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Properties

Label 731.2.a.f
Level $731$
Weight $2$
Character orbit 731.a
Self dual yes
Analytic conductor $5.837$
Analytic rank $0$
Dimension $21$
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] = [731,2,Mod(1,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, 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("731.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 2 q^{2} - q^{3} + 32 q^{4} - 3 q^{5} + q^{6} + 5 q^{7} + 6 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 2 q^{2} - q^{3} + 32 q^{4} - 3 q^{5} + q^{6} + 5 q^{7} + 6 q^{8} + 34 q^{9} + 12 q^{10} + 2 q^{11} - 5 q^{12} + 26 q^{13} - 3 q^{14} + 5 q^{15} + 62 q^{16} - 21 q^{17} - 10 q^{18} + 16 q^{19} - 27 q^{20} + 36 q^{21} - 14 q^{22} - q^{23} + 15 q^{24} + 40 q^{25} - 3 q^{26} - 16 q^{27} + 25 q^{28} + 15 q^{29} + 38 q^{30} + 18 q^{31} + 14 q^{32} + 14 q^{33} - 2 q^{34} - 5 q^{35} + 73 q^{36} + 12 q^{37} + 19 q^{38} - 11 q^{39} + 41 q^{40} - 45 q^{42} + 21 q^{43} - 34 q^{44} - 18 q^{45} + 28 q^{46} - q^{47} - 36 q^{48} + 40 q^{49} + 24 q^{50} + q^{51} + 23 q^{52} + 39 q^{53} - 83 q^{54} - 2 q^{55} - 10 q^{56} + 3 q^{57} + 19 q^{58} + 4 q^{59} - 35 q^{60} + 50 q^{61} + 5 q^{62} - 37 q^{63} + 120 q^{64} - 8 q^{65} - 37 q^{66} + 16 q^{67} - 32 q^{68} + 33 q^{69} - q^{70} + q^{71} - 54 q^{72} + 15 q^{73} + 52 q^{74} + 11 q^{75} - 15 q^{76} + 13 q^{77} - 100 q^{78} + 56 q^{79} - 100 q^{80} + 97 q^{81} - 11 q^{82} + 61 q^{84} + 3 q^{85} + 2 q^{86} - 8 q^{87} - 56 q^{88} + 5 q^{89} - 69 q^{90} - 4 q^{91} - 27 q^{92} + 17 q^{93} - 47 q^{94} - 9 q^{95} + 81 q^{96} - 28 q^{97} + 18 q^{98} - 19 q^{99}+O(q^{100}) \) 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 −2.79909 −2.92541 5.83488 −1.76305 8.18847 −2.49895 −10.7342 5.55801 4.93492
1.2 −2.71140 −0.220213 5.35167 −0.654741 0.597084 1.40135 −9.08771 −2.95151 1.77526
1.3 −2.48059 2.78565 4.15334 −4.06907 −6.91006 4.55433 −5.34155 4.75983 10.0937
1.4 −1.94262 3.44146 1.77376 0.802251 −6.68543 −1.20527 0.439504 8.84362 −1.55846
1.5 −1.79843 −0.497156 1.23435 −2.09348 0.894100 2.50164 1.37698 −2.75284 3.76498
1.6 −1.62264 −3.32613 0.632968 2.56619 5.39712 −2.29244 2.21820 8.06315 −4.16401
1.7 −1.58029 −0.808715 0.497302 3.43972 1.27800 2.59985 2.37469 −2.34598 −5.43574
1.8 −0.899791 0.468379 −1.19038 −0.603312 −0.421443 −2.56518 2.87067 −2.78062 0.542854
1.9 −0.387858 1.84237 −1.84957 2.41846 −0.714576 4.70313 1.49308 0.394314 −0.938020
1.10 −0.172452 −0.293380 −1.97026 −4.21864 0.0505941 −4.31664 0.684680 −2.91393 0.727513
1.11 0.169978 −1.63219 −1.97111 −3.37745 −0.277437 0.890456 −0.675002 −0.335949 −0.574094
1.12 0.549823 2.96872 −1.69769 −0.0524046 1.63227 −0.375563 −2.03308 5.81330 −0.0288132
1.13 0.715531 −2.68508 −1.48801 1.60306 −1.92126 −4.82676 −2.49578 4.20967 1.14704
1.14 1.10090 −1.62202 −0.788018 2.17965 −1.78568 4.08725 −3.06933 −0.369054 2.39957
1.15 1.35548 2.29949 −0.162684 4.25442 3.11690 −1.26534 −2.93147 2.28766 5.76676
1.16 2.02811 1.50881 2.11322 −1.59736 3.06003 3.53856 0.229620 −0.723487 −3.23961
1.17 2.03648 −3.27527 2.14723 −3.88495 −6.67000 −1.33230 0.299836 7.72736 −7.91160
1.18 2.29170 0.857983 3.25189 2.44469 1.96624 2.38002 2.86896 −2.26387 5.60249
1.19 2.68194 2.28817 5.19279 −3.20246 6.13674 1.43094 8.56287 2.23574 −8.58881
1.20 2.72856 −2.69681 5.44504 0.220730 −7.35840 2.22641 9.39999 4.27277 0.602275
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \( +1 \)
\(43\) \( -1 \)

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 731.2.a.f ✓ 21
3.b odd 2 1 6579.2.a.u 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.a.f ✓ 21 1.a even 1 1 trivial
6579.2.a.u 21 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 2 T_{2}^{20} - 35 T_{2}^{19} + 68 T_{2}^{18} + 515 T_{2}^{17} - 966 T_{2}^{16} - 4148 T_{2}^{15} + \cdots - 192 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\). Copy content Toggle raw display