Properties

Label 1110.2.a.r
Level $1110$
Weight $2$
Character orbit 1110.a
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 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.r 2
3.b odd 2 1 3330.2.a.bc 2
4.b odd 2 1 8880.2.a.bp 2
5.b even 2 1 5550.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.r 2 1.a even 1 1 trivial
3330.2.a.bc 2 3.b odd 2 1
5550.2.a.bw 2 5.b even 2 1
8880.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\):

\( T_{7} + 1 \)
\( T_{11}^{2} + 4 T_{11} - 13 \)
\( T_{13}^{2} + T_{13} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -13 + 4 T + T^{2} \)
$13$ \( -38 + T + T^{2} \)
$17$ \( ( 5 + T )^{2} \)
$19$ \( -2 + 3 T + T^{2} \)
$23$ \( 16 + 9 T + T^{2} \)
$29$ \( -38 + T + T^{2} \)
$31$ \( -2 + 3 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -32 - 5 T + T^{2} \)
$43$ \( 4 + 13 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( -59 + 6 T + T^{2} \)
$59$ \( -52 + 8 T + T^{2} \)
$61$ \( -94 + 7 T + T^{2} \)
$67$ \( 64 - 18 T + T^{2} \)
$71$ \( -8 + 6 T + T^{2} \)
$73$ \( 16 + 9 T + T^{2} \)
$79$ \( -32 - 12 T + T^{2} \)
$83$ \( 86 + 19 T + T^{2} \)
$89$ \( -38 + T + T^{2} \)
$97$ \( 52 - 19 T + T^{2} \)
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