Properties

Label 1040.2.f.g
Level $1040$
Weight $2$
Character orbit 1040.f
Analytic conductor $8.304$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} + ( - \beta_1 - 1) q^{9} - \beta_{4} q^{11} - \beta_{7} q^{13} + ( - \beta_{8} + \beta_{6} - \beta_{3} + \beta_1 + 2) q^{15} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4}) q^{17} - \beta_{9} q^{19} + (\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{21} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{23} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{25} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}) q^{27} + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_1 + 2) q^{29} + ( - \beta_{9} - \beta_{8} - \beta_{7}) q^{31} + (\beta_{6} - \beta_{5} + \beta_{2} + 2 \beta_1 + 2) q^{33} + ( - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} - \beta_{2}) q^{35} + (\beta_{2} - 2) q^{37} + (2 \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{39} + ( - \beta_{9} - \beta_{4} + 2 \beta_{3}) q^{41} + ( - \beta_{9} + \beta_{4} - \beta_{3}) q^{43} + (\beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1) q^{45} + \beta_{2} q^{47} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1) q^{49} + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{2} + 2 \beta_1 + 4) q^{51} + ( - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{53} + (\beta_{9} - 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_1 - 2) q^{55} + ( - \beta_{6} + \beta_{5} - \beta_{2} - 2) q^{57} + (\beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{4}) q^{59} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{2} - \beta_1 - 2) q^{61} + (\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 2 \beta_1 - 4) q^{63} + ( - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{65} + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_1) q^{69} + (2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{71} + (\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{8} + 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{75} + (\beta_{9} - 3 \beta_{4} - 2 \beta_{3}) q^{77} + (2 \beta_{8} - 2 \beta_{7} - 2 \beta_1 - 4) q^{79} + ( - \beta_{8} + \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 3 \beta_1 + 5) q^{81} + (\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} + 2 \beta_1) q^{83} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{85} + (\beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{87} + ( - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{3}) q^{89} + (\beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 + 2) q^{91} + (\beta_{6} - \beta_{5} - \beta_{2} + 2 \beta_1 + 2) q^{93} + ( - 2 \beta_{8} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{95} + ( - \beta_{8} + \beta_{7} + 2 \beta_1 + 8) q^{97} + (\beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 8\nu^{6} + 8\nu^{4} - 18\nu^{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 12\nu^{6} + 44\nu^{4} + 50\nu^{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 10\nu^{7} - 26\nu^{5} - 16\nu^{3} + 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{9} - 11\nu^{7} - 35\nu^{5} - 33\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} - 5\nu^{8} - 52\nu^{7} - 52\nu^{6} - 152\nu^{5} - 152\nu^{4} - 146\nu^{3} - 146\nu^{2} - 27\nu - 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{9} + 5\nu^{8} - 52\nu^{7} + 52\nu^{6} - 152\nu^{5} + 152\nu^{4} - 146\nu^{3} + 146\nu^{2} - 27\nu + 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\nu^{9} - \nu^{8} - 33\nu^{7} - 11\nu^{6} - 107\nu^{5} - 35\nu^{4} - 117\nu^{3} - 35\nu^{2} - 22\nu - 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -3\nu^{9} + \nu^{8} - 33\nu^{7} + 11\nu^{6} - 107\nu^{5} + 35\nu^{4} - 117\nu^{3} + 35\nu^{2} - 22\nu + 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 6\nu^{9} + 65\nu^{7} + 205\nu^{5} + 217\nu^{3} + 41\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - 3\beta_{2} - \beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{9} + 5\beta_{8} + 5\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} + 6\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{8} + 4\beta_{7} + \beta_{6} - \beta_{5} + 9\beta_{2} + 2\beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -37\beta_{9} - 29\beta_{8} - 29\beta_{7} - 18\beta_{6} - 18\beta_{5} + 7\beta_{4} - 20\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 55\beta_{8} - 55\beta_{7} - 18\beta_{6} + 18\beta_{5} - 109\beta_{2} - 21\beta _1 - 274 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 230\beta_{9} + 175\beta_{8} + 175\beta_{7} + 128\beta_{6} + 128\beta_{5} - 32\beta_{4} + 84\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -179\beta_{8} + 179\beta_{7} + 64\beta_{6} - 64\beta_{5} + 337\beta_{2} + 63\beta _1 + 832 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -1437\beta_{9} - 1079\beta_{8} - 1079\beta_{7} - 844\beta_{6} - 844\beta_{5} + 173\beta_{4} - 430\beta_{3} ) / 4 \) 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
1.35685i
1.56822i
0.549054i
2.50630i
0.341517i
0.341517i
2.50630i
0.549054i
1.56822i
1.35685i
0 3.22659i 0 −0.589653 + 2.15692i 0 1.65488 0 −7.41088 0
129.2 0 2.14926i 0 1.82442 1.29287i 0 −2.12171 0 −1.61930 0
129.3 0 1.57201i 0 2.14575 + 0.629081i 0 4.19743 0 0.528799 0
129.4 0 0.949078i 0 0.108880 + 2.23342i 0 −3.85660 0 2.09925 0
129.5 0 0.773218i 0 −1.98940 1.02093i 0 1.12600 0 2.40213 0
129.6 0 0.773218i 0 −1.98940 + 1.02093i 0 1.12600 0 2.40213 0
129.7 0 0.949078i 0 0.108880 2.23342i 0 −3.85660 0 2.09925 0
129.8 0 1.57201i 0 2.14575 0.629081i 0 4.19743 0 0.528799 0
129.9 0 2.14926i 0 1.82442 + 1.29287i 0 −2.12171 0 −1.61930 0
129.10 0 3.22659i 0 −0.589653 2.15692i 0 1.65488 0 −7.41088 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.10
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.g 10
4.b odd 2 1 520.2.f.b yes 10
5.b even 2 1 1040.2.f.f 10
13.b even 2 1 1040.2.f.f 10
20.d odd 2 1 520.2.f.a 10
20.e even 4 2 2600.2.k.f 20
52.b odd 2 1 520.2.f.a 10
65.d even 2 1 inner 1040.2.f.g 10
260.g odd 2 1 520.2.f.b yes 10
260.p even 4 2 2600.2.k.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.f.a 10 20.d odd 2 1
520.2.f.a 10 52.b odd 2 1
520.2.f.b yes 10 4.b odd 2 1
520.2.f.b yes 10 260.g odd 2 1
1040.2.f.f 10 5.b even 2 1
1040.2.f.f 10 13.b even 2 1
1040.2.f.g 10 1.a even 1 1 trivial
1040.2.f.g 10 65.d even 2 1 inner
2600.2.k.f 20 20.e even 4 2
2600.2.k.f 20 260.p even 4 2

Hecke kernels

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

\( T_{3}^{10} + 19T_{3}^{8} + 112T_{3}^{6} + 256T_{3}^{4} + 224T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{5} - T_{7}^{4} - 20T_{7}^{3} + 16T_{7}^{2} + 64T_{7} - 64 \) Copy content Toggle raw display
\( T_{37}^{5} + 11T_{37}^{4} + 28T_{37}^{3} - 32T_{37}^{2} - 128T_{37} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 19 T^{8} + 112 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{10} - 3 T^{9} + 5 T^{8} - 12 T^{7} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{5} - T^{4} - 20 T^{3} + 16 T^{2} + 64 T - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 64 T^{8} + 1152 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} - 7 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 107 T^{8} + 4064 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$19$ \( T^{10} + 92 T^{8} + 2688 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{10} + 44 T^{8} + 608 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{5} - 4 T^{4} - 56 T^{3} + 320 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 108 T^{8} + 2496 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( (T^{5} + 11 T^{4} + 28 T^{3} - 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 208 T^{8} + 13824 T^{6} + \cdots + 4194304 \) Copy content Toggle raw display
$43$ \( T^{10} + 211 T^{8} + \cdots + 40246336 \) Copy content Toggle raw display
$47$ \( (T^{5} + T^{4} - 20 T^{3} - 16 T^{2} + 64 T + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 364 T^{8} + \cdots + 294191104 \) Copy content Toggle raw display
$59$ \( T^{10} + 316 T^{8} + \cdots + 59228416 \) Copy content Toggle raw display
$61$ \( (T^{5} + 8 T^{4} - 120 T^{3} - 672 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 8 T^{4} - 80 T^{3} - 704 T^{2} + \cdots + 10496)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 475 T^{8} + \cdots + 583319104 \) Copy content Toggle raw display
$73$ \( (T^{5} + 2 T^{4} - 104 T^{3} + 112 T^{2} + \cdots - 6752)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 14 T^{4} - 144 T^{3} + \cdots + 83968)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 2 T^{4} - 312 T^{3} + \cdots + 132736)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 524 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$97$ \( (T^{5} - 36 T^{4} + 416 T^{3} - 1552 T^{2} + \cdots + 4736)^{2} \) Copy content Toggle raw display
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