Properties

Label 490.2.i.f
Level $490$
Weight $2$
Character orbit 490.i
Analytic conductor $3.913$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{6} - \beta_{5}) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{6} + \beta_{4} q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{6} - \beta_{5}) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{6} + \beta_{4} q^{8} + 3 \beta_{2} q^{9} + (\beta_{6} - \beta_{2} - \beta_1 + 1) q^{10} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{11} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{12} + (\beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{13} + (2 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} + 3) q^{15} - \beta_{2} q^{16} + 2 \beta_1 q^{17} - 3 \beta_1 q^{18} + ( - \beta_{7} + \beta_{6} - \beta_{3} - 4 \beta_{2}) q^{19} + ( - \beta_{7} + \beta_{5} + \beta_{4} + 1) q^{20} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{22} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{23} + (\beta_{6} - \beta_{5}) q^{24} + (2 \beta_{6} - 2 \beta_{5} + \beta_1) q^{25} + (\beta_{7} - \beta_{6} + \beta_{3} - 2 \beta_{2}) q^{26} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2) q^{29} + ( - 2 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{30} + ( - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2} + 4) q^{31} + \beta_1 q^{32} + ( - 12 \beta_{4} + 12 \beta_1) q^{33} - 2 q^{34} + 3 q^{36} + ( - 2 \beta_{4} + 2 \beta_1) q^{37} + (\beta_{6} + \beta_{5} + 4 \beta_1) q^{38} + (2 \beta_{6} - 2 \beta_{5} + 6 \beta_{2} - 6) q^{39} + (\beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{40} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 6) q^{41} + (2 \beta_{7} - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3}) q^{43} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{44} + ( - 3 \beta_{5} + 3 \beta_{2} - 3 \beta_1 - 3) q^{45} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} - 2) q^{46} + (2 \beta_{7} + 2 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} - 4 \beta_1) q^{47} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{48} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 1) q^{50} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3}) q^{51} + ( - \beta_{6} - \beta_{5} + 2 \beta_1) q^{52} + (2 \beta_{6} + 2 \beta_{5} - 6 \beta_1) q^{53} + ( - 6 \beta_{4} - 4 \beta_{3} + 6) q^{55} + ( - 4 \beta_{7} + 4 \beta_{5} + 6 \beta_{4} + 4 \beta_{3}) q^{57} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{58} + (\beta_{6} - \beta_{5} - 4 \beta_{2} + 4) q^{59} + ( - 2 \beta_{5} - 3 \beta_{2} + 3 \beta_1 + 3) q^{60} + (\beta_{7} - \beta_{6} + \beta_{3} + 6 \beta_{2}) q^{61} + (2 \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{62} - q^{64} + (2 \beta_{7} + 2 \beta_{6} + 5 \beta_{4} - 2 \beta_{3} + \beta_{2} - 5 \beta_1) q^{65} + (12 \beta_{2} - 12) q^{66} + 8 \beta_1 q^{67} + ( - 2 \beta_{4} + 2 \beta_1) q^{68} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - 12) q^{69} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - 6) q^{71} + (3 \beta_{4} - 3 \beta_1) q^{72} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{73} + (2 \beta_{2} - 2) q^{74} + ( - \beta_{7} + \beta_{6} + 12 \beta_{4} - \beta_{3} - 12 \beta_1) q^{75} + ( - \beta_{7} + \beta_{5} - \beta_{3} - 4) q^{76} + ( - 2 \beta_{7} + 2 \beta_{5} - 6 \beta_{4} + 2 \beta_{3}) q^{78} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{79} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{80} + ( - 9 \beta_{2} + 9) q^{81} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_{4} + 2 \beta_{3} - 6 \beta_1) q^{82} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{83} + (2 \beta_{4} - 2 \beta_{3} - 2) q^{85} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} + 4 \beta_{2}) q^{86} + (2 \beta_{6} + 2 \beta_{5} - 12 \beta_1) q^{87} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{88} + 10 \beta_{2} q^{89} + ( - 3 \beta_{4} + 3 \beta_{3} + 3) q^{90} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{92} + ( - 4 \beta_{7} - 4 \beta_{6} - 12 \beta_{4} + 4 \beta_{3} + 12 \beta_1) q^{93} + ( - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2} + 4) q^{94} + (2 \beta_{6} + 4 \beta_{5} - \beta_{2} + 7 \beta_1 + 1) q^{95} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{96} + ( - 4 \beta_{7} + 4 \beta_{5} - 6 \beta_{4} + 4 \beta_{3}) q^{97} + ( - 6 \beta_{7} + 6 \beta_{5} - 6 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{5} + 12 q^{9} + 4 q^{10} + 24 q^{15} - 4 q^{16} - 16 q^{19} + 8 q^{20} - 8 q^{26} - 16 q^{29} - 12 q^{30} + 16 q^{31} - 16 q^{34} + 24 q^{36} - 24 q^{39} - 4 q^{40} + 48 q^{41} - 12 q^{45} - 8 q^{46} - 8 q^{50} + 48 q^{55} + 16 q^{59} + 12 q^{60} + 24 q^{61} - 8 q^{64} + 4 q^{65} - 48 q^{66} - 96 q^{69} - 48 q^{71} - 8 q^{74} - 32 q^{76} + 8 q^{79} + 4 q^{80} + 36 q^{81} - 16 q^{85} + 16 q^{86} + 40 q^{89} + 24 q^{90} + 16 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.866025 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i −2.03906 0.917738i 2.44949 0 1.00000i 1.50000 2.59808i 1.30701 + 1.81431i
79.2 −0.866025 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 1.30701 1.81431i −2.44949 0 1.00000i 1.50000 2.59808i −2.03906 + 0.917738i
79.3 0.866025 + 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i 0.917738 2.03906i −2.44949 0 1.00000i 1.50000 2.59808i 1.81431 1.30701i
79.4 0.866025 + 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 1.81431 + 1.30701i 2.44949 0 1.00000i 1.50000 2.59808i 0.917738 + 2.03906i
459.1 −0.866025 + 0.500000i −2.12132 1.22474i 0.500000 0.866025i −2.03906 + 0.917738i 2.44949 0 1.00000i 1.50000 + 2.59808i 1.30701 1.81431i
459.2 −0.866025 + 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 1.30701 + 1.81431i −2.44949 0 1.00000i 1.50000 + 2.59808i −2.03906 0.917738i
459.3 0.866025 0.500000i −2.12132 1.22474i 0.500000 0.866025i 0.917738 + 2.03906i −2.44949 0 1.00000i 1.50000 + 2.59808i 1.81431 + 1.30701i
459.4 0.866025 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 1.81431 1.30701i 2.44949 0 1.00000i 1.50000 + 2.59808i 0.917738 2.03906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.4
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.f 8
5.b even 2 1 inner 490.2.i.f 8
7.b odd 2 1 490.2.i.c 8
7.c even 3 1 490.2.c.e 4
7.c even 3 1 inner 490.2.i.f 8
7.d odd 6 1 70.2.c.a 4
7.d odd 6 1 490.2.i.c 8
21.g even 6 1 630.2.g.g 4
28.f even 6 1 560.2.g.e 4
35.c odd 2 1 490.2.i.c 8
35.i odd 6 1 70.2.c.a 4
35.i odd 6 1 490.2.i.c 8
35.j even 6 1 490.2.c.e 4
35.j even 6 1 inner 490.2.i.f 8
35.k even 12 1 350.2.a.g 2
35.k even 12 1 350.2.a.h 2
35.l odd 12 1 2450.2.a.bl 2
35.l odd 12 1 2450.2.a.bq 2
56.j odd 6 1 2240.2.g.j 4
56.m even 6 1 2240.2.g.i 4
84.j odd 6 1 5040.2.t.t 4
105.p even 6 1 630.2.g.g 4
105.w odd 12 1 3150.2.a.bs 2
105.w odd 12 1 3150.2.a.bt 2
140.s even 6 1 560.2.g.e 4
140.x odd 12 1 2800.2.a.bl 2
140.x odd 12 1 2800.2.a.bm 2
280.ba even 6 1 2240.2.g.i 4
280.bk odd 6 1 2240.2.g.j 4
420.be odd 6 1 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 7.d odd 6 1
70.2.c.a 4 35.i odd 6 1
350.2.a.g 2 35.k even 12 1
350.2.a.h 2 35.k even 12 1
490.2.c.e 4 7.c even 3 1
490.2.c.e 4 35.j even 6 1
490.2.i.c 8 7.b odd 2 1
490.2.i.c 8 7.d odd 6 1
490.2.i.c 8 35.c odd 2 1
490.2.i.c 8 35.i odd 6 1
490.2.i.f 8 1.a even 1 1 trivial
490.2.i.f 8 5.b even 2 1 inner
490.2.i.f 8 7.c even 3 1 inner
490.2.i.f 8 35.j even 6 1 inner
560.2.g.e 4 28.f even 6 1
560.2.g.e 4 140.s even 6 1
630.2.g.g 4 21.g even 6 1
630.2.g.g 4 105.p even 6 1
2240.2.g.i 4 56.m even 6 1
2240.2.g.i 4 280.ba even 6 1
2240.2.g.j 4 56.j odd 6 1
2240.2.g.j 4 280.bk odd 6 1
2450.2.a.bl 2 35.l odd 12 1
2450.2.a.bq 2 35.l odd 12 1
2800.2.a.bl 2 140.x odd 12 1
2800.2.a.bm 2 140.x odd 12 1
3150.2.a.bs 2 105.w odd 12 1
3150.2.a.bt 2 105.w odd 12 1
5040.2.t.t 4 84.j odd 6 1
5040.2.t.t 4 420.be odd 6 1

Hecke kernels

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

\( T_{3}^{4} - 6T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{11}^{4} + 24T_{11}^{2} + 576 \) Copy content Toggle raw display
\( T_{19}^{4} + 8T_{19}^{3} + 54T_{19}^{2} + 80T_{19} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} + 54 T^{2} + 80 T + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 56 T^{6} + 2736 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 20)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} + 72 T^{2} + 64 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 80 T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 80 T^{6} + 6336 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} - 120 T^{6} + 14256 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{3} + 54 T^{2} - 80 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + 114 T^{2} - 360 T + 900)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 64 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} - 56 T^{6} + 2736 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T + 100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 264 T^{2} + 3600)^{2} \) Copy content Toggle raw display
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