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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(149,925)] 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("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-20,0,-12,0,0,-12,0,-10] 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.60703296077824.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 20x^{8} + 142x^{6} + 420x^{4} + 457x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 185)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{8} + \beta_{2}) q^{3} + (\beta_{3} - 2) q^{4} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{6} + ( - 2 \beta_{8} - \beta_{4} + \cdots + \beta_1) q^{7} + ( - 2 \beta_{2} + \beta_1) q^{8}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 20x^{8} + 142x^{6} + 420x^{4} + 457x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 21\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 12\nu^{4} + 39\nu^{2} + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 53\nu^{2} + 36 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 15\nu^{6} + 71\nu^{4} + 113\nu^{2} + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 17\nu^{7} + 97\nu^{5} + 207\nu^{3} + 118\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} + 23\nu^{7} + 175\nu^{5} + 489\nu^{3} + 328\nu ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 2\beta_{5} - 7\beta_{3} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{4} - 20\beta_{2} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{6} + 28\beta_{5} + 45\beta_{3} - 132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{9} - 4\beta_{8} - 52\beta_{4} + 166\beta_{2} - 177\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8\beta_{7} + 218\beta_{6} - 278\beta_{5} - 291\beta_{3} + 822 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -68\beta_{9} + 92\beta_{8} + 496\beta_{4} - 1296\beta_{2} + 1113\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
2.72362i
2.47408i
2.15510i
1.13359i
0.728950i
− 0.728950i
− 1.13359i
− 2.15510i
− 2.47408i
− 2.72362i
− 2.72362i − 2.29298i −5.41809 0 −6.24519 − 3.82710i 9.30957i −2.25774 0
149.2 − 2.47408i − 2.38679i −4.12105 0 −5.90509 4.78404i 5.24765i −2.69675 0
149.3 − 2.15510i 1.38311i −2.64446 0 2.98075 2.62521i 1.38887i 1.08699 0
149.4 − 1.13359i 1.10563i 0.714970 0 1.25333 2.46164i − 3.07767i 1.77758 0
149.5 − 0.728950i 2.62871i 1.46863 0 1.91620 − 2.55244i − 2.52846i −3.91009 0
149.6 0.728950i − 2.62871i 1.46863 0 1.91620 2.55244i 2.52846i −3.91009 0
149.7 1.13359i − 1.10563i 0.714970 0 1.25333 − 2.46164i 3.07767i 1.77758 0
149.8 2.15510i − 1.38311i −2.64446 0 2.98075 − 2.62521i − 1.38887i 1.08699 0
149.9 2.47408i 2.38679i −4.12105 0 −5.90509 − 4.78404i − 5.24765i −2.69675 0
149.10 2.72362i 2.29298i −5.41809 0 −6.24519 3.82710i − 9.30957i −2.25774 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 149.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.b.f 10
5.b even 2 1 inner 925.2.b.f 10
5.c odd 4 1 185.2.a.e ✓ 5
5.c odd 4 1 925.2.a.f 5
15.e even 4 1 1665.2.a.p 5
15.e even 4 1 8325.2.a.ch 5
20.e even 4 1 2960.2.a.w 5
35.f even 4 1 9065.2.a.k 5
185.h odd 4 1 6845.2.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.e ✓ 5 5.c odd 4 1
925.2.a.f 5 5.c odd 4 1
925.2.b.f 10 1.a even 1 1 trivial
925.2.b.f 10 5.b even 2 1 inner
1665.2.a.p 5 15.e even 4 1
2960.2.a.w 5 20.e even 4 1
6845.2.a.f 5 185.h odd 4 1
8325.2.a.ch 5 15.e even 4 1
9065.2.a.k 5 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\):

\( T_{2}^{10} + 20T_{2}^{8} + 142T_{2}^{6} + 420T_{2}^{4} + 457T_{2}^{2} + 144 \) Copy content Toggle raw display
\( T_{3}^{10} + 21T_{3}^{8} + 164T_{3}^{6} + 580T_{3}^{4} + 896T_{3}^{2} + 484 \) Copy content Toggle raw display
\( T_{7}^{10} + 57T_{7}^{8} + 1192T_{7}^{6} + 11532T_{7}^{4} + 52496T_{7}^{2} + 91204 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 20 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( T^{10} + 21 T^{8} + \cdots + 484 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 57 T^{8} + \cdots + 91204 \) Copy content Toggle raw display
$11$ \( (T^{5} + 5 T^{4} - 8 T^{3} + \cdots + 96)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 72 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{10} + 104 T^{8} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( (T^{5} - 4 T^{4} - 38 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 160 T^{8} + \cdots + 36864 \) Copy content Toggle raw display
$29$ \( (T^{5} - 4 T^{4} + \cdots + 192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots + 324)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$41$ \( (T^{5} + 5 T^{4} - 8 T^{3} + \cdots + 96)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 260 T^{8} + \cdots + 6390784 \) Copy content Toggle raw display
$47$ \( T^{10} + 233 T^{8} + \cdots + 956484 \) Copy content Toggle raw display
$53$ \( T^{10} + 289 T^{8} + \cdots + 278784 \) Copy content Toggle raw display
$59$ \( (T^{5} - 30 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 14 T^{4} + \cdots + 3296)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 119946304 \) Copy content Toggle raw display
$71$ \( (T^{5} + 7 T^{4} + \cdots - 7104)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 409 T^{8} + \cdots + 135424 \) Copy content Toggle raw display
$79$ \( (T^{5} + 28 T^{4} + \cdots + 19508)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 373 T^{8} + \cdots + 23213124 \) Copy content Toggle raw display
$89$ \( (T^{5} + 6 T^{4} + \cdots - 22944)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 27880984576 \) Copy content Toggle raw display
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