# Properties

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

# Learn more

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: 3.3.7057.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 22x + 32$$ x^3 - x^2 - 22*x + 32 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 152) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} - 2 \beta_1 - 2) q^{5} + (\beta_{2} + \beta_1 + 2) q^{7} + (3 \beta_{2} - 4 \beta_1 + 6) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + (-b2 - 2*b1 - 2) * q^5 + (b2 + b1 + 2) * q^7 + (3*b2 - 4*b1 + 6) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} - 2 \beta_1 - 2) q^{5} + (\beta_{2} + \beta_1 + 2) q^{7} + (3 \beta_{2} - 4 \beta_1 + 6) q^{9} + (5 \beta_{2} - 2 \beta_1 - 36) q^{11} + ( - 2 \beta_{2} + 9 \beta_1 - 10) q^{13} + ( - 4 \beta_{2} - 2 \beta_1 + 42) q^{15} + (2 \beta_{2} + 8 \beta_1 + 3) q^{17} + 19 q^{19} + (3 \beta_1 - 38) q^{21} + ( - 14 \beta_{2} + 5 \beta_1 + 110) q^{23} + ( - 15 \beta_{2} + 20 \beta_1 + 59) q^{25} + ( - \beta_{2} + 16 \beta_1 - 59) q^{27} + ( - 11 \beta_1 + 46) q^{29} + (6 \beta_{2} - 6 \beta_1 + 138) q^{31} + (18 \beta_{2} + 22 \beta_1 - 116) q^{33} + (9 \beta_{2} - 10 \beta_1 - 114) q^{35} + (18 \beta_{2} + 54 \beta_1 - 40) q^{37} + (50 \beta_{2} - 17 \beta_1 + 38) q^{39} + (14 \beta_{2} - 128) q^{41} + (19 \beta_{2} - 38 \beta_1 - 150) q^{43} + ( - 15 \beta_{2} + 40 \beta_1 + 148) q^{45} + (53 \beta_{2} - 48 \beta_1 - 84) q^{47} + ( - 4 \beta_{2} + 4 \beta_1 - 266) q^{49} + (25 \beta_{2} - 99) q^{51} + (62 \beta_{2} - 67 \beta_1 + 82) q^{53} + (49 \beta_{2} + 112 \beta_1 + 12) q^{55} + ( - 19 \beta_{2} - 19) q^{57} + ( - 13 \beta_{2} + 12 \beta_1 + 67) q^{59} + (45 \beta_{2} + 34 \beta_1 + 188) q^{61} + (23 \beta_{2} - 30 \beta_1 - 28) q^{63} + (54 \beta_{2} - 78 \beta_1 - 530) q^{65} + (57 \beta_{2} + 68 \beta_1 + 63) q^{67} + ( - 62 \beta_{2} - 61 \beta_1 + 318) q^{69} + ( - 18 \beta_{2} + 98 \beta_1 - 312) q^{71} + ( - 104 \beta_{2} - 64 \beta_1 - 175) q^{73} + (51 \beta_{2} - 80 \beta_1 + 341) q^{75} + ( - 31 \beta_{2} - 68 \beta_1 + 34) q^{77} + (8 \beta_{2} - 150 \beta_1 + 148) q^{79} + (44 \beta_{2} + 88 \beta_1 - 135) q^{81} + (168 \beta_{2} + 58 \beta_1 + 204) q^{83} + (55 \beta_{2} - 78 \beta_1 - 646) q^{85} + ( - 90 \beta_{2} + 11 \beta_1 - 2) q^{87} + ( - 30 \beta_{2} - 2 \beta_1 - 442) q^{89} + ( - 53 \beta_{2} + 52 \beta_1 + 241) q^{91} + ( - 174 \beta_{2} + 30 \beta_1 - 306) q^{93} + ( - 19 \beta_{2} - 38 \beta_1 - 38) q^{95} + (226 \beta_{2} - 88 \beta_1 - 258) q^{97} + (33 \beta_{2} + 104 \beta_1 + 424) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + (-b2 - 2*b1 - 2) * q^5 + (b2 + b1 + 2) * q^7 + (3*b2 - 4*b1 + 6) * q^9 + (5*b2 - 2*b1 - 36) * q^11 + (-2*b2 + 9*b1 - 10) * q^13 + (-4*b2 - 2*b1 + 42) * q^15 + (2*b2 + 8*b1 + 3) * q^17 + 19 * q^19 + (3*b1 - 38) * q^21 + (-14*b2 + 5*b1 + 110) * q^23 + (-15*b2 + 20*b1 + 59) * q^25 + (-b2 + 16*b1 - 59) * q^27 + (-11*b1 + 46) * q^29 + (6*b2 - 6*b1 + 138) * q^31 + (18*b2 + 22*b1 - 116) * q^33 + (9*b2 - 10*b1 - 114) * q^35 + (18*b2 + 54*b1 - 40) * q^37 + (50*b2 - 17*b1 + 38) * q^39 + (14*b2 - 128) * q^41 + (19*b2 - 38*b1 - 150) * q^43 + (-15*b2 + 40*b1 + 148) * q^45 + (53*b2 - 48*b1 - 84) * q^47 + (-4*b2 + 4*b1 - 266) * q^49 + (25*b2 - 99) * q^51 + (62*b2 - 67*b1 + 82) * q^53 + (49*b2 + 112*b1 + 12) * q^55 + (-19*b2 - 19) * q^57 + (-13*b2 + 12*b1 + 67) * q^59 + (45*b2 + 34*b1 + 188) * q^61 + (23*b2 - 30*b1 - 28) * q^63 + (54*b2 - 78*b1 - 530) * q^65 + (57*b2 + 68*b1 + 63) * q^67 + (-62*b2 - 61*b1 + 318) * q^69 + (-18*b2 + 98*b1 - 312) * q^71 + (-104*b2 - 64*b1 - 175) * q^73 + (51*b2 - 80*b1 + 341) * q^75 + (-31*b2 - 68*b1 + 34) * q^77 + (8*b2 - 150*b1 + 148) * q^79 + (44*b2 + 88*b1 - 135) * q^81 + (168*b2 + 58*b1 + 204) * q^83 + (55*b2 - 78*b1 - 646) * q^85 + (-90*b2 + 11*b1 - 2) * q^87 + (-30*b2 - 2*b1 - 442) * q^89 + (-53*b2 + 52*b1 + 241) * q^91 + (-174*b2 + 30*b1 - 306) * q^93 + (-19*b2 - 38*b1 - 38) * q^95 + (226*b2 - 88*b1 - 258) * q^97 + (33*b2 + 104*b1 + 424) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9}+O(q^{10})$$ 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9 $$3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9} - 103 q^{11} - 32 q^{13} + 122 q^{15} + 11 q^{17} + 57 q^{19} - 114 q^{21} + 316 q^{23} + 162 q^{25} - 178 q^{27} + 138 q^{29} + 420 q^{31} - 330 q^{33} - 333 q^{35} - 102 q^{37} + 164 q^{39} - 370 q^{41} - 431 q^{43} + 429 q^{45} - 199 q^{47} - 802 q^{49} - 272 q^{51} + 308 q^{53} + 85 q^{55} - 76 q^{57} + 188 q^{59} + 609 q^{61} - 61 q^{63} - 1536 q^{65} + 246 q^{67} + 892 q^{69} - 954 q^{71} - 629 q^{73} + 1074 q^{75} + 71 q^{77} + 452 q^{79} - 361 q^{81} + 780 q^{83} - 1883 q^{85} - 96 q^{87} - 1356 q^{89} + 670 q^{91} - 1092 q^{93} - 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100})$$ 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9 - 103 * q^11 - 32 * q^13 + 122 * q^15 + 11 * q^17 + 57 * q^19 - 114 * q^21 + 316 * q^23 + 162 * q^25 - 178 * q^27 + 138 * q^29 + 420 * q^31 - 330 * q^33 - 333 * q^35 - 102 * q^37 + 164 * q^39 - 370 * q^41 - 431 * q^43 + 429 * q^45 - 199 * q^47 - 802 * q^49 - 272 * q^51 + 308 * q^53 + 85 * q^55 - 76 * q^57 + 188 * q^59 + 609 * q^61 - 61 * q^63 - 1536 * q^65 + 246 * q^67 + 892 * q^69 - 954 * q^71 - 629 * q^73 + 1074 * q^75 + 71 * q^77 + 452 * q^79 - 361 * q^81 + 780 * q^83 - 1883 * q^85 - 96 * q^87 - 1356 * q^89 + 670 * q^91 - 1092 * q^93 - 133 * q^95 - 548 * q^97 + 1305 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 22x + 32$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} + 3\nu - 16 ) / 2$$ (v^2 + 3*v - 16) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{2} - \nu - 14 ) / 2$$ (v^2 - v - 14) / 2
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 2$$ (-b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} + \beta _1 + 29 ) / 2$$ (3*b2 + b1 + 29) / 2

## 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
 −4.86867 4.36181 1.50686
0 −8.28632 0 −2.38427 0 5.83529 0 41.6631 0
1.2 0 −1.33180 0 −18.4427 0 10.3872 0 −25.2263 0
1.3 0 5.61812 0 13.8269 0 −9.22252 0 4.56325 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.r 3
4.b odd 2 1 1216.4.a.w 3
8.b even 2 1 152.4.a.c 3
8.d odd 2 1 304.4.a.g 3
24.h odd 2 1 1368.4.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.c 3 8.b even 2 1
304.4.a.g 3 8.d odd 2 1
1216.4.a.r 3 1.a even 1 1 trivial
1216.4.a.w 3 4.b odd 2 1
1368.4.a.d 3 24.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{3} + 4T_{3}^{2} - 43T_{3} - 62$$ T3^3 + 4*T3^2 - 43*T3 - 62 $$T_{5}^{3} + 7T_{5}^{2} - 244T_{5} - 608$$ T5^3 + 7*T5^2 - 244*T5 - 608

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 4 T^{2} - 43 T - 62$$
$5$ $$T^{3} + 7 T^{2} - 244 T - 608$$
$7$ $$T^{3} - 7 T^{2} - 89 T + 559$$
$11$ $$T^{3} + 103 T^{2} + 2212 T - 22156$$
$13$ $$T^{3} + 32 T^{2} - 3677 T - 131420$$
$17$ $$T^{3} - 11 T^{2} - 3417 T - 32173$$
$19$ $$(T - 19)^{3}$$
$23$ $$T^{3} - 316 T^{2} + 23147 T + 242336$$
$29$ $$T^{3} - 138 T^{2} + 419 T + 345766$$
$31$ $$T^{3} - 420 T^{2} + 55584 T - 2336256$$
$37$ $$T^{3} + 102 T^{2} + \cdots - 15565400$$
$41$ $$T^{3} + 370 T^{2} + 36160 T + 707456$$
$43$ $$T^{3} + 431 T^{2} - 20508 T - 5420480$$
$47$ $$T^{3} + 199 T^{2} + \cdots - 45306112$$
$53$ $$T^{3} - 308 T^{2} - 340901 T - 6630640$$
$59$ $$T^{3} - 188 T^{2} - 2195 T + 1077242$$
$61$ $$T^{3} - 609 T^{2} + \cdots + 50618548$$
$67$ $$T^{3} - 246 T^{2} + \cdots + 96246632$$
$71$ $$T^{3} + 954 T^{2} + \cdots - 237273448$$
$73$ $$T^{3} + 629 T^{2} + \cdots - 417051529$$
$79$ $$T^{3} - 452 T^{2} + \cdots + 601611824$$
$83$ $$T^{3} - 780 T^{2} + \cdots + 1048786960$$
$89$ $$T^{3} + 1356 T^{2} + \cdots + 71672320$$
$97$ $$T^{3} + 548 T^{2} + \cdots - 2035482752$$
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