↑

Properties

Label 252.2.b.d
Level $252$
Weight $2$
Character orbit 252.b
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(55,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.b (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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2312.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5} + ( - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5} + ( - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{3} + 2) q^{8} + ( - \beta_{3} - \beta_{2} + 2) q^{10} + (\beta_{2} - \beta_1) q^{11} + (2 \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{14} + (\beta_{3} + 2 \beta_1 - 2) q^{16} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{17} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{19} + ( - 2 \beta_{3} + 2 \beta_1) q^{20} + (\beta_{3} - \beta_{2} + 2) q^{22} + ( - \beta_{2} + \beta_1) q^{23} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{25} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{26} + ( - \beta_{2} + 2 \beta_1 + 4) q^{28} + 2 q^{29} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{32} + (3 \beta_{3} - \beta_{2} + 2) q^{34} + ( - 3 \beta_{2} - \beta_1 - 2) q^{35} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{37} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{38} + ( - 2 \beta_{3} + 2 \beta_{2} + 4) q^{40} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{41} + ( - 3 \beta_{3} + \beta_{2} - 4 \beta_1) q^{43} + (2 \beta_1 - 4) q^{44} + ( - \beta_{3} + \beta_{2} - 2) q^{46} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{47}+ \cdots + ( - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + 8 q^{10} + 9 q^{14} - 7 q^{16} - 12 q^{19} + 4 q^{20} + 6 q^{22} - 8 q^{25} + 12 q^{26} + 17 q^{28} + 8 q^{29} - 9 q^{32} + 4 q^{34} - 12 q^{35} - 12 q^{37} - 20 q^{38} + 20 q^{40} - 14 q^{44} - 6 q^{46} - 8 q^{47} + 8 q^{49} - 19 q^{50} - 28 q^{52} - 16 q^{53} - 4 q^{55} - q^{56} + 2 q^{58} + 16 q^{59} + q^{64} - 8 q^{65} - 32 q^{68} - 24 q^{70} + 14 q^{74} - 20 q^{76} + 8 q^{77} + 36 q^{80} - 20 q^{82} - 8 q^{83} + 20 q^{85} + 30 q^{86} - 2 q^{88} + 16 q^{91} + 14 q^{92} + 32 q^{94} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.780776 − 1.17915i
−0.780776 + 1.17915i
1.28078 − 0.599676i
1.28078 + 0.599676i
−0.780776 − 1.17915i 0 −0.780776 + 1.84130i 1.69614i 0 −2.56155 − 0.662153i 2.78078 − 0.516994i 0 2.00000 − 1.32431i
55.2 −0.780776 + 1.17915i 0 −0.780776 − 1.84130i − 1.69614i 0 −2.56155 + 0.662153i 2.78078 + 0.516994i 0 2.00000 + 1.32431i
55.3 1.28078 − 0.599676i 0 1.28078 − 1.53610i 3.33513i 0 1.56155 + 2.13578i 0.719224 − 2.73546i 0 2.00000 + 4.27156i
55.4 1.28078 + 0.599676i 0 1.28078 + 1.53610i − 3.33513i 0 1.56155 − 2.13578i 0.719224 + 2.73546i 0 2.00000 − 4.27156i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.b.d 4
3.b odd 2 1 84.2.b.b yes 4
4.b odd 2 1 252.2.b.e 4
7.b odd 2 1 252.2.b.e 4
8.b even 2 1 4032.2.b.j 4
8.d odd 2 1 4032.2.b.n 4
12.b even 2 1 84.2.b.a ✓ 4
21.c even 2 1 84.2.b.a ✓ 4
21.g even 6 2 588.2.o.c 8
21.h odd 6 2 588.2.o.a 8
24.f even 2 1 1344.2.b.f 4
24.h odd 2 1 1344.2.b.e 4
28.d even 2 1 inner 252.2.b.d 4
56.e even 2 1 4032.2.b.j 4
56.h odd 2 1 4032.2.b.n 4
84.h odd 2 1 84.2.b.b yes 4
84.j odd 6 2 588.2.o.a 8
84.n even 6 2 588.2.o.c 8
168.e odd 2 1 1344.2.b.e 4
168.i even 2 1 1344.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a ✓ 4 12.b even 2 1
84.2.b.a ✓ 4 21.c even 2 1
84.2.b.b yes 4 3.b odd 2 1
84.2.b.b yes 4 84.h odd 2 1
252.2.b.d 4 1.a even 1 1 trivial
252.2.b.d 4 28.d even 2 1 inner
252.2.b.e 4 4.b odd 2 1
252.2.b.e 4 7.b odd 2 1
588.2.o.a 8 21.h odd 6 2
588.2.o.a 8 84.j odd 6 2
588.2.o.c 8 21.g even 6 2
588.2.o.c 8 84.n even 6 2
1344.2.b.e 4 24.h odd 2 1
1344.2.b.e 4 168.e odd 2 1
1344.2.b.f 4 24.f even 2 1
1344.2.b.f 4 168.i even 2 1
4032.2.b.j 4 8.b even 2 1
4032.2.b.j 4 56.e even 2 1
4032.2.b.n 4 8.d odd 2 1
4032.2.b.n 4 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{4} + 14T_{5}^{2} + 32 \) Copy content Toggle raw display
\( T_{11}^{4} + 10T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{19}^{2} + 6T_{19} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 14T^{2} + 32 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 10T^{2} + 8 \) Copy content Toggle raw display
$13$ \( T^{4} + 40T^{2} + 128 \) Copy content Toggle raw display
$17$ \( T^{4} + 46T^{2} + 512 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 10T^{2} + 8 \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 62T^{2} + 128 \) Copy content Toggle raw display
$43$ \( T^{4} + 148T^{2} + 5408 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 52)^{2} \) Copy content Toggle raw display
$59$ \( (T - 4)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 112T^{2} + 2048 \) Copy content Toggle raw display
$67$ \( T^{4} + 124T^{2} + 512 \) Copy content Toggle raw display
$71$ \( T^{4} + 170T^{2} + 2312 \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 512 \) Copy content Toggle raw display
$79$ \( T^{4} + 28T^{2} + 128 \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 64)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 62T^{2} + 128 \) Copy content Toggle raw display
$97$ \( T^{4} + 184T^{2} + 8192 \) Copy content Toggle raw display
show more
show less