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Properties

Label 228.3.l.b
Level $228$
Weight $3$
Character orbit 228.l
Analytic conductor $6.213$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [228,3,Mod(145,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.l (of order \(6\), degree \(2\), 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: \(6.21255002741\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9} + 16 q^{11} + ( - 9 \zeta_{6} + 18) q^{13} + ( - 2 \zeta_{6} + 4) q^{15} + (22 \zeta_{6} - 22) q^{17} + 19 q^{19} + ( - \zeta_{6} - 1) q^{21} - 40 \zeta_{6} q^{23} + 21 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{27} + ( - 10 \zeta_{6} + 20) q^{29} + (58 \zeta_{6} - 29) q^{31} + (16 \zeta_{6} + 16) q^{33} + (2 \zeta_{6} - 2) q^{35} + ( - 18 \zeta_{6} + 9) q^{37} + 27 q^{39} + ( - 8 \zeta_{6} - 8) q^{41} + ( - 49 \zeta_{6} + 49) q^{43} + 6 q^{45} - 46 \zeta_{6} q^{47} - 48 q^{49} + (22 \zeta_{6} - 44) q^{51} + (28 \zeta_{6} - 56) q^{53} + ( - 32 \zeta_{6} + 32) q^{55} + (19 \zeta_{6} + 19) q^{57} + ( - 38 \zeta_{6} - 38) q^{59} + 97 \zeta_{6} q^{61} - 3 \zeta_{6} q^{63} + ( - 36 \zeta_{6} + 18) q^{65} + (15 \zeta_{6} - 30) q^{67} + ( - 80 \zeta_{6} + 40) q^{69} + ( - 28 \zeta_{6} - 28) q^{71} + (35 \zeta_{6} - 35) q^{73} + (42 \zeta_{6} - 21) q^{75} - 16 q^{77} + ( - 51 \zeta_{6} - 51) q^{79} + (9 \zeta_{6} - 9) q^{81} - 146 q^{83} + 44 \zeta_{6} q^{85} + 30 q^{87} + (22 \zeta_{6} - 44) q^{89} + (9 \zeta_{6} - 18) q^{91} + (87 \zeta_{6} - 87) q^{93} + ( - 38 \zeta_{6} + 38) q^{95} + (36 \zeta_{6} + 36) q^{97} + 48 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 32 q^{11} + 27 q^{13} + 6 q^{15} - 22 q^{17} + 38 q^{19} - 3 q^{21} - 40 q^{23} + 21 q^{25} + 30 q^{29} + 48 q^{33} - 2 q^{35} + 54 q^{39} - 24 q^{41} + 49 q^{43} + 12 q^{45} - 46 q^{47} - 96 q^{49} - 66 q^{51} - 84 q^{53} + 32 q^{55} + 57 q^{57} - 114 q^{59} + 97 q^{61} - 3 q^{63} - 45 q^{67} - 84 q^{71} - 35 q^{73} - 32 q^{77} - 153 q^{79} - 9 q^{81} - 292 q^{83} + 44 q^{85} + 60 q^{87} - 66 q^{89} - 27 q^{91} - 87 q^{93} + 38 q^{95} + 108 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\) \(115\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 1.50000 − 0.866025i 0 1.00000 + 1.73205i 0 −1.00000 0 1.50000 − 2.59808i 0
217.1 0 1.50000 + 0.866025i 0 1.00000 − 1.73205i 0 −1.00000 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.3.l.b ✓ 2
3.b odd 2 1 684.3.y.a 2
4.b odd 2 1 912.3.be.c 2
19.d odd 6 1 inner 228.3.l.b ✓ 2
57.f even 6 1 684.3.y.a 2
76.f even 6 1 912.3.be.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.b ✓ 2 1.a even 1 1 trivial
228.3.l.b ✓ 2 19.d odd 6 1 inner
684.3.y.a 2 3.b odd 2 1
684.3.y.a 2 57.f even 6 1
912.3.be.c 2 4.b odd 2 1
912.3.be.c 2 76.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\):

\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 27T + 243 \) Copy content Toggle raw display
$17$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$29$ \( T^{2} - 30T + 300 \) Copy content Toggle raw display
$31$ \( T^{2} + 2523 \) Copy content Toggle raw display
$37$ \( T^{2} + 243 \) Copy content Toggle raw display
$41$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$43$ \( T^{2} - 49T + 2401 \) Copy content Toggle raw display
$47$ \( T^{2} + 46T + 2116 \) Copy content Toggle raw display
$53$ \( T^{2} + 84T + 2352 \) Copy content Toggle raw display
$59$ \( T^{2} + 114T + 4332 \) Copy content Toggle raw display
$61$ \( T^{2} - 97T + 9409 \) Copy content Toggle raw display
$67$ \( T^{2} + 45T + 675 \) Copy content Toggle raw display
$71$ \( T^{2} + 84T + 2352 \) Copy content Toggle raw display
$73$ \( T^{2} + 35T + 1225 \) Copy content Toggle raw display
$79$ \( T^{2} + 153T + 7803 \) Copy content Toggle raw display
$83$ \( (T + 146)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 66T + 1452 \) Copy content Toggle raw display
$97$ \( T^{2} - 108T + 3888 \) Copy content Toggle raw display
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