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Properties

Label 322.2.o.b
Level $322$
Weight $2$
Character orbit 322.o
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(5,322)] 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("322.5"); 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([55, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.o (of order \(66\), 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 \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9} - 27 q^{12} - 11 q^{14} + 8 q^{16} + 66 q^{17} + 12 q^{18} - 66 q^{21} - 18 q^{23} - 6 q^{24} - 2 q^{25} + 6 q^{26} - 22 q^{28} - 16 q^{29}+ \cdots - 220 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0.723734 − 0.690079i −0.618301 + 3.20805i 0.0475819 − 0.998867i 2.71652 + 2.13629i 1.76632 + 2.74845i 2.21316 − 1.44980i −0.654861 − 0.755750i −7.12418 − 2.85209i 3.44025 − 0.328504i
5.2 0.723734 − 0.690079i −0.290209 + 1.50575i 0.0475819 − 0.998867i −0.209478 − 0.164735i 0.829051 + 1.29003i 0.478644 + 2.60210i −0.654861 − 0.755750i 0.602045 + 0.241023i −0.265286 + 0.0253318i
5.3 0.723734 − 0.690079i −0.219533 + 1.13904i 0.0475819 − 0.998867i −0.936223 − 0.736254i 0.627147 + 0.975860i 0.155114 − 2.64120i −0.654861 − 0.755750i 1.53588 + 0.614873i −1.18565 + 0.113216i
5.4 0.723734 − 0.690079i 0.141475 − 0.734043i 0.0475819 − 0.998867i 1.50572 + 1.18411i −0.404157 − 0.628881i 2.62746 + 0.310594i −0.654861 − 0.755750i 2.26630 + 0.907289i 1.90687 − 0.182084i
5.5 0.723734 − 0.690079i 0.223563 − 1.15995i 0.0475819 − 0.998867i −2.96167 − 2.32908i −0.638660 − 0.993774i −1.92017 + 1.82015i −0.654861 − 0.755750i 1.48959 + 0.596343i −3.75071 + 0.358150i
5.6 0.723734 − 0.690079i 0.340605 − 1.76722i 0.0475819 − 0.998867i −0.793626 − 0.624115i −0.973017 − 1.51404i −0.911188 − 2.48390i −0.654861 − 0.755750i −0.221966 − 0.0888616i −1.00506 + 0.0959718i
5.7 0.723734 − 0.690079i 0.410973 − 2.13233i 0.0475819 − 0.998867i 3.20038 + 2.51681i −1.17404 − 1.82684i −2.40619 − 1.10010i −0.654861 − 0.755750i −1.59282 − 0.637669i 4.05302 − 0.387017i
5.8 0.723734 − 0.690079i 0.640458 − 3.32301i 0.0475819 − 0.998867i −0.463735 − 0.364685i −1.82962 − 2.84694i 1.94583 + 1.79269i −0.654861 − 0.755750i −7.84710 − 3.14151i −0.587283 + 0.0560787i
17.1 −0.995472 + 0.0950560i −2.26151 − 2.37180i 0.981929 − 0.189251i 2.68625 + 1.38486i 2.47672 + 2.14609i 2.03190 − 1.69451i −0.959493 + 0.281733i −0.368277 + 7.73108i −2.80573 − 1.12324i
17.2 −0.995472 + 0.0950560i −1.67171 − 1.75324i 0.981929 − 0.189251i −3.49473 − 1.80166i 1.83079 + 1.58639i 2.30654 − 1.29610i −0.959493 + 0.281733i −0.136486 + 2.86520i 3.65017 + 1.46131i
17.3 −0.995472 + 0.0950560i −1.02117 − 1.07097i 0.981929 − 0.189251i 0.591499 + 0.304939i 1.11835 + 0.969052i −2.51796 − 0.812315i −0.959493 + 0.281733i 0.0385527 − 0.809321i −0.617807 − 0.247332i
17.4 −0.995472 + 0.0950560i −0.855603 − 0.897331i 0.981929 − 0.189251i −2.48995 − 1.28366i 0.937026 + 0.811937i −1.18187 + 2.36710i −0.959493 + 0.281733i 0.0696000 − 1.46108i 2.60070 + 1.04116i
17.5 −0.995472 + 0.0950560i 0.322555 + 0.338286i 0.981929 − 0.189251i 0.680578 + 0.350862i −0.353250 − 0.306093i −0.141341 − 2.64197i −0.959493 + 0.281733i 0.132350 − 2.77837i −0.710848 − 0.284581i
17.6 −0.995472 + 0.0950560i 0.519459 + 0.544793i 0.981929 − 0.189251i 2.76848 + 1.42725i −0.568893 − 0.492948i 2.09567 + 1.61498i −0.959493 + 0.281733i 0.115784 − 2.43061i −2.89162 − 1.15763i
17.7 −0.995472 + 0.0950560i 1.77845 + 1.86518i 0.981929 − 0.189251i −1.66354 − 0.857616i −1.94769 − 1.68768i 2.61594 − 0.396066i −0.959493 + 0.281733i −0.173283 + 3.63765i 1.73753 + 0.695603i
17.8 −0.995472 + 0.0950560i 2.19647 + 2.30360i 0.981929 − 0.189251i 1.78887 + 0.922229i −2.40550 − 2.08438i −2.58627 + 0.557860i −0.959493 + 0.281733i −0.339309 + 7.12298i −1.86844 − 0.748010i
19.1 −0.995472 − 0.0950560i −2.26151 + 2.37180i 0.981929 + 0.189251i 2.68625 − 1.38486i 2.47672 − 2.14609i 2.03190 + 1.69451i −0.959493 − 0.281733i −0.368277 − 7.73108i −2.80573 + 1.12324i
19.2 −0.995472 − 0.0950560i −1.67171 + 1.75324i 0.981929 + 0.189251i −3.49473 + 1.80166i 1.83079 − 1.58639i 2.30654 + 1.29610i −0.959493 − 0.281733i −0.136486 − 2.86520i 3.65017 − 1.46131i
19.3 −0.995472 − 0.0950560i −1.02117 + 1.07097i 0.981929 + 0.189251i 0.591499 − 0.304939i 1.11835 − 0.969052i −2.51796 + 0.812315i −0.959493 − 0.281733i 0.0385527 + 0.809321i −0.617807 + 0.247332i
19.4 −0.995472 − 0.0950560i −0.855603 + 0.897331i 0.981929 + 0.189251i −2.48995 + 1.28366i 0.937026 − 0.811937i −1.18187 − 2.36710i −0.959493 − 0.281733i 0.0696000 + 1.46108i 2.60070 − 1.04116i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.o.b ✓ 160
7.d odd 6 1 inner 322.2.o.b ✓ 160
23.d odd 22 1 inner 322.2.o.b ✓ 160
161.o even 66 1 inner 322.2.o.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.o.b ✓ 160 1.a even 1 1 trivial
322.2.o.b ✓ 160 7.d odd 6 1 inner
322.2.o.b ✓ 160 23.d odd 22 1 inner
322.2.o.b ✓ 160 161.o even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} + 6 T_{3}^{159} + 24 T_{3}^{158} + 72 T_{3}^{157} + 118 T_{3}^{156} + 93 T_{3}^{155} + \cdots + 39501840932401 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display