Properties

Label 9.9.b.a
Level 9
Weight 9
Character orbit 9.b
Analytic conductor 3.666
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(3.66640749055\)
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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + 238 q^{4} \) \( + 233 \beta q^{5} \) \( + 1652 q^{7} \) \( + 494 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + 238 q^{4} \) \( + 233 \beta q^{5} \) \( + 1652 q^{7} \) \( + 494 \beta q^{8} \) \( -4194 q^{10} \) \( -5036 \beta q^{11} \) \( -26272 q^{13} \) \( + 1652 \beta q^{14} \) \( + 52036 q^{16} \) \( + 3579 \beta q^{17} \) \( + 46640 q^{19} \) \( + 55454 \beta q^{20} \) \( + 90648 q^{22} \) \( -77332 \beta q^{23} \) \( -586577 q^{25} \) \( -26272 \beta q^{26} \) \( + 393176 q^{28} \) \( -144953 \beta q^{29} \) \( -196444 q^{31} \) \( + 178500 \beta q^{32} \) \( -64422 q^{34} \) \( + 384916 \beta q^{35} \) \( + 2819414 q^{37} \) \( + 46640 \beta q^{38} \) \( -2071836 q^{40} \) \( -166507 \beta q^{41} \) \( -2213464 q^{43} \) \( -1198568 \beta q^{44} \) \( + 1391976 q^{46} \) \( + 384892 \beta q^{47} \) \( -3035697 q^{49} \) \( -586577 \beta q^{50} \) \( -6252736 q^{52} \) \( + 1222983 \beta q^{53} \) \( + 21120984 q^{55} \) \( + 816088 \beta q^{56} \) \( + 2609154 q^{58} \) \( + 2768264 \beta q^{59} \) \( -17405302 q^{61} \) \( -196444 \beta q^{62} \) \( + 10108216 q^{64} \) \( -6121376 \beta q^{65} \) \( -14322664 q^{67} \) \( + 851802 \beta q^{68} \) \( -6928488 q^{70} \) \( -3687708 \beta q^{71} \) \( -8906992 q^{73} \) \( + 2819414 \beta q^{74} \) \( + 11100320 q^{76} \) \( -8319472 \beta q^{77} \) \( + 32758844 q^{79} \) \( + 12124388 \beta q^{80} \) \( + 2997126 q^{82} \) \( + 20055628 \beta q^{83} \) \( -15010326 q^{85} \) \( -2213464 \beta q^{86} \) \( + 44780112 q^{88} \) \( -13891371 \beta q^{89} \) \( -43401344 q^{91} \) \( -18405016 \beta q^{92} \) \( -6928056 q^{94} \) \( + 10867120 \beta q^{95} \) \( -24451744 q^{97} \) \( -3035697 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 476q^{4} \) \(\mathstrut +\mathstrut 3304q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 476q^{4} \) \(\mathstrut +\mathstrut 3304q^{7} \) \(\mathstrut -\mathstrut 8388q^{10} \) \(\mathstrut -\mathstrut 52544q^{13} \) \(\mathstrut +\mathstrut 104072q^{16} \) \(\mathstrut +\mathstrut 93280q^{19} \) \(\mathstrut +\mathstrut 181296q^{22} \) \(\mathstrut -\mathstrut 1173154q^{25} \) \(\mathstrut +\mathstrut 786352q^{28} \) \(\mathstrut -\mathstrut 392888q^{31} \) \(\mathstrut -\mathstrut 128844q^{34} \) \(\mathstrut +\mathstrut 5638828q^{37} \) \(\mathstrut -\mathstrut 4143672q^{40} \) \(\mathstrut -\mathstrut 4426928q^{43} \) \(\mathstrut +\mathstrut 2783952q^{46} \) \(\mathstrut -\mathstrut 6071394q^{49} \) \(\mathstrut -\mathstrut 12505472q^{52} \) \(\mathstrut +\mathstrut 42241968q^{55} \) \(\mathstrut +\mathstrut 5218308q^{58} \) \(\mathstrut -\mathstrut 34810604q^{61} \) \(\mathstrut +\mathstrut 20216432q^{64} \) \(\mathstrut -\mathstrut 28645328q^{67} \) \(\mathstrut -\mathstrut 13856976q^{70} \) \(\mathstrut -\mathstrut 17813984q^{73} \) \(\mathstrut +\mathstrut 22200640q^{76} \) \(\mathstrut +\mathstrut 65517688q^{79} \) \(\mathstrut +\mathstrut 5994252q^{82} \) \(\mathstrut -\mathstrut 30020652q^{85} \) \(\mathstrut +\mathstrut 89560224q^{88} \) \(\mathstrut -\mathstrut 86802688q^{91} \) \(\mathstrut -\mathstrut 13856112q^{94} \) \(\mathstrut -\mathstrut 48903488q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
8.2 4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
\(n\): 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 \(S_{9}^{\mathrm{new}}(9, [\chi])\).