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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,5,Mod(97,784)] 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("784.97"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 784.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-180,0,270] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.0420510577\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 28)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - 30) q^{9} + (\beta_{4} + 45) q^{11} + (\beta_{3} + 21 \beta_{2} + 5 \beta_1) q^{13} + ( - 2 \beta_{5} - 3 \beta_{4} + 81) q^{15} + (\beta_{3} + \beta_{2} + 18 \beta_1) q^{17}+ \cdots + (3 \beta_{5} + 12 \beta_{4} - 1350) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 180 q^{9} + 270 q^{11} + 486 q^{15} + 486 q^{23} - 3756 q^{25} - 540 q^{29} - 4710 q^{37} + 13176 q^{39} + 948 q^{43} - 1782 q^{51} - 12582 q^{53} - 6894 q^{57} - 15336 q^{65} + 3318 q^{67} - 2268 q^{71}+ \cdots - 8100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 18x^{4} + 81x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{3} + 63\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 13\nu^{3} + 18\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{5} + 131\nu^{3} + 456\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{4} + 147\nu^{2} - 252 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\nu^{4} + 231\nu^{2} + 252 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} - 9\beta_{2} - \beta_1 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} - 252 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 27\beta_{2} + 7\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} + 22\beta_{4} + 2268 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33\beta_{3} - 241\beta_{2} - 85\beta_1 ) / 14 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
− 2.78499i
− 3.17656i
0.391571i
− 0.391571i
3.17656i
2.78499i
0 − 15.7854i 0 28.7598i 0 0 0 −168.178 0
97.2 0 − 9.01997i 0 − 31.0914i 0 0 0 −0.359775 0
97.3 0 − 1.56925i 0 44.2628i 0 0 0 78.5375 0
97.4 0 1.56925i 0 − 44.2628i 0 0 0 78.5375 0
97.5 0 9.01997i 0 31.0914i 0 0 0 −0.359775 0
97.6 0 15.7854i 0 − 28.7598i 0 0 0 −168.178 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 97.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.5.c.e 6
4.b odd 2 1 196.5.b.a 6
7.b odd 2 1 inner 784.5.c.e 6
7.c even 3 1 112.5.s.c 6
7.d odd 6 1 112.5.s.c 6
28.d even 2 1 196.5.b.a 6
28.f even 6 1 28.5.h.a ✓ 6
28.f even 6 1 196.5.h.c 6
28.g odd 6 1 28.5.h.a ✓ 6
28.g odd 6 1 196.5.h.c 6
84.j odd 6 1 252.5.z.f 6
84.n even 6 1 252.5.z.f 6
140.p odd 6 1 700.5.s.a 6
140.s even 6 1 700.5.s.a 6
140.w even 12 2 700.5.o.a 12
140.x odd 12 2 700.5.o.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.5.h.a ✓ 6 28.f even 6 1
28.5.h.a ✓ 6 28.g odd 6 1
112.5.s.c 6 7.c even 3 1
112.5.s.c 6 7.d odd 6 1
196.5.b.a 6 4.b odd 2 1
196.5.b.a 6 28.d even 2 1
196.5.h.c 6 28.f even 6 1
196.5.h.c 6 28.g odd 6 1
252.5.z.f 6 84.j odd 6 1
252.5.z.f 6 84.n even 6 1
700.5.o.a 12 140.w even 12 2
700.5.o.a 12 140.x odd 12 2
700.5.s.a 6 140.p odd 6 1
700.5.s.a 6 140.s even 6 1
784.5.c.e 6 1.a even 1 1 trivial
784.5.c.e 6 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 333T_{3}^{4} + 21087T_{3}^{2} + 49923 \) acting on \(S_{5}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 333 T^{4} + \cdots + 49923 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 1566504603 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 135 T^{2} + \cdots + 43821)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 65933832204288 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 86645980184427 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 163848759776883 \) Copy content Toggle raw display
$23$ \( (T^{3} - 243 T^{2} + \cdots - 43118811)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 270 T^{2} + \cdots + 401089968)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 81\!\cdots\!63 \) Copy content Toggle raw display
$37$ \( (T^{3} + 2355 T^{2} + \cdots - 1396235791)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 30\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{3} - 474 T^{2} + \cdots - 2925826856)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 12\!\cdots\!27 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6291 T^{2} + \cdots - 1861224399)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 40\!\cdots\!87 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 15\!\cdots\!47 \) Copy content Toggle raw display
$67$ \( (T^{3} - 1659 T^{2} + \cdots - 5511616579)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 1134 T^{2} + \cdots - 66080643048)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 51\!\cdots\!67 \) Copy content Toggle raw display
$79$ \( (T^{3} + 7773 T^{2} + \cdots - 57207887659)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 12\!\cdots\!07 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
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