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Properties

Label 3211.2.a.q
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 −2.79349 −3.12390 5.80357 0.980067 8.72656 −2.98760 −10.6252 6.75874 −2.73780
1.2 −2.67898 1.03542 5.17693 3.67669 −2.77386 −4.64008 −8.51094 −1.92791 −9.84978
1.3 −2.56444 2.52373 4.57635 −2.40670 −6.47196 1.53509 −6.60690 3.36923 6.17183
1.4 −2.54752 −0.391996 4.48985 −0.375555 0.998616 −0.868582 −6.34294 −2.84634 0.956734
1.5 −2.35237 −2.88605 3.53367 3.56944 6.78907 4.11595 −3.60776 5.32927 −8.39667
1.6 −2.32064 2.76703 3.38537 −1.24367 −6.42129 −3.95921 −3.21495 4.65647 2.88610
1.7 −2.29369 1.17898 3.26104 −2.51481 −2.70422 −3.03279 −2.89243 −1.61001 5.76820
1.8 −2.05958 −1.55900 2.24188 −1.10956 3.21088 4.98908 −0.498165 −0.569526 2.28524
1.9 −1.39601 3.22772 −0.0511438 3.78448 −4.50595 −2.89474 2.86343 7.41820 −5.28318
1.10 −1.36274 0.267768 −0.142937 3.28069 −0.364898 3.09573 2.92027 −2.92830 −4.47073
1.11 −1.17499 3.13268 −0.619388 −2.00431 −3.68089 4.38182 3.07777 6.81370 2.35505
1.12 −1.04929 −2.39377 −0.898998 −4.28712 2.51175 0.0332575 3.04188 2.73013 4.49842
1.13 −0.815365 0.979660 −1.33518 2.21684 −0.798780 1.04073 2.71939 −2.04027 −1.80754
1.14 −0.757215 −0.248261 −1.42663 1.35324 0.187987 −3.87566 2.59469 −2.93837 −1.02470
1.15 −0.721557 −2.38282 −1.47936 1.36356 1.71934 −3.10408 2.51055 2.67783 −0.983889
1.16 −0.299944 −3.05373 −1.91003 −3.17906 0.915947 1.08923 1.17279 6.32526 0.953539
1.17 −0.197799 0.0825846 −1.96088 0.981617 −0.0163351 2.07289 0.783457 −2.99318 −0.194163
1.18 0.373227 2.59752 −1.86070 −3.63956 0.969466 −2.29275 −1.44092 3.74711 −1.35838
1.19 0.463356 2.91541 −1.78530 2.08982 1.35087 −0.967457 −1.75394 5.49960 0.968329
1.20 0.914444 2.88812 −1.16379 1.25273 2.64103 5.08196 −2.89311 5.34126 1.14555
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.q ✓ 33
13.b even 2 1 3211.2.a.r yes 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3211.2.a.q ✓ 33 1.a even 1 1 trivial
3211.2.a.r yes 33 13.b even 2 1

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