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Properties

Label 29.18.a.b
Level $29$
Weight $18$
Character orbit 29.a
Self dual yes
Analytic conductor $53.134$
Analytic rank $0$
Dimension $21$
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] = [29,18,Mod(1,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(53.1344053299\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 256 q^{2} + 23966 q^{3} + 1452522 q^{4} + 998272 q^{5} + 3411526 q^{6} + 2193368 q^{7} - 138137226 q^{8} + 1264832799 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 256 q^{2} + 23966 q^{3} + 1452522 q^{4} + 998272 q^{5} + 3411526 q^{6} + 2193368 q^{7} - 138137226 q^{8} + 1264832799 q^{9} - 224469478 q^{10} + 1203139534 q^{11} - 5164251122 q^{12} + 3854339312 q^{13} + 25262272904 q^{14} + 28324474306 q^{15} + 196520815922 q^{16} + 76444714794 q^{17} + 75758949126 q^{18} + 246497292428 q^{19} - 46900976670 q^{20} + 360937126704 q^{21} - 275001533522 q^{22} + 213498528140 q^{23} - 451123453870 q^{24} + 3898884886997 q^{25} - 3609347694206 q^{26} - 2718903745978 q^{27} - 5946174617200 q^{28} + 10505174672181 q^{29} - 20237658929454 q^{30} + 16670029895798 q^{31} - 42141001912046 q^{32} - 7157109761394 q^{33} + 12785761151136 q^{34} + 46677934312888 q^{35} + 132137824374868 q^{36} + 53445659988410 q^{37} + 76581637956388 q^{38} + 79233849032530 q^{39} + 193617444734146 q^{40} - 20814769309298 q^{41} + 76690667258352 q^{42} + 185498647364454 q^{43} + 315429066899678 q^{44} - 486270821438526 q^{45} + 261474367677132 q^{46} + 389503471719450 q^{47} - 101509672247630 q^{48} + 730079062141437 q^{49} + 14\!\cdots\!54 q^{50}+ \cdots - 95\!\cdots\!64 q^{99}+O(q^{100}) \) 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 −716.767 6161.78 382683. −44516.1 −4.41656e6 −2.28029e7 −1.80347e8 −9.11726e7 3.19077e7
1.2 −687.348 −21372.9 341375. −1.16997e6 1.46906e7 −9.37855e6 −1.44551e8 3.27660e8 8.04176e8
1.3 −645.655 14883.2 285798. 954733. −9.60943e6 1.80737e7 −9.98999e7 9.23704e7 −6.16428e8
1.4 −553.336 −346.147 175108. 641880. 191536. −1.77073e7 −2.43669e7 −1.29020e8 −3.55175e8
1.5 −361.395 9478.77 −465.313 215546. −3.42559e6 −1.26142e7 4.75370e7 −3.92930e7 −7.78973e7
1.6 −332.840 −18225.8 −20289.6 −318485. 6.06628e6 9.82044e6 5.03792e7 2.03040e8 1.06005e8
1.7 −275.201 7065.32 −55336.6 −620891. −1.94438e6 2.72618e7 5.12998e7 −7.92215e7 1.70870e8
1.8 −237.844 −14184.6 −74502.2 1.22166e6 3.37373e6 1.05564e7 4.88946e7 7.20633e7 −2.90566e8
1.9 −65.9873 22515.6 −126718. 342266. −1.48575e6 1.63719e7 1.70108e7 3.77814e8 −2.25852e7
1.10 −39.6552 13037.0 −129499. −1.72282e6 −516984. −1.48112e7 1.03330e7 4.08230e7 6.83187e7
1.11 −3.42944 −6754.54 −131060. 511876. 23164.3 −1.86320e7 898966. −8.35164e7 −1.75545e6
1.12 59.1674 15524.2 −127571. 1.59085e6 918527. −2.66210e7 −1.53032e7 1.11861e8 9.41262e7
1.13 217.779 −424.636 −83644.2 −874312. −92476.9 3.28654e6 −4.67607e7 −1.28960e8 −1.90407e8
1.14 290.453 −16308.4 −46709.2 805745. −4.73681e6 2.28724e7 −5.16370e7 1.36823e8 2.34031e8
1.15 327.881 6727.77 −23565.9 1.45971e6 2.20591e6 2.18126e7 −5.07029e7 −8.38773e7 4.78610e8
1.16 361.094 −2259.76 −683.052 −433238. −815984. −1.41185e7 −4.75760e7 −1.24034e8 −1.56440e8
1.17 460.739 20203.5 81208.7 −1.66304e6 9.30854e6 1.23855e7 −2.29740e7 2.79040e8 −7.66230e8
1.18 500.684 −16927.2 119612. −1.40156e6 −8.47517e6 −1.54199e7 −5.73773e6 1.57390e8 −7.01740e8
1.19 616.905 17761.1 249500. 768235. 1.09569e7 1.64486e6 7.30585e7 1.86316e8 4.73928e8
1.20 652.859 4140.75 295153. 358296. 2.70332e6 1.05214e7 1.07122e8 −1.11994e8 2.33917e8
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)

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 29.18.a.b ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.18.a.b ✓ 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 256 T_{2}^{20} - 2069749 T_{2}^{19} + 559124270 T_{2}^{18} + 1763203907568 T_{2}^{17} + \cdots + 35\!\cdots\!00 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display