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Properties

Label 197.4.a.b
Level $197$
Weight $4$
Character orbit 197.a
Self dual yes
Analytic conductor $11.623$
Analytic rank $0$
Dimension $27$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,4,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 197.a (trivial)

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: yes
Analytic conductor: \(11.6233762711\)
Analytic rank: \(0\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 27 q + 4 q^{2} + 32 q^{3} + 128 q^{4} + 29 q^{5} + 36 q^{6} + 122 q^{7} + 27 q^{8} + 287 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q + 4 q^{2} + 32 q^{3} + 128 q^{4} + 29 q^{5} + 36 q^{6} + 122 q^{7} + 27 q^{8} + 287 q^{9} + 127 q^{10} + 98 q^{11} + 256 q^{12} + 193 q^{13} + 113 q^{14} + 194 q^{15} + 672 q^{16} + 124 q^{17} + 61 q^{18} + 535 q^{19} + 279 q^{20} + 188 q^{21} + 165 q^{22} + 344 q^{23} + 165 q^{24} + 910 q^{25} - 44 q^{26} + 1382 q^{27} + 1038 q^{28} + 346 q^{29} + 115 q^{30} + 599 q^{31} + 100 q^{32} + 794 q^{33} + 886 q^{34} + 12 q^{35} + 1714 q^{36} + 1428 q^{37} + 1108 q^{38} + 568 q^{39} + 1206 q^{40} + 549 q^{41} + 355 q^{42} + 2091 q^{43} + 586 q^{44} + 1167 q^{45} + 469 q^{46} + 1270 q^{47} + 2258 q^{48} + 2165 q^{49} - 353 q^{50} - 128 q^{51} + 1000 q^{52} - 208 q^{53} - 2945 q^{54} + 810 q^{55} - 5592 q^{56} - 1006 q^{57} + 287 q^{58} + 603 q^{59} - 2211 q^{60} - 314 q^{61} - 1347 q^{62} + 2416 q^{63} + 5 q^{64} - 1958 q^{65} - 8706 q^{66} + 4372 q^{67} - 6709 q^{68} - 306 q^{69} - 4329 q^{70} - 871 q^{71} - 7589 q^{72} + 942 q^{73} - 5237 q^{74} - 1530 q^{75} - 2470 q^{76} - 1868 q^{77} - 10179 q^{78} - 197 q^{79} - 1882 q^{80} + 1083 q^{81} - 961 q^{82} - 799 q^{83} - 3140 q^{84} - 1444 q^{85} - 7066 q^{86} - 794 q^{87} - 2409 q^{88} - 572 q^{89} - 10939 q^{90} + 3872 q^{91} - 6154 q^{92} - 1506 q^{93} + 1231 q^{94} - 756 q^{95} - 10975 q^{96} + 6335 q^{97} - 5330 q^{98} + 1940 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.48735 9.71574 22.1111 4.58132 −53.3137 25.8976 −77.4324 67.3955 −25.1393
1.2 −5.12187 −1.50664 18.2336 −16.6923 7.71684 18.4353 −52.4151 −24.7300 85.4958
1.3 −4.87065 −7.68695 15.7232 2.76237 37.4404 26.4058 −37.6171 32.0892 −13.4545
1.4 −4.64024 5.04740 13.5318 −14.3374 −23.4211 −19.5904 −25.6688 −1.52374 66.5290
1.5 −4.19736 −5.11666 9.61786 18.9633 21.4765 9.71359 −6.79075 −0.819797 −79.5957
1.6 −3.62545 −4.72863 5.14388 −3.40441 17.1434 −27.4348 10.3547 −4.64009 12.3425
1.7 −3.50700 8.13145 4.29904 12.2669 −28.5170 −29.1324 12.9793 39.1205 −43.0198
1.8 −3.11501 4.75612 1.70330 9.27815 −14.8154 28.7970 19.6143 −4.37935 −28.9015
1.9 −1.89943 −3.98313 −4.39218 11.9589 7.56566 −5.63514 23.5380 −11.1347 −22.7151
1.10 −1.75778 0.628838 −4.91021 −15.9710 −1.10536 −26.2224 22.6933 −26.6046 28.0735
1.11 −1.18786 −0.801428 −6.58899 −18.4442 0.951984 18.5342 17.3297 −26.3577 21.9092
1.12 −0.598548 9.95206 −7.64174 −0.534457 −5.95679 −3.08049 9.36233 72.0436 0.319898
1.13 0.246877 −1.02548 −7.93905 3.77286 −0.253168 −11.5484 −3.93498 −25.9484 0.931430
1.14 0.270197 8.02851 −7.92699 20.5119 2.16928 14.4694 −4.30343 37.4570 5.54226
1.15 0.407397 5.93866 −7.83403 −12.2475 2.41939 27.0857 −6.45074 8.26769 −4.98957
1.16 0.598148 −6.46754 −7.64222 −12.9216 −3.86854 −16.8113 −9.35636 14.8290 −7.72901
1.17 1.52338 −2.88724 −5.67933 21.0824 −4.39834 21.0908 −20.8388 −18.6639 32.1164
1.18 2.25077 −8.70988 −2.93404 0.262181 −19.6039 −3.36659 −24.6100 48.8621 0.590109
1.19 3.02359 5.94916 1.14207 8.38076 17.9878 4.69758 −20.7355 8.39253 25.3400
1.20 3.07527 −6.54222 1.45729 −17.7121 −20.1191 33.6154 −20.1206 15.8007 −54.4694
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(197\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.4.a.b ✓ 27
3.b odd 2 1 1773.4.a.d 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.4.a.b ✓ 27 1.a even 1 1 trivial
1773.4.a.d 27 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} - 4 T_{2}^{26} - 164 T_{2}^{25} + 647 T_{2}^{24} + 11748 T_{2}^{23} - 45616 T_{2}^{22} + \cdots - 5812256768 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(197))\). Copy content Toggle raw display