# Properties

 Label 1323.4.a.ba Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $4$ Inner twists $2$

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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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 4$$ x^4 - 9*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1) q^{2} + ( - \beta_{3} + 6) q^{4} + (3 \beta_{2} + 4 \beta_1) q^{5} + ( - 13 \beta_{2} - 5 \beta_1) q^{8}+O(q^{10})$$ q + (-b2 - b1) * q^2 + (-b3 + 6) * q^4 + (3*b2 + 4*b1) * q^5 + (-13*b2 - 5*b1) * q^8 $$q + ( - \beta_{2} - \beta_1) q^{2} + ( - \beta_{3} + 6) q^{4} + (3 \beta_{2} + 4 \beta_1) q^{5} + ( - 13 \beta_{2} - 5 \beta_1) q^{8} + (3 \beta_{3} - 50) q^{10} + (11 \beta_{2} - 6 \beta_1) q^{11} + ( - 6 \beta_{3} + 14) q^{13} + ( - 5 \beta_{3} + 70) q^{16} + (2 \beta_{2} + 4 \beta_1) q^{17} + ( - 4 \beta_{3} - 45) q^{19} + (71 \beta_{2} + 39 \beta_1) q^{20} + (11 \beta_{3} - 18) q^{22} + ( - 43 \beta_{2} - 8 \beta_1) q^{23} + ( - 8 \beta_{3} + 64) q^{25} + ( - 104 \beta_{2} - 56 \beta_1) q^{26} + (28 \beta_{2} + 22 \beta_1) q^{29} + ( - 10 \beta_{3} - 107) q^{31} + ( - 41 \beta_{2} - 65 \beta_1) q^{32} + (2 \beta_{3} - 44) q^{34} + ( - 6 \beta_{3} + 327) q^{37} + ( - 15 \beta_{2} + 17 \beta_1) q^{38} + (47 \beta_{3} - 338) q^{40} + ( - 111 \beta_{2} - 22 \beta_1) q^{41} + ( - 2 \beta_{3} + 28) q^{43} + (95 \beta_{2} + 143 \beta_1) q^{44} + ( - 43 \beta_{3} + 322) q^{46} + (6 \beta_{2} - 114 \beta_1) q^{47} + ( - 184 \beta_{2} - 120 \beta_1) q^{50} + ( - 56 \beta_{3} + 960) q^{52} + (20 \beta_{2} + 148 \beta_1) q^{53} + ( - 50 \beta_{3} - 113) q^{55} + (28 \beta_{3} - 344) q^{58} + (74 \beta_{2} + 94 \beta_1) q^{59} + (8 \beta_{3} + 672) q^{61} + ( - 43 \beta_{2} + 37 \beta_1) q^{62} + ( - \beta_{3} + 206) q^{64} + (360 \beta_{2} + 146 \beta_1) q^{65} + ( - 38 \beta_{3} + 342) q^{67} + (58 \beta_{2} + 26 \beta_1) q^{68} + (61 \beta_{2} - 136 \beta_1) q^{71} + (10 \beta_{3} + 104) q^{73} + ( - 417 \beta_{2} - 369 \beta_1) q^{74} + (17 \beta_{3} + 314) q^{76} + ( - 26 \beta_{3} + 292) q^{79} + (475 \beta_{2} + 355 \beta_1) q^{80} + ( - 111 \beta_{3} + 842) q^{82} + ( - 4 \beta_{2} - 158 \beta_1) q^{83} + ( - 4 \beta_{3} + 178) q^{85} + ( - 58 \beta_{2} - 42 \beta_1) q^{86} + (7 \beta_{3} - 1570) q^{88} + ( - 87 \beta_{2} - 110 \beta_1) q^{89} + ( - 623 \beta_{2} - 559 \beta_1) q^{92} + (6 \beta_{3} + 876) q^{94} + (77 \beta_{2} - 120 \beta_1) q^{95} + ( - 8 \beta_{3} + 152) q^{97}+O(q^{100})$$ q + (-b2 - b1) * q^2 + (-b3 + 6) * q^4 + (3*b2 + 4*b1) * q^5 + (-13*b2 - 5*b1) * q^8 + (3*b3 - 50) * q^10 + (11*b2 - 6*b1) * q^11 + (-6*b3 + 14) * q^13 + (-5*b3 + 70) * q^16 + (2*b2 + 4*b1) * q^17 + (-4*b3 - 45) * q^19 + (71*b2 + 39*b1) * q^20 + (11*b3 - 18) * q^22 + (-43*b2 - 8*b1) * q^23 + (-8*b3 + 64) * q^25 + (-104*b2 - 56*b1) * q^26 + (28*b2 + 22*b1) * q^29 + (-10*b3 - 107) * q^31 + (-41*b2 - 65*b1) * q^32 + (2*b3 - 44) * q^34 + (-6*b3 + 327) * q^37 + (-15*b2 + 17*b1) * q^38 + (47*b3 - 338) * q^40 + (-111*b2 - 22*b1) * q^41 + (-2*b3 + 28) * q^43 + (95*b2 + 143*b1) * q^44 + (-43*b3 + 322) * q^46 + (6*b2 - 114*b1) * q^47 + (-184*b2 - 120*b1) * q^50 + (-56*b3 + 960) * q^52 + (20*b2 + 148*b1) * q^53 + (-50*b3 - 113) * q^55 + (28*b3 - 344) * q^58 + (74*b2 + 94*b1) * q^59 + (8*b3 + 672) * q^61 + (-43*b2 + 37*b1) * q^62 + (-b3 + 206) * q^64 + (360*b2 + 146*b1) * q^65 + (-38*b3 + 342) * q^67 + (58*b2 + 26*b1) * q^68 + (61*b2 - 136*b1) * q^71 + (10*b3 + 104) * q^73 + (-417*b2 - 369*b1) * q^74 + (17*b3 + 314) * q^76 + (-26*b3 + 292) * q^79 + (475*b2 + 355*b1) * q^80 + (-111*b3 + 842) * q^82 + (-4*b2 - 158*b1) * q^83 + (-4*b3 + 178) * q^85 + (-58*b2 - 42*b1) * q^86 + (7*b3 - 1570) * q^88 + (-87*b2 - 110*b1) * q^89 + (-623*b2 - 559*b1) * q^92 + (6*b3 + 876) * q^94 + (77*b2 - 120*b1) * q^95 + (-8*b3 + 152) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 26 q^{4}+O(q^{10})$$ 4 * q + 26 * q^4 $$4 q + 26 q^{4} - 206 q^{10} + 68 q^{13} + 290 q^{16} - 172 q^{19} - 94 q^{22} + 272 q^{25} - 408 q^{31} - 180 q^{34} + 1320 q^{37} - 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 3952 q^{52} - 352 q^{55} - 1432 q^{58} + 2672 q^{61} + 826 q^{64} + 1444 q^{67} + 396 q^{73} + 1222 q^{76} + 1220 q^{79} + 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 3492 q^{94} + 624 q^{97}+O(q^{100})$$ 4 * q + 26 * q^4 - 206 * q^10 + 68 * q^13 + 290 * q^16 - 172 * q^19 - 94 * q^22 + 272 * q^25 - 408 * q^31 - 180 * q^34 + 1320 * q^37 - 1446 * q^40 + 116 * q^43 + 1374 * q^46 + 3952 * q^52 - 352 * q^55 - 1432 * q^58 + 2672 * q^61 + 826 * q^64 + 1444 * q^67 + 396 * q^73 + 1222 * q^76 + 1220 * q^79 + 3590 * q^82 + 720 * q^85 - 6294 * q^88 + 3492 * q^94 + 624 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 5\nu ) / 2$$ (v^3 - 5*v) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 11\nu ) / 2$$ (v^3 - 11*v) / 2 $$\beta_{3}$$ $$=$$ $$3\nu^{2} - 14$$ 3*v^2 - 14
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 3$$ (-b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 14 ) / 3$$ (b3 + 14) / 3 $$\nu^{3}$$ $$=$$ $$( -5\beta_{2} + 11\beta_1 ) / 3$$ (-5*b2 + 11*b1) / 3

## 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
 −0.684742 2.92081 −2.92081 0.684742
−5.15688 0 18.5934 17.0220 0 0 −54.6288 0 −87.7802
1.2 −1.55133 0 −5.59339 9.81086 0 0 21.0878 0 −15.2198
1.3 1.55133 0 −5.59339 −9.81086 0 0 −21.0878 0 −15.2198
1.4 5.15688 0 18.5934 −17.0220 0 0 54.6288 0 −87.7802
 $$n$$: 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.ba 4
3.b odd 2 1 inner 1323.4.a.ba 4
7.b odd 2 1 189.4.a.l 4
21.c even 2 1 189.4.a.l 4

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

 $$T_{2}^{4} - 29T_{2}^{2} + 64$$ T2^4 - 29*T2^2 + 64 $$T_{5}^{4} - 386T_{5}^{2} + 27889$$ T5^4 - 386*T5^2 + 27889 $$T_{13}^{2} - 34T_{13} - 4976$$ T13^2 - 34*T13 - 4976

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 29T^{2} + 64$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 386 T^{2} + 27889$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 5906 T^{2} + 4592449$$
$13$ $$(T^{2} - 34 T - 4976)^{2}$$
$17$ $$(T^{2} - 180)^{2}$$
$19$ $$(T^{2} + 86 T - 491)^{2}$$
$23$ $$T^{4} - 40986 T^{2} + 363016809$$
$29$ $$T^{4} - 18404 T^{2} + 2849344$$
$31$ $$(T^{2} + 204 T - 4221)^{2}$$
$37$ $$(T^{2} - 660 T + 103635)^{2}$$
$41$ $$T^{4} + \cdots + 15513948025$$
$43$ $$(T^{2} - 58 T + 256)^{2}$$
$47$ $$T^{4} + \cdots + 8950673664$$
$53$ $$T^{4} + \cdots + 43477254144$$
$59$ $$T^{4} + \cdots + 8087045184$$
$61$ $$(T^{2} - 1336 T + 436864)^{2}$$
$67$ $$(T^{2} - 722 T - 80864)^{2}$$
$71$ $$T^{4} - 848826 T^{2} + 68112009$$
$73$ $$(T^{2} - 198 T - 4824)^{2}$$
$79$ $$(T^{2} - 610 T - 5840)^{2}$$
$83$ $$T^{4} + \cdots + 43147598400$$
$89$ $$T^{4} + \cdots + 15083032969$$
$97$ $$(T^{2} - 312 T + 14976)^{2}$$
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