# Properties

 Label 1323.4.a.s 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: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{4} + \beta q^{5} - 9 \beta q^{8} +O(q^{10})$$ q + b * q^2 - q^4 + b * q^5 - 9*b * q^8 $$q + \beta q^{2} - q^{4} + \beta q^{5} - 9 \beta q^{8} + 7 q^{10} - 7 \beta q^{11} + 26 q^{13} - 55 q^{16} + 24 \beta q^{17} + 35 q^{19} - \beta q^{20} - 49 q^{22} + 39 \beta q^{23} - 118 q^{25} + 26 \beta q^{26} + 2 \beta q^{29} + 75 q^{31} + 17 \beta q^{32} + 168 q^{34} - 111 q^{37} + 35 \beta q^{38} - 63 q^{40} - 181 \beta q^{41} - 328 q^{43} + 7 \beta q^{44} + 273 q^{46} - 144 \beta q^{47} - 118 \beta q^{50} - 26 q^{52} - 48 \beta q^{53} - 49 q^{55} + 14 q^{58} - 48 \beta q^{59} + 152 q^{61} + 75 \beta q^{62} + 559 q^{64} + 26 \beta q^{65} + 202 q^{67} - 24 \beta q^{68} - 399 \beta q^{71} - 672 q^{73} - 111 \beta q^{74} - 35 q^{76} - 988 q^{79} - 55 \beta q^{80} - 1267 q^{82} + 54 \beta q^{83} + 168 q^{85} - 328 \beta q^{86} + 441 q^{88} - 365 \beta q^{89} - 39 \beta q^{92} - 1008 q^{94} + 35 \beta q^{95} + 492 q^{97} +O(q^{100})$$ q + b * q^2 - q^4 + b * q^5 - 9*b * q^8 + 7 * q^10 - 7*b * q^11 + 26 * q^13 - 55 * q^16 + 24*b * q^17 + 35 * q^19 - b * q^20 - 49 * q^22 + 39*b * q^23 - 118 * q^25 + 26*b * q^26 + 2*b * q^29 + 75 * q^31 + 17*b * q^32 + 168 * q^34 - 111 * q^37 + 35*b * q^38 - 63 * q^40 - 181*b * q^41 - 328 * q^43 + 7*b * q^44 + 273 * q^46 - 144*b * q^47 - 118*b * q^50 - 26 * q^52 - 48*b * q^53 - 49 * q^55 + 14 * q^58 - 48*b * q^59 + 152 * q^61 + 75*b * q^62 + 559 * q^64 + 26*b * q^65 + 202 * q^67 - 24*b * q^68 - 399*b * q^71 - 672 * q^73 - 111*b * q^74 - 35 * q^76 - 988 * q^79 - 55*b * q^80 - 1267 * q^82 + 54*b * q^83 + 168 * q^85 - 328*b * q^86 + 441 * q^88 - 365*b * q^89 - 39*b * q^92 - 1008 * q^94 + 35*b * q^95 + 492 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 14 q^{10} + 52 q^{13} - 110 q^{16} + 70 q^{19} - 98 q^{22} - 236 q^{25} + 150 q^{31} + 336 q^{34} - 222 q^{37} - 126 q^{40} - 656 q^{43} + 546 q^{46} - 52 q^{52} - 98 q^{55} + 28 q^{58} + 304 q^{61} + 1118 q^{64} + 404 q^{67} - 1344 q^{73} - 70 q^{76} - 1976 q^{79} - 2534 q^{82} + 336 q^{85} + 882 q^{88} - 2016 q^{94} + 984 q^{97}+O(q^{100})$$ 2 * q - 2 * q^4 + 14 * q^10 + 52 * q^13 - 110 * q^16 + 70 * q^19 - 98 * q^22 - 236 * q^25 + 150 * q^31 + 336 * q^34 - 222 * q^37 - 126 * q^40 - 656 * q^43 + 546 * q^46 - 52 * q^52 - 98 * q^55 + 28 * q^58 + 304 * q^61 + 1118 * q^64 + 404 * q^67 - 1344 * q^73 - 70 * q^76 - 1976 * q^79 - 2534 * q^82 + 336 * q^85 + 882 * q^88 - 2016 * q^94 + 984 * 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
 −2.64575 2.64575
−2.64575 0 −1.00000 −2.64575 0 0 23.8118 0 7.00000
1.2 2.64575 0 −1.00000 2.64575 0 0 −23.8118 0 7.00000
 $$n$$: 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.s 2
3.b odd 2 1 inner 1323.4.a.s 2
7.b odd 2 1 189.4.a.g 2
21.c even 2 1 189.4.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.g 2 7.b odd 2 1
189.4.a.g 2 21.c even 2 1
1323.4.a.s 2 1.a even 1 1 trivial
1323.4.a.s 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{2} - 7$$ T2^2 - 7 $$T_{5}^{2} - 7$$ T5^2 - 7 $$T_{13} - 26$$ T13 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 7$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 7$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 343$$
$13$ $$(T - 26)^{2}$$
$17$ $$T^{2} - 4032$$
$19$ $$(T - 35)^{2}$$
$23$ $$T^{2} - 10647$$
$29$ $$T^{2} - 28$$
$31$ $$(T - 75)^{2}$$
$37$ $$(T + 111)^{2}$$
$41$ $$T^{2} - 229327$$
$43$ $$(T + 328)^{2}$$
$47$ $$T^{2} - 145152$$
$53$ $$T^{2} - 16128$$
$59$ $$T^{2} - 16128$$
$61$ $$(T - 152)^{2}$$
$67$ $$(T - 202)^{2}$$
$71$ $$T^{2} - 1114407$$
$73$ $$(T + 672)^{2}$$
$79$ $$(T + 988)^{2}$$
$83$ $$T^{2} - 20412$$
$89$ $$T^{2} - 932575$$
$97$ $$(T - 492)^{2}$$