Properties

Label 98.10.a.h
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 4037 x + 70980\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -16 q^{2} + ( 24 + \beta_{1} ) q^{3} + 256 q^{4} + ( -363 - 5 \beta_{1} + \beta_{2} ) q^{5} + ( -384 - 16 \beta_{1} ) q^{6} -4096 q^{8} + ( 3024 - 25 \beta_{1} - 9 \beta_{2} ) q^{9} +O(q^{10})\) \( q -16 q^{2} + ( 24 + \beta_{1} ) q^{3} + 256 q^{4} + ( -363 - 5 \beta_{1} + \beta_{2} ) q^{5} + ( -384 - 16 \beta_{1} ) q^{6} -4096 q^{8} + ( 3024 - 25 \beta_{1} - 9 \beta_{2} ) q^{9} + ( 5808 + 80 \beta_{1} - 16 \beta_{2} ) q^{10} + ( -894 - 164 \beta_{1} + 37 \beta_{2} ) q^{11} + ( 6144 + 256 \beta_{1} ) q^{12} + ( -6212 - 565 \beta_{1} - \beta_{2} ) q^{13} + ( -117900 - 768 \beta_{1} + 141 \beta_{2} ) q^{15} + 65536 q^{16} + ( 7347 + 1115 \beta_{1} + 167 \beta_{2} ) q^{17} + ( -48384 + 400 \beta_{1} + 144 \beta_{2} ) q^{18} + ( 406870 + 234 \beta_{1} - 273 \beta_{2} ) q^{19} + ( -92928 - 1280 \beta_{1} + 256 \beta_{2} ) q^{20} + ( 14304 + 2624 \beta_{1} - 592 \beta_{2} ) q^{22} + ( 658134 + 12847 \beta_{1} + 652 \beta_{2} ) q^{23} + ( -98304 - 4096 \beta_{1} ) q^{24} + ( 291964 + 13342 \beta_{1} - 1022 \beta_{2} ) q^{25} + ( 99392 + 9040 \beta_{1} + 16 \beta_{2} ) q^{26} + ( -966294 - 9584 \beta_{1} - 639 \beta_{2} ) q^{27} + ( -864336 + 12191 \beta_{1} + 947 \beta_{2} ) q^{29} + ( 1886400 + 12288 \beta_{1} - 2256 \beta_{2} ) q^{30} + ( -3117008 + 29420 \beta_{1} - 2455 \beta_{2} ) q^{31} -1048576 q^{32} + ( -3596661 - 16908 \beta_{1} + 5028 \beta_{2} ) q^{33} + ( -117552 - 17840 \beta_{1} - 2672 \beta_{2} ) q^{34} + ( 774144 - 6400 \beta_{1} - 2304 \beta_{2} ) q^{36} + ( 8596349 - 24804 \beta_{1} - 1062 \beta_{2} ) q^{37} + ( -6509920 - 3744 \beta_{1} + 4368 \beta_{2} ) q^{38} + ( -12654570 + 22123 \beta_{1} + 4989 \beta_{2} ) q^{39} + ( 1486848 + 20480 \beta_{1} - 4096 \beta_{2} ) q^{40} + ( -55836 + 32649 \beta_{1} + 9093 \beta_{2} ) q^{41} + ( 968372 - 40716 \beta_{1} + 1524 \beta_{2} ) q^{43} + ( -228864 - 41984 \beta_{1} + 9472 \beta_{2} ) q^{44} + ( -12474432 - 73503 \beta_{1} + 765 \beta_{2} ) q^{45} + ( -10530144 - 205552 \beta_{1} - 10432 \beta_{2} ) q^{46} + ( 16096512 - 77040 \beta_{1} - 24693 \beta_{2} ) q^{47} + ( 1572864 + 65536 \beta_{1} ) q^{48} + ( -4671424 - 213472 \beta_{1} + 16352 \beta_{2} ) q^{50} + ( 25097382 - 155838 \beta_{1} + 5997 \beta_{2} ) q^{51} + ( -1590272 - 144640 \beta_{1} - 256 \beta_{2} ) q^{52} + ( -34034487 + 67746 \beta_{1} + 15204 \beta_{2} ) q^{53} + ( 15460704 + 153344 \beta_{1} + 10224 \beta_{2} ) q^{54} + ( 76224414 + 417361 \beta_{1} - 22820 \beta_{2} ) q^{55} + ( 14543043 + 572854 \beta_{1} - 28314 \beta_{2} ) q^{57} + ( 13829376 - 195056 \beta_{1} - 15152 \beta_{2} ) q^{58} + ( 48066516 - 32087 \beta_{1} + 11500 \beta_{2} ) q^{59} + ( -30182400 - 196608 \beta_{1} + 36096 \beta_{2} ) q^{60} + ( 93531535 - 417556 \beta_{1} + 75290 \beta_{2} ) q^{61} + ( 49872128 - 470720 \beta_{1} + 39280 \beta_{2} ) q^{62} + 16777216 q^{64} + ( 62378796 + 388893 \beta_{1} - 72243 \beta_{2} ) q^{65} + ( 57546576 + 270528 \beta_{1} - 80448 \beta_{2} ) q^{66} + ( -56768836 + 546197 \beta_{1} - 142306 \beta_{2} ) q^{67} + ( 1880832 + 285440 \beta_{1} + 42752 \beta_{2} ) q^{68} + ( 301068657 - 395169 \beta_{1} - 53031 \beta_{2} ) q^{69} + ( 156350460 - 721994 \beta_{1} + 14686 \beta_{2} ) q^{71} + ( -12386304 + 102400 \beta_{1} + 36864 \beta_{2} ) q^{72} + ( 204996025 + 1088182 \beta_{1} + 61354 \beta_{2} ) q^{73} + ( -137541584 + 396864 \beta_{1} + 16992 \beta_{2} ) q^{74} + ( 300779664 + 302506 \beta_{1} - 218190 \beta_{2} ) q^{75} + ( 104158720 + 59904 \beta_{1} - 69888 \beta_{2} ) q^{76} + ( 202473120 - 353968 \beta_{1} - 79824 \beta_{2} ) q^{78} + ( -65132926 + 1794499 \beta_{1} + 252532 \beta_{2} ) q^{79} + ( -23789568 - 327680 \beta_{1} + 65536 \beta_{2} ) q^{80} + ( -295753365 + 410747 \beta_{1} + 202059 \beta_{2} ) q^{81} + ( 893376 - 522384 \beta_{1} - 145488 \beta_{2} ) q^{82} + ( 357629064 - 1569286 \beta_{1} + 275042 \beta_{2} ) q^{83} + ( 137340879 + 624674 \beta_{1} + 125444 \beta_{2} ) q^{85} + ( -15493952 + 651456 \beta_{1} - 24384 \beta_{2} ) q^{86} + ( 250444206 - 2077245 \beta_{1} - 18807 \beta_{2} ) q^{87} + ( 3661824 + 671744 \beta_{1} - 151552 \beta_{2} ) q^{88} + ( 269662485 + 2845428 \beta_{1} + 411600 \beta_{2} ) q^{89} + ( 199590912 + 1176048 \beta_{1} - 12240 \beta_{2} ) q^{90} + ( 168482304 + 3288832 \beta_{1} + 166912 \beta_{2} ) q^{92} + ( 572684343 - 2962838 \beta_{1} - 500460 \beta_{2} ) q^{93} + ( -257544192 + 1232640 \beta_{1} + 395088 \beta_{2} ) q^{94} + ( -601138542 - 4448333 \beta_{1} + 454450 \beta_{2} ) q^{95} + ( -25165824 - 1048576 \beta_{1} ) q^{96} + ( 753753784 - 1333445 \beta_{1} - 323297 \beta_{2} ) q^{97} + ( -435538134 - 2808357 \beta_{1} - 93411 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 48 q^{2} + 71 q^{3} + 768 q^{4} - 1085 q^{5} - 1136 q^{6} - 12288 q^{8} + 9106 q^{9} + O(q^{10}) \) \( 3 q - 48 q^{2} + 71 q^{3} + 768 q^{4} - 1085 q^{5} - 1136 q^{6} - 12288 q^{8} + 9106 q^{9} + 17360 q^{10} - 2555 q^{11} + 18176 q^{12} - 18070 q^{13} - 353073 q^{15} + 196608 q^{16} + 20759 q^{17} - 145696 q^{18} + 1220649 q^{19} - 277760 q^{20} + 40880 q^{22} + 1960903 q^{23} - 290816 q^{24} + 863572 q^{25} + 289120 q^{26} - 2888659 q^{27} - 2606146 q^{29} + 5649168 q^{30} - 9377989 q^{31} - 3145728 q^{32} - 10778103 q^{33} - 332144 q^{34} + 2331136 q^{36} + 25814913 q^{37} - 19530384 q^{38} - 37990822 q^{39} + 4444160 q^{40} - 209250 q^{41} + 2944308 q^{43} - 654080 q^{44} - 37350558 q^{45} - 31374448 q^{46} + 48391269 q^{47} + 4653056 q^{48} - 13817152 q^{50} + 75441987 q^{51} - 4625920 q^{52} - 102186411 q^{53} + 46218544 q^{54} + 228278701 q^{55} + 43084589 q^{57} + 41698336 q^{58} + 144220135 q^{59} - 90386688 q^{60} + 280936871 q^{61} + 150047824 q^{62} + 50331648 q^{64} + 186819738 q^{65} + 172449648 q^{66} - 170710399 q^{67} + 5314304 q^{68} + 903654171 q^{69} + 469758688 q^{71} - 37298176 q^{72} + 613838539 q^{73} - 413038608 q^{74} + 902254676 q^{75} + 312486144 q^{76} + 607853152 q^{78} - 197445809 q^{79} - 71106560 q^{80} - 887872901 q^{81} + 3348000 q^{82} + 1074181436 q^{83} + 411272519 q^{85} - 47108928 q^{86} + 753428670 q^{87} + 10465280 q^{88} + 805730427 q^{89} + 597608928 q^{90} + 501991168 q^{92} + 1721516327 q^{93} - 774260304 q^{94} - 1799421743 q^{95} - 74448896 q^{96} + 2262918094 q^{97} - 1303712634 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4037 x + 70980\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{2} + 20 \nu - 5397 \)\()/21\)
\(\beta_{2}\)\(=\)\((\)\( 10 \nu^{2} + 688 \nu - 27153 \)\()/21\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 5 \beta_{1} + 8\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(-5 \beta_{2} + 172 \beta_{1} + 37739\)\()/14\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
19.2603
52.2593
−70.5196
−16.0000 −179.327 256.000 168.288 2869.24 0 −4096.00 12475.3 −2692.61
1.2 −16.0000 76.8690 256.000 1092.26 −1229.90 0 −4096.00 −13774.2 −17476.2
1.3 −16.0000 173.458 256.000 −2345.55 −2775.34 0 −4096.00 10404.8 37528.8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.a.h 3
7.b odd 2 1 98.10.a.g 3
7.c even 3 2 98.10.c.l 6
7.d odd 6 2 14.10.c.b 6
21.g even 6 2 126.10.g.e 6
28.f even 6 2 112.10.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.c.b 6 7.d odd 6 2
98.10.a.g 3 7.b odd 2 1
98.10.a.h 3 1.a even 1 1 trivial
98.10.c.l 6 7.c even 3 2
112.10.i.a 6 28.f even 6 2
126.10.g.e 6 21.g even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 71 T_{3}^{2} - 31557 T_{3} + 2391075 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + T )^{3} \)
$3$ \( 2391075 - 31557 T - 71 T^{2} + T^{3} \)
$5$ \( 431145855 - 2772861 T + 1085 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 55717735209129 - 4091349717 T + 2555 T^{2} + T^{3} \)
$13$ \( -368974841338200 - 10506518276 T + 18070 T^{2} + T^{3} \)
$17$ \( -3429291415002165 - 107901437877 T - 20759 T^{2} + T^{3} \)
$19$ \( 19237666690422725 + 319328427843 T - 1220649 T^{2} + T^{3} \)
$23$ \( 9895961116236687219 - 5248576102245 T - 1960903 T^{2} + T^{3} \)
$29$ \( -1136027665699119432 - 4849005669540 T + 2606146 T^{2} + T^{3} \)
$31$ \( -\)\(19\!\cdots\!53\)\( - 13312107107093 T + 9377989 T^{2} + T^{3} \)
$37$ \( -\)\(47\!\cdots\!87\)\( + 198895703491659 T - 25814913 T^{2} + T^{3} \)
$41$ \( -\)\(12\!\cdots\!24\)\( - 231960363901092 T + 209250 T^{2} + T^{3} \)
$43$ \( 85536171172243734592 - 57381514505424 T - 2944308 T^{2} + T^{3} \)
$47$ \( \)\(46\!\cdots\!85\)\( - 864694984242357 T - 48391269 T^{2} + T^{3} \)
$53$ \( \)\(10\!\cdots\!89\)\( + 2777652765212499 T + 102186411 T^{2} + T^{3} \)
$59$ \( -\)\(94\!\cdots\!25\)\( + 6588757817916699 T - 144220135 T^{2} + T^{3} \)
$61$ \( \)\(17\!\cdots\!55\)\( + 7296507676617883 T - 280936871 T^{2} + T^{3} \)
$67$ \( -\)\(54\!\cdots\!95\)\( - 47591489138815141 T + 170710399 T^{2} + T^{3} \)
$71$ \( -\)\(12\!\cdots\!40\)\( + 55775894095701696 T - 469758688 T^{2} + T^{3} \)
$73$ \( \)\(49\!\cdots\!79\)\( + 77028316914358739 T - 613838539 T^{2} + T^{3} \)
$79$ \( -\)\(27\!\cdots\!85\)\( - 246746715036248597 T + 197445809 T^{2} + T^{3} \)
$83$ \( \)\(87\!\cdots\!00\)\( + 126447385266753648 T - 1074181436 T^{2} + T^{3} \)
$89$ \( \)\(10\!\cdots\!75\)\( - 458222954702204493 T - 805730427 T^{2} + T^{3} \)
$97$ \( -\)\(14\!\cdots\!40\)\( + 1399099087623178268 T - 2262918094 T^{2} + T^{3} \)
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