Properties

Label 370.2.m.a
Level $370$
Weight $2$
Character orbit 370.m
Analytic conductor $2.954$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(159,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.159");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.m (of order \(6\), degree \(2\), 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: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{5} + (\beta_{3} + \beta_{2} - \beta_1) q^{6} + ( - 2 \beta_{2} - 2) q^{7} + q^{8} + (\beta_{3} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{5} + (\beta_{3} + \beta_{2} - \beta_1) q^{6} + ( - 2 \beta_{2} - 2) q^{7} + q^{8} + (\beta_{3} - 2 \beta_1 + 1) q^{9} + (\beta_{3} + 2 \beta_{2}) q^{10} + ( - \beta_{3} - \beta_{2} + 1) q^{12} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 3) q^{13}+ \cdots + ( - 5 \beta_{2} + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 6 q^{5} - 12 q^{7} + 4 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 6 q^{5} - 12 q^{7} + 4 q^{8} + q^{9} + 3 q^{10} + 3 q^{12} - 5 q^{13} - q^{15} - 2 q^{16} - 3 q^{17} + q^{18} + 3 q^{20} + 6 q^{21} - 12 q^{23} - 3 q^{24} - 2 q^{25} + 10 q^{26} + 12 q^{28} + 11 q^{30} - 2 q^{32} - 3 q^{34} + 18 q^{35} - 2 q^{36} + 2 q^{37} - 9 q^{39} - 6 q^{40} + 6 q^{42} - 2 q^{43} + 15 q^{45} + 6 q^{46} + 10 q^{49} + q^{50} - 5 q^{52} + 9 q^{53} - 9 q^{54} - 12 q^{56} + 21 q^{58} - 30 q^{59} - 10 q^{60} - 9 q^{61} - 21 q^{62} + 4 q^{64} - 9 q^{65} + 6 q^{67} + 6 q^{68} - 24 q^{69} + 6 q^{71} + q^{72} + 20 q^{74} + 18 q^{75} + 9 q^{78} + 6 q^{79} + 3 q^{80} + 4 q^{81} - 24 q^{83} - 12 q^{84} + 21 q^{85} + q^{86} + 16 q^{87} - 39 q^{89} - 18 q^{90} + 30 q^{91} + 6 q^{92} - 27 q^{93} + 6 q^{94} + 3 q^{96} - 14 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
−0.500000 0.866025i −2.18614 1.26217i −0.500000 + 0.866025i −1.50000 1.65831i 2.52434i −3.00000 1.73205i 1.00000 1.68614 + 2.92048i −0.686141 + 2.12819i
159.2 −0.500000 0.866025i 0.686141 + 0.396143i −0.500000 + 0.866025i −1.50000 + 1.65831i 0.792287i −3.00000 1.73205i 1.00000 −1.18614 2.05446i 2.18614 + 0.469882i
249.1 −0.500000 + 0.866025i −2.18614 + 1.26217i −0.500000 0.866025i −1.50000 + 1.65831i 2.52434i −3.00000 + 1.73205i 1.00000 1.68614 2.92048i −0.686141 2.12819i
249.2 −0.500000 + 0.866025i 0.686141 0.396143i −0.500000 0.866025i −1.50000 1.65831i 0.792287i −3.00000 + 1.73205i 1.00000 −1.18614 + 2.05446i 2.18614 0.469882i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.m.a 4
5.b even 2 1 370.2.m.b yes 4
37.e even 6 1 370.2.m.b yes 4
185.l even 6 1 inner 370.2.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.m.a 4 1.a even 1 1 trivial
370.2.m.a 4 185.l even 6 1 inner
370.2.m.b yes 4 5.b even 2 1
370.2.m.b yes 4 37.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{3} + T_{3}^{2} - 6T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 79T^{2} + 1156 \) Copy content Toggle raw display
$31$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 33T^{2} + 1089 \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 74)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} - 9 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 30 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( T^{4} + 9 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$83$ \( T^{4} + 24 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{4} + 39 T^{3} + \cdots + 15376 \) Copy content Toggle raw display
$97$ \( (T^{2} + 7 T - 62)^{2} \) Copy content Toggle raw display
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