# Properties

 Label 98.4.c.b 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -12 \zeta_{6} q^{5} + 4 q^{6} + 8 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -12 \zeta_{6} q^{5} + 4 q^{6} + 8 q^{8} + 23 \zeta_{6} q^{9} + ( -24 + 24 \zeta_{6} ) q^{10} + ( -48 + 48 \zeta_{6} ) q^{11} -8 \zeta_{6} q^{12} -56 q^{13} + 24 q^{15} -16 \zeta_{6} q^{16} + ( -114 + 114 \zeta_{6} ) q^{17} + ( 46 - 46 \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} + 48 q^{20} + 96 q^{22} + 120 \zeta_{6} q^{23} + ( -16 + 16 \zeta_{6} ) q^{24} + ( -19 + 19 \zeta_{6} ) q^{25} + 112 \zeta_{6} q^{26} -100 q^{27} -54 q^{29} -48 \zeta_{6} q^{30} + ( 236 - 236 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -96 \zeta_{6} q^{33} + 228 q^{34} -92 q^{36} -146 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( 112 - 112 \zeta_{6} ) q^{39} -96 \zeta_{6} q^{40} -126 q^{41} -376 q^{43} -192 \zeta_{6} q^{44} + ( 276 - 276 \zeta_{6} ) q^{45} + ( 240 - 240 \zeta_{6} ) q^{46} -12 \zeta_{6} q^{47} + 32 q^{48} + 38 q^{50} -228 \zeta_{6} q^{51} + ( 224 - 224 \zeta_{6} ) q^{52} + ( -174 + 174 \zeta_{6} ) q^{53} + 200 \zeta_{6} q^{54} + 576 q^{55} -4 q^{57} + 108 \zeta_{6} q^{58} + ( 138 - 138 \zeta_{6} ) q^{59} + ( -96 + 96 \zeta_{6} ) q^{60} + 380 \zeta_{6} q^{61} -472 q^{62} + 64 q^{64} + 672 \zeta_{6} q^{65} + ( -192 + 192 \zeta_{6} ) q^{66} + ( 484 - 484 \zeta_{6} ) q^{67} -456 \zeta_{6} q^{68} -240 q^{69} + 576 q^{71} + 184 \zeta_{6} q^{72} + ( -1150 + 1150 \zeta_{6} ) q^{73} + ( -292 + 292 \zeta_{6} ) q^{74} -38 \zeta_{6} q^{75} -8 q^{76} -224 q^{78} -776 \zeta_{6} q^{79} + ( -192 + 192 \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} + 252 \zeta_{6} q^{82} -378 q^{83} + 1368 q^{85} + 752 \zeta_{6} q^{86} + ( 108 - 108 \zeta_{6} ) q^{87} + ( -384 + 384 \zeta_{6} ) q^{88} -390 \zeta_{6} q^{89} -552 q^{90} -480 q^{92} + 472 \zeta_{6} q^{93} + ( -24 + 24 \zeta_{6} ) q^{94} + ( 24 - 24 \zeta_{6} ) q^{95} -64 \zeta_{6} q^{96} + 1330 q^{97} -1104 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} - 4q^{4} - 12q^{5} + 8q^{6} + 16q^{8} + 23q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} - 4q^{4} - 12q^{5} + 8q^{6} + 16q^{8} + 23q^{9} - 24q^{10} - 48q^{11} - 8q^{12} - 112q^{13} + 48q^{15} - 16q^{16} - 114q^{17} + 46q^{18} + 2q^{19} + 96q^{20} + 192q^{22} + 120q^{23} - 16q^{24} - 19q^{25} + 112q^{26} - 200q^{27} - 108q^{29} - 48q^{30} + 236q^{31} - 32q^{32} - 96q^{33} + 456q^{34} - 184q^{36} - 146q^{37} + 4q^{38} + 112q^{39} - 96q^{40} - 252q^{41} - 752q^{43} - 192q^{44} + 276q^{45} + 240q^{46} - 12q^{47} + 64q^{48} + 76q^{50} - 228q^{51} + 224q^{52} - 174q^{53} + 200q^{54} + 1152q^{55} - 8q^{57} + 108q^{58} + 138q^{59} - 96q^{60} + 380q^{61} - 944q^{62} + 128q^{64} + 672q^{65} - 192q^{66} + 484q^{67} - 456q^{68} - 480q^{69} + 1152q^{71} + 184q^{72} - 1150q^{73} - 292q^{74} - 38q^{75} - 16q^{76} - 448q^{78} - 776q^{79} - 192q^{80} - 421q^{81} + 252q^{82} - 756q^{83} + 2736q^{85} + 752q^{86} + 108q^{87} - 384q^{88} - 390q^{89} - 1104q^{90} - 960q^{92} + 472q^{93} - 24q^{94} + 24q^{95} - 64q^{96} + 2660q^{97} - 2208q^{99} + O(q^{100})$$

## 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 −1.00000 + 1.73205i −2.00000 + 3.46410i −6.00000 10.3923i 4.00000 0 8.00000 11.5000 + 19.9186i −12.0000 + 20.7846i
79.1 −1.00000 + 1.73205i −1.00000 1.73205i −2.00000 3.46410i −6.00000 + 10.3923i 4.00000 0 8.00000 11.5000 19.9186i −12.0000 20.7846i
 $$n$$: 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.b 2
3.b odd 2 1 882.4.g.v 2
7.b odd 2 1 98.4.c.c 2
7.c even 3 1 98.4.a.e 1
7.c even 3 1 inner 98.4.c.b 2
7.d odd 6 1 14.4.a.b 1
7.d odd 6 1 98.4.c.c 2
21.c even 2 1 882.4.g.p 2
21.g even 6 1 126.4.a.d 1
21.g even 6 1 882.4.g.p 2
21.h odd 6 1 882.4.a.b 1
21.h odd 6 1 882.4.g.v 2
28.f even 6 1 112.4.a.e 1
28.g odd 6 1 784.4.a.h 1
35.i odd 6 1 350.4.a.f 1
35.j even 6 1 2450.4.a.i 1
35.k even 12 2 350.4.c.g 2
56.j odd 6 1 448.4.a.k 1
56.m even 6 1 448.4.a.g 1
77.i even 6 1 1694.4.a.b 1
84.j odd 6 1 1008.4.a.r 1
91.s odd 6 1 2366.4.a.c 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$4 + 2 T + T^{2}$$
$5$ $$144 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$2304 + 48 T + T^{2}$$
$13$ $$( 56 + T )^{2}$$
$17$ $$12996 + 114 T + T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$14400 - 120 T + T^{2}$$
$29$ $$( 54 + T )^{2}$$
$31$ $$55696 - 236 T + T^{2}$$
$37$ $$21316 + 146 T + T^{2}$$
$41$ $$( 126 + T )^{2}$$
$43$ $$( 376 + T )^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$30276 + 174 T + T^{2}$$
$59$ $$19044 - 138 T + T^{2}$$
$61$ $$144400 - 380 T + T^{2}$$
$67$ $$234256 - 484 T + T^{2}$$
$71$ $$( -576 + T )^{2}$$
$73$ $$1322500 + 1150 T + T^{2}$$
$79$ $$602176 + 776 T + T^{2}$$
$83$ $$( 378 + T )^{2}$$
$89$ $$152100 + 390 T + T^{2}$$
$97$ $$( -1330 + T )^{2}$$