# Properties

 Label 1216.4.a.bh Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 2x^{6} - 130x^{5} + 212x^{4} + 4589x^{3} - 4178x^{2} - 46788x + 7848$$ x^7 - 2*x^6 - 130*x^5 + 212*x^4 + 4589*x^3 - 4178*x^2 - 46788*x + 7848 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + ( - \beta_{5} + 1) q^{5} + ( - \beta_{3} - 4) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 + 12) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + (-b5 + 1) * q^5 + (-b3 - 4) * q^7 + (-b5 + b4 + b3 + 2*b1 + 12) * q^9 $$q + (\beta_1 + 1) q^{3} + ( - \beta_{5} + 1) q^{5} + ( - \beta_{3} - 4) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 + 12) q^{9} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 11) q^{11} + ( - \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 - 6) q^{13} + ( - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{15} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 3) q^{17} + 19 q^{19} + (3 \beta_{6} + 4 \beta_{5} + \beta_{4} - 6 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 14) q^{21} + (2 \beta_{6} - 11 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 25) q^{23} + (2 \beta_{6} + \beta_{5} + \beta_{4} - 4 \beta_{3} - \beta_{2} + 13 \beta_1 + 2) q^{25} + ( - 4 \beta_{6} - 7 \beta_{5} + \beta_{4} + 7 \beta_{3} - 6 \beta_{2} + 10 \beta_1 + 67) q^{27} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - \beta_1 + 11) q^{29} + ( - 4 \beta_{6} - 7 \beta_{5} + \beta_{4} + 5 \beta_{3} - 6 \beta_{2} + 13 \beta_1 - 14) q^{31} + (\beta_{5} - \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + 33 \beta_1 - 14) q^{33} + (3 \beta_{6} + \beta_{5} - 5 \beta_{4} - 8 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 44) q^{35} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 8 \beta_{2} + 12 \beta_1 - 27) q^{37} + (7 \beta_{6} - \beta_{5} + 4 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 11 \beta_1 - 46) q^{39} + ( - \beta_{6} + \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 4 \beta_{2} + 26 \beta_1 + 23) q^{41} + (9 \beta_{6} - 15 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 172) q^{43} + ( - 5 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} + 12 \beta_{3} + 6 \beta_{2} + \cdots + 30) q^{45}+ \cdots + (22 \beta_{6} - 9 \beta_{5} + 23 \beta_{4} + 10 \beta_{3} - 5 \beta_{2} + \cdots + 795) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 + (-b5 + 1) * q^5 + (-b3 - 4) * q^7 + (-b5 + b4 + b3 + 2*b1 + 12) * q^9 + (b5 + b4 + b2 - b1 + 11) * q^11 + (-b6 + b5 - b3 - b1 - 6) * q^13 + (-b6 - 3*b5 - b4 + b3 + 2*b2 + 1) * q^15 + (-b6 + 2*b5 - 2*b4 - b3 + 3*b2 + 6*b1 - 3) * q^17 + 19 * q^19 + (3*b6 + 4*b5 + b4 - 6*b3 + 4*b2 - 12*b1 - 14) * q^21 + (2*b6 - 11*b5 + 4*b4 + 3*b3 - 2*b2 + 2*b1 + 25) * q^23 + (2*b6 + b5 + b4 - 4*b3 - b2 + 13*b1 + 2) * q^25 + (-4*b6 - 7*b5 + b4 + 7*b3 - 6*b2 + 10*b1 + 67) * q^27 + (-2*b6 - 2*b5 + b4 + 4*b3 - 2*b2 - b1 + 11) * q^29 + (-4*b6 - 7*b5 + b4 + 5*b3 - 6*b2 + 13*b1 - 14) * q^31 + (b5 - b4 - 5*b3 - 6*b2 + 33*b1 - 14) * q^33 + (3*b6 + b5 - 5*b4 - 8*b3 + 5*b2 - 8*b1 + 44) * q^35 + (-b6 + b5 - b4 - b3 - 8*b2 + 12*b1 - 27) * q^37 + (7*b6 - b5 + 4*b4 - 5*b3 + 2*b2 - 11*b1 - 46) * q^39 + (-b6 + b5 + 5*b4 - 5*b3 - 4*b2 + 26*b1 + 23) * q^41 + (9*b6 - 15*b5 + 5*b4 - 2*b3 - 3*b2 - 8*b1 + 172) * q^43 + (-5*b6 - 7*b5 - 6*b4 + 12*b3 + 6*b2 + 5*b1 + 30) * q^45 + (-2*b6 + 9*b5 - 6*b4 - 4*b3 - 4*b1 - 131) * q^47 + (-b6 - 14*b5 - 2*b4 + 13*b3 + 5*b2 + 26*b1 + 82) * q^49 + (5*b6 - 24*b5 + 4*b4 + 2*b3 + 8*b2 - 14*b1 + 282) * q^51 + (9*b6 - 12*b5 + b4 - 2*b3 - 18*b2 + 40*b1 - 2) * q^53 + (-3*b6 + 3*b5 - 10*b4 + 4*b3 + 18*b2 - 11*b1 - 126) * q^55 + (19*b1 + 19) * q^57 + (-13*b6 + 7*b5 - 5*b4 + b3 - 4*b2 + 19*b1 + 232) * q^59 + (17*b6 + 13*b5 + 12*b4 - 16*b3 + 18*b2 - 7*b1 + 62) * q^61 + (9*b6 + 57*b5 - 18*b4 - 40*b3 + 12*b2 - 39*b1 - 382) * q^63 + (5*b6 + 10*b5 - 8*b4 - 4*b3 + 8*b2 - 7*b1 - 29) * q^65 + (-15*b6 - 12*b5 - 20*b4 + 4*b3 + 6*b2 + 8*b1 + 318) * q^67 + (-24*b6 - 11*b5 - 10*b4 + 19*b3 - 6*b2 + 86*b1 + 65) * q^69 + (-3*b6 - 32*b5 + 12*b4 + 16*b3 - 36*b2 + 31*b1 - 57) * q^71 + (4*b6 + 14*b5 - 8*b4 + b3 + 17*b2 - 51*b1 - 34) * q^73 + (8*b6 + 31*b5 + 15*b4 - 19*b3 + 10*b2 - 12*b1 + 419) * q^75 + (-7*b6 + 39*b5 - 20*b4 - 14*b3 + 10*b2 + 53*b1 - 102) * q^77 + (-13*b6 + 17*b5 + 5*b4 + 11*b3 + 6*b2 - 4*b1 - 87) * q^79 + (-10*b6 - 36*b5 - 6*b4 + 48*b3 - 18*b2 + 64*b1 + 119) * q^81 + (b6 + 32*b5 - 8*b4 - 14*b3 - 30*b2 + 15*b1 + 329) * q^83 + (13*b5 + 4*b4 - 20*b3 + 40*b2 + 66*b1 + 115) * q^85 + (-6*b6 - 27*b5 + 4*b4 + 31*b3 - 16*b2 + 50*b1 - 9) * q^87 + (-17*b6 + 21*b5 + 15*b4 + 5*b3 - 18*b2 - 34*b1 - 19) * q^89 + (-7*b6 + 13*b5 - 11*b4 + 21*b3 + 10*b2 + b1 + 586) * q^91 + (-4*b6 - 58*b5 + 26*b4 + 66*b3 - 10*b2 + 24*b1 + 456) * q^93 + (-19*b5 + 19) * q^95 + (-b6 - 6*b5 + 26*b4 + 42*b3 - 22*b2 - 17*b1 + 213) * q^97 + (22*b6 - 9*b5 + 23*b4 + 10*b3 - 5*b2 - 45*b1 + 795) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 9 q^{3} + 5 q^{5} - 28 q^{7} + 86 q^{9}+O(q^{10})$$ 7 * q + 9 * q^3 + 5 * q^5 - 28 * q^7 + 86 * q^9 $$7 q + 9 q^{3} + 5 q^{5} - 28 q^{7} + 86 q^{9} + 77 q^{11} - 43 q^{13} - 6 q^{17} + 133 q^{19} - 111 q^{21} + 159 q^{23} + 44 q^{25} + 471 q^{27} + 69 q^{29} - 90 q^{31} - 30 q^{33} + 297 q^{35} - 164 q^{37} - 339 q^{39} + 214 q^{41} + 1167 q^{43} + 201 q^{45} - 909 q^{47} + 597 q^{49} + 1903 q^{51} + 51 q^{53} - 901 q^{55} + 171 q^{57} + 1663 q^{59} + 463 q^{61} - 2629 q^{63} - 192 q^{65} + 2203 q^{67} + 581 q^{69} - 404 q^{71} - 308 q^{73} + 2979 q^{75} - 537 q^{77} - 596 q^{79} + 879 q^{81} + 2398 q^{83} + 963 q^{85} - 23 q^{87} - 176 q^{89} + 4123 q^{91} + 3120 q^{93} + 95 q^{95} + 1444 q^{97} + 5479 q^{99}+O(q^{100})$$ 7 * q + 9 * q^3 + 5 * q^5 - 28 * q^7 + 86 * q^9 + 77 * q^11 - 43 * q^13 - 6 * q^17 + 133 * q^19 - 111 * q^21 + 159 * q^23 + 44 * q^25 + 471 * q^27 + 69 * q^29 - 90 * q^31 - 30 * q^33 + 297 * q^35 - 164 * q^37 - 339 * q^39 + 214 * q^41 + 1167 * q^43 + 201 * q^45 - 909 * q^47 + 597 * q^49 + 1903 * q^51 + 51 * q^53 - 901 * q^55 + 171 * q^57 + 1663 * q^59 + 463 * q^61 - 2629 * q^63 - 192 * q^65 + 2203 * q^67 + 581 * q^69 - 404 * q^71 - 308 * q^73 + 2979 * q^75 - 537 * q^77 - 596 * q^79 + 879 * q^81 + 2398 * q^83 + 963 * q^85 - 23 * q^87 - 176 * q^89 + 4123 * q^91 + 3120 * q^93 + 95 * q^95 + 1444 * q^97 + 5479 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 2x^{6} - 130x^{5} + 212x^{4} + 4589x^{3} - 4178x^{2} - 46788x + 7848$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 92\nu^{5} - 146\nu^{4} - 10320\nu^{3} + 8069\nu^{2} + 233100\nu - 70812 ) / 10944$$ (v^6 + 92*v^5 - 146*v^4 - 10320*v^3 + 8069*v^2 + 233100*v - 70812) / 10944 $$\beta_{3}$$ $$=$$ $$( -11\nu^{6} + 14\nu^{5} + 1492\nu^{4} - 1050\nu^{3} - 48517\nu^{2} + 3180\nu + 219420 ) / 10944$$ (-11*v^6 + 14*v^5 + 1492*v^4 - 1050*v^3 - 48517*v^2 + 3180*v + 219420) / 10944 $$\beta_{4}$$ $$=$$ $$( -5\nu^{6} - 4\nu^{5} + 274\nu^{4} + 72\nu^{3} + 18479\nu^{2} + 3684\nu - 515988 ) / 10944$$ (-5*v^6 - 4*v^5 + 274*v^4 + 72*v^3 + 18479*v^2 + 3684*v - 515988) / 10944 $$\beta_{5}$$ $$=$$ $$( -8\nu^{6} + 5\nu^{5} + 883\nu^{4} - 489\nu^{3} - 20491\nu^{2} + 3432\nu + 59652 ) / 5472$$ (-8*v^6 + 5*v^5 + 883*v^4 - 489*v^3 - 20491*v^2 + 3432*v + 59652) / 5472 $$\beta_{6}$$ $$=$$ $$( \nu^{6} - 22\nu^{5} - 32\nu^{4} + 2106\nu^{3} - 4813\nu^{2} - 31380\nu + 80124 ) / 1824$$ (v^6 - 22*v^5 - 32*v^4 + 2106*v^3 - 4813*v^2 - 31380*v + 80124) / 1824
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} + 38$$ -b5 + b4 + b3 + 38 $$\nu^{3}$$ $$=$$ $$-4\beta_{6} - 4\beta_{5} - 2\beta_{4} + 4\beta_{3} - 6\beta_{2} + 61\beta _1 + 6$$ -4*b6 - 4*b5 - 2*b4 + 4*b3 - 6*b2 + 61*b1 + 6 $$\nu^{4}$$ $$=$$ $$6\beta_{6} - 95\beta_{5} + 77\beta_{4} + 107\beta_{3} + 6\beta_{2} - 22\beta _1 + 2296$$ 6*b6 - 95*b5 + 77*b4 + 107*b3 + 6*b2 - 22*b1 + 2296 $$\nu^{5}$$ $$=$$ $$-446\beta_{6} - 418\beta_{5} - 254\beta_{4} + 430\beta_{3} - 552\beta_{2} + 4307\beta _1 - 20$$ -446*b6 - 418*b5 - 254*b4 + 430*b3 - 552*b2 + 4307*b1 - 20 $$\nu^{6}$$ $$=$$ $$628\beta_{6} - 8625\beta_{5} + 5901\beta_{4} + 9273\beta_{3} + 684\beta_{2} - 3036\beta _1 + 163166$$ 628*b6 - 8625*b5 + 5901*b4 + 9273*b3 + 684*b2 - 3036*b1 + 163166

## 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
 −8.94968 −5.20371 −3.90450 0.165732 4.53243 6.44524 8.91449
0 −7.94968 0 0.0633525 0 −19.8268 0 36.1974 0
1.2 0 −4.20371 0 −3.63602 0 8.86224 0 −9.32886 0
1.3 0 −2.90450 0 12.8220 0 11.9961 0 −18.5639 0
1.4 0 1.16573 0 −9.90212 0 −23.9754 0 −25.6411 0
1.5 0 5.53243 0 15.3317 0 23.3717 0 3.60773 0
1.6 0 7.44524 0 −18.2823 0 6.49029 0 28.4317 0
1.7 0 9.91449 0 8.60335 0 −34.9181 0 71.2970 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.bh 7
4.b odd 2 1 1216.4.a.be 7
8.b even 2 1 608.4.a.i 7
8.d odd 2 1 608.4.a.l yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.i 7 8.b even 2 1
608.4.a.l yes 7 8.d odd 2 1
1216.4.a.be 7 4.b odd 2 1
1216.4.a.bh 7 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{7} - 9T_{3}^{6} - 97T_{3}^{5} + 797T_{3}^{4} + 2516T_{3}^{3} - 15424T_{3}^{2} - 26144T_{3} + 46208$$ T3^7 - 9*T3^6 - 97*T3^5 + 797*T3^4 + 2516*T3^3 - 15424*T3^2 - 26144*T3 + 46208 $$T_{5}^{7} - 5T_{5}^{6} - 447T_{5}^{5} + 2537T_{5}^{4} + 46234T_{5}^{3} - 193600T_{5}^{2} - 1101184T_{5} + 70528$$ T5^7 - 5*T5^6 - 447*T5^5 + 2537*T5^4 + 46234*T5^3 - 193600*T5^2 - 1101184*T5 + 70528

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$T^{7} - 9 T^{6} - 97 T^{5} + \cdots + 46208$$
$5$ $$T^{7} - 5 T^{6} - 447 T^{5} + \cdots + 70528$$
$7$ $$T^{7} + 28 T^{6} + \cdots + 267674508$$
$11$ $$T^{7} - 77 T^{6} + \cdots - 8913577312$$
$13$ $$T^{7} + 43 T^{6} + \cdots - 3972040064$$
$17$ $$T^{7} + 6 T^{6} + \cdots + 1272145656338$$
$19$ $$(T - 19)^{7}$$
$23$ $$T^{7} - 159 T^{6} + \cdots + 43343115459584$$
$29$ $$T^{7} - 69 T^{6} + \cdots - 53253690048$$
$31$ $$T^{7} + \cdots - 312246972080128$$
$37$ $$T^{7} + 164 T^{6} + \cdots - 1907389311488$$
$41$ $$T^{7} + \cdots - 460009274261504$$
$43$ $$T^{7} + \cdots - 372474759986432$$
$47$ $$T^{7} + \cdots + 829054396295168$$
$53$ $$T^{7} - 51 T^{6} + \cdots + 27\!\cdots\!48$$
$59$ $$T^{7} - 1663 T^{6} + \cdots + 56\!\cdots\!76$$
$61$ $$T^{7} - 463 T^{6} + \cdots - 24\!\cdots\!64$$
$67$ $$T^{7} - 2203 T^{6} + \cdots - 42\!\cdots\!76$$
$71$ $$T^{7} + 404 T^{6} + \cdots - 68\!\cdots\!76$$
$73$ $$T^{7} + 308 T^{6} + \cdots - 10\!\cdots\!54$$
$79$ $$T^{7} + 596 T^{6} + \cdots + 34\!\cdots\!96$$
$83$ $$T^{7} - 2398 T^{6} + \cdots + 16\!\cdots\!92$$
$89$ $$T^{7} + 176 T^{6} + \cdots - 25\!\cdots\!36$$
$97$ $$T^{7} - 1444 T^{6} + \cdots + 26\!\cdots\!56$$