Properties

Label 637.2.h.f
Level $637$
Weight $2$
Character orbit 637.h
Analytic conductor $5.086$
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] = [637,2,Mod(165,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (of order \(3\), degree \(2\), not 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(\beta_{3}\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
0.381966 −0.190983 0.330792i −1.85410 0.190983 + 0.330792i −0.0729490 0.126351i 0 −1.47214 1.42705 2.47172i 0.0729490 + 0.126351i
165.2 2.61803 −1.30902 2.26728i 4.85410 1.30902 + 2.26728i −3.42705 5.93583i 0 7.47214 −1.92705 + 3.33775i 3.42705 + 5.93583i
471.1 0.381966 −0.190983 + 0.330792i −1.85410 0.190983 0.330792i −0.0729490 + 0.126351i 0 −1.47214 1.42705 + 2.47172i 0.0729490 0.126351i
471.2 2.61803 −1.30902 + 2.26728i 4.85410 1.30902 2.26728i −3.42705 + 5.93583i 0 7.47214 −1.92705 3.33775i 3.42705 5.93583i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.h.f 4
7.b odd 2 1 637.2.h.g 4
7.c even 3 1 637.2.f.c 4
7.c even 3 1 637.2.g.c 4
7.d odd 6 1 91.2.f.a 4
7.d odd 6 1 637.2.g.b 4
13.c even 3 1 637.2.g.c 4
21.g even 6 1 819.2.o.c 4
28.f even 6 1 1456.2.s.h 4
91.g even 3 1 637.2.f.c 4
91.h even 3 1 inner 637.2.h.f 4
91.h even 3 1 8281.2.a.bb 2
91.k even 6 1 8281.2.a.n 2
91.l odd 6 1 1183.2.a.c 2
91.m odd 6 1 91.2.f.a 4
91.n odd 6 1 637.2.g.b 4
91.v odd 6 1 637.2.h.g 4
91.v odd 6 1 1183.2.a.g 2
91.ba even 12 2 1183.2.c.c 4
273.bf even 6 1 819.2.o.c 4
364.br even 6 1 1456.2.s.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 7.d odd 6 1
91.2.f.a 4 91.m odd 6 1
637.2.f.c 4 7.c even 3 1
637.2.f.c 4 91.g even 3 1
637.2.g.b 4 7.d odd 6 1
637.2.g.b 4 91.n odd 6 1
637.2.g.c 4 7.c even 3 1
637.2.g.c 4 13.c even 3 1
637.2.h.f 4 1.a even 1 1 trivial
637.2.h.f 4 91.h even 3 1 inner
637.2.h.g 4 7.b odd 2 1
637.2.h.g 4 91.v odd 6 1
819.2.o.c 4 21.g even 6 1
819.2.o.c 4 273.bf even 6 1
1183.2.a.c 2 91.l odd 6 1
1183.2.a.g 2 91.v odd 6 1
1183.2.c.c 4 91.ba even 12 2
1456.2.s.h 4 28.f even 6 1
1456.2.s.h 4 364.br even 6 1
8281.2.a.n 2 91.k even 6 1
8281.2.a.bb 2 91.h even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 3T_{3}^{3} + 8T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 3T_{5}^{3} + 8T_{5}^{2} - 3T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 11)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$37$ \( (T - 4)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
$47$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$59$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + \cdots + 13456 \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 21 T + 79)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 31 T^{3} + \cdots + 52441 \) Copy content Toggle raw display
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