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Properties

Label 57.3.l.b
Level $57$
Weight $3$
Character orbit 57.l
Analytic conductor $1.553$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,3,Mod(5,57)]
 
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([9, 16]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.l (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: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 12 q^{3} - 18 q^{4} + 6 q^{6} - 6 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 12 q^{3} - 18 q^{4} + 6 q^{6} - 6 q^{7} - 24 q^{9} + 18 q^{10} - 39 q^{12} - 66 q^{13} - 21 q^{15} - 126 q^{16} + 84 q^{18} + 96 q^{19} + 120 q^{21} + 132 q^{22} - 222 q^{24} + 24 q^{25} - 114 q^{27} + 168 q^{28} - 270 q^{30} - 6 q^{31} - 132 q^{33} - 30 q^{34} + 42 q^{36} - 120 q^{37} + 84 q^{39} - 234 q^{40} + 435 q^{42} + 90 q^{43} + 240 q^{45} + 114 q^{46} + 567 q^{48} + 324 q^{49} + 270 q^{51} + 300 q^{52} + 465 q^{54} - 234 q^{55} - 222 q^{57} - 144 q^{58} - 126 q^{60} - 750 q^{61} + 63 q^{63} - 516 q^{64} - 1140 q^{66} - 198 q^{67} - 120 q^{69} - 618 q^{70} - 624 q^{72} - 1086 q^{73} + 774 q^{75} + 252 q^{76} + 177 q^{78} + 690 q^{79} - 1056 q^{81} + 660 q^{82} - 69 q^{84} + 966 q^{85} - 483 q^{87} - 462 q^{88} - 474 q^{90} + 1158 q^{91} - 9 q^{93} + 48 q^{94} + 714 q^{96} - 324 q^{97} + 798 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.12972 − 3.10387i −1.21201 + 2.74427i −5.29358 + 4.44184i −6.01022 + 7.16270i 9.88710 + 0.661685i −3.00034 − 5.19675i 8.32497 + 4.80643i −6.06205 − 6.65219i 29.0219 + 10.5631i
5.2 −1.09382 − 3.00525i −1.53549 − 2.57726i −4.77090 + 4.00326i 1.45078 − 1.72897i −6.06575 + 7.43360i 0.103009 + 0.178417i 6.17070 + 3.56266i −4.28453 + 7.91472i −6.78288 − 2.46877i
5.3 −0.925756 − 2.54349i 2.50381 + 1.65256i −2.54816 + 2.13816i 3.40242 − 4.05485i 1.88535 − 7.89828i 0.519973 + 0.900619i −1.57903 − 0.911653i 3.53811 + 8.27537i −13.4633 − 4.90024i
5.4 −0.462070 − 1.26953i 2.02887 − 2.20991i 1.66599 − 1.39793i −2.23787 + 2.66699i −3.74302 − 1.55457i 1.02557 + 1.77634i −7.22452 − 4.17108i −0.767376 − 8.96723i 4.41986 + 1.60870i
5.5 −0.323559 − 0.888970i −2.90352 + 0.754709i 2.37860 − 1.99588i 2.89548 − 3.45070i 1.61037 + 2.33695i −4.61272 − 7.98946i −5.82101 − 3.36076i 7.86083 − 4.38262i −4.00443 − 1.45749i
5.6 0.323559 + 0.888970i 2.47029 + 1.70226i 2.37860 − 1.99588i −2.89548 + 3.45070i −0.713971 + 2.74679i −4.61272 − 7.98946i 5.82101 + 3.36076i 3.20465 + 8.41013i −4.00443 − 1.45749i
5.7 0.462070 + 1.26953i −1.15068 − 2.77055i 1.66599 − 1.39793i 2.23787 − 2.66699i 2.98559 − 2.74101i 1.02557 + 1.77634i 7.22452 + 4.17108i −6.35187 + 6.37603i 4.41986 + 1.60870i
5.8 0.925756 + 2.54349i −2.91802 + 0.696543i −2.54816 + 2.13816i −3.40242 + 4.05485i −4.47302 − 6.77713i 0.519973 + 0.900619i 1.57903 + 0.911653i 8.02966 − 4.06505i −13.4633 − 4.90024i
5.9 1.09382 + 3.00525i 2.32436 − 1.89666i −4.77090 + 4.00326i −1.45078 + 1.72897i 8.24238 + 4.91069i 0.103009 + 0.178417i −6.17070 − 3.56266i 1.80534 − 8.81707i −6.78288 − 2.46877i
5.10 1.12972 + 3.10387i 0.200324 + 2.99330i −5.29358 + 4.44184i 6.01022 − 7.16270i −9.06452 + 4.00337i −3.00034 − 5.19675i −8.32497 − 4.80643i −8.91974 + 1.19926i 29.0219 + 10.5631i
17.1 −3.27297 − 0.577112i 1.60029 − 2.53753i 6.62048 + 2.40966i 0.0213852 + 0.0587554i −6.70214 + 7.38171i −4.52481 − 7.83720i −8.76518 − 5.06058i −3.87814 − 8.12158i −0.0360846 − 0.204646i
17.2 −2.89794 − 0.510985i −2.92924 − 0.647739i 4.37817 + 1.59352i −0.0239176 − 0.0657130i 8.15776 + 3.37390i 3.71261 + 6.43043i −1.67977 − 0.969815i 8.16087 + 3.79476i 0.0357333 + 0.202654i
17.3 −2.08248 − 0.367197i 2.55197 + 1.57717i 0.443120 + 0.161283i 1.33747 + 3.67467i −4.73530 − 4.22149i 1.14864 + 1.98951i 6.46164 + 3.73063i 4.02510 + 8.04976i −1.43593 − 8.14354i
17.4 −1.49626 − 0.263831i −0.915797 + 2.85680i −1.58958 − 0.578559i −2.42476 − 6.66197i 2.12399 − 4.03291i −3.32542 − 5.75980i 7.48895 + 4.32374i −7.32263 − 5.23250i 1.87044 + 10.6078i
17.5 −0.317661 − 0.0560121i 1.96130 − 2.27009i −3.66100 − 1.33249i −2.70319 − 7.42696i −0.750180 + 0.611261i 4.29691 + 7.44246i 2.20570 + 1.27346i −1.30661 − 8.90465i 0.442698 + 2.51066i
17.6 0.317661 + 0.0560121i −1.89503 + 2.32570i −3.66100 − 1.33249i 2.70319 + 7.42696i −0.732242 + 0.632639i 4.29691 + 7.44246i −2.20570 − 1.27346i −1.81776 − 8.81452i 0.442698 + 2.51066i
17.7 1.49626 + 0.263831i 2.65437 − 1.39796i −1.58958 − 0.578559i 2.42476 + 6.66197i 4.34046 − 1.39141i −3.32542 − 5.75980i −7.48895 − 4.32374i 5.09140 − 7.42143i 1.87044 + 10.6078i
17.8 2.08248 + 0.367197i 1.99635 + 2.23933i 0.443120 + 0.161283i −1.33747 − 3.67467i 3.33508 + 5.39641i 1.14864 + 1.98951i −6.46164 − 3.73063i −1.02917 + 8.94096i −1.43593 − 8.14354i
17.9 2.89794 + 0.510985i −1.14656 − 2.77226i 4.37817 + 1.59352i 0.0239176 + 0.0657130i −1.90606 − 8.61970i 3.71261 + 6.43043i 1.67977 + 0.969815i −6.37082 + 6.35709i 0.0357333 + 0.202654i
17.10 3.27297 + 0.577112i −2.22109 + 2.01662i 6.62048 + 2.40966i −0.0213852 − 0.0587554i −8.43338 + 5.31850i −4.52481 − 7.83720i 8.76518 + 5.06058i 0.866517 − 8.95819i −0.0360846 − 0.204646i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.e even 9 1 inner
57.l 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.l.b ✓ 60
3.b odd 2 1 inner 57.3.l.b ✓ 60
19.e even 9 1 inner 57.3.l.b ✓ 60
57.l odd 18 1 inner 57.3.l.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.l.b ✓ 60 1.a even 1 1 trivial
57.3.l.b ✓ 60 3.b odd 2 1 inner
57.3.l.b ✓ 60 19.e even 9 1 inner
57.3.l.b ✓ 60 57.l odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 9 T_{2}^{58} + 126 T_{2}^{56} - 1983 T_{2}^{54} - 25104 T_{2}^{52} - 284040 T_{2}^{50} + \cdots + 15635088699129 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display