# Properties

 Label 3528.1.dh.a Level $3528$ Weight $1$ Character orbit 3528.dh Analytic conductor $1.761$ Analytic rank $0$ Dimension $12$ Projective image $D_{14}$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3528.dh (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: $$\Q(\zeta_{28})$$ Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{14}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{14} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{28} q^{2} + \zeta_{28}^{2} q^{4} + ( \zeta_{28}^{9} + \zeta_{28}^{11} ) q^{5} + \zeta_{28}^{4} q^{7} + \zeta_{28}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{28} q^{2} + \zeta_{28}^{2} q^{4} + ( \zeta_{28}^{9} + \zeta_{28}^{11} ) q^{5} + \zeta_{28}^{4} q^{7} + \zeta_{28}^{3} q^{8} + ( \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{10} + ( -\zeta_{28}^{3} + \zeta_{28}^{13} ) q^{11} + \zeta_{28}^{5} q^{14} + \zeta_{28}^{4} q^{16} + ( \zeta_{28}^{11} + \zeta_{28}^{13} ) q^{20} + ( -1 - \zeta_{28}^{4} ) q^{22} + ( -\zeta_{28}^{4} - \zeta_{28}^{6} - \zeta_{28}^{8} ) q^{25} + \zeta_{28}^{6} q^{28} + ( \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{29} + ( \zeta_{28}^{4} + \zeta_{28}^{10} ) q^{31} + \zeta_{28}^{5} q^{32} + ( -\zeta_{28} + \zeta_{28}^{13} ) q^{35} + ( -1 + \zeta_{28}^{12} ) q^{40} + ( -\zeta_{28} - \zeta_{28}^{5} ) q^{44} + \zeta_{28}^{8} q^{49} + ( -\zeta_{28}^{5} - \zeta_{28}^{7} - \zeta_{28}^{9} ) q^{50} -2 \zeta_{28}^{9} q^{53} + ( 1 - \zeta_{28}^{8} - \zeta_{28}^{10} - \zeta_{28}^{12} ) q^{55} + \zeta_{28}^{7} q^{56} + ( 1 + \zeta_{28}^{12} ) q^{58} + ( \zeta_{28}^{5} + \zeta_{28}^{11} ) q^{62} + \zeta_{28}^{6} q^{64} + ( -1 - \zeta_{28}^{2} ) q^{70} + ( -\zeta_{28}^{2} + \zeta_{28}^{10} ) q^{73} + ( -\zeta_{28}^{3} - \zeta_{28}^{7} ) q^{77} + ( -\zeta_{28}^{4} + \zeta_{28}^{10} ) q^{79} + ( -\zeta_{28} + \zeta_{28}^{13} ) q^{80} + ( \zeta_{28}^{5} + \zeta_{28}^{7} ) q^{83} + ( -\zeta_{28}^{2} - \zeta_{28}^{6} ) q^{88} + ( -\zeta_{28}^{2} - \zeta_{28}^{12} ) q^{97} + \zeta_{28}^{9} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 2q^{4} - 2q^{7} + O(q^{10})$$ $$12q + 2q^{4} - 2q^{7} - 2q^{16} - 10q^{22} + 2q^{25} + 2q^{28} - 14q^{40} - 2q^{49} + 14q^{55} + 10q^{58} + 2q^{64} - 14q^{70} + 4q^{79} - 4q^{88} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$\zeta_{28}^{6}$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −0.974928 − 0.222521i 0.974928 + 0.222521i −0.974928 + 0.222521i 0.974928 − 0.222521i −0.433884 − 0.900969i 0.433884 + 0.900969i −0.781831 + 0.623490i 0.781831 − 0.623490i −0.781831 − 0.623490i 0.781831 + 0.623490i −0.433884 + 0.900969i 0.433884 − 0.900969i
−0.974928 0.222521i 0 0.900969 + 0.433884i 1.21572 1.52446i 0 0.623490 + 0.781831i −0.781831 0.623490i 0 −1.52446 + 1.21572i
181.2 0.974928 + 0.222521i 0 0.900969 + 0.433884i −1.21572 + 1.52446i 0 0.623490 + 0.781831i 0.781831 + 0.623490i 0 −1.52446 + 1.21572i
1189.1 −0.974928 + 0.222521i 0 0.900969 0.433884i 1.21572 + 1.52446i 0 0.623490 0.781831i −0.781831 + 0.623490i 0 −1.52446 1.21572i
1189.2 0.974928 0.222521i 0 0.900969 0.433884i −1.21572 1.52446i 0 0.623490 0.781831i 0.781831 0.623490i 0 −1.52446 1.21572i
1693.1 −0.433884 0.900969i 0 −0.623490 + 0.781831i −0.193096 + 0.846011i 0 −0.222521 0.974928i 0.974928 + 0.222521i 0 0.846011 0.193096i
1693.2 0.433884 + 0.900969i 0 −0.623490 + 0.781831i 0.193096 0.846011i 0 −0.222521 0.974928i −0.974928 0.222521i 0 0.846011 0.193096i
2197.1 −0.781831 + 0.623490i 0 0.222521 0.974928i −1.40881 + 0.678448i 0 −0.900969 0.433884i 0.433884 + 0.900969i 0 0.678448 1.40881i
2197.2 0.781831 0.623490i 0 0.222521 0.974928i 1.40881 0.678448i 0 −0.900969 0.433884i −0.433884 0.900969i 0 0.678448 1.40881i
2701.1 −0.781831 0.623490i 0 0.222521 + 0.974928i −1.40881 0.678448i 0 −0.900969 + 0.433884i 0.433884 0.900969i 0 0.678448 + 1.40881i
2701.2 0.781831 + 0.623490i 0 0.222521 + 0.974928i 1.40881 + 0.678448i 0 −0.900969 + 0.433884i −0.433884 + 0.900969i 0 0.678448 + 1.40881i
3205.1 −0.433884 + 0.900969i 0 −0.623490 0.781831i −0.193096 0.846011i 0 −0.222521 + 0.974928i 0.974928 0.222521i 0 0.846011 + 0.193096i
3205.2 0.433884 0.900969i 0 −0.623490 0.781831i 0.193096 + 0.846011i 0 −0.222521 + 0.974928i −0.974928 + 0.222521i 0 0.846011 + 0.193096i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3205.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
8.b even 2 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner
392.r odd 14 1 inner
1176.bo even 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.dh.a 12
3.b odd 2 1 inner 3528.1.dh.a 12
8.b even 2 1 inner 3528.1.dh.a 12
24.h odd 2 1 CM 3528.1.dh.a 12
49.f odd 14 1 inner 3528.1.dh.a 12
147.k even 14 1 inner 3528.1.dh.a 12
392.r odd 14 1 inner 3528.1.dh.a 12
1176.bo even 14 1 inner 3528.1.dh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.dh.a 12 1.a even 1 1 trivial
3528.1.dh.a 12 3.b odd 2 1 inner
3528.1.dh.a 12 8.b even 2 1 inner
3528.1.dh.a 12 24.h odd 2 1 CM
3528.1.dh.a 12 49.f odd 14 1 inner
3528.1.dh.a 12 147.k even 14 1 inner
3528.1.dh.a 12 392.r odd 14 1 inner
3528.1.dh.a 12 1176.bo even 14 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3528, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$49 + 98 T^{2} + 49 T^{4} - 14 T^{6} + 14 T^{8} + T^{12}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$1 - 2 T^{2} + 25 T^{4} + 6 T^{6} + 2 T^{8} - 4 T^{10} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$1 + 5 T^{2} + 18 T^{4} - 29 T^{6} + 16 T^{8} - 4 T^{10} + T^{12}$$
$31$ $$( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2}$$
$37$ $$T^{12}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$4096 - 1024 T^{2} + 256 T^{4} - 64 T^{6} + 16 T^{8} - 4 T^{10} + T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$( 7 - 7 T + 7 T^{3} + T^{6} )^{2}$$
$79$ $$( 1 - 2 T - T^{2} + T^{3} )^{4}$$
$83$ $$49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12}$$
$89$ $$T^{12}$$
$97$ $$( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2}$$