Properties

Label 4680.2.g.j
Level $4680$
Weight $2$
Character orbit 4680.g
Analytic conductor $37.370$
Analytic rank $0$
Dimension $6$
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] = [4680,2,Mod(2521,4680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4680.2521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.g (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: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 520)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{5} + \beta_{4} + \beta_{3}) q^{11} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{13} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{19} + ( - 2 \beta_{2} + \beta_1 - 4) q^{23} - q^{25} + ( - 3 \beta_{2} - 3 \beta_1 + 1) q^{29} + ( - 5 \beta_{4} + 3 \beta_{3}) q^{31} + ( - \beta_{2} + \beta_1 - 1) q^{35} + (\beta_{5} + \beta_{4} - 3 \beta_{3}) q^{37} + (4 \beta_{5} + 2 \beta_{4}) q^{41} + (3 \beta_1 - 6) q^{43} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{47} + (2 \beta_1 + 3) q^{49} + ( - 2 \beta_{2} - 4 \beta_1) q^{53} + (\beta_{2} + 2 \beta_1 - 1) q^{55} + ( - \beta_{4} + \beta_{3}) q^{59} + ( - 3 \beta_{2} - \beta_1 - 7) q^{61} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{65} + (\beta_{5} + \beta_{4} + 3 \beta_{3}) q^{67} + ( - 6 \beta_{5} + 5 \beta_{4} - \beta_{3}) q^{71} + ( - 5 \beta_{5} + 3 \beta_{4} - \beta_{3}) q^{73} + ( - 2 \beta_{2} + 6 \beta_1 - 4) q^{77} + ( - 6 \beta_{2} + 2 \beta_1 + 2) q^{79} + (\beta_{5} - 7 \beta_{4} + 7 \beta_{3}) q^{83} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{85} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{89} + (2 \beta_{5} + 6 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{91} + ( - \beta_{2} + 2 \beta_1 - 1) q^{95} + ( - 6 \beta_{5} + 2 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{13} - 8 q^{17} - 20 q^{23} - 6 q^{25} + 12 q^{29} - 4 q^{35} - 36 q^{43} + 18 q^{49} + 4 q^{53} - 8 q^{55} - 36 q^{61} - 2 q^{65} - 20 q^{77} + 24 q^{79} + 8 q^{91} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2521.1
1.45161 1.45161i
0.403032 0.403032i
−0.854638 + 0.854638i
−0.854638 0.854638i
0.403032 + 0.403032i
1.45161 + 1.45161i
0 0 0 1.00000i 0 2.90321i 0 0 0
2521.2 0 0 0 1.00000i 0 0.806063i 0 0 0
2521.3 0 0 0 1.00000i 0 1.70928i 0 0 0
2521.4 0 0 0 1.00000i 0 1.70928i 0 0 0
2521.5 0 0 0 1.00000i 0 0.806063i 0 0 0
2521.6 0 0 0 1.00000i 0 2.90321i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2521.6
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 4680.2.g.j 6
3.b odd 2 1 520.2.k.a 6
12.b even 2 1 1040.2.k.b 6
13.b even 2 1 inner 4680.2.g.j 6
15.d odd 2 1 2600.2.k.b 6
15.e even 4 1 2600.2.f.c 6
15.e even 4 1 2600.2.f.d 6
39.d odd 2 1 520.2.k.a 6
39.f even 4 1 6760.2.a.u 3
39.f even 4 1 6760.2.a.v 3
156.h even 2 1 1040.2.k.b 6
195.e odd 2 1 2600.2.k.b 6
195.s even 4 1 2600.2.f.c 6
195.s even 4 1 2600.2.f.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.k.a 6 3.b odd 2 1
520.2.k.a 6 39.d odd 2 1
1040.2.k.b 6 12.b even 2 1
1040.2.k.b 6 156.h even 2 1
2600.2.f.c 6 15.e even 4 1
2600.2.f.c 6 195.s even 4 1
2600.2.f.d 6 15.e even 4 1
2600.2.f.d 6 195.s even 4 1
2600.2.k.b 6 15.d odd 2 1
2600.2.k.b 6 195.e odd 2 1
4680.2.g.j 6 1.a even 1 1 trivial
4680.2.g.j 6 13.b even 2 1 inner
6760.2.a.u 3 39.f even 4 1
6760.2.a.v 3 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{6} + 12T_{7}^{4} + 32T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 52T_{11}^{4} + 692T_{11}^{2} + 2116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{6} + 52 T^{4} + 692 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + 7 T^{4} + 64 T^{3} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} + 4 T^{2} - 16 T - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 32 T^{4} + 156 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( (T^{3} + 10 T^{2} + 20 T - 26)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} - 72 T + 428)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 172 T^{4} + 7028 T^{2} + \cdots + 13924 \) Copy content Toggle raw display
$37$ \( T^{6} + 40 T^{4} + 80 T^{2} + 16 \) Copy content Toggle raw display
$41$ \( T^{6} + 188 T^{4} + 9264 T^{2} + \cdots + 40000 \) Copy content Toggle raw display
$43$ \( (T^{3} + 18 T^{2} + 72 T + 54)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 108 T^{4} + 2592 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$53$ \( (T^{3} - 2 T^{2} - 92 T + 200)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 8 T^{4} + 12 T^{2} + 4 \) Copy content Toggle raw display
$61$ \( (T^{3} + 18 T^{2} + 68 T + 52)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 52 T^{4} + 656 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{6} + 336 T^{4} + 28652 T^{2} + \cdots + 75076 \) Copy content Toggle raw display
$73$ \( T^{6} + 200 T^{4} + 10000 T^{2} + \cdots + 71824 \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} - 64 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 372 T^{4} + 34736 T^{2} + \cdots + 929296 \) Copy content Toggle raw display
$89$ \( T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$97$ \( T^{6} + 268 T^{4} + 20144 T^{2} + \cdots + 462400 \) Copy content Toggle raw display
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