Properties

Label 2100.2.q.i
Level $2100$
Weight $2$
Character orbit 2100.q
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + \beta_1 q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + \beta_1 q^{7} + \beta_{2} q^{9} + (3 \beta_{2} + \beta_1 + 3) q^{11} + q^{13} + ( - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{19} + (\beta_{3} + \beta_1) q^{21} - 2 \beta_{2} q^{23} - q^{27} + ( - 2 \beta_{3} + 2) q^{29} + ( - 4 \beta_{2} - 4) q^{31} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{33} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{37} + (\beta_{2} + 1) q^{39} + ( - \beta_{3} - 3) q^{41} + (2 \beta_{3} - 2) q^{43} + (3 \beta_{3} - 5 \beta_{2} + 3 \beta_1) q^{47} + 7 \beta_{2} q^{49} + (\beta_{3} - \beta_{2} + \beta_1) q^{51} + ( - 5 \beta_{2} + \beta_1 - 5) q^{53} + ( - \beta_{3} + 2) q^{57} + ( - 7 \beta_{2} - 3 \beta_1 - 7) q^{59} + (4 \beta_{3} - \beta_{2} + 4 \beta_1) q^{61} + \beta_{3} q^{63} + 3 \beta_1 q^{67} + 2 q^{69} + ( - 2 \beta_{3} - 2) q^{71} + (\beta_{2} + 2 \beta_1 + 1) q^{73} + (3 \beta_{3} + 7 \beta_{2} + 3 \beta_1) q^{77} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{79} + ( - \beta_{2} - 1) q^{81} + ( - \beta_{3} + 1) q^{83} + (2 \beta_{2} + 2 \beta_1 + 2) q^{87} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{89} + \beta_1 q^{91} - 4 \beta_{2} q^{93} + ( - 2 \beta_{3} + 9) q^{97} + (\beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{9} + 6 q^{11} + 4 q^{13} - 2 q^{17} + 4 q^{19} + 4 q^{23} - 4 q^{27} + 8 q^{29} - 8 q^{31} - 6 q^{33} - 6 q^{37} + 2 q^{39} - 12 q^{41} - 8 q^{43} + 10 q^{47} - 14 q^{49} + 2 q^{51} - 10 q^{53} + 8 q^{57} - 14 q^{59} + 2 q^{61} + 8 q^{69} - 8 q^{71} + 2 q^{73} - 14 q^{77} + 8 q^{79} - 2 q^{81} + 4 q^{83} + 4 q^{87} + 2 q^{89} + 8 q^{93} + 36 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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.

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
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0 0.500000 0.866025i 0 0 0 −1.32288 + 2.29129i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 0 0 1.32288 2.29129i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −1.32288 2.29129i 0 −0.500000 + 0.866025i 0
1801.2 0 0.500000 + 0.866025i 0 0 0 1.32288 + 2.29129i 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
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.i yes 4
5.b even 2 1 2100.2.q.g 4
5.c odd 4 2 2100.2.bc.h 8
7.c even 3 1 inner 2100.2.q.i yes 4
35.j even 6 1 2100.2.q.g 4
35.l odd 12 2 2100.2.bc.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.g 4 5.b even 2 1
2100.2.q.g 4 35.j even 6 1
2100.2.q.i yes 4 1.a even 1 1 trivial
2100.2.q.i yes 4 7.c even 3 1 inner
2100.2.bc.h 8 5.c odd 4 2
2100.2.bc.h 8 35.l odd 12 2

Hecke kernels

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

\( T_{11}^{4} - 6T_{11}^{3} + 34T_{11}^{2} - 12T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 34 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 24)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + 138 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} + 82 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + 210 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + 115 T^{2} + \cdots + 12321 \) Copy content Toggle raw display
$67$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 24)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + 31 T^{2} + 54 T + 729 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + 55 T^{2} - 72 T + 81 \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + 66 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$97$ \( (T^{2} - 18 T + 53)^{2} \) Copy content Toggle raw display
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