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Properties

Label 29.16.b.a
Level $29$
Weight $16$
Character orbit 29.b
Analytic conductor $41.381$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [29,16,Mod(28,29)]
 
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([1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.28");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 29.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(41.3811164790\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9} + 133305618 q^{13} + 5626041364 q^{16} - 30737731548 q^{20} - 51638088984 q^{22} - 23459433564 q^{23} - 13473060100 q^{24} + 169887741474 q^{25} + 281303298768 q^{28} - 85550328684 q^{29} - 681215606256 q^{30} + 831111242422 q^{33} - 449988200584 q^{34} + 726838987044 q^{35} + 1809260484664 q^{36} - 2518300733088 q^{38} - 5363921425320 q^{42} - 16561773855556 q^{45} + 29824615981340 q^{49} + 1184881612900 q^{51} + 21527128606228 q^{52} - 40200435711486 q^{53} + 9043904345168 q^{54} + 42099004809572 q^{57} - 3461494533632 q^{58} - 50458797940572 q^{59} - 298531808710416 q^{62} + 159779590145904 q^{63} - 71569159267548 q^{64} + 92095395748902 q^{65} + 130146715692752 q^{67} - 178710878083152 q^{71} - 205323946615296 q^{74} + 13818320315976 q^{78} + 857820862108188 q^{80} + 126746036597568 q^{81} + 249211917251112 q^{82} - 541736282848188 q^{83} + 630538772195064 q^{86} - 633552108095260 q^{87} + 969723837884556 q^{88} - 962583563732444 q^{91} + 22\!\cdots\!64 q^{92}+ \cdots + 40\!\cdots\!64 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 − 350.198i 3847.09i −89870.4 106507. 1.34724e6 −2.76298e6 1.99971e7i −451190. − 3.72984e7i
28.2 − 327.159i − 2163.79i −74264.8 235169. −707903. 2.51260e6 1.35760e7i 9.66691e6 − 7.69374e7i
28.3 − 317.689i − 3423.11i −68158.0 −176108. −1.08748e6 98456.3 1.12430e7i 2.63125e6 5.59476e7i
28.4 − 297.362i 5034.99i −55656.3 −122226. 1.49722e6 2.97082e6 6.80612e6i −1.10022e7 3.63455e7i
28.5 − 285.356i − 6609.41i −48660.2 173971. −1.88604e6 −3.97639e6 4.53494e6i −2.93355e7 − 4.96437e7i
28.6 − 255.661i 4338.48i −32594.8 −284737. 1.10918e6 −3.50246e6 − 44288.6i −4.47354e6 7.27962e7i
28.7 − 240.042i 2267.47i −24852.3 195163. 544290. 168543. − 1.90009e6i 9.20747e6 − 4.68475e7i
28.8 − 231.586i − 885.668i −20863.9 −97507.9 −205108. 415555. − 2.75682e6i 1.35645e7 2.25814e7i
28.9 − 202.124i − 5797.90i −8086.11 172967. −1.17189e6 3.88134e6 − 4.98880e6i −1.92667e7 − 3.49608e7i
28.10 − 195.247i 7180.80i −5353.24 274602. 1.40203e6 −964261. − 5.35264e6i −3.72150e7 − 5.36151e7i
28.11 − 169.905i − 817.971i 3900.43 58678.2 −138977. −2.72949e6 − 6.23013e6i 1.36798e7 − 9.96969e6i
28.12 − 150.477i − 6169.52i 10124.7 −189529. −928371. 634911. − 6.45436e6i −2.37141e7 2.85197e7i
28.13 − 118.325i 6445.47i 18767.1 −115158. 762661. 493995. − 6.09791e6i −2.71951e7 1.36261e7i
28.14 − 101.334i 71.6384i 22499.4 −295367. 7259.42 1.06773e6 − 5.60048e6i 1.43438e7 2.99307e7i
28.15 − 84.1317i 2443.60i 25689.9 22406.3 205584. 3.47198e6 − 4.91816e6i 8.37775e6 − 1.88508e6i
28.16 − 68.3669i − 2973.41i 28094.0 302546. −203283. −153099. − 4.16094e6i 5.50774e6 − 2.06841e7i
28.17 − 14.2479i 3944.08i 32565.0 −125301. 56194.9 −3.40529e6 − 930860.i −1.20682e6 1.78527e6i
28.18 − 10.1196i − 4604.16i 32665.6 98965.0 −46592.0 −632277. − 662159.i −6.84936e6 − 1.00148e6i
28.19 10.1196i 4604.16i 32665.6 98965.0 −46592.0 −632277. 662159.i −6.84936e6 1.00148e6i
28.20 14.2479i − 3944.08i 32565.0 −125301. 56194.9 −3.40529e6 930860.i −1.20682e6 − 1.78527e6i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.16.b.a ✓ 36
29.b even 2 1 inner 29.16.b.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.16.b.a ✓ 36 1.a even 1 1 trivial
29.16.b.a ✓ 36 29.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(29, [\chi])\).