↑

Properties

Label 720.2.bm.f
Level $720$
Weight $2$
Character orbit 720.bm
Analytic conductor $5.749$
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] = [720,2,Mod(109,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bm (of order \(4\), degree \(2\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.534694406811304329216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} + (\beta_{8} - \beta_{7} + \beta_{4} - 1) q^{4} + (\beta_{9} - \beta_{7}) q^{5} + (\beta_{13} - \beta_{11} + \cdots - \beta_{3}) q^{7}+ \cdots + (\beta_{14} + \beta_{13} - \beta_{11}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} + (\beta_{8} - \beta_{7} + \beta_{4} - 1) q^{4} + (\beta_{9} - \beta_{7}) q^{5} + (\beta_{13} - \beta_{11} + \cdots - \beta_{3}) q^{7}+ \cdots + (2 \beta_{15} + 3 \beta_{14} + \cdots + \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 8 q^{5} - 12 q^{10} - 8 q^{11} + 4 q^{14} + 16 q^{16} - 8 q^{19} + 4 q^{20} - 32 q^{26} + 16 q^{29} + 16 q^{31} + 48 q^{34} + 24 q^{35} + 24 q^{40} + 8 q^{44} - 28 q^{46} + 16 q^{49} - 24 q^{50} + 56 q^{56} + 24 q^{59} - 16 q^{64} + 20 q^{70} - 88 q^{76} + 16 q^{79} - 16 q^{80} + 28 q^{86} - 16 q^{91} + 12 q^{94} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} - 6\nu^{10} - 28\nu^{8} - 20\nu^{6} + 96\nu^{4} - 160\nu^{2} + 256 ) / 576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} - 6\nu^{10} - 28\nu^{8} - 20\nu^{6} + 96\nu^{4} + 416\nu^{2} - 320 ) / 576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 2\nu^{7} + 4\nu^{5} - 20\nu^{3} - 32\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 14\nu^{12} - 6\nu^{10} + 44\nu^{8} + 76\nu^{6} - 48\nu^{4} - 448\nu^{2} + 64 ) / 576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} + 6\nu^{10} - 4\nu^{8} + 52\nu^{6} + 48\nu^{4} - 112\nu^{2} - 32 ) / 288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} + 10\nu^{10} + 4\nu^{8} - 20\nu^{6} + 32\nu^{4} - 96\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{14} + 2\nu^{12} - 6\nu^{10} + 28\nu^{8} + 44\nu^{6} - 48\nu^{4} - 32\nu^{2} - 64 ) / 576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{14} - 2\nu^{12} - 18\nu^{10} + 20\nu^{8} + 4\nu^{6} + 144\nu^{4} + 128\nu^{2} - 704 ) / 576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{15} + 2\nu^{13} + 6\nu^{11} + 4\nu^{9} - 12\nu^{7} + 48\nu^{3} + 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7\nu^{15} + 2\nu^{13} - 54\nu^{11} - 20\nu^{9} + 44\nu^{7} + 144\nu^{5} + 256\nu^{3} - 1792\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} + 2\nu^{13} + 2\nu^{11} - 4\nu^{9} - 4\nu^{7} - 16\nu^{5} + 32\nu^{3} + 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -11\nu^{15} + 2\nu^{13} + 30\nu^{11} + 28\nu^{9} - 124\nu^{7} - 192\nu^{5} - 224\nu^{3} + 896\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -11\nu^{15} - 10\nu^{13} + 30\nu^{11} + 4\nu^{9} + 68\nu^{7} - 336\nu^{5} - 128\nu^{3} + 896\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{15} + \nu^{13} + 2\nu^{11} + 2\nu^{9} - 4\nu^{7} - 44\nu^{5} + 8\nu^{3} + 96\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} + \nu^{13} - 2\nu^{11} - 6\nu^{9} + 4\nu^{7} + 20\nu^{5} + 8\nu^{3} - 128\nu ) / 96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{12} + \beta_{10} + \beta_{9} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - 3\beta_{14} + 2\beta_{11} + \beta_{9} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{6} + 4\beta_{5} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{13} - 8\beta_{12} + 2\beta_{11} - 2\beta_{10} + 2\beta_{9} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6\beta_{8} + 6\beta_{7} + 4\beta_{6} - 2\beta_{5} - 2\beta_{2} - 6\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -8\beta_{15} + 6\beta_{14} - 6\beta_{13} - 10\beta_{12} - 6\beta_{11} - 2\beta_{10} + 8\beta_{9} + 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -4\beta_{8} - 8\beta_{7} + 8\beta_{6} + 8\beta_{5} - 4\beta_{4} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16\beta_{15} + 12\beta_{13} - 16\beta_{11} - 12\beta_{10} + 20\beta_{9} - 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12\beta_{8} + 24\beta_{7} - 4\beta_{6} + 8\beta_{5} - 36\beta_{4} - 36\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20\beta_{15} + 24\beta_{14} - 20\beta_{13} - 20\beta_{12} + 20\beta_{11} - 20\beta_{10} + 12\beta_{9} + 32\beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 24\beta_{8} - 72\beta_{7} - 16\beta_{6} + 8\beta_{5} - 32\beta_{2} - 32\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 24\beta_{15} + 72\beta_{14} - 72\beta_{13} - 64\beta_{12} - 72\beta_{11} - 80\beta_{10} - 24\beta_{9} - 32\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.238945 − 1.39388i
−0.841995 − 1.13624i
−1.32661 − 0.490008i
−1.40501 − 0.161069i
1.40501 + 0.161069i
1.32661 + 0.490008i
0.841995 + 1.13624i
−0.238945 + 1.39388i
0.238945 + 1.39388i
−0.841995 + 1.13624i
−1.32661 + 0.490008i
−1.40501 + 0.161069i
1.40501 − 0.161069i
1.32661 − 0.490008i
0.841995 − 1.13624i
−0.238945 − 1.39388i
−1.39388 + 0.238945i 0 1.88581 − 0.666123i 2.16225 − 0.569800i 0 −3.84853 −2.46943 + 1.37910i 0 −2.87777 + 1.31089i
109.2 −1.13624 − 0.841995i 0 0.582088 + 1.91342i 1.17216 + 1.90421i 0 3.61392 0.949697 − 2.66422i 0 0.271479 − 3.15060i
109.3 −0.490008 − 1.32661i 0 −1.51978 + 1.30010i 2.16225 − 0.569800i 0 1.09033 2.46943 + 1.37910i 0 −1.81542 − 2.58926i
109.4 −0.161069 − 1.40501i 0 −1.94811 + 0.452606i −1.90421 − 1.17216i 0 −1.71452 0.949697 + 2.66422i 0 −1.34019 + 2.86424i
109.5 0.161069 + 1.40501i 0 −1.94811 + 0.452606i 1.17216 + 1.90421i 0 1.71452 −0.949697 − 2.66422i 0 −2.48664 + 1.95361i
109.6 0.490008 + 1.32661i 0 −1.51978 + 1.30010i 0.569800 − 2.16225i 0 −1.09033 −2.46943 − 1.37910i 0 3.14767 − 0.303617i
109.7 1.13624 + 0.841995i 0 0.582088 + 1.91342i −1.90421 − 1.17216i 0 −3.61392 −0.949697 + 2.66422i 0 −1.17669 − 2.93520i
109.8 1.39388 − 0.238945i 0 1.88581 − 0.666123i 0.569800 − 2.16225i 0 3.84853 2.46943 − 1.37910i 0 0.277574 − 3.15007i
469.1 −1.39388 − 0.238945i 0 1.88581 + 0.666123i 2.16225 + 0.569800i 0 −3.84853 −2.46943 − 1.37910i 0 −2.87777 − 1.31089i
469.2 −1.13624 + 0.841995i 0 0.582088 − 1.91342i 1.17216 − 1.90421i 0 3.61392 0.949697 + 2.66422i 0 0.271479 + 3.15060i
469.3 −0.490008 + 1.32661i 0 −1.51978 − 1.30010i 2.16225 + 0.569800i 0 1.09033 2.46943 − 1.37910i 0 −1.81542 + 2.58926i
469.4 −0.161069 + 1.40501i 0 −1.94811 − 0.452606i −1.90421 + 1.17216i 0 −1.71452 0.949697 − 2.66422i 0 −1.34019 − 2.86424i
469.5 0.161069 − 1.40501i 0 −1.94811 − 0.452606i 1.17216 − 1.90421i 0 1.71452 −0.949697 + 2.66422i 0 −2.48664 − 1.95361i
469.6 0.490008 − 1.32661i 0 −1.51978 − 1.30010i 0.569800 + 2.16225i 0 −1.09033 −2.46943 + 1.37910i 0 3.14767 + 0.303617i
469.7 1.13624 − 0.841995i 0 0.582088 − 1.91342i −1.90421 + 1.17216i 0 −3.61392 −0.949697 − 2.66422i 0 −1.17669 + 2.93520i
469.8 1.39388 + 0.238945i 0 1.88581 + 0.666123i 0.569800 + 2.16225i 0 3.84853 2.46943 + 1.37910i 0 0.277574 + 3.15007i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bm.f 16
3.b odd 2 1 80.2.q.c ✓ 16
5.b even 2 1 inner 720.2.bm.f 16
12.b even 2 1 320.2.q.c 16
15.d odd 2 1 80.2.q.c ✓ 16
15.e even 4 2 400.2.l.i 16
16.e even 4 1 inner 720.2.bm.f 16
24.f even 2 1 640.2.q.f 16
24.h odd 2 1 640.2.q.e 16
48.i odd 4 1 80.2.q.c ✓ 16
48.i odd 4 1 640.2.q.e 16
48.k even 4 1 320.2.q.c 16
48.k even 4 1 640.2.q.f 16
60.h even 2 1 320.2.q.c 16
60.l odd 4 2 1600.2.l.h 16
80.q even 4 1 inner 720.2.bm.f 16
120.i odd 2 1 640.2.q.e 16
120.m even 2 1 640.2.q.f 16
240.t even 4 1 320.2.q.c 16
240.t even 4 1 640.2.q.f 16
240.z odd 4 1 1600.2.l.h 16
240.bb even 4 1 400.2.l.i 16
240.bd odd 4 1 1600.2.l.h 16
240.bf even 4 1 400.2.l.i 16
240.bm odd 4 1 80.2.q.c ✓ 16
240.bm odd 4 1 640.2.q.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.c ✓ 16 3.b odd 2 1
80.2.q.c ✓ 16 15.d odd 2 1
80.2.q.c ✓ 16 48.i odd 4 1
80.2.q.c ✓ 16 240.bm odd 4 1
320.2.q.c 16 12.b even 2 1
320.2.q.c 16 48.k even 4 1
320.2.q.c 16 60.h even 2 1
320.2.q.c 16 240.t even 4 1
400.2.l.i 16 15.e even 4 2
400.2.l.i 16 240.bb even 4 1
400.2.l.i 16 240.bf even 4 1
640.2.q.e 16 24.h odd 2 1
640.2.q.e 16 48.i odd 4 1
640.2.q.e 16 120.i odd 2 1
640.2.q.e 16 240.bm odd 4 1
640.2.q.f 16 24.f even 2 1
640.2.q.f 16 48.k even 4 1
640.2.q.f 16 120.m even 2 1
640.2.q.f 16 240.t even 4 1
720.2.bm.f 16 1.a even 1 1 trivial
720.2.bm.f 16 5.b even 2 1 inner
720.2.bm.f 16 16.e even 4 1 inner
720.2.bm.f 16 80.q even 4 1 inner
1600.2.l.h 16 60.l odd 4 2
1600.2.l.h 16 240.z odd 4 1
1600.2.l.h 16 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{8} - 32T_{7}^{6} + 312T_{7}^{4} - 896T_{7}^{2} + 676 \) Copy content Toggle raw display
\( T_{11}^{8} + 4T_{11}^{7} + 8T_{11}^{6} - 8T_{11}^{5} + 136T_{11}^{4} + 464T_{11}^{3} + 800T_{11}^{2} - 160T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{8} + 224T_{13}^{4} + 7744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 32 T^{6} + \cdots + 676)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 224 T^{4} + 7744)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 28 T^{2} + 88)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots + 59536)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 32 T^{6} + 168 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + \cdots + 208)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 1536 T^{4} + 147456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 192 T^{6} + \cdots + 219024)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 9721171216 \) Copy content Toggle raw display
$47$ \( (T^{8} + 80 T^{6} + \cdots + 27556)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 4672 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 576)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36804120336 \) Copy content Toggle raw display
$71$ \( (T^{8} + 256 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 456 T^{6} + \cdots + 48776256)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + \cdots - 368)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{8} + 208 T^{6} + \cdots + 135424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 200 T^{6} + \cdots + 2383936)^{2} \) Copy content Toggle raw display
show more
show less