Properties

Label 15.5.d.a
Level 15
Weight 5
Character orbit 15.d
Self dual yes
Analytic conductor 1.551
Analytic rank 0
Dimension 1
CM discriminant -15
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 15.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 7q^{2} + 9q^{3} + 33q^{4} + 25q^{5} - 63q^{6} - 119q^{8} + 81q^{9} + O(q^{10}) \) \( q - 7q^{2} + 9q^{3} + 33q^{4} + 25q^{5} - 63q^{6} - 119q^{8} + 81q^{9} - 175q^{10} + 297q^{12} + 225q^{15} + 305q^{16} - 382q^{17} - 567q^{18} - 238q^{19} + 825q^{20} + 98q^{23} - 1071q^{24} + 625q^{25} + 729q^{27} - 1575q^{30} - 1918q^{31} - 231q^{32} + 2674q^{34} + 2673q^{36} + 1666q^{38} - 2975q^{40} + 2025q^{45} - 686q^{46} - 4222q^{47} + 2745q^{48} + 2401q^{49} - 4375q^{50} - 3438q^{51} + 1778q^{53} - 5103q^{54} - 2142q^{57} + 7425q^{60} + 6482q^{61} + 13426q^{62} - 3263q^{64} - 12606q^{68} + 882q^{69} - 9639q^{72} + 5625q^{75} - 7854q^{76} - 2878q^{79} + 7625q^{80} + 6561q^{81} + 9938q^{83} - 9550q^{85} - 14175q^{90} + 3234q^{92} - 17262q^{93} + 29554q^{94} - 5950q^{95} - 2079q^{96} - 16807q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0
−7.00000 9.00000 33.0000 25.0000 −63.0000 0 −119.000 81.0000 −175.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.d.a 1
3.b odd 2 1 15.5.d.b yes 1
4.b odd 2 1 240.5.c.a 1
5.b even 2 1 15.5.d.b yes 1
5.c odd 4 2 75.5.c.e 2
12.b even 2 1 240.5.c.b 1
15.d odd 2 1 CM 15.5.d.a 1
15.e even 4 2 75.5.c.e 2
20.d odd 2 1 240.5.c.b 1
60.h even 2 1 240.5.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.d.a 1 1.a even 1 1 trivial
15.5.d.a 1 15.d odd 2 1 CM
15.5.d.b yes 1 3.b odd 2 1
15.5.d.b yes 1 5.b even 2 1
75.5.c.e 2 5.c odd 4 2
75.5.c.e 2 15.e even 4 2
240.5.c.a 1 4.b odd 2 1
240.5.c.a 1 60.h even 2 1
240.5.c.b 1 12.b even 2 1
240.5.c.b 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 7 \) acting on \(S_{5}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 7 T + 16 T^{2} \)
$3$ \( 1 - 9 T \)
$5$ \( 1 - 25 T \)
$7$ \( ( 1 - 49 T )( 1 + 49 T ) \)
$11$ \( ( 1 - 121 T )( 1 + 121 T ) \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( 1 + 382 T + 83521 T^{2} \)
$19$ \( 1 + 238 T + 130321 T^{2} \)
$23$ \( 1 - 98 T + 279841 T^{2} \)
$29$ \( ( 1 - 841 T )( 1 + 841 T ) \)
$31$ \( 1 + 1918 T + 923521 T^{2} \)
$37$ \( ( 1 - 1369 T )( 1 + 1369 T ) \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( ( 1 - 1849 T )( 1 + 1849 T ) \)
$47$ \( 1 + 4222 T + 4879681 T^{2} \)
$53$ \( 1 - 1778 T + 7890481 T^{2} \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( 1 - 6482 T + 13845841 T^{2} \)
$67$ \( ( 1 - 4489 T )( 1 + 4489 T ) \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( ( 1 - 5329 T )( 1 + 5329 T ) \)
$79$ \( 1 + 2878 T + 38950081 T^{2} \)
$83$ \( 1 - 9938 T + 47458321 T^{2} \)
$89$ \( ( 1 - 7921 T )( 1 + 7921 T ) \)
$97$ \( ( 1 - 9409 T )( 1 + 9409 T ) \)
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