37.4		12	15.2	even	4
80.15.p.c.17.3		12	20.7	even	4
80.15.p.c.33.3		12	4.3	odd	2