# Properties

 Label 3332.1.ce.b Level $3332$ Weight $1$ Character orbit 3332.ce Analytic conductor $1.663$ Analytic rank $0$ Dimension $16$ Projective image $D_{16}$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.ce (of order $$48$$, degree $$16$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + ( - \zeta_{48}^{10} - \zeta_{48}) q^{5} - \zeta_{48}^{3} q^{8} - \zeta_{48}^{7} q^{9} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 + (-z^10 - z) * q^5 - z^3 * q^8 - z^7 * q^9 $$q - \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + ( - \zeta_{48}^{10} - \zeta_{48}) q^{5} - \zeta_{48}^{3} q^{8} - \zeta_{48}^{7} q^{9} + (\zeta_{48}^{11} + \zeta_{48}^{2}) q^{10} + (\zeta_{48}^{12} - 1) q^{13} + \zeta_{48}^{4} q^{16} + \zeta_{48}^{17} q^{17} + \zeta_{48}^{8} q^{18} + ( - \zeta_{48}^{12} - \zeta_{48}^{3}) q^{20} + (\zeta_{48}^{20} + \zeta_{48}^{11} + \zeta_{48}^{2}) q^{25} + ( - \zeta_{48}^{13} + \zeta_{48}) q^{26} + ( - \zeta_{48}^{21} - \zeta_{48}^{6}) q^{29} - \zeta_{48}^{5} q^{32} - \zeta_{48}^{18} q^{34} - \zeta_{48}^{9} q^{36} + (\zeta_{48}^{16} + \zeta_{48}^{7}) q^{37} + (\zeta_{48}^{13} + \zeta_{48}^{4}) q^{40} + (\zeta_{48}^{12} + \zeta_{48}^{9}) q^{41} + (\zeta_{48}^{17} + \zeta_{48}^{8}) q^{45} + ( - \zeta_{48}^{21} - \zeta_{48}^{12} - \zeta_{48}^{3}) q^{50} + (\zeta_{48}^{14} - \zeta_{48}^{2}) q^{52} + ( - \zeta_{48}^{20} - \zeta_{48}^{14}) q^{53} + (\zeta_{48}^{22} + \zeta_{48}^{7}) q^{58} + ( - \zeta_{48}^{4} + \zeta_{48}) q^{61} + \zeta_{48}^{6} q^{64} + ( - \zeta_{48}^{22} - \zeta_{48}^{13} + \zeta_{48}^{10} + \zeta_{48}) q^{65} + \zeta_{48}^{19} q^{68} + \zeta_{48}^{10} q^{72} + ( - \zeta_{48}^{14} + \zeta_{48}^{5}) q^{73} + ( - \zeta_{48}^{17} - \zeta_{48}^{8}) q^{74} + ( - \zeta_{48}^{14} - \zeta_{48}^{5}) q^{80} + \zeta_{48}^{14} q^{81} + ( - \zeta_{48}^{13} - \zeta_{48}^{10}) q^{82} + ( - \zeta_{48}^{18} + \zeta_{48}^{3}) q^{85} - \zeta_{48}^{22} q^{89} + ( - \zeta_{48}^{18} - \zeta_{48}^{9}) q^{90} + (\zeta_{48}^{18} - \zeta_{48}^{9}) q^{97} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 + (-z^10 - z) * q^5 - z^3 * q^8 - z^7 * q^9 + (z^11 + z^2) * q^10 + (z^12 - 1) * q^13 + z^4 * q^16 + z^17 * q^17 + z^8 * q^18 + (-z^12 - z^3) * q^20 + (z^20 + z^11 + z^2) * q^25 + (-z^13 + z) * q^26 + (-z^21 - z^6) * q^29 - z^5 * q^32 - z^18 * q^34 - z^9 * q^36 + (z^16 + z^7) * q^37 + (z^13 + z^4) * q^40 + (z^12 + z^9) * q^41 + (z^17 + z^8) * q^45 + (-z^21 - z^12 - z^3) * q^50 + (z^14 - z^2) * q^52 + (-z^20 - z^14) * q^53 + (z^22 + z^7) * q^58 + (-z^4 + z) * q^61 + z^6 * q^64 + (-z^22 - z^13 + z^10 + z) * q^65 + z^19 * q^68 + z^10 * q^72 + (-z^14 + z^5) * q^73 + (-z^17 - z^8) * q^74 + (-z^14 - z^5) * q^80 + z^14 * q^81 + (-z^13 - z^10) * q^82 + (-z^18 + z^3) * q^85 - z^22 * q^89 + (-z^18 - z^9) * q^90 + (z^18 - z^9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q - 16 q^{13} + 8 q^{18} - 8 q^{37} + 8 q^{45} - 8 q^{74}+O(q^{100})$$ 16 * q - 16 * q^13 + 8 * q^18 - 8 * q^37 + 8 * q^45 - 8 * q^74

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-\zeta_{48}^{15}$$ $$-\zeta_{48}^{16}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.793353 − 0.608761i −0.793353 + 0.608761i 0.130526 + 0.991445i 0.130526 − 0.991445i 0.991445 + 0.130526i −0.991445 + 0.130526i −0.991445 − 0.130526i −0.130526 + 0.991445i 0.793353 − 0.608761i 0.991445 − 0.130526i 0.608761 + 0.793353i −0.130526 − 0.991445i 0.793353 + 0.608761i −0.608761 + 0.793353i −0.608761 − 0.793353i 0.608761 − 0.793353i
0.793353 + 0.608761i 0 0.258819 + 0.965926i −0.172572 + 0.349942i 0 0 −0.382683 + 0.923880i −0.130526 0.991445i −0.349942 + 0.172572i
215.1 0.793353 0.608761i 0 0.258819 0.965926i −0.172572 0.349942i 0 0 −0.382683 0.923880i −0.130526 + 0.991445i −0.349942 0.172572i
227.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i 0.128293 1.95737i 0 0 0.382683 + 0.923880i 0.793353 + 0.608761i −1.95737 + 0.128293i
411.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i 0.128293 + 1.95737i 0 0 0.382683 0.923880i 0.793353 0.608761i −1.95737 0.128293i
607.1 −0.991445 0.130526i 0 0.965926 + 0.258819i −1.25026 1.09645i 0 0 −0.923880 0.382683i −0.608761 0.793353i 1.09645 + 1.25026i
619.1 0.991445 0.130526i 0 0.965926 0.258819i 0.732626 + 0.835400i 0 0 0.923880 0.382683i 0.608761 0.793353i 0.835400 + 0.732626i
1195.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i 0.732626 0.835400i 0 0 0.923880 + 0.382683i 0.608761 + 0.793353i 0.835400 0.732626i
1391.1 0.130526 0.991445i 0 −0.965926 0.258819i 0.389345 0.0255190i 0 0 −0.382683 + 0.923880i −0.793353 + 0.608761i 0.0255190 0.389345i
1587.1 −0.793353 + 0.608761i 0 0.258819 0.965926i −1.75928 + 0.867580i 0 0 0.382683 + 0.923880i 0.130526 0.991445i 0.867580 1.75928i
1795.1 −0.991445 + 0.130526i 0 0.965926 0.258819i −1.25026 + 1.09645i 0 0 −0.923880 + 0.382683i −0.608761 + 0.793353i 1.09645 1.25026i
1979.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i 0.357164 1.05217i 0 0 0.923880 0.382683i −0.991445 0.130526i −1.05217 + 0.357164i
2187.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i 0.389345 + 0.0255190i 0 0 −0.382683 0.923880i −0.793353 0.608761i 0.0255190 + 0.389345i
2383.1 −0.793353 0.608761i 0 0.258819 + 0.965926i −1.75928 0.867580i 0 0 0.382683 0.923880i 0.130526 + 0.991445i 0.867580 + 1.75928i
2579.1 0.608761 0.793353i 0 −0.258819 0.965926i 1.57469 0.534534i 0 0 −0.923880 0.382683i 0.991445 0.130526i 0.534534 1.57469i
3155.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i 1.57469 + 0.534534i 0 0 −0.923880 + 0.382683i 0.991445 + 0.130526i 0.534534 + 1.57469i
3167.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i 0.357164 + 1.05217i 0 0 0.923880 + 0.382683i −0.991445 + 0.130526i −1.05217 0.357164i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3167.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.c even 3 1 inner
28.g odd 6 1 inner
119.p even 16 1 inner
119.s even 48 1 inner
476.bf odd 16 1 inner
476.bk odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.ce.b 16
4.b odd 2 1 CM 3332.1.ce.b 16
7.b odd 2 1 3332.1.ce.d 16
7.c even 3 1 3332.1.bn.a 8
7.c even 3 1 inner 3332.1.ce.b 16
7.d odd 6 1 3332.1.bn.c yes 8
7.d odd 6 1 3332.1.ce.d 16
17.e odd 16 1 3332.1.ce.d 16
28.d even 2 1 3332.1.ce.d 16
28.f even 6 1 3332.1.bn.c yes 8
28.f even 6 1 3332.1.ce.d 16
28.g odd 6 1 3332.1.bn.a 8
28.g odd 6 1 inner 3332.1.ce.b 16
68.i even 16 1 3332.1.ce.d 16
119.p even 16 1 inner 3332.1.ce.b 16
119.s even 48 1 3332.1.bn.a 8
119.s even 48 1 inner 3332.1.ce.b 16
119.t odd 48 1 3332.1.bn.c yes 8
119.t odd 48 1 3332.1.ce.d 16
476.bf odd 16 1 inner 3332.1.ce.b 16
476.bk odd 48 1 3332.1.bn.a 8
476.bk odd 48 1 inner 3332.1.ce.b 16
476.bm even 48 1 3332.1.bn.c yes 8
476.bm even 48 1 3332.1.ce.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.bn.a 8 7.c even 3 1
3332.1.bn.a 8 28.g odd 6 1
3332.1.bn.a 8 119.s even 48 1
3332.1.bn.a 8 476.bk odd 48 1
3332.1.bn.c yes 8 7.d odd 6 1
3332.1.bn.c yes 8 28.f even 6 1
3332.1.bn.c yes 8 119.t odd 48 1
3332.1.bn.c yes 8 476.bm even 48 1
3332.1.ce.b 16 1.a even 1 1 trivial
3332.1.ce.b 16 4.b odd 2 1 CM
3332.1.ce.b 16 7.c even 3 1 inner
3332.1.ce.b 16 28.g odd 6 1 inner
3332.1.ce.b 16 119.p even 16 1 inner
3332.1.ce.b 16 119.s even 48 1 inner
3332.1.ce.b 16 476.bf odd 16 1 inner
3332.1.ce.b 16 476.bk odd 48 1 inner
3332.1.ce.d 16 7.b odd 2 1
3332.1.ce.d 16 7.d odd 6 1
3332.1.ce.d 16 17.e odd 16 1
3332.1.ce.d 16 28.d even 2 1
3332.1.ce.d 16 28.f even 6 1
3332.1.ce.d 16 68.i even 16 1
3332.1.ce.d 16 119.t odd 48 1
3332.1.ce.d 16 476.bm even 48 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - 2 T_{5}^{12} - 16 T_{5}^{11} + 40 T_{5}^{10} - 8 T_{5}^{9} + 2 T_{5}^{8} - 32 T_{5}^{7} + 216 T_{5}^{6} - 288 T_{5}^{5} + 268 T_{5}^{4} - 96 T_{5}^{3} + 24 T_{5}^{2} - 16 T_{5} + 4$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - T^{8} + 1$$
$3$ $$T^{16}$$
$5$ $$T^{16} - 2 T^{12} - 16 T^{11} + 40 T^{10} + \cdots + 4$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$(T^{2} + 2 T + 2)^{8}$$
$17$ $$T^{16} - T^{8} + 1$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$(T^{8} + 8 T^{5} + 2 T^{4} + 12 T^{2} - 8 T + 2)^{2}$$
$31$ $$T^{16}$$
$37$ $$T^{16} + 8 T^{15} + 36 T^{14} + 112 T^{13} + \cdots + 4$$
$41$ $$(T^{8} + 4 T^{6} + 6 T^{4} - 8 T^{3} + 4 T^{2} + \cdots + 2)^{2}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$(T^{8} - 2 T^{6} + 8 T^{5} + 2 T^{4} - 8 T^{3} + \cdots + 4)^{2}$$
$59$ $$T^{16}$$
$61$ $$T^{16} - 4 T^{14} + 10 T^{12} + 8 T^{11} + \cdots + 4$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$T^{16} - 16 T^{13} - 2 T^{12} + 88 T^{10} + \cdots + 4$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$(T^{8} - 16 T^{4} + 256)^{2}$$
$97$ $$(T^{8} + 2 T^{4} + 16 T^{3} + 20 T^{2} + \cdots + 2)^{2}$$