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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,7,Mod(17,54)] 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("54.17"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 54.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.4229205155\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - 32 \beta_{2} q^{4} + ( - \beta_{7} - 24 \beta_{2} - 48) q^{5} + (\beta_{10} + \beta_{8} + 2 \beta_{7} + \cdots + 40) q^{7} + (32 \beta_{4} + 32 \beta_{3}) q^{8} + (2 \beta_{10} - 2 \beta_{9} + \cdots - \beta_1) q^{10}+ \cdots + (254 \beta_{11} - 1132 \beta_{10} + \cdots + 302352) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 192 q^{4} - 432 q^{5} + 240 q^{7} - 378 q^{11} + 1680 q^{13} + 4752 q^{14} - 6144 q^{16} - 2820 q^{19} - 13824 q^{20} - 3600 q^{22} + 76248 q^{23} + 8094 q^{25} + 15360 q^{28} - 97092 q^{29} + 21480 q^{31}+ \cdots - 38874 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7 \nu^{10} - 806378 \nu^{8} - 198775097 \nu^{6} - 12957175728 \nu^{4} - 57552855264 \nu^{2} + 7369432971840 ) / 18521148960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 119 \nu^{11} - 59366 \nu^{9} - 10447223 \nu^{7} - 794976432 \nu^{5} - 25420007664 \nu^{3} + \cdots - 25705589760 ) / 51411179520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79977 \nu^{11} - 234158 \nu^{10} + 22976562 \nu^{9} - 160687484 \nu^{8} + \cdots - 10\!\cdots\!40 ) / 10520012609280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79977 \nu^{11} + 234158 \nu^{10} + 22976562 \nu^{9} + 160687484 \nu^{8} + \cdots + 10\!\cdots\!40 ) / 10520012609280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1650933 \nu^{11} + 524294672 \nu^{10} + 355215438 \nu^{9} + 168117019424 \nu^{8} + \cdots + 14\!\cdots\!80 ) / 42080050437120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1650933 \nu^{11} + 524294672 \nu^{10} - 355215438 \nu^{9} + 168117019424 \nu^{8} + \cdots + 14\!\cdots\!80 ) / 42080050437120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2804005 \nu^{11} + 2835456 \nu^{10} - 937140106 \nu^{9} - 243106272 \nu^{8} + \cdots - 78\!\cdots\!80 ) / 4675561159680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2804005 \nu^{11} - 2835456 \nu^{10} - 937140106 \nu^{9} + 243106272 \nu^{8} + \cdots + 78\!\cdots\!80 ) / 4675561159680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24333735 \nu^{11} + 53593072 \nu^{10} - 8404314654 \nu^{9} + 14407397248 \nu^{8} + \cdots - 17\!\cdots\!00 ) / 21040025218560 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24333735 \nu^{11} - 53593072 \nu^{10} - 8404314654 \nu^{9} - 14407397248 \nu^{8} + \cdots + 17\!\cdots\!00 ) / 21040025218560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 169346043 \nu^{11} - 7952 \nu^{10} + 57570639966 \nu^{9} + 916045408 \nu^{8} + \cdots - 83\!\cdots\!40 ) / 42080050437120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} - 5\beta_{4} - 5\beta_{3} - 36\beta_{2} - 18 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 5 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + 42 \beta_{4} + \cdots - 3330 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 32 \beta_{11} + 95 \beta_{10} + 95 \beta_{9} - 130 \beta_{8} - 130 \beta_{7} - 107 \beta_{6} + \cdots + 5670 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 977 \beta_{10} - 977 \beta_{9} - 266 \beta_{8} + 266 \beta_{7} + 205 \beta_{6} + 205 \beta_{5} + \cdots + 299826 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5000 \beta_{11} - 8731 \beta_{10} - 8731 \beta_{9} + 9266 \beta_{8} + 9266 \beta_{7} + 14143 \beta_{6} + \cdots - 986022 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 151881 \beta_{10} + 151881 \beta_{9} + 29178 \beta_{8} - 29178 \beta_{7} - 32913 \beta_{6} + \cdots - 32744394 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 731688 \beta_{11} + 902903 \beta_{10} + 902903 \beta_{9} - 643402 \beta_{8} - 643402 \beta_{7} + \cdots + 150200550 ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 22070137 \beta_{10} - 22070137 \beta_{9} - 3058234 \beta_{8} + 3058234 \beta_{7} + 4884461 \beta_{6} + \cdots + 3980601810 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 105940840 \beta_{11} - 105108331 \beta_{10} - 105108331 \beta_{9} + 40008722 \beta_{8} + \cdots - 21849040278 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3127309105 \beta_{10} + 3127309105 \beta_{9} + 323447242 \beta_{8} - 323447242 \beta_{7} + \cdots - 514655128170 ) / 54 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15181664872 \beta_{11} + 13272884855 \beta_{10} + 13272884855 \beta_{9} - 1746634090 \beta_{8} + \cdots + 3115054097910 ) / 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
8.88570i
− 3.87527i
− 8.15670i
4.28281i
− 11.8022i
7.20150i
− 8.88570i
3.87527i
8.15670i
− 4.28281i
11.8022i
− 7.20150i
−4.89898 + 2.82843i 0 16.0000 − 27.7128i −202.253 − 116.771i 0 95.5752 + 165.541i 181.019i 0 1321.11
17.2 −4.89898 + 2.82843i 0 16.0000 − 27.7128i −1.59771 − 0.922438i 0 6.34411 + 10.9883i 181.019i 0 10.4362
17.3 −4.89898 + 2.82843i 0 16.0000 − 27.7128i 95.8504 + 55.3393i 0 −163.169 − 282.617i 181.019i 0 −626.092
17.4 4.89898 − 2.82843i 0 16.0000 − 27.7128i −156.951 − 90.6160i 0 104.306 + 180.663i − 181.019i 0 −1025.20
17.5 4.89898 − 2.82843i 0 16.0000 − 27.7128i 9.39126 + 5.42205i 0 322.041 + 557.792i − 181.019i 0 61.3435
17.6 4.89898 − 2.82843i 0 16.0000 − 27.7128i 39.5602 + 22.8401i 0 −245.097 − 424.521i − 181.019i 0 258.406
35.1 −4.89898 − 2.82843i 0 16.0000 + 27.7128i −202.253 + 116.771i 0 95.5752 − 165.541i − 181.019i 0 1321.11
35.2 −4.89898 − 2.82843i 0 16.0000 + 27.7128i −1.59771 + 0.922438i 0 6.34411 − 10.9883i − 181.019i 0 10.4362
35.3 −4.89898 − 2.82843i 0 16.0000 + 27.7128i 95.8504 − 55.3393i 0 −163.169 + 282.617i − 181.019i 0 −626.092
35.4 4.89898 + 2.82843i 0 16.0000 + 27.7128i −156.951 + 90.6160i 0 104.306 − 180.663i 181.019i 0 −1025.20
35.5 4.89898 + 2.82843i 0 16.0000 + 27.7128i 9.39126 − 5.42205i 0 322.041 − 557.792i 181.019i 0 61.3435
35.6 4.89898 + 2.82843i 0 16.0000 + 27.7128i 39.5602 − 22.8401i 0 −245.097 + 424.521i 181.019i 0 258.406
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.7.d.a 12
3.b odd 2 1 18.7.d.a ✓ 12
4.b odd 2 1 432.7.q.b 12
9.c even 3 1 18.7.d.a ✓ 12
9.c even 3 1 162.7.b.c 12
9.d odd 6 1 inner 54.7.d.a 12
9.d odd 6 1 162.7.b.c 12
12.b even 2 1 144.7.q.c 12
36.f odd 6 1 144.7.q.c 12
36.h even 6 1 432.7.q.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a ✓ 12 3.b odd 2 1
18.7.d.a ✓ 12 9.c even 3 1
54.7.d.a 12 1.a even 1 1 trivial
54.7.d.a 12 9.d odd 6 1 inner
144.7.q.c 12 12.b even 2 1
144.7.q.c 12 36.f odd 6 1
162.7.b.c 12 9.c even 3 1
162.7.b.c 12 9.d odd 6 1
432.7.q.b 12 4.b odd 2 1
432.7.q.b 12 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(54, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 32 T^{2} + 1024)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 54\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 49\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
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