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Properties

Label 2009.4.a.j
Level $2009$
Weight $4$
Character orbit 2009.a
Self dual yes
Analytic conductor $118.535$
Analytic rank $1$
Dimension $36$
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] = [2009,4,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, 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("2009.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2009.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: \(118.534837202\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 287)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 5 q^{2} - 6 q^{3} + 117 q^{4} + 4 q^{5} - 12 q^{6} - 39 q^{8} + 236 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 5 q^{2} - 6 q^{3} + 117 q^{4} + 4 q^{5} - 12 q^{6} - 39 q^{8} + 236 q^{9} - 12 q^{10} - 140 q^{11} + 186 q^{12} - 72 q^{13} - 366 q^{15} - 15 q^{16} - 2 q^{17} - 212 q^{18} + 30 q^{19} - 334 q^{20} - 346 q^{22} - 314 q^{23} + 106 q^{24} + 570 q^{25} + 303 q^{26} - 204 q^{27} - 356 q^{29} - 357 q^{30} - 4 q^{31} - 532 q^{32} - 30 q^{33} - 364 q^{34} + 113 q^{36} - 1398 q^{37} - 264 q^{38} - 1348 q^{39} + 26 q^{40} + 1476 q^{41} - 1072 q^{43} - 1507 q^{44} - 1132 q^{45} - 1356 q^{46} - 622 q^{47} + 1724 q^{48} - 1426 q^{50} - 668 q^{51} - 877 q^{52} - 412 q^{53} - 1814 q^{54} + 1114 q^{55} - 4082 q^{57} - 1309 q^{58} - 620 q^{59} - 3724 q^{60} + 774 q^{61} + 1665 q^{62} - 3285 q^{64} - 1036 q^{65} - 1056 q^{66} - 2972 q^{67} - 1525 q^{68} + 3304 q^{69} - 3540 q^{71} - 821 q^{72} - 60 q^{73} - 2043 q^{74} + 450 q^{75} + 2171 q^{76} - 1136 q^{78} - 5190 q^{79} - 1564 q^{80} + 284 q^{81} - 205 q^{82} + 1656 q^{83} - 5064 q^{85} - 782 q^{86} - 1940 q^{87} - 4232 q^{88} - 1196 q^{89} + 8030 q^{90} - 4618 q^{92} + 698 q^{93} - 35 q^{94} - 1968 q^{95} - 7926 q^{96} + 3862 q^{97} - 5964 q^{99}+O(q^{100}) \) 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 −5.24204 1.46828 19.4790 −7.63398 −7.69676 0 −60.1731 −24.8442 40.0176
1.2 −5.05226 6.21075 17.5254 9.48476 −31.3783 0 −48.1246 11.5734 −47.9195
1.3 −4.84147 8.89523 15.4398 −19.8253 −43.0660 0 −36.0196 52.1251 95.9835
1.4 −4.67229 −4.57802 13.8303 8.79022 21.3898 0 −27.2411 −6.04177 −41.0705
1.5 −4.65790 −6.35474 13.6960 −11.2439 29.5997 0 −26.5313 13.3827 52.3728
1.6 −4.30190 −0.345300 10.5063 17.1344 1.48545 0 −10.7821 −26.8808 −73.7105
1.7 −3.98055 −1.09910 7.84477 16.1194 4.37501 0 0.617915 −25.7920 −64.1642
1.8 −3.61690 −9.85524 5.08200 −11.4029 35.6455 0 10.5541 70.1257 41.2432
1.9 −3.59229 3.87874 4.90453 −10.9640 −13.9335 0 11.1198 −11.9554 39.3859
1.10 −2.84821 −7.14998 0.112273 18.5031 20.3646 0 22.4659 24.1223 −52.7005
1.11 −2.56286 −4.35049 −1.43176 −5.73556 11.1497 0 24.1723 −8.07327 14.6994
1.12 −2.29878 8.78314 −2.71563 2.02920 −20.1905 0 24.6328 50.1435 −4.66467
1.13 −2.25528 8.55708 −2.91373 −4.20502 −19.2986 0 24.6135 46.2236 9.48347
1.14 −1.68929 −2.43988 −5.14630 3.14907 4.12166 0 22.2079 −21.0470 −5.31970
1.15 −1.64556 −1.90121 −5.29212 −11.4725 3.12855 0 21.8730 −23.3854 18.8787
1.16 −1.48336 7.02935 −5.79964 10.3252 −10.4271 0 20.4698 22.4118 −15.3160
1.17 −0.706634 2.07976 −7.50067 −16.5960 −1.46963 0 10.9533 −22.6746 11.7273
1.18 −0.598281 −8.38024 −7.64206 −3.61677 5.01374 0 9.35835 43.2285 2.16385
1.19 −0.140258 5.16208 −7.98033 −0.516531 −0.724025 0 2.24138 −0.352951 0.0724479
1.20 0.388024 0.580020 −7.84944 19.1871 0.225062 0 −6.14996 −26.6636 7.44506
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(41\) \(-1\)

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 2009.4.a.j 36
7.b odd 2 1 2009.4.a.k 36
7.c even 3 2 287.4.e.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.4.e.a ✓ 72 7.c even 3 2
2009.4.a.j 36 1.a even 1 1 trivial
2009.4.a.k 36 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{36} + 5 T_{2}^{35} - 190 T_{2}^{34} - 952 T_{2}^{33} + 16346 T_{2}^{32} + 82211 T_{2}^{31} + \cdots - 32865404963328 \) Copy content Toggle raw display
\( T_{3}^{36} + 6 T_{3}^{35} - 586 T_{3}^{34} - 3412 T_{3}^{33} + 154674 T_{3}^{32} + 877416 T_{3}^{31} + \cdots - 34\!\cdots\!24 \) Copy content Toggle raw display