Properties

Label 720.6.f.n
Level $720$
Weight $6$
Character orbit 720.f
Analytic conductor $115.476$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 40)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{5} + (\beta_{6} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{5} + (\beta_{6} - \beta_1) q^{7} + (\beta_{4} - \beta_1 - 92) q^{11} + (\beta_{7} - 4 \beta_{6} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + ( - 2 \beta_{7} - \beta_{3} - 14 \beta_{2} + 9 \beta_1) q^{17} + (4 \beta_{7} + \beta_{4} + 4 \beta_{2} + 19 \beta_1 - 172) q^{19} + ( - 4 \beta_{7} + 9 \beta_{6} - 10 \beta_{3} + 6 \beta_{2} + \beta_1) q^{23} + (3 \beta_{7} + 8 \beta_{6} + \beta_{5} - 4 \beta_{4} + 9 \beta_{3} - 25 \beta_{2} + \cdots - 267) q^{25}+ \cdots + (86 \beta_{7} - 488 \beta_{6} + 49 \beta_{3} - 486 \beta_{2} + 107 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 736 q^{11} - 1376 q^{19} - 2136 q^{25} - 5872 q^{29} - 4224 q^{31} + 19232 q^{35} - 23600 q^{41} - 45000 q^{49} - 15008 q^{55} + 91680 q^{59} + 123856 q^{61} + 72064 q^{65} - 125632 q^{71} - 43264 q^{79} - 293760 q^{85} + 41904 q^{89} + 487616 q^{91} + 442592 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -176\nu^{7} - 66\nu^{6} - 7372\nu^{5} - 2232\nu^{4} - 86042\nu^{3} - 19752\nu^{2} - 203676\nu - 29412 ) / 639 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -182\nu^{7} - 7420\nu^{5} - 82106\nu^{3} - 160458\nu ) / 213 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -656\nu^{7} + 66\nu^{6} - 26548\nu^{5} + 2232\nu^{4} - 295142\nu^{3} + 19752\nu^{2} - 646692\nu + 29412 ) / 639 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -176\nu^{7} - 282\nu^{6} - 7372\nu^{5} - 17592\nu^{4} - 86042\nu^{3} - 259752\nu^{2} - 203676\nu - 390564 ) / 639 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -176\nu^{7} + 1710\nu^{6} - 7372\nu^{5} + 59688\nu^{4} - 86042\nu^{3} + 457848\nu^{2} - 203676\nu - 144180 ) / 639 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 974\nu^{7} - 66\nu^{6} + 40168\nu^{5} - 2232\nu^{4} + 454172\nu^{3} - 19752\nu^{2} + 991374\nu - 29412 ) / 639 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1426\nu^{7} - 330\nu^{6} + 59120\nu^{5} - 11160\nu^{4} + 676528\nu^{3} - 98760\nu^{2} + 1499754\nu - 147060 ) / 639 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -24\beta_{7} + 72\beta_{6} - 9\beta_{3} + 64\beta_{2} + 39\beta_1 ) / 3840 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -34\beta_{7} - 13\beta_{5} - 3\beta_{4} - 34\beta_{2} - 154\beta _1 - 19680 ) / 1920 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} - 18\beta_{6} - 3\beta_{3} - 8\beta_{2} - 15\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 790\beta_{7} + 283\beta_{5} - 87\beta_{4} + 790\beta_{2} + 3754\beta _1 + 365280 ) / 1920 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9048\beta_{7} + 30888\beta_{6} + 10329\beta_{3} + 11104\beta_{2} + 24681\beta_1 ) / 3840 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -230\beta_{7} - 71\beta_{5} + 48\beta_{4} - 230\beta_{2} - 1127\beta _1 - 91488 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 173496\beta_{7} - 673128\beta_{6} - 304899\beta_{3} - 224896\beta_{2} - 499251\beta_1 ) / 3840 \) 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
0.0965878i
0.0965878i
3.98753i
3.98753i
1.64654i
1.64654i
4.73066i
4.73066i
0 0 0 −46.7401 30.6653i 0 179.876i 0 0 0
289.2 0 0 0 −46.7401 + 30.6653i 0 179.876i 0 0 0
289.3 0 0 0 −23.4238 50.7575i 0 10.2635i 0 0 0
289.4 0 0 0 −23.4238 + 50.7575i 0 10.2635i 0 0 0
289.5 0 0 0 13.1588 54.3309i 0 146.828i 0 0 0
289.6 0 0 0 13.1588 + 54.3309i 0 146.828i 0 0 0
289.7 0 0 0 53.0051 17.7613i 0 188.968i 0 0 0
289.8 0 0 0 53.0051 + 17.7613i 0 188.968i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
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.n 8
3.b odd 2 1 80.6.c.d 8
4.b odd 2 1 360.6.f.b 8
5.b even 2 1 inner 720.6.f.n 8
12.b even 2 1 40.6.c.a 8
15.d odd 2 1 80.6.c.d 8
15.e even 4 1 400.6.a.z 4
15.e even 4 1 400.6.a.ba 4
20.d odd 2 1 360.6.f.b 8
24.f even 2 1 320.6.c.j 8
24.h odd 2 1 320.6.c.i 8
60.h even 2 1 40.6.c.a 8
60.l odd 4 1 200.6.a.j 4
60.l odd 4 1 200.6.a.k 4
120.i odd 2 1 320.6.c.i 8
120.m even 2 1 320.6.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 12.b even 2 1
40.6.c.a 8 60.h even 2 1
80.6.c.d 8 3.b odd 2 1
80.6.c.d 8 15.d odd 2 1
200.6.a.j 4 60.l odd 4 1
200.6.a.k 4 60.l odd 4 1
320.6.c.i 8 24.h odd 2 1
320.6.c.i 8 120.i odd 2 1
320.6.c.j 8 24.f even 2 1
320.6.c.j 8 120.m even 2 1
360.6.f.b 8 4.b odd 2 1
360.6.f.b 8 20.d odd 2 1
400.6.a.z 4 15.e even 4 1
400.6.a.ba 4 15.e even 4 1
720.6.f.n 8 1.a even 1 1 trivial
720.6.f.n 8 5.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 \(S_{6}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{8} + 89728T_{7}^{6} + 2632172640T_{7}^{4} + 25184325932032T_{7}^{2} + 2623778518100224 \) Copy content Toggle raw display
\( T_{11}^{4} + 368T_{11}^{3} - 468000T_{11}^{2} - 126641408T_{11} + 37397137664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{8} + 89728 T^{6} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{4} + 368 T^{3} + \cdots + 37397137664)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 1391872 T^{6} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} + 5097984 T^{6} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + 688 T^{3} + \cdots + 1042985883904)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 16814848 T^{6} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2936 T^{3} + \cdots - 97147517576176)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2112 T^{3} + \cdots + 244229603328000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 302977024 T^{6} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + 11800 T^{3} + \cdots - 296811236945008)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 892855872 T^{6} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + 871407744 T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + 2162178304 T^{6} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{4} - 45840 T^{3} + \cdots - 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 61928 T^{3} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 1519081792 T^{6} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} + 62816 T^{3} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 7443893248 T^{6} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{4} + 21632 T^{3} + \cdots - 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 24752427328 T^{6} + \cdots + 64\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} - 20952 T^{3} + \cdots - 93\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 23434806784 T^{6} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
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