Properties

Label 1155.2.c.d
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.0.350464.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-\beta_{2} + 5 \beta_{1} - 7\)
\(\nu^{5}\)\(=\)\(-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9\)

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
1.45161 + 1.45161i
0.403032 0.403032i
−0.854638 + 0.854638i
−0.854638 0.854638i
0.403032 + 0.403032i
1.45161 1.45161i
2.21432i 1.00000i −2.90321 0.311108 + 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 0.688892i
694.2 1.67513i 1.00000i −0.806063 −1.48119 + 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 + 2.48119i
694.3 0.539189i 1.00000i 1.70928 2.17009 + 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 1.17009i
694.4 0.539189i 1.00000i 1.70928 2.17009 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 + 1.17009i
694.5 1.67513i 1.00000i −0.806063 −1.48119 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 2.48119i
694.6 2.21432i 1.00000i −2.90321 0.311108 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 + 0.688892i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.6
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.d 6
5.b even 2 1 inner 1155.2.c.d 6
5.c odd 4 1 5775.2.a.bs 3
5.c odd 4 1 5775.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.d 6 1.a even 1 1 trivial
1155.2.c.d 6 5.b even 2 1 inner
5775.2.a.bs 3 5.c odd 4 1
5775.2.a.bv 3 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 \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{6} + 8 T_{2}^{4} + 16 T_{2}^{2} + 4 \)
\( T_{13}^{6} + 28 T_{13}^{4} + 164 T_{13}^{2} + 100 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 12 T^{4} - 28 T^{6} + 48 T^{8} - 64 T^{10} + 64 T^{12} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 15 T^{4} - 50 T^{5} + 125 T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( 1 - T )^{6} \)
$13$ \( 1 - 50 T^{2} + 1243 T^{4} - 19712 T^{6} + 210067 T^{8} - 1428050 T^{10} + 4826809 T^{12} \)
$17$ \( 1 - 83 T^{2} + 3150 T^{4} - 68783 T^{6} + 910350 T^{8} - 6932243 T^{10} + 24137569 T^{12} \)
$19$ \( ( 1 - 9 T + 68 T^{2} - 337 T^{3} + 1292 T^{4} - 3249 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 111 T^{2} + 5558 T^{4} - 162563 T^{6} + 2940182 T^{8} - 31062351 T^{10} + 148035889 T^{12} \)
$29$ \( ( 1 - 23 T + 260 T^{2} - 1759 T^{3} + 7540 T^{4} - 19343 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( ( 1 - 4 T + 61 T^{2} - 250 T^{3} + 1891 T^{4} - 3844 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 - 114 T^{2} + 6267 T^{4} - 249568 T^{6} + 8579523 T^{8} - 213654354 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 + 16 T + 193 T^{2} + 1362 T^{3} + 7913 T^{4} + 26896 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 139 T^{2} + 11654 T^{4} - 595707 T^{6} + 21548246 T^{8} - 475213339 T^{10} + 6321363049 T^{12} \)
$47$ \( 1 - 94 T^{2} + 6263 T^{4} - 289872 T^{6} + 13834967 T^{8} - 458690014 T^{10} + 10779215329 T^{12} \)
$53$ \( 1 - 243 T^{2} + 27470 T^{4} - 1839911 T^{6} + 77163230 T^{8} - 1917386883 T^{10} + 22164361129 T^{12} \)
$59$ \( ( 1 + T + 128 T^{2} + 203 T^{3} + 7552 T^{4} + 3481 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 + T + 106 T^{2} - 71 T^{3} + 6466 T^{4} + 3721 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( 1 + 58 T^{2} + 13735 T^{4} + 511756 T^{6} + 61656415 T^{8} + 1168765018 T^{10} + 90458382169 T^{12} \)
$71$ \( ( 1 + 8 T + 219 T^{2} + 1134 T^{3} + 15549 T^{4} + 40328 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 358 T^{2} + 57215 T^{4} - 5315604 T^{6} + 304898735 T^{8} - 10166570278 T^{10} + 151334226289 T^{12} \)
$79$ \( ( 1 - 16 T + 285 T^{2} - 2530 T^{3} + 22515 T^{4} - 99856 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 431 T^{2} + 82158 T^{4} - 8840003 T^{6} + 565986462 T^{8} - 20454536351 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 - 13 T + 156 T^{2} - 1833 T^{3} + 13884 T^{4} - 102973 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 239 T^{2} + 37702 T^{4} - 4451063 T^{6} + 354738118 T^{8} - 21158498159 T^{10} + 832972004929 T^{12} \)
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