Properties

Label 1155.1.e.c
Level 1155
Weight 1
Character orbit 1155.e
Analytic conductor 0.576
Analytic rank 0
Dimension 4
Projective image \(D_{6}\)
CM disc. -231
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.576420089591\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.46690875.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{2} \) \( + \zeta_{12}^{3} q^{3} \) \( + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4} \) \( -\zeta_{12} q^{5} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6} \) \( -\zeta_{12}^{3} q^{7} \) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{2} \) \( + \zeta_{12}^{3} q^{3} \) \( + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4} \) \( -\zeta_{12} q^{5} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6} \) \( -\zeta_{12}^{3} q^{7} \) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{8} \) \(- q^{9}\) \( + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{10} \) \(- q^{11}\) \( + ( -\zeta_{12} - \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{12} \) \( -\zeta_{12}^{3} q^{13} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{14} \) \( -\zeta_{12}^{4} q^{15} \) \(+ q^{16}\) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{18} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{19} \) \( + ( \zeta_{12} + \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{20} \) \(+ q^{21}\) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{22} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{24} \) \( + \zeta_{12}^{2} q^{25} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{26} \) \( -\zeta_{12}^{3} q^{27} \) \( + ( \zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{28} \) \(- q^{29}\) \( + ( 1 + \zeta_{12}^{2} ) q^{30} \) \( -\zeta_{12}^{3} q^{33} \) \( + \zeta_{12}^{4} q^{35} \) \( + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{36} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{37} \) \( + ( -\zeta_{12} - 2 \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{38} \) \(+ q^{39}\) \( + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{40} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{42} \) \( + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{44} \) \( + \zeta_{12} q^{45} \) \( + \zeta_{12}^{3} q^{47} \) \( + \zeta_{12}^{3} q^{48} \) \(- q^{49}\) \( + ( -1 + \zeta_{12}^{4} ) q^{50} \) \( + ( \zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{52} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{54} \) \( + \zeta_{12} q^{55} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{56} \) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{57} \) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{58} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{59} \) \( + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{60} \) \( + \zeta_{12}^{3} q^{63} \) \(+ q^{64}\) \( + \zeta_{12}^{4} q^{65} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{66} \) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{67} \) \( + ( -1 - \zeta_{12}^{2} ) q^{70} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{72} \) \( + \zeta_{12}^{3} q^{73} \) \( + ( -2 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{74} \) \( + \zeta_{12}^{5} q^{75} \) \( + ( 2 \zeta_{12} - 2 \zeta_{12}^{5} ) q^{76} \) \( + \zeta_{12}^{3} q^{77} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{78} \) \( -\zeta_{12} q^{80} \) \(+ q^{81}\) \( + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{84} \) \( -\zeta_{12}^{3} q^{87} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{88} \) \( + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{90} \) \(- q^{91}\) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{94} \) \( + ( 1 + \zeta_{12}^{2} ) q^{95} \) \( + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{98} \) \(+ q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 8q^{36} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 8q^{44} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 6q^{50} \) \(\mathstrut -\mathstrut 4q^{60} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 6q^{70} \) \(\mathstrut -\mathstrut 12q^{74} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 8q^{84} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 6q^{95} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1154.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
1.73205i 1.00000i −2.00000 −0.866025 + 0.500000i −1.73205 1.00000i 1.73205i −1.00000 0.866025 + 1.50000i
1154.2 1.73205i 1.00000i −2.00000 0.866025 0.500000i 1.73205 1.00000i 1.73205i −1.00000 −0.866025 1.50000i
1154.3 1.73205i 1.00000i −2.00000 0.866025 + 0.500000i 1.73205 1.00000i 1.73205i −1.00000 −0.866025 + 1.50000i
1154.4 1.73205i 1.00000i −2.00000 −0.866025 0.500000i −1.73205 1.00000i 1.73205i −1.00000 0.866025 1.50000i
\(n\): 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
231.h Odd 1 CM by \(\Q(\sqrt{-231}) \) yes
5.b Even 1 yes
7.b Odd 1 yes
33.d Even 1 yes
35.c Odd 1 yes
165.d Even 1 yes
1155.e Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1155, [\chi])\):

\(T_{2}^{2} \) \(\mathstrut +\mathstrut 3 \)
\(T_{29} \) \(\mathstrut +\mathstrut 1 \)