Properties

Label 1155.2.c.a
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} + q^{4} + ( -2 + i ) q^{5} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} + q^{4} + ( -2 + i ) q^{5} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} + ( -1 - 2 i ) q^{10} - q^{11} + i q^{12} + 4 i q^{13} - q^{14} + ( -1 - 2 i ) q^{15} - q^{16} + 2 i q^{17} -i q^{18} + ( -2 + i ) q^{20} - q^{21} -i q^{22} -6 i q^{23} -3 q^{24} + ( 3 - 4 i ) q^{25} -4 q^{26} -i q^{27} + i q^{28} -4 q^{29} + ( 2 - i ) q^{30} + 2 q^{31} + 5 i q^{32} -i q^{33} -2 q^{34} + ( -1 - 2 i ) q^{35} - q^{36} + 4 i q^{37} -4 q^{39} + ( -3 - 6 i ) q^{40} -10 q^{41} -i q^{42} -4 i q^{43} - q^{44} + ( 2 - i ) q^{45} + 6 q^{46} -i q^{48} - q^{49} + ( 4 + 3 i ) q^{50} -2 q^{51} + 4 i q^{52} + 6 i q^{53} + q^{54} + ( 2 - i ) q^{55} -3 q^{56} -4 i q^{58} -4 q^{59} + ( -1 - 2 i ) q^{60} -2 q^{61} + 2 i q^{62} -i q^{63} -7 q^{64} + ( -4 - 8 i ) q^{65} + q^{66} -8 i q^{67} + 2 i q^{68} + 6 q^{69} + ( 2 - i ) q^{70} -8 q^{71} -3 i q^{72} -4 q^{74} + ( 4 + 3 i ) q^{75} -i q^{77} -4 i q^{78} + 14 q^{79} + ( 2 - i ) q^{80} + q^{81} -10 i q^{82} + 14 i q^{83} - q^{84} + ( -2 - 4 i ) q^{85} + 4 q^{86} -4 i q^{87} -3 i q^{88} + 12 q^{89} + ( 1 + 2 i ) q^{90} -4 q^{91} -6 i q^{92} + 2 i q^{93} -5 q^{96} + 14 i q^{97} -i q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 4q^{5} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} - 4q^{5} - 2q^{6} - 2q^{9} - 2q^{10} - 2q^{11} - 2q^{14} - 2q^{15} - 2q^{16} - 4q^{20} - 2q^{21} - 6q^{24} + 6q^{25} - 8q^{26} - 8q^{29} + 4q^{30} + 4q^{31} - 4q^{34} - 2q^{35} - 2q^{36} - 8q^{39} - 6q^{40} - 20q^{41} - 2q^{44} + 4q^{45} + 12q^{46} - 2q^{49} + 8q^{50} - 4q^{51} + 2q^{54} + 4q^{55} - 6q^{56} - 8q^{59} - 2q^{60} - 4q^{61} - 14q^{64} - 8q^{65} + 2q^{66} + 12q^{69} + 4q^{70} - 16q^{71} - 8q^{74} + 8q^{75} + 28q^{79} + 4q^{80} + 2q^{81} - 2q^{84} - 4q^{85} + 8q^{86} + 24q^{89} + 2q^{90} - 8q^{91} - 10q^{96} + 2q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 −2.00000 1.00000i −1.00000 1.00000i 3.00000i −1.00000 −1.00000 + 2.00000i
694.2 1.00000i 1.00000i 1.00000 −2.00000 + 1.00000i −1.00000 1.00000i 3.00000i −1.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.c.a 2
5.b even 2 1 inner 1155.2.c.a 2
5.c odd 4 1 5775.2.a.h 1
5.c odd 4 1 5775.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.a 2 1.a even 1 1 trivial
1155.2.c.a 2 5.b even 2 1 inner
5775.2.a.h 1 5.c odd 4 1
5775.2.a.s 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{13}^{2} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 58 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 70 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{2} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 30 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 12 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T^{2} + 9409 T^{4} \)
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