Properties

Label 2450.4.a.r
Level $2450$
Weight $4$
Character orbit 2450.a
Self dual yes
Analytic conductor $144.555$
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] = [2450,4,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.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: \(144.554679514\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 5 q^{3} + 4 q^{4} - 10 q^{6} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 5 q^{3} + 4 q^{4} - 10 q^{6} - 8 q^{8} - 2 q^{9} - q^{11} + 20 q^{12} + 7 q^{13} + 16 q^{16} - 51 q^{17} + 4 q^{18} - 30 q^{19} + 2 q^{22} + 50 q^{23} - 40 q^{24} - 14 q^{26} - 145 q^{27} + 79 q^{29} + 212 q^{31} - 32 q^{32} - 5 q^{33} + 102 q^{34} - 8 q^{36} + 190 q^{37} + 60 q^{38} + 35 q^{39} + 308 q^{41} - 422 q^{43} - 4 q^{44} - 100 q^{46} + 121 q^{47} + 80 q^{48} - 255 q^{51} + 28 q^{52} - 664 q^{53} + 290 q^{54} - 150 q^{57} - 158 q^{58} - 628 q^{59} + 684 q^{61} - 424 q^{62} + 64 q^{64} + 10 q^{66} - 1056 q^{67} - 204 q^{68} + 250 q^{69} + 744 q^{71} + 16 q^{72} + 726 q^{73} - 380 q^{74} - 120 q^{76} - 70 q^{78} - 407 q^{79} - 671 q^{81} - 616 q^{82} + 644 q^{83} + 844 q^{86} + 395 q^{87} + 8 q^{88} + 880 q^{89} + 200 q^{92} + 1060 q^{93} - 242 q^{94} - 160 q^{96} - 1351 q^{97} + 2 q^{99}+O(q^{100}) \) 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 5.00000 4.00000 0 −10.0000 0 −8.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 2450.4.a.r 1
5.b even 2 1 490.4.a.j 1
7.b odd 2 1 350.4.a.c 1
35.c odd 2 1 70.4.a.e 1
35.f even 4 2 350.4.c.k 2
35.i odd 6 2 490.4.e.c 2
35.j even 6 2 490.4.e.g 2
105.g even 2 1 630.4.a.b 1
140.c even 2 1 560.4.a.f 1
280.c odd 2 1 2240.4.a.h 1
280.n even 2 1 2240.4.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.e 1 35.c odd 2 1
350.4.a.c 1 7.b odd 2 1
350.4.c.k 2 35.f even 4 2
490.4.a.j 1 5.b even 2 1
490.4.e.c 2 35.i odd 6 2
490.4.e.g 2 35.j even 6 2
560.4.a.f 1 140.c even 2 1
630.4.a.b 1 105.g even 2 1
2240.4.a.h 1 280.c odd 2 1
2240.4.a.bc 1 280.n even 2 1
2450.4.a.r 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} - 5 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{19} + 30 \) Copy content Toggle raw display
\( T_{23} - 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 5 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 7 \) Copy content Toggle raw display
$17$ \( T + 51 \) Copy content Toggle raw display
$19$ \( T + 30 \) Copy content Toggle raw display
$23$ \( T - 50 \) Copy content Toggle raw display
$29$ \( T - 79 \) Copy content Toggle raw display
$31$ \( T - 212 \) Copy content Toggle raw display
$37$ \( T - 190 \) Copy content Toggle raw display
$41$ \( T - 308 \) Copy content Toggle raw display
$43$ \( T + 422 \) Copy content Toggle raw display
$47$ \( T - 121 \) Copy content Toggle raw display
$53$ \( T + 664 \) Copy content Toggle raw display
$59$ \( T + 628 \) Copy content Toggle raw display
$61$ \( T - 684 \) Copy content Toggle raw display
$67$ \( T + 1056 \) Copy content Toggle raw display
$71$ \( T - 744 \) Copy content Toggle raw display
$73$ \( T - 726 \) Copy content Toggle raw display
$79$ \( T + 407 \) Copy content Toggle raw display
$83$ \( T - 644 \) Copy content Toggle raw display
$89$ \( T - 880 \) Copy content Toggle raw display
$97$ \( T + 1351 \) Copy content Toggle raw display
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