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Properties

Label 6223.2.a.p
Level $6223$
Weight $2$
Character orbit 6223.a
Self dual yes
Analytic conductor $49.691$
Analytic rank $0$
Dimension $38$
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] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 6223 = 7^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6223.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [38,-2,11,38,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(49.6909051778\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: no (minimal twist has level 889)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{2} + 11 q^{3} + 38 q^{4} + 16 q^{5} + 11 q^{6} + 47 q^{9} + 12 q^{10} - 2 q^{11} + 30 q^{12} + 21 q^{13} + 7 q^{15} + 46 q^{16} + 58 q^{17} - 13 q^{18} + 17 q^{19} + 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 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 −2.79997 2.93029 5.83986 3.88842 −8.20473 0 −10.7515 5.58658 −10.8875
1.2 −2.59227 −1.48791 4.71984 −2.92107 3.85707 0 −7.05054 −0.786114 7.57220
1.3 −2.49062 0.897479 4.20317 −0.525060 −2.23528 0 −5.48724 −2.19453 1.30772
1.4 −2.45917 −0.580047 4.04750 2.82005 1.42643 0 −5.03514 −2.66355 −6.93496
1.5 −2.40700 −0.841410 3.79367 4.15395 2.02528 0 −4.31737 −2.29203 −9.99856
1.6 −2.20103 3.12886 2.84454 −2.57615 −6.88671 0 −1.85887 6.78973 5.67019
1.7 −2.08520 −2.29235 2.34805 −0.933672 4.78000 0 −0.725759 2.25486 1.94689
1.8 −2.04311 −0.499376 2.17431 −2.15463 1.02028 0 −0.356134 −2.75062 4.40215
1.9 −1.80084 −3.27481 1.24303 1.98237 5.89741 0 1.36319 7.72436 −3.56994
1.10 −1.47469 1.78718 0.174697 −2.55693 −2.63553 0 2.69175 0.194014 3.77067
1.11 −1.45409 −1.33511 0.114369 3.16583 1.94137 0 2.74187 −1.21748 −4.60340
1.12 −1.31336 2.15446 −0.275086 −2.87032 −2.82958 0 2.98801 1.64168 3.76976
1.13 −1.10143 3.33909 −0.786855 1.46251 −3.67776 0 3.06952 8.14949 −1.61085
1.14 −0.738217 −2.41551 −1.45504 −2.60719 1.78317 0 2.55057 2.83467 1.92467
1.15 −0.635960 0.215746 −1.59556 −3.24565 −0.137206 0 2.28663 −2.95345 2.06410
1.16 −0.615166 −2.48422 −1.62157 3.80329 1.52820 0 2.22787 3.17132 −2.33966
1.17 −0.583418 3.17679 −1.65962 1.76616 −1.85339 0 2.13509 7.09199 −1.03041
1.18 −0.527092 2.21448 −1.72217 1.85969 −1.16724 0 1.96193 1.90392 −0.980227
1.19 −0.494281 1.68478 −1.75569 −0.676418 −0.832754 0 1.85636 −0.161523 0.334341
1.20 −0.0419437 −1.92360 −1.99824 −0.820012 0.0806829 0 0.167701 0.700229 0.0343944
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(127\) \( -1 \)

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 6223.2.a.p 38
7.b odd 2 1 6223.2.a.o 38
7.c even 3 2 889.2.f.c ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.f.c ✓ 76 7.c even 3 2
6223.2.a.o 38 7.b odd 2 1
6223.2.a.p 38 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6223))\):

\( T_{2}^{38} + 2 T_{2}^{37} - 55 T_{2}^{36} - 110 T_{2}^{35} + 1372 T_{2}^{34} + 2750 T_{2}^{33} + \cdots + 213 \) Copy content Toggle raw display
\( T_{3}^{38} - 11 T_{3}^{37} - 20 T_{3}^{36} + 630 T_{3}^{35} - 941 T_{3}^{34} - 15429 T_{3}^{33} + \cdots - 50092 \) Copy content Toggle raw display
\( T_{5}^{38} - 16 T_{5}^{37} + 7 T_{5}^{36} + 1159 T_{5}^{35} - 4586 T_{5}^{34} - 34379 T_{5}^{33} + \cdots + 4636506252 \) Copy content Toggle raw display