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Properties

Label 57.3.k.a
Level $57$
Weight $3$
Character orbit 57.k
Analytic conductor $1.553$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,3,Mod(10,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.k (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55313750685\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 48 x^{16} + 936 x^{14} + 9539 x^{12} + 54576 x^{10} + 176517 x^{8} + 313396 x^{6} + 277917 x^{4} + \cdots + 8427 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{14} - \beta_{13} + \beta_{9} + \cdots + 1) q^{2}+ \cdots + 3 \beta_{14} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} - \beta_{13} + \beta_{9} + \cdots + 1) q^{2}+ \cdots + (6 \beta_{17} - 15 \beta_{16} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8} - 78 q^{10} + 15 q^{11} + 36 q^{12} + 36 q^{13} - 39 q^{14} - 18 q^{15} - 3 q^{16} - 30 q^{17} + 54 q^{19} - 30 q^{20} - 27 q^{21} + 132 q^{22} + 69 q^{23} + 72 q^{24} + 138 q^{25} + 48 q^{26} - 81 q^{27} - 246 q^{28} - 162 q^{29} + 72 q^{31} - 21 q^{32} - 63 q^{33} - 285 q^{34} + 54 q^{35} + 9 q^{36} - 204 q^{38} - 18 q^{39} - 51 q^{40} + 30 q^{41} + 171 q^{42} + 402 q^{43} + 471 q^{44} - 9 q^{45} - 99 q^{46} - 105 q^{47} - 72 q^{48} + 66 q^{49} + 567 q^{50} + 153 q^{51} - 3 q^{52} - 36 q^{53} - 27 q^{54} - 15 q^{55} + 45 q^{57} - 48 q^{58} - 180 q^{59} - 207 q^{60} + 93 q^{61} + 189 q^{62} - 9 q^{63} - 183 q^{64} - 891 q^{65} - 324 q^{66} - 354 q^{67} + 153 q^{68} - 36 q^{69} + 372 q^{70} + 144 q^{71} - 54 q^{72} - 453 q^{73} - 489 q^{74} - 150 q^{76} - 36 q^{77} + 153 q^{78} - 96 q^{79} + 144 q^{80} + 249 q^{82} - 99 q^{83} + 135 q^{84} - 573 q^{85} - 33 q^{86} + 207 q^{87} + 360 q^{88} + 795 q^{89} + 117 q^{90} + 414 q^{91} + 285 q^{92} + 306 q^{93} + 198 q^{95} - 306 q^{96} - 483 q^{97} - 39 q^{98} + 117 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 48 x^{16} + 936 x^{14} + 9539 x^{12} + 54576 x^{10} + 176517 x^{8} + 313396 x^{6} + 277917 x^{4} + \cdots + 8427 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 528 \nu^{16} + 23835 \nu^{14} + 428502 \nu^{12} + 3898957 \nu^{10} + 18838485 \nu^{8} + \cdots + 1939005 ) / 348512 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 528 \nu^{16} + 23835 \nu^{14} + 428502 \nu^{12} + 3898957 \nu^{10} + 18838485 \nu^{8} + \cdots + 1939005 ) / 348512 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4233970 \nu^{17} + 67591377 \nu^{16} + 195138573 \nu^{15} + 3064072623 \nu^{14} + \cdots + 211497508855 ) / 25619465632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4233970 \nu^{17} - 67591377 \nu^{16} + 195138573 \nu^{15} - 3064072623 \nu^{14} + \cdots - 211497508855 ) / 25619465632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4126339 \nu^{16} - 187667349 \nu^{14} - 3391398671 \nu^{12} - 30889278138 \nu^{10} + \cdots - 11315191157 ) / 483386144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10451946 \nu^{17} + 228795117 \nu^{16} + 411996102 \nu^{15} + 10517126603 \nu^{14} + \cdots + 1270873451663 ) / 51238931264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10451946 \nu^{17} - 228795117 \nu^{16} + 411996102 \nu^{15} - 10517126603 \nu^{14} + \cdots - 1270873451663 ) / 51238931264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12195 \nu^{17} - 557376 \nu^{15} - 10151265 \nu^{13} - 93617499 \nu^{11} - 458909599 \nu^{9} + \cdots + 9235568 ) / 18471136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53286777 \nu^{17} - 353878721 \nu^{16} - 2443278565 \nu^{15} - 16074514743 \nu^{14} + \cdots - 1176416942823 ) / 51238931264 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 61754717 \nu^{17} + 218695967 \nu^{16} - 2833555711 \nu^{15} + 9946369497 \nu^{14} + \cdots + 702182993849 ) / 51238931264 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37599049 \nu^{17} - 53182479 \nu^{16} - 1665590132 \nu^{15} - 2444264229 \nu^{14} + \cdots - 184568256767 ) / 25619465632 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 37599049 \nu^{17} + 53182479 \nu^{16} - 1665590132 \nu^{15} + 2444264229 \nu^{14} + \cdots + 184568256767 ) / 25619465632 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50676912 \nu^{17} - 8091987 \nu^{16} + 2290992111 \nu^{15} - 350290886 \nu^{14} + \cdots - 22869932374 ) / 25619465632 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 103185005 \nu^{17} + 85284526 \nu^{16} + 4700470019 \nu^{15} + 3870953792 \nu^{14} + \cdots + 184278644044 ) / 51238931264 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 54910882 \nu^{17} - 75683364 \nu^{16} - 2486130684 \nu^{15} - 3414363509 \nu^{14} + \cdots - 259986906861 ) / 25619465632 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 111652945 \nu^{17} + 49898228 \nu^{16} + 5090747165 \nu^{15} + 2257191454 \nu^{14} + \cdots + 238716373666 ) / 51238931264 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 183438319 \nu^{17} - 218695967 \nu^{16} - 8369280891 \nu^{15} - 9946369497 \nu^{14} + \cdots - 599705131321 ) / 51238931264 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{16} + \beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + 4 \beta_{8} + 2 \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{16} - \beta_{15} + 3 \beta_{14} - \beta_{13} - 14 \beta_{10} + 14 \beta_{9} - \beta_{7} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{17} - 38 \beta_{16} - 20 \beta_{15} - 38 \beta_{14} + 20 \beta_{13} - 12 \beta_{12} - 12 \beta_{11} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 86 \beta_{16} + 13 \beta_{15} - 86 \beta_{14} + 13 \beta_{13} + 9 \beta_{12} - 9 \beta_{11} + 169 \beta_{10} + \cdots - 452 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 60 \beta_{17} + 567 \beta_{16} + 320 \beta_{15} + 567 \beta_{14} - 320 \beta_{13} + 121 \beta_{12} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1543 \beta_{16} - 122 \beta_{15} + 1543 \beta_{14} - 122 \beta_{13} - 229 \beta_{12} + 229 \beta_{11} + \cdots + 4919 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1122 \beta_{17} - 7811 \beta_{16} - 4666 \beta_{15} - 7811 \beta_{14} + 4666 \beta_{13} - 1153 \beta_{12} + \cdots + 606 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 23433 \beta_{16} + 817 \beta_{15} - 23433 \beta_{14} + 817 \beta_{13} + 4074 \beta_{12} - 4074 \beta_{11} + \cdots - 55244 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 17670 \beta_{17} + 103578 \beta_{16} + 64701 \beta_{15} + 103578 \beta_{14} - 64701 \beta_{13} + \cdots - 1413 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 329095 \beta_{16} - 816 \beta_{15} + 329095 \beta_{14} - 816 \beta_{13} - 62973 \beta_{12} + \cdots + 634845 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 256274 \beta_{17} - 1344751 \beta_{16} - 870407 \beta_{15} - 1344751 \beta_{14} + 870407 \beta_{13} + \cdots - 41287 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4431334 \beta_{16} - 102040 \beta_{15} - 4431334 \beta_{14} - 102040 \beta_{13} + 906228 \beta_{12} + \cdots - 7432313 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3549828 \beta_{17} + 17242502 \beta_{16} + 11483820 \beta_{15} + 17242502 \beta_{14} - 11483820 \beta_{13} + \cdots + 1075586 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 58221702 \beta_{16} + 2542956 \beta_{15} + 58221702 \beta_{14} + 2542956 \beta_{13} - 12509044 \beta_{12} + \cdots + 88349901 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 47801988 \beta_{17} - 219422544 \beta_{16} - 149548591 \beta_{15} - 219422544 \beta_{14} + \cdots - 18759623 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
1.45784i
− 1.57028i
− 3.14274i
− 2.85524i
0.358663i
3.09812i
2.85524i
− 0.358663i
− 3.09812i
− 1.60177i
0.707729i
3.54770i
− 1.45784i
1.57028i
3.14274i
1.60177i
− 0.707729i
− 3.54770i
−1.43569 + 0.253151i 1.11334 + 1.32683i −1.76165 + 0.641190i 3.63888 + 1.32445i −1.93430 − 1.62307i −6.51740 + 11.2885i 7.41696 − 4.28219i −0.520945 + 2.95442i −5.55959 − 0.980306i
10.2 1.54643 − 0.272677i 1.11334 + 1.32683i −1.44168 + 0.524730i 4.30798 + 1.56798i 2.08350 + 1.74826i 3.52548 − 6.10631i −7.52600 + 4.34514i −0.520945 + 2.95442i 7.08954 + 1.25008i
10.3 3.09500 − 0.545732i 1.11334 + 1.32683i 5.52242 − 2.00999i −8.12052 − 2.95563i 4.16988 + 3.49894i −1.67445 + 2.90024i 5.10816 − 2.94920i −0.520945 + 2.95442i −26.7460 − 4.71603i
13.1 −1.83532 − 2.18724i 0.592396 − 1.62760i −0.721060 + 4.08934i −0.943682 − 5.35188i −4.64718 + 1.69144i −1.78953 + 3.09955i 0.376890 − 0.217598i −2.29813 − 1.92836i −9.97392 + 11.8865i
13.2 0.230544 + 0.274751i 0.592396 − 1.62760i 0.672255 − 3.81255i 0.125095 + 0.709447i 0.583757 − 0.212470i 1.07822 − 1.86753i 2.44493 − 1.41158i −2.29813 − 1.92836i −0.166082 + 0.197929i
13.3 1.99143 + 2.37330i 0.592396 − 1.62760i −0.972139 + 5.51328i 0.0525425 + 0.297983i 5.04248 − 1.83531i −1.79984 + 3.11741i −4.28839 + 2.47590i −2.29813 − 1.92836i −0.602568 + 0.718112i
22.1 −1.83532 + 2.18724i 0.592396 + 1.62760i −0.721060 − 4.08934i −0.943682 + 5.35188i −4.64718 − 1.69144i −1.78953 − 3.09955i 0.376890 + 0.217598i −2.29813 + 1.92836i −9.97392 − 11.8865i
22.2 0.230544 − 0.274751i 0.592396 + 1.62760i 0.672255 + 3.81255i 0.125095 − 0.709447i 0.583757 + 0.212470i 1.07822 + 1.86753i 2.44493 + 1.41158i −2.29813 + 1.92836i −0.166082 − 0.197929i
22.3 1.99143 − 2.37330i 0.592396 + 1.62760i −0.972139 − 5.51328i 0.0525425 − 0.297983i 5.04248 + 1.83531i −1.79984 − 3.11741i −4.28839 − 2.47590i −2.29813 + 1.92836i −0.602568 − 0.718112i
34.1 −0.547838 + 1.50517i −1.70574 + 0.300767i 1.09876 + 0.921971i −3.62469 + 3.04147i 0.481760 − 2.73220i −1.80909 + 3.13344i −7.53836 + 4.35228i 2.81908 − 1.02606i −2.59220 − 7.12201i
34.2 0.242057 − 0.665047i −1.70574 + 0.300767i 2.68048 + 2.24919i 5.34756 − 4.48713i −0.212862 + 1.20720i −0.504300 + 0.873473i 4.59629 − 2.65367i 2.81908 − 1.02606i −1.68974 − 4.64252i
34.3 1.21338 − 3.33374i −1.70574 + 0.300767i −6.57738 − 5.51907i −0.783175 + 0.657162i −1.06703 + 6.05144i 4.99091 − 8.64451i −14.0905 + 8.13515i 2.81908 − 1.02606i 1.24052 + 3.40830i
40.1 −1.43569 − 0.253151i 1.11334 − 1.32683i −1.76165 − 0.641190i 3.63888 − 1.32445i −1.93430 + 1.62307i −6.51740 − 11.2885i 7.41696 + 4.28219i −0.520945 − 2.95442i −5.55959 + 0.980306i
40.2 1.54643 + 0.272677i 1.11334 − 1.32683i −1.44168 − 0.524730i 4.30798 − 1.56798i 2.08350 − 1.74826i 3.52548 + 6.10631i −7.52600 − 4.34514i −0.520945 − 2.95442i 7.08954 − 1.25008i
40.3 3.09500 + 0.545732i 1.11334 − 1.32683i 5.52242 + 2.00999i −8.12052 + 2.95563i 4.16988 − 3.49894i −1.67445 − 2.90024i 5.10816 + 2.94920i −0.520945 − 2.95442i −26.7460 + 4.71603i
52.1 −0.547838 − 1.50517i −1.70574 − 0.300767i 1.09876 − 0.921971i −3.62469 − 3.04147i 0.481760 + 2.73220i −1.80909 − 3.13344i −7.53836 − 4.35228i 2.81908 + 1.02606i −2.59220 + 7.12201i
52.2 0.242057 + 0.665047i −1.70574 − 0.300767i 2.68048 − 2.24919i 5.34756 + 4.48713i −0.212862 − 1.20720i −0.504300 − 0.873473i 4.59629 + 2.65367i 2.81908 + 1.02606i −1.68974 + 4.64252i
52.3 1.21338 + 3.33374i −1.70574 − 0.300767i −6.57738 + 5.51907i −0.783175 − 0.657162i −1.06703 − 6.05144i 4.99091 + 8.64451i −14.0905 − 8.13515i 2.81908 + 1.02606i 1.24052 − 3.40830i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.k.a ✓ 18
3.b odd 2 1 171.3.ba.c 18
19.f odd 18 1 inner 57.3.k.a ✓ 18
57.j even 18 1 171.3.ba.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.k.a ✓ 18 1.a even 1 1 trivial
57.3.k.a ✓ 18 19.f odd 18 1 inner
171.3.ba.c 18 3.b odd 2 1
171.3.ba.c 18 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 9 T_{2}^{17} + 42 T_{2}^{16} - 126 T_{2}^{15} + 273 T_{2}^{14} - 759 T_{2}^{13} + \cdots + 8427 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 9 T^{17} + \cdots + 8427 \) Copy content Toggle raw display
$3$ \( (T^{6} + 9 T^{3} + 27)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} - 69 T^{16} + \cdots + 37662769 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 97026643081 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 548969445429409 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 42\!\cdots\!43 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 219095266685689 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 59\!\cdots\!83 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 51\!\cdots\!07 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 16\!\cdots\!23 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 14\!\cdots\!87 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 42\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 73\!\cdots\!67 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 52\!\cdots\!07 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 23\!\cdots\!67 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 20\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 35\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 65\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 41\!\cdots\!47 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 24\!\cdots\!63 \) Copy content Toggle raw display
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