Properties

Label 228.3.j.b
Level $228$
Weight $3$
Character orbit 228.j
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(7,228)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.j (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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \zeta_{6} + 2) q^{3} + 4 q^{4} - 4 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 4) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + 8 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \zeta_{6} + 2) q^{3} + 4 q^{4} - 4 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 4) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + 8 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} - 8 \zeta_{6} q^{10} + ( - 4 \zeta_{6} + 2) q^{11} + ( - 4 \zeta_{6} + 8) q^{12} + ( - 7 \zeta_{6} + 7) q^{13} + ( - 4 \zeta_{6} + 2) q^{14} + ( - 4 \zeta_{6} - 4) q^{15} + 16 q^{16} + 32 \zeta_{6} q^{17} + ( - 6 \zeta_{6} + 6) q^{18} - 19 q^{19} - 16 \zeta_{6} q^{20} - 3 \zeta_{6} q^{21} + ( - 8 \zeta_{6} + 4) q^{22} + ( - 4 \zeta_{6} - 4) q^{23} + ( - 8 \zeta_{6} + 16) q^{24} + ( - 9 \zeta_{6} + 9) q^{25} + ( - 14 \zeta_{6} + 14) q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 8 \zeta_{6} + 4) q^{28} + (52 \zeta_{6} - 52) q^{29} + ( - 8 \zeta_{6} - 8) q^{30} + (14 \zeta_{6} - 7) q^{31} + 32 q^{32} - 6 \zeta_{6} q^{33} + 64 \zeta_{6} q^{34} + (4 \zeta_{6} - 8) q^{35} + ( - 12 \zeta_{6} + 12) q^{36} - q^{37} - 38 q^{38} + ( - 14 \zeta_{6} + 7) q^{39} - 32 \zeta_{6} q^{40} + 44 \zeta_{6} q^{41} - 6 \zeta_{6} q^{42} + (23 \zeta_{6} - 46) q^{43} + ( - 16 \zeta_{6} + 8) q^{44} - 12 q^{45} + ( - 8 \zeta_{6} - 8) q^{46} + ( - 30 \zeta_{6} - 30) q^{47} + ( - 16 \zeta_{6} + 32) q^{48} + 46 q^{49} + ( - 18 \zeta_{6} + 18) q^{50} + (32 \zeta_{6} + 32) q^{51} + ( - 28 \zeta_{6} + 28) q^{52} + ( - 44 \zeta_{6} + 44) q^{53} + ( - 12 \zeta_{6} + 6) q^{54} + (8 \zeta_{6} - 16) q^{55} + ( - 16 \zeta_{6} + 8) q^{56} + (19 \zeta_{6} - 38) q^{57} + (104 \zeta_{6} - 104) q^{58} + (18 \zeta_{6} - 36) q^{59} + ( - 16 \zeta_{6} - 16) q^{60} + (11 \zeta_{6} - 11) q^{61} + (28 \zeta_{6} - 14) q^{62} + ( - 3 \zeta_{6} - 3) q^{63} + 64 q^{64} - 28 q^{65} - 12 \zeta_{6} q^{66} + ( - 29 \zeta_{6} - 29) q^{67} + 128 \zeta_{6} q^{68} - 12 q^{69} + (8 \zeta_{6} - 16) q^{70} + (26 \zeta_{6} - 52) q^{71} + ( - 24 \zeta_{6} + 24) q^{72} - 5 \zeta_{6} q^{73} - 2 q^{74} + ( - 18 \zeta_{6} + 9) q^{75} - 76 q^{76} - 6 q^{77} + ( - 28 \zeta_{6} + 14) q^{78} + ( - 43 \zeta_{6} + 86) q^{79} - 64 \zeta_{6} q^{80} - 9 \zeta_{6} q^{81} + 88 \zeta_{6} q^{82} + (152 \zeta_{6} - 76) q^{83} - 12 \zeta_{6} q^{84} + ( - 128 \zeta_{6} + 128) q^{85} + (46 \zeta_{6} - 92) q^{86} + (104 \zeta_{6} - 52) q^{87} + ( - 32 \zeta_{6} + 16) q^{88} + (130 \zeta_{6} - 130) q^{89} - 24 q^{90} + ( - 7 \zeta_{6} - 7) q^{91} + ( - 16 \zeta_{6} - 16) q^{92} + 21 \zeta_{6} q^{93} + ( - 60 \zeta_{6} - 60) q^{94} + 76 \zeta_{6} q^{95} + ( - 32 \zeta_{6} + 64) q^{96} - 74 \zeta_{6} q^{97} + 92 q^{98} + ( - 6 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} - 4 q^{5} + 6 q^{6} + 16 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} - 4 q^{5} + 6 q^{6} + 16 q^{8} + 3 q^{9} - 8 q^{10} + 12 q^{12} + 7 q^{13} - 12 q^{15} + 32 q^{16} + 32 q^{17} + 6 q^{18} - 38 q^{19} - 16 q^{20} - 3 q^{21} - 12 q^{23} + 24 q^{24} + 9 q^{25} + 14 q^{26} - 52 q^{29} - 24 q^{30} + 64 q^{32} - 6 q^{33} + 64 q^{34} - 12 q^{35} + 12 q^{36} - 2 q^{37} - 76 q^{38} - 32 q^{40} + 44 q^{41} - 6 q^{42} - 69 q^{43} - 24 q^{45} - 24 q^{46} - 90 q^{47} + 48 q^{48} + 92 q^{49} + 18 q^{50} + 96 q^{51} + 28 q^{52} + 44 q^{53} - 24 q^{55} - 57 q^{57} - 104 q^{58} - 54 q^{59} - 48 q^{60} - 11 q^{61} - 9 q^{63} + 128 q^{64} - 56 q^{65} - 12 q^{66} - 87 q^{67} + 128 q^{68} - 24 q^{69} - 24 q^{70} - 78 q^{71} + 24 q^{72} - 5 q^{73} - 4 q^{74} - 152 q^{76} - 12 q^{77} + 129 q^{79} - 64 q^{80} - 9 q^{81} + 88 q^{82} - 12 q^{84} + 128 q^{85} - 138 q^{86} - 130 q^{89} - 48 q^{90} - 21 q^{91} - 48 q^{92} + 21 q^{93} - 180 q^{94} + 76 q^{95} + 96 q^{96} - 74 q^{97} + 184 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 + 0.866025i
0.500000 0.866025i
2.00000 1.50000 0.866025i 4.00000 −2.00000 3.46410i 3.00000 1.73205i 1.73205i 8.00000 1.50000 2.59808i −4.00000 6.92820i
163.1 2.00000 1.50000 + 0.866025i 4.00000 −2.00000 + 3.46410i 3.00000 + 1.73205i 1.73205i 8.00000 1.50000 + 2.59808i −4.00000 + 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g 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.j.b yes 2
4.b odd 2 1 228.3.j.a 2
19.c even 3 1 228.3.j.a 2
76.g odd 6 1 inner 228.3.j.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.j.a 2 4.b odd 2 1
228.3.j.a 2 19.c even 3 1
228.3.j.b yes 2 1.a even 1 1 trivial
228.3.j.b yes 2 76.g odd 6 1 inner

Hecke kernels

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

\( T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{23}^{2} + 12T_{23} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$17$ \( T^{2} - 32T + 1024 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$29$ \( T^{2} + 52T + 2704 \) Copy content Toggle raw display
$31$ \( T^{2} + 147 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 44T + 1936 \) Copy content Toggle raw display
$43$ \( T^{2} + 69T + 1587 \) Copy content Toggle raw display
$47$ \( T^{2} + 90T + 2700 \) Copy content Toggle raw display
$53$ \( T^{2} - 44T + 1936 \) Copy content Toggle raw display
$59$ \( T^{2} + 54T + 972 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} + 87T + 2523 \) Copy content Toggle raw display
$71$ \( T^{2} + 78T + 2028 \) Copy content Toggle raw display
$73$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$79$ \( T^{2} - 129T + 5547 \) Copy content Toggle raw display
$83$ \( T^{2} + 17328 \) Copy content Toggle raw display
$89$ \( T^{2} + 130T + 16900 \) Copy content Toggle raw display
$97$ \( T^{2} + 74T + 5476 \) Copy content Toggle raw display
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