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Properties

Label 6223.2.a.r
Level $6223$
Weight $2$
Character orbit 6223.a
Self dual yes
Analytic conductor $49.691$
Analytic rank $0$
Dimension $40$
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] = [6223,2,Mod(1,6223)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(49.6909051778\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} + 6 q^{3} + 39 q^{4} + 24 q^{5} - 9 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} + 6 q^{3} + 39 q^{4} + 24 q^{5} - 9 q^{8} + 42 q^{9} + 15 q^{10} + 4 q^{11} + 29 q^{12} + 7 q^{13} + 8 q^{15} + 29 q^{16} + 56 q^{17} - 12 q^{18} + 6 q^{19} + 45 q^{20} + 3 q^{22} - 7 q^{23} + 2 q^{24} + 48 q^{25} + 32 q^{26} + 54 q^{27} - 14 q^{29} + 2 q^{30} - 2 q^{31} - 31 q^{32} + 70 q^{33} - 5 q^{34} + 67 q^{36} + 45 q^{38} - 15 q^{39} + 15 q^{40} + 38 q^{41} + 27 q^{43} - 24 q^{44} + 66 q^{45} - 24 q^{46} + 69 q^{47} + 34 q^{48} + 26 q^{50} - 26 q^{51} + 41 q^{52} - 8 q^{53} + 8 q^{54} + 24 q^{55} + 12 q^{57} - 21 q^{58} + 45 q^{59} - 10 q^{60} - 5 q^{61} + 27 q^{62} + 65 q^{64} + q^{65} + 52 q^{66} + 11 q^{67} + 78 q^{68} + 47 q^{69} + 15 q^{71} - 95 q^{72} + 64 q^{73} - 23 q^{74} + 4 q^{75} - 31 q^{76} + 76 q^{78} - 12 q^{79} + 110 q^{80} + 72 q^{81} - 38 q^{82} + 103 q^{83} + 30 q^{85} - 45 q^{86} + 17 q^{87} + 2 q^{88} + 78 q^{89} + 75 q^{90} - 51 q^{92} - 85 q^{93} + 48 q^{94} - 15 q^{95} - 128 q^{96} + 65 q^{97} + 65 q^{99}+O(q^{100}) \) 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 −2.73783 3.09982 5.49572 −2.24078 −8.48678 0 −9.57068 6.60889 6.13487
1.2 −2.68664 0.656933 5.21805 1.45033 −1.76494 0 −8.64575 −2.56844 −3.89652
1.3 −2.67543 −2.21467 5.15792 3.77035 5.92518 0 −8.44878 1.90474 −10.0873
1.4 −2.60936 2.27618 4.80875 0.761982 −5.93936 0 −7.32905 2.18098 −1.98829
1.5 −2.24401 −2.80584 3.03558 −1.28157 6.29634 0 −2.32386 4.87274 2.87586
1.6 −2.15797 −1.79179 2.65682 0.285167 3.86663 0 −1.41741 0.210520 −0.615381
1.7 −1.89455 1.68668 1.58933 0.447365 −3.19551 0 0.778030 −0.155104 −0.847556
1.8 −1.82190 −1.06573 1.31931 −1.64925 1.94165 0 1.24014 −1.86422 3.00476
1.9 −1.79801 0.144112 1.23286 4.04161 −0.259116 0 1.37934 −2.97923 −7.26687
1.10 −1.77970 3.07662 1.16735 3.89409 −5.47548 0 1.48188 6.46560 −6.93033
1.11 −1.73616 2.58223 1.01424 −1.04688 −4.48316 0 1.71143 3.66791 1.81755
1.12 −1.60118 −2.10158 0.563789 −3.69203 3.36501 0 2.29964 1.41662 5.91162
1.13 −1.28300 −2.26059 −0.353920 1.66860 2.90033 0 3.02007 2.11025 −2.14080
1.14 −1.14887 −0.105959 −0.680100 −0.464834 0.121733 0 3.07908 −2.98877 0.534033
1.15 −1.09299 0.661876 −0.805363 0.633566 −0.723427 0 3.06625 −2.56192 −0.692484
1.16 −1.01110 −0.720309 −0.977683 −2.17925 0.728302 0 3.01073 −2.48115 2.20344
1.17 −0.783388 2.52435 −1.38630 3.31043 −1.97755 0 2.65279 3.37237 −2.59335
1.18 −0.628033 1.21971 −1.60557 −2.87801 −0.766017 0 2.26442 −1.51232 1.80749
1.19 −0.156733 3.33897 −1.97543 −0.705840 −0.523328 0 0.623083 8.14873 0.110629
1.20 −0.138409 0.525573 −1.98084 2.27925 −0.0727438 0 0.550983 −2.72377 −0.315468
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

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.r 40
7.b odd 2 1 6223.2.a.q 40
7.d odd 6 2 889.2.f.d ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.f.d ✓ 80 7.d odd 6 2
6223.2.a.q 40 7.b odd 2 1
6223.2.a.r 40 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6223))\):

\( T_{2}^{40} + 3 T_{2}^{39} - 55 T_{2}^{38} - 167 T_{2}^{37} + 1381 T_{2}^{36} + 4255 T_{2}^{35} + \cdots + 789 \) Copy content Toggle raw display
\( T_{3}^{40} - 6 T_{3}^{39} - 63 T_{3}^{38} + 420 T_{3}^{37} + 1764 T_{3}^{36} - 13397 T_{3}^{35} + \cdots - 16527 \) Copy content Toggle raw display
\( T_{5}^{40} - 24 T_{5}^{39} + 164 T_{5}^{38} + 556 T_{5}^{37} - 12110 T_{5}^{36} + 31411 T_{5}^{35} + \cdots + 25258724 \) Copy content Toggle raw display