# Properties

 Label 833.2.e.b Level $833$ Weight $2$ Character orbit 833.e Analytic conductor $6.652$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$833 = 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 833.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.65153848837$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} + 3 q^{8} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^2 + (-z + 1) * q^4 + 2*z * q^5 + 3 * q^8 + 3*z * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} + 3 q^{8} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} - 2 q^{13} + \zeta_{6} q^{16} + (\zeta_{6} - 1) q^{17} + (3 \zeta_{6} - 3) q^{18} + 4 \zeta_{6} q^{19} + 2 q^{20} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 2 \zeta_{6} q^{26} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + ( - 5 \zeta_{6} + 5) q^{32} - q^{34} + 3 q^{36} + 2 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} + 6 \zeta_{6} q^{40} - 6 q^{41} + 4 q^{43} + (6 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 4) q^{46} + q^{50} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} + 6 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + 10 \zeta_{6} q^{61} - 4 q^{62} + 7 q^{64} - 4 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} + \zeta_{6} q^{68} - 4 q^{71} + 9 \zeta_{6} q^{72} + ( - 6 \zeta_{6} + 6) q^{73} + (2 \zeta_{6} - 2) q^{74} + 4 q^{76} - 12 \zeta_{6} q^{79} + (2 \zeta_{6} - 2) q^{80} + (9 \zeta_{6} - 9) q^{81} - 6 \zeta_{6} q^{82} - 4 q^{83} - 2 q^{85} + 4 \zeta_{6} q^{86} - 10 \zeta_{6} q^{89} - 6 q^{90} - 4 q^{92} + (8 \zeta_{6} - 8) q^{95} + 2 q^{97} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^4 + 2*z * q^5 + 3 * q^8 + 3*z * q^9 + (2*z - 2) * q^10 - 2 * q^13 + z * q^16 + (z - 1) * q^17 + (3*z - 3) * q^18 + 4*z * q^19 + 2 * q^20 - 4*z * q^23 + (-z + 1) * q^25 - 2*z * q^26 + 6 * q^29 + (4*z - 4) * q^31 + (-5*z + 5) * q^32 - q^34 + 3 * q^36 + 2*z * q^37 + (4*z - 4) * q^38 + 6*z * q^40 - 6 * q^41 + 4 * q^43 + (6*z - 6) * q^45 + (-4*z + 4) * q^46 + q^50 + (2*z - 2) * q^52 + (6*z - 6) * q^53 + 6*z * q^58 + (-12*z + 12) * q^59 + 10*z * q^61 - 4 * q^62 + 7 * q^64 - 4*z * q^65 + (4*z - 4) * q^67 + z * q^68 - 4 * q^71 + 9*z * q^72 + (-6*z + 6) * q^73 + (2*z - 2) * q^74 + 4 * q^76 - 12*z * q^79 + (2*z - 2) * q^80 + (9*z - 9) * q^81 - 6*z * q^82 - 4 * q^83 - 2 * q^85 + 4*z * q^86 - 10*z * q^89 - 6 * q^90 - 4 * q^92 + (8*z - 8) * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 2 q^{5} + 6 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 + q^4 + 2 * q^5 + 6 * q^8 + 3 * q^9 $$2 q + q^{2} + q^{4} + 2 q^{5} + 6 q^{8} + 3 q^{9} - 2 q^{10} - 4 q^{13} + q^{16} - q^{17} - 3 q^{18} + 4 q^{19} + 4 q^{20} - 4 q^{23} + q^{25} - 2 q^{26} + 12 q^{29} - 4 q^{31} + 5 q^{32} - 2 q^{34} + 6 q^{36} + 2 q^{37} - 4 q^{38} + 6 q^{40} - 12 q^{41} + 8 q^{43} - 6 q^{45} + 4 q^{46} + 2 q^{50} - 2 q^{52} - 6 q^{53} + 6 q^{58} + 12 q^{59} + 10 q^{61} - 8 q^{62} + 14 q^{64} - 4 q^{65} - 4 q^{67} + q^{68} - 8 q^{71} + 9 q^{72} + 6 q^{73} - 2 q^{74} + 8 q^{76} - 12 q^{79} - 2 q^{80} - 9 q^{81} - 6 q^{82} - 8 q^{83} - 4 q^{85} + 4 q^{86} - 10 q^{89} - 12 q^{90} - 8 q^{92} - 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + q^2 + q^4 + 2 * q^5 + 6 * q^8 + 3 * q^9 - 2 * q^10 - 4 * q^13 + q^16 - q^17 - 3 * q^18 + 4 * q^19 + 4 * q^20 - 4 * q^23 + q^25 - 2 * q^26 + 12 * q^29 - 4 * q^31 + 5 * q^32 - 2 * q^34 + 6 * q^36 + 2 * q^37 - 4 * q^38 + 6 * q^40 - 12 * q^41 + 8 * q^43 - 6 * q^45 + 4 * q^46 + 2 * q^50 - 2 * q^52 - 6 * q^53 + 6 * q^58 + 12 * q^59 + 10 * q^61 - 8 * q^62 + 14 * q^64 - 4 * q^65 - 4 * q^67 + q^68 - 8 * q^71 + 9 * q^72 + 6 * q^73 - 2 * q^74 + 8 * q^76 - 12 * q^79 - 2 * q^80 - 9 * q^81 - 6 * q^82 - 8 * q^83 - 4 * q^85 + 4 * q^86 - 10 * q^89 - 12 * q^90 - 8 * q^92 - 8 * q^95 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/833\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$785$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 + 1.73205i 0 0 3.00000 1.50000 + 2.59808i −1.00000 + 1.73205i
324.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 1.73205i 0 0 3.00000 1.50000 2.59808i −1.00000 1.73205i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.e.b 2
7.b odd 2 1 833.2.e.a 2
7.c even 3 1 17.2.a.a 1
7.c even 3 1 inner 833.2.e.b 2
7.d odd 6 1 833.2.a.a 1
7.d odd 6 1 833.2.e.a 2
21.g even 6 1 7497.2.a.l 1
21.h odd 6 1 153.2.a.c 1
28.g odd 6 1 272.2.a.b 1
35.j even 6 1 425.2.a.d 1
35.l odd 12 2 425.2.b.b 2
56.k odd 6 1 1088.2.a.h 1
56.p even 6 1 1088.2.a.i 1
77.h odd 6 1 2057.2.a.e 1
84.n even 6 1 2448.2.a.o 1
91.r even 6 1 2873.2.a.c 1
105.o odd 6 1 3825.2.a.d 1
119.j even 6 1 289.2.a.a 1
119.n even 12 2 289.2.b.a 2
119.q even 24 4 289.2.c.a 4
119.t odd 48 8 289.2.d.d 8
133.r odd 6 1 6137.2.a.b 1
140.p odd 6 1 6800.2.a.n 1
161.f odd 6 1 8993.2.a.a 1
168.s odd 6 1 9792.2.a.n 1
168.v even 6 1 9792.2.a.i 1
357.q odd 6 1 2601.2.a.g 1
476.o odd 6 1 4624.2.a.d 1
595.z even 6 1 7225.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 7.c even 3 1
153.2.a.c 1 21.h odd 6 1
272.2.a.b 1 28.g odd 6 1
289.2.a.a 1 119.j even 6 1
289.2.b.a 2 119.n even 12 2
289.2.c.a 4 119.q even 24 4
289.2.d.d 8 119.t odd 48 8
425.2.a.d 1 35.j even 6 1
425.2.b.b 2 35.l odd 12 2
833.2.a.a 1 7.d odd 6 1
833.2.e.a 2 7.b odd 2 1
833.2.e.a 2 7.d odd 6 1
833.2.e.b 2 1.a even 1 1 trivial
833.2.e.b 2 7.c even 3 1 inner
1088.2.a.h 1 56.k odd 6 1
1088.2.a.i 1 56.p even 6 1
2057.2.a.e 1 77.h odd 6 1
2448.2.a.o 1 84.n even 6 1
2601.2.a.g 1 357.q odd 6 1
2873.2.a.c 1 91.r even 6 1
3825.2.a.d 1 105.o odd 6 1
4624.2.a.d 1 476.o odd 6 1
6137.2.a.b 1 133.r odd 6 1
6800.2.a.n 1 140.p odd 6 1
7225.2.a.g 1 595.z even 6 1
7497.2.a.l 1 21.g even 6 1
8993.2.a.a 1 161.f odd 6 1
9792.2.a.i 1 168.v even 6 1
9792.2.a.n 1 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(833, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{3}$$ T3 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} + T + 1$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} + 12T + 144$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 10T + 100$$
$97$ $$(T - 2)^{2}$$