# Properties

 Label 1368.2.e.b Level $1368$ Weight $2$ Character orbit 1368.e Analytic conductor $10.924$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{4} + ( -1 + 2 \beta_{3} ) q^{7} + ( \beta_{1} - 2 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{4} + ( -1 + 2 \beta_{3} ) q^{7} + ( \beta_{1} - 2 \beta_{2} ) q^{8} + 4 q^{11} + ( -\beta_{1} - \beta_{2} ) q^{13} + ( -\beta_{1} + 4 \beta_{2} ) q^{14} + ( -3 - \beta_{3} ) q^{16} + q^{17} + ( 4 + \beta_{1} - \beta_{2} ) q^{19} + 4 \beta_{1} q^{22} + ( 1 - 2 \beta_{3} ) q^{23} + 5 q^{25} + ( -3 + \beta_{3} ) q^{26} + ( 7 + \beta_{3} ) q^{28} + ( \beta_{1} + \beta_{2} ) q^{29} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{32} + \beta_{1} q^{34} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 + 4 \beta_{1} - \beta_{3} ) q^{38} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{41} + ( 4 - 4 \beta_{3} ) q^{44} + ( \beta_{1} - 4 \beta_{2} ) q^{46} + ( -2 + 4 \beta_{3} ) q^{47} -8 q^{49} + 5 \beta_{1} q^{50} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{52} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 7 \beta_{1} + 2 \beta_{2} ) q^{56} + ( 3 - \beta_{3} ) q^{58} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{59} + ( 12 - 4 \beta_{3} ) q^{62} + ( -7 + 3 \beta_{3} ) q^{64} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{67} + ( 1 - \beta_{3} ) q^{68} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{71} + 3 q^{73} + ( 6 - 2 \beta_{3} ) q^{74} + ( 4 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -4 + 8 \beta_{3} ) q^{77} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{79} + ( 4 + 4 \beta_{3} ) q^{82} + 8 q^{83} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 8 \beta_{2} ) q^{89} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{91} + ( -7 - \beta_{3} ) q^{92} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{94} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{97} -8 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + O(q^{10})$$ $$4 q + 2 q^{4} + 16 q^{11} - 14 q^{16} + 4 q^{17} + 16 q^{19} + 20 q^{25} - 10 q^{26} + 30 q^{28} - 6 q^{38} + 8 q^{44} - 32 q^{49} + 10 q^{58} + 40 q^{62} - 22 q^{64} + 2 q^{68} + 12 q^{73} + 20 q^{74} + 8 q^{76} + 24 q^{82} + 32 q^{83} - 30 q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} + \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} + 3 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + \beta_{1} - 2$$

## 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
 0.809017 − 1.40126i 0.809017 + 1.40126i −0.309017 + 0.535233i −0.309017 − 0.535233i
−1.11803 0.866025i 0 0.500000 + 1.93649i 0 0 3.87298i 1.11803 2.59808i 0 0
379.2 −1.11803 + 0.866025i 0 0.500000 1.93649i 0 0 3.87298i 1.11803 + 2.59808i 0 0
379.3 1.11803 0.866025i 0 0.500000 1.93649i 0 0 3.87298i −1.11803 2.59808i 0 0
379.4 1.11803 + 0.866025i 0 0.500000 + 1.93649i 0 0 3.87298i −1.11803 + 2.59808i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 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.b 4
3.b odd 2 1 152.2.b.b 4
4.b odd 2 1 5472.2.e.b 4
8.b even 2 1 5472.2.e.b 4
8.d odd 2 1 inner 1368.2.e.b 4
12.b even 2 1 608.2.b.b 4
19.b odd 2 1 inner 1368.2.e.b 4
24.f even 2 1 152.2.b.b 4
24.h odd 2 1 608.2.b.b 4
57.d even 2 1 152.2.b.b 4
76.d even 2 1 5472.2.e.b 4
152.b even 2 1 inner 1368.2.e.b 4
152.g odd 2 1 5472.2.e.b 4
228.b odd 2 1 608.2.b.b 4
456.l odd 2 1 152.2.b.b 4
456.p even 2 1 608.2.b.b 4

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 15$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 15 + T^{2} )^{2}$$
$11$ $$( -4 + T )^{4}$$
$13$ $$( -5 + T^{2} )^{2}$$
$17$ $$( -1 + T )^{4}$$
$19$ $$( 19 - 8 T + T^{2} )^{2}$$
$23$ $$( 15 + T^{2} )^{2}$$
$29$ $$( -5 + T^{2} )^{2}$$
$31$ $$( -80 + T^{2} )^{2}$$
$37$ $$( -20 + T^{2} )^{2}$$
$41$ $$( 48 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 60 + T^{2} )^{2}$$
$53$ $$( -45 + T^{2} )^{2}$$
$59$ $$( 75 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 147 + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( -3 + T )^{4}$$
$79$ $$( -80 + T^{2} )^{2}$$
$83$ $$( -8 + T )^{4}$$
$89$ $$( 192 + T^{2} )^{2}$$
$97$ $$( 48 + T^{2} )^{2}$$
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