Properties

Label 1155.2.a.t
Level 1155
Weight 2
Character orbit 1155.a
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 3
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: \(3\)
Coefficient field: 3.3.316.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.81361
0.470683
2.34292
−1.81361 1.00000 1.28917 1.00000 −1.81361 −1.00000 1.28917 1.00000 −1.81361
1.2 0.470683 1.00000 −1.77846 1.00000 0.470683 −1.00000 −1.77846 1.00000 0.470683
1.3 2.34292 1.00000 3.48929 1.00000 2.34292 −1.00000 3.48929 1.00000 2.34292
\(n\): 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.t 3
3.b odd 2 1 3465.2.a.bb 3
5.b even 2 1 5775.2.a.bq 3
7.b odd 2 1 8085.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.t 3 1.a even 1 1 trivial
3465.2.a.bb 3 3.b odd 2 1
5775.2.a.bq 3 5.b even 2 1
8085.2.a.bl 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 2 \)
\( T_{13}^{3} - 8 T_{13}^{2} + 14 T_{13} + 4 \)
\( T_{17}^{3} - 8 T_{17}^{2} + 5 T_{17} + 46 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} - 2 T^{3} + 4 T^{4} - 4 T^{5} + 8 T^{6} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( ( 1 - T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 - 8 T + 53 T^{2} - 204 T^{3} + 689 T^{4} - 1352 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 8 T + 56 T^{2} - 226 T^{3} + 952 T^{4} - 2312 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 2 T + 26 T^{2} + 120 T^{3} + 494 T^{4} + 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 6 T + 74 T^{2} + 268 T^{3} + 1702 T^{4} + 3174 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 10 T + 116 T^{2} - 602 T^{3} + 3364 T^{4} - 8410 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 2 T + 79 T^{2} + 92 T^{3} + 2449 T^{4} + 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 8 T + 53 T^{2} - 300 T^{3} + 1961 T^{4} - 10952 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 4 T + 73 T^{2} - 396 T^{3} + 2993 T^{4} - 6724 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 86 T^{2} - 44 T^{3} + 3698 T^{4} + 79507 T^{6} \)
$47$ \( 1 - 2 T + 127 T^{2} - 156 T^{3} + 5969 T^{4} - 4418 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 152 T^{2} + 2 T^{3} + 8056 T^{4} + 148877 T^{6} \)
$59$ \( 1 + 12 T + 186 T^{2} + 1340 T^{3} + 10974 T^{4} + 41772 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 4 T + 132 T^{2} - 262 T^{3} + 8052 T^{4} - 14884 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 173 T^{2} - 16 T^{3} + 11591 T^{4} + 300763 T^{6} \)
$71$ \( 1 + 18 T + 255 T^{2} + 2284 T^{3} + 18105 T^{4} + 90738 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 14 T + 267 T^{2} - 2060 T^{3} + 19491 T^{4} - 74606 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 2 T - 9 T^{2} + 292 T^{3} - 711 T^{4} - 12482 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 6 T + 114 T^{2} - 1288 T^{3} + 9462 T^{4} - 41334 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 10 T + 200 T^{2} + 1778 T^{3} + 17800 T^{4} + 79210 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 10 T + 84 T^{2} - 118 T^{3} + 8148 T^{4} - 94090 T^{5} + 912673 T^{6} \)
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