Properties

Label 1040.2.bg.n
Level $1040$
Weight $2$
Character orbit 1040.bg
Analytic conductor $8.304$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(577,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.bg (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.619810816.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{3} + (\beta_{7} + \beta_{6}) q^{5} + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{3} + (\beta_{7} + \beta_{6}) q^{5} + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{7} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 2 q^{5} - 6 q^{11} - 2 q^{13} + 2 q^{15} - 16 q^{17} - 14 q^{19} - 12 q^{21} - 14 q^{23} + 12 q^{25} - 12 q^{27} - 2 q^{31} - 8 q^{33} + 24 q^{35} + 6 q^{39} + 16 q^{41} - 6 q^{43} + 6 q^{45} - 24 q^{49} - 24 q^{53} - 10 q^{55} - 40 q^{57} - 22 q^{59} + 20 q^{61} - 16 q^{63} + 12 q^{67} - 4 q^{69} + 10 q^{71} - 4 q^{73} + 30 q^{75} + 24 q^{77} - 20 q^{81} - 20 q^{85} - 16 q^{87} - 28 q^{89} - 20 q^{91} + 2 q^{95} + 12 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.252709 0.252709i
1.18254 + 1.18254i
0.561103 + 0.561103i
−1.49094 1.49094i
−0.252709 + 0.252709i
1.18254 1.18254i
0.561103 0.561103i
−1.49094 + 1.49094i
0 −0.725850 + 0.725850i 0 2.23127 0.146426i 0 4.24997i 0 1.94628i 0
577.2 0 0.240275 0.240275i 0 −1.60536 + 1.55654i 0 3.95872i 0 2.88454i 0
577.3 0 1.33000 1.33000i 0 −1.45220 1.70032i 0 1.61845i 0 0.537789i 0
577.4 0 2.15558 2.15558i 0 1.82630 + 1.29021i 0 1.90970i 0 6.29303i 0
593.1 0 −0.725850 0.725850i 0 2.23127 + 0.146426i 0 4.24997i 0 1.94628i 0
593.2 0 0.240275 + 0.240275i 0 −1.60536 1.55654i 0 3.95872i 0 2.88454i 0
593.3 0 1.33000 + 1.33000i 0 −1.45220 + 1.70032i 0 1.61845i 0 0.537789i 0
593.4 0 2.15558 + 2.15558i 0 1.82630 1.29021i 0 1.90970i 0 6.29303i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.bg.n 8
4.b odd 2 1 65.2.k.b yes 8
5.c odd 4 1 1040.2.cd.n 8
12.b even 2 1 585.2.w.e 8
13.d odd 4 1 1040.2.cd.n 8
20.d odd 2 1 325.2.k.b 8
20.e even 4 1 65.2.f.b 8
20.e even 4 1 325.2.f.b 8
52.b odd 2 1 845.2.k.b 8
52.f even 4 1 65.2.f.b 8
52.f even 4 1 845.2.f.b 8
52.i odd 6 2 845.2.o.c 16
52.j odd 6 2 845.2.o.d 16
52.l even 12 2 845.2.t.c 16
52.l even 12 2 845.2.t.d 16
60.l odd 4 1 585.2.n.e 8
65.k even 4 1 inner 1040.2.bg.n 8
156.l odd 4 1 585.2.n.e 8
260.l odd 4 1 325.2.k.b 8
260.l odd 4 1 845.2.k.b 8
260.p even 4 1 845.2.f.b 8
260.s odd 4 1 65.2.k.b yes 8
260.u even 4 1 325.2.f.b 8
260.be odd 12 2 845.2.o.d 16
260.bg even 12 2 845.2.t.d 16
260.bj even 12 2 845.2.t.c 16
260.bl odd 12 2 845.2.o.c 16
780.bn even 4 1 585.2.w.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 20.e even 4 1
65.2.f.b 8 52.f even 4 1
65.2.k.b yes 8 4.b odd 2 1
65.2.k.b yes 8 260.s odd 4 1
325.2.f.b 8 20.e even 4 1
325.2.f.b 8 260.u even 4 1
325.2.k.b 8 20.d odd 2 1
325.2.k.b 8 260.l odd 4 1
585.2.n.e 8 60.l odd 4 1
585.2.n.e 8 156.l odd 4 1
585.2.w.e 8 12.b even 2 1
585.2.w.e 8 780.bn even 4 1
845.2.f.b 8 52.f even 4 1
845.2.f.b 8 260.p even 4 1
845.2.k.b 8 52.b odd 2 1
845.2.k.b 8 260.l odd 4 1
845.2.o.c 16 52.i odd 6 2
845.2.o.c 16 260.bl odd 12 2
845.2.o.d 16 52.j odd 6 2
845.2.o.d 16 260.be odd 12 2
845.2.t.c 16 52.l even 12 2
845.2.t.c 16 260.bj even 12 2
845.2.t.d 16 52.l even 12 2
845.2.t.d 16 260.bg even 12 2
1040.2.bg.n 8 1.a even 1 1 trivial
1040.2.bg.n 8 65.k even 4 1 inner
1040.2.cd.n 8 5.c odd 4 1
1040.2.cd.n 8 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{8} - 6T_{3}^{7} + 18T_{3}^{6} - 20T_{3}^{5} + 8T_{3}^{4} + 4T_{3}^{3} + 32T_{3}^{2} - 16T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 40T_{7}^{6} + 504T_{7}^{4} + 2096T_{7}^{2} + 2704 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} + 18T_{11}^{6} + 8T_{11}^{5} - 4T_{11}^{4} - 12T_{11}^{3} + 32T_{11}^{2} - 16T_{11} + 4 \) Copy content Toggle raw display
\( T_{19}^{8} + 14T_{19}^{7} + 98T_{19}^{6} + 396T_{19}^{5} + 1004T_{19}^{4} + 1524T_{19}^{3} + 1352T_{19}^{2} + 520T_{19} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 40 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 16 T^{7} + \cdots + 13456 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( T^{8} + 14 T^{7} + \cdots + 40804 \) Copy content Toggle raw display
$29$ \( T^{8} + 44 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 16900 \) Copy content Toggle raw display
$37$ \( T^{8} + 140 T^{6} + \cdots + 336400 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots + 13456 \) Copy content Toggle raw display
$43$ \( T^{8} + 6 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$47$ \( T^{8} + 160 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + \cdots + 19600 \) Copy content Toggle raw display
$59$ \( T^{8} + 22 T^{7} + \cdots + 119716 \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} + \cdots - 3628)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots - 148)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 10 T^{7} + \cdots + 1223236 \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} + \cdots + 740)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 292 T^{6} + \cdots + 13719616 \) Copy content Toggle raw display
$83$ \( T^{8} + 416 T^{6} + \cdots + 54346384 \) Copy content Toggle raw display
$89$ \( T^{8} + 28 T^{7} + \cdots + 1795600 \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} + \cdots + 3704)^{2} \) Copy content Toggle raw display
show more
show less