Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 0 1.00000 1.50000 + 2.59808i −0.500000 + 0.866025i
471.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 0 1.00000 1.50000 2.59808i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.e.c 2
7.b odd 2 1 490.2.e.d 2
7.c even 3 1 490.2.a.h 1
7.c even 3 1 inner 490.2.e.c 2
7.d odd 6 1 70.2.a.a 1
7.d odd 6 1 490.2.e.d 2
21.g even 6 1 630.2.a.d 1
21.h odd 6 1 4410.2.a.b 1
28.f even 6 1 560.2.a.d 1
28.g odd 6 1 3920.2.a.t 1
35.i odd 6 1 350.2.a.b 1
35.j even 6 1 2450.2.a.l 1
35.k even 12 2 350.2.c.b 2
35.l odd 12 2 2450.2.c.k 2
56.j odd 6 1 2240.2.a.n 1
56.m even 6 1 2240.2.a.q 1
77.i even 6 1 8470.2.a.j 1
84.j odd 6 1 5040.2.a.bm 1
105.p even 6 1 3150.2.a.bj 1
105.w odd 12 2 3150.2.g.c 2
140.s even 6 1 2800.2.a.m 1
140.x odd 12 2 2800.2.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 7.d odd 6 1
350.2.a.b 1 35.i odd 6 1
350.2.c.b 2 35.k even 12 2
490.2.a.h 1 7.c even 3 1
490.2.e.c 2 1.a even 1 1 trivial
490.2.e.c 2 7.c even 3 1 inner
490.2.e.d 2 7.b odd 2 1
490.2.e.d 2 7.d odd 6 1
560.2.a.d 1 28.f even 6 1
630.2.a.d 1 21.g even 6 1
2240.2.a.n 1 56.j odd 6 1
2240.2.a.q 1 56.m even 6 1
2450.2.a.l 1 35.j even 6 1
2450.2.c.k 2 35.l odd 12 2
2800.2.a.m 1 140.s even 6 1
2800.2.g.n 2 140.x odd 12 2
3150.2.a.bj 1 105.p even 6 1
3150.2.g.c 2 105.w odd 12 2
3920.2.a.t 1 28.g odd 6 1
4410.2.a.b 1 21.h odd 6 1
5040.2.a.bm 1 84.j odd 6 1
8470.2.a.j 1 77.i even 6 1

Hecke kernels

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

\( T_{3} \)
\( T_{11}^{2} + 4 T_{11} + 16 \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( ( -6 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 64 - 8 T + T^{2} \)
$37$ \( 100 - 10 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 196 + 14 T + T^{2} \)
$67$ \( 144 - 12 T + T^{2} \)
$71$ \( ( 16 + T )^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 64 - 8 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 100 - 10 T + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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