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Properties

Label 1003.4.a.e
Level $1003$
Weight $4$
Character orbit 1003.a
Self dual yes
Analytic conductor $59.179$
Analytic rank $0$
Dimension $61$
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] = [1003,4,Mod(1,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, 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("1003.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1003.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: \(59.1789157358\)
Analytic rank: \(0\)
Dimension: \(61\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 61 q + 13 q^{2} + 30 q^{3} + 259 q^{4} + 128 q^{5} + 9 q^{6} + 116 q^{7} + 81 q^{8} + 609 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 61 q + 13 q^{2} + 30 q^{3} + 259 q^{4} + 128 q^{5} + 9 q^{6} + 116 q^{7} + 81 q^{8} + 609 q^{9} + 100 q^{10} + 142 q^{11} + 336 q^{12} + 240 q^{13} + 117 q^{14} + 246 q^{15} + 1079 q^{16} + 1037 q^{17} + 416 q^{18} + 126 q^{19} + 1185 q^{20} + 80 q^{21} - 46 q^{22} + 492 q^{23} - 111 q^{24} + 1815 q^{25} + 31 q^{26} + 1158 q^{27} + 1283 q^{28} + 764 q^{29} - 589 q^{30} + 582 q^{31} + 781 q^{32} + 216 q^{33} + 221 q^{34} + 1120 q^{35} + 3028 q^{36} + 1440 q^{37} + 1701 q^{38} + 188 q^{39} + 1142 q^{40} + 2004 q^{41} - 139 q^{42} - 242 q^{43} + 2291 q^{44} + 3490 q^{45} + 2027 q^{46} + 2272 q^{47} + 3271 q^{48} + 4267 q^{49} + 1494 q^{50} + 510 q^{51} + 2169 q^{52} + 2520 q^{53} + 593 q^{54} + 1148 q^{55} + 653 q^{56} + 2050 q^{57} - 374 q^{58} + 3599 q^{59} + 1544 q^{60} + 3238 q^{61} + 1461 q^{62} + 4472 q^{63} + 6021 q^{64} + 792 q^{65} - 1056 q^{66} + 1812 q^{67} + 4403 q^{68} - 608 q^{69} - 1601 q^{70} + 1534 q^{71} + 1703 q^{72} + 2308 q^{73} + 765 q^{74} + 3772 q^{75} + 3143 q^{76} + 5838 q^{77} + 4613 q^{78} + 1676 q^{79} + 9817 q^{80} + 7737 q^{81} + 935 q^{82} - 732 q^{83} + 2092 q^{84} + 2176 q^{85} + 108 q^{86} - 520 q^{87} - 492 q^{88} + 4910 q^{89} + 2268 q^{90} + 2092 q^{91} + 5525 q^{92} + 3316 q^{93} + 900 q^{94} + 5254 q^{95} - 2698 q^{96} + 5452 q^{97} + 5731 q^{98} + 3726 q^{99}+O(q^{100}) \) 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 −5.64707 8.76816 23.8894 16.8821 −49.5145 23.2837 −89.7287 49.8807 −95.3342
1.2 −5.39389 3.49263 21.0940 −0.0777995 −18.8389 −22.2118 −70.6278 −14.8015 0.419642
1.3 −4.95915 −6.57614 16.5932 15.2034 32.6121 29.3394 −42.6149 16.2456 −75.3959
1.4 −4.95221 −7.27189 16.5244 −13.6656 36.0119 −5.57965 −42.2146 25.8804 67.6747
1.5 −4.92352 0.708892 16.2410 2.89853 −3.49024 −1.37345 −40.5748 −26.4975 −14.2710
1.6 −4.87917 4.55996 15.8063 −4.68413 −22.2488 −20.4913 −38.0885 −6.20674 22.8547
1.7 −4.73076 −5.39289 14.3801 −0.882937 25.5125 11.1833 −30.1827 2.08325 4.17696
1.8 −4.65377 6.02515 13.6576 −11.2945 −28.0397 28.4581 −26.3291 9.30241 52.5619
1.9 −4.33340 −7.61394 10.7784 16.3178 32.9943 −26.7374 −12.0399 30.9721 −70.7116
1.10 −3.91155 9.70610 7.30020 7.75576 −37.9659 14.1825 2.73731 67.2085 −30.3370
1.11 −3.65592 −1.46093 5.36576 −17.5273 5.34105 20.5614 9.63058 −24.8657 64.0786
1.12 −3.61678 4.79165 5.08108 19.3549 −17.3303 −15.2124 10.5571 −4.04009 −70.0024
1.13 −3.59235 6.93715 4.90498 −14.1022 −24.9207 −20.6426 11.1184 21.1241 50.6602
1.14 −3.44837 −1.74501 3.89125 22.0834 6.01746 30.7411 14.1685 −23.9549 −76.1517
1.15 −3.27995 3.15409 2.75806 4.91426 −10.3453 17.4070 17.1933 −17.0517 −16.1185
1.16 −2.96166 −3.21349 0.771415 11.0174 9.51726 11.6574 21.4086 −16.6735 −32.6298
1.17 −2.90966 −0.997604 0.466136 −12.6232 2.90269 −10.6102 21.9210 −26.0048 36.7293
1.18 −2.63067 −7.80626 −1.07957 −5.22480 20.5357 −19.9752 23.8854 33.9377 13.7447
1.19 −2.25108 9.69428 −2.93265 13.8947 −21.8226 −28.5293 24.6102 66.9791 −31.2780
1.20 −2.01127 1.48416 −3.95478 −3.09310 −2.98504 −2.40774 24.0443 −24.7973 6.22107
See all 61 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.61
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(59\) \(-1\)

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 1003.4.a.e ✓ 61
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.4.a.e ✓ 61 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{61} - 13 T_{2}^{60} - 289 T_{2}^{59} + 4393 T_{2}^{58} + 36857 T_{2}^{57} - 697677 T_{2}^{56} + \cdots + 37\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1003))\). Copy content Toggle raw display