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Properties

Label 192.4.f.c
Level $192$
Weight $4$
Character orbit 192.f
Analytic conductor $11.328$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,4,Mod(95,192)]
 
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, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.95");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.f (of order \(2\), degree \(1\), 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: \(11.3283667211\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1731891456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{5} - \beta_{3} + 3 \beta_1) q^{5} + \beta_{7} q^{7} + ( - 6 \beta_{6} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{5} - \beta_{3} + 3 \beta_1) q^{5} + \beta_{7} q^{7} + ( - 6 \beta_{6} - \beta_{4}) q^{9} + (\beta_{5} + \beta_{3} - 4 \beta_{2}) q^{11} + ( - 6 \beta_{5} - 6 \beta_{3} - \beta_{2}) q^{13} + (3 \beta_{7} + 3 \beta_{6} + 4 \beta_{4} - 36) q^{15} + (8 \beta_{7} + 4 \beta_{6}) q^{17} + (9 \beta_{5} - 9 \beta_{3} + 8 \beta_1) q^{19} + ( - \beta_{5} - 3 \beta_{3} + \cdots - 13 \beta_1) q^{21}+ \cdots + ( - 27 \beta_{5} - 75 \beta_{3} - 216 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 288 q^{15} + 960 q^{23} + 584 q^{25} - 408 q^{33} + 1248 q^{39} - 2688 q^{47} + 1512 q^{49} - 2136 q^{57} + 192 q^{63} - 192 q^{71} + 1520 q^{73} - 3384 q^{81} - 3744 q^{87} + 7872 q^{95} + 2368 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 25\nu^{5} - 145\nu^{3} + 544\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{6} + 65\nu^{4} - 585\nu^{2} + 776 ) / 130 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} + 74\nu^{6} + 65\nu^{5} - 650\nu^{4} - 585\nu^{3} + 3770\nu^{2} + 2336\nu - 5456 ) / 1040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6\nu^{6} + 891 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} + 74\nu^{6} - 65\nu^{5} - 650\nu^{4} + 585\nu^{3} + 3770\nu^{2} - 2336\nu - 5456 ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{7} + 65\nu^{5} - 377\nu^{3} + 256\nu ) / 208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 153\nu^{7} - 1105\nu^{5} + 8905\nu^{3} - 4352\nu ) / 2080 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + \beta_{6} - 6\beta_{5} + 6\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{5} - 2\beta_{4} - 6\beta_{3} - 15\beta_{2} + 54 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{7} + 17\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -18\beta_{5} + 6\beta_{4} - 18\beta_{3} - 29\beta_{2} - 98 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 58\beta_{7} + 137\beta_{6} + 174\beta_{5} - 174\beta_{3} + 216\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 65\beta_{4} - 891 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -362\beta_{7} - 961\beta_{6} + 1086\beta_{5} - 1086\beta_{3} + 1560\beta_1 ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
−2.21837 − 1.28078i
−2.21837 + 1.28078i
−1.35234 + 0.780776i
−1.35234 − 0.780776i
1.35234 − 0.780776i
1.35234 + 0.780776i
2.21837 + 1.28078i
2.21837 − 1.28078i
0 −4.43674 − 2.70469i 0 19.2658 0 − 13.3693i 0 12.3693 + 24.0000i 0
95.2 0 −4.43674 + 2.70469i 0 19.2658 0 13.3693i 0 12.3693 − 24.0000i 0
95.3 0 −2.70469 − 4.43674i 0 −4.98293 0 11.3693i 0 −12.3693 + 24.0000i 0
95.4 0 −2.70469 + 4.43674i 0 −4.98293 0 − 11.3693i 0 −12.3693 − 24.0000i 0
95.5 0 2.70469 − 4.43674i 0 4.98293 0 − 11.3693i 0 −12.3693 − 24.0000i 0
95.6 0 2.70469 + 4.43674i 0 4.98293 0 11.3693i 0 −12.3693 + 24.0000i 0
95.7 0 4.43674 − 2.70469i 0 −19.2658 0 13.3693i 0 12.3693 − 24.0000i 0
95.8 0 4.43674 + 2.70469i 0 −19.2658 0 − 13.3693i 0 12.3693 + 24.0000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.4.f.c ✓ 8
3.b odd 2 1 192.4.f.d yes 8
4.b odd 2 1 192.4.f.d yes 8
8.b even 2 1 inner 192.4.f.c ✓ 8
8.d odd 2 1 192.4.f.d yes 8
12.b even 2 1 inner 192.4.f.c ✓ 8
16.e even 4 2 768.4.c.t 8
16.f odd 4 2 768.4.c.u 8
24.f even 2 1 inner 192.4.f.c ✓ 8
24.h odd 2 1 192.4.f.d yes 8
48.i odd 4 2 768.4.c.u 8
48.k even 4 2 768.4.c.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.f.c ✓ 8 1.a even 1 1 trivial
192.4.f.c ✓ 8 8.b even 2 1 inner
192.4.f.c ✓ 8 12.b even 2 1 inner
192.4.f.c ✓ 8 24.f even 2 1 inner
192.4.f.d yes 8 3.b odd 2 1
192.4.f.d yes 8 4.b odd 2 1
192.4.f.d yes 8 8.d odd 2 1
192.4.f.d yes 8 24.h odd 2 1
768.4.c.t 8 16.e even 4 2
768.4.c.t 8 48.k even 4 2
768.4.c.u 8 16.f odd 4 2
768.4.c.u 8 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{4} - 396T_{5}^{2} + 9216 \) Copy content Toggle raw display
\( T_{23}^{2} - 240T_{23} + 4608 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 846 T^{4} + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 396 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 308 T^{2} + 23104)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1836 T^{2} + 665856)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 3696 T^{2} + 3326976)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 20736 T^{2} + 84934656)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 12012 T^{2} + 5089536)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 240 T + 4608)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 57132 T^{2} + 269485056)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 88532 T^{2} + 1950812224)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 56688 T^{2} + 608707584)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 48384 T^{2} + 21233664)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 130572 T^{2} + 4143239424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 672 T + 73728)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 243468 T^{2} + 10729645056)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 237708 T^{2} + 1979894016)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 648816 T^{2} + 727704576)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 605292 T^{2} + 36698298624)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 48 T - 244224)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 380 T - 260108)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1449812 T^{2} + 239715993664)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1090188 T^{2} + 258340425984)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1976832 T^{2} + 455722205184)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 592 T - 236132)^{4} \) Copy content Toggle raw display
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