Properties

Label 9.5.b.a
Level 9
Weight 5
Character orbit 9.b
Analytic conductor 0.930
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 9.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.930329667755\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -2 q^{4} \) \( -7 \beta q^{5} \) \( -28 q^{7} \) \( + 14 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -2 q^{4} \) \( -7 \beta q^{5} \) \( -28 q^{7} \) \( + 14 \beta q^{8} \) \( + 126 q^{10} \) \( + 4 \beta q^{11} \) \( -112 q^{13} \) \( -28 \beta q^{14} \) \( -284 q^{16} \) \( -21 \beta q^{17} \) \( + 560 q^{19} \) \( + 14 \beta q^{20} \) \( -72 q^{22} \) \( + 188 \beta q^{23} \) \( -257 q^{25} \) \( -112 \beta q^{26} \) \( + 56 q^{28} \) \( -233 \beta q^{29} \) \( -364 q^{31} \) \( -60 \beta q^{32} \) \( + 378 q^{34} \) \( + 196 \beta q^{35} \) \( -826 q^{37} \) \( + 560 \beta q^{38} \) \( + 1764 q^{40} \) \( -427 \beta q^{41} \) \( + 1736 q^{43} \) \( -8 \beta q^{44} \) \( -3384 q^{46} \) \( -308 \beta q^{47} \) \( -1617 q^{49} \) \( -257 \beta q^{50} \) \( + 224 q^{52} \) \( + 423 \beta q^{53} \) \( + 504 q^{55} \) \( -392 \beta q^{56} \) \( + 4194 q^{58} \) \( + 1064 \beta q^{59} \) \( + 2618 q^{61} \) \( -364 \beta q^{62} \) \( -3464 q^{64} \) \( + 784 \beta q^{65} \) \( -3784 q^{67} \) \( + 42 \beta q^{68} \) \( -3528 q^{70} \) \( -2028 \beta q^{71} \) \( + 6608 q^{73} \) \( -826 \beta q^{74} \) \( -1120 q^{76} \) \( -112 \beta q^{77} \) \( -4276 q^{79} \) \( + 1988 \beta q^{80} \) \( + 7686 q^{82} \) \( + 28 \beta q^{83} \) \( -2646 q^{85} \) \( + 1736 \beta q^{86} \) \( -1008 q^{88} \) \( + 1029 \beta q^{89} \) \( + 3136 q^{91} \) \( -376 \beta q^{92} \) \( + 5544 q^{94} \) \( -3920 \beta q^{95} \) \( -5824 q^{97} \) \( -1617 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 56q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 56q^{7} \) \(\mathstrut +\mathstrut 252q^{10} \) \(\mathstrut -\mathstrut 224q^{13} \) \(\mathstrut -\mathstrut 568q^{16} \) \(\mathstrut +\mathstrut 1120q^{19} \) \(\mathstrut -\mathstrut 144q^{22} \) \(\mathstrut -\mathstrut 514q^{25} \) \(\mathstrut +\mathstrut 112q^{28} \) \(\mathstrut -\mathstrut 728q^{31} \) \(\mathstrut +\mathstrut 756q^{34} \) \(\mathstrut -\mathstrut 1652q^{37} \) \(\mathstrut +\mathstrut 3528q^{40} \) \(\mathstrut +\mathstrut 3472q^{43} \) \(\mathstrut -\mathstrut 6768q^{46} \) \(\mathstrut -\mathstrut 3234q^{49} \) \(\mathstrut +\mathstrut 448q^{52} \) \(\mathstrut +\mathstrut 1008q^{55} \) \(\mathstrut +\mathstrut 8388q^{58} \) \(\mathstrut +\mathstrut 5236q^{61} \) \(\mathstrut -\mathstrut 6928q^{64} \) \(\mathstrut -\mathstrut 7568q^{67} \) \(\mathstrut -\mathstrut 7056q^{70} \) \(\mathstrut +\mathstrut 13216q^{73} \) \(\mathstrut -\mathstrut 2240q^{76} \) \(\mathstrut -\mathstrut 8552q^{79} \) \(\mathstrut +\mathstrut 15372q^{82} \) \(\mathstrut -\mathstrut 5292q^{85} \) \(\mathstrut -\mathstrut 2016q^{88} \) \(\mathstrut +\mathstrut 6272q^{91} \) \(\mathstrut +\mathstrut 11088q^{94} \) \(\mathstrut -\mathstrut 11648q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.41421i
1.41421i
4.24264i 0 −2.00000 29.6985i 0 −28.0000 59.3970i 0 126.000
8.2 4.24264i 0 −2.00000 29.6985i 0 −28.0000 59.3970i 0 126.000
\(n\): 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 \(S_{5}^{\mathrm{new}}(9, [\chi])\).