Properties

Label 21.6.e.b
Level $21$
Weight $6$
Character orbit 21.e
Analytic conductor $3.368$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
Defining polynomial: \(x^{4} - x^{3} - 20 x^{2} - 21 x + 441\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} - \beta_{3} ) q^{2} -9 \beta_{1} q^{3} + ( 31 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{4} + ( 13 + 13 \beta_{1} - 7 \beta_{3} ) q^{5} + ( 9 - 9 \beta_{2} ) q^{6} + ( -84 + 7 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} ) q^{7} + ( -185 + 5 \beta_{2} ) q^{8} + ( -81 - 81 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} - \beta_{3} ) q^{2} -9 \beta_{1} q^{3} + ( 31 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{4} + ( 13 + 13 \beta_{1} - 7 \beta_{3} ) q^{5} + ( 9 - 9 \beta_{2} ) q^{6} + ( -84 + 7 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} ) q^{7} + ( -185 + 5 \beta_{2} ) q^{8} + ( -81 - 81 \beta_{1} ) q^{9} + ( 447 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{10} + ( -569 \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 279 + 279 \beta_{1} - 27 \beta_{3} ) q^{12} + ( 458 - 9 \beta_{2} ) q^{13} + ( 343 - 518 \beta_{1} - 21 \beta_{2} + 98 \beta_{3} ) q^{14} + ( 117 - 63 \beta_{2} ) q^{15} + ( 497 + 497 \beta_{1} + 99 \beta_{3} ) q^{16} + ( -236 \beta_{1} + 148 \beta_{2} - 148 \beta_{3} ) q^{17} + ( -81 \beta_{1} - 81 \beta_{2} + 81 \beta_{3} ) q^{18} + ( -1142 - 1142 \beta_{1} + 27 \beta_{3} ) q^{19} + ( -1705 + 277 \beta_{2} ) q^{20} + ( 63 + 819 \beta_{1} + 63 \beta_{2} + 63 \beta_{3} ) q^{21} + ( 507 - 567 \beta_{2} ) q^{22} + ( -644 - 644 \beta_{1} + 308 \beta_{3} ) q^{23} + ( 1665 \beta_{1} + 45 \beta_{2} - 45 \beta_{3} ) q^{24} + ( 82 \beta_{1} + 231 \beta_{2} - 231 \beta_{3} ) q^{25} + ( 1016 + 1016 \beta_{1} - 476 \beta_{3} ) q^{26} -729 q^{27} + ( 2387 - 1519 \beta_{1} - 490 \beta_{2} + 35 \beta_{3} ) q^{28} + ( -1131 - 45 \beta_{2} ) q^{29} + ( 4023 + 4023 \beta_{1} - 243 \beta_{3} ) q^{30} + ( 1763 \beta_{1} + 768 \beta_{2} - 768 \beta_{3} ) q^{31} + ( 279 \beta_{1} + 459 \beta_{2} - 459 \beta_{3} ) q^{32} + ( -5121 - 5121 \beta_{1} - 9 \beta_{3} ) q^{33} + ( -8940 + 60 \beta_{2} ) q^{34} + ( 1855 - 4130 \beta_{1} - 231 \beta_{2} + 728 \beta_{3} ) q^{35} + ( 2511 - 243 \beta_{2} ) q^{36} + ( 9982 + 9982 \beta_{1} + 855 \beta_{3} ) q^{37} + ( -2816 \beta_{1} - 1196 \beta_{2} + 1196 \beta_{3} ) q^{38} + ( -4122 \beta_{1} - 81 \beta_{2} + 81 \beta_{3} ) q^{39} + ( -4575 - 4575 \beta_{1} + 1395 \beta_{3} ) q^{40} + ( -6852 - 846 \beta_{2} ) q^{41} + ( -4662 - 7749 \beta_{1} + 693 \beta_{2} + 189 \beta_{3} ) q^{42} + ( -364 + 2043 \beta_{2} ) q^{43} + ( 17453 + 17453 \beta_{1} - 1673 \beta_{3} ) q^{44} + ( -1053 \beta_{1} - 567 \beta_{2} + 567 \beta_{3} ) q^{45} + ( -19740 \beta_{1} - 1260 \beta_{2} + 1260 \beta_{3} ) q^{46} + ( 11278 + 11278 \beta_{1} - 604 \beta_{3} ) q^{47} + ( 4473 + 891 \beta_{2} ) q^{48} + ( -2107 - 1225 \beta_{1} + 1225 \beta_{2} - 2450 \beta_{3} ) q^{49} + ( -14404 + 544 \beta_{2} ) q^{50} + ( -2124 - 2124 \beta_{1} - 1332 \beta_{3} ) q^{51} + ( 15872 \beta_{1} + 1680 \beta_{2} - 1680 \beta_{3} ) q^{52} + ( -14951 \beta_{1} - 1751 \beta_{2} + 1751 \beta_{3} ) q^{53} + ( -729 - 729 \beta_{1} + 729 \beta_{3} ) q^{54} + ( 6963 - 3963 \beta_{2} ) q^{55} + ( 17710 + 3045 \beta_{1} + 875 \beta_{2} - 2625 \beta_{3} ) q^{56} + ( -10278 + 243 \beta_{2} ) q^{57} + ( 1659 + 1659 \beta_{1} + 1041 \beta_{3} ) q^{58} + ( 22507 \beta_{1} - 3917 \beta_{2} + 3917 \beta_{3} ) q^{59} + ( 15345 \beta_{1} + 2493 \beta_{2} - 2493 \beta_{3} ) q^{60} + ( -22298 - 22298 \beta_{1} - 2544 \beta_{3} ) q^{61} + ( -49379 + 3299 \beta_{2} ) q^{62} + ( 7371 + 6804 \beta_{1} + 1134 \beta_{2} - 567 \beta_{3} ) q^{63} + ( -12833 + 4365 \beta_{2} ) q^{64} + ( 9860 + 9860 \beta_{1} - 3386 \beta_{3} ) q^{65} + ( -4563 \beta_{1} - 5103 \beta_{2} + 5103 \beta_{3} ) q^{66} + ( 17612 \beta_{1} - 4461 \beta_{2} + 4461 \beta_{3} ) q^{67} + ( -20212 - 20212 \beta_{1} + 4324 \beta_{3} ) q^{68} + ( -5796 + 2772 \beta_{2} ) q^{69} + ( 20307 - 28959 \beta_{1} - 5586 \beta_{2} - 861 \beta_{3} ) q^{70} + ( 50346 - 1404 \beta_{2} ) q^{71} + ( 14985 + 14985 \beta_{1} - 405 \beta_{3} ) q^{72} + ( -16912 \beta_{1} + 5247 \beta_{2} - 5247 \beta_{3} ) q^{73} + ( -43028 \beta_{1} + 8272 \beta_{2} - 8272 \beta_{3} ) q^{74} + ( 738 + 738 \beta_{1} - 2079 \beta_{3} ) q^{75} + ( 40424 - 4344 \beta_{2} ) q^{76} + ( 4851 + 52213 \beta_{1} + 3892 \beta_{2} + 4067 \beta_{3} ) q^{77} + ( 9144 - 4284 \beta_{2} ) q^{78} + ( 12649 + 12649 \beta_{1} + 6834 \beta_{3} ) q^{79} + ( -36505 \beta_{1} + 1499 \beta_{2} - 1499 \beta_{3} ) q^{80} + 6561 \beta_{1} q^{81} + ( 45600 + 45600 \beta_{1} + 5160 \beta_{3} ) q^{82} + ( 31539 + 1899 \beta_{2} ) q^{83} + ( -13671 - 35154 \beta_{1} - 4095 \beta_{2} + 4410 \beta_{3} ) q^{84} + ( -61164 + 1308 \beta_{2} ) q^{85} + ( -127030 - 127030 \beta_{1} + 4450 \beta_{3} ) q^{86} + ( 10179 \beta_{1} - 405 \beta_{2} + 405 \beta_{3} ) q^{87} + ( 104955 \beta_{1} + 2655 \beta_{2} - 2655 \beta_{3} ) q^{88} + ( -14726 - 14726 \beta_{1} - 130 \beta_{3} ) q^{89} + ( 36207 - 2187 \beta_{2} ) q^{90} + ( -42378 - 4606 \beta_{1} - 2450 \beta_{2} + 6475 \beta_{3} ) q^{91} + ( 77252 - 12404 \beta_{2} ) q^{92} + ( 15867 + 15867 \beta_{1} - 6912 \beta_{3} ) q^{93} + ( 48726 \beta_{1} + 12486 \beta_{2} - 12486 \beta_{3} ) q^{94} + ( -26564 \beta_{1} - 8534 \beta_{2} + 8534 \beta_{3} ) q^{95} + ( 2511 + 2511 \beta_{1} - 4131 \beta_{3} ) q^{96} + ( -4387 + 1017 \beta_{2} ) q^{97} + ( -76832 + 73843 \beta_{1} + 3675 \beta_{2} - 343 \beta_{3} ) q^{98} + ( -46089 - 81 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + 18q^{3} - 65q^{4} + 33q^{5} + 54q^{6} - 350q^{7} - 750q^{8} - 162q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + 18q^{3} - 65q^{4} + 33q^{5} + 54q^{6} - 350q^{7} - 750q^{8} - 162q^{9} - 921q^{10} + 1137q^{11} + 585q^{12} + 1850q^{13} + 2352q^{14} + 594q^{15} + 895q^{16} + 324q^{17} + 243q^{18} - 2311q^{19} - 7374q^{20} - 1575q^{21} + 3162q^{22} - 1596q^{23} - 3375q^{24} - 395q^{25} + 2508q^{26} - 2916q^{27} + 13531q^{28} - 4434q^{29} + 8289q^{30} - 4294q^{31} - 1017q^{32} - 10233q^{33} - 35880q^{34} + 15414q^{35} + 10530q^{36} + 19109q^{37} + 6828q^{38} + 8325q^{39} - 10545q^{40} - 25716q^{41} - 4725q^{42} - 5542q^{43} + 36579q^{44} + 2673q^{45} + 40740q^{46} + 23160q^{47} + 16110q^{48} - 5978q^{49} - 58704q^{50} - 2916q^{51} - 33424q^{52} + 31653q^{53} - 2187q^{54} + 35778q^{55} + 65625q^{56} - 41598q^{57} + 2277q^{58} - 41097q^{59} - 33183q^{60} - 42052q^{61} - 204114q^{62} + 14175q^{63} - 60062q^{64} + 23106q^{65} + 14229q^{66} - 30763q^{67} - 44748q^{68} - 28728q^{69} + 151179q^{70} + 204192q^{71} + 30375q^{72} + 28577q^{73} + 77784q^{74} + 3555q^{75} + 170384q^{76} - 96873q^{77} + 45144q^{78} + 18464q^{79} + 71511q^{80} - 13122q^{81} + 86040q^{82} + 122358q^{83} + 19404q^{84} - 247272q^{85} - 258510q^{86} - 19953q^{87} - 212565q^{88} - 29322q^{89} + 149202q^{90} - 161875q^{91} + 333816q^{92} + 38646q^{93} - 109938q^{94} + 61662q^{95} + 9153q^{96} - 19582q^{97} - 462021q^{98} - 184194q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 20 x^{2} - 21 x + 441\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 20 \nu^{2} - 20 \nu - 441 \)\()/420\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 41 \nu \)\()/21\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 20 \nu - 41 \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 61 \beta_{1} + 62\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(40 \beta_{3} - 20 \beta_{2} + 20 \beta_{1} + 103\)\()/3\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
4.19493 + 1.84460i
−3.69493 2.71062i
4.19493 1.84460i
−3.69493 + 2.71062i
−3.19493 5.53379i 4.50000 7.79423i −4.41520 + 7.64735i −19.3645 33.5404i −57.5088 −87.5000 + 95.6596i −148.051 −40.5000 70.1481i −123.737 + 214.318i
4.2 4.69493 + 8.13186i 4.50000 7.79423i −28.0848 + 48.6443i 35.8645 + 62.1192i 84.5088 −87.5000 95.6596i −226.949 −40.5000 70.1481i −336.763 + 583.291i
16.1 −3.19493 + 5.53379i 4.50000 + 7.79423i −4.41520 7.64735i −19.3645 + 33.5404i −57.5088 −87.5000 95.6596i −148.051 −40.5000 + 70.1481i −123.737 214.318i
16.2 4.69493 8.13186i 4.50000 + 7.79423i −28.0848 48.6443i 35.8645 62.1192i 84.5088 −87.5000 + 95.6596i −226.949 −40.5000 + 70.1481i −336.763 583.291i
\(n\): 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.b 4
3.b odd 2 1 63.6.e.c 4
4.b odd 2 1 336.6.q.e 4
7.b odd 2 1 147.6.e.l 4
7.c even 3 1 inner 21.6.e.b 4
7.c even 3 1 147.6.a.i 2
7.d odd 6 1 147.6.a.k 2
7.d odd 6 1 147.6.e.l 4
21.g even 6 1 441.6.a.s 2
21.h odd 6 1 63.6.e.c 4
21.h odd 6 1 441.6.a.t 2
28.g odd 6 1 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 1.a even 1 1 trivial
21.6.e.b 4 7.c even 3 1 inner
63.6.e.c 4 3.b odd 2 1
63.6.e.c 4 21.h odd 6 1
147.6.a.i 2 7.c even 3 1
147.6.a.k 2 7.d odd 6 1
147.6.e.l 4 7.b odd 2 1
147.6.e.l 4 7.d odd 6 1
336.6.q.e 4 4.b odd 2 1
336.6.q.e 4 28.g odd 6 1
441.6.a.s 2 21.g even 6 1
441.6.a.t 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3 T_{2}^{3} + 69 T_{2}^{2} + 180 T_{2} + 3600 \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3600 + 180 T + 69 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( ( 81 - 9 T + T^{2} )^{2} \)
$5$ \( 7717284 + 91674 T + 3867 T^{2} - 33 T^{3} + T^{4} \)
$7$ \( ( 16807 + 175 T + T^{2} )^{2} \)
$11$ \( 104412996900 - 367398810 T + 969639 T^{2} - 1137 T^{3} + T^{4} \)
$13$ \( ( 208864 - 925 T + T^{2} )^{2} \)
$17$ \( 1788317798400 + 433278720 T + 1442256 T^{2} - 324 T^{3} + T^{4} \)
$19$ \( 1663584040000 + 2980727800 T + 4050921 T^{2} + 2311 T^{3} + T^{4} \)
$23$ \( 27756881510400 - 8408494080 T + 7815696 T^{2} + 1596 T^{3} + T^{4} \)
$29$ \( ( 1102716 + 2217 T + T^{2} )^{2} \)
$31$ \( 1030855275094225 - 137867178890 T + 50545371 T^{2} + 4294 T^{3} + T^{4} \)
$37$ \( 2096006540522896 - 874851371876 T + 319371717 T^{2} - 19109 T^{3} + T^{4} \)
$41$ \( ( -3221280 + 12858 T + T^{2} )^{2} \)
$43$ \( ( -257902490 + 2771 T + T^{2} )^{2} \)
$47$ \( 12406975550652816 - 2579713748640 T + 424998996 T^{2} - 23160 T^{3} + T^{4} \)
$53$ \( 3554489549811600 - 1887137299620 T + 942292869 T^{2} - 31653 T^{3} + T^{4} \)
$59$ \( 283933372527504144 - 21898700344836 T + 2221817397 T^{2} + 41097 T^{3} + T^{4} \)
$61$ \( 1537789558915600 + 1649054882320 T + 1729156044 T^{2} + 42052 T^{3} + T^{4} \)
$67$ \( 1004438694601272100 - 30831198187070 T + 1948579059 T^{2} + 30763 T^{3} + T^{4} \)
$71$ \( ( 2483190108 - 102096 T + T^{2} )^{2} \)
$73$ \( 2279025242240470084 + 43141098817006 T + 2326289007 T^{2} - 28577 T^{3} + T^{4} \)
$79$ \( 7964059539255172369 + 52106636539168 T + 3162985833 T^{2} - 18464 T^{3} + T^{4} \)
$83$ \( ( 711231498 - 61179 T + T^{2} )^{2} \)
$89$ \( 45750170959266816 + 6271767496512 T + 645886788 T^{2} + 29322 T^{3} + T^{4} \)
$97$ \( ( -40418570 + 9791 T + T^{2} )^{2} \)
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