Properties

Label 490.4.c.c
Level $490$
Weight $4$
Character orbit 490.c
Analytic conductor $28.911$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.0.43197465600.1
Defining polynomial: \( x^{6} - 16x^{3} + 1521x^{2} - 624x + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{3} q^{2} + ( - \beta_{5} + 2 \beta_{3}) q^{3} - 4 q^{4} + ( - 3 \beta_{3} - \beta_{2} + 3) q^{5} + ( - 2 \beta_1 - 4) q^{6} - 8 \beta_{3} q^{8} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 4 \beta_1 - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{3} q^{2} + ( - \beta_{5} + 2 \beta_{3}) q^{3} - 4 q^{4} + ( - 3 \beta_{3} - \beta_{2} + 3) q^{5} + ( - 2 \beta_1 - 4) q^{6} - 8 \beta_{3} q^{8} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 4 \beta_1 - 25) q^{9} + ( - 2 \beta_{4} + 6 \beta_{3} + 6) q^{10} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 6) q^{11} + (4 \beta_{5} - 8 \beta_{3}) q^{12} + ( - 5 \beta_{5} + 12 \beta_{3}) q^{13} + ( - 10 \beta_{5} - 3 \beta_{4} - 9 \beta_{3} - \beta_{2} + 5 \beta_1 + 17) q^{15} + 16 q^{16} + ( - 4 \beta_{5} - 3 \beta_{4} - 52 \beta_{3} + \beta_{2} - \beta_1) q^{17} + (8 \beta_{5} + 6 \beta_{4} - 50 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{18} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 15 \beta_1 + 56) q^{19} + (12 \beta_{3} + 4 \beta_{2} - 12) q^{20} + (4 \beta_{5} + 6 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{22} + (14 \beta_{5} - 6 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{23} + (8 \beta_1 + 16) q^{24} + (5 \beta_{5} + 4 \beta_{4} - 71 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 82) q^{25} + ( - 10 \beta_1 - 24) q^{26} + (28 \beta_{5} + 21 \beta_{4} - 154 \beta_{3} - 7 \beta_{2} + 7 \beta_1) q^{27} + ( - 3 \beta_{5} - 3 \beta_{4} - 9 \beta_{2} + 2 \beta_1 - 28) q^{29} + ( - 10 \beta_{5} - 2 \beta_{4} + 34 \beta_{3} + 6 \beta_{2} - 20 \beta_1 + 18) q^{30} + (4 \beta_{5} + 4 \beta_{4} + 12 \beta_{2} + 10 \beta_1 - 64) q^{31} + 32 \beta_{3} q^{32} + (20 \beta_{5} + 15 \beta_{4} - 50 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{33} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 8 \beta_1 + 104) q^{34} + ( - 4 \beta_{5} - 4 \beta_{4} - 12 \beta_{2} + 16 \beta_1 + 100) q^{36} + (16 \beta_{5} + 12 \beta_{4} - 38 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{37} + (30 \beta_{5} + 6 \beta_{4} + 112 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{38} + (5 \beta_{5} + 5 \beta_{4} + 15 \beta_{2} - 22 \beta_1 - 264) q^{39} + (8 \beta_{4} - 24 \beta_{3} - 24) q^{40} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} + 30 \beta_1 - 10) q^{41} + ( - 6 \beta_{5} + 30 \beta_{4} + 60 \beta_{3} - 10 \beta_{2} + 10 \beta_1) q^{43} + ( - 4 \beta_{5} - 4 \beta_{4} - 12 \beta_{2} + 8 \beta_1 - 24) q^{44} + ( - 30 \beta_{5} - 11 \beta_{4} + 211 \beta_{3} + 16 \beta_{2} + \cdots - 325) q^{45}+ \cdots + ( - 8 \beta_{5} - 8 \beta_{4} - 24 \beta_{2} + 146 \beta_1 + 1052) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{4} + 16 q^{5} - 28 q^{6} - 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{4} + 16 q^{5} - 28 q^{6} - 152 q^{9} + 36 q^{10} + 38 q^{11} + 110 q^{15} + 96 q^{16} + 312 q^{19} - 64 q^{20} + 112 q^{24} + 486 q^{25} - 164 q^{26} - 182 q^{29} + 80 q^{30} - 340 q^{31} + 620 q^{34} + 608 q^{36} - 1598 q^{39} - 144 q^{40} + 12 q^{41} - 152 q^{44} - 1988 q^{45} + 200 q^{46} + 856 q^{50} - 238 q^{51} + 1876 q^{54} - 1636 q^{55} + 1180 q^{59} - 440 q^{60} - 704 q^{61} - 384 q^{64} + 586 q^{65} + 620 q^{66} + 4500 q^{69} + 1448 q^{71} + 472 q^{74} + 1960 q^{75} - 1248 q^{76} - 2074 q^{79} + 256 q^{80} + 4814 q^{81} - 82 q^{85} - 864 q^{86} - 2096 q^{89} - 2608 q^{90} + 316 q^{94} + 100 q^{95} - 448 q^{96} + 6556 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 16x^{3} + 1521x^{2} - 624x + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\nu^{5} - 1513\nu^{4} + 1209\nu^{3} - 248\nu^{2} - 1500719 ) / 177765 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -29\nu^{5} - 223\nu^{4} - 1131\nu^{3} + 232\nu^{2} - 25395\nu - 193529 ) / 25395 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1521\nu^{5} + 312\nu^{4} + 64\nu^{3} - 12168\nu^{2} + 2310945\nu - 475064 ) / 474040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5343\nu^{5} - 1096\nu^{4} + 3248\nu^{3} + 110464\nu^{2} - 7779335\nu + 1641032 ) / 203160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38337\nu^{5} + 7864\nu^{4} + 13768\nu^{3} - 780736\nu^{2} + 58721705\nu - 12071288 ) / 1422120 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{2} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{5} + 3\beta_{4} + 125\beta_{3} - \beta_{2} + \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39\beta_{5} + 39\beta_{4} - 40\beta_{3} - 78\beta_{2} + 78\beta _1 + 40 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{5} - 31\beta_{4} - 93\beta_{2} - 492\beta _1 - 4875 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -626\beta_{5} - 602\beta_{4} + 512\beta_{3} - 301\beta_{2} + 325\beta _1 + 512 \) 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
−4.51508 + 4.51508i
4.30950 4.30950i
0.205574 0.205574i
0.205574 + 0.205574i
4.30950 + 4.30950i
−4.51508 4.51508i
2.00000i 10.2673i −4.00000 8.27790 + 7.51508i −20.5347 0 8.00000i −78.4181 15.0302 16.5558i
99.2 2.00000i 3.17488i −4.00000 −11.1034 1.30950i −6.34975 0 8.00000i 16.9202 −2.61901 + 22.2068i
99.3 2.00000i 6.44221i −4.00000 10.8255 + 2.79443i 12.8844 0 8.00000i −14.5021 5.58885 21.6510i
99.4 2.00000i 6.44221i −4.00000 10.8255 2.79443i 12.8844 0 8.00000i −14.5021 5.58885 + 21.6510i
99.5 2.00000i 3.17488i −4.00000 −11.1034 + 1.30950i −6.34975 0 8.00000i 16.9202 −2.61901 22.2068i
99.6 2.00000i 10.2673i −4.00000 8.27790 7.51508i −20.5347 0 8.00000i −78.4181 15.0302 + 16.5558i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 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.c 6
5.b even 2 1 inner 490.4.c.c 6
5.c odd 4 1 2450.4.a.cf 3
5.c odd 4 1 2450.4.a.cg 3
7.b odd 2 1 70.4.c.b 6
21.c even 2 1 630.4.g.j 6
28.d even 2 1 560.4.g.e 6
35.c odd 2 1 70.4.c.b 6
35.f even 4 1 350.4.a.w 3
35.f even 4 1 350.4.a.x 3
105.g even 2 1 630.4.g.j 6
140.c even 2 1 560.4.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.b 6 7.b odd 2 1
70.4.c.b 6 35.c odd 2 1
350.4.a.w 3 35.f even 4 1
350.4.a.x 3 35.f even 4 1
490.4.c.c 6 1.a even 1 1 trivial
490.4.c.c 6 5.b even 2 1 inner
560.4.g.e 6 28.d even 2 1
560.4.g.e 6 140.c even 2 1
630.4.g.j 6 21.c even 2 1
630.4.g.j 6 105.g even 2 1
2450.4.a.cf 3 5.c odd 4 1
2450.4.a.cg 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{6} + 157T_{3}^{4} + 5856T_{3}^{2} + 44100 \) Copy content Toggle raw display
\( T_{19}^{3} - 156T_{19}^{2} - 8046T_{19} + 1196840 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 157 T^{4} + 5856 T^{2} + \cdots + 44100 \) Copy content Toggle raw display
$5$ \( T^{6} - 16 T^{5} - 115 T^{4} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 19 T^{2} - 1560 T - 600)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 4077 T^{4} + \cdots + 829555204 \) Copy content Toggle raw display
$17$ \( T^{6} + 12729 T^{4} + \cdots + 69288976 \) Copy content Toggle raw display
$19$ \( (T^{3} - 156 T^{2} - 8046 T + 1196840)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 45500 T^{4} + \cdots + 2480625000000 \) Copy content Toggle raw display
$29$ \( (T^{3} + 91 T^{2} - 11096 T + 204564)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 170 T^{2} - 25688 T - 2033232)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 80172 T^{4} + \cdots + 16972878400 \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} - 74460 T + 7243272)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 125041775755264 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 169390641480256 \) Copy content Toggle raw display
$53$ \( T^{6} + 492124 T^{4} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{3} - 590 T^{2} - 143278 T + 88782692)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 352 T^{2} - 83370 T + 3430824)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 546404 T^{4} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} - 724 T^{2} - 210420 T + 55149600)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 1394268 T^{4} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{3} + 1037 T^{2} - 212824 T - 276622388)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 1558444 T^{4} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1048 T^{2} - 984844 T - 210826480)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 2918281 T^{4} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
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