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Properties

Label 1369.4.a.j
Level $1369$
Weight $4$
Character orbit 1369.a
Self dual yes
Analytic conductor $80.774$
Analytic rank $1$
Dimension $48$
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] = [1369,4,Mod(1,1369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1369, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1369.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1369 = 37^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1369.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: \(80.7736147979\)
Analytic rank: \(1\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 37)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 36 q^{3} + 144 q^{4} - 84 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 36 q^{3} + 144 q^{4} - 84 q^{7} + 216 q^{9} - 114 q^{10} - 270 q^{11} - 270 q^{12} + 192 q^{16} - 666 q^{21} + 732 q^{25} - 378 q^{26} - 414 q^{27} + 240 q^{28} + 840 q^{30} - 648 q^{33} - 432 q^{34} - 852 q^{36} - 2526 q^{38} - 522 q^{40} - 3096 q^{41} - 1662 q^{44} - 2970 q^{46} - 2370 q^{47} - 4326 q^{48} - 120 q^{49} - 4326 q^{53} - 1398 q^{58} - 2790 q^{62} + 390 q^{63} - 3024 q^{64} - 1842 q^{65} + 2844 q^{67} - 1320 q^{70} - 2640 q^{71} - 7002 q^{73} - 5604 q^{75} + 2904 q^{77} + 462 q^{78} - 6000 q^{81} - 4476 q^{83} - 17148 q^{84} + 1452 q^{85} - 5856 q^{86} - 13956 q^{90} - 7782 q^{95} - 6198 q^{99}+O(q^{100}) \) 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 −5.48114 −7.08299 22.0429 14.2317 38.8228 14.2241 −76.9711 23.1687 −78.0061
1.2 −5.04931 −4.01068 17.4955 −2.79713 20.2512 −5.49936 −47.9458 −10.9144 14.1236
1.3 −4.82266 −1.61879 15.2581 11.8941 7.80690 −7.40402 −35.0033 −24.3795 −57.3613
1.4 −4.79749 −0.590330 15.0160 −14.8234 2.83211 −27.0744 −33.6590 −26.6515 71.1150
1.5 −4.58104 6.56918 12.9859 4.45040 −30.0937 −33.3511 −22.8405 16.1541 −20.3874
1.6 −4.39058 2.87222 11.2772 2.27020 −12.6107 −10.2332 −14.3886 −18.7503 −9.96750
1.7 −4.35983 −9.25319 11.0081 −10.3016 40.3423 23.2539 −13.1148 58.6216 44.9134
1.8 −3.91654 −1.24464 7.33927 −2.81231 4.87467 31.4446 2.58777 −25.4509 11.0145
1.9 −3.69017 1.90867 5.61739 −19.1507 −7.04333 0.846801 8.79226 −23.3570 70.6695
1.10 −3.64989 6.87514 5.32168 12.3784 −25.0935 4.24364 9.77558 20.2675 −45.1796
1.11 −3.62665 6.93606 5.15261 8.72206 −25.1547 5.83445 10.3265 21.1090 −31.6319
1.12 −3.15154 −1.88890 1.93217 −14.5669 5.95292 5.94184 19.1230 −23.4321 45.9080
1.13 −3.09545 5.08903 1.58184 −7.30439 −15.7529 −0.861867 19.8671 −1.10173 22.6104
1.14 −2.86753 −6.12352 0.222724 3.17381 17.5594 14.6045 22.3016 10.4975 −9.10099
1.15 −2.39494 −9.25275 −2.26425 19.2902 22.1598 −15.5696 24.5823 58.6133 −46.1990
1.16 −2.04469 −4.54048 −3.81924 16.8594 9.28388 1.93495 24.1667 −6.38403 −34.4723
1.17 −1.88104 −7.86382 −4.46168 4.41605 14.7922 25.3649 23.4410 34.8396 −8.30678
1.18 −1.16951 0.650071 −6.63224 6.45507 −0.760266 26.1017 17.1126 −26.5774 −7.54928
1.19 −1.16133 −4.61101 −6.65132 10.1196 5.35489 −27.0357 17.0150 −5.73854 −11.7522
1.20 −1.11167 −3.42317 −6.76419 −19.9955 3.80543 −25.5427 16.4129 −15.2819 22.2284
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.48
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1369.4.a.j 48
37.b even 2 1 inner 1369.4.a.j 48
37.i odd 36 2 37.4.h.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.h.a ✓ 48 37.i odd 36 2
1369.4.a.j 48 1.a even 1 1 trivial
1369.4.a.j 48 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 264 T_{2}^{46} + 32400 T_{2}^{44} - 2455432 T_{2}^{42} + 128734638 T_{2}^{40} + \cdots + 190720961150976 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1369))\). Copy content Toggle raw display