Properties

Label 2169.2.a.d
Level $2169$
Weight $2$
Character orbit 2169.a
Self dual yes
Analytic conductor $17.320$
Analytic rank $1$
Dimension $5$
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] = [2169,2,Mod(1,2169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2169.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.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: \(17.3195521984\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.24217.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 723)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{2} - \beta_{4} q^{4} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_{4} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 2 \beta_{2} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{2} - \beta_{4} q^{4} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_{4} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 2 \beta_{2} - \beta_1 + 1) q^{8} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{10} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} - 4) q^{13} + (4 \beta_{4} - \beta_{2} - 3 \beta_1) q^{14} + (\beta_{3} + \beta_1 - 2) q^{16} + (\beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{4} - \beta_{3} - 3 \beta_1 - 3) q^{19} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{20} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{22} + ( - \beta_{4} - 2 \beta_{2} - 3 \beta_1 - 2) q^{23} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{25} + (2 \beta_{4} + \beta_{3} - 5 \beta_{2} - 4 \beta_1) q^{26} + (3 \beta_{4} - 2 \beta_{3} - \beta_1 - 3) q^{28} + (4 \beta_{3} - 1) q^{29} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{31} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 1) q^{32} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 1) q^{34} + (4 \beta_{2} + 5 \beta_1 - 1) q^{35} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{37} + ( - 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 4 \beta_1) q^{38} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{40} + ( - \beta_{4} - 4 \beta_{2} + \beta_1 + 2) q^{41} + ( - 3 \beta_{4} - 4 \beta_{3} - \beta_{2} + 5 \beta_1 + 1) q^{43} + (3 \beta_{4} - \beta_{3} + 3 \beta_{2} - 2) q^{44} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{46} + ( - 2 \beta_{4} + 3 \beta_{2} + 2 \beta_1 + 3) q^{47} + ( - 3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{49} + (2 \beta_{4} + \beta_{2} + 3 \beta_1 - 2) q^{50} + (2 \beta_{4} + 2 \beta_{3} + \beta_1 + 2) q^{52} + ( - 2 \beta_{4} + 4 \beta_{3} + \beta_{2} + 6 \beta_1 + 1) q^{53} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{55} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{56} + (\beta_{4} + 4 \beta_{3} - 5 \beta_{2} - \beta_1 + 4) q^{58} + (4 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{59} + (2 \beta_{4} + 5 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{61} + (6 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} - 9 \beta_1 - 3) q^{62} + (5 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{64} + ( - 6 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 9) q^{65} + (\beta_{4} - 3 \beta_{3} + 5 \beta_{2} + \beta_1 - 3) q^{67} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{68} + (6 \beta_{4} + \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 4) q^{70} + ( - 2 \beta_{4} + 5 \beta_{2} + 5 \beta_1) q^{71} + (7 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 2) q^{73} + (4 \beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{74} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 5) q^{76} + ( - 4 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 2) q^{77} + (2 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 3) q^{79} + ( - 3 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 7 \beta_1 - 3) q^{80} + ( - 3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{82} + ( - 6 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 1) q^{83} + (4 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 9 \beta_1) q^{85} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 7 \beta_1 - 2) q^{86} + (4 \beta_{4} - 2 \beta_{2} + 4 \beta_1 + 3) q^{88} + (3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 1) q^{89} + ( - \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{91} + (3 \beta_{3} + \beta_1 + 5) q^{92} + ( - 5 \beta_{4} - \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 5) q^{94} + ( - \beta_{4} - \beta_{3} - 6 \beta_{2} - 11 \beta_1 - 8) q^{95} + (2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 7 \beta_1 - 4) q^{97} + ( - 8 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{10} - 2 q^{11} - 20 q^{13} - 4 q^{14} - 12 q^{16} + 4 q^{17} - 7 q^{19} - 3 q^{20} - 4 q^{22} - 6 q^{23} + 3 q^{25} - 6 q^{26} - 19 q^{28} - 5 q^{29} - 31 q^{31} - 7 q^{32} - 5 q^{34} - 7 q^{35} - 17 q^{37} - 2 q^{38} + 10 q^{40} + 2 q^{41} - q^{43} - 10 q^{44} - 7 q^{46} + 21 q^{47} + 6 q^{49} - 18 q^{50} + 4 q^{52} - q^{53} - 25 q^{55} - 25 q^{56} + 10 q^{58} - 3 q^{59} - q^{61} - 23 q^{62} + 7 q^{64} - 35 q^{65} - 9 q^{67} - 2 q^{68} + 18 q^{70} + 4 q^{71} - 6 q^{73} - 24 q^{74} + 21 q^{76} + 18 q^{77} - 17 q^{79} - 11 q^{80} - 9 q^{82} + q^{83} + 6 q^{86} - 5 q^{88} + 3 q^{89} + 6 q^{91} + 23 q^{92} + 35 q^{94} - 28 q^{95} - 30 q^{97} + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 5\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 5\nu^{2} + \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} - \nu^{3} - 9\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} + 5\beta_{3} - 4\beta _1 + 8 \) 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
−1.96003
−0.369680
2.17442
0.878095
−0.722813
−1.55183 0 0.408192 3.35190 0 −3.36822 2.47022 0 −5.20160
1.2 −1.11854 0 −0.748862 1.84170 0 −0.620818 3.07472 0 −2.06002
1.3 0.774909 0 −1.39952 3.26823 0 2.57394 −2.63432 0 2.53258
1.4 1.67128 0 0.793189 −1.36778 0 −0.915094 −2.01692 0 −2.28595
1.5 2.22418 0 2.94700 0.905942 0 −4.66981 2.10630 0 2.01498
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(241\) \(-1\)

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 2169.2.a.d 5
3.b odd 2 1 723.2.a.d 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.2.a.d 5 3.b odd 2 1
2169.2.a.d 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 2T_{2}^{4} - 4T_{2}^{3} + 7T_{2}^{2} + 4T_{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} - 4 T^{3} + 7 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 8 T^{4} + 18 T^{3} + T^{2} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{5} + 7 T^{4} + 4 T^{3} - 45 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$11$ \( T^{5} + 2 T^{4} - 43 T^{3} + \cdots + 1241 \) Copy content Toggle raw display
$13$ \( T^{5} + 20 T^{4} + 149 T^{3} + \cdots + 305 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} - 31 T^{3} + 64 T^{2} + \cdots + 157 \) Copy content Toggle raw display
$19$ \( T^{5} + 7 T^{4} - 51 T^{3} - 222 T^{2} + \cdots + 149 \) Copy content Toggle raw display
$23$ \( T^{5} + 6 T^{4} - 40 T^{3} - 139 T^{2} + \cdots + 629 \) Copy content Toggle raw display
$29$ \( T^{5} + 5 T^{4} - 70 T^{3} + \cdots + 2161 \) Copy content Toggle raw display
$31$ \( T^{5} + 31 T^{4} + 348 T^{3} + \cdots - 3923 \) Copy content Toggle raw display
$37$ \( T^{5} + 17 T^{4} + 83 T^{3} + \cdots - 485 \) Copy content Toggle raw display
$41$ \( T^{5} - 2 T^{4} - 98 T^{3} + \cdots + 3673 \) Copy content Toggle raw display
$43$ \( T^{5} + T^{4} - 174 T^{3} - 239 T^{2} + \cdots - 7603 \) Copy content Toggle raw display
$47$ \( T^{5} - 21 T^{4} + 141 T^{3} + \cdots + 131 \) Copy content Toggle raw display
$53$ \( T^{5} + T^{4} - 247 T^{3} - 21 T^{2} + \cdots - 9719 \) Copy content Toggle raw display
$59$ \( T^{5} + 3 T^{4} - 66 T^{3} + \cdots - 1285 \) Copy content Toggle raw display
$61$ \( T^{5} + T^{4} - 157 T^{3} - 119 T^{2} + \cdots + 2069 \) Copy content Toggle raw display
$67$ \( T^{5} + 9 T^{4} - 104 T^{3} + \cdots + 557 \) Copy content Toggle raw display
$71$ \( T^{5} - 4 T^{4} - 91 T^{3} + \cdots - 3307 \) Copy content Toggle raw display
$73$ \( T^{5} + 6 T^{4} - 247 T^{3} + \cdots - 41549 \) Copy content Toggle raw display
$79$ \( T^{5} + 17 T^{4} - 43 T^{3} + \cdots + 991 \) Copy content Toggle raw display
$83$ \( T^{5} - T^{4} - 201 T^{3} - 897 T^{2} + \cdots + 1579 \) Copy content Toggle raw display
$89$ \( T^{5} - 3 T^{4} - 161 T^{3} + \cdots - 21487 \) Copy content Toggle raw display
$97$ \( T^{5} + 30 T^{4} + 100 T^{3} + \cdots + 3463 \) Copy content Toggle raw display
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