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Properties

Label 1503.4.a.d
Level $1503$
Weight $4$
Character orbit 1503.a
Self dual yes
Analytic conductor $88.680$
Analytic rank $1$
Dimension $23$
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] = [1503,4,Mod(1,1503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1503, 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("1503.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1503 = 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1503.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: \(88.6798707386\)
Analytic rank: \(1\)
Dimension: \(23\)
Twist minimal: no (minimal twist has level 501)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q - 11 q^{2} + 103 q^{4} - 48 q^{5} + 28 q^{7} - 147 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q - 11 q^{2} + 103 q^{4} - 48 q^{5} + 28 q^{7} - 147 q^{8} + 43 q^{10} - 204 q^{11} + 34 q^{13} - 101 q^{14} + 587 q^{16} - 368 q^{17} + 448 q^{19} - 543 q^{20} + 307 q^{22} - 406 q^{23} + 863 q^{25} - 335 q^{26} + 115 q^{28} - 556 q^{29} + 776 q^{31} - 1402 q^{32} + 672 q^{34} - 730 q^{35} + 270 q^{37} - 238 q^{38} + 694 q^{40} - 816 q^{41} - 84 q^{43} - 2035 q^{44} - 327 q^{46} - 1420 q^{47} + 1131 q^{49} - 430 q^{50} - 828 q^{52} - 1430 q^{53} - 758 q^{55} + 433 q^{56} - 2233 q^{58} - 3110 q^{59} + 278 q^{61} - 2044 q^{62} + 859 q^{64} - 2332 q^{65} + 802 q^{67} - 1384 q^{68} - 4411 q^{70} - 1696 q^{71} - 1048 q^{73} - 225 q^{74} - 2804 q^{76} - 986 q^{77} + 460 q^{79} - 4026 q^{80} - 3401 q^{82} - 1886 q^{83} + 360 q^{85} - 633 q^{86} - 116 q^{88} - 6404 q^{89} + 3304 q^{91} - 5135 q^{92} - 1860 q^{94} - 796 q^{95} + 1040 q^{97} - 5905 q^{98}+O(q^{100}) \) 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 −5.47578 0 21.9841 −2.14520 0 16.4179 −76.5739 0 11.7466
1.2 −5.37117 0 20.8494 −18.0951 0 −34.9079 −69.0163 0 97.1916
1.3 −4.99247 0 16.9247 7.44760 0 −9.19941 −44.5565 0 −37.1819
1.4 −4.89366 0 15.9479 13.6242 0 23.0458 −38.8943 0 −66.6719
1.5 −4.50458 0 12.2912 −15.0468 0 16.4765 −19.3301 0 67.7797
1.6 −4.45747 0 11.8691 −8.14975 0 1.52132 −17.2462 0 36.3273
1.7 −2.70059 0 −0.706831 −12.7334 0 31.5507 23.5136 0 34.3875
1.8 −2.62629 0 −1.10260 −15.4507 0 −7.76153 23.9061 0 40.5780
1.9 −2.41543 0 −2.16568 18.6586 0 −9.26627 24.5545 0 −45.0687
1.10 −2.02544 0 −3.89758 4.51905 0 23.6818 24.0979 0 −9.15308
1.11 −0.812553 0 −7.33976 −6.32913 0 −26.3940 12.4644 0 5.14275
1.12 −0.490116 0 −7.75979 10.5833 0 7.33541 7.72412 0 −5.18704
1.13 −0.335985 0 −7.88711 17.4355 0 −32.7201 5.33784 0 −5.85808
1.14 0.545825 0 −7.70208 −21.1014 0 11.5716 −8.57058 0 −11.5177
1.15 1.14753 0 −6.68318 −1.62556 0 34.8572 −16.8493 0 −1.86538
1.16 1.73679 0 −4.98355 −17.1109 0 −17.2874 −22.5497 0 −29.7181
1.17 2.02030 0 −3.91838 10.6458 0 −7.92176 −24.0787 0 21.5077
1.18 2.56370 0 −1.42744 −4.16167 0 11.7571 −24.1691 0 −10.6693
1.19 4.04216 0 8.33906 −15.6905 0 10.6774 1.37055 0 −63.4234
1.20 4.06478 0 8.52242 2.61397 0 16.3991 2.12352 0 10.6252
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(167\) \(1\)

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 1503.4.a.d 23
3.b odd 2 1 501.4.a.d ✓ 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.4.a.d ✓ 23 3.b odd 2 1
1503.4.a.d 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + 11 T_{2}^{22} - 83 T_{2}^{21} - 1249 T_{2}^{20} + 1908 T_{2}^{19} + 59247 T_{2}^{18} + \cdots + 617766912 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1503))\). Copy content Toggle raw display