Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,10,-9,10,-3,-9,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.1372077370\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: 10.10.4074747968929.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 13x^{8} + 8x^{7} + 57x^{6} - 11x^{5} - 96x^{4} - 15x^{3} + 51x^{2} + 19x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{9} + \beta_1) q^{5} + (\beta_1 - 1) q^{6} - q^{7} + q^{8} + (\beta_{9} + \beta_{7} + \cdots - 2 \beta_1) q^{9} + ( - \beta_{9} + \beta_1) q^{10}+ \cdots + (\beta_{9} + \beta_{7} - 2 \beta_{6} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 9 q^{3} + 10 q^{4} - 3 q^{5} - 9 q^{6} - 10 q^{7} + 10 q^{8} + 5 q^{9} - 3 q^{10} + 8 q^{11} - 9 q^{12} - 15 q^{13} - 10 q^{14} + 21 q^{15} + 10 q^{16} - 9 q^{17} + 5 q^{18} + q^{19}+ \cdots + 25 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 13x^{8} + 8x^{7} + 57x^{6} - 11x^{5} - 96x^{4} - 15x^{3} + 51x^{2} + 19x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 17\nu^{7} + 4\nu^{6} + 95\nu^{5} - 37\nu^{4} - 194\nu^{3} + 85\nu^{2} + 105\nu - 30 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{9} - 22 \nu^{8} - 138 \nu^{7} + 213 \nu^{6} + 513 \nu^{5} - 576 \nu^{4} - 711 \nu^{3} + \cdots - 30 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12 \nu^{9} + 19 \nu^{8} + 145 \nu^{7} - 182 \nu^{6} - 579 \nu^{5} + 478 \nu^{4} + 882 \nu^{3} + \cdots + 1 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15 \nu^{9} + 26 \nu^{8} + 176 \nu^{7} - 248 \nu^{6} - 677 \nu^{5} + 649 \nu^{4} + 988 \nu^{3} + \cdots + 1 ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20 \nu^{9} - 31 \nu^{8} - 242 \nu^{7} + 294 \nu^{6} + 965 \nu^{5} - 759 \nu^{4} - 1442 \nu^{3} + \cdots + 6 ) / 11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23 \nu^{9} + 38 \nu^{8} + 273 \nu^{7} - 360 \nu^{6} - 1063 \nu^{5} + 930 \nu^{4} + 1559 \nu^{3} + \cdots + 16 ) / 11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26 \nu^{9} + 44 \nu^{8} + 310 \nu^{7} - 423 \nu^{6} - 1216 \nu^{5} + 1127 \nu^{4} + 1799 \nu^{3} + \cdots + 43 ) / 11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43 \nu^{9} - 72 \nu^{8} - 508 \nu^{7} + 685 \nu^{6} + 1962 \nu^{5} - 1787 \nu^{4} - 2830 \nu^{3} + \cdots - 61 ) / 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{7} - \beta_{5} - \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} + 9\beta_{7} - 6\beta_{6} - \beta_{5} - 8\beta_{4} - 7\beta_{3} - \beta_{2} - \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + \beta_{8} + 17\beta_{7} - 2\beta_{6} - 11\beta_{5} - 2\beta_{4} - 10\beta_{3} + \beta_{2} + 21\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45 \beta_{9} + \beta_{8} + 64 \beta_{7} - 35 \beta_{6} - 10 \beta_{5} - 53 \beta_{4} - 43 \beta_{3} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 67 \beta_{9} + 14 \beta_{8} + 121 \beta_{7} - 21 \beta_{6} - 85 \beta_{5} - 23 \beta_{4} - 73 \beta_{3} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 287 \beta_{9} + 21 \beta_{8} + 425 \beta_{7} - 206 \beta_{6} - 84 \beta_{5} - 335 \beta_{4} - 262 \beta_{3} + \cdots + 229 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 472 \beta_{9} + 139 \beta_{8} + 822 \beta_{7} - 164 \beta_{6} - 591 \beta_{5} - 200 \beta_{4} + \cdots - 32 \) 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
−2.37885
−1.72935
−1.15060
−0.700616
−0.664082
0.0468464
0.985245
1.54802
2.46032
2.58307
1.00000 −3.37885 1.00000 −1.49205 −3.37885 −1.00000 1.00000 8.41663 −1.49205
1.2 1.00000 −2.72935 1.00000 −3.61116 −2.72935 −1.00000 1.00000 4.44937 −3.61116
1.3 1.00000 −2.15060 1.00000 −0.522140 −2.15060 −1.00000 1.00000 1.62506 −0.522140
1.4 1.00000 −1.70062 1.00000 −2.96270 −1.70062 −1.00000 1.00000 −0.107906 −2.96270
1.5 1.00000 −1.66408 1.00000 0.264223 −1.66408 −1.00000 1.00000 −0.230830 0.264223
1.6 1.00000 −0.953154 1.00000 0.0978227 −0.953154 −1.00000 1.00000 −2.09150 0.0978227
1.7 1.00000 −0.0147545 1.00000 4.16631 −0.0147545 −1.00000 1.00000 −2.99978 4.16631
1.8 1.00000 0.548020 1.00000 0.970940 0.548020 −1.00000 1.00000 −2.69967 0.970940
1.9 1.00000 1.46032 1.00000 1.11649 1.46032 −1.00000 1.00000 −0.867474 1.11649
1.10 1.00000 1.58307 1.00000 −1.02774 1.58307 −1.00000 1.00000 −0.493895 −1.02774
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(23\) \( -1 \)

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 7406.2.a.bo 10
23.b odd 2 1 7406.2.a.bp 10
23.d odd 22 2 322.2.i.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.c ✓ 20 23.d odd 22 2
7406.2.a.bo 10 1.a even 1 1 trivial
7406.2.a.bp 10 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):

\( T_{3}^{10} + 9T_{3}^{9} + 23T_{3}^{8} - 12T_{3}^{7} - 125T_{3}^{6} - 103T_{3}^{5} + 158T_{3}^{4} + 219T_{3}^{3} - 12T_{3}^{2} - 68T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{10} + 3T_{5}^{9} - 18T_{5}^{8} - 62T_{5}^{7} + 19T_{5}^{6} + 143T_{5}^{5} + 9T_{5}^{4} - 96T_{5}^{3} - 9T_{5}^{2} + 12T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{10} - 8 T_{11}^{9} + 144 T_{11}^{7} - 300 T_{11}^{6} - 502 T_{11}^{5} + 2119 T_{11}^{4} + \cdots - 439 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{9} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 8 T^{9} + \cdots - 439 \) Copy content Toggle raw display
$13$ \( T^{10} + 15 T^{9} + \cdots + 42481 \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + \cdots - 40217 \) Copy content Toggle raw display
$19$ \( T^{10} - T^{9} + \cdots - 69497 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 21 T^{9} + \cdots + 178441 \) Copy content Toggle raw display
$31$ \( T^{10} + 30 T^{9} + \cdots + 1103057 \) Copy content Toggle raw display
$37$ \( T^{10} - 25 T^{9} + \cdots - 258103 \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} + \cdots - 14647313 \) Copy content Toggle raw display
$43$ \( T^{10} - 36 T^{9} + \cdots + 65281667 \) Copy content Toggle raw display
$47$ \( T^{10} + 34 T^{9} + \cdots - 27697 \) Copy content Toggle raw display
$53$ \( T^{10} - 23 T^{9} + \cdots - 36871 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 213245791 \) Copy content Toggle raw display
$61$ \( T^{10} - 2 T^{9} + \cdots - 35604097 \) Copy content Toggle raw display
$67$ \( T^{10} - T^{9} + \cdots - 16320809 \) Copy content Toggle raw display
$71$ \( T^{10} + 36 T^{9} + \cdots - 2671657 \) Copy content Toggle raw display
$73$ \( T^{10} + 15 T^{9} + \cdots + 7126967 \) Copy content Toggle raw display
$79$ \( T^{10} - 9 T^{9} + \cdots + 26960263 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 1236283487 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 157247771 \) Copy content Toggle raw display
$97$ \( T^{10} + 44 T^{9} + \cdots - 44300807 \) Copy content Toggle raw display
show more
show less