Properties

Label 4600.2.e.q
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.3356224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{3} + ( - \beta_{5} - 2 \beta_{3}) q^{7} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{3} + ( - \beta_{5} - 2 \beta_{3}) q^{7} + ( - \beta_{4} - 1) q^{9} + (\beta_{4} + 1) q^{11} + (2 \beta_{3} - \beta_1) q^{13} + (2 \beta_{5} + \beta_{3} - 3 \beta_1) q^{17} + (2 \beta_{4} - \beta_{2} - 2) q^{19} + ( - \beta_{4} - 3 \beta_{2} - 1) q^{21} - \beta_{3} q^{23} + ( - 2 \beta_{5} + \beta_{3} + 2 \beta_1) q^{27} + (5 \beta_{4} + \beta_{2}) q^{29} + ( - \beta_{4} - 2 \beta_{2} + 4) q^{31} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{33} + ( - 2 \beta_{5} + 6 \beta_1) q^{37} + ( - 2 \beta_{4} + 2 \beta_{2} - 3) q^{39} + (4 \beta_{4} - \beta_{2} - 3) q^{41} + (4 \beta_{3} + 4 \beta_1) q^{43} + ( - \beta_{5} + \beta_1) q^{47} + ( - \beta_{4} - 6 \beta_{2}) q^{49} + ( - 4 \beta_{4} + 3 \beta_{2} + 2) q^{51} + (4 \beta_{5} + 2 \beta_{3}) q^{53} + (3 \beta_{5} + 3 \beta_{3} - 5 \beta_1) q^{57} + ( - 6 \beta_{4} - 4 \beta_{2} - 2) q^{59} + ( - 3 \beta_{4} - 4 \beta_{2} - 3) q^{61} + (\beta_{5} + \beta_{3} - \beta_1) q^{63} + ( - 6 \beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{67} + ( - \beta_{2} + 1) q^{69} + ( - 5 \beta_{2} + 5) q^{71} + (\beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{73} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{77} + (2 \beta_{4} - 2 \beta_{2} - 4) q^{79} + ( - \beta_{4} - \beta_{2} - 8) q^{81} + ( - 2 \beta_{5} + 6 \beta_{3} + 2 \beta_1) q^{83} + ( - \beta_{5} + 2 \beta_{3} - 9 \beta_1) q^{87} + ( - 4 \beta_{4} - 4 \beta_{2} + 8) q^{89} + (\beta_{4} + 2 \beta_{2} + 3) q^{91} + ( - 2 \beta_{5} + 9 \beta_{3}) q^{93} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{97} + ( - 2 \beta_{4} + \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89} + 22 q^{91} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} + 16x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 7\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 7\beta_{3} + 16\beta_1 \) 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
2.11491i
0.254102i
1.86081i
1.86081i
0.254102i
2.11491i
0 2.47283i 0 0 0 0.527166i 0 −3.11491 0
4049.2 0 1.93543i 0 0 0 4.93543i 0 −0.745898 0
4049.3 0 1.46260i 0 0 0 1.53740i 0 0.860806 0
4049.4 0 1.46260i 0 0 0 1.53740i 0 0.860806 0
4049.5 0 1.93543i 0 0 0 4.93543i 0 −0.745898 0
4049.6 0 2.47283i 0 0 0 0.527166i 0 −3.11491 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.6
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.q 6
5.b even 2 1 inner 4600.2.e.q 6
5.c odd 4 1 920.2.a.i 3
5.c odd 4 1 4600.2.a.v 3
15.e even 4 1 8280.2.a.bl 3
20.e even 4 1 1840.2.a.q 3
20.e even 4 1 9200.2.a.ci 3
40.i odd 4 1 7360.2.a.bw 3
40.k even 4 1 7360.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.i 3 5.c odd 4 1
1840.2.a.q 3 20.e even 4 1
4600.2.a.v 3 5.c odd 4 1
4600.2.e.q 6 1.a even 1 1 trivial
4600.2.e.q 6 5.b even 2 1 inner
7360.2.a.bw 3 40.i odd 4 1
7360.2.a.cf 3 40.k even 4 1
8280.2.a.bl 3 15.e even 4 1
9200.2.a.ci 3 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{6} + 12T_{3}^{4} + 44T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{6} + 27T_{7}^{4} + 65T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{6} + 20T_{13}^{4} + 52T_{13}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 27 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} - 3 T^{2} - T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 20 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} + 87 T^{4} + \cdots + 21904 \) Copy content Toggle raw display
$19$ \( (T^{3} + 7 T^{2} + \cdots - 106)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - T^{2} - 90 T + 148)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 10 T^{2} + \cdots + 53)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 260 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( (T^{3} + 10 T^{2} + \cdots - 481)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 176 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( T^{6} + 13 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{6} + 204 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{3} + 10 T^{2} + \cdots - 1184)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 13 T^{2} + \cdots - 214)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 356 T^{4} + \cdots + 1401856 \) Copy content Toggle raw display
$71$ \( (T^{3} - 10 T^{2} + \cdots + 875)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 117 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
$79$ \( (T^{3} + 14 T^{2} + \cdots - 224)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 136 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{3} - 20 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 71 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
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