Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1035,4,Mod(1,1035)] 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("1035.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1035.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,0,24,-45,0,-12] 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: \(61.0669768559\)
Analytic rank: \(1\)
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: \( x^{9} - 48x^{7} + 713x^{5} - 8x^{4} - 3438x^{3} + 488x^{2} + 3732x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 13 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - 5 q^{5} + ( - \beta_{6} - \beta_{4} + 2 \beta_1 - 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - 5 \beta_1 q^{10} + (\beta_{5} + \beta_{4} + \cdots - 3 \beta_1) q^{11}+ \cdots + (6 \beta_{8} - 32 \beta_{6} + \cdots - 206) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 24 q^{4} - 45 q^{5} - 12 q^{7} + 2 q^{11} - 118 q^{13} + 206 q^{14} + 28 q^{16} - 50 q^{19} - 120 q^{20} - 314 q^{22} + 207 q^{23} + 225 q^{25} + 212 q^{26} - 170 q^{28} + 532 q^{29} - 292 q^{31}+ \cdots - 1804 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 48x^{7} + 713x^{5} - 8x^{4} - 3438x^{3} + 488x^{2} + 3732x + 288 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 18\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{8} + 123 \nu^{7} + 39 \nu^{6} - 5229 \nu^{5} - 794 \nu^{4} + 60950 \nu^{3} + 15612 \nu^{2} + \cdots - 45984 ) / 5184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{8} - 69 \nu^{7} + 1095 \nu^{6} + 2259 \nu^{5} - 23434 \nu^{4} - 13610 \nu^{3} + \cdots - 258144 ) / 10368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13 \nu^{8} - 15 \nu^{7} - 651 \nu^{6} + 441 \nu^{5} + 10178 \nu^{4} - 1790 \nu^{3} - 52332 \nu^{2} + \cdots + 53472 ) / 3456 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49 \nu^{8} - 21 \nu^{7} + 2343 \nu^{6} + 819 \nu^{5} - 33290 \nu^{4} - 5914 \nu^{3} + 129084 \nu^{2} + \cdots - 31872 ) / 2592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35 \nu^{8} - 15 \nu^{7} + 1581 \nu^{6} + 585 \nu^{5} - 21094 \nu^{4} - 6446 \nu^{3} + 79428 \nu^{2} + \cdots - 36096 ) / 1296 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + 2\beta_{7} + 2\beta_{6} - 2\beta_{5} - 2\beta_{3} + 26\beta_{2} + 202 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 12\beta_{6} - 4\beta_{5} - 4\beta_{4} + 31\beta_{3} - 2\beta_{2} + 370\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -43\beta_{8} + 78\beta_{7} + 58\beta_{6} - 58\beta_{5} - 82\beta_{3} + 616\beta_{2} - 28\beta _1 + 4196 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{8} - 78 \beta_{7} - 504 \beta_{6} - 176 \beta_{5} - 128 \beta_{4} + 815 \beta_{3} + \cdots - 132 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1375 \beta_{8} + 2318 \beta_{7} + 1430 \beta_{6} - 1406 \beta_{5} - 12 \beta_{4} - 2514 \beta_{3} + \cdots + 91852 \) 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
−4.93964
−3.83291
−2.81004
−1.04285
−0.0784181
1.49264
2.19239
4.18193
4.83690
−4.93964 0 16.4001 −5.00000 0 −20.8112 −41.4933 0 24.6982
1.2 −3.83291 0 6.69123 −5.00000 0 −12.2906 5.01639 0 19.1646
1.3 −2.81004 0 −0.103678 −5.00000 0 17.8981 22.7717 0 14.0502
1.4 −1.04285 0 −6.91247 −5.00000 0 −20.3904 15.5514 0 5.21424
1.5 −0.0784181 0 −7.99385 −5.00000 0 −10.1232 1.25421 0 0.392091
1.6 1.49264 0 −5.77202 −5.00000 0 25.8231 −20.5567 0 −7.46321
1.7 2.19239 0 −3.19341 −5.00000 0 −1.42421 −24.5404 0 −10.9620
1.8 4.18193 0 9.48855 −5.00000 0 −5.85194 6.22500 0 −20.9097
1.9 4.83690 0 15.3956 −5.00000 0 15.1704 35.7717 0 −24.1845
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( -1 \)

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 1035.4.a.s ✓ 9
3.b odd 2 1 1035.4.a.t yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1035.4.a.s ✓ 9 1.a even 1 1 trivial
1035.4.a.t yes 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\):

\( T_{2}^{9} - 48T_{2}^{7} + 713T_{2}^{5} - 8T_{2}^{4} - 3438T_{2}^{3} + 488T_{2}^{2} + 3732T_{2} + 288 \) Copy content Toggle raw display
\( T_{7}^{9} + 12 T_{7}^{8} - 1106 T_{7}^{7} - 15800 T_{7}^{6} + 337669 T_{7}^{5} + 5902148 T_{7}^{4} + \cdots - 3085331328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 48 T^{7} + \cdots + 288 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( (T + 5)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 3085331328 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 95625107968 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 23436984268800 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 66838166343936 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T - 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 99\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 70\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 24\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 11\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 39\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 90\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 48\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
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