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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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 120q^{3} + 1788q^{4} - 5940q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 120q^{3} + 1788q^{4} - 5940q^{9} - 6936q^{12} + 23160q^{15} + 5844q^{16} - 17556q^{22} - 34704q^{25} + 7560q^{27} + 131328q^{31} + 28644q^{33} - 436080q^{34} - 302832q^{36} + 1072488q^{37} - 100296q^{42} - 1854252q^{45} + 3172548q^{48} - 1169304q^{49} - 4401936q^{55} - 2920440q^{58} + 1981452q^{60} + 14578596q^{64} + 2895156q^{66} + 7729200q^{67} - 4614108q^{69} - 6924504q^{70} - 4375800q^{75} - 26237448q^{78} + 22367340q^{81} - 23600232q^{82} - 25179660q^{88} + 9380640q^{91} + 25396908q^{93} + 72948360q^{97} + 53650080q^{99} + O(q^{100}) \)

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.

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 \( T_{2}^{12} - 1215 T_{2}^{10} + 553254 T_{2}^{8} - 118801512 T_{2}^{6} + 12291842016 T_{2}^{4} - 543457517184 T_{2}^{2} + \)6005462031360

'>\(60\!\cdots\!60\)\( \) acting on \(S_{8}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database