Properties

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

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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: \(\Q(\sqrt{193}) \)
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{193})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + 9 q^{3} + (3 \beta + 17) q^{4} - 36 q^{5} + ( - 9 \beta - 9) q^{6} + (9 \beta - 129) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + 9 q^{3} + (3 \beta + 17) q^{4} - 36 q^{5} + ( - 9 \beta - 9) q^{6} + (9 \beta - 129) q^{8} + 81 q^{9} + (36 \beta + 36) q^{10} + (8 \beta + 236) q^{11} + (27 \beta + 153) q^{12} + ( - 72 \beta - 612) q^{13} - 324 q^{15} + (15 \beta - 847) q^{16} + (216 \beta - 576) q^{17} + ( - 81 \beta - 81) q^{18} + (288 \beta - 36) q^{19} + ( - 108 \beta - 612) q^{20} + ( - 252 \beta - 620) q^{22} + ( - 56 \beta - 224) q^{23} + (81 \beta - 1161) q^{24} - 1829 q^{25} + (756 \beta + 4068) q^{26} + 729 q^{27} + (640 \beta + 2866) q^{29} + (324 \beta + 324) q^{30} + (576 \beta - 5256) q^{31} + (529 \beta + 4255) q^{32} + (72 \beta + 2124) q^{33} + (144 \beta - 9792) q^{34} + (243 \beta + 1377) q^{36} + ( - 1056 \beta + 6090) q^{37} + ( - 540 \beta - 13788) q^{38} + ( - 648 \beta - 5508) q^{39} + ( - 324 \beta + 4644) q^{40} + ( - 216 \beta - 10368) q^{41} + ( - 2400 \beta - 1932) q^{43} + (868 \beta + 5164) q^{44} - 2916 q^{45} + (336 \beta + 2912) q^{46} + ( - 3456 \beta - 2232) q^{47} + (135 \beta - 7623) q^{48} + (1829 \beta + 1829) q^{50} + (1944 \beta - 5184) q^{51} + ( - 3276 \beta - 20772) q^{52} + ( - 1184 \beta + 1702) q^{53} + ( - 729 \beta - 729) q^{54} + ( - 288 \beta - 8496) q^{55} + (2592 \beta - 324) q^{57} + ( - 4146 \beta - 33586) q^{58} + ( - 864 \beta - 14436) q^{59} + ( - 972 \beta - 5508) q^{60} + ( - 4680 \beta + 10980) q^{61} + (4104 \beta - 22392) q^{62} + ( - 5793 \beta - 2543) q^{64} + (2592 \beta + 22032) q^{65} + ( - 2268 \beta - 5580) q^{66} + (6480 \beta - 13580) q^{67} + (2592 \beta + 21312) q^{68} + ( - 504 \beta - 2016) q^{69} + (2552 \beta - 47416) q^{71} + (729 \beta - 10449) q^{72} + (1872 \beta - 29232) q^{73} + ( - 3978 \beta + 44598) q^{74} - 16461 q^{75} + (5652 \beta + 40860) q^{76} + (6804 \beta + 36612) q^{78} + ( - 6480 \beta - 24808) q^{79} + ( - 540 \beta + 30492) q^{80} + 6561 q^{81} + (10800 \beta + 20736) q^{82} + ( - 9504 \beta + 40428) q^{83} + ( - 7776 \beta + 20736) q^{85} + (6732 \beta + 117132) q^{86} + (5760 \beta + 25794) q^{87} + (1164 \beta - 26988) q^{88} + (3672 \beta - 63432) q^{89} + (2916 \beta + 2916) q^{90} + ( - 1792 \beta - 11872) q^{92} + (5184 \beta - 47304) q^{93} + (9144 \beta + 168120) q^{94} + ( - 10368 \beta + 1296) q^{95} + (4761 \beta + 38295) q^{96} + ( - 19584 \beta - 8136) q^{97} + (648 \beta + 19116) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 18 q^{3} + 37 q^{4} - 72 q^{5} - 27 q^{6} - 249 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 18 q^{3} + 37 q^{4} - 72 q^{5} - 27 q^{6} - 249 q^{8} + 162 q^{9} + 108 q^{10} + 480 q^{11} + 333 q^{12} - 1296 q^{13} - 648 q^{15} - 1679 q^{16} - 936 q^{17} - 243 q^{18} + 216 q^{19} - 1332 q^{20} - 1492 q^{22} - 504 q^{23} - 2241 q^{24} - 3658 q^{25} + 8892 q^{26} + 1458 q^{27} + 6372 q^{29} + 972 q^{30} - 9936 q^{31} + 9039 q^{32} + 4320 q^{33} - 19440 q^{34} + 2997 q^{36} + 11124 q^{37} - 28116 q^{38} - 11664 q^{39} + 8964 q^{40} - 20952 q^{41} - 6264 q^{43} + 11196 q^{44} - 5832 q^{45} + 6160 q^{46} - 7920 q^{47} - 15111 q^{48} + 5487 q^{50} - 8424 q^{51} - 44820 q^{52} + 2220 q^{53} - 2187 q^{54} - 17280 q^{55} + 1944 q^{57} - 71318 q^{58} - 29736 q^{59} - 11988 q^{60} + 17280 q^{61} - 40680 q^{62} - 10879 q^{64} + 46656 q^{65} - 13428 q^{66} - 20680 q^{67} + 45216 q^{68} - 4536 q^{69} - 92280 q^{71} - 20169 q^{72} - 56592 q^{73} + 85218 q^{74} - 32922 q^{75} + 87372 q^{76} + 80028 q^{78} - 56096 q^{79} + 60444 q^{80} + 13122 q^{81} + 52272 q^{82} + 71352 q^{83} + 33696 q^{85} + 240996 q^{86} + 57348 q^{87} - 52812 q^{88} - 123192 q^{89} + 8748 q^{90} - 25536 q^{92} - 89424 q^{93} + 345384 q^{94} - 7776 q^{95} + 81351 q^{96} - 35856 q^{97} + 38880 q^{99}+O(q^{100}) \) 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
7.44622
−6.44622
−8.44622 9.00000 39.3387 −36.0000 −76.0160 0 −61.9840 81.0000 304.064
1.2 5.44622 9.00000 −2.33867 −36.0000 49.0160 0 −187.016 81.0000 −196.064
\(n\): 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.j yes 2
3.b odd 2 1 441.6.a.r 2
7.b odd 2 1 147.6.a.h 2
7.c even 3 2 147.6.e.m 4
7.d odd 6 2 147.6.e.n 4
21.c even 2 1 441.6.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 7.b odd 2 1
147.6.a.j yes 2 1.a even 1 1 trivial
147.6.e.m 4 7.c even 3 2
147.6.e.n 4 7.d odd 6 2
441.6.a.q 2 21.c even 2 1
441.6.a.r 2 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} - 46 \) Copy content Toggle raw display
\( T_{5} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 46 \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( (T + 36)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 480T + 54512 \) Copy content Toggle raw display
$13$ \( T^{2} + 1296 T + 169776 \) Copy content Toggle raw display
$17$ \( T^{2} + 936 T - 2032128 \) Copy content Toggle raw display
$19$ \( T^{2} - 216 T - 3990384 \) Copy content Toggle raw display
$23$ \( T^{2} + 504T - 87808 \) Copy content Toggle raw display
$29$ \( T^{2} - 6372 T - 9612604 \) Copy content Toggle raw display
$31$ \( T^{2} + 9936 T + 8672832 \) Copy content Toggle raw display
$37$ \( T^{2} - 11124 T - 22869468 \) Copy content Toggle raw display
$41$ \( T^{2} + 20952 T + 107495424 \) Copy content Toggle raw display
$43$ \( T^{2} + 6264 T - 268110576 \) Copy content Toggle raw display
$47$ \( T^{2} + 7920 T - 560613312 \) Copy content Toggle raw display
$53$ \( T^{2} - 2220 T - 66407452 \) Copy content Toggle raw display
$59$ \( T^{2} + 29736 T + 185038992 \) Copy content Toggle raw display
$61$ \( T^{2} - 17280 T - 982141200 \) Copy content Toggle raw display
$67$ \( T^{2} + 20680 T - 1919121200 \) Copy content Toggle raw display
$71$ \( T^{2} + 92280 T + 1814661632 \) Copy content Toggle raw display
$73$ \( T^{2} + 56592 T + 631577088 \) Copy content Toggle raw display
$79$ \( T^{2} + 56096 T - 1239346496 \) Copy content Toggle raw display
$83$ \( T^{2} - 71352 T - 3085453296 \) Copy content Toggle raw display
$89$ \( T^{2} + 123192 T + 3143484288 \) Copy content Toggle raw display
$97$ \( T^{2} + 35856 T - 18184056768 \) Copy content Toggle raw display
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