# Properties

 Label 98.10.a.d Level $98$ Weight $10$ Character orbit 98.a Self dual yes Analytic conductor $50.474$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{43})$$ Defining polynomial: $$x^{2} - 43$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 8\sqrt{43}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -16 q^{2} + \beta q^{3} + 256 q^{4} + 35 \beta q^{5} -16 \beta q^{6} -4096 q^{8} -16931 q^{9} +O(q^{10})$$ $$q -16 q^{2} + \beta q^{3} + 256 q^{4} + 35 \beta q^{5} -16 \beta q^{6} -4096 q^{8} -16931 q^{9} -560 \beta q^{10} + 9380 q^{11} + 256 \beta q^{12} + 3413 \beta q^{13} + 96320 q^{15} + 65536 q^{16} + 1866 \beta q^{17} + 270896 q^{18} + 10727 \beta q^{19} + 8960 \beta q^{20} -150080 q^{22} -978936 q^{23} -4096 \beta q^{24} + 1418075 q^{25} -54608 \beta q^{26} -36614 \beta q^{27} -4317214 q^{29} -1541120 q^{30} + 152014 \beta q^{31} -1048576 q^{32} + 9380 \beta q^{33} -29856 \beta q^{34} -4334336 q^{36} + 2714394 q^{37} -171632 \beta q^{38} + 9392576 q^{39} -143360 \beta q^{40} + 171606 \beta q^{41} -34755692 q^{43} + 2401280 q^{44} -592585 \beta q^{45} + 15662976 q^{46} + 804002 \beta q^{47} + 65536 \beta q^{48} -22689200 q^{50} + 5135232 q^{51} + 873728 \beta q^{52} + 68067926 q^{53} + 585824 \beta q^{54} + 328300 \beta q^{55} + 29520704 q^{57} + 69075424 q^{58} -652537 \beta q^{59} + 24657920 q^{60} -3161153 \beta q^{61} -2432224 \beta q^{62} + 16777216 q^{64} + 328740160 q^{65} -150080 \beta q^{66} + 242944420 q^{67} + 477696 \beta q^{68} -978936 \beta q^{69} -94292464 q^{71} + 69349376 q^{72} + 2541664 \beta q^{73} -43430304 q^{74} + 1418075 \beta q^{75} + 2746112 \beta q^{76} -150281216 q^{78} + 677625160 q^{79} + 2293760 \beta q^{80} + 232491145 q^{81} -2745696 \beta q^{82} + 8412459 \beta q^{83} + 179733120 q^{85} + 556091072 q^{86} -4317214 \beta q^{87} -38420480 q^{88} + 10584880 \beta q^{89} + 9481360 \beta q^{90} -250607616 q^{92} + 418342528 q^{93} -12864032 \beta q^{94} + 1033224640 q^{95} -1048576 \beta q^{96} -24189050 \beta q^{97} -158812780 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{2} + 512 q^{4} - 8192 q^{8} - 33862 q^{9} + O(q^{10})$$ $$2 q - 32 q^{2} + 512 q^{4} - 8192 q^{8} - 33862 q^{9} + 18760 q^{11} + 192640 q^{15} + 131072 q^{16} + 541792 q^{18} - 300160 q^{22} - 1957872 q^{23} + 2836150 q^{25} - 8634428 q^{29} - 3082240 q^{30} - 2097152 q^{32} - 8668672 q^{36} + 5428788 q^{37} + 18785152 q^{39} - 69511384 q^{43} + 4802560 q^{44} + 31325952 q^{46} - 45378400 q^{50} + 10270464 q^{51} + 136135852 q^{53} + 59041408 q^{57} + 138150848 q^{58} + 49315840 q^{60} + 33554432 q^{64} + 657480320 q^{65} + 485888840 q^{67} - 188584928 q^{71} + 138698752 q^{72} - 86860608 q^{74} - 300562432 q^{78} + 1355250320 q^{79} + 464982290 q^{81} + 359466240 q^{85} + 1112182144 q^{86} - 76840960 q^{88} - 501215232 q^{92} + 836685056 q^{93} + 2066449280 q^{95} - 317625560 q^{99} + O(q^{100})$$

## 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
 −6.55744 6.55744
−16.0000 −52.4595 256.000 −1836.08 839.352 0 −4096.00 −16931.0 29377.3
1.2 −16.0000 52.4595 256.000 1836.08 −839.352 0 −4096.00 −16931.0 −29377.3
 $$n$$: 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.d 2
7.b odd 2 1 inner 98.10.a.d 2
7.c even 3 2 98.10.c.i 4
7.d odd 6 2 98.10.c.i 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2752$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + T )^{2}$$
$3$ $$-2752 + T^{2}$$
$5$ $$-3371200 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -9380 + T )^{2}$$
$13$ $$-32056861888 + T^{2}$$
$17$ $$-9582342912 + T^{2}$$
$19$ $$-316668591808 + T^{2}$$
$23$ $$( 978936 + T )^{2}$$
$29$ $$( 4317214 + T )^{2}$$
$31$ $$-63593921051392 + T^{2}$$
$37$ $$( -2714394 + T )^{2}$$
$41$ $$-81042600137472 + T^{2}$$
$43$ $$( 34755692 + T )^{2}$$
$47$ $$-1778945682443008 + T^{2}$$
$53$ $$( -68067926 + T )^{2}$$
$59$ $$-1171814084087488 + T^{2}$$
$61$ $$-27500428572453568 + T^{2}$$
$67$ $$( -242944420 + T )^{2}$$
$71$ $$( 94292464 + T )^{2}$$
$73$ $$-17778073806241792 + T^{2}$$
$79$ $$( -677625160 + T )^{2}$$
$83$ $$-194757571606226112 + T^{2}$$
$89$ $$-308333212058828800 + T^{2}$$
$97$ $$-1610223105011680000 + T^{2}$$