Properties

Label 21.6.e.c
Level $21$
Weight $6$
Character orbit 21.e
Analytic conductor $3.368$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + 9 \beta_{2} q^{3} + (\beta_{6} + \beta_{3} + 18 \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{5} + ( - 9 \beta_{6} + 9) q^{6} + (\beta_{7} + 6 \beta_{6} + \beta_{5} + \beta_{4} + 45 \beta_{2} - 2 \beta_1 + 53) q^{7} + ( - 2 \beta_{7} - 27 \beta_{6} + 2 \beta_{5} + \beta_{4} + 37) q^{8} + ( - 81 \beta_{2} - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + 9 \beta_{2} q^{3} + (\beta_{6} + \beta_{3} + 18 \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{5} + ( - 9 \beta_{6} + 9) q^{6} + (\beta_{7} + 6 \beta_{6} + \beta_{5} + \beta_{4} + 45 \beta_{2} - 2 \beta_1 + 53) q^{7} + ( - 2 \beta_{7} - 27 \beta_{6} + 2 \beta_{5} + \beta_{4} + 37) q^{8} + ( - 81 \beta_{2} - 81) q^{9} + (23 \beta_{6} - \beta_{3} + 76 \beta_{2} - 23 \beta_1) q^{10} + ( - 3 \beta_{7} + 8 \beta_{6} + 9 \beta_{3} + 107 \beta_{2} - 8 \beta_1) q^{11} + ( - 9 \beta_{4} - 9 \beta_{3} - 162 \beta_{2} + 9 \beta_1 - 153) q^{12} + (9 \beta_{7} + 76 \beta_{6} - 9 \beta_{5} - \beta_{4} + 97) q^{13} + ( - 4 \beta_{7} - 83 \beta_{6} + 9 \beta_{4} + \beta_{3} + 113 \beta_{2} + \cdots + 299) q^{14}+ \cdots + (243 \beta_{7} - 648 \beta_{6} - 243 \beta_{5} + 729 \beta_{4} + \cdots + 7938) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - 36 q^{3} - 69 q^{4} + 54 q^{6} + 258 q^{7} + 246 q^{8} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - 36 q^{3} - 69 q^{4} + 54 q^{6} + 258 q^{7} + 246 q^{8} - 324 q^{9} - 283 q^{10} - 402 q^{11} - 621 q^{12} + 924 q^{13} + 1926 q^{14} - 3273 q^{16} - 276 q^{17} - 243 q^{18} - 510 q^{19} + 9438 q^{20} - 3564 q^{21} + 2750 q^{22} - 6900 q^{23} - 1107 q^{24} - 2814 q^{25} + 15138 q^{26} + 5832 q^{27} - 26221 q^{28} + 1080 q^{29} - 2547 q^{30} + 6410 q^{31} - 15519 q^{32} - 3618 q^{33} + 42288 q^{34} - 33108 q^{35} + 11178 q^{36} - 15250 q^{37} + 41250 q^{38} - 4158 q^{39} + 8547 q^{40} + 8616 q^{41} - 16281 q^{42} + 58396 q^{43} - 70743 q^{44} - 61800 q^{46} + 15060 q^{47} + 58914 q^{48} - 64252 q^{49} - 14604 q^{50} - 2484 q^{51} + 47476 q^{52} - 13692 q^{53} - 2187 q^{54} + 146248 q^{55} - 15921 q^{56} + 9180 q^{57} - 52309 q^{58} - 34830 q^{59} - 42471 q^{60} + 5364 q^{61} + 32058 q^{62} + 11178 q^{63} - 146974 q^{64} - 66864 q^{65} - 12375 q^{66} + 5994 q^{67} + 58272 q^{68} + 124200 q^{69} - 4307 q^{70} + 178536 q^{71} - 9963 q^{72} - 59638 q^{73} + 185442 q^{74} - 25326 q^{75} + 42616 q^{76} - 75660 q^{77} - 272484 q^{78} + 44062 q^{79} + 33381 q^{80} - 26244 q^{81} - 57596 q^{82} - 416892 q^{83} + 63036 q^{84} + 72648 q^{85} + 136968 q^{86} - 4860 q^{87} - 87597 q^{88} + 77520 q^{89} + 45846 q^{90} + 104722 q^{91} + 316512 q^{92} + 57690 q^{93} + 73722 q^{94} + 221376 q^{95} - 139671 q^{96} - 377260 q^{97} + 382479 q^{98} + 65124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8905 \nu^{7} + 366059 \nu^{6} - 1191820 \nu^{5} + 33153695 \nu^{4} - 47979268 \nu^{3} + 3262309295 \nu^{2} - 141627885 \nu + 307508418 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1257 \nu^{7} - 279647 \nu^{6} + 388990 \nu^{5} - 26314019 \nu^{4} - 14452616 \nu^{3} - 2382173465 \nu^{2} - 51985031 \nu - 251390874 ) / 156968070 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22795 \nu^{7} + 56368 \nu^{6} + 2163175 \nu^{5} + 2122285 \nu^{4} + 222588334 \nu^{3} + 7196875 \nu^{2} + 20796090 \nu - 31645054854 ) / 706356315 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 341623 \nu^{7} - 8468248 \nu^{6} + 27571040 \nu^{5} - 966332554 \nu^{4} + 1109931296 \nu^{3} - 75468829240 \nu^{2} + \cdots - 235881275616 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25511 \nu^{7} + 22795 \nu^{6} - 2420915 \nu^{5} - 2375153 \nu^{4} - 232964210 \nu^{3} - 8054375 \nu^{2} - 23273922 \nu - 54979470 ) / 706356315 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15589754 \nu^{7} - 23777815 \nu^{6} + 1539409445 \nu^{5} + 516927917 \nu^{4} + 141447402185 \nu^{3} - 70438948615 \nu^{2} + \cdots + 24613669710 ) / 9888988410 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{3} + 49\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 91\beta_{6} + 2\beta_{5} - 2\beta_{4} - 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 99\beta_{4} - 99\beta_{3} - 4505\beta_{2} - 181\beta _1 - 4406 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 196\beta_{7} + 8707\beta_{6} - 286\beta_{3} - 8624\beta_{2} - 8707\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 376\beta_{7} + 25333\beta_{6} - 376\beta_{5} + 9581\beta_{4} + 421300 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -18786\beta_{5} + 36042\beta_{4} + 36042\beta_{3} + 1222350\beta_{2} + 841891\beta _1 + 1186308 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−4.61193 7.98809i
−0.874091 1.51397i
0.895402 + 1.55088i
5.09061 + 8.81720i
−4.61193 + 7.98809i
−0.874091 + 1.51397i
0.895402 1.55088i
5.09061 8.81720i
−5.11193 8.85412i −4.50000 + 7.79423i −36.2636 + 62.8104i −11.8764 20.5705i 92.0147 30.6840 + 125.958i 414.344 −40.5000 70.1481i −121.423 + 210.310i
4.2 −1.37409 2.37999i −4.50000 + 7.79423i 12.2237 21.1722i −29.1836 50.5475i 24.7336 −21.4366 127.857i −155.128 −40.5000 70.1481i −80.2019 + 138.914i
4.3 0.395402 + 0.684857i −4.50000 + 7.79423i 15.6873 27.1712i 52.0958 + 90.2327i −7.11724 −7.12980 + 129.446i 50.1170 −40.5000 70.1481i −41.1977 + 71.3564i
4.4 4.59061 + 7.95118i −4.50000 + 7.79423i −26.1475 + 45.2888i −11.0358 19.1146i −82.6311 126.882 + 26.6059i −186.333 −40.5000 70.1481i 101.322 175.495i
16.1 −5.11193 + 8.85412i −4.50000 7.79423i −36.2636 62.8104i −11.8764 + 20.5705i 92.0147 30.6840 125.958i 414.344 −40.5000 + 70.1481i −121.423 210.310i
16.2 −1.37409 + 2.37999i −4.50000 7.79423i 12.2237 + 21.1722i −29.1836 + 50.5475i 24.7336 −21.4366 + 127.857i −155.128 −40.5000 + 70.1481i −80.2019 138.914i
16.3 0.395402 0.684857i −4.50000 7.79423i 15.6873 + 27.1712i 52.0958 90.2327i −7.11724 −7.12980 129.446i 50.1170 −40.5000 + 70.1481i −41.1977 71.3564i
16.4 4.59061 7.95118i −4.50000 7.79423i −26.1475 45.2888i −11.0358 + 19.1146i −82.6311 126.882 26.6059i −186.333 −40.5000 + 70.1481i 101.322 + 175.495i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.4
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.c 8
3.b odd 2 1 63.6.e.e 8
4.b odd 2 1 336.6.q.j 8
7.b odd 2 1 147.6.e.o 8
7.c even 3 1 inner 21.6.e.c 8
7.c even 3 1 147.6.a.m 4
7.d odd 6 1 147.6.a.l 4
7.d odd 6 1 147.6.e.o 8
21.g even 6 1 441.6.a.v 4
21.h odd 6 1 63.6.e.e 8
21.h odd 6 1 441.6.a.w 4
28.g odd 6 1 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 1.a even 1 1 trivial
21.6.e.c 8 7.c even 3 1 inner
63.6.e.e 8 3.b odd 2 1
63.6.e.e 8 21.h odd 6 1
147.6.a.l 4 7.d odd 6 1
147.6.a.m 4 7.c even 3 1
147.6.e.o 8 7.b odd 2 1
147.6.e.o 8 7.d odd 6 1
336.6.q.j 8 4.b odd 2 1
336.6.q.j 8 28.g odd 6 1
441.6.a.v 4 21.g even 6 1
441.6.a.w 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{7} + 103T_{2}^{6} + 90T_{2}^{5} + 9190T_{2}^{4} + 16260T_{2}^{3} + 53772T_{2}^{2} - 37944T_{2} + 41616 \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + 103 T^{6} + \cdots + 41616 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 7657 T^{6} + \cdots + 10164899803536 \) Copy content Toggle raw display
$7$ \( T^{8} - 258 T^{7} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + 402 T^{7} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} - 462 T^{3} + \cdots + 149501563456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 276 T^{7} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + 510 T^{7} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{8} + 6900 T^{7} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} - 540 T^{3} + \cdots + 408027025117872)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 6410 T^{7} + \cdots + 75\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{8} + 15250 T^{7} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4308 T^{3} + \cdots - 18\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 29198 T^{3} + \cdots - 991662745581932)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 15060 T^{7} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + 13692 T^{7} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + 34830 T^{7} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} - 5364 T^{7} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} - 5994 T^{7} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} - 89268 T^{3} + \cdots + 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 59638 T^{7} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} - 44062 T^{7} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{4} + 208446 T^{3} + \cdots - 41\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 77520 T^{7} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{4} + 188630 T^{3} + \cdots - 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
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