Properties

Label 15.12.a.a
Level $15$
Weight $12$
Character orbit 15.a
Self dual yes
Analytic conductor $11.525$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.5251477084\)
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

\(f(q)\) \(=\) \( q - 56 q^{2} - 243 q^{3} + 1088 q^{4} + 3125 q^{5} + 13608 q^{6} + 27984 q^{7} + 53760 q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 56 q^{2} - 243 q^{3} + 1088 q^{4} + 3125 q^{5} + 13608 q^{6} + 27984 q^{7} + 53760 q^{8} + 59049 q^{9} - 175000 q^{10} - 112028 q^{11} - 264384 q^{12} - 1096922 q^{13} - 1567104 q^{14} - 759375 q^{15} - 5238784 q^{16} - 249566 q^{17} - 3306744 q^{18} - 13712420 q^{19} + 3400000 q^{20} - 6800112 q^{21} + 6273568 q^{22} + 41395728 q^{23} - 13063680 q^{24} + 9765625 q^{25} + 61427632 q^{26} - 14348907 q^{27} + 30446592 q^{28} - 4533850 q^{29} + 42525000 q^{30} - 265339008 q^{31} + 183271424 q^{32} + 27222804 q^{33} + 13975696 q^{34} + 87450000 q^{35} + 64245312 q^{36} - 212136946 q^{37} + 767895520 q^{38} + 266552046 q^{39} + 168000000 q^{40} - 1266969958 q^{41} + 380806272 q^{42} + 14129548 q^{43} - 121886464 q^{44} + 184528125 q^{45} - 2318160768 q^{46} - 2657273336 q^{47} + 1273024512 q^{48} - 1194222487 q^{49} - 546875000 q^{50} + 60644538 q^{51} - 1193451136 q^{52} + 2402699278 q^{53} + 803538792 q^{54} - 350087500 q^{55} + 1504419840 q^{56} + 3332118060 q^{57} + 253895600 q^{58} + 7498737220 q^{59} - 826200000 q^{60} - 4064828858 q^{61} + 14858984448 q^{62} + 1652427216 q^{63} + 465829888 q^{64} - 3427881250 q^{65} - 1524477024 q^{66} + 6871514244 q^{67} - 271527808 q^{68} - 10059161904 q^{69} - 4897200000 q^{70} - 13283734648 q^{71} + 3174474240 q^{72} - 28875844262 q^{73} + 11879668976 q^{74} - 2373046875 q^{75} - 14919112960 q^{76} - 3134991552 q^{77} - 14926914576 q^{78} + 27100302240 q^{79} - 16371200000 q^{80} + 3486784401 q^{81} + 70950317648 q^{82} - 34365255132 q^{83} - 7398521856 q^{84} - 779893750 q^{85} - 791254688 q^{86} + 1101725550 q^{87} - 6022625280 q^{88} - 63500412630 q^{89} - 10333575000 q^{90} - 30696265248 q^{91} + 45038552064 q^{92} + 64477378944 q^{93} + 148807306816 q^{94} - 42851312500 q^{95} - 44534956032 q^{96} + 19634495234 q^{97} + 66876459272 q^{98} - 6615141372 q^{99}+O(q^{100}) \) 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
−56.0000 −243.000 1088.00 3125.00 13608.0 27984.0 53760.0 59049.0 −175000.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 15.12.a.a 1
3.b odd 2 1 45.12.a.b 1
4.b odd 2 1 240.12.a.e 1
5.b even 2 1 75.12.a.b 1
5.c odd 4 2 75.12.b.b 2
15.d odd 2 1 225.12.a.a 1
15.e even 4 2 225.12.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.12.a.a 1 1.a even 1 1 trivial
45.12.a.b 1 3.b odd 2 1
75.12.a.b 1 5.b even 2 1
75.12.b.b 2 5.c odd 4 2
225.12.a.a 1 15.d odd 2 1
225.12.b.b 2 15.e even 4 2
240.12.a.e 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 56 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 56 \) Copy content Toggle raw display
$3$ \( T + 243 \) Copy content Toggle raw display
$5$ \( T - 3125 \) Copy content Toggle raw display
$7$ \( T - 27984 \) Copy content Toggle raw display
$11$ \( T + 112028 \) Copy content Toggle raw display
$13$ \( T + 1096922 \) Copy content Toggle raw display
$17$ \( T + 249566 \) Copy content Toggle raw display
$19$ \( T + 13712420 \) Copy content Toggle raw display
$23$ \( T - 41395728 \) Copy content Toggle raw display
$29$ \( T + 4533850 \) Copy content Toggle raw display
$31$ \( T + 265339008 \) Copy content Toggle raw display
$37$ \( T + 212136946 \) Copy content Toggle raw display
$41$ \( T + 1266969958 \) Copy content Toggle raw display
$43$ \( T - 14129548 \) Copy content Toggle raw display
$47$ \( T + 2657273336 \) Copy content Toggle raw display
$53$ \( T - 2402699278 \) Copy content Toggle raw display
$59$ \( T - 7498737220 \) Copy content Toggle raw display
$61$ \( T + 4064828858 \) Copy content Toggle raw display
$67$ \( T - 6871514244 \) Copy content Toggle raw display
$71$ \( T + 13283734648 \) Copy content Toggle raw display
$73$ \( T + 28875844262 \) Copy content Toggle raw display
$79$ \( T - 27100302240 \) Copy content Toggle raw display
$83$ \( T + 34365255132 \) Copy content Toggle raw display
$89$ \( T + 63500412630 \) Copy content Toggle raw display
$97$ \( T - 19634495234 \) Copy content Toggle raw display
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