Properties

Label 52.2.l.a
Level $52$
Weight $2$
Character orbit 52.l
Analytic conductor $0.415$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [52,2,Mod(7,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 52.l (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(0.415222090511\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{10} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{16} + ( - 4 \zeta_{12}^{2} + \zeta_{12} - 4) q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{20} + ( - \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{25} + ( - 5 \zeta_{12}^{2} + \zeta_{12} + 5) q^{26} + (5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 5 \zeta_{12}) q^{29} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 5) q^{34} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - 6 \zeta_{12} + 7) q^{37} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 6) q^{40} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 5 \zeta_{12} - 9) q^{41} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{45} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{49} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{50} + (4 \zeta_{12}^{3} - 6) q^{52} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 7) q^{53} + ( - 10 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{58} + ( - 12 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 6 \zeta_{12} + 5) q^{61} + 8 \zeta_{12}^{3} q^{64} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \zeta_{12} - 1) q^{65} + ( - 8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 8 \zeta_{12}) q^{68} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + (8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{73} + (14 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 7 \zeta_{12} + 5) q^{74} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{80} - 9 \zeta_{12}^{2} q^{81} + (9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{82} + (11 \zeta_{12}^{3} - \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{85} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{89} + ( - 9 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{90} + (5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{97} + (7 \zeta_{12}^{2} - 7 \zeta_{12} - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9} - 6 q^{10} + 4 q^{13} + 8 q^{16} - 24 q^{17} + 12 q^{18} + 12 q^{20} + 10 q^{26} + 4 q^{29} + 8 q^{32} - 4 q^{34} + 26 q^{37} - 24 q^{40} - 28 q^{41} - 16 q^{50} - 24 q^{52} + 28 q^{53} - 26 q^{58} + 10 q^{61} + 12 q^{65} + 4 q^{68} + 12 q^{72} + 22 q^{73} + 10 q^{74} + 24 q^{80} - 18 q^{81} + 54 q^{82} - 42 q^{85} - 6 q^{89} - 10 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.36603 + 0.366025i 0 1.73205 1.00000i 2.36603 + 2.36603i 0 0 −2.00000 + 2.00000i −1.50000 2.59808i −4.09808 2.36603i
11.1 0.366025 1.36603i 0 −1.73205 1.00000i 0.633975 + 0.633975i 0 0 −2.00000 + 2.00000i −1.50000 + 2.59808i 1.09808 0.633975i
15.1 −1.36603 0.366025i 0 1.73205 + 1.00000i 2.36603 2.36603i 0 0 −2.00000 2.00000i −1.50000 + 2.59808i −4.09808 + 2.36603i
19.1 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 0.633975 0.633975i 0 0 −2.00000 2.00000i −1.50000 2.59808i 1.09808 + 0.633975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.f odd 12 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.l.a 4
3.b odd 2 1 468.2.cb.d 4
4.b odd 2 1 CM 52.2.l.a 4
8.b even 2 1 832.2.bu.d 4
8.d odd 2 1 832.2.bu.d 4
12.b even 2 1 468.2.cb.d 4
13.b even 2 1 676.2.l.d 4
13.c even 3 1 676.2.f.e 4
13.c even 3 1 676.2.l.c 4
13.d odd 4 1 676.2.l.c 4
13.d odd 4 1 676.2.l.e 4
13.e even 6 1 676.2.f.d 4
13.e even 6 1 676.2.l.e 4
13.f odd 12 1 inner 52.2.l.a 4
13.f odd 12 1 676.2.f.d 4
13.f odd 12 1 676.2.f.e 4
13.f odd 12 1 676.2.l.d 4
39.k even 12 1 468.2.cb.d 4
52.b odd 2 1 676.2.l.d 4
52.f even 4 1 676.2.l.c 4
52.f even 4 1 676.2.l.e 4
52.i odd 6 1 676.2.f.d 4
52.i odd 6 1 676.2.l.e 4
52.j odd 6 1 676.2.f.e 4
52.j odd 6 1 676.2.l.c 4
52.l even 12 1 inner 52.2.l.a 4
52.l even 12 1 676.2.f.d 4
52.l even 12 1 676.2.f.e 4
52.l even 12 1 676.2.l.d 4
104.u even 12 1 832.2.bu.d 4
104.x odd 12 1 832.2.bu.d 4
156.v odd 12 1 468.2.cb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.a 4 1.a even 1 1 trivial
52.2.l.a 4 4.b odd 2 1 CM
52.2.l.a 4 13.f odd 12 1 inner
52.2.l.a 4 52.l even 12 1 inner
468.2.cb.d 4 3.b odd 2 1
468.2.cb.d 4 12.b even 2 1
468.2.cb.d 4 39.k even 12 1
468.2.cb.d 4 156.v odd 12 1
676.2.f.d 4 13.e even 6 1
676.2.f.d 4 13.f odd 12 1
676.2.f.d 4 52.i odd 6 1
676.2.f.d 4 52.l even 12 1
676.2.f.e 4 13.c even 3 1
676.2.f.e 4 13.f odd 12 1
676.2.f.e 4 52.j odd 6 1
676.2.f.e 4 52.l even 12 1
676.2.l.c 4 13.c even 3 1
676.2.l.c 4 13.d odd 4 1
676.2.l.c 4 52.f even 4 1
676.2.l.c 4 52.j odd 6 1
676.2.l.d 4 13.b even 2 1
676.2.l.d 4 13.f odd 12 1
676.2.l.d 4 52.b odd 2 1
676.2.l.d 4 52.l even 12 1
676.2.l.e 4 13.d odd 4 1
676.2.l.e 4 13.e even 6 1
676.2.l.e 4 52.f even 4 1
676.2.l.e 4 52.i odd 6 1
832.2.bu.d 4 8.b even 2 1
832.2.bu.d 4 8.d odd 2 1
832.2.bu.d 4 104.u even 12 1
832.2.bu.d 4 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(52, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + 18 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 87 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 26 T^{3} + 233 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$41$ \( T^{4} + 28 T^{3} + 365 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T + 37)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + 183 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
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