Properties

Label 324.2.h.b
Level 324
Weight 2
Character orbit 324.h
Analytic conductor 2.587
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 4 \beta_{2} q^{10} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} -2 \beta_{3} q^{17} + ( -3 + 6 \beta_{2} ) q^{19} + 4 \beta_{3} q^{20} + ( 4 - 8 \beta_{2} ) q^{22} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{23} + 3 \beta_{2} q^{25} + ( \beta_{1} - \beta_{3} ) q^{26} + ( -2 - 2 \beta_{2} ) q^{28} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{29} + ( 2 + 2 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 4 - 4 \beta_{2} ) q^{34} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{35} - q^{37} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{38} + ( -8 + 8 \beta_{2} ) q^{40} -4 \beta_{1} q^{41} + ( 4 - 2 \beta_{2} ) q^{43} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{44} + ( 8 - 4 \beta_{2} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -4 + 4 \beta_{2} ) q^{49} + 3 \beta_{3} q^{50} + 2 q^{52} + 4 \beta_{3} q^{53} + ( 8 - 16 \beta_{2} ) q^{55} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{56} + 8 q^{58} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{59} -11 \beta_{2} q^{61} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{62} -8 q^{64} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -7 - 7 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{68} + ( -4 - 4 \beta_{2} ) q^{70} - q^{73} -\beta_{1} q^{74} + ( -12 + 6 \beta_{2} ) q^{76} + 6 \beta_{1} q^{77} + ( -2 + \beta_{2} ) q^{79} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{80} -8 \beta_{2} q^{82} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{83} + ( 8 - 8 \beta_{2} ) q^{85} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{86} + ( 16 - 8 \beta_{2} ) q^{88} -2 \beta_{3} q^{89} + ( -1 + 2 \beta_{2} ) q^{91} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 4 - 8 \beta_{2} ) q^{94} + ( -6 \beta_{1} + 12 \beta_{3} ) q^{95} + 13 \beta_{2} q^{97} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 6q^{7} + O(q^{10}) \) \( 4q + 4q^{4} - 6q^{7} + 8q^{10} + 2q^{13} - 8q^{16} + 6q^{25} - 12q^{28} + 12q^{31} + 8q^{34} - 4q^{37} - 16q^{40} + 12q^{43} + 24q^{46} - 8q^{49} + 8q^{52} + 32q^{58} - 22q^{61} - 32q^{64} - 42q^{67} - 24q^{70} - 4q^{73} - 36q^{76} - 6q^{79} - 16q^{82} + 16q^{85} + 48q^{88} + 26q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −2.44949 + 1.41421i 0 −1.50000 0.866025i 2.82843i 0 2.00000 3.46410i
107.2 1.22474 0.707107i 0 1.00000 1.73205i 2.44949 1.41421i 0 −1.50000 0.866025i 2.82843i 0 2.00000 3.46410i
215.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −2.44949 1.41421i 0 −1.50000 + 0.866025i 2.82843i 0 2.00000 + 3.46410i
215.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.44949 + 1.41421i 0 −1.50000 + 0.866025i 2.82843i 0 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.h.b 4
3.b odd 2 1 inner 324.2.h.b 4
4.b odd 2 1 324.2.h.a 4
9.c even 3 1 108.2.b.b 4
9.c even 3 1 324.2.h.a 4
9.d odd 6 1 108.2.b.b 4
9.d odd 6 1 324.2.h.a 4
12.b even 2 1 324.2.h.a 4
36.f odd 6 1 108.2.b.b 4
36.f odd 6 1 inner 324.2.h.b 4
36.h even 6 1 108.2.b.b 4
36.h even 6 1 inner 324.2.h.b 4
72.j odd 6 1 1728.2.c.d 4
72.l even 6 1 1728.2.c.d 4
72.n even 6 1 1728.2.c.d 4
72.p odd 6 1 1728.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.b 4 9.c even 3 1
108.2.b.b 4 9.d odd 6 1
108.2.b.b 4 36.f odd 6 1
108.2.b.b 4 36.h even 6 1
324.2.h.a 4 4.b odd 2 1
324.2.h.a 4 9.c even 3 1
324.2.h.a 4 9.d odd 6 1
324.2.h.a 4 12.b even 2 1
324.2.h.b 4 1.a even 1 1 trivial
324.2.h.b 4 3.b odd 2 1 inner
324.2.h.b 4 36.f odd 6 1 inner
324.2.h.b 4 36.h even 6 1 inner
1728.2.c.d 4 72.j odd 6 1
1728.2.c.d 4 72.l even 6 1
1728.2.c.d 4 72.n even 6 1
1728.2.c.d 4 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 \(S_{2}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} - 8 T_{5}^{2} + 64 \)
\( T_{7}^{2} + 3 T_{7} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2}( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 22 T^{2} - 45 T^{4} - 11638 T^{6} + 279841 T^{8} \)
$29$ \( 1 + 26 T^{2} - 165 T^{4} + 21866 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{4} \)
$41$ \( 1 + 50 T^{2} + 819 T^{4} + 84050 T^{6} + 2825761 T^{8} \)
$43$ \( ( 1 - 6 T + 55 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 70 T^{2} + 2691 T^{4} - 154630 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 5 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 + T + 73 T^{2} )^{4} \)
$79$ \( ( 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 70 T^{2} - 1989 T^{4} - 482230 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 - 170 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} )^{2} \)
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