# Properties

 Label 819.2.c.b Level $819$ Weight $2$ Character orbit 819.c Analytic conductor $6.540$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(64,819)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("819.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{5} - \beta_{3}) q^{5} + \beta_{3} q^{7} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{8}+O(q^{10})$$ q + (b4 - b3) * q^2 + (-b2 - b1 - 1) * q^4 + (-b5 - b3) * q^5 + b3 * q^7 + (-2*b5 + 2*b3) * q^8 $$q + (\beta_{4} - \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{5} - \beta_{3}) q^{5} + \beta_{3} q^{7} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{8} + \beta_1 q^{10} + ( - \beta_{4} + 3 \beta_{3}) q^{11} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{13} + (\beta_{2} + 1) q^{14} + (2 \beta_{2} + 2) q^{16} + (2 \beta_{2} - \beta_1 + 2) q^{17} + ( - 4 \beta_{5} + \beta_{4}) q^{19} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3}) q^{20} + (3 \beta_{2} + \beta_1 + 5) q^{22} + ( - 3 \beta_{2} + \beta_1 - 2) q^{23} + ( - \beta_{2} + 3 \beta_1 + 1) q^{25} + (\beta_{5} + \beta_{4} - 5 \beta_{3} - \beta_1 - 2) q^{26} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{28} + (2 \beta_{2} + 2 \beta_1 + 3) q^{29} + (\beta_{5} + 2 \beta_{4} + 3 \beta_{3}) q^{31} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{32} + (\beta_{5} + \beta_{4} - 5 \beta_{3}) q^{34} + ( - \beta_1 + 1) q^{35} + ( - 3 \beta_{4} - 3 \beta_{3}) q^{37} + (4 \beta_{2} + 3 \beta_1 + 2) q^{38} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{40} + (4 \beta_{4} - 2 \beta_{3}) q^{41} + ( - 2 \beta_{2} + 2 \beta_1 - 5) q^{43} + (4 \beta_{5} + 4 \beta_{4} - 6 \beta_{3}) q^{44} + ( - 2 \beta_{5} - \beta_{4} + 7 \beta_{3}) q^{46} + ( - 2 \beta_{5} + \beta_{4} + 6 \beta_{3}) q^{47} - q^{49} + (2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{50} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{52} + (\beta_{2} + \beta_1) q^{53} + ( - 3 \beta_1 + 2) q^{55} + ( - 2 \beta_1 - 2) q^{56} + (4 \beta_{5} + 5 \beta_{4} - 9 \beta_{3}) q^{58} + (3 \beta_{5} - \beta_{4} + \beta_{3}) q^{59} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{61} + (2 \beta_{2} - 3 \beta_1 - 2) q^{62} + 4 \beta_{2} q^{64} + ( - 4 \beta_{5} + \beta_{4} - 3 \beta_{3} + 1) q^{65} + (2 \beta_{5} - 4 \beta_{3}) q^{67} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{68} - \beta_{5} q^{70} + ( - 2 \beta_{5} - 5 \beta_{4} - 3 \beta_{3}) q^{71} + ( - 4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{73} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{74} + ( - \beta_{5} + 7 \beta_{4} - 13 \beta_{3}) q^{76} + ( - \beta_{2} - 3) q^{77} + (2 \beta_{2} + 4 \beta_1 + 5) q^{79} - 2 \beta_{5} q^{80} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{82} + (5 \beta_{5} - 4 \beta_{4} + \beta_{3}) q^{83} + ( - 4 \beta_{5} + \beta_{4} - 3 \beta_{3}) q^{85} + ( - 3 \beta_{4} + 7 \beta_{3}) q^{86} + ( - 4 \beta_{2} - 6 \beta_1 - 8) q^{88} + (2 \beta_{5} + 3 \beta_{4}) q^{89} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{91} + (3 \beta_{2} + 5 \beta_1 + 7) q^{92} + (8 \beta_{2} + \beta_1 + 6) q^{94} + ( - 4 \beta_{2} + 8 \beta_1 - 11) q^{95} + (2 \beta_{5} - 9 \beta_{4} + 8 \beta_{3}) q^{97} + ( - \beta_{4} + \beta_{3}) q^{98}+O(q^{100})$$ q + (b4 - b3) * q^2 + (-b2 - b1 - 1) * q^4 + (-b5 - b3) * q^5 + b3 * q^7 + (-2*b5 + 2*b3) * q^8 + b1 * q^10 + (-b4 + 3*b3) * q^11 + (b4 + 2*b2 - b1 + 2) * q^13 + (b2 + 1) * q^14 + (2*b2 + 2) * q^16 + (2*b2 - b1 + 2) * q^17 + (-4*b5 + b4) * q^19 + (-b5 + b4 - 3*b3) * q^20 + (3*b2 + b1 + 5) * q^22 + (-3*b2 + b1 - 2) * q^23 + (-b2 + 3*b1 + 1) * q^25 + (b5 + b4 - 5*b3 - b1 - 2) * q^26 + (b5 + b4 - b3) * q^28 + (2*b2 + 2*b1 + 3) * q^29 + (b5 + 2*b4 + 3*b3) * q^31 + (-2*b5 + 2*b4 - 2*b3) * q^32 + (b5 + b4 - 5*b3) * q^34 + (-b1 + 1) * q^35 + (-3*b4 - 3*b3) * q^37 + (4*b2 + 3*b1 + 2) * q^38 + (-2*b2 + 2*b1 - 4) * q^40 + (4*b4 - 2*b3) * q^41 + (-2*b2 + 2*b1 - 5) * q^43 + (4*b5 + 4*b4 - 6*b3) * q^44 + (-2*b5 - b4 + 7*b3) * q^46 + (-2*b5 + b4 + 6*b3) * q^47 - q^49 + (2*b5 + 4*b4 - 2*b3) * q^50 + (-b5 - b4 + 3*b3 - 2*b2 - 4*b1 - 4) * q^52 + (b2 + b1) * q^53 + (-3*b1 + 2) * q^55 + (-2*b1 - 2) * q^56 + (4*b5 + 5*b4 - 9*b3) * q^58 + (3*b5 - b4 + b3) * q^59 + (-2*b2 - 2*b1 + 4) * q^61 + (2*b2 - 3*b1 - 2) * q^62 + 4*b2 * q^64 + (-4*b5 + b4 - 3*b3 + 1) * q^65 + (2*b5 - 4*b3) * q^67 + (-2*b2 - 4*b1 - 4) * q^68 - b5 * q^70 + (-2*b5 - 5*b4 - 3*b3) * q^71 + (-4*b5 - 3*b4 + 4*b3) * q^73 + (-3*b2 + 3*b1 + 3) * q^74 + (-b5 + 7*b4 - 13*b3) * q^76 + (-b2 - 3) * q^77 + (2*b2 + 4*b1 + 5) * q^79 - 2*b5 * q^80 + (-2*b2 - 4*b1 - 10) * q^82 + (5*b5 - 4*b4 + b3) * q^83 + (-4*b5 + b4 - 3*b3) * q^85 + (-3*b4 + 7*b3) * q^86 + (-4*b2 - 6*b1 - 8) * q^88 + (2*b5 + 3*b4) * q^89 + (b5 - 2*b4 + 2*b3 + b2) * q^91 + (3*b2 + 5*b1 + 7) * q^92 + (8*b2 + b1 + 6) * q^94 + (-4*b2 + 8*b1 - 11) * q^95 + (2*b5 - 9*b4 + 8*b3) * q^97 + (-b4 + b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{4}+O(q^{10})$$ 6 * q - 4 * q^4 $$6 q - 4 q^{4} + 8 q^{13} + 4 q^{14} + 8 q^{16} + 8 q^{17} + 24 q^{22} - 6 q^{23} + 8 q^{25} - 12 q^{26} + 14 q^{29} + 6 q^{35} + 4 q^{38} - 20 q^{40} - 26 q^{43} - 6 q^{49} - 20 q^{52} - 2 q^{53} + 12 q^{55} - 12 q^{56} + 28 q^{61} - 16 q^{62} - 8 q^{64} + 6 q^{65} - 20 q^{68} + 24 q^{74} - 16 q^{77} + 26 q^{79} - 56 q^{82} - 40 q^{88} - 2 q^{91} + 36 q^{92} + 20 q^{94} - 58 q^{95}+O(q^{100})$$ 6 * q - 4 * q^4 + 8 * q^13 + 4 * q^14 + 8 * q^16 + 8 * q^17 + 24 * q^22 - 6 * q^23 + 8 * q^25 - 12 * q^26 + 14 * q^29 + 6 * q^35 + 4 * q^38 - 20 * q^40 - 26 * q^43 - 6 * q^49 - 20 * q^52 - 2 * q^53 + 12 * q^55 - 12 * q^56 + 28 * q^61 - 16 * q^62 - 8 * q^64 + 6 * q^65 - 20 * q^68 + 24 * q^74 - 16 * q^77 + 26 * q^79 - 56 * q^82 - 40 * q^88 - 2 * q^91 + 36 * q^92 + 20 * q^94 - 58 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

## Character values

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

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.403032 + 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 − 0.403032i
2.48119i 0 −4.15633 0.675131i 0 1.00000i 5.35026i 0 1.67513
64.2 1.17009i 0 0.630898 0.460811i 0 1.00000i 3.07838i 0 0.539189
64.3 0.688892i 0 1.52543 3.21432i 0 1.00000i 2.42864i 0 −2.21432
64.4 0.688892i 0 1.52543 3.21432i 0 1.00000i 2.42864i 0 −2.21432
64.5 1.17009i 0 0.630898 0.460811i 0 1.00000i 3.07838i 0 0.539189
64.6 2.48119i 0 −4.15633 0.675131i 0 1.00000i 5.35026i 0 1.67513
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.6 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 819.2.c.b 6
3.b odd 2 1 91.2.c.a 6
12.b even 2 1 1456.2.k.c 6
13.b even 2 1 inner 819.2.c.b 6
21.c even 2 1 637.2.c.d 6
21.g even 6 2 637.2.r.d 12
21.h odd 6 2 637.2.r.e 12
39.d odd 2 1 91.2.c.a 6
39.f even 4 1 1183.2.a.h 3
39.f even 4 1 1183.2.a.j 3
156.h even 2 1 1456.2.k.c 6
273.g even 2 1 637.2.c.d 6
273.o odd 4 1 8281.2.a.be 3
273.o odd 4 1 8281.2.a.bi 3
273.w odd 6 2 637.2.r.e 12
273.ba even 6 2 637.2.r.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 3.b odd 2 1
91.2.c.a 6 39.d odd 2 1
637.2.c.d 6 21.c even 2 1
637.2.c.d 6 273.g even 2 1
637.2.r.d 12 21.g even 6 2
637.2.r.d 12 273.ba even 6 2
637.2.r.e 12 21.h odd 6 2
637.2.r.e 12 273.w odd 6 2
819.2.c.b 6 1.a even 1 1 trivial
819.2.c.b 6 13.b even 2 1 inner
1183.2.a.h 3 39.f even 4 1
1183.2.a.j 3 39.f even 4 1
1456.2.k.c 6 12.b even 2 1
1456.2.k.c 6 156.h even 2 1
8281.2.a.be 3 273.o odd 4 1
8281.2.a.bi 3 273.o odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 8T_{2}^{4} + 12T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 8 T^{4} + 12 T^{2} + 4$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 11 T^{4} + 7 T^{2} + 1$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$T^{6} + 28 T^{4} + 164 T^{2} + \cdots + 100$$
$13$ $$T^{6} - 8 T^{5} + 7 T^{4} + 64 T^{3} + \cdots + 2197$$
$17$ $$(T^{3} - 4 T^{2} - 8 T + 34)^{2}$$
$19$ $$T^{6} + 119 T^{4} + 3867 T^{2} + \cdots + 37249$$
$23$ $$(T^{3} + 3 T^{2} - 25 T - 79)^{2}$$
$29$ $$(T^{3} - 7 T^{2} - 21 T - 5)^{2}$$
$31$ $$T^{6} + 83 T^{4} + 1791 T^{2} + \cdots + 4225$$
$37$ $$T^{6} + 108 T^{4} + 1620 T^{2} + \cdots + 2916$$
$41$ $$T^{6} + 108 T^{4} + 2864 T^{2} + \cdots + 1600$$
$43$ $$(T^{3} + 13 T^{2} + 35 T - 17)^{2}$$
$47$ $$T^{6} + 151 T^{4} + 5819 T^{2} + \cdots + 18769$$
$53$ $$(T^{3} + T^{2} - 9 T - 13)^{2}$$
$59$ $$T^{6} + 68 T^{4} + 816 T^{2} + \cdots + 2704$$
$61$ $$(T^{3} - 14 T^{2} + 28 T + 152)^{2}$$
$67$ $$T^{6} + 80 T^{4} + 1408 T^{2} + \cdots + 256$$
$71$ $$T^{6} + 304 T^{4} + 27836 T^{2} + \cdots + 792100$$
$73$ $$T^{6} + 263 T^{4} + 7723 T^{2} + \cdots + 961$$
$79$ $$(T^{3} - 13 T^{2} - 37 T + 185)^{2}$$
$83$ $$T^{6} + 227 T^{4} + 13095 T^{2} + \cdots + 26569$$
$89$ $$T^{6} + 119 T^{4} + 4387 T^{2} + \cdots + 51529$$
$97$ $$T^{6} + 575 T^{4} + 96115 T^{2} + \cdots + 4765489$$