Properties

Label 15.5.d.c
Level 15
Weight 5
Character orbit 15.d
Analytic conductor 1.551
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{10}, \sqrt{-26})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( \beta_{1} + \beta_{2} ) q^{3} \) \( -6 q^{4} \) \( + ( 2 \beta_{2} - \beta_{3} ) q^{5} \) \( + ( 15 + \beta_{3} ) q^{6} \) \( + ( -2 \beta_{1} + \beta_{2} ) q^{7} \) \( -22 \beta_{2} q^{8} \) \( + ( -36 + 3 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( \beta_{1} + \beta_{2} ) q^{3} \) \( -6 q^{4} \) \( + ( 2 \beta_{2} - \beta_{3} ) q^{5} \) \( + ( 15 + \beta_{3} ) q^{6} \) \( + ( -2 \beta_{1} + \beta_{2} ) q^{7} \) \( -22 \beta_{2} q^{8} \) \( + ( -36 + 3 \beta_{3} ) q^{9} \) \( + ( 20 - 10 \beta_{1} + 5 \beta_{2} ) q^{10} \) \( -4 \beta_{3} q^{11} \) \( + ( -6 \beta_{1} - 6 \beta_{2} ) q^{12} \) \( + ( 32 \beta_{1} - 16 \beta_{2} ) q^{13} \) \( -2 \beta_{3} q^{14} \) \( + ( 30 - 15 \beta_{1} + 66 \beta_{2} + 2 \beta_{3} ) q^{15} \) \( -124 q^{16} \) \( + 88 \beta_{2} q^{17} \) \( + ( 30 \beta_{1} - 51 \beta_{2} ) q^{18} \) \( + 308 q^{19} \) \( + ( -12 \beta_{2} + 6 \beta_{3} ) q^{20} \) \( + ( 117 - 3 \beta_{3} ) q^{21} \) \( + ( -40 \beta_{1} + 20 \beta_{2} ) q^{22} \) \( -131 \beta_{2} q^{23} \) \( + ( -330 - 22 \beta_{3} ) q^{24} \) \( + ( -545 - 40 \beta_{1} + 20 \beta_{2} ) q^{25} \) \( + 32 \beta_{3} q^{26} \) \( + ( 9 \beta_{1} - 234 \beta_{2} ) q^{27} \) \( + ( 12 \beta_{1} - 6 \beta_{2} ) q^{28} \) \( + 8 \beta_{3} q^{29} \) \( + ( 585 + 20 \beta_{1} + 20 \beta_{2} - 15 \beta_{3} ) q^{30} \) \( + 32 q^{31} \) \( + 228 \beta_{2} q^{32} \) \( + ( -60 \beta_{1} + 264 \beta_{2} ) q^{33} \) \( + 880 q^{34} \) \( + ( -117 \beta_{2} - 4 \beta_{3} ) q^{35} \) \( + ( 216 - 18 \beta_{3} ) q^{36} \) \( + ( 168 \beta_{1} - 84 \beta_{2} ) q^{37} \) \( + 308 \beta_{2} q^{38} \) \( + ( -1872 + 48 \beta_{3} ) q^{39} \) \( + ( -440 + 220 \beta_{1} - 110 \beta_{2} ) q^{40} \) \( -86 \beta_{3} q^{41} \) \( + ( -30 \beta_{1} + 132 \beta_{2} ) q^{42} \) \( + ( -338 \beta_{1} + 169 \beta_{2} ) q^{43} \) \( + 24 \beta_{3} q^{44} \) \( + ( 1755 + 60 \beta_{1} - 102 \beta_{2} + 36 \beta_{3} ) q^{45} \) \( -1310 q^{46} \) \( -773 \beta_{2} q^{47} \) \( + ( -124 \beta_{1} - 124 \beta_{2} ) q^{48} \) \( + 2167 q^{49} \) \( + ( -545 \beta_{2} - 40 \beta_{3} ) q^{50} \) \( + ( 1320 + 88 \beta_{3} ) q^{51} \) \( + ( -192 \beta_{1} + 96 \beta_{2} ) q^{52} \) \( + 448 \beta_{2} q^{53} \) \( + ( -2295 + 9 \beta_{3} ) q^{54} \) \( + ( -2340 - 80 \beta_{1} + 40 \beta_{2} ) q^{55} \) \( + 44 \beta_{3} q^{56} \) \( + ( 308 \beta_{1} + 308 \beta_{2} ) q^{57} \) \( + ( 80 \beta_{1} - 40 \beta_{2} ) q^{58} \) \( -164 \beta_{3} q^{59} \) \( + ( -180 + 90 \beta_{1} - 396 \beta_{2} - 12 \beta_{3} ) q^{60} \) \( -928 q^{61} \) \( + 32 \beta_{2} q^{62} \) \( + ( 72 \beta_{1} + 315 \beta_{2} ) q^{63} \) \( + 4264 q^{64} \) \( + ( 1872 \beta_{2} + 64 \beta_{3} ) q^{65} \) \( + ( 2340 - 60 \beta_{3} ) q^{66} \) \( + ( 338 \beta_{1} - 169 \beta_{2} ) q^{67} \) \( -528 \beta_{2} q^{68} \) \( + ( -1965 - 131 \beta_{3} ) q^{69} \) \( + ( -1170 - 40 \beta_{1} + 20 \beta_{2} ) q^{70} \) \( + 192 \beta_{3} q^{71} \) \( + ( -660 \beta_{1} + 1122 \beta_{2} ) q^{72} \) \( + ( 552 \beta_{1} - 276 \beta_{2} ) q^{73} \) \( + 168 \beta_{3} q^{74} \) \( + ( 2340 - 545 \beta_{1} - 545 \beta_{2} - 60 \beta_{3} ) q^{75} \) \( -1848 q^{76} \) \( -468 \beta_{2} q^{77} \) \( + ( 480 \beta_{1} - 2112 \beta_{2} ) q^{78} \) \( + 8 q^{79} \) \( + ( -248 \beta_{2} + 124 \beta_{3} ) q^{80} \) \( + ( -3969 - 216 \beta_{3} ) q^{81} \) \( + ( -860 \beta_{1} + 430 \beta_{2} ) q^{82} \) \( -1403 \beta_{2} q^{83} \) \( + ( -702 + 18 \beta_{3} ) q^{84} \) \( + ( 1760 - 880 \beta_{1} + 440 \beta_{2} ) q^{85} \) \( -338 \beta_{3} q^{86} \) \( + ( 120 \beta_{1} - 528 \beta_{2} ) q^{87} \) \( + ( 880 \beta_{1} - 440 \beta_{2} ) q^{88} \) \( + 384 \beta_{3} q^{89} \) \( + ( -720 + 360 \beta_{1} + 1575 \beta_{2} + 60 \beta_{3} ) q^{90} \) \( + 3744 q^{91} \) \( + 786 \beta_{2} q^{92} \) \( + ( 32 \beta_{1} + 32 \beta_{2} ) q^{93} \) \( -7730 q^{94} \) \( + ( 616 \beta_{2} - 308 \beta_{3} ) q^{95} \) \( + ( 3420 + 228 \beta_{3} ) q^{96} \) \( + ( 328 \beta_{1} - 164 \beta_{2} ) q^{97} \) \( + 2167 \beta_{2} q^{98} \) \( + ( 7020 + 144 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 24q^{4} \) \(\mathstrut +\mathstrut 60q^{6} \) \(\mathstrut -\mathstrut 144q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 24q^{4} \) \(\mathstrut +\mathstrut 60q^{6} \) \(\mathstrut -\mathstrut 144q^{9} \) \(\mathstrut +\mathstrut 80q^{10} \) \(\mathstrut +\mathstrut 120q^{15} \) \(\mathstrut -\mathstrut 496q^{16} \) \(\mathstrut +\mathstrut 1232q^{19} \) \(\mathstrut +\mathstrut 468q^{21} \) \(\mathstrut -\mathstrut 1320q^{24} \) \(\mathstrut -\mathstrut 2180q^{25} \) \(\mathstrut +\mathstrut 2340q^{30} \) \(\mathstrut +\mathstrut 128q^{31} \) \(\mathstrut +\mathstrut 3520q^{34} \) \(\mathstrut +\mathstrut 864q^{36} \) \(\mathstrut -\mathstrut 7488q^{39} \) \(\mathstrut -\mathstrut 1760q^{40} \) \(\mathstrut +\mathstrut 7020q^{45} \) \(\mathstrut -\mathstrut 5240q^{46} \) \(\mathstrut +\mathstrut 8668q^{49} \) \(\mathstrut +\mathstrut 5280q^{51} \) \(\mathstrut -\mathstrut 9180q^{54} \) \(\mathstrut -\mathstrut 9360q^{55} \) \(\mathstrut -\mathstrut 720q^{60} \) \(\mathstrut -\mathstrut 3712q^{61} \) \(\mathstrut +\mathstrut 17056q^{64} \) \(\mathstrut +\mathstrut 9360q^{66} \) \(\mathstrut -\mathstrut 7860q^{69} \) \(\mathstrut -\mathstrut 4680q^{70} \) \(\mathstrut +\mathstrut 9360q^{75} \) \(\mathstrut -\mathstrut 7392q^{76} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut -\mathstrut 15876q^{81} \) \(\mathstrut -\mathstrut 2808q^{84} \) \(\mathstrut +\mathstrut 7040q^{85} \) \(\mathstrut -\mathstrut 2880q^{90} \) \(\mathstrut +\mathstrut 14976q^{91} \) \(\mathstrut -\mathstrut 30920q^{94} \) \(\mathstrut +\mathstrut 13680q^{96} \) \(\mathstrut +\mathstrut 28080q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(8\) \(x^{2}\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 26 \nu \)\()/9\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu \)\()/9\)
\(\beta_{3}\)\(=\)\( 3 \nu^{2} + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(12\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/3\)

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
−1.58114 2.54951i
−1.58114 + 2.54951i
1.58114 2.54951i
1.58114 + 2.54951i
−3.16228 −4.74342 7.64853i −6.00000 −6.32456 24.1868i 15.0000 + 24.1868i 15.2971i 69.5701 −36.0000 + 72.5603i 20.0000 + 76.4853i
14.2 −3.16228 −4.74342 + 7.64853i −6.00000 −6.32456 + 24.1868i 15.0000 24.1868i 15.2971i 69.5701 −36.0000 72.5603i 20.0000 76.4853i
14.3 3.16228 4.74342 7.64853i −6.00000 6.32456 + 24.1868i 15.0000 24.1868i 15.2971i −69.5701 −36.0000 72.5603i 20.0000 + 76.4853i
14.4 3.16228 4.74342 + 7.64853i −6.00000 6.32456 24.1868i 15.0000 + 24.1868i 15.2971i −69.5701 −36.0000 + 72.5603i 20.0000 76.4853i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
5.b Even 1 yes
15.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 10 \) acting on \(S_{5}^{\mathrm{new}}(15, [\chi])\).