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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [635,2,Mod(1,635)] 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("635.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(635, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 635 = 5 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 635.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.07050052835\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 13x^{7} + 25x^{6} + 52x^{5} - 99x^{4} - 60x^{3} + 132x^{2} - 16x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{8} q^{3} + ( - \beta_1 + 3) q^{4} - q^{5} + (\beta_{7} + \beta_{6} + \cdots - \beta_{4}) q^{6} + ( - \beta_{7} + 1) q^{7} + (\beta_{8} + \beta_{7} + \beta_{3} + \cdots + 2) q^{8}+ \cdots + ( - \beta_{8} + 2 \beta_{7} + \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{2} + 3 q^{3} + 25 q^{4} - 9 q^{5} + q^{6} + 12 q^{7} + 12 q^{8} + 8 q^{9} - 3 q^{10} + 17 q^{12} + 9 q^{13} - 4 q^{14} - 3 q^{15} + 13 q^{16} + q^{17} + 14 q^{18} + 6 q^{19} - 25 q^{20} - 12 q^{21}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 13x^{7} + 25x^{6} + 52x^{5} - 99x^{4} - 60x^{3} + 132x^{2} - 16x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 11\nu^{5} + 19\nu^{4} + 32\nu^{3} - 41\nu^{2} - 18\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 2\nu^{7} + 11\nu^{6} - 19\nu^{5} - 32\nu^{4} + 41\nu^{3} + 14\nu^{2} - 4\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 13\nu^{6} + 21\nu^{5} + 56\nu^{4} - 59\nu^{3} - 88\nu^{2} + 40\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 17\nu^{6} + 3\nu^{5} + 90\nu^{4} - 27\nu^{3} - 150\nu^{2} + 56\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} + 2\nu^{7} + 13\nu^{6} - 25\nu^{5} - 52\nu^{4} + 99\nu^{3} + 60\nu^{2} - 124\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - \nu^{7} - 13\nu^{6} + 10\nu^{5} + 53\nu^{4} - 29\nu^{3} - 69\nu^{2} + 28\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} + \nu^{7} + 13\nu^{6} - 10\nu^{5} - 53\nu^{4} + 29\nu^{3} + 73\nu^{2} - 28\nu - 20 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{8} + 8\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{8} + 11\beta_{7} + 9\beta_{6} + 10\beta_{5} - \beta_{4} + \beta_{3} - 11\beta_{2} + 19\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 48\beta_{8} + 61\beta_{7} + 12\beta_{6} + \beta_{5} + 7\beta_{4} + 11\beta_{3} - 14\beta_{2} + \beta _1 + 137 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 82\beta_{8} + 100\beta_{7} + 72\beta_{6} + 80\beta_{5} - 16\beta_{4} + 14\beta_{3} - 94\beta_{2} + 101\beta _1 + 134 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 333 \beta_{8} + 461 \beta_{7} + 114 \beta_{6} + 22 \beta_{5} + 32 \beta_{4} + 94 \beta_{3} - 142 \beta_{2} + \cdots + 910 \) 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
−1.71227
0.572215
1.19685
2.73150
2.31219
1.59007
−0.285681
−1.89448
−2.51038
−2.59080 0.356688 4.71227 −1.00000 −0.924110 1.71145 −7.02696 −2.87277 2.59080
1.2 −2.10423 3.12921 2.42778 −1.00000 −6.58458 1.54336 −0.900157 6.79197 2.10423
1.3 −1.95017 −1.24593 1.80315 −1.00000 2.42977 4.81349 0.383882 −1.44766 1.95017
1.4 −1.50616 −0.822536 0.268505 −1.00000 1.23887 −1.63853 2.60790 −2.32343 1.50616
1.5 1.63945 −2.78856 0.687812 −1.00000 −4.57172 2.44235 −2.15127 4.77607 −1.63945
1.6 1.84660 2.21676 1.40993 −1.00000 4.09346 0.254931 −1.08962 1.91401 −1.84660
1.7 2.29906 1.76160 3.28568 −1.00000 4.05002 3.15679 2.95586 0.103225 −2.29906
1.8 2.62573 −1.67196 4.89448 −1.00000 −4.39013 3.08290 7.60015 −0.204539 −2.62573
1.9 2.74051 2.06473 5.51038 −1.00000 5.65842 −3.36673 9.62021 1.26313 −2.74051
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(127\) \( -1 \)

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 635.2.a.f ✓ 9
3.b odd 2 1 5715.2.a.n 9
5.b even 2 1 3175.2.a.l 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
635.2.a.f ✓ 9 1.a even 1 1 trivial
3175.2.a.l 9 5.b even 2 1
5715.2.a.n 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(635))\):

\( T_{2}^{9} - 3T_{2}^{8} - 17T_{2}^{7} + 52T_{2}^{6} + 103T_{2}^{5} - 323T_{2}^{4} - 263T_{2}^{3} + 852T_{2}^{2} + 236T_{2} - 802 \) Copy content Toggle raw display
\( T_{3}^{9} - 3T_{3}^{8} - 13T_{3}^{7} + 39T_{3}^{6} + 50T_{3}^{5} - 146T_{3}^{4} - 85T_{3}^{3} + 185T_{3}^{2} + 71T_{3} - 43 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 3 T^{8} + \cdots - 802 \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots - 43 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 12 T^{8} + \cdots - 425 \) Copy content Toggle raw display
$11$ \( T^{9} - 56 T^{7} + \cdots + 487 \) Copy content Toggle raw display
$13$ \( T^{9} - 9 T^{8} + \cdots - 2016 \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} + \cdots - 10496 \) Copy content Toggle raw display
$19$ \( T^{9} - 6 T^{8} + \cdots - 284 \) Copy content Toggle raw display
$23$ \( T^{9} - 3 T^{8} + \cdots - 17 \) Copy content Toggle raw display
$29$ \( T^{9} - 4 T^{8} + \cdots + 128 \) Copy content Toggle raw display
$31$ \( T^{9} - 2 T^{8} + \cdots + 3864100 \) Copy content Toggle raw display
$37$ \( T^{9} - 31 T^{8} + \cdots + 4127872 \) Copy content Toggle raw display
$41$ \( T^{9} + 29 T^{8} + \cdots - 2241612 \) Copy content Toggle raw display
$43$ \( T^{9} - 20 T^{8} + \cdots - 17408 \) Copy content Toggle raw display
$47$ \( T^{9} - 6 T^{8} + \cdots + 1430144 \) Copy content Toggle raw display
$53$ \( T^{9} - 19 T^{8} + \cdots + 150748712 \) Copy content Toggle raw display
$59$ \( T^{9} + 2 T^{8} + \cdots - 142448128 \) Copy content Toggle raw display
$61$ \( T^{9} - 10 T^{8} + \cdots - 77859 \) Copy content Toggle raw display
$67$ \( T^{9} - 33 T^{8} + \cdots - 9122416 \) Copy content Toggle raw display
$71$ \( T^{9} - 10 T^{8} + \cdots + 151115697 \) Copy content Toggle raw display
$73$ \( T^{9} - 22 T^{8} + \cdots + 883232 \) Copy content Toggle raw display
$79$ \( T^{9} - 9 T^{8} + \cdots + 97369 \) Copy content Toggle raw display
$83$ \( T^{9} - 19 T^{8} + \cdots + 1841308 \) Copy content Toggle raw display
$89$ \( T^{9} + 33 T^{8} + \cdots + 75385216 \) Copy content Toggle raw display
$97$ \( T^{9} - 40 T^{8} + \cdots + 104069268 \) Copy content Toggle raw display
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