Properties

Label 2450.4.a.i
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 14)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{6} - 8 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{6} - 8 q^{8} - 23 q^{9} + 48 q^{11} - 8 q^{12} + 56 q^{13} + 16 q^{16} - 114 q^{17} + 46 q^{18} - 2 q^{19} - 96 q^{22} + 120 q^{23} + 16 q^{24} - 112 q^{26} + 100 q^{27} - 54 q^{29} - 236 q^{31} - 32 q^{32} - 96 q^{33} + 228 q^{34} - 92 q^{36} - 146 q^{37} + 4 q^{38} - 112 q^{39} - 126 q^{41} + 376 q^{43} + 192 q^{44} - 240 q^{46} - 12 q^{47} - 32 q^{48} + 228 q^{51} + 224 q^{52} - 174 q^{53} - 200 q^{54} + 4 q^{57} + 108 q^{58} - 138 q^{59} - 380 q^{61} + 472 q^{62} + 64 q^{64} + 192 q^{66} + 484 q^{67} - 456 q^{68} - 240 q^{69} + 576 q^{71} + 184 q^{72} - 1150 q^{73} + 292 q^{74} - 8 q^{76} + 224 q^{78} + 776 q^{79} + 421 q^{81} + 252 q^{82} + 378 q^{83} - 752 q^{86} + 108 q^{87} - 384 q^{88} + 390 q^{89} + 480 q^{92} + 472 q^{93} + 24 q^{94} + 64 q^{96} - 1330 q^{97} - 1104 q^{99}+O(q^{100}) \) 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 −2.00000 4.00000 0 4.00000 0 −8.00000 −23.0000 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.i 1
5.b even 2 1 98.4.a.e 1
7.b odd 2 1 350.4.a.f 1
15.d odd 2 1 882.4.a.b 1
20.d odd 2 1 784.4.a.h 1
35.c odd 2 1 14.4.a.b 1
35.f even 4 2 350.4.c.g 2
35.i odd 6 2 98.4.c.c 2
35.j even 6 2 98.4.c.b 2
105.g even 2 1 126.4.a.d 1
105.o odd 6 2 882.4.g.v 2
105.p even 6 2 882.4.g.p 2
140.c even 2 1 112.4.a.e 1
280.c odd 2 1 448.4.a.k 1
280.n even 2 1 448.4.a.g 1
385.h even 2 1 1694.4.a.b 1
420.o odd 2 1 1008.4.a.r 1
455.h odd 2 1 2366.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 35.c odd 2 1
98.4.a.e 1 5.b even 2 1
98.4.c.b 2 35.j even 6 2
98.4.c.c 2 35.i odd 6 2
112.4.a.e 1 140.c even 2 1
126.4.a.d 1 105.g even 2 1
350.4.a.f 1 7.b odd 2 1
350.4.c.g 2 35.f even 4 2
448.4.a.g 1 280.n even 2 1
448.4.a.k 1 280.c odd 2 1
784.4.a.h 1 20.d odd 2 1
882.4.a.b 1 15.d odd 2 1
882.4.g.p 2 105.p even 6 2
882.4.g.v 2 105.o odd 6 2
1008.4.a.r 1 420.o odd 2 1
1694.4.a.b 1 385.h even 2 1
2366.4.a.c 1 455.h odd 2 1
2450.4.a.i 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{11} - 48 \) Copy content Toggle raw display
\( T_{19} + 2 \) Copy content Toggle raw display
\( T_{23} - 120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 48 \) Copy content Toggle raw display
$13$ \( T - 56 \) Copy content Toggle raw display
$17$ \( T + 114 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T + 54 \) Copy content Toggle raw display
$31$ \( T + 236 \) Copy content Toggle raw display
$37$ \( T + 146 \) Copy content Toggle raw display
$41$ \( T + 126 \) Copy content Toggle raw display
$43$ \( T - 376 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 174 \) Copy content Toggle raw display
$59$ \( T + 138 \) Copy content Toggle raw display
$61$ \( T + 380 \) Copy content Toggle raw display
$67$ \( T - 484 \) Copy content Toggle raw display
$71$ \( T - 576 \) Copy content Toggle raw display
$73$ \( T + 1150 \) Copy content Toggle raw display
$79$ \( T - 776 \) Copy content Toggle raw display
$83$ \( T - 378 \) Copy content Toggle raw display
$89$ \( T - 390 \) Copy content Toggle raw display
$97$ \( T + 1330 \) Copy content Toggle raw display
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