Properties

Label 1449.2.a.m
Level $1449$
Weight $2$
Character orbit 1449.a
Self dual yes
Analytic conductor $11.570$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1449,2,Mod(1,1449)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1449.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1449.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: \(11.5703232529\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (2 \beta_1 + 1) q^{4} + (\beta_1 - 1) q^{5} - q^{7} + (\beta_{2} + 3 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (2 \beta_1 + 1) q^{4} + (\beta_1 - 1) q^{5} - q^{7} + (\beta_{2} + 3 \beta_1 + 2) q^{8} + (\beta_1 + 1) q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11} + 2 \beta_1 q^{13} + ( - \beta_{2} - \beta_1) q^{14} + (4 \beta_{2} + 4 \beta_1 + 3) q^{16} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + (2 \beta_{2} + \beta_1 + 3) q^{20} + ( - \beta_{2} - 3 \beta_1 - 3) q^{22} - q^{23} + (\beta_{2} - \beta_1 - 2) q^{25} + (2 \beta_{2} + 4 \beta_1 + 2) q^{26} + ( - 2 \beta_1 - 1) q^{28} + (\beta_{2} - \beta_1 + 1) q^{29} + (4 \beta_{2} + \beta_1 + 5) q^{31} + (\beta_{2} + 5 \beta_1 + 8) q^{32} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{34} + ( - \beta_1 + 1) q^{35} + ( - 2 \beta_{2} - 2) q^{37} + (6 \beta_{2} + 6 \beta_1 - 2) q^{38} + (2 \beta_{2} + 3 \beta_1 + 3) q^{40} + ( - 4 \beta_1 + 6) q^{41} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{43} + ( - 3 \beta_{2} - 7 \beta_1 - 3) q^{44} + ( - \beta_{2} - \beta_1) q^{46} + (2 \beta_{2} - \beta_1 - 5) q^{47} + q^{49} + ( - 4 \beta_{2} - 4 \beta_1 + 1) q^{50} + (4 \beta_{2} + 6 \beta_1 + 8) q^{52} + ( - 4 \beta_{2} - 2) q^{53} - 2 \beta_1 q^{55} + ( - \beta_{2} - 3 \beta_1 - 2) q^{56} + ( - \beta_{2} - \beta_1 + 1) q^{58} + (6 \beta_{2} + \beta_1 + 3) q^{59} + ( - 2 \beta_{2} - 5 \beta_1 + 5) q^{61} + (2 \beta_{2} + 7 \beta_1 + 9) q^{62} + (4 \beta_{2} + 10 \beta_1 + 1) q^{64} + (2 \beta_{2} + 4) q^{65} + (5 \beta_{2} + \beta_1 + 5) q^{67} + ( - 2 \beta_{2} - 5 \beta_1 - 5) q^{68} + ( - \beta_1 - 1) q^{70} + ( - 4 \beta_{2} - 4 \beta_1) q^{71} - 6 \beta_1 q^{73} + ( - 2 \beta_1 - 4) q^{74} + (2 \beta_{2} + 6 \beta_1 + 14) q^{76} + (\beta_{2} + \beta_1 + 1) q^{77} + ( - \beta_{2} - 5 \beta_1 + 7) q^{79} + (7 \beta_1 + 1) q^{80} + (2 \beta_{2} - 2 \beta_1 - 4) q^{82} + ( - 8 \beta_{2} - 2 \beta_1 - 6) q^{83} + ( - \beta_{2} - \beta_1 - 1) q^{85} + ( - 6 \beta_1 - 10) q^{86} + ( - 5 \beta_{2} - 11 \beta_1 - 7) q^{88} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{89} - 2 \beta_1 q^{91} + ( - 2 \beta_1 - 1) q^{92} + ( - 8 \beta_{2} - 7 \beta_1 + 3) q^{94} + (4 \beta_{2} + 4) q^{95} + (2 \beta_{2} + 3 \beta_1 + 1) q^{97} + (\beta_{2} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{7} + 9 q^{8} + 4 q^{10} - 4 q^{11} + 2 q^{13} - q^{14} + 13 q^{16} - 4 q^{17} + 8 q^{19} + 10 q^{20} - 12 q^{22} - 3 q^{23} - 7 q^{25} + 10 q^{26} - 5 q^{28} + 2 q^{29} + 16 q^{31} + 29 q^{32} - 6 q^{34} + 2 q^{35} - 6 q^{37} + 12 q^{40} + 14 q^{41} - 8 q^{43} - 16 q^{44} - q^{46} - 16 q^{47} + 3 q^{49} - q^{50} + 30 q^{52} - 6 q^{53} - 2 q^{55} - 9 q^{56} + 2 q^{58} + 10 q^{59} + 10 q^{61} + 34 q^{62} + 13 q^{64} + 12 q^{65} + 16 q^{67} - 20 q^{68} - 4 q^{70} - 4 q^{71} - 6 q^{73} - 14 q^{74} + 48 q^{76} + 4 q^{77} + 16 q^{79} + 10 q^{80} - 14 q^{82} - 20 q^{83} - 4 q^{85} - 36 q^{86} - 32 q^{88} - 4 q^{89} - 2 q^{91} - 5 q^{92} + 2 q^{94} + 12 q^{95} + 6 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

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
0.311108
−1.48119
2.17009
−1.90321 0 1.62222 −0.688892 0 −1.00000 0.719004 0 1.31111
1.2 0.193937 0 −1.96239 −2.48119 0 −1.00000 −0.768452 0 −0.481194
1.3 2.70928 0 5.34017 1.17009 0 −1.00000 9.04945 0 3.17009
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(1\)

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 1449.2.a.m 3
3.b odd 2 1 161.2.a.c 3
12.b even 2 1 2576.2.a.v 3
15.d odd 2 1 4025.2.a.j 3
21.c even 2 1 1127.2.a.f 3
69.c even 2 1 3703.2.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.c 3 3.b odd 2 1
1127.2.a.f 3 21.c even 2 1
1449.2.a.m 3 1.a even 1 1 trivial
2576.2.a.v 3 12.b even 2 1
3703.2.a.c 3 69.c even 2 1
4025.2.a.j 3 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1449))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 2T_{5}^{2} - 2T_{5} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} - 2 T - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 4T^{2} - 4 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 8 T - 4 \) Copy content Toggle raw display
$31$ \( T^{3} - 16 T^{2} + 26 T + 338 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 4 T - 40 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 12 T + 152 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 40 T - 304 \) Copy content Toggle raw display
$47$ \( T^{3} + 16 T^{2} + 62 T + 10 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 52 T - 248 \) Copy content Toggle raw display
$59$ \( T^{3} - 10 T^{2} - 102 T + 970 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} - 46 T + 494 \) Copy content Toggle raw display
$67$ \( T^{3} - 16 T^{2} - 8 T + 676 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 80 T - 64 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 108 T - 216 \) Copy content Toggle raw display
$79$ \( T^{3} - 16 T^{2} + 8 T + 428 \) Copy content Toggle raw display
$83$ \( T^{3} + 20 T^{2} - 104 T - 2672 \) Copy content Toggle raw display
$89$ \( T^{3} + 4 T^{2} - 114 T + 278 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} - 22 T - 2 \) Copy content Toggle raw display
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