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Properties

Label 112.8.i.d
Level $112$
Weight $8$
Character orbit 112.i
Analytic conductor $34.987$
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] = [112,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
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,-27] 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: \(34.9871228542\)
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: \( x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4}\cdot 7^{5} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{3} + 5 \beta_1 - 5) q^{3} + (\beta_{6} - \beta_{3} + 49 \beta_1) q^{5} + (\beta_{6} + 4 \beta_{5} + \beta_{4} + \cdots + 64) q^{7} + (\beta_{9} + \beta_{8} - 3 \beta_{6} + \cdots - 1) q^{9}+ \cdots + ( - 1815 \beta_{9} - 5686 \beta_{8} + \cdots + 11958415) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 41255617984 \nu^{9} + 107667922597 \nu^{8} - 14045699887767 \nu^{7} + \cdots + 50\!\cdots\!00 ) / 63\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14873845281605 \nu^{9} + \cdots - 11\!\cdots\!40 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18878237161999 \nu^{9} + 177453228114559 \nu^{8} + \cdots + 23\!\cdots\!00 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11636545172827 \nu^{9} + 9656168590565 \nu^{8} + \cdots + 43\!\cdots\!20 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8370298711069 \nu^{9} + 7698009493391 \nu^{8} + \cdots + 10\!\cdots\!56 ) / 83\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 118726983973433 \nu^{9} - 653443179493277 \nu^{8} + \cdots - 14\!\cdots\!00 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 351418300605379 \nu^{9} + \cdots - 36\!\cdots\!80 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\!\cdots\!64 \nu^{9} + \cdots - 21\!\cdots\!00 ) / 83\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!11 \nu^{9} + \cdots + 10\!\cdots\!60 ) / 20\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{9} - 5 \beta_{8} - \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 40 \beta_{3} + \cdots + 3 ) / 504 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 16 \beta_{9} + 38 \beta_{8} - 5 \beta_{7} - 4 \beta_{6} - 1277 \beta_{5} - 15 \beta_{4} + \cdots - 69369 ) / 504 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 95 \beta_{9} + 416 \beta_{8} - 113 \beta_{7} - 190 \beta_{6} - 6029 \beta_{5} + 276 \beta_{4} + \cdots - 128535 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3991 \beta_{9} - 3671 \beta_{8} - 2554 \beta_{7} + 1252 \beta_{6} - 3991 \beta_{5} - 1437 \beta_{4} + \cdots - 1437 ) / 504 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 134972 \beta_{9} + 30698 \beta_{8} + 217807 \beta_{7} - 491812 \beta_{6} + 7536523 \beta_{5} + \cdots + 215226507 ) / 504 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 171205 \beta_{9} - 1419524 \beta_{8} + 880967 \beta_{7} - 342410 \beta_{6} + 63335039 \beta_{5} + \cdots + 2811598671 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 78916229 \beta_{9} - 153780005 \beta_{8} - 24954592 \beta_{7} + 221720128 \beta_{6} + \cdots + 103870821 ) / 504 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1065092422 \beta_{9} + 1208137316 \beta_{8} - 993569975 \beta_{7} + 1673224172 \beta_{6} + \cdots - 3130977766083 ) / 504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 870652005 \beta_{9} + 6542734464 \beta_{8} - 2400715227 \beta_{7} - 1741304010 \beta_{6} + \cdots - 5023199128205 ) / 56 \) 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
−1.38371 + 2.39666i
1.14872 − 1.98964i
−1.17613 + 2.03713i
−8.10827 + 14.0439i
10.0194 − 17.3541i
−1.38371 − 2.39666i
1.14872 + 1.98964i
−1.17613 − 2.03713i
−8.10827 − 14.0439i
10.0194 + 17.3541i
0 −37.9099 − 65.6620i 0 274.614 − 475.646i 0 −907.484 − 4.07000i 0 −1780.83 + 3084.49i 0
65.2 0 −29.3690 − 50.8685i 0 −209.191 + 362.329i 0 505.918 + 753.386i 0 −631.572 + 1093.92i 0
65.3 0 2.50059 + 4.33114i 0 89.0278 − 154.201i 0 795.793 − 436.183i 0 1080.99 − 1872.34i 0
65.4 0 9.22936 + 15.9857i 0 −88.1161 + 152.621i 0 −836.746 − 351.283i 0 923.138 − 1598.92i 0
65.5 0 42.0490 + 72.8309i 0 58.1649 − 100.745i 0 276.518 + 864.338i 0 −2442.73 + 4230.93i 0
81.1 0 −37.9099 + 65.6620i 0 274.614 + 475.646i 0 −907.484 + 4.07000i 0 −1780.83 − 3084.49i 0
81.2 0 −29.3690 + 50.8685i 0 −209.191 − 362.329i 0 505.918 − 753.386i 0 −631.572 − 1093.92i 0
81.3 0 2.50059 − 4.33114i 0 89.0278 + 154.201i 0 795.793 + 436.183i 0 1080.99 + 1872.34i 0
81.4 0 9.22936 − 15.9857i 0 −88.1161 − 152.621i 0 −836.746 + 351.283i 0 923.138 + 1598.92i 0
81.5 0 42.0490 − 72.8309i 0 58.1649 + 100.745i 0 276.518 − 864.338i 0 −2442.73 − 4230.93i 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 112.8.i.d 10
4.b odd 2 1 28.8.e.a ✓ 10
7.c even 3 1 inner 112.8.i.d 10
12.b even 2 1 252.8.k.c 10
28.d even 2 1 196.8.e.f 10
28.f even 6 1 196.8.a.e 5
28.f even 6 1 196.8.e.f 10
28.g odd 6 1 28.8.e.a ✓ 10
28.g odd 6 1 196.8.a.d 5
84.n even 6 1 252.8.k.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.8.e.a ✓ 10 4.b odd 2 1
28.8.e.a ✓ 10 28.g odd 6 1
112.8.i.d 10 1.a even 1 1 trivial
112.8.i.d 10 7.c even 3 1 inner
196.8.a.d 5 28.g odd 6 1
196.8.a.e 5 28.f even 6 1
196.8.e.f 10 28.d even 2 1
196.8.e.f 10 28.f even 6 1
252.8.k.c 10 12.b even 2 1
252.8.k.c 10 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 27 T_{3}^{9} + 8683 T_{3}^{8} + 202998 T_{3}^{7} + 60752893 T_{3}^{6} + \cdots + 11\!\cdots\!29 \) acting on \(S_{8}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 70\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 23\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 51\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 63\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 63\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 90\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 47\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 93\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 76\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 21\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 78\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 34\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 47\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 58\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 95\!\cdots\!40)^{2} \) Copy content Toggle raw display
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