Properties

Label 98.10.a.k
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{4817})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 2411 x^{2} + 2412 x + 1444802\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -16 q^{2} + ( 5 \beta_{1} + \beta_{3} ) q^{3} + 256 q^{4} + ( 104 \beta_{1} - 7 \beta_{3} ) q^{5} + ( -80 \beta_{1} - 16 \beta_{3} ) q^{6} -4096 q^{8} + ( 3978 - 9 \beta_{2} ) q^{9} +O(q^{10})\) \( q -16 q^{2} + ( 5 \beta_{1} + \beta_{3} ) q^{3} + 256 q^{4} + ( 104 \beta_{1} - 7 \beta_{3} ) q^{5} + ( -80 \beta_{1} - 16 \beta_{3} ) q^{6} -4096 q^{8} + ( 3978 - 9 \beta_{2} ) q^{9} + ( -1664 \beta_{1} + 112 \beta_{3} ) q^{10} + ( 2387 + 37 \beta_{2} ) q^{11} + ( 1280 \beta_{1} + 256 \beta_{3} ) q^{12} + ( -182 \beta_{1} + 809 \beta_{3} ) q^{13} + ( -104328 - 76 \beta_{2} ) q^{15} + 65536 q^{16} + ( -43071 \beta_{1} - 248 \beta_{3} ) q^{17} + ( -63648 + 144 \beta_{2} ) q^{18} + ( -31639 \beta_{1} - 3679 \beta_{3} ) q^{19} + ( 26624 \beta_{1} - 1792 \beta_{3} ) q^{20} + ( -38192 - 592 \beta_{2} ) q^{22} + ( -709184 - 692 \beta_{2} ) q^{23} + ( -20480 \beta_{1} - 4096 \beta_{3} ) q^{24} + ( 241536 + 1505 \beta_{2} ) q^{25} + ( 2912 \beta_{1} - 12944 \beta_{3} ) q^{26} + ( 118548 \beta_{1} - 11736 \beta_{3} ) q^{27} + ( 4167493 + 1087 \beta_{2} ) q^{29} + ( 1669248 + 1216 \beta_{2} ) q^{30} + ( -100134 \beta_{1} - 41130 \beta_{3} ) q^{31} -1048576 q^{32} + ( -798254 \beta_{1} - 13930 \beta_{3} ) q^{33} + ( 689136 \beta_{1} + 3968 \beta_{3} ) q^{34} + ( 1018368 - 2304 \beta_{2} ) q^{36} + ( 806593 - 9021 \beta_{2} ) q^{37} + ( 506224 \beta_{1} + 58864 \beta_{3} ) q^{38} + ( 17277642 - 3054 \beta_{2} ) q^{39} + ( -425984 \beta_{1} + 28672 \beta_{3} ) q^{40} + ( 1430791 \beta_{1} + 56652 \beta_{3} ) q^{41} + ( -19120051 + 9147 \beta_{2} ) q^{43} + ( 611072 + 9472 \beta_{2} ) q^{44} + ( -904500 \beta_{1} + 66969 \beta_{3} ) q^{45} + ( 11346944 + 11072 \beta_{2} ) q^{46} + ( -2664762 \beta_{1} + 163054 \beta_{3} ) q^{47} + ( 327680 \beta_{1} + 65536 \beta_{3} ) q^{48} + ( -3864576 - 24080 \beta_{2} ) q^{50} + ( -24315399 + 44063 \beta_{2} ) q^{51} + ( -46592 \beta_{1} + 207104 \beta_{3} ) q^{52} + ( 3427028 - 58226 \beta_{2} ) q^{53} + ( -1896768 \beta_{1} + 187776 \beta_{3} ) q^{54} + ( 5667564 \beta_{1} - 406504 \beta_{3} ) q^{55} + ( -92889423 + 46355 \beta_{2} ) q^{57} + ( -66679888 - 17392 \beta_{2} ) q^{58} + ( 3105293 \beta_{1} - 314395 \beta_{3} ) q^{59} + ( -26707968 - 19456 \beta_{2} ) q^{60} + ( -6621532 \beta_{1} - 340041 \beta_{3} ) q^{61} + ( 1602144 \beta_{1} + 658080 \beta_{3} ) q^{62} + 16777216 q^{64} + ( -128932797 - 91073 \beta_{2} ) q^{65} + ( 12772064 \beta_{1} + 222880 \beta_{3} ) q^{66} + ( -203605194 + 69906 \beta_{2} ) q^{67} + ( -11026176 \beta_{1} - 63488 \beta_{3} ) q^{68} + ( 11606804 \beta_{1} - 404012 \beta_{3} ) q^{69} + ( -331167186 - 30534 \beta_{2} ) q^{71} + ( -16293888 + 36864 \beta_{2} ) q^{72} + ( 24266301 \beta_{1} - 1351834 \beta_{3} ) q^{73} + ( -12905488 + 144336 \beta_{2} ) q^{74} + ( -31747305 \beta_{1} - 422169 \beta_{3} ) q^{75} + ( -8099584 \beta_{1} - 941824 \beta_{3} ) q^{76} + ( -276442272 + 48864 \beta_{2} ) q^{78} + ( -167102794 - 40838 \beta_{2} ) q^{79} + ( 6815744 \beta_{1} - 458752 \beta_{3} ) q^{80} + ( -277826922 + 105543 \beta_{2} ) q^{81} + ( -22892656 \beta_{1} - 906432 \beta_{3} ) q^{82} + ( -22750843 \beta_{1} + 1981881 \beta_{3} ) q^{83} + ( -414816241 - 273969 \beta_{2} ) q^{85} + ( 305920816 - 146352 \beta_{2} ) q^{86} + ( -2964574 \beta_{1} + 3688126 \beta_{3} ) q^{87} + ( -9777152 - 151552 \beta_{2} ) q^{88} + ( -69656107 \beta_{1} - 1118280 \beta_{3} ) q^{89} + ( 14472000 \beta_{1} - 1071504 \beta_{3} ) q^{90} + ( -181551104 - 177152 \beta_{2} ) q^{92} + ( -926644374 + 264654 \beta_{2} ) q^{93} + ( 42636192 \beta_{1} - 2608864 \beta_{3} ) q^{94} + ( 244297172 + 186896 \beta_{2} ) q^{95} + ( -5242880 \beta_{1} - 1048576 \beta_{3} ) q^{96} + ( 89705213 \beta_{1} + 79754 \beta_{3} ) q^{97} + ( -697895415 + 125703 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9} + O(q^{10}) \) \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9} + 9548 q^{11} - 417312 q^{15} + 262144 q^{16} - 254592 q^{18} - 152768 q^{22} - 2836736 q^{23} + 966144 q^{25} + 16669972 q^{29} + 6676992 q^{30} - 4194304 q^{32} + 4073472 q^{36} + 3226372 q^{37} + 69110568 q^{39} - 76480204 q^{43} + 2444288 q^{44} + 45387776 q^{46} - 15458304 q^{50} - 97261596 q^{51} + 13708112 q^{53} - 371557692 q^{57} - 266719552 q^{58} - 106831872 q^{60} + 67108864 q^{64} - 515731188 q^{65} - 814420776 q^{67} - 1324668744 q^{71} - 65175552 q^{72} - 51621952 q^{74} - 1105769088 q^{78} - 668411176 q^{79} - 1111307688 q^{81} - 1659264964 q^{85} + 1223683264 q^{86} - 39108608 q^{88} - 726204416 q^{92} - 3706577496 q^{93} + 977188688 q^{95} - 2791581660 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 2411 x^{2} + 2412 x + 1444802\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 2419 \nu - 1210 \)\()/687\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 6 \nu^{2} + 14456 \nu - 7229 \)\()/229\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 1029 \nu^{2} - 2240 \nu - 1242178 \)\()/687\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 6 \beta_{1} + 21\)\()/42\)
\(\nu^{2}\)\(=\)\((\)\(28 \beta_{3} + \beta_{2} + 8 \beta_{1} + 50673\)\()/42\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{3} + 173 \beta_{2} - 3096 \beta_{1} + 10857\)\()/6\)

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
−32.7881
33.7881
36.6165
−35.6165
−16.0000 −191.777 256.000 −33.5898 3068.43 0 −4096.00 17095.5 537.437
1.2 −16.0000 −102.682 256.000 2094.80 1642.91 0 −4096.00 −9139.47 −33516.8
1.3 −16.0000 102.682 256.000 −2094.80 −1642.91 0 −4096.00 −9139.47 33516.8
1.4 −16.0000 191.777 256.000 33.5898 −3068.43 0 −4096.00 17095.5 −537.437
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.a.k 4
7.b odd 2 1 inner 98.10.a.k 4
7.c even 3 2 98.10.c.m 8
7.d odd 6 2 98.10.c.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.k 4 1.a even 1 1 trivial
98.10.a.k 4 7.b odd 2 1 inner
98.10.c.m 8 7.c even 3 2
98.10.c.m 8 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 47322 T_{3}^{2} + 387774864 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + T )^{4} \)
$3$ \( 387774864 - 47322 T^{2} + T^{4} \)
$5$ \( 4951092496 - 4389322 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -2902464824 - 4774 T + T^{2} )^{2} \)
$13$ \( \)\(20\!\cdots\!96\)\( - 28441135314 T^{2} + T^{4} \)
$17$ \( \)\(32\!\cdots\!76\)\( - 364177565476 T^{2} + T^{4} \)
$19$ \( \)\(42\!\cdots\!44\)\( - 760834631722 T^{2} + T^{4} \)
$23$ \( ( -514307412752 + 1418368 T + T^{2} )^{2} \)
$29$ \( ( 14857994423056 - 8334986 T + T^{2} )^{2} \)
$31$ \( \)\(12\!\cdots\!84\)\( - 74580188894856 T^{2} + T^{4} \)
$37$ \( ( -172221385813328 - 1613186 T + T^{2} )^{2} \)
$41$ \( \)\(15\!\cdots\!36\)\( - 524653229945812 T^{2} + T^{4} \)
$43$ \( ( 187841499446728 + 38240102 T + T^{2} )^{2} \)
$47$ \( \)\(26\!\cdots\!56\)\( - 2630861410847464 T^{2} + T^{4} \)
$53$ \( ( -7190189657832788 - 6854056 T + T^{2} )^{2} \)
$59$ \( \)\(12\!\cdots\!24\)\( - 6371386772799514 T^{2} + T^{4} \)
$61$ \( \)\(24\!\cdots\!96\)\( - 13170727678904314 T^{2} + T^{4} \)
$67$ \( ( 31073956702009344 + 407210388 T + T^{2} )^{2} \)
$71$ \( ( 107691169551243264 + 662334372 T + T^{2} )^{2} \)
$73$ \( \)\(45\!\cdots\!24\)\( - 201160054502509972 T^{2} + T^{4} \)
$79$ \( ( 24380563916903968 + 334205588 T + T^{2} )^{2} \)
$83$ \( \)\(89\!\cdots\!96\)\( - 280763948299304794 T^{2} + T^{4} \)
$89$ \( \)\(19\!\cdots\!84\)\( - 989995707342536644 T^{2} + T^{4} \)
$97$ \( \)\(62\!\cdots\!56\)\( - 1576090761528077764 T^{2} + T^{4} \)
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