↑

Properties

Label 1503.4.a.g
Level $1503$
Weight $4$
Character orbit 1503.a
Self dual yes
Analytic conductor $88.680$
Analytic rank $1$
Dimension $41$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(41\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 41 q - 12 q^{2} + 170 q^{4} - 60 q^{5} - 20 q^{7} - 144 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 41 q - 12 q^{2} + 170 q^{4} - 60 q^{5} - 20 q^{7} - 144 q^{8} + 14 q^{10} - 132 q^{11} + 80 q^{13} - 168 q^{14} + 718 q^{16} - 442 q^{17} - 104 q^{19} - 600 q^{20} + 182 q^{22} - 552 q^{23} + 1287 q^{25} - 260 q^{26} + 134 q^{28} - 464 q^{29} - 444 q^{31} - 1344 q^{32} - 276 q^{34} - 560 q^{35} + 20 q^{37} - 760 q^{38} + 192 q^{40} - 1066 q^{41} + 616 q^{43} - 1584 q^{44} - 404 q^{46} - 1796 q^{47} + 1937 q^{49} - 4776 q^{50} - 66 q^{52} - 3016 q^{53} + 388 q^{55} - 7344 q^{56} + 182 q^{58} - 1926 q^{59} - 194 q^{61} - 1488 q^{62} + 4854 q^{64} - 6348 q^{65} + 1144 q^{67} - 6704 q^{68} + 564 q^{70} - 2272 q^{71} - 1750 q^{73} - 5086 q^{74} - 1580 q^{76} - 4264 q^{77} + 434 q^{79} - 7542 q^{80} - 1326 q^{82} - 4170 q^{83} + 488 q^{85} - 10680 q^{86} - 108 q^{88} - 4584 q^{89} - 760 q^{91} - 10674 q^{92} - 3888 q^{94} - 4100 q^{95} + 1506 q^{97} - 4116 q^{98}+O(q^{100}) \) 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 −5.50503 0 22.3054 17.8497 0 15.2769 −78.7514 0 −98.2634
1.2 −5.48857 0 22.1244 −12.6742 0 19.8092 −77.5230 0 69.5631
1.3 −5.34235 0 20.5407 −19.6767 0 13.6950 −66.9968 0 105.120
1.4 −5.25407 0 19.6052 −8.03670 0 −21.1696 −60.9747 0 42.2254
1.5 −4.85294 0 15.5511 0.817004 0 26.2414 −36.6449 0 −3.96487
1.6 −4.79549 0 14.9968 −8.68952 0 −1.50718 −33.5529 0 41.6705
1.7 −4.36403 0 11.0447 18.9686 0 2.40952 −13.2873 0 −82.7797
1.8 −4.00352 0 8.02818 17.1592 0 −22.5817 −0.112816 0 −68.6970
1.9 −3.88955 0 7.12859 −18.8379 0 24.3676 3.38940 0 73.2708
1.10 −3.61020 0 5.03353 −18.9367 0 −34.0229 10.7096 0 68.3653
1.11 −3.28220 0 2.77285 5.36858 0 −34.8013 17.1566 0 −17.6208
1.12 −3.24345 0 2.51995 7.68819 0 −15.4302 17.7743 0 −24.9362
1.13 −3.00860 0 1.05167 0.826935 0 18.5755 20.9047 0 −2.48792
1.14 −2.50422 0 −1.72890 7.89227 0 −10.2825 24.3633 0 −19.7639
1.15 −2.23553 0 −3.00242 −15.7505 0 24.0474 24.5962 0 35.2107
1.16 −2.06152 0 −3.75013 −3.13827 0 13.4388 24.2231 0 6.46960
1.17 −2.05137 0 −3.79188 16.9452 0 6.14852 24.1895 0 −34.7609
1.18 −1.57038 0 −5.53391 −5.90366 0 −34.9747 21.2534 0 9.27098
1.19 −1.36345 0 −6.14100 −20.7452 0 −9.91660 19.2806 0 28.2851
1.20 −0.383939 0 −7.85259 −0.0961682 0 17.6451 6.08642 0 0.0369227
See all 41 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.41
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.g ✓ 41
3.b odd 2 1 1503.4.a.h yes 41
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1503.4.a.g ✓ 41 1.a even 1 1 trivial
1503.4.a.h yes 41 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{41} + 12 T_{2}^{40} - 177 T_{2}^{39} - 2588 T_{2}^{38} + 12769 T_{2}^{37} + 254364 T_{2}^{36} + \cdots - 22426815430656 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1503))\). Copy content Toggle raw display