Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

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

Self dual: no
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{2} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{8} + 3 q^{9} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{2} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{8} + 3 q^{9} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{10} -4 q^{11} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{13} + q^{16} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} -3 \zeta_{8}^{2} q^{18} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{20} + 4 \zeta_{8}^{2} q^{22} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{26} + 4 q^{29} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{2} q^{32} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{34} -3 q^{36} + 6 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{38} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{40} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{41} -12 \zeta_{8}^{2} q^{43} + 4 q^{44} + ( 6 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{45} + ( 3 + 4 \zeta_{8}^{2} ) q^{50} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{52} -12 \zeta_{8}^{2} q^{53} + ( -8 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} -4 \zeta_{8}^{2} q^{58} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{61} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{62} - q^{64} + ( -9 + 3 \zeta_{8}^{2} ) q^{65} + 12 \zeta_{8}^{2} q^{67} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{68} + 8 q^{71} + 3 \zeta_{8}^{2} q^{72} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{73} + 6 q^{74} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{76} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} + 9 q^{81} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{82} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{83} + ( -9 + 3 \zeta_{8}^{2} ) q^{85} -12 q^{86} -4 \zeta_{8}^{2} q^{88} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + ( 3 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{90} + ( 4 + 12 \zeta_{8}^{2} ) q^{95} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{97} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 12q^{9} - 16q^{11} + 4q^{16} - 16q^{25} + 16q^{29} - 12q^{36} + 16q^{44} + 12q^{50} - 4q^{64} - 36q^{65} + 32q^{71} + 24q^{74} + 36q^{81} - 36q^{85} - 48q^{86} + 16q^{95} - 48q^{99} + O(q^{100}) \)

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.

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 0 −1.00000 −0.707107 2.12132i 0 0 1.00000i 3.00000 −2.12132 + 0.707107i
99.2 1.00000i 0 −1.00000 0.707107 + 2.12132i 0 0 1.00000i 3.00000 2.12132 0.707107i
99.3 1.00000i 0 −1.00000 −0.707107 + 2.12132i 0 0 1.00000i 3.00000 −2.12132 0.707107i
99.4 1.00000i 0 −1.00000 0.707107 2.12132i 0 0 1.00000i 3.00000 2.12132 + 0.707107i
\(n\): 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.g 4
5.b even 2 1 inner 490.2.c.g 4
5.c odd 4 1 2450.2.a.bi 2
5.c odd 4 1 2450.2.a.bo 2
7.b odd 2 1 inner 490.2.c.g 4
7.c even 3 2 490.2.i.d 8
7.d odd 6 2 490.2.i.d 8
35.c odd 2 1 inner 490.2.c.g 4
35.f even 4 1 2450.2.a.bi 2
35.f even 4 1 2450.2.a.bo 2
35.i odd 6 2 490.2.i.d 8
35.j even 6 2 490.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.g 4 1.a even 1 1 trivial
490.2.c.g 4 5.b even 2 1 inner
490.2.c.g 4 7.b odd 2 1 inner
490.2.c.g 4 35.c odd 2 1 inner
490.2.i.d 8 7.c even 3 2
490.2.i.d 8 7.d odd 6 2
490.2.i.d 8 35.i odd 6 2
490.2.i.d 8 35.j even 6 2
2450.2.a.bi 2 5.c odd 4 1
2450.2.a.bi 2 35.f even 4 1
2450.2.a.bo 2 5.c odd 4 1
2450.2.a.bo 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 \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3} \)
\( T_{11} + 4 \)
\( T_{19}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 + T )^{4} \)
$13$ \( ( 18 + T^{2} )^{2} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( ( -32 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( ( -2 + T^{2} )^{2} \)
$43$ \( ( 144 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 144 + T^{2} )^{2} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( -50 + T^{2} )^{2} \)
$67$ \( ( 144 + T^{2} )^{2} \)
$71$ \( ( -8 + T )^{4} \)
$73$ \( ( 18 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( ( 288 + T^{2} )^{2} \)
$89$ \( ( -18 + T^{2} )^{2} \)
$97$ \( ( 18 + T^{2} )^{2} \)
show more
show less