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 disc. -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 \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0553962164023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.333.1
Artin image size \(16\)
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
111.d Odd 1 CM by \(\Q(\sqrt{-111}) \) yes
3.b Odd 1 yes
37.b Even 1 yes

Hecke kernels

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