Properties

Label 1575.4.a.h
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

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: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 4 q^{4} - 7 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 4 q^{4} - 7 q^{7} - 24 q^{8} + 21 q^{11} + 24 q^{13} - 14 q^{14} - 16 q^{16} + 22 q^{17} + 16 q^{19} + 42 q^{22} + 25 q^{23} + 48 q^{26} + 28 q^{28} - 167 q^{29} + 10 q^{31} + 160 q^{32} + 44 q^{34} - 133 q^{37} + 32 q^{38} + 168 q^{41} - 97 q^{43} - 84 q^{44} + 50 q^{46} + 400 q^{47} + 49 q^{49} - 96 q^{52} + 182 q^{53} + 168 q^{56} - 334 q^{58} - 488 q^{59} + 28 q^{61} + 20 q^{62} + 448 q^{64} - 967 q^{67} - 88 q^{68} + 285 q^{71} - 838 q^{73} - 266 q^{74} - 64 q^{76} - 147 q^{77} - 469 q^{79} + 336 q^{82} + 406 q^{83} - 194 q^{86} - 504 q^{88} - 324 q^{89} - 168 q^{91} - 100 q^{92} + 800 q^{94} - 114 q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 0 −4.00000 0 0 −7.00000 −24.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.h 1
3.b odd 2 1 525.4.a.d 1
5.b even 2 1 1575.4.a.d 1
15.d odd 2 1 525.4.a.f yes 1
15.e even 4 2 525.4.d.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 3.b odd 2 1
525.4.a.f yes 1 15.d odd 2 1
525.4.d.e 2 15.e even 4 2
1575.4.a.d 1 5.b even 2 1
1575.4.a.h 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{11} - 21 \) Copy content Toggle raw display
\( T_{13} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T - 21 \) Copy content Toggle raw display
$13$ \( T - 24 \) Copy content Toggle raw display
$17$ \( T - 22 \) Copy content Toggle raw display
$19$ \( T - 16 \) Copy content Toggle raw display
$23$ \( T - 25 \) Copy content Toggle raw display
$29$ \( T + 167 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T + 133 \) Copy content Toggle raw display
$41$ \( T - 168 \) Copy content Toggle raw display
$43$ \( T + 97 \) Copy content Toggle raw display
$47$ \( T - 400 \) Copy content Toggle raw display
$53$ \( T - 182 \) Copy content Toggle raw display
$59$ \( T + 488 \) Copy content Toggle raw display
$61$ \( T - 28 \) Copy content Toggle raw display
$67$ \( T + 967 \) Copy content Toggle raw display
$71$ \( T - 285 \) Copy content Toggle raw display
$73$ \( T + 838 \) Copy content Toggle raw display
$79$ \( T + 469 \) Copy content Toggle raw display
$83$ \( T - 406 \) Copy content Toggle raw display
$89$ \( T + 324 \) Copy content Toggle raw display
$97$ \( T + 114 \) Copy content Toggle raw display
show more
show less