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Properties

Label 48.17.e.b
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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Show commands: Magma / PariGP / SageMath

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: \( x^{4} + 3814x^{2} + 2981440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 3)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 513) q^{3} + (15 \beta_{3} - 45 \beta_{2} + 10 \beta_1) q^{5} + ( - 217 \beta_{3} - 651 \beta_{2} + \cdots + 785386) q^{7}+ \cdots + ( - 729 \beta_{3} + 135 \beta_{2} + \cdots + 4654665) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 513) q^{3} + (15 \beta_{3} - 45 \beta_{2} + 10 \beta_1) q^{5} + ( - 217 \beta_{3} - 651 \beta_{2} + \cdots + 785386) q^{7}+ \cdots + (374876825841 \beta_{3} + \cdots - 47\!\cdots\!40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2052 q^{3} + 3141544 q^{7} + 18618660 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2052 q^{3} + 3141544 q^{7} + 18618660 q^{9} - 1580730424 q^{13} - 6829958880 q^{15} + 56117116360 q^{19} + 124455437064 q^{21} - 8074048700 q^{25} + 317983667652 q^{27} - 2471781156248 q^{31} - 3610697951520 q^{33} + 370563213896 q^{37} - 7022170227384 q^{39} + 28065022062664 q^{43} + 18795326443200 q^{45} + 29478262537164 q^{49} + 82841575222656 q^{51} - 290253653236800 q^{55} - 335129108488344 q^{57} + 362269793083208 q^{61} + 266698363786344 q^{63} + 26774405363464 q^{67} + 58823173290816 q^{69} + 317628887539976 q^{73} + 15\!\cdots\!00 q^{75}+ \cdots - 18\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3814x^{2} + 2981440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 48\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} + 132\nu^{2} - 6558\nu + 251724 ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{3} + 396\nu^{2} + 20730\nu + 755172 ) / 22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} - \beta _1 - 68652 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 88\beta_{3} - 264\beta_{2} - 3367\beta_1 ) / 72 \) 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
− 33.1293i
33.1293i
52.1196i
− 52.1196i
0 −4343.70 − 4917.21i 0 − 482304.i 0 −5.53804e6 0 −5.31128e6 + 4.27178e7i 0
17.2 0 −4343.70 + 4917.21i 0 482304.i 0 −5.53804e6 0 −5.31128e6 − 4.27178e7i 0
17.3 0 5369.70 − 3770.02i 0 − 276758.i 0 7.10881e6 0 1.46206e7 − 4.04878e7i 0
17.4 0 5369.70 + 3770.02i 0 276758.i 0 7.10881e6 0 1.46206e7 + 4.04878e7i 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.b 4
3.b odd 2 1 inner 48.17.e.b 4
4.b odd 2 1 3.17.b.a ✓ 4
12.b even 2 1 3.17.b.a ✓ 4
20.d odd 2 1 75.17.c.d 4
20.e even 4 2 75.17.d.b 8
60.h even 2 1 75.17.c.d 4
60.l odd 4 2 75.17.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.17.b.a ✓ 4 4.b odd 2 1
3.17.b.a ✓ 4 12.b even 2 1
48.17.e.b 4 1.a even 1 1 trivial
48.17.e.b 4 3.b odd 2 1 inner
75.17.c.d 4 20.d odd 2 1
75.17.c.d 4 60.h even 2 1
75.17.d.b 8 20.e even 4 2
75.17.d.b 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 309212805600T_{5}^{2} + 17817390680833497600000 \) 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
$5$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + \cdots - 39368833865900)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots + 53\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 28\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 32\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 38\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 79\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 26\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 77\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots - 28\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 92\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 20\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 81\!\cdots\!20)^{2} \) Copy content Toggle raw display
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