Properties

Label 112.9.s.b
Level $112$
Weight $9$
Character orbit 112.s
Analytic conductor $45.626$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,9,Mod(17,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.17");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.6264043268\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + 4552713792 x^{3} + 25685059104 x^{2} + \cdots + 63214027776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 5 \beta_1 + 6) q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + 56 \beta_1 - 112) q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 50 \beta_1 - 178) q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 378 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 5 \beta_1 + 6) q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + 56 \beta_1 - 112) q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 50 \beta_1 - 178) q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 378 \beta_1 + 3) q^{9} + ( - 3 \beta_{9} + \beta_{8} - 8 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + \cdots - 742) q^{11}+ \cdots + (2340 \beta_{9} - 2340 \beta_{8} - 29004 \beta_{7} + \cdots + 11749365) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 81 q^{3} - 837 q^{5} - 1526 q^{7} + 1902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 81 q^{3} - 837 q^{5} - 1526 q^{7} + 1902 q^{9} - 3705 q^{11} - 76134 q^{15} + 78003 q^{17} + 96741 q^{19} - 153153 q^{21} - 208533 q^{23} + 367978 q^{25} + 754764 q^{29} + 1053717 q^{31} - 1032993 q^{33} + 1306389 q^{35} - 998075 q^{37} - 1431900 q^{39} - 738292 q^{43} - 7432758 q^{45} - 710883 q^{47} + 13288114 q^{49} + 2571909 q^{51} + 10501461 q^{53} - 2744514 q^{57} + 37089081 q^{59} - 8180481 q^{61} - 47152518 q^{63} + 21459108 q^{65} - 48020189 q^{67} + 31918236 q^{71} - 133345593 q^{73} + 119504178 q^{75} + 188477625 q^{77} - 53590181 q^{79} + 173295063 q^{81} - 157179282 q^{85} + 413284806 q^{87} - 241368273 q^{89} - 420709128 q^{91} + 137961999 q^{93} - 347126775 q^{95} + 117796500 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + 4552713792 x^{3} + 25685059104 x^{2} + \cdots + 63214027776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\!\cdots\!92 \nu^{9} + \cdots + 42\!\cdots\!80 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!95 \nu^{9} + \cdots - 13\!\cdots\!16 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!87 \nu^{9} + \cdots + 67\!\cdots\!92 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!16 \nu^{9} + \cdots + 35\!\cdots\!08 ) / 36\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53\!\cdots\!07 \nu^{9} + \cdots - 92\!\cdots\!88 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!37 \nu^{9} + \cdots + 18\!\cdots\!20 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 64\!\cdots\!02 \nu^{9} + \cdots - 18\!\cdots\!56 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83\!\cdots\!77 \nu^{9} + \cdots + 69\!\cdots\!20 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!43 \nu^{9} + \cdots + 84\!\cdots\!52 ) / 22\!\cdots\!68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} - 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 25\beta_{3} - 13\beta_{2} - 17\beta _1 + 30 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16 \beta_{9} - 23 \beta_{8} - 47 \beta_{7} + 39 \beta_{6} + 23 \beta_{5} + 47 \beta_{4} - 499 \beta_{3} - 1021 \beta_{2} - 96255 \beta _1 - 498 ) / 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1857 \beta_{9} - 1857 \beta_{8} + 2185 \beta_{7} + 376 \beta_{6} - 376 \beta_{5} + 4370 \beta_{4} + 23764 \beta_{3} - 27854 \beta_{2} - 328 \beta _1 - 3094340 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 62551 \beta_{9} - 25040 \beta_{8} + 206046 \beta_{7} - 62551 \beta_{6} - 87591 \beta_{5} + 103023 \beta_{4} + 2198383 \beta_{3} + 1130467 \beta_{2} + 166598071 \beta _1 - 167769010 ) / 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1031736 \beta_{9} + 3046569 \beta_{8} + 4798825 \beta_{7} - 4078305 \beta_{6} - 3046569 \beta_{5} - 4798825 \beta_{4} + 55476845 \beta_{3} + 114000259 \beta_{2} + \cdots + 56771158 ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 191587143 \beta_{9} + 191587143 \beta_{8} - 225528271 \beta_{7} - 50945168 \beta_{6} + 50945168 \beta_{5} - 451056542 \beta_{4} - 2327275836 \beta_{3} + \cdots + 358537009892 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6622946953 \beta_{9} + 2337530360 \beta_{8} - 21085302418 \beta_{7} + 6622946953 \beta_{6} + 8960477313 \beta_{5} - 10542651209 \beta_{4} + \cdots + 16543071944622 ) / 168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 110369207632 \beta_{9} - 309539717399 \beta_{8} - 494220385071 \beta_{7} + 419908925031 \beta_{6} + 309539717399 \beta_{5} + 494220385071 \beta_{4} + \cdots - 5777178183538 ) / 168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 19661294346273 \beta_{9} - 19661294346273 \beta_{8} + 23136427328297 \beta_{7} + 5152654242744 \beta_{6} - 5152654242744 \beta_{5} + \cdots - 36\!\cdots\!92 ) / 168 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
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
−9.33129 + 16.1623i
−0.957903 + 1.65914i
−4.78762 + 8.29240i
23.4172 40.5598i
−7.84041 + 13.5800i
−9.33129 16.1623i
−0.957903 1.65914i
−4.78762 8.29240i
23.4172 + 40.5598i
−7.84041 13.5800i
0 −83.9248 48.4540i 0 336.492 194.274i 0 −2329.92 + 579.874i 0 1415.08 + 2450.99i 0
17.2 0 −67.4293 38.9303i 0 −616.084 + 355.696i 0 2382.47 + 297.754i 0 −249.357 431.899i 0
17.3 0 25.9870 + 15.0036i 0 485.304 280.190i 0 622.564 2318.88i 0 −2830.28 4902.20i 0
17.4 0 80.8255 + 46.6646i 0 327.308 188.971i 0 894.589 + 2228.12i 0 1074.68 + 1861.40i 0
17.5 0 85.0416 + 49.0988i 0 −951.520 + 549.361i 0 −2332.69 568.626i 0 1540.88 + 2668.88i 0
33.1 0 −83.9248 + 48.4540i 0 336.492 + 194.274i 0 −2329.92 579.874i 0 1415.08 2450.99i 0
33.2 0 −67.4293 + 38.9303i 0 −616.084 355.696i 0 2382.47 297.754i 0 −249.357 + 431.899i 0
33.3 0 25.9870 15.0036i 0 485.304 + 280.190i 0 622.564 + 2318.88i 0 −2830.28 + 4902.20i 0
33.4 0 80.8255 46.6646i 0 327.308 + 188.971i 0 894.589 2228.12i 0 1074.68 1861.40i 0
33.5 0 85.0416 49.0988i 0 −951.520 549.361i 0 −2332.69 + 568.626i 0 1540.88 2668.88i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.9.s.b 10
4.b odd 2 1 28.9.h.a 10
7.d odd 6 1 inner 112.9.s.b 10
12.b even 2 1 252.9.z.c 10
28.d even 2 1 196.9.h.a 10
28.f even 6 1 28.9.h.a 10
28.f even 6 1 196.9.b.a 10
28.g odd 6 1 196.9.b.a 10
28.g odd 6 1 196.9.h.a 10
84.j odd 6 1 252.9.z.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.9.h.a 10 4.b odd 2 1
28.9.h.a 10 28.f even 6 1
112.9.s.b 10 1.a even 1 1 trivial
112.9.s.b 10 7.d odd 6 1 inner
196.9.b.a 10 28.f even 6 1
196.9.b.a 10 28.g odd 6 1
196.9.h.a 10 28.d even 2 1
196.9.h.a 10 28.g odd 6 1
252.9.z.c 10 12.b even 2 1
252.9.z.c 10 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 81 T_{3}^{9} - 14073 T_{3}^{8} + 1317060 T_{3}^{7} + 169089741 T_{3}^{6} - 14981776281 T_{3}^{5} - 709991029161 T_{3}^{4} + 73631981184264 T_{3}^{3} + \cdots + 43\!\cdots\!27 \) acting on \(S_{9}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 81 T^{9} + \cdots + 43\!\cdots\!27 \) Copy content Toggle raw display
$5$ \( T^{10} + 837 T^{9} + \cdots + 41\!\cdots\!75 \) Copy content Toggle raw display
$7$ \( T^{10} + 1526 T^{9} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{10} + 3705 T^{9} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{10} + 4263799464 T^{8} + \cdots + 73\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{10} - 78003 T^{9} + \cdots + 25\!\cdots\!43 \) Copy content Toggle raw display
$19$ \( T^{10} - 96741 T^{9} + \cdots + 38\!\cdots\!47 \) Copy content Toggle raw display
$23$ \( T^{10} + 208533 T^{9} + \cdots + 30\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{5} - 377382 T^{4} + \cdots + 35\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 1053717 T^{9} + \cdots + 18\!\cdots\!07 \) Copy content Toggle raw display
$37$ \( T^{10} + 998075 T^{9} + \cdots + 81\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{10} + 45924266952552 T^{8} + \cdots + 31\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( (T^{5} + 369146 T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 710883 T^{9} + \cdots + 30\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{10} - 10501461 T^{9} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{10} - 37089081 T^{9} + \cdots + 16\!\cdots\!83 \) Copy content Toggle raw display
$61$ \( T^{10} + 8180481 T^{9} + \cdots + 62\!\cdots\!03 \) Copy content Toggle raw display
$67$ \( T^{10} + 48020189 T^{9} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{5} - 15959118 T^{4} + \cdots + 10\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 133345593 T^{9} + \cdots + 56\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( T^{10} + 53590181 T^{9} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 56\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + 241368273 T^{9} + \cdots + 32\!\cdots\!23 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 24\!\cdots\!88 \) Copy content Toggle raw display
show more
show less