# Properties

 Label 1216.4.a.y Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12$$ x^5 - 2*x^4 - 28*x^3 - 8*x^2 + 73*x - 12 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{3} + \beta_{2} - 1) q^{5} + (\beta_{4} - 2 \beta_{2} - 1) q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + (-b3 + b2 - 1) * q^5 + (b4 - 2*b2 - 1) * q^7 + (-b3 + 2*b2 + b1 - 1) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{3} + \beta_{2} - 1) q^{5} + (\beta_{4} - 2 \beta_{2} - 1) q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{9} + (2 \beta_{4} + \beta_{3} - \beta_1 - 3) q^{11} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 14) q^{13} + ( - 2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 15) q^{15} + ( - 4 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} - 2 \beta_1 - 13) q^{17} + 19 q^{19} + ( - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 45) q^{21} + ( - \beta_{4} + \beta_{3} - 13 \beta_{2} - 8) q^{23} + ( - 3 \beta_{3} + 10 \beta_{2} + 3 \beta_1 - 18) q^{25} + (9 \beta_{3} + 14 \beta_{2} - 3 \beta_1 - 16) q^{27} + ( - 5 \beta_{4} + 6 \beta_{3} + 16 \beta_{2} - 11 \beta_1 + 19) q^{29} + ( - 6 \beta_{4} + 5 \beta_{3} - 25 \beta_{2} + 5 \beta_1 + 27) q^{31} + ( - 6 \beta_{4} + \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 15) q^{33} + ( - 10 \beta_{4} + 3 \beta_{3} + 16 \beta_{2} - \beta_1 + 5) q^{35} + (2 \beta_{4} - 9 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 87) q^{37} + (13 \beta_{4} + \beta_{3} - 21 \beta_{2} + 6 \beta_1 - 66) q^{39} + (16 \beta_{4} + 11 \beta_{3} + 27 \beta_{2} - 13 \beta_1 - 171) q^{41} + ( - 2 \beta_{4} + 15 \beta_{3} + 17 \beta_1 + 29) q^{43} + (6 \beta_{4} - 5 \beta_{3} + 33 \beta_{2} + 4 \beta_1 + 111) q^{45} + (14 \beta_{4} + 21 \beta_{3} - 5 \beta_{2} - 18 \beta_1 - 139) q^{47} + ( - 16 \beta_{4} + 20 \beta_{3} + 10 \beta_{2} + 6 \beta_1 - 86) q^{49} + ( - 10 \beta_{3} + 23 \beta_{2} - 22 \beta_1 - 121) q^{51} + (7 \beta_{4} - 2 \beta_{3} + 16 \beta_{2} + 11 \beta_1 + 179) q^{53} + ( - 14 \beta_{4} + \beta_{3} + 9 \beta_{2} + 12 \beta_1 - 3) q^{55} + ( - 19 \beta_{2} - 19) q^{57} + ( - 4 \beta_{4} - 7 \beta_{3} - 18 \beta_{2} + 21 \beta_1 - 30) q^{59} + (4 \beta_{4} - 13 \beta_{3} + 57 \beta_{2} - 10 \beta_1 + 205) q^{61} + ( - 8 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 27) q^{63} + (22 \beta_{4} - 8 \beta_{3} + 10 \beta_{2} - 6 \beta_1 - 328) q^{65} + (36 \beta_{4} + 2 \beta_{3} - 57 \beta_{2} + 22 \beta_1 - 151) q^{67} + (5 \beta_{4} - 19 \beta_{3} + 27 \beta_{2} + 14 \beta_1 + 330) q^{69} + ( - 2 \beta_{4} - 26 \beta_{3} + 66 \beta_{2} - 6 \beta_1 - 176) q^{71} + (24 \beta_{4} + 14 \beta_{3} + 34 \beta_{2} - 2 \beta_1 - 285) q^{73} + (31 \beta_{3} - 28 \beta_{2} - 13 \beta_1 - 214) q^{75} + ( - 14 \beta_{4} + 29 \beta_{3} + 39 \beta_{2} - 10 \beta_1 + 279) q^{77} + ( - 10 \beta_{4} + 15 \beta_{3} + 71 \beta_{2} + 35 \beta_1 - 41) q^{79} + (12 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} - 26 \beta_1 - 379) q^{81} + ( - 14 \beta_{4} - 50 \beta_{3} - 46 \beta_1 - 444) q^{83} + (24 \beta_{4} - 3 \beta_{3} - 87 \beta_{2} + 8 \beta_1 + 381) q^{85} + (5 \beta_{4} - 71 \beta_{3} + 97 \beta_{2} - 20 \beta_1 - 410) q^{87} + ( - 6 \beta_{4} - 49 \beta_{3} + 15 \beta_{2} + 73 \beta_1 - 87) q^{89} + (60 \beta_{4} - 37 \beta_{3} - 116 \beta_{2} + 7 \beta_1 - 346) q^{91} + (38 \beta_{4} - 34 \beta_{3} - 14 \beta_{2} + 34 \beta_1 + 574) q^{93} + ( - 19 \beta_{3} + 19 \beta_{2} - 19) q^{95} + ( - 48 \beta_{4} - 36 \beta_{3} + 26 \beta_{2} + 28 \beta_1 - 298) q^{97} + ( - 28 \beta_{4} - 29 \beta_{3} + 17 \beta_1 - 111) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + (-b3 + b2 - 1) * q^5 + (b4 - 2*b2 - 1) * q^7 + (-b3 + 2*b2 + b1 - 1) * q^9 + (2*b4 + b3 - b1 - 3) * q^11 + (-b4 + 3*b3 + b2 + 2*b1 + 14) * q^13 + (-2*b4 + 3*b3 - 3*b2 - 3*b1 - 15) * q^15 + (-4*b4 - 4*b3 + 8*b2 - 2*b1 - 13) * q^17 + 19 * q^19 + (-3*b4 + 2*b3 + 3*b1 + 45) * q^21 + (-b4 + b3 - 13*b2 - 8) * q^23 + (-3*b3 + 10*b2 + 3*b1 - 18) * q^25 + (9*b3 + 14*b2 - 3*b1 - 16) * q^27 + (-5*b4 + 6*b3 + 16*b2 - 11*b1 + 19) * q^29 + (-6*b4 + 5*b3 - 25*b2 + 5*b1 + 27) * q^31 + (-6*b4 + b3 + 9*b2 + 3*b1 - 15) * q^33 + (-10*b4 + 3*b3 + 16*b2 - b1 + 5) * q^35 + (2*b4 - 9*b3 + 7*b2 - 3*b1 + 87) * q^37 + (13*b4 + b3 - 21*b2 + 6*b1 - 66) * q^39 + (16*b4 + 11*b3 + 27*b2 - 13*b1 - 171) * q^41 + (-2*b4 + 15*b3 + 17*b1 + 29) * q^43 + (6*b4 - 5*b3 + 33*b2 + 4*b1 + 111) * q^45 + (14*b4 + 21*b3 - 5*b2 - 18*b1 - 139) * q^47 + (-16*b4 + 20*b3 + 10*b2 + 6*b1 - 86) * q^49 + (-10*b3 + 23*b2 - 22*b1 - 121) * q^51 + (7*b4 - 2*b3 + 16*b2 + 11*b1 + 179) * q^53 + (-14*b4 + b3 + 9*b2 + 12*b1 - 3) * q^55 + (-19*b2 - 19) * q^57 + (-4*b4 - 7*b3 - 18*b2 + 21*b1 - 30) * q^59 + (4*b4 - 13*b3 + 57*b2 - 10*b1 + 205) * q^61 + (-8*b4 - b3 - 3*b2 + 4*b1 - 27) * q^63 + (22*b4 - 8*b3 + 10*b2 - 6*b1 - 328) * q^65 + (36*b4 + 2*b3 - 57*b2 + 22*b1 - 151) * q^67 + (5*b4 - 19*b3 + 27*b2 + 14*b1 + 330) * q^69 + (-2*b4 - 26*b3 + 66*b2 - 6*b1 - 176) * q^71 + (24*b4 + 14*b3 + 34*b2 - 2*b1 - 285) * q^73 + (31*b3 - 28*b2 - 13*b1 - 214) * q^75 + (-14*b4 + 29*b3 + 39*b2 - 10*b1 + 279) * q^77 + (-10*b4 + 15*b3 + 71*b2 + 35*b1 - 41) * q^79 + (12*b4 + 8*b3 + 2*b2 - 26*b1 - 379) * q^81 + (-14*b4 - 50*b3 - 46*b1 - 444) * q^83 + (24*b4 - 3*b3 - 87*b2 + 8*b1 + 381) * q^85 + (5*b4 - 71*b3 + 97*b2 - 20*b1 - 410) * q^87 + (-6*b4 - 49*b3 + 15*b2 + 73*b1 - 87) * q^89 + (60*b4 - 37*b3 - 116*b2 + 7*b1 - 346) * q^91 + (38*b4 - 34*b3 - 14*b2 + 34*b1 + 574) * q^93 + (-19*b3 + 19*b2 - 19) * q^95 + (-48*b4 - 36*b3 + 26*b2 + 28*b1 - 298) * q^97 + (-28*b4 - 29*b3 + 17*b1 - 111) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 6 q^{3} - 5 q^{5} - 7 q^{7} - 5 q^{9}+O(q^{10})$$ 5 * q - 6 * q^3 - 5 * q^5 - 7 * q^7 - 5 * q^9 $$5 q - 6 q^{3} - 5 q^{5} - 7 q^{7} - 5 q^{9} - 13 q^{11} + 72 q^{13} - 72 q^{15} - 59 q^{17} + 95 q^{19} + 224 q^{21} - 52 q^{23} - 86 q^{25} - 54 q^{27} + 128 q^{29} + 110 q^{31} - 68 q^{33} + 45 q^{35} + 436 q^{37} - 356 q^{39} - 804 q^{41} + 143 q^{43} + 579 q^{45} - 661 q^{47} - 406 q^{49} - 570 q^{51} + 898 q^{53} - 17 q^{55} - 114 q^{57} - 196 q^{59} + 1079 q^{61} - 143 q^{63} - 1632 q^{65} - 832 q^{67} + 1644 q^{69} - 834 q^{71} - 1375 q^{73} - 1054 q^{75} + 1473 q^{77} - 154 q^{79} - 1859 q^{81} - 2224 q^{83} + 1807 q^{85} - 2004 q^{87} - 542 q^{89} - 1890 q^{91} + 2788 q^{93} - 95 q^{95} - 1528 q^{97} - 601 q^{99}+O(q^{100})$$ 5 * q - 6 * q^3 - 5 * q^5 - 7 * q^7 - 5 * q^9 - 13 * q^11 + 72 * q^13 - 72 * q^15 - 59 * q^17 + 95 * q^19 + 224 * q^21 - 52 * q^23 - 86 * q^25 - 54 * q^27 + 128 * q^29 + 110 * q^31 - 68 * q^33 + 45 * q^35 + 436 * q^37 - 356 * q^39 - 804 * q^41 + 143 * q^43 + 579 * q^45 - 661 * q^47 - 406 * q^49 - 570 * q^51 + 898 * q^53 - 17 * q^55 - 114 * q^57 - 196 * q^59 + 1079 * q^61 - 143 * q^63 - 1632 * q^65 - 832 * q^67 + 1644 * q^69 - 834 * q^71 - 1375 * q^73 - 1054 * q^75 + 1473 * q^77 - 154 * q^79 - 1859 * q^81 - 2224 * q^83 + 1807 * q^85 - 2004 * q^87 - 542 * q^89 - 1890 * q^91 + 2788 * q^93 - 95 * q^95 - 1528 * q^97 - 601 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{4} - 7\nu^{3} - 15\nu^{2} + 177\nu + 24 ) / 22$$ (v^4 - 7*v^3 - 15*v^2 + 177*v + 24) / 22 $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 7\nu^{3} - 15\nu^{2} + 89\nu + 68 ) / 22$$ (v^4 - 7*v^3 - 15*v^2 + 89*v + 68) / 22 $$\beta_{3}$$ $$=$$ $$( 5\nu^{4} - 13\nu^{3} - 119\nu^{2} - 17\nu + 153 ) / 11$$ (5*v^4 - 13*v^3 - 119*v^2 - 17*v + 153) / 11 $$\beta_{4}$$ $$=$$ $$( -3\nu^{4} - \nu^{3} + 111\nu^{2} + 151\nu - 270 ) / 11$$ (-3*v^4 - v^3 + 111*v^2 + 151*v - 270) / 11
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 2 ) / 4$$ (-b2 + b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + \beta_{3} - 5\beta_{2} + \beta _1 + 25 ) / 2$$ (b4 + b3 - 5*b2 + b1 + 25) / 2 $$\nu^{3}$$ $$=$$ $$( 4\beta_{4} + 6\beta_{3} - 61\beta_{2} + 25\beta _1 + 176 ) / 4$$ (4*b4 + 6*b3 - 61*b2 + 25*b1 + 176) / 4 $$\nu^{4}$$ $$=$$ $$( 29\beta_{4} + 36\beta_{3} - 200\beta_{2} + 58\beta _1 + 766 ) / 2$$ (29*b4 + 36*b3 - 200*b2 + 58*b1 + 766) / 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
 1.35870 0.169415 −3.54663 −2.31928 6.33779
0 −7.68562 0 15.2627 0 −2.79440 0 32.0687 0
1.2 0 −4.75519 0 −10.5762 0 −30.4413 0 −4.38817 0
1.3 0 −2.55330 0 −7.40075 0 10.4962 0 −20.4807 0
1.4 0 3.67449 0 7.12795 0 −0.511313 0 −13.4981 0
1.5 0 5.31962 0 −9.41363 0 16.2508 0 1.29831 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.y 5
4.b odd 2 1 1216.4.a.bd 5
8.b even 2 1 608.4.a.h yes 5
8.d odd 2 1 608.4.a.e 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.e 5 8.d odd 2 1
608.4.a.h yes 5 8.b even 2 1
1216.4.a.y 5 1.a even 1 1 trivial
1216.4.a.bd 5 4.b odd 2 1

## Hecke kernels

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

 $$T_{3}^{5} + 6T_{3}^{4} - 47T_{3}^{3} - 228T_{3}^{2} + 496T_{3} + 1824$$ T3^5 + 6*T3^4 - 47*T3^3 - 228*T3^2 + 496*T3 + 1824 $$T_{5}^{5} + 5T_{5}^{4} - 257T_{5}^{3} - 1825T_{5}^{2} + 10428T_{5} + 80160$$ T5^5 + 5*T5^4 - 257*T5^3 - 1825*T5^2 + 10428*T5 + 80160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + 6 T^{4} - 47 T^{3} + \cdots + 1824$$
$5$ $$T^{5} + 5 T^{4} - 257 T^{3} + \cdots + 80160$$
$7$ $$T^{5} + 7 T^{4} - 630 T^{3} + \cdots + 7419$$
$11$ $$T^{5} + 13 T^{4} - 1945 T^{3} + \cdots + 122580$$
$13$ $$T^{5} - 72 T^{4} - 3293 T^{3} + \cdots - 527040$$
$17$ $$T^{5} + 59 T^{4} + \cdots - 297647925$$
$19$ $$(T - 19)^{5}$$
$23$ $$T^{5} + 52 T^{4} + \cdots + 304895232$$
$29$ $$T^{5} - 128 T^{4} + \cdots - 106742590368$$
$31$ $$T^{5} - 110 T^{4} + \cdots - 53136006400$$
$37$ $$T^{5} - 436 T^{4} + \cdots - 4385970560$$
$41$ $$T^{5} + 804 T^{4} + \cdots - 855201369600$$
$43$ $$T^{5} - 143 T^{4} + \cdots - 197931733952$$
$47$ $$T^{5} + 661 T^{4} + \cdots - 2970610340352$$
$53$ $$T^{5} - 898 T^{4} + \cdots + 173906788200$$
$59$ $$T^{5} + 196 T^{4} + \cdots + 131722274520$$
$61$ $$T^{5} - 1079 T^{4} + \cdots + 2650589198500$$
$67$ $$T^{5} + 832 T^{4} + \cdots + 823051244968$$
$71$ $$T^{5} + 834 T^{4} + \cdots - 679526470240$$
$73$ $$T^{5} + 1375 T^{4} + \cdots + 2709187037875$$
$79$ $$T^{5} + 154 T^{4} + \cdots - 17190104220000$$
$83$ $$T^{5} + \cdots - 297963542605824$$
$89$ $$T^{5} + \cdots + 414712220295168$$
$97$ $$T^{5} + 1528 T^{4} + \cdots + 212723907840$$
show more
show less