Properties

Label 4050.2.a.v
Level $4050$
Weight $2$
Character orbit 4050.a
Self dual yes
Analytic conductor $32.339$
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] = [4050,2,Mod(1,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - 2 q^{7} + q^{8} + 3 q^{11} - 2 q^{13} - 2 q^{14} + q^{16} - 3 q^{17} - q^{19} + 3 q^{22} - 6 q^{23} - 2 q^{26} - 2 q^{28} - 6 q^{29} - 4 q^{31} + q^{32} - 3 q^{34} + 4 q^{37} - q^{38} - 9 q^{41} + q^{43} + 3 q^{44} - 6 q^{46} - 6 q^{47} - 3 q^{49} - 2 q^{52} + 12 q^{53} - 2 q^{56} - 6 q^{58} - 3 q^{59} + 8 q^{61} - 4 q^{62} + q^{64} - 5 q^{67} - 3 q^{68} + 12 q^{71} - 11 q^{73} + 4 q^{74} - q^{76} - 6 q^{77} - 4 q^{79} - 9 q^{82} + 12 q^{83} + q^{86} + 3 q^{88} - 6 q^{89} + 4 q^{91} - 6 q^{92} - 6 q^{94} - 5 q^{97} - 3 q^{98}+O(q^{100}) \) 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 1.00000 0 0 −2.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 4050.2.a.v 1
3.b odd 2 1 4050.2.a.c 1
5.b even 2 1 162.2.a.b 1
5.c odd 4 2 4050.2.c.r 2
9.c even 3 2 1350.2.e.c 2
9.d odd 6 2 450.2.e.i 2
15.d odd 2 1 162.2.a.c 1
15.e even 4 2 4050.2.c.c 2
20.d odd 2 1 1296.2.a.f 1
35.c odd 2 1 7938.2.a.i 1
40.e odd 2 1 5184.2.a.p 1
40.f even 2 1 5184.2.a.q 1
45.h odd 6 2 18.2.c.a 2
45.j even 6 2 54.2.c.a 2
45.k odd 12 4 1350.2.j.a 4
45.l even 12 4 450.2.j.e 4
60.h even 2 1 1296.2.a.g 1
105.g even 2 1 7938.2.a.x 1
120.i odd 2 1 5184.2.a.r 1
120.m even 2 1 5184.2.a.o 1
180.n even 6 2 144.2.i.c 2
180.p odd 6 2 432.2.i.b 2
315.q odd 6 2 2646.2.e.c 2
315.r even 6 2 2646.2.e.b 2
315.u even 6 2 882.2.h.b 2
315.v odd 6 2 882.2.h.c 2
315.z even 6 2 882.2.f.d 2
315.bg odd 6 2 2646.2.f.g 2
315.bn odd 6 2 2646.2.h.i 2
315.bo even 6 2 2646.2.h.h 2
315.bq even 6 2 882.2.e.g 2
315.br odd 6 2 882.2.e.i 2
360.z odd 6 2 1728.2.i.f 2
360.bd even 6 2 576.2.i.a 2
360.bh odd 6 2 576.2.i.g 2
360.bk even 6 2 1728.2.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 45.h odd 6 2
54.2.c.a 2 45.j even 6 2
144.2.i.c 2 180.n even 6 2
162.2.a.b 1 5.b even 2 1
162.2.a.c 1 15.d odd 2 1
432.2.i.b 2 180.p odd 6 2
450.2.e.i 2 9.d odd 6 2
450.2.j.e 4 45.l even 12 4
576.2.i.a 2 360.bd even 6 2
576.2.i.g 2 360.bh odd 6 2
882.2.e.g 2 315.bq even 6 2
882.2.e.i 2 315.br odd 6 2
882.2.f.d 2 315.z even 6 2
882.2.h.b 2 315.u even 6 2
882.2.h.c 2 315.v odd 6 2
1296.2.a.f 1 20.d odd 2 1
1296.2.a.g 1 60.h even 2 1
1350.2.e.c 2 9.c even 3 2
1350.2.j.a 4 45.k odd 12 4
1728.2.i.e 2 360.bk even 6 2
1728.2.i.f 2 360.z odd 6 2
2646.2.e.b 2 315.r even 6 2
2646.2.e.c 2 315.q odd 6 2
2646.2.f.g 2 315.bg odd 6 2
2646.2.h.h 2 315.bo even 6 2
2646.2.h.i 2 315.bn odd 6 2
4050.2.a.c 1 3.b odd 2 1
4050.2.a.v 1 1.a even 1 1 trivial
4050.2.c.c 2 15.e even 4 2
4050.2.c.r 2 5.c odd 4 2
5184.2.a.o 1 120.m even 2 1
5184.2.a.p 1 40.e odd 2 1
5184.2.a.q 1 40.f even 2 1
5184.2.a.r 1 120.i odd 2 1
7938.2.a.i 1 35.c odd 2 1
7938.2.a.x 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display
\( T_{41} + 9 \) 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
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 5 \) Copy content Toggle raw display
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