Properties

Label 95.1.d.b
Level 95
Weight 1
Character orbit 95.d
Self dual Yes
Analytic conductor 0.047
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -95
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 95 = 5 \cdot 19 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 95.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0474111762001\)
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 4.2.475.1
Artin image size \(16\)
Artin image $D_8$
Artin field Galois closure of 8.2.4286875.1

$q$-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.

\(f(q)\) \(=\) \( q\) \( -\beta q^{2} \) \( + \beta q^{3} \) \(+ q^{4}\) \(- q^{5}\) \( -2 q^{6} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta q^{2} \) \( + \beta q^{3} \) \(+ q^{4}\) \(- q^{5}\) \( -2 q^{6} \) \(+ q^{9}\) \( + \beta q^{10} \) \( + \beta q^{12} \) \( -\beta q^{13} \) \( -\beta q^{15} \) \(- q^{16}\) \( -\beta q^{18} \) \(- q^{19}\) \(- q^{20}\) \(+ q^{25}\) \( + 2 q^{26} \) \( + 2 q^{30} \) \( + \beta q^{32} \) \(+ q^{36}\) \( + \beta q^{37} \) \( + \beta q^{38} \) \( -2 q^{39} \) \(- q^{45}\) \( -\beta q^{48} \) \(+ q^{49}\) \( -\beta q^{50} \) \( -\beta q^{52} \) \( + \beta q^{53} \) \( -\beta q^{57} \) \( -\beta q^{60} \) \(- q^{64}\) \( + \beta q^{65} \) \( -\beta q^{67} \) \( -2 q^{74} \) \( + \beta q^{75} \) \(- q^{76}\) \( + 2 \beta q^{78} \) \(+ q^{80}\) \(- q^{81}\) \( + \beta q^{90} \) \(+ q^{95}\) \( + 2 q^{96} \) \( + \beta q^{97} \) \( -\beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 4q^{74} \) \(\mathstrut -\mathstrut 2q^{76} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut 4q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1
1.41421
−1.41421
−1.41421 1.41421 1.00000 −1.00000 −2.00000 0 0 1.00000 1.41421
94.2 1.41421 −1.41421 1.00000 −1.00000 −2.00000 0 0 1.00000 −1.41421
\(n\): 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
95.d Odd 1 CM by \(\Q(\sqrt{-95}) \) yes
5.b Even 1 yes
19.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \) acting on \(S_{1}^{\mathrm{new}}(95, [\chi])\).