Properties

Label 2100.2.k.f
Level $2100$
Weight $2$
Character orbit 2100.k
Analytic conductor $16.769$
Analytic rank $0$
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: \(0\)
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.

\(f(q)\) \(=\) \( q -i q^{3} -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} -i q^{7} - q^{9} + q^{11} + 4 i q^{13} -2 i q^{17} + 4 q^{19} - q^{21} + 7 i q^{23} + i q^{27} + 9 q^{29} -2 q^{31} -i q^{33} -i q^{37} + 4 q^{39} + 8 q^{41} -9 i q^{43} -4 i q^{47} - q^{49} -2 q^{51} + 6 i q^{53} -4 i q^{57} -4 q^{59} + 4 q^{61} + i q^{63} -9 i q^{67} + 7 q^{69} + 5 q^{71} + 10 i q^{73} -i q^{77} + 15 q^{79} + q^{81} -6 i q^{83} -9 i q^{87} -8 q^{89} + 4 q^{91} + 2 i q^{93} -10 i q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{11} + 8q^{19} - 2q^{21} + 18q^{29} - 4q^{31} + 8q^{39} + 16q^{41} - 2q^{49} - 4q^{51} - 8q^{59} + 8q^{61} + 14q^{69} + 10q^{71} + 30q^{79} + 2q^{81} - 16q^{89} + 8q^{91} - 2q^{99} + O(q^{100}) \)

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.f 2
3.b odd 2 1 6300.2.k.h 2
5.b even 2 1 inner 2100.2.k.f 2
5.c odd 4 1 2100.2.a.g 1
5.c odd 4 1 2100.2.a.m yes 1
15.d odd 2 1 6300.2.k.h 2
15.e even 4 1 6300.2.a.f 1
15.e even 4 1 6300.2.a.x 1
20.e even 4 1 8400.2.a.x 1
20.e even 4 1 8400.2.a.bv 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.g 1 5.c odd 4 1
2100.2.a.m yes 1 5.c odd 4 1
2100.2.k.f 2 1.a even 1 1 trivial
2100.2.k.f 2 5.b even 2 1 inner
6300.2.a.f 1 15.e even 4 1
6300.2.a.x 1 15.e even 4 1
6300.2.k.h 2 3.b odd 2 1
6300.2.k.h 2 15.d odd 2 1
8400.2.a.x 1 20.e even 4 1
8400.2.a.bv 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 \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11} - 1 \)
\( T_{13}^{2} + 16 \)
\( T_{17}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 3 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 9 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 73 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 5 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 78 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 4 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 53 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 5 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 15 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 8 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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