Properties

Label 48.15.e.c
Level $48$
Weight $15$
Character orbit 48.e
Analytic conductor $59.678$
Analytic rank $0$
Dimension $4$
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] = [48,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.17");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(59.6779047129\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 440x^{2} + 48015 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{9}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 12)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 537) q^{3} + (\beta_{2} + \beta_1) q^{5} + (5 \beta_{3} - 250 \beta_1 - 427322) q^{7} + (12 \beta_{3} + 45 \beta_{2} + \cdots - 1434231) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 537) q^{3} + (\beta_{2} + \beta_1) q^{5} + (5 \beta_{3} - 250 \beta_1 - 427322) q^{7} + (12 \beta_{3} + 45 \beta_{2} + \cdots - 1434231) q^{9}+ \cdots + ( - 293507469 \beta_{3} + \cdots - 81336548066400) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2148 q^{3} - 1709288 q^{7} - 5736924 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2148 q^{3} - 1709288 q^{7} - 5736924 q^{9} - 93505048 q^{13} - 24235200 q^{15} + 1349200696 q^{19} - 3572752344 q^{21} - 4043764700 q^{25} - 7381750212 q^{27} - 46585213736 q^{31} - 60689217600 q^{33} + 300873217064 q^{37} + 96914866776 q^{39} - 246448758152 q^{43} - 1275658243200 q^{45} + 1654942475340 q^{49} + 4102471929600 q^{51} - 7505039836800 q^{55} - 5979467701752 q^{57} + 8008332264296 q^{61} + 12097408557528 q^{63} - 38294908213448 q^{67} - 27868623868800 q^{69} + 12721406693576 q^{73} + 13261586187900 q^{75} + 29803403331928 q^{79} - 23892053776956 q^{81} - 28369656691200 q^{85} - 58743848116800 q^{87} + 2127612761456 q^{91} + 130412478704232 q^{93} - 301625619131512 q^{97} - 325346192265600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 440x^{2} + 48015 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu^{3} + 60\nu^{2} + 1350\nu + 13200 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -42\nu^{3} - 60\nu^{2} - 3690\nu - 13200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 300\nu^{3} - 6720\nu^{2} + 67500\nu - 1478400 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 27\beta_{2} + 139\beta_1 ) / 155520 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 50\beta _1 - 2138400 ) / 9720 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{3} - 1215\beta_{2} - 2671\beta_1 ) / 31104 \) Copy content Toggle raw display

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
15.4797i
15.4797i
14.1555i
14.1555i
0 −1714.29 1358.01i 0 97311.2i 0 526279. 0 1.09458e6 + 4.65604e6i 0
17.2 0 −1714.29 + 1358.01i 0 97311.2i 0 526279. 0 1.09458e6 4.65604e6i 0
17.3 0 640.285 2091.17i 0 68988.7i 0 −1.38092e6 0 −3.96304e6 2.67789e6i 0
17.4 0 640.285 + 2091.17i 0 68988.7i 0 −1.38092e6 0 −3.96304e6 + 2.67789e6i 0
\(n\): e.g. 2-40 or 990-1000
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 48.15.e.c 4
3.b odd 2 1 inner 48.15.e.c 4
4.b odd 2 1 12.15.c.b 4
12.b even 2 1 12.15.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.15.c.b 4 4.b odd 2 1
12.15.c.b 4 12.b even 2 1
48.15.e.c 4 1.a even 1 1 trivial
48.15.e.c 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 14228913600T_{5}^{2} + 45069410819119104000 \) acting on \(S_{15}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 22876792454961 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + 854644 T - 726750508316)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots + 448093831556644)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 36\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots + 55\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 37\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 81\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 91\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 37\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
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