Properties

Label 1155.2.l.d
Level 1155
Weight 2
Character orbit 1155.l
Analytic conductor 9.223
Analytic rank 0
Dimension 4
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.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−0.414214 1.00000 1.41421i −1.82843 1.00000i −0.414214 + 0.585786i 1.00000i 1.58579 −1.00000 2.82843i 0.414214i
1121.2 −0.414214 1.00000 + 1.41421i −1.82843 1.00000i −0.414214 0.585786i 1.00000i 1.58579 −1.00000 + 2.82843i 0.414214i
1121.3 2.41421 1.00000 1.41421i 3.82843 1.00000i 2.41421 3.41421i 1.00000i 4.41421 −1.00000 2.82843i 2.41421i
1121.4 2.41421 1.00000 + 1.41421i 3.82843 1.00000i 2.41421 + 3.41421i 1.00000i 4.41421 −1.00000 + 2.82843i 2.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d 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.l.d yes 4
3.b odd 2 1 1155.2.l.a 4
11.b odd 2 1 1155.2.l.a 4
33.d even 2 1 inner 1155.2.l.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.l.a 4 3.b odd 2 1
1155.2.l.a 4 11.b odd 2 1
1155.2.l.d yes 4 1.a even 1 1 trivial
1155.2.l.d yes 4 33.d even 2 1 inner

Hecke kernels

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

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{17}^{2} + 8 T_{17} + 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \)
$3$ \( ( 1 - 2 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( 1 + 14 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 + 8 T + 42 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \)
$19$ \( 1 - 52 T^{2} + 1270 T^{4} - 18772 T^{6} + 130321 T^{8} \)
$23$ \( 1 - 68 T^{2} + 2086 T^{4} - 35972 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 26 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 4 T + 58 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 4 T + 54 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 76 T^{2} + 3094 T^{4} - 140524 T^{6} + 3418801 T^{8} \)
$47$ \( 1 - 44 T^{2} + 2854 T^{4} - 97196 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 188 T^{2} + 14326 T^{4} - 528092 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 100 T^{2} + 4854 T^{4} - 348100 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 100 T^{2} + 7894 T^{4} - 372100 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 - 4 T + 66 T^{2} - 268 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 148 T^{2} + 10950 T^{4} - 746068 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 - 2 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{4} \)
$89$ \( 1 - 284 T^{2} + 35494 T^{4} - 2249564 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 + 16 T + 250 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2} \)
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