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Properties

Label 52.2.f.a
Level $52$
Weight $2$
Character orbit 52.f
Analytic conductor $0.415$
Analytic rank $0$
Dimension $2$
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(31,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 52.f (of order \(4\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{5} + (2 i - 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{5} + (2 i - 2) q^{8} + 3 q^{9} - 6 i q^{10} + (3 i - 2) q^{13} - 4 q^{16} - 2 i q^{17} + (3 i + 3) q^{18} + ( - 6 i + 6) q^{20} + 13 i q^{25} + (i - 5) q^{26} + 4 q^{29} + ( - 4 i - 4) q^{32} + ( - 2 i + 2) q^{34} + 6 i q^{36} + ( - 5 i + 5) q^{37} + 12 q^{40} + ( - i - 1) q^{41} + ( - 9 i - 9) q^{45} - 7 i q^{49} + (13 i - 13) q^{50} + ( - 4 i - 6) q^{52} - 14 q^{53} + (4 i + 4) q^{58} + 10 q^{61} - 8 i q^{64} + ( - 3 i + 15) q^{65} + 4 q^{68} + (6 i - 6) q^{72} + (11 i - 11) q^{73} + 10 q^{74} + (12 i + 12) q^{80} + 9 q^{81} - 2 i q^{82} + (6 i - 6) q^{85} + ( - 3 i + 3) q^{89} - 18 i q^{90} + (5 i + 5) q^{97} + ( - 7 i + 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{5} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 6 q^{5} - 4 q^{8} + 6 q^{9} - 4 q^{13} - 8 q^{16} + 6 q^{18} + 12 q^{20} - 10 q^{26} + 8 q^{29} - 8 q^{32} + 4 q^{34} + 10 q^{37} + 24 q^{40} - 2 q^{41} - 18 q^{45} - 26 q^{50} - 12 q^{52} - 28 q^{53} + 8 q^{58} + 20 q^{61} + 30 q^{65} + 8 q^{68} - 12 q^{72} - 22 q^{73} + 20 q^{74} + 24 q^{80} + 18 q^{81} - 12 q^{85} + 6 q^{89} + 10 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
− 1.00000i
1.00000i
1.00000 − 1.00000i 0 − 2.00000i −3.00000 + 3.00000i 0 0 −2.00000 − 2.00000i 3.00000 6.00000i
47.1 1.00000 + 1.00000i 0 2.00000i −3.00000 − 3.00000i 0 0 −2.00000 + 2.00000i 3.00000 − 6.00000i
\(n\): 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.d odd 4 1 inner
52.f even 4 1 inner

Twists

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

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 14)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 22T + 242 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
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