Properties

Label 4600.2.e.d
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
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] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (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: \(36.7311849298\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 3 i q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{3} + 3 i q^{7} - q^{9} + 5 q^{11} - 5 i q^{13} + 4 i q^{17} - q^{19} - 6 q^{21} - i q^{23} + 4 i q^{27} - 9 q^{29} - 2 q^{31} + 10 i q^{33} + 2 i q^{37} + 10 q^{39} + 3 q^{41} - 7 i q^{43} + 12 i q^{47} - 2 q^{49} - 8 q^{51} + 12 i q^{53} - 2 i q^{57} + 6 q^{59} - 10 q^{61} - 3 i q^{63} + 8 i q^{67} + 2 q^{69} + 2 q^{71} + i q^{73} + 15 i q^{77} + 11 q^{79} - 11 q^{81} + 9 i q^{83} - 18 i q^{87} + 14 q^{89} + 15 q^{91} - 4 i q^{93} + 16 i q^{97} - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 10 q^{11} - 2 q^{19} - 12 q^{21} - 18 q^{29} - 4 q^{31} + 20 q^{39} + 6 q^{41} - 4 q^{49} - 16 q^{51} + 12 q^{59} - 20 q^{61} + 4 q^{69} + 4 q^{71} + 22 q^{79} - 22 q^{81} + 28 q^{89} + 30 q^{91} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
1.00000i
1.00000i
0 2.00000i 0 0 0 3.00000i 0 −1.00000 0
4049.2 0 2.00000i 0 0 0 3.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4600.2.e.d 2
5.b even 2 1 inner 4600.2.e.d 2
5.c odd 4 1 4600.2.a.d 1
5.c odd 4 1 4600.2.a.m yes 1
20.e even 4 1 9200.2.a.j 1
20.e even 4 1 9200.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.d 1 5.c odd 4 1
4600.2.a.m yes 1 5.c odd 4 1
4600.2.e.d 2 1.a even 1 1 trivial
4600.2.e.d 2 5.b even 2 1 inner
9200.2.a.j 1 20.e even 4 1
9200.2.a.bc 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 49 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1 \) Copy content Toggle raw display
$79$ \( (T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 256 \) Copy content Toggle raw display
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