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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,10,Mod(1,21)] 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("21.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,9] 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: \(10.8157525594\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2353}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 588 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{2353})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 5) q^{2} - 81 q^{3} + ( - 9 \beta + 101) q^{4} + (70 \beta + 550) q^{5} + (81 \beta - 405) q^{6} - 2401 q^{7} + (375 \beta + 3237) q^{8} + 6561 q^{9} + ( - 270 \beta - 38410) q^{10} + (550 \beta - 73148) q^{11}+ \cdots + (3608550 \beta - 479924028) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} - 162 q^{3} + 193 q^{4} + 1170 q^{5} - 729 q^{6} - 4802 q^{7} + 6849 q^{8} + 13122 q^{9} - 77090 q^{10} - 145746 q^{11} - 15633 q^{12} + 86528 q^{13} - 21609 q^{14} - 94770 q^{15} - 509183 q^{16}+ \cdots - 956239506 q^{99}+O(q^{100}) \) 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
24.7539
−23.7539
−19.7539 −81.0000 −121.785 2282.77 1600.06 −2401.00 12519.7 6561.00 −45093.5
1.2 28.7539 −81.0000 314.785 −1112.77 −2329.06 −2401.00 −5670.70 6561.00 −31996.5
\(n\): 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 21.10.a.b ✓ 2
3.b odd 2 1 63.10.a.c 2
4.b odd 2 1 336.10.a.m 2
7.b odd 2 1 147.10.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.a.b ✓ 2 1.a even 1 1 trivial
63.10.a.c 2 3.b odd 2 1
147.10.a.d 2 7.b odd 2 1
336.10.a.m 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 9T_{2} - 568 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 9T - 568 \) Copy content Toggle raw display
$3$ \( (T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 1170 T - 2540200 \) Copy content Toggle raw display
$7$ \( (T + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 5132528504 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 1864912348 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 288898506792 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 12186222736 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 1016626003344 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 23456377117508 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 22088820805632 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 46813520369772 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 409048772180000 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 660121537363952 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 290934877476096 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 197034132205404 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 86\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 69\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 517610480788736 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 29\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 40\!\cdots\!32 \) Copy content Toggle raw display
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