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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,10,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.4472576783\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 15) q^{2} + (30 \beta + 58) q^{4} + ( - 70 \beta - 564) q^{5} + 2401 q^{7} + (4 \beta - 3540) q^{8} + (1614 \beta + 32610) q^{10} + ( - 1718 \beta - 36642) q^{11} + (528 \beta + 70550) q^{13}+ \cdots + ( - 5764801 \beta - 86472015) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 30 q^{2} + 116 q^{4} - 1128 q^{5} + 4802 q^{7} - 7080 q^{8} + 65220 q^{10} - 73284 q^{11} + 141100 q^{13} - 72030 q^{14} + 44048 q^{16} + 101784 q^{17} + 481744 q^{19} - 1514424 q^{20} + 2284680 q^{22}+ \cdots - 172944030 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.78709
−8.78709
−33.5742 0 615.225 −1864.19 0 2401.00 −3465.70 0 62588.7
1.2 3.57418 0 −499.225 736.192 0 2401.00 −3614.30 0 2631.28
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.10.a.b 2
3.b odd 2 1 21.10.a.c ✓ 2
12.b even 2 1 336.10.a.l 2
21.c even 2 1 147.10.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.a.c ✓ 2 3.b odd 2 1
63.10.a.b 2 1.a even 1 1 trivial
147.10.a.e 2 21.c even 2 1
336.10.a.l 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 30T_{2} - 120 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 30T - 120 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1128 T - 1372404 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 73284 T + 324360384 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 4881122020 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 73285474836 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 55133856304 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 2917833651264 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 5355274808556 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 7175316130304 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 1422306192956 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 579780377334684 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 6134871382480 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 48\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 89\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 17\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 99\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 39\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 54\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 49\!\cdots\!84 \) Copy content Toggle raw display
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