Properties

Label 2028.4.b.a
Level $2028$
Weight $4$
Character orbit 2028.b
Analytic conductor $119.656$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2028,4,Mod(337,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2028.b (of order \(2\), degree \(1\), not 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: \(119.655873492\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + 3 \beta q^{5} - 2 \beta q^{7} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 3 \beta q^{5} - 2 \beta q^{7} + 9 q^{9} + 18 \beta q^{11} - 9 \beta q^{15} - 66 q^{17} - 28 \beta q^{19} + 6 \beta q^{21} - 96 q^{23} + 89 q^{25} - 27 q^{27} + 222 q^{29} - 130 \beta q^{31} - 54 \beta q^{33} + 24 q^{35} - 53 \beta q^{37} + 45 \beta q^{41} - 44 q^{43} + 27 \beta q^{45} + 84 \beta q^{47} + 327 q^{49} + 198 q^{51} + 30 q^{53} - 216 q^{55} + 84 \beta q^{57} + 174 \beta q^{59} - 346 q^{61} - 18 \beta q^{63} + 128 \beta q^{67} + 288 q^{69} + 84 \beta q^{71} - 407 \beta q^{73} - 267 q^{75} + 144 q^{77} + 200 q^{79} + 81 q^{81} - 618 \beta q^{83} - 198 \beta q^{85} - 666 q^{87} + 159 \beta q^{89} + 390 \beta q^{93} + 336 q^{95} + 251 \beta q^{97} + 162 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 18 q^{9} - 132 q^{17} - 192 q^{23} + 178 q^{25} - 54 q^{27} + 444 q^{29} + 48 q^{35} - 88 q^{43} + 654 q^{49} + 396 q^{51} + 60 q^{53} - 432 q^{55} - 692 q^{61} + 576 q^{69} - 534 q^{75} + 288 q^{77} + 400 q^{79} + 162 q^{81} - 1332 q^{87} + 672 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 −3.00000 0 6.00000i 0 4.00000i 0 9.00000 0
337.2 0 −3.00000 0 6.00000i 0 4.00000i 0 9.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.4.b.a 2
13.b even 2 1 inner 2028.4.b.a 2
13.d odd 4 1 156.4.a.a 1
13.d odd 4 1 2028.4.a.a 1
39.f even 4 1 468.4.a.b 1
52.f even 4 1 624.4.a.h 1
104.j odd 4 1 2496.4.a.n 1
104.m even 4 1 2496.4.a.e 1
156.l odd 4 1 1872.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.a 1 13.d odd 4 1
468.4.a.b 1 39.f even 4 1
624.4.a.h 1 52.f even 4 1
1872.4.a.j 1 156.l odd 4 1
2028.4.a.a 1 13.d odd 4 1
2028.4.b.a 2 1.a even 1 1 trivial
2028.4.b.a 2 13.b even 2 1 inner
2496.4.a.e 1 104.m even 4 1
2496.4.a.n 1 104.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 36 \) acting on \(S_{4}^{\mathrm{new}}(2028, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 66)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3136 \) Copy content Toggle raw display
$23$ \( (T + 96)^{2} \) Copy content Toggle raw display
$29$ \( (T - 222)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 67600 \) Copy content Toggle raw display
$37$ \( T^{2} + 11236 \) Copy content Toggle raw display
$41$ \( T^{2} + 8100 \) Copy content Toggle raw display
$43$ \( (T + 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 28224 \) Copy content Toggle raw display
$53$ \( (T - 30)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 121104 \) Copy content Toggle raw display
$61$ \( (T + 346)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 65536 \) Copy content Toggle raw display
$71$ \( T^{2} + 28224 \) Copy content Toggle raw display
$73$ \( T^{2} + 662596 \) Copy content Toggle raw display
$79$ \( (T - 200)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1527696 \) Copy content Toggle raw display
$89$ \( T^{2} + 101124 \) Copy content Toggle raw display
$97$ \( T^{2} + 252004 \) Copy content Toggle raw display
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