Properties

Label 324.5.g.d
Level 324
Weight 5
Character orbit 324.g
Analytic conductor 33.492
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \zeta_{12} q^{5} -5 \zeta_{12}^{2} q^{7} +O(q^{10})\) \( q + 9 \zeta_{12} q^{5} -5 \zeta_{12}^{2} q^{7} + ( -117 \zeta_{12} + 117 \zeta_{12}^{3} ) q^{11} + ( 34 - 34 \zeta_{12}^{2} ) q^{13} -450 \zeta_{12}^{3} q^{17} -64 q^{19} + 612 \zeta_{12} q^{23} -544 \zeta_{12}^{2} q^{25} + ( -1062 \zeta_{12} + 1062 \zeta_{12}^{3} ) q^{29} + ( 697 - 697 \zeta_{12}^{2} ) q^{31} -45 \zeta_{12}^{3} q^{35} -748 q^{37} + 684 \zeta_{12} q^{41} -2618 \zeta_{12}^{2} q^{43} + ( 2646 \zeta_{12} - 2646 \zeta_{12}^{3} ) q^{47} + ( 2376 - 2376 \zeta_{12}^{2} ) q^{49} + 1071 \zeta_{12}^{3} q^{53} -1053 q^{55} -5814 \zeta_{12} q^{59} -6404 \zeta_{12}^{2} q^{61} + ( 306 \zeta_{12} - 306 \zeta_{12}^{3} ) q^{65} + ( 5218 - 5218 \zeta_{12}^{2} ) q^{67} + 6570 \zeta_{12}^{3} q^{71} -4519 q^{73} + 585 \zeta_{12} q^{77} -7502 \zeta_{12}^{2} q^{79} + ( 5481 \zeta_{12} - 5481 \zeta_{12}^{3} ) q^{83} + ( 4050 - 4050 \zeta_{12}^{2} ) q^{85} -8874 \zeta_{12}^{3} q^{89} -170 q^{91} -576 \zeta_{12} q^{95} -10571 \zeta_{12}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{7} + O(q^{10}) \) \( 4q - 10q^{7} + 68q^{13} - 256q^{19} - 1088q^{25} + 1394q^{31} - 2992q^{37} - 5236q^{43} + 4752q^{49} - 4212q^{55} - 12808q^{61} + 10436q^{67} - 18076q^{73} - 15004q^{79} + 8100q^{85} - 680q^{91} - 21142q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 −7.79423 4.50000i 0 −2.50000 4.33013i 0 0 0
53.2 0 0 0 7.79423 + 4.50000i 0 −2.50000 4.33013i 0 0 0
269.1 0 0 0 −7.79423 + 4.50000i 0 −2.50000 + 4.33013i 0 0 0
269.2 0 0 0 7.79423 4.50000i 0 −2.50000 + 4.33013i 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 inner
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 324.5.g.d 4
3.b odd 2 1 inner 324.5.g.d 4
9.c even 3 1 108.5.c.c 2
9.c even 3 1 inner 324.5.g.d 4
9.d odd 6 1 108.5.c.c 2
9.d odd 6 1 inner 324.5.g.d 4
36.f odd 6 1 432.5.e.d 2
36.h even 6 1 432.5.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.c.c 2 9.c even 3 1
108.5.c.c 2 9.d odd 6 1
324.5.g.d 4 1.a even 1 1 trivial
324.5.g.d 4 3.b odd 2 1 inner
324.5.g.d 4 9.c even 3 1 inner
324.5.g.d 4 9.d odd 6 1 inner
432.5.e.d 2 36.f odd 6 1
432.5.e.d 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} - 81 T_{5}^{2} + 6561 \)
\( T_{7}^{2} + 5 T_{7} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 1169 T^{2} + 975936 T^{4} + 456640625 T^{6} + 152587890625 T^{8} \)
$7$ \( ( 1 + 5 T - 2376 T^{2} + 12005 T^{3} + 5764801 T^{4} )^{2} \)
$11$ \( 1 + 15593 T^{2} + 28782768 T^{4} + 3342498031433 T^{6} + 45949729863572161 T^{8} \)
$13$ \( ( 1 - 34 T - 27405 T^{2} - 971074 T^{3} + 815730721 T^{4} )^{2} \)
$17$ \( ( 1 + 35458 T^{2} + 6975757441 T^{4} )^{2} \)
$19$ \( ( 1 + 64 T + 130321 T^{2} )^{4} \)
$23$ \( 1 + 185138 T^{2} - 44034906237 T^{4} + 14498339192953778 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} \)
$29$ \( 1 + 286718 T^{2} - 418039201437 T^{4} + 143429651031351998 T^{6} + \)\(25\!\cdots\!21\)\( T^{8} \)
$31$ \( ( 1 - 697 T - 437712 T^{2} - 643694137 T^{3} + 852891037441 T^{4} )^{2} \)
$37$ \( ( 1 + 748 T + 1874161 T^{2} )^{4} \)
$41$ \( 1 + 5183666 T^{2} + 18885467970435 T^{4} + 41391185422736737586 T^{6} + \)\(63\!\cdots\!41\)\( T^{8} \)
$43$ \( ( 1 + 2618 T + 3435123 T^{2} + 8950421018 T^{3} + 11688200277601 T^{4} )^{2} \)
$47$ \( 1 + 2758046 T^{2} - 16204468923645 T^{4} + 65672623932323279006 T^{6} + \)\(56\!\cdots\!21\)\( T^{8} \)
$53$ \( ( 1 - 14633921 T^{2} + 62259690411361 T^{4} )^{2} \)
$59$ \( 1 - 9567874 T^{2} - 55286224724445 T^{4} - \)\(14\!\cdots\!54\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} \)
$61$ \( ( 1 + 6404 T + 27165375 T^{2} + 88668765764 T^{3} + 191707312997281 T^{4} )^{2} \)
$67$ \( ( 1 - 5218 T + 7076403 T^{2} - 105148549378 T^{3} + 406067677556641 T^{4} )^{2} \)
$71$ \( ( 1 - 7658462 T^{2} + 645753531245761 T^{4} )^{2} \)
$73$ \( ( 1 + 4519 T + 28398241 T^{2} )^{4} \)
$79$ \( ( 1 + 7502 T + 17329923 T^{2} + 292203507662 T^{3} + 1517108809906561 T^{4} )^{2} \)
$83$ \( 1 + 64875281 T^{2} + 1956509852689920 T^{4} + \)\(14\!\cdots\!21\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 46736606 T^{2} + 3936588805702081 T^{4} )^{2} \)
$97$ \( ( 1 + 10571 T + 23216760 T^{2} + 935843029451 T^{3} + 7837433594376961 T^{4} )^{2} \)
show more
show less