Properties

Label 3528.1.cg.a
Level $3528$
Weight $1$
Character orbit 3528.cg
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.cg (of order \(6\), degree \(2\), 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: \(\Z[a_1, a_2]\)
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 $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2059.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 0 1.00000 −0.500000 0.866025i 0
3235.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 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 \(\Q(\sqrt{-2}) \)
9.c even 3 1 inner
72.p 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.cg.a 2
7.b odd 2 1 72.1.p.a 2
7.c even 3 1 3528.1.ba.a 2
7.c even 3 1 3528.1.ce.b 2
7.d odd 6 1 3528.1.ba.b 2
7.d odd 6 1 3528.1.ce.a 2
8.d odd 2 1 CM 3528.1.cg.a 2
9.c even 3 1 inner 3528.1.cg.a 2
21.c even 2 1 216.1.p.a 2
28.d even 2 1 288.1.t.a 2
35.c odd 2 1 1800.1.bk.d 2
35.f even 4 2 1800.1.ba.b 4
56.e even 2 1 72.1.p.a 2
56.h odd 2 1 288.1.t.a 2
56.k odd 6 1 3528.1.ba.a 2
56.k odd 6 1 3528.1.ce.b 2
56.m even 6 1 3528.1.ba.b 2
56.m even 6 1 3528.1.ce.a 2
63.g even 3 1 3528.1.ce.b 2
63.h even 3 1 3528.1.ba.a 2
63.k odd 6 1 3528.1.ce.a 2
63.l odd 6 1 72.1.p.a 2
63.l odd 6 1 648.1.b.b 1
63.o even 6 1 216.1.p.a 2
63.o even 6 1 648.1.b.a 1
63.t odd 6 1 3528.1.ba.b 2
72.p odd 6 1 inner 3528.1.cg.a 2
84.h odd 2 1 864.1.t.a 2
112.j even 4 2 2304.1.o.c 4
112.l odd 4 2 2304.1.o.c 4
168.e odd 2 1 216.1.p.a 2
168.i even 2 1 864.1.t.a 2
252.s odd 6 1 864.1.t.a 2
252.s odd 6 1 2592.1.b.a 1
252.bi even 6 1 288.1.t.a 2
252.bi even 6 1 2592.1.b.b 1
280.n even 2 1 1800.1.bk.d 2
280.y odd 4 2 1800.1.ba.b 4
315.bg odd 6 1 1800.1.bk.d 2
315.cb even 12 2 1800.1.ba.b 4
504.ba odd 6 1 3528.1.ce.b 2
504.be even 6 1 72.1.p.a 2
504.be even 6 1 648.1.b.b 1
504.bf even 6 1 3528.1.ba.b 2
504.bn odd 6 1 288.1.t.a 2
504.bn odd 6 1 2592.1.b.b 1
504.cc even 6 1 864.1.t.a 2
504.cc even 6 1 2592.1.b.a 1
504.ce odd 6 1 3528.1.ba.a 2
504.co odd 6 1 216.1.p.a 2
504.co odd 6 1 648.1.b.a 1
504.cz even 6 1 3528.1.ce.a 2
1008.dk even 12 2 2304.1.o.c 4
1008.dn odd 12 2 2304.1.o.c 4
2520.dy even 6 1 1800.1.bk.d 2
2520.hw odd 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.b odd 2 1
72.1.p.a 2 56.e even 2 1
72.1.p.a 2 63.l odd 6 1
72.1.p.a 2 504.be even 6 1
216.1.p.a 2 21.c even 2 1
216.1.p.a 2 63.o even 6 1
216.1.p.a 2 168.e odd 2 1
216.1.p.a 2 504.co odd 6 1
288.1.t.a 2 28.d even 2 1
288.1.t.a 2 56.h odd 2 1
288.1.t.a 2 252.bi even 6 1
288.1.t.a 2 504.bn odd 6 1
648.1.b.a 1 63.o even 6 1
648.1.b.a 1 504.co odd 6 1
648.1.b.b 1 63.l odd 6 1
648.1.b.b 1 504.be even 6 1
864.1.t.a 2 84.h odd 2 1
864.1.t.a 2 168.i even 2 1
864.1.t.a 2 252.s odd 6 1
864.1.t.a 2 504.cc even 6 1
1800.1.ba.b 4 35.f even 4 2
1800.1.ba.b 4 280.y odd 4 2
1800.1.ba.b 4 315.cb even 12 2
1800.1.ba.b 4 2520.hw odd 12 2
1800.1.bk.d 2 35.c odd 2 1
1800.1.bk.d 2 280.n even 2 1
1800.1.bk.d 2 315.bg odd 6 1
1800.1.bk.d 2 2520.dy even 6 1
2304.1.o.c 4 112.j even 4 2
2304.1.o.c 4 112.l odd 4 2
2304.1.o.c 4 1008.dk even 12 2
2304.1.o.c 4 1008.dn odd 12 2
2592.1.b.a 1 252.s odd 6 1
2592.1.b.a 1 504.cc even 6 1
2592.1.b.b 1 252.bi even 6 1
2592.1.b.b 1 504.bn odd 6 1
3528.1.ba.a 2 7.c even 3 1
3528.1.ba.a 2 56.k odd 6 1
3528.1.ba.a 2 63.h even 3 1
3528.1.ba.a 2 504.ce odd 6 1
3528.1.ba.b 2 7.d odd 6 1
3528.1.ba.b 2 56.m even 6 1
3528.1.ba.b 2 63.t odd 6 1
3528.1.ba.b 2 504.bf even 6 1
3528.1.ce.a 2 7.d odd 6 1
3528.1.ce.a 2 56.m even 6 1
3528.1.ce.a 2 63.k odd 6 1
3528.1.ce.a 2 504.cz even 6 1
3528.1.ce.b 2 7.c even 3 1
3528.1.ce.b 2 56.k odd 6 1
3528.1.ce.b 2 63.g even 3 1
3528.1.ce.b 2 504.ba odd 6 1
3528.1.cg.a 2 1.a even 1 1 trivial
3528.1.cg.a 2 8.d odd 2 1 CM
3528.1.cg.a 2 9.c even 3 1 inner
3528.1.cg.a 2 72.p 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 \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

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

Hecke characteristic polynomials

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