Properties

Label 15.3.f.a
Level 15
Weight 3
Character orbit 15.f
Analytic conductor 0.409
Analytic rank 0
Dimension 4
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.40872039654\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{2} \) \( + \beta_{3} q^{3} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} \) \( + ( -3 + \beta_{1} - \beta_{3} ) q^{6} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{7} \) \( + ( 3 - 3 \beta_{2} + \beta_{3} ) q^{8} \) \( -3 \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{2} \) \( + \beta_{3} q^{3} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} \) \( + ( -3 + \beta_{1} - \beta_{3} ) q^{6} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{7} \) \( + ( 3 - 3 \beta_{2} + \beta_{3} ) q^{8} \) \( -3 \beta_{2} q^{9} \) \( + ( 1 + 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{10} \) \( + ( 4 + 3 \beta_{1} - 3 \beta_{3} ) q^{11} \) \( + ( 6 - \beta_{1} + 6 \beta_{2} ) q^{12} \) \( + ( -8 + 8 \beta_{2} - 2 \beta_{3} ) q^{13} \) \( + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{14} \) \( + ( 6 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{15} \) \( + ( -5 - 4 \beta_{1} + 4 \beta_{3} ) q^{16} \) \( + ( -10 - 6 \beta_{1} - 10 \beta_{2} ) q^{17} \) \( + ( -3 + 3 \beta_{2} - 3 \beta_{3} ) q^{18} \) \( + ( 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{19} \) \( + ( -9 + 7 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{20} \) \( + ( -6 - \beta_{1} + \beta_{3} ) q^{21} \) \( + ( 5 - 2 \beta_{1} + 5 \beta_{2} ) q^{22} \) \( + ( 14 - 14 \beta_{2} + 2 \beta_{3} ) q^{23} \) \( + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{24} \) \( + ( 4 - 2 \beta_{1} + 3 \beta_{2} - 14 \beta_{3} ) q^{25} \) \( + ( 22 - 10 \beta_{1} + 10 \beta_{3} ) q^{26} \) \( + 3 \beta_{1} q^{27} \) \( + ( 11 - 11 \beta_{2} - 2 \beta_{3} ) q^{28} \) \( + ( 7 \beta_{1} + 18 \beta_{2} + 7 \beta_{3} ) q^{29} \) \( + ( -6 + 8 \beta_{1} - 12 \beta_{2} + \beta_{3} ) q^{30} \) \( + ( -4 + 6 \beta_{1} - 6 \beta_{3} ) q^{31} \) \( + ( -19 + 7 \beta_{1} - 19 \beta_{2} ) q^{32} \) \( + ( -9 + 9 \beta_{2} + 4 \beta_{3} ) q^{33} \) \( + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{34} \) \( + ( -10 - 5 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} ) q^{35} \) \( + ( 3 - 6 \beta_{1} + 6 \beta_{3} ) q^{36} \) \( + ( 16 - 18 \beta_{1} + 16 \beta_{2} ) q^{37} \) \( + ( -24 + 24 \beta_{2} - 18 \beta_{3} ) q^{38} \) \( + ( -8 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{39} \) \( + ( 12 - 6 \beta_{1} + 9 \beta_{2} + 8 \beta_{3} ) q^{40} \) \( + ( -14 + 6 \beta_{1} - 6 \beta_{3} ) q^{41} \) \( + ( 3 - 4 \beta_{1} + 3 \beta_{2} ) q^{42} \) \( + ( -2 + 2 \beta_{2} + 20 \beta_{3} ) q^{43} \) \( + ( -5 \beta_{1} - 32 \beta_{2} - 5 \beta_{3} ) q^{44} \) \( + ( 9 + 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{45} \) \( + ( -34 + 16 \beta_{1} - 16 \beta_{3} ) q^{46} \) \( + ( 32 + 10 \beta_{1} + 32 \beta_{2} ) q^{47} \) \( + ( 12 - 12 \beta_{2} - 5 \beta_{3} ) q^{48} \) \( + ( 4 \beta_{1} - 35 \beta_{2} + 4 \beta_{3} ) q^{49} \) \( + ( 41 - 8 \beta_{1} - 13 \beta_{2} + 19 \beta_{3} ) q^{50} \) \( + ( 18 + 10 \beta_{1} - 10 \beta_{3} ) q^{51} \) \( + ( -20 + 34 \beta_{1} - 20 \beta_{2} ) q^{52} \) \( + ( 14 - 14 \beta_{2} + 12 \beta_{3} ) q^{53} \) \( + ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{54} \) \( + ( -31 - 2 \beta_{1} + 3 \beta_{2} + 16 \beta_{3} ) q^{55} \) \( + ( 5 \beta_{1} - 5 \beta_{3} ) q^{56} \) \( + ( -18 + 6 \beta_{1} - 18 \beta_{2} ) q^{57} \) \( + ( -3 + 3 \beta_{2} + 4 \beta_{3} ) q^{58} \) \( + ( -31 \beta_{1} + 36 \beta_{2} - 31 \beta_{3} ) q^{59} \) \( + ( -21 - 17 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{60} \) \( + ( 50 - 18 \beta_{1} + 18 \beta_{3} ) q^{61} \) \( + ( 22 - 16 \beta_{1} + 22 \beta_{2} ) q^{62} \) \( + ( 3 - 3 \beta_{2} - 6 \beta_{3} ) q^{63} \) \( + ( -10 \beta_{1} + 79 \beta_{2} - 10 \beta_{3} ) q^{64} \) \( + ( -28 + 14 \beta_{1} - 26 \beta_{2} - 22 \beta_{3} ) q^{65} \) \( + ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{66} \) \( + ( -50 + 4 \beta_{1} - 50 \beta_{2} ) q^{67} \) \( + ( -26 + 26 \beta_{2} + 34 \beta_{3} ) q^{68} \) \( + ( 14 \beta_{1} - 6 \beta_{2} + 14 \beta_{3} ) q^{69} \) \( + ( -15 + 5 \beta_{2} - 10 \beta_{3} ) q^{70} \) \( -68 q^{71} \) \( + ( -9 + 3 \beta_{1} - 9 \beta_{2} ) q^{72} \) \( + ( 19 - 19 \beta_{2} - 48 \beta_{3} ) q^{73} \) \( + ( 34 \beta_{1} - 86 \beta_{2} + 34 \beta_{3} ) q^{74} \) \( + ( 6 - 3 \beta_{1} + 42 \beta_{2} + 4 \beta_{3} ) q^{75} \) \( + ( 78 - 18 \beta_{1} + 18 \beta_{3} ) q^{76} \) \( + ( 22 + 14 \beta_{1} + 22 \beta_{2} ) q^{77} \) \( + ( 30 - 30 \beta_{2} + 22 \beta_{3} ) q^{78} \) \( + ( -10 \beta_{1} - 10 \beta_{3} ) q^{79} \) \( + ( 41 + 2 \beta_{1} - 3 \beta_{2} - 21 \beta_{3} ) q^{80} \) \( -9 q^{81} \) \( + ( 32 - 26 \beta_{1} + 32 \beta_{2} ) q^{82} \) \( + ( -4 + 4 \beta_{2} - 14 \beta_{3} ) q^{83} \) \( + ( 11 \beta_{1} + 6 \beta_{2} + 11 \beta_{3} ) q^{84} \) \( + ( 58 + 36 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{85} \) \( + ( -56 + 18 \beta_{1} - 18 \beta_{3} ) q^{86} \) \( + ( -21 - 18 \beta_{1} - 21 \beta_{2} ) q^{87} \) \( + ( 3 - 3 \beta_{2} - 14 \beta_{3} ) q^{88} \) \( + ( 36 \beta_{1} - 6 \beta_{2} + 36 \beta_{3} ) q^{89} \) \( + ( -24 + 12 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{90} \) \( + ( -4 - 14 \beta_{1} + 14 \beta_{3} ) q^{91} \) \( + ( 26 - 58 \beta_{1} + 26 \beta_{2} ) q^{92} \) \( + ( -18 + 18 \beta_{2} - 4 \beta_{3} ) q^{93} \) \( + ( 22 \beta_{1} - 34 \beta_{2} + 22 \beta_{3} ) q^{94} \) \( + ( 36 - 18 \beta_{1} - 48 \beta_{2} + 24 \beta_{3} ) q^{95} \) \( + ( -21 + 19 \beta_{1} - 19 \beta_{3} ) q^{96} \) \( + ( -5 - 16 \beta_{1} - 5 \beta_{2} ) q^{97} \) \( + ( -47 + 47 \beta_{2} - 43 \beta_{3} ) q^{98} \) \( + ( -9 \beta_{1} - 12 \beta_{2} - 9 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 16q^{11} \) \(\mathstrut +\mathstrut 24q^{12} \) \(\mathstrut -\mathstrut 32q^{13} \) \(\mathstrut +\mathstrut 24q^{15} \) \(\mathstrut -\mathstrut 20q^{16} \) \(\mathstrut -\mathstrut 40q^{17} \) \(\mathstrut -\mathstrut 12q^{18} \) \(\mathstrut -\mathstrut 36q^{20} \) \(\mathstrut -\mathstrut 24q^{21} \) \(\mathstrut +\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 56q^{23} \) \(\mathstrut +\mathstrut 16q^{25} \) \(\mathstrut +\mathstrut 88q^{26} \) \(\mathstrut +\mathstrut 44q^{28} \) \(\mathstrut -\mathstrut 24q^{30} \) \(\mathstrut -\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 76q^{32} \) \(\mathstrut -\mathstrut 36q^{33} \) \(\mathstrut -\mathstrut 40q^{35} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 64q^{37} \) \(\mathstrut -\mathstrut 96q^{38} \) \(\mathstrut +\mathstrut 48q^{40} \) \(\mathstrut -\mathstrut 56q^{41} \) \(\mathstrut +\mathstrut 12q^{42} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 36q^{45} \) \(\mathstrut -\mathstrut 136q^{46} \) \(\mathstrut +\mathstrut 128q^{47} \) \(\mathstrut +\mathstrut 48q^{48} \) \(\mathstrut +\mathstrut 164q^{50} \) \(\mathstrut +\mathstrut 72q^{51} \) \(\mathstrut -\mathstrut 80q^{52} \) \(\mathstrut +\mathstrut 56q^{53} \) \(\mathstrut -\mathstrut 124q^{55} \) \(\mathstrut -\mathstrut 72q^{57} \) \(\mathstrut -\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 84q^{60} \) \(\mathstrut +\mathstrut 200q^{61} \) \(\mathstrut +\mathstrut 88q^{62} \) \(\mathstrut +\mathstrut 12q^{63} \) \(\mathstrut -\mathstrut 112q^{65} \) \(\mathstrut +\mathstrut 24q^{66} \) \(\mathstrut -\mathstrut 200q^{67} \) \(\mathstrut -\mathstrut 104q^{68} \) \(\mathstrut -\mathstrut 60q^{70} \) \(\mathstrut -\mathstrut 272q^{71} \) \(\mathstrut -\mathstrut 36q^{72} \) \(\mathstrut +\mathstrut 76q^{73} \) \(\mathstrut +\mathstrut 24q^{75} \) \(\mathstrut +\mathstrut 312q^{76} \) \(\mathstrut +\mathstrut 88q^{77} \) \(\mathstrut +\mathstrut 120q^{78} \) \(\mathstrut +\mathstrut 164q^{80} \) \(\mathstrut -\mathstrut 36q^{81} \) \(\mathstrut +\mathstrut 128q^{82} \) \(\mathstrut -\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 232q^{85} \) \(\mathstrut -\mathstrut 224q^{86} \) \(\mathstrut -\mathstrut 84q^{87} \) \(\mathstrut +\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 96q^{90} \) \(\mathstrut -\mathstrut 16q^{91} \) \(\mathstrut +\mathstrut 104q^{92} \) \(\mathstrut -\mathstrut 72q^{93} \) \(\mathstrut +\mathstrut 144q^{95} \) \(\mathstrut -\mathstrut 84q^{96} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut -\mathstrut 188q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{3}\)

Character Values

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

\(n\) \(7\) \(11\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−2.22474 2.22474i 1.22474 1.22474i 5.89898i 2.67423 + 4.22474i −5.44949 −1.44949 1.44949i 4.22474 4.22474i 3.00000i 3.44949 15.3485i
7.2 0.224745 + 0.224745i −1.22474 + 1.22474i 3.89898i −4.67423 + 1.77526i −0.550510 3.44949 + 3.44949i 1.77526 1.77526i 3.00000i −1.44949 0.651531i
13.1 −2.22474 + 2.22474i 1.22474 + 1.22474i 5.89898i 2.67423 4.22474i −5.44949 −1.44949 + 1.44949i 4.22474 + 4.22474i 3.00000i 3.44949 + 15.3485i
13.2 0.224745 0.224745i −1.22474 1.22474i 3.89898i −4.67423 1.77526i −0.550510 3.44949 3.44949i 1.77526 + 1.77526i 3.00000i −1.44949 + 0.651531i
\(n\): 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.c Odd 1 yes

Hecke kernels

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