↑

Properties

Label 6300.2.d.b
Level $6300$
Weight $2$
Character orbit 6300.d
Analytic conductor $50.306$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6300,2,Mod(3401,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6300.3401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.d (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: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1260)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{7} + \beta_1 q^{11} + (\beta_{2} + \beta_1) q^{13} - 2 \beta_{3} q^{17} + ( - \beta_{2} - \beta_1) q^{19} + \beta_{2} q^{23} + (2 \beta_{2} + 3 \beta_1) q^{29} + ( - \beta_{2} + \beta_1) q^{31}+ \cdots + (3 \beta_{2} + 7 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{37} + 16 q^{41} - 16 q^{43} - 8 q^{47} + 12 q^{49} - 24 q^{59} + 8 q^{77} - 40 q^{83} - 8 q^{89} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 4\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3401.1
− 2.28825i
2.28825i
0.874032i
− 0.874032i
0 0 0 0 0 −2.23607 − 1.41421i 0 0 0
3401.2 0 0 0 0 0 −2.23607 + 1.41421i 0 0 0
3401.3 0 0 0 0 0 2.23607 − 1.41421i 0 0 0
3401.4 0 0 0 0 0 2.23607 + 1.41421i 0 0 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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.d.b 4
3.b odd 2 1 6300.2.d.a 4
5.b even 2 1 1260.2.d.a ✓ 4
5.c odd 4 2 6300.2.f.c 8
7.b odd 2 1 6300.2.d.a 4
15.d odd 2 1 1260.2.d.b yes 4
15.e even 4 2 6300.2.f.a 8
20.d odd 2 1 5040.2.f.b 4
21.c even 2 1 inner 6300.2.d.b 4
35.c odd 2 1 1260.2.d.b yes 4
35.f even 4 2 6300.2.f.a 8
60.h even 2 1 5040.2.f.d 4
105.g even 2 1 1260.2.d.a ✓ 4
105.k odd 4 2 6300.2.f.c 8
140.c even 2 1 5040.2.f.d 4
420.o odd 2 1 5040.2.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.d.a ✓ 4 5.b even 2 1
1260.2.d.a ✓ 4 105.g even 2 1
1260.2.d.b yes 4 15.d odd 2 1
1260.2.d.b yes 4 35.c odd 2 1
5040.2.f.b 4 20.d odd 2 1
5040.2.f.b 4 420.o odd 2 1
5040.2.f.d 4 60.h even 2 1
5040.2.f.d 4 140.c even 2 1
6300.2.d.a 4 3.b odd 2 1
6300.2.d.a 4 7.b odd 2 1
6300.2.d.b 4 1.a even 1 1 trivial
6300.2.d.b 4 21.c even 2 1 inner
6300.2.f.a 8 15.e even 4 2
6300.2.f.a 8 35.f even 4 2
6300.2.f.c 8 5.c odd 4 2
6300.2.f.c 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\):

\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{37}^{2} + 12T_{37} + 16 \) Copy content Toggle raw display
\( T_{41}^{2} - 8T_{41} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 6T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 116T^{2} + 484 \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 64 \) Copy content Toggle raw display
$37$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 84T^{2} + 484 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 96T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 84T^{2} + 1444 \) Copy content Toggle raw display
$73$ \( T^{4} + 280T^{2} + 1600 \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 20 T + 80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 376T^{2} + 64 \) Copy content Toggle raw display
show more
show less