Properties

Label 3528.1.cg.d
Level $3528$
Weight $1$
Character orbit 3528.cg
Analytic conductor $1.761$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.254016.3

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{5} q^{5} -\zeta_{12}^{3} q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{5} q^{5} -\zeta_{12}^{3} q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{9} - q^{10} -\zeta_{12}^{4} q^{11} + \zeta_{12}^{4} q^{12} -\zeta_{12}^{5} q^{13} + \zeta_{12} q^{15} + \zeta_{12}^{4} q^{16} - q^{17} -\zeta_{12}^{5} q^{18} + q^{19} + \zeta_{12} q^{20} + \zeta_{12}^{5} q^{22} -\zeta_{12}^{5} q^{23} -\zeta_{12}^{5} q^{24} - q^{26} - q^{27} -\zeta_{12} q^{29} -\zeta_{12}^{2} q^{30} -\zeta_{12}^{5} q^{32} + q^{33} + \zeta_{12} q^{34} - q^{36} + \zeta_{12}^{3} q^{37} -\zeta_{12} q^{38} + \zeta_{12} q^{39} -\zeta_{12}^{2} q^{40} + \zeta_{12}^{2} q^{41} -\zeta_{12}^{4} q^{43} + q^{44} + \zeta_{12}^{3} q^{45} - q^{46} - q^{48} -\zeta_{12}^{2} q^{51} + \zeta_{12} q^{52} -\zeta_{12}^{3} q^{53} + \zeta_{12} q^{54} -\zeta_{12}^{3} q^{55} + \zeta_{12}^{2} q^{57} + \zeta_{12}^{2} q^{58} + \zeta_{12}^{3} q^{60} - q^{64} -\zeta_{12}^{4} q^{65} -\zeta_{12} q^{66} -\zeta_{12}^{2} q^{68} + \zeta_{12} q^{69} + \zeta_{12} q^{72} + q^{73} -\zeta_{12}^{4} q^{74} + \zeta_{12}^{2} q^{76} -\zeta_{12}^{2} q^{78} -2 \zeta_{12} q^{79} + \zeta_{12}^{3} q^{80} -\zeta_{12}^{2} q^{81} -\zeta_{12}^{3} q^{82} -\zeta_{12}^{4} q^{83} + \zeta_{12}^{5} q^{85} + \zeta_{12}^{5} q^{86} -\zeta_{12}^{3} q^{87} -\zeta_{12} q^{88} + q^{89} -\zeta_{12}^{4} q^{90} + \zeta_{12} q^{92} -\zeta_{12}^{5} q^{95} + \zeta_{12} q^{96} + \zeta_{12}^{4} q^{97} + \zeta_{12}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 2q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 2q^{4} - 2q^{9} - 4q^{10} + 2q^{11} - 2q^{12} - 2q^{16} - 4q^{17} + 4q^{19} - 4q^{26} - 4q^{27} - 2q^{30} + 4q^{33} - 4q^{36} - 2q^{40} + 2q^{41} + 2q^{43} + 4q^{44} - 4q^{46} - 4q^{48} - 2q^{51} + 2q^{57} + 2q^{58} - 4q^{64} + 2q^{65} - 2q^{68} + 4q^{73} + 2q^{74} + 2q^{76} - 2q^{78} - 2q^{81} + 2q^{83} + 4q^{89} + 2q^{90} - 2q^{97} + 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\) \(-1\) \(-1\)

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 0 1.00000i −0.500000 + 0.866025i −1.00000
2059.2 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0 1.00000i −0.500000 + 0.866025i −1.00000
3235.1 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 0 1.00000i −0.500000 0.866025i −1.00000
3235.2 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0 1.00000i −0.500000 0.866025i −1.00000
\(n\): 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 inner
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.d 4
7.b odd 2 1 3528.1.cg.c 4
7.c even 3 1 504.1.ba.a 4
7.c even 3 1 504.1.ce.a yes 4
7.d odd 6 1 3528.1.ba.d 4
7.d odd 6 1 3528.1.ce.c 4
8.d odd 2 1 inner 3528.1.cg.d 4
9.c even 3 1 inner 3528.1.cg.d 4
21.h odd 6 1 1512.1.ba.a 4
21.h odd 6 1 1512.1.ce.a 4
28.g odd 6 1 2016.1.bi.a 4
28.g odd 6 1 2016.1.cm.a 4
56.e even 2 1 3528.1.cg.c 4
56.k odd 6 1 504.1.ba.a 4
56.k odd 6 1 504.1.ce.a yes 4
56.m even 6 1 3528.1.ba.d 4
56.m even 6 1 3528.1.ce.c 4
56.p even 6 1 2016.1.bi.a 4
56.p even 6 1 2016.1.cm.a 4
63.g even 3 1 504.1.ce.a yes 4
63.h even 3 1 504.1.ba.a 4
63.j odd 6 1 1512.1.ba.a 4
63.k odd 6 1 3528.1.ce.c 4
63.l odd 6 1 3528.1.cg.c 4
63.n odd 6 1 1512.1.ce.a 4
63.t odd 6 1 3528.1.ba.d 4
72.p odd 6 1 inner 3528.1.cg.d 4
168.v even 6 1 1512.1.ba.a 4
168.v even 6 1 1512.1.ce.a 4
252.u odd 6 1 2016.1.bi.a 4
252.bl odd 6 1 2016.1.cm.a 4
504.w even 6 1 2016.1.cm.a 4
504.ba odd 6 1 504.1.ce.a yes 4
504.be even 6 1 3528.1.cg.c 4
504.bf even 6 1 3528.1.ba.d 4
504.bt even 6 1 1512.1.ba.a 4
504.ce odd 6 1 504.1.ba.a 4
504.cq even 6 1 2016.1.bi.a 4
504.cy even 6 1 1512.1.ce.a 4
504.cz even 6 1 3528.1.ce.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.ba.a 4 7.c even 3 1
504.1.ba.a 4 56.k odd 6 1
504.1.ba.a 4 63.h even 3 1
504.1.ba.a 4 504.ce odd 6 1
504.1.ce.a yes 4 7.c even 3 1
504.1.ce.a yes 4 56.k odd 6 1
504.1.ce.a yes 4 63.g even 3 1
504.1.ce.a yes 4 504.ba odd 6 1
1512.1.ba.a 4 21.h odd 6 1
1512.1.ba.a 4 63.j odd 6 1
1512.1.ba.a 4 168.v even 6 1
1512.1.ba.a 4 504.bt even 6 1
1512.1.ce.a 4 21.h odd 6 1
1512.1.ce.a 4 63.n odd 6 1
1512.1.ce.a 4 168.v even 6 1
1512.1.ce.a 4 504.cy even 6 1
2016.1.bi.a 4 28.g odd 6 1
2016.1.bi.a 4 56.p even 6 1
2016.1.bi.a 4 252.u odd 6 1
2016.1.bi.a 4 504.cq even 6 1
2016.1.cm.a 4 28.g odd 6 1
2016.1.cm.a 4 56.p even 6 1
2016.1.cm.a 4 252.bl odd 6 1
2016.1.cm.a 4 504.w even 6 1
3528.1.ba.d 4 7.d odd 6 1
3528.1.ba.d 4 56.m even 6 1
3528.1.ba.d 4 63.t odd 6 1
3528.1.ba.d 4 504.bf even 6 1
3528.1.ce.c 4 7.d odd 6 1
3528.1.ce.c 4 56.m even 6 1
3528.1.ce.c 4 63.k odd 6 1
3528.1.ce.c 4 504.cz even 6 1
3528.1.cg.c 4 7.b odd 2 1
3528.1.cg.c 4 56.e even 2 1
3528.1.cg.c 4 63.l odd 6 1
3528.1.cg.c 4 504.be even 6 1
3528.1.cg.d 4 1.a even 1 1 trivial
3528.1.cg.d 4 8.d odd 2 1 inner
3528.1.cg.d 4 9.c even 3 1 inner
3528.1.cg.d 4 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 \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5}^{4} - T_{5}^{2} + 1 \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( 1 - T^{2} + T^{4} \)
$17$ \( ( 1 + T )^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( 1 - T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( ( 1 - T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -1 + T )^{4} \)
$79$ \( 16 - 4 T^{2} + T^{4} \)
$83$ \( ( 1 - T + T^{2} )^{2} \)
$89$ \( ( -1 + T )^{4} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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