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Properties

Label 650.2.c.e
Level $650$
Weight $2$
Character orbit 650.c
Analytic conductor $5.190$
Analytic rank $0$
Dimension $6$
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] = [650,2,Mod(649,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.c (of order \(2\), degree \(1\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.126157824.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + (\beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + (\beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 3) q^{9} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{11} + \beta_1 q^{12} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{3} - 1) q^{14} + q^{16} - 3 \beta_{2} q^{17} + (\beta_{5} - \beta_{4} + \beta_{3} + 3) q^{18} + ( - 2 \beta_{2} - \beta_1) q^{19} + (\beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{21}+ \cdots + ( - \beta_{5} - \beta_{4} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{7} - 6 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{7} - 6 q^{8} - 18 q^{9} - 6 q^{14} + 6 q^{16} + 18 q^{18} + 6 q^{28} + 18 q^{29} - 6 q^{32} + 24 q^{33} - 18 q^{36} - 24 q^{37} + 12 q^{39} - 6 q^{47} - 6 q^{56} + 36 q^{57} - 18 q^{58} - 30 q^{61} - 30 q^{63} + 6 q^{64} - 24 q^{66} + 18 q^{67} - 48 q^{69} + 18 q^{72} + 12 q^{73} + 24 q^{74} - 12 q^{78} + 36 q^{79} + 54 q^{81} + 30 q^{83} + 24 q^{91} + 12 q^{93} + 6 q^{94} - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 18x^{4} + 81x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 9\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + \nu^{4} + 15\nu^{3} + 11\nu^{2} + 52\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - \nu^{4} + 15\nu^{3} - 11\nu^{2} + 52\nu - 12 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} - 9\beta_{4} + 11\beta_{3} + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 2\beta_{4} - 30\beta_{2} + 83\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
− 3.10548i
− 2.88202i
− 0.223462i
0.223462i
2.88202i
3.10548i
−1.00000 − 3.10548i 1.00000 0 3.10548i 4.10548 −1.00000 −6.64402 0
649.2 −1.00000 − 2.88202i 1.00000 0 2.88202i −1.88202 −1.00000 −5.30604 0
649.3 −1.00000 − 0.223462i 1.00000 0 0.223462i 0.776538 −1.00000 2.95006 0
649.4 −1.00000 0.223462i 1.00000 0 − 0.223462i 0.776538 −1.00000 2.95006 0
649.5 −1.00000 2.88202i 1.00000 0 − 2.88202i −1.88202 −1.00000 −5.30604 0
649.6 −1.00000 3.10548i 1.00000 0 − 3.10548i 4.10548 −1.00000 −6.64402 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.c.e 6
5.b even 2 1 650.2.c.f 6
5.c odd 4 1 650.2.d.c ✓ 6
5.c odd 4 1 650.2.d.d yes 6
13.b even 2 1 650.2.c.f 6
65.d even 2 1 inner 650.2.c.e 6
65.f even 4 1 8450.2.a.cd 3
65.f even 4 1 8450.2.a.ce 3
65.h odd 4 1 650.2.d.c ✓ 6
65.h odd 4 1 650.2.d.d yes 6
65.k even 4 1 8450.2.a.bq 3
65.k even 4 1 8450.2.a.br 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.c.e 6 1.a even 1 1 trivial
650.2.c.e 6 65.d even 2 1 inner
650.2.c.f 6 5.b even 2 1
650.2.c.f 6 13.b even 2 1
650.2.d.c ✓ 6 5.c odd 4 1
650.2.d.c ✓ 6 65.h odd 4 1
650.2.d.d yes 6 5.c odd 4 1
650.2.d.d yes 6 65.h odd 4 1
8450.2.a.bq 3 65.k even 4 1
8450.2.a.br 3 65.k even 4 1
8450.2.a.cd 3 65.f even 4 1
8450.2.a.ce 3 65.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{6} + 18T_{3}^{4} + 81T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 6T_{7} + 6 \) Copy content Toggle raw display
\( T_{37}^{3} + 12T_{37}^{2} + 18T_{37} - 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 18 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 3 T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 63 T^{4} + \cdots + 2025 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} + 30 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( T^{6} + 84 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( (T^{3} - 9 T^{2} + \cdots + 240)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 129 T^{4} + \cdots + 3600 \) Copy content Toggle raw display
$37$ \( (T^{3} + 12 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 162 T^{4} + \cdots + 129600 \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$47$ \( (T^{3} + 3 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 129 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( T^{6} + 369 T^{4} + \cdots + 1498176 \) Copy content Toggle raw display
$61$ \( (T^{3} + 15 T^{2} + \cdots - 226)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 9 T^{2} + \cdots + 123)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} + \cdots + 900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 18 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 15 T^{2} + 45 T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 342 T^{4} + \cdots + 93636 \) Copy content Toggle raw display
$97$ \( (T^{3} + 30 T^{2} + \cdots + 624)^{2} \) Copy content Toggle raw display
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