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Properties

Label 112.9.s.a
Level $112$
Weight $9$
Character orbit 112.s
Analytic conductor $45.626$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 7)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + 7 \beta_{2} + 14) q^{3} + (2 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + \cdots - 70) q^{5}+ \cdots + (57 \beta_{6} + 42 \beta_{5} + \cdots + 99) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + 7 \beta_{2} + 14) q^{3} + (2 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + \cdots - 70) q^{5}+ \cdots + ( - 141009 \beta_{7} - 1624701 \beta_{6} + \cdots - 84030501) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 84 q^{3} - 840 q^{5} + 140 q^{7} + 396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 84 q^{3} - 840 q^{5} + 140 q^{7} + 396 q^{9} - 1784 q^{11} - 131808 q^{15} - 141456 q^{17} + 257544 q^{19} + 474012 q^{21} + 348940 q^{23} - 557752 q^{25} + 4983176 q^{29} + 2376696 q^{31} - 5719140 q^{33} - 2491272 q^{35} + 492740 q^{37} + 2850372 q^{39} - 4448432 q^{43} + 3328164 q^{45} - 2704128 q^{47} - 3811024 q^{49} + 350856 q^{51} + 2281460 q^{53} - 43638408 q^{57} - 25291140 q^{59} + 59368764 q^{61} + 60081756 q^{63} + 16923396 q^{65} + 107108 q^{67} + 82809760 q^{71} + 116758404 q^{73} - 79832424 q^{75} - 58244200 q^{77} + 50628092 q^{79} - 96868872 q^{81} + 119231208 q^{85} - 26702676 q^{87} - 2322516 q^{89} + 151794552 q^{91} - 57693204 q^{93} + 172787052 q^{95} - 672244008 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40665038848 \nu^{7} + 933800395536 \nu^{6} - 23107427488768 \nu^{5} + \cdots - 24\!\cdots\!71 ) / 78\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4986016178489 \nu^{7} + 729234428373612 \nu^{6} + \cdots + 58\!\cdots\!33 ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 398294048 \nu^{7} - 412145664 \nu^{6} - 226325883218 \nu^{5} + 234196900224 \nu^{4} + \cdots - 34\!\cdots\!96 ) / 16\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 142450897959062 \nu^{7} + 415028421151641 \nu^{6} + \cdots + 22\!\cdots\!49 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35983628949817 \nu^{7} - 17429303220966 \nu^{6} + \cdots + 87\!\cdots\!11 ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43918482590713 \nu^{7} - 39288196594836 \nu^{6} + \cdots + 59\!\cdots\!21 ) / 21\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{4} - \beta_{3} + 296\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 50\beta_{6} - 28\beta_{5} - 547\beta_{4} + 882 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 592\beta_{6} + 592\beta_{3} - 161165\beta_{2} + 886\beta _1 - 161165 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4728 \beta_{7} - 16576 \beta_{6} + 33152 \beta_{5} + 310933 \beta_{4} - 4728 \beta_{3} + \cdots - 310933 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -338161\beta_{7} - 321697\beta_{6} - 8232\beta_{5} - 671263\beta_{4} + 91505156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6008350\beta_{6} - 9419116\beta_{5} + 3410766\beta_{3} - 692305026\beta_{2} + 177419623\beta _1 - 692305026 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\) \(1\)

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
−2.23583 − 3.87257i
11.6264 + 20.1376i
−12.1698 − 21.0787i
2.77916 + 4.81365i
−2.23583 + 3.87257i
11.6264 − 20.1376i
−12.1698 + 21.0787i
2.77916 − 4.81365i
0 −58.8837 − 33.9965i 0 586.541 − 338.640i 0 168.652 − 2395.07i 0 −968.977 − 1678.32i 0
17.2 0 −31.2153 − 18.0222i 0 −305.183 + 176.198i 0 −2236.80 + 872.649i 0 −2630.90 − 4556.86i 0
17.3 0 7.68317 + 4.43588i 0 −538.352 + 310.818i 0 −207.497 + 2392.02i 0 −3241.15 − 5613.83i 0
17.4 0 124.416 + 71.8315i 0 −163.006 + 94.1113i 0 2345.65 + 512.580i 0 7039.03 + 12192.0i 0
33.1 0 −58.8837 + 33.9965i 0 586.541 + 338.640i 0 168.652 + 2395.07i 0 −968.977 + 1678.32i 0
33.2 0 −31.2153 + 18.0222i 0 −305.183 − 176.198i 0 −2236.80 − 872.649i 0 −2630.90 + 4556.86i 0
33.3 0 7.68317 − 4.43588i 0 −538.352 − 310.818i 0 −207.497 − 2392.02i 0 −3241.15 + 5613.83i 0
33.4 0 124.416 − 71.8315i 0 −163.006 − 94.1113i 0 2345.65 − 512.580i 0 7039.03 − 12192.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
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.a 8
4.b odd 2 1 7.9.d.a ✓ 8
7.d odd 6 1 inner 112.9.s.a 8
12.b even 2 1 63.9.m.b 8
28.d even 2 1 49.9.d.c 8
28.f even 6 1 7.9.d.a ✓ 8
28.f even 6 1 49.9.b.a 8
28.g odd 6 1 49.9.b.a 8
28.g odd 6 1 49.9.d.c 8
84.j odd 6 1 63.9.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.9.d.a ✓ 8 4.b odd 2 1
7.9.d.a ✓ 8 28.f even 6 1
49.9.b.a 8 28.f even 6 1
49.9.b.a 8 28.g odd 6 1
49.9.d.c 8 28.d even 2 1
49.9.d.c 8 28.g odd 6 1
63.9.m.b 8 12.b even 2 1
63.9.m.b 8 84.j odd 6 1
112.9.s.a 8 1.a even 1 1 trivial
112.9.s.a 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 84 T_{3}^{7} - 9792 T_{3}^{6} + 1020096 T_{3}^{5} + 156051981 T_{3}^{4} + \cdots + 9756895701609 \) acting on \(S_{9}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 9756895701609 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 39\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 21\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 98\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 47\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 25\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 40\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 63\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 31\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
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