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Properties

Label 2100.2.q.l
Level $2100$
Weight $2$
Character orbit 2100.q
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1201,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.q (of order \(3\), degree \(2\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.17819046144.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 10x^{6} + 8x^{5} + 38x^{4} - 4x^{3} + 16x^{2} + 4x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{6} - \beta_1) q^{7} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{6} - \beta_1) q^{7} + ( - \beta_{4} - 1) q^{9} + ( - \beta_{6} + \beta_{5} + \beta_{4}) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{13} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{7} - \beta_{5} + \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 2 q^{7} - 4 q^{9} - 4 q^{11} - 4 q^{13} + 2 q^{17} - 4 q^{19} - 2 q^{21} - 6 q^{23} + 8 q^{27} - 12 q^{29} - 4 q^{33} - 4 q^{37} + 2 q^{39} + 24 q^{41} + 32 q^{43} - 2 q^{47} - 4 q^{49} + 2 q^{51} - 20 q^{53} + 8 q^{57} + 14 q^{59} - 16 q^{61} + 4 q^{63} - 18 q^{67} + 12 q^{69} - 28 q^{71} + 8 q^{73} + 10 q^{77} + 8 q^{79} - 4 q^{81} - 20 q^{83} + 6 q^{87} + 8 q^{89} - 20 q^{91} - 20 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 10x^{6} + 8x^{5} + 38x^{4} - 4x^{3} + 16x^{2} + 4x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} + 78\nu^{6} - 137\nu^{5} + 1009\nu^{4} + 820\nu^{3} + 2992\nu^{2} - 786\nu + 712 ) / 450 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} + 14\nu^{6} - 81\nu^{5} - 108\nu^{4} - 390\nu^{3} - 54\nu^{2} - 18\nu - 44 ) / 150 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 31\nu^{6} + 124\nu^{5} + 7\nu^{4} + 310\nu^{3} - 434\nu^{2} + 122\nu - 124 ) / 150 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -52\nu^{7} + 117\nu^{6} - 493\nu^{5} - 424\nu^{4} - 1270\nu^{3} + 788\nu^{2} + 696\nu - 682 ) / 450 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -58\nu^{7} + 168\nu^{6} - 697\nu^{5} + 29\nu^{4} - 1780\nu^{3} + 1502\nu^{2} - 1716\nu - 478 ) / 450 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 9\nu^{6} + 43\nu^{5} + 19\nu^{4} + 154\nu^{3} - 50\nu^{2} + 72\nu - 20 ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 3\beta_{4} - 2\beta_{3} - \beta_{2} + 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 2\beta_{6} + \beta_{5} - 9\beta_{3} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} - 12\beta_{6} + 12\beta_{5} + 26\beta_{4} + 10\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{7} - 10\beta_{6} + 30\beta_{5} + 76\beta_{4} + 104\beta_{3} + 40\beta_{2} - 104\beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 144\beta_{7} + 114\beta_{6} - 30\beta_{5} + 354\beta_{3} + 30\beta_{2} + 292 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 384\beta_{7} + 498\beta_{6} - 498\beta_{5} - 978\beta_{4} - 384\beta_{2} + 1258\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
0.350883 − 0.607748i
−0.234240 + 0.405716i
1.75161 − 3.03388i
−0.868255 + 1.50386i
0.350883 + 0.607748i
−0.234240 − 0.405716i
1.75161 + 3.03388i
−0.868255 − 1.50386i
0 −0.500000 + 0.866025i 0 0 0 −2.30641 + 1.29633i 0 −0.500000 − 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 0 0 −0.687541 − 2.55486i 0 −0.500000 − 0.866025i 0
1201.3 0 −0.500000 + 0.866025i 0 0 0 −0.618546 + 2.57243i 0 −0.500000 − 0.866025i 0
1201.4 0 −0.500000 + 0.866025i 0 0 0 2.61250 + 0.418148i 0 −0.500000 − 0.866025i 0
1801.1 0 −0.500000 − 0.866025i 0 0 0 −2.30641 − 1.29633i 0 −0.500000 + 0.866025i 0
1801.2 0 −0.500000 − 0.866025i 0 0 0 −0.687541 + 2.55486i 0 −0.500000 + 0.866025i 0
1801.3 0 −0.500000 − 0.866025i 0 0 0 −0.618546 − 2.57243i 0 −0.500000 + 0.866025i 0
1801.4 0 −0.500000 − 0.866025i 0 0 0 2.61250 − 0.418148i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1201.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.q.l 8
5.b even 2 1 2100.2.q.m 8
5.c odd 4 2 420.2.bb.a ✓ 16
7.c even 3 1 inner 2100.2.q.l 8
15.e even 4 2 1260.2.bm.c 16
20.e even 4 2 1680.2.di.e 16
35.f even 4 2 2940.2.bb.i 16
35.j even 6 1 2100.2.q.m 8
35.k even 12 2 2940.2.k.g 8
35.k even 12 2 2940.2.bb.i 16
35.l odd 12 2 420.2.bb.a ✓ 16
35.l odd 12 2 2940.2.k.f 8
105.x even 12 2 1260.2.bm.c 16
140.w even 12 2 1680.2.di.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bb.a ✓ 16 5.c odd 4 2
420.2.bb.a ✓ 16 35.l odd 12 2
1260.2.bm.c 16 15.e even 4 2
1260.2.bm.c 16 105.x even 12 2
1680.2.di.e 16 20.e even 4 2
1680.2.di.e 16 140.w even 12 2
2100.2.q.l 8 1.a even 1 1 trivial
2100.2.q.l 8 7.c even 3 1 inner
2100.2.q.m 8 5.b even 2 1
2100.2.q.m 8 35.j even 6 1
2940.2.k.f 8 35.l odd 12 2
2940.2.k.g 8 35.k even 12 2
2940.2.bb.i 16 35.f even 4 2
2940.2.bb.i 16 35.k even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{8} + 4T_{11}^{7} + 22T_{11}^{6} + 20T_{11}^{5} + 110T_{11}^{4} + 20T_{11}^{3} + 568T_{11}^{2} - 308T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 24T_{13}^{2} + 75 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 24 T^{2} + 75)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 25281 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + \cdots + 1098)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 102 T^{6} + \cdots + 1212201 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 30625 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 42 T^{2} + \cdots + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + \cdots + 499)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 2 T^{7} + \cdots + 379456 \) Copy content Toggle raw display
$53$ \( T^{8} + 20 T^{7} + \cdots + 9216 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} + \cdots + 142611364 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + \cdots + 3984016 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots - 8802)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 8 T^{7} + \cdots + 3530641 \) Copy content Toggle raw display
$79$ \( T^{8} - 8 T^{7} + \cdots + 1089 \) Copy content Toggle raw display
$83$ \( (T^{4} + 10 T^{3} + \cdots + 5282)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 36264484 \) Copy content Toggle raw display
$97$ \( (T^{4} + 10 T^{3} + \cdots + 844)^{2} \) Copy content Toggle raw display
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