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Properties

Label 4003.2.a.b
Level $4003$
Weight $2$
Character orbit 4003.a
Self dual yes
Analytic conductor $31.964$
Analytic rank $1$
Dimension $152$
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] = [4003,2,Mod(1,4003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4003.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: \(31.9641159291\)
Analytic rank: \(1\)
Dimension: \(152\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 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 −2.81629 1.48630 5.93147 2.03150 −4.18584 −2.55919 −11.0722 −0.790918 −5.72129
1.2 −2.78616 −2.78181 5.76271 −2.35951 7.75059 2.45659 −10.4835 4.73848 6.57398
1.3 −2.74603 −0.470182 5.54067 −1.60967 1.29113 −2.65994 −9.72280 −2.77893 4.42020
1.4 −2.71591 −1.21901 5.37616 2.34595 3.31073 3.38769 −9.16936 −1.51401 −6.37139
1.5 −2.71344 −2.64650 5.36273 −1.98460 7.18110 −4.86510 −9.12456 4.00395 5.38509
1.6 −2.70039 3.13054 5.29213 −0.910228 −8.45368 0.335007 −8.89004 6.80026 2.45797
1.7 −2.68690 0.420690 5.21946 −4.00007 −1.13035 −1.97783 −8.65037 −2.82302 10.7478
1.8 −2.64207 1.02501 4.98054 2.32673 −2.70814 −0.493904 −7.87479 −1.94936 −6.14739
1.9 −2.60205 1.48054 4.77066 1.31197 −3.85243 3.44490 −7.20939 −0.808005 −3.41380
1.10 −2.59568 2.00123 4.73753 −2.55331 −5.19456 4.30357 −7.10575 1.00494 6.62756
1.11 −2.57418 1.68195 4.62641 −3.90988 −4.32965 1.64141 −6.76085 −0.171031 10.0647
1.12 −2.54789 −3.05680 4.49172 1.69745 7.78837 −1.95167 −6.34863 6.34402 −4.32491
1.13 −2.45287 −0.154053 4.01656 −1.18015 0.377872 4.09823 −4.94635 −2.97627 2.89476
1.14 −2.44543 −1.15615 3.98012 −3.94364 2.82729 3.67660 −4.84225 −1.66331 9.64388
1.15 −2.43441 1.77953 3.92633 −0.360928 −4.33209 −1.53438 −4.68946 0.166714 0.878646
1.16 −2.42958 −1.44088 3.90288 0.473210 3.50074 −4.22120 −4.62321 −0.923870 −1.14970
1.17 −2.41802 1.98760 3.84680 1.18517 −4.80604 −3.98093 −4.46560 0.950540 −2.86577
1.18 −2.37503 −2.00580 3.64079 0.380635 4.76385 3.69515 −3.89692 1.02324 −0.904022
1.19 −2.36790 −2.62074 3.60697 3.23023 6.20567 1.16162 −3.80516 3.86830 −7.64888
1.20 −2.36704 −1.77609 3.60290 −2.33381 4.20407 1.88575 −3.79413 0.154480 5.52424
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.152
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4003\) \(1\)

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 4003.2.a.b ✓ 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4003.2.a.b ✓ 152 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{152} + 22 T_{2}^{151} + 21 T_{2}^{150} - 3036 T_{2}^{149} - 18712 T_{2}^{148} + 177403 T_{2}^{147} + \cdots - 67141 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\). Copy content Toggle raw display