Properties

Label 37.2.f.b
Level $37$
Weight $2$
Character orbit 37.f
Analytic conductor $0.295$
Analytic rank $0$
Dimension $6$
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(7,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
0.939693 0.342020i
1.26604 0.460802i −1.76604 0.642788i −0.141559 + 0.118782i 0.233956 + 1.32683i −2.53209 −0.120615 0.684040i −1.47178 + 2.54920i 0.407604 + 0.342020i 0.907604 + 1.57202i
9.1 −0.439693 2.49362i −0.0603074 + 0.342020i −4.14543 + 1.50881i 1.93969 + 1.62760i 0.879385 −2.34730 1.96962i 3.05303 + 5.28801i 2.70574 + 0.984808i 3.20574 5.55250i
12.1 0.673648 0.565258i −1.17365 0.984808i −0.213011 + 1.20805i 0.826352 0.300767i −1.34730 −3.53209 + 1.28558i 1.41875 + 2.45734i −0.113341 0.642788i 0.386659 0.669713i
16.1 1.26604 + 0.460802i −1.76604 + 0.642788i −0.141559 0.118782i 0.233956 1.32683i −2.53209 −0.120615 + 0.684040i −1.47178 2.54920i 0.407604 0.342020i 0.907604 1.57202i
33.1 −0.439693 + 2.49362i −0.0603074 0.342020i −4.14543 1.50881i 1.93969 1.62760i 0.879385 −2.34730 + 1.96962i 3.05303 5.28801i 2.70574 0.984808i 3.20574 + 5.55250i
34.1 0.673648 + 0.565258i −1.17365 + 0.984808i −0.213011 1.20805i 0.826352 + 0.300767i −1.34730 −3.53209 1.28558i 1.41875 2.45734i −0.113341 + 0.642788i 0.386659 + 0.669713i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.f.b 6
3.b odd 2 1 333.2.x.a 6
4.b odd 2 1 592.2.bc.c 6
5.b even 2 1 925.2.p.a 6
5.c odd 4 2 925.2.bc.b 12
37.f even 9 1 inner 37.2.f.b 6
37.f even 9 1 1369.2.a.i 3
37.h even 18 1 1369.2.a.l 3
37.i odd 36 2 1369.2.b.e 6
111.p odd 18 1 333.2.x.a 6
148.p odd 18 1 592.2.bc.c 6
185.x even 18 1 925.2.p.a 6
185.bd odd 36 2 925.2.bc.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.f.b 6 1.a even 1 1 trivial
37.2.f.b 6 37.f even 9 1 inner
333.2.x.a 6 3.b odd 2 1
333.2.x.a 6 111.p odd 18 1
592.2.bc.c 6 4.b odd 2 1
592.2.bc.c 6 148.p odd 18 1
925.2.p.a 6 5.b even 2 1
925.2.p.a 6 185.x even 18 1
925.2.bc.b 12 5.c odd 4 2
925.2.bc.b 12 185.bd odd 36 2
1369.2.a.i 3 37.f even 9 1
1369.2.a.l 3 37.h even 18 1
1369.2.b.e 6 37.i odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(37, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$29$ \( T^{6} - 9 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} + 6 T - 53)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 12 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} - 15 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( (T^{3} + 9 T^{2} + 24 T + 19)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{6} - 21 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$67$ \( T^{6} - 24 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$71$ \( T^{6} + 12 T^{5} + \cdots + 1418481 \) Copy content Toggle raw display
$73$ \( (T^{3} - 39 T - 89)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} + \cdots + 751689 \) Copy content Toggle raw display
$89$ \( T^{6} + 24 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} + 27 T^{5} + \cdots + 128881 \) Copy content Toggle raw display
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