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

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-35}, \sqrt{165})\)
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.

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

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
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 Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
165.d even 2 1 RM by \(\Q(\sqrt{165}) \)
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
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.b yes 2
3.b odd 2 1 1155.1.e.a 2
5.b even 2 1 inner 1155.1.e.b yes 2
7.b odd 2 1 inner 1155.1.e.b yes 2
11.b odd 2 1 1155.1.e.a 2
15.d odd 2 1 1155.1.e.a 2
21.c even 2 1 1155.1.e.a 2
33.d even 2 1 inner 1155.1.e.b yes 2
35.c odd 2 1 CM 1155.1.e.b yes 2
55.d odd 2 1 1155.1.e.a 2
77.b even 2 1 1155.1.e.a 2
105.g even 2 1 1155.1.e.a 2
165.d even 2 1 RM 1155.1.e.b yes 2
231.h odd 2 1 CM 1155.1.e.b yes 2
385.h even 2 1 1155.1.e.a 2
1155.e odd 2 1 inner 1155.1.e.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.1.e.a 2 3.b odd 2 1
1155.1.e.a 2 11.b odd 2 1
1155.1.e.a 2 15.d odd 2 1
1155.1.e.a 2 21.c even 2 1
1155.1.e.a 2 55.d odd 2 1
1155.1.e.a 2 77.b even 2 1
1155.1.e.a 2 105.g even 2 1
1155.1.e.a 2 385.h even 2 1
1155.1.e.b yes 2 1.a even 1 1 trivial
1155.1.e.b yes 2 5.b even 2 1 inner
1155.1.e.b yes 2 7.b odd 2 1 inner
1155.1.e.b yes 2 33.d even 2 1 inner
1155.1.e.b yes 2 35.c odd 2 1 CM
1155.1.e.b yes 2 165.d even 2 1 RM
1155.1.e.b yes 2 231.h odd 2 1 CM
1155.1.e.b yes 2 1155.e odd 2 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{29} + 2 \)

Hecke Characteristic Polynomials

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