# Properties

 Label 1216.2.a.w Level $1216$ Weight $2$ Character orbit 1216.a Self dual yes Analytic conductor $9.710$ Analytic rank $1$ 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] = [1216,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$9.70980888579$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + \beta_1 q^{5} + ( - \beta_{3} + \beta_{2}) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + b1 * q^5 + (-b3 + b2) * q^7 + (b2 + 2) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + \beta_1 q^{5} + ( - \beta_{3} + \beta_{2}) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_1 - 2) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{13} + (2 \beta_{3} - 2 \beta_1 - 2) q^{15} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{17} - q^{19} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{21} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{25} + (\beta_{2} - 3) q^{27} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{29} + ( - 2 \beta_{3} - 2) q^{31} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{33} + (2 \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{3} - 4) q^{37} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{39} + (2 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{2} - \beta_1) q^{43} + ( - 2 \beta_{3} + 3 \beta_1 + 2) q^{45} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{47} + (2 \beta_{3} - 5 \beta_{2} + \beta_1) q^{49} + ( - 3 \beta_{2} + 4 \beta_1 + 1) q^{51} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{53} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 10) q^{55} + (\beta_{2} + 1) q^{57} + ( - \beta_{2} + 2 \beta_1 - 5) q^{59} + (2 \beta_{2} + \beta_1 + 2) q^{61} + (\beta_1 + 2) q^{63} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 6) q^{65} + (3 \beta_{2} - 2 \beta_1 - 1) q^{67} + (3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{69} + (2 \beta_{3} - 2 \beta_{2} - 8) q^{71} + (2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{73} + ( - 3 \beta_{2} - 4 \beta_1 - 11) q^{75} + (2 \beta_{3} - 4 \beta_{2} - \beta_1) q^{77} + (2 \beta_{3} + 2 \beta_{2} + 4) q^{79} - 7 q^{81} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{83} + ( - 6 \beta_{2} - \beta_1 - 8) q^{85} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{87} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 6) q^{89} + (4 \beta_{3} - 3 \beta_{2} - 3) q^{91} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 6) q^{93} - \beta_1 q^{95} + ( - 2 \beta_1 - 2) q^{97} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 6) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + b1 * q^5 + (-b3 + b2) * q^7 + (b2 + 2) * q^9 + (-b1 - 2) * q^11 + (b3 + b2 - b1 - 2) * q^13 + (2*b3 - 2*b1 - 2) * q^15 + (2*b3 - b2 - b1 + 1) * q^17 - q^19 + (-b3 + b2 - b1 - 2) * q^21 + (b3 - b2 + b1 - 2) * q^23 + (-2*b3 + 2*b2 + b1 + 5) * q^25 + (b2 - 3) * q^27 + (b3 + b2 - b1 - 2) * q^29 + (-2*b3 - 2) * q^31 + (-2*b3 + 2*b2 + 2*b1 + 4) * q^33 + (2*b2 + b1) * q^35 + (-2*b3 - 4) * q^37 + (-b3 + b2 + 3*b1 - 2) * q^39 + (2*b2 - 2*b1) * q^41 + (2*b2 - b1) * q^43 + (-2*b3 + 3*b1 + 2) * q^45 + (-2*b3 - 2*b2 + b1 - 2) * q^47 + (2*b3 - 5*b2 + b1) * q^49 + (-3*b2 + 4*b1 + 1) * q^51 + (b3 - 3*b2 + b1 - 2) * q^53 + (2*b3 - 2*b2 - 3*b1 - 10) * q^55 + (b2 + 1) * q^57 + (-b2 + 2*b1 - 5) * q^59 + (2*b2 + b1 + 2) * q^61 + (b1 + 2) * q^63 + (-2*b3 - 4*b2 - 2*b1 - 6) * q^65 + (3*b2 - 2*b1 - 1) * q^67 + (3*b3 + b2 - b1 + 2) * q^69 + (2*b3 - 2*b2 - 8) * q^71 + (2*b3 + b2 - b1 + 3) * q^73 + (-3*b2 - 4*b1 - 11) * q^75 + (2*b3 - 4*b2 - b1) * q^77 + (2*b3 + 2*b2 + 4) * q^79 - 7 * q^81 + (-2*b3 + 2*b2 + 2*b1) * q^83 + (-6*b2 - b1 - 8) * q^85 + (-b3 + b2 + 3*b1 - 2) * q^87 + (2*b3 - 6*b2 + 2*b1 - 6) * q^89 + (4*b3 - 3*b2 - 3) * q^91 + (-2*b3 + 4*b2 - 2*b1 + 6) * q^93 - b1 * q^95 + (-2*b1 - 2) * q^97 + (2*b3 - 2*b2 - 3*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 $$4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37} - 12 q^{39} - 2 q^{41} - 3 q^{43} + 7 q^{45} - 3 q^{47} + 7 q^{49} + 6 q^{51} - 4 q^{53} - 35 q^{55} + 2 q^{57} - 20 q^{59} + 3 q^{61} + 7 q^{63} - 12 q^{65} - 8 q^{67} + 4 q^{69} - 30 q^{71} + 9 q^{73} - 34 q^{75} + 7 q^{77} + 10 q^{79} - 28 q^{81} - 4 q^{83} - 19 q^{85} - 12 q^{87} - 16 q^{89} - 10 q^{91} + 20 q^{93} + q^{95} - 6 q^{97} - 19 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu$$ v^3 - v^2 - 3*v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5\nu + 1$$ v^3 - v^2 - 5*v + 1
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta _1 + 1 ) / 2$$ (-b3 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + \beta _1 + 7 ) / 2$$ (-b3 + 2*b2 + b1 + 7) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + \beta_{2} + 3\beta _1 + 5$$ -2*b3 + b2 + 3*b1 + 5

## 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
 −1.69353 2.69353 1.32973 −0.329727
0 −2.56155 0 −2.64453 0 −0.180969 0 3.56155 0
1.2 0 −2.56155 0 4.20608 0 1.74252 0 3.56155 0
1.3 0 1.56155 0 −3.40617 0 2.50407 0 −0.561553 0
1.4 0 1.56155 0 0.844614 0 −5.06562 0 −0.561553 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 1216.2.a.w 4
4.b odd 2 1 1216.2.a.x 4
8.b even 2 1 608.2.a.j yes 4
8.d odd 2 1 608.2.a.i 4
24.f even 2 1 5472.2.a.bt 4
24.h odd 2 1 5472.2.a.bs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.i 4 8.d odd 2 1
608.2.a.j yes 4 8.b even 2 1
1216.2.a.w 4 1.a even 1 1 trivial
1216.2.a.x 4 4.b odd 2 1
5472.2.a.bs 4 24.h odd 2 1
5472.2.a.bt 4 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{5}^{4} + T_{5}^{3} - 18T_{5}^{2} - 24T_{5} + 32$$ T5^4 + T5^3 - 18*T5^2 - 24*T5 + 32 $$T_{7}^{4} + T_{7}^{3} - 17T_{7}^{2} + 19T_{7} + 4$$ T7^4 + T7^3 - 17*T7^2 + 19*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + T - 4)^{2}$$
$5$ $$T^{4} + T^{3} - 18 T^{2} - 24 T + 32$$
$7$ $$T^{4} + T^{3} - 17 T^{2} + 19 T + 4$$
$11$ $$T^{4} + 7 T^{3} - 28 T + 16$$
$13$ $$T^{4} + 10 T^{3} + 7 T^{2} - 158 T - 344$$
$17$ $$T^{4} - 5 T^{3} - 51 T^{2} + 141 T + 698$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 8 T^{3} - 17 T^{2} - 132 T + 64$$
$29$ $$T^{4} + 10 T^{3} + 7 T^{2} - 158 T - 344$$
$31$ $$T^{4} + 6 T^{3} - 56 T^{2} - 416 T - 512$$
$37$ $$T^{4} + 14 T^{3} + 4 T^{2} + \cdots - 1504$$
$41$ $$T^{4} + 2 T^{3} - 72 T^{2} - 192 T + 512$$
$43$ $$T^{4} + 3 T^{3} - 32 T^{2} - 96 T - 64$$
$47$ $$T^{4} + 3 T^{3} - 108 T^{2} - 176 T - 64$$
$53$ $$T^{4} + 4 T^{3} - 69 T^{2} - 384 T - 508$$
$59$ $$T^{4} + 20 T^{3} + 85 T^{2} + \cdots - 152$$
$61$ $$T^{4} - 3 T^{3} - 66 T^{2} + 28 T + 344$$
$67$ $$T^{4} + 8 T^{3} - 75 T^{2} - 602 T - 824$$
$71$ $$T^{4} + 30 T^{3} + 268 T^{2} + \cdots - 2432$$
$73$ $$T^{4} - 9 T^{3} - 47 T^{2} + 309 T + 2$$
$79$ $$T^{4} - 10 T^{3} - 100 T^{2} + \cdots - 3136$$
$83$ $$T^{4} + 4 T^{3} - 116 T^{2} + \cdots - 1024$$
$89$ $$T^{4} + 16 T^{3} - 204 T^{2} + \cdots - 15296$$
$97$ $$T^{4} + 6 T^{3} - 60 T^{2} - 88 T + 608$$