Properties

Label 2100.2.bi.f
Level 2100
Weight 2
Character orbit 2100.bi
Analytic conductor 16.769
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 0.866025i 0 0 0 −2.00000 1.73205i 0 1.50000 2.59808i 0
1601.1 0 1.50000 + 0.866025i 0 0 0 −2.00000 + 1.73205i 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
21.g 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.bi.f 2
3.b odd 2 1 2100.2.bi.e 2
5.b even 2 1 84.2.k.a 2
5.c odd 4 2 2100.2.bo.a 4
7.d odd 6 1 2100.2.bi.e 2
15.d odd 2 1 84.2.k.b yes 2
15.e even 4 2 2100.2.bo.f 4
20.d odd 2 1 336.2.bc.d 2
21.g even 6 1 inner 2100.2.bi.f 2
35.c odd 2 1 588.2.k.d 2
35.i odd 6 1 84.2.k.b yes 2
35.i odd 6 1 588.2.f.a 2
35.j even 6 1 588.2.f.c 2
35.j even 6 1 588.2.k.c 2
35.k even 12 2 2100.2.bo.f 4
45.h odd 6 1 2268.2.w.a 2
45.h odd 6 1 2268.2.bm.f 2
45.j even 6 1 2268.2.w.f 2
45.j even 6 1 2268.2.bm.a 2
60.h even 2 1 336.2.bc.b 2
105.g even 2 1 588.2.k.c 2
105.o odd 6 1 588.2.f.a 2
105.o odd 6 1 588.2.k.d 2
105.p even 6 1 84.2.k.a 2
105.p even 6 1 588.2.f.c 2
105.w odd 12 2 2100.2.bo.a 4
140.p odd 6 1 2352.2.k.a 2
140.s even 6 1 336.2.bc.b 2
140.s even 6 1 2352.2.k.d 2
315.q odd 6 1 2268.2.bm.f 2
315.u even 6 1 2268.2.w.f 2
315.bn odd 6 1 2268.2.w.a 2
315.bq even 6 1 2268.2.bm.a 2
420.ba even 6 1 2352.2.k.d 2
420.be odd 6 1 336.2.bc.d 2
420.be odd 6 1 2352.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a 2 5.b even 2 1
84.2.k.a 2 105.p even 6 1
84.2.k.b yes 2 15.d odd 2 1
84.2.k.b yes 2 35.i odd 6 1
336.2.bc.b 2 60.h even 2 1
336.2.bc.b 2 140.s even 6 1
336.2.bc.d 2 20.d odd 2 1
336.2.bc.d 2 420.be odd 6 1
588.2.f.a 2 35.i odd 6 1
588.2.f.a 2 105.o odd 6 1
588.2.f.c 2 35.j even 6 1
588.2.f.c 2 105.p even 6 1
588.2.k.c 2 35.j even 6 1
588.2.k.c 2 105.g even 2 1
588.2.k.d 2 35.c odd 2 1
588.2.k.d 2 105.o odd 6 1
2100.2.bi.e 2 3.b odd 2 1
2100.2.bi.e 2 7.d odd 6 1
2100.2.bi.f 2 1.a even 1 1 trivial
2100.2.bi.f 2 21.g even 6 1 inner
2100.2.bo.a 4 5.c odd 4 2
2100.2.bo.a 4 105.w odd 12 2
2100.2.bo.f 4 15.e even 4 2
2100.2.bo.f 4 35.k even 12 2
2268.2.w.a 2 45.h odd 6 1
2268.2.w.a 2 315.bn odd 6 1
2268.2.w.f 2 45.j even 6 1
2268.2.w.f 2 315.u even 6 1
2268.2.bm.a 2 45.j even 6 1
2268.2.bm.a 2 315.bq even 6 1
2268.2.bm.f 2 45.h odd 6 1
2268.2.bm.f 2 315.q odd 6 1
2352.2.k.a 2 140.p odd 6 1
2352.2.k.a 2 420.be odd 6 1
2352.2.k.d 2 140.s even 6 1
2352.2.k.d 2 420.ba even 6 1

Hecke kernels

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

\( T_{11}^{2} + 9 T_{11} + 27 \)
\( T_{13} \)
\( T_{19}^{2} - 3 T_{19} + 3 \)
\( T_{37}^{2} - 7 T_{37} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ 1
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} \)
$23$ \( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( 1 - 34 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 21 T + 220 T^{2} - 1533 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 146 T^{2} + 9409 T^{4} \)
show more
show less