Properties

Label 1155.2.c.d
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 6
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.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut +\mathstrut \) \(2\) \(x^{4}\mathstrut +\mathstrut \) \(2\) \(x^{3}\mathstrut +\mathstrut \) \(4\) \(x^{2}\mathstrut -\mathstrut \) \(4\) \(x\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(7\)
\(\nu^{5}\)\(=\)\(-\)\(8\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(9\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
1.45161 + 1.45161i
0.403032 0.403032i
−0.854638 + 0.854638i
−0.854638 0.854638i
0.403032 + 0.403032i
1.45161 1.45161i
2.21432i 1.00000i −2.90321 0.311108 + 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 0.688892i
694.2 1.67513i 1.00000i −0.806063 −1.48119 + 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 + 2.48119i
694.3 0.539189i 1.00000i 1.70928 2.17009 + 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 1.17009i
694.4 0.539189i 1.00000i 1.70928 2.17009 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 + 1.17009i
694.5 1.67513i 1.00000i −0.806063 −1.48119 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 2.48119i
694.6 2.21432i 1.00000i −2.90321 0.311108 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 + 0.688892i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

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

\(T_{2}^{6} \) \(\mathstrut +\mathstrut 8 T_{2}^{4} \) \(\mathstrut +\mathstrut 16 T_{2}^{2} \) \(\mathstrut +\mathstrut 4 \)
\(T_{13}^{6} \) \(\mathstrut +\mathstrut 28 T_{13}^{4} \) \(\mathstrut +\mathstrut 164 T_{13}^{2} \) \(\mathstrut +\mathstrut 100 \)