# Properties

 Label 772.1.u.a Level $772$ Weight $1$ Character orbit 772.u Analytic conductor $0.385$ Analytic rank $0$ Dimension $16$ Projective image $D_{48}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$772 = 2^{2} \cdot 193$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 772.u (of order $$48$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.385278189753$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{48}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{48} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{48}^{19} q^{2} -\zeta_{48}^{14} q^{4} + ( -\zeta_{48}^{13} - \zeta_{48}^{22} ) q^{5} -\zeta_{48}^{9} q^{8} + \zeta_{48}^{12} q^{9} +O(q^{10})$$ $$q -\zeta_{48}^{19} q^{2} -\zeta_{48}^{14} q^{4} + ( -\zeta_{48}^{13} - \zeta_{48}^{22} ) q^{5} -\zeta_{48}^{9} q^{8} + \zeta_{48}^{12} q^{9} + ( -\zeta_{48}^{8} - \zeta_{48}^{17} ) q^{10} + ( \zeta_{48}^{19} + \zeta_{48}^{20} ) q^{13} -\zeta_{48}^{4} q^{16} + ( \zeta_{48}^{13} + \zeta_{48}^{16} ) q^{17} + \zeta_{48}^{7} q^{18} + ( -\zeta_{48}^{3} - \zeta_{48}^{12} ) q^{20} + ( -\zeta_{48}^{2} - \zeta_{48}^{11} - \zeta_{48}^{20} ) q^{25} + ( \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{26} + ( \zeta_{48}^{4} - \zeta_{48}^{5} ) q^{29} + \zeta_{48}^{23} q^{32} + ( \zeta_{48}^{8} + \zeta_{48}^{11} ) q^{34} + \zeta_{48}^{2} q^{36} + ( -\zeta_{48}^{10} - \zeta_{48}^{21} ) q^{37} + ( -\zeta_{48}^{7} + \zeta_{48}^{22} ) q^{40} + ( \zeta_{48}^{6} - \zeta_{48}^{23} ) q^{41} + ( \zeta_{48} + \zeta_{48}^{10} ) q^{45} -\zeta_{48}^{16} q^{49} + ( -\zeta_{48}^{6} - \zeta_{48}^{15} + \zeta_{48}^{21} ) q^{50} + ( \zeta_{48}^{9} + \zeta_{48}^{10} ) q^{52} + ( -\zeta_{48} - \zeta_{48}^{6} ) q^{53} + ( -1 - \zeta_{48}^{23} ) q^{58} + ( -\zeta_{48}^{15} + \zeta_{48}^{22} ) q^{61} + \zeta_{48}^{18} q^{64} + ( \zeta_{48}^{8} + \zeta_{48}^{9} + \zeta_{48}^{17} + \zeta_{48}^{18} ) q^{65} + ( \zeta_{48}^{3} + \zeta_{48}^{6} ) q^{68} -\zeta_{48}^{21} q^{72} + ( -\zeta_{48}^{18} + \zeta_{48}^{23} ) q^{73} + ( -\zeta_{48}^{5} - \zeta_{48}^{16} ) q^{74} + ( -\zeta_{48}^{2} + \zeta_{48}^{17} ) q^{80} - q^{81} + ( \zeta_{48} - \zeta_{48}^{18} ) q^{82} + ( \zeta_{48}^{2} + \zeta_{48}^{5} + \zeta_{48}^{11} + \zeta_{48}^{14} ) q^{85} + ( -\zeta_{48}^{12} + \zeta_{48}^{21} ) q^{89} + ( \zeta_{48}^{5} - \zeta_{48}^{20} ) q^{90} + ( \zeta_{48}^{3} + \zeta_{48}^{7} ) q^{97} -\zeta_{48}^{11} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{10} - 8q^{17} + 8q^{34} + 8q^{49} - 16q^{58} + 8q^{65} + 8q^{74} - 16q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/772\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$387$$ $$\chi(n)$$ $$-\zeta_{48}^{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 −0.991445 + 0.130526i 0.608761 + 0.793353i −0.130526 − 0.991445i 0.130526 + 0.991445i −0.608761 − 0.793353i 0.991445 − 0.130526i 0.130526 − 0.991445i 0.793353 + 0.608761i 0.991445 + 0.130526i −0.608761 + 0.793353i −0.793353 + 0.608761i 0.793353 − 0.608761i 0.608761 − 0.793353i −0.991445 − 0.130526i −0.793353 − 0.608761i −0.130526 + 0.991445i
−0.793353 0.608761i 0 0.258819 + 0.965926i 0.835400 0.732626i 0 0 0.382683 0.923880i 1.00000i −1.10876 + 0.0726721i
131.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i −1.05217 0.357164i 0 0 0.382683 0.923880i 1.00000i 0.491445 0.996552i
147.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i 0.0255190 0.389345i 0 0 0.923880 + 0.382683i 1.00000i 0.293353 + 0.257264i
239.1 0.608761 0.793353i 0 −0.258819 0.965926i −1.95737 0.128293i 0 0 −0.923880 0.382683i 1.00000i −1.29335 + 1.47479i
255.1 0.130526 0.991445i 0 −0.965926 0.258819i 0.534534 1.57469i 0 0 −0.382683 + 0.923880i 1.00000i −1.49144 0.735499i
327.1 0.793353 + 0.608761i 0 0.258819 + 0.965926i 1.09645 + 1.25026i 0 0 −0.382683 + 0.923880i 1.00000i 0.108761 + 1.65938i
407.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i −1.95737 + 0.128293i 0 0 −0.923880 + 0.382683i 1.00000i −1.29335 1.47479i
531.1 −0.991445 + 0.130526i 0 0.965926 0.258819i 0.867580 1.75928i 0 0 −0.923880 + 0.382683i 1.00000i −0.630526 + 1.85747i
543.1 0.793353 0.608761i 0 0.258819 0.965926i 1.09645 1.25026i 0 0 −0.382683 0.923880i 1.00000i 0.108761 1.65938i
551.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i 0.534534 + 1.57469i 0 0 −0.382683 0.923880i 1.00000i −1.49144 + 0.735499i
575.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i −0.349942 + 0.172572i 0 0 0.923880 + 0.382683i 1.00000i −0.369474 + 0.125419i
583.1 −0.991445 0.130526i 0 0.965926 + 0.258819i 0.867580 + 1.75928i 0 0 −0.923880 0.382683i 1.00000i −0.630526 1.85747i
607.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i −1.05217 + 0.357164i 0 0 0.382683 + 0.923880i 1.00000i 0.491445 + 0.996552i
615.1 −0.793353 + 0.608761i 0 0.258819 0.965926i 0.835400 + 0.732626i 0 0 0.382683 + 0.923880i 1.00000i −1.10876 0.0726721i
627.1 0.991445 0.130526i 0 0.965926 0.258819i −0.349942 0.172572i 0 0 0.923880 0.382683i 1.00000i −0.369474 0.125419i
751.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i 0.0255190 + 0.389345i 0 0 0.923880 0.382683i 1.00000i 0.293353 0.257264i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 751.1 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 $$\Q(\sqrt{-1})$$
193.k even 48 1 inner
772.u odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 772.1.u.a 16
4.b odd 2 1 CM 772.1.u.a 16
193.k even 48 1 inner 772.1.u.a 16
772.u odd 48 1 inner 772.1.u.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
772.1.u.a 16 1.a even 1 1 trivial
772.1.u.a 16 4.b odd 2 1 CM
772.1.u.a 16 193.k even 48 1 inner
772.1.u.a 16 772.u odd 48 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(772, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{8} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$4 + 16 T + 40 T^{2} + 96 T^{3} + 140 T^{4} - 48 T^{5} + 40 T^{6} + 192 T^{7} + 2 T^{8} - 8 T^{9} + 88 T^{10} - 2 T^{12} + 16 T^{13} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 + 8 T + 56 T^{2} - 16 T^{3} + 86 T^{4} - 112 T^{5} + 44 T^{6} + 112 T^{7} + 18 T^{8} + 24 T^{9} - 16 T^{10} - 16 T^{11} + 10 T^{12} - 4 T^{14} + T^{16}$$
$17$ $$4 + 16 T + 8 T^{2} + 196 T^{4} + 560 T^{5} + 840 T^{6} + 1024 T^{7} + 1106 T^{8} + 1016 T^{9} + 784 T^{10} + 504 T^{11} + 266 T^{12} + 112 T^{13} + 36 T^{14} + 8 T^{15} + T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$1 - 8 T + 56 T^{2} + 16 T^{3} + 86 T^{4} + 112 T^{5} + 44 T^{6} - 112 T^{7} + 18 T^{8} - 24 T^{9} - 16 T^{10} + 16 T^{11} + 10 T^{12} - 4 T^{14} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$1 + 16 T + 84 T^{2} + 144 T^{3} + 268 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 5 T^{8} + 8 T^{9} - 20 T^{10} - 32 T^{11} - 2 T^{12} + T^{16}$$
$41$ $$1 - 8 T + 24 T^{2} + 144 T^{3} + 274 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 5 T^{8} - 16 T^{9} - 20 T^{10} + 16 T^{11} + 4 T^{12} + T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$1 + 8 T + 40 T^{2} - 96 T^{3} + 146 T^{4} - 24 T^{5} - 152 T^{6} + 96 T^{7} + 5 T^{8} + 8 T^{9} + 52 T^{10} + 4 T^{12} + 8 T^{13} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$1 + 16 T + 84 T^{2} + 144 T^{3} + 268 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 5 T^{8} + 8 T^{9} - 20 T^{10} - 32 T^{11} - 2 T^{12} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$1 - 8 T + 40 T^{2} + 96 T^{3} + 146 T^{4} + 24 T^{5} - 152 T^{6} - 96 T^{7} + 5 T^{8} - 8 T^{9} + 52 T^{10} + 4 T^{12} - 8 T^{13} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$( 2 - 8 T + 4 T^{2} + 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$97$ $$1 - 24 T^{4} + 191 T^{8} + 24 T^{12} + T^{16}$$