↑

Properties

Label 48.11.e.e
Level $48$
Weight $11$
Character orbit 48.e
Analytic conductor $30.497$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,11,Mod(17,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.17");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 48.e (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: \(30.4971481283\)
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: \( x^{10} - 532 x^{8} - 1350 x^{7} + 106101 x^{6} + 516780 x^{5} - 8879077 x^{4} - 65126430 x^{3} + \cdots + 6338653425 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{72}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{2} + 7 \beta_1 + 545) q^{7} + ( - \beta_{8} - 2 \beta_1 - 2894) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{2} + 7 \beta_1 + 545) q^{7} + ( - \beta_{8} - 2 \beta_1 - 2894) q^{9} + ( - \beta_{8} + \beta_{7} + \cdots - 4 \beta_1) q^{11}+ \cdots + ( - 3204 \beta_{9} + 14458 \beta_{8} + \cdots - 4350990363) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 22 q^{3} + 5436 q^{7} - 28934 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 22 q^{3} + 5436 q^{7} - 28934 q^{9} - 124508 q^{13} + 627808 q^{15} + 4893484 q^{19} - 3929724 q^{21} - 17742214 q^{25} + 3536326 q^{27} + 4251484 q^{31} - 2965600 q^{33} + 89985156 q^{37} - 52569188 q^{39} - 159987316 q^{43} + 39125824 q^{45} + 301480958 q^{49} - 387377536 q^{51} + 852340544 q^{55} - 970086764 q^{57} - 101460764 q^{61} - 733153572 q^{63} + 3014528044 q^{67} - 3501669184 q^{69} + 4920922036 q^{73} - 5355440986 q^{75} + 7631690012 q^{79} - 7700105942 q^{81} + 18713636096 q^{85} - 19781179104 q^{87} + 17913072600 q^{91} - 24272938652 q^{93} + 37861379156 q^{97} - 43508497216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 532 x^{8} - 1350 x^{7} + 106101 x^{6} + 516780 x^{5} - 8879077 x^{4} - 65126430 x^{3} + \cdots + 6338653425 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18\!\cdots\!50 \nu^{9} + \cdots + 14\!\cdots\!45 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!86 \nu^{9} + \cdots - 13\!\cdots\!65 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24\!\cdots\!86 \nu^{9} + \cdots - 12\!\cdots\!05 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!02 \nu^{9} + \cdots + 44\!\cdots\!25 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\!\cdots\!26 \nu^{9} + \cdots + 34\!\cdots\!90 ) / 11\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 98\!\cdots\!22 \nu^{9} + \cdots + 70\!\cdots\!35 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!82 \nu^{9} + \cdots - 30\!\cdots\!45 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!04 \nu^{9} + \cdots + 16\!\cdots\!65 ) / 11\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!74 \nu^{9} + \cdots - 97\!\cdots\!65 ) / 23\!\cdots\!70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 12 \beta_{9} - 84 \beta_{8} - 42 \beta_{7} - 451 \beta_{6} + 153 \beta_{5} - 9 \beta_{4} + \cdots + 6952 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 36 \beta_{9} + 204 \beta_{8} + 282 \beta_{7} - 2209 \beta_{6} - 12 \beta_{5} - 171 \beta_{4} + \cdots + 141238657 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3960 \beta_{9} - 22248 \beta_{8} - 7560 \beta_{7} - 138752 \beta_{6} + 107667 \beta_{5} + \cdots + 1076590331 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4860 \beta_{9} + 49692 \beta_{8} + 40110 \beta_{7} - 541039 \beta_{6} + 277347 \beta_{5} + \cdots + 18804905998 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 689088 \beta_{9} - 1806528 \beta_{8} - 273084 \beta_{7} - 25870238 \beta_{6} + 26444277 \beta_{5} + \cdots + 267505592483 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3621456 \beta_{9} + 15097392 \beta_{8} + 6901452 \beta_{7} - 124170322 \beta_{6} + \cdots + 2815290317785 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 76736604 \beta_{9} + 53024892 \beta_{8} + 45901578 \beta_{7} - 2509543057 \beta_{6} + \cdots + 26993459009293 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1206500580 \beta_{9} + 4582835700 \beta_{8} + 1569431658 \beta_{7} - 27544870813 \beta_{6} + \cdots + 442122123818926 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 38668338648 \beta_{9} + 104203343160 \beta_{8} + 40889194416 \beta_{7} - 972283313204 \beta_{6} + \cdots + 98\!\cdots\!81 ) / 2654208 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
14.5165 − 1.41421i
14.5165 + 1.41421i
10.3554 + 1.41421i
10.3554 − 1.41421i
−8.61800 + 1.41421i
−8.61800 − 1.41421i
−11.0996 − 1.41421i
−11.0996 + 1.41421i
−5.15423 + 1.41421i
−5.15423 − 1.41421i
0 −202.837 − 133.814i 0 265.153i 0 −17613.7 0 23236.7 + 54284.8i 0
17.2 0 −202.837 + 133.814i 0 − 265.153i 0 −17613.7 0 23236.7 − 54284.8i 0
17.3 0 −171.197 − 172.454i 0 − 2896.26i 0 28955.7 0 −431.864 + 59047.4i 0
17.4 0 −171.197 + 172.454i 0 2896.26i 0 28955.7 0 −431.864 − 59047.4i 0
17.5 0 27.8503 − 241.399i 0 5802.43i 0 −13554.8 0 −57497.7 − 13446.0i 0
17.6 0 27.8503 + 241.399i 0 − 5802.43i 0 −13554.8 0 −57497.7 + 13446.0i 0
17.7 0 126.767 − 207.314i 0 − 1673.71i 0 12918.9 0 −26909.2 − 52561.2i 0
17.8 0 126.767 + 207.314i 0 1673.71i 0 12918.9 0 −26909.2 + 52561.2i 0
17.9 0 230.417 − 77.1815i 0 − 3573.67i 0 −7988.14 0 47135.0 − 35567.9i 0
17.10 0 230.417 + 77.1815i 0 3573.67i 0 −7988.14 0 47135.0 + 35567.9i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.11.e.e 10
3.b odd 2 1 inner 48.11.e.e 10
4.b odd 2 1 24.11.e.a ✓ 10
8.b even 2 1 192.11.e.i 10
8.d odd 2 1 192.11.e.j 10
12.b even 2 1 24.11.e.a ✓ 10
24.f even 2 1 192.11.e.j 10
24.h odd 2 1 192.11.e.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.11.e.a ✓ 10 4.b odd 2 1
24.11.e.a ✓ 10 12.b even 2 1
48.11.e.e 10 1.a even 1 1 trivial
48.11.e.e 10 3.b odd 2 1 inner
192.11.e.i 10 8.b even 2 1
192.11.e.i 10 24.h odd 2 1
192.11.e.j 10 8.d odd 2 1
192.11.e.j 10 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 57699232 T_{5}^{8} + 977168596572160 T_{5}^{6} + \cdots + 71\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots + 71\!\cdots\!44)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 68\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 60\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 51\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 33\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 93\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 15\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 43\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 31\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 22\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
show more
show less