Properties

Label 1859.2.a.g
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: 3.3.361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + \beta_1 q^{3} + (\beta_1 + 2) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{2} - 2) q^{7} + ( - 2 \beta_{2} - \beta_1 - 1) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + \beta_1 q^{3} + (\beta_1 + 2) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{2} - 2) q^{7} + ( - 2 \beta_{2} - \beta_1 - 1) q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{2} - \beta_1 - 2) q^{10} - q^{11} + (\beta_{2} + 2 \beta_1 + 4) q^{12} + (3 \beta_{2} + \beta_1 - 4) q^{14} + (\beta_{2} + 2 \beta_1 - 3) q^{15} + (\beta_{2} + 2 \beta_1) q^{16} + ( - \beta_1 - 2) q^{17} + ( - 2 \beta_1 - 1) q^{18} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{19} + (3 \beta_{2} + 2 \beta_1 - 3) q^{20} + (\beta_{2} - 3) q^{21} + (\beta_{2} + \beta_1 - 1) q^{22} + ( - 2 \beta_{2} - \beta_1 + 2) q^{23} + ( - 3 \beta_{2} - 5 \beta_1 + 2) q^{24} + ( - \beta_{2} - \beta_1) q^{25} + (\beta_{2} - 3) q^{27} + (3 \beta_{2} - 7) q^{28} + (\beta_{2} + 4 \beta_1 - 3) q^{29} - 7 q^{30} + (2 \beta_{2} - 1) q^{31} + (\beta_{2} - \beta_1 - 2) q^{32} - \beta_1 q^{33} + (4 \beta_{2} + 3 \beta_1 - 1) q^{34} + ( - 3 \beta_{2} - \beta_1 + 5) q^{35} + (3 \beta_{2} + 3 \beta_1 - 1) q^{36} + (\beta_{2} + 2 \beta_1 - 5) q^{37} + (3 \beta_{2} + 4 \beta_1 + 8) q^{38} - 7 q^{40} + ( - 4 \beta_{2} - 3 \beta_1 + 2) q^{41} + (4 \beta_{2} + 2 \beta_1 - 5) q^{42} + ( - 5 \beta_{2} - 2 \beta_1 + 1) q^{43} + ( - \beta_1 - 2) q^{44} + ( - \beta_1 + 5) q^{45} + ( - 2 \beta_{2} + \beta_1 + 7) q^{46} + (3 \beta_{2} - \beta_1 + 2) q^{47} + (3 \beta_{2} + 2 \beta_1 + 5) q^{48} + ( - 5 \beta_{2} - \beta_1 + 2) q^{49} + (\beta_{2} + 2 \beta_1 + 3) q^{50} + ( - \beta_{2} - 2 \beta_1 - 4) q^{51} + (\beta_{2} + 2 \beta_1 - 4) q^{53} + (4 \beta_{2} + 2 \beta_1 - 5) q^{54} - \beta_{2} q^{55} + (4 \beta_{2} + 2 \beta_1 - 5) q^{56} + ( - 5 \beta_{2} - 3 \beta_1 - 6) q^{57} + ( - 4 \beta_{2} - 2 \beta_1 - 9) q^{58} + (\beta_{2} + 2 \beta_1 - 11) q^{59} + (5 \beta_{2} + 3 \beta_1 - 1) q^{60} + ( - 3 \beta_{2} - 3 \beta_1 - 4) q^{61} + (3 \beta_{2} - \beta_1 - 5) q^{62} + ( - 2 \beta_{2} - \beta_1 + 3) q^{63} + (3 \beta_{2} - 2 \beta_1 - 3) q^{64} + (2 \beta_{2} + \beta_1 + 1) q^{66} + (\beta_{2} - \beta_1 - 5) q^{67} + ( - \beta_{2} - 4 \beta_1 - 8) q^{68} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{69} + ( - 6 \beta_{2} - \beta_1 + 12) q^{70} + (\beta_{2} + \beta_1 + 3) q^{71} + ( - 2 \beta_{2} - \beta_1 - 8) q^{72} + 2 q^{73} + (2 \beta_{2} + 2 \beta_1 - 9) q^{74} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{75} + ( - 9 \beta_{2} - 9 \beta_1 - 4) q^{76} + ( - \beta_{2} + 2) q^{77} + (2 \beta_{2} + 4 \beta_1 + 2) q^{79} + (\beta_{2} + 3 \beta_1 - 1) q^{80} + ( - 2 \beta_{2} - \beta_1 - 6) q^{81} + (5 \beta_1 + 13) q^{82} + (\beta_{2} + 4 \beta_1 + 3) q^{83} + (3 \beta_{2} - \beta_1 - 9) q^{84} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{85} + ( - 2 \beta_{2} + 6 \beta_1 + 13) q^{86} + (5 \beta_{2} - \beta_1 + 13) q^{87} + (2 \beta_{2} + \beta_1 + 1) q^{88} + ( - \beta_{2} + 6) q^{89} + ( - 3 \beta_{2} - 4 \beta_1 + 6) q^{90} + ( - 7 \beta_{2} - 4 \beta_1 + 6) q^{92} + (2 \beta_{2} + 3 \beta_1 - 6) q^{93} + (3 \beta_{2} - 4 \beta_1 - 3) q^{94} + ( - 4 \beta_1 - 1) q^{95} - 7 q^{96} + (4 \beta_1 - 5) q^{97} + ( - 5 \beta_{2} + 4 \beta_1 + 13) q^{98} + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{3} + 7 q^{4} + q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{3} + 7 q^{4} + q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 3 q^{11} + 15 q^{12} - 8 q^{14} - 6 q^{15} + 3 q^{16} - 7 q^{17} - 5 q^{18} - 2 q^{19} - 4 q^{20} - 8 q^{21} - q^{22} + 3 q^{23} - 2 q^{24} - 2 q^{25} - 8 q^{27} - 18 q^{28} - 4 q^{29} - 21 q^{30} - q^{31} - 6 q^{32} - q^{33} + 4 q^{34} + 11 q^{35} + 3 q^{36} - 12 q^{37} + 31 q^{38} - 21 q^{40} - q^{41} - 9 q^{42} - 4 q^{43} - 7 q^{44} + 14 q^{45} + 20 q^{46} + 8 q^{47} + 20 q^{48} + 12 q^{50} - 15 q^{51} - 9 q^{53} - 9 q^{54} - q^{55} - 9 q^{56} - 26 q^{57} - 33 q^{58} - 30 q^{59} + 5 q^{60} - 18 q^{61} - 13 q^{62} + 6 q^{63} - 8 q^{64} + 6 q^{66} - 15 q^{67} - 29 q^{68} + q^{69} + 29 q^{70} + 11 q^{71} - 27 q^{72} + 6 q^{73} - 23 q^{74} - 7 q^{75} - 30 q^{76} + 5 q^{77} + 12 q^{79} + q^{80} - 21 q^{81} + 44 q^{82} + 14 q^{83} - 25 q^{84} + 4 q^{85} + 43 q^{86} + 43 q^{87} + 6 q^{88} + 17 q^{89} + 11 q^{90} + 7 q^{92} - 13 q^{93} - 10 q^{94} - 7 q^{95} - 21 q^{96} - 11 q^{97} + 38 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 7 \) : 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
2.28514
−2.50702
1.22188
−2.50702 2.28514 4.28514 1.22188 −5.72889 −0.778124 −5.72889 2.22188 −3.06327
1.2 1.22188 −2.50702 −0.507019 2.28514 −3.06327 0.285142 −3.06327 3.28514 2.79216
1.3 2.28514 1.22188 3.22188 −2.50702 2.79216 −4.50702 2.79216 −1.50702 −5.72889
\(n\): 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.g 3
13.b even 2 1 1859.2.a.f 3
13.e even 6 2 143.2.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.b 6 13.e even 6 2
1859.2.a.f 3 13.b even 2 1
1859.2.a.g 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 6T_{2} + 7 \) Copy content Toggle raw display
\( T_{7}^{3} + 5T_{7}^{2} + 2T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$7$ \( T^{3} + 5 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} + \cdots + 77 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} + \cdots - 49 \) Copy content Toggle raw display
$31$ \( T^{3} + T^{2} + \cdots + 31 \) Copy content Toggle raw display
$37$ \( T^{3} + 12 T^{2} + \cdots - 31 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} + \cdots + 31 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} + \cdots - 581 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} + \cdots + 259 \) Copy content Toggle raw display
$53$ \( T^{3} + 9 T^{2} + \cdots - 49 \) Copy content Toggle raw display
$59$ \( T^{3} + 30 T^{2} + \cdots + 791 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + \cdots + 7 \) Copy content Toggle raw display
$67$ \( T^{3} + 15 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$71$ \( T^{3} - 11 T^{2} + \cdots - 31 \) Copy content Toggle raw display
$73$ \( (T - 2)^{3} \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} + \cdots + 88 \) Copy content Toggle raw display
$83$ \( T^{3} - 14 T^{2} + \cdots + 341 \) Copy content Toggle raw display
$89$ \( T^{3} - 17 T^{2} + \cdots - 151 \) Copy content Toggle raw display
$97$ \( T^{3} + 11 T^{2} + \cdots - 7 \) Copy content Toggle raw display
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