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Properties

Label 197.12.a.b
Level $197$
Weight $12$
Character orbit 197.a
Self dual yes
Analytic conductor $151.364$
Analytic rank $0$
Dimension $92$
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] = [197,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(151.363606570\)
Analytic rank: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 64 q^{2} + 2420 q^{3} + 98304 q^{4} + 16604 q^{5} + 46656 q^{6} + 305891 q^{7} + 234027 q^{8} + 5900444 q^{9} + 1074277 q^{10} + 595928 q^{11} + 4956160 q^{12} + 7463810 q^{13} + 4915769 q^{14} + 6749159 q^{15} + 109051904 q^{16} + 9869683 q^{17} + 15324721 q^{18} + 71500013 q^{19} + 79804779 q^{20} + 55741034 q^{21} + 120367289 q^{22} + 38597564 q^{23} + 104015637 q^{24} + 1039976880 q^{25} + 51802726 q^{26} + 665312351 q^{27} + 713160630 q^{28} - 2827541 q^{29} - 43013765 q^{30} + 939775728 q^{31} + 723479980 q^{32} + 978717002 q^{33} + 1063528738 q^{34} + 843197112 q^{35} + 6613742458 q^{36} + 2009031770 q^{37} + 918086794 q^{38} + 2279970607 q^{39} + 3093842646 q^{40} - 30014736 q^{41} + 3159549307 q^{42} + 6320365127 q^{43} + 3019176802 q^{44} + 5538230697 q^{45} + 4344764621 q^{46} + 2853606373 q^{47} + 15616060178 q^{48} + 31617715853 q^{49} - 28784136763 q^{50} - 7216660253 q^{51} + 19141188860 q^{52} + 3389732468 q^{53} + 14059623295 q^{54} + 23123173075 q^{55} + 75592019872 q^{56} + 27508365203 q^{57} + 42079603023 q^{58} + 40032802875 q^{59} + 99482549469 q^{60} + 31816702886 q^{61} + 35401555571 q^{62} + 79170604993 q^{63} + 168463042533 q^{64} + 50313809737 q^{65} + 93218912814 q^{66} + 129966589578 q^{67} + 73640879491 q^{68} - 9850635033 q^{69} + 62387700391 q^{70} + 32189826123 q^{71} + 47439469795 q^{72} + 60612500511 q^{73} - 110473527245 q^{74} + 48823210110 q^{75} + 48170982110 q^{76} - 9381499362 q^{77} - 251751253299 q^{78} + 40812909551 q^{79} - 36932560002 q^{80} + 294126355776 q^{81} + 29914326881 q^{82} + 112077199076 q^{83} - 242466547988 q^{84} - 65194167314 q^{85} - 384596985360 q^{86} + 30701235703 q^{87} + 102229734783 q^{88} - 47595308310 q^{89} - 957657692329 q^{90} + 217575098757 q^{91} - 762245602306 q^{92} - 47762776839 q^{93} - 251435906373 q^{94} - 84935429021 q^{95} - 1089429573223 q^{96} + 432665748880 q^{97} - 404245603446 q^{98} + 9542377031 q^{99}+O(q^{100}) \) 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 −89.8122 408.684 6018.23 −5687.18 −36704.8 −1069.14 −356575. −10124.4 510778.
1.2 −87.1440 786.167 5546.07 12634.2 −68509.7 −52867.9 −304836. 440912. −1.10099e6
1.3 −86.5721 −287.043 5446.73 −10768.4 24849.9 −50234.8 −294236. −94753.4 932242.
1.4 −84.9113 −568.693 5161.93 −8080.39 48288.5 44471.4 −264408. 146264. 686117.
1.5 −84.8742 339.556 5155.63 2009.94 −28819.5 −56259.8 −263758. −61848.8 −170592.
1.6 −79.8918 101.497 4334.69 1864.60 −8108.77 52018.3 −182688. −166845. −148966.
1.7 −79.8310 −674.325 4324.98 3933.54 53832.0 −6505.44 −181774. 277567. −314018.
1.8 −78.5767 504.204 4126.30 −5592.54 −39618.7 −45440.8 −163306. 77074.3 439443.
1.9 −78.1317 593.673 4056.56 6158.78 −46384.7 72514.6 −156932. 175301. −481196.
1.10 −75.5952 −693.315 3666.63 10252.9 52411.3 78779.8 −122361. 303539. −775068.
1.11 −73.7285 331.029 3387.90 2290.71 −24406.3 −50357.4 −98788.6 −67567.1 −168891.
1.12 −72.3048 −558.542 3179.98 7098.19 40385.2 −17185.4 −81847.3 134822. −513233.
1.13 −69.7386 286.356 2815.48 −13059.6 −19970.1 78397.4 −53522.7 −95147.0 910755.
1.14 −67.2111 −95.0378 2469.33 1940.57 6387.60 −29829.0 −28318.0 −168115. −130428.
1.15 −66.4070 −356.598 2361.89 −8882.65 23680.6 −71949.2 −20844.3 −49984.8 589870.
1.16 −63.4914 −424.047 1983.16 564.056 26923.4 62228.3 4117.09 2669.21 −35812.7
1.17 −60.3928 −648.662 1599.29 −13683.9 39174.5 −8398.03 27098.6 243615. 826410.
1.18 −59.6321 54.3091 1507.99 −3681.72 −3238.57 14583.9 32202.1 −174198. 219549.
1.19 −58.1801 −202.649 1336.93 5488.34 11790.1 63266.7 41370.2 −136080. −319313.
1.20 −57.9178 751.961 1306.47 5857.83 −43551.9 −5863.08 42947.8 388299. −339273.
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
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 197.12.a.b ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.12.a.b ✓ 92 1.a even 1 1 trivial