Properties

Label 147.6.e.i
Level $147$
Weight $6$
Character orbit 147.e
Analytic conductor $23.576$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 78 \zeta_{6} q^{5} - 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 6 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 78 \zeta_{6} q^{5} - 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} + (468 \zeta_{6} - 468) q^{10} + (444 \zeta_{6} - 444) q^{11} - 36 \zeta_{6} q^{12} + 442 q^{13} - 702 q^{15} + 1136 \zeta_{6} q^{16} + (126 \zeta_{6} - 126) q^{17} + ( - 486 \zeta_{6} + 486) q^{18} + 2684 \zeta_{6} q^{19} - 312 q^{20} - 2664 q^{22} - 4200 \zeta_{6} q^{23} + (1512 \zeta_{6} - 1512) q^{24} + (2959 \zeta_{6} - 2959) q^{25} + 2652 \zeta_{6} q^{26} + 729 q^{27} - 5442 q^{29} - 4212 \zeta_{6} q^{30} + ( - 80 \zeta_{6} + 80) q^{31} + (1440 \zeta_{6} - 1440) q^{32} - 3996 \zeta_{6} q^{33} - 756 q^{34} + 324 q^{36} + 5434 \zeta_{6} q^{37} + (16104 \zeta_{6} - 16104) q^{38} + (3978 \zeta_{6} - 3978) q^{39} + 13104 \zeta_{6} q^{40} - 7962 q^{41} - 11524 q^{43} - 1776 \zeta_{6} q^{44} + ( - 6318 \zeta_{6} + 6318) q^{45} + ( - 25200 \zeta_{6} + 25200) q^{46} - 13920 \zeta_{6} q^{47} - 10224 q^{48} - 17754 q^{50} - 1134 \zeta_{6} q^{51} + (1768 \zeta_{6} - 1768) q^{52} + ( - 9594 \zeta_{6} + 9594) q^{53} + 4374 \zeta_{6} q^{54} - 34632 q^{55} - 24156 q^{57} - 32652 \zeta_{6} q^{58} + ( - 27492 \zeta_{6} + 27492) q^{59} + ( - 2808 \zeta_{6} + 2808) q^{60} + 49478 \zeta_{6} q^{61} + 480 q^{62} + 27712 q^{64} + 34476 \zeta_{6} q^{65} + ( - 23976 \zeta_{6} + 23976) q^{66} + ( - 59356 \zeta_{6} + 59356) q^{67} - 504 \zeta_{6} q^{68} + 37800 q^{69} + 32040 q^{71} - 13608 \zeta_{6} q^{72} + (61846 \zeta_{6} - 61846) q^{73} + (32604 \zeta_{6} - 32604) q^{74} - 26631 \zeta_{6} q^{75} - 10736 q^{76} - 23868 q^{78} + 65776 \zeta_{6} q^{79} + (88608 \zeta_{6} - 88608) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 47772 \zeta_{6} q^{82} - 40188 q^{83} - 9828 q^{85} - 69144 \zeta_{6} q^{86} + ( - 48978 \zeta_{6} + 48978) q^{87} + (74592 \zeta_{6} - 74592) q^{88} - 7974 \zeta_{6} q^{89} + 37908 q^{90} + 16800 q^{92} + 720 \zeta_{6} q^{93} + ( - 83520 \zeta_{6} + 83520) q^{94} + (209352 \zeta_{6} - 209352) q^{95} - 12960 \zeta_{6} q^{96} + 143662 q^{97} + 35964 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} - 9 q^{3} - 4 q^{4} + 78 q^{5} - 108 q^{6} + 336 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} - 9 q^{3} - 4 q^{4} + 78 q^{5} - 108 q^{6} + 336 q^{8} - 81 q^{9} - 468 q^{10} - 444 q^{11} - 36 q^{12} + 884 q^{13} - 1404 q^{15} + 1136 q^{16} - 126 q^{17} + 486 q^{18} + 2684 q^{19} - 624 q^{20} - 5328 q^{22} - 4200 q^{23} - 1512 q^{24} - 2959 q^{25} + 2652 q^{26} + 1458 q^{27} - 10884 q^{29} - 4212 q^{30} + 80 q^{31} - 1440 q^{32} - 3996 q^{33} - 1512 q^{34} + 648 q^{36} + 5434 q^{37} - 16104 q^{38} - 3978 q^{39} + 13104 q^{40} - 15924 q^{41} - 23048 q^{43} - 1776 q^{44} + 6318 q^{45} + 25200 q^{46} - 13920 q^{47} - 20448 q^{48} - 35508 q^{50} - 1134 q^{51} - 1768 q^{52} + 9594 q^{53} + 4374 q^{54} - 69264 q^{55} - 48312 q^{57} - 32652 q^{58} + 27492 q^{59} + 2808 q^{60} + 49478 q^{61} + 960 q^{62} + 55424 q^{64} + 34476 q^{65} + 23976 q^{66} + 59356 q^{67} - 504 q^{68} + 75600 q^{69} + 64080 q^{71} - 13608 q^{72} - 61846 q^{73} - 32604 q^{74} - 26631 q^{75} - 21472 q^{76} - 47736 q^{78} + 65776 q^{79} - 88608 q^{80} - 6561 q^{81} - 47772 q^{82} - 80376 q^{83} - 19656 q^{85} - 69144 q^{86} + 48978 q^{87} - 74592 q^{88} - 7974 q^{89} + 75816 q^{90} + 33600 q^{92} + 720 q^{93} + 83520 q^{94} - 209352 q^{95} - 12960 q^{96} + 287324 q^{97} + 71928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
3.00000 + 5.19615i −4.50000 + 7.79423i −2.00000 + 3.46410i 39.0000 + 67.5500i −54.0000 0 168.000 −40.5000 70.1481i −234.000 + 405.300i
79.1 3.00000 5.19615i −4.50000 7.79423i −2.00000 3.46410i 39.0000 67.5500i −54.0000 0 168.000 −40.5000 + 70.1481i −234.000 405.300i
\(n\): 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.i 2
7.b odd 2 1 147.6.e.j 2
7.c even 3 1 147.6.a.b 1
7.c even 3 1 inner 147.6.e.i 2
7.d odd 6 1 21.6.a.a 1
7.d odd 6 1 147.6.e.j 2
21.g even 6 1 63.6.a.d 1
21.h odd 6 1 441.6.a.j 1
28.f even 6 1 336.6.a.r 1
35.i odd 6 1 525.6.a.d 1
35.k even 12 2 525.6.d.b 2
84.j odd 6 1 1008.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 7.d odd 6 1
63.6.a.d 1 21.g even 6 1
147.6.a.b 1 7.c even 3 1
147.6.e.i 2 1.a even 1 1 trivial
147.6.e.i 2 7.c even 3 1 inner
147.6.e.j 2 7.b odd 2 1
147.6.e.j 2 7.d odd 6 1
336.6.a.r 1 28.f even 6 1
441.6.a.j 1 21.h odd 6 1
525.6.a.d 1 35.i odd 6 1
525.6.d.b 2 35.k even 12 2
1008.6.a.c 1 84.j odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 6T_{2} + 36 \) Copy content Toggle raw display
\( T_{5}^{2} - 78T_{5} + 6084 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 78T + 6084 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 444T + 197136 \) Copy content Toggle raw display
$13$ \( (T - 442)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$19$ \( T^{2} - 2684 T + 7203856 \) Copy content Toggle raw display
$23$ \( T^{2} + 4200 T + 17640000 \) Copy content Toggle raw display
$29$ \( (T + 5442)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 80T + 6400 \) Copy content Toggle raw display
$37$ \( T^{2} - 5434 T + 29528356 \) Copy content Toggle raw display
$41$ \( (T + 7962)^{2} \) Copy content Toggle raw display
$43$ \( (T + 11524)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 13920 T + 193766400 \) Copy content Toggle raw display
$53$ \( T^{2} - 9594 T + 92044836 \) Copy content Toggle raw display
$59$ \( T^{2} - 27492 T + 755810064 \) Copy content Toggle raw display
$61$ \( T^{2} - 49478 T + 2448072484 \) Copy content Toggle raw display
$67$ \( T^{2} - 59356 T + 3523134736 \) Copy content Toggle raw display
$71$ \( (T - 32040)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 61846 T + 3824927716 \) Copy content Toggle raw display
$79$ \( T^{2} - 65776 T + 4326482176 \) Copy content Toggle raw display
$83$ \( (T + 40188)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 7974 T + 63584676 \) Copy content Toggle raw display
$97$ \( (T - 143662)^{2} \) Copy content Toggle raw display
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