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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(145,432)] 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("432.145"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
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: \( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 14 \beta_1 - 14) q^{5} + (\beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{7} + (\beta_{7} - 3 \beta_{6} + 6 \beta_{4} + \cdots - 6) q^{11} + (4 \beta_{6} + \beta_{4} - 4 \beta_{2} + \cdots - 1600) q^{13}+ \cdots + (1884 \beta_{7} + 968 \beta_{6} + \cdots - 10346) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 54 q^{5} + 44 q^{7} + 2172 q^{11} - 6398 q^{13} + 51972 q^{17} - 90712 q^{19} - 2028 q^{23} - 173446 q^{25} - 283098 q^{29} + 812 q^{31} - 1492992 q^{35} + 441688 q^{37} - 610704 q^{41} + 84380 q^{43}+ \cdots - 22003112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 1336\nu^{5} + 539944\nu^{3} + 62785035\nu + 22961400 ) / 45922800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27\nu^{4} + 18981\nu^{2} + 2846070 ) / 245 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 2081\nu^{4} - 675545\nu^{2} - 66902715 ) / 5145 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1989 \nu^{7} - 31240 \nu^{6} - 2048124 \nu^{5} - 32505220 \nu^{4} - 644054976 \nu^{3} + \cdots - 1044940043400 ) / 160729800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 3595\nu^{4} - 1071559\nu^{2} - 113802195 ) / 5145 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1503 \nu^{7} + 1586268 \nu^{5} + 2952180 \nu^{4} + 529813512 \nu^{3} + 2075382540 \nu^{2} + \cdots + 311189293800 ) / 53576600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37141 \nu^{7} - 124960 \nu^{6} - 39404896 \nu^{5} - 112307800 \nu^{4} - 13321965064 \nu^{3} + \cdots - 3555180571800 ) / 321459600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 10\beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_{3} - 5\beta_{2} + 2\beta _1 - 3 ) / 486 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{5} + 14\beta_{3} + 7\beta_{2} - 54101 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1018 \beta_{7} - 3452 \beta_{6} + 509 \beta_{5} + 1366 \beta_{4} - 683 \beta_{3} + 1726 \beta_{2} + \cdots + 88278 ) / 486 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4921\beta_{5} - 9842\beta_{3} - 3451\beta_{2} + 20956583 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 502874 \beta_{7} + 1358446 \beta_{6} - 251437 \beta_{5} - 1266788 \beta_{4} + 633394 \beta_{3} + \cdots - 98473359 ) / 486 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -918631\beta_{5} + 1698347\beta_{3} + 408786\beta_{2} - 2983538168 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 247746742 \beta_{7} - 578847518 \beta_{6} + 123873371 \beta_{5} + 703725124 \beta_{4} + \cdots + 72924345897 ) / 486 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
− 20.7149i
13.4379i
22.3446i
− 15.0676i
20.7149i
− 13.4379i
− 22.3446i
15.0676i
0 0 0 −218.032 + 377.643i 0 −239.730 − 415.224i 0 0 0
145.2 0 0 0 −103.252 + 178.837i 0 808.604 + 1400.54i 0 0 0
145.3 0 0 0 47.1766 − 81.7123i 0 101.366 + 175.572i 0 0 0
145.4 0 0 0 247.107 − 428.002i 0 −648.241 − 1122.79i 0 0 0
289.1 0 0 0 −218.032 − 377.643i 0 −239.730 + 415.224i 0 0 0
289.2 0 0 0 −103.252 − 178.837i 0 808.604 − 1400.54i 0 0 0
289.3 0 0 0 47.1766 + 81.7123i 0 101.366 − 175.572i 0 0 0
289.4 0 0 0 247.107 + 428.002i 0 −648.241 + 1122.79i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.8.i.b 8
3.b odd 2 1 144.8.i.b 8
4.b odd 2 1 54.8.c.b 8
9.c even 3 1 inner 432.8.i.b 8
9.d odd 6 1 144.8.i.b 8
12.b even 2 1 18.8.c.b ✓ 8
36.f odd 6 1 54.8.c.b 8
36.f odd 6 1 162.8.a.i 4
36.h even 6 1 18.8.c.b ✓ 8
36.h even 6 1 162.8.a.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.8.c.b ✓ 8 12.b even 2 1
18.8.c.b ✓ 8 36.h even 6 1
54.8.c.b 8 4.b odd 2 1
54.8.c.b 8 36.f odd 6 1
144.8.i.b 8 3.b odd 2 1
144.8.i.b 8 9.d odd 6 1
162.8.a.h 4 36.h even 6 1
162.8.a.i 4 36.f odd 6 1
432.8.i.b 8 1.a even 1 1 trivial
432.8.i.b 8 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 54 T_{5}^{7} + 244431 T_{5}^{6} + 33030990 T_{5}^{5} + 55374420825 T_{5}^{4} + \cdots + 17\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 21\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 57\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 14\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 32\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 73\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 72\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 25\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
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