Properties

 Label 1216.2.h.b Level $1216$ Weight $2$ Character orbit 1216.h Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.h (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{7} -3 q^{9} + ( 2 \beta_{1} - \beta_{3} ) q^{11} + ( -3 - \beta_{2} ) q^{17} + ( -\beta_{1} - 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{23} + ( 10 - \beta_{2} ) q^{25} + ( -8 \beta_{1} - 5 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{43} + ( 3 - 3 \beta_{2} ) q^{45} + ( -6 \beta_{1} + \beta_{3} ) q^{47} + ( -6 + 3 \beta_{2} ) q^{49} -7 \beta_{3} q^{55} + ( 7 + \beta_{2} ) q^{61} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{63} + ( -7 + 3 \beta_{2} ) q^{73} + ( -7 + \beta_{2} ) q^{77} + 9 q^{81} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -11 - 3 \beta_{2} ) q^{85} + ( 10 \beta_{1} + \beta_{3} ) q^{95} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} - 12q^{9} + O(q^{10})$$ $$4q - 2q^{5} - 12q^{9} - 14q^{17} + 38q^{25} + 6q^{45} - 18q^{49} + 30q^{61} - 22q^{73} - 26q^{77} + 36q^{81} - 50q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 15$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 9 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 5 \beta_{1} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} - 2 \beta_{1} + 7$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1215.1
 −1.63746 + 1.52274i −1.63746 − 1.52274i 2.13746 − 0.656712i 2.13746 + 0.656712i
0 0 0 −4.27492 0 4.77753i 0 −3.00000 0
1215.2 0 0 0 −4.27492 0 4.77753i 0 −3.00000 0
1215.3 0 0 0 3.27492 0 0.418627i 0 −3.00000 0
1215.4 0 0 0 3.27492 0 0.418627i 0 −3.00000 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 CM by $$\Q(\sqrt{-19})$$
4.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 1216.2.h.b 4
4.b odd 2 1 inner 1216.2.h.b 4
8.b even 2 1 304.2.h.c 4
8.d odd 2 1 304.2.h.c 4
19.b odd 2 1 CM 1216.2.h.b 4
24.f even 2 1 2736.2.k.j 4
24.h odd 2 1 2736.2.k.j 4
76.d even 2 1 inner 1216.2.h.b 4
152.b even 2 1 304.2.h.c 4
152.g odd 2 1 304.2.h.c 4
456.l odd 2 1 2736.2.k.j 4
456.p even 2 1 2736.2.k.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.c 4 8.b even 2 1
304.2.h.c 4 8.d odd 2 1
304.2.h.c 4 152.b even 2 1
304.2.h.c 4 152.g odd 2 1
1216.2.h.b 4 1.a even 1 1 trivial
1216.2.h.b 4 4.b odd 2 1 inner
1216.2.h.b 4 19.b odd 2 1 CM
1216.2.h.b 4 76.d even 2 1 inner
2736.2.k.j 4 24.f even 2 1
2736.2.k.j 4 24.h odd 2 1
2736.2.k.j 4 456.l odd 2 1
2736.2.k.j 4 456.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 $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}$$ $$T_{5}^{2} + T_{5} - 14$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -14 + T + T^{2} )^{2}$$
$7$ $$4 + 23 T^{2} + T^{4}$$
$11$ $$196 + 47 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -2 + 7 T + T^{2} )^{2}$$
$19$ $$( 19 + T^{2} )^{2}$$
$23$ $$( 76 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$1764 + 87 T^{2} + T^{4}$$
$47$ $$14884 + 263 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 42 - 15 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -98 + 11 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( 76 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$