Properties

Label 48.16.c.b
Level $48$
Weight $16$
Character orbit 48.c
Analytic conductor $68.493$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(68.4928824480\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 32461x^{2} + 32462x + 263623935 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{7} \)
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.

\(f(q)\) \(=\) \( q + ( - 17 \beta_{2} + \beta_1) q^{3} + 5 \beta_{3} q^{5} - 10213 \beta_{2} q^{7} + (909 \beta_{3} - 3829275) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 17 \beta_{2} + \beta_1) q^{3} + 5 \beta_{3} q^{5} - 10213 \beta_{2} q^{7} + (909 \beta_{3} - 3829275) q^{9} + ( - 5553 \beta_{2} + 33318 \beta_1) q^{11} + 254445334 q^{13} + ( - 957375 \beta_{2} - 99990 \beta_1) q^{15} + 48556 \beta_{3} q^{17} - 2971927 \beta_{2} q^{19} + (275751 \beta_{3} - 5514468498) q^{21} + ( - 1239498 \beta_{2} + 7436988 \beta_1) q^{23} + 24731780525 q^{25} + ( - 108953100 \beta_{2} - 22007457 \beta_1) q^{27} + 10804075 \beta_{3} q^{29} + 667596871 \beta_{2} q^{31} + (15143031 \beta_{3} + 175246549488) q^{33} + (10110870 \beta_{2} - 60665220 \beta_1) q^{35} - 178908455266 q^{37} + ( - 4325570678 \beta_{2} + 254445334 \beta_1) q^{39} - 34816182 \beta_{3} q^{41} + 3190423037 \beta_{2} q^{43} + ( - 19146375 \beta_{3} - 1051858003680) q^{45} + (100037340 \beta_{2} - 600224040 \beta_1) q^{47} + 1401862493899 q^{49} + ( - 9297260100 \beta_{2} - 971022888 \beta_1) q^{51} - 857577711 \beta_{3} q^{53} - 32453064720 \beta_{2} q^{55} + (80242029 \beta_{3} - 1604680095942) q^{57} + (45898659 \beta_{2} - 275391954 \beta_1) q^{59} - 14469775333882 q^{61} + (40946541741 \beta_{2} - 11028936996 \beta_1) q^{63} + 1272226670 \beta_{3} q^{65} - 488388156279 \beta_{2} q^{67} + (3380111046 \beta_{3} + 39117188474208) q^{69} + (6008770962 \beta_{2} - 36052625772 \beta_1) q^{71} + 66786962361658 q^{73} + ( - 420440268925 \beta_{2} + 24731780525 \beta_1) q^{75} + 9187471818 \beta_{3} q^{77} - 1080270581713 \beta_{2} q^{79} + ( - 6961621950 \beta_{3} - 176564438043399) q^{81} + (30048734529 \beta_{2} - 180292407174 \beta_1) q^{83} - 56187037653120 q^{85} + ( - 2068710260625 \beta_{2} - 216059891850 \beta_1) q^{87} - 36895061026 \beta_{3} q^{89} - 2598650196142 \beta_{2} q^{91} + ( - 18025115517 \beta_{3} + 360466260108966) q^{93} + (2942207730 \beta_{2} - 17653246380 \beta_1) q^{95} - 375170793938030 q^{97} + ( - 5878703202021 \beta_{2} - 127583784450 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 15317100 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 15317100 q^{9} + 1017781336 q^{13} - 22057873992 q^{21} + 98927122100 q^{25} + 700986197952 q^{33} - 715633821064 q^{37} - 4207432014720 q^{45} + 5607449975596 q^{49} - 6418720383768 q^{57} - 57879101335528 q^{61} + 156468753896832 q^{69} + 267147849446632 q^{73} - 706257752173596 q^{81} - 224748150612480 q^{85} + 14\!\cdots\!64 q^{93}+ \cdots - 15\!\cdots\!20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 32461x^{2} + 32462x + 263623935 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 24\nu^{3} - 36\nu^{2} - 779082\nu + 389547 ) / 21649 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 72\nu^{3} - 108\nu^{2} - 1168200\nu + 584118 ) / 21649 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 36\nu^{2} - 36\nu - 584316 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 3\beta _1 + 27 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + 2\beta_{2} - 6\beta _1 + 1753002 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{3} + 129853\beta_{2} - 194718\beta _1 + 5258952 ) / 216 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
127.913 + 1.65831i
127.913 1.65831i
−126.913 + 1.65831i
−126.913 1.65831i
0 −2293.43 3014.81i 0 76064.4i 0 1.82913e6i 0 −3.82928e6 + 1.38285e7i 0
47.2 0 −2293.43 + 3014.81i 0 76064.4i 0 1.82913e6i 0 −3.82928e6 1.38285e7i 0
47.3 0 2293.43 3014.81i 0 76064.4i 0 1.82913e6i 0 −3.82928e6 1.38285e7i 0
47.4 0 2293.43 + 3014.81i 0 76064.4i 0 1.82913e6i 0 −3.82928e6 + 1.38285e7i 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
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 48.16.c.b 4
3.b odd 2 1 inner 48.16.c.b 4
4.b odd 2 1 inner 48.16.c.b 4
12.b even 2 1 inner 48.16.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.16.c.b 4 1.a even 1 1 trivial
48.16.c.b 4 3.b odd 2 1 inner
48.16.c.b 4 4.b odd 2 1 inner
48.16.c.b 4 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 5785797600 \) acting on \(S_{16}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 205891132094649 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5785797600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 3345699016044)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 58\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( (T - 254445334)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 54\!\cdots\!44)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 29\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T + 178908455266)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 28\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 39\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T + 14469775333882)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 76\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 68\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T - 66786962361658)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 37\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 31\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T + 375170793938030)^{4} \) Copy content Toggle raw display
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