Properties

Label 370.2.l.a
Level $370$
Weight $2$
Character orbit 370.l
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(11,370)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.l (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: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{3} - \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{3} - \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} - q^{10} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 3) q^{11} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{12} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \cdots + 6) q^{13}+ \cdots + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{7} - 4 q^{10} + 12 q^{11} + 18 q^{13} + 6 q^{15} - 2 q^{16} + 6 q^{17} - 12 q^{19} + 6 q^{21} + 6 q^{22} + 6 q^{24} + 2 q^{25} - 8 q^{26} - 2 q^{28} - 6 q^{33} - 2 q^{34} - 6 q^{35} - 2 q^{37} + 16 q^{38} + 12 q^{39} - 2 q^{40} + 2 q^{41} + 6 q^{42} + 6 q^{44} + 6 q^{46} - 20 q^{47} + 6 q^{49} + 18 q^{52} - 6 q^{53} - 18 q^{54} - 6 q^{55} - 6 q^{56} - 24 q^{57} + 4 q^{58} - 30 q^{59} - 12 q^{61} + 2 q^{62} - 4 q^{64} + 4 q^{65} - 16 q^{67} - 18 q^{69} - 2 q^{70} - 4 q^{71} + 12 q^{73} - 12 q^{76} + 12 q^{77} - 18 q^{78} + 18 q^{81} - 24 q^{83} + 12 q^{84} + 4 q^{85} - 10 q^{86} - 12 q^{87} - 24 q^{89} + 6 q^{91} - 6 q^{92} - 6 q^{93} - 6 q^{94} - 8 q^{95} + 6 q^{96} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0.866025 + 0.500000i 1.73205i −0.366025 + 0.633975i 1.00000i 0 −1.00000
11.2 0.866025 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i −0.866025 0.500000i 1.73205i 1.36603 2.36603i 1.00000i 0 −1.00000
101.1 −0.866025 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0.866025 0.500000i 1.73205i −0.366025 0.633975i 1.00000i 0 −1.00000
101.2 0.866025 + 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.73205i 1.36603 + 2.36603i 1.00000i 0 −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.e 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.l.a 4
37.e even 6 1 inner 370.2.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.l.a 4 1.a even 1 1 trivial
370.2.l.a 4 37.e even 6 1 inner

Hecke kernels

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

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} + 6T_{7}^{2} + 4T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 18 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( T^{4} + 98T^{2} + 2209 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( T^{4} + 74T^{2} + 169 \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T + 22)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$59$ \( T^{4} + 30 T^{3} + \cdots + 4356 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$73$ \( (T^{2} - 6 T - 66)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$83$ \( T^{4} + 24 T^{3} + \cdots + 17424 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$97$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
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