Properties

Label 48.17.e.c
Level $48$
Weight $17$
Character orbit 48.e
Analytic conductor $77.916$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.17");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 17 \)
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: \(77.9157810512\)
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: \( x^{4} + 15630x^{2} + 12922000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{11}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 3105) q^{3} + ( - \beta_{3} - \beta_{2} - 8 \beta_1) q^{5} + (77 \beta_{3} - 462 \beta_1 + 262570) q^{7} + ( - 657 \beta_{3} + 27 \beta_{2} + \cdots - 2530359) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 3105) q^{3} + ( - \beta_{3} - \beta_{2} - 8 \beta_1) q^{5} + (77 \beta_{3} - 462 \beta_1 + 262570) q^{7} + ( - 657 \beta_{3} + 27 \beta_{2} + \cdots - 2530359) q^{9}+ \cdots + (72908835723 \beta_{3} + \cdots - 22\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12420 q^{3} + 1050280 q^{7} - 10121436 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12420 q^{3} + 1050280 q^{7} - 10121436 q^{9} - 274863160 q^{13} + 1065273120 q^{15} + 41348739784 q^{19} - 55600393464 q^{21} - 525638756540 q^{25} + 457426227780 q^{27} + 1712336487784 q^{31} - 953298720 q^{33} + 5258670683720 q^{37} - 9161332213176 q^{39} - 18970169729720 q^{43} + 30786788111040 q^{45} - 51073893429300 q^{49} - 65388537498240 q^{51} + 328085114005440 q^{55} + 103623957688680 q^{57} - 984197053056184 q^{61} - 733717575221400 q^{63} + 798011457001480 q^{67} + 20\!\cdots\!80 q^{69}+ \cdots - 88\!\cdots\!40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 15630x^{2} + 12922000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 54\nu^{2} + 6210\nu + 422010 ) / 115 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 216\nu^{3} + 216\nu^{2} + 3004560\nu + 1688040 ) / 115 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -648\nu^{2} + 37260\nu - 5064120 ) / 115 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 12\beta_1 ) / 972 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -115\beta_{3} + 690\beta _1 - 7596180 ) / 972 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27590\beta_{3} + 1035\beta_{2} - 335220\beta_1 ) / 1944 \) 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
121.467i
121.467i
29.5942i
29.5942i
0 −153.398 6559.21i 0 100768.i 0 4.77871e6 0 −4.29997e7 + 2.01234e6i 0
17.2 0 −153.398 + 6559.21i 0 100768.i 0 4.77871e6 0 −4.29997e7 2.01234e6i 0
17.3 0 6363.40 1598.09i 0 746888.i 0 −4.25357e6 0 3.79389e7 2.03386e7i 0
17.4 0 6363.40 + 1598.09i 0 746888.i 0 −4.25357e6 0 3.79389e7 + 2.03386e7i 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 48.17.e.c 4
3.b odd 2 1 inner 48.17.e.c 4
4.b odd 2 1 12.17.c.b 4
12.b even 2 1 12.17.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.17.c.b 4 4.b odd 2 1
12.17.c.b 4 12.b even 2 1
48.17.e.c 4 1.a even 1 1 trivial
48.17.e.c 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 567995159520T_{5}^{2} + 5664396744104716800000 \) acting on \(S_{17}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + \cdots - 20326571202476)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 40\!\cdots\!56)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 91\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 60\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 26\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 24\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 61\!\cdots\!64)^{2} \) Copy content Toggle raw display
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