Properties

Label 324.4.e.b
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: no (minimal twist has level 108)
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.

\(f(q)\) \(=\) \( q - 9 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 9 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + (63 \zeta_{6} - 63) q^{11} + 28 \zeta_{6} q^{13} + 72 q^{17} + 98 q^{19} - 126 \zeta_{6} q^{23} + ( - 44 \zeta_{6} + 44) q^{25} + ( - 126 \zeta_{6} + 126) q^{29} + 259 \zeta_{6} q^{31} - 9 q^{35} + 386 q^{37} + 450 \zeta_{6} q^{41} + ( - 34 \zeta_{6} + 34) q^{43} + ( - 54 \zeta_{6} + 54) q^{47} + 342 \zeta_{6} q^{49} - 693 q^{53} + 567 q^{55} - 180 \zeta_{6} q^{59} + ( - 280 \zeta_{6} + 280) q^{61} + ( - 252 \zeta_{6} + 252) q^{65} + 586 \zeta_{6} q^{67} + 504 q^{71} + 161 q^{73} + 63 \zeta_{6} q^{77} + (440 \zeta_{6} - 440) q^{79} + (999 \zeta_{6} - 999) q^{83} - 648 \zeta_{6} q^{85} + 882 q^{89} + 28 q^{91} - 882 \zeta_{6} q^{95} + ( - 721 \zeta_{6} + 721) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{5} + q^{7} - 63 q^{11} + 28 q^{13} + 144 q^{17} + 196 q^{19} - 126 q^{23} + 44 q^{25} + 126 q^{29} + 259 q^{31} - 18 q^{35} + 772 q^{37} + 450 q^{41} + 34 q^{43} + 54 q^{47} + 342 q^{49} - 1386 q^{53} + 1134 q^{55} - 180 q^{59} + 280 q^{61} + 252 q^{65} + 586 q^{67} + 1008 q^{71} + 322 q^{73} + 63 q^{77} - 440 q^{79} - 999 q^{83} - 648 q^{85} + 1764 q^{89} + 56 q^{91} - 882 q^{95} + 721 q^{97}+O(q^{100}) \) 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 −4.50000 + 7.79423i 0 0.500000 + 0.866025i 0 0 0
217.1 0 0 0 −4.50000 7.79423i 0 0.500000 0.866025i 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.b 2
3.b odd 2 1 324.4.e.g 2
9.c even 3 1 108.4.a.d yes 1
9.c even 3 1 inner 324.4.e.b 2
9.d odd 6 1 108.4.a.a 1
9.d odd 6 1 324.4.e.g 2
36.f odd 6 1 432.4.a.l 1
36.h even 6 1 432.4.a.c 1
72.j odd 6 1 1728.4.a.y 1
72.l even 6 1 1728.4.a.z 1
72.n even 6 1 1728.4.a.g 1
72.p odd 6 1 1728.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.a 1 9.d odd 6 1
108.4.a.d yes 1 9.c even 3 1
324.4.e.b 2 1.a even 1 1 trivial
324.4.e.b 2 9.c even 3 1 inner
324.4.e.g 2 3.b odd 2 1
324.4.e.g 2 9.d odd 6 1
432.4.a.c 1 36.h even 6 1
432.4.a.l 1 36.f odd 6 1
1728.4.a.g 1 72.n even 6 1
1728.4.a.h 1 72.p odd 6 1
1728.4.a.y 1 72.j odd 6 1
1728.4.a.z 1 72.l even 6 1

Hecke kernels

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

\( T_{5}^{2} + 9T_{5} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) 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
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 63T + 3969 \) Copy content Toggle raw display
$13$ \( T^{2} - 28T + 784 \) Copy content Toggle raw display
$17$ \( (T - 72)^{2} \) Copy content Toggle raw display
$19$ \( (T - 98)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$29$ \( T^{2} - 126T + 15876 \) Copy content Toggle raw display
$31$ \( T^{2} - 259T + 67081 \) Copy content Toggle raw display
$37$ \( (T - 386)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 450T + 202500 \) Copy content Toggle raw display
$43$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$47$ \( T^{2} - 54T + 2916 \) Copy content Toggle raw display
$53$ \( (T + 693)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 180T + 32400 \) Copy content Toggle raw display
$61$ \( T^{2} - 280T + 78400 \) Copy content Toggle raw display
$67$ \( T^{2} - 586T + 343396 \) Copy content Toggle raw display
$71$ \( (T - 504)^{2} \) Copy content Toggle raw display
$73$ \( (T - 161)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 440T + 193600 \) Copy content Toggle raw display
$83$ \( T^{2} + 999T + 998001 \) Copy content Toggle raw display
$89$ \( (T - 882)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 721T + 519841 \) Copy content Toggle raw display
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