Properties

Label 5070.2.b.c
Level $5070$
Weight $2$
Character orbit 5070.b
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5070,2,Mod(1351,5070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5070.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{3} - q^{4} - i q^{5} + i q^{6} + i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{3} - q^{4} - i q^{5} + i q^{6} + i q^{8} + q^{9} - q^{10} + q^{12} + i q^{15} + q^{16} + 6 q^{17} - i q^{18} + i q^{20} + 4 q^{23} - i q^{24} - q^{25} - q^{27} - 10 q^{29} + q^{30} - i q^{32} - 6 i q^{34} - q^{36} + 6 i q^{37} + q^{40} + 2 i q^{41} + 4 q^{43} - i q^{45} - 4 i q^{46} - q^{48} + 7 q^{49} + i q^{50} - 6 q^{51} - 6 q^{53} + i q^{54} + 10 i q^{58} - i q^{60} + 6 q^{61} - q^{64} + 4 i q^{67} - 6 q^{68} - 4 q^{69} + 16 i q^{71} + i q^{72} + 2 i q^{73} + 6 q^{74} + q^{75} - i q^{80} + q^{81} + 2 q^{82} + 4 i q^{83} - 6 i q^{85} - 4 i q^{86} + 10 q^{87} + 6 i q^{89} - q^{90} - 4 q^{92} + i q^{96} + 14 i q^{97} - 7 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{16} + 12 q^{17} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 20 q^{29} + 2 q^{30} - 2 q^{36} + 2 q^{40} + 8 q^{43} - 2 q^{48} + 14 q^{49} - 12 q^{51} - 12 q^{53} + 12 q^{61} - 2 q^{64} - 12 q^{68} - 8 q^{69} + 12 q^{74} + 2 q^{75} + 2 q^{81} + 4 q^{82} + 20 q^{87} - 2 q^{90} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.c 2
13.b even 2 1 inner 5070.2.b.c 2
13.d odd 4 1 390.2.a.a 1
13.d odd 4 1 5070.2.a.s 1
39.f even 4 1 1170.2.a.m 1
52.f even 4 1 3120.2.a.q 1
65.f even 4 1 1950.2.e.l 2
65.g odd 4 1 1950.2.a.y 1
65.k even 4 1 1950.2.e.l 2
156.l odd 4 1 9360.2.a.bn 1
195.j odd 4 1 5850.2.e.p 2
195.n even 4 1 5850.2.a.m 1
195.u odd 4 1 5850.2.e.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.a 1 13.d odd 4 1
1170.2.a.m 1 39.f even 4 1
1950.2.a.y 1 65.g odd 4 1
1950.2.e.l 2 65.f even 4 1
1950.2.e.l 2 65.k even 4 1
3120.2.a.q 1 52.f even 4 1
5070.2.a.s 1 13.d odd 4 1
5070.2.b.c 2 1.a even 1 1 trivial
5070.2.b.c 2 13.b even 2 1 inner
5850.2.a.m 1 195.n even 4 1
5850.2.e.p 2 195.j odd 4 1
5850.2.e.p 2 195.u odd 4 1
9360.2.a.bn 1 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 256 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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