LMFDB - Newform orbit 1224.2.f.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(613,1224)]

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, 0]))

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

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

chi := DirichletCharacter("1224.613");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1224.f (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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{14} - 2 x^{13} + x^{12} + 2 x^{11} - 10 x^{10} + 8 x^{9} + 10 x^{8} - 20 x^{7} + 20 x^{6} + \cdots + 128$ x^14 - 2x^13 + x^12 + 2x^11 - 10x^10 + 8x^9 + 10x^8 - 20x^7 + 20x^6 + 32x^5 - 80x^4 + 32x^3 + 32x^2 - 128x + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{7}$
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 $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{4} q^{2} - \beta_{2} q^{4} - \beta_{3} q^{5} + ( - \beta_{13} - \beta_{8} + \beta_1) q^{7} - \beta_{11} q^{8} + ( - \beta_{5} + \beta_{3} - 1) q^{10} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_{3}) q^{11}+ \cdots + ( - 2 \beta_{13} + 2 \beta_{12} + \cdots - 2 \beta_1) q^{98}+O(q^{100})$ q + b4 * q^2 - b2 * q^4 - b3 * q^5 + (-b13 - b8 + b1) * q^7 - b11 * q^8 + (-b5 + b3 - 1) * q^10 + (-b10 + b7 - b6 - b4 - b3) * q^11 + (-b10 + b6 + b5 - b4 + b3 - b2 + b1) * q^13 + (b13 - b12 + b11 + b10 + b8 + b7 - 2b6 + b5 + b4 - b3 - b1 + 1) q^14 + (b9 + b8 - b7 + b6 + b4 - b1 + 3) * q^16 + q^17 + (b13 - b10 - b8 + b7 - b6 - b4 - b3 + b1) * q^19 + (b13 - b10 - b6 + b5 - b4 - b3 + 1) * q^20 + (b13 - b9 - b6 - 2b5 - b4 + b2) q^22 + (-b13 - 2b11 - b10 + b9 - b8 + b7 + b6 + 2b4 + 2) * q^23 + (b13 - b10 + b9 + b8 - b6 + b5 + 2b4 + b2 - b1 + 3) q^25 + (-b13 + 2b10 - b9 - b8 + b5 + b4 + 1) q^26 + (b12 - b11 - b10 - b9 + b7 + 2b6 - b5 - 2b4 + b1 - 3) * q^28 + (2b12 - 2b10 - 2b8 - 2b5 + 2b3 + 2b2) * q^29 + (b13 - b11 - b10 + b9 + b8 - b7 + b6 - 2b5 + b1) q^31 + (-b13 + b12 - b11 - b10 + b9 - 2b8 + b6 - b5 + 2b4 + b3 + 2b1) q^32 + b4 * q^34 + (-2b12 + b11 + b10 - b9 + 2b8 + b7 - b6 + 2b5 - 2b3 - 2b2) q^35 + (b10 + b7 + b5 + b4 - b2 + b1) * q^37 + (-b12 + b11 + b10 - 3b9 + b8 + b7 - b6 + b5 + b3 - b2 - b1 - 1) q^38 + (-b13 - b12 + b10 - b9 + b6 - b5 + b3) * q^40 + (b13 - b9 + b8 + 2b6 - 2b5 - 5b4 + 2b2 + 2b1 - 4) q^41 + (b13 - b10 - b8 - b7 + b6 - b4 + b3 + b1) * q^43 + (2b13 - b12 + b11 - 2b10 + b8 - b7 - b6 - b1) * q^44 + (-b13 + 2b12 + b9 - b6 + 5b4 - b2 - 2b1 + 6) q^46 + (-b11 - b10 + b9 + b7 + b6 + 2b2 - 4) q^47 + (-2b13 - 2b8 + 2b7 + 2b5 + 2b2 + 1) q^49 + (-b13 - b9 - b8 + 2b6 - b5 - 2b2 + 2b1 + 1) q^50 + (-b13 + b10 + 2b9 + 2b8 + b6 + b5 + 3b4 - b3 - 2b1 - 1) * q^52 + (2b11 - 2b9 + b7 - b6 + 2b4 + 2b3) * q^53 + (2b13 + 2b11 - b9 + 2b8 - 3b6 + 3b5 + b4 + b2 - 2b1) * q^55 + (b12 + b11 + b10 - b9 - 3b7 - b5 + 2b3 - b1 - 1) * q^56 + (-2b12 + 2b11 - 2b9 - 4b6 + 2b5 - 2b3 - 2b2) q^58 + (-b10 + b7 + b5 - b4 + 2b3 - b2 + b1) q^59 + (-2b12 + 2b11 + 2b10 - 2b9 + 2b8 - b7 - b6 + 2b4 + 2b3 - 2b1) * q^61 + (2b13 - b12 + b11 - b10 - b9 - b7 + b5 + 2b3 - 2b2 + b1 + 3) q^62 + (b13 - 2b12 + 2b11 + 2b10 - 2b9 + 2b8 + b7 - 3b6 + 4b5 + 2b4 - b3 - 4b2 - b1 + 3) q^64 + (b13 + 2b11 - b9 + b8 - 4b6 + 4b5 + 3b4 - 2b1 + 6) q^65 + (-2b11 + 3b10 + 2b9 + b7 - 2b6 - b5 + b4 - 4b3 + b2 - b1) q^67 - b2 * q^68 + (b13 + 2b12 - 2b11 + b9 - b8 + 2b6 - 4b5 - b4 + b3 + 2b2 - 2) q^70 + (b13 - b11 - b9 + b8 + b6 - b5 - b4 - b2 + 2) * q^71 + (b10 - 2b9 + 2b7 + b6 + b5 - 3b4 + b2 - b1 - 2) q^73 + (2b10 - 2b9 + 2b8 + 2b6 + 2b5 - 2b2 - 2) * q^74 + (2b12 - 2b11 - b10 + 3b9 - b8 - 2b7 + b6 - 3b5 - 2b3 + 2b2 - 2b1 - 1) * q^76 + (-2b13 + 2b12 - b10 + 2b7 + b6 + b5 + b4 - 2b3 - b2 + b1) * q^77 + (b13 - b11 - b10 - 3b9 + b8 + b7 - b6 + 2b5 + b1) * q^79 + (b13 + 2b12 - 2b10 + 3b9 - b8 - 2b6 + 3b4 - b3 + 2b2 - 2b1 + 2) q^80 + (2b13 + 4b11 - 2b9 - 2b7 + b5 - 2b4 + 3b3 + 2b2 - 3) q^82 + (-2b13 + 2b12 - b11 + b9 - 2b7 + 5b6 + b5 + b4 - b2 + b1) * q^83 - b3 * q^85 + (-2b13 - b12 + b11 + b10 - b9 - b8 + b7 - b6 + 3b5 + 2b4 + b3 - b2 - b1 + 1) q^86 + (-b13 - b11 - 2b10 - 4b8 + b7 - b6 - 2b5 - 2b4 + b3 + b1 + 1) * q^88 + (2b13 + 2b11 + b10 - 2b9 + 2b8 - 2b7 - 3b6 + b5 + b4 - 3b2 - b1 + 2) q^89 + (2b12 - b11 - 5b10 + b9 - 2b8 - b7 + 3b6 - 4b4 + 2b3 + 2b1) q^91 + (-b12 - 3b11 + b8 - b7 + b6 - 2b5 + 2b4 - 2b3 - 4b2 + 3b1 - 2) * q^92 + (-b13 + 2b12 + 2b11 - b9 - b8 - 3b4 + b3 + 2) q^94 + (-b11 - b10 - 2b9 + b7 + b5 - b4 + b2 + 2b1 - 4) * q^95 + (-2b13 - 2b11 - b10 - 2b9 - 2b8 + 2b7 + b6 + b5 - b4 + b2 + 3b1 - 2) * q^97 + (-2b13 + 2b12 + 2b11 + 2b10 + 2b7 - 2b6 + 3b4 - 2b3 + 2b2 - 2b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$14 q - 4 q^{2} - 2 q^{4} + 12 q^{7} + 2 q^{8} - 8 q^{10} - 4 q^{14} + 26 q^{16} + 14 q^{17} + 16 q^{22} + 24 q^{23} + 6 q^{25} + 24 q^{26} - 16 q^{28} + 4 q^{31} + 6 q^{32} - 4 q^{34} - 4 q^{38} + 28 q^{40}+ \cdots - 12 q^{98}+O(q^{100})$ 14 * q - 4 * q^2 - 2 * q^4 + 12 * q^7 + 2 * q^8 - 8 * q^10 - 4 * q^14 + 26 * q^16 + 14 * q^17 + 16 * q^22 + 24 * q^23 + 6 * q^25 + 24 * q^26 - 16 * q^28 + 4 * q^31 + 6 * q^32 - 4 * q^34 - 4 * q^38 + 28 * q^40 - 4 * q^41 - 28 * q^44 + 40 * q^46 - 56 * q^47 + 22 * q^49 + 44 * q^50 - 44 * q^52 - 48 * q^55 - 16 * q^56 - 8 * q^58 + 28 * q^62 + 10 * q^64 + 24 * q^65 - 2 * q^68 - 8 * q^70 + 40 * q^71 - 4 * q^73 - 20 * q^74 - 32 * q^76 + 4 * q^79 - 36 * q^80 - 48 * q^82 + 4 * q^86 + 40 * q^88 - 4 * q^89 - 4 * q^92 + 40 * q^94 - 32 * q^95 + 20 * q^97 - 12 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{14} - 2 x^{13} + x^{12} + 2 x^{11} - 10 x^{10} + 8 x^{9} + 10 x^{8} - 20 x^{7} + 20 x^{6} + \cdots + 128$ x^14 - 2*x^13 + x^12 + 2*x^11 - 10*x^10 + 8*x^9 + 10*x^8 - 20*x^7 + 20*x^6 + 32*x^5 - 80*x^4 + 32*x^3 + 32*x^2 - 128*x + 128 :

$\beta_{1}$	$=$	$( -\nu^{12} + 3\nu^{10} + 2\nu^{8} + 8\nu^{7} - 6\nu^{6} - 16\nu^{5} + 4\nu^{4} - 16\nu^{3} - 8\nu^{2} + 32\nu + 48 ) / 16$ (-v^12 + 3v^10 + 2v^8 + 8v^7 - 6v^6 - 16v^5 + 4v^4 - 16v^3 - 8v^2 + 32*v + 48) / 16
$\beta_{2}$	$=$	$( - \nu^{13} + \nu^{12} + \nu^{11} - 3 \nu^{10} + 8 \nu^{9} + 2 \nu^{8} - 18 \nu^{7} + 10 \nu^{6} + \cdots + 96 ) / 32$ (-v^13 + v^12 + v^11 - 3v^10 + 8v^9 + 2v^8 - 18v^7 + 10v^6 - 52v^4 + 48v^3 + 48v^2 - 64*v + 96) / 32
$\beta_{3}$	$=$	$( - 2 \nu^{13} + \nu^{12} - 7 \nu^{10} + 10 \nu^{9} + 6 \nu^{8} - 20 \nu^{7} + 18 \nu^{6} + \cdots - 128 \nu ) / 32$ (-2v^13 + v^12 - 7v^10 + 10v^9 + 6v^8 - 20v^7 + 18v^6 + 28v^5 - 60v^4 + 16v^3 + 32v^2 - 128*v) / 32
$\beta_{4}$	$=$	$( \nu^{13} + \nu^{12} - 3 \nu^{11} + 5 \nu^{10} + 6 \nu^{9} - 14 \nu^{8} + 6 \nu^{7} + 18 \nu^{6} + \cdots + 160 ) / 32$ (v^13 + v^12 - 3v^11 + 5v^10 + 6v^9 - 14v^8 + 6v^7 + 18v^6 - 52v^5 - 4v^4 + 72v^3 - 64v^2 + 32*v + 160) / 32
$\beta_{5}$	$=$	$( - 5 \nu^{12} + 7 \nu^{10} - 12 \nu^{9} + 14 \nu^{8} + 36 \nu^{7} - 42 \nu^{6} - 16 \nu^{5} + 52 \nu^{4} + \cdots - 96 ) / 32$ (-5v^12 + 7v^10 - 12v^9 + 14v^8 + 36v^7 - 42v^6 - 16v^5 + 52v^4 - 120v^3 - 32v^2 + 160*v - 96) / 32
$\beta_{6}$	$=$	$( 3 \nu^{13} - 2 \nu^{12} + \nu^{11} + 10 \nu^{10} - 24 \nu^{9} - 8 \nu^{8} + 26 \nu^{7} - 44 \nu^{6} + \cdots - 224 ) / 32$ (3v^13 - 2v^12 + v^11 + 10v^10 - 24v^9 - 8v^8 + 26v^7 - 44v^6 - 8v^5 + 144v^4 - 72v^3 - 16v^2 + 224v - 224) / 32
$\beta_{7}$	$=$	$( 2 \nu^{13} + \nu^{11} + 4 \nu^{10} - 13 \nu^{9} - 8 \nu^{8} + 6 \nu^{7} - 20 \nu^{6} + 14 \nu^{5} + \cdots - 176 ) / 16$ (2v^13 + v^11 + 4v^10 - 13v^9 - 8v^8 + 6v^7 - 20v^6 + 14v^5 + 88v^4 - 20v^3 + 24v^2 + 80*v - 176) / 16
$\beta_{8}$	$=$	$( 5 \nu^{13} - 2 \nu^{12} + \nu^{11} + 6 \nu^{10} - 30 \nu^{9} - 12 \nu^{8} + 18 \nu^{7} - 36 \nu^{6} + \cdots - 384 ) / 32$ (5v^13 - 2v^12 + v^11 + 6v^10 - 30v^9 - 12v^8 + 18v^7 - 36v^6 + 44v^5 + 168v^4 - 80v^3 + 64v^2 + 64v - 384) / 32
$\beta_{9}$	$=$	$( - 3 \nu^{13} + 2 \nu^{11} - 4 \nu^{10} + 15 \nu^{9} + 16 \nu^{8} - 12 \nu^{7} + 4 \nu^{6} - 10 \nu^{5} + \cdots + 160 ) / 16$ (-3v^13 + 2v^11 - 4v^10 + 15v^9 + 16v^8 - 12v^7 + 4v^6 - 10v^5 - 88v^4 - 4v^3 + 24v^2 - 16v + 160) / 16
$\beta_{10}$	$=$	$( 2 \nu^{13} - 7 \nu^{12} + 6 \nu^{11} + 13 \nu^{10} - 36 \nu^{9} + 18 \nu^{8} + 40 \nu^{7} - 102 \nu^{6} + \cdots - 416 ) / 32$ (2v^13 - 7v^12 + 6v^11 + 13v^10 - 36v^9 + 18v^8 + 40v^7 - 102v^6 + 32v^5 + 188v^4 - 216v^3 + 48v^2 + 288*v - 416) / 32
$\beta_{11}$	$=$	$( - 6 \nu^{13} + 7 \nu^{12} + 8 \nu^{11} - 13 \nu^{10} + 38 \nu^{9} + 22 \nu^{8} - 76 \nu^{7} + \cdots + 352 ) / 32$ (-6v^13 + 7v^12 + 8v^11 - 13v^10 + 38v^9 + 22v^8 - 76v^7 + 14v^6 + 4v^5 - 220v^4 + 112v^3 + 192v^2 - 64*v + 352) / 32
$\beta_{12}$	$=$	$( 8 \nu^{13} - 9 \nu^{12} + 6 \nu^{11} + 27 \nu^{10} - 70 \nu^{9} - 2 \nu^{8} + 80 \nu^{7} - 162 \nu^{6} + \cdots - 768 ) / 32$ (8v^13 - 9v^12 + 6v^11 + 27v^10 - 70v^9 - 2v^8 + 80v^7 - 162v^6 + 28v^5 + 404v^4 - 304v^3 + 64v^2 + 512*v - 768) / 32
$\beta_{13}$	$=$	$( 4 \nu^{13} - 5 \nu^{12} + 13 \nu^{10} - 36 \nu^{9} - 4 \nu^{8} + 48 \nu^{7} - 62 \nu^{6} + 8 \nu^{5} + \cdots - 400 ) / 16$ (4v^13 - 5v^12 + 13v^10 - 36v^9 - 4v^8 + 48v^7 - 62v^6 + 8v^5 + 208v^4 - 144v^3 - 40v^2 + 224v - 400) / 16

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

$\nu$	$=$	$( -\beta_{13} - \beta_{12} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 4$ (-b13 - b12 + b10 + b8 + b7 + b6 + 2*b5 + b4 - b2 - b1 + 1) / 4
$\nu^{2}$	$=$	$( -\beta_{10} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 ) / 2$ (-b10 + b7 + b5 - b4 + b2 + b1) / 2
$\nu^{3}$	$=$	$( \beta_{13} - \beta_{12} + \beta_{10} - \beta_{8} + 3 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 1 ) / 4$ (b13 - b12 + b10 - b8 + 3b7 - 5b6 + 2b5 + b4 - 4b3 + 3b2 - 3b1 + 1) / 4
$\nu^{4}$	$=$	$( 2\beta_{13} - \beta_{10} + 2\beta_{9} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta _1 + 6 ) / 2$ (2b13 - b10 + 2b9 + b7 - 2*b6 - b5 + b4 + b2 - b1 + 6) / 2
$\nu^{5}$	$=$	$( - 5 \beta_{13} - 3 \beta_{12} + 7 \beta_{10} - 4 \beta_{9} + \beta_{8} + 5 \beta_{7} - 3 \beta_{6} + \cdots + 7 ) / 4$ (-5b13 - 3b12 + 7b10 - 4b9 + b8 + 5b7 - 3b6 + 6b5 + 3b4 - 7*b2 - b1 + 7) / 4
$\nu^{6}$	$=$	$( - 4 \beta_{12} + \beta_{10} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots - 6 ) / 2$ (-4b12 + b10 + 2b8 + 3b7 + 2b6 + b5 - 3b4 - 2b3 - b2 + 5*b1 - 6) / 2
$\nu^{7}$	$=$	$( - 3 \beta_{13} + 3 \beta_{12} - 4 \beta_{11} - 7 \beta_{10} - 9 \beta_{8} + 15 \beta_{7} - 13 \beta_{6} + \cdots - 15 ) / 4$ (-3b13 + 3b12 - 4b11 - 7b10 - 9b8 + 15b7 - 13b6 + 6b5 - 15b4 - 16b3 + 3b2 + 5b1 - 15) / 4
$\nu^{8}$	$=$	$( 12 \beta_{13} - 8 \beta_{12} + 8 \beta_{11} + 3 \beta_{10} + 6 \beta_{8} + \beta_{7} - 18 \beta_{6} + \cdots + 14 ) / 2$ (12b13 - 8b12 + 8b11 + 3b10 + 6b8 + b7 - 18b6 + 3b5 + 7b4 - 10b3 - 3b2 - 5*b1 + 14) / 2
$\nu^{9}$	$=$	$( - 5 \beta_{13} + 5 \beta_{12} - 4 \beta_{11} + 7 \beta_{10} + 16 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + \cdots + 15 ) / 4$ (-5b13 + 5b12 - 4b11 + 7b10 + 16b9 + 9b8 + 9b7 - 19b6 + 2b5 + 31b4 + 8b3 - 19b2 - 21*b1 + 15) / 4
$\nu^{10}$	$=$	$( - 4 \beta_{13} - 12 \beta_{12} + 8 \beta_{11} + 17 \beta_{10} - 20 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + \cdots - 26 ) / 2$ (-4b13 - 12b12 + 8b11 + 17b10 - 20b9 + 2b8 + 3b7 - 2b6 + 5b5 + b4 + 6b3 - 17b2 + 25b1 - 26) / 2
$\nu^{11}$	$=$	$( - 27 \beta_{13} + 11 \beta_{12} - 36 \beta_{11} + 9 \beta_{10} + 24 \beta_{9} - 25 \beta_{8} + \cdots - 159 ) / 4$ (-27b13 + 11b12 - 36b11 + 9b10 + 24b9 - 25b8 + 15b7 + 19b6 - 58b5 - 55b4 - 56b3 + 11b2 + 29*b1 - 159) / 4
$\nu^{12}$	$=$	$( 24 \beta_{13} + 24 \beta_{11} - 21 \beta_{10} - 20 \beta_{9} - 14 \beta_{8} + \beta_{7} - 34 \beta_{6} + \cdots - 2 ) / 2$ (24b13 + 24b11 - 21b10 - 20b9 - 14b8 + b7 - 34b6 - 5b5 - 29b4 - 22b3 - 27b2 + 27*b1 - 2) / 2
$\nu^{13}$	$=$	$( 11 \beta_{13} - 27 \beta_{12} + 36 \beta_{11} + 71 \beta_{10} + 24 \beta_{9} + 121 \beta_{8} - 63 \beta_{7} + \cdots + 63 ) / 4$ (11b13 - 27b12 + 36b11 + 71b10 + 24b9 + 121b8 - 63b7 - 83b6 + 10b5 + 151b4 - 56b3 - 107b2 - 125*b1 + 63) / 4

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}

613.1

0.105804	+	1.41025i
0.105804	−	1.41025i
0.548965	+	1.30332i
0.548965	−	1.30332i
−1.17656	−	0.784673i
−1.17656	+	0.784673i
1.37100	+	0.346935i
1.37100	−	0.346935i
−1.40561	+	0.155777i
−1.40561	−	0.155777i
1.38507	−	0.285600i
1.38507	+	0.285600i
0.171324	−	1.40380i
0.171324	+	1.40380i

−1.41025

−

0.105804i

1.97761

0.298419i

−

2.49626i

−1.77008

−2.75735

−

0.630084i

−0.264113

3.52034i

613.2

−1.41025

0.105804i

1.97761

−

0.298419i

2.49626i

−1.77008

−2.75735

0.630084i

−0.264113

−

3.52034i

613.3

−1.30332

−

0.548965i

1.39727

1.43095i

3.97294i

3.04833

−1.03555

−

2.63204i

2.18101

−

5.17800i

613.4

−1.30332

0.548965i

1.39727

−

1.43095i

−

3.97294i

3.04833

−1.03555

2.63204i

2.18101

5.17800i

613.5

−0.784673

−

1.17656i

−0.768577

1.84643i

−

2.31278i

2.94745

2.77551

−

0.544565i

−2.72112

1.81478i

613.6

−0.784673

1.17656i

−0.768577

−

1.84643i

2.31278i

2.94745

2.77551

0.544565i

−2.72112

−

1.81478i

613.7

−0.346935

−

1.37100i

−1.75927

0.951293i

−

1.24794i

−3.59705

1.91457

2.08192i

−1.71093

0.432954i

613.8

−0.346935

1.37100i

−1.75927

−

0.951293i

1.24794i

−3.59705

1.91457

−

2.08192i

−1.71093

−

0.432954i

613.9

0.155777

−

1.40561i

−1.95147

−

0.437924i

0.191498i

1.22930

−0.919543

2.67478i

0.269171

0.0298310i

613.10

0.155777

1.40561i

−1.95147

0.437924i

−

0.191498i

1.22930

−0.919543

−

2.67478i

0.269171

−

0.0298310i

613.11

0.285600

−

1.38507i

−1.83687

−

0.791155i

−

1.39571i

4.88695

−1.62042

2.31824i

−1.93317

−

0.398616i

613.12

0.285600

1.38507i

−1.83687

0.791155i

1.39571i

4.88695

−1.62042

−

2.31824i

−1.93317

0.398616i

613.13

1.40380

−

0.171324i

1.94130

−

0.481007i

1.04568i

−0.744897

2.64278

−

1.00783i

0.179150

1.46793i

613.14

1.40380

0.171324i

1.94130

0.481007i

−

1.04568i

−0.744897

2.64278

1.00783i

0.179150

−

1.46793i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	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.f.f		14
3.b	odd	2	1		408.2.f.d	✓	14
4.b	odd	2	1		4896.2.f.f		14
8.b	even	2	1	inner	1224.2.f.f		14
8.d	odd	2	1		4896.2.f.f		14
12.b	even	2	1		1632.2.f.d		14
24.f	even	2	1		1632.2.f.d		14
24.h	odd	2	1		408.2.f.d	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
408.2.f.d	✓	14	3.b	odd	2	1
408.2.f.d	✓	14	24.h	odd	2	1
1224.2.f.f		14	1.a	even	1	1	trivial
1224.2.f.f		14	8.b	even	2	1	inner
1632.2.f.d		14	12.b	even	2	1
1632.2.f.d		14	24.f	even	2	1
4896.2.f.f		14	4.b	odd	2	1
4896.2.f.f		14	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1224, [\chi])$ :

T_{5}^{14} + 32T_{5}^{12} + 350T_{5}^{10} + 1724T_{5}^{8} + 4057T_{5}^{6} + 4476T_{5}^{4} + 1904T_{5}^{2} + 64

T_{23}^{7} - 12T_{23}^{6} - 42T_{23}^{5} + 720T_{23}^{4} + 373T_{23}^{3} - 12444T_{23}^{2} - 1088T_{23} + 62848

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{14} + 4 T^{13} + \cdots + 128$ T^14 + 4T^13 + 9T^12 + 14T^11 + 10T^10 - 12T^9 - 54T^8 - 100T^7 - 108T^6 - 48T^5 + 80T^4 + 224T^3 + 288T^2 + 256*T + 128
$3$	$T^{14}$ T^14
$5$	$T^{14} + 32 T^{12} + \cdots + 64$ T^14 + 32T^12 + 350T^10 + 1724T^8 + 4057T^6 + 4476T^4 + 1904T^2 + 64
$7$	$(T^{7} - 6 T^{6} + \cdots + 256)^{2}$ (T^7 - 6T^6 - 12T^5 + 104T^4 - 24T^3 - 368T^2 + 128T + 256)^2
$11$	$T^{14} + 92 T^{12} + \cdots + 262144$ T^14 + 92T^12 + 3166T^10 + 51180T^8 + 409129T^6 + 1548288T^4 + 2212864T^2 + 262144
$13$	$T^{14} + 108 T^{12} + \cdots + 11075584$ T^14 + 108T^12 + 4326T^10 + 84476T^8 + 871329T^6 + 4700432T^4 + 11995136T^2 + 11075584
$17$	$(T - 1)^{14}$ (T - 1)^14
$19$	$T^{14} + \cdots + 125798656$ T^14 + 124T^12 + 6214T^10 + 162692T^8 + 2393521T^6 + 19607176T^4 + 81226128T^2 + 125798656
$23$	$(T^{7} - 12 T^{6} + \cdots + 62848)^{2}$ (T^7 - 12T^6 - 42T^5 + 720T^4 + 373T^3 - 12444T^2 - 1088T + 62848)^2
$29$	$T^{14} + \cdots + 177209344$ T^14 + 284T^12 + 30576T^10 + 1517120T^8 + 32507904T^6 + 196313088T^4 + 357564416T^2 + 177209344
$31$	$(T^{7} - 2 T^{6} + \cdots - 1024)^{2}$ (T^7 - 2T^6 - 88T^5 - 112T^4 + 1488T^3 + 4192T^2 + 1792T - 1024)^2
$37$	$T^{14} + \cdots + 268435456$ T^14 + 216T^12 + 16912T^10 + 606720T^8 + 10887168T^6 + 96665600T^4 + 363069440T^2 + 268435456
$41$	$(T^{7} + 2 T^{6} + \cdots - 170584)^{2}$ (T^7 + 2T^6 - 210T^5 - 88T^4 + 11901T^3 - 2274T^2 - 161412T - 170584)^2
$43$	$T^{14} + 172 T^{12} + \cdots + 256$ T^14 + 172T^12 + 9494T^10 + 180180T^8 + 1044641T^6 + 1016776T^4 + 196496T^2 + 256
$47$	$(T^{7} + 28 T^{6} + \cdots + 26624)^{2}$ (T^7 + 28T^6 + 252T^5 + 512T^4 - 3384T^3 - 12896T^2 + 4096T + 26624)^2
$53$	$T^{14} + \cdots + 48020586496$ T^14 + 300T^12 + 36800T^10 + 2367424T^8 + 84963648T^6 + 1660066048T^4 + 15590752256T^2 + 48020586496
$59$	$T^{14} + 248 T^{12} + \cdots + 11075584$ T^14 + 248T^12 + 18672T^10 + 588288T^8 + 8019712T^6 + 44226560T^4 + 85528576T^2 + 11075584
$61$	$T^{14} + \cdots + 3743481856$ T^14 + 496T^12 + 88240T^10 + 6973248T^8 + 261885504T^6 + 4605458944T^4 + 30748943360T^2 + 3743481856
$67$	$T^{14} + \cdots + 84727164829696$ T^14 + 744T^12 + 233232T^10 + 39784192T^8 + 3966134272T^6 + 229099208704T^4 + 6996845199360T^2 + 84727164829696
$71$	$(T^{7} - 20 T^{6} + \cdots + 26624)^{2}$ (T^7 - 20T^6 + 72T^5 + 992T^4 - 9968T^3 + 34496T^2 - 51200T + 26624)^2
$73$	$(T^{7} + 2 T^{6} + \cdots - 8768)^{2}$ (T^7 + 2T^6 - 224T^5 - 848T^4 + 7288T^3 + 16432T^2 - 17184T - 8768)^2
$79$	$(T^{7} - 2 T^{6} + \cdots + 730624)^{2}$ (T^7 - 2T^6 - 392T^5 - 176T^4 + 42640T^3 + 41696T^2 - 1221888T + 730624)^2
$83$	$T^{14} + \cdots + 6964903936$ T^14 + 624T^12 + 134640T^10 + 11575616T^8 + 338243648T^6 + 3564194304T^4 + 9332507648T^2 + 6964903936
$89$	$(T^{7} + 2 T^{6} + \cdots + 1280704)^{2}$ (T^7 + 2T^6 - 376T^5 - 336T^4 + 39640T^3 - 3344T^2 - 1176352T + 1280704)^2
$97$	$(T^{7} - 10 T^{6} + \cdots + 11072)^{2}$ (T^7 - 10T^6 - 216T^5 + 3312T^4 - 12360T^3 - 1456T^2 + 48864T + 11072)^2
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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 613.14
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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