Properties

Label 48.13.g.a
Level $48$
Weight $13$
Character orbit 48.g
Analytic conductor $43.872$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,13,Mod(31,48)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 48.g (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: \(43.8717032293\)
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: \( x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 243 \beta_1 q^{3} + ( - 5 \beta_{2} - 5490) q^{5} + ( - 3 \beta_{3} - 5372 \beta_1) q^{7} - 177147 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 243 \beta_1 q^{3} + ( - 5 \beta_{2} - 5490) q^{5} + ( - 3 \beta_{3} - 5372 \beta_1) q^{7} - 177147 q^{9} + (374 \beta_{3} + 93060 \beta_1) q^{11} + ( - 2250 \beta_{2} - 1452518) q^{13} + ( - 1215 \beta_{3} + 1334070 \beta_1) q^{15} + (4826 \beta_{2} - 8470566) q^{17} + (4974 \beta_{3} + 16087436 \beta_1) q^{19} + (2187 \beta_{2} - 3916188) q^{21} + ( - 21702 \beta_{3} + 55966464 \beta_1) q^{23} + (54900 \beta_{2} + 35537075) q^{25} + 43046721 \beta_1 q^{27} + ( - 119443 \beta_{2} + 10645974) q^{29} + (123861 \beta_{3} - 208303380 \beta_1) q^{31} + ( - 272646 \beta_{2} + 67840740) q^{33} + ( - 10390 \beta_{3} - 120230280 \beta_1) q^{35} + (372708 \beta_{2} + 2350716434) q^{37} + ( - 546750 \beta_{3} + 352961874 \beta_1) q^{39} + (370134 \beta_{2} + 4492220634) q^{41} + (996882 \beta_{3} + 341094852 \beta_1) q^{43} + (885735 \beta_{2} + 972537030) q^{45} + ( - 1486618 \beta_{3} + 6216878808 \beta_1) q^{47} + (96696 \beta_{2} + 13485211441) q^{49} + (1172718 \beta_{3} + 2058347538 \beta_1) q^{51} + ( - 4280187 \beta_{2} + 15136739190) q^{53} + ( - 1587960 \beta_{3} + 18154513080 \beta_1) q^{55} + ( - 3626046 \beta_{2} + 11727740844) q^{57} + (7263544 \beta_{3} + 15536372220 \beta_1) q^{59} + ( - 1069776 \beta_{2} + 42071800418) q^{61} + (531441 \beta_{3} + 951633684 \beta_1) q^{63} + (19615090 \beta_{2} + 120266243820) q^{65} + ( - 19551816 \beta_{3} + 28609418996 \beta_1) q^{67} + (15820758 \beta_{2} + 40799552256) q^{69} + ( - 3537406 \beta_{3} + 12886833744 \beta_1) q^{71} + ( - 30749832 \beta_{2} + 197256657250) q^{73} + (13340700 \beta_{3} - 8635509225 \beta_1) q^{75} + ( - 6864924 \beta_{2} + 35097497424) q^{77} + (38237649 \beta_{3} + 5869754828 \beta_1) q^{79} + 31381059609 q^{81} + ( - 17505770 \beta_{3} + 77882293404 \beta_1) q^{83} + (15858090 \beta_{2} - 194350284180) q^{85} + ( - 29024649 \beta_{3} - 2586971682 \beta_1) q^{87} + ( - 99275060 \beta_{2} - 159667615134) q^{89} + ( - 7729446 \beta_{3} - 59572225304 \beta_1) q^{91} + ( - 90294669 \beta_{2} - 151853164020) q^{93} + (53129920 \beta_{3} + 159919980840 \beta_1) q^{95} + (338673564 \beta_{2} - 140885510750) q^{97} + ( - 66252978 \beta_{3} - 16485299820 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4333\nu^{2} - 4333\nu + 9380946 ) / 9385278 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\nu^{3} + 311928 ) / 4333 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 4\nu^{2} + 34660\nu + 8664 ) / 361 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 207960\beta _1 - 207960 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4333\beta_{2} - 311928 ) / 48 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−32.6599 56.5686i
33.1599 + 57.4347i
−32.6599 + 56.5686i
33.1599 57.4347i
0 420.888i 0 −21286.8 0 7111.90i 0 −177147. 0
31.2 0 420.888i 0 10306.8 0 25721.1i 0 −177147. 0
31.3 0 420.888i 0 −21286.8 0 7111.90i 0 −177147. 0
31.4 0 420.888i 0 10306.8 0 25721.1i 0 −177147. 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.13.g.a 4
3.b odd 2 1 144.13.g.h 4
4.b odd 2 1 inner 48.13.g.a 4
8.b even 2 1 192.13.g.c 4
8.d odd 2 1 192.13.g.c 4
12.b even 2 1 144.13.g.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.13.g.a 4 1.a even 1 1 trivial
48.13.g.a 4 4.b odd 2 1 inner
144.13.g.h 4 3.b odd 2 1
144.13.g.h 4 12.b even 2 1
192.13.g.c 4 8.b even 2 1
192.13.g.c 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 10980T_{5} - 219397500 \) acting on \(S_{13}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 177147)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 10980 T - 219397500)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 48421555459676)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + \cdots - 160721494694748)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots - 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots + 18\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 86\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots + 46\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 73\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 29\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 81\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 72\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
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