# Properties

 Label 1216.4.a.j Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 44$$ x^2 - x - 44 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) 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 $$\beta = \frac{1}{2}(1 + \sqrt{177})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + (2 \beta - 6) q^{5} + (\beta + 28) q^{7} + (\beta + 17) q^{9} +O(q^{10})$$ q - b * q^3 + (2*b - 6) * q^5 + (b + 28) * q^7 + (b + 17) * q^9 $$q - \beta q^{3} + (2 \beta - 6) q^{5} + (\beta + 28) q^{7} + (\beta + 17) q^{9} + ( - 2 \beta - 4) q^{11} + (7 \beta - 10) q^{13} + (4 \beta - 88) q^{15} + ( - 15 \beta - 18) q^{17} + 19 q^{19} + ( - 29 \beta - 44) q^{21} + (13 \beta - 84) q^{23} + ( - 20 \beta + 87) q^{25} + (9 \beta - 44) q^{27} + ( - 29 \beta + 54) q^{29} - 16 \beta q^{31} + (6 \beta + 88) q^{33} + (52 \beta - 80) q^{35} + (16 \beta - 198) q^{37} + (3 \beta - 308) q^{39} + (6 \beta - 398) q^{41} + ( - 48 \beta - 124) q^{43} + (30 \beta - 14) q^{45} + (40 \beta - 120) q^{47} + (57 \beta + 485) q^{49} + (33 \beta + 660) q^{51} + ( - 9 \beta - 194) q^{53} - 152 q^{55} - 19 \beta q^{57} + (71 \beta - 136) q^{59} + ( - 44 \beta + 362) q^{61} + (46 \beta + 520) q^{63} + ( - 48 \beta + 676) q^{65} + (43 \beta + 448) q^{67} + (71 \beta - 572) q^{69} + (22 \beta + 192) q^{71} + ( - \beta + 62) q^{73} + ( - 67 \beta + 880) q^{75} + ( - 62 \beta - 200) q^{77} + (58 \beta + 24) q^{79} + (8 \beta - 855) q^{81} + (6 \beta - 1116) q^{83} + (24 \beta - 1212) q^{85} + ( - 25 \beta + 1276) q^{87} + ( - 10 \beta - 430) q^{89} + (193 \beta + 28) q^{91} + (16 \beta + 704) q^{93} + (38 \beta - 114) q^{95} + ( - 76 \beta - 894) q^{97} + ( - 40 \beta - 156) q^{99} +O(q^{100})$$ q - b * q^3 + (2*b - 6) * q^5 + (b + 28) * q^7 + (b + 17) * q^9 + (-2*b - 4) * q^11 + (7*b - 10) * q^13 + (4*b - 88) * q^15 + (-15*b - 18) * q^17 + 19 * q^19 + (-29*b - 44) * q^21 + (13*b - 84) * q^23 + (-20*b + 87) * q^25 + (9*b - 44) * q^27 + (-29*b + 54) * q^29 - 16*b * q^31 + (6*b + 88) * q^33 + (52*b - 80) * q^35 + (16*b - 198) * q^37 + (3*b - 308) * q^39 + (6*b - 398) * q^41 + (-48*b - 124) * q^43 + (30*b - 14) * q^45 + (40*b - 120) * q^47 + (57*b + 485) * q^49 + (33*b + 660) * q^51 + (-9*b - 194) * q^53 - 152 * q^55 - 19*b * q^57 + (71*b - 136) * q^59 + (-44*b + 362) * q^61 + (46*b + 520) * q^63 + (-48*b + 676) * q^65 + (43*b + 448) * q^67 + (71*b - 572) * q^69 + (22*b + 192) * q^71 + (-b + 62) * q^73 + (-67*b + 880) * q^75 + (-62*b - 200) * q^77 + (58*b + 24) * q^79 + (8*b - 855) * q^81 + (6*b - 1116) * q^83 + (24*b - 1212) * q^85 + (-25*b + 1276) * q^87 + (-10*b - 430) * q^89 + (193*b + 28) * q^91 + (16*b + 704) * q^93 + (38*b - 114) * q^95 + (-76*b - 894) * q^97 + (-40*b - 156) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 10 q^{5} + 57 q^{7} + 35 q^{9}+O(q^{10})$$ 2 * q - q^3 - 10 * q^5 + 57 * q^7 + 35 * q^9 $$2 q - q^{3} - 10 q^{5} + 57 q^{7} + 35 q^{9} - 10 q^{11} - 13 q^{13} - 172 q^{15} - 51 q^{17} + 38 q^{19} - 117 q^{21} - 155 q^{23} + 154 q^{25} - 79 q^{27} + 79 q^{29} - 16 q^{31} + 182 q^{33} - 108 q^{35} - 380 q^{37} - 613 q^{39} - 790 q^{41} - 296 q^{43} + 2 q^{45} - 200 q^{47} + 1027 q^{49} + 1353 q^{51} - 397 q^{53} - 304 q^{55} - 19 q^{57} - 201 q^{59} + 680 q^{61} + 1086 q^{63} + 1304 q^{65} + 939 q^{67} - 1073 q^{69} + 406 q^{71} + 123 q^{73} + 1693 q^{75} - 462 q^{77} + 106 q^{79} - 1702 q^{81} - 2226 q^{83} - 2400 q^{85} + 2527 q^{87} - 870 q^{89} + 249 q^{91} + 1424 q^{93} - 190 q^{95} - 1864 q^{97} - 352 q^{99}+O(q^{100})$$ 2 * q - q^3 - 10 * q^5 + 57 * q^7 + 35 * q^9 - 10 * q^11 - 13 * q^13 - 172 * q^15 - 51 * q^17 + 38 * q^19 - 117 * q^21 - 155 * q^23 + 154 * q^25 - 79 * q^27 + 79 * q^29 - 16 * q^31 + 182 * q^33 - 108 * q^35 - 380 * q^37 - 613 * q^39 - 790 * q^41 - 296 * q^43 + 2 * q^45 - 200 * q^47 + 1027 * q^49 + 1353 * q^51 - 397 * q^53 - 304 * q^55 - 19 * q^57 - 201 * q^59 + 680 * q^61 + 1086 * q^63 + 1304 * q^65 + 939 * q^67 - 1073 * q^69 + 406 * q^71 + 123 * q^73 + 1693 * q^75 - 462 * q^77 + 106 * q^79 - 1702 * q^81 - 2226 * q^83 - 2400 * q^85 + 2527 * q^87 - 870 * q^89 + 249 * q^91 + 1424 * q^93 - 190 * q^95 - 1864 * q^97 - 352 * q^99

## 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
 7.15207 −6.15207
0 −7.15207 0 8.30413 0 35.1521 0 24.1521 0
1.2 0 6.15207 0 −18.3041 0 21.8479 0 10.8479 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.j 2
4.b odd 2 1 1216.4.a.l 2
8.b even 2 1 38.4.a.b 2
8.d odd 2 1 304.4.a.d 2
24.h odd 2 1 342.4.a.k 2
40.f even 2 1 950.4.a.h 2
40.i odd 4 2 950.4.b.g 4
56.h odd 2 1 1862.4.a.b 2
152.g odd 2 1 722.4.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 8.b even 2 1
304.4.a.d 2 8.d odd 2 1
342.4.a.k 2 24.h odd 2 1
722.4.a.i 2 152.g odd 2 1
950.4.a.h 2 40.f even 2 1
950.4.b.g 4 40.i odd 4 2
1216.4.a.j 2 1.a even 1 1 trivial
1216.4.a.l 2 4.b odd 2 1
1862.4.a.b 2 56.h odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} - 44$$ T3^2 + T3 - 44 $$T_{5}^{2} + 10T_{5} - 152$$ T5^2 + 10*T5 - 152

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 44$$
$5$ $$T^{2} + 10T - 152$$
$7$ $$T^{2} - 57T + 768$$
$11$ $$T^{2} + 10T - 152$$
$13$ $$T^{2} + 13T - 2126$$
$17$ $$T^{2} + 51T - 9306$$
$19$ $$(T - 19)^{2}$$
$23$ $$T^{2} + 155T - 1472$$
$29$ $$T^{2} - 79T - 35654$$
$31$ $$T^{2} + 16T - 11264$$
$37$ $$T^{2} + 380T + 24772$$
$41$ $$T^{2} + 790T + 154432$$
$43$ $$T^{2} + 296T - 80048$$
$47$ $$T^{2} + 200T - 60800$$
$53$ $$T^{2} + 397T + 35818$$
$59$ $$T^{2} + 201T - 212964$$
$61$ $$T^{2} - 680T + 29932$$
$67$ $$T^{2} - 939T + 138612$$
$71$ $$T^{2} - 406T + 19792$$
$73$ $$T^{2} - 123T + 3738$$
$79$ $$T^{2} - 106T - 146048$$
$83$ $$T^{2} + 2226 T + 1237176$$
$89$ $$T^{2} + 870T + 184800$$
$97$ $$T^{2} + 1864 T + 613036$$