Properties

Label 370.2.d.c
Level $370$
Weight $2$
Character orbit 370.d
Analytic conductor $2.954$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(221,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.d (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_1 - 1) q^{3} - q^{4} - \beta_{2} q^{5} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{6} + (\beta_{3} + 2) q^{7} - \beta_{2} q^{8} + (2 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_1 - 1) q^{3} - q^{4} - \beta_{2} q^{5} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{6} + (\beta_{3} + 2) q^{7} - \beta_{2} q^{8} + (2 \beta_{3} + 3) q^{9} + q^{10} + ( - 2 \beta_{3} - \beta_1 - 1) q^{11} + (\beta_{3} + \beta_1 + 1) q^{12} - 2 \beta_{5} q^{13} + ( - \beta_{5} + 2 \beta_{2}) q^{14} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{15} + q^{16} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{17} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{18} + (\beta_{5} + \beta_{4} - 3 \beta_{2}) q^{19} + \beta_{2} q^{20} + ( - 4 \beta_{3} - 2 \beta_1 - 4) q^{21} + (2 \beta_{5} + \beta_{4} - \beta_{2}) q^{22} - 2 \beta_{5} q^{23} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{24} - q^{25} - 2 \beta_{3} q^{26} + ( - 4 \beta_{3} - 4) q^{27} + ( - \beta_{3} - 2) q^{28} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{29} + ( - \beta_{3} - \beta_1 - 1) q^{30} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{31} + \beta_{2} q^{32} + (4 \beta_{3} + 8) q^{33} + ( - 2 \beta_{3} + \beta_1 - 3) q^{34} + (\beta_{5} - 2 \beta_{2}) q^{35} + ( - 2 \beta_{3} - 3) q^{36} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{37}+ \cdots + ( - 10 \beta_{3} - 5 \beta_1 - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{4} + 10 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 6 q^{4} + 10 q^{7} + 14 q^{9} + 6 q^{10} - 2 q^{11} + 4 q^{12} + 6 q^{16} - 16 q^{21} - 6 q^{25} + 4 q^{26} - 16 q^{27} - 10 q^{28} - 4 q^{30} + 40 q^{33} - 14 q^{34} - 14 q^{36} - 10 q^{37} + 16 q^{38} - 6 q^{40} + 2 q^{41} + 2 q^{44} + 4 q^{46} - 4 q^{47} - 4 q^{48} - 8 q^{49} + 2 q^{53} - 14 q^{58} - 10 q^{62} + 58 q^{63} - 6 q^{64} - 4 q^{65} + 8 q^{67} + 10 q^{70} - 20 q^{71} - 12 q^{73} - 8 q^{74} + 4 q^{75} - 30 q^{77} + 16 q^{78} + 6 q^{81} - 20 q^{83} + 16 q^{84} + 14 q^{85} - 2 q^{86} + 14 q^{90} - 16 q^{95} - 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.40680 + 0.144584i
−0.671462 + 1.24464i
0.264658 1.38923i
1.40680 0.144584i
−0.671462 1.24464i
0.264658 + 1.38923i
1.00000i −3.10278 −1.00000 1.00000i 3.10278i 3.81361 1.00000i 6.62721 1.00000
221.2 1.00000i −1.14637 −1.00000 1.00000i 1.14637i −0.342923 1.00000i −1.68585 1.00000
221.3 1.00000i 2.24914 −1.00000 1.00000i 2.24914i 1.52932 1.00000i 2.05863 1.00000
221.4 1.00000i −3.10278 −1.00000 1.00000i 3.10278i 3.81361 1.00000i 6.62721 1.00000
221.5 1.00000i −1.14637 −1.00000 1.00000i 1.14637i −0.342923 1.00000i −1.68585 1.00000
221.6 1.00000i 2.24914 −1.00000 1.00000i 2.24914i 1.52932 1.00000i 2.05863 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.d.c 6
3.b odd 2 1 3330.2.h.n 6
4.b odd 2 1 2960.2.p.g 6
5.b even 2 1 1850.2.d.f 6
5.c odd 4 1 1850.2.c.i 6
5.c odd 4 1 1850.2.c.j 6
37.b even 2 1 inner 370.2.d.c 6
111.d odd 2 1 3330.2.h.n 6
148.b odd 2 1 2960.2.p.g 6
185.d even 2 1 1850.2.d.f 6
185.h odd 4 1 1850.2.c.i 6
185.h odd 4 1 1850.2.c.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.d.c 6 1.a even 1 1 trivial
370.2.d.c 6 37.b even 2 1 inner
1850.2.c.i 6 5.c odd 4 1
1850.2.c.i 6 185.h odd 4 1
1850.2.c.j 6 5.c odd 4 1
1850.2.c.j 6 185.h odd 4 1
1850.2.d.f 6 5.b even 2 1
1850.2.d.f 6 185.d even 2 1
2960.2.p.g 6 4.b odd 2 1
2960.2.p.g 6 148.b odd 2 1
3330.2.h.n 6 3.b odd 2 1
3330.2.h.n 6 111.d odd 2 1

Hecke kernels

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

\( T_{3}^{3} + 2T_{3}^{2} - 6T_{3} - 8 \) Copy content Toggle raw display
\( T_{7}^{3} - 5T_{7}^{2} + 4T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{3} + 2 T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 5 T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 36 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{6} + 81 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( T^{6} + 36 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{6} + 36 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{6} + 49 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{6} + 57 T^{4} + \cdots + 6724 \) Copy content Toggle raw display
$37$ \( T^{6} + 10 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( (T^{3} - T^{2} - 56 T + 172)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 33 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{3} + 2 T^{2} - 14 T - 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 104 T + 352)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 96 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{6} + 121 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$67$ \( (T^{3} - 4 T^{2} + \cdots - 124)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 10 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} + \cdots - 344)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 80 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( (T^{3} + 10 T^{2} - 62 T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 256 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$97$ \( T^{6} + 289 T^{4} + \cdots + 29584 \) Copy content Toggle raw display
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