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Properties

Label 1078.2.e.t
Level $1078$
Weight $2$
Character orbit 1078.e
Analytic conductor $8.608$
Analytic rank $0$
Dimension $4$
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] = [1078,2,Mod(67,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1078.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.e (of order \(3\), 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: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} - 2 \beta_1 q^{5} - q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} - 2 \beta_1 q^{5} - q^{8} + (3 \beta_{2} + 3) q^{9} + ( - 2 \beta_{3} - 2 \beta_1) q^{10} + \beta_{2} q^{11} - 3 \beta_{3} q^{13} + ( - \beta_{2} - 1) q^{16} + ( - 2 \beta_{3} - 2 \beta_1) q^{17} + 3 \beta_{2} q^{18} - \beta_1 q^{19} - 2 \beta_{3} q^{20} - q^{22} + ( - 6 \beta_{2} - 6) q^{23} + 3 \beta_{2} q^{25} + 3 \beta_1 q^{26} + 8 q^{29} + ( - \beta_{3} - \beta_1) q^{31} - \beta_{2} q^{32} - 2 \beta_{3} q^{34} - 3 q^{36} + (6 \beta_{2} + 6) q^{37} + ( - \beta_{3} - \beta_1) q^{38} + 2 \beta_1 q^{40} - 6 \beta_{3} q^{41} + 10 q^{43} + ( - \beta_{2} - 1) q^{44} + ( - 6 \beta_{3} - 6 \beta_1) q^{45} - 6 \beta_{2} q^{46} + 5 \beta_1 q^{47} - 3 q^{50} + (3 \beta_{3} + 3 \beta_1) q^{52} + 6 \beta_{2} q^{53} - 2 \beta_{3} q^{55} + (8 \beta_{2} + 8) q^{58} + ( - 10 \beta_{3} - 10 \beta_1) q^{59} + 3 \beta_1 q^{61} - \beta_{3} q^{62} + q^{64} + ( - 12 \beta_{2} - 12) q^{65} - 4 \beta_{2} q^{67} + 2 \beta_1 q^{68} + ( - 3 \beta_{2} - 3) q^{72} + (6 \beta_{3} + 6 \beta_1) q^{73} + 6 \beta_{2} q^{74} - \beta_{3} q^{76} + (2 \beta_{3} + 2 \beta_1) q^{80} + 9 \beta_{2} q^{81} + 6 \beta_1 q^{82} + 5 \beta_{3} q^{83} - 8 q^{85} + (10 \beta_{2} + 10) q^{86} - \beta_{2} q^{88} - 13 \beta_1 q^{89} - 6 \beta_{3} q^{90} + 6 q^{92} + (5 \beta_{3} + 5 \beta_1) q^{94} + 4 \beta_{2} q^{95} + \beta_{3} q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + 6 q^{9} - 2 q^{11} - 2 q^{16} - 6 q^{18} - 4 q^{22} - 12 q^{23} - 6 q^{25} + 32 q^{29} + 2 q^{32} - 12 q^{36} + 12 q^{37} + 40 q^{43} - 2 q^{44} + 12 q^{46} - 12 q^{50} - 12 q^{53} + 16 q^{58} + 4 q^{64} - 24 q^{65} + 8 q^{67} - 6 q^{72} - 12 q^{74} - 18 q^{81} - 32 q^{85} + 20 q^{86} + 2 q^{88} + 24 q^{92} - 8 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.41421 − 2.44949i 0 0 −1.00000 1.50000 + 2.59808i 1.41421 − 2.44949i
67.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.41421 + 2.44949i 0 0 −1.00000 1.50000 + 2.59808i −1.41421 + 2.44949i
177.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.41421 + 2.44949i 0 0 −1.00000 1.50000 − 2.59808i 1.41421 + 2.44949i
177.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.41421 − 2.44949i 0 0 −1.00000 1.50000 − 2.59808i −1.41421 − 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.t 4
7.b odd 2 1 inner 1078.2.e.t 4
7.c even 3 1 1078.2.a.p ✓ 2
7.c even 3 1 inner 1078.2.e.t 4
7.d odd 6 1 1078.2.a.p ✓ 2
7.d odd 6 1 inner 1078.2.e.t 4
21.g even 6 1 9702.2.a.df 2
21.h odd 6 1 9702.2.a.df 2
28.f even 6 1 8624.2.a.bl 2
28.g odd 6 1 8624.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.p ✓ 2 7.c even 3 1
1078.2.a.p ✓ 2 7.d odd 6 1
1078.2.e.t 4 1.a even 1 1 trivial
1078.2.e.t 4 7.b odd 2 1 inner
1078.2.e.t 4 7.c even 3 1 inner
1078.2.e.t 4 7.d odd 6 1 inner
8624.2.a.bl 2 28.f even 6 1
8624.2.a.bl 2 28.g odd 6 1
9702.2.a.df 2 21.g even 6 1
9702.2.a.df 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T - 8)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$61$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 338 T^{2} + 114244 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
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