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Properties

Label 48.8.c.c
Level $48$
Weight $8$
Character orbit 48.c
Analytic conductor $14.994$
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] = [48,8,Mod(47,48)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 48.c (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: \(14.9944812232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 145x^{6} + 20649x^{4} - 54520x^{2} + 141376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + \beta_{3} q^{5} + (5 \beta_{4} - 10 \beta_{2} - 20 \beta_1) q^{7} + (\beta_{7} - \beta_{5} - 3 \beta_{3} + 141) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + \beta_{3} q^{5} + (5 \beta_{4} - 10 \beta_{2} - 20 \beta_1) q^{7} + (\beta_{7} - \beta_{5} - 3 \beta_{3} + 141) q^{9} + ( - \beta_{6} - 5 \beta_{4} + \cdots - 5 \beta_1) q^{11}+ \cdots + (1134 \beta_{6} - 52245 \beta_{4} + \cdots - 450711 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1128 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1128 q^{9} - 34000 q^{13} - 139920 q^{21} - 482072 q^{25} + 267840 q^{33} - 1131920 q^{37} + 3086208 q^{45} + 912344 q^{49} + 8157840 q^{57} - 2881616 q^{61} + 13074048 q^{69} - 30579760 q^{73} + 17580744 q^{81} - 34831872 q^{85} - 6526800 q^{93} - 29582320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 145x^{6} + 20649x^{4} - 54520x^{2} + 141376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -290\nu^{6} + 41298\nu^{4} - 5988210\nu^{2} + 8046776 ) / 970503 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6951 \nu^{7} - 12026 \nu^{6} + 1025567 \nu^{5} + 1748282 \nu^{4} - 146119207 \nu^{3} + \cdots + 326980504 ) / 23292072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11165\nu^{7} - 1589973\nu^{5} + 222782061\nu^{3} - 10885952\nu ) / 11646036 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6951 \nu^{7} + 27532 \nu^{6} + 1025567 \nu^{5} - 3992140 \nu^{4} - 146119207 \nu^{3} + \cdots - 750522320 ) / 11646036 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -21605\nu^{7} - 752\nu^{6} + 3076701\nu^{5} - 438357621\nu^{3} + 21065024\nu - 1084784440 ) / 2911509 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 340383 \nu^{7} - 12026 \nu^{6} + 51078743 \nu^{5} + 1748282 \nu^{4} - 7217493151 \nu^{3} + \cdots + 326980504 ) / 23292072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -43210\nu^{7} + 3008\nu^{6} + 6153402\nu^{5} - 876715242\nu^{3} + 42130048\nu + 4339137760 ) / 2911509 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - 3\beta_{6} - 4\beta_{5} + 36\beta_{4} - 24\beta_{3} + 147\beta_{2} + 36\beta_1 ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} - 24\beta_{4} + 48\beta_{2} - 278\beta _1 + 2320 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -77\beta_{7} - 308\beta_{5} - 3576\beta_{3} ) / 1152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 145\beta_{7} - 290\beta_{5} - 3480\beta_{4} + 6960\beta_{2} - 38806\beta _1 - 324368 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10789 \beta_{7} + 63687 \beta_{6} - 43156 \beta_{5} - 513684 \beta_{4} - 509496 \beta_{3} + \cdots - 513684 \beta_1 ) / 2304 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 20649\beta_{7} - 41298\beta_{5} - 46161040 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1535453 \beta_{7} + 9066543 \beta_{6} + 6141812 \beta_{5} - 73117044 \beta_{4} + \cdots - 73117044 \beta_1 ) / 2304 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.40745 − 0.812591i
1.40745 + 0.812591i
10.3329 + 5.96571i
10.3329 − 5.96571i
−10.3329 − 5.96571i
−10.3329 + 5.96571i
−1.40745 + 0.812591i
−1.40745 − 0.812591i
0 −45.7761 − 9.56815i 0 − 80.5643i 0 − 1049.15i 0 2003.90 + 875.985i 0
47.2 0 −45.7761 + 9.56815i 0 80.5643i 0 1049.15i 0 2003.90 − 875.985i 0
47.3 0 −15.2496 − 44.2092i 0 − 519.882i 0 − 564.173i 0 −1721.90 + 1348.34i 0
47.4 0 −15.2496 + 44.2092i 0 519.882i 0 564.173i 0 −1721.90 − 1348.34i 0
47.5 0 15.2496 − 44.2092i 0 519.882i 0 − 564.173i 0 −1721.90 − 1348.34i 0
47.6 0 15.2496 + 44.2092i 0 − 519.882i 0 564.173i 0 −1721.90 + 1348.34i 0
47.7 0 45.7761 − 9.56815i 0 80.5643i 0 − 1049.15i 0 2003.90 − 875.985i 0
47.8 0 45.7761 + 9.56815i 0 − 80.5643i 0 1049.15i 0 2003.90 + 875.985i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.8.c.c ✓ 8
3.b odd 2 1 inner 48.8.c.c ✓ 8
4.b odd 2 1 inner 48.8.c.c ✓ 8
8.b even 2 1 192.8.c.d 8
8.d odd 2 1 192.8.c.d 8
12.b even 2 1 inner 48.8.c.c ✓ 8
24.f even 2 1 192.8.c.d 8
24.h odd 2 1 192.8.c.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.8.c.c ✓ 8 1.a even 1 1 trivial
48.8.c.c ✓ 8 3.b odd 2 1 inner
48.8.c.c ✓ 8 4.b odd 2 1 inner
48.8.c.c ✓ 8 12.b even 2 1 inner
192.8.c.d 8 8.b even 2 1
192.8.c.d 8 8.d odd 2 1
192.8.c.d 8 24.f even 2 1
192.8.c.d 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 276768T_{5}^{2} + 1754265600 \) acting on \(S_{8}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 22876792454961 \) Copy content Toggle raw display
$5$ \( (T^{4} + 276768 T^{2} + 1754265600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1419000 T^{2} + 350345610000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 524124669966336)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8500 T + 13065124)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 50\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 282980 T - 12768363836)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 54\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 43\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 720404 T + 108631567204)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 29\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 11348670016036)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 77\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 35\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 72186391106300)^{4} \) Copy content Toggle raw display
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