Properties

Label 15.4.b.a
Level 15
Weight 4
Character orbit 15.b
Analytic conductor 0.885
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 15.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.885028650086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + \beta_{2} q^{3} \) \( + ( -5 + \beta_{3} ) q^{4} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} \) \( + ( 5 - \beta_{3} ) q^{6} \) \( + ( 6 \beta_{1} - 4 \beta_{2} ) q^{7} \) \( + ( \beta_{1} + 8 \beta_{2} ) q^{8} \) \( -9 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + \beta_{2} q^{3} \) \( + ( -5 + \beta_{3} ) q^{4} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} \) \( + ( 5 - \beta_{3} ) q^{6} \) \( + ( 6 \beta_{1} - 4 \beta_{2} ) q^{7} \) \( + ( \beta_{1} + 8 \beta_{2} ) q^{8} \) \( -9 q^{9} \) \( + ( 3 - 6 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{10} \) \( + ( -22 + 2 \beta_{3} ) q^{11} \) \( -9 \beta_{1} q^{12} \) \( + ( -6 \beta_{1} + 16 \beta_{2} ) q^{13} \) \( + ( 58 - 2 \beta_{3} ) q^{14} \) \( + ( 13 + 9 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{15} \) \( + ( 13 - \beta_{3} ) q^{16} \) \( + ( -10 \beta_{1} - 12 \beta_{2} ) q^{17} \) \( + 9 \beta_{1} q^{18} \) \( + ( -24 - 8 \beta_{3} ) q^{19} \) \( + ( -102 + 9 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} ) q^{20} \) \( + ( 6 + 6 \beta_{3} ) q^{21} \) \( + ( 30 \beta_{1} + 16 \beta_{2} ) q^{22} \) \( + ( 8 \beta_{1} - 12 \beta_{2} ) q^{23} \) \( + ( -77 + \beta_{3} ) q^{24} \) \( + ( 67 - 24 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} ) q^{25} \) \( + ( 2 - 10 \beta_{3} ) q^{26} \) \( -9 \beta_{2} q^{27} \) \( + ( -18 \beta_{1} - 48 \beta_{2} ) q^{28} \) \( + ( 152 + 14 \beta_{3} ) q^{29} \) \( + ( 102 - 9 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{30} \) \( + ( 24 + 4 \beta_{3} ) q^{31} \) \( + ( -9 \beta_{1} + 56 \beta_{2} ) q^{32} \) \( + ( -18 \beta_{1} - 12 \beta_{2} ) q^{33} \) \( + ( -190 + 22 \beta_{3} ) q^{34} \) \( + ( -70 + 60 \beta_{2} - 10 \beta_{3} ) q^{35} \) \( + ( 45 - 9 \beta_{3} ) q^{36} \) \( + ( -54 \beta_{1} - 24 \beta_{2} ) q^{37} \) \( + ( -8 \beta_{1} - 64 \beta_{2} ) q^{38} \) \( + ( -114 - 6 \beta_{3} ) q^{39} \) \( + ( 101 + 78 \beta_{1} - 16 \beta_{2} + 7 \beta_{3} ) q^{40} \) \( + ( -206 + 4 \beta_{3} ) q^{41} \) \( + ( 18 \beta_{1} + 48 \beta_{2} ) q^{42} \) \( + ( 96 \beta_{1} - 28 \beta_{2} ) q^{43} \) \( + ( 294 - 30 \beta_{3} ) q^{44} \) \( + ( -18 - 9 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} ) q^{45} \) \( + ( 44 + 4 \beta_{3} ) q^{46} \) \( + ( 92 \beta_{1} - 76 \beta_{2} ) q^{47} \) \( + ( 9 \beta_{1} + 8 \beta_{2} ) q^{48} \) \( + ( -29 - 12 \beta_{3} ) q^{49} \) \( + ( -172 - 91 \beta_{1} - 48 \beta_{2} - 4 \beta_{3} ) q^{50} \) \( + ( 158 - 10 \beta_{3} ) q^{51} \) \( + ( -90 \beta_{1} + 48 \beta_{2} ) q^{52} \) \( + ( -82 \beta_{1} + 108 \beta_{2} ) q^{53} \) \( + ( -45 + 9 \beta_{3} ) q^{54} \) \( + ( -228 + 6 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{55} \) \( + ( -10 + 50 \beta_{3} ) q^{56} \) \( + ( 72 \beta_{1} - 64 \beta_{2} ) q^{57} \) \( + ( -96 \beta_{1} + 112 \beta_{2} ) q^{58} \) \( + ( 94 - 2 \beta_{3} ) q^{59} \) \( + ( 27 - 54 \beta_{1} - 72 \beta_{2} + 9 \beta_{3} ) q^{60} \) \( + ( 186 - 32 \beta_{3} ) q^{61} \) \( + ( -8 \beta_{1} + 32 \beta_{2} ) q^{62} \) \( + ( -54 \beta_{1} + 36 \beta_{2} ) q^{63} \) \( + ( 267 - 55 \beta_{3} ) q^{64} \) \( + ( 226 + 108 \beta_{1} - 96 \beta_{2} + 22 \beta_{3} ) q^{65} \) \( + ( -294 + 30 \beta_{3} ) q^{66} \) \( + ( -60 \beta_{1} - 92 \beta_{2} ) q^{67} \) \( + ( 198 \beta_{1} + 80 \beta_{2} ) q^{68} \) \( + ( 68 + 8 \beta_{3} ) q^{69} \) \( + ( 300 + 30 \beta_{1} - 80 \beta_{2} - 60 \beta_{3} ) q^{70} \) \( + ( 12 - 60 \beta_{3} ) q^{71} \) \( + ( -9 \beta_{1} - 72 \beta_{2} ) q^{72} \) \( + ( 108 \beta_{1} + 168 \beta_{2} ) q^{73} \) \( + ( -822 + 78 \beta_{3} ) q^{74} \) \( + ( -132 + 54 \beta_{1} + 37 \beta_{2} - 24 \beta_{3} ) q^{75} \) \( + ( -616 + 8 \beta_{3} ) q^{76} \) \( + ( -108 \beta_{1} - 48 \beta_{2} ) q^{77} \) \( + ( 90 \beta_{1} - 48 \beta_{2} ) q^{78} \) \( + ( -240 + 100 \beta_{3} ) q^{79} \) \( + ( 118 - \beta_{1} - 8 \beta_{2} - 14 \beta_{3} ) q^{80} \) \( + 81 q^{81} \) \( + ( 222 \beta_{1} + 32 \beta_{2} ) q^{82} \) \( + ( -208 \beta_{1} - 60 \beta_{2} ) q^{83} \) \( + ( 522 - 18 \beta_{3} ) q^{84} \) \( + ( -126 - 168 \beta_{1} - 44 \beta_{2} - 2 \beta_{3} ) q^{85} \) \( + ( 1108 - 68 \beta_{3} ) q^{86} \) \( + ( -126 \beta_{1} + 222 \beta_{2} ) q^{87} \) \( + ( -174 \beta_{1} - 112 \beta_{2} ) q^{88} \) \( + ( -534 - 48 \beta_{3} ) q^{89} \) \( + ( -27 + 54 \beta_{1} + 72 \beta_{2} - 9 \beta_{3} ) q^{90} \) \( + ( 444 + 84 \beta_{3} ) q^{91} \) \( + ( 36 \beta_{1} - 64 \beta_{2} ) q^{92} \) \( + ( -36 \beta_{1} + 44 \beta_{2} ) q^{93} \) \( + ( 816 - 16 \beta_{3} ) q^{94} \) \( + ( 688 - 136 \beta_{1} + 192 \beta_{2} + 16 \beta_{3} ) q^{95} \) \( + ( -459 - 9 \beta_{3} ) q^{96} \) \( + ( 240 \beta_{1} + 8 \beta_{2} ) q^{97} \) \( + ( -19 \beta_{1} - 96 \beta_{2} ) q^{98} \) \( + ( 198 - 18 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 18q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 36q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 18q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 36q^{9} \) \(\mathstrut +\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 84q^{11} \) \(\mathstrut +\mathstrut 228q^{14} \) \(\mathstrut +\mathstrut 54q^{15} \) \(\mathstrut +\mathstrut 50q^{16} \) \(\mathstrut -\mathstrut 112q^{19} \) \(\mathstrut -\mathstrut 396q^{20} \) \(\mathstrut +\mathstrut 36q^{21} \) \(\mathstrut -\mathstrut 306q^{24} \) \(\mathstrut +\mathstrut 256q^{25} \) \(\mathstrut -\mathstrut 12q^{26} \) \(\mathstrut +\mathstrut 636q^{29} \) \(\mathstrut +\mathstrut 396q^{30} \) \(\mathstrut +\mathstrut 104q^{31} \) \(\mathstrut -\mathstrut 716q^{34} \) \(\mathstrut -\mathstrut 300q^{35} \) \(\mathstrut +\mathstrut 162q^{36} \) \(\mathstrut -\mathstrut 468q^{39} \) \(\mathstrut +\mathstrut 418q^{40} \) \(\mathstrut -\mathstrut 816q^{41} \) \(\mathstrut +\mathstrut 1116q^{44} \) \(\mathstrut -\mathstrut 54q^{45} \) \(\mathstrut +\mathstrut 184q^{46} \) \(\mathstrut -\mathstrut 140q^{49} \) \(\mathstrut -\mathstrut 696q^{50} \) \(\mathstrut +\mathstrut 612q^{51} \) \(\mathstrut -\mathstrut 162q^{54} \) \(\mathstrut -\mathstrut 864q^{55} \) \(\mathstrut +\mathstrut 60q^{56} \) \(\mathstrut +\mathstrut 372q^{59} \) \(\mathstrut +\mathstrut 126q^{60} \) \(\mathstrut +\mathstrut 680q^{61} \) \(\mathstrut +\mathstrut 958q^{64} \) \(\mathstrut +\mathstrut 948q^{65} \) \(\mathstrut -\mathstrut 1116q^{66} \) \(\mathstrut +\mathstrut 288q^{69} \) \(\mathstrut +\mathstrut 1080q^{70} \) \(\mathstrut -\mathstrut 72q^{71} \) \(\mathstrut -\mathstrut 3132q^{74} \) \(\mathstrut -\mathstrut 576q^{75} \) \(\mathstrut -\mathstrut 2448q^{76} \) \(\mathstrut -\mathstrut 760q^{79} \) \(\mathstrut +\mathstrut 444q^{80} \) \(\mathstrut +\mathstrut 324q^{81} \) \(\mathstrut +\mathstrut 2052q^{84} \) \(\mathstrut -\mathstrut 508q^{85} \) \(\mathstrut +\mathstrut 4296q^{86} \) \(\mathstrut -\mathstrut 2232q^{89} \) \(\mathstrut -\mathstrut 126q^{90} \) \(\mathstrut +\mathstrut 1944q^{91} \) \(\mathstrut +\mathstrut 3232q^{94} \) \(\mathstrut +\mathstrut 2784q^{95} \) \(\mathstrut -\mathstrut 1854q^{96} \) \(\mathstrut +\mathstrut 756q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(21\) \(x^{2}\mathstrut +\mathstrut \) \(100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{3} + 33 \nu \)\()/10\)
\(\beta_{3}\)\(=\)\( 3 \nu^{2} + 32 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(32\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
3.70156i
2.70156i
2.70156i
3.70156i
4.70156i 3.00000i −14.1047 11.1047 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 52.2094i
4.2 1.70156i 3.00000i 5.10469 −8.10469 + 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 + 13.7906i
4.3 1.70156i 3.00000i 5.10469 −8.10469 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 13.7906i
4.4 4.70156i 3.00000i −14.1047 11.1047 + 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 + 52.2094i
\(n\): 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
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(15, [\chi])\).