# Properties

 Label 507.1.o.a Level $507$ Weight $1$ Character orbit 507.o Analytic conductor $0.253$ Analytic rank $0$ Dimension $12$ Projective image $D_{13}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 507.o (of order $$26$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.253025961405$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{26})$$ Defining polynomial: $$x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{13}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{13} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{26}^{7} q^{3} + \zeta_{26}^{2} q^{4} + ( -\zeta_{26}^{9} + \zeta_{26}^{10} ) q^{7} -\zeta_{26} q^{9} +O(q^{10})$$ $$q -\zeta_{26}^{7} q^{3} + \zeta_{26}^{2} q^{4} + ( -\zeta_{26}^{9} + \zeta_{26}^{10} ) q^{7} -\zeta_{26} q^{9} -\zeta_{26}^{9} q^{12} + q^{13} + \zeta_{26}^{4} q^{16} + ( -\zeta_{26} + \zeta_{26}^{12} ) q^{19} + ( -\zeta_{26}^{3} + \zeta_{26}^{4} ) q^{21} -\zeta_{26}^{5} q^{25} + \zeta_{26}^{8} q^{27} + ( -\zeta_{26}^{11} + \zeta_{26}^{12} ) q^{28} + ( \zeta_{26}^{6} + \zeta_{26}^{10} ) q^{31} -\zeta_{26}^{3} q^{36} + ( -\zeta_{26}^{5} - \zeta_{26}^{11} ) q^{37} -\zeta_{26}^{7} q^{39} + ( \zeta_{26}^{2} + \zeta_{26}^{8} ) q^{43} -\zeta_{26}^{11} q^{48} + ( -\zeta_{26}^{5} + \zeta_{26}^{6} - \zeta_{26}^{7} ) q^{49} + \zeta_{26}^{2} q^{52} + ( \zeta_{26}^{6} + \zeta_{26}^{8} ) q^{57} + ( \zeta_{26}^{8} + \zeta_{26}^{12} ) q^{61} + ( \zeta_{26}^{10} - \zeta_{26}^{11} ) q^{63} + \zeta_{26}^{6} q^{64} + ( -\zeta_{26}^{3} + \zeta_{26}^{6} ) q^{67} + ( -\zeta_{26}^{3} + \zeta_{26}^{8} ) q^{73} + \zeta_{26}^{12} q^{75} + ( -\zeta_{26} - \zeta_{26}^{3} ) q^{76} + ( \zeta_{26}^{4} - \zeta_{26}^{5} ) q^{79} + \zeta_{26}^{2} q^{81} + ( -\zeta_{26}^{5} + \zeta_{26}^{6} ) q^{84} + ( -\zeta_{26}^{9} + \zeta_{26}^{10} ) q^{91} + ( 1 + \zeta_{26}^{4} ) q^{93} + 2 \zeta_{26}^{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - q^{3} - q^{4} - 2q^{7} - q^{9} + O(q^{10})$$ $$12q - q^{3} - q^{4} - 2q^{7} - q^{9} - q^{12} + 12q^{13} - q^{16} - 2q^{19} - 2q^{21} - q^{25} - q^{27} - 2q^{28} - 2q^{31} - q^{36} - 2q^{37} - q^{39} - 2q^{43} - q^{48} - 3q^{49} - q^{52} - 2q^{57} - 2q^{61} - 2q^{63} - q^{64} - 2q^{67} - 2q^{73} - q^{75} - 2q^{76} - 2q^{79} - q^{81} - 2q^{84} - 2q^{91} + 11q^{93} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$\zeta_{26}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 −0.568065 + 0.822984i 0.748511 − 0.663123i −0.885456 + 0.464723i 0.970942 − 0.239316i 0.970942 + 0.239316i −0.885456 − 0.464723i 0.748511 + 0.663123i −0.568065 − 0.822984i 0.354605 + 0.935016i −0.120537 − 0.992709i −0.120537 + 0.992709i 0.354605 − 0.935016i
0 0.885456 0.464723i −0.354605 0.935016i 0 0 −1.71945 0.423807i 0 0.568065 0.822984i 0
53.1 0 −0.354605 0.935016i 0.120537 0.992709i 0 0 −0.402877 0.583668i 0 −0.748511 + 0.663123i 0
92.1 0 −0.970942 + 0.239316i 0.568065 0.822984i 0 0 −0.234068 + 1.92773i 0 0.885456 0.464723i 0
131.1 0 0.120537 + 0.992709i 0.885456 0.464723i 0 0 −0.180446 + 0.159861i 0 −0.970942 + 0.239316i 0
209.1 0 0.120537 0.992709i 0.885456 + 0.464723i 0 0 −0.180446 0.159861i 0 −0.970942 0.239316i 0
248.1 0 −0.970942 0.239316i 0.568065 + 0.822984i 0 0 −0.234068 1.92773i 0 0.885456 + 0.464723i 0
287.1 0 −0.354605 + 0.935016i 0.120537 + 0.992709i 0 0 −0.402877 + 0.583668i 0 −0.748511 0.663123i 0
326.1 0 0.885456 + 0.464723i −0.354605 + 0.935016i 0 0 −1.71945 + 0.423807i 0 0.568065 + 0.822984i 0
365.1 0 0.568065 0.822984i −0.748511 + 0.663123i 0 0 1.00599 + 0.527986i 0 −0.354605 0.935016i 0
404.1 0 −0.748511 0.663123i −0.970942 + 0.239316i 0 0 0.530851 + 1.39974i 0 0.120537 + 0.992709i 0
443.1 0 −0.748511 + 0.663123i −0.970942 0.239316i 0 0 0.530851 1.39974i 0 0.120537 0.992709i 0
482.1 0 0.568065 + 0.822984i −0.748511 0.663123i 0 0 1.00599 0.527986i 0 −0.354605 + 0.935016i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 482.1 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 $$\Q(\sqrt{-3})$$
169.g even 13 1 inner
507.o odd 26 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.o.a 12
3.b odd 2 1 CM 507.1.o.a 12
169.g even 13 1 inner 507.1.o.a 12
507.o odd 26 1 inner 507.1.o.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.1.o.a 12 1.a even 1 1 trivial
507.1.o.a 12 3.b odd 2 1 CM
507.1.o.a 12 169.g even 13 1 inner
507.1.o.a 12 507.o odd 26 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(507, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$1 + 7 T + 23 T^{2} + 18 T^{3} + 9 T^{4} - 15 T^{5} + 12 T^{6} + 6 T^{7} + 3 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$( -1 + T )^{12}$$
$17$ $$T^{12}$$
$19$ $$( -1 + 3 T + 6 T^{2} - 4 T^{3} - 5 T^{4} + T^{5} + T^{6} )^{2}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$37$ $$1 + 7 T + 23 T^{2} + 18 T^{3} + 9 T^{4} - 15 T^{5} + 12 T^{6} + 6 T^{7} + 3 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 + 7 T + 23 T^{2} + 18 T^{3} + 9 T^{4} - 15 T^{5} + 12 T^{6} + 6 T^{7} + 3 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 - 6 T + 23 T^{2} - 21 T^{3} - 4 T^{4} - 2 T^{5} - T^{6} + 6 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$67$ $$1 + 7 T + 36 T^{2} + 96 T^{3} + 139 T^{4} + 115 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$71$ $$T^{12}$$
$73$ $$1 + 7 T + 36 T^{2} + 96 T^{3} + 139 T^{4} + 115 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$79$ $$1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$4096 + 2048 T + 1024 T^{2} + 512 T^{3} + 256 T^{4} + 128 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$