Properties

Label 147.6.e.d
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - 5 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 94 \zeta_{6} q^{5} - 45 q^{6} - 195 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 5 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 94 \zeta_{6} q^{5} - 45 q^{6} - 195 q^{8} - 81 \zeta_{6} q^{9} + ( - 470 \zeta_{6} + 470) q^{10} + (52 \zeta_{6} - 52) q^{11} - 63 \zeta_{6} q^{12} + 770 q^{13} + 846 q^{15} + 751 \zeta_{6} q^{16} + (2022 \zeta_{6} - 2022) q^{17} + (405 \zeta_{6} - 405) q^{18} + 1732 \zeta_{6} q^{19} + 658 q^{20} + 260 q^{22} + 576 \zeta_{6} q^{23} + (1755 \zeta_{6} - 1755) q^{24} + (5711 \zeta_{6} - 5711) q^{25} - 3850 \zeta_{6} q^{26} - 729 q^{27} + 5518 q^{29} - 4230 \zeta_{6} q^{30} + ( - 6336 \zeta_{6} + 6336) q^{31} + (2485 \zeta_{6} - 2485) q^{32} + 468 \zeta_{6} q^{33} + 10110 q^{34} - 567 q^{36} + 7338 \zeta_{6} q^{37} + ( - 8660 \zeta_{6} + 8660) q^{38} + ( - 6930 \zeta_{6} + 6930) q^{39} - 18330 \zeta_{6} q^{40} + 3262 q^{41} + 5420 q^{43} + 364 \zeta_{6} q^{44} + ( - 7614 \zeta_{6} + 7614) q^{45} + ( - 2880 \zeta_{6} + 2880) q^{46} + 864 \zeta_{6} q^{47} + 6759 q^{48} + 28555 q^{50} + 18198 \zeta_{6} q^{51} + ( - 5390 \zeta_{6} + 5390) q^{52} + (4182 \zeta_{6} - 4182) q^{53} + 3645 \zeta_{6} q^{54} - 4888 q^{55} + 15588 q^{57} - 27590 \zeta_{6} q^{58} + (11220 \zeta_{6} - 11220) q^{59} + ( - 5922 \zeta_{6} + 5922) q^{60} - 45602 \zeta_{6} q^{61} - 31680 q^{62} + 36457 q^{64} + 72380 \zeta_{6} q^{65} + ( - 2340 \zeta_{6} + 2340) q^{66} + (1396 \zeta_{6} - 1396) q^{67} + 14154 \zeta_{6} q^{68} + 5184 q^{69} + 18720 q^{71} + 15795 \zeta_{6} q^{72} + ( - 46362 \zeta_{6} + 46362) q^{73} + ( - 36690 \zeta_{6} + 36690) q^{74} + 51399 \zeta_{6} q^{75} + 12124 q^{76} - 34650 q^{78} - 97424 \zeta_{6} q^{79} + (70594 \zeta_{6} - 70594) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 16310 \zeta_{6} q^{82} + 81228 q^{83} - 190068 q^{85} - 27100 \zeta_{6} q^{86} + ( - 49662 \zeta_{6} + 49662) q^{87} + ( - 10140 \zeta_{6} + 10140) q^{88} - 3182 \zeta_{6} q^{89} - 38070 q^{90} + 4032 q^{92} - 57024 \zeta_{6} q^{93} + ( - 4320 \zeta_{6} + 4320) q^{94} + (162808 \zeta_{6} - 162808) q^{95} + 22365 \zeta_{6} q^{96} - 4914 q^{97} + 4212 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} + 9 q^{3} + 7 q^{4} + 94 q^{5} - 90 q^{6} - 390 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{2} + 9 q^{3} + 7 q^{4} + 94 q^{5} - 90 q^{6} - 390 q^{8} - 81 q^{9} + 470 q^{10} - 52 q^{11} - 63 q^{12} + 1540 q^{13} + 1692 q^{15} + 751 q^{16} - 2022 q^{17} - 405 q^{18} + 1732 q^{19} + 1316 q^{20} + 520 q^{22} + 576 q^{23} - 1755 q^{24} - 5711 q^{25} - 3850 q^{26} - 1458 q^{27} + 11036 q^{29} - 4230 q^{30} + 6336 q^{31} - 2485 q^{32} + 468 q^{33} + 20220 q^{34} - 1134 q^{36} + 7338 q^{37} + 8660 q^{38} + 6930 q^{39} - 18330 q^{40} + 6524 q^{41} + 10840 q^{43} + 364 q^{44} + 7614 q^{45} + 2880 q^{46} + 864 q^{47} + 13518 q^{48} + 57110 q^{50} + 18198 q^{51} + 5390 q^{52} - 4182 q^{53} + 3645 q^{54} - 9776 q^{55} + 31176 q^{57} - 27590 q^{58} - 11220 q^{59} + 5922 q^{60} - 45602 q^{61} - 63360 q^{62} + 72914 q^{64} + 72380 q^{65} + 2340 q^{66} - 1396 q^{67} + 14154 q^{68} + 10368 q^{69} + 37440 q^{71} + 15795 q^{72} + 46362 q^{73} + 36690 q^{74} + 51399 q^{75} + 24248 q^{76} - 69300 q^{78} - 97424 q^{79} - 70594 q^{80} - 6561 q^{81} - 16310 q^{82} + 162456 q^{83} - 380136 q^{85} - 27100 q^{86} + 49662 q^{87} + 10140 q^{88} - 3182 q^{89} - 76140 q^{90} + 8064 q^{92} - 57024 q^{93} + 4320 q^{94} - 162808 q^{95} + 22365 q^{96} - 9828 q^{97} + 8424 q^{99}+O(q^{100}) \) 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
−2.50000 4.33013i 4.50000 7.79423i 3.50000 6.06218i 47.0000 + 81.4064i −45.0000 0 −195.000 −40.5000 70.1481i 235.000 407.032i
79.1 −2.50000 + 4.33013i 4.50000 + 7.79423i 3.50000 + 6.06218i 47.0000 81.4064i −45.0000 0 −195.000 −40.5000 + 70.1481i 235.000 + 407.032i
\(n\): 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.d 2
7.b odd 2 1 147.6.e.c 2
7.c even 3 1 147.6.a.f 1
7.c even 3 1 inner 147.6.e.d 2
7.d odd 6 1 21.6.a.c 1
7.d odd 6 1 147.6.e.c 2
21.g even 6 1 63.6.a.b 1
21.h odd 6 1 441.6.a.c 1
28.f even 6 1 336.6.a.i 1
35.i odd 6 1 525.6.a.b 1
35.k even 12 2 525.6.d.c 2
84.j odd 6 1 1008.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 7.d odd 6 1
63.6.a.b 1 21.g even 6 1
147.6.a.f 1 7.c even 3 1
147.6.e.c 2 7.b odd 2 1
147.6.e.c 2 7.d odd 6 1
147.6.e.d 2 1.a even 1 1 trivial
147.6.e.d 2 7.c even 3 1 inner
336.6.a.i 1 28.f even 6 1
441.6.a.c 1 21.h odd 6 1
525.6.a.b 1 35.i odd 6 1
525.6.d.c 2 35.k even 12 2
1008.6.a.a 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 \(S_{6}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{2} + 5T_{2} + 25 \) Copy content Toggle raw display
\( T_{5}^{2} - 94T_{5} + 8836 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 94T + 8836 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 52T + 2704 \) Copy content Toggle raw display
$13$ \( (T - 770)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2022 T + 4088484 \) Copy content Toggle raw display
$19$ \( T^{2} - 1732 T + 2999824 \) Copy content Toggle raw display
$23$ \( T^{2} - 576T + 331776 \) Copy content Toggle raw display
$29$ \( (T - 5518)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6336 T + 40144896 \) Copy content Toggle raw display
$37$ \( T^{2} - 7338 T + 53846244 \) Copy content Toggle raw display
$41$ \( (T - 3262)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5420)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 864T + 746496 \) Copy content Toggle raw display
$53$ \( T^{2} + 4182 T + 17489124 \) Copy content Toggle raw display
$59$ \( T^{2} + 11220 T + 125888400 \) Copy content Toggle raw display
$61$ \( T^{2} + 45602 T + 2079542404 \) Copy content Toggle raw display
$67$ \( T^{2} + 1396 T + 1948816 \) Copy content Toggle raw display
$71$ \( (T - 18720)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 46362 T + 2149435044 \) Copy content Toggle raw display
$79$ \( T^{2} + 97424 T + 9491435776 \) Copy content Toggle raw display
$83$ \( (T - 81228)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3182 T + 10125124 \) Copy content Toggle raw display
$97$ \( (T + 4914)^{2} \) Copy content Toggle raw display
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