# Properties

 Label 832.2.f.d Level $832$ Weight $2$ Character orbit 832.f Analytic conductor $6.644$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.64355344817$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 i q^{5} + 3 i q^{7} -2 q^{9} +O(q^{10})$$ $$q + q^{3} + 3 i q^{5} + 3 i q^{7} -2 q^{9} + ( -2 - 3 i ) q^{13} + 3 i q^{15} + 3 q^{17} + 6 i q^{19} + 3 i q^{21} -6 q^{23} -4 q^{25} -5 q^{27} -9 q^{35} -3 i q^{37} + ( -2 - 3 i ) q^{39} + q^{43} -6 i q^{45} + 3 i q^{47} -2 q^{49} + 3 q^{51} + 6 q^{53} + 6 i q^{57} + 6 i q^{59} + 8 q^{61} -6 i q^{63} + ( 9 - 6 i ) q^{65} + 12 i q^{67} -6 q^{69} + 15 i q^{71} + 6 i q^{73} -4 q^{75} + 10 q^{79} + q^{81} -6 i q^{83} + 9 i q^{85} -6 i q^{89} + ( 9 - 6 i ) q^{91} -18 q^{95} -12 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 4 q^{9} - 4 q^{13} + 6 q^{17} - 12 q^{23} - 8 q^{25} - 10 q^{27} - 18 q^{35} - 4 q^{39} + 2 q^{43} - 4 q^{49} + 6 q^{51} + 12 q^{53} + 16 q^{61} + 18 q^{65} - 12 q^{69} - 8 q^{75} + 20 q^{79} + 2 q^{81} + 18 q^{91} - 36 q^{95} + O(q^{100})$$

## 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 3.00000i 0 3.00000i 0 −2.00000 0
129.2 0 1.00000 0 3.00000i 0 3.00000i 0 −2.00000 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
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 832.2.f.d 2
4.b odd 2 1 832.2.f.b 2
8.b even 2 1 26.2.b.a 2
8.d odd 2 1 208.2.f.a 2
13.b even 2 1 inner 832.2.f.d 2
24.f even 2 1 1872.2.c.f 2
24.h odd 2 1 234.2.b.b 2
40.f even 2 1 650.2.d.b 2
40.i odd 4 1 650.2.c.a 2
40.i odd 4 1 650.2.c.d 2
52.b odd 2 1 832.2.f.b 2
56.h odd 2 1 1274.2.d.c 2
56.j odd 6 2 1274.2.n.c 4
56.p even 6 2 1274.2.n.d 4
104.e even 2 1 26.2.b.a 2
104.h odd 2 1 208.2.f.a 2
104.j odd 4 1 338.2.a.b 1
104.j odd 4 1 338.2.a.d 1
104.m even 4 1 2704.2.a.j 1
104.m even 4 1 2704.2.a.k 1
104.r even 6 2 338.2.e.c 4
104.s even 6 2 338.2.e.c 4
104.x odd 12 2 338.2.c.b 2
104.x odd 12 2 338.2.c.f 2
312.b odd 2 1 234.2.b.b 2
312.h even 2 1 1872.2.c.f 2
312.y even 4 1 3042.2.a.g 1
312.y even 4 1 3042.2.a.j 1
520.p even 2 1 650.2.d.b 2
520.bg odd 4 1 650.2.c.a 2
520.bg odd 4 1 650.2.c.d 2
520.bo odd 4 1 8450.2.a.h 1
520.bo odd 4 1 8450.2.a.u 1
728.l odd 2 1 1274.2.d.c 2
728.bv odd 6 2 1274.2.n.c 4
728.cj even 6 2 1274.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 8.b even 2 1
26.2.b.a 2 104.e even 2 1
208.2.f.a 2 8.d odd 2 1
208.2.f.a 2 104.h odd 2 1
234.2.b.b 2 24.h odd 2 1
234.2.b.b 2 312.b odd 2 1
338.2.a.b 1 104.j odd 4 1
338.2.a.d 1 104.j odd 4 1
338.2.c.b 2 104.x odd 12 2
338.2.c.f 2 104.x odd 12 2
338.2.e.c 4 104.r even 6 2
338.2.e.c 4 104.s even 6 2
650.2.c.a 2 40.i odd 4 1
650.2.c.a 2 520.bg odd 4 1
650.2.c.d 2 40.i odd 4 1
650.2.c.d 2 520.bg odd 4 1
650.2.d.b 2 40.f even 2 1
650.2.d.b 2 520.p even 2 1
832.2.f.b 2 4.b odd 2 1
832.2.f.b 2 52.b odd 2 1
832.2.f.d 2 1.a even 1 1 trivial
832.2.f.d 2 13.b even 2 1 inner
1274.2.d.c 2 56.h odd 2 1
1274.2.d.c 2 728.l odd 2 1
1274.2.n.c 4 56.j odd 6 2
1274.2.n.c 4 728.bv odd 6 2
1274.2.n.d 4 56.p even 6 2
1274.2.n.d 4 728.cj even 6 2
1872.2.c.f 2 24.f even 2 1
1872.2.c.f 2 312.h even 2 1
2704.2.a.j 1 104.m even 4 1
2704.2.a.k 1 104.m even 4 1
3042.2.a.g 1 312.y even 4 1
3042.2.a.j 1 312.y even 4 1
8450.2.a.h 1 520.bo odd 4 1
8450.2.a.u 1 520.bo odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(832, [\chi])$$:

 $$T_{3} - 1$$ $$T_{5}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$9 + T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 4 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$9 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$9 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$225 + T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$144 + T^{2}$$