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Properties

Label 1040.2.bg.m
Level $1040$
Weight $2$
Character orbit 1040.bg
Analytic conductor $8.304$
Analytic rank $0$
Dimension $8$
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(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: 8.0.31678304256.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{5} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{5} - \beta_{3} - \beta_1) q^{9} + (\beta_{7} - \beta_{6} + 2 \beta_{4} + \cdots - 2) q^{11}+ \cdots + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 2 q^{5} - 14 q^{11} + 14 q^{13} - 10 q^{15} - 2 q^{19} + 4 q^{21} + 22 q^{23} + 12 q^{25} + 16 q^{27} + 6 q^{31} + 16 q^{33} + 4 q^{35} + 34 q^{39} - 8 q^{41} + 14 q^{43} + 22 q^{45} - 24 q^{49} - 8 q^{53} + 30 q^{55} + 16 q^{57} - 2 q^{59} - 12 q^{61} + 64 q^{63} + 32 q^{65} - 20 q^{67} - 20 q^{69} + 22 q^{71} - 28 q^{73} + 14 q^{75} + 8 q^{77} - 20 q^{81} + 12 q^{85} - 12 q^{87} + 4 q^{89} + 6 q^{95} + 12 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97\nu^{7} - 173\nu^{6} + 220\nu^{5} + 316\nu^{4} + 4010\nu^{3} - 1148\nu^{2} - 1300\nu - 6130 ) / 2462 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 140\nu^{7} - 237\nu^{6} + 216\nu^{5} + 1116\nu^{4} + 5280\nu^{3} - 1530\nu^{2} + 738\nu + 696 ) / 2462 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 174\nu^{7} - 488\nu^{6} + 585\nu^{5} + 1176\nu^{4} + 4452\nu^{3} - 8760\nu^{2} + 2922\nu + 654 ) / 2462 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -382\nu^{7} + 1227\nu^{6} - 1539\nu^{5} - 2412\nu^{4} - 8076\nu^{3} + 22288\nu^{2} - 5566\nu - 1266 ) / 2462 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 452\nu^{7} - 730\nu^{6} + 416\nu^{5} + 4201\nu^{4} + 15640\nu^{3} - 4588\nu^{2} - 5144\nu + 4076 ) / 2462 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 605\nu^{7} - 1244\nu^{6} + 1461\nu^{5} + 4471\nu^{4} + 19300\nu^{3} - 11272\nu^{2} + 12070\nu + 2656 ) / 2462 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 12\beta_{3} - 8\beta_{2} - 12\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} - 8\beta_{6} + 12\beta_{5} + 26\beta_{4} - 12\beta_{2} - 66\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{7} + 66\beta_{5} + 158\beta_{4} - 120\beta_{3} - 120\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66\beta_{7} + 66\beta_{6} + 120\beta_{5} + 270\beta_{4} - 572\beta_{3} + 120\beta_{2} + 270 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(\beta_{4}\) \(1\)

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
2.13456 + 2.13456i
0.575868 + 0.575868i
−0.285451 − 0.285451i
−1.42497 − 1.42497i
2.13456 − 2.13456i
0.575868 − 0.575868i
−0.285451 + 0.285451i
−1.42497 + 1.42497i
0 −2.13456 + 2.13456i 0 −0.666078 + 2.13456i 0 2.84356i 0 − 6.11268i 0
577.2 0 −0.575868 + 0.575868i 0 2.16064 + 0.575868i 0 − 2.48849i 0 2.33675i 0
577.3 0 0.285451 − 0.285451i 0 −2.21777 − 0.285451i 0 − 1.26613i 0 2.83704i 0
577.4 0 1.42497 − 1.42497i 0 1.72321 − 1.42497i 0 4.91106i 0 − 1.06111i 0
593.1 0 −2.13456 − 2.13456i 0 −0.666078 − 2.13456i 0 − 2.84356i 0 6.11268i 0
593.2 0 −0.575868 − 0.575868i 0 2.16064 − 0.575868i 0 2.48849i 0 − 2.33675i 0
593.3 0 0.285451 + 0.285451i 0 −2.21777 + 0.285451i 0 1.26613i 0 − 2.83704i 0
593.4 0 1.42497 + 1.42497i 0 1.72321 + 1.42497i 0 − 4.91106i 0 1.06111i 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.m 8
4.b odd 2 1 260.2.m.c ✓ 8
5.c odd 4 1 1040.2.cd.m 8
12.b even 2 1 2340.2.u.g 8
13.d odd 4 1 1040.2.cd.m 8
20.d odd 2 1 1300.2.m.c 8
20.e even 4 1 260.2.r.c yes 8
20.e even 4 1 1300.2.r.c 8
52.f even 4 1 260.2.r.c yes 8
60.l odd 4 1 2340.2.bp.g 8
65.k even 4 1 inner 1040.2.bg.m 8
156.l odd 4 1 2340.2.bp.g 8
260.l odd 4 1 1300.2.m.c 8
260.s odd 4 1 260.2.m.c ✓ 8
260.u even 4 1 1300.2.r.c 8
780.bn even 4 1 2340.2.u.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.c ✓ 8 4.b odd 2 1
260.2.m.c ✓ 8 260.s odd 4 1
260.2.r.c yes 8 20.e even 4 1
260.2.r.c yes 8 52.f even 4 1
1040.2.bg.m 8 1.a even 1 1 trivial
1040.2.bg.m 8 65.k even 4 1 inner
1040.2.cd.m 8 5.c odd 4 1
1040.2.cd.m 8 13.d odd 4 1
1300.2.m.c 8 20.d odd 2 1
1300.2.m.c 8 260.l odd 4 1
1300.2.r.c 8 20.e even 4 1
1300.2.r.c 8 260.u even 4 1
2340.2.u.g 8 12.b even 2 1
2340.2.u.g 8 780.bn even 4 1
2340.2.bp.g 8 60.l odd 4 1
2340.2.bp.g 8 156.l odd 4 1

Hecke kernels

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

\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} - 8T_{3}^{5} + 32T_{3}^{4} + 20T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 40T_{7}^{6} + 456T_{7}^{4} + 1840T_{7}^{2} + 1936 \) Copy content Toggle raw display
\( T_{11}^{8} + 14T_{11}^{7} + 98T_{11}^{6} + 332T_{11}^{5} + 580T_{11}^{4} + 124T_{11}^{3} + 8T_{11}^{2} - 8T_{11} + 4 \) Copy content Toggle raw display
\( T_{19}^{8} + 2T_{19}^{7} + 2T_{19}^{6} - 48T_{19}^{5} + 708T_{19}^{4} + 276T_{19}^{3} + 288T_{19}^{2} - 7056T_{19} + 86436 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 40 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{8} + 14 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 86436 \) Copy content Toggle raw display
$23$ \( T^{8} - 22 T^{7} + \cdots + 6724 \) Copy content Toggle raw display
$29$ \( T^{8} + 156 T^{6} + \cdots + 810000 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + \cdots + 22500 \) Copy content Toggle raw display
$37$ \( T^{8} + 76 T^{6} + \cdots + 35344 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$43$ \( T^{8} - 14 T^{7} + \cdots + 1085764 \) Copy content Toggle raw display
$47$ \( T^{8} + 400 T^{6} + \cdots + 29637136 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} + \cdots + 86436 \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + \cdots - 1452)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} + \cdots - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 22 T^{7} + \cdots + 46321636 \) Copy content Toggle raw display
$73$ \( (T^{4} + 14 T^{3} + \cdots - 13212)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 340 T^{6} + \cdots + 440896 \) Copy content Toggle raw display
$83$ \( T^{8} + 160 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} + \cdots + 22391824 \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} + \cdots - 4392)^{2} \) Copy content Toggle raw display
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