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Properties

Label 504.2.cs.a
Level $504$
Weight $2$
Character orbit 504.cs
Analytic conductor $4.024$
Analytic rank $0$
Dimension $72$
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] = [504,2,Mod(85,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.cs (of order \(6\), degree \(2\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{6} + 36 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{6} + 36 q^{7} + 6 q^{8} + 20 q^{12} + 40 q^{17} - q^{18} + 2 q^{20} - 12 q^{22} + 12 q^{23} + 10 q^{24} + 36 q^{25} - 42 q^{26} + 4 q^{30} + 5 q^{32} + 8 q^{33} + 6 q^{34} - 18 q^{36} - 25 q^{38} - 4 q^{39} - 9 q^{40} + 24 q^{41} - 7 q^{42} + 12 q^{46} - 53 q^{48} - 36 q^{49} + 41 q^{50} - 9 q^{52} + 12 q^{54} + 3 q^{56} + 4 q^{57} + 9 q^{58} - 50 q^{60} + 60 q^{62} - 6 q^{64} - 40 q^{65} - 2 q^{66} + 23 q^{68} - 56 q^{71} + 19 q^{72} + 22 q^{74} - 24 q^{76} - 54 q^{78} - 6 q^{80} - 4 q^{81} - 48 q^{82} + 10 q^{84} - 39 q^{86} - 76 q^{87} - 12 q^{88} - 88 q^{89} - 77 q^{90} + 11 q^{92} + 24 q^{94} - 24 q^{95} + 59 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −1.41413 + 0.0151305i −1.72706 + 0.131454i 1.99954 − 0.0427932i −1.26304 − 0.729219i 2.44030 − 0.212025i 0.500000 + 0.866025i −2.82697 + 0.0907694i 2.96544 − 0.454056i 1.79715 + 1.01210i
85.2 −1.39778 − 0.214989i 0.0497101 + 1.73134i 1.90756 + 0.601014i 0.261066 + 0.150726i 0.302736 − 2.43071i 0.500000 + 0.866025i −2.53713 − 1.25019i −2.99506 + 0.172130i −0.332507 − 0.266808i
85.3 −1.38589 − 0.281624i 1.31821 − 1.12353i 1.84138 + 0.780600i 2.74638 + 1.58562i −2.14331 + 1.18585i 0.500000 + 0.866025i −2.33211 − 1.60040i 0.475349 − 2.96210i −3.35962 − 2.97094i
85.4 −1.31153 + 0.529035i 0.349968 − 1.69633i 1.44024 − 1.38770i −2.84022 − 1.63980i 0.438422 + 2.40993i 0.500000 + 0.866025i −1.15479 + 2.58195i −2.75505 − 1.18732i 4.59255 + 0.648078i
85.5 −1.27427 − 0.613390i 1.34597 + 1.09012i 1.24750 + 1.56324i −2.33508 − 1.34816i −1.04645 − 2.21471i 0.500000 + 0.866025i −0.630773 − 2.75720i 0.623266 + 2.93454i 2.14857 + 3.15023i
85.6 −1.25626 + 0.649466i 1.65525 − 0.510053i 1.15639 − 1.63180i 1.14688 + 0.662153i −1.74816 + 1.71579i 0.500000 + 0.866025i −0.392930 + 2.80100i 2.47969 − 1.68853i −1.87083 − 0.0869767i
85.7 −1.20390 + 0.742040i −1.50659 + 0.854512i 0.898754 − 1.78668i 3.20115 + 1.84819i 1.17970 − 2.14670i 0.500000 + 0.866025i 0.243781 + 2.81790i 1.53962 − 2.57480i −5.22530 + 0.150350i
85.8 −1.14128 − 0.835149i 1.40449 − 1.01361i 0.605053 + 1.90628i −1.92732 − 1.11274i −2.44944 − 0.0161356i 0.500000 + 0.866025i 0.901492 − 2.68092i 0.945172 − 2.84722i 1.27031 + 2.87955i
85.9 −0.972917 − 1.02637i −0.120790 − 1.72783i −0.106865 + 1.99714i 2.23744 + 1.29179i −1.65588 + 1.80501i 0.500000 + 0.866025i 2.15378 − 1.83337i −2.97082 + 0.417410i −0.850996 − 3.55324i
85.10 −0.930716 + 1.06479i −1.06868 − 1.36306i −0.267536 − 1.98203i 0.220108 + 0.127079i 2.44600 + 0.130709i 0.500000 + 0.866025i 2.35943 + 1.55983i −0.715859 + 2.91334i −0.340170 + 0.116093i
85.11 −0.870602 + 1.11447i −0.119319 + 1.72794i −0.484104 − 1.94053i −0.525102 − 0.303168i −1.82186 − 1.63732i 0.500000 + 0.866025i 2.58413 + 1.14991i −2.97153 − 0.412353i 0.795028 − 0.321274i
85.12 −0.828761 − 1.14593i 1.26004 + 1.18840i −0.626309 + 1.89940i 3.33304 + 1.92433i 0.317546 − 2.42882i 0.500000 + 0.866025i 2.69564 − 0.856447i 0.175418 + 2.99487i −0.557147 − 5.41424i
85.13 −0.589905 + 1.28531i 1.55592 + 0.760985i −1.30402 − 1.51642i −3.48956 − 2.01470i −1.89595 + 1.55093i 0.500000 + 0.866025i 2.71831 − 0.781526i 1.84180 + 2.36807i 4.64802 − 3.29667i
85.14 −0.578023 − 1.29069i −1.26004 − 1.18840i −1.33178 + 1.49210i −3.33304 − 1.92433i −0.805523 + 2.31325i 0.500000 + 0.866025i 2.69564 + 0.856447i 0.175418 + 2.99487i −0.557147 + 5.41424i
85.15 −0.402403 − 1.35576i 0.120790 + 1.72783i −1.67614 + 1.09112i −2.23744 − 1.29179i 2.29391 − 0.859046i 0.500000 + 0.866025i 2.15378 + 1.83337i −2.97082 + 0.417410i −0.850996 + 3.55324i
85.16 −0.305084 + 1.38091i 1.64418 + 0.544667i −1.81385 − 0.842588i 2.23227 + 1.28880i −1.25375 + 2.10431i 0.500000 + 0.866025i 1.71692 − 2.24771i 2.40668 + 1.79106i −2.46075 + 2.68938i
85.17 −0.248810 + 1.39215i −1.73065 − 0.0695272i −1.87619 − 0.692764i 1.22717 + 0.708510i 0.527397 − 2.39204i 0.500000 + 0.866025i 1.43125 − 2.43958i 2.99033 + 0.240655i −1.29169 + 1.53213i
85.18 −0.152619 − 1.40595i −1.40449 + 1.01361i −1.95342 + 0.429150i 1.92732 + 1.11274i 1.63945 + 1.81995i 0.500000 + 0.866025i 0.901492 + 2.68092i 0.945172 − 2.84722i 1.27031 − 2.87955i
85.19 0.105921 − 1.41024i −1.34597 − 1.09012i −1.97756 − 0.298749i 2.33508 + 1.34816i −1.67990 + 1.78267i 0.500000 + 0.866025i −0.630773 + 2.75720i 0.623266 + 2.93454i 2.14857 − 3.15023i
85.20 0.109593 + 1.40996i 0.416131 − 1.68132i −1.97598 + 0.309044i −0.226907 − 0.131005i 2.41620 + 0.402467i 0.500000 + 0.866025i −0.652293 − 2.75218i −2.65367 − 1.39930i 0.159844 − 0.334288i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cs.a ✓ 72
8.b even 2 1 inner 504.2.cs.a ✓ 72
9.c even 3 1 inner 504.2.cs.a ✓ 72
72.n even 6 1 inner 504.2.cs.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cs.a ✓ 72 1.a even 1 1 trivial
504.2.cs.a ✓ 72 8.b even 2 1 inner
504.2.cs.a ✓ 72 9.c even 3 1 inner
504.2.cs.a ✓ 72 72.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 108 T_{5}^{70} + 6402 T_{5}^{68} - 262128 T_{5}^{66} + 8189076 T_{5}^{64} + \cdots + 112392565559296 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display