Properties

Label 1110.6.a.a
Level $1110$
Weight $6$
Character orbit 1110.a
Self dual yes
Analytic conductor $178.026$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(178.026039992\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 9q^{3} + 16q^{4} + 25q^{5} - 36q^{6} + 33q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} - 9q^{3} + 16q^{4} + 25q^{5} - 36q^{6} + 33q^{7} + 64q^{8} + 81q^{9} + 100q^{10} + 39q^{11} - 144q^{12} - 314q^{13} + 132q^{14} - 225q^{15} + 256q^{16} - 1949q^{17} + 324q^{18} + 922q^{19} + 400q^{20} - 297q^{21} + 156q^{22} + 3844q^{23} - 576q^{24} + 625q^{25} - 1256q^{26} - 729q^{27} + 528q^{28} - 8313q^{29} - 900q^{30} + 5985q^{31} + 1024q^{32} - 351q^{33} - 7796q^{34} + 825q^{35} + 1296q^{36} - 1369q^{37} + 3688q^{38} + 2826q^{39} + 1600q^{40} - 13607q^{41} - 1188q^{42} + 847q^{43} + 624q^{44} + 2025q^{45} + 15376q^{46} + 2904q^{47} - 2304q^{48} - 15718q^{49} + 2500q^{50} + 17541q^{51} - 5024q^{52} - 33851q^{53} - 2916q^{54} + 975q^{55} + 2112q^{56} - 8298q^{57} - 33252q^{58} - 2186q^{59} - 3600q^{60} + 19893q^{61} + 23940q^{62} + 2673q^{63} + 4096q^{64} - 7850q^{65} - 1404q^{66} - 18596q^{67} - 31184q^{68} - 34596q^{69} + 3300q^{70} - 17740q^{71} + 5184q^{72} - 44536q^{73} - 5476q^{74} - 5625q^{75} + 14752q^{76} + 1287q^{77} + 11304q^{78} + 79732q^{79} + 6400q^{80} + 6561q^{81} - 54428q^{82} + 36254q^{83} - 4752q^{84} - 48725q^{85} + 3388q^{86} + 74817q^{87} + 2496q^{88} - 57970q^{89} + 8100q^{90} - 10362q^{91} + 61504q^{92} - 53865q^{93} + 11616q^{94} + 23050q^{95} - 9216q^{96} + 85531q^{97} - 62872q^{98} + 3159q^{99} + O(q^{100}) \)

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
4.00000 −9.00000 16.0000 25.0000 −36.0000 33.0000 64.0000 81.0000 100.000
\(n\): 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 1110.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 33 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1110))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( 9 + T \)
$5$ \( -25 + T \)
$7$ \( -33 + T \)
$11$ \( -39 + T \)
$13$ \( 314 + T \)
$17$ \( 1949 + T \)
$19$ \( -922 + T \)
$23$ \( -3844 + T \)
$29$ \( 8313 + T \)
$31$ \( -5985 + T \)
$37$ \( 1369 + T \)
$41$ \( 13607 + T \)
$43$ \( -847 + T \)
$47$ \( -2904 + T \)
$53$ \( 33851 + T \)
$59$ \( 2186 + T \)
$61$ \( -19893 + T \)
$67$ \( 18596 + T \)
$71$ \( 17740 + T \)
$73$ \( 44536 + T \)
$79$ \( -79732 + T \)
$83$ \( -36254 + T \)
$89$ \( 57970 + T \)
$97$ \( -85531 + T \)
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