# Properties

 Label 848.1.x.a Level $848$ Weight $1$ Character orbit 848.x Analytic conductor $0.423$ Analytic rank $0$ Dimension $12$ Projective image $D_{26}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$848 = 2^{4} \cdot 53$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 848.x (of order $$26$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.423207130713$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{26}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{26} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{26} + \zeta_{26}^{11} ) q^{5} -\zeta_{26}^{6} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{26} + \zeta_{26}^{11} ) q^{5} -\zeta_{26}^{6} q^{9} + ( -\zeta_{26}^{3} - \zeta_{26}^{7} ) q^{13} + ( -\zeta_{26}^{5} + \zeta_{26}^{10} ) q^{17} + ( \zeta_{26}^{2} - \zeta_{26}^{9} - \zeta_{26}^{12} ) q^{25} + ( -\zeta_{26}^{8} + \zeta_{26}^{9} ) q^{29} + ( -\zeta_{26}^{2} - \zeta_{26}^{4} ) q^{37} + ( -\zeta_{26}^{10} + \zeta_{26}^{12} ) q^{41} + ( \zeta_{26}^{4} + \zeta_{26}^{7} ) q^{45} + \zeta_{26}^{8} q^{49} + \zeta_{26}^{3} q^{53} + ( \zeta_{26}^{5} + \zeta_{26}^{6} ) q^{61} + ( \zeta_{26} + \zeta_{26}^{4} + \zeta_{26}^{5} + \zeta_{26}^{8} ) q^{65} + ( -\zeta_{26}^{4} - \zeta_{26}^{11} ) q^{73} + \zeta_{26}^{12} q^{81} + ( \zeta_{26}^{3} + \zeta_{26}^{6} - \zeta_{26}^{8} - \zeta_{26}^{11} ) q^{85} + ( -1 - \zeta_{26}^{2} ) q^{89} + ( -1 - \zeta_{26}^{12} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + q^{9} + O(q^{10})$$ $$12q + q^{9} - 2q^{13} - 2q^{17} - q^{25} + 2q^{29} + 2q^{37} - q^{49} + q^{53} - q^{81} - 11q^{89} - 11q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/848\mathbb{Z}\right)^\times$$.

 $$n$$ $$161$$ $$213$$ $$319$$ $$\chi(n)$$ $$-\zeta_{26}^{8}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.748511 − 0.663123i −0.885456 − 0.464723i 0.748511 + 0.663123i −0.120537 − 0.992709i −0.568065 − 0.822984i 0.970942 − 0.239316i −0.568065 + 0.822984i −0.885456 + 0.464723i 0.354605 − 0.935016i −0.120537 + 0.992709i 0.970942 + 0.239316i 0.354605 + 0.935016i
0 0 0 −0.869047 0.329586i 0 0 0 0.354605 0.935016i 0
223.1 0 0 0 0.317391 + 1.28771i 0 0 0 0.970942 0.239316i 0
255.1 0 0 0 −0.869047 + 0.329586i 0 0 0 0.354605 + 0.935016i 0
271.1 0 0 0 1.09148 + 1.23202i 0 0 0 0.748511 0.663123i 0
303.1 0 0 0 0.922670 + 1.75800i 0 0 0 −0.885456 + 0.464723i 0
335.1 0 0 0 −1.85640 0.225408i 0 0 0 −0.120537 + 0.992709i 0
431.1 0 0 0 0.922670 1.75800i 0 0 0 −0.885456 0.464723i 0
559.1 0 0 0 0.317391 1.28771i 0 0 0 0.970942 + 0.239316i 0
623.1 0 0 0 0.393906 + 0.271894i 0 0 0 −0.568065 + 0.822984i 0
751.1 0 0 0 1.09148 1.23202i 0 0 0 0.748511 + 0.663123i 0
767.1 0 0 0 −1.85640 + 0.225408i 0 0 0 −0.120537 0.992709i 0
799.1 0 0 0 0.393906 0.271894i 0 0 0 −0.568065 0.822984i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 799.1 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 $$\Q(\sqrt{-1})$$
53.e even 26 1 inner
212.h odd 26 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 848.1.x.a 12
4.b odd 2 1 CM 848.1.x.a 12
8.b even 2 1 3392.1.bn.a 12
8.d odd 2 1 3392.1.bn.a 12
53.e even 26 1 inner 848.1.x.a 12
212.h odd 26 1 inner 848.1.x.a 12
424.n even 26 1 3392.1.bn.a 12
424.q odd 26 1 3392.1.bn.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
848.1.x.a 12 1.a even 1 1 trivial
848.1.x.a 12 4.b odd 2 1 CM
848.1.x.a 12 53.e even 26 1 inner
848.1.x.a 12 212.h odd 26 1 inner
3392.1.bn.a 12 8.b even 2 1
3392.1.bn.a 12 8.d odd 2 1
3392.1.bn.a 12 424.n even 26 1
3392.1.bn.a 12 424.q odd 26 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(848, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$13 - 26 T + 65 T^{3} + 13 T^{4} + 52 T^{6} + 13 T^{9} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$T^{12}$$
$13$ $$1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$17$ $$1 + 7 T + 36 T^{2} + 96 T^{3} + 139 T^{4} + 115 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$1 + 6 T + 10 T^{2} - 5 T^{3} + 35 T^{4} - 24 T^{5} + 12 T^{6} + 20 T^{7} - 10 T^{8} + 5 T^{9} + 4 T^{10} - 2 T^{11} + T^{12}$$
$31$ $$T^{12}$$
$37$ $$1 - 7 T + 36 T^{2} - 96 T^{3} + 139 T^{4} - 115 T^{5} + 64 T^{6} - 32 T^{7} + 16 T^{8} - 8 T^{9} + 4 T^{10} - 2 T^{11} + T^{12}$$
$41$ $$13 + 13 T + 26 T^{2} - 52 T^{3} + 65 T^{5} - 13 T^{7} + T^{12}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$59$ $$T^{12}$$
$61$ $$13 + 13 T + 26 T^{2} - 52 T^{3} + 65 T^{5} - 13 T^{7} + T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$13 - 26 T + 65 T^{3} + 13 T^{4} + 52 T^{6} + 13 T^{9} + T^{12}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$1 + 6 T + 36 T^{2} + 125 T^{3} + 295 T^{4} + 496 T^{5} + 610 T^{6} + 553 T^{7} + 367 T^{8} + 174 T^{9} + 56 T^{10} + 11 T^{11} + T^{12}$$
$97$ $$1 + 6 T + 36 T^{2} + 125 T^{3} + 295 T^{4} + 496 T^{5} + 610 T^{6} + 553 T^{7} + 367 T^{8} + 174 T^{9} + 56 T^{10} + 11 T^{11} + T^{12}$$