# Properties

 Label 2646.2.h.k Level $2646$ Weight $2$ Character orbit 2646.h Analytic conductor $21.128$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(361,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 4]))

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

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

chi := DirichletCharacter("2646.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.h (of order $$3$$, degree $$2$$, not 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: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - \beta_1 q^{4} + (\beta_{2} - 1) q^{5} + q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 - b1 * q^4 + (b2 - 1) * q^5 + q^8 $$q + (\beta_1 - 1) q^{2} - \beta_1 q^{4} + (\beta_{2} - 1) q^{5} + q^{8} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{10} + ( - \beta_{2} - 2) q^{11} + (2 \beta_1 - 2) q^{13} + (\beta_1 - 1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{17} - 5 \beta_1 q^{19} + ( - \beta_{3} + 2 \beta_1) q^{20} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{22} + (\beta_{2} + 5) q^{23} + ( - 3 \beta_{2} + 4) q^{25} - 2 \beta_1 q^{26} + (2 \beta_{3} - 4 \beta_1) q^{29} - 2 \beta_1 q^{31} - \beta_1 q^{32} + ( - \beta_{3} - \beta_1) q^{34} - 2 \beta_1 q^{37} + 5 q^{38} + (\beta_{2} - 1) q^{40} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 + 8) q^{41} + ( - 3 \beta_{3} + \beta_1) q^{43} + (\beta_{3} + \beta_1) q^{44} + (\beta_{3} - \beta_{2} + 4 \beta_1 - 5) q^{46} + ( - 3 \beta_{3} + 3 \beta_{2} + 7 \beta_1 - 4) q^{50} + 2 q^{52} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{53} - 6 q^{55} + ( - 2 \beta_{2} + 2) q^{58} + (3 \beta_{3} - 3 \beta_1) q^{59} + ( - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 7) q^{61} + 2 q^{62} + q^{64} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{65} + ( - 3 \beta_{3} - 5 \beta_1) q^{67} + (\beta_{2} + 2) q^{68} + ( - 3 \beta_{2} - 3) q^{71} + ( - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 2) q^{73} + 2 q^{74} + (5 \beta_1 - 5) q^{76} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 5) q^{79} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{80} + (\beta_{3} + 7 \beta_1) q^{82} + ( - 4 \beta_{3} + 8 \beta_1) q^{83} + (6 \beta_1 - 6) q^{85} + (3 \beta_{2} + 2) q^{86} + ( - \beta_{2} - 2) q^{88} + ( - 2 \beta_{3} + 10 \beta_1) q^{89} + ( - \beta_{3} - 4 \beta_1) q^{92} + ( - 5 \beta_{3} + 10 \beta_1) q^{95} + ( - 3 \beta_{3} + \beta_1) q^{97}+O(q^{100})$$ q + (b1 - 1) * q^2 - b1 * q^4 + (b2 - 1) * q^5 + q^8 + (b3 - b2 - 2*b1 + 1) * q^10 + (-b2 - 2) * q^11 + (2*b1 - 2) * q^13 + (b1 - 1) * q^16 + (b3 - b2 + b1 - 2) * q^17 - 5*b1 * q^19 + (-b3 + 2*b1) * q^20 + (-b3 + b2 - b1 + 2) * q^22 + (b2 + 5) * q^23 + (-3*b2 + 4) * q^25 - 2*b1 * q^26 + (2*b3 - 4*b1) * q^29 - 2*b1 * q^31 - b1 * q^32 + (-b3 - b1) * q^34 - 2*b1 * q^37 + 5 * q^38 + (b2 - 1) * q^40 + (-b3 + b2 - 7*b1 + 8) * q^41 + (-3*b3 + b1) * q^43 + (b3 + b1) * q^44 + (b3 - b2 + 4*b1 - 5) * q^46 + (-3*b3 + 3*b2 + 7*b1 - 4) * q^50 + 2 * q^52 + (-2*b3 + 2*b2 + 4*b1 - 2) * q^53 - 6 * q^55 + (-2*b2 + 2) * q^58 + (3*b3 - 3*b1) * q^59 + (-3*b3 + 3*b2 - 4*b1 + 7) * q^61 + 2 * q^62 + q^64 + (2*b3 - 2*b2 - 4*b1 + 2) * q^65 + (-3*b3 - 5*b1) * q^67 + (b2 + 2) * q^68 + (-3*b2 - 3) * q^71 + (-3*b3 + 3*b2 + 5*b1 - 2) * q^73 + 2 * q^74 + (5*b1 - 5) * q^76 + (3*b3 - 3*b2 + 2*b1 - 5) * q^79 + (b3 - b2 - 2*b1 + 1) * q^80 + (b3 + 7*b1) * q^82 + (-4*b3 + 8*b1) * q^83 + (6*b1 - 6) * q^85 + (3*b2 + 2) * q^86 + (-b2 - 2) * q^88 + (-2*b3 + 10*b1) * q^89 + (-b3 - 4*b1) * q^92 + (-5*b3 + 10*b1) * q^95 + (-3*b3 + b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} - 6 q^{5} + 4 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 - 6 * q^5 + 4 * q^8 $$4 q - 2 q^{2} - 2 q^{4} - 6 q^{5} + 4 q^{8} + 3 q^{10} - 6 q^{11} - 4 q^{13} - 2 q^{16} - 3 q^{17} - 10 q^{19} + 3 q^{20} + 3 q^{22} + 18 q^{23} + 22 q^{25} - 4 q^{26} - 6 q^{29} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 4 q^{37} + 20 q^{38} - 6 q^{40} + 15 q^{41} - q^{43} + 3 q^{44} - 9 q^{46} - 11 q^{50} + 8 q^{52} - 6 q^{53} - 24 q^{55} + 12 q^{58} - 3 q^{59} + 11 q^{61} + 8 q^{62} + 4 q^{64} + 6 q^{65} - 13 q^{67} + 6 q^{68} - 6 q^{71} - 7 q^{73} + 8 q^{74} - 10 q^{76} - 7 q^{79} + 3 q^{80} + 15 q^{82} + 12 q^{83} - 12 q^{85} + 2 q^{86} - 6 q^{88} + 18 q^{89} - 9 q^{92} + 15 q^{95} - q^{97}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 - 6 * q^5 + 4 * q^8 + 3 * q^10 - 6 * q^11 - 4 * q^13 - 2 * q^16 - 3 * q^17 - 10 * q^19 + 3 * q^20 + 3 * q^22 + 18 * q^23 + 22 * q^25 - 4 * q^26 - 6 * q^29 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 4 * q^37 + 20 * q^38 - 6 * q^40 + 15 * q^41 - q^43 + 3 * q^44 - 9 * q^46 - 11 * q^50 + 8 * q^52 - 6 * q^53 - 24 * q^55 + 12 * q^58 - 3 * q^59 + 11 * q^61 + 8 * q^62 + 4 * q^64 + 6 * q^65 - 13 * q^67 + 6 * q^68 - 6 * q^71 - 7 * q^73 + 8 * q^74 - 10 * q^76 - 7 * q^79 + 3 * q^80 + 15 * q^82 + 12 * q^83 - 12 * q^85 + 2 * q^86 - 6 * q^88 + 18 * q^89 - 9 * q^92 + 15 * q^95 - q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −4.37228 0 0 1.00000 0 2.18614 3.78651i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.37228 0 0 1.00000 0 −0.686141 + 1.18843i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −4.37228 0 0 1.00000 0 2.18614 + 3.78651i
667.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.37228 0 0 1.00000 0 −0.686141 1.18843i
 $$n$$: 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
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.h.k 4
3.b odd 2 1 882.2.h.m 4
7.b odd 2 1 2646.2.h.l 4
7.c even 3 1 378.2.f.c 4
7.c even 3 1 2646.2.e.n 4
7.d odd 6 1 2646.2.e.m 4
7.d odd 6 1 2646.2.f.j 4
9.c even 3 1 2646.2.e.n 4
9.d odd 6 1 882.2.e.l 4
21.c even 2 1 882.2.h.n 4
21.g even 6 1 882.2.e.k 4
21.g even 6 1 882.2.f.k 4
21.h odd 6 1 126.2.f.d 4
21.h odd 6 1 882.2.e.l 4
28.g odd 6 1 3024.2.r.f 4
63.g even 3 1 1134.2.a.n 2
63.g even 3 1 inner 2646.2.h.k 4
63.h even 3 1 378.2.f.c 4
63.i even 6 1 882.2.f.k 4
63.j odd 6 1 126.2.f.d 4
63.k odd 6 1 2646.2.h.l 4
63.k odd 6 1 7938.2.a.bs 2
63.l odd 6 1 2646.2.e.m 4
63.n odd 6 1 882.2.h.m 4
63.n odd 6 1 1134.2.a.k 2
63.o even 6 1 882.2.e.k 4
63.s even 6 1 882.2.h.n 4
63.s even 6 1 7938.2.a.bh 2
63.t odd 6 1 2646.2.f.j 4
84.n even 6 1 1008.2.r.f 4
252.o even 6 1 9072.2.a.bm 2
252.u odd 6 1 3024.2.r.f 4
252.bb even 6 1 1008.2.r.f 4
252.bl odd 6 1 9072.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 21.h odd 6 1
126.2.f.d 4 63.j odd 6 1
378.2.f.c 4 7.c even 3 1
378.2.f.c 4 63.h even 3 1
882.2.e.k 4 21.g even 6 1
882.2.e.k 4 63.o even 6 1
882.2.e.l 4 9.d odd 6 1
882.2.e.l 4 21.h odd 6 1
882.2.f.k 4 21.g even 6 1
882.2.f.k 4 63.i even 6 1
882.2.h.m 4 3.b odd 2 1
882.2.h.m 4 63.n odd 6 1
882.2.h.n 4 21.c even 2 1
882.2.h.n 4 63.s even 6 1
1008.2.r.f 4 84.n even 6 1
1008.2.r.f 4 252.bb even 6 1
1134.2.a.k 2 63.n odd 6 1
1134.2.a.n 2 63.g even 3 1
2646.2.e.m 4 7.d odd 6 1
2646.2.e.m 4 63.l odd 6 1
2646.2.e.n 4 7.c even 3 1
2646.2.e.n 4 9.c even 3 1
2646.2.f.j 4 7.d odd 6 1
2646.2.f.j 4 63.t odd 6 1
2646.2.h.k 4 1.a even 1 1 trivial
2646.2.h.k 4 63.g even 3 1 inner
2646.2.h.l 4 7.b odd 2 1
2646.2.h.l 4 63.k odd 6 1
3024.2.r.f 4 28.g odd 6 1
3024.2.r.f 4 252.u odd 6 1
7938.2.a.bh 2 63.s even 6 1
7938.2.a.bs 2 63.k odd 6 1
9072.2.a.bb 2 252.bl odd 6 1
9072.2.a.bm 2 252.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} - 6$$ T5^2 + 3*T5 - 6 $$T_{11}^{2} + 3T_{11} - 6$$ T11^2 + 3*T11 - 6 $$T_{13}^{2} + 2T_{13} + 4$$ T13^2 + 2*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 3 T - 6)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 3 T - 6)^{2}$$
$13$ $$(T^{2} + 2 T + 4)^{2}$$
$17$ $$T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36$$
$19$ $$(T^{2} + 5 T + 25)^{2}$$
$23$ $$(T^{2} - 9 T + 12)^{2}$$
$29$ $$T^{4} + 6 T^{3} + 60 T^{2} - 144 T + 576$$
$31$ $$(T^{2} + 2 T + 4)^{2}$$
$37$ $$(T^{2} + 2 T + 4)^{2}$$
$41$ $$T^{4} - 15 T^{3} + 177 T^{2} + \cdots + 2304$$
$43$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 6 T^{3} + 60 T^{2} - 144 T + 576$$
$59$ $$T^{4} + 3 T^{3} + 81 T^{2} + \cdots + 5184$$
$61$ $$T^{4} - 11 T^{3} + 165 T^{2} + \cdots + 1936$$
$67$ $$T^{4} + 13 T^{3} + 201 T^{2} + \cdots + 1024$$
$71$ $$(T^{2} + 3 T - 72)^{2}$$
$73$ $$T^{4} + 7 T^{3} + 111 T^{2} + \cdots + 3844$$
$79$ $$T^{4} + 7 T^{3} + 111 T^{2} + \cdots + 3844$$
$83$ $$T^{4} - 12 T^{3} + 240 T^{2} + \cdots + 9216$$
$89$ $$T^{4} - 18 T^{3} + 276 T^{2} + \cdots + 2304$$
$97$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$