Properties

Label 21.3.f.a
Level $21$
Weight $3$
Character orbit 21.f
Analytic conductor $0.572$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.572208555157\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 3 \zeta_{6} q^{2} + ( - \zeta_{6} - 1) q^{3} + (5 \zeta_{6} - 5) q^{4} + ( - 3 \zeta_{6} + 6) q^{5} + (6 \zeta_{6} - 3) q^{6} + (3 \zeta_{6} + 5) q^{7} + 3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{6} q^{2} + ( - \zeta_{6} - 1) q^{3} + (5 \zeta_{6} - 5) q^{4} + ( - 3 \zeta_{6} + 6) q^{5} + (6 \zeta_{6} - 3) q^{6} + (3 \zeta_{6} + 5) q^{7} + 3 q^{8} + 3 \zeta_{6} q^{9} + ( - 9 \zeta_{6} - 9) q^{10} + (15 \zeta_{6} - 15) q^{11} + ( - 5 \zeta_{6} + 10) q^{12} + ( - 16 \zeta_{6} + 8) q^{13} + ( - 24 \zeta_{6} + 9) q^{14} - 9 q^{15} + 11 \zeta_{6} q^{16} + (6 \zeta_{6} + 6) q^{17} + ( - 9 \zeta_{6} + 9) q^{18} + (6 \zeta_{6} - 12) q^{19} + (30 \zeta_{6} - 15) q^{20} + ( - 11 \zeta_{6} - 2) q^{21} + 45 q^{22} + ( - 3 \zeta_{6} - 3) q^{24} + ( - 2 \zeta_{6} + 2) q^{25} + (24 \zeta_{6} - 48) q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + (25 \zeta_{6} - 40) q^{28} - 9 q^{29} + 27 \zeta_{6} q^{30} + ( - 7 \zeta_{6} - 7) q^{31} + ( - 45 \zeta_{6} + 45) q^{32} + ( - 15 \zeta_{6} + 30) q^{33} + ( - 36 \zeta_{6} + 18) q^{34} + ( - 6 \zeta_{6} + 39) q^{35} - 15 q^{36} - 10 \zeta_{6} q^{37} + (18 \zeta_{6} + 18) q^{38} + (24 \zeta_{6} - 24) q^{39} + ( - 9 \zeta_{6} + 18) q^{40} + ( - 12 \zeta_{6} + 6) q^{41} + (39 \zeta_{6} - 33) q^{42} - 74 q^{43} - 75 \zeta_{6} q^{44} + (9 \zeta_{6} + 9) q^{45} + ( - 22 \zeta_{6} + 11) q^{48} + (39 \zeta_{6} + 16) q^{49} - 6 q^{50} - 18 \zeta_{6} q^{51} + (40 \zeta_{6} + 40) q^{52} + (33 \zeta_{6} - 33) q^{53} + (9 \zeta_{6} - 18) q^{54} + (90 \zeta_{6} - 45) q^{55} + (9 \zeta_{6} + 15) q^{56} + 18 q^{57} + 27 \zeta_{6} q^{58} + (9 \zeta_{6} + 9) q^{59} + ( - 45 \zeta_{6} + 45) q^{60} + ( - 52 \zeta_{6} + 104) q^{61} + (42 \zeta_{6} - 21) q^{62} + (24 \zeta_{6} - 9) q^{63} - 91 q^{64} - 72 \zeta_{6} q^{65} + ( - 45 \zeta_{6} - 45) q^{66} + ( - 76 \zeta_{6} + 76) q^{67} + (30 \zeta_{6} - 60) q^{68} + ( - 99 \zeta_{6} - 18) q^{70} + 84 q^{71} + 9 \zeta_{6} q^{72} + ( - 36 \zeta_{6} - 36) q^{73} + (30 \zeta_{6} - 30) q^{74} + (2 \zeta_{6} - 4) q^{75} + ( - 60 \zeta_{6} + 30) q^{76} + (75 \zeta_{6} - 120) q^{77} + 72 q^{78} + 43 \zeta_{6} q^{79} + (33 \zeta_{6} + 33) q^{80} + (9 \zeta_{6} - 9) q^{81} + (18 \zeta_{6} - 36) q^{82} + ( - 138 \zeta_{6} + 69) q^{83} + ( - 10 \zeta_{6} + 65) q^{84} + 54 q^{85} + 222 \zeta_{6} q^{86} + (9 \zeta_{6} + 9) q^{87} + (45 \zeta_{6} - 45) q^{88} + (42 \zeta_{6} - 84) q^{89} + ( - 54 \zeta_{6} + 27) q^{90} + ( - 104 \zeta_{6} + 88) q^{91} + 21 \zeta_{6} q^{93} + (54 \zeta_{6} - 54) q^{95} + (45 \zeta_{6} - 90) q^{96} + (214 \zeta_{6} - 107) q^{97} + ( - 165 \zeta_{6} + 117) q^{98} - 45 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{3} - 5 q^{4} + 9 q^{5} + 13 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{3} - 5 q^{4} + 9 q^{5} + 13 q^{7} + 6 q^{8} + 3 q^{9} - 27 q^{10} - 15 q^{11} + 15 q^{12} - 6 q^{14} - 18 q^{15} + 11 q^{16} + 18 q^{17} + 9 q^{18} - 18 q^{19} - 15 q^{21} + 90 q^{22} - 9 q^{24} + 2 q^{25} - 72 q^{26} - 55 q^{28} - 18 q^{29} + 27 q^{30} - 21 q^{31} + 45 q^{32} + 45 q^{33} + 72 q^{35} - 30 q^{36} - 10 q^{37} + 54 q^{38} - 24 q^{39} + 27 q^{40} - 27 q^{42} - 148 q^{43} - 75 q^{44} + 27 q^{45} + 71 q^{49} - 12 q^{50} - 18 q^{51} + 120 q^{52} - 33 q^{53} - 27 q^{54} + 39 q^{56} + 36 q^{57} + 27 q^{58} + 27 q^{59} + 45 q^{60} + 156 q^{61} + 6 q^{63} - 182 q^{64} - 72 q^{65} - 135 q^{66} + 76 q^{67} - 90 q^{68} - 135 q^{70} + 168 q^{71} + 9 q^{72} - 108 q^{73} - 30 q^{74} - 6 q^{75} - 165 q^{77} + 144 q^{78} + 43 q^{79} + 99 q^{80} - 9 q^{81} - 54 q^{82} + 120 q^{84} + 108 q^{85} + 222 q^{86} + 27 q^{87} - 45 q^{88} - 126 q^{89} + 72 q^{91} + 21 q^{93} - 54 q^{95} - 135 q^{96} + 69 q^{98} - 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 2.59808i −1.50000 0.866025i −2.50000 + 4.33013i 4.50000 2.59808i 5.19615i 6.50000 + 2.59808i 3.00000 1.50000 + 2.59808i −13.5000 7.79423i
19.1 −1.50000 + 2.59808i −1.50000 + 0.866025i −2.50000 4.33013i 4.50000 + 2.59808i 5.19615i 6.50000 2.59808i 3.00000 1.50000 2.59808i −13.5000 + 7.79423i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.f.a 2
3.b odd 2 1 63.3.m.d 2
4.b odd 2 1 336.3.bh.d 2
5.b even 2 1 525.3.o.h 2
5.c odd 4 2 525.3.s.e 4
7.b odd 2 1 147.3.f.a 2
7.c even 3 1 147.3.d.c 2
7.c even 3 1 147.3.f.a 2
7.d odd 6 1 inner 21.3.f.a 2
7.d odd 6 1 147.3.d.c 2
12.b even 2 1 1008.3.cg.a 2
21.c even 2 1 441.3.m.g 2
21.g even 6 1 63.3.m.d 2
21.g even 6 1 441.3.d.a 2
21.h odd 6 1 441.3.d.a 2
21.h odd 6 1 441.3.m.g 2
28.f even 6 1 336.3.bh.d 2
28.f even 6 1 2352.3.f.a 2
28.g odd 6 1 2352.3.f.a 2
35.i odd 6 1 525.3.o.h 2
35.k even 12 2 525.3.s.e 4
84.j odd 6 1 1008.3.cg.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 1.a even 1 1 trivial
21.3.f.a 2 7.d odd 6 1 inner
63.3.m.d 2 3.b odd 2 1
63.3.m.d 2 21.g even 6 1
147.3.d.c 2 7.c even 3 1
147.3.d.c 2 7.d odd 6 1
147.3.f.a 2 7.b odd 2 1
147.3.f.a 2 7.c even 3 1
336.3.bh.d 2 4.b odd 2 1
336.3.bh.d 2 28.f even 6 1
441.3.d.a 2 21.g even 6 1
441.3.d.a 2 21.h odd 6 1
441.3.m.g 2 21.c even 2 1
441.3.m.g 2 21.h odd 6 1
525.3.o.h 2 5.b even 2 1
525.3.o.h 2 35.i odd 6 1
525.3.s.e 4 5.c odd 4 2
525.3.s.e 4 35.k even 12 2
1008.3.cg.a 2 12.b even 2 1
1008.3.cg.a 2 84.j odd 6 1
2352.3.f.a 2 28.f even 6 1
2352.3.f.a 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$7$ \( T^{2} - 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$13$ \( T^{2} + 192 \) Copy content Toggle raw display
$17$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$19$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} + 108 \) Copy content Toggle raw display
$43$ \( (T + 74)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 33T + 1089 \) Copy content Toggle raw display
$59$ \( T^{2} - 27T + 243 \) Copy content Toggle raw display
$61$ \( T^{2} - 156T + 8112 \) Copy content Toggle raw display
$67$ \( T^{2} - 76T + 5776 \) Copy content Toggle raw display
$71$ \( (T - 84)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 108T + 3888 \) Copy content Toggle raw display
$79$ \( T^{2} - 43T + 1849 \) Copy content Toggle raw display
$83$ \( T^{2} + 14283 \) Copy content Toggle raw display
$89$ \( T^{2} + 126T + 5292 \) Copy content Toggle raw display
$97$ \( T^{2} + 34347 \) Copy content Toggle raw display
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