Properties

Label 15.8.a.b
Level 15
Weight 8
Character orbit 15.a
Self dual yes
Analytic conductor 4.686
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q - 13q^{2} - 27q^{3} + 41q^{4} - 125q^{5} + 351q^{6} + 1380q^{7} + 1131q^{8} + 729q^{9} + O(q^{10}) \) \( q - 13q^{2} - 27q^{3} + 41q^{4} - 125q^{5} + 351q^{6} + 1380q^{7} + 1131q^{8} + 729q^{9} + 1625q^{10} - 3304q^{11} - 1107q^{12} + 8506q^{13} - 17940q^{14} + 3375q^{15} - 19951q^{16} - 9994q^{17} - 9477q^{18} + 41236q^{19} - 5125q^{20} - 37260q^{21} + 42952q^{22} + 84120q^{23} - 30537q^{24} + 15625q^{25} - 110578q^{26} - 19683q^{27} + 56580q^{28} + 132802q^{29} - 43875q^{30} - 55800q^{31} + 114595q^{32} + 89208q^{33} + 129922q^{34} - 172500q^{35} + 29889q^{36} + 228170q^{37} - 536068q^{38} - 229662q^{39} - 141375q^{40} - 139670q^{41} + 484380q^{42} - 755492q^{43} - 135464q^{44} - 91125q^{45} - 1093560q^{46} + 836984q^{47} + 538677q^{48} + 1080857q^{49} - 203125q^{50} + 269838q^{51} + 348746q^{52} + 1641650q^{53} + 255879q^{54} + 413000q^{55} + 1560780q^{56} - 1113372q^{57} - 1726426q^{58} - 989656q^{59} + 138375q^{60} - 1658162q^{61} + 725400q^{62} + 1006020q^{63} + 1063993q^{64} - 1063250q^{65} - 1159704q^{66} - 4523844q^{67} - 409754q^{68} - 2271240q^{69} + 2242500q^{70} - 389408q^{71} + 824499q^{72} + 5617330q^{73} - 2966210q^{74} - 421875q^{75} + 1690676q^{76} - 4559520q^{77} + 2985606q^{78} + 3901080q^{79} + 2493875q^{80} + 531441q^{81} + 1815710q^{82} - 9394116q^{83} - 1527660q^{84} + 1249250q^{85} + 9821396q^{86} - 3585654q^{87} - 3736824q^{88} + 2803746q^{89} + 1184625q^{90} + 11738280q^{91} + 3448920q^{92} + 1506600q^{93} - 10880792q^{94} - 5154500q^{95} - 3094065q^{96} + 5099426q^{97} - 14051141q^{98} - 2408616q^{99} + O(q^{100}) \)

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
−13.0000 −27.0000 41.0000 −125.000 351.000 1380.00 1131.00 729.000 1625.00
\(n\): 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.b 1
3.b odd 2 1 45.8.a.e 1
4.b odd 2 1 240.8.a.h 1
5.b even 2 1 75.8.a.b 1
5.c odd 4 2 75.8.b.b 2
15.d odd 2 1 225.8.a.c 1
15.e even 4 2 225.8.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.b 1 1.a even 1 1 trivial
45.8.a.e 1 3.b odd 2 1
75.8.a.b 1 5.b even 2 1
75.8.b.b 2 5.c odd 4 2
225.8.a.c 1 15.d odd 2 1
225.8.b.c 2 15.e even 4 2
240.8.a.h 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 \( T_{2} + 13 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 13 T + 128 T^{2} \)
$3$ \( 1 + 27 T \)
$5$ \( 1 + 125 T \)
$7$ \( 1 - 1380 T + 823543 T^{2} \)
$11$ \( 1 + 3304 T + 19487171 T^{2} \)
$13$ \( 1 - 8506 T + 62748517 T^{2} \)
$17$ \( 1 + 9994 T + 410338673 T^{2} \)
$19$ \( 1 - 41236 T + 893871739 T^{2} \)
$23$ \( 1 - 84120 T + 3404825447 T^{2} \)
$29$ \( 1 - 132802 T + 17249876309 T^{2} \)
$31$ \( 1 + 55800 T + 27512614111 T^{2} \)
$37$ \( 1 - 228170 T + 94931877133 T^{2} \)
$41$ \( 1 + 139670 T + 194754273881 T^{2} \)
$43$ \( 1 + 755492 T + 271818611107 T^{2} \)
$47$ \( 1 - 836984 T + 506623120463 T^{2} \)
$53$ \( 1 - 1641650 T + 1174711139837 T^{2} \)
$59$ \( 1 + 989656 T + 2488651484819 T^{2} \)
$61$ \( 1 + 1658162 T + 3142742836021 T^{2} \)
$67$ \( 1 + 4523844 T + 6060711605323 T^{2} \)
$71$ \( 1 + 389408 T + 9095120158391 T^{2} \)
$73$ \( 1 - 5617330 T + 11047398519097 T^{2} \)
$79$ \( 1 - 3901080 T + 19203908986159 T^{2} \)
$83$ \( 1 + 9394116 T + 27136050989627 T^{2} \)
$89$ \( 1 - 2803746 T + 44231334895529 T^{2} \)
$97$ \( 1 - 5099426 T + 80798284478113 T^{2} \)
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