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Properties

Label 197.4.d.a
Level $197$
Weight $4$
Character orbit 197.d
Analytic conductor $11.623$
Analytic rank $0$
Dimension $294$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,4,Mod(36,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.36");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 197.d (of order \(7\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6233762711\)
Analytic rank: \(0\)
Dimension: \(294\)
Relative dimension: \(49\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 294 q - 5 q^{2} - 5 q^{3} - 161 q^{4} - 5 q^{5} + 58 q^{6} + 15 q^{7} + 115 q^{8} - 446 q^{9} - 211 q^{10} + 93 q^{11} - 47 q^{12} + 23 q^{13} + 656 q^{14} + 343 q^{15} - 1505 q^{16} - 17 q^{17} + 132 q^{18}+ \cdots + 4526 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −4.99354 + 2.40476i −5.23278 − 2.51997i 14.1646 − 17.7619i −12.6486 − 15.8608i 32.1900 16.7036 + 8.04405i −18.1521 + 79.5297i 4.19748 + 5.26348i 101.303 + 48.7848i
36.2 −4.96059 + 2.38889i −7.79565 − 3.75419i 13.9127 − 17.4460i 10.3332 + 12.9574i 47.6394 −26.2094 − 12.6218i −17.5373 + 76.8359i 29.8440 + 37.4232i −82.2124 − 39.5914i
36.3 −4.86777 + 2.34419i 3.69855 + 1.78113i 13.2120 − 16.5673i 7.62651 + 9.56334i −22.1790 13.6794 + 6.58764i −15.8580 + 69.4785i −6.32736 − 7.93426i −59.5424 − 28.6741i
36.4 −4.54516 + 2.18883i 6.85363 + 3.30053i 10.8795 − 13.6425i −1.34248 − 1.68342i −38.3751 −29.8193 − 14.3602i −10.6076 + 46.4748i 19.2445 + 24.1318i 9.78651 + 4.71294i
36.5 −4.23571 + 2.03981i 5.41516 + 2.60780i 8.79251 − 11.0255i −7.52587 − 9.43714i −28.2565 15.8752 + 7.64508i −6.38361 + 27.9684i 5.68910 + 7.13391i 51.1274 + 24.6217i
36.6 −4.14519 + 1.99622i −1.88374 − 0.907162i 8.20981 − 10.2948i −4.59596 − 5.76315i 9.61936 −18.7661 − 9.03730i −5.29039 + 23.1787i −14.1087 − 17.6917i 30.5557 + 14.7148i
36.7 −4.04091 + 1.94600i −1.44300 − 0.694910i 7.55409 − 9.47253i 1.68297 + 2.11038i 7.18331 0.423695 + 0.204041i −4.10768 + 17.9969i −15.2349 − 19.1039i −10.9075 − 5.25278i
36.8 −3.89468 + 1.87558i −3.17499 − 1.52899i 6.66280 − 8.35489i 7.46144 + 9.35635i 15.2333 20.5374 + 9.89028i −2.58398 + 11.3211i −9.09148 − 11.4004i −46.6085 − 22.4454i
36.9 −3.67532 + 1.76994i −7.40057 − 3.56393i 5.38734 − 6.75551i −2.19674 − 2.75463i 33.5074 8.85732 + 4.26546i −0.581514 + 2.54778i 25.2326 + 31.6407i 12.9492 + 6.23602i
36.10 −3.23729 + 1.55900i 2.40034 + 1.15594i 3.06165 − 3.83919i 12.7344 + 15.9684i −9.57271 −16.4771 − 7.93493i 2.47019 − 10.8226i −12.4088 − 15.5601i −66.1197 − 31.8416i
36.11 −3.10199 + 1.49384i 8.84799 + 4.26097i 2.40285 − 3.01308i 5.43559 + 6.81602i −33.8115 3.70763 + 1.78550i 3.17646 − 13.9170i 43.2968 + 54.2925i −27.0432 − 13.0233i
36.12 −3.03249 + 1.46037i 3.37988 + 1.62767i 2.07541 − 2.60249i −13.2921 − 16.6678i −12.6265 −8.24050 − 3.96842i 3.49863 − 15.3285i −8.05991 − 10.1068i 64.6494 + 31.1335i
36.13 −2.71290 + 1.30647i −8.46100 − 4.07460i 0.665075 − 0.833977i −2.95496 − 3.70540i 28.2772 −5.24170 − 2.52427i 4.64553 − 20.3534i 38.1519 + 47.8410i 12.8575 + 6.19185i
36.14 −2.69059 + 1.29572i 6.01269 + 2.89556i 0.572466 − 0.717849i 0.538788 + 0.675619i −19.9295 19.6300 + 9.45330i 4.70603 − 20.6185i 10.9339 + 13.7107i −2.32507 − 1.11969i
36.15 −2.25876 + 1.08776i 2.36915 + 1.14092i −1.06916 + 1.34068i 4.71882 + 5.91721i −6.59238 −13.2401 − 6.37609i 5.41956 − 23.7447i −12.5231 − 15.7034i −17.0952 − 8.23260i
36.16 −2.08511 + 1.00413i −6.32083 − 3.04395i −1.64854 + 2.06720i 11.8661 + 14.8797i 16.2361 6.37684 + 3.07092i 5.48146 − 24.0158i 13.8531 + 17.3712i −39.6833 − 19.1105i
36.17 −2.05041 + 0.987423i −3.59199 − 1.72981i −1.75876 + 2.20541i −8.13502 − 10.2010i 9.07310 −19.7705 − 9.52099i 5.47976 − 24.0084i −6.92408 − 8.68252i 26.7528 + 12.8835i
36.18 −2.04676 + 0.985669i 0.445213 + 0.214403i −1.77022 + 2.21979i −3.24849 − 4.07347i −1.12258 24.4430 + 11.7711i 5.47932 − 24.0065i −16.6820 − 20.9185i 10.6640 + 5.13550i
36.19 −1.75279 + 0.844099i −3.92827 − 1.89175i −2.62815 + 3.29560i −9.59724 − 12.0346i 8.48226 26.9215 + 12.9647i 5.28801 − 23.1683i −4.98167 − 6.24682i 26.9803 + 12.9930i
36.20 −1.26252 + 0.607996i 8.03734 + 3.87058i −3.76363 + 4.71944i −10.7505 − 13.4806i −12.5006 −15.5899 − 7.50770i 4.37677 − 19.1759i 32.7833 + 41.1089i 21.7688 + 10.4833i
See next 80 embeddings (of 294 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.49
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
197.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.4.d.a ✓ 294
197.d even 7 1 inner 197.4.d.a ✓ 294
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.4.d.a ✓ 294 1.a even 1 1 trivial
197.4.d.a ✓ 294 197.d even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(197, [\chi])\).