↑

Properties

Label 197.8.a.b
Level $197$
Weight $8$
Character orbit 197.a
Self dual yes
Analytic conductor $61.540$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 16 q^{2} + 296 q^{3} + 4224 q^{4} + 554 q^{5} + 1200 q^{6} + 4959 q^{7} + 2571 q^{8} + 47384 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 16 q^{2} + 296 q^{3} + 4224 q^{4} + 554 q^{5} + 1200 q^{6} + 4959 q^{7} + 2571 q^{8} + 47384 q^{9} + 16237 q^{10} + 12452 q^{11} + 38656 q^{12} + 36460 q^{13} - 567 q^{14} + 55139 q^{15} + 319488 q^{16} + 13657 q^{17} + 75985 q^{18} + 179089 q^{19} + 78379 q^{20} + 71390 q^{21} + 220057 q^{22} + 159428 q^{23} + 288645 q^{24} + 1250470 q^{25} + 245262 q^{26} + 789647 q^{27} + 775366 q^{28} + 111693 q^{29} + 200635 q^{30} + 1035260 q^{31} + 688764 q^{32} + 535514 q^{33} + 1430866 q^{34} + 698672 q^{35} + 4315642 q^{36} + 1226630 q^{37} + 133058 q^{38} + 1107523 q^{39} + 2427846 q^{40} + 1766062 q^{41} + 693163 q^{42} + 3240035 q^{43} + 2056578 q^{44} + 1462797 q^{45} + 1088877 q^{46} + 3015185 q^{47} + 4531442 q^{48} + 8513327 q^{49} - 8803339 q^{50} - 1200965 q^{51} + 2403172 q^{52} - 2021580 q^{53} - 9268097 q^{54} + 2538479 q^{55} - 13776768 q^{56} - 638299 q^{57} + 3745679 q^{58} + 6845183 q^{59} + 5842749 q^{60} + 6328462 q^{61} + 7784379 q^{62} + 11502133 q^{63} + 28157685 q^{64} + 14369523 q^{65} + 25244790 q^{66} + 20888346 q^{67} + 18152427 q^{68} + 7874901 q^{69} + 23377047 q^{70} + 7085107 q^{71} + 50177539 q^{72} + 21946913 q^{73} + 32718747 q^{74} + 44992638 q^{75} + 52601230 q^{76} + 22159370 q^{77} + 86884893 q^{78} + 9667191 q^{79} + 50245998 q^{80} + 56936064 q^{81} + 55357649 q^{82} + 57432744 q^{83} + 65898868 q^{84} + 32071274 q^{85} + 91529136 q^{86} + 38121559 q^{87} + 95846511 q^{88} + 46839682 q^{89} + 136287359 q^{90} + 81150673 q^{91} + 99073310 q^{92} + 81689463 q^{93} + 55616387 q^{94} + 30995799 q^{95} + 156893033 q^{96} + 72697122 q^{97} + 69757306 q^{98} + 64628303 q^{99}+O(q^{100}) \) 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 −21.9779 −46.0918 355.028 −508.529 1013.00 1271.38 −4989.61 −62.5423 11176.4
1.2 −21.3411 −55.6399 327.443 134.694 1187.42 −274.833 −4256.33 908.798 −2874.52
1.3 −21.1053 19.7965 317.434 534.839 −417.811 1641.16 −3998.06 −1795.10 −11288.0
1.4 −20.8533 −0.827001 306.860 −12.3329 17.2457 246.396 −3729.81 −2186.32 257.181
1.5 −20.7354 83.9180 301.958 −512.036 −1740.07 308.678 −3607.08 4855.22 10617.3
1.6 −20.1726 −28.7735 278.935 480.748 580.436 −1059.65 −3044.76 −1359.09 −9697.95
1.7 −18.5263 88.4348 215.222 377.477 −1638.36 879.714 −1615.90 5633.71 −6993.24
1.8 −17.1767 23.0763 167.038 −418.802 −396.373 419.875 −670.545 −1654.49 7193.63
1.9 −16.5276 69.6348 145.163 −93.2649 −1150.90 825.610 −283.663 2662.01 1541.45
1.10 −16.3073 22.4490 137.928 281.170 −366.082 −559.636 −161.904 −1683.04 −4585.12
1.11 −16.0510 −45.8139 129.635 −397.516 735.360 −164.041 −26.2505 −88.0854 6380.54
1.12 −15.5881 10.3776 114.990 315.670 −161.768 403.171 202.797 −2079.30 −4920.71
1.13 −15.5057 −44.3590 112.428 72.8541 687.820 −1088.37 241.454 −219.276 −1129.66
1.14 −14.9461 56.4076 95.3854 −196.598 −843.073 −1497.90 487.460 994.822 2938.38
1.15 −14.5260 −80.0909 83.0049 −103.281 1163.40 −773.736 653.600 4227.56 1500.26
1.16 −12.9041 78.9621 38.5156 −356.709 −1018.93 −513.898 1154.72 4048.01 4603.01
1.17 −11.4263 −79.0210 2.56123 −431.704 902.921 742.367 1433.31 4057.32 4932.80
1.18 −9.99164 15.9423 −28.1670 −216.554 −159.289 −103.542 1560.37 −1932.84 2163.73
1.19 −8.57214 −8.03142 −54.5184 3.16930 68.8464 1453.35 1564.57 −2122.50 −27.1677
1.20 −8.25431 −14.1669 −59.8663 11.5658 116.938 −664.362 1550.71 −1986.30 −95.4680
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.60
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.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.8.a.b ✓ 60 1.a even 1 1 trivial