Properties

Label 1110.2.a.p
Level $1110$
Weight $2$
Character orbit 1110.a
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.37228
−2.37228
−1.00000 −1.00000 1.00000 1.00000 1.00000 −3.37228 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 2.37228 −1.00000 1.00000 −1.00000
\(n\): 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 1110.2.a.p 2
3.b odd 2 1 3330.2.a.be 2
4.b odd 2 1 8880.2.a.br 2
5.b even 2 1 5550.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.p 2 1.a even 1 1 trivial
3330.2.a.be 2 3.b odd 2 1
5550.2.a.ca 2 5.b even 2 1
8880.2.a.br 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\):

\( T_{7}^{2} + T_{7} - 8 \)
\( T_{11}^{2} - 3 T_{11} - 6 \)
\( T_{13}^{2} + 3 T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( -6 - 3 T + T^{2} \)
$13$ \( -6 + 3 T + T^{2} \)
$17$ \( -6 - 3 T + T^{2} \)
$19$ \( -6 - 3 T + T^{2} \)
$23$ \( -8 - T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -24 + 6 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( -24 - 6 T + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -32 - 2 T + T^{2} \)
$53$ \( -2 - 5 T + T^{2} \)
$59$ \( 48 - 18 T + T^{2} \)
$61$ \( -24 - 6 T + T^{2} \)
$67$ \( -24 + 6 T + T^{2} \)
$71$ \( 16 - 14 T + T^{2} \)
$73$ \( -54 - 9 T + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 4 - 7 T + T^{2} \)
$89$ \( 102 - 21 T + T^{2} \)
$97$ \( -128 - 4 T + T^{2} \)
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