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Properties

Label 112.13.c.c
Level $112$
Weight $13$
Character orbit 112.c
Analytic conductor $102.367$
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] = [112,13,Mod(97,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.97");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 112.c (of order \(2\), degree \(1\), 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: \(102.367307535\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 154710x^{6} + 8245426887x^{4} + 174724076278260x^{2} + 1264170035276291934 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{4}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{6} - 11 \beta_{5}) q^{5} + (\beta_{7} - 4 \beta_{6} + \cdots - 24395) q^{7}+ \cdots + ( - 5 \beta_{4} - 5 \beta_{3} + \cdots - 184863) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{6} - 11 \beta_{5}) q^{5} + (\beta_{7} - 4 \beta_{6} + \cdots - 24395) q^{7}+ \cdots + (2129490 \beta_{4} + \cdots + 225119282058) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 195160 q^{7} - 1478904 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 195160 q^{7} - 1478904 q^{9} + 213840 q^{11} - 65882304 q^{15} - 4449984 q^{21} - 156731760 q^{23} + 191237000 q^{25} + 308853648 q^{29} + 3764734848 q^{35} - 3243600880 q^{37} - 13521315264 q^{39} - 21006302000 q^{43} - 19258758904 q^{49} + 80965832832 q^{51} + 180445637520 q^{53} - 63145962240 q^{57} + 402706514280 q^{63} - 424890168192 q^{65} - 369211259440 q^{67} - 574058144304 q^{71} - 73915435440 q^{77} + 607826610128 q^{79} + 919051941384 q^{81} - 247202260608 q^{85} - 144527421696 q^{91} + 2292312458880 q^{93} + 1053641981376 q^{95} + 1800954256464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 154710x^{6} + 8245426887x^{4} + 174724076278260x^{2} + 1264170035276291934 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 500\nu^{6} + 35855356\nu^{4} + 346467953880\nu^{2} - 3299941733579064 ) / 2344559972559 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 38674502 \nu^{6} - 4173859898274 \nu^{4} + \cdots - 13\!\cdots\!04 ) / 218044077447987 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 985539289 \nu^{7} + 5198341035612 \nu^{6} + 557070330448179 \nu^{5} + \cdots + 12\!\cdots\!48 ) / 51\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 985539289 \nu^{7} + 5198341035612 \nu^{6} - 557070330448179 \nu^{5} + \cdots + 12\!\cdots\!48 ) / 51\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1774768547 \nu^{7} + 176258268664353 \nu^{5} + \cdots + 42\!\cdots\!34 \nu ) / 51\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3573248699 \nu^{7} - 538494861420537 \nu^{5} + \cdots - 29\!\cdots\!26 \nu ) / 25\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7700612389 \nu^{7} + \cdots - 48\!\cdots\!70 \nu ) / 57\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -23\beta_{7} + 135\beta_{6} - 349\beta_{5} + 5\beta_{4} - 5\beta_{3} ) / 10752 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -16\beta_{4} - 16\beta_{3} - 2\beta_{2} - 159\beta _1 - 4331880 ) / 112 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 363199\beta_{7} - 2338479\beta_{6} + 4770773\beta_{5} + 3683\beta_{4} - 3683\beta_{3} ) / 3584 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 616590\beta_{4} + 616590\beta_{3} + 68355\beta_{2} - 1124106\beta _1 + 104220624564 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 19794993669 \beta_{7} + 127358619093 \beta_{6} - 262594807191 \beta_{5} + \cdots + 2187068049 \beta_{3} ) / 3584 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 38672620650 \beta_{4} - 38672620650 \beta_{3} - 4208849811 \beta_{2} + 398289563217 \beta _1 - 56\!\cdots\!80 ) / 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11\!\cdots\!39 \beta_{7} + \cdots - 167084551875483 \beta_{3} ) / 3584 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
− 242.361i
237.947i
− 149.422i
− 130.480i
130.480i
149.422i
− 237.947i
242.361i
0 − 1265.70i 0 − 23188.3i 0 −109682. + 42556.3i 0 −1.07056e6 0
97.2 0 − 1072.00i 0 1140.63i 0 −74038.7 − 91430.6i 0 −617752. 0
97.3 0 − 321.888i 0 − 17149.5i 0 71756.5 + 93232.5i 0 427829. 0
97.4 0 − 102.042i 0 6916.58i 0 14384.7 + 116766.i 0 521029. 0
97.5 0 102.042i 0 − 6916.58i 0 14384.7 − 116766.i 0 521029. 0
97.6 0 321.888i 0 17149.5i 0 71756.5 − 93232.5i 0 427829. 0
97.7 0 1072.00i 0 − 1140.63i 0 −74038.7 + 91430.6i 0 −617752. 0
97.8 0 1265.70i 0 23188.3i 0 −109682. − 42556.3i 0 −1.07056e6 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.13.c.c 8
4.b odd 2 1 14.13.b.a ✓ 8
7.b odd 2 1 inner 112.13.c.c 8
12.b even 2 1 126.13.c.a 8
28.d even 2 1 14.13.b.a ✓ 8
28.f even 6 2 98.13.d.b 16
28.g odd 6 2 98.13.d.b 16
84.h odd 2 1 126.13.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.b.a ✓ 8 4.b odd 2 1
14.13.b.a ✓ 8 28.d even 2 1
98.13.d.b 16 28.f even 6 2
98.13.d.b 16 28.g odd 6 2
112.13.c.c 8 1.a even 1 1 trivial
112.13.c.c 8 7.b odd 2 1 inner
126.13.c.a 8 12.b even 2 1
126.13.c.a 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2865216T_{3}^{6} + 2155787566560T_{3}^{4} + 212887983685229568T_{3}^{2} + 1986183742263160725504 \) acting on \(S_{13}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 24\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
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