Properties

Label 2009.4.a.l
Level $2009$
Weight $4$
Character orbit 2009.a
Self dual yes
Analytic conductor $118.535$
Analytic rank $0$
Dimension $44$
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: \(0\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 3 q^{2} - 6 q^{3} + 213 q^{4} + 4 q^{5} - 12 q^{6} + 57 q^{8} + 452 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 3 q^{2} - 6 q^{3} + 213 q^{4} + 4 q^{5} - 12 q^{6} + 57 q^{8} + 452 q^{9} - 12 q^{10} + 124 q^{11} - 220 q^{12} + 96 q^{13} + 234 q^{15} + 1281 q^{16} - 2 q^{17} + 148 q^{18} - 58 q^{19} + 422 q^{20} + 270 q^{22} + 638 q^{23} + 106 q^{24} + 1570 q^{25} - 327 q^{26} + 48 q^{27} + 224 q^{29} + 1133 q^{30} - 4 q^{31} + 364 q^{32} - 254 q^{33} - 140 q^{34} + 2705 q^{36} + 410 q^{37} - 264 q^{38} + 1460 q^{39} - 26 q^{40} - 1804 q^{41} + 1476 q^{43} + 893 q^{44} + 1724 q^{45} + 1588 q^{46} - 430 q^{47} - 2210 q^{48} + 2710 q^{50} + 1460 q^{51} + 1811 q^{52} + 648 q^{53} - 540 q^{54} - 882 q^{55} + 3734 q^{57} + 1049 q^{58} + 1480 q^{59} + 1076 q^{60} + 1014 q^{61} - 2463 q^{62} + 7595 q^{64} + 468 q^{65} + 2864 q^{66} + 1852 q^{67} + 419 q^{68} - 3248 q^{69} + 4656 q^{71} + 151 q^{72} + 2488 q^{73} + 917 q^{74} + 1538 q^{75} - 987 q^{76} + 5488 q^{78} + 4730 q^{79} + 4484 q^{80} + 6116 q^{81} - 123 q^{82} - 2136 q^{83} + 5796 q^{85} + 646 q^{86} + 2988 q^{87} + 3160 q^{88} - 236 q^{89} - 7902 q^{90} + 4726 q^{92} + 1402 q^{93} - 2485 q^{94} + 5756 q^{95} + 9084 q^{96} - 362 q^{97} + 9816 q^{99}+O(q^{100}) \) 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 −5.56779 −8.65991 23.0003 13.7471 48.2165 0 −83.5184 47.9940 −76.5410
1.2 −5.52545 4.18525 22.5307 6.77679 −23.1254 0 −80.2885 −9.48369 −37.4448
1.3 −5.30261 −1.50705 20.1177 −18.8661 7.99129 0 −64.2554 −24.7288 100.039
1.4 −5.12991 −4.65672 18.3160 8.04761 23.8886 0 −52.9201 −5.31492 −41.2835
1.5 −4.68637 9.79636 13.9621 −2.50471 −45.9094 0 −27.9405 68.9687 11.7380
1.6 −4.44628 −3.60545 11.7694 −7.35509 16.0309 0 −16.7599 −14.0007 32.7028
1.7 −4.31737 −8.67456 10.6397 0.0539338 37.4513 0 −11.3965 48.2480 −0.232852
1.8 −4.11365 8.35880 8.92214 18.0628 −34.3852 0 −3.79335 42.8695 −74.3042
1.9 −4.01943 5.51783 8.15585 −10.7255 −22.1786 0 −0.626446 3.44645 43.1104
1.10 −3.66338 1.94510 5.42033 2.50423 −7.12562 0 9.45029 −23.2166 −9.17393
1.11 −3.42310 0.926042 3.71763 −9.50387 −3.16994 0 14.6590 −26.1424 32.5327
1.12 −3.25509 4.79420 2.59562 8.80361 −15.6056 0 17.5917 −4.01566 −28.6566
1.13 −2.71328 −8.93459 −0.638133 12.0134 24.2420 0 23.4376 52.8269 −32.5957
1.14 −2.57834 −5.14811 −1.35214 −22.1686 13.2736 0 24.1130 −0.496954 57.1582
1.15 −2.25123 −5.39880 −2.93196 4.87704 12.1539 0 24.6104 2.14704 −10.9793
1.16 −1.75078 −0.529464 −4.93476 5.78068 0.926975 0 22.6459 −26.7197 −10.1207
1.17 −1.74060 6.03699 −4.97031 −20.3587 −10.5080 0 22.5761 9.44523 35.4364
1.18 −1.38854 1.80099 −6.07196 19.5363 −2.50075 0 19.5395 −23.7564 −27.1269
1.19 −1.08367 5.78762 −6.82566 13.9821 −6.27188 0 16.0661 6.49657 −15.1520
1.20 −0.149425 −6.65948 −7.97767 12.7223 0.995092 0 2.38746 17.3487 −1.90102
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
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.l 44
7.b odd 2 1 2009.4.a.m 44
7.c even 3 2 287.4.e.b 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.4.e.b 88 7.c even 3 2
2009.4.a.l 44 1.a even 1 1 trivial
2009.4.a.m 44 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{44} - 3 T_{2}^{43} - 278 T_{2}^{42} + 808 T_{2}^{41} + 35734 T_{2}^{40} - 100321 T_{2}^{39} + \cdots + 422240214581248 \) Copy content Toggle raw display
\( T_{3}^{44} + 6 T_{3}^{43} - 802 T_{3}^{42} - 4792 T_{3}^{41} + 295326 T_{3}^{40} + 1744424 T_{3}^{39} + \cdots - 49\!\cdots\!16 \) Copy content Toggle raw display