Properties

Label 1040.2.f.d
Level $1040$
Weight $2$
Character orbit 1040.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{3} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{3} + \beta_1 - 1) q^{9} + (\beta_{3} - \beta_{2} - \beta_1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} - 2 \beta_{2}) q^{15} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{21} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{3} - \beta_1) q^{27} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{29} + (\beta_{3} + \beta_{2} - \beta_1) q^{31} - 4 q^{33} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{35} + ( - 3 \beta_{3} - 3 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{39} - 4 \beta_{2} q^{41} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{3} - \beta_{2} - 5) q^{45} + (\beta_{3} + \beta_1 - 8) q^{47} + (\beta_{3} + \beta_1 + 1) q^{49} + (3 \beta_{3} + 3 \beta_1 - 8) q^{51} + 2 \beta_{2} q^{53} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{55} + (2 \beta_{3} + 2 \beta_1 - 4) q^{57} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{59} + (2 \beta_{3} + 2 \beta_1 - 2) q^{61} - 8 q^{63} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 3) q^{65} + 4 q^{67} + ( - 2 \beta_{3} - 2 \beta_1 + 12) q^{69} - 5 \beta_{2} q^{71} - 10 q^{73} + ( - 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{75} + ( - 4 \beta_{3} + 4 \beta_1) q^{77} + (2 \beta_{3} + 2 \beta_1) q^{79} + (2 \beta_{3} + 2 \beta_1 - 3) q^{81} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{83} + ( - 5 \beta_{2} + \beta_1 + 4) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_1) q^{89} + ( - 3 \beta_{3} + 4 \beta_{2} + \beta_1) q^{91} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{93} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 4) q^{95} + (2 \beta_{3} + 2 \beta_1 - 2) q^{97} + (3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{5} - 2 q^{7} - 2 q^{9} + 4 q^{13} - q^{15} + q^{25} - 12 q^{29} - 16 q^{33} + 15 q^{35} + 2 q^{37} + 12 q^{39} - 18 q^{45} - 30 q^{47} + 6 q^{49} - 26 q^{51} - 20 q^{55} - 12 q^{57} - 4 q^{61} - 32 q^{63} - 15 q^{65} + 16 q^{67} + 44 q^{69} - 40 q^{73} + 15 q^{75} + 4 q^{79} - 8 q^{81} + 12 q^{83} + 17 q^{85} - 2 q^{91} + 12 q^{93} + 18 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + \nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} + \nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} - \nu^{2} + 4\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 3\beta_{2} + \beta _1 + 8 ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
0 2.52434i 0 2.18614 + 0.469882i 0 2.37228 0 −3.37228 0
129.2 0 0.792287i 0 −0.686141 2.12819i 0 −3.37228 0 2.37228 0
129.3 0 0.792287i 0 −0.686141 + 2.12819i 0 −3.37228 0 2.37228 0
129.4 0 2.52434i 0 2.18614 0.469882i 0 2.37228 0 −3.37228 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
65.d 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.f.d 4
4.b odd 2 1 130.2.c.a 4
5.b even 2 1 1040.2.f.c 4
12.b even 2 1 1170.2.f.b 4
13.b even 2 1 1040.2.f.c 4
20.d odd 2 1 130.2.c.b yes 4
20.e even 4 2 650.2.d.e 8
52.b odd 2 1 130.2.c.b yes 4
52.f even 4 2 1690.2.b.d 8
60.h even 2 1 1170.2.f.a 4
65.d even 2 1 inner 1040.2.f.d 4
156.h even 2 1 1170.2.f.a 4
260.g odd 2 1 130.2.c.a 4
260.l odd 4 2 8450.2.a.cn 4
260.p even 4 2 650.2.d.e 8
260.s odd 4 2 8450.2.a.cj 4
260.u even 4 2 1690.2.b.d 8
780.d even 2 1 1170.2.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.c.a 4 4.b odd 2 1
130.2.c.a 4 260.g odd 2 1
130.2.c.b yes 4 20.d odd 2 1
130.2.c.b yes 4 52.b odd 2 1
650.2.d.e 8 20.e even 4 2
650.2.d.e 8 260.p even 4 2
1040.2.f.c 4 5.b even 2 1
1040.2.f.c 4 13.b even 2 1
1040.2.f.d 4 1.a even 1 1 trivial
1040.2.f.d 4 65.d even 2 1 inner
1170.2.f.a 4 60.h even 2 1
1170.2.f.a 4 156.h even 2 1
1170.2.f.b 4 12.b even 2 1
1170.2.f.b 4 780.d even 2 1
1690.2.b.d 8 52.f even 4 2
1690.2.b.d 8 260.u even 4 2
8450.2.a.cj 4 260.s odd 4 2
8450.2.a.cn 4 260.l odd 4 2

Hecke kernels

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

\( T_{3}^{4} + 7T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{37}^{2} - T_{37} - 74 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 4 T^{2} - 15 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 43T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 112T^{2} + 1024 \) Copy content Toggle raw display
$43$ \( T^{4} + 87T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T^{2} + 15 T + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 175T^{2} + 2500 \) Copy content Toggle raw display
$73$ \( (T + 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
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