Properties

Label 1155.2.c.c
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} + ( -\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} + ( -1 - 2 \beta_{2} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{20} - q^{21} + \beta_{5} q^{22} + ( -\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 + \beta_{1} - \beta_{2} ) q^{26} + \beta_{3} q^{27} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{28} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( -1 - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{30} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{31} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{32} + \beta_{3} q^{33} + ( 2 - \beta_{1} ) q^{34} + ( \beta_{1} - \beta_{4} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} ) q^{36} + ( -\beta_{3} - \beta_{4} ) q^{37} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{38} + ( -1 + \beta_{2} ) q^{39} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{40} + ( -3 - \beta_{2} ) q^{41} + \beta_{5} q^{42} + ( -2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} ) q^{44} + ( \beta_{2} + \beta_{5} ) q^{45} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{46} + ( 2 \beta_{4} - \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} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{52} + ( 7 \beta_{3} + 2 \beta_{5} ) q^{53} -\beta_{1} q^{54} + ( \beta_{2} + \beta_{5} ) q^{55} + 2 q^{56} + ( \beta_{3} - 2 \beta_{4} ) q^{57} + ( -5 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{58} + ( 5 - \beta_{1} ) q^{59} + ( 2 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{60} + ( -8 - 3 \beta_{1} - \beta_{2} ) q^{61} + ( -\beta_{3} + 3 \beta_{4} + 5 \beta_{5} ) q^{62} + \beta_{3} q^{63} + ( 4 - 4 \beta_{1} ) q^{64} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{65} -\beta_{1} q^{66} -2 \beta_{3} q^{67} + ( -3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{68} + ( \beta_{1} + \beta_{2} ) q^{69} + ( -1 - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} + ( -3 - 4 \beta_{1} + 3 \beta_{2} ) q^{71} -2 \beta_{3} q^{72} + ( -8 \beta_{3} - 2 \beta_{5} ) q^{73} + ( -1 + \beta_{1} - \beta_{2} ) q^{74} + ( -2 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{75} + ( 3 + \beta_{1} - \beta_{2} ) q^{76} + \beta_{3} q^{77} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{78} + ( -\beta_{1} + 4 \beta_{2} ) q^{79} + ( 4 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{80} + q^{81} + ( -\beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{82} + ( 8 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} ) q^{84} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{85} + ( -5 + 4 \beta_{1} - \beta_{2} ) q^{86} + ( -2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{87} -2 \beta_{3} q^{88} + ( 6 - 2 \beta_{1} - 3 \beta_{2} ) q^{89} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{90} + ( -1 + \beta_{2} ) q^{91} + ( \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{92} + ( 4 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{93} + ( -1 + \beta_{1} + \beta_{2} ) q^{94} + ( 4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{95} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{96} + ( -\beta_{4} + 4 \beta_{5} ) q^{97} + \beta_{5} q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{4} + 2q^{5} - 6q^{9} + O(q^{10}) \) \( 6q - 4q^{4} + 2q^{5} - 6q^{9} - 16q^{10} - 6q^{11} - 8q^{16} - 2q^{19} + 8q^{20} - 6q^{21} + 12q^{24} - 2q^{25} - 4q^{26} + 10q^{29} - 4q^{30} - 16q^{31} + 12q^{34} + 4q^{36} - 8q^{39} - 16q^{41} + 4q^{44} - 2q^{45} - 20q^{46} - 6q^{49} + 8q^{50} - 2q^{51} - 2q^{55} + 12q^{56} + 30q^{59} + 12q^{60} - 46q^{61} + 24q^{64} - 4q^{65} - 2q^{69} - 4q^{70} - 24q^{71} - 4q^{74} - 8q^{75} + 20q^{76} - 8q^{79} + 24q^{80} + 6q^{81} + 4q^{84} + 12q^{85} - 28q^{86} + 42q^{89} + 16q^{90} - 8q^{91} - 8q^{94} + 26q^{95} + 16q^{96} + 6q^{99} + O(q^{100}) \)

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.c 6
5.b even 2 1 inner 1155.2.c.c 6
5.c odd 4 1 5775.2.a.bt 3
5.c odd 4 1 5775.2.a.bu 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.c 6 1.a even 1 1 trivial
1155.2.c.c 6 5.b even 2 1 inner
5775.2.a.bt 3 5.c odd 4 1
5775.2.a.bu 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 \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{6} + 8 T_{2}^{4} + 16 T_{2}^{2} + 4 \)
\( T_{13}^{6} + 12 T_{13}^{4} + 20 T_{13}^{2} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 12 T^{4} - 28 T^{6} + 48 T^{8} - 64 T^{10} + 64 T^{12} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 15 T^{4} - 50 T^{5} + 125 T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( 1 + T )^{6} \)
$13$ \( 1 - 66 T^{2} + 1931 T^{4} - 32288 T^{6} + 326339 T^{8} - 1885026 T^{10} + 4826809 T^{12} \)
$17$ \( 1 - 91 T^{2} + 3614 T^{4} - 80103 T^{6} + 1044446 T^{8} - 7600411 T^{10} + 24137569 T^{12} \)
$19$ \( ( 1 + T + 44 T^{2} + 33 T^{3} + 836 T^{4} + 361 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 119 T^{2} + 6294 T^{4} - 187787 T^{6} + 3329526 T^{8} - 33301079 T^{10} + 148035889 T^{12} \)
$29$ \( ( 1 - 5 T + 80 T^{2} - 289 T^{3} + 2320 T^{4} - 4205 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( ( 1 + 8 T + 65 T^{2} + 282 T^{3} + 2015 T^{4} + 7688 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 - 210 T^{2} + 18779 T^{4} - 915968 T^{6} + 25708451 T^{8} - 393573810 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 + 8 T + 141 T^{2} + 666 T^{3} + 5781 T^{4} + 13448 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 219 T^{2} + 21086 T^{4} - 1162547 T^{6} + 38988014 T^{8} - 748717419 T^{10} + 6321363049 T^{12} \)
$47$ \( 1 - 254 T^{2} + 27975 T^{4} - 1715024 T^{6} + 61796775 T^{8} - 1239438974 T^{10} + 10779215329 T^{12} \)
$53$ \( 1 - 139 T^{2} + 10974 T^{4} - 619087 T^{6} + 30825966 T^{8} - 1096776859 T^{10} + 22164361129 T^{12} \)
$59$ \( ( 1 - 15 T + 248 T^{2} - 1877 T^{3} + 14632 T^{4} - 52215 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 + 23 T + 314 T^{2} + 2919 T^{3} + 19154 T^{4} + 85583 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( ( 1 - 130 T^{2} + 4489 T^{4} )^{3} \)
$71$ \( ( 1 + 12 T + 191 T^{2} + 1402 T^{3} + 13561 T^{4} + 60492 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 214 T^{2} + 26303 T^{4} - 2177460 T^{6} + 140168687 T^{8} - 6077223574 T^{10} + 151334226289 T^{12} \)
$79$ \( ( 1 + 4 T + 193 T^{2} + 670 T^{3} + 15247 T^{4} + 24964 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 199 T^{2} + 21726 T^{4} - 1728427 T^{6} + 149670414 T^{8} - 9444205879 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 - 21 T + 356 T^{2} - 3505 T^{3} + 31684 T^{4} - 166341 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 463 T^{2} + 98830 T^{4} - 12248383 T^{6} + 929891470 T^{8} - 40989057103 T^{10} + 832972004929 T^{12} \)
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