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Properties

Label 1040.2.d.c
Level $1040$
Weight $2$
Character orbit 1040.d
Analytic conductor $8.304$
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] = [1040,2,Mod(209,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, 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("1040.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.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: \(8.30444181021\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3}) q^{3} + (\beta_{4} + \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{3}) q^{3} + (\beta_{4} + \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{2} - \beta_1) q^{9} + (\beta_1 + 2) q^{11} + \beta_{3} q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 2) q^{15}+ \cdots + (8 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 12 q^{11} - 16 q^{15} - 4 q^{21} + 2 q^{25} - 12 q^{29} + 20 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} + 18 q^{49} - 24 q^{51} + 16 q^{55} - 16 q^{59} + 12 q^{61} - 2 q^{65} - 24 q^{69} + 24 q^{71} - 16 q^{75} - 32 q^{79} + 46 q^{81} - 12 q^{85} - 20 q^{89} + 4 q^{91} - 16 q^{95} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.854638 + 0.854638i
1.45161 − 1.45161i
0.403032 + 0.403032i
0.403032 − 0.403032i
1.45161 + 1.45161i
−0.854638 − 0.854638i
0 − 3.17009i 0 0.539189 − 2.17009i 0 − 1.70928i 0 −7.04945 0
209.2 0 − 1.31111i 0 −2.21432 − 0.311108i 0 2.90321i 0 1.28100 0
209.3 0 − 0.481194i 0 1.67513 − 1.48119i 0 − 0.806063i 0 2.76845 0
209.4 0 0.481194i 0 1.67513 + 1.48119i 0 0.806063i 0 2.76845 0
209.5 0 1.31111i 0 −2.21432 + 0.311108i 0 − 2.90321i 0 1.28100 0
209.6 0 3.17009i 0 0.539189 + 2.17009i 0 1.70928i 0 −7.04945 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.6
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 1040.2.d.c 6
4.b odd 2 1 65.2.b.a ✓ 6
5.b even 2 1 inner 1040.2.d.c 6
5.c odd 4 1 5200.2.a.cb 3
5.c odd 4 1 5200.2.a.cj 3
12.b even 2 1 585.2.c.b 6
20.d odd 2 1 65.2.b.a ✓ 6
20.e even 4 1 325.2.a.j 3
20.e even 4 1 325.2.a.k 3
52.b odd 2 1 845.2.b.c 6
52.f even 4 1 845.2.d.a 6
52.f even 4 1 845.2.d.b 6
52.i odd 6 2 845.2.n.g 12
52.j odd 6 2 845.2.n.f 12
52.l even 12 2 845.2.l.d 12
52.l even 12 2 845.2.l.e 12
60.h even 2 1 585.2.c.b 6
60.l odd 4 1 2925.2.a.bf 3
60.l odd 4 1 2925.2.a.bj 3
260.g odd 2 1 845.2.b.c 6
260.p even 4 1 4225.2.a.ba 3
260.p even 4 1 4225.2.a.bh 3
260.u even 4 1 845.2.d.a 6
260.u even 4 1 845.2.d.b 6
260.v odd 6 2 845.2.n.f 12
260.w odd 6 2 845.2.n.g 12
260.bc even 12 2 845.2.l.d 12
260.bc even 12 2 845.2.l.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a ✓ 6 4.b odd 2 1
65.2.b.a ✓ 6 20.d odd 2 1
325.2.a.j 3 20.e even 4 1
325.2.a.k 3 20.e even 4 1
585.2.c.b 6 12.b even 2 1
585.2.c.b 6 60.h even 2 1
845.2.b.c 6 52.b odd 2 1
845.2.b.c 6 260.g odd 2 1
845.2.d.a 6 52.f even 4 1
845.2.d.a 6 260.u even 4 1
845.2.d.b 6 52.f even 4 1
845.2.d.b 6 260.u even 4 1
845.2.l.d 12 52.l even 12 2
845.2.l.d 12 260.bc even 12 2
845.2.l.e 12 52.l even 12 2
845.2.l.e 12 260.bc even 12 2
845.2.n.f 12 52.j odd 6 2
845.2.n.f 12 260.v odd 6 2
845.2.n.g 12 52.i odd 6 2
845.2.n.g 12 260.w odd 6 2
1040.2.d.c 6 1.a even 1 1 trivial
1040.2.d.c 6 5.b even 2 1 inner
2925.2.a.bf 3 60.l odd 4 1
2925.2.a.bj 3 60.l odd 4 1
4225.2.a.ba 3 260.p even 4 1
4225.2.a.bh 3 260.p even 4 1
5200.2.a.cb 3 5.c odd 4 1
5200.2.a.cj 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 12T_{3}^{4} + 20T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} - 6 T^{2} + 8 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 44 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 72 T^{4} + \cdots + 7396 \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 10 T^{2} + \cdots + 26)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 56 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 32 T + 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 128 T^{4} + \cdots + 77284 \) Copy content Toggle raw display
$47$ \( T^{6} + 44 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$53$ \( T^{6} + 144 T^{4} + \cdots + 92416 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} + \cdots - 262)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 220 T^{4} + \cdots + 364816 \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T^{2} + \cdots + 754)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 248 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
$79$ \( (T^{3} + 16 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 180 T^{4} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( (T^{3} + 10 T^{2} + \cdots - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 364 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
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