Properties

Label 381.3.d.a
Level $381$
Weight $3$
Character orbit 381.d
Analytic conductor $10.381$
Analytic rank $0$
Dimension $2$
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] = [381,3,Mod(253,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.253");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 381.d (of order \(2\), degree \(1\), 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: \(10.3814980721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta q^{3} + 5 \beta q^{5} - 2 \beta q^{6} - 4 \beta q^{7} + 8 q^{8} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + \beta q^{3} + 5 \beta q^{5} - 2 \beta q^{6} - 4 \beta q^{7} + 8 q^{8} - 3 q^{9} - 10 \beta q^{10} + 10 q^{11} + 23 q^{13} + 8 \beta q^{14} - 15 q^{15} - 16 q^{16} + 10 q^{17} + 6 q^{18} + 26 q^{19} + 12 q^{21} - 20 q^{22} + 5 \beta q^{23} + 8 \beta q^{24} - 50 q^{25} - 46 q^{26} - 3 \beta q^{27} + 7 \beta q^{29} + 30 q^{30} - 25 q^{31} + 10 \beta q^{33} - 20 q^{34} + 60 q^{35} - 37 q^{37} - 52 q^{38} + 23 \beta q^{39} + 40 \beta q^{40} + 16 q^{41} - 24 q^{42} + 14 \beta q^{43} - 15 \beta q^{45} - 10 \beta q^{46} + 82 q^{47} - 16 \beta q^{48} + q^{49} + 100 q^{50} + 10 \beta q^{51} - 45 \beta q^{53} + 6 \beta q^{54} + 50 \beta q^{55} - 32 \beta q^{56} + 26 \beta q^{57} - 14 \beta q^{58} + 53 \beta q^{59} - 79 q^{61} + 50 q^{62} + 12 \beta q^{63} + 64 q^{64} + 115 \beta q^{65} - 20 \beta q^{66} + 46 \beta q^{67} - 15 q^{69} - 120 q^{70} - 26 q^{71} - 24 q^{72} + 17 q^{73} + 74 q^{74} - 50 \beta q^{75} - 40 \beta q^{77} - 46 \beta q^{78} + 26 q^{79} - 80 \beta q^{80} + 9 q^{81} - 32 q^{82} - 35 \beta q^{83} + 50 \beta q^{85} - 28 \beta q^{86} - 21 q^{87} + 80 q^{88} + 33 \beta q^{89} + 30 \beta q^{90} - 92 \beta q^{91} - 25 \beta q^{93} - 164 q^{94} + 130 \beta q^{95} - 70 \beta q^{97} - 2 q^{98} - 30 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 16 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 16 q^{8} - 6 q^{9} + 20 q^{11} + 46 q^{13} - 30 q^{15} - 32 q^{16} + 20 q^{17} + 12 q^{18} + 52 q^{19} + 24 q^{21} - 40 q^{22} - 100 q^{25} - 92 q^{26} + 60 q^{30} - 50 q^{31} - 40 q^{34} + 120 q^{35} - 74 q^{37} - 104 q^{38} + 32 q^{41} - 48 q^{42} + 164 q^{47} + 2 q^{49} + 200 q^{50} - 158 q^{61} + 100 q^{62} + 128 q^{64} - 30 q^{69} - 240 q^{70} - 52 q^{71} - 48 q^{72} + 34 q^{73} + 148 q^{74} + 52 q^{79} + 18 q^{81} - 64 q^{82} - 42 q^{87} + 160 q^{88} - 328 q^{94} - 4 q^{98} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/381\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(130\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
253.1
0.500000 0.866025i
0.500000 + 0.866025i
−2.00000 1.73205i 0 8.66025i 3.46410i 6.92820i 8.00000 −3.00000 17.3205i
253.2 −2.00000 1.73205i 0 8.66025i 3.46410i 6.92820i 8.00000 −3.00000 17.3205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.3.d.a 2
127.b odd 2 1 inner 381.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.3.d.a 2 1.a even 1 1 trivial
381.3.d.a 2 127.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{3}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 75 \) Copy content Toggle raw display
$7$ \( T^{2} + 48 \) Copy content Toggle raw display
$11$ \( (T - 10)^{2} \) Copy content Toggle raw display
$13$ \( (T - 23)^{2} \) Copy content Toggle raw display
$17$ \( (T - 10)^{2} \) Copy content Toggle raw display
$19$ \( (T - 26)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 75 \) Copy content Toggle raw display
$29$ \( T^{2} + 147 \) Copy content Toggle raw display
$31$ \( (T + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 588 \) Copy content Toggle raw display
$47$ \( (T - 82)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6075 \) Copy content Toggle raw display
$59$ \( T^{2} + 8427 \) Copy content Toggle raw display
$61$ \( (T + 79)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 6348 \) Copy content Toggle raw display
$71$ \( (T + 26)^{2} \) Copy content Toggle raw display
$73$ \( (T - 17)^{2} \) Copy content Toggle raw display
$79$ \( (T - 26)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3675 \) Copy content Toggle raw display
$89$ \( T^{2} + 3267 \) Copy content Toggle raw display
$97$ \( T^{2} + 14700 \) Copy content Toggle raw display
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