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Properties

Label 197.6.a.b
Level $197$
Weight $6$
Character orbit 197.a
Self dual yes
Analytic conductor $31.596$
Analytic rank $0$
Dimension $43$
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] = [197,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(31.5956125032\)
Analytic rank: \(0\)
Dimension: \(43\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 43 q + 12 q^{2} + 98 q^{3} + 752 q^{4} + 146 q^{5} + 144 q^{6} + 831 q^{7} + 699 q^{8} + 3725 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 43 q + 12 q^{2} + 98 q^{3} + 752 q^{4} + 146 q^{5} + 144 q^{6} + 831 q^{7} + 699 q^{8} + 3725 q^{9} + 1033 q^{10} + 1370 q^{11} + 3136 q^{12} + 2948 q^{13} + 929 q^{14} + 1859 q^{15} + 14080 q^{16} + 1545 q^{17} + 6757 q^{18} + 12233 q^{19} + 7419 q^{20} + 3674 q^{21} + 10369 q^{22} + 5564 q^{23} + 6813 q^{24} + 35439 q^{25} + 1578 q^{26} + 33755 q^{27} + 38990 q^{28} - 2405 q^{29} + 1483 q^{30} + 31280 q^{31} + 17700 q^{32} + 39326 q^{33} + 9394 q^{34} + 21436 q^{35} + 53290 q^{36} + 35510 q^{37} + 33614 q^{38} + 22963 q^{39} + 50046 q^{40} + 9866 q^{41} + 22195 q^{42} + 51379 q^{43} + 29746 q^{44} + 72219 q^{45} + 35053 q^{46} + 42481 q^{47} + 75938 q^{48} + 175192 q^{49} + 162531 q^{50} + 154051 q^{51} + 210234 q^{52} + 81848 q^{53} + 320743 q^{54} + 105027 q^{55} + 452104 q^{56} + 208589 q^{57} + 159979 q^{58} + 168687 q^{59} + 279933 q^{60} + 148330 q^{61} + 152995 q^{62} + 250849 q^{63} + 540617 q^{64} + 125561 q^{65} + 376032 q^{66} + 395178 q^{67} + 306925 q^{68} + 150249 q^{69} + 271319 q^{70} + 115563 q^{71} + 352975 q^{72} + 247249 q^{73} + 65989 q^{74} + 460002 q^{75} + 440494 q^{76} + 90902 q^{77} + 201225 q^{78} + 222999 q^{79} + 175770 q^{80} + 406335 q^{81} + 172447 q^{82} + 253668 q^{83} - 274214 q^{84} + 114986 q^{85} - 301722 q^{86} + 9553 q^{87} + 79659 q^{88} + 65396 q^{89} - 177829 q^{90} + 360807 q^{91} - 275338 q^{92} - 200055 q^{93} - 249179 q^{94} + 91983 q^{95} - 1089739 q^{96} + 399386 q^{97} - 57978 q^{98} + 119867 q^{99}+O(q^{100}) \) 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 −11.1452 −0.336383 92.2162 −47.6497 3.74907 −75.1526 −671.124 −242.887 531.067
1.2 −10.7119 19.4903 82.7444 86.8228 −208.777 −107.873 −543.567 136.870 −930.035
1.3 −10.2273 19.3367 72.5968 −29.5603 −197.761 191.672 −415.193 130.908 302.320
1.4 −9.97895 −1.74464 67.5794 45.8747 17.4096 1.40539 −355.044 −239.956 −457.781
1.5 −9.66048 −27.2329 61.3248 −56.3745 263.083 56.3197 −283.292 498.631 544.605
1.6 −8.86713 0.0718698 46.6260 8.58831 −0.637279 −242.621 −129.691 −242.995 −76.1537
1.7 −7.81855 −3.85238 29.1298 −63.0871 30.1200 258.975 22.4409 −228.159 493.250
1.8 −6.90647 23.1830 15.6994 95.8550 −160.113 92.5145 112.580 294.453 −662.020
1.9 −6.59656 −22.5190 11.5147 31.9905 148.548 −73.6589 135.133 264.105 −211.027
1.10 −6.46686 10.1034 9.82031 −95.3169 −65.3375 −49.1923 143.433 −140.921 616.401
1.11 −5.80302 12.9092 1.67501 23.4969 −74.9125 −167.264 175.976 −76.3518 −136.353
1.12 −5.60314 −27.4488 −0.604834 60.7084 153.800 236.038 182.689 510.439 −340.158
1.13 −5.44316 28.1713 −2.37201 14.3332 −153.341 119.246 187.092 550.622 −78.0177
1.14 −5.31138 −21.4514 −3.78923 −29.1356 113.937 124.416 190.090 217.164 154.750
1.15 −5.19771 3.76016 −4.98380 5.57792 −19.5442 75.5206 192.231 −228.861 −28.9924
1.16 −4.71670 27.6997 −9.75271 −79.0194 −130.651 189.011 196.935 524.272 372.711
1.17 −3.68168 −9.39506 −18.4452 31.5081 34.5896 14.4366 185.723 −154.733 −116.003
1.18 −1.56744 −0.438374 −29.5431 79.7142 0.687127 218.717 96.4653 −242.808 −124.947
1.19 −0.834938 25.3960 −31.3029 −87.5439 −21.2041 −171.205 52.8540 401.958 73.0938
1.20 −0.742164 −1.92490 −31.4492 −52.2115 1.42859 −106.825 47.0897 −239.295 38.7495
See all 43 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.43
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 197.6.a.b ✓ 43
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.6.a.b ✓ 43 1.a even 1 1 trivial