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Properties

Label 2009.4.a.h
Level $2009$
Weight $4$
Character orbit 2009.a
Self dual yes
Analytic conductor $118.535$
Analytic rank $1$
Dimension $30$
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: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} - 12 q^{3} + 103 q^{4} - 20 q^{5} - 36 q^{6} - 9 q^{8} + 318 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} - 12 q^{3} + 103 q^{4} - 20 q^{5} - 36 q^{6} - 9 q^{8} + 318 q^{9} - 80 q^{10} + 38 q^{11} + 83 q^{12} - 78 q^{13} + 24 q^{15} + 287 q^{16} - 260 q^{17} - 185 q^{18} - 336 q^{19} - 240 q^{20} - 160 q^{22} - 90 q^{23} - 1112 q^{24} + 606 q^{25} + 55 q^{26} - 432 q^{27} + 130 q^{29} - 674 q^{30} - 1320 q^{31} - 331 q^{32} - 152 q^{33} - 816 q^{34} + 983 q^{36} - 4 q^{37} - 396 q^{38} - 248 q^{39} - 934 q^{40} - 1230 q^{41} - 214 q^{43} + 926 q^{44} - 804 q^{45} - 248 q^{46} - 2262 q^{47} + 568 q^{48} - 543 q^{50} + 204 q^{51} - 650 q^{52} - 522 q^{53} - 3253 q^{54} - 1328 q^{55} - 160 q^{57} + 888 q^{58} - 656 q^{59} + 994 q^{60} - 4300 q^{61} + 728 q^{62} + 1637 q^{64} + 1848 q^{65} + 744 q^{66} + 1642 q^{67} - 4860 q^{68} + 1556 q^{69} - 980 q^{71} - 2248 q^{72} - 1112 q^{73} + 1609 q^{74} - 6916 q^{75} - 3096 q^{76} + 343 q^{78} + 2068 q^{79} + 2440 q^{80} + 3130 q^{81} - 41 q^{82} - 356 q^{83} + 788 q^{85} - 514 q^{86} - 820 q^{87} - 1130 q^{88} - 5560 q^{89} - 2160 q^{90} + 1573 q^{92} + 124 q^{93} + 2377 q^{94} + 580 q^{95} - 9857 q^{96} - 3828 q^{97} - 2870 q^{99}+O(q^{100}) \) 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 −5.44610 3.46823 21.6600 9.65556 −18.8883 0 −74.3935 −14.9714 −52.5851
1.2 −5.25966 9.95056 19.6641 −0.690716 −52.3366 0 −61.3490 72.0137 3.63293
1.3 −4.59222 −5.30191 13.0885 17.8900 24.3475 0 −23.3675 1.11023 −82.1547
1.4 −4.43600 −5.27473 11.6781 −19.0439 23.3987 0 −16.3160 0.822745 84.4787
1.5 −4.33219 −4.25218 10.7679 −6.40888 18.4212 0 −11.9908 −8.91899 27.7645
1.6 −3.69842 7.66955 5.67831 −4.73696 −28.3652 0 8.58659 31.8219 17.5193
1.7 −3.13685 −0.726413 1.83984 8.18759 2.27865 0 19.3235 −26.4723 −25.6833
1.8 −3.13019 −9.57945 1.79808 −15.3925 29.9855 0 19.4132 64.7659 48.1816
1.9 −2.61264 −0.0204702 −1.17409 −12.1971 0.0534812 0 23.9686 −26.9996 31.8666
1.10 −2.48350 6.40015 −1.83221 −4.02052 −15.8948 0 24.4183 13.9619 9.98498
1.11 −1.76668 6.65681 −4.87886 16.8187 −11.7604 0 22.7528 17.3131 −29.7131
1.12 −1.15126 −8.67479 −6.67460 8.17766 9.98692 0 16.8943 48.2520 −9.41460
1.13 −1.00255 0.387093 −6.99489 −15.3378 −0.388082 0 15.0332 −26.8502 15.3769
1.14 −0.941072 −8.32984 −7.11438 21.2675 7.83898 0 14.2237 42.3862 −20.0142
1.15 −0.208351 −3.07702 −7.95659 2.07481 0.641101 0 3.32458 −17.5319 −0.432289
1.16 −0.116020 5.21300 −7.98654 11.3743 −0.604813 0 1.85476 0.175418 −1.31965
1.17 1.32102 −10.0678 −6.25490 −8.86662 −13.2998 0 −18.8311 74.3601 −11.7130
1.18 1.36514 2.26242 −6.13639 −19.1582 3.08852 0 −19.2982 −21.8815 −26.1536
1.19 1.40105 −4.34317 −6.03706 3.78319 −6.08500 0 −19.6666 −8.13684 5.30043
1.20 1.65259 8.44588 −5.26896 2.08319 13.9575 0 −21.9281 44.3328 3.44266
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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.h ✓ 30
7.b odd 2 1 2009.4.a.i yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.4.a.h ✓ 30 1.a even 1 1 trivial
2009.4.a.i yes 30 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{30} - T_{2}^{29} - 171 T_{2}^{28} + 169 T_{2}^{27} + 12973 T_{2}^{26} - 12569 T_{2}^{25} + \cdots + 82043329536 \) Copy content Toggle raw display
\( T_{3}^{30} + 12 T_{3}^{29} - 492 T_{3}^{28} - 6120 T_{3}^{27} + 105467 T_{3}^{26} + \cdots + 84\!\cdots\!04 \) Copy content Toggle raw display