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 discriminant -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 \(10\), degree \(4\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{10}\)
Projective field Galois closure of 10.2.4299150233375353125.2

$q$-expansion

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

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

Character values

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
33.f even 10 1 inner
165.r even 10 1 inner
231.r odd 10 1 inner
1155.ca odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.1.ca.a 8
3.b odd 2 1 1155.1.ca.b yes 8
5.b even 2 1 inner 1155.1.ca.a 8
7.b odd 2 1 inner 1155.1.ca.a 8
11.d odd 10 1 1155.1.ca.b yes 8
15.d odd 2 1 1155.1.ca.b yes 8
21.c even 2 1 1155.1.ca.b yes 8
33.f even 10 1 inner 1155.1.ca.a 8
35.c odd 2 1 CM 1155.1.ca.a 8
55.h odd 10 1 1155.1.ca.b yes 8
77.l even 10 1 1155.1.ca.b yes 8
105.g even 2 1 1155.1.ca.b yes 8
165.r even 10 1 inner 1155.1.ca.a 8
231.r odd 10 1 inner 1155.1.ca.a 8
385.v even 10 1 1155.1.ca.b yes 8
1155.ca odd 10 1 inner 1155.1.ca.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.1.ca.a 8 1.a even 1 1 trivial
1155.1.ca.a 8 5.b even 2 1 inner
1155.1.ca.a 8 7.b odd 2 1 inner
1155.1.ca.a 8 33.f even 10 1 inner
1155.1.ca.a 8 35.c odd 2 1 CM
1155.1.ca.a 8 165.r even 10 1 inner
1155.1.ca.a 8 231.r odd 10 1 inner
1155.1.ca.a 8 1155.ca odd 10 1 inner
1155.1.ca.b yes 8 3.b odd 2 1
1155.1.ca.b yes 8 11.d odd 10 1
1155.1.ca.b yes 8 15.d odd 2 1
1155.1.ca.b yes 8 21.c even 2 1
1155.1.ca.b yes 8 55.h odd 10 1
1155.1.ca.b yes 8 77.l even 10 1
1155.1.ca.b yes 8 105.g even 2 1
1155.1.ca.b yes 8 385.v even 10 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$7$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$11$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$13$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$17$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
$19$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$23$ \( ( 1 + T^{2} )^{8} \)
$29$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$37$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$41$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$43$ \( ( 1 + T^{2} )^{8} \)
$47$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$53$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$59$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$61$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$67$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$73$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$79$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$83$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$89$ \( ( 1 + T^{2} )^{8} \)
$97$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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