# Properties

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

# Related objects

## 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: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 1115 x + 2100$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}\cdot 3\cdot 7$$ 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.

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + ( -78 - \beta_{1} ) q^{3} + 256 q^{4} + ( -246 - 6 \beta_{1} + \beta_{2} ) q^{5} + ( -1248 - 16 \beta_{1} ) q^{6} + 4096 q^{8} + ( 5103 + 252 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + 16 q^{2} + ( -78 - \beta_{1} ) q^{3} + 256 q^{4} + ( -246 - 6 \beta_{1} + \beta_{2} ) q^{5} + ( -1248 - 16 \beta_{1} ) q^{6} + 4096 q^{8} + ( 5103 + 252 \beta_{1} - \beta_{2} ) q^{9} + ( -3936 - 96 \beta_{1} + 16 \beta_{2} ) q^{10} + ( -2475 - 35 \beta_{1} - 51 \beta_{2} ) q^{11} + ( -19968 - 256 \beta_{1} ) q^{12} + ( -32789 + 224 \beta_{1} - 73 \beta_{2} ) q^{13} + ( 123471 + 1281 \beta_{1} + 13 \beta_{2} ) q^{15} + 65536 q^{16} + ( -101526 + 2232 \beta_{1} - 145 \beta_{2} ) q^{17} + ( 81648 + 4032 \beta_{1} - 16 \beta_{2} ) q^{18} + ( -125285 + 1793 \beta_{1} + 343 \beta_{2} ) q^{19} + ( -62976 - 1536 \beta_{1} + 256 \beta_{2} ) q^{20} + ( -39600 - 560 \beta_{1} - 816 \beta_{2} ) q^{22} + ( 758040 + 7021 \beta_{1} - 156 \beta_{2} ) q^{23} + ( -319488 - 4096 \beta_{1} ) q^{24} + ( 47494 - 756 \beta_{1} + 626 \beta_{2} ) q^{25} + ( -524624 + 3584 \beta_{1} - 1168 \beta_{2} ) q^{26} + ( -3567735 - 29259 \beta_{1} + 233 \beta_{2} ) q^{27} + ( -2185725 - 17584 \beta_{1} + 3387 \beta_{2} ) q^{29} + ( 1975536 + 20496 \beta_{1} + 208 \beta_{2} ) q^{30} + ( -2210729 + 21601 \beta_{1} + 729 \beta_{2} ) q^{31} + 1048576 q^{32} + ( 1251999 + 9024 \beta_{1} - 1004 \beta_{2} ) q^{33} + ( -1624416 + 35712 \beta_{1} - 2320 \beta_{2} ) q^{34} + ( 1306368 + 64512 \beta_{1} - 256 \beta_{2} ) q^{36} + ( 7425827 - 14490 \beta_{1} + 4002 \beta_{2} ) q^{37} + ( -2004560 + 28688 \beta_{1} + 5488 \beta_{2} ) q^{38} + ( -1052889 - 5530 \beta_{1} - 1163 \beta_{2} ) q^{39} + ( -1007616 - 24576 \beta_{1} + 4096 \beta_{2} ) q^{40} + ( -11338167 + 29120 \beta_{1} + 4477 \beta_{2} ) q^{41} + ( -20962888 - 50288 \beta_{1} - 14236 \beta_{2} ) q^{43} + ( -633600 - 8960 \beta_{1} - 13056 \beta_{2} ) q^{44} + ( -28849059 - 228384 \beta_{1} - 18155 \beta_{2} ) q^{45} + ( 12128640 + 112336 \beta_{1} - 2496 \beta_{2} ) q^{46} + ( -17543715 + 67943 \beta_{1} + 3931 \beta_{2} ) q^{47} + ( -5111808 - 65536 \beta_{1} ) q^{48} + ( 759904 - 12096 \beta_{1} + 10016 \beta_{2} ) q^{50} + ( -32674131 - 285537 \beta_{1} - 523 \beta_{2} ) q^{51} + ( -8393984 + 57344 \beta_{1} - 18688 \beta_{2} ) q^{52} + ( 3866913 - 487886 \beta_{1} - 2500 \beta_{2} ) q^{53} + ( -57083760 - 468144 \beta_{1} + 3728 \beta_{2} ) q^{54} + ( -62775900 + 405915 \beta_{1} - 15916 \beta_{2} ) q^{55} + ( -26480103 - 189784 \beta_{1} + 8310 \beta_{2} ) q^{57} + ( -34971600 - 281344 \beta_{1} + 54192 \beta_{2} ) q^{58} + ( -4156638 + 382899 \beta_{1} + 97084 \beta_{2} ) q^{59} + ( 31608576 + 327936 \beta_{1} + 3328 \beta_{2} ) q^{60} + ( -53318603 + 318998 \beta_{1} - 22654 \beta_{2} ) q^{61} + ( -35371664 + 345616 \beta_{1} + 11664 \beta_{2} ) q^{62} + 16777216 q^{64} + ( -111244863 + 533652 \beta_{1} - 76971 \beta_{2} ) q^{65} + ( 20031984 + 144384 \beta_{1} - 16064 \beta_{2} ) q^{66} + ( 160205648 - 284095 \beta_{1} + 10814 \beta_{2} ) q^{67} + ( -25990656 + 571392 \beta_{1} - 37120 \beta_{2} ) q^{68} + ( -189196938 - 1978290 \beta_{1} + 4057 \beta_{2} ) q^{69} + ( -12174294 + 623840 \beta_{1} + 63998 \beta_{2} ) q^{71} + ( 20901888 + 1032192 \beta_{1} - 4096 \beta_{2} ) q^{72} + ( 83252491 - 1464324 \beta_{1} - 160486 \beta_{2} ) q^{73} + ( 118813232 - 231840 \beta_{1} + 64032 \beta_{2} ) q^{74} + ( 5470626 + 78416 \beta_{1} + 11138 \beta_{2} ) q^{75} + ( -32072960 + 459008 \beta_{1} + 87808 \beta_{2} ) q^{76} + ( -16846224 - 88480 \beta_{1} - 18608 \beta_{2} ) q^{78} + ( -95137612 + 927073 \beta_{1} + 154876 \beta_{2} ) q^{79} + ( -16121856 - 393216 \beta_{1} + 65536 \beta_{2} ) q^{80} + ( 723195342 + 3696588 \beta_{1} - 5149 \beta_{2} ) q^{81} + ( -181410672 + 465920 \beta_{1} + 71632 \beta_{2} ) q^{82} + ( -382307766 + 742000 \beta_{1} - 74126 \beta_{2} ) q^{83} + ( -398372985 - 174510 \beta_{1} - 351908 \beta_{2} ) q^{85} + ( -335406208 - 804608 \beta_{1} - 227776 \beta_{2} ) q^{86} + ( 472486995 + 5214858 \beta_{1} + 46769 \beta_{2} ) q^{87} + ( -10137600 - 143360 \beta_{1} - 208896 \beta_{2} ) q^{88} + ( -300776163 - 1327780 \beta_{1} + 243136 \beta_{2} ) q^{89} + ( -461584944 - 3654144 \beta_{1} - 290480 \beta_{2} ) q^{90} + ( 194058240 + 1797376 \beta_{1} - 39936 \beta_{2} ) q^{92} + ( -237325281 - 1554406 \beta_{1} + 35452 \beta_{2} ) q^{93} + ( -280699440 + 1087088 \beta_{1} + 62896 \beta_{2} ) q^{94} + ( 294681606 - 3144729 \beta_{1} - 176630 \beta_{2} ) q^{95} + ( -81788928 - 1048576 \beta_{1} ) q^{96} + ( -102649241 + 6520304 \beta_{1} + 385911 \beta_{2} ) q^{97} + ( -209746629 - 2124234 \beta_{1} + 993781 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 48 q^{2} - 233 q^{3} + 768 q^{4} - 733 q^{5} - 3728 q^{6} + 12288 q^{8} + 15058 q^{9} + O(q^{10})$$ $$3 q + 48 q^{2} - 233 q^{3} + 768 q^{4} - 733 q^{5} - 3728 q^{6} + 12288 q^{8} + 15058 q^{9} - 11728 q^{10} - 7339 q^{11} - 59648 q^{12} - 98518 q^{13} + 369119 q^{15} + 196608 q^{16} - 306665 q^{17} + 240928 q^{18} - 377991 q^{19} - 187648 q^{20} - 117424 q^{22} + 2267255 q^{23} - 954368 q^{24} + 142612 q^{25} - 1576288 q^{26} - 10674179 q^{27} - 6542978 q^{29} + 5905904 q^{30} - 6654517 q^{31} + 3145728 q^{32} + 3747977 q^{33} - 4906640 q^{34} + 3854848 q^{36} + 22287969 q^{37} - 6047856 q^{38} - 3151974 q^{39} - 3002368 q^{40} - 34048098 q^{41} - 62824140 q^{43} - 1878784 q^{44} - 86300638 q^{45} + 36276080 q^{46} - 52703019 q^{47} - 15269888 q^{48} + 2281792 q^{50} - 97736333 q^{51} - 25220608 q^{52} + 12091125 q^{53} - 170786864 q^{54} - 188717699 q^{55} - 79258835 q^{57} - 104687648 q^{58} - 12949897 q^{59} + 94494464 q^{60} - 160252153 q^{61} - 106472272 q^{62} + 50331648 q^{64} - 334191270 q^{65} + 59967632 q^{66} + 480890225 q^{67} - 78506240 q^{68} - 565616581 q^{69} - 37210720 q^{71} + 61677568 q^{72} + 251382283 q^{73} + 356607504 q^{74} + 16322324 q^{75} - 96765696 q^{76} - 50431584 q^{78} - 286494785 q^{79} - 48037888 q^{80} + 2165894587 q^{81} - 544769568 q^{82} - 1147591172 q^{83} - 1194592537 q^{85} - 1005186240 q^{86} + 1412199358 q^{87} - 30060544 q^{88} - 901243845 q^{89} - 1380810208 q^{90} + 580417280 q^{92} - 710456889 q^{93} - 843248304 q^{94} + 887366177 q^{95} - 244318208 q^{96} - 314853938 q^{97} - 628109434 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 1115 x + 2100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{2} + 68 \nu - 1515$$$$)/15$$ $$\beta_{2}$$ $$=$$ $$($$$$-28 \nu^{2} + 308 \nu + 20715$$$$)/15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 14 \beta_{1} + 33$$$$)/84$$ $$\nu^{2}$$ $$=$$ $$($$$$-17 \beta_{2} + 77 \beta_{1} + 31254$$$$)/42$$

## 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
 32.9264 1.88624 −33.8126
16.0000 −270.819 256.000 −1369.57 −4333.11 0 4096.00 53660.1 −21913.1
1.2 16.0000 13.9747 256.000 1718.94 223.595 0 4096.00 −19487.7 27503.0
1.3 16.0000 23.8447 256.000 −1082.37 381.515 0 4096.00 −19114.4 −17317.9
 $$n$$: 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

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.i 3
7.b odd 2 1 98.10.a.j 3
7.c even 3 2 98.10.c.k 6
7.d odd 6 2 14.10.c.a 6
21.g even 6 2 126.10.g.f 6
28.f even 6 2 112.10.i.b 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 233 T_{3}^{2} - 9909 T_{3} + 90243$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -16 + T )^{3}$$
$3$ $$90243 - 9909 T + 233 T^{2} + T^{3}$$
$5$ $$-2548114785 - 2732349 T + 733 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-58366007241975 - 5381919141 T + 7339 T^{2} + T^{3}$$
$13$ $$62445940634280 - 8685337220 T + 98518 T^{2} + T^{3}$$
$17$ $$-19724491547956917 - 143470727733 T + 306665 T^{2} + T^{3}$$
$19$ $$36275859228526373 - 297858917325 T + 377991 T^{2} + T^{3}$$
$23$ $$61241855995368819 + 309320186283 T - 2267255 T^{2} + T^{3}$$
$29$ $$-$$$$12\!\cdots\!84$$$$- 16444066423332 T + 6542978 T^{2} + T^{3}$$
$31$ $$-34255130067371021145 + 229935151771 T + 6654517 T^{2} + T^{3}$$
$37$ $$-$$$$21\!\cdots\!27$$$$+ 128312212154955 T - 22287969 T^{2} + T^{3}$$
$41$ $$89\!\cdots\!08$$$$+ 318544865229276 T + 34048098 T^{2} + T^{3}$$
$43$ $$-$$$$54\!\cdots\!00$$$$+ 812958581920560 T + 62824140 T^{2} + T^{3}$$
$47$ $$23\!\cdots\!97$$$$+ 758589113063547 T + 52703019 T^{2} + T^{3}$$
$53$ $$23\!\cdots\!49$$$$- 6659365634347437 T - 12091125 T^{2} + T^{3}$$
$59$ $$13\!\cdots\!55$$$$- 24226405027915605 T + 12949897 T^{2} + T^{3}$$
$61$ $$-$$$$45\!\cdots\!89$$$$+ 4832695972500763 T + 160252153 T^{2} + T^{3}$$
$67$ $$-$$$$37\!\cdots\!83$$$$+ 74658851232780587 T - 480890225 T^{2} + T^{3}$$
$71$ $$38\!\cdots\!16$$$$- 19772597378317632 T + 37210720 T^{2} + T^{3}$$
$73$ $$-$$$$84\!\cdots\!65$$$$- 97305758651115949 T - 251382283 T^{2} + T^{3}$$
$79$ $$14\!\cdots\!75$$$$- 49226347140639365 T + 286494785 T^{2} + T^{3}$$
$83$ $$47\!\cdots\!64$$$$+ 413638076348661360 T + 1147591172 T^{2} + T^{3}$$
$89$ $$-$$$$46\!\cdots\!37$$$$+ 108101752353947187 T + 901243845 T^{2} + T^{3}$$
$97$ $$-$$$$24\!\cdots\!80$$$$- 1522439324115043684 T + 314853938 T^{2} + T^{3}$$