# Properties

 Label 1232.4.a.w Level $1232$ Weight $4$ Character orbit 1232.a Self dual yes Analytic conductor $72.690$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1232.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: $$72.6903531271$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.509800.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 10x^{2} + 5x + 15$$ x^4 - x^3 - 10*x^2 + 5*x + 15 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 77) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 3) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{5} + 7 q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} + 15) q^{9}+O(q^{10})$$ q + (-b2 + 3) * q^3 + (-b3 - b2 + b1 - 5) * q^5 + 7 * q^7 + (-3*b3 - 5*b2 + 15) * q^9 $$q + ( - \beta_{2} + 3) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{5} + 7 q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} + 15) q^{9} - 11 q^{11} + (5 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 31) q^{13} + ( - 11 \beta_{3} - \beta_{2} + \cdots + 10) q^{15}+ \cdots + (33 \beta_{3} + 55 \beta_{2} - 165) q^{99}+O(q^{100})$$ q + (-b2 + 3) * q^3 + (-b3 - b2 + b1 - 5) * q^5 + 7 * q^7 + (-3*b3 - 5*b2 + 15) * q^9 - 11 * q^11 + (5*b3 - 4*b2 - 4*b1 - 31) * q^13 + (-11*b3 - b2 + 8*b1 + 10) * q^15 + (-5*b3 - 6*b2 + 2*b1 - 21) * q^17 + (-6*b3 - 2*b2 - 10*b1 + 38) * q^19 + (-7*b2 + 21) * q^21 + (5*b3 - b2 + 8*b1 - 46) * q^23 + (-b3 - 9*b2 - 6*b1 + 9) * q^25 + (-21*b3 - 13*b2 + 9*b1 + 117) * q^27 + (20*b3 + 12*b2 + 5*b1 - 17) * q^29 + (16*b3 - 33*b2 + 8*b1 + 111) * q^31 + (11*b2 - 33) * q^33 + (-7*b3 - 7*b2 + 7*b1 - 35) * q^35 + (b3 - b2 + 24*b1 + 72) * q^37 + (22*b3 + 44*b2 - 35*b1 + 75) * q^39 + (9*b3 - 14*b2 - 31*b1) * q^41 + (-6*b3 - 36*b2 + 11*b1 + 121) * q^43 + (-46*b3 - 32*b2 + 46*b1 + 122) * q^45 + (7*b3 - 30*b2 - 3*b1 - 12) * q^47 + 49 * q^49 + (-40*b3 - 14*b2 + 25*b1 + 107) * q^51 + (-34*b3 + 4*b2 + 24*b1 - 224) * q^53 + (11*b3 + 11*b2 - 11*b1 + 55) * q^55 + (42*b3 - 82*b2 - 32*b1 + 196) * q^57 + (-6*b3 - 19*b2 + 24*b1 + 303) * q^59 + (-3*b3 - 42*b2 - 48*b1 - 89) * q^61 + (-21*b3 - 35*b2 + 105) * q^63 + (-12*b3 + 112*b2 + 11*b1 - 105) * q^65 + (35*b3 + 61*b2 - 14*b1 + 392) * q^67 + (-41*b3 + 77*b2 + 25*b1 - 117) * q^69 + (7*b3 - 11*b2 - 578) * q^71 + (31*b3 + 2*b2 - 41*b1 - 392) * q^73 + (7*b3 - 38*b2 - 27*b1 + 344) * q^75 - 77 * q^77 + (38*b3 - 54*b2 - 6*b1 + 274) * q^79 + (-54*b3 - 104*b2 + 108*b1 + 255) * q^81 + (102*b3 - 20*b2 - 27*b1 + 531) * q^83 + (-14*b3 - 26*b2 + 3*b1 + 495) * q^85 + (46*b3 + 146*b2 - 35*b1 - 387) * q^87 + (83*b3 - 99*b2 + 29*b1 + 429) * q^89 + (35*b3 - 28*b2 - 28*b1 - 217) * q^91 + (-115*b3 - 89*b2 - 8*b1 + 1454) * q^93 + (-26*b3 - 18*b2 + 140*b1 - 596) * q^95 + (15*b3 - 57*b2 - 37*b1 - 543) * q^97 + (33*b3 + 55*b2 - 165) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{3} - 18 q^{5} + 28 q^{7} + 66 q^{9}+O(q^{10})$$ 4 * q + 12 * q^3 - 18 * q^5 + 28 * q^7 + 66 * q^9 $$4 q + 12 q^{3} - 18 q^{5} + 28 q^{7} + 66 q^{9} - 44 q^{11} - 134 q^{13} + 62 q^{15} - 74 q^{17} + 164 q^{19} + 84 q^{21} - 194 q^{23} + 38 q^{25} + 510 q^{27} - 108 q^{29} + 412 q^{31} - 132 q^{33} - 126 q^{35} + 286 q^{37} + 256 q^{39} - 18 q^{41} + 496 q^{43} + 580 q^{45} - 62 q^{47} + 196 q^{49} + 508 q^{51} - 828 q^{53} + 198 q^{55} + 700 q^{57} + 1224 q^{59} - 350 q^{61} + 462 q^{63} - 396 q^{65} + 1498 q^{67} - 386 q^{69} - 2326 q^{71} - 1630 q^{73} + 1362 q^{75} - 308 q^{77} + 1020 q^{79} + 1128 q^{81} + 1920 q^{83} + 2008 q^{85} - 1640 q^{87} + 1550 q^{89} - 938 q^{91} + 6046 q^{93} - 2332 q^{95} - 2202 q^{97} - 726 q^{99}+O(q^{100})$$ 4 * q + 12 * q^3 - 18 * q^5 + 28 * q^7 + 66 * q^9 - 44 * q^11 - 134 * q^13 + 62 * q^15 - 74 * q^17 + 164 * q^19 + 84 * q^21 - 194 * q^23 + 38 * q^25 + 510 * q^27 - 108 * q^29 + 412 * q^31 - 132 * q^33 - 126 * q^35 + 286 * q^37 + 256 * q^39 - 18 * q^41 + 496 * q^43 + 580 * q^45 - 62 * q^47 + 196 * q^49 + 508 * q^51 - 828 * q^53 + 198 * q^55 + 700 * q^57 + 1224 * q^59 - 350 * q^61 + 462 * q^63 - 396 * q^65 + 1498 * q^67 - 386 * q^69 - 2326 * q^71 - 1630 * q^73 + 1362 * q^75 - 308 * q^77 + 1020 * q^79 + 1128 * q^81 + 1920 * q^83 + 2008 * q^85 - 1640 * q^87 + 1550 * q^89 - 938 * q^91 + 6046 * q^93 - 2332 * q^95 - 2202 * q^97 - 726 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10x^{2} + 5x + 15$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu - 1$$ 4*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7\nu + 3$$ v^3 - v^2 - 7*v + 3 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 11$$ 2*v^2 - 11
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 4$$ (b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 11 ) / 2$$ (b3 + 11) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} + 4\beta_{2} + 7\beta _1 + 17 ) / 4$$ (2*b3 + 4*b2 + 7*b1 + 17) / 4

## 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
 −1.11082 3.18303 −2.66444 1.59222
0 −5.17115 0 −10.0822 0 7.00000 0 −0.259212 0
1.2 0 0.163384 0 −5.36789 0 7.00000 0 −26.9733 0
1.3 0 7.36360 0 −15.4926 0 7.00000 0 27.2227 0
1.4 0 9.64416 0 12.9427 0 7.00000 0 66.0098 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

## 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 1232.4.a.w 4
4.b odd 2 1 77.4.a.c 4
12.b even 2 1 693.4.a.m 4
20.d odd 2 1 1925.4.a.q 4
28.d even 2 1 539.4.a.f 4
44.c even 2 1 847.4.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.c 4 4.b odd 2 1
539.4.a.f 4 28.d even 2 1
693.4.a.m 4 12.b even 2 1
847.4.a.e 4 44.c even 2 1
1232.4.a.w 4 1.a even 1 1 trivial
1925.4.a.q 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1232))$$:

 $$T_{3}^{4} - 12T_{3}^{3} - 15T_{3}^{2} + 370T_{3} - 60$$ T3^4 - 12*T3^3 - 15*T3^2 + 370*T3 - 60 $$T_{5}^{4} + 18T_{5}^{3} - 107T_{5}^{2} - 2960T_{5} - 10852$$ T5^4 + 18*T5^3 - 107*T5^2 - 2960*T5 - 10852

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 12 T^{3} + \cdots - 60$$
$5$ $$T^{4} + 18 T^{3} + \cdots - 10852$$
$7$ $$(T - 7)^{4}$$
$11$ $$(T + 11)^{4}$$
$13$ $$T^{4} + 134 T^{3} + \cdots - 9904192$$
$17$ $$T^{4} + 74 T^{3} + \cdots - 4708304$$
$19$ $$T^{4} - 164 T^{3} + \cdots - 85552320$$
$23$ $$T^{4} + 194 T^{3} + \cdots - 39720496$$
$29$ $$T^{4} + 108 T^{3} + \cdots - 365881040$$
$31$ $$T^{4} + \cdots - 5207968724$$
$37$ $$T^{4} + \cdots + 1094639996$$
$41$ $$T^{4} + 18 T^{3} + \cdots - 410971280$$
$43$ $$T^{4} - 496 T^{3} + \cdots - 998066176$$
$47$ $$T^{4} + 62 T^{3} + \cdots + 463480064$$
$53$ $$T^{4} + 828 T^{3} + \cdots - 394495824$$
$59$ $$T^{4} + \cdots + 1674727140$$
$61$ $$T^{4} + \cdots - 3730099088$$
$67$ $$T^{4} + \cdots - 57482107536$$
$71$ $$T^{4} + \cdots + 109860635344$$
$73$ $$T^{4} + \cdots - 34532794928$$
$79$ $$T^{4} + \cdots - 48577598400$$
$83$ $$T^{4} + \cdots + 42421669632$$
$89$ $$T^{4} + \cdots - 926653158300$$
$97$ $$T^{4} + \cdots - 78194289572$$