↑

Properties

Label 297.3.l.b
Level $297$
Weight $3$
Character orbit 297.l
Analytic conductor $8.093$
Analytic rank $0$
Dimension $64$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [297,3,Mod(28,297)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 9])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("297.28"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,32,0,0,0,0,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} + 40 q^{16} - 72 q^{22} - 242 q^{25} + 160 q^{28} + 22 q^{31} - 116 q^{34} + 174 q^{37} + 240 q^{40} + 320 q^{46} + 72 q^{49} + 440 q^{52} + 178 q^{55} + 228 q^{58} + 300 q^{61} + 240 q^{67}+ \cdots - 192 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −3.48622 + 1.13274i 0 7.63453 − 5.54681i −0.766665 + 2.35955i 0 −2.71772 − 3.74062i −11.7141 + 16.1231i 0 − 9.09434i
28.2 −3.41410 + 1.10931i 0 7.18945 − 5.22344i 2.78589 − 8.57409i 0 5.28260 + 7.27087i −10.3110 + 14.1918i 0 32.3632i
28.3 −2.95304 + 0.959502i 0 4.56376 − 3.31576i −2.12063 + 6.52662i 0 −1.17161 − 1.61258i −2.99516 + 4.12249i 0 − 21.3082i
28.4 −2.17344 + 0.706193i 0 0.989055 − 0.718590i 0.353597 − 1.08826i 0 4.76453 + 6.55781i 3.73085 − 5.13507i 0 2.61497i
28.5 −1.86420 + 0.605715i 0 −0.127721 + 0.0927948i 2.39947 − 7.38482i 0 −7.73432 − 10.6454i 4.79044 − 6.59348i 0 15.2202i
28.6 −1.32011 + 0.428931i 0 −1.67735 + 1.21867i 0.0558517 − 0.171894i 0 2.71308 + 3.73423i 4.95507 − 6.82007i 0 0.250876i
28.7 −0.763996 + 0.248237i 0 −2.71400 + 1.97184i −2.83974 + 8.73981i 0 −0.00743157 − 0.0102287i 3.47270 − 4.77977i 0 − 7.38210i
28.8 −0.600591 + 0.195144i 0 −2.91344 + 2.11674i −0.173185 + 0.533010i 0 −3.36519 − 4.63179i 2.82146 − 3.88341i 0 − 0.353917i
28.9 0.600591 − 0.195144i 0 −2.91344 + 2.11674i 0.173185 − 0.533010i 0 −3.36519 − 4.63179i −2.82146 + 3.88341i 0 − 0.353917i
28.10 0.763996 − 0.248237i 0 −2.71400 + 1.97184i 2.83974 − 8.73981i 0 −0.00743157 − 0.0102287i −3.47270 + 4.77977i 0 − 7.38210i
28.11 1.32011 − 0.428931i 0 −1.67735 + 1.21867i −0.0558517 + 0.171894i 0 2.71308 + 3.73423i −4.95507 + 6.82007i 0 0.250876i
28.12 1.86420 − 0.605715i 0 −0.127721 + 0.0927948i −2.39947 + 7.38482i 0 −7.73432 − 10.6454i −4.79044 + 6.59348i 0 15.2202i
28.13 2.17344 − 0.706193i 0 0.989055 − 0.718590i −0.353597 + 1.08826i 0 4.76453 + 6.55781i −3.73085 + 5.13507i 0 2.61497i
28.14 2.95304 − 0.959502i 0 4.56376 − 3.31576i 2.12063 − 6.52662i 0 −1.17161 − 1.61258i 2.99516 − 4.12249i 0 − 21.3082i
28.15 3.41410 − 1.10931i 0 7.18945 − 5.22344i −2.78589 + 8.57409i 0 5.28260 + 7.27087i 10.3110 − 14.1918i 0 32.3632i
28.16 3.48622 − 1.13274i 0 7.63453 − 5.54681i 0.766665 − 2.35955i 0 −2.71772 − 3.74062i 11.7141 − 16.1231i 0 − 9.09434i
217.1 −2.19894 − 3.02659i 0 −3.08880 + 9.50636i −0.777076 − 0.564579i 0 −1.43576 − 0.466506i 21.3321 − 6.93121i 0 3.59336i
217.2 −1.91199 − 2.63163i 0 −2.03371 + 6.25911i −1.65999 − 1.20606i 0 10.3634 + 3.36727i 7.98545 − 2.59463i 0 6.67447i
217.3 −1.74465 − 2.40130i 0 −1.48638 + 4.57462i 5.42117 + 3.93871i 0 −8.55033 − 2.77817i 2.28665 − 0.742976i 0 − 19.8895i
217.4 −1.41441 − 1.94677i 0 −0.553285 + 1.70284i 5.91381 + 4.29664i 0 4.75262 + 1.54422i −5.05666 + 1.64301i 0 − 17.5900i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.l.b ✓ 64
3.b odd 2 1 inner 297.3.l.b ✓ 64
11.d odd 10 1 inner 297.3.l.b ✓ 64
33.f even 10 1 inner 297.3.l.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.l.b ✓ 64 1.a even 1 1 trivial
297.3.l.b ✓ 64 3.b odd 2 1 inner
297.3.l.b ✓ 64 11.d odd 10 1 inner
297.3.l.b ✓ 64 33.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 48 T_{2}^{62} + 1366 T_{2}^{60} - 30192 T_{2}^{58} + 581223 T_{2}^{56} + \cdots + 22\!\cdots\!56 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display