Properties

Label 1003.4.a.d
Level $1003$
Weight $4$
Character orbit 1003.a
Self dual yes
Analytic conductor $59.179$
Analytic rank $0$
Dimension $59$
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: \(59\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 59 q + 13 q^{2} + 12 q^{3} + 235 q^{4} + 102 q^{5} + 71 q^{6} - 24 q^{7} + 189 q^{8} + 587 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 59 q + 13 q^{2} + 12 q^{3} + 235 q^{4} + 102 q^{5} + 71 q^{6} - 24 q^{7} + 189 q^{8} + 587 q^{9} + 80 q^{11} + 144 q^{12} + 256 q^{13} + 33 q^{14} + 66 q^{15} + 1199 q^{16} - 1003 q^{17} + 616 q^{18} + 50 q^{19} + 1039 q^{20} + 560 q^{21} + 424 q^{22} + 248 q^{23} + 1425 q^{24} + 1737 q^{25} + 1499 q^{26} + 510 q^{27} - 381 q^{28} + 1038 q^{29} + 1325 q^{30} + 158 q^{31} + 1645 q^{32} + 1664 q^{33} - 221 q^{34} + 624 q^{35} + 2764 q^{36} + 258 q^{37} + 933 q^{38} + 352 q^{39} + 428 q^{40} + 1040 q^{41} + 3859 q^{42} + 558 q^{43} + 983 q^{44} + 2296 q^{45} - 231 q^{46} + 2576 q^{47} + 609 q^{48} + 3581 q^{49} + 2214 q^{50} - 204 q^{51} + 2425 q^{52} + 2684 q^{53} + 133 q^{54} + 692 q^{55} + 1583 q^{56} + 502 q^{57} + 1454 q^{58} - 3481 q^{59} - 308 q^{60} - 788 q^{61} + 3221 q^{62} + 684 q^{63} + 5425 q^{64} + 1564 q^{65} + 1872 q^{66} + 1252 q^{67} - 3995 q^{68} + 4552 q^{69} + 3923 q^{70} + 2566 q^{71} + 5791 q^{72} + 2980 q^{73} + 465 q^{74} + 5734 q^{75} - 1453 q^{76} + 7026 q^{77} + 1221 q^{78} + 1632 q^{79} + 8215 q^{80} + 9083 q^{81} + 2133 q^{82} + 9588 q^{83} + 3148 q^{84} - 1734 q^{85} + 1820 q^{86} + 6984 q^{87} + 4354 q^{88} + 7142 q^{89} - 172 q^{90} - 2188 q^{91} - 1471 q^{92} + 7476 q^{93} + 1136 q^{94} + 4322 q^{95} + 8822 q^{96} + 2168 q^{97} + 1849 q^{98} + 4588 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −5.40755 −4.67692 21.2416 −2.87639 25.2907 −4.48444 −71.6049 −5.12642 15.5542
1.2 −5.29240 −7.81243 20.0095 13.3255 41.3466 13.4613 −63.5594 34.0341 −70.5238
1.3 −5.22230 2.89021 19.2724 −13.0151 −15.0935 −12.6756 −58.8681 −18.6467 67.9690
1.4 −5.14462 8.08621 18.4671 13.4813 −41.6004 −35.5728 −53.8490 38.3868 −69.3562
1.5 −4.75874 −0.0136729 14.6456 7.09219 0.0650658 29.9834 −31.6249 −26.9998 −33.7499
1.6 −4.48088 −9.17502 12.0783 14.2952 41.1122 0.116715 −18.2743 57.1811 −64.0553
1.7 −4.37818 5.34433 11.1685 14.2453 −23.3984 16.6347 −13.8722 1.56183 −62.3687
1.8 −4.32381 −2.73962 10.6953 −3.64380 11.8456 −25.6284 −11.6540 −19.4945 15.7551
1.9 −4.24567 −5.30878 10.0257 −2.13367 22.5394 13.4319 −8.60063 1.18317 9.05886
1.10 −3.80251 0.949084 6.45907 −12.8894 −3.60890 12.7659 5.85941 −26.0992 49.0119
1.11 −3.78983 2.25550 6.36280 7.85572 −8.54797 −32.8110 6.20470 −21.9127 −29.7718
1.12 −3.59651 −3.23254 4.93488 21.8542 11.6259 −7.56433 11.0237 −16.5507 −78.5990
1.13 −3.41156 7.82449 3.63872 −11.0434 −26.6937 −12.2760 14.8788 34.2227 37.6752
1.14 −3.15653 9.81890 1.96368 −17.5841 −30.9937 17.4517 19.0538 69.4108 55.5047
1.15 −3.03342 8.31160 1.20166 16.0796 −25.2126 33.0100 20.6222 42.0827 −48.7762
1.16 −2.71515 −7.15011 −0.627970 −17.2227 19.4136 2.81026 23.4262 24.1241 46.7621
1.17 −2.52910 4.08665 −1.60364 −1.65408 −10.3355 −7.18720 24.2886 −10.2993 4.18333
1.18 −2.51005 −4.94455 −1.69965 14.8874 12.4111 −8.40781 24.3466 −2.55144 −37.3682
1.19 −1.74480 −1.69757 −4.95569 14.8388 2.96191 15.0825 22.6050 −24.1183 −25.8906
1.20 −1.69154 2.68732 −5.13868 −11.1598 −4.54573 2.97227 22.2247 −19.7783 18.8772
See all 59 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.59
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.d 59
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.4.a.d 59 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{59} - 13 T_{2}^{58} - 269 T_{2}^{57} + 4097 T_{2}^{56} + 31923 T_{2}^{55} - 605911 T_{2}^{54} - 2105624 T_{2}^{53} + 55909895 T_{2}^{52} + 72729444 T_{2}^{51} - 3610196232 T_{2}^{50} + \cdots + 55\!\cdots\!88 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1003))\). Copy content Toggle raw display