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Properties

Label 48.3.i.a
Level $48$
Weight $3$
Character orbit 48.i
Analytic conductor $1.308$
Analytic rank $0$
Dimension $8$
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] = [48,3,Mod(5,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.i (of order \(4\), degree \(2\), 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: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{4} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{3}) q^{8} + ( - 2 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{4} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{3}) q^{8} + ( - 2 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{9} + (4 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 8 \beta_{2} + \cdots - 4) q^{10}+ \cdots + ( - 6 \beta_{5} + 8 \beta_{4} - 14 \beta_{3} + 16 \beta_{2} - 8 \beta_1 + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 8 q^{4} + 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 8 q^{4} + 16 q^{6} - 56 q^{10} + 56 q^{12} - 96 q^{13} + 112 q^{15} - 64 q^{18} + 16 q^{19} - 32 q^{21} + 16 q^{22} - 48 q^{24} - 68 q^{27} + 64 q^{28} + 56 q^{30} + 72 q^{31} - 64 q^{33} + 32 q^{34} + 104 q^{36} + 112 q^{37} - 24 q^{42} - 240 q^{43} - 112 q^{45} + 224 q^{46} - 64 q^{48} + 328 q^{49} - 32 q^{51} - 112 q^{52} - 168 q^{54} - 56 q^{58} - 336 q^{60} + 208 q^{61} - 104 q^{63} - 320 q^{64} - 80 q^{66} - 232 q^{67} + 112 q^{70} + 160 q^{72} + 324 q^{75} + 448 q^{76} + 152 q^{78} - 136 q^{79} + 184 q^{81} + 176 q^{82} + 64 q^{84} - 112 q^{85} - 64 q^{88} + 392 q^{90} + 152 q^{91} + 64 q^{93} - 368 q^{94} + 512 q^{96} - 480 q^{97} + 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} + 24\nu^{2} + 8\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 2\nu^{5} + 2\nu^{3} - 8\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 2\nu^{5} + 2\nu^{3} + 16\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} + 4\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{5} + 4\nu^{3} - 10\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 8\nu^{6} - 2\nu^{5} + 8\nu^{4} - 10\nu^{3} - 8\nu^{2} - 8\nu + 32 ) / 24 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 4\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 4\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{6} - 2\beta_{5} - 6\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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
−1.38255 + 0.297594i
0.767178 + 1.18804i
−0.767178 − 1.18804i
1.38255 − 0.297594i
−1.38255 − 0.297594i
0.767178 − 1.18804i
−0.767178 + 1.18804i
1.38255 + 0.297594i
−1.68014 − 1.08495i 2.90783 − 0.737922i 1.64575 + 3.64575i 1.57472 − 1.57472i −5.68618 − 1.91505i − 3.64575i 1.19038 − 7.91094i 7.91094 − 4.29150i −4.35425 + 0.937254i
5.2 −0.420861 + 1.95522i −2.77809 − 1.13234i −3.64575 − 1.64575i −6.28651 + 6.28651i 3.38317 − 4.95522i 1.64575i 4.75216 − 6.43560i 6.43560 + 6.29150i −9.64575 − 14.9373i
5.3 0.420861 − 1.95522i 1.13234 + 2.77809i −3.64575 − 1.64575i 6.28651 − 6.28651i 5.90834 − 1.04478i 1.64575i −4.75216 + 6.43560i −6.43560 + 6.29150i −9.64575 − 14.9373i
5.4 1.68014 + 1.08495i 0.737922 − 2.90783i 1.64575 + 3.64575i −1.57472 + 1.57472i 4.39467 − 4.08495i − 3.64575i −1.19038 + 7.91094i −7.91094 − 4.29150i −4.35425 + 0.937254i
29.1 −1.68014 + 1.08495i 2.90783 + 0.737922i 1.64575 − 3.64575i 1.57472 + 1.57472i −5.68618 + 1.91505i 3.64575i 1.19038 + 7.91094i 7.91094 + 4.29150i −4.35425 − 0.937254i
29.2 −0.420861 − 1.95522i −2.77809 + 1.13234i −3.64575 + 1.64575i −6.28651 − 6.28651i 3.38317 + 4.95522i − 1.64575i 4.75216 + 6.43560i 6.43560 − 6.29150i −9.64575 + 14.9373i
29.3 0.420861 + 1.95522i 1.13234 − 2.77809i −3.64575 + 1.64575i 6.28651 + 6.28651i 5.90834 + 1.04478i − 1.64575i −4.75216 − 6.43560i −6.43560 − 6.29150i −9.64575 + 14.9373i
29.4 1.68014 − 1.08495i 0.737922 + 2.90783i 1.64575 − 3.64575i −1.57472 − 1.57472i 4.39467 + 4.08495i 3.64575i −1.19038 − 7.91094i −7.91094 + 4.29150i −4.35425 − 0.937254i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.i.a ✓ 8
3.b odd 2 1 inner 48.3.i.a ✓ 8
4.b odd 2 1 192.3.i.a 8
8.b even 2 1 384.3.i.a 8
8.d odd 2 1 384.3.i.b 8
12.b even 2 1 192.3.i.a 8
16.e even 4 1 inner 48.3.i.a ✓ 8
16.e even 4 1 384.3.i.a 8
16.f odd 4 1 192.3.i.a 8
16.f odd 4 1 384.3.i.b 8
24.f even 2 1 384.3.i.b 8
24.h odd 2 1 384.3.i.a 8
48.i odd 4 1 inner 48.3.i.a ✓ 8
48.i odd 4 1 384.3.i.a 8
48.k even 4 1 192.3.i.a 8
48.k even 4 1 384.3.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.a ✓ 8 1.a even 1 1 trivial
48.3.i.a ✓ 8 3.b odd 2 1 inner
48.3.i.a ✓ 8 16.e even 4 1 inner
48.3.i.a ✓ 8 48.i odd 4 1 inner
192.3.i.a 8 4.b odd 2 1
192.3.i.a 8 12.b even 2 1
192.3.i.a 8 16.f odd 4 1
192.3.i.a 8 48.k even 4 1
384.3.i.a 8 8.b even 2 1
384.3.i.a 8 16.e even 4 1
384.3.i.a 8 24.h odd 2 1
384.3.i.a 8 48.i odd 4 1
384.3.i.b 8 8.d odd 2 1
384.3.i.b 8 16.f odd 4 1
384.3.i.b 8 24.f even 2 1
384.3.i.b 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6272T_{5}^{4} + 153664 \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + 8 T^{4} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 8 T^{6} + 12 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} + 6272 T^{4} + 153664 \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 36)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 2048 T^{4} + 16384 \) Copy content Toggle raw display
$13$ \( (T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 75076)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 920 T^{2} + 103968)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 8 T^{3} + 32 T^{2} + 1728 T + 46656)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1120 T^{2} + 225792)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 614656 T^{4} + \cdots + 29884728384 \) Copy content Toggle raw display
$31$ \( (T^{2} - 18 T + 74)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 664 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 120 T^{3} + 7200 T^{2} + \cdots + 2483776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8144 T^{2} + 14193792)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 18284288 T^{4} + \cdots + 18847994944 \) Copy content Toggle raw display
$59$ \( T^{8} + 3520000 T^{4} + \cdots + 2624400000000 \) Copy content Toggle raw display
$61$ \( (T^{4} - 104 T^{3} + 5408 T^{2} + \cdots + 1004004)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 116 T^{3} + 6728 T^{2} + \cdots + 2155024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 16560 T^{2} + 24668288)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3712 T^{2} + 788544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 34 T - 894)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 135735296 T^{4} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6240 T^{2} + 184832)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 120 T + 3152)^{4} \) Copy content Toggle raw display
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