# 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (-8*z + 8) * q^3 + (4*z - 4) * q^4 - 14*z * q^5 + 16 * q^6 - 8 * q^8 - 37*z * q^9 $$q + 2 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} + ( - 28 \zeta_{6} + 28) q^{10} + ( - 28 \zeta_{6} + 28) q^{11} + 32 \zeta_{6} q^{12} - 18 q^{13} - 112 q^{15} - 16 \zeta_{6} q^{16} + ( - 74 \zeta_{6} + 74) q^{17} + ( - 74 \zeta_{6} + 74) q^{18} + 80 \zeta_{6} q^{19} + 56 q^{20} + 56 q^{22} + 112 \zeta_{6} q^{23} + (64 \zeta_{6} - 64) q^{24} + (71 \zeta_{6} - 71) q^{25} - 36 \zeta_{6} q^{26} - 80 q^{27} + 190 q^{29} - 224 \zeta_{6} q^{30} + ( - 72 \zeta_{6} + 72) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 224 \zeta_{6} q^{33} + 148 q^{34} + 148 q^{36} + 346 \zeta_{6} q^{37} + (160 \zeta_{6} - 160) q^{38} + (144 \zeta_{6} - 144) q^{39} + 112 \zeta_{6} q^{40} - 162 q^{41} - 412 q^{43} + 112 \zeta_{6} q^{44} + (518 \zeta_{6} - 518) q^{45} + (224 \zeta_{6} - 224) q^{46} + 24 \zeta_{6} q^{47} - 128 q^{48} - 142 q^{50} - 592 \zeta_{6} q^{51} + ( - 72 \zeta_{6} + 72) q^{52} + (318 \zeta_{6} - 318) q^{53} - 160 \zeta_{6} q^{54} - 392 q^{55} + 640 q^{57} + 380 \zeta_{6} q^{58} + (200 \zeta_{6} - 200) q^{59} + ( - 448 \zeta_{6} + 448) q^{60} - 198 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} + 252 \zeta_{6} q^{65} + ( - 448 \zeta_{6} + 448) q^{66} + ( - 716 \zeta_{6} + 716) q^{67} + 296 \zeta_{6} q^{68} + 896 q^{69} + 392 q^{71} + 296 \zeta_{6} q^{72} + ( - 538 \zeta_{6} + 538) q^{73} + (692 \zeta_{6} - 692) q^{74} + 568 \zeta_{6} q^{75} - 320 q^{76} - 288 q^{78} - 240 \zeta_{6} q^{79} + (224 \zeta_{6} - 224) q^{80} + ( - 359 \zeta_{6} + 359) q^{81} - 324 \zeta_{6} q^{82} + 1072 q^{83} - 1036 q^{85} - 824 \zeta_{6} q^{86} + ( - 1520 \zeta_{6} + 1520) q^{87} + (224 \zeta_{6} - 224) q^{88} + 810 \zeta_{6} q^{89} - 1036 q^{90} - 448 q^{92} - 576 \zeta_{6} q^{93} + (48 \zeta_{6} - 48) q^{94} + ( - 1120 \zeta_{6} + 1120) q^{95} - 256 \zeta_{6} q^{96} - 1354 q^{97} - 1036 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (-8*z + 8) * q^3 + (4*z - 4) * q^4 - 14*z * q^5 + 16 * q^6 - 8 * q^8 - 37*z * q^9 + (-28*z + 28) * q^10 + (-28*z + 28) * q^11 + 32*z * q^12 - 18 * q^13 - 112 * q^15 - 16*z * q^16 + (-74*z + 74) * q^17 + (-74*z + 74) * q^18 + 80*z * q^19 + 56 * q^20 + 56 * q^22 + 112*z * q^23 + (64*z - 64) * q^24 + (71*z - 71) * q^25 - 36*z * q^26 - 80 * q^27 + 190 * q^29 - 224*z * q^30 + (-72*z + 72) * q^31 + (-32*z + 32) * q^32 - 224*z * q^33 + 148 * q^34 + 148 * q^36 + 346*z * q^37 + (160*z - 160) * q^38 + (144*z - 144) * q^39 + 112*z * q^40 - 162 * q^41 - 412 * q^43 + 112*z * q^44 + (518*z - 518) * q^45 + (224*z - 224) * q^46 + 24*z * q^47 - 128 * q^48 - 142 * q^50 - 592*z * q^51 + (-72*z + 72) * q^52 + (318*z - 318) * q^53 - 160*z * q^54 - 392 * q^55 + 640 * q^57 + 380*z * q^58 + (200*z - 200) * q^59 + (-448*z + 448) * q^60 - 198*z * q^61 + 144 * q^62 + 64 * q^64 + 252*z * q^65 + (-448*z + 448) * q^66 + (-716*z + 716) * q^67 + 296*z * q^68 + 896 * q^69 + 392 * q^71 + 296*z * q^72 + (-538*z + 538) * q^73 + (692*z - 692) * q^74 + 568*z * q^75 - 320 * q^76 - 288 * q^78 - 240*z * q^79 + (224*z - 224) * q^80 + (-359*z + 359) * q^81 - 324*z * q^82 + 1072 * q^83 - 1036 * q^85 - 824*z * q^86 + (-1520*z + 1520) * q^87 + (224*z - 224) * q^88 + 810*z * q^89 - 1036 * q^90 - 448 * q^92 - 576*z * q^93 + (48*z - 48) * q^94 + (-1120*z + 1120) * q^95 - 256*z * q^96 - 1354 * q^97 - 1036 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 8 q^{3} - 4 q^{4} - 14 q^{5} + 32 q^{6} - 16 q^{8} - 37 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 8 * q^3 - 4 * q^4 - 14 * q^5 + 32 * q^6 - 16 * q^8 - 37 * q^9 $$2 q + 2 q^{2} + 8 q^{3} - 4 q^{4} - 14 q^{5} + 32 q^{6} - 16 q^{8} - 37 q^{9} + 28 q^{10} + 28 q^{11} + 32 q^{12} - 36 q^{13} - 224 q^{15} - 16 q^{16} + 74 q^{17} + 74 q^{18} + 80 q^{19} + 112 q^{20} + 112 q^{22} + 112 q^{23} - 64 q^{24} - 71 q^{25} - 36 q^{26} - 160 q^{27} + 380 q^{29} - 224 q^{30} + 72 q^{31} + 32 q^{32} - 224 q^{33} + 296 q^{34} + 296 q^{36} + 346 q^{37} - 160 q^{38} - 144 q^{39} + 112 q^{40} - 324 q^{41} - 824 q^{43} + 112 q^{44} - 518 q^{45} - 224 q^{46} + 24 q^{47} - 256 q^{48} - 284 q^{50} - 592 q^{51} + 72 q^{52} - 318 q^{53} - 160 q^{54} - 784 q^{55} + 1280 q^{57} + 380 q^{58} - 200 q^{59} + 448 q^{60} - 198 q^{61} + 288 q^{62} + 128 q^{64} + 252 q^{65} + 448 q^{66} + 716 q^{67} + 296 q^{68} + 1792 q^{69} + 784 q^{71} + 296 q^{72} + 538 q^{73} - 692 q^{74} + 568 q^{75} - 640 q^{76} - 576 q^{78} - 240 q^{79} - 224 q^{80} + 359 q^{81} - 324 q^{82} + 2144 q^{83} - 2072 q^{85} - 824 q^{86} + 1520 q^{87} - 224 q^{88} + 810 q^{89} - 2072 q^{90} - 896 q^{92} - 576 q^{93} - 48 q^{94} + 1120 q^{95} - 256 q^{96} - 2708 q^{97} - 2072 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 8 * q^3 - 4 * q^4 - 14 * q^5 + 32 * q^6 - 16 * q^8 - 37 * q^9 + 28 * q^10 + 28 * q^11 + 32 * q^12 - 36 * q^13 - 224 * q^15 - 16 * q^16 + 74 * q^17 + 74 * q^18 + 80 * q^19 + 112 * q^20 + 112 * q^22 + 112 * q^23 - 64 * q^24 - 71 * q^25 - 36 * q^26 - 160 * q^27 + 380 * q^29 - 224 * q^30 + 72 * q^31 + 32 * q^32 - 224 * q^33 + 296 * q^34 + 296 * q^36 + 346 * q^37 - 160 * q^38 - 144 * q^39 + 112 * q^40 - 324 * q^41 - 824 * q^43 + 112 * q^44 - 518 * q^45 - 224 * q^46 + 24 * q^47 - 256 * q^48 - 284 * q^50 - 592 * q^51 + 72 * q^52 - 318 * q^53 - 160 * q^54 - 784 * q^55 + 1280 * q^57 + 380 * q^58 - 200 * q^59 + 448 * q^60 - 198 * q^61 + 288 * q^62 + 128 * q^64 + 252 * q^65 + 448 * q^66 + 716 * q^67 + 296 * q^68 + 1792 * q^69 + 784 * q^71 + 296 * q^72 + 538 * q^73 - 692 * q^74 + 568 * q^75 - 640 * q^76 - 576 * q^78 - 240 * q^79 - 224 * q^80 + 359 * q^81 - 324 * q^82 + 2144 * q^83 - 2072 * q^85 - 824 * q^86 + 1520 * q^87 - 224 * q^88 + 810 * q^89 - 2072 * q^90 - 896 * q^92 - 576 * q^93 - 48 * q^94 + 1120 * q^95 - 256 * q^96 - 2708 * q^97 - 2072 * 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 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 $$T_{3}^{2} - 8T_{3} + 64$$ acting on $$S_{4}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

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