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Properties

Label 1638.2.m.h
Level $1638$
Weight $2$
Character orbit 1638.m
Analytic conductor $13.079$
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] = [1638,2,Mod(289,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.m (of order \(3\), degree \(2\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.447703281.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 2 q^{5} + 3 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 2 q^{5} + 3 q^{7} - 8 q^{8} + 2 q^{10} - 4 q^{11} + 3 q^{13} - 3 q^{14} + 8 q^{16} - 4 q^{17} - 4 q^{19} - 2 q^{20} + 4 q^{22} + 8 q^{23} + 2 q^{25} - 3 q^{26} + 3 q^{28} - 2 q^{29} + 14 q^{31} - 8 q^{32} + 4 q^{34} + 22 q^{35} + 12 q^{37} + 4 q^{38} + 2 q^{40} - 12 q^{41} - 4 q^{44} - 8 q^{46} - 7 q^{47} + 5 q^{49} - 2 q^{50} + 3 q^{52} + q^{53} - 25 q^{55} - 3 q^{56} + 2 q^{58} + 32 q^{59} - 4 q^{61} - 14 q^{62} + 8 q^{64} - 10 q^{65} + 19 q^{67} - 4 q^{68} - 22 q^{70} - 20 q^{71} - 7 q^{73} - 12 q^{74} - 4 q^{76} + 24 q^{77} + 24 q^{79} - 2 q^{80} + 12 q^{82} + 64 q^{83} + 15 q^{85} + 4 q^{88} - 22 q^{89} - 38 q^{91} + 8 q^{92} + 7 q^{94} + 56 q^{95} + 11 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 4\nu^{5} + 2\nu^{4} + \nu^{3} - 10\nu^{2} + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} - 2\nu^{3} - \nu^{2} + 6\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - \nu^{6} + 2\nu^{5} + 2\nu^{4} - \nu^{3} - 16\nu^{2} - 8\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - 6\nu^{4} + \nu^{3} + 16\nu^{2} + 4\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - \nu^{6} - 4\nu^{5} + 3\nu^{3} + 8\nu^{2} - 2\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - 6\nu^{4} + \nu^{3} + 24\nu^{2} + 12\nu - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + \nu^{6} - 9\nu^{5} - 6\nu^{4} + 6\nu^{3} + 27\nu^{2} + 4\nu - 40 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - \beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} + \beta_{2} + 4\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{6} - 4\beta_{5} - 3\beta_{4} - 4\beta_{3} - 5\beta_{2} + \beta _1 - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6\beta_{7} - 8\beta_{6} - 5\beta_{5} - 6\beta_{4} + \beta_{3} + 5\beta_{2} - 7\beta _1 - 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} - 11\beta_{6} + 4\beta_{5} + 15\beta_{4} - 11\beta_{3} + 2\beta_{2} + 5\beta _1 - 12 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1 + \beta_{2}\) \(1\)

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
−1.38232 − 0.298668i
1.26359 + 0.635098i
−0.571299 + 1.29368i
1.19003 − 0.764088i
−1.38232 + 0.298668i
1.26359 − 0.635098i
−0.571299 − 1.29368i
1.19003 + 0.764088i
−1.00000 0 1.00000 −1.75410 + 3.03819i 0 −2.63641 + 0.222079i −1.00000 0 1.75410 − 3.03819i
289.2 −1.00000 0 1.00000 −0.611519 + 1.05918i 0 1.15207 − 2.38175i −1.00000 0 0.611519 − 1.05918i
289.3 −1.00000 0 1.00000 0.441221 − 0.764218i 0 0.369922 + 2.61976i −1.00000 0 −0.441221 + 0.764218i
289.4 −1.00000 0 1.00000 0.924396 − 1.60110i 0 2.61442 + 0.405935i −1.00000 0 −0.924396 + 1.60110i
1621.1 −1.00000 0 1.00000 −1.75410 − 3.03819i 0 −2.63641 − 0.222079i −1.00000 0 1.75410 + 3.03819i
1621.2 −1.00000 0 1.00000 −0.611519 − 1.05918i 0 1.15207 + 2.38175i −1.00000 0 0.611519 + 1.05918i
1621.3 −1.00000 0 1.00000 0.441221 + 0.764218i 0 0.369922 − 2.61976i −1.00000 0 −0.441221 − 0.764218i
1621.4 −1.00000 0 1.00000 0.924396 + 1.60110i 0 2.61442 − 0.405935i −1.00000 0 −0.924396 − 1.60110i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.m.h 8
3.b odd 2 1 546.2.j.c ✓ 8
7.c even 3 1 1638.2.p.h 8
13.c even 3 1 1638.2.p.h 8
21.h odd 6 1 546.2.k.c yes 8
39.i odd 6 1 546.2.k.c yes 8
91.h even 3 1 inner 1638.2.m.h 8
273.s odd 6 1 546.2.j.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.c ✓ 8 3.b odd 2 1
546.2.j.c ✓ 8 273.s odd 6 1
546.2.k.c yes 8 21.h odd 6 1
546.2.k.c yes 8 39.i odd 6 1
1638.2.m.h 8 1.a even 1 1 trivial
1638.2.m.h 8 91.h even 3 1 inner
1638.2.p.h 8 7.c even 3 1
1638.2.p.h 8 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2T_{5}^{7} + 11T_{5}^{6} - 6T_{5}^{5} + 50T_{5}^{4} + 65T_{5}^{2} - 28T_{5} + 49 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} - 44 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 23409 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} - 34 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 1121481 \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + \cdots + 219961 \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + 8 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 881721 \) Copy content Toggle raw display
$43$ \( T^{8} + 34 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( T^{8} + 7 T^{7} + \cdots + 3969 \) Copy content Toggle raw display
$53$ \( T^{8} - T^{7} + \cdots + 2283121 \) Copy content Toggle raw display
$59$ \( (T^{4} - 16 T^{3} + \cdots - 1079)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 186624 \) Copy content Toggle raw display
$67$ \( T^{8} - 19 T^{7} + \cdots + 13697401 \) Copy content Toggle raw display
$71$ \( T^{8} + 20 T^{7} + \cdots + 79762761 \) Copy content Toggle raw display
$73$ \( T^{8} + 7 T^{7} + \cdots + 388129 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots + 4313929 \) Copy content Toggle raw display
$83$ \( (T^{4} - 32 T^{3} + \cdots + 2429)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 11 T^{3} + \cdots + 2949)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 11 T^{7} + \cdots + 4239481 \) Copy content Toggle raw display
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