Properties

Label 3.12.a.a
Level $3$
Weight $12$
Character orbit 3.a
Self dual yes
Analytic conductor $2.305$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.30502954168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 78 q^{2} - 243 q^{3} + 4036 q^{4} - 5370 q^{5} - 18954 q^{6} - 27760 q^{7} + 155064 q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 78 q^{2} - 243 q^{3} + 4036 q^{4} - 5370 q^{5} - 18954 q^{6} - 27760 q^{7} + 155064 q^{8} + 59049 q^{9} - 418860 q^{10} + 637836 q^{11} - 980748 q^{12} + 766214 q^{13} - 2165280 q^{14} + 1304910 q^{15} + 3829264 q^{16} + 3084354 q^{17} + 4605822 q^{18} - 19511404 q^{19} - 21673320 q^{20} + 6745680 q^{21} + 49751208 q^{22} + 15312360 q^{23} - 37680552 q^{24} - 19991225 q^{25} + 59764692 q^{26} - 14348907 q^{27} - 112039360 q^{28} + 10751262 q^{29} + 101782980 q^{30} - 50937400 q^{31} - 18888480 q^{32} - 154994148 q^{33} + 240579612 q^{34} + 149071200 q^{35} + 238321764 q^{36} + 664740830 q^{37} - 1521889512 q^{38} - 186190002 q^{39} - 832693680 q^{40} + 898833450 q^{41} + 526163040 q^{42} - 957947188 q^{43} + 2574306096 q^{44} - 317093130 q^{45} + 1194364080 q^{46} - 1555741344 q^{47} - 930511152 q^{48} - 1206709143 q^{49} - 1559315550 q^{50} - 749498022 q^{51} + 3092439704 q^{52} + 3792417030 q^{53} - 1119214746 q^{54} - 3425179320 q^{55} - 4304576640 q^{56} + 4741271172 q^{57} + 838598436 q^{58} + 555306924 q^{59} + 5266616760 q^{60} + 4950420998 q^{61} - 3973117200 q^{62} - 1639200240 q^{63} - 9315634112 q^{64} - 4114569180 q^{65} - 12089543544 q^{66} + 5292399284 q^{67} + 12448452744 q^{68} - 3720903480 q^{69} + 11627553600 q^{70} - 14831086248 q^{71} + 9156374136 q^{72} + 13971005210 q^{73} + 51849784740 q^{74} + 4857867675 q^{75} - 78748026544 q^{76} - 17706327360 q^{77} - 14522820156 q^{78} + 3720542360 q^{79} - 20563147680 q^{80} + 3486784401 q^{81} + 70109009100 q^{82} + 8768454036 q^{83} + 27225564480 q^{84} - 16562980980 q^{85} - 74719880664 q^{86} - 2612556666 q^{87} + 98905401504 q^{88} - 25472769174 q^{89} - 24733264140 q^{90} - 21270100640 q^{91} + 61800684960 q^{92} + 12377788200 q^{93} - 121347824832 q^{94} + 104776239480 q^{95} + 4589900640 q^{96} - 39092494846 q^{97} - 94123313154 q^{98} + 37663577964 q^{99}+O(q^{100}) \) 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 −27760.0 155064. 59049.0 −418860.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

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 3.12.a.a 1
3.b odd 2 1 9.12.a.a 1
4.b odd 2 1 48.12.a.f 1
5.b even 2 1 75.12.a.a 1
5.c odd 4 2 75.12.b.a 2
7.b odd 2 1 147.12.a.c 1
8.b even 2 1 192.12.a.q 1
8.d odd 2 1 192.12.a.g 1
9.c even 3 2 81.12.c.a 2
9.d odd 6 2 81.12.c.e 2
12.b even 2 1 144.12.a.l 1
15.d odd 2 1 225.12.a.f 1
15.e even 4 2 225.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 1.a even 1 1 trivial
9.12.a.a 1 3.b odd 2 1
48.12.a.f 1 4.b odd 2 1
75.12.a.a 1 5.b even 2 1
75.12.b.a 2 5.c odd 4 2
81.12.c.a 2 9.c even 3 2
81.12.c.e 2 9.d odd 6 2
144.12.a.l 1 12.b even 2 1
147.12.a.c 1 7.b odd 2 1
192.12.a.g 1 8.d odd 2 1
192.12.a.q 1 8.b even 2 1
225.12.a.f 1 15.d odd 2 1
225.12.b.a 2 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(\Gamma_0(3))\).

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
$7$ \( T + 27760 \) 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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