Properties

Label 1734.2.a.p
Level $1734$
Weight $2$
Character orbit 1734.a
Self dual yes
Analytic conductor $13.846$
Analytic rank $0$
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] = [1734,2,Mod(1,1734)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1734, 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("1734.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.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: \(13.8460597105\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
−1.53209
−0.347296
1.87939
−1.00000 −1.00000 1.00000 −3.53209 1.00000 2.75877 −1.00000 1.00000 3.53209
1.2 −1.00000 −1.00000 1.00000 −2.34730 1.00000 −4.06418 −1.00000 1.00000 2.34730
1.3 −1.00000 −1.00000 1.00000 −0.120615 1.00000 −1.69459 −1.00000 1.00000 0.120615
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(17\) \(-1\)

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 1734.2.a.p 3
3.b odd 2 1 5202.2.a.bp 3
17.b even 2 1 1734.2.a.q yes 3
17.c even 4 2 1734.2.b.j 6
17.d even 8 4 1734.2.f.n 12
51.c odd 2 1 5202.2.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1734.2.a.p 3 1.a even 1 1 trivial
1734.2.a.q yes 3 17.b even 2 1
1734.2.b.j 6 17.c even 4 2
1734.2.f.n 12 17.d even 8 4
5202.2.a.bm 3 51.c odd 2 1
5202.2.a.bp 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1734))\):

\( T_{5}^{3} + 6T_{5}^{2} + 9T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 9T_{7} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} + 9 T + 1 \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 9 T - 19 \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} - 9 T - 51 \) Copy content Toggle raw display
$13$ \( T^{3} - 6T^{2} + 24 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 24 T - 136 \) Copy content Toggle raw display
$23$ \( (T + 6)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 57T + 163 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 9 T - 3 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 72 T - 296 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 72 T + 136 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 132 T + 856 \) Copy content Toggle raw display
$47$ \( T^{3} - 84T + 296 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 9 T + 71 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} + 45 T - 51 \) Copy content Toggle raw display
$61$ \( T^{3} - 30 T^{2} + 288 T - 872 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} - 36 T + 408 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 216 T + 1752 \) Copy content Toggle raw display
$73$ \( T^{3} - 24 T^{2} + 165 T - 269 \) Copy content Toggle raw display
$79$ \( T^{3} - 147T + 683 \) Copy content Toggle raw display
$83$ \( T^{3} + 15 T^{2} - 9 T - 159 \) Copy content Toggle raw display
$89$ \( T^{3} - 24 T^{2} + 36 T + 1448 \) Copy content Toggle raw display
$97$ \( T^{3} - 12 T^{2} - 63 T + 813 \) Copy content Toggle raw display
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