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Properties

Label 1040.6.a.v
Level $1040$
Weight $6$
Character orbit 1040.a
Self dual yes
Analytic conductor $166.799$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,6,Mod(1,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1040.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(166.799172605\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 1252x^{5} + 1388x^{4} + 394896x^{3} - 722832x^{2} - 18679104x - 35596800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + 25 q^{5} + ( - \beta_{3} + \beta_1 - 13) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 115) q^{9} + ( - \beta_{5} - \beta_{3} + 5 \beta_1 - 74) q^{11} + 169 q^{13} + 25 \beta_1 q^{15}+ \cdots + (4 \beta_{6} - 63 \beta_{5} + \cdots - 9292) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{3} + 175 q^{5} - 86 q^{7} + 807 q^{9} - 508 q^{11} + 1183 q^{13} + 50 q^{15} - 566 q^{17} - 164 q^{19} + 1816 q^{21} + 318 q^{23} + 4375 q^{25} + 2384 q^{27} + 6410 q^{29} - 6208 q^{31} + 10556 q^{33}+ \cdots - 59968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 1252x^{5} + 1388x^{4} + 394896x^{3} - 722832x^{2} - 18679104x - 35596800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 577 \nu^{6} - 18284 \nu^{5} + 61400 \nu^{4} + 2527124 \nu^{3} - 268154232 \nu^{2} + \cdots + 7433895768 ) / 25281816 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1558 \nu^{6} + 77 \nu^{5} + 1981192 \nu^{4} - 1730066 \nu^{3} - 596484780 \nu^{2} + \cdots + 13742016300 ) / 37922724 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1558 \nu^{6} - 77 \nu^{5} - 1981192 \nu^{4} + 1730066 \nu^{3} + 634407504 \nu^{2} + \cdots - 27318351492 ) / 37922724 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2073 \nu^{6} - 29176 \nu^{5} - 2306128 \nu^{4} + 24969844 \nu^{3} + 651507352 \nu^{2} + \cdots - 23636901816 ) / 25281816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3712 \nu^{6} + 134057 \nu^{5} + 996154 \nu^{4} - 81160094 \nu^{3} + 808501836 \nu^{2} + \cdots - 23566528980 ) / 37922724 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 358 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 3\beta_{5} + 4\beta_{4} - 5\beta_{2} + 623\beta _1 + 304 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -42\beta_{6} - 90\beta_{5} + 764\beta_{4} + 650\beta_{3} - 62\beta_{2} + 2476\beta _1 + 222808 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2036\beta_{6} - 4230\beta_{5} + 6900\beta_{4} + 752\beta_{3} - 4602\beta_{2} + 434750\beta _1 + 831196 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 51288 \beta_{6} - 111324 \beta_{5} + 584568 \beta_{4} + 419400 \beta_{3} - 73516 \beta_{2} + \cdots + 154790728 \) 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.

comment: embeddings in the coefficient field
 
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
−24.6069
−24.0849
−4.57223
−2.42853
10.0905
18.2862
29.3158
0 −24.6069 0 25.0000 0 −112.106 0 362.497 0
1.2 0 −24.0849 0 25.0000 0 −28.5352 0 337.081 0
1.3 0 −4.57223 0 25.0000 0 193.140 0 −222.095 0
1.4 0 −2.42853 0 25.0000 0 −143.512 0 −237.102 0
1.5 0 10.0905 0 25.0000 0 186.401 0 −141.181 0
1.6 0 18.2862 0 25.0000 0 −212.281 0 91.3845 0
1.7 0 29.3158 0 25.0000 0 30.8928 0 616.416 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(13\) \( -1 \)

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 1040.6.a.v 7
4.b odd 2 1 260.6.a.f ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.6.a.f ✓ 7 4.b odd 2 1
1040.6.a.v 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 2T_{3}^{6} - 1252T_{3}^{5} + 1388T_{3}^{4} + 394896T_{3}^{3} - 722832T_{3}^{2} - 18679104T_{3} - 35596800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 2 T^{6} + \cdots - 35596800 \) Copy content Toggle raw display
$5$ \( (T - 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 108389941198848 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T - 169)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 16\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 62\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 24\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 25\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 11\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 19\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 42\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 30\!\cdots\!52 \) Copy content Toggle raw display
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