# Properties

 Label 855.2.c.d Level $855$ Weight $2$ Character orbit 855.c Analytic conductor $6.827$ 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] = [855,2,Mod(514,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("855.514");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.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: $$6.82720937282$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.16516096.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 13x^{2} + 1$$ x^6 + 9*x^4 + 13*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{5} - \beta_{3}) q^{5} + (\beta_{4} + \beta_{3}) q^{7} + (\beta_{4} + \beta_{3} + \beta_1) q^{8}+O(q^{10})$$ q - b5 * q^2 + (b2 - 1) * q^4 + (b5 - b3) * q^5 + (b4 + b3) * q^7 + (b4 + b3 + b1) * q^8 $$q - \beta_{5} q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{5} - \beta_{3}) q^{5} + (\beta_{4} + \beta_{3}) q^{7} + (\beta_{4} + \beta_{3} + \beta_1) q^{8} + ( - \beta_{5} - 2 \beta_1 + 2) q^{10} + ( - \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{13} + ( - 2 \beta_{2} + 2) q^{14} + (\beta_{4} - \beta_{3} - 1) q^{16} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{17} + q^{19} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{20} + (2 \beta_{5} + 4 \beta_1) q^{22} + 2 \beta_1 q^{23} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{25} + (\beta_{4} - \beta_{3} - \beta_{2} - 2) q^{26} + ( - 4 \beta_{5} - 2 \beta_1) q^{28} + 6 q^{29} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{31} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{32} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{34} + (\beta_{5} + \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 1) q^{35} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{37} - \beta_{5} q^{38} + (2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{40} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{41} + (\beta_{4} + \beta_{3}) q^{43} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{44} + (2 \beta_{4} - 2 \beta_{3} - 2) q^{46} + (4 \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_1) q^{47} + ( - \beta_{4} + \beta_{3} + 1) q^{49} + ( - \beta_{5} - 4 \beta_{3}) q^{50} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_1) q^{52} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{53} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1 - 5) q^{55} + ( - 2 \beta_{4} + 2 \beta_{3} - 6) q^{56} - 6 \beta_{5} q^{58} + (2 \beta_{2} + 4) q^{59} + ( - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2) q^{61} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 6 \beta_1) q^{62} + ( - 3 \beta_{2} + 1) q^{64} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{65} + (3 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_1) q^{67} + ( - 6 \beta_{5} - 6 \beta_1) q^{68} + (4 \beta_{4} - 2 \beta_{2} + 2) q^{70} + (2 \beta_{2} - 8) q^{71} + ( - 4 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{73} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 6) q^{74} + (\beta_{2} - 1) q^{76} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_1) q^{77} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4) q^{79} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 5) q^{80} + (2 \beta_{4} + 2 \beta_{3} - 6 \beta_1) q^{82} + (4 \beta_{5} - 2 \beta_1) q^{83} + (3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3) q^{85} + ( - 2 \beta_{2} + 2) q^{86} + ( - 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{88} + ( - 2 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + 2) q^{89} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4) q^{91} + ( - 2 \beta_{5} - 4 \beta_1) q^{92} + (4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 6) q^{94} + (\beta_{5} - \beta_{3}) q^{95} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{97} + (\beta_{5} + 4 \beta_1) q^{98}+O(q^{100})$$ q - b5 * q^2 + (b2 - 1) * q^4 + (b5 - b3) * q^5 + (b4 + b3) * q^7 + (b4 + b3 + b1) * q^8 + (-b5 - 2*b1 + 2) * q^10 + (-b4 + b3) * q^11 + (-b5 + b4 + b3 + b1) * q^13 + (-2*b2 + 2) * q^14 + (b4 - b3 - 1) * q^16 + (b4 + b3 + 2*b1) * q^17 + q^19 + (-2*b4 + b2 - 1) * q^20 + (2*b5 + 4*b1) * q^22 + 2*b1 * q^23 + (b5 - b4 - 2*b2 + 2*b1) * q^25 + (b4 - b3 - b2 - 2) * q^26 + (-4*b5 - 2*b1) * q^28 + 6 * q^29 + (2*b4 - 2*b3 + 2*b2) * q^31 + (-b5 + 2*b4 + 2*b3 - 2*b1) * q^32 + (2*b4 - 2*b3 - 2*b2) * q^34 + (b5 + b4 + 2*b2 + 2*b1 + 1) * q^35 + (-b5 - b4 - b3 + b1) * q^37 - b5 * q^38 + (2*b5 + b4 + b3 + 2*b2 + b1 + 2) * q^40 + (2*b4 - 2*b3 + 2*b2 - 2) * q^41 + (b4 + b3) * q^43 + (2*b4 - 2*b3 - 2*b2 + 2) * q^44 + (2*b4 - 2*b3 - 2) * q^46 + (4*b5 - b4 - b3 + 4*b1) * q^47 + (-b4 + b3 + 1) * q^49 + (-b5 - 4*b3) * q^50 + (-3*b5 + b4 + b3 - 3*b1) * q^52 + (-3*b5 - b4 - b3 - b1) * q^53 + (-b5 + b4 + 2*b2 - 2*b1 - 5) * q^55 + (-2*b4 + 2*b3 - 6) * q^56 - 6*b5 * q^58 + (2*b2 + 4) * q^59 + (-3*b4 + 3*b3 - 2*b2 - 2) * q^61 + (-2*b5 + 2*b4 + 2*b3 - 6*b1) * q^62 + (-3*b2 + 1) * q^64 + (b5 + b4 + b3 + 2*b2 - b1 + 4) * q^65 + (3*b5 - 5*b4 - 5*b3 + 3*b1) * q^67 + (-6*b5 - 6*b1) * q^68 + (4*b4 - 2*b2 + 2) * q^70 + (2*b2 - 8) * q^71 + (-4*b5 + 3*b4 + 3*b3 - 2*b1) * q^73 + (b4 - b3 + 3*b2 - 6) * q^74 + (b2 - 1) * q^76 + (-2*b5 - b4 - b3 - 4*b1) * q^77 + (-2*b4 + 2*b3 - 2*b2 + 4) * q^79 + (-b4 + b3 - 2*b2 + 2*b1 + 5) * q^80 + (2*b4 + 2*b3 - 6*b1) * q^82 + (4*b5 - 2*b1) * q^83 + (3*b5 + b4 + 2*b3 + 2*b2 + 3) * q^85 + (-2*b2 + 2) * q^86 + (-4*b5 - 2*b4 - 2*b3 - 2*b1) * q^88 + (-2*b4 + 2*b3 - 8*b2 + 2) * q^89 + (-2*b4 + 2*b3 - 2*b2 - 4) * q^91 + (-2*b5 - 4*b1) * q^92 + (4*b4 - 4*b3 - 2*b2 + 6) * q^94 + (b5 - b3) * q^95 + (b5 - b4 - b3 - b1) * q^97 + (b5 + 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{4} + q^{5}+O(q^{10})$$ 6 * q - 8 * q^4 + q^5 $$6 q - 8 q^{4} + q^{5} + 12 q^{10} - 2 q^{11} + 16 q^{14} - 4 q^{16} + 6 q^{19} - 10 q^{20} + 3 q^{25} - 8 q^{26} + 36 q^{29} + 8 q^{34} + 3 q^{35} + 8 q^{40} - 12 q^{41} + 20 q^{44} - 8 q^{46} + 4 q^{49} + 4 q^{50} - 33 q^{55} - 40 q^{56} + 20 q^{59} - 14 q^{61} + 12 q^{64} + 20 q^{65} + 20 q^{70} - 52 q^{71} - 40 q^{74} - 8 q^{76} + 24 q^{79} + 32 q^{80} + 13 q^{85} + 16 q^{86} + 24 q^{89} - 24 q^{91} + 48 q^{94} + q^{95}+O(q^{100})$$ 6 * q - 8 * q^4 + q^5 + 12 * q^10 - 2 * q^11 + 16 * q^14 - 4 * q^16 + 6 * q^19 - 10 * q^20 + 3 * q^25 - 8 * q^26 + 36 * q^29 + 8 * q^34 + 3 * q^35 + 8 * q^40 - 12 * q^41 + 20 * q^44 - 8 * q^46 + 4 * q^49 + 4 * q^50 - 33 * q^55 - 40 * q^56 + 20 * q^59 - 14 * q^61 + 12 * q^64 + 20 * q^65 + 20 * q^70 - 52 * q^71 - 40 * q^74 - 8 * q^76 + 24 * q^79 + 32 * q^80 + 13 * q^85 + 16 * q^86 + 24 * q^89 - 24 * q^91 + 48 * q^94 + q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9x^{4} + 13x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 8\nu^{2} + 5 ) / 2$$ (v^4 + 8*v^2 + 5) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + \nu^{4} + 10\nu^{3} + 6\nu^{2} + 19\nu - 1 ) / 4$$ (v^5 + v^4 + 10*v^3 + 6*v^2 + 19*v - 1) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 10\nu^{3} - 6\nu^{2} + 19\nu + 1 ) / 4$$ (v^5 - v^4 + 10*v^3 - 6*v^2 + 19*v + 1) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 8\nu^{3} - 7\nu ) / 2$$ (-v^5 - 8*v^3 - 7*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} - 3$$ b4 - b3 + b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - 6\beta_1$$ b5 + b4 + b3 - 6*b1 $$\nu^{4}$$ $$=$$ $$-8\beta_{4} + 8\beta_{3} - 6\beta_{2} + 19$$ -8*b4 + 8*b3 - 6*b2 + 19 $$\nu^{5}$$ $$=$$ $$-10\beta_{5} - 8\beta_{4} - 8\beta_{3} + 41\beta_1$$ -10*b5 - 8*b4 - 8*b3 + 41*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
514.1
 1.30397i − 2.68667i − 0.285442i 0.285442i 2.68667i − 1.30397i
2.41987i 0 −3.85577 2.07772 + 0.826491i 0 3.18676i 4.49073i 0 2.00000 5.02781i
514.2 1.82254i 0 −1.32164 −1.94827 + 1.09737i 0 1.45033i 1.23634i 0 2.00000 + 3.55080i
514.3 0.906968i 0 1.17741 0.370556 + 2.20515i 0 2.59637i 2.88181i 0 2.00000 0.336083i
514.4 0.906968i 0 1.17741 0.370556 2.20515i 0 2.59637i 2.88181i 0 2.00000 + 0.336083i
514.5 1.82254i 0 −1.32164 −1.94827 1.09737i 0 1.45033i 1.23634i 0 2.00000 3.55080i
514.6 2.41987i 0 −3.85577 2.07772 0.826491i 0 3.18676i 4.49073i 0 2.00000 + 5.02781i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 514.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.c.d 6
3.b odd 2 1 95.2.b.b 6
5.b even 2 1 inner 855.2.c.d 6
5.c odd 4 2 4275.2.a.br 6
12.b even 2 1 1520.2.d.h 6
15.d odd 2 1 95.2.b.b 6
15.e even 4 2 475.2.a.j 6
57.d even 2 1 1805.2.b.e 6
60.h even 2 1 1520.2.d.h 6
60.l odd 4 2 7600.2.a.ck 6
285.b even 2 1 1805.2.b.e 6
285.j odd 4 2 9025.2.a.bx 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 3.b odd 2 1
95.2.b.b 6 15.d odd 2 1
475.2.a.j 6 15.e even 4 2
855.2.c.d 6 1.a even 1 1 trivial
855.2.c.d 6 5.b even 2 1 inner
1520.2.d.h 6 12.b even 2 1
1520.2.d.h 6 60.h even 2 1
1805.2.b.e 6 57.d even 2 1
1805.2.b.e 6 285.b even 2 1
4275.2.a.br 6 5.c odd 4 2
7600.2.a.ck 6 60.l odd 4 2
9025.2.a.bx 6 285.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(855, [\chi])$$:

 $$T_{2}^{6} + 10T_{2}^{4} + 27T_{2}^{2} + 16$$ T2^6 + 10*T2^4 + 27*T2^2 + 16 $$T_{11}^{3} + T_{11}^{2} - 16T_{11} - 12$$ T11^3 + T11^2 - 16*T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 10 T^{4} + 27 T^{2} + 16$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{5} - T^{4} + 2 T^{3} - 5 T^{2} + \cdots + 125$$
$7$ $$T^{6} + 19 T^{4} + 104 T^{2} + \cdots + 144$$
$11$ $$(T^{3} + T^{2} - 16 T - 12)^{2}$$
$13$ $$T^{6} + 28 T^{4} + 236 T^{2} + \cdots + 576$$
$17$ $$T^{6} + 59 T^{4} + 1008 T^{2} + \cdots + 5184$$
$19$ $$(T - 1)^{6}$$
$23$ $$T^{6} + 36 T^{4} + 208 T^{2} + \cdots + 64$$
$29$ $$(T - 6)^{6}$$
$31$ $$(T^{3} - 56 T + 128)^{2}$$
$37$ $$T^{6} + 56 T^{4} + 764 T^{2} + \cdots + 1296$$
$41$ $$(T^{3} + 6 T^{2} - 44 T + 24)^{2}$$
$43$ $$T^{6} + 19 T^{4} + 104 T^{2} + \cdots + 144$$
$47$ $$T^{6} + 187 T^{4} + 7464 T^{2} + \cdots + 85264$$
$53$ $$T^{6} + 156 T^{4} + 2476 T^{2} + \cdots + 64$$
$59$ $$(T^{3} - 10 T^{2} + 8 T + 48)^{2}$$
$61$ $$(T^{3} + 7 T^{2} - 104 T - 776)^{2}$$
$67$ $$T^{6} + 340 T^{4} + 28556 T^{2} + \cdots + 484416$$
$71$ $$(T^{3} + 26 T^{2} + 200 T + 432)^{2}$$
$73$ $$T^{6} + 131 T^{4} + 1616 T^{2} + \cdots + 5184$$
$79$ $$(T^{3} - 12 T^{2} - 8 T + 32)^{2}$$
$83$ $$T^{6} + 228 T^{4} + 11728 T^{2} + \cdots + 141376$$
$89$ $$(T^{3} - 12 T^{2} - 284 T + 3456)^{2}$$
$97$ $$T^{6} + 28 T^{4} + 236 T^{2} + \cdots + 576$$