Properties

Label 2100.2.k.d
Level 2100
Weight 2
Character orbit 2100.k
Analytic conductor 16.769
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{3} + i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + i q^{7} - q^{9} - q^{11} -2 i q^{13} -6 q^{19} + q^{21} + i q^{23} + i q^{27} - q^{29} -2 q^{31} + i q^{33} + 7 i q^{37} -2 q^{39} -8 q^{41} -i q^{43} + 2 i q^{47} - q^{49} -14 i q^{53} + 6 i q^{57} -10 q^{59} -i q^{63} + 3 i q^{67} + q^{69} -9 q^{71} -i q^{77} - q^{79} + q^{81} + 2 i q^{83} + i q^{87} -2 q^{89} + 2 q^{91} + 2 i q^{93} + 10 i q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 2q^{11} - 12q^{19} + 2q^{21} - 2q^{29} - 4q^{31} - 4q^{39} - 16q^{41} - 2q^{49} - 20q^{59} + 2q^{69} - 18q^{71} - 2q^{79} + 2q^{81} - 4q^{89} + 4q^{91} + 2q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.k.d 2
3.b odd 2 1 6300.2.k.k 2
5.b even 2 1 inner 2100.2.k.d 2
5.c odd 4 1 2100.2.a.c 1
5.c odd 4 1 2100.2.a.p yes 1
15.d odd 2 1 6300.2.k.k 2
15.e even 4 1 6300.2.a.k 1
15.e even 4 1 6300.2.a.z 1
20.e even 4 1 8400.2.a.h 1
20.e even 4 1 8400.2.a.cp 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.c 1 5.c odd 4 1
2100.2.a.p yes 1 5.c odd 4 1
2100.2.k.d 2 1.a even 1 1 trivial
2100.2.k.d 2 5.b even 2 1 inner
6300.2.a.k 1 15.e even 4 1
6300.2.a.z 1 15.e even 4 1
6300.2.k.k 2 3.b odd 2 1
6300.2.k.k 2 15.d odd 2 1
8400.2.a.h 1 20.e even 4 1
8400.2.a.cp 1 20.e even 4 1

Hecke kernels

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

\( T_{11} + 1 \)
\( T_{13}^{2} + 4 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{2} \)
$19$ \( ( 1 + 6 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 45 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 25 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 85 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 90 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \)
$59$ \( ( 1 + 10 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( 1 - 125 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 9 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{2} \)
$79$ \( ( 1 + T + 79 T^{2} )^{2} \)
$83$ \( 1 - 162 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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