Properties

Label 1425.2.c.k
Level $1425$
Weight $2$
Character orbit 1425.c
Analytic conductor $11.379$
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] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not 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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 285)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} + \beta_{3} q^{6} + ( - \beta_{2} + \beta_1) q^{7} - \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} + \beta_{3} q^{6} + ( - \beta_{2} + \beta_1) q^{7} - \beta_{2} q^{8} - q^{9} + ( - \beta_{3} + 3) q^{11} - \beta_1 q^{12} + ( - \beta_{2} - \beta_1) q^{13} + (\beta_{3} - 3) q^{14} - 5 q^{16} + \beta_{2} q^{18} - q^{19} + (\beta_{3} - 1) q^{21} + ( - 3 \beta_{2} + 3 \beta_1) q^{22} - 2 \beta_{2} q^{23} + \beta_{3} q^{24} + ( - \beta_{3} - 3) q^{26} - \beta_1 q^{27} + (\beta_{2} - \beta_1) q^{28} + (3 \beta_{3} - 3) q^{29} + ( - 4 \beta_{3} + 2) q^{31} + 3 \beta_{2} q^{32} + ( - \beta_{2} + 3 \beta_1) q^{33} + q^{36} + ( - 3 \beta_{2} + \beta_1) q^{37} + \beta_{2} q^{38} + (\beta_{3} + 1) q^{39} + (\beta_{3} + 3) q^{41} + (\beta_{2} - 3 \beta_1) q^{42} + ( - 3 \beta_{2} - \beta_1) q^{43} + (\beta_{3} - 3) q^{44} - 6 q^{46} - 2 \beta_{2} q^{47} - 5 \beta_1 q^{48} + (2 \beta_{3} + 3) q^{49} + (\beta_{2} + \beta_1) q^{52} + (2 \beta_{2} - 6 \beta_1) q^{53} - \beta_{3} q^{54} + (\beta_{3} - 3) q^{56} - \beta_1 q^{57} + (3 \beta_{2} - 9 \beta_1) q^{58} + ( - 2 \beta_{3} - 6) q^{59} + ( - 2 \beta_{3} - 10) q^{61} + ( - 2 \beta_{2} + 12 \beta_1) q^{62} + (\beta_{2} - \beta_1) q^{63} - q^{64} + (3 \beta_{3} - 3) q^{66} - 8 \beta_1 q^{67} + 2 \beta_{3} q^{69} + (6 \beta_{3} + 6) q^{71} + \beta_{2} q^{72} + (4 \beta_{2} - 10 \beta_1) q^{73} + (\beta_{3} - 9) q^{74} + q^{76} + ( - 4 \beta_{2} + 6 \beta_1) q^{77} + ( - \beta_{2} - 3 \beta_1) q^{78} + ( - 4 \beta_{3} + 4) q^{79} + q^{81} + ( - 3 \beta_{2} - 3 \beta_1) q^{82} + (4 \beta_{2} - 6 \beta_1) q^{83} + ( - \beta_{3} + 1) q^{84} + ( - \beta_{3} - 9) q^{86} + (3 \beta_{2} - 3 \beta_1) q^{87} + ( - 3 \beta_{2} + 3 \beta_1) q^{88} + (\beta_{3} - 9) q^{89} - 2 q^{91} + 2 \beta_{2} q^{92} + ( - 4 \beta_{2} + 2 \beta_1) q^{93} - 6 q^{94} - 3 \beta_{3} q^{96} + ( - 3 \beta_{2} + \beta_1) q^{97} + ( - 3 \beta_{2} - 6 \beta_1) q^{98} + (\beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 4 q^{19} - 4 q^{21} - 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{36} + 4 q^{39} + 12 q^{41} - 12 q^{44} - 24 q^{46} + 12 q^{49} - 12 q^{56} - 24 q^{59} - 40 q^{61} - 4 q^{64} - 12 q^{66} + 24 q^{71} - 36 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{81} + 4 q^{84} - 36 q^{86} - 36 q^{89} - 8 q^{91} - 24 q^{94} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
1.73205i 1.00000i −1.00000 0 −1.73205 2.73205i 1.73205i −1.00000 0
799.2 1.73205i 1.00000i −1.00000 0 1.73205 0.732051i 1.73205i −1.00000 0
799.3 1.73205i 1.00000i −1.00000 0 1.73205 0.732051i 1.73205i −1.00000 0
799.4 1.73205i 1.00000i −1.00000 0 −1.73205 2.73205i 1.73205i −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1425.2.c.k 4
5.b even 2 1 inner 1425.2.c.k 4
5.c odd 4 1 285.2.a.e 2
5.c odd 4 1 1425.2.a.o 2
15.e even 4 1 855.2.a.f 2
15.e even 4 1 4275.2.a.t 2
20.e even 4 1 4560.2.a.bh 2
95.g even 4 1 5415.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.e 2 5.c odd 4 1
855.2.a.f 2 15.e even 4 1
1425.2.a.o 2 5.c odd 4 1
1425.2.c.k 4 1.a even 1 1 trivial
1425.2.c.k 4 5.b even 2 1 inner
4275.2.a.t 2 15.e even 4 1
4560.2.a.bh 2 20.e even 4 1
5415.2.a.r 2 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1425, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 20 T + 88)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 296T^{2} + 2704 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 78)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
show more
show less