LMFDB - Newform orbit 1224.2.l.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(1189,1224)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1224.1189");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1224.l (of order $2$ , degree $1$ , minimal)

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:	no
Analytic conductor:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512$ x^18 - x^16 - 2x^14 + 2x^12 - 4x^11 + 4x^10 + 8x^8 - 16x^7 + 16x^6 - 64x^4 - 128*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{8}$
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{5} q^{2} - \beta_{2} q^{4} - \beta_{9} q^{5} + \beta_{15} q^{7} + \beta_{4} q^{8} + \beta_{8} q^{10} - \beta_{12} q^{11} - \beta_{13} q^{13} + (\beta_{16} - \beta_{15} + \cdots - \beta_{10}) q^{14}+ \cdots + (2 \beta_{15} + 2 \beta_{11} + \cdots + 2) q^{98}+O(q^{100})$ q + b5 * q^2 - b2 * q^4 - b9 * q^5 + b15 * q^7 + b4 * q^8 + b8 * q^10 - b12 * q^11 - b13 * q^13 + (b16 - b15 - b13 + b11 - b10) * q^14 + (b10 - b9) * q^16 + (b17 - b16 + b15 + b13 - b11 + b10 - b5) * q^17 + (-b14 + b10 - b5 + b4 - b3 + b2) * q^19 + (b17 + b15 - b14 - b7 - b3) * q^20 + (b17 - b16 + b15 - b14 - b11 + b10 + b6 + b4 - b3 + b2 - b1) * q^22 + (b16 - b15 - b10 - b5 - b4 + b3 - b2 + b1) * q^23 + (b17 - b16 + b15 - b14 - b12 - b8 + b5 - b3 + b2 + 1) * q^25 + (-b17 + b16 - b13 + b12) * q^26 + (-b14 - b11 + b10 + b9) * q^28 + (b17 - b16 + b15 - b14 - b12 - b11 + b9 - b7 - b3 - b2) * q^29 + (b14 + b11 - b10 - b6 - b4 + b3 - b2 + b1) * q^31 + (b12 + b10 + b8 + b7 + b4 - b3 + b2 - b1) * q^32 + (b15 + b14 - b3 + b2) * q^34 + (-b16 + b15 - b14 + b13 - b11 + 2b10 + b6 + b5 - b1) q^35 + (b16 + b14 + b12 - b9 - 2b2 - 2) q^37 + (-b16 + b11 + b10 + b7 - b6 + 2b2 + 2) q^38 + (-b15 - b14 - b13 - b12 + b11 + b9 - b6 - b3 + b2 + b1) * q^40 + (b15 - b11 + b6 - b5) * q^41 + (b17 + b16 - b14 - b11 + b8 + b6 - b3 + b2 - b1) * q^43 + (-b17 + b14 - b13 + b12 + b9 - b7 - b6 - b4 - b2 + b1) * q^44 + (b17 + b15 - b12 - b11 - b10 + 2) * q^46 + (-b17 + b14 + b10 + b7 - b6 - b4) * q^47 + (-b17 - b16 - b11 + b10 + 2b9 + b8 - b6 + b5 - b4 + b1 + 1) q^49 + (-b17 - b16 - b13 - b12 + 2b9 + b5 + 2) q^50 + (-b17 + 2b16 - 2b15 + b14 - b13 + b12 + 2b11 - 2b10 - b9 + b7 - b6 - b4 + 2b3 - b2 + b1) q^52 + (-b16 + b15 + b14 + b13 - b11 + b6 - b5) * q^53 + (-b16 + b13 - b12 - b11 + b10 + b8 + b7 + b4 + b3 + b2) * q^55 + (b15 + b14 - b13 + b11 - b10 - b9 - b8 + b7 - b6) * q^56 + (-b16 + b15 - b14 - b13 - b12 - b8 - b7 + b4 - b3 + b2 + b1) * q^58 + (b16 - b15 + 2b11 - b10 + b8 + b7 - b6 - b5 - b4 + b3 - b2 + b1) q^59 + (b17 + b16 - b12 - b10 - b9 - b7 + b6 + b4 - 2b2 - 2) q^61 + (b16 - b14 - b12 + b6 + b4 - b3 + b2 - b1) * q^62 + (b15 + b14 + b13 + b12 + b11 + b9 - b6 - b4 + b3 - b2 + b1) * q^64 + (b17 - b16 + b15 + b10 + b8 + b4 - 2b3 + 2b2 - b1) * q^65 + (-b16 - b15 + b10 + b8 + b7 + b6 - b5 + b4 - b3 + b2 - b1) * q^67 + (-b15 + b13 + b12 + b9 + b6 - b4 + b3 + b1 + 2) * q^68 + (-b15 + b14 - b13 + b12 + b11 - b10 - b6 + b4 - b3 - b2 - b1 - 2) * q^70 + (-b16 - b15 - b14 + b11 + 2b10 + b8 + b7 - b5) q^71 + (b17 - b15 - b14 + b13 + b10 + b8 - b4) * q^73 + (b16 - b15 + b14 + b13 + b12 + b8 + b7 - 2b5 + b4 + b3 - b2 + b1) q^74 + (-b17 + b15 + b11 + b10 + b9 + b7 + 2b5 - b3 + 2b2 - 2b1) q^76 + (b14 - b13 + b11 - b10 + b8 + b7 + b4 - b3 + b2 - b1) * q^77 + (-2b16 + b15 + b14 + b13 - b11 + b10 - b8 - b7 - b4 + b3 - b2 + b1) q^79 + (-2b16 + 2b15 - 2b14 - b12 + 2b10 + b9 - b8 + b7 + b4 - b3 + 3b2 - b1 + 4) q^80 + (b16 - b15 + b14 - b13 + b12 - 2b10 - b7 - b4 + b3 - b2 + b1 + 2) q^82 + (-b17 + b16 - 3b15 - b14 - 2b13 - b8 - b5 + b3 - b2 + b1) * q^83 + (-b17 - b12 - b9 - 2b8 + b7 + b6 - b5 + b3 + b2 - b1) q^85 + (2b17 - b16 + 2b15 - b11 - b10 + 2b9 - b7 - b6 - 2b4 - 2b2 + 2b1) * q^86 + (-2b17 + 2b16 - 2b15 + b12 + 2b11 - b10 - 2b9 - b8 + b7 + b3 + b2 + b1) q^88 + (b17 - b16 + b15 - b14 - 2b11 + b8 - 2b7 - b5 - b3 - 3b2) q^89 + (-b14 - b13 + b11 - b10 - b8 - b7 - 2b5 - b4 - b3 + b2 - b1) q^91 + (b17 + 2b15 - b13 - b12 - b11 - b10 - b7 + b6 + 2b5 - b4 - b2 + b1) * q^92 + (b15 + b14 + b13 + b12 + b11 - b10 + 2b7 + b6 + b4 + b3 + b2 - b1) q^94 + (-2b17 - b15 - b14 - b13 + b11 + b10 - 2b8 - b6 + b4 + b3 - b2 + b1) * q^95 + (-b17 - 2b15 + b14 + 2b13 - b10 + b7 + b6 - b4 + 2b3 - 2b2) * q^97 + (2b15 + 2b11 - 2b10 - 2b9 - 2b8 + b5 + 2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$18 q + 2 q^{4} + 4 q^{5} - 8 q^{10} + 6 q^{14} + 10 q^{16} + 2 q^{17} + 2 q^{20} + 2 q^{22} + 22 q^{25} - 2 q^{26} - 10 q^{28} - 12 q^{29} + 6 q^{34} - 16 q^{37} + 34 q^{38} - 10 q^{40} - 12 q^{44} + 32 q^{46}+ \cdots + 60 q^{98}+O(q^{100})$ 18 * q + 2 * q^4 + 4 * q^5 - 8 * q^10 + 6 * q^14 + 10 * q^16 + 2 * q^17 + 2 * q^20 + 2 * q^22 + 22 * q^25 - 2 * q^26 - 10 * q^28 - 12 * q^29 + 6 * q^34 - 16 * q^37 + 34 * q^38 - 10 * q^40 - 12 * q^44 + 32 * q^46 - 18 * q^49 + 14 * q^50 + 8 * q^52 - 16 * q^55 + 14 * q^56 - 6 * q^58 - 16 * q^61 + 6 * q^62 + 2 * q^64 + 36 * q^68 - 32 * q^70 + 6 * q^74 + 62 * q^80 + 34 * q^82 + 16 * q^85 - 14 * q^86 + 16 * q^88 - 20 * q^89 + 2 * q^92 + 12 * q^94 + 60 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512$ x^18 - x^16 - 2*x^14 + 2*x^12 - 4*x^11 + 4*x^10 + 8*x^8 - 16*x^7 + 16*x^6 - 64*x^4 - 128*x^2 + 512 :

$\beta_{1}$	$=$	$2\nu$ 2*v
$\beta_{2}$	$=$	$( \nu^{16} - \nu^{14} - 2 \nu^{12} + 2 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + 8 \nu^{6} - 16 \nu^{5} + \cdots - 128 ) / 128$ (v^16 - v^14 - 2v^12 + 2v^10 - 4v^9 + 4v^8 + 8v^6 - 16v^5 + 16v^4 - 64v^2 - 128) / 128
$\beta_{3}$	$=$	$( \nu^{16} - \nu^{14} + 4 \nu^{13} - 2 \nu^{12} + 4 \nu^{11} + 2 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + \cdots - 256 ) / 128$ (v^16 - v^14 + 4v^13 - 2v^12 + 4v^11 + 2v^10 - 4v^9 + 4v^8 + 8v^7 - 8v^6 + 16v^5 - 16v^4 + 96v^3 - 192v^2 - 256) / 128
$\beta_{4}$	$=$	$( - \nu^{17} - 3 \nu^{15} + 6 \nu^{13} + 6 \nu^{11} + 4 \nu^{10} - 12 \nu^{9} + 16 \nu^{8} + \cdots + 384 \nu ) / 256$ (-v^17 - 3v^15 + 6v^13 + 6v^11 + 4v^10 - 12v^9 + 16v^8 - 24v^7 + 16v^6 - 48v^5 + 64v^4 + 384*v) / 256
$\beta_{5}$	$=$	$( - \nu^{17} + \nu^{15} + 2 \nu^{13} - 2 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 8 \nu^{7} + 16 \nu^{6} + \cdots + 128 \nu ) / 256$ (-v^17 + v^15 + 2v^13 - 2v^11 + 4v^10 - 4v^9 - 8v^7 + 16v^6 - 16v^5 + 64v^3 + 128*v) / 256
$\beta_{6}$	$=$	$( - \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 2 \nu^{12} - 2 \nu^{11} - 2 \nu^{9} - 8 \nu^{8} + 12 \nu^{7} + \cdots + 128 ) / 64$ (-v^15 + 2v^14 - 3v^13 + 2v^12 - 2v^11 - 2v^9 - 8v^8 + 12v^7 + 16v^6 - 24v^5 + 32v^4 - 48v^3 + 96v^2 - 64*v + 128) / 64
$\beta_{7}$	$=$	$( - \nu^{17} + 2 \nu^{16} - \nu^{15} + 4 \nu^{14} - 4 \nu^{13} + 2 \nu^{12} + 2 \nu^{11} - 8 \nu^{10} + \cdots - 256 ) / 128$ (-v^17 + 2v^16 - v^15 + 4v^14 - 4v^13 + 2v^12 + 2v^11 - 8v^10 + 8v^9 + 12v^8 - 8v^7 + 48v^6 - 64v^5 + 144v^4 - 128v^3 + 128v^2 - 256) / 128
$\beta_{8}$	$=$	$( 3 \nu^{17} - 6 \nu^{16} + 9 \nu^{15} - 10 \nu^{14} + 6 \nu^{13} - 4 \nu^{12} + 6 \nu^{11} + 8 \nu^{10} + \cdots + 256 ) / 256$ (3v^17 - 6v^16 + 9v^15 - 10v^14 + 6v^13 - 4v^12 + 6v^11 + 8v^10 - 4v^9 - 8v^8 - 8v^7 - 128v^6 + 176v^5 - 416v^4 + 576v^3 - 512v^2 + 128*v + 256) / 256
$\beta_{9}$	$=$	$( 3 \nu^{17} - 6 \nu^{16} + 5 \nu^{15} - 6 \nu^{14} + 2 \nu^{13} - 10 \nu^{11} + 8 \nu^{10} + 4 \nu^{9} + \cdots + 768 ) / 256$ (3v^17 - 6v^16 + 5v^15 - 6v^14 + 2v^13 - 10v^11 + 8v^10 + 4v^9 + 16v^8 + 40v^7 - 64v^6 + 208v^5 - 384v^4 + 256v^3 - 256v^2 - 128v + 768) / 256
$\beta_{10}$	$=$	$( 3 \nu^{17} - 8 \nu^{16} + 5 \nu^{15} - 12 \nu^{14} + 2 \nu^{13} + 12 \nu^{12} - 10 \nu^{11} + \cdots + 1536 ) / 256$ (3v^17 - 8v^16 + 5v^15 - 12v^14 + 2v^13 + 12v^12 - 10v^11 + 20v^10 + 12v^9 - 8v^8 + 72v^7 - 112v^6 + 240v^5 - 480v^4 + 384v^3 - 256v^2 - 128*v + 1536) / 256
$\beta_{11}$	$=$	$( - 2 \nu^{17} + \nu^{16} - 2 \nu^{15} + \nu^{14} + 4 \nu^{13} - 8 \nu^{12} + 8 \nu^{11} + 2 \nu^{10} + \cdots - 384 ) / 128$ (-2v^17 + v^16 - 2v^15 + v^14 + 4v^13 - 8v^12 + 8v^11 + 2v^10 + 4v^9 - 8v^8 + 24v^6 - 112v^5 + 64v^4 - 64v^2 + 512*v - 384) / 128
$\beta_{12}$	$=$	$( - 3 \nu^{17} + 5 \nu^{16} - 5 \nu^{15} + 7 \nu^{14} - 2 \nu^{13} - 6 \nu^{12} + 10 \nu^{11} + \cdots - 768 ) / 128$ (-3v^17 + 5v^16 - 5v^15 + 7v^14 - 2v^13 - 6v^12 + 10v^11 - 2v^10 - 4v^8 - 40v^7 + 104v^6 - 160v^5 + 272v^4 - 256v^3 + 256v^2 + 256v - 768) / 128
$\beta_{13}$	$=$	$( 2 \nu^{17} - 3 \nu^{16} - 2 \nu^{15} + 3 \nu^{14} - 12 \nu^{13} + 14 \nu^{12} - 20 \nu^{11} + \cdots + 1280 ) / 128$ (2v^17 - 3v^16 - 2v^15 + 3v^14 - 12v^13 + 14v^12 - 20v^11 + 10v^10 + 4v^9 + 4v^8 - 72v^6 + 80v^5 - 16v^4 - 192v^3 + 384v^2 - 768v + 1280) / 128
$\beta_{14}$	$=$	$( - 3 \nu^{17} - 2 \nu^{16} + 7 \nu^{15} - 18 \nu^{14} + 26 \nu^{13} - 32 \nu^{12} + 26 \nu^{11} + \cdots - 1280 ) / 256$ (-3v^17 - 2v^16 + 7v^15 - 18v^14 + 26v^13 - 32v^12 + 26v^11 - 8v^10 - 12v^9 + 24v^7 + 64v^6 + 16v^5 - 192v^4 + 768v^3 - 896v^2 + 1920v - 1280) / 256
$\beta_{15}$	$=$	$( 5 \nu^{17} - 4 \nu^{16} - \nu^{15} + 8 \nu^{14} - 22 \nu^{13} + 28 \nu^{12} - 22 \nu^{11} + \cdots + 1792 ) / 256$ (5v^17 - 4v^16 - v^15 + 8v^14 - 22v^13 + 28v^12 - 22v^11 + 20v^10 - 4v^9 + 8v^8 + 56v^7 - 144v^6 + 176v^5 - 96v^4 - 384v^3 + 640v^2 - 1664v + 1792) / 256
$\beta_{16}$	$=$	$( 9 \nu^{17} - 10 \nu^{16} + 3 \nu^{15} - 2 \nu^{14} - 22 \nu^{13} + 40 \nu^{12} - 46 \nu^{11} + \cdots + 3328 ) / 256$ (9v^17 - 10v^16 + 3v^15 - 2v^14 - 22v^13 + 40v^12 - 46v^11 + 40v^10 + 20v^9 + 16v^8 + 88v^7 - 224v^6 + 400v^5 - 384v^4 - 128v^3 + 512v^2 - 2176*v + 3328) / 256
$\beta_{17}$	$=$	$( - 11 \nu^{17} + 14 \nu^{16} - 5 \nu^{15} - 2 \nu^{14} + 30 \nu^{13} - 48 \nu^{12} + 50 \nu^{11} + \cdots - 3840 ) / 256$ (-11v^17 + 14v^16 - 5v^15 - 2v^14 + 30v^13 - 48v^12 + 50v^11 - 24v^10 + 12v^9 + 16v^8 - 56v^7 + 288v^6 - 528v^5 + 512v^4 + 320v^3 - 512v^2 + 2688*v - 3840) / 256

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_1 ) / 2$ (b1) / 2
$\nu^{2}$	$=$	$( \beta_{17} + \beta_{15} - \beta_{11} - \beta_{7} - \beta_{3} - \beta_{2} ) / 2$ (b17 + b15 - b11 - b7 - b3 - b2) / 2
$\nu^{3}$	$=$	$( \beta_{17} + \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_{2} ) / 2$ (b17 + b13 - b11 + b8 + b6 + b3 - b2) / 2
$\nu^{4}$	$=$	$( - \beta_{17} + 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots - \beta_{2} ) / 2$ (-b17 + 2b16 - b15 + 2b14 + b11 - 2b10 - 2b9 + b7 + b3 - b2) / 2
$\nu^{5}$	$=$	$( - \beta_{17} + 2 \beta_{14} + \beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{8} - \beta_{6} + \cdots + 2 \beta_1 ) / 2$ (-b17 + 2b14 + b13 + 2b12 - b11 - b8 - b6 - 2b4 + 3b3 - 3b2 + 2b1) / 2
$\nu^{6}$	$=$	$( - \beta_{17} + 2 \beta_{16} - 3 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{9} + \cdots - 2 \beta_1 ) / 2$ (-b17 + 2b16 - 3b15 - 2b13 + 2b12 - b11 + 2b9 + b7 + 2b6 + 2b4 - b3 + b2 - 2b1) / 2
$\nu^{7}$	$=$	$( \beta_{17} + 4 \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} - 3 \beta_{8} + \cdots - \beta_{2} ) / 2$ (b17 + 4b15 + 2b14 - b13 - 2b12 + b11 - 3b8 + b6 - 2*b4 + b3 - b2) / 2
$\nu^{8}$	$=$	$( 3 \beta_{17} - 2 \beta_{16} + 5 \beta_{15} + 2 \beta_{13} - 2 \beta_{12} - \beta_{11} + 6 \beta_{9} + \cdots - 2 \beta_1 ) / 2$ (3b17 - 2b16 + 5b15 + 2b13 - 2b12 - b11 + 6b9 + 5b7 - 2b6 + 6b4 - b3 + 5b2 - 2*b1) / 2
$\nu^{9}$	$=$	$( \beta_{17} + 8 \beta_{16} - 8 \beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{12} + 5 \beta_{11} + \cdots - 9 \beta_{2} ) / 2$ (b17 + 8b16 - 8b15 - 2b14 - b13 + 2b12 + 5b11 + b8 + 4b7 - 3b6 - 8b5 - 6b4 + b3 - 9b2) / 2
$\nu^{10}$	$=$	$( - 5 \beta_{17} + 14 \beta_{16} - 3 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 10 \beta_{12} + 11 \beta_{11} + \cdots - 16 ) / 2$ (-5b17 + 14b16 - 3b15 + 4b14 - 6b13 + 10b12 + 11b11 - 4b10 - 2b9 + 4b8 + b7 - 2b6 + 16b5 + 2b4 + 3b3 + 9b2 + 2b1 - 16) / 2
$\nu^{11}$	$=$	$( - 3 \beta_{17} + 12 \beta_{15} + 6 \beta_{14} - 9 \beta_{13} + 10 \beta_{12} + \beta_{11} + 8 \beta_{10} + \cdots + 32 ) / 2$ (-3b17 + 12b15 + 6b14 - 9b13 + 10b12 + b11 + 8b10 - 8b9 + 9b8 + 8b7 - 3b6 - 24b5 + 10b4 - 3b3 - 5b2 - 12*b1 + 32) / 2
$\nu^{12}$	$=$	$( - 5 \beta_{17} - 2 \beta_{16} - 7 \beta_{15} - 12 \beta_{14} - 10 \beta_{13} - 6 \beta_{12} - 5 \beta_{11} + \cdots - 48 ) / 2$ (-5b17 - 2b16 - 7b15 - 12b14 - 10b13 - 6b12 - 5b11 + 20b10 - 18b9 + 13b7 + 2b6 + 16b5 + 18b4 + 3b3 + b2 + 10*b1 - 48) / 2
$\nu^{13}$	$=$	$( 5 \beta_{17} + 24 \beta_{16} - 8 \beta_{15} - 10 \beta_{14} - 29 \beta_{13} - 14 \beta_{12} + \beta_{11} + \cdots + 32 ) / 2$ (5b17 + 24b16 - 8b15 - 10b14 - 29b13 - 14b12 + b11 - 24b10 - 19b8 - 28b7 - 7b6 + 24b5 + 18b4 - 11b3 - 45b2 - 12*b1 + 32) / 2
$\nu^{14}$	$=$	$( - 5 \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 12 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 19 \beta_{11} + \cdots + 48 ) / 2$ (-5b17 + 6b16 - 3b15 - 12b14 - 6b13 + 2b12 + 19b11 - 36b10 + 38b9 + 12b8 + 9b7 + 14b6 + 48b5 - 6b4 + 43b3 - 55b2 + 26*b1 + 48) / 2
$\nu^{15}$	$=$	$( - 19 \beta_{17} + 32 \beta_{16} + 20 \beta_{15} + 14 \beta_{14} - 57 \beta_{13} - 14 \beta_{12} + \cdots + 96 ) / 2$ (-19b17 + 32b16 + 20b15 + 14b14 - 57b13 - 14b12 + 25b11 - 40b10 - 24b9 + b8 + 16b7 - 27b6 + 120b5 - 30b4 - 51b3 + 11b2 + 4b1 + 96) / 2
$\nu^{16}$	$=$	$( 59 \beta_{17} - 34 \beta_{16} + 41 \beta_{15} - 52 \beta_{14} + 6 \beta_{13} + 2 \beta_{12} - 77 \beta_{11} + \cdots + 240 ) / 2$ (59b17 - 34b16 + 41b15 - 52b14 + 6b13 + 2b12 - 77b11 + 44b10 - 2b9 - 8b8 - 59b7 - 14b6 + 16b5 - 70b4 + 27b3 + 25b2 + 98*b1 + 240) / 2
$\nu^{17}$	$=$	$( 29 \beta_{17} + 136 \beta_{16} - 80 \beta_{15} - 42 \beta_{14} - 93 \beta_{13} - 14 \beta_{12} + \cdots + 32 ) / 2$ (29b17 + 136b16 - 80b15 - 42b14 - 93b13 - 14b12 - 23b11 - 120b10 + 16b9 + 61b8 - 52b7 + 73b6 - 200b5 + 98b4 - 67b3 + 11b2 + 76*b1 + 32) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$ .

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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}

1189.1

−1.36011	+	0.387425i
−1.36011	−	0.387425i
−1.35042	+	0.419960i
−1.35042	−	0.419960i
−0.943929	+	1.05309i
−0.943929	−	1.05309i
−0.410541	+	1.35331i
−0.410541	−	1.35331i
−0.0535843	+	1.41320i
−0.0535843	−	1.41320i
0.480188	+	1.33020i
0.480188	−	1.33020i
0.937200	+	1.05908i
0.937200	−	1.05908i
1.29188	+	0.575356i
1.29188	−	0.575356i
1.40931	+	0.117654i
1.40931	−	0.117654i

−1.36011

−

0.387425i

1.69980

1.05388i

−0.665591

−

0.998700i

−1.90362

−

2.09194i

0.905277

0.257866i

1189.2

−1.36011

0.387425i

1.69980

−

1.05388i

−0.665591

0.998700i

−1.90362

2.09194i

0.905277

−

0.257866i

1189.3

−1.35042

−

0.419960i

1.64727

1.13424i

3.71339

5.19876i

−1.74817

−

2.22349i

−5.01464

−

1.55948i

1189.4

−1.35042

0.419960i

1.64727

−

1.13424i

3.71339

−

5.19876i

−1.74817

2.22349i

−5.01464

1.55948i

1189.5

−0.943929

−

1.05309i

−0.217998

1.98808i

−1.57273

−

1.68207i

2.29941

−

1.64704i

1.48454

1.65622i

1189.6

−0.943929

1.05309i

−0.217998

−

1.98808i

−1.57273

1.68207i

2.29941

1.64704i

1.48454

−

1.65622i

1189.7

−0.410541

−

1.35331i

−1.66291

1.11118i

−1.09133

4.50716i

2.18647

1.79426i

0.448036

1.47691i

1189.8

−0.410541

1.35331i

−1.66291

−

1.11118i

−1.09133

−

4.50716i

2.18647

−

1.79426i

0.448036

−

1.47691i

1189.9

−0.0535843

−

1.41320i

−1.99426

0.151450i

3.17378

−

2.51539i

0.320890

2.81017i

−0.170065

−

4.48518i

1189.10

−0.0535843

1.41320i

−1.99426

−

0.151450i

3.17378

2.51539i

0.320890

−

2.81017i

−0.170065

4.48518i

1189.11

0.480188

−

1.33020i

−1.53884

−

1.27749i

1.12134

−

0.430012i

−2.43824

1.43352i

0.538452

−

1.49159i

1189.12

0.480188

1.33020i

−1.53884

1.27749i

1.12134

0.430012i

−2.43824

−

1.43352i

0.538452

1.49159i

1189.13

0.937200

−

1.05908i

−0.243310

−

1.98514i

−4.06326

1.33474i

−2.33046

−

1.60279i

−3.80809

4.30332i

1189.14

0.937200

1.05908i

−0.243310

1.98514i

−4.06326

−

1.33474i

−2.33046

1.60279i

−3.80809

−

4.30332i

1189.15

1.29188

−

0.575356i

1.33793

−

1.48659i

2.84912

−

1.38176i

0.873137

−

2.69028i

3.68073

−

1.63925i

1189.16

1.29188

0.575356i

1.33793

1.48659i

2.84912

1.38176i

0.873137

2.69028i

3.68073

1.63925i

1189.17

1.40931

−

0.117654i

1.97232

−

0.331622i

−1.46472

3.26018i

2.74059

−

0.699408i

−2.06424

0.172330i

1189.18

1.40931

0.117654i

1.97232

0.331622i

−1.46472

−

3.26018i

2.74059

0.699408i

−2.06424

−

0.172330i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
136.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1224.2.l.d		18
3.b	odd	2	1		408.2.l.b	yes	18
4.b	odd	2	1		4896.2.l.d		18
8.b	even	2	1		1224.2.l.c		18
8.d	odd	2	1		4896.2.l.c		18
12.b	even	2	1		1632.2.l.a		18
17.b	even	2	1		1224.2.l.c		18
24.f	even	2	1		1632.2.l.b		18
24.h	odd	2	1		408.2.l.a	✓	18
51.c	odd	2	1		408.2.l.a	✓	18
68.d	odd	2	1		4896.2.l.c		18
136.e	odd	2	1		4896.2.l.d		18
136.h	even	2	1	inner	1224.2.l.d		18
204.h	even	2	1		1632.2.l.b		18
408.b	odd	2	1		408.2.l.b	yes	18
408.h	even	2	1		1632.2.l.a		18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
408.2.l.a	✓	18	24.h	odd	2	1
408.2.l.a	✓	18	51.c	odd	2	1
408.2.l.b	yes	18	3.b	odd	2	1
408.2.l.b	yes	18	408.b	odd	2	1
1224.2.l.c		18	8.b	even	2	1
1224.2.l.c		18	17.b	even	2	1
1224.2.l.d		18	1.a	even	1	1	trivial
1224.2.l.d		18	136.h	even	2	1	inner
1632.2.l.a		18	12.b	even	2	1
1632.2.l.a		18	408.h	even	2	1
1632.2.l.b		18	24.f	even	2	1
1632.2.l.b		18	204.h	even	2	1
4896.2.l.c		18	8.d	odd	2	1
4896.2.l.c		18	68.d	odd	2	1
4896.2.l.d		18	4.b	odd	2	1
4896.2.l.d		18	136.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{9} - 2T_{5}^{8} - 26T_{5}^{7} + 44T_{5}^{6} + 197T_{5}^{5} - 166T_{5}^{4} - 648T_{5}^{3} - 60T_{5}^{2} + 552T_{5} + 256$ T5^9 - 2*T5^8 - 26*T5^7 + 44*T5^6 + 197*T5^5 - 166*T5^4 - 648*T5^3 - 60*T5^2 + 552*T5 + 256 acting on $S_{2}^{\mathrm{new}}(1224, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{18} - T^{16} + \cdots + 512$ T^18 - T^16 - 2T^14 + 2T^12 - 4T^11 + 4T^10 + 8T^8 - 16T^7 + 16T^6 - 64T^4 - 128*T^2 + 512
$3$	$T^{18}$ T^18
$5$	$(T^{9} - 2 T^{8} + \cdots + 256)^{2}$ (T^9 - 2T^8 - 26T^7 + 44T^6 + 197T^5 - 166T^4 - 648T^3 - 60T^2 + 552T + 256)^2
$7$	$T^{18} + 72 T^{16} + \cdots + 65536$ T^18 + 72T^16 + 1936T^14 + 24848T^12 + 165760T^10 + 595456T^8 + 1162240T^6 + 1180672T^4 + 536576T^2 + 65536
$11$	$(T^{9} - 54 T^{7} + \cdots - 512)^{2}$ (T^9 - 54T^7 + 12T^6 + 869T^5 - 516T^4 - 4224T^3 + 4832T^2 + 256*T - 512)^2
$13$	$T^{18} + 124 T^{16} + \cdots + 1048576$ T^18 + 124T^16 + 5942T^14 + 141740T^12 + 1831185T^10 + 12872592T^8 + 46004736T^6 + 70639616T^4 + 35454976T^2 + 1048576
$17$	$T^{18} + \cdots + 118587876497$ T^18 - 2T^17 - 7T^16 - 32T^15 + 372T^14 + 328T^13 - 1308T^12 - 36832T^11 + 33534T^10 + 474228T^9 + 570078T^8 - 10644448T^7 - 6426204T^6 + 27394888T^5 + 528186804T^4 - 772402208T^3 - 2872370711T^2 - 13951514882*T + 118587876497
$19$	$T^{18} + \cdots + 5110534144$ T^18 + 172T^16 + 12182T^14 + 463172T^12 + 10331201T^10 + 138447352T^8 + 1092519312T^6 + 4727637248T^4 + 9468067840T^2 + 5110534144
$23$	$T^{18} + \cdots + 57431163904$ T^18 + 208T^16 + 17454T^14 + 767948T^12 + 19481993T^10 + 298115980T^8 + 2781677264T^6 + 15401579344T^4 + 46207416000T^2 + 57431163904
$29$	$(T^{9} + 6 T^{8} + \cdots - 8192)^{2}$ (T^9 + 6T^8 - 96T^7 - 452T^6 + 3064T^5 + 9792T^4 - 41088T^3 - 59392T^2 + 197632T - 8192)^2
$31$	$T^{18} + 192 T^{16} + \cdots + 75759616$ T^18 + 192T^16 + 13776T^14 + 486800T^12 + 9127616T^10 + 88919552T^8 + 400877568T^6 + 678055936T^4 + 405323776T^2 + 75759616
$37$	$(T^{9} + 8 T^{8} + \cdots + 182272)^{2}$ (T^9 + 8T^8 - 120T^7 - 1324T^6 - 424T^5 + 28704T^4 + 62528T^3 - 106176T^2 - 237440T + 182272)^2
$41$	$T^{18} + \cdots + 1474922807296$ T^18 + 268T^16 + 30334T^14 + 1899612T^12 + 72351209T^10 + 1728727296T^8 + 25633919808T^6 + 222664749312T^4 + 981879013376T^2 + 1474922807296
$43$	$T^{18} + \cdots + 22641152425984$ T^18 + 460T^16 + 87302T^14 + 8884596T^12 + 527878865T^10 + 18666572456T^8 + 381732516240T^6 + 4112931754752T^4 + 18655548141568T^2 + 22641152425984
$47$	$(T^{9} - 220 T^{7} + \cdots - 8781824)^{2}$ (T^9 - 220T^7 + 168T^6 + 15768T^5 - 17760T^4 - 452224T^3 + 678400T^2 + 4526080*T - 8781824)^2
$53$	$T^{18} + \cdots + 3561644621824$ T^18 + 468T^16 + 85840T^14 + 8036288T^12 + 416194112T^10 + 12078132992T^8 + 190497352704T^6 + 1488380530688T^4 + 4583668711424T^2 + 3561644621824
$59$	$T^{18} + \cdots + 10417711611904$ T^18 + 584T^16 + 136528T^14 + 16640000T^12 + 1162466304T^10 + 47957118976T^8 + 1148157165568T^6 + 14680342921216T^4 + 77611199889408T^2 + 10417711611904
$61$	$(T^{9} + 8 T^{8} + \cdots - 22528)^{2}$ (T^9 + 8T^8 - 244T^7 - 1428T^6 + 13744T^5 + 60400T^4 - 257792T^3 - 626784T^2 + 1488832T - 22528)^2
$67$	$T^{18} + \cdots + 8875147264$ T^18 + 696T^16 + 185104T^14 + 23781632T^12 + 1499146240T^10 + 39489441792T^8 + 197586911232T^6 + 270823587840T^4 + 101602820096T^2 + 8875147264
$71$	$T^{18} + \cdots + 34530008498176$ T^18 + 684T^16 + 176272T^14 + 21262160T^12 + 1248136896T^10 + 38000423424T^8 + 623708907520T^6 + 5438766157824T^4 + 22915287924736T^2 + 34530008498176
$73$	$T^{18} + \cdots + 3427652337664$ T^18 + 504T^16 + 97152T^14 + 9259520T^12 + 485384768T^10 + 14571400192T^8 + 248628002816T^6 + 2241859223552T^4 + 8500369424384T^2 + 3427652337664
$79$	$T^{18} + \cdots + 223514460160000$ T^18 + 624T^16 + 160944T^14 + 22274960T^12 + 1795715264T^10 + 85471540224T^8 + 2310578601984T^6 + 31632995647488T^4 + 164534891315200T^2 + 223514460160000
$83$	$T^{18} + \cdots + 945638269517824$ T^18 + 976T^16 + 389776T^14 + 82133504T^12 + 9802309184T^10 + 660149347840T^8 + 23513636582400T^6 + 375870435344384T^4 + 1545785101451264T^2 + 945638269517824
$89$	$(T^{9} + 10 T^{8} + \cdots + 192794368)^{2}$ (T^9 + 10T^8 - 372T^7 - 4496T^6 + 32088T^5 + 483920T^4 - 725696T^3 - 17810048T^2 - 810624T + 192794368)^2
$97$	$T^{18} + \cdots + 9088150798336$ T^18 + 920T^16 + 343328T^14 + 65998336T^12 + 6810167872T^10 + 352007434240T^8 + 7283499139072T^6 + 45825032978432T^4 + 45953667039232T^2 + 9088150798336
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1189.18
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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