Properties

Label 1575.4.a.k
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 8 q^{4} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 8 q^{4} + 7 q^{7} - 62 q^{11} + 62 q^{13} + 28 q^{14} - 64 q^{16} + 84 q^{17} + 100 q^{19} - 248 q^{22} - 42 q^{23} + 248 q^{26} + 56 q^{28} + 10 q^{29} - 48 q^{31} - 256 q^{32} + 336 q^{34} + 246 q^{37} + 400 q^{38} + 248 q^{41} - 68 q^{43} - 496 q^{44} - 168 q^{46} + 324 q^{47} + 49 q^{49} + 496 q^{52} + 258 q^{53} + 40 q^{58} - 120 q^{59} + 622 q^{61} - 192 q^{62} - 512 q^{64} - 904 q^{67} + 672 q^{68} + 678 q^{71} + 642 q^{73} + 984 q^{74} + 800 q^{76} - 434 q^{77} + 740 q^{79} + 992 q^{82} + 468 q^{83} - 272 q^{86} - 200 q^{89} + 434 q^{91} - 336 q^{92} + 1296 q^{94} + 1266 q^{97} + 196 q^{98}+O(q^{100}) \) 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
4.00000 0 8.00000 0 0 7.00000 0 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.k 1
3.b odd 2 1 525.4.a.b 1
5.b even 2 1 63.4.a.a 1
15.d odd 2 1 21.4.a.b 1
15.e even 4 2 525.4.d.b 2
20.d odd 2 1 1008.4.a.m 1
35.c odd 2 1 441.4.a.b 1
35.i odd 6 2 441.4.e.n 2
35.j even 6 2 441.4.e.m 2
60.h even 2 1 336.4.a.h 1
105.g even 2 1 147.4.a.g 1
105.o odd 6 2 147.4.e.c 2
105.p even 6 2 147.4.e.b 2
120.i odd 2 1 1344.4.a.w 1
120.m even 2 1 1344.4.a.i 1
420.o odd 2 1 2352.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 15.d odd 2 1
63.4.a.a 1 5.b even 2 1
147.4.a.g 1 105.g even 2 1
147.4.e.b 2 105.p even 6 2
147.4.e.c 2 105.o odd 6 2
336.4.a.h 1 60.h even 2 1
441.4.a.b 1 35.c odd 2 1
441.4.e.m 2 35.j even 6 2
441.4.e.n 2 35.i odd 6 2
525.4.a.b 1 3.b odd 2 1
525.4.d.b 2 15.e even 4 2
1008.4.a.m 1 20.d odd 2 1
1344.4.a.i 1 120.m even 2 1
1344.4.a.w 1 120.i odd 2 1
1575.4.a.k 1 1.a even 1 1 trivial
2352.4.a.l 1 420.o odd 2 1

Hecke kernels

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

\( T_{2} - 4 \) Copy content Toggle raw display
\( T_{11} + 62 \) Copy content Toggle raw display
\( T_{13} - 62 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) 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
$11$ \( T + 62 \) Copy content Toggle raw display
$13$ \( T - 62 \) Copy content Toggle raw display
$17$ \( T - 84 \) Copy content Toggle raw display
$19$ \( T - 100 \) Copy content Toggle raw display
$23$ \( T + 42 \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T + 48 \) Copy content Toggle raw display
$37$ \( T - 246 \) Copy content Toggle raw display
$41$ \( T - 248 \) Copy content Toggle raw display
$43$ \( T + 68 \) Copy content Toggle raw display
$47$ \( T - 324 \) Copy content Toggle raw display
$53$ \( T - 258 \) Copy content Toggle raw display
$59$ \( T + 120 \) Copy content Toggle raw display
$61$ \( T - 622 \) Copy content Toggle raw display
$67$ \( T + 904 \) Copy content Toggle raw display
$71$ \( T - 678 \) Copy content Toggle raw display
$73$ \( T - 642 \) Copy content Toggle raw display
$79$ \( T - 740 \) Copy content Toggle raw display
$83$ \( T - 468 \) Copy content Toggle raw display
$89$ \( T + 200 \) Copy content Toggle raw display
$97$ \( T - 1266 \) Copy content Toggle raw display
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