Properties

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

Learn more

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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{2} + 2) q^{2} + (5 \beta_{3} + 5 \beta_1) q^{3} + 4 \beta_{2} q^{4} - 14 \beta_1 q^{5} + 10 \beta_{3} q^{6} - 8 q^{8} + ( - 23 \beta_{2} - 23) q^{9}+O(q^{10})$$ q + (2*b2 + 2) * q^2 + (5*b3 + 5*b1) * q^3 + 4*b2 * q^4 - 14*b1 * q^5 + 10*b3 * q^6 - 8 * q^8 + (-23*b2 - 23) * q^9 $$q + (2 \beta_{2} + 2) q^{2} + (5 \beta_{3} + 5 \beta_1) q^{3} + 4 \beta_{2} q^{4} - 14 \beta_1 q^{5} + 10 \beta_{3} q^{6} - 8 q^{8} + ( - 23 \beta_{2} - 23) q^{9} + ( - 28 \beta_{3} - 28 \beta_1) q^{10} - 14 \beta_{2} q^{11} - 20 \beta_1 q^{12} + 36 \beta_{3} q^{13} + 140 q^{15} + ( - 16 \beta_{2} - 16) q^{16} + ( - \beta_{3} - \beta_1) q^{17} - 46 \beta_{2} q^{18} - \beta_1 q^{19} - 56 \beta_{3} q^{20} + 28 q^{22} + ( - 140 \beta_{2} - 140) q^{23} + ( - 40 \beta_{3} - 40 \beta_1) q^{24} + 267 \beta_{2} q^{25} - 72 \beta_1 q^{26} + 20 \beta_{3} q^{27} - 286 q^{29} + (280 \beta_{2} + 280) q^{30} + (66 \beta_{3} + 66 \beta_1) q^{31} - 32 \beta_{2} q^{32} + 70 \beta_1 q^{33} - 2 \beta_{3} q^{34} + 92 q^{36} + (38 \beta_{2} + 38) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} - 360 \beta_{2} q^{39} + 112 \beta_1 q^{40} - 89 \beta_{3} q^{41} - 34 q^{43} + (56 \beta_{2} + 56) q^{44} + (322 \beta_{3} + 322 \beta_1) q^{45} - 280 \beta_{2} q^{46} + 370 \beta_1 q^{47} - 80 \beta_{3} q^{48} - 534 q^{50} + (10 \beta_{2} + 10) q^{51} + ( - 144 \beta_{3} - 144 \beta_1) q^{52} - 74 \beta_{2} q^{53} - 40 \beta_1 q^{54} + 196 \beta_{3} q^{55} + 10 q^{57} + ( - 572 \beta_{2} - 572) q^{58} + ( - 307 \beta_{3} - 307 \beta_1) q^{59} + 560 \beta_{2} q^{60} + 10 \beta_1 q^{61} + 132 \beta_{3} q^{62} + 64 q^{64} + (1008 \beta_{2} + 1008) q^{65} + (140 \beta_{3} + 140 \beta_1) q^{66} + 684 \beta_{2} q^{67} + 4 \beta_1 q^{68} - 700 \beta_{3} q^{69} + 588 q^{71} + (184 \beta_{2} + 184) q^{72} + (191 \beta_{3} + 191 \beta_1) q^{73} + 76 \beta_{2} q^{74} - 1335 \beta_1 q^{75} - 4 \beta_{3} q^{76} + 720 q^{78} + ( - 1220 \beta_{2} - 1220) q^{79} + (224 \beta_{3} + 224 \beta_1) q^{80} - 821 \beta_{2} q^{81} + 178 \beta_1 q^{82} + 299 \beta_{3} q^{83} - 28 q^{85} + ( - 68 \beta_{2} - 68) q^{86} + ( - 1430 \beta_{3} - 1430 \beta_1) q^{87} + 112 \beta_{2} q^{88} + 437 \beta_1 q^{89} + 644 \beta_{3} q^{90} + 560 q^{92} + ( - 660 \beta_{2} - 660) q^{93} + (740 \beta_{3} + 740 \beta_1) q^{94} + 28 \beta_{2} q^{95} + 160 \beta_1 q^{96} + 1049 \beta_{3} q^{97} - 322 q^{99}+O(q^{100})$$ q + (2*b2 + 2) * q^2 + (5*b3 + 5*b1) * q^3 + 4*b2 * q^4 - 14*b1 * q^5 + 10*b3 * q^6 - 8 * q^8 + (-23*b2 - 23) * q^9 + (-28*b3 - 28*b1) * q^10 - 14*b2 * q^11 - 20*b1 * q^12 + 36*b3 * q^13 + 140 * q^15 + (-16*b2 - 16) * q^16 + (-b3 - b1) * q^17 - 46*b2 * q^18 - b1 * q^19 - 56*b3 * q^20 + 28 * q^22 + (-140*b2 - 140) * q^23 + (-40*b3 - 40*b1) * q^24 + 267*b2 * q^25 - 72*b1 * q^26 + 20*b3 * q^27 - 286 * q^29 + (280*b2 + 280) * q^30 + (66*b3 + 66*b1) * q^31 - 32*b2 * q^32 + 70*b1 * q^33 - 2*b3 * q^34 + 92 * q^36 + (38*b2 + 38) * q^37 + (-2*b3 - 2*b1) * q^38 - 360*b2 * q^39 + 112*b1 * q^40 - 89*b3 * q^41 - 34 * q^43 + (56*b2 + 56) * q^44 + (322*b3 + 322*b1) * q^45 - 280*b2 * q^46 + 370*b1 * q^47 - 80*b3 * q^48 - 534 * q^50 + (10*b2 + 10) * q^51 + (-144*b3 - 144*b1) * q^52 - 74*b2 * q^53 - 40*b1 * q^54 + 196*b3 * q^55 + 10 * q^57 + (-572*b2 - 572) * q^58 + (-307*b3 - 307*b1) * q^59 + 560*b2 * q^60 + 10*b1 * q^61 + 132*b3 * q^62 + 64 * q^64 + (1008*b2 + 1008) * q^65 + (140*b3 + 140*b1) * q^66 + 684*b2 * q^67 + 4*b1 * q^68 - 700*b3 * q^69 + 588 * q^71 + (184*b2 + 184) * q^72 + (191*b3 + 191*b1) * q^73 + 76*b2 * q^74 - 1335*b1 * q^75 - 4*b3 * q^76 + 720 * q^78 + (-1220*b2 - 1220) * q^79 + (224*b3 + 224*b1) * q^80 - 821*b2 * q^81 + 178*b1 * q^82 + 299*b3 * q^83 - 28 * q^85 + (-68*b2 - 68) * q^86 + (-1430*b3 - 1430*b1) * q^87 + 112*b2 * q^88 + 437*b1 * q^89 + 644*b3 * q^90 + 560 * q^92 + (-660*b2 - 660) * q^93 + (740*b3 + 740*b1) * q^94 + 28*b2 * q^95 + 160*b1 * q^96 + 1049*b3 * q^97 - 322 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 46 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 - 46 * q^9 $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 46 q^{9} + 28 q^{11} + 560 q^{15} - 32 q^{16} + 92 q^{18} + 112 q^{22} - 280 q^{23} - 534 q^{25} - 1144 q^{29} + 560 q^{30} + 64 q^{32} + 368 q^{36} + 76 q^{37} + 720 q^{39} - 136 q^{43} + 112 q^{44} + 560 q^{46} - 2136 q^{50} + 20 q^{51} + 148 q^{53} + 40 q^{57} - 1144 q^{58} - 1120 q^{60} + 256 q^{64} + 2016 q^{65} - 1368 q^{67} + 2352 q^{71} + 368 q^{72} - 152 q^{74} + 2880 q^{78} - 2440 q^{79} + 1642 q^{81} - 112 q^{85} - 136 q^{86} - 224 q^{88} + 2240 q^{92} - 1320 q^{93} - 56 q^{95} - 1288 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 - 46 * q^9 + 28 * q^11 + 560 * q^15 - 32 * q^16 + 92 * q^18 + 112 * q^22 - 280 * q^23 - 534 * q^25 - 1144 * q^29 + 560 * q^30 + 64 * q^32 + 368 * q^36 + 76 * q^37 + 720 * q^39 - 136 * q^43 + 112 * q^44 + 560 * q^46 - 2136 * q^50 + 20 * q^51 + 148 * q^53 + 40 * q^57 - 1144 * q^58 - 1120 * q^60 + 256 * q^64 + 2016 * q^65 - 1368 * q^67 + 2352 * q^71 + 368 * q^72 - 152 * q^74 + 2880 * q^78 - 2440 * q^79 + 1642 * q^81 - 112 * q^85 - 136 * q^86 - 224 * q^88 + 2240 * q^92 - 1320 * q^93 - 56 * q^95 - 1288 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1 - \beta_{2}$$

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
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
1.00000 + 1.73205i −3.53553 + 6.12372i −2.00000 + 3.46410i −9.89949 17.1464i −14.1421 0 −8.00000 −11.5000 19.9186i 19.7990 34.2929i
67.2 1.00000 + 1.73205i 3.53553 6.12372i −2.00000 + 3.46410i 9.89949 + 17.1464i 14.1421 0 −8.00000 −11.5000 19.9186i −19.7990 + 34.2929i
79.1 1.00000 1.73205i −3.53553 6.12372i −2.00000 3.46410i −9.89949 + 17.1464i −14.1421 0 −8.00000 −11.5000 + 19.9186i 19.7990 + 34.2929i
79.2 1.00000 1.73205i 3.53553 + 6.12372i −2.00000 3.46410i 9.89949 17.1464i 14.1421 0 −8.00000 −11.5000 + 19.9186i −19.7990 34.2929i
 $$n$$: 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.c.h 4
3.b odd 2 1 882.4.g.ba 4
7.b odd 2 1 inner 98.4.c.h 4
7.c even 3 1 98.4.a.g 2
7.c even 3 1 inner 98.4.c.h 4
7.d odd 6 1 98.4.a.g 2
7.d odd 6 1 inner 98.4.c.h 4
21.c even 2 1 882.4.g.ba 4
21.g even 6 1 882.4.a.bg 2
21.g even 6 1 882.4.g.ba 4
21.h odd 6 1 882.4.a.bg 2
21.h odd 6 1 882.4.g.ba 4
28.f even 6 1 784.4.a.y 2
28.g odd 6 1 784.4.a.y 2
35.i odd 6 1 2450.4.a.bx 2
35.j even 6 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 7.c even 3 1
98.4.a.g 2 7.d odd 6 1
98.4.c.h 4 1.a even 1 1 trivial
98.4.c.h 4 7.b odd 2 1 inner
98.4.c.h 4 7.c even 3 1 inner
98.4.c.h 4 7.d odd 6 1 inner
784.4.a.y 2 28.f even 6 1
784.4.a.y 2 28.g odd 6 1
882.4.a.bg 2 21.g even 6 1
882.4.a.bg 2 21.h odd 6 1
882.4.g.ba 4 3.b odd 2 1
882.4.g.ba 4 21.c even 2 1
882.4.g.ba 4 21.g even 6 1
882.4.g.ba 4 21.h odd 6 1
2450.4.a.bx 2 35.i odd 6 1
2450.4.a.bx 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 50T_{3}^{2} + 2500$$ acting on $$S_{4}^{\mathrm{new}}(98, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{2}$$
$3$ $$T^{4} + 50T^{2} + 2500$$
$5$ $$T^{4} + 392 T^{2} + 153664$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 14 T + 196)^{2}$$
$13$ $$(T^{2} - 2592)^{2}$$
$17$ $$T^{4} + 2T^{2} + 4$$
$19$ $$T^{4} + 2T^{2} + 4$$
$23$ $$(T^{2} + 140 T + 19600)^{2}$$
$29$ $$(T + 286)^{4}$$
$31$ $$T^{4} + 8712 T^{2} + \cdots + 75898944$$
$37$ $$(T^{2} - 38 T + 1444)^{2}$$
$41$ $$(T^{2} - 15842)^{2}$$
$43$ $$(T + 34)^{4}$$
$47$ $$T^{4} + 273800 T^{2} + \cdots + 74966440000$$
$53$ $$(T^{2} - 74 T + 5476)^{2}$$
$59$ $$T^{4} + 188498 T^{2} + \cdots + 35531496004$$
$61$ $$T^{4} + 200 T^{2} + 40000$$
$67$ $$(T^{2} + 684 T + 467856)^{2}$$
$71$ $$(T - 588)^{4}$$
$73$ $$T^{4} + 72962 T^{2} + \cdots + 5323453444$$
$79$ $$(T^{2} + 1220 T + 1488400)^{2}$$
$83$ $$(T^{2} - 178802)^{2}$$
$89$ $$T^{4} + 381938 T^{2} + \cdots + 145876635844$$
$97$ $$(T^{2} - 2200802)^{2}$$
show more
show less