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Properties

Label 3721.2.a.m
Level $3721$
Weight $2$
Character orbit 3721.a
Self dual yes
Analytic conductor $29.712$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3721,2,Mod(1,3721)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3721, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3721.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3721 = 61^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3721.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: \(29.7123345921\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 61)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} + 42 q^{4} + 14 q^{5} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} + 42 q^{4} + 14 q^{5} + 56 q^{9} + 18 q^{12} + 28 q^{13} + 30 q^{14} + 40 q^{15} + 50 q^{16} + 30 q^{19} + 76 q^{20} + 18 q^{22} + 34 q^{25} + 34 q^{27} + 8 q^{34} + 8 q^{36} + 92 q^{39} + 52 q^{41} - 10 q^{42} - 2 q^{45} - 46 q^{46} + 120 q^{47} + 14 q^{48} + 16 q^{49} - 8 q^{52} + 100 q^{56} + 12 q^{57} - 88 q^{58} + 200 q^{60} - 82 q^{62} - 40 q^{64} + 30 q^{65} - 54 q^{66} + 142 q^{70} - 28 q^{73} + 96 q^{74} + 18 q^{75} + 38 q^{76} + 42 q^{77} + 106 q^{80} + 64 q^{81} + 124 q^{83} + 44 q^{86} - 34 q^{88} + 118 q^{95} - 4 q^{97}+O(q^{100}) \) 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 −2.71502 0.188121 5.37133 2.58245 −0.510753 −0.405428 −9.15322 −2.96461 −7.01141
1.2 −2.60168 3.02645 4.76873 −0.294694 −7.87384 0.111763 −7.20334 6.15939 0.766698
1.3 −2.52041 2.05696 4.35249 3.90025 −5.18439 1.55424 −5.92925 1.23109 −9.83025
1.4 −2.48054 −1.41100 4.15307 3.07306 3.50004 −4.18766 −5.34078 −1.00908 −7.62285
1.5 −2.33196 −1.07075 3.43804 −0.499324 2.49694 −2.39217 −3.35344 −1.85350 1.16440
1.6 −2.32456 −2.89230 3.40357 −2.69005 6.72332 −2.94007 −3.26268 5.36541 6.25318
1.7 −2.07623 −1.53615 2.31072 −1.76184 3.18939 3.05525 −0.645115 −0.640248 3.65798
1.8 −2.03386 1.43539 2.13658 3.50831 −2.91938 −1.70968 −0.277792 −0.939661 −7.13542
1.9 −1.93083 2.43521 1.72811 1.17427 −4.70197 −4.19418 0.524973 2.93023 −2.26731
1.10 −1.85725 −2.18304 1.44939 −2.99319 4.05446 3.30466 1.02263 1.76566 5.55912
1.11 −1.55606 −0.227754 0.421337 2.86428 0.354400 −3.68970 2.45650 −2.94813 −4.45701
1.12 −1.35727 3.07700 −0.157814 2.68721 −4.17633 2.99516 2.92874 6.46794 −3.64727
1.13 −1.15507 −0.502339 −0.665812 −1.60087 0.580237 3.53393 3.07920 −2.74766 1.84911
1.14 −0.904151 3.38917 −1.18251 0.120298 −3.06432 −3.07362 2.87747 8.48644 −0.108768
1.15 −0.787277 1.63872 −1.38020 −4.19867 −1.29013 1.69138 2.66115 −0.314598 3.30551
1.16 −0.646916 −2.55086 −1.58150 2.52506 1.65020 0.745974 2.31693 3.50691 −1.63350
1.17 −0.563606 1.14368 −1.68235 −2.10783 −0.644585 3.89798 2.07539 −1.69200 1.18799
1.18 −0.270904 −3.19996 −1.92661 −1.01311 0.866884 1.50588 1.06374 7.23976 0.274456
1.19 −0.194690 1.26853 −1.96210 −0.439531 −0.246970 −2.01661 0.771380 −1.39083 0.0855724
1.20 −0.0743483 −2.08506 −1.99447 2.16392 0.155021 1.80789 0.296982 1.34749 −0.160884
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(61\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3721.2.a.m 40
61.b even 2 1 inner 3721.2.a.m 40
61.l odd 60 2 61.2.k.a ✓ 40
183.x even 60 2 549.2.bs.e 40
244.w even 60 2 976.2.cl.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.k.a ✓ 40 61.l odd 60 2
549.2.bs.e 40 183.x even 60 2
976.2.cl.c 40 244.w even 60 2
3721.2.a.m 40 1.a even 1 1 trivial
3721.2.a.m 40 61.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 61 T_{2}^{38} + 1705 T_{2}^{36} - 28940 T_{2}^{34} + 333292 T_{2}^{32} - 2756295 T_{2}^{30} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3721))\). Copy content Toggle raw display