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Properties

Label 1638.2.m.i
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: 8.0.6498455769.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 25x^{4} - 3x^{3} + 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{6} - \beta_{2} + \cdots + 1) q^{7}+ \cdots + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{6} - \beta_{2} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{7} - \beta_{5} - 6 \beta_{4} + \cdots + 7) q^{98}+O(q^{100}) \) 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 \( 8 q + 8 q^{2} + 8 q^{4} + 2 q^{5} + 3 q^{7} + 8 q^{8} + 2 q^{10} + 2 q^{11} + 7 q^{13} + 3 q^{14} + 8 q^{16} - 12 q^{17} + 2 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 6 q^{25} + 7 q^{26} + 3 q^{28} - 6 q^{29} + 10 q^{31} + 8 q^{32} - 12 q^{34} - 8 q^{35} + 24 q^{37} + 2 q^{38} + 2 q^{40} + 6 q^{41} - 4 q^{43} + 2 q^{44} + 8 q^{46} + 17 q^{47} + 17 q^{49} + 6 q^{50} + 7 q^{52} - 3 q^{53} + 25 q^{55} + 3 q^{56} - 6 q^{58} - 4 q^{61} + 10 q^{62} + 8 q^{64} - 12 q^{65} - 7 q^{67} - 12 q^{68} - 8 q^{70} - 6 q^{71} - 19 q^{73} + 24 q^{74} + 2 q^{76} + 10 q^{77} + 24 q^{79} + 2 q^{80} + 6 q^{82} - 12 q^{83} - 3 q^{85} - 4 q^{86} + 2 q^{88} - 14 q^{89} + 40 q^{91} + 8 q^{92} + 17 q^{94} - 24 q^{95} - 25 q^{97} + 17 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 25x^{4} - 3x^{3} + 6x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6\nu^{7} + 21\nu^{6} - 14\nu^{5} + 193\nu^{4} + 126\nu^{3} + 532\nu^{2} - 664\nu + 112 ) / 119 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{7} - 6\nu^{6} + 21\nu^{5} + 59\nu^{4} + 66\nu^{3} - 84\nu^{2} - 449\nu - 15 ) / 119 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -15\nu^{7} + 7\nu^{6} - 84\nu^{5} - 66\nu^{4} - 434\nu^{3} - 21\nu^{2} - 6\nu + 315 ) / 119 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -22\nu^{7} + 42\nu^{6} - 147\nu^{5} + 46\nu^{4} - 462\nu^{3} + 588\nu^{2} - 104\nu + 105 ) / 119 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -24\nu^{7} + 69\nu^{6} - 182\nu^{5} + 180\nu^{4} - 402\nu^{3} + 1085\nu^{2} - 81\nu - 6 ) / 119 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} + 14\nu^{5} - \nu^{4} + 54\nu^{3} - 28\nu^{2} + 47\nu - 4 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} - 4\nu^{6} + 28\nu^{5} + 22\nu^{4} + 121\nu^{3} + 7\nu^{2} + 2\nu + 4 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 2\beta_{4} + 2\beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 3\beta_{5} + 7\beta_{4} + \beta_{2} + \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 5\beta_{6} - 5\beta_{4} + 3\beta_{3} - 5\beta_{2} + 10\beta _1 - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{6} + 6\beta_{5} - 17\beta_{4} + 6\beta_{3} - 7\beta_{2} + 3\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{7} - 64\beta_{6} + 30\beta_{5} - 89\beta_{4} - 50\beta_{2} - 32\beta _1 + 57 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 30\beta_{7} - 71\beta_{6} + 71\beta_{4} - 114\beta_{3} + 71\beta_{2} - 142\beta _1 + 324 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 217\beta_{6} - 243\beta_{5} + 890\beta_{4} - 243\beta_{3} + 548\beta_{2} - 217\beta _1 + 243 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\beta_{4}\) \(-\beta_{4}\) \(1\)

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
1.33821 − 2.31784i
−0.186817 + 0.323577i
0.271028 − 0.469434i
−0.922415 + 1.59767i
1.33821 + 2.31784i
−0.186817 − 0.323577i
0.271028 + 0.469434i
−0.922415 − 1.59767i
1.00000 0 1.00000 −0.651388 + 1.12824i 0 −2.36323 − 1.18960i 1.00000 0 −0.651388 + 1.12824i
289.2 1.00000 0 1.00000 −0.651388 + 1.12824i 0 2.21184 + 1.45181i 1.00000 0 −0.651388 + 1.12824i
289.3 1.00000 0 1.00000 1.15139 − 1.99426i 0 −0.964471 − 2.46370i 1.00000 0 1.15139 − 1.99426i
289.4 1.00000 0 1.00000 1.15139 − 1.99426i 0 2.61586 − 0.396592i 1.00000 0 1.15139 − 1.99426i
1621.1 1.00000 0 1.00000 −0.651388 − 1.12824i 0 −2.36323 + 1.18960i 1.00000 0 −0.651388 − 1.12824i
1621.2 1.00000 0 1.00000 −0.651388 − 1.12824i 0 2.21184 − 1.45181i 1.00000 0 −0.651388 − 1.12824i
1621.3 1.00000 0 1.00000 1.15139 + 1.99426i 0 −0.964471 + 2.46370i 1.00000 0 1.15139 + 1.99426i
1621.4 1.00000 0 1.00000 1.15139 + 1.99426i 0 2.61586 + 0.396592i 1.00000 0 1.15139 + 1.99426i
\(n\): 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.i 8
3.b odd 2 1 546.2.j.b ✓ 8
7.c even 3 1 1638.2.p.g 8
13.c even 3 1 1638.2.p.g 8
21.h odd 6 1 546.2.k.d yes 8
39.i odd 6 1 546.2.k.d yes 8
91.h even 3 1 inner 1638.2.m.i 8
273.s odd 6 1 546.2.j.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.b ✓ 8 3.b odd 2 1
546.2.j.b ✓ 8 273.s odd 6 1
546.2.k.d yes 8 21.h odd 6 1
546.2.k.d yes 8 39.i odd 6 1
1638.2.m.i 8 1.a even 1 1 trivial
1638.2.m.i 8 91.h even 3 1 inner
1638.2.p.g 8 7.c even 3 1
1638.2.p.g 8 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 4T_{5}^{2} + 3T_{5} + 9 \) 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
$5$ \( (T^{4} - T^{3} + 4 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{8} - 7 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{3} + 2 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 2 T^{7} + \cdots + 89401 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} - 12 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + \cdots + 829)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 4397409 \) Copy content Toggle raw display
$43$ \( T^{8} + 4 T^{7} + \cdots + 29800681 \) Copy content Toggle raw display
$47$ \( T^{8} - 17 T^{7} + \cdots + 14085009 \) Copy content Toggle raw display
$53$ \( T^{8} + 3 T^{7} + \cdots + 1108809 \) Copy content Toggle raw display
$59$ \( (T^{4} - 135 T^{2} + 4293)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 719104 \) Copy content Toggle raw display
$67$ \( T^{8} + 7 T^{7} + \cdots + 97969 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 227529 \) Copy content Toggle raw display
$73$ \( T^{8} + 19 T^{7} + \cdots + 301401 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots + 860307561 \) Copy content Toggle raw display
$83$ \( (T^{4} + 6 T^{3} + 2 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 7 T^{3} + \cdots - 243)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 25 T^{7} + \cdots + 299209 \) Copy content Toggle raw display
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