Properties

Label 147.12.a.c
Level $147$
Weight $12$
Character orbit 147.a
Self dual yes
Analytic conductor $112.946$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(112.946447542\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 78 q^{2} + 243 q^{3} + 4036 q^{4} + 5370 q^{5} + 18954 q^{6} + 155064 q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 78 q^{2} + 243 q^{3} + 4036 q^{4} + 5370 q^{5} + 18954 q^{6} + 155064 q^{8} + 59049 q^{9} + 418860 q^{10} + 637836 q^{11} + 980748 q^{12} - 766214 q^{13} + 1304910 q^{15} + 3829264 q^{16} - 3084354 q^{17} + 4605822 q^{18} + 19511404 q^{19} + 21673320 q^{20} + 49751208 q^{22} + 15312360 q^{23} + 37680552 q^{24} - 19991225 q^{25} - 59764692 q^{26} + 14348907 q^{27} + 10751262 q^{29} + 101782980 q^{30} + 50937400 q^{31} - 18888480 q^{32} + 154994148 q^{33} - 240579612 q^{34} + 238321764 q^{36} + 664740830 q^{37} + 1521889512 q^{38} - 186190002 q^{39} + 832693680 q^{40} - 898833450 q^{41} - 957947188 q^{43} + 2574306096 q^{44} + 317093130 q^{45} + 1194364080 q^{46} + 1555741344 q^{47} + 930511152 q^{48} - 1559315550 q^{50} - 749498022 q^{51} - 3092439704 q^{52} + 3792417030 q^{53} + 1119214746 q^{54} + 3425179320 q^{55} + 4741271172 q^{57} + 838598436 q^{58} - 555306924 q^{59} + 5266616760 q^{60} - 4950420998 q^{61} + 3973117200 q^{62} - 9315634112 q^{64} - 4114569180 q^{65} + 12089543544 q^{66} + 5292399284 q^{67} - 12448452744 q^{68} + 3720903480 q^{69} - 14831086248 q^{71} + 9156374136 q^{72} - 13971005210 q^{73} + 51849784740 q^{74} - 4857867675 q^{75} + 78748026544 q^{76} - 14522820156 q^{78} + 3720542360 q^{79} + 20563147680 q^{80} + 3486784401 q^{81} - 70109009100 q^{82} - 8768454036 q^{83} - 16562980980 q^{85} - 74719880664 q^{86} + 2612556666 q^{87} + 98905401504 q^{88} + 25472769174 q^{89} + 24733264140 q^{90} + 61800684960 q^{92} + 12377788200 q^{93} + 121347824832 q^{94} + 104776239480 q^{95} - 4589900640 q^{96} + 39092494846 q^{97} + 37663577964 q^{99}+O(q^{100}) \) 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
0
78.0000 243.000 4036.00 5370.00 18954.0 0 155064. 59049.0 418860.
\(n\): 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 147.12.a.c 1
7.b odd 2 1 3.12.a.a 1
21.c even 2 1 9.12.a.a 1
28.d even 2 1 48.12.a.f 1
35.c odd 2 1 75.12.a.a 1
35.f even 4 2 75.12.b.a 2
56.e even 2 1 192.12.a.g 1
56.h odd 2 1 192.12.a.q 1
63.l odd 6 2 81.12.c.a 2
63.o even 6 2 81.12.c.e 2
84.h odd 2 1 144.12.a.l 1
105.g even 2 1 225.12.a.f 1
105.k odd 4 2 225.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 7.b odd 2 1
9.12.a.a 1 21.c even 2 1
48.12.a.f 1 28.d even 2 1
75.12.a.a 1 35.c odd 2 1
75.12.b.a 2 35.f even 4 2
81.12.c.a 2 63.l odd 6 2
81.12.c.e 2 63.o even 6 2
144.12.a.l 1 84.h odd 2 1
147.12.a.c 1 1.a even 1 1 trivial
192.12.a.g 1 56.e even 2 1
192.12.a.q 1 56.h odd 2 1
225.12.a.f 1 105.g even 2 1
225.12.b.a 2 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} - 78 \) Copy content Toggle raw display
\( T_{5} - 5370 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 78 \) Copy content Toggle raw display
$3$ \( T - 243 \) Copy content Toggle raw display
$5$ \( T - 5370 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 637836 \) Copy content Toggle raw display
$13$ \( T + 766214 \) Copy content Toggle raw display
$17$ \( T + 3084354 \) Copy content Toggle raw display
$19$ \( T - 19511404 \) Copy content Toggle raw display
$23$ \( T - 15312360 \) Copy content Toggle raw display
$29$ \( T - 10751262 \) Copy content Toggle raw display
$31$ \( T - 50937400 \) Copy content Toggle raw display
$37$ \( T - 664740830 \) Copy content Toggle raw display
$41$ \( T + 898833450 \) Copy content Toggle raw display
$43$ \( T + 957947188 \) Copy content Toggle raw display
$47$ \( T - 1555741344 \) Copy content Toggle raw display
$53$ \( T - 3792417030 \) Copy content Toggle raw display
$59$ \( T + 555306924 \) Copy content Toggle raw display
$61$ \( T + 4950420998 \) Copy content Toggle raw display
$67$ \( T - 5292399284 \) Copy content Toggle raw display
$71$ \( T + 14831086248 \) Copy content Toggle raw display
$73$ \( T + 13971005210 \) Copy content Toggle raw display
$79$ \( T - 3720542360 \) Copy content Toggle raw display
$83$ \( T + 8768454036 \) Copy content Toggle raw display
$89$ \( T - 25472769174 \) Copy content Toggle raw display
$97$ \( T - 39092494846 \) Copy content Toggle raw display
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