Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,1,0,-50,0,-28] 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: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 5 q^{5} + ( - \beta_{6} - \beta_1 - 3) q^{7} + (\beta_{2} + \beta_1 + 14) q^{9} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 2) q^{11} + (\beta_{6} - \beta_{5} + \beta_{4} + 2) q^{13}+ \cdots + (5 \beta_{9} - 7 \beta_{8} + \cdots - 419) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35}+ \cdots - 4356 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3854981 \nu^{9} + 89269783 \nu^{8} + 723349168 \nu^{7} - 16966937858 \nu^{6} + \cdots + 15927316040736 ) / 649381829040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4287527 \nu^{9} - 18632021 \nu^{8} + 958382440 \nu^{7} + 3713138890 \nu^{6} + \cdots - 14421069607200 ) / 389629097424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 72608749 \nu^{9} + 196190847 \nu^{8} + 15013259412 \nu^{7} - 32852266542 \nu^{6} + \cdots + 115237989620584 ) / 2922218230680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41435197 \nu^{9} - 38587743 \nu^{8} - 8659056120 \nu^{7} + 1706664114 \nu^{6} + \cdots + 8867676107504 ) / 1168887292272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 456055897 \nu^{9} - 514489611 \nu^{8} - 93405512736 \nu^{7} + 29305755546 \nu^{6} + \cdots + 273658774052288 ) / 5844436461360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48402857 \nu^{9} - 20806896 \nu^{8} - 9854440296 \nu^{7} - 3859706262 \nu^{6} + \cdots + 17109836030200 ) / 584443646136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 312403603 \nu^{9} + 511006659 \nu^{8} + 62814279954 \nu^{7} - 50573267814 \nu^{6} + \cdots - 92807654043212 ) / 1461109115340 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 3\beta_{7} - 4\beta_{6} - 2\beta_{5} + \beta_{4} + 77\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{9} - 8 \beta_{8} + 7 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} - 12 \beta_{4} - 3 \beta_{3} + \cdots + 3201 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 126 \beta_{9} + 165 \beta_{8} + 352 \beta_{7} - 539 \beta_{6} - 192 \beta_{5} + 29 \beta_{4} + \cdots + 6340 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 271 \beta_{9} - 1006 \beta_{8} + 1011 \beta_{7} - 1523 \beta_{6} + 1100 \beta_{5} - 2206 \beta_{4} + \cdots + 282923 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12628 \beta_{9} + 20071 \beta_{8} + 35963 \beta_{7} - 65369 \beta_{6} - 16334 \beta_{5} - 5653 \beta_{4} + \cdots + 790558 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 27132 \beta_{9} - 95202 \beta_{8} + 124672 \beta_{7} - 253946 \beta_{6} + 102528 \beta_{5} + \cdots + 26201647 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1197157 \beta_{9} + 2211691 \beta_{8} + 3599343 \beta_{7} - 7540678 \beta_{6} - 1366838 \beta_{5} + \cdots + 90602246 \) 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
−9.58471
−7.77468
−5.32629
−2.66698
−0.575045
0.0622186
2.52979
4.78870
9.27510
10.2719
0 −9.58471 0 −5.00000 0 −21.8969 0 64.8667 0
1.2 0 −7.77468 0 −5.00000 0 16.1209 0 33.4457 0
1.3 0 −5.32629 0 −5.00000 0 −14.9292 0 1.36937 0
1.4 0 −2.66698 0 −5.00000 0 32.5699 0 −19.8872 0
1.5 0 −0.575045 0 −5.00000 0 24.8747 0 −26.6693 0
1.6 0 0.0622186 0 −5.00000 0 −14.4792 0 −26.9961 0
1.7 0 2.52979 0 −5.00000 0 −24.3398 0 −20.6002 0
1.8 0 4.78870 0 −5.00000 0 −0.156416 0 −4.06831 0
1.9 0 9.27510 0 −5.00000 0 −33.0838 0 59.0275 0
1.10 0 10.2719 0 −5.00000 0 7.31988 0 78.5119 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( +1 \)

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 1840.4.a.bb 10
4.b odd 2 1 920.4.a.g ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.g ✓ 10 4.b odd 2 1
1840.4.a.bb 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{10} - T_{3}^{9} - 204 T_{3}^{8} + 42 T_{3}^{7} + 12958 T_{3}^{6} + 5872 T_{3}^{5} - 259871 T_{3}^{4} + \cdots - 43712 \) Copy content Toggle raw display
\( T_{7}^{10} + 28 T_{7}^{9} - 1904 T_{7}^{8} - 56306 T_{7}^{7} + 1019676 T_{7}^{6} + 34617872 T_{7}^{5} + \cdots + 56995990528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots - 43712 \) Copy content Toggle raw display
$5$ \( (T + 5)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 56995990528 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 63035886761000 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T + 23)^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 29\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 37\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 21\!\cdots\!22 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 37\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 25\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 60\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
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