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Properties

Label 2299.4.a.p
Level $2299$
Weight $4$
Character orbit 2299.a
Self dual yes
Analytic conductor $135.645$
Analytic rank $0$
Dimension $22$
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] = [2299,4,Mod(1,2299)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2299, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2299.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2299.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: \(135.645391103\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 11 q^{2} + 7 q^{3} + 95 q^{4} - 17 q^{6} + 25 q^{7} + 105 q^{8} + 205 q^{9} + 124 q^{10} - 70 q^{12} + 46 q^{13} + 38 q^{14} + 52 q^{15} + 515 q^{16} + 284 q^{17} + 223 q^{18} + 418 q^{19} + 86 q^{20}+ \cdots + 709 q^{98}+O(q^{100}) \) 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.

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 −5.19157 −1.87096 18.9524 7.68751 9.71321 18.0081 −56.8604 −23.4995 −39.9103
1.2 −5.06259 −6.63903 17.6298 −13.0064 33.6107 −28.9606 −48.7518 17.0768 65.8463
1.3 −4.43852 6.80992 11.7005 3.17808 −30.2260 18.5062 −16.4247 19.3750 −14.1060
1.4 −3.86777 5.32976 6.95964 −12.9431 −20.6143 −22.5451 4.02387 1.40635 50.0611
1.5 −2.82403 1.56441 −0.0248805 15.1210 −4.41794 −15.4762 22.6625 −24.5526 −42.7021
1.6 −2.19322 9.23440 −3.18979 −10.3281 −20.2531 24.6227 24.5417 58.2742 22.6519
1.7 −1.87154 −4.33923 −4.49733 −20.2949 8.12105 −13.0526 23.3893 −8.17107 37.9828
1.8 −1.75593 −6.28115 −4.91672 8.34309 11.0292 35.4301 22.6808 12.4528 −14.6499
1.9 −0.768521 0.419154 −7.40938 −10.1932 −0.322129 10.4096 11.8424 −26.8243 7.83369
1.10 −0.414400 4.79718 −7.82827 11.0417 −1.98795 −9.77215 6.55923 −3.98702 −4.57566
1.11 0.402649 −9.75295 −7.83787 4.77814 −3.92702 −20.1025 −6.37711 68.1200 1.92391
1.12 0.858493 9.94500 −7.26299 16.1474 8.53772 23.7898 −13.1032 71.9031 13.8624
1.13 1.59152 −6.78232 −5.46707 −5.15999 −10.7942 8.08597 −21.4331 18.9998 −8.21221
1.14 1.63049 5.25391 −5.34152 −3.49440 8.56642 −8.36917 −21.7531 0.603588 −5.69756
1.15 3.13281 0.143257 1.81447 15.3516 0.448795 11.4592 −19.3781 −26.9795 48.0935
1.16 3.44831 −5.82366 3.89087 4.62665 −20.0818 −13.1060 −14.1696 6.91497 15.9542
1.17 3.78735 1.50826 6.34402 −20.4839 5.71230 −17.5528 −6.27177 −24.7252 −77.5798
1.18 4.13578 1.49366 9.10470 −15.4990 6.17746 35.6996 4.56880 −24.7690 −64.1005
1.19 4.62591 8.89109 13.3990 −4.19389 41.1293 −15.6646 24.9753 52.0515 −19.4006
1.20 5.16512 6.12086 18.6785 15.5661 31.6150 20.8137 55.1556 10.4649 80.4008
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(19\) \( -1 \)

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 2299.4.a.p yes 22
11.b odd 2 1 2299.4.a.o ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2299.4.a.o ✓ 22 11.b odd 2 1
2299.4.a.p yes 22 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):

\( T_{2}^{22} - 11 T_{2}^{21} - 75 T_{2}^{20} + 1175 T_{2}^{19} + 1220 T_{2}^{18} - 51224 T_{2}^{17} + \cdots + 298400256 \) Copy content Toggle raw display
\( T_{5}^{22} - 1702 T_{5}^{20} + 556 T_{5}^{19} + 1219708 T_{5}^{18} - 633492 T_{5}^{17} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display