Properties

Label 2100.2.d.i
Level 2100
Weight 2
Character orbit 2100.d
Analytic conductor 16.769
Analytic rank 0
Dimension 4
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{3} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{3} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} -2 \beta_{1} q^{11} -2 \beta_{1} q^{13} + 4 q^{17} -2 \beta_{2} q^{19} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} -2 \beta_{2} q^{23} + ( -5 + \beta_{1} ) q^{27} + 4 \beta_{1} q^{29} -2 \beta_{2} q^{31} + ( 4 - 2 \beta_{1} ) q^{33} -4 \beta_{3} q^{37} + ( 4 - 2 \beta_{1} ) q^{39} + 2 \beta_{3} q^{41} -2 \beta_{3} q^{43} + 6 q^{47} + ( 3 - 2 \beta_{2} ) q^{49} + ( 4 + 4 \beta_{1} ) q^{51} -2 \beta_{2} q^{53} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{57} + 4 \beta_{3} q^{59} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{63} -2 \beta_{3} q^{67} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{69} + 10 \beta_{1} q^{71} + 6 \beta_{1} q^{73} + ( -4 - 2 \beta_{2} ) q^{77} + 12 q^{79} + ( -7 - 4 \beta_{1} ) q^{81} -10 q^{83} + ( -8 + 4 \beta_{1} ) q^{87} + 2 \beta_{3} q^{89} + ( -4 - 2 \beta_{2} ) q^{91} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{93} -6 \beta_{1} q^{97} + ( 8 + 2 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{9} + 16q^{17} + 8q^{21} - 20q^{27} + 16q^{33} + 16q^{39} + 24q^{47} + 12q^{49} + 16q^{51} + 16q^{63} - 16q^{77} + 48q^{79} - 28q^{81} - 40q^{83} - 32q^{87} - 16q^{91} + 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 8 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 4 \beta_{1}\)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
2.28825i
0.874032i
2.28825i
0.874032i
0 1.00000 1.41421i 0 0 0 −2.23607 + 1.41421i 0 −1.00000 2.82843i 0
1301.2 0 1.00000 1.41421i 0 0 0 2.23607 + 1.41421i 0 −1.00000 2.82843i 0
1301.3 0 1.00000 + 1.41421i 0 0 0 −2.23607 1.41421i 0 −1.00000 + 2.82843i 0
1301.4 0 1.00000 + 1.41421i 0 0 0 2.23607 1.41421i 0 −1.00000 + 2.82843i 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
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.i 4
3.b odd 2 1 2100.2.d.f 4
5.b even 2 1 2100.2.d.f 4
5.c odd 4 2 420.2.f.b 8
7.b odd 2 1 2100.2.d.f 4
15.d odd 2 1 inner 2100.2.d.i 4
15.e even 4 2 420.2.f.b 8
20.e even 4 2 1680.2.k.g 8
21.c even 2 1 inner 2100.2.d.i 4
35.c odd 2 1 inner 2100.2.d.i 4
35.f even 4 2 420.2.f.b 8
60.l odd 4 2 1680.2.k.g 8
105.g even 2 1 2100.2.d.f 4
105.k odd 4 2 420.2.f.b 8
140.j odd 4 2 1680.2.k.g 8
420.w even 4 2 1680.2.k.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.f.b 8 5.c odd 4 2
420.2.f.b 8 15.e even 4 2
420.2.f.b 8 35.f even 4 2
420.2.f.b 8 105.k odd 4 2
1680.2.k.g 8 20.e even 4 2
1680.2.k.g 8 60.l odd 4 2
1680.2.k.g 8 140.j odd 4 2
1680.2.k.g 8 420.w even 4 2
2100.2.d.f 4 3.b odd 2 1
2100.2.d.f 4 5.b even 2 1
2100.2.d.f 4 7.b odd 2 1
2100.2.d.f 4 105.g even 2 1
2100.2.d.i 4 1.a even 1 1 trivial
2100.2.d.i 4 15.d odd 2 1 inner
2100.2.d.i 4 21.c even 2 1 inner
2100.2.d.i 4 35.c odd 2 1 inner

Hecke kernels

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

\( T_{11}^{2} + 8 \)
\( T_{13}^{2} + 8 \)
\( T_{17} - 4 \)
\( T_{37}^{2} - 80 \)
\( T_{41}^{2} - 20 \)
\( T_{43}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 2 T + 3 T^{2} )^{2} \)
$5$ 1
$7$ \( 1 - 6 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 18 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 4 T + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 6 T + 19 T^{2} )^{2}( 1 + 6 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 6 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 26 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 22 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 6 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 62 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 66 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 6 T + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 66 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 114 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 74 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 12 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 10 T + 83 T^{2} )^{4} \)
$89$ \( ( 1 + 158 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 122 T^{2} + 9409 T^{4} )^{2} \)
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