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Properties

Label 97.2.k.a
Level $97$
Weight $2$
Character orbit 97.k
Analytic conductor $0.775$
Analytic rank $0$
Dimension $128$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [97,2,Mod(2,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(48)) chi = DirichletCharacter(H, H._module([17])) 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("97.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 97.k (of order \(48\), degree \(16\), 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: \(0.774548899606\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(8\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{2} - 24 q^{3} - 16 q^{4} - 16 q^{5} - 24 q^{6} - 16 q^{7} - 16 q^{8} + 8 q^{9} + 16 q^{10} - 16 q^{11} - 8 q^{12} - 16 q^{13} + 32 q^{14} + 16 q^{15} - 24 q^{16} - 8 q^{17} - 80 q^{18} - 24 q^{19}+ \cdots - 224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.33073 − 0.306846i 1.66551 − 1.27799i 3.40628 + 0.912711i −0.216411 − 0.438839i −4.27401 + 2.46760i 0.693387 − 0.0454470i −3.31527 − 1.37323i 0.364211 − 1.35925i 0.369740 + 1.08922i
2.2 −1.89518 − 0.249505i −2.35730 + 1.80882i 1.59759 + 0.428074i −1.17616 − 2.38503i 4.91880 − 2.83987i 3.87526 − 0.253998i 0.611134 + 0.253140i 1.50857 − 5.63005i 1.63396 + 4.81350i
2.3 −1.80779 − 0.238000i −0.513958 + 0.394374i 1.27960 + 0.342867i 0.489150 + 0.991897i 1.02299 − 0.590622i −4.46562 + 0.292692i 1.13754 + 0.471184i −0.667835 + 2.49239i −0.648207 − 1.90956i
2.4 −0.256803 − 0.0338088i 1.33735 − 1.02618i −1.86705 − 0.500274i −1.64618 − 3.33812i −0.378130 + 0.218313i 2.37756 − 0.155833i 0.941155 + 0.389839i −0.0410073 + 0.153041i 0.309886 + 0.912895i
2.5 0.158865 + 0.0209150i 2.50677 − 1.92351i −1.90705 − 0.510993i 1.51248 + 3.06701i 0.438469 − 0.253150i −3.15644 + 0.206884i −0.588355 − 0.243705i 1.80753 − 6.74581i 0.176134 + 0.518875i
2.6 0.974068 + 0.128239i −2.43883 + 1.87138i −0.999488 − 0.267812i 1.44759 + 2.93541i −2.61557 + 1.51010i 0.913554 − 0.0598775i −2.75460 − 1.14099i 1.66937 − 6.23016i 1.03361 + 3.04493i
2.7 1.54657 + 0.203610i 0.230888 − 0.177167i 0.418576 + 0.112157i −0.0131102 − 0.0265848i 0.393158 − 0.226990i 1.08041 − 0.0708136i −2.25783 − 0.935223i −0.754536 + 2.81597i −0.0148629 − 0.0437847i
2.8 2.63651 + 0.347103i −1.86483 + 1.43094i 4.89884 + 1.31264i −1.51150 − 3.06503i −5.41333 + 3.12539i −2.66138 + 0.174436i 7.54655 + 3.12588i 0.653567 − 2.43915i −2.92121 − 8.60562i
3.1 −2.21047 + 1.69615i 0.609046 + 0.0801824i 1.49160 − 5.56674i 1.61548 − 1.84211i −1.48228 + 0.855794i 1.15399 − 3.39953i 4.01240 + 9.68680i −2.53327 − 0.678788i −0.446484 + 6.81203i
3.2 −1.42517 + 1.09357i −0.384247 − 0.0505870i 0.317569 − 1.18518i −0.731341 + 0.833934i 0.602936 − 0.348105i −1.40879 + 4.15016i −0.531401 − 1.28292i −2.75269 − 0.737581i 0.130318 − 1.98827i
3.3 −1.16349 + 0.892776i 2.92924 + 0.385642i 0.0390183 − 0.145618i −2.02739 + 2.31179i −3.75243 + 2.16647i 0.992445 − 2.92365i −1.03784 − 2.50556i 5.53396 + 1.48282i 0.294929 − 4.49974i
3.4 −1.03969 + 0.797785i −3.02339 − 0.398037i −0.0731357 + 0.272946i 2.32750 − 2.65400i 3.46095 − 1.99818i 0.407692 − 1.20102i −1.14473 − 2.76363i 6.08467 + 1.63038i −0.302561 + 4.61619i
3.5 0.155906 − 0.119631i −2.22857 − 0.293397i −0.507643 + 1.89455i −2.71257 + 3.09310i −0.382548 + 0.220864i 0.785965 − 2.31538i 0.297909 + 0.719216i 1.98267 + 0.531254i −0.0528766 + 0.806742i
3.6 0.197369 − 0.151447i 1.11099 + 0.146265i −0.501620 + 1.87207i 0.732298 − 0.835026i 0.241427 − 0.139388i −0.0793805 + 0.233847i 0.374922 + 0.905141i −1.68487 − 0.451459i 0.0180712 − 0.275713i
3.7 1.49285 − 1.14550i −1.49507 − 0.196829i 0.398779 − 1.48827i 1.59493 − 1.81867i −2.45738 + 1.41877i 0.0473419 − 0.139465i 0.330691 + 0.798360i −0.701291 − 0.187910i 0.297698 − 4.54199i
3.8 1.94052 − 1.48902i 0.232841 + 0.0306541i 1.03083 − 3.84709i −2.33385 + 2.66124i 0.497478 − 0.287219i −0.782817 + 2.30610i −1.85597 − 4.48072i −2.84450 − 0.762182i −0.566253 + 8.63935i
11.1 −0.335606 − 2.54918i −1.83458 + 2.39087i −4.45382 + 1.19340i −2.15137 + 0.730293i 6.71043 + 3.87427i −0.179816 + 0.157694i 2.56902 + 6.20217i −1.57411 − 5.87467i 2.58366 + 5.23914i
11.2 −0.294083 − 2.23378i 0.644162 − 0.839488i −2.97146 + 0.796200i 3.40291 − 1.15513i −2.06467 − 1.19204i −3.05237 + 2.67686i 0.927978 + 2.24034i 0.486661 + 1.81625i −3.58106 − 7.26167i
11.3 −0.129856 − 0.986358i −0.567607 + 0.739719i 0.975813 − 0.261468i 0.354879 − 0.120465i 0.803335 + 0.463806i 1.84879 − 1.62135i −1.14606 − 2.76683i 0.551450 + 2.05804i −0.164905 − 0.334394i
11.4 −0.0400576 − 0.304268i 1.22544 − 1.59703i 1.84088 − 0.493262i −2.23081 + 0.757256i −0.535013 − 0.308890i −1.44599 + 1.26810i −0.458710 − 1.10743i −0.272333 − 1.01636i 0.319769 + 0.648428i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
97.k even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 97.2.k.a ✓ 128
3.b odd 2 1 873.2.bu.d 128
97.k even 48 1 inner 97.2.k.a ✓ 128
97.l odd 96 2 9409.2.a.o 128
291.v odd 48 1 873.2.bu.d 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.2.k.a ✓ 128 1.a even 1 1 trivial
97.2.k.a ✓ 128 97.k even 48 1 inner
873.2.bu.d 128 3.b odd 2 1
873.2.bu.d 128 291.v odd 48 1
9409.2.a.o 128 97.l odd 96 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(97, [\chi])\).