Properties

 Label 2736.2.k.k Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 304) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{7} -2 \beta_{2} q^{11} -\beta_{3} q^{13} -3 q^{17} + ( \beta_{1} + 2 \beta_{2} ) q^{19} -3 \beta_{2} q^{23} -5 q^{25} + \beta_{3} q^{29} + 2 \beta_{1} q^{31} + 2 \beta_{3} q^{37} + 2 \beta_{3} q^{41} + 2 \beta_{2} q^{47} + 4 q^{49} -\beta_{3} q^{53} + 3 \beta_{1} q^{59} -8 q^{61} -\beta_{1} q^{67} -6 \beta_{1} q^{71} -11 q^{73} -6 q^{77} -2 \beta_{1} q^{79} -4 \beta_{2} q^{83} + 4 \beta_{3} q^{89} + 3 \beta_{1} q^{91} -2 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 12q^{17} - 20q^{25} + 16q^{49} - 32q^{61} - 44q^{73} - 24q^{77} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/7$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} + 7$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 14 \nu$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 1.32288 + 2.29129i −1.32288 − 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
0 0 0 0 0 1.73205i 0 0 0
2431.2 0 0 0 0 0 1.73205i 0 0 0
2431.3 0 0 0 0 0 1.73205i 0 0 0
2431.4 0 0 0 0 0 1.73205i 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
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.k 4
3.b odd 2 1 304.2.h.b 4
4.b odd 2 1 inner 2736.2.k.k 4
12.b even 2 1 304.2.h.b 4
19.b odd 2 1 inner 2736.2.k.k 4
24.f even 2 1 1216.2.h.c 4
24.h odd 2 1 1216.2.h.c 4
57.d even 2 1 304.2.h.b 4
76.d even 2 1 inner 2736.2.k.k 4
228.b odd 2 1 304.2.h.b 4
456.l odd 2 1 1216.2.h.c 4
456.p even 2 1 1216.2.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.b 4 3.b odd 2 1
304.2.h.b 4 12.b even 2 1
304.2.h.b 4 57.d even 2 1
304.2.h.b 4 228.b odd 2 1
1216.2.h.c 4 24.f even 2 1
1216.2.h.c 4 24.h odd 2 1
1216.2.h.c 4 456.l odd 2 1
1216.2.h.c 4 456.p even 2 1
2736.2.k.k 4 1.a even 1 1 trivial
2736.2.k.k 4 4.b odd 2 1 inner
2736.2.k.k 4 19.b odd 2 1 inner
2736.2.k.k 4 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} + 3$$ $$T_{11}^{2} + 12$$ $$T_{31}^{2} - 28$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 3 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( 21 + T^{2} )^{2}$$
$17$ $$( 3 + T )^{4}$$
$19$ $$361 + 10 T^{2} + T^{4}$$
$23$ $$( 27 + T^{2} )^{2}$$
$29$ $$( 21 + T^{2} )^{2}$$
$31$ $$( -28 + T^{2} )^{2}$$
$37$ $$( 84 + T^{2} )^{2}$$
$41$ $$( 84 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 12 + T^{2} )^{2}$$
$53$ $$( 21 + T^{2} )^{2}$$
$59$ $$( -63 + T^{2} )^{2}$$
$61$ $$( 8 + T )^{4}$$
$67$ $$( -7 + T^{2} )^{2}$$
$71$ $$( -252 + T^{2} )^{2}$$
$73$ $$( 11 + T )^{4}$$
$79$ $$( -28 + T^{2} )^{2}$$
$83$ $$( 48 + T^{2} )^{2}$$
$89$ $$( 336 + T^{2} )^{2}$$
$97$ $$( 84 + T^{2} )^{2}$$