Properties

Label 2100.2.q.k
Level $2100$
Weight $2$
Character orbit 2100.q
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1) q^{7} + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1) q^{7} + \beta_1 q^{9} + (\beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_{3} + \beta_{2}) q^{17} - 7 \beta_1 q^{19} + ( - \beta_{2} + 1) q^{21} + ( - \beta_{3} + 2 \beta_{2}) q^{23} - q^{27} + (2 \beta_{3} - \beta_{2} - 6) q^{29} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{33} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{37} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{39} + ( - 2 \beta_{3} + \beta_{2}) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{43} + 6 \beta_1 q^{47} + ( - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 5) q^{49} + ( - \beta_{3} + 2 \beta_{2}) q^{51} + (2 \beta_{3} + 2 \beta_{2}) q^{53} + 7 q^{57} + (\beta_{3} + \beta_{2} + 6 \beta_1 + 6) q^{59} + ( - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{61} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{63} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{67} + ( - 2 \beta_{3} + \beta_{2}) q^{69} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{71} + (\beta_{3} + \beta_{2} + 5 \beta_1 + 5) q^{73} + (2 \beta_{3} - \beta_{2} - 12 \beta_1 - 6) q^{77} + 11 \beta_1 q^{79} + ( - \beta_1 - 1) q^{81} + (2 \beta_{3} - \beta_{2} + 6) q^{83} + (\beta_{3} + \beta_{2} - 6 \beta_1 - 6) q^{87} + (\beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{89} + ( - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 12) q^{91} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1) q^{93} + (4 \beta_{3} - 2 \beta_{2} - 8) q^{97} + ( - 2 \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{7} - 2 q^{9} + 4 q^{13} + 14 q^{19} + 4 q^{21} - 4 q^{27} - 24 q^{29} + 2 q^{31} - 2 q^{37} + 2 q^{39} + 4 q^{43} - 12 q^{47} + 10 q^{49} + 28 q^{57} + 12 q^{59} + 8 q^{61} + 2 q^{63} - 2 q^{67} + 10 q^{73} - 22 q^{79} - 2 q^{81} + 24 q^{83} - 12 q^{87} + 12 q^{89} + 38 q^{91} - 2 q^{93} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0 0.500000 0.866025i 0 0 0 −1.62132 + 2.09077i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 0 0 2.62132 0.358719i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −1.62132 2.09077i 0 −0.500000 + 0.866025i 0
1801.2 0 0.500000 + 0.866025i 0 0 0 2.62132 + 0.358719i 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.k 4
5.b even 2 1 420.2.q.d 4
5.c odd 4 2 2100.2.bc.f 8
7.c even 3 1 inner 2100.2.q.k 4
15.d odd 2 1 1260.2.s.e 4
20.d odd 2 1 1680.2.bg.t 4
35.c odd 2 1 2940.2.q.q 4
35.i odd 6 1 2940.2.a.p 2
35.i odd 6 1 2940.2.q.q 4
35.j even 6 1 420.2.q.d 4
35.j even 6 1 2940.2.a.r 2
35.l odd 12 2 2100.2.bc.f 8
105.o odd 6 1 1260.2.s.e 4
105.o odd 6 1 8820.2.a.bk 2
105.p even 6 1 8820.2.a.bf 2
140.p odd 6 1 1680.2.bg.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 5.b even 2 1
420.2.q.d 4 35.j even 6 1
1260.2.s.e 4 15.d odd 2 1
1260.2.s.e 4 105.o odd 6 1
1680.2.bg.t 4 20.d odd 2 1
1680.2.bg.t 4 140.p odd 6 1
2100.2.q.k 4 1.a even 1 1 trivial
2100.2.q.k 4 7.c even 3 1 inner
2100.2.bc.f 8 5.c odd 4 2
2100.2.bc.f 8 35.l odd 12 2
2940.2.a.p 2 35.i odd 6 1
2940.2.a.r 2 35.j even 6 1
2940.2.q.q 4 35.c odd 2 1
2940.2.q.q 4 35.i odd 6 1
8820.2.a.bf 2 105.p even 6 1
8820.2.a.bk 2 105.o odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 18T_{11}^{2} + 324 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 17 \) 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
$7$ \( T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 17)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + 75 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 17)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 120 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 10 T^{3} + 93 T^{2} - 70 T + 49 \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T + 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$97$ \( (T^{2} + 16 T - 8)^{2} \) Copy content Toggle raw display
show more
show less