# Properties

 Label 832.4.f.e Level $832$ Weight $4$ Character orbit 832.f Analytic conductor $49.090$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$832 = 2^{6} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 832.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$49.0895891248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + 9 i q^{5} -15 i q^{7} -26 q^{9} +O(q^{10})$$ $$q + q^{3} + 9 i q^{5} -15 i q^{7} -26 q^{9} -48 i q^{11} + ( -26 + 39 i ) q^{13} + 9 i q^{15} -45 q^{17} -6 i q^{19} -15 i q^{21} + 162 q^{23} + 44 q^{25} -53 q^{27} + 144 q^{29} + 264 i q^{31} -48 i q^{33} + 135 q^{35} + 303 i q^{37} + ( -26 + 39 i ) q^{39} -192 i q^{41} + 97 q^{43} -234 i q^{45} -111 i q^{47} + 118 q^{49} -45 q^{51} + 414 q^{53} + 432 q^{55} -6 i q^{57} + 522 i q^{59} -376 q^{61} + 390 i q^{63} + ( -351 - 234 i ) q^{65} + 36 i q^{67} + 162 q^{69} + 357 i q^{71} + 1098 i q^{73} + 44 q^{75} -720 q^{77} -830 q^{79} + 649 q^{81} + 438 i q^{83} -405 i q^{85} + 144 q^{87} + 438 i q^{89} + ( 585 + 390 i ) q^{91} + 264 i q^{93} + 54 q^{95} -852 i q^{97} + 1248 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 52 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 52 q^{9} - 52 q^{13} - 90 q^{17} + 324 q^{23} + 88 q^{25} - 106 q^{27} + 288 q^{29} + 270 q^{35} - 52 q^{39} + 194 q^{43} + 236 q^{49} - 90 q^{51} + 828 q^{53} + 864 q^{55} - 752 q^{61} - 702 q^{65} + 324 q^{69} + 88 q^{75} - 1440 q^{77} - 1660 q^{79} + 1298 q^{81} + 288 q^{87} + 1170 q^{91} + 108 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$703$$ $$769$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 0
129.2 0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.4.f.e 2
4.b odd 2 1 832.4.f.c 2
8.b even 2 1 13.4.b.a 2
8.d odd 2 1 208.4.f.b 2
13.b even 2 1 inner 832.4.f.e 2
24.h odd 2 1 117.4.b.a 2
40.f even 2 1 325.4.c.b 2
40.i odd 4 1 325.4.d.a 2
40.i odd 4 1 325.4.d.b 2
52.b odd 2 1 832.4.f.c 2
104.e even 2 1 13.4.b.a 2
104.h odd 2 1 208.4.f.b 2
104.j odd 4 1 169.4.a.b 1
104.j odd 4 1 169.4.a.c 1
104.r even 6 2 169.4.e.d 4
104.s even 6 2 169.4.e.d 4
104.x odd 12 2 169.4.c.b 2
104.x odd 12 2 169.4.c.c 2
312.b odd 2 1 117.4.b.a 2
312.y even 4 1 1521.4.a.d 1
312.y even 4 1 1521.4.a.i 1
520.p even 2 1 325.4.c.b 2
520.bg odd 4 1 325.4.d.a 2
520.bg odd 4 1 325.4.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 8.b even 2 1
13.4.b.a 2 104.e even 2 1
117.4.b.a 2 24.h odd 2 1
117.4.b.a 2 312.b odd 2 1
169.4.a.b 1 104.j odd 4 1
169.4.a.c 1 104.j odd 4 1
169.4.c.b 2 104.x odd 12 2
169.4.c.c 2 104.x odd 12 2
169.4.e.d 4 104.r even 6 2
169.4.e.d 4 104.s even 6 2
208.4.f.b 2 8.d odd 2 1
208.4.f.b 2 104.h odd 2 1
325.4.c.b 2 40.f even 2 1
325.4.c.b 2 520.p even 2 1
325.4.d.a 2 40.i odd 4 1
325.4.d.a 2 520.bg odd 4 1
325.4.d.b 2 40.i odd 4 1
325.4.d.b 2 520.bg odd 4 1
832.4.f.c 2 4.b odd 2 1
832.4.f.c 2 52.b odd 2 1
832.4.f.e 2 1.a even 1 1 trivial
832.4.f.e 2 13.b even 2 1 inner
1521.4.a.d 1 312.y even 4 1
1521.4.a.i 1 312.y even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(832, [\chi])$$:

 $$T_{3} - 1$$ $$T_{5}^{2} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$81 + T^{2}$$
$7$ $$225 + T^{2}$$
$11$ $$2304 + T^{2}$$
$13$ $$2197 + 52 T + T^{2}$$
$17$ $$( 45 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( -162 + T )^{2}$$
$29$ $$( -144 + T )^{2}$$
$31$ $$69696 + T^{2}$$
$37$ $$91809 + T^{2}$$
$41$ $$36864 + T^{2}$$
$43$ $$( -97 + T )^{2}$$
$47$ $$12321 + T^{2}$$
$53$ $$( -414 + T )^{2}$$
$59$ $$272484 + T^{2}$$
$61$ $$( 376 + T )^{2}$$
$67$ $$1296 + T^{2}$$
$71$ $$127449 + T^{2}$$
$73$ $$1205604 + T^{2}$$
$79$ $$( 830 + T )^{2}$$
$83$ $$191844 + T^{2}$$
$89$ $$191844 + T^{2}$$
$97$ $$725904 + T^{2}$$