# Properties

 Label 98.10.c.f 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{2} + 6q^{3} - 256q^{4} - 560q^{5} + 192q^{6} - 8192q^{8} + 19647q^{9} + O(q^{10})$$ $$2q + 16q^{2} + 6q^{3} - 256q^{4} - 560q^{5} + 192q^{6} - 8192q^{8} + 19647q^{9} + 8960q^{10} + 54152q^{11} + 1536q^{12} - 226344q^{13} - 6720q^{15} - 65536q^{16} - 6262q^{17} - 314352q^{18} - 257078q^{19} + 286720q^{20} + 1732864q^{22} + 266000q^{23} - 24576q^{24} + 1639525q^{25} - 1810752q^{26} + 471960q^{27} + 3149428q^{29} - 53760q^{30} + 4637484q^{31} + 1048576q^{32} - 324912q^{33} - 200384q^{34} - 10059264q^{36} + 11946238q^{37} + 4113248q^{38} - 679032q^{39} + 2293760q^{40} + 43818252q^{41} + 55041184q^{43} + 13862912q^{44} + 11002320q^{45} - 4256000q^{46} - 52927836q^{47} - 786432q^{48} + 52464800q^{50} + 37572q^{51} + 28972032q^{52} - 16221222q^{53} + 3775680q^{54} - 60650240q^{55} - 3084936q^{57} + 25195424q^{58} + 140509618q^{59} + 860160q^{60} + 202963560q^{61} + 148399488q^{62} + 33554432q^{64} + 63376320q^{65} + 5198592q^{66} - 153734572q^{67} - 1603072q^{68} + 3192000q^{69} + 559311872q^{71} - 80474112q^{72} + 404022830q^{73} - 191139808q^{74} - 9837150q^{75} + 131623936q^{76} - 21729024q^{78} + 130689816q^{79} - 36700160q^{80} - 385296021q^{81} + 350546016q^{82} + 840268028q^{83} + 7013440q^{85} + 440329472q^{86} + 9448284q^{87} - 221806592q^{88} + 469542390q^{89} + 352074240q^{90} - 136192000q^{92} - 27824904q^{93} + 846845376q^{94} - 143963680q^{95} - 6291456q^{96} - 1745003380q^{97} + 2127848688q^{99} + O(q^{100})$$

## 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.f 2
7.b odd 2 1 98.10.c.e 2
7.c even 3 1 14.10.a.a 1
7.c even 3 1 inner 98.10.c.f 2
7.d odd 6 1 98.10.a.a 1
7.d odd 6 1 98.10.c.e 2
21.h odd 6 1 126.10.a.e 1
28.g odd 6 1 112.10.a.b 1
35.j even 6 1 350.10.a.c 1
35.l odd 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.c even 3 1
98.10.a.a 1 7.d odd 6 1
98.10.c.e 2 7.b odd 2 1
98.10.c.e 2 7.d odd 6 1
98.10.c.f 2 1.a even 1 1 trivial
98.10.c.f 2 7.c even 3 1 inner
112.10.a.b 1 28.g odd 6 1
126.10.a.e 1 21.h odd 6 1
350.10.a.c 1 35.j even 6 1
350.10.c.b 2 35.l odd 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

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