Properties

Label 24.4.f.a
Level 24
Weight 4
Character orbit 24.f
Analytic conductor 1.416
Analytic rank 0
Dimension 2
CM disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 24.f (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.41604584014\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + 2 \beta q^{2} \) \( + ( 5 + \beta ) q^{3} \) \( -8 q^{4} \) \( + ( -4 + 10 \beta ) q^{6} \) \( -16 \beta q^{8} \) \( + ( 23 + 10 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + 2 \beta q^{2} \) \( + ( 5 + \beta ) q^{3} \) \( -8 q^{4} \) \( + ( -4 + 10 \beta ) q^{6} \) \( -16 \beta q^{8} \) \( + ( 23 + 10 \beta ) q^{9} \) \( -50 \beta q^{11} \) \( + ( -40 - 8 \beta ) q^{12} \) \( + 64 q^{16} \) \( + 76 \beta q^{17} \) \( + ( -40 + 46 \beta ) q^{18} \) \( -106 q^{19} \) \( + 200 q^{22} \) \( + ( 32 - 80 \beta ) q^{24} \) \( -125 q^{25} \) \( + ( 95 + 73 \beta ) q^{27} \) \( + 128 \beta q^{32} \) \( + ( 100 - 250 \beta ) q^{33} \) \( -304 q^{34} \) \( + ( -184 - 80 \beta ) q^{36} \) \( -212 \beta q^{38} \) \( + 40 \beta q^{41} \) \( + 290 q^{43} \) \( + 400 \beta q^{44} \) \( + ( 320 + 64 \beta ) q^{48} \) \( + 343 q^{49} \) \( -250 \beta q^{50} \) \( + ( -152 + 380 \beta ) q^{51} \) \( + ( -292 + 190 \beta ) q^{54} \) \( + ( -530 - 106 \beta ) q^{57} \) \( -230 \beta q^{59} \) \( -512 q^{64} \) \( + ( 1000 + 200 \beta ) q^{66} \) \( -70 q^{67} \) \( -608 \beta q^{68} \) \( + ( 320 - 368 \beta ) q^{72} \) \( -430 q^{73} \) \( + ( -625 - 125 \beta ) q^{75} \) \( + 848 q^{76} \) \( + ( 329 + 460 \beta ) q^{81} \) \( -160 q^{82} \) \( -482 \beta q^{83} \) \( + 580 \beta q^{86} \) \( -1600 q^{88} \) \( + 940 \beta q^{89} \) \( + ( -256 + 640 \beta ) q^{96} \) \( + 1910 q^{97} \) \( + 686 \beta q^{98} \) \( + ( 1000 - 1150 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut -\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 46q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut -\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 46q^{9} \) \(\mathstrut -\mathstrut 80q^{12} \) \(\mathstrut +\mathstrut 128q^{16} \) \(\mathstrut -\mathstrut 80q^{18} \) \(\mathstrut -\mathstrut 212q^{19} \) \(\mathstrut +\mathstrut 400q^{22} \) \(\mathstrut +\mathstrut 64q^{24} \) \(\mathstrut -\mathstrut 250q^{25} \) \(\mathstrut +\mathstrut 190q^{27} \) \(\mathstrut +\mathstrut 200q^{33} \) \(\mathstrut -\mathstrut 608q^{34} \) \(\mathstrut -\mathstrut 368q^{36} \) \(\mathstrut +\mathstrut 580q^{43} \) \(\mathstrut +\mathstrut 640q^{48} \) \(\mathstrut +\mathstrut 686q^{49} \) \(\mathstrut -\mathstrut 304q^{51} \) \(\mathstrut -\mathstrut 584q^{54} \) \(\mathstrut -\mathstrut 1060q^{57} \) \(\mathstrut -\mathstrut 1024q^{64} \) \(\mathstrut +\mathstrut 2000q^{66} \) \(\mathstrut -\mathstrut 140q^{67} \) \(\mathstrut +\mathstrut 640q^{72} \) \(\mathstrut -\mathstrut 860q^{73} \) \(\mathstrut -\mathstrut 1250q^{75} \) \(\mathstrut +\mathstrut 1696q^{76} \) \(\mathstrut +\mathstrut 658q^{81} \) \(\mathstrut -\mathstrut 320q^{82} \) \(\mathstrut -\mathstrut 3200q^{88} \) \(\mathstrut -\mathstrut 512q^{96} \) \(\mathstrut +\mathstrut 3820q^{97} \) \(\mathstrut +\mathstrut 2000q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
1.41421i
1.41421i
2.82843i 5.00000 1.41421i −8.00000 0 −4.00000 14.1421i 0 22.6274i 23.0000 14.1421i 0
11.2 2.82843i 5.00000 + 1.41421i −8.00000 0 −4.00000 + 14.1421i 0 22.6274i 23.0000 + 14.1421i 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
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
3.b Odd 1 yes
24.f Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5} \) acting on \(S_{4}^{\mathrm{new}}(24, [\chi])\).