Properties

Label 2320.2.d.g
Level $2320$
Weight $2$
Character orbit 2320.d
Analytic conductor $18.525$
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] = [2320,2,Mod(929,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.929");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.d (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: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.84345856.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 13x^{4} + 41x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_1 q^{5} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_1 q^{5} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} + \beta_1 - 2) q^{9} + (\beta_{5} + \beta_1 + 2) q^{11} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{5} + 2 \beta_{3} - 1) q^{15} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{17} + (\beta_{5} + \beta_{2} + 3) q^{19} + (2 \beta_{5} + 2 \beta_1 - 2) q^{21} + ( - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{25} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{27} + q^{29} + ( - \beta_{5} - \beta_1 - 4) q^{31} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{33} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{35} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{37} + (4 \beta_{5} + \beta_{2} + 3 \beta_1 - 1) q^{39} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{43} + (\beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 - 6) q^{45} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{47} + (2 \beta_{5} + 2 \beta_1 + 1) q^{49} + ( - 2 \beta_{5} - 2 \beta_{2} - 8) q^{51} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{53} + (2 \beta_{5} + \beta_{3} - \beta_1 - 3) q^{55} + ( - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{57} + 10 q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{61} + ( - 2 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{63} + (\beta_{5} + 3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 6) q^{65} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{67} + 2 q^{69} - 2 q^{71} + (\beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{73} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{75} + ( - \beta_{4} - 3 \beta_{3}) q^{77} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{79} + ( - 2 \beta_{5} - 2 \beta_{2} + 1) q^{81} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{83} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 5) q^{85} + \beta_{3} q^{87} + (2 \beta_{5} + 2 \beta_1 + 6) q^{89} + (2 \beta_{5} - 2 \beta_{2} + 4 \beta_1 - 6) q^{91} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{93} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{95} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{97} + ( - 5 \beta_{5} - 3 \beta_{2} - 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61} + 13 q^{65} + 12 q^{69} - 12 q^{71} - 9 q^{75} + 2 q^{79} + 10 q^{81} - 26 q^{85} + 32 q^{89} - 40 q^{91} + 28 q^{95} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 13x^{4} + 41x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{4} + 14\nu^{3} + 6\nu^{2} + 47\nu - 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 14\nu^{3} - 6\nu^{2} + 47\nu + 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 12\nu^{3} - 33\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 12\nu^{3} + 37\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 14\nu^{3} + 10\nu^{2} - 47\nu + 15 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{4} - 5\beta_{3} + 2\beta_{2} + 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{5} - 8\beta_{2} + 2\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 51\beta_{4} + 23\beta_{3} - 24\beta_{2} - 24\beta _1 - 24 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1
2.30229i
0.156785i
2.77035i
2.77035i
0.156785i
2.30229i
0 2.89028i 0 2.17686 + 0.511167i 0 3.91261i 0 −5.35371 0
929.2 0 2.56387i 0 1.28672 1.82876i 0 1.09364i 0 −3.57344 0
929.3 0 0.269894i 0 −1.96358 1.06975i 0 1.86960i 0 2.92716 0
929.4 0 0.269894i 0 −1.96358 + 1.06975i 0 1.86960i 0 2.92716 0
929.5 0 2.56387i 0 1.28672 + 1.82876i 0 1.09364i 0 −3.57344 0
929.6 0 2.89028i 0 2.17686 0.511167i 0 3.91261i 0 −5.35371 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.6
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 2320.2.d.g 6
4.b odd 2 1 145.2.b.c 6
5.b even 2 1 inner 2320.2.d.g 6
12.b even 2 1 1305.2.c.h 6
20.d odd 2 1 145.2.b.c 6
20.e even 4 2 725.2.a.l 6
60.h even 2 1 1305.2.c.h 6
60.l odd 4 2 6525.2.a.bt 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.c 6 4.b odd 2 1
145.2.b.c 6 20.d odd 2 1
725.2.a.l 6 20.e even 4 2
1305.2.c.h 6 12.b even 2 1
1305.2.c.h 6 60.h even 2 1
2320.2.d.g 6 1.a even 1 1 trivial
2320.2.d.g 6 5.b even 2 1 inner
6525.2.a.bt 6 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2320, [\chi])\):

\( T_{3}^{6} + 15T_{3}^{4} + 56T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 20T_{7}^{4} + 76T_{7}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 15 T^{4} + 56 T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} - T^{4} + 14 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + 76 T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} - 6 T + 38)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 59 T^{4} + 1048 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{4} + 380 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$19$ \( (T^{3} - 8 T^{2} + 6 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 56 T^{4} + 60 T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 1)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} + 11 T^{2} + 26 T - 22)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 20 T^{4} + 76 T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 127 T^{4} + 1400 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$47$ \( T^{6} + 63 T^{4} + 1304 T^{2} + \cdots + 8836 \) Copy content Toggle raw display
$53$ \( T^{6} + 187 T^{4} + 7640 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$59$ \( (T - 10)^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 64 T - 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 140 T^{4} + 6316 T^{2} + \cdots + 92416 \) Copy content Toggle raw display
$71$ \( (T + 2)^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 188 T^{4} + 8556 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$79$ \( (T^{3} - T^{2} - 54 T - 98)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 48 T^{4} + 380 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$89$ \( (T^{3} - 16 T^{2} + 28 T + 304)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 476 T^{4} + 52396 T^{2} + \cdots + 7744 \) Copy content Toggle raw display
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