Properties

Label 490.4.c.e
Level $490$
Weight $4$
Character orbit 490.c
Analytic conductor $28.911$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 185x^{10} + 12748x^{8} + 405460x^{6} + 5908496x^{4} + 33016000x^{2} + 60840000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{7} + \beta_{4}) q^{3} - 4 q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{3} - 2) q^{6} - 4 \beta_{7} q^{8} + (\beta_{3} + \beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + (\beta_{7} + \beta_{4}) q^{3} - 4 q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{3} - 2) q^{6} - 4 \beta_{7} q^{8} + (\beta_{3} + \beta_1 - 5) q^{9} + ( - \beta_{10} - \beta_{7} + 1) q^{10} + ( - \beta_{3} - \beta_{2} - \beta_1 + 5) q^{11} + ( - 4 \beta_{7} - 4 \beta_{4}) q^{12} + (\beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4}) q^{13} + ( - \beta_{11} + \beta_{9} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 8) q^{15}+ \cdots + (\beta_{10} + \beta_{9} - 13 \beta_{6} + 13 \beta_{5} + 49 \beta_{3} - 9 \beta_{2} + \cdots - 705) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} - 8 q^{5} - 28 q^{6} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{4} - 8 q^{5} - 28 q^{6} - 62 q^{9} + 12 q^{10} + 62 q^{11} + 86 q^{15} + 192 q^{16} + 186 q^{19} + 32 q^{20} + 112 q^{24} - 126 q^{25} - 236 q^{26} - 338 q^{29} - 28 q^{30} - 652 q^{31} + 272 q^{34} + 248 q^{36} + 868 q^{39} - 48 q^{40} - 396 q^{41} - 248 q^{44} + 664 q^{45} - 376 q^{46} + 160 q^{50} + 1448 q^{51} + 1540 q^{54} - 298 q^{55} + 1336 q^{59} - 344 q^{60} - 314 q^{61} - 768 q^{64} - 1862 q^{65} - 1600 q^{66} - 90 q^{69} + 2216 q^{71} + 1012 q^{74} - 4550 q^{75} - 744 q^{76} + 1772 q^{79} - 128 q^{80} - 1228 q^{81} + 2282 q^{85} - 396 q^{86} + 6094 q^{89} - 100 q^{90} + 3604 q^{94} - 1166 q^{95} - 448 q^{96} - 8546 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 185x^{10} + 12748x^{8} + 405460x^{6} + 5908496x^{4} + 33016000x^{2} + 60840000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{10} - 1331\nu^{8} - 32112\nu^{6} + 1082536\nu^{4} + 40381408\nu^{2} + 167652000 ) / 529200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{10} - 631\nu^{8} - 33981\nu^{6} - 725074\nu^{4} - 5385040\nu^{2} - 12402000 ) / 44100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -53\nu^{11} - 8765\nu^{9} - 511584\nu^{7} - 12654320\nu^{5} - 124631048\nu^{3} - 326805600\nu ) / 22932000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 469 \nu^{11} + 4290 \nu^{10} + 80395 \nu^{9} + 737490 \nu^{8} + 4984182 \nu^{7} + 45611280 \nu^{6} + 136581760 \nu^{5} + 1229242560 \nu^{4} + \cdots + 40345344000 ) / 137592000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 469 \nu^{11} - 4290 \nu^{10} + 80395 \nu^{9} - 737490 \nu^{8} + 4984182 \nu^{7} - 45611280 \nu^{6} + 136581760 \nu^{5} - 1229242560 \nu^{4} + \cdots - 40345344000 ) / 137592000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\nu^{11} + 8765\nu^{9} + 511584\nu^{7} + 12654320\nu^{5} + 124631048\nu^{3} + 349737600\nu ) / 11466000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -128\nu^{11} - 16335\nu^{9} - 510039\nu^{7} + 5166700\nu^{5} + 324703332\nu^{3} + 1474639200\nu ) / 11466000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 225 \nu^{11} + 1924 \nu^{10} + 34995 \nu^{9} + 292864 \nu^{8} + 1842210 \nu^{7} + 14793948 \nu^{6} + 37724400 \nu^{5} + 274726816 \nu^{4} + \cdots + 1575194400 ) / 13759200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 225 \nu^{11} + 1924 \nu^{10} - 34995 \nu^{9} + 292864 \nu^{8} - 1842210 \nu^{7} + 14793948 \nu^{6} - 37724400 \nu^{5} + 274726816 \nu^{4} + \cdots + 1575194400 ) / 13759200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -201\nu^{11} - 35885\nu^{9} - 2341348\nu^{7} - 67921560\nu^{5} - 821150696\nu^{3} - 2546759200\nu ) / 7644000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 2\beta_{10} + 2\beta_{9} + \beta_{8} - 36\beta_{7} - 3\beta_{6} - 3\beta_{5} - 46\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} - \beta_{9} + 13\beta_{6} - 13\beta_{5} - 15\beta_{3} + 13\beta_{2} - 63\beta _1 + 1469 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 149 \beta_{11} + 157 \beta_{10} - 157 \beta_{9} - 29 \beta_{8} + 2638 \beta_{7} + 321 \beta_{6} + 321 \beta_{5} + 2446 \beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 215\beta_{10} + 215\beta_{9} - 1409\beta_{6} + 1409\beta_{5} + 1905\beta_{3} - 1193\beta_{2} + 3803\beta _1 - 79749 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13677 \beta_{11} - 11175 \beta_{10} + 11175 \beta_{9} - 303 \beta_{8} - 192326 \beta_{7} - 24417 \beta_{6} - 24417 \beta_{5} - 141466 \beta_{4} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 25011 \beta_{10} - 25011 \beta_{9} + 114849 \beta_{6} - 114849 \beta_{5} - 175185 \beta_{3} + 83385 \beta_{2} - 234091 \beta _1 + 4653661 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1080973 \beta_{11} + 775739 \beta_{10} - 775739 \beta_{9} + 132767 \beta_{8} + 13794318 \beta_{7} + 1680537 \beta_{6} + 1680537 \beta_{5} + 8628346 \beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2300275 \beta_{10} + 2300275 \beta_{9} - 8504113 \beta_{6} + 8504113 \beta_{5} + 14159985 \beta_{3} - 5375641 \beta_{2} + 14694075 \beta _1 - 284277197 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 79974653 \beta_{11} - 53205043 \beta_{10} + 53205043 \beta_{9} - 14459407 \beta_{8} - 973114606 \beta_{7} - 111824601 \beta_{6} - 111824601 \beta_{5} - 543865786 \beta_{4} \) 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
8.18765i
5.44232i
2.06759i
2.20031i
5.49484i
7.00242i
7.00242i
5.49484i
2.20031i
2.06759i
5.44232i
8.18765i
2.00000i 9.18765i −4.00000 −8.58556 + 7.16157i −18.3753 0 8.00000i −57.4130 14.3231 + 17.1711i
99.2 2.00000i 6.44232i −4.00000 6.24768 9.27181i −12.8846 0 8.00000i −14.5035 −18.5436 12.4954i
99.3 2.00000i 3.06759i −4.00000 9.26709 + 6.25468i −6.13518 0 8.00000i 17.5899 12.5094 18.5342i
99.4 2.00000i 1.20031i −4.00000 −11.1276 1.08430i 2.40061 0 8.00000i 25.5593 −2.16860 + 22.2553i
99.5 2.00000i 4.49484i −4.00000 3.34202 + 10.6692i 8.98967 0 8.00000i 6.79645 21.3383 6.68404i
99.6 2.00000i 6.00242i −4.00000 −3.14360 10.7293i 12.0048 0 8.00000i −9.02909 −21.4586 + 6.28719i
99.7 2.00000i 6.00242i −4.00000 −3.14360 + 10.7293i 12.0048 0 8.00000i −9.02909 −21.4586 6.28719i
99.8 2.00000i 4.49484i −4.00000 3.34202 10.6692i 8.98967 0 8.00000i 6.79645 21.3383 + 6.68404i
99.9 2.00000i 1.20031i −4.00000 −11.1276 + 1.08430i 2.40061 0 8.00000i 25.5593 −2.16860 22.2553i
99.10 2.00000i 3.06759i −4.00000 9.26709 6.25468i −6.13518 0 8.00000i 17.5899 12.5094 + 18.5342i
99.11 2.00000i 6.44232i −4.00000 6.24768 + 9.27181i −12.8846 0 8.00000i −14.5035 −18.5436 + 12.4954i
99.12 2.00000i 9.18765i −4.00000 −8.58556 7.16157i −18.3753 0 8.00000i −57.4130 14.3231 17.1711i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.12
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.e 12
5.b even 2 1 inner 490.4.c.e 12
5.c odd 4 1 2450.4.a.cw 6
5.c odd 4 1 2450.4.a.cx 6
7.b odd 2 1 490.4.c.f 12
7.d odd 6 2 70.4.i.a 24
35.c odd 2 1 490.4.c.f 12
35.f even 4 1 2450.4.a.cv 6
35.f even 4 1 2450.4.a.cy 6
35.i odd 6 2 70.4.i.a 24
35.k even 12 2 350.4.e.n 12
35.k even 12 2 350.4.e.o 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.i.a 24 7.d odd 6 2
70.4.i.a 24 35.i odd 6 2
350.4.e.n 12 35.k even 12 2
350.4.e.o 12 35.k even 12 2
490.4.c.e 12 1.a even 1 1 trivial
490.4.c.e 12 5.b even 2 1 inner
490.4.c.f 12 7.b odd 2 1
490.4.c.f 12 35.c odd 2 1
2450.4.a.cv 6 35.f even 4 1
2450.4.a.cw 6 5.c odd 4 1
2450.4.a.cx 6 5.c odd 4 1
2450.4.a.cy 6 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 \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{12} + 193T_{3}^{10} + 13302T_{3}^{8} + 413878T_{3}^{6} + 5835817T_{3}^{4} + 31585449T_{3}^{2} + 34574400 \) Copy content Toggle raw display
\( T_{19}^{6} - 93T_{19}^{5} - 10611T_{19}^{4} + 120401T_{19}^{3} + 17245338T_{19}^{2} + 170214204T_{19} - 1453282440 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 193 T^{10} + \cdots + 34574400 \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{11} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 31 T^{5} - 3033 T^{4} + \cdots + 286379520)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 22989 T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + 36294 T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} - 93 T^{5} - 10611 T^{4} + \cdots - 1453282440)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 72122 T^{10} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{6} + 169 T^{5} + \cdots - 1796409740592)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 326 T^{5} + \cdots - 495991036080)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 284703 T^{10} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} + 198 T^{5} + \cdots - 1146896772372)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 485343 T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + 634983 T^{10} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + 1105483 T^{10} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{6} - 668 T^{5} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 157 T^{5} + \cdots + 12462758786850)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 1366301 T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} - 1108 T^{5} + \cdots + 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 1718370 T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} - 886 T^{5} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 5010391 T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} - 3047 T^{5} + \cdots - 106054121644030)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 4439728 T^{10} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
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