# 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: $$x^{4} - 2x^{3} - 2411x^{2} + 2412x + 1444802$$ 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.

 $$f(q)$$ $$=$$ $$q - 16 q^{2} + (\beta_{3} + 5 \beta_1) q^{3} + 256 q^{4} + ( - 7 \beta_{3} + 104 \beta_1) q^{5} + ( - 16 \beta_{3} - 80 \beta_1) q^{6} - 4096 q^{8} + ( - 9 \beta_{2} + 3978) q^{9}+O(q^{10})$$ q - 16 * q^2 + (b3 + 5*b1) * q^3 + 256 * q^4 + (-7*b3 + 104*b1) * q^5 + (-16*b3 - 80*b1) * q^6 - 4096 * q^8 + (-9*b2 + 3978) * q^9 $$q - 16 q^{2} + (\beta_{3} + 5 \beta_1) q^{3} + 256 q^{4} + ( - 7 \beta_{3} + 104 \beta_1) q^{5} + ( - 16 \beta_{3} - 80 \beta_1) q^{6} - 4096 q^{8} + ( - 9 \beta_{2} + 3978) q^{9} + (112 \beta_{3} - 1664 \beta_1) q^{10} + (37 \beta_{2} + 2387) q^{11} + (256 \beta_{3} + 1280 \beta_1) q^{12} + (809 \beta_{3} - 182 \beta_1) q^{13} + ( - 76 \beta_{2} - 104328) q^{15} + 65536 q^{16} + ( - 248 \beta_{3} - 43071 \beta_1) q^{17} + (144 \beta_{2} - 63648) q^{18} + ( - 3679 \beta_{3} - 31639 \beta_1) q^{19} + ( - 1792 \beta_{3} + 26624 \beta_1) q^{20} + ( - 592 \beta_{2} - 38192) q^{22} + ( - 692 \beta_{2} - 709184) q^{23} + ( - 4096 \beta_{3} - 20480 \beta_1) q^{24} + (1505 \beta_{2} + 241536) q^{25} + ( - 12944 \beta_{3} + 2912 \beta_1) q^{26} + ( - 11736 \beta_{3} + 118548 \beta_1) q^{27} + (1087 \beta_{2} + 4167493) q^{29} + (1216 \beta_{2} + 1669248) q^{30} + ( - 41130 \beta_{3} - 100134 \beta_1) q^{31} - 1048576 q^{32} + ( - 13930 \beta_{3} - 798254 \beta_1) q^{33} + (3968 \beta_{3} + 689136 \beta_1) q^{34} + ( - 2304 \beta_{2} + 1018368) q^{36} + ( - 9021 \beta_{2} + 806593) q^{37} + (58864 \beta_{3} + 506224 \beta_1) q^{38} + ( - 3054 \beta_{2} + 17277642) q^{39} + (28672 \beta_{3} - 425984 \beta_1) q^{40} + (56652 \beta_{3} + 1430791 \beta_1) q^{41} + (9147 \beta_{2} - 19120051) q^{43} + (9472 \beta_{2} + 611072) q^{44} + (66969 \beta_{3} - 904500 \beta_1) q^{45} + (11072 \beta_{2} + 11346944) q^{46} + (163054 \beta_{3} - 2664762 \beta_1) q^{47} + (65536 \beta_{3} + 327680 \beta_1) q^{48} + ( - 24080 \beta_{2} - 3864576) q^{50} + (44063 \beta_{2} - 24315399) q^{51} + (207104 \beta_{3} - 46592 \beta_1) q^{52} + ( - 58226 \beta_{2} + 3427028) q^{53} + (187776 \beta_{3} - 1896768 \beta_1) q^{54} + ( - 406504 \beta_{3} + 5667564 \beta_1) q^{55} + (46355 \beta_{2} - 92889423) q^{57} + ( - 17392 \beta_{2} - 66679888) q^{58} + ( - 314395 \beta_{3} + 3105293 \beta_1) q^{59} + ( - 19456 \beta_{2} - 26707968) q^{60} + ( - 340041 \beta_{3} - 6621532 \beta_1) q^{61} + (658080 \beta_{3} + 1602144 \beta_1) q^{62} + 16777216 q^{64} + ( - 91073 \beta_{2} - 128932797) q^{65} + (222880 \beta_{3} + 12772064 \beta_1) q^{66} + (69906 \beta_{2} - 203605194) q^{67} + ( - 63488 \beta_{3} - 11026176 \beta_1) q^{68} + ( - 404012 \beta_{3} + 11606804 \beta_1) q^{69} + ( - 30534 \beta_{2} - 331167186) q^{71} + (36864 \beta_{2} - 16293888) q^{72} + ( - 1351834 \beta_{3} + 24266301 \beta_1) q^{73} + (144336 \beta_{2} - 12905488) q^{74} + ( - 422169 \beta_{3} - 31747305 \beta_1) q^{75} + ( - 941824 \beta_{3} - 8099584 \beta_1) q^{76} + (48864 \beta_{2} - 276442272) q^{78} + ( - 40838 \beta_{2} - 167102794) q^{79} + ( - 458752 \beta_{3} + 6815744 \beta_1) q^{80} + (105543 \beta_{2} - 277826922) q^{81} + ( - 906432 \beta_{3} - 22892656 \beta_1) q^{82} + (1981881 \beta_{3} - 22750843 \beta_1) q^{83} + ( - 273969 \beta_{2} - 414816241) q^{85} + ( - 146352 \beta_{2} + 305920816) q^{86} + (3688126 \beta_{3} - 2964574 \beta_1) q^{87} + ( - 151552 \beta_{2} - 9777152) q^{88} + ( - 1118280 \beta_{3} - 69656107 \beta_1) q^{89} + ( - 1071504 \beta_{3} + 14472000 \beta_1) q^{90} + ( - 177152 \beta_{2} - 181551104) q^{92} + (264654 \beta_{2} - 926644374) q^{93} + ( - 2608864 \beta_{3} + 42636192 \beta_1) q^{94} + (186896 \beta_{2} + 244297172) q^{95} + ( - 1048576 \beta_{3} - 5242880 \beta_1) q^{96} + (79754 \beta_{3} + 89705213 \beta_1) q^{97} + (125703 \beta_{2} - 697895415) q^{99}+O(q^{100})$$ q - 16 * q^2 + (b3 + 5*b1) * q^3 + 256 * q^4 + (-7*b3 + 104*b1) * q^5 + (-16*b3 - 80*b1) * q^6 - 4096 * q^8 + (-9*b2 + 3978) * q^9 + (112*b3 - 1664*b1) * q^10 + (37*b2 + 2387) * q^11 + (256*b3 + 1280*b1) * q^12 + (809*b3 - 182*b1) * q^13 + (-76*b2 - 104328) * q^15 + 65536 * q^16 + (-248*b3 - 43071*b1) * q^17 + (144*b2 - 63648) * q^18 + (-3679*b3 - 31639*b1) * q^19 + (-1792*b3 + 26624*b1) * q^20 + (-592*b2 - 38192) * q^22 + (-692*b2 - 709184) * q^23 + (-4096*b3 - 20480*b1) * q^24 + (1505*b2 + 241536) * q^25 + (-12944*b3 + 2912*b1) * q^26 + (-11736*b3 + 118548*b1) * q^27 + (1087*b2 + 4167493) * q^29 + (1216*b2 + 1669248) * q^30 + (-41130*b3 - 100134*b1) * q^31 - 1048576 * q^32 + (-13930*b3 - 798254*b1) * q^33 + (3968*b3 + 689136*b1) * q^34 + (-2304*b2 + 1018368) * q^36 + (-9021*b2 + 806593) * q^37 + (58864*b3 + 506224*b1) * q^38 + (-3054*b2 + 17277642) * q^39 + (28672*b3 - 425984*b1) * q^40 + (56652*b3 + 1430791*b1) * q^41 + (9147*b2 - 19120051) * q^43 + (9472*b2 + 611072) * q^44 + (66969*b3 - 904500*b1) * q^45 + (11072*b2 + 11346944) * q^46 + (163054*b3 - 2664762*b1) * q^47 + (65536*b3 + 327680*b1) * q^48 + (-24080*b2 - 3864576) * q^50 + (44063*b2 - 24315399) * q^51 + (207104*b3 - 46592*b1) * q^52 + (-58226*b2 + 3427028) * q^53 + (187776*b3 - 1896768*b1) * q^54 + (-406504*b3 + 5667564*b1) * q^55 + (46355*b2 - 92889423) * q^57 + (-17392*b2 - 66679888) * q^58 + (-314395*b3 + 3105293*b1) * q^59 + (-19456*b2 - 26707968) * q^60 + (-340041*b3 - 6621532*b1) * q^61 + (658080*b3 + 1602144*b1) * q^62 + 16777216 * q^64 + (-91073*b2 - 128932797) * q^65 + (222880*b3 + 12772064*b1) * q^66 + (69906*b2 - 203605194) * q^67 + (-63488*b3 - 11026176*b1) * q^68 + (-404012*b3 + 11606804*b1) * q^69 + (-30534*b2 - 331167186) * q^71 + (36864*b2 - 16293888) * q^72 + (-1351834*b3 + 24266301*b1) * q^73 + (144336*b2 - 12905488) * q^74 + (-422169*b3 - 31747305*b1) * q^75 + (-941824*b3 - 8099584*b1) * q^76 + (48864*b2 - 276442272) * q^78 + (-40838*b2 - 167102794) * q^79 + (-458752*b3 + 6815744*b1) * q^80 + (105543*b2 - 277826922) * q^81 + (-906432*b3 - 22892656*b1) * q^82 + (1981881*b3 - 22750843*b1) * q^83 + (-273969*b2 - 414816241) * q^85 + (-146352*b2 + 305920816) * q^86 + (3688126*b3 - 2964574*b1) * q^87 + (-151552*b2 - 9777152) * q^88 + (-1118280*b3 - 69656107*b1) * q^89 + (-1071504*b3 + 14472000*b1) * q^90 + (-177152*b2 - 181551104) * q^92 + (264654*b2 - 926644374) * q^93 + (-2608864*b3 + 42636192*b1) * q^94 + (186896*b2 + 244297172) * q^95 + (-1048576*b3 - 5242880*b1) * q^96 + (79754*b3 + 89705213*b1) * q^97 + (125703*b2 - 697895415) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9}+O(q^{10})$$ 4 * q - 64 * q^2 + 1024 * q^4 - 16384 * q^8 + 15912 * q^9 $$4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9} + 9548 q^{11} - 417312 q^{15} + 262144 q^{16} - 254592 q^{18} - 152768 q^{22} - 2836736 q^{23} + 966144 q^{25} + 16669972 q^{29} + 6676992 q^{30} - 4194304 q^{32} + 4073472 q^{36} + 3226372 q^{37} + 69110568 q^{39} - 76480204 q^{43} + 2444288 q^{44} + 45387776 q^{46} - 15458304 q^{50} - 97261596 q^{51} + 13708112 q^{53} - 371557692 q^{57} - 266719552 q^{58} - 106831872 q^{60} + 67108864 q^{64} - 515731188 q^{65} - 814420776 q^{67} - 1324668744 q^{71} - 65175552 q^{72} - 51621952 q^{74} - 1105769088 q^{78} - 668411176 q^{79} - 1111307688 q^{81} - 1659264964 q^{85} + 1223683264 q^{86} - 39108608 q^{88} - 726204416 q^{92} - 3706577496 q^{93} + 977188688 q^{95} - 2791581660 q^{99}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} + 2419\nu - 1210 ) / 687$$ (-2*v^3 + 3*v^2 + 2419*v - 1210) / 687 $$\beta_{2}$$ $$=$$ $$( -4\nu^{3} + 6\nu^{2} + 14456\nu - 7229 ) / 229$$ (-4*v^3 + 6*v^2 + 14456*v - 7229) / 229 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 1029\nu^{2} - 2240\nu - 1242178 ) / 687$$ (v^3 + 1029*v^2 - 2240*v - 1242178) / 687
 $$\nu$$ $$=$$ $$( \beta_{2} - 6\beta _1 + 21 ) / 42$$ (b2 - 6*b1 + 21) / 42 $$\nu^{2}$$ $$=$$ $$( 28\beta_{3} + \beta_{2} + 8\beta _1 + 50673 ) / 42$$ (28*b3 + b2 + 8*b1 + 50673) / 42 $$\nu^{3}$$ $$=$$ $$( 6\beta_{3} + 173\beta_{2} - 3096\beta _1 + 10857 ) / 6$$ (6*b3 + 173*b2 - 3096*b1 + 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 $$T_{3}^{4} - 47322T_{3}^{2} + 387774864$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 16)^{4}$$
$3$ $$T^{4} - 47322 T^{2} + \cdots + 387774864$$
$5$ $$T^{4} - 4389322 T^{2} + \cdots + 4951092496$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 4774 T - 2902464824)^{2}$$
$13$ $$T^{4} - 28441135314 T^{2} + \cdots + 20\!\cdots\!96$$
$17$ $$T^{4} - 364177565476 T^{2} + \cdots + 32\!\cdots\!76$$
$19$ $$T^{4} - 760834631722 T^{2} + \cdots + 42\!\cdots\!44$$
$23$ $$(T^{2} + 1418368 T - 514307412752)^{2}$$
$29$ $$(T^{2} - 8334986 T + 14857994423056)^{2}$$
$31$ $$T^{4} - 74580188894856 T^{2} + \cdots + 12\!\cdots\!84$$
$37$ $$(T^{2} - 1613186 T - 172221385813328)^{2}$$
$41$ $$T^{4} - 524653229945812 T^{2} + \cdots + 15\!\cdots\!36$$
$43$ $$(T^{2} + 38240102 T + 187841499446728)^{2}$$
$47$ $$T^{4} + \cdots + 26\!\cdots\!56$$
$53$ $$(T^{2} - 6854056 T - 71\!\cdots\!88)^{2}$$
$59$ $$T^{4} + \cdots + 12\!\cdots\!24$$
$61$ $$T^{4} + \cdots + 24\!\cdots\!96$$
$67$ $$(T^{2} + 407210388 T + 31\!\cdots\!44)^{2}$$
$71$ $$(T^{2} + 662334372 T + 10\!\cdots\!64)^{2}$$
$73$ $$T^{4} + \cdots + 45\!\cdots\!24$$
$79$ $$(T^{2} + 334205588 T + 24\!\cdots\!68)^{2}$$
$83$ $$T^{4} + \cdots + 89\!\cdots\!96$$
$89$ $$T^{4} + \cdots + 19\!\cdots\!84$$
$97$ $$T^{4} + \cdots + 62\!\cdots\!56$$
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