Properties

Label 370.2.e.f
Level $370$
Weight $2$
Character orbit 370.e
Analytic conductor $2.954$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2696112.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{5} q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{4} + 1) q^{5} - \beta_{3} q^{6} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{5} - \beta_{4} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{5} q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{4} + 1) q^{5} - \beta_{3} q^{6} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{5} - \beta_{4} + \cdots + 2 \beta_1) q^{9}+ \cdots + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8} - 5 q^{9} - 6 q^{10} + 10 q^{11} - 3 q^{13} + 4 q^{14} - 3 q^{16} + 2 q^{17} - 5 q^{18} + 9 q^{19} + 3 q^{20} - 18 q^{21} - 5 q^{22} - 22 q^{23} - 3 q^{25} + 6 q^{26} + 12 q^{27} - 2 q^{28} - 4 q^{29} - 24 q^{31} - 3 q^{32} - 4 q^{33} + 2 q^{34} + 2 q^{35} + 10 q^{36} + 9 q^{37} - 18 q^{38} - 2 q^{39} + 3 q^{40} - 12 q^{41} - 18 q^{42} - 16 q^{43} - 5 q^{44} - 10 q^{45} + 11 q^{46} + 34 q^{47} - 11 q^{49} - 3 q^{50} + 76 q^{51} - 3 q^{52} - 6 q^{53} - 6 q^{54} + 5 q^{55} - 2 q^{56} - 14 q^{57} + 2 q^{58} + 13 q^{59} + 26 q^{61} + 12 q^{62} - 8 q^{63} + 6 q^{64} + 3 q^{65} + 8 q^{66} - 8 q^{67} - 4 q^{68} + 4 q^{69} + 2 q^{70} - 10 q^{71} - 5 q^{72} + 4 q^{73} + 9 q^{74} + 9 q^{76} - 16 q^{77} - 2 q^{78} - 16 q^{79} - 6 q^{80} + q^{81} + 24 q^{82} + 10 q^{83} + 36 q^{84} + 4 q^{85} + 8 q^{86} + 20 q^{87} + 10 q^{88} + 27 q^{89} + 5 q^{90} + 18 q^{91} + 11 q^{92} - 12 q^{93} - 17 q^{94} - 9 q^{95} + 12 q^{97} - 11 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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
0.235342 + 0.407624i
−0.906803 1.57063i
1.17146 + 2.02903i
0.235342 0.407624i
−0.906803 + 1.57063i
1.17146 2.02903i
−0.500000 + 0.866025i −1.38923 2.40621i −0.500000 0.866025i 0.500000 + 0.866025i 2.77846 −1.65389 2.86462i 1.00000 −2.35991 + 4.08749i −1.00000
121.2 −0.500000 + 0.866025i 0.144584 + 0.250427i −0.500000 0.866025i 0.500000 + 0.866025i −0.289169 −1.26222 2.18623i 1.00000 1.45819 2.52566i −1.00000
121.3 −0.500000 + 0.866025i 1.24464 + 2.15579i −0.500000 0.866025i 0.500000 + 0.866025i −2.48929 1.91611 + 3.31879i 1.00000 −1.59828 + 2.76830i −1.00000
211.1 −0.500000 0.866025i −1.38923 + 2.40621i −0.500000 + 0.866025i 0.500000 0.866025i 2.77846 −1.65389 + 2.86462i 1.00000 −2.35991 4.08749i −1.00000
211.2 −0.500000 0.866025i 0.144584 0.250427i −0.500000 + 0.866025i 0.500000 0.866025i −0.289169 −1.26222 + 2.18623i 1.00000 1.45819 + 2.52566i −1.00000
211.3 −0.500000 0.866025i 1.24464 2.15579i −0.500000 + 0.866025i 0.500000 0.866025i −2.48929 1.91611 3.31879i 1.00000 −1.59828 2.76830i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
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.f 6
37.c even 3 1 inner 370.2.e.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.f 6 1.a even 1 1 trivial
370.2.e.f 6 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 \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{6} + 7T_{3}^{4} - 4T_{3}^{3} + 49T_{3}^{2} - 14T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} + 18T_{7}^{4} + 36T_{7}^{3} + 260T_{7}^{2} + 448T_{7} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 7 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + \cdots + 30976 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{3} + 11 T^{2} + \cdots + 34)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 28 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 12 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 9 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$43$ \( (T^{3} + 8 T^{2} - 11 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 17 T^{2} + 722)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$59$ \( T^{6} - 13 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$61$ \( T^{6} - 26 T^{5} + \cdots + 179776 \) Copy content Toggle raw display
$67$ \( T^{6} + 8 T^{5} + \cdots + 719104 \) Copy content Toggle raw display
$71$ \( T^{6} + 10 T^{5} + \cdots + 952576 \) Copy content Toggle raw display
$73$ \( (T^{3} - 2 T^{2} + \cdots + 176)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 16 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{6} - 10 T^{5} + \cdots + 123904 \) Copy content Toggle raw display
$89$ \( T^{6} - 27 T^{5} + \cdots + 46656 \) Copy content Toggle raw display
$97$ \( (T^{3} - 6 T^{2} + \cdots + 872)^{2} \) Copy content Toggle raw display
show more
show less