Properties

Label 48.10.c.b
Level $48$
Weight $10$
Character orbit 48.c
Analytic conductor $24.722$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,10,Mod(47,48)]
 
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, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 48.c (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: \(24.7217201359\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-51}, \sqrt{534})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 1041x^{2} + 1042x + 298675 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{3} q^{5} + 77 \beta_{2} q^{7} + (3 \beta_{3} + 18765) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{3} q^{5} + 77 \beta_{2} q^{7} + (3 \beta_{3} + 18765) q^{9} + ( - 9 \beta_{2} + 54 \beta_1) q^{11} + 112318 q^{13} + ( - 2187 \beta_{2} + 306 \beta_1) q^{15} - 236 \beta_{3} q^{17} + 3899 \beta_{2} q^{19} + (693 \beta_{3} - 212058) q^{21} + ( - 2178 \beta_{2} + 13068 \beta_1) q^{23} - 1968571 q^{25} + (6561 \beta_{2} + 17847 \beta_1) q^{27} - 1175 \beta_{3} q^{29} - 64343 \beta_{2} q^{31} + (81 \beta_{3} + 1038096) q^{33} + ( - 23562 \beta_{2} + 141372 \beta_1) q^{35} + 12067382 q^{37} + 112318 \beta_1 q^{39} + 7734 \beta_{3} q^{41} + 132623 \beta_{2} q^{43} + ( - 18765 \beta_{3} + 11765088) q^{45} + ( - 52020 \beta_{2} + 312120 \beta_1) q^{47} - 57617189 q^{49} + ( - 516132 \beta_{2} + 72216 \beta_1) q^{51} + 19059 \beta_{3} q^{53} - 115344 \beta_{2} q^{55} + (35091 \beta_{3} - 10737846) q^{57} + (105723 \beta_{2} - 634338 \beta_1) q^{59} + 3669518 q^{61} + (1515591 \beta_{2} - 424116 \beta_1) q^{63} - 112318 \beta_{3} q^{65} + 74091 \beta_{2} q^{67} + (19602 \beta_{3} + 251219232) q^{69} + (325386 \beta_{2} - 1952316 \beta_1) q^{71} - 76753334 q^{73} - 1968571 \beta_1 q^{75} + 37422 \beta_{3} q^{77} - 1367599 \beta_{2} q^{79} + (112590 \beta_{3} + 316829961) q^{81} + (733329 \beta_{2} - 4399974 \beta_1) q^{83} - 925520256 q^{85} + ( - 2569725 \beta_{2} + 359550 \beta_1) q^{87} + 108578 \beta_{3} q^{89} + 8648486 \beta_{2} q^{91} + ( - 579087 \beta_{3} + 177200622) q^{93} + ( - 1193094 \beta_{2} + 7158564 \beta_1) q^{95} - 641848430 q^{97} + (177147 \beta_{2} + 1013310 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 75060 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 75060 q^{9} + 449272 q^{13} - 848232 q^{21} - 7874284 q^{25} + 4152384 q^{33} + 48269528 q^{37} + 47060352 q^{45} - 230468756 q^{49} - 42951384 q^{57} + 14678072 q^{61} + 1004876928 q^{69} - 307013336 q^{73} + 1267319844 q^{81} - 3702081024 q^{85} + 708802488 q^{93} - 2567393720 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 1041x^{2} + 1042x + 298675 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{3} - 12\nu^{2} - 3960\nu + 1982 ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 12\nu^{2} - 12\nu - 6252 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta _1 + 6258 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 243\beta_{2} + 662\beta _1 + 6256 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−22.6084 3.57071i
−22.6084 + 3.57071i
23.6084 3.57071i
23.6084 + 3.57071i
0 −138.651 21.4243i 0 1980.33i 0 9898.02i 0 18765.0 + 5940.98i 0
47.2 0 −138.651 + 21.4243i 0 1980.33i 0 9898.02i 0 18765.0 5940.98i 0
47.3 0 138.651 21.4243i 0 1980.33i 0 9898.02i 0 18765.0 5940.98i 0
47.4 0 138.651 + 21.4243i 0 1980.33i 0 9898.02i 0 18765.0 + 5940.98i 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
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.10.c.b 4
3.b odd 2 1 inner 48.10.c.b 4
4.b odd 2 1 inner 48.10.c.b 4
8.b even 2 1 192.10.c.b 4
8.d odd 2 1 192.10.c.b 4
12.b even 2 1 inner 48.10.c.b 4
24.f even 2 1 192.10.c.b 4
24.h odd 2 1 192.10.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.10.c.b 4 1.a even 1 1 trivial
48.10.c.b 4 3.b odd 2 1 inner
48.10.c.b 4 4.b odd 2 1 inner
48.10.c.b 4 12.b even 2 1 inner
192.10.c.b 4 8.b even 2 1
192.10.c.b 4 8.d odd 2 1
192.10.c.b 4 24.f even 2 1
192.10.c.b 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3921696 \) acting on \(S_{10}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 37530 T^{2} + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3921696)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 97970796)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 56057184)^{2} \) Copy content Toggle raw display
$13$ \( (T - 112318)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 218422780416)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 251201169324)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3282932923776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 5414391540000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 68409717728076)^{2} \) Copy content Toggle raw display
$37$ \( (T - 12067382)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 234575289346176)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 290638324771596)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 77\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T - 3669518)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 90708106067244)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 73\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T + 76753334)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 30\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 46\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T + 641848430)^{4} \) Copy content Toggle raw display
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