Properties

Label 364.2.a.d
Level $364$
Weight $2$
Character orbit 364.a
Self dual yes
Analytic conductor $2.907$
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] = [364,2,Mod(1,364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(364, 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("364.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 364.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: \(2.90655463357\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - \beta q^{5} + q^{7} + (2 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} - \beta q^{5} + q^{7} + (2 \beta + 1) q^{9} + ( - \beta + 3) q^{11} + q^{13} + ( - \beta - 3) q^{15} + (3 \beta + 3) q^{17} + ( - 3 \beta + 2) q^{19} + (\beta + 1) q^{21} + 3 q^{23} - 2 q^{25} + 4 q^{27} + ( - 2 \beta - 3) q^{29} + ( - 3 \beta - 4) q^{31} + 2 \beta q^{33} - \beta q^{35} + ( - 3 \beta - 1) q^{37} + (\beta + 1) q^{39} - 2 \beta q^{41} - 7 q^{43} + ( - \beta - 6) q^{45} + ( - \beta + 6) q^{47} + q^{49} + (6 \beta + 12) q^{51} + (4 \beta - 3) q^{53} + ( - 3 \beta + 3) q^{55} + ( - \beta - 7) q^{57} + 6 \beta q^{59} - 10 q^{61} + (2 \beta + 1) q^{63} - \beta q^{65} + (6 \beta - 4) q^{67} + (3 \beta + 3) q^{69} + ( - \beta - 3) q^{71} + (3 \beta - 4) q^{73} + ( - 2 \beta - 2) q^{75} + ( - \beta + 3) q^{77} + (6 \beta - 1) q^{79} + ( - 2 \beta + 1) q^{81} + ( - 3 \beta - 6) q^{83} + ( - 3 \beta - 9) q^{85} + ( - 5 \beta - 9) q^{87} + ( - 3 \beta + 6) q^{89} + q^{91} + ( - 7 \beta - 13) q^{93} + ( - 2 \beta + 9) q^{95} + (3 \beta + 8) q^{97} + (5 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} - 6 q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} + 6 q^{23} - 4 q^{25} + 8 q^{27} - 6 q^{29} - 8 q^{31} - 2 q^{37} + 2 q^{39} - 14 q^{43} - 12 q^{45} + 12 q^{47} + 2 q^{49} + 24 q^{51} - 6 q^{53} + 6 q^{55} - 14 q^{57} - 20 q^{61} + 2 q^{63} - 8 q^{67} + 6 q^{69} - 6 q^{71} - 8 q^{73} - 4 q^{75} + 6 q^{77} - 2 q^{79} + 2 q^{81} - 12 q^{83} - 18 q^{85} - 18 q^{87} + 12 q^{89} + 2 q^{91} - 26 q^{93} + 18 q^{95} + 16 q^{97} - 6 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 −0.732051 0 1.73205 0 1.00000 0 −2.46410 0
1.2 0 2.73205 0 −1.73205 0 1.00000 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 364.2.a.d 2
3.b odd 2 1 3276.2.a.n 2
4.b odd 2 1 1456.2.a.n 2
5.b even 2 1 9100.2.a.o 2
7.b odd 2 1 2548.2.a.l 2
7.c even 3 2 2548.2.j.k 4
7.d odd 6 2 2548.2.j.n 4
8.b even 2 1 5824.2.a.bh 2
8.d odd 2 1 5824.2.a.bq 2
13.b even 2 1 4732.2.a.l 2
13.d odd 4 2 4732.2.g.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.a.d 2 1.a even 1 1 trivial
1456.2.a.n 2 4.b odd 2 1
2548.2.a.l 2 7.b odd 2 1
2548.2.j.k 4 7.c even 3 2
2548.2.j.n 4 7.d odd 6 2
3276.2.a.n 2 3.b odd 2 1
4732.2.a.l 2 13.b even 2 1
4732.2.g.h 4 13.d odd 4 2
5824.2.a.bh 2 8.b even 2 1
5824.2.a.bq 2 8.d odd 2 1
9100.2.a.o 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(364))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 23 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 12 \) Copy content Toggle raw display
$43$ \( (T + 7)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 33 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 39 \) Copy content Toggle raw display
$59$ \( T^{2} - 108 \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 107 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 37 \) Copy content Toggle raw display
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