# Properties

 Label 1368.1.bw.a Level $1368$ Weight $1$ Character orbit 1368.bw Analytic conductor $0.683$ Analytic rank $0$ Dimension $8$ Projective image $S_{4}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1368.bw (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.682720937282$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.623808.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} - q^{7} + \zeta_{24}^{3} q^{8} +O(q^{10})$$ q - z^5 * q^2 + z^10 * q^4 - q^7 + z^3 * q^8 $$q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} - q^{7} + \zeta_{24}^{3} q^{8} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{11} + \zeta_{24}^{10} q^{13} + \zeta_{24}^{5} q^{14} - \zeta_{24}^{8} q^{16} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{17} - \zeta_{24}^{6} q^{19} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{22} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{23} + \zeta_{24}^{4} q^{25} + \zeta_{24}^{3} q^{26} - \zeta_{24}^{10} q^{28} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{29} + q^{31} - \zeta_{24} q^{32} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{34} + \zeta_{24}^{6} q^{37} + \zeta_{24}^{11} q^{38} - \zeta_{24}^{2} q^{43} + (\zeta_{24}^{7} - \zeta_{24}) q^{44} + ( - \zeta_{24}^{6} - 1) q^{46} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{47} - \zeta_{24}^{9} q^{50} - \zeta_{24}^{8} q^{52} + (\zeta_{24}^{7} + \zeta_{24}) q^{53} - \zeta_{24}^{3} q^{56} + (\zeta_{24}^{6} - 1) q^{58} - \zeta_{24}^{10} q^{61} - \zeta_{24}^{5} q^{62} + \zeta_{24}^{6} q^{64} - \zeta_{24}^{10} q^{67} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{68} - \zeta_{24}^{8} q^{73} - \zeta_{24}^{11} q^{74} + \zeta_{24}^{4} q^{76} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{77} + \zeta_{24}^{8} q^{79} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{83} + \zeta_{24}^{7} q^{86} + (\zeta_{24}^{6} + 1) q^{88} - \zeta_{24}^{10} q^{91} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{92} + ( - \zeta_{24}^{6} - 1) q^{94} + \zeta_{24}^{8} q^{97} +O(q^{100})$$ q - z^5 * q^2 + z^10 * q^4 - q^7 + z^3 * q^8 + (-z^9 + z^3) * q^11 + z^10 * q^13 + z^5 * q^14 - z^8 * q^16 + (-z^11 + z^5) * q^17 - z^6 * q^19 + (-z^8 - z^2) * q^22 + (-z^7 + z) * q^23 + z^4 * q^25 + z^3 * q^26 - z^10 * q^28 + (-z^7 - z) * q^29 + q^31 - z * q^32 + (-z^10 - z^4) * q^34 + z^6 * q^37 + z^11 * q^38 - z^2 * q^43 + (z^7 - z) * q^44 + (-z^6 - 1) * q^46 + (-z^7 + z) * q^47 - z^9 * q^50 - z^8 * q^52 + (z^7 + z) * q^53 - z^3 * q^56 + (z^6 - 1) * q^58 - z^10 * q^61 - z^5 * q^62 + z^6 * q^64 - z^10 * q^67 + (z^9 - z^3) * q^68 - z^8 * q^73 - z^11 * q^74 + z^4 * q^76 + (z^9 - z^3) * q^77 + z^8 * q^79 + (z^9 - z^3) * q^83 + z^7 * q^86 + (z^6 + 1) * q^88 - z^10 * q^91 + (z^11 + z^5) * q^92 + (-z^6 - 1) * q^94 + z^8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{7}+O(q^{10})$$ 8 * q - 8 * q^7 $$8 q - 8 q^{7} + 4 q^{16} + 4 q^{22} + 4 q^{25} + 8 q^{31} - 4 q^{34} - 8 q^{46} + 4 q^{52} - 8 q^{58} + 4 q^{73} + 4 q^{76} - 4 q^{79} + 8 q^{88} - 8 q^{94} - 8 q^{97}+O(q^{100})$$ 8 * q - 8 * q^7 + 4 * q^16 + 4 * q^22 + 4 * q^25 + 8 * q^31 - 4 * q^34 - 8 * q^46 + 4 * q^52 - 8 * q^58 + 4 * q^73 + 4 * q^76 - 4 * q^79 + 8 * q^88 - 8 * q^94 - 8 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{24}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1
 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 −1.00000 −0.707107 + 0.707107i 0 0
125.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.00000 0.707107 + 0.707107i 0 0
125.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.00000 −0.707107 0.707107i 0 0
125.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −1.00000 0.707107 0.707107i 0 0
197.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 −1.00000 −0.707107 0.707107i 0 0
197.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −1.00000 0.707107 0.707107i 0 0
197.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 −1.00000 −0.707107 + 0.707107i 0 0
197.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −1.00000 0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.4 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 inner
8.b even 2 1 inner
19.c even 3 1 inner
24.h odd 2 1 inner
57.h odd 6 1 inner
152.p even 6 1 inner
456.x odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.bw.a 8
3.b odd 2 1 inner 1368.1.bw.a 8
8.b even 2 1 inner 1368.1.bw.a 8
19.c even 3 1 inner 1368.1.bw.a 8
24.h odd 2 1 inner 1368.1.bw.a 8
57.h odd 6 1 inner 1368.1.bw.a 8
152.p even 6 1 inner 1368.1.bw.a 8
456.x odd 6 1 inner 1368.1.bw.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.bw.a 8 1.a even 1 1 trivial
1368.1.bw.a 8 3.b odd 2 1 inner
1368.1.bw.a 8 8.b even 2 1 inner
1368.1.bw.a 8 19.c even 3 1 inner
1368.1.bw.a 8 24.h odd 2 1 inner
1368.1.bw.a 8 57.h odd 6 1 inner
1368.1.bw.a 8 152.p even 6 1 inner
1368.1.bw.a 8 456.x odd 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{4} + 1$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T + 1)^{8}$$
$11$ $$(T^{2} - 2)^{4}$$
$13$ $$(T^{4} - T^{2} + 1)^{2}$$
$17$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$19$ $$(T^{2} + 1)^{4}$$
$23$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$29$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$31$ $$(T - 1)^{8}$$
$37$ $$(T^{2} + 1)^{4}$$
$41$ $$T^{8}$$
$43$ $$(T^{4} - T^{2} + 1)^{2}$$
$47$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$53$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$59$ $$T^{8}$$
$61$ $$(T^{4} - T^{2} + 1)^{2}$$
$67$ $$(T^{4} - T^{2} + 1)^{2}$$
$71$ $$T^{8}$$
$73$ $$(T^{2} - T + 1)^{4}$$
$79$ $$(T^{2} + T + 1)^{4}$$
$83$ $$(T^{2} - 2)^{4}$$
$89$ $$T^{8}$$
$97$ $$(T^{2} + 2 T + 4)^{4}$$