# Properties

 Label 98.4.c.f Level $98$ Weight $4$ Character orbit 98.c Analytic conductor $5.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 98.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.78218718056$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( 8 - 8 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -14 \zeta_{6} q^{5} + 16 q^{6} -8 q^{8} -37 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( 8 - 8 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -14 \zeta_{6} q^{5} + 16 q^{6} -8 q^{8} -37 \zeta_{6} q^{9} + ( 28 - 28 \zeta_{6} ) q^{10} + ( 28 - 28 \zeta_{6} ) q^{11} + 32 \zeta_{6} q^{12} -18 q^{13} -112 q^{15} -16 \zeta_{6} q^{16} + ( 74 - 74 \zeta_{6} ) q^{17} + ( 74 - 74 \zeta_{6} ) q^{18} + 80 \zeta_{6} q^{19} + 56 q^{20} + 56 q^{22} + 112 \zeta_{6} q^{23} + ( -64 + 64 \zeta_{6} ) q^{24} + ( -71 + 71 \zeta_{6} ) q^{25} -36 \zeta_{6} q^{26} -80 q^{27} + 190 q^{29} -224 \zeta_{6} q^{30} + ( 72 - 72 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -224 \zeta_{6} q^{33} + 148 q^{34} + 148 q^{36} + 346 \zeta_{6} q^{37} + ( -160 + 160 \zeta_{6} ) q^{38} + ( -144 + 144 \zeta_{6} ) q^{39} + 112 \zeta_{6} q^{40} -162 q^{41} -412 q^{43} + 112 \zeta_{6} q^{44} + ( -518 + 518 \zeta_{6} ) q^{45} + ( -224 + 224 \zeta_{6} ) q^{46} + 24 \zeta_{6} q^{47} -128 q^{48} -142 q^{50} -592 \zeta_{6} q^{51} + ( 72 - 72 \zeta_{6} ) q^{52} + ( -318 + 318 \zeta_{6} ) q^{53} -160 \zeta_{6} q^{54} -392 q^{55} + 640 q^{57} + 380 \zeta_{6} q^{58} + ( -200 + 200 \zeta_{6} ) q^{59} + ( 448 - 448 \zeta_{6} ) q^{60} -198 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} + 252 \zeta_{6} q^{65} + ( 448 - 448 \zeta_{6} ) q^{66} + ( 716 - 716 \zeta_{6} ) q^{67} + 296 \zeta_{6} q^{68} + 896 q^{69} + 392 q^{71} + 296 \zeta_{6} q^{72} + ( 538 - 538 \zeta_{6} ) q^{73} + ( -692 + 692 \zeta_{6} ) q^{74} + 568 \zeta_{6} q^{75} -320 q^{76} -288 q^{78} -240 \zeta_{6} q^{79} + ( -224 + 224 \zeta_{6} ) q^{80} + ( 359 - 359 \zeta_{6} ) q^{81} -324 \zeta_{6} q^{82} + 1072 q^{83} -1036 q^{85} -824 \zeta_{6} q^{86} + ( 1520 - 1520 \zeta_{6} ) q^{87} + ( -224 + 224 \zeta_{6} ) q^{88} + 810 \zeta_{6} q^{89} -1036 q^{90} -448 q^{92} -576 \zeta_{6} q^{93} + ( -48 + 48 \zeta_{6} ) q^{94} + ( 1120 - 1120 \zeta_{6} ) q^{95} -256 \zeta_{6} q^{96} -1354 q^{97} -1036 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 8q^{3} - 4q^{4} - 14q^{5} + 32q^{6} - 16q^{8} - 37q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 8q^{3} - 4q^{4} - 14q^{5} + 32q^{6} - 16q^{8} - 37q^{9} + 28q^{10} + 28q^{11} + 32q^{12} - 36q^{13} - 224q^{15} - 16q^{16} + 74q^{17} + 74q^{18} + 80q^{19} + 112q^{20} + 112q^{22} + 112q^{23} - 64q^{24} - 71q^{25} - 36q^{26} - 160q^{27} + 380q^{29} - 224q^{30} + 72q^{31} + 32q^{32} - 224q^{33} + 296q^{34} + 296q^{36} + 346q^{37} - 160q^{38} - 144q^{39} + 112q^{40} - 324q^{41} - 824q^{43} + 112q^{44} - 518q^{45} - 224q^{46} + 24q^{47} - 256q^{48} - 284q^{50} - 592q^{51} + 72q^{52} - 318q^{53} - 160q^{54} - 784q^{55} + 1280q^{57} + 380q^{58} - 200q^{59} + 448q^{60} - 198q^{61} + 288q^{62} + 128q^{64} + 252q^{65} + 448q^{66} + 716q^{67} + 296q^{68} + 1792q^{69} + 784q^{71} + 296q^{72} + 538q^{73} - 692q^{74} + 568q^{75} - 640q^{76} - 576q^{78} - 240q^{79} - 224q^{80} + 359q^{81} - 324q^{82} + 2144q^{83} - 2072q^{85} - 824q^{86} + 1520q^{87} - 224q^{88} + 810q^{89} - 2072q^{90} - 896q^{92} - 576q^{93} - 48q^{94} + 1120q^{95} - 256q^{96} - 2708q^{97} - 2072q^{99} + O(q^{100})$$

## 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
1.00000 + 1.73205i 4.00000 6.92820i −2.00000 + 3.46410i −7.00000 12.1244i 16.0000 0 −8.00000 −18.5000 32.0429i 14.0000 24.2487i
79.1 1.00000 1.73205i 4.00000 + 6.92820i −2.00000 3.46410i −7.00000 + 12.1244i 16.0000 0 −8.00000 −18.5000 + 32.0429i 14.0000 + 24.2487i
 $$n$$: 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 98.4.c.f 2
3.b odd 2 1 882.4.g.k 2
7.b odd 2 1 98.4.c.d 2
7.c even 3 1 98.4.a.a 1
7.c even 3 1 inner 98.4.c.f 2
7.d odd 6 1 14.4.a.a 1
7.d odd 6 1 98.4.c.d 2
21.c even 2 1 882.4.g.b 2
21.g even 6 1 126.4.a.h 1
21.g even 6 1 882.4.g.b 2
21.h odd 6 1 882.4.a.i 1
21.h odd 6 1 882.4.g.k 2
28.f even 6 1 112.4.a.a 1
28.g odd 6 1 784.4.a.s 1
35.i odd 6 1 350.4.a.l 1
35.j even 6 1 2450.4.a.bo 1
35.k even 12 2 350.4.c.b 2
56.j odd 6 1 448.4.a.b 1
56.m even 6 1 448.4.a.o 1
77.i even 6 1 1694.4.a.g 1
84.j odd 6 1 1008.4.a.s 1
91.s odd 6 1 2366.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 7.d odd 6 1
98.4.a.a 1 7.c even 3 1
98.4.c.d 2 7.b odd 2 1
98.4.c.d 2 7.d odd 6 1
98.4.c.f 2 1.a even 1 1 trivial
98.4.c.f 2 7.c even 3 1 inner
112.4.a.a 1 28.f even 6 1
126.4.a.h 1 21.g even 6 1
350.4.a.l 1 35.i odd 6 1
350.4.c.b 2 35.k even 12 2
448.4.a.b 1 56.j odd 6 1
448.4.a.o 1 56.m even 6 1
784.4.a.s 1 28.g odd 6 1
882.4.a.i 1 21.h odd 6 1
882.4.g.b 2 21.c even 2 1
882.4.g.b 2 21.g even 6 1
882.4.g.k 2 3.b odd 2 1
882.4.g.k 2 21.h odd 6 1
1008.4.a.s 1 84.j odd 6 1
1694.4.a.g 1 77.i even 6 1
2366.4.a.h 1 91.s odd 6 1
2450.4.a.bo 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 8 T_{3} + 64$$ acting on $$S_{4}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$64 - 8 T + T^{2}$$
$5$ $$196 + 14 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$784 - 28 T + T^{2}$$
$13$ $$( 18 + T )^{2}$$
$17$ $$5476 - 74 T + T^{2}$$
$19$ $$6400 - 80 T + T^{2}$$
$23$ $$12544 - 112 T + T^{2}$$
$29$ $$( -190 + T )^{2}$$
$31$ $$5184 - 72 T + T^{2}$$
$37$ $$119716 - 346 T + T^{2}$$
$41$ $$( 162 + T )^{2}$$
$43$ $$( 412 + T )^{2}$$
$47$ $$576 - 24 T + T^{2}$$
$53$ $$101124 + 318 T + T^{2}$$
$59$ $$40000 + 200 T + T^{2}$$
$61$ $$39204 + 198 T + T^{2}$$
$67$ $$512656 - 716 T + T^{2}$$
$71$ $$( -392 + T )^{2}$$
$73$ $$289444 - 538 T + T^{2}$$
$79$ $$57600 + 240 T + T^{2}$$
$83$ $$( -1072 + T )^{2}$$
$89$ $$656100 - 810 T + T^{2}$$
$97$ $$( 1354 + T )^{2}$$