Properties

Label 15.3.c.a
Level 15
Weight 3
Character orbit 15.c
Analytic conductor 0.409
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.40872039654\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + ( -2 - \beta ) q^{3} \) \(- q^{4}\) \( -\beta q^{5} \) \( + ( 5 - 2 \beta ) q^{6} \) \( -6 q^{7} \) \( + 3 \beta q^{8} \) \( + ( -1 + 4 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + ( -2 - \beta ) q^{3} \) \(- q^{4}\) \( -\beta q^{5} \) \( + ( 5 - 2 \beta ) q^{6} \) \( -6 q^{7} \) \( + 3 \beta q^{8} \) \( + ( -1 + 4 \beta ) q^{9} \) \( + 5 q^{10} \) \( -2 \beta q^{11} \) \( + ( 2 + \beta ) q^{12} \) \( + 16 q^{13} \) \( -6 \beta q^{14} \) \( + ( -5 + 2 \beta ) q^{15} \) \( -19 q^{16} \) \( -2 \beta q^{17} \) \( + ( -20 - \beta ) q^{18} \) \( -2 q^{19} \) \( + \beta q^{20} \) \( + ( 12 + 6 \beta ) q^{21} \) \( + 10 q^{22} \) \( -6 \beta q^{23} \) \( + ( 15 - 6 \beta ) q^{24} \) \( -5 q^{25} \) \( + 16 \beta q^{26} \) \( + ( 22 - 7 \beta ) q^{27} \) \( + 6 q^{28} \) \( + 14 \beta q^{29} \) \( + ( -10 - 5 \beta ) q^{30} \) \( -18 q^{31} \) \( -7 \beta q^{32} \) \( + ( -10 + 4 \beta ) q^{33} \) \( + 10 q^{34} \) \( + 6 \beta q^{35} \) \( + ( 1 - 4 \beta ) q^{36} \) \( -16 q^{37} \) \( -2 \beta q^{38} \) \( + ( -32 - 16 \beta ) q^{39} \) \( + 15 q^{40} \) \( -28 \beta q^{41} \) \( + ( -30 + 12 \beta ) q^{42} \) \( + 16 q^{43} \) \( + 2 \beta q^{44} \) \( + ( 20 + \beta ) q^{45} \) \( + 30 q^{46} \) \( + 22 \beta q^{47} \) \( + ( 38 + 19 \beta ) q^{48} \) \( -13 q^{49} \) \( -5 \beta q^{50} \) \( + ( -10 + 4 \beta ) q^{51} \) \( -16 q^{52} \) \( -2 \beta q^{53} \) \( + ( 35 + 22 \beta ) q^{54} \) \( -10 q^{55} \) \( -18 \beta q^{56} \) \( + ( 4 + 2 \beta ) q^{57} \) \( -70 q^{58} \) \( -2 \beta q^{59} \) \( + ( 5 - 2 \beta ) q^{60} \) \( + 82 q^{61} \) \( -18 \beta q^{62} \) \( + ( 6 - 24 \beta ) q^{63} \) \( -41 q^{64} \) \( -16 \beta q^{65} \) \( + ( -20 - 10 \beta ) q^{66} \) \( + 24 q^{67} \) \( + 2 \beta q^{68} \) \( + ( -30 + 12 \beta ) q^{69} \) \( -30 q^{70} \) \( + 56 \beta q^{71} \) \( + ( -60 - 3 \beta ) q^{72} \) \( -74 q^{73} \) \( -16 \beta q^{74} \) \( + ( 10 + 5 \beta ) q^{75} \) \( + 2 q^{76} \) \( + 12 \beta q^{77} \) \( + ( 80 - 32 \beta ) q^{78} \) \( + 138 q^{79} \) \( + 19 \beta q^{80} \) \( + ( -79 - 8 \beta ) q^{81} \) \( + 140 q^{82} \) \( + 42 \beta q^{83} \) \( + ( -12 - 6 \beta ) q^{84} \) \( -10 q^{85} \) \( + 16 \beta q^{86} \) \( + ( 70 - 28 \beta ) q^{87} \) \( + 30 q^{88} \) \( -48 \beta q^{89} \) \( + ( -5 + 20 \beta ) q^{90} \) \( -96 q^{91} \) \( + 6 \beta q^{92} \) \( + ( 36 + 18 \beta ) q^{93} \) \( -110 q^{94} \) \( + 2 \beta q^{95} \) \( + ( -35 + 14 \beta ) q^{96} \) \( -166 q^{97} \) \( -13 \beta q^{98} \) \( + ( 40 + 2 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 10q^{10} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 32q^{13} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut -\mathstrut 38q^{16} \) \(\mathstrut -\mathstrut 40q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 24q^{21} \) \(\mathstrut +\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 30q^{24} \) \(\mathstrut -\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 44q^{27} \) \(\mathstrut +\mathstrut 12q^{28} \) \(\mathstrut -\mathstrut 20q^{30} \) \(\mathstrut -\mathstrut 36q^{31} \) \(\mathstrut -\mathstrut 20q^{33} \) \(\mathstrut +\mathstrut 20q^{34} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 32q^{37} \) \(\mathstrut -\mathstrut 64q^{39} \) \(\mathstrut +\mathstrut 30q^{40} \) \(\mathstrut -\mathstrut 60q^{42} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut +\mathstrut 40q^{45} \) \(\mathstrut +\mathstrut 60q^{46} \) \(\mathstrut +\mathstrut 76q^{48} \) \(\mathstrut -\mathstrut 26q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut -\mathstrut 32q^{52} \) \(\mathstrut +\mathstrut 70q^{54} \) \(\mathstrut -\mathstrut 20q^{55} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 140q^{58} \) \(\mathstrut +\mathstrut 10q^{60} \) \(\mathstrut +\mathstrut 164q^{61} \) \(\mathstrut +\mathstrut 12q^{63} \) \(\mathstrut -\mathstrut 82q^{64} \) \(\mathstrut -\mathstrut 40q^{66} \) \(\mathstrut +\mathstrut 48q^{67} \) \(\mathstrut -\mathstrut 60q^{69} \) \(\mathstrut -\mathstrut 60q^{70} \) \(\mathstrut -\mathstrut 120q^{72} \) \(\mathstrut -\mathstrut 148q^{73} \) \(\mathstrut +\mathstrut 20q^{75} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut +\mathstrut 160q^{78} \) \(\mathstrut +\mathstrut 276q^{79} \) \(\mathstrut -\mathstrut 158q^{81} \) \(\mathstrut +\mathstrut 280q^{82} \) \(\mathstrut -\mathstrut 24q^{84} \) \(\mathstrut -\mathstrut 20q^{85} \) \(\mathstrut +\mathstrut 140q^{87} \) \(\mathstrut +\mathstrut 60q^{88} \) \(\mathstrut -\mathstrut 10q^{90} \) \(\mathstrut -\mathstrut 192q^{91} \) \(\mathstrut +\mathstrut 72q^{93} \) \(\mathstrut -\mathstrut 220q^{94} \) \(\mathstrut -\mathstrut 70q^{96} \) \(\mathstrut -\mathstrut 332q^{97} \) \(\mathstrut +\mathstrut 80q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
2.23607i
2.23607i
2.23607i −2.00000 + 2.23607i −1.00000 2.23607i 5.00000 + 4.47214i −6.00000 6.70820i −1.00000 8.94427i 5.00000
11.2 2.23607i −2.00000 2.23607i −1.00000 2.23607i 5.00000 4.47214i −6.00000 6.70820i −1.00000 + 8.94427i 5.00000
\(n\): 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

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(15, [\chi])\).