Properties

Label 720.6.f.m
Level $720$
Weight $6$
Character orbit 720.f
Analytic conductor $115.476$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,6,Mod(289,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.289");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.f (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: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 373x^{4} + 33732x^{2} + 186624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 60)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 6) q^{5} + (\beta_{5} - \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 6) q^{5} + (\beta_{5} - \beta_{2} + \beta_1) q^{7} + (\beta_{4} + \beta_{2} + 49) q^{11} + ( - 3 \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{13} + (5 \beta_{5} + 7 \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 7) q^{17} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 12 \beta_{2} - 2 \beta_1 + 996) q^{19} + ( - 16 \beta_{5} - 2 \beta_{3} + 12 \beta_{2} - 11 \beta_1 + 2) q^{23} + ( - 15 \beta_{5} + 4 \beta_{4} + \beta_{3} - \beta_{2} + 16 \beta_1 + 1014) q^{25} + ( - 3 \beta_{5} + 3 \beta_{3} - 21 \beta_{2} + 3 \beta_1 + 2661) q^{29} + (10 \beta_{5} + 8 \beta_{4} - 10 \beta_{3} + 78 \beta_{2} - 10 \beta_1 - 70) q^{31} + (50 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + 93 \beta_1 + 3335) q^{35} + ( - 19 \beta_{5} + 21 \beta_{3} + 61 \beta_{2} - 132 \beta_1 - 21) q^{37} + ( - 14 \beta_{5} + 4 \beta_{4} + 14 \beta_{3} - 94 \beta_{2} + 14 \beta_1 - 2692) q^{41} + ( - 64 \beta_{5} - 72 \beta_{3} - 80 \beta_{2} - 193 \beta_1 + 72) q^{43} + (146 \beta_{5} + 46 \beta_{3} - 54 \beta_{2} - 363 \beta_1 - 46) q^{47} + ( - 38 \beta_{5} - 16 \beta_{4} + 38 \beta_{3} - 282 \beta_{2} + \cdots - 4515) q^{49}+ \cdots + (256 \beta_{5} - 84 \beta_{3} - 424 \beta_{2} - 2946 \beta_1 + 84) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 38 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 38 q^{5} + 296 q^{11} + 6000 q^{19} + 6054 q^{25} + 15924 q^{29} - 264 q^{31} + 20096 q^{35} - 16340 q^{41} - 27654 q^{49} + 26088 q^{55} + 92456 q^{59} + 6252 q^{61} + 52440 q^{65} - 160800 q^{71} - 128952 q^{79} - 177864 q^{85} - 76060 q^{89} + 98400 q^{91} + 232800 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 373x^{4} + 33732x^{2} + 186624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 805\nu^{3} - 114516\nu ) / 7416 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43\nu^{5} + 972\nu^{4} + 12367\nu^{3} + 248508\nu^{2} + 1030788\nu + 7722864 ) / 133488 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -49\nu^{5} - 972\nu^{4} - 17197\nu^{3} - 248508\nu^{2} - 383004\nu - 7656120 ) / 66744 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -43\nu^{5} - 4212\nu^{4} - 12367\nu^{3} - 186948\nu^{2} - 1030788\nu + 77373792 ) / 133488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -139\nu^{5} + 324\nu^{4} - 45151\nu^{3} + 82836\nu^{2} - 3347604\nu + 2574288 ) / 44496 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{3} + 6\beta_{2} - 2\beta _1 - 3 ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 3\beta_{4} - \beta_{3} + 10\beta_{2} - \beta _1 - 2490 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12\beta_{5} - 187\beta_{3} - 386\beta_{2} - 182\beta _1 + 187 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{5} - 767\beta_{4} - 19\beta_{3} - 634\beta_{2} - 19\beta _1 + 477978 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9660\beta_{5} + 36019\beta_{3} + 81698\beta_{2} + 74534\beta _1 - 36019 ) / 20 \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
11.7229i
11.7229i
15.1546i
15.1546i
2.43168i
2.43168i
0 0 0 −54.4661 12.5876i 0 12.5817i 0 0 0
289.2 0 0 0 −54.4661 + 12.5876i 0 12.5817i 0 0 0
289.3 0 0 0 20.3651 52.0602i 0 121.510i 0 0 0
289.4 0 0 0 20.3651 + 52.0602i 0 121.510i 0 0 0
289.5 0 0 0 53.1009 17.4726i 0 222.092i 0 0 0
289.6 0 0 0 53.1009 + 17.4726i 0 222.092i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.6.f.m 6
3.b odd 2 1 240.6.f.d 6
4.b odd 2 1 180.6.d.d 6
5.b even 2 1 inner 720.6.f.m 6
12.b even 2 1 60.6.d.a 6
15.d odd 2 1 240.6.f.d 6
20.d odd 2 1 180.6.d.d 6
20.e even 4 1 900.6.a.w 3
20.e even 4 1 900.6.a.x 3
60.h even 2 1 60.6.d.a 6
60.l odd 4 1 300.6.a.i 3
60.l odd 4 1 300.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.d.a 6 12.b even 2 1
60.6.d.a 6 60.h even 2 1
180.6.d.d 6 4.b odd 2 1
180.6.d.d 6 20.d odd 2 1
240.6.f.d 6 3.b odd 2 1
240.6.f.d 6 15.d odd 2 1
300.6.a.i 3 60.l odd 4 1
300.6.a.j 3 60.l odd 4 1
720.6.f.m 6 1.a even 1 1 trivial
720.6.f.m 6 5.b even 2 1 inner
900.6.a.w 3 20.e even 4 1
900.6.a.x 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{6} + 64248T_{7}^{4} + 738417168T_{7}^{2} + 115284695296 \) Copy content Toggle raw display
\( T_{11}^{3} - 148T_{11}^{2} - 480532T_{11} + 78696304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 38 T^{5} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{6} + 64248 T^{4} + \cdots + 115284695296 \) Copy content Toggle raw display
$11$ \( (T^{3} - 148 T^{2} - 480532 T + 78696304)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 1081800 T^{4} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{6} + 5928408 T^{4} + \cdots + 50\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{3} - 3000 T^{2} + 316800 T + 829440000)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 13854768 T^{4} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{3} - 7962 T^{2} + \cdots - 12509144064)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 132 T^{2} + \cdots - 233735426816)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 171675912 T^{4} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{3} + 8170 T^{2} + \cdots - 323508673000)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 836774448 T^{4} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} + 1500491088 T^{4} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{6} + 138197592 T^{4} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{3} - 46228 T^{2} + \cdots + 7848174247024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 3126 T^{2} + \cdots + 20932589363512)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 6887557008 T^{4} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{3} + 80400 T^{2} + \cdots + 13864753152000)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 7132671072 T^{4} + \cdots + 57\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{3} + 64476 T^{2} + \cdots - 33510082240512)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 11536133808 T^{4} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{3} + 38030 T^{2} + \cdots - 129133988519000)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 43563893952 T^{4} + \cdots + 97\!\cdots\!04 \) Copy content Toggle raw display
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