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Properties

Label 2028.4.b.c
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

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2028,4,Mod(337,2028)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,0,0,0,0,18,0,0,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(119.655873492\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content 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 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + 3 q^{3} + 9 \beta q^{5} + 4 \beta q^{7} + 9 q^{9} + 18 \beta q^{11} + 27 \beta q^{15} - 18 q^{17} + 50 \beta q^{19} + 12 \beta q^{21} - 72 q^{23} - 199 q^{25} + 27 q^{27} - 234 q^{29} + 8 \beta q^{31}+ \cdots + 162 \beta q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 18 q^{9} - 36 q^{17} - 144 q^{23} - 398 q^{25} + 54 q^{27} - 468 q^{29} - 288 q^{35} - 904 q^{43} + 558 q^{49} - 108 q^{51} + 828 q^{53} - 1296 q^{55} + 844 q^{61} - 432 q^{69} - 1194 q^{75}+ \cdots - 3600 q^{95}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 − 18.0000i 0 − 8.00000i 0 9.00000 0
337.2 0 3.00000 0 18.0000i 0 8.00000i 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.c 2
13.b even 2 1 inner 2028.4.b.c 2
13.d odd 4 1 12.4.a.a ✓ 1
13.d odd 4 1 2028.4.a.c 1
39.f even 4 1 36.4.a.a 1
52.f even 4 1 48.4.a.a 1
65.f even 4 1 300.4.d.e 2
65.g odd 4 1 300.4.a.b 1
65.k even 4 1 300.4.d.e 2
91.i even 4 1 588.4.a.c 1
91.z odd 12 2 588.4.i.d 2
91.bb even 12 2 588.4.i.e 2
104.j odd 4 1 192.4.a.f 1
104.m even 4 1 192.4.a.l 1
117.y odd 12 2 324.4.e.h 2
117.z even 12 2 324.4.e.a 2
143.g even 4 1 1452.4.a.d 1
156.l odd 4 1 144.4.a.g 1
195.j odd 4 1 900.4.d.c 2
195.n even 4 1 900.4.a.g 1
195.u odd 4 1 900.4.d.c 2
208.l even 4 1 768.4.d.j 2
208.m odd 4 1 768.4.d.g 2
208.r odd 4 1 768.4.d.g 2
208.s even 4 1 768.4.d.j 2
260.l odd 4 1 1200.4.f.d 2
260.s odd 4 1 1200.4.f.d 2
260.u even 4 1 1200.4.a.be 1
273.o odd 4 1 1764.4.a.b 1
273.cb odd 12 2 1764.4.k.o 2
273.cd even 12 2 1764.4.k.b 2
312.w odd 4 1 576.4.a.a 1
312.y even 4 1 576.4.a.b 1
364.p odd 4 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a ✓ 1 13.d odd 4 1
36.4.a.a 1 39.f even 4 1
48.4.a.a 1 52.f even 4 1
144.4.a.g 1 156.l odd 4 1
192.4.a.f 1 104.j odd 4 1
192.4.a.l 1 104.m even 4 1
300.4.a.b 1 65.g odd 4 1
300.4.d.e 2 65.f even 4 1
300.4.d.e 2 65.k even 4 1
324.4.e.a 2 117.z even 12 2
324.4.e.h 2 117.y odd 12 2
576.4.a.a 1 312.w odd 4 1
576.4.a.b 1 312.y even 4 1
588.4.a.c 1 91.i even 4 1
588.4.i.d 2 91.z odd 12 2
588.4.i.e 2 91.bb even 12 2
768.4.d.g 2 208.m odd 4 1
768.4.d.g 2 208.r odd 4 1
768.4.d.j 2 208.l even 4 1
768.4.d.j 2 208.s even 4 1
900.4.a.g 1 195.n even 4 1
900.4.d.c 2 195.j odd 4 1
900.4.d.c 2 195.u odd 4 1
1200.4.a.be 1 260.u even 4 1
1200.4.f.d 2 260.l odd 4 1
1200.4.f.d 2 260.s odd 4 1
1452.4.a.d 1 143.g even 4 1
1764.4.a.b 1 273.o odd 4 1
1764.4.k.b 2 273.cd even 12 2
1764.4.k.o 2 273.cb odd 12 2
2028.4.a.c 1 13.d odd 4 1
2028.4.b.c 2 1.a even 1 1 trivial
2028.4.b.c 2 13.b even 2 1 inner
2352.4.a.bk 1 364.p odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 324 \) 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
$5$ \( T^{2} + 324 \) Copy content Toggle raw display
$7$ \( T^{2} + 64 \) Copy content Toggle raw display
$11$ \( T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10000 \) Copy content Toggle raw display
$23$ \( (T + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T + 234)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 256 \) Copy content Toggle raw display
$37$ \( T^{2} + 51076 \) Copy content Toggle raw display
$41$ \( T^{2} + 8100 \) Copy content Toggle raw display
$43$ \( (T + 452)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 186624 \) Copy content Toggle raw display
$53$ \( (T - 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 467856 \) Copy content Toggle raw display
$61$ \( (T - 422)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 110224 \) Copy content Toggle raw display
$71$ \( T^{2} + 129600 \) Copy content Toggle raw display
$73$ \( T^{2} + 676 \) Copy content Toggle raw display
$79$ \( (T - 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1411344 \) Copy content Toggle raw display
$89$ \( T^{2} + 396900 \) Copy content Toggle raw display
$97$ \( T^{2} + 1110916 \) Copy content Toggle raw display
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