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Properties

Label 945.2.cl.a
Level $945$
Weight $2$
Character orbit 945.cl
Analytic conductor $7.546$
Analytic rank $0$
Dimension $840$
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] = [945,2,Mod(164,945)] 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("945.164"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(945, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1, 9, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cl (of order \(18\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(840\)
Relative dimension: \(140\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 6 q^{4} - 9 q^{5} - 18 q^{6} - 12 q^{9} - 12 q^{11} - 6 q^{14} - 18 q^{16} - 18 q^{20} - 30 q^{21} - 18 q^{24} - 3 q^{25} - 42 q^{29} - 21 q^{30} - 18 q^{31} + 36 q^{34} + 18 q^{35} - 42 q^{36}+ \cdots - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1 −2.64397 + 0.962325i −0.618652 − 1.61780i 4.53239 − 3.80313i −2.14428 − 0.634076i 3.19254 + 3.68206i −1.28661 + 2.31185i −5.51000 + 9.54359i −2.23454 + 2.00171i 6.27960 − 0.387021i
164.2 −2.57075 + 0.935675i 1.62751 + 0.592644i 4.20116 − 3.52519i 1.31963 − 1.80515i −4.73843 0.000721851i −2.63281 − 0.261393i −4.76595 + 8.25487i 2.29755 + 1.92906i −1.70341 + 5.87533i
164.3 −2.54060 + 0.924703i 0.0346710 + 1.73170i 4.06748 − 3.41302i −1.28479 − 1.83011i −1.68940 − 4.36751i 2.45826 + 0.978251i −4.47416 + 7.74947i −2.99760 + 0.120080i 4.95646 + 3.46152i
164.4 −2.52720 + 0.919827i 0.283972 + 1.70861i 4.00859 − 3.36361i −0.783203 + 2.09442i −2.28928 − 4.05681i −0.948469 − 2.46990i −4.34720 + 7.52958i −2.83872 + 0.970395i 0.0528103 − 6.01344i
164.5 −2.48051 + 0.902830i 1.22402 − 1.22547i 3.80572 − 3.19338i 2.23036 + 0.159650i −1.92979 + 4.14487i 0.0269241 + 2.64561i −3.91733 + 6.78501i −0.00355513 − 3.00000i −5.67656 + 1.61762i
164.6 −2.42629 + 0.883098i 1.68421 − 0.404265i 3.57494 − 2.99973i −2.04190 − 0.911408i −3.72938 + 2.46819i −0.247245 − 2.63417i −3.44278 + 5.96307i 2.67314 − 1.36173i 5.75909 + 0.408148i
164.7 −2.40949 + 0.876981i 1.57723 + 0.715785i 3.50444 − 2.94057i 0.723633 + 2.11574i −4.42804 − 0.341474i 2.51534 + 0.820410i −3.30094 + 5.71740i 1.97530 + 2.25791i −3.59905 − 4.46323i
164.8 −2.40712 + 0.876119i 0.431623 − 1.67741i 3.49454 − 2.93227i −0.246975 + 2.22239i 0.430644 + 4.41587i −1.45064 − 2.21261i −3.28115 + 5.68312i −2.62740 − 1.44802i −1.35258 − 5.56593i
164.9 −2.40313 + 0.874669i −1.08875 − 1.34708i 3.47792 − 2.91832i 1.36945 + 1.76766i 3.79467 + 2.28490i 2.61877 − 0.376886i −3.24797 + 5.62565i −0.629228 + 2.93327i −4.83708 − 3.05011i
164.10 −2.38791 + 0.869127i −1.72713 + 0.130428i 3.41463 − 2.86521i −2.00703 + 0.985823i 4.01087 − 1.81255i 2.12013 − 1.58273i −3.12242 + 5.40819i 2.96598 − 0.450532i 3.93578 − 4.09841i
164.11 −2.36799 + 0.861880i 0.751798 − 1.56038i 3.33247 − 2.79628i 1.04214 − 1.97837i −0.435390 + 4.34294i 1.88227 − 1.85932i −2.96126 + 5.12905i −1.86960 − 2.34619i −0.762669 + 5.58297i
164.12 −2.31452 + 0.842416i −1.73066 − 0.0692943i 3.11524 − 2.61400i 1.44500 − 1.70645i 4.06403 − 1.29756i 1.47459 + 2.19672i −2.54515 + 4.40834i 2.99040 + 0.239850i −1.90695 + 5.16690i
164.13 −2.31295 + 0.841846i −1.30870 + 1.13459i 3.10896 − 2.60873i −0.669303 − 2.13355i 2.07182 − 3.72598i −2.63719 + 0.212653i −2.53334 + 4.38787i 0.425411 − 2.96968i 3.34419 + 4.37135i
164.14 −2.27238 + 0.827080i −1.05409 + 1.37437i 2.94758 − 2.47331i 2.19486 − 0.427300i 1.25859 − 3.99491i 0.611912 − 2.57402i −2.23418 + 3.86972i −0.777776 − 2.89742i −4.63416 + 2.78632i
164.15 −2.11073 + 0.768243i −1.17190 + 1.27541i 2.33289 − 1.95753i −1.57344 + 1.58880i 1.49373 − 3.59234i −0.526661 + 2.59280i −1.17406 + 2.03353i −0.253324 − 2.98929i 2.10051 − 4.56231i
164.16 −2.10941 + 0.767763i −0.701997 − 1.58341i 2.32807 − 1.95348i 2.23149 + 0.142944i 2.69649 + 2.80110i −2.04202 − 1.68231i −1.16626 + 2.02001i −2.01440 + 2.22310i −4.81689 + 1.41173i
164.17 −2.09496 + 0.762503i 1.18088 + 1.26710i 2.27536 − 1.90925i −2.20650 + 0.362415i −3.44005 − 1.75410i −0.629195 + 2.56985i −1.08157 + 1.87333i −0.211066 + 2.99257i 4.34619 − 2.44171i
164.18 −2.08145 + 0.757587i 1.45359 − 0.941852i 2.22642 − 1.86819i −0.223281 + 2.22489i −2.31204 + 3.06164i −2.35203 + 1.21159i −1.00384 + 1.73871i 1.22583 − 2.73813i −1.22080 − 4.80017i
164.19 −2.06504 + 0.751615i 0.229901 + 1.71673i 2.16740 − 1.81866i 1.78971 + 1.34050i −1.76507 − 3.37232i −2.12154 + 1.58084i −0.911261 + 1.57835i −2.89429 + 0.789355i −4.70337 − 1.42302i
164.20 −2.00971 + 0.731473i −1.63110 − 0.582691i 1.97177 − 1.65451i −1.57880 − 1.58348i 3.70424 − 0.0220651i −1.75435 − 1.98047i −0.613768 + 1.06308i 2.32094 + 1.90085i 4.33119 + 2.02747i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
189.ba even 18 1 inner
945.cl even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.cl.a ✓ 840
5.b even 2 1 inner 945.2.cl.a ✓ 840
7.d odd 6 1 945.2.cq.a yes 840
27.f odd 18 1 945.2.cq.a yes 840
35.i odd 6 1 945.2.cq.a yes 840
135.n odd 18 1 945.2.cq.a yes 840
189.ba even 18 1 inner 945.2.cl.a ✓ 840
945.cl even 18 1 inner 945.2.cl.a ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cl.a ✓ 840 1.a even 1 1 trivial
945.2.cl.a ✓ 840 5.b even 2 1 inner
945.2.cl.a ✓ 840 189.ba even 18 1 inner
945.2.cl.a ✓ 840 945.cl even 18 1 inner
945.2.cq.a yes 840 7.d odd 6 1
945.2.cq.a yes 840 27.f odd 18 1
945.2.cq.a yes 840 35.i odd 6 1
945.2.cq.a yes 840 135.n odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).