↑

Properties

Label 192.13.g.d
Level $192$
Weight $13$
Character orbit 192.g
Analytic conductor $175.487$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,13,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \(175.486812917\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 10954 x^{10} + 45014993 x^{8} + 82985051536 x^{6} + 60318292405112 x^{4} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{102}\cdot 3^{28} \)
Twist minimal: no (minimal twist has level 96)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + (\beta_1 - 1430) q^{5} + (\beta_{9} - 8 \beta_{7} - 18 \beta_{6}) q^{7} - 177147 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + (\beta_1 - 1430) q^{5} + (\beta_{9} - 8 \beta_{7} - 18 \beta_{6}) q^{7} - 177147 q^{9} + (11 \beta_{8} - 484 \beta_{7} - 33 \beta_{6}) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots - 387090) q^{13}+ \cdots + ( - 1948617 \beta_{8} + \cdots + 5845851 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 17160 q^{5} - 2125764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 17160 q^{5} - 2125764 q^{9} - 4645080 q^{13} - 9378120 q^{17} - 16971120 q^{21} - 1015689948 q^{25} + 1774091160 q^{29} - 1025790480 q^{33} + 12228790920 q^{37} - 20957039304 q^{41} + 3039842520 q^{45} - 51766481268 q^{49} + 23389252632 q^{53} + 29147881104 q^{57} - 54489777720 q^{61} + 108400739088 q^{65} - 108868510080 q^{69} + 571141695576 q^{73} - 690293085504 q^{77} + 376572715308 q^{81} - 424438284624 q^{85} - 789948684648 q^{89} + 897357557808 q^{93} - 411208276968 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 10954 x^{10} + 45014993 x^{8} + 82985051536 x^{6} + 60318292405112 x^{4} + \cdots + 11\!\cdots\!44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33\!\cdots\!31 \nu^{10} + \cdots + 23\!\cdots\!72 ) / 43\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57\!\cdots\!37 \nu^{10} + \cdots + 28\!\cdots\!72 ) / 43\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32\!\cdots\!37 \nu^{10} + \cdots - 10\!\cdots\!24 ) / 72\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 33\!\cdots\!75 \nu^{10} + \cdots + 26\!\cdots\!20 ) / 43\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69\!\cdots\!51 \nu^{10} + \cdots - 54\!\cdots\!08 ) / 54\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9037596007696 \nu^{11} + \cdots - 57\!\cdots\!28 \nu ) / 24\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 834372532935 \nu^{11} + \cdots - 14\!\cdots\!64 \nu ) / 69\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!33 \nu^{11} + \cdots + 14\!\cdots\!04 \nu ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 78\!\cdots\!37 \nu^{11} + \cdots + 13\!\cdots\!04 \nu ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48\!\cdots\!63 \nu^{11} + \cdots - 17\!\cdots\!04 \nu ) / 41\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!67 \nu^{11} + \cdots + 48\!\cdots\!04 \nu ) / 24\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 32\beta_{11} - 780\beta_{10} + 3044\beta_{9} - 4764\beta_{8} + 788\beta_{7} - 9511\beta_{6} ) / 23887872 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 57\beta_{5} - 48\beta_{4} - 1462\beta_{3} + 2634\beta_{2} + 144360\beta _1 - 3634274304 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 79072 \beta_{11} + 2319096 \beta_{10} - 7592680 \beta_{9} + 12653448 \beta_{8} + \cdots - 368690113 \beta_{6} ) / 23887872 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 240939 \beta_{5} + 358416 \beta_{4} + 5370214 \beta_{3} - 6173694 \beta_{2} + \cdots + 9940052090880 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 146921504 \beta_{11} - 7228864248 \beta_{10} + 22590093800 \beta_{9} + \cdots + 1717082344415 \beta_{6} ) / 23887872 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 268330281 \beta_{5} - 542777136 \beta_{4} - 6353343821 \beta_{3} + 4599220602 \beta_{2} + \cdots - 92\!\cdots\!52 ) / 663552 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 162273829792 \beta_{11} + 22693282910040 \beta_{10} - 67703644664584 \beta_{9} + \cdots - 67\!\cdots\!35 \beta_{6} ) / 23887872 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2374184852079 \beta_{5} + 6571230370512 \beta_{4} + 66121338291974 \beta_{3} + \cdots + 79\!\cdots\!80 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 353126375659744 \beta_{11} + \cdots + 24\!\cdots\!95 \beta_{6} ) / 23887872 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 63\!\cdots\!11 \beta_{5} + \cdots - 23\!\cdots\!64 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 35\!\cdots\!32 \beta_{11} + \cdots - 83\!\cdots\!55 \beta_{6} ) / 23887872 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
− 56.9675i
52.8983i
− 48.2513i
− 5.64695i
8.12355i
49.8439i
56.9675i
− 52.8983i
48.2513i
5.64695i
− 8.12355i
− 49.8439i
0 − 420.888i 0 −23272.4 0 − 32609.3i 0 −177147. 0
127.2 0 − 420.888i 0 −11100.9 0 144967.i 0 −177147. 0
127.3 0 − 420.888i 0 −2746.31 0 − 22891.2i 0 −177147. 0
127.4 0 − 420.888i 0 6584.98 0 − 186936.i 0 −177147. 0
127.5 0 − 420.888i 0 10618.7 0 194209.i 0 −177147. 0
127.6 0 − 420.888i 0 11335.9 0 − 116900.i 0 −177147. 0
127.7 0 420.888i 0 −23272.4 0 32609.3i 0 −177147. 0
127.8 0 420.888i 0 −11100.9 0 − 144967.i 0 −177147. 0
127.9 0 420.888i 0 −2746.31 0 22891.2i 0 −177147. 0
127.10 0 420.888i 0 6584.98 0 186936.i 0 −177147. 0
127.11 0 420.888i 0 10618.7 0 − 194209.i 0 −177147. 0
127.12 0 420.888i 0 11335.9 0 116900.i 0 −177147. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.13.g.d 12
4.b odd 2 1 inner 192.13.g.d 12
8.b even 2 1 96.13.g.b ✓ 12
8.d odd 2 1 96.13.g.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.13.g.b ✓ 12 8.b even 2 1
96.13.g.b ✓ 12 8.d odd 2 1
192.13.g.d 12 1.a even 1 1 trivial
192.13.g.d 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 8580 T_{5}^{5} - 441691188 T_{5}^{4} - 315730950560 T_{5}^{3} + \cdots - 56\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 177147)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 92\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 62\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 16\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 66\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 78\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 66\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 35\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 62\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 15\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 78\!\cdots\!96)^{2} \) Copy content Toggle raw display
show more
show less