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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,6,Mod(31,112)] 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("112.31"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 112.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.9629878191\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{6}\cdot 7^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1 + 1) q^{3} + ( - \beta_{6} - \beta_{2} + 2 \beta_1 + 3) q^{5} + (\beta_{3} - \beta_{2} + 27 \beta_1 + 15) q^{7} + ( - \beta_{12} + \beta_{11} + \cdots + 76 \beta_1) q^{9} + ( - \beta_{13} + \beta_{10} + \cdots - 18) q^{11}+ \cdots + ( - 39 \beta_{13} - 220 \beta_{12} + \cdots + 22849) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 9 q^{3} + 33 q^{5} + 28 q^{7} - 538 q^{9} - 333 q^{11} + 801 q^{17} + 2135 q^{19} + 2017 q^{21} - 2667 q^{23} + 5434 q^{25} - 17910 q^{27} + 684 q^{29} - 3119 q^{31} + 29013 q^{33} + 2247 q^{35}+ \cdots + 124833 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\!\cdots\!69 \nu^{13} + \cdots + 70\!\cdots\!76 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 88\!\cdots\!03 \nu^{13} + \cdots + 51\!\cdots\!68 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!53 \nu^{13} + \cdots + 99\!\cdots\!12 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!57 \nu^{13} + \cdots - 48\!\cdots\!52 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!17 \nu^{13} + \cdots - 59\!\cdots\!20 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52\!\cdots\!71 \nu^{13} + \cdots + 88\!\cdots\!24 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59\!\cdots\!83 \nu^{13} + \cdots + 18\!\cdots\!08 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!01 \nu^{13} + \cdots - 27\!\cdots\!56 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30\!\cdots\!19 \nu^{13} + \cdots + 62\!\cdots\!16 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!07 \nu^{13} + \cdots - 16\!\cdots\!24 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 34\!\cdots\!25 \nu^{13} + \cdots + 42\!\cdots\!44 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!77 \nu^{13} + \cdots + 91\!\cdots\!68 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{13} + \cdots - 40\!\cdots\!12 ) / 78\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 14 \beta_{13} + 7 \beta_{10} + \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 8 \beta_{6} + 231 \beta_{5} + \cdots - 1 ) / 504 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 77 \beta_{13} + 126 \beta_{12} + 77 \beta_{10} + 254 \beta_{9} + 223 \beta_{8} + 272 \beta_{7} + \cdots - 98702 ) / 504 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1141 \beta_{13} - 336 \beta_{11} - 2282 \beta_{10} - 1471 \beta_{9} + 361 \beta_{8} + 732 \beta_{7} + \cdots + 226129 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 139286 \beta_{13} - 66402 \beta_{12} + 66402 \beta_{11} + 69643 \beta_{10} - 13061 \beta_{9} + \cdots - 90385 ) / 504 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2163245 \beta_{13} + 1098216 \beta_{12} + 2163245 \beta_{10} + 4115960 \beta_{9} + 3364003 \beta_{8} + \cdots - 767447438 ) / 504 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17390331 \beta_{13} - 13248186 \beta_{11} - 34780662 \beta_{10} - 33817841 \beta_{9} + \cdots + 8994921755 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2931230414 \beta_{13} - 882704592 \beta_{12} + 882704592 \beta_{11} + 1465615207 \beta_{10} + \cdots - 1803643717 ) / 504 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 37413983579 \beta_{13} + 26068077378 \beta_{12} + 37413983579 \beta_{10} + 78473445458 \beta_{9} + \cdots - 17785045049942 ) / 504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 338182418623 \beta_{13} - 215973097368 \beta_{11} - 676364837246 \beta_{10} - 588086991633 \beta_{9} + \cdots + 148416627744183 ) / 168 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 52935985048226 \beta_{13} - 17830890690606 \beta_{12} + 17830890690606 \beta_{11} + \cdots - 32296503570787 ) / 504 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 707831842633559 \beta_{13} + 461614992385680 \beta_{12} + 707831842633559 \beta_{10} + \cdots - 31\!\cdots\!58 ) / 504 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 62\!\cdots\!25 \beta_{13} + \cdots + 28\!\cdots\!93 ) / 168 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 99\!\cdots\!94 \beta_{13} + \cdots - 60\!\cdots\!31 ) / 504 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
7.14120 + 12.3689i
8.45653 + 14.6471i
−2.79973 − 4.84928i
5.49560 + 9.51866i
−1.05545 − 1.82809i
−13.2422 − 22.9362i
−2.49593 − 4.32307i
7.14120 − 12.3689i
8.45653 − 14.6471i
−2.79973 + 4.84928i
5.49560 − 9.51866i
−1.05545 + 1.82809i
−13.2422 + 22.9362i
−2.49593 + 4.32307i
0 −11.5627 + 20.0272i 0 −75.7872 + 43.7557i 0 −90.8476 − 92.4863i 0 −145.892 − 252.692i 0
31.2 0 −9.30172 + 16.1110i 0 16.3894 − 9.46243i 0 83.8668 + 98.8603i 0 −51.5439 − 89.2767i 0
31.3 0 −4.77385 + 8.26855i 0 88.0882 − 50.8577i 0 −12.6490 − 129.023i 0 75.9208 + 131.499i 0
31.4 0 3.33086 − 5.76923i 0 0.990405 − 0.571810i 0 −129.618 − 2.46714i 0 99.3107 + 172.011i 0
31.5 0 4.47608 − 7.75280i 0 −62.1336 + 35.8729i 0 115.750 − 58.3859i 0 81.4294 + 141.040i 0
31.6 0 6.90723 − 11.9637i 0 −4.24780 + 2.45247i 0 −30.4454 + 126.016i 0 26.0804 + 45.1725i 0
31.7 0 15.4241 − 26.7153i 0 53.2006 − 30.7154i 0 77.9434 − 103.595i 0 −354.305 − 613.675i 0
47.1 0 −11.5627 − 20.0272i 0 −75.7872 − 43.7557i 0 −90.8476 + 92.4863i 0 −145.892 + 252.692i 0
47.2 0 −9.30172 − 16.1110i 0 16.3894 + 9.46243i 0 83.8668 − 98.8603i 0 −51.5439 + 89.2767i 0
47.3 0 −4.77385 − 8.26855i 0 88.0882 + 50.8577i 0 −12.6490 + 129.023i 0 75.9208 − 131.499i 0
47.4 0 3.33086 + 5.76923i 0 0.990405 + 0.571810i 0 −129.618 + 2.46714i 0 99.3107 − 172.011i 0
47.5 0 4.47608 + 7.75280i 0 −62.1336 − 35.8729i 0 115.750 + 58.3859i 0 81.4294 − 141.040i 0
47.6 0 6.90723 + 11.9637i 0 −4.24780 − 2.45247i 0 −30.4454 − 126.016i 0 26.0804 − 45.1725i 0
47.7 0 15.4241 + 26.7153i 0 53.2006 + 30.7154i 0 77.9434 + 103.595i 0 −354.305 + 613.675i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.6.p.c yes 14
4.b odd 2 1 112.6.p.b ✓ 14
7.c even 3 1 784.6.f.c 14
7.d odd 6 1 112.6.p.b ✓ 14
7.d odd 6 1 784.6.f.d 14
28.f even 6 1 inner 112.6.p.c yes 14
28.f even 6 1 784.6.f.c 14
28.g odd 6 1 784.6.f.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.6.p.b ✓ 14 4.b odd 2 1
112.6.p.b ✓ 14 7.d odd 6 1
112.6.p.c yes 14 1.a even 1 1 trivial
112.6.p.c yes 14 28.f even 6 1 inner
784.6.f.c 14 7.c even 3 1
784.6.f.c 14 28.f even 6 1
784.6.f.d 14 7.d odd 6 1
784.6.f.d 14 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 9 T_{3}^{13} + 1160 T_{3}^{12} - 1311 T_{3}^{11} + 933393 T_{3}^{10} + \cdots + 10\!\cdots\!29 \) acting on \(S_{6}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 17\!\cdots\!43 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 49\!\cdots\!27 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 66\!\cdots\!87 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 29\!\cdots\!29 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 11\!\cdots\!75 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 38\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 91\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 29\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 25\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 43\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 19\!\cdots\!75 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 25\!\cdots\!23 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 85\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 48\!\cdots\!67 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 28\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 24\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 15\!\cdots\!83 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 66\!\cdots\!88 \) Copy content Toggle raw display
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