Newform orbit 111.5.d.b

• Label: 111.5.d.b
• Level: $111$
• Weight: $5$
• Character orbit: 111.d
• Self dual: yes
• Analytic conductor: $11.474$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -111
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	111.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.4740659023\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{37}) \)
Defining polynomial:	\( x^{2} - x - 9 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + 9 q^{3} + 21 q^{4} + 2 \beta q^{5} - 9 \beta q^{6} + 50 q^{7} - 5 \beta q^{8} + 81 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 42 q^{4} + 100 q^{7} + 162 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

110.1

3.54138
−2.54138

−6.08276

9.00000

21.0000

12.1655

−54.7449

50.0000

−30.4138

81.0000

−74.0000

110.2

6.08276

9.00000

21.0000

−12.1655

54.7449

50.0000

30.4138

81.0000

−74.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
111.d	odd	2	1	CM by \(\Q(\sqrt{-111}) \)
3.b	odd	2	1	inner
37.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	111.5.d.b	✓	2
3.b	odd	2	1	inner	111.5.d.b	✓	2
37.b	even	2	1	inner	111.5.d.b	✓	2
111.d	odd	2	1	CM	111.5.d.b	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
111.5.d.b	✓	2	1.a	even	1	1	trivial
111.5.d.b	✓	2	3.b	odd	2	1	inner
111.5.d.b	✓	2	37.b	even	2	1	inner
111.5.d.b	✓	2	111.d	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 37 \) acting on \(S_{5}^{\mathrm{new}}(111, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 37 \)
$3$	\( (T - 9)^{2} \)
$5$	\( T^{2} - 148 \)
$7$	\( (T - 50)^{2} \)
$11$	\( T^{2} \)