# Properties

 Label 723.1.bc.a Level $723$ Weight $1$ Character orbit 723.bc Analytic conductor $0.361$ Analytic rank $0$ Dimension $16$ Projective image $D_{40}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$723 = 3 \cdot 241$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 723.bc (of order $$40$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.360824004134$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{40}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{40} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{40}^{13} q^{3} + \zeta_{40}^{10} q^{4} + ( -\zeta_{40} - \zeta_{40}^{2} ) q^{7} -\zeta_{40}^{6} q^{9} +O(q^{10})$$ $$q + \zeta_{40}^{13} q^{3} + \zeta_{40}^{10} q^{4} + ( -\zeta_{40} - \zeta_{40}^{2} ) q^{7} -\zeta_{40}^{6} q^{9} -\zeta_{40}^{3} q^{12} + ( \zeta_{40}^{6} + \zeta_{40}^{15} ) q^{13} - q^{16} + ( \zeta_{40}^{17} - \zeta_{40}^{18} ) q^{19} + ( -\zeta_{40}^{14} - \zeta_{40}^{15} ) q^{21} + \zeta_{40}^{14} q^{25} -\zeta_{40}^{19} q^{27} + ( -\zeta_{40}^{11} - \zeta_{40}^{12} ) q^{28} + ( -\zeta_{40}^{4} - \zeta_{40}^{9} ) q^{31} -\zeta_{40}^{16} q^{36} + ( \zeta_{40} + \zeta_{40}^{18} ) q^{37} + ( -\zeta_{40}^{8} + \zeta_{40}^{19} ) q^{39} + ( \zeta_{40}^{5} + \zeta_{40}^{12} ) q^{43} -\zeta_{40}^{13} q^{48} + ( \zeta_{40}^{2} + \zeta_{40}^{3} + \zeta_{40}^{4} ) q^{49} + ( -\zeta_{40}^{5} + \zeta_{40}^{16} ) q^{52} + ( -\zeta_{40}^{10} + \zeta_{40}^{11} ) q^{57} + ( \zeta_{40}^{9} - \zeta_{40}^{17} ) q^{61} + ( \zeta_{40}^{7} + \zeta_{40}^{8} ) q^{63} -\zeta_{40}^{10} q^{64} + ( \zeta_{40}^{7} + \zeta_{40}^{11} ) q^{67} + ( \zeta_{40}^{3} + \zeta_{40}^{16} ) q^{73} -\zeta_{40}^{7} q^{75} + ( -\zeta_{40}^{7} + \zeta_{40}^{8} ) q^{76} + ( -\zeta_{40}^{5} + \zeta_{40}^{9} ) q^{79} + \zeta_{40}^{12} q^{81} + ( \zeta_{40}^{4} + \zeta_{40}^{5} ) q^{84} + ( -\zeta_{40}^{7} - \zeta_{40}^{8} - \zeta_{40}^{16} - \zeta_{40}^{17} ) q^{91} + ( \zeta_{40}^{2} - \zeta_{40}^{17} ) q^{93} + ( -\zeta_{40}^{13} + \zeta_{40}^{19} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{16} - 4q^{28} - 4q^{31} + 4q^{36} + 4q^{39} + 4q^{43} + 4q^{49} - 4q^{52} - 4q^{63} - 4q^{73} - 4q^{76} + 4q^{81} + 4q^{84} + 8q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/723\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$242$$ $$\chi(n)$$ $$\zeta_{40}^{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.987688 − 0.156434i −0.453990 − 0.891007i 0.453990 − 0.891007i 0.891007 + 0.453990i −0.891007 − 0.453990i −0.453990 + 0.891007i 0.453990 + 0.891007i 0.987688 + 0.156434i −0.156434 − 0.987688i 0.156434 − 0.987688i −0.987688 + 0.156434i 0.891007 − 0.453990i −0.891007 + 0.453990i 0.987688 − 0.156434i −0.156434 + 0.987688i 0.156434 + 0.987688i
0 0.453990 0.891007i 1.00000i 0 0 0.0366318 0.152583i 0 −0.587785 0.809017i 0
41.1 0 0.156434 0.987688i 1.00000i 0 0 1.04178 + 0.0819895i 0 −0.951057 0.309017i 0
47.1 0 −0.156434 0.987688i 1.00000i 0 0 0.133795 + 1.70002i 0 −0.951057 + 0.309017i 0
116.1 0 0.987688 0.156434i 1.00000i 0 0 −1.47879 1.26301i 0 0.951057 0.309017i 0
125.1 0 −0.987688 + 0.156434i 1.00000i 0 0 0.303221 0.355026i 0 0.951057 0.309017i 0
194.1 0 0.156434 + 0.987688i 1.00000i 0 0 1.04178 0.0819895i 0 −0.951057 + 0.309017i 0
200.1 0 −0.156434 + 0.987688i 1.00000i 0 0 0.133795 1.70002i 0 −0.951057 0.309017i 0
236.1 0 −0.453990 + 0.891007i 1.00000i 0 0 −1.93874 0.465451i 0 −0.587785 0.809017i 0
302.1 0 −0.891007 + 0.453990i 1.00000i 0 0 1.10749 + 0.678671i 0 0.587785 0.809017i 0
320.1 0 0.891007 + 0.453990i 1.00000i 0 0 0.794622 + 1.29671i 0 0.587785 + 0.809017i 0
434.1 0 0.453990 + 0.891007i 1.00000i 0 0 0.0366318 + 0.152583i 0 −0.587785 + 0.809017i 0
455.1 0 0.987688 + 0.156434i 1.00000i 0 0 −1.47879 + 1.26301i 0 0.951057 + 0.309017i 0
509.1 0 −0.987688 0.156434i 1.00000i 0 0 0.303221 + 0.355026i 0 0.951057 + 0.309017i 0
530.1 0 −0.453990 0.891007i 1.00000i 0 0 −1.93874 + 0.465451i 0 −0.587785 + 0.809017i 0
644.1 0 −0.891007 0.453990i 1.00000i 0 0 1.10749 0.678671i 0 0.587785 + 0.809017i 0
662.1 0 0.891007 0.453990i 1.00000i 0 0 0.794622 1.29671i 0 0.587785 0.809017i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 662.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
241.o even 40 1 inner
723.bc odd 40 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 723.1.bc.a 16
3.b odd 2 1 CM 723.1.bc.a 16
241.o even 40 1 inner 723.1.bc.a 16
723.bc odd 40 1 inner 723.1.bc.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.1.bc.a 16 1.a even 1 1 trivial
723.1.bc.a 16 3.b odd 2 1 CM
723.1.bc.a 16 241.o even 40 1 inner
723.1.bc.a 16 723.bc odd 40 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(723, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$1 - 8 T + 68 T^{2} - 256 T^{3} + 557 T^{4} - 628 T^{5} + 298 T^{6} + 32 T^{7} - 100 T^{8} + 32 T^{9} + 16 T^{10} - 16 T^{11} + 2 T^{12} + 4 T^{13} - 2 T^{14} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 - 12 T + 68 T^{2} - 144 T^{3} + 222 T^{4} - 92 T^{5} + 118 T^{6} - 72 T^{7} - 80 T^{8} + 8 T^{9} + 6 T^{10} + 16 T^{11} + 7 T^{12} - 4 T^{13} - 2 T^{14} + T^{16}$$
$17$ $$T^{16}$$
$19$ $$1 - 8 T + 58 T^{2} - 136 T^{3} + 82 T^{4} + 52 T^{5} + 28 T^{6} + 72 T^{7} + 75 T^{8} + 52 T^{9} + 16 T^{10} - 16 T^{11} + 2 T^{12} + 4 T^{13} - 2 T^{14} + T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$16 + 32 T + 16 T^{2} - 32 T^{3} - 56 T^{4} - 112 T^{5} - 128 T^{6} - 16 T^{7} + 156 T^{8} + 160 T^{9} + 112 T^{10} + 64 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$37$ $$1 - 12 T + 58 T^{2} - 104 T^{3} + 237 T^{4} - 112 T^{5} - 42 T^{6} + 208 T^{7} - 80 T^{8} + 8 T^{9} + 86 T^{10} - 24 T^{11} + 2 T^{12} + 16 T^{13} - 2 T^{14} + T^{16}$$
$41$ $$T^{16}$$
$43$ $$1 - 12 T + 76 T^{2} - 188 T^{3} + 174 T^{4} + 72 T^{5} - 158 T^{6} + 156 T^{7} - 64 T^{8} - 60 T^{9} + 82 T^{10} - 64 T^{11} + 39 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$625 + 125 T^{4} + 150 T^{8} - 20 T^{12} + T^{16}$$
$67$ $$625 + 125 T^{4} + 150 T^{8} - 20 T^{12} + T^{16}$$
$71$ $$T^{16}$$
$73$ $$1 - 8 T - 4 T^{2} + 68 T^{3} + 269 T^{4} + 328 T^{5} + 382 T^{6} + 284 T^{7} + 156 T^{8} + 60 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$79$ $$1 + 4 T^{4} + 46 T^{8} - 11 T^{12} + T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16}$$