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Properties

Label 504.2.cs.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{6} - 36 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{6} - 36 q^{7} + 6 q^{8} - 8 q^{12} - 40 q^{17} - 21 q^{18} + 12 q^{20} + 12 q^{22} + 12 q^{23} - 12 q^{24} + 36 q^{25} - 14 q^{26} - 60 q^{30} - 15 q^{32} + 8 q^{33} + 6 q^{34} + 18 q^{36} - 3 q^{38} - 20 q^{39} + 21 q^{40} - 32 q^{41} - 13 q^{42} - 64 q^{44} + 12 q^{46} + 29 q^{48} - 36 q^{49} + 5 q^{50} - 9 q^{52} + 30 q^{54} - 3 q^{56} + 4 q^{57} + 9 q^{58} + 34 q^{60} - 12 q^{62} - 54 q^{64} + 40 q^{65} + 120 q^{66} + 55 q^{68} - 56 q^{71} + 15 q^{72} - 22 q^{74} + 12 q^{76} + 62 q^{78} + 94 q^{80} - 4 q^{81} + 12 q^{82} + 4 q^{84} - 3 q^{86} - 28 q^{87} - 12 q^{88} + 88 q^{89} - 83 q^{90} + 55 q^{92} - 18 q^{94} - 40 q^{95} - 83 q^{96}+O(q^{100}) \) 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 −1.40280 + 0.179343i −0.897491 + 1.48139i 1.93567 − 0.503164i 0.991981 + 0.572721i 0.993320 − 2.23904i −0.500000 − 0.866025i −2.62511 + 1.05299i −1.38902 − 2.65906i −1.49426 − 0.625505i
85.2 −1.40215 − 0.184326i 1.61889 + 0.615780i 1.93205 + 0.516905i 2.02215 + 1.16749i −2.15643 − 1.16182i −0.500000 − 0.866025i −2.61374 − 1.08091i 2.24163 + 1.99376i −2.62016 − 2.00973i
85.3 −1.35959 − 0.389240i 0.885523 − 1.48857i 1.69698 + 1.05841i −0.313644 − 0.181082i −1.78336 + 1.67917i −0.500000 − 0.866025i −1.89523 − 2.09955i −1.43170 − 2.63633i 0.355944 + 0.368281i
85.4 −1.35007 + 0.421096i 0.234769 + 1.71607i 1.64536 − 1.13701i −3.20237 − 1.84889i −1.03958 − 2.21794i −0.500000 − 0.866025i −1.74255 + 2.22790i −2.88977 + 0.805757i 5.10197 + 1.14762i
85.5 −1.30114 − 0.554117i −1.71503 − 0.242257i 1.38591 + 1.44196i 3.59330 + 2.07459i 2.09724 + 1.26553i −0.500000 − 0.866025i −1.00424 − 2.64415i 2.88262 + 0.830955i −3.52580 − 4.69043i
85.6 −1.28916 + 0.581429i −1.31449 − 1.12788i 1.32388 − 1.49911i 1.50625 + 0.869635i 2.35037 + 0.689740i −0.500000 − 0.866025i −0.835071 + 2.70234i 0.455768 + 2.96518i −2.44743 − 0.245323i
85.7 −1.14498 − 0.830079i −1.41639 + 0.996919i 0.621938 + 1.90084i −3.39378 − 1.95940i 2.44925 + 0.0342651i −0.500000 − 0.866025i 0.865744 − 2.69267i 1.01230 − 2.82405i 2.25934 + 5.06056i
85.8 −1.14188 + 0.834337i −1.71270 − 0.258185i 0.607763 − 1.90542i −1.96075 − 1.13204i 2.17110 − 1.13415i −0.500000 − 0.866025i 0.895772 + 2.68283i 2.86668 + 0.884386i 3.18343 − 0.343276i
85.9 −0.972690 + 1.02658i 0.821098 + 1.52506i −0.107748 − 1.99710i 2.72052 + 1.57069i −2.36427 − 0.640483i −0.500000 − 0.866025i 2.15499 + 1.83194i −1.65160 + 2.50444i −4.25867 + 1.26504i
85.10 −0.908808 + 1.08354i 1.55167 − 0.769621i −0.348135 − 1.96947i −1.05668 − 0.610074i −0.576254 + 2.38074i −0.500000 − 0.866025i 2.45039 + 1.41265i 1.81537 − 2.38840i 1.62136 − 0.590518i
85.11 −0.889632 − 1.09934i 1.72874 + 0.107029i −0.417109 + 1.95602i −0.436219 − 0.251851i −1.42028 − 1.99570i −0.500000 − 0.866025i 2.52141 − 1.28159i 2.97709 + 0.370051i 0.111204 + 0.703609i
85.12 −0.808719 − 1.16016i −0.339544 − 1.69844i −0.691947 + 1.87649i 0.642635 + 0.371025i −1.69587 + 1.76749i −0.500000 − 0.866025i 2.73662 − 0.714781i −2.76942 + 1.15339i −0.0892617 − 1.04561i
85.13 −0.600369 − 1.28045i 0.339544 + 1.69844i −1.27911 + 1.53749i −0.642635 − 0.371025i 1.97092 − 1.45446i −0.500000 − 0.866025i 2.73662 + 0.714781i −2.76942 + 1.15339i −0.0892617 + 1.04561i
85.14 −0.507243 − 1.32012i −1.72874 − 0.107029i −1.48541 + 1.33924i 0.436219 + 0.251851i 0.735600 + 2.33643i −0.500000 − 0.866025i 2.52141 + 1.28159i 2.97709 + 0.370051i 0.111204 − 0.703609i
85.15 −0.462760 + 1.33636i −1.16981 + 1.27732i −1.57171 − 1.23683i −0.563250 − 0.325193i −1.16562 − 2.15438i −0.500000 − 0.866025i 2.38016 − 1.52801i −0.263093 − 2.98844i 0.695224 − 0.602218i
85.16 −0.451845 + 1.34009i −0.169829 − 1.72370i −1.59167 − 1.21102i −2.48308 − 1.43361i 2.38665 + 0.551261i −0.500000 − 0.866025i 2.34207 − 1.58579i −2.94232 + 0.585469i 3.04313 − 2.67978i
85.17 −0.146382 − 1.40662i 1.41639 − 0.996919i −1.95714 + 0.411806i 3.39378 + 1.95940i −1.60962 − 1.84638i −0.500000 − 0.866025i 0.865744 + 2.69267i 1.01230 − 2.82405i 2.25934 − 5.06056i
85.18 0.0728331 + 1.41234i 1.64907 − 0.529693i −1.98939 + 0.205730i 1.76518 + 1.01913i 0.868211 + 2.29046i −0.500000 − 0.866025i −0.435453 − 2.79471i 2.43885 − 1.74700i −1.31079 + 2.56725i
85.19 0.106383 + 1.41021i −1.33989 − 1.09759i −1.97737 + 0.300044i 0.882573 + 0.509554i 1.40529 − 2.00628i −0.500000 − 0.866025i −0.633481 − 2.75657i 0.590596 + 2.94129i −0.624686 + 1.29882i
85.20 0.158116 + 1.40535i 0.757390 + 1.55768i −1.95000 + 0.444416i −2.00933 − 1.16009i −2.06932 + 1.31069i −0.500000 − 0.866025i −0.932885 − 2.67015i −1.85272 + 2.35954i 1.31262 − 3.00724i
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.b ✓ 72
8.b even 2 1 inner 504.2.cs.b ✓ 72
9.c even 3 1 inner 504.2.cs.b ✓ 72
72.n even 6 1 inner 504.2.cs.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cs.b ✓ 72 1.a even 1 1 trivial
504.2.cs.b ✓ 72 8.b even 2 1 inner
504.2.cs.b ✓ 72 9.c even 3 1 inner
504.2.cs.b ✓ 72 72.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 108 T_{5}^{70} + 6450 T_{5}^{68} - 265392 T_{5}^{66} + 8305620 T_{5}^{64} + \cdots + 17\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display