Properties

Label 9.3.d.a
Level 9
Weight 3
Character orbit 9.d
Analytic conductor 0.245
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 9.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.245232237924\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \zeta_{6} ) q^{2} \) \( + ( -3 + 3 \zeta_{6} ) q^{3} \) \( -\zeta_{6} q^{4} \) \( + ( 4 - 2 \zeta_{6} ) q^{5} \) \( + ( 6 - 3 \zeta_{6} ) q^{6} \) \( + ( -2 + 2 \zeta_{6} ) q^{7} \) \( + ( -5 + 10 \zeta_{6} ) q^{8} \) \( -9 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \zeta_{6} ) q^{2} \) \( + ( -3 + 3 \zeta_{6} ) q^{3} \) \( -\zeta_{6} q^{4} \) \( + ( 4 - 2 \zeta_{6} ) q^{5} \) \( + ( 6 - 3 \zeta_{6} ) q^{6} \) \( + ( -2 + 2 \zeta_{6} ) q^{7} \) \( + ( -5 + 10 \zeta_{6} ) q^{8} \) \( -9 \zeta_{6} q^{9} \) \( -6 q^{10} \) \( + ( -1 - \zeta_{6} ) q^{11} \) \( + 3 q^{12} \) \( + 4 \zeta_{6} q^{13} \) \( + ( 4 - 2 \zeta_{6} ) q^{14} \) \( + ( -6 + 12 \zeta_{6} ) q^{15} \) \( + ( 11 - 11 \zeta_{6} ) q^{16} \) \( + ( 9 - 18 \zeta_{6} ) q^{17} \) \( + ( -9 + 18 \zeta_{6} ) q^{18} \) \( + 11 q^{19} \) \( + ( -2 - 2 \zeta_{6} ) q^{20} \) \( -6 \zeta_{6} q^{21} \) \( + 3 \zeta_{6} q^{22} \) \( + ( -32 + 16 \zeta_{6} ) q^{23} \) \( + ( -15 - 15 \zeta_{6} ) q^{24} \) \( + ( -13 + 13 \zeta_{6} ) q^{25} \) \( + ( 4 - 8 \zeta_{6} ) q^{26} \) \( + 27 q^{27} \) \( + 2 q^{28} \) \( + ( 26 + 26 \zeta_{6} ) q^{29} \) \( + ( 18 - 18 \zeta_{6} ) q^{30} \) \( -32 \zeta_{6} q^{31} \) \( + ( 18 - 9 \zeta_{6} ) q^{32} \) \( + ( 6 - 3 \zeta_{6} ) q^{33} \) \( + ( -27 + 27 \zeta_{6} ) q^{34} \) \( + ( -4 + 8 \zeta_{6} ) q^{35} \) \( + ( -9 + 9 \zeta_{6} ) q^{36} \) \( -34 q^{37} \) \( + ( -11 - 11 \zeta_{6} ) q^{38} \) \( -12 q^{39} \) \( + 30 \zeta_{6} q^{40} \) \( + ( -14 + 7 \zeta_{6} ) q^{41} \) \( + ( -6 + 12 \zeta_{6} ) q^{42} \) \( + ( 61 - 61 \zeta_{6} ) q^{43} \) \( + ( -1 + 2 \zeta_{6} ) q^{44} \) \( + ( -18 - 18 \zeta_{6} ) q^{45} \) \( + 48 q^{46} \) \( + ( -28 - 28 \zeta_{6} ) q^{47} \) \( + 33 \zeta_{6} q^{48} \) \( + 45 \zeta_{6} q^{49} \) \( + ( 26 - 13 \zeta_{6} ) q^{50} \) \( + ( 27 + 27 \zeta_{6} ) q^{51} \) \( + ( 4 - 4 \zeta_{6} ) q^{52} \) \( + ( -27 - 27 \zeta_{6} ) q^{54} \) \( -6 q^{55} \) \( + ( -10 - 10 \zeta_{6} ) q^{56} \) \( + ( -33 + 33 \zeta_{6} ) q^{57} \) \( -78 \zeta_{6} q^{58} \) \( + ( 58 - 29 \zeta_{6} ) q^{59} \) \( + ( 12 - 6 \zeta_{6} ) q^{60} \) \( + ( -56 + 56 \zeta_{6} ) q^{61} \) \( + ( -32 + 64 \zeta_{6} ) q^{62} \) \( + 18 q^{63} \) \( -71 q^{64} \) \( + ( 8 + 8 \zeta_{6} ) q^{65} \) \( -9 q^{66} \) \( + 31 \zeta_{6} q^{67} \) \( + ( -18 + 9 \zeta_{6} ) q^{68} \) \( + ( 48 - 96 \zeta_{6} ) q^{69} \) \( + ( 12 - 12 \zeta_{6} ) q^{70} \) \( + ( -18 + 36 \zeta_{6} ) q^{71} \) \( + ( 90 - 45 \zeta_{6} ) q^{72} \) \( + 65 q^{73} \) \( + ( 34 + 34 \zeta_{6} ) q^{74} \) \( -39 \zeta_{6} q^{75} \) \( -11 \zeta_{6} q^{76} \) \( + ( 4 - 2 \zeta_{6} ) q^{77} \) \( + ( 12 + 12 \zeta_{6} ) q^{78} \) \( + ( -38 + 38 \zeta_{6} ) q^{79} \) \( + ( 22 - 44 \zeta_{6} ) q^{80} \) \( + ( -81 + 81 \zeta_{6} ) q^{81} \) \( + 21 q^{82} \) \( + ( -28 - 28 \zeta_{6} ) q^{83} \) \( + ( -6 + 6 \zeta_{6} ) q^{84} \) \( -54 \zeta_{6} q^{85} \) \( + ( -122 + 61 \zeta_{6} ) q^{86} \) \( + ( -156 + 78 \zeta_{6} ) q^{87} \) \( + ( 15 - 15 \zeta_{6} ) q^{88} \) \( + ( 72 - 144 \zeta_{6} ) q^{89} \) \( + 54 \zeta_{6} q^{90} \) \( -8 q^{91} \) \( + ( 16 + 16 \zeta_{6} ) q^{92} \) \( + 96 q^{93} \) \( + 84 \zeta_{6} q^{94} \) \( + ( 44 - 22 \zeta_{6} ) q^{95} \) \( + ( -27 + 54 \zeta_{6} ) q^{96} \) \( + ( 115 - 115 \zeta_{6} ) q^{97} \) \( + ( 45 - 90 \zeta_{6} ) q^{98} \) \( + ( -9 + 18 \zeta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 12q^{10} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut 6q^{12} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 11q^{16} \) \(\mathstrut +\mathstrut 22q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 3q^{22} \) \(\mathstrut -\mathstrut 48q^{23} \) \(\mathstrut -\mathstrut 45q^{24} \) \(\mathstrut -\mathstrut 13q^{25} \) \(\mathstrut +\mathstrut 54q^{27} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 78q^{29} \) \(\mathstrut +\mathstrut 18q^{30} \) \(\mathstrut -\mathstrut 32q^{31} \) \(\mathstrut +\mathstrut 27q^{32} \) \(\mathstrut +\mathstrut 9q^{33} \) \(\mathstrut -\mathstrut 27q^{34} \) \(\mathstrut -\mathstrut 9q^{36} \) \(\mathstrut -\mathstrut 68q^{37} \) \(\mathstrut -\mathstrut 33q^{38} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 30q^{40} \) \(\mathstrut -\mathstrut 21q^{41} \) \(\mathstrut +\mathstrut 61q^{43} \) \(\mathstrut -\mathstrut 54q^{45} \) \(\mathstrut +\mathstrut 96q^{46} \) \(\mathstrut -\mathstrut 84q^{47} \) \(\mathstrut +\mathstrut 33q^{48} \) \(\mathstrut +\mathstrut 45q^{49} \) \(\mathstrut +\mathstrut 39q^{50} \) \(\mathstrut +\mathstrut 81q^{51} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 81q^{54} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 30q^{56} \) \(\mathstrut -\mathstrut 33q^{57} \) \(\mathstrut -\mathstrut 78q^{58} \) \(\mathstrut +\mathstrut 87q^{59} \) \(\mathstrut +\mathstrut 18q^{60} \) \(\mathstrut -\mathstrut 56q^{61} \) \(\mathstrut +\mathstrut 36q^{63} \) \(\mathstrut -\mathstrut 142q^{64} \) \(\mathstrut +\mathstrut 24q^{65} \) \(\mathstrut -\mathstrut 18q^{66} \) \(\mathstrut +\mathstrut 31q^{67} \) \(\mathstrut -\mathstrut 27q^{68} \) \(\mathstrut +\mathstrut 12q^{70} \) \(\mathstrut +\mathstrut 135q^{72} \) \(\mathstrut +\mathstrut 130q^{73} \) \(\mathstrut +\mathstrut 102q^{74} \) \(\mathstrut -\mathstrut 39q^{75} \) \(\mathstrut -\mathstrut 11q^{76} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 36q^{78} \) \(\mathstrut -\mathstrut 38q^{79} \) \(\mathstrut -\mathstrut 81q^{81} \) \(\mathstrut +\mathstrut 42q^{82} \) \(\mathstrut -\mathstrut 84q^{83} \) \(\mathstrut -\mathstrut 6q^{84} \) \(\mathstrut -\mathstrut 54q^{85} \) \(\mathstrut -\mathstrut 183q^{86} \) \(\mathstrut -\mathstrut 234q^{87} \) \(\mathstrut +\mathstrut 15q^{88} \) \(\mathstrut +\mathstrut 54q^{90} \) \(\mathstrut -\mathstrut 16q^{91} \) \(\mathstrut +\mathstrut 48q^{92} \) \(\mathstrut +\mathstrut 192q^{93} \) \(\mathstrut +\mathstrut 84q^{94} \) \(\mathstrut +\mathstrut 66q^{95} \) \(\mathstrut +\mathstrut 115q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 0.866025i −1.50000 + 2.59808i −0.500000 0.866025i 3.00000 1.73205i 4.50000 2.59808i −1.00000 + 1.73205i 8.66025i −4.50000 7.79423i −6.00000
5.1 −1.50000 + 0.866025i −1.50000 2.59808i −0.500000 + 0.866025i 3.00000 + 1.73205i 4.50000 + 2.59808i −1.00000 1.73205i 8.66025i −4.50000 + 7.79423i −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
9.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(9, [\chi])\).