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Properties

Label 1343.2.a.e
Level $1343$
Weight $2$
Character orbit 1343.a
Self dual yes
Analytic conductor $10.724$
Analytic rank $0$
Dimension $37$
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] = [1343,2,Mod(1,1343)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1343, 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("1343.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1343 = 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1343.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: \(10.7239089915\)
Analytic rank: \(0\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 6 q^{2} + 2 q^{3} + 48 q^{4} + 3 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 6 q^{2} + 2 q^{3} + 48 q^{4} + 3 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 61 q^{9} + 11 q^{10} - q^{11} + 3 q^{12} + 25 q^{13} - 11 q^{14} - 4 q^{15} + 62 q^{16} + 37 q^{17} + 23 q^{18} + 15 q^{19} - 14 q^{20} + 16 q^{21} + 35 q^{22} + 7 q^{23} - 18 q^{24} + 98 q^{25} + 21 q^{26} - 7 q^{27} + 49 q^{28} - q^{29} - 9 q^{30} - 3 q^{31} + 33 q^{32} + 48 q^{33} + 6 q^{34} + 11 q^{35} + 36 q^{36} + 48 q^{37} - 11 q^{38} + 7 q^{39} + 61 q^{40} - 6 q^{41} + 48 q^{42} + 54 q^{43} - 25 q^{44} + 2 q^{46} - 5 q^{47} - 49 q^{48} + 100 q^{49} + 31 q^{50} + 2 q^{51} + 23 q^{52} + 16 q^{53} - 39 q^{54} + 30 q^{55} - 21 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 21 q^{60} + 19 q^{61} + 34 q^{62} + 6 q^{63} + 44 q^{64} + 66 q^{66} + 49 q^{67} + 48 q^{68} + 10 q^{69} - 65 q^{70} - 32 q^{71} - 12 q^{72} + 85 q^{73} + 20 q^{74} - 37 q^{75} + 81 q^{76} + 28 q^{77} - 63 q^{78} - 37 q^{79} - 75 q^{80} + 93 q^{81} - 58 q^{82} + 21 q^{83} + 49 q^{84} + 3 q^{85} - 21 q^{86} - 7 q^{87} + 117 q^{88} + 33 q^{89} + 19 q^{90} + 6 q^{91} - 41 q^{92} + 15 q^{93} + 36 q^{94} - 45 q^{95} - 17 q^{96} + 47 q^{97} - 25 q^{98} - 65 q^{99}+O(q^{100}) \) 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 −2.71221 −1.70430 5.35611 −3.67465 4.62243 2.46225 −9.10249 −0.0953550 9.96645
1.2 −2.64364 2.83927 4.98883 −2.97323 −7.50600 −3.49369 −7.90139 5.06144 7.86015
1.3 −2.54083 −1.99941 4.45582 −1.04037 5.08016 1.91575 −6.23982 0.997645 2.64339
1.4 −2.44313 0.464768 3.96891 3.84341 −1.13549 5.19980 −4.81030 −2.78399 −9.38998
1.5 −2.22416 2.09140 2.94690 3.69605 −4.65160 −0.328488 −2.10605 1.37393 −8.22062
1.6 −2.18376 −1.19036 2.76883 2.26055 2.59947 −3.09823 −1.67893 −1.58304 −4.93652
1.7 −2.16314 1.02770 2.67916 −4.08143 −2.22307 4.13438 −1.46912 −1.94382 8.82869
1.8 −1.90850 3.08436 1.64236 −0.289320 −5.88648 4.57938 0.682563 6.51327 0.552166
1.9 −1.76024 −3.07971 1.09843 −1.20294 5.42101 −1.01720 1.58698 6.48459 2.11746
1.10 −1.75643 0.962561 1.08504 −0.388069 −1.69067 −2.77132 1.60707 −2.07348 0.681614
1.11 −1.16033 −1.72383 −0.653623 1.27713 2.00022 2.88644 3.07909 −0.0284036 −1.48190
1.12 −1.06200 2.66299 −0.872146 2.95944 −2.82811 2.84469 3.05023 4.09153 −3.14294
1.13 −0.534865 −2.97279 −1.71392 −0.695797 1.59004 0.394684 1.98644 5.83746 0.372157
1.14 −0.493939 −1.47197 −1.75602 4.18832 0.727066 −2.79482 1.85525 −0.833294 −2.06878
1.15 −0.485709 1.42006 −1.76409 −4.00439 −0.689736 −4.88124 1.82825 −0.983436 1.94497
1.16 −0.417531 0.625454 −1.82567 −0.242652 −0.261147 −3.41506 1.59734 −2.60881 0.101315
1.17 −0.360663 2.59059 −1.86992 1.06524 −0.934331 −0.475581 1.39574 3.71116 −0.384192
1.18 0.0384527 −3.40100 −1.99852 3.56924 −0.130778 4.62687 −0.153754 8.56683 0.137247
1.19 0.285316 −0.301530 −1.91859 −3.66474 −0.0860312 −0.0125082 −1.11804 −2.90908 −1.04561
1.20 0.638191 2.05115 −1.59271 0.661511 1.30902 4.94053 −2.29284 1.20721 0.422170
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(79\) \(1\)

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 1343.2.a.e ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1343.2.a.e ✓ 37 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 6 T_{2}^{36} - 43 T_{2}^{35} + 320 T_{2}^{34} + 715 T_{2}^{33} - 7673 T_{2}^{32} + \cdots + 6255 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1343))\). Copy content Toggle raw display