Properties

Label 370.2.e.e
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: \(\Q(\sqrt{-3}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{2} - 1) q^{5} + q^{6} + ( - 2 \beta_{2} + \beta_1 - 2) q^{7} - q^{8} - 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{2} - 1) q^{5} + q^{6} + ( - 2 \beta_{2} + \beta_1 - 2) q^{7} - q^{8} - 2 \beta_{2} q^{9} - q^{10} + ( - \beta_{3} + 2) q^{11} - \beta_{2} q^{12} + (\beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} - 2) q^{14} - \beta_{2} q^{15} + \beta_{2} q^{16} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{18} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{19} + \beta_{2} q^{20} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{21} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{22} + \beta_{3} q^{23} + ( - \beta_{2} - 1) q^{24} + \beta_{2} q^{25} + (\beta_{3} + 1) q^{26} + 5 q^{27} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{28} + (2 \beta_{3} + 2) q^{29} + ( - \beta_{2} - 1) q^{30} + (\beta_{3} + 1) q^{31} + (\beta_{2} + 1) q^{32} + (2 \beta_{2} + \beta_1 + 2) q^{33} + ( - 2 \beta_{2} - \beta_1 - 2) q^{34} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{35} - 2 q^{36} + (3 \beta_{2} + \beta_1 - 3) q^{37} + ( - 2 \beta_{3} - 2) q^{38} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{39} + (\beta_{2} + 1) q^{40} + (\beta_{2} + 2 \beta_1 + 1) q^{41} + ( - 2 \beta_{2} + \beta_1 - 2) q^{42} + (2 \beta_{3} + 1) q^{43} + ( - 2 \beta_{2} - \beta_1 - 2) q^{44} - 2 q^{45} + (\beta_{3} + \beta_1) q^{46} + (\beta_{3} + 2) q^{47} - q^{48} + ( - 4 \beta_{3} + 7 \beta_{2} - 4 \beta_1) q^{49} + (\beta_{2} + 1) q^{50} + ( - \beta_{3} + 2) q^{51} + (\beta_{3} - \beta_{2} + \beta_1) q^{52} + ( - \beta_{3} + 11 \beta_{2} - \beta_1) q^{53} - 5 \beta_{2} q^{54} + ( - 2 \beta_{2} - \beta_1 - 2) q^{55} + (2 \beta_{2} - \beta_1 + 2) q^{56} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{57} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{58} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{59} - q^{60} + (2 \beta_{2} - 2 \beta_1 + 2) q^{61} + (\beta_{3} - \beta_{2} + \beta_1) q^{62} + ( - 2 \beta_{3} - 4) q^{63} + q^{64} + (\beta_{3} - \beta_{2} + \beta_1) q^{65} + ( - \beta_{3} + 2) q^{66} + (2 \beta_{2} - 2 \beta_1 + 2) q^{67} + (\beta_{3} - 2) q^{68} - \beta_1 q^{69} + (2 \beta_{2} - \beta_1 + 2) q^{70} + 2 \beta_1 q^{71} + 2 \beta_{2} q^{72} + (3 \beta_{3} - 6) q^{73} + ( - \beta_{3} + 6 \beta_{2} + 3) q^{74} - q^{75} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{76} + (6 \beta_{2} + 6) q^{77} + (\beta_{2} - \beta_1 + 1) q^{78} + (2 \beta_{2} + 2) q^{79} + q^{80} + ( - \beta_{2} - 1) q^{81} + ( - 2 \beta_{3} + 1) q^{82} - 4 \beta_{2} q^{83} + ( - \beta_{3} - 2) q^{84} + (\beta_{3} - 2) q^{85} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{86} + (2 \beta_{2} - 2 \beta_1 + 2) q^{87} + (\beta_{3} - 2) q^{88} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{89} + 2 \beta_{2} q^{90} + (3 \beta_{3} - 12 \beta_{2} + 3 \beta_1) q^{91} + \beta_1 q^{92} + (\beta_{2} - \beta_1 + 1) q^{93} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{95} + \beta_{2} q^{96} + ( - 2 \beta_{3} - 6) q^{97} + (7 \beta_{2} - 4 \beta_1 + 7) q^{98} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 8 q^{11} + 2 q^{12} + 2 q^{13} - 8 q^{14} + 2 q^{15} - 2 q^{16} + 4 q^{17} - 4 q^{18} - 4 q^{19} - 2 q^{20} + 4 q^{21} + 4 q^{22} - 2 q^{24} - 2 q^{25} + 4 q^{26} + 20 q^{27} - 4 q^{28} + 8 q^{29} - 2 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{33} - 4 q^{34} - 4 q^{35} - 8 q^{36} - 18 q^{37} - 8 q^{38} - 2 q^{39} + 2 q^{40} + 2 q^{41} - 4 q^{42} + 4 q^{43} - 4 q^{44} - 8 q^{45} + 8 q^{47} - 4 q^{48} - 14 q^{49} + 2 q^{50} + 8 q^{51} + 2 q^{52} - 22 q^{53} + 10 q^{54} - 4 q^{55} + 4 q^{56} + 4 q^{57} + 4 q^{58} - 4 q^{59} - 4 q^{60} + 4 q^{61} + 2 q^{62} - 16 q^{63} + 4 q^{64} + 2 q^{65} + 8 q^{66} + 4 q^{67} - 8 q^{68} + 4 q^{70} - 4 q^{72} - 24 q^{73} - 4 q^{75} - 4 q^{76} + 12 q^{77} + 2 q^{78} + 4 q^{79} + 4 q^{80} - 2 q^{81} + 4 q^{82} + 8 q^{83} - 8 q^{84} - 8 q^{85} + 2 q^{86} + 4 q^{87} - 8 q^{88} + 8 q^{89} - 4 q^{90} + 24 q^{91} + 2 q^{93} + 4 q^{94} - 4 q^{95} - 2 q^{96} - 24 q^{97} + 14 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{3} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−1.58114 2.73861i
1.58114 + 2.73861i
−1.58114 + 2.73861i
1.58114 2.73861i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −2.58114 4.47066i −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 0.581139 + 1.00656i −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 −2.58114 + 4.47066i −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 0.581139 1.00656i −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.e 4
37.c even 3 1 inner 370.2.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.e 4 1.a even 1 1 trivial
370.2.e.e 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
\( T_{7}^{4} + 4T_{7}^{3} + 22T_{7}^{2} - 24T_{7} + 36 \) 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
$7$ \( T^{4} + 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 9)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 18 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 39)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 4 T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 22 T^{3} + \cdots + 12321 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} + 40T^{2} + 1600 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T - 54)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$97$ \( (T^{2} + 12 T - 4)^{2} \) Copy content Toggle raw display
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