Properties

Label 1040.2.da.a
Level $1040$
Weight $2$
Character orbit 1040.da
Analytic conductor $8.304$
Analytic rank $0$
Dimension $4$
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.da (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + \zeta_{12}^{3} q^{5} - 3 \zeta_{12} q^{7} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + \zeta_{12}^{3} q^{5} - 3 \zeta_{12} q^{7} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{9} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{11} + (\zeta_{12}^{2} + 3) q^{13} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{15} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{17} + (2 \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{19} + (3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{21} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{23} - q^{25} + 4 q^{27} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{29} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{31} + ( - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{33} + ( - 3 \zeta_{12}^{2} + 3) q^{35} + (6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{37} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{39} + (6 \zeta_{12}^{2} - 12) q^{41} + ( - 2 \zeta_{12}^{2} + 2) q^{43} + (2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{45} + 3 \zeta_{12}^{3} q^{47} + 2 \zeta_{12}^{2} q^{49} - 6 q^{51} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 3) q^{53} - 3 \zeta_{12}^{2} q^{55} + (9 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5) q^{57} + ( - 6 \zeta_{12}^{2} - 6) q^{59} + (6 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{61} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12} - 12) q^{63} + (4 \zeta_{12}^{3} - \zeta_{12}) q^{65} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{69} + 6 \zeta_{12} q^{71} + ( - 3 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5) q^{73} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{75} + 9 q^{77} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 1) q^{79} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12}) q^{81} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{83} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{85} + (16 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 8 \zeta_{12} + 12) q^{87} + ( - 12 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 12 \zeta_{12} - 6) q^{89} + ( - 3 \zeta_{12}^{3} - 9 \zeta_{12}) q^{91} + (2 \zeta_{12}^{2} - 4) q^{93} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{95} + (7 \zeta_{12}^{2} - 3 \zeta_{12} + 7) q^{97} + ( - 3 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{9} + 14 q^{13} - 6 q^{15} - 6 q^{17} + 12 q^{19} + 12 q^{23} - 4 q^{25} + 16 q^{27} + 12 q^{29} - 18 q^{33} + 6 q^{35} + 18 q^{37} - 4 q^{39} - 36 q^{41} + 4 q^{43} + 12 q^{45} + 4 q^{49} - 24 q^{51} - 12 q^{53} - 6 q^{55} - 36 q^{59} - 2 q^{61} - 36 q^{63} + 2 q^{75} + 36 q^{77} - 4 q^{79} - 2 q^{81} - 18 q^{85} + 24 q^{87} - 18 q^{89} - 12 q^{93} + 6 q^{95} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{12}^{2}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −1.36603 + 2.36603i 0 1.00000i 0 2.59808 1.50000i 0 −2.23205 3.86603i 0
641.2 0 0.366025 0.633975i 0 1.00000i 0 −2.59808 + 1.50000i 0 1.23205 + 2.13397i 0
881.1 0 −1.36603 2.36603i 0 1.00000i 0 2.59808 + 1.50000i 0 −2.23205 + 3.86603i 0
881.2 0 0.366025 + 0.633975i 0 1.00000i 0 −2.59808 1.50000i 0 1.23205 2.13397i 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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.da.a 4
4.b odd 2 1 130.2.l.a 4
12.b even 2 1 1170.2.bs.c 4
13.e even 6 1 inner 1040.2.da.a 4
20.d odd 2 1 650.2.m.a 4
20.e even 4 1 650.2.n.a 4
20.e even 4 1 650.2.n.b 4
52.b odd 2 1 1690.2.l.g 4
52.f even 4 1 1690.2.e.l 4
52.f even 4 1 1690.2.e.n 4
52.i odd 6 1 130.2.l.a 4
52.i odd 6 1 1690.2.d.f 4
52.j odd 6 1 1690.2.d.f 4
52.j odd 6 1 1690.2.l.g 4
52.l even 12 1 1690.2.a.j 2
52.l even 12 1 1690.2.a.m 2
52.l even 12 1 1690.2.e.l 4
52.l even 12 1 1690.2.e.n 4
156.r even 6 1 1170.2.bs.c 4
260.w odd 6 1 650.2.m.a 4
260.bc even 12 1 8450.2.a.bf 2
260.bc even 12 1 8450.2.a.bm 2
260.bg even 12 1 650.2.n.a 4
260.bg even 12 1 650.2.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 4.b odd 2 1
130.2.l.a 4 52.i odd 6 1
650.2.m.a 4 20.d odd 2 1
650.2.m.a 4 260.w odd 6 1
650.2.n.a 4 20.e even 4 1
650.2.n.a 4 260.bg even 12 1
650.2.n.b 4 20.e even 4 1
650.2.n.b 4 260.bg even 12 1
1040.2.da.a 4 1.a even 1 1 trivial
1040.2.da.a 4 13.e even 6 1 inner
1170.2.bs.c 4 12.b even 2 1
1170.2.bs.c 4 156.r even 6 1
1690.2.a.j 2 52.l even 12 1
1690.2.a.m 2 52.l even 12 1
1690.2.d.f 4 52.i odd 6 1
1690.2.d.f 4 52.j odd 6 1
1690.2.e.l 4 52.f even 4 1
1690.2.e.l 4 52.l even 12 1
1690.2.e.n 4 52.f even 4 1
1690.2.e.n 4 52.l even 12 1
1690.2.l.g 4 52.b odd 2 1
1690.2.l.g 4 52.j odd 6 1
8450.2.a.bf 2 260.bc even 12 1
8450.2.a.bm 2 260.bc even 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 7 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} - 18 T^{3} + 99 T^{2} + 162 T + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$89$ \( T^{4} + 18 T^{3} - 9 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{4} - 42 T^{3} + 726 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
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