Properties

Label 1110.2.a.q
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: \(\Q(\sqrt{113}) \)
Defining polynomial: \(x^{2} - x - 28\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{113})\). We also show the integral \(q\)-expansion of the trace form.

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

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
5.81507
−4.81507
−1.00000 1.00000 1.00000 1.00000 −1.00000 3.00000 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 3.00000 −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.q 2
3.b odd 2 1 3330.2.a.bf 2
4.b odd 2 1 8880.2.a.bh 2
5.b even 2 1 5550.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.q 2 1.a even 1 1 trivial
3330.2.a.bf 2 3.b odd 2 1
5550.2.a.bx 2 5.b even 2 1
8880.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\):

\( T_{7} - 3 \)
\( T_{11} + 1 \)
\( T_{13}^{2} - 3 T_{13} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -26 - 3 T + T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( -22 - 5 T + T^{2} \)
$23$ \( -28 - T + T^{2} \)
$29$ \( -26 + 3 T + T^{2} \)
$31$ \( -26 - 3 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -28 - T + T^{2} \)
$43$ \( -28 - T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( -113 + T^{2} \)
$59$ \( ( -2 + T )^{2} \)
$61$ \( -26 - 3 T + T^{2} \)
$67$ \( -104 + 6 T + T^{2} \)
$71$ \( -112 + 2 T + T^{2} \)
$73$ \( -28 + T + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( -26 - 3 T + T^{2} \)
$89$ \( 14 - 13 T + T^{2} \)
$97$ \( -28 - T + T^{2} \)
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