Properties

Label 1045.2.a.h
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_1) q^{8} + (\beta_{6} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_1) q^{8} + (\beta_{6} - \beta_{3} + 1) q^{9} - \beta_1 q^{10} + q^{11} + (\beta_{6} + 2 \beta_1 + 2) q^{12} + (2 \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{14} - \beta_{5} q^{15} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{18} - q^{19} + ( - \beta_{2} - 1) q^{20} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{21} + \beta_1 q^{22} + ( - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} + 4) q^{24} + q^{25} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{26} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{27} + (2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_1 + 1) q^{28} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{29} + (\beta_{6} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{30} + (\beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{31} + (\beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 1) q^{32} + \beta_{5} q^{33} + ( - 3 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{34} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{35} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{36} + ( - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 4) q^{37} - \beta_1 q^{38} + (2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 5) q^{39} + ( - \beta_{6} - \beta_{5} - \beta_{3} - \beta_1) q^{40} + (2 \beta_{6} + \beta_{4} - \beta_1 + 3) q^{41} + (3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 + 5) q^{42} + (\beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{43} + (\beta_{2} + 1) q^{44} + ( - \beta_{6} + \beta_{3} - 1) q^{45} + ( - 2 \beta_{6} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_1 + 4) q^{46} + (3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_1 - 1) q^{47} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{48} + (3 \beta_{6} - 2 \beta_{5} - \beta_{2} + 3) q^{49} + \beta_1 q^{50} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{51} + (2 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{52} + (\beta_{4} + 4 \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{53} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{54} - q^{55} + (\beta_{6} - 4 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 7) q^{56} - \beta_{5} q^{57} + (2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 4) q^{58} + (2 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{59} + ( - \beta_{6} - 2 \beta_1 - 2) q^{60} + (3 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 4 \beta_1 + 4) q^{61} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 6) q^{62} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{63} + (2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{64} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{65} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{66} + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1) q^{67} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1) q^{68} + ( - 4 \beta_{6} - 4 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{69} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{70} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{71} + ( - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 3 \beta_1) q^{72} + (3 \beta_{6} - 3 \beta_{5} - \beta_{3} + 2 \beta_{2} + 6) q^{73} + ( - 4 \beta_{6} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 7 \beta_1 - 1) q^{74} + \beta_{5} q^{75} + ( - \beta_{2} - 1) q^{76} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{77} + ( - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} + 5 \beta_1 - 5) q^{78} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{79} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{80} + ( - 4 \beta_{6} + 3 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 6) q^{81} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} + 4 \beta_1 - 2) q^{82} + (2 \beta_{6} - \beta_{4} - \beta_{2} - \beta_1 + 7) q^{83} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{84} + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{85} + ( - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{86} + (3 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{87} + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_1) q^{88} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{89} + (\beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{90} + ( - 3 \beta_{6} - 5 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{91} + ( - \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 6) q^{92} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{93} + ( - \beta_{6} + \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 1) q^{94} + q^{95} + (\beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - \beta_1 - 5) q^{96} + ( - 2 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{97} + (\beta_{6} - 4 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{98} + (\beta_{6} - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} + 11\nu^{4} + 2\nu^{3} - 33\nu^{2} - 10\nu + 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{6} + \nu^{5} + 20\nu^{4} - 5\nu^{3} - 52\nu^{2} - \nu + 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{6} + \nu^{5} + 30\nu^{4} - 3\nu^{3} - 78\nu^{2} - 10\nu + 41 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\nu^{6} - \nu^{5} - 41\nu^{4} + 2\nu^{3} + 111\nu^{2} + 15\nu - 62 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{6} + 7\beta_{5} + 3\beta_{4} + 9\beta_{3} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} - 9\beta_{5} + 11\beta_{4} + 12\beta_{3} + 44\beta_{2} + 11\beta _1 + 76 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.40300
−1.54354
−1.08185
0.719047
0.745312
1.97792
2.58611
−2.40300 −1.17935 3.77440 −1.00000 2.83397 −2.15598 −4.26389 −1.60914 2.40300
1.2 −1.54354 2.58921 0.382516 −1.00000 −3.99655 −3.41636 2.49665 3.70400 1.54354
1.3 −1.08185 −0.870874 −0.829591 −1.00000 0.942160 4.02863 3.06121 −2.24158 1.08185
1.4 0.719047 0.163114 −1.48297 −1.00000 0.117287 1.05678 −2.50442 −2.97339 −0.719047
1.5 0.745312 −2.05058 −1.44451 −1.00000 −1.52832 −3.86782 −2.56724 1.20489 −0.745312
1.6 1.97792 2.65412 1.91218 −1.00000 5.24965 1.06653 −0.173707 4.04436 −1.97792
1.7 2.58611 1.69436 4.68797 −1.00000 4.38180 2.28823 6.95140 −0.129144 −2.58611
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.a.h 7
3.b odd 2 1 9405.2.a.bd 7
5.b even 2 1 5225.2.a.m 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.h 7 1.a even 1 1 trivial
5225.2.a.m 7 5.b even 2 1
9405.2.a.bd 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 8T_{2}^{4} + 27T_{2}^{3} - 16T_{2}^{2} - 18T_{2} + 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 10 T^{5} + 8 T^{4} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( T^{7} - 3 T^{6} - 7 T^{5} + 20 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + T^{6} - 27 T^{5} - 18 T^{4} + \cdots + 296 \) Copy content Toggle raw display
$11$ \( (T - 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} - 49 T^{5} - 55 T^{4} + \cdots - 1184 \) Copy content Toggle raw display
$17$ \( T^{7} - T^{6} - 61 T^{5} + 141 T^{4} + \cdots + 1096 \) Copy content Toggle raw display
$19$ \( (T + 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + 8 T^{6} - 38 T^{5} + \cdots - 7724 \) Copy content Toggle raw display
$29$ \( T^{7} - 11 T^{6} + T^{5} + 253 T^{4} + \cdots - 368 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} - 83 T^{5} + \cdots + 8300 \) Copy content Toggle raw display
$37$ \( T^{7} + 17 T^{6} - 10 T^{5} + \cdots - 20896 \) Copy content Toggle raw display
$41$ \( T^{7} - 17 T^{6} + 80 T^{5} + \cdots - 1328 \) Copy content Toggle raw display
$43$ \( T^{7} + 3 T^{6} - 159 T^{5} + \cdots + 2872 \) Copy content Toggle raw display
$47$ \( T^{7} - 14 T^{6} - 82 T^{5} + \cdots + 35972 \) Copy content Toggle raw display
$53$ \( T^{7} - 7 T^{6} - 263 T^{5} + \cdots + 1211488 \) Copy content Toggle raw display
$59$ \( T^{7} - 35 T^{6} + 357 T^{5} + \cdots - 106388 \) Copy content Toggle raw display
$61$ \( T^{7} - 17 T^{6} - 103 T^{5} + \cdots + 1052 \) Copy content Toggle raw display
$67$ \( T^{7} - 4 T^{6} - 133 T^{5} + \cdots - 1748 \) Copy content Toggle raw display
$71$ \( T^{7} - 10 T^{6} - 110 T^{5} + \cdots + 14492 \) Copy content Toggle raw display
$73$ \( T^{7} - 22 T^{6} + 69 T^{5} + \cdots + 85096 \) Copy content Toggle raw display
$79$ \( T^{7} - 11 T^{6} - 106 T^{5} + \cdots + 19664 \) Copy content Toggle raw display
$83$ \( T^{7} - 39 T^{6} + 533 T^{5} + \cdots + 2216 \) Copy content Toggle raw display
$89$ \( T^{7} - 18 T^{6} - 81 T^{5} + \cdots - 20716 \) Copy content Toggle raw display
$97$ \( T^{7} + 4 T^{6} - 415 T^{5} + \cdots - 6460112 \) Copy content Toggle raw display
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