Properties

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

Related objects

Downloads

Learn more

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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 31 \zeta_{6} + 31) q^{4} + 34 \zeta_{6} q^{5} - 9 q^{6} - 63 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 31 \zeta_{6} + 31) q^{4} + 34 \zeta_{6} q^{5} - 9 q^{6} - 63 q^{8} - 81 \zeta_{6} q^{9} + ( - 34 \zeta_{6} + 34) q^{10} + ( - 340 \zeta_{6} + 340) q^{11} - 279 \zeta_{6} q^{12} + 454 q^{13} + 306 q^{15} - 929 \zeta_{6} q^{16} + ( - 798 \zeta_{6} + 798) q^{17} + (81 \zeta_{6} - 81) q^{18} - 892 \zeta_{6} q^{19} + 1054 q^{20} - 340 q^{22} + 3192 \zeta_{6} q^{23} + (567 \zeta_{6} - 567) q^{24} + ( - 1969 \zeta_{6} + 1969) q^{25} - 454 \zeta_{6} q^{26} - 729 q^{27} - 8242 q^{29} - 306 \zeta_{6} q^{30} + ( - 2496 \zeta_{6} + 2496) q^{31} + (2945 \zeta_{6} - 2945) q^{32} - 3060 \zeta_{6} q^{33} - 798 q^{34} - 2511 q^{36} - 9798 \zeta_{6} q^{37} + (892 \zeta_{6} - 892) q^{38} + ( - 4086 \zeta_{6} + 4086) q^{39} - 2142 \zeta_{6} q^{40} + 19834 q^{41} - 17236 q^{43} - 10540 \zeta_{6} q^{44} + ( - 2754 \zeta_{6} + 2754) q^{45} + ( - 3192 \zeta_{6} + 3192) q^{46} - 8928 \zeta_{6} q^{47} - 8361 q^{48} - 1969 q^{50} - 7182 \zeta_{6} q^{51} + ( - 14074 \zeta_{6} + 14074) q^{52} + (150 \zeta_{6} - 150) q^{53} + 729 \zeta_{6} q^{54} + 11560 q^{55} - 8028 q^{57} + 8242 \zeta_{6} q^{58} + ( - 42396 \zeta_{6} + 42396) q^{59} + ( - 9486 \zeta_{6} + 9486) q^{60} - 14758 \zeta_{6} q^{61} - 2496 q^{62} - 26783 q^{64} + 15436 \zeta_{6} q^{65} + (3060 \zeta_{6} - 3060) q^{66} + ( - 1676 \zeta_{6} + 1676) q^{67} - 24738 \zeta_{6} q^{68} + 28728 q^{69} + 14568 q^{71} + 5103 \zeta_{6} q^{72} + (78378 \zeta_{6} - 78378) q^{73} + (9798 \zeta_{6} - 9798) q^{74} - 17721 \zeta_{6} q^{75} - 27652 q^{76} - 4086 q^{78} + 2272 \zeta_{6} q^{79} + ( - 31586 \zeta_{6} + 31586) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 19834 \zeta_{6} q^{82} - 37764 q^{83} + 27132 q^{85} + 17236 \zeta_{6} q^{86} + (74178 \zeta_{6} - 74178) q^{87} + (21420 \zeta_{6} - 21420) q^{88} + 117286 \zeta_{6} q^{89} - 2754 q^{90} + 98952 q^{92} - 22464 \zeta_{6} q^{93} + (8928 \zeta_{6} - 8928) q^{94} + ( - 30328 \zeta_{6} + 30328) q^{95} + 26505 \zeta_{6} q^{96} + 10002 q^{97} - 27540 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 9 q^{3} + 31 q^{4} + 34 q^{5} - 18 q^{6} - 126 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 9 q^{3} + 31 q^{4} + 34 q^{5} - 18 q^{6} - 126 q^{8} - 81 q^{9} + 34 q^{10} + 340 q^{11} - 279 q^{12} + 908 q^{13} + 612 q^{15} - 929 q^{16} + 798 q^{17} - 81 q^{18} - 892 q^{19} + 2108 q^{20} - 680 q^{22} + 3192 q^{23} - 567 q^{24} + 1969 q^{25} - 454 q^{26} - 1458 q^{27} - 16484 q^{29} - 306 q^{30} + 2496 q^{31} - 2945 q^{32} - 3060 q^{33} - 1596 q^{34} - 5022 q^{36} - 9798 q^{37} - 892 q^{38} + 4086 q^{39} - 2142 q^{40} + 39668 q^{41} - 34472 q^{43} - 10540 q^{44} + 2754 q^{45} + 3192 q^{46} - 8928 q^{47} - 16722 q^{48} - 3938 q^{50} - 7182 q^{51} + 14074 q^{52} - 150 q^{53} + 729 q^{54} + 23120 q^{55} - 16056 q^{57} + 8242 q^{58} + 42396 q^{59} + 9486 q^{60} - 14758 q^{61} - 4992 q^{62} - 53566 q^{64} + 15436 q^{65} - 3060 q^{66} + 1676 q^{67} - 24738 q^{68} + 57456 q^{69} + 29136 q^{71} + 5103 q^{72} - 78378 q^{73} - 9798 q^{74} - 17721 q^{75} - 55304 q^{76} - 8172 q^{78} + 2272 q^{79} + 31586 q^{80} - 6561 q^{81} - 19834 q^{82} - 75528 q^{83} + 54264 q^{85} + 17236 q^{86} - 74178 q^{87} - 21420 q^{88} + 117286 q^{89} - 5508 q^{90} + 197904 q^{92} - 22464 q^{93} - 8928 q^{94} + 30328 q^{95} + 26505 q^{96} + 20004 q^{97} - 55080 q^{99}+O(q^{100}) \) 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
−0.500000 0.866025i 4.50000 7.79423i 15.5000 26.8468i 17.0000 + 29.4449i −9.00000 0 −63.0000 −40.5000 70.1481i 17.0000 29.4449i
79.1 −0.500000 + 0.866025i 4.50000 + 7.79423i 15.5000 + 26.8468i 17.0000 29.4449i −9.00000 0 −63.0000 −40.5000 + 70.1481i 17.0000 + 29.4449i
\(n\): 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.f 2
7.b odd 2 1 147.6.e.e 2
7.c even 3 1 21.6.a.b 1
7.c even 3 1 inner 147.6.e.f 2
7.d odd 6 1 147.6.a.e 1
7.d odd 6 1 147.6.e.e 2
21.g even 6 1 441.6.a.d 1
21.h odd 6 1 63.6.a.c 1
28.g odd 6 1 336.6.a.l 1
35.j even 6 1 525.6.a.c 1
35.l odd 12 2 525.6.d.d 2
84.n even 6 1 1008.6.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 7.c even 3 1
63.6.a.c 1 21.h odd 6 1
147.6.a.e 1 7.d odd 6 1
147.6.e.e 2 7.b odd 2 1
147.6.e.e 2 7.d odd 6 1
147.6.e.f 2 1.a even 1 1 trivial
147.6.e.f 2 7.c even 3 1 inner
336.6.a.l 1 28.g odd 6 1
441.6.a.d 1 21.g even 6 1
525.6.a.c 1 35.j even 6 1
525.6.d.d 2 35.l odd 12 2
1008.6.a.t 1 84.n even 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 34T_{5} + 1156 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 340T + 115600 \) Copy content Toggle raw display
$13$ \( (T - 454)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 798T + 636804 \) Copy content Toggle raw display
$19$ \( T^{2} + 892T + 795664 \) Copy content Toggle raw display
$23$ \( T^{2} - 3192 T + 10188864 \) Copy content Toggle raw display
$29$ \( (T + 8242)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2496 T + 6230016 \) Copy content Toggle raw display
$37$ \( T^{2} + 9798 T + 96000804 \) Copy content Toggle raw display
$41$ \( (T - 19834)^{2} \) Copy content Toggle raw display
$43$ \( (T + 17236)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8928 T + 79709184 \) Copy content Toggle raw display
$53$ \( T^{2} + 150T + 22500 \) Copy content Toggle raw display
$59$ \( T^{2} - 42396 T + 1797420816 \) Copy content Toggle raw display
$61$ \( T^{2} + 14758 T + 217798564 \) Copy content Toggle raw display
$67$ \( T^{2} - 1676 T + 2808976 \) Copy content Toggle raw display
$71$ \( (T - 14568)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 78378 T + 6143110884 \) Copy content Toggle raw display
$79$ \( T^{2} - 2272 T + 5161984 \) Copy content Toggle raw display
$83$ \( (T + 37764)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 117286 T + 13756005796 \) Copy content Toggle raw display
$97$ \( (T - 10002)^{2} \) Copy content Toggle raw display
show more
show less