Properties

Label 546.2.a.h
Level $546$
Weight $2$
Character orbit 546.a
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
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] = [546,2,Mod(1,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, 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("546.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - \beta q^{5} + q^{6} + q^{7} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - \beta q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + \beta q^{10} + ( - \beta + 2) q^{11} - q^{12} - q^{13} - q^{14} + \beta q^{15} + q^{16} + (\beta - 4) q^{17} - q^{18} + ( - \beta + 2) q^{19} - \beta q^{20} - q^{21} + (\beta - 2) q^{22} + (\beta - 2) q^{23} + q^{24} + (\beta + 9) q^{25} + q^{26} - q^{27} + q^{28} + (\beta + 4) q^{29} - \beta q^{30} + 8 q^{31} - q^{32} + (\beta - 2) q^{33} + ( - \beta + 4) q^{34} - \beta q^{35} + q^{36} - \beta q^{37} + (\beta - 2) q^{38} + q^{39} + \beta q^{40} + (2 \beta - 2) q^{41} + q^{42} + ( - \beta + 2) q^{43} + ( - \beta + 2) q^{44} - \beta q^{45} + ( - \beta + 2) q^{46} - q^{48} + q^{49} + ( - \beta - 9) q^{50} + ( - \beta + 4) q^{51} - q^{52} + 10 q^{53} + q^{54} + ( - \beta + 14) q^{55} - q^{56} + (\beta - 2) q^{57} + ( - \beta - 4) q^{58} + 8 q^{59} + \beta q^{60} + ( - \beta - 8) q^{61} - 8 q^{62} + q^{63} + q^{64} + \beta q^{65} + ( - \beta + 2) q^{66} + (2 \beta + 4) q^{67} + (\beta - 4) q^{68} + ( - \beta + 2) q^{69} + \beta q^{70} - q^{72} - 3 \beta q^{73} + \beta q^{74} + ( - \beta - 9) q^{75} + ( - \beta + 2) q^{76} + ( - \beta + 2) q^{77} - q^{78} + (2 \beta + 4) q^{79} - \beta q^{80} + q^{81} + ( - 2 \beta + 2) q^{82} + ( - 2 \beta + 4) q^{83} - q^{84} + (3 \beta - 14) q^{85} + (\beta - 2) q^{86} + ( - \beta - 4) q^{87} + (\beta - 2) q^{88} - 14 q^{89} + \beta q^{90} - q^{91} + (\beta - 2) q^{92} - 8 q^{93} + ( - \beta + 14) q^{95} + q^{96} + (4 \beta - 2) q^{97} - q^{98} + ( - \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + q^{10} + 3 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{14} + q^{15} + 2 q^{16} - 7 q^{17} - 2 q^{18} + 3 q^{19} - q^{20} - 2 q^{21} - 3 q^{22} - 3 q^{23} + 2 q^{24} + 19 q^{25} + 2 q^{26} - 2 q^{27} + 2 q^{28} + 9 q^{29} - q^{30} + 16 q^{31} - 2 q^{32} - 3 q^{33} + 7 q^{34} - q^{35} + 2 q^{36} - q^{37} - 3 q^{38} + 2 q^{39} + q^{40} - 2 q^{41} + 2 q^{42} + 3 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 2 q^{48} + 2 q^{49} - 19 q^{50} + 7 q^{51} - 2 q^{52} + 20 q^{53} + 2 q^{54} + 27 q^{55} - 2 q^{56} - 3 q^{57} - 9 q^{58} + 16 q^{59} + q^{60} - 17 q^{61} - 16 q^{62} + 2 q^{63} + 2 q^{64} + q^{65} + 3 q^{66} + 10 q^{67} - 7 q^{68} + 3 q^{69} + q^{70} - 2 q^{72} - 3 q^{73} + q^{74} - 19 q^{75} + 3 q^{76} + 3 q^{77} - 2 q^{78} + 10 q^{79} - q^{80} + 2 q^{81} + 2 q^{82} + 6 q^{83} - 2 q^{84} - 25 q^{85} - 3 q^{86} - 9 q^{87} - 3 q^{88} - 28 q^{89} + q^{90} - 2 q^{91} - 3 q^{92} - 16 q^{93} + 27 q^{95} + 2 q^{96} - 2 q^{98} + 3 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
−1.00000 −1.00000 1.00000 −4.27492 1.00000 1.00000 −1.00000 1.00000 4.27492
1.2 −1.00000 −1.00000 1.00000 3.27492 1.00000 1.00000 −1.00000 1.00000 −3.27492
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(7\) \( -1 \)
\(13\) \( +1 \)

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 546.2.a.h 2
3.b odd 2 1 1638.2.a.y 2
4.b odd 2 1 4368.2.a.bh 2
7.b odd 2 1 3822.2.a.bm 2
13.b even 2 1 7098.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.h 2 1.a even 1 1 trivial
1638.2.a.y 2 3.b odd 2 1
3822.2.a.bm 2 7.b odd 2 1
4368.2.a.bh 2 4.b odd 2 1
7098.2.a.bu 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5}^{2} + T_{5} - 14 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 12 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 3T - 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 17T + 58 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3T - 126 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 48 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 228 \) Copy content Toggle raw display
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