Properties

Label 111.1.d.b
Level 111
Weight 1
Character orbit 111.d
Self dual yes
Analytic conductor 0.055
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -111
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0553962164023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.4107.1
Artin image $D_8$
Artin field Galois closure of 8.0.4102893.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + q^{4} + \beta q^{5} + \beta q^{6} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + q^{4} + \beta q^{5} + \beta q^{6} + q^{9} -2 q^{10} - q^{12} -\beta q^{15} - q^{16} -\beta q^{17} -\beta q^{18} + \beta q^{20} + \beta q^{23} + q^{25} - q^{27} -\beta q^{29} + 2 q^{30} + \beta q^{32} + 2 q^{34} + q^{36} - q^{37} + \beta q^{45} -2 q^{46} + q^{48} - q^{49} -\beta q^{50} + \beta q^{51} + \beta q^{54} + 2 q^{58} -\beta q^{59} -\beta q^{60} - q^{64} -\beta q^{68} -\beta q^{69} + \beta q^{74} - q^{75} -\beta q^{80} + q^{81} -2 q^{85} + \beta q^{87} + \beta q^{89} -2 q^{90} + \beta q^{92} -\beta q^{96} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} + 2q^{9} - 4q^{10} - 2q^{12} - 2q^{16} + 2q^{25} - 2q^{27} + 4q^{30} + 4q^{34} + 2q^{36} - 2q^{37} - 4q^{46} + 2q^{48} - 2q^{49} + 4q^{58} - 2q^{64} - 2q^{75} + 2q^{81} - 4q^{85} - 4q^{90} + O(q^{100}) \)

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
1.41421
−1.41421
−1.41421 −1.00000 1.00000 1.41421 1.41421 0 0 1.00000 −2.00000
110.2 1.41421 −1.00000 1.00000 −1.41421 −1.41421 0 0 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 111.1.d.b 2
3.b odd 2 1 inner 111.1.d.b 2
4.b odd 2 1 1776.1.n.c 2
5.b even 2 1 2775.1.h.b 2
5.c odd 4 2 2775.1.b.b 4
9.c even 3 2 2997.1.n.d 4
9.d odd 6 2 2997.1.n.d 4
12.b even 2 1 1776.1.n.c 2
15.d odd 2 1 2775.1.h.b 2
15.e even 4 2 2775.1.b.b 4
37.b even 2 1 inner 111.1.d.b 2
111.d odd 2 1 CM 111.1.d.b 2
148.b odd 2 1 1776.1.n.c 2
185.d even 2 1 2775.1.h.b 2
185.h odd 4 2 2775.1.b.b 4
333.n odd 6 2 2997.1.n.d 4
333.q even 6 2 2997.1.n.d 4
444.g even 2 1 1776.1.n.c 2
555.b odd 2 1 2775.1.h.b 2
555.n even 4 2 2775.1.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.b 2 1.a even 1 1 trivial
111.1.d.b 2 3.b odd 2 1 inner
111.1.d.b 2 37.b even 2 1 inner
111.1.d.b 2 111.d odd 2 1 CM
1776.1.n.c 2 4.b odd 2 1
1776.1.n.c 2 12.b even 2 1
1776.1.n.c 2 148.b odd 2 1
1776.1.n.c 2 444.g even 2 1
2775.1.b.b 4 5.c odd 4 2
2775.1.b.b 4 15.e even 4 2
2775.1.b.b 4 185.h odd 4 2
2775.1.b.b 4 555.n even 4 2
2775.1.h.b 2 5.b even 2 1
2775.1.h.b 2 15.d odd 2 1
2775.1.h.b 2 185.d even 2 1
2775.1.h.b 2 555.b odd 2 1
2997.1.n.d 4 9.c even 3 2
2997.1.n.d 4 9.d odd 6 2
2997.1.n.d 4 333.n odd 6 2
2997.1.n.d 4 333.q even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(111, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$13$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$17$ \( 1 + T^{4} \)
$19$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$23$ \( 1 + T^{4} \)
$29$ \( 1 + T^{4} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( 1 + T^{4} \)
$61$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$67$ \( ( 1 + T^{2} )^{2} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( 1 + T^{4} \)
$97$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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