Properties

Label 1155.1.ca.a
Level 1155
Weight 1
Character orbit 1155.ca
Analytic conductor 0.576
Analytic rank 0
Dimension 8
Projective image \(D_{10}\)
CM disc. -35
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.ca (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.576420089591\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{10}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{10} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{20}^{5} q^{3} \) \( + \zeta_{20}^{4} q^{4} \) \( -\zeta_{20}^{3} q^{5} \) \( + \zeta_{20}^{9} q^{7} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + \zeta_{20}^{5} q^{3} \) \( + \zeta_{20}^{4} q^{4} \) \( -\zeta_{20}^{3} q^{5} \) \( + \zeta_{20}^{9} q^{7} \) \(- q^{9}\) \( -\zeta_{20}^{2} q^{11} \) \( + \zeta_{20}^{9} q^{12} \) \( + ( -\zeta_{20} + \zeta_{20}^{3} ) q^{13} \) \( -\zeta_{20}^{8} q^{15} \) \( + \zeta_{20}^{8} q^{16} \) \( + ( -\zeta_{20} + \zeta_{20}^{5} ) q^{17} \) \( -\zeta_{20}^{7} q^{20} \) \( -\zeta_{20}^{4} q^{21} \) \( + \zeta_{20}^{6} q^{25} \) \( -\zeta_{20}^{5} q^{27} \) \( -\zeta_{20}^{3} q^{28} \) \( + ( \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{29} \) \( -\zeta_{20}^{7} q^{33} \) \( + \zeta_{20}^{2} q^{35} \) \( -\zeta_{20}^{4} q^{36} \) \( + ( -\zeta_{20}^{6} + \zeta_{20}^{8} ) q^{39} \) \( -\zeta_{20}^{6} q^{44} \) \( + \zeta_{20}^{3} q^{45} \) \( + ( \zeta_{20}^{3} - \zeta_{20}^{9} ) q^{47} \) \( -\zeta_{20}^{3} q^{48} \) \( -\zeta_{20}^{8} q^{49} \) \( + ( -1 - \zeta_{20}^{6} ) q^{51} \) \( + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{52} \) \( + \zeta_{20}^{5} q^{55} \) \( + \zeta_{20}^{2} q^{60} \) \( -\zeta_{20}^{9} q^{63} \) \( -\zeta_{20}^{2} q^{64} \) \( + ( \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{65} \) \( + ( -\zeta_{20}^{5} + \zeta_{20}^{9} ) q^{68} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{4} ) q^{71} \) \( + ( \zeta_{20} - \zeta_{20}^{7} ) q^{73} \) \( -\zeta_{20} q^{75} \) \( + \zeta_{20} q^{77} \) \( + ( \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{79} \) \( + \zeta_{20} q^{80} \) \(+ q^{81}\) \( + ( \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{83} \) \( -\zeta_{20}^{8} q^{84} \) \( + ( \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{85} \) \( + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{87} \) \( + ( 1 - \zeta_{20}^{2} ) q^{91} \) \( + ( \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{97} \) \( + \zeta_{20}^{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 10q^{51} \) \(\mathstrut +\mathstrut 2q^{60} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-\zeta_{20}^{4}\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
314.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0 1.00000i −0.809017 + 0.587785i −0.951057 0.309017i 0 0.587785 + 0.809017i 0 −1.00000 0
314.2 0 1.00000i −0.809017 + 0.587785i 0.951057 + 0.309017i 0 −0.587785 0.809017i 0 −1.00000 0
524.1 0 1.00000i 0.309017 + 0.951057i 0.587785 + 0.809017i 0 0.951057 0.309017i 0 −1.00000 0
524.2 0 1.00000i 0.309017 + 0.951057i −0.587785 0.809017i 0 −0.951057 + 0.309017i 0 −1.00000 0
629.1 0 1.00000i −0.809017 0.587785i 0.951057 0.309017i 0 −0.587785 + 0.809017i 0 −1.00000 0
629.2 0 1.00000i −0.809017 0.587785i −0.951057 + 0.309017i 0 0.587785 0.809017i 0 −1.00000 0
734.1 0 1.00000i 0.309017 0.951057i −0.587785 + 0.809017i 0 −0.951057 0.309017i 0 −1.00000 0
734.2 0 1.00000i 0.309017 0.951057i 0.587785 0.809017i 0 0.951057 + 0.309017i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 734.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
35.c Odd 1 CM by \(\Q(\sqrt{-35}) \) yes
5.b Even 1 yes
7.b Odd 1 yes
33.f Even 1 yes
165.r Even 1 yes
231.r Odd 1 yes
1155.ca Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{29}^{4} \) \(\mathstrut -\mathstrut 2 T_{29}^{3} \) \(\mathstrut +\mathstrut 4 T_{29}^{2} \) \(\mathstrut -\mathstrut 3 T_{29} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(1155, [\chi])\).