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Properties

Label 380.2.i.a
Level $380$
Weight $2$
Character orbit 380.i
Analytic conductor $3.034$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(121,380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("380.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 4 q^{7} - \zeta_{6} q^{9} - 3 q^{11} - 6 \zeta_{6} q^{13} + 2 \zeta_{6} q^{15} + (2 \zeta_{6} - 2) q^{17} + (3 \zeta_{6} + 2) q^{19} + ( - 8 \zeta_{6} + 8) q^{21} + \cdots + 3 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} - 8 q^{7} - q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 7 q^{19} + 8 q^{21} - 4 q^{23} - q^{25} - 8 q^{27} - q^{29} - 10 q^{31} + 6 q^{33} - 4 q^{35} - 8 q^{37} + 24 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −1.00000 − 1.73205i 0 0.500000 + 0.866025i 0 −4.00000 0 −0.500000 + 0.866025i 0
201.1 0 −1.00000 + 1.73205i 0 0.500000 − 0.866025i 0 −4.00000 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.i.a ✓ 2
3.b odd 2 1 3420.2.t.b 2
4.b odd 2 1 1520.2.q.g 2
5.b even 2 1 1900.2.i.b 2
5.c odd 4 2 1900.2.s.b 4
19.c even 3 1 inner 380.2.i.a ✓ 2
19.c even 3 1 7220.2.a.e 1
19.d odd 6 1 7220.2.a.a 1
57.h odd 6 1 3420.2.t.b 2
76.g odd 6 1 1520.2.q.g 2
95.i even 6 1 1900.2.i.b 2
95.m odd 12 2 1900.2.s.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a ✓ 2 1.a even 1 1 trivial
380.2.i.a ✓ 2 19.c even 3 1 inner
1520.2.q.g 2 4.b odd 2 1
1520.2.q.g 2 76.g odd 6 1
1900.2.i.b 2 5.b even 2 1
1900.2.i.b 2 95.i even 6 1
1900.2.s.b 4 5.c odd 4 2
1900.2.s.b 4 95.m odd 12 2
3420.2.t.b 2 3.b odd 2 1
3420.2.t.b 2 57.h odd 6 1
7220.2.a.a 1 19.d odd 6 1
7220.2.a.e 1 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( (T + 5)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T - 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
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