Properties

Label 490.2.c.f
Level $490$
Weight $2$
Character orbit 490.c
Analytic conductor $3.913$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,2,Mod(99,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("490.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.c (of order \(2\), degree \(1\), 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: \(3.91266969904\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} + (\zeta_{8}^{3} + \zeta_{8}) q^{3} - q^{4} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + (\zeta_{8}^{3} - \zeta_{8}) q^{6} - \zeta_{8}^{2} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{2} + (\zeta_{8}^{3} + \zeta_{8}) q^{3} - q^{4} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + (\zeta_{8}^{3} - \zeta_{8}) q^{6} - \zeta_{8}^{2} q^{8} + q^{9} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{12} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{13} + (3 \zeta_{8}^{2} + 1) q^{15} + q^{16} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{17} + \zeta_{8}^{2} q^{18} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{19} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{20} - 6 \zeta_{8}^{2} q^{23} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{24} + ( - 3 \zeta_{8}^{2} + 4) q^{25} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{26} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{27} - 6 q^{29} + (\zeta_{8}^{2} - 3) q^{30} + (6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{31} + \zeta_{8}^{2} q^{32} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{34} - q^{36} + 6 \zeta_{8}^{2} q^{37} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{38} - 6 q^{39} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{40} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{41} - 12 \zeta_{8}^{2} q^{43} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{45} + 6 q^{46} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{47} + (\zeta_{8}^{3} + \zeta_{8}) q^{48} + (4 \zeta_{8}^{2} + 3) q^{50} - 8 q^{51} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{52} + 6 \zeta_{8}^{2} q^{53} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{54} + 6 \zeta_{8}^{2} q^{57} - 6 \zeta_{8}^{2} q^{58} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{59} + ( - 3 \zeta_{8}^{2} - 1) q^{60} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{61} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{62} - q^{64} + (9 \zeta_{8}^{2} + 3) q^{65} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{68} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{69} + 6 q^{71} - \zeta_{8}^{2} q^{72} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{73} - 6 q^{74} + (\zeta_{8}^{3} + 7 \zeta_{8}) q^{75} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{76} - 6 \zeta_{8}^{2} q^{78} + 10 q^{79} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{80} - 5 q^{81} + (6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{82} + ( - 11 \zeta_{8}^{3} - 11 \zeta_{8}) q^{83} + (12 \zeta_{8}^{2} + 4) q^{85} + 12 q^{86} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{87} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{90} + 6 \zeta_{8}^{2} q^{92} - 12 \zeta_{8}^{2} q^{93} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{94} + ( - 3 \zeta_{8}^{2} + 9) q^{95} + (\zeta_{8}^{3} - \zeta_{8}) q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{9} + 4 q^{15} + 4 q^{16} + 16 q^{25} - 24 q^{29} - 12 q^{30} - 4 q^{36} - 24 q^{39} + 24 q^{46} + 12 q^{50} - 32 q^{51} - 4 q^{60} - 4 q^{64} + 12 q^{65} + 24 q^{71} - 24 q^{74} + 40 q^{79} - 20 q^{81} + 16 q^{85} + 48 q^{86} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i 1.41421i −1.00000 2.12132 + 0.707107i −1.41421 0 1.00000i 1.00000 0.707107 2.12132i
99.2 1.00000i 1.41421i −1.00000 −2.12132 0.707107i 1.41421 0 1.00000i 1.00000 −0.707107 + 2.12132i
99.3 1.00000i 1.41421i −1.00000 −2.12132 + 0.707107i 1.41421 0 1.00000i 1.00000 −0.707107 2.12132i
99.4 1.00000i 1.41421i −1.00000 2.12132 0.707107i −1.41421 0 1.00000i 1.00000 0.707107 + 2.12132i
\(n\): 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
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.c.f 4
5.b even 2 1 inner 490.2.c.f 4
5.c odd 4 1 2450.2.a.bk 2
5.c odd 4 1 2450.2.a.bp 2
7.b odd 2 1 inner 490.2.c.f 4
7.c even 3 2 490.2.i.e 8
7.d odd 6 2 490.2.i.e 8
35.c odd 2 1 inner 490.2.c.f 4
35.f even 4 1 2450.2.a.bk 2
35.f even 4 1 2450.2.a.bp 2
35.i odd 6 2 490.2.i.e 8
35.j even 6 2 490.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.f 4 1.a even 1 1 trivial
490.2.c.f 4 5.b even 2 1 inner
490.2.c.f 4 7.b odd 2 1 inner
490.2.c.f 4 35.c odd 2 1 inner
490.2.i.e 8 7.c even 3 2
490.2.i.e 8 7.d odd 6 2
490.2.i.e 8 35.i odd 6 2
490.2.i.e 8 35.j even 6 2
2450.2.a.bk 2 5.c odd 4 1
2450.2.a.bk 2 35.f even 4 1
2450.2.a.bp 2 5.c odd 4 1
2450.2.a.bp 2 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T - 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 242)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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