# Properties

 Label 1792.2.m.d Level $1792$ Weight $2$ Character orbit 1792.m Analytic conductor $14.309$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} -\beta_{6} q^{7} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} -\beta_{6} q^{7} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{9} + ( -\beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{13} -\beta_{1} q^{15} + ( 1 - \beta_{2} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{19} + ( 1 - \beta_{2} ) q^{21} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{27} + ( 1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{29} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{33} + ( -\beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( -1 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{39} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} ) q^{41} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{6} + 2 \beta_{7} ) q^{43} + ( -1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{45} + ( 5 - \beta_{2} + \beta_{3} ) q^{47} - q^{49} + ( 2 - \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{51} + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{6} + \beta_{7} ) q^{53} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{55} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{57} + ( -8 + 2 \beta_{2} - \beta_{4} + 6 \beta_{6} + \beta_{7} ) q^{59} + ( 2 - \beta_{1} - 2 \beta_{3} + 3 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{61} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{63} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 6 + 2 \beta_{1} - 4 \beta_{3} + 6 \beta_{6} + 2 \beta_{7} ) q^{67} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{71} + ( 4 - 4 \beta_{2} - 4 \beta_{3} + 6 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -6 + 4 \beta_{2} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{75} + ( 2 - \beta_{1} - 2 \beta_{5} - \beta_{7} ) q^{77} + ( -\beta_{1} + 5 \beta_{2} - 5 \beta_{3} + \beta_{4} + \beta_{5} ) q^{79} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( -2 + 3 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{83} + ( -\beta_{1} + \beta_{7} ) q^{85} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{87} + ( 6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} ) q^{89} + ( -2 - \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{91} + ( 2 - 3 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{93} + ( 10 + 6 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{95} + ( 9 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{97} + ( -3 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} + 4q^{5} + O(q^{10})$$ $$8q + 4q^{3} + 4q^{5} - 8q^{11} - 12q^{13} + 8q^{17} + 4q^{19} + 4q^{21} - 8q^{27} + 8q^{31} - 16q^{33} - 4q^{35} + 8q^{37} - 24q^{43} - 12q^{45} + 40q^{47} - 8q^{49} + 24q^{51} - 16q^{53} - 52q^{59} + 20q^{61} - 24q^{65} + 32q^{67} - 8q^{69} - 28q^{75} + 8q^{77} + 16q^{81} - 12q^{83} - 12q^{91} + 40q^{93} + 80q^{95} + 72q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} + 19 \nu^{2} - 12 \nu + 4$$ $$\beta_{2}$$ $$=$$ $$\nu^{7} - 3 \nu^{6} + 11 \nu^{5} - 17 \nu^{4} + 26 \nu^{3} - 19 \nu^{2} + 13 \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 28 \nu^{4} + 43 \nu^{3} - 43 \nu^{2} + 29 \nu - 8$$ $$\beta_{4}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 59 \nu^{5} - 102 \nu^{4} + 146 \nu^{3} - 120 \nu^{2} + 65 \nu - 14$$ $$\beta_{5}$$ $$=$$ $$-5 \nu^{7} + 17 \nu^{6} - 59 \nu^{5} + 103 \nu^{4} - 148 \nu^{3} + 127 \nu^{2} - 71 \nu + 19$$ $$\beta_{6}$$ $$=$$ $$-8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31$$ $$\beta_{7}$$ $$=$$ $$-10 \nu^{7} + 35 \nu^{6} - 123 \nu^{5} + 220 \nu^{4} - 325 \nu^{3} + 285 \nu^{2} - 166 \nu + 42$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{7} + 5 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{7} + 11 \beta_{6} - 3 \beta_{5} - \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 5 \beta_{1} + 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} - 13 \beta_{6} - 5 \beta_{5} - 10 \beta_{4} - 7 \beta_{3} + 18 \beta_{2} - 10 \beta_{1} + 18$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$50 \beta_{7} - 67 \beta_{6} + 11 \beta_{5} - 9 \beta_{4} + 38 \beta_{3} + 26 \beta_{2} + 18 \beta_{1} - 48$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{7} - 7 \beta_{6} + 30 \beta_{5} + 33 \beta_{4} + 72 \beta_{3} - 61 \beta_{2} + 72 \beta_{1} - 122$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.5 + 2.10607i 0.5 + 0.0297061i 0.5 − 1.44392i 0.5 − 0.691860i 0.5 − 2.10607i 0.5 − 0.0297061i 0.5 + 1.44392i 0.5 + 0.691860i
0 −0.898966 + 0.898966i 0 −0.786578 0.786578i 0 1.00000i 0 1.38372i 0
449.2 0 −0.236813 + 0.236813i 0 2.98593 + 2.98593i 0 1.00000i 0 2.88784i 0
449.3 0 1.23681 1.23681i 0 −0.571717 0.571717i 0 1.00000i 0 0.0594122i 0
449.4 0 1.89897 1.89897i 0 0.372364 + 0.372364i 0 1.00000i 0 4.21215i 0
1345.1 0 −0.898966 0.898966i 0 −0.786578 + 0.786578i 0 1.00000i 0 1.38372i 0
1345.2 0 −0.236813 0.236813i 0 2.98593 2.98593i 0 1.00000i 0 2.88784i 0
1345.3 0 1.23681 + 1.23681i 0 −0.571717 + 0.571717i 0 1.00000i 0 0.0594122i 0
1345.4 0 1.89897 + 1.89897i 0 0.372364 0.372364i 0 1.00000i 0 4.21215i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1345.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.m.d yes 8
4.b odd 2 1 1792.2.m.b yes 8
8.b even 2 1 1792.2.m.a 8
8.d odd 2 1 1792.2.m.c yes 8
16.e even 4 1 1792.2.m.a 8
16.e even 4 1 inner 1792.2.m.d yes 8
16.f odd 4 1 1792.2.m.b yes 8
16.f odd 4 1 1792.2.m.c yes 8
32.g even 8 1 7168.2.a.t 4
32.g even 8 1 7168.2.a.x 4
32.h odd 8 1 7168.2.a.s 4
32.h odd 8 1 7168.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.a 8 8.b even 2 1
1792.2.m.a 8 16.e even 4 1
1792.2.m.b yes 8 4.b odd 2 1
1792.2.m.b yes 8 16.f odd 4 1
1792.2.m.c yes 8 8.d odd 2 1
1792.2.m.c yes 8 16.f odd 4 1
1792.2.m.d yes 8 1.a even 1 1 trivial
1792.2.m.d yes 8 16.e even 4 1 inner
7168.2.a.s 4 32.h odd 8 1
7168.2.a.t 4 32.g even 8 1
7168.2.a.w 4 32.h odd 8 1
7168.2.a.x 4 32.g even 8 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{8} - 4 T_{3}^{7} + 8 T_{3}^{6} - 8 T_{3}^{3} + 32 T_{3}^{2} + 16 T_{3} + 4$$ $$T_{5}^{8} - 4 T_{5}^{7} + 8 T_{5}^{6} + 24 T_{5}^{5} + 32 T_{5}^{4} + 8 T_{5}^{3} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 + 16 T + 32 T^{2} - 8 T^{3} + 8 T^{6} - 4 T^{7} + T^{8}$$
$5$ $$4 + 8 T^{3} + 32 T^{4} + 24 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$256 - 1536 T + 4608 T^{2} + 1408 T^{3} + 224 T^{4} - 32 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$13$ $$3844 + 2480 T + 800 T^{2} + 184 T^{3} + 320 T^{4} + 208 T^{5} + 72 T^{6} + 12 T^{7} + T^{8}$$
$17$ $$( 8 + 16 T - 4 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$19$ $$31684 + 48416 T + 36992 T^{2} - 16152 T^{3} + 3488 T^{4} - 24 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$23$ $$16 + 2272 T^{2} + 792 T^{4} + 56 T^{6} + T^{8}$$
$29$ $$16 - 512 T + 8192 T^{2} + 6656 T^{3} + 2696 T^{4} + 128 T^{5} + T^{8}$$
$31$ $$( 8 - 32 T - 44 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$37$ $$258064 - 268224 T + 139392 T^{2} - 35744 T^{3} + 4616 T^{4} - 48 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$41$ $$61504 + 125504 T^{2} + 8736 T^{4} + 184 T^{6} + T^{8}$$
$43$ $$984064 + 317440 T + 51200 T^{2} + 5888 T^{3} + 5120 T^{4} + 1664 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$47$ $$( 392 - 400 T + 140 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$53$ $$80656 + 45440 T + 12800 T^{2} + 1344 T^{3} + 968 T^{4} + 480 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$59$ $$21104836 + 16905920 T + 6771200 T^{2} + 1534872 T^{3} + 223136 T^{4} + 21384 T^{5} + 1352 T^{6} + 52 T^{7} + T^{8}$$
$61$ $$40934404 - 15867040 T + 3075200 T^{2} - 301560 T^{3} + 17696 T^{4} - 1080 T^{5} + 200 T^{6} - 20 T^{7} + T^{8}$$
$67$ $$20647936 - 5816320 T + 819200 T^{2} - 43008 T^{3} + 15488 T^{4} - 3840 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$71$ $$5837056 + 1673216 T^{2} + 44320 T^{4} + 384 T^{6} + T^{8}$$
$73$ $$2027776 + 1834752 T^{2} + 44896 T^{4} + 368 T^{6} + T^{8}$$
$79$ $$( 19088 - 128 T - 288 T^{2} + T^{4} )^{2}$$
$83$ $$9604 - 76048 T + 301088 T^{2} + 50040 T^{3} + 4160 T^{4} - 16 T^{5} + 72 T^{6} + 12 T^{7} + T^{8}$$
$89$ $$24760576 + 3843072 T^{2} + 96032 T^{4} + 576 T^{6} + T^{8}$$
$97$ $$( -3064 - 960 T + 388 T^{2} - 36 T^{3} + T^{4} )^{2}$$