Properties

Label 147.6.a.k
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{249})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + 9 q^{3} + (3 \beta + 31) q^{4} + (7 \beta + 13) q^{5} + ( - 9 \beta - 9) q^{6} + ( - 5 \beta - 185) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + 9 q^{3} + (3 \beta + 31) q^{4} + (7 \beta + 13) q^{5} + ( - 9 \beta - 9) q^{6} + ( - 5 \beta - 185) q^{8} + 81 q^{9} + ( - 27 \beta - 447) q^{10} + (\beta - 569) q^{11} + (27 \beta + 279) q^{12} + ( - 9 \beta - 458) q^{13} + (63 \beta + 117) q^{15} + (99 \beta - 497) q^{16} + ( - 148 \beta + 236) q^{17} + ( - 81 \beta - 81) q^{18} + ( - 27 \beta - 1142) q^{19} + (277 \beta + 1705) q^{20} + (567 \beta + 507) q^{22} + (308 \beta + 644) q^{23} + ( - 45 \beta - 1665) q^{24} + (231 \beta + 82) q^{25} + (476 \beta + 1016) q^{26} + 729 q^{27} + (45 \beta - 1131) q^{29} + ( - 243 \beta - 4023) q^{30} + ( - 768 \beta - 1763) q^{31} + (459 \beta + 279) q^{32} + (9 \beta - 5121) q^{33} + (60 \beta + 8940) q^{34} + (243 \beta + 2511) q^{36} + (855 \beta - 9982) q^{37} + (1196 \beta + 2816) q^{38} + ( - 81 \beta - 4122) q^{39} + ( - 1395 \beta - 4575) q^{40} + ( - 846 \beta + 6852) q^{41} + ( - 2043 \beta - 364) q^{43} + ( - 1673 \beta - 17453) q^{44} + (567 \beta + 1053) q^{45} + ( - 1260 \beta - 19740) q^{46} + (604 \beta + 11278) q^{47} + (891 \beta - 4473) q^{48} + ( - 544 \beta - 14404) q^{50} + ( - 1332 \beta + 2124) q^{51} + ( - 1680 \beta - 15872) q^{52} + ( - 1751 \beta - 14951) q^{53} + ( - 729 \beta - 729) q^{54} + ( - 3963 \beta - 6963) q^{55} + ( - 243 \beta - 10278) q^{57} + (1041 \beta - 1659) q^{58} + (3917 \beta - 22507) q^{59} + (2493 \beta + 15345) q^{60} + (2544 \beta - 22298) q^{61} + (3299 \beta + 49379) q^{62} + ( - 4365 \beta - 12833) q^{64} + ( - 3386 \beta - 9860) q^{65} + (5103 \beta + 4563) q^{66} + ( - 4461 \beta + 17612) q^{67} + ( - 4324 \beta - 20212) q^{68} + (2772 \beta + 5796) q^{69} + (1404 \beta + 50346) q^{71} + ( - 405 \beta - 14985) q^{72} + ( - 5247 \beta + 16912) q^{73} + (8272 \beta - 43028) q^{74} + (2079 \beta + 738) q^{75} + ( - 4344 \beta - 40424) q^{76} + (4284 \beta + 9144) q^{78} + (6834 \beta - 12649) q^{79} + ( - 1499 \beta + 36505) q^{80} + 6561 q^{81} + ( - 5160 \beta + 45600) q^{82} + (1899 \beta - 31539) q^{83} + ( - 1308 \beta - 61164) q^{85} + (4450 \beta + 127030) q^{86} + (405 \beta - 10179) q^{87} + (2655 \beta + 104955) q^{88} + (130 \beta - 14726) q^{89} + ( - 2187 \beta - 36207) q^{90} + (12404 \beta + 77252) q^{92} + ( - 6912 \beta - 15867) q^{93} + ( - 12486 \beta - 48726) q^{94} + ( - 8534 \beta - 26564) q^{95} + (4131 \beta + 2511) q^{96} + (1017 \beta + 4387) q^{97} + (81 \beta - 46089) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9} - 921 q^{10} - 1137 q^{11} + 585 q^{12} - 925 q^{13} + 297 q^{15} - 895 q^{16} + 324 q^{17} - 243 q^{18} - 2311 q^{19} + 3687 q^{20} + 1581 q^{22} + 1596 q^{23} - 3375 q^{24} + 395 q^{25} + 2508 q^{26} + 1458 q^{27} - 2217 q^{29} - 8289 q^{30} - 4294 q^{31} + 1017 q^{32} - 10233 q^{33} + 17940 q^{34} + 5265 q^{36} - 19109 q^{37} + 6828 q^{38} - 8325 q^{39} - 10545 q^{40} + 12858 q^{41} - 2771 q^{43} - 36579 q^{44} + 2673 q^{45} - 40740 q^{46} + 23160 q^{47} - 8055 q^{48} - 29352 q^{50} + 2916 q^{51} - 33424 q^{52} - 31653 q^{53} - 2187 q^{54} - 17889 q^{55} - 20799 q^{57} - 2277 q^{58} - 41097 q^{59} + 33183 q^{60} - 42052 q^{61} + 102057 q^{62} - 30031 q^{64} - 23106 q^{65} + 14229 q^{66} + 30763 q^{67} - 44748 q^{68} + 14364 q^{69} + 102096 q^{71} - 30375 q^{72} + 28577 q^{73} - 77784 q^{74} + 3555 q^{75} - 85192 q^{76} + 22572 q^{78} - 18464 q^{79} + 71511 q^{80} + 13122 q^{81} + 86040 q^{82} - 61179 q^{83} - 123636 q^{85} + 258510 q^{86} - 19953 q^{87} + 212565 q^{88} - 29322 q^{89} - 74601 q^{90} + 166908 q^{92} - 38646 q^{93} - 109938 q^{94} - 61662 q^{95} + 9153 q^{96} + 9791 q^{97} - 92097 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
8.38987
−7.38987
−9.38987 9.00000 56.1696 71.7291 −84.5088 0 −226.949 81.0000 −673.526
1.2 6.38987 9.00000 8.83040 −38.7291 57.5088 0 −148.051 81.0000 −247.474
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.a.k 2
3.b odd 2 1 441.6.a.s 2
7.b odd 2 1 147.6.a.i 2
7.c even 3 2 147.6.e.l 4
7.d odd 6 2 21.6.e.b 4
21.c even 2 1 441.6.a.t 2
21.g even 6 2 63.6.e.c 4
28.f even 6 2 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 7.d odd 6 2
63.6.e.c 4 21.g even 6 2
147.6.a.i 2 7.b odd 2 1
147.6.a.k 2 1.a even 1 1 trivial
147.6.e.l 4 7.c even 3 2
336.6.q.e 4 28.f even 6 2
441.6.a.s 2 3.b odd 2 1
441.6.a.t 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{2} + 3T_{2} - 60 \) Copy content Toggle raw display
\( T_{5}^{2} - 33T_{5} - 2778 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 60 \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 33T - 2778 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1137 T + 323130 \) Copy content Toggle raw display
$13$ \( T^{2} + 925T + 208864 \) Copy content Toggle raw display
$17$ \( T^{2} - 324 T - 1337280 \) Copy content Toggle raw display
$19$ \( T^{2} + 2311 T + 1289800 \) Copy content Toggle raw display
$23$ \( T^{2} - 1596 T - 5268480 \) Copy content Toggle raw display
$29$ \( T^{2} + 2217 T + 1102716 \) Copy content Toggle raw display
$31$ \( T^{2} + 4294 T - 32106935 \) Copy content Toggle raw display
$37$ \( T^{2} + 19109 T + 45782164 \) Copy content Toggle raw display
$41$ \( T^{2} - 12858 T - 3221280 \) Copy content Toggle raw display
$43$ \( T^{2} + 2771 T - 257902490 \) Copy content Toggle raw display
$47$ \( T^{2} - 23160 T + 111386604 \) Copy content Toggle raw display
$53$ \( T^{2} + 31653 T + 59619540 \) Copy content Toggle raw display
$59$ \( T^{2} + 41097 T - 532853988 \) Copy content Toggle raw display
$61$ \( T^{2} + 42052 T + 39214660 \) Copy content Toggle raw display
$67$ \( T^{2} - 30763 T - 1002216890 \) Copy content Toggle raw display
$71$ \( T^{2} - 102096 T + 2483190108 \) Copy content Toggle raw display
$73$ \( T^{2} - 28577 T - 1509644078 \) Copy content Toggle raw display
$79$ \( T^{2} + 18464 T - 2822066537 \) Copy content Toggle raw display
$83$ \( T^{2} + 61179 T + 711231498 \) Copy content Toggle raw display
$89$ \( T^{2} + 29322 T + 213892896 \) Copy content Toggle raw display
$97$ \( T^{2} - 9791 T - 40418570 \) Copy content Toggle raw display
show more
show less