# Properties

 Label 1216.4.a.bc Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# 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: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 20x^{3} + 24x^{2} + 2x - 4$$ x^5 - x^4 - 20*x^3 + 24*x^2 + 2*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{3} + (\beta_1 - 5) q^{5} + ( - 3 \beta_{4} - \beta_{2} - \beta_1 + 5) q^{7} + (\beta_{4} + 3 \beta_{2} - 2 \beta_1 + 29) q^{9}+O(q^{10})$$ q + (b3 + 1) * q^3 + (b1 - 5) * q^5 + (-3*b4 - b2 - b1 + 5) * q^7 + (b4 + 3*b2 - 2*b1 + 29) * q^9 $$q + (\beta_{3} + 1) q^{3} + (\beta_1 - 5) q^{5} + ( - 3 \beta_{4} - \beta_{2} - \beta_1 + 5) q^{7} + (\beta_{4} + 3 \beta_{2} - 2 \beta_1 + 29) q^{9} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 12) q^{11} + (5 \beta_{4} - \beta_{2} - 40) q^{13} + (\beta_{4} - 9 \beta_{3} - 7 \beta_{2} + 2 \beta_1 - 9) q^{15} + (5 \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_1 - 14) q^{17} + 19 q^{19} + ( - 8 \beta_{4} - \beta_{3} + 16 \beta_{2} + 4 \beta_1 + 7) q^{21} + ( - 15 \beta_{4} + 13 \beta_{2} - 6 \beta_1 + 8) q^{23} + ( - 10 \beta_{4} + \beta_{3} - \beta_{2} - 13 \beta_1 - 1) q^{25} + ( - 13 \beta_{4} + 24 \beta_{3} + 3 \beta_{2} - 22 \beta_1) q^{27} + (6 \beta_{4} - 13 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{29} + (17 \beta_{4} - 9 \beta_{3} - 11 \beta_{2} - 147) q^{31} + (27 \beta_{4} - 33 \beta_{3} - 13 \beta_{2} + 22 \beta_1 + 47) q^{33} + ( - 2 \beta_{4} + 15 \beta_{3} + 9 \beta_{2} + 21 \beta_1 - 64) q^{35} + (19 \beta_{4} - 21 \beta_{3} - 7 \beta_{2} - 12 \beta_1 - 41) q^{37} + (25 \beta_{4} - 34 \beta_{3} - 7 \beta_{2} + 6 \beta_1 - 26) q^{39} + ( - \beta_{4} - 15 \beta_{3} + 9 \beta_{2} + 24 \beta_1 + 13) q^{41} + (12 \beta_{4} + \beta_{3} - 9 \beta_{2} - 17 \beta_1 + 76) q^{43} + (32 \beta_{4} - 34 \beta_{3} - 22 \beta_{2} + 37 \beta_1 - 347) q^{45} + (32 \beta_{4} - 10 \beta_{3} + 28 \beta_{2} - \beta_1 - 7) q^{47} + ( - 9 \beta_{4} - 33 \beta_{3} - 20 \beta_{2} + 5 \beta_1 + 203) q^{49} + ( - 2 \beta_{4} + 17 \beta_{3} - 18 \beta_{2} - 24 \beta_1 - 71) q^{51} + ( - 20 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 197) q^{53} + ( - 32 \beta_{4} + 20 \beta_{3} + 14 \beta_{2} - 33 \beta_1 + 291) q^{55} + (19 \beta_{3} + 19) q^{57} + (61 \beta_{4} - 54 \beta_{3} - 45 \beta_{2} + 8 \beta_1 + 96) q^{59} + ( - 28 \beta_{3} - 30 \beta_{2} - 23 \beta_1 - 297) q^{61} + ( - 28 \beta_{4} + 40 \beta_{3} - 36 \beta_{2} - 59 \beta_1 - 279) q^{63} + (12 \beta_{4} + 8 \beta_{2} - 48 \beta_1 + 84) q^{65} + ( - 28 \beta_{4} - 29 \beta_{3} + 42 \beta_{2} + 22 \beta_1 - 183) q^{67} + ( - 131 \beta_{4} + 54 \beta_{3} + 33 \beta_{2} - 90 \beta_1 - 50) q^{69} + ( - 16 \beta_{4} + 22 \beta_{3} + 32 \beta_{2} + 46 \beta_1 + 316) q^{71} + ( - 17 \beta_{4} - 67 \beta_{3} - 20 \beta_{2} + 19 \beta_1 + 528) q^{73} + ( - 47 \beta_{4} + 26 \beta_{3} + 117 \beta_{2} - 22 \beta_1 + 90) q^{75} + ( - 42 \beta_{4} - 32 \beta_{3} + 52 \beta_{2} + 33 \beta_1 - 27) q^{77} + ( - 77 \beta_{4} + 43 \beta_{3} - 13 \beta_{2} + 24 \beta_1 - 107) q^{79} + ( - 92 \beta_{4} + 50 \beta_{3} + 162 \beta_{2} - 56 \beta_1 + 587) q^{81} + ( - 90 \beta_{4} - 42 \beta_{3} + 16 \beta_{2} - 16 \beta_1 + 110) q^{83} + (36 \beta_{4} - 42 \beta_{3} - 24 \beta_{2} - 25 \beta_1 - 111) q^{85} + (29 \beta_{4} + 16 \beta_{3} - 25 \beta_{2} + 46 \beta_1 - 680) q^{87} + ( - 63 \beta_{4} - 79 \beta_{3} - 59 \beta_{2} + 6 \beta_1 + 207) q^{89} + (117 \beta_{4} - 34 \beta_{3} + 61 \beta_{2} - 14 \beta_1 - 662) q^{91} + (114 \beta_{4} - 148 \beta_{3} - 28 \beta_{2} + 84 \beta_1 - 564) q^{93} + (19 \beta_1 - 95) q^{95} + (34 \beta_{4} - 16 \beta_{3} - 96 \beta_{2} + 50 \beta_1 + 104) q^{97} + (108 \beta_{4} - 33 \beta_{3} - 187 \beta_{2} + 107 \beta_1 - 1426) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^3 + (b1 - 5) * q^5 + (-3*b4 - b2 - b1 + 5) * q^7 + (b4 + 3*b2 - 2*b1 + 29) * q^9 + (2*b4 + b3 - 3*b2 + 3*b1 - 12) * q^11 + (5*b4 - b2 - 40) * q^13 + (b4 - 9*b3 - 7*b2 + 2*b1 - 9) * q^15 + (5*b4 - b3 + 4*b2 - b1 - 14) * q^17 + 19 * q^19 + (-8*b4 - b3 + 16*b2 + 4*b1 + 7) * q^21 + (-15*b4 + 13*b2 - 6*b1 + 8) * q^23 + (-10*b4 + b3 - b2 - 13*b1 - 1) * q^25 + (-13*b4 + 24*b3 + 3*b2 - 22*b1) * q^27 + (6*b4 - 13*b3 - 4*b2 - 2*b1 - 1) * q^29 + (17*b4 - 9*b3 - 11*b2 - 147) * q^31 + (27*b4 - 33*b3 - 13*b2 + 22*b1 + 47) * q^33 + (-2*b4 + 15*b3 + 9*b2 + 21*b1 - 64) * q^35 + (19*b4 - 21*b3 - 7*b2 - 12*b1 - 41) * q^37 + (25*b4 - 34*b3 - 7*b2 + 6*b1 - 26) * q^39 + (-b4 - 15*b3 + 9*b2 + 24*b1 + 13) * q^41 + (12*b4 + b3 - 9*b2 - 17*b1 + 76) * q^43 + (32*b4 - 34*b3 - 22*b2 + 37*b1 - 347) * q^45 + (32*b4 - 10*b3 + 28*b2 - b1 - 7) * q^47 + (-9*b4 - 33*b3 - 20*b2 + 5*b1 + 203) * q^49 + (-2*b4 + 17*b3 - 18*b2 - 24*b1 - 71) * q^51 + (-20*b4 - 3*b3 + 2*b2 + 4*b1 - 197) * q^53 + (-32*b4 + 20*b3 + 14*b2 - 33*b1 + 291) * q^55 + (19*b3 + 19) * q^57 + (61*b4 - 54*b3 - 45*b2 + 8*b1 + 96) * q^59 + (-28*b3 - 30*b2 - 23*b1 - 297) * q^61 + (-28*b4 + 40*b3 - 36*b2 - 59*b1 - 279) * q^63 + (12*b4 + 8*b2 - 48*b1 + 84) * q^65 + (-28*b4 - 29*b3 + 42*b2 + 22*b1 - 183) * q^67 + (-131*b4 + 54*b3 + 33*b2 - 90*b1 - 50) * q^69 + (-16*b4 + 22*b3 + 32*b2 + 46*b1 + 316) * q^71 + (-17*b4 - 67*b3 - 20*b2 + 19*b1 + 528) * q^73 + (-47*b4 + 26*b3 + 117*b2 - 22*b1 + 90) * q^75 + (-42*b4 - 32*b3 + 52*b2 + 33*b1 - 27) * q^77 + (-77*b4 + 43*b3 - 13*b2 + 24*b1 - 107) * q^79 + (-92*b4 + 50*b3 + 162*b2 - 56*b1 + 587) * q^81 + (-90*b4 - 42*b3 + 16*b2 - 16*b1 + 110) * q^83 + (36*b4 - 42*b3 - 24*b2 - 25*b1 - 111) * q^85 + (29*b4 + 16*b3 - 25*b2 + 46*b1 - 680) * q^87 + (-63*b4 - 79*b3 - 59*b2 + 6*b1 + 207) * q^89 + (117*b4 - 34*b3 + 61*b2 - 14*b1 - 662) * q^91 + (114*b4 - 148*b3 - 28*b2 + 84*b1 - 564) * q^93 + (19*b1 - 95) * q^95 + (34*b4 - 16*b3 - 96*b2 + 50*b1 + 104) * q^97 + (108*b4 - 33*b3 - 187*b2 + 107*b1 - 1426) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9}+O(q^{10})$$ 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9 $$5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9} - 67 q^{11} - 191 q^{13} - 36 q^{15} - 52 q^{17} + 95 q^{19} + 29 q^{21} + 35 q^{23} - 2 q^{25} - 27 q^{27} + 33 q^{29} - 694 q^{31} + 298 q^{33} - 387 q^{35} - 108 q^{37} - 31 q^{39} + 54 q^{41} + 427 q^{43} - 1699 q^{45} + 79 q^{47} + 1033 q^{49} - 363 q^{51} - 1025 q^{53} + 1431 q^{55} + 57 q^{57} + 649 q^{59} - 1413 q^{61} - 1449 q^{63} + 548 q^{65} - 915 q^{67} - 407 q^{69} + 1444 q^{71} + 2682 q^{73} + 465 q^{75} - 169 q^{77} - 836 q^{79} + 2925 q^{81} + 502 q^{83} - 373 q^{85} - 3491 q^{87} + 996 q^{89} - 2919 q^{91} - 2492 q^{93} - 513 q^{95} + 424 q^{97} - 7249 q^{99}+O(q^{100})$$ 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9 - 67 * q^11 - 191 * q^13 - 36 * q^15 - 52 * q^17 + 95 * q^19 + 29 * q^21 + 35 * q^23 - 2 * q^25 - 27 * q^27 + 33 * q^29 - 694 * q^31 + 298 * q^33 - 387 * q^35 - 108 * q^37 - 31 * q^39 + 54 * q^41 + 427 * q^43 - 1699 * q^45 + 79 * q^47 + 1033 * q^49 - 363 * q^51 - 1025 * q^53 + 1431 * q^55 + 57 * q^57 + 649 * q^59 - 1413 * q^61 - 1449 * q^63 + 548 * q^65 - 915 * q^67 - 407 * q^69 + 1444 * q^71 + 2682 * q^73 + 465 * q^75 - 169 * q^77 - 836 * q^79 + 2925 * q^81 + 502 * q^83 - 373 * q^85 - 3491 * q^87 + 996 * q^89 - 2919 * q^91 - 2492 * q^93 - 513 * q^95 + 424 * q^97 - 7249 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 20x^{3} + 24x^{2} + 2x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + \nu^{3} + 18\nu^{2} - 24\nu + 14 ) / 2$$ (-v^4 + v^3 + 18*v^2 - 24*v + 14) / 2 $$\beta_{2}$$ $$=$$ $$-\nu^{4} + 20\nu^{2} - 2\nu - 8$$ -v^4 + 20*v^2 - 2*v - 8 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 20\nu^{2} + 6\nu + 7$$ v^4 - 20*v^2 + 6*v + 7 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 20\nu^{2} + 24\nu + 2$$ v^4 - v^3 - 20*v^2 + 24*v + 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 1 ) / 4$$ (b3 + b2 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} - 2\beta _1 + 16 ) / 2$$ (-b4 - 2*b1 + 16) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{4} + 11\beta_{3} + 9\beta_{2} - 1 ) / 2$$ (-2*b4 + 11*b3 + 9*b2 - 1) / 2 $$\nu^{4}$$ $$=$$ $$( -20\beta_{4} - \beta_{3} - 3\beta_{2} - 40\beta _1 + 303 ) / 2$$ (-20*b4 - b3 - 3*b2 - 40*b1 + 303) / 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
 1.14402 −4.53955 −0.389032 0.457551 4.32701
0 −9.59869 0 −0.0570048 0 −24.6071 0 65.1348 0
1.2 0 −6.71806 0 −17.1669 0 31.2505 0 18.1323 0
1.3 0 2.66179 0 7.98961 0 27.0741 0 −19.9149 0
1.4 0 6.60208 0 −1.58045 0 −19.8743 0 16.5875 0
1.5 0 10.0529 0 −16.1852 0 6.15678 0 74.0603 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.bc 5
4.b odd 2 1 1216.4.a.z 5
8.b even 2 1 608.4.a.f 5
8.d odd 2 1 608.4.a.g yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.f 5 8.b even 2 1
608.4.a.g yes 5 8.d odd 2 1
1216.4.a.z 5 4.b odd 2 1
1216.4.a.bc 5 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{5} - 3T_{3}^{4} - 140T_{3}^{3} + 384T_{3}^{2} + 4256T_{3} - 11392$$ T3^5 - 3*T3^4 - 140*T3^3 + 384*T3^2 + 4256*T3 - 11392 $$T_{5}^{5} + 27T_{5}^{4} + 53T_{5}^{3} - 2199T_{5}^{2} - 3634T_{5} - 200$$ T5^5 + 27*T5^4 + 53*T5^3 - 2199*T5^2 - 3634*T5 - 200

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 3 T^{4} - 140 T^{3} + \cdots - 11392$$
$5$ $$T^{5} + 27 T^{4} + 53 T^{3} + \cdots - 200$$
$7$ $$T^{5} - 20 T^{4} - 1174 T^{3} + \cdots - 2547508$$
$11$ $$T^{5} + 67 T^{4} - 1415 T^{3} + \cdots + 42711800$$
$13$ $$T^{5} + 191 T^{4} + \cdots - 38142080$$
$17$ $$T^{5} + 52 T^{4} - 6732 T^{3} + \cdots - 9857070$$
$19$ $$(T - 19)^{5}$$
$23$ $$T^{5} - 35 T^{4} + \cdots - 5659603968$$
$29$ $$T^{5} - 33 T^{4} + \cdots + 1528050752$$
$31$ $$T^{5} + 694 T^{4} + \cdots + 1546708480$$
$37$ $$T^{5} + 108 T^{4} + \cdots + 264959360$$
$41$ $$T^{5} - 54 T^{4} + \cdots + 397858568960$$
$43$ $$T^{5} - 427 T^{4} + \cdots - 331567864112$$
$47$ $$T^{5} - 79 T^{4} + \cdots - 905506740224$$
$53$ $$T^{5} + 1025 T^{4} + \cdots - 210846255280$$
$59$ $$T^{5} - 649 T^{4} + \cdots - 34775624292320$$
$61$ $$T^{5} + 1413 T^{4} + \cdots - 29987676575940$$
$67$ $$T^{5} + 915 T^{4} + \cdots + 24348069855968$$
$71$ $$T^{5} - 1444 T^{4} + \cdots + 35176103924480$$
$73$ $$T^{5} - 2682 T^{4} + \cdots + 71728412586510$$
$79$ $$T^{5} + 836 T^{4} + \cdots - 31064499767680$$
$83$ $$T^{5} - 502 T^{4} + \cdots - 8567040270336$$
$89$ $$T^{5} - 996 T^{4} + \cdots - 13823629765888$$
$97$ $$T^{5} + \cdots - 461524239069440$$