# 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 $4$

# Related objects

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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{12} - 1215 T_{2}^{10} + 553254 T_{2}^{8} - 118801512 T_{2}^{6} + 12291842016 T_{2}^{4} - 543457517184 T_{2}^{2} + 6005462031360$$ acting on $$S_{8}^{\mathrm{new}}(33, [\chi])$$.