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Properties

Label 637.2.f.g
Level $637$
Weight $2$
Character orbit 637.f
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(295,637)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.295");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.f (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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1485512441856.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 24x^{6} + 455x^{4} + 2904x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + \beta_{6} q^{3} + \beta_{2} q^{4} + ( - \beta_{7} + \beta_1) q^{5} - \beta_{5} q^{6} - 3 q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + \beta_{6} q^{3} + \beta_{2} q^{4} + ( - \beta_{7} + \beta_1) q^{5} - \beta_{5} q^{6} - 3 q^{8} + \beta_{2} q^{9} + \beta_{7} q^{10} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{11} + (\beta_{6} - \beta_{5}) q^{12} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{13} + (\beta_{4} + \beta_{3} + \beta_{2} - 1) q^{15} + ( - \beta_{2} + 1) q^{16} + ( - 2 \beta_{5} + \beta_1) q^{17} - q^{18} + 2 \beta_{5} q^{19} + \beta_1 q^{20} + (\beta_{4} - \beta_{2}) q^{22} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} + 3) q^{23} - 3 \beta_{6} q^{24} + ( - \beta_{3} + 7) q^{25} + (\beta_{6} - \beta_1) q^{26} + (4 \beta_{6} - 4 \beta_{5}) q^{27} + (\beta_{4} + \beta_{3} - 4 \beta_{2} + 4) q^{29} + ( - \beta_{4} - \beta_{2}) q^{30} + (\beta_{6} - \beta_{5}) q^{31} - 5 \beta_{2} q^{32} + ( - 2 \beta_{5} + 2 \beta_1) q^{33} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_1) q^{34} + (\beta_{2} - 1) q^{36} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 2) q^{37} + (2 \beta_{6} - 2 \beta_{5}) q^{38} + ( - \beta_{4} + \beta_{2} - 2) q^{39} + (3 \beta_{7} - 3 \beta_1) q^{40} + ( - \beta_{7} + 5 \beta_{6}) q^{41} + ( - \beta_{4} - 3 \beta_{2}) q^{43} + ( - \beta_{3} - 1) q^{44} + \beta_1 q^{45} + (\beta_{4} + 3 \beta_{2}) q^{46} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{47} + \beta_{5} q^{48} + (\beta_{4} + \beta_{3} + 7 \beta_{2} - 7) q^{50} + (\beta_{3} + 3) q^{51} + (\beta_{7} + \beta_{5} - \beta_1) q^{52} + (2 \beta_{3} - 3) q^{53} - 4 \beta_{6} q^{54} - 11 \beta_{6} q^{55} - 4 q^{57} + ( - \beta_{4} + 4 \beta_{2}) q^{58} + (3 \beta_{5} + 2 \beta_1) q^{59} + (\beta_{3} - 1) q^{60} + ( - 3 \beta_{5} - \beta_1) q^{61} - \beta_{6} q^{62} + 7 q^{64} + (\beta_{4} + 12 \beta_{2} - 11) q^{65} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{66} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{67} + (\beta_{7} - 2 \beta_{6}) q^{68} + (2 \beta_{5} + 2 \beta_1) q^{69} - 6 \beta_{2} q^{71} - 3 \beta_{2} q^{72} + (\beta_{7} + 6 \beta_{6} - 6 \beta_{5} - \beta_1) q^{73} + ( - \beta_{4} + 2 \beta_{2}) q^{74} + (2 \beta_{7} + 6 \beta_{6}) q^{75} + 2 \beta_{6} q^{76} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{78} + ( - \beta_{3} - 7) q^{79} - \beta_{7} q^{80} + ( - 5 \beta_{2} + 5) q^{81} + ( - 5 \beta_{5} + \beta_1) q^{82} + (7 \beta_{6} - 7 \beta_{5}) q^{83} + ( - \beta_{4} + 10 \beta_{2}) q^{85} + ( - \beta_{3} + 3) q^{86} + (5 \beta_{5} - 2 \beta_1) q^{87} + (3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 3) q^{88} + ( - 4 \beta_{7} + \beta_{6}) q^{89} + (\beta_{7} - \beta_1) q^{90} + ( - \beta_{3} + 3) q^{92} + ( - 2 \beta_{2} + 2) q^{93} + 2 \beta_{6} q^{94} + (2 \beta_{4} + 2 \beta_{2}) q^{95} + ( - 5 \beta_{6} + 5 \beta_{5}) q^{96} + 3 \beta_{5} q^{97} + ( - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 4 q^{4} - 24 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 4 q^{4} - 24 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{15} + 4 q^{16} - 8 q^{18} - 4 q^{22} + 12 q^{23} + 56 q^{25} + 16 q^{29} - 4 q^{30} - 20 q^{32} - 4 q^{36} + 8 q^{37} - 12 q^{39} - 12 q^{43} - 8 q^{44} + 12 q^{46} - 28 q^{50} + 24 q^{51} - 24 q^{53} - 32 q^{57} + 16 q^{58} - 8 q^{60} + 56 q^{64} - 40 q^{65} - 4 q^{67} - 24 q^{71} - 12 q^{72} + 8 q^{74} - 56 q^{79} + 20 q^{81} + 40 q^{85} + 24 q^{86} + 12 q^{88} + 24 q^{92} + 8 q^{93} + 8 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 24x^{6} + 455x^{4} + 2904x^{2} + 14641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\nu^{6} + 455\nu^{4} + 10920\nu^{2} + 69696 ) / 55055 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 2556 ) / 455 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -167\nu^{6} - 5460\nu^{4} - 75985\nu^{2} - 484968 ) / 55055 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3011\nu ) / 5005 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -191\nu^{7} - 5915\nu^{5} - 86905\nu^{3} - 554664\nu ) / 605605 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\nu^{7} + 455\nu^{5} + 10920\nu^{3} + 69696\nu ) / 55055 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 12\beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{7} + 11\beta_{6} - 11\beta_{5} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -24\beta_{4} - 167\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -191\beta_{7} - 264\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -455\beta_{3} + 2556 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5005\beta_{5} + 3011\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
−1.34203 − 2.32446i
2.04914 + 3.54921i
−2.04914 − 3.54921i
1.34203 + 2.32446i
−1.34203 + 2.32446i
2.04914 − 3.54921i
−2.04914 + 3.54921i
1.34203 − 2.32446i
−0.500000 + 0.866025i −0.707107 + 1.22474i 0.500000 + 0.866025i −2.68406 −0.707107 − 1.22474i 0 −3.00000 0.500000 + 0.866025i 1.34203 − 2.32446i
295.2 −0.500000 + 0.866025i −0.707107 + 1.22474i 0.500000 + 0.866025i 4.09827 −0.707107 − 1.22474i 0 −3.00000 0.500000 + 0.866025i −2.04914 + 3.54921i
295.3 −0.500000 + 0.866025i 0.707107 − 1.22474i 0.500000 + 0.866025i −4.09827 0.707107 + 1.22474i 0 −3.00000 0.500000 + 0.866025i 2.04914 − 3.54921i
295.4 −0.500000 + 0.866025i 0.707107 − 1.22474i 0.500000 + 0.866025i 2.68406 0.707107 + 1.22474i 0 −3.00000 0.500000 + 0.866025i −1.34203 + 2.32446i
393.1 −0.500000 − 0.866025i −0.707107 − 1.22474i 0.500000 − 0.866025i −2.68406 −0.707107 + 1.22474i 0 −3.00000 0.500000 − 0.866025i 1.34203 + 2.32446i
393.2 −0.500000 − 0.866025i −0.707107 − 1.22474i 0.500000 − 0.866025i 4.09827 −0.707107 + 1.22474i 0 −3.00000 0.500000 − 0.866025i −2.04914 − 3.54921i
393.3 −0.500000 − 0.866025i 0.707107 + 1.22474i 0.500000 − 0.866025i −4.09827 0.707107 − 1.22474i 0 −3.00000 0.500000 − 0.866025i 2.04914 + 3.54921i
393.4 −0.500000 − 0.866025i 0.707107 + 1.22474i 0.500000 − 0.866025i 2.68406 0.707107 − 1.22474i 0 −3.00000 0.500000 − 0.866025i −1.34203 − 2.32446i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 393.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
91.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.f.g ✓ 8
7.b odd 2 1 inner 637.2.f.g ✓ 8
7.c even 3 1 637.2.g.h 8
7.c even 3 1 637.2.h.k 8
7.d odd 6 1 637.2.g.h 8
7.d odd 6 1 637.2.h.k 8
13.c even 3 1 inner 637.2.f.g ✓ 8
13.c even 3 1 8281.2.a.bv 4
13.e even 6 1 8281.2.a.bn 4
91.g even 3 1 637.2.h.k 8
91.h even 3 1 637.2.g.h 8
91.m odd 6 1 637.2.h.k 8
91.n odd 6 1 inner 637.2.f.g ✓ 8
91.n odd 6 1 8281.2.a.bv 4
91.t odd 6 1 8281.2.a.bn 4
91.v odd 6 1 637.2.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.g ✓ 8 1.a even 1 1 trivial
637.2.f.g ✓ 8 7.b odd 2 1 inner
637.2.f.g ✓ 8 13.c even 3 1 inner
637.2.f.g ✓ 8 91.n odd 6 1 inner
637.2.g.h 8 7.c even 3 1
637.2.g.h 8 7.d odd 6 1
637.2.g.h 8 91.h even 3 1
637.2.g.h 8 91.v odd 6 1
637.2.h.k 8 7.c even 3 1
637.2.h.k 8 7.d odd 6 1
637.2.h.k 8 91.g even 3 1
637.2.h.k 8 91.m odd 6 1
8281.2.a.bn 4 13.e even 6 1
8281.2.a.bn 4 91.t odd 6 1
8281.2.a.bv 4 13.c even 3 1
8281.2.a.bv 4 91.n odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} - 24T_{5}^{2} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 24 T^{2} + 121)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 26 T^{2} - 44 T + 484)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 20 T^{6} + 231 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 32 T^{6} + 975 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} + 50 T^{2} + 84 T + 196)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + 71 T^{2} + 56 T + 49)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + 35 T^{2} + 76 T + 361)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 104 T^{6} + 9975 T^{4} + \cdots + 707281 \) Copy content Toggle raw display
$43$ \( (T^{4} + 6 T^{3} + 50 T^{2} - 84 T + 196)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 83)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + 156 T^{6} + 24140 T^{4} + \cdots + 38416 \) Copy content Toggle raw display
$61$ \( T^{8} + 72 T^{6} + 5015 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$67$ \( (T^{4} + 2 T^{3} + 26 T^{2} - 44 T + 484)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 192 T^{2} + 5329)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 26)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} + 372 T^{6} + \cdots + 1097199376 \) Copy content Toggle raw display
$97$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
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