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Properties

Label 197.3.f.a
Level $197$
Weight $3$
Character orbit 197.f
Analytic conductor $5.368$
Analytic rank $0$
Dimension $384$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,3,Mod(20,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.20");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 197.f (of order \(28\), degree \(12\), 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: \(5.36786120790\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(32\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 12 q^{2} - 12 q^{3} - 14 q^{4} - 4 q^{5} - 14 q^{7} + 48 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 12 q^{2} - 12 q^{3} - 14 q^{4} - 4 q^{5} - 14 q^{7} + 48 q^{8} - 14 q^{9} - 14 q^{10} + 32 q^{11} - 26 q^{12} - 16 q^{13} - 68 q^{14} - 14 q^{15} + 70 q^{16} + 16 q^{17} + 180 q^{18} - 128 q^{20} + 6 q^{21} - 14 q^{22} - 62 q^{23} + 150 q^{24} - 14 q^{25} - 168 q^{26} + 6 q^{27} + 420 q^{28} + 140 q^{29} - 98 q^{30} + 50 q^{31} + 142 q^{32} + 38 q^{34} + 78 q^{35} - 2260 q^{36} - 64 q^{37} + 2 q^{38} - 14 q^{39} - 626 q^{40} - 126 q^{41} + 410 q^{42} - 14 q^{43} - 154 q^{44} + 544 q^{45} + 136 q^{46} - 364 q^{47} + 1114 q^{48} - 18 q^{49} - 140 q^{50} - 138 q^{51} - 214 q^{52} + 2 q^{53} + 90 q^{54} - 14 q^{55} + 26 q^{56} + 506 q^{57} - 364 q^{58} - 350 q^{59} + 348 q^{60} + 434 q^{61} + 756 q^{62} - 1122 q^{63} + 1330 q^{64} - 126 q^{65} - 742 q^{66} + 98 q^{67} + 1016 q^{68} + 454 q^{69} - 62 q^{70} - 682 q^{71} + 908 q^{72} - 32 q^{73} - 880 q^{74} - 78 q^{75} + 470 q^{76} - 410 q^{77} - 1626 q^{78} + 1100 q^{79} - 810 q^{80} + 1118 q^{81} - 86 q^{82} + 784 q^{84} - 1068 q^{85} - 632 q^{86} - 478 q^{87} - 274 q^{88} - 430 q^{89} + 308 q^{90} - 1570 q^{91} - 1022 q^{92} + 190 q^{94} + 1366 q^{95} - 462 q^{96} - 602 q^{97} + 1412 q^{98} - 662 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −0.435755 − 3.86743i 2.95944 + 0.333448i −10.8674 + 2.48042i −1.52759 − 2.43114i − 11.5907i −10.9320 + 8.71802i 9.18674 + 26.2542i −0.127271 − 0.0290489i −8.73662 + 6.96722i
20.2 −0.419508 − 3.72324i 2.90493 + 0.327307i −9.78678 + 2.23377i 4.78832 + 7.62057i − 10.9530i 8.80031 − 7.01801i 7.47253 + 21.3553i −0.442873 − 0.101083i 26.3644 − 21.0249i
20.3 −0.414952 − 3.68280i −4.30059 − 0.484560i −9.49114 + 2.16629i 1.58670 + 2.52521i 16.0393i −0.116216 + 0.0926788i 7.02019 + 20.0626i 9.48591 + 2.16510i 8.64146 − 6.89134i
20.4 −0.371930 − 3.30097i −2.56336 − 0.288822i −6.85837 + 1.56538i −2.94574 − 4.68813i 8.56901i −0.501926 + 0.400272i 3.32954 + 9.51528i −2.28693 − 0.521977i −14.3798 + 11.4675i
20.5 −0.342570 − 3.04039i 0.502201 + 0.0565845i −5.22690 + 1.19301i −1.18537 − 1.88650i − 1.54627i 5.27485 − 4.20656i 1.37566 + 3.93140i −8.52535 − 1.94585i −5.32963 + 4.25024i
20.6 −0.319435 − 2.83506i 4.80794 + 0.541725i −4.03584 + 0.921154i −2.57594 − 4.09958i − 13.8039i 3.34211 − 2.66524i 0.131571 + 0.376009i 14.0485 + 3.20647i −10.7997 + 8.61250i
20.7 −0.297386 − 2.63937i −0.843629 − 0.0950542i −2.97813 + 0.679738i 3.14437 + 5.00424i 2.25492i −6.48064 + 5.16814i −0.829245 − 2.36985i −8.07168 − 1.84231i 12.2730 − 9.78736i
20.8 −0.262848 − 2.33284i −0.767376 − 0.0864625i −1.47335 + 0.336283i 2.37800 + 3.78457i 1.81289i −0.926942 + 0.739212i −1.92969 − 5.51475i −8.19296 − 1.86999i 8.20375 − 6.54227i
20.9 −0.238617 − 2.11778i 4.98319 + 0.561471i −0.528356 + 0.120594i 3.30351 + 5.25750i − 10.6873i −6.87622 + 5.48360i −2.43408 − 6.95619i 15.7426 + 3.59314i 10.3460 − 8.25065i
20.10 −0.234182 − 2.07842i −5.36165 − 0.604113i −0.365283 + 0.0833735i 3.47891 + 5.53666i 11.2852i 3.60638 − 2.87599i −2.50438 − 7.15712i 19.6080 + 4.47540i 10.6928 − 8.52723i
20.11 −0.203015 − 1.80181i −4.26999 − 0.481113i 0.694418 − 0.158496i −3.99552 − 6.35884i 7.79138i 8.06220 − 6.42939i −2.82202 − 8.06486i 9.22703 + 2.10601i −10.6462 + 8.49010i
20.12 −0.163026 − 1.44689i 0.999880 + 0.112659i 1.83279 − 0.418322i −3.86158 − 6.14567i − 1.46509i −6.20051 + 4.94474i −2.82767 − 8.08101i −7.78728 − 1.77740i −8.26259 + 6.58920i
20.13 −0.110851 − 0.983829i 3.27838 + 0.369385i 2.94408 − 0.671967i 1.96731 + 3.13096i − 3.26631i 3.16882 − 2.52705i −2.29543 − 6.55997i 1.83698 + 0.419279i 2.86225 − 2.28257i
20.14 −0.0979222 − 0.869084i −3.64248 − 0.410409i 3.15399 − 0.719878i −0.0627753 − 0.0999063i 3.20581i −7.30411 + 5.82484i −2.08991 − 5.97261i 4.32487 + 0.987123i −0.0806799 + 0.0643401i
20.15 −0.0797429 − 0.707737i 0.378635 + 0.0426619i 3.40518 − 0.777210i −0.451619 − 0.718748i − 0.271376i 7.59800 − 6.05920i −1.76252 − 5.03699i −8.63281 − 1.97038i −0.472671 + 0.376943i
20.16 0.00409705 + 0.0363623i −4.18203 − 0.471201i 3.89841 − 0.889786i 0.530491 + 0.844271i − 0.153999i 0.477052 − 0.380436i 0.0966694 + 0.276265i 8.49298 + 1.93847i −0.0285262 + 0.0227489i
20.17 0.0450830 + 0.400123i 4.90548 + 0.552715i 3.74165 − 0.854006i −3.62530 − 5.76964i 1.98771i −1.55739 + 1.24198i 1.04235 + 2.97885i 14.9839 + 3.41998i 2.14512 − 1.71068i
20.18 0.0564146 + 0.500693i −1.43240 − 0.161393i 3.65220 − 0.833591i 5.14385 + 8.18640i − 0.726298i 3.28095 − 2.61647i 1.28907 + 3.68395i −6.74863 − 1.54033i −3.80869 + 3.03733i
20.19 0.0829492 + 0.736194i −2.01235 − 0.226738i 3.36461 − 0.767950i −2.14168 − 3.40847i − 1.50029i 0.423154 − 0.337454i 1.82320 + 5.21042i −4.77620 − 1.09014i 2.33165 − 1.85943i
20.20 0.115261 + 1.02297i 1.88088 + 0.211925i 2.86652 − 0.654265i 1.86461 + 2.96751i 1.94852i −8.57510 + 6.83842i 2.35971 + 6.74367i −5.28154 − 1.20548i −2.82077 + 2.24949i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
197.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.3.f.a ✓ 384
197.f odd 28 1 inner 197.3.f.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.3.f.a ✓ 384 1.a even 1 1 trivial
197.3.f.a ✓ 384 197.f odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(197, [\chi])\).