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Properties

Label 2299.4.a.o
Level $2299$
Weight $4$
Character orbit 2299.a
Self dual yes
Analytic conductor $135.645$
Analytic rank $1$
Dimension $22$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2299,4,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [22,-11,7,95,0] 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: \(135.645391103\)
Analytic rank: \(1\)
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.

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 −5.40331 −6.30540 21.1957 −6.56988 34.0700 −15.4912 −71.3005 12.7581 35.4991
1.2 −5.20636 −6.71618 19.1062 20.3257 34.9668 32.7143 −57.8228 18.1071 −105.823
1.3 −5.16512 6.12086 18.6785 15.5661 −31.6150 −20.8137 −55.1556 10.4649 −80.4008
1.4 −4.62591 8.89109 13.3990 −4.19389 −41.1293 15.6646 −24.9753 52.0515 19.4006
1.5 −4.13578 1.49366 9.10470 −15.4990 −6.17746 −35.6996 −4.56880 −24.7690 64.1005
1.6 −3.78735 1.50826 6.34402 −20.4839 −5.71230 17.5528 6.27177 −24.7252 77.5798
1.7 −3.44831 −5.82366 3.89087 4.62665 20.0818 13.1060 14.1696 6.91497 −15.9542
1.8 −3.13281 0.143257 1.81447 15.3516 −0.448795 −11.4592 19.3781 −26.9795 −48.0935
1.9 −1.63049 5.25391 −5.34152 −3.49440 −8.56642 8.36917 21.7531 0.603588 5.69756
1.10 −1.59152 −6.78232 −5.46707 −5.15999 10.7942 −8.08597 21.4331 18.9998 8.21221
1.11 −0.858493 9.94500 −7.26299 16.1474 −8.53772 −23.7898 13.1032 71.9031 −13.8624
1.12 −0.402649 −9.75295 −7.83787 4.77814 3.92702 20.1025 6.37711 68.1200 −1.92391
1.13 0.414400 4.79718 −7.82827 11.0417 1.98795 9.77215 −6.55923 −3.98702 4.57566
1.14 0.768521 0.419154 −7.40938 −10.1932 0.322129 −10.4096 −11.8424 −26.8243 −7.83369
1.15 1.75593 −6.28115 −4.91672 8.34309 −11.0292 −35.4301 −22.6808 12.4528 14.6499
1.16 1.87154 −4.33923 −4.49733 −20.2949 −8.12105 13.0526 −23.3893 −8.17107 −37.9828
1.17 2.19322 9.23440 −3.18979 −10.3281 20.2531 −24.6227 −24.5417 58.2742 −22.6519
1.18 2.82403 1.56441 −0.0248805 15.1210 4.41794 15.4762 −22.6625 −24.5526 42.7021
1.19 3.86777 5.32976 6.95964 −12.9431 20.6143 22.5451 −4.02387 1.40635 −50.0611
1.20 4.43852 6.80992 11.7005 3.17808 30.2260 −18.5062 16.4247 19.3750 14.1060
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
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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.o ✓ 22
11.b odd 2 1 2299.4.a.p yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2299.4.a.o ✓ 22 1.a even 1 1 trivial
2299.4.a.p yes 22 11.b odd 2 1

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