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 discriminant -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 \(2\), degree \(1\), minimal)

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: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 8q^{4} - 4q^{9} - 4q^{11} + 2q^{15} + 4q^{16} + 4q^{21} + 2q^{25} - 4q^{29} + 6q^{30} - 2q^{35} + 8q^{36} + 4q^{39} + 8q^{44} - 4q^{49} - 6q^{50} - 4q^{60} + 4q^{64} - 2q^{65} - 6q^{70} - 12q^{74} + 4q^{81} - 8q^{84} - 4q^{91} + 6q^{95} + 4q^{99} + O(q^{100}) \)

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 Parity Ord Mult Type
1.a even 1 1 trivial
231.h odd 2 1 CM by \(\Q(\sqrt{-231}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
33.d even 2 1 inner
35.c odd 2 1 inner
165.d even 2 1 inner
1155.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.1.e.c 4
3.b odd 2 1 1155.1.e.d yes 4
5.b even 2 1 inner 1155.1.e.c 4
7.b odd 2 1 inner 1155.1.e.c 4
11.b odd 2 1 1155.1.e.d yes 4
15.d odd 2 1 1155.1.e.d yes 4
21.c even 2 1 1155.1.e.d yes 4
33.d even 2 1 inner 1155.1.e.c 4
35.c odd 2 1 inner 1155.1.e.c 4
55.d odd 2 1 1155.1.e.d yes 4
77.b even 2 1 1155.1.e.d yes 4
105.g even 2 1 1155.1.e.d yes 4
165.d even 2 1 inner 1155.1.e.c 4
231.h odd 2 1 CM 1155.1.e.c 4
385.h even 2 1 1155.1.e.d yes 4
1155.e odd 2 1 inner 1155.1.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.1.e.c 4 1.a even 1 1 trivial
1155.1.e.c 4 5.b even 2 1 inner
1155.1.e.c 4 7.b odd 2 1 inner
1155.1.e.c 4 33.d even 2 1 inner
1155.1.e.c 4 35.c odd 2 1 inner
1155.1.e.c 4 165.d even 2 1 inner
1155.1.e.c 4 231.h odd 2 1 CM
1155.1.e.c 4 1155.e odd 2 1 inner
1155.1.e.d yes 4 3.b odd 2 1
1155.1.e.d yes 4 11.b odd 2 1
1155.1.e.d yes 4 15.d odd 2 1
1155.1.e.d yes 4 21.c even 2 1
1155.1.e.d yes 4 55.d odd 2 1
1155.1.e.d yes 4 77.b even 2 1
1155.1.e.d yes 4 105.g even 2 1
1155.1.e.d yes 4 385.h even 2 1

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$17$ \( ( 1 + T^{2} )^{4} \)
$19$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{2} )^{4} \)
$29$ \( ( 1 + T + T^{2} )^{4} \)
$31$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$37$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$41$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$43$ \( ( 1 + T^{2} )^{4} \)
$47$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$53$ \( ( 1 + T^{2} )^{4} \)
$59$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$61$ \( ( 1 + T^{2} )^{4} \)
$67$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$71$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$73$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$79$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$83$ \( ( 1 + T^{2} )^{4} \)
$89$ \( ( 1 + T^{2} )^{4} \)
$97$ \( ( 1 + T^{2} )^{4} \)
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