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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(79,112)] 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("112.79"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-9] 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: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 485 x^{8} - 1910 x^{7} + 81837 x^{6} - 238847 x^{5} + 5758115 x^{4} + \cdots + 1406445775 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3} - 1) q^{3} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{5} + (\beta_{8} + 14 \beta_{4} + \beta_{3} - 8) q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 14) q^{9} + (\beta_{9} + \beta_{6} - 2 \beta_{5} + \cdots - 23) q^{11}+ \cdots + ( - 38 \beta_{9} + 29 \beta_{8} + \cdots - 2373) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 9 q^{3} + 9 q^{5} - 10 q^{7} + 74 q^{9} - 189 q^{11} + 84 q^{13} - 435 q^{17} - 357 q^{19} - 61 q^{21} - 1269 q^{23} - 776 q^{25} + 660 q^{29} - 969 q^{31} + 1759 q^{33} + 1521 q^{35} - 583 q^{37}+ \cdots - 43356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 485 x^{8} - 1910 x^{7} + 81837 x^{6} - 238847 x^{5} + 5758115 x^{4} + \cdots + 1406445775 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35311 \nu^{8} + 141244 \nu^{7} - 15758286 \nu^{6} + 46780504 \nu^{5} - 2231822483 \nu^{4} + \cdots - 1160856437455 ) / 6842113830 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 125771 \nu^{9} + 21400794 \nu^{8} - 133637836 \nu^{7} + 8376176564 \nu^{6} + \cdots + 353780183135050 ) / 16574147291835 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 251542 \nu^{9} - 1131939 \nu^{8} + 100597076 \nu^{7} - 346807384 \nu^{6} + \cdots + 8247831963700 ) / 16574147291835 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 555803725 \nu^{9} + 1519093554 \nu^{8} + 241953470414 \nu^{7} + 902656285874 \nu^{6} + \cdots + 11\!\cdots\!60 ) / 15\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 247621049 \nu^{9} + 3075310907 \nu^{8} - 114565136048 \nu^{7} + 1241017769922 \nu^{6} + \cdots + 11\!\cdots\!15 ) / 519323281810830 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 338927184 \nu^{9} - 840469407 \nu^{8} + 149663610038 \nu^{7} - 286909801402 \nu^{6} + \cdots - 36\!\cdots\!95 ) / 519323281810830 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 338927184 \nu^{9} - 2209875249 \nu^{8} + 155141233406 \nu^{7} - 765672208924 \nu^{6} + \cdots + 13\!\cdots\!35 ) / 519323281810830 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 253066739 \nu^{9} - 284702637 \nu^{8} + 108311972338 \nu^{7} - 19282705232 \nu^{6} + \cdots + 28\!\cdots\!85 ) / 222567120776070 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} - 2\beta_{4} - 3\beta_{3} + \beta_{2} - \beta _1 - 93 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 17 \beta_{5} - 250 \beta_{4} + \beta_{3} + \cdots + 50 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 176 \beta_{9} + 236 \beta_{8} - 46 \beta_{7} - 198 \beta_{6} - 34 \beta_{5} - 39 \beta_{4} + \cdots + 13301 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1922 \beta_{9} - 16 \beta_{8} + 756 \beta_{7} + 982 \beta_{6} + 3546 \beta_{5} + 59854 \beta_{4} + \cdots - 7210 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28070 \beta_{9} - 45698 \beta_{8} + 14047 \beta_{7} + 36837 \beta_{6} + 10723 \beta_{5} + 82806 \beta_{4} + \cdots - 2218219 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 427156 \beta_{9} - 66928 \beta_{8} - 182703 \beta_{7} - 196688 \beta_{6} - 610134 \beta_{5} + \cdots + 754378 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4344524 \beta_{9} + 8410388 \beta_{8} - 3339020 \beta_{7} - 6880864 \beta_{6} - 2490656 \beta_{5} + \cdots + 392577176 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 88520368 \beta_{9} + 24433564 \beta_{8} + 38658116 \beta_{7} + 36619360 \beta_{6} + 100087460 \beta_{5} + \cdots - 16177828 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 − 12.4230i
0.500000 − 6.63333i
0.500000 − 3.92097i
0.500000 + 8.26373i
0.500000 + 13.8476i
0.500000 + 12.4230i
0.500000 + 6.63333i
0.500000 + 3.92097i
0.500000 − 8.26373i
0.500000 − 13.8476i
0 −11.5086 + 6.64452i 0 −15.0876 + 26.1326i 0 5.01564 + 48.7426i 0 47.7992 − 82.7906i 0
79.2 0 −6.49463 + 3.74968i 0 24.4965 − 42.4292i 0 −29.5189 + 39.1106i 0 −12.3799 + 21.4425i 0
79.3 0 −4.14566 + 2.39350i 0 −1.24690 + 2.15969i 0 23.3384 − 43.0850i 0 −29.0423 + 50.3028i 0
79.4 0 6.40660 − 3.69885i 0 −10.1759 + 17.6251i 0 −48.6746 − 5.63742i 0 −13.1370 + 22.7540i 0
79.5 0 11.2423 − 6.49076i 0 6.51387 − 11.2823i 0 44.8395 + 19.7590i 0 43.7600 − 75.7946i 0
95.1 0 −11.5086 − 6.64452i 0 −15.0876 − 26.1326i 0 5.01564 − 48.7426i 0 47.7992 + 82.7906i 0
95.2 0 −6.49463 − 3.74968i 0 24.4965 + 42.4292i 0 −29.5189 − 39.1106i 0 −12.3799 − 21.4425i 0
95.3 0 −4.14566 − 2.39350i 0 −1.24690 − 2.15969i 0 23.3384 + 43.0850i 0 −29.0423 − 50.3028i 0
95.4 0 6.40660 + 3.69885i 0 −10.1759 − 17.6251i 0 −48.6746 + 5.63742i 0 −13.1370 − 22.7540i 0
95.5 0 11.2423 + 6.49076i 0 6.51387 + 11.2823i 0 44.8395 − 19.7590i 0 43.7600 + 75.7946i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 79.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.5.r.a ✓ 10
4.b odd 2 1 112.5.r.b yes 10
7.c even 3 1 112.5.r.b yes 10
7.c even 3 1 784.5.d.k 10
7.d odd 6 1 784.5.d.l 10
28.f even 6 1 784.5.d.l 10
28.g odd 6 1 inner 112.5.r.a ✓ 10
28.g odd 6 1 784.5.d.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.r.a ✓ 10 1.a even 1 1 trivial
112.5.r.a ✓ 10 28.g odd 6 1 inner
112.5.r.b yes 10 4.b odd 2 1
112.5.r.b yes 10 7.c even 3 1
784.5.d.k 10 7.c even 3 1
784.5.d.k 10 28.g odd 6 1
784.5.d.l 10 7.d odd 6 1
784.5.d.l 10 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 9 T_{3}^{9} - 199 T_{3}^{8} - 2034 T_{3}^{7} + 35821 T_{3}^{6} + 361347 T_{3}^{5} + \cdots + 2098966203 \) acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 2098966203 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 955504295001 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!23 \) Copy content Toggle raw display
$13$ \( (T^{5} - 42 T^{4} + \cdots - 118492281376)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 29\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 71\!\cdots\!03 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 57\!\cdots\!43 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 13475415904992)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 12\!\cdots\!23 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 42\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 69\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 11\!\cdots\!27 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 97\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 95\!\cdots\!87 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 51\!\cdots\!27 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 20\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 63\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 40\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 24\!\cdots\!04)^{2} \) Copy content Toggle raw display
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