Properties

Label 380.2.y.a
Level $380$
Weight $2$
Character orbit 380.y
Analytic conductor $3.034$
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] = [380,2,Mod(217,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.y (of order \(12\), degree \(4\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{3} + ( - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{7} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{3} + ( - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{7} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{9} + q^{11} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{13} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{15} + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{17} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}) q^{19} + ( - 4 \zeta_{12}^{2} - 4) q^{21} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{25} + (5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{29} + ( - 2 \zeta_{12}^{2} + 1) q^{31} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{33} + (6 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{35} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{37} - 6 \zeta_{12}^{3} q^{39} + (2 \zeta_{12}^{2} + 2) q^{41} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{43} + ( - 3 \zeta_{12}^{3} - 6) q^{45} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{47} + \zeta_{12}^{3} q^{49} + ( - 2 \zeta_{12}^{2} + 4) q^{51} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{53} + ( - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{55} + ( - \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 7) q^{57} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{59} - 13 \zeta_{12}^{2} q^{61} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{63} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12} - 1) q^{65} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 10) q^{67} + (7 \zeta_{12}^{2} + 7) q^{71} + ( - 10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{73} + (\zeta_{12}^{3} + 14 \zeta_{12}^{2} - 2 \zeta_{12} - 7) q^{75} + ( - 2 \zeta_{12}^{3} - 2) q^{77} + (14 \zeta_{12}^{3} - 7 \zeta_{12}) q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} + (4 \zeta_{12}^{3} - 4) q^{83} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12}) q^{85} + (15 \zeta_{12}^{3} + 15) q^{87} + (9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{89} + (4 \zeta_{12}^{2} - 8) q^{91} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{93} + ( - 5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{95} + ( - 16 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{97} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 2 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 2 q^{5} - 8 q^{7} + 4 q^{11} + 6 q^{13} - 6 q^{15} + 2 q^{17} - 24 q^{21} + 6 q^{25} + 6 q^{33} - 12 q^{35} + 12 q^{41} - 8 q^{43} - 24 q^{45} + 6 q^{47} + 12 q^{51} + 18 q^{53} + 2 q^{55} + 12 q^{57} - 26 q^{61} - 12 q^{63} + 30 q^{67} + 42 q^{71} + 20 q^{73} - 8 q^{77} - 18 q^{81} - 16 q^{83} - 6 q^{85} + 60 q^{87} - 24 q^{91} + 6 q^{93} - 32 q^{95} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(\zeta_{12}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 2.36603 0.633975i 0 −1.23205 1.86603i 0 −2.00000 2.00000i 0 2.59808 1.50000i 0
293.1 0 0.633975 + 2.36603i 0 2.23205 + 0.133975i 0 −2.00000 + 2.00000i 0 −2.59808 + 1.50000i 0
297.1 0 0.633975 2.36603i 0 2.23205 0.133975i 0 −2.00000 2.00000i 0 −2.59808 1.50000i 0
373.1 0 2.36603 + 0.633975i 0 −1.23205 + 1.86603i 0 −2.00000 + 2.00000i 0 2.59808 + 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.y.a 4
5.c odd 4 1 inner 380.2.y.a 4
19.d odd 6 1 inner 380.2.y.a 4
95.l even 12 1 inner 380.2.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.y.a 4 1.a even 1 1 trivial
380.2.y.a 4 5.c odd 4 1 inner
380.2.y.a 4 19.d odd 6 1 inner
380.2.y.a 4 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6T_{3}^{3} + 18T_{3}^{2} - 36T_{3} + 36 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - T^{2} - 10 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 37T^{2} + 361 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 576 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$53$ \( T^{4} - 18 T^{3} + 162 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 30 T^{3} + 450 T^{2} + \cdots + 22500 \) Copy content Toggle raw display
$71$ \( (T^{2} - 21 T + 147)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 40000 \) Copy content Toggle raw display
$79$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$83$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 243 T^{2} + 59049 \) Copy content Toggle raw display
$97$ \( T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 147456 \) Copy content Toggle raw display
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