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Properties

Label 1369.2.a.o
Level $1369$
Weight $2$
Character orbit 1369.a
Self dual yes
Analytic conductor $10.932$
Analytic rank $0$
Dimension $27$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1369,2,Mod(1,1369)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1369, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1369.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1369 = 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1369.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [27,9,-1,25,17,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.9315200367\)
Analytic rank: \(0\)
Dimension: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q + 9 q^{2} - q^{3} + 25 q^{4} + 17 q^{5} + 10 q^{6} - 3 q^{7} + 21 q^{8} + 20 q^{9} - 11 q^{10} - 5 q^{11} - 10 q^{12} + 15 q^{13} + 25 q^{14} + 23 q^{15} + 13 q^{16} + 26 q^{17} + 15 q^{18} + 27 q^{19}+ \cdots + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.46536 −2.58193 4.07799 1.93880 6.36538 −1.09829 −5.12299 3.66636 −4.77983
1.2 −2.27635 −0.878687 3.18175 3.11945 2.00020 2.60464 −2.69008 −2.22791 −7.10095
1.3 −2.17626 −0.134844 2.73611 0.493276 0.293456 −3.91530 −1.60198 −2.98182 −1.07350
1.4 −1.78026 −0.264277 1.16931 −0.759212 0.470481 0.0964160 1.47884 −2.93016 1.35159
1.5 −1.63132 1.29788 0.661191 4.07724 −2.11725 2.52094 2.18402 −1.31552 −6.65127
1.6 −1.60545 −1.45646 0.577470 0.100180 2.33828 2.17714 2.28380 −0.878712 −0.160833
1.7 −1.48106 2.16471 0.193544 2.23277 −3.20607 −3.28353 2.67547 1.68596 −3.30688
1.8 −0.938003 −3.28011 −1.12015 0.685175 3.07676 −0.676819 2.92671 7.75915 −0.642697
1.9 −0.870268 1.53326 −1.24263 −2.76332 −1.33435 −5.01094 2.82196 −0.649118 2.40483
1.10 −0.284658 −2.27963 −1.91897 2.88820 0.648915 2.30497 1.11557 2.19670 −0.822149
1.11 −0.113306 3.06761 −1.98716 2.00299 −0.347577 −2.76693 0.451768 6.41020 −0.226950
1.12 −0.102627 2.31494 −1.98947 2.44822 −0.237576 3.25582 0.409428 2.35896 −0.251254
1.13 0.187256 0.225157 −1.96494 −3.33783 0.0421622 −1.65840 −0.742460 −2.94930 −0.625030
1.14 0.526565 −2.50816 −1.72273 −0.0684348 −1.32071 −3.53356 −1.96026 3.29085 −0.0360353
1.15 0.581223 −1.17520 −1.66218 −0.135564 −0.683054 −3.40406 −2.12854 −1.61890 −0.0787931
1.16 0.749275 1.82487 −1.43859 −1.90093 1.36733 −0.420122 −2.57645 0.330146 −1.42432
1.17 0.966457 1.18143 −1.06596 3.99413 1.14180 0.596515 −2.96312 −1.60423 3.86015
1.18 1.54004 −2.34046 0.371724 −2.97938 −3.60440 1.91365 −2.50761 2.47775 −4.58837
1.19 1.81380 −2.67610 1.28987 −2.99386 −4.85391 −2.99597 −1.28804 4.16150 −5.43027
1.20 1.92532 2.94967 1.70685 1.06786 5.67904 −3.23936 −0.564411 5.70053 2.05598
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(37\) \( -1 \)

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 1369.2.a.o yes 27
37.b even 2 1 1369.2.a.n ✓ 27
37.d odd 4 2 1369.2.b.h 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1369.2.a.n ✓ 27 37.b even 2 1
1369.2.a.o yes 27 1.a even 1 1 trivial
1369.2.b.h 54 37.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1369))\):

\( T_{2}^{27} - 9 T_{2}^{26} + T_{2}^{25} + 215 T_{2}^{24} - 470 T_{2}^{23} - 1956 T_{2}^{22} + 7238 T_{2}^{21} + \cdots - 19 \) Copy content Toggle raw display
\( T_{3}^{27} + T_{3}^{26} - 50 T_{3}^{25} - 44 T_{3}^{24} + 1099 T_{3}^{23} + 833 T_{3}^{22} - 14001 T_{3}^{21} + \cdots - 1369 \) Copy content Toggle raw display