# Properties

 Label 1216.4.a.x 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: 3.3.3221.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 2$$ x^3 - x^2 - 9*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 2) q^{3} + ( - 2 \beta_{2} + \beta_1 - 1) q^{5} + (2 \beta_1 - 11) q^{7} + (6 \beta_{2} + 3 \beta_1 + 19) q^{9}+O(q^{10})$$ q + (b2 + 2) * q^3 + (-2*b2 + b1 - 1) * q^5 + (2*b1 - 11) * q^7 + (6*b2 + 3*b1 + 19) * q^9 $$q + (\beta_{2} + 2) q^{3} + ( - 2 \beta_{2} + \beta_1 - 1) q^{5} + (2 \beta_1 - 11) q^{7} + (6 \beta_{2} + 3 \beta_1 + 19) q^{9} + ( - \beta_1 + 9) q^{11} + ( - 2 \beta_{2} + 7 \beta_1 + 38) q^{13} + ( - 5 \beta_{2} - 7 \beta_1 - 80) q^{15} + ( - 6 \beta_{2} - 43) q^{17} - 19 q^{19} + ( - 3 \beta_{2} - 2 \beta_1 - 10) q^{21} + ( - 4 \beta_{2} + 11 \beta_1 - 62) q^{23} + ( - 6 \beta_{2} + 17 \beta_1 + 66) q^{25} + (28 \beta_{2} + 15 \beta_1 + 254) q^{27} + (23 \beta_{2} - 2 \beta_1 + 106) q^{29} + (23 \beta_{2} + 3 \beta_1 - 38) q^{31} + (5 \beta_{2} + \beta_1 + 12) q^{33} + (2 \beta_{2} - 11 \beta_1 + 79) q^{35} + ( - \beta_{2} + 27 \beta_1 - 4) q^{37} + (58 \beta_{2} - 13 \beta_1 + 34) q^{39} + ( - 23 \beta_{2} + 7 \beta_1 + 240) q^{41} + (4 \beta_{2} + 37 \beta_1 + 185) q^{43} + ( - 74 \beta_{2} - 35 \beta_1 - 385) q^{45} + (28 \beta_{2} + 57 \beta_1 + 73) q^{47} + ( - 8 \beta_{2} - 64 \beta_1 - 38) q^{49} + ( - 67 \beta_{2} - 18 \beta_1 - 338) q^{51} + (9 \beta_{2} + 50 \beta_1 - 84) q^{53} + ( - 8 \beta_{2} + 9 \beta_1 - 43) q^{55} + ( - 19 \beta_{2} - 38) q^{57} + (100 \beta_{2} - 9 \beta_1 - 36) q^{59} + ( - 28 \beta_{2} - 75 \beta_1 - 53) q^{61} + ( - 30 \beta_{2} - 61 \beta_1 + 139) q^{63} + ( - 144 \beta_{2} + 56 \beta_1 + 356) q^{65} + (65 \beta_{2} - 58 \beta_1 + 172) q^{67} + ( - 34 \beta_{2} - 23 \beta_1 - 226) q^{69} + (34 \beta_{2} - 78 \beta_1 - 306) q^{71} + ( - 34 \beta_{2} + 14 \beta_1 + 61) q^{73} + (110 \beta_{2} - 35 \beta_1 - 18) q^{75} + (4 \beta_{2} + 39 \beta_1 - 191) q^{77} + ( - 89 \beta_{2} - 75 \beta_1 + 26) q^{79} + (264 \beta_{2} - 12 \beta_1 + 1261) q^{81} + ( - 80 \beta_{2} + 42 \beta_1 + 8) q^{83} + (92 \beta_{2} + 11 \beta_1 + 511) q^{85} + (190 \beta_{2} + 71 \beta_1 + 1166) q^{87} + (39 \beta_{2} - 21 \beta_1 + 594) q^{89} + ( - 22 \beta_{2} - 59 \beta_1 + 202) q^{91} + (66 \beta_{2} + 66 \beta_1 + 908) q^{93} + (38 \beta_{2} - 19 \beta_1 + 19) q^{95} + (14 \beta_{2} - 68 \beta_1 - 744) q^{97} + (36 \beta_{2} + 41 \beta_1 - 3) q^{99}+O(q^{100})$$ q + (b2 + 2) * q^3 + (-2*b2 + b1 - 1) * q^5 + (2*b1 - 11) * q^7 + (6*b2 + 3*b1 + 19) * q^9 + (-b1 + 9) * q^11 + (-2*b2 + 7*b1 + 38) * q^13 + (-5*b2 - 7*b1 - 80) * q^15 + (-6*b2 - 43) * q^17 - 19 * q^19 + (-3*b2 - 2*b1 - 10) * q^21 + (-4*b2 + 11*b1 - 62) * q^23 + (-6*b2 + 17*b1 + 66) * q^25 + (28*b2 + 15*b1 + 254) * q^27 + (23*b2 - 2*b1 + 106) * q^29 + (23*b2 + 3*b1 - 38) * q^31 + (5*b2 + b1 + 12) * q^33 + (2*b2 - 11*b1 + 79) * q^35 + (-b2 + 27*b1 - 4) * q^37 + (58*b2 - 13*b1 + 34) * q^39 + (-23*b2 + 7*b1 + 240) * q^41 + (4*b2 + 37*b1 + 185) * q^43 + (-74*b2 - 35*b1 - 385) * q^45 + (28*b2 + 57*b1 + 73) * q^47 + (-8*b2 - 64*b1 - 38) * q^49 + (-67*b2 - 18*b1 - 338) * q^51 + (9*b2 + 50*b1 - 84) * q^53 + (-8*b2 + 9*b1 - 43) * q^55 + (-19*b2 - 38) * q^57 + (100*b2 - 9*b1 - 36) * q^59 + (-28*b2 - 75*b1 - 53) * q^61 + (-30*b2 - 61*b1 + 139) * q^63 + (-144*b2 + 56*b1 + 356) * q^65 + (65*b2 - 58*b1 + 172) * q^67 + (-34*b2 - 23*b1 - 226) * q^69 + (34*b2 - 78*b1 - 306) * q^71 + (-34*b2 + 14*b1 + 61) * q^73 + (110*b2 - 35*b1 - 18) * q^75 + (4*b2 + 39*b1 - 191) * q^77 + (-89*b2 - 75*b1 + 26) * q^79 + (264*b2 - 12*b1 + 1261) * q^81 + (-80*b2 + 42*b1 + 8) * q^83 + (92*b2 + 11*b1 + 511) * q^85 + (190*b2 + 71*b1 + 1166) * q^87 + (39*b2 - 21*b1 + 594) * q^89 + (-22*b2 - 59*b1 + 202) * q^91 + (66*b2 + 66*b1 + 908) * q^93 + (38*b2 - 19*b1 + 19) * q^95 + (14*b2 - 68*b1 - 744) * q^97 + (36*b2 + 41*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 5 q^{3} - 2 q^{5} - 35 q^{7} + 48 q^{9}+O(q^{10})$$ 3 * q + 5 * q^3 - 2 * q^5 - 35 * q^7 + 48 * q^9 $$3 q + 5 q^{3} - 2 q^{5} - 35 q^{7} + 48 q^{9} + 28 q^{11} + 109 q^{13} - 228 q^{15} - 123 q^{17} - 57 q^{19} - 25 q^{21} - 193 q^{23} + 187 q^{25} + 719 q^{27} + 297 q^{29} - 140 q^{31} + 30 q^{33} + 246 q^{35} - 38 q^{37} + 57 q^{39} + 736 q^{41} + 514 q^{43} - 1046 q^{45} + 134 q^{47} - 42 q^{49} - 929 q^{51} - 311 q^{53} - 130 q^{55} - 95 q^{57} - 199 q^{59} - 56 q^{61} + 508 q^{63} + 1156 q^{65} + 509 q^{67} - 621 q^{69} - 874 q^{71} + 203 q^{73} - 129 q^{75} - 616 q^{77} + 242 q^{79} + 3531 q^{81} + 62 q^{83} + 1430 q^{85} + 3237 q^{87} + 1764 q^{89} + 687 q^{91} + 2592 q^{93} + 38 q^{95} - 2178 q^{97} - 86 q^{99}+O(q^{100})$$ 3 * q + 5 * q^3 - 2 * q^5 - 35 * q^7 + 48 * q^9 + 28 * q^11 + 109 * q^13 - 228 * q^15 - 123 * q^17 - 57 * q^19 - 25 * q^21 - 193 * q^23 + 187 * q^25 + 719 * q^27 + 297 * q^29 - 140 * q^31 + 30 * q^33 + 246 * q^35 - 38 * q^37 + 57 * q^39 + 736 * q^41 + 514 * q^43 - 1046 * q^45 + 134 * q^47 - 42 * q^49 - 929 * q^51 - 311 * q^53 - 130 * q^55 - 95 * q^57 - 199 * q^59 - 56 * q^61 + 508 * q^63 + 1156 * q^65 + 509 * q^67 - 621 * q^69 - 874 * q^71 + 203 * q^73 - 129 * q^75 - 616 * q^77 + 242 * q^79 + 3531 * q^81 + 62 * q^83 + 1430 * q^85 + 3237 * q^87 + 1764 * q^89 + 687 * q^91 + 2592 * q^93 + 38 * q^95 - 2178 * q^97 - 86 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{2} + 3\nu + 5$$ -v^2 + 3*v + 5 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 2 ) / 4$$ (b2 + b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} - \beta _1 + 26 ) / 4$$ (3*b2 - b1 + 26) / 4

## 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
 0.218090 −2.66246 3.44437
0 −4.73435 0 18.0754 0 0.213413 0 −4.58596 0
1.2 0 −0.573746 0 −5.92862 0 −31.1522 0 −26.6708 0
1.3 0 10.3081 0 −14.1468 0 −4.06119 0 79.2568 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.x 3
4.b odd 2 1 1216.4.a.q 3
8.b even 2 1 152.4.a.b 3
8.d odd 2 1 304.4.a.j 3
24.h odd 2 1 1368.4.a.e 3

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

 $$T_{3}^{3} - 5T_{3}^{2} - 52T_{3} - 28$$ T3^3 - 5*T3^2 - 52*T3 - 28 $$T_{5}^{3} + 2T_{5}^{2} - 279T_{5} - 1516$$ T5^3 + 2*T5^2 - 279*T5 - 1516

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 5 T^{2} - 52 T - 28$$
$5$ $$T^{3} + 2 T^{2} - 279 T - 1516$$
$7$ $$T^{3} + 35 T^{2} + 119 T - 27$$
$11$ $$T^{3} - 28 T^{2} + 189 T - 358$$
$13$ $$T^{3} - 109 T^{2} + 408 T + 113456$$
$17$ $$T^{3} + 123 T^{2} + 2871 T + 6637$$
$19$ $$(T + 19)^{3}$$
$23$ $$T^{3} + 193 T^{2} + 3432 T - 246848$$
$29$ $$T^{3} - 297 T^{2} - 2036 T + 1167636$$
$31$ $$T^{3} + 140 T^{2} - 27184 T - 3668096$$
$37$ $$T^{3} + 38 T^{2} - 51860 T + 3429384$$
$41$ $$T^{3} - 736 T^{2} + 147788 T - 7266624$$
$43$ $$T^{3} - 514 T^{2} + \cdots + 25097948$$
$47$ $$T^{3} - 134 T^{2} + \cdots + 58888776$$
$53$ $$T^{3} + 311 T^{2} + \cdots + 13619792$$
$59$ $$T^{3} + 199 T^{2} + \cdots - 117609444$$
$61$ $$T^{3} + 56 T^{2} + \cdots - 120500582$$
$67$ $$T^{3} - 509 T^{2} + \cdots + 177822064$$
$71$ $$T^{3} + 874 T^{2} + \cdots - 112230216$$
$73$ $$T^{3} - 203 T^{2} - 62253 T + 474103$$
$79$ $$T^{3} - 242 T^{2} + \cdots + 201599456$$
$83$ $$T^{3} - 62 T^{2} - 456448 T - 83648992$$
$89$ $$T^{3} - 1764 T^{2} + \cdots - 127322496$$
$97$ $$T^{3} + 2178 T^{2} + \cdots + 99903104$$