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Properties

Label 3211.2.a.r
Level $3211$
Weight $2$
Character orbit 3211.a
Self dual yes
Analytic conductor $25.640$
Analytic rank $0$
Dimension $33$
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] = [3211,2,Mod(1,3211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3211, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3211.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3211 = 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3211.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: \(25.6399640890\)
Analytic rank: \(0\)
Dimension: \(33\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q + q^{2} + 7 q^{3} + 43 q^{4} - 5 q^{5} - 4 q^{6} - 6 q^{7} + 12 q^{8} + 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q + q^{2} + 7 q^{3} + 43 q^{4} - 5 q^{5} - 4 q^{6} - 6 q^{7} + 12 q^{8} + 58 q^{9} - 4 q^{10} - 12 q^{11} + 2 q^{12} + 38 q^{14} + 7 q^{15} + 55 q^{16} + 21 q^{17} - 5 q^{18} - 33 q^{19} - 22 q^{20} + 29 q^{21} - 8 q^{22} + 33 q^{23} - 23 q^{24} + 58 q^{25} + 16 q^{27} - q^{28} + 88 q^{29} - 8 q^{30} - q^{31} + 40 q^{32} - 5 q^{33} + 53 q^{34} + 6 q^{35} + 119 q^{36} + 5 q^{37} - q^{38} - 4 q^{40} + 4 q^{41} + 13 q^{42} - 5 q^{43} - 43 q^{44} + 37 q^{45} - q^{46} - 25 q^{47} + 3 q^{48} + 65 q^{49} - 11 q^{50} + 11 q^{51} + 107 q^{53} - 44 q^{54} - 23 q^{55} + 98 q^{56} - 7 q^{57} + 11 q^{58} + 15 q^{59} + 177 q^{60} + 27 q^{61} - 8 q^{62} - 44 q^{63} + 72 q^{64} + 55 q^{66} - 16 q^{67} + 33 q^{68} + 84 q^{69} - 72 q^{70} + 25 q^{71} - 44 q^{72} + 11 q^{73} + 74 q^{74} - 20 q^{75} - 43 q^{76} + 104 q^{77} + 46 q^{79} - 22 q^{80} + 145 q^{81} - 31 q^{82} - 53 q^{83} + 123 q^{84} - 43 q^{85} + 26 q^{86} + 12 q^{87} - 31 q^{88} + 18 q^{89} - 46 q^{90} + 123 q^{92} - 20 q^{93} - 20 q^{94} + 5 q^{95} - 159 q^{96} + 56 q^{97} + 124 q^{98} + q^{99}+O(q^{100}) \) 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 −2.73155 −1.17826 5.46137 −1.59634 3.21849 −2.57537 −9.45491 −1.61169 4.36047
1.2 −2.66016 3.43911 5.07644 2.73416 −9.14857 −0.125544 −8.18383 8.82747 −7.27330
1.3 −2.51167 2.12774 4.30847 1.74188 −5.34416 1.58266 −5.79810 1.52726 −4.37501
1.4 −2.38122 −2.92141 3.67021 −3.95620 6.95653 −1.80429 −3.97715 5.53464 9.42059
1.5 −2.15580 2.66014 2.64748 −2.31052 −5.73474 −0.105789 −1.39584 4.07636 4.98103
1.6 −2.01825 −3.44368 2.07333 2.28229 6.95021 −2.34105 −0.148005 8.85895 −4.60623
1.7 −2.00311 1.61400 2.01244 −2.36073 −3.23302 −3.38277 −0.0249159 −0.394998 4.72879
1.8 −1.87826 −1.28279 1.52788 −3.13402 2.40941 −0.797206 0.886768 −1.35446 5.88653
1.9 −1.69895 0.0690255 0.886448 2.16421 −0.117271 −4.70021 1.89187 −2.99524 −3.67690
1.10 −1.54668 −1.27875 0.392224 2.90104 1.97783 1.30417 2.48672 −1.36479 −4.48698
1.11 −1.06873 1.08421 −0.857815 3.53150 −1.15873 4.54676 3.05423 −1.82448 −3.77422
1.12 −1.04741 −1.37971 −0.902933 0.521235 1.44512 −2.56788 3.04056 −1.09639 −0.545947
1.13 −0.932806 −0.0667323 −1.12987 −3.72964 0.0622483 3.77930 2.91956 −2.99555 3.47903
1.14 −0.914444 2.88812 −1.16379 −1.25273 −2.64103 −5.08196 2.89311 5.34126 1.14555
1.15 −0.463356 2.91541 −1.78530 −2.08982 −1.35087 0.967457 1.75394 5.49960 0.968329
1.16 −0.373227 2.59752 −1.86070 3.63956 −0.969466 2.29275 1.44092 3.74711 −1.35838
1.17 0.197799 0.0825846 −1.96088 −0.981617 0.0163351 −2.07289 −0.783457 −2.99318 −0.194163
1.18 0.299944 −3.05373 −1.91003 3.17906 −0.915947 −1.08923 −1.17279 6.32526 0.953539
1.19 0.721557 −2.38282 −1.47936 −1.36356 −1.71934 3.10408 −2.51055 2.67783 −0.983889
1.20 0.757215 −0.248261 −1.42663 −1.35324 −0.187987 3.87566 −2.59469 −2.93837 −1.02470
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(19\) \(1\)

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 3211.2.a.r yes 33
13.b even 2 1 3211.2.a.q ✓ 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3211.2.a.q ✓ 33 13.b even 2 1
3211.2.a.r yes 33 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{33} - T_{2}^{32} - 54 T_{2}^{31} + 49 T_{2}^{30} + 1319 T_{2}^{29} - 1075 T_{2}^{28} + \cdots - 24128 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\). Copy content Toggle raw display