Properties

Label 650.2.j.f
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: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} - q^{4} + (\beta_{3} - 1) q^{6} - 3 q^{7} + \beta_1 q^{8} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} - q^{4} + (\beta_{3} - 1) q^{6} - 3 q^{7} + \beta_1 q^{8} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{9} + (2 \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{2} + \beta_1) q^{12} + (2 \beta_1 + 3) q^{13} + 3 \beta_1 q^{14} + q^{16} + (\beta_{2} + \beta_1 + 2) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{18} + (2 \beta_1 + 2) q^{19} + ( - 3 \beta_{2} + 3 \beta_1) q^{21} + (2 \beta_{3} - \beta_1 - 1) q^{22} + (2 \beta_{3} - \beta_1 - 1) q^{23} + ( - \beta_{3} + 1) q^{24} + ( - 3 \beta_1 + 2) q^{26} + ( - \beta_{3} - 5 \beta_1 + 6) q^{27} + 3 q^{28} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{29} + ( - \beta_1 + 1) q^{31} - \beta_1 q^{32} + (\beta_{3} - \beta_{2} + 11 \beta_1 - 1) q^{33} + (\beta_{3} - 2 \beta_1 + 1) q^{34} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{36} - 3 q^{37} + ( - 2 \beta_1 + 2) q^{38} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 2) q^{39} + ( - 2 \beta_{3} + 4 \beta_1 - 2) q^{41} + ( - 3 \beta_{3} + 3) q^{42} + ( - 3 \beta_{3} - \beta_1 + 4) q^{43} + ( - 2 \beta_{2} + \beta_1 - 1) q^{44} + ( - 2 \beta_{2} + \beta_1 - 1) q^{46} + (\beta_{3} + \beta_{2} - \beta_1 + 6) q^{47} + (\beta_{2} - \beta_1) q^{48} + 2 q^{49} + ( - \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{51} + ( - 2 \beta_1 - 3) q^{52} + ( - 4 \beta_{2} - \beta_1 - 5) q^{53} + (\beta_{2} - 6 \beta_1 - 5) q^{54} - 3 \beta_1 q^{56} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{57} + (\beta_{3} + \beta_{2} - \beta_1 + 3) q^{58} + (2 \beta_{3} + 2 \beta_1 - 4) q^{59} + 2 q^{61} + ( - \beta_1 - 1) q^{62} + ( - 3 \beta_{3} + 3 \beta_{2} - 9 \beta_1 + 3) q^{63} - q^{64} + ( - \beta_{3} - \beta_{2} + \beta_1 + 11) q^{66} + (3 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 3) q^{67} + ( - \beta_{2} - \beta_1 - 2) q^{68} + ( - \beta_{3} - \beta_{2} + \beta_1 + 11) q^{69} + (\beta_{3} + 4 \beta_1 - 5) q^{71} + (\beta_{3} + \beta_{2} - \beta_1 - 3) q^{72} + 4 \beta_1 q^{73} + 3 \beta_1 q^{74} + ( - 2 \beta_1 - 2) q^{76} + ( - 6 \beta_{2} + 3 \beta_1 - 3) q^{77} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{78} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{79} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 2) q^{81} + (2 \beta_{2} + 2 \beta_1 + 4) q^{82} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{83} + (3 \beta_{2} - 3 \beta_1) q^{84} + (3 \beta_{2} - 4 \beta_1 - 1) q^{86} + ( - 2 \beta_{3} + 5 \beta_1 - 3) q^{87} + ( - 2 \beta_{3} + \beta_1 + 1) q^{88} + (2 \beta_{3} + 2 \beta_1 - 4) q^{89} + ( - 6 \beta_1 - 9) q^{91} + ( - 2 \beta_{3} + \beta_1 + 1) q^{92} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{93} + (\beta_{3} - \beta_{2} - 6 \beta_1 - 1) q^{94} + (\beta_{3} - 1) q^{96} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{97} - 2 \beta_1 q^{98} + ( - 6 \beta_{3} - 8 \beta_1 + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{4} - 2 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{4} - 2 q^{6} - 12 q^{7} + 2 q^{12} + 12 q^{13} + 4 q^{16} + 6 q^{17} + 12 q^{18} + 8 q^{19} + 6 q^{21} + 2 q^{24} + 8 q^{26} + 22 q^{27} + 12 q^{28} + 4 q^{31} + 6 q^{34} - 12 q^{37} + 8 q^{38} - 2 q^{39} - 12 q^{41} + 6 q^{42} + 10 q^{43} + 24 q^{47} - 2 q^{48} + 8 q^{49} - 12 q^{52} - 12 q^{53} - 22 q^{54} + 12 q^{58} - 12 q^{59} + 8 q^{61} - 4 q^{62} - 4 q^{64} + 44 q^{66} - 6 q^{68} + 44 q^{69} - 18 q^{71} - 12 q^{72} - 8 q^{76} - 10 q^{78} - 8 q^{81} + 12 q^{82} - 12 q^{83} - 6 q^{84} - 10 q^{86} - 16 q^{87} - 12 q^{89} - 36 q^{91} - 4 q^{93} - 2 q^{96} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−1.65831 0.500000i
1.65831 0.500000i
−1.65831 + 0.500000i
1.65831 + 0.500000i
1.00000i −2.15831 + 2.15831i −1.00000 0 −2.15831 2.15831i −3.00000 1.00000i 6.31662i 0
307.2 1.00000i 1.15831 1.15831i −1.00000 0 1.15831 + 1.15831i −3.00000 1.00000i 0.316625i 0
343.1 1.00000i −2.15831 2.15831i −1.00000 0 −2.15831 + 2.15831i −3.00000 1.00000i 6.31662i 0
343.2 1.00000i 1.15831 + 1.15831i −1.00000 0 1.15831 1.15831i −3.00000 1.00000i 0.316625i 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.f 4
5.b even 2 1 130.2.j.d yes 4
5.c odd 4 1 130.2.g.d 4
5.c odd 4 1 650.2.g.g 4
13.d odd 4 1 650.2.g.g 4
15.d odd 2 1 1170.2.w.e 4
15.e even 4 1 1170.2.m.e 4
20.d odd 2 1 1040.2.cd.i 4
20.e even 4 1 1040.2.bg.k 4
65.f even 4 1 inner 650.2.j.f 4
65.g odd 4 1 130.2.g.d 4
65.k even 4 1 130.2.j.d yes 4
195.j odd 4 1 1170.2.w.e 4
195.n even 4 1 1170.2.m.e 4
260.s odd 4 1 1040.2.cd.i 4
260.u even 4 1 1040.2.bg.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.d 4 5.c odd 4 1
130.2.g.d 4 65.g odd 4 1
130.2.j.d yes 4 5.b even 2 1
130.2.j.d yes 4 65.k even 4 1
650.2.g.g 4 5.c odd 4 1
650.2.g.g 4 13.d odd 4 1
650.2.j.f 4 1.a even 1 1 trivial
650.2.j.f 4 65.f even 4 1 inner
1040.2.bg.k 4 20.e even 4 1
1040.2.bg.k 4 260.u even 4 1
1040.2.cd.i 4 20.d odd 2 1
1040.2.cd.i 4 260.s odd 4 1
1170.2.m.e 4 15.e even 4 1
1170.2.m.e 4 195.n even 4 1
1170.2.w.e 4 15.d odd 2 1
1170.2.w.e 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 \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 10T_{3} + 25 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} - 10 T + 25 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 3)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 484 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 18 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 484 \) Copy content Toggle raw display
$29$ \( T^{4} + 40T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 72 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12 T + 25)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 4900 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 72 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 248T^{2} + 5476 \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1225 \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 216T^{2} + 8100 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 72 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$97$ \( T^{4} + 200T^{2} + 9604 \) Copy content Toggle raw display
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