Embedded newform 50.4.d.a.11.1

• Label: 50.4.d.a.11.1
• Level: $50$
• Weight: $4$
• Character: 50.11
• Analytic conductor: $2.950$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95009550029\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 6*x^11 + 78*x^10 - 335*x^9 + 1991*x^8 - 6020*x^7 + 20827*x^6 - 42752*x^5 + 86341*x^4 - 107815*x^3 + 97905*x^2 - 50215*x + 11005
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 5^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			11.1
Root			\(0.500000 - 3.48876i\) of defining polynomial
Character	\(\chi\)	\(=\)	50.11
Dual form			50.4.d.a.41.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61803 - 1.17557i) q^{2} +(-1.33712 + 4.11522i) q^{3} +(1.23607 - 3.80423i) q^{4} +(10.6469 + 3.41218i) q^{5} +(2.67423 + 8.23044i) q^{6} +17.3088 q^{7} +(-2.47214 - 7.60845i) q^{8} +(6.69632 + 4.86516i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + q^{3} - 12 q^{4} + 20 q^{5} - 2 q^{6} + 58 q^{7} + 24 q^{8} - 26 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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
							\(3\)	−1.33712	+	4.11522i	−0.257328	+	0.791974i	0.736034	+	0.676944i	\(0.236696\pi\)
−0.993362	+	0.115030i	\(0.963304\pi\)
\(5\)	10.6469	+	3.41218i	0.952290	+	0.305195i
							\(7\)	17.3088			0.934589			0.467295	−	0.884102i	\(-0.345229\pi\)
0.467295	+	0.884102i	\(0.345229\pi\)
\(11\)	−21.0492	+	15.2931i	−0.576960	+	0.419186i	−0.837627	−	0.546243i	\(-0.816058\pi\)
							0.260667	+	0.965429i	\(0.416058\pi\)
\(13\)	−54.1436	−	39.3376i	−1.15513	−	0.839254i	−0.165979	−	0.986129i	\(-0.553078\pi\)
							−0.989155	+	0.146875i	\(0.953078\pi\)
\(17\)	−20.1609	−	62.0488i	−0.287631	−	0.885238i	−0.985598	−	0.169107i	\(-0.945912\pi\)
							0.697966	−	0.716131i	\(-0.254088\pi\)
\(19\)	−2.96374	−	9.12145i	−0.0357857	−	0.110137i	0.931568	−	0.363567i	\(-0.118441\pi\)
							−0.967354	+	0.253430i	\(0.918441\pi\)
\(23\)	−89.1539	+	64.7741i	−0.808255	+	0.587232i	−0.913324	−	0.407233i	\(-0.866494\pi\)
							0.105069	+	0.994465i	\(0.466494\pi\)
\(29\)	73.5566	−	226.384i	0.471004	−	1.44960i	−0.380268	−	0.924876i	\(-0.624168\pi\)
							0.851272	−	0.524725i	\(-0.175832\pi\)
\(31\)	0.922132	+	2.83803i	0.00534257	+	0.0164428i	0.953692	−	0.300784i	\(-0.0972483\pi\)
							−0.948350	+	0.317227i	\(0.897248\pi\)
\(37\)	−255.618	−	185.717i	−1.13577	−	0.825182i	−0.149242	−	0.988801i	\(-0.547683\pi\)
							−0.986524	+	0.163619i	\(0.947683\pi\)
\(41\)	355.587	+	258.349i	1.35447	+	0.984083i	0.998775	+	0.0494818i	\(0.0157570\pi\)
							0.355698	+	0.934601i	\(0.384243\pi\)
\(43\)	132.682			0.470554			0.235277	−	0.971928i	\(-0.424400\pi\)
							0.235277	+	0.971928i	\(0.424400\pi\)
\(47\)	−48.8583	+	150.370i	−0.151632	+	0.466676i	−0.997804	−	0.0662342i	\(-0.978902\pi\)
							0.846172	+	0.532910i	\(0.178902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.4.d.a.11.1	✓	12
5.2	odd	4		250.4.e.a.199.6		24
5.3	odd	4		250.4.e.a.199.1		24
5.4	even	2		250.4.d.a.51.3		12
25.4	even	10		1250.4.a.e.1.5		6
25.9	even	10		250.4.d.a.201.3		12
25.12	odd	20		250.4.e.a.49.1		24
25.13	odd	20		250.4.e.a.49.6		24
25.16	even	5	inner	50.4.d.a.41.1	yes	12
25.21	even	5		1250.4.a.d.1.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.d.a.11.1	✓	12	1.1	even	1	trivial
50.4.d.a.41.1	yes	12	25.16	even	5	inner
250.4.d.a.51.3		12	5.4	even	2
250.4.d.a.201.3		12	25.9	even	10
250.4.e.a.49.1		24	25.12	odd	20
250.4.e.a.49.6		24	25.13	odd	20
250.4.e.a.199.1		24	5.3	odd	4
250.4.e.a.199.6		24	5.2	odd	4
1250.4.a.d.1.2		6	25.21	even	5
1250.4.a.e.1.5		6	25.4	even	10