Properties

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

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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.x (of order \(4\), degree \(2\), minimal)

Newform invariants

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

$q$-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.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{9} -3 q^{11} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{13} -6 \zeta_{24}^{3} q^{17} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{19} + ( -1 - 2 \zeta_{24}^{4} ) q^{21} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{23} + \zeta_{24}^{3} q^{27} -9 \zeta_{24}^{6} q^{29} + ( 4 - 8 \zeta_{24}^{4} ) q^{31} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{33} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{37} + 2 \zeta_{24}^{6} q^{39} + ( 7 \zeta_{24} + 7 \zeta_{24}^{5} ) q^{43} -6 \zeta_{24}^{3} q^{47} + ( -8 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{49} + 6 q^{51} + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{57} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{59} + ( -4 + 8 \zeta_{24}^{4} ) q^{61} + ( 3 \zeta_{24} - \zeta_{24}^{5} ) q^{63} + ( 7 \zeta_{24}^{3} - 14 \zeta_{24}^{7} ) q^{67} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{69} + 3 q^{71} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{73} + ( -3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{77} -\zeta_{24}^{6} q^{79} - q^{81} + ( 12 \zeta_{24} - 12 \zeta_{24}^{5} ) q^{83} + 9 \zeta_{24}^{3} q^{87} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{89} + ( 2 + 4 \zeta_{24}^{4} ) q^{91} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{93} -4 \zeta_{24}^{3} q^{97} + 3 \zeta_{24}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{11} - 16q^{21} + 48q^{51} + 24q^{71} - 8q^{81} + 32q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{24}^{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1357.1
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
−0.965926 0.258819i
0 −0.707107 0.707107i 0 0 0 0.189469 2.63896i 0 1.00000i 0
1357.2 0 −0.707107 0.707107i 0 0 0 2.63896 0.189469i 0 1.00000i 0
1357.3 0 0.707107 + 0.707107i 0 0 0 −2.63896 + 0.189469i 0 1.00000i 0
1357.4 0 0.707107 + 0.707107i 0 0 0 −0.189469 + 2.63896i 0 1.00000i 0
1693.1 0 −0.707107 + 0.707107i 0 0 0 0.189469 + 2.63896i 0 1.00000i 0
1693.2 0 −0.707107 + 0.707107i 0 0 0 2.63896 + 0.189469i 0 1.00000i 0
1693.3 0 0.707107 0.707107i 0 0 0 −2.63896 0.189469i 0 1.00000i 0
1693.4 0 0.707107 0.707107i 0 0 0 −0.189469 2.63896i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1693.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.x.a 8
5.b even 2 1 inner 2100.2.x.a 8
5.c odd 4 2 inner 2100.2.x.a 8
7.b odd 2 1 inner 2100.2.x.a 8
35.c odd 2 1 inner 2100.2.x.a 8
35.f even 4 2 inner 2100.2.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.x.a 8 1.a even 1 1 trivial
2100.2.x.a 8 5.b even 2 1 inner
2100.2.x.a 8 5.c odd 4 2 inner
2100.2.x.a 8 7.b odd 2 1 inner
2100.2.x.a 8 35.c odd 2 1 inner
2100.2.x.a 8 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} + 3 \) acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ 1
$7$ \( 1 - 94 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{8} \)
$13$ \( ( 1 + 146 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 26 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 697 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 23 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 14 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 529 T^{4} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{8} \)
$43$ \( ( 1 + 23 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 1054 T^{4} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{4}( 1 + 14 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 8809 T^{4} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{8} \)
$73$ \( ( 1 - 24 T + 288 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 157 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 13294 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 12866 T^{4} + 88529281 T^{8} )^{2} \)
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