Properties

Label 37.2.c.a
Level $37$
Weight $2$
Character orbit 37.c
Analytic conductor $0.295$
Analytic rank $0$
Dimension $2$
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(10,37)]
 
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([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.c (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: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 1.73205i −3.00000 1.50000 2.59808i 1.00000
26.1 −0.500000 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 + 1.73205i −3.00000 1.50000 + 2.59808i 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 37.2.c.a 2
3.b odd 2 1 333.2.f.a 2
4.b odd 2 1 592.2.i.c 2
5.b even 2 1 925.2.e.a 2
5.c odd 4 2 925.2.o.a 4
37.c even 3 1 inner 37.2.c.a 2
37.c even 3 1 1369.2.a.d 1
37.e even 6 1 1369.2.a.b 1
37.g odd 12 2 1369.2.b.b 2
111.i odd 6 1 333.2.f.a 2
148.i odd 6 1 592.2.i.c 2
185.n even 6 1 925.2.e.a 2
185.s odd 12 2 925.2.o.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.c.a 2 1.a even 1 1 trivial
37.2.c.a 2 37.c even 3 1 inner
333.2.f.a 2 3.b odd 2 1
333.2.f.a 2 111.i odd 6 1
592.2.i.c 2 4.b odd 2 1
592.2.i.c 2 148.i odd 6 1
925.2.e.a 2 5.b even 2 1
925.2.e.a 2 185.n even 6 1
925.2.o.a 4 5.c odd 4 2
925.2.o.a 4 185.s odd 12 2
1369.2.a.b 1 37.e even 6 1
1369.2.a.d 1 37.c even 3 1
1369.2.b.b 2 37.g odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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