Properties

Label 1155.2.i.a
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: 8.0.303595776.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + \beta_{4} q^{3} + ( -2 - \beta_{7} ) q^{4} -\beta_{4} q^{5} + ( \beta_{3} - \beta_{6} ) q^{6} + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{7} + ( -\beta_{1} + 3 \beta_{5} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + \beta_{4} q^{3} + ( -2 - \beta_{7} ) q^{4} -\beta_{4} q^{5} + ( \beta_{3} - \beta_{6} ) q^{6} + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{7} + ( -\beta_{1} + 3 \beta_{5} ) q^{8} - q^{9} + ( -\beta_{3} + \beta_{6} ) q^{10} + ( -2 \beta_{1} + \beta_{5} ) q^{11} + ( -\beta_{2} - 2 \beta_{4} ) q^{12} + ( 2 - \beta_{2} - 4 \beta_{4} ) q^{14} + q^{15} + ( 4 + \beta_{7} ) q^{16} + ( -2 \beta_{3} + 3 \beta_{6} ) q^{17} + ( -\beta_{1} + \beta_{5} ) q^{18} + ( -4 \beta_{3} + \beta_{6} ) q^{19} + ( \beta_{2} + 2 \beta_{4} ) q^{20} + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{21} + ( 6 + \beta_{7} ) q^{22} + ( -1 - 3 \beta_{7} ) q^{23} + ( -3 \beta_{3} + \beta_{6} ) q^{24} - q^{25} -\beta_{4} q^{27} + ( -5 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{28} + ( -\beta_{1} - 2 \beta_{5} ) q^{29} + ( \beta_{1} - \beta_{5} ) q^{30} -2 \beta_{2} q^{31} + ( 3 \beta_{1} - \beta_{5} ) q^{32} + ( -\beta_{3} + 2 \beta_{6} ) q^{33} + ( 2 \beta_{2} + 10 \beta_{4} ) q^{34} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{35} + ( 2 + \beta_{7} ) q^{36} + ( 4 - 2 \beta_{7} ) q^{37} + ( 4 \beta_{2} + 10 \beta_{4} ) q^{38} + ( 3 \beta_{3} - \beta_{6} ) q^{40} + ( -2 \beta_{3} + 6 \beta_{6} ) q^{41} + ( 4 + 2 \beta_{4} + \beta_{7} ) q^{42} + ( -3 \beta_{1} + 2 \beta_{5} ) q^{43} + ( 3 \beta_{1} - 7 \beta_{5} ) q^{44} + \beta_{4} q^{45} + ( -4 \beta_{1} + 10 \beta_{5} ) q^{46} + ( 2 \beta_{2} - 6 \beta_{4} ) q^{47} + ( \beta_{2} + 4 \beta_{4} ) q^{48} + ( 1 + 4 \beta_{4} + 2 \beta_{7} ) q^{49} + ( -\beta_{1} + \beta_{5} ) q^{50} + ( 3 \beta_{1} - 2 \beta_{5} ) q^{51} + ( -5 - \beta_{7} ) q^{53} + ( -\beta_{3} + \beta_{6} ) q^{54} + ( \beta_{3} - 2 \beta_{6} ) q^{55} + ( 3 \beta_{2} + 8 \beta_{4} - 2 \beta_{7} ) q^{56} + ( \beta_{1} - 4 \beta_{5} ) q^{57} + ( -2 - 2 \beta_{7} ) q^{58} + ( -\beta_{2} - 5 \beta_{4} ) q^{59} + ( -2 - \beta_{7} ) q^{60} + ( -2 \beta_{3} - 3 \beta_{6} ) q^{61} + ( -6 \beta_{3} + 2 \beta_{6} ) q^{62} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{63} + \beta_{7} q^{64} + ( \beta_{2} + 6 \beta_{4} ) q^{66} + ( 2 + 2 \beta_{7} ) q^{67} + ( 12 \beta_{3} - 6 \beta_{6} ) q^{68} + ( -3 \beta_{2} - \beta_{4} ) q^{69} + ( -4 - 2 \beta_{4} - \beta_{7} ) q^{70} + ( 10 + 2 \beta_{7} ) q^{71} + ( \beta_{1} - 3 \beta_{5} ) q^{72} + 4 \beta_{3} q^{73} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{74} -\beta_{4} q^{75} + ( 14 \beta_{3} - 12 \beta_{6} ) q^{76} + ( -5 + \beta_{2} + 6 \beta_{4} + \beta_{7} ) q^{77} + ( -4 \beta_{1} + 6 \beta_{5} ) q^{79} + ( -\beta_{2} - 4 \beta_{4} ) q^{80} + q^{81} + ( 2 \beta_{2} + 16 \beta_{4} ) q^{82} + ( -4 \beta_{3} - \beta_{6} ) q^{83} + ( 3 \beta_{1} + 2 \beta_{3} - 5 \beta_{5} ) q^{84} + ( -3 \beta_{1} + 2 \beta_{5} ) q^{85} + ( 10 + 2 \beta_{7} ) q^{86} + ( 2 \beta_{3} + \beta_{6} ) q^{87} + ( -8 - 5 \beta_{7} ) q^{88} + ( -\beta_{2} + 5 \beta_{4} ) q^{89} + ( \beta_{3} - \beta_{6} ) q^{90} + ( 26 + 4 \beta_{7} ) q^{92} + 2 \beta_{7} q^{93} + 4 \beta_{6} q^{94} + ( -\beta_{1} + 4 \beta_{5} ) q^{95} + ( \beta_{3} - 3 \beta_{6} ) q^{96} + ( \beta_{2} - 5 \beta_{4} ) q^{97} + ( 3 \beta_{1} + 4 \beta_{3} - 7 \beta_{5} - 4 \beta_{6} ) q^{98} + ( 2 \beta_{1} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 12q^{4} - 8q^{9} + O(q^{10}) \) \( 8q - 12q^{4} - 8q^{9} + 16q^{14} + 8q^{15} + 28q^{16} + 44q^{22} + 4q^{23} - 8q^{25} + 12q^{36} + 40q^{37} + 28q^{42} - 36q^{53} + 8q^{56} - 8q^{58} - 12q^{60} - 4q^{64} + 8q^{67} - 28q^{70} + 72q^{71} - 44q^{77} + 8q^{81} + 72q^{86} - 44q^{88} + 192q^{92} - 8q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27 \)\()/36\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu \)\()/54\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu \)\()/216\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153 \)\()/72\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{5} + \beta_{1} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - 4 \beta_{4} - \beta_{3}\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{7} - 6 \beta_{5} + 5 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{6} + 15 \beta_{4} + 16 \beta_{3} - \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(8 \beta_{5} - 16 \beta_{1} - 5\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{6} + 35 \beta_{4} - 48 \beta_{3} - 13 \beta_{2}\)\()/2\)

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
−0.396143 1.68614i
0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 1.18614i
1.26217 + 1.18614i
−1.26217 1.18614i
0.396143 1.68614i
−0.396143 + 1.68614i
2.52434i 1.00000i −4.37228 1.00000i −2.52434 −2.52434 + 0.792287i 5.98844i −1.00000 2.52434
76.2 2.52434i 1.00000i −4.37228 1.00000i 2.52434 2.52434 + 0.792287i 5.98844i −1.00000 −2.52434
76.3 0.792287i 1.00000i 1.37228 1.00000i −0.792287 −0.792287 + 2.52434i 2.67181i −1.00000 0.792287
76.4 0.792287i 1.00000i 1.37228 1.00000i 0.792287 0.792287 + 2.52434i 2.67181i −1.00000 −0.792287
76.5 0.792287i 1.00000i 1.37228 1.00000i 0.792287 0.792287 2.52434i 2.67181i −1.00000 −0.792287
76.6 0.792287i 1.00000i 1.37228 1.00000i −0.792287 −0.792287 2.52434i 2.67181i −1.00000 0.792287
76.7 2.52434i 1.00000i −4.37228 1.00000i 2.52434 2.52434 0.792287i 5.98844i −1.00000 −2.52434
76.8 2.52434i 1.00000i −4.37228 1.00000i −2.52434 −2.52434 0.792287i 5.98844i −1.00000 2.52434
\(n\): 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.a 8
7.b odd 2 1 inner 1155.2.i.a 8
11.b odd 2 1 inner 1155.2.i.a 8
77.b even 2 1 inner 1155.2.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.i.a 8 1.a even 1 1 trivial
1155.2.i.a 8 7.b odd 2 1 inner
1155.2.i.a 8 11.b odd 2 1 inner
1155.2.i.a 8 77.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 7 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} - 4 T^{6} + 16 T^{8} )^{2} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 1 - 34 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{8} \)
$17$ \( ( 1 + 17 T^{2} + 576 T^{4} + 4913 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 3 T^{2} + 320 T^{4} - 1083 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - T - 28 T^{2} - 23 T^{3} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 73 T^{2} + 2808 T^{4} - 61393 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 56 T^{2} + 2574 T^{4} - 53816 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 10 T + 66 T^{2} - 370 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 40 T^{2} + 2574 T^{4} - 67240 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 121 T^{2} + 7284 T^{4} - 223729 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 24 T^{2} - 1906 T^{4} - 53016 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 9 T + 118 T^{2} + 477 T^{3} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 - 179 T^{2} + 14304 T^{4} - 623099 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 121 T^{2} + 7464 T^{4} + 450241 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 2 T + 102 T^{2} - 134 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 18 T + 190 T^{2} - 1278 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 98 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 132 T^{2} + 8390 T^{4} - 823812 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 205 T^{2} + 23616 T^{4} + 1412245 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 279 T^{2} + 34304 T^{4} - 2209959 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 311 T^{2} + 42000 T^{4} - 2926199 T^{6} + 88529281 T^{8} )^{2} \)
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