Properties

Label 1110.4.a.a
Level $1110$
Weight $4$
Character orbit 1110.a
Self dual yes
Analytic conductor $65.492$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(65.4921201064\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} + q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} + q^{7} - 8q^{8} + 9q^{9} + 10q^{10} - 47q^{11} + 12q^{12} + 35q^{13} - 2q^{14} - 15q^{15} + 16q^{16} + 55q^{17} - 18q^{18} - 125q^{19} - 20q^{20} + 3q^{21} + 94q^{22} + 213q^{23} - 24q^{24} + 25q^{25} - 70q^{26} + 27q^{27} + 4q^{28} + 6q^{29} + 30q^{30} - 98q^{31} - 32q^{32} - 141q^{33} - 110q^{34} - 5q^{35} + 36q^{36} - 37q^{37} + 250q^{38} + 105q^{39} + 40q^{40} - 40q^{41} - 6q^{42} - 188q^{43} - 188q^{44} - 45q^{45} - 426q^{46} + 304q^{47} + 48q^{48} - 342q^{49} - 50q^{50} + 165q^{51} + 140q^{52} + 341q^{53} - 54q^{54} + 235q^{55} - 8q^{56} - 375q^{57} - 12q^{58} - 518q^{59} - 60q^{60} - 382q^{61} + 196q^{62} + 9q^{63} + 64q^{64} - 175q^{65} + 282q^{66} - 578q^{67} + 220q^{68} + 639q^{69} + 10q^{70} + 882q^{71} - 72q^{72} - 713q^{73} + 74q^{74} + 75q^{75} - 500q^{76} - 47q^{77} - 210q^{78} + 288q^{79} - 80q^{80} + 81q^{81} + 80q^{82} - 849q^{83} + 12q^{84} - 275q^{85} + 376q^{86} + 18q^{87} + 376q^{88} - 1263q^{89} + 90q^{90} + 35q^{91} + 852q^{92} - 294q^{93} - 608q^{94} + 625q^{95} - 96q^{96} - 1756q^{97} + 684q^{98} - 423q^{99} + O(q^{100}) \)

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
−2.00000 3.00000 4.00000 −5.00000 −6.00000 1.00000 −8.00000 9.00000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

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 1110.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.4.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( 5 + T \)
$7$ \( -1 + T \)
$11$ \( 47 + T \)
$13$ \( -35 + T \)
$17$ \( -55 + T \)
$19$ \( 125 + T \)
$23$ \( -213 + T \)
$29$ \( -6 + T \)
$31$ \( 98 + T \)
$37$ \( 37 + T \)
$41$ \( 40 + T \)
$43$ \( 188 + T \)
$47$ \( -304 + T \)
$53$ \( -341 + T \)
$59$ \( 518 + T \)
$61$ \( 382 + T \)
$67$ \( 578 + T \)
$71$ \( -882 + T \)
$73$ \( 713 + T \)
$79$ \( -288 + T \)
$83$ \( 849 + T \)
$89$ \( 1263 + T \)
$97$ \( 1756 + T \)
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