Properties

 Label 2736.3.o.b Level $2736$ Weight $3$ Character orbit 2736.o Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.o (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$74.5506003290$$ 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 $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

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 2736.3.o.b 2
3.b odd 2 1 304.3.e.f 2
4.b odd 2 1 1368.3.o.a 2
12.b even 2 1 152.3.e.a 2
19.b odd 2 1 inner 2736.3.o.b 2
24.f even 2 1 1216.3.e.d 2
24.h odd 2 1 1216.3.e.c 2
57.d even 2 1 304.3.e.f 2
76.d even 2 1 1368.3.o.a 2
228.b odd 2 1 152.3.e.a 2
456.l odd 2 1 1216.3.e.d 2
456.p even 2 1 1216.3.e.c 2

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} + 7$$ $$T_{7} + 11$$

Hecke characteristic polynomials

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