Properties

Label 21.6.e.a
Level $21$
Weight $6$
Character orbit 21.e
Analytic conductor $3.368$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.36806021607\)
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + 9 \zeta_{6} q^{3} + 28 \zeta_{6} q^{4} + ( -11 + 11 \zeta_{6} ) q^{5} + 18 q^{6} + ( 126 + 7 \zeta_{6} ) q^{7} + 120 q^{8} + ( -81 + 81 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + 9 \zeta_{6} q^{3} + 28 \zeta_{6} q^{4} + ( -11 + 11 \zeta_{6} ) q^{5} + 18 q^{6} + ( 126 + 7 \zeta_{6} ) q^{7} + 120 q^{8} + ( -81 + 81 \zeta_{6} ) q^{9} + 22 \zeta_{6} q^{10} -269 \zeta_{6} q^{11} + ( -252 + 252 \zeta_{6} ) q^{12} -308 q^{13} + ( 266 - 252 \zeta_{6} ) q^{14} -99 q^{15} + ( -656 + 656 \zeta_{6} ) q^{16} -1896 \zeta_{6} q^{17} + 162 \zeta_{6} q^{18} + ( 164 - 164 \zeta_{6} ) q^{19} -308 q^{20} + ( -63 + 1197 \zeta_{6} ) q^{21} -538 q^{22} + ( 3264 - 3264 \zeta_{6} ) q^{23} + 1080 \zeta_{6} q^{24} + 3004 \zeta_{6} q^{25} + ( -616 + 616 \zeta_{6} ) q^{26} -729 q^{27} + ( -196 + 3724 \zeta_{6} ) q^{28} + 2417 q^{29} + ( -198 + 198 \zeta_{6} ) q^{30} -2841 \zeta_{6} q^{31} + 5152 \zeta_{6} q^{32} + ( 2421 - 2421 \zeta_{6} ) q^{33} -3792 q^{34} + ( -1463 + 1386 \zeta_{6} ) q^{35} -2268 q^{36} + ( 11328 - 11328 \zeta_{6} ) q^{37} -328 \zeta_{6} q^{38} -2772 \zeta_{6} q^{39} + ( -1320 + 1320 \zeta_{6} ) q^{40} -16856 q^{41} + ( 2268 + 126 \zeta_{6} ) q^{42} -7894 q^{43} + ( 7532 - 7532 \zeta_{6} ) q^{44} -891 \zeta_{6} q^{45} -6528 \zeta_{6} q^{46} + ( -21102 + 21102 \zeta_{6} ) q^{47} -5904 q^{48} + ( 15827 + 1813 \zeta_{6} ) q^{49} + 6008 q^{50} + ( 17064 - 17064 \zeta_{6} ) q^{51} -8624 \zeta_{6} q^{52} + 29691 \zeta_{6} q^{53} + ( -1458 + 1458 \zeta_{6} ) q^{54} + 2959 q^{55} + ( 15120 + 840 \zeta_{6} ) q^{56} + 1476 q^{57} + ( 4834 - 4834 \zeta_{6} ) q^{58} + 8163 \zeta_{6} q^{59} -2772 \zeta_{6} q^{60} + ( -15166 + 15166 \zeta_{6} ) q^{61} -5682 q^{62} + ( -10773 + 10206 \zeta_{6} ) q^{63} -10688 q^{64} + ( 3388 - 3388 \zeta_{6} ) q^{65} -4842 \zeta_{6} q^{66} + 32078 \zeta_{6} q^{67} + ( 53088 - 53088 \zeta_{6} ) q^{68} + 29376 q^{69} + ( -154 + 2926 \zeta_{6} ) q^{70} -38274 q^{71} + ( -9720 + 9720 \zeta_{6} ) q^{72} -34866 \zeta_{6} q^{73} -22656 \zeta_{6} q^{74} + ( -27036 + 27036 \zeta_{6} ) q^{75} + 4592 q^{76} + ( 1883 - 35777 \zeta_{6} ) q^{77} -5544 q^{78} + ( -13529 + 13529 \zeta_{6} ) q^{79} -7216 \zeta_{6} q^{80} -6561 \zeta_{6} q^{81} + ( -33712 + 33712 \zeta_{6} ) q^{82} -68103 q^{83} + ( -33516 + 31752 \zeta_{6} ) q^{84} + 20856 q^{85} + ( -15788 + 15788 \zeta_{6} ) q^{86} + 21753 \zeta_{6} q^{87} -32280 \zeta_{6} q^{88} + ( 114922 - 114922 \zeta_{6} ) q^{89} -1782 q^{90} + ( -38808 - 2156 \zeta_{6} ) q^{91} + 91392 q^{92} + ( 25569 - 25569 \zeta_{6} ) q^{93} + 42204 \zeta_{6} q^{94} + 1804 \zeta_{6} q^{95} + ( -46368 + 46368 \zeta_{6} ) q^{96} + 154959 q^{97} + ( 35280 - 31654 \zeta_{6} ) q^{98} + 21789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 9q^{3} + 28q^{4} - 11q^{5} + 36q^{6} + 259q^{7} + 240q^{8} - 81q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 9q^{3} + 28q^{4} - 11q^{5} + 36q^{6} + 259q^{7} + 240q^{8} - 81q^{9} + 22q^{10} - 269q^{11} - 252q^{12} - 616q^{13} + 280q^{14} - 198q^{15} - 656q^{16} - 1896q^{17} + 162q^{18} + 164q^{19} - 616q^{20} + 1071q^{21} - 1076q^{22} + 3264q^{23} + 1080q^{24} + 3004q^{25} - 616q^{26} - 1458q^{27} + 3332q^{28} + 4834q^{29} - 198q^{30} - 2841q^{31} + 5152q^{32} + 2421q^{33} - 7584q^{34} - 1540q^{35} - 4536q^{36} + 11328q^{37} - 328q^{38} - 2772q^{39} - 1320q^{40} - 33712q^{41} + 4662q^{42} - 15788q^{43} + 7532q^{44} - 891q^{45} - 6528q^{46} - 21102q^{47} - 11808q^{48} + 33467q^{49} + 12016q^{50} + 17064q^{51} - 8624q^{52} + 29691q^{53} - 1458q^{54} + 5918q^{55} + 31080q^{56} + 2952q^{57} + 4834q^{58} + 8163q^{59} - 2772q^{60} - 15166q^{61} - 11364q^{62} - 11340q^{63} - 21376q^{64} + 3388q^{65} - 4842q^{66} + 32078q^{67} + 53088q^{68} + 58752q^{69} + 2618q^{70} - 76548q^{71} - 9720q^{72} - 34866q^{73} - 22656q^{74} - 27036q^{75} + 9184q^{76} - 32011q^{77} - 11088q^{78} - 13529q^{79} - 7216q^{80} - 6561q^{81} - 33712q^{82} - 136206q^{83} - 35280q^{84} + 41712q^{85} - 15788q^{86} + 21753q^{87} - 32280q^{88} + 114922q^{89} - 3564q^{90} - 79772q^{91} + 182784q^{92} + 25569q^{93} + 42204q^{94} + 1804q^{95} - 46368q^{96} + 309918q^{97} + 38906q^{98} + 43578q^{99} + O(q^{100}) \)

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 4.50000 7.79423i 14.0000 24.2487i −5.50000 9.52628i 18.0000 129.500 6.06218i 120.000 −40.5000 70.1481i 11.0000 19.0526i
16.1 1.00000 1.73205i 4.50000 + 7.79423i 14.0000 + 24.2487i −5.50000 + 9.52628i 18.0000 129.500 + 6.06218i 120.000 −40.5000 + 70.1481i 11.0000 + 19.0526i
\(n\): 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 21.6.e.a 2
3.b odd 2 1 63.6.e.a 2
4.b odd 2 1 336.6.q.b 2
7.b odd 2 1 147.6.e.g 2
7.c even 3 1 inner 21.6.e.a 2
7.c even 3 1 147.6.a.c 1
7.d odd 6 1 147.6.a.d 1
7.d odd 6 1 147.6.e.g 2
21.g even 6 1 441.6.a.h 1
21.h odd 6 1 63.6.e.a 2
21.h odd 6 1 441.6.a.g 1
28.g odd 6 1 336.6.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 1.a even 1 1 trivial
21.6.e.a 2 7.c even 3 1 inner
63.6.e.a 2 3.b odd 2 1
63.6.e.a 2 21.h odd 6 1
147.6.a.c 1 7.c even 3 1
147.6.a.d 1 7.d odd 6 1
147.6.e.g 2 7.b odd 2 1
147.6.e.g 2 7.d odd 6 1
336.6.q.b 2 4.b odd 2 1
336.6.q.b 2 28.g odd 6 1
441.6.a.g 1 21.h odd 6 1
441.6.a.h 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( 81 - 9 T + T^{2} \)
$5$ \( 121 + 11 T + T^{2} \)
$7$ \( 16807 - 259 T + T^{2} \)
$11$ \( 72361 + 269 T + T^{2} \)
$13$ \( ( 308 + T )^{2} \)
$17$ \( 3594816 + 1896 T + T^{2} \)
$19$ \( 26896 - 164 T + T^{2} \)
$23$ \( 10653696 - 3264 T + T^{2} \)
$29$ \( ( -2417 + T )^{2} \)
$31$ \( 8071281 + 2841 T + T^{2} \)
$37$ \( 128323584 - 11328 T + T^{2} \)
$41$ \( ( 16856 + T )^{2} \)
$43$ \( ( 7894 + T )^{2} \)
$47$ \( 445294404 + 21102 T + T^{2} \)
$53$ \( 881555481 - 29691 T + T^{2} \)
$59$ \( 66634569 - 8163 T + T^{2} \)
$61$ \( 230007556 + 15166 T + T^{2} \)
$67$ \( 1028998084 - 32078 T + T^{2} \)
$71$ \( ( 38274 + T )^{2} \)
$73$ \( 1215637956 + 34866 T + T^{2} \)
$79$ \( 183033841 + 13529 T + T^{2} \)
$83$ \( ( 68103 + T )^{2} \)
$89$ \( 13207066084 - 114922 T + T^{2} \)
$97$ \( ( -154959 + T )^{2} \)
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