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Properties

Label 89.2.f.a
Level $89$
Weight $2$
Character orbit 89.f
Analytic conductor $0.711$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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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] = [89,2,Mod(11,89)] 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("89.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(89, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([21])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 89.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.710668577989\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 9 q^{2} - 13 q^{4} - 9 q^{5} - 11 q^{6} - 11 q^{7} - 5 q^{8} - 6 q^{9} + 12 q^{10} + 8 q^{11} - 11 q^{13} - 66 q^{14} - 11 q^{15} - 5 q^{16} + 8 q^{17} + 38 q^{18} - 11 q^{19} + 11 q^{20} - 19 q^{21}+ \cdots + 107 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.48394 − 1.71256i 0.872351 − 0.125425i −0.446146 + 3.10301i 2.82308 − 1.81428i −1.50931 − 1.30783i −1.07923 − 1.67931i 2.16351 − 1.39041i −2.13321 + 0.626368i −7.29634 − 2.14240i
11.2 −1.31415 − 1.51661i −1.98015 + 0.284702i −0.288487 + 2.00647i −1.65151 + 1.06136i 3.03399 + 2.62897i 1.37650 + 2.14187i 0.0457543 − 0.0294045i 0.961445 − 0.282306i 3.77999 + 1.10991i
11.3 −0.382053 − 0.440912i 1.76256 − 0.253417i 0.236190 − 1.64274i −1.04570 + 0.672030i −0.785124 − 0.680313i 0.0536672 + 0.0835078i −1.79613 + 1.15430i 0.163903 − 0.0481262i 0.695818 + 0.204311i
11.4 −0.00273185 − 0.00315273i −2.32529 + 0.334326i 0.284627 − 1.97963i 3.08866 − 1.98496i 0.00740638 + 0.00641766i 0.0818830 + 0.127413i −0.0140376 + 0.00902143i 2.41670 − 0.709608i −0.0146958 − 0.00431508i
11.5 0.842240 + 0.971996i 0.387160 − 0.0556652i 0.0492204 − 0.342335i −0.677335 + 0.435297i 0.380188 + 0.329435i 1.28980 + 2.00696i 2.53814 − 1.63116i −2.73168 + 0.802095i −0.993585 − 0.291743i
11.6 1.48295 + 1.71141i −0.517381 + 0.0743881i −0.445172 + 3.09624i 0.719151 − 0.462170i −0.894558 − 0.775139i −2.30720 − 3.59007i −2.14903 + 1.38110i −2.61633 + 0.768224i 1.85743 + 0.545390i
22.1 −2.22383 − 1.42917i −0.119759 − 0.0546921i 2.07205 + 4.53717i −0.462825 + 3.21902i 0.188159 + 0.292781i 2.54671 + 0.366161i 1.12407 − 7.81806i −1.95323 − 2.25415i 5.62976 − 6.49708i
22.2 −1.12960 − 0.725947i 2.11558 + 0.966153i −0.0818420 − 0.179209i 0.0525329 − 0.365374i −1.68837 − 2.62716i 1.87109 + 0.269022i −0.419835 + 2.92002i 1.57764 + 1.82070i −0.324584 + 0.374589i
22.3 −0.699304 − 0.449415i −2.19264 − 1.00134i −0.543778 − 1.19071i −0.533702 + 3.71198i 1.08330 + 1.68565i −3.21377 − 0.462070i −0.391459 + 2.72266i 1.84038 + 2.12392i 2.04144 − 2.35595i
22.4 0.0470506 + 0.0302376i −1.16192 − 0.530631i −0.829531 − 1.81642i 0.280294 − 1.94949i −0.0386240 − 0.0601002i 2.23576 + 0.321454i 0.0318133 − 0.221266i −0.896092 − 1.03415i 0.0721357 − 0.0832490i
22.5 0.717209 + 0.460922i 1.90233 + 0.868765i −0.528891 − 1.15811i −0.104983 + 0.730173i 0.963935 + 1.49991i −5.03396 − 0.723774i 0.397133 − 2.76212i 0.899524 + 1.03811i −0.411848 + 0.475298i
22.6 1.87305 + 1.20374i −1.05614 − 0.482322i 1.22851 + 2.69006i −0.0474779 + 0.330216i −1.39761 − 2.17473i −0.365320 − 0.0525251i −0.303335 + 2.10974i −1.08179 − 1.24845i −0.486422 + 0.561361i
25.1 −0.349474 − 2.43064i 0.521935 − 1.77755i −3.86692 + 1.13543i 0.567391 + 1.24241i −4.50299 − 0.647432i 1.55636 − 0.710764i 2.07099 + 4.53484i −0.363498 − 0.233606i 2.82158 − 1.81332i
25.2 −0.220044 − 1.53044i −0.308654 + 1.05118i −0.374846 + 0.110065i −1.41670 − 3.10213i 1.67669 + 0.241071i 2.37239 − 1.08343i −1.03368 − 2.26345i 1.51405 + 0.973020i −4.43589 + 2.85078i
25.3 −0.125840 − 0.875240i 0.380201 − 1.29485i 1.16878 − 0.343184i 0.405702 + 0.888363i −1.18115 − 0.169823i −3.78551 + 1.72878i −1.18210 − 2.58844i 0.991685 + 0.637317i 0.726477 − 0.466879i
25.4 0.0849017 + 0.590504i −0.651489 + 2.21877i 1.57750 − 0.463196i −0.296239 − 0.648674i −1.36550 − 0.196330i −1.92972 + 0.881274i 0.903105 + 1.97752i −1.97473 − 1.26908i 0.357893 − 0.230004i
25.5 0.226216 + 1.57337i 0.754338 − 2.56904i −0.505329 + 0.148378i −0.524671 − 1.14887i 4.21269 + 0.605693i −1.15210 + 0.526149i 0.972877 + 2.13030i −3.50718 − 2.25393i 1.68890 − 1.08539i
25.6 0.343734 + 2.39072i −0.270493 + 0.921214i −3.67841 + 1.08008i −0.444850 − 0.974085i −2.29534 − 0.330021i 1.28373 − 0.586259i −1.83985 − 4.02871i 1.74829 + 1.12356i 2.17586 − 1.39834i
44.1 −2.48992 − 0.731108i −0.137920 − 0.214607i 3.98270 + 2.55953i 0.244739 + 0.282444i 0.186508 + 0.635189i 0.818937 − 0.709613i −4.64655 − 5.36241i 1.21921 − 2.66970i −0.402884 − 0.882194i
44.2 −1.28676 − 0.377826i 1.52617 + 2.37476i −0.169514 − 0.108940i 0.528006 + 0.609351i −1.06656 − 3.63237i −1.88467 + 1.63308i 1.93341 + 2.23127i −2.06407 + 4.51968i −0.449187 − 0.983582i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.f even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.2.f.a ✓ 60
3.b odd 2 1 801.2.p.a 60
89.f even 22 1 inner 89.2.f.a ✓ 60
89.g even 44 2 7921.2.a.t 60
267.l odd 22 1 801.2.p.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.2.f.a ✓ 60 1.a even 1 1 trivial
89.2.f.a ✓ 60 89.f even 22 1 inner
801.2.p.a 60 3.b odd 2 1
801.2.p.a 60 267.l odd 22 1
7921.2.a.t 60 89.g even 44 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(89, [\chi])\).