Properties

Label 2100.2.bc.b
Level 2100
Weight 2
Character orbit 2100.bc
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.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} -2 \zeta_{12}^{3} q^{13} + 2 \zeta_{12} q^{17} + 4 \zeta_{12}^{2} q^{19} + ( -1 + 3 \zeta_{12}^{2} ) q^{21} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{3} q^{27} -4 q^{29} + ( -3 + 3 \zeta_{12}^{2} ) q^{31} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{37} + ( -2 + 2 \zeta_{12}^{2} ) q^{39} + 6 q^{41} -\zeta_{12}^{3} q^{43} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -2 \zeta_{12}^{2} q^{51} -2 \zeta_{12} q^{53} -4 \zeta_{12}^{3} q^{57} + ( -6 + 6 \zeta_{12}^{2} ) q^{59} + \zeta_{12}^{2} q^{61} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + 12 \zeta_{12} q^{67} + 8 q^{69} + 10 q^{71} -\zeta_{12} q^{73} + ( 6 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + 7 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 18 \zeta_{12}^{3} q^{83} + 4 \zeta_{12} q^{87} + 10 \zeta_{12}^{2} q^{89} + ( -6 + 4 \zeta_{12}^{2} ) q^{91} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{93} -5 \zeta_{12}^{3} q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} - 4q^{11} + 8q^{19} + 2q^{21} - 16q^{29} - 6q^{31} - 4q^{39} + 24q^{41} - 4q^{49} - 4q^{51} - 12q^{59} + 2q^{61} + 32q^{69} + 40q^{71} + 14q^{79} - 2q^{81} + 20q^{89} - 16q^{91} - 8q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 0.500000i 0 0 0 −1.73205 2.00000i 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 0 0 1.73205 + 2.00000i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −1.73205 + 2.00000i 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 0 0 1.73205 2.00000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.b 4
5.b even 2 1 inner 2100.2.bc.b 4
5.c odd 4 1 2100.2.q.c 2
5.c odd 4 1 2100.2.q.e yes 2
7.c even 3 1 inner 2100.2.bc.b 4
35.j even 6 1 inner 2100.2.bc.b 4
35.l odd 12 1 2100.2.q.c 2
35.l odd 12 1 2100.2.q.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.c 2 5.c odd 4 1
2100.2.q.c 2 35.l odd 12 1
2100.2.q.e yes 2 5.c odd 4 1
2100.2.q.e yes 2 35.l odd 12 1
2100.2.bc.b 4 1.a even 1 1 trivial
2100.2.bc.b 4 5.b even 2 1 inner
2100.2.bc.b 4 7.c even 3 1 inner
2100.2.bc.b 4 35.j even 6 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{2} + T^{4} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 - 7 T^{2} - 1320 T^{4} - 9583 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 85 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 145 T^{2} + 15696 T^{4} + 772705 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \)
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