Properties

Label 738.2.a.k
Level $738$
Weight $2$
Character orbit 738.a
Self dual yes
Analytic conductor $5.893$
Analytic rank $1$
Dimension $2$
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] = [738,2,Mod(1,738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(738, 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("738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.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: \(5.89295966917\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 82)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 \beta q^{5} + ( - \beta - 2) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + 2 \beta q^{5} + ( - \beta - 2) q^{7} - q^{8} - 2 \beta q^{10} - 3 \beta q^{11} + (\beta + 2) q^{14} + q^{16} + ( - 4 \beta - 2) q^{17} + ( - \beta - 4) q^{19} + 2 \beta q^{20} + 3 \beta q^{22} + (2 \beta - 4) q^{23} + 3 q^{25} + ( - \beta - 2) q^{28} + (4 \beta - 4) q^{29} + (2 \beta - 4) q^{31} - q^{32} + (4 \beta + 2) q^{34} + ( - 4 \beta - 4) q^{35} + 6 \beta q^{37} + (\beta + 4) q^{38} - 2 \beta q^{40} + q^{41} + ( - 4 \beta + 4) q^{43} - 3 \beta q^{44} + ( - 2 \beta + 4) q^{46} + (5 \beta + 2) q^{47} + (4 \beta - 1) q^{49} - 3 q^{50} - 12 q^{53} - 12 q^{55} + (\beta + 2) q^{56} + ( - 4 \beta + 4) q^{58} + ( - 2 \beta + 4) q^{59} + 6 q^{61} + ( - 2 \beta + 4) q^{62} + q^{64} + ( - 3 \beta - 4) q^{67} + ( - 4 \beta - 2) q^{68} + (4 \beta + 4) q^{70} + ( - \beta + 2) q^{71} + ( - 4 \beta - 8) q^{73} - 6 \beta q^{74} + ( - \beta - 4) q^{76} + (6 \beta + 6) q^{77} + ( - 3 \beta - 6) q^{79} + 2 \beta q^{80} - q^{82} + ( - 4 \beta - 12) q^{83} + ( - 4 \beta - 16) q^{85} + (4 \beta - 4) q^{86} + 3 \beta q^{88} + (4 \beta + 6) q^{89} + (2 \beta - 4) q^{92} + ( - 5 \beta - 2) q^{94} + ( - 8 \beta - 4) q^{95} + (4 \beta - 2) q^{97} + ( - 4 \beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 4 q^{14} + 2 q^{16} - 4 q^{17} - 8 q^{19} - 8 q^{23} + 6 q^{25} - 4 q^{28} - 8 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} - 8 q^{35} + 8 q^{38} + 2 q^{41} + 8 q^{43} + 8 q^{46} + 4 q^{47} - 2 q^{49} - 6 q^{50} - 24 q^{53} - 24 q^{55} + 4 q^{56} + 8 q^{58} + 8 q^{59} + 12 q^{61} + 8 q^{62} + 2 q^{64} - 8 q^{67} - 4 q^{68} + 8 q^{70} + 4 q^{71} - 16 q^{73} - 8 q^{76} + 12 q^{77} - 12 q^{79} - 2 q^{82} - 24 q^{83} - 32 q^{85} - 8 q^{86} + 12 q^{89} - 8 q^{92} - 4 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
−1.00000 0 1.00000 −2.82843 0 −0.585786 −1.00000 0 2.82843
1.2 −1.00000 0 1.00000 2.82843 0 −3.41421 −1.00000 0 −2.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(41\) \(-1\)

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 738.2.a.k 2
3.b odd 2 1 82.2.a.b 2
4.b odd 2 1 5904.2.a.z 2
12.b even 2 1 656.2.a.e 2
15.d odd 2 1 2050.2.a.h 2
15.e even 4 2 2050.2.c.l 4
21.c even 2 1 4018.2.a.ba 2
24.f even 2 1 2624.2.a.l 2
24.h odd 2 1 2624.2.a.j 2
33.d even 2 1 9922.2.a.i 2
123.b odd 2 1 3362.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.2.a.b 2 3.b odd 2 1
656.2.a.e 2 12.b even 2 1
738.2.a.k 2 1.a even 1 1 trivial
2050.2.a.h 2 15.d odd 2 1
2050.2.c.l 4 15.e even 4 2
2624.2.a.j 2 24.h odd 2 1
2624.2.a.l 2 24.f even 2 1
3362.2.a.m 2 123.b odd 2 1
4018.2.a.ba 2 21.c even 2 1
5904.2.a.z 2 4.b odd 2 1
9922.2.a.i 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(738))\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 18 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( (T - 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 46 \) Copy content Toggle raw display
$53$ \( (T + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 2 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$73$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$83$ \( T^{2} + 24T + 112 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
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