Properties

Label 240.3.c.b
Level $240$
Weight $3$
Character orbit 240.c
Self dual yes
Analytic conductor $6.540$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{3} + 5 q^{5} + 9 q^{9} + O(q^{10}) \) \( q + 3 q^{3} + 5 q^{5} + 9 q^{9} + 15 q^{15} - 14 q^{17} + 22 q^{19} - 34 q^{23} + 25 q^{25} + 27 q^{27} - 2 q^{31} + 45 q^{45} + 14 q^{47} + 49 q^{49} - 42 q^{51} - 86 q^{53} + 66 q^{57} - 118 q^{61} - 102 q^{69} + 75 q^{75} - 98 q^{79} + 81 q^{81} - 154 q^{83} - 70 q^{85} - 6 q^{93} + 110 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0
0 3.00000 0 5.00000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.c.b 1
3.b odd 2 1 240.3.c.a 1
4.b odd 2 1 15.3.d.b yes 1
5.b even 2 1 240.3.c.a 1
5.c odd 4 2 1200.3.l.l 2
8.b even 2 1 960.3.c.a 1
8.d odd 2 1 960.3.c.c 1
12.b even 2 1 15.3.d.a 1
15.d odd 2 1 CM 240.3.c.b 1
15.e even 4 2 1200.3.l.l 2
20.d odd 2 1 15.3.d.a 1
20.e even 4 2 75.3.c.d 2
24.f even 2 1 960.3.c.b 1
24.h odd 2 1 960.3.c.d 1
36.f odd 6 2 405.3.h.a 2
36.h even 6 2 405.3.h.b 2
40.e odd 2 1 960.3.c.b 1
40.f even 2 1 960.3.c.d 1
60.h even 2 1 15.3.d.b yes 1
60.l odd 4 2 75.3.c.d 2
120.i odd 2 1 960.3.c.a 1
120.m even 2 1 960.3.c.c 1
180.n even 6 2 405.3.h.a 2
180.p odd 6 2 405.3.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 12.b even 2 1
15.3.d.a 1 20.d odd 2 1
15.3.d.b yes 1 4.b odd 2 1
15.3.d.b yes 1 60.h even 2 1
75.3.c.d 2 20.e even 4 2
75.3.c.d 2 60.l odd 4 2
240.3.c.a 1 3.b odd 2 1
240.3.c.a 1 5.b even 2 1
240.3.c.b 1 1.a even 1 1 trivial
240.3.c.b 1 15.d odd 2 1 CM
405.3.h.a 2 36.f odd 6 2
405.3.h.a 2 180.n even 6 2
405.3.h.b 2 36.h even 6 2
405.3.h.b 2 180.p odd 6 2
960.3.c.a 1 8.b even 2 1
960.3.c.a 1 120.i odd 2 1
960.3.c.b 1 24.f even 2 1
960.3.c.b 1 40.e odd 2 1
960.3.c.c 1 8.d odd 2 1
960.3.c.c 1 120.m even 2 1
960.3.c.d 1 24.h odd 2 1
960.3.c.d 1 40.f even 2 1
1200.3.l.l 2 5.c odd 4 2
1200.3.l.l 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\):

\( T_{7} \)
\( T_{17} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -5 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( 14 + T \)
$19$ \( -22 + T \)
$23$ \( 34 + T \)
$29$ \( T \)
$31$ \( 2 + T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( -14 + T \)
$53$ \( 86 + T \)
$59$ \( T \)
$61$ \( 118 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( T \)
$79$ \( 98 + T \)
$83$ \( 154 + T \)
$89$ \( T \)
$97$ \( T \)
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