# Properties

 Label 3528.1.ba.d Level $3528$ Weight $1$ Character orbit 3528.ba Analytic conductor $1.761$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

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

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-\zeta_{12}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 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 1.00000i 1.00000i 0 1.00000i −0.500000 + 0.866025i −0.500000 + 0.866025i
67.2 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 1.00000i 1.00000i 0 1.00000i −0.500000 + 0.866025i −0.500000 + 0.866025i
1843.1 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 1.00000i 1.00000i 0 1.00000i −0.500000 0.866025i −0.500000 0.866025i
1843.2 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 1.00000i 1.00000i 0 1.00000i −0.500000 0.866025i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
63.g even 3 1 inner
504.ba 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.ba.d 4
7.b odd 2 1 504.1.ba.a 4
7.c even 3 1 3528.1.ce.c 4
7.c even 3 1 3528.1.cg.c 4
7.d odd 6 1 504.1.ce.a yes 4
7.d odd 6 1 3528.1.cg.d 4
8.d odd 2 1 inner 3528.1.ba.d 4
9.c even 3 1 3528.1.ce.c 4
21.c even 2 1 1512.1.ba.a 4
21.g even 6 1 1512.1.ce.a 4
28.d even 2 1 2016.1.bi.a 4
28.f even 6 1 2016.1.cm.a 4
56.e even 2 1 504.1.ba.a 4
56.h odd 2 1 2016.1.bi.a 4
56.j odd 6 1 2016.1.cm.a 4
56.k odd 6 1 3528.1.ce.c 4
56.k odd 6 1 3528.1.cg.c 4
56.m even 6 1 504.1.ce.a yes 4
56.m even 6 1 3528.1.cg.d 4
63.g even 3 1 inner 3528.1.ba.d 4
63.h even 3 1 3528.1.cg.c 4
63.k odd 6 1 504.1.ba.a 4
63.l odd 6 1 504.1.ce.a yes 4
63.o even 6 1 1512.1.ce.a 4
63.s even 6 1 1512.1.ba.a 4
63.t odd 6 1 3528.1.cg.d 4
72.p odd 6 1 3528.1.ce.c 4
168.e odd 2 1 1512.1.ba.a 4
168.be odd 6 1 1512.1.ce.a 4
252.n even 6 1 2016.1.bi.a 4
252.bi even 6 1 2016.1.cm.a 4
504.u odd 6 1 1512.1.ba.a 4
504.ba odd 6 1 inner 3528.1.ba.d 4
504.be even 6 1 504.1.ce.a yes 4
504.bf even 6 1 3528.1.cg.d 4
504.bn odd 6 1 2016.1.cm.a 4
504.ce odd 6 1 3528.1.cg.c 4
504.co odd 6 1 1512.1.ce.a 4
504.cw odd 6 1 2016.1.bi.a 4
504.cz even 6 1 504.1.ba.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.ba.a 4 7.b odd 2 1
504.1.ba.a 4 56.e even 2 1
504.1.ba.a 4 63.k odd 6 1
504.1.ba.a 4 504.cz even 6 1
504.1.ce.a yes 4 7.d odd 6 1
504.1.ce.a yes 4 56.m even 6 1
504.1.ce.a yes 4 63.l odd 6 1
504.1.ce.a yes 4 504.be even 6 1
1512.1.ba.a 4 21.c even 2 1
1512.1.ba.a 4 63.s even 6 1
1512.1.ba.a 4 168.e odd 2 1
1512.1.ba.a 4 504.u odd 6 1
1512.1.ce.a 4 21.g even 6 1
1512.1.ce.a 4 63.o even 6 1
1512.1.ce.a 4 168.be odd 6 1
1512.1.ce.a 4 504.co odd 6 1
2016.1.bi.a 4 28.d even 2 1
2016.1.bi.a 4 56.h odd 2 1
2016.1.bi.a 4 252.n even 6 1
2016.1.bi.a 4 504.cw odd 6 1
2016.1.cm.a 4 28.f even 6 1
2016.1.cm.a 4 56.j odd 6 1
2016.1.cm.a 4 252.bi even 6 1
2016.1.cm.a 4 504.bn odd 6 1
3528.1.ba.d 4 1.a even 1 1 trivial
3528.1.ba.d 4 8.d odd 2 1 inner
3528.1.ba.d 4 63.g even 3 1 inner
3528.1.ba.d 4 504.ba odd 6 1 inner
3528.1.ce.c 4 7.c even 3 1
3528.1.ce.c 4 9.c even 3 1
3528.1.ce.c 4 56.k odd 6 1
3528.1.ce.c 4 72.p odd 6 1
3528.1.cg.c 4 7.c even 3 1
3528.1.cg.c 4 56.k odd 6 1
3528.1.cg.c 4 63.h even 3 1
3528.1.cg.c 4 504.ce odd 6 1
3528.1.cg.d 4 7.d odd 6 1
3528.1.cg.d 4 56.m even 6 1
3528.1.cg.d 4 63.t odd 6 1
3528.1.cg.d 4 504.bf even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 1$$ $$T_{11} + 1$$ $$T_{17}^{2} + T_{17} + 1$$

## Hecke characteristic polynomials

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