Properties

Label 1155.2.i.b
Level 1155
Weight 2
Character orbit 1155.i
Analytic conductor 9.223
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} -\zeta_{24}^{6} q^{3} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{4} + \zeta_{24}^{6} q^{5} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{6} + ( -\zeta_{24} + 3 \zeta_{24}^{5} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} -\zeta_{24}^{6} q^{3} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{4} + \zeta_{24}^{6} q^{5} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{6} + ( -\zeta_{24} + 3 \zeta_{24}^{5} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{8} - q^{9} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{10} + ( -3 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{11} + ( -1 + 2 \zeta_{24}^{4} ) q^{12} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{13} + ( -2 - 3 \zeta_{24}^{2} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{14} + q^{15} + ( -1 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{16} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{17} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{18} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{19} + ( 1 - 2 \zeta_{24}^{4} ) q^{20} + ( 3 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{21} + ( 1 + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{5} - \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{22} -4 q^{23} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{24} - q^{25} + ( 3 - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{26} + \zeta_{24}^{6} q^{27} + ( -\zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{28} + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{29} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{30} + ( 4 - 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{31} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{32} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{33} + ( -1 + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{34} + ( -3 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{35} + ( 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{36} + ( -4 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{37} + ( -2 + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{38} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{39} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{40} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{41} + ( -1 - \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{42} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{43} + ( \zeta_{24} + 6 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{44} -\zeta_{24}^{6} q^{45} + ( -4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{46} + ( -4 + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{47} + ( -2 + 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{48} + ( -8 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{49} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{50} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{51} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{52} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{53} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{54} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{55} + ( 1 - \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{56} + ( -2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{57} + ( -4 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{58} + ( -4 + 8 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{59} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{60} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{61} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{62} + ( \zeta_{24} - 3 \zeta_{24}^{5} ) q^{63} + ( 4 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{64} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{65} + ( 1 - 3 \zeta_{24} - 3 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{66} + ( -2 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{67} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{68} + 4 \zeta_{24}^{6} q^{69} + ( 1 + \zeta_{24}^{2} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{70} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{71} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{72} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{74} + \zeta_{24}^{6} q^{75} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{76} + ( 3 + 3 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 9 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{77} + ( -3 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{78} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{79} + ( 2 - 4 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{80} + q^{81} + ( 2 - 4 \zeta_{24}^{4} ) q^{82} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{83} + ( -5 \zeta_{24} + \zeta_{24}^{5} ) q^{84} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{85} + ( -3 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{86} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{87} + ( -1 - 3 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{88} + ( -4 + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{89} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{90} + ( 9 - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{91} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{92} + ( -4 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{93} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{94} + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{95} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{96} + ( 4 - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{97} + ( 5 \zeta_{24} - 3 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{98} + ( 3 - \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 24q^{11} - 20q^{14} + 8q^{15} - 8q^{16} + 8q^{22} - 32q^{23} - 8q^{25} - 32q^{37} + 4q^{42} - 4q^{56} - 32q^{58} + 32q^{64} - 16q^{67} - 4q^{70} + 16q^{77} - 24q^{78} + 8q^{81} - 24q^{86} - 8q^{88} + 48q^{91} - 32q^{93} + 24q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.965926 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
1.93185i 1.00000i −1.73205 1.00000i −1.93185 0.189469 2.63896i 0.517638i −1.00000 1.93185
76.2 1.93185i 1.00000i −1.73205 1.00000i 1.93185 −0.189469 2.63896i 0.517638i −1.00000 −1.93185
76.3 0.517638i 1.00000i 1.73205 1.00000i −0.517638 −2.63896 + 0.189469i 1.93185i −1.00000 0.517638
76.4 0.517638i 1.00000i 1.73205 1.00000i 0.517638 2.63896 + 0.189469i 1.93185i −1.00000 −0.517638
76.5 0.517638i 1.00000i 1.73205 1.00000i 0.517638 2.63896 0.189469i 1.93185i −1.00000 −0.517638
76.6 0.517638i 1.00000i 1.73205 1.00000i −0.517638 −2.63896 0.189469i 1.93185i −1.00000 0.517638
76.7 1.93185i 1.00000i −1.73205 1.00000i 1.93185 −0.189469 + 2.63896i 0.517638i −1.00000 −1.93185
76.8 1.93185i 1.00000i −1.73205 1.00000i −1.93185 0.189469 + 2.63896i 0.517638i −1.00000 1.93185
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.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.i.b 8
7.b odd 2 1 inner 1155.2.i.b 8
11.b odd 2 1 inner 1155.2.i.b 8
77.b even 2 1 inner 1155.2.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.i.b 8 1.a even 1 1 trivial
1155.2.i.b 8 7.b odd 2 1 inner
1155.2.i.b 8 11.b odd 2 1 inner
1155.2.i.b 8 77.b even 2 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} )^{2} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 1 - 94 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 6 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 8 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 28 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 + 60 T^{2} + 1574 T^{4} + 21660 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{8} \)
$29$ \( ( 1 - 100 T^{2} + 4134 T^{4} - 84100 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 4 T^{2} - 1146 T^{4} + 3844 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 8 T + 78 T^{2} + 296 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 116 T^{2} + 6294 T^{4} + 194996 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 80 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 60 T^{2} + 2246 T^{4} - 132540 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 94 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 - 68 T^{2} + 1206 T^{4} - 236708 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 98 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 + 4 T + 126 T^{2} + 268 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 94 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 208 T^{2} + 19746 T^{4} + 1108432 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 300 T^{2} + 34934 T^{4} - 1872300 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 164 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 252 T^{2} + 30950 T^{4} - 1996092 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 284 T^{2} + 38214 T^{4} - 2672156 T^{6} + 88529281 T^{8} )^{2} \)
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