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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [48,13,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, 13); 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, 13, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1350] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(43.8717032293\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-26}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
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 \(\beta = 18\sqrt{-26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta + 675) q^{3} + 230 \beta q^{5} - 40250 q^{7} + (4050 \beta + 379809) q^{9} + 12650 \beta q^{11} + 1284050 q^{13} + (155250 \beta - 5812560) q^{15} + 161736 \beta q^{17} - 53343578 q^{19} + ( - 120750 \beta - 27168750) q^{21}+ \cdots + (4804583850 \beta - 431582580000) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1350 q^{3} - 80500 q^{7} + 759618 q^{9} + 2568100 q^{13} - 11625120 q^{15} - 106687156 q^{19} - 54337500 q^{21} - 402977950 q^{25} + 308038950 q^{27} - 133052404 q^{31} - 639381600 q^{33} + 4457452900 q^{37}+ \cdots - 863165160000 q^{99}+O(q^{100}) \) 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
− 5.09902i
5.09902i
0 675.000 − 275.347i 0 − 21109.9i 0 −40250.0 0 379809. − 371719.i 0
17.2 0 675.000 + 275.347i 0 21109.9i 0 −40250.0 0 379809. + 371719.i 0
\(n\): e.g. 2-40 or 80-90
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 48.13.e.b 2
3.b odd 2 1 inner 48.13.e.b 2
4.b odd 2 1 3.13.b.b ✓ 2
8.b even 2 1 192.13.e.c 2
8.d odd 2 1 192.13.e.d 2
12.b even 2 1 3.13.b.b ✓ 2
20.d odd 2 1 75.13.c.c 2
20.e even 4 2 75.13.d.b 4
24.f even 2 1 192.13.e.d 2
24.h odd 2 1 192.13.e.c 2
36.f odd 6 2 81.13.d.c 4
36.h even 6 2 81.13.d.c 4
60.h even 2 1 75.13.c.c 2
60.l odd 4 2 75.13.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.b ✓ 2 4.b odd 2 1
3.13.b.b ✓ 2 12.b even 2 1
48.13.e.b 2 1.a even 1 1 trivial
48.13.e.b 2 3.b odd 2 1 inner
75.13.c.c 2 20.d odd 2 1
75.13.c.c 2 60.h even 2 1
75.13.d.b 4 20.e even 4 2
75.13.d.b 4 60.l odd 4 2
81.13.d.c 4 36.f odd 6 2
81.13.d.c 4 36.h even 6 2
192.13.e.c 2 8.b even 2 1
192.13.e.c 2 24.h odd 2 1
192.13.e.d 2 8.d odd 2 1
192.13.e.d 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 445629600 \) acting on \(S_{13}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 1350 T + 531441 \) Copy content Toggle raw display
$5$ \( T^{2} + 445629600 \) Copy content Toggle raw display
$7$ \( (T + 40250)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1348029540000 \) Copy content Toggle raw display
$13$ \( (T - 1284050)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 220359487855104 \) Copy content Toggle raw display
$19$ \( (T + 53343578)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{2} + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T + 66526202)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2228726450)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 67\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T + 8977216250)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + 16\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T + 40679935918)^{2} \) Copy content Toggle raw display
$67$ \( (T + 121176846650)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T + 60956187550)^{2} \) Copy content Toggle raw display
$79$ \( (T - 252324997702)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T - 653817778850)^{2} \) Copy content Toggle raw display
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