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Properties

Label 5.24.b.a
Level $5$
Weight $24$
Character orbit 5.b
Analytic conductor $16.760$
Analytic rank $0$
Dimension $10$
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] = [5,24,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 5.b (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: \(16.7602018673\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 15644845 x^{8} + 79349217360160 x^{6} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12}\cdot 5^{22} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 18 \beta_1) q^{3} + (\beta_{2} - 4127268) q^{4} + (\beta_{4} + 3 \beta_{3} + \cdots + 12476175) q^{5}+ \cdots + (\beta_{7} - \beta_{6} + \cdots - 18925063137) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 18 \beta_1) q^{3} + (\beta_{2} - 4127268) q^{4} + (\beta_{4} + 3 \beta_{3} + \cdots + 12476175) q^{5}+ \cdots + (640536775164 \beta_{7} + \cdots + 41\!\cdots\!36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 41272680 q^{4} + 124761750 q^{5} - 2262077880 q^{6} - 189250631370 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 41272680 q^{4} + 124761750 q^{5} - 2262077880 q^{6} - 189250631370 q^{9} - 992748199000 q^{10} - 1448637536280 q^{11} + 20750531044440 q^{14} - 14566613457000 q^{15} + 307971806876960 q^{16} + 887626815301400 q^{19} - 23\!\cdots\!00 q^{20}+ \cdots + 41\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 15644845 x^{8} + 79349217360160 x^{6} + \cdots + 34\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 12515876 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2453096257 \nu^{9} + \cdots - 13\!\cdots\!04 \nu ) / 98\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5435304996773 \nu^{9} + \cdots + 59\!\cdots\!28 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16305914990319 \nu^{9} + \cdots + 46\!\cdots\!16 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48917744970957 \nu^{9} + \cdots + 15\!\cdots\!52 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 358791457193443 \nu^{9} + \cdots - 47\!\cdots\!52 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!47 \nu^{9} + \cdots + 22\!\cdots\!92 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{9} + \cdots + 67\!\cdots\!36 ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 12515876 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - \beta_{7} - 41\beta_{4} + 42810\beta_{3} + 18\beta_{2} - 22000955\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2542 \beta_{7} + 121 \beta_{6} + 20126 \beta_{5} + 75687 \beta_{4} - 5084 \beta_{3} + \cdots + 68850192282964 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1066176 \beta_{9} - 9460253 \beta_{8} + 17315805 \beta_{7} + 603561125 \beta_{4} + \cdots + 142114511735535 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24253366134 \beta_{7} - 1608475125 \beta_{6} - 236322854502 \beta_{5} - 765071484075 \beta_{4} + \cdots - 44\!\cdots\!76 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13322991015360 \beta_{9} + 78086888404089 \beta_{8} - 181038084493113 \beta_{7} + \cdots - 10\!\cdots\!19 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20\!\cdots\!86 \beta_{7} + \cdots + 31\!\cdots\!04 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11\!\cdots\!44 \beta_{9} + \cdots + 73\!\cdots\!59 \beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 2780.51i
− 1968.02i
− 1812.37i
− 839.925i
− 224.285i
224.285i
839.925i
1812.37i
1968.02i
2780.51i
− 5561.03i − 421877.i −2.25364e7 7.09941e7 − 8.29504e7i −2.34607e9 7.17617e8i 7.86762e10i −8.38373e10 −4.61289e11 − 3.94800e11i
4.2 − 3936.05i 96287.9i −7.10385e6 −8.57356e6 + 1.08846e8i 3.78993e8 − 1.58872e9i − 5.05685e9i 8.48718e10 4.28422e11 + 3.37459e10i
4.3 − 3624.75i 421266.i −4.75020e6 −7.79198e6 − 1.08905e8i 1.52698e9 4.49913e9i − 1.31883e10i −8.33219e10 −3.94752e11 + 2.82440e10i
4.4 − 1679.85i − 330622.i 5.56671e6 −9.79539e7 − 4.82283e7i −5.55396e8 − 4.39640e9i − 2.34428e10i −1.51680e10 −8.10163e10 + 1.64548e11i
4.5 − 448.570i − 302181.i 8.18739e6 1.05706e8 + 2.73336e7i −1.35549e8 8.28173e9i − 7.43550e9i 2.83002e9 1.22610e10 − 4.74167e10i
4.6 448.570i 302181.i 8.18739e6 1.05706e8 − 2.73336e7i −1.35549e8 − 8.28173e9i 7.43550e9i 2.83002e9 1.22610e10 + 4.74167e10i
4.7 1679.85i 330622.i 5.56671e6 −9.79539e7 + 4.82283e7i −5.55396e8 4.39640e9i 2.34428e10i −1.51680e10 −8.10163e10 − 1.64548e11i
4.8 3624.75i − 421266.i −4.75020e6 −7.79198e6 + 1.08905e8i 1.52698e9 − 4.49913e9i 1.31883e10i −8.33219e10 −3.94752e11 − 2.82440e10i
4.9 3936.05i − 96287.9i −7.10385e6 −8.57356e6 − 1.08846e8i 3.78993e8 1.58872e9i 5.05685e9i 8.48718e10 4.28422e11 − 3.37459e10i
4.10 5561.03i 421877.i −2.25364e7 7.09941e7 + 8.29504e7i −2.34607e9 − 7.17617e8i − 7.86762e10i −8.38373e10 −4.61289e11 + 3.94800e11i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.24.b.a ✓ 10
3.b odd 2 1 45.24.b.b 10
5.b even 2 1 inner 5.24.b.a ✓ 10
5.c odd 4 2 25.24.a.f 10
15.d odd 2 1 45.24.b.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.24.b.a ✓ 10 1.a even 1 1 trivial
5.24.b.a ✓ 10 5.b even 2 1 inner
25.24.a.f 10 5.c odd 4 2
45.24.b.b 10 3.b odd 2 1
45.24.b.b 10 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 91\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 94\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 65\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
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