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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

\(f(q)\) \(=\) \( q - 5q^{2} - q^{3} + 17q^{4} - 5q^{5} + 5q^{6} - 2q^{7} - 45q^{8} - 26q^{9} + O(q^{10}) \) \( q - 5q^{2} - q^{3} + 17q^{4} - 5q^{5} + 5q^{6} - 2q^{7} - 45q^{8} - 26q^{9} + 25q^{10} - 11q^{11} - 17q^{12} - 7q^{13} + 10q^{14} + 5q^{15} + 89q^{16} + 14q^{17} + 130q^{18} + 19q^{19} - 85q^{20} + 2q^{21} + 55q^{22} + 55q^{23} + 45q^{24} + 25q^{25} + 35q^{26} + 53q^{27} - 34q^{28} - 26q^{29} - 25q^{30} + 261q^{31} - 85q^{32} + 11q^{33} - 70q^{34} + 10q^{35} - 442q^{36} - 126q^{37} - 95q^{38} + 7q^{39} + 225q^{40} - 381q^{41} - 10q^{42} + 387q^{43} - 187q^{44} + 130q^{45} - 275q^{46} + 189q^{47} - 89q^{48} - 339q^{49} - 125q^{50} - 14q^{51} - 119q^{52} - 404q^{53} - 265q^{54} + 55q^{55} + 90q^{56} - 19q^{57} + 130q^{58} + 746q^{59} + 85q^{60} + 79q^{61} - 1305q^{62} + 52q^{63} - 287q^{64} + 35q^{65} - 55q^{66} + 537q^{67} + 238q^{68} - 55q^{69} - 50q^{70} - 824q^{71} + 1170q^{72} + 169q^{73} + 630q^{74} - 25q^{75} + 323q^{76} + 22q^{77} - 35q^{78} - 338q^{79} - 445q^{80} + 649q^{81} + 1905q^{82} + 601q^{83} + 34q^{84} - 70q^{85} - 1935q^{86} + 26q^{87} + 495q^{88} - 762q^{89} - 650q^{90} + 14q^{91} + 935q^{92} - 261q^{93} - 945q^{94} - 95q^{95} + 85q^{96} + 866q^{97} + 1695q^{98} + 286q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

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