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Properties

Label 197.12.a.a
Level $197$
Weight $12$
Character orbit 197.a
Self dual yes
Analytic conductor $151.364$
Analytic rank $1$
Dimension $87$
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: \(1\)
Dimension: \(87\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 87 q - 96 q^{2} - 2926 q^{3} + 82944 q^{4} - 20896 q^{5} - 31104 q^{6} - 231933 q^{7} - 257493 q^{8} + 5014709 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 87 q - 96 q^{2} - 2926 q^{3} + 82944 q^{4} - 20896 q^{5} - 31104 q^{6} - 231933 q^{7} - 257493 q^{8} + 5014709 q^{9} - 1525723 q^{10} - 2302990 q^{11} - 5992448 q^{12} - 4417566 q^{13} - 1538119 q^{14} - 11475841 q^{15} + 72351744 q^{16} - 10008315 q^{17} - 31914479 q^{18} - 67161531 q^{19} - 35395221 q^{20} - 9604582 q^{21} - 75470727 q^{22} - 115874668 q^{23} - 134863083 q^{24} + 698180005 q^{25} - 114536538 q^{26} - 597391465 q^{27} - 663668810 q^{28} - 248961329 q^{29} - 431813765 q^{30} - 777973332 q^{31} - 450925140 q^{32} - 899781862 q^{33} - 1208242462 q^{34} - 627415388 q^{35} + 3892764538 q^{36} - 2151605650 q^{37} - 1775908918 q^{38} + 114589831 q^{39} - 4893357354 q^{40} - 2347138756 q^{41} - 1545335045 q^{42} - 5440310313 q^{43} - 2917807262 q^{44} - 5533456803 q^{45} - 3893754419 q^{46} - 8154953963 q^{47} - 6806689006 q^{48} + 18906329648 q^{49} + 18182247991 q^{50} + 4054124079 q^{51} - 823275058 q^{52} - 3269965888 q^{53} - 19032402169 q^{54} - 13688990281 q^{55} - 40263832936 q^{56} - 38080107187 q^{57} - 46087545821 q^{58} - 23845087293 q^{59} - 93472697171 q^{60} - 46282101530 q^{61} - 78865027833 q^{62} - 64002156291 q^{63} - 5976089007 q^{64} - 48080984707 q^{65} - 113038744652 q^{66} - 82711903250 q^{67} - 48519004335 q^{68} - 64623548359 q^{69} - 38346286833 q^{70} - 17437816565 q^{71} + 36958156415 q^{72} - 95370872119 q^{73} + 15210602113 q^{74} - 42166387322 q^{75} + 7376933366 q^{76} + 7483199710 q^{77} + 283269531217 q^{78} - 11045063185 q^{79} + 152129843698 q^{80} + 341321274971 q^{81} + 40429738967 q^{82} + 52892654796 q^{83} + 390062558190 q^{84} + 57091090634 q^{85} + 582356615014 q^{86} + 68877580777 q^{87} + 163219292163 q^{88} + 100730586952 q^{89} + 478753958199 q^{90} - 255358981417 q^{91} + 340625282918 q^{92} + 317595200965 q^{93} + 4819436601 q^{94} + 39858710059 q^{95} + 922299238973 q^{96} - 444995372128 q^{97} + 381061272938 q^{98} - 237904561213 q^{99}+O(q^{100}) \) 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 −88.6400 −20.5819 5809.05 5525.77 1824.38 36575.6 −333379. −176723. −489804.
1.2 −87.4082 −330.577 5592.19 10211.9 28895.1 3759.95 −309791. −67865.8 −892600.
1.3 −83.4232 −464.759 4911.44 704.980 38771.7 −75183.8 −238877. 38853.8 −58811.7
1.4 −83.2998 −824.779 4890.86 −1344.00 68704.0 −23932.5 −236810. 503114. 111955.
1.5 −81.6406 759.281 4617.19 −7790.01 −61988.2 17561.7 −209750. 399361. 635981.
1.6 −80.5320 583.209 4437.40 3620.77 −46967.0 52016.6 −192423. 162986. −291588.
1.7 −80.3186 −228.720 4403.08 −3202.18 18370.5 59878.0 −189156. −124834. 257195.
1.8 −78.3803 171.771 4095.47 −11319.5 −13463.4 25833.1 −160481. −147642. 887222.
1.9 −77.9965 −293.459 4035.45 10021.4 22888.7 −55110.9 −155014. −91029.0 −781632.
1.10 −76.3606 −27.2673 3782.94 −7206.70 2082.15 −38114.7 −132481. −176403. 550308.
1.11 −75.1483 180.715 3599.27 11778.4 −13580.5 −19019.3 −116575. −144489. −885131.
1.12 −69.8803 −199.808 2835.26 −2201.80 13962.7 8077.62 −55013.9 −137224. 153862.
1.13 −69.5078 367.465 2783.34 10345.6 −25541.7 15991.1 −51112.0 −42116.6 −719098.
1.14 −64.7604 −639.551 2145.90 −7648.30 41417.6 39965.6 −6340.32 231879. 495307.
1.15 −64.5566 710.476 2119.56 −6173.51 −45865.9 4458.74 −4619.38 327630. 398541.
1.16 −64.0210 700.704 2050.69 372.774 −44859.7 −62139.1 −172.063 313839. −23865.4
1.17 −61.9744 −578.177 1792.83 −9089.52 35832.2 7742.51 15814.3 157142. 563317.
1.18 −61.6192 548.719 1748.92 −12761.2 −33811.6 −65209.1 18428.9 123946. 786333.
1.19 −57.5730 −788.240 1266.65 −4331.55 45381.3 −65840.4 44984.7 444175. 249381.
1.20 −54.5248 −255.325 924.956 10305.0 13921.5 −64390.2 61233.8 −111956. −561879.
See all 87 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.87
Significant digits:
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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.a ✓ 87
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.12.a.a ✓ 87 1.a even 1 1 trivial