↑

Properties

Label 6048.2.h.h
Level $6048$
Weight $2$
Character orbit 6048.h
Analytic conductor $48.294$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6048,2,Mod(2591,6048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6048.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6048.h (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: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{5} + \beta_{3} q^{7} + (\beta_{13} - \beta_{11}) q^{11} - \beta_{4} q^{13} + ( - \beta_{14} - \beta_{8}) q^{17} + (\beta_{6} - 2 \beta_{3}) q^{19} + ( - 3 \beta_{11} - 2 \beta_{10}) q^{23} + (\beta_{2} + 3) q^{25} + \beta_{9} q^{29} + ( - \beta_{7} - \beta_{3} + 2 \beta_1) q^{31} + \beta_{11} q^{35} + ( - \beta_{4} - \beta_{2} + 2) q^{37} + (\beta_{14} - \beta_{12} + \cdots + \beta_{8}) q^{41}+ \cdots + (\beta_{5} - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{13} + 48 q^{25} + 40 q^{37} - 16 q^{49} - 56 q^{73} - 16 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{14} + 1786\nu^{10} + 188597\nu^{6} + 4585536\nu^{2} ) / 3345408 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{12} + 1406\nu^{8} + 30007\nu^{4} - 455616 ) / 292608 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2485\nu^{14} - 608858\nu^{10} - 35048677\nu^{6} - 685063872\nu^{2} ) / 849733632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 755\nu^{12} + 173854\nu^{8} + 7752299\nu^{4} + 42152256 ) / 8851392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1003\nu^{12} + 211910\nu^{8} + 7466011\nu^{4} + 42762240 ) / 8851392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -25\nu^{14} - 5602\nu^{10} - 242441\nu^{6} - 2339264\nu^{2} ) / 780288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9721\nu^{14} - 2320034\nu^{10} - 120569065\nu^{6} - 1445284032\nu^{2} ) / 283244544 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3577 \nu^{15} - 1002 \nu^{13} - 901346 \nu^{11} - 608820 \nu^{9} - 53043433 \nu^{7} + \cdots - 935501184 \nu ) / 2549200896 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2485\nu^{15} - 608858\nu^{11} - 35048677\nu^{7} - 685063872\nu^{3} - 849733632\nu ) / 849733632 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 58873 \nu^{15} - 148440 \nu^{13} + 13669538 \nu^{11} - 32950320 \nu^{9} + \cdots - 11174367744 \nu ) / 10196803584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 73181 \nu^{15} + 152448 \nu^{13} - 17274922 \nu^{11} + 35385600 \nu^{9} + \cdots + 14916372480 \nu ) / 10196803584 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 73181 \nu^{15} - 152448 \nu^{13} - 17274922 \nu^{11} - 35385600 \nu^{9} + \cdots - 14916372480 \nu ) / 10196803584 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 88693 \nu^{15} + 148440 \nu^{13} - 20975834 \nu^{11} + 32950320 \nu^{9} + \cdots + 21371171328 \nu ) / 10196803584 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 118241 \nu^{15} - 91488 \nu^{13} + 30024946 \nu^{11} - 13610688 \nu^{9} + \cdots + 40990482432 \nu ) / 10196803584 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 118241 \nu^{15} + 91488 \nu^{13} + 30024946 \nu^{11} + 13610688 \nu^{9} + \cdots - 40990482432 \nu ) / 10196803584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{10} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 13\beta_{3} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} - 9\beta_{13} + 2\beta_{12} - 6\beta_{11} - 17\beta_{10} - 9\beta_{9} - 8\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{5} - 8\beta_{4} - 52\beta_{2} - 125 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 30 \beta_{15} + 30 \beta_{14} - 97 \beta_{13} - 22 \beta_{12} - 130 \beta_{11} - 249 \beta_{10} + \cdots + 152 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 160\beta_{7} - 249\beta_{6} + 1253\beta_{3} + 788\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 394 \beta_{15} + 394 \beta_{14} + 1145 \beta_{13} - 282 \beta_{12} + 2030 \beta_{11} + \cdots + 2312 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3457\beta_{5} + 2424\beta_{4} + 7732\beta_{2} + 17197 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5078 \beta_{15} - 5078 \beta_{14} + 14193 \beta_{13} + 3902 \beta_{12} + 28602 \beta_{11} + \cdots - 32504 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -33680\beta_{7} + 46697\beta_{6} - 183333\beta_{3} - 131076\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 65538 \beta_{15} - 65538 \beta_{14} - 180553 \beta_{13} + 53890 \beta_{12} - 387046 \beta_{11} + \cdots - 440936 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 621489\beta_{5} - 452584\beta_{4} - 1246516\beta_{2} - 2788125 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 849550 \beta_{15} + 849550 \beta_{14} - 2328065 \beta_{13} - 731238 \beta_{12} + \cdots + 5877080 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5995392\beta_{7} - 8205145\beta_{6} + 30146533\beta_{3} + 22104052\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 11052026 \beta_{15} + 11052026 \beta_{14} + 30227865 \beta_{13} - 9777770 \beta_{12} + \cdots + 77631944 \beta_{8} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
−0.826679 − 0.826679i
1.53379 + 1.53379i
−1.53379 + 1.53379i
0.826679 − 0.826679i
−2.55743 − 2.55743i
1.85032 + 1.85032i
−1.85032 + 1.85032i
2.55743 − 2.55743i
−1.85032 − 1.85032i
2.55743 + 2.55743i
−2.55743 + 2.55743i
1.85032 − 1.85032i
−1.53379 − 1.53379i
0.826679 + 0.826679i
−0.826679 + 0.826679i
1.53379 − 1.53379i
0 0 0 − 1.93185i 0 − 1.00000i 0 0 0
2591.2 0 0 0 − 1.93185i 0 − 1.00000i 0 0 0
2591.3 0 0 0 − 1.93185i 0 1.00000i 0 0 0
2591.4 0 0 0 − 1.93185i 0 1.00000i 0 0 0
2591.5 0 0 0 − 0.517638i 0 − 1.00000i 0 0 0
2591.6 0 0 0 − 0.517638i 0 − 1.00000i 0 0 0
2591.7 0 0 0 − 0.517638i 0 1.00000i 0 0 0
2591.8 0 0 0 − 0.517638i 0 1.00000i 0 0 0
2591.9 0 0 0 0.517638i 0 − 1.00000i 0 0 0
2591.10 0 0 0 0.517638i 0 − 1.00000i 0 0 0
2591.11 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.12 0 0 0 0.517638i 0 1.00000i 0 0 0
2591.13 0 0 0 1.93185i 0 − 1.00000i 0 0 0
2591.14 0 0 0 1.93185i 0 − 1.00000i 0 0 0
2591.15 0 0 0 1.93185i 0 1.00000i 0 0 0
2591.16 0 0 0 1.93185i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
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 6048.2.h.h ✓ 16
3.b odd 2 1 inner 6048.2.h.h ✓ 16
4.b odd 2 1 inner 6048.2.h.h ✓ 16
12.b even 2 1 inner 6048.2.h.h ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.h ✓ 16 1.a even 1 1 trivial
6048.2.h.h ✓ 16 3.b odd 2 1 inner
6048.2.h.h ✓ 16 4.b odd 2 1 inner
6048.2.h.h ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{4} + 4T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 56T_{11}^{6} + 930T_{11}^{4} - 5240T_{11}^{2} + 5329 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} - 56 T^{6} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 26 T^{2} + 96)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 112 T^{6} + \cdots + 76176)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 116 T^{6} + \cdots + 62001)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 28 T^{2} + 169)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 52 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 136 T^{6} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots - 27)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 160 T^{6} + \cdots + 4761)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 80 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 236 T^{6} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 160 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 212 T^{6} + \cdots + 141376)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 164 T^{2} + \cdots - 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 184 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 352 T^{6} + \cdots + 3884841)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 14 T^{3} + \cdots + 24)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 688 T^{6} + \cdots + 576384064)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 148 T^{6} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 376 T^{6} + \cdots + 2852721)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 384)^{4} \) Copy content Toggle raw display
show more
show less