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Properties

Label 570.2.bf.a
Level $570$
Weight $2$
Character orbit 570.bf
Analytic conductor $4.551$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [570,2,Mod(29,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.bf (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: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 30 q^{15} + 48 q^{19} + 12 q^{25} - 48 q^{39} + 36 q^{45} - 72 q^{46} + 216 q^{49} - 180 q^{51} + 36 q^{54} + 60 q^{55} + 6 q^{60} - 72 q^{61} + 120 q^{64} - 156 q^{66} + 48 q^{79} - 120 q^{81} - 108 q^{84} - 48 q^{85} - 36 q^{90} - 24 q^{91} - 216 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.984808 + 0.173648i −1.70539 − 0.302753i 0.939693 − 0.342020i 0.247201 − 2.22236i 1.73205 + 0.00201678i 4.20953 + 2.43037i −0.866025 + 0.500000i 2.81668 + 1.03262i 0.142463 + 2.23153i
29.2 −0.984808 + 0.173648i −1.64517 + 0.541678i 0.939693 − 0.342020i 2.21549 − 0.302666i 1.52611 − 0.819129i −2.76521 − 1.59649i −0.866025 + 0.500000i 2.41317 − 1.78230i −2.12927 + 0.682783i
29.3 −0.984808 + 0.173648i −1.62527 − 0.598755i 0.939693 − 0.342020i −2.17153 + 0.533326i 1.70455 + 0.307434i 0.260757 + 0.150548i −0.866025 + 0.500000i 2.28299 + 1.94627i 2.04593 − 0.902307i
29.4 −0.984808 + 0.173648i −1.37297 + 1.05591i 0.939693 − 0.342020i 1.98695 + 1.02568i 1.16876 − 1.27828i 3.69762 + 2.13482i −0.866025 + 0.500000i 0.770112 − 2.89947i −2.13487 − 0.665072i
29.5 −0.984808 + 0.173648i −1.23853 − 1.21080i 0.939693 − 0.342020i 1.86795 − 1.22913i 1.42997 + 0.977341i −1.22822 − 0.709111i −0.866025 + 0.500000i 0.0679076 + 2.99923i −1.62614 + 1.53482i
29.6 −0.984808 + 0.173648i −1.20820 + 1.24107i 0.939693 − 0.342020i −1.35867 − 1.77595i 0.974336 − 1.43202i −1.28021 − 0.739129i −0.866025 + 0.500000i −0.0805041 − 2.99892i 1.64642 + 1.51304i
29.7 −0.984808 + 0.173648i −1.03267 + 1.39054i 0.939693 − 0.342020i −1.21793 + 1.87527i 0.775516 − 1.54873i −1.48423 − 0.856923i −0.866025 + 0.500000i −0.867188 − 2.87193i 0.873794 − 2.05827i
29.8 −0.984808 + 0.173648i −0.977341 − 1.42997i 0.939693 − 0.342020i 0.640862 + 2.14226i 1.21080 + 1.23853i 1.22822 + 0.709111i −0.866025 + 0.500000i −1.08961 + 2.79513i −1.00313 − 1.99843i
29.9 −0.984808 + 0.173648i −0.307434 − 1.70455i 0.939693 − 0.342020i −1.32068 − 1.80439i 0.598755 + 1.62527i −0.260757 − 0.150548i −0.866025 + 0.500000i −2.81097 + 1.04807i 1.61394 + 1.54764i
29.10 −0.984808 + 0.173648i −0.00201678 − 1.73205i 0.939693 − 0.342020i −1.23914 + 1.86133i 0.302753 + 1.70539i −4.20953 − 2.43037i −0.866025 + 0.500000i −2.99999 + 0.00698632i 0.897098 − 2.04822i
29.11 −0.984808 + 0.173648i 0.144472 + 1.72602i 0.939693 − 0.342020i −2.21146 − 0.330832i −0.441997 − 1.67471i 2.42894 + 1.40235i −0.866025 + 0.500000i −2.95826 + 0.498722i 2.23531 − 0.0582100i
29.12 −0.984808 + 0.173648i 0.438663 + 1.67558i 0.939693 − 0.342020i 2.09802 − 0.773495i −0.722960 − 1.57395i 1.27073 + 0.733656i −0.866025 + 0.500000i −2.61515 + 1.47003i −1.93183 + 1.12606i
29.13 −0.984808 + 0.173648i 0.819129 − 1.52611i 0.939693 − 0.342020i 1.50261 + 1.65594i −0.541678 + 1.64517i 2.76521 + 1.59649i −0.866025 + 0.500000i −1.65805 − 2.50017i −1.76734 − 1.36986i
29.14 −0.984808 + 0.173648i 0.917943 + 1.46880i 0.939693 − 0.342020i 0.501847 − 2.17902i −1.15905 − 1.28709i −3.34801 − 1.93297i −0.866025 + 0.500000i −1.31476 + 2.69655i −0.115839 + 2.23307i
29.15 −0.984808 + 0.173648i 1.27828 − 1.16876i 0.939693 − 0.342020i 2.18139 + 0.491467i −1.05591 + 1.37297i −3.69762 − 2.13482i −0.866025 + 0.500000i 0.268009 − 2.98800i −2.23359 − 0.105206i
29.16 −0.984808 + 0.173648i 1.28709 + 1.15905i 0.939693 − 0.342020i −1.01621 + 1.99181i −1.46880 − 0.917943i 3.34801 + 1.93297i −0.866025 + 0.500000i 0.313195 + 2.98361i 0.654901 − 2.13801i
29.17 −0.984808 + 0.173648i 1.43202 − 0.974336i 0.939693 − 0.342020i −2.18236 + 0.487123i −1.24107 + 1.20820i 1.28021 + 0.739129i −0.866025 + 0.500000i 1.10134 − 2.79053i 2.06462 − 0.858686i
29.18 −0.984808 + 0.173648i 1.54873 − 0.775516i 0.939693 − 0.342020i 0.272408 − 2.21941i −1.39054 + 1.03267i 1.48423 + 0.856923i −0.866025 + 0.500000i 1.79715 − 2.40214i 0.117127 + 2.23300i
29.19 −0.984808 + 0.173648i 1.57395 + 0.722960i 0.939693 − 0.342020i 1.10999 + 1.94112i −1.67558 − 0.438663i −1.27073 − 0.733656i −0.866025 + 0.500000i 1.95466 + 2.27581i −1.43019 − 1.71888i
29.20 −0.984808 + 0.173648i 1.67471 + 0.441997i 0.939693 − 0.342020i −1.90673 − 1.16807i −1.72602 − 0.144472i −2.42894 − 1.40235i −0.866025 + 0.500000i 2.60928 + 1.48043i 2.08060 + 0.819221i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
19.f odd 18 1 inner
57.j even 18 1 inner
95.o odd 18 1 inner
285.bf even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bf.a ✓ 240
3.b odd 2 1 inner 570.2.bf.a ✓ 240
5.b even 2 1 inner 570.2.bf.a ✓ 240
15.d odd 2 1 inner 570.2.bf.a ✓ 240
19.f odd 18 1 inner 570.2.bf.a ✓ 240
57.j even 18 1 inner 570.2.bf.a ✓ 240
95.o odd 18 1 inner 570.2.bf.a ✓ 240
285.bf even 18 1 inner 570.2.bf.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bf.a ✓ 240 1.a even 1 1 trivial
570.2.bf.a ✓ 240 3.b odd 2 1 inner
570.2.bf.a ✓ 240 5.b even 2 1 inner
570.2.bf.a ✓ 240 15.d odd 2 1 inner
570.2.bf.a ✓ 240 19.f odd 18 1 inner
570.2.bf.a ✓ 240 57.j even 18 1 inner
570.2.bf.a ✓ 240 95.o odd 18 1 inner
570.2.bf.a ✓ 240 285.bf even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(570, [\chi])\).