Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} -3 \zeta_{12}^{2} q^{11} + 5 \zeta_{12}^{3} q^{13} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{18} + ( 5 - 5 \zeta_{12}^{2} ) q^{19} + ( -2 + \zeta_{12}^{3} ) q^{20} -3 \zeta_{12}^{3} q^{22} + 7 \zeta_{12} q^{23} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( -5 + 5 \zeta_{12}^{2} ) q^{26} + 4 q^{29} -2 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} -2 q^{34} -3 q^{36} + \zeta_{12} q^{37} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{38} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{40} -3 q^{41} + 2 \zeta_{12}^{3} q^{43} + ( 3 - 3 \zeta_{12}^{2} ) q^{44} + ( -3 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{45} + 7 \zeta_{12}^{2} q^{46} + 7 \zeta_{12} q^{47} + ( -4 - 3 \zeta_{12}^{3} ) q^{50} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{52} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{53} + ( 6 - 3 \zeta_{12}^{3} ) q^{55} + 4 \zeta_{12} q^{58} + 4 \zeta_{12}^{2} q^{59} + ( 6 - 6 \zeta_{12}^{2} ) q^{61} -2 \zeta_{12}^{3} q^{62} - q^{64} + ( -5 - 10 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{65} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{67} -2 \zeta_{12} q^{68} -6 q^{71} -3 \zeta_{12} q^{72} + ( 16 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{73} + \zeta_{12}^{2} q^{74} + 5 q^{76} + ( 14 - 14 \zeta_{12}^{2} ) q^{79} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} -3 \zeta_{12} q^{82} + 6 \zeta_{12}^{3} q^{83} + ( -2 - 4 \zeta_{12}^{3} ) q^{85} + ( -2 + 2 \zeta_{12}^{2} ) q^{86} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{88} + ( -2 + 2 \zeta_{12}^{2} ) q^{89} + ( -3 - 6 \zeta_{12}^{3} ) q^{90} + 7 \zeta_{12}^{3} q^{92} + 7 \zeta_{12}^{2} q^{94} + ( 5 \zeta_{12} + 10 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{95} -12 \zeta_{12}^{3} q^{97} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{5} - 6 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{4} - 4 q^{5} - 6 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{16} + 10 q^{19} - 8 q^{20} - 6 q^{25} - 10 q^{26} + 16 q^{29} - 4 q^{31} - 8 q^{34} - 12 q^{36} - 2 q^{40} - 12 q^{41} + 6 q^{44} - 12 q^{45} + 14 q^{46} - 16 q^{50} + 24 q^{55} + 8 q^{59} + 12 q^{61} - 4 q^{64} - 10 q^{65} - 24 q^{71} + 2 q^{74} + 20 q^{76} + 28 q^{79} - 4 q^{80} - 18 q^{81} - 8 q^{85} - 4 q^{86} - 4 q^{89} - 12 q^{90} + 14 q^{94} + 20 q^{95} + 36 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i −1.86603 + 1.23205i 0 0 1.00000i −1.50000 + 2.59808i 2.23205 0.133975i
79.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.133975 + 2.23205i 0 0 1.00000i −1.50000 + 2.59808i −1.23205 + 1.86603i
459.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.86603 1.23205i 0 0 1.00000i −1.50000 2.59808i 2.23205 + 0.133975i
459.2 0.866025 0.500000i 0 0.500000 0.866025i −0.133975 2.23205i 0 0 1.00000i −1.50000 2.59808i −1.23205 1.86603i
\(n\): 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.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.i.a 4
5.b even 2 1 inner 490.2.i.a 4
7.b odd 2 1 70.2.i.b 4
7.c even 3 1 490.2.c.d 2
7.c even 3 1 inner 490.2.i.a 4
7.d odd 6 1 70.2.i.b 4
7.d odd 6 1 490.2.c.a 2
21.c even 2 1 630.2.u.a 4
21.g even 6 1 630.2.u.a 4
28.d even 2 1 560.2.bw.d 4
28.f even 6 1 560.2.bw.d 4
35.c odd 2 1 70.2.i.b 4
35.f even 4 1 350.2.e.c 2
35.f even 4 1 350.2.e.j 2
35.i odd 6 1 70.2.i.b 4
35.i odd 6 1 490.2.c.a 2
35.j even 6 1 490.2.c.d 2
35.j even 6 1 inner 490.2.i.a 4
35.k even 12 1 350.2.e.c 2
35.k even 12 1 350.2.e.j 2
35.k even 12 1 2450.2.a.k 1
35.k even 12 1 2450.2.a.ba 1
35.l odd 12 1 2450.2.a.j 1
35.l odd 12 1 2450.2.a.bb 1
105.g even 2 1 630.2.u.a 4
105.p even 6 1 630.2.u.a 4
140.c even 2 1 560.2.bw.d 4
140.s even 6 1 560.2.bw.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 7.b odd 2 1
70.2.i.b 4 7.d odd 6 1
70.2.i.b 4 35.c odd 2 1
70.2.i.b 4 35.i odd 6 1
350.2.e.c 2 35.f even 4 1
350.2.e.c 2 35.k even 12 1
350.2.e.j 2 35.f even 4 1
350.2.e.j 2 35.k even 12 1
490.2.c.a 2 7.d odd 6 1
490.2.c.a 2 35.i odd 6 1
490.2.c.d 2 7.c even 3 1
490.2.c.d 2 35.j even 6 1
490.2.i.a 4 1.a even 1 1 trivial
490.2.i.a 4 5.b even 2 1 inner
490.2.i.a 4 7.c even 3 1 inner
490.2.i.a 4 35.j even 6 1 inner
560.2.bw.d 4 28.d even 2 1
560.2.bw.d 4 28.f even 6 1
560.2.bw.d 4 140.c even 2 1
560.2.bw.d 4 140.s even 6 1
630.2.u.a 4 21.c even 2 1
630.2.u.a 4 21.g even 6 1
630.2.u.a 4 105.g even 2 1
630.2.u.a 4 105.p even 6 1
2450.2.a.j 1 35.l odd 12 1
2450.2.a.k 1 35.k even 12 1
2450.2.a.ba 1 35.k even 12 1
2450.2.a.bb 1 35.l odd 12 1

Hecke kernels

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

\( T_{3} \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{19}^{2} - 5 T_{19} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( ( 25 + T^{2} )^{2} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 25 - 5 T + T^{2} )^{2} \)
$23$ \( 2401 - 49 T^{2} + T^{4} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( 1 - T^{2} + T^{4} \)
$41$ \( ( 3 + T )^{4} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( 2401 - 49 T^{2} + T^{4} \)
$53$ \( 6561 - 81 T^{2} + T^{4} \)
$59$ \( ( 16 - 4 T + T^{2} )^{2} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( 65536 - 256 T^{2} + T^{4} \)
$79$ \( ( 196 - 14 T + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 4 + 2 T + T^{2} )^{2} \)
$97$ \( ( 144 + T^{2} )^{2} \)
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