Properties

Label 2028.4.a.c
Level $2028$
Weight $4$
Character orbit 2028.a
Self dual yes
Analytic conductor $119.656$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2028.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.655873492\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 18q^{5} - 8q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 18q^{5} - 8q^{7} + 9q^{9} - 36q^{11} + 54q^{15} + 18q^{17} + 100q^{19} - 24q^{21} + 72q^{23} + 199q^{25} + 27q^{27} - 234q^{29} + 16q^{31} - 108q^{33} - 144q^{35} + 226q^{37} - 90q^{41} + 452q^{43} + 162q^{45} - 432q^{47} - 279q^{49} + 54q^{51} + 414q^{53} - 648q^{55} + 300q^{57} + 684q^{59} + 422q^{61} - 72q^{63} - 332q^{67} + 216q^{69} + 360q^{71} - 26q^{73} + 597q^{75} + 288q^{77} + 512q^{79} + 81q^{81} + 1188q^{83} + 324q^{85} - 702q^{87} + 630q^{89} + 48q^{93} + 1800q^{95} + 1054q^{97} - 324q^{99} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 3.00000 0 18.0000 0 −8.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.4.a.c 1
13.b even 2 1 12.4.a.a 1
13.d odd 4 2 2028.4.b.c 2
39.d odd 2 1 36.4.a.a 1
52.b odd 2 1 48.4.a.a 1
65.d even 2 1 300.4.a.b 1
65.h odd 4 2 300.4.d.e 2
91.b odd 2 1 588.4.a.c 1
91.r even 6 2 588.4.i.d 2
91.s odd 6 2 588.4.i.e 2
104.e even 2 1 192.4.a.f 1
104.h odd 2 1 192.4.a.l 1
117.n odd 6 2 324.4.e.a 2
117.t even 6 2 324.4.e.h 2
143.d odd 2 1 1452.4.a.d 1
156.h even 2 1 144.4.a.g 1
195.e odd 2 1 900.4.a.g 1
195.s even 4 2 900.4.d.c 2
208.o odd 4 2 768.4.d.j 2
208.p even 4 2 768.4.d.g 2
260.g odd 2 1 1200.4.a.be 1
260.p even 4 2 1200.4.f.d 2
273.g even 2 1 1764.4.a.b 1
273.w odd 6 2 1764.4.k.b 2
273.ba even 6 2 1764.4.k.o 2
312.b odd 2 1 576.4.a.b 1
312.h even 2 1 576.4.a.a 1
364.h even 2 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 13.b even 2 1
36.4.a.a 1 39.d odd 2 1
48.4.a.a 1 52.b odd 2 1
144.4.a.g 1 156.h even 2 1
192.4.a.f 1 104.e even 2 1
192.4.a.l 1 104.h odd 2 1
300.4.a.b 1 65.d even 2 1
300.4.d.e 2 65.h odd 4 2
324.4.e.a 2 117.n odd 6 2
324.4.e.h 2 117.t even 6 2
576.4.a.a 1 312.h even 2 1
576.4.a.b 1 312.b odd 2 1
588.4.a.c 1 91.b odd 2 1
588.4.i.d 2 91.r even 6 2
588.4.i.e 2 91.s odd 6 2
768.4.d.g 2 208.p even 4 2
768.4.d.j 2 208.o odd 4 2
900.4.a.g 1 195.e odd 2 1
900.4.d.c 2 195.s even 4 2
1200.4.a.be 1 260.g odd 2 1
1200.4.f.d 2 260.p even 4 2
1452.4.a.d 1 143.d odd 2 1
1764.4.a.b 1 273.g even 2 1
1764.4.k.b 2 273.w odd 6 2
1764.4.k.o 2 273.ba even 6 2
2028.4.a.c 1 1.a even 1 1 trivial
2028.4.b.c 2 13.d odd 4 2
2352.4.a.bk 1 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2028))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -18 + T \)
$7$ \( 8 + T \)
$11$ \( 36 + T \)
$13$ \( T \)
$17$ \( -18 + T \)
$19$ \( -100 + T \)
$23$ \( -72 + T \)
$29$ \( 234 + T \)
$31$ \( -16 + T \)
$37$ \( -226 + T \)
$41$ \( 90 + T \)
$43$ \( -452 + T \)
$47$ \( 432 + T \)
$53$ \( -414 + T \)
$59$ \( -684 + T \)
$61$ \( -422 + T \)
$67$ \( 332 + T \)
$71$ \( -360 + T \)
$73$ \( 26 + T \)
$79$ \( -512 + T \)
$83$ \( -1188 + T \)
$89$ \( -630 + T \)
$97$ \( -1054 + T \)
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