Properties

Label 490.4.c.d
Level $490$
Weight $4$
Character orbit 490.c
Analytic conductor $28.911$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6654810844696576.6
Defining polynomial: \( x^{8} + 1325x^{4} + 9604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} - 4 q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{6} - 4 \beta_1 q^{8} + ( - \beta_{3} + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} - 4 q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{6} - 4 \beta_1 q^{8} + ( - \beta_{3} + 8) q^{9} + (3 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4}) q^{10} + ( - 3 \beta_{3} - 3) q^{11} - 4 \beta_{6} q^{12} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5}) q^{13} + (2 \beta_{3} + \beta_{2} - 13 \beta_1 + 17) q^{15} + 16 q^{16} + ( - 13 \beta_{7} - 3 \beta_{6} - 13 \beta_{5}) q^{17} + (2 \beta_{2} + 8 \beta_1) q^{18} + ( - \beta_{7} + \beta_{5} + 16 \beta_{4}) q^{19} + (8 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 4 \beta_{4}) q^{20} + (6 \beta_{2} - 3 \beta_1) q^{22} + (10 \beta_{2} - 4 \beta_1) q^{23} + (4 \beta_{7} - 4 \beta_{5} + 4 \beta_{4}) q^{24} + (\beta_{3} - 7 \beta_{2} + \beta_1 - 39) q^{25} + (\beta_{7} - \beta_{5} + 5 \beta_{4}) q^{26} + ( - 7 \beta_{7} + 15 \beta_{6} - 7 \beta_{5}) q^{27} + (\beta_{3} + 107) q^{29} + (2 \beta_{3} - 4 \beta_{2} + 17 \beta_1 + 52) q^{30} + (32 \beta_{7} - 32 \beta_{5} + 10 \beta_{4}) q^{31} + 16 \beta_1 q^{32} + ( - 21 \beta_{7} - 63 \beta_{6} - 21 \beta_{5}) q^{33} + (3 \beta_{7} - 3 \beta_{5} - 23 \beta_{4}) q^{34} + (4 \beta_{3} - 32) q^{36} + ( - 6 \beta_{2} - 96 \beta_1) q^{37} + ( - 34 \beta_{7} - 4 \beta_{6} - 34 \beta_{5}) q^{38} + (\beta_{3} + 47) q^{39} + ( - 12 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 4 \beta_{4}) q^{40} + (20 \beta_{7} - 20 \beta_{5} - 26 \beta_{4}) q^{41} + ( - 8 \beta_{2} + 26 \beta_1) q^{43} + (12 \beta_{3} + 12) q^{44} + ( - 17 \beta_{7} + 3 \beta_{6} - 36 \beta_{5} - 11 \beta_{4}) q^{45} + (20 \beta_{3} + 16) q^{46} + ( - 35 \beta_{7} - 43 \beta_{6} - 35 \beta_{5}) q^{47} + 16 \beta_{6} q^{48} + ( - 14 \beta_{3} - 2 \beta_{2} - 39 \beta_1 - 4) q^{50} + (3 \beta_{3} - 125) q^{51} + ( - 8 \beta_{7} + 4 \beta_{6} - 8 \beta_{5}) q^{52} + (6 \beta_{2} - 54 \beta_1) q^{53} + ( - 15 \beta_{7} + 15 \beta_{5} - 29 \beta_{4}) q^{54} + (3 \beta_{7} + 63 \beta_{6} - 81 \beta_{5} - 6 \beta_{4}) q^{55} + (2 \beta_{2} + 100 \beta_1) q^{57} + ( - 2 \beta_{2} + 107 \beta_1) q^{58} + (51 \beta_{7} - 51 \beta_{5} - 66 \beta_{4}) q^{59} + ( - 8 \beta_{3} - 4 \beta_{2} + 52 \beta_1 - 68) q^{60} + (43 \beta_{7} - 43 \beta_{5} + 104 \beta_{4}) q^{61} + (44 \beta_{7} + 128 \beta_{6} + 44 \beta_{5}) q^{62} - 64 q^{64} + ( - 8 \beta_{3} + \beta_{2} + 47 \beta_1 + 47) q^{65} + (63 \beta_{7} - 63 \beta_{5} + 21 \beta_{4}) q^{66} + (16 \beta_{2} - 148 \beta_1) q^{67} + (52 \beta_{7} + 12 \beta_{6} + 52 \beta_{5}) q^{68} + (104 \beta_{7} - 104 \beta_{5} + 34 \beta_{4}) q^{69} + ( - 18 \beta_{3} - 432) q^{71} + ( - 8 \beta_{2} - 32 \beta_1) q^{72} + ( - 8 \beta_{7} + 190 \beta_{6} - 8 \beta_{5}) q^{73} + ( - 12 \beta_{3} + 384) q^{74} + ( - 64 \beta_{7} - 19 \beta_{6} + 78 \beta_{5} - 22 \beta_{4}) q^{75} + (4 \beta_{7} - 4 \beta_{5} - 64 \beta_{4}) q^{76} + ( - 2 \beta_{2} + 47 \beta_1) q^{78} + ( - 37 \beta_{3} - 367) q^{79} + ( - 32 \beta_{7} - 32 \beta_{6} - 16 \beta_{5} - 16 \beta_{4}) q^{80} + ( - 42 \beta_{3} - 167) q^{81} + (92 \beta_{7} + 80 \beta_{6} + 92 \beta_{5}) q^{82} + ( - 125 \beta_{7} - 106 \beta_{6} - 125 \beta_{5}) q^{83} + (33 \beta_{3} - 16 \beta_{2} - 182 \beta_1 - 467) q^{85} + ( - 16 \beta_{3} - 104) q^{86} + (7 \beta_{7} + 127 \beta_{6} + 7 \beta_{5}) q^{87} + ( - 24 \beta_{2} + 12 \beta_1) q^{88} + ( - 98 \beta_{7} + 98 \beta_{5} - 46 \beta_{4}) q^{89} + (38 \beta_{7} + 38 \beta_{6} + 44 \beta_{5} - 56 \beta_{4}) q^{90} + ( - 40 \beta_{2} + 16 \beta_1) q^{92} + ( - 64 \beta_{2} + 454 \beta_1) q^{93} + (43 \beta_{7} - 43 \beta_{5} - 27 \beta_{4}) q^{94} + (19 \beta_{3} + 47 \beta_{2} + 289 \beta_1 - 526) q^{95} + ( - 16 \beta_{7} + 16 \beta_{5} - 16 \beta_{4}) q^{96} + ( - 205 \beta_{7} - 175 \beta_{6} - 205 \beta_{5}) q^{97} + ( - 18 \beta_{3} + 822) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} + 60 q^{9} - 36 q^{11} + 144 q^{15} + 128 q^{16} - 308 q^{25} + 860 q^{29} + 424 q^{30} - 240 q^{36} + 380 q^{39} + 144 q^{44} + 208 q^{46} - 88 q^{50} - 988 q^{51} - 576 q^{60} - 512 q^{64} + 344 q^{65} - 3528 q^{71} + 3024 q^{74} - 3084 q^{79} - 1504 q^{81} - 3604 q^{85} - 896 q^{86} - 4132 q^{95} + 6504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1325x^{4} + 9604 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 1423\nu^{2} ) / 1911 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{6} - 12319\nu^{2} ) / 1911 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 682 ) / 39 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 1423\nu^{3} + 3822\nu ) / 3822 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{7} - 98\nu^{5} + 31753\nu^{3} - 139454\nu ) / 53508 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\nu^{7} + 98\nu^{5} + 51675\nu^{3} + 139454\nu ) / 53508 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -39\nu^{7} + 98\nu^{5} - 51675\nu^{3} + 85946\nu ) / 53508 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -16\beta_{7} - 7\beta_{6} - 23\beta_{5} - 16\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 39\beta_{3} - 682 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 848\beta_{7} + 273\beta_{6} + 575\beta_{5} - 575\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1423\beta_{2} - 12319\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20857\beta_{7} + 9961\beta_{6} + 30818\beta_{5} + 20857\beta_{4} \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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
1.16183 1.16183i
4.26030 4.26030i
−4.26030 + 4.26030i
−1.16183 + 1.16183i
−1.16183 1.16183i
−4.26030 4.26030i
4.26030 + 4.26030i
1.16183 + 1.16183i
2.00000i 6.02497i −4.00000 −7.18680 + 8.56445i −12.0499 0 8.00000i −9.30030 17.1289 + 14.3736i
99.2 2.00000i 1.64308i −4.00000 −5.90338 9.49474i −3.28615 0 8.00000i 24.3003 −18.9895 + 11.8068i
99.3 2.00000i 1.64308i −4.00000 5.90338 + 9.49474i 3.28615 0 8.00000i 24.3003 18.9895 11.8068i
99.4 2.00000i 6.02497i −4.00000 7.18680 8.56445i 12.0499 0 8.00000i −9.30030 −17.1289 14.3736i
99.5 2.00000i 6.02497i −4.00000 7.18680 + 8.56445i 12.0499 0 8.00000i −9.30030 −17.1289 + 14.3736i
99.6 2.00000i 1.64308i −4.00000 5.90338 9.49474i 3.28615 0 8.00000i 24.3003 18.9895 + 11.8068i
99.7 2.00000i 1.64308i −4.00000 −5.90338 + 9.49474i −3.28615 0 8.00000i 24.3003 −18.9895 11.8068i
99.8 2.00000i 6.02497i −4.00000 −7.18680 8.56445i −12.0499 0 8.00000i −9.30030 17.1289 14.3736i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
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 490.4.c.d 8
5.b even 2 1 inner 490.4.c.d 8
5.c odd 4 1 2450.4.a.cm 4
5.c odd 4 1 2450.4.a.cs 4
7.b odd 2 1 inner 490.4.c.d 8
35.c odd 2 1 inner 490.4.c.d 8
35.f even 4 1 2450.4.a.cm 4
35.f even 4 1 2450.4.a.cs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.4.c.d 8 1.a even 1 1 trivial
490.4.c.d 8 5.b even 2 1 inner
490.4.c.d 8 7.b odd 2 1 inner
490.4.c.d 8 35.c odd 2 1 inner
2450.4.a.cm 4 5.c odd 4 1
2450.4.a.cm 4 35.f even 4 1
2450.4.a.cs 4 5.c odd 4 1
2450.4.a.cs 4 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 \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{4} + 39T_{3}^{2} + 98 \) Copy content Toggle raw display
\( T_{19}^{4} - 20794T_{19}^{2} + 15103008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 39 T^{2} + 98)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 154 T^{6} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9 T - 2520)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 463 T^{2} + 39762)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 11349 T^{2} + 1648928)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 20794 T^{2} + 15103008)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 56788 T^{2} + \cdots + 787139136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 215 T + 11274)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 118648 T^{2} + \cdots + 757694592)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 91764 T^{2} + \cdots + 653313600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 124408 T^{2} + \cdots + 1914072192)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 42400 T^{2} + \cdots + 222845184)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 83381 T^{2} + \cdots + 1636148808)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 44964 T^{2} + 4665600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 805194 T^{2} + \cdots + 80763412608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 872458 T^{2} + \cdots + 190295143200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 329344 T^{2} + \cdots + 406425600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 882 T + 103032)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1498012 T^{2} + \cdots + 195782782752)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 771 T - 237790)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 914954 T^{2} + \cdots + 96227090208)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 1138384 T^{2} + \cdots + 16460236800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2463325 T^{2} + \cdots + 711385920000)^{2} \) Copy content Toggle raw display
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