Properties

Label 1040.2.q.i
Level $1040$
Weight $2$
Character orbit 1040.q
Analytic conductor $8.304$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} - q^{5} - \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} - q^{5} - \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + (4 \zeta_{6} - 1) q^{13} + (\zeta_{6} - 1) q^{15} + 3 \zeta_{6} q^{17} + 5 \zeta_{6} q^{19} - q^{21} + ( - 9 \zeta_{6} + 9) q^{23} + q^{25} + 5 q^{27} + ( - 9 \zeta_{6} + 9) q^{29} - 8 q^{31} - 3 \zeta_{6} q^{33} + \zeta_{6} q^{35} + ( - 7 \zeta_{6} + 7) q^{37} + (\zeta_{6} + 3) q^{39} + (3 \zeta_{6} - 3) q^{41} - \zeta_{6} q^{43} - 2 \zeta_{6} q^{45} + ( - 6 \zeta_{6} + 6) q^{49} + 3 q^{51} + 6 q^{53} + (3 \zeta_{6} - 3) q^{55} + 5 q^{57} + 9 \zeta_{6} q^{59} + \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{63} + ( - 4 \zeta_{6} + 1) q^{65} + ( - 5 \zeta_{6} + 5) q^{67} - 9 \zeta_{6} q^{69} + 9 \zeta_{6} q^{71} + 2 q^{73} + ( - \zeta_{6} + 1) q^{75} - 3 q^{77} - 8 q^{79} + (\zeta_{6} - 1) q^{81} - 3 \zeta_{6} q^{85} - 9 \zeta_{6} q^{87} + (3 \zeta_{6} - 3) q^{89} + ( - 3 \zeta_{6} + 4) q^{91} + (8 \zeta_{6} - 8) q^{93} - 5 \zeta_{6} q^{95} - 17 \zeta_{6} q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} + 5 q^{19} - 2 q^{21} + 9 q^{23} + 2 q^{25} + 10 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} + 7 q^{37} + 7 q^{39} - 3 q^{41} - q^{43} - 2 q^{45} + 6 q^{49} + 6 q^{51} + 12 q^{53} - 3 q^{55} + 10 q^{57} + 9 q^{59} + q^{61} + 2 q^{63} - 2 q^{65} + 5 q^{67} - 9 q^{69} + 9 q^{71} + 4 q^{73} + q^{75} - 6 q^{77} - 16 q^{79} - q^{81} - 3 q^{85} - 9 q^{87} - 3 q^{89} + 5 q^{91} - 8 q^{93} - 5 q^{95} - 17 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0 1.00000 1.73205i 0
321.1 0 0.500000 0.866025i 0 −1.00000 0 −0.500000 0.866025i 0 1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
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.i 2
4.b odd 2 1 260.2.i.a 2
12.b even 2 1 2340.2.q.f 2
13.c even 3 1 inner 1040.2.q.i 2
20.d odd 2 1 1300.2.i.d 2
20.e even 4 2 1300.2.bb.b 4
52.i odd 6 1 3380.2.a.i 1
52.j odd 6 1 260.2.i.a 2
52.j odd 6 1 3380.2.a.f 1
52.l even 12 2 3380.2.f.d 2
156.p even 6 1 2340.2.q.f 2
260.v odd 6 1 1300.2.i.d 2
260.bj even 12 2 1300.2.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.a 2 4.b odd 2 1
260.2.i.a 2 52.j odd 6 1
1040.2.q.i 2 1.a even 1 1 trivial
1040.2.q.i 2 13.c even 3 1 inner
1300.2.i.d 2 20.d odd 2 1
1300.2.i.d 2 260.v odd 6 1
1300.2.bb.b 4 20.e even 4 2
1300.2.bb.b 4 260.bj even 12 2
2340.2.q.f 2 12.b even 2 1
2340.2.q.f 2 156.p even 6 1
3380.2.a.f 1 52.j odd 6 1
3380.2.a.i 1 52.i odd 6 1
3380.2.f.d 2 52.l even 12 2

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{19}^{2} - 5T_{19} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
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