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Properties

Label 322.2.m.b
Level $322$
Weight $2$
Character orbit 322.m
Analytic conductor $2.571$
Analytic rank $0$
Dimension $160$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [322,2,Mod(9,322)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("322.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([22, 30])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [160,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9} + 2 q^{10} + 9 q^{12} + 12 q^{13} - 11 q^{14} - 20 q^{15} + 8 q^{16} + 30 q^{17} - 16 q^{18} + 2 q^{19} - 26 q^{20}+ \cdots + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0.235759 − 0.971812i −1.01715 + 2.93887i −0.888835 − 0.458227i 3.44290 + 1.37833i 2.61623 + 1.68135i −1.72354 + 2.00734i −0.654861 + 0.755750i −5.24421 − 4.12409i 2.15117 − 3.02089i
9.2 0.235759 − 0.971812i −0.728804 + 2.10574i −0.888835 − 0.458227i 0.0766304 + 0.0306782i 1.87456 + 1.20471i 2.09067 − 1.62144i −0.654861 + 0.755750i −1.54483 − 1.21487i 0.0478797 − 0.0672376i
9.3 0.235759 − 0.971812i −0.526198 + 1.52035i −0.888835 − 0.458227i −1.39502 − 0.558482i 1.35344 + 0.869801i −1.65886 − 2.06112i −0.654861 + 0.755750i 0.323581 + 0.254467i −0.871628 + 1.22403i
9.4 0.235759 − 0.971812i −0.270988 + 0.782968i −0.888835 − 0.458227i −2.81356 − 1.12638i 0.697010 + 0.447941i −1.54703 + 2.14632i −0.654861 + 0.755750i 1.81855 + 1.43013i −1.75795 + 2.46869i
9.5 0.235759 − 0.971812i 0.0566868 − 0.163786i −0.888835 − 0.458227i 1.92505 + 0.770673i −0.145804 − 0.0937029i 1.33776 + 2.28263i −0.654861 + 0.755750i 2.33455 + 1.83591i 1.20280 − 1.68909i
9.6 0.235759 − 0.971812i 0.117329 − 0.339001i −0.888835 − 0.458227i 3.25700 + 1.30391i −0.301784 − 0.193945i −1.01171 − 2.44468i −0.654861 + 0.755750i 2.25700 + 1.77493i 2.03502 − 2.85778i
9.7 0.235759 − 0.971812i 0.790008 − 2.28258i −0.888835 − 0.458227i −1.30805 − 0.523663i −2.03198 − 1.30588i −1.85213 + 1.88934i −0.654861 + 0.755750i −2.22789 − 1.75203i −0.817286 + 1.14772i
9.8 0.235759 − 0.971812i 0.951480 − 2.74912i −0.888835 − 0.458227i 1.74443 + 0.698364i −2.44731 − 1.57279i 2.57549 − 0.605703i −0.654861 + 0.755750i −4.29420 − 3.37699i 1.08994 − 1.53061i
25.1 0.0475819 − 0.998867i −2.89782 − 1.16011i −0.995472 − 0.0950560i 0.340081 + 1.40183i −1.29668 + 2.83934i 0.632305 + 2.56908i −0.142315 + 0.989821i 4.88031 + 4.65336i 1.41643 − 0.272994i
25.2 0.0475819 − 0.998867i −1.82051 − 0.728823i −0.995472 − 0.0950560i 0.792140 + 3.26525i −0.814621 + 1.78377i −0.938065 − 2.47387i −0.142315 + 0.989821i 0.611881 + 0.583427i 3.29924 − 0.635876i
25.3 0.0475819 − 0.998867i −1.43191 − 0.573252i −0.995472 − 0.0950560i −0.790592 − 3.25886i −0.640736 + 1.40302i −1.18728 + 2.36439i −0.142315 + 0.989821i −0.449440 − 0.428541i −3.29279 + 0.634633i
25.4 0.0475819 − 0.998867i −0.547945 − 0.219364i −0.995472 − 0.0950560i 0.328236 + 1.35301i −0.245188 + 0.536886i −2.51182 − 0.831108i −0.142315 + 0.989821i −1.91908 − 1.82984i 1.36709 − 0.263485i
25.5 0.0475819 − 0.998867i 0.0359061 + 0.0143747i −0.995472 − 0.0950560i −0.272928 − 1.12502i 0.0160669 − 0.0351815i 2.64127 + 0.153951i −0.142315 + 0.989821i −2.17012 − 2.06920i −1.13673 + 0.219088i
25.6 0.0475819 − 0.998867i 1.23138 + 0.492969i −0.995472 − 0.0950560i −0.578323 − 2.38388i 0.551002 − 1.20653i −0.901962 − 2.48726i −0.142315 + 0.989821i −0.897930 − 0.856175i −2.40870 + 0.464239i
25.7 0.0475819 − 0.998867i 1.23362 + 0.493867i −0.995472 − 0.0950560i 0.559388 + 2.30583i 0.552006 − 1.20872i 2.06645 + 1.65221i −0.142315 + 0.989821i −0.893289 − 0.851749i 2.32983 − 0.449038i
25.8 0.0475819 − 0.998867i 2.63530 + 1.05502i −0.995472 − 0.0950560i 0.632138 + 2.60571i 1.17921 − 2.58212i −2.35864 + 1.19868i −0.142315 + 0.989821i 3.66057 + 3.49034i 2.63283 − 0.507437i
39.1 0.981929 + 0.189251i −0.129433 − 2.71714i 0.928368 + 0.371662i −0.147211 + 0.206729i 0.387127 − 2.69253i 2.54468 − 0.724279i 0.841254 + 0.540641i −4.37966 + 0.418207i −0.183675 + 0.175134i
39.2 0.981929 + 0.189251i −0.0895616 − 1.88013i 0.928368 + 0.371662i 2.49808 − 3.50806i 0.267874 − 1.86310i 0.0372373 + 2.64549i 0.841254 + 0.540641i −0.540450 + 0.0516067i 3.11684 − 2.97190i
39.3 0.981929 + 0.189251i −0.0465947 − 0.978143i 0.928368 + 0.371662i −1.87411 + 2.63183i 0.139362 − 0.969285i 0.306605 + 2.62793i 0.841254 + 0.540641i 2.03182 − 0.194016i −2.33832 + 2.22959i
39.4 0.981929 + 0.189251i −0.0119548 − 0.250962i 0.928368 + 0.371662i 0.931525 − 1.30814i 0.0357562 − 0.248690i −2.50636 − 0.847436i 0.841254 + 0.540641i 2.92358 − 0.279168i 1.16226 − 1.10821i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.m.b ✓ 160
7.c even 3 1 inner 322.2.m.b ✓ 160
23.c even 11 1 inner 322.2.m.b ✓ 160
161.m even 33 1 inner 322.2.m.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.m.b ✓ 160 1.a even 1 1 trivial
322.2.m.b ✓ 160 7.c even 3 1 inner
322.2.m.b ✓ 160 23.c even 11 1 inner
322.2.m.b ✓ 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} + 2 T_{3}^{159} - 2 T_{3}^{158} - 2 T_{3}^{157} - 36 T_{3}^{156} - 83 T_{3}^{155} + \cdots + 13392445265761 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display