Properties

Label 120.1.i.a
Level 120
Weight 1
Character orbit 120.i
Analytic conductor 0.060
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM disc. -15, -24, 40
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 120.i (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.059887801516\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-6}, \sqrt{10})\)
Artin image size \(16\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.3240000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( -i q^{2} \) \( -i q^{3} \) \(- q^{4}\) \( + i q^{5} \) \(- q^{6}\) \( + i q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( -i q^{2} \) \( -i q^{3} \) \(- q^{4}\) \( + i q^{5} \) \(- q^{6}\) \( + i q^{8} \) \(- q^{9}\) \(+ q^{10}\) \( + i q^{12} \) \(+ q^{15}\) \(+ q^{16}\) \( + i q^{18} \) \( -i q^{20} \) \(+ q^{24}\) \(- q^{25}\) \( + i q^{27} \) \( -i q^{30} \) \( -2 q^{31} \) \( -i q^{32} \) \(+ q^{36}\) \(- q^{40}\) \( -i q^{45} \) \( -i q^{48} \) \(+ q^{49}\) \( + i q^{50} \) \( -2 i q^{53} \) \(+ q^{54}\) \(- q^{60}\) \( + 2 i q^{62} \) \(- q^{64}\) \( -i q^{72} \) \( + i q^{75} \) \( + 2 q^{79} \) \( + i q^{80} \) \(+ q^{81}\) \( + 2 i q^{83} \) \(- q^{90}\) \( + 2 i q^{93} \) \(- q^{96}\) \( -i q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{24} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 2q^{40} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 2q^{60} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000
29.2 1.00000i 1.00000i −1.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000
\(n\): 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
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
40.f Even 1 RM by \(\Q(\sqrt{10}) \) yes
3.b Odd 1 yes
5.b Even 1 yes
8.b Even 1 yes
120.i Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(120, [\chi])\).