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Properties

Label 197.6.a.a
Level $197$
Weight $6$
Character orbit 197.a
Self dual yes
Analytic conductor $31.596$
Analytic rank $1$
Dimension $38$
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: \(1\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 8 q^{2} - 100 q^{3} + 512 q^{4} - 154 q^{5} - 216 q^{6} - 737 q^{7} - 261 q^{8} + 2510 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 8 q^{2} - 100 q^{3} + 512 q^{4} - 154 q^{5} - 216 q^{6} - 737 q^{7} - 261 q^{8} + 2510 q^{9} - 1567 q^{10} - 808 q^{11} - 3200 q^{12} - 2460 q^{13} - 1423 q^{14} - 3541 q^{15} + 5120 q^{16} - 2501 q^{17} - 1343 q^{18} - 7983 q^{19} - 6981 q^{20} - 3382 q^{21} - 8023 q^{22} - 7132 q^{23} - 10467 q^{24} + 13564 q^{25} - 7886 q^{26} - 30397 q^{27} - 23730 q^{28} - 12497 q^{29} - 12917 q^{30} - 26380 q^{31} - 18140 q^{32} - 12946 q^{33} - 48406 q^{34} - 12864 q^{35} - 5030 q^{36} - 46630 q^{37} - 15482 q^{38} - 13541 q^{39} - 74754 q^{40} - 23754 q^{41} - 41309 q^{42} - 96541 q^{43} - 39950 q^{44} - 49281 q^{45} - 49587 q^{46} - 63551 q^{47} - 126814 q^{48} + 67147 q^{49} - 158667 q^{50} - 140697 q^{51} - 243408 q^{52} - 109108 q^{53} - 345313 q^{54} - 176321 q^{55} - 439504 q^{56} - 172501 q^{57} - 283105 q^{58} - 129225 q^{59} - 491987 q^{60} - 156006 q^{61} - 257561 q^{62} - 285435 q^{63} - 71443 q^{64} - 214451 q^{65} - 370574 q^{66} - 326834 q^{67} - 244233 q^{68} - 63505 q^{69} - 267713 q^{70} - 118805 q^{71} - 237565 q^{72} - 226965 q^{73} - 82169 q^{74} - 166814 q^{75} - 177290 q^{76} - 74218 q^{77} + 87061 q^{78} - 18825 q^{79} - 154730 q^{80} + 211130 q^{81} - 279743 q^{82} - 191524 q^{83} - 121632 q^{84} + 10830 q^{85} + 188032 q^{86} - 202673 q^{87} - 344937 q^{88} + 22634 q^{89} + 597315 q^{90} - 490747 q^{91} + 409454 q^{92} - 49403 q^{93} - 151313 q^{94} + 135623 q^{95} + 669377 q^{96} - 585150 q^{97} + 726482 q^{98} - 108889 q^{99}+O(q^{100}) \) 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 −11.0038 −15.7404 89.0834 53.4333 173.205 209.271 −628.134 4.76162 −587.969
1.2 −10.0136 16.9316 68.2724 −90.0389 −169.547 −10.8089 −363.218 43.6807 901.615
1.3 −9.75231 −24.1109 63.1076 79.2538 235.137 −142.789 −303.371 338.336 −772.908
1.4 −9.62766 −15.7401 60.6918 −53.9029 151.540 −68.0572 −276.234 4.74987 518.959
1.5 −9.01479 28.8518 49.2665 −22.2762 −260.093 −142.112 −155.654 589.427 200.815
1.6 −8.87189 14.8432 46.7105 31.5238 −131.688 109.462 −130.510 −22.6783 −279.676
1.7 −8.55327 −6.50356 41.1584 49.9916 55.6267 64.4308 −78.3344 −200.704 −427.592
1.8 −7.88082 −8.49014 30.1074 −63.1556 66.9093 38.3135 14.9154 −170.918 497.718
1.9 −7.08529 −5.02957 18.2013 90.5704 35.6359 95.4649 97.7679 −217.703 −641.717
1.10 −6.03212 15.3300 4.38649 −34.4752 −92.4726 −74.5425 166.568 −7.99034 207.959
1.11 −5.71440 −14.9999 0.654416 −103.918 85.7153 −231.118 179.121 −18.0043 593.831
1.12 −4.95553 2.88640 −7.44275 109.673 −14.3036 −146.696 195.460 −234.669 −543.485
1.13 −3.91773 −28.7523 −16.6514 −46.5596 112.644 −206.991 190.603 583.694 182.408
1.14 −3.28867 −10.3441 −21.1847 −14.8884 34.0181 −29.1061 174.907 −136.000 48.9628
1.15 −2.84932 24.3331 −23.8814 56.3388 −69.3327 −117.274 159.224 349.099 −160.527
1.16 −2.34981 11.7619 −26.4784 −45.6671 −27.6383 98.1565 137.413 −104.657 107.309
1.17 −2.23960 −16.6391 −26.9842 79.6153 37.2648 −172.296 132.101 33.8583 −178.306
1.18 −1.78681 −17.1157 −28.8073 −96.1357 30.5825 140.530 108.651 49.9467 171.776
1.19 −1.50045 11.0676 −29.7487 20.0445 −16.6063 171.046 92.6507 −120.508 −30.0757
1.20 −0.225855 20.4337 −31.9490 9.10251 −4.61506 −176.555 14.4432 174.537 −2.05584
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
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.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.6.a.a ✓ 38 1.a even 1 1 trivial