Properties

Label 4650.2.d.bg
Level $4650$
Weight $2$
Character orbit 4650.d
Analytic conductor $37.130$
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] = [4650,2,Mod(3349,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4650.3349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (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: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{65})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 33x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 33x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 17\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 17 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{2} - 17\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3349.1
3.53113i
4.53113i
4.53113i
3.53113i
1.00000i 1.00000i −1.00000 0 1.00000 3.53113i 1.00000i −1.00000 0
3349.2 1.00000i 1.00000i −1.00000 0 1.00000 4.53113i 1.00000i −1.00000 0
3349.3 1.00000i 1.00000i −1.00000 0 1.00000 4.53113i 1.00000i −1.00000 0
3349.4 1.00000i 1.00000i −1.00000 0 1.00000 3.53113i 1.00000i −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4650.2.d.bg 4
5.b even 2 1 inner 4650.2.d.bg 4
5.c odd 4 1 930.2.a.q 2
5.c odd 4 1 4650.2.a.bz 2
15.e even 4 1 2790.2.a.bf 2
20.e even 4 1 7440.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.q 2 5.c odd 4 1
2790.2.a.bf 2 15.e even 4 1
4650.2.a.bz 2 5.c odd 4 1
4650.2.d.bg 4 1.a even 1 1 trivial
4650.2.d.bg 4 5.b even 2 1 inner
7440.2.a.bd 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4650, [\chi])\):

\( T_{7}^{4} + 33T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} - 10 \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} + T_{19} - 16 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 33T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T - 10)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 45T^{2} + 100 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 132T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 73T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 148T^{2} + 3136 \) Copy content Toggle raw display
$53$ \( T^{4} + 37T^{2} + 196 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2 T - 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 56)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 148T^{2} + 3136 \) Copy content Toggle raw display
$71$ \( (T^{2} + 17 T + 56)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 73T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 9 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 5 T - 10)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 260)^{2} \) Copy content Toggle raw display
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