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 $2$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-5}) \)
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
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.

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

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 \(S_{3}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( (T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 20 \) Copy content Toggle raw display
$13$ \( (T - 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 180 \) Copy content Toggle raw display
$29$ \( T^{2} + 980 \) Copy content Toggle raw display
$31$ \( (T + 18)^{2} \) Copy content Toggle raw display
$37$ \( (T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3920 \) Copy content Toggle raw display
$43$ \( (T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2420 \) Copy content Toggle raw display
$53$ \( T^{2} + 20 \) Copy content Toggle raw display
$59$ \( T^{2} + 20 \) Copy content Toggle raw display
$61$ \( (T - 82)^{2} \) Copy content Toggle raw display
$67$ \( (T - 24)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 15680 \) Copy content Toggle raw display
$73$ \( (T + 74)^{2} \) Copy content Toggle raw display
$79$ \( (T - 138)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8820 \) Copy content Toggle raw display
$89$ \( T^{2} + 11520 \) Copy content Toggle raw display
$97$ \( (T + 166)^{2} \) Copy content Toggle raw display
show more
show less