↑

Properties

Label 396.2.j.a
Level $396$
Weight $2$
Character orbit 396.j
Analytic conductor $3.162$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [396,2,Mod(37,396)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("396.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-3,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.16207592004\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{2} - 1) q^{5} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{7} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots - 1) q^{11} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \cdots - 3) q^{13}+ \cdots + ( - 9 \zeta_{10}^{3} + 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{5} - 7 q^{7} + 4 q^{11} - q^{13} + q^{17} + 7 q^{19} - 8 q^{23} + 6 q^{25} + 5 q^{29} - 5 q^{31} + 9 q^{35} - 9 q^{37} + 5 q^{41} + 5 q^{47} - 12 q^{49} - 11 q^{53} + 7 q^{55} - 17 q^{59} - q^{61}+ \cdots + 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/396\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.809017 − 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 − 0.951057i
0 0 0 −1.30902 + 0.951057i 0 −1.19098 − 3.66547i 0 0 0
181.1 0 0 0 −0.190983 + 0.587785i 0 −2.30902 + 1.67760i 0 0 0
289.1 0 0 0 −1.30902 − 0.951057i 0 −1.19098 + 3.66547i 0 0 0
361.1 0 0 0 −0.190983 − 0.587785i 0 −2.30902 − 1.67760i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.2.j.a 4
3.b odd 2 1 44.2.e.a ✓ 4
11.c even 5 1 inner 396.2.j.a 4
11.c even 5 1 4356.2.a.t 2
11.d odd 10 1 4356.2.a.u 2
12.b even 2 1 176.2.m.b 4
15.d odd 2 1 1100.2.n.a 4
15.e even 4 2 1100.2.cb.a 8
24.f even 2 1 704.2.m.d 4
24.h odd 2 1 704.2.m.e 4
33.d even 2 1 484.2.e.c 4
33.f even 10 1 484.2.a.c 2
33.f even 10 1 484.2.e.c 4
33.f even 10 2 484.2.e.d 4
33.h odd 10 1 44.2.e.a ✓ 4
33.h odd 10 1 484.2.a.b 2
33.h odd 10 2 484.2.e.e 4
132.n odd 10 1 1936.2.a.z 2
132.o even 10 1 176.2.m.b 4
132.o even 10 1 1936.2.a.ba 2
165.o odd 10 1 1100.2.n.a 4
165.v even 20 2 1100.2.cb.a 8
264.r odd 10 1 7744.2.a.bo 2
264.t odd 10 1 704.2.m.e 4
264.t odd 10 1 7744.2.a.da 2
264.u even 10 1 7744.2.a.db 2
264.w even 10 1 704.2.m.d 4
264.w even 10 1 7744.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.e.a ✓ 4 3.b odd 2 1
44.2.e.a ✓ 4 33.h odd 10 1
176.2.m.b 4 12.b even 2 1
176.2.m.b 4 132.o even 10 1
396.2.j.a 4 1.a even 1 1 trivial
396.2.j.a 4 11.c even 5 1 inner
484.2.a.b 2 33.h odd 10 1
484.2.a.c 2 33.f even 10 1
484.2.e.c 4 33.d even 2 1
484.2.e.c 4 33.f even 10 1
484.2.e.d 4 33.f even 10 2
484.2.e.e 4 33.h odd 10 2
704.2.m.d 4 24.f even 2 1
704.2.m.d 4 264.w even 10 1
704.2.m.e 4 24.h odd 2 1
704.2.m.e 4 264.t odd 10 1
1100.2.n.a 4 15.d odd 2 1
1100.2.n.a 4 165.o odd 10 1
1100.2.cb.a 8 15.e even 4 2
1100.2.cb.a 8 165.v even 20 2
1936.2.a.z 2 132.n odd 10 1
1936.2.a.ba 2 132.o even 10 1
4356.2.a.t 2 11.c even 5 1
4356.2.a.u 2 11.d odd 10 1
7744.2.a.bo 2 264.r odd 10 1
7744.2.a.bp 2 264.w even 10 1
7744.2.a.da 2 264.t odd 10 1
7744.2.a.db 2 264.u even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(396, [\chi])\):

\( T_{5}^{4} + 3T_{5}^{3} + 4T_{5}^{2} + 2T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 7T_{7}^{3} + 34T_{7}^{2} + 88T_{7} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} - 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{4} + 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$59$ \( T^{4} + 17 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$61$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} - 19 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{4} - 3 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( T^{4} - 31 T^{3} + \cdots + 43681 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 27 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
show more
show less