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Properties

Label 2169.4.a.h
Level $2169$
Weight $4$
Character orbit 2169.a
Self dual yes
Analytic conductor $127.975$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2169,4,Mod(1,2169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2169.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2169.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: \(127.975142802\)
Analytic rank: \(0\)
Dimension: \(66\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 322 q^{4} + 116 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 322 q^{4} + 116 q^{7} + 110 q^{10} + 120 q^{13} + 1650 q^{16} + 292 q^{19} + 396 q^{22} + 2350 q^{25} + 1262 q^{28} + 1644 q^{31} + 998 q^{34} + 540 q^{37} + 804 q^{40} + 1612 q^{43} + 1018 q^{46} + 5686 q^{49} + 552 q^{52} + 2836 q^{55} + 2296 q^{58} + 2508 q^{61} + 15952 q^{64} + 5072 q^{67} + 13096 q^{70} + 4204 q^{73} + 7198 q^{76} + 8280 q^{79} + 8602 q^{82} + 2036 q^{85} + 16338 q^{88} + 9084 q^{91} + 6386 q^{94} + 4140 q^{97}+O(q^{100}) \) 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 −5.62413 0 23.6308 20.5181 0 −23.4784 −87.9099 0 −115.397
1.2 −5.58132 0 23.1511 −9.56317 0 29.9174 −84.5631 0 53.3751
1.3 −5.47450 0 21.9701 −17.7355 0 −17.7491 −76.4793 0 97.0931
1.4 −5.22172 0 19.2663 −4.73115 0 13.0767 −58.8296 0 24.7047
1.5 −5.20827 0 19.1261 12.4626 0 10.9383 −57.9479 0 −64.9088
1.6 −5.15263 0 18.5496 6.85188 0 10.6443 −54.3585 0 −35.3052
1.7 −5.05482 0 17.5512 −8.17783 0 18.1707 −48.2796 0 41.3375
1.8 −4.78601 0 14.9059 −20.2771 0 29.9496 −33.0517 0 97.0466
1.9 −4.72411 0 14.3172 −15.8878 0 −10.6674 −29.8432 0 75.0556
1.10 −4.27128 0 10.2439 6.48795 0 −29.8493 −9.58422 0 −27.7119
1.11 −4.20305 0 9.66564 21.3174 0 −5.85639 −7.00075 0 −89.5981
1.12 −4.11157 0 8.90503 11.1449 0 −25.4579 −3.72109 0 −45.8230
1.13 −3.81077 0 6.52194 −7.04453 0 −2.85345 5.63254 0 26.8450
1.14 −3.77792 0 6.27271 8.07271 0 32.6303 6.52559 0 −30.4981
1.15 −3.74408 0 6.01810 −11.0664 0 26.7234 7.42039 0 41.4336
1.16 −3.42051 0 3.69987 −10.9745 0 −29.3287 14.7086 0 37.5385
1.17 −3.16595 0 2.02324 14.2928 0 −13.1837 18.9221 0 −45.2503
1.18 −3.06546 0 1.39706 8.91870 0 −6.66787 20.2411 0 −27.3399
1.19 −2.83467 0 0.0353689 1.69741 0 25.6188 22.5771 0 −4.81160
1.20 −2.78325 0 −0.253509 15.5015 0 34.6844 22.9716 0 −43.1446
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.66
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(241\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2169.4.a.h ✓ 66
3.b odd 2 1 inner 2169.4.a.h ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2169.4.a.h ✓ 66 1.a even 1 1 trivial
2169.4.a.h ✓ 66 3.b odd 2 1 inner