Properties

Label 21.3.h.b
Level $21$
Weight $3$
Character orbit 21.h
Analytic conductor $0.572$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} -\beta_{1} q^{5} + ( -5 - 2 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} -3 \beta_{3} q^{8} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} -\beta_{1} q^{5} + ( -5 - 2 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} -3 \beta_{3} q^{8} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{9} -5 \beta_{2} q^{10} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{11} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{12} -2 q^{13} + 7 \beta_{3} q^{14} + ( 5 + 2 \beta_{3} ) q^{15} + ( 19 - 19 \beta_{2} ) q^{16} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{17} + ( \beta_{1} + 20 \beta_{2} - \beta_{3} ) q^{18} + ( -16 + 16 \beta_{2} ) q^{19} -\beta_{3} q^{20} + ( 14 - 7 \beta_{1} - 14 \beta_{2} ) q^{21} -25 q^{22} -6 \beta_{1} q^{23} + ( -6 \beta_{1} + 15 \beta_{2} + 6 \beta_{3} ) q^{24} -20 \beta_{2} q^{25} -2 \beta_{1} q^{26} + ( -22 - 7 \beta_{3} ) q^{27} + ( -7 + 7 \beta_{2} ) q^{28} + 7 \beta_{3} q^{29} + ( -10 + 5 \beta_{1} + 10 \beta_{2} ) q^{30} + 3 \beta_{2} q^{31} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{32} + ( 25 + 10 \beta_{1} - 25 \beta_{2} ) q^{33} + 60 q^{34} -7 \beta_{3} q^{35} + ( 1 + 4 \beta_{3} ) q^{36} + ( -12 + 12 \beta_{2} ) q^{37} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{38} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{39} + ( -15 + 15 \beta_{2} ) q^{40} + 14 \beta_{3} q^{41} + ( 14 \beta_{1} - 35 \beta_{2} - 14 \beta_{3} ) q^{42} + 44 q^{43} -5 \beta_{1} q^{44} + ( -\beta_{1} - 20 \beta_{2} + \beta_{3} ) q^{45} -30 \beta_{2} q^{46} + 6 \beta_{1} q^{47} + ( -38 + 19 \beta_{3} ) q^{48} + ( -49 + 49 \beta_{2} ) q^{49} -20 \beta_{3} q^{50} + ( -60 - 24 \beta_{1} + 60 \beta_{2} ) q^{51} -2 \beta_{2} q^{52} + ( 9 \beta_{1} - 9 \beta_{3} ) q^{53} + ( 35 - 22 \beta_{1} - 35 \beta_{2} ) q^{54} + 25 q^{55} + ( 21 \beta_{1} - 21 \beta_{3} ) q^{56} + ( 32 - 16 \beta_{3} ) q^{57} + ( -35 + 35 \beta_{2} ) q^{58} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{59} + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{60} + ( 26 - 26 \beta_{2} ) q^{61} + 3 \beta_{3} q^{62} + ( 7 + 28 \beta_{3} ) q^{63} -41 q^{64} + 2 \beta_{1} q^{65} + ( 25 \beta_{1} + 50 \beta_{2} - 25 \beta_{3} ) q^{66} -52 \beta_{2} q^{67} + 12 \beta_{1} q^{68} + ( 30 + 12 \beta_{3} ) q^{69} + ( 35 - 35 \beta_{2} ) q^{70} + 42 \beta_{3} q^{71} + ( 60 - 3 \beta_{1} - 60 \beta_{2} ) q^{72} -18 \beta_{2} q^{73} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{74} + ( -40 + 20 \beta_{1} + 40 \beta_{2} ) q^{75} -16 q^{76} -35 \beta_{1} q^{77} + ( 10 + 4 \beta_{3} ) q^{78} + ( 79 - 79 \beta_{2} ) q^{79} + ( -19 \beta_{1} + 19 \beta_{3} ) q^{80} + ( 8 \beta_{1} + 79 \beta_{2} - 8 \beta_{3} ) q^{81} + ( -70 + 70 \beta_{2} ) q^{82} -63 \beta_{3} q^{83} + ( 14 - 7 \beta_{3} ) q^{84} -60 q^{85} + 44 \beta_{1} q^{86} + ( 14 \beta_{1} - 35 \beta_{2} - 14 \beta_{3} ) q^{87} + 75 \beta_{2} q^{88} -22 \beta_{1} q^{89} + ( -5 - 20 \beta_{3} ) q^{90} -14 \beta_{2} q^{91} -6 \beta_{3} q^{92} + ( 6 - 3 \beta_{1} - 6 \beta_{2} ) q^{93} + 30 \beta_{2} q^{94} + ( 16 \beta_{1} - 16 \beta_{3} ) q^{95} + ( -35 - 14 \beta_{1} + 35 \beta_{2} ) q^{96} -93 q^{97} + ( -49 \beta_{1} + 49 \beta_{3} ) q^{98} + ( -100 + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 2q^{4} - 20q^{6} + 14q^{7} + 2q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 2q^{4} - 20q^{6} + 14q^{7} + 2q^{9} - 10q^{10} + 4q^{12} - 8q^{13} + 20q^{15} + 38q^{16} + 40q^{18} - 32q^{19} + 28q^{21} - 100q^{22} + 30q^{24} - 40q^{25} - 88q^{27} - 14q^{28} - 20q^{30} + 6q^{31} + 50q^{33} + 240q^{34} + 4q^{36} - 24q^{37} + 8q^{39} - 30q^{40} - 70q^{42} + 176q^{43} - 40q^{45} - 60q^{46} - 152q^{48} - 98q^{49} - 120q^{51} - 4q^{52} + 70q^{54} + 100q^{55} + 128q^{57} - 70q^{58} + 10q^{60} + 52q^{61} + 28q^{63} - 164q^{64} + 100q^{66} - 104q^{67} + 120q^{69} + 70q^{70} + 120q^{72} - 36q^{73} - 80q^{75} - 64q^{76} + 40q^{78} + 158q^{79} + 158q^{81} - 140q^{82} + 56q^{84} - 240q^{85} - 70q^{87} + 150q^{88} - 20q^{90} - 28q^{91} + 12q^{93} + 60q^{94} - 70q^{96} - 372q^{97} - 400q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i 0.936492 2.85008i 0.500000 + 0.866025i 1.93649 + 1.11803i −5.00000 + 4.47214i 3.50000 + 6.06218i 6.70820i −7.24597 5.33816i −2.50000 4.33013i
2.2 1.93649 + 1.11803i −2.93649 0.614017i 0.500000 + 0.866025i −1.93649 1.11803i −5.00000 4.47214i 3.50000 + 6.06218i 6.70820i 8.24597 + 3.60611i −2.50000 4.33013i
11.1 −1.93649 + 1.11803i 0.936492 + 2.85008i 0.500000 0.866025i 1.93649 1.11803i −5.00000 4.47214i 3.50000 6.06218i 6.70820i −7.24597 + 5.33816i −2.50000 + 4.33013i
11.2 1.93649 1.11803i −2.93649 + 0.614017i 0.500000 0.866025i −1.93649 + 1.11803i −5.00000 + 4.47214i 3.50000 6.06218i 6.70820i 8.24597 3.60611i −2.50000 + 4.33013i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h 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.h.b 4
3.b odd 2 1 inner 21.3.h.b 4
4.b odd 2 1 336.3.bn.f 4
7.b odd 2 1 147.3.h.b 4
7.c even 3 1 inner 21.3.h.b 4
7.c even 3 1 147.3.b.d 2
7.d odd 6 1 147.3.b.c 2
7.d odd 6 1 147.3.h.b 4
12.b even 2 1 336.3.bn.f 4
21.c even 2 1 147.3.h.b 4
21.g even 6 1 147.3.b.c 2
21.g even 6 1 147.3.h.b 4
21.h odd 6 1 inner 21.3.h.b 4
21.h odd 6 1 147.3.b.d 2
28.g odd 6 1 336.3.bn.f 4
84.n even 6 1 336.3.bn.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.h.b 4 1.a even 1 1 trivial
21.3.h.b 4 3.b odd 2 1 inner
21.3.h.b 4 7.c even 3 1 inner
21.3.h.b 4 21.h odd 6 1 inner
147.3.b.c 2 7.d odd 6 1
147.3.b.c 2 21.g even 6 1
147.3.b.d 2 7.c even 3 1
147.3.b.d 2 21.h odd 6 1
147.3.h.b 4 7.b odd 2 1
147.3.h.b 4 7.d odd 6 1
147.3.h.b 4 21.c even 2 1
147.3.h.b 4 21.g even 6 1
336.3.bn.f 4 4.b odd 2 1
336.3.bn.f 4 12.b even 2 1
336.3.bn.f 4 28.g odd 6 1
336.3.bn.f 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5 T_{2}^{2} + 25 \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} - 7 T^{4} + 48 T^{6} + 256 T^{8} \)
$3$ \( 1 + 4 T + 7 T^{2} + 36 T^{3} + 81 T^{4} \)
$5$ \( 1 + 45 T^{2} + 1400 T^{4} + 28125 T^{6} + 390625 T^{8} \)
$7$ \( ( 1 - 7 T + 49 T^{2} )^{2} \)
$11$ \( 1 + 117 T^{2} - 952 T^{4} + 1712997 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 + 2 T + 169 T^{2} )^{4} \)
$17$ \( 1 - 142 T^{2} - 63357 T^{4} - 11859982 T^{6} + 6975757441 T^{8} \)
$19$ \( ( 1 + 16 T - 105 T^{2} + 5776 T^{3} + 130321 T^{4} )^{2} \)
$23$ \( ( 1 - 44 T + 1407 T^{2} - 23276 T^{3} + 279841 T^{4} )( 1 + 44 T + 1407 T^{2} + 23276 T^{3} + 279841 T^{4} ) \)
$29$ \( ( 1 - 1437 T^{2} + 707281 T^{4} )^{2} \)
$31$ \( ( 1 - 3 T - 952 T^{2} - 2883 T^{3} + 923521 T^{4} )^{2} \)
$37$ \( ( 1 + 12 T - 1225 T^{2} + 16428 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 - 2382 T^{2} + 2825761 T^{4} )^{2} \)
$43$ \( ( 1 - 44 T + 1849 T^{2} )^{4} \)
$47$ \( 1 + 4238 T^{2} + 13080963 T^{4} + 20680088078 T^{6} + 23811286661761 T^{8} \)
$53$ \( 1 + 5213 T^{2} + 19284888 T^{4} + 41133077453 T^{6} + 62259690411361 T^{8} \)
$59$ \( 1 + 6557 T^{2} + 30876888 T^{4} + 79453536077 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 - 26 T - 3045 T^{2} - 96746 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 + 52 T - 1785 T^{2} + 233428 T^{3} + 20151121 T^{4} )^{2} \)
$71$ \( ( 1 - 1262 T^{2} + 25411681 T^{4} )^{2} \)
$73$ \( ( 1 + 18 T - 5005 T^{2} + 95922 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( ( 1 - 79 T )^{4}( 1 + 79 T + 6241 T^{2} )^{2} \)
$83$ \( ( 1 + 6067 T^{2} + 47458321 T^{4} )^{2} \)
$89$ \( 1 + 13422 T^{2} + 117407843 T^{4} + 842126358702 T^{6} + 3936588805702081 T^{8} \)
$97$ \( ( 1 + 93 T + 9409 T^{2} )^{4} \)
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