# Properties

 Label 1216.4.a.s Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 19) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + ( - \beta_{2} + \beta_1 - 4) q^{5} + (2 \beta_1 - 11) q^{7} + ( - 3 \beta_{2} - 3 \beta_1 + 16) q^{9}+O(q^{10})$$ q - b2 * q^3 + (-b2 + b1 - 4) * q^5 + (2*b1 - 11) * q^7 + (-3*b2 - 3*b1 + 16) * q^9 $$q - \beta_{2} q^{3} + ( - \beta_{2} + \beta_1 - 4) q^{5} + (2 \beta_1 - 11) q^{7} + ( - 3 \beta_{2} - 3 \beta_1 + 16) q^{9} + (\beta_{2} - \beta_1 - 6) q^{11} + (3 \beta_{2} - \beta_1 - 23) q^{13} + (4 \beta_{2} - 7 \beta_1 + 43) q^{15} + (2 \beta_{2} + 12 \beta_1 + 13) q^{17} + 19 q^{19} + (17 \beta_{2} - 8 \beta_1) q^{21} + (7 \beta_{2} + 9 \beta_1 - 33) q^{23} + (15 \beta_{2} - 17 \beta_1 - 23) q^{25} + ( - 7 \beta_{2} + 3 \beta_1 + 129) q^{27} + (15 \beta_{2} + 20 \beta_1 - 124) q^{29} + (6 \beta_{2} - 25 \beta_1 - 57) q^{31} + (6 \beta_{2} + 7 \beta_1 - 43) q^{33} + (25 \beta_{2} - 23 \beta_1 + 130) q^{35} + ( - 22 \beta_{2} - 27 \beta_1 + 95) q^{37} + (29 \beta_{2} + 13 \beta_1 - 129) q^{39} + (4 \beta_{2} + 5 \beta_1 + 319) q^{41} + (55 \beta_{2} + 13 \beta_1 + 176) q^{43} + ( - 25 \beta_{2} + 13 \beta_1 - 64) q^{45} + ( - 9 \beta_{2} - 21 \beta_1 + 18) q^{47} + (16 \beta_{2} - 36 \beta_1 - 50) q^{49} + (29 \beta_{2} - 42 \beta_1 - 86) q^{51} + (5 \beta_{2} - 12 \beta_1 - 278) q^{53} + ( - 5 \beta_{2} + 7 \beta_1 - 62) q^{55} - 19 \beta_{2} q^{57} + (33 \beta_{2} + 43 \beta_1 - 85) q^{59} + ( - 55 \beta_{2} - 39 \beta_1 - 324) q^{61} + (27 \beta_{2} + 29 \beta_1 - 434) q^{63} + (4 \beta_{2} + 4 \beta_1 - 80) q^{65} + ( - 23 \beta_{2} - 26 \beta_1 + 68) q^{67} + (81 \beta_{2} - 15 \beta_1 - 301) q^{69} + (32 \beta_{2} + 2 \beta_1 + 272) q^{71} + (4 \beta_{2} - 86 \beta_1 + 179) q^{73} + (17 \beta_{2} + 113 \beta_1 - 645) q^{75} + ( - 25 \beta_{2} + 3 \beta_1 - 20) q^{77} + (4 \beta_{2} + 105 \beta_1 + 161) q^{79} + ( - 60 \beta_{2} + 48 \beta_1 - 131) q^{81} + ( - 30 \beta_{2} + 50 \beta_1 + 282) q^{83} + (63 \beta_{2} - 45 \beta_1 + 378) q^{85} + (229 \beta_{2} - 35 \beta_1 - 645) q^{87} + ( - 110 \beta_{2} + 29 \beta_1 - 11) q^{89} + ( - 59 \beta_{2} - 15 \beta_1 + 167) q^{91} + (118 \beta_1 - 258) q^{93} + ( - 19 \beta_{2} + 19 \beta_1 - 76) q^{95} + (98 \beta_{2} + 4 \beta_1 - 848) q^{97} + (55 \beta_{2} + 17 \beta_1 - 96) q^{99}+O(q^{100})$$ q - b2 * q^3 + (-b2 + b1 - 4) * q^5 + (2*b1 - 11) * q^7 + (-3*b2 - 3*b1 + 16) * q^9 + (b2 - b1 - 6) * q^11 + (3*b2 - b1 - 23) * q^13 + (4*b2 - 7*b1 + 43) * q^15 + (2*b2 + 12*b1 + 13) * q^17 + 19 * q^19 + (17*b2 - 8*b1) * q^21 + (7*b2 + 9*b1 - 33) * q^23 + (15*b2 - 17*b1 - 23) * q^25 + (-7*b2 + 3*b1 + 129) * q^27 + (15*b2 + 20*b1 - 124) * q^29 + (6*b2 - 25*b1 - 57) * q^31 + (6*b2 + 7*b1 - 43) * q^33 + (25*b2 - 23*b1 + 130) * q^35 + (-22*b2 - 27*b1 + 95) * q^37 + (29*b2 + 13*b1 - 129) * q^39 + (4*b2 + 5*b1 + 319) * q^41 + (55*b2 + 13*b1 + 176) * q^43 + (-25*b2 + 13*b1 - 64) * q^45 + (-9*b2 - 21*b1 + 18) * q^47 + (16*b2 - 36*b1 - 50) * q^49 + (29*b2 - 42*b1 - 86) * q^51 + (5*b2 - 12*b1 - 278) * q^53 + (-5*b2 + 7*b1 - 62) * q^55 - 19*b2 * q^57 + (33*b2 + 43*b1 - 85) * q^59 + (-55*b2 - 39*b1 - 324) * q^61 + (27*b2 + 29*b1 - 434) * q^63 + (4*b2 + 4*b1 - 80) * q^65 + (-23*b2 - 26*b1 + 68) * q^67 + (81*b2 - 15*b1 - 301) * q^69 + (32*b2 + 2*b1 + 272) * q^71 + (4*b2 - 86*b1 + 179) * q^73 + (17*b2 + 113*b1 - 645) * q^75 + (-25*b2 + 3*b1 - 20) * q^77 + (4*b2 + 105*b1 + 161) * q^79 + (-60*b2 + 48*b1 - 131) * q^81 + (-30*b2 + 50*b1 + 282) * q^83 + (63*b2 - 45*b1 + 378) * q^85 + (229*b2 - 35*b1 - 645) * q^87 + (-110*b2 + 29*b1 - 11) * q^89 + (-59*b2 - 15*b1 + 167) * q^91 + (118*b1 - 258) * q^93 + (-19*b2 + 19*b1 - 76) * q^95 + (98*b2 + 4*b1 - 848) * q^97 + (55*b2 + 17*b1 - 96) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 14 q^{5} - 35 q^{7} + 48 q^{9}+O(q^{10})$$ 3 * q - q^3 - 14 * q^5 - 35 * q^7 + 48 * q^9 $$3 q - q^{3} - 14 q^{5} - 35 q^{7} + 48 q^{9} - 16 q^{11} - 65 q^{13} + 140 q^{15} + 29 q^{17} + 57 q^{19} + 25 q^{21} - 101 q^{23} - 37 q^{25} + 377 q^{27} - 377 q^{29} - 140 q^{31} - 130 q^{33} + 438 q^{35} + 290 q^{37} - 371 q^{39} + 956 q^{41} + 570 q^{43} - 230 q^{45} + 66 q^{47} - 98 q^{49} - 187 q^{51} - 817 q^{53} - 198 q^{55} - 19 q^{57} - 265 q^{59} - 988 q^{61} - 1304 q^{63} - 240 q^{65} + 207 q^{67} - 807 q^{69} + 846 q^{71} + 627 q^{73} - 2031 q^{75} - 88 q^{77} + 382 q^{79} - 501 q^{81} + 766 q^{83} + 1242 q^{85} - 1671 q^{87} - 172 q^{89} + 457 q^{91} - 892 q^{93} - 266 q^{95} - 2450 q^{97} - 250 q^{99}+O(q^{100})$$ 3 * q - q^3 - 14 * q^5 - 35 * q^7 + 48 * q^9 - 16 * q^11 - 65 * q^13 + 140 * q^15 + 29 * q^17 + 57 * q^19 + 25 * q^21 - 101 * q^23 - 37 * q^25 + 377 * q^27 - 377 * q^29 - 140 * q^31 - 130 * q^33 + 438 * q^35 + 290 * q^37 - 371 * q^39 + 956 * q^41 + 570 * q^43 - 230 * q^45 + 66 * q^47 - 98 * q^49 - 187 * q^51 - 817 * q^53 - 198 * q^55 - 19 * q^57 - 265 * q^59 - 988 * q^61 - 1304 * q^63 - 240 * q^65 + 207 * q^67 - 807 * q^69 + 846 * q^71 + 627 * q^73 - 2031 * q^75 - 88 * q^77 + 382 * q^79 - 501 * q^81 + 766 * q^83 + 1242 * q^85 - 1671 * q^87 - 172 * q^89 + 457 * q^91 - 892 * q^93 - 266 * q^95 - 2450 * q^97 - 250 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 16x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 11$$ b2 + b1 + 11

## 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
 −3.20905 4.73549 −0.526440
0 −6.71610 0 −18.1342 0 −25.8362 0 18.1060 0
1.2 0 −2.95388 0 1.51710 0 5.94196 0 −18.2746 0
1.3 0 8.66998 0 2.61710 0 −15.1058 0 48.1686 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
$$2$$ $$1$$
$$19$$ $$-1$$

## 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 1216.4.a.s 3
4.b odd 2 1 1216.4.a.u 3
8.b even 2 1 19.4.a.b 3
8.d odd 2 1 304.4.a.i 3
24.h odd 2 1 171.4.a.f 3
40.f even 2 1 475.4.a.f 3
40.i odd 4 2 475.4.b.f 6
56.h odd 2 1 931.4.a.c 3
88.b odd 2 1 2299.4.a.h 3
152.g odd 2 1 361.4.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.b 3 8.b even 2 1
171.4.a.f 3 24.h odd 2 1
304.4.a.i 3 8.d odd 2 1
361.4.a.i 3 152.g odd 2 1
475.4.a.f 3 40.f even 2 1
475.4.b.f 6 40.i odd 4 2
931.4.a.c 3 56.h odd 2 1
1216.4.a.s 3 1.a even 1 1 trivial
1216.4.a.u 3 4.b odd 2 1
2299.4.a.h 3 88.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{3} + T_{3}^{2} - 64T_{3} - 172$$ T3^3 + T3^2 - 64*T3 - 172 $$T_{5}^{3} + 14T_{5}^{2} - 71T_{5} + 72$$ T5^3 + 14*T5^2 - 71*T5 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 64 T - 172$$
$5$ $$T^{3} + 14 T^{2} - 71 T + 72$$
$7$ $$T^{3} + 35 T^{2} + 147 T - 2319$$
$11$ $$T^{3} + 16 T^{2} - 51 T - 1182$$
$13$ $$T^{3} + 65 T^{2} + 744 T - 4848$$
$17$ $$T^{3} - 29 T^{2} - 9225 T - 218619$$
$19$ $$(T - 19)^{3}$$
$23$ $$T^{3} + 101 T^{2} - 4624 T - 378176$$
$29$ $$T^{3} + 377 T^{2} + 8768 T - 4544396$$
$31$ $$T^{3} + 140 T^{2} - 37616 T - 2444352$$
$37$ $$T^{3} - 290 T^{2} + \cdots + 10001448$$
$41$ $$T^{3} - 956 T^{2} + \cdots - 31578144$$
$43$ $$T^{3} - 570 T^{2} + \cdots + 65963504$$
$47$ $$T^{3} - 66 T^{2} - 31311 T + 2940624$$
$53$ $$T^{3} + 817 T^{2} + \cdots + 16824816$$
$59$ $$T^{3} + 265 T^{2} + \cdots - 31557612$$
$61$ $$T^{3} + 988 T^{2} + \cdots - 76875874$$
$67$ $$T^{3} - 207 T^{2} - 59928 T + 7515248$$
$71$ $$T^{3} - 846 T^{2} + 172860 T + 1727928$$
$73$ $$T^{3} - 627 T^{2} + \cdots + 145581839$$
$79$ $$T^{3} - 382 T^{2} + \cdots - 56023488$$
$83$ $$T^{3} - 766 T^{2} + \cdots + 78728352$$
$89$ $$T^{3} + 172 T^{2} + \cdots - 76923456$$
$97$ $$T^{3} + 2450 T^{2} + \cdots + 196438912$$