Properties

Label 18.3.b.a
Level 18
Weight 3
Character orbit 18.b
Analytic conductor 0.490
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 18.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.490464475849\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -2 q^{4} \) \( -3 \beta q^{5} \) \( -4 q^{7} \) \( -2 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -2 q^{4} \) \( -3 \beta q^{5} \) \( -4 q^{7} \) \( -2 \beta q^{8} \) \( + 6 q^{10} \) \( + 12 \beta q^{11} \) \( + 8 q^{13} \) \( -4 \beta q^{14} \) \( + 4 q^{16} \) \( -9 \beta q^{17} \) \( -16 q^{19} \) \( + 6 \beta q^{20} \) \( -24 q^{22} \) \( -12 \beta q^{23} \) \( + 7 q^{25} \) \( + 8 \beta q^{26} \) \( + 8 q^{28} \) \( + 3 \beta q^{29} \) \( + 44 q^{31} \) \( + 4 \beta q^{32} \) \( + 18 q^{34} \) \( + 12 \beta q^{35} \) \( -34 q^{37} \) \( -16 \beta q^{38} \) \( -12 q^{40} \) \( + 33 \beta q^{41} \) \( -40 q^{43} \) \( -24 \beta q^{44} \) \( + 24 q^{46} \) \( -60 \beta q^{47} \) \( -33 q^{49} \) \( + 7 \beta q^{50} \) \( -16 q^{52} \) \( + 27 \beta q^{53} \) \( + 72 q^{55} \) \( + 8 \beta q^{56} \) \( -6 q^{58} \) \( + 24 \beta q^{59} \) \( + 50 q^{61} \) \( + 44 \beta q^{62} \) \( -8 q^{64} \) \( -24 \beta q^{65} \) \( + 8 q^{67} \) \( + 18 \beta q^{68} \) \( -24 q^{70} \) \( -36 \beta q^{71} \) \( -16 q^{73} \) \( -34 \beta q^{74} \) \( + 32 q^{76} \) \( -48 \beta q^{77} \) \( -76 q^{79} \) \( -12 \beta q^{80} \) \( -66 q^{82} \) \( + 84 \beta q^{83} \) \( -54 q^{85} \) \( -40 \beta q^{86} \) \( + 48 q^{88} \) \( + 9 \beta q^{89} \) \( -32 q^{91} \) \( + 24 \beta q^{92} \) \( + 120 q^{94} \) \( + 48 \beta q^{95} \) \( + 176 q^{97} \) \( -33 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 12q^{10} \) \(\mathstrut +\mathstrut 16q^{13} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 32q^{19} \) \(\mathstrut -\mathstrut 48q^{22} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut +\mathstrut 16q^{28} \) \(\mathstrut +\mathstrut 88q^{31} \) \(\mathstrut +\mathstrut 36q^{34} \) \(\mathstrut -\mathstrut 68q^{37} \) \(\mathstrut -\mathstrut 24q^{40} \) \(\mathstrut -\mathstrut 80q^{43} \) \(\mathstrut +\mathstrut 48q^{46} \) \(\mathstrut -\mathstrut 66q^{49} \) \(\mathstrut -\mathstrut 32q^{52} \) \(\mathstrut +\mathstrut 144q^{55} \) \(\mathstrut -\mathstrut 12q^{58} \) \(\mathstrut +\mathstrut 100q^{61} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 48q^{70} \) \(\mathstrut -\mathstrut 32q^{73} \) \(\mathstrut +\mathstrut 64q^{76} \) \(\mathstrut -\mathstrut 152q^{79} \) \(\mathstrut -\mathstrut 132q^{82} \) \(\mathstrut -\mathstrut 108q^{85} \) \(\mathstrut +\mathstrut 96q^{88} \) \(\mathstrut -\mathstrut 64q^{91} \) \(\mathstrut +\mathstrut 240q^{94} \) \(\mathstrut +\mathstrut 352q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
1.41421i 0 −2.00000 4.24264i 0 −4.00000 2.82843i 0 6.00000
17.2 1.41421i 0 −2.00000 4.24264i 0 −4.00000 2.82843i 0 6.00000
\(n\): 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
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(18, [\chi])\).