Properties

 Label 1216.4.a.v Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.35529.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x + 4$$ x^3 - x^2 - 52*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{2} - 3) q^{5} + (2 \beta_{2} + \beta_1 - 15) q^{7} + ( - 3 \beta_{2} + 3 \beta_1 + 19) q^{9}+O(q^{10})$$ q + (b2 + b1) * q^3 + (-b2 - 3) * q^5 + (2*b2 + b1 - 15) * q^7 + (-3*b2 + 3*b1 + 19) * q^9 $$q + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{2} - 3) q^{5} + (2 \beta_{2} + \beta_1 - 15) q^{7} + ( - 3 \beta_{2} + 3 \beta_1 + 19) q^{9} + ( - 3 \beta_{2} - 8 \beta_1 + 29) q^{11} + ( - 7 \beta_{2} + 5 \beta_1 + 2) q^{13} + (2 \beta_{2} - 30) q^{15} + ( - 6 \beta_{2} + 13 \beta_1 + 23) q^{17} + 19 q^{19} + ( - 23 \beta_{2} - 15 \beta_1 + 76) q^{21} + (9 \beta_{2} + 11 \beta_1 - 38) q^{23} + (3 \beta_{2} - 6 \beta_1 - 68) q^{25} + (13 \beta_{2} + 19 \beta_1 - 42) q^{27} + (23 \beta_{2} + 21 \beta_1 + 24) q^{29} + (12 \beta_{2} - 16 \beta_1 + 44) q^{31} + (28 \beta_{2} - 10 \beta_1 - 218) q^{33} + (17 \beta_{2} + 6 \beta_1 - 33) q^{35} + (36 \beta_{2} + 16 \beta_1 + 150) q^{37} + (47 \beta_{2} + 53 \beta_1 - 130) q^{39} + ( - 26 \beta_{2} - 14 \beta_1 - 58) q^{41} + (27 \beta_{2} + 22 \beta_1 - 11) q^{43} + ( - 13 \beta_{2} - 36 \beta_1 + 141) q^{45} + ( - 41 \beta_{2} + 14 \beta_1 - 59) q^{47} + ( - 76 \beta_{2} - 39 \beta_1 + 36) q^{49} + (79 \beta_{2} + 119 \beta_1 + 28) q^{51} + ( - 31 \beta_{2} - 59 \beta_1 - 46) q^{53} + ( - 45 \beta_{2} + 30 \beta_1 - 87) q^{55} + (19 \beta_{2} + 19 \beta_1) q^{57} + (37 \beta_{2} + 25 \beta_1 + 452) q^{59} + (13 \beta_{2} + 78 \beta_1 + 109) q^{61} + (107 \beta_{2} + 28 \beta_1 - 525) q^{63} + (8 \beta_{2} - 72 \beta_1 + 420) q^{65} + ( - 61 \beta_{2} - 23 \beta_1 + 322) q^{67} + ( - 61 \beta_{2} + \beta_1 + 446) q^{69} + (88 \beta_{2} - 50 \beta_1 - 334) q^{71} + (84 \beta_{2} + 167 \beta_1 - 249) q^{73} + ( - 95 \beta_{2} - 113 \beta_1 - 6) q^{75} + (127 \beta_{2} + 104 \beta_1 - 653) q^{77} + (10 \beta_{2} + 70 \beta_1 + 444) q^{79} + (12 \beta_{2} - 48 \beta_1 + 181) q^{81} + (54 \beta_{2} + 26 \beta_1 + 592) q^{83} + (3 \beta_{2} - 114 \beta_1 + 453) q^{85} + ( - 49 \beta_{2} + 81 \beta_1 + 1026) q^{87} + (68 \beta_{2} + 70 \beta_1 + 632) q^{89} + (165 \beta_{2} + 35 \beta_1 - 586) q^{91} + ( - 48 \beta_{2} - 88 \beta_1 + 104) q^{93} + ( - 19 \beta_{2} - 57) q^{95} + (54 \beta_{2} + 96 \beta_1 - 16) q^{97} + ( - 297 \beta_{2} - 146 \beta_1 - 103) q^{99}+O(q^{100})$$ q + (b2 + b1) * q^3 + (-b2 - 3) * q^5 + (2*b2 + b1 - 15) * q^7 + (-3*b2 + 3*b1 + 19) * q^9 + (-3*b2 - 8*b1 + 29) * q^11 + (-7*b2 + 5*b1 + 2) * q^13 + (2*b2 - 30) * q^15 + (-6*b2 + 13*b1 + 23) * q^17 + 19 * q^19 + (-23*b2 - 15*b1 + 76) * q^21 + (9*b2 + 11*b1 - 38) * q^23 + (3*b2 - 6*b1 - 68) * q^25 + (13*b2 + 19*b1 - 42) * q^27 + (23*b2 + 21*b1 + 24) * q^29 + (12*b2 - 16*b1 + 44) * q^31 + (28*b2 - 10*b1 - 218) * q^33 + (17*b2 + 6*b1 - 33) * q^35 + (36*b2 + 16*b1 + 150) * q^37 + (47*b2 + 53*b1 - 130) * q^39 + (-26*b2 - 14*b1 - 58) * q^41 + (27*b2 + 22*b1 - 11) * q^43 + (-13*b2 - 36*b1 + 141) * q^45 + (-41*b2 + 14*b1 - 59) * q^47 + (-76*b2 - 39*b1 + 36) * q^49 + (79*b2 + 119*b1 + 28) * q^51 + (-31*b2 - 59*b1 - 46) * q^53 + (-45*b2 + 30*b1 - 87) * q^55 + (19*b2 + 19*b1) * q^57 + (37*b2 + 25*b1 + 452) * q^59 + (13*b2 + 78*b1 + 109) * q^61 + (107*b2 + 28*b1 - 525) * q^63 + (8*b2 - 72*b1 + 420) * q^65 + (-61*b2 - 23*b1 + 322) * q^67 + (-61*b2 + b1 + 446) * q^69 + (88*b2 - 50*b1 - 334) * q^71 + (84*b2 + 167*b1 - 249) * q^73 + (-95*b2 - 113*b1 - 6) * q^75 + (127*b2 + 104*b1 - 653) * q^77 + (10*b2 + 70*b1 + 444) * q^79 + (12*b2 - 48*b1 + 181) * q^81 + (54*b2 + 26*b1 + 592) * q^83 + (3*b2 - 114*b1 + 453) * q^85 + (-49*b2 + 81*b1 + 1026) * q^87 + (68*b2 + 70*b1 + 632) * q^89 + (165*b2 + 35*b1 - 586) * q^91 + (-48*b2 - 88*b1 + 104) * q^93 + (-19*b2 - 57) * q^95 + (54*b2 + 96*b1 - 16) * q^97 + (-297*b2 - 146*b1 - 103) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 9 q^{5} - 44 q^{7} + 60 q^{9}+O(q^{10})$$ 3 * q + q^3 - 9 * q^5 - 44 * q^7 + 60 * q^9 $$3 q + q^{3} - 9 q^{5} - 44 q^{7} + 60 q^{9} + 79 q^{11} + 11 q^{13} - 90 q^{15} + 82 q^{17} + 57 q^{19} + 213 q^{21} - 103 q^{23} - 210 q^{25} - 107 q^{27} + 93 q^{29} + 116 q^{31} - 664 q^{33} - 93 q^{35} + 466 q^{37} - 337 q^{39} - 188 q^{41} - 11 q^{43} + 387 q^{45} - 163 q^{47} + 69 q^{49} + 203 q^{51} - 197 q^{53} - 231 q^{55} + 19 q^{57} + 1381 q^{59} + 405 q^{61} - 1547 q^{63} + 1188 q^{65} + 943 q^{67} + 1339 q^{69} - 1052 q^{71} - 580 q^{73} - 131 q^{75} - 1855 q^{77} + 1402 q^{79} + 495 q^{81} + 1802 q^{83} + 1245 q^{85} + 3159 q^{87} + 1966 q^{89} - 1723 q^{91} + 224 q^{93} - 171 q^{95} + 48 q^{97} - 455 q^{99}+O(q^{100})$$ 3 * q + q^3 - 9 * q^5 - 44 * q^7 + 60 * q^9 + 79 * q^11 + 11 * q^13 - 90 * q^15 + 82 * q^17 + 57 * q^19 + 213 * q^21 - 103 * q^23 - 210 * q^25 - 107 * q^27 + 93 * q^29 + 116 * q^31 - 664 * q^33 - 93 * q^35 + 466 * q^37 - 337 * q^39 - 188 * q^41 - 11 * q^43 + 387 * q^45 - 163 * q^47 + 69 * q^49 + 203 * q^51 - 197 * q^53 - 231 * q^55 + 19 * q^57 + 1381 * q^59 + 405 * q^61 - 1547 * q^63 + 1188 * q^65 + 943 * q^67 + 1339 * q^69 - 1052 * q^71 - 580 * q^73 - 131 * q^75 - 1855 * q^77 + 1402 * q^79 + 495 * q^81 + 1802 * q^83 + 1245 * q^85 + 3159 * q^87 + 1966 * q^89 - 1723 * q^91 + 224 * q^93 - 171 * q^95 + 48 * q^97 - 455 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 3\nu - 34 ) / 4$$ (v^2 - 3*v - 34) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$4\beta_{2} + 3\beta _1 + 34$$ 4*b2 + 3*b1 + 34

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
 0.0768183 −6.76918 7.69236
0 −8.47932 0 5.55614 0 −32.0355 0 44.8989 0
1.2 0 1.26314 0 −11.0323 0 −5.70454 0 −25.4045 0
1.3 0 8.21618 0 −3.52382 0 −6.26000 0 40.5056 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.v 3
4.b odd 2 1 1216.4.a.t 3
8.b even 2 1 304.4.a.h 3
8.d odd 2 1 76.4.a.b 3
24.f even 2 1 684.4.a.i 3
40.e odd 2 1 1900.4.a.c 3
40.k even 4 2 1900.4.c.c 6
152.b even 2 1 1444.4.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.b 3 8.d odd 2 1
304.4.a.h 3 8.b even 2 1
684.4.a.i 3 24.f even 2 1
1216.4.a.t 3 4.b odd 2 1
1216.4.a.v 3 1.a even 1 1 trivial
1444.4.a.e 3 152.b even 2 1
1900.4.a.c 3 40.e odd 2 1
1900.4.c.c 6 40.k even 4 2

Hecke kernels

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

 $$T_{3}^{3} - T_{3}^{2} - 70T_{3} + 88$$ T3^3 - T3^2 - 70*T3 + 88 $$T_{5}^{3} + 9T_{5}^{2} - 42T_{5} - 216$$ T5^3 + 9*T5^2 - 42*T5 - 216

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 70 T + 88$$
$5$ $$T^{3} + 9 T^{2} - 42 T - 216$$
$7$ $$T^{3} + 44 T^{2} + 419 T + 1144$$
$11$ $$T^{3} - 79 T^{2} - 666 T + 108888$$
$13$ $$T^{3} - 11 T^{2} - 6434 T + 201816$$
$17$ $$T^{3} - 82 T^{2} - 13065 T + 1022082$$
$19$ $$(T - 19)^{3}$$
$23$ $$T^{3} + 103 T^{2} - 3336 T - 235392$$
$29$ $$T^{3} - 93 T^{2} - 32064 T + 2252268$$
$31$ $$T^{3} - 116 T^{2} - 28640 T - 1084416$$
$37$ $$T^{3} - 466 T^{2} + \cdots + 15144328$$
$41$ $$T^{3} + 188 T^{2} - 26556 T - 5041152$$
$43$ $$T^{3} + 11 T^{2} - 45296 T + 2359248$$
$47$ $$T^{3} + 163 T^{2} - 146664 T + 3850752$$
$53$ $$T^{3} + 197 T^{2} + \cdots + 11566104$$
$59$ $$T^{3} - 1381 T^{2} + \cdots - 52865928$$
$61$ $$T^{3} - 405 T^{2} - 223668 T + 847124$$
$67$ $$T^{3} - 943 T^{2} + 83536 T + 1169904$$
$71$ $$T^{3} + 1052 T^{2} + \cdots - 521792928$$
$73$ $$T^{3} + 580 T^{2} + \cdots - 726527962$$
$79$ $$T^{3} - 1402 T^{2} + \cdots - 18139904$$
$83$ $$T^{3} - 1802 T^{2} + \cdots - 91984992$$
$89$ $$T^{3} - 1966 T^{2} + \cdots - 47198784$$
$97$ $$T^{3} - 48 T^{2} - 418356 T - 82010336$$