Embedded newform 150.4.g.b.61.3

• Label: 150.4.g.b.61.3
• Level: $150$
• Weight: $4$
• Character: 150.61
• Analytic conductor: $8.850$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.85028650086\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 4*x^15 + 66*x^14 + 116*x^13 - 15174*x^12 + 66830*x^11 - 253085*x^10 - 2763760*x^9 + 83790040*x^8 - 245751650*x^7 + 550471825*x^6 + 10129042700*x^5 - 118334390050*x^4 + 286942998000*x^3 + 664960189500*x^2 - 7593598503000*x + 48733788520125
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 5^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.3
Root			\(1.36008 + 4.54276i\) of defining polynomial
Character	\(\chi\)	\(=\)	150.61
Dual form			150.4.g.b.91.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61803 + 1.17557i) q^{2} +(0.927051 - 2.85317i) q^{3} +(1.23607 - 3.80423i) q^{4} +(0.900748 - 11.1440i) q^{5} +(1.85410 + 5.70634i) q^{6} -31.6588 q^{7} +(2.47214 + 7.60845i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{2} - 12 q^{3} - 16 q^{4} + 15 q^{5} - 24 q^{6} + 46 q^{7} - 32 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61803	+	1.17557i	−0.572061	+	0.415627i
							\(3\)	0.927051	−	2.85317i	0.178411	−	0.549093i
\(5\)	0.900748	−	11.1440i	0.0805654	−	0.996749i
							\(7\)	−31.6588			−1.70942			−0.854708	−	0.519109i	\(-0.826264\pi\)
−0.854708	+	0.519109i	\(0.826264\pi\)
\(11\)	−32.3404	+	23.4967i	−0.886454	+	0.644047i	−0.934951	−	0.354777i	\(-0.884557\pi\)
							0.0484970	+	0.998823i	\(0.484557\pi\)
\(13\)	57.7437	+	41.9533i	1.23194	+	0.895058i	0.997034	−	0.0769602i	\(-0.0245215\pi\)
							0.234907	+	0.972018i	\(0.424521\pi\)
\(17\)	38.0021	+	116.958i	0.542169	+	1.66862i	0.727628	+	0.685972i	\(0.240623\pi\)
							−0.185460	+	0.982652i	\(0.559377\pi\)
\(19\)	−5.21701	−	16.0563i	−0.0629928	−	0.193872i	0.914607	−	0.404344i	\(-0.132500\pi\)
							−0.977600	+	0.210472i	\(0.932500\pi\)
\(23\)	−37.8694	+	27.5138i	−0.343318	+	0.249435i	−0.746061	−	0.665878i	\(-0.768057\pi\)
							0.402742	+	0.915313i	\(0.368057\pi\)
\(29\)	−56.3174	+	173.327i	−0.360617	+	1.10986i	0.592064	+	0.805891i	\(0.298313\pi\)
							−0.952681	+	0.303973i	\(0.901687\pi\)
\(31\)	−89.9717	−	276.905i	−0.521271	−	1.60431i	−0.771574	−	0.636140i	\(-0.780530\pi\)
							0.250303	−	0.968168i	\(-0.419470\pi\)
\(37\)	−200.744	−	145.849i	−0.891950	−	0.648040i	0.0444355	−	0.999012i	\(-0.485851\pi\)
							−0.936386	+	0.350972i	\(0.885851\pi\)
\(41\)	−149.266	−	108.448i	−0.568572	−	0.413092i	0.266014	−	0.963969i	\(-0.414293\pi\)
							−0.834586	+	0.550877i	\(0.814293\pi\)
\(43\)	−113.319			−0.401884			−0.200942	−	0.979603i	\(-0.564400\pi\)
							−0.200942	+	0.979603i	\(0.564400\pi\)
\(47\)	−68.2785	+	210.140i	−0.211903	+	0.652171i	0.787456	+	0.616371i	\(0.211398\pi\)
							−0.999359	+	0.0357997i	\(0.988602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.g.b.61.3	✓	16
25.16	even	5	inner	150.4.g.b.91.3	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.g.b.61.3	✓	16	1.1	even	1	trivial
150.4.g.b.91.3	yes	16	25.16	even	5	inner