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Properties

Label 2009.4.a.m
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

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 −5.56779 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.m 44
7.b odd 2 1 2009.4.a.l 44
7.d odd 6 2 287.4.e.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.4.e.b ✓ 88 7.d odd 6 2
2009.4.a.l 44 7.b odd 2 1
2009.4.a.m 44 1.a even 1 1 trivial

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