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Properties

Label 570.2.bi.a
Level $570$
Weight $2$
Character orbit 570.bi
Analytic conductor $4.551$
Analytic rank $0$
Dimension $480$
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(17,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 9, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.bi (of order \(36\), degree \(12\), 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: \(480\)
Relative dimension: \(40\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 36 q^{15} - 48 q^{18} + 24 q^{22} - 24 q^{25} + 72 q^{33} + 24 q^{43} - 36 q^{45} + 24 q^{51} - 120 q^{55} + 108 q^{57} - 48 q^{60} - 48 q^{61} - 36 q^{63} - 24 q^{66} + 48 q^{67} - 48 q^{70} - 48 q^{78} - 144 q^{81} - 48 q^{85} - 96 q^{87} - 168 q^{90} - 144 q^{91} - 228 q^{93} + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.573576 − 0.819152i −1.73167 + 0.0360929i −0.342020 + 0.939693i 1.13574 + 1.92616i 1.02281 + 1.39780i 0.166944 − 0.623042i 0.965926 − 0.258819i 2.99739 − 0.125002i 0.926389 − 2.03514i
17.2 −0.573576 − 0.819152i −1.66345 + 0.482617i −0.342020 + 0.939693i −0.967388 − 2.01598i 1.34945 + 1.08580i −1.11042 + 4.14413i 0.965926 − 0.258819i 2.53416 − 1.60562i −1.09652 + 1.94875i
17.3 −0.573576 − 0.819152i −1.61147 − 0.634960i −0.342020 + 0.939693i 1.56453 − 1.59758i 0.404170 + 1.68423i 0.997670 − 3.72335i 0.965926 − 0.258819i 2.19365 + 2.04644i −2.20604 − 0.365253i
17.4 −0.573576 − 0.819152i −1.42141 + 0.989747i −0.342020 + 0.939693i 2.23252 − 0.125940i 1.62604 + 0.596654i −0.138460 + 0.516742i 0.965926 − 0.258819i 1.04080 − 2.81367i −1.38368 − 1.75654i
17.5 −0.573576 − 0.819152i −1.16927 − 1.27781i −0.342020 + 0.939693i −0.0995518 − 2.23385i −0.376053 + 1.69073i −0.252738 + 0.943230i 0.965926 − 0.258819i −0.265597 + 2.98822i −1.77276 + 1.36283i
17.6 −0.573576 − 0.819152i −0.949742 − 1.44844i −0.342020 + 0.939693i 0.0415911 + 2.23568i −0.641746 + 1.60878i 0.955789 − 3.56705i 0.965926 − 0.258819i −1.19598 + 2.75130i 1.80751 − 1.31640i
17.7 −0.573576 − 0.819152i −0.879843 + 1.49194i −0.342020 + 0.939693i −1.99640 + 1.00718i 1.72678 − 0.135014i 0.945547 − 3.52883i 0.965926 − 0.258819i −1.45175 − 2.62534i 1.97012 + 1.05766i
17.8 −0.573576 − 0.819152i −0.749446 + 1.56152i −0.342020 + 0.939693i −0.948601 + 2.02488i 1.70898 − 0.281738i −1.34429 + 5.01697i 0.965926 − 0.258819i −1.87666 − 2.34054i 2.20278 − 0.384377i
17.9 −0.573576 − 0.819152i −0.580543 + 1.63186i −0.342020 + 0.939693i 0.846732 − 2.06955i 1.66973 − 0.460444i 0.562193 − 2.09813i 0.965926 − 0.258819i −2.32594 − 1.89473i −2.18094 + 0.493443i
17.10 −0.573576 − 0.819152i −0.377684 − 1.69037i −0.342020 + 0.939693i −2.19306 − 0.436449i −1.16804 + 1.27894i −0.411556 + 1.53595i 0.965926 − 0.258819i −2.71471 + 1.27685i 0.900369 + 2.04679i
17.11 −0.573576 − 0.819152i 0.0306508 − 1.73178i −0.342020 + 0.939693i 1.63662 + 1.52364i −1.43617 + 0.968200i −0.718669 + 2.68211i 0.965926 − 0.258819i −2.99812 − 0.106161i 0.309371 − 2.21456i
17.12 −0.573576 − 0.819152i 0.530463 + 1.64882i −0.342020 + 0.939693i 1.62022 + 1.54107i 1.04637 − 1.38025i −0.100091 + 0.373543i 0.965926 − 0.258819i −2.43722 + 1.74928i 0.333051 − 2.21113i
17.13 −0.573576 − 0.819152i 0.721746 + 1.57451i −0.342020 + 0.939693i −2.18122 − 0.492217i 0.875787 − 1.49432i 0.145303 − 0.542278i 0.965926 − 0.258819i −1.95817 + 2.27279i 0.847896 + 2.06908i
17.14 −0.573576 − 0.819152i 0.986006 − 1.42401i −0.342020 + 0.939693i −1.55782 + 1.60412i −1.73203 + 0.00908702i −0.120110 + 0.448257i 0.965926 − 0.258819i −1.05558 − 2.80816i 2.20755 + 0.356004i
17.15 −0.573576 − 0.819152i 1.09670 − 1.34062i −0.342020 + 0.939693i −1.34788 − 1.78416i −1.72721 − 0.129414i 1.18632 − 4.42739i 0.965926 − 0.258819i −0.594516 − 2.94050i −0.688389 + 2.12747i
17.16 −0.573576 − 0.819152i 1.16230 − 1.28416i −0.342020 + 0.939693i 1.22020 − 1.87380i −1.71859 − 0.215538i −0.489365 + 1.82633i 0.965926 − 0.258819i −0.298119 − 2.98515i −2.23480 + 0.0752363i
17.17 −0.573576 − 0.819152i 1.47068 + 0.914939i −0.342020 + 0.939693i 1.79088 − 1.33893i −0.0940712 − 1.72949i −0.768695 + 2.86881i 0.965926 − 0.258819i 1.32577 + 2.69116i −2.12400 − 0.699027i
17.18 −0.573576 − 0.819152i 1.67662 + 0.434671i −0.342020 + 0.939693i 2.18570 + 0.471924i −0.605609 − 1.62273i 0.724038 − 2.70214i 0.965926 − 0.258819i 2.62212 + 1.45756i −0.867089 − 2.06111i
17.19 −0.573576 − 0.819152i 1.72809 + 0.117004i −0.342020 + 0.939693i −2.18895 − 0.456614i −0.895350 − 1.48268i −0.960813 + 3.58580i 0.965926 − 0.258819i 2.97262 + 0.404389i 0.881494 + 2.05499i
17.20 −0.573576 − 0.819152i 1.73128 − 0.0515684i −0.342020 + 0.939693i −0.793861 + 2.09040i −1.03527 − 1.38861i 0.731406 − 2.72965i 0.965926 − 0.258819i 2.99468 − 0.178559i 2.16770 − 0.548713i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
19.e even 9 1 inner
57.l odd 18 1 inner
95.q odd 36 1 inner
285.bi even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bi.a ✓ 480
3.b odd 2 1 inner 570.2.bi.a ✓ 480
5.c odd 4 1 inner 570.2.bi.a ✓ 480
15.e even 4 1 inner 570.2.bi.a ✓ 480
19.e even 9 1 inner 570.2.bi.a ✓ 480
57.l odd 18 1 inner 570.2.bi.a ✓ 480
95.q odd 36 1 inner 570.2.bi.a ✓ 480
285.bi even 36 1 inner 570.2.bi.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bi.a ✓ 480 1.a even 1 1 trivial
570.2.bi.a ✓ 480 3.b odd 2 1 inner
570.2.bi.a ✓ 480 5.c odd 4 1 inner
570.2.bi.a ✓ 480 15.e even 4 1 inner
570.2.bi.a ✓ 480 19.e even 9 1 inner
570.2.bi.a ✓ 480 57.l odd 18 1 inner
570.2.bi.a ✓ 480 95.q odd 36 1 inner
570.2.bi.a ✓ 480 285.bi even 36 1 inner

Hecke kernels

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