# Properties

 Label 1216.4.a.p Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 18$$ x^2 - x - 18 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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 5) q^{3} + ( - 3 \beta + 6) q^{5} + ( - 4 \beta + 11) q^{7} + ( - 9 \beta + 16) q^{9}+O(q^{10})$$ q + (-b + 5) * q^3 + (-3*b + 6) * q^5 + (-4*b + 11) * q^7 + (-9*b + 16) * q^9 $$q + ( - \beta + 5) q^{3} + ( - 3 \beta + 6) q^{5} + ( - 4 \beta + 11) q^{7} + ( - 9 \beta + 16) q^{9} + ( - \beta - 8) q^{11} + (13 \beta - 15) q^{13} + ( - 18 \beta + 84) q^{15} + (2 \beta - 41) q^{17} + 19 q^{19} + ( - 27 \beta + 127) q^{21} + (13 \beta - 43) q^{23} + ( - 27 \beta + 73) q^{25} + ( - 25 \beta + 107) q^{27} + ( - 21 \beta + 9) q^{29} + ( - 44 \beta - 84) q^{31} + (4 \beta - 22) q^{33} + ( - 45 \beta + 282) q^{35} + ( - 28 \beta - 82) q^{37} + (67 \beta - 309) q^{39} + ( - 10 \beta - 20) q^{41} + ( - 7 \beta + 342) q^{43} + ( - 75 \beta + 582) q^{45} + ( - 71 \beta + 230) q^{47} + ( - 72 \beta + 66) q^{49} + (49 \beta - 241) q^{51} + (17 \beta + 601) q^{53} + (21 \beta + 6) q^{55} + ( - 19 \beta + 95) q^{57} + ( - 25 \beta - 131) q^{59} + (111 \beta - 212) q^{61} + ( - 127 \beta + 824) q^{63} + (84 \beta - 792) q^{65} + (77 \beta + 573) q^{67} + (95 \beta - 449) q^{69} + (116 \beta - 158) q^{71} + (184 \beta + 97) q^{73} + ( - 181 \beta + 851) q^{75} + (25 \beta - 16) q^{77} + ( - 58 \beta - 646) q^{79} + (36 \beta + 553) q^{81} + ( - 194 \beta - 238) q^{83} + (129 \beta - 354) q^{85} + ( - 93 \beta + 423) q^{87} + (188 \beta - 212) q^{89} + (151 \beta - 1101) q^{91} + ( - 92 \beta + 372) q^{93} + ( - 57 \beta + 114) q^{95} + ( - 102 \beta + 698) q^{97} + (65 \beta + 34) q^{99}+O(q^{100})$$ q + (-b + 5) * q^3 + (-3*b + 6) * q^5 + (-4*b + 11) * q^7 + (-9*b + 16) * q^9 + (-b - 8) * q^11 + (13*b - 15) * q^13 + (-18*b + 84) * q^15 + (2*b - 41) * q^17 + 19 * q^19 + (-27*b + 127) * q^21 + (13*b - 43) * q^23 + (-27*b + 73) * q^25 + (-25*b + 107) * q^27 + (-21*b + 9) * q^29 + (-44*b - 84) * q^31 + (4*b - 22) * q^33 + (-45*b + 282) * q^35 + (-28*b - 82) * q^37 + (67*b - 309) * q^39 + (-10*b - 20) * q^41 + (-7*b + 342) * q^43 + (-75*b + 582) * q^45 + (-71*b + 230) * q^47 + (-72*b + 66) * q^49 + (49*b - 241) * q^51 + (17*b + 601) * q^53 + (21*b + 6) * q^55 + (-19*b + 95) * q^57 + (-25*b - 131) * q^59 + (111*b - 212) * q^61 + (-127*b + 824) * q^63 + (84*b - 792) * q^65 + (77*b + 573) * q^67 + (95*b - 449) * q^69 + (116*b - 158) * q^71 + (184*b + 97) * q^73 + (-181*b + 851) * q^75 + (25*b - 16) * q^77 + (-58*b - 646) * q^79 + (36*b + 553) * q^81 + (-194*b - 238) * q^83 + (129*b - 354) * q^85 + (-93*b + 423) * q^87 + (188*b - 212) * q^89 + (151*b - 1101) * q^91 + (-92*b + 372) * q^93 + (-57*b + 114) * q^95 + (-102*b + 698) * q^97 + (65*b + 34) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 9 q^{3} + 9 q^{5} + 18 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q + 9 * q^3 + 9 * q^5 + 18 * q^7 + 23 * q^9 $$2 q + 9 q^{3} + 9 q^{5} + 18 q^{7} + 23 q^{9} - 17 q^{11} - 17 q^{13} + 150 q^{15} - 80 q^{17} + 38 q^{19} + 227 q^{21} - 73 q^{23} + 119 q^{25} + 189 q^{27} - 3 q^{29} - 212 q^{31} - 40 q^{33} + 519 q^{35} - 192 q^{37} - 551 q^{39} - 50 q^{41} + 677 q^{43} + 1089 q^{45} + 389 q^{47} + 60 q^{49} - 433 q^{51} + 1219 q^{53} + 33 q^{55} + 171 q^{57} - 287 q^{59} - 313 q^{61} + 1521 q^{63} - 1500 q^{65} + 1223 q^{67} - 803 q^{69} - 200 q^{71} + 378 q^{73} + 1521 q^{75} - 7 q^{77} - 1350 q^{79} + 1142 q^{81} - 670 q^{83} - 579 q^{85} + 753 q^{87} - 236 q^{89} - 2051 q^{91} + 652 q^{93} + 171 q^{95} + 1294 q^{97} + 133 q^{99}+O(q^{100})$$ 2 * q + 9 * q^3 + 9 * q^5 + 18 * q^7 + 23 * q^9 - 17 * q^11 - 17 * q^13 + 150 * q^15 - 80 * q^17 + 38 * q^19 + 227 * q^21 - 73 * q^23 + 119 * q^25 + 189 * q^27 - 3 * q^29 - 212 * q^31 - 40 * q^33 + 519 * q^35 - 192 * q^37 - 551 * q^39 - 50 * q^41 + 677 * q^43 + 1089 * q^45 + 389 * q^47 + 60 * q^49 - 433 * q^51 + 1219 * q^53 + 33 * q^55 + 171 * q^57 - 287 * q^59 - 313 * q^61 + 1521 * q^63 - 1500 * q^65 + 1223 * q^67 - 803 * q^69 - 200 * q^71 + 378 * q^73 + 1521 * q^75 - 7 * q^77 - 1350 * q^79 + 1142 * q^81 - 670 * q^83 - 579 * q^85 + 753 * q^87 - 236 * q^89 - 2051 * q^91 + 652 * q^93 + 171 * q^95 + 1294 * q^97 + 133 * 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
 4.77200 −3.77200
0 0.227998 0 −8.31601 0 −8.08801 0 −26.9480 0
1.2 0 8.77200 0 17.3160 0 26.0880 0 49.9480 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.p 2
4.b odd 2 1 1216.4.a.g 2
8.b even 2 1 304.4.a.c 2
8.d odd 2 1 38.4.a.c 2
24.f even 2 1 342.4.a.h 2
40.e odd 2 1 950.4.a.e 2
40.k even 4 2 950.4.b.i 4
56.e even 2 1 1862.4.a.e 2
152.b even 2 1 722.4.a.f 2

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

## Hecke kernels

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

 $$T_{3}^{2} - 9T_{3} + 2$$ T3^2 - 9*T3 + 2 $$T_{5}^{2} - 9T_{5} - 144$$ T5^2 - 9*T5 - 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 9T + 2$$
$5$ $$T^{2} - 9T - 144$$
$7$ $$T^{2} - 18T - 211$$
$11$ $$T^{2} + 17T + 54$$
$13$ $$T^{2} + 17T - 3012$$
$17$ $$T^{2} + 80T + 1527$$
$19$ $$(T - 19)^{2}$$
$23$ $$T^{2} + 73T - 1752$$
$29$ $$T^{2} + 3T - 8046$$
$31$ $$T^{2} + 212T - 24096$$
$37$ $$T^{2} + 192T - 5092$$
$41$ $$T^{2} + 50T - 1200$$
$43$ $$T^{2} - 677T + 113688$$
$47$ $$T^{2} - 389T - 54168$$
$53$ $$T^{2} - 1219 T + 366216$$
$59$ $$T^{2} + 287T + 9186$$
$61$ $$T^{2} + 313T - 200366$$
$67$ $$T^{2} - 1223 T + 265728$$
$71$ $$T^{2} + 200T - 235572$$
$73$ $$T^{2} - 378T - 582151$$
$79$ $$T^{2} + 1350 T + 394232$$
$83$ $$T^{2} + 670T - 574632$$
$89$ $$T^{2} + 236T - 631104$$
$97$ $$T^{2} - 1294 T + 228736$$