Properties

Label 36.2.b.a
Level 36
Weight 2
Character orbit 36.b
Analytic conductor 0.287
Analytic rank 0
Dimension 2
CM disc. -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 36.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta q^{2} -2 q^{4} -\beta q^{5} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -2 q^{4} -\beta q^{5} -2 \beta q^{8} + 2 q^{10} -4 q^{13} + 4 q^{16} + 5 \beta q^{17} + 2 \beta q^{20} + 3 q^{25} -4 \beta q^{26} -7 \beta q^{29} + 4 \beta q^{32} -10 q^{34} + 2 q^{37} -4 q^{40} -\beta q^{41} + 7 q^{49} + 3 \beta q^{50} + 8 q^{52} + 5 \beta q^{53} + 14 q^{58} -10 q^{61} -8 q^{64} + 4 \beta q^{65} -10 \beta q^{68} -16 q^{73} + 2 \beta q^{74} -4 \beta q^{80} + 2 q^{82} + 10 q^{85} -13 \beta q^{89} + 8 q^{97} + 7 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} + 4q^{10} - 8q^{13} + 8q^{16} + 6q^{25} - 20q^{34} + 4q^{37} - 8q^{40} + 14q^{49} + 16q^{52} + 28q^{58} - 20q^{61} - 16q^{64} - 32q^{73} + 4q^{82} + 20q^{85} + 16q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
1.41421i
1.41421i
1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 2.00000
35.2 1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 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
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
3.b Odd 1 yes
12.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(36, [\chi])\).