Properties

Label 147.6.e.l
Level $147$
Weight $6$
Character orbit 147.e
Analytic conductor $23.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5764215125\)
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: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 + 1) q^{2} + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + (7 \beta_{3} - 13 \beta_1 - 13) q^{5} + (9 \beta_{2} - 9) q^{6} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10}) \) 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} + (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} + ( - 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} + ( - 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} + ( - 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} + ( - 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} + (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} + (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} + (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} + (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} + ( - 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} + ( - 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} + (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} + ( - 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} + ( - 81 \beta_{2} - 46089) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 18 q^{3} - 65 q^{4} - 33 q^{5} - 54 q^{6} - 750 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} - 18 q^{3} - 65 q^{4} - 33 q^{5} - 54 q^{6} - 750 q^{8} - 162 q^{9} + 921 q^{10} + 1137 q^{11} - 585 q^{12} - 1850 q^{13} + 594 q^{15} + 895 q^{16} - 324 q^{17} + 243 q^{18} + 2311 q^{19} + 7374 q^{20} + 3162 q^{22} - 1596 q^{23} + 3375 q^{24} - 395 q^{25} - 2508 q^{26} + 2916 q^{27} - 4434 q^{29} + 8289 q^{30} + 4294 q^{31} - 1017 q^{32} + 10233 q^{33} + 35880 q^{34} + 10530 q^{36} + 19109 q^{37} - 6828 q^{38} + 8325 q^{39} + 10545 q^{40} + 25716 q^{41} - 5542 q^{43} + 36579 q^{44} - 2673 q^{45} + 40740 q^{46} - 23160 q^{47} - 16110 q^{48} - 58704 q^{50} - 2916 q^{51} + 33424 q^{52} + 31653 q^{53} + 2187 q^{54} - 35778 q^{55} - 41598 q^{57} + 2277 q^{58} + 41097 q^{59} - 33183 q^{60} + 42052 q^{61} + 204114 q^{62} - 60062 q^{64} + 23106 q^{65} - 14229 q^{66} - 30763 q^{67} + 44748 q^{68} + 28728 q^{69} + 204192 q^{71} + 30375 q^{72} - 28577 q^{73} + 77784 q^{74} - 3555 q^{75} - 170384 q^{76} + 45144 q^{78} + 18464 q^{79} - 71511 q^{80} - 13122 q^{81} - 86040 q^{82} - 122358 q^{83} - 247272 q^{85} - 258510 q^{86} + 19953 q^{87} - 212565 q^{88} + 29322 q^{89} - 149202 q^{90} + 333816 q^{92} + 38646 q^{93} + 109938 q^{94} + 61662 q^{95} - 9153 q^{96} + 19582 q^{97} - 184194 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
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 0 −148.051 −40.5000 70.1481i 123.737 214.318i
67.2 4.69493 + 8.13186i −4.50000 + 7.79423i −28.0848 + 48.6443i −35.8645 62.1192i −84.5088 0 −226.949 −40.5000 70.1481i 336.763 583.291i
79.1 −3.19493 + 5.53379i −4.50000 7.79423i −4.41520 7.64735i 19.3645 33.5404i 57.5088 0 −148.051 −40.5000 + 70.1481i 123.737 + 214.318i
79.2 4.69493 8.13186i −4.50000 7.79423i −28.0848 48.6443i −35.8645 + 62.1192i −84.5088 0 −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 147.6.e.l 4
7.b odd 2 1 21.6.e.b 4
7.c even 3 1 147.6.a.k 2
7.c even 3 1 inner 147.6.e.l 4
7.d odd 6 1 21.6.e.b 4
7.d odd 6 1 147.6.a.i 2
21.c even 2 1 63.6.e.c 4
21.g even 6 1 63.6.e.c 4
21.g even 6 1 441.6.a.t 2
21.h odd 6 1 441.6.a.s 2
28.d even 2 1 336.6.q.e 4
28.f even 6 1 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 7.b odd 2 1
21.6.e.b 4 7.d odd 6 1
63.6.e.c 4 21.c even 2 1
63.6.e.c 4 21.g even 6 1
147.6.a.i 2 7.d odd 6 1
147.6.a.k 2 7.c even 3 1
147.6.e.l 4 1.a even 1 1 trivial
147.6.e.l 4 7.c even 3 1 inner
336.6.q.e 4 28.d even 2 1
336.6.q.e 4 28.f even 6 1
441.6.a.s 2 21.h odd 6 1
441.6.a.t 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{4} - 3T_{2}^{3} + 69T_{2}^{2} + 180T_{2} + 3600 \) Copy content Toggle raw display
\( T_{5}^{4} + 33T_{5}^{3} + 3867T_{5}^{2} - 91674T_{5} + 7717284 \) 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
$7$ \( T^{4} \) 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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