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Properties

Label 2009.4.a.n
Level $2009$
Weight $4$
Character orbit 2009.a
Self dual yes
Analytic conductor $118.535$
Analytic rank $1$
Dimension $60$
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: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 2 q^{2} - 24 q^{3} + 230 q^{4} - 40 q^{5} - 72 q^{6} - 18 q^{8} + 572 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 2 q^{2} - 24 q^{3} + 230 q^{4} - 40 q^{5} - 72 q^{6} - 18 q^{8} + 572 q^{9} - 160 q^{10} - 100 q^{11} - 646 q^{12} - 156 q^{13} + 64 q^{15} + 1294 q^{16} - 136 q^{17} + 106 q^{18} - 848 q^{19} - 480 q^{20} - 236 q^{22} - 268 q^{23} - 864 q^{24} + 1500 q^{25} - 1150 q^{26} - 864 q^{27} - 276 q^{29} + 20 q^{30} - 2480 q^{31} + 18 q^{32} - 752 q^{33} - 1632 q^{34} + 2638 q^{36} + 152 q^{37} - 456 q^{38} - 448 q^{39} - 1972 q^{40} + 2460 q^{41} - 380 q^{43} - 560 q^{44} - 1800 q^{45} - 136 q^{46} - 1668 q^{47} - 5360 q^{48} - 430 q^{50} - 680 q^{51} - 1872 q^{52} - 1012 q^{53} - 1318 q^{54} - 6144 q^{55} + 1112 q^{57} - 596 q^{58} - 1888 q^{59} + 2284 q^{60} - 3176 q^{61} - 3440 q^{62} + 7210 q^{64} - 664 q^{65} - 2112 q^{66} + 660 q^{67} + 312 q^{68} - 7528 q^{69} - 2168 q^{71} - 1004 q^{72} - 3504 q^{73} + 286 q^{74} - 3112 q^{75} - 9008 q^{76} + 570 q^{78} + 1872 q^{79} - 4480 q^{80} + 3796 q^{81} + 82 q^{82} - 4600 q^{83} + 72 q^{85} - 816 q^{86} - 3480 q^{87} - 4884 q^{88} - 2600 q^{89} - 4320 q^{90} - 2810 q^{92} + 3376 q^{93} - 7610 q^{94} - 5672 q^{95} - 7294 q^{96} - 8648 q^{97} - 7620 q^{99}+O(q^{100}) \) 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 −5.49786 2.52638 22.2265 −1.76575 −13.8897 0 −78.2152 −20.6174 9.70787
1.2 −5.33246 8.03659 20.4351 13.4123 −42.8548 0 −66.3096 37.5867 −71.5204
1.3 −5.29714 −8.61912 20.0597 −18.7572 45.6567 0 −63.8820 47.2892 99.3594
1.4 −5.18542 −5.95392 18.8886 18.5974 30.8736 0 −56.4620 8.44915 −96.4352
1.5 −5.08467 −8.85003 17.8539 9.49670 44.9995 0 −50.1039 51.3230 −48.2876
1.6 −5.08058 0.507104 17.8123 −13.5498 −2.57638 0 −49.8519 −26.7428 68.8407
1.7 −4.82815 −1.12572 15.3110 −7.07953 5.43512 0 −35.2986 −25.7328 34.1810
1.8 −4.66327 −6.89357 13.7460 2.31451 32.1465 0 −26.7953 20.5213 −10.7932
1.9 −4.10843 6.89455 8.87916 −16.3865 −28.3257 0 −3.61198 20.5348 67.3228
1.10 −4.08931 7.07895 8.72244 1.42150 −28.9480 0 −2.95427 23.1115 −5.81297
1.11 −3.99014 6.15813 7.92125 10.2830 −24.5718 0 0.314211 10.9226 −41.0305
1.12 −3.81033 2.35464 6.51862 16.4970 −8.97195 0 5.64455 −21.4557 −62.8589
1.13 −3.56350 −5.16516 4.69851 4.50228 18.4060 0 11.7649 −0.321131 −16.0439
1.14 −3.49060 0.191137 4.18427 −0.318964 −0.667182 0 13.3192 −26.9635 1.11337
1.15 −3.09138 0.499673 1.55662 −19.7649 −1.54468 0 19.9189 −26.7503 61.1008
1.16 −2.70807 10.0009 −0.666347 −16.3679 −27.0833 0 23.4691 73.0187 44.3254
1.17 −2.67118 −8.92227 −0.864788 −15.7867 23.8330 0 23.6795 52.6070 42.1690
1.18 −2.42047 −3.67223 −2.14132 −8.32094 8.88852 0 24.5468 −13.5147 20.1406
1.19 −2.27328 −4.10084 −2.83221 16.2838 9.32234 0 24.6246 −10.1831 −37.0177
1.20 −2.18585 3.77251 −3.22204 −1.48830 −8.24616 0 24.5297 −12.7682 3.25321
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.60
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.n ✓ 60
7.b odd 2 1 2009.4.a.o yes 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.4.a.n ✓ 60 1.a even 1 1 trivial
2009.4.a.o yes 60 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{60} - 2 T_{2}^{59} - 353 T_{2}^{58} + 704 T_{2}^{57} + 58689 T_{2}^{56} - 116674 T_{2}^{55} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
\( T_{3}^{60} + 24 T_{3}^{59} - 808 T_{3}^{58} - 23280 T_{3}^{57} + 281662 T_{3}^{56} + \cdots + 18\!\cdots\!12 \) Copy content Toggle raw display