Properties

Label 1155.2.a.u
Level 1155
Weight 2
Character orbit 1155.a
Self dual Yes
Analytic conductor 9.223
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: 4.4.13448.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(7\) \(x^{2}\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)

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
−2.58874
−0.546295
0.546295
2.58874
−2.58874 1.00000 4.70156 −1.00000 −2.58874 −1.00000 −6.99364 1.00000 2.58874
1.2 −0.546295 1.00000 −1.70156 −1.00000 −0.546295 −1.00000 2.02214 1.00000 0.546295
1.3 0.546295 1.00000 −1.70156 −1.00000 0.546295 −1.00000 −2.02214 1.00000 −0.546295
1.4 2.58874 1.00000 4.70156 −1.00000 2.58874 −1.00000 6.99364 1.00000 −2.58874
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{2}^{4} \) \(\mathstrut -\mathstrut 7 T_{2}^{2} \) \(\mathstrut +\mathstrut 2 \)
\(T_{13}^{4} \) \(\mathstrut -\mathstrut 8 T_{13}^{3} \) \(\mathstrut -\mathstrut 2 T_{13}^{2} \) \(\mathstrut +\mathstrut 72 T_{13} \) \(\mathstrut +\mathstrut 40 \)
\(T_{17}^{2} \) \(\mathstrut +\mathstrut T_{17} \) \(\mathstrut -\mathstrut 10 \)