Properties

Label 1575.4.a.e
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 7)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 7 q^{4} + 7 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 7 q^{4} + 7 q^{7} + 15 q^{8} + 8 q^{11} - 28 q^{13} - 7 q^{14} + 41 q^{16} + 54 q^{17} - 110 q^{19} - 8 q^{22} + 48 q^{23} + 28 q^{26} - 49 q^{28} + 110 q^{29} + 12 q^{31} - 161 q^{32} - 54 q^{34} + 246 q^{37} + 110 q^{38} - 182 q^{41} - 128 q^{43} - 56 q^{44} - 48 q^{46} + 324 q^{47} + 49 q^{49} + 196 q^{52} - 162 q^{53} + 105 q^{56} - 110 q^{58} - 810 q^{59} - 488 q^{61} - 12 q^{62} - 167 q^{64} - 244 q^{67} - 378 q^{68} + 768 q^{71} + 702 q^{73} - 246 q^{74} + 770 q^{76} + 56 q^{77} + 440 q^{79} + 182 q^{82} - 1302 q^{83} + 128 q^{86} + 120 q^{88} - 730 q^{89} - 196 q^{91} - 336 q^{92} - 324 q^{94} - 294 q^{97} - 49 q^{98}+O(q^{100}) \) 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
−1.00000 0 −7.00000 0 0 7.00000 15.0000 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.e 1
3.b odd 2 1 175.4.a.b 1
5.b even 2 1 63.4.a.b 1
15.d odd 2 1 7.4.a.a 1
15.e even 4 2 175.4.b.b 2
20.d odd 2 1 1008.4.a.c 1
21.c even 2 1 1225.4.a.j 1
35.c odd 2 1 441.4.a.i 1
35.i odd 6 2 441.4.e.e 2
35.j even 6 2 441.4.e.h 2
60.h even 2 1 112.4.a.f 1
105.g even 2 1 49.4.a.b 1
105.o odd 6 2 49.4.c.c 2
105.p even 6 2 49.4.c.b 2
120.i odd 2 1 448.4.a.i 1
120.m even 2 1 448.4.a.e 1
165.d even 2 1 847.4.a.b 1
195.e odd 2 1 1183.4.a.b 1
255.h odd 2 1 2023.4.a.a 1
420.o odd 2 1 784.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 15.d odd 2 1
49.4.a.b 1 105.g even 2 1
49.4.c.b 2 105.p even 6 2
49.4.c.c 2 105.o odd 6 2
63.4.a.b 1 5.b even 2 1
112.4.a.f 1 60.h even 2 1
175.4.a.b 1 3.b odd 2 1
175.4.b.b 2 15.e even 4 2
441.4.a.i 1 35.c odd 2 1
441.4.e.e 2 35.i odd 6 2
441.4.e.h 2 35.j even 6 2
448.4.a.e 1 120.m even 2 1
448.4.a.i 1 120.i odd 2 1
784.4.a.g 1 420.o odd 2 1
847.4.a.b 1 165.d even 2 1
1008.4.a.c 1 20.d odd 2 1
1183.4.a.b 1 195.e odd 2 1
1225.4.a.j 1 21.c even 2 1
1575.4.a.e 1 1.a even 1 1 trivial
2023.4.a.a 1 255.h odd 2 1

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 8 \) Copy content Toggle raw display
\( T_{13} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) 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
$11$ \( T - 8 \) Copy content Toggle raw display
$13$ \( T + 28 \) Copy content Toggle raw display
$17$ \( T - 54 \) Copy content Toggle raw display
$19$ \( T + 110 \) Copy content Toggle raw display
$23$ \( T - 48 \) Copy content Toggle raw display
$29$ \( T - 110 \) Copy content Toggle raw display
$31$ \( T - 12 \) Copy content Toggle raw display
$37$ \( T - 246 \) Copy content Toggle raw display
$41$ \( T + 182 \) Copy content Toggle raw display
$43$ \( T + 128 \) Copy content Toggle raw display
$47$ \( T - 324 \) Copy content Toggle raw display
$53$ \( T + 162 \) Copy content Toggle raw display
$59$ \( T + 810 \) Copy content Toggle raw display
$61$ \( T + 488 \) Copy content Toggle raw display
$67$ \( T + 244 \) Copy content Toggle raw display
$71$ \( T - 768 \) Copy content Toggle raw display
$73$ \( T - 702 \) Copy content Toggle raw display
$79$ \( T - 440 \) Copy content Toggle raw display
$83$ \( T + 1302 \) Copy content Toggle raw display
$89$ \( T + 730 \) Copy content Toggle raw display
$97$ \( T + 294 \) Copy content Toggle raw display
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