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Properties

Label 1003.2.a.j
Level $1003$
Weight $2$
Character orbit 1003.a
Self dual yes
Analytic conductor $8.009$
Analytic rank $0$
Dimension $22$
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] = [1003,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(8.00899532273\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 5 q^{2} + 7 q^{3} + 25 q^{4} + 19 q^{5} + 5 q^{6} + 3 q^{7} + 21 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 5 q^{2} + 7 q^{3} + 25 q^{4} + 19 q^{5} + 5 q^{6} + 3 q^{7} + 21 q^{8} + 31 q^{9} + 8 q^{11} + 20 q^{12} + 14 q^{13} + 5 q^{14} + 15 q^{15} + 23 q^{16} - 22 q^{17} + 6 q^{18} + 6 q^{19} + 43 q^{20} + 8 q^{21} - 8 q^{22} + 15 q^{23} - 9 q^{24} + 33 q^{25} + 9 q^{26} + 25 q^{27} + 11 q^{28} - q^{29} - 51 q^{30} - 9 q^{31} + 37 q^{32} + 21 q^{33} - 5 q^{34} + 29 q^{35} + 30 q^{36} - 2 q^{37} + 39 q^{38} + 4 q^{39} + 4 q^{40} + 21 q^{41} - 65 q^{42} + q^{43} + 17 q^{44} + 65 q^{45} - 39 q^{46} + 37 q^{47} + 15 q^{48} + 25 q^{49} - 48 q^{50} - 7 q^{51} + 7 q^{52} + 69 q^{53} + 13 q^{54} + 10 q^{55} - 33 q^{56} - 4 q^{57} + 4 q^{58} + 22 q^{59} + 18 q^{60} - 29 q^{61} + 29 q^{62} + 7 q^{63} - 3 q^{64} + 25 q^{65} - 16 q^{66} - 10 q^{67} - 25 q^{68} + 26 q^{69} + 29 q^{70} + 3 q^{71} + 53 q^{72} - 4 q^{73} + 13 q^{74} - 8 q^{75} - 13 q^{76} + 71 q^{77} + 11 q^{78} - 20 q^{79} - 9 q^{80} + 42 q^{81} + 11 q^{82} + 24 q^{83} - 92 q^{84} - 19 q^{85} - 10 q^{86} - 4 q^{87} + 2 q^{88} + 40 q^{89} - 78 q^{90} - 31 q^{91} - 39 q^{92} + 53 q^{93} + 32 q^{94} + 42 q^{95} - 36 q^{96} + 13 q^{97} - 15 q^{98} - 64 q^{99}+O(q^{100}) \) 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 −2.50776 1.17790 4.28884 3.99303 −2.95389 3.24590 −5.73985 −1.61255 −10.0135
1.2 −2.28467 0.0353698 3.21973 −0.992800 −0.0808085 −0.288469 −2.78667 −2.99875 2.26822
1.3 −2.18647 3.36438 2.78064 4.25307 −7.35611 −0.847258 −1.70683 8.31907 −9.29919
1.4 −2.02992 −1.59232 2.12059 −2.38274 3.23228 −2.16746 −0.244779 −0.464532 4.83678
1.5 −1.46853 2.37414 0.156587 0.0990345 −3.48650 4.77589 2.70711 2.63653 −0.145435
1.6 −1.37758 −3.34566 −0.102265 1.06817 4.60892 2.01707 2.89604 8.19343 −1.47150
1.7 −1.25705 −2.18501 −0.419822 4.45642 2.74667 −4.53135 3.04184 1.77428 −5.60195
1.8 −0.921671 −0.437600 −1.15052 −1.68123 0.403323 −3.39099 2.90375 −2.80851 1.54954
1.9 −0.607530 2.06889 −1.63091 2.69382 −1.25692 −3.19752 2.20589 1.28032 −1.63658
1.10 −0.0191532 1.86649 −1.99963 −3.05402 −0.0357493 −1.55451 0.0766058 0.483795 0.0584943
1.11 0.296053 −0.947860 −1.91235 1.42120 −0.280617 3.81838 −1.15826 −2.10156 0.420750
1.12 0.324383 2.83368 −1.89478 2.69264 0.919198 3.71446 −1.26340 5.02975 0.873446
1.13 0.369502 −1.38817 −1.86347 −1.75953 −0.512933 1.96987 −1.42756 −1.07297 −0.650150
1.14 1.00923 −2.57166 −0.981445 −2.22902 −2.59541 −3.29508 −3.00898 3.61346 −2.24961
1.15 1.49676 −1.13573 0.240278 3.48120 −1.69992 −1.05586 −2.63387 −1.71011 5.21050
1.16 1.71910 3.15409 0.955303 −1.04866 5.42219 1.98044 −1.79594 6.94828 −1.80276
1.17 2.05162 1.73406 2.20914 2.98180 3.55764 −0.740180 0.429083 0.00697358 6.11752
1.18 2.12841 2.12034 2.53015 1.18258 4.51297 0.555884 1.12837 1.49586 2.51702
1.19 2.29289 −0.141514 3.25733 2.10475 −0.324476 3.17762 2.88291 −2.97997 4.82596
1.20 2.59194 −2.74602 4.71816 3.63003 −7.11751 3.18482 7.04532 4.54060 9.40883
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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 1003.2.a.j ✓ 22
3.b odd 2 1 9027.2.a.s 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.j ✓ 22 1.a even 1 1 trivial
9027.2.a.s 22 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\( T_{2}^{22} - 5 T_{2}^{21} - 22 T_{2}^{20} + 138 T_{2}^{19} + 171 T_{2}^{18} - 1607 T_{2}^{17} + \cdots + 12 \) Copy content Toggle raw display
\( T_{3}^{22} - 7 T_{3}^{21} - 24 T_{3}^{20} + 260 T_{3}^{19} + 65 T_{3}^{18} - 3924 T_{3}^{17} + \cdots - 128 \) Copy content Toggle raw display