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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [84,6,Mod(1,84)] 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("84.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 84.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,18] 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: yes
Analytic conductor: \(13.4722408643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{505}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\beta = 3\sqrt{505}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( - \beta + 39) q^{5} + 49 q^{7} + 81 q^{9} + (5 \beta + 87) q^{11} + (6 \beta + 104) q^{13} + ( - 9 \beta + 351) q^{15} + ( - 19 \beta + 741) q^{17} + (36 \beta + 176) q^{19} + 441 q^{21}+ \cdots + (405 \beta + 7047) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9} + 174 q^{11} + 208 q^{13} + 702 q^{15} + 1482 q^{17} + 352 q^{19} + 882 q^{21} + 3354 q^{23} + 5882 q^{25} + 1458 q^{27} + 276 q^{29} + 6520 q^{31} + 1566 q^{33}+ \cdots + 14094 q^{99}+O(q^{100}) \) 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
11.7361
−10.7361
0 9.00000 0 −28.4166 0 49.0000 0 81.0000 0
1.2 0 9.00000 0 106.417 0 49.0000 0 81.0000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 84.6.a.d ✓ 2
3.b odd 2 1 252.6.a.e 2
4.b odd 2 1 336.6.a.u 2
7.b odd 2 1 588.6.a.g 2
7.c even 3 2 588.6.i.h 4
7.d odd 6 2 588.6.i.n 4
12.b even 2 1 1008.6.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.d ✓ 2 1.a even 1 1 trivial
252.6.a.e 2 3.b odd 2 1
336.6.a.u 2 4.b odd 2 1
588.6.a.g 2 7.b odd 2 1
588.6.i.h 4 7.c even 3 2
588.6.i.n 4 7.d odd 6 2
1008.6.a.bf 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 78T_{5} - 3024 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(84))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 78T - 3024 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 174T - 106056 \) Copy content Toggle raw display
$13$ \( T^{2} - 208T - 152804 \) Copy content Toggle raw display
$17$ \( T^{2} - 1482 T - 1091664 \) Copy content Toggle raw display
$19$ \( T^{2} - 352 T - 5859344 \) Copy content Toggle raw display
$23$ \( T^{2} - 3354 T + 807984 \) Copy content Toggle raw display
$29$ \( T^{2} - 276 T - 45430956 \) Copy content Toggle raw display
$31$ \( T^{2} - 6520 T + 4737280 \) Copy content Toggle raw display
$37$ \( T^{2} - 13864 T + 43962124 \) Copy content Toggle raw display
$41$ \( T^{2} + 12930 T + 41246280 \) Copy content Toggle raw display
$43$ \( T^{2} - 12712 T - 25049264 \) Copy content Toggle raw display
$47$ \( T^{2} + 28116 T + 182337984 \) Copy content Toggle raw display
$53$ \( T^{2} + 46992 T + 492995196 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1037582784 \) Copy content Toggle raw display
$61$ \( T^{2} + 13148 T - 365832524 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1367827744 \) Copy content Toggle raw display
$71$ \( T^{2} + 66042 T + 931452336 \) Copy content Toggle raw display
$73$ \( T^{2} - 60496 T + 842785084 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2170625216 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 1671979536 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 6775324920 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 11297373964 \) Copy content Toggle raw display
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