Properties

Label 21.3.d.a
Level $21$
Weight $3$
Character orbit 21.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{4} + ( 4 - 8 \zeta_{6} ) q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} + ( -3 + 8 \zeta_{6} ) q^{7} -7 q^{8} -3 q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{4} + ( 4 - 8 \zeta_{6} ) q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} + ( -3 + 8 \zeta_{6} ) q^{7} -7 q^{8} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{10} + 10 q^{11} + ( 3 - 6 \zeta_{6} ) q^{12} + ( -4 + 8 \zeta_{6} ) q^{13} + ( -3 + 8 \zeta_{6} ) q^{14} + 12 q^{15} + 5 q^{16} -3 q^{18} + ( 12 - 24 \zeta_{6} ) q^{19} + ( -12 + 24 \zeta_{6} ) q^{20} + ( -13 + 2 \zeta_{6} ) q^{21} + 10 q^{22} -14 q^{23} + ( 7 - 14 \zeta_{6} ) q^{24} -23 q^{25} + ( -4 + 8 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 9 - 24 \zeta_{6} ) q^{28} -38 q^{29} + 12 q^{30} + ( -16 + 32 \zeta_{6} ) q^{31} + 33 q^{32} + ( -10 + 20 \zeta_{6} ) q^{33} + ( 52 - 8 \zeta_{6} ) q^{35} + 9 q^{36} + 26 q^{37} + ( 12 - 24 \zeta_{6} ) q^{38} -12 q^{39} + ( -28 + 56 \zeta_{6} ) q^{40} + ( 40 - 80 \zeta_{6} ) q^{41} + ( -13 + 2 \zeta_{6} ) q^{42} + 26 q^{43} -30 q^{44} + ( -12 + 24 \zeta_{6} ) q^{45} -14 q^{46} + ( -16 + 32 \zeta_{6} ) q^{47} + ( -5 + 10 \zeta_{6} ) q^{48} + ( -55 + 16 \zeta_{6} ) q^{49} -23 q^{50} + ( 12 - 24 \zeta_{6} ) q^{52} + 10 q^{53} + ( 3 - 6 \zeta_{6} ) q^{54} + ( 40 - 80 \zeta_{6} ) q^{55} + ( 21 - 56 \zeta_{6} ) q^{56} + 36 q^{57} -38 q^{58} + ( -44 + 88 \zeta_{6} ) q^{59} -36 q^{60} + ( -20 + 40 \zeta_{6} ) q^{61} + ( -16 + 32 \zeta_{6} ) q^{62} + ( 9 - 24 \zeta_{6} ) q^{63} + 13 q^{64} + 48 q^{65} + ( -10 + 20 \zeta_{6} ) q^{66} + 74 q^{67} + ( 14 - 28 \zeta_{6} ) q^{69} + ( 52 - 8 \zeta_{6} ) q^{70} -62 q^{71} + 21 q^{72} + ( 24 - 48 \zeta_{6} ) q^{73} + 26 q^{74} + ( 23 - 46 \zeta_{6} ) q^{75} + ( -36 + 72 \zeta_{6} ) q^{76} + ( -30 + 80 \zeta_{6} ) q^{77} -12 q^{78} -46 q^{79} + ( 20 - 40 \zeta_{6} ) q^{80} + 9 q^{81} + ( 40 - 80 \zeta_{6} ) q^{82} + ( -52 + 104 \zeta_{6} ) q^{83} + ( 39 - 6 \zeta_{6} ) q^{84} + 26 q^{86} + ( 38 - 76 \zeta_{6} ) q^{87} -70 q^{88} + ( 24 - 48 \zeta_{6} ) q^{89} + ( -12 + 24 \zeta_{6} ) q^{90} + ( -52 + 8 \zeta_{6} ) q^{91} + 42 q^{92} -48 q^{93} + ( -16 + 32 \zeta_{6} ) q^{94} -144 q^{95} + ( -33 + 66 \zeta_{6} ) q^{96} + ( -32 + 64 \zeta_{6} ) q^{97} + ( -55 + 16 \zeta_{6} ) q^{98} -30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 6q^{4} + 2q^{7} - 14q^{8} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 6q^{4} + 2q^{7} - 14q^{8} - 6q^{9} + 20q^{11} + 2q^{14} + 24q^{15} + 10q^{16} - 6q^{18} - 24q^{21} + 20q^{22} - 28q^{23} - 46q^{25} - 6q^{28} - 76q^{29} + 24q^{30} + 66q^{32} + 96q^{35} + 18q^{36} + 52q^{37} - 24q^{39} - 24q^{42} + 52q^{43} - 60q^{44} - 28q^{46} - 94q^{49} - 46q^{50} + 20q^{53} - 14q^{56} + 72q^{57} - 76q^{58} - 72q^{60} - 6q^{63} + 26q^{64} + 96q^{65} + 148q^{67} + 96q^{70} - 124q^{71} + 42q^{72} + 52q^{74} + 20q^{77} - 24q^{78} - 92q^{79} + 18q^{81} + 72q^{84} + 52q^{86} - 140q^{88} - 96q^{91} + 84q^{92} - 96q^{93} - 288q^{95} - 94q^{98} - 60q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 6.92820i −7.00000 −3.00000 6.92820i
13.2 1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 + 6.92820i −7.00000 −3.00000 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.d.a 2
3.b odd 2 1 63.3.d.b 2
4.b odd 2 1 336.3.f.a 2
5.b even 2 1 525.3.h.a 2
5.c odd 4 2 525.3.e.a 4
7.b odd 2 1 inner 21.3.d.a 2
7.c even 3 1 147.3.f.b 2
7.c even 3 1 147.3.f.d 2
7.d odd 6 1 147.3.f.b 2
7.d odd 6 1 147.3.f.d 2
8.b even 2 1 1344.3.f.c 2
8.d odd 2 1 1344.3.f.b 2
12.b even 2 1 1008.3.f.d 2
21.c even 2 1 63.3.d.b 2
21.g even 6 1 441.3.m.d 2
21.g even 6 1 441.3.m.f 2
21.h odd 6 1 441.3.m.d 2
21.h odd 6 1 441.3.m.f 2
28.d even 2 1 336.3.f.a 2
35.c odd 2 1 525.3.h.a 2
35.f even 4 2 525.3.e.a 4
56.e even 2 1 1344.3.f.b 2
56.h odd 2 1 1344.3.f.c 2
84.h odd 2 1 1008.3.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 1.a even 1 1 trivial
21.3.d.a 2 7.b odd 2 1 inner
63.3.d.b 2 3.b odd 2 1
63.3.d.b 2 21.c even 2 1
147.3.f.b 2 7.c even 3 1
147.3.f.b 2 7.d odd 6 1
147.3.f.d 2 7.c even 3 1
147.3.f.d 2 7.d odd 6 1
336.3.f.a 2 4.b odd 2 1
336.3.f.a 2 28.d even 2 1
441.3.m.d 2 21.g even 6 1
441.3.m.d 2 21.h odd 6 1
441.3.m.f 2 21.g even 6 1
441.3.m.f 2 21.h odd 6 1
525.3.e.a 4 5.c odd 4 2
525.3.e.a 4 35.f even 4 2
525.3.h.a 2 5.b even 2 1
525.3.h.a 2 35.c odd 2 1
1008.3.f.d 2 12.b even 2 1
1008.3.f.d 2 84.h odd 2 1
1344.3.f.b 2 8.d odd 2 1
1344.3.f.b 2 56.e even 2 1
1344.3.f.c 2 8.b even 2 1
1344.3.f.c 2 56.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + 4 T^{2} )^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - 2 T^{2} + 625 T^{4} \)
$7$ \( 1 - 2 T + 49 T^{2} \)
$11$ \( ( 1 - 10 T + 121 T^{2} )^{2} \)
$13$ \( 1 - 290 T^{2} + 28561 T^{4} \)
$17$ \( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \)
$19$ \( 1 - 290 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 + 14 T + 529 T^{2} )^{2} \)
$29$ \( ( 1 + 38 T + 841 T^{2} )^{2} \)
$31$ \( 1 - 1154 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 - 26 T + 1369 T^{2} )^{2} \)
$41$ \( 1 + 1438 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 - 26 T + 1849 T^{2} )^{2} \)
$47$ \( 1 - 3650 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 - 10 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 1154 T^{2} + 12117361 T^{4} \)
$61$ \( 1 - 6242 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 - 74 T + 4489 T^{2} )^{2} \)
$71$ \( ( 1 + 62 T + 5041 T^{2} )^{2} \)
$73$ \( 1 - 8930 T^{2} + 28398241 T^{4} \)
$79$ \( ( 1 + 46 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 5666 T^{2} + 47458321 T^{4} \)
$89$ \( 1 - 14114 T^{2} + 62742241 T^{4} \)
$97$ \( 1 - 15746 T^{2} + 88529281 T^{4} \)
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