Properties

Label 2100.2.x.b
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: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{7} q^{7} -\beta_{3} q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + \beta_{7} q^{7} -\beta_{3} q^{9} -2 q^{11} + 2 \beta_{4} q^{13} + 4 \beta_{5} q^{17} -2 \beta_{6} q^{19} + \beta_{1} q^{21} -2 \beta_{2} q^{23} -\beta_{5} q^{27} + 6 \beta_{3} q^{29} + 2 \beta_{1} q^{31} -2 \beta_{4} q^{33} -2 \beta_{3} q^{39} + 4 \beta_{1} q^{41} + 2 \beta_{2} q^{43} -8 \beta_{5} q^{47} -7 \beta_{3} q^{49} + 4 q^{51} -2 \beta_{2} q^{53} + 2 \beta_{7} q^{57} -4 \beta_{6} q^{59} + 4 \beta_{1} q^{61} + \beta_{2} q^{63} + 2 \beta_{7} q^{67} -2 \beta_{6} q^{69} -2 q^{71} -14 \beta_{4} q^{73} -2 \beta_{7} q^{77} + 4 \beta_{3} q^{79} - q^{81} + 6 \beta_{5} q^{87} + 2 \beta_{1} q^{91} + 2 \beta_{2} q^{93} -2 \beta_{5} q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 16q^{11} + 32q^{51} - 16q^{71} - 8q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{4} + 1 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 11 \nu \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 7 \nu^{3} \)\()/24\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{2} \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 13 \nu^{3} \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 5 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} + 9 \beta_{3}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{7} - 13 \beta_{5}\)\()/2\)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{3}\) \(-1\)

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
0.581861 1.28897i
−1.28897 + 0.581861i
1.28897 0.581861i
−0.581861 + 1.28897i
0.581861 + 1.28897i
−1.28897 0.581861i
1.28897 + 0.581861i
−0.581861 1.28897i
0 −0.707107 0.707107i 0 0 0 −1.87083 1.87083i 0 1.00000i 0
1357.2 0 −0.707107 0.707107i 0 0 0 1.87083 + 1.87083i 0 1.00000i 0
1357.3 0 0.707107 + 0.707107i 0 0 0 −1.87083 1.87083i 0 1.00000i 0
1357.4 0 0.707107 + 0.707107i 0 0 0 1.87083 + 1.87083i 0 1.00000i 0
1693.1 0 −0.707107 + 0.707107i 0 0 0 −1.87083 + 1.87083i 0 1.00000i 0
1693.2 0 −0.707107 + 0.707107i 0 0 0 1.87083 1.87083i 0 1.00000i 0
1693.3 0 0.707107 0.707107i 0 0 0 −1.87083 + 1.87083i 0 1.00000i 0
1693.4 0 0.707107 0.707107i 0 0 0 1.87083 1.87083i 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.b 8
5.b even 2 1 inner 2100.2.x.b 8
5.c odd 4 2 inner 2100.2.x.b 8
7.b odd 2 1 inner 2100.2.x.b 8
35.c odd 2 1 inner 2100.2.x.b 8
35.f even 4 2 inner 2100.2.x.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.x.b 8 1.a even 1 1 trivial
2100.2.x.b 8 5.b even 2 1 inner
2100.2.x.b 8 5.c odd 4 2 inner
2100.2.x.b 8 7.b odd 2 1 inner
2100.2.x.b 8 35.c odd 2 1 inner
2100.2.x.b 8 35.f even 4 2 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{8} \)
$13$ \( ( 1 + 146 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 254 T^{4} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 734 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 34 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 30 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 334 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 3518 T^{4} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 466 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 6 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 10 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 + 2258 T^{4} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{8} \)
$73$ \( ( 1 - 8158 T^{4} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 + 17282 T^{4} + 88529281 T^{8} )^{2} \)
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