Properties

Label 24.3.e.a
Level 24
Weight 3
Character orbit 24.e
Analytic conductor 0.654
Analytic rank 0
Dimension 2
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 24.e (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta ) q^{3} \) \( -2 \beta q^{5} \) \( -6 q^{7} \) \( + ( -7 + 2 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta ) q^{3} \) \( -2 \beta q^{5} \) \( -6 q^{7} \) \( + ( -7 + 2 \beta ) q^{9} \) \( -2 \beta q^{11} \) \( + 10 q^{13} \) \( + ( 16 - 2 \beta ) q^{15} \) \( + 8 \beta q^{17} \) \( + 2 q^{19} \) \( + ( -6 - 6 \beta ) q^{21} \) \( + 4 \beta q^{23} \) \( -7 q^{25} \) \( + ( -23 - 5 \beta ) q^{27} \) \( -6 \beta q^{29} \) \( -22 q^{31} \) \( + ( 16 - 2 \beta ) q^{33} \) \( + 12 \beta q^{35} \) \( -6 q^{37} \) \( + ( 10 + 10 \beta ) q^{39} \) \( -12 \beta q^{41} \) \( + 82 q^{43} \) \( + ( 32 + 14 \beta ) q^{45} \) \( -24 \beta q^{47} \) \( -13 q^{49} \) \( + ( -64 + 8 \beta ) q^{51} \) \( + 22 \beta q^{53} \) \( -32 q^{55} \) \( + ( 2 + 2 \beta ) q^{57} \) \( -26 \beta q^{59} \) \( -86 q^{61} \) \( + ( 42 - 12 \beta ) q^{63} \) \( -20 \beta q^{65} \) \( + 2 q^{67} \) \( + ( -32 + 4 \beta ) q^{69} \) \( + 44 \beta q^{71} \) \( + 82 q^{73} \) \( + ( -7 - 7 \beta ) q^{75} \) \( + 12 \beta q^{77} \) \( + 10 q^{79} \) \( + ( 17 - 28 \beta ) q^{81} \) \( + 26 \beta q^{83} \) \( + 128 q^{85} \) \( + ( 48 - 6 \beta ) q^{87} \) \( + 12 \beta q^{89} \) \( -60 q^{91} \) \( + ( -22 - 22 \beta ) q^{93} \) \( -4 \beta q^{95} \) \( -94 q^{97} \) \( + ( 32 + 14 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut 20q^{13} \) \(\mathstrut +\mathstrut 32q^{15} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 12q^{21} \) \(\mathstrut -\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 46q^{27} \) \(\mathstrut -\mathstrut 44q^{31} \) \(\mathstrut +\mathstrut 32q^{33} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 20q^{39} \) \(\mathstrut +\mathstrut 164q^{43} \) \(\mathstrut +\mathstrut 64q^{45} \) \(\mathstrut -\mathstrut 26q^{49} \) \(\mathstrut -\mathstrut 128q^{51} \) \(\mathstrut -\mathstrut 64q^{55} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 172q^{61} \) \(\mathstrut +\mathstrut 84q^{63} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut -\mathstrut 64q^{69} \) \(\mathstrut +\mathstrut 164q^{73} \) \(\mathstrut -\mathstrut 14q^{75} \) \(\mathstrut +\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 34q^{81} \) \(\mathstrut +\mathstrut 256q^{85} \) \(\mathstrut +\mathstrut 96q^{87} \) \(\mathstrut -\mathstrut 120q^{91} \) \(\mathstrut -\mathstrut 44q^{93} \) \(\mathstrut -\mathstrut 188q^{97} \) \(\mathstrut +\mathstrut 64q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
0 1.00000 2.82843i 0 5.65685i 0 −6.00000 0 −7.00000 5.65685i 0
17.2 0 1.00000 + 2.82843i 0 5.65685i 0 −6.00000 0 −7.00000 + 5.65685i 0
\(n\): 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 \(S_{3}^{\mathrm{new}}(24, [\chi])\).