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Properties

Label 48.3.e.b
Level $48$
Weight $3$
Character orbit 48.e
Analytic conductor $1.308$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [48,3,Mod(17,48)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} + 2 \beta q^{5} + 6 q^{7} + ( - 2 \beta - 7) q^{9} - 2 \beta q^{11} + 10 q^{13} + ( - 2 \beta - 16) q^{15} - 8 \beta q^{17} - 2 q^{19} + (6 \beta - 6) q^{21} + 4 \beta q^{23} - 7 q^{25} + \cdots + (14 \beta - 32) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 12 q^{7} - 14 q^{9} + 20 q^{13} - 32 q^{15} - 4 q^{19} - 12 q^{21} - 14 q^{25} + 46 q^{27} + 44 q^{31} + 32 q^{33} - 12 q^{37} - 20 q^{39} - 164 q^{43} + 64 q^{45} - 26 q^{49} + 128 q^{51}+ \cdots - 64 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
− 1.41421i
1.41421i
0 −1.00000 − 2.82843i 0 − 5.65685i 0 6.00000 0 −7.00000 + 5.65685i 0
17.2 0 −1.00000 + 2.82843i 0 5.65685i 0 6.00000 0 −7.00000 − 5.65685i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.e.b 2
3.b odd 2 1 inner 48.3.e.b 2
4.b odd 2 1 24.3.e.a ✓ 2
5.b even 2 1 1200.3.l.n 2
5.c odd 4 2 1200.3.c.i 4
8.b even 2 1 192.3.e.d 2
8.d odd 2 1 192.3.e.c 2
9.c even 3 2 1296.3.q.e 4
9.d odd 6 2 1296.3.q.e 4
12.b even 2 1 24.3.e.a ✓ 2
15.d odd 2 1 1200.3.l.n 2
15.e even 4 2 1200.3.c.i 4
16.e even 4 2 768.3.h.c 4
16.f odd 4 2 768.3.h.d 4
20.d odd 2 1 600.3.l.b 2
20.e even 4 2 600.3.c.a 4
24.f even 2 1 192.3.e.c 2
24.h odd 2 1 192.3.e.d 2
28.d even 2 1 1176.3.d.a 2
36.f odd 6 2 648.3.m.d 4
36.h even 6 2 648.3.m.d 4
48.i odd 4 2 768.3.h.c 4
48.k even 4 2 768.3.h.d 4
60.h even 2 1 600.3.l.b 2
60.l odd 4 2 600.3.c.a 4
84.h odd 2 1 1176.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a ✓ 2 4.b odd 2 1
24.3.e.a ✓ 2 12.b even 2 1
48.3.e.b 2 1.a even 1 1 trivial
48.3.e.b 2 3.b odd 2 1 inner
192.3.e.c 2 8.d odd 2 1
192.3.e.c 2 24.f even 2 1
192.3.e.d 2 8.b even 2 1
192.3.e.d 2 24.h odd 2 1
600.3.c.a 4 20.e even 4 2
600.3.c.a 4 60.l odd 4 2
600.3.l.b 2 20.d odd 2 1
600.3.l.b 2 60.h even 2 1
648.3.m.d 4 36.f odd 6 2
648.3.m.d 4 36.h even 6 2
768.3.h.c 4 16.e even 4 2
768.3.h.c 4 48.i odd 4 2
768.3.h.d 4 16.f odd 4 2
768.3.h.d 4 48.k even 4 2
1176.3.d.a 2 28.d even 2 1
1176.3.d.a 2 84.h odd 2 1
1200.3.c.i 4 5.c odd 4 2
1200.3.c.i 4 15.e even 4 2
1200.3.l.n 2 5.b even 2 1
1200.3.l.n 2 15.d odd 2 1
1296.3.q.e 4 9.c even 3 2
1296.3.q.e 4 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 32 \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 32 \) Copy content Toggle raw display
$7$ \( (T - 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( (T - 10)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 512 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 128 \) Copy content Toggle raw display
$29$ \( T^{2} + 288 \) Copy content Toggle raw display
$31$ \( (T - 22)^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1152 \) Copy content Toggle raw display
$43$ \( (T + 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4608 \) Copy content Toggle raw display
$53$ \( T^{2} + 3872 \) Copy content Toggle raw display
$59$ \( T^{2} + 5408 \) Copy content Toggle raw display
$61$ \( (T + 86)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 15488 \) Copy content Toggle raw display
$73$ \( (T - 82)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5408 \) Copy content Toggle raw display
$89$ \( T^{2} + 1152 \) Copy content Toggle raw display
$97$ \( (T + 94)^{2} \) Copy content Toggle raw display
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