# Properties

 Label 1368.2.e.d Level $1368$ Weight $2$ Character orbit 1368.e Analytic conductor $10.924$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.9235349965$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.207360000.1 Defining polynomial: $$x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{5} ) q^{2} + ( -1 - \beta_{4} ) q^{4} + 2 \beta_{4} q^{7} + 2 \beta_{5} q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{5} ) q^{2} + ( -1 - \beta_{4} ) q^{4} + 2 \beta_{4} q^{7} + 2 \beta_{5} q^{8} -\beta_{7} q^{11} + 2 \beta_{3} q^{13} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{14} + ( -2 + 2 \beta_{4} ) q^{16} + 2 \beta_{7} q^{17} + ( -2 - \beta_{6} ) q^{19} + ( -\beta_{3} + \beta_{6} ) q^{22} + ( 2 \beta_{2} + \beta_{7} ) q^{23} + 5 q^{25} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{26} + ( 6 - 2 \beta_{4} ) q^{28} -\beta_{5} q^{29} + 4 \beta_{3} q^{31} -4 \beta_{1} q^{32} + ( 2 \beta_{3} - 2 \beta_{6} ) q^{34} + 2 \beta_{3} q^{37} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{38} + ( -2 \beta_{1} + \beta_{5} ) q^{41} -2 \beta_{2} q^{44} + ( -3 \beta_{3} - \beta_{6} ) q^{46} + ( 2 \beta_{2} + \beta_{7} ) q^{47} -5 q^{49} + ( 5 \beta_{1} - 5 \beta_{5} ) q^{50} + ( -2 \beta_{3} - 2 \beta_{6} ) q^{52} -3 \beta_{5} q^{53} + ( 8 \beta_{1} - 4 \beta_{5} ) q^{56} + ( 1 - \beta_{4} ) q^{58} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{59} + 6 \beta_{4} q^{61} + ( 4 \beta_{2} + 4 \beta_{7} ) q^{62} + 8 q^{64} + 2 \beta_{6} q^{67} + 4 \beta_{2} q^{68} -8 \beta_{5} q^{71} -12 q^{73} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{74} + ( 2 - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{76} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{77} + 2 \beta_{3} q^{79} + ( 3 + \beta_{4} ) q^{82} -5 \beta_{7} q^{83} + 4 \beta_{3} q^{88} + ( -14 \beta_{1} + 7 \beta_{5} ) q^{89} + 4 \beta_{6} q^{91} + ( -2 \beta_{2} - 4 \beta_{7} ) q^{92} + ( -3 \beta_{3} - \beta_{6} ) q^{94} + 4 \beta_{6} q^{97} + ( -5 \beta_{1} + 5 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} + O(q^{10})$$ $$8 q - 8 q^{4} - 16 q^{16} - 16 q^{19} + 40 q^{25} + 48 q^{28} - 40 q^{49} + 8 q^{58} + 64 q^{64} - 96 q^{73} + 16 q^{76} + 24 q^{82} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - 104 \nu$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 232 \nu$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 72$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} + 16 \nu^{4} + 96 \nu^{2} + 40$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 8 \nu$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 6 \nu^{4} - 28 \nu^{2} - 12$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} + 16 \nu^{5} + 88 \nu^{3} + 8 \nu$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{4} - \beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 4 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{6} - 7 \beta_{4} - 3 \beta_{3} - 7$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{7} + 22 \beta_{5} - 10 \beta_{2} - 22 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$32 \beta_{3} + 72$$ $$\nu^{7}$$ $$=$$ $$52 \beta_{2} + 116 \beta_{1}$$

## Character values

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

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 −0.437016 + 0.756934i 1.14412 − 1.98168i −0.437016 − 0.756934i 1.14412 + 1.98168i −1.14412 − 1.98168i 0.437016 + 0.756934i −1.14412 + 1.98168i 0.437016 − 0.756934i
−0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i 2.82843 0 0
379.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i 2.82843 0 0
379.3 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i 2.82843 0 0
379.4 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i 2.82843 0 0
379.5 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i −2.82843 0 0
379.6 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i −2.82843 0 0
379.7 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i −2.82843 0 0
379.8 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i −2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.8 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.d odd 2 1 inner
19.b odd 2 1 inner
24.f even 2 1 inner
57.d even 2 1 inner
152.b even 2 1 inner
456.l odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.e.d 8
3.b odd 2 1 inner 1368.2.e.d 8
4.b odd 2 1 5472.2.e.d 8
8.b even 2 1 5472.2.e.d 8
8.d odd 2 1 inner 1368.2.e.d 8
12.b even 2 1 5472.2.e.d 8
19.b odd 2 1 inner 1368.2.e.d 8
24.f even 2 1 inner 1368.2.e.d 8
24.h odd 2 1 5472.2.e.d 8
57.d even 2 1 inner 1368.2.e.d 8
76.d even 2 1 5472.2.e.d 8
152.b even 2 1 inner 1368.2.e.d 8
152.g odd 2 1 5472.2.e.d 8
228.b odd 2 1 5472.2.e.d 8
456.l odd 2 1 inner 1368.2.e.d 8
456.p even 2 1 5472.2.e.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.e.d 8 1.a even 1 1 trivial
1368.2.e.d 8 3.b odd 2 1 inner
1368.2.e.d 8 8.d odd 2 1 inner
1368.2.e.d 8 19.b odd 2 1 inner
1368.2.e.d 8 24.f even 2 1 inner
1368.2.e.d 8 57.d even 2 1 inner
1368.2.e.d 8 152.b even 2 1 inner
1368.2.e.d 8 456.l odd 2 1 inner
5472.2.e.d 8 4.b odd 2 1
5472.2.e.d 8 8.b even 2 1
5472.2.e.d 8 12.b even 2 1
5472.2.e.d 8 24.h odd 2 1
5472.2.e.d 8 76.d even 2 1
5472.2.e.d 8 152.g odd 2 1
5472.2.e.d 8 228.b odd 2 1
5472.2.e.d 8 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1368, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 12 + T^{2} )^{4}$$
$11$ $$( -10 + T^{2} )^{4}$$
$13$ $$( -20 + T^{2} )^{4}$$
$17$ $$( -40 + T^{2} )^{4}$$
$19$ $$( 19 + 4 T + T^{2} )^{4}$$
$23$ $$( 30 + T^{2} )^{4}$$
$29$ $$( -2 + T^{2} )^{4}$$
$31$ $$( -80 + T^{2} )^{4}$$
$37$ $$( -20 + T^{2} )^{4}$$
$41$ $$( 6 + T^{2} )^{4}$$
$43$ $$T^{8}$$
$47$ $$( 30 + T^{2} )^{4}$$
$53$ $$( -18 + T^{2} )^{4}$$
$59$ $$( 24 + T^{2} )^{4}$$
$61$ $$( 108 + T^{2} )^{4}$$
$67$ $$( 60 + T^{2} )^{4}$$
$71$ $$( -128 + T^{2} )^{4}$$
$73$ $$( 12 + T )^{8}$$
$79$ $$( -20 + T^{2} )^{4}$$
$83$ $$( -250 + T^{2} )^{4}$$
$89$ $$( 294 + T^{2} )^{4}$$
$97$ $$( 240 + T^{2} )^{4}$$