Properties

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

Related objects

Downloads

Learn more about

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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + 2 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + ( -1 + 3 \zeta_{6} ) q^{21} -8 \zeta_{6} q^{23} - q^{27} + 4 q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} -9 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{39} + 6 q^{41} + q^{43} -6 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} -4 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} + ( 2 + \zeta_{6} ) q^{63} + ( 12 - 12 \zeta_{6} ) q^{67} -8 q^{69} + 10 q^{71} + ( 1 - \zeta_{6} ) q^{73} + ( 2 - 6 \zeta_{6} ) q^{77} -7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -18 q^{83} + ( 4 - 4 \zeta_{6} ) q^{87} -10 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} + 3 \zeta_{6} q^{93} -5 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 4q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 4q^{7} - q^{9} - 2q^{11} + 4q^{13} + 2q^{17} - 4q^{19} + q^{21} - 8q^{23} - 2q^{27} + 8q^{29} - 3q^{31} + 2q^{33} - 9q^{37} + 2q^{39} + 12q^{41} + 2q^{43} - 6q^{47} + 2q^{49} - 2q^{51} + 2q^{53} - 8q^{57} + 6q^{59} + q^{61} + 5q^{63} + 12q^{67} - 16q^{69} + 20q^{71} + q^{73} - 2q^{77} - 7q^{79} - q^{81} - 36q^{83} + 4q^{87} - 10q^{89} - 8q^{91} + 3q^{93} - 10q^{97} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0 0 −2.00000 + 1.73205i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −2.00000 1.73205i 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.e yes 2
5.b even 2 1 2100.2.q.c 2
5.c odd 4 2 2100.2.bc.b 4
7.c even 3 1 inner 2100.2.q.e yes 2
35.j even 6 1 2100.2.q.c 2
35.l odd 12 2 2100.2.bc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.c 2 5.b even 2 1
2100.2.q.c 2 35.j even 6 1
2100.2.q.e yes 2 1.a even 1 1 trivial
2100.2.q.e yes 2 7.c even 3 1 inner
2100.2.bc.b 4 5.c odd 4 2
2100.2.bc.b 4 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 \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + T^{2} \)
$5$ 1
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \)
$37$ \( 1 + 9 T + 44 T^{2} + 333 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{2} \)
$73$ \( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 18 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 5 T + 97 T^{2} )^{2} \)
show more
show less