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Properties

Label 650.2.j.g
Level $650$
Weight $2$
Character orbit 650.j
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(307,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(4))
 
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("650.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.j (of order \(4\), 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 \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} - q^{4} + (\beta_{3} - \beta_{2} + 1) q^{6} - q^{7} - \beta_{2} q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} - q^{4} + (\beta_{3} - \beta_{2} + 1) q^{6} - q^{7} - \beta_{2} q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{11} + \beta_1 q^{12} + ( - 2 \beta_{3} + 2 \beta_1 - 1) q^{13} - \beta_{2} q^{14} + q^{16} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{17} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{18} + (2 \beta_{2} - 4 \beta_1 + 2) q^{19} + \beta_1 q^{21} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{22} + ( - 2 \beta_{3} + 3 \beta_{2} - 3) q^{23} + ( - \beta_{3} + \beta_{2} - 1) q^{24} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{26} + ( - 3 \beta_{3} + 2 \beta_{2} - 2) q^{27} + q^{28} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{29} + (5 \beta_{2} - 5) q^{31} + \beta_{2} q^{32} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{33} + ( - 3 \beta_{3} - \beta_{2} + 1) q^{34} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{36} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{37} + (4 \beta_{3} - 2 \beta_{2} + 2) q^{38} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{39} + ( - 6 \beta_{3} + 2 \beta_{2} - 2) q^{41} + ( - \beta_{3} + \beta_{2} - 1) q^{42} + ( - 9 \beta_{3} + 4 \beta_{2} - 4) q^{43} + (3 \beta_{2} - 2 \beta_1 + 3) q^{44} + ( - \beta_{2} - 2 \beta_1 - 1) q^{46} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{47} - \beta_1 q^{48} - 6 q^{49} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{51} + (2 \beta_{3} - 2 \beta_1 + 1) q^{52} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{53} + (\beta_{2} - 3 \beta_1 + 1) q^{54} + \beta_{2} q^{56} + ( - 2 \beta_{3} + 6 \beta_{2} + \cdots - 2) q^{57}+ \cdots + (6 \beta_{3} + 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{4} + 2 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{4} + 2 q^{6} - 4 q^{7} - 8 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{16} - 10 q^{17} + 4 q^{18} + 2 q^{21} + 8 q^{22} - 8 q^{23} - 2 q^{24} - 2 q^{27} + 4 q^{28} - 20 q^{31} + 10 q^{34} + 4 q^{37} + 10 q^{39} + 4 q^{41} - 2 q^{42} + 2 q^{43} + 8 q^{44} - 8 q^{46} + 8 q^{47} - 2 q^{48} - 24 q^{49} - 4 q^{52} - 20 q^{53} - 2 q^{54} - 20 q^{58} + 12 q^{59} + 8 q^{61} - 20 q^{62} - 4 q^{64} + 4 q^{66} + 10 q^{68} + 20 q^{69} + 10 q^{71} - 4 q^{72} + 8 q^{77} + 14 q^{78} + 8 q^{81} + 4 q^{82} - 12 q^{83} - 2 q^{84} + 2 q^{86} + 16 q^{87} - 8 q^{88} - 20 q^{89} - 4 q^{91} + 8 q^{92} + 20 q^{93} + 2 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
− 1.00000i −1.36603 + 1.36603i −1.00000 0 1.36603 + 1.36603i −1.00000 1.00000i − 0.732051i 0
307.2 − 1.00000i 0.366025 − 0.366025i −1.00000 0 −0.366025 − 0.366025i −1.00000 1.00000i 2.73205i 0
343.1 1.00000i −1.36603 − 1.36603i −1.00000 0 1.36603 − 1.36603i −1.00000 − 1.00000i 0.732051i 0
343.2 1.00000i 0.366025 + 0.366025i −1.00000 0 −0.366025 + 0.366025i −1.00000 − 1.00000i − 2.73205i 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
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.j.g 4
5.b even 2 1 130.2.j.e yes 4
5.c odd 4 1 130.2.g.e ✓ 4
5.c odd 4 1 650.2.g.f 4
13.d odd 4 1 650.2.g.f 4
15.d odd 2 1 1170.2.w.d 4
15.e even 4 1 1170.2.m.d 4
20.d odd 2 1 1040.2.cd.k 4
20.e even 4 1 1040.2.bg.i 4
65.f even 4 1 inner 650.2.j.g 4
65.g odd 4 1 130.2.g.e ✓ 4
65.k even 4 1 130.2.j.e yes 4
195.j odd 4 1 1170.2.w.d 4
195.n even 4 1 1170.2.m.d 4
260.s odd 4 1 1040.2.cd.k 4
260.u even 4 1 1040.2.bg.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.e ✓ 4 5.c odd 4 1
130.2.g.e ✓ 4 65.g odd 4 1
130.2.j.e yes 4 5.b even 2 1
130.2.j.e yes 4 65.k even 4 1
650.2.g.f 4 5.c odd 4 1
650.2.g.f 4 13.d odd 4 1
650.2.j.g 4 1.a even 1 1 trivial
650.2.j.g 4 65.f even 4 1 inner
1040.2.bg.i 4 20.e even 4 1
1040.2.bg.i 4 260.u even 4 1
1040.2.cd.k 4 20.d odd 2 1
1040.2.cd.k 4 260.s odd 4 1
1170.2.m.d 4 15.e even 4 1
1170.2.m.d 4 195.n even 4 1
1170.2.w.d 4 15.d odd 2 1
1170.2.w.d 4 195.j odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 576 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T - 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 20 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
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