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Properties

Label 1620.3.o.e
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: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 540)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{5} + (\beta_{6} + 5 \beta_{2} - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{5} + (\beta_{6} + 5 \beta_{2} - 5) q^{7} + ( - \beta_{3} - 3 \beta_1) q^{11} + ( - \beta_{5} + 7 \beta_{2}) q^{13} + (5 \beta_{7} + 5 \beta_{3}) q^{17} + ( - 2 \beta_{6} + 2 \beta_{5} - 1) q^{19} + (7 \beta_{7} + 6 \beta_{4} - 6 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + (\beta_{3} - 3 \beta_1) q^{29} + ( - 2 \beta_{5} + 7 \beta_{2}) q^{31} + ( - 5 \beta_{7} - 5 \beta_{4} - 5 \beta_{3}) q^{35} + ( - 6 \beta_{6} + 6 \beta_{5} - 16) q^{37} + (3 \beta_{7} - 15 \beta_{4} + 15 \beta_1) q^{41} + (7 \beta_{6} + 29 \beta_{2} - 29) q^{43} + ( - 4 \beta_{3} - 6 \beta_1) q^{47} + ( - 10 \beta_{5} - 21 \beta_{2}) q^{49} + (\beta_{7} - 30 \beta_{4} + \beta_{3}) q^{53} + ( - \beta_{6} + \beta_{5} + 15) q^{55} + ( - 17 \beta_{7} - 9 \beta_{4} + 9 \beta_1) q^{59} + (12 \beta_{6} - \beta_{2} + 1) q^{61} + (5 \beta_{3} - 7 \beta_1) q^{65} - 14 \beta_{2} q^{67} + ( - 15 \beta_{7} - 21 \beta_{4} - 15 \beta_{3}) q^{71} + ( - 3 \beta_{6} + 3 \beta_{5} - 7) q^{73} + ( - 20 \beta_{7} - 24 \beta_{4} + 24 \beta_1) q^{77} + ( - 12 \beta_{6} - 67 \beta_{2} + 67) q^{79} + ( - 11 \beta_{3} - 48 \beta_1) q^{83} - 5 \beta_{5} q^{85} + (3 \beta_{7} - 51 \beta_{4} + 3 \beta_{3}) q^{89} + (2 \beta_{6} - 2 \beta_{5} + 10) q^{91} + (10 \beta_{7} - \beta_{4} + \beta_1) q^{95} + (2 \beta_{6} + 2 \beta_{2} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 29\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 39\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -27\nu^{6} + 72\nu^{4} - 168\nu^{2} + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -21\nu^{6} + 72\nu^{4} - 168\nu^{2} + 63 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{7} + 48\nu^{5} - 120\nu^{3} + 45\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 9\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 3\beta_{4} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + 7\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} + 15\beta_{4} - 15\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{6} - 4\beta_{5} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -29\beta_{3} - 39\beta_1 ) / 6 \) 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
1.40126 − 0.809017i
0.535233 − 0.309017i
−1.40126 + 0.809017i
−0.535233 + 0.309017i
1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 − 0.809017i
−0.535233 − 0.309017i
0 0 0 −1.93649 − 1.11803i 0 −5.85410 − 10.1396i 0 0 0
701.2 0 0 0 −1.93649 − 1.11803i 0 0.854102 + 1.47935i 0 0 0
701.3 0 0 0 1.93649 + 1.11803i 0 −5.85410 − 10.1396i 0 0 0
701.4 0 0 0 1.93649 + 1.11803i 0 0.854102 + 1.47935i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −5.85410 + 10.1396i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 0.854102 − 1.47935i 0 0 0
1241.3 0 0 0 1.93649 − 1.11803i 0 −5.85410 + 10.1396i 0 0 0
1241.4 0 0 0 1.93649 − 1.11803i 0 0.854102 − 1.47935i 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.e 8
3.b odd 2 1 inner 1620.3.o.e 8
9.c even 3 1 540.3.g.d ✓ 4
9.c even 3 1 inner 1620.3.o.e 8
9.d odd 6 1 540.3.g.d ✓ 4
9.d odd 6 1 inner 1620.3.o.e 8
36.f odd 6 1 2160.3.l.e 4
36.h even 6 1 2160.3.l.e 4
45.h odd 6 1 2700.3.g.n 4
45.j even 6 1 2700.3.g.n 4
45.k odd 12 1 2700.3.b.g 4
45.k odd 12 1 2700.3.b.l 4
45.l even 12 1 2700.3.b.g 4
45.l even 12 1 2700.3.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.d ✓ 4 9.c even 3 1
540.3.g.d ✓ 4 9.d odd 6 1
1620.3.o.e 8 1.a even 1 1 trivial
1620.3.o.e 8 3.b odd 2 1 inner
1620.3.o.e 8 9.c even 3 1 inner
1620.3.o.e 8 9.d odd 6 1 inner
2160.3.l.e 4 36.f odd 6 1
2160.3.l.e 4 36.h even 6 1
2700.3.b.g 4 45.k odd 12 1
2700.3.b.g 4 45.l even 12 1
2700.3.b.l 4 45.k odd 12 1
2700.3.b.l 4 45.l even 12 1
2700.3.g.n 4 45.h odd 6 1
2700.3.g.n 4 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 10T_{7}^{3} + 120T_{7}^{2} - 200T_{7} + 400 \) 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
$7$ \( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 108 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$13$ \( (T^{4} - 14 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 225)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 179)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 4640470641 \) Copy content Toggle raw display
$29$ \( T^{8} - 108 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$31$ \( (T^{4} - 14 T^{3} + \cdots + 17161)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T - 1364)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1187960484096 \) Copy content Toggle raw display
$43$ \( (T^{4} + 58 T^{3} + \cdots + 1860496)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 648 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( (T^{4} + 9018 T^{2} + 20169081)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 23255696077056 \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} + \cdots + 41977441)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 14 T + 196)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8460 T^{2} + 32400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T - 356)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 134 T^{3} + \cdots + 3964081)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{4} + 26172 T^{2} + 167029776)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 30976)^{2} \) Copy content Toggle raw display
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