Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(121,370)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.e (of order \(3\), 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: \(0\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} - 2 \zeta_{6} q^{7} - q^{8} + ( - 2 \zeta_{6} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} - 2 \zeta_{6} q^{7} - q^{8} + ( - 2 \zeta_{6} + 2) q^{9} + q^{10} + (\zeta_{6} - 1) q^{12} - 5 \zeta_{6} q^{13} - 2 q^{14} + ( - \zeta_{6} + 1) q^{15} + (\zeta_{6} - 1) q^{16} - 2 \zeta_{6} q^{18} - 2 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{20} + (2 \zeta_{6} - 2) q^{21} + \zeta_{6} q^{24} + (\zeta_{6} - 1) q^{25} - 5 q^{26} - 5 q^{27} + (2 \zeta_{6} - 2) q^{28} + 6 q^{29} - \zeta_{6} q^{30} - q^{31} + \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{35} - 2 q^{36} + (3 \zeta_{6} + 4) q^{37} - 2 q^{38} + (5 \zeta_{6} - 5) q^{39} - \zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + 2 \zeta_{6} q^{42} - q^{43} + 2 q^{45} - 6 q^{47} + q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + \zeta_{6} q^{50} + (5 \zeta_{6} - 5) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + (5 \zeta_{6} - 5) q^{54} + 2 \zeta_{6} q^{56} + (2 \zeta_{6} - 2) q^{57} + ( - 6 \zeta_{6} + 6) q^{58} + ( - 12 \zeta_{6} + 12) q^{59} - q^{60} + 10 \zeta_{6} q^{61} + (\zeta_{6} - 1) q^{62} - 4 q^{63} + q^{64} + ( - 5 \zeta_{6} + 5) q^{65} + 4 \zeta_{6} q^{67} - 2 \zeta_{6} q^{70} + 12 \zeta_{6} q^{71} + (2 \zeta_{6} - 2) q^{72} + 8 q^{73} + ( - 4 \zeta_{6} + 7) q^{74} + q^{75} + (2 \zeta_{6} - 2) q^{76} + 5 \zeta_{6} q^{78} + 4 \zeta_{6} q^{79} - q^{80} - \zeta_{6} q^{81} + 9 q^{82} + ( - 12 \zeta_{6} + 12) q^{83} + 2 q^{84} + (\zeta_{6} - 1) q^{86} - 6 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{89} + ( - 2 \zeta_{6} + 2) q^{90} + (10 \zeta_{6} - 10) q^{91} + \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{94} + ( - 2 \zeta_{6} + 2) q^{95} + ( - \zeta_{6} + 1) q^{96} + 2 q^{97} - 3 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - q^{12} - 5 q^{13} - 4 q^{14} + q^{15} - q^{16} - 2 q^{18} - 2 q^{19} + q^{20} - 2 q^{21} + q^{24} - q^{25} - 10 q^{26} - 10 q^{27} - 2 q^{28} + 12 q^{29} - q^{30} - 2 q^{31} + q^{32} + 2 q^{35} - 4 q^{36} + 11 q^{37} - 4 q^{38} - 5 q^{39} - q^{40} + 9 q^{41} + 2 q^{42} - 2 q^{43} + 4 q^{45} - 12 q^{47} + 2 q^{48} + 3 q^{49} + q^{50} - 5 q^{52} + 9 q^{53} - 5 q^{54} + 2 q^{56} - 2 q^{57} + 6 q^{58} + 12 q^{59} - 2 q^{60} + 10 q^{61} - q^{62} - 8 q^{63} + 2 q^{64} + 5 q^{65} + 4 q^{67} - 2 q^{70} + 12 q^{71} - 2 q^{72} + 16 q^{73} + 10 q^{74} + 2 q^{75} - 2 q^{76} + 5 q^{78} + 4 q^{79} - 2 q^{80} - q^{81} + 18 q^{82} + 12 q^{83} + 4 q^{84} - q^{86} - 6 q^{87} - 6 q^{89} + 2 q^{90} - 10 q^{91} + q^{93} - 6 q^{94} + 2 q^{95} + q^{96} + 4 q^{97} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −1.00000 1.73205i −1.00000 1.00000 1.73205i 1.00000
211.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i −1.00000 −1.00000 + 1.73205i −1.00000 1.00000 + 1.73205i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.e.b 2
37.c even 3 1 inner 370.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.b 2 1.a even 1 1 trivial
370.2.e.b 2 37.c even 3 1 inner

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$73$ \( (T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
show more
show less