Properties

Label 37.2.e.a
Level $37$
Weight $2$
Character orbit 37.e
Analytic conductor $0.295$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,2,Mod(11,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.e (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: \(0.295446487479\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{5} + \cdots + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{5} + \cdots + (14 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \cdots + 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{4} - 6 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{4} - 6 q^{5} - 2 q^{9} - 8 q^{10} - 12 q^{11} + 2 q^{12} + 12 q^{13} + 6 q^{15} + 2 q^{16} + 6 q^{17} + 12 q^{18} - 6 q^{19} + 6 q^{20} + 12 q^{21} + 6 q^{22} - 18 q^{24} + 4 q^{25} - 16 q^{27} + 2 q^{30} - 12 q^{33} - 4 q^{34} - 24 q^{35} + 4 q^{36} + 2 q^{37} - 12 q^{38} + 12 q^{39} + 12 q^{40} - 6 q^{41} - 12 q^{42} + 6 q^{44} - 2 q^{46} + 12 q^{47} + 4 q^{48} - 10 q^{49} + 24 q^{50} - 12 q^{52} + 12 q^{53} + 6 q^{55} + 36 q^{56} + 12 q^{57} - 4 q^{58} - 12 q^{59} - 6 q^{61} + 6 q^{62} + 48 q^{63} - 28 q^{64} - 12 q^{65} + 18 q^{67} - 12 q^{70} - 12 q^{71} - 36 q^{72} - 40 q^{75} + 6 q^{76} + 12 q^{77} - 12 q^{78} - 18 q^{79} - 2 q^{81} + 6 q^{83} - 24 q^{84} + 4 q^{85} + 6 q^{87} + 18 q^{89} - 8 q^{90} + 6 q^{92} - 36 q^{93} + 18 q^{94} + 18 q^{95} + 30 q^{96} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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 1.36603 2.36603i −0.500000 + 0.866025i 0.232051 + 0.133975i 2.73205i −1.73205 + 3.00000i 3.00000i −2.23205 3.86603i −0.267949
11.2 0.866025 0.500000i −0.366025 + 0.633975i −0.500000 + 0.866025i −3.23205 1.86603i 0.732051i 1.73205 3.00000i 3.00000i 1.23205 + 2.13397i −3.73205
27.1 −0.866025 0.500000i 1.36603 + 2.36603i −0.500000 0.866025i 0.232051 0.133975i 2.73205i −1.73205 3.00000i 3.00000i −2.23205 + 3.86603i −0.267949
27.2 0.866025 + 0.500000i −0.366025 0.633975i −0.500000 0.866025i −3.23205 + 1.86603i 0.732051i 1.73205 + 3.00000i 3.00000i 1.23205 2.13397i −3.73205
\(n\): 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 37.2.e.a 4
3.b odd 2 1 333.2.s.b 4
4.b odd 2 1 592.2.w.c 4
5.b even 2 1 925.2.n.a 4
5.c odd 4 1 925.2.m.a 4
5.c odd 4 1 925.2.m.b 4
37.c even 3 1 1369.2.b.d 4
37.e even 6 1 inner 37.2.e.a 4
37.e even 6 1 1369.2.b.d 4
37.g odd 12 1 1369.2.a.g 2
37.g odd 12 1 1369.2.a.h 2
111.h odd 6 1 333.2.s.b 4
148.j odd 6 1 592.2.w.c 4
185.l even 6 1 925.2.n.a 4
185.r odd 12 1 925.2.m.a 4
185.r odd 12 1 925.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.e.a 4 1.a even 1 1 trivial
37.2.e.a 4 37.e even 6 1 inner
333.2.s.b 4 3.b odd 2 1
333.2.s.b 4 111.h odd 6 1
592.2.w.c 4 4.b odd 2 1
592.2.w.c 4 148.j odd 6 1
925.2.m.a 4 5.c odd 4 1
925.2.m.a 4 185.r odd 12 1
925.2.m.b 4 5.c odd 4 1
925.2.m.b 4 185.r odd 12 1
925.2.n.a 4 5.b even 2 1
925.2.n.a 4 185.l even 6 1
1369.2.a.g 2 37.g odd 12 1
1369.2.a.h 2 37.g odd 12 1
1369.2.b.d 4 37.c even 3 1
1369.2.b.d 4 37.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + \cdots + 7744 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 18 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$97$ \( T^{4} + 78T^{2} + 1089 \) Copy content Toggle raw display
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