Properties

Label 2100.2.bo.a
Level 2100
Weight 2
Character orbit 2100.bo
Analytic conductor 16.769
Analytic rank 1
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.bo (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -3 - 3 \zeta_{12}^{2} ) q^{11} -3 \zeta_{12} q^{17} + ( -2 + \zeta_{12}^{2} ) q^{19} + ( -4 - \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -1 - \zeta_{12}^{2} ) q^{31} + 9 \zeta_{12}^{3} q^{33} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{37} -6 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -3 + 6 \zeta_{12}^{2} ) q^{51} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + 3 \zeta_{12} q^{57} + ( 3 - 3 \zeta_{12}^{2} ) q^{59} + ( -14 + 7 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} + 5 \zeta_{12} q^{67} + ( 9 - 9 \zeta_{12}^{2} ) q^{69} + ( 6 - 12 \zeta_{12}^{2} ) q^{71} + ( 7 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{73} + ( -15 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} -\zeta_{12}^{2} q^{79} -9 \zeta_{12}^{2} q^{81} + 12 \zeta_{12}^{3} q^{83} + 9 \zeta_{12}^{2} q^{89} + 3 \zeta_{12}^{3} q^{93} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{97} + ( 18 - 9 \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 18q^{11} - 6q^{19} - 18q^{21} - 6q^{31} - 24q^{41} - 4q^{49} + 6q^{59} - 42q^{61} + 18q^{69} - 2q^{79} - 18q^{81} + 18q^{89} + 54q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1349.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 1.50000i 0 0 0 1.73205 + 2.00000i 0 −1.50000 2.59808i 0
1349.2 0 0.866025 1.50000i 0 0 0 −1.73205 2.00000i 0 −1.50000 2.59808i 0
1949.1 0 −0.866025 1.50000i 0 0 0 1.73205 2.00000i 0 −1.50000 + 2.59808i 0
1949.2 0 0.866025 + 1.50000i 0 0 0 −1.73205 + 2.00000i 0 −1.50000 + 2.59808i 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
21.g even 6 1 inner
105.p 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.bo.a 4
3.b odd 2 1 2100.2.bo.f 4
5.b even 2 1 inner 2100.2.bo.a 4
5.c odd 4 1 84.2.k.a 2
5.c odd 4 1 2100.2.bi.f 2
7.d odd 6 1 2100.2.bo.f 4
15.d odd 2 1 2100.2.bo.f 4
15.e even 4 1 84.2.k.b yes 2
15.e even 4 1 2100.2.bi.e 2
20.e even 4 1 336.2.bc.d 2
21.g even 6 1 inner 2100.2.bo.a 4
35.f even 4 1 588.2.k.d 2
35.i odd 6 1 2100.2.bo.f 4
35.k even 12 1 84.2.k.b yes 2
35.k even 12 1 588.2.f.a 2
35.k even 12 1 2100.2.bi.e 2
35.l odd 12 1 588.2.f.c 2
35.l odd 12 1 588.2.k.c 2
45.k odd 12 1 2268.2.w.f 2
45.k odd 12 1 2268.2.bm.a 2
45.l even 12 1 2268.2.w.a 2
45.l even 12 1 2268.2.bm.f 2
60.l odd 4 1 336.2.bc.b 2
105.k odd 4 1 588.2.k.c 2
105.p even 6 1 inner 2100.2.bo.a 4
105.w odd 12 1 84.2.k.a 2
105.w odd 12 1 588.2.f.c 2
105.w odd 12 1 2100.2.bi.f 2
105.x even 12 1 588.2.f.a 2
105.x even 12 1 588.2.k.d 2
140.w even 12 1 2352.2.k.a 2
140.x odd 12 1 336.2.bc.b 2
140.x odd 12 1 2352.2.k.d 2
315.bs even 12 1 2268.2.bm.f 2
315.bu odd 12 1 2268.2.bm.a 2
315.bw odd 12 1 2268.2.w.f 2
315.cg even 12 1 2268.2.w.a 2
420.bp odd 12 1 2352.2.k.d 2
420.br even 12 1 336.2.bc.d 2
420.br even 12 1 2352.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a 2 5.c odd 4 1
84.2.k.a 2 105.w odd 12 1
84.2.k.b yes 2 15.e even 4 1
84.2.k.b yes 2 35.k even 12 1
336.2.bc.b 2 60.l odd 4 1
336.2.bc.b 2 140.x odd 12 1
336.2.bc.d 2 20.e even 4 1
336.2.bc.d 2 420.br even 12 1
588.2.f.a 2 35.k even 12 1
588.2.f.a 2 105.x even 12 1
588.2.f.c 2 35.l odd 12 1
588.2.f.c 2 105.w odd 12 1
588.2.k.c 2 35.l odd 12 1
588.2.k.c 2 105.k odd 4 1
588.2.k.d 2 35.f even 4 1
588.2.k.d 2 105.x even 12 1
2100.2.bi.e 2 15.e even 4 1
2100.2.bi.e 2 35.k even 12 1
2100.2.bi.f 2 5.c odd 4 1
2100.2.bi.f 2 105.w odd 12 1
2100.2.bo.a 4 1.a even 1 1 trivial
2100.2.bo.a 4 5.b even 2 1 inner
2100.2.bo.a 4 21.g even 6 1 inner
2100.2.bo.a 4 105.p even 6 1 inner
2100.2.bo.f 4 3.b odd 2 1
2100.2.bo.f 4 7.d odd 6 1
2100.2.bo.f 4 15.d odd 2 1
2100.2.bo.f 4 35.i odd 6 1
2268.2.w.a 2 45.l even 12 1
2268.2.w.a 2 315.cg even 12 1
2268.2.w.f 2 45.k odd 12 1
2268.2.w.f 2 315.bw odd 12 1
2268.2.bm.a 2 45.k odd 12 1
2268.2.bm.a 2 315.bu odd 12 1
2268.2.bm.f 2 45.l even 12 1
2268.2.bm.f 2 315.bs even 12 1
2352.2.k.a 2 140.w even 12 1
2352.2.k.a 2 420.br even 12 1
2352.2.k.d 2 140.x odd 12 1
2352.2.k.d 2 420.bp odd 12 1

Hecke kernels

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

\( T_{11}^{2} + 9 T_{11} + 27 \)
\( T_{13} \)
\( T_{19}^{2} + 3 T_{19} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 29 T^{2} )^{4} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 79 T^{2} + 3432 T^{4} - 221911 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 - 34 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( 1 + T^{2} - 5328 T^{4} + 5329 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 146 T^{2} + 9409 T^{4} )^{2} \)
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