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Properties

Label 669.2.i.a
Level $669$
Weight $2$
Character orbit 669.i
Analytic conductor $5.342$
Analytic rank $0$
Dimension $648$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [669,2,Mod(4,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(74)) chi = DirichletCharacter(H, H._module([0, 46])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 669 = 3 \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 669.i (of order \(37\), degree \(36\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [648] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.34199189522\)
Analytic rank: \(0\)
Dimension: \(648\)
Relative dimension: \(18\) over \(\Q(\zeta_{37})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{37}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 648 q + q^{2} + 18 q^{3} - 17 q^{4} + 4 q^{5} - q^{6} + 4 q^{7} + 9 q^{8} - 18 q^{9} + 8 q^{10} + 6 q^{11} + 17 q^{12} - 2 q^{13} + 12 q^{14} - 4 q^{15} - 7 q^{16} + 8 q^{17} + q^{18} - 66 q^{19} + 14 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.51246 + 0.887710i −0.372856 + 0.927889i 3.96837 − 3.20424i −2.24573 − 3.64729i 0.113092 − 2.66228i 0.412183 − 3.21880i −4.33174 + 7.03519i −0.721956 − 0.691939i 8.88004 + 7.17013i
4.2 −2.39659 + 0.846770i −0.372856 + 0.927889i 3.47057 − 2.80229i 1.57885 + 2.56421i 0.107876 − 2.53950i 0.438089 − 3.42110i −3.27929 + 5.32590i −0.721956 − 0.691939i −5.95514 − 4.80844i
4.3 −2.04994 + 0.724288i −0.372856 + 0.927889i 2.12157 − 1.71305i −0.554185 − 0.900054i 0.0922726 − 2.17217i −0.114368 + 0.893114i −0.828521 + 1.34560i −0.721956 − 0.691939i 1.78794 + 1.44366i
4.4 −1.89340 + 0.668982i −0.372856 + 0.927889i 1.58137 − 1.27686i 0.155467 + 0.252495i 0.0852267 − 2.00630i 0.0558488 − 0.436131i −0.0342308 + 0.0555943i −0.721956 − 0.691939i −0.463277 − 0.374070i
4.5 −1.35947 + 0.480331i −0.372856 + 0.927889i 0.0613669 − 0.0495503i −1.23679 − 2.00867i 0.0611930 − 1.44053i −0.492570 + 3.84655i 1.45230 − 2.35868i −0.721956 − 0.691939i 2.64620 + 2.13666i
4.6 −0.992643 + 0.350723i −0.372856 + 0.927889i −0.693738 + 0.560155i 1.51216 + 2.45591i 0.0446813 − 1.05183i −0.283003 + 2.21001i 1.59614 − 2.59229i −0.721956 − 0.691939i −2.36238 − 1.90749i
4.7 −0.871922 + 0.308070i −0.372856 + 0.927889i −0.890730 + 0.719214i 1.42047 + 2.30699i 0.0392473 − 0.923913i 0.275779 − 2.15360i 1.52478 − 2.47640i −0.721956 − 0.691939i −1.94925 − 1.57391i
4.8 −0.834832 + 0.294965i −0.372856 + 0.927889i −0.946131 + 0.763948i −1.22823 − 1.99478i 0.0375778 − 0.884611i 0.597467 − 4.66570i 1.49298 − 2.42475i −0.721956 − 0.691939i 1.61376 + 1.30302i
4.9 −0.337033 + 0.119082i −0.372856 + 0.927889i −1.45666 + 1.17617i −0.484594 − 0.787030i 0.0151707 − 0.357130i 0.111523 − 0.870898i 0.725713 − 1.17863i −0.721956 − 0.691939i 0.257045 + 0.207549i
4.10 0.153732 − 0.0543169i −0.372856 + 0.927889i −1.53539 + 1.23974i −1.34968 − 2.19202i −0.00691984 + 0.162898i 0.0153152 − 0.119599i −0.339671 + 0.551660i −0.721956 − 0.691939i −0.326553 − 0.263673i
4.11 0.173139 − 0.0611740i −0.372856 + 0.927889i −1.52984 + 1.23526i −0.169781 − 0.275742i −0.00779341 + 0.183463i −0.312492 + 2.44030i −0.381865 + 0.620188i −0.721956 − 0.691939i −0.0462641 − 0.0373556i
4.12 0.617044 − 0.218016i −0.372856 + 0.927889i −1.22286 + 0.987389i 1.32191 + 2.14691i −0.0277747 + 0.653837i −0.367115 + 2.86685i −1.22553 + 1.99039i −0.721956 − 0.691939i 1.28374 + 1.03654i
4.13 1.29048 − 0.455957i −0.372856 + 0.927889i −0.0986224 + 0.0796320i −0.799103 − 1.29783i −0.0580878 + 1.36743i 0.461598 − 3.60469i −1.52616 + 2.47865i −0.721956 − 0.691939i −1.62298 − 1.31047i
4.14 1.61999 − 0.572380i −0.372856 + 0.927889i 0.740690 − 0.598066i 0.661781 + 1.07480i −0.0729199 + 1.71659i −0.0685235 + 0.535110i −0.944072 + 1.53327i −0.721956 − 0.691939i 1.68728 + 1.36238i
4.15 1.73410 − 0.612698i −0.372856 + 0.927889i 1.07565 − 0.868523i 1.94371 + 3.15678i −0.0780562 + 1.83750i −0.158479 + 1.23759i −0.595432 + 0.967044i −0.721956 − 0.691939i 5.30475 + 4.28329i
4.16 1.99733 − 0.705701i −0.372856 + 0.927889i 1.93524 − 1.56260i −0.358816 − 0.582754i −0.0899046 + 2.11643i −0.647697 + 5.05796i 0.541267 − 0.879073i −0.721956 − 0.691939i −1.12792 − 0.910735i
4.17 2.28064 − 0.805803i −0.372856 + 0.927889i 2.99595 − 2.41906i 1.57258 + 2.55403i −0.102657 + 2.41663i 0.459069 − 3.58494i 2.34701 − 3.81179i −0.721956 − 0.691939i 5.64453 + 4.55764i
4.18 2.43895 − 0.861735i −0.372856 + 0.927889i 3.64981 − 2.94701i −1.26220 − 2.04995i −0.109783 + 2.58438i 0.0928182 − 0.724830i 3.64968 − 5.92746i −0.721956 − 0.691939i −4.84496 − 3.91203i
7.1 −1.80550 − 2.05145i −0.985616 + 0.169001i −0.694570 + 5.42400i −0.0330679 − 0.0223500i 2.12623 + 1.71681i −0.873185 + 1.73186i 7.85277 − 5.30755i 0.942877 − 0.333140i 0.0138543 + 0.108190i
7.2 −1.57799 − 1.79294i −0.985616 + 0.169001i −0.470561 + 3.67468i 2.77692 + 1.87687i 1.85830 + 1.50047i −0.437519 + 0.867768i 3.37333 − 2.27997i 0.942877 − 0.333140i −1.01682 − 7.94052i
See next 80 embeddings (of 648 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
223.e even 37 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 669.2.i.a ✓ 648
223.e even 37 1 inner 669.2.i.a ✓ 648
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
669.2.i.a ✓ 648 1.a even 1 1 trivial
669.2.i.a ✓ 648 223.e even 37 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{648} - T_{2}^{647} + 27 T_{2}^{646} - 31 T_{2}^{645} + 432 T_{2}^{644} - 558 T_{2}^{643} + \cdots + 79\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(669, [\chi])\). Copy content Toggle raw display