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: \(\Q(\sqrt{-3}) \)
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.

\(f(q)\) \(=\) \( q -9 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q -9 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + ( -63 + 63 \zeta_{6} ) q^{11} + 28 \zeta_{6} q^{13} + 72 q^{17} + 98 q^{19} -126 \zeta_{6} q^{23} + ( 44 - 44 \zeta_{6} ) q^{25} + ( 126 - 126 \zeta_{6} ) q^{29} + 259 \zeta_{6} q^{31} -9 q^{35} + 386 q^{37} + 450 \zeta_{6} q^{41} + ( 34 - 34 \zeta_{6} ) q^{43} + ( 54 - 54 \zeta_{6} ) q^{47} + 342 \zeta_{6} q^{49} -693 q^{53} + 567 q^{55} -180 \zeta_{6} q^{59} + ( 280 - 280 \zeta_{6} ) q^{61} + ( 252 - 252 \zeta_{6} ) q^{65} + 586 \zeta_{6} q^{67} + 504 q^{71} + 161 q^{73} + 63 \zeta_{6} q^{77} + ( -440 + 440 \zeta_{6} ) q^{79} + ( -999 + 999 \zeta_{6} ) q^{83} -648 \zeta_{6} q^{85} + 882 q^{89} + 28 q^{91} -882 \zeta_{6} q^{95} + ( 721 - 721 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 9q^{5} + q^{7} + O(q^{10}) \) \( 2q - 9q^{5} + q^{7} - 63q^{11} + 28q^{13} + 144q^{17} + 196q^{19} - 126q^{23} + 44q^{25} + 126q^{29} + 259q^{31} - 18q^{35} + 772q^{37} + 450q^{41} + 34q^{43} + 54q^{47} + 342q^{49} - 1386q^{53} + 1134q^{55} - 180q^{59} + 280q^{61} + 252q^{65} + 586q^{67} + 1008q^{71} + 322q^{73} + 63q^{77} - 440q^{79} - 999q^{83} - 648q^{85} + 1764q^{89} + 56q^{91} - 882q^{95} + 721q^{97} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 9 T_{5} + 81 \)
\( T_{7}^{2} - T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4} \)
$7$ \( 1 - T - 342 T^{2} - 343 T^{3} + 117649 T^{4} \)
$11$ \( 1 + 63 T + 2638 T^{2} + 83853 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 28 T - 1413 T^{2} - 61516 T^{3} + 4826809 T^{4} \)
$17$ \( ( 1 - 72 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 98 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 126 T + 3709 T^{2} + 1533042 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 126 T - 8513 T^{2} - 3073014 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 259 T + 37290 T^{2} - 7715869 T^{3} + 887503681 T^{4} \)
$37$ \( ( 1 - 386 T + 50653 T^{2} )^{2} \)
$41$ \( 1 - 450 T + 133579 T^{2} - 31014450 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 34 T - 78351 T^{2} - 2703238 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 54 T - 100907 T^{2} - 5606442 T^{3} + 10779215329 T^{4} \)
$53$ \( ( 1 + 693 T + 148877 T^{2} )^{2} \)
$59$ \( 1 + 180 T - 172979 T^{2} + 36968220 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 280 T - 148581 T^{2} - 63554680 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 586 T + 42633 T^{2} - 176247118 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 504 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 161 T + 389017 T^{2} )^{2} \)
$79$ \( 1 + 440 T - 299439 T^{2} + 216937160 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 999 T + 426214 T^{2} + 571215213 T^{3} + 326940373369 T^{4} \)
$89$ \( ( 1 - 882 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 721 T - 392832 T^{2} - 658037233 T^{3} + 832972004929 T^{4} \)
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