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Properties

Label 228.3.l.c
Level $228$
Weight $3$
Character orbit 228.l
Analytic conductor $6.213$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [228,3,Mod(145,228)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,2,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.21255002741\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + 11 q^{7} + 3 \zeta_{6} q^{9} - 8 q^{11} + (3 \zeta_{6} - 6) q^{13} + ( - 2 \zeta_{6} + 4) q^{15} + ( - 26 \zeta_{6} + 26) q^{17} + 19 q^{19} + (11 \zeta_{6} + 11) q^{21}+ \cdots - 24 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 2 q^{5} + 22 q^{7} + 3 q^{9} - 16 q^{11} - 9 q^{13} + 6 q^{15} + 26 q^{17} + 38 q^{19} + 33 q^{21} + 32 q^{23} + 21 q^{25} - 42 q^{29} - 24 q^{33} + 22 q^{35} - 18 q^{39} - 24 q^{41} - 47 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 11.0000 0 1.50000 − 2.59808i 0
217.1 0 1.50000 + 0.866025i 0 1.00000 − 1.73205i 0 11.0000 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.c ✓ 2
3.b odd 2 1 684.3.y.b 2
4.b odd 2 1 912.3.be.b 2
19.d odd 6 1 inner 228.3.l.c ✓ 2
57.f even 6 1 684.3.y.b 2
76.f even 6 1 912.3.be.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.c ✓ 2 1.a even 1 1 trivial
228.3.l.c ✓ 2 19.d odd 6 1 inner
684.3.y.b 2 3.b odd 2 1
684.3.y.b 2 57.f even 6 1
912.3.be.b 2 4.b odd 2 1
912.3.be.b 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
\( T_{7} - 11 \) 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
$7$ \( (T - 11)^{2} \) Copy content Toggle raw display
$11$ \( (T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$17$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 32T + 1024 \) Copy content Toggle raw display
$29$ \( T^{2} + 42T + 588 \) Copy content Toggle raw display
$31$ \( T^{2} + 2883 \) Copy content Toggle raw display
$37$ \( T^{2} + 2187 \) Copy content Toggle raw display
$41$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$43$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$47$ \( T^{2} + 70T + 4900 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$59$ \( T^{2} + 186T + 11532 \) Copy content Toggle raw display
$61$ \( T^{2} + 35T + 1225 \) Copy content Toggle raw display
$67$ \( T^{2} - 27T + 243 \) Copy content Toggle raw display
$71$ \( T^{2} + 228T + 17328 \) Copy content Toggle raw display
$73$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$79$ \( T^{2} + 45T + 675 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 138T + 6348 \) Copy content Toggle raw display
$97$ \( T^{2} + 36T + 432 \) Copy content Toggle raw display
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