Properties

Label 4900.2.e.q
Level $4900$
Weight $2$
Character orbit 4900.e
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4900,2,Mod(2549,4900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, 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("4900.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (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: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 980)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{3} - 2 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{3} - 2 \beta_{3} q^{9} + (2 \beta_{3} - 1) q^{11} + (\beta_{2} + 5 \beta_1) q^{13} + (\beta_{2} - 5 \beta_1) q^{17} + ( - 4 \beta_{3} - 2) q^{19} + ( - \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} - \beta_1) q^{27} + (4 \beta_{3} - 1) q^{29} + ( - \beta_{3} - 6) q^{31} + (\beta_{2} + 3 \beta_1) q^{33} + (\beta_{2} - 2 \beta_1) q^{37} + ( - 6 \beta_{3} - 7) q^{39} + ( - \beta_{3} - 2) q^{41} + (4 \beta_{2} - 6 \beta_1) q^{43} + ( - 7 \beta_{2} - \beta_1) q^{47} + (4 \beta_{3} + 3) q^{51} + (3 \beta_{2} + 8 \beta_1) q^{53} + ( - 6 \beta_{2} - 10 \beta_1) q^{57} + ( - \beta_{3} + 2) q^{59} + ( - 2 \beta_{3} - 8) q^{61} + (5 \beta_{2} - 4 \beta_1) q^{67} + (3 \beta_{3} + 4) q^{69} + ( - 6 \beta_{3} - 2) q^{71} + (2 \beta_{2} + 8 \beta_1) q^{73} + (10 \beta_{3} + 1) q^{79} + ( - 6 \beta_{3} - 1) q^{81} - 8 \beta_1 q^{83} + (3 \beta_{2} + 7 \beta_1) q^{87} - 12 \beta_{3} q^{89} + ( - 7 \beta_{2} - 8 \beta_1) q^{93} + (9 \beta_{2} - 3 \beta_1) q^{97} + (2 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} - 8 q^{19} - 4 q^{29} - 24 q^{31} - 28 q^{39} - 8 q^{41} + 12 q^{51} + 8 q^{59} - 32 q^{61} + 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 2.41421i 0 0 0 0 0 −2.82843 0
2549.2 0 0.414214i 0 0 0 0 0 2.82843 0
2549.3 0 0.414214i 0 0 0 0 0 2.82843 0
2549.4 0 2.41421i 0 0 0 0 0 −2.82843 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 4900.2.e.q 4
5.b even 2 1 inner 4900.2.e.q 4
5.c odd 4 1 980.2.a.j 2
5.c odd 4 1 4900.2.a.z 2
7.b odd 2 1 4900.2.e.r 4
15.e even 4 1 8820.2.a.bg 2
20.e even 4 1 3920.2.a.bx 2
35.c odd 2 1 4900.2.e.r 4
35.f even 4 1 980.2.a.k yes 2
35.f even 4 1 4900.2.a.x 2
35.k even 12 2 980.2.i.k 4
35.l odd 12 2 980.2.i.l 4
105.k odd 4 1 8820.2.a.bl 2
140.j odd 4 1 3920.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 5.c odd 4 1
980.2.a.k yes 2 35.f even 4 1
980.2.i.k 4 35.k even 12 2
980.2.i.l 4 35.l odd 12 2
3920.2.a.bo 2 140.j odd 4 1
3920.2.a.bx 2 20.e even 4 1
4900.2.a.x 2 35.f even 4 1
4900.2.a.z 2 5.c odd 4 1
4900.2.e.q 4 1.a even 1 1 trivial
4900.2.e.q 4 5.b even 2 1 inner
4900.2.e.r 4 7.b odd 2 1
4900.2.e.r 4 35.c odd 2 1
8820.2.a.bg 2 15.e even 4 1
8820.2.a.bl 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4900, [\chi])\):

\( T_{3}^{4} + 6T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 7 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 28 \) Copy content Toggle raw display
\( T_{31}^{2} + 12T_{31} + 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 7)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$17$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 31)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 198T^{2} + 9409 \) Copy content Toggle raw display
$53$ \( T^{4} + 164T^{2} + 2116 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 132T^{2} + 1156 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 199)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 342 T^{2} + 23409 \) Copy content Toggle raw display
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