↑

Properties

Label 330.2.u.a
Level $330$
Weight $2$
Character orbit 330.u
Analytic conductor $2.635$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [330,2,Mod(7,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 5, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.u (of order \(20\), degree \(8\), 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: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{5} + 16 q^{7} - 4 q^{10} - 4 q^{11} - 4 q^{15} + 12 q^{16} + 20 q^{17} + 24 q^{19} + 20 q^{20} - 12 q^{22} - 8 q^{23} + 12 q^{24} + 20 q^{25} - 8 q^{26} + 4 q^{28} - 4 q^{29} - 8 q^{31} + 8 q^{33} - 36 q^{35} + 12 q^{36} + 44 q^{37} + 40 q^{38} + 12 q^{39} + 8 q^{40} + 20 q^{41} - 16 q^{42} - 24 q^{43} + 4 q^{44} - 20 q^{46} - 44 q^{47} - 80 q^{49} - 48 q^{50} - 40 q^{51} - 20 q^{52} - 36 q^{53} - 48 q^{54} - 24 q^{55} + 8 q^{56} - 40 q^{57} + 4 q^{58} - 12 q^{60} - 40 q^{61} - 48 q^{62} - 16 q^{63} - 32 q^{65} - 16 q^{66} + 40 q^{67} - 20 q^{68} - 20 q^{69} + 36 q^{70} + 36 q^{73} + 16 q^{74} + 32 q^{75} - 52 q^{77} + 32 q^{79} + 12 q^{80} + 12 q^{81} - 8 q^{82} - 48 q^{83} - 12 q^{84} + 96 q^{85} - 16 q^{86} + 4 q^{87} - 4 q^{88} - 12 q^{90} - 20 q^{91} + 68 q^{92} + 72 q^{93} + 24 q^{94} + 56 q^{95} - 168 q^{97} - 64 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.987688 + 0.156434i −0.453990 − 0.891007i 0.951057 − 0.309017i −1.68341 − 1.47177i 0.587785 + 0.809017i 4.48144 + 2.28341i −0.891007 + 0.453990i −0.587785 + 0.809017i 1.89292 + 1.19031i
7.2 −0.987688 + 0.156434i −0.453990 − 0.891007i 0.951057 − 0.309017i 1.00148 − 1.99926i 0.587785 + 0.809017i −4.02901 − 2.05288i −0.891007 + 0.453990i −0.587785 + 0.809017i −0.676398 + 2.13131i
7.3 −0.987688 + 0.156434i −0.453990 − 0.891007i 0.951057 − 0.309017i 2.22967 − 0.169023i 0.587785 + 0.809017i 2.21628 + 1.12925i −0.891007 + 0.453990i −0.587785 + 0.809017i −2.17578 + 0.515739i
7.4 0.987688 − 0.156434i 0.453990 + 0.891007i 0.951057 − 0.309017i −1.82626 + 1.29026i 0.587785 + 0.809017i 2.17264 + 1.10702i 0.891007 − 0.453990i −0.587785 + 0.809017i −1.60193 + 1.56007i
7.5 0.987688 − 0.156434i 0.453990 + 0.891007i 0.951057 − 0.309017i 1.47176 + 1.68342i 0.587785 + 0.809017i 0.0261104 + 0.0133039i 0.891007 − 0.453990i −0.587785 + 0.809017i 1.71699 + 1.43246i
7.6 0.987688 − 0.156434i 0.453990 + 0.891007i 0.951057 − 0.309017i 1.70887 − 1.44214i 0.587785 + 0.809017i −0.631390 − 0.321709i 0.891007 − 0.453990i −0.587785 + 0.809017i 1.46223 − 1.69171i
13.1 −0.891007 + 0.453990i 0.987688 + 0.156434i 0.587785 − 0.809017i −1.74269 + 1.40108i −0.951057 + 0.309017i −0.171084 − 1.08018i −0.156434 + 0.987688i 0.951057 + 0.309017i 0.916674 − 2.03954i
13.2 −0.891007 + 0.453990i 0.987688 + 0.156434i 0.587785 − 0.809017i 1.39507 + 1.74751i −0.951057 + 0.309017i 0.526761 + 3.32584i −0.156434 + 0.987688i 0.951057 + 0.309017i −2.03637 − 0.923695i
13.3 −0.891007 + 0.453990i 0.987688 + 0.156434i 0.587785 − 0.809017i 2.16998 + 0.539605i −0.951057 + 0.309017i −0.726828 − 4.58901i −0.156434 + 0.987688i 0.951057 + 0.309017i −2.17844 + 0.504360i
13.4 0.891007 − 0.453990i −0.987688 − 0.156434i 0.587785 − 0.809017i −1.55378 + 1.60803i −0.951057 + 0.309017i −0.802616 − 5.06752i 0.156434 − 0.987688i 0.951057 + 0.309017i −0.654401 + 2.13817i
13.5 0.891007 − 0.453990i −0.987688 − 0.156434i 0.587785 − 0.809017i −0.169260 + 2.22965i −0.951057 + 0.309017i 0.617156 + 3.89657i 0.156434 − 0.987688i 0.951057 + 0.309017i 0.861430 + 2.06348i
13.6 0.891007 − 0.453990i −0.987688 − 0.156434i 0.587785 − 0.809017i 2.07626 − 0.830158i −0.951057 + 0.309017i 0.320542 + 2.02382i 0.156434 − 0.987688i 0.951057 + 0.309017i 1.47307 − 1.68228i
73.1 −0.156434 − 0.987688i 0.891007 − 0.453990i −0.951057 + 0.309017i −2.23336 + 0.109924i −0.587785 − 0.809017i 1.66666 − 3.27101i 0.453990 + 0.891007i 0.587785 − 0.809017i 0.457946 + 2.18867i
73.2 −0.156434 − 0.987688i 0.891007 − 0.453990i −0.951057 + 0.309017i 0.723916 + 2.11564i −0.587785 − 0.809017i −0.778803 + 1.52849i 0.453990 + 0.891007i 0.587785 − 0.809017i 1.97635 − 1.04596i
73.3 −0.156434 − 0.987688i 0.891007 − 0.453990i −0.951057 + 0.309017i 1.66882 − 1.48830i −0.587785 − 0.809017i 0.949592 − 1.86368i 0.453990 + 0.891007i 0.587785 − 0.809017i −1.73104 − 1.41545i
73.4 0.156434 + 0.987688i −0.891007 + 0.453990i −0.951057 + 0.309017i −2.21854 − 0.279466i −0.587785 − 0.809017i 1.87801 − 3.68580i −0.453990 − 0.891007i 0.587785 − 0.809017i −0.0710300 − 2.23494i
73.5 0.156434 + 0.987688i −0.891007 + 0.453990i −0.951057 + 0.309017i 0.215253 + 2.22568i −0.587785 − 0.809017i 0.0152534 − 0.0299364i −0.453990 − 0.891007i 0.587785 − 0.809017i −2.16461 + 0.560777i
73.6 0.156434 + 0.987688i −0.891007 + 0.453990i −0.951057 + 0.309017i 0.941800 − 2.02806i −0.587785 − 0.809017i 0.505353 − 0.991810i −0.453990 − 0.891007i 0.587785 − 0.809017i 2.15042 + 0.612947i
127.1 −0.891007 − 0.453990i 0.987688 − 0.156434i 0.587785 + 0.809017i −1.74269 − 1.40108i −0.951057 − 0.309017i −0.171084 + 1.08018i −0.156434 − 0.987688i 0.951057 − 0.309017i 0.916674 + 2.03954i
127.2 −0.891007 − 0.453990i 0.987688 − 0.156434i 0.587785 + 0.809017i 1.39507 − 1.74751i −0.951057 − 0.309017i 0.526761 − 3.32584i −0.156434 − 0.987688i 0.951057 − 0.309017i −2.03637 + 0.923695i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.u.a ✓ 48
3.b odd 2 1 990.2.bh.a 48
5.c odd 4 1 330.2.u.b yes 48
11.d odd 10 1 330.2.u.b yes 48
15.e even 4 1 990.2.bh.b 48
33.f even 10 1 990.2.bh.b 48
55.l even 20 1 inner 330.2.u.a ✓ 48
165.u odd 20 1 990.2.bh.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.u.a ✓ 48 1.a even 1 1 trivial
330.2.u.a ✓ 48 55.l even 20 1 inner
330.2.u.b yes 48 5.c odd 4 1
330.2.u.b yes 48 11.d odd 10 1
990.2.bh.a 48 3.b odd 2 1
990.2.bh.a 48 165.u odd 20 1
990.2.bh.b 48 15.e even 4 1
990.2.bh.b 48 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} - 16 T_{7}^{47} + 168 T_{7}^{46} - 1248 T_{7}^{45} + 6953 T_{7}^{44} - 26804 T_{7}^{43} + \cdots + 2823434496 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\). Copy content Toggle raw display