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Properties

Label 1620.3.o.f
Level $1620$
Weight $3$
Character orbit 1620.o
Analytic conductor $44.142$
Analytic rank $0$
Dimension $8$
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] = [1620,3,Mod(701,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.o (of order \(6\), 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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{7} - 4 \beta_{2} + 4) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{7} - 4 \beta_{2} + 4) q^{7} + (6 \beta_{5} - \beta_{4}) q^{11} + (\beta_{6} - 2 \beta_{2}) q^{13} - 4 \beta_{3} q^{17} + (2 \beta_{7} - 2 \beta_{6} + 8) q^{19} + (2 \beta_{4} - 2 \beta_{3} + 6 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + (6 \beta_{5} - 8 \beta_{4}) q^{29} + (2 \beta_{6} - 2 \beta_{2}) q^{31} + (4 \beta_{5} - 5 \beta_{3} + 4 \beta_1) q^{35} + ( - 3 \beta_{7} + 3 \beta_{6} - 34) q^{37} + ( - 3 \beta_{4} + 3 \beta_{3} - 24 \beta_1) q^{41} + (2 \beta_{7} + 20 \beta_{2} - 20) q^{43} + (12 \beta_{5} + 14 \beta_{4}) q^{47} + ( - 8 \beta_{6} - 57 \beta_{2}) q^{49} + (24 \beta_{5} + 10 \beta_{3} + 24 \beta_1) q^{53} + (\beta_{7} - \beta_{6} - 30) q^{55} + (17 \beta_{4} - 17 \beta_{3} - 18 \beta_1) q^{59} + (6 \beta_{7} - 10 \beta_{2} + 10) q^{61} + (2 \beta_{5} + 5 \beta_{4}) q^{65} + 76 \beta_{2} q^{67} + ( - 12 \beta_{5} + 12 \beta_{3} - 12 \beta_1) q^{71} + ( - 6 \beta_{7} + 6 \beta_{6} + 38) q^{73} + ( - 34 \beta_{4} + 34 \beta_{3} - 42 \beta_1) q^{77} + ( - 6 \beta_{7} + 50 \beta_{2} - 50) q^{79} + (24 \beta_{5} - 2 \beta_{4}) q^{83} - 4 \beta_{6} q^{85} + (12 \beta_{5} + 21 \beta_{3} + 12 \beta_1) q^{89} + ( - 2 \beta_{7} + 2 \beta_{6} + 82) q^{91} + ( - 10 \beta_{4} + 10 \beta_{3} + 8 \beta_1) q^{95} + ( - 2 \beta_{7} - 106 \beta_{2} + 106) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} + 7\nu^{5} - 35\nu^{3} + 81\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 13\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} + 49\nu^{5} + 133\nu^{3} + 684\nu ) / 63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - 7\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 19\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{5} - 7\beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -29\beta_{6} - 13\beta_{4} + 13\beta_{3} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
−1.40294 + 1.01575i
1.40294 − 1.01575i
−0.178197 + 1.72286i
0.178197 − 1.72286i
−1.40294 − 1.01575i
1.40294 + 1.01575i
−0.178197 − 1.72286i
0.178197 + 1.72286i
0 0 0 −1.93649 − 1.11803i 0 −2.74342 − 4.75174i 0 0 0
701.2 0 0 0 −1.93649 − 1.11803i 0 6.74342 + 11.6799i 0 0 0
701.3 0 0 0 1.93649 + 1.11803i 0 −2.74342 − 4.75174i 0 0 0
701.4 0 0 0 1.93649 + 1.11803i 0 6.74342 + 11.6799i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −2.74342 + 4.75174i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 6.74342 − 11.6799i 0 0 0
1241.3 0 0 0 1.93649 − 1.11803i 0 −2.74342 + 4.75174i 0 0 0
1241.4 0 0 0 1.93649 − 1.11803i 0 6.74342 − 11.6799i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.o.f 8
3.b odd 2 1 inner 1620.3.o.f 8
9.c even 3 1 180.3.g.a ✓ 4
9.c even 3 1 inner 1620.3.o.f 8
9.d odd 6 1 180.3.g.a ✓ 4
9.d odd 6 1 inner 1620.3.o.f 8
36.f odd 6 1 720.3.l.c 4
36.h even 6 1 720.3.l.c 4
45.h odd 6 1 900.3.g.d 4
45.j even 6 1 900.3.g.d 4
45.k odd 12 2 900.3.b.b 8
45.l even 12 2 900.3.b.b 8
72.j odd 6 1 2880.3.l.b 4
72.l even 6 1 2880.3.l.f 4
72.n even 6 1 2880.3.l.b 4
72.p odd 6 1 2880.3.l.f 4
180.n even 6 1 3600.3.l.n 4
180.p odd 6 1 3600.3.l.n 4
180.v odd 12 2 3600.3.c.k 8
180.x even 12 2 3600.3.c.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a ✓ 4 9.c even 3 1
180.3.g.a ✓ 4 9.d odd 6 1
720.3.l.c 4 36.f odd 6 1
720.3.l.c 4 36.h even 6 1
900.3.b.b 8 45.k odd 12 2
900.3.b.b 8 45.l even 12 2
900.3.g.d 4 45.h odd 6 1
900.3.g.d 4 45.j even 6 1
1620.3.o.f 8 1.a even 1 1 trivial
1620.3.o.f 8 3.b odd 2 1 inner
1620.3.o.f 8 9.c even 3 1 inner
1620.3.o.f 8 9.d odd 6 1 inner
2880.3.l.b 4 72.j odd 6 1
2880.3.l.b 4 72.n even 6 1
2880.3.l.f 4 72.l even 6 1
2880.3.l.f 4 72.p odd 6 1
3600.3.c.k 8 180.v odd 12 2
3600.3.c.k 8 180.x even 12 2
3600.3.l.n 4 180.n even 6 1
3600.3.l.n 4 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 8T_{7}^{3} + 138T_{7}^{2} + 592T_{7} + 5476 \) acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\). 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
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 396 T^{6} + \cdots + 688747536 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots + 7396)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T - 296)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 504 T^{6} + \cdots + 136048896 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 892616806656 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 126736)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 68 T + 346)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 54575510850576 \) Copy content Toggle raw display
$43$ \( (T^{4} + 40 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 62171080298496 \) Copy content Toggle raw display
$53$ \( (T^{4} + 9360 T^{2} + 1166400)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 164627478364176 \) Copy content Toggle raw display
$61$ \( (T^{4} - 20 T^{3} + \cdots + 9859600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 76 T + 5776)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6624 T^{2} + 3504384)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 76 T - 1796)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 100 T^{3} + \cdots + 547600)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 62171080298496 \) Copy content Toggle raw display
$89$ \( (T^{4} + 17316 T^{2} + 52099524)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 212 T^{3} + \cdots + 118287376)^{2} \) Copy content Toggle raw display
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