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Properties

Label 112.6.i.g
Level $112$
Weight $6$
Character orbit 112.i
Analytic conductor $17.963$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,6,Mod(65,112)] 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("112.65"); 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([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,13] 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
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: \( x^{10} - 3 x^{9} - 119 x^{8} - 521 x^{7} - 898 x^{6} + 27806 x^{5} + 657990 x^{4} + 3648839 x^{3} + \cdots + 92895579 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 3 \beta_1 + 3) q^{3} + ( - \beta_{6} + 6 \beta_1) q^{5} + ( - \beta_{8} + 13 \beta_1 + 16) q^{7} + (\beta_{7} - 3 \beta_{6} + \cdots + 47 \beta_1) q^{9} + ( - 3 \beta_{9} + 4 \beta_{8} + \cdots - 72) q^{11}+ \cdots + ( - 872 \beta_{9} + 323 \beta_{8} + \cdots + 61297) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 13 q^{3} - 31 q^{5} + 92 q^{7} - 230 q^{9} - 351 q^{11} - 108 q^{13} - 1214 q^{15} - 111 q^{17} + 1035 q^{19} - 1365 q^{21} + 3639 q^{23} - 1540 q^{25} - 7214 q^{27} - 1468 q^{29} + 7677 q^{31} + 7439 q^{33}+ \cdots + 600308 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 119 x^{8} - 521 x^{7} - 898 x^{6} + 27806 x^{5} + 657990 x^{4} + 3648839 x^{3} + \cdots + 92895579 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17151069986588 \nu^{9} + 80250234146853 \nu^{8} + \cdots - 28\!\cdots\!55 ) / 70\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83\!\cdots\!81 \nu^{9} + \cdots - 11\!\cdots\!41 ) / 16\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\!\cdots\!99 \nu^{9} + \cdots + 49\!\cdots\!28 ) / 44\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36\!\cdots\!86 \nu^{9} + \cdots - 22\!\cdots\!73 ) / 16\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!65 \nu^{9} + \cdots - 30\!\cdots\!37 ) / 44\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60\!\cdots\!44 \nu^{9} + \cdots + 13\!\cdots\!09 ) / 62\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42\!\cdots\!22 \nu^{9} + \cdots + 47\!\cdots\!71 ) / 44\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\!\cdots\!65 \nu^{9} + \cdots - 35\!\cdots\!58 ) / 48\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!20 \nu^{9} + \cdots - 58\!\cdots\!27 ) / 16\!\cdots\!91 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 5 \beta_{8} + 4 \beta_{7} + \beta_{6} - 8 \beta_{5} - 35 \beta_{4} - 10 \beta_{3} + \cdots + 50 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{9} + 58 \beta_{8} + 80 \beta_{7} - 58 \beta_{6} - 73 \beta_{5} + 413 \beta_{4} + \cdots + 5770 ) / 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 667 \beta_{9} + 1003 \beta_{8} + 584 \beta_{7} - 1147 \beta_{6} - 712 \beta_{5} + 1001 \beta_{4} + \cdots + 27076 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 596 \beta_{9} + 3176 \beta_{8} + 2181 \beta_{7} - 2470 \beta_{6} - 2248 \beta_{5} + 2527 \beta_{4} + \cdots + 272392 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9160 \beta_{9} + 93130 \beta_{8} + 64235 \beta_{7} - 80866 \beta_{6} - 102217 \beta_{5} + \cdots + 8232142 ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 224486 \beta_{9} + 1303424 \beta_{8} + 968353 \beta_{7} - 925010 \beta_{6} - 1226285 \beta_{5} + \cdots + 78493982 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 472211 \beta_{9} + 2156285 \beta_{8} + 1387834 \beta_{7} - 1994093 \beta_{6} - 1942958 \beta_{5} + \cdots + 140378726 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8115460 \beta_{9} + 57664528 \beta_{8} + 33650740 \beta_{7} - 43496094 \beta_{6} - 50530232 \beta_{5} + \cdots + 4068170496 ) / 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 73974505 \beta_{9} + 959974858 \beta_{8} + 569474060 \beta_{7} - 696937600 \beta_{6} + \cdots + 71737018801 ) / 84 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.63676 − 1.62554i
0.261205 + 8.00832i
11.5439 + 0.371202i
−6.81880 − 2.23591i
−5.12305 − 3.65205i
1.63676 + 1.62554i
0.261205 − 8.00832i
11.5439 − 0.371202i
−6.81880 + 2.23591i
−5.12305 + 3.65205i
0 −9.28458 − 16.0814i 0 −28.9597 + 50.1597i 0 128.340 − 18.3243i 0 −50.9068 + 88.1732i 0
65.2 0 −7.26579 − 12.5847i 0 22.4113 − 38.8175i 0 −96.4534 + 86.6241i 0 15.9167 − 27.5685i 0
65.3 0 3.11280 + 5.39153i 0 −4.70275 + 8.14541i 0 −105.969 − 74.6834i 0 102.121 − 176.879i 0
65.4 0 7.34084 + 12.7147i 0 36.0801 − 62.4926i 0 111.685 − 65.8284i 0 13.7240 − 23.7707i 0
65.5 0 12.5967 + 21.8182i 0 −40.3290 + 69.8518i 0 8.39664 + 129.370i 0 −195.855 + 339.231i 0
81.1 0 −9.28458 + 16.0814i 0 −28.9597 − 50.1597i 0 128.340 + 18.3243i 0 −50.9068 − 88.1732i 0
81.2 0 −7.26579 + 12.5847i 0 22.4113 + 38.8175i 0 −96.4534 − 86.6241i 0 15.9167 + 27.5685i 0
81.3 0 3.11280 − 5.39153i 0 −4.70275 − 8.14541i 0 −105.969 + 74.6834i 0 102.121 + 176.879i 0
81.4 0 7.34084 − 12.7147i 0 36.0801 + 62.4926i 0 111.685 + 65.8284i 0 13.7240 + 23.7707i 0
81.5 0 12.5967 − 21.8182i 0 −40.3290 − 69.8518i 0 8.39664 − 129.370i 0 −195.855 − 339.231i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.6.i.g 10
4.b odd 2 1 56.6.i.a ✓ 10
7.c even 3 1 inner 112.6.i.g 10
7.c even 3 1 784.6.a.bj 5
7.d odd 6 1 784.6.a.bm 5
12.b even 2 1 504.6.s.d 10
28.d even 2 1 392.6.i.p 10
28.f even 6 1 392.6.a.i 5
28.f even 6 1 392.6.i.p 10
28.g odd 6 1 56.6.i.a ✓ 10
28.g odd 6 1 392.6.a.l 5
84.n even 6 1 504.6.s.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.i.a ✓ 10 4.b odd 2 1
56.6.i.a ✓ 10 28.g odd 6 1
112.6.i.g 10 1.a even 1 1 trivial
112.6.i.g 10 7.c even 3 1 inner
392.6.a.i 5 28.f even 6 1
392.6.a.l 5 28.g odd 6 1
392.6.i.p 10 28.d even 2 1
392.6.i.p 10 28.f even 6 1
504.6.s.d 10 12.b even 2 1
504.6.s.d 10 84.n even 6 1
784.6.a.bj 5 7.c even 3 1
784.6.a.bm 5 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 13 T_{3}^{9} + 807 T_{3}^{8} - 3142 T_{3}^{7} + 390805 T_{3}^{6} - 1914555 T_{3}^{5} + \cdots + 386099434161 \) acting on \(S_{6}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 386099434161 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 20\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 13\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T^{5} + 54 T^{4} + \cdots + 173219419488)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 22\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 36\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 36\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 69\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 92\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 27\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 60\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 15\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 53\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 79\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
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