Properties

Label 1040.2.da.d
Level $1040$
Weight $2$
Character orbit 1040.da
Analytic conductor $8.304$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{6} + \beta_1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{6} + \beta_1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 2) q^{9} + (\beta_{7} + \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{13} + (\beta_{3} + \beta_1 - 1) q^{15} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 2) q^{17} + ( - 2 \beta_{6} + \beta_{5} + 1) q^{19} + (2 \beta_{6} - \beta_{5} + 5 \beta_{4} + \beta_{3} - \beta_1 - 3) q^{21} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{23} - q^{25} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_1 - 4) q^{27} + (4 \beta_{7} + 4 \beta_{6} + 2 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{29} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{31} + (3 \beta_{5} - 5 \beta_{4} - 2) q^{33} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1) q^{35} + (\beta_{7} + 3 \beta_{4} + \beta_{2} - \beta_1 - 6) q^{37} + ( - 2 \beta_{5} + 4 \beta_{4} - \beta_{2} + \beta_1 + 3) q^{39} + ( - \beta_{7} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{41} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 8 \beta_1) q^{43} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{45} + (2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1) q^{47} + (8 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{49} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{51} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 9) q^{53} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 1) q^{55} + (2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{57} + (2 \beta_{4} + 2) q^{59} + (2 \beta_{6} - \beta_{5} - 7 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 6) q^{61} + (2 \beta_{3} + 7 \beta_1 - 2) q^{63} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{65} + (4 \beta_{4} - 4 \beta_{3} - 8 \beta_1 - 4) q^{67} + ( - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{4} + 2 \beta_1 + 4) q^{69} + 6 \beta_{6} q^{71} + ( - 3 \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 5) q^{73} + ( - \beta_{7} - \beta_{4} + \beta_{2}) q^{75} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{77} + (4 \beta_{6} - 3 \beta_{2} + \beta_1 - 1) q^{79} + ( - 5 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} - 10 \beta_{4} - \beta_{3} + 5 \beta_{2} + \cdots + 1) q^{81}+ \cdots + ( - 4 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 13 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 6 q^{15} + 6 q^{17} + 6 q^{19} - 12 q^{23} - 8 q^{25} - 40 q^{27} - 42 q^{33} - 30 q^{37} + 40 q^{39} + 12 q^{41} - 4 q^{43} - 12 q^{45} + 8 q^{49} + 60 q^{53} - 6 q^{55} + 24 q^{59} + 26 q^{61} - 12 q^{63} + 6 q^{65} - 24 q^{67} + 12 q^{69} - 2 q^{75} - 60 q^{77} - 20 q^{79} - 28 q^{81} - 18 q^{85} - 48 q^{87} - 24 q^{89} + 66 q^{91} + 48 q^{93} - 6 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - \nu^{5} - 3\nu^{4} + 5\nu^{3} + 3\nu^{2} - 12\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 4\nu^{4} - 2\nu^{3} - 6\nu^{2} + 11\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} - 3\nu^{5} - 7\nu^{4} + 11\nu^{3} + 7\nu^{2} - 27\nu + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 22\nu^{6} + 13\nu^{5} + 32\nu^{4} - 47\nu^{3} - 30\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 7\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{5} + 7\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} + 9\beta_{2} + 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{7} - 12\beta_{6} - 3\beta_{5} - 10\beta_{4} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\) \(1\)

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
0.665665 + 1.24775i
1.40994 + 0.109843i
1.20036 0.747754i
−1.27597 0.609843i
0.665665 1.24775i
1.40994 0.109843i
1.20036 + 0.747754i
−1.27597 + 0.609843i
0 −0.913419 + 1.58209i 0 1.00000i 0 −3.45632 + 1.99551i 0 −0.168669 0.292144i 0
641.2 0 −0.300098 + 0.519785i 0 1.00000i 0 −1.24653 + 0.719687i 0 1.31988 + 2.28610i 0
641.3 0 0.547394 0.948114i 0 1.00000i 0 3.45632 1.99551i 0 0.900720 + 1.56009i 0
641.4 0 1.66612 2.88581i 0 1.00000i 0 1.24653 0.719687i 0 −4.05193 7.01815i 0
881.1 0 −0.913419 1.58209i 0 1.00000i 0 −3.45632 1.99551i 0 −0.168669 + 0.292144i 0
881.2 0 −0.300098 0.519785i 0 1.00000i 0 −1.24653 0.719687i 0 1.31988 2.28610i 0
881.3 0 0.547394 + 0.948114i 0 1.00000i 0 3.45632 + 1.99551i 0 0.900720 1.56009i 0
881.4 0 1.66612 + 2.88581i 0 1.00000i 0 1.24653 + 0.719687i 0 −4.05193 + 7.01815i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.4
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.d 8
4.b odd 2 1 130.2.l.b 8
12.b even 2 1 1170.2.bs.g 8
13.e even 6 1 inner 1040.2.da.d 8
20.d odd 2 1 650.2.m.c 8
20.e even 4 1 650.2.n.d 8
20.e even 4 1 650.2.n.e 8
52.b odd 2 1 1690.2.l.j 8
52.f even 4 1 1690.2.e.s 8
52.f even 4 1 1690.2.e.t 8
52.i odd 6 1 130.2.l.b 8
52.i odd 6 1 1690.2.d.k 8
52.j odd 6 1 1690.2.d.k 8
52.j odd 6 1 1690.2.l.j 8
52.l even 12 1 1690.2.a.t 4
52.l even 12 1 1690.2.a.u 4
52.l even 12 1 1690.2.e.s 8
52.l even 12 1 1690.2.e.t 8
156.r even 6 1 1170.2.bs.g 8
260.w odd 6 1 650.2.m.c 8
260.bc even 12 1 8450.2.a.ci 4
260.bc even 12 1 8450.2.a.cm 4
260.bg even 12 1 650.2.n.d 8
260.bg even 12 1 650.2.n.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.b 8 4.b odd 2 1
130.2.l.b 8 52.i odd 6 1
650.2.m.c 8 20.d odd 2 1
650.2.m.c 8 260.w odd 6 1
650.2.n.d 8 20.e even 4 1
650.2.n.d 8 260.bg even 12 1
650.2.n.e 8 20.e even 4 1
650.2.n.e 8 260.bg even 12 1
1040.2.da.d 8 1.a even 1 1 trivial
1040.2.da.d 8 13.e even 6 1 inner
1170.2.bs.g 8 12.b even 2 1
1170.2.bs.g 8 156.r even 6 1
1690.2.a.t 4 52.l even 12 1
1690.2.a.u 4 52.l even 12 1
1690.2.d.k 8 52.i odd 6 1
1690.2.d.k 8 52.j odd 6 1
1690.2.e.s 8 52.f even 4 1
1690.2.e.s 8 52.l even 12 1
1690.2.e.t 8 52.f even 4 1
1690.2.e.t 8 52.l even 12 1
1690.2.l.j 8 52.b odd 2 1
1690.2.l.j 8 52.j odd 6 1
8450.2.a.ci 4 260.bc even 12 1
8450.2.a.cm 4 260.bc even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 10T_{3}^{6} + 4T_{3}^{5} + 40T_{3}^{4} - 8T_{3}^{3} + 40T_{3}^{2} + 16T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + 10 T^{6} + 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{6} + 291 T^{4} + \cdots + 1089 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} - 6 T^{6} + 108 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} - 2 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + 54 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + 6 T^{6} + 36 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + 108 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{8} + 96 T^{6} + 7104 T^{4} + \cdots + 4460544 \) Copy content Toggle raw display
$31$ \( T^{8} + 144 T^{6} + 3324 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{8} + 30 T^{7} + 390 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + 12 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$43$ \( T^{8} + 4 T^{7} + 136 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + 11718 T^{4} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( (T^{4} - 30 T^{3} + 294 T^{2} - 882 T - 579)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 26 T^{7} + 454 T^{6} + \cdots + 394384 \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + 96 T^{6} + \cdots + 9437184 \) Copy content Toggle raw display
$71$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 264 T^{6} + 15228 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} - 54 T^{2} - 260 T - 188)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 480 T^{6} + 67932 T^{4} + \cdots + 4511376 \) Copy content Toggle raw display
$89$ \( T^{8} + 24 T^{7} + 210 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} - 306 T^{6} + \cdots + 18558864 \) Copy content Toggle raw display
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