Properties

Label 324.4.e.e
Level 324
Weight 4
Character orbit 324.e
Analytic conductor 19.117
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

Related objects

Downloads

Learn more about

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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 37 - 37 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 37 - 37 \zeta_{6} ) q^{7} + 19 \zeta_{6} q^{13} -163 q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} -308 \zeta_{6} q^{31} + 323 q^{37} + ( 520 - 520 \zeta_{6} ) q^{43} -1026 \zeta_{6} q^{49} + ( -719 + 719 \zeta_{6} ) q^{61} + 127 \zeta_{6} q^{67} -919 q^{73} + ( 1387 - 1387 \zeta_{6} ) q^{79} + 703 q^{91} + ( 523 - 523 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 37q^{7} + O(q^{10}) \) \( 2q + 37q^{7} + 19q^{13} - 326q^{19} + 125q^{25} - 308q^{31} + 646q^{37} + 520q^{43} - 1026q^{49} - 719q^{61} + 127q^{67} - 1838q^{73} + 1387q^{79} + 1406q^{91} + 523q^{97} + O(q^{100}) \)

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 0 0 18.5000 + 32.0429i 0 0 0
217.1 0 0 0 0 0 18.5000 32.0429i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.e.e 2
3.b odd 2 1 CM 324.4.e.e 2
9.c even 3 1 108.4.a.b 1
9.c even 3 1 inner 324.4.e.e 2
9.d odd 6 1 108.4.a.b 1
9.d odd 6 1 inner 324.4.e.e 2
36.f odd 6 1 432.4.a.h 1
36.h even 6 1 432.4.a.h 1
72.j odd 6 1 1728.4.a.o 1
72.l even 6 1 1728.4.a.r 1
72.n even 6 1 1728.4.a.o 1
72.p odd 6 1 1728.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.b 1 9.c even 3 1
108.4.a.b 1 9.d odd 6 1
324.4.e.e 2 1.a even 1 1 trivial
324.4.e.e 2 3.b odd 2 1 CM
324.4.e.e 2 9.c even 3 1 inner
324.4.e.e 2 9.d odd 6 1 inner
432.4.a.h 1 36.f odd 6 1
432.4.a.h 1 36.h even 6 1
1728.4.a.o 1 72.j odd 6 1
1728.4.a.o 1 72.n even 6 1
1728.4.a.r 1 72.l even 6 1
1728.4.a.r 1 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 \(S_{4}^{\mathrm{new}}(324, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} - 37 T_{7} + 1369 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 125 T^{2} + 15625 T^{4} \)
$7$ \( ( 1 - 20 T + 343 T^{2} )( 1 - 17 T + 343 T^{2} ) \)
$11$ \( 1 - 1331 T^{2} + 1771561 T^{4} \)
$13$ \( ( 1 - 89 T + 2197 T^{2} )( 1 + 70 T + 2197 T^{2} ) \)
$17$ \( ( 1 + 4913 T^{2} )^{2} \)
$19$ \( ( 1 + 163 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 12167 T^{2} + 148035889 T^{4} \)
$29$ \( 1 - 24389 T^{2} + 594823321 T^{4} \)
$31$ \( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \)
$37$ \( ( 1 - 323 T + 50653 T^{2} )^{2} \)
$41$ \( 1 - 68921 T^{2} + 4750104241 T^{4} \)
$43$ \( ( 1 - 449 T + 79507 T^{2} )( 1 - 71 T + 79507 T^{2} ) \)
$47$ \( 1 - 103823 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 + 148877 T^{2} )^{2} \)
$59$ \( 1 - 205379 T^{2} + 42180533641 T^{4} \)
$61$ \( ( 1 - 182 T + 226981 T^{2} )( 1 + 901 T + 226981 T^{2} ) \)
$67$ \( ( 1 - 1007 T + 300763 T^{2} )( 1 + 880 T + 300763 T^{2} ) \)
$71$ \( ( 1 + 357911 T^{2} )^{2} \)
$73$ \( ( 1 + 919 T + 389017 T^{2} )^{2} \)
$79$ \( ( 1 - 884 T + 493039 T^{2} )( 1 - 503 T + 493039 T^{2} ) \)
$83$ \( 1 - 571787 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 + 704969 T^{2} )^{2} \)
$97$ \( ( 1 - 1853 T + 912673 T^{2} )( 1 + 1330 T + 912673 T^{2} ) \)
show more
show less