Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,18,49,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
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: \( x^{8} - x^{7} - 56x^{6} + 38x^{5} + 975x^{4} - 101x^{3} - 5352x^{2} - 4096x + 1632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + 2) q^{3} + (\beta_{2} + 6) q^{4} - 5 q^{5} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 + 1) q^{6} + (\beta_{4} - \beta_{3} + 5) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 4) q^{8}+ \cdots + (9 \beta_{7} - 37 \beta_{6} + \cdots + 419) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 18 q^{3} + 49 q^{4} - 40 q^{5} + 4 q^{6} + 38 q^{7} - 39 q^{8} + 58 q^{9} + 5 q^{10} + 84 q^{11} + 122 q^{12} + 24 q^{13} - 56 q^{14} - 90 q^{15} + 245 q^{16} - 235 q^{18} + 22 q^{19} - 245 q^{20}+ \cdots + 3356 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 56x^{6} + 38x^{5} + 975x^{4} - 101x^{3} - 5352x^{2} - 4096x + 1632 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} - 48\nu^{5} + 58\nu^{4} + 651\nu^{3} - 761\nu^{2} - 2164\nu + 704 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{6} + 5\nu^{5} - 81\nu^{4} - 175\nu^{3} + 787\nu^{2} + 1420\nu - 218 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 9\nu^{6} + 68\nu^{5} - 382\nu^{4} - 1311\nu^{3} + 3949\nu^{2} + 6964\nu - 2296 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 48\nu^{5} - 54\nu^{4} - 647\nu^{3} + 653\nu^{2} + 2096\nu - 408 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 27\nu^{6} + 376\nu^{5} - 1206\nu^{4} - 5877\nu^{3} + 12987\nu^{2} + 25148\nu - 8608 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 22\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - \beta_{5} + \beta_{4} + 9\beta_{3} + 28\beta_{2} - 5\beta _1 + 300 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} + 2\beta_{6} + 42\beta_{5} - 32\beta_{4} + 24\beta_{3} - 36\beta_{2} + 521\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} + 76\beta_{6} - 58\beta_{5} + 38\beta_{4} + 392\beta_{3} + 743\beta_{2} - 290\beta _1 + 7030 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -182\beta_{7} + 56\beta_{6} + 1365\beta_{5} - 905\beta_{4} + 411\beta_{3} - 1197\beta_{2} + 12850\beta _1 - 1680 \) Copy content Toggle raw display

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

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
4.91331
4.83075
3.79130
0.289794
−1.63349
−1.74738
−4.11126
−5.33303
−4.91331 10.2724 16.1406 −5.00000 −50.4717 21.7147 −39.9972 78.5231 24.5665
1.2 −4.83075 −1.17308 15.3362 −5.00000 5.66685 −17.9463 −35.4393 −25.6239 24.1538
1.3 −3.79130 −5.21292 6.37398 −5.00000 19.7638 18.4051 6.16475 0.174523 18.9565
1.4 −0.289794 2.72824 −7.91602 −5.00000 −0.790629 29.7500 4.61237 −19.5567 1.44897
1.5 1.63349 9.29997 −5.33171 −5.00000 15.1914 −29.4708 −21.7772 59.4894 −8.16745
1.6 1.74738 0.322521 −4.94666 −5.00000 0.563566 −7.55169 −22.6227 −26.8960 −8.73690
1.7 4.11126 −3.82687 8.90247 −5.00000 −15.7333 15.9229 3.71030 −12.3551 −20.5563
1.8 5.33303 5.58969 20.4412 −5.00000 29.8100 7.17613 66.3491 4.24466 −26.6651
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(17\) \( +1 \)

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 1445.4.a.p 8
17.b even 2 1 1445.4.a.o 8
17.c even 4 2 85.4.d.b ✓ 16
51.f odd 4 2 765.4.g.c 16
85.f odd 4 2 425.4.c.e 16
85.i odd 4 2 425.4.c.f 16
85.j even 4 2 425.4.d.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.d.b ✓ 16 17.c even 4 2
425.4.c.e 16 85.f odd 4 2
425.4.c.f 16 85.i odd 4 2
425.4.d.g 16 85.j even 4 2
765.4.g.c 16 51.f odd 4 2
1445.4.a.o 8 17.b even 2 1
1445.4.a.p 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):

\( T_{2}^{8} + T_{2}^{7} - 56T_{2}^{6} - 38T_{2}^{5} + 975T_{2}^{4} + 101T_{2}^{3} - 5352T_{2}^{2} + 4096T_{2} + 1632 \) Copy content Toggle raw display
\( T_{3}^{8} - 18T_{3}^{7} + 25T_{3}^{6} + 866T_{3}^{5} - 2282T_{3}^{4} - 11472T_{3}^{3} + 22840T_{3}^{2} + 27988T_{3} - 10996 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 1632 \) Copy content Toggle raw display
$3$ \( T^{8} - 18 T^{7} + \cdots - 10996 \) Copy content Toggle raw display
$5$ \( (T + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 5426287360 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 378947148240 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 1309766957360 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 747381242730240 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 11759213394608 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 471947310680540 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 18\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 20\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 79\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 22\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 32\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 63\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 97\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 24\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
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