Properties

Label 3528.1.ce.a
Level $3528$
Weight $1$
Character orbit 3528.ce
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.ce (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.648.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.99574272.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{2} + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + q^{2} + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{6} + q^{8} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{12} + q^{16} + \zeta_{6} q^{17} -\zeta_{6} q^{18} -\zeta_{6}^{2} q^{19} -\zeta_{6}^{2} q^{22} + \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} + q^{32} + \zeta_{6} q^{33} + \zeta_{6} q^{34} -\zeta_{6} q^{36} -\zeta_{6}^{2} q^{38} -\zeta_{6}^{2} q^{41} + \zeta_{6} q^{43} -\zeta_{6}^{2} q^{44} + \zeta_{6}^{2} q^{48} + \zeta_{6}^{2} q^{50} - q^{51} + q^{54} + \zeta_{6} q^{57} - q^{59} + q^{64} + \zeta_{6} q^{66} - q^{67} + \zeta_{6} q^{68} -\zeta_{6} q^{72} + \zeta_{6} q^{73} -\zeta_{6} q^{75} -\zeta_{6}^{2} q^{76} + \zeta_{6}^{2} q^{81} -\zeta_{6}^{2} q^{82} -2 \zeta_{6} q^{83} + \zeta_{6} q^{86} -\zeta_{6}^{2} q^{88} + 2 \zeta_{6}^{2} q^{89} + \zeta_{6}^{2} q^{96} + \zeta_{6} q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{6} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{6} + 2q^{8} - q^{9} + q^{11} - q^{12} + 2q^{16} + q^{17} - q^{18} + q^{19} + q^{22} - q^{24} - q^{25} + 2q^{27} + 2q^{32} + q^{33} + q^{34} - q^{36} + q^{38} + q^{41} + q^{43} + q^{44} - q^{48} - q^{50} - 2q^{51} + 2q^{54} + q^{57} - 2q^{59} + 2q^{64} + q^{66} - 2q^{67} + q^{68} - q^{72} + q^{73} - q^{75} + q^{76} - q^{81} + q^{82} - 2q^{83} + q^{86} + q^{88} - 2q^{89} - q^{96} + q^{97} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(\zeta_{6}^{2}\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2419.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i 0 1.00000 −0.500000 + 0.866025i 0
3019.1 1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i 0 1.00000 −0.500000 0.866025i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
63.h even 3 1 inner
504.ce odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.ce.a 2
7.b odd 2 1 3528.1.ce.b 2
7.c even 3 1 72.1.p.a 2
7.c even 3 1 3528.1.ba.b 2
7.d odd 6 1 3528.1.ba.a 2
7.d odd 6 1 3528.1.cg.a 2
8.d odd 2 1 CM 3528.1.ce.a 2
9.c even 3 1 3528.1.ba.b 2
21.h odd 6 1 216.1.p.a 2
28.g odd 6 1 288.1.t.a 2
35.j even 6 1 1800.1.bk.d 2
35.l odd 12 2 1800.1.ba.b 4
56.e even 2 1 3528.1.ce.b 2
56.k odd 6 1 72.1.p.a 2
56.k odd 6 1 3528.1.ba.b 2
56.m even 6 1 3528.1.ba.a 2
56.m even 6 1 3528.1.cg.a 2
56.p even 6 1 288.1.t.a 2
63.g even 3 1 72.1.p.a 2
63.h even 3 1 648.1.b.b 1
63.h even 3 1 inner 3528.1.ce.a 2
63.j odd 6 1 648.1.b.a 1
63.k odd 6 1 3528.1.cg.a 2
63.l odd 6 1 3528.1.ba.a 2
63.n odd 6 1 216.1.p.a 2
63.t odd 6 1 3528.1.ce.b 2
72.p odd 6 1 3528.1.ba.b 2
84.n even 6 1 864.1.t.a 2
112.u odd 12 2 2304.1.o.c 4
112.w even 12 2 2304.1.o.c 4
168.s odd 6 1 864.1.t.a 2
168.v even 6 1 216.1.p.a 2
252.o even 6 1 864.1.t.a 2
252.u odd 6 1 2592.1.b.b 1
252.bb even 6 1 2592.1.b.a 1
252.bl odd 6 1 288.1.t.a 2
280.bi odd 6 1 1800.1.bk.d 2
280.br even 12 2 1800.1.ba.b 4
315.bo even 6 1 1800.1.bk.d 2
315.ch odd 12 2 1800.1.ba.b 4
504.w even 6 1 288.1.t.a 2
504.ba odd 6 1 72.1.p.a 2
504.be even 6 1 3528.1.ba.a 2
504.bf even 6 1 3528.1.ce.b 2
504.bi odd 6 1 2592.1.b.a 1
504.bt even 6 1 648.1.b.a 1
504.ce odd 6 1 648.1.b.b 1
504.ce odd 6 1 inner 3528.1.ce.a 2
504.cq even 6 1 2592.1.b.b 1
504.cy even 6 1 216.1.p.a 2
504.cz even 6 1 3528.1.cg.a 2
504.db odd 6 1 864.1.t.a 2
1008.do even 12 2 2304.1.o.c 4
1008.dz odd 12 2 2304.1.o.c 4
2520.gj odd 6 1 1800.1.bk.d 2
2520.jd even 12 2 1800.1.ba.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 7.c even 3 1
72.1.p.a 2 56.k odd 6 1
72.1.p.a 2 63.g even 3 1
72.1.p.a 2 504.ba odd 6 1
216.1.p.a 2 21.h odd 6 1
216.1.p.a 2 63.n odd 6 1
216.1.p.a 2 168.v even 6 1
216.1.p.a 2 504.cy even 6 1
288.1.t.a 2 28.g odd 6 1
288.1.t.a 2 56.p even 6 1
288.1.t.a 2 252.bl odd 6 1
288.1.t.a 2 504.w even 6 1
648.1.b.a 1 63.j odd 6 1
648.1.b.a 1 504.bt even 6 1
648.1.b.b 1 63.h even 3 1
648.1.b.b 1 504.ce odd 6 1
864.1.t.a 2 84.n even 6 1
864.1.t.a 2 168.s odd 6 1
864.1.t.a 2 252.o even 6 1
864.1.t.a 2 504.db odd 6 1
1800.1.ba.b 4 35.l odd 12 2
1800.1.ba.b 4 280.br even 12 2
1800.1.ba.b 4 315.ch odd 12 2
1800.1.ba.b 4 2520.jd even 12 2
1800.1.bk.d 2 35.j even 6 1
1800.1.bk.d 2 280.bi odd 6 1
1800.1.bk.d 2 315.bo even 6 1
1800.1.bk.d 2 2520.gj odd 6 1
2304.1.o.c 4 112.u odd 12 2
2304.1.o.c 4 112.w even 12 2
2304.1.o.c 4 1008.do even 12 2
2304.1.o.c 4 1008.dz odd 12 2
2592.1.b.a 1 252.bb even 6 1
2592.1.b.a 1 504.bi odd 6 1
2592.1.b.b 1 252.u odd 6 1
2592.1.b.b 1 504.cq even 6 1
3528.1.ba.a 2 7.d odd 6 1
3528.1.ba.a 2 56.m even 6 1
3528.1.ba.a 2 63.l odd 6 1
3528.1.ba.a 2 504.be even 6 1
3528.1.ba.b 2 7.c even 3 1
3528.1.ba.b 2 9.c even 3 1
3528.1.ba.b 2 56.k odd 6 1
3528.1.ba.b 2 72.p odd 6 1
3528.1.ce.a 2 1.a even 1 1 trivial
3528.1.ce.a 2 8.d odd 2 1 CM
3528.1.ce.a 2 63.h even 3 1 inner
3528.1.ce.a 2 504.ce odd 6 1 inner
3528.1.ce.b 2 7.b odd 2 1
3528.1.ce.b 2 56.e even 2 1
3528.1.ce.b 2 63.t odd 6 1
3528.1.ce.b 2 504.bf even 6 1
3528.1.cg.a 2 7.d odd 6 1
3528.1.cg.a 2 56.m even 6 1
3528.1.cg.a 2 63.k odd 6 1
3528.1.cg.a 2 504.cz even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{17}^{2} - T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 1 - T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( 1 - T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 1 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 1 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 4 + 2 T + T^{2} \)
$89$ \( 4 + 2 T + T^{2} \)
$97$ \( 1 - T + T^{2} \)
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