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Properties

Label 2028.2.q.i
Level $2028$
Weight $2$
Character orbit 2028.q
Analytic conductor $16.194$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2028,2,Mod(361,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(16.1936615299\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - 1) q^{3} + ( - 2 \beta_{11} - \beta_{10} + \cdots - 2 \beta_{2}) q^{5} + (2 \beta_{11} + 3 \beta_{10} + \cdots - \beta_1) q^{7} - \beta_{7} q^{9} + ( - 3 \beta_{6} + \beta_{2} + \beta_1) q^{11}+ \cdots + ( - \beta_{11} - 3 \beta_{10} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 6 q^{9} + 6 q^{17} - 28 q^{23} + 4 q^{25} + 12 q^{27} - 20 q^{29} + 28 q^{35} - 10 q^{43} + 10 q^{49} - 12 q^{51} - 12 q^{53} + 14 q^{55} + 2 q^{61} - 28 q^{69} - 2 q^{75} - 84 q^{77} - 44 q^{79}+ \cdots - 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.385418 − 0.222521i
−1.56052 + 0.900969i
1.07992 − 0.623490i
−1.07992 + 0.623490i
1.56052 − 0.900969i
−0.385418 + 0.222521i
−0.385418 − 0.222521i
1.56052 + 0.900969i
−1.07992 − 0.623490i
1.07992 + 0.623490i
−1.56052 − 0.900969i
0.385418 + 0.222521i
0 −0.500000 − 0.866025i 0 − 3.04892i 0 4.37249 + 2.52446i 0 −0.500000 + 0.866025i 0
361.2 0 −0.500000 − 0.866025i 0 − 1.69202i 0 −0.266717 − 0.153989i 0 −0.500000 + 0.866025i 0
361.3 0 −0.500000 − 0.866025i 0 − 1.35690i 0 −0.556945 − 0.321552i 0 −0.500000 + 0.866025i 0
361.4 0 −0.500000 − 0.866025i 0 1.35690i 0 0.556945 + 0.321552i 0 −0.500000 + 0.866025i 0
361.5 0 −0.500000 − 0.866025i 0 1.69202i 0 0.266717 + 0.153989i 0 −0.500000 + 0.866025i 0
361.6 0 −0.500000 − 0.866025i 0 3.04892i 0 −4.37249 − 2.52446i 0 −0.500000 + 0.866025i 0
1837.1 0 −0.500000 + 0.866025i 0 − 3.04892i 0 −4.37249 + 2.52446i 0 −0.500000 − 0.866025i 0
1837.2 0 −0.500000 + 0.866025i 0 − 1.69202i 0 0.266717 − 0.153989i 0 −0.500000 − 0.866025i 0
1837.3 0 −0.500000 + 0.866025i 0 − 1.35690i 0 0.556945 − 0.321552i 0 −0.500000 − 0.866025i 0
1837.4 0 −0.500000 + 0.866025i 0 1.35690i 0 −0.556945 + 0.321552i 0 −0.500000 − 0.866025i 0
1837.5 0 −0.500000 + 0.866025i 0 1.69202i 0 −0.266717 + 0.153989i 0 −0.500000 − 0.866025i 0
1837.6 0 −0.500000 + 0.866025i 0 3.04892i 0 4.37249 − 2.52446i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
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.i 12
13.b even 2 1 inner 2028.2.q.i 12
13.c even 3 1 2028.2.b.g 6
13.c even 3 1 inner 2028.2.q.i 12
13.d odd 4 1 2028.2.i.j 6
13.d odd 4 1 2028.2.i.k 6
13.e even 6 1 2028.2.b.g 6
13.e even 6 1 inner 2028.2.q.i 12
13.f odd 12 1 2028.2.a.k ✓ 3
13.f odd 12 1 2028.2.a.l yes 3
13.f odd 12 1 2028.2.i.j 6
13.f odd 12 1 2028.2.i.k 6
39.h odd 6 1 6084.2.b.q 6
39.i odd 6 1 6084.2.b.q 6
39.k even 12 1 6084.2.a.z 3
39.k even 12 1 6084.2.a.ba 3
52.l even 12 1 8112.2.a.ca 3
52.l even 12 1 8112.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2028.2.a.k ✓ 3 13.f odd 12 1
2028.2.a.l yes 3 13.f odd 12 1
2028.2.b.g 6 13.c even 3 1
2028.2.b.g 6 13.e even 6 1
2028.2.i.j 6 13.d odd 4 1
2028.2.i.j 6 13.f odd 12 1
2028.2.i.k 6 13.d odd 4 1
2028.2.i.k 6 13.f odd 12 1
2028.2.q.i 12 1.a even 1 1 trivial
2028.2.q.i 12 13.b even 2 1 inner
2028.2.q.i 12 13.c even 3 1 inner
2028.2.q.i 12 13.e even 6 1 inner
6084.2.a.z 3 39.k even 12 1
6084.2.a.ba 3 39.k even 12 1
6084.2.b.q 6 39.h odd 6 1
6084.2.b.q 6 39.i odd 6 1
8112.2.a.ca 3 52.l even 12 1
8112.2.a.cd 3 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 \(S_{2}^{\mathrm{new}}(2028, [\chi])\):

\( T_{5}^{6} + 14T_{5}^{4} + 49T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{12} - 26T_{7}^{10} + 663T_{7}^{8} - 336T_{7}^{6} + 143T_{7}^{4} - 13T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{12} - 21T_{11}^{10} + 343T_{11}^{8} - 1960T_{11}^{6} + 8575T_{11}^{4} - 4802T_{11}^{2} + 2401 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 14 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 26 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 21 T^{10} + \cdots + 2401 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 3 T^{5} + 13 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 89 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$23$ \( (T^{6} + 14 T^{5} + \cdots + 49)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 10 T^{5} + \cdots + 169)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 110 T^{4} + \cdots + 32761)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 110 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} - 117 T^{10} + \cdots + 47458321 \) Copy content Toggle raw display
$43$ \( (T^{6} + 5 T^{5} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 161 T^{4} + \cdots + 90601)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 3 T^{2} + \cdots - 643)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1529041063936 \) Copy content Toggle raw display
$61$ \( (T^{6} - T^{5} + 73 T^{4} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2418052990081 \) Copy content Toggle raw display
$71$ \( T^{12} - 138 T^{10} + \cdots + 25411681 \) Copy content Toggle raw display
$73$ \( (T^{6} + 210 T^{4} + \cdots + 8281)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 11 T^{2} + \cdots + 13)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 362 T^{4} + \cdots + 841)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 896295799441 \) Copy content Toggle raw display
$97$ \( T^{12} - 251 T^{10} + \cdots + 707281 \) Copy content Toggle raw display
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