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Properties

Label 720.4.w.d
Level $720$
Weight $4$
Character orbit 720.w
Analytic conductor $42.481$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,4,Mod(17,720)]
 
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, 0, 2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.w (of order \(4\), degree \(2\), 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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 45)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{7}) q^{5} + ( - \beta_{2} - 2 \beta_1 - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{7}) q^{5} + ( - \beta_{2} - 2 \beta_1 - 2) q^{7} + ( - 4 \beta_{11} - 3 \beta_{10} + \cdots - 3 \beta_{6}) q^{11}+ \cdots + ( - 25 \beta_{5} - 25 \beta_{3} + \cdots + 231) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{7} - 108 q^{13} - 144 q^{25} + 1248 q^{31} + 828 q^{37} + 96 q^{43} + 1512 q^{55} + 96 q^{61} - 1632 q^{67} + 3972 q^{73} - 1752 q^{85} - 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{10} + 56\nu^{8} - 634\nu^{6} - 3248\nu^{4} - 83197\nu^{2} - 418024 ) / 182400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1029\nu^{10} - 22972\nu^{8} + 158738\nu^{6} - 2560724\nu^{4} + 22729229\nu^{2} + 78379688 ) / 9051600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 197\nu^{10} - 2580\nu^{8} - 13006\nu^{6} - 58220\nu^{4} - 1342803\nu^{2} + 25801640 ) / 1206880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5723\nu^{10} - 110024\nu^{8} + 167406\nu^{6} - 2434608\nu^{4} + 44277523\nu^{2} + 559571096 ) / 18103200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -47187\nu^{10} + 385336\nu^{8} - 382814\nu^{6} + 49174112\nu^{4} + 226650613\nu^{2} - 139488344 ) / 36206400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 113621 \nu^{11} + 12179272 \nu^{9} - 76074438 \nu^{7} - 253832976 \nu^{5} + \cdots - 93476563288 \nu ) / 12889478400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 398131 \nu^{11} + 26089272 \nu^{9} - 283134218 \nu^{7} - 151683376 \nu^{5} + \cdots - 106435875688 \nu ) / 25778956800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 632611 \nu^{11} - 15626248 \nu^{9} + 84278742 \nu^{7} - 878674416 \nu^{5} + \cdots + 155395648792 \nu ) / 25778956800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6773\nu^{11} + 146104\nu^{9} - 577306\nu^{7} + 5223568\nu^{5} - 79774373\nu^{3} - 554840616\nu ) / 226131200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1095\nu^{11} + 7908\nu^{9} + 18534\nu^{7} + 745404\nu^{5} + 8928141\nu^{3} - 11971272\nu ) / 21482464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1093919 \nu^{11} - 7804552 \nu^{9} + 15466318 \nu^{7} - 1380835984 \nu^{5} + \cdots - 28260515592 \nu ) / 12889478400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{11} - \beta_{10} - \beta_{9} + 6\beta_{8} + 3\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 9\beta_{4} - 10\beta_{3} + 3\beta_{2} + 64\beta _1 + 64 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{11} + 53\beta_{10} - 137\beta_{9} + 12\beta_{8} + 6\beta_{7} + 57\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{5} + 93\beta_{4} - 134\beta_{3} - 135\beta_{2} + 1136 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -849\beta_{11} - 615\beta_{10} - 1259\beta_{9} + 6\beta_{8} - 348\beta_{7} + 933\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -338\beta_{5} + 111\beta_{4} - 1606\beta_{3} - 1035\beta_{2} - 5824\beta _1 + 25216 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7749\beta_{11} - 2133\beta_{10} - 12439\beta_{9} + 5940\beta_{8} - 6918\beta_{7} + 15591\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -126\beta_{5} + 2913\beta_{4} - 7294\beta_{3} - 2035\beta_{2} + 20480\beta _1 + 129968 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -102159\beta_{11} + 24551\beta_{10} - 220021\beta_{9} + 88602\beta_{8} - 38724\beta_{7} + 199131\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -15406\beta_{5} + 210609\beta_{4} - 353338\beta_{3} - 167733\beta_{2} + 856384\beta _1 + 4399744 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1634235 \beta_{11} - 305227 \beta_{10} - 3762665 \beta_{9} + 776748 \beta_{8} + \cdots + 2769081 \beta_{6} ) / 24 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
2.02004 + 2.30794i
−0.347140 + 2.27426i
−3.78139 − 0.0336790i
3.78139 + 0.0336790i
0.347140 − 2.27426i
−2.02004 − 2.30794i
2.02004 − 2.30794i
−0.347140 − 2.27426i
−3.78139 + 0.0336790i
3.78139 − 0.0336790i
0.347140 + 2.27426i
−2.02004 + 2.30794i
0 0 0 −9.93887 + 5.12043i 0 −15.0557 − 15.0557i 0 0 0
17.2 0 0 0 −8.38994 + 7.38978i 0 10.8652 + 10.8652i 0 0 0
17.3 0 0 0 −0.572391 − 11.1657i 0 −1.80948 − 1.80948i 0 0 0
17.4 0 0 0 0.572391 + 11.1657i 0 −1.80948 − 1.80948i 0 0 0
17.5 0 0 0 8.38994 − 7.38978i 0 10.8652 + 10.8652i 0 0 0
17.6 0 0 0 9.93887 − 5.12043i 0 −15.0557 − 15.0557i 0 0 0
593.1 0 0 0 −9.93887 − 5.12043i 0 −15.0557 + 15.0557i 0 0 0
593.2 0 0 0 −8.38994 − 7.38978i 0 10.8652 − 10.8652i 0 0 0
593.3 0 0 0 −0.572391 + 11.1657i 0 −1.80948 + 1.80948i 0 0 0
593.4 0 0 0 0.572391 − 11.1657i 0 −1.80948 + 1.80948i 0 0 0
593.5 0 0 0 8.38994 + 7.38978i 0 10.8652 − 10.8652i 0 0 0
593.6 0 0 0 9.93887 + 5.12043i 0 −15.0557 + 15.0557i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.w.d 12
3.b odd 2 1 inner 720.4.w.d 12
4.b odd 2 1 45.4.f.a ✓ 12
5.c odd 4 1 inner 720.4.w.d 12
12.b even 2 1 45.4.f.a ✓ 12
15.e even 4 1 inner 720.4.w.d 12
20.d odd 2 1 225.4.f.c 12
20.e even 4 1 45.4.f.a ✓ 12
20.e even 4 1 225.4.f.c 12
60.h even 2 1 225.4.f.c 12
60.l odd 4 1 45.4.f.a ✓ 12
60.l odd 4 1 225.4.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.f.a ✓ 12 4.b odd 2 1
45.4.f.a ✓ 12 12.b even 2 1
45.4.f.a ✓ 12 20.e even 4 1
45.4.f.a ✓ 12 60.l odd 4 1
225.4.f.c 12 20.d odd 2 1
225.4.f.c 12 20.e even 4 1
225.4.f.c 12 60.h even 2 1
225.4.f.c 12 60.l odd 4 1
720.4.w.d 12 1.a even 1 1 trivial
720.4.w.d 12 3.b odd 2 1 inner
720.4.w.d 12 5.c odd 4 1 inner
720.4.w.d 12 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 12T_{7}^{5} + 72T_{7}^{4} - 2560T_{7}^{3} + 97344T_{7}^{2} + 369408T_{7} + 700928 \) acting on \(S_{4}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 12 T^{5} + \cdots + 700928)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 4344 T^{4} + \cdots + 2390999552)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 54 T^{5} + \cdots + 40033352)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + 26640 T^{4} + \cdots + 604661760000)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 3276052522632)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 312 T^{2} + \cdots - 333184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 414 T^{5} + \cdots + 431026413512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 12363096915968)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 24 T^{2} + \cdots - 70019072)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 25\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 826210599600128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 35\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
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