Properties

Label 48.13.g.b
Level $48$
Weight $13$
Character orbit 48.g
Analytic conductor $43.872$
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] = [48,13,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: \(\Q(\sqrt{-3}, \sqrt{-2803})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 700x^{2} - 701x + 491401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{11} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 5598) q^{5} + ( - 5 \beta_{3} - 63 \beta_1) q^{7} - 177147 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 5598) q^{5} + ( - 5 \beta_{3} - 63 \beta_1) q^{7} - 177147 q^{9} + ( - 198 \beta_{3} + 2442 \beta_1) q^{11} + (18 \beta_{2} + 2226490) q^{13} + ( - 729 \beta_{3} + 5679 \beta_1) q^{15} + ( - 178 \beta_{2} + 18136890) q^{17} + (2914 \beta_{3} - 47286 \beta_1) q^{19} + ( - 1215 \beta_{2} + 11258676) q^{21} + (9270 \beta_{3} + 102426 \beta_1) q^{23} + (11196 \beta_{2} + 179527283) q^{25} - 177147 \beta_1 q^{27} + ( - 18985 \beta_{2} - 182102778) q^{29} + ( - 90205 \beta_{3} + 1330785 \beta_1) q^{31} + ( - 48114 \beta_{2} - 428695740) q^{33} + (18342 \beta_{3} + 2333058 \beta_1) q^{35} + (196380 \beta_{2} - 681143470) q^{37} + ( - 13122 \beta_{3} + 2227948 \beta_1) q^{39} + ( - 148350 \beta_{2} - 1791030150) q^{41} + (750430 \beta_{3} + 8008326 \beta_1) q^{43} + ( - 177147 \beta_{2} - 991668906) q^{45} + ( - 485910 \beta_{3} + 25132542 \beta_1) q^{47} + (154440 \beta_{2} + 9856316593) q^{49} + (129762 \beta_{3} + 18122472 \beta_1) q^{51} + (131871 \beta_{2} + 8154817830) q^{53} + ( - 2872584 \beta_{3} + 120425184 \beta_1) q^{55} + (708102 \beta_{2} + 8319216780) q^{57} + (746568 \beta_{3} + 135469668 \beta_1) q^{59} + ( - 2482560 \beta_{2} - 32684060446) q^{61} + (885735 \beta_{3} + 11160261 \beta_1) q^{63} + (2327254 \beta_{2} + 19525836492) q^{65} + (5716680 \beta_{3} + 179960508 \beta_1) q^{67} + (2252610 \beta_{2} - 18326920032) q^{69} + (6276078 \beta_{3} + 358260018 \beta_1) q^{71} + ( - 6082488 \beta_{2} - 42624362270) q^{73} + ( - 8161884 \beta_{3} + 180434159 \beta_1) q^{75} + (117612 \beta_{2} - 102223004400) q^{77} + ( - 4352441 \beta_{3} + 240436989 \beta_1) q^{79} + 31381059609 q^{81} + ( - 28062630 \beta_{3} + 138076386 \beta_1) q^{83} + (17140446 \beta_{2} + 31695516108) q^{85} + (13840065 \beta_{3} - 183640563 \beta_1) q^{87} + ( - 7633820 \beta_{2} + 503055597474) q^{89} + ( - 10298474 \beta_{3} - 91925694 \beta_1) q^{91} + ( - 21919815 \beta_{2} - 233969065380) q^{93} + (50548032 \beta_{3} - 1836755832 \beta_1) q^{95} + (19661076 \beta_{2} + 372854702050) q^{97} + (35075106 \beta_{3} - 432592974 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 22392 q^{5} - 708588 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 22392 q^{5} - 708588 q^{9} + 8905960 q^{13} + 72547560 q^{17} + 45034704 q^{21} + 718109132 q^{25} - 728411112 q^{29} - 1714782960 q^{33} - 2724573880 q^{37} - 7164120600 q^{41} - 3966675624 q^{45} + 39425266372 q^{49} + 32619271320 q^{53} + 33276867120 q^{57} - 130736241784 q^{61} + 78103345968 q^{65} - 73307680128 q^{69} - 170497449080 q^{73} - 408892017600 q^{77} + 125524238436 q^{81} + 126782064432 q^{85} + 2012222389896 q^{89} - 935876261520 q^{93} + 1491418808200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 700x^{2} - 701x + 491401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 243\nu^{3} + 170100\nu^{2} - 170100\nu - 59790393 ) / 245350 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -432\nu^{3} + 432\nu^{2} + 605232\nu + 151416 ) / 701 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -21627\nu^{3} + 2700\nu^{2} - 2700\nu + 21784977 ) / 35050 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -9\beta_{3} + 9\beta_{2} - 7\beta _1 + 1944 ) / 7776 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{3} + 9\beta_{2} + 11207\beta _1 + 2723544 ) / 7776 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1575\beta_{3} + 175\beta _1 + 1021572 ) / 972 \) 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
−22.6751 + 13.6689i
23.1751 12.8028i
−22.6751 13.6689i
23.1751 + 12.8028i
0 420.888i 0 −14209.3 0 83928.6i 0 −177147. 0
31.2 0 420.888i 0 25405.3 0 30429.0i 0 −177147. 0
31.3 0 420.888i 0 −14209.3 0 83928.6i 0 −177147. 0
31.4 0 420.888i 0 25405.3 0 30429.0i 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.b 4
3.b odd 2 1 144.13.g.f 4
4.b odd 2 1 inner 48.13.g.b 4
8.b even 2 1 192.13.g.b 4
8.d odd 2 1 192.13.g.b 4
12.b even 2 1 144.13.g.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.13.g.b 4 1.a even 1 1 trivial
48.13.g.b 4 4.b odd 2 1 inner
144.13.g.f 4 3.b odd 2 1
144.13.g.f 4 12.b even 2 1
192.13.g.b 4 8.b even 2 1
192.13.g.b 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 11196T_{5} - 360992700 \) 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
$5$ \( (T^{2} - 11196 T - 360992700)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 65\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots + 4830142701604)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + \cdots + 316516185520164)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots + 59\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 99\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 23\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 12\!\cdots\!04)^{2} \) Copy content Toggle raw display
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