↑

Properties

Label 5472.2.g.b
Level $5472$
Weight $2$
Character orbit 5472.g
Analytic conductor $43.694$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5472,2,Mod(2737,5472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5472.2737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.g (of order \(2\), degree \(1\), not 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: \(43.6941399860\)
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: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 152)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{4}) q^{5} + (\beta_{12} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{4}) q^{5} + (\beta_{12} + 1) q^{7} + \beta_{8} q^{11} + (\beta_{4} + \beta_{3}) q^{13} + ( - \beta_{12} + \beta_{11} - \beta_{9}) q^{17} - \beta_{4} q^{19} + (\beta_{14} - \beta_{13} + \beta_{7}) q^{23} + ( - \beta_{15} - 2 \beta_{9} - 2) q^{25} + ( - \beta_{8} - \beta_{6} + \cdots - \beta_{2}) q^{29}+ \cdots + (2 \beta_{15} + \beta_{14} - \beta_{13} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} + 3 \nu^{13} + 4 \nu^{12} - 4 \nu^{11} + 14 \nu^{9} + 24 \nu^{8} + 40 \nu^{7} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{15} + 4 \nu^{14} + 11 \nu^{13} - 2 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 14 \nu^{9} + 76 \nu^{8} + \cdots + 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{14} + \nu^{13} + \nu^{12} - \nu^{11} - 3 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} + 30 \nu^{7} + \cdots + 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} + 8 \nu^{14} + 11 \nu^{13} + 10 \nu^{12} - 12 \nu^{11} + 4 \nu^{10} + 14 \nu^{9} + \cdots + 1152 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{15} - 20 \nu^{14} - 5 \nu^{13} - 26 \nu^{12} + 20 \nu^{11} + 20 \nu^{10} - 82 \nu^{9} + \cdots - 2432 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7 \nu^{15} - 12 \nu^{14} - 17 \nu^{13} - 10 \nu^{12} + 4 \nu^{11} + 4 \nu^{10} - 42 \nu^{9} + \cdots - 1408 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{15} - \nu^{14} + 2 \nu^{13} - 3 \nu^{12} - 2 \nu^{11} + 12 \nu^{10} - 4 \nu^{9} + 34 \nu^{8} + \cdots - 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{15} + 5 \nu^{14} + 4 \nu^{13} + 3 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 8 \nu^{9} + 30 \nu^{8} + \cdots + 384 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9 \nu^{15} + 10 \nu^{14} - 15 \nu^{13} + 28 \nu^{12} - 44 \nu^{10} + 42 \nu^{9} - 168 \nu^{8} + \cdots + 1536 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{15} + 48 \nu^{14} + 33 \nu^{13} + 62 \nu^{12} - 68 \nu^{11} + 4 \nu^{10} + 106 \nu^{9} + \cdots + 5504 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{15} + 7 \nu^{14} - 9 \nu^{13} + 23 \nu^{12} - 2 \nu^{11} - 28 \nu^{10} + 30 \nu^{9} + \cdots + 1280 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15 \nu^{15} + 16 \nu^{14} - 17 \nu^{13} + 50 \nu^{12} + 4 \nu^{11} - 68 \nu^{10} + 70 \nu^{9} + \cdots + 2688 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9 \nu^{15} - 11 \nu^{14} + 13 \nu^{13} - 31 \nu^{12} + 40 \nu^{10} - 54 \nu^{9} + 170 \nu^{8} + \cdots - 1728 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 21 \nu^{15} - 22 \nu^{14} + 27 \nu^{13} - 64 \nu^{12} + 100 \nu^{10} - 98 \nu^{9} + 368 \nu^{8} + \cdots - 3584 ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13 \nu^{15} - 15 \nu^{14} + 17 \nu^{13} - 39 \nu^{12} + 60 \nu^{10} - 70 \nu^{9} + 226 \nu^{8} + \cdots - 2240 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{12} + \beta_{7} - \beta_{6} + \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{6} + 3 \beta_{4} + \cdots - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{6} - 2 \beta_{5} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} - \beta_{6} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} + 4 \beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} + 3 \beta_{9} - 2 \beta_{7} + \cdots - 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{9} + \cdots - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6 \beta_{15} - 2 \beta_{14} + 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + \beta_{9} - 4 \beta_{8} + \cdots - 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 7 \beta_{12} - 11 \beta_{11} + 9 \beta_{10} + 5 \beta_{9} + \cdots + 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 14 \beta_{15} + 14 \beta_{14} + 3 \beta_{12} - \beta_{11} + \beta_{10} - 11 \beta_{9} - 16 \beta_{8} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6 \beta_{15} + 8 \beta_{14} - 10 \beta_{13} + 3 \beta_{12} + 5 \beta_{11} - 3 \beta_{10} - 31 \beta_{9} + \cdots + 33 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2 \beta_{15} + 14 \beta_{14} - 4 \beta_{13} + 31 \beta_{12} - 21 \beta_{11} - 23 \beta_{10} + 29 \beta_{9} + \cdots + 39 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 10 \beta_{15} - 20 \beta_{14} - 14 \beta_{13} + 15 \beta_{12} - 27 \beta_{11} + 9 \beta_{10} - 3 \beta_{9} + \cdots - 47 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 30 \beta_{15} + 18 \beta_{14} + 32 \beta_{13} + 11 \beta_{12} + 51 \beta_{11} - 43 \beta_{10} + \cdots - 113 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 30 \beta_{15} + 120 \beta_{14} + 6 \beta_{13} + 43 \beta_{12} + 29 \beta_{11} + 29 \beta_{10} + \cdots + 41 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 90 \beta_{15} + 14 \beta_{14} - 100 \beta_{13} - 65 \beta_{12} + 3 \beta_{11} - 23 \beta_{10} + \cdots - 177 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2053\) \(3745\) \(4447\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2737.1
−0.182898 − 1.40234i
1.33852 + 0.456455i
−0.889165 + 1.09972i
0.340606 − 1.37258i
−0.466170 − 1.33517i
−1.40771 + 0.135487i
1.12629 − 0.855255i
1.14052 − 0.836196i
1.14052 + 0.836196i
1.12629 + 0.855255i
−1.40771 − 0.135487i
−0.466170 + 1.33517i
0.340606 + 1.37258i
−0.889165 − 1.09972i
1.33852 − 0.456455i
−0.182898 + 1.40234i
0 0 0 − 4.04855i 0 3.59283 0 0 0
2737.2 0 0 0 − 3.36827i 0 −4.47116 0 0 0
2737.3 0 0 0 − 3.13887i 0 0.535658 0 0 0
2737.4 0 0 0 − 2.13486i 0 −3.29464 0 0 0
2737.5 0 0 0 − 2.10882i 0 2.73436 0 0 0
2737.6 0 0 0 − 1.66222i 0 1.99556 0 0 0
2737.7 0 0 0 − 1.51356i 0 −0.580162 0 0 0
2737.8 0 0 0 − 0.594041i 0 3.48756 0 0 0
2737.9 0 0 0 0.594041i 0 3.48756 0 0 0
2737.10 0 0 0 1.51356i 0 −0.580162 0 0 0
2737.11 0 0 0 1.66222i 0 1.99556 0 0 0
2737.12 0 0 0 2.10882i 0 2.73436 0 0 0
2737.13 0 0 0 2.13486i 0 −3.29464 0 0 0
2737.14 0 0 0 3.13887i 0 0.535658 0 0 0
2737.15 0 0 0 3.36827i 0 −4.47116 0 0 0
2737.16 0 0 0 4.04855i 0 3.59283 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2737.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5472.2.g.b 16
3.b odd 2 1 608.2.c.b 16
4.b odd 2 1 1368.2.g.b 16
8.b even 2 1 inner 5472.2.g.b 16
8.d odd 2 1 1368.2.g.b 16
12.b even 2 1 152.2.c.b ✓ 16
24.f even 2 1 152.2.c.b ✓ 16
24.h odd 2 1 608.2.c.b 16
48.i odd 4 1 4864.2.a.bn 8
48.i odd 4 1 4864.2.a.bp 8
48.k even 4 1 4864.2.a.bo 8
48.k even 4 1 4864.2.a.bq 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.c.b ✓ 16 12.b even 2 1
152.2.c.b ✓ 16 24.f even 2 1
608.2.c.b 16 3.b odd 2 1
608.2.c.b 16 24.h odd 2 1
1368.2.g.b 16 4.b odd 2 1
1368.2.g.b 16 8.d odd 2 1
4864.2.a.bn 8 48.i odd 4 1
4864.2.a.bo 8 48.k even 4 1
4864.2.a.bp 8 48.i odd 4 1
4864.2.a.bq 8 48.k even 4 1
5472.2.g.b 16 1.a even 1 1 trivial
5472.2.g.b 16 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 52 T_{5}^{14} + 1078 T_{5}^{12} + 11532 T_{5}^{10} + 68929 T_{5}^{8} + 233080 T_{5}^{6} + \cdots + 82944 \) acting on \(S_{2}^{\mathrm{new}}(5472, [\chi])\). 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
$5$ \( T^{16} + 52 T^{14} + \cdots + 82944 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots - 313)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 84 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( T^{16} + 112 T^{14} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} - 124 T^{6} + \cdots + 155824)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 248 T^{14} + \cdots + 8620096 \) Copy content Toggle raw display
$31$ \( (T^{8} + 8 T^{7} + \cdots + 7552)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 144 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + \cdots - 3072)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 204 T^{14} + \cdots + 1401856 \) Copy content Toggle raw display
$47$ \( (T^{8} - 12 T^{7} + \cdots + 34432)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 132199142464 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2722334976 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10615398064384 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4865341504 \) Copy content Toggle raw display
$71$ \( (T^{8} - 24 T^{7} + \cdots - 9153024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 316 T^{6} + \cdots + 6637913)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 24 T^{7} + \cdots - 70912)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22968008704 \) Copy content Toggle raw display
$89$ \( (T^{8} - 8 T^{7} + \cdots - 400128)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 16 T^{7} + \cdots + 8192)^{2} \) Copy content Toggle raw display
show more
show less