Properties

Label 490.2.e.i
Level $490$
Weight $2$
Character orbit 490.e
Analytic conductor $3.913$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{6} - q^{8} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{6} - q^{8} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{9} -\beta_{2} q^{10} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{12} + ( -2 + 2 \beta_{3} ) q^{13} + ( 2 - \beta_{3} ) q^{15} + ( -1 - \beta_{2} ) q^{16} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{17} + ( -4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{18} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{19} + q^{20} + ( -2 - 2 \beta_{3} ) q^{22} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{24} + \beta_{2} q^{25} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{26} + ( 8 - 8 \beta_{3} ) q^{27} + ( -2 + 2 \beta_{3} ) q^{29} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{30} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{31} -\beta_{2} q^{32} + 2 \beta_{1} q^{33} + ( -4 - \beta_{3} ) q^{34} + ( 3 - 4 \beta_{3} ) q^{36} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{38} + ( -6 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{39} + ( 1 + \beta_{2} ) q^{40} + ( 4 + 5 \beta_{3} ) q^{41} + ( -6 + 2 \beta_{3} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{45} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{47} + ( 2 - \beta_{3} ) q^{48} - q^{50} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{51} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{53} + ( 8 + 8 \beta_{1} + 8 \beta_{2} ) q^{54} + ( 2 + 2 \beta_{3} ) q^{55} + 2 q^{57} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -\beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{59} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{60} + ( 2 - 8 \beta_{1} + 2 \beta_{2} ) q^{61} -2 \beta_{3} q^{62} + q^{64} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{66} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{67} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{68} -4 q^{69} + ( 4 + 6 \beta_{3} ) q^{71} + ( 3 + 4 \beta_{1} + 3 \beta_{2} ) q^{72} + ( -\beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{73} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{75} + ( 2 + \beta_{3} ) q^{76} + ( 8 - 6 \beta_{3} ) q^{78} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( 12 \beta_{1} + 23 \beta_{2} + 12 \beta_{3} ) q^{81} + ( 4 - 5 \beta_{1} + 4 \beta_{2} ) q^{82} + ( 2 + 3 \beta_{3} ) q^{83} + ( 4 + \beta_{3} ) q^{85} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{86} + ( -6 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{88} -9 \beta_{1} q^{89} + ( -3 + 4 \beta_{3} ) q^{90} + ( -4 - 2 \beta_{3} ) q^{92} + ( 4 + 4 \beta_{1} + 4 \beta_{2} ) q^{93} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{95} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{96} + ( -12 - 3 \beta_{3} ) q^{97} + ( -10 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} - 8q^{6} - 4q^{8} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} - 8q^{6} - 4q^{8} - 6q^{9} + 2q^{10} - 4q^{11} - 4q^{12} - 8q^{13} + 8q^{15} - 2q^{16} - 8q^{17} + 6q^{18} - 4q^{19} + 4q^{20} - 8q^{22} + 8q^{23} + 4q^{24} - 2q^{25} - 4q^{26} + 32q^{27} - 8q^{29} + 4q^{30} + 2q^{32} - 16q^{34} + 12q^{36} + 4q^{37} + 4q^{38} + 16q^{39} + 2q^{40} + 16q^{41} - 24q^{43} - 4q^{44} - 6q^{45} - 8q^{46} - 16q^{47} + 8q^{48} - 4q^{50} - 12q^{51} + 4q^{52} + 4q^{53} + 16q^{54} + 8q^{55} + 8q^{57} - 4q^{58} - 20q^{59} - 4q^{60} + 4q^{61} + 4q^{64} + 4q^{65} + 8q^{67} - 8q^{68} - 16q^{69} + 16q^{71} + 6q^{72} + 16q^{73} - 4q^{74} - 4q^{75} + 8q^{76} + 32q^{78} + 8q^{79} - 2q^{80} - 46q^{81} + 8q^{82} + 8q^{83} + 16q^{85} - 12q^{86} + 16q^{87} + 4q^{88} - 12q^{90} - 16q^{92} + 8q^{93} + 16q^{94} - 4q^{95} + 4q^{96} - 48q^{97} - 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0.500000 + 0.866025i −1.70711 + 2.95680i −0.500000 + 0.866025i −0.500000 0.866025i −3.41421 0 −1.00000 −4.32843 7.49706i 0.500000 0.866025i
361.2 0.500000 + 0.866025i −0.292893 + 0.507306i −0.500000 + 0.866025i −0.500000 0.866025i −0.585786 0 −1.00000 1.32843 + 2.30090i 0.500000 0.866025i
471.1 0.500000 0.866025i −1.70711 2.95680i −0.500000 0.866025i −0.500000 + 0.866025i −3.41421 0 −1.00000 −4.32843 + 7.49706i 0.500000 + 0.866025i
471.2 0.500000 0.866025i −0.292893 0.507306i −0.500000 0.866025i −0.500000 + 0.866025i −0.585786 0 −1.00000 1.32843 2.30090i 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.i 4
7.b odd 2 1 490.2.e.j 4
7.c even 3 1 490.2.a.m yes 2
7.c even 3 1 inner 490.2.e.i 4
7.d odd 6 1 490.2.a.l 2
7.d odd 6 1 490.2.e.j 4
21.g even 6 1 4410.2.a.by 2
21.h odd 6 1 4410.2.a.bt 2
28.f even 6 1 3920.2.a.ca 2
28.g odd 6 1 3920.2.a.bm 2
35.i odd 6 1 2450.2.a.bs 2
35.j even 6 1 2450.2.a.bn 2
35.k even 12 2 2450.2.c.w 4
35.l odd 12 2 2450.2.c.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 7.d odd 6 1
490.2.a.m yes 2 7.c even 3 1
490.2.e.i 4 1.a even 1 1 trivial
490.2.e.i 4 7.c even 3 1 inner
490.2.e.j 4 7.b odd 2 1
490.2.e.j 4 7.d odd 6 1
2450.2.a.bn 2 35.j even 6 1
2450.2.a.bs 2 35.i odd 6 1
2450.2.c.t 4 35.l odd 12 2
2450.2.c.w 4 35.k even 12 2
3920.2.a.bm 2 28.g odd 6 1
3920.2.a.ca 2 28.f even 6 1
4410.2.a.bt 2 21.h odd 6 1
4410.2.a.by 2 21.g even 6 1

Hecke kernels

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

\( T_{3}^{4} + 4 T_{3}^{3} + 14 T_{3}^{2} + 8 T_{3} + 4 \)
\( T_{11}^{4} + 4 T_{11}^{3} + 20 T_{11}^{2} - 16 T_{11} + 16 \)
\( T_{13}^{2} + 4 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( ( -4 + 4 T + T^{2} )^{2} \)
$17$ \( 196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( 64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( ( -4 + 4 T + T^{2} )^{2} \)
$31$ \( 64 + 8 T^{2} + T^{4} \)
$37$ \( 784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( -34 - 8 T + T^{2} )^{2} \)
$43$ \( ( 28 + 12 T + T^{2} )^{2} \)
$47$ \( 3136 + 896 T + 200 T^{2} + 16 T^{3} + T^{4} \)
$53$ \( 4624 + 272 T + 84 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( 9604 + 1960 T + 302 T^{2} + 20 T^{3} + T^{4} \)
$61$ \( 15376 + 496 T + 140 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( 256 + 128 T + 80 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( -56 - 8 T + T^{2} )^{2} \)
$73$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$79$ \( 64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4} \)
$83$ \( ( -14 - 4 T + T^{2} )^{2} \)
$89$ \( 26244 + 162 T^{2} + T^{4} \)
$97$ \( ( 126 + 24 T + T^{2} )^{2} \)
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