↑

Properties

Label 48.12.c.b
Level $48$
Weight $12$
Character orbit 48.c
Analytic conductor $36.880$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(36.8804726669\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} + 1026 x^{6} - 3064 x^{5} + 14948113 x^{4} - 29891124 x^{3} + 24696259780 x^{2} + \cdots + 586313338523535 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{19} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{6} - 6 \beta_{2}) q^{7} + ( - \beta_{4} - \beta_1 - 9171) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{6} - 6 \beta_{2}) q^{7} + ( - \beta_{4} - \beta_1 - 9171) q^{9} + ( - \beta_{7} - 89 \beta_{2}) q^{11} + (6 \beta_{4} - \beta_{3} + 165142) q^{13} + (3 \beta_{7} + 45 \beta_{6} + \cdots - 23 \beta_{2}) q^{15}+ \cdots + (80190 \beta_{7} + \cdots - 17129556 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 73368 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 73368 q^{9} + 1321136 q^{13} + 7795440 q^{21} - 5818520 q^{25} - 126577728 q^{33} - 20962064 q^{37} + 388575360 q^{45} + 1262960600 q^{49} - 5239861488 q^{57} + 5523680560 q^{61} - 15608204928 q^{69} + 20816625872 q^{73} - 74779754040 q^{81} + 158152527360 q^{85} - 269277464016 q^{93} + 317919410960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} + 1026 x^{6} - 3064 x^{5} + 14948113 x^{4} - 29891124 x^{3} + 24696259780 x^{2} + \cdots + 586313338523535 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 176 \nu^{6} - 528 \nu^{5} - 6197288 \nu^{4} + 12395456 \nu^{3} + 384062652 \nu^{2} + \cdots - 43309347947460 ) / 20791566243 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 128 \nu^{7} + 448 \nu^{6} - 131104 \nu^{5} + 326640 \nu^{4} - 1913195144 \nu^{3} + \cdots + 1579365025839 ) / 2541865828329 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31264 \nu^{6} + 93792 \nu^{5} - 19220672 \nu^{4} + 38285024 \nu^{3} + 1782715939632 \nu^{2} + \cdots + 87064576761456 ) / 6930522081 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30736 \nu^{6} - 92208 \nu^{5} + 37812536 \nu^{4} - 75471392 \nu^{3} + 461621026656 \nu^{2} + \cdots + 615463201413144 ) / 20791566243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1664 \nu^{7} + 5824 \nu^{6} - 1704352 \nu^{5} + 4246320 \nu^{4} - 24871536872 \nu^{3} + \cdots - 18\!\cdots\!34 ) / 2541865828329 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30112 \nu^{7} - 105392 \nu^{6} - 313531552 \nu^{5} + 784092360 \nu^{4} + \cdots + 920608152265728 ) / 19487638017189 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 198976 \nu^{7} - 696416 \nu^{6} + 6402528992 \nu^{5} - 16004581440 \nu^{4} + \cdots - 26\!\cdots\!73 ) / 17793060798303 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 13\beta_{2} + 729 ) / 1458 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 27\beta_{4} + 9\beta_{3} - 26\beta_{2} + 81\beta _1 - 742122 ) / 2916 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1701 \beta_{7} - 33534 \beta_{6} + 58 \beta_{5} + 81 \beta_{4} + 27 \beta_{3} - 1407481 \beta_{2} + \cdots - 2227824 ) / 5832 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6804 \beta_{7} - 134136 \beta_{6} + 224 \beta_{5} + 178794 \beta_{4} - 12573 \beta_{3} + \cdots - 84131679672 ) / 11664 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40397049 \beta_{7} - 523700478 \beta_{6} - 121352321 \beta_{5} + 893430 \beta_{4} + \cdots - 420643544256 ) / 23328 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 121225167 \beta_{7} - 1570430754 \beta_{6} - 364058075 \beta_{5} + 12099851658 \beta_{4} + \cdots - 172062194615280 ) / 23328 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 121422328209 \beta_{7} + 5070256093116 \beta_{6} - 550095485971 \beta_{5} + 84692708352 \beta_{4} + \cdots - 12\!\cdots\!80 ) / 46656 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−61.0241 + 33.6977i
−61.0241 − 33.6977i
−29.1780 + 63.5607i
−29.1780 − 63.5607i
30.1780 + 63.5607i
30.1780 − 63.5607i
62.0241 + 33.6977i
62.0241 − 33.6977i
0 −369.145 − 202.186i 0 6336.01i 0 − 51032.1i 0 95388.5 + 149272.i 0
47.2 0 −369.145 + 202.186i 0 − 6336.01i 0 51032.1i 0 95388.5 − 149272.i 0
47.3 0 −178.068 − 381.364i 0 − 7678.92i 0 32165.7i 0 −113730. + 135818.i 0
47.4 0 −178.068 + 381.364i 0 7678.92i 0 − 32165.7i 0 −113730. − 135818.i 0
47.5 0 178.068 − 381.364i 0 7678.92i 0 32165.7i 0 −113730. − 135818.i 0
47.6 0 178.068 + 381.364i 0 − 7678.92i 0 − 32165.7i 0 −113730. + 135818.i 0
47.7 0 369.145 − 202.186i 0 − 6336.01i 0 − 51032.1i 0 95388.5 − 149272.i 0
47.8 0 369.145 + 202.186i 0 6336.01i 0 51032.1i 0 95388.5 + 149272.i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
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 48.12.c.b ✓ 8
3.b odd 2 1 inner 48.12.c.b ✓ 8
4.b odd 2 1 inner 48.12.c.b ✓ 8
8.b even 2 1 192.12.c.b 8
8.d odd 2 1 192.12.c.b 8
12.b even 2 1 inner 48.12.c.b ✓ 8
24.f even 2 1 192.12.c.b 8
24.h odd 2 1 192.12.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.12.c.b ✓ 8 1.a even 1 1 trivial
48.12.c.b ✓ 8 3.b odd 2 1 inner
48.12.c.b ✓ 8 4.b odd 2 1 inner
48.12.c.b ✓ 8 12.b even 2 1 inner
192.12.c.b 8 8.b even 2 1
192.12.c.b 8 8.d odd 2 1
192.12.c.b 8 24.f even 2 1
192.12.c.b 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 99110880T_{5}^{2} + 2367186861312000 \) acting on \(S_{12}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 98\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( (T^{4} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 3514919011100)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 32\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 84\!\cdots\!00)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 61\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 37\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 27\!\cdots\!16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 52\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 18\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 48\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 18\!\cdots\!00)^{4} \) Copy content Toggle raw display
show more
show less