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Properties

Label 197.8.a.a
Level $197$
Weight $8$
Character orbit 197.a
Self dual yes
Analytic conductor $61.540$
Analytic rank $1$
Dimension $55$
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] = [197,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(61.5398500204\)
Analytic rank: \(1\)
Dimension: \(55\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 55 q - 24 q^{2} - 298 q^{3} + 3264 q^{4} - 946 q^{5} - 960 q^{6} - 6017 q^{7} - 5109 q^{8} + 36449 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 55 q - 24 q^{2} - 298 q^{3} + 3264 q^{4} - 946 q^{5} - 960 q^{6} - 6017 q^{7} - 5109 q^{8} + 36449 q^{9} - 9763 q^{10} - 11506 q^{11} - 37376 q^{12} - 33844 q^{13} - 33495 q^{14} - 25861 q^{15} + 176128 q^{16} - 55125 q^{17} - 69815 q^{18} - 205015 q^{19} - 209621 q^{20} - 76786 q^{21} - 184567 q^{22} - 132580 q^{23} - 126075 q^{24} + 703595 q^{25} - 802 q^{26} - 942457 q^{27} - 980794 q^{28} - 180975 q^{29} - 231365 q^{30} - 752200 q^{31} - 458116 q^{32} - 1189462 q^{33} - 534334 q^{34} - 501828 q^{35} + 2216122 q^{36} - 1812550 q^{37} - 1732590 q^{38} - 316133 q^{39} - 2564154 q^{40} + 387642 q^{41} - 1974005 q^{42} - 3120525 q^{43} - 1010046 q^{44} - 4004703 q^{45} - 2804563 q^{46} - 1968319 q^{47} - 5200654 q^{48} + 3219122 q^{49} + 5080367 q^{50} + 4015335 q^{51} - 105242 q^{52} + 793560 q^{53} + 9980039 q^{54} - 2193645 q^{55} + 7180088 q^{56} - 848029 q^{57} - 4318541 q^{58} - 4206985 q^{59} - 3767411 q^{60} - 3233922 q^{61} - 4798113 q^{62} - 16370607 q^{63} - 3153855 q^{64} - 5234873 q^{65} - 27799028 q^{66} - 29330378 q^{67} - 38894327 q^{68} - 12421645 q^{69} - 42976929 q^{70} - 20675277 q^{71} - 57843841 q^{72} - 21269365 q^{73} - 32764695 q^{74} - 51817010 q^{75} - 59377690 q^{76} - 16275190 q^{77} - 85089407 q^{78} - 28033705 q^{79} - 75304702 q^{80} - 923221 q^{81} - 54988201 q^{82} - 35573360 q^{83} - 80450394 q^{84} - 33089314 q^{85} - 71770618 q^{86} - 38591063 q^{87} - 62367981 q^{88} - 22496216 q^{89} - 138633489 q^{90} - 69344213 q^{91} - 112457146 q^{92} - 38701181 q^{93} - 49016719 q^{94} - 35589281 q^{95} - 173964883 q^{96} - 59838870 q^{97} - 95514862 q^{98} - 85438141 q^{99}+O(q^{100}) \) 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 −22.2510 45.1613 367.105 28.2712 −1004.88 −292.983 −5320.33 −147.455 −629.060
1.2 −20.7456 69.7029 302.379 71.1733 −1446.03 −755.780 −3617.59 2671.50 −1476.53
1.3 −20.3297 −92.2509 285.297 240.282 1875.43 528.959 −3197.81 6323.23 −4884.87
1.4 −20.2556 −80.5531 282.289 −368.076 1631.65 −1402.81 −3125.22 4301.80 7455.60
1.5 −19.8304 −24.8621 265.245 −146.760 493.027 −946.144 −2721.63 −1568.87 2910.32
1.6 −19.4258 18.3231 249.363 −401.739 −355.942 −1624.25 −2357.58 −1851.26 7804.13
1.7 −19.2901 35.5299 244.107 −116.493 −685.374 1300.96 −2239.71 −924.628 2247.16
1.8 −18.3103 −51.5977 207.268 98.1266 944.771 976.314 −1451.42 475.327 −1796.73
1.9 −17.5947 61.4535 181.573 339.311 −1081.26 −1066.95 −942.604 1589.54 −5970.08
1.10 −16.3599 −65.1308 139.647 −162.360 1065.53 1195.64 −190.548 2055.02 2656.19
1.11 −15.4291 −4.35926 110.056 −253.520 67.2593 265.166 276.855 −2168.00 3911.57
1.12 −14.7987 −34.4797 91.0011 280.810 510.255 1213.61 547.535 −998.149 −4155.62
1.13 −13.5137 −76.4894 54.6202 518.006 1033.65 363.757 991.632 3663.62 −7000.18
1.14 −12.7849 15.4183 35.4535 305.354 −197.121 −880.474 1183.20 −1949.28 −3903.92
1.15 −11.5822 68.2819 6.14643 6.41923 −790.852 1174.18 1411.33 2475.42 −74.3486
1.16 −11.5335 −59.4702 5.02171 254.185 685.899 −1419.31 1418.37 1349.70 −2931.65
1.17 −10.6427 44.6209 −14.7320 347.881 −474.889 835.820 1519.06 −195.974 −3702.41
1.18 −10.6203 −31.9061 −15.2088 −403.368 338.853 −734.185 1520.92 −1169.00 4283.90
1.19 −10.4867 67.6647 −18.0290 −106.711 −709.579 −371.212 1531.36 2391.51 1119.04
1.20 −8.94046 32.9611 −48.0682 −459.188 −294.687 1720.14 1574.13 −1100.57 4105.35
See all 55 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.55
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 197.8.a.a ✓ 55
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.8.a.a ✓ 55 1.a even 1 1 trivial