Properties

Label 5.6.b.a
Level $5$
Weight $6$
Character orbit 5.b
Analytic conductor $0.802$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5,6,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), 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: \(0.801919099065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 3 \beta q^{3} - 12 q^{4} + ( - 5 \beta - 45) q^{5} + 132 q^{6} + 9 \beta q^{7} - 20 \beta q^{8} - 153 q^{9} + (45 \beta - 220) q^{10} + 252 q^{11} - 36 \beta q^{12} + 18 \beta q^{13} + \cdots - 38556 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{4} - 90 q^{5} + 264 q^{6} - 306 q^{9} - 440 q^{10} + 504 q^{11} + 792 q^{14} + 1320 q^{15} - 2528 q^{16} - 440 q^{19} + 1080 q^{20} - 2376 q^{21} + 5280 q^{24} + 1850 q^{25} + 1584 q^{26} - 13860 q^{29}+ \cdots - 77112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 1.65831i
0.500000 1.65831i
6.63325i 19.8997i −12.0000 −45.0000 33.1662i 132.000 59.6992i 132.665i −153.000 −220.000 + 298.496i
4.2 6.63325i 19.8997i −12.0000 −45.0000 + 33.1662i 132.000 59.6992i 132.665i −153.000 −220.000 298.496i
\(n\): 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 5.6.b.a 2
3.b odd 2 1 45.6.b.b 2
4.b odd 2 1 80.6.c.a 2
5.b even 2 1 inner 5.6.b.a 2
5.c odd 4 2 25.6.a.c 2
7.b odd 2 1 245.6.b.a 2
8.b even 2 1 320.6.c.f 2
8.d odd 2 1 320.6.c.g 2
12.b even 2 1 720.6.f.f 2
15.d odd 2 1 45.6.b.b 2
15.e even 4 2 225.6.a.n 2
20.d odd 2 1 80.6.c.a 2
20.e even 4 2 400.6.a.t 2
35.c odd 2 1 245.6.b.a 2
40.e odd 2 1 320.6.c.g 2
40.f even 2 1 320.6.c.f 2
60.h even 2 1 720.6.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 1.a even 1 1 trivial
5.6.b.a 2 5.b even 2 1 inner
25.6.a.c 2 5.c odd 4 2
45.6.b.b 2 3.b odd 2 1
45.6.b.b 2 15.d odd 2 1
80.6.c.a 2 4.b odd 2 1
80.6.c.a 2 20.d odd 2 1
225.6.a.n 2 15.e even 4 2
245.6.b.a 2 7.b odd 2 1
245.6.b.a 2 35.c odd 2 1
320.6.c.f 2 8.b even 2 1
320.6.c.f 2 40.f even 2 1
320.6.c.g 2 8.d odd 2 1
320.6.c.g 2 40.e odd 2 1
400.6.a.t 2 20.e even 4 2
720.6.f.f 2 12.b even 2 1
720.6.f.f 2 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 44 \) Copy content Toggle raw display
$3$ \( T^{2} + 396 \) Copy content Toggle raw display
$5$ \( T^{2} + 90T + 3125 \) Copy content Toggle raw display
$7$ \( T^{2} + 3564 \) Copy content Toggle raw display
$11$ \( (T - 252)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 14256 \) Copy content Toggle raw display
$17$ \( T^{2} + 475904 \) Copy content Toggle raw display
$19$ \( (T + 220)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5926316 \) Copy content Toggle raw display
$29$ \( (T + 6930)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6752)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 195150384 \) Copy content Toggle raw display
$41$ \( (T + 198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 174636 \) Copy content Toggle raw display
$47$ \( T^{2} + 111096524 \) Copy content Toggle raw display
$53$ \( T^{2} + 33918896 \) Copy content Toggle raw display
$59$ \( (T + 24660)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5698)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1904462604 \) Copy content Toggle raw display
$71$ \( (T - 53352)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5030030016 \) Copy content Toggle raw display
$79$ \( (T - 51920)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3824406476 \) Copy content Toggle raw display
$89$ \( (T + 9990)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 10251546624 \) Copy content Toggle raw display
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