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Properties

Label 28.2.d.a
Level $28$
Weight $2$
Character orbit 28.d
Analytic conductor $0.224$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,2,Mod(27,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 28.d (of order \(2\), degree \(1\), 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: \(0.223581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 2) q^{4} + (2 \beta - 1) q^{7} + (\beta + 2) q^{8} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 2) q^{4} + (2 \beta - 1) q^{7} + (\beta + 2) q^{8} - 3 q^{9} + ( - 4 \beta + 2) q^{11} + ( - \beta + 4) q^{14} + ( - 3 \beta + 2) q^{16} + 3 \beta q^{18} + (2 \beta - 8) q^{22} + (4 \beta - 2) q^{23} + 5 q^{25} + ( - 3 \beta - 2) q^{28} - 2 q^{29} + (\beta - 6) q^{32} + ( - 3 \beta + 6) q^{36} + 6 q^{37} + ( - 4 \beta + 2) q^{43} + (6 \beta + 4) q^{44} + ( - 2 \beta + 8) q^{46} - 7 q^{49} - 5 \beta q^{50} - 10 q^{53} + (5 \beta - 6) q^{56} + 2 \beta q^{58} + ( - 6 \beta + 3) q^{63} + (5 \beta + 2) q^{64} + (12 \beta - 6) q^{67} + (4 \beta - 2) q^{71} + ( - 3 \beta - 6) q^{72} - 6 \beta q^{74} + 14 q^{77} + ( - 12 \beta + 6) q^{79} + 9 q^{81} + (2 \beta - 8) q^{86} + ( - 10 \beta + 12) q^{88} + ( - 6 \beta - 4) q^{92} + 7 \beta q^{98} + (12 \beta - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{4} + 5 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 3 q^{4} + 5 q^{8} - 6 q^{9} + 7 q^{14} + q^{16} + 3 q^{18} - 14 q^{22} + 10 q^{25} - 7 q^{28} - 4 q^{29} - 11 q^{32} + 9 q^{36} + 12 q^{37} + 14 q^{44} + 14 q^{46} - 14 q^{49} - 5 q^{50} - 20 q^{53} - 7 q^{56} + 2 q^{58} + 9 q^{64} - 15 q^{72} - 6 q^{74} + 28 q^{77} + 18 q^{81} - 14 q^{86} + 14 q^{88} - 14 q^{92} + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
0.500000 + 1.32288i
0.500000 − 1.32288i
−0.500000 − 1.32288i 0 −1.50000 + 1.32288i 0 0 2.64575i 2.50000 + 1.32288i −3.00000 0
27.2 −0.500000 + 1.32288i 0 −1.50000 − 1.32288i 0 0 − 2.64575i 2.50000 − 1.32288i −3.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
4.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.2.d.a ✓ 2
3.b odd 2 1 252.2.b.a 2
4.b odd 2 1 inner 28.2.d.a ✓ 2
5.b even 2 1 700.2.g.a 2
5.c odd 4 2 700.2.c.d 4
7.b odd 2 1 CM 28.2.d.a ✓ 2
7.c even 3 2 196.2.f.b 4
7.d odd 6 2 196.2.f.b 4
8.b even 2 1 448.2.f.b 2
8.d odd 2 1 448.2.f.b 2
12.b even 2 1 252.2.b.a 2
16.e even 4 2 1792.2.e.b 4
16.f odd 4 2 1792.2.e.b 4
20.d odd 2 1 700.2.g.a 2
20.e even 4 2 700.2.c.d 4
21.c even 2 1 252.2.b.a 2
24.f even 2 1 4032.2.b.e 2
24.h odd 2 1 4032.2.b.e 2
28.d even 2 1 inner 28.2.d.a ✓ 2
28.f even 6 2 196.2.f.b 4
28.g odd 6 2 196.2.f.b 4
35.c odd 2 1 700.2.g.a 2
35.f even 4 2 700.2.c.d 4
56.e even 2 1 448.2.f.b 2
56.h odd 2 1 448.2.f.b 2
84.h odd 2 1 252.2.b.a 2
112.j even 4 2 1792.2.e.b 4
112.l odd 4 2 1792.2.e.b 4
140.c even 2 1 700.2.g.a 2
140.j odd 4 2 700.2.c.d 4
168.e odd 2 1 4032.2.b.e 2
168.i even 2 1 4032.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a ✓ 2 1.a even 1 1 trivial
28.2.d.a ✓ 2 4.b odd 2 1 inner
28.2.d.a ✓ 2 7.b odd 2 1 CM
28.2.d.a ✓ 2 28.d even 2 1 inner
196.2.f.b 4 7.c even 3 2
196.2.f.b 4 7.d odd 6 2
196.2.f.b 4 28.f even 6 2
196.2.f.b 4 28.g odd 6 2
252.2.b.a 2 3.b odd 2 1
252.2.b.a 2 12.b even 2 1
252.2.b.a 2 21.c even 2 1
252.2.b.a 2 84.h odd 2 1
448.2.f.b 2 8.b even 2 1
448.2.f.b 2 8.d odd 2 1
448.2.f.b 2 56.e even 2 1
448.2.f.b 2 56.h odd 2 1
700.2.c.d 4 5.c odd 4 2
700.2.c.d 4 20.e even 4 2
700.2.c.d 4 35.f even 4 2
700.2.c.d 4 140.j odd 4 2
700.2.g.a 2 5.b even 2 1
700.2.g.a 2 20.d odd 2 1
700.2.g.a 2 35.c odd 2 1
700.2.g.a 2 140.c even 2 1
1792.2.e.b 4 16.e even 4 2
1792.2.e.b 4 16.f odd 4 2
1792.2.e.b 4 112.j even 4 2
1792.2.e.b 4 112.l odd 4 2
4032.2.b.e 2 24.f even 2 1
4032.2.b.e 2 24.h odd 2 1
4032.2.b.e 2 168.e odd 2 1
4032.2.b.e 2 168.i even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 28 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 28 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 28 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 252 \) Copy content Toggle raw display
$71$ \( T^{2} + 28 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 252 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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