Properties

Label 1155.2.a.q
Level 1155
Weight 2
Character orbit 1155.a
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.44949
2.44949
−2.44949 1.00000 4.00000 1.00000 −2.44949 1.00000 −4.89898 1.00000 −2.44949
1.2 2.44949 1.00000 4.00000 1.00000 2.44949 1.00000 4.89898 1.00000 2.44949
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1155.2.a.q 2
3.b odd 2 1 3465.2.a.y 2
5.b even 2 1 5775.2.a.bh 2
7.b odd 2 1 8085.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.q 2 1.a even 1 1 trivial
3465.2.a.y 2 3.b odd 2 1
5775.2.a.bh 2 5.b even 2 1
8085.2.a.be 2 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{2} - 6 \)
\( T_{13}^{2} - 4 T_{13} - 2 \)
\( T_{17} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 - 4 T + 24 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 - 6 T + 31 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 61 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 4 T + 12 T^{2} - 124 T^{3} + 961 T^{4} \)
$37$ \( 1 - 4 T + 72 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 76 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 10 T + 105 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 12 T + 124 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 3 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 6 T + 121 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 14 T + 147 T^{2} + 854 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 8 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 16 T + 186 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T + 120 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T + 133 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 2 T + 189 T^{2} + 194 T^{3} + 9409 T^{4} \)
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