Properties

Label 1449.2.a.r
Level $1449$
Weight $2$
Character orbit 1449.a
Self dual yes
Analytic conductor $11.570$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(5\)
Coefficient field: 5.5.2147108.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{4} + 1) q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{4} + 1) q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{10} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + (\beta_{4} + \beta_{3} - 1) q^{13} - \beta_1 q^{14} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + 1) q^{16} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{17} + ( - 2 \beta_1 + 2) q^{19} + (3 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 1) q^{20} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{22} + q^{23} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{25} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{26} + (\beta_{2} + 2) q^{28} + (3 \beta_{2} - \beta_1) q^{29} + ( - \beta_{3} + 6) q^{31} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{32} + (3 \beta_{4} + 4 \beta_{2} - 2 \beta_1 + 1) q^{34} + (\beta_{4} + 1) q^{35} + (2 \beta_{3} + 2 \beta_1) q^{37} + (2 \beta_{2} - 2 \beta_1 + 8) q^{38} + ( - 3 \beta_{4} - 6 \beta_{2} + 2 \beta_1 - 9) q^{40} + (\beta_{4} - \beta_{3} - 1) q^{41} + (2 \beta_{4} - 2) q^{43} + (3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 5) q^{44} - \beta_1 q^{46} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{47} + q^{49} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 3) q^{50} + (3 \beta_{4} + \beta_{3} + \beta_{2} - 4) q^{52} + (2 \beta_{2} - 4) q^{53} + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{55} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{56} + ( - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 4) q^{58} + (\beta_{4} + 2 \beta_1 - 5) q^{59} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 3) q^{61} + (\beta_{4} - \beta_{2} - 6 \beta_1) q^{62} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 5) q^{64} + ( - 2 \beta_{4} + 2 \beta_1 + 4) q^{65} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_1 - 3) q^{67} + ( - 5 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{68} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{70} + ( - \beta_{4} - 3 \beta_{3} - 1) q^{71} + ( - \beta_{4} - \beta_{3} - 4 \beta_{2} + 1) q^{73} + ( - 2 \beta_{4} - 8) q^{74} + ( - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_1 + 4) q^{76} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 7) q^{79} + (3 \beta_{4} + 6 \beta_{2} + 8 \beta_1 - 3) q^{80} + ( - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{82} + ( - 2 \beta_{4} - 2) q^{83} + (\beta_{4} + \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 1) q^{85} + ( - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{86} + ( - 5 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 3) q^{88} + (\beta_{4} + 2 \beta_{3} - 2 \beta_1 + 5) q^{89} + (\beta_{4} + \beta_{3} - 1) q^{91} + (\beta_{2} + 2) q^{92} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 8) q^{94} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{95} + ( - 3 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 3) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + \nu^{3} - 10\nu^{2} - 5\nu + 19 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 9\beta_{2} + 21 \) 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
2.69017
2.11948
1.23828
−1.50216
−2.54577
−2.69017 0 5.23702 3.51109 0 1.00000 −8.70812 0 −9.44544
1.2 −2.11948 0 2.49221 −2.40920 0 1.00000 −1.04322 0 5.10626
1.3 −1.23828 0 −0.466664 1.86253 0 1.00000 3.05442 0 −2.30633
1.4 1.50216 0 0.256481 3.82405 0 1.00000 −2.61904 0 5.74433
1.5 2.54577 0 4.48096 −2.78847 0 1.00000 6.31597 0 −7.09882
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.r 5
3.b odd 2 1 161.2.a.d 5
12.b even 2 1 2576.2.a.bd 5
15.d odd 2 1 4025.2.a.p 5
21.c even 2 1 1127.2.a.h 5
69.c even 2 1 3703.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.d 5 3.b odd 2 1
1127.2.a.h 5 21.c even 2 1
1449.2.a.r 5 1.a even 1 1 trivial
2576.2.a.bd 5 12.b even 2 1
3703.2.a.j 5 69.c even 2 1
4025.2.a.p 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1449))\):

\( T_{2}^{5} + 2T_{2}^{4} - 9T_{2}^{3} - 17T_{2}^{2} + 16T_{2} + 27 \) Copy content Toggle raw display
\( T_{5}^{5} - 4T_{5}^{4} - 14T_{5}^{3} + 54T_{5}^{2} + 52T_{5} - 168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} - 9 T^{3} - 17 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 4 T^{4} - 14 T^{3} + 54 T^{2} + \cdots - 168 \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} - 28 T^{3} + 148 T^{2} + \cdots + 48 \) Copy content Toggle raw display
$13$ \( T^{5} + 6 T^{4} - 9 T^{3} - 46 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$17$ \( T^{5} - 12 T^{4} + 6 T^{3} + \cdots + 1536 \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} - 28 T^{3} + 96 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 4 T^{4} - 111 T^{3} + \cdots + 1452 \) Copy content Toggle raw display
$31$ \( T^{5} - 30 T^{4} + 347 T^{3} + \cdots - 5206 \) Copy content Toggle raw display
$37$ \( T^{5} - 4 T^{4} - 76 T^{3} + 376 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$41$ \( T^{5} + 6 T^{4} - 29 T^{3} - 146 T^{2} + \cdots + 456 \) Copy content Toggle raw display
$43$ \( T^{5} + 12 T^{4} - 24 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{5} + 10 T^{4} - 125 T^{3} + \cdots - 11142 \) Copy content Toggle raw display
$53$ \( T^{5} + 16 T^{4} + 52 T^{3} + \cdots + 480 \) Copy content Toggle raw display
$59$ \( T^{5} + 22 T^{4} + 118 T^{3} + \cdots - 1440 \) Copy content Toggle raw display
$61$ \( T^{5} + 18 T^{4} + 34 T^{3} - 438 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$67$ \( T^{5} + 2 T^{4} - 300 T^{3} + \cdots + 61936 \) Copy content Toggle raw display
$71$ \( T^{5} + 4 T^{4} - 101 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$73$ \( T^{5} + 2 T^{4} - 197 T^{3} + \cdots - 27656 \) Copy content Toggle raw display
$79$ \( T^{5} - 30 T^{4} + 308 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$83$ \( T^{5} + 8 T^{4} - 56 T^{3} + \cdots + 5376 \) Copy content Toggle raw display
$89$ \( T^{5} - 20 T^{4} + 66 T^{3} + \cdots + 4704 \) Copy content Toggle raw display
$97$ \( T^{5} + 12 T^{4} - 114 T^{3} + \cdots + 4120 \) Copy content Toggle raw display
show more
show less