Properties

Label 24.5.e.a
Level $24$
Weight $5$
Character orbit 24.e
Analytic conductor $2.481$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,5,Mod(17,24)]
 
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, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 24.e (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: \(2.48087911401\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{2} - 4 \beta_1 + 6) q^{7} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{2} - 4 \beta_1 + 6) q^{7} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 25) q^{9} + (4 \beta_{3} - 3 \beta_{2} + 6 \beta_1) q^{11} + (4 \beta_{2} + 16 \beta_1 - 62) q^{13} + ( - 12 \beta_{3} - 5 \beta_{2} + 6 \beta_1 - 136) q^{15} + (8 \beta_{3} - 4 \beta_{2} + 8 \beta_1) q^{17} + (\beta_{2} + 4 \beta_1 + 206) q^{19} + ( - 9 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 306) q^{21} + (16 \beta_{3} + 14 \beta_{2} - 28 \beta_1) q^{23} + ( - 20 \beta_{2} - 80 \beta_1 - 511) q^{25} + (12 \beta_{3} + 23 \beta_{2} - 23 \beta_1 - 385) q^{27} + (3 \beta_{3} + 11 \beta_{2} - 22 \beta_1) q^{29} + (11 \beta_{2} + 44 \beta_1 + 950) q^{31} + (15 \beta_{3} - 41 \beta_{2} - 18 \beta_1 + 632) q^{33} + ( - 60 \beta_{3} - 14 \beta_{2} + 28 \beta_1) q^{35} + (28 \beta_{2} + 112 \beta_1 - 702) q^{37} + (36 \beta_{3} - 12 \beta_{2} + 62 \beta_1 - 1186) q^{39} + ( - 66 \beta_{3} - 50 \beta_{2} + 100 \beta_1) q^{41} + ( - 7 \beta_{2} - 28 \beta_1 - 242) q^{43} + (69 \beta_{3} + 85 \beta_{2} + 110 \beta_1 + 688) q^{45} + ( - 24 \beta_{3} - 20 \beta_{2} + 40 \beta_1) q^{47} + ( - 12 \beta_{2} - 48 \beta_1 - 493) q^{49} + (12 \beta_{3} - 76 \beta_{2} - 24 \beta_1 + 928) q^{51} + ( - 59 \beta_{3} + 101 \beta_{2} - 202 \beta_1) q^{53} + ( - 52 \beta_{2} - 208 \beta_1 + 1168) q^{55} + (9 \beta_{3} - 3 \beta_{2} - 206 \beta_1 - 518) q^{57} + (112 \beta_{3} + 57 \beta_{2} - 114 \beta_1) q^{59} + (44 \beta_{2} + 176 \beta_1 + 2146) q^{61} + (81 \beta_{2} - 288 \beta_1 - 1098) q^{63} + (202 \beta_{3} + 18 \beta_{2} - 36 \beta_1) q^{65} + (5 \beta_{2} + 20 \beta_1 - 3778) q^{67} + ( - 174 \beta_{3} - 86 \beta_{2} + 84 \beta_1 - 1840) q^{69} + ( - 136 \beta_{3} - 102 \beta_{2} + 204 \beta_1) q^{71} + ( - 48 \beta_{2} - 192 \beta_1 + 1378) q^{73} + ( - 180 \beta_{3} + 60 \beta_{2} + 511 \beta_1 + 6751) q^{75} + (138 \beta_{3} - 182 \beta_{2} + 364 \beta_1) q^{77} + (139 \beta_{2} + 556 \beta_1 - 266) q^{79} + ( - 174 \beta_{3} - 50 \beta_{2} + 500 \beta_1 - 3647) q^{81} + (308 \beta_{3} + 47 \beta_{2} - 94 \beta_1) q^{83} + ( - 112 \beta_{2} - 448 \beta_1 + 704) q^{85} + ( - 108 \beta_{3} + 9 \beta_{2} + 66 \beta_1 - 1752) q^{87} + ( - 258 \beta_{3} - 30 \beta_{2} + 60 \beta_1) q^{89} + (86 \beta_{2} + 344 \beta_1 - 7860) q^{91} + (99 \beta_{3} - 33 \beta_{2} - 950 \beta_1 - 4382) q^{93} + (272 \beta_{3} + 226 \beta_{2} - 452 \beta_1) q^{95} + (100 \beta_{2} + 400 \beta_1 + 8114) q^{97} + (24 \beta_{3} - 143 \beta_{2} - 778 \beta_1 + 9136) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 24 q^{7} + 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 24 q^{7} + 100 q^{9} - 248 q^{13} - 544 q^{15} + 824 q^{19} + 1224 q^{21} - 2044 q^{25} - 1540 q^{27} + 3800 q^{31} + 2528 q^{33} - 2808 q^{37} - 4744 q^{39} - 968 q^{43} + 2752 q^{45} - 1972 q^{49} + 3712 q^{51} + 4672 q^{55} - 2072 q^{57} + 8584 q^{61} - 4392 q^{63} - 15112 q^{67} - 7360 q^{69} + 5512 q^{73} + 27004 q^{75} - 1064 q^{79} - 14588 q^{81} + 2816 q^{85} - 7008 q^{87} - 31440 q^{91} - 17528 q^{93} + 32456 q^{97} + 36544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} - 4\nu^{2} + 28\nu - 4 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{3} + 40\nu^{2} - 44 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\nu^{3} - 48\nu^{2} + 112\nu - 48 ) / 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{3} + 2\beta_{2} + 8\beta _1 + 24 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + 5\beta_{2} - 4\beta _1 + 36 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{3} + 4\beta_{2} - 20\beta _1 + 48 ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
2.30278 1.41421i
2.30278 + 1.41421i
−1.30278 + 1.41421i
−1.30278 1.41421i
0 −8.21110 3.68481i 0 44.7363i 0 −37.2666 0 53.8444 + 60.5126i 0
17.2 0 −8.21110 + 3.68481i 0 44.7363i 0 −37.2666 0 53.8444 60.5126i 0
17.3 0 6.21110 6.51323i 0 16.4520i 0 49.2666 0 −3.84441 80.9087i 0
17.4 0 6.21110 + 6.51323i 0 16.4520i 0 49.2666 0 −3.84441 + 80.9087i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.5.e.a 4
3.b odd 2 1 inner 24.5.e.a 4
4.b odd 2 1 48.5.e.c 4
5.b even 2 1 600.5.l.a 4
5.c odd 4 2 600.5.c.a 8
8.b even 2 1 192.5.e.f 4
8.d odd 2 1 192.5.e.e 4
9.c even 3 2 648.5.m.e 8
9.d odd 6 2 648.5.m.e 8
12.b even 2 1 48.5.e.c 4
15.d odd 2 1 600.5.l.a 4
15.e even 4 2 600.5.c.a 8
24.f even 2 1 192.5.e.e 4
24.h odd 2 1 192.5.e.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.e.a 4 1.a even 1 1 trivial
24.5.e.a 4 3.b odd 2 1 inner
48.5.e.c 4 4.b odd 2 1
48.5.e.c 4 12.b even 2 1
192.5.e.e 4 8.d odd 2 1
192.5.e.e 4 24.f even 2 1
192.5.e.f 4 8.b even 2 1
192.5.e.f 4 24.h odd 2 1
600.5.c.a 8 5.c odd 4 2
600.5.c.a 8 15.e even 4 2
600.5.l.a 4 5.b even 2 1
600.5.l.a 4 15.d odd 2 1
648.5.m.e 8 9.c even 3 2
648.5.m.e 8 9.d odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} - 42 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{4} + 2272 T^{2} + 541696 \) Copy content Toggle raw display
$7$ \( (T^{2} - 12 T - 1836)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 43744 T^{2} + \cdots + 25240576 \) Copy content Toggle raw display
$13$ \( (T^{2} + 124 T - 26108)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 122368 T^{2} + \cdots + 975437824 \) Copy content Toggle raw display
$19$ \( (T^{2} - 412 T + 40564)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 485248 T^{2} + \cdots + 15447506944 \) Copy content Toggle raw display
$29$ \( T^{4} + 227808 T^{2} + \cdots + 12680561664 \) Copy content Toggle raw display
$31$ \( (T^{2} - 1900 T + 675988)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1404 T - 974844)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6966144 T^{2} + \cdots + 1432636637184 \) Copy content Toggle raw display
$43$ \( (T^{2} + 484 T - 33164)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1027584 T^{2} + \cdots + 55228760064 \) Copy content Toggle raw display
$53$ \( T^{4} + 28706272 T^{2} + \cdots + 22497339114496 \) Copy content Toggle raw display
$59$ \( T^{4} + 14492128 T^{2} + \cdots + 1354747012096 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4292 T + 981124)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7556 T + 14226484)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 29260672 T^{2} + \cdots + 23483250786304 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2756 T - 2414204)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 532 T - 36098156)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 90476512 T^{2} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 919970046296064 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16228 T + 47116996)^{2} \) Copy content Toggle raw display
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