# Properties

 Label 98.10.c.e Level $98$ Weight $10$ Character orbit 98.c 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.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$50.4735119441$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 \zeta_{6} q^{2} + ( -6 + 6 \zeta_{6} ) q^{3} + ( -256 + 256 \zeta_{6} ) q^{4} + 560 \zeta_{6} q^{5} -96 q^{6} -4096 q^{8} + 19647 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 16 \zeta_{6} q^{2} + ( -6 + 6 \zeta_{6} ) q^{3} + ( -256 + 256 \zeta_{6} ) q^{4} + 560 \zeta_{6} q^{5} -96 q^{6} -4096 q^{8} + 19647 \zeta_{6} q^{9} + ( -8960 + 8960 \zeta_{6} ) q^{10} + ( 54152 - 54152 \zeta_{6} ) q^{11} -1536 \zeta_{6} q^{12} + 113172 q^{13} -3360 q^{15} -65536 \zeta_{6} q^{16} + ( 6262 - 6262 \zeta_{6} ) q^{17} + ( -314352 + 314352 \zeta_{6} ) q^{18} + 257078 \zeta_{6} q^{19} -143360 q^{20} + 866432 q^{22} + 266000 \zeta_{6} q^{23} + ( 24576 - 24576 \zeta_{6} ) q^{24} + ( 1639525 - 1639525 \zeta_{6} ) q^{25} + 1810752 \zeta_{6} q^{26} -235980 q^{27} + 1574714 q^{29} -53760 \zeta_{6} q^{30} + ( -4637484 + 4637484 \zeta_{6} ) q^{31} + ( 1048576 - 1048576 \zeta_{6} ) q^{32} + 324912 \zeta_{6} q^{33} + 100192 q^{34} -5029632 q^{36} + 11946238 \zeta_{6} q^{37} + ( -4113248 + 4113248 \zeta_{6} ) q^{38} + ( -679032 + 679032 \zeta_{6} ) q^{39} -2293760 \zeta_{6} q^{40} -21909126 q^{41} + 27520592 q^{43} + 13862912 \zeta_{6} q^{44} + ( -11002320 + 11002320 \zeta_{6} ) q^{45} + ( -4256000 + 4256000 \zeta_{6} ) q^{46} + 52927836 \zeta_{6} q^{47} + 393216 q^{48} + 26232400 q^{50} + 37572 \zeta_{6} q^{51} + ( -28972032 + 28972032 \zeta_{6} ) q^{52} + ( -16221222 + 16221222 \zeta_{6} ) q^{53} -3775680 \zeta_{6} q^{54} + 30325120 q^{55} -1542468 q^{57} + 25195424 \zeta_{6} q^{58} + ( -140509618 + 140509618 \zeta_{6} ) q^{59} + ( 860160 - 860160 \zeta_{6} ) q^{60} -202963560 \zeta_{6} q^{61} -74199744 q^{62} + 16777216 q^{64} + 63376320 \zeta_{6} q^{65} + ( -5198592 + 5198592 \zeta_{6} ) q^{66} + ( -153734572 + 153734572 \zeta_{6} ) q^{67} + 1603072 \zeta_{6} q^{68} -1596000 q^{69} + 279655936 q^{71} -80474112 \zeta_{6} q^{72} + ( -404022830 + 404022830 \zeta_{6} ) q^{73} + ( -191139808 + 191139808 \zeta_{6} ) q^{74} + 9837150 \zeta_{6} q^{75} -65811968 q^{76} -10864512 q^{78} + 130689816 \zeta_{6} q^{79} + ( 36700160 - 36700160 \zeta_{6} ) q^{80} + ( -385296021 + 385296021 \zeta_{6} ) q^{81} -350546016 \zeta_{6} q^{82} -420134014 q^{83} + 3506720 q^{85} + 440329472 \zeta_{6} q^{86} + ( -9448284 + 9448284 \zeta_{6} ) q^{87} + ( -221806592 + 221806592 \zeta_{6} ) q^{88} -469542390 \zeta_{6} q^{89} -176037120 q^{90} -68096000 q^{92} -27824904 \zeta_{6} q^{93} + ( -846845376 + 846845376 \zeta_{6} ) q^{94} + ( -143963680 + 143963680 \zeta_{6} ) q^{95} + 6291456 \zeta_{6} q^{96} + 872501690 q^{97} + 1063924344 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{2} - 6 q^{3} - 256 q^{4} + 560 q^{5} - 192 q^{6} - 8192 q^{8} + 19647 q^{9} + O(q^{10})$$ $$2 q + 16 q^{2} - 6 q^{3} - 256 q^{4} + 560 q^{5} - 192 q^{6} - 8192 q^{8} + 19647 q^{9} - 8960 q^{10} + 54152 q^{11} - 1536 q^{12} + 226344 q^{13} - 6720 q^{15} - 65536 q^{16} + 6262 q^{17} - 314352 q^{18} + 257078 q^{19} - 286720 q^{20} + 1732864 q^{22} + 266000 q^{23} + 24576 q^{24} + 1639525 q^{25} + 1810752 q^{26} - 471960 q^{27} + 3149428 q^{29} - 53760 q^{30} - 4637484 q^{31} + 1048576 q^{32} + 324912 q^{33} + 200384 q^{34} - 10059264 q^{36} + 11946238 q^{37} - 4113248 q^{38} - 679032 q^{39} - 2293760 q^{40} - 43818252 q^{41} + 55041184 q^{43} + 13862912 q^{44} - 11002320 q^{45} - 4256000 q^{46} + 52927836 q^{47} + 786432 q^{48} + 52464800 q^{50} + 37572 q^{51} - 28972032 q^{52} - 16221222 q^{53} - 3775680 q^{54} + 60650240 q^{55} - 3084936 q^{57} + 25195424 q^{58} - 140509618 q^{59} + 860160 q^{60} - 202963560 q^{61} - 148399488 q^{62} + 33554432 q^{64} + 63376320 q^{65} - 5198592 q^{66} - 153734572 q^{67} + 1603072 q^{68} - 3192000 q^{69} + 559311872 q^{71} - 80474112 q^{72} - 404022830 q^{73} - 191139808 q^{74} + 9837150 q^{75} - 131623936 q^{76} - 21729024 q^{78} + 130689816 q^{79} + 36700160 q^{80} - 385296021 q^{81} - 350546016 q^{82} - 840268028 q^{83} + 7013440 q^{85} + 440329472 q^{86} - 9448284 q^{87} - 221806592 q^{88} - 469542390 q^{89} - 352074240 q^{90} - 136192000 q^{92} - 27824904 q^{93} - 846845376 q^{94} - 143963680 q^{95} + 6291456 q^{96} + 1745003380 q^{97} + 2127848688 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
8.00000 + 13.8564i −3.00000 + 5.19615i −128.000 + 221.703i 280.000 + 484.974i −96.0000 0 −4096.00 9823.50 + 17014.8i −4480.00 + 7759.59i
79.1 8.00000 13.8564i −3.00000 5.19615i −128.000 221.703i 280.000 484.974i −96.0000 0 −4096.00 9823.50 17014.8i −4480.00 7759.59i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.c.e 2
7.b odd 2 1 98.10.c.f 2
7.c even 3 1 98.10.a.a 1
7.c even 3 1 inner 98.10.c.e 2
7.d odd 6 1 14.10.a.a 1
7.d odd 6 1 98.10.c.f 2
21.g even 6 1 126.10.a.e 1
28.f even 6 1 112.10.a.b 1
35.i odd 6 1 350.10.a.c 1
35.k even 12 2 350.10.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 7.d odd 6 1
98.10.a.a 1 7.c even 3 1
98.10.c.e 2 1.a even 1 1 trivial
98.10.c.e 2 7.c even 3 1 inner
98.10.c.f 2 7.b odd 2 1
98.10.c.f 2 7.d odd 6 1
112.10.a.b 1 28.f even 6 1
126.10.a.e 1 21.g even 6 1
350.10.a.c 1 35.i odd 6 1
350.10.c.b 2 35.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 6 T_{3} + 36$$ acting on $$S_{10}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 16 T + T^{2}$$
$3$ $$36 + 6 T + T^{2}$$
$5$ $$313600 - 560 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$2932439104 - 54152 T + T^{2}$$
$13$ $$( -113172 + T )^{2}$$
$17$ $$39212644 - 6262 T + T^{2}$$
$19$ $$66089098084 - 257078 T + T^{2}$$
$23$ $$70756000000 - 266000 T + T^{2}$$
$29$ $$( -1574714 + T )^{2}$$
$31$ $$21506257850256 + 4637484 T + T^{2}$$
$37$ $$142712602352644 - 11946238 T + T^{2}$$
$41$ $$( 21909126 + T )^{2}$$
$43$ $$( -27520592 + T )^{2}$$
$47$ $$2801355823642896 - 52927836 T + T^{2}$$
$53$ $$263128043173284 + 16221222 T + T^{2}$$
$59$ $$19742952750505924 + 140509618 T + T^{2}$$
$61$ $$41194206687873600 + 202963560 T + T^{2}$$
$67$ $$23634318628023184 + 153734572 T + T^{2}$$
$71$ $$( -279655936 + T )^{2}$$
$73$ $$163234447161208900 + 404022830 T + T^{2}$$
$79$ $$17079828006113856 - 130689816 T + T^{2}$$
$83$ $$( 420134014 + T )^{2}$$
$89$ $$220470056006912100 + 469542390 T + T^{2}$$
$97$ $$( -872501690 + T )^{2}$$