# Properties

 Label 33.14.d.b Level $33$ Weight $14$ Character orbit 33.d Analytic conductor $35.386$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,14,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, 14, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("33.32");

S:= CuspForms(chi, 14);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$14$$ 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: $$35.3862065541$$ Analytic rank: $$0$$ Dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 696 q^{3} + 212988 q^{4} - 421992 q^{9}+O(q^{10})$$ 48 * q + 696 * q^3 + 212988 * q^4 - 421992 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 696 q^{3} + 212988 q^{4} - 421992 q^{9} - 29009832 q^{12} - 139692624 q^{15} + 451061268 q^{16} + 129381612 q^{22} + 1445681664 q^{25} - 7187342760 q^{27} + 996102576 q^{31} + 2284129824 q^{33} - 28861389456 q^{34} - 32483333376 q^{36} + 82716128736 q^{37} + 73491925752 q^{42} + 82265889240 q^{45} - 144718860732 q^{48} - 497220837312 q^{49} + 772810576128 q^{55} - 1320507819672 q^{58} + 485643633852 q^{60} - 666380650140 q^{64} + 81296681364 q^{66} + 2685776621328 q^{67} + 367837906728 q^{69} + 1525248006024 q^{70} - 3597395983416 q^{75} - 7585996724712 q^{78} + 2480971630104 q^{81} + 4112019989784 q^{82} + 15737831112276 q^{88} - 24762746990496 q^{91} + 27438142662024 q^{93} - 12947335390848 q^{97} - 13419956389536 q^{99}+O(q^{100})$$ 48 * q + 696 * q^3 + 212988 * q^4 - 421992 * q^9 - 29009832 * q^12 - 139692624 * q^15 + 451061268 * q^16 + 129381612 * q^22 + 1445681664 * q^25 - 7187342760 * q^27 + 996102576 * q^31 + 2284129824 * q^33 - 28861389456 * q^34 - 32483333376 * q^36 + 82716128736 * q^37 + 73491925752 * q^42 + 82265889240 * q^45 - 144718860732 * q^48 - 497220837312 * q^49 + 772810576128 * q^55 - 1320507819672 * q^58 + 485643633852 * q^60 - 666380650140 * q^64 + 81296681364 * q^66 + 2685776621328 * q^67 + 367837906728 * q^69 + 1525248006024 * q^70 - 3597395983416 * q^75 - 7585996724712 * q^78 + 2480971630104 * q^81 + 4112019989784 * q^82 + 15737831112276 * q^88 - 24762746990496 * q^91 + 27438142662024 * q^93 - 12947335390848 * q^97 - 13419956389536 * 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 −170.839 −925.430 859.013i 20993.9 8677.01i 158099. + 146753.i 329762.i −2.18706e6 118517. + 1.58991e6i 1.48237e6i
32.2 −170.839 −925.430 + 859.013i 20993.9 8677.01i 158099. 146753.i 329762.i −2.18706e6 118517. 1.58991e6i 1.48237e6i
32.3 −163.385 1052.54 697.490i 18502.6 19154.3i −171968. + 113959.i 232054.i −1.68459e6 621338. 1.46827e6i 3.12952e6i
32.4 −163.385 1052.54 + 697.490i 18502.6 19154.3i −171968. 113959.i 232054.i −1.68459e6 621338. + 1.46827e6i 3.12952e6i
32.5 −146.356 −138.713 1255.02i 13228.2 28563.5i 20301.6 + 183681.i 547807.i −737084. −1.55584e6 + 348176.i 4.18045e6i
32.6 −146.356 −138.713 + 1255.02i 13228.2 28563.5i 20301.6 183681.i 547807.i −737084. −1.55584e6 348176.i 4.18045e6i
32.7 −139.894 486.512 1165.17i 11378.3 64265.5i −68060.1 + 163001.i 236833.i −445747. −1.12094e6 1.13374e6i 8.99035e6i
32.8 −139.894 486.512 + 1165.17i 11378.3 64265.5i −68060.1 163001.i 236833.i −445747. −1.12094e6 + 1.13374e6i 8.99035e6i
32.9 −131.377 −1221.71 318.993i 9068.04 56962.3i 160505. + 41908.4i 78540.7i −115092. 1.39081e6 + 779430.i 7.48357e6i
32.10 −131.377 −1221.71 + 318.993i 9068.04 56962.3i 160505. 41908.4i 78540.7i −115092. 1.39081e6 779430.i 7.48357e6i
32.11 −103.446 1260.72 + 70.0278i 2509.08 37467.6i −130417. 7244.09i 159657.i 587875. 1.58452e6 + 176571.i 3.87587e6i
32.12 −103.446 1260.72 70.0278i 2509.08 37467.6i −130417. + 7244.09i 159657.i 587875. 1.58452e6 176571.i 3.87587e6i
32.13 −100.350 304.827 + 1225.32i 1878.02 6402.02i −30589.3 122960.i 527405.i 633604. −1.40848e6 + 747020.i 642439.i
32.14 −100.350 304.827 1225.32i 1878.02 6402.02i −30589.3 + 122960.i 527405.i 633604. −1.40848e6 747020.i 642439.i
32.15 −99.4888 −899.927 885.694i 1706.02 14437.0i 89532.6 + 88116.7i 7425.56i 645282. 25413.7 + 1.59412e6i 1.43632e6i
32.16 −99.4888 −899.927 + 885.694i 1706.02 14437.0i 89532.6 88116.7i 7425.56i 645282. 25413.7 1.59412e6i 1.43632e6i
32.17 −53.3141 696.893 + 1052.93i −5349.61 39833.9i −37154.2 56136.0i 214863.i 721958. −623002. + 1.46756e6i 2.12371e6i
32.18 −53.3141 696.893 1052.93i −5349.61 39833.9i −37154.2 + 56136.0i 214863.i 721958. −623002. 1.46756e6i 2.12371e6i
32.19 −42.4148 −1228.16 293.180i −6392.98 21829.9i 52092.0 + 12435.2i 493447.i 618619. 1.42241e6 + 720142.i 925910.i
32.20 −42.4148 −1228.16 + 293.180i −6392.98 21829.9i 52092.0 12435.2i 493447.i 618619. 1.42241e6 720142.i 925910.i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.48 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 33.14.d.b 48
3.b odd 2 1 inner 33.14.d.b 48
11.b odd 2 1 inner 33.14.d.b 48
33.d even 2 1 inner 33.14.d.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.14.d.b 48 1.a even 1 1 trivial
33.14.d.b 48 3.b odd 2 1 inner
33.14.d.b 48 11.b odd 2 1 inner
33.14.d.b 48 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 151551 T_{2}^{22} + 9967864638 T_{2}^{20} - 373579513970400 T_{2}^{18} + \cdots + 28\!\cdots\!00$$ acting on $$S_{14}^{\mathrm{new}}(33, [\chi])$$.