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Properties

Label 1040.2.q.r
Level $1040$
Weight $2$
Character orbit 1040.q
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(81,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.q (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: 8.0.649638144.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + (\beta_{5} - \beta_{3} - \beta_{2} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} + 4 \beta_{4} + 3 \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 4 q^{7} - 6 q^{9} - 2 q^{11} + 4 q^{13} - 8 q^{17} - 2 q^{19} + 24 q^{21} + 10 q^{23} + 8 q^{25} + 36 q^{27} - 10 q^{29} + 4 q^{31} - 12 q^{33} - 4 q^{35} + 6 q^{37} - 6 q^{39} - 6 q^{41} + 10 q^{43} - 6 q^{45} + 28 q^{47} - 24 q^{49} - 36 q^{51} - 2 q^{55} - 48 q^{57} + 2 q^{59} + 20 q^{61} - 30 q^{63} + 4 q^{65} + 14 q^{67} - 6 q^{69} - 18 q^{71} + 12 q^{73} + 76 q^{77} - 12 q^{79} - 12 q^{81} - 4 q^{83} - 8 q^{85} - 18 q^{87} - 12 q^{89} - 14 q^{91} - 24 q^{93} - 2 q^{95} - 4 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 62\nu^{3} + 79\nu - 26 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} - 13\nu^{6} - 57\nu^{5} + 221\nu^{4} + 245\nu^{3} - 1001\nu^{2} - 122\nu + 1300 ) / 364 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{7} + 114\nu^{5} - 91\nu^{4} - 490\nu^{3} + 637\nu^{2} + 790\nu - 1183 ) / 364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 39\nu^{6} - 16\nu^{5} + 481\nu^{4} - 126\nu^{3} - 1911\nu^{2} + 673\nu + 2262 ) / 364 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{7} + 52\nu^{6} + 114\nu^{5} - 611\nu^{4} - 490\nu^{3} + 2457\nu^{2} + 790\nu - 2743 ) / 364 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 29\nu^{5} + 95\nu^{3} + 26\nu^{2} - 68\nu - 78 ) / 52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{4} + 7\beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 4\beta_{6} - 17\beta_{5} + 22\beta_{4} + 35\beta_{3} + 7\beta_{2} - 7\beta _1 + 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 46\beta_{7} - 41\beta_{6} - 106\beta_{5} + 89\beta_{4} + 154\beta_{3} + 134\beta_{2} - 35\beta _1 + 85 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{7} + 17\beta_{6} + 10\beta_{5} - 37\beta_{4} + 10\beta_{3} + 10\beta_{2} + 35\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 413\beta_{7} - 247\beta_{6} - 509\beta_{5} + 277\beta_{4} + 539\beta_{3} + 1051\beta_{2} - 214\beta _1 + 566 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
2.34138 − 0.500000i
1.42055 + 0.500000i
−2.34138 − 0.500000i
−1.42055 + 0.500000i
2.34138 + 0.500000i
1.42055 − 0.500000i
−2.34138 + 0.500000i
−1.42055 − 0.500000i
0 −1.60370 − 2.77769i 0 1.00000 0 −2.23205 + 3.86603i 0 −3.64372 + 6.31111i 0
81.2 0 −0.277260 − 0.480228i 0 1.00000 0 1.23205 − 2.13397i 0 1.34625 − 2.33178i 0
81.3 0 0.737676 + 1.27769i 0 1.00000 0 −2.23205 + 3.86603i 0 0.411667 − 0.713029i 0
81.4 0 1.14329 + 1.98023i 0 1.00000 0 1.23205 − 2.13397i 0 −1.11420 + 1.92986i 0
321.1 0 −1.60370 + 2.77769i 0 1.00000 0 −2.23205 − 3.86603i 0 −3.64372 − 6.31111i 0
321.2 0 −0.277260 + 0.480228i 0 1.00000 0 1.23205 + 2.13397i 0 1.34625 + 2.33178i 0
321.3 0 0.737676 − 1.27769i 0 1.00000 0 −2.23205 − 3.86603i 0 0.411667 + 0.713029i 0
321.4 0 1.14329 − 1.98023i 0 1.00000 0 1.23205 + 2.13397i 0 −1.11420 − 1.92986i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.q.r 8
4.b odd 2 1 520.2.q.h ✓ 8
13.c even 3 1 inner 1040.2.q.r 8
52.i odd 6 1 6760.2.a.ba 4
52.j odd 6 1 520.2.q.h ✓ 8
52.j odd 6 1 6760.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.q.h ✓ 8 4.b odd 2 1
520.2.q.h ✓ 8 52.j odd 6 1
1040.2.q.r 8 1.a even 1 1 trivial
1040.2.q.r 8 13.c even 3 1 inner
6760.2.a.ba 4 52.i odd 6 1
6760.2.a.bb 4 52.j odd 6 1

Hecke kernels

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

\( T_{3}^{8} + 9T_{3}^{6} - 12T_{3}^{5} + 75T_{3}^{4} - 54T_{3}^{3} + 90T_{3}^{2} + 36T_{3} + 36 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 15T_{7}^{2} - 22T_{7} + 121 \) Copy content Toggle raw display
\( T_{11}^{8} + 2T_{11}^{7} + 34T_{11}^{6} + 128T_{11}^{5} + 1159T_{11}^{4} + 3104T_{11}^{3} + 6706T_{11}^{2} + 6674T_{11} + 5041 \) Copy content Toggle raw display
\( T_{19}^{8} + 2T_{19}^{7} + 46T_{19}^{6} - 136T_{19}^{5} + 1675T_{19}^{4} - 1240T_{19}^{3} + 2230T_{19}^{2} + 962T_{19} + 1369 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 9 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 5041 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 676 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 1369 \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} + \cdots + 913936 \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} - 42 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$41$ \( T^{8} + 6 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$43$ \( T^{8} - 10 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 75 T^{2} + \cdots + 1284)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 2 T^{7} + \cdots + 57395776 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + \cdots + 141752836 \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{8} + 18 T^{7} + \cdots + 318096 \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots - 936)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + \cdots + 1536)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} - 66 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 6 T^{3} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + \cdots + 20164 \) Copy content Toggle raw display
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