Properties

Label 1040.2.k.c
Level $1040$
Weight $2$
Character orbit 1040.k
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{4} + 1) q^{9} + (\beta_{5} - 2 \beta_{2}) q^{11} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{13} - \beta_1 q^{15} + (\beta_{5} - 2 \beta_1) q^{19} + (2 \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{4} - \beta_{3} + 2) q^{23} - q^{25} - 2 q^{27} + ( - \beta_{4} - 2 \beta_{3} - 2) q^{29} + (\beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{31} + (\beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{33} + \beta_{4} q^{35} + (\beta_{5} + 4 \beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{39} + (2 \beta_{5} + 2 \beta_{2}) q^{41} + ( - 3 \beta_{3} - 2) q^{43} + (\beta_{5} - \beta_{2} + \beta_1) q^{45} + ( - \beta_{5} - 4 \beta_{2} + 3 \beta_1) q^{47} + ( - 2 \beta_{4} - 2 \beta_{3} - 1) q^{49} + (2 \beta_{4} - 2 \beta_{3} - 2) q^{53} + (\beta_{4} + \beta_{3} - 2) q^{55} + (3 \beta_{5} - 10 \beta_{2} + \beta_1) q^{57} + ( - \beta_{5} + 8 \beta_{2}) q^{59} + (3 \beta_{4} - 2) q^{61} + ( - \beta_{5} - 8 \beta_{2} + \beta_1) q^{63} + ( - \beta_{5} - \beta_{3} - 1) q^{65} + (3 \beta_{5} - 4 \beta_{2} + \beta_1) q^{67} + ( - \beta_{4} - 6 \beta_{3}) q^{69} + ( - 3 \beta_{5} - 6 \beta_1) q^{71} + ( - \beta_{5} + 8 \beta_{2} - 3 \beta_1) q^{73} + \beta_{3} q^{75} - 6 q^{77} + (2 \beta_{4} - 4 \beta_{3} + 4) q^{79} + (3 \beta_{4} + 2 \beta_{3} - 3) q^{81} + (\beta_{5} + 4 \beta_{2} + 3 \beta_1) q^{83} + ( - 2 \beta_{4} + 6) q^{87} + (2 \beta_{5} + 8 \beta_{2} + 6 \beta_1) q^{89} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 2) q^{91} + ( - \beta_{5} + 6 \beta_{2} - \beta_1) q^{93} + (\beta_{4} + 3 \beta_{3}) q^{95} + ( - 2 \beta_{5} - 6 \beta_{2} - 2 \beta_1) q^{97} + (\beta_{5} - 8 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{9} + 12 q^{23} - 6 q^{25} - 12 q^{27} - 12 q^{29} - 12 q^{39} - 12 q^{43} - 6 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{77} + 24 q^{79} - 18 q^{81} + 36 q^{87} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 6\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 22\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 2\beta_{3} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 20\beta_{2} + 38\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
2.60168i
2.60168i
0.339877i
0.339877i
2.26180i
2.26180i
0 −2.60168 0 1.00000i 0 2.76873i 0 3.76873 0
961.2 0 −2.60168 0 1.00000i 0 2.76873i 0 3.76873 0
961.3 0 0.339877 0 1.00000i 0 3.88448i 0 −2.88448 0
961.4 0 0.339877 0 1.00000i 0 3.88448i 0 −2.88448 0
961.5 0 2.26180 0 1.00000i 0 1.11575i 0 2.11575 0
961.6 0 2.26180 0 1.00000i 0 1.11575i 0 2.11575 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.k.c 6
4.b odd 2 1 260.2.f.a 6
12.b even 2 1 2340.2.c.d 6
13.b even 2 1 inner 1040.2.k.c 6
20.d odd 2 1 1300.2.f.e 6
20.e even 4 1 1300.2.d.c 6
20.e even 4 1 1300.2.d.d 6
52.b odd 2 1 260.2.f.a 6
52.f even 4 1 3380.2.a.m 3
52.f even 4 1 3380.2.a.n 3
156.h even 2 1 2340.2.c.d 6
260.g odd 2 1 1300.2.f.e 6
260.p even 4 1 1300.2.d.c 6
260.p even 4 1 1300.2.d.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 4.b odd 2 1
260.2.f.a 6 52.b odd 2 1
1040.2.k.c 6 1.a even 1 1 trivial
1040.2.k.c 6 13.b even 2 1 inner
1300.2.d.c 6 20.e even 4 1
1300.2.d.c 6 260.p even 4 1
1300.2.d.d 6 20.e even 4 1
1300.2.d.d 6 260.p even 4 1
1300.2.f.e 6 20.d odd 2 1
1300.2.f.e 6 260.g odd 2 1
2340.2.c.d 6 12.b even 2 1
2340.2.c.d 6 156.h even 2 1
3380.2.a.m 3 52.f even 4 1
3380.2.a.n 3 52.f even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 6 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + 144 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{4} - 16 T^{3} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 96 T^{4} + 2304 T^{2} + \cdots + 12996 \) Copy content Toggle raw display
$23$ \( (T^{3} - 6 T^{2} - 30 T + 174)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 12 T - 84)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 60 T^{4} + 936 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$37$ \( T^{6} + 144 T^{4} + 6048 T^{2} + \cdots + 63504 \) Copy content Toggle raw display
$41$ \( T^{6} + 108 T^{4} + 2160 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$43$ \( (T^{3} + 6 T^{2} - 42 T - 46)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 216 T^{4} + 12528 T^{2} + \cdots + 219024 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 84 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 216 T^{4} + 12528 T^{2} + \cdots + 171396 \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} - 96 T - 532)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 240 T^{4} + 16416 T^{2} + \cdots + 345744 \) Copy content Toggle raw display
$71$ \( T^{6} + 432 T^{4} + 46656 T^{2} + \cdots + 492804 \) Copy content Toggle raw display
$73$ \( T^{6} + 288 T^{4} + 8640 T^{2} + \cdots + 63504 \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} - 144 T + 1696)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 144 T^{4} + 6048 T^{2} + \cdots + 63504 \) Copy content Toggle raw display
$89$ \( T^{6} + 576 T^{4} + 96768 T^{2} + \cdots + 4064256 \) Copy content Toggle raw display
$97$ \( T^{6} + 204 T^{4} + 2736 T^{2} + \cdots + 576 \) Copy content Toggle raw display
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