Properties

Label 363.6.a.c
Level $363$
Weight $6$
Character orbit 363.a
Self dual yes
Analytic conductor $58.219$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.2193265921\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} - 9q^{3} - 28q^{4} + 46q^{5} - 18q^{6} - 148q^{7} - 120q^{8} + 81q^{9} + O(q^{10}) \) \( q + 2q^{2} - 9q^{3} - 28q^{4} + 46q^{5} - 18q^{6} - 148q^{7} - 120q^{8} + 81q^{9} + 92q^{10} + 252q^{12} - 574q^{13} - 296q^{14} - 414q^{15} + 656q^{16} + 722q^{17} + 162q^{18} - 2160q^{19} - 1288q^{20} + 1332q^{21} - 2536q^{23} + 1080q^{24} - 1009q^{25} - 1148q^{26} - 729q^{27} + 4144q^{28} - 4650q^{29} - 828q^{30} + 5032q^{31} + 5152q^{32} + 1444q^{34} - 6808q^{35} - 2268q^{36} + 8118q^{37} - 4320q^{38} + 5166q^{39} - 5520q^{40} + 5138q^{41} + 2664q^{42} - 8304q^{43} + 3726q^{45} - 5072q^{46} + 24728q^{47} - 5904q^{48} + 5097q^{49} - 2018q^{50} - 6498q^{51} + 16072q^{52} - 28746q^{53} - 1458q^{54} + 17760q^{56} + 19440q^{57} - 9300q^{58} - 5860q^{59} + 11592q^{60} + 53658q^{61} + 10064q^{62} - 11988q^{63} - 10688q^{64} - 26404q^{65} + 30908q^{67} - 20216q^{68} + 22824q^{69} - 13616q^{70} - 69648q^{71} - 9720q^{72} + 18446q^{73} + 16236q^{74} + 9081q^{75} + 60480q^{76} + 10332q^{78} + 25300q^{79} + 30176q^{80} + 6561q^{81} + 10276q^{82} + 17556q^{83} - 37296q^{84} + 33212q^{85} - 16608q^{86} + 41850q^{87} + 132570q^{89} + 7452q^{90} + 84952q^{91} + 71008q^{92} - 45288q^{93} + 49456q^{94} - 99360q^{95} - 46368q^{96} + 70658q^{97} + 10194q^{98} + O(q^{100}) \)

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
2.00000 −9.00000 −28.0000 46.0000 −18.0000 −148.000 −120.000 81.0000 92.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

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 363.6.a.c 1
3.b odd 2 1 1089.6.a.d 1
11.b odd 2 1 33.6.a.a 1
33.d even 2 1 99.6.a.b 1
44.c even 2 1 528.6.a.i 1
55.d odd 2 1 825.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 11.b odd 2 1
99.6.a.b 1 33.d even 2 1
363.6.a.c 1 1.a even 1 1 trivial
528.6.a.i 1 44.c even 2 1
825.6.a.b 1 55.d odd 2 1
1089.6.a.d 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 9 + T \)
$5$ \( -46 + T \)
$7$ \( 148 + T \)
$11$ \( T \)
$13$ \( 574 + T \)
$17$ \( -722 + T \)
$19$ \( 2160 + T \)
$23$ \( 2536 + T \)
$29$ \( 4650 + T \)
$31$ \( -5032 + T \)
$37$ \( -8118 + T \)
$41$ \( -5138 + T \)
$43$ \( 8304 + T \)
$47$ \( -24728 + T \)
$53$ \( 28746 + T \)
$59$ \( 5860 + T \)
$61$ \( -53658 + T \)
$67$ \( -30908 + T \)
$71$ \( 69648 + T \)
$73$ \( -18446 + T \)
$79$ \( -25300 + T \)
$83$ \( -17556 + T \)
$89$ \( -132570 + T \)
$97$ \( -70658 + T \)
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