Properties

Label 21.2.e.a
Level 21
Weight 2
Character orbit 21.e
Analytic conductor 0.168
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.167685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{2} -\zeta_{6} q^{3} -2 \zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + 2 q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{2} -\zeta_{6} q^{3} -2 \zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + 2 q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} + 4 \zeta_{6} q^{10} + 2 \zeta_{6} q^{11} + ( -2 + 2 \zeta_{6} ) q^{12} + q^{13} + ( 4 - 6 \zeta_{6} ) q^{14} -2 q^{15} + ( 4 - 4 \zeta_{6} ) q^{16} -2 \zeta_{6} q^{18} + ( -1 + \zeta_{6} ) q^{19} -4 q^{20} + ( 1 + 2 \zeta_{6} ) q^{21} -4 q^{22} + \zeta_{6} q^{25} + ( -2 + 2 \zeta_{6} ) q^{26} + q^{27} + ( 2 + 4 \zeta_{6} ) q^{28} + 4 q^{29} + ( 4 - 4 \zeta_{6} ) q^{30} -9 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{33} + ( -4 + 6 \zeta_{6} ) q^{35} + 2 q^{36} + ( -3 + 3 \zeta_{6} ) q^{37} -2 \zeta_{6} q^{38} -\zeta_{6} q^{39} -10 q^{41} + ( -6 + 2 \zeta_{6} ) q^{42} + 5 q^{43} + ( 4 - 4 \zeta_{6} ) q^{44} + 2 \zeta_{6} q^{45} + ( 6 - 6 \zeta_{6} ) q^{47} -4 q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} -2 q^{50} -2 \zeta_{6} q^{52} -12 \zeta_{6} q^{53} + ( -2 + 2 \zeta_{6} ) q^{54} + 4 q^{55} + q^{57} + ( -8 + 8 \zeta_{6} ) q^{58} + 12 \zeta_{6} q^{59} + 4 \zeta_{6} q^{60} + ( -10 + 10 \zeta_{6} ) q^{61} + 18 q^{62} + ( 2 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{66} + 5 \zeta_{6} q^{67} + ( -4 - 8 \zeta_{6} ) q^{70} -6 q^{71} + 3 \zeta_{6} q^{73} -6 \zeta_{6} q^{74} + ( 1 - \zeta_{6} ) q^{75} + 2 q^{76} + ( -2 - 4 \zeta_{6} ) q^{77} + 2 q^{78} + ( 1 - \zeta_{6} ) q^{79} -8 \zeta_{6} q^{80} -\zeta_{6} q^{81} + ( 20 - 20 \zeta_{6} ) q^{82} + 6 q^{83} + ( 4 - 6 \zeta_{6} ) q^{84} + ( -10 + 10 \zeta_{6} ) q^{86} -4 \zeta_{6} q^{87} + ( -16 + 16 \zeta_{6} ) q^{89} -4 q^{90} + ( -3 + \zeta_{6} ) q^{91} + ( -9 + 9 \zeta_{6} ) q^{93} + 12 \zeta_{6} q^{94} + 2 \zeta_{6} q^{95} + ( 8 - 8 \zeta_{6} ) q^{96} -6 q^{97} + ( -6 + 16 \zeta_{6} ) q^{98} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + 4q^{6} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + 4q^{6} - 5q^{7} - q^{9} + 4q^{10} + 2q^{11} - 2q^{12} + 2q^{13} + 2q^{14} - 4q^{15} + 4q^{16} - 2q^{18} - q^{19} - 8q^{20} + 4q^{21} - 8q^{22} + q^{25} - 2q^{26} + 2q^{27} + 8q^{28} + 8q^{29} + 4q^{30} - 9q^{31} + 8q^{32} + 2q^{33} - 2q^{35} + 4q^{36} - 3q^{37} - 2q^{38} - q^{39} - 20q^{41} - 10q^{42} + 10q^{43} + 4q^{44} + 2q^{45} + 6q^{47} - 8q^{48} + 11q^{49} - 4q^{50} - 2q^{52} - 12q^{53} - 2q^{54} + 8q^{55} + 2q^{57} - 8q^{58} + 12q^{59} + 4q^{60} - 10q^{61} + 36q^{62} + q^{63} - 16q^{64} + 2q^{65} + 4q^{66} + 5q^{67} - 16q^{70} - 12q^{71} + 3q^{73} - 6q^{74} + q^{75} + 4q^{76} - 8q^{77} + 4q^{78} + q^{79} - 8q^{80} - q^{81} + 20q^{82} + 12q^{83} + 2q^{84} - 10q^{86} - 4q^{87} - 16q^{89} - 8q^{90} - 5q^{91} - 9q^{93} + 12q^{94} + 2q^{95} + 8q^{96} - 12q^{97} + 4q^{98} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i 1.00000 + 1.73205i 2.00000 −2.50000 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i
16.1 −1.00000 + 1.73205i −0.500000 0.866025i −1.00000 1.73205i 1.00000 1.73205i 2.00000 −2.50000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.2.e.a 2
3.b odd 2 1 63.2.e.b 2
4.b odd 2 1 336.2.q.f 2
5.b even 2 1 525.2.i.e 2
5.c odd 4 2 525.2.r.e 4
7.b odd 2 1 147.2.e.a 2
7.c even 3 1 inner 21.2.e.a 2
7.c even 3 1 147.2.a.c 1
7.d odd 6 1 147.2.a.b 1
7.d odd 6 1 147.2.e.a 2
8.b even 2 1 1344.2.q.m 2
8.d odd 2 1 1344.2.q.c 2
9.c even 3 1 567.2.g.a 2
9.c even 3 1 567.2.h.f 2
9.d odd 6 1 567.2.g.f 2
9.d odd 6 1 567.2.h.a 2
12.b even 2 1 1008.2.s.d 2
21.c even 2 1 441.2.e.e 2
21.g even 6 1 441.2.a.a 1
21.g even 6 1 441.2.e.e 2
21.h odd 6 1 63.2.e.b 2
21.h odd 6 1 441.2.a.b 1
28.d even 2 1 2352.2.q.c 2
28.f even 6 1 2352.2.a.w 1
28.f even 6 1 2352.2.q.c 2
28.g odd 6 1 336.2.q.f 2
28.g odd 6 1 2352.2.a.d 1
35.i odd 6 1 3675.2.a.c 1
35.j even 6 1 525.2.i.e 2
35.j even 6 1 3675.2.a.a 1
35.l odd 12 2 525.2.r.e 4
56.j odd 6 1 9408.2.a.bz 1
56.k odd 6 1 1344.2.q.c 2
56.k odd 6 1 9408.2.a.cv 1
56.m even 6 1 9408.2.a.k 1
56.p even 6 1 1344.2.q.m 2
56.p even 6 1 9408.2.a.bg 1
63.g even 3 1 567.2.h.f 2
63.h even 3 1 567.2.g.a 2
63.j odd 6 1 567.2.g.f 2
63.n odd 6 1 567.2.h.a 2
84.j odd 6 1 7056.2.a.m 1
84.n even 6 1 1008.2.s.d 2
84.n even 6 1 7056.2.a.bp 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 1.a even 1 1 trivial
21.2.e.a 2 7.c even 3 1 inner
63.2.e.b 2 3.b odd 2 1
63.2.e.b 2 21.h odd 6 1
147.2.a.b 1 7.d odd 6 1
147.2.a.c 1 7.c even 3 1
147.2.e.a 2 7.b odd 2 1
147.2.e.a 2 7.d odd 6 1
336.2.q.f 2 4.b odd 2 1
336.2.q.f 2 28.g odd 6 1
441.2.a.a 1 21.g even 6 1
441.2.a.b 1 21.h odd 6 1
441.2.e.e 2 21.c even 2 1
441.2.e.e 2 21.g even 6 1
525.2.i.e 2 5.b even 2 1
525.2.i.e 2 35.j even 6 1
525.2.r.e 4 5.c odd 4 2
525.2.r.e 4 35.l odd 12 2
567.2.g.a 2 9.c even 3 1
567.2.g.a 2 63.h even 3 1
567.2.g.f 2 9.d odd 6 1
567.2.g.f 2 63.j odd 6 1
567.2.h.a 2 9.d odd 6 1
567.2.h.a 2 63.n odd 6 1
567.2.h.f 2 9.c even 3 1
567.2.h.f 2 63.g even 3 1
1008.2.s.d 2 12.b even 2 1
1008.2.s.d 2 84.n even 6 1
1344.2.q.c 2 8.d odd 2 1
1344.2.q.c 2 56.k odd 6 1
1344.2.q.m 2 8.b even 2 1
1344.2.q.m 2 56.p even 6 1
2352.2.a.d 1 28.g odd 6 1
2352.2.a.w 1 28.f even 6 1
2352.2.q.c 2 28.d even 2 1
2352.2.q.c 2 28.f even 6 1
3675.2.a.a 1 35.j even 6 1
3675.2.a.c 1 35.i odd 6 1
7056.2.a.m 1 84.j odd 6 1
7056.2.a.bp 1 84.n even 6 1
9408.2.a.k 1 56.m even 6 1
9408.2.a.bg 1 56.p even 6 1
9408.2.a.bz 1 56.j odd 6 1
9408.2.a.cv 1 56.k odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(21, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
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