# Properties

 Label 1296.4.a.u Level $1296$ Weight $4$ Character orbit 1296.a Self dual yes Analytic conductor $76.466$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1296.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: $$76.4664753674$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 8) q^{5} + ( - 3 \beta - 2) q^{7}+O(q^{10})$$ q + (-b + 8) * q^5 + (-3*b - 2) * q^7 $$q + ( - \beta + 8) q^{5} + ( - 3 \beta - 2) q^{7} + (8 \beta - 37) q^{11} + ( - 15 \beta + 2) q^{13} + (9 \beta + 45) q^{17} + (27 \beta + 25) q^{19} + (19 \beta - 26) q^{23} + ( - 15 \beta - 53) q^{25} + (\beta - 26) q^{29} + ( - 3 \beta - 20) q^{31} + ( - 19 \beta + 8) q^{35} + (54 \beta - 52) q^{37} + (98 \beta + 17) q^{41} + (6 \beta - 47) q^{43} + ( - 91 \beta - 154) q^{47} + (21 \beta - 267) q^{49} + ( - 162 \beta + 108) q^{53} + (93 \beta - 360) q^{55} + (136 \beta - 467) q^{59} + ( - 105 \beta + 272) q^{61} + ( - 107 \beta + 136) q^{65} + ( - 66 \beta - 461) q^{67} + ( - 144 \beta - 612) q^{71} + (243 \beta - 349) q^{73} + (71 \beta - 118) q^{77} + ( - 309 \beta + 556) q^{79} + (107 \beta - 460) q^{83} + (18 \beta + 288) q^{85} + (72 \beta - 234) q^{89} + (69 \beta + 356) q^{91} + (164 \beta - 16) q^{95} + (102 \beta + 317) q^{97}+O(q^{100})$$ q + (-b + 8) * q^5 + (-3*b - 2) * q^7 + (8*b - 37) * q^11 + (-15*b + 2) * q^13 + (9*b + 45) * q^17 + (27*b + 25) * q^19 + (19*b - 26) * q^23 + (-15*b - 53) * q^25 + (b - 26) * q^29 + (-3*b - 20) * q^31 + (-19*b + 8) * q^35 + (54*b - 52) * q^37 + (98*b + 17) * q^41 + (6*b - 47) * q^43 + (-91*b - 154) * q^47 + (21*b - 267) * q^49 + (-162*b + 108) * q^53 + (93*b - 360) * q^55 + (136*b - 467) * q^59 + (-105*b + 272) * q^61 + (-107*b + 136) * q^65 + (-66*b - 461) * q^67 + (-144*b - 612) * q^71 + (243*b - 349) * q^73 + (71*b - 118) * q^77 + (-309*b + 556) * q^79 + (107*b - 460) * q^83 + (18*b + 288) * q^85 + (72*b - 234) * q^89 + (69*b + 356) * q^91 + (164*b - 16) * q^95 + (102*b + 317) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 15 q^{5} - 7 q^{7}+O(q^{10})$$ 2 * q + 15 * q^5 - 7 * q^7 $$2 q + 15 q^{5} - 7 q^{7} - 66 q^{11} - 11 q^{13} + 99 q^{17} + 77 q^{19} - 33 q^{23} - 121 q^{25} - 51 q^{29} - 43 q^{31} - 3 q^{35} - 50 q^{37} + 132 q^{41} - 88 q^{43} - 399 q^{47} - 513 q^{49} + 54 q^{53} - 627 q^{55} - 798 q^{59} + 439 q^{61} + 165 q^{65} - 988 q^{67} - 1368 q^{71} - 455 q^{73} - 165 q^{77} + 803 q^{79} - 813 q^{83} + 594 q^{85} - 396 q^{89} + 781 q^{91} + 132 q^{95} + 736 q^{97}+O(q^{100})$$ 2 * q + 15 * q^5 - 7 * q^7 - 66 * q^11 - 11 * q^13 + 99 * q^17 + 77 * q^19 - 33 * q^23 - 121 * q^25 - 51 * q^29 - 43 * q^31 - 3 * q^35 - 50 * q^37 + 132 * q^41 - 88 * q^43 - 399 * q^47 - 513 * q^49 + 54 * q^53 - 627 * q^55 - 798 * q^59 + 439 * q^61 + 165 * q^65 - 988 * q^67 - 1368 * q^71 - 455 * q^73 - 165 * q^77 + 803 * q^79 - 813 * q^83 + 594 * q^85 - 396 * q^89 + 781 * q^91 + 132 * q^95 + 736 * 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
 3.37228 −2.37228
0 0 0 4.62772 0 −12.1168 0 0 0
1.2 0 0 0 10.3723 0 5.11684 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.4.a.u 2
3.b odd 2 1 1296.4.a.i 2
4.b odd 2 1 81.4.a.d 2
9.c even 3 2 144.4.i.c 4
9.d odd 6 2 432.4.i.c 4
12.b even 2 1 81.4.a.a 2
20.d odd 2 1 2025.4.a.g 2
36.f odd 6 2 9.4.c.a 4
36.h even 6 2 27.4.c.a 4
60.h even 2 1 2025.4.a.n 2
180.p odd 6 2 225.4.e.b 4
180.x even 12 4 225.4.k.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 36.f odd 6 2
27.4.c.a 4 36.h even 6 2
81.4.a.a 2 12.b even 2 1
81.4.a.d 2 4.b odd 2 1
144.4.i.c 4 9.c even 3 2
225.4.e.b 4 180.p odd 6 2
225.4.k.b 8 180.x even 12 4
432.4.i.c 4 9.d odd 6 2
1296.4.a.i 2 3.b odd 2 1
1296.4.a.u 2 1.a even 1 1 trivial
2025.4.a.g 2 20.d odd 2 1
2025.4.a.n 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 15T_{5} + 48$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1296))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 15T + 48$$
$7$ $$T^{2} + 7T - 62$$
$11$ $$T^{2} + 66T + 561$$
$13$ $$T^{2} + 11T - 1826$$
$17$ $$T^{2} - 99T + 1782$$
$19$ $$T^{2} - 77T - 4532$$
$23$ $$T^{2} + 33T - 2706$$
$29$ $$T^{2} + 51T + 642$$
$31$ $$T^{2} + 43T + 388$$
$37$ $$T^{2} + 50T - 23432$$
$41$ $$T^{2} - 132T - 74877$$
$43$ $$T^{2} + 88T + 1639$$
$47$ $$T^{2} + 399T - 28518$$
$53$ $$T^{2} - 54T - 215784$$
$59$ $$T^{2} + 798T + 6609$$
$61$ $$T^{2} - 439T - 42776$$
$67$ $$T^{2} + 988T + 208099$$
$71$ $$T^{2} + 1368 T + 296784$$
$73$ $$T^{2} + 455T - 435398$$
$79$ $$T^{2} - 803T - 626516$$
$83$ $$T^{2} + 813T + 70788$$
$89$ $$T^{2} + 396T - 3564$$
$97$ $$T^{2} - 736T + 49591$$