Properties

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

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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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} - 7 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + q^{4} - 7 q^{7} + 21 q^{8} + 6 q^{11} - 41 q^{13} + 21 q^{14} - 71 q^{16} + 27 q^{17} - 4 q^{19} - 18 q^{22} + 75 q^{23} + 123 q^{26} - 7 q^{28} + 123 q^{29} - 205 q^{31} + 45 q^{32} - 81 q^{34} + 262 q^{37} + 12 q^{38} - 57 q^{41} - 407 q^{43} + 6 q^{44} - 225 q^{46} - 60 q^{47} + 49 q^{49} - 41 q^{52} + 327 q^{53} - 147 q^{56} - 369 q^{58} - 33 q^{59} - 427 q^{61} + 615 q^{62} + 433 q^{64} + 628 q^{67} + 27 q^{68} - 300 q^{71} - 98 q^{73} - 786 q^{74} - 4 q^{76} - 42 q^{77} + 686 q^{79} + 171 q^{82} + 1401 q^{83} + 1221 q^{86} + 126 q^{88} - 714 q^{89} + 287 q^{91} + 75 q^{92} + 180 q^{94} - 494 q^{97} - 147 q^{98}+O(q^{100}) \) 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
−3.00000 0 1.00000 0 0 −7.00000 21.0000 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.a 1
3.b odd 2 1 525.4.a.h yes 1
5.b even 2 1 1575.4.a.j 1
15.d odd 2 1 525.4.a.c 1
15.e even 4 2 525.4.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.c 1 15.d odd 2 1
525.4.a.h yes 1 3.b odd 2 1
525.4.d.d 2 15.e even 4 2
1575.4.a.a 1 1.a even 1 1 trivial
1575.4.a.j 1 5.b even 2 1

Hecke kernels

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

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} + 41 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) 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
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T + 41 \) Copy content Toggle raw display
$17$ \( T - 27 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 75 \) Copy content Toggle raw display
$29$ \( T - 123 \) Copy content Toggle raw display
$31$ \( T + 205 \) Copy content Toggle raw display
$37$ \( T - 262 \) Copy content Toggle raw display
$41$ \( T + 57 \) Copy content Toggle raw display
$43$ \( T + 407 \) Copy content Toggle raw display
$47$ \( T + 60 \) Copy content Toggle raw display
$53$ \( T - 327 \) Copy content Toggle raw display
$59$ \( T + 33 \) Copy content Toggle raw display
$61$ \( T + 427 \) Copy content Toggle raw display
$67$ \( T - 628 \) Copy content Toggle raw display
$71$ \( T + 300 \) Copy content Toggle raw display
$73$ \( T + 98 \) Copy content Toggle raw display
$79$ \( T - 686 \) Copy content Toggle raw display
$83$ \( T - 1401 \) Copy content Toggle raw display
$89$ \( T + 714 \) Copy content Toggle raw display
$97$ \( T + 494 \) Copy content Toggle raw display
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