Properties

Label 2100.2.bc.g
Level 2100
Weight 2
Character orbit 2100.bc
Analytic conductor 16.769
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{24}^{2} q^{3} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{2} q^{3} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{4} q^{9} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{11} + 5 \zeta_{24}^{6} q^{13} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} - \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{19} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{21} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{23} -\zeta_{24}^{6} q^{27} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{29} + ( 4 - 4 \zeta_{24}^{4} ) q^{31} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{33} + ( -5 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{37} + ( 5 - 5 \zeta_{24}^{4} ) q^{39} + ( -6 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{41} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{43} + ( 3 \zeta_{24} - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{47} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( 3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{51} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{53} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{57} + ( 6 - 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{59} + \zeta_{24}^{4} q^{61} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{63} + ( -\zeta_{24}^{2} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + 9 \zeta_{24}^{7} ) q^{67} -6 q^{69} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{71} + ( -5 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{73} + ( 3 \zeta_{24} - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 5 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{79} + ( -1 + \zeta_{24}^{4} ) q^{81} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{83} + ( -6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{87} + ( 3 \zeta_{24} + 12 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{89} + ( -5 + 10 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{91} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{93} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{97} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{9} - 4q^{19} - 4q^{21} + 16q^{31} + 20q^{39} - 48q^{41} - 20q^{49} + 24q^{59} + 4q^{61} - 48q^{69} + 20q^{79} - 4q^{81} + 48q^{89} - 20q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\zeta_{24}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0 −0.866025 0.500000i 0 0 0 −0.358719 + 2.62132i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 2.09077 1.62132i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −2.09077 + 1.62132i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 0.358719 2.62132i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −0.358719 2.62132i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 2.09077 + 1.62132i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −2.09077 1.62132i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 0.358719 + 2.62132i 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.g 8
5.b even 2 1 inner 2100.2.bc.g 8
5.c odd 4 1 2100.2.q.f 4
5.c odd 4 1 2100.2.q.j yes 4
7.c even 3 1 inner 2100.2.bc.g 8
35.j even 6 1 inner 2100.2.bc.g 8
35.l odd 12 1 2100.2.q.f 4
35.l odd 12 1 2100.2.q.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.f 4 5.c odd 4 1
2100.2.q.f 4 35.l odd 12 1
2100.2.q.j yes 4 5.c odd 4 1
2100.2.q.j yes 4 35.l odd 12 1
2100.2.bc.g 8 1.a even 1 1 trivial
2100.2.bc.g 8 5.b even 2 1 inner
2100.2.bc.g 8 7.c even 3 1 inner
2100.2.bc.g 8 35.j even 6 1 inner

Hecke kernels

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

\( T_{11}^{4} + 18 T_{11}^{2} + 324 \)
\( T_{13}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ 1
$7$ \( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 16 T^{2} - 33 T^{4} + 4624 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 2 T - 17 T^{2} - 34 T^{3} + 4 T^{4} - 646 T^{5} - 6137 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 14 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 4 T^{2} - 906 T^{4} + 7396 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 80 T^{2} + 2974 T^{4} - 79360 T^{6} - 7279805 T^{8} - 175306240 T^{10} + 14512171294 T^{12} + 862337226320 T^{14} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 88 T^{2} + 4935 T^{4} + 247192 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 12744 T^{5} + 27848 T^{6} - 2464548 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{4}( 1 + 13 T + 61 T^{2} )^{4} \)
$67$ \( 1 - 58 T^{2} - 5807 T^{4} - 11194 T^{6} + 48855124 T^{8} - 50249866 T^{10} - 117017559647 T^{12} - 5246586165802 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 + 70 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( 1 + 98 T^{2} + 3745 T^{4} - 470302 T^{6} - 45250076 T^{8} - 2506239358 T^{10} + 106351412545 T^{12} + 14830754176322 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 10 T - 65 T^{2} - 70 T^{3} + 13084 T^{4} - 5530 T^{5} - 405665 T^{6} - 4930390 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 224 T^{2} + 23730 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 24 T + 272 T^{2} - 3024 T^{3} + 33231 T^{4} - 269136 T^{5} + 2154512 T^{6} - 16919256 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} )^{2} \)
show more
show less