Properties

Label 1610.2.a.k
Level $1610$
Weight $2$
Character orbit 1610.a
Self dual yes
Analytic conductor $12.856$
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] = [1610,2,Mod(1,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, 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("1610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.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: \(12.8559147254\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} - q^{5} + (\beta_{2} + 1) q^{6} + q^{7} - q^{8} + (\beta_{2} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} - q^{5} + (\beta_{2} + 1) q^{6} + q^{7} - q^{8} + (\beta_{2} - \beta_1 + 3) q^{9} + q^{10} + (\beta_{2} + \beta_1) q^{11} + ( - \beta_{2} - 1) q^{12} - 4 q^{13} - q^{14} + (\beta_{2} + 1) q^{15} + q^{16} + ( - \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} + \beta_1 - 3) q^{18} + (2 \beta_1 - 4) q^{19} - q^{20} + ( - \beta_{2} - 1) q^{21} + ( - \beta_{2} - \beta_1) q^{22} + q^{23} + (\beta_{2} + 1) q^{24} + q^{25} + 4 q^{26} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{27} + q^{28} + (2 \beta_{2} + 4) q^{29} + ( - \beta_{2} - 1) q^{30} + (3 \beta_{2} - 2 \beta_1 - 1) q^{31} - q^{32} + (2 \beta_{2} - 6) q^{33} + (\beta_{2} + \beta_1) q^{34} - q^{35} + (\beta_{2} - \beta_1 + 3) q^{36} + (\beta_{2} - \beta_1 + 4) q^{37} + ( - 2 \beta_1 + 4) q^{38} + (4 \beta_{2} + 4) q^{39} + q^{40} + (2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (\beta_{2} + 1) q^{42} + (\beta_{2} - 3) q^{43} + (\beta_{2} + \beta_1) q^{44} + ( - \beta_{2} + \beta_1 - 3) q^{45} - q^{46} + (3 \beta_{2} + \beta_1) q^{47} + ( - \beta_{2} - 1) q^{48} + q^{49} - q^{50} + ( - 2 \beta_{2} + 6) q^{51} - 4 q^{52} + ( - 5 \beta_{2} + \beta_1) q^{53} + (2 \beta_{2} - 2 \beta_1 + 4) q^{54} + ( - \beta_{2} - \beta_1) q^{55} - q^{56} + (8 \beta_{2} - 2 \beta_1 + 2) q^{57} + ( - 2 \beta_{2} - 4) q^{58} + ( - 5 \beta_{2} + 3 \beta_1) q^{59} + (\beta_{2} + 1) q^{60} - 2 q^{61} + ( - 3 \beta_{2} + 2 \beta_1 + 1) q^{62} + (\beta_{2} - \beta_1 + 3) q^{63} + q^{64} + 4 q^{65} + ( - 2 \beta_{2} + 6) q^{66} + (\beta_{2} - 3) q^{67} + ( - \beta_{2} - \beta_1) q^{68} + ( - \beta_{2} - 1) q^{69} + q^{70} - 4 q^{71} + ( - \beta_{2} + \beta_1 - 3) q^{72} + ( - 2 \beta_{2} - \beta_1 - 11) q^{73} + ( - \beta_{2} + \beta_1 - 4) q^{74} + ( - \beta_{2} - 1) q^{75} + (2 \beta_1 - 4) q^{76} + (\beta_{2} + \beta_1) q^{77} + ( - 4 \beta_{2} - 4) q^{78} + (\beta_1 - 9) q^{79} - q^{80} + (5 \beta_{2} - \beta_1 + 3) q^{81} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{82} + (\beta_{2} - 3 \beta_1 + 2) q^{83} + ( - \beta_{2} - 1) q^{84} + (\beta_{2} + \beta_1) q^{85} + ( - \beta_{2} + 3) q^{86} + ( - 4 \beta_{2} + 2 \beta_1 - 14) q^{87} + ( - \beta_{2} - \beta_1) q^{88} + ( - 2 \beta_1 + 10) q^{89} + (\beta_{2} - \beta_1 + 3) q^{90} - 4 q^{91} + q^{92} + ( - 3 \beta_{2} + 5 \beta_1 - 12) q^{93} + ( - 3 \beta_{2} - \beta_1) q^{94} + ( - 2 \beta_1 + 4) q^{95} + (\beta_{2} + 1) q^{96} + (5 \beta_{2} - 3 \beta_1 - 4) q^{97} - q^{98} + (3 \beta_{2} - \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + 3 q^{10} - 2 q^{12} - 12 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} - 7 q^{18} - 10 q^{19} - 3 q^{20} - 2 q^{21} + 3 q^{23} + 2 q^{24} + 3 q^{25} + 12 q^{26} - 8 q^{27} + 3 q^{28} + 10 q^{29} - 2 q^{30} - 8 q^{31} - 3 q^{32} - 20 q^{33} - 3 q^{35} + 7 q^{36} + 10 q^{37} + 10 q^{38} + 8 q^{39} + 3 q^{40} + 2 q^{41} + 2 q^{42} - 10 q^{43} - 7 q^{45} - 3 q^{46} - 2 q^{47} - 2 q^{48} + 3 q^{49} - 3 q^{50} + 20 q^{51} - 12 q^{52} + 6 q^{53} + 8 q^{54} - 3 q^{56} - 4 q^{57} - 10 q^{58} + 8 q^{59} + 2 q^{60} - 6 q^{61} + 8 q^{62} + 7 q^{63} + 3 q^{64} + 12 q^{65} + 20 q^{66} - 10 q^{67} - 2 q^{69} + 3 q^{70} - 12 q^{71} - 7 q^{72} - 32 q^{73} - 10 q^{74} - 2 q^{75} - 10 q^{76} - 8 q^{78} - 26 q^{79} - 3 q^{80} + 3 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{84} + 10 q^{86} - 36 q^{87} + 28 q^{89} + 7 q^{90} - 12 q^{91} + 3 q^{92} - 28 q^{93} + 2 q^{94} + 10 q^{95} + 2 q^{96} - 20 q^{97} - 3 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
−1.81361
2.34292
0.470683
−1.00000 −3.10278 1.00000 −1.00000 3.10278 1.00000 −1.00000 6.62721 1.00000
1.2 −1.00000 −1.14637 1.00000 −1.00000 1.14637 1.00000 −1.00000 −1.68585 1.00000
1.3 −1.00000 2.24914 1.00000 −1.00000 −2.24914 1.00000 −1.00000 2.05863 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 1610.2.a.k 3
5.b even 2 1 8050.2.a.bn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1610.2.a.k 3 1.a even 1 1 trivial
8050.2.a.bn 3 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1610))\):

\( T_{3}^{3} + 2T_{3}^{2} - 6T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{3} - 28T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 6 T - 8 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 28T + 16 \) Copy content Toggle raw display
$13$ \( (T + 4)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 28T - 16 \) Copy content Toggle raw display
$19$ \( T^{3} + 10 T^{2} - 28 T - 344 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 4 T + 88 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 74 T - 524 \) Copy content Toggle raw display
$37$ \( T^{3} - 10 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 68 T + 8 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} + 26 T + 16 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 96 T + 304 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 160 T + 688 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 220 T + 1936 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} + 26 T + 16 \) Copy content Toggle raw display
$71$ \( (T + 4)^{3} \) Copy content Toggle raw display
$73$ \( T^{3} + 32 T^{2} + 286 T + 484 \) Copy content Toggle raw display
$79$ \( T^{3} + 26 T^{2} + 210 T + 496 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} - 128 T + 608 \) Copy content Toggle raw display
$89$ \( T^{3} - 28 T^{2} + 200 T - 64 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} - 108 T - 2624 \) Copy content Toggle raw display
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