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Properties

Label 197.4.h.a
Level $197$
Weight $4$
Character orbit 197.h
Analytic conductor $11.623$
Analytic rank $0$
Dimension $2016$
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(4,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(98))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 197.h (of order \(98\), degree \(42\), 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: \(2016\)
Relative dimension: \(48\) over \(\Q(\zeta_{98})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{98}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2016 q - 42 q^{2} - 42 q^{3} - 42 q^{5} - 119 q^{6} - 98 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2016 q - 42 q^{2} - 42 q^{3} - 42 q^{5} - 119 q^{6} - 98 q^{8} - 42 q^{9} - 252 q^{10} - 140 q^{11} + 14 q^{12} - 42 q^{13} - 1722 q^{15} + 168 q^{16} - 42 q^{17} + 147 q^{18} - 567 q^{19} - 49 q^{20} - 42 q^{21} - 1386 q^{22} - 42 q^{23} + 1246 q^{24} - 42 q^{25} + 1134 q^{26} - 504 q^{27} + 525 q^{28} - 3269 q^{29} - 847 q^{30} - 966 q^{31} - 994 q^{32} - 1631 q^{33} + 798 q^{34} - 42 q^{35} - 4207 q^{36} - 2149 q^{37} + 14 q^{38} - 252 q^{39} - 686 q^{40} - 392 q^{41} + 1176 q^{42} - 378 q^{43} + 12894 q^{44} + 3759 q^{45} + 2520 q^{46} + 28 q^{47} + 3654 q^{48} + 2646 q^{49} + 4914 q^{50} + 14 q^{51} - 3703 q^{52} - 42 q^{53} + 4305 q^{54} - 4130 q^{55} + 2989 q^{56} - 11585 q^{57} + 13286 q^{58} - 3136 q^{59} - 4914 q^{60} - 1050 q^{61} + 3836 q^{62} - 2198 q^{63} - 2562 q^{64} + 2359 q^{65} - 4683 q^{66} + 3528 q^{67} - 49 q^{68} - 49 q^{69} + 5208 q^{70} - 8792 q^{71} + 4928 q^{72} + 630 q^{73} + 266 q^{74} + 1232 q^{75} - 7903 q^{76} - 49 q^{77} - 4788 q^{78} - 1512 q^{79} - 490 q^{80} + 1078 q^{81} - 42 q^{82} - 5607 q^{83} + 10535 q^{84} - 19824 q^{85} - 7833 q^{86} - 49 q^{87} - 2373 q^{88} - 196 q^{89} - 8484 q^{90} + 14084 q^{91} - 1554 q^{92} - 2639 q^{93} + 7784 q^{94} + 7651 q^{95} + 15211 q^{96} - 1442 q^{97} + 18788 q^{98} - 14763 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.59899 − 0.179549i −0.0719909 + 0.0375576i 23.3328 + 1.49802i 10.6042 − 3.14727i 0.409819 − 0.197359i 0.978044 + 30.4990i −85.7634 − 8.27349i −15.4434 + 22.1392i −59.9379 + 15.7176i
4.2 −5.54205 − 0.177723i 5.62389 − 2.93398i 22.6991 + 1.45733i −18.2604 + 5.41960i −31.6893 + 15.2608i −0.478938 − 14.9350i −81.3865 − 7.85125i 7.57278 − 10.8561i 102.163 − 26.7904i
4.3 −5.14681 − 0.165048i −8.31459 + 4.33772i 18.4789 + 1.18638i −18.6014 + 5.52079i 43.5096 − 20.9531i 0.806502 + 25.1497i −53.9063 − 5.20028i 34.8695 − 49.9880i 96.6490 − 25.3443i
4.4 −5.02297 − 0.161077i −6.04395 + 3.15313i 17.2207 + 1.10561i 6.03871 − 1.79226i 30.8665 − 14.8645i −0.745364 − 23.2432i −46.3023 − 4.46673i 11.1400 − 15.9701i −30.6209 + 8.02976i
4.5 −4.84119 − 0.155248i 6.75489 − 3.52402i 15.4294 + 0.990603i 11.4670 − 3.40334i −33.2488 + 16.0118i −0.497056 − 15.5000i −35.9727 − 3.47024i 17.7626 − 25.4641i −56.0421 + 14.6960i
4.6 −4.65231 − 0.149190i 0.203946 − 0.106399i 13.6382 + 0.875598i −0.723623 + 0.214768i −0.964694 + 0.464572i −0.222342 − 6.93345i −26.2527 − 2.53257i −15.4169 + 22.1013i 3.39856 − 0.891207i
4.7 −4.47074 − 0.143368i −1.40966 + 0.735421i 11.9834 + 0.769358i −10.4142 + 3.09087i 6.40768 − 3.08577i −0.296520 − 9.24657i −17.8453 − 1.72151i −14.0008 + 20.0713i 47.0022 − 12.3254i
4.8 −4.27036 − 0.136942i 7.00949 − 3.65685i 10.2337 + 0.657023i 3.98425 − 1.18251i −30.4338 + 14.6562i 0.192674 + 6.00830i −9.58897 − 0.925036i 20.3133 − 29.1206i −17.1761 + 4.50411i
4.9 −3.94494 − 0.126506i −6.13710 + 3.20172i 7.56297 + 0.485559i 13.4744 − 3.99914i 24.6155 − 11.8542i 0.569670 + 17.7644i 1.65580 + 0.159733i 11.9658 − 17.1539i −53.6618 + 14.0718i
4.10 −3.92131 − 0.125749i 6.45836 − 3.36932i 7.37726 + 0.473636i −9.26748 + 2.75054i −25.7489 + 12.4000i 1.08979 + 33.9836i 2.37259 + 0.228881i 14.9110 − 21.3760i 36.6865 − 9.62033i
4.11 −3.52515 − 0.113045i −2.28495 + 1.19206i 4.43036 + 0.284438i −12.5509 + 3.72505i 8.18955 − 3.94388i 0.507844 + 15.8364i 12.4998 + 1.20584i −11.6472 + 16.6971i 44.6650 − 11.7125i
4.12 −3.34417 − 0.107241i 2.09981 − 1.09547i 3.18843 + 0.204704i 14.6220 − 4.33973i −7.13962 + 3.43826i 0.325213 + 10.1413i 16.0027 + 1.54376i −12.2380 + 17.5441i −49.3638 + 12.9447i
4.13 −2.78217 − 0.0892188i −1.87897 + 0.980259i −0.251057 − 0.0161184i 19.6548 − 5.83345i 5.31508 − 2.55961i −0.869697 − 27.1203i 22.8629 + 2.20556i −12.8775 + 18.4609i −55.2035 + 14.4761i
4.14 −2.63628 − 0.0845404i 6.79613 − 3.54553i −1.04073 − 0.0668170i −18.6621 + 5.53881i −18.2162 + 8.77248i −0.458603 − 14.3009i 23.7416 + 2.29032i 18.1694 − 26.0471i 49.6667 − 13.0242i
4.15 −2.61985 − 0.0840136i 3.80323 − 1.98414i −1.12699 − 0.0723550i −2.11039 + 0.626353i −10.1306 + 4.87864i −1.12669 − 35.1345i 23.8192 + 2.29781i −4.91940 + 7.05233i 5.58154 − 1.46365i
4.16 −2.52646 − 0.0810187i −7.43397 + 3.87830i −1.60712 − 0.103180i −9.38322 + 2.78489i 19.0959 − 9.19608i −0.593600 − 18.5106i 24.1806 + 2.33267i 24.7755 − 35.5176i 23.9320 − 6.27570i
4.17 −2.29111 − 0.0734715i −7.17037 + 3.74078i −2.73977 − 0.175899i 1.02306 − 0.303639i 16.7030 − 8.04372i −0.0377790 − 1.17809i 24.5178 + 2.36520i 21.9736 − 31.5008i −2.36625 + 0.620505i
4.18 −2.01740 − 0.0646941i −3.84065 + 2.00366i −3.91784 − 0.251534i 0.429442 − 0.127456i 7.87776 − 3.79373i 1.01593 + 31.6804i 23.9605 + 2.31144i −4.71123 + 6.75391i −0.874603 + 0.229348i
4.19 −1.87681 − 0.0601858i 2.36417 − 1.23339i −4.46475 − 0.286647i −9.67924 + 2.87275i −4.51135 + 2.17255i 0.0490453 + 1.52941i 23.3151 + 2.24918i −11.3791 + 16.3128i 18.3390 − 4.80906i
4.20 −1.33337 − 0.0427585i 2.89683 − 1.51127i −6.20752 − 0.398536i 5.75746 − 1.70878i −3.92716 + 1.89122i 0.493668 + 15.3944i 18.8830 + 1.82162i −9.33949 + 13.3889i −7.74988 + 2.03226i
See next 80 embeddings (of 2016 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
197.h even 98 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.4.h.a ✓ 2016
197.h even 98 1 inner 197.4.h.a ✓ 2016
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.4.h.a ✓ 2016 1.a even 1 1 trivial
197.4.h.a ✓ 2016 197.h even 98 1 inner

Hecke kernels

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