Properties

Label 1155.2.l.b
Level 1155
Weight 2
Character orbit 1155.l
Analytic conductor 9.223
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.l (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \zeta_{8}^{2} \)
\(\beta_{2}\)\(=\)\( \zeta_{8}^{3} + \zeta_{8} \)
\(\beta_{3}\)\(=\)\( -\zeta_{8}^{3} + \zeta_{8} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{8}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\zeta_{8}^{2}\)\(=\)\(\beta_{1}\)
\(\zeta_{8}^{3}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−1.41421 1.00000 1.41421i 0 1.00000i −1.41421 + 2.00000i 1.00000i 2.82843 −1.00000 2.82843i 1.41421i
1121.2 −1.41421 1.00000 + 1.41421i 0 1.00000i −1.41421 2.00000i 1.00000i 2.82843 −1.00000 + 2.82843i 1.41421i
1121.3 1.41421 1.00000 1.41421i 0 1.00000i 1.41421 2.00000i 1.00000i −2.82843 −1.00000 2.82843i 1.41421i
1121.4 1.41421 1.00000 + 1.41421i 0 1.00000i 1.41421 + 2.00000i 1.00000i −2.82843 −1.00000 + 2.82843i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
33.d Even 1 yes

Hecke kernels

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

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{17}^{2} \) \(\mathstrut +\mathstrut 6 T_{17} \) \(\mathstrut +\mathstrut 1 \)