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Properties

Label 197.2.e.b
Level $197$
Weight $2$
Character orbit 197.e
Analytic conductor $1.573$
Analytic rank $0$
Dimension $90$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,2,Mod(6,197)]
 
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([13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 197.e (of order \(14\), 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: \(1.57305291982\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(15\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 90 q - 7 q^{2} + 25 q^{4} - 14 q^{5} - 18 q^{6} + 6 q^{7} - 28 q^{8} + 7 q^{9} + 10 q^{10} - 14 q^{11} - 21 q^{12} + 35 q^{15} - 5 q^{16} - 21 q^{18} - 20 q^{19} - 14 q^{21} + 10 q^{22} - 40 q^{23} - 26 q^{24}+ \cdots - 63 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −2.63359 + 0.601099i −2.91456 − 0.665230i 4.77252 − 2.29832i −1.31218 + 2.72478i 8.07562 −0.552295 + 2.41976i −6.96337 + 5.55310i 5.34923 + 2.57605i 1.81789 − 7.96469i
6.2 −2.62401 + 0.598912i 0.715047 + 0.163205i 4.72478 − 2.27533i 1.15577 − 2.39998i −1.97403 0.549496 − 2.40750i −6.82654 + 5.44398i −2.21825 − 1.06825i −1.59537 + 6.98976i
6.3 −1.94832 + 0.444691i 2.93408 + 0.669685i 1.79626 − 0.865034i 0.116141 − 0.241169i −6.01433 −0.836257 + 3.66388i 0.00984237 − 0.00784903i 5.45745 + 2.62817i −0.119034 + 0.521521i
6.4 −1.86511 + 0.425700i −0.519018 − 0.118463i 1.49550 − 0.720193i −0.0420268 + 0.0872695i 1.01846 0.0269378 − 0.118022i 0.508728 − 0.405697i −2.44756 − 1.17868i 0.0412341 − 0.180658i
6.5 −1.51063 + 0.344792i 0.198794 + 0.0453735i 0.361186 − 0.173938i −1.59912 + 3.32061i −0.315949 0.203799 − 0.892902i 1.93722 − 1.54488i −2.66545 − 1.28361i 1.27076 − 5.56758i
6.6 −0.999333 + 0.228091i 2.16685 + 0.494570i −0.855297 + 0.411890i 0.808978 − 1.67986i −2.27822 0.918600 − 4.02465i 2.36358 − 1.88490i 1.74775 + 0.841672i −0.425277 + 1.86326i
6.7 −0.540806 + 0.123436i −2.94459 − 0.672083i −1.52470 + 0.734258i −0.282052 + 0.585688i 1.67541 0.165956 − 0.727101i 1.60132 − 1.27701i 5.51599 + 2.65636i 0.0802411 − 0.351559i
6.8 0.113305 − 0.0258612i −1.20266 − 0.274499i −1.78977 + 0.861907i 0.228077 − 0.473606i −0.143367 0.759053 − 3.32563i −0.362229 + 0.288868i −1.33187 − 0.641394i 0.0135943 − 0.0595605i
6.9 0.219262 − 0.0500452i −0.121858 − 0.0278132i −1.75637 + 0.845821i −0.403983 + 0.838879i −0.0281107 −0.928396 + 4.06757i −0.694445 + 0.553801i −2.68883 − 1.29487i −0.0465963 + 0.204152i
6.10 0.375832 − 0.0857812i 2.55923 + 0.584128i −1.66805 + 0.803289i −1.18481 + 2.46028i 1.01195 0.0979929 − 0.429335i −1.16079 + 0.925696i 3.50556 + 1.68819i −0.234243 + 1.02629i
6.11 1.05771 − 0.241414i 2.20553 + 0.503397i −0.741477 + 0.357077i 1.41127 − 2.93053i 2.45432 −0.324895 + 1.42346i −2.39449 + 1.90954i 1.90803 + 0.918858i 0.785234 − 3.44033i
6.12 1.50872 − 0.344355i −1.34485 − 0.306954i 0.355708 − 0.171300i 1.40968 − 2.92723i −2.13471 0.549058 − 2.40558i −1.94212 + 1.54879i −0.988495 − 0.476034i 1.11880 − 4.90178i
6.13 1.99054 − 0.454329i 1.08804 + 0.248338i 1.95391 − 0.940954i −0.647835 + 1.34524i 2.27862 0.529429 − 2.31958i 0.269259 − 0.214727i −1.58074 − 0.761246i −0.678362 + 2.97210i
6.14 2.27480 − 0.519209i 0.0683280 + 0.0155954i 3.10321 − 1.49443i −0.134017 + 0.278288i 0.163530 −0.594171 + 2.60323i 2.63477 − 2.10116i −2.69848 − 1.29952i −0.160371 + 0.702633i
6.15 2.68066 − 0.611843i −2.88837 − 0.659251i 5.00965 − 2.41252i −0.109337 + 0.227041i −8.14609 0.435692 − 1.90889i 7.65363 − 6.10356i 5.20516 + 2.50667i −0.154182 + 0.675516i
19.1 −2.12061 + 1.69113i 2.59668 + 2.07078i 1.19203 − 5.22263i 0.988538 − 0.225627i −9.00853 1.90869 − 2.39342i 3.95062 + 8.20355i 1.78704 + 7.82955i −1.71474 + 2.15022i
19.2 −1.94361 + 1.54998i −1.98096 − 1.57976i 0.930147 − 4.07524i −0.192469 + 0.0439299i 6.29880 0.548213 − 0.687438i 2.35144 + 4.88282i 0.760986 + 3.33410i 0.305995 − 0.383705i
19.3 −1.63334 + 1.30255i −0.0124583 − 0.00993517i 0.526135 − 2.30515i 1.56168 − 0.356443i 0.0332897 0.877717 − 1.10062i 0.330331 + 0.685939i −0.667506 − 2.92454i −2.08647 + 2.61635i
19.4 −1.49452 + 1.19184i 0.834842 + 0.665764i 0.368062 − 1.61258i −2.72869 + 0.622806i −2.04117 −0.654078 + 0.820188i −0.286925 − 0.595806i −0.413844 − 1.81317i 3.33579 − 4.18295i
19.5 −0.994678 + 0.793229i −1.61766 − 1.29004i −0.0848704 + 0.371841i 1.17096 − 0.267264i 2.63234 −0.391521 + 0.490952i −1.31455 − 2.72968i 0.285053 + 1.24890i −0.952726 + 1.19468i
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
197.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.2.e.b ✓ 90
197.e even 14 1 inner 197.2.e.b ✓ 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.2.e.b ✓ 90 1.a even 1 1 trivial
197.2.e.b ✓ 90 197.e even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{90} + 7 T_{2}^{89} - 3 T_{2}^{88} - 126 T_{2}^{87} - 72 T_{2}^{86} + 1589 T_{2}^{85} + \cdots + 199927 \) acting on \(S_{2}^{\mathrm{new}}(197, [\chi])\). Copy content Toggle raw display