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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [56,5,Mod(41,56)] 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("56.41"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 56.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.78871793270\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 112x^{6} + 3122x^{4} + 21888x^{2} + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{5} + 7) q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} - 31) q^{9} + (\beta_{6} - \beta_{5} - 18) q^{11} + (\beta_{7} - \beta_{5} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - 53 \beta_{6} - 25 \beta_{5} + \cdots + 9422) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 56 q^{7} - 248 q^{9} - 144 q^{11} + 64 q^{15} - 320 q^{21} + 1200 q^{23} + 648 q^{25} - 1392 q^{29} - 384 q^{35} - 496 q^{37} - 704 q^{39} + 5872 q^{43} + 5896 q^{49} - 4992 q^{51} + 1680 q^{53} - 9472 q^{57}+ \cdots + 75376 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 112x^{6} + 3122x^{4} + 21888x^{2} + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -40\nu^{7} + 8372\nu^{5} + 1111180\nu^{3} + 20424402\nu ) / 892269 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -104\nu^{6} - 12224\nu^{4} - 340096\nu^{2} - 1231488 ) / 42489 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2992 \nu^{7} - 6972 \nu^{6} + 325528 \nu^{5} - 682206 \nu^{4} + 8490020 \nu^{3} - 13327734 \nu^{2} + \cdots - 33550902 ) / 892269 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2992 \nu^{7} + 6972 \nu^{6} + 325528 \nu^{5} + 682206 \nu^{4} + 8490020 \nu^{3} + 13327734 \nu^{2} + \cdots + 33550902 ) / 892269 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2992 \nu^{7} - 4788 \nu^{6} + 325528 \nu^{5} - 425502 \nu^{4} + 8490020 \nu^{3} - 2616642 \nu^{2} + \cdots + 92244474 ) / 892269 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -16336\nu^{7} - 1815520\nu^{5} - 48244112\nu^{3} - 246346956\nu ) / 892269 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} - 112 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} + 11\beta_{5} + 11\beta_{4} + 12\beta_{2} - 218\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -69\beta_{6} - 13\beta_{5} + 82\beta_{4} - 152\beta_{3} + 6300 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -385\beta_{7} - 1055\beta_{5} - 1055\beta_{4} - 594\beta_{2} + 14346\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9680\beta_{6} + 3056\beta_{5} - 12736\beta_{4} + 25923\beta_{3} - 843200 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 61075\beta_{7} + 169526\beta_{5} + 169526\beta_{4} + 61152\beta_{2} - 2021746\beta_1 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
− 8.57711i
− 5.31877i
− 3.05383i
− 0.904426i
0.904426i
3.05383i
5.31877i
8.57711i
0 − 17.1542i 0 − 0.823997i 0 41.3762 − 26.2490i 0 −213.267 0
41.2 0 − 10.6375i 0 − 20.2944i 0 −43.0244 + 23.4500i 0 −32.1573 0
41.3 0 − 6.10767i 0 37.0396i 0 48.5391 − 6.70476i 0 43.6964 0
41.4 0 − 1.80885i 0 19.7871i 0 −18.8909 + 45.2121i 0 77.7281 0
41.5 0 1.80885i 0 − 19.7871i 0 −18.8909 − 45.2121i 0 77.7281 0
41.6 0 6.10767i 0 − 37.0396i 0 48.5391 + 6.70476i 0 43.6964 0
41.7 0 10.6375i 0 20.2944i 0 −43.0244 − 23.4500i 0 −32.1573 0
41.8 0 17.1542i 0 0.823997i 0 41.3762 + 26.2490i 0 −213.267 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 41.8
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 56.5.c.a ✓ 8
3.b odd 2 1 504.5.f.a 8
4.b odd 2 1 112.5.c.d 8
7.b odd 2 1 inner 56.5.c.a ✓ 8
7.c even 3 2 392.5.o.b 16
7.d odd 6 2 392.5.o.b 16
8.b even 2 1 448.5.c.h 8
8.d odd 2 1 448.5.c.g 8
12.b even 2 1 1008.5.f.l 8
21.c even 2 1 504.5.f.a 8
28.d even 2 1 112.5.c.d 8
56.e even 2 1 448.5.c.g 8
56.h odd 2 1 448.5.c.h 8
84.h odd 2 1 1008.5.f.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.5.c.a ✓ 8 1.a even 1 1 trivial
56.5.c.a ✓ 8 7.b odd 2 1 inner
112.5.c.d 8 4.b odd 2 1
112.5.c.d 8 28.d even 2 1
392.5.o.b 16 7.c even 3 2
392.5.o.b 16 7.d odd 6 2
448.5.c.g 8 8.d odd 2 1
448.5.c.g 8 56.e even 2 1
448.5.c.h 8 8.b even 2 1
448.5.c.h 8 56.h odd 2 1
504.5.f.a 8 3.b odd 2 1
504.5.f.a 8 21.c even 2 1
1008.5.f.l 8 12.b even 2 1
1008.5.f.l 8 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(56, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 448 T^{6} + \cdots + 4064256 \) Copy content Toggle raw display
$5$ \( T^{8} + 2176 T^{6} + \cdots + 150209536 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} + 72 T^{3} + \cdots + 111399568)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} - 600 T^{3} + \cdots - 1425819248)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 696 T^{3} + \cdots + 17690649616)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{4} + 248 T^{3} + \cdots - 749057293808)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2936 T^{3} + \cdots - 233113325936)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 4387301644816)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 456715562640016)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 21632183493872)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 339367970051312)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
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