Properties

Label 525.4.a.d
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
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] = [525,4,Mod(1,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, 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("525.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(30.9760027530\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 6 q^{6} - 7 q^{7} + 24 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 6 q^{6} - 7 q^{7} + 24 q^{8} + 9 q^{9} - 21 q^{11} - 12 q^{12} + 24 q^{13} + 14 q^{14} - 16 q^{16} - 22 q^{17} - 18 q^{18} + 16 q^{19} - 21 q^{21} + 42 q^{22} - 25 q^{23} + 72 q^{24} - 48 q^{26} + 27 q^{27} + 28 q^{28} + 167 q^{29} + 10 q^{31} - 160 q^{32} - 63 q^{33} + 44 q^{34} - 36 q^{36} - 133 q^{37} - 32 q^{38} + 72 q^{39} - 168 q^{41} + 42 q^{42} - 97 q^{43} + 84 q^{44} + 50 q^{46} - 400 q^{47} - 48 q^{48} + 49 q^{49} - 66 q^{51} - 96 q^{52} - 182 q^{53} - 54 q^{54} - 168 q^{56} + 48 q^{57} - 334 q^{58} + 488 q^{59} + 28 q^{61} - 20 q^{62} - 63 q^{63} + 448 q^{64} + 126 q^{66} - 967 q^{67} + 88 q^{68} - 75 q^{69} - 285 q^{71} + 216 q^{72} - 838 q^{73} + 266 q^{74} - 64 q^{76} + 147 q^{77} - 144 q^{78} - 469 q^{79} + 81 q^{81} + 336 q^{82} - 406 q^{83} + 84 q^{84} + 194 q^{86} + 501 q^{87} - 504 q^{88} + 324 q^{89} - 168 q^{91} + 100 q^{92} + 30 q^{93} + 800 q^{94} - 480 q^{96} - 114 q^{97} - 98 q^{98} - 189 q^{99}+O(q^{100}) \) 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 3.00000 −4.00000 0 −6.00000 −7.00000 24.0000 9.00000 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 525.4.a.d 1
3.b odd 2 1 1575.4.a.h 1
5.b even 2 1 525.4.a.f yes 1
5.c odd 4 2 525.4.d.e 2
15.d odd 2 1 1575.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 1.a even 1 1 trivial
525.4.a.f yes 1 5.b even 2 1
525.4.d.e 2 5.c odd 4 2
1575.4.a.d 1 15.d odd 2 1
1575.4.a.h 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{11} + 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 3 \) 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
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