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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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