Properties

Label 324.3.f.b
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
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.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + (5 \zeta_{6} - 10) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + (5 \zeta_{6} - 10) q^{7} - 8 q^{8} + (14 \zeta_{6} - 14) q^{10} + (5 \zeta_{6} - 10) q^{11} + (20 \zeta_{6} - 20) q^{13} + ( - 10 \zeta_{6} + 20) q^{14} + 16 q^{16} + 8 q^{17} + (12 \zeta_{6} - 6) q^{19} + ( - 28 \zeta_{6} + 28) q^{20} + ( - 10 \zeta_{6} + 20) q^{22} + ( - 2 \zeta_{6} - 2) q^{23} - 24 \zeta_{6} q^{25} + ( - 40 \zeta_{6} + 40) q^{26} + (20 \zeta_{6} - 40) q^{28} + 10 \zeta_{6} q^{29} + (31 \zeta_{6} + 31) q^{31} - 32 q^{32} - 16 q^{34} + (70 \zeta_{6} - 35) q^{35} - 10 q^{37} + ( - 24 \zeta_{6} + 12) q^{38} + (56 \zeta_{6} - 56) q^{40} + (50 \zeta_{6} - 50) q^{41} + ( - 10 \zeta_{6} + 20) q^{43} + (20 \zeta_{6} - 40) q^{44} + (4 \zeta_{6} + 4) q^{46} + (50 \zeta_{6} - 100) q^{47} + ( - 26 \zeta_{6} + 26) q^{49} + 48 \zeta_{6} q^{50} + (80 \zeta_{6} - 80) q^{52} + 47 q^{53} + (70 \zeta_{6} - 35) q^{55} + ( - 40 \zeta_{6} + 80) q^{56} - 20 \zeta_{6} q^{58} + ( - 20 \zeta_{6} - 20) q^{59} + 64 \zeta_{6} q^{61} + ( - 62 \zeta_{6} - 62) q^{62} + 64 q^{64} + 140 \zeta_{6} q^{65} + ( - 50 \zeta_{6} - 50) q^{67} + 32 q^{68} + ( - 140 \zeta_{6} + 70) q^{70} - 55 q^{73} + 20 q^{74} + (48 \zeta_{6} - 24) q^{76} + ( - 75 \zeta_{6} + 75) q^{77} + ( - 4 \zeta_{6} + 8) q^{79} + ( - 112 \zeta_{6} + 112) q^{80} + ( - 100 \zeta_{6} + 100) q^{82} + (17 \zeta_{6} - 34) q^{83} + ( - 56 \zeta_{6} + 56) q^{85} + (20 \zeta_{6} - 40) q^{86} + ( - 40 \zeta_{6} + 80) q^{88} - 10 q^{89} + ( - 200 \zeta_{6} + 100) q^{91} + ( - 8 \zeta_{6} - 8) q^{92} + ( - 100 \zeta_{6} + 200) q^{94} + (42 \zeta_{6} + 42) q^{95} + 25 \zeta_{6} q^{97} + (52 \zeta_{6} - 52) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 15 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 15 q^{7} - 16 q^{8} - 14 q^{10} - 15 q^{11} - 20 q^{13} + 30 q^{14} + 32 q^{16} + 16 q^{17} + 28 q^{20} + 30 q^{22} - 6 q^{23} - 24 q^{25} + 40 q^{26} - 60 q^{28} + 10 q^{29} + 93 q^{31} - 64 q^{32} - 32 q^{34} - 20 q^{37} - 56 q^{40} - 50 q^{41} + 30 q^{43} - 60 q^{44} + 12 q^{46} - 150 q^{47} + 26 q^{49} + 48 q^{50} - 80 q^{52} + 94 q^{53} + 120 q^{56} - 20 q^{58} - 60 q^{59} + 64 q^{61} - 186 q^{62} + 128 q^{64} + 140 q^{65} - 150 q^{67} + 64 q^{68} - 110 q^{73} + 40 q^{74} + 75 q^{77} + 12 q^{79} + 112 q^{80} + 100 q^{82} - 51 q^{83} + 56 q^{85} - 60 q^{86} + 120 q^{88} - 20 q^{89} - 24 q^{92} + 300 q^{94} + 126 q^{95} + 25 q^{97} - 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
−2.00000 0 4.00000 3.50000 + 6.06218i 0 −7.50000 4.33013i −8.00000 0 −7.00000 12.1244i
271.1 −2.00000 0 4.00000 3.50000 6.06218i 0 −7.50000 + 4.33013i −8.00000 0 −7.00000 + 12.1244i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.b 2
3.b odd 2 1 324.3.f.i 2
4.b odd 2 1 324.3.f.h 2
9.c even 3 1 108.3.d.b yes 2
9.c even 3 1 324.3.f.h 2
9.d odd 6 1 108.3.d.a 2
9.d odd 6 1 324.3.f.c 2
12.b even 2 1 324.3.f.c 2
36.f odd 6 1 108.3.d.b yes 2
36.f odd 6 1 inner 324.3.f.b 2
36.h even 6 1 108.3.d.a 2
36.h even 6 1 324.3.f.i 2
72.j odd 6 1 1728.3.g.a 2
72.l even 6 1 1728.3.g.a 2
72.n even 6 1 1728.3.g.f 2
72.p odd 6 1 1728.3.g.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 9.d odd 6 1
108.3.d.a 2 36.h even 6 1
108.3.d.b yes 2 9.c even 3 1
108.3.d.b yes 2 36.f odd 6 1
324.3.f.b 2 1.a even 1 1 trivial
324.3.f.b 2 36.f odd 6 1 inner
324.3.f.c 2 9.d odd 6 1
324.3.f.c 2 12.b even 2 1
324.3.f.h 2 4.b odd 2 1
324.3.f.h 2 9.c even 3 1
324.3.f.i 2 3.b odd 2 1
324.3.f.i 2 36.h even 6 1
1728.3.g.a 2 72.j odd 6 1
1728.3.g.a 2 72.l even 6 1
1728.3.g.f 2 72.n even 6 1
1728.3.g.f 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} - 7T_{5} + 49 \) Copy content Toggle raw display
\( T_{7}^{2} + 15T_{7} + 75 \) Copy content Toggle raw display
\( T_{11}^{2} + 15T_{11} + 75 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$11$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$13$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$17$ \( (T - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 108 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$31$ \( T^{2} - 93T + 2883 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 50T + 2500 \) Copy content Toggle raw display
$43$ \( T^{2} - 30T + 300 \) Copy content Toggle raw display
$47$ \( T^{2} + 150T + 7500 \) Copy content Toggle raw display
$53$ \( (T - 47)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 60T + 1200 \) Copy content Toggle raw display
$61$ \( T^{2} - 64T + 4096 \) Copy content Toggle raw display
$67$ \( T^{2} + 150T + 7500 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 55)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$83$ \( T^{2} + 51T + 867 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 25T + 625 \) Copy content Toggle raw display
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