# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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