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Properties

Label 33.8.d.b
Level $33$
Weight $8$
Character orbit 33.d
Analytic conductor $10.309$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,8,Mod(32,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.32");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.d (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: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 120 q^{3} + 1788 q^{4} - 5940 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 120 q^{3} + 1788 q^{4} - 5940 q^{9} - 6936 q^{12} + 23160 q^{15} + 5844 q^{16} - 17556 q^{22} - 34704 q^{25} + 7560 q^{27} + 131328 q^{31} + 28644 q^{33} - 436080 q^{34} - 302832 q^{36} + 1072488 q^{37} - 100296 q^{42} - 1854252 q^{45} + 3172548 q^{48} - 1169304 q^{49} - 4401936 q^{55} - 2920440 q^{58} + 1981452 q^{60} + 14578596 q^{64} + 2895156 q^{66} + 7729200 q^{67} - 4614108 q^{69} - 6924504 q^{70} - 4375800 q^{75} - 26237448 q^{78} + 22367340 q^{81} - 23600232 q^{82} - 25179660 q^{88} + 9380640 q^{91} + 25396908 q^{93} + 72948360 q^{97} + 53650080 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −21.2791 13.2650 − 44.8446i 324.799 116.081i −282.268 + 954.252i − 73.6975i −4187.71 −1835.08 − 1189.73i − 2470.10i
32.2 −21.2791 13.2650 + 44.8446i 324.799 − 116.081i −282.268 − 954.252i 73.6975i −4187.71 −1835.08 + 1189.73i 2470.10i
32.3 −17.1890 −43.8101 − 16.3606i 167.460 − 90.8965i 753.051 + 281.222i 1506.19i −678.279 1651.66 + 1433.52i 1562.42i
32.4 −17.1890 −43.8101 + 16.3606i 167.460 90.8965i 753.051 − 281.222i − 1506.19i −678.279 1651.66 − 1433.52i − 1562.42i
32.5 −15.1340 46.1737 − 7.41559i 101.039 − 382.367i −698.794 + 112.228i 606.007i 408.027 2077.02 − 684.810i 5786.75i
32.6 −15.1340 46.1737 + 7.41559i 101.039 382.367i −698.794 − 112.228i − 606.007i 408.027 2077.02 + 684.810i − 5786.75i
32.7 −10.8692 −0.0827838 − 46.7653i −9.86124 − 348.238i 0.899791 + 508.300i − 1046.71i 1498.44 −2186.99 + 7.74282i 3785.06i
32.8 −10.8692 −0.0827838 + 46.7653i −9.86124 348.238i 0.899791 − 508.300i 1046.71i 1498.44 −2186.99 − 7.74282i − 3785.06i
32.9 −10.1755 −20.0376 − 42.2551i −24.4601 419.841i 203.892 + 429.965i − 43.4849i 1551.35 −1383.99 + 1693.38i − 4272.07i
32.10 −10.1755 −20.0376 + 42.2551i −24.4601 − 419.841i 203.892 − 429.965i 43.4849i 1551.35 −1383.99 − 1693.38i 4272.07i
32.11 −4.00282 34.4918 − 31.5803i −111.977 109.310i −138.065 + 126.410i 1222.62i 960.586 192.374 − 2178.52i − 437.547i
32.12 −4.00282 34.4918 + 31.5803i −111.977 − 109.310i −138.065 − 126.410i − 1222.62i 960.586 192.374 + 2178.52i 437.547i
32.13 4.00282 34.4918 − 31.5803i −111.977 109.310i 138.065 − 126.410i − 1222.62i −960.586 192.374 − 2178.52i 437.547i
32.14 4.00282 34.4918 + 31.5803i −111.977 − 109.310i 138.065 + 126.410i 1222.62i −960.586 192.374 + 2178.52i − 437.547i
32.15 10.1755 −20.0376 − 42.2551i −24.4601 419.841i −203.892 − 429.965i 43.4849i −1551.35 −1383.99 + 1693.38i 4272.07i
32.16 10.1755 −20.0376 + 42.2551i −24.4601 − 419.841i −203.892 + 429.965i − 43.4849i −1551.35 −1383.99 − 1693.38i − 4272.07i
32.17 10.8692 −0.0827838 − 46.7653i −9.86124 − 348.238i −0.899791 − 508.300i 1046.71i −1498.44 −2186.99 + 7.74282i − 3785.06i
32.18 10.8692 −0.0827838 + 46.7653i −9.86124 348.238i −0.899791 + 508.300i − 1046.71i −1498.44 −2186.99 − 7.74282i 3785.06i
32.19 15.1340 46.1737 − 7.41559i 101.039 − 382.367i 698.794 − 112.228i − 606.007i −408.027 2077.02 − 684.810i − 5786.75i
32.20 15.1340 46.1737 + 7.41559i 101.039 382.367i 698.794 + 112.228i 606.007i −408.027 2077.02 + 684.810i 5786.75i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.8.d.b ✓ 24
3.b odd 2 1 inner 33.8.d.b ✓ 24
11.b odd 2 1 inner 33.8.d.b ✓ 24
33.d even 2 1 inner 33.8.d.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.d.b ✓ 24 1.a even 1 1 trivial
33.8.d.b ✓ 24 3.b odd 2 1 inner
33.8.d.b ✓ 24 11.b odd 2 1 inner
33.8.d.b ✓ 24 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 1215 T_{2}^{10} + 553254 T_{2}^{8} - 118801512 T_{2}^{6} + 12291842016 T_{2}^{4} + \cdots + 6005462031360 \) acting on \(S_{8}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display