Properties

Label 2028.2.q.h
Level $2028$
Weight $2$
Character orbit 2028.q
Analytic conductor $16.194$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1936615299\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{3} + 4 \zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12}^{2} q^{3} + 4 \zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{11} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{15} + ( 2 - 2 \zeta_{12}^{2} ) q^{17} -2 \zeta_{12} q^{19} + 2 \zeta_{12}^{3} q^{21} -11 q^{25} - q^{27} + 6 \zeta_{12}^{2} q^{29} + 10 \zeta_{12}^{3} q^{31} -4 \zeta_{12} q^{33} + ( -8 + 8 \zeta_{12}^{2} ) q^{35} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{37} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{41} + ( 4 - 4 \zeta_{12}^{2} ) q^{43} -4 \zeta_{12} q^{45} -4 \zeta_{12}^{3} q^{47} -3 \zeta_{12}^{2} q^{49} + 2 q^{51} -10 q^{53} -16 \zeta_{12}^{2} q^{55} -2 \zeta_{12}^{3} q^{57} + 8 \zeta_{12} q^{59} + ( 14 - 14 \zeta_{12}^{2} ) q^{61} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{67} + 16 \zeta_{12} q^{71} -10 \zeta_{12}^{3} q^{73} -11 \zeta_{12}^{2} q^{75} -8 q^{77} -16 q^{79} -\zeta_{12}^{2} q^{81} + 8 \zeta_{12} q^{85} + ( -6 + 6 \zeta_{12}^{2} ) q^{87} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -10 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{93} + ( 8 - 8 \zeta_{12}^{2} ) q^{95} -2 \zeta_{12} q^{97} -4 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 2q^{9} + 4q^{17} - 44q^{25} - 4q^{27} + 12q^{29} - 16q^{35} + 8q^{43} - 6q^{49} + 8q^{51} - 40q^{53} - 32q^{55} + 28q^{61} - 22q^{75} - 32q^{77} - 64q^{79} - 2q^{81} - 12q^{87} + 16q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0.500000 + 0.866025i 0 4.00000i 0 −1.73205 1.00000i 0 −0.500000 + 0.866025i 0
361.2 0 0.500000 + 0.866025i 0 4.00000i 0 1.73205 + 1.00000i 0 −0.500000 + 0.866025i 0
1837.1 0 0.500000 0.866025i 0 4.00000i 0 1.73205 1.00000i 0 −0.500000 0.866025i 0
1837.2 0 0.500000 0.866025i 0 4.00000i 0 −1.73205 + 1.00000i 0 −0.500000 0.866025i 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
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.2.q.h 4
13.b even 2 1 inner 2028.2.q.h 4
13.c even 3 1 2028.2.b.a 2
13.c even 3 1 inner 2028.2.q.h 4
13.d odd 4 1 2028.2.i.e 2
13.d odd 4 1 2028.2.i.g 2
13.e even 6 1 2028.2.b.a 2
13.e even 6 1 inner 2028.2.q.h 4
13.f odd 12 1 156.2.a.a 1
13.f odd 12 1 2028.2.a.c 1
13.f odd 12 1 2028.2.i.e 2
13.f odd 12 1 2028.2.i.g 2
39.h odd 6 1 6084.2.b.j 2
39.i odd 6 1 6084.2.b.j 2
39.k even 12 1 468.2.a.d 1
39.k even 12 1 6084.2.a.b 1
52.l even 12 1 624.2.a.e 1
52.l even 12 1 8112.2.a.bi 1
65.o even 12 1 3900.2.h.b 2
65.s odd 12 1 3900.2.a.m 1
65.t even 12 1 3900.2.h.b 2
91.bc even 12 1 7644.2.a.k 1
104.u even 12 1 2496.2.a.o 1
104.x odd 12 1 2496.2.a.bc 1
117.w odd 12 1 4212.2.i.l 2
117.x even 12 1 4212.2.i.b 2
117.bb odd 12 1 4212.2.i.l 2
117.bc even 12 1 4212.2.i.b 2
156.v odd 12 1 1872.2.a.s 1
312.bo even 12 1 7488.2.a.c 1
312.bq odd 12 1 7488.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 13.f odd 12 1
468.2.a.d 1 39.k even 12 1
624.2.a.e 1 52.l even 12 1
1872.2.a.s 1 156.v odd 12 1
2028.2.a.c 1 13.f odd 12 1
2028.2.b.a 2 13.c even 3 1
2028.2.b.a 2 13.e even 6 1
2028.2.i.e 2 13.d odd 4 1
2028.2.i.e 2 13.f odd 12 1
2028.2.i.g 2 13.d odd 4 1
2028.2.i.g 2 13.f odd 12 1
2028.2.q.h 4 1.a even 1 1 trivial
2028.2.q.h 4 13.b even 2 1 inner
2028.2.q.h 4 13.c even 3 1 inner
2028.2.q.h 4 13.e even 6 1 inner
2496.2.a.o 1 104.u even 12 1
2496.2.a.bc 1 104.x odd 12 1
3900.2.a.m 1 65.s odd 12 1
3900.2.h.b 2 65.o even 12 1
3900.2.h.b 2 65.t even 12 1
4212.2.i.b 2 117.x even 12 1
4212.2.i.b 2 117.bc even 12 1
4212.2.i.l 2 117.w odd 12 1
4212.2.i.l 2 117.bb odd 12 1
6084.2.a.b 1 39.k even 12 1
6084.2.b.j 2 39.h odd 6 1
6084.2.b.j 2 39.i odd 6 1
7488.2.a.c 1 312.bo even 12 1
7488.2.a.d 1 312.bq odd 12 1
7644.2.a.k 1 91.bc even 12 1
8112.2.a.bi 1 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2028, [\chi])\):

\( T_{5}^{2} + 16 \)
\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)
\( T_{11}^{4} - 16 T_{11}^{2} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 16 + T^{2} )^{2} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( 256 - 16 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 4 - 2 T + T^{2} )^{2} \)
$19$ \( 16 - 4 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 36 - 6 T + T^{2} )^{2} \)
$31$ \( ( 100 + T^{2} )^{2} \)
$37$ \( 10000 - 100 T^{2} + T^{4} \)
$41$ \( 4096 - 64 T^{2} + T^{4} \)
$43$ \( ( 16 - 4 T + T^{2} )^{2} \)
$47$ \( ( 16 + T^{2} )^{2} \)
$53$ \( ( 10 + T )^{4} \)
$59$ \( 4096 - 64 T^{2} + T^{4} \)
$61$ \( ( 196 - 14 T + T^{2} )^{2} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( 65536 - 256 T^{2} + T^{4} \)
$73$ \( ( 100 + T^{2} )^{2} \)
$79$ \( ( 16 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( 256 - 16 T^{2} + T^{4} \)
$97$ \( 16 - 4 T^{2} + T^{4} \)
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