Properties

Label 1155.2.c.a
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 −2.00000 1.00000i −1.00000 1.00000i 3.00000i −1.00000 −1.00000 + 2.00000i
694.2 1.00000i 1.00000i 1.00000 −2.00000 + 1.00000i −1.00000 1.00000i 3.00000i −1.00000 −1.00000 2.00000i
\(n\): 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
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 \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\(T_{2}^{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{13}^{2} \) \(\mathstrut +\mathstrut 16 \)