Properties

Label 24.3.h.c
Level $24$
Weight $3$
Character orbit 24.h
Analytic conductor $0.654$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,3,Mod(5,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 24.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + (\beta_{3} - 3) q^{4} + 4 \beta_{2} q^{5} + (\beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{6} + 4 q^{7} + (4 \beta_{2} - 2 \beta_1) q^{8} + ( - 2 \beta_{2} - 4 \beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + (\beta_{3} - 3) q^{4} + 4 \beta_{2} q^{5} + (\beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{6} + 4 q^{7} + (4 \beta_{2} - 2 \beta_1) q^{8} + ( - 2 \beta_{2} - 4 \beta_1 - 5) q^{9} + ( - 4 \beta_{3} - 4) q^{10} - 6 \beta_{2} q^{11} + (3 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 7) q^{12} + 4 \beta_{3} q^{13} + 4 \beta_1 q^{14} + (4 \beta_{2} + 8 \beta_1 - 8) q^{15} + ( - 6 \beta_{3} + 2) q^{16} + (4 \beta_{2} + 8 \beta_1) q^{17} + ( - 2 \beta_{3} - 5 \beta_1 + 14) q^{18} + 2 \beta_{3} q^{19} + ( - 16 \beta_{2} - 8 \beta_1) q^{20} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{21} + (6 \beta_{3} + 6) q^{22} + ( - 8 \beta_{2} - 16 \beta_1) q^{23} + ( - 2 \beta_{3} + 12 \beta_{2} + 10 \beta_1 - 10) q^{24} + 7 q^{25} + (16 \beta_{2} + 4 \beta_1) q^{26} + (\beta_{3} + 19 \beta_{2}) q^{27} + (4 \beta_{3} - 12) q^{28} - 12 \beta_{2} q^{29} + (4 \beta_{3} - 8 \beta_1 - 28) q^{30} - 4 q^{31} + ( - 24 \beta_{2} - 4 \beta_1) q^{32} + ( - 6 \beta_{2} - 12 \beta_1 + 12) q^{33} + (4 \beta_{3} - 28) q^{34} + 16 \beta_{2} q^{35} + ( - 5 \beta_{3} - 8 \beta_{2} + 12 \beta_1 + 15) q^{36} - 20 \beta_{3} q^{37} + (8 \beta_{2} + 2 \beta_1) q^{38} + (4 \beta_{2} + 8 \beta_1 + 28) q^{39} + (8 \beta_{3} + 40) q^{40} + (8 \beta_{2} + 16 \beta_1) q^{41} + (4 \beta_{3} - 16 \beta_{2} - 4 \beta_1 + 4) q^{42} - 2 \beta_{3} q^{43} + (24 \beta_{2} + 12 \beta_1) q^{44} + (16 \beta_{3} - 20 \beta_{2}) q^{45} + ( - 8 \beta_{3} + 56) q^{46} + ( - 2 \beta_{3} - 8 \beta_{2} - 12 \beta_1 - 42) q^{48} - 33 q^{49} + 7 \beta_1 q^{50} + (8 \beta_{3} - 28 \beta_{2}) q^{51} + ( - 12 \beta_{3} - 28) q^{52} + 36 \beta_{2} q^{53} + ( - 19 \beta_{3} + 4 \beta_{2} + \beta_1 - 19) q^{54} - 48 q^{55} + (16 \beta_{2} - 8 \beta_1) q^{56} + (2 \beta_{2} + 4 \beta_1 + 14) q^{57} + (12 \beta_{3} + 12) q^{58} - 34 \beta_{2} q^{59} + ( - 8 \beta_{3} + 16 \beta_{2} - 24 \beta_1 + 24) q^{60} + 36 \beta_{3} q^{61} - 4 \beta_1 q^{62} + ( - 8 \beta_{2} - 16 \beta_1 - 20) q^{63} + (20 \beta_{3} + 36) q^{64} + ( - 16 \beta_{2} - 32 \beta_1) q^{65} + ( - 6 \beta_{3} + 12 \beta_1 + 42) q^{66} - 18 \beta_{3} q^{67} + (16 \beta_{2} - 24 \beta_1) q^{68} + ( - 16 \beta_{3} + 56 \beta_{2}) q^{69} + ( - 16 \beta_{3} - 16) q^{70} + (24 \beta_{2} + 48 \beta_1) q^{71} + (20 \beta_{3} - 20 \beta_{2} + 10 \beta_1 - 28) q^{72} - 6 q^{73} + ( - 80 \beta_{2} - 20 \beta_1) q^{74} + ( - 7 \beta_{3} - 7 \beta_{2}) q^{75} + ( - 6 \beta_{3} - 14) q^{76} - 24 \beta_{2} q^{77} + (4 \beta_{3} + 28 \beta_1 - 28) q^{78} + 124 q^{79} + (32 \beta_{2} + 48 \beta_1) q^{80} + (20 \beta_{2} + 40 \beta_1 - 31) q^{81} + (8 \beta_{3} - 56) q^{82} + 2 \beta_{2} q^{83} + (12 \beta_{3} + 16 \beta_{2} + 8 \beta_1 + 28) q^{84} - 32 \beta_{3} q^{85} + ( - 8 \beta_{2} - 2 \beta_1) q^{86} + ( - 12 \beta_{2} - 24 \beta_1 + 24) q^{87} + ( - 12 \beta_{3} - 60) q^{88} + ( - 28 \beta_{2} - 56 \beta_1) q^{89} + (20 \beta_{3} + 64 \beta_{2} + 16 \beta_1 + 20) q^{90} + 16 \beta_{3} q^{91} + ( - 32 \beta_{2} + 48 \beta_1) q^{92} + (4 \beta_{3} + 4 \beta_{2}) q^{93} + ( - 8 \beta_{2} - 16 \beta_1) q^{95} + ( - 4 \beta_{3} - 8 \beta_{2} - 44 \beta_1 + 44) q^{96} + 118 q^{97} - 33 \beta_1 q^{98} + ( - 24 \beta_{3} + 30 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} + 4 q^{6} + 16 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} + 4 q^{6} + 16 q^{7} - 20 q^{9} - 16 q^{10} + 28 q^{12} - 32 q^{15} + 8 q^{16} + 56 q^{18} + 24 q^{22} - 40 q^{24} + 28 q^{25} - 48 q^{28} - 112 q^{30} - 16 q^{31} + 48 q^{33} - 112 q^{34} + 60 q^{36} + 112 q^{39} + 160 q^{40} + 16 q^{42} + 224 q^{46} - 168 q^{48} - 132 q^{49} - 112 q^{52} - 76 q^{54} - 192 q^{55} + 56 q^{57} + 48 q^{58} + 96 q^{60} - 80 q^{63} + 144 q^{64} + 168 q^{66} - 64 q^{70} - 112 q^{72} - 24 q^{73} - 56 q^{76} - 112 q^{78} + 496 q^{79} - 124 q^{81} - 224 q^{82} + 112 q^{84} + 96 q^{87} - 240 q^{88} + 80 q^{90} + 176 q^{96} + 472 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 2\beta_1 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.707107 1.87083i
−0.707107 + 1.87083i
0.707107 1.87083i
0.707107 + 1.87083i
−0.707107 1.87083i −1.41421 2.64575i −3.00000 + 2.64575i 5.65685 −3.94975 + 4.51658i 4.00000 7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 10.5830i
5.2 −0.707107 + 1.87083i −1.41421 + 2.64575i −3.00000 2.64575i 5.65685 −3.94975 4.51658i 4.00000 7.07107 3.74166i −5.00000 7.48331i −4.00000 + 10.5830i
5.3 0.707107 1.87083i 1.41421 + 2.64575i −3.00000 2.64575i −5.65685 5.94975 0.774923i 4.00000 −7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 + 10.5830i
5.4 0.707107 + 1.87083i 1.41421 2.64575i −3.00000 + 2.64575i −5.65685 5.94975 + 0.774923i 4.00000 −7.07107 3.74166i −5.00000 7.48331i −4.00000 10.5830i
\(n\): 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
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.3.h.c 4
3.b odd 2 1 inner 24.3.h.c 4
4.b odd 2 1 96.3.h.c 4
8.b even 2 1 inner 24.3.h.c 4
8.d odd 2 1 96.3.h.c 4
12.b even 2 1 96.3.h.c 4
16.e even 4 2 768.3.e.i 4
16.f odd 4 2 768.3.e.l 4
24.f even 2 1 96.3.h.c 4
24.h odd 2 1 inner 24.3.h.c 4
48.i odd 4 2 768.3.e.i 4
48.k even 4 2 768.3.e.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.c 4 1.a even 1 1 trivial
24.3.h.c 4 3.b odd 2 1 inner
24.3.h.c 4 8.b even 2 1 inner
24.3.h.c 4 24.h odd 2 1 inner
96.3.h.c 4 4.b odd 2 1
96.3.h.c 4 8.d odd 2 1
96.3.h.c 4 12.b even 2 1
96.3.h.c 4 24.f even 2 1
768.3.e.i 4 16.e even 4 2
768.3.e.i 4 48.i odd 4 2
768.3.e.l 4 16.f odd 4 2
768.3.e.l 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 32 \) acting on \(S_{3}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 10T^{2} + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$7$ \( (T - 4)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 224)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 896)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2592)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2312)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 9072)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2268)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8064)^{2} \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( (T - 124)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10976)^{2} \) Copy content Toggle raw display
$97$ \( (T - 118)^{4} \) Copy content Toggle raw display
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