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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.408720396540\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.3.c.a 2
3.b odd 2 1 inner 15.3.c.a 2
4.b odd 2 1 240.3.l.b 2
5.b even 2 1 75.3.c.e 2
5.c odd 4 2 75.3.d.b 4
8.b even 2 1 960.3.l.c 2
8.d odd 2 1 960.3.l.b 2
9.c even 3 2 405.3.i.b 4
9.d odd 6 2 405.3.i.b 4
12.b even 2 1 240.3.l.b 2
15.d odd 2 1 75.3.c.e 2
15.e even 4 2 75.3.d.b 4
20.d odd 2 1 1200.3.l.g 2
20.e even 4 2 1200.3.c.f 4
24.f even 2 1 960.3.l.b 2
24.h odd 2 1 960.3.l.c 2
60.h even 2 1 1200.3.l.g 2
60.l odd 4 2 1200.3.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 1.a even 1 1 trivial
15.3.c.a 2 3.b odd 2 1 inner
75.3.c.e 2 5.b even 2 1
75.3.c.e 2 15.d odd 2 1
75.3.d.b 4 5.c odd 4 2
75.3.d.b 4 15.e even 4 2
240.3.l.b 2 4.b odd 2 1
240.3.l.b 2 12.b even 2 1
405.3.i.b 4 9.c even 3 2
405.3.i.b 4 9.d odd 6 2
960.3.l.b 2 8.d odd 2 1
960.3.l.b 2 24.f even 2 1
960.3.l.c 2 8.b even 2 1
960.3.l.c 2 24.h odd 2 1
1200.3.c.f 4 20.e even 4 2
1200.3.c.f 4 60.l odd 4 2
1200.3.l.g 2 20.d odd 2 1
1200.3.l.g 2 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(15, [\chi])\).