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Properties

Label 1375.4.a.a
Level $1375$
Weight $4$
Character orbit 1375.a
Self dual yes
Analytic conductor $81.128$
Analytic rank $1$
Dimension $24$
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] = [1375,4,Mod(1,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, 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("1375.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1375.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: \(81.1276262579\)
Analytic rank: \(1\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 90 q^{4} - 50 q^{6} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 90 q^{4} - 50 q^{6} + 32 q^{9} + 264 q^{11} - 388 q^{14} - 30 q^{16} - 150 q^{19} - 638 q^{21} - 628 q^{24} - 478 q^{26} - 1106 q^{29} - 868 q^{31} - 368 q^{34} - 124 q^{36} - 662 q^{39} - 1440 q^{41} + 990 q^{44} - 670 q^{46} + 72 q^{49} - 1262 q^{51} - 316 q^{54} - 2728 q^{56} - 3168 q^{59} - 442 q^{61} - 3576 q^{64} - 550 q^{66} - 470 q^{69} - 4208 q^{71} - 1054 q^{74} - 1810 q^{76} - 3642 q^{79} - 6756 q^{81} + 698 q^{84} - 446 q^{86} - 7784 q^{89} - 9214 q^{91} + 224 q^{94} + 190 q^{96} + 352 q^{99}+O(q^{100}) \) 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 −5.13680 −2.91957 18.3867 0 14.9973 32.8852 −53.3543 −18.4761 0
1.2 −4.80198 7.41062 15.0590 0 −35.5857 19.4709 −33.8974 27.9172 0
1.3 −4.64801 6.58761 13.6040 0 −30.6193 −15.0499 −26.0476 16.3966 0
1.4 −4.40001 −0.101947 11.3601 0 0.448566 −14.1172 −14.7846 −26.9896 0
1.5 −3.70271 −3.08564 5.71003 0 11.4252 −11.3119 8.47908 −17.4788 0
1.6 −3.53863 −7.33630 4.52189 0 25.9604 16.5303 12.3078 26.8214 0
1.7 −3.37606 4.30066 3.39779 0 −14.5193 8.34229 15.5373 −8.50434 0
1.8 −2.26809 −7.51000 −2.85575 0 17.0334 9.49896 24.6219 29.4001 0
1.9 −2.10696 0.218458 −3.56073 0 −0.460283 −6.06480 24.3580 −26.9523 0
1.10 −1.80539 6.46475 −4.74057 0 −11.6714 5.92315 23.0017 14.7929 0
1.11 −0.288880 2.61339 −7.91655 0 −0.754955 20.2723 4.59797 −20.1702 0
1.12 −0.184406 6.80023 −7.96599 0 −1.25400 −34.9541 2.94422 19.2432 0
1.13 0.184406 −6.80023 −7.96599 0 −1.25400 34.9541 −2.94422 19.2432 0
1.14 0.288880 −2.61339 −7.91655 0 −0.754955 −20.2723 −4.59797 −20.1702 0
1.15 1.80539 −6.46475 −4.74057 0 −11.6714 −5.92315 −23.0017 14.7929 0
1.16 2.10696 −0.218458 −3.56073 0 −0.460283 6.06480 −24.3580 −26.9523 0
1.17 2.26809 7.51000 −2.85575 0 17.0334 −9.49896 −24.6219 29.4001 0
1.18 3.37606 −4.30066 3.39779 0 −14.5193 −8.34229 −15.5373 −8.50434 0
1.19 3.53863 7.33630 4.52189 0 25.9604 −16.5303 −12.3078 26.8214 0
1.20 3.70271 3.08564 5.71003 0 11.4252 11.3119 −8.47908 −17.4788 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.4.a.a ✓ 24
5.b even 2 1 inner 1375.4.a.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.4.a.a ✓ 24 1.a even 1 1 trivial
1375.4.a.a ✓ 24 5.b even 2 1 inner