Properties

Label 324.4.e.h
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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
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.

\(f(q)\) \(=\) \( q + 18 \zeta_{6} q^{5} + ( -8 + 8 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 18 \zeta_{6} q^{5} + ( -8 + 8 \zeta_{6} ) q^{7} + ( -36 + 36 \zeta_{6} ) q^{11} + 10 \zeta_{6} q^{13} + 18 q^{17} -100 q^{19} -72 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} + ( 234 - 234 \zeta_{6} ) q^{29} + 16 \zeta_{6} q^{31} -144 q^{35} -226 q^{37} -90 \zeta_{6} q^{41} + ( -452 + 452 \zeta_{6} ) q^{43} + ( -432 + 432 \zeta_{6} ) q^{47} + 279 \zeta_{6} q^{49} + 414 q^{53} -648 q^{55} + 684 \zeta_{6} q^{59} + ( -422 + 422 \zeta_{6} ) q^{61} + ( -180 + 180 \zeta_{6} ) q^{65} -332 \zeta_{6} q^{67} -360 q^{71} + 26 q^{73} -288 \zeta_{6} q^{77} + ( -512 + 512 \zeta_{6} ) q^{79} + ( 1188 - 1188 \zeta_{6} ) q^{83} + 324 \zeta_{6} q^{85} -630 q^{89} -80 q^{91} -1800 \zeta_{6} q^{95} + ( 1054 - 1054 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 18q^{5} - 8q^{7} + O(q^{10}) \) \( 2q + 18q^{5} - 8q^{7} - 36q^{11} + 10q^{13} + 36q^{17} - 200q^{19} - 72q^{23} - 199q^{25} + 234q^{29} + 16q^{31} - 288q^{35} - 452q^{37} - 90q^{41} - 452q^{43} - 432q^{47} + 279q^{49} + 828q^{53} - 1296q^{55} + 684q^{59} - 422q^{61} - 180q^{65} - 332q^{67} - 720q^{71} + 52q^{73} - 288q^{77} - 512q^{79} + 1188q^{83} + 324q^{85} - 1260q^{89} - 160q^{91} - 1800q^{95} + 1054q^{97} + O(q^{100}) \)

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 9.00000 15.5885i 0 −4.00000 6.92820i 0 0 0
217.1 0 0 0 9.00000 + 15.5885i 0 −4.00000 + 6.92820i 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.h 2
3.b odd 2 1 324.4.e.a 2
9.c even 3 1 12.4.a.a 1
9.c even 3 1 inner 324.4.e.h 2
9.d odd 6 1 36.4.a.a 1
9.d odd 6 1 324.4.e.a 2
36.f odd 6 1 48.4.a.a 1
36.h even 6 1 144.4.a.g 1
45.h odd 6 1 900.4.a.g 1
45.j even 6 1 300.4.a.b 1
45.k odd 12 2 300.4.d.e 2
45.l even 12 2 900.4.d.c 2
63.g even 3 1 588.4.i.d 2
63.h even 3 1 588.4.i.d 2
63.i even 6 1 1764.4.k.o 2
63.j odd 6 1 1764.4.k.b 2
63.k odd 6 1 588.4.i.e 2
63.l odd 6 1 588.4.a.c 1
63.n odd 6 1 1764.4.k.b 2
63.o even 6 1 1764.4.a.b 1
63.s even 6 1 1764.4.k.o 2
63.t odd 6 1 588.4.i.e 2
72.j odd 6 1 576.4.a.b 1
72.l even 6 1 576.4.a.a 1
72.n even 6 1 192.4.a.f 1
72.p odd 6 1 192.4.a.l 1
99.h odd 6 1 1452.4.a.d 1
117.t even 6 1 2028.4.a.c 1
117.y odd 12 2 2028.4.b.c 2
144.v odd 12 2 768.4.d.j 2
144.x even 12 2 768.4.d.g 2
180.p odd 6 1 1200.4.a.be 1
180.x even 12 2 1200.4.f.d 2
252.bi even 6 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 9.c even 3 1
36.4.a.a 1 9.d odd 6 1
48.4.a.a 1 36.f odd 6 1
144.4.a.g 1 36.h even 6 1
192.4.a.f 1 72.n even 6 1
192.4.a.l 1 72.p odd 6 1
300.4.a.b 1 45.j even 6 1
300.4.d.e 2 45.k odd 12 2
324.4.e.a 2 3.b odd 2 1
324.4.e.a 2 9.d odd 6 1
324.4.e.h 2 1.a even 1 1 trivial
324.4.e.h 2 9.c even 3 1 inner
576.4.a.a 1 72.l even 6 1
576.4.a.b 1 72.j odd 6 1
588.4.a.c 1 63.l odd 6 1
588.4.i.d 2 63.g even 3 1
588.4.i.d 2 63.h even 3 1
588.4.i.e 2 63.k odd 6 1
588.4.i.e 2 63.t odd 6 1
768.4.d.g 2 144.x even 12 2
768.4.d.j 2 144.v odd 12 2
900.4.a.g 1 45.h odd 6 1
900.4.d.c 2 45.l even 12 2
1200.4.a.be 1 180.p odd 6 1
1200.4.f.d 2 180.x even 12 2
1452.4.a.d 1 99.h odd 6 1
1764.4.a.b 1 63.o even 6 1
1764.4.k.b 2 63.j odd 6 1
1764.4.k.b 2 63.n odd 6 1
1764.4.k.o 2 63.i even 6 1
1764.4.k.o 2 63.s even 6 1
2028.4.a.c 1 117.t even 6 1
2028.4.b.c 2 117.y odd 12 2
2352.4.a.bk 1 252.bi even 6 1

Hecke kernels

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

\( T_{5}^{2} - 18 T_{5} + 324 \)
\( T_{7}^{2} + 8 T_{7} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 324 - 18 T + T^{2} \)
$7$ \( 64 + 8 T + T^{2} \)
$11$ \( 1296 + 36 T + T^{2} \)
$13$ \( 100 - 10 T + T^{2} \)
$17$ \( ( -18 + T )^{2} \)
$19$ \( ( 100 + T )^{2} \)
$23$ \( 5184 + 72 T + T^{2} \)
$29$ \( 54756 - 234 T + T^{2} \)
$31$ \( 256 - 16 T + T^{2} \)
$37$ \( ( 226 + T )^{2} \)
$41$ \( 8100 + 90 T + T^{2} \)
$43$ \( 204304 + 452 T + T^{2} \)
$47$ \( 186624 + 432 T + T^{2} \)
$53$ \( ( -414 + T )^{2} \)
$59$ \( 467856 - 684 T + T^{2} \)
$61$ \( 178084 + 422 T + T^{2} \)
$67$ \( 110224 + 332 T + T^{2} \)
$71$ \( ( 360 + T )^{2} \)
$73$ \( ( -26 + T )^{2} \)
$79$ \( 262144 + 512 T + T^{2} \)
$83$ \( 1411344 - 1188 T + T^{2} \)
$89$ \( ( 630 + T )^{2} \)
$97$ \( 1110916 - 1054 T + T^{2} \)
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