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Properties

Label 29.10.b.a
Level $29$
Weight $10$
Character orbit 29.b
Analytic conductor $14.936$
Analytic rank $0$
Dimension $22$
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] = [29,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.28");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(14.9360392488\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 5804 q^{4} - 1374 q^{5} - 8304 q^{6} - 4956 q^{7} - 112244 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 5804 q^{4} - 1374 q^{5} - 8304 q^{6} - 4956 q^{7} - 112244 q^{9} - 244222 q^{13} + 1246804 q^{16} - 1658748 q^{20} + 822328 q^{22} - 874956 q^{23} + 8668172 q^{24} + 5307748 q^{25} - 620352 q^{28} - 2425374 q^{29} - 8942448 q^{30} + 10134274 q^{33} - 37785784 q^{34} - 20790348 q^{35} + 34550680 q^{36} - 30663552 q^{38} + 56872008 q^{42} - 43877176 q^{45} - 131743922 q^{49} - 6194732 q^{51} + 342496580 q^{52} + 34886610 q^{53} + 116488784 q^{54} - 308361676 q^{57} + 342193888 q^{58} + 175799052 q^{59} - 484313328 q^{62} - 190643424 q^{63} - 419498924 q^{64} - 149739966 q^{65} - 508277640 q^{67} + 263144256 q^{71} + 435201408 q^{74} + 1065897336 q^{78} + 2990464236 q^{80} - 129895134 q^{81} - 527065064 q^{82} + 1555989756 q^{83} - 3422424120 q^{86} + 2176720604 q^{87} - 387386068 q^{88} - 1493579244 q^{91} - 1262849472 q^{92} + 2042413382 q^{93} + 166226488 q^{94} - 6686432820 q^{96}+O(q^{100}) \) 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 − 43.1405i − 202.470i −1349.11 1768.78 −8734.66 −2939.41 36113.2i −21311.1 − 76306.1i
28.2 − 39.9552i 171.812i −1084.42 917.921 6864.79 835.627 22871.2i −9836.35 − 36675.8i
28.3 − 37.8978i − 104.304i −924.245 −2682.94 −3952.89 4793.67 15623.2i 8803.72 101678.i
28.4 − 34.5490i 59.9553i −681.634 −636.317 2071.40 −7630.94 5860.68i 16088.4 21984.1i
28.5 − 27.8229i − 61.9183i −262.115 1301.46 −1722.75 8294.02 − 6952.53i 15849.1 − 36210.3i
28.6 − 25.3186i 226.177i −129.033 −952.179 5726.51 6455.42 − 9696.20i −31473.3 24107.9i
28.7 − 20.3455i − 80.4219i 98.0611 535.165 −1636.22 −10636.4 − 12412.0i 13215.3 − 10888.2i
28.8 − 19.5556i − 267.976i 129.577 −595.738 −5240.44 −893.851 − 12546.4i −52128.1 11650.0i
28.9 − 10.3669i 137.845i 404.527 −2029.73 1429.03 −4871.43 − 9501.57i 681.666 21042.0i
28.10 − 9.11441i 171.314i 428.927 2218.10 1561.43 −1871.46 − 8576.00i −9665.49 − 20216.7i
28.11 − 6.67374i − 77.6458i 467.461 −531.517 −518.188 5986.73 − 6536.67i 13654.1 3547.20i
28.12 6.67374i 77.6458i 467.461 −531.517 −518.188 5986.73 6536.67i 13654.1 − 3547.20i
28.13 9.11441i − 171.314i 428.927 2218.10 1561.43 −1871.46 8576.00i −9665.49 20216.7i
28.14 10.3669i − 137.845i 404.527 −2029.73 1429.03 −4871.43 9501.57i 681.666 − 21042.0i
28.15 19.5556i 267.976i 129.577 −595.738 −5240.44 −893.851 12546.4i −52128.1 − 11650.0i
28.16 20.3455i 80.4219i 98.0611 535.165 −1636.22 −10636.4 12412.0i 13215.3 10888.2i
28.17 25.3186i − 226.177i −129.033 −952.179 5726.51 6455.42 9696.20i −31473.3 − 24107.9i
28.18 27.8229i 61.9183i −262.115 1301.46 −1722.75 8294.02 6952.53i 15849.1 36210.3i
28.19 34.5490i − 59.9553i −681.634 −636.317 2071.40 −7630.94 − 5860.68i 16088.4 − 21984.1i
28.20 37.8978i 104.304i −924.245 −2682.94 −3952.89 4793.67 − 15623.2i 8803.72 − 101678.i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.22
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 29.10.b.a ✓ 22
29.b even 2 1 inner 29.10.b.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.b.a ✓ 22 1.a even 1 1 trivial
29.10.b.a ✓ 22 29.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(29, [\chi])\).