Properties

Label 1155.2.i.a
Level 1155
Weight 2
Character orbit 1155.i
Analytic conductor 9.223
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} - \beta_{5} ) q^{2} \) \( + \beta_{4} q^{3} \) \( + ( -2 - \beta_{7} ) q^{4} \) \( -\beta_{4} q^{5} \) \( + ( \beta_{3} - \beta_{6} ) q^{6} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{7} \) \( + ( -\beta_{1} + 3 \beta_{5} ) q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} - \beta_{5} ) q^{2} \) \( + \beta_{4} q^{3} \) \( + ( -2 - \beta_{7} ) q^{4} \) \( -\beta_{4} q^{5} \) \( + ( \beta_{3} - \beta_{6} ) q^{6} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{7} \) \( + ( -\beta_{1} + 3 \beta_{5} ) q^{8} \) \(- q^{9}\) \( + ( -\beta_{3} + \beta_{6} ) q^{10} \) \( + ( -2 \beta_{1} + \beta_{5} ) q^{11} \) \( + ( -\beta_{2} - 2 \beta_{4} ) q^{12} \) \( + ( 2 - \beta_{2} - 4 \beta_{4} ) q^{14} \) \(+ q^{15}\) \( + ( 4 + \beta_{7} ) q^{16} \) \( + ( -2 \beta_{3} + 3 \beta_{6} ) q^{17} \) \( + ( -\beta_{1} + \beta_{5} ) q^{18} \) \( + ( -4 \beta_{3} + \beta_{6} ) q^{19} \) \( + ( \beta_{2} + 2 \beta_{4} ) q^{20} \) \( + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{21} \) \( + ( 6 + \beta_{7} ) q^{22} \) \( + ( -1 - 3 \beta_{7} ) q^{23} \) \( + ( -3 \beta_{3} + \beta_{6} ) q^{24} \) \(- q^{25}\) \( -\beta_{4} q^{27} \) \( + ( -5 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{28} \) \( + ( -\beta_{1} - 2 \beta_{5} ) q^{29} \) \( + ( \beta_{1} - \beta_{5} ) q^{30} \) \( -2 \beta_{2} q^{31} \) \( + ( 3 \beta_{1} - \beta_{5} ) q^{32} \) \( + ( -\beta_{3} + 2 \beta_{6} ) q^{33} \) \( + ( 2 \beta_{2} + 10 \beta_{4} ) q^{34} \) \( + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{35} \) \( + ( 2 + \beta_{7} ) q^{36} \) \( + ( 4 - 2 \beta_{7} ) q^{37} \) \( + ( 4 \beta_{2} + 10 \beta_{4} ) q^{38} \) \( + ( 3 \beta_{3} - \beta_{6} ) q^{40} \) \( + ( -2 \beta_{3} + 6 \beta_{6} ) q^{41} \) \( + ( 4 + 2 \beta_{4} + \beta_{7} ) q^{42} \) \( + ( -3 \beta_{1} + 2 \beta_{5} ) q^{43} \) \( + ( 3 \beta_{1} - 7 \beta_{5} ) q^{44} \) \( + \beta_{4} q^{45} \) \( + ( -4 \beta_{1} + 10 \beta_{5} ) q^{46} \) \( + ( 2 \beta_{2} - 6 \beta_{4} ) q^{47} \) \( + ( \beta_{2} + 4 \beta_{4} ) q^{48} \) \( + ( 1 + 4 \beta_{4} + 2 \beta_{7} ) q^{49} \) \( + ( -\beta_{1} + \beta_{5} ) q^{50} \) \( + ( 3 \beta_{1} - 2 \beta_{5} ) q^{51} \) \( + ( -5 - \beta_{7} ) q^{53} \) \( + ( -\beta_{3} + \beta_{6} ) q^{54} \) \( + ( \beta_{3} - 2 \beta_{6} ) q^{55} \) \( + ( 3 \beta_{2} + 8 \beta_{4} - 2 \beta_{7} ) q^{56} \) \( + ( \beta_{1} - 4 \beta_{5} ) q^{57} \) \( + ( -2 - 2 \beta_{7} ) q^{58} \) \( + ( -\beta_{2} - 5 \beta_{4} ) q^{59} \) \( + ( -2 - \beta_{7} ) q^{60} \) \( + ( -2 \beta_{3} - 3 \beta_{6} ) q^{61} \) \( + ( -6 \beta_{3} + 2 \beta_{6} ) q^{62} \) \( + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{63} \) \( + \beta_{7} q^{64} \) \( + ( \beta_{2} + 6 \beta_{4} ) q^{66} \) \( + ( 2 + 2 \beta_{7} ) q^{67} \) \( + ( 12 \beta_{3} - 6 \beta_{6} ) q^{68} \) \( + ( -3 \beta_{2} - \beta_{4} ) q^{69} \) \( + ( -4 - 2 \beta_{4} - \beta_{7} ) q^{70} \) \( + ( 10 + 2 \beta_{7} ) q^{71} \) \( + ( \beta_{1} - 3 \beta_{5} ) q^{72} \) \( + 4 \beta_{3} q^{73} \) \( + ( 2 \beta_{1} + 2 \beta_{5} ) q^{74} \) \( -\beta_{4} q^{75} \) \( + ( 14 \beta_{3} - 12 \beta_{6} ) q^{76} \) \( + ( -5 + \beta_{2} + 6 \beta_{4} + \beta_{7} ) q^{77} \) \( + ( -4 \beta_{1} + 6 \beta_{5} ) q^{79} \) \( + ( -\beta_{2} - 4 \beta_{4} ) q^{80} \) \(+ q^{81}\) \( + ( 2 \beta_{2} + 16 \beta_{4} ) q^{82} \) \( + ( -4 \beta_{3} - \beta_{6} ) q^{83} \) \( + ( 3 \beta_{1} + 2 \beta_{3} - 5 \beta_{5} ) q^{84} \) \( + ( -3 \beta_{1} + 2 \beta_{5} ) q^{85} \) \( + ( 10 + 2 \beta_{7} ) q^{86} \) \( + ( 2 \beta_{3} + \beta_{6} ) q^{87} \) \( + ( -8 - 5 \beta_{7} ) q^{88} \) \( + ( -\beta_{2} + 5 \beta_{4} ) q^{89} \) \( + ( \beta_{3} - \beta_{6} ) q^{90} \) \( + ( 26 + 4 \beta_{7} ) q^{92} \) \( + 2 \beta_{7} q^{93} \) \( + 4 \beta_{6} q^{94} \) \( + ( -\beta_{1} + 4 \beta_{5} ) q^{95} \) \( + ( \beta_{3} - 3 \beta_{6} ) q^{96} \) \( + ( \beta_{2} - 5 \beta_{4} ) q^{97} \) \( + ( 3 \beta_{1} + 4 \beta_{3} - 7 \beta_{5} - 4 \beta_{6} ) q^{98} \) \( + ( 2 \beta_{1} - \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 28q^{16} \) \(\mathstrut +\mathstrut 44q^{22} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 40q^{37} \) \(\mathstrut +\mathstrut 28q^{42} \) \(\mathstrut -\mathstrut 36q^{53} \) \(\mathstrut +\mathstrut 8q^{56} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut -\mathstrut 12q^{60} \) \(\mathstrut -\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 28q^{70} \) \(\mathstrut +\mathstrut 72q^{71} \) \(\mathstrut -\mathstrut 44q^{77} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut 72q^{86} \) \(\mathstrut -\mathstrut 44q^{88} \) \(\mathstrut +\mathstrut 192q^{92} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(5\) \(x^{6}\mathstrut +\mathstrut \) \(16\) \(x^{4}\mathstrut +\mathstrut \) \(45\) \(x^{2}\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27 \)\()/36\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu \)\()/54\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu \)\()/216\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153 \)\()/72\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)\()/2\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\)
\(\nu^{4}\)\(=\)\((\)\(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{7}\)\(=\)\((\)\(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\) \(\beta_{4}\mathstrut -\mathstrut \) \(48\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\) \(\beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.396143 1.68614i
0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 1.18614i
1.26217 + 1.18614i
−1.26217 1.18614i
0.396143 1.68614i
−0.396143 + 1.68614i
2.52434i 1.00000i −4.37228 1.00000i −2.52434 −2.52434 + 0.792287i 5.98844i −1.00000 2.52434
76.2 2.52434i 1.00000i −4.37228 1.00000i 2.52434 2.52434 + 0.792287i 5.98844i −1.00000 −2.52434
76.3 0.792287i 1.00000i 1.37228 1.00000i −0.792287 −0.792287 + 2.52434i 2.67181i −1.00000 0.792287
76.4 0.792287i 1.00000i 1.37228 1.00000i 0.792287 0.792287 + 2.52434i 2.67181i −1.00000 −0.792287
76.5 0.792287i 1.00000i 1.37228 1.00000i 0.792287 0.792287 2.52434i 2.67181i −1.00000 −0.792287
76.6 0.792287i 1.00000i 1.37228 1.00000i −0.792287 −0.792287 2.52434i 2.67181i −1.00000 0.792287
76.7 2.52434i 1.00000i −4.37228 1.00000i 2.52434 2.52434 0.792287i 5.98844i −1.00000 −2.52434
76.8 2.52434i 1.00000i −4.37228 1.00000i −2.52434 −2.52434 0.792287i 5.98844i −1.00000 2.52434
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 yes
11.b Odd 1 yes
77.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 7 T_{2}^{2} \) \(\mathstrut +\mathstrut 4 \) acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\).