Properties

Label 2100.1.bn.c
Level $2100$
Weight $1$
Character orbit 2100.bn
Analytic conductor $1.048$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2100.bn (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.588.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{7} - \zeta_{6} q^{9} + q^{13} + \zeta_{6} q^{19} + q^{21} - q^{27} - \zeta_{6}^{2} q^{31} - \zeta_{6} q^{37} - \zeta_{6}^{2} q^{39} + q^{43} + \zeta_{6}^{2} q^{49} + q^{57} - \zeta_{6} q^{61} - \zeta_{6}^{2} q^{63} + \zeta_{6}^{2} q^{67} + \zeta_{6}^{2} q^{73} + \zeta_{6} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{91} - \zeta_{6} q^{93} - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{7} - q^{9} + 2 q^{13} + q^{19} + 2 q^{21} - 2 q^{27} + q^{31} - q^{37} + q^{39} + 2 q^{43} - q^{49} + 2 q^{57} - 2 q^{61} + q^{63} - q^{67} - q^{73} + q^{79} - q^{81} + q^{91} - q^{93} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 0.500000 0.866025i 0 −0.500000 + 0.866025i 0
1901.1 0 0.500000 0.866025i 0 0 0 0.500000 + 0.866025i 0 −0.500000 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.1.bn.c 2
3.b odd 2 1 CM 2100.1.bn.c 2
5.b even 2 1 84.1.p.a 2
5.c odd 4 2 2100.1.bh.a 4
7.c even 3 1 inner 2100.1.bn.c 2
15.d odd 2 1 84.1.p.a 2
15.e even 4 2 2100.1.bh.a 4
20.d odd 2 1 336.1.bn.a 2
21.h odd 6 1 inner 2100.1.bn.c 2
35.c odd 2 1 588.1.p.a 2
35.i odd 6 1 588.1.c.a 1
35.i odd 6 1 588.1.p.a 2
35.j even 6 1 84.1.p.a 2
35.j even 6 1 588.1.c.b 1
35.l odd 12 2 2100.1.bh.a 4
40.e odd 2 1 1344.1.bn.a 2
40.f even 2 1 1344.1.bn.b 2
45.h odd 6 1 2268.1.m.a 2
45.h odd 6 1 2268.1.bh.b 2
45.j even 6 1 2268.1.m.a 2
45.j even 6 1 2268.1.bh.b 2
60.h even 2 1 336.1.bn.a 2
105.g even 2 1 588.1.p.a 2
105.o odd 6 1 84.1.p.a 2
105.o odd 6 1 588.1.c.b 1
105.p even 6 1 588.1.c.a 1
105.p even 6 1 588.1.p.a 2
105.x even 12 2 2100.1.bh.a 4
120.i odd 2 1 1344.1.bn.b 2
120.m even 2 1 1344.1.bn.a 2
140.c even 2 1 2352.1.bn.a 2
140.p odd 6 1 336.1.bn.a 2
140.p odd 6 1 2352.1.d.a 1
140.s even 6 1 2352.1.d.b 1
140.s even 6 1 2352.1.bn.a 2
280.bf even 6 1 1344.1.bn.b 2
280.bi odd 6 1 1344.1.bn.a 2
315.r even 6 1 2268.1.m.a 2
315.v odd 6 1 2268.1.bh.b 2
315.bo even 6 1 2268.1.bh.b 2
315.br odd 6 1 2268.1.m.a 2
420.o odd 2 1 2352.1.bn.a 2
420.ba even 6 1 336.1.bn.a 2
420.ba even 6 1 2352.1.d.a 1
420.be odd 6 1 2352.1.d.b 1
420.be odd 6 1 2352.1.bn.a 2
840.cg odd 6 1 1344.1.bn.b 2
840.cv even 6 1 1344.1.bn.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 5.b even 2 1
84.1.p.a 2 15.d odd 2 1
84.1.p.a 2 35.j even 6 1
84.1.p.a 2 105.o odd 6 1
336.1.bn.a 2 20.d odd 2 1
336.1.bn.a 2 60.h even 2 1
336.1.bn.a 2 140.p odd 6 1
336.1.bn.a 2 420.ba even 6 1
588.1.c.a 1 35.i odd 6 1
588.1.c.a 1 105.p even 6 1
588.1.c.b 1 35.j even 6 1
588.1.c.b 1 105.o odd 6 1
588.1.p.a 2 35.c odd 2 1
588.1.p.a 2 35.i odd 6 1
588.1.p.a 2 105.g even 2 1
588.1.p.a 2 105.p even 6 1
1344.1.bn.a 2 40.e odd 2 1
1344.1.bn.a 2 120.m even 2 1
1344.1.bn.a 2 280.bi odd 6 1
1344.1.bn.a 2 840.cv even 6 1
1344.1.bn.b 2 40.f even 2 1
1344.1.bn.b 2 120.i odd 2 1
1344.1.bn.b 2 280.bf even 6 1
1344.1.bn.b 2 840.cg odd 6 1
2100.1.bh.a 4 5.c odd 4 2
2100.1.bh.a 4 15.e even 4 2
2100.1.bh.a 4 35.l odd 12 2
2100.1.bh.a 4 105.x even 12 2
2100.1.bn.c 2 1.a even 1 1 trivial
2100.1.bn.c 2 3.b odd 2 1 CM
2100.1.bn.c 2 7.c even 3 1 inner
2100.1.bn.c 2 21.h odd 6 1 inner
2268.1.m.a 2 45.h odd 6 1
2268.1.m.a 2 45.j even 6 1
2268.1.m.a 2 315.r even 6 1
2268.1.m.a 2 315.br odd 6 1
2268.1.bh.b 2 45.h odd 6 1
2268.1.bh.b 2 45.j even 6 1
2268.1.bh.b 2 315.v odd 6 1
2268.1.bh.b 2 315.bo even 6 1
2352.1.d.a 1 140.p odd 6 1
2352.1.d.a 1 420.ba even 6 1
2352.1.d.b 1 140.s even 6 1
2352.1.d.b 1 420.be odd 6 1
2352.1.bn.a 2 140.c even 2 1
2352.1.bn.a 2 140.s even 6 1
2352.1.bn.a 2 420.o odd 2 1
2352.1.bn.a 2 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(2100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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