Properties

 Label 1323.4.a.u Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.a (trivial)

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: yes Analytic conductor: $$78.0595269376$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$\beta = \sqrt{21}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 13 q^{4} + \beta q^{5} - 5 \beta q^{8} +O(q^{10})$$ q - b * q^2 + 13 * q^4 + b * q^5 - 5*b * q^8 $$q - \beta q^{2} + 13 q^{4} + \beta q^{5} - 5 \beta q^{8} - 21 q^{10} - 5 \beta q^{11} - 44 q^{13} + q^{16} + 14 \beta q^{17} + 49 q^{19} + 13 \beta q^{20} + 105 q^{22} + 17 \beta q^{23} - 104 q^{25} + 44 \beta q^{26} + 48 \beta q^{29} - 191 q^{31} + 39 \beta q^{32} - 294 q^{34} + 29 q^{37} - 49 \beta q^{38} - 105 q^{40} - 81 \beta q^{41} + 386 q^{43} - 65 \beta q^{44} - 357 q^{46} - 14 \beta q^{47} + 104 \beta q^{50} - 572 q^{52} + 28 \beta q^{53} - 105 q^{55} - 1008 q^{58} - 154 \beta q^{59} - 128 q^{61} + 191 \beta q^{62} - 827 q^{64} - 44 \beta q^{65} - 988 q^{67} + 182 \beta q^{68} + 177 \beta q^{71} - 266 q^{73} - 29 \beta q^{74} + 637 q^{76} + 146 q^{79} + \beta q^{80} + 1701 q^{82} + 312 \beta q^{83} + 294 q^{85} - 386 \beta q^{86} + 525 q^{88} + 175 \beta q^{89} + 221 \beta q^{92} + 294 q^{94} + 49 \beta q^{95} - 152 q^{97} +O(q^{100})$$ q - b * q^2 + 13 * q^4 + b * q^5 - 5*b * q^8 - 21 * q^10 - 5*b * q^11 - 44 * q^13 + q^16 + 14*b * q^17 + 49 * q^19 + 13*b * q^20 + 105 * q^22 + 17*b * q^23 - 104 * q^25 + 44*b * q^26 + 48*b * q^29 - 191 * q^31 + 39*b * q^32 - 294 * q^34 + 29 * q^37 - 49*b * q^38 - 105 * q^40 - 81*b * q^41 + 386 * q^43 - 65*b * q^44 - 357 * q^46 - 14*b * q^47 + 104*b * q^50 - 572 * q^52 + 28*b * q^53 - 105 * q^55 - 1008 * q^58 - 154*b * q^59 - 128 * q^61 + 191*b * q^62 - 827 * q^64 - 44*b * q^65 - 988 * q^67 + 182*b * q^68 + 177*b * q^71 - 266 * q^73 - 29*b * q^74 + 637 * q^76 + 146 * q^79 + b * q^80 + 1701 * q^82 + 312*b * q^83 + 294 * q^85 - 386*b * q^86 + 525 * q^88 + 175*b * q^89 + 221*b * q^92 + 294 * q^94 + 49*b * q^95 - 152 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 26 q^{4}+O(q^{10})$$ 2 * q + 26 * q^4 $$2 q + 26 q^{4} - 42 q^{10} - 88 q^{13} + 2 q^{16} + 98 q^{19} + 210 q^{22} - 208 q^{25} - 382 q^{31} - 588 q^{34} + 58 q^{37} - 210 q^{40} + 772 q^{43} - 714 q^{46} - 1144 q^{52} - 210 q^{55} - 2016 q^{58} - 256 q^{61} - 1654 q^{64} - 1976 q^{67} - 532 q^{73} + 1274 q^{76} + 292 q^{79} + 3402 q^{82} + 588 q^{85} + 1050 q^{88} + 588 q^{94} - 304 q^{97}+O(q^{100})$$ 2 * q + 26 * q^4 - 42 * q^10 - 88 * q^13 + 2 * q^16 + 98 * q^19 + 210 * q^22 - 208 * q^25 - 382 * q^31 - 588 * q^34 + 58 * q^37 - 210 * q^40 + 772 * q^43 - 714 * q^46 - 1144 * q^52 - 210 * q^55 - 2016 * q^58 - 256 * q^61 - 1654 * q^64 - 1976 * q^67 - 532 * q^73 + 1274 * q^76 + 292 * q^79 + 3402 * q^82 + 588 * q^85 + 1050 * q^88 + 588 * q^94 - 304 * q^97

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.79129 −1.79129
−4.58258 0 13.0000 4.58258 0 0 −22.9129 0 −21.0000
1.2 4.58258 0 13.0000 −4.58258 0 0 22.9129 0 −21.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.u 2
3.b odd 2 1 inner 1323.4.a.u 2
7.b odd 2 1 189.4.a.h 2
21.c even 2 1 189.4.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.h 2 7.b odd 2 1
189.4.a.h 2 21.c even 2 1
1323.4.a.u 2 1.a even 1 1 trivial
1323.4.a.u 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{2} - 21$$ T2^2 - 21 $$T_{5}^{2} - 21$$ T5^2 - 21 $$T_{13} + 44$$ T13 + 44

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 21$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 21$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 525$$
$13$ $$(T + 44)^{2}$$
$17$ $$T^{2} - 4116$$
$19$ $$(T - 49)^{2}$$
$23$ $$T^{2} - 6069$$
$29$ $$T^{2} - 48384$$
$31$ $$(T + 191)^{2}$$
$37$ $$(T - 29)^{2}$$
$41$ $$T^{2} - 137781$$
$43$ $$(T - 386)^{2}$$
$47$ $$T^{2} - 4116$$
$53$ $$T^{2} - 16464$$
$59$ $$T^{2} - 498036$$
$61$ $$(T + 128)^{2}$$
$67$ $$(T + 988)^{2}$$
$71$ $$T^{2} - 657909$$
$73$ $$(T + 266)^{2}$$
$79$ $$(T - 146)^{2}$$
$83$ $$T^{2} - 2044224$$
$89$ $$T^{2} - 643125$$
$97$ $$(T + 152)^{2}$$