Properties

Label 370.2.e.d
Level $370$
Weight $2$
Character orbit 370.e
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(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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + \beta_1 q^{3} - \beta_1 q^{4} - \beta_1 q^{5} - q^{6} - \beta_{2} q^{7} + q^{8} + ( - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + \beta_1 q^{3} - \beta_1 q^{4} - \beta_1 q^{5} - q^{6} - \beta_{2} q^{7} + q^{8} + ( - 2 \beta_1 + 2) q^{9} + q^{10} + ( - \beta_{3} - 2) q^{11} + ( - \beta_1 + 1) q^{12} + ( - \beta_{2} - \beta_1) q^{13} + \beta_{3} q^{14} + ( - \beta_1 + 1) q^{15} + (\beta_1 - 1) q^{16} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{17} + 2 \beta_1 q^{18} + 2 \beta_1 q^{19} + (\beta_1 - 1) q^{20} + (\beta_{3} - \beta_{2}) q^{21} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{22} + ( - \beta_{3} - 2) q^{23} + \beta_1 q^{24} + (\beta_1 - 1) q^{25} + (\beta_{3} + 1) q^{26} + 5 q^{27} + ( - \beta_{3} + \beta_{2}) q^{28} - 2 q^{29} + \beta_1 q^{30} + (\beta_{3} + 5) q^{31} - \beta_1 q^{32} + ( - \beta_{2} - 2 \beta_1) q^{33} + (\beta_{2} - 2 \beta_1) q^{34} + ( - \beta_{3} + \beta_{2}) q^{35} - 2 q^{36} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{37} - 2 q^{38} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{39} - \beta_1 q^{40} + (2 \beta_{2} + 5 \beta_1) q^{41} + \beta_{2} q^{42} + ( - 2 \beta_{3} - 3) q^{43} + (\beta_{2} + 2 \beta_1) q^{44} - 2 q^{45} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{46} + \beta_{3} q^{47} - q^{48} + (5 \beta_1 - 5) q^{49} - \beta_1 q^{50} + (\beta_{3} - 2) q^{51} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{52} + ( - \beta_{3} + \beta_{2} - 9 \beta_1 + 9) q^{53} + (5 \beta_1 - 5) q^{54} + (\beta_{2} + 2 \beta_1) q^{55} - \beta_{2} q^{56} + (2 \beta_1 - 2) q^{57} + ( - 2 \beta_1 + 2) q^{58} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{59} - q^{60} + ( - 2 \beta_{2} - 2 \beta_1) q^{61} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{62} - 2 \beta_{3} q^{63} + q^{64} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{65} + (\beta_{3} + 2) q^{66} + ( - 2 \beta_{2} - 8 \beta_1) q^{67} + ( - \beta_{3} + 2) q^{68} + ( - \beta_{2} - 2 \beta_1) q^{69} - \beta_{2} q^{70} + (2 \beta_{2} - 8 \beta_1) q^{71} + ( - 2 \beta_1 + 2) q^{72} + ( - \beta_{3} + 6) q^{73} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{74} - q^{75} + ( - 2 \beta_1 + 2) q^{76} + (2 \beta_{2} + 12 \beta_1) q^{77} + (\beta_{2} + \beta_1) q^{78} + (2 \beta_{2} + 8 \beta_1) q^{79} + q^{80} - \beta_1 q^{81} + ( - 2 \beta_{3} - 5) q^{82} + ( - 4 \beta_{3} + 4 \beta_{2} + \cdots + 4) q^{83}+ \cdots + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} + 4 q^{10} - 8 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{15} - 2 q^{16} - 4 q^{17} + 4 q^{18} + 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 2 q^{24} - 2 q^{25} + 4 q^{26} + 20 q^{27} - 8 q^{29} + 2 q^{30} + 20 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{34} - 8 q^{36} - 2 q^{37} - 8 q^{38} + 2 q^{39} - 2 q^{40} + 10 q^{41} - 12 q^{43} + 4 q^{44} - 8 q^{45} + 4 q^{46} - 4 q^{48} - 10 q^{49} - 2 q^{50} - 8 q^{51} - 2 q^{52} + 18 q^{53} - 10 q^{54} + 4 q^{55} - 4 q^{57} + 4 q^{58} + 4 q^{59} - 4 q^{60} - 4 q^{61} - 10 q^{62} + 4 q^{64} - 2 q^{65} + 8 q^{66} - 16 q^{67} + 8 q^{68} - 4 q^{69} - 16 q^{71} + 4 q^{72} + 24 q^{73} + 4 q^{74} - 4 q^{75} + 4 q^{76} + 24 q^{77} + 2 q^{78} + 16 q^{79} + 4 q^{80} - 2 q^{81} - 20 q^{82} + 8 q^{83} + 8 q^{85} + 6 q^{86} - 4 q^{87} - 8 q^{88} + 4 q^{89} + 4 q^{90} - 24 q^{91} + 4 q^{92} + 10 q^{93} + 4 q^{95} + 2 q^{96} - 8 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 6 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 −1.73205 3.00000i 1.00000 1.00000 1.73205i 1.00000
121.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 1.73205 + 3.00000i 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 −1.73205 + 3.00000i 1.00000 1.00000 + 1.73205i 1.00000
211.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 1.73205 3.00000i 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.d 4
37.c even 3 1 inner 370.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.d 4 1.a even 1 1 trivial
370.2.e.d 4 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
\( T_{7}^{4} + 12T_{7}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 13)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T - 39)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 18 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{4} - 8 T^{3} + \cdots + 30976 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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