Properties

Label 48.9.g.b
Level $48$
Weight $9$
Character orbit 48.g
Analytic conductor $19.554$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,9,Mod(31,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.g (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: \(19.5541732829\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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 \(\beta = 9\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta q^{3} + 726 q^{5} - 196 \beta q^{7} - 2187 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta q^{3} + 726 q^{5} - 196 \beta q^{7} - 2187 q^{9} + 852 \beta q^{11} + 39034 q^{13} - 2178 \beta q^{15} - 65814 q^{17} - 8356 \beta q^{19} - 142884 q^{21} - 32208 \beta q^{23} + 136451 q^{25} + 6561 \beta q^{27} + 202062 q^{29} - 76700 \beta q^{31} + 621108 q^{33} - 142296 \beta q^{35} - 1876030 q^{37} - 117102 \beta q^{39} + 3091050 q^{41} - 145228 \beta q^{43} - 1587762 q^{45} + 407784 \beta q^{47} - 3570287 q^{49} + 197442 \beta q^{51} - 1066482 q^{53} + 618552 \beta q^{55} - 6091524 q^{57} - 369732 \beta q^{59} + 17154194 q^{61} + 428652 \beta q^{63} + 28338684 q^{65} + 1759476 \beta q^{67} - 23479632 q^{69} + 2555328 \beta q^{71} - 53286014 q^{73} - 409353 \beta q^{75} + 40579056 q^{77} - 1171996 \beta q^{79} + 4782969 q^{81} - 499668 \beta q^{83} - 47780964 q^{85} - 606186 \beta q^{87} + 86667234 q^{89} - 7650664 \beta q^{91} - 55914300 q^{93} - 6066456 \beta q^{95} - 73901822 q^{97} - 1863324 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1452 q^{5} - 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1452 q^{5} - 4374 q^{9} + 78068 q^{13} - 131628 q^{17} - 285768 q^{21} + 272902 q^{25} + 404124 q^{29} + 1242216 q^{33} - 3752060 q^{37} + 6182100 q^{41} - 3175524 q^{45} - 7140574 q^{49} - 2132964 q^{53} - 12183048 q^{57} + 34308388 q^{61} + 56677368 q^{65} - 46959264 q^{69} - 106572028 q^{73} + 81158112 q^{77} + 9565938 q^{81} - 95561928 q^{85} + 173334468 q^{89} - 111828600 q^{93} - 147803644 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 46.7654i 0 726.000 0 3055.34i 0 −2187.00 0
31.2 0 46.7654i 0 726.000 0 3055.34i 0 −2187.00 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.9.g.b 2
3.b odd 2 1 144.9.g.c 2
4.b odd 2 1 inner 48.9.g.b 2
8.b even 2 1 192.9.g.a 2
8.d odd 2 1 192.9.g.a 2
12.b even 2 1 144.9.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.g.b 2 1.a even 1 1 trivial
48.9.g.b 2 4.b odd 2 1 inner
144.9.g.c 2 3.b odd 2 1
144.9.g.c 2 12.b even 2 1
192.9.g.a 2 8.b even 2 1
192.9.g.a 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 726 \) acting on \(S_{9}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2187 \) Copy content Toggle raw display
$5$ \( (T - 726)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9335088 \) Copy content Toggle raw display
$11$ \( T^{2} + 176394672 \) Copy content Toggle raw display
$13$ \( (T - 39034)^{2} \) Copy content Toggle raw display
$17$ \( (T + 65814)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16966924848 \) Copy content Toggle raw display
$23$ \( T^{2} + 252077329152 \) Copy content Toggle raw display
$29$ \( (T - 202062)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1429542270000 \) Copy content Toggle raw display
$37$ \( (T + 1876030)^{2} \) Copy content Toggle raw display
$41$ \( (T - 3091050)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5125154792112 \) Copy content Toggle raw display
$47$ \( T^{2} + 40407933129408 \) Copy content Toggle raw display
$53$ \( (T + 1066482)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 33218525693232 \) Copy content Toggle raw display
$61$ \( (T - 17154194)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 752268658081968 \) Copy content Toggle raw display
$71$ \( T^{2} + 15\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( (T + 53286014)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 333778633635888 \) Copy content Toggle raw display
$83$ \( T^{2} + 60669350784432 \) Copy content Toggle raw display
$89$ \( (T - 86667234)^{2} \) Copy content Toggle raw display
$97$ \( (T + 73901822)^{2} \) Copy content Toggle raw display
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