Properties

Label 324.4.e.c
Level $324$
Weight $4$
Character orbit 324.e
Analytic conductor $19.117$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1166188419\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 3 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 24 \zeta_{6} + 24) q^{11} + 25 \zeta_{6} q^{13} - 21 q^{17} - 52 q^{19} - 168 \zeta_{6} q^{23} + ( - 116 \zeta_{6} + 116) q^{25} + ( - 177 \zeta_{6} + 177) q^{29} + 124 \zeta_{6} q^{31} - 12 q^{35} - 265 q^{37} - 426 \zeta_{6} q^{41} + ( - 160 \zeta_{6} + 160) q^{43} + ( - 540 \zeta_{6} + 540) q^{47} + 327 \zeta_{6} q^{49} - 258 q^{53} - 72 q^{55} - 528 \zeta_{6} q^{59} + ( - 505 \zeta_{6} + 505) q^{61} + ( - 75 \zeta_{6} + 75) q^{65} + 244 \zeta_{6} q^{67} + 204 q^{71} - 397 q^{73} - 96 \zeta_{6} q^{77} + (200 \zeta_{6} - 200) q^{79} + ( - 540 \zeta_{6} + 540) q^{83} + 63 \zeta_{6} q^{85} - 453 q^{89} + 100 q^{91} + 156 \zeta_{6} q^{95} + (290 \zeta_{6} - 290) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} + 4 q^{7} + 24 q^{11} + 25 q^{13} - 42 q^{17} - 104 q^{19} - 168 q^{23} + 116 q^{25} + 177 q^{29} + 124 q^{31} - 24 q^{35} - 530 q^{37} - 426 q^{41} + 160 q^{43} + 540 q^{47} + 327 q^{49} - 516 q^{53} - 144 q^{55} - 528 q^{59} + 505 q^{61} + 75 q^{65} + 244 q^{67} + 408 q^{71} - 794 q^{73} - 96 q^{77} - 200 q^{79} + 540 q^{83} + 63 q^{85} - 906 q^{89} + 200 q^{91} + 156 q^{95} - 290 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −1.50000 + 2.59808i 0 2.00000 + 3.46410i 0 0 0
217.1 0 0 0 −1.50000 2.59808i 0 2.00000 3.46410i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.e.c 2
3.b odd 2 1 324.4.e.f 2
9.c even 3 1 324.4.a.b yes 1
9.c even 3 1 inner 324.4.e.c 2
9.d odd 6 1 324.4.a.a 1
9.d odd 6 1 324.4.e.f 2
36.f odd 6 1 1296.4.a.e 1
36.h even 6 1 1296.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.4.a.a 1 9.d odd 6 1
324.4.a.b yes 1 9.c even 3 1
324.4.e.c 2 1.a even 1 1 trivial
324.4.e.c 2 9.c even 3 1 inner
324.4.e.f 2 3.b odd 2 1
324.4.e.f 2 9.d odd 6 1
1296.4.a.d 1 36.h even 6 1
1296.4.a.e 1 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T + 576 \) Copy content Toggle raw display
$13$ \( T^{2} - 25T + 625 \) Copy content Toggle raw display
$17$ \( (T + 21)^{2} \) Copy content Toggle raw display
$19$ \( (T + 52)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 168T + 28224 \) Copy content Toggle raw display
$29$ \( T^{2} - 177T + 31329 \) Copy content Toggle raw display
$31$ \( T^{2} - 124T + 15376 \) Copy content Toggle raw display
$37$ \( (T + 265)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 426T + 181476 \) Copy content Toggle raw display
$43$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$47$ \( T^{2} - 540T + 291600 \) Copy content Toggle raw display
$53$ \( (T + 258)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 528T + 278784 \) Copy content Toggle raw display
$61$ \( T^{2} - 505T + 255025 \) Copy content Toggle raw display
$67$ \( T^{2} - 244T + 59536 \) Copy content Toggle raw display
$71$ \( (T - 204)^{2} \) Copy content Toggle raw display
$73$ \( (T + 397)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 200T + 40000 \) Copy content Toggle raw display
$83$ \( T^{2} - 540T + 291600 \) Copy content Toggle raw display
$89$ \( (T + 453)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 290T + 84100 \) Copy content Toggle raw display
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