# Properties

 Label 98.10.a.f Level $98$ Weight $10$ Character orbit 98.a Self dual yes Analytic conductor $50.474$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{106})$$ Defining polynomial: $$x^{2} - 106$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{106}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 3 \beta q^{3} + 256 q^{4} -31 \beta q^{5} + 48 \beta q^{6} + 4096 q^{8} -4419 q^{9} +O(q^{10})$$ $$q + 16 q^{2} + 3 \beta q^{3} + 256 q^{4} -31 \beta q^{5} + 48 \beta q^{6} + 4096 q^{8} -4419 q^{9} -496 \beta q^{10} -27644 q^{11} + 768 \beta q^{12} + 2023 \beta q^{13} -157728 q^{15} + 65536 q^{16} -13458 \beta q^{17} -70704 q^{18} + 8341 \beta q^{19} -7936 \beta q^{20} -442304 q^{22} -535704 q^{23} + 12288 \beta q^{24} -323269 q^{25} + 32368 \beta q^{26} -72306 \beta q^{27} -2591006 q^{29} -2523648 q^{30} -148054 \beta q^{31} + 1048576 q^{32} -82932 \beta q^{33} -215328 \beta q^{34} -1131264 q^{36} -17807910 q^{37} + 133456 \beta q^{38} + 10293024 q^{39} -126976 \beta q^{40} + 741426 \beta q^{41} -8759756 q^{43} -7076864 q^{44} + 136989 \beta q^{45} -8571264 q^{46} + 662246 \beta q^{47} + 196608 \beta q^{48} -5172304 q^{50} -68474304 q^{51} + 517888 \beta q^{52} -83488490 q^{53} -1156896 \beta q^{54} + 856964 \beta q^{55} + 42439008 q^{57} -41456096 q^{58} + 1980149 \beta q^{59} -40378368 q^{60} + 1210325 \beta q^{61} -2368864 \beta q^{62} + 16777216 q^{64} -106361248 q^{65} -1326912 \beta q^{66} -185492924 q^{67} -3445248 \beta q^{68} -1607112 \beta q^{69} + 370270672 q^{71} -18100224 q^{72} -5973664 \beta q^{73} -284926560 q^{74} -969807 \beta q^{75} + 2135296 \beta q^{76} + 164688384 q^{78} -519907256 q^{79} -2031616 \beta q^{80} -280913751 q^{81} + 11862816 \beta q^{82} -7257663 \beta q^{83} + 707567808 q^{85} -140156096 q^{86} -7773018 \beta q^{87} -113229824 q^{88} + 7224592 \beta q^{89} + 2191824 \beta q^{90} -137140224 q^{92} -753298752 q^{93} + 10595936 \beta q^{94} -438536416 q^{95} + 3145728 \beta q^{96} + 18957890 \beta q^{97} + 122158836 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{2} + 512 q^{4} + 8192 q^{8} - 8838 q^{9} + O(q^{10})$$ $$2 q + 32 q^{2} + 512 q^{4} + 8192 q^{8} - 8838 q^{9} - 55288 q^{11} - 315456 q^{15} + 131072 q^{16} - 141408 q^{18} - 884608 q^{22} - 1071408 q^{23} - 646538 q^{25} - 5182012 q^{29} - 5047296 q^{30} + 2097152 q^{32} - 2262528 q^{36} - 35615820 q^{37} + 20586048 q^{39} - 17519512 q^{43} - 14153728 q^{44} - 17142528 q^{46} - 10344608 q^{50} - 136948608 q^{51} - 166976980 q^{53} + 84878016 q^{57} - 82912192 q^{58} - 80756736 q^{60} + 33554432 q^{64} - 212722496 q^{65} - 370985848 q^{67} + 740541344 q^{71} - 36200448 q^{72} - 569853120 q^{74} + 329376768 q^{78} - 1039814512 q^{79} - 561827502 q^{81} + 1415135616 q^{85} - 280312192 q^{86} - 226459648 q^{88} - 274280448 q^{92} - 1506597504 q^{93} - 877072832 q^{95} + 244317672 q^{99} + O(q^{100})$$

## 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
 −10.2956 10.2956
16.0000 −123.548 256.000 1276.66 −1976.76 0 4096.00 −4419.00 20426.5
1.2 16.0000 123.548 256.000 −1276.66 1976.76 0 4096.00 −4419.00 −20426.5
 $$n$$: 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.f 2
7.b odd 2 1 inner 98.10.a.f 2
7.c even 3 2 98.10.c.g 4
7.d odd 6 2 98.10.c.g 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 15264$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -16 + T )^{2}$$
$3$ $$-15264 + T^{2}$$
$5$ $$-1629856 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 27644 + T )^{2}$$
$13$ $$-6940929184 + T^{2}$$
$17$ $$-307175727744 + T^{2}$$
$19$ $$-117994588576 + T^{2}$$
$23$ $$( 535704 + T )^{2}$$
$29$ $$( 2591006 + T )^{2}$$
$31$ $$-37176297809536 + T^{2}$$
$37$ $$( 17807910 + T )^{2}$$
$41$ $$-932312422855296 + T^{2}$$
$43$ $$( 8759756 + T )^{2}$$
$47$ $$-743814320619136 + T^{2}$$
$53$ $$( 83488490 + T )^{2}$$
$59$ $$-6649999145492896 + T^{2}$$
$61$ $$-2484447683140000 + T^{2}$$
$67$ $$( 185492924 + T )^{2}$$
$71$ $$( -370270672 + T )^{2}$$
$73$ $$-60521186047983616 + T^{2}$$
$79$ $$( 519907256 + T )^{2}$$
$83$ $$-89334548087781024 + T^{2}$$
$89$ $$-88522261344722944 + T^{2}$$
$97$ $$-609545102155561600 + T^{2}$$