Embedded newform 51.4.i.a.5.7

• Label: 51.4.i.a.5.7
• Level: $51$
• Weight: $4$
• Character: 51.5
• Analytic conductor: $3.009$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	51.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.00909741029\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	51.5
Dual form			51.4.i.a.41.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39521 + 0.577915i) q^{2} +(5.18290 - 0.370861i) q^{3} +(-4.04423 + 4.04423i) q^{4} +(6.67638 + 1.32802i) q^{5} +(-7.01690 + 3.51270i) q^{6} +(3.58738 + 18.0350i) q^{7} +(7.92864 - 19.1414i) q^{8} +(26.7249 - 3.84427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39521	+	0.577915i	−0.493281	+	0.204324i	−0.615435	−	0.788187i	\(-0.711020\pi\)
							0.122154	+	0.992511i	\(0.461020\pi\)
\(3\)	5.18290	−	0.370861i	0.997450	−	0.0713722i
							\(5\)	6.67638	+	1.32802i	0.597154	+	0.118781i	0.484407	−	0.874843i	\(-0.339035\pi\)
0.112746	+	0.993624i	\(0.464035\pi\)
\(7\)	3.58738	+	18.0350i	0.193700	+	0.973798i	0.948243	+	0.317546i	\(0.102859\pi\)
							−0.754542	+	0.656251i	\(0.772141\pi\)
\(11\)	−15.2764	+	22.8628i	−0.418729	+	0.626672i	−0.979535	−	0.201275i	\(-0.935491\pi\)
							0.560806	+	0.827947i	\(0.310491\pi\)
\(13\)	52.7421	+	52.7421i	1.12523	+	1.12523i	0.990942	+	0.134292i	\(0.0428760\pi\)
							0.134292	+	0.990942i	\(0.457124\pi\)
\(17\)	−17.0163	−	67.9959i	−0.242768	−	0.970084i
							\(19\)	−39.3719	−	95.0522i	−0.475396	−	1.14771i	−0.961746	−	0.273944i	\(-0.911672\pi\)
0.486349	−	0.873765i	\(-0.338328\pi\)
\(23\)	−80.8374	−	54.0138i	−0.732860	−	0.489681i	0.132280	−	0.991212i	\(-0.457770\pi\)
							−0.865139	+	0.501531i	\(0.832770\pi\)
\(29\)	37.4203	−	188.125i	0.239613	−	1.20462i	−0.654251	−	0.756277i	\(-0.727016\pi\)
							0.893864	−	0.448338i	\(-0.147984\pi\)
\(31\)	60.4937	−	40.4206i	0.350483	−	0.234186i	−0.367849	−	0.929886i	\(-0.619906\pi\)
							0.718332	+	0.695700i	\(0.244906\pi\)
\(37\)	85.6992	+	128.258i	0.380780	+	0.569878i	0.971512	−	0.236990i	\(-0.0761609\pi\)
							−0.590732	+	0.806868i	\(0.701161\pi\)
\(41\)	−188.690	+	37.5328i	−0.718742	+	0.142967i	−0.540893	−	0.841091i	\(-0.681914\pi\)
							−0.177848	+	0.984058i	\(0.556914\pi\)
\(43\)	−91.5356	+	220.987i	−0.324629	+	0.783724i	0.674344	+	0.738417i	\(0.264427\pi\)
							−0.998973	+	0.0453068i	\(0.985573\pi\)
\(47\)	227.341	−	227.341i	0.705556	−	0.705556i	−0.260041	−	0.965598i	\(-0.583736\pi\)
							0.965598	+	0.260041i	\(0.0837361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.5.7	✓	128
3.2	odd	2	inner	51.4.i.a.5.10	yes	128
17.7	odd	16	inner	51.4.i.a.41.10	yes	128
51.41	even	16	inner	51.4.i.a.41.7	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.5.7	✓	128	1.1	even	1	trivial
51.4.i.a.5.10	yes	128	3.2	odd	2	inner
51.4.i.a.41.7	yes	128	51.41	even	16	inner
51.4.i.a.41.10	yes	128	17.7	odd	16	inner