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Properties

Label 197.2.h.a
Level $197$
Weight $2$
Character orbit 197.h
Analytic conductor $1.573$
Analytic rank $0$
Dimension $672$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.57305291982\)
Analytic rank: \(0\)
Dimension: \(672\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 42 q^{2} - 42 q^{3} - 56 q^{4} - 42 q^{5} - 21 q^{6} - 56 q^{7} - 28 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 672 q - 42 q^{2} - 42 q^{3} - 56 q^{4} - 42 q^{5} - 21 q^{6} - 56 q^{7} - 28 q^{8} - 42 q^{9} - 56 q^{10} - 28 q^{11} - 28 q^{12} - 42 q^{13} + 70 q^{15} - 56 q^{16} - 42 q^{17} - 21 q^{18} - 7 q^{19} - 49 q^{20} - 42 q^{21} - 84 q^{22} - 42 q^{23} - 56 q^{24} - 42 q^{25} - 42 q^{26} + 119 q^{28} + 119 q^{29} + 77 q^{30} - 70 q^{31} - 56 q^{32} - 49 q^{33} - 70 q^{34} - 42 q^{35} - 301 q^{36} + 7 q^{37} - 28 q^{38} - 63 q^{39} - 56 q^{40} - 28 q^{41} + 112 q^{42} - 28 q^{43} + 126 q^{44} - 56 q^{45} - 56 q^{46} - 7 q^{47} + 42 q^{48} - 42 q^{49} - 182 q^{50} - 70 q^{51} + 49 q^{52} - 42 q^{53} - 63 q^{54} - 84 q^{55} + 77 q^{56} + 175 q^{57} + 126 q^{58} - 42 q^{59} + 84 q^{60} - 14 q^{61} - 56 q^{62} + 7 q^{63} - 14 q^{64} - 133 q^{65} + 63 q^{66} - 112 q^{67} - 49 q^{68} - 49 q^{69} + 168 q^{70} + 91 q^{71} + 392 q^{72} - 56 q^{73} - 56 q^{74} - 91 q^{75} - 161 q^{76} - 49 q^{77} + 84 q^{78} + 42 q^{79} - 70 q^{80} - 14 q^{81} - 42 q^{82} - 35 q^{83} + 245 q^{84} + 322 q^{85} + 63 q^{86} - 49 q^{87} - 133 q^{88} + 28 q^{89} + 196 q^{90} - 126 q^{91} - 7 q^{93} - 210 q^{94} + 84 q^{95} - 49 q^{96} - 42 q^{97} + 280 q^{98} + 259 q^{99}+O(q^{100}) \) 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 −2.48438 − 0.0796692i 0.733651 − 0.382745i 4.16990 + 0.267717i 3.41043 − 1.01220i −1.85316 + 0.892435i −0.0933948 − 2.91239i −5.38995 − 0.519962i −1.32460 + 1.89891i −8.55344 + 2.24297i
4.2 −2.46912 − 0.0791800i 1.01072 − 0.527293i 4.09441 + 0.262870i −1.88441 + 0.559284i −2.53735 + 1.22192i 0.0874837 + 2.72806i −5.17083 − 0.498823i −0.972829 + 1.39462i 4.69714 − 1.23173i
4.3 −2.23573 − 0.0716955i −2.58744 + 1.34986i 2.99746 + 0.192443i 1.85478 − 0.550490i 5.88159 − 2.83242i 0.0326190 + 1.01718i −2.23462 − 0.215571i 3.15634 − 4.52485i −4.18626 + 1.09777i
4.4 −1.75886 − 0.0564032i −1.46200 + 0.762723i 1.09451 + 0.0702702i −3.76944 + 1.11875i 2.61447 − 1.25906i −0.154052 − 4.80390i 1.58213 + 0.152626i −0.160662 + 0.230321i 6.69302 − 1.75512i
4.5 −1.58661 − 0.0508795i 3.00454 − 1.56747i 0.518856 + 0.0333117i −1.18982 + 0.353131i −4.84679 + 2.33409i −0.0447258 − 1.39471i 2.33866 + 0.225607i 4.85396 − 6.95852i 1.90574 − 0.499745i
4.6 −1.33631 − 0.0428530i −0.967825 + 0.504914i −0.211994 − 0.0136105i 0.517220 − 0.153508i 1.31495 − 0.633249i 0.00760859 + 0.237264i 2.94435 + 0.284038i −1.03460 + 1.48318i −0.697746 + 0.182971i
4.7 −0.718993 − 0.0230567i 0.693613 − 0.361857i −1.47947 − 0.0949852i −1.65735 + 0.491892i −0.507046 + 0.244181i 0.100126 + 3.12229i 2.49362 + 0.240556i −1.36619 + 1.95854i 1.20296 − 0.315454i
4.8 −0.431724 − 0.0138445i −1.15502 + 0.602572i −1.80970 − 0.116186i 2.70765 − 0.803617i 0.506991 − 0.244154i 0.0227637 + 0.709855i 1.63958 + 0.158168i −0.745377 + 1.06855i −1.18008 + 0.309454i
4.9 −0.0545187 − 0.00174831i 1.63079 − 0.850780i −1.99292 − 0.127950i 1.65668 − 0.491693i −0.0903958 + 0.0435323i −0.124611 − 3.88583i 0.217017 + 0.0209354i 0.219288 − 0.314366i −0.0911794 + 0.0239101i
4.10 0.723652 + 0.0232061i −2.36317 + 1.23286i −1.47276 − 0.0945541i 0.332877 − 0.0987962i −1.73872 + 0.837325i −0.158779 − 4.95130i −2.50493 − 0.241647i 2.34826 − 3.36641i 0.243180 − 0.0637693i
4.11 0.990936 + 0.0317774i 2.04585 − 1.06732i −1.01495 − 0.0651617i 1.62499 − 0.482288i 2.06122 − 0.992633i 0.0921623 + 2.87396i −2.97740 − 0.287227i 1.32999 − 1.90664i 1.62559 − 0.426279i
4.12 1.02446 + 0.0328524i −1.30409 + 0.680343i −0.947453 − 0.0608285i −2.32350 + 0.689602i −1.35834 + 0.654141i 0.0568363 + 1.77236i −3.00913 − 0.290287i −0.478567 + 0.686062i −2.40298 + 0.630137i
4.13 1.91435 + 0.0613896i 0.516877 − 0.269655i 1.66509 + 0.106902i 1.03412 − 0.306923i 1.00604 − 0.484483i −0.0104517 − 0.325921i −0.631963 − 0.0609647i −1.52190 + 2.18176i 1.99852 − 0.524074i
4.14 1.96514 + 0.0630184i 2.20952 − 1.15271i 1.86193 + 0.119540i −3.51840 + 1.04424i 4.41467 − 2.12599i −0.0407428 − 1.27051i −0.262707 − 0.0253431i 1.83691 − 2.63335i −6.97998 + 1.83037i
4.15 2.05500 + 0.0659000i −2.67118 + 1.39355i 2.22281 + 0.142709i 2.61768 − 0.776914i −5.58111 + 2.68772i 0.149948 + 4.67594i 0.465360 + 0.0448928i 3.47684 − 4.98432i 5.43054 − 1.42406i
4.16 2.66038 + 0.0853132i −0.906974 + 0.473168i 5.07445 + 0.325790i −1.24548 + 0.369652i −2.45326 + 1.18143i −0.0466036 − 1.45327i 8.17327 + 0.788465i −1.11764 + 1.60222i −3.34499 + 0.877159i
7.1 −1.93241 + 1.99537i −0.313596 + 0.190104i −0.183197 − 5.71277i 1.69226 + 2.26748i 0.226668 − 0.993097i 0.277139 − 0.268394i 7.64054 + 6.93894i −1.32541 + 2.54057i −7.79461 − 1.00500i
7.2 −1.57782 + 1.62922i 1.63650 − 0.992058i −0.100766 − 3.14226i −1.25967 − 1.68785i −0.965816 + 4.23152i 0.486560 − 0.471207i 1.92052 + 1.74416i 0.306351 − 0.587217i 4.73742 + 0.610822i
7.3 −1.38244 + 1.42748i −2.28722 + 1.38653i −0.0624630 − 1.94782i 0.447688 + 0.599861i 1.18270 − 5.18174i −3.53302 + 3.42154i −0.0752796 − 0.0683669i 1.92131 − 3.68278i −1.47519 − 0.190205i
7.4 −1.36347 + 1.40790i −1.88684 + 1.14381i −0.0590143 − 1.84028i −1.20083 − 1.60900i 0.962282 − 4.21603i 3.02946 − 2.93387i −0.230355 − 0.209202i 0.864241 − 1.65659i 3.90260 + 0.503185i
See next 80 embeddings (of 672 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
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 197.2.h.a ✓ 672
197.h even 98 1 inner 197.2.h.a ✓ 672
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.2.h.a ✓ 672 1.a even 1 1 trivial
197.2.h.a ✓ 672 197.h even 98 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(197, [\chi])\).