Properties

Label 1045.4.a.a
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{2} - q^{3} + 17 q^{4} - 5 q^{5} + 5 q^{6} - 2 q^{7} - 45 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} - q^{3} + 17 q^{4} - 5 q^{5} + 5 q^{6} - 2 q^{7} - 45 q^{8} - 26 q^{9} + 25 q^{10} - 11 q^{11} - 17 q^{12} - 7 q^{13} + 10 q^{14} + 5 q^{15} + 89 q^{16} + 14 q^{17} + 130 q^{18} + 19 q^{19} - 85 q^{20} + 2 q^{21} + 55 q^{22} + 55 q^{23} + 45 q^{24} + 25 q^{25} + 35 q^{26} + 53 q^{27} - 34 q^{28} - 26 q^{29} - 25 q^{30} + 261 q^{31} - 85 q^{32} + 11 q^{33} - 70 q^{34} + 10 q^{35} - 442 q^{36} - 126 q^{37} - 95 q^{38} + 7 q^{39} + 225 q^{40} - 381 q^{41} - 10 q^{42} + 387 q^{43} - 187 q^{44} + 130 q^{45} - 275 q^{46} + 189 q^{47} - 89 q^{48} - 339 q^{49} - 125 q^{50} - 14 q^{51} - 119 q^{52} - 404 q^{53} - 265 q^{54} + 55 q^{55} + 90 q^{56} - 19 q^{57} + 130 q^{58} + 746 q^{59} + 85 q^{60} + 79 q^{61} - 1305 q^{62} + 52 q^{63} - 287 q^{64} + 35 q^{65} - 55 q^{66} + 537 q^{67} + 238 q^{68} - 55 q^{69} - 50 q^{70} - 824 q^{71} + 1170 q^{72} + 169 q^{73} + 630 q^{74} - 25 q^{75} + 323 q^{76} + 22 q^{77} - 35 q^{78} - 338 q^{79} - 445 q^{80} + 649 q^{81} + 1905 q^{82} + 601 q^{83} + 34 q^{84} - 70 q^{85} - 1935 q^{86} + 26 q^{87} + 495 q^{88} - 762 q^{89} - 650 q^{90} + 14 q^{91} + 935 q^{92} - 261 q^{93} - 945 q^{94} - 95 q^{95} + 85 q^{96} + 866 q^{97} + 1695 q^{98} + 286 q^{99}+O(q^{100}) \) 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
0
−5.00000 −1.00000 17.0000 −5.00000 5.00000 −2.00000 −45.0000 −26.0000 25.0000
\(n\): e.g. 2-40 or 990-1000
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 1045.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 7 \) Copy content Toggle raw display
$17$ \( T - 14 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T - 55 \) Copy content Toggle raw display
$29$ \( T + 26 \) Copy content Toggle raw display
$31$ \( T - 261 \) Copy content Toggle raw display
$37$ \( T + 126 \) Copy content Toggle raw display
$41$ \( T + 381 \) Copy content Toggle raw display
$43$ \( T - 387 \) Copy content Toggle raw display
$47$ \( T - 189 \) Copy content Toggle raw display
$53$ \( T + 404 \) Copy content Toggle raw display
$59$ \( T - 746 \) Copy content Toggle raw display
$61$ \( T - 79 \) Copy content Toggle raw display
$67$ \( T - 537 \) Copy content Toggle raw display
$71$ \( T + 824 \) Copy content Toggle raw display
$73$ \( T - 169 \) Copy content Toggle raw display
$79$ \( T + 338 \) Copy content Toggle raw display
$83$ \( T - 601 \) Copy content Toggle raw display
$89$ \( T + 762 \) Copy content Toggle raw display
$97$ \( T - 866 \) Copy content Toggle raw display
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