Properties

Label 2100.1.bh.a
Level $2100$
Weight $1$
Character orbit 2100.bh
Analytic conductor $1.048$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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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.bh (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{3} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{3} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{2} q^{9} -\zeta_{12}^{3} q^{13} -\zeta_{12}^{2} q^{19} + q^{21} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{4} q^{31} -\zeta_{12}^{5} q^{37} + \zeta_{12}^{4} q^{39} -\zeta_{12}^{3} q^{43} -\zeta_{12}^{4} q^{49} + \zeta_{12}^{3} q^{57} -2 \zeta_{12}^{2} q^{61} -\zeta_{12} q^{63} -\zeta_{12} q^{67} + \zeta_{12} q^{73} -\zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{2} q^{91} + \zeta_{12}^{5} q^{93} -2 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} - 2q^{19} + 4q^{21} + 2q^{31} - 2q^{39} + 2q^{49} - 4q^{61} - 2q^{79} - 2q^{81} + 2q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 0.500000i 0 0 0 −0.866025 0.500000i 0 0.500000 0.866025i 0
149.2 0 0.866025 0.500000i 0 0 0 0.866025 + 0.500000i 0 0.500000 0.866025i 0
1649.1 0 −0.866025 0.500000i 0 0 0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i 0
1649.2 0 0.866025 + 0.500000i 0 0 0 0.866025 0.500000i 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 \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o 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.bh.a 4
3.b odd 2 1 CM 2100.1.bh.a 4
5.b even 2 1 inner 2100.1.bh.a 4
5.c odd 4 1 84.1.p.a 2
5.c odd 4 1 2100.1.bn.c 2
7.c even 3 1 inner 2100.1.bh.a 4
15.d odd 2 1 inner 2100.1.bh.a 4
15.e even 4 1 84.1.p.a 2
15.e even 4 1 2100.1.bn.c 2
20.e even 4 1 336.1.bn.a 2
21.h odd 6 1 inner 2100.1.bh.a 4
35.f even 4 1 588.1.p.a 2
35.j even 6 1 inner 2100.1.bh.a 4
35.k even 12 1 588.1.c.a 1
35.k even 12 1 588.1.p.a 2
35.l odd 12 1 84.1.p.a 2
35.l odd 12 1 588.1.c.b 1
35.l odd 12 1 2100.1.bn.c 2
40.i odd 4 1 1344.1.bn.b 2
40.k even 4 1 1344.1.bn.a 2
45.k odd 12 1 2268.1.m.a 2
45.k odd 12 1 2268.1.bh.b 2
45.l even 12 1 2268.1.m.a 2
45.l even 12 1 2268.1.bh.b 2
60.l odd 4 1 336.1.bn.a 2
105.k odd 4 1 588.1.p.a 2
105.o odd 6 1 inner 2100.1.bh.a 4
105.w odd 12 1 588.1.c.a 1
105.w odd 12 1 588.1.p.a 2
105.x even 12 1 84.1.p.a 2
105.x even 12 1 588.1.c.b 1
105.x even 12 1 2100.1.bn.c 2
120.q odd 4 1 1344.1.bn.a 2
120.w even 4 1 1344.1.bn.b 2
140.j odd 4 1 2352.1.bn.a 2
140.w even 12 1 336.1.bn.a 2
140.w even 12 1 2352.1.d.a 1
140.x odd 12 1 2352.1.d.b 1
140.x odd 12 1 2352.1.bn.a 2
280.br even 12 1 1344.1.bn.a 2
280.bt odd 12 1 1344.1.bn.b 2
315.bt odd 12 1 2268.1.m.a 2
315.bv even 12 1 2268.1.m.a 2
315.bx even 12 1 2268.1.bh.b 2
315.ch odd 12 1 2268.1.bh.b 2
420.w even 4 1 2352.1.bn.a 2
420.bp odd 12 1 336.1.bn.a 2
420.bp odd 12 1 2352.1.d.a 1
420.br even 12 1 2352.1.d.b 1
420.br even 12 1 2352.1.bn.a 2
840.dc even 12 1 1344.1.bn.b 2
840.dp odd 12 1 1344.1.bn.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 5.c odd 4 1
84.1.p.a 2 15.e even 4 1
84.1.p.a 2 35.l odd 12 1
84.1.p.a 2 105.x even 12 1
336.1.bn.a 2 20.e even 4 1
336.1.bn.a 2 60.l odd 4 1
336.1.bn.a 2 140.w even 12 1
336.1.bn.a 2 420.bp odd 12 1
588.1.c.a 1 35.k even 12 1
588.1.c.a 1 105.w odd 12 1
588.1.c.b 1 35.l odd 12 1
588.1.c.b 1 105.x even 12 1
588.1.p.a 2 35.f even 4 1
588.1.p.a 2 35.k even 12 1
588.1.p.a 2 105.k odd 4 1
588.1.p.a 2 105.w odd 12 1
1344.1.bn.a 2 40.k even 4 1
1344.1.bn.a 2 120.q odd 4 1
1344.1.bn.a 2 280.br even 12 1
1344.1.bn.a 2 840.dp odd 12 1
1344.1.bn.b 2 40.i odd 4 1
1344.1.bn.b 2 120.w even 4 1
1344.1.bn.b 2 280.bt odd 12 1
1344.1.bn.b 2 840.dc even 12 1
2100.1.bh.a 4 1.a even 1 1 trivial
2100.1.bh.a 4 3.b odd 2 1 CM
2100.1.bh.a 4 5.b even 2 1 inner
2100.1.bh.a 4 7.c even 3 1 inner
2100.1.bh.a 4 15.d odd 2 1 inner
2100.1.bh.a 4 21.h odd 6 1 inner
2100.1.bh.a 4 35.j even 6 1 inner
2100.1.bh.a 4 105.o odd 6 1 inner
2100.1.bn.c 2 5.c odd 4 1
2100.1.bn.c 2 15.e even 4 1
2100.1.bn.c 2 35.l odd 12 1
2100.1.bn.c 2 105.x even 12 1
2268.1.m.a 2 45.k odd 12 1
2268.1.m.a 2 45.l even 12 1
2268.1.m.a 2 315.bt odd 12 1
2268.1.m.a 2 315.bv even 12 1
2268.1.bh.b 2 45.k odd 12 1
2268.1.bh.b 2 45.l even 12 1
2268.1.bh.b 2 315.bx even 12 1
2268.1.bh.b 2 315.ch odd 12 1
2352.1.d.a 1 140.w even 12 1
2352.1.d.a 1 420.bp odd 12 1
2352.1.d.b 1 140.x odd 12 1
2352.1.d.b 1 420.br even 12 1
2352.1.bn.a 2 140.j odd 4 1
2352.1.bn.a 2 140.x odd 12 1
2352.1.bn.a 2 420.w even 4 1
2352.1.bn.a 2 420.br even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2100, [\chi])\).

Hecke characteristic polynomials

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