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Properties

Label 961.2.k.a
Level $961$
Weight $2$
Character orbit 961.k
Analytic conductor $7.674$
Analytic rank $0$
Dimension $4920$
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] = [961,2,Mod(5,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(186)) chi = DirichletCharacter(H, H._module([88])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.k (of order \(93\), degree \(60\), 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: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(4920\)
Relative dimension: \(82\) over \(\Q(\zeta_{93})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{93}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 4920 q - 58 q^{2} - 64 q^{3} - 226 q^{4} - 60 q^{5} - 56 q^{6} - 60 q^{7} - 56 q^{8} + 18 q^{9} - 61 q^{10} - 64 q^{11} - 72 q^{12} - 64 q^{13} - 65 q^{14} + 66 q^{15} - 234 q^{16} - 68 q^{17} - 70 q^{18}+ \cdots - 86 q^{99}+O(q^{100}) \) 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 −2.10337 − 1.80568i −0.288292 − 3.40558i 0.860816 + 5.61911i −0.132804 − 0.0796949i −5.54302 + 7.68377i −1.27034 − 1.89254i 5.40301 − 8.66833i −8.55757 + 1.45930i 0.135432 + 0.407430i
5.2 −2.04802 − 1.75817i 0.258993 + 3.05948i 0.800382 + 5.22462i −2.08380 − 1.25048i 4.84866 − 6.72123i 1.13821 + 1.69569i 4.69103 − 7.52606i −6.33601 + 1.08046i 2.06912 + 6.22468i
5.3 −2.04201 − 1.75301i 0.0800879 + 0.946076i 0.793911 + 5.18237i 0.610803 + 0.366540i 1.49494 − 2.07229i 0.839904 + 1.25128i 4.61642 − 7.40635i 2.06866 − 0.352764i −0.604719 − 1.81922i
5.4 −2.01438 − 1.72929i −0.111118 − 1.31263i 0.764437 + 4.98998i 2.88302 + 1.73009i −2.04609 + 2.83630i 1.61292 + 2.40291i 4.28063 − 6.86764i 1.24666 − 0.212590i −2.81569 − 8.47065i
5.5 −2.01430 − 1.72922i 0.0950259 + 1.12254i 0.764353 + 4.98943i −0.621354 − 0.372871i 1.74971 − 2.42545i −2.66184 − 3.96558i 4.27968 − 6.86611i 1.70625 − 0.290962i 0.606819 + 1.82554i
5.6 −2.00348 − 1.71993i −0.0956978 − 1.13048i 0.752911 + 4.91475i −3.68236 − 2.20976i −1.75261 + 2.42948i 0.0940630 + 0.140134i 4.15115 − 6.65990i 1.68849 − 0.287935i 3.57689 + 10.7606i
5.7 −1.85608 − 1.59340i −0.137950 − 1.62960i 0.603283 + 3.93802i −0.705510 − 0.423373i −2.34055 + 3.24448i 1.55965 + 2.32354i 2.56717 − 4.11864i 0.320755 − 0.0546975i 0.634886 + 1.90997i
5.8 −1.85459 − 1.59212i 0.235232 + 2.77879i 0.601828 + 3.92852i 2.70151 + 1.62116i 3.98790 − 5.52804i 1.28956 + 1.92116i 2.55268 − 4.09540i −4.70902 + 0.803018i −2.42913 − 7.30772i
5.9 −1.78072 − 1.52870i −0.0831804 − 0.982607i 0.531191 + 3.46743i 1.66490 + 0.999096i −1.35399 + 1.87691i −1.19061 − 1.77375i 1.87192 − 3.00321i 1.99871 − 0.340835i −1.43740 − 4.32424i
5.10 −1.67966 − 1.44194i −0.0837641 − 0.989503i 0.439212 + 2.86703i −2.47742 − 1.48668i −1.28611 + 1.78281i −1.08373 − 1.61453i 1.05443 − 1.69167i 1.98521 − 0.338533i 2.01751 + 6.06943i
5.11 −1.67939 − 1.44171i −0.222135 − 2.62407i 0.438970 + 2.86545i −0.0342144 − 0.0205319i −3.41010 + 4.72709i 1.12782 + 1.68021i 1.05238 − 1.68839i −3.87909 + 0.661492i 0.0278583 + 0.0838081i
5.12 −1.61700 − 1.38815i 0.0645110 + 0.762066i 0.384874 + 2.51233i 1.88615 + 1.13187i 0.953546 − 1.32181i −0.570214 − 0.849498i 0.610575 − 0.979577i 2.38073 − 0.405979i −1.47870 − 4.44848i
5.13 −1.60071 − 1.37417i −0.186174 − 2.19927i 0.371090 + 2.42235i 2.42546 + 1.45551i −2.72415 + 3.77623i −2.19381 − 3.26831i 0.502843 − 0.806738i −1.84482 + 0.314592i −1.88236 − 5.66284i
5.14 −1.59990 − 1.37347i 0.236187 + 2.79007i 0.370404 + 2.41787i −2.11621 − 1.26993i 3.45419 − 4.78821i −1.83967 − 2.74071i 0.497537 − 0.798224i −4.77137 + 0.813651i 1.64152 + 4.93830i
5.15 −1.55856 − 1.33798i 0.163503 + 1.93146i 0.336065 + 2.19372i −1.26605 − 0.759751i 2.32943 − 3.22906i 1.39962 + 2.08514i 0.238290 − 0.382300i −0.746496 + 0.127298i 0.956688 + 2.87807i
5.16 −1.43236 − 1.22964i 0.0738978 + 0.872952i 0.236782 + 1.54563i −1.69123 − 1.01490i 0.967568 − 1.34125i −0.608867 − 0.907082i −0.435707 + 0.699028i 2.20073 − 0.375284i 1.17449 + 3.53329i
5.17 −1.41888 − 1.21807i 0.0200290 + 0.236602i 0.226677 + 1.47967i −2.57781 − 1.54693i 0.259779 − 0.360107i 2.63507 + 3.92569i −0.497621 + 0.798359i 2.90173 − 0.494825i 1.77334 + 5.33488i
5.18 −1.33727 − 1.14801i 0.262593 + 3.10201i 0.167510 + 1.09345i 0.880210 + 0.528209i 3.20996 − 4.44967i −1.56993 − 2.33886i −0.833259 + 1.33684i −6.59617 + 1.12483i −0.570688 − 1.71684i
5.19 −1.32395 − 1.13658i −0.0311209 − 0.367630i 0.158189 + 1.03260i 1.26767 + 0.760721i −0.376637 + 0.522096i −0.180905 − 0.269510i −0.881773 + 1.41467i 2.82313 − 0.481421i −0.813715 − 2.44796i
5.20 −1.21654 − 1.04436i 0.0310500 + 0.366792i 0.0864144 + 0.564084i 3.54479 + 2.12721i 0.345291 − 0.478644i 2.48456 + 3.70147i −1.21222 + 1.94483i 2.82374 − 0.481525i −2.09079 − 6.28989i
See next 80 embeddings (of 4920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.82
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
961.k even 93 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.k.a ✓ 4920
961.k even 93 1 inner 961.2.k.a ✓ 4920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.k.a ✓ 4920 1.a even 1 1 trivial
961.2.k.a ✓ 4920 961.k even 93 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(961, [\chi])\).