Properties

Label 1859.2.a.e
Level $1859$
Weight $2$
Character orbit 1859.a
Self dual yes
Analytic conductor $14.844$
Analytic rank $1$
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] = [1859,2,Mod(1,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, 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("1859.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.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: \(14.8441897358\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + 3 q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + 3 q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{2} + 1) q^{10} - q^{11} - \beta_1 q^{12} + (\beta_{2} - \beta_1) q^{14} + ( - 3 \beta_1 - 2) q^{15} - q^{16} + (\beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{2} - 1) q^{18} + (\beta_{2} + \beta_1 - 2) q^{19} + (\beta_{2} + 1) q^{20} + (\beta_{2} - 2 \beta_1 + 2) q^{21} + q^{22} - 2 \beta_1 q^{23} + 3 \beta_1 q^{24} + (2 \beta_1 + 4) q^{25} + 2 q^{27} + (\beta_{2} - \beta_1) q^{28} + (2 \beta_{2} - \beta_1 - 1) q^{29} + (3 \beta_1 + 2) q^{30} + (2 \beta_{2} - 4 \beta_1 - 2) q^{31} - 5 q^{32} - \beta_1 q^{33} + ( - \beta_{2} - \beta_1 + 1) q^{34} + ( - \beta_{2} - \beta_1 + 6) q^{35} + ( - \beta_{2} - 1) q^{36} + (\beta_{2} - 4 \beta_1 - 1) q^{37} + ( - \beta_{2} - \beta_1 + 2) q^{38} + ( - 3 \beta_{2} - 3) q^{40} + ( - 2 \beta_{2} + 3 \beta_1 - 1) q^{41} + ( - \beta_{2} + 2 \beta_1 - 2) q^{42} + ( - 2 \beta_1 - 2) q^{43} + q^{44} + ( - 2 \beta_1 - 9) q^{45} + 2 \beta_1 q^{46} + (2 \beta_{2} + \beta_1 + 2) q^{47} - \beta_1 q^{48} + ( - \beta_{2} - 2 \beta_1 + 1) q^{49} + ( - 2 \beta_1 - 4) q^{50} + (\beta_{2} + \beta_1 + 6) q^{51} + (2 \beta_{2} - 2 \beta_1 + 1) q^{53} - 2 q^{54} + (\beta_{2} + 1) q^{55} + ( - 3 \beta_{2} + 3 \beta_1) q^{56} + (\beta_{2} + 6) q^{57} + ( - 2 \beta_{2} + \beta_1 + 1) q^{58} + ( - 2 \beta_{2} - \beta_1 + 6) q^{59} + (3 \beta_1 + 2) q^{60} + ( - 3 \beta_1 + 3) q^{61} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{62} + (\beta_{2} + \beta_1 - 6) q^{63} + 7 q^{64} + \beta_1 q^{66} + (2 \beta_{2} + \beta_1 - 4) q^{67} + ( - \beta_{2} - \beta_1 + 1) q^{68} + ( - 2 \beta_{2} - 8) q^{69} + (\beta_{2} + \beta_1 - 6) q^{70} - 6 q^{71} + (3 \beta_{2} + 3) q^{72} + ( - 5 \beta_{2} + \beta_1 - 3) q^{73} + ( - \beta_{2} + 4 \beta_1 + 1) q^{74} + (2 \beta_{2} + 4 \beta_1 + 8) q^{75} + ( - \beta_{2} - \beta_1 + 2) q^{76} + (\beta_{2} - \beta_1) q^{77} + ( - \beta_{2} - 3 \beta_1 - 4) q^{79} + (\beta_{2} + 1) q^{80} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{81} + (2 \beta_{2} - 3 \beta_1 + 1) q^{82} + (\beta_{2} + 3 \beta_1) q^{83} + ( - \beta_{2} + 2 \beta_1 - 2) q^{84} + (2 \beta_{2} - 5 \beta_1 - 9) q^{85} + (2 \beta_1 + 2) q^{86} + ( - \beta_{2} + 3 \beta_1) q^{87} - 3 q^{88} + ( - 3 \beta_{2} - 6) q^{89} + (2 \beta_1 + 9) q^{90} + 2 \beta_1 q^{92} + ( - 4 \beta_{2} + 2 \beta_1 - 12) q^{93} + ( - 2 \beta_{2} - \beta_1 - 2) q^{94} + (3 \beta_{2} - 5 \beta_1 - 8) q^{95} - 5 \beta_1 q^{96} + ( - \beta_{2} + 2 \beta_1 - 6) q^{97} + (\beta_{2} + 2 \beta_1 - 1) q^{98} + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{4} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{4} - 3 q^{5} + 9 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} - 6 q^{15} - 3 q^{16} - 3 q^{17} - 3 q^{18} - 6 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{22} + 12 q^{25} + 6 q^{27} - 3 q^{29} + 6 q^{30} - 6 q^{31} - 15 q^{32} + 3 q^{34} + 18 q^{35} - 3 q^{36} - 3 q^{37} + 6 q^{38} - 9 q^{40} - 3 q^{41} - 6 q^{42} - 6 q^{43} + 3 q^{44} - 27 q^{45} + 6 q^{47} + 3 q^{49} - 12 q^{50} + 18 q^{51} + 3 q^{53} - 6 q^{54} + 3 q^{55} + 18 q^{57} + 3 q^{58} + 18 q^{59} + 6 q^{60} + 9 q^{61} + 6 q^{62} - 18 q^{63} + 21 q^{64} - 12 q^{67} + 3 q^{68} - 24 q^{69} - 18 q^{70} - 18 q^{71} + 9 q^{72} - 9 q^{73} + 3 q^{74} + 24 q^{75} + 6 q^{76} - 12 q^{79} + 3 q^{80} - 9 q^{81} + 3 q^{82} - 6 q^{84} - 27 q^{85} + 6 q^{86} - 9 q^{88} - 18 q^{89} + 27 q^{90} - 36 q^{93} - 6 q^{94} - 24 q^{95} - 18 q^{97} - 3 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.26180
−0.339877
2.60168
−1.00000 −2.26180 −1.00000 −2.11575 2.26180 −3.37755 3.00000 2.11575 2.11575
1.2 −1.00000 −0.339877 −1.00000 2.88448 0.339877 3.54461 3.00000 −2.88448 −2.88448
1.3 −1.00000 2.60168 −1.00000 −3.76873 −2.60168 −0.167055 3.00000 3.76873 3.76873
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(13\) \( +1 \)

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 1859.2.a.e 3
13.b even 2 1 1859.2.a.h 3
13.e even 6 2 143.2.e.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.a 6 13.e even 6 2
1859.2.a.e 3 1.a even 1 1 trivial
1859.2.a.h 3 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 12T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 6T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} + 3 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$7$ \( T^{3} - 12T - 2 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} + \cdots - 49 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + \cdots - 66 \) Copy content Toggle raw display
$23$ \( T^{3} - 24T + 16 \) Copy content Toggle raw display
$29$ \( T^{3} + 3 T^{2} + \cdots + 57 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} + \cdots - 536 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} + \cdots - 279 \) Copy content Toggle raw display
$41$ \( T^{3} + 3 T^{2} + \cdots + 73 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} + \cdots - 24 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} + \cdots + 122 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} + \cdots + 63 \) Copy content Toggle raw display
$59$ \( T^{3} - 18 T^{2} + \cdots + 182 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} + \cdots + 189 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} + \cdots - 202 \) Copy content Toggle raw display
$71$ \( (T + 6)^{3} \) Copy content Toggle raw display
$73$ \( T^{3} + 9 T^{2} + \cdots - 2483 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} + \cdots + 22 \) Copy content Toggle raw display
$83$ \( T^{3} - 84T - 294 \) Copy content Toggle raw display
$89$ \( T^{3} + 18T^{2} - 756 \) Copy content Toggle raw display
$97$ \( T^{3} + 18 T^{2} + \cdots + 116 \) Copy content Toggle raw display
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