Properties

Label 15.8.a.a
Level 15
Weight 8
Character orbit 15.a
Self dual yes
Analytic conductor 4.686
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.68577538226\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 22q^{2} + 27q^{3} + 356q^{4} - 125q^{5} - 594q^{6} - 420q^{7} - 5016q^{8} + 729q^{9} + O(q^{10}) \) \( q - 22q^{2} + 27q^{3} + 356q^{4} - 125q^{5} - 594q^{6} - 420q^{7} - 5016q^{8} + 729q^{9} + 2750q^{10} - 2944q^{11} + 9612q^{12} - 11006q^{13} + 9240q^{14} - 3375q^{15} + 64784q^{16} - 16546q^{17} - 16038q^{18} - 25364q^{19} - 44500q^{20} - 11340q^{21} + 64768q^{22} - 5880q^{23} - 135432q^{24} + 15625q^{25} + 242132q^{26} + 19683q^{27} - 149520q^{28} + 163042q^{29} + 74250q^{30} - 201600q^{31} - 783200q^{32} - 79488q^{33} + 364012q^{34} + 52500q^{35} + 259524q^{36} + 120530q^{37} + 558008q^{38} - 297162q^{39} + 627000q^{40} - 115910q^{41} + 249480q^{42} - 19148q^{43} - 1048064q^{44} - 91125q^{45} + 129360q^{46} + 841016q^{47} + 1749168q^{48} - 647143q^{49} - 343750q^{50} - 446742q^{51} - 3918136q^{52} + 501890q^{53} - 433026q^{54} + 368000q^{55} + 2106720q^{56} - 684828q^{57} - 3586924q^{58} - 1586176q^{59} - 1201500q^{60} - 372962q^{61} + 4435200q^{62} - 306180q^{63} + 8938048q^{64} + 1375750q^{65} + 1748736q^{66} + 4561044q^{67} - 5890376q^{68} - 158760q^{69} - 1155000q^{70} + 1512832q^{71} - 3656664q^{72} - 1522910q^{73} - 2651660q^{74} + 421875q^{75} - 9029584q^{76} + 1236480q^{77} + 6537564q^{78} + 4231920q^{79} - 8098000q^{80} + 531441q^{81} + 2550020q^{82} - 1854204q^{83} - 4037040q^{84} + 2068250q^{85} + 421256q^{86} + 4402134q^{87} + 14767104q^{88} - 6888174q^{89} + 2004750q^{90} + 4622520q^{91} - 2093280q^{92} - 5443200q^{93} - 18502352q^{94} + 3170500q^{95} - 21146400q^{96} + 3700034q^{97} + 14237146q^{98} - 2146176q^{99} + O(q^{100}) \)

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
−22.0000 27.0000 356.000 −125.000 −594.000 −420.000 −5016.00 729.000 2750.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 15.8.a.a 1
3.b odd 2 1 45.8.a.g 1
4.b odd 2 1 240.8.a.c 1
5.b even 2 1 75.8.a.c 1
5.c odd 4 2 75.8.b.a 2
15.d odd 2 1 225.8.a.a 1
15.e even 4 2 225.8.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.a 1 1.a even 1 1 trivial
45.8.a.g 1 3.b odd 2 1
75.8.a.c 1 5.b even 2 1
75.8.b.a 2 5.c odd 4 2
225.8.a.a 1 15.d odd 2 1
225.8.b.a 2 15.e even 4 2
240.8.a.c 1 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 22 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 22 T + 128 T^{2} \)
$3$ \( 1 - 27 T \)
$5$ \( 1 + 125 T \)
$7$ \( 1 + 420 T + 823543 T^{2} \)
$11$ \( 1 + 2944 T + 19487171 T^{2} \)
$13$ \( 1 + 11006 T + 62748517 T^{2} \)
$17$ \( 1 + 16546 T + 410338673 T^{2} \)
$19$ \( 1 + 25364 T + 893871739 T^{2} \)
$23$ \( 1 + 5880 T + 3404825447 T^{2} \)
$29$ \( 1 - 163042 T + 17249876309 T^{2} \)
$31$ \( 1 + 201600 T + 27512614111 T^{2} \)
$37$ \( 1 - 120530 T + 94931877133 T^{2} \)
$41$ \( 1 + 115910 T + 194754273881 T^{2} \)
$43$ \( 1 + 19148 T + 271818611107 T^{2} \)
$47$ \( 1 - 841016 T + 506623120463 T^{2} \)
$53$ \( 1 - 501890 T + 1174711139837 T^{2} \)
$59$ \( 1 + 1586176 T + 2488651484819 T^{2} \)
$61$ \( 1 + 372962 T + 3142742836021 T^{2} \)
$67$ \( 1 - 4561044 T + 6060711605323 T^{2} \)
$71$ \( 1 - 1512832 T + 9095120158391 T^{2} \)
$73$ \( 1 + 1522910 T + 11047398519097 T^{2} \)
$79$ \( 1 - 4231920 T + 19203908986159 T^{2} \)
$83$ \( 1 + 1854204 T + 27136050989627 T^{2} \)
$89$ \( 1 + 6888174 T + 44231334895529 T^{2} \)
$97$ \( 1 - 3700034 T + 80798284478113 T^{2} \)
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