Properties

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

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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: no (minimal twist has level 420)
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.

\(f(q)\) \(=\) \( q -\zeta_{24}^{2} q^{3} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{2} q^{3} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{4} q^{9} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{11} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{13} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{17} -7 \zeta_{24}^{4} q^{19} + ( 1 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{6} q^{27} + ( 6 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{29} + ( 1 + 6 \zeta_{24}^{3} - \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{33} + ( -3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{37} + ( -1 + 3 \zeta_{24}^{3} + \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{39} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{41} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{43} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{47} + ( -5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( -3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{51} + ( -6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{53} + 7 \zeta_{24}^{6} q^{57} + ( -6 - 3 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{59} + ( -6 \zeta_{24} + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{61} + ( 2 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{63} + ( -\zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{67} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{69} + ( -9 \zeta_{24} - 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{71} + ( -5 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} - 12 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{77} + 11 \zeta_{24}^{4} 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}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{87} + ( -3 \zeta_{24} - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{89} + ( 12 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{91} + ( -6 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{93} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{97} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{9} - 28q^{19} + 8q^{21} + 48q^{29} + 4q^{31} - 4q^{39} - 20q^{49} - 24q^{59} + 16q^{61} + 44q^{79} - 4q^{81} - 24q^{89} + 76q^{91} + O(q^{100}) \)

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 −2.09077 1.62132i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 0.358719 + 2.62132i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −0.358719 2.62132i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 2.09077 + 1.62132i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −2.09077 + 1.62132i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 0.358719 2.62132i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −0.358719 + 2.62132i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 2.09077 1.62132i 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.f 8
5.b even 2 1 inner 2100.2.bc.f 8
5.c odd 4 1 420.2.q.d 4
5.c odd 4 1 2100.2.q.k 4
7.c even 3 1 inner 2100.2.bc.f 8
15.e even 4 1 1260.2.s.e 4
20.e even 4 1 1680.2.bg.t 4
35.f even 4 1 2940.2.q.q 4
35.j even 6 1 inner 2100.2.bc.f 8
35.k even 12 1 2940.2.a.p 2
35.k even 12 1 2940.2.q.q 4
35.l odd 12 1 420.2.q.d 4
35.l odd 12 1 2100.2.q.k 4
35.l odd 12 1 2940.2.a.r 2
105.w odd 12 1 8820.2.a.bf 2
105.x even 12 1 1260.2.s.e 4
105.x even 12 1 8820.2.a.bk 2
140.w even 12 1 1680.2.bg.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 5.c odd 4 1
420.2.q.d 4 35.l odd 12 1
1260.2.s.e 4 15.e even 4 1
1260.2.s.e 4 105.x even 12 1
1680.2.bg.t 4 20.e even 4 1
1680.2.bg.t 4 140.w even 12 1
2100.2.q.k 4 5.c odd 4 1
2100.2.q.k 4 35.l odd 12 1
2100.2.bc.f 8 1.a even 1 1 trivial
2100.2.bc.f 8 5.b even 2 1 inner
2100.2.bc.f 8 7.c even 3 1 inner
2100.2.bc.f 8 35.j even 6 1 inner
2940.2.a.p 2 35.k even 12 1
2940.2.a.r 2 35.l odd 12 1
2940.2.q.q 4 35.f even 4 1
2940.2.q.q 4 35.k even 12 1
8820.2.a.bf 2 105.w odd 12 1
8820.2.a.bk 2 105.x even 12 1

Hecke kernels

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

\( T_{11}^{4} + 18 T_{11}^{2} + 324 \)
\( T_{13}^{4} + 38 T_{13}^{2} + 289 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ 1
$7$ \( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 315 T^{4} - 2366 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 16 T^{2} - 33 T^{4} + 4624 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 12 T + 76 T^{2} - 348 T^{3} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 2 T + 13 T^{2} + 142 T^{3} - 1004 T^{4} + 4402 T^{5} + 12493 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( 1 + 110 T^{2} + 6409 T^{4} + 324830 T^{6} + 13948420 T^{8} + 444692270 T^{10} + 12011497849 T^{12} + 282229904990 T^{14} + 3512479453921 T^{16} \)
$41$ \( ( 1 + 64 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 134 T^{2} + 8115 T^{4} - 247766 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 34 T^{2} - 1653 T^{4} + 95506 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 12744 T^{5} + 27848 T^{6} + 2464548 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 27328 T^{5} - 7442 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 + 230 T^{2} + 30769 T^{4} + 3025190 T^{6} + 232161940 T^{8} + 13580077910 T^{10} + 620029842049 T^{12} + 20805427898870 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 20 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( 1 + 206 T^{2} + 22969 T^{4} + 1814654 T^{6} + 124424404 T^{8} + 9670291166 T^{10} + 652279197529 T^{12} + 31174850615534 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 224 T^{2} + 23730 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 12 T - 52 T^{2} + 216 T^{3} + 17679 T^{4} + 19224 T^{5} - 411892 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 116 T^{2} + 3750 T^{4} - 1091444 T^{6} + 88529281 T^{8} )^{2} \)
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