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Properties

Label 197.10.a.a
Level $197$
Weight $10$
Character orbit 197.a
Self dual yes
Analytic conductor $101.462$
Analytic rank $1$
Dimension $71$
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] = [197,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(101.462059724\)
Analytic rank: \(1\)
Dimension: \(71\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 71 q - 32 q^{2} - 892 q^{3} + 16896 q^{4} - 2329 q^{5} - 10272 q^{6} - 37846 q^{7} - 24933 q^{8} + 419903 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 71 q - 32 q^{2} - 892 q^{3} + 16896 q^{4} - 2329 q^{5} - 10272 q^{6} - 37846 q^{7} - 24933 q^{8} + 419903 q^{9} - 138907 q^{10} - 143074 q^{11} - 496640 q^{12} - 433821 q^{13} - 130143 q^{14} - 670126 q^{15} + 3670016 q^{16} - 226712 q^{17} - 1140455 q^{18} - 4200657 q^{19} - 2522261 q^{20} - 2222620 q^{21} - 5109807 q^{22} - 1576984 q^{23} - 7953747 q^{24} + 20408644 q^{25} - 3606210 q^{26} - 22380322 q^{27} - 28281314 q^{28} - 1600554 q^{29} + 1211083 q^{30} - 27408409 q^{31} - 19842828 q^{32} - 26002786 q^{33} - 33310606 q^{34} - 25745764 q^{35} + 75847738 q^{36} - 56498684 q^{37} - 8523742 q^{38} - 67092176 q^{39} - 69825234 q^{40} - 41524361 q^{41} - 29804525 q^{42} - 103636613 q^{43} - 93059022 q^{44} - 41029851 q^{45} - 35065939 q^{46} - 155481346 q^{47} - 310890046 q^{48} + 274445845 q^{49} - 532508319 q^{50} - 401628576 q^{51} - 638463912 q^{52} - 235002576 q^{53} - 485779441 q^{54} - 274178598 q^{55} - 174143696 q^{56} + 10220750 q^{57} - 54794577 q^{58} - 309204685 q^{59} + 349852813 q^{60} - 172553358 q^{61} + 313207667 q^{62} - 291588072 q^{63} + 1670833437 q^{64} + 394656726 q^{65} + 2405207530 q^{66} - 1034845512 q^{67} + 2160039119 q^{68} + 472060058 q^{69} + 1282221119 q^{70} + 515265617 q^{71} + 1722271619 q^{72} - 667322970 q^{73} + 1111943967 q^{74} - 304507118 q^{75} - 1476894506 q^{76} - 347407724 q^{77} - 99563531 q^{78} - 338475893 q^{79} - 652382330 q^{80} + 1981631951 q^{81} - 2682827483 q^{82} - 2423465319 q^{83} - 1042585800 q^{84} - 2101960924 q^{85} - 2597172884 q^{86} - 2135063354 q^{87} - 5542447545 q^{88} - 2765884440 q^{89} - 6736154529 q^{90} - 6821909968 q^{91} - 4167412114 q^{92} - 4756429190 q^{93} - 5116706209 q^{94} - 4295841932 q^{95} - 14387243887 q^{96} - 6239794299 q^{97} - 8201372742 q^{98} - 6380320552 q^{99}+O(q^{100}) \) 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 −43.6171 −175.865 1390.45 −847.533 7670.72 −3078.38 −38315.6 11245.4 36967.0
1.2 −43.4377 12.8549 1374.83 −2210.23 −558.386 −3816.57 −37479.5 −19517.8 96007.3
1.3 −43.0275 6.58470 1339.37 1186.50 −283.323 −12199.7 −35599.5 −19639.6 −51052.1
1.4 −42.9605 78.7960 1333.60 2236.45 −3385.11 958.546 −35296.4 −13474.2 −96078.9
1.5 −42.8620 210.134 1325.15 1176.05 −9006.77 4058.46 −34853.1 24473.5 −50407.8
1.6 −40.8757 −50.9073 1158.82 −210.651 2080.87 2177.02 −26439.2 −17091.4 8610.49
1.7 −37.3812 −258.991 885.355 −1331.44 9681.38 10098.3 −13956.5 47393.1 49770.7
1.8 −36.4427 87.0222 816.070 967.185 −3171.32 −967.647 −11081.1 −12110.1 −35246.8
1.9 −35.7043 −240.764 762.797 2021.67 8596.29 −11544.2 −8954.52 38284.1 −72182.2
1.10 −33.9082 120.976 637.767 −1477.52 −4102.08 2679.70 −4264.53 −5047.78 50100.2
1.11 −31.8115 226.712 499.972 −1449.41 −7212.06 9874.56 382.633 31715.5 46108.1
1.12 −30.8011 −107.364 436.705 −558.513 3306.94 −8318.45 2319.16 −8155.88 17202.8
1.13 −30.3815 −93.2662 411.034 −2695.73 2833.56 5782.46 3067.50 −10984.4 81900.2
1.14 −29.9863 −185.209 387.177 872.853 5553.74 8968.76 3742.98 14619.5 −26173.6
1.15 −28.2265 196.080 284.733 2569.90 −5534.64 −11219.9 6414.94 18764.3 −72539.3
1.16 −27.9326 −217.666 268.231 −392.118 6079.98 569.248 6809.10 27695.5 10952.9
1.17 −27.8171 240.637 261.789 −2452.49 −6693.83 −4713.20 6960.13 38223.4 68221.0
1.18 −27.2613 −2.80973 231.177 1123.40 76.5969 −8481.66 7655.60 −19675.1 −30625.2
1.19 −25.1553 61.9332 120.787 1012.30 −1557.95 8568.72 9841.06 −15847.3 −25464.8
1.20 −24.2760 −268.553 77.3221 968.528 6519.37 1482.52 10552.2 52437.5 −23511.9
See all 71 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.71
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 197.10.a.a ✓ 71
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.10.a.a ✓ 71 1.a even 1 1 trivial