Properties

Label 370.2.b.c
Level $370$
Weight $2$
Character orbit 370.b
Analytic conductor $2.954$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(149,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.b (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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) 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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} + (\beta_1 - 2) q^{5} - \beta_{3} q^{6} + (\beta_{2} + 2 \beta_1) q^{7} - \beta_1 q^{8} - 3 q^{9} + ( - 2 \beta_1 - 1) q^{10} + 2 \beta_{3} q^{11} - \beta_{2} q^{12}+ \cdots - 6 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 8 q^{5} - 12 q^{9} - 4 q^{10} - 8 q^{14} + 4 q^{16} - 24 q^{19} + 8 q^{20} - 24 q^{21} + 12 q^{25} + 16 q^{26} - 16 q^{31} - 8 q^{35} + 12 q^{36} + 4 q^{40} + 8 q^{41} + 24 q^{45} - 16 q^{46}+ \cdots + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.00000i 2.44949i −1.00000 −2.00000 1.00000i −2.44949 4.44949i 1.00000i −3.00000 −1.00000 + 2.00000i
149.2 1.00000i 2.44949i −1.00000 −2.00000 1.00000i 2.44949 0.449490i 1.00000i −3.00000 −1.00000 + 2.00000i
149.3 1.00000i 2.44949i −1.00000 −2.00000 + 1.00000i 2.44949 0.449490i 1.00000i −3.00000 −1.00000 2.00000i
149.4 1.00000i 2.44949i −1.00000 −2.00000 + 1.00000i −2.44949 4.44949i 1.00000i −3.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.b.c 4
3.b odd 2 1 3330.2.d.m 4
5.b even 2 1 inner 370.2.b.c 4
5.c odd 4 1 1850.2.a.s 2
5.c odd 4 1 1850.2.a.v 2
15.d odd 2 1 3330.2.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.c 4 1.a even 1 1 trivial
370.2.b.c 4 5.b even 2 1 inner
1850.2.a.s 2 5.c odd 4 1
1850.2.a.v 2 5.c odd 4 1
3330.2.d.m 4 3.b odd 2 1
3330.2.d.m 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{4} + 20T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12 T + 30)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{4} + 200T^{2} + 8464 \) Copy content Toggle raw display
$59$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 140T^{2} + 3364 \) Copy content Toggle raw display
$71$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 300 T^{2} + 19044 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 60)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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