# Properties

 Label 1368.3.o.a Level $1368$ Weight $3$ Character orbit 1368.o Analytic conductor $37.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$37.2753001645$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ 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 $$\beta = 4\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -7 q^{5} + 11 q^{7} +O(q^{10})$$ $$q -7 q^{5} + 11 q^{7} -3 q^{11} + 2 \beta q^{13} + 17 q^{17} -19 q^{19} -2 q^{23} + 24 q^{25} -7 \beta q^{29} -\beta q^{31} -77 q^{35} + 7 \beta q^{37} + 7 \beta q^{41} -21 q^{43} + 5 q^{47} + 72 q^{49} + \beta q^{53} + 21 q^{55} + 6 \beta q^{59} + 23 q^{61} -14 \beta q^{65} -7 \beta q^{67} + 16 \beta q^{71} + 39 q^{73} -33 q^{77} + 17 \beta q^{79} + 6 q^{83} -119 q^{85} + 21 \beta q^{89} + 22 \beta q^{91} + 133 q^{95} + 30 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 14q^{5} + 22q^{7} + O(q^{10})$$ $$2q - 14q^{5} + 22q^{7} - 6q^{11} + 34q^{17} - 38q^{19} - 4q^{23} + 48q^{25} - 154q^{35} - 42q^{43} + 10q^{47} + 144q^{49} + 42q^{55} + 46q^{61} + 78q^{73} - 66q^{77} + 12q^{83} - 238q^{85} + 266q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 − 1.41421i 1.41421i
0 0 0 −7.00000 0 11.0000 0 0 0
721.2 0 0 0 −7.00000 0 11.0000 0 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.3.o.a 2
3.b odd 2 1 152.3.e.a 2
4.b odd 2 1 2736.3.o.b 2
12.b even 2 1 304.3.e.f 2
19.b odd 2 1 inner 1368.3.o.a 2
24.f even 2 1 1216.3.e.c 2
24.h odd 2 1 1216.3.e.d 2
57.d even 2 1 152.3.e.a 2
76.d even 2 1 2736.3.o.b 2
228.b odd 2 1 304.3.e.f 2
456.l odd 2 1 1216.3.e.c 2
456.p even 2 1 1216.3.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.a 2 3.b odd 2 1
152.3.e.a 2 57.d even 2 1
304.3.e.f 2 12.b even 2 1
304.3.e.f 2 228.b odd 2 1
1216.3.e.c 2 24.f even 2 1
1216.3.e.c 2 456.l odd 2 1
1216.3.e.d 2 24.h odd 2 1
1216.3.e.d 2 456.p even 2 1
1368.3.o.a 2 1.a even 1 1 trivial
1368.3.o.a 2 19.b odd 2 1 inner
2736.3.o.b 2 4.b odd 2 1
2736.3.o.b 2 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 7$$ acting on $$S_{3}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 7 + T )^{2}$$
$7$ $$( -11 + T )^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$128 + T^{2}$$
$17$ $$( -17 + T )^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$( 2 + T )^{2}$$
$29$ $$1568 + T^{2}$$
$31$ $$32 + T^{2}$$
$37$ $$1568 + T^{2}$$
$41$ $$1568 + T^{2}$$
$43$ $$( 21 + T )^{2}$$
$47$ $$( -5 + T )^{2}$$
$53$ $$32 + T^{2}$$
$59$ $$1152 + T^{2}$$
$61$ $$( -23 + T )^{2}$$
$67$ $$1568 + T^{2}$$
$71$ $$8192 + T^{2}$$
$73$ $$( -39 + T )^{2}$$
$79$ $$9248 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$14112 + T^{2}$$
$97$ $$28800 + T^{2}$$