Properties

Label 24.2.d.a
Level 24
Weight 2
Character orbit 24.d
Analytic conductor 0.192
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.191640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 + 1.00000i −2.00000 2.00000 2.00000i −1.00000 2.00000 2.00000i
13.2 −1.00000 + 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 1.00000i −2.00000 2.00000 + 2.00000i −1.00000 2.00000 + 2.00000i
\(n\): 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
8.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(24, [\chi])\).