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Properties

Label 650.2.e.e
Level $650$
Weight $2$
Character orbit 650.e
Analytic conductor $5.190$
Analytic rank $0$
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] = [650,2,Mod(451,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, 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("650.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.e (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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{3} + \beta_{2} q^{4} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{3} + \beta_{2} q^{4} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{9} + (2 \beta_{2} + \beta_1 + 2) q^{11} - \beta_{3} q^{12} + (\beta_{2} - 2 \beta_1 + 1) q^{13} + ( - \beta_{3} + 3) q^{14} + ( - \beta_{2} - 1) q^{16} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{17} + \beta_{3} q^{18} + ( - 2 \beta_{3} - 3 \beta_{2} + \cdots + 2) q^{19}+ \cdots + ( - 2 \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + q^{3} - 2 q^{4} + q^{6} - 5 q^{7} + 4 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + q^{3} - 2 q^{4} + q^{6} - 5 q^{7} + 4 q^{8} - q^{9} + 5 q^{11} - 2 q^{12} + 10 q^{14} - 2 q^{16} + 5 q^{17} + 2 q^{18} + 8 q^{19} + 8 q^{21} + 5 q^{22} + 4 q^{23} + q^{24} - 8 q^{27} - 5 q^{28} - 3 q^{29} - 20 q^{31} - 2 q^{32} + 4 q^{33} - 10 q^{34} - q^{36} + 11 q^{37} - 16 q^{38} - 13 q^{39} - 2 q^{41} - 4 q^{42} + q^{43} - 10 q^{44} + 4 q^{46} + 34 q^{47} + q^{48} - 5 q^{49} - 34 q^{51} - 20 q^{53} + 4 q^{54} - 5 q^{56} - 18 q^{57} - 3 q^{58} + 6 q^{59} + 6 q^{61} + 10 q^{62} + 4 q^{63} + 4 q^{64} - 8 q^{66} - 4 q^{67} + 5 q^{68} - 15 q^{69} + 14 q^{71} - q^{72} - 32 q^{73} + 11 q^{74} + 8 q^{76} - 38 q^{77} + 26 q^{78} + 30 q^{79} + 14 q^{81} - 2 q^{82} + 24 q^{83} - 4 q^{84} - 2 q^{86} + 8 q^{87} + 5 q^{88} - 11 q^{89} + 26 q^{91} - 8 q^{92} - 18 q^{93} - 17 q^{94} - 2 q^{96} + 27 q^{97} - 5 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.15139 − 1.99426i
−0.651388 + 1.12824i
1.15139 + 1.99426i
−0.651388 − 1.12824i
−0.500000 + 0.866025i −0.651388 + 1.12824i −0.500000 − 0.866025i 0 −0.651388 − 1.12824i −2.15139 − 3.72631i 1.00000 0.651388 + 1.12824i 0
451.2 −0.500000 + 0.866025i 1.15139 − 1.99426i −0.500000 − 0.866025i 0 1.15139 + 1.99426i −0.348612 − 0.603814i 1.00000 −1.15139 − 1.99426i 0
601.1 −0.500000 − 0.866025i −0.651388 − 1.12824i −0.500000 + 0.866025i 0 −0.651388 + 1.12824i −2.15139 + 3.72631i 1.00000 0.651388 − 1.12824i 0
601.2 −0.500000 − 0.866025i 1.15139 + 1.99426i −0.500000 + 0.866025i 0 1.15139 − 1.99426i −0.348612 + 0.603814i 1.00000 −1.15139 + 1.99426i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.e.e ✓ 4
5.b even 2 1 650.2.e.g yes 4
5.c odd 4 2 650.2.o.h 8
13.c even 3 1 inner 650.2.e.e ✓ 4
13.c even 3 1 8450.2.a.bi 2
13.e even 6 1 8450.2.a.ba 2
65.l even 6 1 8450.2.a.bl 2
65.n even 6 1 650.2.e.g yes 4
65.n even 6 1 8450.2.a.bd 2
65.q odd 12 2 650.2.o.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.e.e ✓ 4 1.a even 1 1 trivial
650.2.e.e ✓ 4 13.c even 3 1 inner
650.2.e.g yes 4 5.b even 2 1
650.2.e.g yes 4 65.n even 6 1
650.2.o.h 8 5.c odd 4 2
650.2.o.h 8 65.q odd 12 2
8450.2.a.ba 2 13.e even 6 1
8450.2.a.bd 2 65.n even 6 1
8450.2.a.bi 2 13.c even 3 1
8450.2.a.bl 2 65.l even 6 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{3} + 22T_{7}^{2} + 15T_{7} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{4} - 5 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 5 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 11 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$41$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$47$ \( (T^{2} - 17 T + 69)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 10 T - 27)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( T^{4} - 14 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( (T + 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 53)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T - 81)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 11 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{4} - 27 T^{3} + \cdots + 32041 \) Copy content Toggle raw display
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