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 \(6\), degree \(2\), minimal)

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.3.d.a 2
3.b odd 2 1 27.3.d.a 2
4.b odd 2 1 144.3.q.a 2
5.b even 2 1 225.3.j.a 2
5.c odd 4 2 225.3.i.a 4
7.b odd 2 1 441.3.r.a 2
7.c even 3 1 441.3.j.a 2
7.c even 3 1 441.3.n.b 2
7.d odd 6 1 441.3.j.b 2
7.d odd 6 1 441.3.n.a 2
8.b even 2 1 576.3.q.b 2
8.d odd 2 1 576.3.q.a 2
9.c even 3 1 27.3.d.a 2
9.c even 3 1 81.3.b.a 2
9.d odd 6 1 inner 9.3.d.a 2
9.d odd 6 1 81.3.b.a 2
12.b even 2 1 432.3.q.a 2
15.d odd 2 1 675.3.j.a 2
15.e even 4 2 675.3.i.a 4
24.f even 2 1 1728.3.q.b 2
24.h odd 2 1 1728.3.q.a 2
36.f odd 6 1 432.3.q.a 2
36.f odd 6 1 1296.3.e.a 2
36.h even 6 1 144.3.q.a 2
36.h even 6 1 1296.3.e.a 2
45.h odd 6 1 225.3.j.a 2
45.j even 6 1 675.3.j.a 2
45.k odd 12 2 675.3.i.a 4
45.l even 12 2 225.3.i.a 4
63.i even 6 1 441.3.n.a 2
63.j odd 6 1 441.3.n.b 2
63.n odd 6 1 441.3.j.a 2
63.o even 6 1 441.3.r.a 2
63.s even 6 1 441.3.j.b 2
72.j odd 6 1 576.3.q.b 2
72.l even 6 1 576.3.q.a 2
72.n even 6 1 1728.3.q.a 2
72.p odd 6 1 1728.3.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 1.a even 1 1 trivial
9.3.d.a 2 9.d odd 6 1 inner
27.3.d.a 2 3.b odd 2 1
27.3.d.a 2 9.c even 3 1
81.3.b.a 2 9.c even 3 1
81.3.b.a 2 9.d odd 6 1
144.3.q.a 2 4.b odd 2 1
144.3.q.a 2 36.h even 6 1
225.3.i.a 4 5.c odd 4 2
225.3.i.a 4 45.l even 12 2
225.3.j.a 2 5.b even 2 1
225.3.j.a 2 45.h odd 6 1
432.3.q.a 2 12.b even 2 1
432.3.q.a 2 36.f odd 6 1
441.3.j.a 2 7.c even 3 1
441.3.j.a 2 63.n odd 6 1
441.3.j.b 2 7.d odd 6 1
441.3.j.b 2 63.s even 6 1
441.3.n.a 2 7.d odd 6 1
441.3.n.a 2 63.i even 6 1
441.3.n.b 2 7.c even 3 1
441.3.n.b 2 63.j odd 6 1
441.3.r.a 2 7.b odd 2 1
441.3.r.a 2 63.o even 6 1
576.3.q.a 2 8.d odd 2 1
576.3.q.a 2 72.l even 6 1
576.3.q.b 2 8.b even 2 1
576.3.q.b 2 72.j odd 6 1
675.3.i.a 4 15.e even 4 2
675.3.i.a 4 45.k odd 12 2
675.3.j.a 2 15.d odd 2 1
675.3.j.a 2 45.j even 6 1
1296.3.e.a 2 36.f odd 6 1
1296.3.e.a 2 36.h even 6 1
1728.3.q.a 2 24.h odd 2 1
1728.3.q.a 2 72.n even 6 1
1728.3.q.b 2 24.f even 2 1
1728.3.q.b 2 72.p odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(9, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} \)
$3$ \( 1 + 3 T + 9 T^{2} \)
$5$ \( 1 - 6 T + 37 T^{2} - 150 T^{3} + 625 T^{4} \)
$7$ \( ( 1 - 11 T + 49 T^{2} )( 1 + 13 T + 49 T^{2} ) \)
$11$ \( 1 + 3 T + 124 T^{2} + 363 T^{3} + 14641 T^{4} \)
$13$ \( 1 - 4 T - 153 T^{2} - 676 T^{3} + 28561 T^{4} \)
$17$ \( 1 - 335 T^{2} + 83521 T^{4} \)
$19$ \( ( 1 - 11 T + 361 T^{2} )^{2} \)
$23$ \( 1 + 48 T + 1297 T^{2} + 25392 T^{3} + 279841 T^{4} \)
$29$ \( 1 - 78 T + 2869 T^{2} - 65598 T^{3} + 707281 T^{4} \)
$31$ \( 1 + 32 T + 63 T^{2} + 30752 T^{3} + 923521 T^{4} \)
$37$ \( ( 1 + 34 T + 1369 T^{2} )^{2} \)
$41$ \( 1 + 21 T + 1828 T^{2} + 35301 T^{3} + 2825761 T^{4} \)
$43$ \( ( 1 - 83 T + 1849 T^{2} )( 1 + 22 T + 1849 T^{2} ) \)
$47$ \( 1 + 84 T + 4561 T^{2} + 185556 T^{3} + 4879681 T^{4} \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \)
$59$ \( 1 - 87 T + 6004 T^{2} - 302847 T^{3} + 12117361 T^{4} \)
$61$ \( 1 + 56 T - 585 T^{2} + 208376 T^{3} + 13845841 T^{4} \)
$67$ \( 1 - 31 T - 3528 T^{2} - 139159 T^{3} + 20151121 T^{4} \)
$71$ \( 1 - 9110 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 - 65 T + 5329 T^{2} )^{2} \)
$79$ \( 1 + 38 T - 4797 T^{2} + 237158 T^{3} + 38950081 T^{4} \)
$83$ \( 1 + 84 T + 9241 T^{2} + 578676 T^{3} + 47458321 T^{4} \)
$89$ \( 1 - 290 T^{2} + 62742241 T^{4} \)
$97$ \( 1 - 115 T + 3816 T^{2} - 1082035 T^{3} + 88529281 T^{4} \)
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