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Properties

Label 1003.4.a.b
Level $1003$
Weight $4$
Character orbit 1003.a
Self dual yes
Analytic conductor $59.179$
Analytic rank $1$
Dimension $52$
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: \(1\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{2} - 19 q^{3} + 216 q^{4} - 95 q^{5} + 42 q^{6} - 50 q^{7} - 144 q^{8} + 309 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{2} - 19 q^{3} + 216 q^{4} - 95 q^{5} + 42 q^{6} - 50 q^{7} - 144 q^{8} + 309 q^{9} + 17 q^{10} - 29 q^{11} - 323 q^{12} - 99 q^{13} - 193 q^{14} - 35 q^{15} + 380 q^{16} - 884 q^{17} + 66 q^{18} - 498 q^{19} - 282 q^{20} - 151 q^{21} - 635 q^{22} + 94 q^{23} - 258 q^{24} + 873 q^{25} - 1182 q^{26} - 964 q^{27} - 731 q^{28} + 58 q^{29} - 1388 q^{30} - 910 q^{31} - 560 q^{32} - 951 q^{33} + 136 q^{34} - 869 q^{35} + 638 q^{36} - 385 q^{37} - 1639 q^{38} + 90 q^{39} + 173 q^{40} - 1388 q^{41} + 448 q^{42} - 1368 q^{43} - 762 q^{44} - 2193 q^{45} - 425 q^{46} - 2868 q^{47} - 3834 q^{48} + 1504 q^{49} - 1614 q^{50} + 323 q^{51} - 2110 q^{52} - 3939 q^{53} - 553 q^{54} - 677 q^{55} - 1267 q^{56} + 79 q^{57} + 420 q^{58} + 3068 q^{59} - 5209 q^{60} + 186 q^{61} - 1579 q^{62} - 3019 q^{63} + 530 q^{64} - 2650 q^{65} - 4133 q^{66} - 896 q^{67} - 3672 q^{68} - 5630 q^{69} - 2408 q^{70} - 2180 q^{71} - 3455 q^{72} - 1476 q^{73} - 520 q^{74} + 3345 q^{75} - 4009 q^{76} - 5295 q^{77} - 1917 q^{78} - 1921 q^{79} - 1062 q^{80} - 632 q^{81} - 2801 q^{82} - 2863 q^{83} + 1245 q^{84} + 1615 q^{85} - 3974 q^{86} - 3716 q^{87} - 3589 q^{88} - 3042 q^{89} - 2627 q^{90} + 569 q^{91} - 8801 q^{92} - 4993 q^{93} - 16 q^{94} - 1383 q^{95} - 6431 q^{96} - 265 q^{97} - 10132 q^{98} - 2211 q^{99}+O(q^{100}) \) 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 −5.42538 1.74374 21.4348 9.78853 −9.46048 0.0197408 −72.8889 −23.9594 −53.1065
1.2 −5.35633 −5.36808 20.6902 −6.02482 28.7532 −34.2676 −67.9731 1.81633 32.2709
1.3 −5.24484 −4.99628 19.5083 −13.5214 26.2047 23.5097 −60.3593 −2.03722 70.9177
1.4 −5.20645 −0.708838 19.1071 18.4206 3.69053 2.60025 −57.8288 −26.4975 −95.9057
1.5 −4.99829 7.03909 16.9829 −10.3707 −35.1834 25.7179 −44.8990 22.5488 51.8356
1.6 −4.70215 −9.40302 14.1102 4.96671 44.2144 −20.7018 −28.7311 61.4168 −23.3542
1.7 −4.63182 0.573343 13.4537 −18.1679 −2.65562 −3.70954 −25.2607 −26.6713 84.1503
1.8 −4.39884 −8.18392 11.3498 −10.8546 35.9998 35.8290 −14.7352 39.9766 47.7476
1.9 −4.34409 8.44221 10.8711 3.72854 −36.6738 −9.48649 −12.4725 44.2710 −16.1971
1.10 −4.16936 5.49086 9.38355 −19.9625 −22.8934 −26.3293 −5.76850 3.14953 83.2308
1.11 −3.88503 −1.19638 7.09348 11.2101 4.64797 −23.2412 3.52187 −25.5687 −43.5516
1.12 −3.62457 −7.60225 5.13748 12.5310 27.5549 13.5650 10.3754 30.7942 −45.4196
1.13 −3.40428 4.59544 3.58913 2.89743 −15.6442 −6.15333 15.0158 −5.88197 −9.86367
1.14 −3.27457 2.63494 2.72279 1.94734 −8.62828 27.1586 17.2806 −20.0571 −6.37670
1.15 −3.26212 −9.19078 2.64144 −6.09278 29.9814 −13.3008 17.4803 57.4705 19.8754
1.16 −3.23801 −1.71989 2.48472 −8.97858 5.56902 16.7105 17.8585 −24.0420 29.0727
1.17 −2.81170 −4.88069 −0.0943360 −2.44298 13.7230 2.14928 22.7589 −3.17888 6.86894
1.18 −2.57124 6.37799 −1.38874 17.4906 −16.3993 −2.97843 24.1407 13.6788 −44.9724
1.19 −2.31316 −2.61880 −2.64931 −15.3691 6.05769 −14.8321 24.6335 −20.1419 35.5511
1.20 −2.19853 8.11991 −3.16647 −3.73660 −17.8519 −25.9615 24.5498 38.9329 8.21502
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.52
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.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.4.a.b ✓ 52 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 8 T_{2}^{51} - 284 T_{2}^{50} - 2352 T_{2}^{49} + 37657 T_{2}^{48} + 324088 T_{2}^{47} + \cdots + 15\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1003))\). Copy content Toggle raw display