# Properties

 Label 832.4.f.c 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: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - 3 \beta q^{5} - 5 \beta q^{7} - 26 q^{9} +O(q^{10})$$ q - q^3 - 3*b * q^5 - 5*b * q^7 - 26 * q^9 $$q - q^{3} - 3 \beta q^{5} - 5 \beta q^{7} - 26 q^{9} - 16 \beta q^{11} + ( - 13 \beta - 26) q^{13} + 3 \beta q^{15} - 45 q^{17} - 2 \beta q^{19} + 5 \beta q^{21} - 162 q^{23} + 44 q^{25} + 53 q^{27} + 144 q^{29} + 88 \beta q^{31} + 16 \beta q^{33} - 135 q^{35} - 101 \beta q^{37} + (13 \beta + 26) q^{39} + 64 \beta q^{41} - 97 q^{43} + 78 \beta q^{45} - 37 \beta q^{47} + 118 q^{49} + 45 q^{51} + 414 q^{53} - 432 q^{55} + 2 \beta q^{57} + 174 \beta q^{59} - 376 q^{61} + 130 \beta q^{63} + (78 \beta - 351) q^{65} + 12 \beta q^{67} + 162 q^{69} + 119 \beta q^{71} - 366 \beta q^{73} - 44 q^{75} - 720 q^{77} + 830 q^{79} + 649 q^{81} + 146 \beta q^{83} + 135 \beta q^{85} - 144 q^{87} - 146 \beta q^{89} + (130 \beta - 585) q^{91} - 88 \beta q^{93} - 54 q^{95} + 284 \beta q^{97} + 416 \beta q^{99} +O(q^{100})$$ q - q^3 - 3*b * q^5 - 5*b * q^7 - 26 * q^9 - 16*b * q^11 + (-13*b - 26) * q^13 + 3*b * q^15 - 45 * q^17 - 2*b * q^19 + 5*b * q^21 - 162 * q^23 + 44 * q^25 + 53 * q^27 + 144 * q^29 + 88*b * q^31 + 16*b * q^33 - 135 * q^35 - 101*b * q^37 + (13*b + 26) * q^39 + 64*b * q^41 - 97 * q^43 + 78*b * q^45 - 37*b * q^47 + 118 * q^49 + 45 * q^51 + 414 * q^53 - 432 * q^55 + 2*b * q^57 + 174*b * q^59 - 376 * q^61 + 130*b * q^63 + (78*b - 351) * q^65 + 12*b * q^67 + 162 * q^69 + 119*b * q^71 - 366*b * q^73 - 44 * q^75 - 720 * q^77 + 830 * q^79 + 649 * q^81 + 146*b * q^83 + 135*b * q^85 - 144 * q^87 - 146*b * q^89 + (130*b - 585) * q^91 - 88*b * q^93 - 54 * q^95 + 284*b * q^97 + 416*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 52 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 52 * q^9 $$2 q - 2 q^{3} - 52 q^{9} - 52 q^{13} - 90 q^{17} - 324 q^{23} + 88 q^{25} + 106 q^{27} + 288 q^{29} - 270 q^{35} + 52 q^{39} - 194 q^{43} + 236 q^{49} + 90 q^{51} + 828 q^{53} - 864 q^{55} - 752 q^{61} - 702 q^{65} + 324 q^{69} - 88 q^{75} - 1440 q^{77} + 1660 q^{79} + 1298 q^{81} - 288 q^{87} - 1170 q^{91} - 108 q^{95}+O(q^{100})$$ 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

## 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.c 2
4.b odd 2 1 832.4.f.e 2
8.b even 2 1 208.4.f.b 2
8.d odd 2 1 13.4.b.a 2
13.b even 2 1 inner 832.4.f.c 2
24.f even 2 1 117.4.b.a 2
40.e odd 2 1 325.4.c.b 2
40.k even 4 1 325.4.d.a 2
40.k even 4 1 325.4.d.b 2
52.b odd 2 1 832.4.f.e 2
104.e even 2 1 208.4.f.b 2
104.h odd 2 1 13.4.b.a 2
104.m even 4 1 169.4.a.b 1
104.m even 4 1 169.4.a.c 1
104.n odd 6 2 169.4.e.d 4
104.p odd 6 2 169.4.e.d 4
104.u even 12 2 169.4.c.b 2
104.u even 12 2 169.4.c.c 2
312.h even 2 1 117.4.b.a 2
312.w odd 4 1 1521.4.a.d 1
312.w odd 4 1 1521.4.a.i 1
520.b odd 2 1 325.4.c.b 2
520.bc even 4 1 325.4.d.a 2
520.bc even 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.d odd 2 1
13.4.b.a 2 104.h odd 2 1
117.4.b.a 2 24.f even 2 1
117.4.b.a 2 312.h even 2 1
169.4.a.b 1 104.m even 4 1
169.4.a.c 1 104.m even 4 1
169.4.c.b 2 104.u even 12 2
169.4.c.c 2 104.u even 12 2
169.4.e.d 4 104.n odd 6 2
169.4.e.d 4 104.p odd 6 2
208.4.f.b 2 8.b even 2 1
208.4.f.b 2 104.e even 2 1
325.4.c.b 2 40.e odd 2 1
325.4.c.b 2 520.b odd 2 1
325.4.d.a 2 40.k even 4 1
325.4.d.a 2 520.bc even 4 1
325.4.d.b 2 40.k even 4 1
325.4.d.b 2 520.bc even 4 1
832.4.f.c 2 1.a even 1 1 trivial
832.4.f.c 2 13.b even 2 1 inner
832.4.f.e 2 4.b odd 2 1
832.4.f.e 2 52.b odd 2 1
1521.4.a.d 1 312.w odd 4 1
1521.4.a.i 1 312.w odd 4 1

## Hecke kernels

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

 $$T_{3} + 1$$ T3 + 1 $$T_{5}^{2} + 81$$ T5^2 + 81

## Hecke characteristic polynomials

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