Properties

Label 54.9.b.c
Level $54$
Weight $9$
Character orbit 54.b
Analytic conductor $21.998$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,9,Mod(53,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 54.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.9984449433\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} - 128 q^{4} + (51 \beta_{2} - 11 \beta_1) q^{5} + ( - 7 \beta_{3} + 77) q^{7} + 256 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} - 128 q^{4} + (51 \beta_{2} - 11 \beta_1) q^{5} + ( - 7 \beta_{3} + 77) q^{7} + 256 \beta_{2} q^{8} + ( - 22 \beta_{3} + 3264) q^{10} + ( - 870 \beta_{2} + 409 \beta_1) q^{11} + (106 \beta_{3} - 11170) q^{13} + ( - 154 \beta_{2} + 448 \beta_1) q^{14} + 16384 q^{16} + ( - 324 \beta_{2} + 3098 \beta_1) q^{17} + (946 \beta_{3} - 14128) q^{19} + ( - 6528 \beta_{2} + 1408 \beta_1) q^{20} + (818 \beta_{3} - 55680) q^{22} + (27222 \beta_{2} + 1220 \beta_1) q^{23} + (1122 \beta_{3} + 219184) q^{25} + (22340 \beta_{2} - 6784 \beta_1) q^{26} + (896 \beta_{3} - 9856) q^{28} + ( - 85848 \beta_{2} - 2 \beta_1) q^{29} + (7305 \beta_{3} - 593857) q^{31} - 32768 \beta_{2} q^{32} + (6196 \beta_{3} - 20736) q^{34} + (60060 \beta_{2} - 12271 \beta_1) q^{35} + (14618 \beta_{3} + 1299332) q^{37} + (28256 \beta_{2} - 60544 \beta_1) q^{38} + (2816 \beta_{3} - 417792) q^{40} + (54870 \beta_{2} - 28052 \beta_1) q^{41} + (28038 \beta_{3} - 688774) q^{43} + (111360 \beta_{2} - 52352 \beta_1) q^{44} + (2440 \beta_{3} + 1742208) q^{46} + ( - 17268 \beta_{2} + 136770 \beta_1) q^{47} + ( - 1078 \beta_{3} - 4615800) q^{49} + ( - 438368 \beta_{2} - 71808 \beta_1) q^{50} + ( - 13568 \beta_{3} + 1429760) q^{52} + (344133 \beta_{2} + 379789 \beta_1) q^{53} + ( - 30429 \beta_{3} + 4699611) q^{55} + (19712 \beta_{2} - 57344 \beta_1) q^{56} + ( - 4 \beta_{3} - 5494272) q^{58} + ( - 1017804 \beta_{2} + 455762 \beta_1) q^{59} + ( - 95208 \beta_{3} + 9210596) q^{61} + (1187714 \beta_{2} - 467520 \beta_1) q^{62} - 2097152 q^{64} + ( - 1419684 \beta_{2} + 295862 \beta_1) q^{65} + ( - 61242 \beta_{3} - 13042594) q^{67} + (41472 \beta_{2} - 396544 \beta_1) q^{68} + ( - 24542 \beta_{3} + 3843840) q^{70} + (6071274 \beta_{2} + 38798 \beta_1) q^{71} + ( - 145926 \beta_{3} + 5610233) q^{73} + ( - 2598664 \beta_{2} - 935552 \beta_1) q^{74} + ( - 121088 \beta_{3} + 1808384) q^{76} + ( - 2154117 \beta_{2} + 226373 \beta_1) q^{77} + ( - 138764 \beta_{3} - 12786610) q^{79} + (835584 \beta_{2} - 180224 \beta_1) q^{80} + ( - 56104 \beta_{3} + 3511680) q^{82} + ( - 11181894 \beta_{2} + 462483 \beta_1) q^{83} + ( - 161562 \beta_{3} + 25371630) q^{85} + (1377548 \beta_{2} - 1794432 \beta_1) q^{86} + ( - 104704 \beta_{3} + 7127040) q^{88} + (17830278 \beta_{2} + 114002 \beta_1) q^{89} + (86352 \beta_{3} - 18169466) q^{91} + ( - 3484416 \beta_{2} - 156160 \beta_1) q^{92} + (273540 \beta_{3} - 1105152) q^{94} + ( - 8306502 \beta_{2} + 1699280 \beta_1) q^{95} + (487336 \beta_{3} - 67338805) q^{97} + (9231600 \beta_{2} + 68992 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 512 q^{4} + 308 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 512 q^{4} + 308 q^{7} + 13056 q^{10} - 44680 q^{13} + 65536 q^{16} - 56512 q^{19} - 222720 q^{22} + 876736 q^{25} - 39424 q^{28} - 2375428 q^{31} - 82944 q^{34} + 5197328 q^{37} - 1671168 q^{40} - 2755096 q^{43} + 6968832 q^{46} - 18463200 q^{49} + 5719040 q^{52} + 18798444 q^{55} - 21977088 q^{58} + 36842384 q^{61} - 8388608 q^{64} - 52170376 q^{67} + 15375360 q^{70} + 22440932 q^{73} + 7233536 q^{76} - 51146440 q^{79} + 14046720 q^{82} + 101486520 q^{85} + 28508160 q^{88} - 72677864 q^{91} - 4420608 q^{94} - 269355220 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 27\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{8}^{3} + 4\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -108\zeta_{8}^{3} + 108\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + 27\beta_{2} ) / 216 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 27 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + 27\beta_{2} ) / 216 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
11.3137i 0 −128.000 8.50043i 0 −992.145 1448.15i 0 −96.1714
53.2 11.3137i 0 −128.000 585.500i 0 1146.15 1448.15i 0 6624.17
53.3 11.3137i 0 −128.000 585.500i 0 1146.15 1448.15i 0 6624.17
53.4 11.3137i 0 −128.000 8.50043i 0 −992.145 1448.15i 0 −96.1714
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.9.b.c 4
3.b odd 2 1 inner 54.9.b.c 4
4.b odd 2 1 432.9.e.h 4
9.c even 3 2 162.9.d.e 8
9.d odd 6 2 162.9.d.e 8
12.b even 2 1 432.9.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.c 4 1.a even 1 1 trivial
54.9.b.c 4 3.b odd 2 1 inner
162.9.d.e 8 9.c even 3 2
162.9.d.e 8 9.d odd 6 2
432.9.e.h 4 4.b odd 2 1
432.9.e.h 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 342882T_{5}^{2} + 24770529 \) acting on \(S_{9}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 342882 T^{2} + 24770529 \) Copy content Toggle raw display
$7$ \( (T^{2} - 154 T - 1137143)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 95\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{2} + 22340 T - 137344508)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{2} + 28256 T - 20677000064)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 55\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{2} + 1187714 T - 892186510751)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 3296601588848)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 17864418046556)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 126623053147376)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 82615668195844)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 465280990005839)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 285693688560188)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 10\!\cdots\!63)^{2} \) Copy content Toggle raw display
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