# Properties

 Label 1216.4.a.ba Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ 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: $$0$$ 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: $$x^{5} - 106x^{3} - 401x^{2} + 356x + 2112$$ x^5 - 106*x^3 - 401*x^2 + 356*x + 2112 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + 4) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_1 + 16) q^{9}+O(q^{10})$$ q - b2 * q^3 + (-b2 - b1) * q^5 + (-b3 + b2 + 4) * q^7 + (-b4 + b3 - b1 + 16) * q^9 $$q - \beta_{2} q^{3} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + 4) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_1 + 16) q^{9} + ( - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \beta_1) q^{11} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 18) q^{13} + ( - 5 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 39) q^{15} + (\beta_{4} + \beta_{3} - 7 \beta_{2} - 2 \beta_1 + 49) q^{17} - 19 q^{19} + (\beta_{4} + 4 \beta_{3} + 7 \beta_{2} - 2 \beta_1 - 65) q^{21} + (2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 7 \beta_1 - 8) q^{23} + ( - 4 \beta_{4} - 6 \beta_{3} - 11 \beta_{2} - \beta_1 + 113) q^{25} + ( - 5 \beta_{4} - 7 \beta_{3} - 10 \beta_{2} - 5 \beta_1 + 39) q^{27} + ( - 3 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 43) q^{29} + (11 \beta_{4} - \beta_{3} - 11 \beta_{2} + 3 \beta_1 - 37) q^{31} + ( - \beta_{4} + 15 \beta_{3} + 5 \beta_{2} - 19 \beta_1 + 131) q^{33} + (18 \beta_{4} + 6 \beta_{3} - 7 \beta_{2} - 5 \beta_1 - 4) q^{35} + (11 \beta_{4} - \beta_{3} - 11 \beta_{2} + 3 \beta_1 - 115) q^{37} + ( - 4 \beta_{4} - 9 \beta_{3} - 12 \beta_{2} - 9 \beta_1 - 26) q^{39} + (9 \beta_{4} - 9 \beta_{3} + 11 \beta_{2} + 7 \beta_1 + 39) q^{41} + (4 \beta_{4} + 4 \beta_{3} + 9 \beta_{2} + 23 \beta_1 + 14) q^{43} + ( - 18 \beta_{4} - 49 \beta_{2} - 13 \beta_1 + 114) q^{45} + (2 \beta_{4} + 4 \beta_{3} - 15 \beta_{2} + 7 \beta_1 - 204) q^{47} + (3 \beta_{4} - 17 \beta_{3} + 13 \beta_{2} + 2 \beta_1 + 94) q^{49} + ( - 14 \beta_{4} - 2 \beta_{3} - 53 \beta_{2} - 2 \beta_1 + 294) q^{51} + (11 \beta_{4} - 8 \beta_{3} - 9 \beta_{2} + 2 \beta_1 + 15) q^{53} + ( - 2 \beta_{4} + 32 \beta_{3} - 55 \beta_{2} - 25 \beta_1 - 112) q^{55} + 19 \beta_{2} q^{57} + ( - 3 \beta_{4} + 15 \beta_{3} + 32 \beta_{2} - 9 \beta_1 + 201) q^{59} + (16 \beta_{4} + 4 \beta_{3} - 15 \beta_{2} - 21 \beta_1 + 56) q^{61} + ( - 4 \beta_{3} + \beta_{2} + 21 \beta_1 - 350) q^{63} + ( - 14 \beta_{4} + 12 \beta_{3} - 40 \beta_{2} + 24 \beta_1 - 14) q^{65} + (12 \beta_{4} - 28 \beta_{3} - 7 \beta_{2} + 10 \beta_1 + 72) q^{67} + (28 \beta_{4} + 21 \beta_{3} + 44 \beta_{2} + 21 \beta_1 + 50) q^{69} + (10 \beta_{4} - 4 \beta_{3} - 40 \beta_{2} - 2 \beta_1 - 238) q^{71} + (43 \beta_{4} + 15 \beta_{3} - 35 \beta_{2} + 4 \beta_1 + 3) q^{73} + ( - 19 \beta_{4} + 39 \beta_{3} - 84 \beta_{2} - 55 \beta_1 + 421) q^{75} + ( - 32 \beta_{4} - 6 \beta_{3} + 97 \beta_{2} + 45 \beta_1 + 218) q^{77} + ( - 15 \beta_{4} - 15 \beta_{3} - 99 \beta_{2} + 5 \beta_1 + 49) q^{79} + ( - 8 \beta_{4} + 8 \beta_{3} - 12 \beta_{2} - 44 \beta_1 - 71) q^{81} + (8 \beta_{4} + 10 \beta_{3} - 52 \beta_{2} - 6 \beta_1 + 136) q^{83} + ( - 40 \beta_{4} - 32 \beta_{3} - 45 \beta_{2} - 59 \beta_1 + 686) q^{85} + ( - 32 \beta_{4} + 23 \beta_{3} + 76 \beta_{2} - 53 \beta_1 + 122) q^{87} + ( - 51 \beta_{4} - 39 \beta_{3} + 27 \beta_{2} - 25 \beta_1 + 131) q^{89} + ( - 15 \beta_{4} + 23 \beta_{3} - 32 \beta_{2} + 49 \beta_1 - 475) q^{91} + (12 \beta_{4} + 22 \beta_{3} + 150 \beta_{2} + 58 \beta_1 + 232) q^{93} + (19 \beta_{2} + 19 \beta_1) q^{95} + ( - 6 \beta_{4} + 10 \beta_{3} - 84 \beta_{2} + 32 \beta_1 + 584) q^{97} + ( - 18 \beta_{4} - 64 \beta_{3} - 243 \beta_{2} - 21 \beta_1 + 60) q^{99}+O(q^{100})$$ q - b2 * q^3 + (-b2 - b1) * q^5 + (-b3 + b2 + 4) * q^7 + (-b4 + b3 - b1 + 16) * q^9 + (-2*b4 - 2*b3 - 3*b2 + b1) * q^11 + (-2*b4 + b3 + 2*b2 - b1 - 18) * q^13 + (-5*b4 - b3 - b2 - 3*b1 + 39) * q^15 + (b4 + b3 - 7*b2 - 2*b1 + 49) * q^17 - 19 * q^19 + (b4 + 4*b3 + 7*b2 - 2*b1 - 65) * q^21 + (2*b4 - b3 - 2*b2 + 7*b1 - 8) * q^23 + (-4*b4 - 6*b3 - 11*b2 - b1 + 113) * q^25 + (-5*b4 - 7*b3 - 10*b2 - 5*b1 + 39) * q^27 + (-3*b4 - 6*b3 - 5*b2 - 6*b1 - 43) * q^29 + (11*b4 - b3 - 11*b2 + 3*b1 - 37) * q^31 + (-b4 + 15*b3 + 5*b2 - 19*b1 + 131) * q^33 + (18*b4 + 6*b3 - 7*b2 - 5*b1 - 4) * q^35 + (11*b4 - b3 - 11*b2 + 3*b1 - 115) * q^37 + (-4*b4 - 9*b3 - 12*b2 - 9*b1 - 26) * q^39 + (9*b4 - 9*b3 + 11*b2 + 7*b1 + 39) * q^41 + (4*b4 + 4*b3 + 9*b2 + 23*b1 + 14) * q^43 + (-18*b4 - 49*b2 - 13*b1 + 114) * q^45 + (2*b4 + 4*b3 - 15*b2 + 7*b1 - 204) * q^47 + (3*b4 - 17*b3 + 13*b2 + 2*b1 + 94) * q^49 + (-14*b4 - 2*b3 - 53*b2 - 2*b1 + 294) * q^51 + (11*b4 - 8*b3 - 9*b2 + 2*b1 + 15) * q^53 + (-2*b4 + 32*b3 - 55*b2 - 25*b1 - 112) * q^55 + 19*b2 * q^57 + (-3*b4 + 15*b3 + 32*b2 - 9*b1 + 201) * q^59 + (16*b4 + 4*b3 - 15*b2 - 21*b1 + 56) * q^61 + (-4*b3 + b2 + 21*b1 - 350) * q^63 + (-14*b4 + 12*b3 - 40*b2 + 24*b1 - 14) * q^65 + (12*b4 - 28*b3 - 7*b2 + 10*b1 + 72) * q^67 + (28*b4 + 21*b3 + 44*b2 + 21*b1 + 50) * q^69 + (10*b4 - 4*b3 - 40*b2 - 2*b1 - 238) * q^71 + (43*b4 + 15*b3 - 35*b2 + 4*b1 + 3) * q^73 + (-19*b4 + 39*b3 - 84*b2 - 55*b1 + 421) * q^75 + (-32*b4 - 6*b3 + 97*b2 + 45*b1 + 218) * q^77 + (-15*b4 - 15*b3 - 99*b2 + 5*b1 + 49) * q^79 + (-8*b4 + 8*b3 - 12*b2 - 44*b1 - 71) * q^81 + (8*b4 + 10*b3 - 52*b2 - 6*b1 + 136) * q^83 + (-40*b4 - 32*b3 - 45*b2 - 59*b1 + 686) * q^85 + (-32*b4 + 23*b3 + 76*b2 - 53*b1 + 122) * q^87 + (-51*b4 - 39*b3 + 27*b2 - 25*b1 + 131) * q^89 + (-15*b4 + 23*b3 - 32*b2 + 49*b1 - 475) * q^91 + (12*b4 + 22*b3 + 150*b2 + 58*b1 + 232) * q^93 + (19*b2 + 19*b1) * q^95 + (-6*b4 + 10*b3 - 84*b2 + 32*b1 + 584) * q^97 + (-18*b4 - 64*b3 - 243*b2 - 21*b1 + 60) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{3} + 22 q^{7} + 83 q^{9}+O(q^{10})$$ 5 * q - 2 * q^3 + 22 * q^7 + 83 * q^9 $$5 q - 2 q^{3} + 22 q^{7} + 83 q^{9} - 6 q^{11} - 82 q^{13} + 204 q^{15} + 234 q^{17} - 95 q^{19} - 308 q^{21} - 60 q^{23} + 549 q^{25} + 190 q^{27} - 210 q^{29} - 224 q^{31} + 704 q^{33} - 42 q^{35} - 614 q^{37} - 132 q^{39} + 194 q^{41} + 38 q^{43} + 516 q^{45} - 1066 q^{47} + 489 q^{49} + 1382 q^{51} + 42 q^{53} - 618 q^{55} + 38 q^{57} + 1090 q^{59} + 276 q^{61} - 1790 q^{63} - 184 q^{65} + 314 q^{67} + 268 q^{69} - 1276 q^{71} - 106 q^{73} + 2066 q^{75} + 1226 q^{77} + 52 q^{79} - 283 q^{81} + 580 q^{83} + 3498 q^{85} + 900 q^{87} + 810 q^{89} - 2522 q^{91} + 1332 q^{93} + 2694 q^{97} - 126 q^{99}+O(q^{100})$$ 5 * q - 2 * q^3 + 22 * q^7 + 83 * q^9 - 6 * q^11 - 82 * q^13 + 204 * q^15 + 234 * q^17 - 95 * q^19 - 308 * q^21 - 60 * q^23 + 549 * q^25 + 190 * q^27 - 210 * q^29 - 224 * q^31 + 704 * q^33 - 42 * q^35 - 614 * q^37 - 132 * q^39 + 194 * q^41 + 38 * q^43 + 516 * q^45 - 1066 * q^47 + 489 * q^49 + 1382 * q^51 + 42 * q^53 - 618 * q^55 + 38 * q^57 + 1090 * q^59 + 276 * q^61 - 1790 * q^63 - 184 * q^65 + 314 * q^67 + 268 * q^69 - 1276 * q^71 - 106 * q^73 + 2066 * q^75 + 1226 * q^77 + 52 * q^79 - 283 * q^81 + 580 * q^83 + 3498 * q^85 + 900 * q^87 + 810 * q^89 - 2522 * q^91 + 1332 * q^93 + 2694 * q^97 - 126 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 106x^{3} - 401x^{2} + 356x + 2112$$ :

 $$\beta_{1}$$ $$=$$ $$( -11\nu^{4} + 16\nu^{3} + 1278\nu^{2} + 2011\nu - 11880 ) / 372$$ (-11*v^4 + 16*v^3 + 1278*v^2 + 2011*v - 11880) / 372 $$\beta_{2}$$ $$=$$ $$( 25\nu^{4} - 104\nu^{3} - 2262\nu^{2} - 749\nu + 15840 ) / 372$$ (25*v^4 - 104*v^3 - 2262*v^2 - 749*v + 15840) / 372 $$\beta_{3}$$ $$=$$ $$( -6\nu^{4} + 20\nu^{3} + 559\nu^{2} + 615\nu - 3256 ) / 31$$ (-6*v^4 + 20*v^3 + 559*v^2 + 615*v - 3256) / 31 $$\beta_{4}$$ $$=$$ $$( 9\nu^{4} - 30\nu^{3} - 854\nu^{2} - 721\nu + 5535 ) / 31$$ (9*v^4 - 30*v^3 - 854*v^2 - 721*v + 5535) / 31
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4$$ (b4 + b3 - b2 + b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( 5\beta_{4} + \beta_{3} - 13\beta_{2} + 13\beta _1 + 181 ) / 4$$ (5*b4 + b3 - 13*b2 + 13*b1 + 181) / 4 $$\nu^{3}$$ $$=$$ $$( 52\beta_{4} + 33\beta_{3} - 101\beta_{2} + 65\beta _1 + 558 ) / 2$$ (52*b4 + 33*b3 - 101*b2 + 65*b1 + 558) / 2 $$\nu^{4}$$ $$=$$ $$( 915\beta_{4} + 395\beta_{3} - 1987\beta_{2} + 1747\beta _1 + 18515 ) / 4$$ (915*b4 + 395*b3 - 1987*b2 + 1747*b1 + 18515) / 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
 2.15594 −6.76535 −3.22412 11.6790 −3.84548
0 −8.62694 0 −4.10677 0 −11.2105 0 47.4241 0
1.2 0 −5.24575 0 −18.7154 0 28.3947 0 0.517848 0
1.3 0 −2.49563 0 15.7941 0 30.5820 0 −20.7718 0
1.4 0 5.36679 0 −12.8065 0 −14.4426 0 1.80242 0
1.5 0 9.00153 0 19.8345 0 −11.3235 0 54.0275 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.ba 5
4.b odd 2 1 1216.4.a.bb 5
8.b even 2 1 152.4.a.d 5
8.d odd 2 1 304.4.a.k 5
24.h odd 2 1 1368.4.a.m 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.d 5 8.b even 2 1
304.4.a.k 5 8.d odd 2 1
1216.4.a.ba 5 1.a even 1 1 trivial
1216.4.a.bb 5 4.b odd 2 1
1368.4.a.m 5 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{5} + 2T_{3}^{4} - 107T_{3}^{3} - 244T_{3}^{2} + 2236T_{3} + 5456$$ T3^5 + 2*T3^4 - 107*T3^3 - 244*T3^2 + 2236*T3 + 5456 $$T_{5}^{5} - 587T_{5}^{3} - 1006T_{5}^{2} + 80568T_{5} + 308352$$ T5^5 - 587*T5^3 - 1006*T5^2 + 80568*T5 + 308352

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + 2 T^{4} - 107 T^{3} + \cdots + 5456$$
$5$ $$T^{5} - 587 T^{3} - 1006 T^{2} + \cdots + 308352$$
$7$ $$T^{5} - 22 T^{4} - 860 T^{3} + \cdots + 1592040$$
$11$ $$T^{5} + 6 T^{4} - 6899 T^{3} + \cdots + 91281200$$
$13$ $$T^{5} + 82 T^{4} - 2101 T^{3} + \cdots + 19005984$$
$17$ $$T^{5} - 234 T^{4} + \cdots + 607064382$$
$19$ $$(T + 19)^{5}$$
$23$ $$T^{5} + 60 T^{4} + \cdots + 4457888256$$
$29$ $$T^{5} + 210 T^{4} + \cdots - 7027069704$$
$31$ $$T^{5} + 224 T^{4} + \cdots + 376353439744$$
$37$ $$T^{5} + 614 T^{4} + \cdots + 345578559712$$
$41$ $$T^{5} - 194 T^{4} + \cdots - 344930807808$$
$43$ $$T^{5} - 38 T^{4} + \cdots - 1008111272576$$
$47$ $$T^{5} + 1066 T^{4} + \cdots - 399004393472$$
$53$ $$T^{5} - 42 T^{4} + \cdots + 115161670944$$
$59$ $$T^{5} - 1090 T^{4} + \cdots + 530364944$$
$61$ $$T^{5} - 276 T^{4} + \cdots - 185141518840$$
$67$ $$T^{5} - 314 T^{4} + \cdots - 4670019292608$$
$71$ $$T^{5} + 1276 T^{4} + \cdots - 79175200896$$
$73$ $$T^{5} + 106 T^{4} + \cdots + 38395259106486$$
$79$ $$T^{5} - 52 T^{4} + \cdots - 23640142664704$$
$83$ $$T^{5} - 580 T^{4} + \cdots - 1385360658176$$
$89$ $$T^{5} - 810 T^{4} + \cdots - 9913402008576$$
$97$ $$T^{5} - 2694 T^{4} + \cdots - 17678567904768$$