Properties

Label 24.4.f.b
Level $24$
Weight $4$
Character orbit 24.f
Analytic conductor $1.416$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,4,Mod(11,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, 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("24.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 24.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: \(1.41604584014\)
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: \( x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} - 2) q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{7} - \beta_{6} + \beta_{3} - \beta_1) q^{5} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2} - 2) q^{7} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{8} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} - 2) q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{7} - \beta_{6} + \beta_{3} - \beta_1) q^{5} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2} - 2) q^{7} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{8} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 6) q^{9} + (\beta_{7} - \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + 2 \beta_{2} - 1) q^{10} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{11} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{12} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} - 4 \beta_{2}) q^{13} + ( - 3 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 5) q^{14} + (\beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 10 \beta_1 + 2) q^{15} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 36) q^{16} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 4) q^{17} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + \cdots + 27) q^{18}+ \cdots + ( - 57 \beta_{5} - 39 \beta_{4} + 24 \beta_{3} + 24 \beta_1 + 255) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 20 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} + 20 q^{4} - 48 q^{9} - 24 q^{10} + 36 q^{12} - 280 q^{16} + 264 q^{18} + 184 q^{19} - 176 q^{22} + 264 q^{24} + 296 q^{25} - 324 q^{27} + 528 q^{28} - 624 q^{30} - 264 q^{33} - 176 q^{34} - 516 q^{36} - 1248 q^{40} + 1320 q^{42} - 152 q^{43} + 1440 q^{46} + 1080 q^{48} - 952 q^{49} + 1056 q^{51} + 2112 q^{52} - 1584 q^{54} + 1176 q^{57} - 2616 q^{58} - 2640 q^{60} - 1360 q^{64} + 792 q^{66} - 1496 q^{67} + 3696 q^{70} + 2640 q^{72} + 1072 q^{73} - 708 q^{75} + 1912 q^{76} - 3696 q^{78} - 504 q^{81} - 2816 q^{82} - 4224 q^{84} - 1232 q^{88} + 4104 q^{90} + 3168 q^{91} + 4800 q^{94} + 4752 q^{96} - 3872 q^{97} + 2112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 10\nu^{5} + 120\nu^{3} - 384\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 4\nu^{5} + 10\nu^{4} + 8\nu^{3} - 56\nu^{2} - 160\nu + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 10\nu^{4} - 8\nu^{3} - 56\nu^{2} + 160\nu + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 2\nu^{5} + 44\nu^{4} + 24\nu^{3} - 16\nu^{2} - 192\nu + 2048 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 2\nu^{5} - 44\nu^{4} + 24\nu^{3} + 16\nu^{2} - 192\nu - 2048 ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{2} - 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + 18\beta_{5} - 18\beta_{4} + 4\beta_{3} - 36\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{7} + 20\beta_{6} - 44\beta_{5} - 44\beta_{4} - 36\beta_{2} - 208 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -100\beta_{7} - 100\beta_{6} + 60\beta_{5} - 60\beta_{4} + 56\beta_{3} - 216\beta _1 + 60 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−2.58576 1.14624i
−2.58576 + 1.14624i
−1.95291 2.04601i
−1.95291 + 2.04601i
1.95291 2.04601i
1.95291 + 2.04601i
2.58576 1.14624i
2.58576 + 1.14624i
−2.58576 1.14624i 1.37228 + 5.01167i 5.37228 + 5.92778i 12.2683 2.19618 14.5319i 14.0624i −7.09677 21.4857i −23.2337 + 13.7548i −31.7228 14.0624i
11.2 −2.58576 + 1.14624i 1.37228 5.01167i 5.37228 5.92778i 12.2683 2.19618 + 14.5319i 14.0624i −7.09677 + 21.4857i −23.2337 13.7548i −31.7228 + 14.0624i
11.3 −1.95291 2.04601i −4.37228 2.80770i −0.372281 + 7.99133i −13.1715 2.79411 + 14.4289i 26.9490i 17.0773 14.8447i 11.2337 + 24.5521i 25.7228 + 26.9490i
11.4 −1.95291 + 2.04601i −4.37228 + 2.80770i −0.372281 7.99133i −13.1715 2.79411 14.4289i 26.9490i 17.0773 + 14.8447i 11.2337 24.5521i 25.7228 26.9490i
11.5 1.95291 2.04601i −4.37228 2.80770i −0.372281 7.99133i 13.1715 −14.2832 + 3.46254i 26.9490i −17.0773 14.8447i 11.2337 + 24.5521i 25.7228 26.9490i
11.6 1.95291 + 2.04601i −4.37228 + 2.80770i −0.372281 + 7.99133i 13.1715 −14.2832 3.46254i 26.9490i −17.0773 + 14.8447i 11.2337 24.5521i 25.7228 + 26.9490i
11.7 2.58576 1.14624i 1.37228 + 5.01167i 5.37228 5.92778i −12.2683 9.29295 + 11.3860i 14.0624i 7.09677 21.4857i −23.2337 + 13.7548i −31.7228 + 14.0624i
11.8 2.58576 + 1.14624i 1.37228 5.01167i 5.37228 + 5.92778i −12.2683 9.29295 11.3860i 14.0624i 7.09677 + 21.4857i −23.2337 13.7548i −31.7228 14.0624i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 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 24.4.f.b 8
3.b odd 2 1 inner 24.4.f.b 8
4.b odd 2 1 96.4.f.b 8
8.b even 2 1 96.4.f.b 8
8.d odd 2 1 inner 24.4.f.b 8
12.b even 2 1 96.4.f.b 8
16.e even 4 2 768.4.c.v 16
16.f odd 4 2 768.4.c.v 16
24.f even 2 1 inner 24.4.f.b 8
24.h odd 2 1 96.4.f.b 8
48.i odd 4 2 768.4.c.v 16
48.k even 4 2 768.4.c.v 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.b 8 1.a even 1 1 trivial
24.4.f.b 8 3.b odd 2 1 inner
24.4.f.b 8 8.d odd 2 1 inner
24.4.f.b 8 24.f even 2 1 inner
96.4.f.b 8 4.b odd 2 1
96.4.f.b 8 8.b even 2 1
96.4.f.b 8 12.b even 2 1
96.4.f.b 8 24.h odd 2 1
768.4.c.v 16 16.e even 4 2
768.4.c.v 16 16.f odd 4 2
768.4.c.v 16 48.i odd 4 2
768.4.c.v 16 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 324T_{5}^{2} + 26112 \) acting on \(S_{4}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 10 T^{6} + 120 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( (T^{4} + 6 T^{3} + 30 T^{2} + 162 T + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 324 T^{2} + 26112)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 924 T^{2} + 143616)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 484 T^{2} + 352)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5808 T^{2} + 574464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2464 T^{2} + 90112)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 46 T - 3464)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 17472 T^{2} + 1671168)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 43620 T^{2} + \cdots + 448108032)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 20988 T^{2} + 41505024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 77616 T^{2} + \cdots + 1494180864)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 193600 T^{2} + \cdots + 3151126528)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 38 T - 49832)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 321792 T^{2} + \cdots + 427819008)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 177156 T^{2} + \cdots + 7033554432)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 21604 T^{2} + 296032)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 550704 T^{2} + \cdots + 18820015104)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 374 T + 32296)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 654912 T^{2} + \cdots + 103614087168)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 268 T - 704876)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1028412 T^{2} + \cdots + 99653562624)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 396484 T^{2} + \cdots + 2128431712)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2644576 T^{2} + \cdots + 147293673472)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 968 T - 33044)^{4} \) Copy content Toggle raw display
show more
show less