# Properties

 Label 90.6.c.d Level $90$ Weight $6$ Character orbit 90.c Analytic conductor $14.435$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 90.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4345437832$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9x^{2} + 25$$ x^4 - 9*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} - 16 q^{4} + ( - 15 \beta_{2} + 5 \beta_1) q^{5} - \beta_{3} q^{7} - 32 \beta_{2} q^{8}+O(q^{10})$$ q + 2*b2 * q^2 - 16 * q^4 + (-15*b2 + 5*b1) * q^5 - b3 * q^7 - 32*b2 * q^8 $$q + 2 \beta_{2} q^{2} - 16 q^{4} + ( - 15 \beta_{2} + 5 \beta_1) q^{5} - \beta_{3} q^{7} - 32 \beta_{2} q^{8} + ( - 10 \beta_{3} + 140) q^{10} + ( - 37 \beta_{2} - 74 \beta_1) q^{11} + 63 \beta_{3} q^{13} + ( - 4 \beta_{2} - 8 \beta_1) q^{14} + 256 q^{16} - 583 \beta_{2} q^{17} + 2244 q^{19} + (240 \beta_{2} - 80 \beta_1) q^{20} + 148 \beta_{3} q^{22} + 250 \beta_{2} q^{23} + (175 \beta_{3} + 675) q^{25} + (252 \beta_{2} + 504 \beta_1) q^{26} + 16 \beta_{3} q^{28} + ( - 27 \beta_{2} - 54 \beta_1) q^{29} + 3856 q^{31} + 512 \beta_{2} q^{32} + 4664 q^{34} + (415 \beta_{2} + 70 \beta_1) q^{35} + 401 \beta_{3} q^{37} + 4488 \beta_{2} q^{38} + (160 \beta_{3} - 2240) q^{40} + ( - 550 \beta_{2} - 1100 \beta_1) q^{41} + 304 \beta_{3} q^{43} + (592 \beta_{2} + 1184 \beta_1) q^{44} - 2000 q^{46} - 9950 \beta_{2} q^{47} + 16503 q^{49} + (2050 \beta_{2} + 1400 \beta_1) q^{50} - 1008 \beta_{3} q^{52} + 573 \beta_{2} q^{53} + ( - 1295 \beta_{3} - 28120) q^{55} + (64 \beta_{2} + 128 \beta_1) q^{56} + 108 \beta_{3} q^{58} + ( - 2143 \beta_{2} - 4286 \beta_1) q^{59} - 38158 q^{61} + 7712 \beta_{2} q^{62} - 4096 q^{64} + ( - 26145 \beta_{2} - 4410 \beta_1) q^{65} - 2078 \beta_{3} q^{67} + 9328 \beta_{2} q^{68} + ( - 140 \beta_{3} - 3040) q^{70} + (954 \beta_{2} + 1908 \beta_1) q^{71} - 3980 \beta_{3} q^{73} + (1604 \beta_{2} + 3208 \beta_1) q^{74} - 35904 q^{76} - 5624 \beta_{2} q^{77} + 20664 q^{79} + ( - 3840 \beta_{2} + 1280 \beta_1) q^{80} + 2200 \beta_{3} q^{82} + 48484 \beta_{2} q^{83} + (2915 \beta_{3} - 40810) q^{85} + (1216 \beta_{2} + 2432 \beta_1) q^{86} - 2368 \beta_{3} q^{88} + (3424 \beta_{2} + 6848 \beta_1) q^{89} + 19152 q^{91} - 4000 \beta_{2} q^{92} + 79600 q^{94} + ( - 33660 \beta_{2} + 11220 \beta_1) q^{95} + 334 \beta_{3} q^{97} + 33006 \beta_{2} q^{98}+O(q^{100})$$ q + 2*b2 * q^2 - 16 * q^4 + (-15*b2 + 5*b1) * q^5 - b3 * q^7 - 32*b2 * q^8 + (-10*b3 + 140) * q^10 + (-37*b2 - 74*b1) * q^11 + 63*b3 * q^13 + (-4*b2 - 8*b1) * q^14 + 256 * q^16 - 583*b2 * q^17 + 2244 * q^19 + (240*b2 - 80*b1) * q^20 + 148*b3 * q^22 + 250*b2 * q^23 + (175*b3 + 675) * q^25 + (252*b2 + 504*b1) * q^26 + 16*b3 * q^28 + (-27*b2 - 54*b1) * q^29 + 3856 * q^31 + 512*b2 * q^32 + 4664 * q^34 + (415*b2 + 70*b1) * q^35 + 401*b3 * q^37 + 4488*b2 * q^38 + (160*b3 - 2240) * q^40 + (-550*b2 - 1100*b1) * q^41 + 304*b3 * q^43 + (592*b2 + 1184*b1) * q^44 - 2000 * q^46 - 9950*b2 * q^47 + 16503 * q^49 + (2050*b2 + 1400*b1) * q^50 - 1008*b3 * q^52 + 573*b2 * q^53 + (-1295*b3 - 28120) * q^55 + (64*b2 + 128*b1) * q^56 + 108*b3 * q^58 + (-2143*b2 - 4286*b1) * q^59 - 38158 * q^61 + 7712*b2 * q^62 - 4096 * q^64 + (-26145*b2 - 4410*b1) * q^65 - 2078*b3 * q^67 + 9328*b2 * q^68 + (-140*b3 - 3040) * q^70 + (954*b2 + 1908*b1) * q^71 - 3980*b3 * q^73 + (1604*b2 + 3208*b1) * q^74 - 35904 * q^76 - 5624*b2 * q^77 + 20664 * q^79 + (-3840*b2 + 1280*b1) * q^80 + 2200*b3 * q^82 + 48484*b2 * q^83 + (2915*b3 - 40810) * q^85 + (1216*b2 + 2432*b1) * q^86 - 2368*b3 * q^88 + (3424*b2 + 6848*b1) * q^89 + 19152 * q^91 - 4000*b2 * q^92 + 79600 * q^94 + (-33660*b2 + 11220*b1) * q^95 + 334*b3 * q^97 + 33006*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 64 q^{4}+O(q^{10})$$ 4 * q - 64 * q^4 $$4 q - 64 q^{4} + 560 q^{10} + 1024 q^{16} + 8976 q^{19} + 2700 q^{25} + 15424 q^{31} + 18656 q^{34} - 8960 q^{40} - 8000 q^{46} + 66012 q^{49} - 112480 q^{55} - 152632 q^{61} - 16384 q^{64} - 12160 q^{70} - 143616 q^{76} + 82656 q^{79} - 163240 q^{85} + 76608 q^{91} + 318400 q^{94}+O(q^{100})$$ 4 * q - 64 * q^4 + 560 * q^10 + 1024 * q^16 + 8976 * q^19 + 2700 * q^25 + 15424 * q^31 + 18656 * q^34 - 8960 * q^40 - 8000 * q^46 + 66012 * q^49 - 112480 * q^55 - 152632 * q^61 - 16384 * q^64 - 12160 * q^70 - 143616 * q^76 + 82656 * q^79 - 163240 * q^85 + 76608 * q^91 + 318400 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 24\nu ) / 5$$ (-v^3 + 24*v) / 5 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 8\nu ) / 5$$ (-2*v^3 + 8*v) / 5 $$\beta_{3}$$ $$=$$ $$8\nu^{2} - 36$$ 8*v^2 - 36
 $$\nu$$ $$=$$ $$( -\beta_{2} + 2\beta_1 ) / 8$$ (-b2 + 2*b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 36 ) / 8$$ (b3 + 36) / 8 $$\nu^{3}$$ $$=$$ $$-3\beta_{2} + \beta_1$$ -3*b2 + b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −2.17945 + 0.500000i 2.17945 + 0.500000i −2.17945 − 0.500000i 2.17945 − 0.500000i
4.00000i 0 −16.0000 −43.5890 + 35.0000i 0 17.4356i 64.0000i 0 140.000 + 174.356i
19.2 4.00000i 0 −16.0000 43.5890 + 35.0000i 0 17.4356i 64.0000i 0 140.000 174.356i
19.3 4.00000i 0 −16.0000 −43.5890 35.0000i 0 17.4356i 64.0000i 0 140.000 174.356i
19.4 4.00000i 0 −16.0000 43.5890 35.0000i 0 17.4356i 64.0000i 0 140.000 + 174.356i
 $$n$$: 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.6.c.d 4
3.b odd 2 1 inner 90.6.c.d 4
4.b odd 2 1 720.6.f.j 4
5.b even 2 1 inner 90.6.c.d 4
5.c odd 4 1 450.6.a.z 2
5.c odd 4 1 450.6.a.be 2
12.b even 2 1 720.6.f.j 4
15.d odd 2 1 inner 90.6.c.d 4
15.e even 4 1 450.6.a.z 2
15.e even 4 1 450.6.a.be 2
20.d odd 2 1 720.6.f.j 4
60.h even 2 1 720.6.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.c.d 4 1.a even 1 1 trivial
90.6.c.d 4 3.b odd 2 1 inner
90.6.c.d 4 5.b even 2 1 inner
90.6.c.d 4 15.d odd 2 1 inner
450.6.a.z 2 5.c odd 4 1
450.6.a.z 2 15.e even 4 1
450.6.a.be 2 5.c odd 4 1
450.6.a.be 2 15.e even 4 1
720.6.f.j 4 4.b odd 2 1
720.6.f.j 4 12.b even 2 1
720.6.f.j 4 20.d odd 2 1
720.6.f.j 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 304$$ acting on $$S_{6}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 16)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 1350 T^{2} + \cdots + 9765625$$
$7$ $$(T^{2} + 304)^{2}$$
$11$ $$(T^{2} - 416176)^{2}$$
$13$ $$(T^{2} + 1206576)^{2}$$
$17$ $$(T^{2} + 1359556)^{2}$$
$19$ $$(T - 2244)^{4}$$
$23$ $$(T^{2} + 250000)^{2}$$
$29$ $$(T^{2} - 221616)^{2}$$
$31$ $$(T - 3856)^{4}$$
$37$ $$(T^{2} + 48883504)^{2}$$
$41$ $$(T^{2} - 91960000)^{2}$$
$43$ $$(T^{2} + 28094464)^{2}$$
$47$ $$(T^{2} + 396010000)^{2}$$
$53$ $$(T^{2} + 1313316)^{2}$$
$59$ $$(T^{2} - 1396104496)^{2}$$
$61$ $$(T + 38158)^{4}$$
$67$ $$(T^{2} + 1312697536)^{2}$$
$71$ $$(T^{2} - 276675264)^{2}$$
$73$ $$(T^{2} + 4815481600)^{2}$$
$79$ $$(T - 20664)^{4}$$
$83$ $$(T^{2} + 9402793024)^{2}$$
$89$ $$(T^{2} - 3564027904)^{2}$$
$97$ $$(T^{2} + 33913024)^{2}$$