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Properties

Label 3528.2.s.h
Level $3528$
Weight $2$
Character orbit 3528.s
Analytic conductor $28.171$
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] = [3528,2,Mod(361,3528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3528.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0,0,0,0,0,0,4,0,0,0,-6,0,-4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.1712218331\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{5} + 2 q^{13} + (6 \zeta_{6} - 6) q^{17} - 4 \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 6 q^{29} + (8 \zeta_{6} - 8) q^{31} + 10 \zeta_{6} q^{37} - 10 q^{41} + \cdots - 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{13} - 6 q^{17} - 4 q^{19} - 4 q^{23} + q^{25} - 12 q^{29} - 8 q^{31} + 10 q^{37} - 20 q^{41} + 24 q^{43} + 8 q^{47} + 6 q^{53} - 4 q^{59} - 10 q^{61} - 4 q^{65} - 12 q^{67} - 8 q^{71}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 −1.00000 − 1.73205i 0 0 0 0 0
3313.1 0 0 0 −1.00000 + 1.73205i 0 0 0 0 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.s.h 2
3.b odd 2 1 1176.2.q.j 2
7.b odd 2 1 3528.2.s.v 2
7.c even 3 1 3528.2.a.w 1
7.c even 3 1 inner 3528.2.s.h 2
7.d odd 6 1 504.2.a.b 1
7.d odd 6 1 3528.2.s.v 2
12.b even 2 1 2352.2.q.j 2
21.c even 2 1 1176.2.q.b 2
21.g even 6 1 168.2.a.b ✓ 1
21.g even 6 1 1176.2.q.b 2
21.h odd 6 1 1176.2.a.a 1
21.h odd 6 1 1176.2.q.j 2
28.f even 6 1 1008.2.a.e 1
28.g odd 6 1 7056.2.a.br 1
56.j odd 6 1 4032.2.a.be 1
56.m even 6 1 4032.2.a.bj 1
84.h odd 2 1 2352.2.q.o 2
84.j odd 6 1 336.2.a.c 1
84.j odd 6 1 2352.2.q.o 2
84.n even 6 1 2352.2.a.q 1
84.n even 6 1 2352.2.q.j 2
105.p even 6 1 4200.2.a.i 1
105.w odd 12 2 4200.2.t.m 2
168.s odd 6 1 9408.2.a.cy 1
168.v even 6 1 9408.2.a.bc 1
168.ba even 6 1 1344.2.a.c 1
168.be odd 6 1 1344.2.a.n 1
336.bo even 12 2 5376.2.c.bd 2
336.br odd 12 2 5376.2.c.f 2
420.be odd 6 1 8400.2.a.bx 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b ✓ 1 21.g even 6 1
336.2.a.c 1 84.j odd 6 1
504.2.a.b 1 7.d odd 6 1
1008.2.a.e 1 28.f even 6 1
1176.2.a.a 1 21.h odd 6 1
1176.2.q.b 2 21.c even 2 1
1176.2.q.b 2 21.g even 6 1
1176.2.q.j 2 3.b odd 2 1
1176.2.q.j 2 21.h odd 6 1
1344.2.a.c 1 168.ba even 6 1
1344.2.a.n 1 168.be odd 6 1
2352.2.a.q 1 84.n even 6 1
2352.2.q.j 2 12.b even 2 1
2352.2.q.j 2 84.n even 6 1
2352.2.q.o 2 84.h odd 2 1
2352.2.q.o 2 84.j odd 6 1
3528.2.a.w 1 7.c even 3 1
3528.2.s.h 2 1.a even 1 1 trivial
3528.2.s.h 2 7.c even 3 1 inner
3528.2.s.v 2 7.b odd 2 1
3528.2.s.v 2 7.d odd 6 1
4032.2.a.be 1 56.j odd 6 1
4032.2.a.bj 1 56.m even 6 1
4200.2.a.i 1 105.p even 6 1
4200.2.t.m 2 105.w odd 12 2
5376.2.c.f 2 336.br odd 12 2
5376.2.c.bd 2 336.bo even 12 2
7056.2.a.br 1 28.g odd 6 1
8400.2.a.bx 1 420.be odd 6 1
9408.2.a.bc 1 168.v even 6 1
9408.2.a.cy 1 168.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 4T_{23} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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