Properties

Label 48.5.e.c
Level $48$
Weight $5$
Character orbit 48.e
Analytic conductor $4.962$
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] = [48,5,Mod(17,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, 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("48.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\), degree \(1\), not 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: \(4.96175822802\)
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: no (minimal twist has level 24)
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.

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

\(\beta_{1}\)\(=\) \( ( 8\nu^{3} + 72\nu^{2} - 224\nu - 12 ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{3} + 36\nu^{2} + 28\nu - 48 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\nu^{3} - 48\nu^{2} + 112\nu - 48 ) / 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{3} + 8\beta_{2} - 2\beta _1 + 24 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4\beta_{2} + \beta _1 + 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{3} + 4\beta_{2} + 8\beta _1 + 48 ) / 24 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
−1.30278 + 1.41421i
−1.30278 1.41421i
2.30278 1.41421i
2.30278 + 1.41421i
0 −6.21110 6.51323i 0 16.4520i 0 −49.2666 0 −3.84441 + 80.9087i 0
17.2 0 −6.21110 + 6.51323i 0 16.4520i 0 −49.2666 0 −3.84441 80.9087i 0
17.3 0 8.21110 3.68481i 0 44.7363i 0 37.2666 0 53.8444 60.5126i 0
17.4 0 8.21110 + 3.68481i 0 44.7363i 0 37.2666 0 53.8444 + 60.5126i 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 48.5.e.c 4
3.b odd 2 1 inner 48.5.e.c 4
4.b odd 2 1 24.5.e.a 4
8.b even 2 1 192.5.e.e 4
8.d odd 2 1 192.5.e.f 4
12.b even 2 1 24.5.e.a 4
20.d odd 2 1 600.5.l.a 4
20.e even 4 2 600.5.c.a 8
24.f even 2 1 192.5.e.f 4
24.h odd 2 1 192.5.e.e 4
36.f odd 6 2 648.5.m.e 8
36.h even 6 2 648.5.m.e 8
60.h even 2 1 600.5.l.a 4
60.l odd 4 2 600.5.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.e.a 4 4.b odd 2 1
24.5.e.a 4 12.b even 2 1
48.5.e.c 4 1.a even 1 1 trivial
48.5.e.c 4 3.b odd 2 1 inner
192.5.e.e 4 8.b even 2 1
192.5.e.e 4 24.h odd 2 1
192.5.e.f 4 8.d odd 2 1
192.5.e.f 4 24.f even 2 1
600.5.c.a 8 20.e even 4 2
600.5.c.a 8 60.l odd 4 2
600.5.l.a 4 20.d odd 2 1
600.5.l.a 4 60.h even 2 1
648.5.m.e 8 36.f odd 6 2
648.5.m.e 8 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2272T_{5}^{2} + 541696 \) acting on \(S_{5}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

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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