Properties

Label 1155.1.e.b
Level 1155
Weight 1
Character orbit 1155.e
Analytic conductor 0.576
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM disc. -35, -231, 165
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: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-35}, \sqrt{165})\)
Artin image size \(16\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.1634180625.2

$q$-expansion

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

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

Character Values

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

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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
1.00000i
1.00000i
0 1.00000i 1.00000 1.00000i 0 1.00000i 0 −1.00000 0
1154.2 0 1.00000i 1.00000 1.00000i 0 1.00000i 0 −1.00000 0
\(n\): 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
35.c Odd 1 CM by \(\Q(\sqrt{-35}) \) yes
165.d Even 1 RM by \(\Q(\sqrt{165}) \) 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
1155.e Odd 1 yes