# Properties

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

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{22})$$ Defining polynomial: $$x^{4} + 22x^{2} + 484$$ x^4 + 22*x^2 + 484 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} + 8 q^{8} + ( - 61 \beta_{2} - 61) q^{9}+O(q^{10})$$ q + (-2*b2 - 2) * q^2 + (b3 + b1) * q^3 + 4*b2 * q^4 + b1 * q^5 - 2*b3 * q^6 + 8 * q^8 + (-61*b2 - 61) * q^9 $$q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} + 8 q^{8} + ( - 61 \beta_{2} - 61) q^{9} + ( - 2 \beta_{3} - 2 \beta_1) q^{10} + 20 \beta_{2} q^{11} - 4 \beta_1 q^{12} + 7 \beta_{3} q^{13} - 88 q^{15} + ( - 16 \beta_{2} - 16) q^{16} + ( - 6 \beta_{3} - 6 \beta_1) q^{17} + 122 \beta_{2} q^{18} + \beta_1 q^{19} + 4 \beta_{3} q^{20} + 40 q^{22} + ( - 48 \beta_{2} - 48) q^{23} + (8 \beta_{3} + 8 \beta_1) q^{24} - 37 \beta_{2} q^{25} + 14 \beta_1 q^{26} - 34 \beta_{3} q^{27} - 166 q^{29} + (176 \beta_{2} + 176) q^{30} + (22 \beta_{3} + 22 \beta_1) q^{31} + 32 \beta_{2} q^{32} - 20 \beta_1 q^{33} + 12 \beta_{3} q^{34} + 244 q^{36} + (78 \beta_{2} + 78) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} - 616 \beta_{2} q^{39} + 8 \beta_1 q^{40} + 42 \beta_{3} q^{41} + 436 q^{43} + ( - 80 \beta_{2} - 80) q^{44} + ( - 61 \beta_{3} - 61 \beta_1) q^{45} + 96 \beta_{2} q^{46} + 22 \beta_1 q^{47} - 16 \beta_{3} q^{48} - 74 q^{50} + (528 \beta_{2} + 528) q^{51} + ( - 28 \beta_{3} - 28 \beta_1) q^{52} + 62 \beta_{2} q^{53} - 68 \beta_1 q^{54} + 20 \beta_{3} q^{55} - 88 q^{57} + (332 \beta_{2} + 332) q^{58} + (71 \beta_{3} + 71 \beta_1) q^{59} - 352 \beta_{2} q^{60} + 29 \beta_1 q^{61} - 44 \beta_{3} q^{62} + 64 q^{64} + ( - 616 \beta_{2} - 616) q^{65} + (40 \beta_{3} + 40 \beta_1) q^{66} + 580 \beta_{2} q^{67} + 24 \beta_1 q^{68} - 48 \beta_{3} q^{69} - 544 q^{71} + ( - 488 \beta_{2} - 488) q^{72} + (64 \beta_{3} + 64 \beta_1) q^{73} - 156 \beta_{2} q^{74} + 37 \beta_1 q^{75} + 4 \beta_{3} q^{76} - 1232 q^{78} + (680 \beta_{2} + 680) q^{79} + ( - 16 \beta_{3} - 16 \beta_1) q^{80} + 1345 \beta_{2} q^{81} + 84 \beta_1 q^{82} + 21 \beta_{3} q^{83} + 528 q^{85} + ( - 872 \beta_{2} - 872) q^{86} + ( - 166 \beta_{3} - 166 \beta_1) q^{87} + 160 \beta_{2} q^{88} - 160 \beta_1 q^{89} + 122 \beta_{3} q^{90} + 192 q^{92} + ( - 1936 \beta_{2} - 1936) q^{93} + ( - 44 \beta_{3} - 44 \beta_1) q^{94} + 88 \beta_{2} q^{95} - 32 \beta_1 q^{96} - 70 \beta_{3} q^{97} + 1220 q^{99}+O(q^{100})$$ q + (-2*b2 - 2) * q^2 + (b3 + b1) * q^3 + 4*b2 * q^4 + b1 * q^5 - 2*b3 * q^6 + 8 * q^8 + (-61*b2 - 61) * q^9 + (-2*b3 - 2*b1) * q^10 + 20*b2 * q^11 - 4*b1 * q^12 + 7*b3 * q^13 - 88 * q^15 + (-16*b2 - 16) * q^16 + (-6*b3 - 6*b1) * q^17 + 122*b2 * q^18 + b1 * q^19 + 4*b3 * q^20 + 40 * q^22 + (-48*b2 - 48) * q^23 + (8*b3 + 8*b1) * q^24 - 37*b2 * q^25 + 14*b1 * q^26 - 34*b3 * q^27 - 166 * q^29 + (176*b2 + 176) * q^30 + (22*b3 + 22*b1) * q^31 + 32*b2 * q^32 - 20*b1 * q^33 + 12*b3 * q^34 + 244 * q^36 + (78*b2 + 78) * q^37 + (-2*b3 - 2*b1) * q^38 - 616*b2 * q^39 + 8*b1 * q^40 + 42*b3 * q^41 + 436 * q^43 + (-80*b2 - 80) * q^44 + (-61*b3 - 61*b1) * q^45 + 96*b2 * q^46 + 22*b1 * q^47 - 16*b3 * q^48 - 74 * q^50 + (528*b2 + 528) * q^51 + (-28*b3 - 28*b1) * q^52 + 62*b2 * q^53 - 68*b1 * q^54 + 20*b3 * q^55 - 88 * q^57 + (332*b2 + 332) * q^58 + (71*b3 + 71*b1) * q^59 - 352*b2 * q^60 + 29*b1 * q^61 - 44*b3 * q^62 + 64 * q^64 + (-616*b2 - 616) * q^65 + (40*b3 + 40*b1) * q^66 + 580*b2 * q^67 + 24*b1 * q^68 - 48*b3 * q^69 - 544 * q^71 + (-488*b2 - 488) * q^72 + (64*b3 + 64*b1) * q^73 - 156*b2 * q^74 + 37*b1 * q^75 + 4*b3 * q^76 - 1232 * q^78 + (680*b2 + 680) * q^79 + (-16*b3 - 16*b1) * q^80 + 1345*b2 * q^81 + 84*b1 * q^82 + 21*b3 * q^83 + 528 * q^85 + (-872*b2 - 872) * q^86 + (-166*b3 - 166*b1) * q^87 + 160*b2 * q^88 - 160*b1 * q^89 + 122*b3 * q^90 + 192 * q^92 + (-1936*b2 - 1936) * q^93 + (-44*b3 - 44*b1) * q^94 + 88*b2 * q^95 - 32*b1 * q^96 - 70*b3 * q^97 + 1220 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 122 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 122 * q^9 $$4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 122 q^{9} - 40 q^{11} - 352 q^{15} - 32 q^{16} - 244 q^{18} + 160 q^{22} - 96 q^{23} + 74 q^{25} - 664 q^{29} + 352 q^{30} - 64 q^{32} + 976 q^{36} + 156 q^{37} + 1232 q^{39} + 1744 q^{43} - 160 q^{44} - 192 q^{46} - 296 q^{50} + 1056 q^{51} - 124 q^{53} - 352 q^{57} + 664 q^{58} + 704 q^{60} + 256 q^{64} - 1232 q^{65} - 1160 q^{67} - 2176 q^{71} - 976 q^{72} + 312 q^{74} - 4928 q^{78} + 1360 q^{79} - 2690 q^{81} + 2112 q^{85} - 1744 q^{86} - 320 q^{88} + 768 q^{92} - 3872 q^{93} - 176 q^{95} + 4880 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 122 * q^9 - 40 * q^11 - 352 * q^15 - 32 * q^16 - 244 * q^18 + 160 * q^22 - 96 * q^23 + 74 * q^25 - 664 * q^29 + 352 * q^30 - 64 * q^32 + 976 * q^36 + 156 * q^37 + 1232 * q^39 + 1744 * q^43 - 160 * q^44 - 192 * q^46 - 296 * q^50 + 1056 * q^51 - 124 * q^53 - 352 * q^57 + 664 * q^58 + 704 * q^60 + 256 * q^64 - 1232 * q^65 - 1160 * q^67 - 2176 * q^71 - 976 * q^72 + 312 * q^74 - 4928 * q^78 + 1360 * q^79 - 2690 * q^81 + 2112 * q^85 - 1744 * q^86 - 320 * q^88 + 768 * q^92 - 3872 * q^93 - 176 * q^95 + 4880 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 22x^{2} + 484$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 22$$ (v^2) / 22 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 11$$ (v^3) / 11
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$22\beta_{2}$$ 22*b2 $$\nu^{3}$$ $$=$$ $$11\beta_{3}$$ 11*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
 2.34521 + 4.06202i −2.34521 − 4.06202i 2.34521 − 4.06202i −2.34521 + 4.06202i
−1.00000 1.73205i −4.69042 + 8.12404i −2.00000 + 3.46410i 4.69042 + 8.12404i 18.7617 0 8.00000 −30.5000 52.8275i 9.38083 16.2481i
67.2 −1.00000 1.73205i 4.69042 8.12404i −2.00000 + 3.46410i −4.69042 8.12404i −18.7617 0 8.00000 −30.5000 52.8275i −9.38083 + 16.2481i
79.1 −1.00000 + 1.73205i −4.69042 8.12404i −2.00000 3.46410i 4.69042 8.12404i 18.7617 0 8.00000 −30.5000 + 52.8275i 9.38083 + 16.2481i
79.2 −1.00000 + 1.73205i 4.69042 + 8.12404i −2.00000 3.46410i −4.69042 + 8.12404i −18.7617 0 8.00000 −30.5000 + 52.8275i −9.38083 16.2481i
 $$n$$: 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.g 4
3.b odd 2 1 882.4.g.bi 4
7.b odd 2 1 inner 98.4.c.g 4
7.c even 3 1 98.4.a.h 2
7.c even 3 1 inner 98.4.c.g 4
7.d odd 6 1 98.4.a.h 2
7.d odd 6 1 inner 98.4.c.g 4
21.c even 2 1 882.4.g.bi 4
21.g even 6 1 882.4.a.w 2
21.g even 6 1 882.4.g.bi 4
21.h odd 6 1 882.4.a.w 2
21.h odd 6 1 882.4.g.bi 4
28.f even 6 1 784.4.a.z 2
28.g odd 6 1 784.4.a.z 2
35.i odd 6 1 2450.4.a.bs 2
35.j even 6 1 2450.4.a.bs 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 88T_{3}^{2} + 7744$$ acting on $$S_{4}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$T^{4} + 88T^{2} + 7744$$
$5$ $$T^{4} + 88T^{2} + 7744$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 20 T + 400)^{2}$$
$13$ $$(T^{2} - 4312)^{2}$$
$17$ $$T^{4} + 3168 T^{2} + \cdots + 10036224$$
$19$ $$T^{4} + 88T^{2} + 7744$$
$23$ $$(T^{2} + 48 T + 2304)^{2}$$
$29$ $$(T + 166)^{4}$$
$31$ $$T^{4} + 42592 T^{2} + \cdots + 1814078464$$
$37$ $$(T^{2} - 78 T + 6084)^{2}$$
$41$ $$(T^{2} - 155232)^{2}$$
$43$ $$(T - 436)^{4}$$
$47$ $$T^{4} + 42592 T^{2} + \cdots + 1814078464$$
$53$ $$(T^{2} + 62 T + 3844)^{2}$$
$59$ $$T^{4} + 443608 T^{2} + \cdots + 196788057664$$
$61$ $$T^{4} + 74008 T^{2} + \cdots + 5477184064$$
$67$ $$(T^{2} + 580 T + 336400)^{2}$$
$71$ $$(T + 544)^{4}$$
$73$ $$T^{4} + 360448 T^{2} + \cdots + 129922760704$$
$79$ $$(T^{2} - 680 T + 462400)^{2}$$
$83$ $$(T^{2} - 38808)^{2}$$
$89$ $$T^{4} + 2252800 T^{2} + \cdots + 5075107840000$$
$97$ $$(T^{2} - 431200)^{2}$$