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: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 + 1) q^{2} - 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + ( - 7 \beta_{3} + 13 \beta_1 + 13) q^{5} + ( - 9 \beta_{2} + 9) q^{6} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 - 84) q^{7} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 + 1) q^{2} - 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + ( - 7 \beta_{3} + 13 \beta_1 + 13) q^{5} + ( - 9 \beta_{2} + 9) q^{6} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 - 84) q^{7} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9} + ( - 27 \beta_{3} + 27 \beta_{2} + 447 \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} - 569 \beta_1) q^{11} + ( - 27 \beta_{3} + 279 \beta_1 + 279) q^{12} + ( - 9 \beta_{2} + 458) q^{13} + (98 \beta_{3} - 21 \beta_{2} - 518 \beta_1 + 343) q^{14} + ( - 63 \beta_{2} + 117) q^{15} + (99 \beta_{3} + 497 \beta_1 + 497) q^{16} + ( - 148 \beta_{3} + 148 \beta_{2} - 236 \beta_1) q^{17} + (81 \beta_{3} - 81 \beta_{2} - 81 \beta_1) q^{18} + (27 \beta_{3} - 1142 \beta_1 - 1142) q^{19} + (277 \beta_{2} - 1705) q^{20} + (63 \beta_{3} + 63 \beta_{2} + 819 \beta_1 + 63) q^{21} + ( - 567 \beta_{2} + 507) q^{22} + (308 \beta_{3} - 644 \beta_1 - 644) q^{23} + ( - 45 \beta_{3} + 45 \beta_{2} + 1665 \beta_1) q^{24} + ( - 231 \beta_{3} + 231 \beta_{2} + 82 \beta_1) q^{25} + ( - 476 \beta_{3} + 1016 \beta_1 + 1016) q^{26} - 729 q^{27} + (35 \beta_{3} - 490 \beta_{2} - 1519 \beta_1 + 2387) q^{28} + ( - 45 \beta_{2} - 1131) q^{29} + ( - 243 \beta_{3} + 4023 \beta_1 + 4023) q^{30} + ( - 768 \beta_{3} + 768 \beta_{2} + 1763 \beta_1) q^{31} + ( - 459 \beta_{3} + 459 \beta_{2} + 279 \beta_1) q^{32} + ( - 9 \beta_{3} - 5121 \beta_1 - 5121) q^{33} + (60 \beta_{2} - 8940) q^{34} + (728 \beta_{3} - 231 \beta_{2} - 4130 \beta_1 + 1855) q^{35} + ( - 243 \beta_{2} + 2511) q^{36} + (855 \beta_{3} + 9982 \beta_1 + 9982) q^{37} + (1196 \beta_{3} - 1196 \beta_{2} - 2816 \beta_1) q^{38} + (81 \beta_{3} - 81 \beta_{2} - 4122 \beta_1) q^{39} + (1395 \beta_{3} - 4575 \beta_1 - 4575) q^{40} + ( - 846 \beta_{2} - 6852) q^{41} + (189 \beta_{3} + 693 \beta_{2} - 7749 \beta_1 - 4662) q^{42} + (2043 \beta_{2} - 364) q^{43} + ( - 1673 \beta_{3} + 17453 \beta_1 + 17453) q^{44} + (567 \beta_{3} - 567 \beta_{2} - 1053 \beta_1) q^{45} + (1260 \beta_{3} - 1260 \beta_{2} - 19740 \beta_1) q^{46} + ( - 604 \beta_{3} + 11278 \beta_1 + 11278) q^{47} + (891 \beta_{2} + 4473) q^{48} + ( - 2450 \beta_{3} + 1225 \beta_{2} - 1225 \beta_1 - 2107) q^{49} + (544 \beta_{2} - 14404) q^{50} + ( - 1332 \beta_{3} - 2124 \beta_1 - 2124) q^{51} + ( - 1680 \beta_{3} + 1680 \beta_{2} + 15872 \beta_1) q^{52} + (1751 \beta_{3} - 1751 \beta_{2} - 14951 \beta_1) q^{53} + (729 \beta_{3} - 729 \beta_1 - 729) q^{54} + ( - 3963 \beta_{2} + 6963) q^{55} + ( - 2625 \beta_{3} + 875 \beta_{2} + 3045 \beta_1 + 17710) q^{56} + (243 \beta_{2} - 10278) q^{57} + (1041 \beta_{3} + 1659 \beta_1 + 1659) q^{58} + (3917 \beta_{3} - 3917 \beta_{2} + 22507 \beta_1) q^{59} + ( - 2493 \beta_{3} + 2493 \beta_{2} + 15345 \beta_1) q^{60} + ( - 2544 \beta_{3} - 22298 \beta_1 - 22298) q^{61} + (3299 \beta_{2} - 49379) q^{62} + ( - 567 \beta_{3} + 1134 \beta_{2} + 6804 \beta_1 + 7371) q^{63} + (4365 \beta_{2} - 12833) q^{64} + ( - 3386 \beta_{3} + 9860 \beta_1 + 9860) q^{65} + (5103 \beta_{3} - 5103 \beta_{2} - 4563 \beta_1) q^{66} + (4461 \beta_{3} - 4461 \beta_{2} + 17612 \beta_1) q^{67} + (4324 \beta_{3} - 20212 \beta_1 - 20212) q^{68} + (2772 \beta_{2} - 5796) q^{69} + ( - 861 \beta_{3} - 5586 \beta_{2} - 28959 \beta_1 + 20307) q^{70} + ( - 1404 \beta_{2} + 50346) q^{71} + ( - 405 \beta_{3} + 14985 \beta_1 + 14985) q^{72} + ( - 5247 \beta_{3} + 5247 \beta_{2} - 16912 \beta_1) q^{73} + ( - 8272 \beta_{3} + 8272 \beta_{2} - 43028 \beta_1) q^{74} + ( - 2079 \beta_{3} + 738 \beta_1 + 738) q^{75} + ( - 4344 \beta_{2} + 40424) q^{76} + (4067 \beta_{3} + 3892 \beta_{2} + 52213 \beta_1 + 4851) q^{77} + ( - 4284 \beta_{2} + 9144) q^{78} + (6834 \beta_{3} + 12649 \beta_1 + 12649) q^{79} + ( - 1499 \beta_{3} + 1499 \beta_{2} - 36505 \beta_1) q^{80} + 6561 \beta_1 q^{81} + (5160 \beta_{3} + 45600 \beta_1 + 45600) q^{82} + (1899 \beta_{2} + 31539) q^{83} + (4410 \beta_{3} - 4095 \beta_{2} - 35154 \beta_1 - 13671) q^{84} + (1308 \beta_{2} - 61164) q^{85} + (4450 \beta_{3} - 127030 \beta_1 - 127030) q^{86} + (405 \beta_{3} - 405 \beta_{2} + 10179 \beta_1) q^{87} + ( - 2655 \beta_{3} + 2655 \beta_{2} + 104955 \beta_1) q^{88} + ( - 130 \beta_{3} - 14726 \beta_1 - 14726) q^{89} + ( - 2187 \beta_{2} + 36207) q^{90} + (6475 \beta_{3} - 2450 \beta_{2} - 4606 \beta_1 - 42378) q^{91} + ( - 12404 \beta_{2} + 77252) q^{92} + ( - 6912 \beta_{3} + 15867 \beta_1 + 15867) q^{93} + ( - 12486 \beta_{3} + 12486 \beta_{2} + 48726 \beta_1) q^{94} + (8534 \beta_{3} - 8534 \beta_{2} - 26564 \beta_1) q^{95} + ( - 4131 \beta_{3} + 2511 \beta_1 + 2511) q^{96} + (1017 \beta_{2} - 4387) q^{97} + ( - 343 \beta_{3} + 3675 \beta_{2} + 73843 \beta_1 - 76832) q^{98} + ( - 81 \beta_{2} - 46089) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 18 q^{3} - 65 q^{4} + 33 q^{5} + 54 q^{6} - 350 q^{7} - 750 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 18 q^{3} - 65 q^{4} + 33 q^{5} + 54 q^{6} - 350 q^{7} - 750 q^{8} - 162 q^{9} - 921 q^{10} + 1137 q^{11} + 585 q^{12} + 1850 q^{13} + 2352 q^{14} + 594 q^{15} + 895 q^{16} + 324 q^{17} + 243 q^{18} - 2311 q^{19} - 7374 q^{20} - 1575 q^{21} + 3162 q^{22} - 1596 q^{23} - 3375 q^{24} - 395 q^{25} + 2508 q^{26} - 2916 q^{27} + 13531 q^{28} - 4434 q^{29} + 8289 q^{30} - 4294 q^{31} - 1017 q^{32} - 10233 q^{33} - 35880 q^{34} + 15414 q^{35} + 10530 q^{36} + 19109 q^{37} + 6828 q^{38} + 8325 q^{39} - 10545 q^{40} - 25716 q^{41} - 4725 q^{42} - 5542 q^{43} + 36579 q^{44} + 2673 q^{45} + 40740 q^{46} + 23160 q^{47} + 16110 q^{48} - 5978 q^{49} - 58704 q^{50} - 2916 q^{51} - 33424 q^{52} + 31653 q^{53} - 2187 q^{54} + 35778 q^{55} + 65625 q^{56} - 41598 q^{57} + 2277 q^{58} - 41097 q^{59} - 33183 q^{60} - 42052 q^{61} - 204114 q^{62} + 14175 q^{63} - 60062 q^{64} + 23106 q^{65} + 14229 q^{66} - 30763 q^{67} - 44748 q^{68} - 28728 q^{69} + 151179 q^{70} + 204192 q^{71} + 30375 q^{72} + 28577 q^{73} + 77784 q^{74} + 3555 q^{75} + 170384 q^{76} - 96873 q^{77} + 45144 q^{78} + 18464 q^{79} + 71511 q^{80} - 13122 q^{81} + 86040 q^{82} + 122358 q^{83} + 19404 q^{84} - 247272 q^{85} - 258510 q^{86} - 19953 q^{87} - 212565 q^{88} - 29322 q^{89} + 149202 q^{90} - 161875 q^{91} + 333816 q^{92} + 38646 q^{93} - 109938 q^{94} + 61662 q^{95} + 9153 q^{96} - 19582 q^{97} - 462021 q^{98} - 184194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 41\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 20\nu - 41 ) / 20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \( T_{2}^{4} - 3T_{2}^{3} + 69T_{2}^{2} + 180T_{2} + 3600 \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 69 T^{2} + \cdots + 3600 \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 33 T^{3} + 3867 T^{2} + \cdots + 7717284 \) Copy content Toggle raw display
$7$ \( (T^{2} + 175 T + 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 1137 T^{3} + \cdots + 104412996900 \) Copy content Toggle raw display
$13$ \( (T^{2} - 925 T + 208864)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 324 T^{3} + \cdots + 1788317798400 \) Copy content Toggle raw display
$19$ \( T^{4} + 2311 T^{3} + \cdots + 1663584040000 \) Copy content Toggle raw display
$23$ \( T^{4} + 1596 T^{3} + \cdots + 27756881510400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2217 T + 1102716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 4294 T^{3} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} - 19109 T^{3} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12858 T - 3221280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2771 T - 257902490)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 23160 T^{3} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} - 31653 T^{3} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + 41097 T^{3} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{4} + 42052 T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + 30763 T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} - 102096 T + 2483190108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 28577 T^{3} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} - 18464 T^{3} + \cdots + 79\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} - 61179 T + 711231498)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 29322 T^{3} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 9791 T - 40418570)^{2} \) Copy content Toggle raw display
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