Properties

 Label 1216.2.h.d Level $1216$ Weight $2$ Character orbit 1216.h Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM no 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: $$8$$ Coefficient field: 8.0.14453810176.1 Defining polynomial: $$x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( 1 + \beta_{1} ) q^{5} -\beta_{7} q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( 1 + \beta_{1} ) q^{5} -\beta_{7} q^{7} + ( 1 + \beta_{1} ) q^{9} + \beta_{2} q^{11} -\beta_{6} q^{13} + ( 2 \beta_{3} - \beta_{4} ) q^{15} - q^{17} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -\beta_{5} + \beta_{6} ) q^{21} + ( \beta_{2} + \beta_{7} ) q^{23} + \beta_{1} q^{25} + ( -\beta_{3} - \beta_{4} ) q^{27} + ( \beta_{5} + \beta_{6} ) q^{29} -\beta_{4} q^{31} -\beta_{5} q^{33} + ( \beta_{2} - 2 \beta_{7} ) q^{35} -\beta_{5} q^{37} + ( \beta_{2} + 3 \beta_{7} ) q^{39} -\beta_{5} q^{41} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{43} + ( 5 + \beta_{1} ) q^{45} + 3 \beta_{2} q^{47} + ( -4 - 4 \beta_{1} ) q^{49} -\beta_{3} q^{51} + ( \beta_{5} - \beta_{6} ) q^{53} + ( -\beta_{2} - 2 \beta_{7} ) q^{55} + ( 4 + 3 \beta_{1} + \beta_{5} ) q^{57} + ( \beta_{3} - 3 \beta_{4} ) q^{59} + ( -7 - 3 \beta_{1} ) q^{61} + ( \beta_{2} - 2 \beta_{7} ) q^{63} + 2 \beta_{5} q^{65} + ( \beta_{3} + 2 \beta_{4} ) q^{67} -\beta_{6} q^{69} + ( -2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( 3 - 2 \beta_{1} ) q^{73} + ( \beta_{3} - \beta_{4} ) q^{75} + ( -5 - 3 \beta_{1} ) q^{77} + ( 4 \beta_{3} - 3 \beta_{4} ) q^{79} + ( -7 - 2 \beta_{1} ) q^{81} + ( 2 \beta_{2} + 4 \beta_{7} ) q^{83} + ( -1 - \beta_{1} ) q^{85} + ( -3 \beta_{2} - \beta_{7} ) q^{87} + ( \beta_{5} - 2 \beta_{6} ) q^{89} + ( 7 \beta_{3} - \beta_{4} ) q^{91} + 2 \beta_{1} q^{93} + ( \beta_{2} + 4 \beta_{3} + 2 \beta_{7} ) q^{95} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{97} + ( -\beta_{2} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{5} + 4q^{9} + O(q^{10})$$ $$8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 3 \nu^{4} + 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} + 2 \nu^{2}$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 2 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 6 \nu^{3} + 4 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 6 \nu^{3} + 20 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 2 \nu^{3} - 12 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + \nu^{4} + 4 \nu^{2} + 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{2} - \beta_{1} - 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} + \beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{4} + 5 \beta_{3}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{7} - 5 \beta_{2} + \beta_{1} - 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{6} - 2 \beta_{5} + \beta_{4} - 9 \beta_{3}$$$$)/2$$

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
 −0.331077 − 1.37491i −0.331077 + 1.37491i 1.06789 + 0.927153i 1.06789 − 0.927153i −1.06789 − 0.927153i −1.06789 + 0.927153i 0.331077 + 1.37491i 0.331077 − 1.37491i
0 −2.35829 0 2.56155 0 4.15286i 0 2.56155 0
1215.2 0 −2.35829 0 2.56155 0 4.15286i 0 2.56155 0
1215.3 0 −1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.4 0 −1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.5 0 1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.6 0 1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.7 0 2.35829 0 2.56155 0 4.15286i 0 2.56155 0
1215.8 0 2.35829 0 2.56155 0 4.15286i 0 2.56155 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1215.8 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 1216.2.h.d 8
4.b odd 2 1 inner 1216.2.h.d 8
8.b even 2 1 76.2.d.a 8
8.d odd 2 1 76.2.d.a 8
19.b odd 2 1 inner 1216.2.h.d 8
24.f even 2 1 684.2.f.b 8
24.h odd 2 1 684.2.f.b 8
76.d even 2 1 inner 1216.2.h.d 8
152.b even 2 1 76.2.d.a 8
152.g odd 2 1 76.2.d.a 8
456.l odd 2 1 684.2.f.b 8
456.p even 2 1 684.2.f.b 8

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

 $$T_{3}^{4} - 7 T_{3}^{2} + 8$$ $$T_{5}^{2} - T_{5} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 8 - 7 T^{2} + T^{4} )^{2}$$
$5$ $$( -4 - T + T^{2} )^{4}$$
$7$ $$( 13 + 18 T^{2} + T^{4} )^{2}$$
$11$ $$( 52 + 15 T^{2} + T^{4} )^{2}$$
$13$ $$( 416 + 41 T^{2} + T^{4} )^{2}$$
$17$ $$( 1 + T )^{8}$$
$19$ $$130321 - 5776 T^{2} + 718 T^{4} - 16 T^{6} + T^{8}$$
$23$ $$( 52 + 19 T^{2} + T^{4} )^{2}$$
$29$ $$( 104 + 73 T^{2} + T^{4} )^{2}$$
$31$ $$( 32 - 20 T^{2} + T^{4} )^{2}$$
$37$ $$( 416 + 44 T^{2} + T^{4} )^{2}$$
$41$ $$( 416 + 44 T^{2} + T^{4} )^{2}$$
$43$ $$( 208 + 123 T^{2} + T^{4} )^{2}$$
$47$ $$( 4212 + 135 T^{2} + T^{4} )^{2}$$
$53$ $$( 104 + 97 T^{2} + T^{4} )^{2}$$
$59$ $$( 5408 - 175 T^{2} + T^{4} )^{2}$$
$61$ $$( -8 + 11 T + T^{2} )^{4}$$
$67$ $$( 8 - 95 T^{2} + T^{4} )^{2}$$
$71$ $$( 512 - 124 T^{2} + T^{4} )^{2}$$
$73$ $$( -1 - 8 T + T^{2} )^{4}$$
$79$ $$( 11552 - 244 T^{2} + T^{4} )^{2}$$
$83$ $$( 13312 + 236 T^{2} + T^{4} )^{2}$$
$89$ $$( 6656 + 232 T^{2} + T^{4} )^{2}$$
$97$ $$( 1664 + 292 T^{2} + T^{4} )^{2}$$