Properties

Label 28.3.b.a
Level $28$
Weight $3$
Character orbit 28.b
Analytic conductor $0.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,3,Mod(13,28)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 28.b (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: \(0.762944740209\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - \beta q^{5} + ( - \beta + 5) q^{7} - 15 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - \beta q^{5} + ( - \beta + 5) q^{7} - 15 q^{9} - 6 q^{11} + \beta q^{13} + 24 q^{15} - 4 \beta q^{17} + 5 \beta q^{19} + (5 \beta + 24) q^{21} - 30 q^{23} + q^{25} - 6 \beta q^{27} - 6 q^{29} - 6 \beta q^{33} + ( - 5 \beta - 24) q^{35} + 10 q^{37} - 24 q^{39} + 10 \beta q^{41} + 10 q^{43} + 15 \beta q^{45} - 4 \beta q^{47} + ( - 10 \beta + 1) q^{49} + 96 q^{51} + 90 q^{53} + 6 \beta q^{55} - 120 q^{57} - 5 \beta q^{59} + 5 \beta q^{61} + (15 \beta - 75) q^{63} + 24 q^{65} - 70 q^{67} - 30 \beta q^{69} + 42 q^{71} - 22 \beta q^{73} + \beta q^{75} + (6 \beta - 30) q^{77} + 74 q^{79} + 9 q^{81} + 13 \beta q^{83} - 96 q^{85} - 6 \beta q^{87} + 30 \beta q^{89} + (5 \beta + 24) q^{91} + 120 q^{95} + 16 \beta q^{97} + 90 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{7} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{7} - 30 q^{9} - 12 q^{11} + 48 q^{15} + 48 q^{21} - 60 q^{23} + 2 q^{25} - 12 q^{29} - 48 q^{35} + 20 q^{37} - 48 q^{39} + 20 q^{43} + 2 q^{49} + 192 q^{51} + 180 q^{53} - 240 q^{57} - 150 q^{63} + 48 q^{65} - 140 q^{67} + 84 q^{71} - 60 q^{77} + 148 q^{79} + 18 q^{81} - 192 q^{85} + 48 q^{91} + 240 q^{95} + 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
2.44949i
2.44949i
0 4.89898i 0 4.89898i 0 5.00000 + 4.89898i 0 −15.0000 0
13.2 0 4.89898i 0 4.89898i 0 5.00000 4.89898i 0 −15.0000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.3.b.a 2
3.b odd 2 1 252.3.d.c 2
4.b odd 2 1 112.3.c.b 2
5.b even 2 1 700.3.d.a 2
5.c odd 4 2 700.3.h.a 4
7.b odd 2 1 inner 28.3.b.a 2
7.c even 3 2 196.3.h.b 4
7.d odd 6 2 196.3.h.b 4
8.b even 2 1 448.3.c.d 2
8.d odd 2 1 448.3.c.c 2
12.b even 2 1 1008.3.f.c 2
21.c even 2 1 252.3.d.c 2
21.g even 6 2 1764.3.z.i 4
21.h odd 6 2 1764.3.z.i 4
28.d even 2 1 112.3.c.b 2
28.f even 6 2 784.3.s.d 4
28.g odd 6 2 784.3.s.d 4
35.c odd 2 1 700.3.d.a 2
35.f even 4 2 700.3.h.a 4
56.e even 2 1 448.3.c.c 2
56.h odd 2 1 448.3.c.d 2
84.h odd 2 1 1008.3.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.b.a 2 1.a even 1 1 trivial
28.3.b.a 2 7.b odd 2 1 inner
112.3.c.b 2 4.b odd 2 1
112.3.c.b 2 28.d even 2 1
196.3.h.b 4 7.c even 3 2
196.3.h.b 4 7.d odd 6 2
252.3.d.c 2 3.b odd 2 1
252.3.d.c 2 21.c even 2 1
448.3.c.c 2 8.d odd 2 1
448.3.c.c 2 56.e even 2 1
448.3.c.d 2 8.b even 2 1
448.3.c.d 2 56.h odd 2 1
700.3.d.a 2 5.b even 2 1
700.3.d.a 2 35.c odd 2 1
700.3.h.a 4 5.c odd 4 2
700.3.h.a 4 35.f even 4 2
784.3.s.d 4 28.f even 6 2
784.3.s.d 4 28.g odd 6 2
1008.3.f.c 2 12.b even 2 1
1008.3.f.c 2 84.h odd 2 1
1764.3.z.i 4 21.g even 6 2
1764.3.z.i 4 21.h odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 24 \) Copy content Toggle raw display
$5$ \( T^{2} + 24 \) Copy content Toggle raw display
$7$ \( T^{2} - 10T + 49 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24 \) Copy content Toggle raw display
$17$ \( T^{2} + 384 \) Copy content Toggle raw display
$19$ \( T^{2} + 600 \) Copy content Toggle raw display
$23$ \( (T + 30)^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2400 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 384 \) Copy content Toggle raw display
$53$ \( (T - 90)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 600 \) Copy content Toggle raw display
$61$ \( T^{2} + 600 \) Copy content Toggle raw display
$67$ \( (T + 70)^{2} \) Copy content Toggle raw display
$71$ \( (T - 42)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11616 \) Copy content Toggle raw display
$79$ \( (T - 74)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4056 \) Copy content Toggle raw display
$89$ \( T^{2} + 21600 \) Copy content Toggle raw display
$97$ \( T^{2} + 6144 \) Copy content Toggle raw display
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