Properties

Label 57.2.i.b
Level $57$
Weight $2$
Character orbit 57.i
Analytic conductor $0.455$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{5} - \beta_{2} - \beta_1 + 1) q^{2} - \beta_{6} q^{3} + (2 \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{11} - \beta_{8} + \beta_{6} - 2 \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{10} + \beta_{6} - \beta_{4} - \beta_1) q^{6} + ( - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_1 - 2) q^{7} + (\beta_{10} + \beta_{8} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_1) q^{8} - \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_{5} - \beta_{2} - \beta_1 + 1) q^{2} - \beta_{6} q^{3} + (2 \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{11} - \beta_{8} + \beta_{6} - 2 \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{10} + \beta_{6} - \beta_{4} - \beta_1) q^{6} + ( - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_1 - 2) q^{7} + (\beta_{10} + \beta_{8} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_1) q^{8} - \beta_{9} q^{9} + (\beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{10}+ \cdots + ( - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} + 6 q^{5} - 3 q^{6} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 3 q^{4} + 6 q^{5} - 3 q^{6} - 9 q^{7} + 3 q^{8} - 18 q^{10} - 9 q^{11} - 6 q^{12} - 3 q^{13} - 3 q^{14} + 6 q^{15} + 33 q^{16} + 12 q^{17} - 9 q^{19} - 18 q^{20} - 24 q^{22} + 9 q^{23} + 30 q^{24} - 12 q^{25} - 6 q^{27} + 30 q^{28} - 6 q^{30} - 3 q^{32} - 9 q^{33} + 21 q^{34} + 30 q^{35} - 3 q^{36} - 12 q^{37} + 18 q^{38} + 6 q^{39} - 3 q^{40} - 18 q^{41} - 3 q^{42} + 15 q^{43} + 15 q^{44} - 9 q^{45} + 27 q^{46} - 9 q^{47} - 12 q^{48} - 9 q^{49} + 9 q^{50} - 15 q^{51} - 27 q^{52} - 30 q^{53} - 3 q^{54} + 15 q^{55} - 78 q^{56} - 3 q^{57} - 24 q^{58} - 30 q^{59} - 15 q^{60} + 21 q^{61} + 3 q^{62} + 9 q^{63} - 21 q^{64} + 21 q^{65} + 30 q^{66} + 3 q^{67} - 21 q^{68} - 6 q^{69} + 18 q^{70} - 30 q^{71} + 30 q^{72} - 24 q^{73} + 39 q^{74} + 30 q^{75} - 60 q^{76} + 96 q^{77} + 9 q^{78} + 24 q^{79} + 42 q^{80} - 33 q^{82} + 3 q^{83} + 3 q^{84} + 3 q^{85} + 51 q^{86} + 15 q^{87} + 42 q^{88} + 3 q^{89} - 9 q^{90} + 24 q^{91} + 69 q^{92} - 6 q^{93} + 18 q^{94} - 54 q^{95} - 42 q^{96} - 27 q^{97} - 3 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 ) / 218 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 ) / 218 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 ) / 218 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 ) / 218 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 ) / 218 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 ) / 218 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 ) / 218 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 ) / 218 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 ) / 218 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 ) / 218 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta _1 + 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta _1 + 77 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta _1 - 121 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta _1 - 535 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta _1 + 257 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta _1 + 3359 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta _1 + 1451 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta _1 - 18736 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(-\beta_{3} + \beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 1.80139i
0.500000 0.168222i
0.500000 + 1.74095i
0.500000 2.42499i
0.500000 1.74095i
0.500000 + 2.42499i
0.500000 0.677980i
0.500000 + 1.96356i
0.500000 + 1.80139i
0.500000 + 0.168222i
0.500000 + 0.677980i
0.500000 1.96356i
−2.23121 0.812094i 0.173648 + 0.984808i 2.78672 + 2.33833i 2.16379 1.81563i 0.412311 2.33833i 1.05756 1.83175i −1.94440 3.36780i −0.939693 + 0.342020i −6.30233 + 2.29386i
4.2 0.791518 + 0.288089i 0.173648 + 0.984808i −0.988583 0.829520i 1.30800 1.09754i −0.146267 + 0.829520i −1.96517 + 3.40377i −1.38582 2.40031i −0.939693 + 0.342020i 1.35149 0.491903i
16.1 −0.958464 0.804247i −0.939693 + 0.342020i −0.0754558 0.427931i 0.755623 4.28535i 1.17573 + 0.427931i 0.898157 + 1.55565i −1.52303 + 2.63796i 0.766044 0.642788i −4.17072 + 3.49965i
16.2 1.22451 + 1.02748i −0.939693 + 0.342020i 0.0964003 + 0.546713i −0.174371 + 0.988909i −1.50208 0.546713i −1.28482 2.22537i 1.15479 2.00015i 0.766044 0.642788i −1.22961 + 1.03176i
25.1 −0.958464 + 0.804247i −0.939693 0.342020i −0.0754558 + 0.427931i 0.755623 + 4.28535i 1.17573 0.427931i 0.898157 1.55565i −1.52303 2.63796i 0.766044 + 0.642788i −4.17072 3.49965i
25.2 1.22451 1.02748i −0.939693 0.342020i 0.0964003 0.546713i −0.174371 0.988909i −1.50208 + 0.546713i −1.28482 + 2.22537i 1.15479 + 2.00015i 0.766044 + 0.642788i −1.22961 1.03176i
28.1 −0.458021 2.59757i 0.766044 0.642788i −4.65819 + 1.69544i 1.91800 + 0.698096i −2.02055 1.69544i −1.30802 + 2.26556i 3.89993 + 6.75488i 0.173648 0.984808i 0.934866 5.30189i
28.2 0.131669 + 0.746734i 0.766044 0.642788i 1.33911 0.487396i −2.97104 1.08137i 0.580856 + 0.487396i −1.89771 + 3.28694i 1.29853 + 2.24912i 0.173648 0.984808i 0.416301 2.36096i
43.1 −2.23121 + 0.812094i 0.173648 0.984808i 2.78672 2.33833i 2.16379 + 1.81563i 0.412311 + 2.33833i 1.05756 + 1.83175i −1.94440 + 3.36780i −0.939693 0.342020i −6.30233 2.29386i
43.2 0.791518 0.288089i 0.173648 0.984808i −0.988583 + 0.829520i 1.30800 + 1.09754i −0.146267 0.829520i −1.96517 3.40377i −1.38582 + 2.40031i −0.939693 0.342020i 1.35149 + 0.491903i
55.1 −0.458021 + 2.59757i 0.766044 + 0.642788i −4.65819 1.69544i 1.91800 0.698096i −2.02055 + 1.69544i −1.30802 2.26556i 3.89993 6.75488i 0.173648 + 0.984808i 0.934866 + 5.30189i
55.2 0.131669 0.746734i 0.766044 + 0.642788i 1.33911 + 0.487396i −2.97104 + 1.08137i 0.580856 0.487396i −1.89771 3.28694i 1.29853 2.24912i 0.173648 + 0.984808i 0.416301 + 2.36096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.i.b 12
3.b odd 2 1 171.2.u.e 12
4.b odd 2 1 912.2.bo.j 12
19.e even 9 1 inner 57.2.i.b 12
19.e even 9 1 1083.2.a.q 6
19.f odd 18 1 1083.2.a.p 6
57.j even 18 1 3249.2.a.bg 6
57.l odd 18 1 171.2.u.e 12
57.l odd 18 1 3249.2.a.bh 6
76.l odd 18 1 912.2.bo.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.i.b 12 1.a even 1 1 trivial
57.2.i.b 12 19.e even 9 1 inner
171.2.u.e 12 3.b odd 2 1
171.2.u.e 12 57.l odd 18 1
912.2.bo.j 12 4.b odd 2 1
912.2.bo.j 12 76.l odd 18 1
1083.2.a.p 6 19.f odd 18 1
1083.2.a.q 6 19.e even 9 1
3249.2.a.bg 6 57.j even 18 1
3249.2.a.bh 6 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3 T_{2}^{11} + 6 T_{2}^{10} + 8 T_{2}^{9} - 9 T_{2}^{8} - 9 T_{2}^{7} + 99 T_{2}^{6} + 18 T_{2}^{5} - 36 T_{2}^{4} - 64 T_{2}^{3} + 96 T_{2}^{2} - 96 T_{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + 6 T^{10} + 8 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + 24 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + 66 T^{10} + \cdots + 145161 \) Copy content Toggle raw display
$11$ \( T^{12} + 9 T^{11} + 75 T^{10} + \cdots + 207936 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} - 6 T^{10} - 21 T^{9} + \cdots + 3249 \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + 18 T^{10} + \cdots + 41062464 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} + 72 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 9 T^{11} - 18 T^{10} + \cdots + 2483776 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{10} - 125 T^{9} + \cdots + 87616 \) Copy content Toggle raw display
$31$ \( T^{12} + 36 T^{10} - 20 T^{9} + \cdots + 185761 \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} - 84 T^{4} - 862 T^{3} + \cdots + 2467)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 18 T^{11} + 183 T^{10} + \cdots + 87616 \) Copy content Toggle raw display
$43$ \( T^{12} - 15 T^{11} + \cdots + 587917009 \) Copy content Toggle raw display
$47$ \( T^{12} + 9 T^{11} + 63 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$53$ \( T^{12} + 30 T^{11} + 531 T^{10} + \cdots + 1871424 \) Copy content Toggle raw display
$59$ \( T^{12} + 30 T^{11} + 498 T^{10} + \cdots + 1617984 \) Copy content Toggle raw display
$61$ \( T^{12} - 21 T^{11} + 285 T^{10} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{11} + \cdots + 1254293056 \) Copy content Toggle raw display
$71$ \( T^{12} + 30 T^{11} + \cdots + 241118784 \) Copy content Toggle raw display
$73$ \( T^{12} + 24 T^{11} + \cdots + 1056705049 \) Copy content Toggle raw display
$79$ \( T^{12} - 24 T^{11} + \cdots + 4606201161 \) Copy content Toggle raw display
$83$ \( T^{12} - 3 T^{11} + \cdots + 75809912896 \) Copy content Toggle raw display
$89$ \( T^{12} - 3 T^{11} + \cdots + 21486869056 \) Copy content Toggle raw display
$97$ \( T^{12} + 27 T^{11} + 606 T^{10} + \cdots + 128881 \) Copy content Toggle raw display
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