# Properties

 Label 819.2.j.c Level $819$ Weight $2$ Character orbit 819.j Analytic conductor $6.540$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (3 \beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + (2 \beta_{3} - 1) q^{7} + ( - 4 \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (-b3 - b1 - 1) * q^2 + (3*b2 + 3*b1) * q^4 + (-b3 + 2*b1 - 1) * q^5 + (2*b3 - 1) * q^7 + (-4*b2 + 1) * q^8 $$q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (3 \beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + (2 \beta_{3} - 1) q^{7} + ( - 4 \beta_{2} + 1) q^{8} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{10} + 3 \beta_{3} q^{11} - q^{13} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{14} + ( - 5 \beta_{3} - 3 \beta_1 - 5) q^{16} + (5 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{17} + ( - 3 \beta_{3} - 3) q^{19} + (3 \beta_{2} - 6) q^{20} + ( - 3 \beta_{2} + 3) q^{22} + ( - 5 \beta_{3} - 2 \beta_1 - 5) q^{23} + (\beta_{3} + \beta_1 + 1) q^{26} + ( - 3 \beta_{2} - 9 \beta_1) q^{28} + (4 \beta_{2} + 2) q^{29} + 5 \beta_{3} q^{31} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{32} + (3 \beta_{2} + 1) q^{34} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 + 3) q^{35} + (5 \beta_{3} - 6 \beta_1 + 5) q^{37} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{38} + (7 \beta_{3} + 6 \beta_1 + 7) q^{40} + ( - 4 \beta_{2} - 2) q^{41} - 8 q^{43} - 9 \beta_1 q^{44} + (7 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{46} + ( - \beta_{3} - 4 \beta_1 - 1) q^{47} + ( - 8 \beta_{3} - 3) q^{49} + ( - 3 \beta_{2} - 3 \beta_1) q^{52} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{53} + (6 \beta_{2} + 3) q^{55} + (2 \beta_{3} + 12 \beta_{2} + 8 \beta_1 - 1) q^{56} + (2 \beta_{3} + 6 \beta_1 + 2) q^{58} + ( - 5 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{59} + ( - 3 \beta_{3} - 3) q^{61} + ( - 5 \beta_{2} + 5) q^{62} + ( - 6 \beta_{2} - 1) q^{64} + (\beta_{3} - 2 \beta_1 + 1) q^{65} - 3 \beta_{3} q^{67} + (12 \beta_{3} - 3 \beta_1 + 12) q^{68} + (3 \beta_{3} + 3 \beta_{2} + 9 \beta_1 + 2) q^{70} + ( - 8 \beta_{2} - 4) q^{71} + (7 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{73} + (\beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{74} - 9 \beta_{2} q^{76} + ( - 9 \beta_{3} - 6) q^{77} + ( - 7 \beta_{3} + 6 \beta_1 - 7) q^{79} + ( - \beta_{3} - 13 \beta_{2} - 13 \beta_1) q^{80} + ( - 2 \beta_{3} - 6 \beta_1 - 2) q^{82} + (6 \beta_{2} + 13) q^{85} + (8 \beta_{3} + 8 \beta_1 + 8) q^{86} + (3 \beta_{3} + 12 \beta_{2} + 12 \beta_1) q^{88} + ( - \beta_{3} + 2 \beta_1 - 1) q^{89} + ( - 2 \beta_{3} + 1) q^{91} + ( - 21 \beta_{2} + 6) q^{92} + (5 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{94} + (3 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{95} + ( - 12 \beta_{2} - 10) q^{97} + (3 \beta_{3} + 8 \beta_{2} + 3 \beta_1 - 5) q^{98}+O(q^{100})$$ q + (-b3 - b1 - 1) * q^2 + (3*b2 + 3*b1) * q^4 + (-b3 + 2*b1 - 1) * q^5 + (2*b3 - 1) * q^7 + (-4*b2 + 1) * q^8 + (-b3 - 3*b2 - 3*b1) * q^10 + 3*b3 * q^11 - q^13 + (b3 - 2*b2 + b1 + 3) * q^14 + (-5*b3 - 3*b1 - 5) * q^16 + (5*b3 - 4*b2 - 4*b1) * q^17 + (-3*b3 - 3) * q^19 + (3*b2 - 6) * q^20 + (-3*b2 + 3) * q^22 + (-5*b3 - 2*b1 - 5) * q^23 + (b3 + b1 + 1) * q^26 + (-3*b2 - 9*b1) * q^28 + (4*b2 + 2) * q^29 + 5*b3 * q^31 + (6*b3 + 3*b2 + 3*b1) * q^32 + (3*b2 + 1) * q^34 + (b3 + 4*b2 - 2*b1 + 3) * q^35 + (5*b3 - 6*b1 + 5) * q^37 + (3*b3 + 3*b2 + 3*b1) * q^38 + (7*b3 + 6*b1 + 7) * q^40 + (-4*b2 - 2) * q^41 - 8 * q^43 - 9*b1 * q^44 + (7*b3 + 9*b2 + 9*b1) * q^46 + (-b3 - 4*b1 - 1) * q^47 + (-8*b3 - 3) * q^49 + (-3*b2 - 3*b1) * q^52 + (b3 + 4*b2 + 4*b1) * q^53 + (6*b2 + 3) * q^55 + (2*b3 + 12*b2 + 8*b1 - 1) * q^56 + (2*b3 + 6*b1 + 2) * q^58 + (-5*b3 + 4*b2 + 4*b1) * q^59 + (-3*b3 - 3) * q^61 + (-5*b2 + 5) * q^62 + (-6*b2 - 1) * q^64 + (b3 - 2*b1 + 1) * q^65 - 3*b3 * q^67 + (12*b3 - 3*b1 + 12) * q^68 + (3*b3 + 3*b2 + 9*b1 + 2) * q^70 + (-8*b2 - 4) * q^71 + (7*b3 - 6*b2 - 6*b1) * q^73 + (b3 + 7*b2 + 7*b1) * q^74 - 9*b2 * q^76 + (-9*b3 - 6) * q^77 + (-7*b3 + 6*b1 - 7) * q^79 + (-b3 - 13*b2 - 13*b1) * q^80 + (-2*b3 - 6*b1 - 2) * q^82 + (6*b2 + 13) * q^85 + (8*b3 + 8*b1 + 8) * q^86 + (3*b3 + 12*b2 + 12*b1) * q^88 + (-b3 + 2*b1 - 1) * q^89 + (-2*b3 + 1) * q^91 + (-21*b2 + 6) * q^92 + (5*b3 + 9*b2 + 9*b1) * q^94 + (3*b3 - 6*b2 - 6*b1) * q^95 + (-12*b2 - 10) * q^97 + (3*b3 + 8*b2 + 3*b1 - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} - 3 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10})$$ 4 * q - 3 * q^2 - 3 * q^4 - 8 * q^7 + 12 * q^8 $$4 q - 3 q^{2} - 3 q^{4} - 8 q^{7} + 12 q^{8} + 5 q^{10} - 6 q^{11} - 4 q^{13} + 15 q^{14} - 13 q^{16} - 6 q^{17} - 6 q^{19} - 30 q^{20} + 18 q^{22} - 12 q^{23} + 3 q^{26} - 3 q^{28} - 10 q^{31} - 15 q^{32} - 2 q^{34} + 4 q^{37} - 9 q^{38} + 20 q^{40} - 32 q^{43} - 9 q^{44} - 23 q^{46} - 6 q^{47} + 4 q^{49} + 3 q^{52} - 6 q^{53} - 24 q^{56} + 10 q^{58} + 6 q^{59} - 6 q^{61} + 30 q^{62} + 8 q^{64} + 6 q^{67} + 21 q^{68} + 5 q^{70} - 8 q^{73} - 9 q^{74} + 18 q^{76} - 6 q^{77} - 8 q^{79} + 15 q^{80} - 10 q^{82} + 40 q^{85} + 24 q^{86} - 18 q^{88} + 8 q^{91} + 66 q^{92} - 19 q^{94} - 16 q^{97} - 39 q^{98}+O(q^{100})$$ 4 * q - 3 * q^2 - 3 * q^4 - 8 * q^7 + 12 * q^8 + 5 * q^10 - 6 * q^11 - 4 * q^13 + 15 * q^14 - 13 * q^16 - 6 * q^17 - 6 * q^19 - 30 * q^20 + 18 * q^22 - 12 * q^23 + 3 * q^26 - 3 * q^28 - 10 * q^31 - 15 * q^32 - 2 * q^34 + 4 * q^37 - 9 * q^38 + 20 * q^40 - 32 * q^43 - 9 * q^44 - 23 * q^46 - 6 * q^47 + 4 * q^49 + 3 * q^52 - 6 * q^53 - 24 * q^56 + 10 * q^58 + 6 * q^59 - 6 * q^61 + 30 * q^62 + 8 * q^64 + 6 * q^67 + 21 * q^68 + 5 * q^70 - 8 * q^73 - 9 * q^74 + 18 * q^76 - 6 * q^77 - 8 * q^79 + 15 * q^80 - 10 * q^82 + 40 * q^85 + 24 * q^86 - 18 * q^88 + 8 * q^91 + 66 * q^92 - 19 * q^94 - 16 * q^97 - 39 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 1 ) / 2$$ (v^3 + 1) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_1$$ b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 1$$ 2*b2 - 1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−1.30902 2.26728i 0 −2.42705 + 4.20378i 1.11803 + 1.93649i 0 −2.00000 + 1.73205i 7.47214 0 2.92705 5.06980i
235.2 −0.190983 0.330792i 0 0.927051 1.60570i −1.11803 1.93649i 0 −2.00000 + 1.73205i −1.47214 0 −0.427051 + 0.739674i
352.1 −1.30902 + 2.26728i 0 −2.42705 4.20378i 1.11803 1.93649i 0 −2.00000 1.73205i 7.47214 0 2.92705 + 5.06980i
352.2 −0.190983 + 0.330792i 0 0.927051 + 1.60570i −1.11803 + 1.93649i 0 −2.00000 1.73205i −1.47214 0 −0.427051 0.739674i
 $$n$$: 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 819.2.j.c 4
3.b odd 2 1 91.2.e.b 4
7.c even 3 1 inner 819.2.j.c 4
7.c even 3 1 5733.2.a.v 2
7.d odd 6 1 5733.2.a.w 2
12.b even 2 1 1456.2.r.j 4
21.c even 2 1 637.2.e.h 4
21.g even 6 1 637.2.a.e 2
21.g even 6 1 637.2.e.h 4
21.h odd 6 1 91.2.e.b 4
21.h odd 6 1 637.2.a.f 2
39.d odd 2 1 1183.2.e.d 4
84.n even 6 1 1456.2.r.j 4
273.w odd 6 1 1183.2.e.d 4
273.w odd 6 1 8281.2.a.z 2
273.ba even 6 1 8281.2.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 3.b odd 2 1
91.2.e.b 4 21.h odd 6 1
637.2.a.e 2 21.g even 6 1
637.2.a.f 2 21.h odd 6 1
637.2.e.h 4 21.c even 2 1
637.2.e.h 4 21.g even 6 1
819.2.j.c 4 1.a even 1 1 trivial
819.2.j.c 4 7.c even 3 1 inner
1183.2.e.d 4 39.d odd 2 1
1183.2.e.d 4 273.w odd 6 1
1456.2.r.j 4 12.b even 2 1
1456.2.r.j 4 84.n even 6 1
5733.2.a.v 2 7.c even 3 1
5733.2.a.w 2 7.d odd 6 1
8281.2.a.z 2 273.w odd 6 1
8281.2.a.ba 2 273.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{3} + 8T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + 8 T^{2} + 3 T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 5T^{2} + 25$$
$7$ $$(T^{2} + 4 T + 7)^{2}$$
$11$ $$(T^{2} + 3 T + 9)^{2}$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121$$
$19$ $$(T^{2} + 3 T + 9)^{2}$$
$23$ $$T^{4} + 12 T^{3} + 113 T^{2} + \cdots + 961$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T^{2} + 5 T + 25)^{2}$$
$37$ $$T^{4} - 4 T^{3} + 57 T^{2} + \cdots + 1681$$
$41$ $$(T^{2} - 20)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121$$
$53$ $$T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121$$
$59$ $$T^{4} - 6 T^{3} + 47 T^{2} + 66 T + 121$$
$61$ $$(T^{2} + 3 T + 9)^{2}$$
$67$ $$(T^{2} - 3 T + 9)^{2}$$
$71$ $$(T^{2} - 80)^{2}$$
$73$ $$T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841$$
$79$ $$T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 5T^{2} + 25$$
$97$ $$(T^{2} + 8 T - 164)^{2}$$