Properties

Label 48.16.c.c
Level $48$
Weight $16$
Character orbit 48.c
Analytic conductor $68.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \( x^{4} - 231x^{2} + 227529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} - 13 \beta_{3} q^{5} + 203 \beta_1 q^{7} + (243 \beta_{3} + 13294773) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} - 13 \beta_{3} q^{5} + 203 \beta_1 q^{7} + (243 \beta_{3} + 13294773) q^{9} + (18810 \beta_{2} - 9405 \beta_1) q^{11} - 166384010 q^{13} + ( - 56394 \beta_{2} - 711243 \beta_1) q^{15} + 14500 \beta_{3} q^{17} - 4493455 \beta_1 q^{19} + (49329 \beta_{3} - 213989202) q^{21} + (4239684 \beta_{2} - 2119842 \beta_1) q^{23} - 52882380595 q^{25} + ( - 12240639 \beta_{2} + 26589546 \beta_1) q^{27} + 4280893 \beta_{3} q^{29} + 119007655 \beta_1 q^{31} + ( - 2285415 \beta_{3} - 259988810400) q^{33} + ( - 22895964 \beta_{2} + 11447982 \beta_1) q^{35} - 312995380930 q^{37} + (166384010 \beta_{2} - 166384010 \beta_1) q^{39} + 57869094 \beta_{3} q^{41} - 1763464699 \beta_1 q^{43} + ( - 172832049 \beta_{3} + 1558937689920) q^{45} + (121793256 \beta_{2} - 60896628 \beta_1) q^{47} + 4660681893931 q^{49} + (62901000 \beta_{2} + 793309500 \beta_1) q^{51} + 448425735 \beta_{3} q^{53} + 13908866400 \beta_1 q^{55} + ( - 1091909565 \beta_{3} + 4736703692970) q^{57} + (6339546450 \beta_{2} - 3169773225 \beta_1) q^{59} + 215703610022 q^{61} + (427978404 \beta_{2} + 2484849717 \beta_1) q^{63} + 2162992130 \beta_{3} q^{65} - 48072226623 \beta_1 q^{67} + ( - 515121606 \beta_{3} - 58600233898560) q^{69} + (25479295020 \beta_{2} - 12739647510 \beta_1) q^{71} - 93057038642630 q^{73} + (52882380595 \beta_{2} - 52882380595 \beta_1) q^{75} - 927878490 \beta_{3} q^{77} + 197814320015 \beta_1 q^{79} + (6461259678 \beta_{3} + 147610846148409) q^{81} + (90468825894 \beta_{2} - 45234412947 \beta_1) q^{83} + 93023030880000 q^{85} + (18570513834 \beta_{2} + 234211936923 \beta_1) q^{87} - 33492783646 \beta_{3} q^{89} - 33775954030 \beta_1 q^{91} + (28918860165 \beta_{3} - 125450015395770) q^{93} + (506807802540 \beta_{2} - 253403901270 \beta_1) q^{95} + 11\!\cdots\!10 q^{97}+ \cdots + (250074680130 \beta_{2} - 385026150465 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 53179092 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 53179092 q^{9} - 665536040 q^{13} - 855956808 q^{21} - 211529522380 q^{25} - 1039955241600 q^{33} - 1251981523720 q^{37} + 6235750759680 q^{45} + 18642727575724 q^{49} + 18946814771880 q^{57} + 862814440088 q^{61} - 234400935594240 q^{69} - 372228154570520 q^{73} + 590443384593636 q^{81} + 372092123520000 q^{85} - 501800061583080 q^{93} + 46\!\cdots\!40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 231x^{2} + 227529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{3} - 1476\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -15\nu^{3} + 7758\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 48\nu^{2} - 5544 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{2} - 5\beta_1 ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5544 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -82\beta_{2} - 431\beta_1 ) / 72 \) 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
17.2119 + 13.4443i
17.2119 13.4443i
−17.2119 + 13.4443i
−17.2119 13.4443i
0 −3717.77 725.994i 0 288791.i 0 294753.i 0 1.32948e7 + 5.39816e6i 0
47.2 0 −3717.77 + 725.994i 0 288791.i 0 294753.i 0 1.32948e7 5.39816e6i 0
47.3 0 3717.77 725.994i 0 288791.i 0 294753.i 0 1.32948e7 5.39816e6i 0
47.4 0 3717.77 + 725.994i 0 288791.i 0 294753.i 0 1.32948e7 + 5.39816e6i 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.c 4
3.b odd 2 1 inner 48.16.c.c 4
4.b odd 2 1 inner 48.16.c.c 4
12.b even 2 1 inner 48.16.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.16.c.c 4 1.a even 1 1 trivial
48.16.c.c 4 3.b odd 2 1 inner
48.16.c.c 4 4.b odd 2 1 inner
48.16.c.c 4 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 83399958720 \) 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
$5$ \( (T^{2} + 83399958720)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 86879616012)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T + 166384010)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 90\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T + 312995380930)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 65\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T - 215703610022)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 48\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T + 93057038642630)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 55\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T - 11\!\cdots\!10)^{4} \) Copy content Toggle raw display
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