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Properties

Label 637.2.g.b
Level $637$
Weight $2$
Character orbit 637.g
Analytic conductor $5.086$
Analytic rank $1$
Dimension $4$
CM no
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(263,637)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.263");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.g (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: \(1\)
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.

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

Display 100 coefficients 10 coefficients

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
0.809017 − 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 − 0.535233i
−1.30902 + 2.26728i −2.61803 −2.42705 − 4.20378i −1.30902 − 2.26728i 3.42705 − 5.93583i 0 7.47214 3.85410 6.85410
263.2 −0.190983 + 0.330792i −0.381966 0.927051 + 1.60570i −0.190983 − 0.330792i 0.0729490 − 0.126351i 0 −1.47214 −2.85410 0.145898
373.1 −1.30902 − 2.26728i −2.61803 −2.42705 + 4.20378i −1.30902 + 2.26728i 3.42705 + 5.93583i 0 7.47214 3.85410 6.85410
373.2 −0.190983 − 0.330792i −0.381966 0.927051 − 1.60570i −0.190983 + 0.330792i 0.0729490 + 0.126351i 0 −1.47214 −2.85410 0.145898
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.g.b 4
7.b odd 2 1 637.2.g.c 4
7.c even 3 1 91.2.f.a ✓ 4
7.c even 3 1 637.2.h.g 4
7.d odd 6 1 637.2.f.c 4
7.d odd 6 1 637.2.h.f 4
13.c even 3 1 637.2.h.g 4
21.h odd 6 1 819.2.o.c 4
28.g odd 6 1 1456.2.s.h 4
91.g even 3 1 inner 637.2.g.b 4
91.g even 3 1 1183.2.a.g 2
91.h even 3 1 91.2.f.a ✓ 4
91.m odd 6 1 637.2.g.c 4
91.m odd 6 1 8281.2.a.bb 2
91.n odd 6 1 637.2.h.f 4
91.p odd 6 1 8281.2.a.n 2
91.u even 6 1 1183.2.a.c 2
91.v odd 6 1 637.2.f.c 4
91.bd odd 12 2 1183.2.c.c 4
273.s odd 6 1 819.2.o.c 4
364.bi odd 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.c even 3 1
91.2.f.a ✓ 4 91.h even 3 1
637.2.f.c 4 7.d odd 6 1
637.2.f.c 4 91.v odd 6 1
637.2.g.b 4 1.a even 1 1 trivial
637.2.g.b 4 91.g even 3 1 inner
637.2.g.c 4 7.b odd 2 1
637.2.g.c 4 91.m odd 6 1
637.2.h.f 4 7.d odd 6 1
637.2.h.f 4 91.n odd 6 1
637.2.h.g 4 7.c even 3 1
637.2.h.g 4 13.c even 3 1
819.2.o.c 4 21.h odd 6 1
819.2.o.c 4 273.s odd 6 1
1183.2.a.c 2 91.u even 6 1
1183.2.a.g 2 91.g even 3 1
1183.2.c.c 4 91.bd odd 12 2
1456.2.s.h 4 28.g odd 6 1
1456.2.s.h 4 364.bi odd 6 1
8281.2.a.n 2 91.p odd 6 1
8281.2.a.bb 2 91.m odd 6 1

Hecke kernels

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

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

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 1)^{2} \) 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
$11$ \( (T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T - 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 20T^{2} + 400 \) 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
$37$ \( (T^{2} + 4 T + 16)^{2} \) 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
$59$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$61$ \( (T + 6)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T - 9)^{2} \) 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
$89$ \( T^{4} + 21 T^{3} + \cdots + 6241 \) Copy content Toggle raw display
$97$ \( T^{4} + 31 T^{3} + \cdots + 52441 \) Copy content Toggle raw display
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