# Properties

 Label 1216.4.a.i Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{93})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 23$$ x^2 - x - 23 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{93}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + (2 \beta + 2) q^{5} + ( - \beta - 22) q^{7} - 26 q^{9}+O(q^{10})$$ q - q^3 + (2*b + 2) * q^5 + (-b - 22) * q^7 - 26 * q^9 $$q - q^{3} + (2 \beta + 2) q^{5} + ( - \beta - 22) q^{7} - 26 q^{9} + ( - 2 \beta + 20) q^{11} + (3 \beta + 10) q^{13} + ( - 2 \beta - 2) q^{15} + ( - 8 \beta + 23) q^{17} + 19 q^{19} + (\beta + 22) q^{21} + ( - 5 \beta + 110) q^{23} + (8 \beta + 251) q^{25} + 53 q^{27} + (19 \beta + 42) q^{29} + ( - 18 \beta + 42) q^{31} + (2 \beta - 20) q^{33} + ( - 46 \beta - 230) q^{35} + (6 \beta - 152) q^{37} + ( - 3 \beta - 10) q^{39} + ( - 20 \beta + 84) q^{41} + ( - 2 \beta - 186) q^{43} + ( - 52 \beta - 52) q^{45} + ( - 24 \beta - 292) q^{47} + (44 \beta + 234) q^{49} + (8 \beta - 23) q^{51} + ( - 33 \beta - 242) q^{53} + (36 \beta - 332) q^{55} - 19 q^{57} + (4 \beta - 537) q^{59} + ( - 14 \beta - 44) q^{61} + (26 \beta + 572) q^{63} + (26 \beta + 578) q^{65} + (28 \beta + 715) q^{67} + (5 \beta - 110) q^{69} + (2 \beta - 924) q^{71} + ( - 8 \beta - 647) q^{73} + ( - 8 \beta - 251) q^{75} + (24 \beta - 254) q^{77} + ( - 78 \beta - 416) q^{79} + 649 q^{81} + (90 \beta + 264) q^{83} + (30 \beta - 1442) q^{85} + ( - 19 \beta - 42) q^{87} + ( - 74 \beta + 922) q^{89} + ( - 76 \beta - 499) q^{91} + (18 \beta - 42) q^{93} + (38 \beta + 38) q^{95} + ( - 52 \beta - 182) q^{97} + (52 \beta - 520) q^{99}+O(q^{100})$$ q - q^3 + (2*b + 2) * q^5 + (-b - 22) * q^7 - 26 * q^9 + (-2*b + 20) * q^11 + (3*b + 10) * q^13 + (-2*b - 2) * q^15 + (-8*b + 23) * q^17 + 19 * q^19 + (b + 22) * q^21 + (-5*b + 110) * q^23 + (8*b + 251) * q^25 + 53 * q^27 + (19*b + 42) * q^29 + (-18*b + 42) * q^31 + (2*b - 20) * q^33 + (-46*b - 230) * q^35 + (6*b - 152) * q^37 + (-3*b - 10) * q^39 + (-20*b + 84) * q^41 + (-2*b - 186) * q^43 + (-52*b - 52) * q^45 + (-24*b - 292) * q^47 + (44*b + 234) * q^49 + (8*b - 23) * q^51 + (-33*b - 242) * q^53 + (36*b - 332) * q^55 - 19 * q^57 + (4*b - 537) * q^59 + (-14*b - 44) * q^61 + (26*b + 572) * q^63 + (26*b + 578) * q^65 + (28*b + 715) * q^67 + (5*b - 110) * q^69 + (2*b - 924) * q^71 + (-8*b - 647) * q^73 + (-8*b - 251) * q^75 + (24*b - 254) * q^77 + (-78*b - 416) * q^79 + 649 * q^81 + (90*b + 264) * q^83 + (30*b - 1442) * q^85 + (-19*b - 42) * q^87 + (-74*b + 922) * q^89 + (-76*b - 499) * q^91 + (18*b - 42) * q^93 + (38*b + 38) * q^95 + (-52*b - 182) * q^97 + (52*b - 520) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} - 44 q^{7} - 52 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^5 - 44 * q^7 - 52 * q^9 $$2 q - 2 q^{3} + 4 q^{5} - 44 q^{7} - 52 q^{9} + 40 q^{11} + 20 q^{13} - 4 q^{15} + 46 q^{17} + 38 q^{19} + 44 q^{21} + 220 q^{23} + 502 q^{25} + 106 q^{27} + 84 q^{29} + 84 q^{31} - 40 q^{33} - 460 q^{35} - 304 q^{37} - 20 q^{39} + 168 q^{41} - 372 q^{43} - 104 q^{45} - 584 q^{47} + 468 q^{49} - 46 q^{51} - 484 q^{53} - 664 q^{55} - 38 q^{57} - 1074 q^{59} - 88 q^{61} + 1144 q^{63} + 1156 q^{65} + 1430 q^{67} - 220 q^{69} - 1848 q^{71} - 1294 q^{73} - 502 q^{75} - 508 q^{77} - 832 q^{79} + 1298 q^{81} + 528 q^{83} - 2884 q^{85} - 84 q^{87} + 1844 q^{89} - 998 q^{91} - 84 q^{93} + 76 q^{95} - 364 q^{97} - 1040 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^5 - 44 * q^7 - 52 * q^9 + 40 * q^11 + 20 * q^13 - 4 * q^15 + 46 * q^17 + 38 * q^19 + 44 * q^21 + 220 * q^23 + 502 * q^25 + 106 * q^27 + 84 * q^29 + 84 * q^31 - 40 * q^33 - 460 * q^35 - 304 * q^37 - 20 * q^39 + 168 * q^41 - 372 * q^43 - 104 * q^45 - 584 * q^47 + 468 * q^49 - 46 * q^51 - 484 * q^53 - 664 * q^55 - 38 * q^57 - 1074 * q^59 - 88 * q^61 + 1144 * q^63 + 1156 * q^65 + 1430 * q^67 - 220 * q^69 - 1848 * q^71 - 1294 * q^73 - 502 * q^75 - 508 * q^77 - 832 * q^79 + 1298 * q^81 + 528 * q^83 - 2884 * q^85 - 84 * q^87 + 1844 * q^89 - 998 * q^91 - 84 * q^93 + 76 * q^95 - 364 * q^97 - 1040 * 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
 −4.32183 5.32183
0 −1.00000 0 −17.2873 0 −12.3563 0 −26.0000 0
1.2 0 −1.00000 0 21.2873 0 −31.6437 0 −26.0000 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.i 2
4.b odd 2 1 1216.4.a.n 2
8.b even 2 1 608.4.a.d yes 2
8.d odd 2 1 608.4.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.c 2 8.d odd 2 1
608.4.a.d yes 2 8.b even 2 1
1216.4.a.i 2 1.a even 1 1 trivial
1216.4.a.n 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{5}^{2} - 4T_{5} - 368$$ T5^2 - 4*T5 - 368

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 4T - 368$$
$7$ $$T^{2} + 44T + 391$$
$11$ $$T^{2} - 40T + 28$$
$13$ $$T^{2} - 20T - 737$$
$17$ $$T^{2} - 46T - 5423$$
$19$ $$(T - 19)^{2}$$
$23$ $$T^{2} - 220T + 9775$$
$29$ $$T^{2} - 84T - 31809$$
$31$ $$T^{2} - 84T - 28368$$
$37$ $$T^{2} + 304T + 19756$$
$41$ $$T^{2} - 168T - 30144$$
$43$ $$T^{2} + 372T + 34224$$
$47$ $$T^{2} + 584T + 31696$$
$53$ $$T^{2} + 484T - 42713$$
$59$ $$T^{2} + 1074 T + 286881$$
$61$ $$T^{2} + 88T - 16292$$
$67$ $$T^{2} - 1430 T + 438313$$
$71$ $$T^{2} + 1848 T + 853404$$
$73$ $$T^{2} + 1294 T + 412657$$
$79$ $$T^{2} + 832T - 392756$$
$83$ $$T^{2} - 528T - 683604$$
$89$ $$T^{2} - 1844 T + 340816$$
$97$ $$T^{2} + 364T - 218348$$