Properties

 Label 98.4.c.a 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 9 \zeta_{6} q^{5} + 10 q^{6} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^2 + (5*z - 5) * q^3 + (4*z - 4) * q^4 - 9*z * q^5 + 10 * q^6 + 8 * q^8 + 2*z * q^9 $$q - 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 9 \zeta_{6} q^{5} + 10 q^{6} + 8 q^{8} + 2 \zeta_{6} q^{9} + (18 \zeta_{6} - 18) q^{10} + ( - 57 \zeta_{6} + 57) q^{11} - 20 \zeta_{6} q^{12} + 70 q^{13} + 45 q^{15} - 16 \zeta_{6} q^{16} + ( - 51 \zeta_{6} + 51) q^{17} + ( - 4 \zeta_{6} + 4) q^{18} + 5 \zeta_{6} q^{19} + 36 q^{20} - 114 q^{22} - 69 \zeta_{6} q^{23} + (40 \zeta_{6} - 40) q^{24} + ( - 44 \zeta_{6} + 44) q^{25} - 140 \zeta_{6} q^{26} - 145 q^{27} + 114 q^{29} - 90 \zeta_{6} q^{30} + ( - 23 \zeta_{6} + 23) q^{31} + (32 \zeta_{6} - 32) q^{32} + 285 \zeta_{6} q^{33} - 102 q^{34} - 8 q^{36} + 253 \zeta_{6} q^{37} + ( - 10 \zeta_{6} + 10) q^{38} + (350 \zeta_{6} - 350) q^{39} - 72 \zeta_{6} q^{40} + 42 q^{41} - 124 q^{43} + 228 \zeta_{6} q^{44} + ( - 18 \zeta_{6} + 18) q^{45} + (138 \zeta_{6} - 138) q^{46} + 201 \zeta_{6} q^{47} + 80 q^{48} - 88 q^{50} + 255 \zeta_{6} q^{51} + (280 \zeta_{6} - 280) q^{52} + ( - 393 \zeta_{6} + 393) q^{53} + 290 \zeta_{6} q^{54} - 513 q^{55} - 25 q^{57} - 228 \zeta_{6} q^{58} + ( - 219 \zeta_{6} + 219) q^{59} + (180 \zeta_{6} - 180) q^{60} - 709 \zeta_{6} q^{61} - 46 q^{62} + 64 q^{64} - 630 \zeta_{6} q^{65} + ( - 570 \zeta_{6} + 570) q^{66} + (419 \zeta_{6} - 419) q^{67} + 204 \zeta_{6} q^{68} + 345 q^{69} - 96 q^{71} + 16 \zeta_{6} q^{72} + (313 \zeta_{6} - 313) q^{73} + ( - 506 \zeta_{6} + 506) q^{74} + 220 \zeta_{6} q^{75} - 20 q^{76} + 700 q^{78} - 461 \zeta_{6} q^{79} + (144 \zeta_{6} - 144) q^{80} + ( - 671 \zeta_{6} + 671) q^{81} - 84 \zeta_{6} q^{82} + 588 q^{83} - 459 q^{85} + 248 \zeta_{6} q^{86} + (570 \zeta_{6} - 570) q^{87} + ( - 456 \zeta_{6} + 456) q^{88} - 1017 \zeta_{6} q^{89} - 36 q^{90} + 276 q^{92} + 115 \zeta_{6} q^{93} + ( - 402 \zeta_{6} + 402) q^{94} + ( - 45 \zeta_{6} + 45) q^{95} - 160 \zeta_{6} q^{96} + 1834 q^{97} + 114 q^{99} +O(q^{100})$$ q - 2*z * q^2 + (5*z - 5) * q^3 + (4*z - 4) * q^4 - 9*z * q^5 + 10 * q^6 + 8 * q^8 + 2*z * q^9 + (18*z - 18) * q^10 + (-57*z + 57) * q^11 - 20*z * q^12 + 70 * q^13 + 45 * q^15 - 16*z * q^16 + (-51*z + 51) * q^17 + (-4*z + 4) * q^18 + 5*z * q^19 + 36 * q^20 - 114 * q^22 - 69*z * q^23 + (40*z - 40) * q^24 + (-44*z + 44) * q^25 - 140*z * q^26 - 145 * q^27 + 114 * q^29 - 90*z * q^30 + (-23*z + 23) * q^31 + (32*z - 32) * q^32 + 285*z * q^33 - 102 * q^34 - 8 * q^36 + 253*z * q^37 + (-10*z + 10) * q^38 + (350*z - 350) * q^39 - 72*z * q^40 + 42 * q^41 - 124 * q^43 + 228*z * q^44 + (-18*z + 18) * q^45 + (138*z - 138) * q^46 + 201*z * q^47 + 80 * q^48 - 88 * q^50 + 255*z * q^51 + (280*z - 280) * q^52 + (-393*z + 393) * q^53 + 290*z * q^54 - 513 * q^55 - 25 * q^57 - 228*z * q^58 + (-219*z + 219) * q^59 + (180*z - 180) * q^60 - 709*z * q^61 - 46 * q^62 + 64 * q^64 - 630*z * q^65 + (-570*z + 570) * q^66 + (419*z - 419) * q^67 + 204*z * q^68 + 345 * q^69 - 96 * q^71 + 16*z * q^72 + (313*z - 313) * q^73 + (-506*z + 506) * q^74 + 220*z * q^75 - 20 * q^76 + 700 * q^78 - 461*z * q^79 + (144*z - 144) * q^80 + (-671*z + 671) * q^81 - 84*z * q^82 + 588 * q^83 - 459 * q^85 + 248*z * q^86 + (570*z - 570) * q^87 + (-456*z + 456) * q^88 - 1017*z * q^89 - 36 * q^90 + 276 * q^92 + 115*z * q^93 + (-402*z + 402) * q^94 + (-45*z + 45) * q^95 - 160*z * q^96 + 1834 * q^97 + 114 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 9 q^{5} + 20 q^{6} + 16 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 5 * q^3 - 4 * q^4 - 9 * q^5 + 20 * q^6 + 16 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 9 q^{5} + 20 q^{6} + 16 q^{8} + 2 q^{9} - 18 q^{10} + 57 q^{11} - 20 q^{12} + 140 q^{13} + 90 q^{15} - 16 q^{16} + 51 q^{17} + 4 q^{18} + 5 q^{19} + 72 q^{20} - 228 q^{22} - 69 q^{23} - 40 q^{24} + 44 q^{25} - 140 q^{26} - 290 q^{27} + 228 q^{29} - 90 q^{30} + 23 q^{31} - 32 q^{32} + 285 q^{33} - 204 q^{34} - 16 q^{36} + 253 q^{37} + 10 q^{38} - 350 q^{39} - 72 q^{40} + 84 q^{41} - 248 q^{43} + 228 q^{44} + 18 q^{45} - 138 q^{46} + 201 q^{47} + 160 q^{48} - 176 q^{50} + 255 q^{51} - 280 q^{52} + 393 q^{53} + 290 q^{54} - 1026 q^{55} - 50 q^{57} - 228 q^{58} + 219 q^{59} - 180 q^{60} - 709 q^{61} - 92 q^{62} + 128 q^{64} - 630 q^{65} + 570 q^{66} - 419 q^{67} + 204 q^{68} + 690 q^{69} - 192 q^{71} + 16 q^{72} - 313 q^{73} + 506 q^{74} + 220 q^{75} - 40 q^{76} + 1400 q^{78} - 461 q^{79} - 144 q^{80} + 671 q^{81} - 84 q^{82} + 1176 q^{83} - 918 q^{85} + 248 q^{86} - 570 q^{87} + 456 q^{88} - 1017 q^{89} - 72 q^{90} + 552 q^{92} + 115 q^{93} + 402 q^{94} + 45 q^{95} - 160 q^{96} + 3668 q^{97} + 228 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 5 * q^3 - 4 * q^4 - 9 * q^5 + 20 * q^6 + 16 * q^8 + 2 * q^9 - 18 * q^10 + 57 * q^11 - 20 * q^12 + 140 * q^13 + 90 * q^15 - 16 * q^16 + 51 * q^17 + 4 * q^18 + 5 * q^19 + 72 * q^20 - 228 * q^22 - 69 * q^23 - 40 * q^24 + 44 * q^25 - 140 * q^26 - 290 * q^27 + 228 * q^29 - 90 * q^30 + 23 * q^31 - 32 * q^32 + 285 * q^33 - 204 * q^34 - 16 * q^36 + 253 * q^37 + 10 * q^38 - 350 * q^39 - 72 * q^40 + 84 * q^41 - 248 * q^43 + 228 * q^44 + 18 * q^45 - 138 * q^46 + 201 * q^47 + 160 * q^48 - 176 * q^50 + 255 * q^51 - 280 * q^52 + 393 * q^53 + 290 * q^54 - 1026 * q^55 - 50 * q^57 - 228 * q^58 + 219 * q^59 - 180 * q^60 - 709 * q^61 - 92 * q^62 + 128 * q^64 - 630 * q^65 + 570 * q^66 - 419 * q^67 + 204 * q^68 + 690 * q^69 - 192 * q^71 + 16 * q^72 - 313 * q^73 + 506 * q^74 + 220 * q^75 - 40 * q^76 + 1400 * q^78 - 461 * q^79 - 144 * q^80 + 671 * q^81 - 84 * q^82 + 1176 * q^83 - 918 * q^85 + 248 * q^86 - 570 * q^87 + 456 * q^88 - 1017 * q^89 - 72 * q^90 + 552 * q^92 + 115 * q^93 + 402 * q^94 + 45 * q^95 - 160 * q^96 + 3668 * q^97 + 228 * q^99

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 −2.50000 + 4.33013i −2.00000 + 3.46410i −4.50000 7.79423i 10.0000 0 8.00000 1.00000 + 1.73205i −9.00000 + 15.5885i
79.1 −1.00000 + 1.73205i −2.50000 4.33013i −2.00000 3.46410i −4.50000 + 7.79423i 10.0000 0 8.00000 1.00000 1.73205i −9.00000 15.5885i
 $$n$$: 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.a 2
3.b odd 2 1 882.4.g.u 2
7.b odd 2 1 14.4.c.a 2
7.c even 3 1 98.4.a.f 1
7.c even 3 1 inner 98.4.c.a 2
7.d odd 6 1 14.4.c.a 2
7.d odd 6 1 98.4.a.d 1
21.c even 2 1 126.4.g.d 2
21.g even 6 1 126.4.g.d 2
21.g even 6 1 882.4.a.f 1
21.h odd 6 1 882.4.a.c 1
21.h odd 6 1 882.4.g.u 2
28.d even 2 1 112.4.i.a 2
28.f even 6 1 112.4.i.a 2
28.f even 6 1 784.4.a.p 1
28.g odd 6 1 784.4.a.c 1
35.c odd 2 1 350.4.e.e 2
35.f even 4 2 350.4.j.b 4
35.i odd 6 1 350.4.e.e 2
35.i odd 6 1 2450.4.a.q 1
35.j even 6 1 2450.4.a.d 1
35.k even 12 2 350.4.j.b 4
56.e even 2 1 448.4.i.e 2
56.h odd 2 1 448.4.i.b 2
56.j odd 6 1 448.4.i.b 2
56.m even 6 1 448.4.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 7.b odd 2 1
14.4.c.a 2 7.d odd 6 1
98.4.a.d 1 7.d odd 6 1
98.4.a.f 1 7.c even 3 1
98.4.c.a 2 1.a even 1 1 trivial
98.4.c.a 2 7.c even 3 1 inner
112.4.i.a 2 28.d even 2 1
112.4.i.a 2 28.f even 6 1
126.4.g.d 2 21.c even 2 1
126.4.g.d 2 21.g even 6 1
350.4.e.e 2 35.c odd 2 1
350.4.e.e 2 35.i odd 6 1
350.4.j.b 4 35.f even 4 2
350.4.j.b 4 35.k even 12 2
448.4.i.b 2 56.h odd 2 1
448.4.i.b 2 56.j odd 6 1
448.4.i.e 2 56.e even 2 1
448.4.i.e 2 56.m even 6 1
784.4.a.c 1 28.g odd 6 1
784.4.a.p 1 28.f even 6 1
882.4.a.c 1 21.h odd 6 1
882.4.a.f 1 21.g even 6 1
882.4.g.u 2 3.b odd 2 1
882.4.g.u 2 21.h odd 6 1
2450.4.a.d 1 35.j even 6 1
2450.4.a.q 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 5T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(98, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 5T + 25$$
$5$ $$T^{2} + 9T + 81$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 57T + 3249$$
$13$ $$(T - 70)^{2}$$
$17$ $$T^{2} - 51T + 2601$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} + 69T + 4761$$
$29$ $$(T - 114)^{2}$$
$31$ $$T^{2} - 23T + 529$$
$37$ $$T^{2} - 253T + 64009$$
$41$ $$(T - 42)^{2}$$
$43$ $$(T + 124)^{2}$$
$47$ $$T^{2} - 201T + 40401$$
$53$ $$T^{2} - 393T + 154449$$
$59$ $$T^{2} - 219T + 47961$$
$61$ $$T^{2} + 709T + 502681$$
$67$ $$T^{2} + 419T + 175561$$
$71$ $$(T + 96)^{2}$$
$73$ $$T^{2} + 313T + 97969$$
$79$ $$T^{2} + 461T + 212521$$
$83$ $$(T - 588)^{2}$$
$89$ $$T^{2} + 1017 T + 1034289$$
$97$ $$(T - 1834)^{2}$$