Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + 9q^{3} - 31q^{4} - 92q^{5} - 9q^{6} + 26q^{7} + 63q^{8} + 81q^{9} + O(q^{10}) \) \( q - q^{2} + 9q^{3} - 31q^{4} - 92q^{5} - 9q^{6} + 26q^{7} + 63q^{8} + 81q^{9} + 92q^{10} - 279q^{12} + 692q^{13} - 26q^{14} - 828q^{15} + 929q^{16} + 1442q^{17} - 81q^{18} - 2160q^{19} + 2852q^{20} + 234q^{21} - 1582q^{23} + 567q^{24} + 5339q^{25} - 692q^{26} + 729q^{27} - 806q^{28} + 5526q^{29} + 828q^{30} + 4792q^{31} - 2945q^{32} - 1442q^{34} - 2392q^{35} - 2511q^{36} - 10194q^{37} + 2160q^{38} + 6228q^{39} - 5796q^{40} + 10622q^{41} - 234q^{42} - 8580q^{43} - 7452q^{45} + 1582q^{46} - 2362q^{47} + 8361q^{48} - 16131q^{49} - 5339q^{50} + 12978q^{51} - 21452q^{52} - 30804q^{53} - 729q^{54} + 1638q^{56} - 19440q^{57} - 5526q^{58} + 6416q^{59} + 25668q^{60} - 42096q^{61} - 4792q^{62} + 2106q^{63} - 26783q^{64} - 63664q^{65} - 28444q^{67} - 44702q^{68} - 14238q^{69} + 2392q^{70} + 45690q^{71} + 5103q^{72} + 18374q^{73} + 10194q^{74} + 48051q^{75} + 66960q^{76} - 6228q^{78} + 105214q^{79} - 85468q^{80} + 6561q^{81} - 10622q^{82} - 62292q^{83} - 7254q^{84} - 132664q^{85} + 8580q^{86} + 49734q^{87} - 72246q^{89} + 7452q^{90} + 17992q^{91} + 49042q^{92} + 43128q^{93} + 2362q^{94} + 198720q^{95} - 26505q^{96} + 79262q^{97} + 16131q^{98} + O(q^{100}) \)

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
−1.00000 9.00000 −31.0000 −92.0000 −9.00000 26.0000 63.0000 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.b 1
3.b odd 2 1 1089.6.a.h 1
11.b odd 2 1 33.6.a.b 1
33.d even 2 1 99.6.a.a 1
44.c even 2 1 528.6.a.a 1
55.d odd 2 1 825.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.b 1 11.b odd 2 1
99.6.a.a 1 33.d even 2 1
363.6.a.b 1 1.a even 1 1 trivial
528.6.a.a 1 44.c even 2 1
825.6.a.a 1 55.d odd 2 1
1089.6.a.h 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -9 + T \)
$5$ \( 92 + T \)
$7$ \( -26 + T \)
$11$ \( T \)
$13$ \( -692 + T \)
$17$ \( -1442 + T \)
$19$ \( 2160 + T \)
$23$ \( 1582 + T \)
$29$ \( -5526 + T \)
$31$ \( -4792 + T \)
$37$ \( 10194 + T \)
$41$ \( -10622 + T \)
$43$ \( 8580 + T \)
$47$ \( 2362 + T \)
$53$ \( 30804 + T \)
$59$ \( -6416 + T \)
$61$ \( 42096 + T \)
$67$ \( 28444 + T \)
$71$ \( -45690 + T \)
$73$ \( -18374 + T \)
$79$ \( -105214 + T \)
$83$ \( 62292 + T \)
$89$ \( 72246 + T \)
$97$ \( -79262 + T \)
show more
show less