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Properties

Label 112.10.i.d
Level $112$
Weight $10$
Character orbit 112.i
Analytic conductor $57.684$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,10,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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, 10, names="a")
 
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, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0] 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: \(57.6840136504\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - x^{11} + 7779 x^{10} + 365650 x^{9} + 45150527 x^{8} + 2129694927 x^{7} + 167292926543 x^{6} + \cdots + 68\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{4}\cdot 7^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{3} + 161 \beta_1 - 161) q^{5} + ( - 5 \beta_{4} - 6 \beta_{3} + \cdots - 337) q^{7} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots - 3603) q^{9}+ \cdots + ( - 24542 \beta_{11} + \cdots - 679317726) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 966 q^{5} - 7696 q^{7} - 21620 q^{9} + 47640 q^{11} + 103432 q^{13} + 400280 q^{15} - 402234 q^{17} + 519960 q^{19} - 1025858 q^{21} + 683124 q^{23} - 1134656 q^{25} + 7641648 q^{27} + 4252680 q^{29}+ \cdots - 8151941296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 7779 x^{10} + 365650 x^{9} + 45150527 x^{8} + 2129694927 x^{7} + 167292926543 x^{6} + \cdots + 68\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 95\!\cdots\!99 \nu^{11} + \cdots - 18\!\cdots\!12 ) / 41\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!11 \nu^{11} + \cdots + 48\!\cdots\!72 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!29 \nu^{11} + \cdots - 46\!\cdots\!96 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!60 \nu^{11} + \cdots + 32\!\cdots\!76 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!13 \nu^{11} + \cdots + 37\!\cdots\!52 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!31 \nu^{11} + \cdots + 89\!\cdots\!64 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\!\cdots\!60 \nu^{11} + \cdots - 62\!\cdots\!24 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!78 \nu^{11} + \cdots - 44\!\cdots\!16 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27\!\cdots\!22 \nu^{11} + \cdots - 11\!\cdots\!72 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!52 \nu^{11} + \cdots - 14\!\cdots\!64 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 56\!\cdots\!03 \nu^{11} + \cdots + 19\!\cdots\!40 ) / 75\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{8} - 6 \beta_{7} - 6 \beta_{6} + \beta_{5} + 46 \beta_{4} + 46 \beta_{3} + \cdots + 1 ) / 672 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{11} - 27 \beta_{9} - 85 \beta_{8} - 198 \beta_{6} + 47 \beta_{5} - 12 \beta_{4} + \cdots - 1742353 ) / 672 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3865 \beta_{11} + 4025 \beta_{10} - 3813 \beta_{9} - 7994 \beta_{8} + 20082 \beta_{7} + \cdots - 62735928 ) / 672 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 206225 \beta_{11} + 284845 \beta_{10} - 9680 \beta_{9} + 137285 \beta_{8} + 1085430 \beta_{7} + \cdots - 196545 ) / 672 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 183200 \beta_{11} + 17483341 \beta_{9} + 52266823 \beta_{8} + 79650126 \beta_{6} - 20741301 \beta_{5} + \cdots + 408605081131 ) / 672 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 373557249 \beta_{11} - 512000149 \beta_{10} + 375267877 \beta_{9} + 882136142 \beta_{8} + \cdots + 10917437300620 ) / 224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 84521064253 \beta_{11} - 111227452585 \beta_{10} - 1729350852 \beta_{9} - 56085325069 \beta_{8} + \cdots + 86250415105 ) / 672 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 137808050840 \beta_{11} - 5991228994365 \beta_{9} - 17835878932255 \beta_{8} + \cdots - 16\!\cdots\!87 ) / 672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 441053116074481 \beta_{11} + 598637533156001 \beta_{10} - 424743310472481 \beta_{9} + \cdots - 11\!\cdots\!32 ) / 672 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 30\!\cdots\!31 \beta_{11} + \cdots - 31\!\cdots\!67 ) / 672 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!52 \beta_{11} + \cdots + 63\!\cdots\!63 ) / 672 \) Copy content Toggle raw display

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

Character values

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

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

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
37.0347 + 64.1459i
−19.1194 − 33.1157i
−15.6595 − 27.1230i
−16.4889 − 28.5596i
34.9935 + 60.6105i
−20.2604 − 35.0921i
37.0347 − 64.1459i
−19.1194 + 33.1157i
−15.6595 + 27.1230i
−16.4889 + 28.5596i
34.9935 − 60.6105i
−20.2604 + 35.0921i
0 −119.425 − 206.850i 0 −595.320 + 1031.12i 0 −3212.82 − 5480.09i 0 −18683.1 + 32360.1i 0
65.2 0 −60.0993 − 104.095i 0 501.604 − 868.804i 0 2319.36 + 5913.90i 0 2617.65 − 4533.90i 0
65.3 0 −22.5288 − 39.0211i 0 184.048 − 318.781i 0 2043.68 − 6014.73i 0 8826.40 − 15287.8i 0
65.4 0 23.7645 + 41.1613i 0 −1137.36 + 1969.96i 0 −4970.86 + 3955.27i 0 8712.00 − 15089.6i 0
65.5 0 82.4523 + 142.812i 0 1030.62 − 1785.08i 0 −6318.25 − 658.292i 0 −3755.26 + 6504.30i 0
65.6 0 95.8363 + 165.993i 0 −466.595 + 808.167i 0 6290.89 − 882.239i 0 −8527.68 + 14770.4i 0
81.1 0 −119.425 + 206.850i 0 −595.320 − 1031.12i 0 −3212.82 + 5480.09i 0 −18683.1 − 32360.1i 0
81.2 0 −60.0993 + 104.095i 0 501.604 + 868.804i 0 2319.36 − 5913.90i 0 2617.65 + 4533.90i 0
81.3 0 −22.5288 + 39.0211i 0 184.048 + 318.781i 0 2043.68 + 6014.73i 0 8826.40 + 15287.8i 0
81.4 0 23.7645 − 41.1613i 0 −1137.36 − 1969.96i 0 −4970.86 − 3955.27i 0 8712.00 + 15089.6i 0
81.5 0 82.4523 − 142.812i 0 1030.62 + 1785.08i 0 −6318.25 + 658.292i 0 −3755.26 − 6504.30i 0
81.6 0 95.8363 − 165.993i 0 −466.595 − 808.167i 0 6290.89 + 882.239i 0 −8527.68 − 14770.4i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
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 112.10.i.d 12
4.b odd 2 1 28.10.e.a ✓ 12
7.c even 3 1 inner 112.10.i.d 12
12.b even 2 1 252.10.k.e 12
28.d even 2 1 196.10.e.h 12
28.f even 6 1 196.10.a.e 6
28.f even 6 1 196.10.e.h 12
28.g odd 6 1 28.10.e.a ✓ 12
28.g odd 6 1 196.10.a.f 6
84.n even 6 1 252.10.k.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.10.e.a ✓ 12 4.b odd 2 1
28.10.e.a ✓ 12 28.g odd 6 1
112.10.i.d 12 1.a even 1 1 trivial
112.10.i.d 12 7.c even 3 1 inner
196.10.a.e 6 28.f even 6 1
196.10.a.f 6 28.g odd 6 1
196.10.e.h 12 28.d even 2 1
196.10.e.h 12 28.f even 6 1
252.10.k.e 12 12.b even 2 1
252.10.k.e 12 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 69859 T_{3}^{10} - 2547216 T_{3}^{9} + 3830582206 T_{3}^{8} - 93595977216 T_{3}^{7} + \cdots + 37\!\cdots\!61 \) acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 37\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 77\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 36\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 99\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 89\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 20\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 53\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 43\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 29\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 89\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 23\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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